1. Introduction

Adaptive optics (AO) corrects for aberrations that are introduced when light propagates through a medium. The aberrations are measured by a wavefront sensor and corrected by a deformable mirror (DM) in real-time. AO is used in astronomy, microscopy, telecommunications, military applications and elsewhere to improve performance.

In conventional AO the arrangement of components corresponds to optical feedback where the wavefront sensor measures the effect of the DM on the aberrations. This is described as closed-loop AO [1, 2]. Since the effect on aberrations is constantly monitored, such an arrangement can accommodate an inaccurate device whose other characteristics can be considered as more important. For example, with piezo-electric actuators, hysteresis is known to cause inaccuracy [3], whereas with MEMS devices there are actuator stroke limitations that result in inter-actuator coupling and non-linear surface response although there is no comparably significant hysteresis [4].

There are now alternative forms of AO being designed for astronomical instrumentation where the DM is operated in a feed-forward situation [5, 6]. In this scenario the wavefront sensor does not observe the correction applied by the DM. For feed-forward or ‘open-loop’ AO, it therefore becomes necessary to have an accurate device for wavefront aberration compensation. Alternatively one can obtain accuracy of the surface shape by introducing additional monitoring of the surface. This monitoring can be internal, where each actuator is controlled using position feedback [7], or external via a figure sensor [8]. Either choice is more costly and complex as it introduces one or more additional sensors, potentially another light source which must be isolated from other optical sensors, and is an additional burden on the real-time control system.

There are other applications where a unique response of the DM surface to actuator commands is crucial. Fast wavefront sensing attempts sensing magnitudes of selected Zernicke modes with frequencies up to 1 MHz. One technique [9] uses a DM in the production of holograms, for calibration and for testing. It is important that the DM at all times uniquely reproduces selected Zernike modes and their superpositions [10].

The main purpose of this work is to test the stability (accuracy) of an ALPAO deformable mirror (DM241-25) as currently sold by the ALPAO SaS company of France, over timescales of several hours. This is the most important property of a DM to be used in a feed-forward operation. We also investigate certain aspects of the linearity and repeatability of the DM, but we do not study hysteresis. The DM has a continuous reflective surface, the shape of which is controlled by magnetic actuators. General characteristics of this type of DM are large stroke, good linearity and low hysteresis [11]. However, due to the presence of a polymer in the mechanical link between the actuators and the reflective surface, the mirror surface may be prone to creep [12]. Creep is a slow and time delayed deformation of a material as a response to mechanical stress applied.

We demonstrate how feed-forward control of the ALPAO DM can be achieved using a priori calibration measurements that maintain a particular surface shape within a few nanometers of root-mean-square (RMS) over several hours. This allows the use of such a mirror in open-loop AO over long integration times with minimal error in the PSF due to intrinsic mirror instability.

2. Experimental setup

The deformable mirror we tested is a Hi-Speed-DM241-25 from the ALPAO SaS company, which has 241 actuators in a 17 × 17 array with an actuator pitch of 2.5 mm and a pupil diameter of 37.5 mm. We monitored its surface using a Fizeau interferometer (PTI-02 from Zygo), which was interfaced using Trioptics μShape software v5.6. The DM was controlled using Python and a Python module SUDS was used for communication with the interferometer μShape software.

The DM was mounted in front of the interferometer at a distance of about 20 cm. To improve the robustness of the recorded phase-maps we placed a linear polarizer between the interferometer and the DM. The polarizer deforms the wavefront slightly by inducing a defocus term with RMS of about 20 nm. Although this would not affect the conclusions of this paper, it was calibrated out for all relevant measurements presented, by using a flat mirror as a reference.

Recording a phase-map with the interferometer takes approximately 7 seconds. The piston and the tip-tilt terms are subtracted from all retrieved phase maps before further analysis. The interferometer phase-maps are 2-D arrays of about 102 000 integer values in units of Å and measure the mirror surface shape. (The wavefront error resulting from an imperfect DM surface will be twice the surface imperfection.) They exhibit an intrinsic RMS resolution of about 6–8 nm, which we estimated from the difference of two consecutive phase-maps of the same surface. For the results given in section 4.3.2, we re-binned the phase-maps by averaging over squares of 20 × 20 phase-map values. This increased the resolution to about 2–3 nm.

