Intelligent single-shot full-field characterization over femtosecond pulses

Domain: physics
Published: 2025-11-25
Editorial outcome: minor_revision

Abstract
Conventional approaches to femtosecond (fs) pulse characterization depend on nonlinearities without exception, thereby impeding their utilization in measuring weak fs pulses. Their reliance on iterative phase retrieval or spectrometer-based readout further prevents single-shot full-field characterization over high-repetition-rate fs pulse trains. Here, we present linear spectral shearing interferometry (LSSI) based intelligent single-shot full-field characterization (ISFC) towards high-repetition-rate fs pulse trains, where both intensity and phase of fs pulses are reconstructed in single shot from temporal interferograms through a well-trained fully-connected neural network. The spectral shear in LSSI is created completely by linear effects enabling measurement of weak fs pulses. We validate the proposed method on a megahertz fs pulse train with picojoule-level energy. Moreover, the switching dynamics of a programmable spectral filter are successfully resolved with the proposed method at a megahertz frame rate. We anticipate LSSI-based ISFC to be a particularly powerful tool for characterizing weak ultraviolet fs pulses, and even attosecond pulses with high repetition rates.

Introduction
Accurate characterization over femtosecond (fs) pulses is vital to various utilizations of fs lasers, incorporating ultrafast spectroscopy, high-precision ranging, nonlinear optics, and attosecond science. A subtle idea for fs pulses characterization is to split the pulse under characterization into two arms sent to a nonlinear crystal for optical autocorrelation, which can only shed approximate temporal information of the pulse. Frequency-resolved optical gating (FROG) is then proposed to spectrally resolve the optical autocorrelation signal, and a two-dimensional spectrogram (i.e., the FROG trace) is recorded. The iterative-Fourier-transform algorithm can be applied for retrieving both the intensity and phase (i.e., full-field characterization) of the pulse from the measured FROG trace. Time lens and dispersive Fourier transform (DFT) are utilized together for full-field characterization over picosecond pulses by performing the iterative Gerchberg-Saxton algorithm between temporally magnified pulse profiles and real-time spectra. Another prominent approach to non-iterative full-field characterization over fs pulses and even attosecond pulses is spectral phase interferometry for direct electric-field reconstruction (SPIDER), where the spectral shear is created by sum-frequency generation in a very sophisticated setup. In recent years, phase-preserving nonlinear autocorrelation and nonlinear-ultrafast-temporal gate-based direct sampling have been demonstrated to deliver full-field characterization over fs pulses. However, the above methods all rely on nonlinearities, thus the demand on the pulse energy is non-negligible, especially when it comes to single-shot measurement (e.g., the microjoule level for FROG). On the other hand, most of the above methods require iterative algorithms for phase retrieval, preventing single-shot full-field characterization over fs pulse trains with high repetition rates. SPIDER delivers non-iterative phase retrieval, but the spectral interferogram readout through the spectrometer imposes restrictions on its measurement speed.

Linear spectral shearing interferometry (LSSI) is utilized for characterization over ultrashort pulses and also delivers direct phase retrieval. The spectral shear in LSSI is introduced by mapping the temporal delay to a spectral frequency difference through DFT, and the measurable pulse energy can be substantially reduced due to the lack of nonlinearities. An implicit but imperative advantage of LSSI is that the interferogram generated in spectral shearing interferometry of a pulse train is converted to consecutive temporal interferograms, thereby allowing single-shot characterization. Pulse reconstruction through the Fourier-transform-based method (FTM) in LSSI extremely depends on precise calibration over the time delay of the interferometer and dispersion parameters of DFT, which fluctuate subjected to environmental disturbances. The nonlinear spectral-to-temporal mapping induced by inevitable high-order dispersions can further deteriorate the performance of the FTM when reconstructing pulses with shorter durations. Therefore, accurate single-shot full-field characterization over fs pulses using LSSI in an open environment remains a tough challenge. Until now, there is no single-shot full-field characterization over fs pulse trains with high repetition rates and low energy, therefore limiting the widespread applications of fs pulse characterization.

Here, we combine LSSI and artificial intelligence to provide an effective solution, LSSI-based intelligent single-shot full-field characterization (ISFC), to achieve single-shot full-field characterization over fs pulse train with the high repetition rate and low energy, where fs pulses are reconstructed in single shot from the acquired temporal interferograms in LSSI through a fully-connected neural network. Data-driven ISFC can learn the delay and dispersion values and even their fluctuations through massive data, thereby waiving the laborious calibration of these parameters. Numerical simulations indicate high-order dispersions evoked by nonlinear spectral-to-temporal mapping barely impair pulse reconstruction of ISFC, and ISFC allows to use smaller shear values than the FTM, thereby lowering requirements on the sequential temporal acquisition device (e.g., the bandwidth, the sampling rate).

We experimentally validate LSSI-based ISFC for characterizing a homemade programmable fs pulse laser source. It turns out LSSI-based ISFC realizes single-shot full-field characterization over various fs pulses at the average pulse energy of merely ~296.6 pJ. Moreover, ISFC provides decent pulse reconstruction under low sampling rates. Given the sampling rate of 3.13 GSa/s, the switching dynamics of the programmable spectral filter are captured and resolved by ISFC in a single-shot full-field manner at the frame rate of ~32.4 MHz. Note that nonlinearities generation in the ultraviolet band is rather difficult due to the strong absorption of common nonlinear crystals in the ultraviolet band. Hence, LSSI-based ISFC, entirely relying on linear effects, is very promising in characterizing weak ultraviolet fs pulses and even attosecond pulses. On the other hand, LSSI-based ISFC could serve as a powerful tool for capturing fast, non-repeatable events at high frame rates.

Results
ISFC over fs pulses via LSSI
Figure 1 conceptually illustrates the LSSI-based single-shot full-field characterization over fs pulses via conventional FTM and ISFC. The pulse under characterization is first sent to an interferometer, where spectral interference of the pulse under characterization and its time-delayed replica occurs. Then, the time delay is converted to a spectral shear through the temporal-to-spectral mapping built on DFT. The LSSI consists of the interferometer and DFT. Unlike the spectral shear created by nonlinearities in SPIDER, the spectral shear here is created by two linear effects only, interferometry and dispersion. Since the LSSI only relies on linear effects, the order of the two elements can be swapped. When the spectral shear is introduced, the other subtle but imperative effect of DFT is that the spectral fringes are also mapped as temporal interferograms. As a result, consecutive temporal interferograms can be acquired by a high-speed real-time oscilloscope after photodetection, thereby enabling single-shot pulse characterization. In LSSI, the spectral shear created by the time delay and DFT can be expressed as

Δω = Δt / (β2 L)

where Δt is the delay, β2 and L are the group velocity dispersion and length of the dispersion medium, respectively. The interferogram after photodetection is derived as

Iint(t) = |A(ω)e^{jφ(ω)} + A(ω+Δω)e^{jφ(ω+Δω)}e^{jωΔt}|^2

where A(ω) and A(ω+Δω) are the spectral amplitude of the original pulse and the shear pulse, respectively. Δφ(ω) is the spectral phase difference between the original pulse and the shear pulse. The spectral shear is very small compared to the spectral bandwidth of the pulse under characterization. Hence, A(ω+Δω) approximately equals to A(ω) and the spectral phase can be retrieved by

φ(ω) = ∫ Δφ(ω) dω / Δω

Therefore, each sideband of Iint(t) centered at the shear includes the desired spectral amplitude and phase information. Conventional FTM first transforms the temporal interferogram back to the spectral domain via the temporal-to-spectral mapping built on DFT, where accurate calibration over dispersion values of DFT is required. Then, one sideband centered at Δt or -Δt is fetched via inverse Fourier transform, and the linear phase term ω·Δt is removed by using the calibrated delay of the interferometer. Thus, the spectral amplitude A(ω) and phase derivative Δφ(ω) can be obtained. Spectral phase φ(ω) can be calculated via integration on Δφ(ω) and thus the spectral field of the pulse under characterization is reconstructed. The corresponding temporal field can be obtained by performing inverse Fourier transform on the reconstructed spectral field.

However, due to the wide spectrum of the fs pulse, high-order dispersions, especially the third-order dispersion (TOD), induced nonlinear temporal-to-spectral mapping is non-negligible. Hence, precise TOD measurement of the dispersive medium is vital to regulate the nonlinear temporal-to-spectral mapping when using the FTM to reconstruct fs pulses. On the other hand, owing to the integration operation in phase retrieval, the FTM is extremely sensitive to shear calibration. The bias in shear calibration leads to a linear slope in the spectral phase derivative Δφ(ω) and the linear slope is further converted to a quadratic phase term (i.e., linear chirp) in the retrieved spectral phase φ(ω), thereby distorting the temporal pulse shape. Precise shear calibration requires accurate measurement of delay and group velocity dispersion. Tracking delay in real time is proposed to alleviate the errors in the calibration of the group velocity dispersion, the fluctuations of which are non-negligible in an open environment. Therefore, accurate full-field characterization over fs pulses using LSSI via the FTM in an open environment remains a tough challenge.

To circumvent the drawbacks of the FTM, we propose ISFC, where a fully-connected neural network (see Methods for details) reconstructs full-field information of the pulse under characterization from the temporal interferogram in single shot. The delay, dispersion values, and even fluctuations of these parameters of an LSSI system are included in massive data, which can be learnt by ISFC. Hence, data-driven ISFC omits the laborious calibration of the delay and dispersion values. ISFC even manifests decent robustness against fluctuations of key parameters. ISFC is designed to output the interleaved real and imaginary parts of the temporal pulse, and the temporal field can be readily reconstructed. We found that the interleaved sequencing outperformed concatenating the intensity and phase. Through the Fourier transform, the spectral field of the pulse can also be reconstructed. It is equivalent for ISFC to predict the spectral field first and then obtain the temporal field via the Fourier transform.

Numerical simulations of ISFC over fs pulses via LSSI
We construct a numerical model to emulate the LSSI system and generate simulation datasets for training ISFC (see Methods for details). Since LSSI only relies on linear effects, nonlinearities are not considered in numerical simulations for efficiency improvement. Figure 2a shows the pulse duration retrieval accuracy in the simulation dataset, where TOD in DFT is considered. The FTM without TOD compensation (i.e., FTM wo TODC), where TOD is not considered in spectral interferogram reconstruction, performs worst in the duration retrieval accuracy as expected. Because the nonlinear spectral-to-temporal mapping effect induced by TOD is more severe on shorter pulses with wider spectra, it is obvious that the shorter pulses suffer larger errors. After compensating for the nonlinear mapping induced by TOD in spectral interferogram reconstruction, the situation is well alleviated as shown in the duration retrieval results of the FTM. When the pulse becomes even shorter, the conventional TOD compensating method cannot completely compensate for the TOD-induced nonlinear mapping effect. As shown in Equation (S1) in Supplementary Information, the high-order dispersions induced nonlinear spectral-to-temporal mapping can be expressed as a polynomial combination between the temporal and spectral coordinates, with the coefficients determined by the dispersion parameters. Thus, the nonlinear polynomial relation can be accurately learnt by ISFC through training on massive data, waiving precise calibration on the high-order dispersion parameters. As a result, ISFC performs perfectly in the entire pulse duration range. Almost all the retrieved durations of ISFC lie in the 10% error range (i.e., indicated by the shadow area). The mean absolute errors (MAEs) of the retrieved durations on the test set of FTM without TOD compensation, FTM, and ISFC are, respectively, 336.83 fs, 127.23 fs, and 8.85 fs. The proposed ISFC is ~14 times better than the conventional FTM in terms of duration retrieval accuracy. Figure 2b demonstrates the reconstructed results of a 130 fs pulse by the three algorithms. In the temporal pulse retrieved by FTM without TOD compensation, the tail oscillations induced by TOD are obvious. With TOD compensation, the tail oscillations are substantially mitigated, and the pulse duration is much closer to the actual duration, but both the temporal intensity and retrieved phase are still far from the ground truth. The reconstructed pulse via ISFC is consistent with the ground truth in both temporal and spectral domains, thereby validating the salient performance of ISFC in compensating the nonlinear spectral-to-temporal mapping effect induced by high-order dispersions.

