1. Introduction

The National Ignition Facility (NIF) in the United States achieved a breakthrough in controlled nuclear fusion in December 2022, when it produced more energy output than laser input for the first time [1]. This milestone marks a significant step towards harnessing fusion energy, which has the potential to provide clean, safe, and abundant power for humanity [2]. The NIF laser system consisting of 192 beams requires precise focusing of lasers for fusion reactions by using optical components [3,4]. However, the wavefront distortion resulting from imperfections and thermal effects of optical components has become one of the major problems in the NIF [5–7]. High-accuracy online wavefront distortion measurement is vital for the performance, efficiency, and reliable operation of high power laser facilities [8,9].

In order to evaluate wavefront distortion, various wavefront sensing methods with distinct characteristics have been proposed. The Shack-Hartmann wavefront sensor (SHWFS) is widely used in high power laser facilities, including the NIF in the United States, the Laser MégaJoule Facility in France, and the Shen Guang-III Facility in China [10–14]. In these facilities, the SHWFS is used to monitor the wavefront distortions caused by thermal effects and optical imperfections in the laser system [15]. In principle, the SHWFS divides the incident beam as sub-beams through a piece of the microlens array and samples sub-wavefronts by a CCD. The original wavefront is reconstructed from the slopes of sampled sub-wavefronts through zonal or modal approaches. In high power laser facilities, the SHWFS mainly focuses on the wavefront distortion measurement of the whole system [16]. In practice, the wavefront distortion of the system should be first measured as a reference wavefront by the reference light without any distortions (i.e. the beam generated from a single-mode fiber laser). However, this calibration process is time-consuming in the high power laser facility, which consists of hundreds of beamlines [17]. Another effective method to measure the wavefront distortion for high power laser facilities is the near-field profile acquisition (NIA) method, which is proposed by Deen Wang et.al. in 2020 [18]. This method uses a cross mark (CM), which comprises a vertical nontransparent stripe and a transverse nontransparent stripe, at the focal plane of the optical route to directly obtain wavefront slope from the output near-field profile power spectral density (PSD) image detected by a CCD camera. Compared to traditional wavefront sensing methods, the NIA wavefront sensing method could directly acquire the wavefront distortion without using reference wavefront. Furthermore, Xuewei Deng et.al. improves the wavefront retrieval accuracy of the NIA method by flexibly dividing the near-field into more sub-apertures based on a CM inserted at the Fourier plane [19]. However, due to the one-by-one sub-beams scanning process during the near-field profile acquisition, the wavefront sensing speeds of the NIA methods are also limited [18,19].

Recently, deep-learning technology has been found to be applicable to wavefront sensing. The essence of the deep-learning-based wavefront sensing method is to use massive data to establish the relationship between CCD images and wavefront distributions [20–24]. Guo et al. predict low-order Zernike coefficients from the calculated x and y spot displacements, which are based on the back-propagation (BP) neural network [25]. Nishizaki et al. presented a deep-learning wavefront sensing method to estimate the first 32 Zernike coefficients of the wavefront with a single-intensity image and verified the feasibility with an overexposure, defocused, or scattered image [26]. Swanson et al. reconstructed the wavefront based on x and y slopes through a U-net and long-short-term memory (LSTM) network [27]. Hu et al. used a convolutional neural network (CNN) to directly predict the Zernike coefficients from the output spot array images, which improves the measurement precision for the high-order aberration [28]. Hu et al. utilized a customized U-net architecture to perform a direct reconstruction of the wavefront distribution, resulting in a reduced wavefront error [29]. Du Bose et al. present a wavefront reconstruction network named intensity/slopes network (ISNet) to reconstruct the wavefront from the intensity of sub-apertures and the wavefront slopes, improving the accuracy of wavefront reconstruction in the case of non-uniform illumination [30]. The above work provided new approaches for accelerating wavefront retrieval by using deep-learning technology, which expands the application range of conventional wavefront sensing methods. However, the conventional deep-learning-liked methods exhibit some limitations in achieving high precision in wavefront reconstruction, caused by the sole concatenation of same-level skip connections within these networks.

