What is the fast and slow axis of a retarder and how do they differ?

In a birefringent material, such as a retarder, the fast axis is the axis through which the light travels faster. For a retarder, the fast axis is typically labeled and marks the axis on the retarder that is used as a reference for whichever desired effect is needed. For a ½ wave (½λ) retarder, the orientation of the fast axis is what determines the orientation of the linearly polarized light emitting from the retarder. For instance, if you rotate a ½ wave retarder 45° with respect to the linear polarized light entering the retarder, then the light emitted by the retarder will be rotated 90° from the incident polarized light. The slow axis in a retarder is the axis through which the light travels slower.