## Linearization of the response of a 91-actuator magnetic liquid deformable mirror

Optics Express, Vol. 18, Issue 8, pp. 8239-8250 (2010)

http://dx.doi.org/10.1364/OE.18.008239

Acrobat PDF (391 KB)

### Abstract

We present the experimental performance of a 91-actuator deformable mirror made of a magnetic liquid (ferrofluid) using a new technique that linearizes the response of the mirror by superposing a uniform magnetic field to the one produced by the actuators. We demonstrate linear driving of the mirror using influence functions, measured with a Fizeau interferometer, by producing the first 36 Zernikes polynomials. Based on our measurements, we predict achievable mean PV wavefront amplitudes of up to 30 µm having RMS residuals of λ/10 at 632.8 nm. Linear combination of Zernikes and over-time repeatability are also demonstrated.

© 2010 OSA

## 1. Introduction

1. J. Liang, D. R. Williams, and D. T. Miller, “Supernormal Vision and High-Resolution Retinal Imaging Through Adaptive Optics,” J. Opt. Soc. Am. A **14**(11), 2884–2892 (1997). [CrossRef]

4. P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong, and A. Ritcey, “Deformable mirrors based on magnetic liquids,” Proc. SPIE **5490**, 1493–1501 (2004). [CrossRef]

5. E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva, and H. Yockell-Lelievre, “Nanoengineered Astronomical Optics,” Astron. Astrophys. **419**(2), 777–782 (2004). [CrossRef]

6. G. Vdovin, “Closed-loop adaptive optical system with a liquid mirror,” Opt. Lett. **34**(4), 524–526 (2009). [CrossRef] [PubMed]

7. D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express **15**(26), 18190–18199 (2007). [CrossRef] [PubMed]

8. D. Brousseau, E. F. Borra, S. Thibault, A. M. Ritcey, J. Parent, O. Seddiki, J.-P. Dery, L. Faucher, J. Vassallo, and A. Naderian, “Wavefront correction with a ferrofluid deformable mirror: experimental results and recent developments,” Proc. SPIE **7015**, 70153J (2008). [CrossRef]

7. D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express **15**(26), 18190–18199 (2007). [CrossRef] [PubMed]

9. A. Iqbal and F. B. Amara, “Modeling of a Magnetic-Fluid Deformable Mirror for Retinal Imaging Adaptive Optics Systems,” Int. J. Optomechatronics **1**(2), 180–208 (2007). [CrossRef]

10. A. Iqbal and F. B. Amara, “Modeling and experimental evaluation of a circular magnetic-fluid deformable mirror,” International Journal of Optomechatronics **2**(2), 126–143 (2008). [CrossRef]

9. A. Iqbal and F. B. Amara, “Modeling of a Magnetic-Fluid Deformable Mirror for Retinal Imaging Adaptive Optics Systems,” Int. J. Optomechatronics **1**(2), 180–208 (2007). [CrossRef]

10. A. Iqbal and F. B. Amara, “Modeling and experimental evaluation of a circular magnetic-fluid deformable mirror,” International Journal of Optomechatronics **2**(2), 126–143 (2008). [CrossRef]

11. J. Parent, E. F. Borra, D. Brousseau, A. M. Ritcey, J.-P. Déry, and S. Thibault, “Dynamic response of ferrofluidic deformable mirrors,” Appl. Opt. **48**(1), 1–6 (2009). [CrossRef]

## 2. Theory

### 2.1 Linearization of the response of FDMs

*h*produced on the ferrofluid is proportional to the square of the total magnetic field at the ferrofluid-air interface [7

7. D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express **15**(26), 18190–18199 (2007). [CrossRef] [PubMed]

9. A. Iqbal and F. B. Amara, “Modeling of a Magnetic-Fluid Deformable Mirror for Retinal Imaging Adaptive Optics Systems,” Int. J. Optomechatronics **1**(2), 180–208 (2007). [CrossRef]

10. A. Iqbal and F. B. Amara, “Modeling and experimental evaluation of a circular magnetic-fluid deformable mirror,” International Journal of Optomechatronics **2**(2), 126–143 (2008). [CrossRef]

*h*on the liquid surface caused by one actuator is then given by,where

*b*is the magnetic field of the actuator,

*B*

_{0}the external and uniform magnetic field with

*B*

_{0}>>

*b*, and

*k*is a constant that depends on the physical properties of the liquid. The magnetic field

