Methods

In this section a brief overview over the background and assumptions of all implemented methods is given.

General Remarks

Each method is implemented as a RetrievePulsesMETHOD class in pulsedjax.core.base_classes_methods.py. These classes define how input data is processed and prepared for the retrieval process. Most importantly they define how the nonlinear signal field is calculated.
The final useable retrieval algorithms are implemented as child-classes using multiple inheritence Algorithm(AlgorithmBASE, RetrievePulsesMETHOD).
The retrieval takes place on the provided frequency axis (not angular frequency) of the measured trace. This axis needs to be broad enough to avoid interference of fundamental and SHG/THG signals as well as to include the range of the pulse and gate-pulse spetra if those are provided.

FROG

Frequency-Resolved-Optical-Gating (FROG [2]) can be understood as a spectrally resolved auto- or cross-correlation measurement. Its nonlinear signal is defined as:

\(S(\tau, t) = E(t)\cdot G(A(\tau, t))\)

where \(E(t)=A(t)\) in the case of an autocorrelation setup and G() refers to the nonlinear effect dependent gate-function. Thus a measurement consists of a pulse \(A(t)\) which is shifted over \(E(t)\) in the time domain. The spectrum of a nonlinear signal at each delay yields the measurement trace \(T(\tau,\omega) = |S(\tau, t)|^2\).
The above description assumes a non-collinear setup. However FROG can be performed collinearly, which yields an interferometric setup. Thus the name i-FROG. In this case te nonlinear signal is defined as:

\(S(\tau, t) = (E(t) + A(\tau, t))\cdot G(E(t) + A(\tau, t))\)

In the case of cross-correlation setup \(\left(E(t)\neq A(t)\right)\), the pulse \(A(t)\) is not necessarily known. If it is known it can be provided via algorithm.get_gate_pulse(). In additioncross_correlation=True has to be specified.
If \(A(t)\) is unknown a cross_correlation="doubleblind" retrieval can be performed. However in such a case serious ambiguities exist, such that the provision of additional constraints is necessary for most algorithms. The spectra of \(E(t)\) and \(A(t)\) can be provided as such constraints via algorithm.use_measured_spectrum().
The available algorithms for FROG-retrieval are described in Algorithms and Method Specific Algorithms.
For i-FROG the algorithms Vanilla, LSGPA, CPCGPA and PIE are not available, due to the requirements of these algorithms.

Chirp-Scan

Contrary to FROG a Chirp-Scan [3, 4, 5] is a single pulse method, such that by definition no cross-correlation and no interferometric version exist. It works by applying an arbitrary (but known) higher order spectral phase to the pulse. For each applied phase the spectrum of a nonlinear process is recorded. Yielding a trace of the form \(T(\theta, \omega) = |S(\theta, \omega)|^2\). The signal field is defined as:

\(S(\theta, t) = E(\theta, t) \cdot G(E(\theta, t))\)

Most commonly the spectral phase is applied via material dispersion. However any known spectral phase (e.g. from a pulse shaper) can be used. In a retrieval the phase_type has to be specified, either as "material" or one of the predefined functions in pulsedjax.core.phase_matrix_funcs.py. Alternatively a custom function can be function provided phase_type = myfunction as outlined below.

def myfunction(measurement_info, chirp_parameters):
    """ 
    Calculates a phase matrix.
    Args:
        measurement_info (Pytree): holds measurement data and parameters, needs to contain the shift-values in appropriate units
        chirp_parameters (tuple): the parameters which are expected by phase_func

    Returns:
        jnp.array, the calculated phase matrix
    """
    
    z_arr, omega = measurement_info.z_arr, 2*jnp.pi*measurement_info.frequency
    phase_matrix = phase_func(omega[jnp.newaxis,:], z_arr[:, jnp.newaxis], chirp_parameters)
    return phase_matrix

In addition to phase_type the chirp_parameters have to provided. These contain any static input parameter which may be used to calculate the spectral phase. For example in the case of phase_type="material", the chirp_parameters need to be an instance of refractiveindex.RefractiveIndexMaterial.
The available algorithms for Chirp-Scan-retrieval are described in Algorithms and Method Specific Algorithms.

