1. Introduction

1.1. Motivation for developing a MEMS DM model

Micro-Electro-Mechanical-System (MEMS) deformable mirrors (DMs) offer a large actuator density, allowing high count DMs to be deployed in small size beams. Therefore, MEMS DMs are an attractive technology for Adaptive Optics (AO) systems and are particularly well suited for high contrast imaging systems and Multi-Object AO (MOAO) instruments. Successful application in these AO architectures relies on a thorough modeling of the DM response to the control system commands.

1.1.1. High-contrast imaging: SCExAO

Currently, more than 500 exoplanets have been discovered utilizing primarily the radial velocity technique [1]. Direct imaging instruments using a coronagraph, coupled with the full on-axis corrective capability of Extreme-AO (ExAO) systems, allow the probing of the inner parts of planetary systems. This will prove invaluable in the attempt to understand their formation and evolution. The two major challenges for these instruments are the extremely high contrast between the parent star and the orbiting planet, and their small angular separation. An efficient coronagraph separates the planet’s light from the much brighter starlight of the parent. The dynamic boiling speckles created by the stellar images make planet detection impossible without the help of a wavefront correction device. MEMS DMs are used both for the Gemini Planet Imager (GPI) [2] and the Subaru Coronagraphic Extreme AO project (SCExAO) [3]. This paper only focuses on SCExAO where the MEMS DM is used for both wavefront sensing and correction.

SCExAO is a high performance instrument utilizing a Phase-Induced Amplitude Apodization Coronagraph (PIAAC) [4,5]. It simultaneously provides high contrast, small inner working angle, high throughput, low chromaticity, full 360 degree discovery space and full 1λ/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.

The SCExAO active wavefront control consists of a high-order wavefront sensor (WFS) and a focal plane WFS [8], coupled with the MEMS DM. A detailed description of SCExAO wavefront control architecture is given in [9]. SCExAO uses coherent light modulation in focal plane introduced by the DM, for both wavefront 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, 9, 10]. It is, therefore, critical to have a good model of the DM otherwise the estimation of wavefront error is poor. The physical properties of the DM need to be well known, as well as its position in the beam. The development of the model presented in this paper was motivated by the need to provide accurate DM control of the SCExAO wavefront.

1.1.2. Classical AO and multi-object AO

The accurate open-loop control of a DM is highly desirable for MOAO instruments [11, 12] and can also improve closed-loop (CL) performance for classical CL AO systems (by allowing an increase in operational speed). The development of a high accuracy DM model is thus an important step toward the progression of future AO technologies.

1.2. Overview of MEMS deformable mirror technology

Typically, MEMS DMs [13, 14] are composed of a thin silicon membrane with a highly reflective metallic coating, supported by an array of electrostatic micro-actuators. The vertical motion of each actuator occurs when a positive voltage is applied to the actuator top plate while the base plate stays grounded. The voltage difference between the two plates induces the creation of an electrostatic force attracting the two plates toward each other. While the base plate is fixed, the actuator top plate get deflected toward the base plate by a quantity which is approximately proportional to the square of the applied voltage. Each actuator top plate is attached through a rigid and incompressible post to the mirror membrane, resulting in the membrane deformation.

The state-of-the-art MEMS devices have up to 4096 actuators with a maximum actuator stroke ranging from 2μm to 8μm. 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].

Compared to other DM technologies (e.g piezo-stack, magnetic) that exhibit bi-directional actuator motion, MEMS DM actuators can only be “pulled” in one direction. However, each actuator has low inertia and, therefore, it can be positioned along its total stroke with great accuracy and at high frequency (e.g in the kHz range).

The DM utilized in this work was made by BMC and had a continuous membrane with 1024 actuators. The actuator pitch was 340 microns and the membrane was coated with a thin layer of Cr-Au (chrome-gold alloy). The array of actuators contained one defective actuator located at position [6,22]. This actuator did not respond properly to the applied voltage command. However, when the same voltage was sent to the whole array, the defective actuator followed the array motion up to approximately 90 V.

