## Abstract

Frequency-resolved optical gating (FROG) is a very popular technique for complete characterization of ultrashort laser pulses. In FROG, a reconstruction algorithm retrieves the pulse amplitude and phase from a measured spectrogram; yet, current FROG reconstruction algorithms require and exhibit several restricting features that weaken FROG performances. For example, the delay step must correspond to the spectral bandwidth measured with a large enough SNR—a condition that limits the temporal resolution of the reconstructed pulse, obscures measurements of weak broadband pulses, and makes measurements of broadband mid-IR pulses hard and slow because the spectrograms become huge due to the geometrical time-smearing effect. We develop a new approach for FROG reconstruction based on ptychography (a scanning coherent diffraction imaging technique), that removes many of the algorithmic restrictions. The ptychographic reconstruction algorithm is significantly faster and more robust to noise than current FROG algorithms that are based on generalized projections (GP). We demonstrate, numerically and experimentally, that ptychographic reconstruction works well with very partial spectrograms, e.g., spectrograms with a reduced number of measured delays and spectrograms that have been substantially spectrally filtered. In addition, we implement the ptychographic approach to blind second harmonic generation (SHG) FROG and demonstrate robust and complete characterization of two unknown pulses from a single measured spectrogram and the power spectrum of only one of the pulses. We believe that the ptychography-based approach will become the standard reconstruction procedure in FROG and related diagnostics methods, allowing successful reconstructions from so far unreconstructable spectrograms.

© 2016 Optical Society of America

## Corrections

Pavel Sidorenko, Oren Lahav, Zohar Avnat, and Oren Cohen, "Ptychographic reconstruction algorithm for frequency resolved optical gating: super-resolution and extreme robustness: erratum," Optica**4**, 1388-1389 (2017)

https://www.osapublishing.org/optica/abstract.cfm?uri=optica-4-11-1388

## 1. INTRODUCTION

Frequency-resolved optical gating (FROG) is probably the most commonly used method for full characterization (i.e., amplitude and phase) of ultrashort optical pulses [1,2]. A FROG apparatus produces a two-dimensional (2D) intensity diagram (spectrogram), also known as a FROG trace, of an input pulse by interacting the pulse with its delayed replica in a nonlinear optical medium, e.g., a second harmonic generation (SHG) crystal [3]. Current FROG reconstruction procedures [4–6] are based on 2D phase retrieval algorithms [7,8], somewhat similar to the approach used in 2D coherent diffraction imaging [9]. These generalized projections (GP) algorithms [4–6] require Fourier relation between the frequency and delay axes of the measured spectrogram: $\mathrm{\Delta}\omega \mathrm{\Delta}t=1/N$, where $\mathrm{\Delta}t$ is the delay step, $\mathrm{\Delta}\omega $ is the spectral resolution, and $N$ is the number of delay steps as well as the number of frequencies measured for each delay. The reconstruction resolution of the GP algorithms is limited by the spectrogram delay step, which according to the above Fourier relation must correspond to the spectral bandwidth of the trace [5]. This algorithmically imposed coupling between the reconstruction resolution, delay step, and measured spectral bandwidth with a large enough SNR weakens FROG performances in several ways. For example, it dictates that the bandwidths of the nonlinear medium and the spectrometer must be large enough that they do not filter the spectrogram. Indeed, this is the reason why crystals of SHG FROG for measuring ultrashort pulses are very thin: such crystals support broad-bandwidth phase matching. Unfortunately, the price for using thin crystals is reduced conversion efficiency and SNR. This feature limits the application of FROG in measurements of weak broad-bandwidth laser pulses. A hardware-based method was suggested to effectively increase the phase-matching bandwidth of SHG FROG [10], but it requires mechanical scanning of the SHG crystal orientation, which significantly increases the measurement acquisition time, and it still provides limited bandwidth improvement. Another problem that results from the imposed Fourier relation is that the spectrograms of ultrashort mid-IR pulses are often huge (mostly due to the geometrical time-smearing effect [11]), resulting in computationally very slow reconstruction algorithms [12]. Also, an algorithm that will work with partial spectrograms (i.e., spectrograms that do not conform to the Fourier relation) may allow us to significantly reduce the number of scanning steps in the FROG apparatus, yielding much faster measurements. Finally, there is always a motivation to increase the robustness of FROG reconstruction algorithms to noise.