The interferometer beam and the polarizer aperture were smaller than the DM (33 mm vs. 37.5 mm), hence we could only monitor about 75% of the DM surface. The impact of DM edge effects has not been investigated within this paper, but we do not believe they impact the conclusions presented here that describe a fully-controlled section of the DM.

We used a temperature and humidity logger Testo-175-H2 to monitor the environmental conditions. The logger was positioned on the bench at a distance of about 15 cm from the DM. The conditions are monitored only, and not controlled. The temperature range during the measurements presented in this paper was between 21°C and 26.5°C, and the relative humidity was between 28% and 55%.

2.1. Definitions

Throughout this paper we use bold capital letters, e.g. A, to denote actuator commands, which are 1-D arrays of 241 values. Interferometer phase-maps are denoted with the calligraphic capital letters, e.g. 𝒜.

3. Stability, repeatability and linearity

The DM exhibits excellent surface stability over time under certain conditions, and excellent repeatability and linearity of surface response to changed actuator commands. These properties are crucial for the performance of the correction procedure described in section 4. In this section we present some of the measurements we performed to test them.

3.1. Stability over time

We tested the stability of the DM by performing the following experiment. We flattened the DM with an iterative procedure, using interferometer phase-maps as feedback to adapt the actuator commands. Once this procedure converged, we stored the actuator commands obtained, F. We then monitored the DM shape over the next hours: every 6 minutes we set the actuator commands to F for about 7 seconds to take a phase-map of the DM surface. Between these measurements we added random values to F with a frequency of about 100 Hz to simulate the real use of the DM. The random values were evenly distributed in the range [−0.05, 0.05] (full range of valid actuator commands is [−1.0, 1.0]), which superimposed a shape with peak-to-valley (PTV) of about 550–850 nm (RMS 110–190 nm) on top of the existing DM shape.

We repeated this test several times. The top blue line in Fig. 1 shows the RMS of the DM surface shape against time for the longest of the experiments. The RMS changed by about 15 nm in 113 hours. During the experiment the temperature varied by 2.9°C and the relative humidity by 10.8%. As one can see from different features of the blue lines in Fig. 1, both have an effect on the shape of the DM surface.

 

Fig. 1 The blue curve at the top shows the RMS of the DM surface after it had been flattened, monitored over several days. For the red curve the DM was replaced with a flat mirror to observe the impact of the measurement method. The temperature and the relative humidity are plotted underneath for both measurements.

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To investigate the impact of instability within the experimental setup on the observed variation, the test was repeated using a calibration flat mirror in place of the DM. The red line in Fig. 1 shows that the RMS change of the blue line is to a large degree due to the DM surface and not due to instrumental drift.

3.2. Repeatability of surface response to actuator commands

We demonstrate the repeatability of DM surface response to actuator commands with the following experiment. We set actuator commands to a set of values A for 6 hours. Then we set actuator commands to a different set of values B. We recorded the shape of the DM surface before and after the change. We repeated these measurements several times over the period of 13 days. Between some of the measurements we performed other tests with the DM. We found that the RMS values of the corresponding DM shapes 𝒜 and ℬ varied by about 20 nm between different measurements and they seem to depend on the history of DM before each measurement.

However, the RMS of the difference between shape 𝒜 and shape ℬ was found to be repeatable, as shown in Fig. 2. The RMS of the difference is about 343 nm and covers a range of 6.4 nm. A large amount of this spread is caused by variations in the ambient temperature (2.9°C): one can see a clear correlation between temperature and measured RMS. The Pearson correlation coefficient is 0.95 and the rate of change is 1.9 nm/°C. From this result we estimate that with stable temperature and relative humidity the RMS value would vary by less than 1 nm (0.3%).

 

Fig. 2 RMS of the difference between shapes 𝒜 and ℬ after changing the actuator commands from A to B. The measurement was performed 14 times to test repeatability. One can see a correlation between the RMS values and the temperature. Unlike in Fig. 1, the effect of relative humidity (RH, different marker styles) is not clear.

