Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 205

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 (), 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 θ₁. It is partly reflected in the medium #1 at an angle 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 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^z polarized waves are summarized below:

(5.2.2)

(5.2.3)

(5.2.4)

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 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:

(5.2.6)

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 (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 the following expressions for the reflection () and transmission () coefficients of the perpendicular (TE) polarized obliquely incident plane waves:

(5.2.8)

Equations (5.2c,d) are known as the Fresnel's Equations of the TE‐polarized waves. They describe the ratio of the reflected and transmitted electric fields to that of the incident electric field. As the reflection and transmission coefficients are complex quantities, they describe both the relative amplitude and phase shifts between the waves. The above equations show that if both media are identical; there is no reflection, Γ_⊥ = 0; and η₁ = η₂, θ₁ = θ₂, leading to total transmission τ_⊥ = 1. It is also noted that τ_⊥^TE = 1 + Γ_⊥^TE.

The total field component in the medium #1 is a summation of the incident and reflected fields, given by equations while in the medium #2, only a refracted field, given by equation (5.2.4) exists. These field equations are summarized below:

Medium #1

(5.2.9)

Medium #2

(5.2.10)

In medium #1, due to the superposition of incident and reflected waves, interference occurs. Following the process used in equation (5.1.7), the computed total wave is partly traveling wave along the y‐axis, while along the negative x‐axis, it is partly standing wave. However, the minima of the standing wave do not reach zero levels as Γ_⊥ ≠ − 1, like a PEC. Although, the wave is traveling along the interface in the y‐direction, still the wave is not the surface wave, as in the x‐direction the field is not confined to the surface. Under certain conditions, discussed in subsection (5.3.2), propagation of the surface wave is possible. The surface wave is further discussed in section (7.5) of chapter 7.