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Just like in my question. How to check laser beam polarization? What's the easiest experiment to do that will help me understend what type of polarization is present in my laser beam?

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What you want to do is measure the Stokes Parameters.

The Stokes vector is defined by $\vec{S}= \left(\begin{array}{c}S_{0} \\ S_{1} \\ S_{2} \\ S_{3}\end{array}\right)$

Where $S_0$ represents the total intensity of the beam, $S_1$ represents linear polarization i.e Vertical and Horizontal polarization. $S_2$ represents linear polarization along $45^\circ$ and lastly, $S_3$ represents circularly polarized light.

From this you can determine every degree of freedom your polarization can possibly have by using the appropriate stokes vector. https://en.wikipedia.org/wiki/Stokes_parameters

The easiest way to experimentally determine the Stokes parameter is straight forward and requires a half waveplate and a polarizer. You just need to find the following intensities:

$I(0^\circ,0^\circ) = \frac{1}{2}(S_0+S_1)$

$I(45^\circ,0^\circ) = \frac{1}{2}(S_0+S_2)$

$I(90^\circ,0^\circ) = \frac{1}{2}(S_0-S_1)$

$I(45^\circ,90^\circ) = \frac{1}{2}(S_0+S_3)$

Where the first argument is the polarizer angle and the second argument is the half-wave plate angle. Now you can solve for any parameter.

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