2012-01-11

Polarisation (waves)




Polarization (waves)

Polarization (also polarisation) is a property of certain types of waves that describes the orientation of their oscillations. Electromagnetic waves, such as light, and gravitational waves exhibit polarization; acoustic waves (sound waves) in a gas or liquid do not have polarization because the direction of vibration and direction of propagation are the same.

By convention, the polarization of light is described by specifying the orientation of the wave's electric field at a point in space over one period of the oscillation. When light travels in free space, in most cases it propagates as a transverse wave—the polarization is perpendicular to the wave's direction of travel. In this case, the electric field may be oriented in a single direction (linear polarization), or it may rotate as the wave travels (circular or elliptical polarization). In the latter cases, the oscillations can rotate either towards the right or towards the left in the direction of travel. Depending on which rotation is present in a given wave it is called the wave's chirality or handedness. In general the polarization of an electromagnetic (EM) wave is a complex issue. For instance in a waveguide such as an optical fiber, or for radially polarized beams in free space, the description of the wave's polarization is more complicated, as the fields can have longitudinal as well as transverse components. Such EM waves are either TM or hybrid modes.

For longitudinal waves such as sound waves in fluids, the direction of oscillation is by definition along the direction of travel, so there is no polarization. In a solid medium, however, sound waves can be transverse. In this case, the polarization is associated with the direction of the shear stress in the plane perpendicular to the propagation direction. This is important in seismology.

Polarization is significant in areas of science and technology dealing with wave propagation, such as optics, seismology, telecommunications and radar science. The polarization of light can be measured with a polarimeter. A polarizer is a device that affects polarization.

Theory

Basics: plane waves

The simplest manifestation of polarization to visualize is that of a plane wave, which is a good approximation of most light waves (a plane wave is a wave with infinitely long and wide wavefronts). For plane waves Maxwell's equations, specifically Gauss's laws, impose the transversality requirement that the electric and magnetic field be perpendicular to the direction of propagation and to each other. Conventionally, when considering polarization, the electric field vector is described and the magnetic field is ignored since it is perpendicular to the electric field and proportional to it. The electric field vector of a plane wave may be arbitrarily divided into two perpendicular components labeled x and y (with z indicating the direction of travel). For a simple harmonic wave, where the amplitude of the electric vector varies in a sinusoidal manner in time, the two components have exactly the same frequency. However, these components have two other defining characteristics that can differ. First, the two components may not have the same amplitude. Second, the two components may not have the same phase, that is they may not reach their maxima and minima at the same time. Mathematically, the electric field of a plane wave can be written as,

\vec{E}(\vec{r},t) = \mathrm{Re} \left[\left(A_{x}, A_{y}\cdot e^{i\phi}, 0 \right) e^{i(kz - \omega t)} \right]

or alternatively,

\vec{E}(\vec{r},t) = (A_{x}\cdot \cos(kz - \omega t), A_{y}\cdot \cos(kz - \omega t + \phi), 0)

where Ax and Ay are the amplitudes of the x and y directions and ϕ is the relative phase between the two components.

Polarization state

The shape traced out in a fixed plane by the electric vector as such a plane wave passes over it (a Lissajous figure) is a description of the polarization state. The following figures show some examples of the evolution of the electric field vector (black), with time(the vertical axes), at a particular point in space, along with its x and y components (red/left and blue/right), and the path traced by the tip of the vector in the plane (yellow in figure 1&3, purple in figure 2): The same evolution would occur when looking at the electric field at a particular time while evolving the point in space, along the direction opposite to propagation.

Linear polarization diagram
Linear
Circular polarization diagram
Circular
Elliptical polarization diagram
Elliptical


In the leftmost figure above, the two orthogonal (perpendicular) components are in phase. In this case the ratio of the strengths of the two components is constant, so the direction of the electric vector (the vector sum of these two components) is constant. Since the tip of the vector traces out a single line in the plane, this special case is called linear polarization. The direction of this line depends on the relative amplitudes of the two components.

In the middle figure, the two orthogonal components have exactly the same amplitude and are exactly ninety degrees out of phase. In this case one component is zero when the other component is at maximum or minimum amplitude. There are two possible phase relationships that satisfy this requirement: the x component can be ninety degrees ahead of the y component or it can be ninety degrees behind the y component. In this special case the electric vector traces out a circle in the plane, so this special case is called circular polarization. The direction the field rotates in depends on which of the two phase relationships exists. These cases are called right-hand circular polarization and left-hand circular polarization, depending on which way the electric vector rotates and the chosen convention.

Another case is when the two components are not in phase and either do not have the same amplitude or are not ninety degrees out of phase, though their phase offset and their amplitude ratio are constant. This kind of polarization is called elliptical polarization because the electric vector traces out an ellipse in the plane (the polarization ellipse). This is shown in the above figure on the right.

