## A Zernike mode decomposition decoupling control algorithm for dual deformable mirrors adaptive optics system |

Optics Express, Vol. 21, Issue 20, pp. 23885-23895 (2013)

http://dx.doi.org/10.1364/OE.21.023885

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### Abstract

A simple but effective decoupling control algorithm based on Zernike mode decomposition for adaptive optics systems with dual deformable mirrors is proposed. One of the two deformable mirrors is characterized with a large stroke (woofer) and the other with high spatial resolutions (tweeter). The algorithm works as follows: wavefront gradient vector is decoupled using the Zernike modes at first, and then the control vector for the woofer is generated with low order Zernike coefficients to eliminate high order modes. At the same time the control vector for the tweeter is reset by a constraint matrix in order to avoid coupling error accumulation. Simulation indicates the algorithm could get better performance compared with traditional Zernike mode decomposition control algorithms. Experiments demonstrate that this algorithm can effectively compensate for phase distortions and significantly suppress the coupling between the woofer and tweeter.

© 2013 Optical Society of America

## 1. Introduction

2. D. C. Chen, S. M. Jones, D. A. Silva, and S. S. Olivier, “High-resolution adaptive optics scanning laser ophthalmoscope with dual deformable mirrors,” J. Opt. Soc. Am. A **24**(5), 1305–1312 (2007). [CrossRef] [PubMed]

5. R. Zawadzki, S. Choi, S. M. Jones, S. S. Oliver, and J. S. Werner, “Adaptive optics-optical coherence tomography: optimizing visualizarion of microscopic retinal structures in three dimensions,” J. Opt. Soc. Am. A **24**(5), 1373–1383 (2007). [CrossRef]

6. B. Cense, E. Koperda, J. M. Brown, O. P. Kocaoglu, W. Gao, R. S. Jonnal, and D. T. Miller, “Volumetric retinal imaging with ultrahigh-resolution spectral-domain optical coherence tomography and adaptive optics using two broadband light sources,” Opt. Express **17**(5), 4095–4111 (2009). [CrossRef] [PubMed]

1. S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt. **45**(12), 2638–2642 (2006). [CrossRef] [PubMed]

7. X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express **20**(20), 22143–22157 (2012). [CrossRef] [PubMed]

8. C. Li, N. Sredar, K. M. Ivers, H. Queener, and J. Porter, “A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system,” Opt. Express **18**(16), 16671–16684 (2010). [CrossRef] [PubMed]

9. J. F. Lavigne and J. P. Véran, “Woofer-tweeter control in an adaptive optics system using a Fourier reconstructor,” J. Opt. Soc. Am. A **25**(9), 2271–2279 (2008). [CrossRef] [PubMed]

10. P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer-tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A **27**(11), A145–A156 (2010). [CrossRef] [PubMed]

11. R. Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, “Distributed modal command for a two-deformable-mirror adaptive optics system,” Appl. Opt. **46**(20), 4329–4340 (2007). [CrossRef] [PubMed]

12. W. Zou, X. Qi, and S. A. Burns, “Wavefront-aberration sorting and correction for a dual-deformable-mirror adaptive-optics system,” Opt. Lett. **33**(22), 2602–2604 (2008). [CrossRef] [PubMed]

15. W. Zou and S. A. Burns, “Testing of Lagrange multiplier damped least-squares control algorithm for woofer-tweeter adaptive optics,” Appl. Opt. **51**(9), 1198–1208 (2012). [CrossRef] [PubMed]

1. S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt. **45**(12), 2638–2642 (2006). [CrossRef] [PubMed]

## 2. Principle of the Zernike decomposition decoupling control algorithm

*g*could be written as:where

*Z*is the slope response matrix from Zernike modes to the Shack-Hartmann wavefront sensor, and

*a*is the coefficient vector of the Zernike modes.

*a*could be calculated as:where

*Z*is the pseudo-inverse matrix of

^{+}*Z*. The woofer only corrects the lower order modes, and the remaining modes are assigned to the tweeter. Then the lower order mode coefficients

*a*could be written as:where

_{w}*I*is a p × p diagonal matrix, p is the number of Zernike modes used for reconstruction. If the woofer is assigned to correct the i

_{w}^{th}mode,

*I*is one, and the other elements of

_{w}(i,i)*I*are zeros. For example, if the woofer is used to correct defocus and astigmatism,

_{w}*I*= diag(0 0 1 1 1 0 … 0).

