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When a ray of light incidents on a Birefringence crystal, it will split into two rays. One of the rays obeys the Snell's Law's of refraction (i.e. for constant refractive index)and another one doesn't obey that law (i.e.medium becomes anisotropic for that ray). what will be the frequencies of these two rays? Has one of them same frequency as that of input ray? Are they completely different from each other in frequencies?

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The two rays have different polarizations and different wavelengths (within the crystal), but the frequencies do not change.

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