# Gaussian Pulses

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: pulses with a temporal intensity profile which has a Gaussian shape

More general term: light pulses

Ultrashort pulses from mode-locked lasers, for example, often have a temporal shape (i.e., shape of the curve showing optical power versus time) which can be approximately described with a Gaussian function:

$$P(t) = {P_{\rm{p}}}\exp \left[ { - 4\ln 2{{\left( {\frac{t}{{{\tau _{\rm{p}}}}}} \right)}^2}} \right] = {P_{\rm{p}}}\;{2^{ - {{\left( {2t/{\tau _{\rm{p}}}} \right)}^2}}}$$where <$\tau$> is the full width at half-maximum (FWHM) pulse duration.

In many cases, Gaussian pulses have no chirp, i.e., are transform-limited. In that case, the spectral width (optical bandwidth, taken as full width at half maximum) is

$$\Delta \nu \approx \frac{{0.44}}{{{\tau _{\rm{p}}}}}$$which means that the time–bandwidth product is ≈ 0.44. For a conversion of the optical bandwidth in terms of wavelength, see the article on bandwidth.

Compared with a sech^{2}-shaped pulse, a Gaussian pulse with the same width at half-maximum has somewhat weaker wings:

The peak power of a Gaussian pulse is ≈ 0.94 times the pulse energy divided by the FWHM pulse duration.

The Gaussian pulse shape is typical for pulses from actively mode-locked lasers; it results e.g. from the Haus master equation in simple cases. However, it is also found in various other situations. Generally, the obtained pulse shape can depend on many factors, such as chromatic dispersion and optical nonlinearities in the laser resonator.

Gaussian pulses should not be confused with Gaussian beams; the latter have a Gaussian *spatial* intensity profile, while Gaussian pulses have a Gaussian *temporal* profile. Of course, one can also have both at the same time.

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## Questions and Comments from Users

2022-12-12

Possibly it is a typo in the second part of the equation, where it says 2^(-2 t / tau)^{2} ? The Gaussian pulse should essentially be an exp-Function, right?

The author's answer:

It is an exponential function – here just with base 2 instead of the Euler number.

2023-11-28

How do you obtain the value 0.44 for the time bandwidth product?

The author's answer:

Just take a Fourier transform of a Gaussian profile, gets its FWHM and multiple it with that of the original profile.65

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2021-02-08

May I use a <$P(t)$> formula for describing a Gaussian beam?

The author's answer:

Sure; only, the pulse may not be fully characterized with that e.g. in terms of the time-dependent optical phase and all the spatial aspects.