The following equations definitions of Stoke-vector and Mueller-matrix in polanalyser.
Note that, these equations are automatically generated by polanalyser's sympy feature.
s = [ s 0 s 1 s 2 s 3 ] \mathbf{s} = \left[\begin{matrix}s_{0} \\
s_{1} \\
s_{2} \\
s_{3}\end{matrix}\right] s = s 0 s 1 s 2 s 3 0.5 a t a n ( s 2 s 1 ) 0.5 \mathrm{atan}{\left(\frac{s_{2}}{s_{1}} \right)} 0.5 atan ( s 1 s 2 ) s 1 2 + s 2 2 + s 3 2 s 0 \frac{\sqrt{s_{1}^{2} + s_{2}^{2} + s_{3}^{2}}}{s_{0}} s 0 s 1 2 + s 2 2 + s 3 2 s 1 2 + s 2 2 s 0 \frac{\sqrt{s_{1}^{2} + s_{2}^{2}}}{s_{0}} s 0 s 1 2 + s 2 2 ∣ s 3 ∣ s 0 \frac{\left|{s_{3}}\right|}{s_{0}} s 0 ∣ s 3 ∣ 0.5 a t a n ( s 3 s 1 2 + s 2 2 ) 0.5 \mathrm{atan}{\left(\frac{s_{3}}{\sqrt{s_{1}^{2} + s_{2}^{2}}} \right)} 0.5 atan ( s 1 2 + s 2 2 s 3 ) M = [ m 00 m 01 m 02 m 03 m 10 m 11 m 12 m 13 m 20 m 21 m 22 m 23 m 30 m 31 m 32 m 33 ] \mathbf{M} = \left[\begin{matrix}m_{00} & m_{01} & m_{02} & m_{03} \\
m_{10} & m_{11} & m_{12} & m_{13} \\
m_{20} & m_{21} & m_{22} & m_{23} \\
m_{30} & m_{31} & m_{32} & m_{33}\end{matrix}\right] M = m 00 m 10 m 20 m 30 m 01 m 11 m 21 m 31 m 02 m 12 m 22 m 32 m 03 m 13 m 23 m 33 [ 0.5 0.5 cos ( 2 θ ) 0.5 sin ( 2 θ ) 0 0.5 cos ( 2 θ ) 0.5 cos 2 ( 2 θ ) 0.5 sin ( 2 θ ) cos ( 2 θ ) 0 0.5 sin ( 2 θ ) 0.5 sin ( 2 θ ) cos ( 2 θ ) 0.5 sin 2 ( 2 θ ) 0 0 0 0 0 ] \left[\begin{matrix}0.5 & 0.5 \cos{\left(2 \theta \right)} & 0.5 \sin{\left(2 \theta \right)} & 0 \\
0.5 \cos{\left(2 \theta \right)} & 0.5 \cos^{2}{\left(2 \theta \right)} & 0.5 \sin{\left(2 \theta \right)} \cos{\left(2 \theta \right)} & 0 \\
0.5 \sin{\left(2 \theta \right)} & 0.5 \sin{\left(2 \theta \right)} \cos{\left(2 \theta \right)} & 0.5 \sin^{2}{\left(2 \theta \right)} & 0 \\
0 & 0 & 0 & 0\end{matrix}\right] 0.5 0.5 cos ( 2 θ ) 0.5 sin ( 2 θ ) 0 0.5 cos ( 2 θ ) 0.5 cos 2 ( 2 θ ) 0.5 sin ( 2 θ ) cos ( 2 θ ) 0 0.5 sin ( 2 θ ) 0.5 sin ( 2 θ ) cos ( 2 θ ) 0.5 sin 2 ( 2 θ ) 0 0 0 0 0 [ 1 0 0 0 0 cos ( 2 θ ) sin ( 2 θ ) 0 0 − sin ( 2 θ ) cos ( 2 θ ) 0 0 0 0 1 ] \left[\begin{matrix}1 & 0 & 0 & 0 \\
0 & \cos{\left(2 \theta \right)} & \sin{\left(2 \theta \right)} & 0 \\
