## Fast iterative algorithm (FIA) for controlling MEMS deformable mirrors: principle and laboratory demonstration |

Optics Express, Vol. 19, Issue 22, pp. 21271-21294 (2011)

http://dx.doi.org/10.1364/OE.19.021271

Acrobat PDF (1619 KB)

### Abstract

We present a fast and high accuracy iterative algorithm to control Micro-Electro-Mechanical-System (MEMS) deformable mirrors (DMs) for open-loop (OL) adaptive optics (AO) applications. Our approach relies on a simple physical model for the forces applied on DM actuators and membrane, defined by a small number of parameters that we measure in an experimental setup. The algorithm iteratively applies forces and updates actuator displacements, allowing real-time utilization in an Extreme-AO system (control rate ⩾ Khz). Our measurements show that it reproduces Kolmogorov type phase screens with an error equal to **7.3**% of the rms of the desired phase (**1.6**% of the peak-to-valley of the desired phase). This performance corresponds to an improvement of a factor three compared to the standard quadratic model (common relation between voltage and actuator displacement). Originally developed for the DM control of the Subaru Coronagraphic Extreme-AO (SCExAO) project, the algorithm is also suitable for Multi-Object AO systems.

© 2011 OSA

## 1. Introduction

### 1.1. Motivation for developing a MEMS DM model

#### 1.1.1. High-contrast imaging: SCExAO

2. B. Macintosh, J. R. Graham, D. W. Palmer, R. Doyon, J. Dunn, D. T. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, A. Sivaramakrishnan, J. K. Wallace, B. Bauman, D. A. Erickson, C. Marois, L. A. Poyneer, and R. Soummer, “The Gemini Planet Imager: from science to design to construction,” Proc. SPIE Adaptive Optics Systems7015, 701518 (2008).

3. O. Guyon, E. A. Pluzhnik, F. Martinache, J. Totems, S. Tanaka, T. Matsuo, C. Blain, and R. Belikov, “High contrast imaging and wavefront control with a PIAA coronagraph: laboratory system validation,” Publ. Astron. Soc. Pac. **122**, 71–84 (2010). [CrossRef]

4. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. **404**, 379–387 (2003). [CrossRef]

5. O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet Iimaging with a phase-induced amplitude apodization aoronagraph. I. principle,” Astrophys. J. **622**, 744–758 (2005). [CrossRef]

*λ*/D angular resolution. SCExAO stands on the Infra Red (IR) Nasmyth platform of Subaru Telescope, between AO188 [6] and HiCIAO [7]. The AO188 output beam contains approximately 200nm of residual wavefront error. To improve the AO correction provided by AO188 and perform additional coronagraphic suppression of starlight, SCExAO utilizes a 1020-actuator MEMS DM.

8. O. Guyon, F. Martinache, R. Belikov, and R. Soummer, “High performance PIAA coronagraphy with complex amplitude focal plane masks,” Astrophys. J. Suppl. **190**(2), 220–232 (2010). [CrossRef]

**sensing**and

**correction**. In this scheme, the DM is used to introduce known aberrations (speckles in the focal plane) which interfere with existing speckles. By monitoring the interference between the pre-existing speckles and the speckles added on purpose by the DM, it is possible to reconstruct the complex amplitude (amplitude and phase) of the focal plane speckles. Thus, the DM is used for wavefront sensing, in a scheme akin to phase diversity. A detailed description of this process can be found in [3

3. O. Guyon, E. A. Pluzhnik, F. Martinache, J. Totems, S. Tanaka, T. Matsuo, C. Blain, and R. Belikov, “High contrast imaging and wavefront control with a PIAA coronagraph: laboratory system validation,” Publ. Astron. Soc. Pac. **122**, 71–84 (2010). [CrossRef]

10. A. Give’on, S. Shaklan, and B. Kern, “Electric field conjugation-based wavefront correction algorithm for high contrast imaging systems - experimental results,” Proceedings of the conference In the Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar Disks in the 21st Century (2007).

