Software compensation method for achieving high stability of Alpao deformable mirrors

Urban Bitenc

doi:10.1364/OE.25.004368

1. Introduction

Adaptive optics (AO; first proposed by [1], for an overview see [2]) 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 our paper from 2014 [3] we discussed the importance of the stability of the DM surface for AO systems where the DM is used in a feed-forward operation [4, 5]. Unlike in the so-called closed-loop AO [6], the wavefront sensor does not observe the correction applied by the DM. Therefore it becomes crucial, that the relationship between the DM actuator commands and the DM surface shape is very well understood. One of the basic requirements is that for the same actuator commands the DM surface is repeatably the same and stable over long periods of time (in case of astronomical observations, over several hours). In microscopy [7] and ophthalmic applications [8,9], high stability is important for achieving optimal performance of wave-front sensor-less AO systems [10], as is for calibration and compensation of non-common-path aberrations in ophthalmic [11] or in astronomical [12] applications.

In [3] we performed detailed tests of a DM manufactured by Alpao. The DM has a continuous reflective surface, the shape of which is controlled by magnetic actuators. We observed excellent linearity and under certain circumstances also excellent stability and repeatability of DM surface shapes. However, we also observed a significant instability of the DM surface in situations where the DM surface has had some shape for several hours, and then this shape is changed. After the change, the DM surface exhibits a creep behavior for several hours. Creep is a slow and time delayed deformation of a material as a response to mechanical stress applied. This behavior has been observed by other researchers [12–14]. We presented a proof-of-concept software compensation for it, showing that very high stability is achievable.

In this paper we extend that proof-of-concept solution to a general use case. We demonstrate that by using this generalized creep compensation, the stability of the DM surface becomes satisfactory for a large variety of use cases.

We also report an observation of another source of surface instability of the DM. Our tests suggest that this is likely related to a thermal warming of the DM as a consequence of the electrical current flowing through the actuator coils. The resulting surface aberration can reach values up to 200 nm RMS. Similarly to creep, this effect is repeatable and predictable. We developed a software compensation for it and demonstrated that this compensation significantly improves the stability of the DM surface.

These compensation methods could be applicable also to other DMs of similar technology and with similar characteristics, for example from Imagine Eyes.

The rest of this paper is structured as follows. In Section 2 we describe the experimental setup used. In Section 3 we give details on the thermal effect and its compensation. In Section 4 we present the general creep compensation algorithm and demonstrate its performance. We conclude in Section 5.

2. Experimental setup

The deformable mirror we tested is a DM277 from the Alpao SaS company, which has 277 actuators in a 19 × 19 array with an actuator pitch of 1.5 mm and a pupil diameter of 24.5 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.

Recording a phase-map with the interferometer takes approximately 6 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 274 × 274 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 resolution of about 2–3 nm RMS, which we estimated from the difference of two consecutive phase-maps of the same surface.

2.1. Definitions

Throughout this paper we use bold capital letters, e.g. A, to denote actuator commands, which are 1-D arrays of 277 values. Interferometer phase-maps are denoted with the calligraphic capital letters, e.g. 𝒜. A central dot · in the equations represents the multiplication (and not the scalar product of vectors); while hence not strictly needed, it is added to improve readability. By “DM rest shape” we denote the shape of the DM surface when no electrical currents are running through actuator coils. This shape is reproduced by sending so called “-offset” commands to the DM actuators. The “offset” commands are provided by the manufacturer.

3. Thermal effect

We observed an effect which we interpret as a change of DM shape due to an increased internal temperature of the DM, which is a consequence of increased electric currents flowing through the actuator coils. This change of DM shape could be due to bi-morfal bending of the DM structure or a similar effect.

3.1. Observation and investigation of the thermal effect

The thermal effect is demonstrated in Fig. 1. The DM is left for several hours to stabilize in its rest shape, with very low currents running through the coils. The first three images show that the RMS of the phase-maps change by a few nano-meters in two minutes; a large part of that is the interferometer repeatability error. The fourth image shows the change in DM rest shape after random commands have been applied to all actuators for 2 minutes: the DM shape has changed by about 100 nm RMS. We observed and investigated this effect on two different DMs. The amount and the shape of the surface change were similar for both.

Fig. 1 Phase-maps demonstrating the shape and the amount of the thermal effect. The initial DM shape is subtracted from all four images to demonstrate the deviation from the initial shape. The left three images demonstrate the stability of the DM surface over 2 minutes if the actuator commands are kept constant. The fourth image exhibits the change in DM shape after random commands (evenly distributed within [−0.5, 0.5], changed once per second) have been applied to actuators for two minutes. The “ripples” seen in the phase-maps are the remnants of the interferometer fringes (measurement “noise”).