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3.3. Linearity of surface response to actuator commands

To test the linearity of surface response to actuator commands the same two sets of actuator commands A and B were used as in section 3.2. We set the actuator commands to B + w(A − B) for 6 hours, then we set the actuator commands to B and recorded the phase-maps of the DM surface before and after the change. We performed this test for different values of parameter w. Figure 3 shows the RMS of the difference between the shapes before and after the change of actuator commands. The data exhibit excellent linearity. We fit the data points with a straight line, RMS = k · w + n. The fit yields an offset n = 0.03 nm, which is consistent with zero. The average residual is 0.34 nm and the plotted residuals do not exhibit any structure.

 

Fig. 3 Linear response of the DM surface to actuator commands. See text for definition of w. The data points are fitted with a straight line. The fit parameter n agrees with 0 and the residuals shown underneath exhibit no structure. The temperatures when taking these data was between 21.8°C and 24.6°C. Measurements for w = ±1.0 were repeated several times.

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4. Creep behavior

We have shown in Fig. 1 that over long periods of time a fixed set of actuator commands repeatedly results in DM surface shapes that are very similar to each other. Next we performed an experiment to show that this only holds once the DM surface has stabilized. We tested the surface stability after a significant change of actuator commands.

4.1. Observing surface creep

We set actuator commands to A for 6 hours. Then we changed actuator commands to B and monitored the shape of the DM over the next 6 hours. The corresponding surface shapes are denoted by 𝒜 and ℬ. The moment when actuator commands are changed from A to B is called “the flipping point” for convenience.

Shapes 𝒜 and ℬ are dominated by low-order spatial modes and are described in more detail in Sec. 4.3.2. This experiment simulates two situations one encounters in practice. The first one is turning the DM on and starting using it. The shape of the DM surface when used will be different from the shape when the DM is switched off, because the equilibrium surface shape (the mirror rest shape) for this type of DM is not flat. The second situation appears when the DM compensates for non-common path errors within an AO system. When these aberrations change (e.g. due to environmental conditions) and the system is re-calibrated, the time-averaged shape of the DM surface also changes to adapt to the altered aberrations. By “time-averaged” we mean averaging over a few seconds, in which the high frequency commands correcting for e.g. atmospheric turbulences would average out.

The monitoring of the DM shape after the flipping point was exactly the same as described in Sec. 3.1: every 6 minutes we set the actuator commands to B and took a phase-map of the DM surface with the interferometer; between the measurements we added random commands with a frequency of about 100 Hz to B to simulate the real use of the DM.

Figure 4 shows a typical example of the RMS of the surface shape after the flipping point. One can see that the surface shape is not stable but changes significantly with time.

 

Fig. 4 The RMS of the DM surface shape changes by 80–90 nm in 6 hours, exhibiting a creep-like behavior of the DM surface. The three curves were obtained as described in the text, using the same actuator commands A and B to test the repeatability. Their shapes are rather similar but they demonstrate some dependency on the environmental conditions. The solid lines represent the results of a model fitted to the data points.

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We observed this behavior for many different choices of actuator commands A and B and in several other experiments similar to the one described here. We conclude that if the actuator commands are kept constant at values A for a long time (several minutes or more) and are then changed to values B, the shape of the DM surface after the change will not be constant but will drift away from the initial shape ℬ. This also holds if we add random values to A or B at 100 Hz such that A and B are the time-averaged actuator commands.

4.2. Understanding the details of surface creep

It is useful to define the following observable to quantify the creep:

Quantity q is a relative measure for the amount of creep: it quantifies the change of shape ℬ compared to the difference between shapes 𝒜 and ℬ. The green curve in Fig. 6 (labeled as “calculated x₁(t)”) shows q(t) for the data presented in Fig. 4.

We measured q for several different choices of actuator commands A and B and found that changing either of them can have a significant impact on q. For different choices of A and B, we see the largest variations in q just after the flipping point. At the first measurement after the flipping point the quotient q(t = 6 min) spans the range between 0.05 and 0.18. Afterwards the amount of creep becomes similar for all cases and in the next six hours q increases by another 0.13 to 0.17. The total change in q in several cases exceeded the value of 0.3 which would seriously affect the performance of an AO system using such a DM in open loop.