To evaluate the performance on retrieving the temporal envelope, the duration normalized mean absolute error (NMAE) is introduced and defined as

Γerr = 1/N * Σ |τn - τ^n| / τn

where τn and τ^n are the retrieved and actual pulse duration for the n-th sample in the test set, respectively, and N is the size of the test set. Normalized root-mean-squared error (NRMSE) is used to quantitatively assess the temporal phase retrieval ability, and it is defined as

Φerr = sqrt(1/N * Σ Σ |ϕn(k) - ϕdn(k)|^2 / K * ϕeffn^2)

where ϕn(k) and ϕdn(k) denotes the k-th point in the retrieved and actual temporal phase curve of the n-th sample in the test set, respectively, K is the number of points in the phase curve. ϕeffn represents the effective phase range of the n-th actual temporal phase curve in the test set, which is the absolute phase range in the effective area. The effective area for calculating ϕeffn is defined as the 20-dB range below the temporal intensity peak. Because there is little energy outside the defined effective area, the corresponding phase is considered trivial to the pulse. Due to thermal and mechanical disturbances, fluctuations of key parameters in an LSSI system are inevitable in the real scene. To evaluate the robustness against random fluctuations of key parameters in LSSI, several simulation datasets with different levels of shear jitter are generated, where high-order dispersions in DFT are ignored. As shown in Fig. 3a, the duration NMAE of ISFC grows more slowly than that of FTM as the shear jitter becomes stronger, indicating ISFC is more robust against shear jitters than FTM.

On the other hand, the shear value is crucial to pulse reconstruction and the subsequent acquisition device. Since the temporal fringe frequency is the shear when high-order dispersions are neglected, the temporal interferogram with the smaller shear can be sampled with a lower-bandwidth and lower-sampling-rate analog-to-digital converter (ADC). Given a fixed dispersive medium, the small shear value corresponds to the small delay, which may be not enough to separate the pulse under measurement and its delayed replica in the time domain. Therefore, the choice of the shear value should consider both the bandwidth and sampling rate of the acquisition system and the approximate pulse duration of the pulse under measurement. The optimal shear should be the minimal one that allows accurate pulse reconstruction. We evaluate the performance of both pulse reconstruction algorithms in various shear values via generating simulation datasets with different delays, where the group velocity dispersion remains fixed and high-order dispersions are still ignored. As shown in Fig. 3b, ISFC distinctly outperforms FTM in all the tested delays, and it signifies that ISFC allows to use smaller shear, thereby lowering the demand on the bandwidth and sampling rate of the acquisition device. Figure 3d shows the retrieved pulse duration results of FTM under different delays. When the delay is 5 ps, small pulse durations are well retrieved since high-order dispersions are not considered. However, as shown in Fig. 3c, the delay is not large enough to avoid overlapping between sidebands and DC components after performing inverse Fourier transform on reconstructed interferograms for wider pulses, thereby resulting in terrible sideband extraction in FTM. As a result, accurate retrieval of pulse durations above ~500 fs is not guaranteed as shown at the top of Fig. 3d. This phenomenon becomes more severe when decreasing the delay to 4 ps. Further reducing the delay to 3 ps even deteriorates the performance of pulses with small durations. In contrast, ISFC accurately retrieves the pulse durations under the three different delays as shown in Fig. 3e. Additional simulations reveal that ISFC achieves significantly better pulse reconstruction performance than FTM when using the lower sampling rate and the worse quantization resolution for temporal interferogram acquisition (see Supplementary Information). Furthermore, ISFC reveals stronger resistance against noises than FTM in simulations (see Supplementary Information).

Experimental validation of ISFC over fs pulses via LSSI
Figure 4 shows the experimental setup for validating ISFC over fs pulses via LSSI. A 32.4 MHz programmable fs laser is built using a homemade C-band fs laser followed by a programmable spectral filter. One branch of the shaped pulses is measured by a commercial FROG for label collection, and the other is sent to the LSSI-based ISFC. In the LSSI, the pulse under characterization is split into two arms by a fiber coupler, generating the original pulse and its time-delayed replica. Then, the pulse and its time-delayed replica are combined and propagate through a spool of dispersion compensation fiber (DCF) to transfer the delay to a spectral shear. The fiber system before the DCF is polarization-maintaining for better interference. A photodetector converts the temporal interferograms to the electrical domain and is then acquired by an oscilloscope. To apply FTM in the experiments for pulse reconstruction, the shear frequency in experiments is calculated to be ~2.13 GHz by calibrating the delay of the interferometer and dispersion values of the DCF (see Methods and Supplementary Information).

Figure 5a shows the pulse duration retrieval accuracy comparison over the test set. The superiority of ISFC over FTM is further magnified in experiments, manifesting decent pulse duration retrieval from 146 fs to 3.21 ps. The duration MAE on the experimental test set of ISFC is 41.32 fs, which is even smaller than the pre-set temporal resolution of ~70 fs of the FROG in experiments. However, the duration MAE on the experimental test set of FTM reaches 8.16 ps. As shown in Fig. 5c, the corresponding duration NMAE of ISFC and FTM are ~0.07 and 17.12 respectively. In terms of the phase retrieval accuracy, the phase NRMSE of ISFC is 0.0256, which is ~17 folds better than that of FTM. The most responsible reason for the terrible performance of FTM is its inherent sensitivity to the shear calibration, which has been unveiled in previous simulations. Experiments are carried out in an open environment. Thus, during the long time of experimental data acquisition time due to the slow updating rate of the FROG (i.e., updating every ~7 s), the delay and dispersion parameters in LSSI determining the shear might vary along the environmental disturbances (e.g., thermal fluctuations and mechanical vibrations). However, the shear value used in FTM for reconstructing diverse pulses is unchanged, leading to awful pulse reconstruction. Another reason behind the terrible performance of FTM is that the shear frequency is too small. As validated in the simulation results shown in Fig. 3, the small shear prevents effective sideband extraction, thus leading to terrible pulse reconstruction of FTM. Additionally, non-negligible nonlinearities in fiber links of LSSI also contribute to the awful pulse reconstruction, as the analysis in the Discussion.

ISFC bypasses the shear calibration by learning the temporal interferogram and its corresponding pulse directly. Shear variations are encoded in different consecutive temporal interferograms and learnt and further adaptively compensated by ISFC. Figure 5b demonstrates accurate pulse reconstruction of ISFC generalized to various pulses. The top row of Fig. 5b shows the accurate reconstruction of a narrow pulse with a duration of 146 fs, and ISFC also generalizes to pulses with negative chirp or positive chirp as shown in the two middle rows of Fig. 5b. Moreover, ISFC can even reconstruct pulses with sophisticated shapes as shown in the bottom row of Fig. 5b, where oscillations in the temporal tail are perfectly retrieved. In the region of energy distribution, all the reconstructed temporal phase align swell with FROG measurements. The spectra shown in the right column of Fig. 5b are obtained by performing the Fourier transform on the corresponding temporal predictions. The spectral intensities and phase reconstructed by ISFC are consistent with FROG measurements.

Powerful single-shot pulse reconstruction of ISFC enables resolving the switching dynamics of the programmable spectral filter with a frame rate as high as the repetition rate of the pulse train. The oscilloscope records about 5 million consecutive frames of interferograms with the sampling rate of 3.13 GSa/s, and each frame is fed to ISFC for single-shot pulse reconstruction. To accommodate the low sampling rate, ISFC is retrained using a dataset with the sampling rate of 3.13 GSa/s, which is obtained via down-sampling the original-100-GSa/s temporal interferograms in the digital domain. Figure 6a shows a switching process of the programmable spectral filter resolved by ISFC in single shot. The left subplot illustrates the intensity dynamics during switching, which starts from the frame of ~1.2 million and terminates at the frame of ~2.7 million. The entire switching process completes within 46.3 ms, conforming to the claimed response time of the programmable spectral filter (less than 100 ms). The pulse first broadens as the switching process initiates, then the peak position of the pulse continues shifting and settles on the main-lobe position of the final pulse. The sidelobe of the final pulse originates from the pulse narrowing dynamics observed at the frame of ~2 million. The pulse duration variation during switching can also be recognized in the phase evolution, as shown in the right panel of Fig. 6a. The change in the width of the range, where the phase is meaningful, is synchronized with the change in the pulse duration. The phase of the sidelobe and the main lobe of the final pulse almost remains steady after the sidelobe settles down. Figure 6b shows the reconstructed pulses of the start frame, the 3 middle frames, and the final frame. Unfortunately, since there is no way to realize full-field characterization over fs pulse train at MHz frame rate, the ISFC resolved full-field switching dynamics cannot be cross validated. However, the validity of the resolved switching dynamics can be guaranteed by for the following two reasons. First, the reconstructed start and final pulses align well with the FROG measurements in both temporal and spectral domains as shown in Fig. 6b. Second, the evolutionary process is consecutive as evidenced in the temporal intensity evolution. Another reconstructed switching dynamics of the programmable spectral filter is demonstrated in Supplementary Information. ISFC provides a viable solution to capture ultrafast events in a single-shot and full-field manner at a frame rate up to MHz. As a result, the LSSI-based ISFC can find utilizations in diverse scenes, such as ultrafast spectroscopy and full-field characterization of ultrafast laser dynamics.

By down-sampling experimental temporal interferograms in the digital domain, the sampling rate dependence of ISFC is evaluated and shown in Fig. 7a. Both duration NMAE and phase NRMSE increase as the sampling rate decreases, as expected. Nevertheless, the pulse retrieval performance is acceptable under the sampling rate of 3.13 GSa/s with the duration NMAE of 0.097 and the phase NRMSE of 0.034. The pulse retrieval performance sharply deteriorates when further reducing the sampling rate to 1.56 GSa/s, as evidenced by the duration NMAE drastically increasing by 104.1% compared to the sampling rate of 100 GSa/s. An empirical deduction drawn from the simulation and experimental results indicates that the minimal sampling rate supported by ISFC lies between the actual shear value and its corresponding Nyquist sampling rate (see Supplementary Information).

