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Abstract
Frequency-resolved optical gating (FROG) counts among the most used methods to characterize complex femtosecond pulses. The multishot FROG experiment, studied in this work, relies on varying a delay between two replicas of the measured pulse, where the delay accuracy can suffer from delay line imperfections, setup instability, or minimization of the acquisition time. We present a detailed study on the effect of the delay line jitter on the pulse retrieval. We carried out simulations with the jitter values ranging from high-precision delay lines (100 nm) up to extremely unstable measurements (>1000 nm). For three sets of pulses, we quantified criteria assuring reliable reconstruction, using ptychographic algorithm, of a complex pulse based on the experimentally available FROG trace error. We observe that the effect of the jitter scales together with the spectral bandwidth. However, the pulse reconstruction is relatively robust against the jitter and, even for a severe distortion of the FROG trace (e.g., a jitter of 500 nm for broadband pulses), the main features of all pulses are retrieved with high fidelity. Our results provide guidance for the limitations based on the delay imperfections in the FROG experiment.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Femtosecond laser pulses make it possible to access ultrafast processes in a variety of materials and physical phenomena. Although the majority of experiments aim at obtaining simple pulses, which are as short as possible, in many cases, it is, on the contrary, crucial to reliably measure a complex pulse temporal profile. As an example, we can list the characterization of various super continuum generation systems producing complex pulses [1,2], study of speckle patterns formed by the propagation of coherent light through a scattering medium motivated by the fields of medical imaging and communication [3], or the field of coherent control of chemical reactions, where the optimized pulse shape brings the information about the ongoing processes [4].
A number of experimental techniques have been developed to retrieve the pulse shape. One of the most used is the concept of frequency-resolved optical gating (FROG) [5] with various implementations which use second harmonic generation (SHG) or many other non-resonant nonlinear processes to reconstruct the pulse waveform. In the FROG experiment, the nonlinear signal arises due to a mutual interaction of two replicas created from the measured pulse. Spectra of the nonlinear signal acquired for various delays between the replicas form a FROG trace. The trace can be consequently employed to iteratively reconstruct the pulse amplitude and phase in time and spectrum.
Owing to its widespread usage and relative simplicity, FROG is a quite well studied and understood technique, including the imperfections in the experiment, such as excessive amount of noise in the FROG trace [6,7], limited phase-match bandwidth of the nonlinear process [8,9], limited detector spectral resolution [10], or instabilities of the pulse train in multiple shot measurements [11,12]. However, the aspect of error in the delay between the two pulse replicas has so far escaped attention.
Several works have focused on the jitter in the cross-correlation measurements, e.g., in the FROG-CRAB experiment, where the jitter typically arises due to different origin and distinct beam pathways of the two interacting pulses. An example can be the work of Murari et al. [10] who investigated the effect of the relative jitter between the XUV and IR pulse generating a FROG-CRAB spectrogram. An analogous investigation was carried out by He Wang et al. [13]. They were interested in the effect of time jitter of the NIR pulse which resulted from slight variations due to the mechanical vibrations of the different trajectories of the pulses, leading to broadening of the FROG-CRAB trace in the delay axis.
In this article, we investigate the effect of the delay line jitter on the reconstruction of complex pulses in a standard FROG experiment. Unlike previous publications, we study pulse retrieval from a dataset, where the FROG trace is not broadened, yet the measured spectra are acquired for delays suffering from a certain error, i.e., jitter. This work is motivated by the fact that relaxed demands on the delay line precision can provide us with a faster experiment, since the delay line settling time can be shortened. In the case of extremely short acquisition time, it is potentially possible to acquire the FROG trace for short femtosecond pulses even in the sweep mode, where the delay line is continuously moving at a constant speed [14]. At the same time, lower demands on the delay line precision introduce the possibility to reduce the cost of the delay line.
We study the effect of the delay line jitter via an extensive series of simulations of multishot experiments on three sets of pulses, which differ in their spectral bandwidth and pulse waveform complexity. We quantify the variation of error in pulse shape retrieval and error (difference) between the measured and the retrieved FROG trace for the delay line imperfections. We evaluate the dependence of both errors on the delay line jitter, which varied from a nearly ideal delay line (100 nm) up to an extremely inaccurate positioning (>1000 nm). Since the FROG error is always experimentally accessible, our results also provide guidance on judging the pulse reconstruction quality when the delay line jitter is the dominant source of noise.
