Multi-perturbation stochastic parallel gradient descent method for wavefront correction

Kenan Wu; Yang Sun; Ying Huai; Shuqin Jia; Xi Chen; Yuqi Jin

doi:10.1364/OE.23.002933

1. Introduction

Adaptive optics (AO) systems without wave-front sensor (WFS) are widely used in microscopy [1–3], laser beam shaping [4–6], optical tweezers [7], and optical ðbre coupling [8]. The operation of these adaptive systems is usually based upon optimization methods that update the control signal of the phase corrector according to the measured quality metric. The stochastic parallel gradient descent (SPGD) algorithm put forward by Vorontsov [9] is one of the most intensively discussed control method for AO without WFS. Previous works have demonstrated that SPGD is suitable for high-resolution adaptive systems [9, 10].

Convergence speed of the optimization process is important for practical applications. Several improved techniques based on SPGD have been put forward. For example, the decoupled SPGD [11] reorganize the actuators into decoupled groups with its own quality-metric. Each group is optimized with its own metric based on the conventional SPGD. The asynchronous SPGD [12] regroup the correctors according to their time response behavior, and different groups are asynchronously controlled by the conventional SPGD. For each single group in these techniques, or for the conventional SPGD, approaches to improve the optimization (convergence) speed remain an important issue.

According to Vorontsov [9], two issues relate closely to the convergence performance of SPGD. One is the expectation of absolute value of the quality-metric perturbation $〈 | δ J | 〉$ . When the dependence on the length of the control vector perturbation $‖ δ u ‖$ is excluded, the larger $〈 | δ J | 〉$ is, the steeper the metric descent is. The other issue relates to the SPGD with global coupling in which the phase correction can be represented as the linear superposition of modal bases [13]. The choice of modal basis set has a strong impact on the convergence performance [14].

In this paper, an improved SPGD, namely the multi-perturbation SPGD is put forward based on improvements related to the above two issue. Adaptive system based on the multi-perturbation SPGD can operate in two modes – the fast descent mode and the modal basis updating mode. The two operation modes utilize different control methods and enhance the conventional SPGD in different ways. The fast descent mode provides faster convergence than the conventional SPGD. The modal basis updating mode can optimize the modal basis set for SPGD with global coupling.

This work is organized as follows. Principle and control methods of the multi-perturbation SPGD are described in Sec. 2. Experimental results as well as discussion are given in Sec. 3. Conclusions are made in Sec. 4.

2. Multi-perturbation stochastic parallel gradient descent technique

A schematic of the adaptive system based on multi-perturbation stochastic parallel gradient descent technique is shown in Fig. 1. The incoming beam with distorted wavefront is split into N sub-beams by the BSs. Each sub-beam is modulated by a wavefront corrector (WC). Finally, the corrected sub-beams are detected independently and their performance metrics are obtained. In this paper, phase of the incoming beam is denoted as $φ (r)$ . Phase introduced by the wave corrector WC_n is $ψ^{n} (r) = \sum_{l = 1}^{L} u_{l}^{n} W_{l}^{n} (r)$ . $W_{l}^{n} (r)$ can be a single actuator influence function $S_{j}^{i} (r)$ or equals to a combination of actuator influence functions. The latter case is often referred to as the global coupling [9] or modal bases control [13]. The residue phase of the nth sub-beam is $ϕ^{n} (r) = φ (r) + ψ^{n} (r)$ . The measured quality metric for the nth sub-beam $J^{n} = J (u^{n})$ is a function of the control parameter vector $u^{n} = {u_{1}^{n}, ..., u_{l}^{n}, ..., u_{L}^{n}}$ . When a perturbation vector $δ u^{n} = {δ u_{1}^{n}, ..., δ u_{l}^{n}, ..., δ u_{L}^{n}}$ is imposed on the wave corrector WC_n, the corresponding change of the quality metric is $δ J^{n} = J (u^{n} + δ u^{n}) - J (u^{n})$ .

Fig. 1 Schematic of adaptive system based on multi-perturbation SPGD. The incoming beam with distortion is split into N sub-beams. BS_n represent beam splitter in the nth sub-beam. WC_n indicate wave corrector in the nth sub-beam. (n = 1, 2 … N.).

