Polarization (waves)
Polarization, or polarisation, is a property of transverse waves which 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. One example of a polarized transverse wave is vibrations traveling along a taut string, for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves in solids.
An electromagnetic wave such as light consists of a coupled oscillating electric field and magnetic field that are always perpendicular to each other. Different states of polarization correspond to different relationships between the directions of the fields and the direction of propagation. In linear polarization, the electric and magnetic fields each oscillate in a single direction, perpendicular to one another. In circular or elliptical polarization, the fields rotate around the beam's direction of travel at a constant rate. The rotation can be either in the right-hand or in the left-hand direction.
Light or other electromagnetic radiation from many sources, such as the sun, flames, and incandescent lamps, consists of short wave trains with an equal mixture of polarizations; this is called unpolarized light. Polarized light can be produced by passing unpolarized light through a polarizer, which allows waves of only one polarization to pass through. The most common optical materials do not affect the polarization of light, but some materials—those that exhibit birefringence, dichroism, or optical activity—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, electromagnetic waves can also be viewed as streams of particles called photons. When viewed in this way, the polarization of an electromagnetic wave is determined by a quantum mechanical property of photons called their spin. A photon has one of two possible spins: it can either spin in a right hand sense or a left hand sense about its direction of travel. Circularly polarized electromagnetic waves are composed of photons with only one type of spin, either right- or left-hand. Linearly polarized waves consist of photons that are in a superposition of right and left circularly polarized states, with equal amplitude and phases synchronized to give oscillation in a plane.
Polarization is an important parameter in areas of science dealing with transverse waves, such as optics, seismology, radio, and microwaves. Especially impacted are technologies such as lasers, wireless and optical fiber telecommunications, and radar.
Introduction
Wave propagation and polarization
Most sources of light are classified as incoherent and unpolarized because they consist of a random mixture of waves having different spatial characteristics, frequencies, phases, and polarization states. However, for understanding electromagnetic waves and polarization in particular, it is easier to just consider coherent plane waves; these are sinusoidal waves of one particular direction, 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. Incoherent states can be modeled stochastically as a weighted combination of such uncorrelated waves with some distribution of frequencies, phases, and polarizations.Transverse electromagnetic waves
, traveling in free space or another homogeneous isotropic non-attenuating medium, are properly described as transverse waves, meaning that a plane wave's electric field vector and magnetic field are each in some direction perpendicular to the direction of wave propagation; and 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 of optical frequency , let us take the direction of propagation as the axis. Being a transverse wave the and fields must then contain components only in the and directions whereas. Using complex notation, the instantaneous physical electric and magnetic fields are given by the real parts of the complex quantities occurring in the following equations. As a function of time and spatial position these complex fields can be written as:and
where is the wavelength and is the period of the wave. Here,,, and are complex numbers. In the second more compact form, as these equations are customarily expressed, these factors are described using the wavenumber and angular frequency . In a more general formulation with propagation restricted to the direction, then the spatial dependence is replaced by where is called the wave vector, the magnitude of which is the wavenumber.
Thus the leading vectors and each contain up to two nonzero components describing the amplitude and phase of the wave's and polarization components. For a given medium with a characteristic impedance, is related to by:
In a dielectric, is real and has the value, where is the refractive index and is the impedance of free space. The impedance will be complex in a conducting medium. Note that given that relationship, the dot product of and must be zero:
indicating that these vectors are orthogonal, as expected.
Knowing the propagation direction and, one can just as well specify the wave in terms of just and describing the electric field. The vector containing and is known as a Jones vector. 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 of the light wave is proportional to the sum of the squared magnitudes of the two electric field components:
However, the wave's state of polarization is only dependent on the ratio of to. So let us just consider waves whose ; this happens to correspond to an intensity of about in free space. And because the absolute phase of a wave is unimportant in discussing its polarization state, let us stipulate that the phase of is zero; in other words is a real number while may be complex. Under these restrictions, and can be represented as follows:
where the polarization state is now fully parameterized by the value of and the relative phase.
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, but 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 medium the electric or magnetic field may have longitudinal as well as transverse components. In those cases the electric displacement and magnetic flux density still obey the above geometry but due to anisotropy in the electric susceptibility, now given by a tensor, the direction of may differ from that of . Even in isotropic media, so-called inhomogeneous waves can be launched into a medium whose refractive index has a significant imaginary part such as metals; these fields are also not strictly transverse. Surface waves or waves propagating in a waveguide are generally transverse waves, but might be described as an electric or magnetic transverse mode, 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 longitudinal.
For longitudinal waves such as sound waves in fluids, 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 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.
Polarization state
Polarization can be defined in terms of pure polarization states with only a coherent sinusoidal wave at one optical frequency. The vector in the adjacent diagram might describe the oscillation of the electric field emitted by a single-mode laser. The field oscillates in the -plane, along the page, with the wave propagating in the 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. The linear polarization at 45° can also be viewed as the addition of a horizontally linearly polarized wave and a vertically polarized wave of the same amplitude.
Now if one were to introduce a phase shift 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°, and the amplitudes are the same, then circular polarization is produced. Circular polarization can be created by sending linearly polarized light through a quarter-wave plate oriented at 45° to the linear polarization to create two components of the same amplitude with the required phase shift. The superposition of the original and phase-shifted components causes a rotating electric field vector, which 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, depending on the relative phases of the components. These correspond to distinct polarization states, such as the two circular polarizations shown above.
The orientation of the and axes used in this description is arbitrary. The choice of such a coordinate system and viewing the polarization ellipse in terms of the and polarization components, corresponds to the definition of the Jones vector in terms of those basis polarizations. Axes are selected to suit a particular problem, such as being in the plane of incidence. Since there are separate reflection coefficients for the linear polarizations in and orthogonal to the plane of incidence, that choice greatly simplifies the calculation of a wave's reflection from a surface.
Any pair of orthogonal polarization states may be used as basis functions, not just linear polarizations. For instance, choosing right and left circular polarizations as basis functions simplifies the solution of problems involving circular birefringence or circular dichroism.