## Multichannel-Hadamard calibration of high-order adaptive optics systems |

Optics Express, Vol. 22, Issue 11, pp. 13792-13803 (2014)

http://dx.doi.org/10.1364/OE.22.013792

Acrobat PDF (919 KB)

### Abstract

we present a novel technique of calibrating the interaction matrix for high-order adaptive optics systems, called the multichannel-Hadamard method. In this method, the deformable mirror actuators are firstly divided into a series of channels according to their coupling relationship, and then the voltage-oriented Hadamard method is applied to these channels. Taking the 595-element adaptive optics system as an example, the procedure is described in detail. The optimal channel dividing is discussed and tested by numerical simulation. The proposed method is also compared with the voltage-oriented Hadamard only method and the multichannel only method by experiments. Results show that the multichannel-Hadamard method can produce significant improvement on interaction matrix measurement.

© 2014 Optical Society of America

## 1. Introduction

2. H. W. Babcock, “Adaptive optics revisited,” Science **249**(4966), 253–257 (1990). [CrossRef] [PubMed]

3. C. Boyer, V. Michau, and G. Rousset, “Adaptive optics: Interaction matrix measurements and real time control algorithms for the COME-ON project,” Proc. SPIE **1237**, 406–421 (1990). [CrossRef]

4. W. Jiang and H. Li, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” Proc. SPIE **1271**, 82–93 (1990). [CrossRef]

5. A. Bouchez, R. Dekany, J. Angione, C. Baranec, M. Britton, K. Bui, R. Burruss, J. Cromer, S. Guiwits, J. Henning, J. Hickey, D. Mckenna, A. Moore, J. Roberts, T. Trinh, M. Troy, T. Truong, and V. Velur, “The PALM-3000 high-order adaptive optics system for Palomar Observatory,” Proc. SPIE **7015**, 70150Z (2008). [CrossRef]

7. S. J. Thomas, L. Poyneer, R. De Rosa, B. Macintosh, D. Dillon, J. K. Wallace, D. Palmer, D. Gavel, B. Bauman, L. Saddlemyer, and S. Goodsell, “Integration and test of the Gemini Planet Imager,” Proc. SPIE **8149**, 814903 (2011). [CrossRef]

8. S. Oberti, F. Quirós-Pacheco, S. Esposito, R. Muradore, R. Arsenault, E. Fedrigo, M. Kasper, J. Kolb, E. Marchetti, A. Riccardi, C. Soenke, and S. Stroebele, “Large DM AO systems: synthetic IM or calibration on sky?” Proc. SPIE **6272**, 627220 (2006). [CrossRef]

9. M. Kasper, E. Fedrigo, D. P. Looze, H. Bonnet, L. Ivanescu, and S. Oberti, “Fast calibration of high-order adaptive optics systems,” J. Opt. Soc. Am. A **21**(6), 1004–1008 (2004). [CrossRef] [PubMed]

10. S. Oberti, H. Bonnet, E. Fedrigo, L. Ivanescu, M. Kasper, and J. Paufique, “Calibration of a curvature sensor/bimorph mirror AO system: interaction matrix measurement on MACAO systems,” Proc. SPIE **5490**, 139–150 (2004). [CrossRef]

11. A. Riccardi, R. Briguglio, E. Pinna, G. Agapito, F. Quiros-Pacheco, and S. Esposito, “Calibration strategy of the pyramid wavefront sensor module of ERIS with the VLT deformable secondary mirror,” Proc. SPIE **8447**, 84475M (2012). [CrossRef]

13. W. Zou and S. A. Burns, “High-accuracy wavefront control for retinal imaging with Adaptive-Influence-Matrix Adaptive Optics,” Opt. Express **17**(22), 20167–20177 (2009). [CrossRef] [PubMed]

14. F. Wildi and G. Brusa, “Determining the interaction matrix using starlight,” Proc. SPIE **5490**, 164–173 (2004). [CrossRef]

15. S. Esposito, R. Tubbs, A. Puglisi, S. Oberti, A. Tozzi, M. Xompero, and D. Zanotti, “High SNR measurement of interaction matrix on-sky and in lab,” Proc. SPIE **6272**, 62721C (2006). [CrossRef]

16. E. Pinna, F. Quiros-Pacheco, A. Riccardi, R. Briguglio, A. Puglisi, L. Busoni, C. Arcidiacono, J. Argomedo, M. Xompero, E. Marchetti, and S. Esposito, “First on-sky calibration of a high order adaptive optics system,” Proc. SPIE **8447**, 84472B (2012). [CrossRef]

