Introduction To Modern Planar Transmission Lines. Anand K. Verma

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by equations (4.5.35b) and (4.5.39b). The uniform plane wave in an unbounded conducting medium is still TEM‐type. However, field components H_z and E_y are not in‐phase. These are in‐phase in a dielectric medium shown in Fig. (4.9a). The power transported per unit area, in a conducting medium, in the x‐direction is also given by the following expression:

      (4.5.42)

      The power transmitted through the conducting lossy medium is a complex quantity. Its real part gives the power that comes out from the medium of length x, whereas the imaginary part gives stored energy due to the field penetration in the conductor. The input power density available at x = 0 is . The power density after traveling distance x in a highly conducting medium is

      (4.5.43)

      The field decreases by a factor e^−αx_, whereas the wave travels through a lossy medium. If the wave travels a distance x = δ = 1/α, known as the skin depth, the field is decreased by 1/e of its initial strength, i.e. approximately 37% of its initial field strength. However, the power decreases at a faster rate, i.e. by the factor e^−2αx. If the initial power density is S₀, the power density at distance x is

(4.5.44)

      The attenuation constant α in the above equation is used from equation (4.5.35b). The power loss of wave traveling a distance x is computed after computing the power loss at unit distance x = 1m:

      (4.5.45)

In the above equation, the value of e is 2.71828. The power loss is about 9 dB per skin‐depth. The attenuation constant α of a lossy medium is defined by equation (4.5.44a) as follows:

      (4.5.46)

4.6 Polarization of EM‐waves

      The uniform plane wave in the unbounded medium is the TEM‐type wave. The monochromatic EM‐wave is characterized by amplitude, phase, and polarization states. The microwave to optical wave devices can appropriately manipulate these characteristics to steer the EM‐waves in the desired direction with shaped wavefront. The modern metasurfaces, discussed in sections (22.5) and (22.6) of chapter 22, can achieve such controls on the reflected and transmitted waves.

      In general, both and fields have two orthogonal field components in a plane normal to the direction of propagation, the x‐direction, as shown in Fig. (4.9a and b). The field components are in the (y‐z)‐plane. For the TEM mode, it is possible to get either (E_y, H_z) or (E_z, H_y) pair of fields. Both pairs of field components can also exist. The orientation of the electric field component and the movement of the tip of the resultant E‐field determine the polarization of the EM‐wave. Thus, the (E_y, H_z) pair of the EM‐wave is called a y‐polarized wave, as only the E_y component of wave exists. The (E_z, H_y) pair of the EM‐wave is called the z‐polarized wave. The (y‐z)‐plane is known as the plane of polarization. It is normal to the direction of propagation, i.e. the x‐axis. Both these polarizations are linear polarization.

Figure (4.9a and c) show that for the EM‐wave propagating in the x‐direction, the tip of the E_y field component moves along the y‐axis from +E₀ to 0 to −E₀. The movement and rotation of the tip of the ‐vector could be seen using the instantaneous ‐vector of the wave propagating in the x‐direction. In Fig. (4.9c), the path of the tip of the E_y‐field vector, in the plane of polarization, traces a line with respect to time. The linear trace demonstrates the vertical linear polarization. However, if both E_y and E_z field components are present, any of the three kinds of wave polarizations can be obtained: (i) linear polarization, (ii) circular polarization, and (iii) elliptical polarization. These polarizations are shown in Fig. (4.10a–c). The type of wave polarization depends upon the magnitude and the relative phase of the orthogonal E_y and E_z field components. The polarization states are briefly discussed below. Finally, the Jones vector and Jones matrix descriptions are summarized to describe elegantly the polarization states and their control by the polarizing devices.

Figure 4.10 Type of polarizations.

      4.6.1 Linear Polarization

      Figure (4.10a) shows E_y and E_z field components of the EM‐wave propagating in the x‐direction. The E‐electric field vector in the (y‐z)‐plane could be written in the phasor form as follows:

(4.6.1)

The e^jωt time‐harmonic factor is suppressed in the above equation (4.6.1a). Equation (4.6.1b) shows the magnitude of the E‐field, and equation (4.6.1c) computes its inclination with respect to the y‐axis. For y‐polarized wave, E_0z = 0, and for the z‐polarized wave, E_0y = 0. In general, the field components E_0y and E_0z are complex quantities. For the in‐phase field components, these are expressed as E_0y=|E_0y| e^jφ and E_0z=|E_0z| e^jφ. The instantaneous field components are considered to trace the movement of the tip of the ‐vector:

      (4.6.2)

      In the above equations, both field components have an identical phase (ωt + φ). Figure

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