“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
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 TEz polarized waves are summarized below:
(5.2.3)
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
Equations (5.2.1) and (5.2.5) show that the wavevector
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:
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 (
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
Medium #2
In
