Stokes parameters

The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation. They were defined by George Gabriel Stokes in 1852,[1][2] as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of polarization (p), and the shape parameters of the polarization ellipse. The effect of an optical system on the polarization of light can be determined by constructing the Stokes vector for the input light and applying Mueller calculus, to obtain the Stokes vector of the light leaving the system. The parameters were named after Stokes first by Subrahamanyan Chandrasekhar.[3]

Definitions

Polarisation ellipse2
Polarisation ellipse, showing the relationship to the Poincaré sphere parameters ψ and χ.
Poincaré sphere
The Poincaré sphere is the parametrisation of the last three Stokes' parameters in spherical coordinates.

The relationship of the Stokes parameters S0, S1, S2, S3 to intensity and polarization ellipse parameters is shown in the equations below and the figure at right.

Here , and are the spherical coordinates of the three-dimensional vector of cartesian coordinates . is the total intensity of the beam, and is the degree of polarization, constrained by . The factor of two before represents the fact that any polarization ellipse is indistinguishable from one rotated by 180°, while the factor of two before indicates that an ellipse is indistinguishable from one with the semi-axis lengths swapped accompanied by a 90° rotation. The phase information of the polarized light is not recorded in the Stokes parameters. The four Stokes parameters are sometimes denoted I, Q, U and V, respectively.

If given the Stokes parameters one can solve for the spherical coordinates with the following equations:

Stokes vectors

The Stokes parameters are often combined into a vector, known as the Stokes vector:

The Stokes vector spans the space of unpolarized, partially polarized, and fully polarized light. For comparison, the Jones vector only spans the space of fully polarized light, but is more useful for problems involving coherent light. The four Stokes parameters are not a preferred coordinate system of the space, but rather were chosen because they can be easily measured or calculated.

Note that there is an ambiguous sign for the component depending on the physical convention used. In practice, there are two separate conventions used, either defining the Stokes parameters when looking down the beam towards the source (opposite the direction of light propagation) or looking down the beam away from the source (coincident with the direction of light propagation). These two conventions result in different signs for , and a convention must be chosen and adhered to.

Examples

Below are shown some Stokes vectors for common states of polarization of light.

Linearly polarized (horizontal)
Linearly polarized (vertical)
Linearly polarized (+45°)
Linearly polarized (−45°)
Right-hand circularly polarized
Left-hand circularly polarized
Unpolarized

Alternate explanation

Polarisation ellipse

A monochromatic plane wave is specified by its propagation vector, , and the complex amplitudes of the electric field, and , in a basis . The pair is called a Jones vector. Alternatively, one may specify the propagation vector, the phase, , and the polarization state, , where is the curve traced out by the electric field as a function of time in a fixed plane. The most familiar polarization states are linear and circular, which are degenerate cases of the most general state, an ellipse.

One way to describe polarization is by giving the semi-major and semi-minor axes of the polarization ellipse, its orientation, and the sense of rotation (See the above figure). The Stokes parameters , , , and , provide an alternative description of the polarization state which is experimentally convenient because each parameter corresponds to a sum or difference of measurable intensities. The next figure shows examples of the Stokes parameters in degenerate states.

StokesParameters

Definitions

The Stokes parameters are defined by

where the subscripts refer to three different bases of the space of Jones vectors: the standard Cartesian basis (), a Cartesian basis rotated by 45° (), and a circular basis (). The circular basis is defined so that .

The symbols ⟨⋅⟩ represent expectation values. The light can be viewed as a random variable taking values in the space C2 of Jones vectors . Any given measurement yields a specific wave (with a specific phase, polarization ellipse, and magnitude), but it keeps flickering and wobbling between different outcomes. The expectation values are various averages of these outcomes. Intense, but unpolarized light will have I > 0 but Q = U = V = 0, reflecting that no polarization type predominates. A convincing waveform is depicted at the article on coherence.

The opposite would be perfectly polarized light which, in addition, has a fixed, nonvarying amplitude -- a pure sine curve. This is represented by a random variable with only a single possible value, say . In this case one may replace the brackets by absolute value bars, obtaining a well-defined quadratic map

from the Jones vectors to the corresponding Stokes vectors; more convenient forms are given below. The map takes its image in the cone defined by |I |2 = |Q |2 + |U |2 + |V |2, where the purity of the state satisfies p = 1 (see below).

