Linear-Chirp Instability Measurement of Ultrashort Laser Pulse Trains Using Convolutional Neural Networks

1. Introduction

Ultrashort laser pulses have been extensively employed to investigate transient phenomena occurring on the femtosecond and attosecond timescales. They have also been utilized in medicine and industry for surgical, imaging, and fabrication purposes, among many others. To understand and, more importantly, control the underlying ultrafast dynamics in various applications, a knowledge of the exact electric field of the interacting pulse (i.e., its amplitude and phase) is often crucial. This objective has been addressed by the development of full-characterization techniques such as frequency-resolved optical gating (FROG). Most of these techniques operate in multi-shot mode and inevitably assume perfect shot-to-shot stability (i.e., no variations in the electric field of typically hundreds to millions of pulses in the train during the measurement time). However, even modern laser sources, especially amplified systems, used in ultrafast science are prone to instability caused by various ambient and intrinsic factors.

In the presence of instability, what is observed in a particular measurement is an “average effect” due to the interactions of many distinct pulses with the system under study, which makes the interpretation of experimental results and the manipulation of desired ultrafast processes challenging, if not impossible! Therefore, evaluating the presence, nature, and degree of shot-to-shot instability is often necessary. Two types of instabilities have been mainly studied: random-intensity-and-phase instability (RPI) and linear-chirp instability (LCI). This paper proposes a novel technique for measuring the LCI, which is essential for applications where linear chirp (second order dispersion) noticeably influences the interaction dynamics, such as generation of high-order harmonics (HHG) and isolated attosecond pulses (IAP) [1,2,3], quantum coherent control [4,5,6,7], relativistic electron beam generation [8,9], externally seeded ultrafast free electron lasers [10,11,12,13], inertial confinement fusion [14,15,16], supercontinuum generation [17,18], and ultrafast spectroscopy [19,20,21,22,23].

The instability problem has been a well-known challenge in ultrafast optics since the 1960s, when intensity autocorrelation was shown to yield a considerably shorter pulse duration than the average duration of pulses present in an unstable pulse train, due to the emergence of a narrow “coherent artifact”[24]. Full-characterization techniques are also susceptible to this issue and have been found to have wildly varying abilities to discern instability. For example, in the presence of RPI, the spectral phase interferometry for direct electric-field reconstruction (SPIDER) technique measures only the coherent artifact, with minimal indication of the instability [25,26]. Intensity autocorrelation actually does better in some sense, with coherent artifact at least sitting atop a broad background, which actually yields an approximate average pulse duration (not available in SPIDER in such cases); although separating it from the background can be difficult when the instability is slight, again yielding an underestimated pulse duration. FROG, on the other hand, does much better and has been demonstrated to provide, not only the approximate average parameters of the pulse, but also a strong indicator of the presence of instability, including LCI [27]. Indeed, recently, it has also been shown that FROG provides a quantitative measure of RPI [28].

FROG traces of pulse trains with RPI exhibit a coherent artifact, a narrow peak at the center of the trace, analogous to what is observed in the autocorrelation of such pulses. One approach to evaluate instability is to isolate this coherent artifact contribution from the remainder of the FROG trace using a modified generalized projections (GP) algorithm based on time domain extended ptychographic iterative engine (tdePIE) technique. The relative strength of the two contributions provides an insight into the degree of RPI in the pulse train [29]. Another approach is to utilize standard FROG algorithms for evaluating instability. These algorithms iteratively find a single pulse corresponding to the measured trace and use this pulse to generate a retrieved trace. If the pulse train is unstable, a considerable discrepancy between the measured and retrieved traces can arise, because the measured trace is an average formed by integrating over many distinct pulses and it is virtually impossible to find a single pulse that can result in the exact same trace. Algorithm stagnation also leads to discrepancy between the measured and retrieved traces, which makes it unfeasible to rely on such a discrepancy as an instability metric using conventional FROG algorithms. Recently, however, the RANA (Retrieved-Amplitude N-grid Algorithmic) approach [30,31] was introduced to achieve 100% reliability in FROG pulse retrieval for stable pulses. In contrast to conventional algorithms, the RANA approach does not suffer from stagnation, enabling the use of discrepancy as a reliable measure of instability. This has been demonstrated in Second Harmonic Generation (SHG) and Transient-Grating (TG) / Polarization-Gating (PG) variants of the FROG techniques for pulse trains with RPI [32,33], and in SHG FROG in the presence of LCI [27]. More recently, the R parameter, which is a generalized 2D version of the well-known statistical “Runs” test, was introduced as a quantitative measure of RPI based on the discrepancy obtained using the RANA approach [28].

