Multiplexed FROG

Gil Ilan Haham; Pavel Sidorenko; Oren Lahav; Oren Cohen

doi:10.1364/OE.25.033007

1. Introduction

Frequency resolved optical gating (FROG) is a widely used method for full characterization (i.e. both amplitude and phase) of ultrashort optical pulses [1, 2]. A FROG apparatus produces a two-dimensional intensity diagram, also known as FROG trace, by spectrally resolving the nonlinear interaction of the probed pulse with its delayed replica. In addition to the standard FROG, several FROG variants have been developed for different applications. Examples include Grating-eliminated no-nonsense observation of ultrafast incident laser light e-fields (GRENOUILLE) that can characterize ultrashort laser pulses in a single shot [3], Cross-reference FROG (XFROG) [4], in which the measured pulse interacts with a (known) reference pulse, blind-FROG [5] where two unknown pulses are reconstructed simultaneously from their joint trace, and FROG for complete reconstruction of attosecond bursts (FROG CRAB) [6].

In FROG and its variants, the pulses are reconstructed algorithmically from the measured FROG traces. Until recently, the prevailing reconstruction algorithms were based on 2D phase retrieval algorithms relying on principal component generalized projections (PCGP) [7, 8]. This approach, however, imposes restrictions on the measured FROG traces that weaken the performances of FROG reconstruction algorithms (see introduction in [9]). For example, it requires a Fourier relation between the measured frequency and delay axes of the FROG traces: $Δ ω Δ t = 1 / N$ where $Δ t$ is the delay step, $Δ ω$ is the spectral resolution and N is the number of delay steps as well as the number of frequencies measured for each delay. Recently, an improved approach for the pulse reconstruction problem that is based on ptychography – a powerful coherent diffraction imaging technique [10, 11] – was adapted and implemented in XFROG [12, 13], FROG-CRAB [14], FROG [9], and interferometric FROG [15]. Time-domain ptychography (TDP) does not require any relation between the delay and frequency axes in the measured traces [9, 13, 14]. Thus, it enables reconstruction from significantly partial traces (FROG traces are considered “complete” when they conform to the Fourier relation requirements of PCGP-based reconstruction algorithms [9]). Indeed, fine reconstructions were demonstrated from rectangular traces; with a small number of delay steps in FROG-CRAB [14] and FROG [9] as well as with a lower frequency resolution in FROG [9]. Notably, the validity and conditions for FROG reconstruction from incomplete FROG traces was recently analyzed mathematically [16].

Until recently, all FROG techniques could characterize pulses only from their coherent measurements. That is, it was critical that a FROG trace corresponds to a coherent interaction between the probed pulse and its delayed replica (FROG and GRENOUILLE), reference pulse (XFROG and FROG-CRAB) or another probed pulse (blind FROG). Inspired by multi-state reconstruction in diffractive imaging [17], partially coherent FROG was proposed and implemented numerically in FROG-CRAB [18]. In that work, the FROG-CRAB PCGP algorithm was modified to account for delay jitter between the attosecond and reference pulses during the measurement process (under assumption that the waveforms of the pulses are fixed). Finally, it is worth mentioning that in imaging, the multi-state reconstruction approach was exploited for various applications, including coherent mode decomposition of the probe beam [17, 19], recovering the spectral response of the investigated sample [20, 21] and ultrahigh speed imaging [22].

Here we propose and demonstrate, numerically and experimentally, multiplexed FROG, that characterizes the pulses in pulse bursts. In multiplexed FROG, the pulses are characterized from a single trace (a multiplexed FROG trace) which corresponds to the incoherent sum of the ordinary FROG traces of each of the individual pulses. The reconstruction algorithm is based on the ptychographic FROG algorithm [9]. In addition, we propose and demonstrate numerically multiplexed blind FROG, where the pulses in a burst are characterized (including their order in the burst) by interacting each pulse with its neighboring pulses (rather than itself). Implementing multiplexed FROG can be used for reducing the scanning range in the characterization of pulse bursts and for characterization of pulses in isolated bursts using GRENOUILLE with an effective delay range that is much shorter than the burst duration. For example, it may be useful for exploring ultrafast dynamics in mode-locked lasers [23], Kerr micro-cavities [24], as well as for coherent control [25].