Unpolarized light

Most sources of electromagnetic radiation contain a large number of atoms or molecules that emit light. The orientation of the electric fields produced by these emitters may not be correlated, in which case the light is said to be unpolarized. If there is partial correlation between the emitters, the light is partially polarized. If the polarization is consistent across the spectrum of the source, partially polarized light can be described as a superposition of a completely unpolarized component, and a completely polarized one. One may then describe the light in terms of the degree of polarization, and the parameters of the polarization ellipse.

Parameterization

Polarisation ellipse2.svg

For ease of visualization, polarization states are often specified in terms of the polarization ellipse, specifically its orientation and elongation. A common parameterization uses the orientation angle, ψ, the angle between the major semi-axis of the ellipse and the x-axis (also known as tilt angle or azimuth angle) and the ellipticity, ε, the major-to-minor-axis ratio (also known as the axial ratio). An ellipticity of zero or infinity corresponds to linear polarization and an ellipticity of 1 corresponds to circular polarization. The ellipticity angle, χ = arccot ε= arctan 1/ε, is also commonly used. An example is shown in the diagram to the right. An alternative to the ellipticity or ellipticity angle is the eccentricity, however unlike the azimuth angle and ellipticity angle, the latter has no obvious geometrical interpretation in terms of the Poincaré sphere (see below).

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 vector (the Jones vector):

\mathbf{e} = \begin{bmatrix} a_1 e^{i \theta_1} \\ a_2 e^{i \theta_2} \end{bmatrix} .

Here a1 and a2 denote the amplitude of the wave in the two components of the electric field vector, while θ1 and θ2 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. The basis vectors used to represent the Jones vector need not represent linear polarization states (i.e. be real). In general any two orthogonal states can be used, where an orthogonal vector pair is formally defined as one having a zero inner product. 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.

Regardless of whether polarization ellipses are 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 the equatorial coordinate system 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) corresponding to due north.

S and P Polarization

Another coordinate system frequently used relates to the plane made by the propagation direction and a vector perpendicular to the plane of a reflecting surface. This is known as the plane of incidence. 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). Light with a p-like electric field is said to be p-polarized, pi-polarized, tangential plane polarized, or is said to be a transverse-magnetic (TM) wave. Light with an s-like electric field is s-polarized, also known as sigma-polarized or sagittal plane polarized, or it can be called a transverse-electric (TE) wave. However, there is no universal convention in this TE and TM naming scheme, and certain authors do refer to light with p-like electric field as TE and light with s-like electric field as TM.

Parameterization of incoherent or partially polarized radiation

In the case of partially-polarized radiation, the Jones vector varies in time and space in a way that differs from the constant rate of phase rotation of monochromatic, purely-polarized waves. In this case, the wave field is likely stochastic, and only statistical information can be gathered about the variations and correlations between components of the electric field. This information is embodied in the coherency matrix:

\mathbf{\Psi} = \left\langle\mathbf{e} \mathbf{e}^\dagger \right\rangle\,
=\left\langle\begin{bmatrix} e_1 e_1^* & e_1 e_2^* \\ e_2 e_1^* & e_2 e_2^* \end{bmatrix} \right\rangle
=\left\langle\begin{bmatrix} a_1^2 & a_1 a_2 e^{i (\theta_1-\theta_2)} \\ a_1 a_2 e^{-i (\theta_1-\theta_2)}& a_2^2 \end{bmatrix} \right\rangle

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 measure the coherence of a spectral decomposition of the signal, while the Wolf 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 matrices, corresponding to the eigenvectors 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.

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, introduced by George Gabriel Stokes in 1852. The relationship of the Stokes parameters to intensity and polarization ellipse parameters is shown in the equations and figure below.

Propagation, reflection and scattering

In a vacuum, the components of the electric field propagate at the speed of light, so that the phase of the wave varies in space and time while the polarization state does not. That is,

\mathbf{e}(z+\Delta z,t+\Delta t) = \mathbf{e}(z, t) e^{i k (c\Delta t - \Delta z)},

where k is the wavenumber and positive z is the direction of propagation. As noted above, the physical electric vector is the real part of the Jones vector. When electromagnetic waves interact with matter, their propagation is altered. If this depends on the polarization states of the waves, then their polarization may also be altered.

In many types of media, electromagnetic waves may be decomposed into two orthogonal components that encounter different propagation effects. A similar situation occurs in the signal processing paths of detection systems that record the electric field directly. Such effects are most easily characterized in the form of a complex 2×2 transformation matrix called the Jones matrix:

\mathbf{e'} = \mathbf{J}\mathbf{e}.

