# Curious formula for elliptical polarisation

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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?