Theory

The polarization of light can be described by a Jones vector, which is of the form.

(1)
\begin{align} \left(\begin{array}{c} E_x\\ E_y \end{array} \right) \end{align}

By convention the first element of a Jones vector is chosen to be real. Thus the second element gives the relative phase between the x and y components of the electric field. In general we can write the a normalized Jones vector as.

(2)
\begin{align} \frac{1}{\sqrt{2}} \left(\begin{array}{c} 1\\ e^{i\phi} \end{array} \right) \end{align}

A table of useful Jones Vectors is given below.

Polarization Jones Vector Relative Phase
Horizontal (linear in x) $\left(\begin{array}{c} 1\\ 0 \end{array} \right)$ 0
Vertical (linear in y) $\left(\begin{array}{c} 0\\1 \end{array} \right)$ 0
Diagonal (linear at 45 degrees) $\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1\\ 1 \end{array} \right)$ 0
Right Hand Circular $\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1\\ -i \end{array} \right)$ $\frac{\pi}{2}$
Left Hand Circular $\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1\\ i \end{array} \right)$ $-\frac{\pi}{2}$

The effect of an optical element on the polarization of light is given by the corresponding Jones matrices. The Jones matrix describing a waveplate with a phase restardance of $\phi_R$ is given by.

(3)
\begin{align} M = \left(\begin{array}{cc} 1 & 0\\ 0 & e^{-i\phi_R} \end{array} \right) \end{align}

If the waveplate is rotated about the optical axis by an angle $\theta$ then the Jones matrix is given by.

(4)
\begin{align} M^{'} = R(-\theta)M R(\theta) \end{align}

Where, the rotation matrix is of the form.

(5)
\begin{align} \left(\begin{array}{cc} cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array} \right) \end{align}

Thus we obtain the Jones matrix for a waveplate of arbitreaty phase retardance at an angle $\theta$ to the optical axis.

(6)
\begin{align} M^{'} = \left(\begin{array}{cc} cos^2\theta +sin^2\theta e^{-i\phi_R} & cos\theta sin\theta -cos\theta sin\theta e^{-i\phi_R} \\ cos\theta sin\theta -cos\theta sin\theta e^{-i\phi_R} & sin^2\theta + cos^2\theta e^{-i\phi_R} \end{array} \right) \end{align}
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