*B*

_{0}being uniform and

*b*very small compared to

*B*

_{0}, the only term that contributes to the local deformation is the term 2

*bB*

_{0}. This has the effect of linearizing the response of the actuators (

*h*directly proportional to

*b*) and also amplifies the maximum amplitude the mirror can produce (

*h*directly proportional to

*B*

_{0}). This method was demonstrated experimentally by Iqbal and Amara [9

**1**(2), 180–208 (2007). [CrossRef]

**2**(2), 126–143 (2008). [CrossRef]

8. D. Brousseau, E. F. Borra, S. Thibault, A. M. Ritcey, J. Parent, O. Seddiki, J.-P. Dery, L. Faucher, J. Vassallo, and A. Naderian, “Wavefront correction with a ferrofluid deformable mirror: experimental results and recent developments,” Proc. SPIE **7015**, 70153J (2008). [CrossRef]

**2**(2), 126–143 (2008). [CrossRef]

^{th}order derivative with respect to the position near the center of the coil [12

12. R. S. Caprari, “Optimal current loop systems for producing uniform magnetic fields,” Meas. Sci. Technol. **6**(5), 593–597 (1995). [CrossRef]

**2**(2), 126–143 (2008). [CrossRef]

12. R. S. Caprari, “Optimal current loop systems for producing uniform magnetic fields,” Meas. Sci. Technol. **6**(5), 593–597 (1995). [CrossRef]

*N*is the number of turns of the middle coil,

*R*is the radius of the middle coil and

*I*is the current supplied to the Maxwell coil.

### 2.2 Control of the FDM

**w**

*produced by the deformable mirror is then given by a linear combination of the individual response functions of the actuators in the matrix form [13*

_{m}13. K. E. Moore and G. N. Lawrence, “Zonal model of an adaptive mirror,” Appl. Opt. **29**(31), 4622–4628 (1990). [CrossRef] [PubMed]

**H**is called the influence matrix and

**a**is a vector made of the control signals of the actuators. For the case of a FDM, the control signals are the currents supplied to the actuators. Each column of the control matrix

**H**represents the response function of a single actuator shifted to its corresponding location, while each row of

**H**corresponds to a single wavefront data sample. The matrix

**H**is usually rectangular and needs to have more rows (data samples) than columns (actuators). To obtain a given targeted wavefront

**w**, the solution for minimum variance of Eq. (3) gives the following vector of control signals to supply to the deformable mirror [14

14. J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. **32**, 2431–2438 (1993). [CrossRef] [PubMed]

**t**denotes the transpose matrix operation. Assuming that the mean wavefront error is zero, the total squared error between the targeted wavefront

**w**and the wavefront

**w**

*produced by the deformable mirror is:where*

_{m}**ε**is the wavefront error:

*N*and is given by

## 3. The 91-actuator deformable mirror

## 4. Results

### 4.1 Driving the FDM

15. M. M-Hernandez, M. Servin, D. M-Hernandez, and G. Paez, “Wavefront fitting using Gaussian functions,” Opt. Commun. **163**, 259–269 (1999). [CrossRef]

*x*and

*y*profiles of an actuator influence function recorded using the HASO wavefront sensor. Apart from slight differences in amplitudes, all actuators, including the ones at the pupil edge, produce identical influence function profiles. This is easily explained as there is any physical constraint on the mirror surface at the edges of the actuators region. As in [7

**15**(26), 18190–18199 (2007). [CrossRef] [PubMed]

**H**was constructed from the recorded influence functions of the 91 actuators. The required signals (currents) to produce the first 36 Zernike polynomials following the OSA numbering scheme [16] were computed from Eq. (4) using singular value decomposition. Each Zernike polynomial was targeted to a PV wavefront amplitude of 4 µm. The wavefront error

**ε**was calculated for each Zernike using Eq. (6). Currents to reproduce this residual wavefront error were then computed using Eq. (4). The Zernikes were then further optimized by subtracting this current vector from the currents of the first run using a gain of 0.6. A third iteration on the currents was found to be unnecessary since the residuals stopped to decrease. Figure 8 shows Zernikes Z

_{2}

^{0}to Z

_{4}

^{0}after the second current iteration. In principle, these iterations should only be done once for each term. Once the best current vector of each Zernike is found and unless the ferrofluid is replaced, it should be possible to keep these currents as a reference set to reproduce Zernikes of different amplitudes by scaling their appropriate current vector. Measurements of a scaled defocus appearing further down in this section confirm this (see Fig. 10 ). Section 4.3 also confirms this by showing the overtime repeatability of some Zernikes using the optimized current vectors.