Reconstruction algorithms that work well even with partial spectrograms were recently developed for several characterization techniques of ultrashort laser pulses [13–18], but not for FROG. Especially prominent is the recent adaptation of the general principle of ptychography, a scanning coherent diffraction imaging method [19], to diagnostics of femtosecond pulses using cross-correlation FROG (XFROG) [15,17] and attosecond pulses from FROG-CRAB (frequency-resolved optical gating for complete reconstruction of attosecond bursts) measurements [14]. These works demonstrated the superb robustness of the ptychographic-based reconstruction approach, both in terms of SNR and the use of only partial spectrograms. Specifically, reconstruction from a reduced number of delay steps was demonstrated. However, the temporal resolution of the recovered field was still limited in these works by the measured spectral bandwidth (reconstruction with spectral filtering was not considered). Overall, the implementation of the ptychographic-based approach to pulse diagnostic techniques (XFROG and FROG-CRAB) in which the unknown pulse interacts with another pulse that is fully or partially known was shown to be very successful. However, so far this approach has not been adapted to techniques like FROG, in which the unknown pulse interacts with its exact replica and therefore the reconstruction problem is more difficult.

Here, we propose and demonstrate experimentally a ptychography-based pulse reconstruction algorithm for FROG that does not require any prior information on the pulse. We show that the proposed algorithm outperforms the conventional GP FROG pulse recovery method in terms of noise robustness. Moreover, we also demonstrate that our algorithm can successfully recover pulses from much fewer measurements than are needed in current pulse recovery algorithms, which are based on the GP principle. We demonstrate, numerically and experimentally, recovery of a pulse even when the delay or frequency axis is under-sampled considerably or truncated by a low-pass filter. We also demonstrate that if additional information about the pulse is known in advance, e.g., its power spectrum, then successful pulse recovery is possible even from a very small number of measurements. Finally, we explore the application of the ptychographic approach to blind FROG, where two unknown pulses are characterized simultaneously. We found that the two pulses can be completely and reliably characterized from a single spectrogram and the power spectrum of one of the pulses.

## 2. RECONSTRUCTION PROCEDURE

Our method is based on ptychography, which is a very powerful coherent diffraction imaging (CDI) technique that has recently gained remarkable momentum in optical microscopy in the visible, extreme ultraviolet, and x ray spectral regions [19]. In ptychography, a complex-valued object is scanned in a step-wise fashion through a localized coherent beam. In each scanning step, the intensity of the diffraction pattern of the object is measured, typically in a Fraunhofer plane. The measured diffraction pattern is associated with the magnitude of the Fourier transform of the illuminated part of the object. Critically, the spatial support of the illuminating spot must be bigger than the step size, so neighboring diffraction patterns will result from different, but overlapping, regions of the object. The set of measured diffraction patterns is used for reconstructing the complex-valued transmittance of the object. Ptychography exhibits several advantages over “conventional” phase retrieval schemes, including significant improvement in the robustness to noise, no requirement for prior information (e.g., support) on the object and probe beam, no loss of information due to the beam stopping, and generally faster and more reliable reconstruction algorithms [20]. Moreover, the uniqueness in one- [21] and two- [22] dimensional ptychography is guaranteed, provided that the illuminating beam is known. Ptychography has been proven to be useful in cases where the illumination beam is known [20,22] and also when the illumination beam is unknown. In the latter case, the unknown illumination is reconstructed together with the sought object [23,24].

