A polarizer or polariser is an optical filter that lets light of a specific polarization pass and blocks waves of additional polarizations. It can convert a beam of light of undefined or mixed polarisation into a beam of well-defined polarization, that's polarized light. The common types of polarizers are linear polarizers and circular polarizers. Polarizers are used in a large number of optical techniques and instruments, and polarizing filters find applications in photography and liquid crystal display technology. Polarizers can additionally be made for additional types of electromagnetic waves besides light, such as radio waves, microwaves, and X-rays.

Linear polarizers

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

Absorptive polarizers

Wire-grid polarizer

A wire-grid polarizer converts an unpolarized beam into one with a single linear polarization. Coloured arrows depict the electric field vector. The diagonally polarised waves additionally contribute to the transmitted polarization. Their vertical components are transmitted, while the horizontal components are absorbed and reflected. (This isn't clearly shown.)

The simplest linear polarizer in concept is the wire-grid polarizer, which consists of a regular array of fine parallel metallic wires, placed in a plane perpendicular to the incident beam. Electromagnetic waves which have a component of their electric fields aligned parallel to the wires induce the movement of electrons along the length of the wires. Since the electrons are free to move in this direction, the polarizer behaves in a similar manner to the surface of a metal when reflecting light, and the wave is reflected backwards along the incident beam (minus a small amount of energy lost to joule heating of the wire).

For waves with electric fields perpendicular to the wires, the electrons can't move quite far across the width of each wire; therefore, little energy is reflected, and the incident wave is able to pass through the grid. Since electric field components parallel to the wires are reflected, the transmitted wave has an electric field purely in the direction perpendicular to the wires, and is thus linearly polarized. Note that the polarisation direction is perpendicular to the wires; the notion that waves "slip through" the gaps between the wires is wrong.

For practical use, the separation distance between the wires must be less than the wavelength of the radiation, and the wire width should be a small fraction of this distance. This means that wire-grid polarizers generally work best for microwaves and for far- and mid-infrared light. Notwithstanding using advanced lithographic techniques, quite tight pitch metallic grids can be made which polarise visible light to a useful degree. Since the degree of polarization depends little on wavelength and angle of incidence, they're used for broad-band applications such as projection.

Other absorptive polarizers

Certain crystals, due to the effects described by crystal optics, show dichroism, preferential absorption of light which is polarised in particular directions. They can therefore be used as linear polarizers. The best known crystal of this type is tourmaline. Notwithstanding this crystal is seldom used as a polarizer, after the dichroic effect is strongly wavelength dependent and the crystal appears coloured. Herapathite is additionally dichroic, and isn't strongly coloured, but is difficult to grow in large crystals.

A Polaroid polarising philtre functions similarly on an atomic scale to the wire-grid polarizer. It was originally made of microscopic herapathite crystals. Its current H-sheet form is made from polyvinyl alcohol (PVA) plastic with an iodine doping. Stretching of the sheet throughout manufacture causes the PVA chains to align in one particular direction. Valence electrons from the iodine dopant are able to move linearly along the polymer chains, but not transverse to them. So incident light polarised parallel to the chains is absorbed by the sheet; light polarised perpendicularly to the chains is transmitted. The durability and practicality of Polaroid makes it the most common type of polarizer in use, for example for sunglasses, photographic filters, and liquid crystal displays. It is additionally much cheaper than additional types of polarizer.

A modern type of absorptive polarizer is made of elongated silver nanoparticles embedded in thin (≤0.5 mm) glass plates. These polarizers are more durable, and can polarise light much better than plastic Polaroid film, achieving polarisation ratios as high as 100,000:1 and absorption of correctly polarised light as low as 1.5%. Such glass polarizers perform best for short-wavelength infrared light, and are widely used in optical fibre communications.

Beam-splitting polarizers

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

Unlike absorptive polarizers, beam splitting polarizers don't need to absorb and dissipate the energy of the rejected polarisation state, and so they're more suitable for use with high intensity beams such as laser light. True polarising beamsplitters are additionally useful where the two polarisation components are to be analysed or used simultaneously.

Polarization by reflection

A stack of plates at Brewster's angle to a beam reflects off a fraction of the s-polarized light at each surface, leaving a p-polarized beam. Full polarisation at Brewster's angle requires a large number of more plates than shown. The arrows indicate the direction of the electrical field, not the magnetic field, which is perpendicular to the electric field

When light reflects at an angle from an interface between two transparent materials, the reflectivity is different for light polarised in the plane of incidence and light polarised perpendicular to it. Light polarised in the plane is said to be p-polarized, while that polarised 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 linear 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 sixteen percent of the s-polarized light present in the beam is reflected for each air-to-glass or glass-to-air transition. It takes a large number of plates to achieve even mediocre polarisation of the transmitted beam with this approach. For a stack of 10 plates (20 reflections), about three percent (= (1-0.16)20) of the s-polarized light is transmitted. The reflected beam, while fully polarized, is spread out and might not be quite useful.

