- Email: [email protected]
A decoupling control algorithm for Woofer–Tweeter adaptive optics system in generalized irregular pupil region
Optics Communications 472 (2020) 125856
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A decoupling control algorithm for Woofer–Tweeter adaptive optics system in generalized irregular pupil region Tao Cheng a,c , ZhenXing Xu a,b,c , Kangjian Yang a,c , Shuai Wang a,c , Xin He a,c , Ping Yang a,c ,∗, Bin Xu a,c ,∗ a
Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China University of Chinese Academy of Sciences, Beijing 100049, China c Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China b
ARTICLE
INFO
Keywords: Decoupling control algorithm Adaptive optics Generalized irregular pupil region
ABSTRACT We proposed a decoupling control algorithm to accomplish the application of woofer–tweeter (W–T) adaptive optics system in generalized irregular pupil region. This algorithm firstly modifies the response matrix of woofer and tweeter by setting corresponding value at some sub-apertures of Shack–Hartmann(S–H) sensor to zero and constructs a slope-based orthogonal basis used to distribute different spatial frequency aberration to dual deformable mirrors (DMs). Then a limited condition is used to calculate the pseudo-inverse matrix of modified response matrix of dual DMs through single value decomposition. Finally, to restrain tweeter generating crosscoupling shape with woofer, a constraint matrix used to reset control vector of tweeter is constructed with the slope-based orthogonal basis and modified response matrix of tweeter. Numeral simulation of W–T adaptive optics system is built to verify the performance of this algorithm and results demonstrate that this algorithm has a good performance to control the W–T adaptive optics system in generalized irregular pupil region, and achieve better performance to suppress the cross-coupling between woofer and tweeter than Zernike decomposition algorithm. At the same time, the experimental results for the W–T AO system demonstrated that this algorithm is practical for the AO system with dual DMs in generalized irregular pupil region.
1. Introduction Through many researchers have done a lot of study about large stroke and high spatial frequency deformable mirror (DM), the application of this DM still need solve many technological problems [1]. So the woofer–tweeter(W–T) adaptive optics system shown in Fig. 1 is really an efficient solution to simultaneously compensate phase distortion with a large amplitude and high spatial frequency [2–4]: a single (S– H) sensor is used to measure the wave-front aberration , and distribute the large stroke and low spatial frequency aberration to woofer and low stroke and high spatial frequency aberration to tweeter. It is well known that woofer and tweeter will generate opposite compensation which will make the dual DMs correct the same aberration using a bigger stroke and may turn the system instable [5]. To make the two DMs work cooperatively, several algorithms have been proposed to suppress the cross-coupling of dual DMs, such as Zernike mode decomposition algorithm [6–8], Fourier mode decomposition algorithm [9], wavelet mode decomposition algorithm [10], Lagrangemultiplier damped least-squares algorithms [11–13], and direct slopebased correction algorithm [14–17]. According to the references [8,15], Zernike mode decomposition algorithm and slope-based decoupling
control algorithm can achieve better performance to suppress correlation error of dual DMs than other algorithms. The reason for the two algorithms restraining the cross-coupling error of dual DMs well is that an orthogonal mode in regular region such as circular or rectangle is constructed firstly, which can be used to distribute different spatial frequency aberration to dual DMs and construct constraint matrix to reset tweeter control vector. However, due to the fluctuation of light intensity in many applications such as laser beam cleaning, a few of sub-apertures of (S–H) sensor has a lack of light or are saturated, it will make the pupil be a generalized irregular geometric region because the information of these sub-apertures will reduce the accuracy of wavefront reconstruction. Unfortunately, the mathematical basis of the mode decoupling algorithm is the orthogonality of the Zernike mode on the unit circle. However, the orthogonality of Zernike mode will be destroyed, so that Zernike mode requires Schmitt orthogonalization onto irregular pupil region if continuing to use Zernike mode for decoupling control, it will be a Complex calculation process and not flexible. To slope-based decoupling control algorithm, some actuators of woofer and tweeter will be not valid in generalized irregular pupil region,
∗ Corresponding authors at: Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China. E-mail addresses: [email protected] (P. Yang), [email protected] (B. Xu).
https://doi.org/10.1016/j.optcom.2020.125856 Received 16 January 2020; Received in revised form 23 March 2020; Accepted 30 March 2020 Available online 4 April 2020 0030-4018/© 2020 Published by Elsevier B.V.
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
of 𝑀𝑊 is orthogonal and corresponding to eigen mode of woofer. What is more, the eigen mode of the woofer is similar with the Zernike mode, of which the spatial frequency is gradually increasing with the mode’s order increasing. So wave-front slop distributed to woofer can be calculated with Eqs. (5) and (6). + 𝑟 = 𝑀𝑊 𝑔
(5)
𝑔𝑊 = 𝑀𝑤 𝐼𝑊 𝑟
(6)
where, 𝑔 is the residual wave-front slope, 𝑟(𝑚1 × 1) is the coefficient + of 𝑀𝑤 , 𝑀𝑊 is the pseudo-inverse matrix of 𝑀𝑤 and 𝐼𝑊 is a diagonal matrix used to assign wave-front slope 𝑔𝑊 to woofer by set 𝐼𝑊 (𝑖, 𝑖) to one or zero. In order to calculate the error voltage 𝑣𝑊 with 𝑔𝑊 , we should get the pseudo-inverse matrix 𝑅+ with Eqs. (7)–(9). 𝑊 _𝑛𝑒𝑤
Fig. 1. The woofer–tweeter adaptive optics system.
(7)
[𝑈 , 𝑆, 𝑉 ] = 𝑆𝑉 𝐷(𝑅𝑊 _𝑛𝑒𝑤 ),
slope-based orthogonal basis constructed from woofer will be inapplicable so that constraint matrix is no longer capable of suppressing cross-coupling of woofer and tweeter. In this paper, in order to control woofer and tweeter simultaneously in generalized irregular pupil region, we proposed a decoupling control algorithm. This algorithm firstly set corresponding value of response matrix of dual DMs at some sub-apertures to zero according to light intensity, secondly construct a slope-based orthogonal basis to distribute different spatial frequency aberration to two deformable mirrors (DMs) with the modified response matrix of woofer, then modify the reconstruction matrix of woofer and tweeter with a limited condition to calculate the control vector of dual DMs, finally reset the control vector of tweeter to restrain tweeter generating opposite compensation with woofer. This paper is organized as follows. Section 2 describes the principle of the decoupling algorithm in generalized irregular pupil region. Section 3 establishes a numerical model for the W–T adaptive optics system to test the validity of this algorithm in theory, and makes a detail comparison with our algorithm and Zernike mode decomposition algorithm. Section 4 established a realistic W–T AO system to test the performance of our algorithm in generalized irregular pupil region. Section 5 is the conclusion.