Based on the error values and their comparison to the effect of noise in the measurement, we can assign a feasible value of the delay line jitter with respect to the bandwidth-limited (BWL) length of the pulse spectrum. Therefore, the presented results can serve as a guideline for limiting the delay jitter in FROG experiments.
2. Simulations
2.1 Pulse shapes
We studied the effect of delay line jitter on three sets of pulses. All of them were created by applying random variations to the spectral phase dependence of a pulse featuring the Gaussian spectrum. Each set differs in the spectral bandwidth and complexity. This approach corresponds to the case, where a spectrum provided by a laser system is modulated by a pulse shaper or by any other means. The main parameters of the sets are summarized in Table 1.
Table 1. Characteristics of the three sets of input pulses. λ0 is the pulse central wavelength, Δλ is the FWHM of the spectral intensity, ΔtGauss is the FWHM of the temporal intensity of the transform limited Gaussian pulse corresponding to the spectral width, if no phase variations were applied, N is the size of the square FROG trace grid, dtGRID is the time interval of the grid (the frequency interval is related by the Fourier transform dν=1/N.dt), # of pulses denotes number of different pulses in the set.
Set 1 is taken from the work of Sidorenko et al. [15], which simulates pulses with very broad bandwidth. The pulses correspond to the few-cycle pulses with respect to their spectral bandwidth. However, due to their phase modulation, they are stretched in time. Here, the random spectral chirp has been adjusted so that the temporal support of the pulse is shorter than 200 fs. Set 2 is formed by pulses with the same shapes as in Set 1; the pulses have, however, been stretched four times in time, i.e., they feature a four times narrower spectral bandwidth. Lastly, Set 3 has been derived from the experimental conditions in our laser laboratory. We applied highly modulated spectral chirps in order to get pulses randomly modulated in time. Hence, the temporal shape of the pulse intensity in Set 3 is more complex compared to Set 2, leading to approximately 2.5-fold increased temporal support. Examples of a pulse for each set are shown in the insets of Fig. 5.
2.2 Jitter simulation
In all the presented cases, calculations were run on all three complete sets of laser pulses. For each pulse from a set, we simulated an ideal FROG trace, which was consequently distorted by the simulation of randomly displaced delay line. It is worth stressing that our calculations assume delay line position, thus also the displacement, being constant throughout the single spectrum acquisition.
We take into account two physical models, which we will denote as random jitter and random-walk jitter. The random jitter corresponds to the delay fluctuations, where the jitter for each data point is independent of the previous one, following a normal distribution with standard deviation бjitter. This case corresponds to the situation, where the delay line is mechanically unstable and its imperfections cause the delay to fluctuate. For instance, the front/rear tilt of 5 µrad of a retroreflector located 50 mm above the delay line will shift the delay by 250 nm.
The random-walk model, on the other hand, simulates the behaviour of a delay line, which is scanning through a set of positions and the error arises due to the sensor and controller limitations. In this case, the displacement from the ideal position depends on the previous data point, because it is calculated by a shift from the previous delay line location, where the shift is burdened by a random error following a normal distribution with standard deviation бjitter. To simulate realistically the delay line behaviour, it is restricted to (i) move only in the scanning direction, (ii) displacements bigger than a certain value (4 бjitter) are not allowed, as such case is expected to be corrected by a controller.
Fig. 1. Example of a displacement of a correct delay line position in case of a) random jitter b) random-walk jitter, both with бjitter = 500 nm. The red line is the correct delay line position set in the experiment, the blue lines correspond to the real jitter-influenced delay line position.
The examples of both types of jitter are shown in Fig. 1. For the sake of simplicity, the Figs. 2–5 show only the data for the random jitter, findings from the data experiencing random-walk jitter are presented in Fig. 6. The values of the бjitter for particular simulations were selected as 0, 100, 250, 500, 1000, and 1500 nm.
