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Polarization (
also Also or ALSO may refer to: *Advanced Life Support in Obstetrics Advanced Life Support in Obstetrics (ALSO) is a program that was developed by the American Academy of Family Physicians (AAFP). This course helps physicians, certified nurse midwi ...
polarisation) is a property applying to
transverse waves In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
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 A string is the vibrating element that produces sound in string instruments such as the guitar, harp, piano (piano wire), and members of the violin family. Strings are lengths of a flexible material that a musical instrument holds under tens ...
. 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 wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves ...
s, such as
sound wave In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
s 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 wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ligh ...
s such as
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 terahe ...
and
radio wave Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum, typically with frequencies of 300 gigahertz (GHz) and below. At 300 GHz, the corresponding wavelength is 1 mm (shor ...
s,
gravitational wave Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s, and transverse sound waves (
shear wave __NOTOC__ In seismology and other areas involving elastic waves, S waves, secondary waves, or shear waves (sometimes called elastic S waves) are a type of elastic wave and are one of the two main types of elastic body waves, so named because th ...
s) in solids. An
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ligh ...
such as light consists of a coupled oscillating
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field f ...
and
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
which are always perpendicular to each other; by convention, the "polarization" of electromagnetic waves refers to the direction of the electric field. In
linear polarization In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. The term ''linear polarizati ...
, the fields oscillate in a single direction. In
circular Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...
or
elliptical polarization In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An ellip ...
, 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 In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
sense with respect to the direction of wave travel, it is called
right circular polarization In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to th ...
, while if the fields rotate in a left hand sense, it is called
left circular polarization In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to th ...
. Light or other electromagnetic radiation from many sources, such as the sun, flames, and
incandescent lamp An incandescent light bulb, incandescent lamp or incandescent light globe is an electric light with a wire filament heated until it glows. The filament is enclosed in a glass bulb with a vacuum or inert gas to protect the filament from oxida ...
s, 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 A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well- ...
, which allows waves of only one polarization to pass through. The most common optical materials do not affect the polarization of light, however, some materials—those that exhibit
birefringence Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefringe ...
,
dichroism In optics, a dichroic material is either one which causes visible light to be split up into distinct beams of different wavelengths (colours) (not to be confused with dispersion), or one in which light rays having different polarizations are ab ...
, or
optical activity Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circula ...
—affect light differently depending on its polarization. Some of these are used to make polarizing filters. Light also becomes partially polarized when it reflects at an angle from a surface. According to
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qu ...
, electromagnetic waves can also be viewed as streams of particles called
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always m ...
s. When viewed in this way, the polarization of an electromagnetic wave is determined by a quantum mechanical property of photons called their
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
. A photon has one of two possible spins: it can either spin in a
right hand In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
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 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, ultraviole ...
,
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
,
radio Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a transmit ...
, and
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency rang ...
s. Especially impacted are technologies such as
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
s, wireless and optical fiber
telecommunications Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
, and
radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weat ...
.


Introduction


Wave propagation and polarization

Most sources of light are classified as incoherent and unpolarized (or only "partially polarized") because they consist of a random mixture of waves having different spatial characteristics, frequencies (wavelengths), phases, and polarization states. However, for understanding electromagnetic waves and polarization in particular, it is easier to just consider coherent
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
s; these are sinusoidal waves of one particular direction (or
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
), frequency, phase, and polarization state. Characterizing an optical system in relation to a plane wave with those given parameters can then be used to predict its response to a more general case, since a wave with any specified spatial structure can be decomposed into a combination of plane waves (its so-called angular spectrum). Incoherent states can be modeled stochastically as a weighted combination of such uncorrelated waves with some
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...
of frequencies (its ''
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
''), phases, and polarizations.


Transverse electromagnetic waves

Electromagnetic waves In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ligh ...
(such as light), traveling in free space or another
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, ...
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
non-attenuating medium, are properly described as
transverse waves In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
, meaning that a plane wave's electric field vector E and magnetic field H are each in some direction perpendicular to (or "transverse" to) the direction of wave propagation; E and H are also perpendicular to each other. By convention, the "polarization" direction of an electromagnetic wave is given by its electric field vector. Considering a monochromatic
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
of optical frequency ''f'' (light of vacuum wavelength λ has a frequency of ''f = c/λ'' where ''c'' is the speed of light), let us take the direction of propagation as the ''z'' axis. Being a transverse wave the E and H fields must then contain components only in the ''x'' and ''y'' directions whereas ''Ez = Hz = 0''. Using
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
(or
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
) notation, the instantaneous physical electric and magnetic fields are given by the
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s of the complex quantities occurring in the following equations. As a function of time ''t'' and spatial position ''z'' (since for a plane wave in the +''z'' direction the fields have no dependence on ''x'' or ''y'') these complex fields can be written as: : \vec(z, t) = \begin e_x \\ e_y \\ 0 \end\; e^ = \begin e_x \\ e_y \\ 0 \end\; e^ and : \vec(z, t) = \begin h_x \\ h_y \\ 0 \end\; e^ = \begin h_x \\ h_y \\ 0 \end\; e^ where λ = λ/''n'' is the wavelength ''in the medium'' (whose
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
is ''n'') and is the period of the wave. Here ''ex'', ''ey'', ''hx'', and ''hy'' are complex numbers. In the second more compact form, as these equations are customarily expressed, these factors are described using the
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
k = 2\pi n / \lambda _0 and
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
(or "radian frequency") \omega = 2\pi f. In a more general formulation with propagation ''not'' restricted to the ''+z'' direction, then the spatial dependence ''kz'' is replaced by \vec \cdot \vec where \vec is called the
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
, the magnitude of which is the wavenumber. Thus the leading vectors e and h each contain up to two nonzero (complex) components describing the amplitude and phase of the wave's ''x'' and ''y'' polarization components (again, there can be no ''z'' polarization component for a transverse wave in the +''z'' direction). For a given medium with a
characteristic impedance The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in ...
\eta, h is related to e by: :h_y = \frac and :h_x = -\frac. In a dielectric, ''η'' is real and has the value ''η''0/''n'', where ''n'' is the refractive index and ''η''0 is the impedance of free space. The impedance will be complex in a conducting medium. Note that given that relationship, the dot product of E and H must be zero: : \vec\left(\vec, t\right) \cdot \vec\left(\vec, t\right) = e_x h_x + e_y h_y + e_z h_z = e_x \left(-\frac\right) + e_y \left(\frac\right) + 0 \cdot 0 = 0 indicating that these vectors are
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
(at right angles to each other), as expected. So knowing the propagation direction (+''z'' in this case) and η, one can just as well specify the wave in terms of just ''ex'' and ''ey'' describing the electric field. The vector containing ''ex'' and ''ey'' (but without the ''z'' component which is necessarily zero for a transverse wave) is known as a
Jones vector In optics, polarized light can be described using the Jones calculus, discovered by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by ''Jones matrices''. When light crosses an opt ...
. In addition to specifying the polarization state of the wave, a general Jones vector also specifies the overall magnitude and phase of that wave. Specifically, the
intensity Intensity may refer to: In colloquial use *Strength (disambiguation) *Amplitude * Level (disambiguation) * Magnitude (disambiguation) In physical sciences Physics *Intensity (physics), power per unit area (W/m2) *Field strength of electric, ma ...
of the light wave is proportional to the sum of the squared magnitudes of the two electric field components: : I = \left(\left, e_x\^2 + \left, e_y\^2\right) \, \frac however the wave's ''state of polarization'' is only dependent on the (complex) ''ratio'' of ''ey'' to ''ex''. So let us just consider waves whose '', ex, 2 + , ey, 2 = 1''; this happens to correspond to an intensity of about .00133
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Watt ...
s per square meter in free space (where \eta = \eta_0). And since the absolute phase of a wave is unimportant in discussing its polarization state, let us stipulate that the phase of ''ex'' is zero, in other words ''ex'' is a real number while ''ey'' may be complex. Under these restrictions, ''ex'' and ''ey'' can be represented as follows: :e_x = \sqrt :e_y = \sqrt\, e^ where the polarization state is now fully parameterized by the value of ''Q'' (such that −1 < ''Q'' < 1) and the relative phase \phi.