⎧ ⎪𝑆 (𝑖, 𝑖) , 𝑆𝑙𝑖𝑚𝑖𝑡 (𝑖, 𝑖) = ⎨ ⎪0, ⎩
𝑖𝑓 (𝑆(𝑖, 𝑖) ≥ 𝑖𝑓 (𝑆(𝑖, 𝑖) <
1 ) 𝑐𝑜𝑛𝑑𝑙𝑖𝑚𝑖𝑡 1 ), 𝑐𝑜𝑛𝑑𝑙𝑖𝑚𝑖𝑡
𝑅+ = 𝑉 𝑆𝑙𝑖𝑚𝑖𝑡 𝑈 𝑇 𝑊 _𝑛𝑒𝑤
of woofer can be calculated with Eqs. (10) and (11), 𝑣𝑊 = 𝑅+ 𝑔 𝑊 _𝑛𝑒𝑤 𝑤
(10)
𝑉𝑊 (𝑘 + 1) = 𝑝𝑖_𝑎 ∗ 𝑉𝑊 (𝑘) + 𝑝𝑖_𝑏 ∗ 𝑣𝑊
(11)
where, 𝑝𝑖_𝑎 and 𝑝𝑖_𝑏 are the parameters of PI controller. 𝑝𝑖_𝑎 is usually set around 0.98 and 𝑝𝑖_𝑏 is usually set around 0.2 in the adaptive optics system. After the correction by the woofer, the residual wave-front slope 𝑔𝑇 compensated by tweeter DM can be solved as following, 𝑔𝑇 = 𝑔 − 𝑔𝑊
where, is the pseudo-inverse matrix of 𝑅𝑇 _𝑛𝑒𝑤 . The control vector 𝑉𝑇 is also solved with a PI controller as, 𝑉𝑇 (𝑘 + 1) = 𝑝𝑖_𝑎 ∗ 𝑉𝑇 (𝑘) + 𝑝𝑖_𝑏 ∗ 𝑣𝑇
(14)
Though the control vector of woofer and tweeter is obtained as above, cross-coupling of dual DMs must be restrained to reduce the waste of stroke of dual DMs and guarantee the stability of W–T adaptive optics system during the correction process. So a constraint matrix 𝑅𝑐 is constructed with 𝑅𝑇 _𝑛𝑒𝑤 and 𝑀𝑊 as, ∑2𝑛 𝑅𝑇 _𝑛𝑒𝑤 (𝑥, 𝑖)𝑀𝑊 (𝑥, 𝑗) 𝑅𝑐 (𝑖, 𝑗) = 𝑘 ∑𝑥=1 (15) 2𝑛 𝑥=1 𝑀𝑊 (𝑥, 𝑗)𝑀𝑊 (𝑥, 𝑗)
where, 𝐼𝑖 is the sum of light intensity at the ith sub-aperture, 𝑇𝑚𝑖𝑛 and 𝑇𝑚𝑎𝑥 are respectively the maximum and minimum thresholds, ( ) 𝑅𝑊 𝑖𝑥 , 𝑗 is value in the 𝑥 direction at the ith sub-aperture of the jth actuator of woofer. Eq. (1) shows that the response value of the jth actuator at one sub-aperture is valid when the sums of light intensity in this sub-aperture is between the maximum and minimum thresholds. At the moment, we can also get 𝑅𝑇 _𝑛𝑒𝑤 through Eq. (1). After getting 𝑅𝑊 _𝑛𝑒𝑤 , a slope-based orthogonal basis corresponding to generalized irregular geometric region can be calculated as following,
(4)
(13)
𝑅+ 𝑇 _𝑛𝑒𝑤
In a W–T adaptive optics system, the response matrix 𝑅𝑊 and 𝑅𝑇 are measured with S–H sensor. When some sub-apertures of S–H sensor has a lack of light or are saturated due to the fluctuation of light intensity, the value of 𝑅𝑊 at some sub-apertures should be set at zero according to the light intensity as following [18,19], { ( ) ( ) 𝑅𝑊 𝑖𝑥 , 𝑗 , 𝑖𝑓 (𝑇𝑚𝑖𝑛 ≤ 𝐼𝑖 ≤ 𝑇𝑚𝑎𝑥 ) (1) 𝑅𝑊 _𝑛𝑒𝑤 𝑖𝑥 , 𝑗 = 0, 𝑖𝑓 (𝐼𝑖 ≥ 𝑇𝑚𝑎𝑥 , or 𝐼𝑖 ≤ 𝑇𝑚𝑖𝑛 )
𝑀𝑊 = 𝑅𝑊 _𝑛𝑒𝑤 𝑈
(12)
At the same time, the error voltage 𝑣𝑇 of tweeter is calculated with Eq. (13). 𝑣𝑇 = 𝑅+ 𝑔 𝑇 _𝑛𝑒𝑤 𝑇
(2)
(9)
A new singular value matrix 𝑆𝑙𝑖𝑚𝑖𝑡 is calculated with a limited condition 𝑐𝑜𝑛𝑑𝑙𝑖𝑚𝑖𝑡 as Eq. (8), here the limited condition 𝑐𝑜𝑛𝑑𝑙𝑖𝑚𝑖𝑡 is used to guarantee the stability of the matrix 𝑅+ . Then control vector 𝑉𝑊 𝑊 _𝑛𝑒𝑤
2. Principle
𝐶𝑤 = 𝑅𝑇𝑊 _𝑛𝑒𝑤 𝑅𝑊 _𝑛𝑒𝑤 ∕2𝑛 [ ] 𝑈 , 𝑆𝑀 , 𝑈 𝑇 = 𝑆𝑉 𝐷(𝐶𝑊 )
(8)
where, 𝑅𝑐 (𝑖, 𝑗) is the relation coefficient between the ith column of 𝑅𝑇 _𝑛𝑒𝑤 and the jth column of 𝑀𝑊 , 𝑘 is an empirical parameter used to adjust the constraint intensity to the tweeter DM, and it should be set according to the relation between influence slope matrix 𝑅𝑇 _𝑛𝑒𝑤 and orthogonal basis 𝑀𝑊 . A larger 𝑘 will enhance the elimination of low order components in the surface shape of the tweeter, but the correction qualities of higher order phase aberrations will be decreased. Thus 𝑘 should be carefully chosen to make a better compromise. With the constraint matrix 𝑅𝑐 , the control vector 𝑉𝑇 of tweeter can be modified by Eqs. (16) and (17), [ ] [ ] 𝐼 𝑉 𝑉𝑇 = 𝐶𝑉𝑇 = 𝑇 (16) 𝑅𝐶 0 [ ]+ [ ] [ ] 𝐼 𝑉𝑇 𝑉 𝑉𝑇′ = = 𝐶+ 𝑇 (17) 𝑅𝐶 0 0
(3)
where, n is the number of Hartmann’s aperture, 𝐶𝑤 (𝑚1 × 𝑚1) describes the correlation of influence slopes of different actuator. Through singular value decomposition of 𝐶𝑤 as Eq. (3), a slope-based orthogonal basis 𝑀𝑊 (2𝑛×𝑚1) can be achieved with Eq. (4). Note that the column vector 2
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 2. Configuration of DM’s actuators and the S–H sensor’s sub-apertures: (a) Woofer; (b) Tweeter. Fig. 4. Singular values of the eigen mode of woofer.
distributed to woofer does not contain the eigen mode of woofer which order is greater than 16. Next, we consider three different cases to verify the validity of our algorithm: (1) half of pupil region lacking of light; (2) a rectangular region lacking of light; (3) a circular region lacking of light. The initial aberration in three different pupil regions are shown in Fig. 5(a1), (a2), and (a3). Then, according to the singular value shown in Fig. 4, we isolate the first ten order eigen mode in the initial aberration and assign them to woofer, and assign the residual aberration to tweeter. In three different pupil regions, the correction of woofer are shown in Fig. 5(b1), (b2) and (b3), the correction of tweeter are shown in Fig. 5(c1), (c2) and (c3). And the residual mean square (RMS) of wave-front aberration are respectively 0.026𝜆, 0.047𝜆 and 0.048𝜆 as shown in Fig. 5(d1), (d2) and (d3). The results of Fig. 5 indicate that our algorithm can make woofer correct low spatial frequency aberration and tweeter correct high spatial frequency aberration respectively in irregular pupil region. Note that, the limit condition described as Eq. (8) and used to calculate the reconstruction matrix of dual DMs is set to 10. Finally, in order to verify the performance of our algorithm to restrain the cross-coupling of dual DMs, we project the correction of woofer and tweeter to the eigen mode of woofer and a compensation ratio 𝑐 is defined as Eq. (18) to evaluate the cross-coupling of dual DMs,
Fig. 3. Eigen mode corresponding to column vector of 𝑀𝑊 .
