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I have read in many places that orthogonally polarized light beams do not interfere. However, I also know that orthogonal vectors, such as force, do affect each other and give a resulting force. So, since amplitude is a vector, shouldn't the amplitudes of orthogonally polarized beams add up in some way and produce a resulting amplitude? If so, could this be called interference?

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Personally I find statements like "differently polarized beams don't interfere" to be imprecise, but I can understand where the writer is coming from.

The interference of waves of like polarization (or of longitudinal waves) is characterized by differing average power at different points, including places where the average power is reduced from the value from either beam alone. That is there is destructive interferences as well as constructive interference.

The interference of beams of perpendicular polarization never exhibits places where the average power falls below that of either beam, because the amplitude of the wave is the hypotenuse of a right triangle the sides of which are the amplitudes of the two beams (possibly with a phase-shift relative one another). That is to say that here there is only constructive interference (though in varying degrees).

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