Jones Calculus - Jones Matrices

Jones Matrices

The Jones matrices are the operators that act on the Jones Vectors as listed above. These matrices are implemented by various optical elements such as lenses, beam splitters, mirrors, etc. The following table gives examples of Jones matrices for polarizers:

Optical element Corresponding Jones matrix
Linear polarizer with axis of transmission horizontal

\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
Linear polarizer with axis of transmission vertical

\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
Linear polarizer with axis of transmission at ±45° with the horizontal

\frac{1}{2} \begin{pmatrix} 1 & \pm 1 \\ \pm 1 & 1 \end{pmatrix}
Right circular polarizer

\frac{1}{2} \begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}

Left circular polarizer

\frac{1}{2} \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}
Linear polarizer with axis of transmission at angle with the horizontal. (Shown construction from rotating up from the horizontal into the polarizing element, the polarizing element, and then rotating back down into the horizontal.)

\begin{pmatrix} \cos^2(\theta) & \cos(\theta)\sin(\theta) \\ \sin(\theta)\cos(\theta) & \sin^2(\theta) \end{pmatrix} =
\begin{pmatrix} \cos(-\theta) & \sin(-\theta) \\ -\sin(-\theta) & \cos(-\theta) \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix}

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