To further investigate the nature of this creep behavior we performed the following experiment. We set the actuator commands to A for 6 hours to stabilize. We then flattened the DM in an iterative process of 9 steps that, for technical reasons, took about 16 minutes rather than an expected ≈70 seconds. Afterwards we monitored the DM shape over the next 6 hours in the same way as described earlier in this section. The DM shapes before flattening, just after it was flattened and 6 hours later are shown in Fig. 5. One can see that the DM shape 6 hours after flattening (Fig. 5(c)) is very similar to the DM shape before flattening (Fig. 5(a)), just with an opposite sign and with a smaller amplitude. We performed this experiment for several different actuator commands A and observed similar behavior for all of them.

 

Fig. 5 Phase-map of the DM surface before flattening (a), immediately after flattening (b) and six hours after flattening (c). This demonstrates the key property of the surface creep: the final surface shape (c) is similar to the shape before flattening (a) but reversed in sign.

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4.3. Compensating for creep

The amount of creep we observe would seriously affect the performance of an AO system. The DM surface only stabilizes several hours after a significant change. We attempted shortening the stabilisation time by applying a series of interchangeably overshooting and undershooting steps when approaching commands B, a technique commonly used to cure hysteresis. This was not successful and this supports the concept of physical creep.

4.3.1. Developing the compensation procedure

We showed that the DM surface response to the change of actuator commands is very repeatable (Sec. 3.2) and that the creep behavior is also very repeatable and hence predictable (Fig. 4). This makes it possible to compensate in software for the creep behavior of the DM surface, by applying adapted actuator commands during the stabilisation period of 6 hours after the flipping point.

The result presented in Fig. 5 suggests that the DM surface can be kept stable by adding a fraction x of the difference A − B to actuator commands B. Therefore we construct the creep compensation in the following form:

The correction factor x(t) is obtained with an iterative procedure. The first iteration’s correction factor, x₁(t), equals q(t) as defined in (1). We repeat the experiment described in 4.1 but with adapting the actuator commands B(t) according to (2). As the actuator commands are being adapted, the material experiences a different force and the actual creep differs from the one measured with constant B. Hence the creep compensation using the first iteration’s x(t) is over-correcting. One measures the residual changes in surface shape and calculates the next iteration’s correction factors, using the following formula:

Index i denotes the iteration number and q_i(t) are calculated according to (1) from phase-maps obtained with correction factors x_i(t). The first iteration starts with x₀(t) = 0.0 for all t (i.e. no correction).

Figure 6 shows the convergence of x_i(t) in three iterations. We anticipated the over-correction in the first iteration and to speed up the convergence we modified the calculated x₁(t) by making an estimate (blue line). The second iteration’s correction factors (red curve) were used for all the creep compensated results presented in this paper. The third iteration’s correction factors (black line) are also plotted to show convergence.

When compensating for creep we adapt the actuator commands every 6 minutes, just before taking a phase-map of the DM surface. One might wish to adapt them more often or even “continuously”, in which case one would extract x(t) between the measured points by interpolating or from a model fitted to the measured points.

 

Fig. 6 Three iterations of correction factor x_i(t) as defined in (3). The second iteration’s correction factors (red curve) were used for all the creep compensated results presented in this paper. The third iteration’s correction factors (black line) are plotted to show convergence. After t = 3 hours the red and the black curve start diverging due to different environmental conditions.

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Note that this compensating procedure only depends on A (the time-averaged actuator commands before t = 0), on B (the new actuator commands at t = 0) and on the precalculated correction factors x(t). No feedback from the interferometer about the current shape of the DM surface is used when compensating for the creep; in other words, the creep compensation is applied in a completely open-loop manner, using an a priori calibration.

