1. Introduction

Optical pulse characterization is a necessary step of all ultrafast spectroscopy experiments performed prior to any measurement of dynamics in media of interest. The most precise and reliable pulse characterization techniques are considered those that allow for pulse retrieval from two-dimensional (2D) datasets acquired, for example, by frequency-resolved optical gating (FROG) [1,2], multiphoton intrapulse interference phase scan (MIIPS) [3], or dispersion scan (D-Scan) [4]. This reliability stems from the over-determination of the corresponding pulse-retrieval problems: a much greater number of points (of the order of $\sim N^2$) representing a spectrally- and time-resolved 2D dataset is mapped onto a much lesser number of only $\sim N$ points describing pulse electric field as a one-dimensional (1D) function of time. This over-determination of the 2D pulse retrieval problem is supported by the fact that it is possible, for example, to retrieve pulses from only a few slices of a measured 2D FROG map [5,6].

It is still, nevertheless, useful to be able to reliably reconstruct electric fields of ultrashort pulses from 1D datasets due to faster data-acquisition times (especially, when the delay stage operates in a sweeping configuration [7]) and potentially simpler experimental settings. Due to the fundamental theorem of algebra, however, pulse retrieval from a single 1D dataset is doomed to failure, and therefore additional information is required [1,8].

Amongst 1D pulse retrieval approaches, spectral phase interferometry for direct electric-field reconstruction (SPIDER) [9] is arguably the most successful technique capable of unambiguous pulse retrieval. In this approach, the spectral phase of the pulse to be characterized is encoded into a measured spectral interferogram generated by the two spectrally sheared pulses (obtained by nonlinear mixing of the two replica pulses to be characterized with a significantly chirped reference pulse). For successful pulse reconstruction, additional constraints such as calibration phase and pulse spectrum are used in the pulse-retrieval procedure. The experimental implementation of the conventional SPIDER method, however, is complex, involves two delay stages and requires highly precise alignment. Other methods based on measuring triple intensity correlations [10] also uniquely yield pulse shapes, but also at an expense of increased complexity.

Acquiring 1D time-domain traces of intensity correlations between two pulses only, on the other hand, is very easy to implement experimentally (as only one delay stage is required), and numerous attempts to reconstruct pulse shapes from such traces were previously made [11–18]. Peatross et al. [18] were able to reconstruct electric fields of ultrashort pulses from intensity autocorrelation traces using the Gerchberg-Saxton algorithm [19]. They note, however, that their approach suffers from non-trivial ambiguities as more than one set of electric-field spectral amplitudes and phases can lead to essentially the same autocorrelation trace. Moreover, their algorithm does not outperform the iterative FROG algorithm [1] or recently reported common pulse retrieval algorithm (COPRA) [20], and can stagnate. Later, Chung and Weiner [21], numerically proved that significantly different pulse shapes can lead to exactly the same intensity autocorrelation.

It was also shown previously that in cases where different pulse shapes lead to same intensity autocorrelation traces, interferometric autocorrelation traces are similar but not equivalent [21]. This is a consequence of the fact that interferometric correlations are also sensitive to the phase of electric field, which inspired attempts to find sufficient number of constraints to solve the corresponding 1D pulse-retrieval problem. Diels et al. [11,12] suggested that laser-pulse spectrum together with second-order intensity and interferometric autocorrelation traces (acquired, for example, by mixing two pulses in a nonlinear crystal via second-harmonic generation) are sufficient to reconstruct pulse electric field, although no proof was given that their iterative algorithm yielded unique solutions. Later, a technique named as the Femto-nitpicker was introduced [13,17]. This technique was capable of retrieving pulse shapes from interferometric autocorrelation traces, which were supplemented by interferometric cross-correlation (IXC) measurements obtained when a thick glass block was inserted into one of the interferometer’s arms. However, the algorithm assumed only broadening effects of the glass block and the accuracy of their iterative algorithm was lower for thinner glass blocks.

