Polarization (waves)

Polarization in electrodynamics

In electrodynamics, polarization is a property of waves, such as light and other electromagnetic radiation. Unlike more familiar wave phenomena such as waves on water or waves propagating on a string, electromagnetic waves are three dimensional, and it is this multidimensional nature that gives rise to the phenomenon of polarization.

Take the case of a simple plane wave, which is a good approximation to most light waves. The plane of the wave is perpendicular to the direction the wave is propagating in. Simply because the plane is two dimensional the electric vector in the plane at a point in space has, in general, two orthogonal components. Call these the x and y components. For a simple harmonic wave, where the amplitude of the electric vector varies in a sinusoidal manner, the two components have exactly the same frequency. 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 in the fixed plane we are talking about.

Consider first the special case where the two orthogonal components are in phase. In this case the direction of the electric vector in the plane, the vector sum of these two components, will always fall on a single line in the plane. We call this special case linear polarization. The direction of this line will depend on the relative amplitude of the two components. This direction can be in any angle in the plane, but the direction never varies.

Now consider another special case, where 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. Notice that there are two possible phase relationships that satisfy this requirenemt. 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 direction of the vector sum electric vector in the plane will rotate in a circle. We call this special case circular polarization. The direction of rotation will depend on which of the two phase relationships exists. We call these cases right hand circular polarization or left hand circular polarization, depending on which way the electic vector rotates.

All the other cases, that is where the two components are not in phase and either do not have the same amplitude or are not ninety degrees out of phase are called eliptical polarization because the direction of the electric vector in the plane will trace out an elipse.

The linear and circular cases are limiting cases of eliptical polarization. In the first case one of two the axes of the elipse has zero length, and we have linear polarization. In the second case the two eliptical axes are equal and we have one of the two circular polarizations.

For circular polarization, it is also useful to consider how the direction of the electric vector varies along the direction of propagation at a single instant of time. While in the plane the direction rotates in a circle, along the propagation axis the direction describes a helix. The pitch of the helix is one wavelength, and the screw sense is either right handed or left handed. Visualizing this spatial variation in the direction of the electric field is useful in understanding how circularly polarized light can interact differently with helical molecular conformations, depending on whether the electic field and the molecule helix sense are the same or opposite. This is part of the phenomenon of circular dichroism.

Polarization of visible light can be observed using a polarizing filter (the lenses of polaroid sunglasses will work). While viewing through the filter, rotate it, and if linear or eliptically polarized light is present the degree of illumination will change. The blue sky is polarized because of the nature of the scattering phenomenon that produces the color. An easy first phenomenon to observe is at sunset to view the horizon at a 90° angle from the sunset.

Common sources of light, such as the Sun and the electric light bulb emit what is known as unpolarized light. More specialised sources, such as certain kinds of discharge tubes and lasers, produce polarized light. The difference between these two types of light is caused by the behaviour of the electromagnetic fields that make up the light.

As described by Maxwell's equations, light is a transverse wave made up of an interacting electric field E and a magnetic field B. The oscillations of these two interacting fields cause the fields to self-propagate in a certain direction, at the speed of light. In most cases, the directions of the electric field, the magnetic field, and the direction of propagation of the light are all mutually perpendicular. That is, both the E and B fields oscillate in a direction at right angles to the direction that the light is moving, and also at right angles to each other.

(In optics, it is usual to define the polarization in terms of the direction of the electric field, and disregard the magnetic field since it is almost always perpendicular to the electric field.)

If the direction of oscillation of the electric field E is fixed, the light wave is said to be linearly polarized.

The direction of polarization is arbitrary with respect to the light itself. It is usual to label the two linear polarization states in accordance with some other external reference. For example, the terms horizontally and vertically polarized are generally used when light is propagating in free space. If the light is interacting with a surface, such as a mirror, lens or some other interface between two media, the terms s- and p-polarized are used. For example, consider the following:

           |       /
           |      /
           |     /
           |    /
           |   /
           |  /
           | /
           |/
  ======================

In the above diagram, a light ray is reflecting off a mirror at some angle. If the electric field of the light is oscillating perpendicular to the plane of the diagram, the light is termed s-polarized. If it is oscillating in the plane of the diagram, it is termed p-polarized. Other terms used for s-polarization are sigma-polarized and sagittal plane polarised. Similarly, p-polarized light is also referred to as pi-polarised and tangential plane polarized.

