Curious formula for elliptical polarisation

[DrDu]

Recently, I have been playing with polarisation microscopy and the measuring of elliptical polarisation. Standard treatments, like that in Born and Wolf, are usually a mayhem of all kinds of trigonometric functions. Now I derived a nice relation, which I didn't find in literature, although I am quite sure that it has been derived numerous times. Here it is:

Suppose we are given a Jones vector $(a, b)^T$ characterizing the polarization direction in the plane perpendicular to the wavevektor. In general, $a$ and $b$ are complex numbers.
The elliptical polarisation is determined by the ratio $b/a$ or equivalently the complex azimuth $w= \arctan(b/a)$. Usually, one uses the real azimuth $\psi$ and the ellipticity $\theta$ to characterize the polarisation state.

Now the two descriptions are related as $w=1/2(2\psi+\mathrm{gd}(2i\theta))$ where "gd" stands for a little known function called the Gudermannian function:
https://en.wikipedia.org/wiki/Gudermannian_function
whose most useful definition here is $\mathrm{gd}(x)=2\arctan⁡(\tanh⁡(x/2))$

It is clear that $\psi$ is the the real azimuth if $\theta=0$ and also for $\psi=0$ one finds easily that
$|b/a| = \tan(\theta)$. Forming the density matrix, one can calculate the Stokes parameters and gets the correct spherical coordinates in terms of $2\psi$ and $2\theta$ for the location on the Poincaré sphere.

Can anybody point me to a reference for this formula?

 

[Andy Resnick]

Not sure which formula you are referring to, but if you mean w=1/2(2\psi+\mathrm{gd}(2i\theta)), a similar formula appears in Azzam and Bashara's "Ellipsometry and Polarized light", eqn 1.79 (using your variables):

w = [tan(ψ)+i tan(θ)]/[1-i tan(ψ)tan(θ)]

 

[DrDu]

Not sure which formula you are referring to, but if you mean w=1/2(2\psi+\mathrm{gd}(2i\theta)), a similar formula appears in Azzam and Bashara's "Ellipsometry and Polarized light", eqn 1.79 (using your variables):

w = [tan(ψ)+i tan(θ)]/[1-i tan(ψ)tan(θ)]

Thank you! Yes, this is clearly equivalent (supposing you forgot a tangent on the LHS). Writing $i\tan(\theta)=\tan (\arctan(i \tan(\theta)))=\tan (i\mathrm{artanh}(\tan(\theta)))$ one can use the addition theorem for the tangent to obtain $w= \psi + i \mathrm{artanh}(\tan(\theta))$. The advantage of the latter formula is that you have a direct split of w into real and imaginary part. Usually, it is easy to derive expressions for linearly polarized light and most of them still hold when the azimuth becomes complex. I'll have a look at this book.