The inference time of ISFC for reconstructing one pulse is crucial to implementing online real-time single-shot full-field characterization over high-repetition-rate pulse trains. Figure 7b shows the evaluated inference time for per-pulse characterization using ISFC on both a CPU (Intel Xeon Gold 6146) and a GPU (Nvidia RTX 3090) accelerated by compute unified devices architecture (CUDA) under different batch sizes. For each batch size, 1000 tests are executed, and the average inference time is illustrated in Fig. 7b. As batch size increases, the average inference time for per pulse characterization decreases due to parallel computing supported by ISFC. The advantage of the GPU in parallel computing over the CPU is magnified as the batch size increases. The average inference time for per-pulse characterization on the GPU gradually converges to 2.41 us and it represents the supported highest repetition rate of allowing online single-shot full-field characterization via ISFC theoretically reaches ~414 kHz, which could be further improved to MHz level through a more powerful GPU or even ASIC. As a comparison, the FROG in experiments refreshes the pulse characterization result every ~7 s with the FROG trace size of 256×256, which is astonishingly 3-million-folds slower than ISFC.

Methods
Numerical simulation of LSSI
In numerical simulations, the temporal resolution is 10 fs and the time window is 20 ns. To generate pulses with asymmetric profiles, the fs pulse under characterization is synthesized by two randomly chirped Gaussian pulses, which can be expressed as

E1(t) = A(t) * exp(-(1+j*C11) * (t/T01)^2 + j*C12 * (t/T01)^3)
E2(t) = A(t) * exp(-(1+j*C21) * (t/T02)^2 + j*C22 * (t/T02)^3)
E(t) = E1(t) + α * E2(t-τ)

C11, C12, C21, C22 are random chirp parameter of two sub-pulses, T01 and T02 are random pulse durations of two sub-pulses. A random small delay τ is introduced between the two sub-pulses to create the asymmetric profile and α is the random attenuation factor. As a result, E1(t) serve as the main part and E2(t-τ) is the secondary part of the synthesized fs pulse.

The fs pulse under characterization is centered at 1030 nm and first goes through the interferometer. Datasets with different delays ranging from 1 ps to 5 ps are generated. During the generation of the data with delay jitter, the delay is jittered randomly within the allowed maximal jitter magnitude for each pulse. By combining the raw pulse and its delayed replica, the spectral fringe patterns can be obtained. Then, DFT is utilized to introduce a spectral shear between the two pulses and map the spectral fringes into time simultaneously. DFT is performed by a 3-km SM980 fiber with the group velocity dispersion of about -47.8 ps/nm/km at 1030 nm. We ignore nonlinearities in the fiber for faster data generation, and the dispersion can be modeled by a spectral phase filter as follows

EDFT(ω) = F(E(t) + E(t-Δt)) * exp(j * (β2/2 * L * ω^2 + β3/6 * L * ω^3))

β2, β3 are the group velocity dispersion and TOD of the fiber respectively and L is the fiber length. EDFT(ω) is the spectral field after DFT. The shear value calculated via Eq. (1) is ~9.85 GHz. After signal acquisition, including photodetection, and bandwidth limit (a low-pass filter with a bandwidth of 20 GHz and super-Gaussian rolling off), the temporal interferogram containing full-field information of the fs pulse under characterization is obtained. We generate 60,000 sets of data as the basic simulation dataset, and the duration of 60,000 pulses ranges from 50 fs to 1.4 ps. Other variant simulation datasets can be readily obtained by downsampling the temporal interferogram of the basic simulation dataset with different sampling rates, quantizing it with different resolutions, and adding different levels of Gaussian random noise to it. 54,000 sets of data are used for training, 3000 sets of data are used for validation, and the rest 3000 sets of data are used for testing.

Experimental details and ISFC training
The programmable fs laser is built using a homemade C-band fs laser with a repetition rate of 32.4 MHz, followed by a programmable spectral filter (Finisar WaveShaper 4000B). A commercial FROG (Mesa Photonics FROGScan Ultra) is used for label collection. The temporal resolution of the FROG is set to ~70 fs with 256 scanning points. The FROG updates the measurement result every ~7 s, and the updating time becomes larger when more scanning points are used. The claimed group velocity dispersion of the DCF is -340 ps/nm. The dispersion amount has to satisfy the far-field Fraunhofer condition described by the inequation below for realizing DFT, where tFWHM is the temporal width. β2 and L are the group velocity dispersion and length of the dispersive medium. It sets a lower bound for the dispersion amount and the wider pulses require larger group velocity dispersion to satisfy far-field Fraunhofer condition. On the other hand, when time delay is fixed, the dispersion determines the shear frequency and too small dispersion causes the very high shear frequency. As a result, the demands on the sequential acquisition system become quite high and are very hard to be met.

tFWHM^2 / (2π * β2 * L) << 1

The dispersed signal reaches nanosecond-level width in time domain (see Supplementary Fig. 3) and it can be readily acquired by the high-speed real-time acquisition system. The bandwidth of the photodetector (EOT ET-3500F) is >15 GHz. A real-time oscilloscope (Tektronix DPO70000SX Series) is used for interferogram acquisition at the sampling rate of 100 GSa/s and the bandwidth of 33 GHz. 70,000 samples are preserved in the experiments for training ISFC, with 63000 samples for training and the rest 7000 samples for testing. Each pulse in the dataset is created by imposing a random phase curve generated by the WaveShaper. The delay of the Mach-Zehnder interferometer is ~5.8 ps calibrated with an amplified spontaneous emission source, and the group velocity dispersion and the TOD are respectively ~-339.65 ps/nm and ~-1.89 ps^3 calibrated via temporal-to-spectral mapping built by DFT (see Supplementary Information).