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We investigated the time-evolution of this effect. In Fig. 2, the first two minutes demonstrate the stability of the DM in its rest shape. Afterwards, we started applying random commands to all actuators for five minutes; they were evenly distributed in the range [−0.5, 0.5] and were changed several times per second. During these five minutes, we returned the DM to its rest shape for a few seconds every minute to measure its shape. The plot (points at t = 3,4,5,6,7 min) shows how the DM rest shape gradually changes; after about 3 minutes the DM stabilizes in its new shape and the curve reaches a plateau. The amplitude of the thermal effect is about 110 nm RMS in this case. At t = 7 min we stopped applying the random commands; we returned the DM to its rest shape and kept monitoring its evolution. The rest shape approaches its initial shape (to within 10 nm RMS) in about 2–3 minutes. The latter evolution of the DM shape is described well by an exponential curve with a time constant τ = 30.1 s.

Fig. 2 The temporal evolution of the thermal effect. The black line in the lower part shows typical actuator commands applied. The blue circles show the RMS of the surface change due to the thermal effect. The green line is a fit to the measured points. The red dots demonstrate the improvement in the DM surface stability, achieved by the software compensation described in the text.

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The amplitude of the thermal effect is given by the height of the plateau seen in Fig. 2. The height of this plateau depends on the amplitude of the random commands that caused the effect, as shown in Fig. 3. The value shown on the x-axis of this plot is the sum of squares of actuator commands, p:

p = \sum_{i = 1}^{N} {(c_{i} + o_{i})}^{2}

N is the number of the DM actuators and c_i is the command applied on the i-th actuator and o_i is the offset command for that actuator. The underlying reasoning is that (a) the electrical current through the actuator coil is proportional to the corresponding actuator command, and (b) that the dissipated power is proportionate to the electrical current squared.

Fig. 3 The amplitude of the thermal effect as a function of the electrical power dissipated in the actuator coils. The measured values (blue circles) are very well approximated by a linear function (line). Additional patterns of commands were tested to demonstrate that the thermal effect amplitude only depends on p: random commands multiplied by $\sqrt{2}$ but applied only on every other step (green x), an alternating constant added to all actuators (stars) and an alternating defocus term (+).

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For the majority of use-cases the amplitude of the thermal deformation will stay below 100 nm RMS. This includes a defocus mode spanning over half of the available actuator stroke, or 70% of the actuator stroke for coma. However, for a more extensive use of the DM, exploiting the full range of actuator stroke, the amplitude of the thermal deformation could reach 400 nm RMS.

Figure 3 exhibits an excellent linear relation between the amplitude of the thermal effect and the dissipated power p. The fitted line crosses the y-axis at the value of 0.21 nm; furthermore, over a range of 100 nm, the measurement points are typically within 0.2 nm RMS from the fitted line. This confirms the validity of our assumptions. Due to the resolution of the interferometer, the lowest points with RMS < 12 nm do not follow the straight line and are excluded from the fit.

Our tests show that the change in DM shape due to the thermal effect is repeatable within 3–4%. We performed tests applying other forms of actuator commands instead of random commands: adding a constant to all commands, varying the amplitude of random commands, adding a defocus term across the DM and changing the frequency of updating the commands. The thermal shapes observed were always very similar - within noise, and had similar amplitudes for the same p. The results shown in Fig. 3 demonstrate that the thermal effect is repeatable, independent of the details of the actuator commands (within 5%) and is well characterized by a single number, p, averaged over a few minutes.

Finally, in Fig. 2 we can see that at time t = 10 min, the rest shape still differs from the initial shape by about 10 nm RMS (blue points). We investigated the shape and the temporal evolution of this difference and both suggest that this is due to creep (Section 4). We estimate that up to about 10 nm of the RMS deviation shown in Fig. 2 could be due to this “second-order” creep effect. Since the contribution is so small and the results of the thermal compensation are satisfactory, we did not make further effort to disentangle both effects.

3.2. Compensation of the thermal effect

Based on the findings described above, we have developed a software compensation for the thermal effect. By monitoring the actuator commands over the past, the algorithm makes an assumption about the amount of the thermal effect. The algorithm then mitigates this effect by applying actuator commands that compensate for it.

Conceptually, the amplitude of the thermal effect is given by the following expression which integrates contributions over all the past:

P = \int_{- \infty}^{0} p (t) e^{t / τ} d t

Here, p(t) is the power dissipated in the DM at time t (see Eq. (1)) and τ is the time constant determined by fit shown in Fig. 2. Note that t < 0.