Naganuma et al. suggested that for successful 1D pulse retrieval it is sufficient to measure intensity autocorrelation, second-harmonic interferogram, and field (first-order) interferogram [14]. Experimentally, they have also supplemented interferometric autocorrelation measurements with those performed with a piece of glass inserted before the interferometer [15] or in one of its arms [16]. The iterative algorithm tended, however, to stagnate, and their experimental approach did not find applications [8]. Later reports by Nicholson et al. [22,23] introduced the PICASO (Phase and Intensity from Correlation And Spectrum Only) method, capable of retrieving pulses from the pulse spectrum, phase-response of the dispersive element, and interferometric cross-correlation trace acquired with an unbalanced Michelson interferometer (where a piece of glass was used in one of its arms). We notice that in this method different initial guesses for phase greatly affected the time required for the pulse retrieval. Most recently, PENGUIN (Phase-Enabled Nonlinear Gating with Unbalanced Intensity) has been demonstrated, capable of retrieving pulses from interferometric cross-correlations, where the two pulses differed in intensity [24]. In this case, the resulting cross-correlation trace can be effectively decomposed into a Newton’s binomial, containing terms corresponding to cross-correlations between various field combinations. Some of the terms were shown to carry phase information in this case. However, this method is applicable only to low-dispersion pulses close to their Fourier-transform limit.

The listed attempts, nevertheless, suggest that such pulse characteristics as the amount of chirp, pulse amplitude time profile and frequency variation within the pulse all leave their fingerprints on interferometric correlation traces. It is, therefore, desirable to be able to use these fingerprints to reconstruct pulse shapes in a reliable and consistent manner from measurements with the simplest experimental arrangement. Convolutional neural networks (CNNs) [25–27] were proven to be extremely powerful in recognizing fingerprints and patterns in large non-trivially complicated datasets, and therefore they could also aid pulse retrieval from 1D interferometric correlation traces.

Previously, neural networks were shown to be successful in pulse retrieval from 2D datasets: FROG traces [28–31], D-Scan traces [32], speckles at the output of a multi-mode fiber [33], and streak traces characterizing attosecond pulses [34]. However, neural-network-assisted reconstruction of ultrashort pulses from 1D interferometric correlation datasets, to the best of our knowledge, have not been demonstrated yet, and our proof-of-principle work aims to fill this gap, as summarized below.

Following developments described above, here we investigate, with the help of CNNs, the question of ambiguity of the retrieval of ultrashort pulse shapes from interferometric correlation traces. We consider a more general case of interferometric cross-correlation trace (with an arbitrary unknown reference pulse) supplemented with spectra of both pulses and an interferometric cross-correlation trace obtained when the pulse to be characterized is affected by a glass plate. This supplemental data serves as the set of additional constraints to the 1D pulse retrieval problem. We synthesize a large dataset to train a neural network, and apply the trained model to the experimental data obtained using the dispersionless transmission-grating-based Michelson interferometer [35]. We demonstrate that it is possible to unambiguously retrieve electric fields of ultrashort pulses from the above-mentioned 1D datasets in as short as $\sim$1.1 milliseconds. Importantly, no other knowledge than spectrum (which partially overlaps the spectrum of the pulse to be characterized) is required of the reference pulse; instead, we treat the interferometric cross-correlation measurement (without the glass plate) itself as a reference measurement.

2. Data preparation for neural-network training

Numerical interferometric intensity cross-correlation traces ($I_{\textrm {IXC}}$) are generated via the expression

 

Fig. 1. Generated spectra and spectral phases of (a) the reference pulse and (b) the pulse to be characterized; time-domain electric fields of (c) the reference pulse, (d) the pulse to be characterized, and (f) the pulse to be characterized passed through a 2-mm fused silica glass plate; interferometric cross-correlation (IXC) traces with the 2-mm fused silica plate (e) out and (g) in.

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In our work, we exploit sensitivity of machine-learning models to the presence of ambiguities in training datasets [36,37]: a simple neural network, for example, will converge well if the dataset used for its training is unambiguous, and the same neural network will not be able to make good predictions if the dataset contains ambiguities. Therefore, we make sure that all ambiguities are completely eliminated from the dataset. During simulations, we avoid all trivial ambiguities that may be present in correlation traces [1]: (i) the direction-of-time ambiguity is intrinsically absent by the nature of cross-correlation measurements, where time symmetry is broken; (ii) the addition-of-constant-spectral-phase ambiguity is eliminated by shifting spectral phase so that it would correspond to 0 radians at the spectral peak; (iii) the time-shift ambiguity is avoided by subtracting the linear components of the spectral phases from overall phase profiles.