Non-transverse waves

In addition to transverse waves, there are many wave motions where the oscillation is not limited to directions perpendicular to the direction of propagation. These cases are far beyond the scope of the current article which concentrates on transverse waves (such as most electromagnetic waves in bulk media), however one should be aware of cases where the polarization of a coherent wave cannot be described simply using a Jones vector, as we have just done. Just considering electromagnetic waves, we note that the preceding discussion strictly applies to plane waves in a homogeneous isotropic non-attenuating medium, whereas in an
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
medium (such as birefringent crystals as discussed below) the electric or magnetic field may have longitudinal as well as transverse components. In those cases the
electric displacement In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
''D'' and
magnetic flux density A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
''B'' still obey the above geometry but due to anisotropy in the
electric susceptibility In electricity (electromagnetism), the electric susceptibility (\chi_; Latin: ''susceptibilis'' "receptive") is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied ...
(or in the
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
), now given by a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
, the direction of ''E'' (or ''H'') may differ from that of ''D'' (or ''B''). Even in isotropic media, so-called inhomogeneous waves can be launched into a medium whose refractive index has a significant imaginary part (or " extinction coefficient") such as metals; these fields are also not strictly transverse.
Surface wave In physics, a surface wave is a mechanical wave that propagates along the interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occur within liquids, at th ...
s or waves propagating in a
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
(such as an
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means t ...
) are generally ''not'' transverse waves, but might be described as an electric or magnetic
transverse mode A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's propagation direction. Transverse modes occur in radio waves and microwav ...
, or a hybrid mode. Even in free space, longitudinal field components can be generated in focal regions, where the plane wave approximation breaks down. An extreme example is radially or tangentially polarized light, at the focus of which the electric or magnetic field respectively is ''entirely'' longitudinal (along the direction of propagation). For
longitudinal wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves ...
s such as
sound wave In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
s in
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s, the direction of oscillation is by definition along the direction of travel, so the issue of polarization is normally not even mentioned. On the other hand, sound waves in a bulk
solid Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural r ...
can be transverse as well as longitudinal, for a total of three polarization components. In this case, the transverse polarization is associated with the direction of the shear stress and displacement in directions perpendicular to the propagation direction, while the longitudinal polarization describes compression of the solid and vibration along the direction of propagation. The differential propagation of transverse and longitudinal polarizations is important in
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
.


Polarization state

Polarization is best understood by initially considering only pure polarization states, and only a coherent sinusoidal wave at some optical frequency. The vector in the adjacent diagram might describe the oscillation of the electric field emitted by a single-mode laser (whose oscillation frequency would be typically 1015 times faster). The field oscillates in the ''x-y'' plane, along the page, with the wave propagating in the ''z'' direction, perpendicular to the page. The first two diagrams below trace the electric field vector over a complete cycle for linear polarization at two different orientations; these are each considered a distinct ''state of polarization'' (SOP). Note that the linear polarization at 45° can also be viewed as the addition of a horizontally linearly polarized wave (as in the leftmost figure) and a vertically polarized wave of the same amplitude ''in the same phase''.
Now if one were to introduce a
phase shift In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it ...
in between those horizontal and vertical polarization components, one would generally obtain elliptical polarization as is shown in the third figure. When the phase shift is exactly ±90°, then ''circular polarization'' is produced (fourth and fifth figures). Thus is circular polarization created in practice, starting with linearly polarized light and employing a
quarter-wave plate A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization direction of linearly polarized ligh ...
to introduce such a phase shift. The result of two such phase-shifted components in causing a rotating electric field vector is depicted in the animation on the right. Note that circular or elliptical polarization can involve either a clockwise or counterclockwise rotation of the field. These correspond to distinct polarization states, such as the two circular polarizations shown above. Of course the orientation of the ''x'' and ''y'' axes used in this description is arbitrary. The choice of such a coordinate system and viewing the polarization ellipse in terms of the ''x'' and ''y'' polarization components, corresponds to the definition of the Jones vector (below) in terms of those
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
polarizations. One would typically choose axes to suit a particular problem such as ''x'' being in the plane of incidence. Since there are separate reflection coefficients for the linear polarizations in and orthogonal to the plane of incidence (''p'' and ''s'' polarizations, see below), that choice greatly simplifies the calculation of a wave's reflection from a surface. Moreover, one can use as basis functions ''any'' pair of
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
polarization states, not just linear polarizations. For instance, choosing right and left circular polarizations as basis functions simplifies the solution of problems involving circular birefringence (optical activity) or circular dichroism.


Polarization ellipse

Consider a purely polarized monochromatic wave. If one were to plot the electric field vector over one cycle of oscillation, an ellipse would generally be obtained, as is shown in the figure, corresponding to a particular state of
elliptical polarization In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An ellip ...
. Note that linear polarization and circular polarization can be seen as special cases of elliptical polarization. A polarization state can then be described in relation to the geometrical parameters of the ellipse, and its "handedness", that is, whether the rotation around the ellipse is clockwise or counter clockwise. One parameterization of the elliptical figure specifies the orientation angle ''ψ'', defined as the angle between the major axis of the ellipse and the ''x''-axis along with the ellipticity ''ε'' = ''a/b'', the ratio of the ellipse's major to minor axis. (also known as the
axial ratio Axial ratio, for any structure or shape with two or more axes, is the ratio of the length (or magnitude) of those axes to each other - the longer axis divided by the shorter. In ''chemistry'' or ''materials science'', the axial ratio (symbol P) i ...
). The ellipticity parameter is an alternative parameterization of an ellipse's
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-center, in geometry * Eccentricity (graph theory) of a v ...
e=\sqrt, or the ellipticity angle, \chi=\arctan b/a =\arctan 1/\varepsilon as is shown in the figure. The angle ''χ'' is also significant in that the latitude (angle from the equator) of the polarization state as represented on the Poincaré sphere (see below) is equal to ±2''χ''. The special cases of linear and circular polarization correspond to an ellipticity ''ε'' of infinity and unity (or ''χ'' of zero and 45°) respectively.


Jones vector

Full information on a completely polarized state is also provided by the amplitude and phase of oscillations in two components of the electric field vector in the plane of polarization. This representation was used above to show how different states of polarization are possible. The amplitude and phase information can be conveniently represented as a two-dimensional
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
vector (the
Jones vector In optics, polarized light can be described using the Jones calculus, discovered by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by ''Jones matrices''. When light crosses an opt ...
): : \mathbf = \begin a_1 e^ \\ a_2 e^ \end. Here a_1 and a_2 denote the amplitude of the wave in the two components of the electric field vector, while \theta_1 and \theta_2 represent the phases. The product of a Jones vector with a complex number of unit modulus gives a different Jones vector representing the same ellipse, and thus the same state of polarization. The physical electric field, as the real part of the Jones vector, would be altered but the polarization state itself is independent of
absolute phase Absolute phase refers to the phase of a waveform relative to some standard (strictly speaking, phase is always relative). To the extent that this standard is accepted by all parties, one can speak of an absolute phase in a particular field of applic ...
. The
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
vectors used to represent the Jones vector need not represent linear polarization states (i.e. be
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
). In general any two orthogonal states can be used, where an orthogonal vector pair is formally defined as one having a zero
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. A common choice is left and right circular polarizations, for example to model the different propagation of waves in two such components in circularly birefringent media (see below) or signal paths of coherent detectors sensitive to circular polarization.


