1. Introduction

1.1. Background

1.1.1. MEMS deformable mirrors for astronomy

In the past decade, the astronomical community’s interest in Micro-Electro-Mechanical-System (MEMS) technologies has increased with the large number of MEMS studies conducted [1–3]. These studies revealed the potential for astronomical adaptive optics (AO) applications. MEMS Deformable Mirrors (DM) [5] present the ability to organize individual micro actuators into arrays that perform a macroscopic function. A flat, high quality mirror can be obtained by placing a reflective membrane above the actuator array. A voltage applied between the actuator and its ground pad creates an electrostatic force that deflects the actuator toward its pad, thus moving the top membrane attached to the actuator.

Compared to other DM technologies (e.g piezo-stack, magnetic) that exhibit bi-directional actuator motion, a MEMS DM actuator can only be “pulled” (through electrostatic force) 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). Further beneficial characteristics have also been demonstrated: a highly repeatable actuator position and a negligible hysteresis [3, 4].

 

Fig. 1: (a) Non-additivity of the influence functions of two neighboring actuators and (b) compensation of the non-additivity when the actuators are set on a push-pull configuration (one up, one down).

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A wide spectrum of AO instruments are currently under development and many are intended for use on the next generation of Extremely Large Telescopes or ELTs. Increasing the size of the primary mirror will increase the distortion of the image due to atmospheric turbulence. Combined with the needs of each specific AO instrument, this elevates the requirement for the DM specifications to very high levels. MEMS DMs fulfil many of the demanding requirements which accompany the next generation of AO systems; such as: speed, compact size, a large number of actuators with a relatively large stroke, no hysteresis and a reasonable fabrication cost. A 1024-actuator MEMS DM is commercially available and has been extensively tested by several research teams [6, 7]. The current state of the art is a 4096-actuator MEMS DM [8] (with an active aperture of approximately 20 mm) designed to be implemented on the Gemini Planet Imager [9].

1.1.2. New scientific goals and custom AO systems

The instrument specifications are defined by new scientific goals, such as the direct imaging of exoplanets or the observation of faint galaxies and nearby star-forming regions. For the observation of faint galaxies, the small number of photons received (long exposure time necessary for each target) combined with the galaxy’s distribution [spread over a large Field of View (FOV)] make the classical Single Conjugate Adaptive Optics systems (SCAO) inefficient.

Multi-Object Spectroscopy (MOS) is a technique which relies on the insertion of several apertures in a wide FOV (each one of them dedicated to a specific target) and the simultaneous measurement of the spectra of each object. To reach the angular resolution necessary to obtain relevant information on each target, it needs to be coupled with a Multi-Object Adaptive Optics system (MOAO) [10]. An MOAO system is designed and optimised to provide a sharp correction on several small areas spread over a large field of view (~5–10’). Instead of correcting the whole FOV, it consists of several sub-AO systems (with small FOV of approximately 2” each) working in parallel. However, in this configuration, the guide star light directed toward the wavefront sensor is independent of the science target light, directed toward the science camera. Therefore, close-loop control is not possible and open-loop control must be used.

The quest for the detection and the characterisation of exoplanets (from giant hot Jupiters to Earth-like planets) is at the center of many exciting research developments [9, 11, 12]. To meet the high performance coronagraphic needs, an AO system needs to be combined with the coronagraph. Classical SCAO systems had to evolve toward the so-called Extreme Adaptive Optics (ExAO) systems optimised to provide a high Strehl correction on axis over a small field of view. MEMS DMs have the ability to reach the desired ExAO performances [13, 14]. To improve the overall performance, new coronagraph designs [15, 16] also include MEMS DMs directly inside the coronagraphic path to correct for the residual aberrations due to alignment errors and to cancel speckles in the planet search area. In such cases, the choice of driving the DM with open-loop control is more appropriate.

For all these applications, it is likely that the DM will be driven using an open-loop (OL) control architecture. Because of the lack of feedback, an exact knowledge of the mirror shape in response to a set of control commands is needed to achieve the scientific requirement of image-sharpening (in the nm range). Much of the research in this area focuses on the development of an accurate physical model [17–21].

 

Fig. 2: (a) Diagram of the experimental setup and (b) DM’s active area (light green square). The interferometer mask is set to cover the 17 by 17 array of actuators.