#### 1.1.2. Classical AO and multi-object AO

11. R. Conan, C. Bradley, O. Lardière, C. Blain, K. Venn, D. Andersen, L. Simard, J.-P. Véran, G. Herriot, D. Loop, T. Usuda, S. Oya, Y. Hayano, H. Terada, and M. Akiyama, “Raven: a harbinger of multi-object adaptive optics-based instruments at the Subaru telescope,” Proc. SPIE Adaptive Optics Systems II7736, 77360T (2010).

12. S. Eikenberry, D. Andersen, R. Guzman, J. Bally, S. Cuevas, M. Fletcher, R. Gardhouse, D. Gavel, A. Gonzalez, N. Gruel, F. Hamann, S. Hamner, R. Julian, J. Julian, D. Koo, E. Lada, B. Leckie, J. A. Lopez, R. Pello, J. Perez, W. Rambold, C. Roman, A. Sarajedini, J. Tan, K. Venn, J.-P. Véran, and J. Ziegert, “IRMOS: the near-infrared multi-object spectrograph for the TMT,” Proc. SPIE Ground-based and Airborne Instrumentation for Astronomy6269, 62695W (2006).

### 1.2. Overview of MEMS deformable mirror technology

14. J. A. Perreault, T. Bifano, B. M. Levine, and M. Harenstein, “Adaptive optic correction using microelectromechnical deformable mirrors,” Opt. Eng. **41**(3), 561–566 (2002). [CrossRef]

*μ*m to 8

*μ*m. Segmented and continuous facesheet versions are both available from Boston Micromachines Corporation (BMC) [15] and IRIS AO [16]. These mirrors have many advantages, for example: sub-nanometre repeatability, stability and hysteresis, not to mention the low weight/size and finally the low cost per actuator (the cost is approximately ten times less per actuator than other more conventional DM technologies) [13].

### 1.3. Deformable mirror modeling—previous works

*npx*is the number of pixels for each image,

*measured*is the value (at pixel i) of the phase screen measured by the system (interferometer or wavefront sensor...) and

_{i}*model*is the value (at pixel i) of the phase screen estimated by the model.

_{i}17. C. Hom, P. Dean, and S. Winzer, “Simulating electrostrictive DM: I. non-linear static analysis,” Smart Mater. Struct. **18**, 691–699 (1999). [CrossRef]

18. C. R. Vogel and Q. Yang, “Modeling, simulation, and open-loop control of a continuous facesheet MEMS deformable mirror,” J. Opt. Soc. Am. A **23**(5), 1074–1081 (2006). [CrossRef]

19. C. R. Vogel, G. Tyler, Y. Lu, T. Bifano, R. Conan, and C. Blain, “Modeling and parameter estimation for point-actuated continuous-facesheet deformable mirrors,” *J. Opt. Soc. Am. A*27(11), A56–A63 (2010). [CrossRef]

21. J. B. Stewart, A. Diouf, Y. Zhou, and T. Bifano, “Open-Loop control of MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. **24**(12), 3827–3833 (2007). [CrossRef]

21. J. B. Stewart, A. Diouf, Y. Zhou, and T. Bifano, “Open-Loop control of MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. **24**(12), 3827–3833 (2007). [CrossRef]

19. C. R. Vogel, G. Tyler, Y. Lu, T. Bifano, R. Conan, and C. Blain, “Modeling and parameter estimation for point-actuated continuous-facesheet deformable mirrors,” *J. Opt. Soc. Am. A*27(11), A56–A63 (2010). [CrossRef]

21. J. B. Stewart, A. Diouf, Y. Zhou, and T. Bifano, “Open-Loop control of MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. **24**(12), 3827–3833 (2007). [CrossRef]

23. E. Laag, D. Gavel, and M. Ammons, “Open-loop woofer-tweeter control on the LAO multi-conjugate adaptive optics testbed,” in *Adaptive Optics for industry and medecine*, C. Dainty (Imperial College Press) 143–148 (2008). [CrossRef]

24. D. Guzmán, F. J. Juez, F. S. Lasheras, R. Myers, and L. Young, “Deformable mirror model for open-loop adaptive optics using multivariate adaptive regression splines,” Opt. Express **18**(7), 6492–6505 (2010). [CrossRef] [PubMed]