2. Multiplexed FROG traces

In this section, we shall first define what a multiplexed FROG trace is and then discuss how it can be measured using current FROG apparatuses. For simplicity, we present multiplexed FROG using second harmonic generation (SHG). We assume a burst of N different pulses, where the complex envelope of the n-th pulse in the train is given by $E^{(n)} (t)$ . The N-multiplexed FROG trace of this burst is given by

I_{M - F R O G}^{N} (ω, τ) = {\sum_{n = 1}^{N} | \int E^{(n)} (t) E^{(n)} (t - τ) e^{i ω t} d t |}^{2}

Where τ is the delay between the pulse and its replica. Notably, a multiplexed FROG trace with

N = 1

corresponds to an ordinary FROG trace. Figure 1 shows an example of such a burst with

N = 3

and its multiplexed FROG trace. The pulse burst is displayed in Fig. 1(a). Figure 1(b) shows the three ordinary FROG traces of the three individual pulses. The multiplexed FROG trace of the burst, which corresponds to the sum of the three ordinary FROG traces in Fig. 1(b), is shown in Fig. 1(c). In multiplexed FROG, the pulses in the burst are reconstructed from the single measured multiplexed FROG trace. Since the multiplexed FROG trace does not include data about the time intervals between the pulses or their order, these quantities remain unknown.

Fig. 1 Principle of the multiplexed FROG trace. (a) A burst of three pulses. The duration of each pulse is much shorter than the time interval between them. (b) SHG-FROG traces which are generated from each pulse separately. (c) A multiplexed FROG trace, that corresponds to the sum of the three individual traces depicted in (b).

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Before moving on to the reconstruction scheme, it is worth mentioning methods for measuring multiplexed FROG traces. In a standard scanning FROG device, a multiplexed FROG trace can be measured by setting the delay range such that each pulse interacts only with its replica (this scheme is used in section 4.2). Using GRENOUILLE, multiplexed FROG traces are obtained when the device’s delay-range is smaller than the interval between pulses in the burst. Also, the spectral resolution of the measurements, $Δ ω$ , should be larger than one over the inter-pulse interval, hence interference fringes between two pulses’ signals are indiscernible.

3. Reconstruction procedure

In this section, we describe the recovery algorithm of multiplexed FROG. Our algorithm is based on the ptychographic reconstruction algorithm for FROG (without soft thresholding) [9], which in turn relies on the extended ptychographic iterative engine (ePIE) [26]. We modified it for multiplexed FROG in a similar fashion that ePIE is modified to Multistate Ptychographic Algorithm (MsPA) [17].

A convenient discrete form of Eq. (1) is

I_{j}^{N} (ω) = {\sum_{n = 1}^{N} | F (χ_{j}^{(n)} (t)) |}^{2},

where

χ_{j}^{(n)} (t) = E^{(n)} (t) E^{(n)} (t - j Δ τ)

,

j = 1, ..., J

is a running step delay index,

Δ τ

is the delay step, and

F

stands for the Fourier transform operator. Our iterative recovery algorithm starts with an initial guess for the set of the N unknown pulses, where N is assumed to be known (the case where N is unknown apriori is discussed in section 5). The initial pulses have a Gaussian spectrum with a random, smooth spectral phase. Each iteration of the recovery algorithm starts with producing a new array of the integers

j = 1, ..., J

, yet in a random order,

s (j)

. This array is used for randomizing the order in which the step delays are updated within one iteration of the algorithm [26]. Therefore, within each iteration, the recovered pulse is modified

J

times in a random order using an internal loop (with running index j). We now describe the sequence of steps to obtain the j-th modification within the internal loop, i.e., the steps to yield the updated recovered fields

E_{j + 1}^{(n)} (t)

from the fields

E_{j}^{(n)} (t)

and the measured spectrum at time delay

s (j)

,

I_{s (j)}^{N} (ω)

. First, the SHG signals of the fields are calculated:

ψ_{j}^{(n)} (t) = E^{(n)} (t) E^{(n)} (t - s (j) Δ τ) .

