05-10-2023-1724 - retroreflector

A retroreflector (sometimes called a retroflector or cataphote) is a device or surface that reflects radiation (usually light) back to its source with minimum scattering. This works at a wide range of angle of incidence, unlike a planar mirror, which does this only if the mirror is exactly perpendicular to the wave front, having a zero angle of incidence. Being directed, the retroflector's reflection is brighter than that of a diffuse reflector. Corner reflectors and cat's eye reflectors are the most used kinds. 

https://en.wikipedia.org/wiki/Retroreflector

 A large relatively thin retroreflector can be formed by combining many small corner reflectors, using the standard hexagonal tiling. 

https://en.wikipedia.org/wiki/Retroreflector

Corner reflector

Working principle of a corner reflector

Comparison of the effect of corner (1) and spherical (2) retroreflectors on three light rays. Reflective surfaces are drawn in dark blue.

Main article: Corner reflector

A set of three mutually perpendicular reflective surfaces, placed to form the internal corner of a cube, work as a retroreflector. The three corresponding normal vectors of the corner's sides form a basis (x, y, z) in which to represent the direction of an arbitrary incoming ray, [a, b, c]. When the ray reflects from the first side, say x, the ray's x-component, a, is reversed to −a, while the y- and z-components are unchanged. Therefore, as the ray reflects first from side x then side y and finally from side z the ray direction goes from [a, b, c] to [−a, b, c] to [−a, −b, c] to [−a, −b, −c] and it leaves the corner with all three components of its direction exactly reversed.

Corner reflectors occur in two varieties. In the more common form, the corner is literally the truncated corner of a cube of transparent material such as conventional optical glass. In this structure, the reflection is achieved either by total internal reflection or silvering of the outer cube surfaces. The second form uses mutually perpendicular flat mirrors bracketing an air space. These two types have similar optical properties.

A large relatively thin retroreflector can be formed by combining many small corner reflectors, using the standard hexagonal tiling. 

https://en.wikipedia.org/wiki/Retroreflector

Cat's eye

Eyeshine from retroreflectors of the transparent sphere type is clearly visible in this cat's eyes

Main article: Cat's eye (road)

Another common type of retroreflector consists of refracting optical elements with a reflective surface, arranged so that the focal surface of the refractive element coincides with the reflective surface, typically a transparent sphere and (optionally) a spherical mirror. In the paraxial approximation, this effect can be achieved with lowest divergence with a single transparent sphere when the refractive index of the material is exactly one plus the refractive index n_i of the medium from which the radiation is incident (n_i is around 1 for air). In that case, the sphere surface behaves as a concave spherical mirror with the required curvature for retroreflection. In practice, the optimal index of refraction may be lower than n_i + 1 ≅ 2 due to several factors. For one, it is sometimes preferable to have an imperfect, slightly divergent retroreflection, as in the case of road signs, where the illumination and observation angles are different. Due to spherical aberration, there also exists a radius from the centerline at which incident rays are focused at the center of the rear surface of the sphere. Finally, high index materials have higher Fresnel reflection coefficients, so the efficiency of coupling of the light from the ambient into the sphere decreases as the index becomes higher. Commercial retroreflective beads thus vary in index from around 1.5 (common forms of glass) up to around 1.9 (commonly barium titanate glass).

The spherical aberration problem with the spherical cat's eye can be solved in various ways, one being a spherically symmetrical index gradient within the sphere, such as in the Luneburg lens design. Practically, this can be approximated by a concentric sphere system.^[2]

Because the back-side reflection for an uncoated sphere is imperfect, it is fairly common to add a metallic coating to the back half of retroreflective spheres to increase the reflectance, but this implies that the retroreflection only works when the sphere is oriented in a particular direction.

An alternative form of the cat's eye retroreflector uses a normal lens focused onto a curved mirror rather than a transparent sphere, though this type is much more limited in the range of incident angles that it retroreflects.

The term cat's eye derives from the resemblance of the cat's eye retroreflector to the optical system that produces the well-known phenomenon of "glowing eyes" or eyeshine in cats and other vertebrates (which are only reflecting light, rather than actually glowing). The combination of the eye's lens and the cornea form the refractive converging system, while the tapetum lucidum behind the retina forms the spherical concave mirror. Because the function of the eye is to form an image on the retina, an eye focused on a distant object has a focal surface that approximately follows the reflective tapetum lucidum structure,^{[citation needed]} which is the condition required to form a good retroreflection.

This type of retroreflector can consist of many small versions of these structures incorporated in a thin sheet or in paint. In the case of paint containing glass beads, the paint adheres the beads to the surface where retroreflection is required and the beads protrude, their diameter being about twice the thickness of the paint.

Phase-conjugate mirror

A third, much less common way of producing a retroreflector is to use the nonlinear optical phenomenon of phase conjugation. This technique is used in advanced optical systems such as high-power lasers and optical transmission lines. Phase-conjugate mirrors^[3] reflects an incoming wave so that the reflected wave exactly follows the path it has previously taken, and require a comparatively expensive and complex apparatus, as well as large quantities of power (as nonlinear optical processes can be efficient only at high enough intensities). However, phase-conjugate mirrors have an inherently much greater accuracy in the direction of the retroreflection, which in passive elements is limited by the mechanical accuracy of the construction. 

https://en.wikipedia.org/wiki/Retroreflector

 

"Aura" around the shadow of a hot-air balloon, caused by retroreflection from dewdrops

 

"Aura" around the shadow of a hot-air balloon, caused by retroreflection from dewdrops

 https://en.wikipedia.org/wiki/Retroreflector#/media/File:Balloon_shadow.jpg

 

Free-space optical communication

Modulated retroreflectors, in which the reflectance is changed over time by some means, are the subject of research and development for free-space optical communications networks. The basic concept of such systems is that a low-power remote system, such as a sensor mote, can receive an optical signal from a base station and reflect the modulated signal back to the base station. Since the base station supplies the optical power, this allows the remote system to communicate without excessive power consumption. Modulated retroreflectors also exist in the form of modulated phase-conjugate mirrors (PCMs). In the latter case, a "time-reversed" wave is generated by the PCM with temporal encoding of the phase-conjugate wave (see, e.g., SciAm, Oct. 1990, "The Photorefractive Effect," David M. Pepper, et al.).

Inexpensive corner-aiming retroreflectors are used in user-controlled technology as optical datalink devices. Aiming is done at night, and the necessary retroreflector area depends on aiming distance and ambient lighting from street lamps. The optical receiver itself behaves as a weak retroreflector because it contains a large, precisely focused lens that detects illuminated objects in its focal plane. This allows aiming without a retroreflector for short ranges.

Other uses

Retroreflectors are used in the following example applications:

• In common (non-SLR) digital cameras, the sensor system is often retroreflective. Researchers have used this property to demonstrate a system to prevent unauthorized photographs by detecting digital cameras and beaming a highly focused beam of light into the lens.^[25]
• In movie screens to allow for high brilliance under dark conditions.^[26]
• Digital compositing programs and chroma key environments use retroreflection to replace traditional lit backdrops in composite work as they provide a more solid colour without requiring that the backdrop be lit separately.^[27]
• In Longpath-DOAS systems retroreflectors are used to reflect the light emitted from a lightsource back into a telescope. It is then spectrally analyzed to obtain information about the trace gas content of the air between the telescope and the retro reflector.
• Barcode labels can be printed on retroreflective material to increase the range of scanning up to 50 feet.^[28]
• In a form of 3D display; where a retro-reflective sheeting and a set of projectors is used to project stereoscopic images back to user's eye. The use of mobile projectors and positional tracking mounted on user's spectacles frame allows the illusion of a hologram to be created for computer generated imagery.^[29][30][31]
• Flashlight fish of the family Anomalopidae have natural retroreflectors. See tapetum lucidum.

https://en.wikipedia.org/wiki/Retroreflector

 

Heiligenschein (German: [ˈhaɪlɪɡn̩ˌʃaɪn]; lit. 'halo, aureola') is an optical phenomenon in which a bright spot appears around the shadow of the viewer's head in the presence of dew. In photogrammetry and remote sensing, it is more commonly known as the hotspot. It is also occasionally known as Cellini's halo after the Italian artist and writer Benvenuto Cellini (1500–1571), who described the phenomenon in his memoirs in 1562.^[1]

Nearly spherical dew droplets act as lenses to focus the light onto the surface behind them. When this light scatters or reflects off that surface, the same lens re-focuses that light into the direction from which it came. This configuration is sometimes called a cat's eye retroreflector. Any retroreflective surface is brightest around the antisolar point.

Opposition surge by other particles than water and the glory in water vapour are similar effects caused by different mechanisms. 

Heiligenschein, or hotspot, around the shadow of a hot-air balloon cast on a field of standing crops (Oxfordshire, England)

 https://en.wikipedia.org/wiki/Heiligenschein

 

A 22° halo around the Sun, observed over Bretton Woods, New Hampshire, USA on February 13, 2021

https://en.wikipedia.org/wiki/Halo_(optical_phenomenon)

Optical phenomena are any observable events that result from the interaction of light and matter.

All optical phenomena coincide with quantum phenomena.^[1] Common optical phenomena are often due to the interaction of light from the sun or moon with the atmosphere, clouds, water, dust, and other particulates. One common example is the rainbow, when light from the sun is reflected and refracted by water droplets. Some phenomena, such as the green ray, are so rare they are sometimes thought to be mythical.^[2] Others, such as Fata Morganas, are commonplace in favored locations.

Other phenomena are simply interesting aspects of optics, or optical effects. For instance, the colors generated by a prism are often shown in classrooms. 

A 22° halo around the Moon in Atherton, California

 https://en.wikipedia.org/wiki/Optical_phenomena

 

A Fata Morgana (Italian: [ˈfaːta morˈɡaːna]) is a complex form of superior mirage visible in a narrow band right above the horizon. The term Fata Morgana is the Italian translation of "Morgan the Fairy" (Morgan le Fay of Arthurian legend). These mirages are often seen in the Italian Strait of Messina, and were described as fairy castles in the air or false land conjured by her magic.

Fata Morgana mirages significantly distort the object or objects on which they are based, often such that the object is completely unrecognizable. A Fata Morgana may be seen on land or at sea, in polar regions, or in deserts. It may involve almost any kind of distant object, including boats, islands, and the coastline. Often, a Fata Morgana changes rapidly. The mirage comprises several inverted (upside down) and erect (right-side up) images that are stacked on top of one another. Fata Morgana mirages also show alternating compressed and stretched zones.^[1]

The optical phenomenon occurs because rays of light bend when they pass through air layers of different temperatures in a steep thermal inversion where an atmospheric duct has formed.^[1] In calm weather, a layer of significantly warmer air may rest over colder dense air, forming an atmospheric duct that acts like a refracting lens, producing a series of both inverted and erect images. A Fata Morgana requires a duct to be present; thermal inversion alone is not enough to produce this kind of mirage. While a thermal inversion often takes place without there being an atmospheric duct, an atmospheric duct cannot exist without there first being a thermal inversion. 

A Fata Morgana seen over the Baltic Sea, 2016. The mirage consists of multiple upright and inverted images over the original object

A Fata Morgana of a cargo ship seen off the coast of Oceanside, California

 Schematic diagram explaining the Fata Morgana mirage

A sequence of a Fata Morgana of the Farallon Islands as seen from San Francisco

The above sequence as an animation.

 

 https://en.wikipedia.org/wiki/Fata_Morgana_(mirage)

 

Observing a Fata Morgana

Schematic diagram explaining the Fata Morgana mirage

A Fata Morgana is most commonly seen in polar regions, especially over large sheets of ice that have a uniform low temperature. It may, however, be observed in almost any area. In polar regions the Fata Morgana phenomenon is observed on relatively cold days. In deserts, over oceans, and over lakes, however, a Fata Morgana may be observed on hot days.

To generate the Fata Morgana phenomenon, the thermal inversion has to be strong enough that the curvature of the light rays within the inversion layer is stronger than the curvature of the Earth.^[1] Under these conditions, the rays bend and create arcs. An observer needs to be within or below an atmospheric duct in order to be able to see a Fata Morgana.^[2] Fata Morgana may be observed from any altitude within the Earth's atmosphere, from sea level up to mountaintops, and even including the view from airplanes.^[3][4]

A sequence of a Fata Morgana of the Farallon Islands as seen from San Francisco

The above sequence as an animation.

A Fata Morgana may be described as a very complex superior mirage with more than three distorted erect and inverted images.^[1] Because of the constantly changing conditions of the atmosphere, a Fata Morgana may change in various ways within just a few seconds of time, including changing to become a straightforward superior mirage. The sequential image here shows sixteen photographic frames of a mirage of the Farallon Islands as seen from San Francisco; the images were all taken on the same day. In the first fourteen frames, elements of the Fata Morgana mirage display alternations of compressed and stretched zones.^[1] The last two frames were photographed a few hours later, around sunset time. At that point in time, the air was cooler while the ocean was probably a little bit warmer, which caused the thermal inversion to be not as extreme as it was few hours before. A mirage was still present at that point, but it was not so complex as a few hours before sunset: the mirage was no longer a Fata Morgana, but instead had become a simple superior mirage.

Fata Morgana mirages are visible to the naked eye, but in order to be able to see the detail within them, it is best to view them through binoculars, a telescope, or as is the case in the images here, through a telephoto lens. Gabriel Gruber (1740–1805) and Tobias Gruber [sl] (1744–1806), who observed Fata Morgana above Lake Cerknica, were the first to study it in a laboratory setting. 

https://en.wikipedia.org/wiki/Fata_Morgana_(mirage)

 

Etymology

La Fata Morgana ("The Fairy Morgana") is the Italian name of Morgan le Fay, also known as Morgana and other variants, who was described as a powerful sorceress in the Arthurian legend. As her name indicates, the figure of Morgan appears to have been originally a fairy figure rather than a human woman. The early works featuring Morgan do not elaborate on her nature, other than describing her role as that of a fairy or magician. Later, she was described as a King Arthur's half-sister and an enchantress.^[5] After King Arthur's final battle at Camlann, Morgan takes her half-brother Arthur to Avalon.^[6] In medieval times, suggestions for the location of Avalon included the other side of the Earth at the antipodes, Sicily, and other locations in the Mediterranean.^[7] Legends claimed that sirens in the waters around Sicily lured the unwary to their death. Morgan is associated not only with Sicily's Mount Etna (the supposedly hollow mountain locally identified as Avalon since the 12th century^[8]), but also with sirens. In a medieval French Arthurian romance of the 13th century, Floriant et Florete, she is called "mistress of the fairies of the salt sea" (La mestresse [des] fées de la mer salée).^[9] Ever since that time, Fata Morgana has been associated with Sicily in the Italian folklore and literature.^[10] For example, a local legend connects Morgan and her magical mirages with Roger I of Sicily and the Norman conquest of the island from the Arabs.^[11][12]

An 1844 drawing entitled The Fata Morgana, As Observed in the Harbour of Messina

Walter Charleton, in his 1654 treatise "Physiologia Epicuro-Gassendo-Charltoniana", devotes several pages to the description of the Morgana of Rhegium, in the Strait of Messina (Book III, Chap. II, Sect. II). He records that a similar phenomenon was reported in Africa by Diodorus Siculus, a Greek historian writing in the first century BC, and that the Rhegium Fata Morgana was described by Damascius, a Greek philosopher of the sixth century AD. In addition, Charleton tells us that Athanasius Kircher described the Rhegium mirage in his book of travels.

An early mention of the term Fata Morgana in English, in 1818, referred to such a mirage noticed in the Strait of Messina, between Calabria and Sicily.^[13]

• Fata Morgana, phr. : It. : a peculiar mirage occasionally seen on the coasts of the Straits of Messina, locally attributed to a fay Morgana. Hence, metaph. any illusory appearance. 1818 In mountainous regions, deceptions of sight, Fata Morgana, &c., are more common: In E. Burl's Lett. N. Scotl., Vol. II. p. in (1818).

 

https://en.wikipedia.org/wiki/Fata_Morgana_(mirage)

 

 

Famous legends and observations

The Flying Dutchman

Main article: Flying Dutchman

The Flying Dutchman, according to folklore, is a ghost ship that can never go home, and is doomed to sail the seven seas forever. The Flying Dutchman is usually spotted from afar, sometimes seen to be glowing with ghostly light. One of the possible explanations of the origin of the Flying Dutchman legend is a Fata Morgana mirage seen at sea.^[14]

A nineteenth-century book illustration, showing enlarged superior mirages; mirages can never be so far above the horizon, and a superior mirage can never increase the length of an object as shown on the right

A Fata Morgana superior mirage of a ship can take many different forms. Even when the boat in the mirage does not seem to be suspended in the air, it still looks ghostly, and unusual, and what is even more important, it is ever-changing in its appearance. Sometimes a Fata Morgana causes a ship to appear to float inside the waves, at other times an inverted ship appears to sail above its real companion.

In fact, with a Fata Morgana it can be hard to say which individual segment of the mirage is real and which is not real: when a real ship is out of sight because it is below the horizon line, a Fata Morgana can cause the image of it to be elevated, and then everything which is seen by the observer is a mirage. On the other hand, if the real ship is still above the horizon, the image of it can be duplicated many times and elaborately distorted by a Fata Morgana.

• 

  The appearance of two ships changing owing to the Fata Morgana phenomenon: the four frames in the first column are of ship No. 1, and the four frames in the second column are of ship No. 2

• 

  In this mirage, at least three separate images of a boat are visible. The real one at the bottom and the uppermost one are in the upright position, whereas the one in the middle is inverted

• 

  Superimposed detail from six frames of a view showing how the miraged image of a ship changes from one moment to the next

Phantom islands

A Fata Morgana of the sea surface and sun glitter, with a boat at the left hand side of the image

In the 19th and early 20th centuries, Fata Morgana mirages may have played a role in a number of unrelated "discoveries" of arctic and antarctic land masses which were later shown not to exist.^{[citation needed]} Icebergs frozen into the pack ice, or the uneven surface of the ice itself, may have contributed to the illusion of distant land features.

Sannikov Land

Main article: Sannikov Land

Yakov Sannikov and Matvei Gedenschtrom claimed to have seen a land mass north of Kotelny Island during their 1809–1810 cartographic expedition to the New Siberian Islands. Sannikov reported this sighting of a "new land" in 1811, and the supposed island was named after him.^[15] Three-quarters of a century later, in 1886, Baron Eduard Toll, a Baltic German explorer in Russian service, reported observing Sannikov Land during another expedition to the New Siberian Islands. In 1900, he would lead still another expedition to the region, which had among its objectives the location and exploration of Sannikov Land.^[16] The expedition was unsuccessful in this respect.^[17] Toll and three others were lost after they departed their ship, which was stuck in ice for the winter, and embarked on a risky expedition by dog sled.^[18] In 1937, the Soviet icebreaker Sadko also tried and failed to find Sannikov Land.^[19] Some historians and geographers have theorised that the land mass that Sannikov and Toll saw was actually Fata Morganas of Bennett Island.^[15]

Croker Mountains

In 1818, Sir John Ross led an expedition to discover the long-sought-after Northwest Passage. When he reached Lancaster Sound in Canada, he sighted, in the distance, a land mass with mountains, directly ahead in the ship's course. He named the mountain range the Croker Mountains,^[20] after First Secretary to the Admiralty John Wilson Croker, and ordered the ship to turn around and return to England. Several of his officers protested, including First Mate William Edward Parry and Edward Sabine, but they could not dissuade him.^[21] The account of Ross's voyage, published a year later, brought to light this disagreement, and the ensuing controversy over the existence of the Croker Mountains ruined Ross's reputation. The year after Ross's expedition, in 1819, Parry was given command of his own Arctic expedition, and proved Ross wrong by continuing west beyond where Ross had turned back, and sailing through the supposed location of the Croker Mountains. The mountain range that had caused Ross to abandon his mission had been a mirage.

Ross made two errors. First, he refused to listen to the counsel of his officers, who may have been more familiar with mirages than he was. Second, his attempt to honour Croker by naming a mountain range after him backfired when the mountains turned out to be non-existent. Ross could not obtain ships, or funds, from the government for his subsequent expeditions, and was forced to rely on private backers instead.^[22]

New South Greenland

Main article: New South Greenland

Benjamin Morrell reported that, in March 1823, while on a voyage to the Antarctic and southern Pacific Ocean, he had explored what he thought was the east coast of New South Greenland.^[23] The west coast of New South Greenland had been explored two years earlier by Robert Johnson, who had given the land its name.^[24] This name was not adopted, however, and the area, which is the northern part of the Antarctic Peninsula, is now known as Graham Land.^[25] Morrell's reported position was actually far to the east of Graham Land.^[26] Searches for the land that Morrell claimed to have explored would continue into the early 20th century before New South Greenland's existence was conclusively disproven. Why Morrell reported exploring a non-existent land is unclear, but one possibility is that he mistook a Fata Morgana for actual land.^[27]

Crocker Land

Main article: Crocker Land Expedition

Robert Peary claimed to have seen, while on a 1906 Arctic expedition, a land mass in the distance. He said that it was north-west from the highest point of Cape Thomas Hubbard, which is situated in what is now the northern Canadian territory of Nunavut, and he estimated it to be 210 km (130 miles) away, at about 83 degrees N, longitude 100 degrees W. He named it Crocker Land, after George Crocker of the Peary Arctic Club.^[28] As Peary's diary contradicts his public claim that he had sighted land,^[29] it is now believed that Crocker Land was a fraudulent invention of Peary,^[30] created in an unsuccessful attempt to secure further funding from Crocker.

In 1913, unaware that Crocker Land was merely an invention, Donald Baxter MacMillan organised the Crocker Land Expedition, which set out to reach and explore the supposed land mass. On 21 April, the members of the expedition did, in fact, see what appeared to be a huge island on the north-western horizon. As MacMillan later said, "Hills, valleys, snow-capped peaks extending through at least one hundred and twenty degrees of the horizon". Piugaattoq, a member of the expedition and an Inuit hunter with 20 years of experience of the area, explained that it was just an illusion. He called it poo-jok, which means 'mist'. However, MacMillan insisted that they press on, even though it was late in the season and the sea ice was breaking up. For five days they went on, following the mirage. Finally, on 27 April, after they had covered some 200 km (125 miles) of dangerous sea ice, MacMillan was forced to admit that Piugaattoq was right—the land that they had sighted was in fact a mirage (probably a Fata Morgana). Later, MacMillan wrote:

from Four Years in the White North^[31]

The day was exceptionally clear, not a cloud or trace of mist; if land could be seen, now was our time. Yes, there it was! It could even be seen without a glass, extending from southwest true to north-northeast. Our powerful glasses, however, brought out more clearly the dark background in contrast with the white, the whole resembling hills, valleys and snow-capped peaks to such a degree that, had we not been out on the frozen sea for 150 miles [240 km], we would have staked our lives upon its reality. Our judgment then, as now, is that this was a mirage or loom of the sea ice.

The expedition collected interesting samples, but is still considered to be a failure and a very expensive mistake. The final cost was $100,000 (equivalent to $2,000,000 in 2021).^[32]

Hy Brasil

Hy Brasil is an island that was said to appear once every few years off the coast of Co. Kerry, Ireland. Hy Brasil has been drawn on ancient maps as a perfectly circular island with a river running directly through it.

Lake Ontario

Lake Ontario is said to be famous for mirages, with opposite shorelines becoming clearly visible during the events.^[33]

In July 1866, mirages of boats and islands were seen from Kingston, Ontario.^[34]

A Mirage – The atmospheric phenomenon known as "mirage" might have been observed on Sunday evening between 6 and 7 o'clock, by looking towards the lake. The line beyond which this phenomenon was observable seemed to strike from about the middle portion of Amherst Island across to the southeast, for while the lower half of the island presented its usual appearance, the upper half was unnaturally distorted and thrown upward in columnar shape with an apparent height of two to three hundred feet. The upper line or cloud from this elevation stretched southward, upon which was thrown the image of objects. A barque sailing in front of this cloud presented a double appearance. While she appeared slightly distorted on the surface of the water, her image was inverted upon the background of the cloud referred to, and both blending together produced a curious sight. At the same time the ship and its shadow were again repeated in a more shadowy form, but distinct, in the foreground, the base being a line of smooth water. Another bark whose hull was entirely below the horizon, the topsails alone being visible, had its hull shadowed on this foreground, but no inversion in this case could be observed. It may be added that these optical phenomena in regard to the vessels could only be seen with the aid of a telescope, for the nearest vessel was at the time fully sixteen miles [26 km] distant. The phenomena lasted over an hour, the illusion changing every moment in its character.

Here the described mirages of vessels "could only be seen with the aid of a telescope". It is often the case when observing a Fata Morgana that one needs to use a telescope or binoculars to really make out the mirage. The "cloud" that the article mentions a few times probably refers to a duct.

On 25 August 1894, Scientific American described a "remarkable mirage" seen by the citizens of Buffalo, New York.^[35][36]

Mirage of the Canadian coast as seen from Rochester, New York on 16 April 1871

The people of Buffalo, N.Y., were treated to a remarkable mirage, between ten and eleven o'clock, on the morning of 16 August, [1894]. It was the city of Toronto with its harbor and small island to the south of the city. Toronto is fifty-six miles [90 km] from Buffalo, but the church spires could be counted with the greatest ease. The mirage took in the whole breadth of Lake Ontario, Charlotte, the suburbs of Rochester, being recognized as a projection east of Toronto. A side–wheel steamer could be seen traveling in a line from Charlotte to Toronto Bay. Two dark objects were at last found to be the steamers of the New York Central plying between Lewiston and Toronto. A sail-boat was also visible and disappeared suddenly. Slowly the mirage began to fade away, to the disappointment of thousands who crowded the roofs of houses and office buildings. A bank of clouds was the cause of the disappearance of the mirage. A close examination of the map showed the mirage did not cause the slightest distortion, the gradual rise of the city from the water being rendered perfectly. It is estimated that at least 20,000 spectators saw the novel spectacle.

This mirage is what is known as that of the third order; that is, the object looms up far above the level and not inverted, as with mirages of the first and second orders, but appearing like a perfect landscape far away in the sky.

Scientific American, 25 August 1894.

This description might refer to looming owing to inversion rather than to an actual mirage.

McMurdo Sound and Antarctica

From McMurdo Station in Antarctica, Fata Morganas are often seen during the Antarctic spring and summer, across McMurdo Sound.^[37][38][39] An Antarctic Fata Morgana, seen from a C-47 transport flight, was recounted:

We were going along smoothly and all of a sudden a mountain peak seemed to rise up out of nowhere up ahead. We looked again and it was gone. A couple of minutes later it popped up again rising some 300 feet higher than our altitude. We never seemed to get any closer to it. The peak just kept popping up and down, getting higher and higher and higher every time it reappeared.

Rear Adm. Fred E. Bakutis, commanding the Antarctic Navy Support Activities^[37]

UFOs

Main article: Unidentified flying object

A Fata Morgana distorting the images of distant boats beyond recognition

Fata Morgana mirages may continue to trick some observers and are still sometimes mistaken for otherworldly objects such as UFOs.^[40] A Fata Morgana can display an object that is located below the astronomical horizon as an apparent object hovering in the sky. A Fata Morgana can also magnify such an object vertically and make it look absolutely unrecognizable.

Some UFOs which are seen on radar may also be due to Fata Morgana mirages. Official UFO investigations in France indicate:

As is well known, atmospheric ducting is the explanation for certain optical mirages, and in particular the arctic illusion called "fata morgana" where distant ocean or surface ice, which is essentially flat, appears to the viewer in the form of vertical columns and spires, or "castles in the air".

People often assume that mirages occur only rarely. This may be true of optical mirages, but conditions for radar mirages are more common, due to the role played by water vapor which strongly affects the atmospheric refractivity in relation to radio waves. Since clouds are closely associated with high levels of water vapor, optical mirages due to water vapor are often rendered undetectable by the accompanying opaque cloud. On the other hand, radar propagation is essentially unaffected by the water droplets of the cloud so that changes in water vapor content with altitude are very effective in producing atmospheric ducting and radar mirages.^[41]

Australia

Fata Morgana mirages could explain the mysterious Australian Min Min light phenomenon.^[42] This would also explain the way in which the legend has changed over time: The first reports were of a stationary light, which in a Fata Morgana effect would be an image of a campfire. In more recent reports this has changed to moving lights, which in an inversion reflection such as Fata Morgana would be headlights over the horizon being reflected by the inversion.

Greenland

Fata Morgana Land is a phantom island in the Arctic, reported first in 1907. After an unfruitful search, it was deemed to be Tobias Island.^[43]

In literature

A Fata Morgana is usually associated with something mysterious, something that never could be approached.^[44]

An unrealistic 1886 drawing of a "Fata Morgana" mirage in a desert

O sweet illusions of song
That tempt me everywhere,
In the lonely fields, and the throng
Of the crowded thoroughfare!

I approach and ye vanish away,
I grasp you, and ye are gone;
But ever by night and by day,
The melody soundeth on.

As the weary traveler sees
In desert or prairie vast,
Blue lakes, overhung with trees
That a pleasant shadow cast;

Fair towns with turrets high,
And shining roofs of gold,
That vanish as he draws nigh,
Like mists together rolled—

So I wander and wander along,
And forever before me gleams
The shining city of song,
In the beautiful land of dreams.

But when I would enter the gate
Of that golden atmosphere,
It is gone, and I wonder and wait
For the vision to reappear.

— Henry Wadsworth Longfellow^[45], Fata Morgana (1873)

In the lines, "the weary traveller sees / In desert or prairie vast, / Blue lakes, overhung with trees / That a pleasant shadow cast", because of the mention of blue lakes, it is clear that the author is actually describing not a Fata Morgana, but rather a common inferior or desert mirage. The 1886 drawing shown here of a "Fata Morgana" in a desert might have been an imaginative illustration for the poem, but in reality no mirage ever looks like this. Andy Young writes, "They're always confined to a narrow strip of sky—less than a finger's width at arm's length—at the horizon."^[1]

The 18th-century poet Christoph Martin Wieland wrote about "Fata Morgana's castles in the air". The idea of castles in the air was probably so irresistible that many languages still use the phrase Fata Morgana to describe a mirage.^[9]

In the book Thunder Below! about the submarine USS Barb, the crew sees a Fata Morgana (called an "arctic mirage" in the book) of four ships trapped in the ice. As they try to approach the ships the mirage vanishes.^[46]

The Fata Morgana is briefly mentioned in the 1936 H. P. Lovecraft horror novel At the Mountains of Madness, in which the narrator states: "On many occasions the curious atmospheric effects enchanted me vastly; these including a strikingly vivid mirage—the first I had ever seen—in which distant bergs became the battlements of unimaginable cosmic castles."

See also

• Atmospheric optics
• Brocken spectre
• Fata Morgana (1971 film)
• Green flash
• Looming and similar refraction phenomena
• Mirage of astronomical objects
• Summerland (2020 film)

https://en.wikipedia.org/wiki/Fata_Morgana_(mirage)

A Brocken spectre (British English; American spelling: Brocken specter; German: Brockengespenst), also called Brocken bow, mountain spectre, or spectre of the Brocken is the magnified (and apparently enormous) shadow of an observer cast in mid air upon any type of cloud opposite a strong light source. Additionally, if the cloud consists of water droplets backscattered, a bright area called Heiligenschein, and halo-like rings of rainbow coloured light called a glory can be seen around the head or apperature silhouette of the spectre. Typically the spectre appears in sunlight opposite the sun's direction at the antisolar point.

The phenomenon can appear on any misty mountainside, cloud bank, or be seen from an aircraft, but the frequent fogs and low-altitude accessibility of the Brocken, a peak in the Harz Mountains in Germany, have created a local legend from which the phenomenon draws its name. The Brocken spectre was observed and described by Johann Silberschlag in 1780, and has often been recorded in literature about the region.

 

A Brocken spectre within glory rings

 

 

Occurrence

A semi-artificial Brocken spectre created by standing in front of the headlight of a car, on a foggy night.

The "spectre" appears when the sun shines from behind the observer, who is looking down from a ridge or peak into mist or fog.^[1] The light projects their shadow through the mist, often in a triangular shape due to perspective.^[2] The apparent magnification of size of the shadow is an optical illusion that occurs when the observer judges their shadow on relatively nearby clouds to be at the same distance as faraway land objects seen through gaps in the clouds, or when there are no reference points by which to judge its size. The shadow also falls on water droplets of varying distances from the eye, confusing depth perception. The ghost can appear to move (sometimes suddenly) because of the movement of the cloud layer and variations in density within the cloud. 

 

References in popular culture and the arts

Samuel Taylor Coleridge's poem "Constancy to an Ideal Object" concludes with an image of the Brocken spectre:

And art thou nothing? Such thou art, as when
The woodman winding westward up the glen
At wintry dawn, where o'er the sheep-track's maze
The viewless snow-mist weaves a glist'ning haze,
Sees full before him, gliding without tread,
An image with a glory round its head;
The enamoured rustic worships its fair hues,
Nor knows he makes the shadow he pursues!

Another night Brocken spectre created by headlights of a car.

Lewis Carroll's "Phantasmagoria" includes a line about a Spectre who "...tried the Brocken business first/but caught a sort of chill/so came to England to be nursed/and here it took the form of thirst/which he complains of still."

Stanisław Lem's Fiasco (1986) has a reference to the "Brocken Specter": "He was alone. He had been chasing himself. Not a common phenomenon, but known even on Earth. The Brocken Specter in the Alps, for example." The situation, of pursuing one's self, via a natural illusion is a repeated theme in Lem. A scene of significance in his book The Investigation (1975) depicts a detective who, within the confines of a snowy, dead-end alley, confronts a man who turns out to be the detective's own reflection, "The stranger... was himself. He was standing in front of a huge mirrored wall marking the end of the arcade."

In The Radiant Warrior (1989), part of Leo Frankowski's Conrad Stargard series, the protagonist uses the Brocken Spectre to instill confidence in his recruits.

The Brocken spectre is a key trope in Paul Beatty's The White Boy Shuffle (1996), in which a character, Nicholas Scoby, declares that his dream (he specifically calls it a "Dream and a half, really") is to see his glory through a Brocken spectre (69).

In James Hogg's novel The Private Memoirs and Confessions of a Justified Sinner (1824) the Brocken spectre is used to suggest psychological horror.

Shadow of an aeroplane cast by the sun on nearby clouds

Brocken spectre observed from an aeroplane.