Next we describe our ptychography-based pulse recovery algorithm for FROG (specifically, SHG FROG) in detail. An SHG FROG trace [3] is given by

Second, the SHG signal is Fourier transformed, and its modulus is replaced by the square root of the $s(j)$-ordered measured spectrum. Importantly, we replace only the part that was measured reliably. Other spectral components are not changed ($\mathrm{\Omega}$ marks the set of reliably measured spectral components):

In addition, we suggest applying a soft thresholding procedure (a well-known procedure for solving linear inverse problems [26]) for the subset of unknown frequencies (${\mathrm{\Omega}}^{C}$ i.e., $\omega \not\subset \mathrm{\Omega}$) for both the real and imaginary parts of the signal by

Finally, the pulse is updated with a weight function based on the complex conjugate of ${E}_{j}^{*}(t)$ according to

In Eq. (8), $\alpha $ is a real parameter that controls the strength of the update. Crucially, in our algorithm a new $\alpha $ is selected randomly in each iteration (its probability is distributed uniformly in the range [0.1, 0.5]). The iterations continue until the stopping criteria (the difference between the measured and calculated FROG traces is smaller than the SNR) is reached or until the predefined maximal number of iterations is reached.

Our algorithm is based on the extended ptychographical iterative engine (ePIE) [24] that is commonly used in ptychography. We adopted it for FROG through three modifications. First, we replaced the illuminating probe beam by the delayed replica of the sought pulse, i.e., by $E(t-j\mathrm{\Delta}t)$. Second, we introduced a randomly varying $\alpha $. We found that convergence with a fixed $\alpha $ requires a very large number of runs of the algorithm with random initial guesses ($\mathrm{typically}>100$), while with a randomly varying $\alpha $, we found out that $\sim 95\%$ of the random initial guesses converge. Intuitively, the randomization of $\alpha $ significantly reduces the probability of stagnation in a local minimum (the advantages of using random ${\alpha}^{\prime}s$ in phase retrieval algorithms are discussed in Ref. [29]). A third optional modification is the replacement of only part of the power spectrum in the second step of the algorithm, i.e., Eq. (4), with or without the soft thresholding procedure. This modification allows us to reconstruct pulses at high resolutions from incomplete spectrograms (as shown below in Fig. 2). More modifications are presented below to allow the utilization of the additional measured information (power spectrum) and modification for the ptychographic blind FROG reconstruction.

## 3. RESULTS

#### A. Numerical Results with Complete Spectrograms

Next, we characterize the performances of our ptychographic-based FROG algorithm as a function of the SNR and compare it with a commonly used GP-based reconstruction FROG algorithm: the Principal Component Generalized Projections Algorithm (PCGPA) [30,31]. To this end, we numerically produced a set of 100 laser pulses that all conform to a Gaussian power spectrum that is centered at 800 nm. Its spectral bandwidth is 213 nm. Each pulse ($N=128$ grid points) is produced by applying a random spectral chirp to the above power spectrum (on the condition that the support of the pulse is $\le 200\text{\hspace{0.17em}}\mathrm{fs}$). The calculated FROG traces are 128 by 128 points with equally spaced delays, $\mathrm{\Delta}t=1.57\text{\hspace{0.17em}}\mathrm{fs}$, and spanning the same frequency window (i.e., the product of the delay and spectral resolutions is $\frac{1}{N}=\frac{1}{128}$). White Gaussian noise $\sigma $ is added to the simulated FROG trace at different SNR values, defined by: $\mathrm{SNR}=20\text{\hspace{0.17em}}\mathrm{log}$ ($\Vert I\_\mathrm{FROG}\Vert /\Vert \sigma \Vert $), where $\Vert \xb7\Vert $ stands for l_{2} norm. Next, we recovered the set of 100 unknown pulses using both the ptychographic-based and PCGPA algorithms, each one with 10 independent runs (the initial pulse in PCGPA was random), where each run is limited to 1000 iterations, and chose the best solution according to the stopping criteria (minimal NMSE between the simulated and recovered traces). In order to characterize the quality of the reconstructions, one needs to take into account that SHG FROG suffers from the following ambiguities: trivial time shift, time direction, and global phase. Thus, we removed the above first two ambiguities from the reconstructed fields and then evaluated the error of the reconstructions using the following angle,

