How to Measure Ultrashort Pulse Duration: Autocorrelation, FROG and SPIDER

Scope

This article covers the measurement of optical pulses shorter than a few picoseconds, which are too fast for any photodiode and oscilloscope. It gives the procedure for the intensity autocorrelator, the most common instrument, with its calibration and the assumptions behind the number it reports; the check against the optical spectrum; and the methods that recover the full pulse field when the duration alone is not enough. The principles are summarized in the autocorrelation and time-bandwidth product entries.

Choosing a method

The intensity autocorrelator

The beam is split in two, one copy passes a variable delay, and the two are recombined in a nonlinear crystal that generates the second harmonic. In the noncollinear (background-free) geometry the two beams cross at a small angle, and only the second harmonic produced by one photon from each beam emerges along the bisector, so its power as a function of delay is the intensity autocorrelation, with no background. The collinear geometry gives the same autocorrelation on a background of one third of the peak.

Procedure

1. Align the two arms so the beams overlap in the crystal, and set the crystal's angle for phase matching at the laser wavelength.
2. Find zero delay by scanning until the second-harmonic signal appears, then center the scan on it.
3. Calibrate the delay axis. For a retroreflector on a translation stage, the delay changes by twice the stage displacement divided by $c$: 1 μm of travel is 6.67 fs. Check it by moving the stage a known distance and observing the shift of the trace, or, in a scanning autocorrelator, by inserting a plate of known thickness and group index into one arm.
4. Record the trace and fit it with the autocorrelation of an assumed shape; read the full width at half maximum.
5. Divide by the deconvolution factor for that shape (below) to obtain the pulse duration.
6. Record the spectrum at the same time with a spectrometer and compute the time-bandwidth product.

The deconvolution factor

The autocorrelation is symmetric and wider than the pulse, and the ratio of the two widths depends on the pulse shape:

The shape cannot be determined from the intensity autocorrelation alone: very different pulses, including asymmetric ones and pulses with satellites or a pedestal, can give similar traces. The reported duration is therefore a number conditional on the assumed shape, which should be stated with it. For mode-locked lasers operating in the soliton regime, sech² is the usual choice.

Checking against the spectrum

The spectrum sets the shortest duration the pulse could have. With the pulse width $\tau$ and the spectral width $\Delta\nu$, both at full width at half maximum,

$$\tau\,\Delta\nu \;\geq\; K,$$

with $K$ from the table above. Converting a measured spectral width in wavelength, $\Delta\nu = c\,\Delta\lambda/\lambda^2$.

Worked example. An 800 nm laser gives an autocorrelation of 150 fs full width at half maximum. Assuming sech², the pulse is 150/1.543 = 97.2 fs. A transform-limited sech² pulse of that duration needs a spectral width of 0.315/97.2 fs = 3.24 THz, or 6.9 nm at 800 nm. If the measured spectrum is 6.9 nm wide, the pulse is at or near its transform limit. If it is 15 nm wide, the product is about 2.2 times the minimum, and the pulse is either chirped, and could be shortened by a compressor, or not sech²-shaped, and only a phase-resolving method can say which.

Interferometric autocorrelation

In a collinear autocorrelator with the delay scanned finely enough to resolve the optical fringes, the second-harmonic signal oscillates at the optical period. For an unchirped pulse the envelope of the fringes reaches eight times the background at zero delay; a chirped pulse fills in the fringes in the wings while keeping the 8:1 ratio at the center. The trace shows chirp qualitatively, and is useful for tuning a compressor by eye, but it does not give the phase uniquely.

FROG, SPIDER and d-scan

FROG (frequency-resolved optical gating) replaces the detector of an autocorrelator with a spectrometer, recording the second-harmonic spectrum at each delay. The resulting spectrogram is inverted by an iterative algorithm to give the pulse's intensity and phase. The retrieval checks itself: the spectrogram computed from the retrieved field should match the measured one, and its marginals should agree with the independently measured spectrum. Second-harmonic FROG cannot tell the direction of time, so the sign of a chirp needs one extra measurement, such as adding known dispersion.

SPIDER (spectral phase interferometry for direct electric-field reconstruction) interferes two copies of the pulse that are shifted in frequency against each other and reads the spectral phase from the fringes directly, without iteration. It suits fast, repeated measurements but needs careful calibration of the spectral shear and the delay between the copies.

Dispersion scan records the second-harmonic spectrum while a known amount of dispersion is added in steps, typically with glass wedges in a compressor, and retrieves the phase from the resulting map. It is common with few-cycle pulses, whose compressors already contain the wedges.

Common failure modes

Crystal too thick. The phase-matching bandwidth of the crystal must cover the pulse's spectrum, or the short-wavelength and long-wavelength parts are converted unequally and the trace is distorted. Pulses of tens of femtoseconds need crystals tens of micrometres thick, and the shortest pulses need thinner still.

Dispersion in the instrument. Beamsplitters, lenses and windows in the autocorrelator add dispersion and lengthen the pulse before it is measured; reflective optics and thin beamsplitters limit this. The pulse is also lengthened by every optic between the laser and the instrument, so it must be measured where it is used.

Uncalibrated delay. An error in the delay axis scales the result directly; calibrate it rather than relying on a nominal stage speed.

Averaging over unstable pulses. A scanning instrument averages over many pulses. A laser that produces a train of varying pulses, or a noise burst, gives a trace with a narrow coherence spike on a broad pedestal, and the spike's width says nothing about the pulse duration.

Shape not stated. A duration given without the assumed shape cannot be compared with another.

References: J.-C. Diels and 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); C. Iaconis and I. A. Walmsley, "Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses," Optics Letters 23, 792 (1998); M. Miranda et al., "Simultaneous compression and characterization of ultrashort laser pulses using chirped mirrors and glass wedges," Optics Express 20, 688 (2012).