0 & - \sin{\left(2 \theta \right)} & \cos{\left(2 \theta \right)} & 0 \\
0 & 0 & 0 & 1\end{matrix}\right] 1 0 0 0 0 cos ( 2 θ ) − sin ( 2 θ ) 0 0 sin ( 2 θ ) cos ( 2 θ ) 0 0 0 0 1 [ 1 0 0 0 0 sin 2 ( 2 θ ) cos ( δ ) + cos 2 ( 2 θ ) − sin ( 2 θ ) cos ( δ ) cos ( 2 θ ) + sin ( 2 θ ) cos ( 2 θ ) − sin ( δ ) sin ( 2 θ ) 0 − sin ( 2 θ ) cos ( δ ) cos ( 2 θ ) + sin ( 2 θ ) cos ( 2 θ ) sin 2 ( 2 θ ) + cos ( δ ) cos 2 ( 2 θ ) sin ( δ ) cos ( 2 θ ) 0 sin ( δ ) sin ( 2 θ ) − sin ( δ ) cos ( 2 θ ) cos ( δ ) ] \left[\begin{matrix}1 & 0 & 0 & 0 \\
0 & \sin^{2}{\left(2 \theta \right)} \cos{\left(\delta \right)} + \cos^{2}{\left(2 \theta \right)} & - \sin{\left(2 \theta \right)} \cos{\left(\delta \right)} \cos{\left(2 \theta \right)} + \sin{\left(2 \theta \right)} \cos{\left(2 \theta \right)} & - \sin{\left(\delta \right)} \sin{\left(2 \theta \right)} \\
0 & - \sin{\left(2 \theta \right)} \cos{\left(\delta \right)} \cos{\left(2 \theta \right)} + \sin{\left(2 \theta \right)} \cos{\left(2 \theta \right)} & \sin^{2}{\left(2 \theta \right)} + \cos{\left(\delta \right)} \cos^{2}{\left(2 \theta \right)} & \sin{\left(\delta \right)} \cos{\left(2 \theta \right)} \\
0 & \sin{\left(\delta \right)} \sin{\left(2 \theta \right)} & - \sin{\left(\delta \right)} \cos{\left(2 \theta \right)} & \cos{\left(\delta \right)}\end{matrix}\right] 1 0 0 0 0 sin 2 ( 2 θ ) cos ( δ ) + cos 2 ( 2 θ ) − sin ( 2 θ ) cos ( δ ) cos ( 2 θ ) + sin ( 2 θ ) cos ( 2 θ ) sin ( δ ) sin ( 2 θ ) 0 − sin ( 2 θ ) cos ( δ ) cos ( 2 θ ) + sin ( 2 θ ) cos ( 2 θ ) sin 2 ( 2 θ ) + cos ( δ ) cos 2 ( 2 θ ) − sin ( δ ) cos ( 2 θ ) 0 − sin ( δ ) sin ( 2 θ ) sin ( δ ) cos ( 2 θ ) cos ( δ ) [ 1 0 0 0 0 cos 2 ( 2 θ ) sin ( 2 θ ) cos ( 2 θ ) − sin ( 2 θ ) 0 sin ( 2 θ ) cos ( 2 θ ) sin 2 ( 2 θ ) cos ( 2 θ ) 0 sin ( 2 θ ) − cos ( 2 θ ) 0 ] \left[\begin{matrix}1 & 0 & 0 & 0 \\
0 & \cos^{2}{\left(2 \theta \right)} & \sin{\left(2 \theta \right)} \cos{\left(2 \theta \right)} & - \sin{\left(2 \theta \right)} \\
0 & \sin{\left(2 \theta \right)} \cos{\left(2 \theta \right)} & \sin^{2}{\left(2 \theta \right)} & \cos{\left(2 \theta \right)} \\
