Themis An4 new polarimetric analyzer is a dual beam full-Stokes analyzer. The “modulator” part is made out of two quarter-wave plates and is quite similar in its principle to our former An2 (it actually uses the very same sets of crystaline retardance plates) or An3 (this version, while identical in the principle, was using twisted nematics LCDs retarders). The “selector” part is somehow different because, unlike before, now we want to send a full field down to any imaging or spectral camera.

The modulator is made of 2 quarter waves retarders. To underestand its principle, we need to compute its Mueller matrix and apply it to an unpolarized input Stokes vector. Each quarter wave will depend on

The Mueller matrix of a pure retarder of angle $\varphi$, with its slow axis aligned with the local reference frame is:

$$M_0(\varphi) = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & \cos\!\left(\mathrm{\varphi}\right) & - \sin\!\left(\mathrm{\varphi}\right)\\ 0 & 0 & \sin\!\left(\mathrm{\varphi}\right) & \cos\!\left(\mathrm{\varphi}\right) \end{array}\right)$$
In this reference frame, the quarter-wave plate is: $$M_0(\pi/2) = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{array}\right)$$

and the half-wave plate in the same reference frame is: $$M_0(\pi)= M_0(\pi/2). M_0(\pi/2)=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{array}\right)$$ We note that 2 aligned quarter-wave plates equals one half-wave. The Mueller matrix of a rotation $\theta$ of the reference frame is: $$\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & \cos\!\left(2\, \mathrm{\theta}\right) & \sin\!\left(2\, \mathrm{\theta}\right) & 0\\ 0 & - \sin\!\left(2\, \mathrm{\theta}\right) & \cos\!\left(2\, \mathrm{\theta}\right) & 0\\ 0 & 0 & 0 & 1 \end{array}\right)$$