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Why do uniaxial crystals produce two refractive indices?

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Uniaxial crystals produce two refractive indices because they are anisotropic media that exhibit double refraction, a phenomenon where a single ray of unpolarized light is split into two polarized components vibrating in mutually perpendicular planes.12

These two components are known as the ordinary ray and the extraordinary ray. The difference in refractive indices arises from the crystal's internal structure:1
* The ordinary ray experiences a constant refractive index in all directions.1
* The extraordinary ray has a refractive index that varies depending on the direction of travel relative to the crystal's optical axis.1

Specifically, uniaxial crystals—which include the tetragonal, hexagonal, and rhombohedral systems—yield one refractive index for light vibrating parallel to the optical axis and another for light vibrating perpendicular to it. Because the speed of light within the medium depends on its refractive index, these two rays travel at different speeds, causing the light to bend at different angles. This distinction vanishes only when light travels directly along the crystal's optic axis.312