Curious formula for elliptical polarisation

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DrDu
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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?
 

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Andy Resnick
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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)]
 
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DrDu
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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.
 

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