A solar glory and brocken spectre from Crib Goch in 2008

Carl Jung in Memories, Dreams, Reflections wrote:

... I had a dream which both frightened and encouraged me. It was night in some unknown place, and I was making slow and painful headway against a mighty wind. Dense fog was flying along everywhere. I had my hands cupped around a tiny light which threatened to go out at any moment... Suddenly I had the feeling that something was coming up behind me. I looked back, and saw a gigantic black figure following me... When I awoke I realized at once that the figure was a "specter of the Brocken," my own shadow on the swirling mists, brought into being by the little light I was carrying.^[3]

In Gravity's Rainbow, Geli Tripping and Slothrop make "god-shadows" from a Harz precipice, as Walpurgisnacht wanes to dawn. Additionally, the French–Canadian quadruple agent Rémy Marathe muses episodically about the possibility of witnessing the fabled spectre on the mountains of Tucson in David Foster Wallace's novel Infinite Jest.

The explorer Eric Shipton saw a Brocken Spectre during his first ascent of Nelion on Mount Kenya with Percy Wyn-Harris and Gustav Sommerfelt in 1929. He wrote:

Then the towering buttresses of Batian and Nelion appeared; the rays of the setting sun broke through and, in the east, sharply defined, a great circle of rainbow colours framed our own silhouettes. It was the only perfect Brocken Spectre I have ever seen.^[4]

The progressive metal band Fates Warning makes numerous references to the Brocken Spectre in both their debut album title Night on Bröcken and in lyrics on a subsequent song called "The Sorceress" from the album Awaken the Guardian that read "Through the Brocken Spectre rose a luring Angel."

The design of Kriemhild Gretchen, a Witch in the anime series Puella Magi Madoka Magica, may have been inspired by the Brocken spectre.^[5]

In Charles Dickens's Little Dorrit, Book II Chapter 23, Flora Finching, in the course of one of her typically free-associative babbles to Mr Clennam, says " ... ere yet Mr F appeared a misty shadow on the horizon paying attentions like the well-known spectre of some place in Germany beginning with a B ... "

"Brocken Spectre" is the title of a track on David Tipper's 2010 album Broken Soul Jamboree.

In the manga and anime Tensou Sentai Goseiger, Brocken Spectres were one of the enemies that Gosei Angels must face.

In the manga One Piece, Brocken spectres make an appearance in the Skypiea story arc.

In the anime Detective Conan, Brocken spectres are mentioned in episode 348 and episode 546 as well.

In "The Problem of Pain" by C.S. Lewis the Brocken spectre is mentioned in the chapter "Heaven".

In chapter 12 of Whose Body? (Lord Peter Wimsey) by Dorothy L. Sayers.

See also

• Diffraction
• Earth's shadow, the shadow that the Earth itself casts on its atmosphere
• Am Fear Liath Mòr, "Big Grey Man" in Scottish Gaelic, a supposed supernatural being found on Scotland's second-highest peak, Ben Macdhui
• Dark Watchers, supposed supernatural beings seen along the crest of the Santa Lucia Mountains, in California
• Gegenschein
• Glory (optical phenomenon)
• Heiligenschein, an optical phenomenon which creates a bright spot around the shadow of the viewer's head
• Opposition surge, the brightening of a rough surface, or an object with many particles, when illuminated from directly behind the observer

References

1. 
   

McKenzie, Steven (17 February 2015). "Shades of grey: What is the brocken spectre". BBC News Online. Retrieved 3 January 2020.

"Brocken spectre". atoptics.co.uk.

Jung, Carl; Jaffé, Aniela (1989). Memories, Dreams, Reflections. Vintage. p. 88.

Shipton, Eric (1969). That Untravelled World. Hodder and Stoughton. pp. 55–56.

5. "Kriemhild Gretchen". puella-magi.net.

Further reading

• Shenstone, A.G (1954). "The Brocken Spectre". Science. 119 (3094): 511–512. Bibcode:1954Sci...119..511S. doi:10.1126/science.119.3094.511. PMID 17842741.
• Goodrich, Samuel Griswold (1851). Peter Parley's Wonders of the Sea and Sky. Archived from the original on 2007-12-25. Retrieved 2007-07-26.
• Minnaert, M. (1954). The Nature of Light and Colour in the Open Air (Paperback). Dover Books on Earth Sciences. [New York] Dover Publications. ISBN 9780486201962.
• Greenler, R (1980). Rainbows, Halos, and Glories. Cambridge University Press.
• Dunlop, Storm (2002). The Weather Identification Handbook. Harper Collins UK. p. 141. ISBN 1-58574-857-9.

External links

Wikimedia Commons has media related to Brocken spectres.

• "What are Brocken Spectres and How Do They Form?", an article on the Online Fellwalking Club page (dead link, 2012 archived version)
• A Cairngorm example, from the Universities Space Research Association
• See a picture and a YouTube videoclip taken by Michael Elcock here [1]
• Snowdon walker captures rare Brocken spectre, BBC News Online, 3 January 2020
• Brocken Spectre panorama
• "Time-lapse video of Brocken specter cast by Mt. Tamalpais fire lookout in Marin County California."

Categories:

• Optical illusions
• Atmospheric optical phenomena

https://en.wikipedia.org/wiki/Brocken_spectre

While mirages are the best known atmospheric refraction phenomena, looming and similar refraction phenomena do not produce mirages. Mirages show an extra image or images of the miraged object, while looming, towering, stooping, and sinking do not. No inverted image is present in those phenomena either. Depending on atmospheric conditions, the objects can appear to be elevated or lowered, stretched or stooped. These phenomena can occur together, changing the appearance of different parts of the objects in different ways. Sometimes these phenomena can occur together with a true mirage.^[1][2]

The distant boats appear to be floating in the sky as a result of looming and other refraction phenomena.

 https://en.wikipedia.org/wiki/Looming_and_similar_refraction_phenomena

 

The green flash and green ray are meteorological optical phenomena that sometimes occur transiently around the moment of sunset or sunrise. When the conditions are right, a distinct green spot is briefly visible above the Sun's upper limb; the green appearance usually lasts for no more than two seconds. Rarely, the green flash can resemble a green ray shooting up from the sunset or sunrise point.

Green flashes occur because the Earth's atmosphere can cause the light from the Sun to separate, or refract, into different colors. Green flashes are a group of similar phenomena that stem from slightly different causes, and therefore, some types of green flashes are more common than others.^[1] 

The development of a green flash at sunset in San Francisco

 

A green flash in Santa Cruz, California

 https://en.wikipedia.org/wiki/Green_flash

 

Atmospheric optical phenomena

This section is an excerpt from List of atmospheric optical phenomena.[edit]

A circumzenithal arc over Grand Forks, North Dakota

The Belt of Venus over Paranal Observatory atop Cerro Paranal in the Atacama Desert, northern Chile^[3]

Atmospheric optical phenomena include:

• Afterglow
• Airglow
• Alexander's band, the dark region between the two bows of a double rainbow.
• Alpenglow
• Anthelion
• Anticrepuscular rays
• Aurora
• Auroral light (northern and southern lights, aurora borealis and aurora australis)
• Belt of Venus
• Brocken Spectre
• Circumhorizontal arc
• Circumzenithal arc
• Cloud iridescence
• Crepuscular rays
• Earth's shadow
• Earthquake lights
• Glories
• Green flash
• Halos, of Sun or Moon, including sun dogs
• Haze
• Heiligenschein or halo effect, partly caused by the opposition effect
• Ice blink
• Light pillar
• Lightning
• Mirages (including Fata Morgana)
• Monochrome Rainbow
• Moon dog
• Moonbow
• Nacreous cloud/Polar stratospheric cloud
• Rainbow
• Subsun
• Sun dog
• Tangent arc
• Tyndall effect
• Upper-atmospheric lightning, including red sprites, Blue jets, and ELVES
• Water sky

A double rainbow at Minsi Lake, Pennsylvania

A sun pillar in Finistère, Brittany

Atmospheric optical phenomenon

Non-atmospheric optical phenomena

Green flash appears above the solar disc for a second or so. One such occurrence was taken from Cerro Paranal.

• Dichromatism
• Gegenschein
• Iridescence
• Opposition effect
• Shadow
  • Shade
  • Silhouette
• Sylvanshine
• Zodiacal light

Other optical effects

• Asterism, star gems such as star sapphire or star ruby
• Aura, a phenomenon in which gas or dust surrounding an object luminesces or reflects light from the object
• Aventurescence, also called the Schiller effect, spangled gems such as aventurine quartz and sunstone
• Baily's beads, grains of sunlight visible in total solar eclipses.
• camera obscura
• Cathodoluminescence
• Caustics
• Chatoyancy, cat's eye gems such as chrysoberyl cat's eye or aquamarine cat's eye
• Chromatic polarization
• Diffraction, the apparent bending and spreading of light waves when they meet an obstruction
• Dispersion
• Double refraction or birefringence of calcite and other minerals
• Double-slit experiment
• Electroluminescence
• Evanescent wave
• Fluorescence, also called luminescence or photoluminescence
• Mie scattering (Why clouds are white)
• Metamerism as of alexandrite
• Moiré pattern
• Newton's rings
• Phosphorescence
• Pleochroism gems or crystals, which seem "many-colored"
• Polarized light-related phenomena such as double refraction, or Haidinger's brush
• Rayleigh scattering (Why the sky is blue, sunsets are red, and associated phenomena)
• Reflection
• Refraction
• Sonoluminescence
  • Shrimpoluminescence
• Synchrotron radiation
• The separation of light into colors by a prism
• Triboluminescence
• Thomson scattering
• Total internal reflection
• Twisted light
• Umov effect
• Zeeman effect
• The ability of light to travel through space or through a vacuum.

Entoptic phenomena

Main article: Entoptic phenomenon

• Diffraction of light through the eyelashes
• Haidinger's brush
• Monocular diplopia (or polyplopia) from reflections at boundaries between the various ocular media
• Phosphenes from stimulation other than by light (e.g., mechanical, electrical) of the rod cells and cones of the eye or of other neurons of the visual system
• Purkinje images.

Optical illusions

Main article: Optical illusion

• The unusually large size of the Moon as it rises and sets, the moon illusion
• The shape of the sky, the sky bowl

Unexplained phenomena

See also: Forteana, Will-o'-the-wisp, and Unidentified flying object

Some phenomena are yet to be conclusively explained and may possibly be some form of optical phenomena. Some^{[weasel words]} consider many of these "mysteries" to simply be local tourist attractions that are not worthy of thorough investigation.^[4]

• Hessdalen lights^[5]
• Min Min lights^[6]
• Light of Saratoga^[7]
• Naga fireballs^[8]

See also

• List of optical topics
• Optics

https://en.wikipedia.org/wiki/Optical_phenomena

In folklore, a will-o'-the-wisp, will-o'-wisp or ignis fatuus (Latin for 'giddy flame'),^[1] plural ignes fatui, is an atmospheric ghost light seen by travellers at night, especially over bogs, swamps or marshes. The phenomenon is known in English folk belief, English folklore and much of European folklore by a variety of names, including jack-o'-lantern, friar's lantern and hinkypunk, and is said to mislead travellers by resembling a flickering lamp or lantern.^[2] In literature, will-o'-the-wisp metaphorically refers to a hope or goal that leads one on, but is impossible to reach, or something one finds strange or sinister.^[3]

Wills-o'-the-wisp appear in folk tales and traditional legends of numerous countries and cultures; notable wills-o'-the-wisp include St. Louis Light in Saskatchewan, the Spooklight in Southwestern Missouri and Northeastern Oklahoma, the Marfa lights of Texas, the Naga fireballs on the Mekong in Thailand, the Paulding Light in Upper Peninsula of Michigan and the Hessdalen light in Norway.

In urban legends, folklore and superstition, wills-o'-the-wisp are typically attributed to ghosts, fairies or elemental spirits. Modern science explains the light aspect as natural phenomena such as bioluminescence or chemiluminescence, caused by the oxidation of phosphine (PH₃), diphosphane (P₂H₄) and methane (CH₄) produced by organic decay. 

The Will o' the Wisp and the Snake by Hermann Hendrich (1854–1931)

 https://en.wikipedia.org/wiki/Will-o%27-the-wisp

 An optical vortex (also known as a photonic quantum vortex, screw dislocation or phase singularity) is a zero of an optical field; a point of zero intensity. The term is also used to describe a beam of light that has such a zero in it. The study of these phenomena is known as singular optics. 

Diagram of different modes, four of which are optical vortices. Columns show the helical structures, phase-front and intensity of the beams

 https://en.wikipedia.org/wiki/Optical_vortex

 

Synchrotron radiation (also known as magnetobremsstrahlung radiation) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (a ⊥ v). It is produced artificially in some types of particle accelerators, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over a large portion of the electromagnetic spectrum.^[1]

Pictorial representation of the radiation emission process by a source moving around a Schwarzschild black hole in a de Sitter universe.

Synchrotron radiation is similar to bremsstrahlung radiation, which is emitted by a charged particle when the acceleration is parallel to the direction of motion. The general term for radiation emitted by particles in a magnetic field is gyromagnetic radiation, for which synchrotron radiation is the ultra-relativistic special case. Radiation emitted by charged particles moving non-relativistically in a magnetic field is called cyclotron emission.^[2] For particles in the mildly relativistic range (≈85% of the speed of light), the emission is termed gyro-synchrotron radiation.^[3]

In astrophysics, synchrotron emission occurs, for instance, due to ultra-relativistic motion of a charged particle around a black hole.^[4] When the source follows a circular geodesic around the black hole, the synchrotron radiation occurs for orbits close to the photosphere where the motion is in the ultra-relativistic regime. 

https://en.wikipedia.org/wiki/Synchrotron_radiation

 

Sonoluminescence is the emission of light from imploding bubbles in a liquid when excited by sound.

Sonoluminescence was first discovered in 1934 at the University of Cologne. It occurs when a sound wave of sufficient intensity induces a gaseous cavity within a liquid to collapse quickly, emitting a burst of light. The phenomenon can be observed in stable single-bubble sonoluminescence (SBSL) and multi-bubble sonoluminescence (MBSL). In 1960, Peter Jarman proposed that sonoluminescence is thermal in origin and might arise from microshocks within collapsing cavities. Later experiments revealed that the temperature inside the bubble during SBSL could reach up to 12,000 kelvins. The exact mechanism behind sonoluminescence remains unknown, with various hypotheses including hotspot, bremsstrahlung , and collision-induced radiation. Some researchers^{[citation needed]} have even speculated that temperatures in sonoluminescing systems could reach millions of kelvins, potentially causing thermonuclear fusion. The phenomenon has also been observed in nature, with the pistol shrimp being the first known instance of an animal producing light through sonoluminescence.

History

The sonoluminescence effect was first discovered at the University of Cologne in 1934 as a result of work on sonar.^[1] Hermann Frenzel and H. Schultes put an ultrasound transducer in a tank of photographic developer fluid. They hoped to speed up the development process. Instead, they noticed tiny dots on the film after developing and realized that the bubbles in the fluid were emitting light with the ultrasound turned on.^[2] It was too difficult to analyze the effect in early experiments because of the complex environment of a large number of short-lived bubbles. This phenomenon is now referred to as multi-bubble sonoluminescence (MBSL).

In 1960, Peter Jarman from Imperial College of London proposed the most reliable theory of sonoluminescence phenomenon. He concluded that sonoluminescence is basically thermal in origin and that it might possibly arise from microshocks with the collapsing cavities.^[3]

In 1989, an experimental advance was introduced which produced stable single-bubble sonoluminescence (SBSL).^{[citation needed]} In single-bubble sonoluminescence, a single bubble trapped in an acoustic standing wave emits a pulse of light with each compression of the bubble within the standing wave. This technique allowed a more systematic study of the phenomenon, because it isolated the complex effects into one stable, predictable bubble. It was realized that the temperature inside the bubble was hot enough to melt steel, as seen in an experiment done in 2012; the temperature inside the bubble as it collapsed reached about 12,000 kelvins.^[4] Interest in sonoluminescence was renewed when an inner temperature of such a bubble well above one million kelvins was postulated.^[5] This temperature is thus far not conclusively proven; rather, recent experiments indicate temperatures around 20,000 K (19,700 °C; 35,500 °F).^[6]

Properties

Long exposure image of multi-bubble sonoluminescence created by a high-intensity ultrasonic horn immersed in a beaker of liquid

Sonoluminescence can occur when a sound wave of sufficient intensity induces a gaseous cavity within a liquid to collapse quickly. This cavity may take the form of a pre-existing bubble, or may be generated through a process known as cavitation. Sonoluminescence in the laboratory can be made to be stable, so that a single bubble will expand and collapse over and over again in a periodic fashion, emitting a burst of light each time it collapses. For this to occur, a standing acoustic wave is set up within a liquid, and the bubble will sit at a pressure anti-node of the standing wave. The frequencies of resonance depend on the shape and size of the container in which the bubble is contained.

Some facts about sonoluminescence:^{[citation needed]}

• The light that flashes from the bubbles last between 35 and a few hundred picoseconds long, with peak intensities of the order of 1–10 mW.
• The bubbles are very small when they emit light—about 1 micrometer in diameter—depending on the ambient fluid (e.g., water) and the gas content of the bubble (e.g., atmospheric air).
• Single-bubble sonoluminescence pulses can have very stable periods and positions. In fact, the frequency of light flashes can be more stable than the rated frequency stability of the oscillator making the sound waves driving them. However, the stability analyses of the bubble show that the bubble itself undergoes significant geometric instabilities, due to, for example, the Bjerknes forces and Rayleigh–Taylor instabilities.
• The addition of a small amount of noble gas (such as helium, argon, or xenon) to the gas in the bubble increases the intensity of the emitted light.

Spectral measurements have given bubble temperatures in the range from 2300 K to 5100 K, the exact temperatures depending on experimental conditions including the composition of the liquid and gas.^[7] Detection of very high bubble temperatures by spectral methods is limited due to the opacity of liquids to short wavelength light characteristic of very high temperatures.

A study describes a method of determining temperatures based on the formation of plasmas. Using argon bubbles in sulfuric acid, the data shows the presence of ionized molecular oxygen O₂⁺, sulfur monoxide, and atomic argon populating high-energy excited states, which confirms a hypothesis that the bubbles have a hot plasma core.^[8] The ionization and excitation energy of dioxygenyl cations, which they observed, is 18 electronvolts. From this they conclude the core temperatures reach at least 20,000 kelvins^[6]—hotter than the surface of the sun. 

https://en.wikipedia.org/wiki/Sonoluminescence

Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequency do not change as a result of the scattering.^[1] This limit is valid as long as the photon energy is much smaller than the mass energy of the particle: ${\displaystyle \nu \ll m{c}^2/h}$, or equivalently, if the wavelength of the light is much greater than the Compton wavelength of the particle (e.g., for electrons, longer wavelengths than hard x-rays).  

Light–matter interaction
Low-energy phenomena:
Photoelectric effect
Mid-energy phenomena:
Thomson scattering
Compton scattering
High-energy phenomena:
Pair production
Photodisintegration
Photofission

https://en.wikipedia.org/wiki/Thomson_scattering

Polarization (also polarisation) is a property of transverse waves which specifies the geometrical orientation of the oscillations.^{[1][2][3][4][5]} In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave.^[4] A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves,^[6] and transverse sound waves (shear waves) in solids.

An electromagnetic wave such as light consists of a coupled oscillating electric field and magnetic field which are always perpendicular to each other; by convention, the "polarization" of electromagnetic waves refers to the direction of the electric field. In linear polarization, the fields oscillate in a single direction. In circular or elliptical polarization, the fields rotate at a constant rate in a plane as the wave travels, either in the right-hand or in the left-hand direction.

Light or other electromagnetic radiation from many sources, such as the sun, flames, and incandescent lamps, consists of short wave trains with an equal mixture of polarizations; this is called unpolarized light. Polarized light can be produced by passing unpolarized light through a polarizer, which allows waves of only one polarization to pass through. The most common optical materials do not affect the polarization of light, but some materials—those that exhibit birefringence, dichroism, or optical activity—affect light differently depending on its polarization. Some of these are used to make polarizing filters. Light also becomes partially polarized when it reflects at an angle from a surface.

According to quantum mechanics, electromagnetic waves can also be viewed as streams of particles called photons. When viewed in this way, the polarization of an electromagnetic wave is determined by a quantum mechanical property of photons called their spin.^[7][8] A photon has one of two possible spins: it can either spin in a right hand sense or a left hand sense about its direction of travel. Circularly polarized electromagnetic waves are composed of photons with only one type of spin, either right- or left-hand. Linearly polarized waves consist of photons that are in a superposition of right and left circularly polarized states, with equal amplitude and phases synchronized to give oscillation in a plane.^[8]

Polarization is an important parameter in areas of science dealing with transverse waves, such as optics, seismology, radio, and microwaves. Especially impacted are technologies such as lasers, wireless and optical fiber telecommunications, and radar.

Introduction

Wave propagation and polarization

cross linear polarized

Most sources of light are classified as incoherent and unpolarized (or only "partially polarized") because they consist of a random mixture of waves having different spatial characteristics, frequencies (wavelengths), phases, and polarization states. However, for understanding electromagnetic waves and polarization in particular, it is easier to just consider coherent plane waves; these are sinusoidal waves of one particular direction (or wavevector), frequency, phase, and polarization state. Characterizing an optical system in relation to a plane wave with those given parameters can then be used to predict its response to a more general case, since a wave with any specified spatial structure can be decomposed into a combination of plane waves (its so-called angular spectrum). Incoherent states can be modeled stochastically as a weighted combination of such uncorrelated waves with some distribution of frequencies (its spectrum), phases, and polarizations.

Transverse electromagnetic waves

A "vertically polarized" electromagnetic wave of wavelength λ has its electric field vector E (red) oscillating in the vertical direction. The magnetic field B (or H) is always at right angles to it (blue), and both are perpendicular to the direction of propagation (z).

Electromagnetic waves (such as light), traveling in free space or another homogeneous isotropic non-attenuating medium, are properly described as transverse waves, meaning that a plane wave's electric field vector E and magnetic field H are each in some direction perpendicular to (or "transverse" to) the direction of wave propagation; E and H are also perpendicular to each other. By convention, the "polarization" direction of an electromagnetic wave is given by its electric field vector. Considering a monochromatic plane wave of optical frequency f (light of vacuum wavelength λ has a frequency of f = c/λ where c is the speed of light), let us take the direction of propagation as the z axis. Being a transverse wave the E and H fields must then contain components only in the x and y directions whereas E_z = H_z = 0. Using complex (or phasor) notation, the instantaneous physical electric and magnetic fields are given by the real parts of the complex quantities occurring in the following equations. As a function of time t and spatial position z (since for a plane wave in the +z direction the fields have no dependence on x or y) these complex fields can be written as:

${\displaystyle \overrightarrow{E}(z,t)=\left[\begin{array}{c}{e}_x\\ e_y\\ 0\end{array}\right]e^{i2\pi \left(\frac{z}{\lambda }-\frac{t}{T}\right)}=\left[\begin{array}{c}{e}_x\\ e_y\\ 0\end{array}\right]e^{i(kz-\omega t)}}$

and

${\displaystyle \overrightarrow{H}(z,t)=\left[\begin{array}{c}{h}_x\\ h_y\\ 0\end{array}\right]e^{i2\pi \left(\frac{z}{\lambda }-\frac{t}{T}\right)}=\left[\begin{array}{c}{h}_x\\ h_y\\ 0\end{array}\right]e^{i(kz-\omega t)}}$

where λ = λ₀/n is the wavelength in the medium (whose refractive index is n) and T = 1/f is the period of the wave. Here e_x, e_y, h_x, and h_y are complex numbers. In the second more compact form, as these equations are customarily expressed, these factors are described using the wavenumber ${\displaystyle k=2\pi n/{\lambda }_0}$ and angular frequency (or "radian frequency") ${\displaystyle \omega =2\pi f}$. In a more general formulation with propagation not restricted to the +z direction, then the spatial dependence kz is replaced by ${\displaystyle \overrightarrow{k}\cdot \overrightarrow{r}}$ where ${\displaystyle \overrightarrow{k}}$ is called the wave vector, the magnitude of which is the wavenumber.

Thus the leading vectors e and h each contain up to two nonzero (complex) components describing the amplitude and phase of the wave's x and y polarization components (again, there can be no z polarization component for a transverse wave in the +z direction). For a given medium with a characteristic impedance ${\displaystyle \eta }$, h is related to e by:

${\displaystyle h_y=\frac{e_x}{\eta }}$

and

${\displaystyle h_x=-\frac{e_y}{\eta }}$.

In a dielectric, η is real and has the value η₀/n, where n is the refractive index and η₀ is the impedance of free space. The impedance will be complex in a conducting medium.^{[clarification needed]} Note that given that relationship, the dot product of E and H must be zero:^{[dubious – discuss]}

${\displaystyle \overrightarrow{E}\left(\overrightarrow{r},t\right)\cdot \overrightarrow{H}\left(\overrightarrow{r},t\right)=e_x{h}_x+e_y{h}_y+e_z{h}_z=e_x\left(-\frac{e_y}{\eta }\right)+e_y\left(\frac{e_x}{\eta }\right)+0\cdot 0=0}$

indicating that these vectors are orthogonal (at right angles to each other), as expected.

So knowing the propagation direction (+z in this case) and η, one can just as well specify the wave in terms of just e_x and e_y describing the electric field. The vector containing e_x and e_y (but without the z component which is necessarily zero for a transverse wave) is known as a Jones vector. In addition to specifying the polarization state of the wave, a general Jones vector also specifies the overall magnitude and phase of that wave. Specifically, the intensity of the light wave is proportional to the sum of the squared magnitudes of the two electric field components:

${\displaystyle I=\left({\left|e_x\right|}^2+{\left|e_y\right|}^2\right)\frac{1}{2\eta }}$

However, the wave's state of polarization is only dependent on the (complex) ratio of e_y to e_x. So let us just consider waves whose |e_x|² + |e_y|² = 1; this happens to correspond to an intensity of about .00133 watts per square meter in free space (where ${\displaystyle \eta =}$ ${\displaystyle {\eta }_0}$). And since the absolute phase of a wave is unimportant in discussing its polarization state, let us stipulate that the phase of e_x is zero, in other words e_x is a real number while e_y may be complex. Under these restrictions, e_x and e_y can be represented as follows:

${\displaystyle e_x=\sqrt{\frac{1+Q}{2}}}$

${\displaystyle e_y=\sqrt{\frac{1-Q}{2}}{e}^{i\varphi }}$

where the polarization state is now fully parameterized by the value of Q (such that −1 < Q < 1) and the relative phase ${\displaystyle \varphi }$.

Non-transverse waves

In addition to transverse waves, there are many wave motions where the oscillation is not limited to directions perpendicular to the direction of propagation. These cases are far beyond the scope of the current article which concentrates on transverse waves (such as most electromagnetic waves in bulk media), but one should be aware of cases where the polarization of a coherent wave cannot be described simply using a Jones vector, as we have just done.

Just considering electromagnetic waves, we note that the preceding discussion strictly applies to plane waves in a homogeneous isotropic non-attenuating medium, whereas in an anisotropic medium (such as birefringent crystals as discussed below) the electric or magnetic field may have longitudinal as well as transverse components. In those cases the electric displacement D and magnetic flux density B^{[clarification needed]} still obey the above geometry but due to anisotropy in the electric susceptibility (or in the magnetic permeability), now given by a tensor, the direction of E (or H) may differ from that of D (or B). Even in isotropic media, so-called inhomogeneous waves can be launched into a medium whose refractive index has a significant imaginary part (or "extinction coefficient") such as metals;^{[clarification needed]} these fields are also not strictly transverse.^{[9]: 179–184 [10]: 51–52} Surface waves or waves propagating in a waveguide (such as an optical fiber) are generally not transverse waves, but might be described as an electric or magnetic transverse mode, or a hybrid mode.

Even in free space, longitudinal field components can be generated in focal regions, where the plane wave approximation breaks down. An extreme example is radially or tangentially polarized light, at the focus of which the electric or magnetic field respectively is entirely longitudinal (along the direction of propagation).^[11]

For longitudinal waves such as sound waves in fluids, the direction of oscillation is by definition along the direction of travel, so the issue of polarization is normally not even mentioned. On the other hand, sound waves in a bulk solid can be transverse as well as longitudinal, for a total of three polarization components. In this case, the transverse polarization is associated with the direction of the shear stress and displacement in directions perpendicular to the propagation direction, while the longitudinal polarization describes compression of the solid and vibration along the direction of propagation. The differential propagation of transverse and longitudinal polarizations is important in seismology.

Polarization state

Further information: Linear polarization, Circular polarization, and Elliptical polarization

Electric field oscillation

Polarization is best understood by initially considering only pure polarization states, and only a coherent sinusoidal wave at some optical frequency. The vector in the adjacent diagram might describe the oscillation of the electric field emitted by a single-mode laser (whose oscillation frequency would be typically 10¹⁵ times faster). The field oscillates in the x-y plane, along the page, with the wave propagating in the z direction, perpendicular to the page. The first two diagrams below trace the electric field vector over a complete cycle for linear polarization at two different orientations; these are each considered a distinct state of polarization (SOP). Note that the linear polarization at 45° can also be viewed as the addition of a horizontally linearly polarized wave (as in the leftmost figure) and a vertically polarized wave of the same amplitude in the same phase.

Animation showing four different polarization states and three orthogonal projections.

A circularly polarized wave as a sum of two linearly polarized components 90° out of phase

Now if one were to introduce a phase shift in between those horizontal and vertical polarization components, one would generally obtain elliptical polarization^[12] as is shown in the third figure. When the phase shift is exactly ±90°, then circular polarization is produced (fourth and fifth figures). Thus is circular polarization created in practice, starting with linearly polarized light and employing a quarter-wave plate to introduce such a phase shift. The result of two such phase-shifted components in causing a rotating electric field vector is depicted in the animation on the right. Note that circular or elliptical polarization can involve either a clockwise or counterclockwise rotation of the field. These correspond to distinct polarization states, such as the two circular polarizations shown above.

Of course the orientation of the x and y axes used in this description is arbitrary. The choice of such a coordinate system and viewing the polarization ellipse in terms of the x and y polarization components, corresponds to the definition of the Jones vector (below) in terms of those basis polarizations. One would typically choose axes to suit a particular problem such as x being in the plane of incidence. Since there are separate reflection coefficients for the linear polarizations in and orthogonal to the plane of incidence (p and s polarizations, see below), that choice greatly simplifies the calculation of a wave's reflection from a surface.

Moreover, one can use as basis functions any pair of orthogonal polarization states, not just linear polarizations. For instance, choosing right and left circular polarizations as basis functions simplifies the solution of problems involving circular birefringence (optical activity) or circular dichroism.

Polarization ellipse

Main article: Polarization ellipse

Consider a purely polarized monochromatic wave. If one were to plot the electric field vector over one cycle of oscillation, an ellipse would generally be obtained, as is shown in the figure, corresponding to a particular state of elliptical polarization. Note that linear polarization and circular polarization can be seen as special cases of elliptical polarization.

A polarization state can then be described in relation to the geometrical parameters of the ellipse, and its "handedness", that is, whether the rotation around the ellipse is clockwise or counter clockwise. One parameterization of the elliptical figure specifies the orientation angle ψ, defined as the angle between the major axis of the ellipse and the x-axis^[13] along with the ellipticity ε = a/b, the ratio of the ellipse's major to minor axis.^[14][15][16] (also known as the axial ratio). The ellipticity parameter is an alternative parameterization of an ellipse's eccentricity $e=\sqrt{1-b^2/a^2},$ or the ellipticity angle, $\chi =\arctan b/a$ $=\arctan 1/\epsilon$ as is shown in the figure.^[13] The angle χ is also significant in that the latitude (angle from the equator) of the polarization state as represented on the Poincaré sphere (see below) is equal to ±2χ. The special cases of linear and circular polarization correspond to an ellipticity ε of infinity and unity (or χ of zero and 45°) respectively.

Jones vector

Main article: Jones vector

Full information on a completely polarized state is also provided by the amplitude and phase of oscillations in two components of the electric field vector in the plane of polarization. This representation was used above to show how different states of polarization are possible. The amplitude and phase information can be conveniently represented as a two-dimensional complex vector (the Jones vector):

${\displaystyle \mathbf{e}=\left[\begin{array}{c}{a}_1{e}^{i{\theta }_1}\\ a_2{e}^{i{\theta }_2}\end{array}\right].}$

Here ${\displaystyle a_1}$ and ${\displaystyle a_2}$ denote the amplitude of the wave in the two components of the electric field vector, while ${\displaystyle {\theta }_1}$ and ${\displaystyle {\theta }_2}$ represent the phases. The product of a Jones vector with a complex number of unit modulus gives a different Jones vector representing the same ellipse, and thus the same state of polarization. The physical electric field, as the real part of the Jones vector, would be altered but the polarization state itself is independent of absolute phase. The basis vectors used to represent the Jones vector need not represent linear polarization states (i.e. be real). In general any two orthogonal states can be used, where an orthogonal vector pair is formally defined as one having a zero inner product. A common choice is left and right circular polarizations, for example to model the different propagation of waves in two such components in circularly birefringent media (see below) or signal paths of coherent detectors sensitive to circular polarization.

Coordinate frame

Regardless of whether polarization state is represented using geometric parameters or Jones vectors, implicit in the parameterization is the orientation of the coordinate frame. This permits a degree of freedom, namely rotation about the propagation direction. When considering light that is propagating parallel to the surface of the Earth, the terms "horizontal" and "vertical" polarization are often used, with the former being associated with the first component of the Jones vector, or zero azimuth angle. On the other hand, in astronomy the equatorial coordinate system is generally used instead, with the zero azimuth (or position angle, as it is more commonly called in astronomy to avoid confusion with the horizontal coordinate system) corresponding to due north.

s and p designations

See also: Fresnel equations § S and P polarizations

0:31

Electromagnetic vectors for $\mathbf{\text{E}}$, $\mathbf{\text{B}}$ and $\mathbf{\text{k}}$ with $\mathbf{\text{E}}=\mathbf{\text{E}}(x,y)$ along with 3 planar projections and a deformation surface of total electric field. The light is always s-polarized in the xy plane. $\theta$ is the polar angle of $\mathbf{\text{k}}$ and ${\phi }_E$ is the azimuthal angle of $\mathbf{\text{E}}$.

Another coordinate system frequently used relates to the plane of incidence. This is the plane made by the incoming propagation direction and the vector perpendicular to the plane of an interface, in other words, the plane in which the ray travels before and after reflection or refraction. The component of the electric field parallel to this plane is termed p-like (parallel) and the component perpendicular to this plane is termed s-like (from senkrecht, German for perpendicular). Polarized light with its electric field along the plane of incidence is thus denoted p-polarized, while light whose electric field is normal to the plane of incidence is called s-polarized. P polarization is commonly referred to as transverse-magnetic (TM), and has also been termed pi-polarized or tangential plane polarized. S polarization is also called transverse-electric (TE), as well as sigma-polarized or sagittal plane polarized.