1.3. Deformable mirror modeling—previous works

The modelling of deformable mirrors, for open-loop control architectures, has been an ongoing research topic for more than a decade. To estimate the efficiency of a control system, the common metric used is the root-mean-square (rms) error. The rms (also named standard deviation) is the square root of the variance for the set of measured minus model phase screens and is computed as:

In 1999, Hom et al. [17], first presented a non-linear model for an electrostrictive DM and obtained approximately 40 nm rms error when correcting a given wavefront. In 2006, a method relying on a coupled system of non-linear partial differential equations (to model the membrane flexure) combined with algebraic equations (dedicated to model the array of actuators underlying the membrane) was developed by Vogel et al. [18]. In 2010, Vogel et al. [19], proposed a refined variant of this method and applied it to both a 140-actuator MEMS DM and a 57-actuator piezo-stack CILAS DM. In this work, several array configurations were experimented (such as (a) similar bias applied to all actuators, (b) central actuator set to various voltage while the rest of the actuators are maintained to a mid-bias voltage and (c) central array of three by three actuators set to various voltages while the rest of the actuators are maintained to a mid-bias voltage) and resulted in less than 10 nm error for a Peak-to-Valley (PV) difference in deflection across the DM for a total deflection of approximately 500 nm.

Morzinski [20] and Stewart [21] reported less than 15 nm rms error for initial phase screens of respectively 500 nm PV and 1500 nm PV. In [21], the performance evaluation was done by applying, (i) influence functions and (ii) focus term (Zernike polynomial), to a 140-actuator MEMS DM. In [20], it was a 1024-actuator MEMS DM and the shape applied correspond to (i) Sinusoidal functions, (ii) Gaussian functions (similar to an actuator influence function) and, (iii) Kolmogorov functions. The works presented in [19–21] all used different variation of the thin plate equation. A detailed comparison of the three methods is given in the “Introduction” section of Vogel et al., 2010. In 2008, both Andersen [22], and Laag [23], reported 10 to 20 nm rms error on an ALPAO DM. In 2010, two new approaches emerged from Guzmán [24] and Blain [25].

Guzmán [26] reported 12.7% rms residual error (relating the residual error rms to the rms value of the desired wavefront) and 3.1% (relating the residual error rms to PV excursion of the desired wavefront correction) over several thousand phase screens of random shape for a fourteen by nine active array of a 1024-actuator MEMS DM [26]. Guzmán model is based on non-parametric estimation techniques [24]. Two different non-parametric techniques were evaluated in [26] (a) Multivariate Adaptive Regression Splines (MARS) and (b) Artificial Neural Networks (ANN). The model, purely mathematical, needs a preliminary training step using a large (few thousand) data set of interferometric phase maps and presents the benefit of a similar model strategy regardless of the type of mirror in operation. Blain reported a mean 11% rms residual error (relating the residual error rms to the rms value of the desired wavefront) over one hundred Kolmogorov type phase screens applied to an eighteen by eighteen active array of a 1024-actuator MEMS DM. This model was based on the preliminary calibration of the DM (utilizing the measurements of the actuator influence function and the stroke-voltage relationship).

To analyse the performance of the model presented in this paper, the selected figure of merit consisted of the evaluation of the residual error rms obtained when the DM responds to the application of voltage maps that match Kolmogorov type phase screens. A similar phase screen approach was utilized by Morzinski and Guzmán, thus their work will be used for comparative performance evaluation in Sec. 4.

The remainder of the paper is structured as follows: description of the FIA algorithm, calibration procedure, experimental results and future work.

2. Modeling MEMS DMs with the fast iterative algorithm

2.1. General description of the method

The model, illustrated in Fig. 1, provided an accurate estimate of the DM shape without employing computationally intensive techniques such as those derived from thin plate theory [18, 20, 27]. The model was based on an iterative algorithm (Block [B], Fig. 1) which utilizes an input voltage map (Block [A], Fig. 1) and converges toward an output phase map (Block [C], Fig. 1). This work extends an initial approach [28] and also experimentally evaluates the improved model on a test bench developed at the University of Victoria (UVic) AO Laboratory.

 

Fig. 1 Overview of the model organisation with the preliminary calibration procedure and the detail of the iterative algorithm.

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The iterative algorithm relied on three force equations that represented the active forces during operation of the DM. As shown in Fig. 2, the forces were:

 

Fig. 2 Schematic of the forces acting on two neighbouring actuators. The DM membrane (in yellow) is attached to the top actuator plate through rigid posts. The electrostatic force F_elec and the restoring force F_restoring are always acting in opposition. The mechanical coupling force F_mec, depends on the relative position of the actuators to each other. In this schematic, the left actuator is lower than the right actuator. The mechanical coupling through the membrane results on the left actuator being pulled up (by the right actuator) and the right actuator is being pulled down (by the left actuator).

