1. Introduction

Over the last 50 years, ultrafast lasers have become ubiquitous tools in many fields of science, industry, and medicine [1–3]. Their extremely short pulse durations and high peak powers make them ideal for diverse applications such as eye surgery [1], laser machining [2], and particle acceleration [3–5]. However, the extreme time scales of ultrafast pulses, typically less than 100 femtoseconds, have historically made measuring their amplitude and phase profiles difficult. To address this problem, numerous techniques have been developed to measure both the intensity and phase of ultrafast pulses. Commonly implemented techniques include, but are not limited to, frequency resolved optical gating (FROG) [6], spectral phase interferometry for direct electric-field reconstruction (SPIDER) [7], multiphoton intrapulse interference phase scan (MIIPS) [8], spectral phase and amplitude retrieval and compensation (SPARC) [9], and dispersion scan (d-scan) [10–12].

Recently, d-scan has emerged as an experimentally simple alternative to other pulse measurement techniques. In d-scan, the pulse compressor itself is used to iteratively vary the dispersion of the pulse, and at each dispersion setting a second harmonic (SH) spectrum is recorded to generate a 2D trace. This makes d-scan an extremely simple technique to implement experimentally as the only components needed in addition to the dispersive element are a focusing mirror, a nonlinear medium, and a spectrometer. D-scan has been demonstrated in a number of experimental configurations [10–18] and recently Escoto et al. compared the performance of various phase retrieval algorithms for d-scan [19]. Additionally, a review of phase retrieval algorithms and experimental configurations for d-scan has recently been published [12].

A useful way of categorizing phase retrieval algorithms is to place them into one of two classes, Class I and Class II. Class I algorithms use a forward propagation model (FPM) to simulate the propagation of a guess pulse through the measurement system. The simulated guess trace is then compared to the measured trace, and various methods are used to adjust the guess pulse to minimize the error between the simulated and measured traces. Class II algorithms follow the same FPM, but apply a data constraint (projection). The FPM-constrained guess pulse is then improved by applying a second constraint (projection) after a backwards propagation model (BPM) is used. Early successful methods for retrieving the pulse from a d-scan trace followed the first class of algorithms and used the Nelder-Mead or machine-learning methods to iteratively improve the guess pulse [10,11]. A concrete example of a Class II type algorithm is the principal components generalized projections (PCGP) algorithm for second harmonic generation (SHG) FROG with two identical pulses (i.e. not TREEFROG [20]). In this case, the algorithm uses a FPM, data constraint (measured FROG trace), BPM, and then applies a mathematical constraint (mathematical model for the SHG process) to iteratively update the guess pulse [21]. More recently, ptychographic algorithms have been also been developed to address the phase retrieval problem for SHG FROG with two identical pulses [22].

One of the difficulties shared by many second harmonic generation (SHG) phase retrieval algorithms that is that a square root operation must be performed to recover the fundamental field. This can lead to an ambiguity in the sign of the fundamental field as either the positive or negative root must be chosen, which is equivalent to choosing a branch in the complex plane. When the wrong root is chosen, retrieval algorithms can stagnate at a solution for the field that is incorrect. This is particularly troublesome for techniques such as d-scan, SPARC, or MIIPS, where the square root operation is performed in the time domain, but the true fundamental field is retrieved in the spectral domain. One of the advances we present in this article is a method to resolve this square-root ambiguity [15,23] in the context of Class II algorithms wherein no additional algorithmic steps are needed [16,17]. Our approach is easily extended to higher-order nonlinearities, such as third-harmonic generation (THG). Following a presentation of our preliminary results [16,17], the THG extension of the algorithm has recently been experimentally demonstrated [18].

