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Polarized light ENCYCLOPEDIA ARTICLES Dynamic nuclear polarization Contents [Hide] The creation of assemblies of nuclei - Types of polarized light - Analyzing devices whose spin axes are not oriented at - Law of Malus - Retardation theory - Linear polarizing devices - Bibliography random, and which are in a steady - Polarization by scattering state that is not a state of thermal...... - Production of polarized light Read Article

View all 3 related articles... Light which has its electric vector oriented in a predictable fashion with respect to the propagation direction. In unpolarized light, the vector is oriented in a random, unpredictable fashion. Even in short time intervals, it appears to RESEARCH UPDATES be oriented in all directions with equal probability. Most light sources seem to be partially polarized so that some Frozen light fraction of the light is polarized and the remainder unpolarized. It is actually more difficult to produce a completely unpolarized beam of light than one which is completely polarized. Ultraslow light was first observed by L. V. Hau and her colleagues in 1998. The polarization of light differs from its other properties in that human sense organs are essentially unable to detect Light pulses were slowed in a the presence of polarization. The Polaroid Corporation with its polarizing sunglasses and camera filters has made Bose-Einstein condensate of sodium to millions of people conscious of phenomena associated with polarization. Light from a rainbow is completely linearly only...... Read Update polarized; that is, the electric vector lies in a plane. The possessor of polarizing sunglasses discovers that with such glasses, the light from a section of the rainbow is extinguished. View all 2 related updates...

According to all available theoretical and experimental evidence, it is the electric vector rather than the magnetic MULTIMEDIA vector of a light wave that is responsible for all the effects of polarization and other observed phenomena associated ANIMATION with light. Therefore, the electric vector of a light wave, for all practical purposes, can be identified as the light vector. See also: Crystal optics; Electromagnetic radiation; Light; Polarization of waves The Eye: Structure and Function

Types of polarized light NEWS Polarized light is classified according to the orientation of the electric vector. In linearly polarized light, the electric March 14, 2010: Supertwisty light vector remains in a plane containing the propagation direction. For monochromatic light, the amplitude of the vector proposed changes sinusoidally with time. In circularly polarized light, the tip of the electric vector describes a circular helix about the propagation direction. The amplitude of the vector is constant. The frequency of rotation is equal to the View all 15 related news... frequency of the light. In elliptically polarized light, the vector also rotates about the propagation direction, but the amplitude of the vector changes so that the projection of the vector on a plane at right angles to the propagation BIOGRAPHIES direction describes an ellipse. Brewster, David

These different types of polarized light can all be broken down into two linear components at right angles to each View all 16 related biographies... other. These are defined by Eqs. (1) and (2), (1) Q&A

(2) Q: Does the color of light affect its where A and A are the amplitudes, ! and ! are the phases, " is 2# times the frequency, and t is the time. For speed in a vacuum? x y x y linearly polarized light Eqs. (3) hold. For circularly polarized light Eqs. (4) hold. For elliptically polarized light Eqs. (5) Read the Answer hold. View all 2 related Q&As... (3) (4) DICTIONARY

(5) - polarized light

In the last case, it is always possible to find a set of orthogonal axes inclined at an angle $ to x and y along which the components will be E% and E% , such that Eqs. (6) hold. x y (6)

In this new system, the x% and y% amplitudes will be the major and minor axes a and b of the ellipse described by the light vector and $ will be the angle of orientation of the ellipse axes with respect to the original coordinate system. The relationships between the different quantities can be written as in Eqs. (7) and (8). The terms are defined by Eqs. (9)–(12). (7)

(8)

(9) (10)

(11)

(12)

These same types of polarized light can also be broken down into right and left circular components or into two orthogonal elliptical components. These different vector bases are useful in different physical situations.

One of the simplest ways of producing linearly polarized light is by reflection from a dielectric surface. At a particular angle of incidence, the reflectivity for light whose electric vector is in the plane of incidence becomes zero. The reflected light is thus linearly polarized at right angles to the plane of incidence. This fact was discovered by E. Malus in 1808. Brewster's law shows that at the polarizing angle the refracted ray makes an angle of 90° with the reflected ray. By combining this relationship with Snell's law of refraction, it is found that Eq. (13) holds, (13) where & is the Brewster angle of incidence and n and n are the refractive indices of the two interfaced media (Fig. B 1 2 1). This provides a simple way of measuring refractive indices. See also: Refraction of waves

Fig. 1 Illustration of the law of Malus.