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Fig. 3: Stroke-voltage relationship plots for the 324 actuators. In this figure, the x axis represents the squared voltages and the y axis represents the stroke (in nm). All actuators have a maximum stroke of approximately 800 nm excepts for the actuator coupled with the defective actuator which only has a maximum stroke of 400 nm.

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1.2. Goal of this paper

The work presented here was motivated by an attempt to find answers to the following questions: Which types of AO systems require the use of a model for MEMS DMs? Which MEMS properties need to be known to develop this model? What accuracy is needed for this model? In order to help the ongoing research focused on the development of MEMS models, a companion paper [7] was focused on the characterisation of the actuator influence function non-additivities (non-linear coupling between neighbouring actuators) and stroke-voltage relationship (SVR). In this former study, it was observed that when neighbouring actuators are set in a push-pull configuration (one up, one down, relative to a bias half-stroke position), the influence function non-additivities are compensated. These results are recalled in Fig. 1.

Figure 1(a) presents the membrane deflection obtained in two cases. First, two neighbouring actuators are pulled consecutively to 0V, 50V and 100V while the rest of the actuators are maintained at a bias voltage of 150V. The influence functions obtained for both actuators are numerically summed for 0V, 50V, and 100V and a transversal cut of the result is plotted and tagged “sum” on the figure’s textbox. Then actuators are pulled simultaneously to 0V, 100V and 150V while the other actuators are maintained at a bias voltage of 150V. These measurements (transversal cut) are tagged as “poke2” on the figure’s textbox. Figure 1(a) shows that as the relative voltage (the difference between the neighbouring actuator’s voltage and the rest of the membrane voltage) increases, the membrane deflection created by simultaneously pulling both actuators (tagged “poke2” in the textbox) separates from the numerical sum of their influence functions (tagged “sum” on the textbox).

On Fig. 1(b), the red and green plots represent the transversal cut of the influence functions of two neighbouring actuators. First, the measurement are taken consecutively, one actuator is pushed up, then the neighbouring actuator is pulled down (while the rest of the actuators are maintained at a bias voltage of 100V). Then, the sum of both actuators influence function (dark blue plot tagged “sum” in the figure textbox), is plotted beside the measurement obtained when both actuators are set simultaneously on the push-pull configuration (light blue plot tagged “poke2” in the figure). It can be seen that in this particular configuration, the non-additivity of the two neighbouring actuator’s influence function becomes negligible (the dark and light blue plots are on top of each other).

From this previous observation emerged a new research path: what accuracy can be reached (best residual rms) using an open-loop control strategy based on the modelling of the influence functions additivity? Indeed, with a thorough characterisation process, the control command (voltage maps) can be computed by using the information obtained from both the actuator stroke-voltage relationship and the actuator influence functions.

In this paper, we will rely on the following two assumptions: first, for a typical turbulent phase screen, there is more power at low spatial frequencies than at high spatial frequencies. Second, the statistical distribution of actuators compensating for a phase screen type Kolmogorov is such that the number of actuators on the “up” position is roughly equal to the number of actuators on the “down” position. Thus, from the plot presented in Fig. 1(b), the non-additivity of the influence functions are considered negligible.

We first present the experimental apparatus and the data collection process in Sec. 2. The results of the actuator SVR and influence function calibration are introduced in Sec. 3. Section 4 focuses on the performance obtained when the DM is controlled in an open-loop fashion using only the knowledge acquired from Sec. 3. The residual rms is minimised through the optimised utilisation of the calibration data. Finally, a statistical study is performed over 100 computer generated phase screens simulating the atmospheric turbulence as seen by a 30 metre diameter telescope.

 

Fig. 4: (a) Normalised influence function for actuator # 171 and (b) transversal cut along x axis, y axis and the main diagonal of influence function # 171. The interferometer spatial resolution is 6.2 pixels per actuator. The slight asymmetry observed in (a) is due to a pixelisation effect.

