Experimental and numerical study on double wedge shock/shock interaction controlled by a single-pulse plasma synthetic jet

1.   Introduction

High-energy particles generated by accelerators are increasingly used in materials science, chemistry, biology, medicine and particle physics [1]. However, traditional radio frequency (RF) accelerators are limited by the ionization breakdown threshold of materials, typically achieving acceleration gradients below 100 MV m^–1 and are expensive to build [2]. In contrast, laser wakefield acceleration (LWFA) utilizes intense laser pulses to excite plasma waves, generating strong electric fields [3‒5]. Their acceleration gradient far surpasses that of RF accelerators, enabling the production of GeV-level electron beams within centimeter scales [6].

In the past two decades, remarkable achievements have been made in the field of laser-driven plasma acceleration, such as generating high-energy, low-emittance electron beams [7‒9]. Many theoretical and experimental investigations have been conducted to achieve these objectives [10, 11]. Recent studies have explored the use of chirped laser pulses to enhance laser wakefield and direct laser acceleration of electrons in the plasma-bubble regime, highlighting their advantages in electron trapping and energy gain [12]. Significant progress has also been made in generating multi-pC and multi-GeV electron beams in a centimeter-scale gas cell [13]. Additionally, the optimization process of key parameters for electron acceleration provides empirical support for external electron injection and optimization strategies. However, challenges remain in producing stable particle beams that meet the requirements for applications such as free electron lasers and as a compact injector for synchrotrons [14]. Most LWFAs currently operate in the self-injection regime [15‒17], where the plasma wave naturally captures and accelerates electrons from the background plasma. While this approach is widely used, it often produces electron beams with a relatively broad energy spread and limited controllability [18], making it less ideal for applications that demand high beam quality. One approach to improve the quality of the electron beam is through external injection in LWFA [19]. This method involves injecting high-quality electron beams generated by conventional particle accelerators into plasma waves at precise moments for further acceleration. By accurately controlling the spatiotemporal matching of laser pulses and electron beams during the injection process, efficient acceleration of the electron beams can be achieved. Additionally, external injection can more efficiently achieve a high-energy beam, as the injected beam starts at a higher energy level compared with self-injection schemes [20]. The external injection method combines the advantages of traditional accelerators and plasma accelerators, allowing for the acceleration of electron beams to the expected energy while preserving beam quality [21].

In nonlinear external injection LWFA, the generation of high-quality electron beams requires matching the relative position between laser and electron beams during the injection process [22]. Simultaneously, the acceleration process needs to balance multiple physical effects of laser nonlinearity coupling and plasma parameters. The interactions among multiple parameters lead to uncertainty in the final acceleration results [23, 24]. Machine learning, especially Bayesian optimization, has demonstrated significant effectiveness in addressing complex multi-objective optimization challenges across various fields, including plasma physics, acceleration science and light source applications. Its ability to efficiently explore parameter spaces and manage uncertainties positions it as an invaluable tool for optimizing the performance of LWFA systems [25‒29]. However, there is still limited detail on how to optimize laser and plasma parameters in the specific context of externally injected electron beams. Here we employ Bayesian optimization methods to manage uncertainty under multiple parameters and identify globally optimal solutions within certain parameter ranges [30].

The dynamics of an externally injected electron beam is simulated using a combination of the ASTRA code [31] and CSRtrack code [32], considering the effects of space charge and coherent synchrotron radiation [33]. The LWFA process is based on particle-in-cell simulations [34], integrated with multi-objective function Bayesian optimization [35]. Initially, we optimize the rising edge parameters of the plasma based on the injected electron beams with a bunch charge of 10 pC to ensure that external electrons are effectively injected into the plasma. Subsequently, by controlling laser parameters, plasma parameters and the relative position between the laser and the injected electron beams, we optimize the acceleration process of the externally injected electron beams in the wakefield, ensuring that the electron beams reach the expected energy without charge loss. Finally, we achieve high-quality electron beams with an energy of 1.5 GeV, normalized transverse emittance of 0.5 mm·mrad and a relative energy spread of 0.5% at 10 pC. These beam parameters meet the requirements for an injector based on the RF linear accelerator of the Wuhan Advanced Light Source, i.e. an energy spread below 0.5% and emittance better than 1 mm·mrad at 1.5 GeV [36], indicating that the electron beams at the exit of the externally injected laser plasma can potentially be used for accumulative injection into the fourth-generation synchrotron radiation light sources such as the Wuhan Advanced Light Source.