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As introduced in Sec. 1, the adaptive system based on multi-perturbation SPGD can operate in two modes – the fast descent mode and the modal basis updating mode. The fast descent mode provides faster convergence than the conventional SPGD. The modal basis updating mode can optimize the modal basis set ${W_{l}^{n} (r)}$ of the phase introduced by the wavefront corrector. Details of control methods of the two operation modes are given below.

2.1 Control method of the fast descent mode

The adaptive system works under an iterative procedure. The ith iteration includes the following steps:

1. All wave correctors start with the same control parameter $u^{n} = u^{(i)}$ , quality metric $J^{n} = J (u^{n})$ corresponding to the current control-parameter $u^{n} = u^{(i)}$ is measured for each sub-beam.
2. Wave-front perturbation $δ ψ^{n} (r) = \sum_{l = 1}^{L} δ u_{l}^{n} W_{l}^{n} (r)$ is imposed on wave corrector WC_n with use of the generated control-parameter perturbation vector $δ u^{n} = {δ u_{1}^{n}, ..., δ u_{l}^{n}, ..., δ u_{L}^{n}}$ for $n = 1, ..., N$ . The generated vectors ${δ u^{n}}$ should be orthogonal to each other.
3. Quality metric corresponding to the perturbed wave front is measured and the quality metric perturbation $δ J^{n} = J (u^{n} + δ u^{n}) - J (u^{n})$ is calculated for each sub-beam.
4. Generate the estimation ${\hat{J}}_{l}$ of the true gradient components $\frac{\partial J (u^{(i)})}{\partial u_{l}}$ as ${\hat{J}}_{l} = δ J^{'} δ {u^{'}}_{l}$ , where $δ {u^{'}}_{l}$ denotes the lth component of the virtual perturbation $δ u^{'}$ defined as $δ u^{'} = \frac{\sum_{n = 1}^{N} s i g n (δ J^{n}) δ u^{n}}{‖ \sum_{n = 1}^{N} s i g n (δ J^{n}) δ u^{n} ‖} \cdot σ .$
And the corresponding quality metric increment is estimated as
$\begin{array}{l} δ J^{'} = J (u^{(i)} + δ u^{'}) - J (u^{(i)}) = \nabla J (u^{(i)}) \cdot \frac{\sum_{n = 1}^{N} s i g n (δ J^{n}) δ u^{n}}{‖ \sum_{n = 1}^{N} s i g n (δ J^{n}) δ u^{n} ‖} \cdot σ + O (σ), \\ = \frac{\sum_{n = 1}^{N} | δ J^{n} |}{‖ \sum_{n = 1}^{N} s i g n (δ J^{n}) δ u^{n} ‖} \cdot σ + O (σ) \end{array}$
where $s i g n (\cdot)$ denotes the signal function and $σ$ is the step length of the virtual perturbation $δ u^{'}$ . The last equation is true because of $δ J^{n} = \nabla J (u^{(i)}) \cdot δ u^{n} + O (σ)$ .
5. Update the wave-front control-parameters for all wave correctors as $u^{n} = u^{(i + 1)} = u^{(i)} - μ δ J^{'} δ u^{'}, n = 1, ..., N .$

Note that in each step, measuring of the quality metric or wavefront perturbing is executed simultaneously for each sub-beam. The time consumption for executing a single step in the multi-perturbation SPGD is the same as that in the conventional SPGD. However, the convergence rate is improved. This can be proved analytically as below. As pointed out by Voronsov [9], convergence speed in SPGD type algorithms depends on the expectation of absolute value of the quality-metric perturbation $〈 | δ J | 〉$ , where $〈 \cdot 〉$ means ensemble averaging over phase distortions and perturbations. For small perturbations $δ u = {δ u_{1}, ..., δ u_{l}, ..., δ u_{L}}$ , the measured metric perturbation $δ J \approx \nabla J (u) \cdot δ u$ is approximately proportional to $‖ δ u ‖$ . To exclude the dependence of $〈 | δ J | 〉$ on $‖ δ u ‖$ , we compare $〈 | δ J | 〉$ with the same $‖ δ u ‖ = σ$ for different optimization methods.