The consecutive temporal interferograms are first segmented into separate frames with a frame length of 30 ns. Cross-correlation is applied among different frames corresponding to different fs pulses to align these frames horizontally. The input dimension of the fully-connected neural network in ISFC is determined by the sampling rate, and the input dimension is 93 under the sampling rate of 3.13 GSa/s. Three hidden layers with dimensions of 512, 256, and 512, respectively are used. The output dimension is 512 representing the real and imaginary parts of the pulse. Layer normalization is used to alleviate internal covariate shifts between two layers and outperforms batch normalization in training. The neural network is implemented using PyTorch, and key hyperparameters (e.g., the learning rate of 2.85e-4, training epochs of 500, the dropout ratio of 4.089e-2, and the batch size of 128) are optimized by Optuna. The optimizer is Adam and the cosine annealing learning rate strategy is applied for better convergence.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript reports on a novel method for single-shot complete characterization of high-repetition-rate femtosecond pulse trains. In particular, the authors combine linear spectral shearing interferometry (LSSI) with a machine learning technique to provide an innovative solution to this challenging task. Although spectral shearing interferometry alone enables single-shot complete characterization in principle, its reliance on precise calibration and its poor robustness to perturbations have hindered practical applications. The authors prove that the two inherent drawbacks of LSSI can be overcome by a well-trained fully-connected neural network in both numerical simulations and experiments. Interestingly, the machine learning model helps to lower the requirements on the sampling device. The proposed method is promising to become a widely used characterization method for femtosecond pulses. I think this work is of considerable interest to the community and worth publishing in Nature Communications, but I recommend the authors consider the following points to improve the manuscript:\n1. Why does the output of the neural network need to be arranged in an interleaved format between the real and imaginary parts? How does the performance compare when simply concatenating intensity and phase instead? Moreover, since the model is currently trained in a supervised manner, could the authors clarify whether its performance might be improved with unsupervised or self-supervised training?\n2. The author mention that the temporal resolution of the FROG is 70 fs, but the duration MAE on the experimental test set of ISFC shown in Fig. 5 is 41.32 fs, which is even smaller than the temporal resolution of the FROG, could the authors explain this discrepancy?\n3. Figure 6 shows the switching dynamics of the programmable spectral filter as resolved by ISFC in single shot. How can the authors ensure that the reconstructed intermediate frames are accurate? Are there any validation methods or ground-truth comparisons?\n4. I suggest that the authors provide a more thorough discussion on the required sampling rate. Could it be further reduced? What are the theoretical and practical limitations that constrain this rate?", "reviewer_2": "The Authors present a fully connected neural network–based linear spectral shearing interferometry (LSSI) technique for single-shot, full-field characterization of femtosecond pulses. This approach enables real-time pulse characterization without the use of a spectrometer and, importantly, allows for the measurement of weak femtosecond pulses. By reconstructing ultrafast pulses without relying on nonlinear effects, the method provides a direct and robust means of characterization. The manuscript has novelty and clearly written, and addresses an important issue in ultrafast optics. However, I would like the authors to address a few of major and minor issues before the decision.\n1. Key application related\n1.1 The authors emphasize that the key application of the proposed technique is the characterization of weak ultraviolet femtosecond pulses, and even attosecond pulses at high repetition rates. Has there been any previous research demonstrating the applicability of LSSI methods to weak ultraviolet or attosecond pulse measurements?\n1.2 Is it feasible to induce a precise linear spectral shearing effect in the UV and EUV ranges of femtosecond pulses? A discussion of the practical challenges and limitations would strengthen the manuscript.\n1.3 To my understanding, direct gating methods and FROG-based techniques have been previously employed for such applications. Could the authors clarify what specific advantages their proposed method offers compared to these established approaches?\n1.4 It would be better to show the LSSI system (optical layout) used in these experiments. (In supplement)\n2. Complete Characterization of Ultrafast Pulses\n2.1 In this work, the authors trained the LSSI model and performed experimental demonstrations using pulses with durations above 40 fs. However, ultrafast pulses are typically defined as having pulse widths below 10 fs. (Ti:Sapphire laser) Is it feasible to demonstrate the method to this short(10 fs) pulse regime?\n2.2 What is the minimum pulse duration that can be fully characterized using the LSSI-based method, both experimentally and with the AI-enhanced model? Please clarify the lower limit of applicability.\n2.3 Defining the critical group delay dispersion (GDD) limit is essential for establishing the reliability of any ultrafast pulse characterization technique. As pulse durations become shorter, GDD and higher-order phase effects (e.g., third-order dispersion) become increasingly significant. In this manuscript, no critical GDD limit is presented, partly because the pulses used are relatively long. Could the authors test their approach using the shortest pulse duration available from their laser system to evaluate and define the critical GDD limit?\n3. Training of the AI Model\n3.1 The authors briefly describe the training procedure in the Methods section. However, providing more detailed information about the model (such as the optimizer used, the number of layers, the number of epochs, and other relevant hyperparameters) would greatly enhance the readers’ understanding.\n3.2 The choice of a fully connected architecture is mentioned but not justified. Could the authors explain why this architecture was selected and what advantages it offers compared to alternatives (e.g., CNNs, RNNs, or hybrid models)?\n3.3 The value of delta_w (Δω) is expected to influence both the experimental results and the training performance of the AI model. Was this parameter fixed during the training process? If so, please clarify how it was chosen.\n3.4 More detailed information about the preparation of training data would also improve the clarity of the manuscript. Could the authors describe in greater detail the data generation and preprocessing steps used to train the AI model?\n4. Comparison between FTM and ISFC\n4.1 Figures 2 and 5 present a comparison between the FTM and ISFC methods. The results indicate that ISFC is highly effective for LSSI-based pulse reconstruction, whereas the FTM method appears to perform very poorly. This raises concerns, as it is difficult to accept that FTM-based LSSI could be useful at all prior to the introduction of the AI-based approach. Could the authors elaborate on this point? Does the FTM method inherently suffer from such low accuracy, or might this comparison exaggerate its limitations? If the former is true, is it then an appropriate baseline for comparison?\n4.2 In Figure 5, the authors compare ISFC with FROG. Why was a more detailed statistical analysis of the FROG results not performed? Such the analysis helps the robustness and generalizability of the proposed method.\n5. Switching Dynamics\n5.1 Figure 6 presents the switching dynamics of the programmable spectral filter. The results show excellent agreement between ISFC and FROG, which strongly supports the validity of the characterization. However, it would be helpful if the authors could clarify in which specific applications or scenarios such switching characteristics could be practically utilized.\n5.2 What is the shortest pulse duration for which this switching dynamic can be reliably demonstrated? Please provide a discussion of the limits.", "reviewer_3": "In the manuscript, the authors demonstrate the intelligent single-shot full-field characterization of femtosecond pulses based on linear spectral shearing interferometry. Since it is a linear process, the work significantly reduced laser pulse energy compared with methods that employ nonlinear process for laser pulse measurements such as SPIDER and FROG. By employing machine learning, the work obviously reduced the affect by noise. The manuscript is well written and the results are interesting. Before I recommend the manuscript accept by Nature Communications, I would like to ask the authors to consider my concerns below for further improvement of the manuscript.\n1. In the manuscript, the signal is detected employing photo detector and oscilloscope. But is the sampling speed enough since the measured pulse is femtosecond and the dispersed pulse duration is in ps.\n2. What is the pulse measurement accuracy and how short the pulse can be measured? For even shorter pulses, GDD, TOD and FOD are all important for the pulse measurement, but the work only measured up to TOD.\n3. In the manuscript, the authors claimed that the method can measure attosecond pulses. Is it possible since the measurement accuracy requires is quite high. How reliable is it? Is there any suitable dispersion materials can provide sufficient dispersion?\n4. What is the requirement on training data accuracy, which is quite important for characterizing ultrashort laser pulses with high reliability?\n5. How much amount dispersion is required for the measurement, considering the temporal resolution of detector and oscilloscope.\n6. In the work, sampling rate is crucial thus high repetition rate is required. What is the lowest repetition rate that can be accepted. What is the limitations of the method.\n7. Can the method measure electric waveform of the laser pulses, which really requires high accuracy and resolution, and there are works have reported such as TIPTOE, FROG and SPIDER."}, "author_response": "We sincerely thank the reviewers for their careful reviewing of the manuscript entitled “Intelligent single-shot full-field characterization over femtosecond pulses” by Guoqing Pu, Chao Luo, Weisheng Hu, and Lilin Yi. We appreciate their invaluable comments. We have revised the manuscript according to the comments of the reviewers. The revised manuscript is provided. For your convenience, we provide this response letter with the reviewer’s comments in italic, with our revisions underlined and highlighted in yellow. In the following, we answer the comments raised by the reviewer in detail.\nComments from Reviewer #1：\nThis manuscript reports on a novel method for single-shot complete characterization of high-repetition-rate femtosecond pulse trains. In particular, the authors combine linear spectral shearing interferometry (LSSI) with a machine learning technique to provide an innovative solution to this challenging task. Although spectral shearing interferometry alone enables single-shot complete characterization in principle, its reliance on precise calibration and its poor robustness to perturbations have hindered practical applications. The authors prove that the two inherent drawbacks of LSSI can be overcome by a well-trained fully-connected neural network in both numerical simulations and experiments. Interestingly, the machine learning model helps to lower the requirements on the sampling device. The proposed method is promising to become a widely used characterization method for femtosecond pulses. I think this work is of considerable interest to the community and worth publishing in Nature Communications, but I recommend the authors consider the following points to improve the manuscript:\nComment 1:\nWhy does the output of the neural network need to be arranged in an interleaved format between the real and imaginary parts? How does the performance compare when simply concatenating intensity and phase instead? Moreover, since the model is currently trained in a supervised manner, could the authors clarify whether its performance might be improved with unsupervised or self-supervised training?\nResponse 1:\nWe sincerely thank the reviewer for highly praising our works and raising the questions. We design the neural network to output both real and imaginary parts to realize full-field reconstruction over fs pulses with only one network instead of two (e.g., one for predicting the magnitude and the other for prediction the phase). As shown in Table R1, we have tried to train with the concatenating intensity and phase, but the performance is inferior. This may be caused by the large value difference between the magnitude and phase as shown in Fig. R1. The discontinuous point between the magnitude and phase also enhances the training difficulty.\nSupervised learning relies on numerous labelled data. As a result, for an identical task, the performance of supervised learning is usually better than unsupervised or self-supervised learning. However, obtaining numerous labelled data can be costly in experiments. Therefore, to realize unsupervised or self-supervised learning with decent performance is valuable and we are working on it. We have revised the manuscript accordingly as follows.