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The cumulative response of theses forces to the application of an input set of voltages resulted in the actuators’ final vertical displacement. The sum of all forces must equal zero for the system to reach an equilibrium state. Because the simultaneity of the force action cannot be reproduced with numerical simulation, the model utilised an 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.

To match the model to a given DM and a given experimental setup, a set of “model coefficients” must be optimized during a preliminary calibration procedure which consisted of two main steps. First, a set of ten different volt maps were applied to the DM and an interferometer was used to measure the ten DM shapes obtained in response to these voltages. Secondly, an optimization algorithm was used to identify the value of the model coefficients that resulted in the best match between the ten previously measured DM shapes and the ten simulated output phase maps of the model. The optimization of the model coefficients was critical to obtain accurate results.

2.2. Definition of the model’s forces

The model relied on force equations translating the physical property of the DM actuators and the DM membrane. These forces are detailed below.

2.2.1. Electrostatic force (F_elec)

The “parallel plate” electrostatic actuator is one of the most basic and common MEMS devices. Each actuator was modeled as a capacitor with a fixed grounded plate electrode coupled to a movable top actuator plate electrode. The capacity can be written as:

The difference of potential, or voltage, between the two plates was noted V and the potential energy can be written as

Finally, F_elec was given by the following relation,

The variables of Eq. (6) had the following physical values:

As a result, k_e was estimated to be k_e = 5.1153.10⁻¹⁹ F.m.

When the DM was at rest (no applied voltages), the initial distance between the two actuator plates, g₀, 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)

Each square actuator plate was rigidly connected to the substrate along two of its edges, and was suspended above an addressable electrode [13]. When a voltage was applied, the top actuator plate was deflected toward the fixed substrate plate and was subjected to a bending load. The material will naturally tend to return to an equilibrium state, creating a restoring force F_restoring. MEMS actuators can only be pulled in one direction (toward the fixed actuator base), so F_restoring will always be in opposition to F_elec.

F_restoring was modelled using the classical spring model and can be written as

2.2.3. Inter-actuator mechanical coupling force (F_mec)

When a voltage was applied to an actuator, F_elec and F_restoring 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 F_elec and F_restoring, the vertical displacement of the adjacent actuators needed to be estimated to accurately predict the vertical displacement of a given actuator in the array.

The mechanical coupling between neighbouring actuators, F_mec, 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] and illustrated in Fig. 2 of that paper confirmed that this is a reasonable assumption. The displacement of a given actuator will be minimized if the neighbouring actuators have lower displacements and amplified if the neighbouring actuators have larger displacements. Any configuration in between is possible. Therefore, F_mec can either be opposite to F_elec or in the direction of F_elec.

Again, the classical spring model was chosen to represent the interaction between neighbouring actuators and took the form shown in Eq. (9),

2.2.4. Equilibrium

When the system is at equilibrium, the sum of all active forces must be equal to zero:

Using Eq. (8) and Eq. (10), dp can be written as:

2.2.5. Model input-output

The model input was a set of actuator voltages to apply to the array of actuators. Using the equations presented in Sec. 2.2 and the DM coefficients (Sec. 2.3), the iterative algorithm generated a matrix of actuator vertical displacements (having the same dimension as the input voltage matrix) named hereafter the 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).

The actuator influence function is the characteristic shape of the mirror response to the action of a single actuator. Influence functions can be represented using Gaussian functions. To go from the displacements matrix to the phase map, the algorithm first performed a proper normalization of the influence function to ensure that the volume under the influence function was equal to the area corresponding to an actuator pushed to the nominal displacement. This was critical to ensure that if all actuators were pushed to 1 micron, the mean value of the membrane shape obtained was actually 1 micron. The final phase map (constituting the output of the model) was then constructed by the cumulative sum of the multiplication, point by point, of the displacement matrix by the normalized influence function. The geometrical coefficients (Sec. 2.3) were also integrated in this step to provide the capability to add geometrical transformations in the final image.

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

It was important to match the model to a specific DM and its associated optical arrangement. Therefore, the model coefficients were separated in two groups named “DM coefficients” and “geometrical coefficients”. Both set of coefficients presented in Table 1 must be optimized during the calibration using a Markov Chain Monte Carlo (MCMC) algorithm.

