Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 219
5.3.2 Critical Angle
ОглавлениеAt 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.