A more useful polarised 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 polarisation of the transmitted beam, at the expense of decreased overall transmission. For angles of incidence steeper than 80° the polarisation of the transmitted beam can approach one hundred percent with as few as four plates, although the transmitted intensity is quite low in this case. Adding more plates and reducing the angle allows a better compromise between transmission and polarisation to be achieved.

Birefringent polarizers

Other linear 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 polarisation states, though not in linear polarisation states except for certain propagation directions relative to the crystal axis. The two rays additionally experience differing refractive indices in the crystal.

A Nicol prism

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 polarisation states. Total internal reflection of the o-ray occurs at the balsam interface, after 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 quite high purity of polarised 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 aren't true polarising beamsplitters after only the transmitted beam is fully polarized.

A Wollaston prism

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 polarised 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 polarised beams with perpendicular polarizations. The Nomarski prism is a variant of the Wollaston prism, which is widely used in differential interference contrast microscopy.

Thin film polarizers

Thin-film linear polarizers are glass substrates on which a special optical coating is applied. Either Brewster's angle reflections or 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's cemented to a second wedge to form a cube with the film cutting diagonally across the centre (one form of this is the quite common MacNeille cube). Thin-film polarizers generally don't perform as well as Glan-type polarizers, but they're 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 additional properties

Polarization of light.
In this picture, θ1θ0 = θi.
If you have 3 linear filters, hold two philtres crossed to block the light with your clumsy hand and use your clever hand to insert third at 45°.

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

where I0 is the initial intensity, and θi is the angle between the light's initial polarisation direction 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 is 1/2, the transmission coefficient becomes

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

If two polarizers are placed one after another (the second polarizer is generally called an analyzer), the mutual angle between their polarising 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 isn't exactly zero (for example, crossed Polaroid sheets appear slightly blue in colour). If a transparent object is placed between the crossed polarizers, any polarisation effects present in the sample (such as birefringence) will be shown as an increase in transmission. This effect is used in polarimetry to measure the optical activity of a sample.

Real polarizers are additionally not perfect blockers of the polarisation orthogonal to their polarisation 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.

In X-ray the Malus’ law (relativistic form):

where - frequency of the polarised radiation falling on the polarizer, - frequency of the radiation passes through polarizer, - Compton wavelength of electron, - speed of light in vacuum.

Circular polarizers

Circular polarizers, additionally referred to as circular polarising filters, can be used to create circularly polarized light or alternatively to selectively absorb or pass clockwise and counter-clockwise circularly polarized light. They are used as polarizing philtres in photography to reduce oblique reflections from non-metallic surfaces, and are the lenses of the 3D glasses worn for viewing a few stereoscopic movies (notably, the RealD 3D variety), where the polarisation of light is used to distinguish which image should be seen by the left and right eye.

Creating circularly polarised light

Circular polarizer creating left-handed circularly polarised light. It is considered left-handed as viewed from the receiver and right-handed as viewed from the source.

There are several ways to create circularly polarised light, the cheapest and most common involves placing a quarter-wave plate after a linear polarizer and directing unpolarized light through the linear polarizer. The linearly polarised light leaving the linear polarizer is transformed into circularly polarised light by the quarter wave plate. The transmission axis of the linear polarizer needs to be half way (45°) between the fast and slow axes of the quarter-wave plate.
In the arrangement above, the transmission axis of the linear polarizer is at a positive 45° angle relative to the right horizontal and is represented with an orange line. The quarter-wave plate has a horizontal slow axis and a vertical fast axis and they're additionally represented using orange lines. In this instance the unpolarized light entering the linear polarizer is displayed as a single wave whose amplitude and angle of linear polarisation are suddenly changing.
When one attempts to pass unpolarized light through the linear polarizer, only light that has its electric field at the positive 45° angle leaves the linear polarizer and enters the quarter-wave plate. In the illustration, the three wavelengths of unpolarized light represented would be transformed into the three wavelengths of linearly polarised light on the additional side of the linear polarizer.

Linearly polarised light, represented using components, entering a quarter-wave plate. The blue and green curves are projections of the red line on the vertical and horizontal planes respectively.