Second, the SHG signals are Fourier transformed, and their moduli are divided by their intensity sum and multiplied by the square root of the measured spectral intensity:

Φ_{j}^{(n)} (ω) = \sqrt{I_{s (j)}^{N}} \frac{F [ψ_{j}^{(n)} (t)]}{\sqrt{\sum_{n} {| F [ψ_{j}^{(n)} (t)] |}^{2}}} .

Third, the updated SHG signals are calculated by

{\tilde{ψ}}_{j}^{(n)} (t) = F^{- 1} [Φ_{j}^{(n)} (ω)] .

Finally, the pulses are updated with a weight function based on the complex conjugates of

E_{j}^{(n)} (t)

according to

E_{j + 1}^{(n)} (t) = E_{j}^{(n)} (t) + α \frac{E_{j}^{(n) *} (t - s (j) Δ τ)}{{| E_{j}^{(n)} (t - s (j) Δ τ) |}_{\max}^{2}} ({\tilde{ψ}}_{j}^{(n)} (t) - ψ_{j}^{(n)} (t)) .

In Eq. (6),

α

is a real parameter that controls the strength of the update. Crucially, in our algorithm a new

α

is selected randomly in each iteration (empirically, we found a uniform probability distribution in the range [0.1, 0.5] to give the best results). Unless other constraints are available, Eq. (6) is the final step in each iteration of the internal loop. Often, however, the power spectra of the pulses can be measured quite accurately. As in [9], the spectra can be useful for improving the performances of the reconstruction algorithm. In this case, these measurements are imposed on the fields obtained in Eq. (6) by replacing their calculated spectral amplitudes with the square root of the measured power spectra -

{\hat{S}}_{j}^{(n)}

:

{\hat{E}}_{j + 1}^{(n)} (ω) = \sqrt{{\hat{S}}_{j}^{(n)}} \frac{F [E_{j + 1}^{(n)} (t)]}{| F [E_{j + 1}^{(n)} (t)] |} .

Finally, by projecting the result of Eq. (7) back to the temporal domain, we obtain the output of the iteration:

{\hat{E}}_{j + 1}^{(n)} (t) = F^{- 1} [{\hat{E}}_{j + 1}^{(n)} (ω)] .

Iterations continue until the stopping criteria (the difference between the measured and calculated FROG traces is smaller than the SNR) is met or until the predefined maximal number of iterations is reached.

4. Results

4.1 Numerical results

Next, we explore the performance of multiplexed FROG numerically. To this end, we numerically produced a set of 100 laser pulses that all conform to a Gaussian power spectrum that is centered at 800nm and its spectral bandwidth is 50nm. Each pulse (N = 256 grid points) is produced by applying a random spectral phase to the above power spectrum, conditioned that the FWHM of the pulse is smaller than 38 fs (~4 times the transform limited FWHM). The calculated FROG traces for each pulse are 256 by 256 points with equally spaced delays, $Δ t = 2.63 fs$ , and spanning the same frequency window (i.e., the product of the delay and spectral resolutions is $1 / N = 1 / 256$ ). We then generated 6 sets of 50 multiplexed FROG traces that correspond to pulse-bursts with 1-6 pulses. White-Gaussian noise $σ$ is added to the simulated multiplexed FROG traces, at different SNR values, defined by SNR = $20 \log (‖ I_{FROG} ‖ / ‖ σ ‖)$ , where $‖ \cdot ‖$ stands for $l_{2}$ norm. While Pulse reconstruction from an incomplete traces was previously demonstrated in [9], in this work we consider complete traces.

We recovered the pulses from the noisy multiplexed traces using our reconstruction algorithm. To characterize the quality of the reconstructions, one needs to consider that SHG FROG suffers from the following trivial ambiguities: time shift, time reversal with phase conjugation, and global phase. We removed the above first two ambiguities from the reconstructed fields by minimizing the difference between the original and reconstructed pulses under translation and conjugate reflection transformations. We then evaluated the error of the reconstructions using the following angle,

δ (E, \hat{E}) = ar \cos (\frac{| 〈 \hat{E} (t) | E (t) 〉 |}{\sqrt{〈 \hat{E} (t) | \hat{E} (t) 〉 〈 E (t) | E (t) 〉}}),

which is insensitive to the global phase (i.e., the third ambiguity of the field). In Eq. (9),