In general the Jones matrix of a medium depends on the frequency of the waves.

For propagation effects in two orthogonal modes, the Jones matrix can be written as

\mathbf{J} = \mathbf{T} \begin{bmatrix} g_1 & 0 \\ 0 & g_2 \end{bmatrix} \mathbf{T}^{-1},

where g1 and g2 are complex numbers representing the change in amplitude and phase caused in each of the two propagation modes, and T is a unitary matrix representing a change of basis from these propagation modes to the linear system used for the Jones vectors. For those media in which the amplitudes are unchanged but a differential phase delay occurs, the Jones matrix is unitary, while those affecting amplitude without phase have Hermitian Jones matrices. In fact, since any matrix may be written as the product of unitary and positive Hermitian matrices, any sequence of linear propagation effects, no matter how complex, can be written as the product of these two basic types of transformations.

Birefringence diagram

Paths taken by vectors in the Poincaré sphere under birefringence. The propagation modes (rotation axes) are shown with red, blue, and yellow lines, the initial vectors by thick black lines, and the paths they take by colored ellipses (which represent circles in three dimensions).

Media in which the two modes accrue a differential delay are called birefringent. Well known manifestations of this effect appear in optical wave plates/retarders (linear modes) and in Faraday rotation/optical rotation (circular modes). An easily visualized example is one where the propagation modes are linear, and the incoming radiation is linearly polarized at a 45° angle to the modes. As the phase difference starts to appear, the polarization becomes elliptical, eventually changing to purely circular polarization (90° phase difference), then to elliptical and eventually linear polarization (180° phase) with an azimuth angle perpendicular to the original direction, 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 in the Poincaré sphere about the axis defined by the propagation modes (this is a consequence of the isomorphism of SU(2) with SO(3)). Examples for linear (blue), circular (red), and elliptical (yellow) birefringence 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, plane waves will exit the material with a significantly different propagation direction, due to refraction. For example, this is the case with macroscopic crystals of calcite, which present the viewer with two offset, orthogonally polarized images of whatever is viewed through them. It was this effect that provided the first discovery of polarization, by Erasmus Bartholinus in 1669. In addition, the phase shift, and thus the change in polarization state, is usually frequency dependent, which, in combination with dichroism, often gives rise to bright colors and rainbow-like effects.

Media in which the amplitude of waves propagating in one of the modes is reduced are called dichroic. Devices that block nearly all of the radiation in one mode are known as polarizing filters or simply "polarizers". In terms of the Stokes parameters, the total intensity is reduced while vectors in the Poincaré sphere are "dragged" towards the direction of the favored mode. Mathematically, under the treatment of the Stokes parameters as a Minkowski 4-vector, the transformation is a scaled Lorentz boost (due to the isomorphism of SL(2,C) and the restricted Lorentz group, SO(3,1)). Just as the Lorentz transformation preserves the proper time, the quantity det Ψ = S0 − S1 − S2 − S3 is invariant within a multiplicative scalar constant under Jones matrix transformations (dichroic and/or birefringent).

In birefringent and dichroic media, in addition to writing a Jones matrix for the net effect of passing through a particular path in a given medium, the evolution of the polarization state along that path can be characterized as the (matrix) product of an infinite series of infinitesimal steps, each operating on the state produced by all earlier matrices. In a uniform medium each step is the same, and one may write

\mathbf{J} = Je^{\mathbf{D}},

where J is an overall (real) gain/loss factor. Here D is a traceless matrix such that αDe gives the derivative of e with respect to z. If D is Hermitian the effect is dichroism, while a unitary matrix models birefringence. The matrix D can be expressed as a linear combination of the Pauli matrices, where real coefficients give Hermitian matrices and imaginary coefficients give unitary matrices. The Jones matrix in each case may therefore be written with the convenient construction

\begin{matrix} \mathbf{J_b} &=& J_be^{\beta \mathbf{\sigma}\cdot\mathbf{\hat{n}}} & \text{and} & \mathbf{J_r} &=& J_re^{\phi i\mathbf{\sigma}\cdot\mathbf{\hat{m}}}, \end{matrix}

where σ is a 3-vector composed of the Pauli matrices (used here as generators for the Lie group SL(2,C)) and n and m are real 3-vectors on the Poincaré sphere corresponding to one of the propagation modes of the medium. The effects in that space correspond to a Lorentz boost of velocity parameter 2β along the given direction, or a rotation of angle 2φ about the given axis. These transformations may also be written as biquaternions (quaternions with complex elements), where the elements are related to the Jones matrix in the same way that the Stokes parameters are related to the coherency matrix. They may then be applied in pre- and post-multiplication to the quaternion representation of the coherency matrix, with the usual exploitation of the quaternion exponential for performing rotations and boosts taking a form equivalent to the matrix exponential equations above. (See Quaternion rotation)

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. These effects are treated by the Fresnel equations. Part of the wave is transmitted and part is reflected, with the ratio depending on angle of incidence and the angle of refraction. In addition, if the plane of the reflecting surface is not aligned with the plane of propagation of the wave, the polarization of the two parts is altered. In general, the Jones matrices of the reflection and transmission are real and diagonal, making the effect similar to that of a simple linear polarizer. For unpolarized light striking a surface at a certain optimum angle of incidence known as Brewster's angle, the reflected wave will be completely s-polarized.