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Law of Malus If linearly polarized light is incident on a dielectric surface at Brewster's angle (the polarizing angle), then the reflectivity of the surface will depend on the angle between the incident electric vector and the plane of incidence. When the vector is in the plane of incidence, the reflectivity will vanish. To compute the complete relationship, the incident light vector A is broken into components, one vibrating in the plane of incidence and one at right angles to the plane, as in Eqs. (14) and (15),

(14)

(15) where & is the angle between the light vector and a plane perpendicular to the plane of incidence. Since the component in the plane of incidence is not reflected, the reflected ray can be written as Eq. (16),

(16)

where r is the reflectivity at Brewster's angle. The intensity is given by Eq. (17).

(17)

This is the mathematical statement of the law of Malus (Fig. 1).

Linear polarizing devices The angle & can be considered as the angle between the transmitting axes of a pair of linear polarizers. When the polarizers are parallel, they are transparent. When they are crossed, the combination is opaque. The first polarizers were glass plates inclined so that the incident light was at Brewster's angle. Such polarizers are quite inefficient since only a small percentage of the incident light is reflected as polarized light. More efficient polarizers can be constructed.

Dichroic crystals Certain natural materials absorb linearly polarized light of one vibration direction much more strongly than light vibrating at right angles. Such materials are termed dichroic. For a description of them. See also: Dichroism

Tourmaline is one of the best-known dichroic crystals, and tourmaline plates were used as polarizers for many years. A pair was usually mounted in what were known as tourmaline tongs.

Birefringent crystals Other natural materials exist in which the velocity of light depends on the vibration direction. These materials are called birefringent. The simplest of these structures are crystals in which there is one direction of propagation for which the light velocity is independent of its state of polarization. These are termed uniaxial crystals, and the propagation direction mentioned is called the optic axis. For all other propagation directions, the electric vector can be split into two components, one lying in a plane containing the optic axis and the other at right angles. The light velocity or refractive index for these two waves is different. See also: Birefringence One of the best-known of these birefringent crystals is transparent calcite (Iceland spar), and a series of polarizers have been made from this substance. W. Nicol (1829) invented the Nicol prism, which is made of two pieces of calcite cemented together as in Fig. 2. The cement is Canada balsam, in which the wave velocity is intermediate between the velocity in calcite for the fast and the slow ray. The angle at which the light strikes the boundary is such that for one ray the angle of incidence is greater than the critical angle for total reflection. Thus the rhomb is transparent for only one polarization direction.

Fig. 2 Nicol prism. The ray for which Snell's law holds is called the ordinary ray.

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Canada balsam is not completely transparent in the ultraviolet at wavelengths shorter than 400 nanometers. Furthermore, large pieces of calcite material are exceedingly rare. A series of polarizers has been made using quartz, which is transparent in the ultraviolet and which is more commonly available in large pieces. Because of the small difference between the refractive indices of quartz and Canada balsam, a Nicol prism of quartz would be tremendously long for a given linear aperture.

A different type of polarizer, made of quartz, was invented by W. H. Wollaston and is shown in Fig. 3. Here the vibration directions are different in the two pieces so that the two rays are deviated as they pass through the material. The incoming light beam is thus separated into two oppositely linearly polarized beams which have an angular separation between them. By using appropriate optical stops (obstacles that restrict light rays) in the system, it is possible to select either beam.

Fig. 3 Wollaston prism.

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In the Wollaston prism, both beams are deviated; since the quartz produces dispersion, each beam is spread into a spectrum. This is not the case of a prism which was invented by A. Rochon. Here the two pieces are arranged as in Fig. 4. One beam proceeds undeviated through the device and is thus achromatic. See also: Optical prism

Fig. 4 Rochon prism.

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Sheet polarizers A third mechanism for obtaining polarized light is the Polaroid sheet polarizer invented by E. H. Land. Sheet polarizers fall into three types. The first is a microcrystalline polarizer in which small crystals of a dichroic material are oriented parallel to each other in a plastic medium. Typical microcrystals, such as needle-shaped quinine iodosulfate, are embedded in a viscous plastic and are oriented by extruding the material through a slit.

The second type depends for its dichroism on a property of an iodine-in-water solution. The iodine appears to form a linear high polymer. If the iodine is put on a transparent oriented sheet of material such as polyvinyl alcohol (PVA), the iodine chains apparently line themselves parallel to the PVA molecules and the resulting dyed sheet is strongly dichroic. A third type of sheet polarizer depends for its dichroism directly on the molecules of the plastic itself. This plastic consists of oriented polyvinylene. Because these polarizers are commercially available and can be obtained in large sheets, many experiments involving polarized light have been performed which would have been quite difficult with the reflection polarizers or the birefringent crystal polarizer.

