First, I have a question about supereposition of the plane waves - whether the direction of all such plane wave is same, i.e. ##\vec{n}## is in some direction. If not, I think it would be ##\vec{E}(\vec{x}, t)=\int\mathbf{\mathfrak{E}}(\vec{k}')e^{i\vec{k}'\cdot\vec{x}-i\omega t}d^3k##. Besides...