The next figure shows how the signs of the Stokes parameters are determined by the helicity and the orientation of the semi-major axis of the polarization ellipse.

StokesParamSign1

Representations in fixed bases

In a fixed () basis, the Stokes parameters when using an increasing phase convention are

while for , they are

and for , they are

Properties

For purely monochromatic coherent radiation, it follows from the above equations that

whereas for the whole (non-coherent) beam radiation, the Stokes parameters are defined as averaged quantities, and the previous equation becomes an inequality:[4]

However, we can define a total polarization intensity , so that

where is the total polarization fraction.

Let us define the complex intensity of linear polarization to be

Under a rotation of the polarization ellipse, it can be shown that and are invariant, but

With these properties, the Stokes parameters may be thought of as constituting three generalized intensities:

where is the total intensity, is the intensity of circular polarization, and is the intensity of linear polarization. The total intensity of polarization is , and the orientation and sense of rotation are given by

Since and , we have

Relation to the polarization ellipse

In terms of the parameters of the polarization ellipse, the Stokes parameters are

Inverting the previous equation gives

Relationship to Hermitian operators and quantum mixed states

From a geometric and algebraic point of view, the Stokes parameters stand in one-to-one correspondence with the closed, convex, 4-real-dimensional cone of nonnegative Hermitian operators on the Hilbert space C2. The parameter I serves as the trace of the operator, whereas the entries of the matrix of the operator are simple linear functions of the four parameters I, Q, U, V, serving as coefficients in a linear combination of the Stokes operators. The eigenvalues and eigenvectors of the operator can be calculated from the polarization ellipse parameters I, p, ψ, χ.

The Stokes parameters with I set equal to 1 (i.e. the trace 1 operators) are in one-to-one correspondence with the closed unit 3-dimensional ball of mixed states (or density operators) of the quantum space C2, whose boundary is the Bloch sphere. The Jones vectors correspond to the underlying space C2, that is, the (unnormalized) pure states of the same system. Note that phase information is lost when passing from a pure state |φ⟩ to the corresponding mixed state |φ⟩⟨φ|, just as it is lost when passing from a Jones vector to the corresponding Stokes vector.

See also

Notes

  1. ^ Stokes, G. G. (1852). On the composition and resolution of streams of polarized light from different sources. Transactions of the Cambridge Philosophical Society, 9, 399.
  2. ^ S. Chandrasekhar 'Radiative Transfer, Dover Publications, New York, 1960, ISBN 0-486-60590-6, page 25
  3. ^ "S. Chandrasekhar - Session II". Oral History Interviews. AIP. 18 May 1977.
  4. ^ H. C. van de Hulst Light scattering by small particles, Dover Publications, New York, 1981, ISBN 0-486-64228-3, page 42

References

Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space. The space of pure states of a quantum system is given by the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space). For a two-dimensional Hilbert space, this is simply the complex projective line ℂℙ1. This is the Bloch sphere.

The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors and , respectively, which in turn might correspond e.g. to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states. The Bloch sphere may be generalized to an n-level quantum system, but then the visualization is less useful.

For historical reasons, in optics the Bloch sphere is also known as the Poincaré sphere and specifically represents different types of polarizations. Six common polarization types exist and are called Jones Vectors. Indeed Henri Poincaré was the first to suggest the use of this kind of geometrical representation at the end of 19th century, as a three-dimensional representation of Stokes parameters.

The natural metric on the Bloch sphere is the Fubini–Study metric. The mapping from the unit 3-sphere in the two-dimensional state space ℂ2 to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere.

Circular polarization

In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electric field of the wave has a constant magnitude but its direction rotates with time at a steady rate in a plane perpendicular to the direction of the wave.

In electrodynamics the strength and direction of an electric field is defined by its electric field vector. In the case of a circularly polarized wave, as seen in the accompanying animation, the tip of the electric field vector, at a given point in space, describes a circle as time progresses. At any instant of time, the electric field vector of the wave describes a helix along the direction of propagation. A circularly polarized wave can be in one of two possible states, right circular polarization in which the electric field vector rotates in a right-hand sense with respect to the direction of propagation, and left circular polarization in which the vector rotates in a left-hand sense.

Circular polarization is a limiting case of the more general condition of elliptical polarization. The other special case is the easier-to-understand linear polarization.

The phenomenon of polarization arises as a consequence of the fact that light behaves as a two-dimensional transverse wave.