In contrast to the important case of retrieval for a single-repeatable pulse (e.g. Fig. 1(a)), an issue inherent in experimental pulse measurement is the presence of multiple pulses in the pulse train that make up a multi-exposure d-scan trace. Figures 1(b)–1(d) illustrate several situations that can arise. When there are two or more coherent (spectrally interfering) pulses in the train that can be spectrally resolved (Fig. 1(b)), the complete, interfering field can be retrieved as a single macro pulse. For long pulse delay (or for other reasons such as polarization) where the spectral interference cannot be resolved, we have mutually-incoherent pulses (modes) whose traces add in intensity (Fig. 1(c)). We refer to such traces as multi-modal. Standard phase retrieval algorithms often fail to retrieve any of the pulses in multi-modal traces due to the assumption that the trace is generated from a single coherent pulse. Recently, an algorithm was developed to retrieve all the mutually-incoherent pulses from multi-modal FROG traces [24]. The preceding examples assumed the pulse train is stable. The challenge of pulse retrieval when the input pulse train is unstable is also important and has generated considerable interest [25–27]. An example of an unstable pulse train and its effect on the d-scan trace is shown in Fig. 1(d). One objective is to ensure that a misleading retrieval is not obtained as a result of the instability: for example, algorithms have been developed for FROG that attempt to address this issue in the case where one of the retrieved modes is the coherent artifact [27]. Another objective is to quantify the level of instability by detecting the presence of pulses that add incoherently in the trace. A recent example in the context of d-scan is Escoto et al., where several approaches were taken with a Class I differential evolution algorithm to arrive at a measure of pulse train fidelity [28].

 

Fig. 1. Examples of four types of ultrafast pulse trains with repetition rates of 1/T along with corresponding representative d-scan traces. a) Ideal, stable pulse train with no secondary pulses. b) Coherent double-pulse train where a distinct and repeatable pair of pulses are generated at the repetition rate of the laser. The delay, $\Delta t$, between the primary and secondary pulses is short enough that their spectral interference is within the resolution of the spectrometer $d\omega$ and is thus visible in the d-scan trace. c) Stable, multi-modal pulse train where a distinct and repeatable set of pulses are generated at the repetition rate of the laser. The delay between the primary and secondary pulses leads to unresolvable spectral interference. In this example, the pulses are sufficiently different in dispersion which partially separates their contributions to the d-scan trace. d) Unstable, partially-coherent pulse train. Each pulse in the train is unique which leads to a smooth trace, stretched along the dispersion axis due to the incoherent overlap of different partial traces.

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Here, we present a Class II root preserving phase retrieval algorithm for d-scan, inspired by ptychography [29,30], as a solution to both the square root problem and the multi-modal problem presented above. This is illustrated by Figs. 2(a)–2(d), which is an example of a retrieval of a simulated single-mode coherent double pulse (such as in Fig. 1(b)) using the root preserving ptychographic algorithm (RPPA) to near machine precision. The accuracy of this algorithm is shown in Fig. 2(e), which shows a comparison between the retrieval errors for this simulation when RPPA was used to retrieve the pulse (blue) versus when a previously demonstrated non-root-preserving ptychographic algorithm (PDPA) [15] was used (red). In black, the retrieval error for the PDPA is shown for a similar simulation where the second pulse was not present in the trace.

 

Fig. 2. Simulated coherent double pulse representative of Fig. 1(b) retrieved to near machine precision ($\delta = 2\times 10^{-15}$). The delay between the pulses is $\Delta t = 150$ fs. a) Simulated d-scan trace b) Retrieved d-scan trace c) Simulated (blue) and retrieved (red-dashed) intensity profiles d) Simulated (blue) and retrieved (red-dashed) spectrum of the pulses as well as the simulated (purple) and retrieved (yellow-dashed) spectral phases e) Retrieval errors $\delta$ calculated as the normalized root mean square error (NRMSE) between the simulated and retrieved d-scan traces (blue). For comparison, the error when using a previously demonstrated ptychographic algorithm (PDPA) [15] is shown for the same double pulse trace (red) and the corresponding single pulse trace (black).

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Our paper is organized as follows. Section 2 describes the RPPA algorithm, including the more general case of multi-mode retrieval (MM-RPPA), which is applied to d-scan for the first time. MM-RPPA utilizes the multi-modal ptychographic imaging phase retrieval concepts found in [31], and demonstrated for FROG in [24], to solve for a finite number of unique modes found in the pulse train. Section 3 decribes how RPPA uses Newton’s method [16–18] to unambiguously determine the correct sign for the field amplitude. The performance of RPPA is further explored both numerically and experimentally using SHG grating-compressor d-scan in Sections 4 and 5.. It is important to note that the advances of RPPA and MM-RPPA can be applied not only to other approaches to varying the dispersion (e.g. chirp scan) but also to other pulse measurement techniques that ptychographic algorithms can be applied to (e.g. ptychographic FROG or multiplexed FROG) [22,24]. The effect of noise and pulse train instabilities on the retrieval accuracy of the algorithm is addressed in Supplement 1.

