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Abstract
To resolve cooperative control issues of dual wavefront correctors in generalized irregular pupil regions, we propose a decoupling control algorithm based on numerical orthogonal polynomials (NOP). The proposed algorithm firstly deduces NOP from Zernike polynomials in generalized irregular pupil regions. Then, according to wavefront restoration, different spatial frequency aberrations to different wavefront correctors are assigned precisely. Finally, the algorithm calculates and eliminates the cross-coupling between dual wavefront correctors. As observed in numerical simulations and experiments based on a typical woofer-tweeter (W-T) adaptive optics system, NOP decoupling control algorithm restrains the cross-coupling between woofer and tweeter in generalized irregular pupil regions. Moreover, there are obvious advantages over Zernike polynomials decoupling control algorithm in cross-coupling suppression for various scenarios in irregular pupil regions and restoration orders.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Adaptive optics (AO) is an essential technology for detecting and correcting wavefront aberrations in real-time [1]. Nowadays, growing number of applications start to use AO systems to correct large amplitude aberrations with lower spatial frequency and small aberrations with higher spatial frequency, such as astronomical observation [2], atmospheric corrections [3], and beam clean-up [4]. Unfortunately, limited by the spatial frequency and actuator stroke of the wavefront corrector [5], it is challenging for AO systems with single wavefront corrector to compensate aberrations which has large amplitude and high spatial frequency simultaneously. Accordingly, a solution to the problem is provided by using dual wavefront correctors [6,7]. For example, the W-T system is a typical double deformable mirrors (DMs) AO system [7], whose woofer is a mirror with large stroke and low spatial frequency and tweeter provides high spatial frequency and small stroke. With appropriate control methods, low order and large amplitude wavefront aberrations can be assigned to the woofer for correcting, while the tweeter is responsible for the residual wavefront aberrations with high orders and small amplitude. However, there is cross-coupling between the dual DMs in the W-T system, which often lead to aberrations correction disorder. Such as, the coupling suppression is tested in the Ref. [8] using Lagrange-multiplier (LM)-based damped least-squares (DLS) control method. Although the coupling between the dual DMs is suppressed by this method, it is not eliminated, which leads to worse stability performance of the AO system. Besides, coupling suppression is not employed in the Ref. [9], which caused a waste of available stroke of the dual DMs.
Therefore, in order to eliminate the cross-coupling, several decoupling control algorithms for dual DMs are proposed and proved, including mode decomposition control algorithm [10,11], zonal decoupling control algorithm [12,13], direct slop-based decoupling control algorithm [14–16], and so on. It is worth highlighting that Zernike mode decomposition control algorithm and slope-based decoupling control algorithm demonstrate great effectiveness in suppressing the correlation error of dual DMs within unit circle pupil, among the aforementioned algorithms, according to Ref. [10,15]. However, causing by non-ideality of system device and light intensity distribution, irregular pupil regions often exist. For instance, telescopes and other optical systems are annular systems with central shading [17,18]. Besides, light consistently blinks in real optical systems, which is a result of uneven light intensity distribution [19] caused by light source defects or unstable transmission process [20–22]. Where, traditional decoupling control algorithms are no longer applicable. Such as, the orthogonality of Zernike polynomials no longer exists in irregular pupil regions [23]. It prevents the wavefront aberrations from being decomposed orthogonally, resulting in Zernike mode decomposition control algorithm is no longer able to assign accurate wavefront aberrations to the woofer and the tweeter respectively. Moreover, the cross-coupling between the dual DMs is not suppressed validly any more. In addition, slope-based decoupling control algorithm is improved to apply to irregular pupil regions in Ref. [24]. However, according to the experimental results, the aberrations with different spatial frequency distribution require manual selection of the woofer eigen mode, which is a complicated and inflexible process. Thus, it is necessary to develop decoupling control algorithms for generalized irregular pupil regions in W-T systems.
In summary, since Zernike polynomials can characterize aberrations conveniently, it is reasonable to realize the assignment of aberrations with different spatial frequency in this approach. Therefore, a decoupling control algorithm based on NOP is proposed in this paper by modifying Zernike polynomials in generalized irregular pupil regions. Firstly, this algorithm constructs the NOP corresponding to the irregular pupil region. Then, the wavefront aberrations assigned to woofer and tweeter are calculated. Finally, to keep the tweeter from generating components coupled to the woofer, the NOP is applied to derive a coupling constraint matrix to modify the tweeter voltage control signal.
The paper is organized as follows: The principle of the decoupling control algorithm based on the NOP is presented in Section 2. The validity of NOP decoupling algorithm is proved in theory by a numerical simulation model established in section 3 and compared with Zernike polynomials decoupling control algorithm. In Section 4, the feasibility of the proposed algorithm is verified by a real W-T AO system, and compared with Zernike polynomials decoupling control algorithm. Section 5 is the conclusion.