To implement Eq. (2) in software, we use the following approximation:

P_{i + 1} = e^{- Δ t / τ} \cdot P_{i} + (1 - e^{- Δ t / τ}) \cdot p (C_{i})

P_i+1 is calculated each time a user wants to apply new commands to the DM, the initial value being P₀ = 0. C_i are the commands that were last applied to the DM and Δt is the length of time since the last actuator commands were applied.

When user wants to apply new commands, C_requested, the actual commands sent to the DM, C_i+1, are obtained with the following formula:

C_{i + 1} = C_{requested} - \frac{P_{i + 1}}{P_{NORM}} \cdot C_{THERMAL}

C_THERMAL are actuator commands that reproduce the thermal shape which is shown in Fig. 1. P_NORM is the factor to normalize the intensity of the compensation; it should be equal to the p used when obtaining the shape for C_THERMAL, but can be tuned to achieve best performance.

We obtained C_THERMAL by recording the DM shape when activating each individual actuator, and solving the inverse problem of generating the thermal shape as a sum of individual actuator contributions.

The performance of the thermal compensation is demonstrated by the red line in Fig. 2. With the compensation, the change in DM shape stays within 15 nm RMS, compared to 110 nm RMS in the absence of compensation.

In summary, the inputs needed by the compensation are: τ, C_THERMAL and P_NORM. These quantities could be obtained for each DM as a part of its factory calibration after manufacturing. The compensation is implemented in such a way that the DM can be used without any knowledge of the details of it. The user only has to choose whether the compensation is applied or not.

4. Generalized creep compensation

In [3] we reported an observation of a creep effect and demonstrated a proof of concept for successfully compensating for it. Here we first shortly summarize those findings and then we develop a general creep compensation that performs well for a general use case.

If commands A have been applied to the DM for a long time (an hour or more) and then at time t = 0 (so called “flipping point”) the commands are changed to B, the DM shape will not be stable over the next hours. The DM membrane remembers shape 𝒜 and over the next hours the difference ℬ − 𝒜 gradually builds up on top of the initial shape ℬ. The effect is repeatable and can be compensated for by continuously adapting the actuator commands, following the equation

B (t) = B (0) - x (t) \cdot [B (0) - A] .

A are the actuator commands applied to the DM before the flipping point, B(0) are the actuator commands at t = 0 and x(t) is a time-dependent correction factor that needs to be determined with a calibration. B(t) are the time-dependent actuator commands applied to the DM after the flipping point; these actuator commands produce a stable DM shape ℬ.

In astronomical AO, new DM commands are applied at 100 Hz – 1kHz to correct for atmospheric aberrations. In such a case, A and B are DM commands averaged over a few seconds, as explained in [3], Sec. 4.1.

4.1. Algorithm

A general use of the DM is illustrated in Fig. 4. Commands B₀ are applied to the DM for a long time. Then, at t₀, commands B₁ are applied. Later, at t₁, commands B₂ are applied, and so on. Finally, at t_N−1, commands B_N are applied. In astronomical AO, B_i could be DM commands averaged over 1 minute, if the DM is correcting for quasi-static aberrations on top of the zero-average atmospheric turbulence. If the differences between these time-points (t_Bi = t_i − t_i−1) and the total time (t_N−1 − t₀) are all short (10 seconds or less), the intermediate shapes can be neglected and the final shape ℬ_𝒩 can be kept stable by applying Eq. (5) with commands B₀ and B_N. If t_Bi are all long (an hour or more), the previous shapes can be neglected and again Eq. (5) can be used to keep each of the shapes ℬ_i stable.

Fig. 4 An illustration of the creep compensated usage of the DM. Before t₀, the DM shape has stabilized at ℬ₀ over several hours, t_B0 → ∞. At point t₀ the user added ΔB_1,0 to the actuator commands to change the DM shape to ℬ₁. This shape is then maintained by the creep compensation mechanism for a length of time t_B1. At point t₁ the user added ΔB_2,1 to the actuator commands to change the DM shape to ℬ₂.

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In the intermediate case, where the differences t_Bi are of the order of minutes, the DM membrane remembers several previous shapes and it is essential to take all of them into account in a general creep compensation formula:

B_{N} (t) = B_{N} (t_{N - 1}) - \sum_{i = 0}^{N - 1} Δ x_{i, N} (t) \cdot [B_{N} - B_{i}] .

The terms used in Eq. (6) are explained in Table 1. This formula and its implementation make use of a small but helpful change in the way the user provides the input actuator commands. Traditionally, the user provides the DM with commands B₀, B₁, B₂, ..., B_N. Instead, in the implementation of the general creep compensation, the user provides the algorithm with the differences of these commands, i.e. B₁ − B₀, B₂ − B₁, ..., B_N − B_N−1. Initially, the user must also provide the actuator commands at which the DM surface has been stable for at least an hour, B₀.