\nTabel R1. Different types of labels and their performance on the experimental test set\nMagnitude & Phase\nReal & Imag interleaved Real & Imag concatenated\nconcatenated\nDuration MAE 41.32 fs 42.43 fs 196.92 fs\nPhase NRMSE 0.0251 0.0253 0.0486\nFigure R1. One magnitude & phase concatenated label\nRevision 1:\nThe last paragraph of ISFC (Intelligent single-shot full-field characterization) over fs pulses via LSSI\nISFC is designed to output the interleaved real and imaginary parts of the temporal pulse, and the temporal field can be readily reconstructed. We found that the interleaved sequencing outperformed concatenating the intensity and phase. Through the Fourier transform, the spectral field of the pulse can also be reconstructed. It is equivalent for ISFC to predict the spectral field first and then obtain the temporal field via the Fourier transform.\nComment 2:\nThe author mention that the temporal resolution of the FROG is 70 fs, but the duration MAE on the experimental test set of ISFC shown in Fig. 5 is 41.32 fs, which is even smaller than the temporal resolution of the FROG, could the authors explain this discrepancy?\nResponse 2:\nWe thank the reviewer for raising this question. 41.32 fs is the mean absolute error (MAE) of pulse durations on the experimental test set. Because ISFC accurately retrieves the pulse duration for most pulses in the test set as shown in Fig. R1, the duration MAE is a statistical characteristic and it could be smaller than the temporal resolution of 70 fs.\nComment 3:\nFigure 6 shows the switching dynamics of the programmable spectral filter as resolved by ISFC in single shot. How can the authors ensure that the reconstructed intermediate frames are accurate? Are there any validation methods or ground-truth comparisons?\nResponse 3:\nWe thank the reviewer for prompting this question. Unfortunately, there is no way to realize full-field characterization over fs pulse train at MHz frame rate. Therefore, there is no ground truth to cross validated the ISFC retrieved switching dynamics. However, the switching dynamics in Fig. 6 is deduced to be correct for the following two reasons. First, the reconstructed start and final pulses align well with the FROG measurements in both temporal and spectral domains. Second, the evolutionary process is consecutive as evidenced in the temporal intensity evolution. We have revised the manuscript accordingly as follows.\nRevision 3:\nThe fourth paragraph of Experimental validation of ISFC over fs pulses via LSSI\nFigure 6b shows the reconstructed pulses of the start frame, the 3 middle frames, and the final frame. Unfortunately, since there is no way to realize full-field characterization over fs pulse train at MHz frame rate, the ISFC resolved full-field switching dynamics cannot be cross validated. However, the validity of the resolved switching dynamics can be guaranteed by for the following two reasons. First, the reconstructed start and final pulses align well with the FROG measurements in both temporal and spectral domains as shown in Fig. 6b. Second, the evolutionary process is consecutive as evidenced in the temporal intensity evolution.\nComment 4:\nI suggest that the authors provide a more thorough discussion on the required sampling rate. Could it be further reduced? What are the theoretical and practical limitations that constrain this rate?\nResponse 4:\nWe thank the reviewer for raising this question. An empirical deduction drawn from the simulation (see Supplementary Information) and experimental results indicates that the minimal sampling rate supported by ISFC lies between the actual shear value and its corresponding Nyquist sampling rate. Though we do not suppose ISFC violates the Nyquist sampling theorem. Because the shear value remains unchanged in simulations and it can be learnt by ISFC through numerous data. When the sampling rate is below the Nyquist sampling rate, ISFC automatically pads the missing but deterministic data points during computation. The shear value in experiments varies due to the environmental disturbances. However, the variation is rather small as the fringe frequencies of different temporal interferograms are all very close to the calibrated shear value of ~2.13 GHz. As a result, the pulse reconstruction performance of ISFC is acceptable under the sampling rate of 3.13 GSa/s. On the other hand, the performance drop induced by using the sampling rate below the Nyquist sampling rate is quite obvious in both simulations and experiments.\nRevision 4:\nThe fifth paragraph of Experimental validation of ISFC over fs pulses via LSSI\nThe pulse retrieval performance sharply deteriorates when further reducing the sampling rate to 1.56 GSa/s, as evidenced by the duration NMAE drastically increasing by 104.1% compared to the sampling rate of 100 GSa/s. An empirical deduction drawn from the simulation and experimental results indicates that the minimal sampling rate supported by ISFC lies between the actual shear value and its corresponding Nyquist sampling rate (see Supplementary Information).\nComments from Reviewer #2:\nThe Authors present a fully connected neural network–based linear spectral shearing interferometry (LSSI) technique for single-shot, full-field characterization of femtosecond pulses. This approach enables real-time pulse characterization without the use of a spectrometer and, importantly, allows for the measurement of weak femtosecond pulses. By reconstructing ultrafast pulses without relying on nonlinear effects, the method provides a direct and robust means of characterization. The manuscript has novelty and clearly written, and addresses an important issue in ultrafast optics. However, I would like the authors to address a few of major and minor issues before the decision.\nComment 1:\n1. Key application related\n1.1 The authors emphasize that the key application of the proposed technique is the characterization of weak ultraviolet femtosecond pulses, and even attosecond pulses at high repetition rates. Has there been any previous research demonstrating the applicability of LSSI methods to weak ultraviolet or attosecond pulse measurements?\n1.2 Is it feasible to induce a precise linear spectral shearing effect in the UV and EUV ranges of femtosecond pulses? A discussion of the practical challenges and limitations would strengthen the manuscript.\n1.3 To my understanding, direct gating methods and FROG-based techniques have been previously employed for such applications. Could the authors clarify what specific advantages their proposed method offers compared to these established approaches?\n1.4 It would be better to show the LSSI system (optical layout) used in these experiments. (In supplement)\nResponse 1.1:\nWe sincerely thank the reviewer for the general comments and proposing the questions. To the best of our knowledge, there is no demonstration of weak ultraviolet or attosecond pulse measurements through LSSI. Due to the lack of ultraviolet fs lasers and attosecond pulse sources in our lab, we also fail to demonstrate it. Technically speaking, dispersion and interferometry is all that LSSI needs to achieve single-shot full-field characterization over weak ultraviolet fs pulse. Attosecond pulses can be measured by spectral shear interferometry [Quéré F, Itatani J, Yudin G L, et al. Attosecond spectral shearing interferometry[J]. Physical Review Letters, 2003, 90(7): 073902] and, therefore, attosecond pulses can also be measured by LSSI.\nResponse 1.2:\nWe thank the reviewer for giving suggestions. Though dispersion is usually easier to obtain than nonlinearities, we suppose the real challenge still lies in dispersion. LSSI requires the dispersive component providing large group velocity dispersion over wide spectral range, covering the spectra of the pulses under measurement. The required dispersion amount can be reduced as the sampling rate increases. Overall, we believe it is feasible to measure UV and even EUV fs pulses via LSSI. We have revised the manuscript accordingly as follows.\nRevision 1.2:\nThe third paragraph of Discussion\nFor ultraviolet wavelength and X-ray band, high-efficiency nonlinearities generation is very challenging due to the strong absorption of conventional nonlinear media on these ultrashort wavelength bandsError! Reference source not found.-Error! Reference source not found.. As a result, characterizing the ultrashort-wavelength fs pulses via FROG or SPIDER is difficult. The LSSI-based ISFC inherently adapts to different wavelengths as long as the corresponding dispersive medium is available, which provides a promising solution to characterize low-energy ultraviolet-wavelength fs pulses in single shot. The implementation of LSSI is flexible enough to support its utilization in various wavelengths. For instance, the spatial Michelson interferometer and diffraction gratings can also constitute the LSSI. Although dispersion is easier to obtain than nonlinearities for shorter wavelengths, the challenge of measuring ultraviolet fs pulses and even attosecond pulses via LSSI-based ISFC still lies in dispersion. LSSI requires the dispersive component providing large group velocity dispersion over a wide spectral range, covering the spectra of the pulses under measurement. Finding or designing the suitable dispersive component becomes a challenge. Certainly, the required dispersion amount can be reduced as the sampling rate increases. The LSSI-based ISFC can be applied to measure even shorter fs pulses accurately as shown in Fig. S5. The shortest pulse, which can be characterized by ISFC, is confined by the bandwidth of the dispersive medium in the LSSI. Therefore, given an ultra-large-bandwidth dispersive medium, single-shot full-field characterization over attosecond pulses via the LSSI-based ISFC can be expected.\nResponse 1.3:\nWe thank the reviewer for raising the question. The proposed LSSI-based ISFC holds two advantages over conventional approaches. First, LSSI-based ISFC can measure weak fs pulses in single shot. Direct gating method and FROG-based techniques certainly can achieve full-field characterization over fs pulses. However, as we summarized in the introduction, all these techniques rely on nonlinearities thereby posing an energy requirement on pulses under measurement. The situation becomes even severer when it comes to single shot characterization. For instance, the required pulse energy of FROG to perform single-shot measurement is at the microjoule level. On the other hand, the proposed LSSI-based ISFC realize single-shot full-field characterization over fs pulses with the energy of ~300 pJ.\nSecond, the proposed LSSI-based ISFC delivers direct and fast pulse reconstruction. Many conventional approaches (e.g., FROG-based techniques, D-scan) requires iterative phase retrieval. As a result, it is quite slow for them to update the measurement results. FROG we used in experiments (MesaPhotonics FROGScan) update the measurement results every ~7 s under the FROG trace size of 256×256 and it could be even slower when we choose larger FROG trace size in the setting (e.g., 512×512, 1024×1024). However, LSSI-based ISFC can reconstruct the pulse in 2.41 us averagely and it represents the supported highest repetition rate of allowing online single-shot full-field characterization via ISFC theoretically reaches ~414 kHz.\nResponse 1.4:\nWe thank the reviewer for the advice. However, the entire optical layout is clearly shown in the upper part of Fig. 4 and we paste it below. The programmable fs laser in the optical layout is built using a homemade C-band fs laser with a repetition rate of 32.4 MHz, followed by a programmable spectral filter (Finisar WaveShaper 4000B). Then, a commercial FROG (MesaPhotonics FROGScan Ultra) is used for label collection. The LSSI system contains a fiber MZI and a spool of DCF for performing DFT. The fiber MZI consists of two fiber couplers and a fiber tunable delay.\nFigure R2. The optical layout of LSSI system.\nComment 2:\n2. Complete Characterization of Ultrafast Pulses\n2.1 In this work, the authors trained the LSSI model and performed experimental demonstrations using pulses with durations above 40 fs. However, ultrafast pulses are typically defined as having pulse widths below 10 fs. (Ti:Sapphire laser) Is it feasible to demonstrate the method to this short(10 fs) pulse regime?\n2.2 What is the minimum pulse duration that can be fully characterized using the LSSI-based method, both experimentally and with the AI-enhanced model? Please clarify the lower limit of applicability.\n2.3 Defining the critical group delay dispersion (GDD) limit is essential for establishing the reliability of any ultrafast pulse characterization technique. As pulse durations become shorter, GDD and higher-order phase effects (e.g., third-order dispersion) become increasingly significant. In this manuscript, no critical GDD limit is presented, partly because the pulses used are relatively long. Could the authors test their approach using the shortest pulse duration available from their laser system to evaluate and define the critical GDD limit?\nResponse 2.1:\nWe thank the reviewer for raising the question. Due to the capacity of our homemade fs laser, we feel sorry we cannot experimentally demonstrate using the LSSI-based ISFC to measure pulses with durations below 10 fs. However, we validate characterization over even shorter fs pulses with the LSSI-based ISFC in simulations and the corresponding pulse reconstruction results are shown in Fig. S5. The average duration MAE and the average phase RMSE over 6000 pulses in the test set are 0.01 fs and 0.071 rad respectively, manifest rather good pulse reconstruction. Actually, the bandwidth of the dispersive medium in the LSSI determines the shortest pulse which can be characterized by ISFC. Therefore, it is feasible to apply the LSSI-based ISFC to measure very short pulses.