2.   Bayesian optimization process

Bayesian optimization is an efficient global optimization technique suitable for optimization of black-box functions that are noisy and expensive to evaluate [37]. It operates by constructing a probabilistic surrogate model of the objective function, which is computationally cheaper than direct evaluations of the objective function, and maximizes an acquisition function to determine the most promising query points. Figure 1 shows the basic steps of Bayesian optimization. Starting from an initial data point, the next most promising evaluation point is selected by building and updating a model to maximize the expected improvement in the objective function. This iterative cycle continually updates the model and selects the next optimal point, gradually optimizing the objective function in the parameter space to enhance optimization accuracy and efficiency [38].

Figure  1.  Basic flow chart of Bayesian optimization.

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In our Bayesian optimization procedure, the tree-structured Parzen estimator (TPE) model is employed to address high-dimensional optimization problems with nine hyperparameters [39]. The specific process of the optimization is as follows:

(i) Select 30 initial sets of hyperparameters and evaluate their corresponding four objective function values. Construct the initial dataset ${D}_{0}={\left\{\left({x}_{i},{y}_{i}\right)\right\}}_{i=1}^{30}$. Here, ${x}_{i}$ represents the i-th set of hyperparameters, while ${y}_{i}$ represents the values of the four objective functions evaluated for that set.

(ii) Train the TPE model using the initial dataset, dividing the hyperparameter space into ‘good’ and ‘bad’ points to build the acquisition function.

(iii) Define the acquisition function as:

where $k$ is a balancing parameter, with $p\left({y}_{\mathrm{g}}\left|x\right.\right)$ and $p\left({y}_{\mathrm{b}}\left|x\right.\right)$ denoting the probability densities of ‘good’ and ‘bad’ points, respectively.

(iv) Randomly sample 24 hyperparameter sets, then select the set ${x}_{\mathrm{n}\mathrm{e}\mathrm{x}\mathrm{t}}$ that maximizes $\mathrm{T}\mathrm{P}\mathrm{E}\left(x\right)$.

(v) Evaluate ${x}_{\mathrm{n}\mathrm{e}\mathrm{x}\mathrm{t}}$ to obtain ${y}_{\mathrm{n}\mathrm{e}\mathrm{x}\mathrm{t}}$, and update the dataset:

(vi) Repeat steps 2–5 until the 300 iterations are complete or convergence criteria are met.

In the process of LWFA using external injection, the following initial beam parameters are required to achieve high-quality electron beams: beam energy of 100 MeV, energy spread below 0.5%, transverse emittance lower than 1 mm·mrad, bunch length less than 1 μm and spot size not exceeding 20 μm. The optimization process needs to balance multiple objectives, such as maximizing acceleration efficiency while minimizing energy spread and transverse emittance. Multi-objective function Bayesian optimization can assist in finding optimal values for laser and plasma parameters, achieving the best choice among these objectives to enhance the overall quality of the final electron beams. Table 1 summarizes the main parameters and their ranges for the simulations of external-injection LWFA based on Bayesian optimization in this study. The selection of parameters is chosen to ensure that the key laser and plasma parameters are feasible for experimental implementation. Among these parameters, laser strength a₀ notably impacts the acceleration mechanism. To significantly enhance the energy gain of the electron beam, we set the range of a₀ between 1 and 4 to ensure nonlinear acceleration characteristics for the electron beam after entering the plasma. Figure 2 shows the plasma density profile along the z direction, where the parameters in the figure are example values. On the rising edge, the plasma density n_e(z) is described by

Table  1.  Main parameters of Bayesian optimization.