25. C. Blain, R. Conan, C. Bradley, and O. Guyon, “Open-loop control demonstration of micro-electro-mechanical system MEMS deformable mirror,” Opt. Express **18**(6), 5433–5448 (2010). [CrossRef] [PubMed]

26. D. Guzmán, F. J. Juez, R. Myers, A. Guesalaga, and F. S. Lasheras, “Modeling a MEMS deformabe mirror using non-parametric estimation techniques,” Opt. Express **18**(20), 21356–21369 (2010). [CrossRef] [PubMed]

26. D. Guzmán, F. J. Juez, R. Myers, A. Guesalaga, and F. S. Lasheras, “Modeling a MEMS deformabe mirror using non-parametric estimation techniques,” Opt. Express **18**(20), 21356–21369 (2010). [CrossRef] [PubMed]

24. D. Guzmán, F. J. Juez, F. S. Lasheras, R. Myers, and L. Young, “Deformable mirror model for open-loop adaptive optics using multivariate adaptive regression splines,” Opt. Express **18**(7), 6492–6505 (2010). [CrossRef] [PubMed]

26. D. Guzmán, F. J. Juez, R. Myers, A. Guesalaga, and F. S. Lasheras, “Modeling a MEMS deformabe mirror using non-parametric estimation techniques,” Opt. Express **18**(20), 21356–21369 (2010). [CrossRef] [PubMed]

## 2. Modeling MEMS DMs with the fast iterative algorithm

### 2.1. General description of the method

18. C. R. Vogel and Q. Yang, “Modeling, simulation, and open-loop control of a continuous facesheet MEMS deformable mirror,” J. Opt. Soc. Am. A **23**(5), 1074–1081 (2006). [CrossRef]

*iterative process*. The displacement of a specific actuator was adjusted in an iterative fashion until all forces applied to the actuator converged to the state of equilibrium.

### 2.2. Definition of the model’s forces

#### 2.2.1. Electrostatic force (*F*_{elec})

_{elec}

*ɛ*

_{0}the permittivity of free space,

*ɛ*the medium dielectric constant,

_{r}*L*and

*w*the actuator plate length and width and

*g*the gap between the two plates.

*V*and the potential energy can be written as thus

*k*was estimated to be

_{e}*k*= 5.1153.10

_{e}^{−19}F.m.

*g*

_{0}, was specified by the manufacturer as 5 microns [29]. However, during the model iterations, the value of

*g*varied with the estimation of the actuator vertical displacement,

*dp*, as

#### 2.2.2. Actuator plate restoring force (*F*_{restoring})

_{restoring}

*F*. MEMS actuators can only be pulled in one direction (toward the fixed actuator base), so

_{restoring}*F*will always be in opposition to

_{restoring}*F*.

_{elec}#### 2.2.3. Inter-actuator mechanical coupling force (*F*_{mec})

_{mec}

*F*and

_{elec}*F*balanced and defined the extent of the vertical displacement for this actuator. As illustrated in Fig. 2, a rigid post connected the actuator plate to the membrane. The membrane acted as a strong connection mechanism between neighbouring actuators. Thus, in addition to

_{restoring}*F*and

_{elec}*F*, the vertical displacement of the adjacent actuators needed to be estimated to accurately predict the vertical displacement of a given actuator in the array.

_{restoring}*F*, can reach up to 30% for some DM technologies but is usually of only few percents for MEMS DMs. The following hypothesis was made: the effect of one actuator is mainly localized to its first eight direct neighbors. The results presented in [24

_{mec}24. D. Guzmán, F. J. Juez, F. S. Lasheras, R. Myers, and L. Young, “Deformable mirror model for open-loop adaptive optics using multivariate adaptive regression splines,” Opt. Express **18**(7), 6492–6505 (2010). [CrossRef] [PubMed]

*F*can either be opposite to

_{mec}*F*or in the direction of

_{elec}*F*.