## 2. Calibration error

8. S. Oberti, F. Quirós-Pacheco, S. Esposito, R. Muradore, R. Arsenault, E. Fedrigo, M. Kasper, J. Kolb, E. Marchetti, A. Riccardi, C. Soenke, and S. Stroebele, “Large DM AO systems: synthetic IM or calibration on sky?” Proc. SPIE **6272**, 627220 (2006). [CrossRef]

*D*

_{0}and the estimated IM is

*D*

_{m}, the calibration error here is defined as

*J*equals 1, while if the estimated IM equals the exact IM, i.e., the perfect calibration performance achieved,

*J*equals 0. Although this metric is different to the one used in [9

9. M. Kasper, E. Fedrigo, D. P. Looze, H. Bonnet, L. Ivanescu, and S. Oberti, “Fast calibration of high-order adaptive optics systems,” J. Opt. Soc. Am. A **21**(6), 1004–1008 (2004). [CrossRef] [PubMed]

## 3. Principle of the multichannel-Hadamard calibration method

### 3.1. Basic principle

*d*, where

_{act}*d*is the spacing of two adjacent actuators in the same line. This rule makes actuators in the same channel almost uncoupled with each other. The detailed description of the channel dividing result is also shown in Fig. 1. The channel dividing result can also be expressed by a channel dividing matrix (Fig. 2) which is defined asHere

_{act}*r*equaling 1.7

*d*is defined as the effective influence radius (EIR), and the circle with center located at the actuator’s position and radius equaling

_{act}*r*is defined to be the effective influence area (EIA) of the actuator. The EIA can be described by an effective influence matrix (EIM) defined as

*S*is 2

_{effect}*n*because both x-slope and y-slope responses are contained in the IM. The EIM of the 595-element AO system is shown in Fig. 3, which indicates that this matrix is a sparse matrix. It is certain that the 9.5% coupling relationship in our system is a bit low, but one can easily adjust the EIR with different actuator coupling as required. If the coupling factor of the AO system increases, the EIR should increase too.

_{sub}*v*is the amplitude of the driving voltage and

_{MO}9. M. Kasper, E. Fedrigo, D. P. Looze, H. Bonnet, L. Ivanescu, and S. Oberti, “Fast calibration of high-order adaptive optics systems,” J. Opt. Soc. Am. A **21**(6), 1004–1008 (2004). [CrossRef] [PubMed]

*v*is the amplitude of the driving voltage and

_{MH}*n*is channel number of the MH method.

_{MH}*n*. Depending on the value of

_{MH}*n*, several channels without any actuators may be necessary to make the channel number

_{MO}*n*satisfy the dimension requirement of a Hadamard matrix. In that case,where the dimension of the block matrix

_{MH}**0**is

*V*is the voltage matrix,

*N*is the measurement noise matrix and

*D*is the relationship matrix that maps the channels’ commands into the WFS measurements which can be estimated by

_{c}*D*into the IM mapping DM actuators’ commands into the WFS measurements, two steps have to be done. Firstly, the slope response of each channel is copied to the columns of the corresponding slope response of the actuators that belong to that channel bySecondly, as the slope response of a channel is composed of the slope response of each actuator in that channel, elements in the slope response that belong to other actuators in the same channel should be set to zeros. This process depends on the EIA of the actuators. As the centre of a subaperture can only be in the EIA of one actuator in the same channel, one can easily determine which subaperture slope response is caused by which actuator in that channel. This procedure can be expressed in mathematics asThus, the matrix

_{cm}*D*is the IM that we need which maps the DM actuators’ commands into the WFS measurements.

_{MH}### 3.2. Error analysis

*N*is the noise matrix of the MH method whose dimension is

_{MH}*N*is the noise matrix of the MO method whose dimension is

_{MO}**21**(6), 1004–1008 (2004). [CrossRef] [PubMed]

*N*is the noise matrix of the HO method whose dimension is

_{HO}*v*is the driving voltage amplitude and

_{HO}*n*. As the slope measurement noise of each subaperture can be considered as a Gaussian white noise, the noise matrix meetswhere

_{HO}*D*with small SNRs are set to zeros. Thus, with Eqs. (1), (11), (14), (16) and (17), the calibration error of the MH method can be approximated bywhere

_{MH}*n*is the total number of elements in

_{D}*S*and

_{effect}*n*is the number of the elements equaling 0. Similarly, it can be inferred that the calibration error of the MO methodand the calibration error of the HO methodWith Eqs. (18), (19) and (20)For a zonal IM, the factor (

