Anand K. Verma - Introduction To Modern Planar Transmission Lines

Both the unbounded medium and transmission line supports the 1D wave propagation. So an unbounded medium could be easily modeled, shown in Fig (5.1b), as a transmission line. The propagation constant of wave on the equivalent transmission line could be treated as identical to that of in the medium. This approach is simple and effective in obtaining the impedance transformation and also impedance matching by using the multilayer dielectric medium. On comparing the wave equations for the E y(x) and H z(x), given in equation ( 4.5.13) of chapter 4, against the voltage and current wave equation ( 2.1.37) of chapter 2, the following equivalences are observed:

(5.1.14)

The solution of the voltage and current waves, given in equations ( 2.1.63) and ( 2.1.64), can be converted to the solution of E yand H z. Finally, the expression of the input impedance of a lossless line of length d, given in equation (2.2.65) of chapter 2, terminated in a load Z L, helps to write the following expression of the input impedance of a dielectric slab of thickness d; and also the expressions for the reflection coefficient, and transmission coefficient at the interface PQ:

Figure 5.1 Normal incidence of TM‐polarized plane wave at the interface of two media.

(5.1.15)

In expression (5.1.15), medium #2 has an infinite extent. In medium #1, the incident wave is located at a distance d from the interface. The transmission line model is used for the multilayer dielectric medium also [B.1–B.4].

The plane wave of any polarization, obliquely incident at the interface of two dielectric media, can be decomposed to the TE and TM polarizations, so the plane wave of any polarization is a linear combination of the TE and TM polarizations. This section considers the reflection and transmission of both polarizations at the boundary of lossless media. However, losses in the media can also be accounted for [J.1].

Figure (5.2a)shows an oblique incidence of the z‐polarized plane wave, i.e. TE zpolarized ray at the interface P‐O‐Q of two media with different electrical characteristics. The interface PQ is along the y ‐axis, and the normal to the interface is along the x ‐axis. The plane x‐o‐y containing the normal to the interface, and the rays of the obliquely incident, reflected and refracted waves, i.e. the wavevectors ( ), is called the plane of incidence . In the present case, the plane of the paper, i.e. the (x − y)‐plane, is the plane of incidence. The incident ray, with the wavevector , strikes the interface at an angle θ 1. It is partly reflected in the medium #1 at an angle and partly refracted in the medium #2 at an angle θ 2. All angles are measured with respect to the normal o‐x. The electric field is perpendicular to the plane of incidence, i.e. in the z ‐direction; so it is called the TE‐polarization . It is also known as the perpendicular , i.e. the s‐polarization as the E incfield is perpendicular to the plane of incidence, or even the σ ‐polarization . The letter “ s ” is from the German word senkrecht (perpendicular). It is noted that the polarization of the incident wave is preserved even after the reflection and refraction at the interface of the natural materials. However, the interface created by the engineered metasurfaces controls and alters the polarization states after reflection and refraction. It is discussed in chapter 22. The magnetic field is in the plane of incidence, and its orientation follows the Poynting vector giving the direction of power‐flow. The position vector and the wavevectors for the incident, reflected, and transmitted (refracted) waves are summarized below:

(5.2.1)

Figure 5.2 Oblique incidence of a plane wave with TE‐polarization at the interface of two media.

Using Fig (5.2a), the incident, reflected, and transmitted field components of the TE zpolarized waves are summarized below:

(5.2.2)

(5.2.3)

(5.2.4)

In equations (5.2.2a,b)– (5.2.4a,b)η 1and η 2are the intrinsic impedance of the medium #1 and #2, respectively; and as both the incident and reflected waves are in the same medium #1. The Poynting vectors for the incident, reflected, and transmitted ( refracted ) fields are obtained from the above equations:

(5.2.5)

Equations (5.2.1)and (5.2.5)show that the wavevector and Poynting vector of the incident, reflected, and transmitted waves are in the same direction; so the waves in both media are the forward waves . The continuity of the tangential components of the and fields, across the interface at x = 0, provides the following expressions: