The average power of solid-state lasers has already reached levels of 100 kW in recent years due to the rapid development of laser diodes, the emergence of advanced thermal management technology, and new processing techniques. However, problems with beam quality control are becoming increasingly prevalent [1]. The thermal effect creates wavefront distortion, which in turn degrades the beam quality. To resolve this issue, researchers have devised technical approaches such as slab, thin disk, and fiber, and achieved substantial progress [2–6].

For an end-pumped, conduction-cooled, high-average-power (HAP) slab laser with a blocky gain medium, dynamic thermally induced wavefront distortion is caused by material boundary effects, uneven cooling conditions, unequal distribution of fluorescence and amplifier spontaneous emission noise (ASE), and uneven distribution of pumping light [7–11].

The existing wavefront aberration suppression methods are categorized as either passive or active aberration correction technology. Passive correction technology can lessen heat effects, but residual aberrations will remain [12]. An adaptive optics (AO) system is primarily used by to suppress wavefront distortion [13, 14]. However, these correction strategies did not account for the possibility that changing the distribution of the pumping light could compensate for the thermal effect within the crystal.

There have been significant advancements in the power, efficiency, life, cost, and other aspects of fiber pumped sources have made in recent years, and its advantages as a high-power solid-state laser pump source have gradually become apparent. When the pump source consists of some fiber pigtails, we can independently manage the output power of a single (group) fiber to alter the intensity distribution of the input surface of the coupling system, thereby achieving a specific customized intensity distribution of the pumping light. This method may be a novel technology for rectifying wave distortion.

This research offers a neural network algorithm whose output is the power matrix of the pump array to thoroughly adjust for the thermal effect at the source. The pump array consists of 100 fiber pumped sources. Once properly trained, a link between the input and output can be formed. The particle swarm optimization (PSO) algorithm enables the rapid solution of the pump array corresponding to the smallest wavefront aberration. The theoretically calculated beam quality is improved from the beam quality factor β 5.34 obtained by the uniformly distributed pump light to the beam quality factor β 1.28 obtained by the specific distribution of the pump light. Finally, the suppression of wavefront distortion is realized by controlling the pump array.

For the purpose of completing the numerical research, this paper is divided into three steps to achieve the optimal wavefront aberration solution. First, a numerical simulation model is built to simulate the corresponding black box of the pump array and the output wavefront aberration. Moreover, the neural network is used to fit or replace the black box. Lastly, using the PSO algorithm, a pump array corresponding to the optimal wavefront aberration is determined in reverse. In the laser amplification process, the thermal effect will be different depending on the characteristics of the pump light, the gain medium and the seed light. It would be complicated to build a simulation model for each of these features. On the one hand, to simplify the calculation, the simulation model can be simplified with a model without considering the laser amplification process because a simple model is enough to verify the feasibility of the neural network algorithm. On the other hand, our future study will be extended to the application of pulse lasers. Because of the energy storage characteristics of a pulse laser in the slab, the lasing process can be considered as an energy non-extracted state and energy extracted state. Ignoring the lasing process can be also considered as a validation of the energy non-extracted state. In this paper, the whole simulation model and experiments are both based on the calculation of the wavefront distortion of a He-Ne laser.

2.1. Numerical Simulation Model and Validation

A beam path in a slab [15, 16] with a length of L is shown in Fig. 1. The signal light is incident from the left end (x, −L/2), and the optical path D (x, y) experienced in the slab can be expressed as Eq. (1):

Figure 1. Schematic diagram of laser beam path in slab.

(1)$$D(x,y)=\frac{1}{\cos \varphi }{\displaystyle \underset{-L/2}{\overset{L/2}{\int }}\{n_0+\frac{\partial n}{\partial T}[T(x^{\text{'}},y,z)-T_0]\text{+}\sigma \frac{\partial \text{n}}{\partial \sigma }\}dz}$$

where x represents the thickness direction of the slab, y represents the width direction, z represents the length direction, n₀ is the refractive index of the material at temperature T₀, ∂n / ∂T is the thermo-optical coefficient of the material, σ is the stress in the material, and ∂n / ∂σ is the corresponding stress optical coefficient tensor. By ignoring the effect of stress on the refractive index, the optical path difference, which is the wavefront distortion due to temperature, can be expressed as Eq. (2):

(2)$$D_{\text{OP}}(x,y)=\frac{1}{\cos \varphi }{\displaystyle \underset{-L/2}{\overset{L/2}{\int }}\{n_0+\frac{\partial n}{\partial T}[T(x^{\text{'}},y,z)-T_0]\}dz}$$

This numerical model can be divided into an optical part and a thermal part. The optical part can use a tracing method to analyze the pump light passing through the slab and obtain the intensity distribution of the pump light. The thermal part can be expressed as simulating the temperature distribution T (x, y, z) inside the slab. Combining Eq. (2), the wave distortion can be obtained.

Figure 2 depicts the optical part of this simulation where the pump array is composed of 1 × 100 fiber-pumped sources. At the entrance of the slab, a beam of equally dispersed pumping light can be obtained by setting the power intensity of the pump array to full load (use 0–1 to indicate its intensity). We place several detectors in the slab as depicted in Fig. 3 to quantify the light intensity inside the crystal by using ZEMAX simulation software. With the long-distance transmission, it has been discovered that the uniform pumping light designed at the entrance will become uneven at the exit. The outcome is depicted in Fig. 4. At the same time, the intensity distribution of the pump light can be obtained. The total power of the pump is 100 W. The pumping area is 4 mm × 15 mm.

Figure 2. Three-dimensional model with Zemax simulation.

Figure 3. Two-dimensional model where some detectors are inserted into the slab.

Figure 4. The intensity distribution of pumping light along the length direction simulated by ZEMAX with the pump consisting of 100 fibers. The maximum intensity is 171.5 W/cm².

Numeral simulations have been performed on thermally induced wavefront distortion by researchers [17–20]. As shown in Eq. (2), thermally induced wavefront distortion is the accumulation of thermally induced refractive index variations along the length direction z. According to the absorption coefficient α of the pump light, after calculating the weighted average of the pumping light distribution along the length direction z, a two-dimensional heat transfer model can be developed. The thermal model is shown in Fig. 5, which is the thermal part of this simulation.

Figure 5. Two-dimensional heat transfer model with boundary conditions.

According to the heat transfer partial differential Eq. (3) [21, 22], the finite volume element integral Eq. (4) is obtained, and the integral derivation of Eq. (6) can be used to obtain the equivalent discrete algebraic equation of the two-dimensional heat transfer partial differential Eq. (6) [21], where a denotes the temperature coefficient, p denotes the controlled volume, and e, w, n, and s denote the grids divided into four directions: East, West, North, and South, respectively.

(3)$$\rho c\frac{\partial T(x,y)}{\partial \tau }=\frac{\partial }{\partial x}(\lambda \frac{\partial T(x,y)}{\partial x})+\frac{\partial }{\partial y}(\lambda \frac{\partial T(x,y)}{\partial y})+S(x,y)$$

(4)$$\begin{array}{l}{\displaystyle {\int }_{\tau }^{\tau +△\tau }{\displaystyle {\int }_w^e{\displaystyle {\int }_s^n\rho c\frac{\partial T(x,y)}{\partial \tau }dxdydt=}}}\\ {\displaystyle {\int }_{\tau }^{\tau +△\tau }{\displaystyle {\int }_w^e{\displaystyle {\int }_s^n(\frac{\partial }{\partial x}(\lambda \frac{\partial T(x,y)}{\partial x})+\frac{\partial }{\partial y}(\lambda \frac{\partial T(x,y)}{\partial y})+S(x,y))}}}dxdydt\end{array}$$

(5)$$S=S_u+T_p{S}_p$$

(6)$$\left\{\begin{array}{l}{\text{a}}_p{T}_p={\text{a}}_e{T}_e+{\text{a}}_w{T}_w+{\text{a}}_n{T}_n{\text{+a}}_s{T}_s+S_u△v+{\text{a}}_p^0{T}_p^0\\ {\text{a}}_p={\text{a}}_e+{\text{a}}_w{\text{+a}}_n+{\text{a}}_s+{\text{a}}_p^0-S_p△v\\ {\text{a}}_p^0=\frac{\rho C△v}{△\tau }\\ {\text{a}}_e={\lambda }_e\frac{△y}{\partial x_e},{\text{a}}_w={\lambda }_w\frac{△y}{\partial x_w},{\text{a}}_n={\lambda }_n\frac{△x}{\partial x_n},{\text{a}}_s={\lambda }_s\frac{△x}{\partial x_s}\\ △v=△x△y\end{array}\right.$$

Density, specific heat capacity, time, temperature, thermal conductivity, and source term are represented as ρ, C, τ, T, λ, and S, respectively. Length L_x, width L_y, thickness L_z and heat transfer coefficient h of the slab used in the simulation model as well as the experimental model are shown in Table 1. S can be divided into the independent source term S_u and the dependent source term S_p, shown in Eq. (5). The initial value of S_u as the independent source term represents the thermal effect of the absorbed pump light. The thermal effect is expressed as the product of the pump light intensity and the absorption coefficient, multiplied by the quantum loss efficiency η [21, 23]. Sp represents the dependent source term, given different initial values depending on the boundary conditions [22].

TABLE 1. The parameters used in Eq. (6).

Parameter	Value
λ	10 W/ (m. K)
T0	293 K
ρ	4,560 kg/m3
LX	0.004 m
LZ	0.16 m
Pump Power P	6000 W
η	11.6%
h	20,000 (W/ m2. K)
Tf	293 K
C	590 J/ (kg. K)
LY	0.042 m
dn/dT	7.3e-6/ K
α	0.1/ cm−1

Using fluorescence light, cooling, and adiabatic condition as three boundary conditions, the finite volume method with an additional source term method can be used to solve the two-dimensional heat transfer differential Eq. (6). The initial temperature of the liquid is T_f. After that, the thermally induced wavefront aberration in the width direction of the central region can be calculated according to Eq. (2). The parameters used in this simulation are shown in Table 1.

We also carried out experiments to verify the theoretical simulations, and the structure of the experiments is depicted in Fig. 6. First, 6,000 W of pump light were injected into the gain medium and the He-Ne laser was injected into the crystal, with a Hartmann detector being used to detect the transmitted wavefront. The initial temperature of the water flow is 293 K, and the water flow rate in the cooling region of the crystal reaches 3 L/min, and the convective heat transfer coefficient can reach 20,000 (W/m².K). The detected results are shown in Figs. 7 and 8. The simulation results are shown in Fig. 8.

Figure 6. Structure of the experimental system. After the pump light passes through the coupling system, the wavefront aberration of the He-Ne laser is measured by a Hartmann wavefront detector.

Figure 7. Wavefront distortion obtained through experiment along the width direction, peak to valley (PV) value = 18.8 μm.

Figure 8. Comparison between simulation results and experiments results. The peak to valley (PV) value (μm) of wavefront distortion distributed along the width direction.

It was found that there are some differences between the results obtained by numerical simulation and the experimental results in Fig. 8. We believe that this may be caused by ignoring the influence of thermal stress in Eq. (1) and the unevenness of cooling conditions and other complex florescent light inside. The liquid will take away energy when it absorbs heat so that the temperature of the fluid is not constant over time. However, there are no effects on the construction of our neural networks. In the theoretical study of wavefront distortion, it is not difficult for us to calculate the wavefront distribution according to the given physical conditions, but it is hard to obtain a specific wavefront distribution with unknown physical conditions. In other words, it is hard for us to do the reverse compute because we do not know which pump light distribution is related with the ideal wavefront. One way to solve this problem can be the use of various optimization algorithms. However, it may be difficult to realize because the calculation consists of a thermal part and optical part with a great calculative scale. Moreover, considering the more complex situation in the experiment, it is more difficult for us to solve the problem by only relying on the optimization algorithms, and the accuracy may be not good enough. On the one hand, in theoretical calculations, neural networks can combine a thermal part and optical part into a single calculation and speed up the calculation. On the other hand, by obtaining the data in the experiment, the neural network can also accurately characterize the relationship between input and output. Therefore, it is necessary for us to use a neural network to replace a simulation model or an experimental model. In this paper, we can obtain the training data through a simulation model to verify the feasibility of the neural network.