Certain effects do not produce linear transformations of the Jones vector, and thus cannot be described with (constant) 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 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 frequently used to study the effects of the scattering of waves from complex surfaces or ensembles of particles.

Polarization in nature, science, and technology

Polarization effects in everyday life

Biology

Many animals are apparently 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 insects, including bees, which use this information to orient their communicative dances. Polarization sensitivity has also been observed in species of octopus, squid, cuttlefish, and mantis shrimp. 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 pigeons, which was assumed to be one of their aids in homing, but research indicates this is a popular myth.

The naked human eye 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. This pattern is very difficult to see, but with practice one can learn to detect polarized light with the naked eye.

Geology

Chemistry

Polarization is principally of importance in chemistry due to circular dichroism and "optical rotation" (circular birefringence) exhibited by optically active (chiral) molecules. It may be measured using polarimetry.

The term "polarization" may also refer to the through-bond (inductive or resonant effect) or through-space influence of a nearby functional group on the electronic properties (e.g. dipole moment) of a covalent bond or atom. This concept is based on the formation of an electric dipole within a molecule, which is related to polarization of electromagnetic waves in infrared spectroscopy. Molecules will absorb infrared light if the frequency of the bond vibration is resonant with (identical to) the incident light frequency, where the molecular vibration at hand produces a change in the dipole moment of the molecule. In some nonlinear optical processes, the direction of an oscillating dipole will dictate the polarization of the emitted electromagnetic radiation, as in vibrational sum frequency generation spectroscopy or similar processes.

Polarized light does interact with anisotropic materials, which is the basis for birefringence. This is usually seen in crystalline materials and is especially useful in geology (see above). The polarized light is "double refracted", as the refractive index is different for horizontally and vertically polarized light in these materials. This is to say, the polarizability of anisotropic materials is not equivalent in all directions. This anisotropy causes changes in the polarization of the incident beam, and is easily observable using cross-polar microscopy or polarimetry. The optical rotation of chiral compounds (as opposed to achiral compounds that form anisotropic crystals), is derived from circular birefringence. Like linear birefringence described above, circular birefringence is the "double refraction" of circular polarized light.

Astronomy

3D movies

Polarization is also used for some 3D movies, in which the images intended for each eye are 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 with suitable polarized filters ensure that each eye receives only the intended image. Historical stereoscopic projection displays used linear polarization encoding because it was inexpensive and offered good separation. Circular polarization makes left-eye/right-eye separation insensitive to the viewing orientation; circular polarization is used in typical 3-D movie exhibition today, such as the system from RealD. Polarized 3-D only works on screens that maintain polarization (such as silver screens); a normal projection screen would cause depolarization which would void the effect.

Communication and radar applications

All radio transmitting and receiving antennas are intrinsically polarized, special use of which is made in radar. Most antennas radiate either horizontal, vertical, or circular polarization although elliptical polarization also exists. The electric field or E-plane determines the polarization or orientation of the radio wave. Vertical polarization is most often used when it is desired to radiate a radio signal in all directions such as widely distributed mobile units. AM and FM radio use vertical polarization, while television uses horizontal polarization. Alternating vertical and horizontal polarization is used on satellite communications (including television satellites), to allow the satellite to carry two separate transmissions on a given frequency, thus doubling the number of customers a single satellite can serve. Electronically controlled birefringent devices are used in combination with polarizing filters as modulators in fiber optics.

Materials science

Navigation

Sky polarization has been exploited in the "", which was used in the 1950s when navigating near the poles of the Earth's magnetic field when neither the sun nor stars were visible (e.g. under daytime cloud or twilight). It has been suggested, controversially, that the Vikings exploited a similar device (the "sunstone") in their extensive expeditions across the North Atlantic in the 9th–11th centuries, before the arrival of the magnetic compass in Europe in the 12th century. Related to the sky compass is the "", invented by Charles Wheatstone in the late 19th century.

Photography

In photography, polarizing filters are used, mostly to improve the appearance of the sky (deeper blue, and clouds more visible):

Other examples of polarization

See also

Notes and references

External links


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