2. Principle
In the W-T AO system, the greatest challenge is to correctly assign various aberrations to the woofer and tweeter according to deformable mirror correction capabilities. Firstly, it is assumed that the light intensity in the pupil region is fully filled up under uniform distribution. Then the residual wavefront slope g(2n×1) detected by an n sub-apertures Hartmann wavefront sensor can be restored using an m-order Zernike polynomials as:
where Z(2n×m) is the wavefront restoration matrix corresponding to Zernike polynomials and a(m×1) denotes the coefficient vector for Zernike polynomials. a can be rewritten as: where Z+ represents the pseudo-inverse matrix of Z. In fact, the non-uniform distribution of light intensity causes a lack of light or saturation of Hartmann sub-apertures, which brings in irregular pupil regions. At this moment, orthogonality of Zernike polynomials loses its effectiveness in irregular pupil regions, which affect the result of wavefront restoration. Therefore, Z should be modified to achieve consistency with the irregular pupil region as:According to the modification process and Ref. [23], we derive the NOP applicable to irregular pupil regions by Znew:
Next, using the pseudo-inverse matrix F+ of F, the coefficients of each order corresponding to NOP are calculated noting from Eq. (2):
Later, the low order mode coefficients in the aberrations are selected, according to the diagonal matrix IW(m×m), they are allocated to the woofer as the low spatial frequency aberrations. For instance, if IW=diag(1,1,1,0,0,0,…), the woofer is responsible for correcting the first 3 orders modes rw of wavefront aberrations:
Then the wavefront aberrations gw to be corrected by woofer will be:
At this time, combining the slope response matrix RW(2n×nw) of woofer with nw actuators, we get the cumulative voltage vw of woofer as:
Where a and b are the control parameters of PI controller. Then, according to Eq. (10), gt corrected by the tweeter can be calculated as:
Thus, the cumulative voltage vt and the driving voltage Vt of the tweeter are:
At this moment, to restrict the cross-coupling between woofer and tweeter, the following coupling constraint matrix Rset(nt×nt) is constructed by Eq. (16):
3. Numeral simulation
3.1 Woofer-tweeter system simulation model
The NOP decoupling control algorithm proposed in this paper mainly focuses on the aberrations distribution and coupling suppression between woofer and tweeter in the AO system for generalized irregular pupil regions. To prove the validity of this algorithm, a woofer-tweeter system is established, using a 19-actuator deformable mirror (DM) as the woofer to correct the low order aberrations and a DM with 127-actuator as the tweeter to correct the high order aberrations. The configuration between the actuators of DM and the sub-apertures of Hartmann wavefront sensor is shown in Fig. 1.
Fig. 1. The configuration between the actuators of DM and the sub-apertures of Hartmann wavefront sensor.
3.2 Validation of the NOP decoupling control algorithm
Firstly, three shading ways are employed to verify the effectiveness of NOP decoupling algorithm: (1) no shading, (2) Semi-circular shading, (3) Irregular shading. And the initial aberrations under the three shading ways are shown in Fig. 2(a1), (a2), and (a3). Then, according to the three shading ways, the new wavefront restoration matrix F can be obtained by Eq. (4)–(7), whose column vectors are verified to be orthogonal in all shading ways, as shown in Fig. 3. Therefore, we can assign the aberration to the dual DMs and restore the wavefront aberration with F.
For the given initial aberrations, the first 3 order modes are isolated by IW=diag(1,1,1,0,0,0,…) and assigned to the woofer, and the residual aberrations are corrected by the tweeter. In three shading ways, results of the woofer correction are shown in Fig. 2(b1), (b2), and (b3). Results of the tweeter correction are shown in Fig. 2(c1), (c2), and (c3). And the root mean square (RMS) of wavefront aberrations are 0.013λ, 0.014λ, and 0.015λ, respectively, shown in Fig. 2(d1), (d2), and (d3).
In addition, we present the aberrations distribution and correction of woofer and tweeter under three different shading ways on the first 10 order modes. From Fig. 4(a), (b), and (c), the first 3 orders of aberrations are corrected by the woofer and have no coefficients from tweeter.
Fig. 4. Coupling suppression performance of the algorithm under different shading ways: (a)No shading, (b) Semi-circular shading, (c) Irregular shading, (d) Curves of RMS and c.
To visualize the decoupling performance, the coupling coefficient c is defined as Eq. (18):
According to Fig. 4, it can be seen that NOP decoupling control algorithm drives the woofer and tweeter to correct the aberration and suppress the coupling efficiently where the pupil region has been changed.
3.3 Comparison with Zernike polynomials decoupling control algorithm
We set up two simulations to compare the performance of Zernike polynomials decoupling control algorithm and NOP decoupling control algorithm: (1) Different modes are taken to correct the aberrations in the semi-circular shading way, (2) Closed-loop correction is accomplished in three different shading ways using the first 48 order modes. For both simulations, we use the woofer to correct the first 3 orders aberrations, while the tweeter takes charge of the rest.