4.3.2. Testing the performance and limitations of the compensation procedure

We implemented the creep compensation and again performed the experiment described in 4.1. We repeated it seven times with the same actuator commands A and B and with the same correction factors x(t) to investigate repeatability and the effects of environmental conditions. Figure 7 shows the results of four out of these seven measurements. The curves are obtained by subtracting the phase-map obtained at t = 6 min from all other phase-maps of that measurement and plotting the RMS of the difference against time. This quantity is much more sensitive to changes of the DM surface shape than the RMS of the surface phase-maps themselves.

 

Fig. 7 Stability of the DM surface with the creep compensation in place. The reference shape is taken at t = 6 min. All curves were obtained with the same commands A and B that were used to extract x(t) and they demonstrate excellent repeatability. The blue curve shows the result in the most varying environmental conditions.

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Figure 7 shows that in the first 6 min after the flipping point the RMS of surface deviation is about 10 nm. In the next 2 hours the RMS of the surface deviation in most cases stays below 4 nm, out of which 2–3 nm are due to the interferometer resolution; in some cases the deviation reaches 6 nm RMS, which we think is due to the environmental conditions. In the uncompensated case, the corresponding surface deviations are 43 nm RMS in the first 6 min after the flipping point and 47 nm RMS in the next 2 hours. For times longer than 2–3 hours after the flipping point the surface deviation is governed by the environmental conditions rather than by the quality of the creep compensation.

Our observations suggest that in a controlled environment (a temperature change less than 0.3°C) and with the correction factor x(t) of a third or fourth iteration, the achievable surface stability after the first 6 min is below 1 nm RMS. Our data also suggest that one would need to add further terms to Eq. (2) if one needed to improve the stability in the first 6 min after the flipping point. However, rather than pursuing the highest achievable surface stability for this particular case of commands A and B, we investigated how useful the proposed compensation technique is in practice.

As described in 4.2, the amount of creep depends on both initial and final actuator commands, A and B. To achieve the best possible correction one would have to prepare the correction factors x(t) for each possible pair of commands A and B that will appear in practice. This is not always possible or practical. We investigated how well the correction factors x(t) compensate for creep if we use them with commands A and B which differ from those that x(t) were extracted from.

By A₀ and B₀ we denote the actuator commands that were used to extract the correction factors. Commands A₀ were obtained by negating the output of function GetOffset(), provided by the manufacturer. They generate the shape the DM obtains when switched off; the shape is shown in Fig. 5(a). Commands B₀ flatten the DM when the surface has stabilized. Apart from A₀ and B₀ we performed tests with the following shapes:

Table 1. Some results of the creep compensation procedure. The change of shape ℬ by 2 hours after the flipping point is presented, as afterwards the change of shape is dominated by environmental conditions which were not controlled. For a few cases we performed the measurement without creep compensation for comparison - these results are given in parentheses.

View Table

We find that the creep compensation with the non-modified x(t) works well in all these cases. For t > 6 min, it compensates for 90% of the creep such that RMS[ℬ(t) − ℬ(6 min)] stays within 16 nm for the first two hours after t > 6 min and below 20 nm by the end of the 6-hour period. For t < 6 min we observe a bigger change in RMS[ℬ], up to 28 nm in 6 minutes. This agrees with the behavior of q(t), described in 4.2: for different choices of A and B the amount of creep differs significantly in the first few minutes after the flipping point, but becomes rather similar later on. (Note that we only monitored the DM surface every 6 minutes; it is likely that the period of big differences in creep is shorter than that.) The results are repeatable and only slightly dependent on environmental conditions. We checked that swapping the initial and final actuator commands (i.e. A = B₀, B = A₀) gives the correction of the same quality as the original commands (i.e. A = A₀, B = B₀). During these tests we noticed that setting the mean value of A and B to 0.0 improves the performance of the creep compensation.

When measuring the correction factor x(t), we set commands A₀ on the DM for 6 hours before the flipping point. In this time the DM surface stabilizes at shape 𝒜₀ - to some extent. In practice this time will be shorter or longer and the drift observed after the flipping point will be slightly different. We repeated the creep compensating test, but setting actuator commands A to A₀ for 1h, 2h, 4h and 12h before the flipping point, rather than 6h, and observed the residual changes in surface change. We find that the differences in creep are small; after t = 6 min the surface changes by at most 10 nm RMS by the end of the 6-hour period.