#### B. Numerical Results with Incomplete Spectrograms

In order to apply the PCGPA and all other current FROG reconstruction methods, the measured FROG trace should consist of $N\times N$ (i.e., $N$ spectral at $N$ time delays) measurements, and the product of the delay step and spectral resolution should be $1/N$ [25]. Also, the bandwidth of the FROG trace should be $\sim 1.4$ times larger than the bandwidth of the pulse power spectrum autocorrelation [25]. We term such FROG traces as complete traces. As discussed in the introduction section, these constraints weaken the performances of FROG and impose hardware limitations in the FROG apparatus. However, the source for these requirements is algorithmic and not fundamental. After all, complete FROG traces are highly redundant: there are many informative measurements in the trace, while the pulse is a vector of only $N$ complex numbers. In addition, the FROG trace depends nonlinearly on the pulse, i.e., it mixes its spectral components. Indeed, we will show below that the ptychographic reconstruction approach can reconstruct pulses from significantly incomplete FROG traces. We define an incompleteness parameter by

Particularly, we will present super-resolution in FROG, where the complete FROG trace and its associated pulse are retrieved from a FROG trace that was spectrally low-pass filtered (such filtering can result from the limited bandwidth of the nonlinear medium or the spectrometer [25,32]). Equivalent to super-resolution in imaging, the bandwidth extrapolation in FROG corresponds to the temporal super-resolution. That is, $q$ times bandwidth extrapolation corresponds to $q$ times super-resolution. The ptychographic approach can reconstruct pulses from their incomplete FROG traces because in each updating step of the reconstructing process, it makes use of a single measured spectrum (measured at one specific delay value). As a result, the delay and spectral axes are completely uncoupled. In addition, the delay step grid does not define the temporal grid (and therefore also the temporal resolution) of the reconstructed pulse. In addition, the delay step does not need to be constant. Moreover, as shown below, the resolution of the reconstructed pulse can be higher than one may expect from the measured spectral bandwidth and Fourier uncertainty. Figure 2 presents typical results of such reconstructions using a pulse from the bank of pulses used for Fig. 1. (An example of a pulse with a very high time bandwidth product (TBP) is presented in Fig. S1 in Supplement 1). Simulated FROG traces of the pulse, without and with 20 dB noise, are presented in Figs. 2(a) and 2(b), respectively. In Figs. 2(c)–2(v), each horizontal panel presents results for a different type of truncation. Importantly, in all the reconstructions presented in this figure, only the incomplete (truncated) noisy FROG traces were fed into the reconstruction algorithm. No prior information about the pulse was assumed. Figures 2(c)–2(g) show reconstructions from incomplete FROG traces that were spectrally truncated by a low-pass filter (LPF). Figure 2(c) shows a FROG trace obtained by filtering the trace in Fig. 2(b) by a window-shaped filter that is centered at $\omega =0$, which conserves the information in only 13 (out of 128) frequencies and nullifies all the other frequencies ($\eta \cong 0.1$). Applying the ptychographic reconstruction on the FROG trace in Fig. 2(c) yields the reconstructed (extrapolated) FROG trace in Fig. 2(d) and the associated reconstructed pulse with amplitude and phase that are shown in Figs. 2(e) and 2(f), respectively, compared with the original pulse. The calculated angle (error) of this reconstruction is $\delta \cong 0.04$. Figure 2(g) shows the angle between the reconstructed and original pulses as a function of the incompleteness parameter, clearly showing that the reconstruction works well until the incompleteness parameter approaches $\eta =0.1$. In the second horizontal panel, the frequency axis was under-sampled. Figure 2(h) shows a FROG trace where 7/8 of the spectral lines of the complete noisy trace in Fig. 2(b) were nullified ($\eta \cong 0.125$). Applying the ptychographic reconstruction on the FROG trace in Fig. 2(h) yields the reconstructed FROG trace in Fig. 2(i) and the associated reconstructed pulse with amplitude and phase that are shown in Figs. 2(j) and 2(k), respectively ($\delta \cong 0.055$). Figure 2(l) shows that the reconstruction is good until the incompleteness parameter approaches $\eta \cong 0.125$. We applied the soft thresholding option [Eqs. (5) and (6)] with $\gamma =5\times {10}^{-6}$ in the reconstructions presented in the first two panels of Fig. 2. Reconstructions without applying soft thresholding are presented in Fig. S2 in Supplement 1. The third horizontal panel presents reconstruction from FROG trace that was truncated by a temporal window-shaped filter that is centered at $\mathrm{\Delta}t=0$. Figure 2(m) shows a FROG trace with 22 delay points ($\eta =0.172$). The reconstructed trace is shown in Fig. 2(n). The original and reconstructed temporal amplitude and phase of the pulse are shown in Figs. 2(o) and 2(p), respectively ($\delta \cong 0.2$). Figure 2(q) shows the reconstruction error as a function of the incompleteness parameter. The bottom panel presents reconstruction from spectrogram with an under-sampled delay axis. Figure 2(r) shows a FROG trace in which only 13 (out of 128) equally spaced delay points were conserved ($\eta \approx 0.1$). The reconstructed FROG trace is shown in Fig. 2(s). The original and reconstructed temporal amplitude and phase are shown in Figs. 2(t) and 2(u), respectively ($\delta \cong 0.04$). Figure 2(v) shows that the reconstruction is good until the incompleteness parameter approaches $\eta \cong 0.1$.