In this paper, we proposed a deep-learning-based single-shot wavefront sensing (SSWFS) method by using near-field profile image and fully-connected retrieval neural network (F-RNN) to accelerate the wavefront distortion measurement in high power laser facility. Compared to other deep-learning-liked methods, the F-RNN has improved the utilization of high-level feature maps from encoder layers to decoder layers and the multi-scale image information. In this method, the F-RNN adopts fully-skip cross connections (FSCC) to extract and learn complex feature maps of near-field profile images for various layers. The architecture of the FSCC could make the network structure more stable and accelerate wavefront retrieval. Based on the proposed F-RNN, the wavefront distortion in high power laser facility could be online measured in high accuracy by using one single-shot near-field profile image. This paper is organized as follows. In Section 2, the wavefront retrieval configuration and principle of the SSWFS method are illustrated. Based on the principle, the network structure of the F-RNN is further proposed. In Section 3, a numerical simulation model is set up. The training process and the prediction results are illustrated. The impact of noise when using the SSWFS method is analyzed. In Section 4, an experiment is carried out and the experimental results are discussed.

2. Configuration and principle

2.1 Wavefront retrieval configuration and principle

Figure 1 shows the configuration for wavefront distortion measurement of the incident beam by using the single-shot wavefront sensing method in high power lasers. In this configuration, before the high power laser working, an ideal laser beam without any distortion is incident and collimated into a parallel beam. The beam passes through a beam splitter (BS) and is introduced distortion through a spatial light modulator (SLM), and then the distorted laser beam passes through a 4-f system, which consists of lens1 (L1) and lens2 (L2). After passing through L2, the beam is collimated and recorded by a near-field CCD (N-CCD, a CCD camera used for detecting the near-field intensity) camera. A CM is inserted at the focal plane of the L1 to serve as a position mark. The focused beam is modulated by the CM. After beam transmission, a near-field profile image records the optical intensity distribution involving convoluted information of the CM and the focus. An F-RNN is built to establish the numerical relationship between the image and wavefront distortion. During operation, a massive amount of wavefront distortions is generated by the SLM, and the corresponding near-field profile images of the incident beam are acquired by the N-CCD camera. Taking the near-field profile image as input and the wavefront distortion introduced by the SLM, the F-RNN can be substantially trained. During the training process, the near-field profile images pass through a complex FSCC network structure consisting of various layers. The image first goes through a contracting path, which includes multiple unpadded convolutional layers, non-linear ReLU layers, and MaxPool layers for downsampling. Each downsampling step doubles the size of the feature map while halving the number of feature channels. Afterward, the feature map undergoes an expansive path for deconvolutional operations. Each deconvolution step doubles the size of the feature map while halving the number of feature channels. After deconvolution, the deconvolved result is concatenated with the corresponding feature map from the downsampled step of the contracting path. The concatenated map is then convolved again by the fully-connected layers. The feature map is transformed into an output wavefront distortion image with a predetermined channel number and size. Finally, the network will finish training until the training objective decreases to a predetermined value. In practice, the SLM is turned off after training and it will not introduce any aberrations once deactivated. After being modulated through the CM, wavefront distortion of the incident beam could be directly obtained through a one-shot N-CCD image and the trained network. By using the SLM to generate different aberrations, a high-quality dataset consisting of a massive quantity of wavefront distortions and near-field profile images is constructed. The dataset could help improve the efficiency of network training and the accuracy of predictions. Compared to the traditional network, the F-RNN utilizes fully-skip cross connections to capture feature maps across multiple scales from various layers. This approach greatly improves the efficiency of wavefront distortion retrieval.

 

Fig. 1. Configuration for wavefront distortion measurement of the incident beam by using the single-shot wavefront sensing method in high power lasers. BS: beam splitter, SLM: spatial light modulator, L1: lens1, L2: lens2, CM: cross mark, N-CCD: near-field CCD, F-RNN: fully-connected retrieval neural network.

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2.2 Network structure of the F-RNN

In the SSWFS, the F-RNN is critical to the single-shot wavefront retrieval of the laser beam distortion. As illustrated in Fig. 2, the F-RNN consists of an FSCC structure unit [Fig. 2(b)], an input unit [Fig. 2(a)] and an output unit [Fig. 2(c)]. The input unit includes the near-field profile images recorded by the N-CCD, while the output unit includes the corresponding distortions generated by the SLM. Figure 2(b) shows the FSCC network architecture, which consists of five stages. Each stage of the FSCC is pairwise connected to extract intricate feature maps of the input images across multiple layers.

 

Fig. 2. Architecture of the F-RNN.