2. Multi-mode root preserving ptychographic algorithm

The field of an ultrafast pulse can generally be represented in the spectral domain by

In d-scan, we aim to characterize both the amplitude $\tilde {E}_0(\omega )$ and spectral phase $\phi (\omega )$ of the pulse by iteratively applying known increments of spectral phase to the pulse and measuring a SH spectrum for each applied spectral phase. In the work presented here, the applied spectral phase comes from adjusting the face-to-face separation $L$ of a two grating pulse compressor. The grating compressor phase is given by [32]

Here, $d$ is the grating groove spacing, $\theta _{i}$ is the incident angle on the first grating, and $c$ is the speed of light in vacuum. By iteratively altering the grating separation and recording an SH spectrum at each separation, a two-dimensional trace is formed with the second harmonic wavelength on the vertical axis and the altered dispersion (i.e. grating separation) on the horizontal axis. That is to say, each column in the d-scan trace represents a SH spectrum that was measured with the pulse compressor set to a different grating separation. More generally, a measured d-scan trace can be the result of the superposition of SH spectra from multiple pulses that are mutually-incoherent. A single SH spectrum measured by the detector, with a grating separation of $L$, in such a trace can be modeled by

MM-RPPA is inspired by ptychography, a coherent diffractive imaging (CDI) technique. In ptychography, a probe beam is iteratively scanned across an object in a series of overlapping spatial positions. At each position, a diffraction pattern is recorded on a detector. An iterative phase retrieval algorithm can then be used to reconstruct the spatial amplitude and phase of both the object and the probe beam [29,30]. The basic idea of our algorithm is inspired by previous ptychographic algorithms applied to both FROG and d-scan [15,22], which were adapted from the extended Ptychographic Iterative Engine (ePIE) to retrieve a single coherent pulse [30]. Our algorithm uses a simple model of the pulse propagation from the compressor to the spectrometer and back, along with a data constraint, to extract the spectral phase and amplitude of the pulse.

MM-RPPA can be broken up into four primary components: the FPM, the modulus constraint (MC), the BPM, and a weighted average over each constrained guess. The algorithm begins by generating a guess field $\tilde {E}_{k,m}(\omega )$ for each of the $M$ modes present in the trace. In this notation, $k$ refers to the current iteration of the main algorithm loop and $m$ refers to one of the $M$ modes present in the trace. A diagram of the algorithm is shown in Fig. 3.

 

Fig. 3. Flowchart of the multi-modal phase retrieval algorithm. The red solid-bordered boxes represent the beginning and end of each iteration of the algorithm. The green dash-bordered boxes represent the FPM for each of column of the trace. The grey circle represents the multi-modal modulus constraint. The blue dash-dot-bordered boxes represent the BPM for each column in the trace.

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The FPM generates the calculated d-scan trace from the current guess by simulating the propagation of the pulse from the output of the pulse compressor, through the doubling crystal and to the spectrometer. This is accomplished by applying the phase from the pulse compressor at each of the $N$ grating separations to $\tilde {E}_{k,m}(\omega )$ simultaneously:

Here, $l$ indicates which of the $N$ grating phases is being applied to the guess field. This new set of fields are then propagated through the doubling crystal, under the assumption that the doubling process is lossless and instantaneous, to produce a set SHG fields in the time domain:

The SHG fields are then propagated through the spectrometer:

In the spectrometer plane, the MC replaces the amplitude of each SHG field $\tilde {\psi }_{k,l,m}(\omega )$ with the amplitude of the measured SHG spectrum, $\tilde {T}_l$ at the $l^{th}$ grating separation while the phase of the guess is preserved:

This modulus constraint forces the updated SHG fields to be constrained by the amplitude of the measured SHG spectrum while enforcing the notion that the trace is formed by the superposition of multiple incoherent modes. It can easily be seen that in the presence of a single mode in the trace, that this modulus constraint reduces to ones shown previously in the literature [15,22].

