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Wednesday, May 10, 2023

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.

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

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 ni of the medium from which the radiation is incident (ni 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 ni + 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

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"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

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 Schematic diagram explaining the Fata Morgana mirage

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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

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.

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

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

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

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

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

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

References


  • 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.

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

    Further reading

    External links

    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

    Non-atmospheric optical phenomena

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

    Other optical effects

    Entoptic phenomena

    Optical illusions

    Unexplained phenomena

    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]

    See also

    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 (av). 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: , 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).  

    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 Ez = Hz = 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:

    and

    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 ex, ey, hx, and hy are complex numbers. In the second more compact form, as these equations are customarily expressed, these factors are described using the wavenumber and angular frequency (or "radian frequency") . In a more general formulation with propagation not restricted to the +z direction, then the spatial dependence kz is replaced by where 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 , h is related to e by:

    and

    .

    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 ]

    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 ex and ey describing the electric field. The vector containing ex and ey (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:

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

    where the polarization state is now fully parameterized by the value of Q (such that −1 < Q < 1) and the relative phase .

    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

    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.

    Polarisation state - Linear polarization parallel to x axis.svg
    Polarisation state - Linear polarization oriented at +45deg.svg

    Polarisation state - Right-elliptical polarization A.svg

    Polarisation state - Right-circular polarization.svg

    Polarisation state - Left-circular polarization.svg

    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

    Polarisation ellipse2.svg

    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 or the ellipticity angle, 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

    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):

    Here and denote the amplitude of the wave in the two components of the electric field vector, while and 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

    0:31
    Electromagnetic vectors for , and with along with 3 planar projections and a deformation surface of total electric field. The light is always s-polarized in the xy plane. is the polar angle of and is the azimuthal angle of .

    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

    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:

    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 . 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:

    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

    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

    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

    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

    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

    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

    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

    References

    Cited references


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  • 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.

    1. 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

     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

    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

    See also

     

    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 Nth bright ring is given by

    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

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

    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


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

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

    Further reading

    External links

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

     

    In mathematics, physics, and art, moiré patterns (UK: /ˈmwɑːr/ MWAR-ay, US: /mwɑːˈr/ 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

    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

    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

    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

    that is
    The distance d between the middle of a pale zone and a dark zone is
    the distance between the middle of two dark zones, which is also the distance between two pale zones, is
    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

    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 , the distance increment Δx per intensity cycle (the wavelength) obtains when k Δx = 2π, or Δx = /k.

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

    such that k1k2.

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

    where it is easily shown that

    and

    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:

    that is:
    thus

    Effect on curved lines

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

    thus

    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

    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

    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

     

     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. 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 no (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 no. 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 no, whereas the refractive index of the extraordinary ray will be in between no and ne, depending on the ray direction as described by the index ellipsoid. The magnitude of the difference is quantified by the birefringence:[verification needed]

    The propagation (as well as reflection coefficient) of the ordinary ray is simply described by no 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 no and ne). 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 ne) 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 nα, nβ and nγ. 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, nα, nβ and nγ, along three coordinate axes; in this case two are equal. So if nα = nβ corresponding to the x and y axes, then the extraordinary index is nγ 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 ne is greater than the ordinary index no. Negative birefringence means that Δn = neno 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:

    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.

    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:

     

     

     

     

    (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:

     

     

     

     

    (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:

     

     

     

     

    (3a)

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

     

     

     

     

    (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 ikx to find:

     

     

     

     

    (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 , eq. 3a then becomes:

     

     

     

     

    (4a)

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

     

     

     

     

    (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:

     

     

     

     

    (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

     

     

     

     

    (5a)

    where Ex, Ey, Ez are the components of E (at any given position in space and time) and kx, ky, kz are the components of k. Rearranging, we can write (and similarly for the y and z components of eq. 4a)

     

     

     

     

    (5b)

     

     

     

     

    (5c)

     

     

     

     

    (5d)

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

     

     

     

     

    (6)

    Evaluating the determinant of eq. 6, and rearranging the terms according to the powers of , the constant terms cancel. After eliminating the common factor from the remaining terms, we obtain

     

     

     

     

    (7)

    In the case of a uniaxial material, choosing the optic axis to be in the z direction so that nx = ny = no and nz = ne, this expression can be factored into

     

     

     

     

    (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 no 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 no and ne, 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), nx, ny and nz 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

    Notes


    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:

     

    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:

    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

    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


  • 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

    1. 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

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