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In contrast to the conventional architecture, the FSCC is structured to extract and assimilate multi-scale feature information across diverse stages, thereby significantly enhancing the wavefront retrieval detail. Figure 2(d) illustrates the detailed architecture of the five stages in the FSCC unit. In STAGE 0, a near-field profile image of 2048 × 2048 pixels is input. The feature extraction is performed by a 7 × 7 convolutional kernel with a stride of 2, resulting in a halving of the feature map resolutions. Subsequently, four MaxPool layers are applied to further reduce the feature map resolution. From STAGE 1 to STAGE 3, the number of channels in the input feature map is doubled while its spatial dimensions are halved. Specifically, each stage consists of one downsampling block and two residual blocks. When concatenating different stages, the parameters, including the size of the convolution and pooling windows and the stride in the convolution and pooling layers, are specifically set to unify the sizes of the feature maps from different layers. The downsampling block sets the initial convolution stride to 2, downsampling the feature map and reducing its spatial dimensions. In STAGE 4, the obtained feature vectors are concatenated into a 2048 × 7 × 1 feature map, downscaled to 2048 × 1 × 1 dimensions using grouped convolution layer (kernel size:7 × 1, groups: 512). There are a large number of 3 × 3 convolution operations in STAGE1 to STAGE4. But as many more layers are added to the network for F-RNN, we cannot afford to have so much of our GPU RAM wasted on those expensive 3 × 3 convolutions, so BottleNeck Blocks (BNB) are used. A BNB uses a 1 × 1 convolution to reduce channels of the input before performing the expensive 3 × 3 convolution, then using another 1 × 1 to project it back into the original shape. The BNB1 has four adjustable parameters: C, W, C1, and S, representing the input shape (C, W, C1, S). The BNB2 has two adjustable parameters: C and W, representing the input shape (C, W, W). The input with shape (C, W, W) is set as x. The three convolutional blocks on the left side of BNB2 are set as the function F(x). After adding them together (F(x)+x) and passing through one ReLU activation function, the output of BNB2 can be obtained, which still has the shape (C, W, W), i.e., project the shape back into the original shape. The BNB1 corresponds to the case where the input x and the output F(x) have different channel numbers. Compared to BNB2, BNB1 has an additional convolutional layer on the right side, denoted as the function G(x). This added convolutional layer transforms x into G(x), matching the dimensions of the input and output channels, i.e., G(x) has the same number of channels as F(x), enabling their summation as F(x)+G(x). In BNB1 and BNB2, the convolution-related parameters are adjusted to ensure that the input and output feature map dimensions of the residual blocks are consistent, enabling element-wise addition and avoiding the issues of gradient vanishing and degradation in deep-learning networks. After the feature maps are extracted by the five stages of multi-scale layer connections in the FSCC architecture, the fully-connected layers output the predicted wavefronts of 2048 × 2048 pixels.

Figure 3 shows the comparison of the SSWFS and the traditional NIA methods proposed by Deen Wang [Figs. 3(a1) to 3(a4)] and Xuewei Deng [Figs. 3(b1) to 3(b4)]. The traditional NIA methods require one-by-one segmentation and scanning of each sub-beam in a practical high power laser facility, and the accuracy of wavefront retrieval will be affected by the dividing number of sub-aperture divisions. The inconvenience would take some trouble for the wavefront distortion acquirement, and it is not flexible for high-precision online measurement due to the segmentation and scanning process. Different from the traditional NIA methods, the SSWFS method [Figs. 3(c1) to 3(c2)] does not need to sub-beams dividing and sub-aperture scanning process. The SSWFS method could directly obtain wavefront distortion in high accuracy by using the trained F-RNN based on one near-field profile image acquired by N-CCD. Thus, the wavefront distortion in high power laser facility could be flexibly measured online at high speed.

 

Fig. 3. Comparison of the SSWFS and the traditional NIA methods.

