FROG trace marginals as consistency checks#

The frequency marginal of a FROG-type trace is its integral over delay,

\[M(\omega) \;=\; \int T(\omega,\tau)\,\mathrm{d}\tau ,\]

a one-dimensional function of optical frequency. It is computed from a measured trace as M_meas = trace.sum(axis=1) (delay is axis 1 in croak’s (Nω, Nτ) convention). From an independently measured fundamental spectrum \(S(\omega)=|\tilde E(\omega)|^2\) we can predict \(M(\omega)\) — or rigorously bound it — and compare, before running a retrieval. A mismatch is a cheap quality-control (QC) signal: it flags calibration errors, spectral-response problems, scatter and detector nonlinearity that a retrieval would otherwise silently absorb.

These checks are implemented in croak.marginal_checks and surfaced in the GUI’s Marginal check stage (see the GUI guide); the how-to recipe is Marginal consistency checks.

Important

The identities below assume a thin, local interaction with coincident replicas, complete delay support, a matching fundamental-spectrum reference plane and a calibrated spectral response. Dispersion, split gates, chromatic spatial fields, finite collection and cropping can invalidate these ideal-model predictions. A mismatch is then evidence to investigate, not proof of a detector fault. The transform-limited PG width is a soft reference, not a universal upper bound.

The general result#

For the ideal local kernels considered here, the signal factorises into an undelayed part \(P\) and a delayed gate \(g\),

\[E_\mathrm{sig}(t,\tau) = P(t)\,g(t-\tau), \qquad T(\omega,\tau)=\bigl|\widehat{E_\mathrm{sig}}(\omega,\tau)\bigr|^2 .\]

Integrating over delay collapses the delay-dependent phase to a delta function, leaving a convolution of spectral intensities:

\[M(\omega) \;\propto\; \bigl(|\tilde P|^2 * |\tilde g|^2\bigr)(\omega),\]

where \(\tilde P,\tilde g\) are the Fourier transforms of \(P,g\). Everything below follows from this one identity.

Two consequences are exact for any nonnegative factors (no assumptions on the pulse), because intensity-weighted moments add under convolution:

Centroid: \(\bar\omega[M] = \bar\omega[\,|\tilde P|^2\,] + \bar\omega[\,|\tilde g|^2\,]\)
Variance: \(\sigma^2[M] = \sigma^2[\,|\tilde P|^2\,] + \sigma^2[\,|\tilde g|^2\,]\)

with \(\bar\omega[f]=\frac{\int\omega f}{\int f}\) and \(\sigma^2[f]=\frac{\int(\omega-\bar\omega)^2 f}{\int f}\) (croak.marginal_checks.centroid() and croak.marginal_checks.rms_width()).

Phase-independence criterion. A factor is phase-independent iff it is built from \(E\) or \(E^\ast\) alone:

factor	expression	phase-dependent?	centroid	even?
\(\lvert\tilde E\rvert^2\)	\(S(\omega)\)	no	\(\bar\omega_S\)	no
\(\lvert\widehat{E^\ast}\rvert^2\)	\(S(-\omega)\)	no	\(-\bar\omega_S\)	no
\(\lvert\widehat{\lvert E\rvert^2}\rvert^2\)	intensity spectrum	yes	\(0\) (exact)	yes
\(\lvert\widehat{E^2}\rvert^2\)	SHG spectrum	yes	phase-dependent	no

The marginal is fully phase-independent only when both factors come from the top two rows. Among the common geometries that is SHG only.

process	\(P\)	\(g\)	marginal \(M\propto\)	phase-indep.?	centred at
SHG	\(E\)	\(E\)	\(S * S\)	yes	\(2\bar\omega_S\)
PG	\(E\)	\(\lvert E\rvert^2\)	\(S * \lvert\widehat{\lvert E\rvert^2}\rvert^2\)	no	\(\bar\omega_S\)
SD	\(E^2\)	\(E^\ast\)	\(\lvert\widehat{E^2}\rvert^2 * S(-\omega)\)	no	\(\approx\bar\omega_S\)
TG	\(E\)	\(\lvert E\rvert^2\)	same as PG	no	\(\bar\omega_S\)

TG-FROG ≡ PG

The transient-grating geometry has the \(|E|^2\) gate, so its marginal is identical to PG regardless of which beam carries the delay (convolution commutes). croak.marginal_checks.predicted_marginal() accepts interaction="tg" and treats it as "pg".