Now, in the BPM, the constrained SHG fields are then propagated back to the crystal plane:

At the crystal plane, the the SHG field is effectively propagated back through the crystal to obtain the fundamental fields just after the compressor:

The $max$ operation is taken to improve the stability of the algorithm by avoiding singularities in the field. Equation (9) will be discussed in more detail in Section 3.. The new fundamental fields are then propagated back through the compressor to obtain $N$ new guesses for the field for each mode:

The final step of the algorithm is to generate a single initial guess for each mode for the next iteration of the loop. This must be done as each $\tilde {E}_{k,l,m}'(\omega )$ only contains information from a single column of the trace, and ultimately a single unique solution must be found for the field for each mode. To generate a single new guess for $k+1$ iteration, we apply a weighted average to $\tilde {E}_{k,l,m}'(\omega )$ along $l$:

The averaging step mixes information from all parts of the trace to inform the next iteration of the loop. More accurately, it tends to help enforce the notion that the sub-trace from each mode is generated from a set of identical pulses that only differ by some known dispersion (i.e. that the underlying pulse is not changing from column to column). This leads to vastly improved convergence stability in noisy traces where the edges of the trace might have low signal-to-noise (SNR) ratios, in traces where spectral interference is present due to multiple coherent pulses within a mode, or when the pulse train itself is inherently unstable.

The new guesses for the fields $\tilde {E}_{k+1,m}(\omega )$ are then fed into the FPM as the start of the next iteration of the loop. The loop will repeat until a desired threshold error (discussed later) is met.

3. Newton’s square root method

In the previous section, the purpose of Eq. (9) is to determine the fundamental field from the SHG field. A form of this equation is often directly transferred from similar ptychography algorithms. In diffractive imaging, the probe and object are unique with no ambiguities. However, in the case of the SHG model used here, this equation amounts to taking the square root of the SHG field in the time domain. This square root leads to an issue in the phase retrieval problem due to an ambiguity in the overall sign of the root (or branch in the complex plane):

One of the two possible roots must be chosen for the fundamental field, but without the ability to directly measure the sign of the field, it is for the algorithm to determine which root is correct. Standard phase retrieval algorithms typically solve this problem by simply choosing the principle root. In many cases this is an adequate solution, but it is not necessarily the correct solution. In simulation, we have found that choosing the principal root can lead to stagnation of phase retrieval algorithms in local minima when the true fundamental field is not the principal root of the SHG field. As real fields can be either root, it is prudent to use a phase retrieval algorithm that will naturally choose the correct root without any prior knowledge of the fundamental field and without additional measurements. We assert that Eq. (9) solves this square root problem by not strictly enforcing either of the two roots. Instead, the modulus constraint, in combination with the averaging appearing in Eq. (11), essentially chooses the correct root, and Eq. (9) simply allows that choice to be preserved.

To see this, we can compare Eq. (9) to Newton’s method for calculating the $n^{th}$ root of a number [33]. In Newton’s method, given a number $A$ and an initial guess for its $n^{th}$ root $x_i$, the next guess is given by:

Using this method, either the positive or negative roots can be found by starting with a positive or negative guess, respectively. For $n=2$ in our notation, this is equivalent to:

By adding 0 to the right side of this equation in the form of

This analysis can be extended to other perturbative nonlinearities such as third harmonic generation (THG). Applying the same methodology as above to THG techniques gives

4. Numerical results

4.1 Single-mode simulations

We now demonstrate the performance of RPPA on simulated single-mode d-scan traces in order to evaluate the algorithm’s performance in the typical scenario where a single coherent mode is present in the trace. All traces were simulated on a grid containing 512 spectral points and 500 grating positions evenly spaced over 20 mm. Pulses in the simulations were generated from Gaussian spectra centered at $\lambda _0 = 800$ nm with a full width at half maximum (FWHM) bandwidth of $\Delta \lambda = 31.4$ nm which can support pulses as short as $\Delta \tau = 30$ fs. However, we note that we observed that the algorithm retrieved any physical pulse we simulated. The simulation and retrieval grating phases were each modeled using 1400 lines/mm gratings with Littrow incident angle ($\theta _i = 34.05^{\circ }$). For noiseless traces, simulations were run until the normalized root mean square error (NRMSE)