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3. Simulation and analysis

3.1 Numerical simulation

To validate the single-shot wavefront sensing method, we conducted a numerical simulation consisting of a wavefront training process and prediction process. In the simulation, the wavelength of the input ideal beam is set as 1053 nm. The beam aperture is 360mm × 360 mm. The focal length of each lens is 13 m. The sampling resolution of the imaging plane of N-CCD is 2048 × 2048 pixels. The resolution of the SLM is 2048 × 2048 pixels. The frequency of aberrations changed by the SLM and the frequency of N-CCD exposure are both set as 1 Hz. A CM with a width of 60 samplings is placed at the Fourier plane of the 4-f system while its length is set long enough to cover the entire focal spot. The network is implemented using a PyTorch framework based on Python 3.8 and trained on a local computing server. A batch size of 64 is used as the minimum input, and all datasets are trained for 120 epochs. The optimizer is initialized with a learning rate value of 0.01, which controls the rate at which the network learns from the training data. As the training progresses, a predetermined training objective function threshold of 0.60% is set, and Adam dynamically adjusts the learning rate value based on the network's performance, ensuring that the network is optimized at an appropriate pace and that the optimization process remains stable. Adaptive learning rates and momentum are combined by the Adam optimization algorithm, enabling it to handle sparse gradients and noisy data more effectively [31,32].

During the simulation, an ideal laser beam with no distortion is first collimated into a parallel beam before the high power laser working. The parallel beam then passes through a BS and is subjected to distortion by an SLM. The SLM randomly changes different Zernike coefficients to generate different aberrations. The simulated dataset includes the near-field profile images and the aberrations, which could be calculated using Eq. (1).

Figure 4 presents the trend of the loss function for network training. The plot shows a continuous reduction of the loss function as the training time increases. This trend indicates that the network is learning the mapping between the input images and the corresponding wavefront distortions accurately. Meanwhile, the plot shows that the loss function of the validation also reduces continuously and ultimately converges gradually with the training. The training required to prepare the network would take about 6 hours for 120 epochs with 100,000 datasets. It could be optimized to shorten the training time by improving efficiency and prevent overfitting through various optimization techniques, such as batch normalization and regularization. No overfitting is observed for the training, which suggests that the network could have a good balance between accuracy and generalization ability. The absence of overfitting also indicates that the network could effectively capture the underlying patterns of the training data and avoid learning the noise of the training dataset. Overall, the plot shows that the functional relationship between the input images and the output wavefront distortions could be well-fitted, and the network has learned to accurately predict wavefront distortions from near-field profile images. After training, the SLM is deactivated. Subsequently, when the incident beam is modulated by the CM, the wavefront distortion can be accurately predicted through a single capture of the N-CCD image and the utilization of the trained network.

 

Fig. 4. Training curves plot of F-RNN.

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Table 1 presents the prediction results of the F-RNN trained by dataset-A to dataset-F. The values of Table 1 are the average values of all 100,000 data of each dataset, including the PV value and RMS value in micrometers (µm), the prediction time in milliseconds (ms) and the loss function in percent (%). In this simulation, the PV value and RMS value are two important metrics to evaluate the prediction performance of the wavefront distortions. The predicted PV value and RMS value closely align with the actual values, indicating a higher level of accuracy and better prediction performance of the network. The prediction time for each wavefront distortion is another important metric in practical applications, which indicates the efficiency of network prediction. The loss function represents the residual error between the predicted wavefront distortion and the actual wavefront distortion, which represents the accuracy of network prediction. A lower loss function indicates that the network is better able to capture the underlying patterns and characteristics of the feature maps, and as a result, has better predictive performance. Through comparison, the predicted PV value and RMS value of datasets-A to F are approximately equivalent to actual values. The loss functions of different datasets are proximate to 0.53%. The prediction time of each wavefront distortion is within 0.14 ms, which indicates that the network could accurately predict wavefront distortion in a short period of time.

Table 1. Training results of simulation.

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Table 2 presents a comparison of the training results by using the F-RNN and other conventional architectures of CNN and U-Net on various datasets labeled as dataset-A to dataset-F. For all datasets, the CNN achieves a prediction time ranging from 0.85 ms to 0.90 ms. The U-Net could reach a slightly lower prediction time, ranging from 0.32 ms to 0.38 ms, while the F-RNN exhibits the fastest prediction speed with a prediction time ranging from 0.11 ms to 0.14 ms. Regarding the loss function, which measures the deviation between predicted and actual values, CNN achieves values between 5.29% and 5.48% for all datasets. The U-Net yields lower loss function values, ranging from 3.98% to 4.10%. The F-RNN consistently achieves the highest prediction accuracy with the lowest loss function values ranging from 0.51% to 0.55%. Compared to the conventional architectures of CNN and U-Net, the F-RNN could derive more stable results and accelerate wavefront retrieval.

Table 2. Training results by using different network architectures.