\nRevision 2.1:\nAdd D. Characterization over shorter fs pulses with LSSI-based ISFC in Supplementary Information\nD. Characterization over shorter fs pulses with LSSI-based ISFC\nTo validate the ability of the proposed LSSI-based ISFC in even shorter fs pulses characterization, we generate a shorter-fs-pulse dataset in numerical simulations, which contains 60000 pulses with durations ranging from 8 fs to 50 fs. Due to the wide bandwidth, the fibre length for performing DFT can be reduced to 100 m to shorten the simulation window to 6 ns, and TOD of the fibre is considered. The time delay of the interferometer is fixed at 1 ps thereby corresponding to the shear frequency of ~59.12 GHz. In the dataset, 54000 pulses are used for training, and the rest 6000 pulses are used for testing. We train the ISFC under the sampling rate of 100 GSa/s and the shorter fs pulse reconstruction results are shown in Fig. S5. The average duration MAE and the average phase RMSE over 6000 pulses in the test set are 0.01 fs and 0.071 rad respectively, manifest rather good pulse reconstruction.\nFig. S5 | Characterization over shorter fs pulses with LSSI-based ISFC. a, Pulse duration retrieval results of ISFC over the dataset with TOD, where the dashed line and shadow area respectively denote the ideal situation and the 10% error range. b, Temporal retrieval results (the left column) of ISFC over an 8-fs pulse and a 20-fs pulse from the test set, and corresponding spectral retrieval results (the right column) via performing Fourier transform on the retrieved temporal pulses.\nResponse 2.2:\nWe thank the reviewer for raising the question. The minimal pulse duration which can be measured by the LSSI-based method is independent of the back-end pulse reconstruction algorithm. The minimal pulse duration is determined by the bandwidth of the dispersive medium in the LSSI. We have revised the manuscript as follows.\nRevision 2.2:\nThe third paragraph of Discussion\nThe implementation of LSSI is flexible enough to support its utilization in various wavelengths. For instance, the spatial Michelson interferometer and diffraction gratings can also constitute the LSSI. Although dispersion is easier to obtain than nonlinearities for shorter wavelengths, the challenge of measuring ultraviolet fs pulses and even attosecond pulses via LSSI-based ISFC still lies in dispersion. LSSI requires the dispersive component providing large group velocity dispersion over a wide spectral range, covering the spectra of the pulses under measurement. Finding or designing the suitable dispersive component becomes a challenge. Certainly, the required dispersion amount can be reduced as the sampling rate increases. The LSSI-based ISFC can be applied to measure even shorter fs pulses accurately as shown in Fig. S5. The shortest pulse, which can be characterized by ISFC, is confined by the bandwidth of the dispersive medium in the LSSI. Therefore, given an ultra-large-bandwidth dispersive medium, single-shot full-field characterization over attosecond pulses via the LSSI-based ISFC can be expected.\nResponse 2.3:\nWe thank the reviewer for proposing the question. First, the group velocity dispersion has to satisfy the far-field Fraunhofer condition described by the inequation below for realizing DFT. 𝑡 is the FWHM temporal width. 𝛽 and 𝐿 are the group velocity dispersion and length of the dispersive medium. The inequation sets a lower bound for the dispersion amount. It obvious that the wider pulses require larger group velocity dispersion to satisfy the far-field Fraunhofer condition. In the experimental dataset, the largest pulse duration is 3.21 ps. Thus, the smallest dispersion amount satisfying the far-field Fraunhofer condition is calculated to be ~164 ps2 (under the condition of 𝑡𝐹𝑊𝐻𝑀 2 =0.01), which corresponds to the dispersion of ~290 ps/nm. We use a spool of DCF with the claimed group velocity dispersion of 340 ps/nm, closing to the low bound.\n𝑡 2\n𝐹𝑊𝐻𝑀\n≪1\n2𝜋𝛽 𝐿\n∆𝑡\n∆𝜔=\n𝛽 𝐿\nRevision 2.3:\nExperimental details and ISFC training in Methods.\nThe claimed group velocity dispersion of the DCF is -340 ps/nm. The dispersion amount has to satisfy the far-field Fraunhofer condition described by the inequation below for realizing DFT, where 𝑡 FWHM is the temporal width. 𝛽 and 𝐿 are the group velocity dispersion and length of the dispersive medium. It sets a lower bound for the dispersion amount and the wider pulses require larger group velocity dispersion to satisfy far-field Fraunhofer condition. On the other hand, when time delay is fixed, the dispersion determines the shear frequency and too small dispersion causes the very high shear frequency. As a result, the demands on the sequential acquisition system become quite high and are very hard to be met.\n𝑡 2\n𝐹𝑊𝐻𝑀 ≪1 (10)\n2𝜋𝛽 𝐿\n3. Training of the AI Model\n3.1 The authors briefly describe the training procedure in the Methods section. However, providing more detailed information about the model (such as the optimizer used, the number of layers, the number of epochs, and other relevant hyperparameters) would greatly enhance the readers’ understanding.\n3.2 The choice of a fully connected architecture is mentioned but not justified. Could the authors explain why this architecture was selected and what advantages it offers compared to alternatives (e.g., CNNs, RNNs, or hybrid models)?\n3.3 The value of delta_w (Δω) is expected to influence both the experimental results and the training performance of the AI model. Was this parameter fixed during the training process? If so, please clarify how it was chosen.\n3.4 More detailed information about the preparation of training data would also improve the clarity of the manuscript. Could the authors describe in greater detail the data generation and preprocessing steps used to train the AI model?\nResponse 3.1:\nWe thank the reviewer for the advice. We have augmented the training part in Methods by providing more details of the model and training process.\nRevision 3.1:\nThe third paragraph of Experimental details and ISFC training in Methods\nThe activation function is ReLU, and dropout is applied to alleviate overfitting. The fully-connected neural network is implemented using PyTorch, and key hyperparameters (e.g., the learning rate of 2.85e-4, training epochs of 500, the dropout ratio of 4.089e-2, and the batch size of 128) are optimized by Optuna. The optimizer is Adam and the cosine annealing learning rate strategy is applied for better convergence.\nResponse 3.2:\nWe thank the reviewer for raising the question. Fully-connected neural network provides the densest connection between two layers. The information of the previous layer can be transferred to the next layer without loss. The information transfer between two layers is totally determined by the learnable weights and bias, not the model structure. Therefore, we choose fully-connected neural network to implement ISFC and we have not tried other kinds of models.\nResponse 3.3:\nWe thank the reviewer for raising the question. The spectral shear ∆𝜔 is an imperative parameter in LSSI, which is determined by the time delay of the interferometer and the group velocity dispersion. In LSSI, the spectral shear can be expressed as\n∆𝑡\n∆𝜔= .\n𝛽 𝐿\n2\n∆𝑡 is the delay, 𝛽 and 𝐿 are the group velocity dispersion and length of the dispersion medium respectively.\nThe shear value is crucial to pulse reconstruction and the subsequent acquisition device. First, since the temporal fringe frequency is the shear when high-order dispersions are neglected, the temporal interferogram with the smaller shear can be sampled with a lower-bandwidth and lower-sampling-rate analog-to-digital converter (ADC). Second, given a fixed dispersive medium, the too small shear value corresponds to the too small delay, which is not enough to separate the pulse under measurement and its time-delayed replica in the time domain. As a result, there is overlapping hindering lossless sideband extraction after inverse Fourier transform in FTM, as shown in Fig. 3c. Hence, the optimal shear should be the minimal one that allows accurate pulse reconstruction. Overall, the choice of the shear value should consider both the bandwidth and sampling rate of the acquisition system and the approximate pulse duration of the pulse under measurement. In our experiments, the time delay is ~5.8 ps corresponding to the shear value of ~2.13 GHz. Considering the acquisition system in experiment has the bandwidth of >15 GHz and the sampling rate of 100 GSa/s, the temporal fringes can be restored very well. But the time delay of ~5.8 ps is a little bit small for the widest pulse with the duration of 3.21 ps and it may cause overlapping. Fortunately, ISFC performs better under small time delays than FTM, which is manifested in simulation results shown in Fig. 3b, d, e. We have revised the manuscript accordingly for better understanding.\nRevision 3.3:\nThe second paragraph of Numerical simulations of ISFC over fs pulses via LSSI\nBecause there is little energy outside the defined effective area, the corresponding phase is considered trivial to the pulse. Due to thermal and mechanical disturbances, fluctuations of key parameters in an LSSI system are inevitable in the real scene.\nThe third paragraph of Numerical simulations of ISFC over fs pulses via LSSI\nSince the temporal fringe frequency is the shear when high-order dispersions are neglected, the temporal interferogram with the smaller shear can be sampled with a lower-bandwidth and lower-sampling-rate analog-to-digital converter (ADC). Given a fixed dispersive medium, the small shear value corresponds to the small delay, which may be not enough to separate the pulse under measurement and its delayed replica in the time domain. Therefore, the choice of the shear value should consider both the bandwidth and sampling rate of the acquisition system and the approximate pulse duration of the pulse under measurement. The optimal shear should be the minimal one that allows accurate pulse reconstruction.\nResponse 3.4:\nWe thank the reviewer for the advice. We have appended more detailed information about numerical simulations in the methods section as follows.\nRevision 3.4:\nNumerical simulation of LSSI in Methods\nIn numerical simulations, the temporal resolution is 10 fs and the time window is 20 ns. To generate pulses with asymmetric profiles, the fs pulse under characterization is synthesised by two randomly chirped Gaussian pulses, which can be expressed as\n−(\n1+𝑗𝐶11∙( 𝑡\n)\n2\n+\n𝑗𝐶12∙( 𝑡\n)\n3\n)\n𝐸 (𝑡)=𝐴 (𝑡)𝑒 2 𝑇01 6 𝑇01 , (6)\n1 1\n−(\n1+𝑗𝐶21∙( 𝑡\n)\n2\n+\n𝑗𝐶22∙( 𝑡\n)\n3\n)\n𝐸 (𝑡)=𝐴 (𝑡)𝑒 2 𝑇02 6 𝑇02 , (7)\n2 2\n𝐸(𝑡)=𝐸 (𝑡)+𝛼∙𝐸 (𝑡−𝜏). (8)\n1 2\n𝐶 , 𝐶 , 𝐶 , 𝐶 are random chirp parameter of two sub-pulses, 𝑇 and 𝑇 are random pulse\n11 12 21 22 01 02\ndurations of two sub-pulses. A random small delay 𝜏 is introduced between the two sub-pulses to create the asymmetric profile and 𝛼 is the random attenuation factor. As a result, 𝐸 (𝑡) serve as the main part\n1\nand 𝐸 (𝑡−𝜏) is the secondary part of the synthesised fs pulse.\n𝐸 (𝜔)=ℱ(𝐸(𝑡)+𝐸(𝑡−∆𝑡))∙𝑒\n𝑗( 𝛽\n2\n2𝐿 ∙𝜔2− 𝛽\n6\n3𝐿 ∙𝜔3)\n. (9)\n𝐷𝐹𝑇\n𝛽 , 𝛽 are the group velocity dispersion and TOD of the fibre respectively and 𝐿 is the fibre length.\n2 3\n𝐸 (𝜔) is the spectral field after DFT.\n𝐷𝐹𝑇\nComment 4:\n4. Comparison between FTM and ISFC\n4.1 Figures 2 and 5 present a comparison between the FTM and ISFC methods. The results indicate that ISFC is highly effective for LSSI-based pulse reconstruction, whereas the FTM method appears to perform very poorly. This raises concerns, as it is difficult to accept that FTM-based LSSI could be useful at all prior to the introduction of the AI-based approach. Could the authors elaborate on this point? Does the FTM method inherently suffer from such low accuracy, or might this comparison exaggerate its limitations? If the former is true, is it then an appropriate baseline for comparison?\n4.2 In Figure 5, the authors compare ISFC with FROG. Why was a more detailed statistical analysis of the FROG results not performed? Such the analysis helps the robustness and generalizability of the proposed method.\nResponse 4.1:\nWe thank the reviewer for raising the question. FTM is indeed the standard demodulation algorithm in spectral shearing interferometry [Iaconis C, Walmsley I A. Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses[J]. Optics Letters, 1998, 23(10): 792-794]. Therefore, using FTM as the baseline is totally reasonable. Actually, FTM performs OK on the simulation datasets. Figure R3 shows two pulses well retrieved by FTM in the simulation dataset where the TOD is considered. The two pulses correspond two pink triangles in Fig. 2 in the manuscript.\nFigure R3. Pulses retrieved by FTM in the simulation dataset where the TOD is considered.\nWhen the TOD is absent in data generation, FTM performs even better as two pulses retrieval shown in Fig. R4 and the two pulses actually correspond two pink triangles in the top of Fig. 3d (Delay 5 ps) in the manuscript.\nFigure R4. Pulses retrieved by FTM in the simulation dataset where the TOD is not considered.\nUnfortunately, FTM performs really bad in experiments as shown in Fig. 5a. The most responsible reason for the terrible performance of FTM is its inherent sensitivity to the shear calibration, which has been unveiled in simulation results shown in Fig. 3a. Experiments are carried out in an open environment. Thus, during the long time of experimental data acquisition time due to the slow updating rate of the FROG (i.e., updating every ~7 seconds), the delay and dispersion parameters in LSSI determining the shear might vary along the environmental disturbances (e.g., thermal fluctuations and mechanical vibrations). However, the shear value used in FTM for reconstructing diverse pulses is unchanged, leading to awful pulse reconstruction. Another reason is that the shear frequency is too small. As validated in the simulation results shown in Fig. 3b,d, the small shear prevents effective sideband extraction as illustrated in Fig. 3c, thus leading to terrible pulse reconstruction of FTM in experiments. Additionally, as we analyzed in the Discussion, non-negligible nonlinearities in fibre links of LSSI also contribute to the awful pulse reconstruction.