where, 𝑉𝑇′ is the control vector of tweeter which will not generate cross-coupling shape with woofer. 3. Numeral simulation
𝑐=
3.1. Testing validity
∫ ∅𝑤𝑐 ∅𝑤 𝑑𝑠 ∫ ∅𝑤 ∅𝑤 𝑑𝑠
(18)
where, ∅𝑤 is the aberration distributed to woofer and ∅𝑤𝑐 is the aberration corrected by woofer. The compensation ratio 𝑐 is an indirectly evaluation indicator, it means that compensation ratio 𝑐 will be one when aberration distributed to woofer ∅𝑤 is corrected completely by woofer ∅𝑤𝑐 and woofer do not generate other aberration. At that moment, we can indirectly demonstrate that tweeter do not generate unnecessary cross-coupling compensation with woofer. Fig. 6(d) shows that the compensation ratios 𝑐 of three different irregular pupil regions both are 0.99, it indicates that our algorithm can restrain tweeter generating cross-coupling shape to woofer well in irregular pupil region. In addition, we test the performance of our algorithm without using Eqs. (15) to (17), and the correction results and cross-coupling suppression in three different pupil regions lacking light are respectively shown in Figs. 7 and 8. Though the residual wave-front aberration is similar with Fig. 5, are respectively 0.024𝜆, 0.050𝜆 and 0.049𝜆 as shown Fig. 7(d1), (d2) and (d3), the correction of woofer as shown in Fig. 7(b1), (b2) and (b3) is obviously higher than Fig. 5(b1), (b2) and (b3), and the correction of tweeter as shown in Fig. 7(c1), (c2) and (c3) is obviously higher than Fig. 5(c1), (c2) and (c3). At this moment, Fig. 8(a), (b) and (c) show that woofer and tweeter have obviously cross-coupling compensation in the first ten orders,
In order to verify the validity of the algorithm proposed above, a W– T adaptive optics system is built. Woofer and tweeter respectively have 19-actuators and 127-actuators, and the configuration of dual DMs’ actuators and S–H sensor’s sub-apertures are shown in Fig. 2. We firstly make a hypothesis that half of pupil region is lack of light or saturated. Then a slope-based orthogonal basis 𝑀𝑊 can be constructed with 𝑅𝑊 _𝑛𝑒𝑤 through Eqs. (2)–(4), and the eigen mode corresponding to column vector of 𝑀𝑊 is shown in Fig. 3. Fig. 3 shows that the spatial frequency of eigen mode of woofer is gradually increasing with the mode’s order increasing. So, these modes can be used to assign different spatial frequency aberration to the dual DMs. However, according to the theory about singular value decomposition, Ref. [20] proved that removing some high order of eigen mode of woofer which singular value is very small and almost equal to zero can still take full use of woofer and improve the stability of close-loop control for woofer. Therefore, singular values of eigen mode of woofer should be calculated by Eq. (7) and shown in Fig. 4. Fig. 4 shows that singular value of woofer mode gradually decreases as the order of eigen mode of increases, and some singular values are almost equal to zero when the order of woofer mode is greater than 16. To keep the stability of close-loop control for W–T system, aberration 3
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 5. Correction of proposed algorithm in three different pupil regions.
Fig. 6. Cross-coupling suppression of proposed algorithm: (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
Fig. 8(d) shows that woofer will generate more compensation besides
3.2. Comparison with Zernike decomposition algorithm
aberration assigned to it with the iteration increasing. From Figs. 7 and 8, it can be concluded that although the compensation ability without
Ref. [8] proves that Zernike decomposition algorithm can restrain cross-coupling between woofer and tweeter well in circular pupil region. At this moment, we verify the performance of Zernike decomposition algorithm in three different irregular pupil regions as above. Note that, before calculating the reconstruction matrix of Zernike mode, the
correlation suppression is similar with that of using correlation suppression, lots of cross-coupling between the two DMs make woofer and tweeter consume more stroke to correct a same wave-front aberration than that of using correlation suppression. 4
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 7. Correction of proposed algorithm without using Eqs. (15) to (17) in three different pupil regions.
Fig. 8. Cross-coupling suppression of proposed algorithm without using Eqs. (15) to (17): (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
Fig. 9(c1), (c2) and (c3). It indicates that Zernike decomposition algorithm can also make woofer correct low spatial frequency aberration and tweeter correct high spatial frequency aberration respectively. But, the RMS of wave-front aberration are 0.038𝜆, 0.055𝜆 and 0.056𝜆, compared with our algorithm, Zernike decomposition algorithm has worse correction performance. In addition, we verify the performance of cross-coupling suppression of Zernike decomposition algorithm by projecting the correction of woofer and tweeter onto Zernike mode, and results are shown in Fig. 10. Fig. 10(a), (b) and (c) show that woofer generate cross-coupling with tweeter while it corrects the first five
response matrix of Zernike mode should be modified by set corresponding value at some sub-apertures of S–H sensor to zero. The same initial aberrations as shown in Fig. 5(a1), (a2) and (a3) are used to verify the performance of Zernike decomposition algorithm, and we make woofer correct the first five order Zernike mode of initial aberration and tweeter correct the residual aberration respectively. The results are shown in Fig. 9. In three different pupil regions, the correction of woofer are shown in Fig. 9(b1), (b2) and (b3), the correction of tweeter are shown in 5
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 9. Correction of Zernike decomposition algorithm in three different pupil regions.
Fig. 10. Cross-coupling suppression of Zernike decomposition algorithm: (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
6
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 11. Correction of proposed algorithm in three cases of lacking of light which are random and have different proportions on a circular pupil region.
Fig. 12. Cross-coupling suppression of proposed algorithm under lack of light randomly: (a) Aberration decomposition in case one, (b) aberration decomposition in case two; (c) aberration decomposition in case three; (d) compensation ratio in three different cases.
order Zernike mode aberration. At the same time, the compensation ratios of woofer are respectively 0.88, 0.86 and 0.86 as shown in Fig. 10(d). It is obvious that Zernike decomposition algorithm cannot restrain the cross-coupling between woofer and tweeter as the same as our algorithm in generalized irregular geometric region.
3.3. Performance of lacking of light randomly In order to further prove that our algorithm can control woofer and tweeter in generalized irregular pupil region, we consider three cases of lacking of light which are random and have different proportions on 7
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 13. Experiment system: (a) W–T system; (b) Configurations of sub-aperture (squares) and deformable mirror actuators (small and unfilled circular s); (c) Wave-front sensor structure.
a circular pupil region: (a) Case one, twenty percent of pupil lacking of light; (b) Case two, thirty percent of pupil lacking of light; (c) Case three, forty percent of pupil lacking of light. The correction results of our algorithm are shown in Fig. 11. Fig. 11(a1), Fig. 10(a2) and (a3) show three different initial aberration with different degrees of lack of light. We also isolate the first ten order eigen mode in the initial aberration and assign them to woofer, and assign the residual aberration to tweeter. The correction of woofer are shown in Fig. 11(b1), (b2) and (b3), correction of tweeter are shown in Fig. 11(c1), (c2) and (c3), and the residual mean square (RMS) of wave-front aberration are respectively 0.060𝜆, 0.073𝜆 and 0.077𝜆 as shown in Fig. 11(d1), (d2) and (d3). The results of Fig. 11 clearly prove that our algorithm can make woofer correct low spatial frequency aberration and tweeter correct high spatial frequency aberration respectively in generalized irregular pupil region. What is more, the projection of woofer and tweeter correction to eigen mode as shown Fig. 12(a), (b) and (c) demonstrate that our algorithm can make tweeter do not generate cross-coupling compensation with woofer, and woofer only corrects the aberrations assigned to it because the compensation ratios of three different cases both are 0.99 Therefore, it can be concluded that our algorithm can achieve decoupling control of W–T adaptive optics system in generalized irregular pupil region.
Fig. 14. Lack of light in three different ways.
so both woofer and tweeter used a kind of 59-actuator PZT deformablemirror (maximum actuator stroke ±2.5 μm, inter-actuator stroke 8 mm). Fig. 13(b) shows the actuator geometry of DM and the configuration between the DM and Shack–Hartmann wave-front sensor’s sub-apertures. In order to test the validity of our algorithm, we first get the lack of light in three different ways as shown in Fig. 14: (a) Half of pupil region lacking of light; (b) Rectangular region lacking of light; (c) An approximately circular region lacking of light. Secondly, eigen mode of woofer corresponding to half of pupil region lacking of light is calculated and shown in Fig. 15. There are 31 orders of eigen mode of woofer because many actuators are invalid in half of pupil region lacking light. In addition, Fig. 15 shows that the realistic eigen mode of woofer has different spatial frequency with different order, but is not symmetrical as well as the ideal woofer mode as shown in Fig. 3. The reason making eigen mode of woofer unsymmetrical is that the calibrating center of S–H sensor in realistic system is not equal to the geometric center of the S–H sub-apertures as well as the simulation. This result is consistent with reference [16,21]. However, though eigen mode of woofer is unsymmetrical, they can be still as a reference to allocate the aberration to the two DMs according the different spatial
4. Experimental testing An experimental W–T AO system as shown in Fig. 13 was established to verify the performance of the algorithm proposed in this paper. In this experimental system, the wave-front sensor as shown in Fig. 13(a) contains three CCD cameras which were respectively used to measure near field, far field, and residual wave-front slope as shown in Fig. 13(c). Because this experimental was mainly to test the effectiveness and correlation suppression of the algorithm we proposed, 8
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 15. Eigen mode of woofer under experiment in half of pupil region lacking of light.