Table 1. Descriptive Table of the DM Coefficients and the Geometrical Coefficients

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The use of “global” model coefficients accompanied the assumption that the DM behavior was isotropic. Exploring anisotropy was outside the scope of this paper, and could be a refinement of the work presented here. The residual fit numbers given in Sec. 4.2 were for an isotropic model, and could be improved with a non-isotropic model (which would require additional calibration).

2.3.1. DM coefficients

A first set of eight coefficients described the DM’s intrinsic physical characteristics. One DM coefficient, k_e, 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 k_r, the spring constant, varied with the displacement of the actuator plate in a quadratic fashion as,

Three additional coefficients were dedicated to the inter-actuator mechanical coupling, F_mec, also modelled using a spring. It will be shown in Sec. 3.2 that k_m, the spring constant, varied with the degree of stretching of the membrane. k_m was estimated by examining the relative difference in displacement between an actuator and its eight direct neighbours, or Δdp. k_m also followed a quadratic law,

Finally, k_l was used to take into account the geometrical difference in the lateral distance between the perpendicular neighbour actuators and the diagonal neighbour actuators to the “central” actuator.

2.3.2. Geometrical coefficients

To match the model to a specific experimental setup, another set of eight coefficients (termed the “geometrical coefficients”) were also estimated. One coefficient was dedicated to the transformation from the displacements matrix to the phase map, where the full width at half the maximum (FWHM) of the Gaussian function (used to represent the influence function of the actuator) was set as a free parameter. The seven remaining coefficients took into account any misalignment existing in the system (possible translation in X or Y, possible rotation of the image on the detector). They also took into account the projection angle of the beam onto the DM, the pixel scale, the size of the image on the detector and the size of the actuators.

The vertical displacement of a given actuator was affected by the vertical position of the surrounding actuators. This was due to two coupled phenomena. First, the FWHM of the Gaussian function defined the extent of the impact of an actuator motion over its neighbours. Second, the mechanical coupling between adjacent actuators defined how the vertical position of a given actuator was biased by the vertical position of its neighbours. The model provided an optimal balance between these two phenomena due to the optimisation of both the coefficient related to the Gaussian FWHM (Sigma2) and the coefficients related to F_mec (k_ma, k_mb and k_mc).

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

The iterative algorithm is the center piece of the model and is illustrated in Fig. 3. A set of voltages and the sixteen model coefficients (Table 1) served as input data to the algorithm. After the algorithm initialisation (Block [A], Fig. 1), the algorithm iterations began and can be subdivided in four sub-steps which are described below:

 

Fig. 3 Detail of the iterative algorithm. The model input is a voltage map and after few iterations of Step (i) to (iv), the model converges toward a displacements matrix.

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When Step (iv) was completed, the output value for F_mec was set as the new F_mec input value for the next iteration and the algorithm started back to Step (i).

When the iterations were complete, the output was the 32 by 32 displacement matrix which will be transformed into a phase map during the “Final Transformation” (Block [C], Fig. 1.).

2.4.1. Initial conditions for the iterative algorithm

The initial conditions are given below:

2.4.2. Constraining the maximum actuator displacements

The reflective membrane, laid on top of the actuator array, was clamped to the edge of the DM structure. Although the clamping was located outside the area of interest, it still affected the edge actuators by minimizing their maximum displacement (vertical displacement) in comparison to the maximum displacement achievable by the rest of the actuator array. To take into account this limitation, a constraint was added to the model: the maximum displacement of the edge actuators was set to a smaller value than the maximum displacement of the other “central” actuators of the array.

A systematic verification of the computed value of dp for each actuator was performed in Step (ii) and Step (iv). Any actuator (central or edge) with a computed displacement exceeding the authorized measured value (respectively dp_max or dp_edge) was reset to the maximum value allowed.

3. Preliminary calibration of the model coefficients

3.1. Description of the Markov Chain Monte Carlo algorithm

In order to accurately estimate the optimum value for each coefficient, a MCMC algorithm was implemented. However, for the MCMC to perform efficiently, it was critical to start with initial values that were close to the real values. Thus, an experimental estimation of the model coefficients was first performed (Sec. 3.2). The MCMC algorithm is a variation of the classical Monte Carlo algorithm and has the additional advantage of only performing computations in the regions where the data optimization is improving.

