Linear polarizer and the angle of incidence

Question

It is known that when a beam of lineary polarized light falls perpendicularly on a linear polarizer, the intensity of polarization changes according to Malus' law and the direction of polarization changes as cosine of angle between polarization vector and polarizer vector.

My question is: is anyone familiar with mathematical treatment of how the direction and intensity of polarization changes when the angle of incidence changes?

Answer 1

It depends on the polarizing material. I assume below that it is about layer of absorptive material.

When wave travels through an anisotropic material, its phase shift and absorption is polarization-dependent. So you should be able to treat it in similar way as the Birefringence (giving a complex refraction index to take into account the absorption).

If you want to consider a simple case of very thin polarizer, with the 100% absorption of one polarization, and 0% of the another - it simplifies. Just you need to say if polarization in the direction $z$ (i.e. the perpendicular to the polarizer) is absorbed or not. Of course for other efficiencies it is still relatively simple, as long as you study the case of the very thin polarizer.

If you are interested in analyzing different polarizers (e.g. polarizing cubes, basing on the birefringence and total internal reflection - then you can use the Fresnel equations).