Degree of polarization

Degree of polarization (DOP) is a quantity used to describe the portion of an electromagnetic wave which is polarized. A perfectly polarized wave has a DOP of 100%, whereas an unpolarized wave has a DOP of 0%. A wave which is partially polarized, and therefore can be represented by a superposition of a polarized and unpolarized component, will have a DOP somewhere in between 0 and 100%. DOP is calculated as the fraction of the total power that is carried by the polarized component of the wave.

DOP can be used to map the strain field in materials when considering the DOP of the photoluminescence. The polarization of the photoluminescence is related to the strain in a material by way of the given material's photoelasticity tensor.

DOP is also visualized using the Poincaré sphere representation of a polarized beam. In this representation, DOP is equal to the length of the vector measured from the center of the sphere.

Unpolarized and partially polarized light

This section is an excerpt from Unpolarized light.[edit]

Unpolarized light is light with a random, time-varying polarization. Natural light, like most other common sources of visible light, produced independently by a large number of atoms or molecules whose emissions are uncorrelated. This term is somewhat inexact, since at any instant of time at one location there is a definite plane of polarization; however, it implies that the polarization changes so quickly in time that it will not be measured or relevant to the outcome of an experiment.

Unpolarized light can be produced from the incoherent combination of vertical and horizontal linearly polarized light, or right- and left-handed circularly polarized light.^[17] Conversely, the two constituent linearly polarized states of unpolarized light cannot form an interference pattern, even if rotated into alignment (Fresnel–Arago 3rd law).^[18]

A so-called depolarizer acts on a polarized beam to create one in which the polarization varies so rapidly across the beam that it may be ignored in the intended applications. Conversely, a polarizer acts on an unpolarized beam or arbitrarily polarized beam to create one which is polarized.

Unpolarized light can be described as a mixture of two independent oppositely polarized streams, each with half the intensity.^[19][20] Light is said to be partially polarized when there is more power in one of these streams than the other. At any particular wavelength, partially polarized light can be statistically described as the superposition of a completely unpolarized component and a completely polarized one.^{[21]: 346–347 [22]: 330} One may then describe the light in terms of the degree of polarization and the parameters of the polarized component. That polarized component can be described in terms of a Jones vector or polarization ellipse. However, in order to also describe the degree of polarization, one normally employs Stokes parameters to specify a state of partial polarization.^{[21]: 351, 374–375}

Implications for reflection and propagation

Polarization in wave propagation

In a vacuum, the components of the electric field propagate at the speed of light, so that the phase of the wave varies in space and time while the polarization state does not. That is, the electric field vector e of a plane wave in the +z direction follows:

${\displaystyle \mathbf{e}(z+\mathrm{\Delta }z,t+\mathrm{\Delta }t)=\mathbf{e}(z,t)e^{ik(c\mathrm{\Delta }t-\mathrm{\Delta }z)},}$

where k is the wavenumber. As noted above, the instantaneous electric field is the real part of the product of the Jones vector times the phase factor ${\displaystyle e^{-i\omega t}}$. When an electromagnetic wave interacts with matter, its propagation is altered according to the material's (complex) index of refraction. When the real or imaginary part of that refractive index is dependent on the polarization state of a wave, properties known as birefringence and polarization dichroism (or diattenuation) respectively, then the polarization state of a wave will generally be altered.

In such media, an electromagnetic wave with any given state of polarization may be decomposed into two orthogonally polarized components that encounter different propagation constants. The effect of propagation over a given path on those two components is most easily characterized in the form of a complex 2×2 transformation matrix J known as a Jones matrix:

${\displaystyle {\mathbf{e}}^{\prime }=\mathbf{J}\mathbf{e}.}$

The Jones matrix due to passage through a transparent material is dependent on the propagation distance as well as the birefringence. The birefringence (as well as the average refractive index) will generally be dispersive, that is, it will vary as a function of optical frequency (wavelength). In the case of non-birefringent materials, however, the 2×2 Jones matrix is the identity matrix (multiplied by a scalar phase factor and attenuation factor), implying no change in polarization during propagation.

For propagation effects in two orthogonal modes, the Jones matrix can be written as

${\displaystyle \mathbf{J}=\mathbf{T}\left[\begin{array}{cc}{g}_1& 0\\ 0& g_2\end{array}\right]{\mathbf{T}}^{-1},}$

where g₁ and g₂ are complex numbers describing the phase delay and possibly the amplitude attenuation due to propagation in each of the two polarization eigenmodes. T is a unitary matrix representing a change of basis from these propagation modes to the linear system used for the Jones vectors; in the case of linear birefringence or diattenuation the modes are themselves linear polarization states so T and T⁻¹ can be omitted if the coordinate axes have been chosen appropriately.

Birefringence

Main article: Birefringence

In a birefringent substance, electromagnetic waves of different polarizations travel at different speeds (phase velocities). As a result when unpolarized waves travel through a plate of birefringent material, one polarization component has a shorter wavelength than the other, resulting in a phase difference between the components which increases the further the waves travel through the material. The Jones matrix is a unitary matrix: |g₁| = |g₂| = 1. Media termed diattenuating (or dichroic in the sense of polarization), in which only the amplitudes of the two polarizations are affected differentially, may be described using a Hermitian matrix (generally multiplied by a common phase factor). In fact, since any matrix may be written as the product of unitary and positive Hermitian matrices, light propagation through any sequence of polarization-dependent optical components can be written as the product of these two basic types of transformations.

Color pattern of a plastic box showing stress-induced birefringence when placed in between two crossed polarizers.

In birefringent media there is no attenuation, but two modes accrue a differential phase delay. Well known manifestations of linear birefringence (that is, in which the basis polarizations are orthogonal linear polarizations) appear in optical wave plates/retarders and many crystals. If linearly polarized light passes through a birefringent material, its state of polarization will generally change, unless its polarization direction is identical to one of those basis polarizations. Since the phase shift, and thus the change in polarization state, is usually wavelength-dependent, such objects viewed under white light in between two polarizers may give rise to colorful effects, as seen in the accompanying photograph.

Circular birefringence is also termed optical activity, especially in chiral fluids, or Faraday rotation, when due to the presence of a magnetic field along the direction of propagation. When linearly polarized light is passed through such an object, it will exit still linearly polarized, but with the axis of polarization rotated. A combination of linear and circular birefringence will have as basis polarizations two orthogonal elliptical polarizations; however, the term "elliptical birefringence" is rarely used.

Paths taken by vectors in the Poincaré sphere under birefringence. The propagation modes (rotation axes) are shown with red, blue, and yellow lines, the initial vectors by thick black lines, and the paths they take by colored ellipses (which represent circles in three dimensions).

One can visualize the case of linear birefringence (with two orthogonal linear propagation modes) with an incoming wave linearly polarized at a 45° angle to those modes. As a differential phase starts to accrue, the polarization becomes elliptical, eventually changing to purely circular polarization (90° phase difference), then to elliptical and eventually linear polarization (180° phase) perpendicular to the original polarization, then through circular again (270° phase), then elliptical with the original azimuth angle, and finally back to the original linearly polarized state (360° phase) where the cycle begins anew. In general the situation is more complicated and can be characterized as a rotation in the Poincaré sphere about the axis defined by the propagation modes. Examples for linear (blue), circular (red), and elliptical (yellow) birefringence are shown in the figure on the left. The total intensity and degree of polarization are unaffected. If the path length in the birefringent medium is sufficient, the two polarization components of a collimated beam (or ray) can exit the material with a positional offset, even though their final propagation directions will be the same (assuming the entrance face and exit face are parallel). This is commonly viewed using calcite crystals, which present the viewer with two slightly offset images, in opposite polarizations, of an object behind the crystal. It was this effect that provided the first discovery of polarization, by Erasmus Bartholinus in 1669.

Dichroism

Media in which transmission of one polarization mode is preferentially reduced are called dichroic or diattenuating. Like birefringence, diattenuation can be with respect to linear polarization modes (in a crystal) or circular polarization modes (usually in a liquid).

Devices that block nearly all of the radiation in one mode are known as polarizing filters or simply "polarizers". This corresponds to g₂=0 in the above representation of the Jones matrix. The output of an ideal polarizer is a specific polarization state (usually linear polarization) with an amplitude equal to the input wave's original amplitude in that polarization mode. Power in the other polarization mode is eliminated. Thus if unpolarized light is passed through an ideal polarizer (where g₁=1 and g₂=0) exactly half of its initial power is retained. Practical polarizers, especially inexpensive sheet polarizers, have additional loss so that g₁ < 1. However, in many instances the more relevant figure of merit is the polarizer's degree of polarization or extinction ratio, which involve a comparison of g₁ to g₂. Since Jones vectors refer to waves' amplitudes (rather than intensity), when illuminated by unpolarized light the remaining power in the unwanted polarization will be (g₂/g₁)² of the power in the intended polarization.

Specular reflection

In addition to birefringence and dichroism in extended media, polarization effects describable using Jones matrices can also occur at (reflective) interface between two materials of different refractive index. These effects are treated by the Fresnel equations. Part of the wave is transmitted and part is reflected; for a given material those proportions (and also the phase of reflection) are dependent on the angle of incidence and are different for the s and p polarizations. Therefore, the polarization state of reflected light (even if initially unpolarized) is generally changed.

A stack of plates at Brewster's angle to a beam reflects off a fraction of the s-polarized light at each surface, leaving (after many such plates) a mainly p-polarized beam.

Any light striking a surface at a special angle of incidence known as Brewster's angle, where the reflection coefficient for p polarization is zero, will be reflected with only the s-polarization remaining. This principle is employed in the so-called "pile of plates polarizer" (see figure) in which part of the s polarization is removed by reflection at each Brewster angle surface, leaving only the p polarization after transmission through many such surfaces. The generally smaller reflection coefficient of the p polarization is also the basis of polarized sunglasses; by blocking the s (horizontal) polarization, most of the glare due to reflection from a wet street, for instance, is removed.^{[23]: 348–350}

In the important special case of reflection at normal incidence (not involving anisotropic materials) there is no particular s or p polarization. Both the x and y polarization components are reflected identically, and therefore the polarization of the reflected wave is identical to that of the incident wave. However, in the case of circular (or elliptical) polarization, the handedness of the polarization state is thereby reversed, since by convention this is specified relative to the direction of propagation. The circular rotation of the electric field around the x-y axes called "right-handed" for a wave in the +z direction is "left-handed" for a wave in the -z direction. But in the general case of reflection at a nonzero angle of incidence, no such generalization can be made. For instance, right-circularly polarized light reflected from a dielectric surface at a grazing angle, will still be right-handed (but elliptically) polarized. Linear polarized light reflected from a metal at non-normal incidence will generally become elliptically polarized. These cases are handled using Jones vectors acted upon by the different Fresnel coefficients for the s and p polarization components.

Measurement techniques involving polarization

Some optical measurement techniques are based on polarization. In many other optical techniques polarization is crucial or at least must be taken into account and controlled; such examples are too numerous to mention.

Measurement of stress

Stress in plastic glasses

In engineering, the phenomenon of stress induced birefringence allows for stresses in transparent materials to be readily observed. As noted above and seen in the accompanying photograph, the chromaticity of birefringence typically creates colored patterns when viewed in between two polarizers. As external forces are applied, internal stress induced in the material is thereby observed. Additionally, birefringence is frequently observed due to stresses "frozen in" at the time of manufacture. This is famously observed in cellophane tape whose birefringence is due to the stretching of the material during the manufacturing process.

Ellipsometry

Main article: Ellipsometry

Ellipsometry is a powerful technique for the measurement of the optical properties of a uniform surface. It involves measuring the polarization state of light following specular reflection from such a surface. This is typically done as a function of incidence angle or wavelength (or both). Since ellipsometry relies on reflection, it is not required for the sample to be transparent to light or for its back side to be accessible.

Ellipsometry can be used to model the (complex) refractive index of a surface of a bulk material. It is also very useful in determining parameters of one or more thin film layers deposited on a substrate. Due to their reflection properties, not only are the predicted magnitude of the p and s polarization components, but their relative phase shifts upon reflection, compared to measurements using an ellipsometer. A normal ellipsometer does not measure the actual reflection coefficient (which requires careful photometric calibration of the illuminating beam) but the ratio of the p and s reflections, as well as change of polarization ellipticity (hence the name) induced upon reflection by the surface being studied. In addition to use in science and research, ellipsometers are used in situ to control production processes for instance.^{[24]: 585ff [25]: 632}

Geology

Photomicrograph of a volcanic sand grain; upper picture is plane-polarized light, bottom picture is cross-polarized light, scale box at left-center is 0.25 millimeter.

The property of (linear) birefringence is widespread in crystalline minerals, and indeed was pivotal in the initial discovery of polarization. In mineralogy, this property is frequently exploited using polarization microscopes, for the purpose of identifying minerals. See optical mineralogy for more details.^{[26]: 163–164}

Sound waves in solid materials exhibit polarization. Differential propagation of the three polarizations through the earth is a crucial in the field of seismology. Horizontally and vertically polarized seismic waves (shear waves) are termed SH and SV, while waves with longitudinal polarization (compressional waves) are termed P-waves.^{[27]: 48–50 [28]: 56–57}

Chemistry

We have seen (above) that the birefringence of a type of crystal is useful in identifying it, and thus detection of linear birefringence is especially useful in geology and mineralogy. Linearly polarized light generally has its polarization state altered upon transmission through such a crystal, making it stand out when viewed in between two crossed polarizers, as seen in the photograph, above. Likewise, in chemistry, rotation of polarization axes in a liquid solution can be a useful measurement. In a liquid, linear birefringence is impossible, but there may be circular birefringence when a chiral molecule is in solution. When the right and left handed enantiomers of such a molecule are present in equal numbers (a so-called racemic mixture) then their effects cancel out. However, when there is only one (or a preponderance of one), as is more often the case for organic molecules, a net circular birefringence (or optical activity) is observed, revealing the magnitude of that imbalance (or the concentration of the molecule itself, when it can be assumed that only one enantiomer is present). This is measured using a polarimeter in which polarized light is passed through a tube of the liquid, at the end of which is another polarizer which is rotated in order to null the transmission of light through it.^{[23]: 360–365 [29]}

Astronomy

Main article: Polarization in astronomy

In many areas of astronomy, the study of polarized electromagnetic radiation from outer space is of great importance. Although not usually a factor in the thermal radiation of stars, polarization is also present in radiation from coherent astronomical sources (e.g. hydroxyl or methanol masers), and incoherent sources such as the large radio lobes in active galaxies, and pulsar radio radiation (which may, it is speculated, sometimes be coherent), and is also imposed upon starlight by scattering from interstellar dust. Apart from providing information on sources of radiation and scattering, polarization also probes the interstellar magnetic field via Faraday rotation.^{[30]: 119, 124 [31]: 336–337} The polarization of the cosmic microwave background is being used to study the physics of the very early universe.^[32][33] Synchrotron radiation is inherently polarized. It has been suggested that astronomical sources caused the chirality of biological molecules on Earth.^[34]

Applications and examples

Polarized sunglasses

Effect of a polarizer on reflection from mud flats. In the picture on the left, the horizontally oriented polarizer preferentially transmits those reflections; rotating the polarizer by 90° (right) as one would view using polarized sunglasses blocks almost all specularly reflected sunlight.

One can test whether sunglasses are polarized by looking through two pairs, with one perpendicular to the other. If both are polarized, all light will be blocked.

Unpolarized light, after being reflected by a specular (shiny) surface, generally obtains a degree of polarization. This phenomenon was observed in 1808 by the mathematician Étienne-Louis Malus, after whom Malus's law is named. Polarizing sunglasses exploit this effect to reduce glare from reflections by horizontal surfaces, notably the road ahead viewed at a grazing angle.

Wearers of polarized sunglasses will occasionally observe inadvertent polarization effects such as color-dependent birefringent effects, for example in toughened glass (e.g., car windows) or items made from transparent plastics, in conjunction with natural polarization by reflection or scattering. The polarized light from LCD monitors (see below) is very conspicuous when these are worn.

Sky polarization and photography

Further information: Polarizing filter (Photography)

The effects of a polarizing filter (right image) on the sky in a photograph

Polarization is observed in the light of the sky, as this is due to sunlight scattered by aerosols as it passes through Earth's atmosphere. The scattered light produces the brightness and color in clear skies. This partial polarization of scattered light can be used to darken the sky in photographs, increasing the contrast. This effect is most strongly observed at points on the sky making a 90° angle to the Sun. Polarizing filters use these effects to optimize the results of photographing scenes in which reflection or scattering by the sky is involved.^{[23]: 346–347 [35]: 495–499}

Colored fringes in the Embassy Gardens Sky Pool when viewed through a polarizer, due to stress-induced birefringence in the skylight

Sky polarization has been used for orientation in navigation. The Pfund sky compass was used in the 1950s when navigating near the poles of the Earth's magnetic field when neither the sun nor stars were visible (e.g., under daytime cloud or twilight). It has been suggested, controversially, that the Vikings exploited a similar device (the "sunstone") in their extensive expeditions across the North Atlantic in the 9th–11th centuries, before the arrival of the magnetic compass from Asia to Europe in the 12th century. Related to the sky compass is the "polar clock", invented by Charles Wheatstone in the late 19th century.^{[36]: 67–69}

Display technologies

The principle of liquid-crystal display (LCD) technology relies on the rotation of the axis of linear polarization by the liquid crystal array. Light from the backlight (or the back reflective layer, in devices not including or requiring a backlight) first passes through a linear polarizing sheet. That polarized light passes through the actual liquid crystal layer which may be organized in pixels (for a TV or computer monitor) or in another format such as a seven-segment display or one with custom symbols for a particular product. The liquid crystal layer is produced with a consistent right (or left) handed chirality, essentially consisting of tiny helices. This causes circular birefringence, and is engineered so that there is a 90 degree rotation of the linear polarization state. However, when a voltage is applied across a cell, the molecules straighten out, lessening or totally losing the circular birefringence. On the viewing side of the display is another linear polarizing sheet, usually oriented at 90 degrees from the one behind the active layer. Therefore, when the circular birefringence is removed by the application of a sufficient voltage, the polarization of the transmitted light remains at right angles to the front polarizer, and the pixel appears dark. With no voltage, however, the 90 degree rotation of the polarization causes it to exactly match the axis of the front polarizer, allowing the light through. Intermediate voltages create intermediate rotation of the polarization axis and the pixel has an intermediate intensity. Displays based on this principle are widespread, and now are used in the vast majority of televisions, computer monitors and video projectors, rendering the previous CRT technology essentially obsolete. The use of polarization in the operation of LCD displays is immediately apparent to someone wearing polarized sunglasses, often making the display unreadable.

In a totally different sense, polarization encoding has become the leading (but not sole) method for delivering separate images to the left and right eye in stereoscopic displays used for 3D movies. This involves separate images intended for each eye either projected from two different projectors with orthogonally oriented polarizing filters or, more typically, from a single projector with time multiplexed polarization (a fast alternating polarization device for successive frames). Polarized 3D glasses with suitable polarizing filters ensure that each eye receives only the intended image. Historically such systems used linear polarization encoding because it was inexpensive and offered good separation. However, circular polarization makes separation of the two images insensitive to tilting of the head, and is widely used in 3-D movie exhibition today, such as the system from RealD. Projecting such images requires screens that maintain the polarization of the projected light when viewed in reflection (such as silver screens); a normal diffuse white projection screen causes depolarization of the projected images, making it unsuitable for this application.

Although now obsolete, CRT computer displays suffered from reflection by the glass envelope, causing glare from room lights and consequently poor contrast. Several anti-reflection solutions were employed to ameliorate this problem. One solution utilized the principle of reflection of circularly polarized light. A circular polarizing filter in front of the screen allows for the transmission of (say) only right circularly polarized room light. Now, right circularly polarized light (depending on the convention used) has its electric (and magnetic) field direction rotating clockwise while propagating in the +z direction. Upon reflection, the field still has the same direction of rotation, but now propagation is in the −z direction making the reflected wave left circularly polarized. With the right circular polarization filter placed in front of the reflecting glass, the unwanted light reflected from the glass will thus be in very polarization state that is blocked by that filter, eliminating the reflection problem. The reversal of circular polarization on reflection and elimination of reflections in this manner can be easily observed by looking in a mirror while wearing 3-D movie glasses which employ left- and right-handed circular polarization in the two lenses. Closing one eye, the other eye will see a reflection in which it cannot see itself; that lens appears black. However, the other lens (of the closed eye) will have the correct circular polarization allowing the closed eye to be easily seen by the open one.

Radio transmission and reception

See also: Antenna (radio) § Polarization

All radio (and microwave) antennas used for transmitting or receiving are intrinsically polarized. They transmit in (or receive signals from) a particular polarization, being totally insensitive to the opposite polarization; in certain cases that polarization is a function of direction. Most antennas are nominally linearly polarized, but elliptical and circular polarization is a possibility. As is the convention in optics, the "polarization" of a radio wave is understood to refer to the polarization of its electric field, with the magnetic field being at a 90 degree rotation with respect to it for a linearly polarized wave.

The vast majority of antennas are linearly polarized. In fact it can be shown from considerations of symmetry that an antenna that lies entirely in a plane which also includes the observer, can only have its polarization in the direction of that plane. This applies to many cases, allowing one to easily infer such an antenna's polarization at an intended direction of propagation. So a typical rooftop Yagi or log-periodic antenna with horizontal conductors, as viewed from a second station toward the horizon, is necessarily horizontally polarized. But a vertical "whip antenna" or AM broadcast tower used as an antenna element (again, for observers horizontally displaced from it) will transmit in the vertical polarization. A turnstile antenna with its four arms in the horizontal plane, likewise transmits horizontally polarized radiation toward the horizon. However, when that same turnstile antenna is used in the "axial mode" (upwards, for the same horizontally-oriented structure) its radiation is circularly polarized. At intermediate elevations it is elliptically polarized.

Polarization is important in radio communications because, for instance, if one attempts to use a horizontally polarized antenna to receive a vertically polarized transmission, the signal strength will be substantially reduced (or under very controlled conditions, reduced to nothing). This principle is used in satellite television in order to double the channel capacity over a fixed frequency band. The same frequency channel can be used for two signals broadcast in opposite polarizations. By adjusting the receiving antenna for one or the other polarization, either signal can be selected without interference from the other.

Especially due to the presence of the ground, there are some differences in propagation (and also in reflections responsible for TV ghosting) between horizontal and vertical polarizations. AM and FM broadcast radio usually use vertical polarization, while television uses horizontal polarization. At low frequencies especially, horizontal polarization is avoided. That is because the phase of a horizontally polarized wave is reversed upon reflection by the ground. A distant station in the horizontal direction will receive both the direct and reflected wave, which thus tend to cancel each other. This problem is avoided with vertical polarization. Polarization is also important in the transmission of radar pulses and reception of radar reflections by the same or a different antenna. For instance, back scattering of radar pulses by rain drops can be avoided by using circular polarization. Just as specular reflection of circularly polarized light reverses the handedness of the polarization, as discussed above, the same principle applies to scattering by objects much smaller than a wavelength such as rain drops. On the other hand, reflection of that wave by an irregular metal object (such as an airplane) will typically introduce a change in polarization and (partial) reception of the return wave by the same antenna.

The effect of free electrons in the ionosphere, in conjunction with the earth's magnetic field, causes Faraday rotation, a sort of circular birefringence. This is the same mechanism which can rotate the axis of linear polarization by electrons in interstellar space as mentioned below. The magnitude of Faraday rotation caused by such a plasma is greatly exaggerated at lower frequencies, so at the higher microwave frequencies used by satellites the effect is minimal. However, medium or short wave transmissions received following refraction by the ionosphere are strongly affected. Since a wave's path through the ionosphere and the earth's magnetic field vector along such a path are rather unpredictable, a wave transmitted with vertical (or horizontal) polarization will generally have a resulting polarization in an arbitrary orientation at the receiver.

Circular polarization through an airplane plastic window, 1989

Polarization and vision

Many animals are capable of perceiving some of the components of the polarization of light, e.g., linear horizontally polarized light. This is generally used for navigational purposes, since the linear polarization of sky light is always perpendicular to the direction of the sun. This ability is very common among the insects, including bees, which use this information to orient their communicative dances.^{[36]: 102–103} Polarization sensitivity has also been observed in species of octopus, squid, cuttlefish, and mantis shrimp.^{[36]: 111–112} In the latter case, one species measures all six orthogonal components of polarization, and is believed to have optimal polarization vision.^[37] The rapidly changing, vividly colored skin patterns of cuttlefish, used for communication, also incorporate polarization patterns, and mantis shrimp are known to have polarization selective reflective tissue. Sky polarization was thought to be perceived by pigeons, which was assumed to be one of their aids in homing, but research indicates this is a popular myth.^[38]

The naked human eye is weakly sensitive to polarization, without the need for intervening filters. Polarized light creates a very faint pattern near the center of the visual field, called Haidinger's brush. This pattern is very difficult to see, but with practice one can learn to detect polarized light with the naked eye.^[36]: 118

Angular momentum using circular polarization

It is well known that electromagnetic radiation carries a certain linear momentum in the direction of propagation. In addition, however, light carries a certain angular momentum if it is circularly polarized (or partially so). In comparison with lower frequencies such as microwaves, the amount of angular momentum in light, even of pure circular polarization, compared to the same wave's linear momentum (or radiation pressure) is very small and difficult to even measure. However, it was utilized in an experiment to achieve speeds of up to 600 million revolutions per minute.^[39][40]

See also

• Depolarizer (optics)
• Fluorescence anisotropy
• Glan–Taylor prism
• Kerr effect
• Nicol prism
• Plane of polarization
• Pockels effect
• Polarization rotator
• Polarized light microscopy
• Polarizer
• Polaroid (polarizer)
• Radial polarization
• Rayleigh sky model
• Waveplate

References

Cited references

1. 
   

Shipman, James; Wilson, Jerry D.; Higgins, Charles A. (2015). An Introduction to Physical Science, 14th Ed. Cengage Learning. p. 187. ISBN 978-1-305-54467-3.

Muncaster, Roger (1993). A-level Physics. Nelson Thornes. pp. 465–467. ISBN 0-7487-1584-3.

Singh, Devraj (2015). Fundamentals of Optics, 2nd Ed. PHI Learning Pvt. Ltd. p. 453. ISBN 978-8120351462.

Avadhanulu, M. N. (1992). A Textbook of Engineering Physics. S. Chand Publishing. pp. 198–199. ISBN 8121908175.

Desmarais, Louis (1997). Applied Electro Optics. Pearson Education. pp. 162–163. ISBN 0-13-244182-9.

Le Tiec, A.; Novak, J. (July 2016). "Theory of Gravitational Waves". An Overview of Gravitational Waves. pp. 1–41. arXiv:1607.04202. doi:10.1142/9789813141766_0001. ISBN 978-981-314-175-9. S2CID 119283594.

Lipson, Stephen G.; Lipson, Henry; Tannhauser, David Stefan (1995). Optical Physics. Cambridge University Press. pp. 125–127. ISBN 978-0-521-43631-1.

Waldman, Gary (2002). Introduction to Light: The Physics of Light, Vision, and Color. Courier Corporation. pp. 79–80. ISBN 978-0-486-42118-6.

Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.

Geoffrey New (7 April 2011). Introduction to Nonlinear Optics. Cambridge University Press. ISBN 978-1-139-50076-0.

Dorn, R.; Quabis, S. & Leuchs, G. (Dec 2003). "Sharper Focus for a Radially Polarized Light Beam". Physical Review Letters. 91 (23): 233901. Bibcode:2003PhRvL..91w3901D. doi:10.1103/PhysRevLett.91.233901. PMID 14683185.

Chandrasekhar, Subrahmanyan (1960). Radiative Transfer. Dover. p. 27. ISBN 0-486-60590-6. OCLC 924844798.

Sletten, Mark A.; Mc Laughlin, David J. (2005-04-15). "Radar Polarimetry". In Chang, Kai (ed.). Encyclopedia of RF and Microwave Engineering. John Wiley & Sons, Inc. doi:10.1002/0471654507.eme343. ISBN 978-0-471-65450-6.

Schrank, Helmut E.; Evans, Gary E.; Davis, Daniel (1990). "6 Reflector Antennas" (PDF). In Skolnik, Merrill Ivan (ed.). Radar Handbook (PDF). McGraw-Hill. pp. 6.30, Fig 6.25. ISBN 978-0-07-057913-2. Archived (PDF) from the original on 2022-10-09.

Ishii, T. Koryu, ed. (1995). Handbook of Microwave Technology. Vol. 2: Applications. Elsevier. p. 177. ISBN 978-0-08-053410-7.

Volakis, John (2007). Antenna Engineering Handbook, Fourth Edition. McGraw-Hill. Sec. 26.1. ISBN 9780071475747: Note: in contrast with other authors, this source initially defines ellipticity reciprocally, as the minor-to-major-axis ratio, but then goes on to say that "Although [it] is less than unity, when expressing ellipticity in decibels, the minus sign is frequently omitted for convenience", which essentially reverts to the definition adopted by other authors.

Chipman, R.A.; Lam, W.S.T.; Young, G. (2018). Polarized Light and Optical Systems. Optical Sciences and Applications of Light. CRC Press. ISBN 978-1-4987-0057-3. Retrieved 2023-01-20.

Sharma, K.K. (2006). Optics: Principles and Applications. Elsevier Science. p. 145. ISBN 978-0-08-046391-9. Retrieved 2023-01-20.

Prakash, Hari; Chandra, Naresh (1971). "Density Operator of Unpolarized Radiation". Physical Review A. 4 (2): 796–799. Bibcode:1971PhRvA...4..796P. doi:10.1103/PhysRevA.4.796.

Chandrasekhar, Subrahmanyan (2013). Radiative transfer. Courier. p. 30.

Hecht, Eugene (2002). Optics (4th ed.). United States of America: Addison Wesley. ISBN 0-8053-8566-5.

Bekefi, George; Barrett, Alan (1977). Electromagnetic Vibrations, Waves, and Radiation. USA: MIT Press. ISBN 0-262-52047-8.

Hecht, Eugene (2002). Optics (4th ed.). United States of America: Addison Wesley. ISBN 0-8053-8566-5.

Dennis Goldstein; Dennis H. Goldstein (3 January 2011). Polarized Light, Revised and Expanded. CRC Press. ISBN 978-0-203-91158-7.

Masud Mansuripur (2009). Classical Optics and Its Applications. Cambridge University Press. ISBN 978-0-521-88169-2.

Randy O. Wayne (16 December 2013). Light and Video Microscopy. Academic Press. ISBN 978-0-12-411536-1.

Peter M. Shearer (2009). Introduction to Seismology. Cambridge University Press. ISBN 978-0-521-88210-1.

Seth Stein; Michael Wysession (1 April 2009). An Introduction to Seismology, Earthquakes, and Earth Structure. John Wiley & Sons. ISBN 978-1-4443-1131-0.

Vollhardt, K. Peter C.; Schore, Neil E. (2003). Organic Chemistry: Structure and Function (4th ed.). W. H. Freeman. pp. 169–172. ISBN 978-0-7167-4374-3.

Vlemmings, W. H. T. (Mar 2007). "A review of maser polarization and magnetic fields". Proceedings of the International Astronomical Union. 3 (S242): 37–46. arXiv:0705.0885. Bibcode:2007IAUS..242...37V. doi:10.1017/s1743921307012549.

Hannu Karttunen; Pekka Kröger; Heikki Oja (27 June 2007). Fundamental Astronomy. Springer. ISBN 978-3-540-34143-7.

Boyle, Latham A.; Steinhardt, PJ; Turok, N (2006). "Inflationary predictions for scalar and tensor fluctuations reconsidered". Physical Review Letters. 96 (11): 111301. arXiv:astro-ph/0507455. Bibcode:2006PhRvL..96k1301B. doi:10.1103/PhysRevLett.96.111301. PMID 16605810. S2CID 10424288.

Tegmark, Max (2005). "What does inflation really predict?". Journal of Cosmology and Astroparticle Physics. 0504 (4): 001. arXiv:astro-ph/0410281. Bibcode:2005JCAP...04..001T. doi:10.1088/1475-7516/2005/04/001. S2CID 17250080.

Clark, S. (1999). "Polarised starlight and the handedness of Life". American Scientist. 97 (4): 336–43. Bibcode:1999AmSci..87..336C. doi:10.1511/1999.4.336.

Bekefi, George; Barrett, Alan (1977). Electromagnetic Vibrations, Waves, and Radiation. USA: MIT Press. ISBN 0-262-52047-8.

J. David Pye (13 February 2001). Polarised Light in Science and Nature. CRC Press. ISBN 978-0-7503-0673-7.

Sonja Kleinlogel; Andrew White (2008). "The secret world of shrimps: polarisation vision at its best". PLOS ONE. 3 (5): e2190. arXiv:0804.2162. Bibcode:2008PLoSO...3.2190K. doi:10.1371/journal.pone.0002190. PMC 2377063. PMID 18478095.

Nuboer, J. F. W.; Coemans, M. a. J. M.; Vos Hzn, J. J. (1995-02-01). "No evidence for polarization sensitivity in the pigeon electroretinogram". Journal of Experimental Biology. 198 (2): 325–335. doi:10.1242/jeb.198.2.325. ISSN 0022-0949. PMID 9317897.

"'Fastest spinning object' created". BBC News. 2013-08-28. Retrieved 2019-08-27.

40. Dholakia, Kishan; Mazilu, Michael; Arita, Yoshihiko (August 28, 2013). "Laser-induced rotation and cooling of a trapped microgyroscope in vacuum". Nature Communications. 4: 2374. Bibcode:2013NatCo...4.2374A. doi:10.1038/ncomms3374. hdl:10023/4019. PMC 3763500. PMID 23982323.

General references

• Principles of Optics, 7th edition, M. Born & E. Wolf, Cambridge University, 1999, ISBN 0-521-64222-1.
• Fundamentals of polarized light: a statistical optics approach, C. Brosseau, Wiley, 1998, ISBN 0-471-14302-2.
• Polarized Light, second edition, Dennis Goldstein, Marcel Dekker, 2003, ISBN 0-8247-4053-X.
• Field Guide to Polarization, Edward Collett, SPIE Field Guides vol. FG05, SPIE, 2005, ISBN 0-8194-5868-6.
• Polarization Optics in Telecommunications, Jay N. Damask, Springer 2004, ISBN 0-387-22493-9.
• Polarized Light in Nature, G. P. Können, Translated by G. A. Beerling, Cambridge University, 1985, ISBN 0-521-25862-6.
• Polarised Light in Science and Nature, D. Pye, Institute of Physics, 2001, ISBN 0-7503-0673-4.
• Polarized Light, Production and Use, William A. Shurcliff, Harvard University, 1962.
• Ellipsometry and Polarized Light, R. M. A. Azzam and N. M. Bashara, North-Holland, 1977, ISBN 0-444-87016-4.
• Secrets of the Viking Navigators—How the Vikings used their amazing sunstones and other techniques to cross the open oceans, Leif Karlsen, One Earth Press, 2003.

