= Solution
A detector too slow to resolve a femtosecond field can measure its integrated nonlinear response instead. Split the pulse, delay one replica, overlap both in a thin phase-matched crystal, and select the sum-frequency signal. For negligible depletion and sufficiently broad <phase matching>, the cross-generated field is proportional to $A(t)A(t-\tau)$; integrated signal gives the <second-harmonic intensity autocorrelation>
$$
C(\tau)\propto\int|A(t)A(t-\tau)|^2\,dt=\int I(t)I(t-\tau)\,dt.
$$
A noncollinear geometry can reject each replica's individual second harmonic. The optical response supplies the fast gate while the detector integrates slowly. Crystal thickness, <group-velocity mismatch>, spectral acceptance, beam overlap and delay calibration limit resolution.
For a <Gaussian pulse> specified by its <intensity> profile, the autocorrelation's <full width at half maximum> is $\sqrt2$ times the <intensity> <full width at half maximum>. For a sech-squared <intensity> profile the ratio is about $1.543$. These are model-dependent conversion factors. An <intensity autocorrelation> is symmetric in delay and loses phase information; different pulses can share it. An <interferometric autocorrelation> retains rapid carrier fringes but still does not generally reconstruct an arbitrary complex field uniquely.
<frequency-resolved optical gating> also resolves the gated signal's spectrum, giving a two-dimensional delay-frequency record. For <second-harmonic frequency-resolved optical gating>,
$$
I_{\rm FROG}(\omega,\tau)\propto
\left|\int A(t)A(t-\tau)e^{-i\omega t}\,dt\right|^2,
$$
where $\omega$ is measured relative to the signal's carrier. An iterative retrieval seeks an envelope whose predicted record matches the measurement. Alternatively, <polarization-gating frequency-resolved optical gating> uses the <optical Kerr effect> to gate the probe by a delayed replica's <intensity>, with signal envelope proportional to $A(t)|A(t-\tau)|^2$. The original <FROG (optics)> work introduced nonlinear optical gating plus spectral resolution to characterize arbitrary femtosecond pulses.<Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating by Daniel J. Kane and Rick Trebino>[1993, abstract: nonlinear gating and spectrogram construction]{ref}
Absolute <wave phase> and absolute time origin remain undetermined by these <intensity> records. The SHG record also cannot distinguish $A(t)$ from $A^*(-t)$, so its time-direction ambiguity remains. Retrieval agreement, an independently measured spectrum, adequate signal dynamic range and gate-response calibration are practical consistency checks. A pulse train that changes between shots can yield an averaged record that no single pulse faithfully represents. Thus nonlinear gating solves the detector-speed problem but does not remove <inverse problem> ambiguities or experimental limitations.
\Image[/past-exam-of-the-nonlinear-optics-and-quantum-states-of-light-course-of-the-university-of-cambridge/2014/paper-1-pulse-measurement.png]
{title=Original nonlinear pulse-gating layout and a calculated Gaussian chirp example distinguishing autocorrelation from spectral gating}
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