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Suppose you have source of coherent light, a laser beam, with +45 polarization. A polarizing beam splitter (PBS) splits the beam into its horizontal and vertical components. The second polarizing beam splitter recombines them into a single beam again. If the two paths between the two PBS are of equal length and there is nothing else in them that would cause a phase shift, I'd assume the output polarization is equal to the input polarization: +45.

What is the output polarization if you place an attenuator in the path of the horizontal component? How do you calculate it?

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If the PBS is perfect, the output polarisation is 45°. The electric field of this beam consists of two orthogonal components (say your transmitted and reflected rays) Ey and Ex. For other angles of input polarisation, they will no longer be equal, so recombining them with the second PBS will not give you 45°. If you attenuate one of the two split components, you won't be able to distinguish whether the final 'no 45°' is due to input polarisation rotation or attenuation of one of the components (if you don't control the power). To calculate how the PBS divides the beam use simple trigonometry. NOTE: NOT ON POWER!!! ...but on the electric field amplitude!!!

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If you think about the extreme case where the attenuator blocks all of the horizontal then the output of the second PBS is V. When there is no attenuation it is 45°. For attenuation between these the output will be a linear polarisation between 45° and V.

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