Introduction To Modern Planar Transmission Lines. Anand K. Verma

rel="nofollow" href="#fb3_img_img_6f5143b2-6ca8-57a4-a7bb-9e58a435c949.png" alt="images"/>. In this case, the interface acts like a PMC. Again, the transmitted wave in the medium #2 propagates in the y‐direction at the interface x = 0⁺. It is seen from the field equation (5.2.10) for θ₂ = π/2.

      Case#3: θ₁ > θ_c

Figure (5.5c) shows the case for θ₁ > θ_c. 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 equation (5.2.7c), also from the wavevector k₂ and its x and y‐directed components k_2x and k_2y using equation (5.2.1):

Figure 5.5 Oblique incidence of plane wave at three different angles of incidence.

(5.3.5)

The negative (−) value is selected in further computation as it provides exponentially decaying fields with distance from the interface in the medium #2. The choice of positive (+) value results in the exponentially growing field with the distance that is physically not possible in a usual passive medium. The expressions (5.3.5b,d,e) are used for both the TE and TM polarization to compute the reflection and transmission coefficients, and also the refracted fields in medium #2 for the case θ₁ > θ_c

      TE Polarization

      Equation (5.2.8c,d) for the reflection and transmission coefficients of the obliquely incident TE‐polarized wave, using an equation (5.3.5a) are reduced to the following expression:

      (5.3.6)

The interface surface acts as a PMC surface for the TE‐polarized incident wave under the case θ₁ ≥ θ_c because . Over a certain frequency band, it is realized as the artificial magnetic conductor (AMC) surface. At the critical angle, θ₂ = π/2, the transmitted wave propagates along the y‐direction at the interface x = 0⁺, as shown in Fig (5.5b). For the case θ₁ > θ_c, shown in Fig (5.5c), the interface supports the slow‐wave type surface wave propagation. It is demonstrated below.

      For the case θ₁ > θ_c, the electric and magnetic field component and power flow of the TE‐polarized wave in the medium #2 is obtained from using equation (5.3.5) with equation (5.2.10a):

(5.3.7)

In equation (5.3.7a,b,c), the y‐directed wavevector k_2y and attenuation factor α are given by equation (5.3.5). Equation (5.3.7a,b,c) shows that along the direction normal to the interface, i.e. + x‐direction, the field is exponentially decaying in medium #2; showing the confinement of field near the interface. However, the wave propagates in the y‐direction. It is also evident from the complex Poynting vector giving real power transportation in the y‐direction, while imaginary power shows storages of energy in the x‐directed evanescent field. So the interface supports a surface wave, excited at the interface of two media by the obliquely TE‐polarized incident plane wave at the angle of incidence θ₁ ≥ θ_c. The surface wave, in more detail, is discussed in chapter 7. Its phase velocity along the y‐axis is

(5.3.8)

In equation (5.3.8), as ε_r1 > ε_r2; so , i.e. the phase velocity of the surface wave is less than that of in the medium #2. Therefore, the surface wave is a slow‐wave.

      TM Polarization

      All three cases of the angle of incidence apply to the TM‐polarized obliquely incident plane wave. For θ₁ > θ_c, the reflection and transmission coefficients of the TM‐polarization, given by equation (5.2.28) are reduced to

(5.3.9)

      The electric and magnetic field components of the TM polarization, also the complex Poynting vector in the medium #2 under θ₁ > θ_c, could be obtained using equation (5.3.5), from equation (5.2.18). The results are summarized below:

(5.3.10)

The expression (5.3.9a) shows that there is a total reflection at the interface for θ₁ ≥ θ_c, without any transmission of power from the medium #1 to medium #2. The transmission coefficient shows the presence of the x‐directed evanescent field in the medium #2. Under such conditions, the interface acts as a perfect electrical conductor (PEC); so the interface surface of two media can act as the artificial electric conductor (AEC). However, there is an exponentially decaying field in the medium #2, and the interface supports the surface wave propagation along the interface at x = 0⁺ in the y‐direction. The real power carried in the y-direction