Notice that all $〈 | δ J^{n} | 〉$ $(n = 1, ..., N)$ are the same for different independent sub-beams with the same perturbation statistics, and is equal to the expectation of absolute value of metric perturbation in conventional SPGD (denoted as ${〈 | δ J | 〉}_{c}$ ) with the same perturbation statistics. When the generated control-parameter perturbation vectors ${δ u^{n}}$ are orthogonal to each other, from Eq. (2) we have

〈 | δ J^{'} | 〉 \approx \frac{\sum_{n = 1}^{N} 〈 | δ J^{n} | 〉}{‖ \sum_{n = 1}^{N} s i g n (δ J^{n}) δ u^{n} ‖} \cdot σ = \frac{N {〈 | δ J | 〉}_{c} σ}{\sqrt{N} ‖ δ u^{n} ‖} = \sqrt{N} {〈 | δ J | 〉}_{c} .

Therefore, $〈 | δ J^{'} | 〉$ is approximately $\sqrt{N}$ times of ${〈 | δ J | 〉}_{c}$ , which means the fast descent mode of the multi-perturbation SPGD provides a faster convergence rate than the conventional SPGD.

2.2 Control method of the modal basis updating mode

In the conventional SPGD with global coupling, a modal basis set is used to construct the phase correction. The choice of the modal basis set has a strong influence on the convergence rate [14]. The modal basis updating mode of the multi-perturbation SPGD is able to update the basis set dynamically to improve the convergence rate. Firstly, a group of aberration sources with different characteristics are chosen as candidates of statistically approximation of the actual phase distortion. A corresponding group of candidate modal basis sets ${W^{m} (r)}$ , $(m = 1, ..., M)$ is obtained, where the mth modal basis set $W^{m} (r)$ is obtained by extracting the proper orthogonal decomposition (POD) bases from the phase snapshots of the mth candidate aberration source. Here, candidate aberration sources can be typical ideal models with certain statistics like Kolmogorov turbulence, or certain flow fields described by fluid dynamic models.

After the group of candidate modal basis sets is obtained, the modal basis sets used in the multi-perturbation SPGD are updated under an iterative procedure. A single iteration includes the following steps:

1. Choose the modal basis set for each WC from the group of candidate modal basis sets.
2. Independently execute the conventional SPGD with global coupling for each sub-beam. Denote the basis set used for the nth sub-beam as $W^{n} (r) = {W_{l}^{n} (r)}$ . The convergence performance ${P_{n}}$ of each sub-beam are evaluated, respectively.
3. Repeat step 2 for K snapshots (realizations) of actual phase distortions. Evaluate the average convergence performance ${〈 P_{n} 〉}$ within the correction of the K snapshots for each sub-beam. Here $〈 \cdot 〉$ means average over K snapshots.
4. Keep the basis set corresponding to the best average convergence performance unchanged. Change the basis sets of the other WCs to other basis sets in the group of candidate modal basis sets.

The purpose of convergence performance evaluation is to find the basis set which best statistically approximates the actual phase distortion. One possible definition of $P_{n}$ is the quality-metric $J^{n} = J (u^{n})$ after certain rounds of conventional SPGD. By consistently switching the basis sets corresponding to poor convergence performance to others, the multi-perturbation SPGD is able to update the basis set dynamically to improve the convergence performance.

3. Results and discussion

3.1 Results of the fast descent mode

Experiment was carried out to test the effectiveness of the fast descent mode of the multi-perturbation SPGD technique. In the experiment, a He-Ne laser beam was expanded, collimated, and modulated by a spatial light modulator (SLM) to generate the incoming beam with distortion. The advantage of using SLM for adding phase distortion is it can generate the desired distortions flexibly.

The Kolmogorov turbulence model was used to generate phase distortions of the incoming beam. In this paper, the Kolmogorov turbulence was generated by the Fourier transform method and the Kolmogorov spatial power spectrum

Φ_{n} (k_{x}, k_{y}) = 0.033 C_{n}^{2} {(k_{x}^{2} + k_{y}^{2} + 4 π^{2} L_{0}^{- 2})}^{11 / 6},

where

C_{n}^{2}

is the index structure constant and is described by the Hufnagel model [15],

k_{x}

and

k_{y}

are spatial wave numbers,

L_{0}

is the outer scale length of the turbulence and was set to be 1m in this work. This is a typical value for outer scale of near ground turbulence [16], and was frequently adopted in previous studies [17, 18].