In the illustration toward the right is the electric field of the linearly polarised light just before it enters the quarter-wave plate. The red line and associated field vectors represent how the magnitude and direction of the electric field varies along the direction of travel. For this plane electromagnetic wave, each vector represents the magnitude and direction of the electric field for an entire plane that's perpendicular to the direction of travel. Refer to these two images in the plane wave article to better appreciate this.
Light and all additional electromagnetic waves have a magnetic field which is in phase with, and perpendicular to, the electric field being displayed in these illustrations.
To understand the effect the quarter-wave plate has on the linearly polarised light it is useful think of the light as being divided into two components which are at right angles (orthogonal) to each other. Towards this end, the blue and green lines are projections of the red line onto the vertical and horizontal planes respectively and represent how the electric field changes in the direction of those two planes. The two components have the same amplitude and are in phase.
Because the quarter-wave plate is made of a birefringent material, when in the wave plate, the light travels at different speeds depending on the direction of its electric field. This means that the horizontal component which is along the slow axis of the wave plate will travel at a slower speed than the component that's directed along the vertical fast axis. Initially the two components are in phase, but as the two components travel through the wave plate the horizontal component of the light drifts farther behind that of the vertical. By adjusting the thickness of the wave plate one can control how much the horizontal component is delayed relative to vertical component before the light leaves the wave plate and they begin again to travel at the same speed. When the light leaves the quarter-wave plate the rightward horizontal component will be exactly one quarter of a wavelength behind the vertical component making the light left-hand circularly polarised when viewed from the receiver.

The top image is left-handed/counter-clockwise circularly polarized, as viewed from the receiver. The bottom image is that of linearly polarised light. The blue and green curves are projections of the red lines on the vertical and horizontal planes respectively.

At the top of the illustration toward the right, is the circularly polarised light after it leaves the wave plate, and again directly below it, for comparison purposes, the linearly polarised light that entered the quarter-wave plate. In the upper image, because this is a plane wave, each vector leading from the axis to the helix represents the magnitude and direction of the electric field for an entire plane that's perpendicular to the direction of travel. All the electric field vectors have the same magnitude indicating that the strength of the electric field doesn't change. The direction of the electric field however steadily rotates.
The blue and green lines are projections of the helix onto the vertical and horizontal planes respectively and represent how the electric field changes in the direction of those two planes. Notice how the rightward horizontal component is now one quarter of a wavelength behind the vertical component. It is this quarter of a wavelength phase shift that results in the rotational nature of the electric field. It is significant to note that when the magnitude of one component is at a maximum the magnitude of the additional component is always zero. This is the reason that there are helix vectors which exactly correspond to the maxima of the two components.

Animation of left-handed/counter-clockwise circularly polarised light. (Left-handed as viewed from the receiver.)

In the instance just cited, using the handedness convention used in a large number of optics textbooks, the light is considered left-handed/counter-clockwise circularly polarized. Referring to the accompanying animation, it is considered left-handed because if one points one's left thumb against the direction of travel, ones fingers curl in the direction the electric field rotates as the wave passes a given point in space. The helix additionally forms a left-handed helix in space. Similarly this light is considered counter-clockwise circularly polarised because if a stationary observer faces against the direction of travel, the person will observe its electric field rotate in the counter-clockwise direction as the wave passes a given point in space.

To create right-handed, clockwise circularly polarised light one simply rotates the axis of the quarter-wave plate 90° relative to the linear polarizer. This reverses the fast and slow axes of the wave plate relative to the transmission axis of the linear polarizer reversing which component leads and which component lags.

In trying to appreciate how the quarter-wave plate transforms the linearly polarised light, it is important to realise that the two components discussed aren't entities in and of themselves but are merely mental constructs one uses to help appreciate what's happening. In the case of linearly and circularly polarised light, at each point in space, there's always a single electric field with a distinct vector direction, the quarter-wave plate merely has the effect of transforming this single electric field.

Absorbing and passing circularly polarised light

Circular polarizers can additionally be used to selectively absorb or pass right-handed or left-handed circularly polarised light. It is this feature which is utilised by the 3D glasses in stereoscopic cinemas such as RealD Cinema. A given polarizer which creates one of the two polarizations of light will pass that same polarisation of light when that light is sent through it in the additional direction. In contrast it will block light of the opposite polarization.

Circular polarizer passing left-handed, counter-clockwise circularly polarised light. (Left-handed as viewed from the receiver.)