Even though the retrieved spectral phases of stable and unstable pulse trains are identical in SPIDER measurements, the fringe contrast of corresponding interferograms vary slightly with the amount of LCI. Fringe contrast calculations can be performed by Fourier analysis, and the resulting fringe contrast versus frequency plots can be used to evaluate the degree of instability. This has been demonstrated for pulse trains with RPI and LCI. However, a significant challenge for using this technique is the contribution of many other sources of fringe-contrast degradation in the measured interferogram (e.g., imperfections of the SPIDER setup or imperfect spatial-mode quality). Employing a reference interferogram can circumvent this issue to some extent. But still, time-delay variations between the two replicas need to be strictly controlled for a reliable instability detection [27,34]. As a result, it is probably not practical to use SPIDER to quantitatively determine LCI or RPI.

In d-scan measurements, instability leads to broadening of the trace along the insertion axis and consequently a discrepancy with the retrieved trace. Three approaches have been used to quantify RPI in d-scan measurements including the fidelity measure, the retrieved dispersion of self-calibrating (SC) d-scan, and the mixed-states reconstruction [34,35]. The LCI quantification also has been demonstrated based on the fidelity measure [27] and retrieved dispersion of SC d-scan [36]. In the latter work experimental SC d-scan traces were used to identify and solve the instability issue of a broadband fiber laser. However, to our knowledge, d-scan has not been demonstrated for measuring complex pulses; thus, it remains unclear whether it can be used to draw quantitative conclusions about instabilities involving highly complex random pulses.

Nevertheless, despite significant progress in detecting and quantifying instability, a direct instability meter still does not exist. The term “direct measurement” here refers to obtaining a numerical value that gauges instability rapidly and non-iteratively from the measured trace or interferogram, rather than relying on post-retrieval indicators or complex metrics which may be influenced by other contributing factors. In this work, we demonstrate the first such technique, to our knowledge, based on convolutional neural networks (CNNs). CNNs have already been employed for characterization of ultrashort laser pulses and offer several advantages over traditional algorithms, including much faster speed, better performance at low signal-to-noise ratio, and the possibility of retrieval with a very partial spectrogram [37,38,39]. Despite these benefits, they have not yet been used for evaluating instability. Our proposed technique employs CNNs to measure LCI precisely and in real-time for ultrashort laser pulse trains using their TG/PG FROG traces, with no need for iterative algorithms or additional processing steps. This can be readily implemented for other types of instabilities and for various characterization techniques due to the ease of use and wide availability of deep pre-trained CNNs.

2. Materials and Methods

A transfer learning approach using the GoogLeNet architecture [40] was employed to obtain CNNs trained on simulated TG/PG FROG traces. These FROG traces of ultrashort pulses, centered at 800 nm with a transform-limited pulse duration of 5 fs and a Gaussian envelope, were simulated using a modified version of the script outlined in [41]. The foundational dataset comprised 35 unique base FROG traces, spanning seven distinct linear chirp (second order dispersion) values and five instability levels (degree of linear chirp instability). The linear chirp values were ${\phi }_2=0$ ${fs}^2$, $\pm 4.1 {fs}^2, \pm 8.2 {fs}^2$, or $\pm 12.2 {fs}^2$ (corresponding to a ${\phi }_2^{\tau }=$ $0\%$, $10\%$, $35\%$, or $70\%$ temporal broadening relative to the transform-limited pulse duration). To model linear chirp instability, each FROG trace was generated by integrating the traces of 100 individual pulses with randomly assigned linear chirp values. These values were drawn from normal distributions centered at one of the seven mean linear chirp values mentioned above (${\phi }_2$), with standard deviations of ${\sigma }_{LCI}=$ $0 {fs}^2$, $2.9 {fs}^2, 6.0 {fs}^2$, $8.8 {fs}^2$, or $12.2 {fs}^2$. These standard deviations indicate the spread of linear chirp values of the 100 pulses that form each FROG trace and are therefore a measure of instability of the pulse train. The temporal broadening percentages equivalent to these standard deviations (i.e. 0%, 5%, 20%, 40%, or 70%) are referred to herein as the “degree of linear chirp instability (${\sigma }_{LCI}^{\tau }$)”.