_{0}*n*-

_{D}*n*)/

_{0}*n*is from the EIA definition which makes the elements with low SNRs set to zeros. Obviously, this factor increases with the EIR which is partly determined by the DM coupling. If the EIR is set large enough, the factor will become 1 and

_{D}*n*will equal

_{MH}*n*. In that case, the MH method is the same as the HO method. Therefore, the HO method is a special case of the MH method which especially fits the AO system with a large coupling DM. What’s more, the factors 1/

_{HO}*n*and 1/

_{MH}*n*are brought by the Hadamard matrix pattern of the driving voltage matrix because with a Hadamard matrix pattern, each actuator is driven up and down repeatedly to average the measurement noise. If the channel number

_{HO}*n*becomes larger, the calibration error will be smaller. However, the time used to finish the calibration will be longer like the principle of increasing the push-pull times to average the measurement noise [17

_{MH}17. X. Zhang, C. Arcidiacono, A. R. Conrad, T. M. Herbst, W. Gaessler, T. Bertram, R. Ragazzoni, L. Schreiber, E. Diolaiti, M. Kuerster, P. Bizenberger, D. Meschke, H. W. Rix, C. Rao, L. Mohr, F. Briegel, F. Kittmann, J. Berwein, and J. Trowitzsch, “Calibrating the interaction matrix for the LINC-NIRVANA high layer wavefront sensor,” Opt. Express **20**(7), 8078–8092 (2012). [PubMed]

### 3.3. Optimal EIR

*v*is the amplitude of the driving voltage,

*n*is the number of push-pulls added during the calibration,

*n*is that of the MH method

_{MH}*n*and

*n*equaled 64 here. The results (Fig. 4) show that the optimal EIRs increase almost linearly with

_{MH}*snr*. Besides, the larger the coupling is, the larger the optimal EIR will be if

*snr*is fixed. For the 595-element AO system we used (9.5% coupling), the optimal EIR is about 1.7~1.8

*d*when

_{act}*snr*increases from 5 to 15. That’s why the EIR used in our experiment is 1.7

*d*.

_{act}## 4. Experiment

*n*,

_{MO}*n*and

_{HO}*n*, the driving voltages of these methods should satisfyto keep these methods working in the same calibration condition. With Eqs. (21) and (23),In our experiment,

_{MH}*n*is 19,

_{MO}*n*is 714000 and

_{D}*n*is 699552 soIn addition, the real IM was also required to calculate the calibration error J defined in Eq. (1). Although it couldn’t be really measured, we could use an IM calibrated in a high SNR condition to replace. In our experiment, an IM calibrated with snr equal to 35 was used as the real IM. Sets of ten IMs were taken by using each of the three methods mentioned above with different SNRs. The corresponding RMs were then calculated. The parameter

_{0}^{−5}rad. The simulation was also done on the computer using the real IM and the slope noise collected on the 595-element AO system with the calibration condition the same as the experiment. The experimental and simulation results are shown in Fig. 6. Because of the relatively small coupling value of the DM in our experiment, a great number of elements approaching zeros exist in the IM, which resulted in the calibration error of the MO method smaller than the HO method. However, the MO method isn’t always better than the HO method, especially if a larger coupling DM is used which makes less elements of the IM close to zeros. With the advantages of both the MO and HO methods, the MH method had the smallest error.

*D*was measured with a high SNR about 35 by using the MH method, it was considered as the real IM in our experiment. Therefore,

_{0}*snr*is high, fitting error is the dominant factor that influences the focal plane intensity and the IM error becomes trivial. More importantly, although the SR acquired by using the MH method decreases with

*snr*decreasing, the one acquired by the MH method with

*snr*equivalent to 5 is still larger than those acquired by the MO and HO methods with

*snr*equal to 10. These results can match the corresponding phase error variances shown in Fig. 8, which also indicts the superiority of the MH method. It means that if the aberration is static, the MH method can make a least phase residual variance in the three methods with the same closed-loop iterations and if the aberration is dynamic, the MH method will track the change of the turbulence best, too. The PSDs of the residual phases with

*snr*10 are shown in Fig. 10. As it shows, the MH method performs very well not only in high spatial frequencies but also in low spatial frequencies. Considering the calibration efficiency requirement of high-order AO systems, the MH method is particular useful for high-order AO systems.