Coordinate frame

Regardless of whether polarization state is represented using geometric parameters or Jones vectors, implicit in the parameterization is the orientation of the coordinate frame. This permits a degree of freedom, namely rotation about the propagation direction. When considering light that is propagating parallel to the surface of the Earth, the terms "horizontal" and "vertical" polarization are often used, with the former being associated with the first component of the Jones vector, or zero azimuth angle. On the other hand, in
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...
the
equatorial coordinate system The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fu ...
is generally used instead, with the zero azimuth (or position angle, as it is more commonly called in astronomy to avoid confusion with the
horizontal coordinate system The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane to define two angles: altitude and azimuth. Therefore, the horizontal coordinate system is sometimes called as the ...
) corresponding to due north.


''s'' and ''p'' designations

Another coordinate system frequently used relates to the ''
plane of incidence In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation. (In wave opt ...
''. This is the plane made by the incoming propagation direction and the vector perpendicular to the plane of an interface, in other words, the plane in which the ray travels before and after reflection or refraction. The component of the electric field parallel to this plane is termed ''p-like'' (parallel) and the component perpendicular to this plane is termed ''s-like'' (from ''senkrecht'', German for perpendicular). Polarized light with its electric field along the plane of incidence is thus denoted ''p-polarized'', while light whose electric field is normal to the plane of incidence is called ''s-polarized''. ''P'' polarization is commonly referred to as ''transverse-magnetic'' (TM), and has also been termed ''pi-polarized'' or ''tangential plane polarized''. ''S'' polarization is also called ''transverse-electric'' (TE), as well as ''sigma-polarized'' or ''sagittal plane polarized''.


Unpolarized and partially polarized light


Definition

Natural light, like most other common sources of visible light, is incoherent: radiation is produced independently by a large number of atoms or molecules whose emissions are
uncorrelated In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...
and generally of random polarizations. In this case the light is said to be ''unpolarized''. This term is somewhat inexact, since at any instant of time at one location there is a definite direction to the electric and magnetic fields, however it implies that the polarization changes so quickly in time that it will not be measured or relevant to the outcome of an experiment. A so-called
depolarizer A depolarizer or depolariser, in electrochemistry, according to an IUPAC definition, is a synonym of electroactive substance, i.e., a substance which changes its oxidation state, or partakes in a formation or breaking of chemical bonds, in a ch ...
acts on a polarized beam to create one which is actually ''fully'' polarized at every point, but in which the polarization varies so rapidly across the beam that it may be ignored in the intended applications. Unpolarized light can be described as a mixture of two independent oppositely polarized streams, each with half the intensity. Light is said to be ''partially polarized'' when there is more power in one of these streams than the other. At any particular wavelength, partially polarized light can be statistically described as the superposition of a completely unpolarized component and a completely polarized one. One may then describe the light in terms of the
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 polarize ...
and the parameters of the polarized component. That polarized component can be described in terms of a Jones vector or polarization ellipse, as is detailed above. However, in order to also describe the degree of polarization, one normally employs Stokes parameters (see below) to specify a state of partial polarization.


Motivation

The transmission of plane waves through a homogeneous medium are fully described in terms of Jones vectors and 2×2 Jones matrices. However, in practice there are cases in which all of the light cannot be viewed in such a simple manner due to spatial inhomogeneities or the presence of mutually incoherent waves. So-called depolarization, for instance, cannot be described using Jones matrices. For these cases it is usual instead to use a 4×4 matrix that acts upon the Stokes 4-vector. Such matrices were first used by Paul Soleillet in 1929, although they have come to be known as Mueller matrices. While every Jones matrix has a Mueller matrix, the reverse is not true. Mueller matrices are then used to describe the observed polarization effects of the
scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
of waves from complex surfaces or ensembles of particles, as shall now be presented.


Coherency matrix

The Jones vector perfectly describes the state of polarization ''and phase'' of a single monochromatic wave, representing a pure state of polarization as described above. However any mixture of waves of different polarizations (or even of different frequencies) do ''not'' correspond to a Jones vector. In so-called partially polarized radiation the fields are
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...
, and the variations and correlations between components of the electric field can only be described
statistical Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industri ...
ly. One such representation is the coherency
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
: : \begin \mathbf &= \left\langle \mathbf\mathbf^\dagger \right\rangle \\ &= \left\langle\begin e_1 e_1^* & e_1 e_2^* \\ e_2 e_1^* & e_2 e_2^* \end\right\rangle \\ &= \left\langle\begin a_1^2 & a_1 a_2 e^ \\ a_1 a_2 e^ & a_2^2 \end\right\rangle \end where angular brackets denote averaging over many wave cycles. Several variants of the coherency matrix have been proposed: the Wiener coherency matrix and the spectral coherency matrix of Richard Barakat measure the coherence of a spectral decomposition of the signal, while the
Wolf The wolf (''Canis lupus''; : wolves), also known as the gray wolf or grey wolf, is a large canine native to Eurasia and North America. More than thirty subspecies of ''Canis lupus'' have been recognized, and gray wolves, as popularly u ...
coherency matrix averages over all time/frequencies. The coherency matrix contains all second order statistical information about the polarization. This matrix can be decomposed into the sum of two
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
matrices, corresponding to the
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the coherency matrix, each representing a polarization state that is orthogonal to the other. An alternative decomposition is into completely polarized (zero determinant) and unpolarized (scaled identity matrix) components. In either case, the operation of summing the components corresponds to the incoherent superposition of waves from the two components. The latter case gives rise to the concept of the "degree of polarization"; i.e., the fraction of the total intensity contributed by the completely polarized component.


Stokes parameters

The coherency matrix is not easy to visualize, and it is therefore common to describe 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. An alternative and mathematically convenient description is given by the
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, as a mathematically convenient alternative to the more common description of incoher ...
, introduced by
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University of Cambridge, where he was the L ...
in 1852. The relationship of the Stokes parameters to intensity and polarization ellipse parameters is shown in the equations and figure below. : S_0 = I \, : S_1 = Ip \cos 2\psi \cos 2\chi\, : S_2 = Ip \sin 2\psi \cos 2\chi\, : S_3 = Ip \sin 2\chi\, Here ''Ip'', 2ψ and 2χ are the
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
of the polarization state in the three-dimensional space of the last three Stokes parameters. Note the factors of two before ψ and χ corresponding respectively to the facts that any polarization ellipse is indistinguishable from one rotated by 180°, or one with the semi-axis lengths swapped accompanied by a 90° rotation. The Stokes parameters are sometimes denoted ''I'', ''Q'', ''U'' and ''V''. The four Stokes parameters are enough to describe 2D polarization of a paraxial wave, but not the 3D polarization of a general non-paraxial wave or an evanescent field.


Poincaré sphere

Neglecting the first Stokes parameter ''S''0 (or ''I''), the three other Stokes parameters can be plotted directly in three-dimensional Cartesian coordinates. For a given power in the polarized component given by : P = \sqrt the set of all polarization states are then mapped to points on the surface of the so-called ''Poincaré sphere'' (but of radius ''P''), as shown in the accompanying diagram. Often the total beam power is not of interest, in which case a normalized Stokes vector is used by dividing the Stokes vector by the total intensity ''S''0: :\mathbf = \frac \beginS_0\\S_1\\S_2\\S_3\end. The normalized Stokes vector \mathbf then has unity power (S'_0 = 1) and the three significant Stokes parameters plotted in three dimensions will lie on the unity-radius Poincaré sphere for pure polarization states (where P'_0 = 1). Partially polarized states will lie ''inside'' the Poincaré sphere at a distance of P' = \sqrt from the origin. When the non-polarized component is not of interest, the Stokes vector can be further normalized to obtain : \mathbf = \frac \begin 1\\S'_1\\S'_2\\S'_3 \end = \frac \begin S_0\\S_1\\S_2\\S_3 \end. When plotted, that point will lie on the surface of the unity-radius Poincaré sphere and indicate the state of polarization of the polarized component. Any two antipodal points on the Poincaré sphere refer to orthogonal polarization states. The
overlap Overlap may refer to: * In set theory, an overlap of elements shared between sets is called an intersection, as in a Venn diagram. * In music theory, overlap is a synonym for reinterpretation of a chord at the boundary of two musical phrases * Ove ...
between any two polarization states is dependent solely on the distance between their locations along the sphere. This property, which can only be true when pure polarization states are mapped onto a sphere, is the motivation for the invention of the Poincaré sphere and the use of Stokes parameters, which are thus plotted on (or beneath) it. Note that the IEEE defines RHCP and LHCP the opposite as those used by Physicists. The IEEE 1979 Antenna Standard will show RHCP on the South Pole of the Poincare Sphere. The IEEE defines RHCP using the right hand with thumb pointing in the direction of transmit, and the fingers showing the direction of rotation of the E field with time. The rationale for the opposite conventions used by Physicists and Engineers is that Astronomical Observations are always done with the incoming wave traveling toward the observer, where as for most engineers, they are assumed to be standing behind the transmitter watching the wave traveling away from them. This article is not using the IEEE 1979 Antenna Standard and is not using the +t convention typically used in IEEE work.