_{elec}*dp*(

*i, j*) represented the vertical displacement of the actuator of index (i,j) while

*dp*(

*perp*) corresponded to the vertical displacements of its four direct

_{i}*perpendicular*neighbours and

*dp*(

*diag*) corresponded to the vertical displacements of its four direct

_{i}*diagonal*neighbours. For clarity purpose,

*dp*(

*i, j*) –

*dp*(

*perp*) and

_{i}*dp*(

*i, j*) –

*dp*(

*diag*) will be named Δ

_{i}*dp*(for perpendicular) and Δ

_{p}*dp*(for diagonal) in the following.

_{d}*k*was the spring constant.

_{m}*k*was another model coefficients dedicated to take into account the difference in the

_{l}**lateral**distance between diagonal/perpendicular neighbours actuators to the “central” actuator (see Sec. 2.3).

#### 2.2.4. Equilibrium

#### 2.2.5. Model input-output

*displacements matrix*. The displacement matrix was then transformed into a phase map of size defined by the user (for example, 256 by 256 pixels or 1024 by 1024 pixels).

### 2.3. Description of the model coefficients: DM coefficients and geometrical coefficients

#### 2.3.1. DM coefficients

*k*, was introduced in Eq. (6). Three coefficients were part of Eq. (8). The restoring force was modelled using the spring equation. It will be shown in Sec. 3.2 that

_{e}*k*, the spring constant, varied with the displacement of the actuator plate in a quadratic fashion as, where

_{r}*dp*represented the vertical displacement of the actuator under evaluation. As a result,

*k*,

_{ra}*k*,

_{rb}*k*, were three additional DM coefficients, noted in descending powers of the second order polynomial

_{rc}*k*, to be optimized.

_{r}*F*, also modelled using a spring. It will be shown in Sec. 3.2 that

_{mec}*k*, the spring constant, varied with the degree of stretching of the membrane.

_{m}*k*was estimated by examining the relative difference in displacement between an actuator and its eight direct neighbours, or Δ

_{m}*dp. k*also followed a quadratic law, Here we encompass both Δ

_{m}*dp*and Δ

_{p}*dp*inside Δ

_{d}*dp*.

*k*was used to take into account the geometrical difference in the

_{l}**lateral**distance between the perpendicular neighbour actuators and the diagonal neighbour actuators to the “central” actuator.

#### 2.3.2. Geometrical coefficients

### 2.4. Description of the iterative algorithm used for DM shape computation

- Step (i): For each actuator of the array, the electrostatic force
*F*was computed using_{elec}**Eq. (5)**. - Step (ii): A first estimation of the displacement
*dp*of each actuator was computed using**Eq. (11)**. During the first iteration only,*F*was set equal to zero and the DM was modelled as if it had a segmented membrane. For all following iterations, the value of_{mec}*F*will be the outcome of Step (iii) from the_{mec}**previous**iteration.

*F*was set as the new

_{mec}*F*input value for the next iteration and the algorithm started back to Step (i).

_{mec}#### 2.4.1. Initial conditions for the iterative algorithm

- – The initial gap between the actuator top plate and the fixed actuator base plate,
*g*_{0}, was equal to 5*μ*m [29] - – All forces were equal to zeros (
*F*=_{elec}*F*=_{restoring}*F*= 0)_{mec} - – The displacement of each actuator was equal to zero (dp = 0)
- – Each model coefficient was set to its optimized value determined during the initial calibration.

#### 2.4.2. Constraining the maximum actuator displacements

## 3. Preliminary calibration of the model coefficients

### 3.1. Description of the Markov Chain Monte Carlo algorithm

**Set I**and

**Set II**), and the model’s performance was evaluated for each set. The difference in the rms value, between model and measured, was compared with Set I and Set II. The set which provided the smallest difference corresponded to the best match between the model coefficients and the real DM.

