 Closed-surface Poynting flux of a plane wave

For a real linearly polarized monochromatic vacuum plane wave, the local Poynting vector is proportional to ${\cos }^2(\mathbf{k}\cdot \mathbf{r}-\omega t)\hat{\mathbf{k}}$. Its flux through a closed surface generally oscillates at $2\omega$, with zero cycle average, although particular volume geometries can cancel it identically. For a circularly polarized plane wave of constant amplitude the Poynting vector is spatially and temporally constant, so the net flux through every closed surface is exactly zero. A nonzero constant local energy flux must not be confused with the net closed-surface flux.