Degree Angular Scale Interferometer

The Degree Angular Scale Interferometer (DASI) was a telescope installed at the U.S. National Science Foundation's Amundsen–Scott South Pole Station in Antarctica. It was a 13-element interferometer operating between 26 and 36 GHz (Ka band) in ten bands. The instrument is similar in design to the Cosmic Background Imager (CBI) and the Very Small Array (VSA).

In 2001 The DASI team announced the most detailed measurements of the temperature, or power spectrum of the Cosmic microwave background (CMB). These results contained the first detection of the 2nd and 3rd acoustic peaks in the CMB, which were important evidence for inflation theory. This announcement was done in conjunction with the BOOMERanG and MAXIMA experiment. In 2002 the team reported the first detection of polarization anisotropies in the CMB.In 2005, the vacant DASI mount was used for the QUaD experiment, which was another CMB imager focussed on the E-mode spectrum.

In 2010, the DASI mount was again repurposed for the Keck Array, which also measures CMB polarization anisotropy.

Degree of polarization

Degree of polarization (DOP) is a quantity used to describe the portion of an electromagnetic wave which is polarized. A perfectly polarized wave has a DOP of 100%, whereas an unpolarized wave has a DOP of 0%. A wave which is partially polarized, and therefore can be represented by a superposition of a polarized and unpolarized component, will have a DOP somewhere in between 0 and 100%. DOP is calculated as the fraction of the total power that is carried by the polarised component of the wave.

DOP can be used to map the strain field in materials when considering the DOP of the photoluminescence. The polarization of the photoluminescence is related to the strain in a material by way of the given material's photoelasticity tensor.

DOP is also visualized using the Poincaré sphere representation of a polarized beam. In this representation, DOP is equal to the length of the vector measured from the center of the sphere.

Dominion Radio Astrophysical Observatory

The Dominion Radio Astrophysical Observatory is a research facility founded in 1960 and located south-west of Okanagan Falls, British Columbia, Canada. The site houses four radio telescopes: an interferometric radio telescope, a 26-m single-dish antenna, a solar flux monitor, and the Canadian Hydrogen Intensity Mapping Experiment (CHIME) — as well as support engineering laboratories. The DRAO is operated by the Herzberg Institute of Astrophysics of the National Research Council of the Canadian government. The observatory was named an IEEE Milestone for first radio astronomical observations using VLBI.

Gonodactylus smithii

Gonodactylus smithii, the purple spot mantis shrimp, is a species of mantis shrimp of the smasher type. It is found from New Caledonia to the western part of the Indian Ocean, including Australia's north coast and the Great Barrier Reef.It is the only organism known to simultaneously detect the four linear and two circular polarization components required for Stokes parameters, which yield a full description of polarization. It is thus believed to have optimal polarization vision.The specific epithet smithii is in commemoration of Sir Percy William Bassett-Smith.

Index of optics articles

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. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

List of things named after George Gabriel Stokes

Sir George Stokes (1819–1903) was an Anglo-Irish mathematical physicist whose career left a prolific body of work in mathematics and physics. Below is a collection of some of the things named after him.

Mantis shrimp

Mantis shrimps, or stomatopods, are marine crustaceans of the order Stomatopoda. Some species have specialised calcified "clubs" that can strike with great power, while others have sharp forelimbs used to capture prey. They branched from other members of the class Malacostraca around 400 million years ago. Mantis shrimps typically grow to around 10 cm (3.9 in) in length. A few can reach up to 38 cm (15 in). The largest mantis shrimp ever caught had a length of 46 cm (18 in) and was caught in the Indian River near Fort Pierce, Florida, in the United States. A mantis shrimp's carapace (the bony, thick shell that covers crustaceans and some other species) covers only the rear part of the head and the first four segments of the thorax. Varieties range from shades of brown to vivid colors, as more than 450 species of mantis shrimps are known. They are among the most important predators in many shallow, tropical and subtropical marine habitats. However, despite being common, they are poorly understood, as many species spend most of their lives tucked away in burrows and holes.Called "sea locusts" by ancient Assyrians, "prawn killers" in Australia, and now sometimes referred to as "thumb splitters"—because of the animal's ability to inflict painful gashes if handled incautiously—mantis shrimps have powerful claws that are used to attack and kill prey by spearing, stunning, or dismembering. In captivity, some larger species can break through aquarium glass.