After having eliminated the trivial ambiguities, the only possible ambiguities left are those referred to as non-trivial. In order to avoid non-trivial ambiguities and characterize the second pulse unambiguously (without the knowledge of the reference pulse), we influence its spectral phase in a predictable manner by making it propagate through a glass plate (e.g., fused silica) of certain thickness $d$, dispersion of which is well-documented [38]. As a result of such propagation, additional phase $kd$ is added to the overall spectral phase of the pulse, with the wave vector defined as $k=n(\omega )\omega /c$, where $n(\omega )$ is the frequency-dependent refractive index of the glass, $\omega$ is the angular frequency, and $c$ is the speed of light. The dispersion $n(\omega )$ of the refractive index was estimated with the help of the Sellmeier equation for fused silica (see Supplement 1, Section S2) [38,39]. The resultant electric field of the modified second pulse in the frequency domain was then Fourier-transformed to yield the electric field in the time domain affected by the glass plate (Fig. 1(f)). The corresponding cross-correlation trace obtained with this stretched pulse is shown in Fig. 1(g).

As clearly indicated previously [14,21], minimum three measurements are required to retrieve the pulses from 1D datasets (although the pulse retrieval from only two measurements has also been reported [40]), and adding extra information can only lead to a better convergence of an iterative algorithm and a higher level of its reliability. Here, four 1D datasets – spectra of both pulses and two interferometric cross-correlation traces obtained with the glass plate in and out of the interferometer’s arm – were used to reconstruct the electric field $E_?(t)$ of the pulse of interest. Examples of such 1D traces are shown in Figs. 1(a,b,e,g). The choice of inclusion of spectra of both pulses as additional constraints was motivated by the dependence of the fringe contrast of interferometric cross-correlation traces on the amount of spectral overlap between the two pulses: the greater the spectral overlap between the pulses, the larger the fringe contrast in the corresponding interferometric traces. The spectral content of the pulses and the extent of their spectral overlap, therefore, should serve as an additional guide for the neural network to reconstruct pulses more reliably. The four 1D datasets also imply the simplest experimental setting, in which only a single-element detector for time-resolved measurement, a spectrometer, and one delay stage are used.

Finally, in order to take advantage of CNNs in solving such 1D pulse-retrieval problem, and to exploit their three-dimensional (3D) convolutional filters capable of efficient characterization (encoding) of patterns in input datasets, we transform the four mentioned 1D datasets into 3D data-structures via two transformations referred to as reshaping and stacking. Reshaping was performed using the functionality of NumPy [41]. Shortly, in the process of reshaping, all four 1D datasets (e.g., those shown in Figs. 1(a,b,e,g)) each containing $N$ values are split into $W$ pieces of length $H$, where $N=W\times H$. Each of these pieces are then used to form columns of a matrix representing a 2D image. As a result, four images of size $W\times H$ are formed (Figs. 2(a–d)). In the stacking procedure, we concatenate these four images along the third dimension forming the depth of the 3D data-structure, which we refer here to as the number of channels $C$ following convention in the machine-learning community. Structurally, a single data-sample, therefore, is a 3D data-cube of size $W\times H\times C$ with the number of channels $C$ representing the amount of one-dimensional datasets used to retrieve pulses. Each such 3D data-sample is supplemented by the corresponding electric field of the pulse of interest (e.g., the one shown in Fig. 1(d)), which is referred to as label and serves as a guide for the neural network during the learning period (see more examples of data-samples in Supplement 1, Section S3).

 

Fig. 2. Spectra of the reference pulse (a), and of the pulse to be characterized (c), reshaped into 2D images with dimensions $W\times H$ (see the text for details); interferometric cross-correlation traces with the 2-mm fused silica plate out (b) and in (d), reshaped into 2D images with dimensions $W\times H$. (e) Resultant 2D images stacked to form a three-dimensional data-sample of size $W\times H\times C$ with $W=H=40$ and $C=4$. The colormap in (a,c) is the same as in (b,d) but with white color corresponding to 0 and the darkest shade of red corresponding to 1.

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In our machine-learning experiments, we used 100,000 such data-samples. We split this data into three parts: (i) training dataset consisting of 80,000 data-samples, (ii) development dataset consisting of 19,000 data-samples, and (iii) test dataset consisting of 1,000 data-samples. We used the training dataset to train the neural network, and the accuracy of the prediction of the labels from the development dataset as a feedback parameter to improve our network’s predicting ability, which will be discussed in more detail in the next section.

