A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. Two common types of waveplates are the half-wave plate, which rotates the polarization direction of linearly polarized light, and the quarter-wave plate, which converts between different elliptical polarizations (such as the special case of converting from linearly polarized light to circularly polarized light and vice versa.) Waveplates are constructed out of a birefringent material (such as quartz or mica, or even plastic), for which the index of refraction is different for light that is linearly polarized along one or the other of two certain perpendicular crystal axes. The behavior of a waveplate (that is, whether it is a half-wave plate, a quarter-wave plate, etc.) depends on the thickness of the crystal, the wavelength of light, and the variation of the index of refraction. By appropriate choice of the relationship between these parameters, it is possible to introduce a controlled phase shift between the two polarization components of a light wave, thereby altering its polarization. With an engineered combination of two birefringent materials, an achromatic waveplate can be manufactured such that the spectral response of its phase retardance can be nearly flat. A common use of waveplates—particularly the sensitive-tint (full-wave) and quarter-wave plates—is in optical mineralogy. Addition of plates between the polarizers of a petrographic microscope makes the optical identification of minerals in thin sections of rocks easier, in particular by allowing deduction of the shape and orientation of the optical indicatrices within the visible crystal sections. This alignment can allow discrimination between minerals which otherwise appear very similar in plane polarized and cross polarized light.
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 {\displaystyle n_{\mathrm {o} }} , and the extraordinary axis, with index of refraction n e {\displaystyle n_{\mathrm {e} }} . The ordinary axis is perpendicular to the optic axis. The extraordinary axis is parallel to the optic axis. For a light wave normally incident upon the plate, the polarization component along the ordinary axis travels through the crystal with a speed v o = c n o {\displaystyle \textstyle v_{\mathrm {o} }={c \over n_{\mathrm {o} }}} , while the polarization component along the extraordinary axis travels with a speed v e = c n e {\displaystyle \textstyle v_{\mathrm {e} }={c \over n_{\mathrm {e} }}} . This leads to a phase difference between the two components as they exit the crystal. When n e < n o {\displaystyle n_{\mathrm {e} }<n_{\mathrm {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 {\displaystyle n_{\mathrm {e} }>n_{\mathrm {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 {\displaystyle \Gamma } , that it imparts on the two components, which is related to the birefringence Δ n {\displaystyle \Delta n} and the thickness L {\displaystyle L} of the crystal by the formula
Γ = 2 π Δ n L λ 0 , {\displaystyle \Gamma ={\frac {2\pi \,\Delta n\,L}{\lambda _{0}}},}
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