#### C. Ptychographic FROG: Experimental Results

We demonstrate our ptychographic-based FROG algorithm experimentally, using our homemade SHG FROG. The laser pulse was produced by an ultrafast Ti-sapphire laser system. The measurement was done with a 3 fs delay step and 512 delay points. Hence, the complete FROG trace consists of $512\times 512$ data points [Fig. 3(a)]. We applied the ptychographic-based and PCGPA algorithms and obtained the reconstructed FROG traces in Figs. 3(b) and 3(c): the NMSEs between the measured and reconstructed traces are 0.0351 and 0.0363, respectively. Figures 3(d) and 3(e) present the reconstructed amplitude and phase of the pulse by the ptychographic-based algorithm (blue solid curve) and the PCGPA algorithm (red dashed curve) respectively. The good correspondences between the reconstructions demonstrate that the ptychographic-based algorithm works well also in the experiments. Next, we demonstrate experimentally the capability of the ptychographic-based FROG reconstruction to retrieve the pulse from an incomplete FROG trace. In this case, the PCGPA and other conventional GP-based FROG reconstruction algorithms fail. Thus, in Fig. 4 we compare the reconstruction from the incomplete trace with the ptychographic reconstruction using the full spectrogram in Fig. 3. The top horizontal panel in Fig. 4 presents a reconstruction from a spectrally low-pass-filtered FROG trace. The trace used, with 128 nonzero spectral lines ($\eta \cong 0.25$), is shown in Fig. 4(a). The second horizontal panel shows pulse reconstruction from a spectrally under-sampled FROG trace. Figure 4(e) presents the trace used, where 75% of the spectral lines were nullified ($\eta \cong 0.25$). The third horizontal panel presents reconstruction from temporally filtered trace with 210 delays ($\eta \cong 0.41$). Finally, the forth panel presents reconstruction from temporally under-sampled spectrogram: only each eighth delay step was conserved ($\eta =0.125$) [Fig. 4(m)]. In each panel, the second plot from the left presents the reconstructed FROG traces [for comparison, see measured trace in Fig. 3(a)] from the corresponding incomplete measured trace to its left. The third and the fourth columns show the reconstructed amplitude and phase of the pulse from full (red dashed curve) and corresponding incomplete measured (blue solid curve) traces, respectively. The reconstructions match quite well (note that the deviations in the reconstructed phases are significant only in the region of low amplitudes), demonstrating experimentally the capability of the ptychographic-based reconstruction algorithm to reconstruct the pulses from incomplete spectrograms.