\hat{E} (t)

and

E (t)

are the original and reconstructed fields, respectively, and

〈 f (t) | g (t) 〉 = \int f (t) \bar{g} (t) d t

stands for the inner product. A typical reconstruction of three pulses from a single noisy multiplexed FROG trace is shown in Fig. 2. Three simulated pulses are shown in Figs. 2(a)-2(c) (solid curves), with their calculated FROG traces. The multiplexed trace is shown in Fig. 2(d). White Gaussian noise at 10dB SNR is added to the multiplexed trace (Fig. 2(e)) and this data is fed to the algorithm. Figure 2f shows the recovered trace by the algorithm while, for comparison, the reconstructed pulses are superimposed on the original pulses in Figs. 2(a)-(c) (dashed curves).

Fig. 2 Numerical example of characterization of three pulses from a single multiplexed FROG trace. (a-c) original (solid) and reconstructed (dashed) amplitudes and phases of the three simulated pulses, as well as calculated δ. The inset in each graph is the corresponding FROG traces. (d) Multiplexed FROG trace, corresponding to the sum of the individual FROG traces. (e) The multiplexed trace with white Gaussian noise at 10dB SNR. (f) Reconstructed multiplexed trace.

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Figure 3 shows the mean of the maximal $δ$ in a burst, i.e. $δ_{\max}$ , (mean over 50 realizations of multiplexed FROG traces) between the recovered and original pulses as a function of a number of pulses in the burst. Figures 3(a) and 3(b) present the w/o and w/ the prior power spectrum option (additional steps in Eqs. (7) and (8)), respectively. When the power spectrum is not used, all the reconstructions of bursts with up to three pulses were fine ( $δ_{\max} < 0.3$ for all tested noise values). Applying the power spectrum option yielded fine reconstructions of all the bursts with up to 5 pulses.

Fig. 3 Statistical investigation of multiplexed FROG. Mean (over 50 realizations of multiplexed FROG traces) of the $δ_{\max}$ in a burst as a function of the number of pulses for different noise levels. (a) Without and (b) with using power spectra constraint of the pulses.

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4.2 Experimental results

We demonstrate a two-pulse burst reconstruction from a multiplexed FROG trace experimentally. We produced the burst by splitting a ~35 fs pulse from ultrafast Ti-sapphire laser system and then sending one of the replicas through a pulse-shaper with a phase-only mask. The two pulses were then combined with approximately 5 ps delay between them. We measure the burst using a homemade scanning SHG FROG with a 2 fs delay step and 512 delay points, i.e. the total delay range is much smaller than the time interval between the pulses in the burst. Thus, each pulse in the burst interacts in the SHG crystal only with its replica and not with the other pulse. This way, the measured trace (Fig. 4(a)) corresponds to a multiplexed FROG trace of the burst. For comparison, we also measured the FROG traces of each pulse by blocking the beamline of the other pulse (Figs. 4(b) and 4(c)). The reconstructed pulses from the individual FROG traces (solid curves) and the multiplexed FROG trace (dashed curves) are shown in Figs. 4(e) and 4(f). The angles (errors) between the reconstructed pulses are presented above each pair. This experiment demonstrates the validity of multiplexed FROG (the somewhat large $δ$ is probably due to instabilities in the laser system during the scan).

Fig. 4 Experimental demonstration of multiplexed FROG. (a) Measured multiplexed FROG trace of a two-pulse burst. (b) and (c) Measured FROG traces for each pulse in the burst. (d) Reconstructed multiplexed FROG trace. (e) and (f) Reconstructed pulses using multiplexed FROG (dashed curves) and ordinary FROG (solid curves).