Implications for reflection and propagation


Polarization in wave propagation

In a
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
, the components of the electric field propagate at the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit for ...
, so that the phase of the wave varies in space and time while the polarization state does not. That is, the electric field vector e of a plane wave in the +''z'' direction follows: :\mathbf(z + \Delta z, t + \Delta t) = \mathbf(z, t) e^, where ''k'' is the
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
. As noted above, the instantaneous electric field is the real part of the product of the Jones vector times the phase factor e^. When an electromagnetic wave interacts with matter, its propagation is altered according to the material's (complex)
index of refraction In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
. When the real or imaginary part of that refractive index is dependent on the polarization state of a wave, properties known as
birefringence Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefringe ...
and polarization
dichroism In optics, a dichroic material is either one which causes visible light to be split up into distinct beams of different wavelengths (colours) (not to be confused with dispersion), or one in which light rays having different polarizations are ab ...
(or diattenuation) respectively, then the polarization state of a wave will generally be altered. In such media, an electromagnetic wave with any given state of polarization may be decomposed into two orthogonally polarized components that encounter different
propagation constant The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a cir ...
s. The effect of propagation over a given path on those two components is most easily characterized in the form of a complex 2×2
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Trans ...
matrix J known as a Jones matrix: :\mathbf = \mathbf\mathbf. The Jones matrix due to passage through a transparent material is dependent on the propagation distance as well as the birefringence. The birefringence (as well as the average refractive index) will generally be dispersive, that is, it will vary as a function of optical frequency (wavelength). In the case of non-birefringent materials, however, the 2×2 Jones matrix is the identity matrix (multiplied by a scalar
phase factor For any complex number written in polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on the ...
and attenuation factor), implying no change in polarization during propagation. For propagation effects in two orthogonal modes, the Jones matrix can be written as :\mathbf = \mathbf\begin g_1 & 0 \\ 0 & g_2 \end\mathbf^, where ''g''1 and ''g''2 are complex numbers describing the
phase delay In signal processing, group delay and phase delay are delay times experienced by a signal's various frequency components when the signal passes through a system that is linear time-invariant (LTI), such as a microphone, coaxial cable, amplifier, l ...
and possibly the amplitude attenuation due to propagation in each of the two polarization eigenmodes. T is a
unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose ...
representing a change of basis from these propagation modes to the linear system used for the Jones vectors; in the case of linear birefringence or diattenuation the modes are themselves linear polarization states so T and T−1 can be omitted if the coordinate axes have been chosen appropriately.


Birefringence

In media termed
birefringent Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefring ...
, in which the amplitudes are unchanged but a differential phase delay occurs, the Jones matrix is a
unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose ...
: , ''g''1, = , ''g''2, = 1. Media termed diattenuating (or ''
dichroic In optics, a dichroic material is either one which causes visible light to be split up into distinct beams of different wavelengths (colours) (not to be confused with dispersion), or one in which light rays having different polarizations are ab ...
'' in the sense of polarization), in which only the amplitudes of the two polarizations are affected differentially, may be described using a
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the ...
(generally multiplied by a common phase factor). In fact, since ''any'' matrix may be written as the product of unitary and positive Hermitian matrices, light propagation through any sequence of polarization-dependent optical components can be written as the product of these two basic types of transformations. In birefringent media there is no attenuation, but two modes accrue a differential phase delay. Well known manifestations of linear birefringence (that is, in which the basis polarizations are orthogonal linear polarizations) appear in optical
wave plate A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization direction of linearly polarized light ...
s/retarders and many crystals. If linearly polarized light passes through a birefringent material, its state of polarization will generally change, ''unless'' its polarization direction is identical to one of those basis polarizations. Since the phase shift, and thus the change in polarization state, is usually wavelength-dependent, such objects viewed under white light in between two polarizers may give rise to colorful effects, as seen in the accompanying photograph. Circular birefringence is also termed
optical activity Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circula ...
, especially in
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
fluids, or
Faraday rotation The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect (MOFE), is a physical magneto-optical phenomenon. The Faraday effect causes a polarization rotation which is proportional to the projection of the m ...
, when due to the presence of a magnetic field along the direction of propagation. When linearly polarized light is passed through such an object, it will exit still linearly polarized, but with the axis of polarization rotated. A combination of linear and circular birefringence will have as basis polarizations two orthogonal elliptical polarizations; however, the term "elliptical birefringence" is rarely used. One can visualize the case of linear birefringence (with two orthogonal linear propagation modes) with an incoming wave linearly polarized at a 45° angle to those modes. As a differential phase starts to accrue, the polarization becomes elliptical, eventually changing to purely circular polarization (90° phase difference), then to elliptical and eventually linear polarization (180° phase) perpendicular to the original polarization, then through circular again (270° phase), then elliptical with the original azimuth angle, and finally back to the original linearly polarized state (360° phase) where the cycle begins anew. In general the situation is more complicated and can be characterized as a
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
in the Poincaré sphere about the axis defined by the propagation modes. Examples for linear (blue), circular (red), and elliptical (yellow)
birefringence Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefringe ...
are shown in the figure on the left. The total intensity and degree of polarization are unaffected. If the path length in the birefringent medium is sufficient, the two polarization components of a collimated beam (or
ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gr ...
) can exit the material with a positional offset, even though their final propagation directions will be the same (assuming the entrance face and exit face are parallel). This is commonly viewed using
calcite Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate (CaCO3). It is a very common mineral, particularly as a component of limestone. Calcite defines hardness 3 on the Mohs scale of mineral hardness, based on scratc ...
crystals, which present the viewer with two slightly offset images, in opposite polarizations, of an object behind the crystal. It was this effect that provided the first discovery of polarization, by
Erasmus Bartholinus Rasmus Bartholin (; Latinized: ''Erasmus Bartholinus''; 13 August 1625 – 4 November 1698) was a Danish physician and grammarian. Biography Bartholin was born in Roskilde. He was the son of Caspar Bartholin the Elder (1585–1629) and Ann ...
in 1669.