3. Neural-network architecture and training

The neural-network architecture used in this work was implemented in Python with the help of the PyTorch machine-learning framework [42], and, as mentioned above, is based on CNNs [25–27]. We adopted the simplest conceptual design of CNN with the two main parts: an encoder and a decoder (Fig. 3). The encoder (consisting of convolutional layers) encodes features contained in the training dataset into an abstract feature-space, and during the training stage tries to generalise trends observed in the training dataset. The decoder (consisting of fully-connected layers), in its essence, is a classical neural network consisting of multiple perceptrons [43] (a.k.a. neurons) applied previously, for example, to the problem of hand-written-digit recognition [44]. The decoder guided by the labels in the training dataset uses the information from the output of the encoder in order to retrieve particular information of interest (electric fields in our case). As the neural network learns, it compares its current predictions with these labels, and tries to adjust its learnt parameters by means of the (mini-batch) gradient-descent algorithm [25,26] (we used the Adam optimization algorithm [45] as the corresponding gradient-descent-based optimisation routine). As a feedback for this parameter adjustment procedure, at each iteration the squared $L_2$-norm ($L$) over the training dataset is calculated as

 

Fig. 3. Schematic representation of the convolutional-neural-network architecture using diagrammatic convention used in the deep learning community. Within the "ENCODER", each block represents a convolutional layer (labeled as ’CONV’) with the first block representing the input data-sample. Each next block represents the result of convolution performed on the corresponding previous block. The depth of the sequential blocks changes and corresponds to the number of convolutional filters used on the previous block. The green sub-blocks represent the size of the convolutional filters used. Each block also encapsulates operations of possible batch-normalization and application of an activation function. Within the "DECODER", each block represents a fully-connected layer (labeled as ’FC’) of a classical fully-connected neural network with the last block corresponding to the predicted output.

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We trained the neural network over 1,000 training runs (epochs) on a computer equipped with the Titan RTX graphic processing unit (GPU) with up to 130 teraFLOPS (floating-point operations per second) and 24-gigabyte memory. Training-time rate is estimated to be $\sim$83.3 sec/epoch, while after CNN training is complete the pulse retrieval takes as short as $\sim$1.1 milliseconds.

4. Results and discussion

4.1 Prediction using simulated data

Figure 4 illustrates the results of neural-network predictions on the simulated data. Figure 4(a) shows an example of the neural-network-predicted electric field from the test dataset (see more prediction examples in Supplement 1, Section S5). It is clear that the electric field envelope and phase are captured by the neural network with high accuracy, even though the shape of the pulse is not trivial, with multiple peaks in the electric field amplitude. After 1,000 epochs of training, the $L_2$-norms (Eq. (2)) over the training, development and test datasets were achieved to be as low as $4.98\cdot 10^{-3}$, $5.08\cdot 10^{-3}$, and $4.11\cdot 10^{-3}$, respectively. These numbers reflect the prediction error integrated over 1,600 time-points and averaged over the corresponding dataset so that the average root-mean-square (rms) error per time-point per data-sample is estimated to be of the order of 2$\cdot$10$^{-3}$ (rounded value for all three datasets). Considering the normalization of the total pulse electric field ($|E_{?,i,j}|\leq 1$) this level of rms-error is very small and underlines high prediction accuracy, which in turn signifies that there are no ambiguities present in the generated dataset. This high prediction accuracy is expected as all trivial ambiguities are eliminated by the construction of the dataset and by the nature of cross-correlation traces facilitating unambiguous mapping of the input data to the corresponding pulse electric field. If there were ambiguities in the generated dataset, it would not be possible to achieve good predictions with the current neural-network architecture by design. Figures 4(e,f) show the error calculated from the predictions over the development dataset, which the neural network has not seen during training. There is a steeply-decaying learning curve during the first three epochs of training when the network learns phase and high-amplitude part of the pulse envelope, followed by a gradual decay of the error over the rest of the epochs during which low-amplitude parts (such as wings) are learned by the CNN. This convergence of the error to ever smaller values is an additional indication that there are no distinguishable ambiguities present in our approach of pulse characterization. It is worth noting that this behavior could also result from overfitting of the training dataset. We argue, however, that this is not the case here. First, the achieved rms-errors are very similar between all three datasets (training, development, and test) indicating that our model is able to generalize, which in turn points at the lack of ambiguities and therefore at the uniqueness of the solution (within the accuracy achieved). Second, we explicitly searched for the non-trivially ambiguous pulses (i.e.,different pulse shapes yielding essentially the same interferometric auto-correlation traces) similarly to what was reported previously [21]. Our search algorithm was only able to converge to very similar pulses up to the time-reversal symmetry (see details in Supplement 1, Section S6).

