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Definition: a type of diagrams which visualize optical or other signals
Spectrograms are quite common in acoustics, but also sometimes used in optics, particularly in the context of ultrashort pulses (→ ultrafast optics). The underlying idea is essentially to display a kind of time-dependent spectrum: a Fourier transform is applied to different temporal sections of the signal. Mathematically, this leads to a signal of the form
where E(t) is the signal under investigation (e.g. the electric field of a pulse) and g(t) is a gate function. The temporally narrower the gain function, the higher is the temporal resolution, but also the lower is the spectral resolution.
It is common to use a horizontal time axis and a vertical frequency (or wavelength) axis, and to encode the intensity for each combination of time and frequency with a gray scale or a color scale. The result is quite intuitive e.g. for an up-chirped ultrashort pulse (see Figure 1).
Figure 1: Spectrogram of an ultrashort pulse with a pronounced up-chirp, i.e., with a rising instantaneous frequency. The graph has been generated with the software RP ProPulse.
Despite the intuitive aspects, there are also some non-trivial mathematical subtleties. The spectral intensity is calculated from a Fourier spectrum, but this can only be computed for a finite time interval, not for one moment of time. The frequency resolution is determined by the inverse width of that time interval. Therefore, there is a tradeoff between high frequency resolution and high temporal resolution, and the outcome depends on the choice e.g. of the temporal resolution. A further important detail is that before a fast Fourier transform (FFT) is applied, the data need to be multiplied by some appropriate "window function" in order to avoid artifacts.
The spectrogram shown in Figure 1 is somewhat reminiscent of a graph showing the instantaneous frequency as a function of time. However, a vertical slice of the spectrogram always has some finite width (in that case determined by the pulse duration), whereas the instantaneous frequency at any temporal position is a sharply defined value.
Figure 2: Spectrogram of a supercontinuum, shown with a logarithmic color scale
Figure 2 shows the spectrogram of an ultrashort pulse after passage through a photonic crystal fiber, in which supercontinuum generation occurred as a result of strong nonlinearities, with a strong influence of chromatic dispersion. In order to span a wider range of spectral intensities, a logarithmic color scale has been chosen. The pulse has split up into several pulses (→ soliton fission), as the temporal trace (black curve at the bottom) shows. The lower part of the diagram shows various bright spots representing soliton pulses, which carry a significant part of the overall energy. It is also apparent that there is some unconverted light from the wings of the initial pulse (horizontal line in the middle) and a weak background extending to high optical frequencies. This background is to some extent correlated with the above-mentioned solitons, because it is generated in a phase-matched four-wave mixing process. Both the lowest and the highest frequency components exhibit a larger time delay than the medium frequency components; this is due to the chromatic dispersion of the fiber.
See also: instantaneous frequency, pulse characterization, supercontinuum generation