2.2. Back Propagation (BP) Neural Network

Artificial neural networks, which learn from samples and gradually improve performance to complete tasks, are one of the most widely used machine learning tools [24]. Neural networks have particularly good nonlinear mapping capabilities, and have unique advantages in solving regression and prediction problems in practical engineering. Therefore, a neural network algorithm can be applied to this optical system.

An artificial neural network contains a collection of connected units called artificial neurons, like axons in a biological brain. Each connection (synapse) between neurons can transmit a signal to another neuron. The receiving (post-synaptic) neuron can process the signal and then send a signal to the downstream neuron connected to it. The output y of the neuron can be expressed as

(7)$$\text{Y}=f({\displaystyle \sum _{n=1}^i{\omega }_i{X}_i-\theta })$$

(8)$$f(z)=\frac{1}{1+e^{-z}}$$

where ω represents the weight of neurons connected to the input signal, i represents the number of neurons, and X is the input signal. Each neuron receives input signals from other neurons, and each signal is transmitted through a weighted connection. The neuron adds up these signals to get a total input value, then compares the total input value with the neuron’s threshold θ (analog threshold potential), and finally obtains the final output through an activation function. The active function f can be expressed as Eq. (8). A typical neuron model structure is shown in Fig. 9.

Figure 9. A typical neuron model structure.

The back propagation (BP) algorithm is one of the most widely used algorithms for training neural networks [25]. The feature of such a network is that the signal is transmitted forward, and the error is propagated backward. In the process of forward transfer, the input signal is processed by a hidden layer, and finally reaches the output layer. The neuron state of each layer only affects the neural state of the next layer. If the output layer does not get the expected output, it will switch to BP, which can adjust the network weight and threshold according to the prediction error. At last, the predicted output of the BP neural network is constantly approaching the expected output. A topological structure of BP neural network is shown in Fig. 10.

Figure 10. Topological structure of back propagation (BP) neural network with three layers.

2.3. PSO Algorithm

The particle swarm optimization (PSO) algorithm is a swarm intelligence optimization algorithm in the smart computing field [26, 27]. This algorithm originated from the study of a bird’s predation problems. The PSO algorithm initializes a group of particles in the feasible solution space, and each particle represents a potential optimal solution to the extreme value optimization problem. Three indicators of position, speed and fitness value are used to express the characteristics of each particle. The particle moves in the solution space, and the individual position is updated by tracking the individual extreme value and the group extreme value. Every time a particle updates its position, the fitness value is calculated once, and the position of the extreme value is updated by comparing the fitness value of the new particle with the fitness value of the extreme value. The update equations are as follows:

(9)$$V_{id}^{k+1}=\omega V_{id}^k+c_1{r}_1(P_{id}^k-X_{id}^k)+c_2{r}_2(P_{gd}^k-X_{id}^k)$$

(10)$$X_{id}^{k+1}=X_{id}^k+V_{id}^{k+1}$$

where ω is the inertia weight, k is the current iteration number, V is the velocity of the particle, c is the acceleration factor, r is a random number distributed between 0 and 1.

In this paper, once the neural network has been well trained, the output (wavefront aberration) can be obtained when the input (random pump array) is given with the PSO method. Since the beam path of the signal light inside the gain medium is a zig-zag optical path, optical path difference (OPD) can be considered as the accumulation in the length direction of the optical path. The wavefront aberration in the longitudinal direction can be weighted and averaged as phase information and then substituted into the Fraunhofer far-field diffraction equation, expressed as Eq. (11) to calculate the electric field intensity of the light spot. Next, the spot radius and beam divergence angle θ_f can be calculated from the spot energy distribution. The beam quality factor β can be obtained by Eq. (12) where θ₀ is the far-field diffraction divergence angle obtained as a plane wave input.

(11)$$U(x,y)=\frac{e^{ikz}{e}^{\frac{ik}{2z}(x^2+y^2)}}{i\lambda z}{\displaystyle \int {\displaystyle {\int }_{-\infty }^{\infty }u(x\text{'},y\text{'})e^{-i\frac{2\pi }{\lambda z}(x\text{'}x+y\text{'}y)}dx\text{'}dy\text{'}}}$$

(12)$$\beta ={\theta }_f/{\theta }_0$$

Taking β as the fitness, when the fitness degree tends to converge, the best output (wave distortion) and the best input (pump array) can be obtained. The best fitness means the minimum wavefront distortion in this optical system. We can substitute the obtained pump array back into the optical system to obtain the real output, and then compare it with the best output to verify the accuracy of this PSO algorithm.

