FROG (frequency-resolved optical gating)

Frequency-resolved optical gating (FROG) measures the intensity and the phase of a femtosecond or picosecond pulse as functions of time. It is built like an intensity autocorrelator: the pulse is split, one copy is delayed by $\tau$, and the two are mixed in a nonlinear crystal. The difference is at the detector. A FROG sends the nonlinear signal into a spectrometer and records its spectrum at each delay, so the measurement is a two-dimensional spectrogram of delay and frequency. An iterative algorithm then finds the pulse field that reproduces this spectrogram. Typical traces are sampled on grids of 64 × 64 to 256 × 256 points, and a single measurement of a Ti:sapphire oscillator pulse near 800 nm takes seconds to minutes depending on the averaging.

Principle and the SHG-FROG trace

In the most common version, second-harmonic FROG, the crystal produces second-harmonic light from the product of the two copies. The signal field is $E(t)\,E(t-\tau)$, and the measured trace is

$$I(\omega,\tau) = \left|\int E(t)\,E(t-\tau)\,e^{-i\omega t}\,dt\right|^2 .$$

Integrating the trace over frequency gives back the intensity autocorrelation, so a FROG contains all the information of an autocorrelator and more. The retrieval algorithm alternates between making a trial field consistent with the form of the signal and making it consistent with the measured intensities, until the two agree. The residual difference between measured and retrieved traces, the FROG error, is reported with every result.

The sampling of the trace is tied to the Fourier transform. For an $N \times N$ grid with delay step $\delta\tau$, the frequency spacing is $1/(N\,\delta\tau)$. With $N = 128$ and $\delta\tau = 5$ fs, the spacing is 1.5625 THz and the grid spans 200 THz, enough for a pulse of a few tens of femtoseconds. The delay stage has to be calibrated with the same care as in an autocorrelator: moving a retroreflector by 1 µm changes the delay by 6.67 fs.

What it recovers that autocorrelation cannot

An autocorrelation gives a width, from which a pulse duration is inferred only by assuming a shape. FROG measures the shape, and it measures the spectral phase, which carries the chirp. This matters in practice. A 30 fs transform-limited Gaussian pulse that passes through optics adding 500 fs² of group-delay dispersion broadens to 55 fs, and an autocorrelation of the chirped pulse reports the longer width without saying why. A FROG shows the quadratic spectral phase directly, its size, and whether higher orders such as third-order dispersion are present, which tells the user how to adjust a pulse compressor. Comparing the retrieved duration with the time-bandwidth product limit then serves as an independent check: a 100 fs sech² pulse at 800 nm needs at least 3.15 THz of bandwidth, 6.7 nm.

FROG also has built-in consistency tests. The frequency marginal of the trace must match the independently measured spectrum convolved with itself, and the delay marginal must match the autocorrelation; disagreement usually indicates a phase-matching bandwidth too narrow for the pulse, a misaligned spectrometer, or a poorly calibrated delay.

Ambiguities

Every pulse-measurement method leaves some features of the field undetermined. For SHG-FROG, the trace is symmetric in delay, so a pulse $E(t)$ and its time-reversed conjugate $E^*(-t)$ give identical traces. The sign of a linear chirp is therefore not determined: an up-chirped and a down-chirped pulse look the same, and a pulse with a trailing satellite cannot be distinguished from one with a leading satellite. The usual remedy is a second measurement after adding a known amount of dispersion, such as a glass plate; the width then grows for one sign and shrinks for the other. Polarization-gated and self-diffraction FROG, which use third-order nonlinearities, give asymmetric traces without this ambiguity, at the cost of much lower sensitivity. All versions leave the absolute phase and the arrival time unmeasured, and the relative phase of two pulses that do not overlap in time is poorly constrained.

SPIDER and related methods

Spectral phase interferometry for direct electric-field reconstruction (SPIDER) takes a different route. Two replicas of the pulse, separated by a delay, are upconverted with a strongly chirped copy so that they emerge shifted in frequency by a small shear $\Omega$. Their interference spectrum contains the difference $\varphi(\omega+\Omega)-\varphi(\omega)$, which is extracted by Fourier filtering without iteration and integrated to give the spectral phase. SPIDER is fast enough for real-time display and suits few-cycle pulses, but its accuracy depends on calibrating the shear and the replica delay. The procedures for autocorrelation, FROG, SPIDER and dispersion scan are compared in How to Measure Ultrashort Pulse Duration.

Common questions

What is a good FROG error?

The value depends on grid size and noise, so there is no universal threshold. A useful practice is to compare the error with that obtained from simulated traces with the same noise level, and to check the marginals; a low error with inconsistent marginals is not a trustworthy result.

Can FROG measure picosecond pulses?

Yes, provided the spectrometer resolves the narrow spectrum. A 1 ps pulse at 800 nm has a transform-limited bandwidth of 0.67 nm if sech² and 0.94 nm if Gaussian, which needs a high-resolution spectrometer and a long delay scan.

Why use SHG rather than a third-order process?

Second-harmonic generation is far more efficient, so SHG-FROG works with nanojoule oscillator pulses where third-order versions need microjoules.

References: D. J. Kane, R. Trebino, IEEE J. Quantum Electron. 29, 571 (1993); R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer, 2000); C. Iaconis, I. A. Walmsley, Opt. Lett. 23, 792 (1998); J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006).