applies to a magnetodielectric medium. However, if both media are dielectrics i.e. μr1 = μr2 = 1 then sin θB → ∞. Thus, the Brewster angle does not exist at the interface of two dielectric media. The Brewster angle could exit for the case (b). Such a material is difficult to get in practice. However, the situation is different for the TM‐polarized plane wave.
TM‐Polarization
In the case of the TM‐polarization, again the reflection is zero, i.e.
(5.3.2)
Unlike the TE polarization, the Brewster angle of the TM‐polarized obliquely incident wave exists even at the interface of the nonmagnetic dielectric medium. Thus, the Brewster angle could be used to separate the TE and TM polarizations from the obliquely incident unpolarized wave at the interface of two dielectric media, as only the TE‐polarized wave component will get reflected. A Brewster angle is also called the polarizing angle.
5.3.2 Critical Angle
At the critical angle of incidence (θ1 = θc), complete reflection occurs at the interface. It occurs for both the TE and TM polarizations. The transmission line model helps to understand it. In the case of a terminated line, total reflection occurs for the load impedance ZL = 0, ∞ , ± jX. In the first case, the line is short‐circuited, i.e. terminated in a PEC with εr → ∞, in the second case, the line is open‐circuited, i.e. terminated in a PMC with μr → ∞; in the third case, the line is terminated in a RIS, either inductive or capacitive. The corresponding surface, i.e. the interface, is a PEC, or PMC, or RIS. These surfaces are further discussed in chapter 20 for the artificially engineered periodic surfaces known as the electromagnetic bandgap (EBG) surfaces. By varying the angle of incidence of both polarizations with respect to the critical angle of incidence, the reflection and transmission of waves could be controlled.
Figure (5.3) shows that in the case TM – polarized wave, total reflection at the critical angle, is obtained by taking the load impedance at the interface (x = 0+) zero. Likewise, for the TE – polarization shown in Fig (5.2), total reflection at a critical angle is obtained for the infinite load impedance at the interface. Using equations, the following conditions are obtained: for TM‐polarization, ZL = η2 cos θ2 = 0; for TE polarization, ZL = − η2/ cos θ2 = ∞. Both cases give the following expression for the critical angle θ1 = θc and Snell's Law of refraction:
For the real value of the critical angle θc, permittivities of media are taken as εr1 > εr2. Thus, the critical angle of the interface is a fixed quantity. The angle of refraction θ2 deciding the direction of wave propagation in medium #2 is a function of angle incidence θ1. Figure (5.5a–c), applicable to for both the TM and TE polarizations, consider three cases, θ1 < θc, θ1 = θc, θ1 > θc, of propagation for the obliquely incident plane wave.
Case #1: θ1 < θc
In this case, equation (5.3.3c) provides the real value of the angle θ2 as sinθ2 < 1 corresponding to an angle of incidence θ1. Figure (5.5a) shows the reflection and transmission of the obliquely incident plane wave at the interface for the case εr1 > εr2.
Case#2: θ1 = θc
Figure (5.5b) shows the case for θ1 = θc. The angle of refraction θ2 is obtained on substituting sinθ1 = sin θc in equation (5.3.3c):
(5.3.4)
Figure (5.5b) shows that the refracted wave travels along the interface x = 0+ in the y‐direction and no component of the refracted (transmitted) wave propagates in the medium #2. Equation (5.2.28a) also shows that for the TM‐polarized incident wave, the reflection coefficient is
In the case of the TE polarization of the obliquely incident wave, equation (5.2.27a,b) shows the reflection coefficient
Case#3: θ1 > θc
Figure (5.5c) shows the case for θ1 > θc. Equation (5.3.3c) of Snell's refraction law shows that sinθ2 > 1. Therefore, there is no real solution to the angle θ2. However, the wave propagation in medium #2 requires sinθ2 and cosθ2, not the value of the angle of refraction θ2. The sinθ2 and cosθ2 could be obtained from Snell's Law given by