SHG — the clean case#

\[M_\mathrm{SHG}(\omega) = (S * S)(\omega).\]

Both factors are the measured spectrum, so the marginal is the autoconvolution of the fundamental spectrum (croak.marginal_checks.spectral_autoconvolution()) — completely phase-independent and computable from \(S\) alone. This is the phase-independent check under the thin-local-model and complete-support assumptions above. Disagreement can indicate calibration, spectral response, noise, incomplete support or a model that is inappropriate for the instrument. The three identities are:

Full predicted marginal: \(S*S\), compared shape-for-shape.
Centroid: \(\bar\omega[M] = 2\,\bar\omega_S\).
RMS width: \(\sigma[M] = \sqrt{2}\,\sigma_S\).

Place the prediction on the second-harmonic frequency axis and compare to the measured marginal after consistent wavelength calibration.

PG (and TG) — three checks#

\[M_\mathrm{PG}(\omega) = \bigl(S * \lvert\widehat{\lvert E\rvert^2}\rvert^2\bigr)(\omega).\]

The gate factor \(\lvert\widehat{\lvert E\rvert^2}\rvert^2\) is the modulus-squared FT of the temporal intensity \(|E(t)|^2\). It carries the spectral phase through \(e^{i[\phi(\omega'+\omega)-\phi(\omega')]}\), so you cannot get it from \(S\) alone. Three things are nonetheless usable.

Transform-limited (TL) predicted marginal#

Assume flat phase: \(\tilde E_\mathrm{TL}=\sqrt{S}\). Then \(|E_\mathrm{TL}(t)|^2\) and the gate factor are computable, so the whole marginal is computable from \(S\). Compare its shape to the measured marginal. A mismatch conflates “bad trace” with “pulse is not TL”, so this is a soft check, most useful alongside the centroid and width anchors below.

Centroid — exact, phase-independent#

The gate factor is the modulus-squared FT of a real function, hence even in \(\omega\), hence has centroid exactly \(0\) for any pulse. Therefore

\[\bar\omega[M_\mathrm{PG}] = \bar\omega_S \quad\text{(exact, any phase).}\]

PG/TG sit at the fundamental (not doubled). This is a genuine hard check on the frequency-axis calibration of the trace, independent of the TL assumption.

Width — a two-sided bracket#

Variances add: \(\sigma^2[M_\mathrm{PG}] = \sigma_S^2 + \sigma^2[\,|\widehat{|E|^2}|^2\,]\).

Lower bound (rigorous): \(\sigma[M_\mathrm{PG}] > \sigma_S\). The gate term is strictly positive (it vanishes only for a CW field). A measured marginal narrower than the bare spectrum is unphysical — typically spectral clipping or detector roll-off.
Upper reference (soft): the TL pulse has the most peaked \(|E(t)|^2\), hence the broadest gate factor and the broadest marginal — for smooth phase. Pathological phase can beat \(|E(t)|^2\) faster and exceed it, so this is typical-case, not a theorem. A measured marginal broader than the TL prediction flags additive broadband artefacts (scatter, background, stray light, detector nonlinearity).

So the expected width sits in the bracket \(\bigl[\sigma_S,\ \sigma_\mathrm{TL}\bigr]\) (width_bracket), the interior consistent with some unknown amount of chirp.

SD — partly similar, with weaker anchors#

\[M_\mathrm{SD}(\omega) = \bigl(\lvert\widehat{E^2}\rvert^2 * S(-\omega)\bigr)(\omega).\]

The signal is \(E_\mathrm{sig}=E(t)^2\,E^\ast(t-\tau)\), so the undelayed factor is \(E^2\) and its spectral intensity \(\lvert\widehat{E^2}\rvert^2\) is the SHG spectrum — phase-dependent. The delayed factor contributes only the reflected fundamental spectrum \(S(-\omega)\), which is phase-independent. SD does not give a phase-independent marginal: the phase enters through the SHG-spectrum factor, exactly as it would in an SHG measurement. Only SHG-FROG has the fully clean marginal.

