Sinusoidal plane-wave solutions are particular solutions to the 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.
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
E ( r , t ) = ( E 0 , x cos ( k z − ω t + α x ) E 0 , y cos ( k z − ω t + α y ) 0 ) = E 0 , x cos ( k z − ω t + α x ) x ^ + E 0 , y cos ( k z − ω t + α y ) y ^ {\displaystyle {\begin{aligned}\mathbf {E} (\mathbf {r} ,t)&={\begin{pmatrix}E_{0,x}\cos \left(kz-\omega t+\alpha _{x}\right)\\E_{0,y}\cos \left(kz-\omega t+\alpha _{y}\right)\\0\end{pmatrix}}\\[1ex]&=E_{0,x}\cos \left(kz-\omega t+\alpha _{x}\right)\,{\hat {\mathbf {x} }}\;+\;E_{0,y}\cos \left(kz-\omega t+\alpha _{y}\right)\,{\hat {\mathbf {y} }}\end{aligned}}}
for the electric field and
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