Figure 4 shows the results of two retrievals on highly-structured, simulated single-mode traces generated from identical pulses. The simulated spectral phase content of the pulses in these simulations consist of third and fourth order phase as well as sinusoidal phase described by

Here, $\phi _{3,S} = 3.4\times 10^4\textrm { fs}^3$, $\phi _{4,S} = 1\times 10^6\textrm { fs}^4$, and $\alpha = 75$ fs. The retrieval in Fig. 4(a)–4(d) shows the result of a retrieval on a trace generated from a single pulse with the spectral phase in Eq. (18). In this simulation the trace, temporal intensity, spectrum, and spectral phase were all retrieved to machine precision.

 

Fig. 4. Simulated pulse retrievals of two different single-mode d-scan traces generated from identical pulses. The top row shows a retrieval for an ideal d-scan trace with no noise. The second row shows a retrieval when AWGN is added to the same trace with a SNR of 25. The leftmost column shows the simulated d-scan traces. The middle-left column shows the retrieved traces. The middle-right column shows the simulated (blue) and retrieved (red-dashed) intensity profiles of the pulses. The right-most column shows the simulated (blue) and retrieved (red-dashed) spectra as well as the simulated (purple) and retrieved (yellow-dashed) spectral phases.

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Figure 4(e)–4(h) shows the pulse retrieval on a trace with additive white Gaussian noise (AWGN) added to the trace. The AWGN for this simulation had a signal-to-noise ratio of 25. We will define the signal to noise ratio (SNR) as the peak intensity of the trace divided by the standard deviation of the background noise:

The retrieved trace agrees well with the simulated trace, and the pulse is retrieved extremely accurately. While the spectral phase is very accurately retrieved across the entire bandwidth, some small distortions of the retrieved spectrum can be seen which are the result of the large amount of AWGN in the simulated trace. The NRMSE for this retrieval was $\delta = 0.015$, which is significantly larger than the error in the noiseless simulation. However, as can be seen in Fig. 4(e)–4(h), the error is largely due to the noise on the simulated trace and the retrieved pulse closely matches the simulated one. This simulation is an extreme example of how robust the algorithm is to noise in an experiment. While the SNR in this simulation was 25, it is generally higher than 500 in typical measurements on our laser system. Supplement 1 further explores how imperfections in the d-scan trace and pulse train instabilities can effect the pulse retrieved by the algorithm.

It should be noted that in order to achieve accurate retrievals in the presence of noise, the raw data requires background subtraction or noise suppression before running the retrieval algorithm. The method of background subtraction we have chosen to employ uses a median threshold on the columns of the trace. The threshold is chosen as the standard deviation of the background noise and all values below the threshold multiplied by the median of the column are set to zero.

As a final simulated demonstration of RPPA, Fig. 2 shows a retrieval of a trace generated from two coherent pulses with the above spectral phase but with an additional time delay of $\Delta t = 150$ fs between the two pulses. Just as with the single pulse noiseless trace, temporal intensity, spectrum, and spectral phase were all retrieved to approximately machine precision ($\delta = 2 \times 10^{-15}$). This is expected as the two coherent pulses can be viewed as a single coherent field. Figure 2(e) shows the retrieval error for RPPA as well as PDPA when attempting to retrieve the pulses from the trace in Fig. 2(a). It can be seen that the non-root-preserving PDPA does not retrieve to machine precision even in the absence of noise, or when only a single pulse is present, while RPPA does (an improvement of over ten orders of magnitude). We attribute this to the fact that the method used in PDPA enforces a principal root to retrieve the fundamental field from the SHG signal which tends to stagnate the retrieval in a local minimum. In light of this, RPPA can be seen as a significant improvement in retrieval accuracy and stability over previous d-scan algorithms.