2.3 SHG FROG reconstruction
The FROG traces were reconstructed using the ptychographic algorithm described in [15], following the version in Ref. [16]. This algorithm is known to be able to reconstruct also incomplete data. However, in our case, the data were reconstructed from a full FROG trace, where no cropping in the temporal and spectral support was introduced. Since the reconstruction algorithm is initiated randomly, we carried out 30 independent runs for each reconstruction in order to eliminate non-converging solutions and to see the statistics of the results.
Figure 2 illustrates a typical reconstruction of a FROG trace of a pulse from the Set 1 corrupted by the delay line random jitter (see top left panel). The jitter value бjitter here is 500 nm, corresponding to 3.3 fs in time, which is significant even with respect to the bandwidth-limited length of the pulse spectrum (6.2 fs). Still, the amplitude of the pulse in the bottom panels (time evolution on the left, spectrum on the right) is reconstructed perfectly and the phase shows reasonable agreement. In both bottom panels, the recovered amplitude (dotted blue line), which corresponds to the actual pulse shape, is overlaid almost perfectly by the solid line of the simulated ansatz across the entire pulse.
Fig. 2. Pulse #4 from Set 1: a) simulated FROG trace taking into account a delay line random jitter with бjitter = 500 nm, b) recovered FROG trace, c) and d) reconstructed (dots) and original (line) pulse time and spectrum evolution, respectively. Blue lines denote amplitudes, red phase.
The quality of the FROG reconstruction was significantly enhanced by using the laser spectrum to promote the algorithm convergence. The spectrum is not explicitly required by the FROG reconstruction procedure, the pulse can be reconstructed from the trace even without any prior knowledge. However, since the spectrum is easily experimentally accessible and improves the speed and the result of the reconstruction, all reconstructions here were done by employing the known laser spectrum in the pulse retrieval. Specifically, after the 10-th iteration, the amplitude of the spectrum of the laser pulse was replaced the spectral amplitude of the actual reconstruction of the pulse, while the phase information was retained.
2.4 Error evaluation
To quantitatively evaluate the fidelity of the reconstruction, we used two measures: (i) the rms error of the reconstructed FROG trace with respect to the jitter-distorted FROG trace entered into the algorithm; (ii) the rms intensity error of the retrieved pulse with respect to the original pulse as defined in [5, p.197, p.192].
A commonly used measure to evaluate the FROG reconstruction is the G error [5, p. 160]:
Because the G error depends on the size of the grid N, it would not allow a direct comparison of different sets of pulses, Therefore, throughout the text, we mainly employed a normalized rms FROG error, which we will denote as ${\varepsilon _{FROG}}$. The normalized FROG error was calculated as:
The pulse intensity error, denoted as ${\varepsilon _I}$, was determined as:
In order to illustrate the quality of the pulse reconstruction with respect to the intensity error, we depict in Fig. 3 the pulse waveform of the best and the worst reconstructed case of the reconstruction of 50 pulses from Set 1 for the random jitter бjitter = 500 nm.
Fig. 3. The intensity of the time evolution of the best and the worst retrieved pulses from Set 1 using FROG traces with random jitter бjitter = 500 nm; blue full line stands for the original pulse, red dotted line for the reconstructed pulse. The text shows the G errors, rms intensity and normalized FROG errors as defined in Eqs. (1)–(3), and the rms normalized FROG error of the reconstructed FROG trace in relation to the ideal FROG trace without jitter.
The intensity error of the best case is 0.001 and of the worst case it is 0.014, while the normalized FROG error is 0.23 and 0.19, respectively. The fact that the both, the G error and the normalized FROG error, are fairly large and similar for both cases reveals that it reflects mainly the difference between the corrupted and the correct FROG trace. This is illustrated by the normalized FROG error calculated from the retrieved trace with respect to the ideal trace ${\varepsilon _{FROGideal}}$, which is an order of magnitude lower compared to the normalized FROG error. On the other hand, it is worth stressing that even though the jitter of 500 nm corresponds to the delay of 3.3 fs and the resulting FROG trace is highly distorted [see Fig. 2(a)], the main features of all pulses were captured in good agreement with reality.