0 & \sin{\left(2 \theta \right)} & - \cos{\left(2 \theta \right)} & 0\end{matrix}\right] 1 0 0 0 0 cos 2 ( 2 θ ) sin ( 2 θ ) cos ( 2 θ ) sin ( 2 θ ) 0 sin ( 2 θ ) cos ( 2 θ ) sin 2 ( 2 θ ) − cos ( 2 θ ) 0 − sin ( 2 θ ) cos ( 2 θ ) 0 [ 1 0 0 0 0 − sin 2 ( 2 θ ) + cos 2 ( 2 θ ) 2 sin ( 2 θ ) cos ( 2 θ ) 0 0 2 sin ( 2 θ ) cos ( 2 θ ) sin 2 ( 2 θ ) − cos 2 ( 2 θ ) 0 0 0 0 − 1 ] \left[\begin{matrix}1 & 0 & 0 & 0 \\
0 & - \sin^{2}{\left(2 \theta \right)} + \cos^{2}{\left(2 \theta \right)} & 2 \sin{\left(2 \theta \right)} \cos{\left(2 \theta \right)} & 0 \\
0 & 2 \sin{\left(2 \theta \right)} \cos{\left(2 \theta \right)} & \sin^{2}{\left(2 \theta \right)} - \cos^{2}{\left(2 \theta \right)} & 0 \\
0 & 0 & 0 & -1\end{matrix}\right] 1 0 0 0 0 − sin 2 ( 2 θ ) + cos 2 ( 2 θ ) 2 sin ( 2 θ ) cos ( 2 θ ) 0 0 2 sin ( 2 θ ) cos ( 2 θ ) sin 2 ( 2 θ ) − cos 2 ( 2 θ ) 0 0 0 0 − 1 I ( θ ) = 0.5 s 0 + 0.5 s 1 cos ( 2 θ ) + 0.5 s 2 sin ( 2 θ ) I(\theta) = 0.5 s_{0} + 0.5 s_{1} \cos{\left(2 \theta \right)} + 0.5 s_{2} \sin{\left(2 \theta \right)} I ( θ ) = 0.5 s 0 + 0.5 s 1 cos ( 2 θ ) + 0.5 s 2 sin ( 2 θ ) Polarization camera captures 0, 45, 90, 135 degree linear polarized light.
The intensity of the light through the polarizer is given by the following equations.
I ( 0 ) = 0.5 s 0 + 0.5 s 1 I ( 45 ) = 0.5 s 0 + 0.5 s 2 I ( 90 ) = 0.5 s 0 − 0.5 s 1 I ( 135 ) = 0.5 s 0 − 0.5 s 2 \begin{align*}
I(0) &= 0.5 s_{0} + 0.5 s_{1} \\
I(45) &= 0.5 s_{0} + 0.5 s_{2} \\
I(90) &= 0.5 s_{0} - 0.5 s_{1} \\
I(135) &= 0.5 s_{0} - 0.5 s_{2} \\
\end{align*} I ( 0 ) I ( 45 ) I ( 90 ) I ( 135 ) = 0.5 s 0 + 0.5 s 1 = 0.5 s 0 + 0.5 s 2 = 0.5 s 0 − 0.5 s 1 = 0.5 s 0 − 0.5 s 2 From the observation of the polarization camera, we can estimate the stokes parameters as follows.
s 0 = 0.5 I ( 0 ) + 0.5 I ( 135 ) + 0.5 I ( 45 ) + 0.5 I ( 90 ) s 1 = 1.0 I ( 0 ) − 1.0 I ( 90 ) s 2 = − 1.0 I ( 135 ) + 1.0 I ( 45 ) \begin{align*}
s_0 &= 0.5 I(0) + 0.5 I(135) + 0.5 I(45) + 0.5 I(90) \\
s_1 &= 1.0 I(0) - 1.0 I(90) \\
s_2 &= - 1.0 I(135) + 1.0 I(45) \\
\end{align*} s 0 s 1 s 2 = 0.5 I ( 0 ) + 0.5 I ( 135 ) + 0.5 I ( 45 ) + 0.5 I ( 90 ) = 1.0 I ( 0 ) − 1.0 I ( 90 ) = − 1.0 I ( 135 ) + 1.0 I ( 45 )