Fig. 16. Correction of proposed algorithm under experiment.
frequency. Then, we isolate the first twenty order eigen mode in the initial aberration and assign them to woofer, and assign the residual aberration to tweeter. In three different pupil regions, Fig. 16(b1), (b2) and (b3) show that woofer correct most of components of initial aberration, tweeter correct the residual aberration as shown in Fig. 16(c1), (c2) and (c3). And the residual mean square (RMS) of wave-front aberration are respectively 0.11𝜆, 0.08𝜆 and 0.08𝜆 as shown in Fig. 16(d1), (d2) and (d3). The results of Fig. 15 indicate that our
algorithm can make two woofer and tweeter correct different spatial aberration. In addition, we verify the performance of cross-coupling suppression of our algorithm by projecting correction of woofer and tweeter onto the eigen mode of woofer as shown in Fig. 17. Fig. 17(a), (b) and (c) show that our algorithm can make woofer just correct the aberration distributed to it, and tweeter do not generate cross-coupling compensation onto the first twenty eigen mode of woofer, and the compensation ratio of woofer are always kept around 1 9
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 17. Cross-coupling suppression of proposed algorithm under experiment: (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
Fig. 18. Correction of proposed algorithm without cross-coupling suppression under experiment.
as shown in Fig. 17(d) with iterations increasing. Therefore, according to the result of Figs. 15 and 16, it can be concluded that the algorithm we proposed can simultaneously make the dual DMs correct different spatial frequency aberration and get good correction performance in irregular pupil region. At the same moment, we also made the experiment of our algorithm without using Eqs. (15) to (17) in three different pupil regions lacking light. Compared with Fig. 16(d1), (d2) and (d3), the result of Fig. 18(d1), (d2) and (d3) show that the compensation ability
without correlation suppression is similar with that of using correlation suppression. But the correction of tweeter as shown in Fig. 18(c1), (c2) and (c3) is obvious higher than the result shown in Fig. 16(c1), (c2) and (c3). What is more, we project the correction of woofer and tweeter onto the eigen mode of woofer as shown in Fig. 19(a), (b) and (c), it is obvious that woofer and tweeter generate cross-coupling compensation at the first twenty of eigen mode of woofer. Fig. 19(d) shows that compensation ratio of woofer cannot be convergent with 10
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 19. Cross-coupling suppression of proposed algorithm without cross-coupling suppression under experiment: (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
References
iterations increasing, it means that woofer will generate more compensation besides aberration assigned to it, and tweeter will correct the unnecessary aberration generated by woofer. Therefore, our algorithm is necessary to restrain the cross-coupling between the dual DMs in irregular pupil region.
[1] D.J. Dagel, W.D. Cowan, O.B. Spahn, et al., Large-stroke MEMS deformable mirrors for adaptive optics, J. Microelectromech. Syst. 15 (3) (2006) 572–583. [2] H. Yang, et al., Combinational-deformable-mirror adaptive optics system for compensation of high-order modes of wavefront, Chin. Opt. Lett. 5 (8) (2007) 435–437. [3] G. Liu, H. Yang, Experimental verification of combinational deformable-mirror for phase correction, Chin. Opt. Lett. 5 (10) (2007) 559–562. [4] Z. Li, et al., Combinational-deformable-mirror adaptive optics system for atmospheric compensation in free space communication, Opt. Commun. 320 (2014) 162–168. [5] C. Correia, Henri-François Raynaud, Caroline Kulcsár, et al., Minimum-variance control for woofer–tweeter systems in adaptive optics, J. Opt. Soc. Amer. A 27 (11) (2010) A133–A144. [6] R. Conan, C. Bradley, P. Hampton, et al., Distributed modal command for a two-deformable-mirror adaptive optics system, Appl. Opt. 46 (20) (2007) 4329–4340. [7] B. Xu, J. Wu, J. Hou, et al., Double-deformable-mirror adaptive optics system for phase compensation, Appl. Opt. 45 (12) (2006) 2638–2642. [8] W. Liu, L. Dong, P. Yang, et al., A Zernike mode decomposition decoupling control algorithm for dual deformable mirrors adaptive optics system, Opt. Express 21 (20) (2013) 23885. [9] L. Jean-François, V. Jean-Pierre, Woofer-tweeter control in an adaptive optics system using a Fourier reconstructor, J. Opt. Soc. Amer. A 25 (9) (2008) 2271–2279. [10] P.J. Hampton, P. Agathoklis, R. Conan, et al., Closed-loop control of a woofer– tweeter adaptive optics system using wavelet-based phase reconstruction, J. Opt. Soc. Amer. A 27 (11) (2010) A145–A156. [11] W. Zou, X. Qi, S.A. Burns, Wavefront-aberration sorting and correction for a dual-deformable-mirror adaptive-optics system, Opt. Lett. 33 (22) (2008) 2602–2604. [12] W. Zou, X. Qi, S.A. Burns, Woofer-tweeter adaptive optics scanning laser ophthalmoscopic imaging based on Lagrange-multiplier damped least-squares algorithm, Biomed. Opt. Express 2 (2011) 1986–2004. [13] W. Zou, S.A. Burns, Testing of Lagrange multiplier damped least-squares control algorithm for woofer-tweeter adaptive optics, Appl. Opt. 51 (9) (2012) 1198–1208. [14] C. Li, N. Sredar, K.M. Ivers, H. Queener, J. Porter, A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system, Opt. Express 18 (2010) 16671–16684. [15] T. Cheng, W. Liu, B. Pang, P. Yang, B. Xu, A slope-based decoupling algorithm to simultaneously control dual deformable mirrors in a woofer–tweeter adaptive optics system, Chin. Phys. B 27 (7) (2018) 070704. [16] T. Cheng, W. Liu, K.J. Yang, et al., Testing for a slope-based decoupling algorithm in a woofer-tweeter adaptive optics system, Appl. Opt. 57 (13) (2018) 3357–3364.
5. Conclusion In this paper, we describe a decoupling control algorithm for W–T adaptive optics system in generalized irregular pupil region. Numerical simulation proves that this algorithm can make woofer correct low spatial frequency aberration and tweeter correct the high spatial frequency aberration well, and restrain tweeter generating cross-coupling shape with woofer in three different irregular pupil regions, even in three cases of lacking of light which are random and have different proportions on a circular pupil region. An experiment is established to test the validity of this algorithm in three different irregular pupil regions, the result indicates that our algorithm can make the dual DMs correct different spatial aberration and restrain cross-coupling between the two DMs as well as numerical simulation. In addition, experimental results under no correlation suppression shows that cross-coupling between the two DMs is obvious, and it means woofer and tweeter will consume more stroke to correct a same wave-front aberration than that of using correlation suppression. In a conclusion, we can make bold assumption that our algorithm is valid and practical for the AO system with dual DMs in generalized irregular pupil region. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant No. 60978049), Youth Innovation Promotion Association, China, Scientists of Chinese Academy of Sciences (Grant No. 2012280), High Resolution Light Imaging Camera System Technology in Geostationary Orbit (Grant No. 2016YFB0500204). 11
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856 [20] Li Xinyang, Jang Wenhan, Analysis of real-time modal reconstruction algorithms for adaptive optics system, Proc. SPIE 4493 (2002) 112–121. [21] Yu Ji, Dong Bing, Deformable mirror eigen modes based wavefront error correction method used for space optical remote sensor, Acta Opt. Sin. 34 (12) (2014) 1228001.
[17] W. Liu, L. Dong, P. Yang, et al., Zonal decoupling algorithm for dual deformable mirror adaptive optics system, Chin. Opt. Lett. 14 (2) (2016) 020101. [18] A.D. Castro, L. Sawides, X. Qi, et al., Adaptive optics retinal imaging with automatic detection of the pupil and its boundary in real time using Shack–Hartmann images, Appl. Opt. 56 (24) (2017) 6748. [19] J. Arines, P. Prado, et al., Pupil tracking with a Hartmann-Shack wavefront sensor, J. Biomed. Opt. 15 (3) (2010) 036022.