#### D. Ptychographic FROG with Measured Power Spectrum

The power spectrum of the measured pulse is often available, or it can be measured by a spectrometer. In the current FROG reconstruction algorithms, the directly measured power spectrum may be used for estimating the consistency of the measured trace and the quality of the reconstruction [25]. However, it is not utilized within the reconstruction procedure. Adding the directly measured spectrum as additional constraint in the reconstruction process [33] leads to instabilities due to the required deconvolution; hence, it is not useful [3]. In our ptychographic-based approach, on the other hand, addition of the directly measured spectrum does not lead to instability because we do not perform deconvolution and, as shown above, we can use only part of the FROG trace (the part with a high SNR).

We employ the measured power spectrum in our ptychographic-based reconstruction scheme by projecting the updated pulse in each iteration to the sub-space with the given power spectrum. (Here we assume that the power spectrum is known exactly. The influence of random and systematic errors in the power spectrum should be investigated in the future). We add a fifth step in each iteration to the algorithm described in the method section [the fourth step corresponds to Eq. (8)]. We first replace the power spectrum of the updated pulse by the directly measured power spectrum

We have found that the ptychographic-based FROG algorithm with a power spectrum constraint works very well, even when the FROG trace is significantly spectrally filtered or under-sampled. Numerical results using the same exemplary pulse that was used in Fig. 2 are displayed in Fig. 5. The structure in Fig. 5 is the same as the structures in Figs. 2(c)–2(v). As in Fig. 2, all the reconstructions presented in Fig. 5 use FROG traces that are filtered from the trace shown in Fig. 2(b). The first horizontal panel in Fig. 5 shows pulse reconstruction from a spectrally low-pass filtered trace that conserved only 5 frequencies ($\eta \cong 0.04$, $\delta \cong 0.05$). The second panel presents pulse reconstruction from a spectrally under-sampled trace ($\eta \cong 0.08$, $\delta \cong 0.09$). The third and fourth panels present reconstructions from temporally filtered ($\eta \cong 0.125$, $\delta \cong 0.08$) and temporally under-sampled ($\eta \cong 0.03$, $\delta \cong 0.04$) traces, respectively. Comparing the right columns in Figs. 2 and 5 clearly shows the strength of adding the measured power spectrum to the reconstruction algorithm. More numerical examples of pulse reconstructions from incomplete spectrograms (with and without known power spectra) are shown in Supplement 1. Figure S3 shows pulse reconstructions from traces that were spectrally filtered by a phase-matching filter, while pulse reconstructions from traces that were measured using a relatively low-resolution spectrometer are presented in Fig. S4.