Mueller calculus

Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of light. It was developed in 1943 by Hans Mueller. In this technique, the effect of a particular optical element is represented by a Mueller matrix—a 4×4 matrix that is an overlapping generalization of the Jones matrix.

Optics

Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.Most optical phenomena can be accounted for using the classical electromagnetic description of light. Complete electromagnetic descriptions of light are, however, often difficult to apply in practice. Practical optics is usually done using simplified models. The most common of these, geometric optics, treats light as a collection of rays that travel in straight lines and bend when they pass through or reflect from surfaces. Physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically, the ray-based model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that light waves were in fact electromagnetic radiation.

Some phenomena depend on the fact that light has both wave-like and particle-like properties. Explanation of these effects requires quantum mechanics. When considering light's particle-like properties, the light is modelled as a collection of particles called "photons". Quantum optics deals with the application of quantum mechanics to optical systems.

Optical science is relevant to and studied in many related disciplines including astronomy, various engineering fields, photography, and medicine (particularly ophthalmology and optometry). Practical applications of optics are found in a variety of technologies and everyday objects, including mirrors, lenses, telescopes, microscopes, lasers, and fibre optics.

Polarization (waves)

Polarization (also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves (shear waves) in solids. In some types of transverse waves, the wave displacement is limited to a single direction, so these also do not exhibit polarization; for example, in surface waves in liquids (gravity waves), the wave displacement of the particles is always in a vertical plane.

An electromagnetic wave such as light consists of a coupled oscillating electric field and magnetic field which are always perpendicular; by convention, the "polarization" of electromagnetic waves refers to the direction of the electric field. In linear polarization, the fields oscillate in a single direction. In circular or elliptical polarization, the fields rotate at a constant rate in a plane as the wave travels. The rotation can have two possible directions; if the fields rotate in a right hand sense with respect to the direction of wave travel, it is called right circular polarization, or, if the fields rotate in a left hand sense, it is called left circular polarization.

Light or other electromagnetic radiation from many sources, such as the sun, flames, and incandescent lamps, consists of short wave trains with an equal mixture of polarizations; this is called unpolarized light. Polarized light can be produced by passing unpolarized light through a polarizer, which allows waves of only one polarization to pass through. The most common optical materials (such as glass) are isotropic and do not affect the polarization of light passing through them; however, some materials—those that exhibit birefringence, dichroism, or optical activity—can change the polarization of light. Some of these are used to make polarizing filters. Light is also partially polarized when it reflects from a surface.

According to quantum mechanics, electromagnetic waves can also be viewed as streams of particles called photons. When viewed in this way, the polarization of an electromagnetic wave is determined by a quantum mechanical property of photons called their spin. A photon has one of two possible spins: it can either spin in a right hand sense or a left hand sense about its direction of travel. Circularly polarized electromagnetic waves are composed of photons with only one type of spin, either right- or left-hand. Linearly polarized waves consist of photons that are in a superposition of right and left circularly polarized states, with equal amplitude and phases synchronized to give oscillation in a plane.Polarization is an important parameter in areas of science dealing with transverse waves, such as optics, seismology, radio, and microwaves. Especially impacted are technologies such as lasers, wireless and optical fiber telecommunications, and radar.

Polarization mixing

In optics, polarization mixing refers to changes in the relative strengths of the Stokes parameters caused by reflection or scattering—see vector radiative transfer—or by changes in the radial orientation of the detector.

QUIET

QUIET is an astronomy experiment to study the polarization of the cosmic microwave background radiation. QUIET stands for Q/U Imaging ExperimenT. The Q/U in the name refers to the ability of the telescope to measure the Q and U Stokes parameters simultaneously. QUIET is located at an elevation of 5,080 metres (16,700 feet) at Llano de Chajnantor Observatory in the Chilean Andes. It began observing in late 2008 and finished observing in December 2010.QUIET is the result of an international collaboration that has its origins in the CAPMAP, Cosmic Background Imager (CBI) and QUaD collaborations. The collaboration consists of 7 groups in the United States (the California Institute of Technology, the University of Chicago, Columbia University, the Jet Propulsion Laboratory, the University of Miami, Princeton University and Stanford University), 4 groups in Europe (the University of Manchester, the Max-Planck-Institut für Radioastronomie Bonn, the University of Oslo and the University of Oxford) and one group in Japan (KEK; the first time a Japan group has been involved in CMB studies). Other members of the collaboration are from the University of California, Berkeley, the Goddard Space Flight Center and the Harvard-Smithsonian Center for Astrophysics.

