Physics: Problems and Solutions

Wave plate

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A wave plate or retarder is an optical device that alters the polarization state of a light wave travelling through it.

A wave plate works by shifting the phase of the light wave between two perpendicular polarization components. A typical wave plate is simply a birefringent crystal with a carefully chosen thickness. The crystal is cut so that the extraordinary axis (polarized parallel to the axis of anisotropy) is parallel to the surfaces of the plate. When the extraordinary index is smaller than the ordinary index (polarized perpendicularly to axis of anisotropy), as in calcite, the extraordinary axis is called the fast axis and the ordinary axis is called the slow axis. Light polarized along the fast axis propagates faster than light polarized along the slow axis. Thus, depending on the thickness of the crystal, light with polarization components along both axes will emerge in a different polarization state. The wave plate is characterized by the amount of relative phase \Gamma; that it imparts on the two components, which is related to the birefringence Δn and the thickness L of the crystal by the formula

\Gamma = \frac{2 \pi\, \Delta n\, L}{\lambda}

For instance a quarter-wave plate creates a quarter-wavelength phase shift and can change linearly polarized light to circular and vice versa. This is done by adjusting the plane of the incident light so that it makes 45° angle with the fast axis. This gives ordinary and extraordinary waves with equal amplitude.

The other common type of wave plate is a half-wave plate, which retards one polarization by half a wavelength, or 180 degrees. This type of wave plate rotates the polarization direction of linear polarized light. Wave plates 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 wave plate or polarizer.

Because of dispersion, a simple wave plate will impart a phase difference that depends on the wavelength of the light. Waveplates are thus manufactured to work for a particular range of wavelengths. The dispersion 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 half-wave plate, half a wavelength rather than half plus an integer. This is called a zero-order wave plate. Like lenses, wave plates can also be made achromatic by combining materials with different dispersion.

For a single wave plate changing the wavelength of the light introduces a linear error in the phase. Tilt of the wave plate enters via 1/cos 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 cos, 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 wave-lengths above 1 µm.

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