Pulse duration

Pulse duration is the time over which a laser pulse delivers its energy, conventionally the full width at half maximum (FWHM) of the intensity profile $I(t)$. A Q-switched Nd:YAG laser produces pulses of roughly 5–20 ns, a gain-switched laser diode 10–100 ps, a mode-locked fiber laser about 100 fs to a few picoseconds, and a Kerr-lens mode-locked Ti:sapphire oscillator 10–100 fs. At 800 nm one optical cycle lasts 2.67 fs, so the shortest pulses from such oscillators, near 5 fs, contain fewer than two cycles.

Definitions and pulse shapes

The FWHM is the usual figure because it can be read directly from a trace, but its relation to the underlying shape parameter depends on the shape. Two shapes are standard. A sech² pulse, $I(t) = I_0\,\mathrm{sech}^2(t/T_0)$, is the natural output of soliton-like mode-locked lasers and has $\tau_\text{FWHM} = 1.763\,T_0$. A Gaussian pulse, $I(t) = I_0\exp(-t^2/T_0^2)$, has $\tau_\text{FWHM} = 1.665\,T_0$. Other conventions appear in the literature: the 1/e² full width, the rms width (second moment), and for nanosecond pulses sometimes the 10–90 % rise time. A datasheet value should be read with its definition, because the rms and FWHM widths of the same pulse can differ considerably when the pulse has wings or satellites.

Measurement

Pulses longer than a few tens of picoseconds can be recorded directly with a fast photodiode and a sampling oscilloscope. A 20 GHz detector has a rise time of about 0.35/(20 GHz) = 17.5 ps, and with an oscilloscope of equal rise time the combined response is about 25 ps, so this route is reliable for pulses of roughly 100 ps and longer; streak cameras reach about a picosecond, and the fastest reach below 1 ps.

Shorter pulses are measured against themselves. An intensity autocorrelator records a trace whose FWHM is wider than the pulse by a factor that depends on the assumed shape. The pulse width is obtained by multiplying the autocorrelation width by the deconvolution factor:

$$\tau_\text{p} = 0.648\,\tau_\text{AC}\quad(\mathrm{sech}^2)$$ $$\tau_\text{p} = 0.707\,\tau_\text{AC}\quad(\text{Gaussian})$$

These are the reciprocals of 1.543 and $\sqrt{2}$. A measured 150 fs autocorrelation therefore corresponds to a 97.2 fs sech² pulse or a 106 fs Gaussian pulse; the autocorrelation alone cannot decide between them. Frequency-resolved optical gating and SPIDER remove this assumption by retrieving the full amplitude and phase. The bench procedure for all three methods is described in How to Measure Ultrashort Pulse Duration.

A second check is the time-bandwidth product: for a transform-limited pulse, $\Delta\nu\,\tau_\text{FWHM}$ is 0.315 for sech² and 0.441 for a Gaussian. A measured product well above these values indicates chirp.

Dispersion and the pulse at the sample

A femtosecond pulse lengthens as it passes through glass, because group velocity dispersion delays its spectral components by different amounts. For a Gaussian pulse with initial width $\tau_0$ and accumulated group-delay dispersion $\phi''$,

$$\tau = \tau_0\sqrt{1 + \left(\frac{4\ln 2\;\phi''}{\tau_0^{2}}\right)^{2}}.$$

Fused silica has about 36 fs²/mm at 800 nm, so a 10 mm window adds about 360 fs². A 30 fs pulse leaves it at about 45 fs, while a 100 fs pulse grows by only 0.5 %. The broadening scales as $1/\tau_0$ once it dominates, which is why sub-50 fs pulses need dispersion compensation between the laser and the experiment and why the duration should be measured where the pulse is used.

Where it matters

Together with the pulse energy, the duration sets the peak power, and hence the strength of any nonlinear effect. It also sets the time resolution of pump-probe spectroscopy and, in machining, how far heat diffuses during the pulse. The pulsed laser calculator converts duration, energy and repetition rate into peak power and duty cycle.

Pitfalls

Autocorrelators assume the pulse is clean; a pulse with a long low-level pedestal or a satellite can give a trace whose FWHM looks short while much of the energy sits outside it. A coherence spike on top of a broad autocorrelation indicates a noisy or partially mode-locked pulse train, and the narrow spike measures the coherence time rather than the pulse. Detector bandwidth limits apply to direct measurement: a 50 ps pulse viewed with a 25 ps system response reads about 56 ps if the two add in quadrature, as is approximately true for Gaussian responses.

Common questions

What is the difference between pulse duration and pulse width?

The terms are used interchangeably. Both usually mean the intensity FWHM; "pulse width" occasionally refers to the electrical drive pulse of a diode laser, which can differ from the optical output.

How do you convert an autocorrelation width to pulse duration?

Multiply by 0.648 for an assumed sech² shape or 0.707 for a Gaussian. The choice should match the laser: passively mode-locked oscillators are usually closer to sech², amplified and spectrally shaped pulses closer to Gaussian.

What is the shortest laser pulse?

From laser oscillators, pulses of about 5 fs at 800 nm have been reported from Ti:sapphire. Much shorter pulses, in the attosecond range, are produced by high-harmonic generation driven by amplified ultrafast lasers, at extreme-ultraviolet wavelengths.

References: J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006); A. M. Weiner, Ultrafast Optics (Wiley, 2009); R. Trebino, Frequency-Resolved Optical Gating (Kluwer, 2000); A. E. Siegman, Lasers (University Science Books, 1986).