A wave plate or retarder is an optical

Optics

Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

 device that alters the polarization state of a light

Light

Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

 wave travelling through it.

Operation

A wave plate works by shifting the phase

Phase (waves)

Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

 between two perpendicular polarization components of the light wave. A typical wave plate is simply a birefringent

Birefringence

Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...

 crystal with a carefully chosen orientation and thickness. The crystal is cut so that the extraordinary axis or "optic axis

Optic axis of a crystal

The optic axis of a crystal is the direction in which a ray of transmitted light suffers no birefringence . Due to the internal structure of the crystal , light propagates along the optical axis differently than in other directions...

" is parallel to the surfaces of the plate. Light polarized along this axis travels through the crystal at a different speed than light with the perpendicular polarization, creating a phase difference. When the extraordinary index is smaller than the ordinary index, as in calcite

Calcite

Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate . The other polymorphs are the minerals aragonite and vaterite. Aragonite will change to calcite at 380-470°C, and vaterite is even less stable.-Properties:...

, the extraordinary axis is called the "fast axis" and the perpendicular direction in the plane of the surfaces is called the "slow axis".

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, , 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 is the vacuum wavelength of the light. For instance a quarter-wave plate creates a quarter-wavelength

Wavelength

In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

 phase shift and can change linearly polarized light

Linear polarization

In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation...

 to circular

Circular polarization

In electrodynamics, circular polarization of an electromagnetic wave is a polarization in which the electric field of the passing wave does not change strength but only changes direction in a rotary type manner....

 and vice versa. This is done by adjusting the direction of the linear polarization 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 changes the polarization direction of linear polarized light. Wave plates in general as well as polarizers can be described using the Jones matrix

Jones calculus

In optics, polarized light can be described using the Jones calculus, invented by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices...

 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.

Although the birefringence Δn may vary slightly due to dispersion

Dispersion (optics)

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...

, this is negligible compared to the variation in phase difference according to the wavelength of the light due to the fixed path difference ( 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 wave plate.

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

Quartz

Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...

 and magnesium fluoride

Magnesium fluoride

Magnesium fluoride is an inorganic compound with the formula MgF2. The compound is a white crystalline salt and is transparent over a wide range of wavelengths, with commercial uses in optics.-Production and structure:...

 the refractive index changes in the second decimal place and true zero order plates are common for wave-lengths above 1 µm.

See also

• Crystal optics
  Crystal optics
  Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be...
  
  
• Photoelastic modulator
  Photoelastic modulator
  A photoelastic modulator is an optical device used to modulate the polarization of a light source. The photoelastic effect is used to change the birefringence of the optical element in the photoelastic modulator.PEM was first invented by J...
  
  
• Circular polarization
  Circular polarization
  In electrodynamics, circular polarization of an electromagnetic wave is a polarization in which the electric field of the passing wave does not change strength but only changes direction in a rotary type manner....
  
  
• Polarization rotator
  Polarization rotator
  A polarization rotator is an optical device that rotates the polarization axis of a linearly polarized light beam by an angle of choice. These rotators are either based on the principle of birefringence or on total internal reflection....
  
  

External links

• Waveplates RP photonics Encyclopedia of Laser Physics and Technology
• Polarizers and Waveplates Animation