= Solution
Vacuum <Maxwell's equations> are invariant under spatial rotations. Rotate the preceding coordinate system so that $\mathbf e_z$ becomes the propagation direction $\widehat{\mathbf k}$, and rotate the transverse electric direction to a unit <polarization vector> $\widehat{\mathbf n}$. Dot and cross products transform covariantly under this rotation, so transversality and the field relation are unchanged. With $\omega=c|\mathbf k|$,
$$
\boxed{\mathbf E(\mathbf r,t)=E_0e^{i(\mathbf k\cdot\mathbf r-\omega t)}\widehat{\mathbf n},\qquad
\mathbf B(\mathbf r,t)=\frac{E_0}{c}e^{i(\mathbf k\cdot\mathbf r-\omega t)}(\widehat{\mathbf k}\times\widehat{\mathbf n}),\qquad\widehat{\mathbf k}\cdot\widehat{\mathbf n}=0.}
$$
These complex fields encode real fields through their real parts. The displayed form describes linear <electromagnetic polarization>; a general transverse complex amplitude can describe elliptical or circular <electromagnetic polarization> as well.
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