Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2022; 6(5): 479-488
Published online October 25, 2022 https://doi.org/10.3807/COPP.2022.6.5.479
Copyright © Optical Society of Korea.
Hang Liu^{1,2,3}, Ping He^{1,3}, Juntao Wang^{1,3}, Dan Wang^{1,3}, Jianli Shang^{1,3}
Corresponding author: *shangjianli@outlook.com, ORCID 0000-0002-2410-5611
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
To account for the internal thermal effects of solid-state lasers, a method using a back propagation (BP) neural network integrated with a particle swarm optimization (PSO) algorithm is developed, which is a new wavefront distortion correction technique. In particular, by using a slab laser model, a series of fiber pumped sources are employed to form a controlled array to pump the gain medium, allowing the internal temperature field of the gain medium to be designed by altering the power of each pump source. Furthermore, the BP artificial neural network is employed to construct a nonlinear mapping relationship between the power matrix of the pump array and the thermally induced wavefront aberration. Lastly, the suppression of thermally induced wavefront distortion can be achieved by changing the power matrix of the pump array and obtaining the optimal pump light intensity distribution combined using the PSO algorithm. The minimal beam quality β can be obtained by optimally distributing the pumping light. Compared with the method of designing uniform pumping light into the gain medium, the theoretically computed single pass beam quality β value is optimized from 5.34 to 1.28. In this numerical analysis, experiments are conducted to validate the relationship between the thermally generated wavefront and certain pumping light distributions.
Keywords: Back propagation neural network, Particle swarm optimization algorithm, Pump array, Wavefront distortion
OCIS codes: (140.2020) Diode lasers; (140.3290) Laser arrays; (140.3460) Lasers; (140.3480) Lasers, diode-pumped
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.
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 (
where
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 (
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.
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
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,
Density, specific heat capacity, time, temperature, thermal conductivity, and source term are represented as
TABLE 1 The parameters used in Eq. (6)
Parameter | Value |
---|---|
10 W/ (m. K) | |
293 K | |
4,560 kg/m^{3} | |
0.004 m | |
0.16 m | |
Pump Power P | 6000 W |
11.6% | |
20,000 (W/ m^{2}. K) | |
293 K | |
590 | |
0.042 m | |
7.3e-6/ K | |
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
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/m^{2}.K). The detected results are shown in Figs. 7 and 8. The simulation results are shown in Fig. 8.
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.
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
where
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.
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:
where
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 θ_{0} is the far-field diffraction divergence angle obtained as a plane wave input.
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.
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
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.
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.
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).
Curr. Opt. Photon. 2022; 6(5): 479-488
Published online October 25, 2022 https://doi.org/10.3807/COPP.2022.6.5.479
Copyright © Optical Society of Korea.
Hang Liu^{1,2,3}, Ping He^{1,3}, Juntao Wang^{1,3}, Dan Wang^{1,3}, Jianli Shang^{1,3}
^{1}Institute of Applied Electronics, China Academy of Engineering Physics, Mianyang, Sichuan 621900, China
^{2}Graduate School of China Academy of Engineering Physics, Beijing 100088, China
^{3}The Key Laboratory of Science and Technology on High Energy Laser, China Academy of Engineering Physics, Mianyang, Sichuan 621900, China
Correspondence to:*shangjianli@outlook.com, ORCID 0000-0002-2410-5611
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
To account for the internal thermal effects of solid-state lasers, a method using a back propagation (BP) neural network integrated with a particle swarm optimization (PSO) algorithm is developed, which is a new wavefront distortion correction technique. In particular, by using a slab laser model, a series of fiber pumped sources are employed to form a controlled array to pump the gain medium, allowing the internal temperature field of the gain medium to be designed by altering the power of each pump source. Furthermore, the BP artificial neural network is employed to construct a nonlinear mapping relationship between the power matrix of the pump array and the thermally induced wavefront aberration. Lastly, the suppression of thermally induced wavefront distortion can be achieved by changing the power matrix of the pump array and obtaining the optimal pump light intensity distribution combined using the PSO algorithm. The minimal beam quality β can be obtained by optimally distributing the pumping light. Compared with the method of designing uniform pumping light into the gain medium, the theoretically computed single pass beam quality β value is optimized from 5.34 to 1.28. In this numerical analysis, experiments are conducted to validate the relationship between the thermally generated wavefront and certain pumping light distributions.
Keywords: Back propagation neural network, Particle swarm optimization algorithm, Pump array, Wavefront distortion
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.
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 (
where
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 (
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.
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
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,
Density, specific heat capacity, time, temperature, thermal conductivity, and source term are represented as
TABLE 1. The parameters used in Eq. (6).
Parameter | Value |
---|---|
10 W/ (m. K) | |
293 K | |
4,560 kg/m^{3} | |
0.004 m | |
0.16 m | |
Pump Power P | 6000 W |
11.6% | |
20,000 (W/ m^{2}. K) | |
293 K | |
590 | |
0.042 m | |
7.3e-6/ K | |
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
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/m^{2}.K). The detected results are shown in Figs. 7 and 8. The simulation results are shown in Fig. 8.
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.
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
where
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.
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:
where
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 θ_{0} is the far-field diffraction divergence angle obtained as a plane wave input.
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.
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
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.
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.
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).
TABLE 1 The parameters used in Eq. (6)
Parameter | Value |
---|---|
10 W/ (m. K) | |
293 K | |
4,560 kg/m^{3} | |
0.004 m | |
0.16 m | |
Pump Power P | 6000 W |
11.6% | |
20,000 (W/ m^{2}. K) | |
293 K | |
590 | |
0.042 m | |
7.3e-6/ K | |
0.1/ cm−1 |