The goal of the model was to reproduce, with the highest accuracy, the phase map produced by the DM for a given set of voltage. The model inputs were:

To optimise the value of the model coefficients, the MCMC minimized the difference (residual error rms) between the “modelled” phase map and the original (measured) phase map. During each iteration, two series of sixteen coefficients were randomly selected (named hereafter 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.

Each iteration of the optimisation procedure was organized as follow:

3.2. Experimental estimation of the model coefficients

The MCMC algorithm required a reasonably accurate initial estimate of the coefficient set in order to converge. This was achieved by an experimental characterization that also produced values for dp_max and dp_edge.

The experimental data presented below, and in Sec. 4, were generated using the following experimental setup: the MEMS DM was fixed on a 5-degrees-of-freedom mount (x, y, z, and tip-tilt) and positioned in front of an interferometer (Zygo PTI 250). In this case, the data were measured directly from the interferometer, without the need of using the phase diversity technique used in SCExAO to measure the DM shape. Two filters were also inserted in front of the interferometer window to optimise the fringes contrast and improve the measurement quality.

All voltage maps and phase maps used in the following had a “reference area” and an “active array” (see Fig. 4(a)). The reference area was the outer corona of seven actuators across (actuator 1 to 7 and 26 to 32 for both rows and columns). The central active array was eighteen by eighteen actuators wide. The reference area was set to zero volt at all time and had two purposes:

 

Fig. 4 (a) Location of the “Active array” and “Reference area”. (b) Phase measurement for the array of 32 by 32 actuators pulled to 100 V. The dead actuator appears on the bottom right part of the image as a dark spot. The white dashed square shows the limit of the active array.

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Figure 4(b), shows a phase measurement of the array of 32 by 32 actuators pulled to 100 V. The dead actuator appears on the bottom right part of the image and the white dashed square shows the limit of the selected “active array” of the DM.

The level of precision of the interferometer measurements was verified to ensure that it did not compromise the evaluation of the performance obtained with the model. Hence, the rms error on the interferometer measurements should be small enough compared to the model rms error to be measured to confidently validate the model performances.

To evaluate the interferometer noise, the rms error of the difference of two consecutive measurements of the same phase screen was compared to the phase screen rms. The percentage obtained between the error rms and the phase screen rms was 0.0332% which was roughly a factor one hundred smaller than the percentage obtained for the model performance evaluation. The interferometer precision was thus sufficient to validate the model performances.

3.2.1. Measurement of the DM maximum displacement (dp_max)

The maximum vertical displacement, dp_max, 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 dp_max = 1.6 μm.

 

Fig. 5 (a) Phase measurement of the 32 by 32 array with the active array set to 200 V (in red) and the reference area left to zero volt (in blue) and (b) transversal cut of the phase measurement presented in (a).

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The interferometer measurements became corrupted when the dynamic range of the displacement became too large. This effect was evident in Fig. 5(a) where the reference area appeared split in two regions (dark blue and light blue area on the image).

3.2.2. Measurement of the maximum displacement for the edge actuators (dp_edge)

The motion of the first outer ring of actuators was constrained because the membrane was clamped at the edges. To measure the maximum displacement of the edge actuators, all actuators of the 32 by 32 array were pulled to 200 V while the last five left rows were maintained to 0 V. The zero-volt plateau was required to accurately measure the maximum displacement as it provided a zero reference point for the measurement. Figure 6 shows a cut along the membrane with the five rows of actuators maintained to zero to the right and the lifted-up edge of the mirror to the left. The mean value of the measured displacement on the left edge was dp_edge = 1.3 μm.

 

Fig. 6 Transversal cut of the phase measurement of the edge actuator maximum displacement. The five rows of actuators maintained to zero are to the right and the lifted up edge of the mirror is on the left.

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For this specific experimental setup, the evaluation of dp_edge 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 dp_edge would be an important constraint.

3.2.3. Measurement of the actuator influence function

During the transformation from the displacements matrix to the final phase map, a Gaussian function (defined by its FWHM) was used to represent the actuator influence function. The coefficient corresponding to the FWHM value can be experimentally estimated by measuring the real influence function of one of the actuator. Figure 7 shows a measurement (vertical cut) of the influence function for actuator [16 16] when pulled to 160 V with the rest of the array maintained at 0 V. The FWHM can be estimated to approximately 8 pixels.