If the direction of the electric field E is not fixed, but rotates in a circle as the light propagates, the light is said to be circularly polarized. Two possible independent circular polarization states exist, termed left-hand or right-hand circularly polarized depending on whether the electric field is rotating in a counter-clockwise or clockwise sense, respectively, when looking in the direction of the light propagation.

Individual photons are inherently circularly polarized; this is related to the concept of spin in particle physics.

If the light consists of many incoherent waves with randomly varying polarisation, the light is said to be unpolarized. It is possible to convert unpolarised light to polarised light by using a polarizer. One such device is Polaroid® sheet. This is a sheet of plastic with molecules that are arranged such that they absorb any light passing through it which has an electric field oscillating in a given direction; this has the effect of linearly polarizing the light. Other devices can split an unpolarised beam into two beams of orthogonal linear polarization; They are generally constructed from certain arrangements of prisms and optical coatings.

The angle of polarization of linearly polarised light can be rotated using a device known as a half-wave plate. Similarly, linear polarization can be converted to circular polarization and vice versa with the use of a quarter-wave plate.

A quarter-wave plate is constructed from a birefringent material, that is, in the plane of the plate there are two orthogonal axes and light passing through it propagates at a different speed along one axis than on the other. The thickness of the plate is adjusted so that the net difference in propagation speed is one quarter of a wavelength. If this plate is oriented so that the fast axis is forty five degress to the direction of linear polarization then the light emerging from the other side will have two components of equal amplitude and a ninety degree phase difference, creating circular polarization. Rotating the quarter wave plate ninety degrees will reverse the sense of circular polarization.

Birefringence can be created by straining a normally uniform material. A properly arranged and controlled mechanical oscillator coupled to a strain-free window can convert linearly polarized light of a single color impinging on the window into alternating left and right hand circularly polarized light emerging from the other side. That is, the window can operate as an oscillating quarter wave plate. If this light is then passed through a material which has a circular dichroism at that color, the emerging light will have an amplitude modulation that varies with the frequency of the oscillator driving the quarter wave plate. This amplitude variation can be detected and used to measure the amount of circular dichroism exhibited. This amplitude will depend on the intrinsic property of the material, and upon the amount of material the light passed through, which in turn depends on the concentration of the absorbing substance and its thickness. Although the phenomenon measured this way is delta-absorption, the results are customarily reported in degrees of elipticity through a simple algebraic conversion.

Poincare sphere

The possible polarization states can be mapped to a sphere, with left circular at +z, right circular at -z (the poles), horizontal at +x, vertical at -x, and the diagonals at +y and -y. All other positions along the equator of the sphere are other angles of linear polarization. All remaining points on the sphere are cases of eliptical polarization. Passing through a dichroic wave plate is equivalent to a rotation of the sphere. The amount of amplitude of polarization x that passes through a polarizer that passes y is 1/2 the distance between x and the antipode of y; the intensity is (x·y+1)/2. The spherical mapping is attributed to Henri Poincare, and is discussed extensively in english by Shurcliff.

See also Brewster's angle, Fresnel equations.

Further Reading

• Polarized Light in Nature, G. P. Können, Translated by G. A. Beerling, Cambridge University Press, 1985, hardcover, ISBN 0-521-25862-6
• Optics, Eugene Hecht, Addison Wesley, 4th edition 2002, hardcover, ISBN 0-8053-8566-5
• Theorie mathematique de la lumiere, Henri Poincare, Gauthiers-Villars, Paris, 1892. The original description of the Poincare Sphere.
• Polarized Light, Production and Use William A. Shurcliff, Harvard University Press, 1962.

Polarization in electrostatics

In electrostatics, the polarization is the vector field that results from permanent or induced dipole moments in a dielectric material. The polarization vector P is defined as the dipole moment per unit volume.

In a homogeneous linear and isotropic dielectric medium, the polarization is in aligned with and proportional to the electric field E:

P = ε₀χE ,

where ε₀ is the permittivity of free space, and χ is the electric susceptibility of the medium.

If the polarization P is not proportional to the electric field E, the medium is termed nonlinear and is described by the field of nonlinear optics. If the direction of P is not aligned with E, as in many crystals, the medium is anisotropic and is described by crystal optics.

Polarization in politics

In politics, polarization is the process by which the public opinion divides and goes to the extremes. The moderates lose power. It may be one of the first steps to a civil war.