4.2 Multi-mode simulations

In this section we numerically demonstrate the ability of MM-RPPA to retrieve the spectral phase and amplitude of two mutually-incoherent pulses simultaneously from a single d-scan trace. To do so, two mutually-incoherent pulses, or modes, were simulated on the same trace. This type of simulation is representative of a multi-modal pulse train similar to the one in Fig. 1(c). The first mode was identical to the pulse in the single-mode simulations above. The second mode was the same as the first mode but with additional 75 mm of BK7 glass phase added to the pulse. This thickness of glass was chosen in order to simulate situations where the two modes are individually spectrally coherent, but the spectral interference fringes cannot be resolved by the spectrometer as the delay between the pulses is too long. The two modes were then added incoherently to generate the simulated trace. That is to say that the SHG signal in each column of the trace, $\tilde {I}(\omega )$, is the sum of the individual signals from the two modes:

The results of running the algorithm on this simulated multi-modal trace are show in Fig. 5. In the simulated trace, the sub-trace from each mode can be seen stacked on top of one another. The pulse profiles, spectral phase, and spectrum were all retrieved to machine precision for both modes. While in this example, we chose two pulses that had a large dispersive difference to correspond to the experimental results below, it is important to note that the algorithm can retrieve mutually-incoherent pulses even if their d-scan traces significantly overlap in the overall trace.

 

Fig. 5. Simulated pulse retrievals on a multi-mode trace who’s two pulses differ in dispersion by the dispersion of 75mm of BK7 glass. a) Simulated trace. b) Simulated (blue) and retrieved (red-dashed) pulse for the first mode. c) Simulated (blue) and retrieved (red-dashed) spectrum as well as simulated (purple) and retrieved (yellow-dashed) spectral phase for the first mode. d) Retrieved trace. e) Simulated (blue) and retrieved (red-dashed) pulse for the second mode. f) Simulated (blue) and retrieved (red-dashed) spectrum as well as simulated (purple) and retrieved (yellow-dashed) spectral phase for the second mode.

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5. Experimental results

5.1 Experimental setup

Lastly, we experimentally verify the performance of the retrieval algorithms using a home-built Ti:Sapphire laser system. The output of the regenerative amplifier and transmission grating compressor chain delivers 0.75 mJ pulses at 1 kHz with a $\Delta \lambda = 30.2$ nm FWHM bandwidth centered near $\lambda _0 = 803$ nm. Pulses were focused into a 50 $\mu$m KDP crystal using an f = 500 mm cylindrical mirror. Spectra were collected using a fiber-coupled Ocean Optics Flame spectrometer. A BG-39 bandpass filter was used to suppress fundamental light while collecting SHG spectra and was removed to collect reference fundamental spectra. A white light calibration of the spectral response of the spectrometer and filter was performed, and the phase matching response of the KDP crystal was accounted for to improve the accuracy of all experimental retrievals.

5.2 Single-mode retrieval

We demonstrate RPPA on a near transform-limited pulse from the system described in the previous section. The measured trace contains 500 grating separations over 10mm. A scanning FROG trace was also measured for comparison and a ptychographic algorithm for FROG [22] was used to reconstruct the pulse. The results of this experiment are shown in Fig. 6.

 

Fig. 6. Experimental measurement of the pulse from our home-built laser system. a) The measured d-scan trace. b) The measured FROG trace for comparison. c) The reference measured spectrum (grey), retrieved FROG (blue) and retrieved d-scan (red) spectra, along with retrieved FROG (purple-dashed) and retrieved d-scan (yellow-dashed) spectral phase. d) Retrieved d-scan trace. e) Retrieved FROG trace. f) Retrieved FROG (blue) and d-scan (red) pulses along with retrieved FROG (purple-dashed) and d-scan (yellow-dashed) temporal phases.