\nResponse 4.2:\nWe thank the reviewer for the advice. We add statistics comparison to Fig. 5c.\nRevision 4.2:\nThe second paragraph of Experimental validation of ISFC over fs pulses via LSSI.\nFigure 5a shows the pulse duration retrieval accuracy comparison over the test set. The superiority of ISFC over FTM is further magnified in experiments, manifesting decent pulse duration retrieval from 146 fs to 3.21 ps. The duration MAE on the experimental test set of ISFC is 41.32 fs, which is even smaller than the pre-set temporal resolution of ~70 fs of the FROG in experiments. However, the duration MAE on the experimental test set of FTM reaches 8.16 ps. As shown in Fig. 5c, the corresponding duration NMAE of ISFC and FTM are ~0.07 and 17.12 respectively. In terms of the phase retrieval accuracy, the phase NRMSE of ISFC is 0.0256, which is ~17 folds better than that of FTM.\nFigure 5\nFig. 5 | Comparison of FTM and ISFC for fs pulse reconstruction in experiments. a, Pulse duration retrieval results of FTM and ISFC, where the dashed line and shadow area respectively denote the ideal situation and the 10% error range. b, Accurate pulse reconstruction of ISFC generalizes to the short pulse (top row), pulses with different signs of chirp (middle rows), and the pulse with a sophisticated shape (bottom row). c, Statistics comparison between FTM and ISFC. The duration NMAE of ISFC and FTM are ~0.07 and 17.12 respectively. The phase NRMSE of ISFC and FTM are 0.0256 and 0.4337 respectively.\nComment 5:\n5. Switching Dynamics\n5.1 Figure 6 presents the switching dynamics of the programmable spectral filter. The results show excellent agreement between ISFC and FROG, which strongly supports the validity of the characterization. However, it would be helpful if the authors could clarify in which specific applications or scenarios such switching characteristics could be practically utilized.\n5.2 What is the shortest pulse duration for which this switching dynamic can be reliably demonstrated? Please provide a discussion of the limits.\nResponse 5.1:\nWe thank the reviewer for the advice. First, the proposed method can help to simplify the systems in spectroscopy-related applications. For instance, fs pulses can be used to measure the spectral response of various matters (e.g., gases). For absorption response measurement, we only need to measure the spectra of the fs pulses before and after going through the matters under test. To obtain the phase response is more complex, coherent detection is required. The fs pulses are split into two branches with one branch goes through the matters under test and the other serves as the reference. Thus, the phase difference between the two branches, which is also the phase response of the matters under test, can be demodulated. With the proposed LSSI-based ISFC, the phase of fs pulse itself can be accurately retrieved. Therefore, fs pulses can directly go through the matters under test and do not need to introduce interferometry during testing. The phase of fs pulses before and after going through the matters under test can be retrieved by the proposed method and the phase response under test lies in the change of the fs pulses’ phase. On the other hand, the measure speed can be substantially enhanced by the LSSI-based ISFC. Conventional ultrafast spectroscopy relies on spectrometers for spectra readout, which is quite slow thereby leading to slow measure speed. However, the LSSI-based ISFC retrieve fs pulses from temporal interferograms. As a result, the measure speed is be substantially enhanced and can easily reach Mega frames per second, making the LSSI-based ISFC suitable for recording high-speed events.\nSecond, ultrafast laser dynamics studies enabled by DFT is very popular [1-8]. However, these studies mainly focus on spectral intensity evolution and the phase evolution is unavailable [1-6]. Time lens and dispersive Fourier transform (DFT) are utilized together for full-field characterization over picosecond pulses by performing the iterative Gerchberg-Saxton algorithm between temporally magnified pulse profiles and real-time spectra. Thus, the full-field characterization of transient dissipative soliton dynamics and intracavity incoherent supercontinuum dynamics are realized [7,8]. With the proposed LSSI-based ISFC, the phase evolution is readily available and full-field characterization of ultrafast laser dynamics can be achieved with a linear, simple physical setup.\n[1] Runge A F J, Broderick N G R, Erkintalo M. Observation of soliton explosions in a passively mode-locked fiber laser[J]. Optica, 2015, 2(1): 36-39.\n[2] Herink G, Jalali B, Ropers C, et al. Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate[J]. Nature Photonics, 2016, 10(5): 321-326.\n[3] Herink G, Kurtz F, Jalali B, et al. Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[J]. Science, 2017, 356(6333): 50-54.\n[4] Liu X, Yao X, Cui Y. Real-time observation of the buildup of soliton molecules[J]. Physical Review Letters, 2018, 121(2): 023905.\n[5] Liu X, Pang M. Revealing the buildup dynamics of harmonic mode-locking states in ultrafast lasers[J]. Laser & Photonics Reviews, 2019, 13(9): 1800333.\n[6] Liu X, Popa D, Akhmediev N. Revealing the transition dynamics from Q switching to mode locking in a soliton laser[J]. Physical Review Letters, 2019, 123(9): 093901.\n[7] Ryczkowski P, Närhi M, Billet C, et al. Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser[J]. Nature Photonics, 2018, 12(4): 221-227.\n[8] Meng F, Lapre C, Billet C, et al. Intracavity incoherent supercontinuum dynamics and rogue waves in a broadband dissipative soliton laser[J]. Nature Communications, 2021, 12(1): 5567.\nRevision 5.1:\nThe fourth paragraph of Experimental validation of ISFC over fs pulses via LSSI.\nISFC provides a viable solution to capture ultrafast events in a single-shot and full-field manner at a frame rate up to MHz. As a result, the LSSI-based ISFC can find utilizations in diverse scenes, such as ultrafast spectroscopy25 and full-field characterization of ultrafast laser dynamics12,13,26-28.\nReferences\n25. Maiuri M, Garavelli M, Cerullo G. Ultrafast spectroscopy: State of the art and open challenges[J]. Journal of the American Chemical Society, 2019, 142(1): 3-15.\n26. Herink G, Jalali B, Ropers C, et al. Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate[J]. Nature Photonics, 2016, 10(5): 321-326.\n27. Herink G, Kurtz F, Jalali B, et al. Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[J]. Science, 2017, 356(6333): 50-54.\n28. Liu X, Pang M. Revealing the buildup dynamics of harmonic mode-locking states in ultrafast lasers[J]. Laser & Photonics Reviews, 2019, 13(9): 1800333.\nResponse 5.2:\nWe thank the reviewer for raising the question. The switching dynamics is resolved by the LSSI-based ISFC frame by frame. Concretely, a piece of continuous temporal waveform containing the switching is firstly segmented into numerous frames. Each frame contains a temporal interferogram, which is generated by one fs pulse and its delayed replica during switching. Then, the segmented frames are sent to ISFC to retrieve each fs pulse during switching. Therefore, the shortest pulse duration in switching dynamics is determined by the ability of the proposed LSSI-based ISFC. As explained previously, the minimal pulse duration which can be measured by the LSSI-based method is independent of the back-end pulse reconstruction algorithm. The minimal pulse duration is determined by the bandwidth of the dispersive medium in the LSSI.\nComments from Reviewer #3:\nIn the manuscript, the authors demonstrate the intelligent single-shot full-field characterization of femtosecond pulses based on linear spectral shearing interferometry. Since it is a linear process, the work significantly reduced laser pulse energy compared with methods that employ nonlinear process for laser pulse measurements such as SPIDER and FROG. By employing machine learning, the work obviously reduced the affect by noise. The manuscript is well written and the results are interesting. Before I recommend the manuscript accept by Nature Communications, I would like to ask the authors to consider my concerns below for further improvement of the manuscript.\nComment 1:\nIn the manuscript, the signal is detected employing photo detector and oscilloscope. But is the sampling speed enough since the measured pulse is femtosecond and the dispersed pulse duration is in ps.\nResponse 1:\nWe sincerely thank the reviewer for the general comments and proposing the questions. The dispersed signal is in nanosecond level as shown in the figure below, which can be readily acquired by high-speed photodetector and oscilloscope.\nFigure R5. Top, Spectral interferogram before DFT; Bottom, 161 segmented temporal frames (grey) and their coherent average result (black).\nRevision 1:\nThe second paragraph of Experimental details and ISFC training in Methods.\nThe dispersed signal reaches nanosecond-level width in time domain (see Fig. S3 in Supplementary Information) and it can be readily acquired by the high-speed real-time acquisition system. The bandwidth of the photodetector (EOT ET-3500F) is >15 GHz.\nComment 2:\nWhat is the pulse measurement accuracy and how short the pulse can be measured? For even shorter pulses, GDD, TOD and FOD are all important for the pulse measurement, but the work only measured up to TOD.\nResponse 2:\nWe thank the reviewer for proposing the question. We add statistics comparison to Fig. 5c. In the experimental test dataset with the pulse durations ranging from 146 fs to 3.21 ps, the duration MAEs and the phase NRMSE of ISFC is 41.32 fs and 0.0256 respectively, signifying rather decent pulse reconstruction. The minimal pulse duration which can be measured by the LSSI-based method is independent of the back-end pulse reconstruction algorithm. The minimal pulse duration is determined by the bandwidth of the dispersive medium in the LSSI.\nIndeed, high-order dispersion terms are important for even short fs pulses. However, when the experimental setup is fixed, all dispersion terms can be considered nearly unchanged. Unlike FTM, the proposed ISFC does not need dispersion calibration and it can sense all dispersion terms through training with numerous data. As in our experiments, the dispersion medium is a spool of dispersive compensation fibre and it also contains high-order dispersion terms (i.e., TOD, FOD). But ISFC can still accurately reconstruct fs pulses from temporal interferograms without knowing the exact high-order dispersion values.\nComment 3:\nIn the manuscript, the authors claimed that the method can measure attosecond pulses. Is it possible since the measurement accuracy requires is quite high. How reliable is it? Is there any suitable dispersion materials can provide sufficient dispersion?\nResponse 3:\nWe thank the reviewer for proposing the question. The proposed method can measure attosecond pulses in principle. However, there are two major challenges in achieving it. First, as we mentioned in the manuscript, it requires an ultra-large-bandwidth dispersive medium and we now do not know what kind of material can provide broadband dispersion for attosecond pulse characterization. Second, ISFC is trained in supervising manner currently. As a result, it requires labelled dataset for ISFC training and the measurement accuracy is highly related to the accuracy of the labelled dataset.\nComment 4:\nWhat is the requirement on training data accuracy, which is quite important for characterizing ultrashort laser pulses with high reliability?\nResponse 4:\nWe thank the reviewer for proposing the question. We suppose the training data obtained by a well-calibrated fs pulses measurement system has decent measurement accuracy and is acceptable. The training data of ISFC is generated with a commercial FROG (MesaPhotonics FROGScan), whose parameters are shown below. The temporal resolution is below 2 fs and it is suitable for the experimental dataset ranging from 146 fs to 3.21 ps. During data collection in experiments, the FROG trace errors (i.e., the error between the measured FROG trace and the retrieved FROG trace) of most pulses is less than 2%, which signifying reliable measurements.\nFigure R6. FROG parameters.\nComment 5:\nHow much amount dispersion is required for the measurement, considering the temporal resolution of detector and oscilloscope.\nResponse 5:\nWe thank the reviewer for proposing the question. First, the dispersion has to satisfy the far-field Fraunhofer condition described by the inequation below for realizing DFT. 𝑡 is the FWHM temporal width. 𝛽 and 𝐿 are the group velocity dispersion and length of the dispersive medium. The inequation sets a lower bound for the dispersion amount. It obvious that the wider pulses require larger group velocity dispersion to satisfy the far-field Fraunhofer condition. In the experimental dataset, the largest pulse duration is 3.21 ps. Thus, the smallest dispersion amount satisfying the far-field Fraunhofer condition is calculated to be ~164 ps2 (under the condition of 𝑡𝐹𝑊𝐻𝑀 2 =0.01), which corresponds to the dispersion of ~290 ps/nm. We use a spool of DCF with the claimed group velocity dispersion of 340 ps/nm, closing to the low bound.\n𝑡 2\n𝐹𝑊𝐻𝑀\n≪1\n2𝜋𝛽 𝐿\n∆𝑡\n∆𝜔=\n𝛽 𝐿\nExperimental details and ISFC training in Methods.