Dichroism

Media in which transmission of one polarization mode is preferentially reduced are called ''
dichroic In optics, a dichroic material is either one which causes visible light to be split up into distinct beams of different wavelengths (colours) (not to be confused with dispersion), or one in which light rays having different polarizations are ab ...
'' or ''diattenuating''. Like birefringence, diattenuation can be with respect to linear polarization modes (in a crystal) or circular polarization modes (usually in a liquid). Devices that block nearly all of the radiation in one mode are known as ''polarizing filters'' or simply "
polarizer A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well- ...
s". This corresponds to ''g''2=0 in the above representation of the Jones matrix. The output of an ideal polarizer is a specific polarization state (usually linear polarization) with an amplitude equal to the input wave's original amplitude in that polarization mode. Power in the other polarization mode is eliminated. Thus if unpolarized light is passed through an ideal polarizer (where ''g''1=1 and ''g''2=0) exactly half of its initial power is retained. Practical polarizers, especially inexpensive sheet polarizers, have additional loss so that ''g''1 < 1. However, in many instances the more relevant figure of merit is the polarizer's
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 polarize ...
or
extinction ratio In telecommunications, extinction ratio (''r''e) is the ratio of two optical power levels of a digital signal generated by an optical source, e.g., a laser diode. The extinction ratio may be expressed as a fraction, in dB, or as a percentage. It ...
, which involve a comparison of ''g''1 to ''g''2. Since Jones vectors refer to waves' amplitudes (rather than
intensity Intensity may refer to: In colloquial use *Strength (disambiguation) *Amplitude * Level (disambiguation) * Magnitude (disambiguation) In physical sciences Physics *Intensity (physics), power per unit area (W/m2) *Field strength of electric, ma ...
), when illuminated by unpolarized light the remaining power in the unwanted polarization will be (''g''2/''g''1)2 of the power in the intended polarization.


Specular reflection

In addition to birefringence and dichroism in extended media, polarization effects describable using Jones matrices can also occur at (reflective) interface between two materials of different
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
. These effects are treated by the
Fresnel equations The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresne ...
. Part of the wave is transmitted and part is reflected; for a given material those proportions (and also the phase of reflection) are dependent on the angle of incidence and are different for the ''s'' and ''p'' polarizations. Therefore, the polarization state of reflected light (even if initially unpolarized) is generally changed. Any light striking a surface at a special angle of incidence known as
Brewster's angle Brewster's angle (also known as the polarization angle) is an angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with ''no reflection''. When ''unpolarized'' ligh ...
, where the reflection coefficient for ''p'' polarization is zero, will be reflected with only the ''s''-polarization remaining. This principle is employed in the so-called "pile of plates polarizer" (see figure) in which part of the ''s'' polarization is removed by reflection at each Brewster angle surface, leaving only the ''p'' polarization after transmission through many such surfaces. The generally smaller reflection coefficient of the ''p'' polarization is also the basis of
polarized sunglasses 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 th ...
; by blocking the ''s'' (horizontal) polarization, most of the glare due to reflection from a wet street, for instance, is removed. In the important special case of reflection at normal incidence (not involving anisotropic materials) there is no particular ''s'' or ''p'' polarization. Both the ''x'' and ''y'' polarization components are reflected identically, and therefore the polarization of the reflected wave is identical to that of the incident wave. However, in the case of circular (or elliptical) polarization, the handedness of the polarization state is thereby reversed, since by convention this is specified relative to the direction of propagation. The circular rotation of the electric field around the ''x-y'' axes called "right-handed" for a wave in the ''+z'' direction is "left-handed" for a wave in the ''-z'' direction. But in the general case of reflection at a nonzero angle of incidence, no such generalization can be made. For instance, right-circularly polarized light reflected from a dielectric surface at a grazing angle, will still be right-handed (but elliptically) polarized. Linear polarized light reflected from a metal at non-normal incidence will generally become elliptically polarized. These cases are handled using Jones vectors acted upon by the different Fresnel coefficients for the ''s'' and ''p'' polarization components.


Measurement techniques involving polarization

Some optical measurement techniques are based on polarization. In many other optical techniques polarization is crucial or at least must be taken into account and controlled; such examples are too numerous to mention.


Measurement of stress

In
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, the phenomenon of stress induced birefringence allows for stresses in transparent materials to be readily observed. As noted above and seen in the accompanying photograph, the chromaticity of birefringence typically creates colored patterns when viewed in between two polarizers. As external forces are applied, internal stress induced in the material is thereby observed. Additionally, birefringence is frequently observed due to stresses "frozen in" at the time of manufacture. This is famously observed in
cellophane Cellophane is a thin, transparent sheet made of regenerated cellulose. Its low permeability to air, oils, greases, bacteria, and liquid water makes it useful for food packaging. Cellophane is highly permeable to water vapour, but may be coat ...
tape whose birefringence is due to the stretching of the material during the manufacturing process.


Ellipsometry

Ellipsometry is a powerful technique for the measurement of the optical properties of a uniform surface. It involves measuring the polarization state of light following specular reflection from such a surface. This is typically done as a function of incidence angle or wavelength (or both). Since ellipsometry relies on reflection, it is not required for the sample to be transparent to light or for its back side to be accessible. Ellipsometry can be used to model the (complex) refractive index of a surface of a bulk material. It is also very useful in determining parameters of one or more
thin film A thin film is a layer of material ranging from fractions of a nanometer (monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step in many a ...
layers deposited on a substrate. Due to their reflection properties, not only are the predicted magnitude of the ''p'' and ''s'' polarization components, but their relative phase shifts upon reflection, compared to measurements using an ellipsometer. A normal ellipsometer does not measure the actual reflection coefficient (which requires careful photometric calibration of the illuminating beam) but the ratio of the ''p'' and ''s'' reflections, as well as change of polarization ellipticity (hence the name) induced upon reflection by the surface being studied. In addition to use in science and research, ellipsometers are used
in situ ''In situ'' (; often not italicized in English) is a Latin phrase that translates literally to "on site" or "in position." It can mean "locally", "on site", "on the premises", or "in place" to describe where an event takes place and is used in ...
to control production processes for instance.


Geology

The property of (linear) birefringence is widespread in crystalline
mineral In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. ( ...
s, and indeed was pivotal in the initial discovery of polarization. In
mineralogy Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical) properties of minerals and mineralized artifacts. Specific studies within mineralogy include the proc ...
, this property is frequently exploited using polarization microscopes, for the purpose of identifying minerals. See
optical mineralogy Optical mineralogy is the study of minerals and rocks by measuring their optical properties. Most commonly, rock and mineral samples are prepared as thin sections or grain mounts for study in the laboratory with a petrographic microscope. Optic ...
for more details. Sound waves in solid materials exhibit polarization. Differential propagation of the three polarizations through the earth is a crucial in the field of
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
. Horizontally and vertically polarized seismic waves (
shear waves In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
) are termed SH and SV, while waves with longitudinal polarization (
compressional wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves ...
s) are termed P-waves.


Chemistry

We have seen (above) that the birefringence of a type of crystal is useful in identifying it, and thus detection of linear birefringence is especially useful in
geology Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
and
mineralogy Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical) properties of minerals and mineralized artifacts. Specific studies within mineralogy include the proc ...
. Linearly polarized light generally has its polarization state altered upon transmission through such a crystal, making it stand out when viewed in between two crossed polarizers, as seen in the photograph, above. Likewise, in chemistry, rotation of polarization axes in a liquid solution can be a useful measurement. In a liquid, linear birefringence is impossible, however there may be circular birefringence when a chiral molecule is in solution. When the right and left handed
enantiomers In chemistry, an enantiomer ( /ɪˈnænti.əmər, ɛ-, -oʊ-/ ''ih-NAN-tee-ə-mər''; from Ancient Greek ἐνάντιος ''(enántios)'' 'opposite', and μέρος ''(méros)'' 'part') – also called optical isomer, antipode, or optical ant ...
of such a molecule are present in equal numbers (a so-called
racemic In chemistry, a racemic mixture, or racemate (), is one that has equal amounts of left- and right-handed enantiomers of a chiral molecule or salt. Racemic mixtures are rare in nature, but many compounds are produced industrially as racemates. ...
mixture) then their effects cancel out. However, when there is only one (or a preponderance of one), as is more often the case for
organic molecules In chemistry, organic compounds are generally any chemical compounds that contain carbon-hydrogen or carbon-carbon bonds. Due to carbon's ability to catenate (form chains with other carbon atoms), millions of organic compounds are known. The s ...
, a net circular birefringence (or ''
optical activity Optical rotation, also known as polarization rotation or circular birefringence, is the rotation of the orientation of the plane of polarization about the optical axis of linearly polarized light as it travels through certain materials. Circula ...
'') is observed, revealing the magnitude of that imbalance (or the concentration of the molecule itself, when it can be assumed that only one enantiomer is present). This is measured using a
polarimeter A polarimeter is a scientific instrument used to measure the angle of rotation caused by passing polarized light through an optically active substance.


