# Polarizer

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A polarizer is a device that converts an unpolarized or mixed-polarization beam of electromagnetic waves (e.g., light) into a beam with a single polarization state (usually, a single linear polarization). Polarizers are used in many optical techniques and instruments, and polarizing filters find applications in photography and liquid crystal display technology.

Polarizers can be divided into two general categories: absorptive polarizers, where the unwanted polarization states are absorbed by the device, and beam-splitting polarizers, where the unpolarized beam is split into two beams with opposite polarization states.

## Beam-splitting polarizersEdit

Beam-splitting polarizers split the incident beam into two beams of differing polarization. For an ideal polarizing beamsplitter these would be fully polarized, with orthogonal polarizations. For many common beam-splitting polarizers, however, only one of the two output beams is fully polarized. The other contains a mixture of polarization states.

Unlike absorptive polarizers, beam splitting polarizers do not need to absorb and dissipate the energy of the rejected polarization state, and so they are more suitable for use with high intensity beams such as laser light. True polarizing beamsplitters are also useful where the two polarization components are to be analyzed or used simultaneously.

### Polarization by reflectionEdit

When light reflects at an angle from an interface between two transparent materials, the reflectivity is different for light polarized in the plane of incidence and light polarized perpendicular to it. Light polarized in the plane is said to be p-polarized, while that polarized perpendicular to it is s-polarized. At a special angle known as Brewster's angle, no p-polarized light is reflected from the surface, thus all reflected light must be s-polarized, with an electric field perpendicular to the plane of incidence.

A simple polarizer can be made by tilting a stack of glass plates at Brewster's angle to the beam. Some of the s-polarized light is reflected from each surface of each plate. For a stack of plates, each reflection depletes the incident beam of s-polarized light, leaving a greater fraction of p-polarized light in the transmitted beam at each stage. For visible light in air and typical glass, Brewster's angle is about 57°, and about 16% of the s-polarized light present in the beam is reflected for each air-to-glass or glass-to-air transition. It takes many plates to achieve even mediocre polarization of the transmitted beam with this approach. For a stack of 10 plates (20 reflections), about 3% (= (1-0.16)20) of the s-polarized light is transmitted. The reflected beam, while fully polarized, is spread out and may not be very useful.

A more useful polarized beam can be obtained by tilting the pile of plates at a steeper angle to the incident beam. Counterintuitively, using incident angles greater than Brewster's angle yields a higher degree of polarization of the transmitted beam, at the expense of decreased overall transmission. For angles of incidence steeper than 80° the polarization of the transmitted beam can approach 100% with as few as four plates, although the transmitted intensity is very low in this case[1]. Adding more plates and reducing the angle allows a better compromise between transmission and polarization to be achieved.

### Birefringent polarizersEdit

Other polarizers exploit the birefringent properties of crystals such as quartz and calcite. In these crystals, a beam of unpolarized light incident on their surface is split by refraction into two rays. Snell's law holds for one of these rays, the ordinary or o-ray, but not for the other, the extraordinary or e-ray. In general the two rays will be in different polarization states, though not in linear polarization states except for certain propagation directions relative to the crystal axis. The two rays also experience differing refractive indices in the crystal.

A Nicol prism was an early type of birefringent polarizer, that consists of a crystal of calcite which has been split and rejoined with Canada balsam. The crystal is cut such that the o- and e-rays are in orthogonal linear polarization states. Total internal reflection of the o-ray occurs at the balsam interface, since it experiences a larger refractive index in calcite than in the balsam, and the ray is deflected to the side of the crystal. The e-ray, which sees a smaller refractive index in the calcite, is transmitted through the interface without deflection. Nicol prisms produce a very high purity of polarized light, and were extensively used in microscopy, though in modern use they have been mostly replaced with alternatives such as the Glan-Thompson prism, Glan-Foucault prism, and Glan-Taylor prism. These prisms are not true polarizing beamsplitters since only the transmitted beam is fully polarized.

A Wollaston prism is another birefringent polarizer consisting of two triangular calcite prisms with orthogonal crystal axes that are cemented together. At the internal interface, an unpolarized beam splits into two linearly polarized rays which leave the prism at a divergence angle of 15°-45°. The Rochon and Sénarmont prisms are similar, but use different optical axis orientations in the two prisms. The Sénarmont prism is air spaced, unlike the Wollaston and Rochon prisms. These prisms truly split the beam into two fully polarized beams with perpendicular polarizations. The Nomarski prism is a variant of the Wollaston prism, which is widely used in differential interference contrast microscopy.looll

### Thin film polarizersEdit

Thin-film polarizers are glass substrates on which a special optical coating is applied. Interference effects in the film cause them to act as beam-splitting polarizers. The substrate for the film can either be a plate, which is inserted into the beam at a particular angle, or a wedge of glass that is cemented to a second wedge to form a cube with the film cutting diagonally across the center.

Thin-film polarizers generally do not perform as well as Glan-type polarizers, but they are inexpensive and provide two beams that are about equally well polarized. The cube-type polarizers generally perform better than the plate polarizers. The former are easily confused with Glan-type birefringent polarizers.

## Malus' law and other propertiesEdit

Malus' law, which is named after Etienne-Louis Malus, says that when a perfect polarizer is placed in a polarized beam of light, the intensity, I, of the light that passes through is given by

$I = I_0 \cos^2 \theta_i \quad ,$

where

I0 is the initial intensity,
and θi is the angle between the light's initial plane of polarization and the axis of the polarizer.

A beam of unpolarized light can be thought of as containing a uniform mixture of linear polarizations at all possible angles. Since the average value of $\cos^2 \theta$ is 1/2, the transmission coefficient becomes

$\frac {I}{I_0} = \frac {1}{2}\quad .$

In practice, some light is lost in the polarizer and the actual transmission of unpolarized light will be somewhat lower than this, around 38% for Polaroid-type polarizers but considerably higher (>49.9%) for some birefringent prism types.

If two polarizers are placed one after another (the second polarizer is generally called an analyzer), the mutual angle between their polarizing axes gives the value of θ in Malus' law. If the two axes are orthogonal, the polarizers are crossed and in theory no light is transmitted, though again practically speaking no polarizer is perfect and the transmission is not exactly zero (for example, crossed Polaroid sheets appear slightly blue in colour). If a transparent object is placed between the crossed polarizers, any polarization effects present in the sample (such as birefringence) will be shown as increases in transmission. This effect is used in polarimetry to measure the optical activity of a sample.

Real polarizers are also not perfect blockers of the polarization orthogonal to their polarization axis; the ratio of the transmission of the unwanted component to the wanted component is called the extinction ratio, and varies from around 1:500 for Polaroid to about 1:106 for Glan-Taylor prism polarizers.