Polarization is a property of transverse waves that describes how their oscillations are oriented and how that orientation evolves. In optics, it usually refers to the behavior of the electric field of light in the plane perpendicular to propagation. It distinguishes waves that otherwise have the same frequency and intensity and influences their interaction with materials. Wave polarization is distinct from electric polarization, which concerns the separation or alignment of charges within matter. (openstax.org)
Physical basis and polarization states
For a plane wave of electromagnetic radiation traveling through free space, the electric field and magnetic field are perpendicular both to one another and to the propagation direction. Polarization is conventionally specified using the electric field. Two perpendicular transverse components, together with their amplitudes and relative phase, determine the polarization state. (openstax.org)
For propagation along the (z)-axis, a monochromatic field can be written
[ E_x=A_x\cos(kz-\omega t),\qquad E_y=A_y\cos(kz-\omega t+\delta), ]
where (A_x,A_y) are component amplitudes, (k) is the wave number, (\omega) is angular frequency, and (\delta) is the relative phase. At a fixed position, the electric-field tip generally traces an ellipse. The principal cases are:
- Linear polarization: the field oscillates along a fixed line. This occurs when the components are in phase or opposite in phase, or when one component vanishes.
- Circular polarization: the field tip traces a circle, requiring equal component amplitudes and a relative phase of (+\pi/2) or (-\pi/2).
- Elliptical polarization: the general case, with linear and circular polarization as limiting forms.
Circular and elliptical states have two possible senses of rotation. “Right-handed” and “left-handed” labels depend on the viewing convention, so a complete specification must identify that convention. (labs.phys.utk.edu)
Polarized and unpolarized radiation
A fully polarized beam has a definite relationship between its transverse field components. In unpolarized radiation, fluctuations make the time-averaged polarization statistics independent of the orientation of the transverse reference axes. This does not mean that the instantaneous electric field has no direction; rather, no single stable polarization state describes the averaged radiation. Partially polarized light lies between these cases and can be represented statistically as a combination of polarized and unpolarized contributions. (optics.byu.edu)
The distinction depends on averaging over the measurement’s temporal, spatial, and spectral resolution. Consequently, polarization measurements characterize both the radiation and the conditions under which it is observed. A deterministic description of a monochromatic field is insufficient for a general fluctuating beam. (optics.byu.edu)
Production and manipulation
A polarizer preferentially transmits one polarization component. Absorptive polarizers use direction-dependent absorption; other devices separate components into different outgoing beams. For linearly polarized incident light and an ideal linear analyzer, Malus’s law gives
[ I=I_0\cos^2\theta, ]
where (\theta) is the angle between the incident polarization and the transmission axis. An ideal linear polarizer transmits half the intensity of unpolarized incident light. Two ideal polarizers with perpendicular transmission axes therefore block light transmitted through the first. (openstax.org)
Reflection and refraction also discriminate between polarization components. At Brewster’s angle, the component polarized parallel to the plane of incidence has zero reflected amplitude at an ideal interface between ordinary transparent, nonmagnetic dielectric media. The angle obeys
[ \tan\theta_B=\frac{n_2}{n_1}, ]
with (n_1,n_2) the refractive indices of the incident and transmitting media. Reflected initially unpolarized light is then linearly polarized perpendicular to the plane of incidence. Scattering likewise produces polarization, as occurs in sunlight scattered by the atmosphere. (openstax.org)
Birefringence causes orthogonal polarization components to experience different refractive indices and accumulate different phases. A wave plate exploits this effect. A quarter-wave plate introduces a relative phase of (\pi/2), converting linear polarization into circular polarization when the input direction is at (45^\circ) to its principal axes. A half-wave plate introduces a phase difference of (\pi) and can rotate linear polarization. (thorlabs.com)
Mathematical description and measurement
Jones calculus represents fully polarized light by a two-component vector of complex field amplitudes. A suitable optical element acts through a (2\times2) matrix, allowing sequences of polarizers and wave plates to be analyzed through matrix multiplication. A single Jones vector does not describe partially polarized radiation. (labs.phys.utk.edu)
Stokes parameters provide an intensity-based description applicable to fully, partially, and unpolarized light. Conventionally, (S_0) denotes total intensity, (S_1,S_2) describe linear-polarization contrasts, and (S_3) describes circular-polarization contrast. The degree of polarization is
[ P=\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0}, ]
ranging from zero to one for nonzero intensity. Normalized Stokes coordinates represent fully polarized states on the Poincaré sphere and partially polarized states inside it. Polarimetry determines these quantities using measured intensities through suitable analyzers and retarders. (optics.byu.edu)
Applications and quantum description
Polarizing sunglasses and filters in photography suppress polarized reflections, while liquid-crystal displays control transmitted intensity by changing polarization between polarizers. These applications exploit orientation-dependent transmission rather than simply reducing all incoming light equally. (openstax.org)
In astronomy, polarization carries information beyond intensity. The polarization of the cosmic microwave background, generated through interactions with electrons in the early universe, preserves information about primordial matter distributions. (esa.int)
In quantum mechanics, a photon traveling in a specified direction has two independent polarization states. Horizontal and vertical states can form a basis for a qubit, with arbitrary pure states expressed as their superposition. The squared projection onto an analyzer’s transmitted state gives the transmission probability, reproducing Malus’s law statistically for many identically prepared photons. (cs.middlebury.edu)