3.3.1 Different orders of restoration
The initial aberration in the simulation is shown in Fig. 2(a2), and the wavefront correction is completed by using the first 5, first 10 and first 15 order modes, respectively. The correction results are shown in Fig. 5.
Fig. 5. Correction results with different restoring orders: (a) RMS of residual wavefront, (b) RMS of woofer correction, (c) RMS of tweeter correction, (d) Curves of c.
As shown in Fig. 5(a), algorithm and Zernike polynomials decoupling control algorithm. However, compared with Zernike polynomials decoupling control algorithm, the RMS of residual wavefront has better convergence features when using the NOP decoupling control algorithm during the closed-loop process, shown in Fig. 5(a). Moreover, from Fig. 5(b) and (c), Zernike polynomials decoupling control algorithm has different the RMS of woofer and tweeter correction when the restoration order is varying. Meanwhile, using the NOP decoupling control algorithm, the RMS of woofer and tweeter correction are consistent, shown in Fig. 5(b) and (c). This indicates that the aberration assigned to woofer and tweeter by NOP decoupling control algorithm is always the same when the restoration order is changed. In addition, it is shown in Fig. 5(d) that the coupling coefficients of the NOP decoupling control algorithm are consistent at 0 approximately, while those of the Zernike polynomials decoupling control algorithm are large and changeful.
3.3.2 Different shading ways
The wavefront correction is completed with three different shading ways, which are no shading, semi-circular shading, and irregular shading, shown in Fig. 2(a1), (a2) and (a3), respectively. The correction results are shown in Fig. 6. Presented on Fig. 6(a), the RMS of residual wavefront converge to 0.012λ and the woofer and tweeter have the same correction for both NOP decoupling control algorithm and Zernike polynomials decoupling control algorithm when the no shading way is used. However, in case of the two other shading ways, which are shown in Fig. 6(b) and (c), the RMS of residual wavefront and dual DMs correction converge to stabilization with fewer iterations during the closed-loop process when using NOP decoupling control algorithm compared with Zernike polynomials decoupling control algorithm. Furthermore, shown in Fig. 6(d), the coupling coefficients of the NOP decoupling control algorithm are stable at 0 approximately, while the values of the coupling coefficients of Zernike polynomials decoupling control algorithm are larger at different shading ways.
Fig. 6. Correction results with different shading ways: (a)No shading, (b) Semi-circular shading, (c) Irregular shading, (d) Curves of c.
In conclusion, NOP decoupling control algorithm in the W-T system can correct the aberration well in the irregular pupil region. Moreover, this algorithm has a more stable coupling suppression performance for different restoration orders and different irregular pupil regions than Zernike polynomials decoupling control algorithm. In other words, there is high robustness in NOP decoupling control algorithm.
4. Experiment
4.1 Experimental conditions
To further test the performance of NOP decoupling control algorithm, the W-T AO system experimental platform is built presented on Fig. 7. A light source with a wavelength of 650nm is employed here in Fig. 7(a), and an aberration plate is placed in front of the source. Two 140mm focal length lenses are placed between woofer and tweeter. And the configuration between the actuators of DM and the sub-apertures of Hartmann wavefront sensor is shown in Fig. 7(b). Figure 7(c) shows the structure of wavefront sensor, which contains Hartmann wavefront sensor, near field CCD camera and far field CCD camera. Moreover, the Hartmann wavefront sensor consists of a 12×12 lens array and an eight-bit CCD camera with a target range of 372×372 pixels, where each sub-aperture contains both x-axis and y-axis directions, and the sub-aperture size is 31×31. Both woofer and tweeter are 59-actuator deformable mirrors.
Fig. 7. A W-T AO system: (a) Experimental platform, (b) The configuration between 59-actuators DM and the sub-apertures of Hartmann wavefront sensor, (c) Structure of wavefront sensor.
4.2 Validation of the NOP decoupling control algorithm
To verify the effectiveness of the NOP decoupling algorithm, we employ three shading ways shown in Fig. 8, which are no shading, Semi-circular shading, and Irregular shading. The first 3 modes are isolated and assigned to the woofer, and the residual aberrations are corrected by the tweeter. In three shading ways, results of the woofer correction are shown in Fig. 9(a1), (a2), and (a3). Results of the tweeter correction are shown in Fig. 9(b1), (b2), and (b3). Moreover, the RMS of wavefront aberrations are 0.051λ, 0.063λ, and 0.072λ, respectively, shown in Fig. 9(c1), (c2), and (c3).
