Birefringence and wave plates

Most transparent materials treat light the same way regardless of which direction it's polarised in — one refractive index describes the whole material. A birefringent material, usually one with a crystal structure that isn't the same in every direction (calcite, quartz, many plastics under mechanical stress), doesn't: light polarised along one axis sees a different refractive index than light polarised along the perpendicular axis. Since speed in a medium is c divided by the refractive index, the two polarisation components travel at different speeds through the same physical thickness.

That speed difference is the whole mechanism. If you shine light that's a mix of both polarisation components into a birefringent slab, the two components enter in step but leave out of step, because one of them took longer to cross the same distance. The size of that lag depends on exactly three things: how thick the slab is, how different the two refractive indices are (their difference is what's meant by "the birefringence" of the material), and the wavelength of the light — change any one of those and the phase lag changes with it.

A wave plate exploits this on purpose: it's a birefringent slab cut to precisely the thickness that introduces a chosen fraction of a full cycle of phase lag for one particular wavelength. A quarter-wave plate (a quarter-cycle lag) turns linearly polarised light oriented at 45° to its two axes into circularly polarised light, because the two equal-amplitude components end up 90° out of phase — exactly the condition for circular polarisation. A half-wave plate (a half-cycle lag) instead flips the orientation of linearly polarised light like a mirror reflection about its own axis. Neither of these is a property of the crystal alone; it's a property of the crystal cut to a specific thickness for a specific wavelength, which is why a wave plate designed for one laser's colour doesn't work correctly at another.