12
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A decoupling control algorithm for Woofer–Tweeter adaptive optics system in generalized irregular pupil region Tao Cheng a,c , ZhenXing Xu a,b,c , Kangjian Yang a,c , Shuai Wang a,c , Xin He a,c , Ping Yang a,c ,∗, Bin Xu a,c ,∗ a
Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China University of Chinese Academy of Sciences, Beijing 100049, China c Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China b
ARTICLE
INFO
Keywords: Decoupling control algorithm Adaptive optics Generalized irregular pupil region
ABSTRACT We proposed a decoupling control algorithm to accomplish the application of woofer–tweeter (W–T) adaptive optics system in generalized irregular pupil region. This algorithm firstly modifies the response matrix of woofer and tweeter by setting corresponding value at some sub-apertures of Shack–Hartmann(S–H) sensor to zero and constructs a slope-based orthogonal basis used to distribute different spatial frequency aberration to dual deformable mirrors (DMs). Then a limited condition is used to calculate the pseudo-inverse matrix of modified response matrix of dual DMs through single value decomposition. Finally, to restrain tweeter generating crosscoupling shape with woofer, a constraint matrix used to reset control vector of tweeter is constructed with the slope-based orthogonal basis and modified response matrix of tweeter. Numeral simulation of W–T adaptive optics system is built to verify the performance of this algorithm and results demonstrate that this algorithm has a good performance to control the W–T adaptive optics system in generalized irregular pupil region, and achieve better performance to suppress the cross-coupling between woofer and tweeter than Zernike decomposition algorithm. At the same time, the experimental results for the W–T AO system demonstrated that this algorithm is practical for the AO system with dual DMs in generalized irregular pupil region.
1. Introduction Through many researchers have done a lot of study about large stroke and high spatial frequency deformable mirror (DM), the application of this DM still need solve many technological problems [1]. So the woofer–tweeter(W–T) adaptive optics system shown in Fig. 1 is really an efficient solution to simultaneously compensate phase distortion with a large amplitude and high spatial frequency [2–4]: a single (S– H) sensor is used to measure the wave-front aberration , and distribute the large stroke and low spatial frequency aberration to woofer and low stroke and high spatial frequency aberration to tweeter. It is well known that woofer and tweeter will generate opposite compensation which will make the dual DMs correct the same aberration using a bigger stroke and may turn the system instable [5]. To make the two DMs work cooperatively, several algorithms have been proposed to suppress the cross-coupling of dual DMs, such as Zernike mode decomposition algorithm [6–8], Fourier mode decomposition algorithm [9], wavelet mode decomposition algorithm [10], Lagrangemultiplier damped least-squares algorithms [11–13], and direct slopebased correction algorithm [14–17]. According to the references [8,15], Zernike mode decomposition algorithm and slope-based decoupling
control algorithm can achieve better performance to suppress correlation error of dual DMs than other algorithms. The reason for the two algorithms restraining the cross-coupling error of dual DMs well is that an orthogonal mode in regular region such as circular or rectangle is constructed firstly, which can be used to distribute different spatial frequency aberration to dual DMs and construct constraint matrix to reset tweeter control vector. However, due to the fluctuation of light intensity in many applications such as laser beam cleaning, a few of sub-apertures of (S–H) sensor has a lack of light or are saturated, it will make the pupil be a generalized irregular geometric region because the information of these sub-apertures will reduce the accuracy of wavefront reconstruction. Unfortunately, the mathematical basis of the mode decoupling algorithm is the orthogonality of the Zernike mode on the unit circle. However, the orthogonality of Zernike mode will be destroyed, so that Zernike mode requires Schmitt orthogonalization onto irregular pupil region if continuing to use Zernike mode for decoupling control, it will be a Complex calculation process and not flexible. To slope-based decoupling control algorithm, some actuators of woofer and tweeter will be not valid in generalized irregular pupil region,
∗ Corresponding authors at: Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China. E-mail addresses: [email protected] (P. Yang), [email protected] (B. Xu).
https://doi.org/10.1016/j.optcom.2020.125856 Received 16 January 2020; Received in revised form 23 March 2020; Accepted 30 March 2020 Available online 4 April 2020 0030-4018/© 2020 Published by Elsevier B.V.
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
of 𝑀𝑊 is orthogonal and corresponding to eigen mode of woofer. What is more, the eigen mode of the woofer is similar with the Zernike mode, of which the spatial frequency is gradually increasing with the mode’s order increasing. So wave-front slop distributed to woofer can be calculated with Eqs. (5) and (6). + 𝑟 = 𝑀𝑊 𝑔
(5)
𝑔𝑊 = 𝑀𝑤 𝐼𝑊 𝑟
(6)
where, 𝑔 is the residual wave-front slope, 𝑟(𝑚1 × 1) is the coefficient + of 𝑀𝑤 , 𝑀𝑊 is the pseudo-inverse matrix of 𝑀𝑤 and 𝐼𝑊 is a diagonal matrix used to assign wave-front slope 𝑔𝑊 to woofer by set 𝐼𝑊 (𝑖, 𝑖) to one or zero. In order to calculate the error voltage 𝑣𝑊 with 𝑔𝑊 , we should get the pseudo-inverse matrix 𝑅+ with Eqs. (7)–(9). 𝑊 _𝑛𝑒𝑤
Fig. 1. The woofer–tweeter adaptive optics system.
(7)
[𝑈 , 𝑆, 𝑉 ] = 𝑆𝑉 𝐷(𝑅𝑊 _𝑛𝑒𝑤 ),
slope-based orthogonal basis constructed from woofer will be inapplicable so that constraint matrix is no longer capable of suppressing cross-coupling of woofer and tweeter. In this paper, in order to control woofer and tweeter simultaneously in generalized irregular pupil region, we proposed a decoupling control algorithm. This algorithm firstly set corresponding value of response matrix of dual DMs at some sub-apertures to zero according to light intensity, secondly construct a slope-based orthogonal basis to distribute different spatial frequency aberration to two deformable mirrors (DMs) with the modified response matrix of woofer, then modify the reconstruction matrix of woofer and tweeter with a limited condition to calculate the control vector of dual DMs, finally reset the control vector of tweeter to restrain tweeter generating opposite compensation with woofer. This paper is organized as follows. Section 2 describes the principle of the decoupling algorithm in generalized irregular pupil region. Section 3 establishes a numerical model for the W–T adaptive optics system to test the validity of this algorithm in theory, and makes a detail comparison with our algorithm and Zernike mode decomposition algorithm. Section 4 established a realistic W–T AO system to test the performance of our algorithm in generalized irregular pupil region. Section 5 is the conclusion.