Moreover, Fig. 10(a), (b), and (c) shows the projection of the woofer and tweeter correction at the first 10 order modes for the three shading ways, and it can be seen that tweeter has almost no coefficients on the first 3 orders. From Fig. 10(d), the NOP decoupling control algorithm keeps c stable to 0 approximately, while the RMS of the residual wavefront converges to 0.051λ, 0.063λ, and 0.072λ, respectively. Figure 10 shows that NOP decoupling control algorithm restrains the coupling error effectively between woofer and tweeter for the changing pupil region.
Fig. 10. Coupling suppression performance of the algorithm under different shading ways: (a)No shading, (b) Semi-circular shading, (c) Irregular shading, (d) Curves of RMS and c.
4.3 Comparison with Zernike polynomials decoupling control algorithm
To compare the performance of Zernike polynomials decoupling control algorithm and NOP decoupling control algorithm, two comparison experiments are carried out. There are restoration matrices of different orders to complete the aberrations closed-loop correction in the first experiment where a semi-circular shading way is selected. In the second experiment, the first 48 order modes are taken, applying three shading ways. For both experiments, we use the woofer to correct the first 3 orders aberrations, while the tweeter deals with the rest.
4.3.1 Different orders of restoration
The wavefront correction is completed using the first 5, first 10 and first 15 order modes, respectively, under the semi-circular shading way shown in Fig. 8(b). And the correction results are shown in Fig. 11. Shown in Fig. 11(a), the convergence characteristics of the RMS of the residual wavefront using the NOP decoupling control algorithm is better than that using the Zernike polynomials decoupling control algorithm. Meanwhile, with the different restoring order comparisons, it is demonstrated in Fig. 11(b) and (c) that the convergence process of the dual DMs correction remains the same compared with the Zernike polynomials decoupling control algorithm. Accordingly, coupling coefficients of the NOP decoupling control algorithm are consistent and almost 0 in Fig. 11(d), while those for the closed-loop process are large by using the Zernike polynomials decoupling control algorithm.
Fig. 11. Correction results with different restoring orders: (a) RMS of the residual wavefront, (b) RMS of woofer correction, (c) RMS of tweeter correction, (d) Curves of c.
4.3.2 Different shading ways
The wavefront correction is completed with three shading ways in Fig. 8, which are no shading, semi-circular shading, and irregular shading, respectively. The correction results are shown in Fig. 12. As shown in Fig. 12(a), the RMS of residual wavefront are 0.051λ approximately and the correction of dual DMs remains stable for both the NOP decoupling control algorithm and the Zernike polynomials decoupling control algorithm when the no shading way is used. However, compared with the Zernike polynomials decoupling control algorithm, shown in Fig. 12(b) and (c), the RMS of residual wavefront and DMs correction converge to stabilization with fewer iterations during aberration correction process using the NOP decoupling control algorithm with the two other shading ways, while the coupling coefficients are 0 approximately, shown in Fig. 12(d). In contrast, the RMS of residual wavefront and DMs correction have not converged during iterations and the coupling coefficients are large as well as not converged by using Zernike polynomials decoupling control algorithm, shown in Fig. 12(b), (c), and (d).
Fig. 12. Correction results with different shading ways:(a) No shading, (b) Semi-circular shading, (c) Irregular shading, (d) Curves of c.
In conclusion, NOP decoupling control algorithm in the W-T system can complete closed-loop correction well in irregular pupil regions. Moreover, this algorithm has a stable coupling suppression performance for different restoration orders and different irregular pupil regions. Hence, the robustness of NOP decoupling control algorithm is much better than that of Zernike polynomials decoupling control algorithm. The conclusions of experiments are consistent with those of numerical simulations and confirm the validity of NOP decoupling control algorithm for the W-T AO system. And as illustrated in Ref. [23], the Zernike polynomials lead to problems such as mode confusion due to non-orthogonality in irregular pupil regions. That results in the coupling suppression of the Zernike polynomials decoupling control algorithm are weak and unstable in irregular pupil regions.
5. Conclusion
In conclusion, this paper proposes and discusses a decoupling control algorithm based on NOP for solving the cross-coupling of dual wavefront correctors in the generalized irregular pupil region. The numerical simulations and experiments show that this algorithm can drive woofer and tweeter to correct aberrations with different spatial frequency and restrain cross-coupling under different restoring orders and different shading ways. Meanwhile, NOP decoupling control algorithm demonstrates better performance than Zernike polynomials decoupling control algorithm. In this paper, we have discussed various scenarios under three different irregular pupil regions to verify the feasibility of NOP decoupling control algorithm. In fact, because of the orthogonality of NOP in generalized irregular pupil regions, we can draw a conclusion that the NOP decoupling control algorithm is applicable for the AO system with multiple wavefront correctors in any other generalized irregular pupil regions.
Funding
National Natural Science Foundation of China (61805251, 61875203, 62005285); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2017429); Western Youth Scholar A CAS.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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