Parameter	Range
Laser strength (a0)	1.0‒4.0
Laser beam waist (w0)	25‒45 μm
Laser duration ($\tau$)	25‒45 fs
Plasma density (ne)	(1‒50) ${\times 10}^{21} \, {\mathrm{m} }^{-3}$
Plasma length (plateau)	50‒250 mm
Relative position between laser and electron beam (pos)	80‒150 μm
Plasma rising edge length (zr)	1‒30 mm
Plasma rising edge parameter (lr)	(0.1‒3) ${\times 10}^{-3}$ mm
Laser pulse focus position (zfoc)	0‒10 mm

 | Show Table

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Figure  2.  Plasma density profile. z_r represents the plasma rising edge length and z_foc represents the laser pulse focus position.

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where n₀ is the peak plasma density, z_r is the position where the plasma density reaches n₀ (the endpoint of the rising edge) and l_r is the characteristic length of the rising edge, which controls the rate of density increase.

Laser parameters and plasma parameters determine the electric field strength generated in the plasma, which affects the energy and energy spread of the electron beam at the plasma exit. Polarization in the bubble regime is set to linear with an angle of ${\dfrac{ {\text{π}}}{4}}$ to ensure that the transverse forces remain largely symmetrical during the electron beam acceleration process [40]. The rising edge parameters of the plasma will focus the spot size of the injected electron beam before main acceleration to meet the requirements of the acceleration process, ultimately influencing the emittance and transmission efficiency of the electron beam at the exit of plasma.

After optimization design with traditional accelerators, the externally injected electron beams have an energy of 100 MeV, a relative energy spread of 0.2%, a normalized emittance of 0.46 mm·mrad, a bunch length of 0.8 μm and a spot size of approximately 13 μm. The wakefield acceleration process uses multi-objective Bayesian optimization, with the objective functions including four aspects: minimizing the absolute difference between electron energy at the plasma exit and 1500 MeV, minimizing the ratio of relative energy spread to bunch charge, minimizing the normalized horizontal emittance and minimizing the normalized vertical emittance. Figure 3 illustrates the Bayesian optimization process based on these four objective functions, showing that as the number of iterations increases, especially after the 180th iteration, the four objective functions tend to stabilize. Furthermore, when the beam is accelerated to 1500 MeV, the minimum energy spread stabilizes at 0.2%, while the emittance remains stable at 0.5 mm·mrad. This indicates that the optimization process has reached a steady state and further iterations have minimal impact on the results.

Figure  3.  The evolutions of the four objective functions with the number of iterations: (a) absolute difference between electron energy and 1500 MeV, (b) ratio of relative energy spread to bunch charge, (c) normalized horizontal emittance, (d) normalized vertical emittance.

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Figure 4 shows the influence of nine parameters on the four objective functions during this simulation process. The effect of each parameter is illustrated by its corresponding bar chart. It can be seen that the energy spread of the electron beam at the plasma exit is primarily influenced by the waist radius of the laser pulse, while the emittance of the electron beam is mainly affected by the plasma density and the focal position of the laser pulse. Specifically, when the laser waist is either too small or too large, the intensity distribution within the focal region varies, leading to uneven acceleration of different parts of the electron bunch and resulting in increased energy spread [41]. Various plasma densities lead to different wakefield strengths, which exert varying transverse forces on the electron beam and thus affect its emittance [42]. Moreover, the focal position of the laser pulse is crucial for effective phase matching between the wakefield and the electron beam. Improper phase matching can lead to increased transverse oscillations and a corresponding rise in emittance [43].

Figure  4.  The influence of Bayesian optimization process parameters on the objective functions.