External links

• Feynman's lecture on polarization
• Polarized Light in Nature and Technology
• Polarized Light Digital Image Gallery: Microscopic images made using polarization effects
• Polarization by the University of Colorado Physics 2000: Animated explanation of polarization
• MathPages: The relationship between photon spin and polarization
• A virtual polarization microscope
• Polarization angle in satellite dishes.
• Using polarizers in photography
• Molecular Expressions: Science, Optics and You — Polarization of Light: Interactive Java tutorial
• SPIE technical group on polarization
• Antenna Polarization
• Animations of Linear, Circular and Elliptical Polarizations on YouTube

Authority control: National	Germany Japan

Categories:

• Polarization (waves)
• Electromagnetic radiation
• Antennas (radio)
• Broadcast engineering
• Physical optics

 https://en.wikipedia.org/wiki/Polarization_(physics)

 

Pleochroism of cordierite shown by rotating a polarizing filter on the lens of the camera

Pleochroism of tourmaline shown by rotating a polarizing filter on the lens of the camera

Pleochroism (from Greek πλέων, pléōn, "more" and χρῶμα, khrôma, "color") is an optical phenomenon in which a substance has different colors when observed at different angles, especially with polarized light.^[1]

ackground

Anisotropic crystals will have optical properties that vary with the direction of light. The direction of the electric field determines the polarization of light, and crystals will respond in different ways if this angle is changed. These kinds of crystals have one or two optical axes. If absorption of light varies with the angle relative to the optical axis in a crystal then pleochroism results.^[2]

Anisotropic crystals have double refraction of light where light of different polarizations is bent different amounts by the crystal, and therefore follows different paths through the crystal. The components of a divided light beam follow different paths within the mineral and travel at different speeds. When the mineral is observed at some angle, light following some combination of paths and polarizations will be present, each of which will have had light of different colors absorbed. At another angle, the light passing through the crystal will be composed of another combination of light paths and polarizations, each with their own color. The light passing through the mineral will therefore have different colors when it is viewed from different angles, making the stone seem to be of different colors.

Tetragonal, trigonal, and hexagonal minerals can only show two colors and are called dichroic. Orthorhombic, monoclinic, and triclinic crystals can show three and are trichroic. For example, hypersthene, which has two optical axes, can have a red, yellow, or blue appearance when oriented in three different ways in three-dimensional space.^[3] Isometric minerals cannot exhibit pleochroism.^[1][4] Tourmaline is notable for exhibiting strong pleochroism. Gems are sometimes cut and set either to display pleochroism or to hide it, depending on the colors and their attractiveness.

The pleochroic colors are at their maximum when light is polarized parallel with a principal optical vector. The axes are designated X, Y, and Z for direction, and alpha, beta, and gamma in magnitude of the refractive index. These axes can be determined from the appearance of a crystal in a conoscopic interference pattern. Where there are two optical axes, the acute bisectrix of the axes gives Z for positive minerals and X for negative minerals and the obtuse bisectrix gives the alternative axis (X or Z). Perpendicular to these is the Y axis. The color is measured with the polarization parallel to each direction. An absorption formula records the amount of absorption parallel to each axis in the form of X < Y < Z with the left most having the least absorption and the rightmost the most.^[5]

In mineralogy and gemology

Pleochroism is an extremely useful tool in mineralogy and gemology for mineral and gem identification, since the number of colors visible from different angles can identify the possible crystalline structure of a gemstone or mineral and therefore help to classify it. Minerals that are otherwise very similar often have very different pleochroic color schemes. In such cases, a thin section of the mineral is used and examined under polarized transmitted light with a petrographic microscope. Another device using this property to identify minerals is the dichroscope.^[6]

List of pleochroic minerals

Purple and violet

• Amethyst (very low): different shades of purple
• Andalusite (strong): green-brown / dark red / purple
• Beryl (medium): purple / colorless
• Corundum (high): purple / orange
• Hypersthene (strong): purple / orange
• Spodumene (Kunzite) (strong): purple / clear / pink
• Tourmaline (strong): pale purple / purple
• Putnisite: pale purple / bluish grey

Blue

• Aquamarine (medium): clear / light blue, or light blue / dark blue
• Alexandrite (strong): dark red-purple / orange / green
• Apatite (strong): blue-yellow / blue-colorless
• Benitoite (strong): colorless / dark blue
• Cordierite (aka Iolite) (orthorhombic; very strong): pale yellow / violet / pale blue
• Corundum (strong): dark violet-blue / light blue-green
• Tanzanite See Zoisite
• Topaz (very low): colorless / pale blue / pink
• Tourmaline (strong): dark blue / light blue
• Zoisite (strong): blue / red-purple / yellow-green
• Zircon (strong): blue / clear / gray

Green

• Alexandrite (strong): dark red / orange / green
• Andalusite (strong): brown-green / dark red
• Corundum (strong): green / yellow-green
• Emerald (strong): green / blue-green
• Peridot (low): yellow-green / green / colorless
• Titanite (medium): brown-green / blue-green
• Tourmaline (strong): blue-green / brown-green / yellow-green
• Zircon (low): greenish brown / green
• Kornerupine (strong): green / pale yellowish-brown / reddish-brown
• Hiddenite (strong): blue-green / emerald-green / yellow-green

Yellow

• Citrine (very weak): different shades of pale yellow
• Chrysoberyl (very weak): red-yellow / yellow-green / green
• Corundum (weak): yellow / pale yellow
• Danburite (weak): very pale yellow / pale yellow
• Kasolite (weak): pale yellow / grey
• Orthoclase (weak): different shades of pale yellow
• Phenacite (medium): colorless / yellow-orange
• Spodumene (medium): different shades of pale yellow
• Topaz (medium): tan / yellow / yellow-orange
• Tourmaline (medium): pale yellow / dark yellow
• Zircon (weak): tan / yellow
• Hornblende (strong): light green / dark green / yellow / brown
• Segnitite (weak): pale to medium yellow

Brown and orange

• Corundum (strong): yellow-brown / orange
• Topaz (medium): brown-yellow / dull brown-yellow
• Tourmaline (very low): dark brown / light brown
• Zircon (very weak): brown-red / brown-yellow
• Biotite (medium): brown

Red and pink

• Alexandrite (strong): dark red / orange / green
• Andalusite (strong): dark red / brown-red
• Corundum (strong): violet-red / orange-red
• Morganite (medium): light red / red-violet
• Tourmaline (strong): dark red / light red
• Zircon (medium): purple / red-brown

See also

• Birefringence
• Medieval sunstone

 

https://en.wikipedia.org/wiki/Pleochroism

Newton's rings is a phenomenon in which an interference pattern is created by the reflection of light between two surfaces, typically a spherical surface and an adjacent touching flat surface. It is named after Isaac Newton, who investigated the effect in 1666. When viewed with monochromatic light, Newton's rings appear as a series of concentric, alternating bright and dark rings centered at the point of contact between the two surfaces. When viewed with white light, it forms a concentric ring pattern of rainbow colors because the different wavelengths of light interfere at different thicknesses of the air layer between the surfaces.

History

The phenomenon was first described by Robert Hooke in his 1665 book Micrographia. Its name derives from the mathematician and physicist Sir Isaac Newton, who studied the phenomenon in 1666 while sequestered at home in Lincolnshire in the time of the Great Plague that had shut down Trinity College, Cambridge. He recorded his observations in an essay entitled "Of Colours". The phenomenon became a source of dispute between Newton, who favored a corpuscular nature of light, and Hooke, who favored a wave-like nature of light.^[1] Newton did not publish his analysis until after Hooke's death, as part of his treatise "Opticks" published in 1704.

Theory

Fig. 4: Closeup of a section of the top glass on the optical flat, showing how interference fringes form. At positions where the path length difference is equal to an odd multiple(2n+1) of a half-wavelength (a), the reflected waves reinforce, resulting in a bright spot. At positions where the path length difference is equal to an even multiple (2n)of a half-wavelength (b), (Lambda by 2)the reflected waves cancel, resulting in a dark spot. This results in a pattern of concentric bright and dark rings, interference fringes.

The pattern is created by placing a very slightly convex curved glass on an optical flat glass. The two pieces of glass make contact only at the center. At other points there is a slight air gap between the two surfaces, increasing with radial distance from the center, as shown in Fig. 3.

Consider monochromatic (single color) light incident from the top that reflects from both the bottom surface of the top lens and the top surface of the optical flat below it.^[2] The light passes through the glass lens until it comes to the glass-to-air boundary, where the transmitted light goes from a higher refractive index (n) value to a lower n value. The transmitted light passes through this boundary with no phase change. The reflected light undergoing internal reflection (about 4% of the total) also has no phase change. The light that is transmitted into the air travels a distance, t, before it is reflected at the flat surface below. Reflection at this air-to-glass boundary causes a half-cycle (180°) phase shift because the air has a lower refractive index than the glass. The reflected light at the lower surface returns a distance of (again) t and passes back into the lens. The additional path length is equal to twice the gap between the surfaces. The two reflected rays will interfere according to the total phase change caused by the extra path length 2t and by the half-cycle phase change induced in reflection at the flat surface. When the distance 2t is zero (lens touching optical flat) the waves interfere destructively, hence the central region of the pattern is dark, as shown in Fig. 2.

A similar analysis for illumination of the device from below instead of from above shows that in this case the central portion of the pattern is bright, not dark, as shown in Fig. 1. When the light is not monochromatic, the radial position of the fringe pattern has a "rainbow" appearance, as shown in Fig. 5.

Constructive interference

(Fig. 4a): In areas where the path length difference between the two rays is equal to an odd multiple of half a wavelength (λ/2) of the light waves, the reflected waves will be in phase, so the "troughs" and "peaks" of the waves coincide. Therefore, the waves will reinforce (add) and the resulting reflected light intensity will be greater. As a result, a bright area will be observed there.

Destructive interference

(Fig. 4b): At other locations, where the path length difference is equal to an even multiple of a half-wavelength, the reflected waves will be 180° out of phase, so a "trough" of one wave coincides with a "peak" of the other wave. Therefore, the waves will cancel (subtract) and the resulting light intensity will be weaker or zero. As a result, a dark area will be observed there. Because of the 180° phase reversal due to reflection of the bottom ray, the center where the two pieces touch is dark.

This interference results in a pattern of bright and dark lines or bands called "interference fringes" being observed on the surface. These are similar to contour lines on maps, revealing differences in the thickness of the air gap. The gap between the surfaces is constant along a fringe. The path length difference between two adjacent bright or dark fringes is one wavelength λ of the light, so the difference in the gap between the surfaces is one-half wavelength. Since the wavelength of light is so small, this technique can measure very small departures from flatness. For example, the wavelength of red light is about 700 nm, so using red light the difference in height between two fringes is half that, or 350 nm, about 1/100 the diameter of a human hair. Since the gap between the glasses increases radially from the center, the interference fringes form concentric rings. For glass surfaces that are not spherical, the fringes will not be rings but will have other shapes.

Quantitative Relationships

Fig. 5: Newton's rings seen in two plano-convex lenses with their flat surfaces in contact. One surface is slightly convex, creating the rings. In white light, the rings are rainbow-colored, because the different wavelengths of each color interfere at different locations.

Rainbow-colored Newton's rings on an Agfacolor slide (slightly right of center on the houses and upper right on the mountains).

For illumination from above, with a dark center, the radius of the N^th bright ring is given by

$${\displaystyle r_N={\left[\lambda R\left(N-\frac{1}{2}\right)\right]}^{1/2},}$$

where N is the bright-ring number, R is the radius of curvature of the glass lens the light is passing through, and λ is the wavelength of the light. The above formula is also applicable for dark rings for the ring pattern obtained by transmitted light.

Given the radial distance of a bright ring, r, and a radius of curvature of the lens, R, the air gap between the glass surfaces, t, is given to a good approximation by

$${\displaystyle t=\frac{r^2}{2R},}$$

where the effect of viewing the pattern at an angle oblique to the incident rays is ignored.

Thin-film interference

Main article: Thin-film interference

The phenomenon of Newton's rings is explained on the same basis as thin-film interference, including effects such as "rainbows" seen in thin films of oil on water or in soap bubbles. The difference is that here the "thin film" is a thin layer of air.

References

1. 
   

Westfall, Richard S. (1980). Never at Rest, A Biography of Isaac Newton. Cambridge University Press. p. 171. ISBN 0-521-23143-4.

2. Young, Hugh D.; Freedman, Roger A. (2012). University Physics, 13th Ed. Addison Wesley. p. 1178. ISBN 978-0-321-69686-1.

Further reading

• Airy, G.B. (1833). "VI.On the phænomena of Newton's rings when formed between two transparent substances of different refractive powers". Philosophical Magazine. Series 3. 2 (7): 20–30. doi:10.1080/14786443308647959. ISSN 1941-5966.
• Illueca, C.; Vazquez, C.; Hernandez, C.; Viqueira, V. (1998). "The use of Newton's rings for characterizing ophthalmic lenses". Ophthalmic and Physiological Optics. 18 (4): 360–371. doi:10.1046/j.1475-1313.1998.00366.x. ISSN 0275-5408. PMID 9829108. S2CID 222086863.
• Dobroiu, Adrian; Alexandrescu, Adrian; Apostol, Dan; Nascov, Victor; Damian, Victor S. (2000). "Improved method for processing Newton's rings fringe patterns". In Necsoiu, Teodor; Robu, Maria; Dumitras, Dan C (eds.). SIOEL '99: Sixth Symposium on Optoelectronics. Proceedings. Vol. 4068. pp. 342–347. doi:10.1117/12.378693. ISSN 0277-786X. S2CID 120241545.
• Tolansky, S. (2009). "XIV. New contributions to interferometry. Part II—New interference phenomena with Newton's rings". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 35 (241): 120–136. doi:10.1080/14786444408521466. ISSN 1941-5982.

External links

Wikimedia Commons has media related to Newton's rings.

• Newton's Ring from Eric Weisstein's World of Physics
• Explanation of and expression for Newton's rings
• Newton's rings Video of a simple experiment with two lenses, and Newton's rings on mica observed. (On the website FizKapu.) (in Hungarian)

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https://en.wikipedia.org/wiki/Newton%27s_rings

 

In mathematics, physics, and art, moiré patterns (UK: /ˈmwɑːreɪ/ MWAR-ay, US: /mwɑːˈreɪ/ mwar-AY,^[1] French: [mwaʁe] (listen)) or moiré fringes^[2] are large-scale interference patterns that can be produced when a partially opaque ruled pattern with transparent gaps is overlaid on another similar pattern. For the moiré interference pattern to appear, the two patterns must not be completely identical, but rather displaced, rotated, or have slightly different pitch.

Moiré patterns appear in many situations. In printing, the printed pattern of dots can interfere with the image. In television and digital photography, a pattern on an object being photographed can interfere with the shape of the light sensors to generate unwanted artifacts. They are also sometimes created deliberately – in micrometers they are used to amplify the effects of very small movements.

In physics, its manifestation is wave interference such as that seen in the double-slit experiment and the beat phenomenon in acoustics.

Etymology

The term originates from moire (moiré in its French adjectival form), a type of textile, traditionally made of silk but now also made of cotton or synthetic fiber, with a rippled or "watered" appearance. Moire, or "watered textile", is made by pressing two layers of the textile when wet. The similar but imperfect spacing of the threads creates a characteristic pattern which remains after the fabric dries.

In French, the noun moire is in use from the 17th century, for "watered silk". It was a loan of the English mohair (attested 1610). In French usage, the noun gave rise to the verb moirer, "to produce a watered textile by weaving or pressing", by the 18th century. The adjective moiré formed from this verb is in use from at least 1823.

Pattern formation

Line moiré with slow movement of the revealing layer upward

Shape moiré

Moiré pattern created by overlapping two sets of concentric circles

Moiré patterns are often an artifact of images produced by various digital imaging and computer graphics techniques, for example when scanning a halftone picture or ray tracing a checkered plane (the latter being a special case of aliasing, due to undersampling a fine regular pattern).^[3] This can be overcome in texture mapping through the use of mipmapping and anisotropic filtering.

The drawing on the upper right shows a moiré pattern. The lines could represent fibers in moiré silk, or lines drawn on paper or on a computer screen. The nonlinear interaction of the optical patterns of lines creates a real and visible pattern of roughly parallel dark and light bands, the moiré pattern, superimposed on the lines.^[4]

The moiré effect also occurs between overlapping transparent objects.^[5] For example, an invisible phase mask is made of a transparent polymer with a wavy thickness profile. As light shines through two overlaid masks of similar phase patterns, a broad moiré pattern occurs on a screen some distance away. This phase moiré effect and the classical moiré effect from opaque lines are two ends of a continuous spectrum in optics, which is called the universal moiré effect. The phase moiré effect is the basis for a type of broadband interferometer in x-ray and particle wave applications. It also provides a way to reveal hidden patterns in invisible layers.

Line moiré

Main article: Line moiré

Line moiré is one type of moiré pattern; a pattern that appears when superposing two transparent layers containing correlated opaque patterns. Line moiré is the case when the superposed patterns comprise straight or curved lines. When moving the layer patterns, the moiré patterns transform or move at a faster speed. This effect is called optical moiré speedup.

More complex line moiré patterns are created if the lines are curved or not exactly parallel.

Shape moiré

Main article: Shape moiré

Shape moiré is one type of moiré pattern demonstrating the phenomenon of moiré magnification.^[6][7] 1D shape moiré is the particular simplified case of 2D shape moiré. One-dimensional patterns may appear when superimposing an opaque layer containing tiny horizontal transparent lines on top of a layer containing a complex shape which is periodically repeating along the vertical axis.

Moiré patterns revealing complex shapes, or sequences of symbols embedded in one of the layers (in form of periodically repeated compressed shapes) are created with shape moiré, otherwise called band moiré patterns. One of the most important properties of shape moiré is its ability to magnify tiny shapes along either one or both axes, that is, stretching. A common 2D example of moiré magnification occurs when viewing a chain-link fence through a second chain-link fence of identical design. The fine structure of the design is visible even at great distances.

Calculations

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Moiré of parallel patterns

Geometrical approach

The patterns are superimposed in the mid-width of the figure.

Moiré obtained by the superimposition of two similar patterns rotated by an angle α

Consider two patterns made of parallel and equidistant lines, e.g., vertical lines. The step of the first pattern is p, the step of the second is p + δp, with 0 < δp < p.

If the lines of the patterns are superimposed at the left of the figure, the shift between the lines increases when going to the right. After a given number of lines, the patterns are opposed: the lines of the second pattern are between the lines of the first pattern. If we look from a far distance, we have the feeling of pale zones when the lines are superimposed (there is white between the lines), and of dark zones when the lines are "opposed".

The middle of the first dark zone is when the shift is equal to p/2. The nth line of the second pattern is shifted by n δp compared to the nth line of the first network. The middle of the first dark zone thus corresponds to

$${\displaystyle n\cdot \delta p=\frac{p}{2}}$$

that is

$${\displaystyle n=\frac{p}{2\delta p}.}$$

The distance d between the middle of a pale zone and a dark zone is

$${\displaystyle d=n\cdot (p+\delta p)=\frac{p^2}{2\delta p}+\frac{p}{2}}$$

the distance between the middle of two dark zones, which is also the distance between two pale zones, is

$${\displaystyle 2d=\frac{p^2}{\delta p}+p}$$

From this formula, we can see that:

• the bigger the step, the bigger the distance between the pale and dark zones;
• the bigger the discrepancy δp, the closer the dark and pale zones; a great spacing between dark and pale zones mean that the patterns have very close steps.

The principle of the moiré is similar to the Vernier scale.

Mathematical function approach

Moiré pattern (bottom) created by superimposing two grids (top and middle)

The essence of the moiré effect is the (mainly visual) perception of a distinctly different third pattern which is caused by inexact superimposition of two similar patterns. The mathematical representation of these patterns is not trivially obtained and can seem somewhat arbitrary. In this section we shall give a mathematical example of two parallel patterns whose superimposition forms a moiré pattern, and show one way (of many possible ways) these patterns and the moiré effect can be rendered mathematically.

The visibility of these patterns is dependent on the medium or substrate in which they appear, and these may be opaque (as for example on paper) or transparent (as for example in plastic film). For purposes of discussion we shall assume the two primary patterns are each printed in greyscale ink on a white sheet, where the opacity (e.g., shade of grey) of the "printed" part is given by a value between 0 (white) and 1 (black) inclusive, with 1/2 representing neutral grey. Any value less than 0 or greater than 1 using this grey scale is essentially "unprintable".

We shall also choose to represent the opacity of the pattern resulting from printing one pattern atop the other at a given point on the paper as the average (i.e. the arithmetic mean) of each pattern's opacity at that position, which is half their sum, and, as calculated, does not exceed 1. (This choice is not unique. Any other method to combine the functions that satisfies keeping the resultant function value within the bounds [0,1] will also serve; arithmetic averaging has the virtue of simplicity—with hopefully minimal damage to one's concepts of the printmaking process.)

We now consider the "printing" superimposition of two almost similar, sinusoidally varying, grey-scale patterns to show how they produce a moiré effect in first printing one pattern on the paper, and then printing the other pattern over the first, keeping their coordinate axes in register. We represent the grey intensity in each pattern by a positive opacity function of distance along a fixed direction (say, the x-coordinate) in the paper plane, in the form

$${\displaystyle f=\frac{1+\sin (kx)}{2}}$$

where the presence of 1 keeps the function positive definite, and the division by 2 prevents function values greater than 1.

The quantity k represents the periodic variation (i.e., spatial frequency) of the pattern's grey intensity, measured as the number of intensity cycles per unit distance. Since the sine function is cyclic over argument changes of 2π, the distance increment Δx per intensity cycle (the wavelength) obtains when k Δx = 2π, or Δx = 2π/k.

Consider now two such patterns, where one has a slightly different periodic variation from the other:

$${\displaystyle \begin{array}{rl}{f}_1& =\frac{1+\sin (k_{1}x)}{2}\\ f_2& =\frac{1+\sin (k_{2}x)}{2}\end{array}}$$

such that k₁ ≈ k₂.

The average of these two functions, representing the superimposed printed image, evaluates as follows (see reverse identities here :Prosthaphaeresis ):

$${\displaystyle \begin{array}{rl}{f}_3& =\frac{f_1+f_2}{2}\\ & =\frac{1}{2}+\frac{\sin (k_{1}x)+\sin (k_{2}x)}{4}\\ & =\frac{1+\sin (Ax)\cos (Bx)}{2}\end{array}}$$

where it is easily shown that

$${\displaystyle A=\frac{k_1+k_2}{2}}$$

and

$${\displaystyle B=\frac{k_1-k_2}{2}.}$$

This function average, f₃, clearly lies in the range [0,1]. Since the periodic variation A is the average of and therefore close to k₁ and k₂, the moiré effect is distinctively demonstrated by the sinusoidal envelope "beat" function cos(Bx), whose periodic variation is half the difference of the periodic variations k₁ and k₂ (and evidently much lower in frequency).

Other one-dimensional moiré effects include the classic beat frequency tone which is heard when two pure notes of almost identical pitch are sounded simultaneously. This is an acoustic version of the moiré effect in the one dimension of time: the original two notes are still present—but the listener's perception is of two pitches that are the average of and half the difference of the frequencies of the two notes. Aliasing in sampling of time-varying signals also belongs to this moiré paradigm.

Rotated patterns

Consider two patterns with the same step p, but the second pattern is rotated by an angle α. Seen from afar, we can also see darker and paler lines: the pale lines correspond to the lines of nodes, that is, lines passing through the intersections of the two patterns.

If we consider a cell of the lattice formed, we can see that it is a rhombus with the four sides equal to d = p/sin α; (we have a right triangle whose hypotenuse is d and the side opposite to the angle α is p).

Unit cell of the "net"; "ligne claire" means "pale line".

Effect of changing angle

The pale lines correspond to the small diagonal of the rhombus. As the diagonals are the bisectors of the neighbouring sides, we can see that the pale line makes an angle equal to α/2 with the perpendicular of each pattern's line.

Additionally, the spacing between two pale lines is D, half of the long diagonal. The long diagonal 2D is the hypotenuse of a right triangle and the sides of the right angle are d(1 + cos α) and p. The Pythagorean theorem gives:

$${\displaystyle (2D{)}^2=d^2(1+\cos \alpha {)}^2+p^2}$$

that is:

$${\displaystyle \begin{array}{rl}(2D{)}^2& =\frac{p^2}{{\sin }^2\alpha }(1+\cos \alpha {)}^2+p^2\\ & =p^2\cdot \left(\frac{(1+\cos \alpha {)}^2}{{\sin }^2\alpha }+1\right)\end{array}}$$

thus

$${\displaystyle \begin{array}{rl}(2D{)}^2& =2p^2\cdot \frac{1+\cos \alpha }{{\sin }^2\alpha }\\ D& =\frac{\frac{p}{2}}{\sin \frac{\alpha }{2}}.\end{array}}$$

Effect on curved lines

When α is very small (α < π/6) the following small-angle approximations can be made:

$${\displaystyle \begin{array}{rl}\sin \alpha & \approx \alpha \\ \cos \alpha & \approx 1\end{array}}$$

thus

$${\displaystyle D\approx \frac{p}{\alpha }.}$$

We can see that the smaller α is, the farther apart the pale lines; when both patterns are parallel (α = 0), the spacing between the pale lines is infinite (there is no pale line).

There are thus two ways to determine α: by the orientation of the pale lines and by their spacing

$${\displaystyle \alpha \approx \frac{p}{D}}$$

If we choose to measure the angle, the final error is proportional to the measurement error. If we choose to measure the spacing, the final error is proportional to the inverse of the spacing. Thus, for the small angles, it is best to measure the spacing.

Implications and applications

Printing full-color images

1:30

Warning: audiogenic epileptic seizure hazard. The product of two "beat tracks" of slightly different speeds overlaid, producing an audible moiré pattern; if the beats of one track correspond to where in space a black dot or line exists and the beats of the other track correspond to the points in space where a camera is sampling light, because the frequencies are not exactly the same and aligned perfectly together, beats (or samples) will align closely at some moments in time and far apart at other times. The closer together beats are, the darker it is at that spot; the farther apart, the lighter. The result is periodic in the same way as a graphic moiré pattern. See: phasing.

In graphic arts and prepress, the usual technology for printing full-color images involves the superimposition of halftone screens. These are regular rectangular dot patterns—often four of them, printed in cyan, yellow, magenta, and black. Some kind of moiré pattern is inevitable, but in favorable circumstances the pattern is "tight"; that is, the spatial frequency of the moiré is so high that it is not noticeable. In the graphic arts, the term moiré means an excessively visible moiré pattern. Part of the prepress art consists of selecting screen angles and halftone frequencies which minimize moiré. The visibility of moiré is not entirely predictable. The same set of screens may produce good results with some images, but visible moiré with others.

Television screens and photographs

Strong moiré visible in this photo of a parrot's feathers (more pronounced in the full-size image)

Moiré pattern seen over a cage in the San Francisco Zoo

Moiré patterns are commonly seen on television screens when a person is wearing a shirt or jacket of a particular weave or pattern, such as a houndstooth jacket. This is due to interlaced scanning in televisions and non-film cameras, referred to as interline twitter. As the person moves about, the moiré pattern is quite noticeable. Because of this, newscasters and other professionals who regularly appear on TV are instructed to avoid clothing which could cause the effect.

Photographs of a TV screen taken with a digital camera often exhibit moiré patterns. Since both the TV screen and the digital camera use a scanning technique to produce or to capture pictures with horizontal scan lines, the conflicting sets of lines cause the moiré patterns. To avoid the effect, the digital camera can be aimed at an angle of 30 degrees to the TV screen.

Marine navigation

The moiré effect is used in shoreside beacons called "Inogon leading marks" or "Inogon lights", manufactured by Inogon Licens AB, Sweden, to designate the safest path of travel for ships heading to locks, marinas, ports, etc., or to indicate underwater hazards (such as pipelines or cables). The moiré effect creates arrows that point towards an imaginary line marking the hazard or line of safe passage; as navigators pass over the line, the arrows on the beacon appear to become vertical bands before changing back to arrows pointing in the reverse direction.^[8][9][10] An example can be found in the UK on the eastern shore of Southampton Water, opposite Fawley oil refinery (50°51′21.63″N 1°19′44.77″W).^[11] Similar moiré effect beacons can be used to guide mariners to the centre point of an oncoming bridge; when the vessel is aligned with the centreline, vertical lines are visible. Inogon lights are deployed at airports to help pilots on the ground keep to the centreline while docking on stand.^[12]

Strain measurement

See also: Theory of elasticity, Strain tensor, and Holographic interferometry

Use of the moiré effect in strain measurement: case of uniaxial traction (top) and of pure shear (bottom); the lines of the patterns are initially horizontal in both cases

In manufacturing industries, these patterns are used for studying microscopic strain in materials: by deforming a grid with respect to a reference grid and measuring the moiré pattern, the stress levels and patterns can be deduced. This technique is attractive because the scale of the moiré pattern is much larger than the deflection that causes it, making measurement easier.

The moiré effect can be used in strain measurement: the operator just has to draw a pattern on the object, and superimpose the reference pattern to the deformed pattern on the deformed object.

A similar effect can be obtained by the superposition of a holographic image of the object to the object itself: the hologram is the reference step, and the difference with the object are the deformations, which appear as pale and dark lines.

Image processing

Some image scanner computer programs provide an optional filter, called a "descreen" filter, to remove Moiré-pattern artifacts which would otherwise be produced when scanning printed halftone images to produce digital images.^[13]

Banknotes

Many banknotes exploit the tendency of digital scanners to produce moiré patterns by including fine circular or wavy designs that are likely to exhibit a moiré pattern when scanned and printed.^[14]

Microscopy

In super-resolution microscopy, the moiré pattern can be used to obtain images with a resolution higher than the diffraction limit, using a technique known as structured illumination microscopy.^[2]

In scanning tunneling microscopy, moiré fringes appear if surface atomic layers have a different crystal structure than the bulk crystal. This can for example be due to surface reconstruction of the crystal, or when a thin layer of a second crystal is on the surface, e.g. single-layer,^[15][16] double-layer graphene,^[17] or Van der Waals heterostructure of graphene and hBN,^[18][19] or bismuth and antimony nanostructures.^[20]

In transmission electron microscopy (TEM), translational moiré fringes can be seen as parallel contrast lines formed in phase-contrast TEM imaging by the interference of diffracting crystal lattice planes that are overlapping, and which might have different spacing and/or orientation.^[21] Most of the moiré contrast observations reported in the literature are obtained using high-resolution phase contrast imaging in TEM. However, if probe aberration-corrected high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) imaging is used, more direct interpretation of the crystal structure in terms of atom types and positions is obtained.^[21][22]

Materials science and condensed matter physics

When graphene is grown on the (111) surface of iridium, its long-wavelength height modulation can be thought of as a moiré pattern arising from the superposition of the two mismatched hexagonal lattices.

Moiré pattern arising from the superposition of two graphene lattices twisted by 4°.

In condensed matter physics, the moiré phenomenon is commonly discussed for two-dimensional materials. The effect occurs when there is mismatch between the lattice parameter or angle of the 2D layer and that of the underlying substrate,^[15][16] or another 2D layer, such as in 2D material heterostructures.^[19][20] The phenomenon is exploited as a means of engineering the electronic structure or optical properties of materials,^[23] which some call moiré materials. The often significant changes in electronic properties when twisting two atomic layers and the prospect of electronic applications has led to the name twistronics of this field. A prominent example is in twisted bi-layer graphene, which forms a moiré pattern and at a particular magic angle exhibits superconductivity and other important electronic properties.^[24]

In materials science, known examples exhibiting moiré contrast are thin films^[25] or nanoparticles of MX-type (M = Ti, Nb; X = C, N) overlapping with austenitic matrix. Both phases, MX and the matrix, have face-centered cubic crystal structure and cube-on-cube orientation relationship. However, they have significant lattice misfit of about 20 to 24% (based on the chemical composition of alloy), which produces a moiré effect.^[22]

See also

• Aliasing
• Angle-sensitive pixel
• Barrier grid animation and stereography (kinegram)
• Beat (acoustics)
• Euclid's orchard
• Guardian (sculpture)
• Kell factor
• Lenticular printing
• Moiré Phase Tracking
• Multidimensional sampling

 

 https://en.wikipedia.org/wiki/Moir%C3%A9_pattern

 

The Mie solution to Maxwell's equations (also known as the Lorenz–Mie solution, the Lorenz–Mie–Debye solution or Mie scattering) describes the scattering of an electromagnetic plane wave by a homogeneous sphere. The solution takes the form of an infinite series of spherical multipole partial waves. It is named after Gustav Mie.

The term Mie solution is also used for solutions of Maxwell's equations for scattering by stratified spheres or by infinite cylinders, or other geometries where one can write separate equations for the radial and angular dependence of solutions. The term Mie theory is sometimes used for this collection of solutions and methods; it does not refer to an independent physical theory or law. More broadly, the "Mie scattering" formulas are most useful in situations where the size of the scattering particles is comparable to the wavelength of the light, rather than much smaller or much larger.

Mie scattering (sometimes referred to as a non-molecular scattering or aerosol particle scattering) takes place in the lower 4,500 m (15,000 ft) of the atmosphere, where many essentially spherical particles with diameters approximately equal to the wavelength of the incident ray may be present. Mie scattering theory has no upper size limitation, and converges to the limit of geometric optics for large particles.^[1]

Applications

Mie theory is very important in meteorological optics, where diameter-to-wavelength ratios of the order of unity and larger are characteristic for many problems regarding haze and cloud scattering. A further application is in the characterization of particles by optical scattering measurements. The Mie solution is also important for understanding the appearance of common materials like milk, biological tissue and latex paint.

Atmospheric science

Mie scattering occurs when the diameters of atmospheric particulates are similar to or larger than the wavelengths of the light. Dust, pollen, smoke and microscopic water droplets that form clouds are common causes of Mie scattering. Mie scattering occurs mostly in the lower portions of the atmosphere, where larger particles are more abundant, and dominates in cloudy conditions.

Cancer detection and screening

Mie theory has been used to determine whether scattered light from tissue corresponds to healthy or cancerous cell nuclei using angle-resolved low-coherence interferometry.

Clinical laboratory analysis

Mie theory is a central principle in the application of nephelometric based assays, widely used in medicine to measure various plasma proteins. A wide array of plasma proteins can be detected and quantified by nephelometry.

