## Note: Reading Spectrograms

This guide to modelocked pulse evolutions makes heavy use of spectrograms to illustrate the relevant pulse-shaping processes. For those not familiar with this representation, a brief review is provided here.

A spectrogram, also known as a Wigner distribution, is a way of visualizing a field in the temporal and spectral domains simultaneously. It can be constructed using a short-time Fourier transform: a time window is defined, and then translated across the field. At each position, the Fourier transform of the windowed field is calculated. Collating these windowed power spectra yields the spectrogram. Note that the duration of the time window will jointly define the temporal and spectral resolution of the resulting plot, subject to the uncertainty principle, and should be chosen so as to best resolve the features of interest. For added clarity, sum-out panels can be added to depict the pulse wholly in one domain or the other, as is done here.

**Illustration of how a spectrogram can be constructed:**

**Some simple examples:**

*Group-velocity dispersion (normal sign case):*

*Self-phase modulation (Gaussian initial pulse):*

Due to the uncertainty principle, a single spectrogram is subject to counterbalance between temporal and spectral resolutions. With a small sliding time window to get a good temporal resolution, spectral resolution is lost; with a large time window for a good spectral resolution, temporal resolution is lost. However, there is a solution to this problem by combining multiple spectrograms with different time windows. Please see **the slides** for detail. The **high-resolution spectrogram code** is shared as well. Feel free to use it.