\nThe claimed group velocity dispersion of the DCF is -340 ps/nm. The dispersion amount has to satisfy the far-field Fraunhofer condition described by the inequation below for realizing DFT, where 𝑡 FWHM is the temporal width. 𝛽 and 𝐿 are the group velocity dispersion and length of the dispersive medium. It sets a lower bound for the dispersion amount and the wider pulses require larger group velocity dispersion to satisfy far-field Fraunhofer condition. On the other hand, when time delay is fixed, the dispersion determines the shear frequency and too small dispersion causes the very high shear frequency. As a result, the demands on the sequential acquisition system become quite high and are very hard to be met.\n𝑡 2\n𝐹𝑊𝐻𝑀 ≪1 (10)\n2𝜋𝛽 𝐿\nIn the work, sampling rate is crucial thus high repetition rate is required. What is the lowest repetition rate that can be accepted. What is the limitations of the method.\nResponse 6:\nWe thank the reviewer for proposing the question. The proposed ISFC realizes single-shot full field characterization over fs pulse. Thus, there is no low bound for the repetition rate of fs pulses. However, to guarantee good pulse reconstruction, the stretched interferograms cannot overlap after DFT. Therefore, the upper bound of the repetition rate is limited by the dispersion.\nAs for the sampling rate, an empirical deduction drawn from the simulation (see Supplementary Information) and experimental results indicates that the minimal sampling rate supported by ISFC lies between the actual shear value and its corresponding Nyquist sampling rate. Though we do not suppose ISFC violates the Nyquist sampling theorem. Because the shear value remains unchanged in simulations and it can be learnt by ISFC through numerous data. When the sampling rate is below the Nyquist sampling rate, ISFC automatically pads the missing but deterministic data points during computation. The shear value in experiments varies due to the environmental disturbances. However, the variation is rather small as the fringe frequencies of different temporal interferograms are all very close to the calibrated shear value of ~2.13 GHz. As a result, the pulse reconstruction performance of ISFC is acceptable under the sampling rate of 3.13 GSa/s. On the other hand, the performance drop induced by using the sampling rate below the Nyquist sampling rate is quite obvious in both simulations and experiments.\nRevision 6:\nThe fifth paragraph of Experimental validation of ISFC over fs pulses via LSSI\nThe pulse retrieval performance sharply deteriorates when further reducing the sampling rate to 1.56 GSa/s, as evidenced by the duration NMAE drastically increasing by 104.1% compared to the sampling rate of 100 GSa/s. An empirical deduction drawn from the simulation and experimental results indicates that the minimal sampling rate supported by ISFC lies between the actual shear value and its corresponding Nyquist sampling rate (see Supplementary Information).\nComment 7:\nCan the method measure electric waveform of the laser pulses, which really requires high accuracy and resolution, and there are works have reported such as TIPTOE, FROG and SPIDER.\nResponse 7:\nWe thank the reviewer for proposing the question. The electric field of fs pulses can be expressed as follows\n𝐸 (𝑡)=√𝐼(𝑡)𝑒𝑗(𝜔0𝑡+𝜙𝐶𝐸𝑃+𝜙𝑒𝑛𝑣(𝑡)).\n𝑒\n𝜔 is the central angular frequency. 𝐼(𝑡) and 𝜙 (𝑡) are the intensity envelope and envelope phase\n0 𝑒𝑛𝑣\nof fs pulse, which can be measured by FROG, SPIDER or the LSSI-based ISFC. 𝜙 is the carrier\n𝐶𝐸𝑃\nenvelope phase, which determines the temporal relation of the pulse envelope with respect to the underlying carrier oscillation. Most pulse characterization methods (e.g., FROG, SPIDER) are not able to measure the absolute value of 𝜙 . The absolute value of 𝜙 is not important if the pulse\n𝐶𝐸𝑃 𝐶𝐸𝑃\nenvelope does not significantly vary within one oscillation period. We set 𝜙 =0 and the electric\n𝐶𝐸𝑃\nfield can be retrieved as shown in Fig. R7, where the top plots show the measured intensity envelope and envelope phase of a fs pulse by FROG and the LSSI-based ISFC and the bottom shows the retrieved electric field.\nFigure R7. The retrieved electric field of a fs pulse.\nWe sincerely thank the reviewers for their careful reviewing of the manuscript entitled “Intelligent single-shot full-field characterization over femtosecond pulses” by Guoqing Pu, Chao Luo, Weisheng Hu, and Lilin Yi. We appreciate their invaluable comments. We have revised the manuscript according to the comments of the reviewers. The revised manuscript is provided. For your convenience, we provide this response letter with the reviewer’s comments in italic, with our revisions underlined and highlighted in yellow. In the following, we answer the comments raised by the reviewer in detail.\nComments from Reviewer #1：\nI am satisfied with the revised version and can recommend acceptance.\nResponse:\nWe sincerely thank the reviewer for raising previous questions, which really helps us to improve the quality of the manuscript, and we thank the reviewer feels satisfied with our revision.\nComments from Reviewer #2:\nThe authors have provided a comprehensive and well-prepared rebuttal that addresses the majority of the reviewers&#x2019; comments in a clear and scientifically rigorous manner. The revised manuscript has been substantially improved both in clarity and completeness.\nThe proposed &#x2018;LSSI-based intelligent single-shot full-field characterization (ISFC)&#x2019; represents a technically innovative approach that combines linear spectral shearing interferometry with data-driven reconstruction. The idea of employing a neural network to circumvent calibration instabilities and high-order dispersion issues is novel and potentially impactful.\nHowever, I still have one concern regarding short(~10 fs level) pulse laser with conventional dispersive medium that causing group delay dispersion. (Practical point of view and precision. )\nThe authors claim that ISFC can characterize pulses as short as 8 fs based on simulations including TOD. However, in practice, broadband femtosecond pulses inevitably experience strong dispersion and spectral phase distortion through the DCF or grating used in LSSI. Such high-order dispersion leads to non-linear temporal-to-spectral mapping that may not be easily compensated by data-driven approaches trained on limited ranges of &#x03B2;&#x2082; and &#x03B2;&#x2083;. Therefore, it remains unclear whether the proposed technique can practically characterize sub-20 fs pulses under realistic experimental conditions. The authors are encouraged to discuss this limitation more concretely or provide preliminary experimental verification with broader-bandwidth pulses.\nResponse:\nWe sincerely thank the reviewer for the positive comments on our previous revisions and presenting the professional concerns about the high-order dispersions when the pulse is too short. As the reviewer mentions, broadband femtosecond pulses inevitably experience spectral phase distortion in strong dispersion, where high-order dispersions lead to nonlinear temporal-to-spectral mapping. However, as shown in Equation S1, the high-order dispersions induced nonlinear temporal-to-spectral mapping can be precisely expressed as a polynomial combination between the temporal and spectral coordinates, with the coefficients determined by the dispersion parameters.\n∞\n1 𝛽 𝐿\n𝑓=𝑓 + 𝑡−∑ 𝑘 𝑡𝑘−1. (𝑆1)\n0 2𝜋𝛽 𝐿 2𝜋(𝑘−1)!(𝛽 𝐿)𝑘\n2 2\n𝑘=3\nIn realistic experiments, though high-order dispersion parameters are not easy to be characterized accurately, learning such a polynomial combination through massive data using AI is not a hurdle. Due to the limitation on our laser source, we feel sorry that we cannot provide additional experimental results.\nRevision:\nThe first paragraph of Numerical simulations of ISFC over fs pulses via LSSI.\nWhen the pulse becomes even shorter, the conventional TOD compensating method cannot completely compensate for the TOD-induced nonlinear mapping effect. As shown in Equation (S1) in Supplementary Information, the high-order dispersions induced nonlinear spectral-to-temporal mapping can be expressed as a polynomial combination between the temporal and spectral coordinates, with the coefficients determined by the dispersion parameters. Thus, the nonlinear polynomial relation can be accurately learnt by ISFC through training on massive data, waiving precise calibration on the high-order dispersion parameters. As a result, ISFC performs perfectly in the entire pulse duration range.\nComments from Reviewer #3:\nThe authors as carefully respond to my comments and make revisions to the manuscript. I agree that the manuscript can be accepted now.\nResponse:\nWe sincerely thank the reviewer for raising previous questions, which really helps us to improve the quality of the manuscript, and we thank the reviewer feels satisfied with our revision.\nOther revisions:\nWe have checked the whole manuscript again and we have corrected several typos as below.\nInstitution\nState Key Laboratory of Photonics and Communications, School of Information Science and Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China\nEquation (4)\n1 𝑁 |𝜏 −𝜏̂|\n𝛤 = ∙∑ 𝑛 𝑛 , (4)\n𝑒𝑟𝑟 𝑁 𝜏̂\n𝑛=1 𝑛\nEquation (9)\n𝐸 (𝜔)=ℱ(𝐸(𝑡)+𝐸(𝑡−∆𝑡))∙𝑒\n𝑗( 𝛽\n2\n2𝐿 ∙𝜔2+ 𝛽\n6\n3𝐿 ∙𝜔3)\n. (9)\n𝐷𝐹𝑇"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am satisfied with the revised version and can recommend acceptance.", "reviewer_2": "The authors have provided a comprehensive and well-prepared rebuttal that addresses the majority of the reviewers’ comments in a clear and scientifically rigorous manner. The revised manuscript has been substantially improved both in clarity and completeness.\nThe proposed ‘LSSI-based intelligent single-shot full-field characterization (ISFC)’ represents a technically innovative approach that combines linear spectral shearing interferometry with data-driven reconstruction. The idea of employing a neural network to circumvent calibration instabilities and high-order dispersion issues is novel and potentially impactful.\nHowever, I still have one concern regarding short(~10 fs level) pulse laser with conventional dispersive medium that causing group delay dispersion. (Practical point of view and precision. )\nThe authors claim that ISFC can characterize pulses as short as 8 fs based on simulations including TOD. However, in practice, broadband femtosecond pulses inevitably experience strong dispersion and spectral phase distortion through the DCF or grating used in LSSI. Such high-order dispersion leads to non-linear temporal-to-spectral mapping that may not be easily compensated by data-driven approaches trained on limited ranges of β₂ and β₃. Therefore, it remains unclear whether the proposed technique can practically characterize sub-20 fs pulses under realistic experimental conditions. The authors are encouraged to discuss this limitation more concretely or provide preliminary experimental verification with broader-bandwidth pulses.", "reviewer_3": "The authors as carefully respond to my comments and make revisions to the manuscript. I agree that the manuscript can be accepted now."}, "author_response": "We sincerely thank the reviewer for the positive comments on our previous revisions and presenting the professional concerns about the high-order dispersions when the pulse is too short. As the reviewer mentions, broadband femtosecond pulses inevitably experience spectral phase distortion in strong dispersion, where high-order dispersions lead to nonlinear temporal-to-spectral mapping. However, as shown in Equation S1, the high-order dispersions induced nonlinear temporal-to-spectral mapping can be precisely expressed as a polynomial combination between the temporal and spectral coordinates, with the coefficients determined by the dispersion parameters.\n∞\n1 𝛽 𝐿\n𝑓=𝑓 + 𝑡−∑ 𝑘 𝑡𝑘−1. (𝑆1)\n0 2𝜋𝛽 𝐿 2𝜋(𝑘−1)!(𝛽 𝐿)𝑘\n2 2\n𝑘=3\nIn realistic experiments, though high-order dispersion parameters are not easy to be characterized accurately, learning such a polynomial combination through massive data using AI is not a hurdle. Due to the limitation on our laser source, we feel sorry that we cannot provide additional experimental results.\nRevision:\nThe first paragraph of Numerical simulations of ISFC over fs pulses via LSSI.\nWhen the pulse becomes even shorter, the conventional TOD compensating method cannot completely compensate for the TOD-induced nonlinear mapping effect. As shown in Equation (S1) in Supplementary Information, the high-order dispersions induced nonlinear spectral-to-temporal mapping can be expressed as a polynomial combination between the temporal and spectral coordinates, with the coefficients determined by the dispersion parameters. Thus, the nonlinear polynomial relation can be accurately learnt by ISFC through training on massive data, waiving precise calibration on the high-order dispersion parameters. As a result, ISFC performs perfectly in the entire pulse duration range.\nComments from Reviewer #3:\nThe authors as carefully respond to my comments and make revisions to the manuscript. I agree that the manuscript can be accepted now.\nResponse:\nWe sincerely thank the reviewer for raising previous questions, which really helps us to improve the quality of the manuscript, and we thank the reviewer feels satisfied with our revision.\nOther revisions:\nWe have checked the whole manuscript again and we have corrected several typos as below.\nInstitution\nState Key Laboratory of Photonics and Communications, School of Information Science and Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China\nEquation (4)\n1 𝑁 |𝜏 −𝜏̂|\n𝛤 = ∙∑ 𝑛 𝑛 , (4)\n𝑒𝑟𝑟 𝑁 𝜏̂\n𝑛=1 𝑛\nEquation (9)\n𝐸 (𝜔)=ℱ(𝐸(𝑡)+𝐸(𝑡−∆𝑡))∙𝑒\n𝑗( 𝛽\n2\n2𝐿 ∙𝜔2+ 𝛽\n6\n3𝐿 ∙𝜔3)\n. (9)\n𝐷𝐹𝑇"}]

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