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Frequency-resolved optical gating characterizes an ultrashort pulse using a nonlinear gate and a measured two-dimensional spectrogram. In second-harmonic frequency-resolved optical gating, split the pulse into two replicas, vary their relative delay, overlap them noncollinearly in a thin second-harmonic crystal, and spectrally resolve the cross-generated beam. Its propagation direction separates it from the individual beams' self-generated harmonics. For complex pulse envelope $E(t)$, the ideal trace iswhere $\Omega$ denotes the output frequency offset in the envelope convention. An iterative phase retrieval algorithm seeks a pulse consistent with both this measured Fourier magnitude and the nonlinear product relation. The delay axis provides timing structure, while the resolved spectrum adds information missing from a plain intensity autocorrelation.^[ref]

The measured trace does not fix absolute optical phase or absolute time origin. SHG FROG also has a conjugate time-reversal ambiguity: $E(t)$ and $E^{\ast }(-t)$ produce the same trace. Extra information, such as a known dispersion change and a second measurement, can resolve that direction-of-time ambiguity. Finite phase-matching bandwidth, group-velocity mismatch, spatial chirp, spectrometer calibration, delay error, noise and instability over the scan must be controlled. A thin gate helps avoid temporal blurring, but trades against signal. FROG therefore reconstructs a pulse under an explicit instrument model rather than reading optical phase directly from a single intensity image.

Phase retrieval reconstructs a signal from measurements that discard its complex wave phase. A basic diffraction model measures $I(\xi )=|\hat{f}(\xi ){|}^2$, rather than the complex Fourier transform itself. The lost phase cannot generally be recovered simply by taking a square root of the intensity.

There are immediate ambiguities. Multiplication by a constant $e^{i\theta }$ leaves every magnitude unchanged. A translation multiplies the transform by a unit-modulus phase factor, and conjugate reflection $f(x)\mapsto \overline{f(-x)}$ conjugates its transform. These transformations therefore preserve $I$. They need not exhaust the ambiguities of one-dimensional data.^[ref]

By the convolution theorem, magnitude-squared data determine the autocorrelation of a square-integrable function under suitable inversion hypotheses, rather than automatically determining the function itself. Known support, additional masked measurements or other structural assumptions can reduce the ambiguity; their sufficiency must be checked for the actual measurement model. Frequency-resolved optical gating uses a family of delayed nonlinear spectra, giving more information than a single unmodified Fourier magnitude.

Robert Beinert coauthored an account of Fourier phase retrieval with Tamir Bendory and Yonina C. Eldar. Their analysis separates uniqueness questions from reconstruction algorithms and records the symmetries invisible to Fourier-magnitude measurements. This clarifies why an algorithm producing a signal is insufficient unless the measurement design also makes the signal identifiable.^[ref]

Tamir Bendory coauthored a study of Fourier phase retrieval with Robert Beinert and Yonina C. Eldar. It examines reconstruction from Fourier magnitudes and distinguishes unavoidable ambiguities from genuine uniqueness failures. Global phase, translation and conjugate reflection already preserve basic magnitude data, so meaningful recovery claims must account for these symmetries.^[ref]

Yonina C. Eldar coauthored a study of Fourier phase retrieval with Tamir Bendory and Robert Beinert. It compares basic magnitude measurements with richer designs such as masks or short-time measurements. Extra measurement structure can remove ambiguities that persist in the simplest model, linking experimental design with mathematical identifiability.^[ref]