- The MCMC must start with a set of user-defined initial coefficients, corresponding to Set I. The values of these initial coefficients were defined experimentally.
- Using Set I, the residual rms error, Φ
_{1}, between the modelled phase map and the measured phase map was estimated. - A second set of parameters was randomly picked, and the residual rms error with Set II, Φ
_{2}, was estimated. To generate a new value for each coefficient of Set II, a random number (positive or negative) was added to each coefficient of Set I (each coefficient had a specific number added or subtracted). The range of values accessible to each coefficient was user-defined at the beginning of the algorithm and was based on results found during the initial experimental evaluation of the model coefficients. This range can be adjusted during the optimization. - If R ⩾ 1, Set II gave the best fit between the DM and the model. The algorithm moved toward the “region” of lower residual rms. Set II was saved as the new Set I for the next iteration of the optimisation.
- If R < 1, Set II was not better than Set I. A random number
*U*taken between 0 and 1 was picked (using a uniform distribution). – If U ⩽ R, Set II was still selected as the starting set for the next MCMC iteration, which gave a chance to explore some regions where the residual rms was larger. – If U > R, the MCMC kept Set I as starting set for the next MCMC iteration. - The coefficients of the best set were saved as the new Set I for the next iteration. The loop started back to (ii).

### 3.2. Experimental estimation of the model coefficients

*dp*and

_{max}*dp*.

_{edge}- – To accurately define a zero reference point for the interferometer measurements. Because the piston was not visible, this zero reference point was necessary to improve the accuracy in the estimation of the displacement for each actuator of the active array.
- – To mitigate the effect of the defective actuator (located at row 6, column 22). This actuator followed the motion of the array up to approximately 90 V, then stayed below the other actuators when the voltage was pulled to higher values. The model performance were thus estimated for the active area which contain only valid actuators.

#### 3.2.1. Measurement of the DM maximum displacement (*dp*_{max})

_{max}

*dp*, should be reached when all actuators are pulled to the maximum voltage of the device, 200 V. However, because a reference point was needed to accurately estimate the displacement, only the “active array” was set to 200 V. This will not impact the performance of the model because all voltage maps will have the identical reference area and active array. Figure 5(a), shows the phase measurement of the 32 by 32 array with the active array set to 200 V and (b), shows the vertical cut along the center of the membrane. The maximum displacement was estimated to

_{max}*dp*= 1.6

_{max}*μ*m.

#### 3.2.2. Measurement of the maximum displacement for the edge actuators (*dp*_{edge})

_{edge}

*dp*= 1.3

_{edge}*μ*m.

*dp*was not critical because the seven outer coronas of actuators are constantly maintained at zero volt. To apply this model to a full 32 by 32 active array, an accurate estimation of

_{edge}*dp*would be an important constraint.

_{edge}#### 3.2.3. Measurement of the actuator influence function

#### 3.2.4. Measurement of the DM coefficient *k*_{r}

_{r}

*k*, introduced in the restoring force equation (Eq. (8)), was estimated experimentally by pushing successively the active array to voltages varying from 20 V to 120 V by steps of 20 V while the rest of the actuators (reference area) were maintained to zero. For the actuators located at the center of the active array, the mechanical coupling can be approximated to zero, because all actuators were at the same voltage, thus at the approximative same vertical displacement. For the central actuator, the following assumptions were made:

_{r}*k*with respect to dp is plotted in Fig. 8(b).

_{r}*k*are presented in Table 3.

_{r}#### 3.2.5. Measurement of the DM coefficient *k*_{m}

_{m}

*k*was known for a given

_{r}*dp*, the same process was applied to the membrane spring constant,

*k*. The central active array of actuators was successively pulled from 40 V to 120 V by steps of 20 V while actuator [16,16] was maintained at 40 V.

_{m}*actu16*, was always subjected to the same voltage. Without mechanical coupling, the vertical position of this actuator should not vary when the rest of the active array moves from 40 V to 120 V. The observed increase in the vertical displacement of

*actu16*was due to the action of the height neighbour actuators which were pulling

*actu16*upward through mechanical coupling of the membrane. Figure 9 shows a vertical cut of the array passing on top of

*actu16*for the five different voltages (40 V to 120 V) and the vertical displacement dp for

*actu16*. Δ

*dp*was estimated for each voltage by measuring the difference between the vertical displacement of

*actu16*and the vertical displacement of its eight neighbours. Using the following assumptions:

- –
*k*was estimated from the optimized values of the quadratic fit._{r} - –
*F*was computed with_{elec}*k*= 5.1153.10_{e}^{−19}F.m. - –
*actu16*diagonal neighbours and perpendicular neighbours were considered to be at the same lateral distance from*actu16*(this corresponded to set*k*= 1)._{l} - – Δ
*dp*was considered to be the same for each of the eight neighbours.