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Figure 5 and Table 3 present the prediction results with different distortions. Figure 5(a) shows the actual wavefront distortion from dataset-G, while Fig. 5(b) is the wavefront distortion predicted from the F-RNN trained by dataset-G. Figure 5(d) shows the actual wavefront distortion from dataset-H, while Fig. 5(e) is the wavefront distortion predicted from the F-RNN trained by dataset-H. Figures 5(c) and 5(f) show the difference between the actual wavefront distortion and the predicted wavefront distortion. Figure 5(g) depicts a comparison between the predicted Zernike coefficients ranging from Z₁ to Z₁₅ and the corresponding actual coefficients from dataset-G. The comparison between the predicted Zernike coefficients ranging from Z₁ to Z₁₅ and the corresponding actual coefficients from dataset-H is presented in Fig. 5(h). The predicted Zernike coefficients are closed in proximity to the actual values, which suggests that the network has effectively learned to capture the underlying patterns and relationships within the training data, enabling it to generate accurate and reliable predictions. For dataset-G, the actual PV value is 0.2086µm and the predicted PV value is 0.2032µm, while the actual RMS value is 0.1535µm and the predicted RMS value is 0.1543µm. For dataset-H, the actual and predicted PV values are 5.0504µm and 5.0535µm, and the actual and predicted RMS values are 3.9379µm and 3.9158µm. The predicted values of PV and RMS from datasets-G and datasets-H are approximately equivalent to actual values. The loss function of the two datasets is below 0.56%. The results obtained from this simulation results provide compelling evidence that the network could consistently and accurately predict a wide range of wavefront distortions. Furthermore, the SSWFS method is versatile and could precisely work for predicting wavefronts up to higher Zernike coefficients (e.g., 36 Zernike coefficients). Consequently, the network could meet the rigorous demands associated with the measurement of complex wavefront distortions in high power lasers.

 

Fig. 5. Prediction results with different distortions. Actual wavefront distortion from (a) dataset-G and (d) dataset-H. Predicted wavefront distortion from F-RNN trained by (b) dataset-G and (e) dataset-H; (c) and (f) show the corresponding residual errors relative to the actual wavefront distortion. The curves between predicted Zernike coefficients and actual Zernike coefficients from (g) dataset-G and (h) dataset-H.

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Table 3. Prediction results with different distortions.

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3.2 Noise analysis

In high power laser facilities, noise is one of the main factors affecting the retrieval of wavefront distortion. In order to find the impact of noise when using the SSWFS method, Gaussian noise and Poisson noise are incorporated into the wavefront distortions by the SLM, generating dataset-I and dataset-J each of which contains 100,000 samples respectively. Within dataset-I, the PV value of Gaussian noise ranges from 1λ to 3λ and is divided into ten groups with an average of 10,000 sample pairs collected per group. Within dataset-J, the PV value of Poisson noise ranges from 1λ to 2λ and is similarly divided into ten groups, with 10,000 sample pairs collected per group on average. The standard of Gaussian noise and Poisson noise are set as 0.05 and 0.08, respectively. Within each of the datasets, 99,000 samples are randomly selected as training data, while 1,000 samples are allocated for validation purposes. In each of the datasets, photon noise, and detector readout noise are also incorporated into the wavefront distortions. The inclusion of these noise factors helps to improve the network's ability to handle unexpected fluctuations in the input data. By training the network with these additional factors, it is expected that the network would be better applied to perform reliably in practical high power lasers with various noises. The prediction results with different noises are shown in Fig. 6 and Table 4.

 

Fig. 6. Prediction results with different noises. Actual wavefront distortion from (a) dataset-I and (d) dataset-J. Predicted wavefront distortion from F-RNN trained by (b) dataset-I and (e) dataset-J; (c) and (f) shows the corresponding residual errors relative to the actual wavefront distortion. The curves between predicted Zernike coefficients and actual Zernike coefficients from (g) dataset-I and (h) dataset-J.

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Table 4. Prediction results with different noises.

