Anand K. Verma - Introduction To Modern Planar Transmission Lines

(5.4.3)

Further, in equations (5.4.1b)and (5.4.2b), the superscripts m2f and m2b show the forward and backward moving waves in the medium #2. The tangential E zand H yfield components at the first interface located at x = 0 are continuous:

(5.4.4)

(5.4.5)

The continuity of the y‐components of the field across the interface also provides the phase matching giving the following result:

(5.4.6)

The dispersion relation ( 4.5.29d) of chapter 4provides the following expressions for the propagation constant of propagating wave, in the medium #2 and medium #3, in the x‐axis direction:

(5.4.7)

After canceling the phase‐matching factors in equations (5.4.4)and (5.4.5), the amplitude matching at the interface #1 provides the following expressions:

(5.4.8)

The above equations are solved to get the following set of expressions for and :

(5.4.9)

After cancellation of the phase-matching factors at the interface #2 (at x = d), the continuity of tangential components E zand H yprovide the following expressions:

(5.4.10)

Equations (5.4.10a,b,c)are again solved for and :

(5.4.11)

Equations (5.4.9)and (5.4.11)are equated to get a pair of expressions for the reflection (Γ TE) and transmission (τ TE) coefficients:

(5.4.12)

The above equations are solved to get the following expressions for the reflection and transmission coefficients of the obliquely incident TE‐polarized wave:

(5.4.13)

For the obliquely incident TM‐polarized wave on the three‐layered medium, shown in Fig (5.6a), the process can be repeated to get the following expressions, similar to expressions (5.4.14):

(5.4.14)

The reflection and transmission coefficients of both the TE and TM‐polarized incident waves are identical in form, except that parameters P and Q are different.

For the case of the normally incident TE‐polarized wave, the angles of incidence and refraction are θ 1= θ 2= 0 ∘. Furthermore, ϕ = k 2d, (k 2x= k 2), and P = η 2/η 1. 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− Z 0)/(Z L+ Z 0). At the first interface, corresponding medium impedances are Z L→ η 2and Z 0→ η 1, while at the second interface, these are Z L→ η 1and Z 0→ η 2. The reflection coefficients Γ 12and Γ 23are 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 |Γ 12Γ 23| < < 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 k 1and k 2as k 1→ k 1x→ k 1cos θ 1and k 2→ k 2x→ k 2cos θ 1respectively; 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 k 2= k 1, μ 2= μ 1. 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 k 2d. This is the case of the matched layer seen in equation (5.4.15a)for η 1= η 2. 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.