Magnetic particles

A number of unusual electromagnetic scattering effects occur for magnetic spheres. When the relative permittivity equals the permeability, the back-scatter gain is zero. Also, the scattered radiation is polarized in the same sense as the incident radiation. In the small-particle (or long-wavelength) limit, conditions can occur for zero forward scatter, for complete polarization of scattered radiation in other directions, and for asymmetry of forward scatter to backscatter. The special case in the small-particle limit provides interesting special instances of complete polarization and forward-scatter-to-backscatter asymmetry.^[25]

Metamaterial

Mie theory has been used to design metamaterials. They usually consist of three-dimensional composites of metal or non-metallic inclusions periodically or randomly embedded in a low-permittivity matrix. In such a scheme, the negative constitutive parameters are designed to appear around the Mie resonances of the inclusions: the negative effective permittivity is designed around the resonance of the Mie electric dipole scattering coefficient, whereas negative effective permeability is designed around the resonance of the Mie magnetic dipole scattering coefficient, and doubly negative material (DNG) is designed around the overlap of resonances of Mie electric and magnetic dipole scattering coefficients. The particle usually have the following combinations:

1. one set of magnetodielectric particles with values of relative permittivity and permeability much greater than one and close to each other;
2. two different dielectric particles with equal permittivity but different size;
3. two different dielectric particles with equal size but different permittivity.

In theory, the particles analyzed by Mie theory are commonly spherical but, in practice, particles are usually fabricated as cubes or cylinders for ease of fabrication. To meet the criteria of homogenization, which may be stated in the form that the lattice constant is much smaller than the operating wavelength, the relative permittivity of the dielectric particles should be much greater than 1, e.g. ${\displaystyle {\epsilon }_{\text{r}}>78(38)}$ to achieve negative effective permittivity (permeability).^[26][27][28] 

 

Mie scattering as a function of particle's radius. Along one cycle, the particle diameter changes from 0.1 wavelength to 1 wavelength. The sphere's refractive index is 1.5

 https://en.wikipedia.org/wiki/Mie_scattering

 https://en.wikipedia.org/wiki/Angle-resolved_low-coherence_interferometry

https://en.wikipedia.org/wiki/Codes_for_electromagnetic_scattering_by_spheres

https://en.wikipedia.org/wiki/Synchrotron_radiation

https://en.wikipedia.org/wiki/Optical_vortex

https://en.wikipedia.org/wiki/Thin-film_optics

https://en.wikipedia.org/wiki/Chromatophore#Iridophores_and_leucophores

https://en.wikipedia.org/wiki/Heliography

https://en.wikipedia.org/wiki/Vector_spherical_harmonics

In colorimetry, metamerism is a perceived matching of colors with different (nonmatching) spectral power distributions. Colors that match this way are called metamers.

A spectral power distribution describes the proportion of total light given off (emitted, transmitted, or reflected) by a color sample at each visible wavelength; it defines the complete information about the light coming from the sample. However, the human eye contains only three color receptors (three types of cone cells), which means that all colors are reduced to three sensory quantities, called the tristimulus values. Metamerism occurs because each type of cone responds to the cumulative energy from a broad range of wavelengths, so that different combinations of light across all wavelengths can produce an equivalent receptor response and the same tristimulus values or color sensation. In color science, the set of sensory spectral sensitivity curves is numerically represented by color matching functions.

Sources of metamerism

Metameric matches are quite common, especially in near neutral (grayed or whitish colors) or dark colors. As colors become brighter or more saturated, the range of possible metameric matches (different combinations of light wavelengths) becomes smaller, especially in colors from surface reflectance spectra.

Metameric matches made between two light sources provide the trichromatic basis of colorimetry. The basis for nearly all commercially available color image reproduction processes such as photography, television, printing, and digital imaging, is the ability to make metameric color matches.

Making metameric matches using reflective materials is more complex. The appearance of surface colors is defined by the product of the spectral reflectance curve of the material and the spectral emittance curve of the light source shining on it. As a result, the color of surfaces depends on the light source used to illuminate them.

Metameric failure

The term illuminant metameric failure or illuminant metamerism is sometimes used to describe situations in which two material samples match when viewed under one light source but not another. Most types of fluorescent lights produce an irregular or peaky spectral emittance curve, so that two materials under fluorescent light might not match, even though they are a metameric match to an incandescent "white" light source with a nearly flat or smooth emittance curve. Material colors that match under one source will often appear different under the other. Inkjet printing is particularly susceptible, and inkjet proofs are best viewed under standard 5000K color temperature lighting for color accuracy.^[1]

Normally, material attributes such as translucency, gloss or surface texture are not considered in color matching. However geometric metameric failure or geometric metamerism can occur when two samples match when viewed from one angle, but then fail to match when viewed from a different angle. A common example is the color variation that appears in pearlescent automobile finishes or "metallic" paper; e.g., Kodak Endura Metallic, Fujicolor Crystal Archive Digital Pearl.

Observer metameric failure or observer metamerism can occur because of differences in color vision between observers. The common source of observer metameric failure is colorblindness, but it is also not uncommon among "normal" observers. In all cases, the proportion of long-wavelength-sensitive cones to medium-wavelength-sensitive cones in the retina, the profile of light sensitivity in each type of cone, and the amount of yellowing in the lens and macular pigment of the eye, differs from one person to the next. This alters the relative importance of different wavelengths in a spectral power distribution to each observer's color perception. As a result, two spectrally dissimilar lights or surfaces may produce a color match for one observer but fail to match when viewed by a second observer.

Field-size metameric failure or field-size metamerism occurs because the relative proportions of the three cone types in the retina vary from the center of the visual field to the periphery, so that colors that match when viewed as very small, centrally fixated areas may appear different when presented as large color areas. In many industrial applications, large-field color matches are used to define color tolerances.

Finally, device metamerism comes up due to the lack of consistency of colorimeters of the same or different manufacturers. Colorimeters basically consist of a combination of a matrix of sensor cells and optical filters, which present an unavoidable variance in their measurements. Moreover, devices built by different manufacturers can differ in their construction.^[2]

The difference in the spectral compositions of two metameric stimuli is often referred to as the degree of metamerism. The sensitivity of a metameric match to any changes in the spectral elements that form the colors depend on the degree of metamerism. Two stimuli with a high degree of metamerism are likely to be very sensitive to any changes in the illuminant, material composition, observer, field of view, and so on.

The word metamerism is often used to indicate a metameric failure rather than a match, or used to describe a situation in which a metameric match is easily degraded by a slight change in conditions, such as a change in the illuminant. 

https://en.wikipedia.org/wiki/Metamerism_(color)

 

Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light.^[1] These optically anisotropic materials are said to be birefringent (or birefractive). The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress.

Birefringence is responsible for the phenomenon of double refraction whereby a ray of light, when incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. This effect was first described by Danish scientist Rasmus Bartholin in 1669, who observed it^[2] in calcite, a crystal having one of the strongest birefringences. In the 19th century Augustin-Jean Fresnel described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarization (perpendicular to the direction of the wave vector).^[3][4] Birefringence plays an important role in achieving phase-matching for a number of nonlinear optical processes.

Explanation

Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images.

A mathematical description of wave propagation in a birefringent medium is presented below. Following is a qualitative explanation of the phenomenon.

Uniaxial materials

The simplest type of birefringence is described as uniaxial, meaning that there is a single direction governing the optical anisotropy whereas all directions perpendicular to it (or at a given angle to it) are optically equivalent. Thus rotating the material around this axis does not change its optical behaviour. This special direction is known as the optic axis of the material. Light propagating parallel to the optic axis (whose polarization is always perpendicular to the optic axis) is governed by a refractive index n_o (for "ordinary") regardless of its specific polarization. For rays with any other propagation direction, there is one linear polarization that would be perpendicular to the optic axis, and a ray with that polarization is called an ordinary ray and is governed by the same refractive index value n_o. For a ray propagating in the same direction but with a polarization perpendicular to that of the ordinary ray, the polarization direction will be partly in the direction of the optic axis, and this extraordinary ray will be governed by a different, direction-dependent refractive index. Because the index of refraction depends on the polarization when unpolarized light enters a uniaxial birefringent material, it is split into two beams travelling in different directions, one having the polarization of the ordinary ray and the other the polarization of the extraordinary ray. The ordinary ray will always experience a refractive index of n_o, whereas the refractive index of the extraordinary ray will be in between n_o and n_e, depending on the ray direction as described by the index ellipsoid. The magnitude of the difference is quantified by the birefringence:^{[verification needed]}

${\displaystyle \mathrm{\Delta }n=n_{\mathrm{e}}-n_{\mathrm{o}}.}$

The propagation (as well as reflection coefficient) of the ordinary ray is simply described by n_o as if there were no birefringence involved. The extraordinary ray, as its name suggests, propagates unlike any wave in an isotropic optical material. Its refraction (and reflection) at a surface can be understood using the effective refractive index (a value in between n_o and n_e). Its power flow (given by the Poynting vector) is not exactly in the direction of the wave vector. This causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of calcite as photographed above. Rotating the calcite crystal will cause one of the two images, that of the extraordinary ray, to rotate slightly around that of the ordinary ray, which remains fixed.^{[verification needed]}

When the light propagates either along or orthogonal to the optic axis, such a lateral shift does not occur. In the first case, both polarizations are perpendicular to the optic axis and see the same effective refractive index, so there is no extraordinary ray. In the second case the extraordinary ray propagates at a different phase velocity (corresponding to n_e) but still has the power flow in the direction of the wave vector. A crystal with its optic axis in this orientation, parallel to the optical surface, may be used to create a waveplate, in which there is no distortion of the image but an intentional modification of the state of polarization of the incident wave. For instance, a quarter-wave plate is commonly used to create circular polarization from a linearly polarized source.

Biaxial materials

The case of so-called biaxial crystals is substantially more complex.^[5] These are characterized by three refractive indices corresponding to three principal axes of the crystal. For most ray directions, both polarizations would be classified as extraordinary rays but with different effective refractive indices. Being extraordinary waves, the direction of power flow is not identical to the direction of the wave vector in either case.

The two refractive indices can be determined using the index ellipsoids for given directions of the polarization. Note that for biaxial crystals the index ellipsoid will not be an ellipsoid of revolution ("spheroid") but is described by three unequal principle refractive indices n_α, n_β and n_γ. Thus there is no axis around which a rotation leaves the optical properties invariant (as there is with uniaxial crystals whose index ellipsoid is a spheroid).

Although there is no axis of symmetry, there are two optical axes or binormals which are defined as directions along which light may propagate without birefringence, i.e., directions along which the wavelength is independent of polarization.^[5] For this reason, birefringent materials with three distinct refractive indices are called biaxial. Additionally, there are two distinct axes known as optical ray axes or biradials along which the group velocity of the light is independent of polarization.

Double refraction

When an arbitrary beam of light strikes the surface of a birefringent material at non-normal incidence, the polarization component normal to the optic axis (ordinary ray) and the other linear polarization (extraordinary ray) will be refracted toward somewhat different paths. Natural light, so-called unpolarized light, consists of equal amounts of energy in any two orthogonal polarizations. Even linearly polarized light has some energy in both polarizations, unless aligned along one of the two axes of birefringence. According to Snell's law of refraction, the two angles of refraction are governed by the effective refractive index of each of these two polarizations. This is clearly seen, for instance, in the Wollaston prism which separates incoming light into two linear polarizations using prisms composed of a birefringent material such as calcite.

The different angles of refraction for the two polarization components are shown in the figure at the top of this page, with the optic axis along the surface (and perpendicular to the plane of incidence), so that the angle of refraction is different for the p polarization (the "ordinary ray" in this case, having its electric vector perpendicular to the optic axis) and the s polarization (the "extraordinary ray" in this case, whose electric field polarization includes a component in the direction of the optic axis). In addition, a distinct form of double refraction occurs, even with normal incidence, in cases where the optic axis is not along the refracting surface (nor exactly normal to it); in this case, the dielectric polarization of the birefringent material is not exactly in the direction of the wave's electric field for the extraordinary ray. The direction of power flow (given by the Poynting vector) for this inhomogenous wave is at a finite angle from the direction of the wave vector resulting in an additional separation between these beams. So even in the case of normal incidence, where one would compute the angle of refraction as zero (according to Snell's law, regardless of the effective index of refraction), the energy of the extraordinary ray is propagated at an angle. If exiting the crystal through a face parallel to the incoming face, the direction of both rays will be restored, but leaving a shift between the two beams. This is commonly observed using a piece of calcite cut along its natural cleavage, placed above a paper with writing, as in the above photographs. On the contrary, waveplates specifically have their optic axis along the surface of the plate, so that with (approximately) normal incidence there will be no shift in the image from light of either polarization, simply a relative phase shift between the two light waves.

Terminology

Comparison of positive and negative birefringence : In positive birefringence (figure 1), the ordinary ray (p-polarisation in this case w.r.t. magenta-coloured plane of incidence), perpendicular to optic axis A is the fast ray (F) while the extraordinary ray (s-polarisation in this case and parallel to optic axis A) is the slow ray (S). In negative birefringence (figure 2), it is the reverse.

Much of the work involving polarization preceded the understanding of light as a transverse electromagnetic wave, and this has affected some terminology in use. Isotropic materials have symmetry in all directions and the refractive index is the same for any polarization direction. An anisotropic material is called "birefringent" because it will generally refract a single incoming ray in two directions, which we now understand correspond to the two different polarizations. This is true of either a uniaxial or biaxial material.

In a uniaxial material, one ray behaves according to the normal law of refraction (corresponding to the ordinary refractive index), so an incoming ray at normal incidence remains normal to the refracting surface. As explained above, the other polarization can deviate from normal incidence, which cannot be described using the law of refraction. This thus became known as the extraordinary ray. The terms "ordinary" and "extraordinary" are still applied to the polarization components perpendicular to and not perpendicular to the optic axis respectively, even in cases where no double refraction is involved.

A material is termed uniaxial when it has a single direction of symmetry in its optical behavior, which we term the optic axis. It also happens to be the axis of symmetry of the index ellipsoid (a spheroid in this case). The index ellipsoid could still be described according to the refractive indices, n_α, n_β and n_γ, along three coordinate axes; in this case two are equal. So if n_α = n_β corresponding to the x and y axes, then the extraordinary index is n_γ corresponding to the z axis, which is also called the optic axis in this case.

Materials in which all three refractive indices are different are termed biaxial and the origin of this term is more complicated and frequently misunderstood. In a uniaxial crystal, different polarization components of a beam will travel at different phase velocities, except for rays in the direction of what we call the optic axis. Thus the optic axis has the particular property that rays in that direction do not exhibit birefringence, with all polarizations in such a beam experiencing the same index of refraction. It is very different when the three principal refractive indices are all different; then an incoming ray in any of those principal directions will still encounter two different refractive indices. But it turns out that there are two special directions (at an angle to all of the 3 axes) where the refractive indices for different polarizations are again equal. For this reason, these crystals were designated as biaxial, with the two "axes" in this case referring to ray directions in which propagation does not experience birefringence.

Fast and slow rays

In a birefringent material, a wave consists of two polarization components which generally are governed by different effective refractive indices. The so-called slow ray is the component for which the material has the higher effective refractive index (slower phase velocity), while the fast ray is the one with a lower effective refractive index. When a beam is incident on such a material from air (or any material with a lower refractive index), the slow ray is thus refracted more towards the normal than the fast ray. In the example figure at top of this page, it can be seen that refracted ray with s polarization (with its electric vibration along the direction of the optic axis, thus called the extraordinary ray^[6]) is the slow ray in given scenario.

Using a thin slab of that material at normal incidence, one would implement a waveplate. In this case, there is essentially no spatial separation between the polarizations, the phase of the wave in the parallel polarization (the slow ray) will be retarded with respect to the perpendicular polarization. These directions are thus known as the slow axis and fast axis of the waveplate.

Positive or negative

Uniaxial birefringence is classified as positive when the extraordinary index of refraction n_e is greater than the ordinary index n_o. Negative birefringence means that Δn = n_e − n_o is less than zero.^[7] In other words, the polarization of the fast (or slow) wave is perpendicular to the optic axis when the birefringence of the crystal is positive (or negative, respectively). In the case of biaxial crystals, all three of the principal axes have different refractive indices, so this designation does not apply. But for any defined ray direction one can just as well designate the fast and slow ray polarizations.

Sources of optical birefringence

View from under the Sky Pool, London with coloured fringes due to stress birefringence of partially polarised skylight through a circular polariser

While the best known source of birefringence is the entrance of light into an anisotropic crystal, it can result in otherwise optically isotropic materials in a few ways:

• Stress birefringence results when a normally isotropic solid is stressed and deformed (i.e., stretched or bent) causing a loss of physical isotropy and consequently a loss of isotropy in the material's permittivity tensor;
• Form birefringence, whereby structure elements such as rods, having one refractive index, are suspended in a medium with a different refractive index. When the lattice spacing is much smaller than a wavelength, such a structure is described as a metamaterial;
• By the Pockels or Kerr effect, whereby an applied electric field induces birefringence due to nonlinear optics;
• By the self or forced alignment into thin films of amphiphilic molecules such as lipids, some surfactants or liquid crystals^{[citation needed]};
• Circular birefringence takes place generally not in materials which are anisotropic but rather ones which are chiral. This can include liquids where there is an enantiomeric excess of a chiral molecule, that is, one that has stereo isomers;
• By the Faraday effect, where a longitudinal magnetic field causes some materials to become circularly birefringent (having slightly different indices of refraction for left- and right-handed circular polarizations), similar to optical activity while the field is applied.

Common birefringent materials

Sandwiched in between crossed polarizers, clear polystyrene cutlery exhibits wavelength-dependent birefringence

The best characterized birefringent materials are crystals. Due to their specific crystal structures their refractive indices are well defined. Depending on the symmetry of a crystal structure (as determined by one of the 32 possible crystallographic point groups), crystals in that group may be forced to be isotropic (not birefringent), to have uniaxial symmetry, or neither in which case it is a biaxial crystal. The crystal structures permitting uniaxial and biaxial birefringence are noted in the two tables, below, listing the two or three principal refractive indices (at wavelength 590 nm) of some better-known crystals.^[8]

In addition to induced birefringence while under stress, many plastics obtain permanent birefringence during manufacture due to stresses which are "frozen in" due to mechanical forces present when the plastic is molded or extruded.^[9] For example, ordinary cellophane is birefringent. Polarizers are routinely used to detect stress, either applied or frozen-in, in plastics such as polystyrene and polycarbonate.

Cotton fiber is birefringent because of high levels of cellulosic material in the fibre's secondary cell wall which is directionally aligned with the cotton fibers.

Polarized light microscopy is commonly used in biological tissue, as many biological materials are linearly or circularly birefringent. Collagen, found in cartilage, tendon, bone, corneas, and several other areas in the body, is birefringent and commonly studied with polarized light microscopy.^[10] Some proteins are also birefringent, exhibiting form birefringence.^[11]

Inevitable manufacturing imperfections in optical fiber leads to birefringence, which is one cause of pulse broadening in fiber-optic communications. Such imperfections can be geometrical (lack of circular symmetry), or due to unequal lateral stress applied to the optical fibre. Birefringence is intentionally introduced (for instance, by making the cross-section elliptical) in order to produce polarization-maintaining optical fibers. Birefringence can be induced (or corrected!) in optical fibers through bending them which causes anisotropy in form and stress given the axis around which it is bent and radius of curvature.

In addition to anisotropy in the electric polarizability that we have been discussing, anisotropy in the magnetic permeability could be a source of birefringence. At optical frequencies, there is no measurable magnetic polarizability (μ=μ₀) of natural materials, so this is not an actual source of birefringence at optical wavelengths.

Uniaxial crystals, at 590 nm[8] Material Crystal system no ne Δn barium borate BaB2O4 Trigonal 1.6776 1.5534 −0.1242 beryl Be3Al2(SiO3)6 Hexagonal 1.602 1.557 −0.045 calcite CaCO3 Trigonal 1.658 1.486 −0.172 ice H2O Hexagonal 1.3090 1.3104 +0.0014[12] lithium niobate LiNbO3 Trigonal 2.272 2.187 −0.085 magnesium fluoride MgF2 Tetragonal 1.380 1.385 +0.006 quartz SiO2 Trigonal 1.544 1.553 +0.009 ruby Al2O3 Trigonal 1.770 1.762 −0.008 rutile TiO2 Tetragonal 2.616 2.903 +0.287 sapphire Al2O3 Trigonal 1.768 1.760 −0.008 silicon carbide SiC Hexagonal 2.647 2.693 +0.046 tourmaline (complex silicate) Trigonal 1.669 1.638 −0.031 zircon, high ZrSiO4 Tetragonal 1.960 2.015 +0.055 zircon, low ZrSiO4 Tetragonal 1.920 1.967 +0.047

Biaxial crystals, at 590 nm[8] Material Crystal system nα nβ nγ borax Na2(B4O5)(OH)4·8H2O Monoclinic 1.447 1.469 1.472 epsom salt MgSO4·7H2O Monoclinic 1.433 1.455 1.461 mica, biotite K(Mg,Fe)3(AlSi3O10)(F,OH)2 Monoclinic 1.595 1.640 1.640 mica, muscovite KAl2(AlSi3O10)(F,OH)2 Monoclinic 1.563 1.596 1.601 olivine (Mg,Fe)2SiO4 Orthorhombic 1.640 1.660 1.680 perovskite CaTiO3 Orthorhombic 2.300 2.340 2.380 topaz Al2SiO4(F,OH)2 Orthorhombic 1.618 1.620 1.627 ulexite NaCaB5O6(OH)6·5H2O Triclinic 1.490 1.510 1.520

Measurement

Birefringence and other polarization-based optical effects (such as optical rotation and linear or circular dichroism) can be observed by measuring any change in the polarization of light passing through the material. These measurements are known as polarimetry. Polarized light microscopes, which contain two polarizers that are at 90° to each other on either side of the sample, are used to visualize birefringence, since light that has not been affected by birefringence remains in a polarization that is totally rejected by the second polarizer ("analyzer"). The addition of quarter-wave plates permits examination using circularly polarized light. Determination of the change in polarization state using such an apparatus is the basis of ellipsometry, by which the optical properties of specular surfaces can be gauged through reflection.

Birefringence measurements have been made with phase-modulated systems for examining the transient flow behaviour of fluids.^[13][14] Birefringence of lipid bilayers can be measured using dual-polarization interferometry. This provides a measure of the degree of order within these fluid layers and how this order is disrupted when the layer interacts with other biomolecules.

For the 3D measurement of birefringence, a technique based on holographic tomography [1] can be used.

Applications

Reflective twisted-nematic liquid-crystal display. Light reflected by the surface (6) (or coming from a backlight) is horizontally polarized (5) and passes through the liquid-crystal modulator (3) sandwiched in between transparent layers (2, 4) containing electrodes. Horizontally polarized light is blocked by the vertically oriented polarizer (1), except where its polarization has been rotated by the liquid crystal (3), appearing bright to the viewer.

Birefringence is used in many optical devices. Liquid-crystal displays, the most common sort of flat-panel display, cause their pixels to become lighter or darker through rotation of the polarization (circular birefringence) of linearly polarized light as viewed through a sheet polarizer at the screen's surface. Similarly, light modulators modulate the intensity of light through electrically induced birefringence of polarized light followed by a polarizer. The Lyot filter is a specialized narrowband spectral filter employing the wavelength dependence of birefringence. Waveplates are thin birefringent sheets widely used in certain optical equipment for modifying the polarization state of light passing through it.

Birefringence also plays an important role in second-harmonic generation and other nonlinear optical components, as the crystals used for this purpose are almost always birefringent. By adjusting the angle of incidence, the effective refractive index of the extraordinary ray can be tuned in order to achieve phase matching, which is required for the efficient operation of these devices.

Medicine

Birefringence is utilized in medical diagnostics. One powerful accessory used with optical microscopes is a pair of crossed polarizing filters. Light from the source is polarized in the x direction after passing through the first polarizer, but above the specimen is a polarizer (a so-called analyzer) oriented in the y direction. Therefore, no light from the source will be accepted by the analyzer, and the field will appear dark. Areas of the sample possessing birefringence will generally couple some of the x-polarized light into the y polarization; these areas will then appear bright against the dark background. Modifications to this basic principle can differentiate between positive and negative birefringence.

Gout and pseudogout crystals viewed under a microscope with a red compensator, which slows red light in one orientation (labeled "polarized light axis").^[15] Urate crystals (left image) in gout appear yellow when their long axis is parallel to the slow transmission axis of the red compensator and appear blue when perpendicular. The opposite colors are seen in calcium pyrophosphate dihydrate crystal deposition disease (pseudogout, right image): blue when parallel and yellow when perpendicular.

For instance, needle aspiration of fluid from a gouty joint will reveal negatively birefringent monosodium urate crystals. Calcium pyrophosphate crystals, in contrast, show weak positive birefringence.^[16] Urate crystals appear yellow, and calcium pyrophosphate crystals appear blue when their long axes are aligned parallel to that of a red compensator filter,^[17] or a crystal of known birefringence is added to the sample for comparison.

The birefringence of tissue inside a living human thigh was measured using polarization-sensitive optical coherence tomography at 1310 nm and a single mode fiber in a needle. Skeletal muscle birefringence was Δn = 1.79 × 10⁻³± 0.18×10⁻³, adipose Δn = 0.07 × 10⁻³ ± 0.50 × 10⁻³, superficial aponeurosis Δn = 5.08 × 10⁻³ ± 0.73 × 10⁻³ and interstitial tissue Δn = 0.65 ×10⁻³ ±0.39 ×10⁻³.^[18] These measurements may be important for the development of a less invasive method to diagnose Duchenne muscular dystrophy.

Birefringence can be observed in amyloid plaques such as are found in the brains of Alzheimer's patients when stained with a dye such as Congo Red. Modified proteins such as immunoglobulin light chains abnormally accumulate between cells, forming fibrils. Multiple folds of these fibers line up and take on a beta-pleated sheet conformation. Congo red dye intercalates between the folds and, when observed under polarized light, causes birefringence.

In ophthalmology, binocular retinal birefringence screening of the Henle fibers (photoreceptor axons that go radially outward from the fovea) provides a reliable detection of strabismus and possibly also of anisometropic amblyopia.^[19] In healthy subjects, the maximum retardation induced by the Henle fiber layer is approximately 22 degrees at 840 nm.^[20] Furthermore, scanning laser polarimetry uses the birefringence of the optic nerve fibre layer to indirectly quantify its thickness, which is of use in the assessment and monitoring of glaucoma. Polarization-sensitive optical coherence tomography measurements obtained from healthy human subjects have demonstrated a change in birefringence of the retinal nerve fiber layer as a function of location around the optic nerve head.^[21] The same technology was recently applied in the living human retina to quantify the polarization properties of vessel walls near the optic nerve.^[22]

Birefringence characteristics in sperm heads allow the selection of spermatozoa for intracytoplasmic sperm injection.^[23] Likewise, zona imaging uses birefringence on oocytes to select the ones with highest chances of successful pregnancy.^[24] Birefringence of particles biopsied from pulmonary nodules indicates silicosis.

Dermatologists use dermatoscopes to view skin lesions. Dermoscopes use polarized light, allowing the user to view crystalline structures corresponding to dermal collagen in the skin. These structures may appear as shiny white lines or rosette shapes and are only visible under polarized dermoscopy.

Stress-induced birefringence

Color pattern of a plastic box with "frozen in" mechanical stress placed between two crossed polarizers

Isotropic solids do not exhibit birefringence. When they are under mechanical stress, birefringence results. The stress can be applied externally or is "frozen in" after a birefringent plastic ware is cooled after it is manufactured using injection molding. When such a sample is placed between two crossed polarizers, colour patterns can be observed, because polarization of a light ray is rotated after passing through a birefringent material and the amount of rotation is dependent on wavelength. The experimental method called photoelasticity used for analyzing stress distribution in solids is based on the same principle. There has been recent research on using stress induced birefringence in a glass plate to generate an Optical vortex and full Poincare beams (optical beams that have every possible polarization state across a cross-section).^[25]

Other cases of birefringence

Birefringent rutile observed in different polarizations using a rotating polarizer (or analyzer)

Birefringence is observed in anisotropic elastic materials. In these materials, the two polarizations split according to their effective refractive indices, which are also sensitive to stress.

The study of birefringence in shear waves traveling through the solid Earth (the Earth's liquid core does not support shear waves) is widely used in seismology.^{[citation needed]}

Birefringence is widely used in mineralogy to identify rocks, minerals, and gemstones.^{[citation needed]}

Theory

Surface of the allowed k vectors for a fixed frequency for a biaxial crystal (see eq. 7).

In an isotropic medium (including free space) the so-called electric displacement (D) is just proportional to the electric field (E) according to D = ɛE where the material's permittivity ε is just a scalar (and equal to n²ε₀ where n is the index of refraction). In an anisotropic material exhibiting birefringence, the relationship between D and E must now be described using a tensor equation:

${\displaystyle \mathbf{D}=\mathit{\epsilon }\mathbf{E}}$

(1)

where ε is now a 3 × 3 permittivity tensor. We assume linearity and no magnetic permeability in the medium: μ = μ₀. The electric field of a plane wave of angular frequency ω can be written in the general form:

${\displaystyle \mathbf{E}={\mathbf{E}}_0{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}}$

(2)

where r is the position vector, t is time, and E₀ is a vector describing the electric field at r = 0, t = 0. Then we shall find the possible wave vectors k. By combining Maxwell's equations for ∇ × E and ∇ × H, we can eliminate H = 1/μ₀B to obtain:

${\displaystyle -\mathrm{\nabla }\times \mathrm{\nabla }\times \mathbf{E}={\mu }_0\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\mathbf{D}}$

(3a)

With no free charges, Maxwell's equation for the divergence of D vanishes:

${\displaystyle \mathrm{\nabla }\cdot \mathbf{D}=0}$

(3b)

We can apply the vector identity ∇ × (∇ × A) = ∇(∇ ⋅ A) − ∇²A to the left hand side of eq. 3a, and use the spatial dependence in which each differentiation in x (for instance) results in multiplication by ik_x to find:

${\displaystyle -\mathrm{\nabla }\times \mathrm{\nabla }\times \mathbf{E}=(\mathbf{k}\cdot \mathbf{E})\mathbf{k}-(\mathbf{k}\cdot \mathbf{k})\mathbf{E}}$

(3c)

The right hand side of eq. 3a can be expressed in terms of E through application of the permittivity tensor ε and noting that differentiation in time results in multiplication by −iω, eq. 3a then becomes:

${\displaystyle (-\mathbf{k}\cdot \mathbf{k})\mathbf{E}+(\mathbf{k}\cdot \mathbf{E})\mathbf{k}=-{\mu }_0{\omega }^2(\mathit{\epsilon }\mathbf{E})}$

(4a)

Applying the differentiation rule to eq. 3b we find:

${\displaystyle \mathbf{k}\cdot \mathbf{D}=0}$

(4b)

Eq. 4b indicates that D is orthogonal to the direction of the wavevector k, even though that is no longer generally true for E as would be the case in an isotropic medium. Eq. 4b will not be needed for the further steps in the following derivation.

Finding the allowed values of k for a given ω is easiest done by using Cartesian coordinates with the x, y and z axes chosen in the directions of the symmetry axes of the crystal (or simply choosing z in the direction of the optic axis of a uniaxial crystal), resulting in a diagonal matrix for the permittivity tensor ε:

${\displaystyle \epsilon ={\epsilon }_0\left[\begin{array}{ccc}{n}_x^2& 0& 0\\ 0& n_y^2& 0\\ 0& 0& n_z^2\end{array}\right]}$

(4c)

where the diagonal values are squares of the refractive indices for polarizations along the three principal axes x, y and z. With ε in this form, and substituting in the speed of light c using c² = 1/μ₀ε₀, the x component of the vector equation eq. 4a becomes

${\displaystyle \left(-k_x^2-k_y^2-k_z^2\right)E_x+k_x^2{E}_x+k_x{k}_y{E}_y+k_x{k}_z{E}_z=-\frac{{\omega }^2{n}_x^2}{c^2}{E}_x}$

(5a)

where E_x, E_y, E_z are the components of E (at any given position in space and time) and k_x, k_y, k_z are the components of k. Rearranging, we can write (and similarly for the y and z components of eq. 4a)

${\displaystyle \left(-k_y^2-k_z^2+\frac{{\omega }^2{n}_x^2}{c^2}\right)E_x+k_x{k}_y{E}_y+k_x{k}_z{E}_z=0}$

(5b)

${\displaystyle k_x{k}_y{E}_x+\left(-k_x^2-k_z^2+\frac{{\omega }^2{n}_y^2}{c^2}\right)E_y+k_y{k}_z{E}_z=0}$

(5c)

${\displaystyle k_x{k}_z{E}_x+k_y{k}_z{E}_y+\left(-k_x^2-k_y^2+\frac{{\omega }^2{n}_z^2}{c^2}\right)E_z=0}$

(5d)

This is a set of linear equations in E_x, E_y, E_z, so it can have a nontrivial solution (that is, one other than E = 0) as long as the following determinant is zero:

${\displaystyle \left|\begin{array}{ccc}\left(-k_y^2-k_z^2+\frac{{\omega }^2{n}_x^2}{c^2}\right)& k_x{k}_y& k_x{k}_z\\ k_x{k}_y& \left(-k_x^2-k_z^2+\frac{{\omega }^2{n}_y^2}{c^2}\right)& k_y{k}_z\\ k_x{k}_z& k_y{k}_z& \left(-k_x^2-k_y^2+\frac{{\omega }^2{n}_z^2}{c^2}\right)\end{array}\right|=0}$

(6)

Evaluating the determinant of eq. 6, and rearranging the terms according to the powers of ${\displaystyle \frac{{\omega }^2}{c^2}}$, the constant terms cancel. After eliminating the common factor ${\displaystyle \frac{{\omega }^2}{c^2}}$ from the remaining terms, we obtain

${\displaystyle \frac{{\omega }^4}{c^4}-\frac{{\omega }^2}{c^2}\left(\frac{k_x^2+k_y^2}{n_z^2}+\frac{k_x^2+k_z^2}{n_y^2}+\frac{k_y^2+k_z^2}{n_x^2}\right)+\left(\frac{k_x^2}{n_y^2{n}_z^2}+\frac{k_y^2}{n_x^2{n}_z^2}+\frac{k_z^2}{n_x^2{n}_y^2}\right)\left(k_x^2+k_y^2+k_z^2\right)=0}$

(7)

In the case of a uniaxial material, choosing the optic axis to be in the z direction so that n_x = n_y = n_o and n_z = n_e, this expression can be factored into

${\displaystyle \left(\frac{k_x^2}{n_{\mathrm{o}}^2}+\frac{k_y^2}{n_{\mathrm{o}}^2}+\frac{k_z^2}{n_{\mathrm{o}}^2}-\frac{{\omega }^2}{c^2}\right)\left(\frac{k_x^2}{n_{\mathrm{e}}^2}+\frac{k_y^2}{n_{\mathrm{e}}^2}+\frac{k_z^2}{n_{\mathrm{o}}^2}-\frac{{\omega }^2}{c^2}\right)=0}$

(8)

Setting either of the factors in eq. 8 to zero will define an ellipsoidal surface^{[note 1]} in the space of wavevectors k that are allowed for a given ω. The first factor being zero defines a sphere; this is the solution for so-called ordinary rays, in which the effective refractive index is exactly n_o regardless of the direction of k. The second defines a spheroid symmetric about the z axis. This solution corresponds to the so-called extraordinary rays in which the effective refractive index is in between n_o and n_e, depending on the direction of k. Therefore, for any arbitrary direction of propagation (other than in the direction of the optic axis), two distinct wavevectors k are allowed corresponding to the polarizations of the ordinary and extraordinary rays.