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Figure 6(a) shows the actual wavefront distortion from dataset-I, while Fig. 6(b) is the wavefront distortion predicted from the F-RNN trained by dataset-I. Figure 6(d) displays the actual wavefront distortion from dataset-J, whereas Fig. 6(e) shows the wavefront distortion predicted by the F-RNN, which is trained using dataset-J. Figures 6(c) and 6(f) show the difference between the actual wavefront distortions and predicted wavefront distortions. Figures 6(g) and 6(h) depict a comparison between the predicted Zernike coefficients, ranging from Z₁ to Z₁₅, and their corresponding actual Zernike coefficients from dataset-I and dataset-J. Concerning dataset-I, the actual PV value is 1.5867µm, whereas the predicted PV value is 1.5873µm, and the actual RMS value is 0.4667µm, whereas the predicted RMS value is 0.4692µm. As for dataset-J, the actual and predicted PV values are 1.1882µm and 1.1891µm, while the actual and predicted RMS values are 0.4170µm and 0.4193µm. The table shows that the residual error between the actual and predicted PV values is less than 0.0010µm, while the residual error between actual and predicted RMS values is less than 0.0025µm. The loss function for both datasets is less than 0.55%. This simulation result could demonstrate the network's proficiency in accurately predicting wavefront distortion even under complex conditions, specifically when subjected to both Gaussian and Poisson noise. This performance attests to the network's robustness in reliably predicting wavefront distortions in the presence of noise interference. It is noted that the proposed algorithm will still work if the noise is large. And if other typical types of noises are incorporated into the data, the proposed algorithm will also run well. However, if the noise is too large, the accuracy of wavefront distortion reconstruction will be affected in a certain way.

4. Experiment results

An experiment is carried out to measure the wavefront distortion in a high-power multi-pass amplification laser facility by using the proposed SSWFS as shown in Fig. 7. The laser beamline of the facility contains a pre-amplifier, a cavity amplifier, a power amplifier, a series of the optical lens (from L1 to L4), spatial filters, an N-CCD camera (JAI, SW-4000M-PMCL, 4096 × 4096 pixels), an SHWFS (HASO4 FIRST, Imagine Optic, 32 × 40 micro-lens array), an SLM (Thorlabs, EXULUS-4K1, 3840 × 2160 pixels) and other optical components. In the experiment, the input ideal beam had a wavelength of 1053 nm. The beam aperture is set to 360mm × 360 mm. The aberration modulation frequency of the SLM and the frequency of N-CCD exposure are both set at 1 Hz. The network is constructed using the PyTorch framework in Python 3.8 and trained on a local computing server (Intel Core i9-12900 K CPU, NVIDIA GeForce RTX 3090Ti GPU, 128 GB RAM). A batch size of 64 is utilized as the minimum input, and all datasets are trained for 120 epochs. The optimizer is initialized with a learning rate value of 0.01, which controls the rate at which the network learns from the training data. As the training progresses, a predetermined training objective function threshold is set as 0.60%. The Adam dynamically adapts the learning rate during the training process to achieve the optimization algorithm by continuously adjusting the learning rate from its initial value of 0.01.

 

Fig. 7. Multi-pass amplification optical configuration of the high power laser ICF facility. Within the beamline, the initial low-energy incident beam is injected from the pre-amplifier to the focal plane of the cavity spatial filter. The near-field profile and wavefront distortion of the pre-amplifier are measured by an N-CCD and an SHWFS respectively. PA1 is positioned inside the cavity spatial filter, while PA2 is located at the focal plane of the transport spatial filter.

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During the experiment, prior to the high power laser working, an ideal laser beam devoid of any distortions is directed and collimated to form a parallel beam. The beam is introduced distortion through an SLM, and then the distorted laser beam passes through the pre-amplifier and injects into the main amplifier, which consists of L1, L2, L3, cavity amplifier, and power amplifier. After passing through L3, the beam is collimated and recorded by an N-CCD and an SHWFS. A pinhole-array plate (PA, a wheel disk with several groups of pinholes) is placed at the focal plane of a spatial filter. A CM is inserted at PA1-1 to measure the wavefront distortion of the pre-amplifier to serve as a position mark, while the width and the off-distance of which have negligible impact on the wavefront distortion measurement accuracy in practical high power laser facility [18]. The incident parallel beam from the front-end enters the pre-amplifier, is emitted into PA1, and then is modulated by the CM. After the transmission of the beam, the near-field profile image captures the convoluted optical intensity distribution information of both the CM and the focus. By altering the SLM to induce aberrations in the ideal incident beam, we gathered 100,000 samples for each dataset. These datasets are denoted as dataset-K and dataset-L, respectively. In dataset-K, the PV values range from 0.1λ to 1λ and are evenly partitioned into ten groups with each group containing 10,000 pairs of samples. In dataset-L, the PV values range from 0.1λ to 2λ and are evenly partitioned into ten groups with each group also consisting of 10,000 pairs of samples. By using the near-field profile image as the input and incorporating the wavefront distortions generated by the SLM, the F-RNN is trained using dataset-K and dataset-L.