_{w}*A*could be generated using the well-know digital PI controller as:where

_{w}*pid_a*and

*pid_b*are the coefficients manually tuned for specific adaptive optics systems. Then we can use the transition matrix

*T*from mode coefficient vector to the control vector of woofer to get the control vector

*V*:and the transition matrix

_{w}*T*could be deducted as follow: here

*R*is the pseudo-inverse of the woofer’s response matrix

_{w}^{+}*R*.

_{w}*v*is the control vector of the woofer to correct the low order mode parts of the residual wavefront.

_{w}*I*is a m × m identity matrix, and m is the dimension of the slope vector

*g*, thus the control vector for correcting the higher order parts of the residual wavefront ishere

*R*is the pseudo-inverse of the tweeter’s response matrix

_{t}^{+}*R*. Then the control vector

_{t}*V*is generated as:

_{t}’*V*consists of two parts, one is

_{t}’*V*which would make tweeter correct high order modes, the other is

_{t}*∆V*which make tweeter correct low order modes inducing coupling.

*V*is generated by the high order mode slope, so

_{t}’*∆V*is often far smaller than

*V*. Then we have:where

_{t}*I*is a n × n identity matrix, and n is the number of the tweeter’s actuators. As described in Ref. 1

1. S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt. **45**(12), 2638–2642 (2006). [CrossRef] [PubMed]

*R*is the j

_{m}(i,j)^{th}order Zernike mode coefficient of the i

^{th}woofer’s influence function

*V*,

_{i}(x,y)*k*is an empirical parameter. 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.

*Z*is the Zernike modes assigned to be corrected by the woofer.

_{j}*V*could be calculated from

_{t}*V*using Eq. (17), and low order Zernike modes would be confined in the resultant surface shape.

_{t}’*V*and

_{w}*V*, decoupling control between woofer and tweeter is realized. Different from the traditional Zernike mode decomposition algorithms concentrating on decoupling the gradients of the residual wavefront,

_{t}*V*is constrained by the matrix

_{w}*T*which converts mode coefficient vectors to the control vectors of woofer, and

*V*is reset by the constraint matrix

_{t}*C*in each control iteration, so the coupling error is suppressed and the problem of coupling accumulation could be well solved.

## 3. Numerical simulation

**45**(12), 2638–2642 (2006). [CrossRef] [PubMed]

*(x*is the position of the i

_{i}, y_{i})^{th}actuator, α is the Gaussian coefficient, and

*d*is the distance between every two neighboring actuators. We set ω to 0.1 and α to 2.35 in the numerical model.

**45**(12), 2638–2642 (2006). [CrossRef] [PubMed]

**45**(12), 2638–2642 (2006). [CrossRef] [PubMed]

*S*is the correction of woofer and

_{w}*S*is the correction of tweeter. In mathematics, the coefficient

_{t}*r*is the direction cosine between

*S*and

_{w}*S*, when it is zero that means the

_{t}*S*and

_{w}*S*are orthogonal with each other, and when it equal to one that means the S

_{t}_{w}and S

_{t}are linearly dependent. So the smaller

*r*indicates the smaller coupling between phase compensation of woofer and tweeter.

## 4. Experiment

### 4.1 Woofer is used to correct defocus

*pid_a*is 0.998,

*pid_b*is 0.1, and

*k*is 10000. In the experiment, after corrected only by the woofer to compensate defocus, RMS of the residual wavefront decreased to 0.373λ, as is shown in Fig. 8(b). After corrected only by the tweeter to compensate the aberrations except defocus, RMS of the residual wavefront decreased to 0.568λ, as is shown in Fig. 8(c). At last, the dual deformable mirrors both joined the correction controlled by the proposed decoupling algorithm, and RMS of the residual wavefront is further lowered to 0.021λ, as is shown in Fig. 8(d). Figure 7(a) shows the RMS of the residual wavefront error reduction curves during the correction, and Fig. 7(b) illustrates the composition of the wavefront errors quantified by Zernike mode coefficients. From Fig. 7(b), it is easy to find that the aberrations are split exactly to defocus and high order parts. The woofer almost deals with defocus only, while the other parts of aberration are mainly corrected by the tweeter as expected.

