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Abstract
Chirped pulse amplification (CPA) and subsequent nonlinear optical (NLO) systems constitute the backbone of myriad advancements in semiconductor manufacturing, communications, biology, defense, and beyond. Accurately and efficiently modeling CPA+NLO-based laser systems is challenging because of the complex coupled processes and diverse simulation frameworks. Our modular start-to-end model unlocks the potential for exciting new optimization and inverse design approaches reliant on data-driven machine learning methods, providing a means to create tailored CPA+NLO systems unattainable with current models. To demonstrate this new, to our knowledge, technical capability, we present a study on the LCLS-II photo-injector laser, representative of a high-power and spectro-temporally non-trivial CPA+NLO system.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The 2018 Nobel Prize in Physics was awarded to Donna Strickland and Gerard Mourou for their work on chirped pulse amplification (CPA) of optical pulses [1]. Since then, CPA systems have driven and rapidly expanded the fields of high-power lasers, ultrafast lasers, and nonlinear optics (NLO) and now underpin every realm of our lives with applications in medical and biomedical imaging and treatment [2–8]; precision machining in manufacturing, especially for semiconductors and electronics [9–13]; current and future telecommunications [14–17] and terahertz wireless and quantum communications [18–20]; national defense, security, and remote monitoring [21–25]; as well as basic sciences including fusion sciences [26], novel accelerators [27–29] and energy sciences [30–34], and strong-field physics [35–37]. Nevertheless, the ground-up design of CPA systems is non-trivial, especially when cascaded nonlinear subsystems are involved. Furthermore, optics and photonics are in the midst of a massive transformation as a host of machine learning (ML) techniques merge with the field [38–40], promising to revolutionize current approaches to the design and manufacturing of laser systems across academic, industrial and medical, and defense complexes.
The future of ML-optimized optics and photonics will rely on large amounts of data to train these networks and search algorithms. A comprehensive start-to-end (S2E) software model for these laser systems would be the ideal solution for quickly generating the large quantities of data required. Fortunately, there is a rich history in modeling CPA+NLO systems with Frantz and Nodvik [41] laying the groundwork for modern theory and simulation in laser pulse propagation in amplifiers [42,43]. Early frameworks for nonlinear pulse propagation were based on the unidirectional pulse propagation equation [44] and the generalized nonlinear Schrodinger equation (NLSE) [45,46]. Modeling these equations has required significant development in numerical techniques such as Runge-Kutta, Fourier split-step, and other spectral methods [47–49], leading to current state-of-the-art software packages [50]. However, these state-of-the-art packages focus on application- or framework-specific systems in isolation, are not modularized for scalability and reproducibility, and do not provide a means for reverse engineering and inverse design, inadvertently ignoring crucial interconnected aspects of cascaded nonlinear processes that hamper potential advances in optics and photonics brought about by exploratory research. We present an efficient, modular and generalizable S2E framework that addresses these critical gaps in modeling integrated, sophisticated CPA+NLO systems consisting of a host of spectro-temporal shaping, amplification, and nonlinear conversion stages. This framework is targeted for an emerging renaissance in data-driven computationally intelligent photonics design.
2. Framework design and operation
To demonstrate our model’s robustness, we take a complex CPA+NLO system (Fig. 1), mimicking the drive laser at SLAC National Accelerator Laboratory’s LCLS-II, the world’s most powerful X-ray free-electron laser (XFEL). It comprises a mode-locked oscillator (Fig. 1(a)), pulse shaper (Fig. 1(b)), CPA regenerative amplifier (RA) (Fig. 1(c)), and NLO upconversion (Fig. 1(d)). The LCLS-II photo-injector laser system is configured to use dispersion-controlled nonlinear synthesis (DCNS) [32], an NLO technique that combines noncollinear sum-frequency generation (SFG) and second-harmonic generation (SHG) to render temporally-shaped flat-top ultraviolet (UV) pulses.
Fig. 1. CPA+NLO model overview. Conceptual diagrams of an oscillator a, pulse shaper ($Fe^{i\phi }$) b, regenerative amplifier-based CPA c, and cascaded NLO stage d accompanied by spectral representations of e a mode-locked oscillator, f an example of amplitude and phase shaping applied by the pulse shaper, g amplified spectrally-shaped pulse, and h nonlinear conversion output in the spectrum resulting from $\chi ^{2,3}$ processes. A representation of the time-domain pulse is shown between each block in the simulation.