After training, the SLM is deactivated to cease the introduction of distortions. During high power laser working, after being modulated by the CM, the wavefront distortion of the pre-amplifier could be directly obtained through a one-shot N-CCD image and the trained F-RNN. The SHWFS is employed as a supplementary wavefront sensor to measure the wavefront distortion, with the aim of validating the accuracy of measurement results obtained through the SSWFS method for comprehensive comparative analysis.

The experiment results are shown in Fig. 8 and Table 5. Figure 8(a) shows the wavefront distortion measured by the SHWFS, yielding PV value and RMS value of 0.23 µm and 0.31 µm, respectively. Figure 8(b) gives the wavefront distortion measured by the SSWFS method, yielding PV value and RMS value of 0.24 µm and 0.32 µm, the F-RNN of which is training by dataset-K. Figure 8(c) shows the difference map with a residual RMS value of 0.01 µm and residual PV value of 0.01 µm. In Fig. 8(d), the wavefront distortion is measured by the SHWFS with the PV value and RMS value of 2.63µm and 3.03µm. Figure 8(e) presents the wavefront distortion measured by the SSWFS method with F-RNN trained by dataset-L, resulting in PV value and RMS value of 2.61µm and 3.02µm. The intensity of the wavefront distortion distribution measured by the SSWFS method is closed similar to the distribution measured by the SHWFS method. Figure 8(f) displays the difference map, revealing a residual RMS value of 0.01 µm and a residual PV value of 0.02 µm. Figures 8(g) and 8(h) show the relationships of the Zernike coefficients ranging from Z₁ to Z₁₅ of the measured wavefront distortions by using the SHWFS and SSWFS (trained on dataset-K and dataset-L, respectively). The comparison presents that the Zernike coefficients measured by the SSWFS method are close proximity to the coefficients measured by the SHWFS. The prediction time for each wavefront distortion measurement with the SSWFS method is less than 0.13 ms. The experimental results manifest a negligible discrepancy between the wavefront distortions measured by the SHWFS and the SSWFS method. Note that the SSWFS method leverages a near-field profile image in conjunction with the F-RNN, yielding a substantial reduction in the time required for predicting wavefront distortion. As a result, the SSWFS method presents a single-shot and easy approach to precisely and conveniently measure wavefront distortion in high power lasers. This technique holds the potential for enhancing both efficiency and accuracy in wavefront distortion analysis, making it a valuable tool in the realm of high power laser research and development.

 

Fig. 8. Experiment results. (a) and (b) are the wavefront distortions measured by an SHWFS and SSWFS method, the F-RNN of which is trained by dataset-K. (d) and (e) are wavefront distortions measured by an SHWFS and SSWFS method, the F-RNN of which is trained by dataset-L. (c) and (f) is the residual error between an SHWFS and SSWFS method. (g) shows the relationship of the Zernike coefficients by using the SHWFS and the SSWFS trained on dataset-K, while (h) shows the relationship of the Zernike coefficients trained on dataset-L.

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Table 5. Wavefront distortion measurement results of the experiment data.

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5. Conclusion

In conclusion, a new deep-learning-based online single-shot wavefront sensing method is presented. By using a near-field profile image and F-RNN, the SSWFS method could achieve fast and robust wavefront sensing. The network training process is illustrated and the prediction results are discussed by the numerical simulation, and the measurement accuracy of noise influence is assessed and analyzed in detail. The simulation result provides evidence of the network's proficiency in accurately predicting wavefront distortion, even in conditions where both Gaussian and Poisson noise are present. This performance serves as a testament to the network's robustness in reliably predicting wavefront distortions in the presence of noise interference. An experiment is carried out to measure the wavefront distortion of the same beamline by using the traditional SHWFS and the SSWFS method respectively. In the experiment, the SSWFS method could acquire and reconstruct wavefront distortion less than 0.13 ms. The difference between the wavefront distortion measured by using the SHWFS and the SSWFS method is as small as a PV value of 0.01 µm and an RMS value of 0.01 µm. The simulation and experiment results demonstrate the efficacy of the proposed method in accurately predicting wavefront distortions in high power lasers. Through the analysis of the prediction data, the SSWFS method exhibits high speed and good robustness in wavefront retrieval, which could meet the requirements of online wavefront distortion measurement in high power laser facility.