How the three PG checks carry over:

TL predicted marginal — fully analogous. Set \(\tilde E_\mathrm{TL}=\sqrt S\), form \(E_\mathrm{TL}^2\), take \(|\widehat{E_\mathrm{TL}^2}|^2\), convolve with \(S(-\omega)\). Same soft-check status as PG.
Centroid — weaker. \(\bar\omega[M_\mathrm{SD}] = \bar\omega[\,|\widehat{E^2}|^2\,] - \bar\omega_S\). The SHG-spectrum centroid is not generally \(2\bar\omega_S\); it coincides at TL and, more generally, for a temporally symmetric intensity. For asymmetric chirped pulses the SD marginal centroid drifts off \(\bar\omega_S\), so treat “centred at the fundamental” as a TL/symmetric anchor, not a hard check (centroid_is_hard is False).
Width — messier. Only the lower bound \(\sigma[M_\mathrm{SD}] > \sigma_S\) is rigorous; the SHG-spectrum width depends on phase non-monotonically (for a Gaussian spectrum with pure GDD it is essentially invariant), so the clean PG-style upper bracket is lost — the prediction reports width_bracket = (σ_S, None).
Bonus check (no TL assumption): if an independently measured SHG spectrum is available, plug it in for \(|\widehat{E^2}|^2\): \(M_\mathrm{SD} = (\text{measured SHG spec}) * S(-\omega)\) is then a fully phase-independent prediction, recovering an SHG-grade check for the SD trace.

What to compute, by process#

check	SHG	PG / TG	SD
predicted marginal from \(S\) alone	exact	TL only	TL only (or via measured SHG spec)
centroid	\(2\bar\omega_S\) (hard)	\(\bar\omega_S\) (hard)	\(\approx\bar\omega_S\) at TL/symmetric (soft)
width	\(\sqrt2\,\sigma_S\) (hard)	bracket \([\sigma_S,\ \sigma_\mathrm{TL}]\)	\(>\sigma_S\); TL estimate (weak)
nature of a mismatch	trace error only	trace error or non-TL phase	trace error or phase

Practical points#

Normalise the predicted and measured marginals (unit peak or unit area) before comparing shape; the absolute scale is set by the conversion efficiency and detector gain. predicted_marginal() returns a peak-normalised marginal.
Compare on a common, calibrated frequency axis: the SH axis for SHG, the fundamental axis for PG/TG/SD. A centroid mismatch on that axis is the calibration check.
Width and centroid estimates can be sensitive to noise and cropping; the full-shape overlay is the most sensitive to scatter and detector nonlinearity but also the most affected by the TL assumption for PG/SD. Restrict the moments and the normalisation to a spectral trust region (in the GUI, 190–1000 nm for a measured trace, or the full loaded range for the noise-free simulated/synthetic traces) so out-of-band detector noise cannot dominate.
For the rigorous SHG and PG-centroid checks, report the numeric discrepancy (measured vs predicted) as a QC metric alongside the retrieval error.

Centroid-matching efficiency correction#

A TG/PG detector’s third-order generation/collection efficiency tilts the marginal by a power law \((\lambda/\mu\mathrm{m})^n\) (croak.preprocess.third_order_scale()). Because the marginal centroid is a hard anchor for PG/TG (and the fundamental anchor for SD/SHG), croak.marginal_checks.auto_third_order_exponent() picks the exponent \(n\) whose corrected-marginal centroid lands on the predicted centroid. The tilt shifts the centroid monotonically in \(n\), so the match is at most one root over a non-degenerate positive trace; it is restricted to the trust region so amplified out-of-band noise cannot bias it. This is the physically grounded replacement for fitting the marginal shape to the spectrum.