The illustration above is identical to the previous similar one with the exception that the left-handed circularly polarised light is now approaching the polarizer from the opposite direction and linearly polarised light is exiting the polarizer toward the right.
First note that a quarter-wave plate always transforms circularly polarised light into linearly polarised light. It is only the resulting angle of polarisation of the linearly polarised light that's determined by the orientation of the fast and slow axes of the quarter-wave plate and the handedness of the circularly polarised light. In the illustration, the left-handed circularly polarised light entering the polarizer is transformed into linearly polarised light which has its direction of polarisation along the transmission axis of the linear polarizer and it therefore passes. In contrast right-handed circularly polarised light would have been transformed into linearly polarised light that had its direction of polarisation along the absorbing axis of the linear polarizer, which is at right angles to the transmission axis, and it would have therefore been blocked.

Left-handed/Counter-Clockwise circularly polarised light displayed above linearly polarised light. The blue and green curves are projections of the helix on the vertical and horizontal planes respectively.

To understand this process, refer to the illustration on the right. It is absolutely identical to the earlier illustration even though the circularly polarised light at the top is now considered to be approaching the polarizer from the left. One can observe from the illustration that the leftward horizontal (as observed looking along the direction of travel) component is leading the vertical component and that when the horizontal component is retarded by one quarter of a wavelength it will be transformed into the linearly polarised light illustrated at the bottom and it will pass through the linear polarizer.

There is a relatively straightforward way to appreciate why a polarizer which creates a given handedness of circularly polarised light additionally passes that same handedness of polarised light. First, given the dual usefulness of this image, begin by imagining the circularly polarised light displayed at the top as still leaving the quarter-wave plate and travelling toward the left. Observe that had the horizontal component of the linearly polarised light been retarded by a quarter of wavelength twice, which would amount to a full half wavelength, the result would have been linearly polarised light that was at a right angle to the light that entered. If such orthogonally polarised light were rotated on the horizontal plane and directed back through the linear polarizer section of the circular polarizer it would clearly pass through given its orientation. Now imagine the circularly polarised light which has already passed through the quarter-wave plate once, turned around and directed back toward the circular polarizer again. Let the circularly polarised light illustrated at the top now represent that light. Such light is going to travel through the quarter-wave plate a second time before reaching the linear polarizer and in the process, its horizontal component is going to be retarded a second time by one quarter of a wavelength. Whether that horizontal component is retarded by one quarter of a wavelength in two distinct steps or retarded a full half wavelength all at once, the orientation of the resulting linearly polarised light will be such that it passes through the linear polarizer.

Had it been right-handed, clockwise circularly polarised light approaching the circular polarizer from the left, its horizontal component would have additionally been retarded, however the resulting linearly polarised light would have been polarised along the absorbing axis of the linear polarizer and it wouldn't have passed.

To create a circular polarizer that instead passes right-handed polarised light and absorbs left-handed light, one again rotates the wave plate and linear polarizer 90° relative to each another. It is easy to appreciate that by reversing the positions of the transmitting and absorbing axes of the linear polarizer relative to the quarter-wave plate, one changes which handedness of polarised light gets transmitted and which gets absorbed.

Homogeneous circular polarizer

Homogeneous circular polarizer passing left-handed, counter-clockwise circularly polarised light. (Left-handed as viewed from the receiver.)

A homogeneous circular polarizer passes one handedness of circular polarisation unaltered and blocks the additional handedness. This is similar to the way that a linear polarizer would fully pass one angle of linearly polarised light unaltered, but would fully block any linearly polarised light that was orthogonal to it.

A homogeneous circular polarizer can be created by sandwiching a linear polarizer between two quarter-wave plates. Specifically we take the circular polarizer described previously, which transforms circularly polarised light into linear polarised light, and add to it a second quarter-wave plate rotated 90° relative to the first one.

Generally speaking, and not making direct reference to the above illustration, when either of the two polarizations of circularly polarised light enters the first quarter-wave plate, one of a pair of orthogonal components is retarded by one quarter of a wavelength relative to the other. This creates one of two linear polarizations depending on the handedness the circularly polarised light. The linear polarizer sandwiched between the quarter wave plates is oriented so that it will pass one linear polarisation and block the other. The second quarter-wave plate then takes the linearly polarised light that passes and retards the orthogonal component that wasn't retarded by the previous quarter-wave plate. This brings the two components back into their initial phase relationship, reestablishing the selected circular polarization.
Note that it doesn't matter in which direction one passes the circularly polarised light.

Circular and linear types

Linear polarising philtres were the first types to be used in photography and can still be used for non-reflex and older SLR cameras. Notwithstanding cameras with through-the-lens metering and autofocusing systems - that is, all modern SLR and DSLR - rely on optical elements that pass linearly polarised light. If light entering the camera is already linearly polarized, it can upset the exposure or autofocus systems. Circular polarising philtres cut out linearly polarised light and so can be used to darken skies or remove reflections, but the circular polarised light it passes doesn't impair through-the-lens systems.

Related to circular polarizers