3. Results and discussion
The main objective of this work is to determine how large a jitter in the delay in the FROG experiment is still acceptable for a reliable reconstruction of a complex pulse shape. Therefore, the desired information to obtain is the intensity error, which reflects the similarity of the reconstructed pulse waveform to its original shape. Unfortunately, this information cannot be attained from the experiment. On the contrary, the FROG error, i.e., the difference between the reconstructed and the measured FROG trace, is always available. Therefore, the core of our work consists in finding conditions under which the FROG error can also reflect the actual pulse reconstruction quality.
3.1 Normalized FROG vs. intensity error correlation
We carried out an extensive batch of simulations, where measurements of pulses from all three sets were simulated for a broad range of delay line jitters scaling from an ideal case (бjitter = 0) up to an extremely unreliable delay line behaviour (бjitter = 5000 nm).
First, we focus on pulses from Set 1. The upper panels in Fig. 4 provide an example of the resulting intensity and normalized FROG errors for two representative random jitters. We present here results from all the reconstructions. Different colours and symbols denote different pulses; same symbols of the same colour denote several runs of the reconstruction algorithm on one FROG trace (belonging to one pulse).
Fig. 4. Relation between the intensity and the normalized FROG error for random jitter a) бjitter = 100 nm and b)-c) бjitter = 1000 nm. Top panels show results from all runs of the reconstruction for all retrieved pulses, bottom panels represent results of one pulse reconstructed from 4 different simulated FROG traces (all of them with бjitter = 1000 nm, but different realizations).
The two presented random jitter values in Fig. 4 show significantly different behaviour. For the 100 nm jitter, most of the simulation runs ended up with both intensity error and the normalized FROG error close to zero, while some of them did not converge to the original solution and reached a high error level. Nevertheless, the values of both errors are clearly correlated and, consequently, we can select the best pulse reconstructions based on the normalized FROG error level. This correlation gradually vanishes and for data with the бjitter = 1000 nm, the data points of the two errors form an uncorrelated cloud. In other words, the knowledge of the normalized FROG error cannot be used to judge the quality of the reconstruction and even a large normalized FROG error does not necessarily mean a large discrepancy between the real and the reconstructed pulse and vice versa.
A question of interest is the possibility to improve the reconstruction by carrying out several different measurements. The natural choice would be to average a series of measurement (i.e., FROG traces), which would cause the averaged FROG trace to be smeared in its delay axis. The effect of such averaging has been described in several articles [10,13,17]. Another option consists in the possibility to evaluate each of the measurement individually and to choose the best measurement only according to the reconstruction error. In principle, a certain combination of the random delay errors can cause the FROG trace to converge to the right pulse. This is addressed in the lower panels of Fig. 4, where various colours stand for pulse retrieval from different simulated measurements for random jitter бjitter = 1000 nm. The pulse intensity error varies among the data points by more than an order of magnitude. However, the normalized FROG error has no significant dependence and, therefore, it is unfortunately not possible to judge from the experimental results, which FROG trace gives the best results.
3.2 Effect of random jitter on the best pulse retrieval
The subsequent task is to determine how the pulse reconstruction quality varies with the delay line jitter. In order to avoid non-converging runs, we will focus in this section on the best reconstructions, which were selected from 30 different runs based on the minimal intensity error. We depict the best reconstructions in Fig. 5 for various бjitter values (different colours) for three different sets of pulses (panels a, b, c) – note the logarithmic intensity error axis. Each panel also includes an example of a typical pulse from the set. All data in Fig. 5 correspond to the random jitter.
First, we will compare Set 1 (panel a) with Set 2 (panel b). The first set of pulses are short pulses with a relatively simple structure, and the second set are the same pulse waveforms stretched in time so that they meet the spectral bandwidth reachable with our laser system. Therefore, the only difference between the sets consists in the pulse bandwidth, which is four times lower for Set 2, leading to a four times shorter duration of the BWL pulse for Set 1. In accord with the BWL pulse length, we observe that the intensity error vs. the normalized FROG error for Set 2 follows the results for Set 1, where the same intensity error values are obtained for four times smaller delay line jitters; compare the green cloud in panel b) and the yellow cloud in panel a). Therefore, we can state that the random jitter value has to be accounted relative to the pulse BWL length.