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2. Experimental setup

2.1. Experimental setup

The experimental setup is illustrated by the diagram in Fig. 2(a) and consists of a 1024-actuator Boston Micromachines MEMS DM with 200 volt (14 bit resolution) electronics manufactured by NASA JPL. The actuator pitch is 340 μm and the maximum stroke is given for 2.5 μm. A Zygo PTI 250 interferometer is positioned in front of the DM. The interferometer beam passes through a density filter to improve the fringe contrast. A mask can be set on top of the interferometer beam using the software provided with the interferometer. One computer is dedicated to the Zygo interface (metrology software, interferometer measurements and data transfer to the laboratory data server), while a second computer controls the DM electronics and initiates the interferometer measurements. The DM and interferometer are setup on a vibration isolation optical table. The active area on the DM is a square of 18 by 18 actuators. Aligning the pupil edges (the edges of the interferometer mask) to half of the edge actuators reduces the fitting error (this result will be detailed in Sec. 4.3). In this optimised configuration, the effective area of the DM will become a square of 17 by 17 actuators. For an optimised pupil size of 106 by 106 pixels, the spatial resolution of the interferometer is 6.2 pixels per actuator for a 17 by 17 actuator array. The repeatability of the Zygo measurement was verified and is less than two nm rms. Finally, all results presented along this paper are given in nm or μm “surface” with one nm “phase” equal to two times a nm “surface”.

2.2. Data collection

There are only two DM properties which need to be evaluated to complete the characterisation process necessary for the control strategy presented in this paper: the actuator stroke-voltage relationship (SVR) and the actuator influence function.

The actuator SVR is the relationship between the membrane deflection and the corresponding applied voltage. To determine this relationship, a set of voltages were applied to the actuators, starting at the lowest voltage that can be sent (0 Volts) to the highest voltage that can be supported by the DM actuators. This highest voltage is determined by the DM physical limit (size of the actuator gap) and by the electronics. The DM physical limit is 250V. If an actuator is exposed to a higher voltage, the electrostatic force generated between the actuator plate and the actuator base would bring the actuator plate all the way down to the actuator base. The electronics used for this experiment can provide a maximum voltage of 200V.

The actuator influence function is the characteristic shape of the mirror response to the action of a single actuator. The DM influence functions are measured by sequentially pushing the DM actuators and measuring the resulting shape over the whole membrane.

The MEMS DM being tested has an array of 32 by 32 actuators. However, a large number of actuators on the right side of the mirror are dead or malfunctioning; therefore, the maximum size of the array available for this test was 18 by 18 (a total of 324 actuators). This 18 by 18 array will be subsequently referred to as the DM’s active area. The position of this array with respect to the DM membrane is presented in Fig. 2(b). The actuator situated at the bottom left corner of the array is coupled with an actuator located outside of the array and has a truncated maximum stroke.

To ensure the reliability of the calibration data, a new set of data should to be taken if the experimental setup is modified (optical re-alignement, change in ambient temperature..). This will guarantee up-to-date DM properties necessary to achieve an accurate control of the DM.

3. Deformable mirror characterisation

3.1. Measurement of the actuator stroke-voltage relationship

The precise calibration of each actuator stroke-voltage relationship is a critical step toward the accurate open-loop control of the deformable mirror. The plots for each actuator of the 18 by 18 array are generated by driving the actuator from 0 Volts to 200 Volts in steps of 20 Volts while the rest of the actuators are set to a voltage bias of 140V (this bias voltage corresponds to the mid-stroke, directly measured from an actuator’s stroke-voltage plot).

The stroke-voltage relationship is quadratic and for each actuator k can be stated as:

where the “gain(k)” refers to the slope of the SVR for the actuator k, the “offset(k)” refers to the 0 position offset and “V” corresponds to the voltage applied to the actuator.

The maximum stroke observed is approximately 800 nm. However, the coupled actuator in the bottom left corner, described in Sec. 2.2, has a maximum stroke of only 400 nm. The SVR plots obtained for the 324 actuators are presented on Fig. 3. The x axis represents the squared voltage sent to the actuator and the y axis the corresponding actuator stroke (in nm). The standard deviation of the measurements presented in .Fig. 3 is 23.4 nm rms.

3.2. Measurement of the actuator influence function

The influence functions are measured by releasing, one at a time, the actuators at 0 Volts while the rest of the membrane is pushed at the bias voltage of 140 Volts. For each actuator, the phase is measured by the interferometer and normalised. To normalise the influence function, each measured phase map is divided by the absolute value of the maximum phase point in the phase map. The normalised influence functions are non-unit phase maps with the scale ranging from zero to one. The normalised influence function for actuator # 171 is presented in Fig. 4.