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. W. Hardy, |

2. | H. W. Babcock, “Adaptive optics revisited,” Science |

3. | C. Boyer, V. Michau, and G. Rousset, “Adaptive optics: Interaction matrix measurements and real time control algorithms for the COME-ON project,” Proc. SPIE |

4. | W. Jiang and H. Li, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” Proc. SPIE |

5. | A. Bouchez, R. Dekany, J. Angione, C. Baranec, M. Britton, K. Bui, R. Burruss, J. Cromer, S. Guiwits, J. Henning, J. Hickey, D. Mckenna, A. Moore, J. Roberts, T. Trinh, M. Troy, T. Truong, and V. Velur, “The PALM-3000 high-order adaptive optics system for Palomar Observatory,” Proc. SPIE |

6. | J.-F. Sauvage, T. Fusco, C. Petit, S. Meimon, E. Fedrigo, M. Suarez Valles, M. Kasper, N. Hubin, J.-L. Beuzit, J. Charton, A. Costille, P. Rabou, D. Mouillet, P. Baudoz, T. Buey, A. Sevin, F. Wildi, and K. Dohlen, “SAXO, the eXtreme Adaptive Optics System of SPHERE. Overview and calibration procedure,” Proc. SPIE |

7. | S. J. Thomas, L. Poyneer, R. De Rosa, B. Macintosh, D. Dillon, J. K. Wallace, D. Palmer, D. Gavel, B. Bauman, L. Saddlemyer, and S. Goodsell, “Integration and test of the Gemini Planet Imager,” Proc. SPIE |

8. | S. Oberti, F. Quirós-Pacheco, S. Esposito, R. Muradore, R. Arsenault, E. Fedrigo, M. Kasper, J. Kolb, E. Marchetti, A. Riccardi, C. Soenke, and S. Stroebele, “Large DM AO systems: synthetic IM or calibration on sky?” Proc. SPIE |

9. | M. Kasper, E. Fedrigo, D. P. Looze, H. Bonnet, L. Ivanescu, and S. Oberti, “Fast calibration of high-order adaptive optics systems,” J. Opt. Soc. Am. A |

10. | S. Oberti, H. Bonnet, E. Fedrigo, L. Ivanescu, M. Kasper, and J. Paufique, “Calibration of a curvature sensor/bimorph mirror AO system: interaction matrix measurement on MACAO systems,” Proc. SPIE |

11. | A. Riccardi, R. Briguglio, E. Pinna, G. Agapito, F. Quiros-Pacheco, and S. Esposito, “Calibration strategy of the pyramid wavefront sensor module of ERIS with the VLT deformable secondary mirror,” Proc. SPIE |

12. | S. Meimon, T. Fusco and C. Petit, “An optimized calibration strategy for high order adaptive optics systems: the Slope-Oriented Hadmard Actuation,” in |

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

14. | F. Wildi and G. Brusa, “Determining the interaction matrix using starlight,” Proc. SPIE |

15. | S. Esposito, R. Tubbs, A. Puglisi, S. Oberti, A. Tozzi, M. Xompero, and D. Zanotti, “High SNR measurement of interaction matrix on-sky and in lab,” Proc. SPIE |

16. | E. Pinna, F. Quiros-Pacheco, A. Riccardi, R. Briguglio, A. Puglisi, L. Busoni, C. Arcidiacono, J. Argomedo, M. Xompero, E. Marchetti, and S. Esposito, “First on-sky calibration of a high order adaptive optics system,” Proc. SPIE |

17. | X. Zhang, C. Arcidiacono, A. R. Conrad, T. M. Herbst, W. Gaessler, T. Bertram, R. Ragazzoni, L. Schreiber, E. Diolaiti, M. Kuerster, P. Bizenberger, D. Meschke, H. W. Rix, C. Rao, L. Mohr, F. Briegel, F. Kittmann, J. Berwein, and J. Trowitzsch, “Calibrating the interaction matrix for the LINC-NIRVANA high layer wavefront sensor,” Opt. Express |

**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: March 6, 2014

Revised Manuscript: May 19, 2014

Manuscript Accepted: May 21, 2014

Published: May 30, 2014

**Citation**

Youming Guo, Changhui Rao, Hua Bao, Ang Zhang, Xuejun Zhang, and Kai Wei, "Multichannel-Hadamard calibration of high-order adaptive optics systems," Opt. Express **22**, 13792-13803 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13792