For the mode-locked oscillator, the model can read-in an experimental input spectrum or, alternatively, use a predefined pulse. Here, we use a transform-limited Gaussian pulse, typical of our system. The initial pulse has a temporal pulse width of $\tau _{\text {0,osc}}$ = 93.8 fs centered at $\lambda _{\text {0,osc}}$ = 1035.5 nm with a bandwidth of about $\delta \lambda _{0,\text {osc}}$ = 16.8 nm. This Gaussian profile when checked against our off-the-shelf oscillator has a Pearson correlation coefficient value of 0.982 in the time domain and 0.997 in the wavelength domain, indicating the close match of our system’s oscillator.
This signal then undergoes spectral phase and amplitude shaping. The pulse shaper is modeled as a transfer function (Eq. (1)) in the frequency domain (angular frequency $\omega$). The transfer function offers spectral amplitude shaping as a Gaussian notch with control over peak position ($\lambda _{\text {1,ps}}$ or $\omega _{\text {1,ps}}$), width ($\delta \lambda _{\text {1,ps}}$ or $\delta \omega _{\text {1,ps}}$), and depth ($\kappa$). Spectral phase shaping is controlled with first-order (delay or $\phi _1$), second-order (SOD or $\phi _2$), third-order (TOD or $\phi _3$), and fourth-order (FOD or $\phi _4$) dispersion [51]. Additionally, a super-Gaussian envelope function of width $\delta \lambda _{\text {0,ps}}$ (or $\delta \omega _{\text {0,ps}}) = 2{\times }{\delta }\lambda _{0,\text {osc}}$ and position $\lambda _{\text {0,ps}}$ (or $\delta \omega _{\text {0,ps}}) = \delta \lambda _{\text {0,osc}}$ (see Eq. (1)) is used to create a limited extent unitary function across the entire input spectrum. In Eq. (1), we recenter the frequency domain to have the transfer function peak at zero and thus reduce computational complexity. This shift requires a similar shift in the notch position as seen in the Gaussian describing the notch.
Post-shaping, the signal is amplified. The RA is modeled with the modified Frantz-Nodvik (mFN) equations for working across spectral bandwidth [43]. These equations update the spectral fluence for each successive pass through the amplifier, $J_i{\lambda }$, for N number of passes as follows (Eq. (2)):
The effective material cross-section is given by
These equations are used for both the pumping process and the amplification of the input seed pulse except the pumping process inversion factor is corrected by the upper-state lifetime of the medium, $\tau _{\text {gain}}$, with
where $\Delta t$ is the time duration of the portion of the simulated fluence. For both pumping and seeding, the cross-sections are adjusted to account for a rotated crystal or mixed-polarization operation (parameterized by $\psi$, discussed further in section 3.1) via a weighted average of the emission and absorption cross-section values for each axis of the crystal where $\psi$ is a set that represents these weights. The pump fluence is taken as a narrow peak centered at $\lambda _\text {P}$ with pump power P$_\text {P}$, and the input seed fluence is taken from the input field for the amplifier module and adjusted by the pump and signal mode radius, r. In the current simulation setup, this pumping and seeding process is taken as a single pass for demonstration purposes where the repetition rate of the experimental system is accounted for in the expected pulse energies and expected amplified spectral intensity. In both cases, a starting inversion fraction is required and, if unknown, can become another parameter to explore in reverse engineering a system. However, this formulation can be limited in treating true operation at high repetition rates without further modifying the inversion factor. Additionally, highly-engineered amplifier systems involve other “tricks-of-the-trade” to further improve output signal, including elements like etalons to adjust the interplay between spectrum build-up and material saturation.For our CPA-based RA system, the stretcher and compressor are well matched. Any discrepancies in added dispersion are negligible compared to the amount of added dispersion from the proceeding DCNS NLO process. Furthermore, the amplifier simulation handles spectral intensity, so we recombine the pre-amplifier phase post amplification. Therefore, any phase added by the pulse shaper, stretcher, or any other device will not directly impact the RA simulation, allowing us to continue with the perfectly matched stretcher/compressor phase. Nonetheless, should this not be the case for a specific application or scenario, our framework can accommodate non-perfectly matched dispersion.
