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Chapter 2.7 - Propagation of Light

2.7   PROPAGATION OF LIGHT

As already described, when light enters matter, it is reflected by its surface and refracted
by the matter, and its velocity changes as well as its wavelength, but not its
frequency.


2.7.1   Reflection and Refraction—Snell’s Law


Index of refraction of a transparent medium (nmed) is defined as the ratio of the
speed of monochromatic light in free space, c, over the speed of the same monochromatic
light in a medium (vmed):

 nmed = c/vmed

Then, between two mediums (1 and 2) the following relationships are true:

 n2/n1 = v1/v2

and

 n1 cos β = n2 cos α

where n1, v1, and n2, v2 are the index of refraction and speed of light in the two media,
and α and β are the angle of incidence and angle of refraction, respectively.

When the angle of incidence is very small, cos α = 1 – α2/2 and the cosine equation
becomes

 n1(1 – β2/2) = n2(1 – α2/2)

The index of refraction, or refractive index, for free space has the numerical value
of 1, whereas for other materials it is typically between 1 and 2, and in some cases
greater than 2 or 3.

The reflected portion of monochromatic light is known as Fresnel reflection. The
amount of reflected power as well as the polarization state of the reflected light depends
on the polarization state of the incident light, on the angle of incidence, and
on the refractive index difference.

For normal incidence on a single surface the reflectivity, ρ, is given by the Fresnel
equation:

 ρ = (n – 1)2/(n + 1)2

If the absorption of the material over a length d is A, which is calculated from the
absorption coefficient (absorbed power per centimeter) α, then the internal material
transmittance, τi, is defined as the inverse of the material absorption. For internal
transmittance τi, the external input–output transmittance (taking into account the reflectivity,
ρ, at the surface) is given by

 τ = [(1 – ρ)2τ1]/[1 – ρ2τ12]

The following basic relationships are also useful:

 Speed of light in free space:c = λf
 Speed of light in medium:vmed = λmed f
 Index of refraction:n1/n2 = λ2/λ1

where f is the frequency of light. Both letters f and v are used for frequency. Here,
we use f to eliminate confusion between v (for speed) and v (for frequency).

Snell’s law relates the ratio of the index of refraction with the angle of the incident
i) and refracted (Θt) rays:

 n2/n1 = sin Θi/sin Θt

The critical angle, Θcritical, is the (maximum) angle of incidence of light (from a
material with high to low refractive index) at which light stops being refracted and
is totally reflected. As the angle of incidence approaches the critical angle, the refracted
ray becomes parallel to the surface (without added phase shift) and is said to
be evanescent. Beyond that point, there is no refracted ray. The critical angle depends
on the refractive index and the wavelength of light:

 sin Θcritical = n1/n2

for n1 = 1 (air), then,

 sin Θcritical = 1/n2

In certain cases, a gradual variation of refractive index may take place. When light
rays enter from one side, then rays are refracted such that they may emerge from the
same side from which they entered. This is the case of the natural phenomenon
known as a mirage.

 

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