# Sinusoidal plane-wave solutions of the electromagnetic wave equation

Sinusoidal plane-wave solutions are particular solutions to the electromagnetic wave equation.

The general solution of the electromagnetic wave equation in homogeneous, linear, time-independent media can be written as a linear superposition of plane-waves of different frequencies and polarizations.

The treatment in this article is classical but, because of the generality of Maxwell's equations 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 experiments of Max Planck and the interpretations of those experiments by Albert Einstein. 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 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, circular or elliptical.

Plane waves

The plane sinusoidal solution for an electromagnetic wave traveling in the z direction is (cgs units and SI units)

:

for the electric field and

:

for the magnetic field, where k is the wavenumber,

:$omega_\left\{ \right\}^\left\{ \right\} = c k$

is the angular frequency of the wave, and $c$ is the speed of light. The hats on the vectors indicate unit vectors in the x, y, and z directions.

The plane wave is parameterized by the amplitudes

:$E_x^0 = mid mathbf\left\{E\right\} mid cos heta$

:$E_y^0 = mid mathbf\left\{E\right\} mid sin heta$

and phases

:$alpha_x^\left\{ \right\} , alpha_y$

where

:$heta stackrel\left\{mathrm\left\{def\left\{=\right\} an^\left\{-1\right\} left \left( \left\{ E_y^0 over E_x^0 \right\} ight \right)$.

and

:$mid mathbf\left\{E\right\} mid^2 stackrel\left\{mathrm\left\{def\left\{=\right\} left \left( E_x^0 ight \right)^2 + left \left( E_y^0 ight \right)^2$.

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 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 re-written in complex notation as

:$mathbf\left\{E\right\} \left( mathbf\left\{r\right\} , t \right) = mid mathbf\left\{E\right\} mid mathrm\left\{Re\right\} left \left\{ |psi angle exp left \left[ i left \left( kz-omega t ight \right) ight \right] ight \right\}$

where

:

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

:.

Normalization of the Jones vector

The Jones vector is normalized. The inner product of the vector with itself is

:.

Polarization states

Linear polarization

In general, the wave is linearly polarized when the phase angles $alpha_x^\left\{ \right\} , alpha_y$ are equal,

:$alpha_x = alpha_y stackrel\left\{mathrm\left\{def\left\{=\right\} alpha$.

This represents a wave polarized at an angle $heta$ with respect to the x axis. In that case the Jones vector can be written

:.

Circular polarization

If $alpha_y$ is rotated by $pi / 2$ radians with respect to $alpha_x$ the wave is circularly polarized. The Jones vector is

:

where the plus sign indicates right circular polarization and the minus sign indicates left circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane.

If unit vectors are defined such that

:

and

:

then a circular polarization state can written in the "R-L basis" as

:$|c angle = psi_R |R angle + psi_L |L angle$

where

:$psi_R stackrel\left\{mathrm\left\{def\left\{=\right\} left \left( \left\{cos heta -isin heta over sqrt\left\{2\right\} \right\} ight \right) exp left \left( i alpha_x ight \right) = left \left( \left\{exp\left(-i heta\right) over sqrt\left\{2\right\} \right\} ight \right) exp left \left( i alpha_x ight \right)$

and

:$psi_L stackrel\left\{mathrm\left\{def\left\{=\right\} left \left( \left\{cos heta +isin heta over sqrt\left\{2\right\} \right\} ight \right) exp left \left( i alpha_x ight \right) = left \left( \left\{exp\left(i heta\right) over sqrt\left\{2\right\} \right\} ight \right) exp left \left( i alpha_x ight \right)$.

Any arbitrary state can be written in the R-L basis

:$|psi angle = a_R exp left \left( i alpha_x -i heta ight \right) |R angle + a_L exp left \left( i alpha_x + i heta ight \right) |L angle$

where

:$1 = mid a_R mid^2 + mid a_L mid^2$.

Elliptical polarization

The general case in which the electric field rotates in the x-y plane and has variable magnitude is called elliptical polarization. The state vector is given by

:.

References

*cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|id=ISBN 0-471-30932-X

ee also

*Fourier series
*Theoretical and experimental justification for the Schrödinger equation
*Maxwell's equations
*Electromagnetic wave equation
*Mathematical descriptions of the electromagnetic field
* [http://www.hydrogenlab.de/elektronium/HTML/einleitung_hauptseite_uk.html Polarisation from an atomic transition: linear and curcular]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Electromagnetic wave equation — The electromagnetic wave equation is a second order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. The homogeneous form of the equation, written in terms of either the… …   Wikipedia

• Inhomogeneous electromagnetic wave equation — Localized time varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell s equations can be written in the form of a inhomogeneous electromagnetic wave equation (or often nonhomogeneous electromagnetic… …   Wikipedia

• Monochromatic electromagnetic plane wave — In general relativity, the monochromatic electromagnetic plane wave spacetime is the analog of the monochromatic plane waves known from Maxwell s theory. The precise definition of the solution is a bit complicated, but very instructive. Any exact …   Wikipedia

• Electromagnetic radiation — Electromagnetism Electricity · …   Wikipedia

• Wave — A wave is a disturbance that propagates through space and time, usually with transference of energy. While a mechanical wave exists in a medium (which on deformation is capable of producing elastic restoring forces), waves of electromagnetic… …   Wikipedia

• Photon dynamics in the double-slit experiment — The dynamics of photons in the double slit experiment describes the relationship between classical electromagnetic waves and photons, the quantum counterpart of classical electromagnetic waves, in the context of the double slit experiment. The… …   Wikipedia

• Helmholtz equation — The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation:( abla^2 + k^2) A = 0where abla^2 is the Laplacian, k is a constant, and the unknown function A=A(x, y, z) is defined on n dimensional… …   Wikipedia

• Partial differential equation — A visualisation of a solution to the heat equation on a two dimensional plane In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several… …   Wikipedia

• Evanescent wave — Schematic representation of evanescent waves propagating along a metal dielectric interface. The charge density oscillations, when associated with electromagnetic fields, are called surface plasmon polariton waves. The exponential dependence of… …   Wikipedia

• Circular polarization — The electric field vectors of a traveling circularly polarized electromagnetic wave. In electrodynamics, circular polarization[1] of an electromagnetic wave is a polarization in which the electric field of the passing wave does not change… …   Wikipedia