Schematic of the setup is shown in Fig. 2. In the experiment, the distorted wavefront was split into two sub-beams. We use two SLMs as the wavefront correctors. And each of the corrected sub-beams was focused by a Fourier lens and detected by a photodetector. The He-Ne laser we used was Melles Griot 05-LHP-151. The SLMs were HOLOEYE LC2002. The screen of each SLM has 832 × 624 pixels. The Strehl ratio was used as the quality metric. The Strehl ratio is defined [19] as the ratio of the peak aberrated image intensity from a point source compared to the maximum attainable intensity using an ideal optical system limited only by diffraction over the system's aperture. For a fixed aperture, it is approximately proportional to the output ðeld zero-spectral component squared modulus $J = {| \int A_{0} \exp [i δ (r)] d^{2} r |}^{2}$ , where $A_{0}$ and $δ (r)$ denotes the amplitude and phase of the aberrated wave [12]. Hence in the field of adaptive optics, it is often measured by using a lens with a small pinhole located in front of a photodetector at the lens’s focal plane coordinate origin as shown in Fig. 2 [11].

Fig. 2 Schematic of the experimental setup. Distortion is added to the plane wave by the spatial light modulator SLM_distortion. OBJ: objective lens. CL: collimating lens. BS: beam splitter. Wave corrector WC₁ and WC₂ are also spatial light modulators. f = 400mm is the focal length of the Fourier lenses. The pinhole and photodetector of the first sub-beam can be replaced by the CCD on the translation stage.

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Twenty snapshots of Kolmogorov turbulence were generated with aperture-averaged phase and global tilts removed. Figure 3 shows a typical gray-scale image of the phase-distortion realization. In the setup, the pinhole and photodetector of the first sub-beam can be replaced by a CCD (Mintron MTV. 1881EX) to record the focal spot. The focal spot corresponding to the uncorrected distorted incoming wavefront is shown in Fig. 4(a). The cross line in the focal spot image is created by diffraction from the rectangular pixel array of the SLM. The cross line is static throughout the experiment. The fast descent mode of the multi-perturbation SPGD and the conventional SPGD were compared. Typical corrected focal spots by using the two methods after 100 iterations are shown in Fig. 4(b) and Fig. 4(c), respectively. The averaged quality metric over 20 snapshots versus iteration number is shown in Fig. 5. The multi-perturbation SPGD improved the quality metric to near its optimal value (around 0.8) in approximately 100 iterations and signiðcantly improved beam quality (~100 ms were required for one iteration). The conventional SPGD needs approximately 150 iterations to bring the quality metric to near its optimal value. Typically, the conventional SPGD took 50% more rounds than the multi-perturbation SPGD for reducing the quality metric to the same level. The result conforms with previous analysis in Sec. 2.1.

Fig. 3 Gray-scale image of the Kolmogorov turbulence phase-distortion.

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Fig. 4 Typical focal spots of light wave with Kolmogorov turbulence phase-distortion (a) before correction, (b) after 100 correction cycles by using the fast descent mode of the multi-perturbation SPGD, and (c) after 100 correction cycles by using the conventional SPGD. Insets: the intensity profile along the top-left to bottom-right diagonal line.

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Fig. 5 Curve of averaged quality metric versus iteration number.

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Histograms for the beam quality metric optimization process with different control methods are shown in Fig. 6. The histogram represents the number of adaptive system states Mst as a function of the beam quality metric value J. Data from all 20 trials were used to calculate the histograms in Fig. 6. It can be seen that the probability of achieving better beam quality with the fast descent mode of multi-perturbation SPGD is higher than that with the conventional SPGD. This also means that the fast descent mode of multi-perturbation SPGD can achieve faster convergence rate than the conventional SPGD.

Fig. 6 Histograms for the beam quality metric optimization process with (a) the fast descent mode of the multi-perturbation SPGD and (b) the conventional SPGD.