Astronomy

In many areas of
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...
, the study of polarized electromagnetic radiation from
outer space Outer space, commonly shortened to space, is the expanse that exists beyond Earth and its atmosphere and between celestial bodies. Outer space is not completely empty—it is a near-perfect vacuum containing a low density of particles, pred ...
is of great importance. Although not usually a factor in the
thermal radiation Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) is ...
of
star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...
s, polarization is also present in radiation from coherent astronomical sources (e.g. hydroxyl or methanol
maser A maser (, an acronym for microwave amplification by stimulated emission of radiation) is a device that produces coherent electromagnetic waves through amplification by stimulated emission. The first maser was built by Charles H. Townes, Jame ...
s), and incoherent sources such as the large radio lobes in active galaxies, and pulsar radio radiation (which may, it is speculated, sometimes be coherent), and is also imposed upon starlight by scattering from
interstellar dust Cosmic dust, also called extraterrestrial dust, star dust or space dust, is dust which exists in outer space, or has fallen on Earth. Most cosmic dust particles measure between a few molecules and 0.1 mm (100 micrometers). Larger particles are c ...
. Apart from providing information on sources of radiation and scattering, polarization also probes the interstellar magnetic field via
Faraday rotation The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect (MOFE), is a physical magneto-optical phenomenon. The Faraday effect causes a polarization rotation which is proportional to the projection of the m ...
. The polarization of the
cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all space ...
is being used to study the physics of the very early universe.
Synchrotron radiation Synchrotron radiation (also known as magnetobremsstrahlung radiation) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in ...
is inherently polarized. It has been suggested that astronomical sources caused the
chirality Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
of biological molecules on Earth.


Applications and examples


Polarized sunglasses

Unpolarized light, after being reflected by a specular (shiny) surface, generally obtains a degree of polarization. This phenomenon was observed in 1808 by the mathematician
Étienne-Louis Malus Étienne-Louis Malus (; ; 23 July 1775 – 23 February 1812) was a French officer, engineer, physicist, and mathematician. Malus was born in Paris, France. He participated in Napoleon's expedition into Egypt (1798 to 1801) and was a member ...
, after whom Malus's law is named. Polarizing
sunglasses Sunglasses or sun glasses (informally called shades or sunnies; more names below) are a form of protective eyewear designed primarily to prevent bright sunlight and high-energy visible light from damaging or discomforting the eyes. They can ...
exploit this effect to reduce glare from reflections by horizontal surfaces, notably the road ahead viewed at a grazing angle. Wearers of polarized sunglasses will occasionally observe inadvertent polarization effects such as color-dependent birefringent effects, for example in
toughened glass Tempered or toughened glass is a type of safety glass processed by controlled thermal or chemical treatments to increase its strength compared with normal glass. Tempering puts the outer surfaces into compression and the interior into tensio ...
(e.g., car windows) or items made from transparent
plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptabi ...
s, in conjunction with natural polarization by reflection or scattering. The polarized light from LCD monitors (see below) is very conspicuous when these are worn.


Sky polarization and photography

Polarization is observed in the light of the
sky The sky is an unobstructed view upward from the surface of the Earth. It includes the atmosphere and outer space. It may also be considered a place between the ground and outer space, thus distinct from outer space. In the field of astronomy, ...
, as this is due to sunlight scattered by
aerosols An aerosol is a suspension of fine solid particles or liquid droplets in air or another gas. Aerosols can be natural or anthropogenic. Examples of natural aerosols are fog or mist, dust, forest exudates, and geyser steam. Examples of anthropogen ...
as it passes through
Earth's atmosphere The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing for ...
. The scattered light produces the brightness and color in clear skies. This partial polarization of scattered light can be used to darken the sky in photographs, increasing the contrast. This effect is most strongly observed at points on the sky making a 90° angle to the Sun. Polarizing filters use these effects to optimize the results of photographing scenes in which reflection or scattering by the sky is involved. Sky polarization has been used for orientation in navigation. The
Pfund sky compass Pfund is German for " pound weight" and is also a surname. Notable people with the surname include: * August Herman Pfund (1879–1949), American physicist and spectroscopist * Guillermo Pfund (born 1989), Argentine footballer * Jessica Pfund (born ...
was used in the 1950s when navigating near the poles of the
Earth's magnetic field Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic ...
when neither the
sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
nor
star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...
s were visible (e.g., under daytime cloud or
twilight Twilight is light produced by sunlight scattering in the upper atmosphere, when the Sun is below the horizon, which illuminates the lower atmosphere and the Earth's surface. The word twilight can also refer to the periods of time when this il ...
). It has been suggested, controversially, that the
Viking Vikings ; non, víkingr is the modern name given to seafaring people originally from Scandinavia (present-day Denmark, Norway and Sweden), who from the late 8th to the late 11th centuries raided, pirated, traded and se ...
s exploited a similar device (the "
sunstone Sunstone is a microcline or oligoclase feldspar, which when viewed from certain directions exhibits a spangled appearance. It has been found in Southern Norway, Sweden, various United States localities and on some beaches along the midcoast of ...
") in their extensive expeditions across the
North Atlantic The Atlantic Ocean is the second-largest of the world's five oceans, with an area of about . It covers approximately 20% of Earth's surface and about 29% of its water surface area. It is known to separate the "Old World" of Africa, Europe an ...
in the 9th–11th centuries, before the arrival of the
magnetic compass A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...
from Asia to Europe in the 12th century. Related to the sky compass is the " polar clock", invented by
Charles Wheatstone Sir Charles Wheatstone FRS FRSE DCL LLD (6 February 1802 – 19 October 1875), was an English scientist and inventor of many scientific breakthroughs of the Victorian era, including the English concertina, the stereoscope (a device for dis ...
in the late 19th century.