The DCNS upconversion technique takes advantage of the ability to purposefully mismatch the post-amplification dispersion to obtain 10’s of picosecond flat-top UV pulses [32]. First, the method applies equal and opposite SOD and TOD to the original pulse and a copy of the pulse. This broadens the pulses and forms a triangular-shaped output if the input is Gaussian. The two pulses then undergo noncolinear sum-frequency generation (SFG). There are three outputs from the SFG–two parasitic SHG pulses and the desired flat-top, highly broadened, flat-phase SFG pulse. The SFG output from this process finally undergoes SHG for upconversion to UV.
The SFG and SHG processes are modeled by solving the wave equation with the slowly varying wave approximation [32,52] as shown below
The index of refraction is a combination of the ordinary and extraordinary axes for the crystal where the resultant index is a function of a crystal orientation angle $\theta$ as shown below
These equations are solved numerically using a Fourier split-step method [32,52] and are subject to several constraints. The selected discretization of spatial, temporal, and frequency domains can significantly affect the output. This discretization must be selected carefully and depends on the expected pulse widths, as one must ensure proper signal roll-off for clean Fourier transforms. Physically, the model assumes non-extreme mixing angles and uniform spatial beam since the presented model uses a 1D simulation code, sufficient here given a low B-integral.
For the integrated model, each stage of the simulator renders time- and frequency-resolved electromagnetic fields alongside essential physical characteristics of energy, fluence, and spectral distribution. The relevant parameters for each model are loaded externally, allowing for fast data generation. Table 1 shows the relevant parameters for the S2E model of the photo-injector laser system. Here, as new capabilities of this model, we present three primary use cases: 1) blackbox optimization for parameter estimation of CPA systems; 2) CPA laser tuning; 3) integration of cascaded CPA and NLO stages.
Table 1. Model parameters, symbols, and units
3. Results
3.1 Blackbox optimization
Considered trade secrets, commercial and defense CPA manufacturers do not typically disclose laser design parameters, essentially making their devices blackboxes. While it is not our intention to replicate any commercial products, accurately modeling off-the-shelf devices requires a reasonable understanding of these intrinsic variables since even small variations in some of these parameters–e.g. crystal length and orientation, dopant ion density, and mode radius–significantly alter the amplifier’s collective performance. For instance, increasing the dopant ion concentration by 10%, a typical tolerance for crystal manufacturers, may yield a 10% reduction in amplifier spectral intensity bandwidth and a near doubling in output energy in some CPA systems (see Supplement 1). More generally, the gain medium, concentration of dopant ions, temperature gradients, and mixing of gain crystal axes all significantly affect Stark energy levels, broadening mechanisms, and amplification and emission cross-sections [54–57]. Determining these or even controlling them with high accuracy can be used to mitigate amplification-driven instabilities [57–59], motivating specific material choices or the design of totally new gain media.
In this work, we use a Yb:KGW-based CPA (ytterbium-doped potassium gadolinium tungstate) medium known for its high absorption and gain, low lasing threshold, broad wavelength coverage, and high thermal conductivity [57,60,61]. To estimate the internal amplifier parameters, we perform a blackbox optimization using an exhaustive search through the CPA parameter space. For each set of parameters, we run the amplifier simulation and calculate the root-mean-square error (RMSE) between the simulation spectral intensity output and the experiment spectral intensity profile
where $n$ is the total number of elements in the spectrum. We set an additional minimum output energy threshold of 50 uJ and an error-weighting that punishes solutions that saturate early (see Supplement 1). The RMSE ensures closely matching spectral features while the filtering yields results aligned physically with the system under test. While there are many search algorithms to choose among, here we demonstrate with an exhaustive search as it helps build a fuller picture of the optimized parameter space (see Fig. 2(a)).Fig. 2. Estimating and optimizing CPA performance. Error heat maps from optimization search in a show a subset of searches over beam radius and combined parameter ($\xi$) for crystal orientation parameter ($\psi$) with min and max error results outlined in magenta and purple, respectively. Post-optimization results in b show normalized spectral intensity for input pulse (red), experiment CPA output (black), local min and max error outputs from a, and the global min error output from the entire parameter search as well as the corresponding energy build up (inset, with number of passes purposefully obscured). e and f show the amplified pulse at each pass in the RA overlayed in log scale and c and d show these pulses separated in linear scale for a pump power of 120 W and 240 W, left and right respectively. g shows the corresponding RA energy build-up over the passes.
