A birefringent crystal is a material that has an optic axis with the property that the light has a different index of refraction for light polarized parallel to the axis than it has for light polarized perpendicular to the axis. Light polarized parallel to the axis are called " extraordinary rays " or " extraordinary photons ", while light polarized perpendicular to the axis are called " ordinary rays " or " ordinary photons ". If a linearly polarized wave impinges on the crystal, the extraordinary component of the wave will emerge from the crystal with a different phase than the ordinary component. In mathematical language, if the incident wave is linearly polarized at an angle t h e t a with respect to the optic axis, the incident state vector can be written | ψ ψ ⟩ ⟩ = ( cos ⁡ ⁡ θ θ sin ⁡ ⁡ θ θ ) and the state vector for the emerging wave can be written | ψ ψ ′ ⟩ ⟩ = ( cos ⁡ ⁡ θ θ exp ⁡ ⁡ ( i α α x ) sin ⁡ ⁡ θ θ exp ⁡ ⁡ ( i α α y ) ) = ( exp ⁡ ⁡ ( i α α x ) 0 0 exp ⁡ ⁡ ( i α α y ) ) ( cos ⁡ ⁡ θ θ sin ⁡ ⁡ θ θ ) = d e f U ^ ^ | ψ ψ ⟩ ⟩ .