Secondly, we can compare the case of pulses with nearly identical BWL pulse length, Set 2 (25 fs) and Set 3 (20 fs), which differ in their phase modulation and the resulting complexity. Surprisingly, reconstruction of the more complex pulses with reasonable delay line jitters (below 1000 nm) typically features a slightly lower intensity error. Nevertheless, the overall trend and scales remain similar.
Fig. 5. The effect of random jitter on relation between intensity and normalized FROG error. Only the best solutions from 30 reconstructions are shown for every pulse. Colours depict different бjitter. a) Set 1, b) Set 2, c) Set 3; insets show an example of a pulse. The time interval and the size of the grid are written above the graphs. d) shows an enlargement of this relation for different (coarser and finer) FROG trace divisions for the pulses from Set 3 with бjitter = 1500 nm.
It can be argued that the pulses from different sets were simulated by using different time step Δt and number of points N and, therefore, a direct comparison between the simulations is not meaningful. For this reason, we performed test reconstructions with all 20 pulses from Set 3, where we simulated data with varying parameters Δt and N. The spectral resolution was set to always satisfy the Fourier transform condition dv = 1/(dt.N). The calculations were done for random jitter бjitter = 1500 nm, i.e., the light blue symbols in Fig. 5(c). The data points in Fig. 5(d) represent an average of the five best retrieved pulses, while the error bars scale with their standard deviation. The graph in Fig. 5(d) demonstrates that both errors, the intensity and the normalized FROG, feature very weak dependence on the data sampling. Note that the scale of the graph is magnified for the sake of comparison, since in the scale of panels a)-c) all the points would overlap.
Based on Fig. 5, we can provide guidance on the required random jitter level, which will make it possible to keep the pulse intensity error below a certain level. The most prominent parameter is the desired intensity error level, which depends on the particular experiment and can be more restricting, for instance, in nonlinear spectroscopy, where a variation of the pulse shape can distort the experimental results [18,19].
3.3 Effect of random-walk jitter on the best pulse retrieval
Analogously to the previous sub-section, we carried out a set of simulations, where the delay line imperfections were simulated as a random-walk jitter [see Fig. 1(b)], which caused the delay line position displacement to gradually evolve through the measurement. It is worth noting that depending on the particular random track, the standard deviation of the resulting delay line displacement is 1.2-2× bigger compared to the бjitter parameter.
Fig. 6. a) and b) Relation between intensity and normalized FROG error for measurements corrupted by a random-walk jitter for Set 1 and Set 3, respectively. Only the best solutions from 30 reconstructions are shown for every pulse. Colours depict different бjitter. c) random-walk jitter with бjitter = 100 nm (crosses) and 500 nm (triangles) applied together with noise to the pulses of Set 3. Colours depict different signal-to-noise ratio (SNR).
Figures 6(a) and 6(b) illustrates the effect of the random-walk jitter on the pulses from Set 1 and Set 3, respectively. Those can be compared to the corresponding panels Figs. 5(a) and 5(c) for the random jitter. When we focus on the random-walk jitter with бjitter = 500 nm for Set 1 [violet circles, Fig. 6(a)], we can see that the effect of random-walk jitter on the pulse reconstruction is significantly more pronounced compared to the random jitter [violet circles, Fig. 5(a)]. This can be partly ascribed to the fact that the character of the random walk jitter causes the delay line displacement Δd to feature up to 2-times higher standard deviation, i.e., бjitter = 500 nm leads to standard deviation of the Δd being typically between 600-1000 nm. However, even for the бjitter = 250 nm random-walk jitter [yellow circles, Fig. 6(a)], we can observe that the intensity error is increased compared to that of the random jitter бjitter = 500 nm.
We systematically observed that the gradual drift of the delay line jitter has a significantly worse effect on the pulse reconstruction compared to the randomly fluctuating jitter. This can be ascribed to the fact that the random-walk jitter, unlike the previous case, can displace certain features in the FROG trace along the delay axis and break the symmetry of the SHG FROG with respect to the zero delay.