#### E. Incomplete Spectrograms

We already presented several examples of pulse reconstructions from incomplete FROG traces, both theoretically (Figs. 2 and 5) and experimentally (Fig. 4). Recalling that GP-based algorithms require complete spectrograms, it is natural to ask whether the opportunity for reconstruction from incomplete spectrograms is general or if it is limited to several examples. Thus, we tested our algorithm on many hundreds of different pulses and found that all the pulses could be reconstructed from significantly incomplete FROG traces. Figure 6 presents results from such a test. In this case, we first numerically produced a set of 100 very different pulses (number of grid points $N=128$) by the following procedure. To get a random power spectrum, we Fourier transform a complex function with a Gaussian-shaped amplitude with $\mathrm{FWHM}=150\text{\hspace{0.17em}}\mathrm{fs}$ and a pseudo-random phase (obtained by applying a moving average window of 5 points to a random vector) and then take an absolute value of the obtained complex function. Next, we multiply the obtained power spectrum by a pseudo-random spectral phase. Finally, we Fourier transform the pulse (back to the time domain) and include it in the set if its support is $\le 200\text{\hspace{0.17em}}\mathrm{fs}$. We repeated this procedure until there were 100 pulses in the set. Then, we calculated the complete FROG traces of the pulses and added 20 db white Gaussian noise. We then applied the ptychographic reconstructed algorithm on the filtered noisy FROG traces. In Fig. 6 we present results for a spectral low-pass filter and delay under-sampling that, in our opinion, are the most important for applications. Figures 6(a) and 6(b) present the mean angle of reconstruction as a function of the incompleteness parameter for frequency filtered traces and for delay under-sampling, respectively, with spectral prior information (red solid curve) and without any prior information (blue dashed curve). In our experience, $\delta <=0.1$ corresponds to very good reconstructions [Figs. 6(c)–6(v) show four reconstruction examples with $\delta \sim 0.1$]. Clearly, good reconstructions were obtained even when the FROG traces were significantly filtered, especially if the power spectrum was included as prior information.

#### F. Ptychographic Blind FROG

FROG setups can typically measure spectrograms produced by two different pulses. Indeed, in some experiments, one would like to fully characterize two potentially different ultrashort laser pulses simultaneously [34–36]. This scenario is known as blind FROG [2,37]. Assuming SHG FROG, the measured spectrogram in this case is given by

In this paragraph, we present the list of modifications in the ptychographic-based FROG algorithm (presented in Section 2) in order to apply it to blind FROG case. First, the algorithm starts with the following initial pulses: the gate pulse is transform limited (with the measured power spectrum), and the probe pulse is zero. Second, Eq. (1) is replaced by Eq. (12). The discrete form of Eq. (2) is not changed. Third, the SHG signal is defined now by ${\chi}_{j}(t)=P(t)G(t-j\mathrm{\Delta}t)$. Hence, Eq. (3) is replaced by Eq. (13):

Fourth, Eq. (8) is replaced by Eqs. (14) and (15), where both pulses are now updated in each iteration:Finally, an additional (fifth) step is added to each iteration of the algorithm. The gated pulse is projected to its measured power spectrum,

Next, we characterize the performances of our ptychographic-based blind FROG algorithm as a function of the SNR. We randomly chose 100 pairs of pulses from the set of 100 pulses that was described at the beginning of Section 3.A and was used for calculating Fig. 1. White Gaussian noise $\sigma $ was added to the simulated blind FROG trace at different SNR values. Next, we apply the ptychographic-based blind FROG algorithm to reconstruct the unknown probe and gate pulses from the noisy trace and the power spectrum of the gate pulse. Figures 7(a) and 7(b) show the mean angle (average over 100 pairs) between the reconstructed and original gate and probe pulses, respectively, as a function of the SNR. Figures 7(c)–7(h) display three examples of the recovery of the probe and gate pulses with different SNR values. Clearly, the ptychographic-based blind FROG reconstruction algorithm correctly recovered all pairs of probe-gate pulses up to a noise level defined by the SNR.