### 4.2 Woofer is used to correct 0° astigmatism

*pid_a*is 0.998,

*pid_b*is 0.1, and

*k*is 10000. In the experiment, after corrected only by the woofer to compensate 0° astigmatism, RMS of the residual wavefront decreased to 0.618λ, as is shown in Fig. 10(b). After corrected only by the tweeter to compensate the aberrations except 0° astigmatism, RMS of the residual wavefront decreased to 0.305λ, as is shown in Fig. 10(c). At last, the dual deformable mirrors both joined the correction controlled by the proposed decoupling algorithm, and RMS of the residual wavefront is further lowered to 0.019λ, as is shown in Fig. 10(d). Figure 9(a) shows the RMS of the residual wavefront error reduction curves during the correction, and Fig. 9(b) illustrates the composition of the wavefront errors quantified by Zernike mode coefficients. From Fig. 9(b), it is easy to find that the aberrations are split exactly to 0° astigmatism and other parts. The woofer almost deals with 0° astigmatism only, while the other parts of aberration are mainly corrected by the tweeter as expected.

### 4.3 Woofer is used to correct defocus and astigmatisms

*pid_a*is 0.998,

*pid_b*is 0.1, and

*k*is 20000. In the experiment, after corrected only by the woofer to compensate defocus and astigmatisms, RMS of the residual wavefront decreased to 0.665λ, as is shown in Fig. 12(b). After corrected only by the tweeter to compensate the aberrations except defocus and astigmatisms, RMS of the residual wavefront decreased to 0.120λ, as is shown in Fig. 12(c). At last, the dual deformable mirrors both joined the correction controlled by the proposed decoupling algorithm, and RMS of the residual wavefront is further lowered to 0.031λ, as is shown in Fig. 12(d). Figure 11(a) shows the RMS of the residual wavefront error reduction curves during the correction, and Fig. 11(b) illustrates the composition of the wavefront errors quantified by Zernike mode coefficients. From Fig. 7(b), it is easy to find that the aberrations are split exactly to defocus, astigmatisms and other parts. The woofer almost deals with defocus and astigmatisms only, while the other parts of aberration are mainly corrected by the tweeter as expected.

### 4.4 Coupling test

## 5. Conclusion

## Acknowledgments

## References and links

1. | S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt. |

2. | D. C. Chen, S. M. Jones, D. A. Silva, and S. S. Olivier, “High-resolution adaptive optics scanning laser ophthalmoscope with dual deformable mirrors,” J. Opt. Soc. Am. A |

3. | H. Yang, G. Liu, C. Rao, Y. Zhang, and W. Jiang, “Combinational-deformable-mirror adaptive optics system for compensation of high-order modes of wavefront,” Chin. Opt. Lett. |

4. | W. Zou, X. Qi, and S. A. Burns, “Woofer-tweeter adaptive optics scanning laser ophthalmoscopic imaging based on Lagrange-multiplier damped least-squares algorithm,” Biomed. Opt. Express |

5. | R. Zawadzki, S. Choi, S. M. Jones, S. S. Oliver, and J. S. Werner, “Adaptive optics-optical coherence tomography: optimizing visualizarion of microscopic retinal structures in three dimensions,” J. Opt. Soc. Am. A |

6. | B. Cense, E. Koperda, J. M. Brown, O. P. Kocaoglu, W. Gao, R. S. Jonnal, and D. T. Miller, “Volumetric retinal imaging with ultrahigh-resolution spectral-domain optical coherence tomography and adaptive optics using two broadband light sources,” Opt. Express |

7. | X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express |

8. | C. Li, N. Sredar, K. M. Ivers, H. Queener, and J. Porter, “A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system,” Opt. Express |

9. | J. F. Lavigne and J. P. Véran, “Woofer-tweeter control in an adaptive optics system using a Fourier reconstructor,” J. Opt. Soc. Am. A |

10. | P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer-tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A |

11. | R. Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, “Distributed modal command for a two-deformable-mirror adaptive optics system,” Appl. Opt. |

12. | W. Zou, X. Qi, and S. A. Burns, “Wavefront-aberration sorting and correction for a dual-deformable-mirror adaptive-optics system,” Opt. Lett. |

13. | W. Zou and S. A. Burns, “High-accuracy wavefront control for retinal imaging with Adaptive-Influence-Matrix adaptive optics,” Opt. Express |