_{4}

^{0}and Z

_{6}

^{0}) are in agreement with numerical simulations that use the addition of Gaussian influence functions over a circular pupil. Based on those residual errors, we extrapolated the maximum wavefront amplitude coefficient that would be achievable with the 91-actuator FDM while keeping constant the residual wavefront RMS error to λ/10 at 632.8 nm. The results are given by the bar chart in Fig. 9b. It can be seen that the achievable mean coefficient amplitude over all terms is over 15 µm with some terms even over 35 µm while still keeping a residual error of λ/10. Note that wavefronts having even higher amplitudes and comparable residuals would be possible by increasing the number of actuators and using a higher current in the Maxwell coil.

_{2}

^{0}as a function of a current scaling factor relative to the current vector that was found to produce it with at a PV amplitude of 4 µm. The results show that the linearity of the FDM still holds at large amplitudes. As another high amplitude example, Fig. 11 shows an astigmatism term having a PV amplitude of 20 µm (left) and its residual wavefront error at right. The residual wavefront has a RMS of 0.044 µm. Both Fig. 10 and 11 confirm the large wavefront amplitudes the FDM can achieve.

### 4.2 Combination of Zernikes

_{2}

^{−2}, Z

_{2}

^{0}, Z

_{2}

^{2}, Z

_{3}

^{−1}and Z

_{3}

^{1}having coefficient amplitudes of 0.4, 0.3, 0.3, 0.3 and 0.1 µm respectively. This theoretical wavefront can be seen at the left of Fig. 11.

### 4.3 Repeatability

_{2}

^{−2}, Z

_{2}

^{0}and Z

_{3}

^{−1}were scaled and applied to the actuators four times over the course of three days and the corresponding Zernike coefficient were extracted from the ZYGO measurements. Table 1 presents the results. It can be seen that once an optimized current vector that produces a given Zernike is found, this current vector can be scaled to produce Zernikes (or combination of Zernikes) of varying amplitude, as also demonstrated by Fig. 10, and that this calibration remains constant over time. From the data of Table 1, we computed that the over-time repeatability of the FDM at producing Zernikes having specific targeted coefficient amplitudes is better than 1.5%.

## 5 Conclusion

**15**(26), 18190–18199 (2007). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. Liang, D. R. Williams, and D. T. Miller, “Supernormal Vision and High-Resolution Retinal Imaging Through Adaptive Optics,” J. Opt. Soc. Am. A |

2. | R. El-Agmy, H. Bulte, A. H. Greenaway, and D. Reid, “Adaptive beam profile control using a simulated annealing algorithm,” Opt. Express |

3. | M. Ogasawara, and M. Sato, “The applications of a liquid crystal aberration compensator for the optical disc systems,” in |

4. | P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong, and A. Ritcey, “Deformable mirrors based on magnetic liquids,” Proc. SPIE |

5. | E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva, and H. Yockell-Lelievre, “Nanoengineered Astronomical Optics,” Astron. Astrophys. |

6. | G. Vdovin, “Closed-loop adaptive optical system with a liquid mirror,” Opt. Lett. |

7. | D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express |

8. | D. Brousseau, E. F. Borra, S. Thibault, A. M. Ritcey, J. Parent, O. Seddiki, J.-P. Dery, L. Faucher, J. Vassallo, and A. Naderian, “Wavefront correction with a ferrofluid deformable mirror: experimental results and recent developments,” Proc. SPIE |

9. | A. Iqbal and F. B. Amara, “Modeling of a Magnetic-Fluid Deformable Mirror for Retinal Imaging Adaptive Optics Systems,” Int. J. Optomechatronics |

10. | A. Iqbal and F. B. Amara, “Modeling and experimental evaluation of a circular magnetic-fluid deformable mirror,” International Journal of Optomechatronics |

11. | J. Parent, E. F. Borra, D. Brousseau, A. M. Ritcey, J.-P. Déry, and S. Thibault, “Dynamic response of ferrofluidic deformable mirrors,” Appl. Opt. |

12. | R. S. Caprari, “Optimal current loop systems for producing uniform magnetic fields,” Meas. Sci. Technol. |

13. | K. E. Moore and G. N. Lawrence, “Zonal model of an adaptive mirror,” Appl. Opt. |

