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 theories of Max Planck and the interpretations by Albert Einstein 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.
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.
for the electric field and
for the magnetic field, where k is the wavenumber,
The plane wave is parameterized by the amplitudes
Polarization state 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 rewritten in complex notation as
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
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
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
then a circular polarization state can written in the "R-L basis" as
Any arbitrary state can be written in the R-L basis
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