Characteristics There are several characteristics of linear polarizers which are of interest to the experimenter. First is the transmission for light polarized parallel and perpendicular to the axis of the polarizer; second is the angular field; and third is the linear aperture. A typical sheet polarizer has a transmittance of 48% for light parallel to the axis and 2 ' 10!4% for light perpendicular to the axis at a wavelength of 550 nm. The angular field is 60°, and sheets can be many feet in diameter. The transmittance perpendicular to the axis varies over the angular field.

The Nicol prism has transmittance similar to that of the Polaroid sheeting but a much reduced linear and angular aperture.

Polarization by scattering When an unpolarized light beam is scattered by molecules or small particles, the light observed at right angles to the original beam is polarized. The light vector in the original beam can be considered as driving the equivalent oscillators (nuclei and electrons) in the molecules. There is no longitudinal component in the original light beam. Accordingly, the scattered light observed at right angles to the beam can only be polarized with the electric vector at right angles to the propagation direction of the original beam. In most situations, the scattered light is only partially polarized because of multiple scattering. The best-known example of polarization by scattering is the light of the north sky. The percentage polarization can be quite high in clean country air. A technique was invented for using measurements of sky polarization to determine the position of the Sun when it is below the horizon. See also: Scattering of electromagnetic radiation

Production of polarized light Linear polarizers have already been discussed. Circularly and elliptically polarized light are normally produced by combining a linear polarizer with a wave plate. A Fresnel rhomb can be used to produce circularly polarized light.

Wave plate A plate of material which is linearly birefringent is called a wave plate or retardation sheet. Wave plates have a pair of orthogonal axes which are designated fast and slow. Polarized light with its electric vector parallel to the fast axis travels faster than light polarized parallel to the slow axis. The thickness of the material can be chosen so that for light traversing the plate, there is a definite phase shift between the fast component and the slow component. A plate with a 90° phase shift is termed a quarter-wave plate. The retardation in waves is given by Eq. (18),

(18)

where n ( n is the birefringence; n is the slow index at wavelength ); n is the fast index; and d is the plate s f s f thickness.

Wave plates can be made by preparing X-cut sections of quartz, calcite, or other birefringent crystals. For retardations of less than a few waves, it is easiest to use sheets of oriented plastics or of split mica. A quarter-wave plate for the visible or infrared is easy to fabricate from mica. The plastic wrappers from many American cigarette packages seem to have almost exactly a half-wave retardation from green light. Since mica is not transparent in the ultraviolet, a small retardation in this region is most easily achieved by crossing two quartz plates which differ by the requisite thickness.

Linearly polarized light incident normally on a quarter-wave plate and oriented at 45° to the fast axis can be split into two equal components parallel to the fast and slow axes. These can be represented, before passing through the plate, by Eqs. (19) and (20), (19)

(20) where x and y are parallel to the wave-plate axes. After passing through the plate, the two components can be written as Eqs. (21) and (22),

(21)

(22) where E is now advanced one quarter-wave with respect to E . x y It is possible to visualize the behavior of the light by studying the sketches in Fig. 5, which show the projection on a plane z = 0 at various times. It is apparent that the light vector is of constant amplitude, and that the projection on a plane normal to the propagation direction is a circle. If the linearly polarized light is oriented at (45° to the fast axis, the light vector will revolve in the opposite direction. Thus it is possible with a quarter-wave plate and a linear polarizer to make either right or left circularly polarized light. If the linearly polarized light is at an angle other than 45° to the fastest axis, the transmitted radiation will be elliptically polarized. When circularly polarized light is incident on a quarter-wave plate, the transmitted light is linearly polarized at an angle of 45° to the wave-plate axes. This polarization is independent of the orientation of the wave-plate axis. For elliptically polarized light, the behavior of the quarter-wave plate is much more complicated. However, as was mentioned earlier, the elliptically polarized light can be considered as composed of two linear components parallel to the major and minor axes of the ellipse and with a quarter-wave phase difference between them. If the quarter-wave plate is oriented parallel to the axes of the ellipse, the two transmitted components will either have zero phase difference or a 180° phase difference and will be linearly polarized. At other angles, the transmitted light will still be elliptically polarized, but with different major and minor axes. Similar treatment for a half-wave plate shows that linearly polarized light oriented at an angle of & to the fast axis is transmitted as linearly polarized light oriented at an angle (& to the fast axis.