 

Fig. 4. Actual (solid line) and neural-network-predicted (dashed line) electric field from the test dataset in the case of (a) the noise-free test dataset, and when white Gaussian noise with (b) SNR = 20, (c) SNR = 10, and (d) SNR = 5 was added to the dataset. (e,f) Convergence curves of the prediction error versus the number of epochs. Red dots mark those epochs where new minima of the $L_2$-norm (Eq. (2)) were achieved during the optimization procedure. The horizontal and vertical axes in (f) are given in logarithmic scale.

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To test the robustness of our neural-network architecture to the presence of noise we added white Gaussian noise (see details in Supplement 1, Section S7) to the simulated test dataset, and applied the already-trained network (trained on noise-free training dataset) to the data from the test dataset with three different noise levels. We note that we have not retrained the network and used the model obtained from the noise-free data. Figures 4(b,c,d) show the predicted electric fields when the training dataset was distorted by noise yielding signal-to-noise ratio (SNR) of 20 (Fig. 4(b)), 10 (Fig. 4(c)), and 5 (Fig. 4(d)). In all cases, the accuracy of the predictions is reasonably high, with the $L_2$-norm (Eq. (2)) of the order of $10^{-3}$ (and the average rms per time point of the order of 2$\cdot$10$^{-3}$). In the case of SNR = 20, the $L_2$-norms over the training, development and test datasets were achieved to be as low as $4.98\cdot 10^{-3}$, $5.08\cdot 10^{-3}$, and $4.11\cdot 10^{-3}$, respectively; in the case of SNR = 10, the $L_2$-norms were obtained to be as low as $4.99\cdot 10^{-3}$, $5.10\cdot 10^{-3}$, and $4.12\cdot 10^{-3}$, respectively; and in the case of SNR = 5, the $L_2$-norms were obtained to be as low as $5.13\cdot 10^{-3}$, $5.24\cdot 10^{-3}$, and $4.29\cdot 10^{-3}$, respectively. Although for lower SNRs, the errors have increased, these increases are marginal indicating high resilience of our neural-network model to noise. We observed, however, that in the case of SNR = 2, the neural network fails to make accurate predictions (not shown in the manuscript).

4.2 Prediction using experimental data

Figure 5 summarizes the result of application of the CNN model to measured data. Here we consider a special case of two identical pulses ($\sim$12 fs), which come from the same laser source (lab-built noncollinear optical parametric amplifier). The data was acquired with a table-top spectrometer (Avantes) and the recently-reported fully-symmetric Michelson interferometer [35]. The measured data was used to form the neural-network input data-sample and is shown in Figs. 5(a–c). Since the pulses were identical, we used the same spectrum (Fig. 5(a)) for the first two channels of our input experimental data-sample. The interferometric auto-correlation trace of the pulse to be diagnosed used to form the third channel of the data-sample is shown in Fig. 5(b). For interferometric cross-correlation measurements with a phase-modulated pulse we break the symmetry of our interferometer by inserting a 1-mm-thick fused-silica glass plate in front of the spherical mirror in the interferometer’s second arm (see Supplement 1, Section S8, for more details). The interferometric cross-correlation trace obtained in this case was used to form the fourth channel of the data-sample and is shown in Fig. 5(c). The pulse in the second arm of the interferometer passes through the fused-silica glass plate twice and should therefore be affected similarly to simulation of the numerical dataset.

 

Fig. 5. (a–c) The experimental data used as input to the neural network: (a) spectrum of both pulses in the two arms of the interferometer; (b,c) interferometric cross-correlation trace obtained with the fused silica glass plate (b) out of and (c) in the beam (in one arm of the interferometer). The traces in (b,c) were obtained from corresponding frequency-resolved data by integrating along the frequency axis. (d) Pulses retrieved with the neural network (dashed) and with the COPRA [20] (solid blue). (e) Pulses retrieved with the neural network (dashed) and with the PICASO [22,23] (solid red) methods.

