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I found in an optics text the following statement, which left me somewhat confused:

In general, a natural light beam of intensity $I_0$ can be decomposed into two linearly polarized light beams, both perpendicular to each other, without any phase correlation and equal intensity $I_0/2$.

But I'm not sure where it comes from... Is it a general law? How could it be demonstrated or, at least, reasoned?

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It is indeed a fact, referring to incoherent natural light. It is easy to demonstrate that a polarizing beamsplitter separates such a beam into two beams of orthigonal polarization: just split a beam using a nonpokarizing beamsplitter and then check the polarization of each resulting beam with a polarizing filter.

To prove that the two beams have no phase relationship is a bit more complicated. There may be an easier way, but a very direct way would be to set up a two-beam interferometer using a non-polarizing beamsplitter and adjust the beam paths until you see interference fringes. Stick a half-wave plate in one beam (which will rotate the polarizations of all the photons), and you should still get interference fringes. Then substitute a polarizing beamsplitter for the non-polarizing beamsplitter, and you will find that you can't get interference fringes. The half-wave plate is needed because the two beams need to have the same polarization or they won't form interference fringes. If the polarization is the same and no interference fringes can be formed, the two beams do not have a stationary phase relationship.

An alternative approach is to leave out the half-wave plate and put a circular polarizer at the output of the (polarizing beamsplitter) interferometer. Forming circular polarization by combining two beams of orthogonal polarization is only possible if the two beams have a stationary phase relationship

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