# Total Reflection and Brewster’s Angle¶

Purpose

We first identify total reflection and brewster angle for a dielectric media, then relate them to conductive medium.

## Total Reflection¶

For a perfect dielectric, the conductivity is zero and the permeability is that of free space that is, \(\mu_1=\mu_2=\mu_0\). In this case, Snell’s law reduces to:

where \(n_{12}\) is the relative index of refraction. If \(\epsilon_2 > \epsilon_1\) then \(n_{12} > 1\). Under this situation, for any angle of incidence \(\theta_i\) there is a real angle of transimission \(\theta_t\). On the other hand, if \(\epsilon_2 < \epsilon_1\) then \(\theta_t\) is real only when \(n_{12} \text{sin} \theta_t \leq 1\). Total reflection occurs when \(n_{12}\text{sin} \theta_t > 1\), and indicates that the wave cannot pass through and is entirely reflected. For the reflection from a conductive surface, a total reflection occurs when \(\sigma_1 > \sigma_2\). Fig. 79 illustrate this.

## Brewster’s Angle¶

From derived reflection coefficients for TE mode in Fresnel Equations, the reflection coefficient for perfect dielectric can be written as

With Snell’s law of refraction shown in Eq. (1), above equation can be modified as

Similarly, the reflection coefficient for TM mode can be obtained as

Power reflection coefficient for TE and TM mode can be

Accordingly, power transmission coefficient will be

If \((\theta_t + \theta_i) \rightarrow \pi/2\), then \(\text{tan}(\theta_t + \theta_i) \rightarrow \infty\), and \(r_{TM} \rightarrow 0\). The reflected and refracted waves are normal to one another, and

so that Eq. (1) becomes

The angle that this equation satisfies is known as the Brewster angle. The reflection from a conductive surface, there will be a minimum in \(R_{TM}\), analogous to the Brewster angle, for some particular angle of incidence. No such minimul occurs in \(R_{TE}\). Fig. 80 illustrate this.