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The 1D interferometric cross-correlation traces were derived from 2D interferometric FROG (iFROG) traces by integrating them along the frequency axis. We acquired iFROG traces (instead of 1D correlation traces) to be able to compare the neural-network retrieval results with those obtained using recently reported COPRA pulse retrieval algorithm [20], which operates on frequency-resolved data. The iFROG traces were acquired by recording second-harmonic signals generated in a 5-$\mu$m-thick beta barium borate (BBO) crystal. The traces were recorded over the 600-fs range of time delays (between the two pulses in the two interferometer’s arms), with 0.1 fs time step and 50 ms integration time. The two pulses from the output of the interferometer were focused by an off-axis parabolic mirror (with focal length of 20.3 mm) onto the BBO-crystal. When recording both the fundamental spectrum of the pulses and the second-harmonic spectrum during iFROG measurements, the spectrometer was placed after the nonlinear crystal. The recorded experimental data were pre-processed to match the standard for inputs that the neural network accepts, as defined above (see Supplement 1, Section S9, for more details on the processing of experimental data).

Figure 5(d) shows the pulse field retrieved using our CNN and the COPRA [20]. We obtained second-harmonic-generation FROG (SHG-FROG) by filtering out fringes (along the time axis) from the measured iFROG trace leaving only static Fourier-component, and used the resultant SHG-FROG trace to feed the COPRA algorithm (see Supplement 1, Section S10, for more details). Although the two retrieval results exhibit small mismatches (such as flipped phase at the left wing of the pulse and amplitude variations), the general pulse shape is captured very well: both retrieval approaches show pulses of similar duration and carrier frequency. The apparent mismatches between the two pulse-retrieval results likely have multiple reasons. First of all, we notice that in the measured auto-correlation trace (Fig. 5(b)) the background-to-peak ratio deviates from the theoretical value of 1:8, likely due to small alignment imperfections. This effect was not present in the simulated dataset and therefore may fall slightly outside of the statistical distribution of the numerical dataset used for training the neural network. Further, the insertion of the glass plate in one arm of the interferometer relaxes otherwise-ideal 4f-imaging of the pulses back onto the grating-based beam-splitter, and therefore the interferometric cross-correlation trace may not exactly correspond to the ideal case of pulse propagation through the 2-mm glass plate. Finally, filtering the static component as well as the first and second harmonics constituting 1D correlation traces from noise may have contributed to distortions related to a particular filtering window shape. All of these possible reasons could be taken into account in the process known as data-engineering and will be the ground for our future work.

Despite all of these factors, our model could still retrieve most of the features of the pulse shape indicating that it is possible to characterize ultrashort pulses rapidly and reliably from 1D interferometric traces. The neural network is capable of predicting variations in the pulse shape capturing such details as multiple amplitude peaks, if any, and minor oscillations at the tails of the electric field envelopes. This could be further improved by retraining the network on larger training datasets with some data-samples generated during comprehensive data-engineering (taking into account many more specific cases that can be encountered in various experiments and the general sources of experimental imperfections) as well as by more sophisticated neural-network architectures. Our approach is therefore scalable: various glasses (e.g., BK7) of various thicknesses can be used as well as other frequency ranges and pulse durations, which all could be considered within the statistical distribution of the training data-samples.

Lastly, we directly compare the neural-network retrieval result with that obtained using implementation of the PICASO algorithm [22,23] (see Fig. 5(e)) where the Optuna’s Tree-structured Parzen Estimator [46,47] was employed as an optimizer. Both retrieval approaches use the similar datasets, and not surprisingly largely agree with each other, although PICASO could not retrieve wings of the pulse shape as well as they were retrieved by COPRA or CNN. In our retrieval attempts using PICASO we also observed the tendency of this method to stagnate occasionally.

5. Conclusions

In conclusion, we demonstrated the proof-of-principle approach to ultrashort-optical-pulse characterization from 1D interferometric correlation time traces using a deep neural network. We show that it is enough to measure interferometric cross-correlation between the two pulses, interferometric cross-correlation between the pulses with one of them propagating through a dispersive element, and spectra of both pulses. Generally, no knowledge about the reference pulse other than its spectrum is needed. All of this implies a simple experimental setup with only one delay stage, a single-channel detector, and a spectrometer, which together with milliseconds-long pulse retrieval makes ultrashort pulse characterization a fast procedure. This could be further improved up to real-time pulse characterization technique by smart design of the experimental setup where both cross-correlation traces are acquired together and in a real-time acquisition using sweeping-mode of the delay stages [7].