To see what happens, resolve the incident field into components polarized along the fast and slow axes, as shown. Suppose a linearly polarized wave is normally incident on a waveplate, and its plane of polarization is at an angle θ with respect to the fast axis.
This difference can be noted by calling it a “multiple order quarter waveplate". However, if the frequency was changed, the retardation would change at a rate faster than it would for a plate that had only 1/4 wave retardation. It would not matter, provided it was only used at exactly the optical frequency designed for the waveplate. Since waves repeat themselves every 2π radians, we could subtract an integral number of 2π's or waves, and for example, call the crystal showing 2π(m+1/4) radian a quarter waveplate. The value of Γ in this formula is in radians, but is more common to express in ”wavelengths” or “waves”, with a “full-wave” meaning Γ = 2π, a “half-wave” meaning Γ = π, a “quarter-wave” meaning Γ = π/2, and so forth. The difference between these two-phase shifts is termed the retardation, Γ= 2πf(n slow - n fast)L/c. Thus, the phase shift for the wave in Figure 1 will be φ fast = 2πfn fastL/c, and for the wave in Figure 2, φ slow = 2πfn slowL/c. The propagation phase constant k can be written as 2πfn/c radians per meter, so that a wave of frequency f will experience a phase shift of φ = 2πfnL/c radians in traveling a distance L through the crystal. The difference in the number of wavelengths shown in Figures 1 and 2 (2 2/3 and 4, respectively) would imply a ratio of the two indices of refraction n fast : n slow = 2 : 3, a much larger difference than in typical natural crystals the ratio has been exaggerated for clarity. The optical components that do this “trick” are called birefringent waveplates or retardation plates (or just waveplates or retarders, for short). We can also “use” birefringence to modify the polarization state of light, which is a useful thing to do in many situations.
Unlike dispersion, birefringence can be avoided by using amorphous materials such as glass, or crystals that have simple symmetries, such as NaCl or GaAs. In turn, this results in different refractive indices for different polarizations. The orderly arrangement of atoms in some crystals results in different resonant frequencies for different orientations of the electric vector relative to the crystalline axes. Birefringence is another consequence of such resonant interaction, which is the change in refractive index with the polarization of light. A result of this dependence is the resonant interactions related to material dispersion. The interaction of light with the atoms or molecules of a material is wavelength dependent.