Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 227

For the case of the normally incident TE‐polarized wave, the angles of incidence and refraction are θ₁ = θ₂ = 0^∘. Furthermore, ϕ = k₂d, (k_2x = k₂), and P = η₂/η₁. Under, this condition, equation (5.4.13a,b) for the reflection and transmission coefficients are reduced to the following expressions:

(5.4.15)

The above expressions are valid for the normally incident TM‐polarized waves also.

Figure (5.6b) shows the equivalent transmission line model of the three‐layered medium. The reflection coefficients at both interfaces can be obtained from the expression, Γ = (Z_L − Z₀)/(Z_L + Z₀). At the first interface, corresponding medium impedances are Z_L → η₂ and Z₀ → η₁, while at the second interface, these are Z_L → η₁ and Z₀ → η₂. The reflection coefficients Γ₁₂ and Γ₂₃ are defined at the first and second interfaces. The above equations are rewritten as follows:

(5.4.16)

The approximate expression (5.4.16c) is used for the small reflection at the interfaces, i.e. for |Γ₁₂Γ₂₃| < < 1.

The above expressions for total reflection and transmission of a slab can also be obtained from the theory of multiple reflections [B.5, J.2]. Using the relation (5.4.3), equation (5.4.16) can be recast in the following format also:

(5.4.17)

In the case of the obliquely incident TE waves, in the above equations, we replace the wavevectors k₁ and k₂ as k₁ → k_1x → k₁ cos θ₁ and k₂ → k_2x → k₂ cos θ₁ respectively; where the waves propagate in the x‐direction. The waves are confined in the y‐direction. As a special limiting case, if the electrical parameters of the slab of thickness d and the host medium are identical, the k₂ = k₁, μ₂ = μ₁. It leads to Γ^Nor = 0, i.e. no reflection at the interface, and , i.e. the slab provides a complete transmission with a lagging phase k₂d. This is the case of the matched layer seen in equation (5.4.15a) for η₁ = η₂. The matching applies even to the interface of electrically dissimilar media given by the condition in equation (5.1.4b). The expressions obtained in the present subsection are useful to characterize the metamaterials slab. It is discussed in subsection (21.4.1) of chapter 21.