Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 207

Figure (5.3a) shows an oblique incidence of TM^z‐polarized plane wave ray at the interface of two electrically different media. The electric field of the incident wave is in the (x‐y)‐ plane of incidence, so the incident wave has a parallel polarization as the E^inc field is parallel to the plane of incidence. The parallel polarization, or p‐polarization, is also known as the transverse magnetic (TM) polarization as the z‐oriented magnetic field is normal to the plane of incidence. It is also called π‐polarization.

The visual inspection of the direction of field vectors in Fig (5.3a) shows that the magnetic field component H_z is normal to the (x − y)‐plane, i.e. to the plane of polarization, while the electric field components E_x and E_y are in the plane of polarization. The field components of the incident, reflected, and transmitted (refracted) TM‐polarized obliquely incident wave, as shown in Fig (5.3a), are summarized below:

(5.2.12)

(5.2.13)

Transmitted wave:

(5.2.14)

The continuity of the tangential electric and magnetic field components, across the interface at x = 0, provides the following expression:

(5.2.15)

The Poynting vectors are the same as that of the TE‐polarization. The phase matching, from equation (5.2.15), again provides Snell's reflection and refractions laws as given by equation (5.2.7). However, the amplitude matching provides different expressions for the reflection (Γ_ll^TM), and transmission τ_ll^TM coefficients of the parallel polarized obliquely incident plane wave:

Figure 5.3 Oblique incidence of a plane wave with TM‐polarization at the interface of two media.

(5.2.16)

For normal incidence, equations (5.2.16c,d) reduce to equations (5.1.3a,b). Equations (5.2.16c,d) are known as the Fresnel's Equations of the TM‐polarized waves. It is noted that . The total fields in both the media are summarized below:

Medium #1

The total field is a summation of the incident and reflected fields:

(5.2.17)

Medium #2

(5.2.18)

In medium #1, there is a traveling wave along the y‐axis, whereas, in the direction of normal to the interface, the wave is a standing wave. However, the minima of the standing wave never reach zero levels as , like a PMC. Again, even if the wave travels in the y‐direction along the interface of two media, the traveling wave is not a surface wave, as in the transverse x‐direction, there is no confinement of field near the interface. However, under certain conditions, discussed in subsection (5.3.2), the surface wave could exist along the interface.