Glossary and notation

A quick map between the symbols used in the theory pages and the names used in the code.

Notation

Symbol	Meaning	In the code
\(\tilde E(\omega)\)	complex pulse spectrum (the unknown)	spectrum, ew
\(E(t)\)	complex temporal field	field, et
\(\Omega\) (often \(\omega\) in the API)	centred frequency offset (rad/s)	omega, g.omega
\(\omega_{\rm abs}=\omega_0+\Omega\)	absolute optical angular frequency (rad/s)	omega + omega0
\(\omega_0\)	carrier angular frequency	omega0
\(t\)	time (s)	t, g.t
\(\tau\)	delay (s)	tau, delays
\(s(E,G)\)	instantaneous nonlinear signal	Interaction.signal
\(\psi(\omega,\tau)\)	frequency-domain signal for one delay	signal_single
\(T(\omega,\tau)\)	trace intensity \(\lvert\psi\rvert^2\)	trace, t_meas/t_sim
\(\mu\)	optimal intensity scale factor	mu, compute_mu
\(w_\omega\)	per-frequency weights	weights
\(r\)	weighted residual sum of squares	compute_r
\(R\)	normalised trace error (“FROG error”)	error, compute_R
\(\beta(\omega)\)	slab propagation constant	materials.beta
\(\bar v = \partial L/\partial v^*\)	Wirtinger cotangent	*_bar, psi_bar, ew_bar
\(z_q, w_q\)	depth quadrature nodes/weights	quadrature_nodes_weights

R vs G

The peak-normalised rms trace error \(R=\sqrt{r/(MN\,\max(T_\text{meas}w)^2)}\) is written \(G\) (the “G-error”) in much of the FROG literature and \(R\) throughout croak. They are the same quantity (croak.metrics.compute_R(), result.error); with weights, croak uses the weighted residual and the peak of the weighted measured trace.

Terms

croak

Complete Retrieval by Optimisation and Adjoint Kernels — the package for retrieving a complex pulse from a model of its FROG measurement.

PNPS

Parameterised Nonlinear-Process Spectra — the family of self-referenced measurements (FROG, TDP, d-scan, …) unified by the trace model. The scanned parameter is the delay \(\tau\) in FROG.

FROG

Frequency-Resolved Optical Gating — a delay-scanned nonlinear spectrogram; the canonical PNPS technique.

Interaction / geometry

the nonlinear process defining the signal \(s(E,G)\) — SHG, SD or PG in croak. See Interactions.

TG-FROG

Transient-Grating FROG; for identical coincident gate replicas in the local model, its signal reduces to the PG kernel, so croak retrieves it with interaction="pg".

Test / gate field

the undelayed field \(E\) and the delayed field \(G(t)=E(t-\tau)\) entering an interaction.

Trace error \(R\)

the normalised, dimensionless residual between measured and fitted traces. Interpret it relative to noise, weights, preprocessing and the model. A small value does not by itself establish field accuracy.

Thin medium

the classic FROG approximation of an infinitely thin, dispersionless nonlinear medium — the special case of the dispersive model with \(\beta=0\) and zero thickness.

Dispersive slab / D-COPRA

retrieval that models propagation inside a finite, dispersive medium; “D-COPRA” is COPRA with a material set.

Wirtinger cotangent

\(\bar v \equiv \partial L/\partial v^*\), the conjugate derivative that gradient descent on a real loss of a complex variable requires. See Gradients.

Phase-only retrieval

fixing the spectral amplitude (to a measured spectrum) and retrieving only the phase (phase_only=True).

R_omega

per-frequency intensity scaling — the scale factor \(\mu\) becomes a vector \(\mu_\omega\). See Regularisation.

B-spline phase basis

representing the spectral phase by the control points of a cubic B-spline over the spectrum support (phase_basis="bspline", n_nodes nodes) instead of one value per frequency bin — a low-dimensional, intrinsically smooth parameterisation. See Regularisation and croak._jax_pulse.make_spline_phase_basis().

CMA-ES / evolution strategy

Covariance-Matrix-Adaptation Evolution Strategy — a derivative-free, population-based global optimiser (CMAES, algorithm="cma-es", via the optional evosax backend) used to escape poor local minima. See global retrieval.

Temporal regularisation

a penalty (reg_time, with a time_window in seconds) on the fraction of pulse energy falling outside a time window, suppressing satellites and pedestals during retrieval. See Regularisation.

Chirped mirror

a multilayer mirror that imparts a designed, wavelength-dependent group delay per reflection. Registered into croak.dispersion.MIRRORS and applied by number of bounces; see Materials and chirped mirrors.

Field reference

the plane and spatial definition of the reported complex field. Source, transverse-integrated beamlet, effective and on-axis entrance fields can have different amplitudes and temporal profiles. See conventions.

\(p\), \(\delta\)

the two timing coordinates of the spatial (smearing) model: the gate-pair separation and the probe-to-gate-centroid offset (croak.smearing). For PG the two gate replicas arrive at \(\tau-\delta\mp p/2\) in the probe frame; for SD, \(p\) separates the two unconjugated replicas and \(\delta\) offsets the conjugated arm (see the forward model).

\(p_f\)

the exponent of a declared spectral field conversion \(\widetilde E_{p_f}=(\omega_{\rm abs}/\omega_0)^{p_f}\widetilde E_{\rm int}\), referenced to the carrier \(\omega_0\). It labels a field convention, not a detector-response correction.

Detector response

wavelength-dependent conversion from optical signal to recorded counts. Distinguish it from the nonlinear source factor and the coordinate Jacobian used to convert a density per wavelength to one per frequency.