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The retrieved pulses from both the FROG and the d-scan show excellent agreement, with a retrieved duration of $\tau = 36$ fs FWHM duration. The spectral and temporal phases from the two methods also match each other quite well. Some small differences can be seen between the retrieved spectra for the two methods, and we attribute these largely to the different optics used in the FROG and d-scan devices. However, the retrieved d-scan spectrum matches the measured reference spectrum very well. The NRMSE between the measured and retrieved d-scan traces for this retrieval was $\delta = 1.6\times 10^{-3}$ and the NRMSE between the measured and retrieved FROG traces was $\delta = 4.6\times 10^{-3}$. The d-scan error is approximately an order of magnitude lower than previously reported errors obtained by other d-scan algorithms [15,19].

5.3 Multi-mode retrieval

Next, we demonstrate the ability of MM-RPPA to retrieve multiple mutually-incoherent modes (such as in Fig. 1(c)) simultaneously from a single measured trace experimentally. To generate a multi-mode trace, the timing of the internal Pockels in the regenerative amplifier was adjusted so that two pulses circulated simultaneously, with one of the pulses taking an extra round trip. The two pulses were separated in time by the round trip time of the amplifier, approximately 12 ns, which is well outside the typical temporal range of FROG devices. This is a similar to the above simulated example where both pulses are technically coherent, but do not temporally overlap enough for their spectral interference to be resolved by the spectrometer. A trace, shown in Fig. 7(a), was recorded while both pulses were present in the pulse train. The extra round trip significantly increases the dispersion of the second pulse, causing its SHG signal to shift along the dispersion axis of the trace. This trace was then processed by the multi-modal algorithm to retrieve both pulses in the train simultaneously, as well as their spectra and phase.

 

Fig. 7. Retrieval results using the multi-modal algorithm from an experimentally measured multi-mode d-scan trace. a) Measured multi-mode trace. b) Retrieved pulses and temporal phases for mode 1. c) Retrieved pulses and temporal phases for mode 2. d) Retrieved multi-mode trace. e) Retrieved spectra and spectral phases for mode 1. f) Retrieved spectra and spectral phases for mode 2. The blue and purple lines correspond to the results from the multi-mode algorithm and multi-mode trace, while the red-dashed and yellow-dashed lines correspond to results from the single-mode algorithm used on the comparison single-mode traces. Mode 1 corresponds to the leftmost mode in the trace and mode 2 corresponds to the rightmost mode in the trace.

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A single-mode trace was also measured for each of the two pulses by adjusting the Pockels cell timing so that either $x$ or $x+1$ round trips were taken in the amplifier. The zero position of the compressor was kept constant for all three traces and was chosen to be the point of best compression for the mode that took fewer round trips in the amplifier. The pulses and spectra retrieved from the single-mode traces were used as a reference for the results of the multi-mode retrieval. It should be noted that though we use the single-mode retrievals as a reference for the multi-mode retrieval, we do not expect them to be exactly the same for reasons that will be discussed below. To our knowledge no other measurement technique exists that can measure both pulses simultaneously to compare against. The results of the single-mode and multi-mode retrievals can be seen in Fig. 7(b)–7(f).

It should be noted that the multi-mode trace originally contained 1200 grating separations over a distance of 24 mm, rather than the 500 grating separations over 10 mm that the single-mode traces contained. The trace was then cropped to 501 grating positions so that the two modes were approximately centered on the cropped trace. This removed areas near the edges of the original trace that had low SNR, and ensured that the two modes were approximately equidistant from the edges of the trace. This helps ensure that the NRMSE of the retrieval monotonically decreases (i.e. does not oscillate) and that both modes are accurately retrieved.

It can easily be seen from Fig. 7(a) and 7(d) that the retrieved multi-mode trace matches the measured one quite well by eye, with only minor differences between the two. The NRMSE for this retrieval was $\delta = 1.1\times 10^{-2}$. This is approximately an order of magnitude higher than the error in the single-mode demonstration above, and we attribute this primarily to the fact that the laser becomes more unstable when operating in this multi-pulse regime.

Figure 7(b) show the retrieved pulse and temporal phase for Mode 1 from both the multi-mode and single-mode retrievals and Fig. 7(e) shows the retrieved spectrum and spectral phase for mode 1 from both the single-mode and multi-mode retrievals. The temporal and spectral phase from the multi-mode retrieval match the corresponding ones from the single phase retrieval quite well. Additionally, the temporal profile from each retrieval is nearly identical. There is a small discrepancy between the shape of the retrieved spectra with the multi-mode spectrum being slightly narrower and blue-shifted. We attribute this spectral discrepancy to the difference in gain dynamics in the amplifier when two pulses are present rather than just one. We expect less spectral gain shaping for each pulse when both are present in the amplifier.