Time-Domain-Ptchography

Time-Domain-Ptychography [6, 7] can be understood as a FROG variant. The only difference is that \(A(t)\) is spectrally filtered. The transfer-function of this filter has to be provided. In pulsedjax.core.spectral_filter_funcs.py exemplary functions to generate/evaluate a filter are available.
The filter is provided via spectral_filter. It can be complex valued, which incorporates (material) dispersion. It needs to have the same frequency axis as the measured trace.
For an auto-correlation based TDP, it is necessary to shift the delays such that zero is located at the point of highest intensity of the trace. Otherwise a mismatch in the group-delay of pulse and filtered pulse causes inconsistencies and prohibits convergence.
The available algorithms for TDP-retrieval are described in Algorithms and Method Specific Algorithms.

2D-SI

Two-Dimensional-Spectral-Shearing-Interferometry (2D-SI, [8]) can be understood as inspired by both FROG and SPIDER. It can be performed in an auto- or cross-correlation setup. The nonlinear signal is defined as:

\(S(\tau, t) = E(t)\cdot G(A'(t) + A''(\tau, t))\)

where \(A'\) and \(A''\) are spectrally filtered pulses with different central frequencies. These are overlayed interferometrically. The final nonlinear signal is obtained in a non-collinear setup. The transfer-functions of the spectral filters have to be provided via spectral_filter1 and spectral_filter2. As in TDP their spectral axis must match that of the measured trace.
Usually 2D-SI is performed via Sum-Frequency Generation. In this case Direct-Reconstruction as described in Method-Specific Algorithms can be used to perform the retrieval. If the prelimiaries for this approach are not fullfilled the algorithms described in Algorithms can be used instead.

VAMPIRE

Very advanced method for phase and intensity retrieval of E-fields (VAMPIRE, [9]) has a setup similar to 2D-SI but functions like a FROG. The nonlinear signal is defined as:

\(S(\tau, t) = E(\tau, t) \cdot G(A(t) + A'(t))\)

where \(A'\) is a modified and delayed version of \(A\). Usually the “modification” is achieved via material dispersion. In contrast to 2D-SI no spectral filtering is introduced.
The working idea is to create a duplicate but modified pulse in the interferometer at a fixed delay. The pulse \(E\) is shifted in time over these in order to create a FROG-like trace. This approach resolves any ambiguities which may be present in an analog FROG measurement.
In order for the retrieval to work the interferometer delay has to be set using tau_interferometer, the material has to be provided via refractive_index=refractiveindex.RefractiveIndexMaterial and its thickness needs to be specified in millimeters with material_thickness. The available algorithms for VAMPIRE-retrieval are described in Algorithms and Method Specific Algorithms.

Attosecond Streaking

Attosecond streaking is implementend via a multichannel Strong-Field-Approximation (SFA). This model extends the SFA by summation over multiple channels in a coherent fashion. Thus incoherent signals from multiple channels are not described.
The basic idea of streaking is to ionize an atom or molecule with an EUV-Pulse. The momentum of the created photoelectrons is dressed in the field of a femtosecond pulse. Scanning the delay of the EUV and femtosecond pulse creates a spectrogram-like measurement of the photoelectron spectrum [10]. The SFA allows an analytic description of this process by neglecting the coulomb interaction. It reads:

\(S(\tau, k) = -i \cdot \sum\limits_C \int\limits_{-\infty}^\infty dt\cdot \rho_C(k+A(t)) \cdot E(t,\tau) \cdot e^{-i t\cdot \left(IP_C + \frac{k^2}{2}\right)}\cdot e^{-i\phi(t,k)}\)

\(\phi(t,k)= \int\limits_t^{\infty} dt'\cdot \left(k A(t') + \frac{A(t')^2}{2}\right)\)

The retrieval can either be done by usage of standard FROG algorithms. This necessitates employment of the central-momentum axpproximation which assumes the transition-dipole-matrix-element (DTME) to be constant.
Alternatively the SFA can be implemented numerically and the retrieval achieved by usage of analytic gradients with respect to the EUV-pulse and vectorpotential [11]. In such a formulation the DTME may be provided or reconstructed as part of the retrieval. The latter requires the provision of both the spectrum of the femtosecond and attosecond pulse in order to avoid ambiguities.
Retrieval based on the SFA is available via the pulsedjax.streaking module with the Generalized Projection Algorithm, COPRA and all general optimization algorithms. These are described in Algorithms.