To enhance model generalization, data augmentation was performed by duplicating each base trace 10 times and applying random horizontal and vertical translations alongside random rotations during training. This process yielded 350 traces, referred to as the original noise-free (clean) database (OC-350) in this manuscript. Another dataset was generated by incorporating 10% additive random noise (much more noise than is typically present in such measurements) into the OC-350 dataset, which is referred to as original noisy database (ON-350). The same training options were used for both the classification and regression tasks, including the Adam optimizer, an initial learning rate of $1\times {10}^{-4}$, a mini-batch size of 64, a maximum of 100 epochs, and a validation frequency of 5. For each task, two separate CNNs were trained: one on the OC-350 dataset and the other on the ON-350 dataset. The CNNs were not provided with frequency and delay axes during any of the training performed in this work.

The left and middle panels of Figure 1 illustrate a representative subset of the ON-350 dataset. The left panel displays traces of pulse trains having a zero mean linear chirp (${\phi }_2^{\tau }=$ $0\%$). As the degree of instability (${\sigma }_{LCI}^{\tau }$) increases from 0% to 70%, the FROG traces undergo spectral narrowing and temporal broadening. Furthermore, weak features begin to emerge near the wings of the traces. This becomes more pronounced for pulse trains possessing non-zero mean linear chirp (${\phi }_2^{\tau }=70\%$), as demonstrated in the middle panel of Figure 1. FROG traces generated for pulses with a ${Sech}^2$ intensity profile exhibit the same behavior as shown in the right panel of Figure 1.

3. Results and Discussion

The original dataset (OC-350 or ON-350) was partitioned into training (80%), validation (10%), and test (10%) sets, where the test set served for preliminary assessment of the network. However, since this initial test set was derived from the 35 base traces with minor perturbations (noise, spatial shifts, and rotations), it was deemed insufficient to assess the real-world performance of the trained neural networks. Consequently, a new test dataset comprising 285 traces—which were entirely unseen by the networks during training—was generated. This unseen test dataset incorporated variations across six parameters: center wavelength ($\lambda$), pulse duration ($\tau$), linear chirp (${\phi }_2^{\tau }$), degree of instability (${\sigma }_{LCI}^{\tau }$), noise, and pulse shape (parameter ranges are detailed in Table 1).

To independently investigate the network's performance against variations in laser parameters, noise level, and pulse shape, the unseen dataset was divided into three subsets: 1) ULN-105, which utilizes unseen values of laser parameters $\lambda$, $\tau$, ${\phi }_2^{\tau }$, and ${\sigma }_{LCI}^{\tau }$, and incorporated 10% additive noise for all traces; 2) UNN-160, where various levels of additive noise were applied; and 3) USN-20 which employed a hyperbolic secant squared intensity profile instead of a Gaussian profile and traces are contaminated with 10% additive noise. To this end, all performance metrics presented in this section evaluated the trained CNNs exclusively on the unseen test datasets.

3.1. Classification

The classification CNN (Instability Classification Network 1, ICN-1), trained on the dataset with 10% additive noise (ON-350), performed well on the unseen dataset ULN-105 as shown in Figure 2. The accuracy and off-by-one accuracy were 82% and 100%, respectively. The off-by-one accuracy evaluates whether the network's predicted class falls either exactly on or immediately adjacent ($\pm 1$) to the actual class. ICN-1 exhibited its weakest performance when distinguishing between the first and second classes (0% and 5% instability), which accounted for almost half of the misclassification cases (9% out of the total 18%).

Thereafter, the classification network ICN-1 was tested on traces of ultrashort laser pulses with a ${\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{h}}^2$ intensity profile (USN-20) incorporating 10% additive noise. Since the network had been trained exclusively on traces of pulses with a Gaussian envelope, it exhibited reduced classification capabilities, achieving an accuracy of 60% and an off-by-one accuracy of 90% (see inset of Figure 2). This suggests that a network trained solely on Gaussian pulses exhibits a significantly lower generalization capacity when evaluating different pulse shapes. However, this limitation could possibly be resolved if the network were trained over a much larger dataset composed of various pulse shapes.

The classification network ICN-1 was also tested on the UNN-160 dataset and proved resilient to variations in noise levels. It performed very well at noise levels below that of the training dataset (10%), as shown in the top panel of Figure 3 (10% additive noise is quite high for PG/TG FROG, and noise levels around 5% are usually considered in the literature). The performance of the network remained resilient for up to 14% noise, but the accuracy dropped significantly for noise levels of 16% and higher. The overall accuracy and off-by-one accuracy were 79% and 100%, respectively, for the 140 traces of UNN-140 dataset with noise levels ranging from 2% to 14% in increments of 2% (i.e., the UNN-160 subset excluding the 20 traces with 16% noise). Similar to the trend observed in Figure 2, the lowest performance occurred when distinguishing between the 0% and 5% instability cases. It should be emphasized that the network was trained exclusively on data with 10% noise; if trained at higher noise levels, its performance could potentially improve for traces with 16% or higher levels of additive noise.