*k*can be estimated, Plotting

_{m}*k*with respect to Δ

_{m}*dp*showed a quadratic variation. Figure 9 shows (a) the measurements of the displacement obtained along the vertical cut passing on top of

*actu16*for the five test voltages and (b) the plot of

*k*with respect to Δ

_{m}*dp*and the corresponding quadratic fit.

*k*are presented in Table 4.

_{m}*k*and

_{m}*k*(respectively the spring constant for the inter-actuator mechanical coupling and the actuator top plate restoring force) were actually not of constant value but instead varied following a quadratic law. This was due to the fact that the model is not perfect and uses a limited number of degree of freedom to evaluate the shape of the DM membrane. During the MCMC optimization of the model coefficients, the physical parameters of the DM which are not taken into account by the model will impact the value of the model coefficients. This also explain the difference between the theoretical value (for

_{r}*k*) and the experimental values (for

_{e}*k*,

_{ma–c}*k*and

_{ra–c}*k*) and their optimized counterpart (see Table 5).

_{l}## 4. First laboratory demonstration: evaluation of the model’s performance

### 4.1. Figures of merit

**18**(20), 21356–21369 (2010). [CrossRef] [PubMed]

### 4.2. Performance of the model with Kolmogorov type phase screens

**1.6**% and

**3.2**% (for the “rms error/PV” FOM) and between

**7.3**% and

**14.6**% (for the “rms/rms” FOM).

### 4.3. Performance comparison with previous modelling approaches

**18**(20), 21356–21369 (2010). [CrossRef] [PubMed]

25. C. Blain, R. Conan, C. Bradley, and O. Guyon, “Open-loop control demonstration of micro-electro-mechanical system MEMS deformable mirror,” Opt. Express **18**(6), 5433–5448 (2010). [CrossRef] [PubMed]

*α*a constant.

*α*was optimized using the MCMC algorithm with the set of ten phase screens used for our model performance evaluation. As shown in Table 6, our model allows to improve the DM control performance by approximately a factor three compared to the SQM. As the intensity of the residual coronagraphic speckle is proportional to the squared wavefront error, this represents an improvement of a factor nine on the speckle intensity. The performance presented for the SQM in Table 6 are the mean value over the ten phase screens under test.

## 5. Future work : on-sky demonstration

## 6. Conclusion

**2.3**% residual rms error (for the “rms error/PV” FOM) and

**11.5**% residual rms error (for the “rms/rms” FOM ) (mean values obtained over a set of ten phase screens with a mean PV of 1448 nm). This performance corresponds to an improvement of a factor three compared to the standard quadratic model (common relation between the voltage sent and the actuator vertical displacement). The model is currently being integrated to the SCExAO control architecture, with the purpose of being tested on-sky at the Subaru Telescope during the summer 2011.

## References and links

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3. | O. Guyon, E. A. Pluzhnik, F. Martinache, J. Totems, S. Tanaka, T. Matsuo, C. Blain, and R. Belikov, “High contrast imaging and wavefront control with a PIAA coronagraph: laboratory system validation,” Publ. Astron. Soc. Pac. |

4. | O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. |

5. | O. Guyon, E. A. Pluzhnik, R. Galicher, F. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet Iimaging with a phase-induced amplitude apodization aoronagraph. I. principle,” Astrophys. J. |

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10. | A. Give’on, S. Shaklan, and B. Kern, “Electric field conjugation-based wavefront correction algorithm for high contrast imaging systems - experimental results,” Proceedings of the conference In the Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar Disks in the 21st Century (2007). |

11. | R. Conan, C. Bradley, O. Lardière, C. Blain, K. Venn, D. Andersen, L. Simard, J.-P. Véran, G. Herriot, D. Loop, T. Usuda, S. Oya, Y. Hayano, H. Terada, and M. Akiyama, “Raven: a harbinger of multi-object adaptive optics-based instruments at the Subaru telescope,” Proc. SPIE Adaptive Optics Systems II7736, 77360T (2010). |