For a biaxial material a similar but more complicated condition on the two waves can be described;^[26] the locus of allowed k vectors (the wavevector surface) is a 4th-degree two-sheeted surface, so that in a given direction there are generally two permitted k vectors (and their opposites).^[27] By inspection one can see that eq. 6 is generally satisfied for two positive values of ω. Or, for a specified optical frequency ω and direction normal to the wavefronts k/|k|, it is satisfied for two wavenumbers (or propagation constants) |k| (and thus effective refractive indices) corresponding to the propagation of two linear polarizations in that direction.

When those two propagation constants are equal then the effective refractive index is independent of polarization, and there is consequently no birefringence encountered by a wave traveling in that particular direction. For a uniaxial crystal, this is the optic axis, the ±z direction according to the above construction. But when all three refractive indices (or permittivities), n_x, n_y and n_z are distinct, it can be shown that there are exactly two such directions, where the two sheets of the wave-vector surface touch;^[27] these directions are not at all obvious and do not lie along any of the three principal axes (x, y, z according to the above convention). Historically that accounts for the use of the term "biaxial" for such crystals, as the existence of exactly two such special directions (considered "axes") was discovered well before polarization and birefringence were understood physically. These two special directions are generally not of particular interest; biaxial crystals are rather specified by their three refractive indices corresponding to the three axes of symmetry.

A general state of polarization launched into the medium can always be decomposed into two waves, one in each of those two polarizations, which will then propagate with different wavenumbers |k|. Applying the different phase of propagation to those two waves over a specified propagation distance will result in a generally different net polarization state at that point; this is the principle of the waveplate for instance. With a waveplate, there is no spatial displacement between the two rays as their k vectors are still in the same direction. That is true when each of the two polarizations is either normal to the optic axis (the ordinary ray) or parallel to it (the extraordinary ray).

In the more general case, there is a difference not only in the magnitude but the direction of the two rays. For instance, the photograph through a calcite crystal (top of page) shows a shifted image in the two polarizations; this is due to the optic axis being neither parallel nor normal to the crystal surface. And even when the optic axis is parallel to the surface, this will occur for waves launched at non-normal incidence (as depicted in the explanatory figure). In these cases the two k vectors can be found by solving eq. 6 constrained by the boundary condition which requires that the components of the two transmitted waves' k vectors, and the k vector of the incident wave, as projected onto the surface of the interface, must all be identical. For a uniaxial crystal it will be found that there is not a spatial shift for the ordinary ray (hence its name) which will refract as if the material were non-birefringent with an index the same as the two axes which are not the optic axis. For a biaxial crystal neither ray is deemed "ordinary" nor would generally be refracted according to a refractive index equal to one of the principal axes.

See also

• Cotton–Mouton effect
• Crystal optics
• Dichroism
• Iceland spar
• Index ellipsoid
• John Kerr
• Optical rotation
• Periodic poling
• Pleochroism

Notes

1. 
   

Although related, note that this is not the same as the index ellipsoid.

 

A calcite crystal laid upon a graph paper with blue lines showing the double refraction

 

In this example, optic axis along the surface is shown perpendicular to plane of incidence. Incoming light in the s polarization (which means perpendicular to plane of incidence - and so in this example becomes "parallel polarisation" to optic axis, thus is called extraordinary ray) sees a greater refractive index than light in the p polarization (which becomes ordinary ray because "perpendicular polarisation" to optic axis) and so s polarization ray is undergoing greater refraction on entering and exiting the crystal.

 

Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images.

Sources of optical birefringence

View from under the Sky Pool, London with coloured fringes due to stress birefringence of partially polarised skylight through a circular polariser

While the best known source of birefringence is the entrance of light into an anisotropic crystal, it can result in otherwise optically isotropic materials in a few ways:

• Stress birefringence results when a normally isotropic solid is stressed and deformed (i.e., stretched or bent) causing a loss of physical isotropy and consequently a loss of isotropy in the material's permittivity tensor;
• Form birefringence, whereby structure elements such as rods, having one refractive index, are suspended in a medium with a different refractive index. When the lattice spacing is much smaller than a wavelength, such a structure is described as a metamaterial;
• By the Pockels or Kerr effect, whereby an applied electric field induces birefringence due to nonlinear optics;
• By the self or forced alignment into thin films of amphiphilic molecules such as lipids, some surfactants or liquid crystals^{[citation needed]};
• Circular birefringence takes place generally not in materials which are anisotropic but rather ones which are chiral. This can include liquids where there is an enantiomeric excess of a chiral molecule, that is, one that has stereo isomers;
• By the Faraday effect, where a longitudinal magnetic field causes some materials to become circularly birefringent (having slightly different indices of refraction for left- and right-handed circular polarizations), similar to optical activity while the field is applied.

 

Sandwiched in between crossed polarizers, clear polystyrene cutlery exhibits wavelength-dependent birefringence

Reflective twisted-nematic liquid-crystal display. Light reflected by the surface (6) (or coming from a backlight) is horizontally polarized (5) and passes through the liquid-crystal modulator (3) sandwiched in between transparent layers (2, 4) containing electrodes. Horizontally polarized light is blocked by the vertically oriented polarizer (1), except where its polarization has been rotated by the liquid crystal (3), appearing bright to the viewer.

 

Measurement

Birefringence and other polarization-based optical effects (such as optical rotation and linear or circular dichroism) can be observed by measuring any change in the polarization of light passing through the material. These measurements are known as polarimetry. Polarized light microscopes, which contain two polarizers that are at 90° to each other on either side of the sample, are used to visualize birefringence, since light that has not been affected by birefringence remains in a polarization that is totally rejected by the second polarizer ("analyzer"). The addition of quarter-wave plates permits examination using circularly polarized light. Determination of the change in polarization state using such an apparatus is the basis of ellipsometry, by which the optical properties of specular surfaces can be gauged through reflection.

Birefringence measurements have been made with phase-modulated systems for examining the transient flow behaviour of fluids.^[13][14] Birefringence of lipid bilayers can be measured using dual-polarization interferometry. This provides a measure of the degree of order within these fluid layers and how this order is disrupted when the layer interacts with other biomolecules.

For the 3D measurement of birefringence, a technique based on holographic tomography [1] can be used. 

https://en.wikipedia.org/wiki/Birefringence

 Photoelasticity describes changes in the optical properties of a material under mechanical deformation. It is a property of all dielectric media and is often used to experimentally determine the stress distribution in a material, where it gives a picture of stress distributions around discontinuities in materials. Photoelastic experiments (also informally referred to as photoelasticity) are an important tool for determining critical stress points in a material, and are used for determining stress concentration in irregular geometries. 

https://en.wikipedia.org/wiki/Photoelasticity

 

The optical properties of a material define how it interacts with light. The optical properties of matter are studied in optical physics, a subfield of optics. The optical properties of matter include:

• Refractive index
• Dispersion
• Transmittance and Transmission coefficient
• Absorption
• Scattering
• Turbidity
• Reflectance and Reflectivity (reflection coefficient)
• Albedo
• Perceived color
• Fluorescence
• Phosphorescence
• Photoluminescence
• Optical bistability
• Dichroism
• Birefringence
• Optical activity
• Photosensitivity

A basic distinction is between isotropic materials, which exhibit the same properties regardless of the direction of the light, and anisotropic ones, which exhibit different properties when light passes through them in different directions.

The optical properties of matter can lead to a variety of interesting optical phenomena.

Properties of specific materials

• Optical properties of water and ice
• Optical properties of carbon nanotubes
• Crystal optics

https://en.wikipedia.org/wiki/Optical_properties

 

In optics, optical bistability is an attribute of certain optical devices where two resonant transmissions states are possible and stable, dependent on the input. Optical devices with a feedback mechanism, e.g. a laser, provide two methods of achieving bistability.

• Absorptive bistability utilizes an absorber to block light inversely dependent on the intensity of the source light. The first bistable state resides at a given intensity where no absorber is used. The second state resides at the point where the light intensity overcomes the absorber's ability to block light.
• Refractive bistability utilizes an optical mechanism that changes its refractive index inversely dependent on the intensity of the source light. The first bistable state resides at a given intensity where no optical mechanism is used. The second state resides at the point where a certain light intensity causes the light to resonate to the corresponding refractive index.

This effect is caused by two factors

• Nonlinear atom-field interaction
• Feedback effect of mirror

Important cases that might be regarded are:

• Atomic detuning
• Cooperating factor
• Cavity mistuning

Applications of this phenomenon include its use in optical transmitters, memory elements and pulse shapers.

Optical bistability was first observed within vapor of sodium during 1974.^[1]

Intrinsic bistability

When the feedback mechanism is provided by an internal procedure (not by an external entity like the mirror within the Interferometers), the latter will be known as intrinsic optical bistability.^[2] This process can be seen in nonlinear media containing the nanoparticles through which the effect of surface plasmon resonance can potentially occur.^[3]

References

1. 
   

Gibbs, Hyatt (1985). "Introduction to Optical Bistability". Optical Bistability: Controlling Light With Light. Quantum electronics--principles and applications. Orlando, FL: Academic Press Inc. p. 1. ISBN 978-0122819407. Retrieved June 16, 2021.

Goldstone, J. A., and E. Garmire. "Intrinsic optical bistability in nonlinear media". Physical review letters 53.9 (1984): 910. https://doi.org/10.1103/PhysRevLett.53.910

3. Sharif, Morteza A., et al. "Difference Frequency Generation-based ultralow threshold Optical Bistability in graphene at visible frequencies, an experimental realization". Journal of Molecular Liquids 284 (2019): 92–101. https://doi.org/10.1016/j.molliq.2019.03.167

• Guangsheng He; Song H. Liu (1999). Physics of Nonlinear Optics. World Scientific. pp. 422–. ISBN 978-981-02-3319-8.

Category:

• Optics

https://en.wikipedia.org/wiki/Optical_bistability

 

Surface plasmon resonance (SPR) is a phenomenon that occurs where electrons in a thin metal sheet become excited by light that is directed to the sheet with a particular angle of incidence, and then travel parallel to the sheet. Assuming a constant light source wavelength and that the metal sheet is thin, the angle of incidence that triggers SPR is related to the refractive index of the material and even a small change in the refractive index will cause SPR to not be observed. This makes SPR a possible technique for detecting particular substances (analytes) and SPR biosensors have been developed to detect various important biomarkers^[1]

^{https://en.wikipedia.org/wiki/Surface_plasmon_resonance}

 

Surface plasmon polaritons (SPPs) are electromagnetic waves that travel along a metal–dielectric or metal–air interface, practically in the infrared or visible-frequency. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal ("surface plasmon") and electromagnetic waves in the air or dielectric ("polariton").^[1]

They are a type of surface wave, guided along the interface in much the same way that light can be guided by an optical fiber. SPPs have a shorter wavelength than light in vacuum at the same frequency (photons).^[2] Hence, SPPs can have a higher momentum and local field intensity.^[2] Perpendicular to the interface, they have subwavelength-scale confinement. An SPP will propagate along the interface until its energy is lost either to absorption in the metal or scattering into other directions (such as into free space).

Application of SPPs enables subwavelength optics in microscopy and photolithography beyond the diffraction limit. It also enables the first steady-state micro-mechanical measurement of a fundamental property of light itself: the momentum of a photon in a dielectric medium. Other applications are photonic data storage, light generation, and bio-photonics.^{[3][2][4][5][6]}

^{https://en.wikipedia.org/wiki/Surface_plasmon_polariton}

 

A superlens, or super lens, is a lens which uses metamaterials to go beyond the diffraction limit. The diffraction limit is a feature of conventional lenses and microscopes that limits the fineness of their resolution depending on the illumination wavelength and the numerical aperture NA of the objective lens. Many lens designs have been proposed that go beyond the diffraction limit in some way, but constraints and obstacles face each of them.^[1]

History

In 1873 Ernst Abbe reported that conventional lenses are incapable of capturing some fine details of any given image. The superlens is intended to capture such details. The limitation of conventional lenses has inhibited progress in the biological sciences. This is because a virus or DNA molecule cannot be resolved with the highest powered conventional microscopes. This limitation extends to the minute processes of cellular proteins moving alongside microtubules of a living cell in their natural environments. Additionally, computer chips and the interrelated microelectronics continue to be manufactured to smaller and smaller scales. This requires specialized optical equipment, which is also limited because these use conventional lenses. Hence, the principles governing a superlens show that it has potential for imaging DNA molecules, cellular protein processes, and aiding in the manufacture of even smaller computer chips and microelectronics.^[2][3][4][5]

Furthermore, conventional lenses capture only the propagating light waves. These are waves that travel from a light source or an object to a lens, or the human eye. This can alternatively be studied as the far field. In contrast, a superlens captures propagating light waves and waves that stay on top of the surface of an object, which, alternatively, can be studied as both the far field and the near field.^[6][7]

In the early 20th century the term "superlens" was used by Dennis Gabor to describe something quite different: a compound lenslet array system.^[8]

Theory

The binocular microscope is a conventional optical system. Spatial resolution is confined by a diffraction limit that is a little above 200 nanometers.

Image formation

Schematic depictions and images of commonly used metallic nanoprobes that can be used to see a sample in nanometer resolution. Notice that the tips of the three nanoprobes are 100 nanometers.^[4]

An image of an object can be defined as a tangible or visible representation of the features of that object. A requirement for image formation is interaction with fields of electromagnetic radiation. Furthermore, the level of feature detail, or image resolution, is limited to a length of a wave of radiation. For example, with optical microscopy, image production and resolution depends on the length of a wave of visible light. However, with a superlens, this limitation may be removed, and a new class of image generated.^[9]

Electron beam lithography can overcome this resolution limit. Optical microscopy, on the other hand cannot, being limited to some value just above 200 nanometers.^[4] However, new technologies combined with optical microscopy are beginning to allow increased feature resolution (see sections below).

One definition of being constrained by the resolution barrier, is a resolution cut off at half the wavelength of light. The visible spectrum has a range that extends from 390 nanometers to 750 nanometers. Green light, half way in between, is around 500 nanometers. Microscopy takes into account parameters such as lens aperture, distance from the object to the lens, and the refractive index of the observed material. This combination defines the resolution cutoff, or microscopy optical limit, which tabulates to 200 nanometers. Therefore, conventional lenses, which literally construct an image of an object by using "ordinary" light waves, discard information that produces very fine, and minuscule details of the object that are contained in evanescent waves. These dimensions are less than 200 nanometers. For this reason, conventional optical systems, such as microscopes, have been unable to accurately image very small, nanometer-sized structures or nanometer-sized organisms in vivo, such as individual viruses, or DNA molecules.^[4][5]

The limitations of standard optical microscopy (bright-field microscopy) lie in three areas:

• The technique can only image dark or strongly refracting objects effectively.
• Diffraction limits the object, or cell's, resolution to approximately 200 nanometers.
• Out-of-focus light from points outside the focal plane reduces image clarity.

Live biological cells in particular generally lack sufficient contrast to be studied successfully, because the internal structures of the cell are mostly colorless and transparent. The most common way to increase contrast is to stain the different structures with selective dyes, but often this involves killing and fixing the sample. Staining may also introduce artifacts, apparent structural details that are caused by the processing of the specimen and are thus not a legitimate feature of the specimen.

Conventional lens

DVD (digital versatile disc). A laser is employed for data transfer.

The conventional glass lens is pervasive throughout our society and in the sciences. It is one of the fundamental tools of optics simply because it interacts with various wavelengths of light. At the same time, the wavelength of light can be analogous to the width of a pencil used to draw the ordinary images. The limit intrudes in all kinds of ways. For example, the laser used in a digital video system cannot read details from a DVD that are smaller than the wavelength of the laser. This limits the storage capacity of DVDs.^[10]

Thus, when an object emits or reflects light there are two types of electromagnetic radiation associated with this phenomenon. These are the near field radiation and the far field radiation. As implied by its description, the far field escapes beyond the object. It is then easily captured and manipulated by a conventional glass lens. However, useful (nanometer-sized) resolution details are not observed, because they are hidden in the near field. They remain localized, staying much closer to the light emitting object, unable to travel, and unable to be captured by the conventional lens. Controlling the near field radiation, for high resolution, can be accomplished with a new class of materials not easily obtained in nature. These are unlike familiar solids, such as crystals, which derive their properties from atomic and molecular units. The new material class, termed metamaterials, obtains its properties from its artificially larger structure. This has resulted in novel properties, and novel responses, which allow for details of images that surpass the limitations imposed by the wavelength of light.^[10]

Subwavelength imaging

The "Electrocomposeur" was an electron-beam lithography machine (electron microscope) designed for mask writing. It was developed in the early 1970s and deployed in the mid 1970s.

This has led to the desire to view live biological cell interactions in a real time, natural environment, and the need for subwavelength imaging. Subwavelength imaging can be defined as optical microscopy with the ability to see details of an object or organism below the wavelength of visible light (see discussion in the above sections). In other words, to have the capability to observe, in real time, below 200 nanometers. Optical microscopy is a non-invasive technique and technology because everyday light is the transmission medium. Imaging below the optical limit in optical microscopy (subwavelength) can be engineered for the cellular level, and nanometer level in principle.

For example, in 2007 a technique was demonstrated where a metamaterials-based lens coupled with a conventional optical lens could manipulate visible light to see (nanoscale) patterns that were too small to be observed with an ordinary optical microscope. This has potential applications not only for observing a whole living cell, or for observing cellular processes, such as how proteins and fats move in and out of cells. In the technology domain, it could be used to improve the first steps of photolithography and nanolithography, essential for manufacturing ever smaller computer chips.^[4][11]

Focusing at subwavelength has become a unique imaging technique which allows visualization of features on the viewed object which are smaller than the wavelength of the photons in use. A photon is the minimum unit of light. While previously thought to be physically impossible, subwavelength imaging has been made possible through the development of metamaterials. This is generally accomplished using a layer of metal such as gold or silver a few atoms thick, which acts as a superlens, or by means of 1D and 2D photonic crystals.^[12][13] There is a subtle interplay between propagating waves, evanescent waves, near field imaging and far field imaging discussed in the sections below.^[4][14]

Early subwavelength imaging

Metamaterial lenses (Superlenses) are able to reconstruct nanometer sized images by producing a negative refractive index in each instance. This compensates for the swiftly decaying evanescent waves. Prior to metamaterials, numerous other techniques had been proposed and even demonstrated for creating super-resolution microscopy. As far back as 1928, Irish physicist Edward Hutchinson Synge, is given credit for conceiving and developing the idea for what would ultimately become near-field scanning optical microscopy.^[15][16][17]

In 1974 proposals for two-dimensional fabrication techniques were presented. These proposals included contact imaging to create a pattern in relief, photolithography, electron lithography, X-ray lithography, or ion bombardment, on an appropriate planar substrate.^[18] The shared technological goals of the metamaterial lens and the variety of lithography aim to optically resolve features having dimensions much smaller than that of the vacuum wavelength of the exposing light.^[19][20] In 1981 two different techniques of contact imaging of planar (flat) submicroscopic metal patterns with blue light (400 nm) were demonstrated. One demonstration resulted in an image resolution of 100 nm and the other a resolution of 50 to 70 nm.^[20]

In 1995, John Guerra combined a transparent grating having 50 nm lines and spaces (the "metamaterial") with a conventional microscope immersion objective. The resulting "superlens" resolved a silicon sample also having 50 nm lines and spaces, far beyond the classical diffraction limit imposed by the illumination having 650 nm wavelength in air.^[21]

Since at least 1998 near field optical lithography was designed to create nanometer-scale features. Research on this technology continued as the first experimentally demonstrated negative index metamaterial came into existence in 2000–2001. The effectiveness of electron-beam lithography was also being researched at the beginning of the new millennium for nanometer-scale applications. Imprint lithography was shown to have desirable advantages for nanometer-scaled research and technology.^[19][22]

Advanced deep UV photolithography can now offer sub-100 nm resolution, yet the minimum feature size and spacing between patterns are determined by the diffraction limit of light. Its derivative technologies such as evanescent near-field lithography, near-field interference lithography, and phase-shifting mask lithography were developed to overcome the diffraction limit.^[19]

In the year 2000, John Pendry proposed using a metamaterial lens to achieve nanometer-scaled imaging for focusing below the wavelength of light.^[1][23]

Analysis of the diffraction limit

The original problem of the perfect lens: The general expansion of an EM field emanating from a source consists of both propagating waves and near-field or evanescent waves. An example of a 2-D line source with an electric field which has S-polarization will have plane waves consisting of propagating and evanescent components, which advance parallel to the interface.^[24] As both the propagating and the smaller evanescent waves advance in a direction parallel to the medium interface, evanescent waves decay in the direction of propagation. Ordinary (positive index) optical elements can refocus the propagating components, but the exponentially decaying inhomogeneous components are always lost, leading to the diffraction limit for focusing to an image.^[24]

A superlens is a lens which is capable of subwavelength imaging, allowing for magnification of near field rays. Conventional lenses have a resolution on the order of one wavelength due to the so-called diffraction limit. This limit hinders imaging very small objects, such as individual atoms, which are much smaller than the wavelength of visible light. A superlens is able to beat the diffraction limit. An example is the initial lens described by Pendry, which uses a slab of material with a negative index of refraction as a flat lens. In theory, a perfect lens would be capable of perfect focus – meaning that it could perfectly reproduce the electromagnetic field of the source plane at the image plane.

The diffraction limit as restriction on resolution

The performance limitation of conventional lenses is due to the diffraction limit. Following Pendry (2000), the diffraction limit can be understood as follows. Consider an object and a lens placed along the z-axis so the rays from the object are traveling in the +z direction. The field emanating from the object can be written in terms of its angular spectrum method, as a superposition of plane waves:

${\displaystyle E(x,y,z,t)=\sum _{k_x,k_y}A(k_x,k_y)e^{i\left(k_{z}z+k_{y}y+k_{x}x-\omega t\right)},}$

where ${\displaystyle k_z}$ is a function of ${\displaystyle k_x,k_y}$:

${\displaystyle k_z=\sqrt{\frac{{\omega }^2}{c^2}-\left(k_x^2+k_y^2\right)}.}$

Only the positive square root is taken as the energy is going in the +z direction. All of the components of the angular spectrum of the image for which ${\displaystyle k_z}$ is real are transmitted and re-focused by an ordinary lens. However, if

${\displaystyle k_x^2+k_y^2>\frac{{\omega }^2}{c^2},}$

then ${\displaystyle k_z}$ becomes imaginary, and the wave is an evanescent wave, whose amplitude decays as the wave propagates along the z axis. This results in the loss of the high-angular-frequency components of the wave, which contain information about the high-frequency (small-scale) features of the object being imaged. The highest resolution that can be obtained can be expressed in terms of the wavelength:

${\displaystyle k_{\text{max}}\approx \frac{\omega }{c}=\frac{2\pi }{\lambda },}$

${\displaystyle \mathrm{\Delta }{x}_{\text{min}}\approx \lambda .}$

A superlens overcomes the limit. A Pendry-type superlens has an index of n=−1 (ε=−1, µ=−1), and in such a material, transport of energy in the +z direction requires the z component of the wave vector to have opposite sign:

${\displaystyle k_z^{\prime }=-\sqrt{\frac{{\omega }^2}{c^2}-\left(k_x^2+k_y^2\right)}.}$

For large angular frequencies, the evanescent wave now grows, so with proper lens thickness, all components of the angular spectrum can be transmitted through the lens undistorted. There are no problems with conservation of energy, as evanescent waves carry none in the direction of growth: the Poynting vector is oriented perpendicularly to the direction of growth. For traveling waves inside a perfect lens, the Poynting vector points in direction opposite to the phase velocity.^[3]

Effects of negative index of refraction

a) When a wave strikes a positive refraction index material from a vacuum. b) When a wave strikes a negative-refraction-index material from a vacuum. c) When an object is placed in front of an object with n=−1, light from it is refracted so it focuses once inside the lens and once outside. This allows subwavelength imaging.

Normally, when a wave passes through the interface of two materials, the wave appears on the opposite side of the normal. However, if the interface is between a material with a positive index of refraction and another material with a negative index of refraction, the wave will appear on the same side of the normal. Pendry's idea of a perfect lens is a flat material where n=−1. Such a lens allows near-field rays, which normally decay due to the diffraction limit, to focus once within the lens and once outside the lens, allowing subwavelength imaging.^[25]

Development and construction

Superlens construction was at one time thought to be impossible. In 2000, Pendry claimed that a simple slab of left-handed material would do the job.^[26] The experimental realization of such a lens took, however, some more time, because it is not that easy to fabricate metamaterials with both negative permittivity and permeability. Indeed, no such material exists naturally and construction of the required metamaterials is non-trivial. Furthermore, it was shown that the parameters of the material are extremely sensitive (the index must equal −1); small deviations make the subwavelength resolution unobservable.^[27][28] Due to the resonant nature of metamaterials, on which many (proposed) implementations of superlenses depend, metamaterials are highly dispersive. The sensitive nature of the superlens to the material parameters causes superlenses based on metamaterials to have a limited usable frequency range. This initial theoretical superlens design consisted of a metamaterial that compensated for wave decay and reconstructs images in the near field. Both propagating and evanescent waves could contribute to the resolution of the image.^[1][23][29]

Pendry also suggested that a lens having only one negative parameter would form an approximate superlens, provided that the distances involved are also very small and provided that the source polarization is appropriate. For visible light this is a useful substitute, since engineering metamaterials with a negative permeability at the frequency of visible light is difficult. Metals are then a good alternative as they have negative permittivity (but not negative permeability). Pendry suggested using silver due to its relatively low loss at the predicted wavelength of operation (356 nm). In 2003 Pendry's theory was first experimentally demonstrated^[13] at RF/microwave frequencies. In 2005, two independent groups verified Pendry's lens at UV range, both using thin layers of silver illuminated with UV light to produce "photographs" of objects smaller than the wavelength.^[30][31] Negative refraction of visible light was experimentally verified in an yttrium orthovanadate (YVO₄) bicrystal in 2003.^[32]

It was discovered that a simple superlens design for microwaves could use an array of parallel conducting wires. ^[33] This structure was shown to be able to improve the resolution of MRI imaging.

In 2004, the first superlens with a negative refractive index provided resolution three times better than the diffraction limit and was demonstrated at microwave frequencies.^[34] In 2005, the first near field superlens was demonstrated by N.Fang et al., but the lens did not rely on negative refraction. Instead, a thin silver film was used to enhance the evanescent modes through surface plasmon coupling.^[35][36] Almost at the same time Melville and Blaikie succeeded with a near field superlens. Other groups followed.^[30][37] Two developments in superlens research were reported in 2008.^[38] In the second case, a metamaterial was formed from silver nanowires which were electrochemically deposited in porous aluminium oxide. The material exhibited negative refraction.^[39] The imaging performance of such isotropic negative dielectric constant slab lenses were also analyzed with respect to the slab material and thickness.^[40] Subwavelength imaging opportunities with planar uniaxial anisotropic lenses, where the dielectric tensor components are of the opposite sign, have also been studied as a function of the structure parameters.^[41]

The superlens has not yet been demonstrated at visible or near-infrared frequencies (Nielsen, R. B.; 2010). Furthermore, as dispersive materials, these are limited to functioning at a single wavelength. Proposed solutions are metal–dielectric composites (MDCs)^[42] and multilayer lens structures.^[43] The multi-layer superlens appears to have better subwavelength resolution than the single layer superlens. Losses are less of a concern with the multi-layer system, but so far it appears to be impractical because of impedance mis-match.^[35]

While the evolution of nanofabrication techniques continues to push the limits in fabrication of nanostructures, surface roughness remains an inevitable source of concern in the design of nano-photonic devices. The impact of this surface roughness on the effective dielectric constants and subwavelength image resolution of multilayer metal–insulator stack lenses has also been studied. ^[44]

Perfect lenses

When the world is observed through conventional lenses, the sharpness of the image is determined by and limited to the wavelength of light. Around the year 2000, a slab of negative index metamaterial was theorized to create a lens with capabilities beyond conventional (positive index) lenses. Pendry proposed that a thin slab of negative refractive metamaterial might overcome known problems with common lenses to achieve a "perfect" lens that would focus the entire spectrum, both the propagating as well as the evanescent spectra.^[1][45]

A slab of silver was proposed as the metamaterial. More specifically, such silver thin film can be regarded as a metasurface. As light moves away (propagates) from the source, it acquires an arbitrary phase. Through a conventional lens the phase remains consistent, but the evanescent waves decay exponentially. In the flat metamaterial DNG slab, normally decaying evanescent waves are contrarily amplified. Furthermore, as the evanescent waves are now amplified, the phase is reversed.^[1]

Therefore, a type of lens was proposed, consisting of a metal film metamaterial. When illuminated near its plasma frequency, the lens could be used for superresolution imaging that compensates for wave decay and reconstructs images in the near-field. In addition, both propagating and evanescent waves contribute to the resolution of the image.^[1]

Pendry suggested that left-handed slabs allow "perfect imaging" if they are completely lossless, impedance matched, and their refractive index is −1 relative to the surrounding medium. Theoretically, this would be a breakthrough in that the optical version resolves objects as minuscule as nanometers across. Pendry predicted that Double negative metamaterials (DNG) with a refractive index of n=−1, can act, at least in principle, as a "perfect lens" allowing imaging resolution which is limited not by the wavelength, but rather by material quality.^{[1][46][47][48]}

Other studies concerning the perfect lens

Further research demonstrated that Pendry's theory behind the perfect lens was not exactly correct. The analysis of the focusing of the evanescent spectrum (equations 13–21 in reference^[1]) was flawed. In addition, this applies to only one (theoretical) instance, and that is one particular medium that is lossless, nondispersive and the constituent parameters are defined as:^[45]

ε(ω) / ε₀=µ(ω) / µ₀=−1, which in turn results in a negative refraction of n=−1

However, the final intuitive result of this theory that both the propagating and evanescent waves are focused, resulting in a converging focal point within the slab and another convergence (focal point) beyond the slab turned out to be correct.^[45]

If the DNG metamaterial medium has a large negative index or becomes lossy or dispersive, Pendry's perfect lens effect cannot be realized. As a result, the perfect lens effect does not exist in general. According to FDTD simulations at the time (2001), the DNG slab acts like a converter from a pulsed cylindrical wave to a pulsed beam. Furthermore, in reality (in practice), a DNG medium must be and is dispersive and lossy, which can have either desirable or undesirable effects, depending on the research or application. Consequently, Pendry's perfect lens effect is inaccessible with any metamaterial designed to be a DNG medium.^[45]

Another analysis, in 2002,^[24] of the perfect lens concept showed it to be in error while using the lossless, dispersionless DNG as the subject. This analysis mathematically demonstrated that subtleties of evanescent waves, restriction to a finite slab and absorption had led to inconsistencies and divergencies that contradict the basic mathematical properties of scattered wave fields. For example, this analysis stated that absorption, which is linked to dispersion, is always present in practice, and absorption tends to transform amplified waves into decaying ones inside this medium (DNG).^[24]

A third analysis of Pendry's perfect lens concept, published in 2003,^[49] used the recent demonstration of negative refraction at microwave frequencies^[50] as confirming the viability of the fundamental concept of the perfect lens. In addition, this demonstration was thought to be experimental evidence that a planar DNG metamaterial would refocus the far field radiation of a point source. However, the perfect lens would require significantly different values for permittivity, permeability, and spatial periodicity than the demonstrated negative refractive sample.^[49][50]

This study agrees that any deviation from conditions where ε=µ=−1 results in the normal, conventional, imperfect image that degrades exponentially i.e., the diffraction limit. The perfect lens solution in the absence of losses is again, not practical, and can lead to paradoxical interpretations.^[24]

It was determined that although resonant surface plasmons are undesirable for imaging, these turn out to be essential for recovery of decaying evanescent waves. This analysis discovered that metamaterial periodicity has a significant effect on the recovery of types of evanescent components. In addition, achieving subwavelength resolution is possible with current technologies. Negative refractive indices have been demonstrated in structured metamaterials. Such materials can be engineered to have tunable material parameters, and so achieve the optimal conditions. Losses up to microwave frequencies can be minimized in structures utilizing superconducting elements. Furthermore, consideration of alternate structures may lead to configurations of left-handed materials that can achieve subwavelength focusing. Such structures were being studied at the time.^[24]

An effective approach for the compensation of losses in metamaterials, called plasmon injection scheme, has been recently proposed.^[51] The plasmon injection scheme has been applied theoretically to imperfect negative index flat lenses with reasonable material losses and in the presence of noise^[52][53] as well as hyperlenses.^[54] It has been shown that even imperfect negative index flat lenses assisted with plasmon injection scheme can enable subdiffraction imaging of objects which is otherwise not possible due to the losses and noise. Although plasmon injection scheme was originally conceptualized for plasmonic metamaterials,^[51] the concept is general and applicable to all types electromagnetic modes. The main idea of the scheme is the coherent superposition of the lossy modes in the metamaterial with an appropriately structured external auxiliary field. This auxiliary field accounts for the losses in the metamaterial, hence effectively reduces the losses experienced by the signal beam or object field in the case of a metamaterial lens. The plasmon injection scheme can be implemented either physically^[53] or equivalently through deconvolution post-processing method.^[52][54] However, the physical implementation has shown to be more effective than the deconvolution. Physical construction of convolution and selective amplification of the spatial frequencies within a narrow bandwidth are the keys to the physical implementation of the plasmon injection scheme. This loss compensation scheme is ideally suited especially for metamaterial lenses since it does not require gain medium, nonlinearity, or any interaction with phonons. Experimental demonstration of the plasmon injection scheme has not yet been shown partly because the theory is rather new.

