Waveplate - Principles of Operation

Principles of Operation

A waveplate works by shifting the phase between two perpendicular polarization components of the light wave. A typical waveplate is simply a birefringent crystal with a carefully chosen orientation and thickness. The crystal is cut into a plate, with the orientation of the cut chosen so that the optic axis of the crystal is parallel to the surfaces of the plate. This results in two axes in the plane of the cut: the ordinary axis, with index of refraction n_o, and the extraordinary axis, with index of refraction n_e. For a light wave normally incident upon the plate, polarization component along the ordinary axis travels through the crystal with a speed v_o = c/n_o, while the polarization component along the extraordinary axis travels with a speed v_e = c/n_e. This leads to a phase difference between the two components as they exit the crystal. When n_e < n_o, as in calcite, the extraordinary axis is called the fast axis and the ordinary axis is called the slow axis. For n_e > n_o the situation is reversed.

Depending on the thickness of the crystal, light with polarization components along both axes will emerge in a different polarization state. The waveplate is characterized by the amount of relative phase, Γ, that it imparts on the two components, which is related to the birefringence Δn and the thickness L of the crystal by the formula

where λ₀ is the vacuum wavelength of the light.

Waveplates in general as well as polarizers can be described using the Jones matrix formalism, which uses a vector to represent the polarization state of light and a matrix to represent the linear transformation of a waveplate or polarizer.

Although the birefringence Δn may vary slightly due to dispersion, this is negligible compared to the variation in phase difference according to the wavelength of the light due to the fixed path difference (λ₀ in the denominator in the above equation). Waveplates are thus manufactured to work for a particular range of wavelengths. The phase variation can be minimized by stacking two waveplates that differ by a tiny amount in thickness back-to-back, with the slow axis of one along the fast axis of the other. With this configuration, the relative phase imparted can be, for the case of a quarter-wave plate, one-fourth a wavelength rather than three-fourths or one-fourth plus an integer. This is called a zero-order waveplate.

For a single waveplate changing the wavelength of the light introduces a linear error in the phase. Tilt of the waveplate enters via a factor of 1/cos θ (where θ is the angle of tilt) into the path length and thus only quadratically into the phase. For the extraordinary polarization the tilt also changes the refractive index to the ordinary via a factor of cos θ, so combined with the path length, the phase shift for the extraordinary light due to tilt is zero.

A polarization-independent phase shift of zero order needs a plate with thickness of one wavelength. For calcite the refractive index changes in the first decimal place, so that a true zero order plate is ten times as thick as one wavelength. For quartz and magnesium fluoride the refractive index changes in the second decimal place and true zero order plates are common for wavelengths above 1 µm.