Note that because the actuators are released one at a time, the influence function measurements do not include any non-linear information related to the mechanical coupling between neighbouring actuators.

 

Fig. 5: (a) original sample phase screen φ generated with Matlab, (b) corresponding fitted phase $\tilde{\phi }$ obtained by the multiplication of the influence function F and the stroke coefficients a_k, (c) stroke map (a_k coefficients obtained by projection of the original phase φ onto the influence functions F), (d) voltage map (vertical scale in Volt) obtained with Eq. (6), (e) projection of the original phase screen φ onto the DM (phase φ^m measured by the interferometer) and (f) “open-loop” error or “measurement” error = φ - φ^m. This error map incorporates both the fitting error and the DM error (non-linear effects such as the mechanical coupling between neighbouring actuators). Vertical scales for (a), (b), (c), (e), and (f) are in nm.

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4. Performance evaluation in open-loop control

This section describes the experimental protocol that was followed, starting from the generation of the 100 test phase screens to their final projection onto the DM. To illustrate the results obtained, a sample phase screen among the 100 generated was selected. Figure 5 presents images corresponding to its original phase, its projection onto the influence functions and onto the DM as well as the matching voltage map.

Figure 6 illustrates the open-loop control process and the various errors introduced in the following sections.

4.1. Generation of phase-screens

100 phase screens φ are generated using Matlab. To create phase screens that match the dynamical range of the MEMS DM, the generated phase screens are first numerically rescaled. They are generated using turbulence parameters, which follow a Von Karman statistic. The Fried parameter, r₀, is set to 15 cm. The Fried parameter determines the seeing cell size (this corresponds to the aperture size beyond which increases in diameter provide no further increase in resolution) [22]. The outer scale, L₀, is set to 60 m and corresponds to the size of the largest turbulence cell. Finally, the pupil diameter, D, is set to 30 m. The characterisation of the actuator SVR presented in Sec. 3.1 shows that the DM being tested has a maximum stroke of about 800 nm. With a bias voltage of 140V, the DM stroke varies from -400 nm to +400nm. However, the scale of the raw phase screens generated through Matlab is around 15000 nm. In order to avoid DM stroke saturation, the phase screens are scaled to 88% of the DM maximum stroke. The piston is also removed in order to match the phase screen dynamic range to the actuator dynamic range (±400nm) with no bias voltage.

Figure 5(a), shows a representative phase screen φ. The phase varies approximately from 250 nm to -300 nm.

 

Fig. 6: (a) Diagram of the open-loop control process and (b) error estimation. The original phase screens φ are generated using Matlab. The influence functions F and the SVRs are measured during the DM calibration. The stroke maps (a_k coefficients) are computed using Eq. (4). The voltage maps [obtained using Eq. (6)] are sent to the DM and the interferometer measures the DM membrane deflection (named the measurement phase screen φ^m). φ^m is the projection of the original phase φ onto the DM.

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Fig. 7: (a) Fitting error versus the size of the interferometer mask. When the size of the mask is decreased, the phase screen is rescaled to match the interferometer mask size. As the mask get smaller, the number of actuators available to reproduce the phase screen decreases, and the fitting error increases as a result. The fitting error is minimum when the mask edges are positioned at the center of the edge actuators of the DM’s active area (b) Diagram of the interferometer mask size relative to the first three outer actuator coronas.

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4.2. Projection on influence function: stroke map computation

For a pixel of coordinate (x, y), the phase for the MEMS DM at this pixel is given by:

where a_k represents the unknown stroke coefficients and F_k represents the normalised influence functions, measured and described in Sec. 3.2 (one a_k coefficient per influence function and one influence function per actuator).

Using matrix representation, it is re-written as

The evaluation of the a_k coefficients is done by performing a least square fit (LSF) projection of the phase screen onto the influence functions which corresponds to the minimisation of the Frobenius norm of φ - F · a with respect to a.

