Total Reflection and Brewster’s Angle


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:

(325)\[\frac{\text{sin} \theta_i}{\text{sin} \theta_t} = \frac{k_1}{k_2} = \Big(\frac{\epsilon_2}{\epsilon_1}\Big)^{1/2} = n_{12}\]

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.


Fig. 79 Tranmission angle \(\theta_t\) as a function of the incident angle \(\theta_i\) when \(\sigma_1\) = 1 S/m and \(\sigma_2\) = 0.1 S/m. Magnetic permeability and dielectric permittivitivy assumed to be those of free-space (\(\epsilon = \epsilon_0\) and \(\mu = \mu_0\))

Brewster’s Angle

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

\[r_{TE} = \frac{(\epsilon_1)^{1/2} \text{cos} \theta_i - (\epsilon_2)^{1/2} \text{cos} \theta_t}{(\epsilon_1)^{1/2} \text{cos} \theta_i + (\epsilon_2)^{1/2} \text{cos} \theta_t}\]

With Snell’s law of refraction shown in Eq. (325), above equation can be modified as

\[r_{TE} = \frac{\text{cos} \theta_i \text {sin} \theta_t - \text{cos} \theta_t \text {sin} \theta_i}{\text{cos} \theta_i \text {sin} \theta_t + \text{cos} \theta_t \text {sin} \theta_i} = \frac{\text {sin} (\theta_t - \theta_i)}{\text {sin}(\theta_t + \theta_i)}\]

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

\[r_{TM} = \frac{\text {tan} (\theta_t - \theta_i)}{\text {tan}(\theta_t + \theta_i)}\]

Power reflection coefficient for TE and TM mode can be

\[R_{TE} \equiv |r_{TE}|^2 = \frac{\text {sin}^2 (\theta_t - \theta_i)}{\text {sin}^2(\theta_t + \theta_i)}\]
\[R_{TM} \equiv |r_{TM}|^2 = \frac{\text {tan}^2 (\theta_t - \theta_i)}{\text {tan}^2(\theta_t + \theta_i)}\]

Accordingly, power transmission coefficient will be

\[T_{TE} \equiv 1-|r_{TE}|^2\]
\[T_{TM} \equiv 1-|r_{TM}|^2\]

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

\[\text {sin} \theta_t = \text {sin} (\pi/2 - \theta_i) = \text {cos} \theta_i\]

so that Eq. (325) becomes

\[\text {tan} \theta_i = \Big(\frac{\epsilon_2}{\epsilon_1}\Big)^{1/2} = n_{12}\]

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.


Fig. 80 The power reflection coefficient \(R_{TE}\) and \(R_{TM}\) versus angle of incidence for plane wave at air-earth interface. The conductivity and dielectric permitivity of earth are taken to be 0.01 S/m, and \(\epsilon = \epsilon_0\), respectively. The frequency is 6 x 10 5 Hz.


Fig. 79 and Fig. 80 are generated by the Reflection and Refraction app that you can adjust conductivity of each medium, and obtain corresponding transmission angle, power reflection and transmission coefficients as a function of incident angle. Below link will direct you to the app: