Autocorrelation (pulse measurement)

Pulses shorter than a few picoseconds are faster than any photodiode and oscilloscope, so they are measured against themselves. In an intensity autocorrelator the pulse is split, one copy is delayed by a variable time $\tau$, and the two are recombined in a crystal that generates the second harmonic. The second-harmonic signal is strongest when the copies overlap and falls to zero as the delay separates them, tracing the intensity autocorrelation $A(\tau )=\int I(t)I(t-\tau )dt$ as the delay is scanned. In the background-free, noncollinear geometry only light produced by both beams together reaches the detector.

The autocorrelation is always symmetric and wider than the pulse, and converting its width to a pulse width requires assuming a shape. For a Gaussian pulse the autocorrelation's full width at half maximum is $\sqrt{2}$ = 1.414 times the pulse's; for a sech² pulse, the shape of a soliton and of many mode-locked lasers, it is 1.543 times. A 154.3 fs autocorrelation width therefore means a 100 fs pulse if the shape is sech², and 109 fs if it is Gaussian. The shape cannot be determined from the autocorrelation itself, which is the method's main limitation: different pulses, including asymmetric ones, can give the same trace, and the spectral phase is not measured at all.

The time-bandwidth product supplies a check. For a pulse without chirp, the product of the pulse width and the spectral width (both full width at half maximum, the spectrum in frequency) takes a minimum set by the shape: 0.441 for a Gaussian and 0.315 for sech². A 100 fs sech² pulse at 1560 nm must have at least 3.15 THz of bandwidth, 25.6 nm. Measuring the spectrum alongside the autocorrelation shows whether the pulse is at this limit; a product well above it indicates chirp, usually from dispersion in the optics after the laser, which a prism or grating compressor can remove.

When the pulse shape or phase matters, autocorrelation gives way to methods that recover the complete field. Frequency-resolved optical gating (FROG) records the second-harmonic spectrum at each delay and retrieves amplitude and phase from the resulting spectrogram; spectral phase interferometry (SPIDER) measures the spectral phase directly. An interferometric autocorrelation, which resolves the optical fringes in the same setup, contains some phase information and reveals chirp qualitatively.

The autocorrelator procedure, its calibration and a worked example are in How to Measure Ultrashort Pulse Duration.

References: J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006); R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer, 2000).