Eliminating local convergences in FROG retrieval algorithms

. The complicated electric field structure of ultrashort pulses as characterized by frequency-resolved optical gating (FROG) algorithms often results in local convergence, which does not accurately represent the actual field pulse. We present an efficient and universal test procedure applicable to any FROG retrieval algorithm that allows recognition of the erroneous local convergence. The 100% efficacy of the procedure is demonstrated using Line-Search FROG algorithm, and comparison is given with the performance of a standard extended ptychographic iterative FROG trace retrieval engine


Introduction
Precisely characterizing the electric field of ultrashort laser pulses is of importance to a wide range of industrial and scientific applications.The modifications of the pulse field structure during propagation through media retain the information of the complex material response function that can be extracted by complete electric field characterization.One popular method to achieve this is frequency-resolved optical gating (FROG).The electric field structure is retrieved from FROG measurements with iterative retrieval algorithms solving the twodimensional phase-retrieval problem.The widely used gradient descent solvers such as extended ptychographic iterative engine (ePIE) and principal components generalized projections algorithm (PCGPA) benefit from pre-processing of the measured FROG trace.In particular, removing negative data points resulting from a standard background subtraction to facilitate calculation of the square root of the measurements [1].This can lead to loss of information in low-signal and high-noise traces.Previously, we presented a line-search FROG (LSF) algorithm [2], which preserves information in the measured FROG traces and gives higher veracity of the field extraction in noisy measurements.In this work, we address another common problem of iterative algorithms, the fact that they are susceptible to convergence into local minima, especially when the traces are complicated, and the signal-to-noise ratio in the measured trace is low.

Method and results
PCGPA and ePIE stop the retrieval process when the standard trace-area normalized FROG error, i.e., the RMS of the difference between the calculated and the measured FROG traces, has stagnated or a pre-determined number of iterations have been executed.The stagnation of the error can mean that the algorithm converged to an erroneous local solution.To avoid this in the LSF algorithm we have here implemented a verification procedure.When apparent convergence in the error is reached, the procedure first determines if there is a discernible structure in a trace , which is the difference between measured and retrieved traces.This straightforward step often reveals the presence of remnant non-random field structures not captured in the local solution.This check is performed by first low-pass filtering  followed by a comparison of the average standard deviation in the four corners of the  frame to the maximum value in .It can be reasonably expected that in a properly framed FROG measurement, only noise contributes to the statistics in the corners of the 2D frame.If the maximum value is greater than five times that of the standard deviation, the field structure corresponding to the local solution is perturbed, and retrieval is resumed.We've had success with replacing the temporal intensity of the local solution to a Gaussian intensity with random temporal width whilst retaining the phase of the local solution as a choice of perturbation.Further, the inset shows the low-pass filtered version of , which is the basis for the evaluation.In case a) the retrieval has converged to an incorrect solution.The user can distinguish a difference between the measured and retrieved trace, however, the algorithm determines the retrieval unsuccessful by comparing the peak value in the low-pass filtered  to the standard deviation in the corners.Case b) demonstrates successful convergence, and no discernible structure can be observed in the lowpass filtered .
To demonstrate this in practice, 250 complex random pulses were generated with time-bandwidth products of 10 as described in [3].SHG-FROG traces were simulated with all these pulses and additive Gaussian noise with a standard deviation of 7% of the peak value in the trace, representing a very noisy trace, was added.Then we attempted to retrieve the original traces using the LSF and the ePIE algorithms.The code used for ePIE was the one provided by the authors of [4] with the modification that noise is added to the intensity of the trace rather than the amplitude to better match experimental conditions.
In experimental cases, only the FROG error is available as the true pulse is unknown.When working with simulated data, a comparison of the retrieved pulse with the actual pulse can be performed directly.Due to this, we can evaluate the quality of the retrieved pulse field structure using the angle between the original and retrieved pulse as a metric.The angle is defined as: with   and   being the original and retrieved pulse and ⟨  |  ⟩ representing the inner product [5].
The above-mentioned 250 traces were retrieved once each.For a fair comparison, both algorithms were permitted 10 000 iterations each.The results of the retrieval quality metric are shown in Fig. 2. It is clear that LSF provides a lower mean error angle for successful retrieval than ePIE.Erroneous convergences are also clearly distinguishable with values far above the mean, with LSF showing a 100% success rate for the dataset tested here.

Conclusion
The procedure introduced here substantially increases the field extraction veracity in the LSF algorithm by introducing a convergence state verification step in the retrieval process.The procedure was tested with a 100% success rate using the LSF algorithm.However, the procedure is essentially blind to the specific solver and, because of this, with only minor modifications, can be adapted for implementations to any standard FROG retrieval algorithm.

Fig. 1 .
Fig. 1.Demonstration of the convergence evaluation.The small inset shows the low-pass filtered .a) case where retrieval has converged to an incorrect local solution.b) case for successful retrieval.

Fig. 1 .
Fig.1.demonstrates the evaluation of the convergence performed by displaying the simulated measured trace, retrieved trace, and the difference between the two, .Further, the inset shows the low-pass filtered version of , which is the basis for the evaluation.In case a) the retrieval has converged to an incorrect solution.The user can distinguish a difference between the measured and retrieved trace, however, the algorithm determines the retrieval unsuccessful by comparing the peak value in the low-pass filtered  to the standard deviation in the corners.Case b) demonstrates successful convergence, and no discernible structure can be observed in the lowpass filtered .To demonstrate this in practice, 250 complex random pulses were generated with time-bandwidth products of 10 as described in[3].SHG-FROG traces were simulated with all these pulses and additive Gaussian noise with a standard deviation of 7% of the peak value in the trace, representing a very noisy trace, was added.Then we attempted to retrieve the original traces using the LSF and the ePIE algorithms.The code used for ePIE was the one provided by the authors of[4] with the modification that noise is added to the intensity of the trace rather than the amplitude to better match experimental conditions.In experimental cases, only the FROG error is available as the true pulse is unknown.When working with simulated data, a comparison of the retrieved pulse with the actual pulse can be performed directly.Due to this, we can evaluate the quality of the retrieved pulse field structure using the angle between the original and retrieved pulse as a metric.The angle is defined as:

Fig. 2 .
Fig. 2. Retrieval angle for the two algorithms for each of the 250 traces with angles from LSF in red and ePIE in blue.