When dealing with Maxwell's equations, we often need to switch between $$(\mathbf{r},t)$$- and $$(\mathbf{k},\omega)$$-space, i.e. between space-time and Fourier domain. This is advantageous, since derivatives in space and time are simple products in Fourier domain.

The Fourier transformation, or often short Fourier transform, of the electric field is given by:

\begin{eqnarray*}\mathbf{E}\left(\mathbf{r},t\right)&=&\intop_{-\infty}^{\infty}\mathbf{\overline{E}}\left(\mathbf{r},\omega\right)e^{-i\omega t}d\omega\\\mathbf{\overline{E}}\left(\mathbf{r},\omega\right)&=&\frac{1}{2\pi}\intop_{-\infty}^{\infty}\mathbf{E}\left(\mathbf{r},t\right)e^{i\omega t}dt \end{eqnarray*} and \begin{eqnarray*} \mathbf{E}\left(\mathbf{r},t\right)&=&\intop_{-\infty}^{\infty}\mathbf{E}\left(\mathbf{k},t\right)e^{i\mathbf{kr}}d\mathbf{r}\\\mathbf{E}\left(\mathbf{k},t\right)&=&\frac{1}{2\pi}\intop_{-\infty}^{\infty}\mathbf{E}\left(\mathbf{r},t\right)e^{-i\mathbf{kr}}d\mathbf{k}\end{eqnarray*}

## The Spectrum of a rectangularly shaped Light Pulse

After we have understood the Fourier Transformation of the Heaviside Function, we can now apply our knowledge to the spectrum of a pulsed electric field. This can be done by the Fourier transform of a Heaviside function with underlying oscillating electric field.

## How to use Fourier transformation on an electric field pulse

In this problem you will learn how to apply the Fourier Transformation to a simple pulsed electric field. We will see the definitions applied and some nice physics unfolds!

## Fourier Transformation of the Heaviside Function

The Heaviside step function is very important in physics. It often models a sudden switch-on phenomenon and is therefore present in a lot of integrals. For example, the derivation of the Kramers-Kronig Relations can be significantly simplified once we know the Fourier-Transform $$\bar{\theta}(\omega)$$ of the Heaviside function $$\theta(t)$$. Although the function appears to be quite simple, the calculation of its Fourier transform can be quite challenging. Let's find out!