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### References

- J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998), Chap. 1.
- H. W. Babcock, “Adaptive optics revisited,” Science 249(4966), 253–257 (1990). [CrossRef] [PubMed]
- C. Boyer, V. Michau, G. Rousset, “Adaptive optics: Interaction matrix measurements and real time control algorithms for the COME-ON project,” Proc. SPIE 1237, 406–421 (1990). [CrossRef]
- W. Jiang, H. Li, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” Proc. SPIE 1271, 82–93 (1990). [CrossRef]
- A. Bouchez, R. Dekany, J. Angione, C. Baranec, M. Britton, K. Bui, R. Burruss, J. Cromer, S. Guiwits, J. Henning, J. Hickey, D. Mckenna, A. Moore, J. Roberts, T. Trinh, M. Troy, T. Truong, V. Velur, “The PALM-3000 high-order adaptive optics system for Palomar Observatory,” Proc. SPIE 7015, 70150Z (2008). [CrossRef]
- J.-F. Sauvage, T. Fusco, C. Petit, S. Meimon, E. Fedrigo, M. Suarez Valles, M. Kasper, N. Hubin, J.-L. Beuzit, J. Charton, A. Costille, P. Rabou, D. Mouillet, P. Baudoz, T. Buey, A. Sevin, F. Wildi, K. Dohlen, “SAXO, the eXtreme Adaptive Optics System of SPHERE. Overview and calibration procedure,” Proc. SPIE 7736, 77360F (2010). [CrossRef]
- S. J. Thomas, L. Poyneer, R. De Rosa, B. Macintosh, D. Dillon, J. K. Wallace, D. Palmer, D. Gavel, B. Bauman, L. Saddlemyer, S. Goodsell, “Integration and test of the Gemini Planet Imager,” Proc. SPIE 8149, 814903 (2011). [CrossRef]
- S. Oberti, F. Quirós-Pacheco, S. Esposito, R. Muradore, R. Arsenault, E. Fedrigo, M. Kasper, J. Kolb, E. Marchetti, A. Riccardi, C. Soenke, S. Stroebele, “Large DM AO systems: synthetic IM or calibration on sky?” Proc. SPIE 6272, 627220 (2006). [CrossRef]
- M. Kasper, E. Fedrigo, D. P. Looze, H. Bonnet, L. Ivanescu, S. Oberti, “Fast calibration of high-order adaptive optics systems,” J. Opt. Soc. Am. A 21(6), 1004–1008 (2004). [CrossRef] [PubMed]
- S. Oberti, H. Bonnet, E. Fedrigo, L. Ivanescu, M. Kasper, J. Paufique, “Calibration of a curvature sensor/bimorph mirror AO system: interaction matrix measurement on MACAO systems,” Proc. SPIE 5490, 139–150 (2004). [CrossRef]
- A. Riccardi, R. Briguglio, E. Pinna, G. Agapito, F. Quiros-Pacheco, S. Esposito, “Calibration strategy of the pyramid wavefront sensor module of ERIS with the VLT deformable secondary mirror,” Proc. SPIE 8447, 84475M (2012). [CrossRef]
- S. Meimon, T. Fusco and C. Petit, “An optimized calibration strategy for high order adaptive optics systems: the Slope-Oriented Hadmard Actuation,” in 1st AO4ELT Conference (2010), paper 07009.
- W. Zou, S. A. Burns, “High-accuracy wavefront control for retinal imaging with Adaptive-Influence-Matrix Adaptive Optics,” Opt. Express 17(22), 20167–20177 (2009). [CrossRef] [PubMed]
- F. Wildi, G. Brusa, “Determining the interaction matrix using starlight,” Proc. SPIE 5490, 164–173 (2004). [CrossRef]
- S. Esposito, R. Tubbs, A. Puglisi, S. Oberti, A. Tozzi, M. Xompero, D. Zanotti, “High SNR measurement of interaction matrix on-sky and in lab,” Proc. SPIE 6272, 62721C (2006). [CrossRef]
- E. Pinna, F. Quiros-Pacheco, A. Riccardi, R. Briguglio, A. Puglisi, L. Busoni, C. Arcidiacono, J. Argomedo, M. Xompero, E. Marchetti, S. Esposito, “First on-sky calibration of a high order adaptive optics system,” Proc. SPIE 8447, 84472B (2012). [CrossRef]
- X. Zhang, C. Arcidiacono, A. R. Conrad, T. M. Herbst, W. Gaessler, T. Bertram, R. Ragazzoni, L. Schreiber, E. Diolaiti, M. Kuerster, P. Bizenberger, D. Meschke, H. W. Rix, C. Rao, L. Mohr, F. Briegel, F. Kittmann, J. Berwein, J. Trowitzsch, “Calibrating the interaction matrix for the LINC-NIRVANA high layer wavefront sensor,” Opt. Express 20(7), 8078–8092 (2012). [PubMed]

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