Jones Vector and Jones Matrix

Jones Vector

1. Linearly polarized wave, in x direction

   ${\displaystyle \left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right]}$

2. Linearly polarized wave, plane of polarization making angle θ with x axis

   ${\displaystyle \left[\begin{array}{c}\hfill \cos \theta \hfill \\ \hfill \sin \theta \hfill \end{array}\right]}$

3. Right circularly polarized

   ${\displaystyle \frac{1}{\sqrt{2}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill j\hfill \end{array}\right]}$

4. Left circularly polarized

   ${\displaystyle \frac{1}{\sqrt{2}}\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -j\hfill \end{array}\right]}$

Jones Matrix

1. Linear polarizer with transmission axis pointing in the x direction

   ${\displaystyle \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]}$

2. Linear polarizer with a transmission axis making an angle θ with the x axis

   ${\displaystyle \left[\begin{array}{cc}\hfill {\cos }^2\theta \hfill & \hfill \sin \theta \cos \theta \hfill \\ \hfill \sin \theta \cos \theta \hfill & \hfill {\sin }^2\theta \hfill \end{array}\right]}$

3. Wave Retarder with fast axis along x direction

   ${\displaystyle \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \exp (-j\Gamma )\hfill \end{array}\right]}$

4. Polarization Rotator

   ${\displaystyle \left[\begin{array}{cc}\hfill \cos \theta \hfill & \hfill -\sin \theta \hfill \\ \hfill \sin \theta \hfill & \hfill \cos \theta \hfill \end{array}\right]}$

Coordinate Transformation

1. In a new coordinate system x'-y', with x' direction making an angle θ with the x direction, the Jones vector J' is given by

   ${\displaystyle J\text{'}=R(\theta )J}$

2. The Jones matrix T is similarly transformed into T'

   ${\displaystyle T\text{'}=R(\theta )TR(-\theta )}$

   ${\displaystyle T=R(-\theta )T\text{'}R(\theta )}$