Figures 7(c) and 7(f) show the temporal and spectral components of the retrieval for mode 2. Again, the temporal and spectral phases agree extremely well between the one- and two-pulse retrievals. When mode 2 and mode 1 are present in the amplifier at the same time the retrieved spectrum of mode 2 is red-shifted and broader relative to mode 1 which brings it more in line with the spectrum from the single-mode retrieval as is expected. Additionally, the temporal profiles for mode 2 are slightly different between the two retrieval methods due to the different gain dynamics in the multi-mode measurement. It is more significant for this mode as it contains more energy due to the extra round trip it takes. Lastly, it should be noted that the spectral phases are not affected by the gain dynamics because the propagation in the amplifier is fully linear and they are only affected by the material dispersion of the amplifier itself.

Based on the comparisons against the single-mode retrievals, there is significant evidence that the multi-mode algorithm is accurately retrieving both pulses in the pulse train. Additional verification of the retrieval can be found by analyzing the retrieved incoherent spectrum, spectral marginal, and dispersion marginal. A reference spectrum was measured for the multi-mode pulse train. This spectrum is the incoherent sum of the spectra from the individual pulses

To compare against this measurement, we can simply add the retrieved spectra from the multi-mode retrieval together. Figure 8(a) shows this comparison. While the individual multi-mode spectra do not perfectly match their single-mode counterparts, the retrieved incoherent spectrum matches the measured one extremely well.

 

Fig. 8. Further verification data from the multi-mode retrieval. a) Mutually-incoherent spectrum b) Spectral Marginal c) Dispersion Marginal. Measured quantities shown in blue lines and retrieved quantities shown in red-dashed lines.

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The spectral marginal is the integral of the trace $\tilde {T}$ along the dispersion axis, which is parametrized by the compressor separation here

In Fig. 8(b), it can be seen that the measured and retrieved spectral marginals agree extremely well. Lastly, the dispersion marginal is the integral of the trace along the spectral axis

Figure 8(c) shows a comparison between the measured and retrieved dispersion marginal. The two curves largely agree with only a small difference in the peak heights. The degree to which the incoherent spectra, and the two marginals match gives further evidence that the multi-mode algorithm accurately retrieved both pulses and their spectra from the multi-mode trace.

6. Conclusion

We have demonstrated, both numerically and experimentally, a root preserving ptychographic phase retrieval algorithm (RPPA) that is capable of solving two common problems in ultrafast pulse measurement. First, it is capable of choosing the correct root for the fundamental field with no external knowledge of the sign of the field or ancillary measurements needed. It does this through careful application of Newton’s square root method to the special case of SHG. This method can also be generalized to higher order processes such as THG. Second, RPPA can be extended to solve for multiple mutually-incoherent pulses simultaneously (MM-RPPA). This effectively allows d-scan to measure a finite number of distinct pulses in a single measurement regardless of the time delay between the pulses; a feature previously only demonstrated by multiplexed FROG [24]. This is a powerful tool for diagnosing pulses with different dispersion characteristics in laser systems and the stability of pulse trains [28]. The algorithm was also shown to be robust to both inherent noise in the measurement and instabilities in the pulse train. The ability of the algorithm to retrieve simulated pulses to machine precision and to retrieve experimental pulses with low error shows that our algorithm improves on previous d-scan phase retrieval algorithms [10,11,15], making it well suited for accurately retrieving the full spectral phase and amplitude of a pulse or series of pulses. It should be stated though, that special care must be taken to ensure that the dispersion of the apparatus is at least moderately well known, and that the spectral response of the apparatus (e.g. spectrometer calibration, phase matching) is correctly accounted for to achieve highly accurate retrievals. We conclude that the d-scan measurement technique, in combination with the ptychographic algorithm presented here, is a simple and effective tool for quantitatively diagnosing the pulse train from a laser system, as well as the laser system itself.