# Angle of incidence effects

The reflection and refraction of plane waves at a boundary between two media of differing properties are well known, following Snell's law and the Fresnel formulas. A plane wave incident on a dielectric discontinuity is split into two waves; the transmitted wave proceeding into the second medium and the reflected wave propagating back into the incident medium from which the following relation is derived.

*n _{1} sin θ_{i }*=

*n*=

_{1}sin θ_{r}*n*(2-30)

_{2}sin θ_{t}where θ* _{i}*,

*θ*, and θ

_{r}*are the incident , reflected and transmitted angles, and*

_{t}*n*&

_{1}*n*are the refractive indices of the incident and transmitted mediums. As (θ

_{2}_{r}= θ

_{i}) then the transmitted angle θ

_{t}into the second medium is defined as :

Although this is true for all forms of electromagnetic wave propagation, the dynamic properties of reflected and transmitted waves, such as intensity, phase changes, polarization effects depend entirely upon the specific nature of the wave propagation and the interface conditions. At an uncoated substrate boundary with a plane wave incident at an oblique angle, the electric and magnetic field vectors are split into two polarization components that are parallel (*p*) and perpendicular (*s*) to the incident plane. Both the transmitted and reflected polarization components can be calculated for each orientation separately and then combined to produce a resultant mean polarization effect.

The *p*-wave is also known as a TM wave, (as the magnetic field vector H is transverse to the plane of incidence), and the *s*-wave is alternatively known as a TE wave, (as the electric field vector E is transverse to the plane of incidence). The Fresnel reflection and transmission coefficient formulas for these *s* and *p* polarizations are :

*s-*polarization

*p-*polarization

These formulae give the ratio of the amplitude of the reflected and transmitted waves relative to the amplitude of the incident wave. The total energy reflected from the boundary and transmitted into the substrate is the square of the Fresnel coefficients.

The Fresnel coefficients r* _{s}* , r

*, t*

_{p}*, t*

_{s}*change differently as a function of the angle of incidence, with the reflectance of the*

_{p}*s*wave always being greater than the

*p*wave. The reflectance of the

*p*-polarization falls to zero at a definite angle (Brewster's angle). At this particular angle, the result of the Fresnel reflectance (r

*) and refracted transmission (t*

_{p}*) waves are at an angle of 90° to each other which produces a reflected beam which is plane polarized in the plane of incidence with oscillations parallel to the surface, and electric vector perpendicular to the plane of polarization. The angle at which this occurs is given by θ*

_{p}*= tan*

_{B}^{-1}

*n*/

_{2}*n*, which for Ge, Si, CdTe, ZnSe and ZnS are at angles of 76.0°, 73.6°, 69.5°, 67.4°, and 65.6° respectively.

_{1}### Substrate optical theory

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