 

Fig. 7 Phase measurement (transversal cut) of the influence function for actuator [16,16] when pulled to 160 V while the rest of the array was maintained at 0 V. The FWHM was approximately 8 pixels.

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3.2.4. Measurement of the DM coefficient k_r

The coefficient k_r, 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:

A look up table (Table 2) can be established using Eq. (11).

Table 2. Look-Up Table Established for the Experimental Evaluation of k_r

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Figure 8(a) shows the measurement of the actuator displacement obtained along the central vertical cut of the active array when pulled from 20 V to 120 V. The variation of k_r with respect to dp is plotted in Fig. 8(b).

 

Fig. 8 (a) Phase measurements of the vertical displacement obtained for the central active array of actuators when pulled from 20 V to 120 V and (b) Evolution of k_r with respect to dp and its quadratic fit.

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The coefficients obtained from the quadratic fit of k_r are presented in Table 3.

Table 3. Values Obtained During the Experimental Evaluation of k_r (N.m⁻¹)

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3.2.5. Measurement of the DM coefficient k_m

Once k_r was known for a given dp, the same process was applied to the membrane spring constant, k_m. 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.

In this configuration, actuator [16,16], named 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. Δdp 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:

 

Fig. 9 (a) Measurements of the displacement obtained along the vertical cut passing on top of actu16 for the five test voltages and (b) plots of k_m and its quadratic fit over the range of Δdp.

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Utilizing Eq. (5), (8) and (9), the value of k_m can be estimated,

The coefficients obtained from the second order polynomial fit of k_m are presented in Table 4.

Table 4. Values Obtained During the Experimental Evaluation of k_m (N.m⁻¹)

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As demonstrated above, both k_m and k_r (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 k_e) and the experimental values (for k_ma–c, k_ra–c and k_l) and their optimized counterpart (see Table 5).

Table 5. Model Coefficients Resulting from the MCMC Optimisation; These Coefficients were used to Evaluate the Model’S Performance

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4. First laboratory demonstration: evaluation of the model’s performance

4.1. Figures of merit

In accordance with the techniques of the previously reported works, two figures of merit (FOM) are described to quantify the model’s performance. The two FOM are similar to Guzmán’s [26] metric and are defined below:

4.2. Performance of the model with Kolmogorov type phase screens

The model’s performance were assessed over the active array of 18 by 18 actuators (thus avoiding the defective actuator) and using the experimental setup discussed in Sec. 3.2. This corresponded to a central 100 by 100 pixels phase map. Although Zernike polynomials are commonly adopted to estimate model’s performance, the goal of this experiment was to evaluate the phase residual when the model is confronted to phase aberrations similar to the one encountered with atmospheric turbulence. As a result, a set of ten different voltage maps was generated, corresponding to voltages that would be sent to the DM when trying to compensate for Kolmogorov turbulence. We note that decomposition of Kolmogorov phase screens into Zernike polynomials can be performed to establish correspondence between the two metrics. An offset of 90 V was applied to all voltage maps to set the DM in the mid-range displacement. The sixteen model coefficients were first estimated following the calibration procedure described in Sec. 3.2. Then, the MCMC optimization was accomplished by simultaneously minimizing the residual rms error (Eq. (1)) obtained with five sets of measured phase and their corresponding modelled phase.

The number of measurements (10) presented in this paper is limited. However, providing that (i) the model assumes that all actuators are the same and (ii) there are 18 by 18 actuators and 10 phase maps, the performance are evaluated over a sample of 3240 data points which is much larger than the number of parameters to optimize (16). Furthermore, the ten phase screens used had PV values varying from 1311 nm to 1716 nm and rms values varying from 474 nm rms to 526 nm rms. For each actuator, the 10 displacement values sample approximately 500nm RMS of displacement around the actuator nominal position (chosen here to be 90V). Assuming that the position of each actuator is a function of the voltage applied as well as the voltage applied to its 8 surrounding actuators, the data sample consist of 360 independents 3 by 3 blocks of voltage with the differential voltage between adjacent actuators representing realistic atmospheric properties. When conservatively accounting for inter-actuator couplings, the number of independent measurements (360) is still much larger than the number of parameters (16), and we therefore conclude that the 10 phase screens are sufficient to constrain these parameters. We note that high precision derivation of the 16 parameters, or, equivalently, high precision analysis of the residuals after parametrization of the DM with our model, would require a larger number of phase screens. The goal of this paper is however simply to present first results from a new DM modelling approach, and we only seek to show that the model can be easily implemented and provides a significant improvement over the simple quadratic voltage-displacement law often used for MEMS devices. For these goals, analysis of 10 phase screens is sufficient.