Fig. 5 Projection of light vector of constant amplitude on plane z = 0 for circularly polarized light. (a) t = 0. (b) t = !/4". (c) t = !/2". (d) t = 3!/4".

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Wave plates all possess a different retardation at each wavelength. This appears immediately from Eq. (18). It is conceivable that a substance could have dispersion of birefringence, such as to make the retardation of a plate independent of wavelength. However, no material having such a characteristic has as yet been found.

Fresnel rhomb A quarter-wave retardation can be provided achromatically by the Fresnel rhomb. This device depends on the phase shift which occurs at total internal reflection. When linearly polarized light is totally internally reflected, it experiences a phase shift which depends on the angle of reflection, the refractive index of the material, and the orientation of the plane of polarization. Light polarized in the plane of incidence experiences a phase shift which is different from that of light polarized at right angles to the plane of incidence. Light polarized at an intermediate angle can be split into two components, parallel and at right angles to the plane of incidence, and the two components mathematically combined after reflection.

The phase shifts can be written Eqs. (23) and (24),

(23)

(24)

where ! is the phase shift parallel to the plane of incidence; ! is the phase shift at right angles to the plane of ! " incidence; i is the angle of incidence on the totally reflecting internal surface; and n is the refractive index. The difference ! ( ! reaches a value of about #/4 at an angle of 52° for n = 1.50. Two such reflections give a retardation ! " of #/2. The Fresnel rhomb shown in Fig. 6 is cut so that the incident light is reflected twice at 52°. Accordingly, light polarized at 45° to the principal plane will be split into two equal components which will be shifted a quarter-wave with respect to each other, and the transmitted light will be circularly polarized. Nearly achromatic wave plates can be made by using a series of wave plates in series with their axes oriented at different specific angles with respect to a coordinate system.

Fig. 6 Fresnel rhomb.

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Analyzing devices Polarized light is one of the most useful tools for studying the characteristics of materials. The absorption constant and refractive index of a metal can be calculated by measuring the effect of the metal on polarized light reflected from its surface. See also: Reflection of electromagnetic radiation

The analysis of polarized light can be performed with a variety of different devices. If the light is linearly polarized, it can be extinguished by a linear polarizer and the direction of polarization of the light determined directly from the orientation of the polarizer. If the light is elliptically polarized, it can be analyzed with the combination of a quarter-wave plate and a linear polarizer. Any such combination of polarizer and analyzer is called a polariscope. As explained previously, a quarter-wave plate oriented parallel to one of the axes of the ellipse will transform elliptically polarized light to linearly polarized light. Accordingly, the quarter-wave plate is rotated until the beam is extinguished by the linear polarizer. At this point, the orientation of the quarter-wave plate gives the orientation of the ellipse and the orientation of the polarizer gives the ratio of the major to minor axis. Knowledge of the origin of the elliptically polarized light usually gives the orientation of the components which produced it, and from these various items, the phase shifts and attenuations produced by the experiment can be deduced.

One of the best-known tools for working with polarized light is the Babinet compensator. This device is normally made of two quartz prisms put together in a rhomb. One prism is cut with the optic axis in the plane of incidence of the prism, and the other with the optic axis perpendicular to the plane of incidence. The retardation is a function of distance along the rhomb; it will be zero at the center, varying to positive and negative values in opposite directions along the rhomb. It can be used to cancel the known or unknown retardation of any wave plate.

Retardation theory It is difficult to see intuitively the effect of a series of retardation plates or even of a single plate of general retardation * on light which is normally incident on the plate and which is polarized in a general fashion. This problem is most easily solved algebraically. The single-wave plate is assumed to be oriented normal to the direction of propagation of the light, which is taken to be the z direction of a set of cartesian coordinates. Its fast axis is at an angle $ to the x axis. The incident light can be represented by Eqs. (25) and (26). (25)

(26) A first step is to break the light up into components E and E parallel to the axes of the plate. It is possible to write x" y" Eqs. (27) and (28). These components can also be written as Eqs. (29) and (30). After passing through the plate, the components become Eqs. (31) and (32). (27)

(28)

(29)

(30)

(31)

(32)

In general, it is of interest to compare the output with the input. The transmitted light is thus broken down into components along the original axes. This results in Eqs. (33) and (34). (33) (34)

With this set of equations, it is possible to compute the effect of a wave plate on any form of polarized light.