3.1. The Accuracy of Neural Network Algorithms

According to the methods mentioned above, a total of 10,000 groups of input and output data can be obtained. When performing neural network training, 9,000 sets of data are used as the training set, and 1,000 sets of data are used as the test set. A BP neural network with a double hidden layer is used to fit the input data and output data. The structure of this network is shown in Fig. 11. The matrix size of input is 100 × 1, and the matrix size of output is 92 × 1. The number of hidden layer nodes is 16 and the number of output layer nodes is 92. The structure of the two-layer BP neural network is 100-16-16-92. The relationship between wavefront distortion and pump array can be connected by the neurons with the weight of neurons ω_ij, k_ij, v_ij. The prediction error of the neural network is less than 1%, which can accurately characterize the nonlinear relationship between input and output. The prediction error is shown in Fig. 12.

Figure 11. The back propagation (BP) neural network with double hidden layers.

Figure 12. Prediction error of back propagation (BP) neural network with five in 1,000 sets of data.

3.2. Optimal Results with PSO Algorithm

In practical applications, since it is difficult to obtain a completely ideal wavefront distribution, that is, since it is close to a straight line, we are more inclined to adjust the wavefront to a distribution that is nearly quadratic, and then use a lens to eliminate defocusing. We usually regard the wavefront of the nearly quadratic function as our ideal wavefront. After 30 generations of iteration, the fitness β tends to converge. The global optimal β is about 1.26. The iterative process is shown in Fig. 13. At the same time, the output of the pump array is obtained, as shown in Fig. 14. The total power is 58% of the rated power.

Figure 13. The iterative process of the fitness β.

Figure 14. Output power of the pump array.

Substituting the best input into the simulation experiment platform for checking calculations, the thermally induced wavefront distortion is obtained, as shown in Fig. 15, and the pumping area is expanded to 4 mm × 23 mm. From the experience of our research group, we are more inclined to obtain a nearly quadratic-curvature wavefront distortion distribution, which can be changed to an ideal distribution by using a lens to eliminate defocusing. The result is shown in Fig. 16, and it can be seen that the OPD has dropped significantly. At the same time, substituting the thermally induced wavefront aberration into the Fraunhofer diffraction equation as the phase information to obtain the far-field pattern as shown in Fig. 17, the beam quality factors are calculated to be 5.34 and 1.28, respectively. It can be seen that the beam quality is greatly improved. The fitness obtained by the PSO algorithm is 1.26, which has a certain error compared with the beam quality factor 1.28 obtained by re-substituting the simulation model, but the experimental results are basically consistent.

Figure 15. The optical path difference (OPD) obtained by the uniformly distributed pumping light and the optimized pump array.

Figure 16. The wavefront distortion obtained by the optimized pump array after using a lens to eliminate defocusing.

Figure 17. Normalized far-field diffraction distribution obtained by the uniformly distributed pumping light and the optimized pump array.

The core idea of this paper is to replace the traditional LD stack with a controllable pump array. The intensity distribution of the pumping light can be modulated on a controllable basis. The thermal effect can be comprehensively compensated by adjusting the power of the fiber pumped sources. The beam quality factor has been improved from 5.34 to 1.28. In the simulation, the time required to calculate each set of data is about one minute, so that the total time required is about 166.7 hours. In actual application, the time required to obtain data will be very short, and the time required to obtain each set of input and output is about several milliseconds. Since the training data used in the simulation is obtained by the slab laser, the thermal compensation capability of this algorithm for solid-state lasers such as disk lasers and rod-type lasers need to be further verified. Moreover, this method used to train a laser is a new method to suppress wavefront distortion and has practical significance. Since not all the power is used when we adjust the pump source, the next research will focus on how to maximize the use of power and conduct experiments.

The authors declare no conflicts of interest.

Data available on request from the authors. The data that support the findings of this study are available from the corresponding author, upon reasonable request.

Innovation Development Fund of CAEP (C-2021-CX 20210047).