Introduction To Modern Planar Transmission Lines. Anand K. Verma

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wave is completely reflected by the interface of two dielectric media. This kind of total reflection is superior to that of the total reflection from any metallic surface that always introduces loss. We consider these two special cases of the angle of incidence for both the TE and TM polarizations.

      

      5.3.1 Brewster Angle

      The Brewster angle of incidence could be obtained for both the TE and TM polarizations.

      TE‐Polarization

In the case of the TE‐polarization, at an incident angle θ₁ = θ_B, i.e. at Brewster angle the reflection coefficient . So from equation (5.2.8c), and also using the Snell's refraction law (5.2.7b), the following expression for Brewster angle is obtained:

      (5.3.1)

      The above relation applies to a magnetodielectric medium. However, if both media are dielectrics i.e. μ_r1 = μ_r2 = 1 then sin θ_B → ∞. 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. , at θ₁ = θ_B. Using equation (5.2.16c), and the Snell's refraction law; the Brewster angle is obtained:

      (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 = θ_c), 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 Z_L = 0, ∞ , ± jX. In the first case, the line is short‐circuited, i.e. terminated in a PEC with ε_r → ∞, in the second case, the line is open‐circuited, i.e. terminated in a PMC with μ_r → ∞; 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, Z_L = η₂ cos θ₂ = 0; for TE polarization, Z_L = − η₂/ cos θ₂ = ∞. Both cases give the following expression for the critical angle θ₁ = θ_c and Snell's Law of refraction:

(5.3.3)

For the real value of the critical angle θ_c, permittivities of media are taken as ε_r1 > ε_r2. 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 < θ_c, θ₁ = θ_c, θ₁ > θ_c, of propagation for the obliquely incident plane wave.

      Case #1: θ₁ < θ_c

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 ε_r1 > ε_r2.

      Case#2: θ₁ = θ_c

      Figure (5.5b) shows the case for θ₁ = θ_c. The angle of refraction θ₂ is obtained on substituting sinθ₁ = sin θ_c 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 at the angle of refraction θ₂ = π/2. It demonstrates that under the total reflection condition, occurring at the critical angle θ₁ = θ_c, the interface behaves as a PEC for the TM‐polarized incident wave. However, equation (5.2.28b) provides also a transmission coefficient . It shows that the transmitted wave in the medium #2, confined at the interface, travels along the y‐axis at the interface x = 0⁺. The transmitted field components at an angle θ₂ = π/2 in the medium #2 are given in equation (5.2.18).

In the case of the TE polarization of the obliquely incident wave, equation (5.2.27a,b) shows the reflection coefficient Скачать книгу