Introduction To Modern Planar Transmission Lines. Anand K. Verma

Introduction To Modern Planar Transmission Lines

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].

5.2 Oblique Incidence of Plane Waves

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].

5.2.1 TE (Perpendicular) Polarization Case

Figure (5.2a) shows an oblique incidence of the z‐polarized plane wave, i.e. TE^z polarized 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 ( images ), 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 images , strikes the interface at an angle θ₁. It is partly reflected in the medium #1 at an angle images and partly refracted in the medium #2 at an angle θ₂. 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^inc field 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 images 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) equation

Schematic illustration of oblique incidence of a plane wave with T E-polarization at the interface of two media.

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^z polarized waves are summarized below:

(5.2.2) equation

(5.2.3) equation

(5.2.4) equation

In equations (5.2.2a,b)–(5.2.4a,b) η₁ and η₂ are the intrinsic impedance of the medium #1 and #2, respectively; and images 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) equation

Equations (5.2.1) and (5.2.5) show that the wavevector images and Poynting vector images 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 images and images fields, across the interface at x = 0, provides the following expressions:

(5.2.6) equation

The fields are complex quantities on both the left and right‐hand sides of the interface. To match the fields at the interface, i.e. along the y‐axis, both the phase and amplitude matching are needed. The continuity equation, given by equation (5.2.6a), holds at all points along the interface, i.e. along the y‐axis. To achieve it, the exponential terms, giving phases of the incident, reflected, and refracted waves must be identical. It is known as the phase matching at the interface. The phase‐matching results in the following well‐known Snell's laws of reflection and refraction:

(5.2.7) equation

Equation (5.2.7b) is used for a magnetodielectric medium, whereas equation (5.2.7c) is valid for a dielectric medium. The n₁ and n₂ are the refractive indexes, whereas η₁ and η₂ are the intrinsic impedance of the medium #1, and medium #2, respectively. Moreover, the classical Snell's laws are obtained under the condition of the uniform phase at the interface in the direction of the y‐axis. However, the phase gradient dϕ/dy can be created on an interface of the engineered metasurface. In this case, the classical Snell's laws are modified to obtain the generalized Snell's laws. It is discussed in subsection (22.5.4) of chapter 22.

The amplitude matching of the tangential components of the E and H‐fields at the interface x = 0, from equation (5.2.6), provides

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