The network achieved an accuracy of 80% (off-by-one accuracy of 100%) on the combined ULN-105 and UNN-140 datasets. It should be noted that the accuracy on the initial test dataset (composed of seen parameters) generally ranged from 80% to 90%. The comparable performance between the seen and unseen datasets demonstrates that the network has achieved a high degree of generalization.

3.2. Regression

To serve as an instability meter, the CNN must accept a FROG trace as input and return a single numerical value representing the degree of instability of the pulse train. To achieve this, we modified our classification network to perform regression and directly predict the degree of instability. Two separate regression CNNs were trained: one on the noise-free dataset (Instability Regression Network 1, IRN-1 trained on OC-350) and the other on the dataset with 10% additive noise (Instability Regression Network 2, IRN-2 trained on ON-350). These CNNs were subsequently evaluated on the unseen test datasets ULN-105, ULC-105 (the noise-free version of ULN-105), and UNN-80 (a subset of UNN-160 incorporating only the 2%, 4%, 6%, and 8% noise levels). This specific subset of UNN-160 was selected to ensure a uniform progression of noise levels, facilitating a direct comparison between the networks trained at either 0% or 10% noise levels.

The top panel of Figure 4 shows the predictions obtained using the CNN trained on the noise-free dataset (IRN-1), while the bottom panel displays the results for the CNN trained on the noisy data (IRN-2). The blue crosses indicate the predictions for the ULC-105 and ULN-105 in the top and bottom panels, respectively. As expected, presence of noise degrades model performance. Specifically, the normalized root-mean square error (NRMSE) was 5.7% for the network trained on the noise-free dataset (IRN-1, top panel), compared to 10.0% for the network trained on noisy dataset (IRN-2, bottom panel). This indicates that, when a network is both trained and tested under low-noise conditions, superior prediction precision is achieved.

The magenta dots indicate the predictions of the two networks on the noisy datast UNN-80. It is evident in the figure that the network trained on the dataset with 10% noise (IRN-2) yields a significantly higher precision compared to the network trained on the noise-free dataset (IRN-1). In particular, NRMSE is 10.4% for IRN-2 (bottom panel), while it is significantly higher at 21.5% for IRN-1 (top panel).

These results reveal a trade-off regarding the optimal training noise level required to achieve highest precision. If the network is trained using low-noise data (top panel), it performs exceptionally well under similarly low-noise conditions but exhibits a substantially degraded performance at higher noise levels. Conversely, if higher-noise data is used for training (bottom panel), the network’s predictions for data with comparable-to-training noise level become slightly less precise, but the network demonstrates greater resilience to variations in noise levels. Optimizing network performance could potentially be achieved by training on a composite dataset spanning the full range of expected experimental noise levels which remains a subject for future investigations.

Given these findings, the CNN trained on traces with 10% noise (IRN-2, bottom panel) is a more suitable candidate for experimental measurements. This network yielded a combined NRMSE of 10.2% across a comprehensive unseen dataset including both ULN-105 and UNN-80. This strong performance indicates that the network has successfully mapped the underlying physical features in the FROG traces with various degrees of linear chirp instability.

4. Conclusions

In this work, we demonstrated a novel, deep-learning-based technique for direct measurement of shot-to-shot linear chirp instability of a pulse train using its FROG trace. By eliminating the need for iterative retrieval algorithms, sophisticated instability metrics, and complex pre- or post-processing steps, this method establishes a framework for direct, real-time instability diagnostics. The trained convolutional neural networks achieved high generalization and strong noise resilience when evaluated on entirely unseen datasets across a diverse parameter space. Specifically, the classification network (ICN-1) achieved an accuracy of 80% (with an off-by-one accuracy of 100%), and the regression network (IRN-2) yielded an NRMSE of 10.2%. The findings also reveal a trade-off regarding the optimal noise level for training: networks trained on noise-free data exhibit higher localized precision, whereas the one exposed to a higher fixed noise level exhibit broader operational resilience against various noise values. Future investigations will focus on resolving this limitation by training the network on composite datasets spanning a wide continuum of noise levels. Overall, our proposed linear chirp instability measurement approach offers an accessible, real-time, and robust tool that can be readily extended to other characterization techniques and employed for various types of instabilities.