Display technologies

The principle of
liquid-crystal display A liquid-crystal display (LCD) is a flat-panel display or other electronically modulated optical device that uses the light-modulating properties of liquid crystals combined with polarizers. Liquid crystals do not emit light directly but in ...
(LCD) technology relies on the rotation of the axis of linear polarization by the liquid crystal array. Light from the
backlight A backlight is a form of illumination used in liquid crystal displays (LCDs). As LCDs do not produce light by themselves—unlike, for example, cathode ray tube (CRT), plasma (PDP) or OLED displays—they need illumination ( ambient light or a ...
(or the back reflective layer, in devices not including or requiring a backlight) first passes through a linear polarizing sheet. That polarized light passes through the actual liquid crystal layer which may be organized in pixels (for a TV or computer monitor) or in another format such as a
seven-segment display A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays. Seven-segment displays are widely used in digital clocks, electronic meters, basic ca ...
or one with custom symbols for a particular product. The liquid crystal layer is produced with a consistent right (or left) handed chirality, essentially consisting of tiny
helices A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
. This causes circular birefringence, and is engineered so that there is a 90 degree rotation of the linear polarization state. However, when a voltage is applied across a cell, the molecules straighten out, lessening or totally losing the circular birefringence. On the viewing side of the display is another linear polarizing sheet, usually oriented at 90 degrees from the one behind the active layer. Therefore, when the circular birefringence is removed by the application of a sufficient voltage, the polarization of the transmitted light remains at right angles to the front polarizer, and the pixel appears dark. With no voltage, however, the 90 degree rotation of the polarization causes it to exactly match the axis of the front polarizer, allowing the light through. Intermediate voltages create intermediate rotation of the polarization axis and the pixel has an intermediate intensity. Displays based on this principle are widespread, and now are used in the vast majority of televisions, computer monitors and video projectors, rendering the previous CRT technology essentially obsolete. The use of polarization in the operation of LCD displays is immediately apparent to someone wearing polarized sunglasses, often making the display unreadable. In a totally different sense, polarization encoding has become the leading (but not sole) method for delivering separate images to the left and right eye in
stereoscopic Stereoscopy (also called stereoscopics, or stereo imaging) is a technique for creating or enhancing the illusion of depth in an image by means of stereopsis for binocular vision. The word ''stereoscopy'' derives . Any stereoscopic image is ...
displays used for
3D movies 3D films are motion pictures made to give an illusion of three-dimensional solidity, usually with the help of special glasses worn by viewers. They have existed in some form since 1915, but had been largely relegated to a niche in the motion pict ...
. This involves separate images intended for each eye either projected from two different projectors with orthogonally oriented polarizing filters or, more typically, from a single projector with time multiplexed polarization (a fast alternating polarization device for successive frames).
Polarized 3D glasses A polarized 3D system uses polarization glasses to create the illusion of three-dimensional images by restricting the light that reaches each eye (an example of stereoscopy). To present stereoscopic images and films, two images are projected su ...
with suitable polarizing filters ensure that each eye receives only the intended image. Historically such systems used linear polarization encoding because it was inexpensive and offered good separation. However circular polarization makes separation of the two images insensitive to tilting of the head, and is widely used in 3-D movie exhibition today, such as the system from
RealD RealD Inc. is a private company known for its RealD 3D system, which is used for projecting films in stereoscopic 3D using circularly polarized light. The company was founded in 2003 by Michael V. Lewis and Joshua Greer. The company was take ...
. Projecting such images requires screens that maintain the polarization of the projected light when viewed in reflection (such as
silver screen A silver screen, also known as a silver lenticular screen, is a type of projection screen that was popular in the early years of the motion picture industry and passed into popular usage as a metonym for the cinema industry. The term silver scree ...
s); a normal diffuse white projection screen causes depolarization of the projected images, making it unsuitable for this application. Although now obsolete, CRT computer displays suffered from reflection by the glass envelope, causing glare from room lights and consequently poor contrast. Several anti-reflection solutions were employed to ameliorate this problem. One solution utilized the principle of reflection of circularly polarized light. A circular polarizing filter in front of the screen allows for the transmission of (say) only right circularly polarized room light. Now, right circularly polarized light (depending on the convention used) has its electric (and magnetic) field direction rotating clockwise while propagating in the +z direction. Upon reflection, the field still has the same direction of rotation, but now propagation is in the −z direction making the reflected wave ''left'' circularly polarized. With the right circular polarization filter placed in front of the reflecting glass, the unwanted light reflected from the glass will thus be in very polarization state that is ''blocked'' by that filter, eliminating the reflection problem. The reversal of circular polarization on reflection and elimination of reflections in this manner can be easily observed by looking in a mirror while wearing 3-D movie glasses which employ left- and right-handed circular polarization in the two lenses. Closing one eye, the other eye will see a reflection in which it cannot see itself; that lens appears black. However the other lens (of the closed eye) will have the correct circular polarization allowing the closed eye to be easily seen by the open one.


Radio transmission and reception

All
radio Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a transmit ...
(and microwave)
antennas In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies a ...
used for transmitting or receiving are intrinsically polarized. They transmit in (or receive signals from) a particular polarization, being totally insensitive to the opposite polarization; in certain cases that polarization is a function of direction. Most antennas are nominally linearly polarized, but elliptical and circular polarization is a possibility. As is the convention in optics, the "polarization" of a radio wave is understood to refer to the polarization of its electric field, with the magnetic field being at a 90 degree rotation with respect to it for a linearly polarized wave. The vast majority of antennas are linearly polarized. In fact it can be shown from considerations of symmetry that an antenna that lies entirely in a plane which also includes the observer, can ''only'' have its polarization in the direction of that plane. This applies to many cases, allowing one to easily infer such an antenna's polarization at an intended direction of propagation. So a typical rooftop Yagi or log-periodic antenna with horizontal conductors, as viewed from a second station toward the horizon, is necessarily horizontally polarized. But a vertical "
whip antenna A whip antenna is an antenna consisting of a straight flexible wire or rod. The bottom end of the whip is connected to the radio receiver or transmitter. A whip antenna is a form of monopole antenna. The antenna is designed to be flexible so ...
" or AM broadcast tower used as an antenna element (again, for observers horizontally displaced from it) will transmit in the vertical polarization. A
turnstile antenna A turnstile antenna, or crossed-dipole antenna, is a radio antenna consisting of a set of two identical dipole antennas mounted at right angles to each other and fed in phase quadrature; the two currents applied to the dipoles are 90° out of pha ...
with its four arms in the horizontal plane, likewise transmits horizontally polarized radiation toward the horizon. However, when that same turnstile antenna is used in the "axial mode" (upwards, for the same horizontally-oriented structure) its radiation is circularly polarized. At intermediate elevations it is elliptically polarized. Polarization is important in radio communications because, for instance, if one attempts to use a horizontally polarized antenna to receive a vertically polarized transmission, the signal strength will be substantially reduced (or under very controlled conditions, reduced to nothing). This principle is used in
satellite television Satellite television is a service that delivers television programming to viewers by relaying it from a communications satellite orbiting the Earth directly to the viewer's location. The signals are received via an outdoor parabolic antenna commo ...
in order to double the channel capacity over a fixed frequency band. The same frequency channel can be used for two signals broadcast in opposite polarizations. By adjusting the receiving antenna for one or the other polarization, either signal can be selected without interference from the other. Especially due to the presence of the
ground Ground may refer to: Geology * Land, the surface of the Earth not covered by water * Soil, a mixture of clay, sand and organic matter present on the surface of the Earth Electricity * Ground (electricity), the reference point in an electrical c ...
, there are some differences in propagation (and also in reflections responsible for TV ghosting) between horizontal and vertical polarizations. AM and FM broadcast radio usually use vertical polarization, while television uses horizontal polarization. At low frequencies especially, horizontal polarization is avoided. That is because the phase of a horizontally polarized wave is reversed upon reflection by the ground. A distant station in the horizontal direction will receive both the direct and reflected wave, which thus tend to cancel each other. This problem is avoided with vertical polarization. Polarization is also important in the transmission of
radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weat ...
pulses and reception of radar reflections by the same or a different antenna. For instance, back scattering of radar pulses by rain drops can be avoided by using circular polarization. Just as specular reflection of circularly polarized light reverses the handedness of the polarization, as discussed above, the same principle applies to scattering by objects much smaller than a wavelength such as rain drops. On the other hand, reflection of that wave by an irregular metal object (such as an airplane) will typically introduce a change in polarization and (partial) reception of the return wave by the same antenna. The effect of free electrons in the
ionosphere The ionosphere () is the ionized part of the upper atmosphere of Earth, from about to above sea level, a region that includes the thermosphere and parts of the mesosphere and exosphere. The ionosphere is ionized by solar radiation. It plays an im ...
, in conjunction with the
earth's magnetic field Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic ...
, causes
Faraday rotation The Faraday effect or Faraday rotation, sometimes referred to as the magneto-optic Faraday effect (MOFE), is a physical magneto-optical phenomenon. The Faraday effect causes a polarization rotation which is proportional to the projection of the m ...
, a sort of circular birefringence. This is the same mechanism which can rotate the axis of linear polarization by electrons in
interstellar space Outer space, commonly shortened to space, is the expanse that exists beyond Earth and its atmosphere and between celestial bodies. Outer space is not completely empty—it is a near-perfect vacuum containing a low density of particles, predo ...
as mentioned
below Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred Below ...
. The magnitude of Faraday rotation caused by such a plasma is greatly exaggerated at lower frequencies, so at the higher microwave frequencies used by satellites the effect is minimal. However medium or
short wave Shortwave radio is radio transmission using shortwave (SW) radio frequencies. There is no official definition of the band, but the range always includes all of the high frequency band (HF), which extends from 3 to 30 MHz (100 to 10 me ...
transmissions received following refraction by the ionosphere are strongly affected. Since a wave's path through the ionosphere and the earth's magnetic field vector along such a path are rather unpredictable, a wave transmitted with vertical (or horizontal) polarization will generally have a resulting polarization in an arbitrary orientation at the receiver.


