The PNPS framework#

This page introduces the formalism that croak is built on: parameterised nonlinear-process spectra (PNPS). FROG is the most familiar member of this family, but the same mathematics describes time-domain ptychography (TDP), dispersion scan (d-scan) and related self-referenced techniques.

Self-referenced pulse measurement#

A photodetector or spectrometer only records intensity; the spectral phase of an ultrashort pulse is lost. Self-referenced techniques recover it by gating the pulse with a (delayed, dispersed, …) replica of itself through a nonlinear interaction, then spectrally resolving the result. Scanning a parameter — the delay \(\tau\) in FROG — builds a two-dimensional trace that encodes both the amplitude and the phase.

The trace model#

Let \(\tilde E(\omega)\) be the complex pulse spectrum we wish to retrieve. A nonlinear process produces a signal operator \(S_\delta[\tilde E](t)\) — the time-domain signal field for parameter value \(\delta\). The measured trace is the squared magnitude of its spectrum:

(1)#\[T(\delta, \omega;\, \tilde E) \;=\; \bigl|\,\mathcal{F}\{S_\delta[\tilde E](t)\}(\omega)\,\bigr|^2 .\]

In FROG, \(\delta\) is the delay \(\tau\) and (1) is a 2-D spectrogram in \((\omega, \tau)\). This is the quantity croak’s ForwardModel computes; see The forward model for the discrete, dispersive form.

Note

croak works with the baseband spectrum \(\tilde E(\omega)\) on a centred angular-frequency grid (zero at the carrier). The carrier \(\omega_0\) fixes absolute frequency for dispersion, wavelength conversion and spatial instrument models. See Conventions.

Signal operators for FROG geometries#

The signal operator is built from a test field \(E\) (undelayed) and a gate field \(G(t) = E(t-\tau)\) (delayed). croak implements three standard geometries, each a different instantaneous nonlinearity in the time domain:

Geometry	Signal \(s(t)\)	croak name	Trace carrier
Second-harmonic generation	\(E\,G\)	`"shg"`	\(2\omega_0\)
Self-diffraction	\(E^2\,G^*\)	`"sd"`	\(\omega_0\)
Polarisation gating	\(E\,\lvert G\rvert^2\)	`"pg"`	\(\omega_0\)

The transient-grating (TG-FROG) geometry used in validation, with coincident gate replicas, shares the PG signal kernel \(E\lvert G\rvert^2\), so retrieving TG-FROG data uses interaction="pg". See Interactions for the signal fields and their adjoints.

The standard idealisations, and where they fail#

The simplest model treats the medium as thin, the replicas as spatially uniform, and the detector response as a frequency-dependent scale. Check these assumptions against the pulse bandwidth and instrument:

Finite dispersive medium. The pulse evolves while the nonlinear signal is generated, and contributions from different depths interfere. In the DUV, even micrometre-scale substrates can matter for few-femtosecond pulses.
Crossed focused beams. Relative arrival times vary across the focal spot. The resulting trace depends on the mask geometry, pulse and focusing; a reduced smearing model approximates this dependence under stated assumptions.
Finite collection. An aperture selects a coherent part of the signal. Its effect can depend on delay as well as frequency, so a row-wise spectral calibration need not absorb it.

The forward-model chapter develops these models and their limits. The validation study shows bounded examples in which a small trace residual coexists with a substantially wrong pulse. It also includes cases that remain poorly retrieved with a more detailed model.

Ambiguities#

The map from pulse to trace is not perfectly one-to-one. Every FROG geometry carries trivial ambiguities that retrieval cannot resolve from the trace alone:

an absolute phase offset \(\tilde E \to \tilde E\,e^{i\phi_0}\);
an absolute time/delay origin (a linear spectral phase);
for SHG-FROG, the direction of time (\(E(t) \to E^*(-t)\)).

croak’s results inherit these: compare retrieved pulses up to a constant phase and a time shift. SD, PG and TG do not have the general SHG time-reversal symmetry, although particular fields or incomplete data can remain difficult to distinguish. Several random starts test sensitivity to local minima; they do not prove uniqueness or remove an information ambiguity.

Beyond FROG#

Because the only geometry-specific ingredient is the signal operator \(S_\delta\), the framework extends to any parameterised nonlinear process — d-scan (where \(\delta\) is inserted glass), TDP (where \(\delta\) is a ptychographic position), and others — by swapping the operator while reusing the same least-squares machinery. croak currently ships the three FROG kernels above.