14. | J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. |

15. | M. M-Hernandez, M. Servin, D. M-Hernandez, and G. Paez, “Wavefront fitting using Gaussian functions,” Opt. Commun. |

16. | L. Thibos, R. A. Applegate, J. T. Schweigerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” in |

17. | S. Thibault, 2006 Feb. 14 “Method and System for Characterizing Aspheric Surfaces of Optical Elements.” United States Patent US 6,999,182. |

**OCIS Codes**

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

(220.1000) Optical design and fabrication : Aberration compensation

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: January 13, 2010

Revised Manuscript: March 17, 2010

Manuscript Accepted: March 18, 2010

Published: April 5, 2010

**Citation**

Denis Brousseau, Ermanno F. Borra, Maxime Rochette, and Daniel Bouffard Landry, "Linearization of the response of a 91-actuator magnetic liquid deformable mirror," Opt. Express **18**, 8239-8250 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8239

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### References

- J. Liang, D. R. Williams, and D. T. Miller, “Supernormal Vision and High-Resolution Retinal Imaging Through Adaptive Optics,” J. Opt. Soc. Am. A 14(11), 2884–2892 (1997). [CrossRef]
- R. El-Agmy, H. Bulte, A. H. Greenaway, and D. Reid, “Adaptive beam profile control using a simulated annealing algorithm,” Opt. Express 13(16), 6085–6091 (2005). [CrossRef] [PubMed]
- M. Ogasawara, and M. Sato, “The applications of a liquid crystal aberration compensator for the optical disc systems,” in Adaptive Optics for Industry and Medicine, Ed. J C Dainty, Imperial College Press, London, 369–375 (2008).
- P. Laird, E. F. Borra, R. Bergamesco, J. Gingras, L. Truong, and A. Ritcey, “Deformable mirrors based on magnetic liquids,” Proc. SPIE 5490, 1493–1501 (2004). [CrossRef]
- E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva, and H. Yockell-Lelievre, “Nanoengineered Astronomical Optics,” Astron. Astrophys. 419(2), 777–782 (2004). [CrossRef]
- G. Vdovin, “Closed-loop adaptive optical system with a liquid mirror,” Opt. Lett. 34(4), 524–526 (2009). [CrossRef] [PubMed]
- D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express 15(26), 18190–18199 (2007). [CrossRef] [PubMed]
- D. Brousseau, E. F. Borra, S. Thibault, A. M. Ritcey, J. Parent, O. Seddiki, J.-P. Dery, L. Faucher, J. Vassallo, and A. Naderian, “Wavefront correction with a ferrofluid deformable mirror: experimental results and recent developments,” Proc. SPIE 7015, 70153J (2008). [CrossRef]
- A. Iqbal and F. B. Amara, “Modeling of a Magnetic-Fluid Deformable Mirror for Retinal Imaging Adaptive Optics Systems,” Int. J. Optomechatronics 1(2), 180–208 (2007). [CrossRef]
- A. Iqbal and F. B. Amara, “Modeling and experimental evaluation of a circular magnetic-fluid deformable mirror,” International Journal of Optomechatronics 2(2), 126–143 (2008). [CrossRef]
- J. Parent, E. F. Borra, D. Brousseau, A. M. Ritcey, J.-P. Déry, and S. Thibault, “Dynamic response of ferrofluidic deformable mirrors,” Appl. Opt. 48(1), 1–6 (2009). [CrossRef]
- R. S. Caprari, “Optimal current loop systems for producing uniform magnetic fields,” Meas. Sci. Technol. 6(5), 593–597 (1995). [CrossRef]
- K. E. Moore and G. N. Lawrence, “Zonal model of an adaptive mirror,” Appl. Opt. 29(31), 4622–4628 (1990). [CrossRef] [PubMed]
- J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. 32, 2431–2438 (1993). [CrossRef] [PubMed]
- M. M-Hernandez, M. Servin, D. M-Hernandez, and G. Paez, “Wavefront fitting using Gaussian functions,” Opt. Commun. 163, 259–269 (1999). [CrossRef]
- L. Thibos, R. A. Applegate, J. T. Schweigerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” in OSA Trends in Optics and Photonics35, 232–244 (2000).
- S. Thibault, 2006 Feb. 14 “Method and System for Characterizing Aspheric Surfaces of Optical Elements.” United States Patent US 6,999,182.

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