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To illustrate these relationships, figure 5 presents plots of energy spread as a function of the waist radius of the laser pulse and emittance as a function of plasma density. In figure 5(a), the energy spread reaches a minimum at approximately 41 μm, indicating the optimal waist radius for minimizing energy spread. Figure 5(b) shows a similar trend where the emittance reaches its minimum at around ${n}_{\mathrm{e}}=4\times {10}^{22} \, {\mathrm{m}}^{-3}$. These plots visually represent how variations in these parameters affect beam quality, highlighting the critical factors that need to be optimized to achieve high-performance wakefield acceleration.

Figure  5.  Impact of laser pulse waist radius and plasma density on beam quality: (a) energy spread versus laser pulse waist radius, (b) emittance in both x and y directions versus plasma density.

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To further explore the interaction between laser and plasma parameters, we systematically adjusted the laser strength ${a}_{0}$ and plasma density ${n}_{\mathrm{e}}$ while maintaining other parameters constant, recording the plasma length and electron beam energy spread when the beam energy reached 1500 MeV. According to reference [44], when the electron bunch length is constant, there is a corresponding relationship between the energy spread and chirp. The chirp can be defined as follows:

where $E'_{z}$ is the slope of the electric field in the z-direction, $L$ is the length of electron acceleration, i.e. plasma length, and $\bar{\gamma }$ represents the average relativistic Lorentz factor.

As shown in figure 6, as a₀ increases, $E'_{z}$ also increases, however, due to the decrease in $L$, when the electron beam reaches 1500 MeV, the energy spread ultimately exhibits a decreasing trend, with a minimum occurring at a₀ = 3.7. Conversely, when ${n}_{\mathrm{e}}$ decreases, $E'_{z}$ diminishes as well, and as the energy reaches 1500 MeV, the required $L$ increases, leading to a similar reduction in energy spread. Within this range, a balance point exists that minimizes the energy spread of the electron beam, highlighting the impact of parameter interactions on beam performance.

Figure  6.  The corresponding plasma length and energy spread when the electron beam reaches 1500 MeV: (a) variation of laser strength, (b) variation of plasma density.

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3.   Simulation results

Considering the target values of energy spread and emittance of the electron beam at the plasma exit, the results from the 245th generation are the most appropriate. Figure 7 shows the transverse and longitudinal phase spaces of the electron beam at the plasma exit. Figure 8 presents the variations of different parameters of the injected electron beam with the acceleration distance after entering the plasma. It can be seen that the emittance remains at 0.5 mm·mrad when the injected electron beam is accelerated to 1.5 GeV. Particularly, on the rising edge of the plasma, through transverse modulation of the electron beam, the beta value gradually decreases to meet the requirements of wakefield matching. There is no significant increase in energy spread during the acceleration process. Ultimately, as shown in figure 7(c), an ultrashort electron beam with a bunch length of 0.9 μm and an energy of 1.53 GeV is obtained. The energy spread of the electron beam at the plasma exit is 0.5% with a single-peaked energy distribution, and the energy spectrum accounts for less than 10 MeV. According to figures 7(a) and (b), the transverse emittance in the x and y directions of the electron beam at the plasma exit is evaluated to be 0.54 mm·mrad and 0.52 mm·mrad, respectively. The beam spot size is approximately 5 μm. Since the electron beam exhibits a divergent profile, the transverse size of the beam increases after exiting the plasma, and a set of quadrupole magnets can be added after the plasma to control the spot size. The obtained beam parameters are comparable with the previously reported values [45], while the output energy spread is obviously reduced.

Figure  7.  Phase space distribution of the electron beam at the plasma exit: (a) transverse phase space distribution in x, (b) transverse phase space distribution in y, (c) longitudinal phase space distribution.

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Figure  8.  Evolutions of electron beam parameters with acceleration distance in the plasma: (a) beta function, (b) normalized emittance, (c) energy spread, (d) energy.

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The simulation results show strong feasibility for the required parameters of the laser, plasma and externally injected electron beam. We need 840 TW for the laser power, while several laboratories have successfully achieved this with petawatt-level pulses. The plasma density of $4\times {10}^{22} \, {\mathrm{m}}^{-3}$ can be attained using the gas cell or a discharged plasma source. Additionally, the parameters for the externally injected electron beam have been simulated and the results meet the injection requirements for LWFA.