5. Efficient use of an ALPAO DM in practice

5.1. Obtaining DM surface stability over a long period of time

As described in 4.2 and 4.3.2, the amount of creep depends largely on A and B in the first few minutes after the flipping point and much less afterwards. Apart from that we have shown that even with optimal x(t), in the first 6 min the creep compensation performs slightly worse than afterwards. Hence we recommend the following procedure, which makes it possible to use a single set of correction factors x(t) efficiently in practice:

5.2. Repeatably obtaining the same DM shapes with the same actuator commands

An application mentioned in section 1 uses a DM to generate different superpositions of Zernike shapes; a measurement with each of them typically takes a few seconds. Between these measurements the DM is not used for several minutes [10]. It is important that actuator commands Z_i always result in the same surface shape 𝒵_i and the creep-like behavior of the DM can present a problem.

In such a case, in the few seconds during the measurements the surface shape 𝒵_i will creep only insignificantly and it is most likely not necessary to compensate for that. However, to guarantee repeatability we recommend setting the actuators always to the same values P in between the measurements, when the DM is not required to have a certain shape. As we have demonstrated in section 3.2, actuator commands Z_i will repeatably result in the same surface shape 𝒵_i if the DM surface has stabilized at shape 𝒫. If the periods of actuator commands Z_i are limited to a few seconds, the stabilized shape 𝒫 will not be significantly affected.

Before starting a set of measurements, the DM must be set to P for at least 2 hours to stabilize. One can avoid this waiting time after the DM has been switched on, by choosing P as the negated output of function GetOffset() provided by the manufacturer. These commands reproduce the DM rest shape at which the surface had stabilized while the DM was switched off.

This recipe is only to overcome the effects of the creep-like behavior of the DM. Other effects may also affect repeatability, but this was not studied within the scope of this paper.

6. Conclusions

We have investigated the surface shape stability of a Hi-Speed-DM241-25 deformable mirror from the ALPAO SaS company. After the actuator commands have been kept constant for several hours at values A, the surface shape is stable and its RMS changes by less than 1 nm per hour. If the actuator commands are changed to values B at a point in time (so called flipping point), the DM surface shape afterwards is not stable but exhibits creep-like behavior, drifting away from the shape it had just after the flipping point. The surface reaches its final shape only several hours later. The change in surface RMS due to this creep can exceed 30% of the RMS of the difference in shape before and after the flipping point. This also holds if a high-frequency signal is added to actuator commands such that A and B are the time-averaged (over a few seconds) values.

We investigated this creep-like behavior for A and B that exhibit only low-order spatial modes such as those that would be expected to compensate for non-common path errors within the AO system.

The deformation of the DM surface in time is repeatable and hence predictable. We demonstrate how to compensate for the deformation in software. We continuously adapt the actuator commands after the flipping point in such a way that they counterbalance the deformations of the DM surface.

This technique requires a time-dependent correction factor x(t) that has to be measured in advance, using an interferometer or a wavefront sensor. It depends slightly on A and B and on the environmental conditions. Best results are achieved if the correction factor is used to compensate for creep with the same A and B as it was extracted from. In this case, for the example of actuator commands used in this paper, the compensated surface shape changes by 10 nm RMS in the first minutes after the flipping point and remains stable within 4 nm RMS afterwards. Using the same x(t) with different A and B, the residual change of DM surface shape is within 28 nm RMS in the first minutes after the flipping point, and within 20 nm RMS for the rest of the 6-hour period.

The tests presented were performed on a single example of ALPAO DM. Due to very similar production process the creep compensation procedure is probably applicable to other examples [12]. The numerical values for x(t) used in this paper are available from the authors if the reader wants to try them out with their own DM.

Using the creep compensation procedure presented in this paper results in significantly improved stability of the DM surface after a change of (time-averaged) actuator commands. Without the compensation it takes several hours for the surface to stabilize.

ALPAO think that the creep is related to the presence of polymer in the mechanical link between the actuators and the mirror surface. They are working on the replacement of the polymer with material less subject to creep [12].