Polarization and vision

Many
animal Animals are multicellular, eukaryotic organisms in the biological kingdom Animalia. With few exceptions, animals consume organic material, breathe oxygen, are able to move, can reproduce sexually, and go through an ontogenetic stage in ...
s are capable of perceiving some of the components of the polarization of light, e.g., linear horizontally polarized light. This is generally used for navigational purposes, since the linear polarization of sky light is always perpendicular to the direction of the sun. This ability is very common among the
insect Insects (from Latin ') are pancrustacean hexapod invertebrates of the class Insecta. They are the largest group within the arthropod phylum. Insects have a chitinous exoskeleton, a three-part body (head, thorax and abdomen), three pairs of ...
s, including
bee Bees are winged insects closely related to wasps and ants, known for their roles in pollination and, in the case of the best-known bee species, the western honey bee, for producing honey. Bees are a monophyletic lineage within the superfami ...
s, which use this information to orient their communicative dances. Polarization sensitivity has also been observed in species of
octopus An octopus ( : octopuses or octopodes, see below for variants) is a soft-bodied, eight- limbed mollusc of the order Octopoda (, ). The order consists of some 300 species and is grouped within the class Cephalopoda with squids, cuttlef ...
,
squid True squid are molluscs with an elongated soft body, large eyes, eight arms, and two tentacles in the superorder Decapodiformes, though many other molluscs within the broader Neocoleoidea are also called squid despite not strictly fitting t ...
,
cuttlefish Cuttlefish or cuttles are marine molluscs of the order Sepiida. They belong to the class Cephalopoda which also includes squid, octopuses, and nautiluses. Cuttlefish have a unique internal shell, the cuttlebone, which is used for control o ...
, and
mantis shrimp Mantis shrimp, or stomatopods, are carnivorous marine crustaceans of the order Stomatopoda (). Stomatopods branched off from other members of the class Malacostraca around 340 million years ago. Mantis shrimp typically grow to around in lengt ...
. In the latter case, one species measures all six orthogonal components of polarization, and is believed to have optimal polarization vision. The rapidly changing, vividly colored skin patterns of cuttlefish, used for communication, also incorporate polarization patterns, and mantis shrimp are known to have polarization selective reflective tissue. Sky polarization was thought to be perceived by
pigeon Columbidae () is a bird family consisting of doves and pigeons. It is the only family in the order Columbiformes. These are stout-bodied birds with short necks and short slender bills that in some species feature fleshy ceres. They primarily ...
s, which was assumed to be one of their aids in homing, but research indicates this is a popular myth. The naked
human eye The human eye is a sensory organ, part of the sensory nervous system, that reacts to visible light and allows humans to use visual information for various purposes including seeing things, keeping balance, and maintaining circadian rhythm. T ...
is weakly sensitive to polarization, without the need for intervening filters. Polarized light creates a very faint pattern near the center of the visual field, called
Haidinger's brush Haidinger's brush, more commonly known as Haidinger's brushes is an image produced by the eye, an entoptic phenomenon, first described by Austrian physicist Wilhelm Karl von Haidinger in 1844. Haidinger saw it when he looked through various mine ...
. This pattern is very difficult to see, but with practice one can learn to detect polarized light with the naked eye.


Angular momentum using circular polarization

It is well known that electromagnetic radiation carries a certain linear
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
in the direction of propagation. In addition, however, light carries a certain
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
if it is circularly polarized (or partially so). In comparison with lower frequencies such as microwaves, the amount of angular momentum in light, even of pure circular polarization, compared to the same wave's linear momentum (or
radiation pressure Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
) is very small and difficult to even measure. However it was utilized in an experiment to achieve speeds of up to 600 million revolutions per minute.


See also

*
Depolarizer (optics) A depolarizer or depolariser is an optical device used to scramble the polarization of light. An ideal depolarizer would output randomly polarized light whatever its input, but all practical depolarizers produce pseudo-random output polarization. O ...
*
Fluorescence anisotropy Fluorescence anisotropy or fluorescence polarization is the phenomenon where the light emitted by a fluorophore has unequal intensities along different axes of polarization. Early pioneers in the field include Aleksander Jablonski, Gregorio Weber ...
* Glan–Taylor prism *
Kerr effect The Kerr effect, also called the quadratic electro-optic (QEO) effect, is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index chang ...
*
Nicol prism A Nicol prism is a type of polarizer, an optical device made from calcite crystal used to produce and analyse plane polarized light. It is made in such a way that it eliminates one of the rays by total internal reflection, i.e. the ordinary ra ...
*
Pockels effect The Pockels effect or Pockels electro-optic effect, named after Friedrich Carl Alwin Pockels (who studied the effect in 1893), changes or produces birefringence in an optical medium induced by an electric field. In the Pockels effect, also known a ...
*
Polarization rotator A polarization rotator is an optical device that rotates the polarization axis of a linearly polarized light beam by an angle of choice. Such devices can be based on the Faraday effect, on birefringence, or on total internal reflection. F. J. Du ...
*
Polarized light microscopy Polarized light microscopy can mean any of a number of optical microscopy techniques involving polarized light. Simple techniques include illumination of the sample with polarized light. Directly transmitted light can, optionally, be blocked with ...
*
Polarizer A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well- ...
*
Polaroid (polarizer) A Polaroid synthetic plastic sheet is a brand name product trademarked and produced by the Polaroid Corporation used as a polarizer or polarizing filter. The term “Polaroid” entered the common vocabulary with the early 1960s introduction of p ...
* Radial polarization * Rayleigh sky model *
Waveplate A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. Two common types of waveplates are the ''half-wave plate'', which shifts the polarization direction of linearly polarized light ...


References


Cited references


General references

* ''
Principles of Optics ''Principles of Optics'', colloquially known as ''Born and Wolf'', is an optics textbook written by Max Born and Emil Wolf that was initially published in 1959 by Pergamon Press. After going through six editions with Pergamon Press, the book wa ...
'', 7th edition, M. Born & E. Wolf, Cambridge University, 1999, . * ''Fundamentals of polarized light: a statistical optics approach'', C. Brosseau, Wiley, 1998, . * ''Polarized Light, second edition'', Dennis Goldstein, Marcel Dekker, 2003, . * ''Field Guide to Polarization'', Edward Collett, SPIE Field Guides vol. FG05, SPIE, 2005, . * ''Polarization Optics in Telecommunications'', Jay N. Damask, Springer 2004, . * ''Polarized Light in Nature'', G. P. Können, Translated by G. A. Beerling, Cambridge University, 1985, . * ''Polarised Light in Science and Nature'', D. Pye, Institute of Physics, 2001, . * ''Polarized Light, Production and Use'', William A. Shurcliff, Harvard University, 1962. * ''Ellipsometry and Polarized Light'', R. M. A. Azzam and N. M. Bashara, North-Holland, 1977, . * ''Secrets of the Viking Navigators—How the Vikings used their amazing sunstones and other techniques to cross the open oceans'', Leif Karlsen, One Earth Press, 2003.


External links


Feynman's lecture on polarization

Polarized Light in Nature and Technology


Microscopic images made using polarization effects

Animated explanation of polarization

* ttp://gerdbreitenbach.de/crystal/crystal.html A virtual polarization microscope
Polarization angle in satellite dishes



Molecular Expressions: Science, Optics and You — Polarization of Light
Interactive Java tutorial
SPIE technical group on polarization

Antenna Polarization

Animations of Linear, Circular and Elliptical Polarizations on YouTube
{{Authority control Electromagnetic radiation Antennas (radio) Broadcast engineering Physical optics