12. | S. Eikenberry, D. Andersen, R. Guzman, J. Bally, S. Cuevas, M. Fletcher, R. Gardhouse, D. Gavel, A. Gonzalez, N. Gruel, F. Hamann, S. Hamner, R. Julian, J. Julian, D. Koo, E. Lada, B. Leckie, J. A. Lopez, R. Pello, J. Perez, W. Rambold, C. Roman, A. Sarajedini, J. Tan, K. Venn, J.-P. Véran, and J. Ziegert, “IRMOS: the near-infrared multi-object spectrograph for the TMT,” Proc. SPIE Ground-based and Airborne Instrumentation for Astronomy6269, 62695W (2006). |

13. | T. Bifano, S. Cornelissen, and P. Bierden, “MEMS deformable mirrors in astronomical adaptive optics,” 1st AO4ELT conference, 06003 (2010). |

14. | J. A. Perreault, T. Bifano, B. M. Levine, and M. Harenstein, “Adaptive optic correction using microelectromechnical deformable mirrors,” Opt. Eng. |

15. | |

16. | |

17. | C. Hom, P. Dean, and S. Winzer, “Simulating electrostrictive DM: I. non-linear static analysis,” Smart Mater. Struct. |

18. | C. R. Vogel and Q. Yang, “Modeling, simulation, and open-loop control of a continuous facesheet MEMS deformable mirror,” J. Opt. Soc. Am. A |

19. | C. R. Vogel, G. Tyler, Y. Lu, T. Bifano, R. Conan, and C. Blain, “Modeling and parameter estimation for point-actuated continuous-facesheet deformable mirrors,” |

20. | K. Morzinski, K. B. Harpsoe, D. Gavel, and S. M. Ammons, “The open-loop control of MEMS: modeling and experimental results,” Proc. SPIE MEMS Adaptive Optics6467, 64670G (2007). |

21. | J. B. Stewart, A. Diouf, Y. Zhou, and T. Bifano, “Open-Loop control of MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. |

22. | D. Andersen, M. Fisher, R. Conan, M. Fletcher, and J.-P. Véran, “VOLT: the Victoria Open Loop Testbed,” Proc. SPIE |

23. | E. Laag, D. Gavel, and M. Ammons, “Open-loop woofer-tweeter control on the LAO multi-conjugate adaptive optics testbed,” in |

24. | D. Guzmán, F. J. Juez, F. S. Lasheras, R. Myers, and L. Young, “Deformable mirror model for open-loop adaptive optics using multivariate adaptive regression splines,” Opt. Express |

25. | C. Blain, R. Conan, C. Bradley, and O. Guyon, “Open-loop control demonstration of micro-electro-mechanical system MEMS deformable mirror,” Opt. Express |

26. | D. Guzmán, F. J. Juez, R. Myers, A. Guesalaga, and F. S. Lasheras, “Modeling a MEMS deformabe mirror using non-parametric estimation techniques,” Opt. Express |

27. | G. J. Baker, “A fast high-fidelity model for the deformation of continuous facesheet deformable mirrors,” Proc. SPIE, Adaptive Optics Systems II6272, 627224 (2006). |

28. | C. Blain, O. Guyon, R. Conan, and C. Bradley, “Simple iterative method for open-loop control of MEMS deformable mirrors,” Proc. SPIE Adaptive Optics Systems7015, 701534 (2008). |

29. | Michael Feinberg, Boston Micromachines Corporation - Private communication. |

30. | http://www.naoj.org/staff/guyon/06software.web/01cfits.web/content.html. |

**OCIS Codes**

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

(230.4040) Optical devices : Mirrors

(010.1285) Atmospheric and oceanic optics : Atmospheric correction

(230.4685) Optical devices : Optical microelectromechanical devices

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: May 23, 2011

Revised Manuscript: July 29, 2011

Manuscript Accepted: September 9, 2011

Published: October 12, 2011

**Virtual Issues**

Vol. 6, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Célia Blain, Olivier Guyon, Colin Bradley, and Olivier Lardière, "Fast iterative algorithm (FIA) for controlling MEMS deformable mirrors: principle and laboratory demonstration," Opt. Express **19**, 21271-21294 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21271

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