3.4 Jitter vs. noise effect on the pulse retrieval
In this section, we will put the presented simulations in perspective with the effect of noise on the pulse retrieval. The effect of noise itself on the FROG reconstruction was extensively discussed in the published literature [5,6,10,15]. Here we focus on the interplay between the effect of noise and the delay line imperfections.
Namely, we applied a random-walk jitter with the бjitter values of 100 and 500 nm on FROG traces generated for Set 3. The FROG traces were consequently corrupted with an additive noise following a normal distribution, which simulated a dark-count noise for a high level of background. The noise level was adjusted by the signal-to-noise ratio (SNR) so that the noise standard deviation was set to 1/SNR of the FROG trace electric field amplitude.
The effect of SNR value on the pulse retrieval is illustrated in Fig. 6(c). We observed that with the decreasing SNR, i.e. increasing noise level, the random noise becomes rapidly the dominating source of the normalized FROG error. For the nearly-ideal delay line (бjitter = 100 nm), already the SNR of 100 significantly increases the normalized FROG error. Despite this, the pulse intensity error is dominantly affected by the delay line precision and depends very mildly on the SNR. The benefit of the delay line precision is lost only for the highly noisy dataset (SNR=10). For бjitter = 500 nm, the normalized FROG error is dominated by the jitter error up to SNR = 100 nm [compare violet circles in Fig. 6(b) and blue, red and yellow triangles in Fig. 6(c)].
To put these results into perspective with other reports on the effect of noise, we will evaluate also the commonly used G error. A typical G value in the SHG FROG experiments for a 128×128 points grid is < 0.5% [20], which arises even for the nearly ideal experimental data due to the inevitable noise in the measurement, such as shot noise or a readout noise. The random walk jitter of 100 nm for the SNR of 1000 (very good standard), leads to the typical G values of 0.3%; for the SNR of 10 (very high noise) the G value 8.5%. Our finding is in line with the previous reports [5 p.185], where the FROG reconstruction was demonstrated to be highly robust against the noisy FROG spectra.
3.5 Acceptable delay line jitter
As we showed in the previous sub-section, the delay line jitter is one of the important factors, when it comes to the pulse retrieval quality. It is, therefore, natural to address the question of an acceptable delay line jitter. While it is clear that a more precise delay line provides a better pulse reconstruction, Fig. 3(b) demonstrates that even the intensity error of 0.014 corresponds to a dataset, where all major features of the pulse are retrieved.
When we evaluate the acceptable jitter, it is needed to take into account the spectral bandwidth of the studied pulse. Since the spectral and temporal amplitudes of a pulse electric field are connected via Fourier transform, the spectral bandwidth determines the mean width of the pulse features in time. As an example, by increasing the spectral bandwidth two times, we gain a pulse, where all its intensity fluctuations in time are two times narrower. Therefore, a natural time scale to judge on the effect of delay line jitter is the BWL pulse length, which determines the shortest pulse length attainable by a given spectrum.
We will first turn to the random jitter. If we focus on the G error of the FROG trace, we reach for Set 1 the level of G around 0.5% already for the random jitter of 100 nm (0.7 fs). The margin of 0.5% corresponds, as stated previously, to the nearly-ideal experimental trace [20]. This might indicate that the random jitter could easily become the limiting factor in experiments with short pulses, as its jitter should be ∼ 7 times lower than the FWHM of the BWL pulse. However, the G error calculated between the retrieved trace and the ideal FROG trace reaches the value of 0.5% only for the random jitter featuring бjitter = 500 nm. Hence, the actual demands put on the delay line for the random jitter need not be so strict.
For comparison, see the results for the 500 nm random jitter in Fig. 5(a), where they are clustered around normalized FROG error 0.2 and intensity error 5×10−3. In this case, the jitter value is comparable to the BWL pulse length itself, yet the pulse retrieval captures all the major features in the pulse shape.
However, the random-walk type of the jitter has significantly more pronounced effect on the pulse reconstruction. In order to assure that the pulse intensity error remains in the majority of cases below 0.02, it is required to keep the random jitter value very low. For our model of the delay line behavior, the acceptable level of бjitter for Set 1 was 100 nm, for the Set 3 it was approx. 500 nm.