The minimisation of the norm corresponds to the following matrix operation:

In the following, the a_k coefficients related to a particular phase screen are called the stroke map. Once the stroke maps are obtained , the fitted phase $\tilde{\phi }$ can be reconstructed through the following matrix multiplication,

The process described through Eq. (3) to Eq. (5) is commonly called the projection of the phase screen onto the influence functions. Figure 5(b) presents the fitted phase $\tilde{\phi }$ obtained by the multiplication of the stroke maps and the influence functions F. Figure 5(c) shows the corresponding stroke map, results of the projection of the phase screen onto the influence functions.

Because the phase screens φ have been scaled to fit the DM’s maximum stroke capability, the stroke coefficients a_k also match the DM’s stroke capability. Phase screens φ and influence function phase maps are 106 by 106 pixels. To ensure that the piston was removed properly, the mean value of both the generated phase screens φ and the stroke maps were checked to be equal to 0. In Sec. 4.4, the stroke coefficients will be utilized to generate the voltage maps.

4.3. Fitting error minimisation

The multiplication of the stroke maps by the normalised influence functions gives the fitted phase screens $\tilde{\phi }$. The fitting error is the result of the difference between the original phase screens φ and the fitted phase screens $\tilde{\phi }$ (see Fig. 6). The fitting error gives an estimation of the limits of the DM’s performance due to the limited DM spatial resolution (limited number of actuators available to reproduce a given phase screen). This is the DM’s sampling error. This shows how well the mirror can reproduce a specific phase screen when the only limiting factor is the number of actuators. The fitting error decreases as the number of actuators increases. Non-linear effects (such as the mechanical coupling between neighbouring actuators) are not taken into account in the estimation of the fitting error. This is due to the fact that the influence functions described in Sec. 3.2 are obtained by releasing one actuator at a time, thus the effects due to coupling between neighbouring actuators are not present.

The fitting error can be minimised by carefully choosing the size and position of the pupil projected onto the DM. In this experiment, the pupil is the interferometer mask. Figure 7(a) shows the variation of the fitting error for various interferometer mask sizes. The initial size is 110 by 110 pixels which corresponds to the whole 18 by 18 array of actuators [see also Fig. 7(b) for an illustration of the size and position of the mask onto the DM’s active area]. The mask size is decreased two pixels at a time with the phase screen being rescaled to maintain the initial phase property. The phase screen is not truncated but rescaled over a smaller number of pixels through an interpolation process. The D and r₀ are identical as well as the variance of the phase screen before and after rescaling, only the resolution varies. When the mask size decreases, the number of actuators available to reproduce the phase screen also decreases, and the fitting error increases as a result.

The smallest fitting error is reached when the edges of the mask are positioned at the centers of the actuators located at the edge of the 18 by 18 array [called the first outer actuator corona in Fig. 7(a)]. In this experiment, this corresponds to a mask of 106 by 106 pixels, represented by the red dashed line in Fig. 7(b). Because half of the actuators located on the first outer actuator corona are outside the pupil, the DM area used to correct the turbulent phase screen is now only 17 by 17 actuators.

The minimisation of the fitting error occurs by optimising the size of the projection of the entrance pupil on the DM’s active area and is critical for the design of all optical elements located upstream (between the entrance pupil and the DM).

Note that in Fig. 7(a), it appears that aligning the pupil to have approximately three quarter of the edge actuators in the pupil would generate the smaller fitting error while maximising the size of the pupil (which is desirous in order to maximise the DM’s spatial resolution).

4.4. Generation of the voltage command maps to be sent to the DM

Section 3.1 presents the SVR for each actuator and shows that it follows a quadratic pattern. Each actuator stroke-voltage plot can be fitted to a second order polynomial. Each polynomial fit provides two coefficients corresponding to the “offset” and the “gain” introduced in Eq. (1). To generate the voltage maps, we insert these coefficients into Eq. (1). This equation is then inverted and, for an actuator k, becomes:

where the stroke maps (thus the coefficients a_k) are estimated from the phase screens projection onto the normalised influence functions.

From this point forward, the voltage maps can be obtained following two slightly different paths. First, option A, the mean polynomial coefficients (for the offset and gain) from Eq. (1) are inserted in Eq. (6). Second, option B, each actuator’s individual polynomial coefficient is inserted in Eq. (6). The benefit (improvement in open-loop performance) of employing option B over option A is evaluated in the following sections.