Jones calculus Equations (33) and (34) still become overwhelmingly complicated in any system involving several optical elements. Various methods have been developed to simplify the problem and to make possible some generalizations about systems of elements. One of the most straightforward, proposed by R. C. Jones, involves reducing Eqs. (33) and (34) to matrix form. The Jones calculus for optical systems involves the polarized electric components of the light vector and is distinguished from other methods in that it takes cognizance of the absolute phase of the light wave.

The Jones calculus writes the light vector in the complex form as in Eqs. (35) and (36).

(35)

(36)

Matrix operators are developed for different optical elements. From Eqs. (25)–(34), the operator for a wave plate can be derived directly. See also: Matrix theory

The Jones calculus is ordinarily used in a normalized form which simplifies the matrices to a considerable extent. In this form, the terms involving the actual amplitude and absolute phase of the vectors and operators are factored out of the expressions. The intensity of the light beam is reduced to unity in the normalized vector so that Eq. (37) holds. (37) Under this arrangement, the matrices for various types of operations can be written as Eq. (38).

(38)

This is the operator for a wave plate of retardation * and with axes along x and y. Equation (39) gives the operator for a rotator which rotates linearly polarized light through an angle $.

(39) Equation (40) gives the operator for a perfect linear polarizer parallel to the x axis.

(40)

A wave plate at an angle $ can be represented by Eq. (41). (41) A series of optical elements can be represented by the product of a series of matrices. This simplifies enormously the task of computing the effect of many elements. It is also possible with the Jones calculus to derive a series of general theorems concerning combinations of optical elements. Jones has described three of these, all of which apply only for monochromatic light.

1. An optical system consisting of any number of retardation plates and rotators is optically equivalent to a system containing only two elements, a retardation plate and a rotator.

2. An optical system containing any number of partial polarizers and rotators is optically equivalent to a system containing only two elements—one a partial polarizer and the other a rotator.

3. An optical system containing any number of retardation plates, partial polarizers, and rotators is optically equivalent to a system containing four elements—two retardation plates, one partial polarizer, and one rotator.

As an example of the power of the calculus, a rather specific theorem can be proved. A rotator of any given angle $ can be formed by a sequence of three retardation plates, a quarter-wave plate, a retardation plate at 45° to the quarter-wave plate, and a second quarter-wave plate crossed with the first, as in Eq. (42),

(42)

where + is the angle between the axis of the first quarter-wave plate and the x axis, and * is the retardation of the plate in the middle of the sandwich.

The first simplification arises from the fact that the axis rotations can be done in any order. This reduces Eq. (43) to Eq. (44). Now Eqs. (45) and (46) hold. When the multiplication is carried through, Eq. (47) is obtained.

(43)

(44)

(45)

(46)

(47)

The rotation angle is therefore equal to one-half the phase angle of the retardation. This combination is a true rotator in that the rotation is independent of the azimuth angle of the incident polarized light.

A variable rotator can be made by using a Soleil compensator for the central element. This consists of two quartz wedges joined to form a plane parallel quartz plate.

Mueller matrices In the Jones calculus, the intensity of the light passing through the system must be obtained by calculation from the components of the light vector. A second calculus is frequently used in which the light vector is split into four components. This also uses matrix operators which are termed Mueller matrices. In this calculus, the intensity I of the light is one component of the vector and thus is automatically calculated. The other components of the vector are given by Eqs. (48)–(50).

(48)

(49)

(50) The matrix of a perfect polarizer parallel to the x axis can be written as Eq. (51).

(51)

The calculus can treat unpolarized light directly. Such a light vector is given by Eq. (52).

(52) The vector for light polarized parallel to the x axis is written as expression (53).

(53)

In the same manner as in the Jones calculus, matrices can be derived for retardation plates, rotators, and partial polarizers. This calculus can also be used to derive various general theorems about various optical systems. See also: Faraday effect; Interference of waves; Optical activity; Optical rotatory dispersion Bruce H. Billings Hong Hua Bibliography R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 1977, paper 1987 B. Billings (ed.), Selected Papers on Applications of Polarized Light, 1992 M. Born and E. Wolf, Principles of Optics, 7th ed., 1999 E. Collett, Polarized Light, 1993 F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed., 1976 D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy, 1990

How to cite this article Bruce H. Billings, Hong Hua, "Polarized light," in AccessScience, ©McGraw-Hill Companies, 2008, http://www.accessscience.com See MLA or APA style

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