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Consider the set of all possible superpositions of all possible "plane waves that satisfy Maxwell's equations in free space". Does this set represent all possible solutions of Maxwell's equations in free space? Why?

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Yes, because $\mathbf{E}$ and $\mathbf{B}$, like any fields in spacetime, can be expressed via the Fourier transform as a superposition of plane waves, each having some frequency $\omega$ and wave vector $\mathbf{k}$. The homogeneity and linearity of Maxwell's equations imply that each of these Fourier modes individually satisfies the equations (meaning the only nonzero contributions are from plane waves where $\mathbf{E}$ and $\mathbf{B}$ are transverse, $\omega^2 = c^2 k^2$, etc.). A caveat is that we must understand "plane wave" broadly enough to include "zero modes" in which $\mathbf{E}$ and $\mathbf{B}$ are constant or linear functions of $t$ and $\mathbf{x}$.

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