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5. Deducing the number of pulses in the burst

In the previous sections, it was assumed that the number of pulses in the burst that produced the multiplexed FROG is known. In this section, we present a method to deduce the number of pulses in the burst from the multiplexed FROG trace. The technique is based on applying the multiplexed FROG reconstruction algorithm for $N = 1, 2, 3 \dots$ and identifying the knee in the curve describing the error on the trace reconstruction as a function of the number of pulses. A typical result of this method is displayed in Fig. 5 (the original 3 pulses and multiplexed FROG trace were presented in Figs. 2(a)-2(c) and Fig. 2(e), respectively), which shows the reconstructed trace NMSE (blue X signs) and pulse reconstruction $δ_{\max}$ (brown plus signs) as a function of assumed number of pulses in the burst ( $N = 1, 2, \dots, 5$ ). As expected, the trace reconstruction NMSE is larger for $N < 3$ . For $N \geq 3$ , the trace reconstruction NMSE is flat, at the expected value according to the noise level. Thus, the correct number of pulses corresponds to the knee of this curve. Notably, the quality of the burst reconstruction, $δ_{\max}$ , is minimal only when the correct number of pulses, 3, is assumed, and is otherwise significantly larger. This result is intuitive when the number of assumed pulses is smaller than the correct one ( $N < 3$ in this case). When the number of assumed pulses is larger than the correct one ( $N > 3$ ), one could hope that the algorithm will still converge to a solution in which all the reconstructed pulses match well to the original pulses (i.e. some reconstructed pulses are approximately identical). The fact that the algorithm converges to different solutions (with comparable trace reconstruction NMSE) indicates that in this case the reconstruction problem is not unique.

Fig. 5 Example for deducing the correct number of pulses in the burst. Trace reconstruction NMSE and reconstruction maximal $δ_{\max}$ as a function of the number of assumed pulses (the original burst which contains 3 pulses is shown in Fig. 2). The correct number of pulses is identified by the knee in the curve of reconstructed trace NMSE vs number of pulses.

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6. Multiplexed blind FROG

As shown above, a multiplexed FROG trace contains sufficient information to reconstruct several pulses. However, it does not include the required information needed to reconstruct their order or the inter-pulse delay. To overcome this limitation, we propose multiplexed blind FROG. Here, the trace corresponds to the sum of blind FROG traces, i.e. each trace is obtained by the interaction between a pair of unknown pulses, e.g. consecutive pulses in a burst.

Figure 6 depicts an example for a multiplexed blind FROG trace. The solid red line in Fig. 6(a) shows a burst of three pulses with a time interval T between pulses. The complete FROG trace of this burst is shown in Fig. 6(b). The central region of the trace (around zero delay) corresponds to the multiplexed trace presented in the previous section. The region around delay $τ = T$ (red dashed box in Fig. 6(b)) is a multiplexed blind FROG trace, which is the sum of blind FROG traces due to interactions between the 1st and 2nd pulses and 2nd and 3rd pulses (Figs. 6(c) and 6(d)).

Fig. 6 Multiplexed Blind FROG: concept and numerical demonstration. (a) Each pulse in the burst interacts only with its consecutive pulses. (b) The complete FROG trace of the burst. The multiplexed blind FROG trace corresponds to the side lobe marked with the dashed red box. (c) The blind FROG traces, which are the components of the multiplexed blind FROG. (d) The multiplexed blind FROG trace. (e) The multiplexed blind FROG trace with white Gaussian noise at 10dB SNR. (f-h) original and reconstructed amplitudes and phases of the three simulated pulses.

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To characterize the burst from the multiplexed blind FROG trace, we modify the multiplexed FROG algorithm. First, we replace Eq. (3) by:

ψ_{j}^{(n)} (t) = E^{(n)} (t) E^{(n + 1)} (t - s (j) Δ τ) .

where

n = 1, 2, 3... N - 1

is a running index and N is the number of pulses in the burst. The next modification is different for

n = 2, 3... N - 1

pulses (these pulses interact with two different pulses) and the

n = 1, N

pulses. In the first case, Eq. (6) is replaced by:

\begin{array}{l} E_{j + 1}^{(n)} (t) = E_{j}^{(n)} (t) + α \frac{E_{j}^{(n + 1) *} (t - s (j) Δ τ)}{{| E_{j}^{(n + 1)} (t - s (j) Δ τ) |}_{\max}^{2}} ({\tilde{ψ}}_{j}^{(n)} (t) - ψ_{j}^{(n)} (t)) + \\ α \frac{E_{j}^{(n - 1) *} (t + s (j) Δ τ)}{{| E_{j}^{(n - 1)} (t + s (j) Δ τ) |}_{\max}^{2}} ({\tilde{ψ}}_{j}^{(n - 1)} (t + s (j) Δ τ) - ψ_{j}^{(n - 1)} (t + s (j) Δ τ)) \end{array}