14. | Y. Ning, B. Chen, H. Yu, H. Zhou, H. Yang, C. Guan, C. Rao, and W. Jiang, “Decoupling algorithm of a double-layer bimorph deformable mirror: analysis and experimental test,” Appl. Opt. |

15. | W. Zou and S. A. Burns, “Testing of Lagrange multiplier damped least-squares control algorithm for woofer-tweeter adaptive optics,” Appl. Opt. |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(090.1000) Holography : Aberration compensation

(010.1285) Atmospheric and oceanic optics : Atmospheric correction

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: July 22, 2013

Revised Manuscript: August 30, 2013

Manuscript Accepted: September 12, 2013

Published: September 30, 2013

**Citation**

Wenjin Liu, Lizhi Dong, Ping Yang, Xiang Lei, Hu Yan, and Bing Xu, "A Zernike mode decomposition decoupling control algorithm for dual deformable mirrors adaptive optics system," Opt. Express **21**, 23885-23895 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23885

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### References

- S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt.45(12), 2638–2642 (2006). [CrossRef] [PubMed]
- D. C. Chen, S. M. Jones, D. A. Silva, and S. S. Olivier, “High-resolution adaptive optics scanning laser ophthalmoscope with dual deformable mirrors,” J. Opt. Soc. Am. A24(5), 1305–1312 (2007). [CrossRef] [PubMed]
- H. Yang, G. Liu, C. Rao, Y. Zhang, and W. Jiang, “Combinational-deformable-mirror adaptive optics system for compensation of high-order modes of wavefront,” Chin. Opt. Lett.5, 435–437 (2007).
- W. Zou, X. Qi, and S. A. Burns, “Woofer-tweeter adaptive optics scanning laser ophthalmoscopic imaging based on Lagrange-multiplier damped least-squares algorithm,” Biomed. Opt. Express2(7), 1986–2004 (2011). [CrossRef] [PubMed]
- R. Zawadzki, S. Choi, S. M. Jones, S. S. Oliver, and J. S. Werner, “Adaptive optics-optical coherence tomography: optimizing visualizarion of microscopic retinal structures in three dimensions,” J. Opt. Soc. Am. A24(5), 1373–1383 (2007). [CrossRef]
- B. Cense, E. Koperda, J. M. Brown, O. P. Kocaoglu, W. Gao, R. S. Jonnal, and D. T. Miller, “Volumetric retinal imaging with ultrahigh-resolution spectral-domain optical coherence tomography and adaptive optics using two broadband light sources,” Opt. Express17(5), 4095–4111 (2009). [CrossRef] [PubMed]
- X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express20(20), 22143–22157 (2012). [CrossRef] [PubMed]
- C. Li, N. Sredar, K. M. Ivers, H. Queener, and J. Porter, “A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system,” Opt. Express18(16), 16671–16684 (2010). [CrossRef] [PubMed]
- J. F. Lavigne and J. P. Véran, “Woofer-tweeter control in an adaptive optics system using a Fourier reconstructor,” J. Opt. Soc. Am. A25(9), 2271–2279 (2008). [CrossRef] [PubMed]
- P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer-tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A27(11), A145–A156 (2010). [CrossRef] [PubMed]
- R. Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, “Distributed modal command for a two-deformable-mirror adaptive optics system,” Appl. Opt.46(20), 4329–4340 (2007). [CrossRef] [PubMed]
- W. Zou, X. Qi, and S. A. Burns, “Wavefront-aberration sorting and correction for a dual-deformable-mirror adaptive-optics system,” Opt. Lett.33(22), 2602–2604 (2008). [CrossRef] [PubMed]
- W. Zou and S. A. Burns, “High-accuracy wavefront control for retinal imaging with Adaptive-Influence-Matrix adaptive optics,” Opt. Express17(22), 20167–20177 (2009). [CrossRef] [PubMed]
- Y. Ning, B. Chen, H. Yu, H. Zhou, H. Yang, C. Guan, C. Rao, and W. Jiang, “Decoupling algorithm of a double-layer bimorph deformable mirror: analysis and experimental test,” Appl. Opt.48(17), 3154–3159 (2009). [CrossRef] [PubMed]
- W. Zou and S. A. Burns, “Testing of Lagrange multiplier damped least-squares control algorithm for woofer-tweeter adaptive optics,” Appl. Opt.51(9), 1198–1208 (2012). [CrossRef] [PubMed]

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