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Concept
Sinusoidal plane-wave solutions of the electromagnetic wave equation
Summary
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.
The plane sinusoidal solution for an electromagnetic wave traveling in the z direction is
for the electric field and
for the magnetic field, where k is the wavenumber,
is the angular frequency of the wave, and is the speed of light. The hats on the vectors indicate unit vectors in the x, y, and z directions. r = (x, y, z) is the position vector (in meters).
The plane wave is parameterized by the amplitudes
and phases
where
and
Jones calculus
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.
Official source
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