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input/output relations of the E‐field in both coordinate systems are expressed in terms of their respective Jones matrices:

(4.6.18)

      In the above expression, subscript “Car.” with Jones matric stands for the (y‐z) Cartesian coordinate system; and the subscript “e” with Jones matric stands for the general (e₁‐e₂) coordinate system. In the present case, it is the anticlockwise rotated Cartesian system, as shown in Fig. (4.12).

Using the rotation Jones matrices of equation (4.6.17a), the input and output E‐field vectors could be transformed from the rotated (e₁‐e₂) coordinate system to the Cartesian (y‐z) coordinate system:

      (4.6.19)

On substituting the above equations in equation (4.6.18a), the following expression is obtained:

      (4.6.20)

      On comparing the above equations against equation (4.6.18b), we get the transformed Jones matrix [J]_e of the rotated polarizer in the (e₁‐e₂) coordinate system from the Jones matrix [J]_Car of the original polarizer in the Cartesian (y‐z) coordinate system:

(4.6.21)

The transformation (4.6.21b) transforms the Jones matrix [J]_e describing a polarizer in the rotated (e₁‐e₂) coordinate system to the Jones matrix [J]_Car in the Cartesian coordinate system. The transformation equation (4.6.21b) is obtained by matrix manipulation. However, it could also be obtained independently, as it is done for equation (4.6.21a).

      The use of the coordinate transformation for the polarizer is illustrated below by a few illustrative simple examples. Application of Jones matrix to the more complex polarizing system is available in the reference [B.30].

       Examples:The Jones matrix of a linear polarizer given by equation (4.6.15c) is transformed below to the rotated Jones matrix [J]e = [JLP(θ)] of a linear polarizer that is rotated at an angle θ:(4.6.22) It is noted that the original linear polarizer in the Cartesian system has no cross‐polarization element. However, the rotated linear polarizer has a cross‐polarization element. The above transformation can be applied to the horizontal polarizer (py = 1, pz = 0) rotated at an angle θ, and also to the linear polarizer rotated at an angle θ = 45°, to get the following rotated Jones matrices:(4.6.23) The anisotropic polarizer with cross‐coupling, in the Cartesian system, is given by equation (4.6.14). The polarizer is rotated by an angle θ. The rotated Jones matrix of the anisotropic polarizer is obtained as follows:(4.6.24)

      Jones Matrix for Retarder (Phase Shifter)

The wave retarder also called the waveplate alters the relative phase between two orthogonal field components passing through it. In this respect, it is acting as a phase shifter. The waveplates are designed using the birefringent, i.e. anisotropic material with orthogonal fast‐axis and slow‐axis. The relative permittivity, also the refractive index, of the anisotropic material, has lower value along the fast‐axis and higher value along the slow‐axis, causing relatively slower phase velocity of the EM‐wave propagation along the slow‐axis. The half‐wavelength thick slab, called the half‐wave plate, changes the direction of the linear polarization at its output. Whereas, the quarter‐waveplate, i.e. a quarter‐wavelength thick slab, converts the linearly polarized incident waves into the circularly polarized waves at its output. The waveplates, i.e. the wave retarders, are characterized by the Jones matrices as discussed below.

For the EM‐wave propagating in the x‐direction, the field components at the output of a wave retarder could be written from equation (4.6.11), by normalizing the magnitude of field components to the unity, as follows:

(4.6.25)

The J_R(Δφ) is the Jones matrix of a wave retarder (waveplate) given by equation (4.6.25b). At the input, the incident wave is linearly polarized and at the output of the retarder slab of thickness d, the differential phase Δφ = φ_z − φ_y is the relative phase between the field components and .

      Jones Matrix of Half‐waveplate

      The Jones matrix of a half‐waveplate and the field components at its output are obtained by taking the relative phase Δφ = π:

      (4.6.26)

      It is noted that at the output of the half‐wave plate the phase difference between two field components is 180°. Two field components are in‐phase at the input of the half‐waveplate.

      Jones Matrix of Quarter‐waveplate

      The Jones matrix of a quarter‐wave retarder and also the field components at the output are obtained by taking the relative phase Δφ = − π/2:

      (4.6.27)

      In the above equation, both field components are equal to E₀ = 1. It is noted that at the output of the quarter‐waveplate, the wave is a right‐hand circularly polarized wave. In the case, input wave components are and _, the field components at the output of the quarter‐waveplate, using equation (4.6.25a) are

(4.6.28)

Equation (4.6.28c) shows that an ellipse is traced by the E‐field in the (y‐z)‐plane. In this case, the quarter‐waveplate produces an elliptically polarized wave. However, for the case m = n, it degenerates into the circularly polarized wave. Further,

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