The model’s performance are estimated, in percent, using the two figures of merit defined in Sec. 4.1. Five of the phase screens have been introduced inside the MCMC optimisation algorithm. The five remaining ones were used to check the model’s performance and verify that the model can accurately reproduce a given phase screen once the coefficients have been optimized. The number of iterations necessary for the model to converge to an equilibrium state was low (≤ 15).

The residual error rms varied from 13 nm rms to 20 nm rms resulting in model performances which oscillated between 1.6% and 3.2% (for the “rms error/PV” FOM) and between 7.3% and 14.6% (for the “rms/rms” FOM).

The values of the model coefficients defined during the MCMC optimization step, and used to evaluate the model’s performance, are listed in Table 5. Figure 10. shows five sets of measured phase maps (left) and their corresponding simulated phase maps (middle) as well as the difference between the measure and the model (right). The top two sets have been used to run the model coefficients optimisation.

 

Fig. 10 Left column: measured phase (in m); Middle column: modelled phase (in m); Right column: difference between measured and modelled phase (in m), for five of the ten phase screens used to evaluate the model performances. The top two phase screens have been used in the MCMC algorithm for the model coefficients optimization.

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4.3. Performance comparison with previous modelling approaches

The model performance was compared to Morzinski [20], Guzmán [26] and Blain [25] model performance as they are the ones who tested their model using either Kolomogorov or random type phase screens. The values for this model, given in Table 6, are the mean value over the ten phase screens under test.

Table 6. Performance Comparison between Guzmán, Morzinski, Blain Modeling Approaches and the SQM; The Numbers Shown for this Model and the SQM are the Mean Values Obtained over the Set of Ten Phase Screens under Test

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In addition, the model performance has been compared with the performance obtained using the standard quadratic model (SQM). The SQM gives the relationship between the actuator vertical displacement and the voltage applied to the actuator as,

For the performance evaluation of the SQM, 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

The model, currently running under Matlab, has been implemented in C code and integrated with the SCExAO control system (under Cfits [30]). The system is scheduled for on-sky engineering demonstration in August 2011. The model coefficients must be optimised for the SCExAO optical setup. The SCExAO DM is slightly different from the DM under test at UVic because it has a gold coating optimized for IR wavelengths. The details of SCExAO control architecture and the model performance, on the SCExAO test bench and eventually on-sky, will be presented in an upcoming paper. For the SCExAO experiment, the DM will be correcting residual coronagraphic speckles (similar to Gaussian noise) instead of Kolmogorov type wavefronts.

Additional effort will be focused on optimizing the overall algorithm in order to operate in real-time. With a quick estimation (∼20 operations per iteration, ∼10 iterations for the model to converge and ∼1000 actuators), the model reaches ∼200,000 operations. Assuming new computer technologies available with parallel architectures and with 10 Giga FLOPS capability, one can expect this model to be able to run at several tens of kHz.

6. Conclusion

A model allowing accurate open-loop control of MEMS DMs has been presented. The model’s structure permits real-time utilization in an ExAO system. It relies on an iterative process which input a set of voltages and after few iterations, outputs the phase map produced by the DM.

The model is flexible and allows changes to the parameters without requiring new algorithms development. It allows to determine quickly which parameters matter through trial and error. Therefore, the approach can be readily adapted to other continuous membrane DM technologies. For example, one could replace the electrostatic force equation by the magnetic force equation for a magnetic motion driven DM.

It is critical to match the model to a specific DM and its associated optical setup. Therefore, a set of sixteen DM and geometrical coefficients must be optimized. These coefficients greatly influence the model performance because they are part of the three force equations used to simulate the active forces during operation. To accurately estimate the values for the sixteen coefficients, a Markov Chain Monte Carlo algorithm was used during a preliminary optimization step. This modeling approach is fast because it does not need the use of heavy computational equations such as the plate theory equation previously proposed for that purpose.

The first laboratory demonstration, conducted at the UVic AO Lab showed a promising performance with 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.