Near-field imaging with magnetic wires

A prism composed of high performance Swiss rolls which behaves as a magnetic faceplate, transferring a magnetic field distribution faithfully from the input to the output face.^[55]

Pendry's theoretical lens was designed to focus both propagating waves and the near-field evanescent waves. From permittivity "ε" and magnetic permeability "µ" an index of refraction "n" is derived. The index of refraction determines how light is bent on traversing from one material to another. In 2003, it was suggested that a metamaterial constructed with alternating, parallel, layers of n=−1 materials and n=+1 materials, would be a more effective design for a metamaterial lens. It is an effective medium made up of a multi-layer stack, which exhibits birefringence, n₂=∞, n_x=0. The effective refractive indices are then perpendicular and parallel, respectively.^[55]

Like a conventional lens, the z-direction is along the axis of the roll. The resonant frequency (w₀) – close to 21.3 MHz – is determined by the construction of the roll. Damping is achieved by the inherent resistance of the layers and the lossy part of permittivity.^[55]

Simply put, as the field pattern is transferred from the input to the output face of a slab, so the image information is transported across each layer. This was experimentally demonstrated. To test the two-dimensional imaging performance of the material, an antenna was constructed from a pair of anti-parallel wires in the shape of the letter M. This generated a line of magnetic flux, so providing a characteristic field pattern for imaging. It was placed horizontally, and the material, consisting of 271 Swiss rolls tuned to 21.5 MHz, was positioned on top of it. The material does indeed act as an image transfer device for the magnetic field. The shape of the antenna is faithfully reproduced in the output plane, both in the distribution of the peak intensity, and in the “valleys” that bound the M.^[55]

A consistent characteristic of the very near (evanescent) field is that the electric and magnetic fields are largely decoupled. This allows for nearly independent manipulation of the electric field with the permittivity and the magnetic field with the permeability.^[55]

Furthermore, this is highly anisotropic system. Therefore, the transverse (perpendicular) components of the EM field which radiate the material, that is the wavevector components k_x and k_y, are decoupled from the longitudinal component k_z. So, the field pattern should be transferred from the input to the output face of a slab of material without degradation of the image information.^[55]

Optical superlens with silver metamaterial

In 2003, a group of researchers showed that optical evanescent waves would be enhanced as they passed through a silver metamaterial lens. This was referred to as a diffraction-free lens. Although a coherent, high-resolution, image was not intended, nor achieved, regeneration of the evanescent field was experimentally demonstrated.^[56][57]

By 2003 it was known for decades that evanescent waves could be enhanced by producing excited states at the interface surfaces. However, the use of surface plasmons to reconstruct evanescent components was not tried until Pendry's recent proposal (see "Perfect lens" above). By studying films of varying thickness it has been noted that a rapidly growing transmission coefficient occurs, under the appropriate conditions. This demonstration provided direct evidence that the foundation of superlensing is solid, and suggested the path that will enable the observation of superlensing at optical wavelengths.^[57]

In 2005, a coherent, high-resolution, image was produced (based on the 2003 results). A thinner slab of silver (35 nm) was better for sub–diffraction-limited imaging, which results in one-sixth of the illumination wavelength. This type of lens was used to compensate for wave decay and reconstruct images in the near-field. Prior attempts to create a working superlens used a slab of silver that was too thick.^[23][46]

Objects were imaged as small as 40 nm across. In 2005 the imaging resolution limit for optical microscopes was at about one tenth the diameter of a red blood cell. With the silver superlens this results in a resolution of one hundredth of the diameter of a red blood cell.^[56]

Conventional lenses, whether man-made or natural, create images by capturing the propagating light waves all objects emit and then bending them. The angle of the bend is determined by the index of refraction and has always been positive until the fabrication of artificial negative index materials. Objects also emit evanescent waves that carry details of the object, but are unobtainable with conventional optics. Such evanescent waves decay exponentially and thus never become part of the image resolution, an optics threshold known as the diffraction limit. Breaking this diffraction limit, and capturing evanescent waves are critical to the creation of a 100-percent perfect representation of an object.^[23]

In addition, conventional optical materials suffer a diffraction limit because only the propagating components are transmitted (by the optical material) from a light source.^[23] The non-propagating components, the evanescent waves, are not transmitted.^[24] Moreover, lenses that improve image resolution by increasing the index of refraction are limited by the availability of high-index materials, and point by point subwavelength imaging of electron microscopy also has limitations when compared to the potential of a working superlens. Scanning electron and atomic force microscopes are now used to capture detail down to a few nanometers. However, such microscopes create images by scanning objects point by point, which means they are typically limited to non-living samples, and image capture times can take up to several minutes.^[23]

With current optical microscopes, scientists can only make out relatively large structures within a cell, such as its nucleus and mitochondria. With a superlens, optical microscopes could one day reveal the movements of individual proteins traveling along the microtubules that make up a cell's skeleton, the researchers said. Optical microscopes can capture an entire frame with a single snapshot in a fraction of a second. With superlenses this opens up nanoscale imaging to living materials, which can help biologists better understand cell structure and function in real time.^[23]

Advances of magnetic coupling in the THz and infrared regime provided the realization of a possible metamaterial superlens. However, in the near field, the electric and magnetic responses of materials are decoupled. Therefore, for transverse magnetic (TM) waves, only the permittivity needed to be considered. Noble metals, then become natural selections for superlensing because negative permittivity is easily achieved.^[23]

By designing the thin metal slab so that the surface current oscillations (the surface plasmons) match the evanescent waves from the object, the superlens is able to substantially enhance the amplitude of the field. Superlensing results from the enhancement of evanescent waves by surface plasmons.^[23][56]

The key to the superlens is its ability to significantly enhance and recover the evanescent waves that carry information at very small scales. This enables imaging well below the diffraction limit. No lens is yet able to completely reconstitute all the evanescent waves emitted by an object, so the goal of a 100-percent perfect image will persist. However, many scientists believe that a true perfect lens is not possible because there will always be some energy absorption loss as the waves pass through any known material. In comparison, the superlens image is substantially better than the one created without the silver superlens.^[23]

50-nm flat silver layer

In February 2004, an electromagnetic radiation focusing system, based on a negative index metamaterial plate, accomplished subwavelength imaging in the microwave domain. This showed that obtaining separated images at much less than the wavelength of light is possible.^[58] Also, in 2004, a silver layer was used for sub-micrometre near-field imaging. Super high resolution was not achieved, but this was intended. The silver layer was too thick to allow significant enhancements of evanescent field components.^[30]

In early 2005, feature resolution was achieved with a different silver layer. Though this was not an actual image, it was intended. Dense feature resolution down to 250 nm was produced in a 50 nm thick photoresist using illumination from a mercury lamp. Using simulations (FDTD), the study noted that resolution improvements could be expected for imaging through silver lenses, rather than another method of near field imaging.^[59]

Building on this prior research, super resolution was achieved at optical frequencies using a 50 nm flat silver layer. The capability of resolving an image beyond the diffraction limit, for far-field imaging, is defined here as superresolution.^[30]

The image fidelity is much improved over earlier results of the previous experimental lens stack. Imaging of sub-micrometre features has been greatly improved by using thinner silver and spacer layers, and by reducing the surface roughness of the lens stack. The ability of the silver lenses to image the gratings has been used as the ultimate resolution test, as there is a concrete limit for the ability of a conventional (far field) lens to image a periodic object – in this case the image is a diffraction grating. For normal-incidence illumination the minimum spatial period that can be resolved with wavelength λ through a medium with refractive index n is λ/n. Zero contrast would therefore be expected in any (conventional) far-field image below this limit, no matter how good the imaging resist might be.^[30]

The (super) lens stack here results in a computational result of a diffraction-limited resolution of 243 nm. Gratings with periods from 500 nm down to 170 nm are imaged, with the depth of the modulation in the resist reducing as the grating period reduces. All of the gratings with periods above the diffraction limit (243 nm) are well resolved.^[30] The key results of this experiment are super-imaging of the sub-diffraction limit for 200 nm and 170 nm periods. In both cases the gratings are resolved, even though the contrast is diminished, but this gives experimental confirmation of Pendry's superlensing proposal.^[30]

For further information see Fresnel number and Fresnel diffraction

Negative index GRIN lenses

Gradient Index (GRIN) – The larger range of material response available in metamaterials should lead to improved GRIN lens design. In particular, since the permittivity and permeability of a metamaterial can be adjusted independently, metamaterial GRIN lenses can presumably be better matched to free space. The GRIN lens is constructed by using a slab of NIM with a variable index of refraction in the y direction, perpendicular to the direction of propagation z.^[60]

Far-field superlens

In 2005, a group proposed a theoretical way to overcome the near-field limitation using a new device termed a far-field superlens (FSL), which is a properly designed periodically corrugated metallic slab-based superlens.^[61]

Imaging was experimentally demonstrated in the far field, taking the next step after near-field experiments. The key element is termed as a far-field superlens (FSL) which consists of a conventional superlens and a nanoscale coupler.^[62]

Focusing beyond the diffraction limit with far-field time reversal

An approach is presented for subwavelength focusing of microwaves using both a time-reversal mirror placed in the far field and a random distribution of scatterers placed in the near field of the focusing point.^[63]

Hyperlens

Once capability for near-field imaging was demonstrated, the next step was to project a near-field image into the far-field. This concept, including technique and materials, is dubbed "hyperlens".^[64][65]

In May 2012, calculations showed an ultraviolet (1200-1400 THz) hyperlens can be created using alternating layers of boron nitride and graphene.^[66]

In February 2018, a mid-infrared (~5-25μm) hyperlens was introduced, made from a variably doped indium arsenide multilayer, which offered drastically lower losses.^[67]

The capability of a metamaterial-hyperlens for sub-diffraction-limited imaging is shown below.

Sub-diffraction imaging in the far field

With conventional optical lenses, the far field is a limit that is too distant for evanescent waves to arrive intact. When imaging an object, this limits the optical resolution of lenses to the order of the wavelength of light. These non-propagating waves carry detailed information in the form of high spatial resolution, and overcome limitations. Therefore, projecting image details, normally limited by diffraction into the far field does require recovery of the evanescent waves.^[68]

In essence steps leading up to this investigation and demonstration was the employment of an anisotropic metamaterial with a hyperbolic dispersion. The effect was such that ordinary evanescent waves propagate along the radial direction of the layered metamaterial. On a microscopic level the large spatial frequency waves propagate through coupled surface plasmon excitations between the metallic layers.^[68]

In 2007, just such an anisotropic metamaterial was employed as a magnifying optical hyperlens. The hyperlens consisted of a curved periodic stack of thin silver and alumina (at 35 nanometers thick) deposited on a half-cylindrical cavity, and fabricated on a quartz substrate. The radial and tangential permittivities have different signs.^[68]

Upon illumination, the scattered evanescent field from the object enters the anisotropic medium and propagates along the radial direction. Combined with another effect of the metamaterial, a magnified image at the outer diffraction limit-boundary of the hyperlens occurs. Once the magnified feature is larger than (beyond) the diffraction limit, it can then be imaged with a conventional optical microscope, thus demonstrating magnification and projection of a sub-diffraction-limited image into the far field.^[68]

The hyperlens magnifies the object by transforming the scattered evanescent waves into propagating waves in the anisotropic medium, projecting a spatial resolution high-resolution image into the far field. This type of metamaterials-based lens, paired with a conventional optical lens is therefore able to reveal patterns too small to be discerned with an ordinary optical microscope. In one experiment, the lens was able to distinguish two 35-nanometer lines etched 150 nanometers apart. Without the metamaterials, the microscope showed only one thick line.^[14]

In a control experiment, the line pair object was imaged without the hyperlens. The line pair could not be resolved because of the diffraction limit of the (optical) aperture was limited to 260 nm. Because the hyperlens supports the propagation of a very broad spectrum of wave vectors, it can magnify arbitrary objects with sub-diffraction-limited resolution.^[68]

Although this work appears to be limited by being only a cylindrical hyperlens, the next step is to design a spherical lens. That lens will exhibit three-dimensional capability. Near-field optical microscopy uses a tip to scan an object. In contrast, this optical hyperlens magnifies an image that is sub-diffraction-limited. The magnified sub-diffraction image is then projected into the far field.^[14][68]

The optical hyperlens shows a notable potential for applications, such as real-time biomolecular imaging and nanolithography. Such a lens could be used to watch cellular processes that have been impossible to see. Conversely, it could be used to project an image with extremely fine features onto a photoresist as a first step in photolithography, a process used to make computer chips. The hyperlens also has applications for DVD technology.^[14][68]

In 2010, a spherical hyperlens for two dimensional imaging at visible frequencies was demonstrated experimentally. The spherical hyperlens was based on silver and titanium oxide in alternating layers and had strong anisotropic hyperbolic dispersion allowing super-resolution with visible spectrum. The resolution was 160 nm in the visible spectrum. It will enable biological imaging at the cellular and DNA level, with a strong benefit of magnifying sub-diffraction resolution into far-field. ^[69]

Plasmon-assisted microscopy

See Near-field scanning optical microscope.

Super-imaging in the visible frequency range

In 2007 researchers demonstrated super imaging using materials, which create negative refractive index and lensing is achieved in the visible range.^[46]

Continual improvements in optical microscopy are needed to keep up with the progress in nanotechnology and microbiology. Advancement in spatial resolution is key. Conventional optical microscopy is limited by a diffraction limit which is on the order of 200 nanometers (wavelength). This means that viruses, proteins, DNA molecules and many other samples are hard to observe with a regular (optical) microscope. The lens previously demonstrated with negative refractive index material, a thin planar superlens, does not provide magnification beyond the diffraction limit of conventional microscopes. Therefore, images smaller than the conventional diffraction limit will still be unavailable.^[46]

Another approach achieving super-resolution at visible wavelength is recently developed spherical hyperlens based on silver and titanium oxide alternating layers. It has strong anisotropic hyperbolic dispersion allowing super-resolution with converting evanescent waves into propagating waves. This method is non-fluorescence based super-resolution imaging, which results in real-time imaging without any reconstruction of images and information.^[69]

Super resolution far-field microscopy techniques

By 2008 the diffraction limit has been surpassed and lateral imaging resolutions of 20 to 50 nm have been achieved by several "super-resolution" far-field microscopy techniques, including stimulated emission depletion (STED) and its related RESOLFT (reversible saturable optically linear fluorescent transitions) microscopy; saturated structured illumination microscopy (SSIM) ; stochastic optical reconstruction microscopy (STORM); photoactivated localization microscopy (PALM); and other methods using similar principles.^[70]

Cylindrical superlens via coordinate transformation

This began with a proposal by Pendry, in 2003. Magnifying the image required a new design concept in which the surface of the negatively refracting lens is curved. One cylinder touches another cylinder, resulting in a curved cylindrical lens which reproduced the contents of the smaller cylinder in magnified but undistorted form outside the larger cylinder. Coordinate transformations are required to curve the original perfect lens into the cylindrical, lens structure.^[71]

This was followed by a 36-page conceptual and mathematical proof in 2005, that the cylindrical superlens works in the quasistatic regime. The debate over the perfect lens is discussed first.^[72]

In 2007, a superlens utilizing coordinate transformation was again the subject. However, in addition to image transfer other useful operations were discussed; translation, rotation, mirroring and inversion as well as the superlens effect. Furthermore, elements that perform magnification are described, which are free from geometric aberrations, on both the input and output sides while utilizing free space sourcing (rather than waveguide). These magnifying elements also operate in the near and far field, transferring the image from near field to far field.^[73]

The cylindrical magnifying superlens was experimentally demonstrated in 2007 by two groups, Liu et al.^[68] and Smolyaninov et al.^[46][74]

Nano-optics with metamaterials

Nanohole array as a lens

Work in 2007 demonstrated that a quasi-periodic array of nanoholes, in a metal screen, were able to focus the optical energy of a plane wave to form subwavelength spots (hot spots). The distances for the spots was a few tens of wavelengths on the other side of the array, or, in other words, opposite the side of the incident plane wave. The quasi-periodic array of nanoholes functioned as a light concentrator.^[75]

In June 2008, this was followed by the demonstrated capability of an array of quasi-crystal nanoholes in a metal screen. More than concentrating hot spots, an image of the point source is displayed a few tens of wavelengths from the array, on the other side of the array (the image plane). Also this type of array exhibited a 1 to 1 linear displacement, – from the location of the point source to its respective, parallel, location on the image plane. In other words, from x to x + δx. For example, other point sources were similarly displaced from x' to x' + δx', from x^ to x^ + δx^, and from x^^ to x^^ + δx^^, and so on. Instead of functioning as a light concentrator, this performs the function of conventional lens imaging with a 1 to 1 correspondence, albeit with a point source.^[75]

However, resolution of more complicated structures can be achieved as constructions of multiple point sources. The fine details, and brighter image, that are normally associated with the high numerical apertures of conventional lenses can be reliably produced. Notable applications for this technology arise when conventional optics is not suitable for the task at hand. For example, this technology is better suited for X-ray imaging, or nano-optical circuits, and so forth.^[75]

Nanolens

In 2010, a nano-wire array prototype, described as a three-dimensional (3D) metamaterial-nanolens, consisting of bulk nanowires deposited in a dielectric substrate was fabricated and tested.^[76][77]

The metamaterial nanolens was constructed of millions of nanowires at 20 nanometers in diameter. These were precisely aligned and a packaged configuration was applied. The lens is able to depict a clear, high-resolution image of nano-sized objects because it uses both normal propagating EM radiation, and evanescent waves to construct the image. Super-resolution imaging was demonstrated over a distance of 6 times the wavelength (λ), in the far-field, with a resolution of at least λ/4. This is a significant improvement over previous research and demonstration of other near field and far field imaging, including nanohole arrays discussed below.^[76][77]

Light transmission properties of holey metal films

2009-12. The light transmission properties of holey metal films in the metamaterial limit, where the unit length of the periodic structures is much smaller than the operating wavelength, are analyzed theoretically.^[78]

Transporting an Image through a subwavelength hole

Theoretically it appears possible to transport a complex electromagnetic image through a tiny subwavelength hole with diameter considerably smaller than the diameter of the image, without losing the subwavelength details.^[79]

Nanoparticle imaging – quantum dots

When observing the complex processes in a living cell, significant processes (changes) or details are easy to overlook. This can more easily occur when watching changes that take a long time to unfold and require high-spatial-resolution imaging. However, recent research offers a solution to scrutinize activities that occur over hours or even days inside cells, potentially solving many of the mysteries associated with molecular-scale events occurring in these tiny organisms.^[80]

A joint research team, working at the National Institute of Standards and Technology (NIST) and the National Institute of Allergy and Infectious Diseases (NIAID), has discovered a method of using nanoparticles to illuminate the cellular interior to reveal these slow processes. Nanoparticles, thousands of times smaller than a cell, have a variety of applications. One type of nanoparticle called a quantum dot glows when exposed to light. These semiconductor particles can be coated with organic materials, which are tailored to be attracted to specific proteins within the part of a cell a scientist wishes to examine.^[80]

Notably, quantum dots last longer than many organic dyes and fluorescent proteins that were previously used to illuminate the interiors of cells. They also have the advantage of monitoring changes in cellular processes while most high-resolution techniques like electron microscopy only provide images of cellular processes frozen at one moment. Using quantum dots, cellular processes involving the dynamic motions of proteins, are observable (elucidated).^[80]

The research focused primarily on characterizing quantum dot properties, contrasting them with other imaging techniques. In one example, quantum dots were designed to target a specific type of human red blood cell protein that forms part of a network structure in the cell's inner membrane. When these proteins cluster together in a healthy cell, the network provides mechanical flexibility to the cell so it can squeeze through narrow capillaries and other tight spaces. But when the cell gets infected with the malaria parasite, the structure of the network protein changes.^[80]

Because the clustering mechanism is not well understood, it was decided to examine it with the quantum dots. If a technique could be developed to visualize the clustering, then the progress of a malaria infection could be understood, which has several distinct developmental stages.^[80]

Research efforts revealed that as the membrane proteins bunch up, the quantum dots attached to them are induced to cluster themselves and glow more brightly, permitting real time observation as the clustering of proteins progresses. More broadly, the research discovered that when quantum dots attach themselves to other nanomaterials, the dots' optical properties change in unique ways in each case. Furthermore, evidence was discovered that quantum dot optical properties are altered as the nanoscale environment changes, offering greater possibility of using quantum dots to sense the local biochemical environment inside cells.^[80]

Some concerns remain over toxicity and other properties. However, the overall findings indicate that quantum dots could be a valuable tool to investigate dynamic cellular processes.^[80]

The abstract from the related published research paper states (in part): Results are presented regarding the dynamic fluorescence properties of bioconjugated nanocrystals or quantum dots (QDs) in different chemical and physical environments. A variety of QD samples was prepared and compared: isolated individual QDs, QD aggregates, and QDs conjugated to other nanoscale materials...