Since the exact Yb$^{3+}$ dopant ion concentration, crystal geometry, mode radius, and transmission losses are unknown, we perform a scan over mode radius (0.1 to 0.8 mm), a parameter $\xi$ that combines crystal length and dopant ion density (1E20 to 4E24 m$^{-2}$), and a parameter $\psi$ that represents the percent contribution from each of the three crystal axes and represents the crystal orientation in the amplifier (a-axis 20–100%, b-axis 0–60%, and c-axis 0–60%).
Figure 2(a) shows the error heat maps for the full scan of mode radius, $\xi$, and for four out of the one-hundred total sets of $\psi$. The minimum and maximum errors for each heat map are highlighted. Figure 2(b) then shows the smallest error from these four heat maps (local minimum) and the largest error from these heat maps (local maximum) as well as the lowest error from the entire parameter scan (global minimum). The associated energy build-up for each is shown in the inset. The local and global minima result in spectral distributions resembling the experiment. The global minimum exhibits more gain narrowing and a slight asymmetry in the intensity that more closely matches the experiment.
3.2 CPA optimization
Controlling and tuning CPA output spectral bandwidth (BW), central wavelength, energy, and temporal pulse shape are essential for optimizing ground-up design of amplifiers and for tuning CPA parameters in situ. In an integrated system, these characteristics also affect downstream components. In this use case scenario, the S2E serves as an ideal playground for finding the optimal regimes of operation for the system in question.
It is well-known that pump power and number of passes significantly alter RA-based CPAs but capturing nuanced quantitative trade-offs can be challenging. For instance, taking the Yb:KGW CPA system parameters matched in Section 3.1 via blackbox optimization, we showcase control over bandwidth and pulse energy by tuning the number of passes and pump power applied to an initial 50 nJ seed pulse. 10 amplification passes yield a ~10 nm BW and a peak fluence of about 200 J/m$^2$ after amplification. Doubling the number of passes yields a 30% reduction in BW and a 10-fold increase in peak fluence. Bringing the number of passes to 60 then reduces the BW by another 60%, bringing the total output energy up to 3.5 mJ. Doubling the pump power in the CPA with 60 passes leads to a slight decrease in BW and decreases total energy to 2.9 mJ (Fig. 2(c–g)). This energy reduction with an increase in pump power comes naturally from saturation of the gain medium and the intra-cavity RA losses.
The gain narrowing is more pronounced at higher pump power yielding higher energy pulses with less number of passes (also see Supplement 1, Fig. S2). Yb-based RAs for example often have around 15-40 round trip passes with tunable ranges of repetition rate exceeding 1 MHz, where the pump power can be used to tune and optimize energy while avoiding side effects from many passes, such as limiting repetition rate or accumulating additional nonlinear phase. As seen in Fig. 2(g) at low pump power (120 W) saturation is reached around the 60th pass. At high pump power (240 W), saturation is reached at the 50th pass and amplification efficiency begins to decay thereafter. These are well-understood phenomena and in their own right show how the model framework can generate data with these simulations to inform ML-based search algorithms for CPA system parameter optimization. These results demonstrate a few of the knobs available for tuning the amplifier, especially for in-situ optimization but become even more important when integrated with the rest of the laser system.
3.3 CPA and NLO integration
The S2E model’s full capabilities are shown when combined with cascaded nonlinear processes. Incorporating CPA simulation reveals important, often overlooked subtleties in simulating cascaded processes, in particular the amplified spectrum’s effect on downstream pulse shape. More importantly, connecting pre-amplification pulse shaping with post-amplification upconversion where the shaping does not map linearly, opens the door to ML-based studies reliant on large amounts of data.
Our example system model from LCLS-II contains a pulse shaper, amplifier, and DCNS upconversion with SFG and SHG. Combining a programmable pulse shaper with a CPA system provides added flexibility in temporal and spectral shaping. The phase of the electric field can be used to change the temporal envelope of a pulse, and current technology can access the first four orders of phase with both programmable elements like spatial light modulators [62,63], acousto-optic devices [63,64], and emerging structured photonics [65–67] as well as static methods including prism-based shaping, grating-based shaping, and sculpting of pulses in fiber [32,68–72]. Altering the spectral envelope also affects the temporal pulse envelope, which can be exploited for amplitude pulse shaping and modulation but can also exhibit unintended consequences like spectral narrowing, spectral distortions, and wavelength shifts. Furthermore, applications such as pump lasers for OPCPA systems [73,74] and photo-injector lasers [75] require higher harmonics with specific or tunable pulse shapes. These can be produced via NLO systems proceeding shaping and amplification [76].
