href="#ulink_00193a88-b322-52ef-a8b0-be4fd00897df">Fig. (4.10c). The instantaneous orthogonal E‐field components in the x = 0 plane are given below:
(4.6.7)
Using the above equation and identity
(4.6.8)
The semi‐major axis OA, the semi‐minor axis OB of the ellipse shown in Fig. (4.10c), and the axial ratio (AR) of the polarization ellipse are given below [B.9, B.29]:
The tilt angle θ of the polarization ellipse, i.e. inclination of the major axis OA with y‐axis is [B.9, B.29]:
(4.6.10)
For φ ≠ π/2, the polarization ellipse is inclined with respect to the y‐axis. The linear and circular polarizations are obtained as special cases from the elliptical polarization. For instance, for Ez(t) = 0 the wave is horizontally polarized in the y‐direction. For E0y = E0z = E0 and φ = ± π/2, the LHCP/RHCP wave is obtained as equation (4.6.9) is reduced to an equation of a circle with OA = OB. For the linear polarization, AR is infinity. However, for the circular polarization, AR is unity. In the case E0y = E0z = E0 and φ ≠ π/2, the wave is not circularly polarized and its AR is cotφ/2. For a practical circularly polarized antenna, the axial ratio is frequency‐dependent and its axial ratio bandwidth is defined as the frequency band over which AR ≤ 3dB.
4.6.4 Jones Matrix Description of Polarization States
The polarizing devices change the state of polarization. For instance, the polarizing devices could change the rotation of the linear polarization or convert the linear polarization into circular polarization. The Jones matrix method describes and manipulates the polarization states of the EM‐wave using a 2 × 1 column vector, known as the Jones vector and transfer matrix of the polarizing device, known as the Jones matrix [B.30, B.31]. The Jones matrix is used in chapter‐22 with metasurfaces.
Jones Vector
The polarization state of the EM‐wave propagation in the x‐direction is given by equation (4.6.1). The orthogonal E‐field components can be expressed in the form of the following column vector, known as the Jones vector:
In the above expression, the common phase angle has been absorbed in the propagation factor
In the above expressions, the normalized magnitude of the E‐field components are |Eoy| = |Eoz| = 1.
Jones Matrix
Figure (4.11) shows that the polarizing device could be described by a 2 × 2 transfer matrix, i.e. the Jones matrix [J]. It relates the output of the device to its input. The input could be the incident wave at certain slab/surface, acting as a polarizing device, and the output could be the transmission or reflection of the waves.
Figure 4.11 Polarizing device described by Jones matrix.
The Jones matrix elements are interpreted in the terms of the co‐polarized and cross‐polarized outgoing waves after transmission/reflection from a slab/surface:
where Jyy and Jzz are responsible for the co‐polarized outgoing waves, and Jyz and Jzy account for the presence of cross‐polarized waves at the output. The co‐polarized output waves have the same polarization as that of the incident input waves. Whereas, the cross‐polarized output waves have orthogonal polarization with respect to the polarization of the incident input waves.
Jones Matrix of Linear Polarizer
A linear polarizer allows the transmission of the incoming wave only along the transmission axis of the polarizer and blocks the transmission of the orthogonal polarizations. Jones matrices of the linear polarizers are summarized below:
For instance, at an inclination angle θ = 45°, the linearly polarized wave, given by Jones vector of equation (4.6.12c), is incident on the horizontal polarizer. The polarizer will provide a horizontally polarized wave given by equation (4.6.12a). It could be examined with the help of equation (4.6.13).
Jones Matrix of a Linear Polarizer Rotated at Angle θ with the y‐Axis
Figure (4.12) shows the (y‐z)‐coordinate system and also the (e1‐e2) ‐coordinate system rotated at an angle θ with respect to the y‐axis. The original polarizer, located in the (y‐z)‐coordinate system is described by the