⎧ ⎪𝑆 (𝑖, 𝑖) , 𝑆𝑙𝑖𝑚𝑖𝑡 (𝑖, 𝑖) = ⎨ ⎪0, ⎩
𝑖𝑓 (𝑆(𝑖, 𝑖) ≥ 𝑖𝑓 (𝑆(𝑖, 𝑖) <
1 ) 𝑐𝑜𝑛𝑑𝑙𝑖𝑚𝑖𝑡 1 ), 𝑐𝑜𝑛𝑑𝑙𝑖𝑚𝑖𝑡
𝑅+ = 𝑉 𝑆𝑙𝑖𝑚𝑖𝑡 𝑈 𝑇 𝑊 _𝑛𝑒𝑤
of woofer can be calculated with Eqs. (10) and (11), 𝑣𝑊 = 𝑅+ 𝑔 𝑊 _𝑛𝑒𝑤 𝑤
(10)
𝑉𝑊 (𝑘 + 1) = 𝑝𝑖_𝑎 ∗ 𝑉𝑊 (𝑘) + 𝑝𝑖_𝑏 ∗ 𝑣𝑊
(11)
where, 𝑝𝑖_𝑎 and 𝑝𝑖_𝑏 are the parameters of PI controller. 𝑝𝑖_𝑎 is usually set around 0.98 and 𝑝𝑖_𝑏 is usually set around 0.2 in the adaptive optics system. After the correction by the woofer, the residual wave-front slope 𝑔𝑇 compensated by tweeter DM can be solved as following, 𝑔𝑇 = 𝑔 − 𝑔𝑊
where, is the pseudo-inverse matrix of 𝑅𝑇 _𝑛𝑒𝑤 . The control vector 𝑉𝑇 is also solved with a PI controller as, 𝑉𝑇 (𝑘 + 1) = 𝑝𝑖_𝑎 ∗ 𝑉𝑇 (𝑘) + 𝑝𝑖_𝑏 ∗ 𝑣𝑇
(14)
Though the control vector of woofer and tweeter is obtained as above, cross-coupling of dual DMs must be restrained to reduce the waste of stroke of dual DMs and guarantee the stability of W–T adaptive optics system during the correction process. So a constraint matrix 𝑅𝑐 is constructed with 𝑅𝑇 _𝑛𝑒𝑤 and 𝑀𝑊 as, ∑2𝑛 𝑅𝑇 _𝑛𝑒𝑤 (𝑥, 𝑖)𝑀𝑊 (𝑥, 𝑗) 𝑅𝑐 (𝑖, 𝑗) = 𝑘 ∑𝑥=1 (15) 2𝑛 𝑥=1 𝑀𝑊 (𝑥, 𝑗)𝑀𝑊 (𝑥, 𝑗)
where, 𝐼𝑖 is the sum of light intensity at the ith sub-aperture, 𝑇𝑚𝑖𝑛 and 𝑇𝑚𝑎𝑥 are respectively the maximum and minimum thresholds, ( ) 𝑅𝑊 𝑖𝑥 , 𝑗 is value in the 𝑥 direction at the ith sub-aperture of the jth actuator of woofer. Eq. (1) shows that the response value of the jth actuator at one sub-aperture is valid when the sums of light intensity in this sub-aperture is between the maximum and minimum thresholds. At the moment, we can also get 𝑅𝑇 _𝑛𝑒𝑤 through Eq. (1). After getting 𝑅𝑊 _𝑛𝑒𝑤 , a slope-based orthogonal basis corresponding to generalized irregular geometric region can be calculated as following,
(4)
(13)
𝑅+ 𝑇 _𝑛𝑒𝑤
In a W–T adaptive optics system, the response matrix 𝑅𝑊 and 𝑅𝑇 are measured with S–H sensor. When some sub-apertures of S–H sensor has a lack of light or are saturated due to the fluctuation of light intensity, the value of 𝑅𝑊 at some sub-apertures should be set at zero according to the light intensity as following [18,19], { ( ) ( ) 𝑅𝑊 𝑖𝑥 , 𝑗 , 𝑖𝑓 (𝑇𝑚𝑖𝑛 ≤ 𝐼𝑖 ≤ 𝑇𝑚𝑎𝑥 ) (1) 𝑅𝑊 _𝑛𝑒𝑤 𝑖𝑥 , 𝑗 = 0, 𝑖𝑓 (𝐼𝑖 ≥ 𝑇𝑚𝑎𝑥 , or 𝐼𝑖 ≤ 𝑇𝑚𝑖𝑛 )
𝑀𝑊 = 𝑅𝑊 _𝑛𝑒𝑤 𝑈
(12)
At the same time, the error voltage 𝑣𝑇 of tweeter is calculated with Eq. (13). 𝑣𝑇 = 𝑅+ 𝑔 𝑇 _𝑛𝑒𝑤 𝑇
(2)
(9)
A new singular value matrix 𝑆𝑙𝑖𝑚𝑖𝑡 is calculated with a limited condition 𝑐𝑜𝑛𝑑𝑙𝑖𝑚𝑖𝑡 as Eq. (8), here the limited condition 𝑐𝑜𝑛𝑑𝑙𝑖𝑚𝑖𝑡 is used to guarantee the stability of the matrix 𝑅+ . Then control vector 𝑉𝑊 𝑊 _𝑛𝑒𝑤
2. Principle
𝐶𝑤 = 𝑅𝑇𝑊 _𝑛𝑒𝑤 𝑅𝑊 _𝑛𝑒𝑤 ∕2𝑛 [ ] 𝑈 , 𝑆𝑀 , 𝑈 𝑇 = 𝑆𝑉 𝐷(𝐶𝑊 )
(8)
where, 𝑅𝑐 (𝑖, 𝑗) is the relation coefficient between the ith column of 𝑅𝑇 _𝑛𝑒𝑤 and the jth column of 𝑀𝑊 , 𝑘 is an empirical parameter used to adjust the constraint intensity to the tweeter DM, and it should be set according to the relation between influence slope matrix 𝑅𝑇 _𝑛𝑒𝑤 and orthogonal basis 𝑀𝑊 . A larger 𝑘 will enhance the elimination of low order components in the surface shape of the tweeter, but the correction qualities of higher order phase aberrations will be decreased. Thus 𝑘 should be carefully chosen to make a better compromise. With the constraint matrix 𝑅𝑐 , the control vector 𝑉𝑇 of tweeter can be modified by Eqs. (16) and (17), [ ] [ ] 𝐼 𝑉 𝑉𝑇 = 𝐶𝑉𝑇 = 𝑇 (16) 𝑅𝐶 0 [ ]+ [ ] [ ] 𝐼 𝑉𝑇 𝑉 𝑉𝑇′ = = 𝐶+ 𝑇 (17) 𝑅𝐶 0 0
(3)
where, n is the number of Hartmann’s aperture, 𝐶𝑤 (𝑚1 × 𝑚1) describes the correlation of influence slopes of different actuator. Through singular value decomposition of 𝐶𝑤 as Eq. (3), a slope-based orthogonal basis 𝑀𝑊 (2𝑛×𝑚1) can be achieved with Eq. (4). Note that the column vector 2
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 2. Configuration of DM’s actuators and the S–H sensor’s sub-apertures: (a) Woofer; (b) Tweeter. Fig. 4. Singular values of the eigen mode of woofer.
distributed to woofer does not contain the eigen mode of woofer which order is greater than 16. Next, we consider three different cases to verify the validity of our algorithm: (1) half of pupil region lacking of light; (2) a rectangular region lacking of light; (3) a circular region lacking of light. The initial aberration in three different pupil regions are shown in Fig. 5(a1), (a2), and (a3). Then, according to the singular value shown in Fig. 4, we isolate the first ten order eigen mode in the initial aberration and assign them to woofer, and assign the residual aberration to tweeter. In three different pupil regions, the correction of woofer are shown in Fig. 5(b1), (b2) and (b3), the correction of tweeter are shown in Fig. 5(c1), (c2) and (c3). And the residual mean square (RMS) of wave-front aberration are respectively 0.026𝜆, 0.047𝜆 and 0.048𝜆 as shown in Fig. 5(d1), (d2) and (d3). The results of Fig. 5 indicate that our algorithm can make woofer correct low spatial frequency aberration and tweeter correct high spatial frequency aberration respectively in irregular pupil region. Note that, the limit condition described as Eq. (8) and used to calculate the reconstruction matrix of dual DMs is set to 10. Finally, in order to verify the performance of our algorithm to restrain the cross-coupling of dual DMs, we project the correction of woofer and tweeter to the eigen mode of woofer and a compensation ratio 𝑐 is defined as Eq. (18) to evaluate the cross-coupling of dual DMs,
Fig. 3. Eigen mode corresponding to column vector of 𝑀𝑊 .