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Frequently used merit functions for AO based on optimization methods include Strehl ratio [12], mean radius [20], and sharpness function [11]. For both the proposed method and the conventional SPGD, the descending speed would be changed by choosing different merit functions. However, the introduction and analysis in section 2 uses a general merit function without any specific choice. The only assumption in analysis in Sec. 2.1 is that the merit function is differentiable. Therefore, when the same merit function is used and start at the same point in searching space, the average descending rate of the proposed method is always $\sqrt{N}$ times of that of conventional SPGD. Although Strehl ratio is used in the experiment, one may choose other merit functions that suitable for the application, the fast descent mode of the multi-perturbation SPGD would always provide a faster convergence rate than the conventional SPGD.

Improving the convergence rate is always considered to be important in optimization based AO systems [11, 12]. Previous study has reported improved optimization method with the assistance of WFS [11]. However, there are circumstances when the WFS is invalid. Most of the adaptive microscope systems so far implemented have not employed WFS, because in these applications the reference source is always far from point-like [21]. And under conditions of strong intensity modulation typical for ground-to-ground imaging, laser communication, laser-beam-forming systems, there may be wave-front dislocations (branch points) offering a challenging problem for detection with WFS [10]. Another variant of the proposed method is to perform multiple perturbations sequentially in a single channel. Compared with this strategy, N channels working simultaneously could obviously obtain faster convergence rate. Apparently, using more channels would increase the system complexity. So the proposed method in Sec. 2.1 would be useful for situations when the benefit in the convergence rate overweighs the increase in the system complexity. Situations of this kind include in vivo microscopy, which have a high demand in convergence rate in order to avoid photobleaching and photo-toxic effects [21], and aero-optical applications, in which the bandwidth of the adaptive system is critical to keep up with changes of turbulence [22].

3.2. Results of the modal basis updating mode

As is described in Sec. 2.2, a group of candidate modal basis sets ${W^{m} (r)}$ , $(m = 1, ..., M)$ should be prepared before the adaptive modal updating process started. We used the Kolmogorov turbulence with von Karman spectrum, the non-Kolmogorov turbulence with power spectrum exponent $α =$ 4.5, the anisotropic Kolmogorov turbulence with anisotropy factor $ε =$ 1, and the back-facing step flow with Reynold number Re = 37,000 as turbulence data sources to generate candidate modal basis sets. The Kolmogorov turbulence, the non-Kolmogorov turbulence and the anisotropic Kolmogorov turbulence was generated by the Fourier transform method and corresponding power spectrum [23, 24]. The step flow turbulence phase distortion was generated by hybrid large-eddy / Reynolds-averaged Navier-Stokes (LES/RANS) simulation [25]. The phase distortion of the step flow was obtained based on the simulated density and the Gladstone–Dale law [26].

For each type of distortion, proper orthogonal decomposition (POD) analysis was performed to get the modal basis set. The POD decomposes the series of distortion snapshots into a number of modes that make up an orthonormal basis spanning the entire data set and captures the most energetic and hence largest structures of the ñow in the ðrst few modes [27]. The obtained candidate modal basis sets corresponding to different types of flow are shown in Fig. 7.

Fig. 7 Obtained candidate modal basis sets corresponding to (a) Kolmogorov turbulence with von Karman spectrum, (b) non-Kolmogorov turbulence with power spectrum exponent $α =$ 4.5, (c) anisotropic Kolmogorov turbulence with anisotropy factor $ε =$ 1, and (d) back-facing step flow with Reynold number Re = 37,000.

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Experiment was carried out to demonstrate the effectiveness of the modal basis updating process. Phase distortion of a certain type was used as the actual distortion of the incoming beam. Then the control method in Sec. 2.2 was performed. In the experiment the incoming beam was split into 2 sub-beams. The convergence performance $P_{n}$ of the basis set of WC_n was evaluated by the quality-metric $J^{n} = J (u^{n})$ after 100 rounds of conventional SPGD. After the 100 rounds, the phase distortion of the incoming beam was changed to another snapshot. The overall performance $〈 P_{n} 〉$ corresponding to the chosen basis set of WC_n was obtained by averaging $P_{n}$ over 3 distortion snapshots of the incoming beam.

