Nikiya Anton Bettey 2021: 09-28-2021-1908

Wednesday, September 29, 2021

09-28-2021-1908 - emission angle

Emission angle[edit]

The geometry of the Cherenkov radiation shown for the ideal case of no dispersion.

In the figure on the geometry, the particle (red arrow) travels in a medium with speed $v_{\text{p}}$ such that

{\frac {c}{n}}<v_{\text{p}}<c,

where

c

is speed of light in vacuum, and

n

is the refractive index of the medium. If the medium is water, the condition is

0.75c<v_{\text{p}}<c

, since

n\approx 1.33

for water at 20 °C.

We define the ratio between the speed of the particle and the speed of light as

\beta ={\frac {v_{\text{p}}}{c}}.

The emitted light waves (denoted by blue arrows) travel at speed

v_{\text{em}}={\frac {c}{n}}.

The left corner of the triangle represents the location of the superluminal particle at some initial moment (t = 0). The right corner of the triangle is the location of the particle at some later time t. In the given time t, the particle travels the distance

x_{\text{p}}=v_{\text{p}}t=\beta \,ct

whereas the emitted electromagnetic waves are constricted to travel the distance

x_{\text{em}}=v_{\text{em}}t={\frac {c}{n}}t.

So the emission angle results in

\cos \theta ={\frac {1}{n\beta }}

Arbitrary emission angle[edit]

Cherenkov radiation can also radiate in an arbitrary direction using properly engineered one dimensional metamaterials.^[12] The latter is designed to introduce a gradient of phase retardation along the trajectory of the fast travelling particle ( $d\phi /dx$ ), reversing or steering Cherenkov emission at arbitrary angles given by the generalized relation:

\cos \theta ={\frac {1}{n\beta }}+{\frac {n}{k_{0}}}\cdot {\frac {d\phi }{dx}}

Note that since this ratio is independent of time, one can take arbitrary times and achieve similar triangles. The angle stays the same, meaning that subsequent waves generated between the initial time t = 0 and final time t will form similar triangles with coinciding right endpoints to the one shown.

Reverse Cherenkov effect[edit]

A reverse Cherenkov effect can be experienced using materials called negative-index metamaterials (materials with a subwavelength microstructure that gives them an effective "average" property very different from their constituent materials, in this case having negative permittivity and negative permeability). This means that, when a charged particle (usually electrons) passes through a medium at a speed greater than the phase velocity of light in that medium, that particle emits trailing radiation from its progress through the medium rather than in front of it (as is the case in normal materials with, both permittivity and permeability positive).^[13] One can also obtain such reverse-cone Cherenkov radiation in non-metamaterial periodic media where the periodic structure is on the same scale as the wavelength, so it cannot be treated as an effectively homogeneous metamaterial.^[10]

In a vacuum[edit]

The Cherenkov effect can occur in vacuum.^[14] In a slow-wave structure, like in a traveling-wave tube (TWT), the phase velocity decreases and the velocity of charged particles can exceed the phase velocity while remaining lower than $c$ . In such a system, this effect can be derived from conservation of the energy and momentum where the momentum of a photon should be $p=\hbar \beta$ ( $\beta$ is phase constant)^[15] rather than the de Broglie relation $p=\hbar k$ . This type of radiation (VCR) is used to generate high-power microwaves.^[16]

https://en.wikipedia.org/wiki/Cherenkov_radiation#Emission_angle

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