[Jrot(θ)]. The rotated polarizer is located in the (e1‐e2) coordinate system. The relation between the Jones matrix [J] of unrotated polarizer and Jones matrix [Jrot(θ)] of the rotated polarizer is obtained by the coordinate transformation through the rotation Jones matrix [Rθ(θ)] and the inverse rotation Jones matrix [Rθ(θ)]−1 = [Rθ(−θ)].
Figure (4.12) shows that the unit vectors of two coordinate systems are related through the following transformations:
(4.6.16)
The following rotation Jones matrix [Rθ(θ)] and its inverse [Rθ(−θ)] can be defined from the above relations that could be useful to transform the vectors from one co‐ordinated system to another:
Figure 4.12 The (y–z) and rotated (e1–e2) Coordinate systems.
The above rotation matrices are used to transform both the Jones vector and the Jones matrix from one to another coordinate system. The Jones matrix concept is applied to the polarizing system in two steps:
Transformation of E‐vector Components
The rotation Jones matrix [Rθ(θ)] transforms the vector components from the rotated (e1‐e2) coordinate system to the (y‐z) Cartesian coordinate system. Whereas the inverse rotation Jones matrix [Rθ(−θ)] transforms the vector components from the (y‐z) Cartesian coordinate system back to the rotated (e1‐e2) coordinate system.
Transformation of Jones Matrix of Polarizer
To perform the transformation of the E‐vector components from one to another coordinate system using a polarizer, the Jones matrix describing the polarizer has to be transformed from one to another coordinated system. Thus, the above relations transform the Jones matrix [J] = [J]Car of a linear polarizer, given by equation (4.6.14) in the Cartesian (y‐z) coordinate system, to the Jones matrix [Jrot(θ)] = [J]e of the polarizer in the rotated (e1‐e2) coordinate system.
The input/output relations of the E‐field in both coordinate systems are expressed in terms of their respective Jones matrices:
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:
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
