Glossary entry

Time-bandwidth product

The product of a pulse's duration and its spectral width, both full width at half maximum. For a pulse without chirp it takes a minimum set by the pulse shape, 0.441 for a Gaussian and 0.315 for sech²; a larger measured value means the pulse is chirped and could be compressed.

Lasers & gain Optics fundamentals Lab practiceUpdated September 2026

A pulse of light cannot be both short and spectrally narrow, because its spectrum is the Fourier transform of its field. The time-bandwidth product $\Delta\nu\,\Delta\tau$ , with the duration $\Delta\tau$ and the spectral width $\Delta\nu$ (in frequency) both taken as full widths at half maximum of the intensity, measures how close a pulse comes to the limit. A pulse whose phase is flat across its spectrum is called transform-limited and has the smallest product its shape allows:

Pulse shape	Minimum $\Delta\nu\,\Delta\tau$
Gaussian	0.441 ( $2\ln 2/\pi$ )
sech²	0.315

The sech² shape is the natural output of many passively mode-locked lasers, and the Gaussian is the usual assumption otherwise. The numbers differ only because of the shape and the choice of full width at half maximum; with rms widths the minimum is 1/(4π) for any shape, reached only by a Gaussian.

Typical values

A transform-limited 100 fs Gaussian pulse at 800 nm needs 4.41 THz of bandwidth, 9.42 nm in wavelength; a 1 ps sech² pulse at 1550 nm needs 315 GHz, or 2.52 nm. Converting between the two scales uses $\Delta\lambda = \lambda^2\Delta\nu/c$ , valid while the bandwidth is a small fraction of the carrier frequency.

Chirp

Passing a pulse through glass, fiber or any other dispersive material stretches it without changing its spectrum, because group velocity dispersion delays some frequencies relative to others and leaves the pulse chirped. The product then rises above the limit. A pulse with 10 nm of spectrum at 800 nm, 4.68 THz, could be as short as 94 fs if Gaussian; measured at 150 fs, its product is 0.70, and the excess says a prism or grating compressor, or chirped mirrors, can shorten it back toward 94 fs.

Measurement

The duration comes from an autocorrelation, divided by the deconvolution factor for the assumed shape (1.414 for a Gaussian, 1.543 for sech²), and the spectral width from an optical spectrum analyzer, converted to frequency. Because the autocorrelation needs an assumed shape and discards the phase, the product is a check on consistency rather than a full description; frequency-resolved optical gating (FROG) and spectral phase interferometry (SPIDER) measure the pulse's amplitude and phase directly.

References: A. M. Weiner, Ultrafast Optics (Wiley, 2009), Ch. 3; J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006), Ch. 1; R. Trebino, Frequency-Resolved Optical Gating (Kluwer, 2000).