Finally, we present the experimental blind FROG pulse reconstruction using our ptychographic-based reconstruction algorithm. One laser pulse was produced directly from a Ti-sapphire laser-amplifier system. The second pulse was produced by passing a replica of the first pulse through 5 mm of glass. We measured a blind FROG spectrogram and the power spectrum of the pulse which passed through the glass. For reference, we also measured each pulse separately using FROG. The FROG and blind FROG measurements were done with 2 fs delay and 512 delay points, i.e., each trace consists of $512\times 512$ data points. The measured blind FROG trace is clearly asymmetrical [Fig. 8(a)]. We applied the ptychographic-based blind FROG algorithm and obtained the reconstructed trace in Fig. 8(b) (the NMSE between the measured and reconstructed traces is 0.092). Figures 8(c) and 8(d) show the amplitude and phase, respectively, of the first pulse recovered from the blind FROG (red dotted curve) and FROG (solid blue curve) traces. The angle between those two recovered pulses is $\delta =0.11$. Figures 8(e) and 8(f) show the amplitude and phase, respectively, of the second pulse recovered from the blind FROG (red dotted curve) and FROG (solid blue curve) traces. The angle between those two recovered pulses is $\delta =0.15$. The good correspondence between the blind FROG and FROG reconstructions demonstrates that the ptychographic-based blind FROG algorithm works well also in the experiments.

## 4. CONCLUSIONS

In conclusion, we proposed, characterized, and demonstrated, numerically and experimentally, a ptychographic-based algorithm for a FROG trace inversion. The ptychographic-based algorithm allows us to retrieve the complex pulse from a very small number of delay steps and/or from a spectrally incomplete FROG trace. In addition, we applied the ptychographic approach to blind FROG and discovered that we can robustly recover the two pulses from a single measured blind FROG trace and the power spectrum of one of the pulses. These new algorithmic capabilities will surely open many new opportunities in diagnostics of ultrashort laser pulses. The fact that our procedure can successfully recover pulses from significantly spectrally filtered spectrograms should allow us to measure ultrashort laser pulses with resolutions that are much higher than the corresponding bandwidth of the nonlinear process. For example, this feature can be useful for measuring weak pulses using thick nonlinear crystals that on the one hand yield improved SNR but on the other hand introduce phase-matching spectral filtering. Remarkably, in FROG-CRAB, it may allow the measurement of zeptosecond temporal structures [41,42] where finding a nonlinear process with the required spectral bandwidth is challenging. Notably, the ptychographic-based algorithm can significantly upgrade all FROG devices without any hardware modifications, leading to successful reconstructions of pulses that were so far unmeasurable with those devices.

While we presented here ptychographic SHG FROG, it is straightforward to revise the reconstruction algorithm to FROGs that are based on other type of nonlinearities [only Eqs. (3) and (8) need to be revised]. Figure S5 in Supplement 1 presents results using ptychographic polarization gating (PG) FROG. In addition, it should be possible to implement the ptychographic approach to interferometric FROG (IFROG) [43]. In this case, we expect that the exclusion of the Fourier relation condition in the spectrogram will facilitate the improvement of the IFROG robustness to interferometer drift.

Finally, it is worth noting more directions that may further improve and extend the scope of ptychographic pulse reconstruction algorithms: (1) we expect that utilizing the oversampling in the spectral axes can be useful for better reconstructions (the spectral resolution of most current FROG devices greatly exceeds the measured temporal window, yet this additional information is lost because the current reconstruction of GP-based algorithms requires a Fourier relation between the spectral and delay axes). (2) Algorithmic translation position determination [44] may be used for delay step correction. Its combination with a nonequally spaced delay grid may improve FROG noise robustness and acquisition speed. (3) The blind FROG can be straightforwardly modified to the case where the two pulses are known to have the same power spectrum (but different spectral phases). In this case, the two pulses could be reconstructed from the FROG trace only, i.e., without measuring the power spectrum. (4) An exciting direction is to utilize structure-based prior knowledge on the laser pulses in order to further enhance the resolution and noise robustness [45–47].

We invite researchers to download (for free) our ptychographic SHG FROG and PG-FROG from our website: http://tx.technion.ac.il/~oren/pty_FROG.html.

## Funding

Israeli Centers for Research Excellence (I-CORE), “Circle of Light”; Wolfson Foundation.

See Supplement 1 for supporting content.

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