while in the second case it is replaced by

\begin{array}{l} E_{j + 1}^{(N)} (t) = E_{j}^{(N)} (t) + α \frac{E_{j}^{(N - 1) *} (t + s (j) Δ τ)}{{| E_{j}^{(N - 1)} (t + s (j) Δ τ) |}_{\max}^{2}} \cdot, \\ \cdot ({\tilde{ψ}}_{j}^{(N - 1)} (t + s (j) Δ τ) - ψ_{j}^{(N - 1)} (t + s (j) Δ τ)) \end{array}

E_{j + 1}^{(1)} (t) = E_{j}^{(1)} (t) + α \frac{E_{j}^{(2) *} (t - s (j) Δ τ)}{{| E_{j}^{(2)} (t - s (j) Δ τ) |}_{\max}^{2}} ({\tilde{ψ}}_{j}^{(1)} (t) - ψ_{j}^{(1)} (t)) .

As in the case of regular blind FROG, an added constraint is required for a solution to be unique [9]. Thus, we assumed a known power spectrum and applied the steps described in Eqs. (7) and (8). Applying the multiplexed blind FROG reconstruction algorithm, we reconstructed the 3-pulse burst shown in Fig. 6(a). Figure 6(e) shows the multiplexed blind FROG with 10dB SNR white Gaussian noise. By feeding this spectrogram to the reconstruction algorithm, we retrieved accurately the amplitudes, phases, pulse orders and accurate intervals of the three pulses (Figs. 6(f)-6(h)).

7. Conclusions

In conclusion, we proposed and demonstrated, numerically and experimentally, multiplexed FROG, where multiple pulses are reconstructed from a single trace. Notably, the multiplexed FROG trace contains data only about the self-interaction of the pulses, i.e. it does not contain any information regarding nonlinear interaction between different pulses. In addition, we proposed multiplexed blind FROG, which reconstructs a burst of pulses (in this case including the order of the pulses in the burst) from a FROG trace in which each pulse interacts only with its adjacent pulses. Multiplexed FROG opens new opportunities in diagnostics of ultrashort pulse bursts. Repetitive pulse bursts, which are ordinarily recorded by scanning the full delay range of their self-interaction, can now be characterized from a much shorter scanning range – the length of the individual pulses’ self-interaction. Furthermore, the fact that multiplexed FROG enables reconstruction of several pulses from a single trace may allow measurement of isolated pulse bursts which previously could not be characterized. For example, all GRENOUILLE devices have an inherent trade-off between the temporal resolution and delay range, inhibiting measurement of pulse bursts when the interval between the pulses in the burst is much longer than the temporal features of the pulses. Recording and reconstructing such bursts by a multiplexed FROG trace should be possible when using a GRENOUILLE device designed to record each pulse separately. This approach, combined with a pulse-burst picker, may be utilized for investigating mode-locking build-up in lasers and Kerr micro-cavities [23,27–30], the pulse-pair of dual-comb systems [31, 32] and simultaneously reconstructing the pump and probe pulses in pump-probe experiments [33, 34].

While we presented here multiplexed SHG-FROG traces, it is straightforward to apply this concept to FROG setups that are based on other types of nonlinearities (only Eqs. (3) and (8) need to be revised), or interferometric FROG (IFROG). Notably, in all of these cases, multiplexing can be applied without any hardware modification. Finally, it is worth noting that this method’s scope may be further extended by utilizing structure-based prior knowledge on the laser pulses to further enhance the resolution and noise robustness [35–37].

Funding

Israeli Center of Research Excellence ‘Circle of Light’ (1802/12); The Wolfson Foundation;

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Multiplexed FROG

Abstract

1. Introduction

2. Multiplexed FROG traces

3. Reconstruction procedure

4. Results

4.1 Numerical results

4.2 Experimental results

5. Deducing the number of pulses in the burst

6. Multiplexed blind FROG

7. Conclusions

Funding

References and links

Cited By

Figures (6)

Equations (13)

Optics Express