While some of these applications target flat-top pulses post-upconversion, other applications strive for particular time-domain pulse shapes pre-upconversion, such as wakefield accelerators where a triangular-shaped laser pulse can drive a highly sought-after electron-bunch charge distribution [73]. Figure 3(a) exhibits this pre-amplification triangle pulse (in light red) and Fig. 3(e) shows the corresponding frequency domain phase required for achieving it.
Fig. 3. Pulse shaper, amplifier, and upconverted pulses. Time domain (a–d) and frequency domain (e–h) intensity (solid) and phase (dashed) for pulse shaper (light red), CPA (dark red), and DCNS (purple) simulation outputs for various combinations of SOD, TOD, and spectral notches (each row). (a–d) have a separate time axis for the DCNS signal (top) and pulse shaper and CPA (bottom).
After amplification (in dark red), the time-domain intensity changes drastically from the desired triangle pulse at the output of the pulse shaper, showcasing how full simulation including amplification is necessary in order to understand pulse shape-driven dynamics. The upconverted output pulse from DCNS (purple) remains relatively flat-top but with an asymmetry in the lobe caused by the added phase from the pulse shaper. Shown in Figs. 3(b) and 3(f), flipping the signs of both SOD and TOD reverses the slope of the pulse shaper temporal output, mirrors the amplifier temporal output, and switches the asymmetry’s side for the upconverted temporal pulse. Figures 3(c) and 3(g) show how sensitive the dynamics are. With a 33% reduction in TOD compared to Fig. 3(a), the triangular pulse at the pulse shaper output begins to smooth and reduce its peak sharpness. The amplified signal change is less noticeable since the modified spectrum due to amplification is a more dominant effect. Similarly, the upconverted pulse shape remains relatively unchanged. Finally, in Figs. 3(d) and 3(h), we introduce a spectral notch and use the same amount of added phase as Figs. 3(a) and 3(e). This notch removes some frequencies resulting in additional oscillations in the pulse shaper and amplifier temporal envelopes. The upconverted temporal profile then exhibits a significantly raised flat-top profile of 10 ps in duration, which could be useful in the LCLS-II example case for enhancing X-ray production [32]. More broadly, this section showcases the ability to combine multiple distinct systems together to examine exploratory pulse shaping effects for applications in light by design or remediation of nonlinear pulse distortion [67]. Given its modular design, this framework facilitates swapping in-and-out various CPA and NLO system models, tuning and optimizing those models for the targeted application, and, ultimately, running full S2E simulations to develop a more complete understanding of the system. Nonetheless, during experimental implementation of ML-informed adaptive shaping, users should still employ protection systems, especially for the amplifier, as pulse shapers and other programmable devices can output unexpected transient states during optimization processes or while switching from one discrete set of parameters to the next, potentially damaging downstream components.
4. Conclusions
The advent of the CPA has a track record of dramatic impact on society with high-power laser systems affecting areas as disparate as medical procedures and diagnoses to national defense and security. In order to continue driving progress in the field and to prepare for an age focused around applying data-driven ML to CPA and NLO systems, accurate and efficient fully integrated S2E modeling is crucial. Our S2E framework addresses this critical simulation gap by providing an accurate, modular, and expandable simulator for use cases spanning from reverse engineering and blackbox optimization of hardware components and high-power laser and nonlinear systems, to optimization and tuning of current system designs, and the full-scale simulation of optical systems for data generation. This framework is poised to transform traditional approaches in laser science and engineering into translational discoveries and technologies impacting the design of new photonics and optical devices, impacting light-driven applications in chemical, biological, and medical applications, and accelerating energy sciences in fusion energy sources.
Funding
Basic Energy Sciences (DE-AC02-76SF00515, DE-SC0022559); Small Business Innovation Research and Small Business Technology Transfer (DE-SC0022464); National Science Foundation (2231334); Air Force Office of Scientific Research (FA9550-23-1-0409); National Defense Science and Engineering Graduate (NDSEG Fellowship).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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