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.

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).