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Sinusoidal plane-wave solutions are particular solutions to the electromagnetic wave equation. The general solution of the electromagnetic
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
in homogeneous, linear, time-independent media can be written as a
linear superposition The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So th ...
of plane-waves of different frequencies and polarizations. The treatment in this article is classical but, because of the generality of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
for electrodynamics, the treatment can be converted into the quantum mechanical treatment with only a reinterpretation of classical quantities (aside from the quantum mechanical treatment needed for charge and current densities). The reinterpretation is based on the theories of Max Planck and the interpretations by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
of those theories and of other experiments. The quantum generalization of the classical treatment can be found in the articles on photon polarization and photon dynamics in the double-slit experiment.


Explanation

Experimentally, every light signal can be decomposed into a
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 ...
of frequencies and wavelengths associated with sinusoidal solutions of the wave equation. Polarizing filters can be used to decompose light into its various polarization components. The polarization components can be
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
, circular or elliptical.


Plane waves

The plane
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often i ...
solution for 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) ...
traveling in the z direction is \mathbf ( \mathbf , t ) = \begin E_x^0 \cos \left ( kz-\omega t + \alpha_x \right ) \\ E_y^0 \cos \left ( kz-\omega t + \alpha_y \right ) \\ 0 \end = E_x^0 \cos \left ( kz-\omega t + \alpha_x \right ) \hat \; + \; E_y^0 \cos \left ( kz-\omega t + \alpha_y \right ) \hat for the electric field and c \, \mathbf ( \mathbf , t ) = \hat \times \mathbf ( \mathbf , t ) = \begin -E_y^0 \cos \left ( kz-\omega t + \alpha_y \right ) \\ E_x^0 \cos \left ( kz-\omega t + \alpha_x \right ) \\ 0 \end = -E_y^0 \cos \left ( kz-\omega t + \alpha_y \right ) \hat \; + \; E_x^0 \cos \left ( kz-\omega t + \alpha_x \right ) \hat for the magnetic field, 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 te ...
, \omega = c k \omega is the
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 ti ...
of the wave, and c is 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 fo ...
. The hats on the vectors indicate unit vectors in the x, y, and z directions. is the position vector (in meters). The plane wave is parameterized by the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
s E_x^0 = \left, \mathbf \ \cos \theta E_y^0 = \left, \mathbf \ \sin \theta and phases \alpha_x , \alpha_y where \theta \ \stackrel\ \tan^ \left ( \right ) . and \left, \mathbf \^2 \ \stackrel\ \left ( E_x^0 \right )^2 + \left ( E_y^0 \right )^2 .


Polarization state vector


Jones vector

All the polarization information can be reduced to a single vector, called the Jones vector, in the x-y plane. This vector, while arising from a purely classical treatment of polarization, can be interpreted as a
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
vector. The connection with quantum mechanics is made in the article on photon polarization. The vector emerges from the plane-wave solution. The electric field solution can be rewritten in complex notation as \mathbf ( \mathbf , t ) = , \mathbf, \operatorname\left \psi\rangle e^ \right where , \psi\rangle \ \stackrel\ \begin \psi_x \\ \psi_y \end = \begin \cos(\theta) e^ \\ \sin(\theta) e^ \end is the Jones vector in the x-y plane. The notation for this vector is the bra–ket notation of Dirac, which is normally used in a quantum context. The quantum notation is used here in anticipation of the interpretation of the Jones vector as a quantum state vector.


Dual Jones vector

The Jones vector has a dual given by \langle \psi , \ \stackrel\ \begin \psi_x^* & \psi_y^* \end = \begin \cos(\theta) e^ & \sin(\theta) e^ \end.


Normalization of the Jones vector

A Jones vector represents a specific wave with a specific phase, amplitude and state of polarization. When one is using a Jones vector simply to indicate a state of polarization, then it is customary for it to be normalized. That requires that the
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 ...
of the vector with itself to be unity: \langle \psi , \psi \rangle = \begin \psi_x^* & \psi_y^* \end \begin \psi_x \\ \psi_y \end = 1 . An arbitrary Jones vector can simply be scaled to achieve this property. All normalized Jones vectors represent a wave of the same intensity (within a particular isotropic medium). Even given a normalized Jones vector, multiplication by a pure phase factor will result in a different normalized Jones vector representing the same state of polarization.


Polarization states


Linear polarization

In general, the wave is linearly polarized when the phase angles \alpha_x , \alpha_y are equal, \alpha_x = \alpha_y \ \stackrel\ \alpha . This represents a wave polarized at an angle \theta with respect to the x axis. In that case the Jones vector can be written , \psi\rangle = \begin \cos\theta \\ \sin\theta \end e^ .


Elliptical and circular polarization

The general case in which the electric field is not confined to one direction but rotates in the ''x''-''y'' plane is called elliptical polarization. The state vector is given by , \psi\rangle = \begin \psi_x \\ \psi_y \end = \begin \cos(\theta) e^ \\ \sin(\theta) e^ \end = e^ \begin \cos(\theta) \\ \sin(\theta) e^ \end . In the special case of \Delta\alpha = 0, this reduces to linear polarization. Circular polarization corresponds to the special cases of \theta=\pm\pi/4 with \Delta\alpha=\pi/2. The two circular polarization states are thus given by the Jones vectors: , \psi\rangle = \begin \psi_x \\ \psi_y \end = e^ \frac 1 \sqrt \begin 1 \\ \pm i \end.


See also

*
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
* Transverse mode *
Transverse wave 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 o ...
*
Theoretical and experimental justification for the Schrödinger equation The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relat ...
*
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
* Electromagnetic wave equation * Mathematical descriptions of the electromagnetic field
Polarization from an atomic transition: linear and circular


References

*{{cite book , last=Jackson, first= John D., title=Classical Electrodynamics , edition=3rd, publisher=Wiley, year=1998, isbn=0-471-30932-X Sinusoidal plane-wave solutions of the electromagnetic wave equation Sinusoidal plane-wave solutions of the electromagnetic wave equation Sinusoidal plane-wave solutions of the electromagnetic wave equation