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Birefringent materials have direction-dependent refractive indices, with different values along different crystal axes. Ordinary rays experience refractive index nₒ; extraordinary rays experience nₑ. This anisotropy causes double refraction and enables wave plate construction for polarization control.
You already know that the refractive index of a material determines how fast light travels through it and how much it bends at an interface. In an ordinary isotropic material — glass, water, air — that index is the same regardless of which direction light travels or how it is polarized. Birefringent materials break this symmetry. Their internal crystal structure is anisotropic, meaning the atomic arrangement differs along different axes, so light "feels" a different electrical environment depending on its orientation. The result is not one refractive index but two.
The two indices are called the ordinary index (nₒ) and the extraordinary index (nₑ). Light polarized perpendicular to the crystal's optic axis obeys ordinary refraction — it follows Snell's law as if the crystal were a simple glass of index nₒ. Light polarized parallel to the optic axis (or at some angle to it) travels at a different speed governed by nₑ. Because the two polarization components travel at different speeds, an incident beam can literally split into two separate refracted beams traveling in slightly different directions — the phenomenon called double refraction or double image formation that you can observe by placing a calcite crystal on a page of text.
The speed difference between the two polarizations has a practical consequence: if both components enter the crystal in phase, they exit with a phase difference that depends on the crystal thickness and the size of (nₑ − nₒ). A crystal cut to a precise thickness can introduce a phase shift of exactly π/2 radians (a quarter-wave plate) or exactly π radians (a half-wave plate). These wave plates are the workhorses of polarization optics — a quarter-wave plate converts linearly polarized light into circularly polarized light, while a half-wave plate rotates the polarization direction. LCD screens, optical isolators, and ellipsometers all depend on this birefringence-based phase control.
A useful mental picture: imagine two runners on a track who start at the same position and same pace, but one lane is slightly boggier than the other. After running the same distance, they arrive at different times — that time lag is the phase difference. By choosing the track length (crystal thickness) you choose exactly how far apart they finish. Birefringence is the phenomenon that gives you two lanes with different friction; the crystal cut is what selects the finish-line gap.