4.   Error analysis

In practical experiments and applications, the LWFA system faces various errors that can significantly affect the performance of the accelerator. Instabilities in laser parameters, plasma density and electron beam parameters introduce complex challenges to the acceleration process. For instance, fluctuations in laser strength a₀ and waist radius w₀ can alter the strength and uniformity of the accelerating field, while variations in plasma density may lead to instability in the acceleration field. Additionally, phase errors between the electron beam and the laser wakefield can result in uncertainty in energy gain and a decline in beam quality. These errors, when combined, may significantly degrade accelerator performance. Therefore, it is essential to conduct an in-depth error analysis.

In the error analysis of laser plasma acceleration, we consider five primary sources of error: laser strength a₀, laser pulse duration τ, waist radius w₀, plasma density n_e and the relative position between the laser and the electron beam ‘pos’. Based on the optimal solution obtained through Bayesian optimization, these parameter values were adjusted by 5% both upwards and downwards. Subsequently, 500 datasets were randomly generated, recording changes in the energy, energy spread and emittance of the electron beam at the plasma exit. A detailed error analysis was then conducted. The results are presented in figure 9.

Figure  9.  Error analysis of laser wakefield acceleration: (a) energy, (b) energy spread, (c) horizontal emittance, (d) vertical emittance.

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Based on the results of error analysis, the laser plasma acceleration system demonstrates the following performance metrics for key parameters. Figure 9(a) presents the histogram of the electron beam energy distribution, showing that the electron beam energy exhibits obvious fluctuations, with an root mean square (RMS) value of 58.44 MeV around an ideal value of 1530 MeV. This indicates that some instabilities may lead to a higher field amplitude. The resulting energy spread has a distribution with an RMS value of 0.21%, as shown in figure 9(b), with all results generally being below 1%. For transverse emittance, referring to figures 9(c) and (d), the values are primarily concentrated in the range of 0.5–0.6 mm·mrad, and the fluctuation is 0.06 mm·mrad apart from the ideal value of 0.5 mm·mrad in both the horizontal and vertical directions. These fluctuations in energy spread and emittance are within the acceptable range for injection into synchrotron storage rings. Efforts must be made to increase the energy stability, in order to make the laser plasma wakefield acceleration suitable for such practical applications.

In addition to the previously discussed errors, external injection LWFA faces several limitations and challenges. For instance, the pulse duration is generally less than a few femtoseconds, requiring precise timing synchronization during the injection process to ensure effective coupling with the laser pulse, and the micrometer-scale transverse size requires careful beamline matching to avoid loss of injection efficiency. Furthermore, to prevent increase in the emittance of the electron beam during the acceleration process, the Twiss parameter of the externally injected electron beam must be matched with the laser and plasma parameters.

5.   Conclusions

Bayesian optimization is utilized to effectively find the optimal solution in a high-dimensional parameter space. By matching the plasma rising edge and the injected electron beam, and optimizing laser and plasma parameters, the quality of the electron beam from the LWFA process is significantly improved. We achieve a high-quality electron beam with an energy of 1.5 GeV, a normalized transverse emittance of 0.5 mm·mrad and a relative energy spread of 0.5% at 10 pC. This result meets the beam parameter requirements of accumulative injection at the fourth-generation synchrotron radiation light sources such as the Wuhan Advanced Light Source. Additionally, the rapid convergence of the optimization process and the achieved results demonstrate that Bayesian optimization performs exceptionally well in handling complex physical simulation problems, particularly in the simulation of LWFA. It ensures a high beam quality while significantly reducing the optimization time. Based on error analysis, efforts must be made to control parameter errors in operation so as to improve the beam stabilities, especially for the beam energy. This study emphasizes the importance of parameter optimization, showing that matching of laser and plasma can significantly enhance electron beam quality. These improvements provide a way to quickly achieve high-quality electron beams in external injection LWFA, which has key implications for developing more efficient and compact particle accelerators.