See also

Acoustic metamaterials History of metamaterials Metamaterial absorber Metamaterial antennas Metamaterial cloaking Negative index metamaterials Nonlinear metamaterials Photonic crystal Photonic metamaterials Plasmonic metamaterials Seismic metamaterials https://en.wikipedia.org/wiki/Superlens A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of natural crystals gives rise to X-ray diffraction and that the atomic lattices (crystal structure) of semiconductors affect their conductivity of electrons. Photonic crystals occur in nature in the form of structural coloration and animal reflectors, and, as artificially produced, promise to be useful in a range of applications. Photonic crystals can be fabricated for one, two, or three dimensions. One-dimensional photonic crystals can be made of thin film layers deposited on each other. Two-dimensional ones can be made by photolithography, or by drilling holes in a suitable substrate. Fabrication methods for three-dimensional ones include drilling under different angles, stacking multiple 2-D layers on top of each other, direct laser writing, or, for example, instigating self-assembly of spheres in a matrix and dissolving the spheres. Photonic crystals can, in principle, find uses wherever light must be manipulated. For example, dielectric mirrors are one-dimensional photonic crystals which can produce ultra-high reflectivity mirrors at a specified wavelength. Two-dimensional photonic crystals called photonic-crystal fibers are used for fiber-optic communication, among other applications. Three-dimensional crystals may one day be used in optical computers, and could lead to more efficient photovoltaic cells.[3] Although the energy of light (and all electromagnetic radiation) is quantized in units called photons, the analysis of photonic crystals requires only classical physics. "Photonic" in the name is a reference to photonics, a modern designation for the study of light (optics) and optical engineering. Indeed, the first research into what we now call photonic crystals may have been as early as 1887 when the English physicist Lord Rayleigh experimented with periodic multi-layer dielectric stacks, showing they can effect a photonic band-gap in one dimension. Research interest grew with work in 1987 by Eli Yablonovitch and Sajeev John on periodic optical structures with more than one dimension—now called photonic crystals. Introduction 0:41 Diffraction from a periodic structure as a function of incident wavelength. For some wavelength ranges, the wave is unable to penetrate the structure. Photonic crystals are composed of periodic dielectric, metallo-dielectric—or even superconductor microstructures or nanostructures that affect electromagnetic wave propagation in the same way that the periodic potential in a semiconductor crystal affects the propagation of electrons, determining allowed and forbidden electronic energy bands. Photonic crystals contain regularly repeating regions of high and low refractive index. Light waves may propagate through this structure or propagation may be disallowed, depending on their wavelength. Wavelengths that may propagate in a given direction are called modes, and the ranges of wavelengths which propagate are called bands. Disallowed bands of wavelengths are called photonic band gaps. This gives rise to distinct optical phenomena, such as inhibition of spontaneous emission,[4] high-reflecting omni-directional mirrors, and low-loss-waveguiding. The bandgap of photonic crystals can be understood as the destructive interference of multiple reflections of light propagating in the crystal at each interface between layers of high- and low- refractive index regions, akin to the bandgaps of electrons in solids. The periodicity of the photonic crystal structure must be around or greater than half the wavelength (in the medium) of the light waves in order for interference effects to be exhibited. Visible light ranges in wavelength between about 400 nm (violet) to about 700 nm (red) and the resulting wavelength inside a material requires dividing that by the average index of refraction. The repeating regions of high and low dielectric constant must, therefore, be fabricated at this scale. In one dimension, this is routinely accomplished using the techniques of thin-film deposition. History Photonic crystals have been studied in one form or another since 1887, but no one used the term photonic crystal until over 100 years later—after Eli Yablonovitch and Sajeev John published two milestone papers on photonic crystals in 1987.[4][5] The early history is well-documented in the form of a story when it was identified as one of the landmark developments in physics by the American Physical Society.[6] Before 1987, one-dimensional photonic crystals in the form of periodic multi-layer dielectric stacks (such as the Bragg mirror) were studied extensively. Lord Rayleigh started their study in 1887,[7] by showing that such systems have a one-dimensional photonic band-gap, a spectral range of large reflectivity, known as a stop-band. Today, such structures are used in a diverse range of applications—from reflective coatings to enhancing LED efficiency to highly reflective mirrors in certain laser cavities (see, for example, VCSEL). The pass-bands and stop-bands in photonic crystals were first reduced to practice by Melvin M. Weiner[8] who called those crystals "discrete phase-ordered media." Weiner achieved those results by extending Darwin's[9] dynamical theory for x-ray Bragg diffraction to arbitrary wavelengths, angles of incidence, and cases where the incident wavefront at a lattice plane is scattered appreciably in the forward-scattered direction. A detailed theoretical study of one-dimensional optical structures was performed by Vladimir P. Bykov,[10] who was the first to investigate the effect of a photonic band-gap on the spontaneous emission from atoms and molecules embedded within the photonic structure. Bykov also speculated as to what could happen if two- or three-dimensional periodic optical structures were used.[11] The concept of three-dimensional photonic crystals was then discussed by Ohtaka in 1979,[12] who also developed a formalism for the calculation of the photonic band structure. However, these ideas did not take off until after the publication of two milestone papers in 1987 by Yablonovitch and John. Both these papers concerned high-dimensional periodic optical structures, i.e., photonic crystals. Yablonovitch's main goal was to engineer photonic density of states to control the spontaneous emission of materials embedded in the photonic crystal. John's idea was to use photonic crystals to affect localisation and control of light. After 1987, the number of research papers concerning photonic crystals began to grow exponentially. However, due to the difficulty of fabricating these structures at optical scales (see Fabrication challenges), early studies were either theoretical or in the microwave regime, where photonic crystals can be built on the more accessible centimetre scale. (This fact is due to a property of the electromagnetic fields known as scale invariance. In essence, electromagnetic fields, as the solutions to Maxwell's equations, have no natural length scale—so solutions for centimetre scale structure at microwave frequencies are the same as for nanometre scale structures at optical frequencies.) By 1991, Yablonovitch had demonstrated the first three-dimensional photonic band-gap in the microwave regime.[13] The structure that Yablonovitch was able to produce involved drilling an array of holes in a transparent material, where the holes of each layer form an inverse diamond structure – today it is known as Yablonovite. In 1996, Thomas Krauss demonstrated a two-dimensional photonic crystal at optical wavelengths.[14] This opened the way to fabricate photonic crystals in semiconductor materials by borrowing methods from the semiconductor industry. Pavel Cheben demonstrated a new type of photonic crystal waveguide – subwavelength grating (SWG) waveguide.[15][16] The SWG waveguide operates in subwavelength region, away from the bandgap. It allows the waveguide properties to be controlled directly by the nanoscale engineering of the resulting metamaterial while mitigating wave interference effects. This provided “a missing degree of freedom in photonics”[17] and resolved an important limitation in silicon photonics which was its restricted set of available materials insufficient to achieve complex optical on-chip functions.[18][19] Today, such techniques use photonic crystal slabs, which are two dimensional photonic crystals "etched" into slabs of semiconductor. Total internal reflection confines light to the slab, and allows photonic crystal effects, such as engineering photonic dispersion in the slab. Researchers around the world are looking for ways to use photonic crystal slabs in integrated computer chips, to improve optical processing of communications—both on-chip and between chips.[citation needed] Autocloning fabrication technique, proposed for infrared and visible range photonic crystals by Sato et al. in 2002, utilizes electron-beam lithography and dry etching: lithographically-formed layers of periodic grooves are stacked by regulated sputter deposition and etching, resulting in "stationary corrugations" and periodicity. Titanium dioxide/silica and tantalum pentoxide/silica devices were produced, exploiting their dispersion characteristics and suitability to sputter deposition.[20] Such techniques have yet to mature into commercial applications, but two-dimensional photonic crystals are commercially used in photonic crystal fibres[21] (otherwise known as holey fibres, because of the air holes that run through them). Photonic crystal fibres were first developed by Philip Russell in 1998, and can be designed to possess enhanced properties over (normal) optical fibres. Study has proceeded more slowly in three-dimensional than in two-dimensional photonic crystals. This is because of more difficult fabrication.[21] Three-dimensional photonic crystal fabrication had no inheritable semiconductor industry techniques to draw on. Attempts have been made, however, to adapt some of the same techniques, and quite advanced examples have been demonstrated,[22] for example in the construction of "woodpile" structures constructed on a planar layer-by-layer basis. Another strand of research has tried to construct three-dimensional photonic structures from self-assembly—essentially letting a mixture of dielectric nano-spheres settle from solution into three-dimensionally periodic structures that have photonic band-gaps. Vasily Astratov's group from the Ioffe Institute realized in 1995 that natural and synthetic opals are photonic crystals with an incomplete bandgap.[23] The first demonstration of an "inverse opal" structure with a complete photonic bandgap came in 2000, from researchers at the University of Toronto, Canada, and Institute of Materials Science of Madrid (ICMM-CSIC), Spain.[24] The ever-expanding field of natural photonics, bioinspiration and biomimetics—the study of natural structures to better understand and use them in design—is also helping researchers in photonic crystals.[25][26][27][28] For example, in 2006 a naturally occurring photonic crystal was discovered in the scales of a Brazilian beetle.[29] Analogously, in 2012 a diamond crystal structure was found in a weevil[30][31] and a gyroid-type architecture in a butterfly.[32] More recently, gyroid photonic crystals have been found in the feather barbs of blue-winged leafbirds and are responsible for the bird's shimmery blue coloration.[33] Some publications suggest the feasibility of the complete photonic band gap in the visible range in photonic crystals with optically saturated media that can be implemented by using laser light as an external optical pump.[34] Construction strategies The fabrication method depends on the number of dimensions that the photonic bandgap must exist in. Examples of possible photonic crystal structures in 1, 2 and 3 dimensions Comparison of 1D, 2D and 3D photonic crystal structures (from left to right, respectively). Schematic of a 1D photonic crystal structure, made of alternating layers of a high-dielectric constant material and a low-dielectric constant material. These layers are typically quarter wavelength in thickness. 2D photonic crystal structure in a square array. Schematic of a 2D photonic crystal made of circular holes. A woodpile structured 3D photonic crystal. These structures have a three-dimensional bandgap for all polarizations One-dimensional photonic crystals To produce a one-dimensional photonic crystal, thin film layers of different dielectric constant may be periodically deposited on a surface which leads to a band gap in a particular propagation direction (such as normal to the surface). A Bragg grating is an example of this type of photonic crystal. One-dimensional photonic crystals can include layers of non-linear optical materials in which the non-linear behaviour is accentuated due to field enhancement at wavelengths near a so-called degenerate band edge. This field enhancement (in terms of intensity) can reach ${\displaystyle N^2}$ where N is the total number of layers. However by using layers which include an optically anisotropic material, it has been shown that the field enhancement can reach ${\displaystyle N^4}$, which, in conjunction with non-linear optics, has potential applications such as in the development of an all-optical switch.[35] A one-dimensional photonic crystal can be implemented using repeated alternating layers of a metamaterial and vacuum.[36] If the metamaterial is such that the relative permittivity and permeability follow the same wavelength dependence, then the photonic crystal behaves identically for TE and TM modes, that is, for both s and p polarizations of light incident at an angle. Recently, researchers fabricated a graphene-based Bragg grating (one-dimensional photonic crystal) and demonstrated that it supports excitation of surface electromagnetic waves in the periodic structure by using 633 nm He-Ne laser as the light source.[37] Besides, a novel type of one-dimensional graphene-dielectric photonic crystal has also been proposed. This structure can act as a far-IR filter and can support low-loss surface plasmons for waveguide and sensing applications.[38] 1D photonic crystals doped with bio-active metals (i.e. silver) have been also proposed as sensing devices for bacterial contaminants.[39] Similar planar 1D photonic crystals made of polymers have been used to detect volatile organic compounds vapors in atmosphere.[40][41] In addition to solid-phase photonic crystals, some liquid crystals with defined ordering can demonstrate photonic color.[42] For example, studies have shown several liquid crystals with short- or long-range one-dimensional positional ordering can form photonic structures.[42] Two-dimensional photonic crystals In two dimensions, holes may be drilled in a substrate that is transparent to the wavelength of radiation that the bandgap is designed to block. Triangular and square lattices of holes have been successfully employed. The Holey fiber or photonic crystal fiber can be made by taking cylindrical rods of glass in hexagonal lattice, and then heating and stretching them, the triangle-like airgaps between the glass rods become the holes that confine the modes. Three-dimensional photonic crystals There are several structure types that have been constructed:[43] Spheres in a diamond lattice Yablonovite The woodpile structure – "rods" are repeatedly etched with beam lithography, filled in, and covered with a layer of new material. As the process repeats, the channels etched in each layer are perpendicular to the layer below, and parallel to and out of phase with the channels two layers below. The process repeats until the structure is of the desired height. The fill-in material is then dissolved using an agent that dissolves the fill-in material but not the deposition material. It is generally hard to introduce defects into this structure. Inverse opals or Inverse Colloidal Crystals-Spheres (such as polystyrene or silicon dioxide) can be allowed to deposit into a cubic close packed lattice suspended in a solvent. Then a hardener is introduced that makes a transparent solid out of the volume occupied by the solvent. The spheres are then dissolved with an acid such as Hydrochloric acid. The colloids can be either spherical[24] or nonspherical.[44][45][46][47] contains in excess of 750,000 polymer nanorods.[clarification needed] Light focused on this beam splitter penetrates or is reflected, depending on polarization.[48][49] A photonic crystal fiber. SEM images of US NRL-produced fiber. (left) The diameter of the solid core at the center of the fiber is 5 µm, while (right) the diameter of the holes is 4 µm. Source: http://www.nrl.navy.mil/techtransfer/fs.php?fs_id=97 An SEM image of a self-assembled PMMA photonic crystal in two dimensions Photonic crystal cavities Not only band gap, photonic crystals may have another effect if we partially remove the symmetry through the creation a nanosize cavity. This defect allows you to guide or to trap the light with the same function as nanophotonic resonator and it is characterized by the strong dielectric modulation in the photonic crystals.[50] For the waveguide, the propagation of light depends on the in-plane control provided by the photonic band gap and to the long confinement of light induced by dielectric mismatch. For the light trap, the light is strongly confined in the cavity resulting further interactions with the materials. First, if we put a pulse of light inside the cavity, it will be delayed by nano- or picoseconds and this is proportional to the quality factor of the cavity. Finally, if we put an emitter inside the cavity, the emission light also can be enhanced significantly and or even the resonant coupling can go through Rabi oscillation. This is related with cavity quantum electrodynamics and the interactions are defined by the weak and strong coupling of the emitter and the cavity. The first studies for the cavity in one-dimensional photonic slabs are usually in grating[51] or distributed feedback structures.[52] For two-dimensional photonic crystal cavities,[53][54][55] they are useful to make efficient photonic devices in telecommunication applications as they can provide very high quality factor up to millions with smaller-than-wavelength mode volume. For three-dimensional photonic crystal cavities, several methods have been developed including lithographic layer-by-layer approach,[56] surface ion beam lithography,[57] and micromanipulation technique.[58] All those mentioned photonic crystal cavities that tightly confine light offer very useful functionality for integrated photonic circuits, but it is challenging to produce them in a manner that allows them to be easily relocated.[59] There is no full control with the cavity creation, the cavity location, and the emitter position relative to the maximum field of the cavity while the studies to solve those problems are still ongoing. Movable cavity of nanowire in photonic crystals is one of solutions to tailor this light matter interaction.[60] Fabrication challenges Higher-dimensional photonic crystal fabrication faces two major challenges: Making them with enough precision to prevent scattering losses blurring the crystal properties Designing processes that can robustly mass-produce the crystals One promising fabrication method for two-dimensionally periodic photonic crystals is a photonic-crystal fiber, such as a holey fiber. Using fiber draw techniques developed for communications fiber it meets these two requirements, and photonic crystal fibres are commercially available. Another promising method for developing two-dimensional photonic crystals is the so-called photonic crystal slab. These structures consist of a slab of material—such as silicon—that can be patterned using techniques from the semiconductor industry. Such chips offer the potential to combine photonic processing with electronic processing on a single chip. For three dimensional photonic crystals, various techniques have been used—including photolithography and etching techniques similar to those used for integrated circuits.[22] Some of these techniques are already commercially available. To avoid the complex machinery of nanotechnological methods, some alternate approaches involve growing photonic crystals from colloidal crystals as self-assembled structures. Mass-scale 3D photonic crystal films and fibres can now be produced using a shear-assembly technique that stacks 200–300 nm colloidal polymer spheres into perfect films of fcc lattice. Because the particles have a softer transparent rubber coating, the films can be stretched and molded, tuning the photonic bandgaps and producing striking structural color effects. Computing photonic band structure The photonic band gap (PBG) is essentially the gap between the air-line and the dielectric-line in the dispersion relation of the PBG system. To design photonic crystal systems, it is essential to engineer the location and size of the bandgap by computational modeling using any of the following methods: 0:03 A video simulation of scattering forces and fields in a photonic crystal structure[61] Plane wave expansion method Inverse dispersion method[62] Finite element method Finite difference time domain method Order-n spectral method[63][64] KKR method Bloch wave – MoM method Essentially, these methods solve for the frequencies (normal modes) of the photonic crystal for each value of the propagation direction given by the wave vector, or vice versa. The various lines in the band structure, correspond to the different cases of n, the band index. For an introduction to photonic band structure, see K. Sakoda's [65] and Joannopoulos [50] books. Band structure of a 1D photonic crystal, DBR air-core calculated using plane wave expansion technique with 101 planewaves, for d/a=0.8, and dielectric contrast of 12.250. The plane wave expansion method can be used to calculate the band structure using an eigen formulation of the Maxwell's equations, and thus solving for the eigen frequencies for each of the propagation directions, of the wave vectors. It directly solves for the dispersion diagram. Electric field strength values can also be calculated over the spatial domain of the problem using the eigen vectors of the same problem. For the picture shown to the right, corresponds to the band-structure of a 1D distributed Bragg reflector (DBR) with air-core interleaved with a dielectric material of relative permittivity 12.25, and a lattice period to air-core thickness ratio (d/a) of 0.8, is solved using 101 planewaves over the first irreducible Brillouin zone. The Inverse dispersion method also exploited plane wave expansion but formulates Maxwell’s equation as an eigenproblem for the wave vector k while the frequency ${\displaystyle \omega }$ is considered as a parameter.[62] Thus, it solves the dispersion relation ${\displaystyle k(\omega )}$ instead of ${\displaystyle \omega (k)}$, which plane wave method does. The inverse dispersion method makes it possible to find complex value of the wave vector e.g. in the bandgap, which allows one to distinguish photonic crystals from metamaterial. Besides, the method is ready for the frequency dispersion of the permittivity to be taken into account. To speed calculation of the frequency band structure, the Reduced Bloch Mode Expansion (RBME) method can be used.[66] The RBME method applies "on top" of any of the primary expansion methods mentioned above. For large unit cell models, the RBME method can reduce time for computing the band structure by up to two orders of magnitude. Applications See also: Photonic crystal sensor Photonic crystals are attractive optical materials for controlling and manipulating light flow. One dimensional photonic crystals are already in widespread use, in the form of thin-film optics, with applications from low and high reflection coatings on lenses and mirrors to colour changing paints and inks.[67][68][47] Higher-dimensional photonic crystals are of great interest for both fundamental and applied research, and the two dimensional ones are beginning to find commercial applications. The first commercial products involving two-dimensionally periodic photonic crystals are already available in the form of photonic-crystal fibers, which use a microscale structure to confine light with radically different characteristics compared to conventional optical fiber for applications in nonlinear devices and guiding exotic wavelengths. The three-dimensional counterparts are still far from commercialization but may offer additional features such as optical nonlinearity required for the operation of optical transistors used in optical computers, when some technological aspects such as manufacturability and principal difficulties such as disorder are under control.[69][citation needed] SWG photonic crystal waveguides have facilitated new integrated photonic devices for controlling transmission of light signals in photonic integrated circuits, including fibre-chip couplers, waveguide crossovers, wavelength and mode multiplexers, ultra-fast optical switches, athermal waveguides, biochemical sensors, polarization management circuits, broadband interference couplers, planar waveguide lenses, anisotropic waveguides, nanoantennas and optical phased arrays.[18][70][71] SWG nanophotonic couplers permit highly-efficient and polarization-independent coupling between photonic chips and external devices.[16] They have been adopted for fibre-chip coupling in volume optoelectronic chip manufacturing.[72][73][74] These coupling interfaces are particularly important because every photonic chip needs to be optically connected with the external world and the chips themselves appear in many established and emerging applications, such as 5G networks, data center interconnects, chip-to-chip interconnects, metro- and long-haul telecommunication systems, and automotive navigation. In addition to the foregoing, photonic crystals have been proposed as platforms for the development of solar cells [75] and optical sensors,[76] including chemical sensors and biosensors.[77][78] See also Wikimedia Commons has media related to Photonic crystals. Animal coloration – General appearance of an animal Animal reflectors Colloidal crystal – Ordered array of colloidal particles Left-handed material – Material with a negative refractive index Metamaterial – Materials engineered to have properties that have not yet been found in nature Nanomaterials – Materials whose granular size lies between 1 and 100 nm Nanotechnology – Field of applied science involving control of matter on atomic and (supra)molecular scales Optical medium – Medium through which electromagnetic waves propagate Photonic-crystal fiber – Class of optical fiber based on the properties of photonic crystals Photonic metamaterials – Type of electromagnetic metamaterial Structural coloration – Colour in living creatures caused by interference effects Superlens Superprism Thin-film optics – Branch of optics that deals with very thin structured layers of different materials Tunable metamaterials https://en.wikipedia.org/wiki/Photonic_crystal  Negative-index metamaterial or negative-index material (NIM) is a metamaterial whose refractive index for an electromagnetic wave has a negative value over some frequency range.[1] NIMs are constructed of periodic basic parts called unit cells, which are usually significantly smaller than the wavelength of the externally applied electromagnetic radiation. The unit cells of the first experimentally investigated NIMs were constructed from circuit board material, or in other words, wires and dielectrics. In general, these artificially constructed cells are stacked or planar and configured in a particular repeated pattern to compose the individual NIM. For instance, the unit cells of the first NIMs were stacked horizontally and vertically, resulting in a pattern that was repeated and intended (see below images). Specifications for the response of each unit cell are predetermined prior to construction and are based on the intended response of the entire, newly constructed, material. In other words, each cell is individually tuned to respond in a certain way, based on the desired output of the NIM. The aggregate response is mainly determined by each unit cell's geometry and substantially differs from the response of its constituent materials. In other words, the way the NIM responds is that of a new material, unlike the wires or metals and dielectrics it is made from. Hence, the NIM has become an effective medium. Also, in effect, this metamaterial has become an “ordered macroscopic material, synthesized from the bottom up”, and has emergent properties beyond its components.[2] Metamaterials that exhibit a negative value for the refractive index are often referred to by any of several terminologies: left-handed media or left-handed material (LHM), backward-wave media (BW media), media with negative refractive index, double negative (DNG) metamaterials, and other similar names.[3] Properties and characteristics A split-ring resonator array arranged to produce a negative index of refraction, constructed of copper split-ring resonators and wires mounted on interlocking sheets of fiberglass circuit board. The total array consists of 3 by 20×20 unit cells with overall dimensions of 10×100×100 millimeters.[4][5] The height of 10 millimeters measures a little more than six subdivision marks on the ruler, which is marked in inches. Electrodynamics of media with negative indices of refraction were first studied by Russian theoretical physicist Victor Veselago from Moscow Institute of Physics and Technology in 1967.[6] The proposed left-handed or negative-index materials were theorized to exhibit optical properties opposite to those of glass, air, and other transparent media. Such materials were predicted to exhibit counterintuitive properties like bending or refracting light in unusual and unexpected ways. However, the first practical metamaterial was not constructed until 33 years later and it does produce Veselago's concepts.[1][3][6][7] Currently, negative-index metamaterials are being developed to manipulate electromagnetic radiation in new ways. For example, optical and electromagnetic properties of natural materials are often altered through chemistry. With metamaterials, optical and electromagnetic properties can be engineered by changing the geometry of its unit cells. The unit cells are materials that are ordered in geometric arrangements with dimensions that are fractions of the wavelength of the radiated electromagnetic wave. Each artificial unit responds to the radiation from the source. The collective result is the material's response to the electromagnetic wave that is broader than normal.[1][3][7] Subsequently, transmission is altered by adjusting the shape, size, and configurations of the unit cells. This results in control over material parameters known as permittivity and magnetic permeability. These two parameters (or quantities) determine the propagation of electromagnetic waves in matter. Therefore, controlling the values of permittivity and permeability means that the refractive index can be negative or zero as well as conventionally positive. It all depends on the intended application or desired result. So, optical properties can be expanded beyond the capabilities of lenses, mirrors, and other conventional materials. Additionally, one of the effects most studied is the negative index of refraction.[1][3][6][7] Reverse propagation When a negative index of refraction occurs, propagation of the electromagnetic wave is reversed. Resolution below the diffraction limit becomes possible. This is known as subwavelength imaging. Transmitting a beam of light via an electromagnetically flat surface is another capability. In contrast, conventional materials are usually curved, and cannot achieve resolution below the diffraction limit. Also, reversing the electromagnetic waves in a material, in conjunction with other ordinary materials (including air) could result in minimizing losses that would normally occur.[1][3][6][7] The reverse of the electromagnetic wave, characterized by an antiparallel phase velocity is also an indicator of negative index of refraction.[1][6] Furthermore, negative-index materials are customized composites. In other words, materials are combined with a desired result in mind. Combinations of materials can be designed to achieve optical properties not seen in nature. The properties of the composite material stem from its lattice structure constructed from components smaller than the impinging electromagnetic wavelength separated by distances that are also smaller than the impinging electromagnetic wavelength. Likewise, by fabricating such metamaterials researchers are trying to overcome fundamental limits tied to the wavelength of light.[1][3][7] The unusual and counterintuitive properties currently have practical and commercial use manipulating electromagnetic microwaves in wireless and communication systems. Lastly, research continues in the other domains of the electromagnetic spectrum, including visible light.[7][8] Materials The first actual metamaterials worked in the microwave regime, or centimeter wavelengths, of the electromagnetic spectrum (about 4.3 GHz). It was constructed of split-ring resonators and conducting straight wires (as unit cells). The unit cells were sized from 7 to 10 millimeters. The unit cells were arranged in a two-dimensional (periodic) repeating pattern which produces a crystal-like geometry. Both the unit cells and the lattice spacing were smaller than the radiated electromagnetic wave. This produced the first left-handed material when both the permittivity and permeability of the material were negative. This system relies on the resonant behavior of the unit cells. Below a group of researchers develop an idea for a left-handed metamaterial that does not rely on such resonant behavior. Research in the microwave range continues with split-ring resonators and conducting wires. Research also continues in the shorter wavelengths with this configuration of materials and the unit cell sizes are scaled down. However, at around 200 terahertz issues arise which make using the split ring resonator problematic. "Alternative materials become more suitable for the terahertz and optical regimes." At these wavelengths selection of materials and size limitations become important.[1][4][9][10] For example, in 2007 a 100 nanometer mesh wire design made of silver and woven in a repeating pattern transmitted beams at the 780 nanometer wavelength, the far end of the visible spectrum. The researchers believe this produced a negative refraction of 0.6. Nevertheless, this operates at only a single wavelength like its predecessor metamaterials in the microwave regime. Hence, the challenges are to fabricate metamaterials so that they "refract light at ever-smaller wavelengths" and to develop broad band capabilities.[11][12] Artificial transmission-line-media Main article: Metamaterial antenna In the metamaterial literature, medium or media refers to transmission medium or optical medium. In 2002, a group of researchers came up with the idea that in contrast to materials that depended on resonant behavior, non-resonant phenomena could surpass narrow bandwidth constraints of the wire/split-ring resonator configuration. This idea translated into a type of medium with broader bandwidth abilities, negative refraction, backward waves, and focusing beyond the diffraction limit. They dispensed with split-ring-resonators and instead used a network of L–C loaded transmission lines. In metamaterial literature this became known as artificial transmission-line media. At that time it had the added advantage of being more compact than a unit made of wires and split ring resonators. The network was both scalable (from the megahertz to the tens of gigahertz range) and tunable. It also includes a method for focusing the wavelengths of interest.[13] By 2007 the negative refractive index transmission line was employed as a subwavelength focusing free-space flat lens. That this is a free-space lens is a significant advance. Part of prior research efforts targeted creating a lens that did not need to be embedded in a transmission line.[14] The optical domain Main articles: Photonic metamaterial and plasmonic metamaterial Metamaterial components shrink as research explores shorter wavelengths (higher frequencies) of the electromagnetic spectrum in the infrared and visible spectrums. For example, theory and experiment have investigated smaller horseshoe shaped split ring resonators designed with lithographic techniques,[15][16] as well as paired metal nanorods or nanostrips,[17] and nanoparticles as circuits designed with lumped element models [18] Applications The science of negative-index materials is being matched with conventional devices that broadcast, transmit, shape, or receive electromagnetic signals that travel over cables, wires, or air. The materials, devices and systems that are involved with this work could have their properties altered or heightened. Hence, this is already happening with metamaterial antennas[19] and related devices which are commercially available. Moreover, in the wireless domain these metamaterial apparatuses continue to be researched. Other applications are also being researched. These are electromagnetic absorbers such as radar-microwave absorbers, electrically small resonators, waveguides that can go beyond the diffraction limit, phase compensators, advancements in focusing devices (e.g. microwave lens), and improved electrically small antennas.[20][21][22][23] In the optical frequency regime developing the superlens may allow for imaging below the diffraction limit. Other potential applications for negative-index metamaterials are optical nanolithography, nanotechnology circuitry, as well as a near field superlens (Pendry, 2000) that could be useful for biomedical imaging and subwavelength photolithography.[23] Manipulating permittivity and permeability Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal. To describe any electromagnetic properties of a given achiral material such as an optical lens, there are two significant parameters. These are permittivity, ${\displaystyle {ϵ}_r}$, and permeability, ${\displaystyle {\mu }_r}$, which allow accurate prediction of light waves traveling within materials, and electromagnetic phenomena that occur at the interface between two materials.[24] For example, refraction is an electromagnetic phenomenon which occurs at the interface between two materials. Snell's law states that the relationship between the angle of incidence of a beam of electromagnetic radiation (light) and the resulting angle of refraction rests on the refractive indices, ${\displaystyle n}$, of the two media (materials). The refractive index of an achiral medium is given by ${\displaystyle n=\pm \sqrt{{ϵ}_r{\mu }_r}}$.[25] Hence, it can be seen that the refractive index is dependent on these two parameters. Therefore, if designed or arbitrarily modified values can be inputs for ${\displaystyle {ϵ}_r}$ and ${\displaystyle {\mu }_r}$, then the behavior of propagating electromagnetic waves inside the material can be manipulated at will. This ability then allows for intentional determination of the refractive index.[24] 0:25 Video representing negative refraction of light at uniform planar interface. For example, in 1967, Victor Veselago analytically determined that light will refract in the reverse direction (negatively) at the interface between a material with negative refractive index and a material exhibiting conventional positive refractive index. This extraordinary material was realized on paper with simultaneous negative values for ${\displaystyle {ϵ}_r}$ and ${\displaystyle {\mu }_r}$, and could therefore be termed a double negative material. However, in Veselago's day a material which exhibits double negative parameters simultaneously seemed impossible because no natural materials exist which can produce this effect. Therefore, his work was ignored for three decades.[24] It was nominated for the Nobel Prize later. In general the physical properties of natural materials cause limitations. Most dielectrics only have positive permittivities, ${\displaystyle {ϵ}_r}$ > 0. Metals will exhibit negative permittivity, ${\displaystyle {ϵ}_r}$ < 0 at optical frequencies, and plasmas exhibit negative permittivity values in certain frequency bands. Pendry et al. demonstrated that the plasma frequency can be made to occur in the lower microwave frequencies for metals with a material made of metal rods that replaces the bulk metal. However, in each of these cases permeability remains always positive. At microwave frequencies it is possible for negative μ to occur in some ferromagnetic materials. But the inherent drawback is they are difficult to find above terahertz frequencies. In any case, a natural material that can achieve negative values for permittivity and permeability simultaneously has not been found or discovered. Hence, all of this has led to constructing artificial composite materials known as metamaterials in order to achieve the desired results.[24] Negative index of refraction due to chirality In case of chiral materials, the refractive index ${\displaystyle n}$ depends not only on permittivity ${\displaystyle {ϵ}_r}$ and permeability ${\displaystyle {\mu }_r}$, but also on the chirality parameter ${\displaystyle \kappa }$, resulting in distinct values for left and right circularly polarized waves, given by ${\displaystyle n=\pm \sqrt{{ϵ}_r{\mu }_r}\pm \kappa }$ A negative index will occur for waves of one circular polarization if ${\displaystyle \kappa }$ > ${\displaystyle \sqrt{{ϵ}_r{\mu }_r}}$. In this case, it is not necessary that either or both ${\displaystyle {ϵ}_r}$ and ${\displaystyle {\mu }_r}$ be negative to achieve a negative index of refraction. A negative refractive index due to chirality was predicted by Pendry[26] and Tretyakov et al.,[27] and first observed simultaneously and independently by Plum et al.[28] and Zhang et al.[29] in 2009. Physical properties never before produced in nature Theoretical articles were published in 1996 and 1999 which showed that synthetic materials could be constructed to purposely exhibit a negative permittivity and permeability.[note 1] These papers, along with Veselago's 1967 theoretical analysis of the properties of negative-index materials, provided the background to fabricate a metamaterial with negative effective permittivity and permeability.[30][31][32] See below. A metamaterial developed to exhibit negative-index behavior is typically formed from individual components. Each component responds differently and independently to a radiated electromagnetic wave as it travels through the material. Since these components are smaller than the radiated wavelength it is understood that a macroscopic view includes an effective value for both permittivity and permeability.[30] Composite material In the year 2000, David R. Smith's team of UCSD researchers produced a new class of composite materials by depositing a structure onto a circuit-board substrate consisting of a series of thin copper split-rings and ordinary wire segments strung parallel to the rings. This material exhibited unusual physical properties that had never been observed in nature. These materials obey the laws of physics, but behave differently from normal materials. In essence these negative-index metamaterials were noted for having the ability to reverse many of the physical properties that govern the behavior of ordinary optical materials. One of those unusual properties is the ability to reverse, for the first time, Snell's law of refraction. Until the demonstration of negative refractive index for microwaves by the UCSD team, the material had been unavailable. Advances during the 1990s in fabrication and computation abilities allowed these first metamaterials to be constructed. Thus, the "new" metamaterial was tested for the effects described by Victor Veselago 30 years earlier. Studies of this experiment, which followed shortly thereafter, announced that other effects had occurred.[5][30][31][33] With antiferromagnets and certain types of insulating ferromagnets, effective negative magnetic permeability is achievable when polariton resonance exists. To achieve a negative index of refraction, however, permittivity with negative values must occur within the same frequency range. The artificially fabricated split-ring resonator is a design that accomplishes this, along with the promise of dampening high losses. With this first introduction of the metamaterial, it appears that the losses incurred were smaller than antiferromagnetic, or ferromagnetic materials.[5] When first demonstrated in 2000, the composite material (NIM) was limited to transmitting microwave radiation at frequencies of 4 to 7 gigahertz (4.28–7.49 cm wavelengths). This range is between the frequency of household microwave ovens (~2.45 GHz, 12.23 cm) and military radars (~10 GHz, 3 cm). At demonstrated frequencies, pulses of electromagnetic radiation moving through the material in one direction are composed of constituent waves moving in the opposite direction.[5][33][34] The metamaterial was constructed as a periodic array of copper split ring and wire conducting elements deposited onto a circuit-board substrate. The design was such that the cells, and the lattice spacing between the cells, were much smaller than the radiated electromagnetic wavelength. Hence, it behaves as an effective medium. The material has become notable because its range of (effective) permittivity εeff and permeability μeff values have exceeded those found in any ordinary material. Furthermore, the characteristic of negative (effective) permeability evinced by this medium is particularly notable, because it has not been found in ordinary materials. In addition, the negative values for the magnetic component is directly related to its left-handed nomenclature, and properties (discussed in a section below). The split-ring resonator (SRR), based on the prior 1999 theoretical article, is the tool employed to achieve negative permeability. This first composite metamaterial is then composed of split-ring resonators and electrical conducting posts.[5] Initially, these materials were only demonstrated at wavelengths longer than those in the visible spectrum. In addition, early NIMs were fabricated from opaque materials and usually made of non-magnetic constituents. As an illustration, however, if these materials are constructed at visible frequencies, and a flashlight is shone onto the resulting NIM slab, the material should focus the light at a point on the other side. This is not possible with a sheet of ordinary opaque material.[1][5][33] In 2007, the NIST in collaboration with the Atwater Lab at Caltech created the first NIM active at optical frequencies. More recently (as of 2008), layered "fishnet" NIM materials made of silicon and silver wires have been integrated into optical fibers to create active optical elements.[35][36][37] Simultaneous negative permittivity and permeability Negative permittivity εeff < 0 had already been discovered and realized in metals for frequencies all the way up to the plasma frequency, before the first metamaterial. There are two requirements to achieve a negative value for refraction. First, is to fabricate a material which can produce negative permeability μeff < 0. Second, negative values for both permittivity and permeability must occur simultaneously over a common range of frequencies.[1][30] Therefore, for the first metamaterial, the nuts and bolts are one split-ring resonator electromagnetically combined with one (electric) conducting post. These are designed to resonate at designated frequencies to achieve the desired values. Looking at the make-up of the split ring, the associated magnetic field pattern from the SRR is dipolar. This dipolar behavior is notable because this means it mimics nature's atom, but on a much larger scale, such as in this case at 2.5 millimeters. Atoms exist on the scale of picometers. The splits in the rings create a dynamic where the SRR unit cell can be made resonant at radiated wavelengths much larger than the diameter of the rings. If the rings were closed, a half wavelength boundary would be electromagnetically imposed as a requirement for resonance.[5] The split in the second ring is oriented opposite to the split in the first ring. It is there to generate a large capacitance, which occurs in the small gap. This capacitance substantially decreases the resonant frequency while concentrating the electric field. The individual SRR depicted on the right had a resonant frequency of 4.845 GHz, and the resonance curve, inset in the graph, is also shown. The radiative losses from absorption and reflection are noted to be small, because the unit dimensions are much smaller than the free space, radiated wavelength.[5] When these units or cells are combined into a periodic arrangement, the magnetic coupling between the resonators is strengthened, and a strong magnetic coupling occurs. Properties unique in comparison to ordinary or conventional materials begin to emerge. For one thing, this periodic strong coupling creates a material, which now has an effective magnetic permeability μeff in response to the radiated-incident magnetic field.[5] Composite material passband Graphing the general dispersion curve, a region of propagation occurs from zero up to a lower band edge, followed by a gap, and then an upper passband. The presence of a 400 MHz gap between 4.2 GHz and 4.6 GHz implies a band of frequencies where μeff < 0 occurs. (Please see the image in the previous section) Furthermore, when wires are added symmetrically between the split rings, a passband occurs within the previously forbidden band of the split ring dispersion curves. That this passband occurs within a previously forbidden region indicates that the negative εeff for this region has combined with the negative μeff to allow propagation, which fits with theoretical predictions. Mathematically, the dispersion relation leads to a band with negative group velocity everywhere, and a bandwidth that is independent of the plasma frequency, within the stated conditions.[5] Mathematical modeling and experiment have both shown that periodically arrayed conducting elements (non-magnetic by nature) respond predominantly to the magnetic component of incident electromagnetic fields. The result is an effective medium and negative μeff over a band of frequencies. The permeability was verified to be the region of the forbidden band, where the gap in propagation occurred – from a finite section of material. This was combined with a negative permittivity material, εeff < 0, to form a “left-handed” medium, which formed a propagation band with negative group velocity where previously there was only attenuation. This validated predictions. In addition, a later work determined that this first metamaterial had a range of frequencies over which the refractive index was predicted to be negative for one direction of propagation (see ref #[1]). Other predicted electrodynamic effects were to be investigated in other research.[5] Describing a left-handed material A comparison of refraction in a negative-index metamaterial to that in a conventional material having the same, but positive refractive index. The incident beam θ enters from air and refracts in a normal (θ') or metamaterial (-θ'). From the conclusions in the above section a left-handed material (LHM) can be defined. It is a material which exhibits simultaneous negative values for permittivity, ε, and permeability, μ, in an overlapping frequency region. Since the values are derived from the effects of the composite medium system as a whole, these are defined as effective permittivity, εeff, and effective permeability, μeff. Real values are then derived to denote the value of negative index of refraction, and wave vectors. This means that in practice losses will occur for a given medium used to transmit electromagnetic radiation such as microwave, or infrared frequencies, or visible light – for example. In this instance, real values describe either the amplitude or the intensity of a transmitted wave relative to an incident wave, while ignoring the negligible loss values.[4][5] Isotropic negative index in two dimensions In the above sections first fabricated metamaterial was constructed with resonating elements, which exhibited one direction of incidence and polarization. In other words, this structure exhibited left-handed propagation in one dimension. This was discussed in relation to Veselago's seminal work 33 years earlier (1967). He predicted that intrinsic to a material, which manifests negative values of effective permittivity and permeability, are several types of reversed physics phenomena. Hence, there was then a critical need for a higher-dimensional LHMs to confirm Veselago's theory, as expected. The confirmation would include reversal of Snell's law (index of refraction), along with other reversed phenomena. In the beginning of 2001 the existence of a higher-dimensional structure was reported. It was two-dimensional and demonstrated by both experiment and numerical confirmation. It was an LHM, a composite constructed of wire strips mounted behind the split-ring resonators (SRRs) in a periodic configuration. It was created for the express purpose of being suitable for further experiments to produce the effects predicted by Veselago.[4] Experimental verification of a negative index of refraction Split-ring resonator consisting of an inner square with a split on one side embedded in an outer square with a split on the other side. Split-ring resonators are on the front and right surfaces of the square grid, and single vertical wires are on the back and left surfaces.[5] A theoretical work published in 1967 by Soviet physicist Victor Veselago showed that a refractive index with negative values is possible and that this does not violate the laws of physics. As discussed previously (above), the first metamaterial had a range of frequencies over which the refractive index was predicted to be negative for one direction of propagation. It was reported in May 2000.[1][6][38] In 2001, a team of researchers constructed a prism composed of metamaterials (negative-index metamaterials) to experimentally test for negative refractive index. The experiment used a waveguide to help transmit the proper frequency and isolate the material. This test achieved its goal because it successfully verified a negative index of refraction.[1][6][39][40][41][42][43] The experimental demonstration of negative refractive index was followed by another demonstration, in 2003, of a reversal of Snell's law, or reversed refraction. However, in this experiment negative index of refraction material is in free space from 12.6 to 13.2 GHz. Although the radiated frequency range is about the same, a notable distinction is this experiment is conducted in free space rather than employing waveguides.[44] Furthering the authenticity of negative refraction, the power flow of a wave transmitted through a dispersive left-handed material was calculated and compared to a dispersive right-handed material. The transmission of an incident field, composed of many frequencies, from an isotropic nondispersive material into an isotropic dispersive media is employed. The direction of power flow for both nondispersive and dispersive media is determined by the time-averaged Poynting vector. Negative refraction was shown to be possible for multiple frequency signals by explicit calculation of the Poynting vector in the LHM.[45] Fundamental electromagnetic properties of the NIM In a slab of conventional material with an ordinary refractive index – a right-handed material (RHM) – the wave front is transmitted away from the source. In a NIM the wavefront travels toward the source. However, the magnitude and direction of the flow of energy essentially remains the same in both the ordinary material and the NIM. Since the flow of energy remains the same in both materials (media), the impedance of the NIM matches the RHM. Hence, the sign of the intrinsic impedance is still positive in a NIM.[46][47] Light incident on a left-handed material, or NIM, will bend to the same side as the incident beam, and for Snell's law to hold, the refraction angle should be negative. In a passive metamaterial medium this determines a negative real and imaginary part of the refractive index.[3][46][47] Negative refractive index in left-handed materials The left-handed orientation is shown on the left, and the right-handed on the right. In 1968 Victor Veselago's paper showed that the opposite directions of EM plane waves and the flow of energy was derived from the individual Maxwell curl equations. In ordinary optical materials, the curl equation for the electric field show a "right hand rule" for the directions of the electric field E, the magnetic induction B, and wave propagation, which goes in the direction of wave vector k. However, the direction of energy flow formed by E × H is right-handed only when permeability is greater than zero. This means that when permeability is less than zero, e.g. negative, wave propagation is reversed (determined by k), and contrary to the direction of energy flow. Furthermore, the relations of vectors E, H, and k form a "left-handed" system – and it was Veselago who coined the term "left-handed" (LH) material, which is in wide use today (2011). He contended that an LH material has a negative refractive index and relied on the steady-state solutions of Maxwell's equations as a center for his argument.[48] After a 30-year void, when LH materials were finally demonstrated, it could be said that the designation of negative refractive index is unique to LH systems; even when compared to photonic crystals. Photonic crystals, like many other known systems, can exhibit unusual propagation behavior such as reversal of phase and group velocities. But, negative refraction does not occur in these systems, and not yet realistically in photonic crystals.[48][49][50] Negative refraction at optical frequencies The negative refractive index in the optical range was first demonstrated in 2005 by Shalaev et al. (at the telecom wavelength λ = 1.5 μm)[17] and by Brueck et al. (at λ = 2 μm) at nearly the same time.[51] In 2006, a Caltech team led by Lezec, Dionne, and Atwater achieved negative refraction in the visible spectral regime.[52][53][54] Experimental verification of reversed Cherenkov radiation Besides reversed values for index of refraction, Veselago predicted the occurrence of reversed Cherenkov radiation (also known simply as CR) in a left-handed medium. In 1934 Pavel Cherenkov discovered a coherent radiation that occurs when certain types of media are bombarded by fast moving electron beams. In 1937 a theory built around CR stated that when charged particles, such as electrons, travel through a medium at speeds faster than the speed of light in the medium only then will CR radiate. As the CR occurs, electromagnetic radiation is emitted in a cone shape, fanning out in the forward direction. CR and the 1937 theory has led to a large array of applications in high energy physics. A notable application are the Cherenkov counters. These are used to determine various properties of a charged particle such as its velocity, charge, direction of motion, and energy. These properties are important in the identification of different particles. For example, the counters were applied in the discovery of the antiproton and the J/ψ meson. Six large Cherenkov counters were used in the discovery of the J/ψ meson. It has been difficult to experimentally prove the reversed Cherenkov radiation.[55][56] Other optics with NIMs Theoretical work, along with numerical simulations, began in the early 2000s on the abilities of DNG slabs for subwavelength focusing. The research began with Pendry's proposed "Perfect lens." Several research investigations that followed Pendry's concluded that the "Perfect lens" was possible in theory but impractical. One direction in subwavelength focusing proceeded with the use of negative-index metamaterials, but based on the enhancements for imaging with surface plasmons. In another direction researchers explored paraxial approximations of NIM slabs.[3] Implications of negative refractive materials The existence of negative refractive materials can result in a change in electrodynamic calculations for the case of permeability μ = 1 . A change from a conventional refractive index to a negative value gives incorrect results for conventional calculations, because some properties and effects have been altered. When permeability μ has values other than 1 this affects Snell's law, the Doppler effect, the Cherenkov radiation, Fresnel's equations, and Fermat's principle.[10] The refractive index is basic to the science of optics. Shifting the refractive index to a negative value may be a cause to revisit or reconsider the interpretation of some norms, or basic laws.[23] US patent on left-handed composite media The first US patent for a fabricated metamaterial, titled "Left handed composite media" by David R. Smith, Sheldon Schultz, Norman Kroll and Richard A. Shelby, was issued in 2004. The invention achieves simultaneous negative permittivity and permeability over a common band of frequencies. The material can integrate media which is already composite or continuous, but which will produce negative permittivity and permeability within the same spectrum of frequencies. Different types of continuous or composite may be deemed appropriate when combined for the desired effect. However, the inclusion of a periodic array of conducting elements is preferred. The array scatters electromagnetic radiation at wavelengths longer than the size of the element and lattice spacing. The array is then viewed as an effective medium.[57] See also History of metamaterials Superlens Metamaterial cloaking Photonic metamaterials Metamaterial antenna Nonlinear metamaterials Photonic crystal Seismic metamaterials Split-ring resonator Acoustic metamaterials Metamaterial absorber Metamaterial Plasmonic metamaterials Terahertz metamaterials Tunable metamaterials Transformation optics Theories of cloaking Academic journals Metamaterials Metamaterials books Metamaterials Handbook Metamaterials: Physics and Engineering Explorations https://en.wikipedia.org/wiki/Negative-index_metamaterial

Split-ring resonator Terahertz metamaterials Theories of cloaking Transformation optics Tunable metamaterials Plasmonic lens Academic journals Metamaterials (journal) Metamaterials books Metamaterials Handbook Metamaterials: Physics and Engineering Explorations

Metamaterials scientists Nader Engheta Ulf Leonhardt John Pendry Vladimir Shalaev David R. Smith Sergei Tretyakov (scientist) Richard W. Ziolkowski