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In single mode fiber the light propagates in two orthogonal planes. Input will be linearly polarized light, which state of polarization will be on output and why? And if there will be some different state of polarizatin on output what will happen?

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In standard single-mode fiber, the polarization will tend to drift as the signal propagates (due to slight and varying birefringence of the glass, possibly stress-induced, coupling one polarization to the other).

For short lenths (1 m or so) polarization is typically maintained fairly well. For long distances (1 km or more, I'd guess, but I don't have a lot of experience with this issue) the output polarization is typically pretty well randomized.

There is also readily available "polarization maintaining" (PM) single mode fiber, that is designed to allow a signal to propagate while maintaining its polarization. It does this by having a deliberately introduced birefringence in a well-defined direction.

This does mean that if you launch a signal into PM fiber but not aligned with the preferred axis, it will have strong dispersion between the two polarizations, which could even lead to a single input pulse generating two output pulses.

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It is my experience that in experiments with limited length (<3 meters) of single mode fibers the output polarization given a linear input polarization is elliptical. The reason for this is that the fiber is often bent causing strain in the fiber material leading to birefringent behaviour of the fiber.

I would not call the output polarization randomized as two orthogonal input polarizations can be mapped back using the right combination of a quarter waveplate and a half waveplate. In other words once you find the effective birefringence of your fiber you can retrieve your input polarization state.

A good source for theory on this might be :"Optical Fiber Birefringence Effects – Sources, Utilization and Methods of Suppression" by Petr Drexler and Pavel Fiala.

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