Figure 8 shows a typical modal basis updating process. In this example, the type of incoming beam distortion was anisotropic Kolmogorov turbulence. Evolving process of the quality metric of sub-beam 1 and 2 are shown in the upper and lower half of Fig. 8, respectively. The vertical dashed lines indicate time points when the basis sets of wavefront correctors are updated. History of the chosen basis sets is given in Fig. 8 under quality metric curves of corresponding iteration periods. The basis sets in red font refer to the ones with better convergence performance at that period. It can be seen that the chosen modal basis set with better performance was kept in the modal updating process while the one with poor performance is changed to others. After several periods of modal updating, at least one WC had chosen the basis set of the same type as the incoming beam distortion. The result demonstrated that the multi-perturbation SPGD is able to evolve an optimized basis set for SPGD with global coupling, even when the character of the incoming beam distortion is not known as prior knowledge.

Fig. 8 Typical modal basis updating process. Upper half: Quality metric and modal basis choice of WC₁. Lower half: Quality metric and modal basis choice of WC₂. The phase distortion of the incoming beam is changed every 100 rounds, and the basis sets of wavefront correctors are updated every 300 rounds. The vertical dashed lines indicate time points when the basis sets of wavefront correctors are updated. Red font refers to the basis set with better convergence performance.

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In microscopy, the aberration type of the specimen can be complex [21]. And in aero-optics applications [26], the aberration type of the turbulence would be changed if the relative wind velocity is changed or the aerodynamic profile of the aircraft is transformed. In these circumstances, to obtain detailed information about the wavefront distortion needs complex measurements, and the update of basis in real time is helpful for keeping the modal basis efficient [28]. Previous literature has presented a method to update the modal basis for AO in microscopy [28]. A series of test aberrations is imposed on deformable mirror to generate the optimized modes. When the number of actuators is large, especially in the case of SLM, the number of tests should be large for obtaining a satisfactory mode set. The proposed method in Sec. 2.2, however, follows a different approach. The mode sets are updated based on an evolving strategy. The mode set updating process in the proposed method takes place during the wave front correction process, and it does not need the series of tests beforehand.

4. Conclusion

The multi-perturbation SPGD is presented in this work. In this technique, the incoming beam with distorted wavefront is split into N sub-beams. Each sub-beam is modulated by a wavefront corrector and its performance metric is measured subsequently. Adaptive system based on the multi-perturbation SPGD can operate in two modes – the fast descent mode and the modal basis updating mode. In the fast descent mode, the time consumption for executing a single step in the multi-perturbation SPGD is the same as that in the conventional SPGD. However, the convergence rate is improved. Analysis showed that the expectation of absolute value of the quality-metric perturbation of the fast descent mode of multi-perturbation SPGD $〈 | δ J^{'} | 〉$ is approximately $\sqrt{N}$ times as much as that of the conventional SPGD ${〈 | δ J | 〉}_{c}$ , which means the gradient of the convergence curve of the multi-perturbation SPGD is approximately $\sqrt{N}$ times as much as that of the conventional SPGD. Experimental results proved that the fast descent mode of multi-perturbation SPGD can achieve faster convergence rate than the conventional SPGD. In the modal basis updating mode, a group of candidate modal basis sets was prepared before the adaptive modal basis updating process started. In the modal basis updating process, the basis sets corresponding to poor performance are consistently switching to others. Experimental result showed that after several iterations of modal updating, at least one WC had chosen the basis set of the same type as the incoming beam distortion. The convergence performance with the optimized modal basis set is better. In conclusion, the two operation modes of the multi-perturbation SPGD enhance the conventional SPGD in different ways. The fast descent mode provides faster convergence than the conventional SPGD. The modal basis updating mode can optimize the modal basis set for SPGD with global coupling.

Acknowledgments

The authors thank Yong Lin of Dalian University of Technology for supports on experiments. This work is under project “principle and control method of intracavity flow boundary layer separation of chemical oxygen-iodine lasers” supported by the National Natural Science Foundation of China (Grant No: 61205139). This work is also under projects supported by the National Natural Science Foundation of China (Grant No: 11204300 and 11304311).

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Multi-perturbation stochastic parallel gradient descent method for wavefront correction

Abstract

1. Introduction

2. Multi-perturbation stochastic parallel gradient descent technique

2.1 Control method of the fast descent mode

2.2 Control method of the modal basis updating mode

3. Results and discussion

3.1 Results of the fast descent mode

3.2. Results of the modal basis updating mode

4. Conclusion

Acknowledgments

References and links

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Figures (8)

Equations (5)

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