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Transformation of Jones Matrix of Polarizer
ОглавлениеTo 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)
