- 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 andpolarization s.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 byAlbert Einstein . The quantum generalization of the classical treatment can be found in the articles onPhoton polarization andPhoton 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 ,:
is the
angular frequency of the wave, and is thespeed of light . The hats on the vectors indicateunit vectors in the x, y, and z directions.The plane wave is parameterized by the
amplitude s:
:
and phases
:
where
:.
and
:.
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 aquantum state vector. The connection with quantum mechanics is made in the article onphoton polarization .The vector emerges from the plane-wave solution. The electric field solution can be re-written in complex notation as
:
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 are equal,
:.
This represents a wave polarized at an angle with respect to the x axis. In that case the Jones vector can be written
:.
Circular polarization
If is rotated by radians with respect to 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
:
where
:
and
:.
Any arbitrary state can be written in the R-L basis
:
where
:.
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]
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