Since the actual behaviour of the delay line and the contribution of the random and random-walk jitter can differ from one experimental setup to the other, it is of interest to set a general criterion, which can ensure a reliable pulse reconstruction. Here we suggest using the experimentally available normalized FROG error. Both the data presented in Fig. 5 and Fig. 6 show that setting the normalized FROG error below 0.2 provides us in the vast majority of datasets with pulse reconstruction with intensity error below 0.02 for all types of pulses and jitter examined in our work. Besides, we observed that the value of the FROG error below 0.2 is also a limit which ensures a good pulse reconstruction in our usual SHG FROG experiments. At the same time, it is worth stressing that this criterion can be only applied for the datasets with a low noise level [see Fig. 6(c)]. For the reduced SNR, the normalized FROG error becomes quickly dominated by the FROG signal noise, while the pulse reconstruction quality might not be affected.
4. Conclusions
We presented an extensive set of simulations where we evaluated the effect of the delay line random jitter on FROG reconstruction. Such jitter can result from the necessity to reduce the acquisition time, or it can arise due to imperfections of the delay line, as well as due to instability of the measurement setup. We carried out calculations for jitter values ranging from high-precision delay lines (100 nm, 0.67 fs delay) up to extremely unstable measurements (> 1000 nm, > 6.7 fs). Also, we compared three sets of pulses featuring different bandwidth and waveform complexity.
For all sets of pulses, both the normalized FROG error and the pulse intensity error increase with the increasing delay line jitter. While the normalized FROG error is a good measure of the reconstruction quality for the low delay line jitters, such as 100 nm, this correlation is completely lost for the large jitter values. We observe that the effect of the jitter scales together with the BWL pulse length, which is given by the pulse spectral bandwidth.
We studied two types of jitter behaviour. The randomly displaced delay (random jitter) was acceptable for the reliable pulse retrieval even for the simulations, where the random jitter was comparable to the BWL pulse length. For instance, for the pulses created by the modulation of a spectral phase of a BWL pulse of 6.2 fs, we observe that the FROG trace error introduced by the delay line jitter equals the nearly-ideal noise level from experimental data when the delay line jitter reaches 100 nm. However, a reliable reconstruction can be obtained up to бjitter = 500 nm. Nevertheless, the random-walk jitter, where the delay error gradually randomly varies around the ideal value, had significantly more pronounced effect and the reliable pulse retrieval required бjitter more than 2× lower.
Since each experimental setup can differ in the particular behaviour of the delay line jitter, we propose a general criterion based on the normalized FROG error, which is available for the experimental data. For all our simulations, when the level of normalized FROG error is below 0.2, we observe that the vast majority of the pulse intensity errors does not exceed 0.02 and the reconstructed pulses feature good fidelity. Even for the FROG error well above that value, we might still attain a very well reconstructed pulse. It is, however, difficult to judge the quality from the experimental data, as there can be a number of possible sources of FROG error, such as random noise in the measured data.
A surprising finding was the fact that we obtained better reconstruction of a pulse for the longer and the more complex pulses, which was, beside the error level, also visually apparent by the waveform comparison. This observation is similar to the experiments in [10], where they observed that the presence of multiple sidebands helps the algorithm to reach faster convergence in the case of a temporal jitter between the XUV pulse and a suitably delayed infrared pulse.
We would like to point out that our results were evaluated based on the ptychographical reconstruction, while various other algorithms (e.g., COPRA or PCGPA [21,22]) might differ in their robustness against the FROG trace jitter. This comparison is an interesting question for future work.
The presented results can be used in order to optimize the FROG experiment with respect to the acquisition time and to the experimental setup cost.
Funding
Akademie Věd České Republiky (ERC-CZ/AV-B ERC100431901); Ministerstvo Školství, Mládeže a Tělovýchovy (CZ.02.1.01/0.0/0.0/16_026/0008390).
Acknowledgement
We thank for helpful discussion about ptychographic reconstruction to Pavel Sidorenko.
Disclosures
The authors declare no conflicts of interest.
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