What is the explicit expression of a plane wave in the frequency domain?

Question

A plane wave in the time domain can be written (using notation for an electric field): $$\boldsymbol{E}(\boldsymbol{r},t)=\boldsymbol{E}_0 e^{i(\boldsymbol{k} \boldsymbol{r}-\omega t)}$$ what is the corresponding expression for a plane wave, $\boldsymbol{E}(\boldsymbol{r}, \omega)$, in the frequency domain?

Answer 1

The key for all these formulas is $$\tag{1} \int_{\mathbb R^d} e^{ix\cdot y}\,\frac{dy}{(2\pi)^{\frac d 2}} = \delta(x), $$ where $\delta $ is the $d$-dimensional Dirac distribution. This question is probably concerned with $d=1$ only, but there is no extra difficulty in doing the general case.

Of course, (1) is to be suitably interpreted; it is not an absolutely convergent integral. My favorite reference for this is the book "Quantum field theory: a tourist guide for mathematicians", by Gerry Folland. (There, he states that (1) is to the mathematical physicist as $\int_{-\infty}^\infty e^{-x^2}\, dx =\sqrt{\pi}$ is to the mathematician. I love that quote).

To finally answer the question, we apply (1) to obtain $$ \int_{\mathbb R} E_0e^{ik\cdot x -\omega t} e^{-i ts}\, ds = (2\pi)^\frac 1 2 E_0e^{i k\cdot x}\delta(t+\omega).$$ This is the Fourier transform of the given function, the "plane wave", in the time variable.