Rayleigh sky model

The Rayleigh sky model describes the observed polarization pattern of the daytime sky. Within the atmosphere Rayleigh scattering of light from air molecules, water, dust, and aerosols causes the sky's light to have a defined polarization pattern. The same elastic scattering processes cause the sky to be blue. The polarization is characterized at each wavelength by its degree of polarization, and orientation (the e-vector angle, or scattering angle).

The polarization pattern of the sky is dependent on the celestial position of the sun. While all scattered light is polarized to some extent, light is highly polarized at a scattering angle of 90° from the light source. In most cases the light source is the sun, but the moon creates the same pattern as well. The degree of polarization first increases with increasing distance from the sun, and then decreases away from the sun. Thus, the maximum degree of polarization occurs in a circular band 90° from the sun. In this band, degrees of polarization near 80% are typically reached.

When the sun is located at the zenith, the band of maximal polarization wraps around the horizon. Light from the sky is polarized horizontally along the horizon. During twilight at either the Vernal or Autumnal equinox, the band of maximal polarization is defined by the North-Zenith-South plane, or meridian. In particular, the polarization is vertical at the horizon in the North and South, where the meridian meets the horizon. The polarization at twilight at an equinox is represented by the figure to the right. The red band represents the circle in the North-Zenith-South plane where the sky is highly polarized. The cardinal directions N, E, S, W are shown at 12-o'clock, 9 o'clock, 6 o'clock and 3 o'clock (counter-clockwise around the celestial sphere since the observer is looking up at the sky).

Note that because the polarization pattern is dependent on the sun, it changes not only throughout the day but throughout the year. When the sun sets toward the South, in the winter, the North-Zenith-South plane is offset, with "effective" North actually located somewhat toward the West. Thus if the sun sets at an azimuth of 255° (15° South of West) the polarization pattern will be at its maximum along the horizon at an azimuth of 345° (15° West of North) and 165° (15° East of South).

During a single day, the pattern rotates with the changing position of the sun. At twilight it typically appears about 45 minutes before local sunrise and disappears 45 minutes after local sunset. Once established it is very stable, showing change only in its rotation. It can easily be seen on any given day using polarized sunglasses.

Many animals use the polarization patterns of the sky at twilight and throughout the day as a navigation tool. Because it is determined purely by the position of the sun, it is easily used as a compass for animal orientation. By orienting themselves with respect to the polarization patterns, animals can locate the sun and thus determine the cardinal directions.

Stokes operators

The Stokes operators are the quantum mechanical operators corresponding to the classical Stokes parameters. These matrix operators are identical to the Pauli matrices .

Vector radiative transfer

In spectroscopy and radiometry, vector radiative transfer (VRT) is a method of modelling the propagation of polarized electromagnetic radiation in low density media. In contrast to scalar radiative transfer (RT), which models only the first Stokes component, the intensity, VRT models all four components through vector methods.

For a single frequency, , the VRT equation for a scattering media can be written as follows:

where s is the path, is the propagation vector, K is the extinction matrix, is the absorption vector, B is the Planck function and Z is the scattering phase matrix.

All the coefficient matrices, K, and Z, will vary depending on the density of absorbers/scatterers present and must be calculated from their density-independent quantities, that is the attenuation coefficient vector, , is calculated from the mass absorption coefficient vector times the density of the absorber. Moreover, it is typical for media to have multiple species causing extinction, absorption and scattering, thus these coefficient matrices must be summed up over all the different species.

Extinction is caused both by simple absorption as well as from scattering out of the line-of-sight, , therefore we calculate the extinction matrix from the combination of the absorption vector and the scattering phase matrix:

where I is the identity matrix.

The four-component radiation vector, where I, Q, U and V are the first through fourth elements of the Stokes parameters, respectively, fully describes the polarization state of the electromagnetic radiation. It is this vector-nature that considerably complicates the equation. Absorption will be different for each of the four components, moreover, whenever the radiation is scattered, there can be a complex transfer between the different Stokes components—see polarization mixing—thus the scattering phase function has 4*4=16 components. It is, in fact, a rank-two tensor.

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