where, 𝑉𝑇′ is the control vector of tweeter which will not generate cross-coupling shape with woofer. 3. Numeral simulation
𝑐=
3.1. Testing validity
∫ ∅𝑤𝑐 ∅𝑤 𝑑𝑠 ∫ ∅𝑤 ∅𝑤 𝑑𝑠
(18)
where, ∅𝑤 is the aberration distributed to woofer and ∅𝑤𝑐 is the aberration corrected by woofer. The compensation ratio 𝑐 is an indirectly evaluation indicator, it means that compensation ratio 𝑐 will be one when aberration distributed to woofer ∅𝑤 is corrected completely by woofer ∅𝑤𝑐 and woofer do not generate other aberration. At that moment, we can indirectly demonstrate that tweeter do not generate unnecessary cross-coupling compensation with woofer. Fig. 6(d) shows that the compensation ratios 𝑐 of three different irregular pupil regions both are 0.99, it indicates that our algorithm can restrain tweeter generating cross-coupling shape to woofer well in irregular pupil region. In addition, we test the performance of our algorithm without using Eqs. (15) to (17), and the correction results and cross-coupling suppression in three different pupil regions lacking light are respectively shown in Figs. 7 and 8. Though the residual wave-front aberration is similar with Fig. 5, are respectively 0.024𝜆, 0.050𝜆 and 0.049𝜆 as shown Fig. 7(d1), (d2) and (d3), the correction of woofer as shown in Fig. 7(b1), (b2) and (b3) is obviously higher than Fig. 5(b1), (b2) and (b3), and the correction of tweeter as shown in Fig. 7(c1), (c2) and (c3) is obviously higher than Fig. 5(c1), (c2) and (c3). At this moment, Fig. 8(a), (b) and (c) show that woofer and tweeter have obviously cross-coupling compensation in the first ten orders,
In order to verify the validity of the algorithm proposed above, a W– T adaptive optics system is built. Woofer and tweeter respectively have 19-actuators and 127-actuators, and the configuration of dual DMs’ actuators and S–H sensor’s sub-apertures are shown in Fig. 2. We firstly make a hypothesis that half of pupil region is lack of light or saturated. Then a slope-based orthogonal basis 𝑀𝑊 can be constructed with 𝑅𝑊 _𝑛𝑒𝑤 through Eqs. (2)–(4), and the eigen mode corresponding to column vector of 𝑀𝑊 is shown in Fig. 3. Fig. 3 shows that the spatial frequency of eigen mode of woofer is gradually increasing with the mode’s order increasing. So, these modes can be used to assign different spatial frequency aberration to the dual DMs. However, according to the theory about singular value decomposition, Ref. [20] proved that removing some high order of eigen mode of woofer which singular value is very small and almost equal to zero can still take full use of woofer and improve the stability of close-loop control for woofer. Therefore, singular values of eigen mode of woofer should be calculated by Eq. (7) and shown in Fig. 4. Fig. 4 shows that singular value of woofer mode gradually decreases as the order of eigen mode of increases, and some singular values are almost equal to zero when the order of woofer mode is greater than 16. To keep the stability of close-loop control for W–T system, aberration 3
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 5. Correction of proposed algorithm in three different pupil regions.
Fig. 6. Cross-coupling suppression of proposed algorithm: (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
Fig. 8(d) shows that woofer will generate more compensation besides
3.2. Comparison with Zernike decomposition algorithm
aberration assigned to it with the iteration increasing. From Figs. 7 and 8, it can be concluded that although the compensation ability without
Ref. [8] proves that Zernike decomposition algorithm can restrain cross-coupling between woofer and tweeter well in circular pupil region. At this moment, we verify the performance of Zernike decomposition algorithm in three different irregular pupil regions as above. Note that, before calculating the reconstruction matrix of Zernike mode, the
correlation suppression is similar with that of using correlation suppression, lots of cross-coupling between the two DMs make woofer and tweeter consume more stroke to correct a same wave-front aberration than that of using correlation suppression. 4
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 7. Correction of proposed algorithm without using Eqs. (15) to (17) in three different pupil regions.
Fig. 8. Cross-coupling suppression of proposed algorithm without using Eqs. (15) to (17): (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
Fig. 9(c1), (c2) and (c3). It indicates that Zernike decomposition algorithm can also make woofer correct low spatial frequency aberration and tweeter correct high spatial frequency aberration respectively. But, the RMS of wave-front aberration are 0.038𝜆, 0.055𝜆 and 0.056𝜆, compared with our algorithm, Zernike decomposition algorithm has worse correction performance. In addition, we verify the performance of cross-coupling suppression of Zernike decomposition algorithm by projecting the correction of woofer and tweeter onto Zernike mode, and results are shown in Fig. 10. Fig. 10(a), (b) and (c) show that woofer generate cross-coupling with tweeter while it corrects the first five
response matrix of Zernike mode should be modified by set corresponding value at some sub-apertures of S–H sensor to zero. The same initial aberrations as shown in Fig. 5(a1), (a2) and (a3) are used to verify the performance of Zernike decomposition algorithm, and we make woofer correct the first five order Zernike mode of initial aberration and tweeter correct the residual aberration respectively. The results are shown in Fig. 9. In three different pupil regions, the correction of woofer are shown in Fig. 9(b1), (b2) and (b3), the correction of tweeter are shown in 5
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 9. Correction of Zernike decomposition algorithm in three different pupil regions.
Fig. 10. Cross-coupling suppression of Zernike decomposition algorithm: (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
6
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 11. Correction of proposed algorithm in three cases of lacking of light which are random and have different proportions on a circular pupil region.
Fig. 12. Cross-coupling suppression of proposed algorithm under lack of light randomly: (a) Aberration decomposition in case one, (b) aberration decomposition in case two; (c) aberration decomposition in case three; (d) compensation ratio in three different cases.
order Zernike mode aberration. At the same time, the compensation ratios of woofer are respectively 0.88, 0.86 and 0.86 as shown in Fig. 10(d). It is obvious that Zernike decomposition algorithm cannot restrain the cross-coupling between woofer and tweeter as the same as our algorithm in generalized irregular geometric region.
3.3. Performance of lacking of light randomly In order to further prove that our algorithm can control woofer and tweeter in generalized irregular pupil region, we consider three cases of lacking of light which are random and have different proportions on 7
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 13. Experiment system: (a) W–T system; (b) Configurations of sub-aperture (squares) and deformable mirror actuators (small and unfilled circular s); (c) Wave-front sensor structure.
a circular pupil region: (a) Case one, twenty percent of pupil lacking of light; (b) Case two, thirty percent of pupil lacking of light; (c) Case three, forty percent of pupil lacking of light. The correction results of our algorithm are shown in Fig. 11. Fig. 11(a1), Fig. 10(a2) and (a3) show three different initial aberration with different degrees of lack of light. We also isolate the first ten order eigen mode in the initial aberration and assign them to woofer, and assign the residual aberration to tweeter. The correction of woofer are shown in Fig. 11(b1), (b2) and (b3), correction of tweeter are shown in Fig. 11(c1), (c2) and (c3), and the residual mean square (RMS) of wave-front aberration are respectively 0.060𝜆, 0.073𝜆 and 0.077𝜆 as shown in Fig. 11(d1), (d2) and (d3). The results of Fig. 11 clearly prove that our algorithm can make woofer correct low spatial frequency aberration and tweeter correct high spatial frequency aberration respectively in generalized irregular pupil region. What is more, the projection of woofer and tweeter correction to eigen mode as shown Fig. 12(a), (b) and (c) demonstrate that our algorithm can make tweeter do not generate cross-coupling compensation with woofer, and woofer only corrects the aberrations assigned to it because the compensation ratios of three different cases both are 0.99 Therefore, it can be concluded that our algorithm can achieve decoupling control of W–T adaptive optics system in generalized irregular pupil region.
Fig. 14. Lack of light in three different ways.
so both woofer and tweeter used a kind of 59-actuator PZT deformablemirror (maximum actuator stroke ±2.5 μm, inter-actuator stroke 8 mm). Fig. 13(b) shows the actuator geometry of DM and the configuration between the DM and Shack–Hartmann wave-front sensor’s sub-apertures. In order to test the validity of our algorithm, we first get the lack of light in three different ways as shown in Fig. 14: (a) Half of pupil region lacking of light; (b) Rectangular region lacking of light; (c) An approximately circular region lacking of light. Secondly, eigen mode of woofer corresponding to half of pupil region lacking of light is calculated and shown in Fig. 15. There are 31 orders of eigen mode of woofer because many actuators are invalid in half of pupil region lacking light. In addition, Fig. 15 shows that the realistic eigen mode of woofer has different spatial frequency with different order, but is not symmetrical as well as the ideal woofer mode as shown in Fig. 3. The reason making eigen mode of woofer unsymmetrical is that the calibrating center of S–H sensor in realistic system is not equal to the geometric center of the S–H sub-apertures as well as the simulation. This result is consistent with reference [16,21]. However, though eigen mode of woofer is unsymmetrical, they can be still as a reference to allocate the aberration to the two DMs according the different spatial
4. Experimental testing An experimental W–T AO system as shown in Fig. 13 was established to verify the performance of the algorithm proposed in this paper. In this experimental system, the wave-front sensor as shown in Fig. 13(a) contains three CCD cameras which were respectively used to measure near field, far field, and residual wave-front slope as shown in Fig. 13(c). Because this experimental was mainly to test the effectiveness and correlation suppression of the algorithm we proposed, 8
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 15. Eigen mode of woofer under experiment in half of pupil region lacking of light.
