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Time–bandwidth Product

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

The time–bandwidth product of a light pulse is the product of its temporal duration and spectral width – in other words, the product of its width in the time and frequency domain. In ultrafast laser physics, it is common to specify the full width at half-maximum (FWHM) in both time and frequency domain. The minimum possible time–bandwidth product is obtained for bandwidth-limited pulses. For example, it is ≈ 0.315 for bandwidth-limited sech²-shaped pulses and ≈ 0.44 for Gaussian-shaped pulses. This means that for a given spectral width, there is a lower limit for the pulse duration. This limitation is essentially a property of the Fourier transform. Although it is certainly not a quantum effect, and also holds for a purely classical description of light, it is mathematically similar to the time–energy uncertainty principle in quantum mechanics.

The time–bandwidth product (also sometimes called duration–bandwidth product) is often used for indicating how close a pulse is to the transform limit, i.e., how close the pulse duration is to the limit which is set by its spectral width. This is an aspect of “pulse quality”; bandwidth-limited pulses have the minimum possible time–bandwidth product (somewhat below 0.5), whereas chirped pulses have larger values. Many mode-locked lasers can generate nearly bandwidth-limited pulses, particularly when they are based on soliton mode locking, although some are generating substantially chirped pulses.

Note that some pulses have a temporal and/or spectral pedestal which is much broader than its FWHM; that is essentially ignored when specifying the TBP based on the FWHM duration and spectral width, although a pedestal can also be an important aspect of (degraded) pulse quality. In order to better take that aspect into account, an alternative definition e.g. based on the second moment of power vs. time and power spectral density vs. frequency may be used, but that is not common in laser physics and technology.

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