Introduction To Modern Planar Transmission Lines. Anand K. Verma

the real part of equation (5.3.10). The imaginary part of the Poynting vector shows the stored energy in the evanescent field. Figure (5.5c) shows the surface wave propagation in the y-direction that also occurs in the case of the obliquely incident TE‐polarized waves.

      This subsection shows the existence of a surface wave at the interface of natural media. However, artificially engineered metasurfaces discussed in subsection (22.5.5) of chapter 2 has additional ability to control the surface wave in the desired manner, and also reradiate it as the leaky wave.

5.4 EM‐Waves Incident at Dielectric Slab

      The EM‐waves can strike the slab embedded in a homogeneous medium both normally and obliquely. Both cases of wave incidence and their transmission line models are discussed below. The analysis can be extended to the multilayer medium.

      5.4.1 Oblique Incidence

Figure (5.6a) illustrates the oblique incidence of the TE‐polarized waves on a three‐layered dielectric medium. It is desired to find overall reflection and transmission coefficients of a dielectric slab of thickness d embedded in a homogeneous medium, while the TE‐polarized wave is obliquely incident at the first interface located at the y‐axis. The forward and reflected waves are present in both the media #1 and #2 and finally transmitted to the medium #3. The media #1 and #3 have identical electrical properties.

      Extending the process given in equations (5.2.1)–(5.2.7), the total E and H‐fields in three media are written as follows:

(5.4.1)

Figure 5.6 Plane‐wave incident on a dielectric slab.

(5.4.2)

In equations (5.4.2a,b,c) superscripts m₁, m₂, and m₃ correspond to medium #1, #2, and #3, respectively. The propagations constant and intrinsic impedance in media are

(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_z and H_y field 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 4 provides 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_z and H_y provide 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 Скачать книгу