Fig. 16. Correction of proposed algorithm under experiment.
frequency. Then, we isolate the first twenty order eigen mode in the initial aberration and assign them to woofer, and assign the residual aberration to tweeter. In three different pupil regions, Fig. 16(b1), (b2) and (b3) show that woofer correct most of components of initial aberration, tweeter correct the residual aberration as shown in Fig. 16(c1), (c2) and (c3). And the residual mean square (RMS) of wave-front aberration are respectively 0.11𝜆, 0.08𝜆 and 0.08𝜆 as shown in Fig. 16(d1), (d2) and (d3). The results of Fig. 15 indicate that our
algorithm can make two woofer and tweeter correct different spatial aberration. In addition, we verify the performance of cross-coupling suppression of our algorithm by projecting correction of woofer and tweeter onto the eigen mode of woofer as shown in Fig. 17. Fig. 17(a), (b) and (c) show that our algorithm can make woofer just correct the aberration distributed to it, and tweeter do not generate cross-coupling compensation onto the first twenty eigen mode of woofer, and the compensation ratio of woofer are always kept around 1 9
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 17. Cross-coupling suppression of proposed algorithm under experiment: (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
Fig. 18. Correction of proposed algorithm without cross-coupling suppression under experiment.
as shown in Fig. 17(d) with iterations increasing. Therefore, according to the result of Figs. 15 and 16, it can be concluded that the algorithm we proposed can simultaneously make the dual DMs correct different spatial frequency aberration and get good correction performance in irregular pupil region. At the same moment, we also made the experiment of our algorithm without using Eqs. (15) to (17) in three different pupil regions lacking light. Compared with Fig. 16(d1), (d2) and (d3), the result of Fig. 18(d1), (d2) and (d3) show that the compensation ability
without correlation suppression is similar with that of using correlation suppression. But the correction of tweeter as shown in Fig. 18(c1), (c2) and (c3) is obvious higher than the result shown in Fig. 16(c1), (c2) and (c3). What is more, we project the correction of woofer and tweeter onto the eigen mode of woofer as shown in Fig. 19(a), (b) and (c), it is obvious that woofer and tweeter generate cross-coupling compensation at the first twenty of eigen mode of woofer. Fig. 19(d) shows that compensation ratio of woofer cannot be convergent with 10
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856
Fig. 19. Cross-coupling suppression of proposed algorithm without cross-coupling suppression under experiment: (a) Aberration decomposition corresponding to half of pupil region, (b) aberration decomposition corresponding to rectangular region; (c) aberration decomposition corresponding to circular region; (d) compensation ratio in three different cases.
References
iterations increasing, it means that woofer will generate more compensation besides aberration assigned to it, and tweeter will correct the unnecessary aberration generated by woofer. Therefore, our algorithm is necessary to restrain the cross-coupling between the dual DMs in irregular pupil region.
[1] D.J. Dagel, W.D. Cowan, O.B. Spahn, et al., Large-stroke MEMS deformable mirrors for adaptive optics, J. Microelectromech. Syst. 15 (3) (2006) 572–583. [2] H. Yang, et al., Combinational-deformable-mirror adaptive optics system for compensation of high-order modes of wavefront, Chin. Opt. Lett. 5 (8) (2007) 435–437. [3] G. Liu, H. Yang, Experimental verification of combinational deformable-mirror for phase correction, Chin. Opt. Lett. 5 (10) (2007) 559–562. [4] Z. Li, et al., Combinational-deformable-mirror adaptive optics system for atmospheric compensation in free space communication, Opt. Commun. 320 (2014) 162–168. [5] C. Correia, Henri-François Raynaud, Caroline Kulcsár, et al., Minimum-variance control for woofer–tweeter systems in adaptive optics, J. Opt. Soc. Amer. A 27 (11) (2010) A133–A144. [6] R. Conan, C. Bradley, P. Hampton, et al., Distributed modal command for a two-deformable-mirror adaptive optics system, Appl. Opt. 46 (20) (2007) 4329–4340. [7] B. Xu, J. Wu, J. Hou, et al., Double-deformable-mirror adaptive optics system for phase compensation, Appl. Opt. 45 (12) (2006) 2638–2642. [8] W. Liu, L. Dong, P. Yang, et al., A Zernike mode decomposition decoupling control algorithm for dual deformable mirrors adaptive optics system, Opt. Express 21 (20) (2013) 23885. [9] L. Jean-François, V. Jean-Pierre, Woofer-tweeter control in an adaptive optics system using a Fourier reconstructor, J. Opt. Soc. Amer. A 25 (9) (2008) 2271–2279. [10] P.J. Hampton, P. Agathoklis, R. Conan, et al., Closed-loop control of a woofer– tweeter adaptive optics system using wavelet-based phase reconstruction, J. Opt. Soc. Amer. A 27 (11) (2010) A145–A156. [11] W. Zou, X. Qi, S.A. Burns, Wavefront-aberration sorting and correction for a dual-deformable-mirror adaptive-optics system, Opt. Lett. 33 (22) (2008) 2602–2604. [12] W. Zou, X. Qi, S.A. Burns, Woofer-tweeter adaptive optics scanning laser ophthalmoscopic imaging based on Lagrange-multiplier damped least-squares algorithm, Biomed. Opt. Express 2 (2011) 1986–2004. [13] W. Zou, S.A. Burns, Testing of Lagrange multiplier damped least-squares control algorithm for woofer-tweeter adaptive optics, Appl. Opt. 51 (9) (2012) 1198–1208. [14] C. Li, N. Sredar, K.M. Ivers, H. Queener, J. Porter, A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system, Opt. Express 18 (2010) 16671–16684. [15] T. Cheng, W. Liu, B. Pang, P. Yang, B. Xu, A slope-based decoupling algorithm to simultaneously control dual deformable mirrors in a woofer–tweeter adaptive optics system, Chin. Phys. B 27 (7) (2018) 070704. [16] T. Cheng, W. Liu, K.J. Yang, et al., Testing for a slope-based decoupling algorithm in a woofer-tweeter adaptive optics system, Appl. Opt. 57 (13) (2018) 3357–3364.
5. Conclusion In this paper, we describe a decoupling control algorithm for W–T adaptive optics system in generalized irregular pupil region. Numerical simulation proves that this algorithm can make woofer correct low spatial frequency aberration and tweeter correct the high spatial frequency aberration well, and restrain tweeter generating cross-coupling shape with woofer in three different irregular pupil regions, even in three cases of lacking of light which are random and have different proportions on a circular pupil region. An experiment is established to test the validity of this algorithm in three different irregular pupil regions, the result indicates that our algorithm can make the dual DMs correct different spatial aberration and restrain cross-coupling between the two DMs as well as numerical simulation. In addition, experimental results under no correlation suppression shows that cross-coupling between the two DMs is obvious, and it means woofer and tweeter will consume more stroke to correct a same wave-front aberration than that of using correlation suppression. In a conclusion, we can make bold assumption that our algorithm is valid and practical for the AO system with dual DMs in generalized irregular pupil region. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant No. 60978049), Youth Innovation Promotion Association, China, Scientists of Chinese Academy of Sciences (Grant No. 2012280), High Resolution Light Imaging Camera System Technology in Geostationary Orbit (Grant No. 2016YFB0500204). 11
T. Cheng, Z. Xu, K. Yang et al.
Optics Communications 472 (2020) 125856 [20] Li Xinyang, Jang Wenhan, Analysis of real-time modal reconstruction algorithms for adaptive optics system, Proc. SPIE 4493 (2002) 112–121. [21] Yu Ji, Dong Bing, Deformable mirror eigen modes based wavefront error correction method used for space optical remote sensor, Acta Opt. Sin. 34 (12) (2014) 1228001.
[17] W. Liu, L. Dong, P. Yang, et al., Zonal decoupling algorithm for dual deformable mirror adaptive optics system, Chin. Opt. Lett. 14 (2) (2016) 020101. [18] A.D. Castro, L. Sawides, X. Qi, et al., Adaptive optics retinal imaging with automatic detection of the pupil and its boundary in real time using Shack–Hartmann images, Appl. Opt. 56 (24) (2017) 6748. [19] J. Arines, P. Prado, et al., Pupil tracking with a Hartmann-Shack wavefront sensor, J. Biomed. Opt. 15 (3) (2010) 036022.
12