The comparison of rms performance obtained with these two options will be presented in the following sections. Note that for a given phase screen, the variations between the voltage maps obtained with option A and option B are in the mV. Figure 5(d), presents the voltage map obtained with option A.

4.5. Statistical analysis of DM errors

Once the voltage maps are computed, the voltage commands are sent to the DM. The measured membrane deflections corresponding to the projection of the phase screens onto the mirror are denoted by φ^m. Two sets of data are taken, one with option A and one with option B. Figure 5(e), shows φ^m obtained with option A. Figure 5(f) shows the corresponding measurement or open-loop error.

 

Fig. 8: Histogram representation of the statistical study over 100 generated phase screens. (a) distribution of the original phase screen rms (vary from 56 nm to 155 nm), (b) distribution of the stroke map or a_k coefficients rms, (c), (d), (e) and (f) illustrate the variation between option A and B for the open-loop error rms and the DM error rms. “x-axis” represents the rms wavefront error (nm surface), “y-axis” represents the number of corresponding phase screens.

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The open-loop error corresponds to the difference between the original phase screen φ and the measurement phase screen φ^m, which is the projection of the original phase screen onto the DM. The DM error, corresponding to the difference between the fitted phase $\tilde{\phi }$ and the measurement phase screen φ^m, gives an estimate of the error due to the DM non-linear effects (inter-actuator mechanical coupling). See illustration Fig. 6.

For an initial mean rms phase screen of approximately 97 nm, the mean open-loop error is 16.5 nm rms, the mean fitting error is 13.3 nm rms and the mean DM error is 10.8 nm rms. The DM error is a critical value because it reflects the ability of this modelling approach to predict the shape of the DM. Note that for the 100 phase screens tested, option A and option B give approximately the same performance. Because option A is computationally less expensive that option B, it appears to be a more effective approach.

Figure 8 shows a histogram representation of the rms distribution over the 100 phases tested for the original phase screens, the stroke maps, the DM errors and the open-loop errors. The lack of significant improvement between option B and A is clearly visible.

An overview of this study is presented in Table 1 and Table 2. The values tabulated correspond to the mean value over the 100 phase screens generated and are given in nm rms.

Table 1:. Mean and standard deviation rms of the fitting errors. All values are given in nm.

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Table 2:. Mean and standard deviation rms of the measurement errors. All numbers are given in nm except for the ratio values given in %.

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5. Conclusion

In this paper, a low computational cost approach is used to control a MEMS DM in open-loop. This control strategy assumes that the non-additivites of influence functions, studied in a companion paper [7], are negligible in the case of a DM who’s shape matches a random Kolmogorov turbulence phase screen. This approach appears as a possible solution for imminent R&D in MOAO system design because of its promising results, its simplicity and its low computational cost.

By driving the DM in stroke command instead of voltage command, the relationship between the membrane deflection and the applied voltage, commonly assumed to be linear, is replaced by the calibrated quadratic relationship.

Promising results of low open-loop residual rms errors obtained with a 1024-actuator MEMS deformable mirror are shown. The characterisation process relies on both the thorough stroke-voltage relationship calibration and the actuator influence function measurements.

A statistical study over 100 phase screens is performed. With an optimised characterisation process, the residual open-loop error obtained is 17% and the residual DM error obtained is 11% (of the original phase screen rms). The test phase screens have a mean rms of approximately 97 nm, the mean open-loop error is 16.5 nm rms. With a mean fitting error of 13.3 nm rms, this brings the DM error to 10.8 nm rms. The DM error shows that the modelling approach used in this paper is good to 10.8 nm within the spatial frequency accessible to the DM.

The fitting error is also highlighted as a critical parameter for reducing the residual open-loop rms error. The projection of the entrance pupil onto the DM’s active area will impact the optical design upstream and need to be taken into account in the very early phase of the design. Section 4.3 experimentally shows that a minimised fitting error is obtained when the pupil edges are aligned with the centers of the actuators located at the edges of the DM’s active area.

The University of Victoria AO Lab is currently implementing the next step of this experiment which consists of running the MEMS DM in open-loop in a real-time control setup. The woofer-tweeter AO test bed currently in function was modified. It is now an hybrid AO bench with the tip-tilt and the woofer mirrors controlled in closed-loop while the MEMS DM is controlled in open-loop. The details and results of this experiment will be presented in an upcoming paper.