Help understanding the formula for the total phase shift of a waveplate stack

[AwesomeTrains]

TL;DR Summary

Looking for an explanation for a formula given in a paper for the total phase shift of a stack of relatively rotated waveplates.

I have problems deriving a formula in a paper I'm reading for a project. The paper is about putting a number of waveplates in series rotated relatively to each other to form a tuneable broadband waveplate. For the i-th waveplate the jones matrix is given by:
$$ J_{i}(\delta_i, \Theta_i)=
\begin{bmatrix}
\cos (\delta_i / 2) + i \cos (2 \Theta_i) \sin (\delta_i / 2) & i \sin (2 \Theta_i) \sin (\delta_i / 2) \\
i \sin (2 \Theta_i) \sin (\delta_i / 2) & \cos (\delta_i / 2) - i \cos (2 \Theta_i) \sin (\delta_i / 2)
\end{bmatrix}
$$
where $\delta_i$ is the phase delay caused by the i-th waveplate and $\Theta_i$ is its relative rotation angle.
Now they say that due to symmetry properties of waveplates (https://arxiv.org/pdf/1311.5556.pdf) the jones matrix of the stack of waveplates is given by:
$$
J = \prod_{i} J_i =
\begin{bmatrix}
A & B \\
-B^{*} & A^{*}
\end{bmatrix}
$$
Then they say that the resulting phase shift between principle axes is:
$$
\delta = 2\arctan \sqrt{\frac{|Im A|^2+|Im B|^2}{|Re A|^2 + |Re B|^2}}
$$
How is the last formula derived? Do you have to make assumptions about the initial polarization of the EM wave?

 

[nbryan5]

A:I'm not sure if this is the full answer you are looking for but I think it may help.The Jones Matrix is used to represent a linear transformation of a two-dimensional vector, usually representing the polarization state of an electromagnetic wave.The elements A and B of the matrix you wrote can be written as:A=cos($\delta$/2)+i*cos(2$\theta$)*sin($\delta$/2)B=i*sin(2$\theta$)*sin($\delta$/2)where $\delta$ is the phase shift and $\theta$ is the relative rotation angle.In order to get the phase shift between principle axes, you need to calculate the argument of A, which is the phase shift. This is given by $\phi$=arg(A)=$\arctan$ (Im(A)/Re(A))In the case of A, Im(A)=cos(2$\theta$)*sin($\delta$/2) and Re(A)=cos($\delta$/2). Hence $\phi$=arg(A)=arctan(cos(2$\theta$)*sin($\delta$/2)/cos($\delta$/2)).The resulting phase shift between principle axes is then given by$\delta$=2$\phi$=2$\arctan$ (cos(2$\theta$)*sin($\delta$/2)/cos($\delta$/2))Similarly, the argument of B can be calculated, which is$\psi$=arg(B)=arctan(Im(B)/Re(B))In the case of B, Im(B)=sin(2$\theta$)*sin($\delta$/2) and Re(B)=0. Hence $\psi$=arg(B)=arctan(sin(2$\theta$)*sin($\delta$/2)/0).The resulting phase shift between principle axes is then given by$\delta$=2$\phi$+2$\psi$=2$\arctan$ (cos(2$\theta