Note the following relationship between x(t) and x(t, t_B):

x (t) \equiv x (t, t_{B} = \infty)

Table 1. Explanation of terms and symbols used in Eq. (6)

View Table

4.2. Calibration

The correction factor x(t, t_B) must be determined in advance by a calibration. In this section we first measure x(t) for the case t_B = ∞. Then we measure x(t, t_B) for a range of t_B values between 0.5 min and 20 min. Finally, we demonstrate that to a good approximation the x(t, t_B) values can be calculated from x(t). This fact greatly simplifies the implementation of the creep compensation and greatly simplifies the calibration procedure.

To simplify our study, we did not apply the thermal compensation during creep studies. Instead, we selected commands A and B such that they have a similar p (Eq. (1)). This reduced the impact of thermal effect on x(t, t_B) below 1%.

4.2.1. Calibration for x(t)

The calibration process for the case t_B = ∞ is described in [3], Sec. 4.3.1. We set DM commands to A for 6 hours. Then we change commands to B and - by iterating the process - extract correction factor x(t) that keeps DM shape ℬ(t) stable over the next 6 hours (see [3], Eq. (3)). The procedure is illustrated in the left plot of Fig. 5. This iterative procedure converges in 4 iterations, meaning that the calibration process takes 48 hours.

Fig. 5 An illustration of the DM shapes applied for the measurement of the calibration constants x(t) (left) and x(t, t_B) (right).

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The calibration result is shown in the upper plot of Fig. 6. The points measured are very well described by the following model:

x (t) = \frac{C t^{α}}{t^{β} + D} .

The free parameters are determined by a χ² fit: C = 0.201, α = 0.9025, β = 0.8918, D = 2.258.

Fig. 6 The creep correction factor x. Upper: x(t) measured over 6 hours from a single transition 𝒜 → ℬ. Lower: x(t, t_B) measured over 1 hour from a double transition 𝒜 → ℬ → 𝒜.

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4.2.2. Calibration for x(t, t_B)

The calibration procedure for a finite t_B is illustrated in the right plot of Fig. 5. First, commands A are applied for a long time (at least 4 hours) so that shape 𝒜 stabilizes. Then shape ℬ is applied for a length of time t_B; in order to keep this shape, the creep compensation is activated, meaning that commands B(t) are applied in accordance with Eq. (5). At t₁, shape 𝒜 is applied again with the creep compensation activated. Equivalently to Eq. (3) from [3], in an iterative procedure factors x(t, t_B) are extracted that keep 𝒜 constant for t > t₁:

x_{i + 1} (t, t_{B}) = x_{i} (t, t_{B}) + \frac{RMS [𝒜 (t, t_{B}) - ℬ]}{RMS [𝒜 (t_{1}) - ℬ]} - 1

The underlying reasoning is that any creep observed in the period t > t₁ is there because shape ℬ was applied for a length of time t_B.

Starting with x₀(t, t_B) = 0, the iterative procedure converges in four iterations. The results for a range of values are shown in the lower plot of Fig. 6. As expected, the longer t_B, the stronger the creep effect. With 4 hours for the stabilisation and 1 hour for the measurement of x(t, t_B), this calibration took 184 hours for 9 values of t_B and four iterations.

4.2.3. Relation between x(t) and x(t, t_B)

Mathematically, the creep compensation must perform identically for the following two cases:

transition 𝒜 → ℬ, using Eq. (5),
transition ℬ₀ → ℬ₁ → ℬ₂, where ℬ₁ = ℬ₀ = 𝒜, using Eq. (6).

In other words, if we provide the creep-compensating algorithm with a fake flipping point, where the DM shape does actually not change, Eq. (6) must still compensate for creep correctly. This leads to the following relationship between x(t, t_B) and x(t):

x (t, t_{B}) = x (t) + x (t_{B}) - x (t + t_{B})

We compare the x(t, t_B) values measured as described in Sec. 4.2.2 with values calculated from Eq. (10). Figure 7 shows that the agreement between the two is very good, within 3%. This excellent, highly non-trivial result gives confidence that our understanding of the DM creep is very good and that the creep compensation will perform well. This result also means that the lengthy calibration of x(t, t_B) can be avoided and it greatly simplifies the implementation of x(t, t_B) calculation for a general value of t_B.

Fig. 7 A comparison of x(t, t_B) obtained in two different ways. The full lines are calculated from Eqs. (10) and (8). The dots are measured in the calibration described in Sec. 4.2.2. They agree very well, within 3%.

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4.2.4. Dependency of x(t) on 𝒜 and ℬ

By repeating the calibration with several different shapes 𝒜 and ℬ, we found out that x(t) depends slightly on the shapes 𝒜 and ℬ. The x(t) values measured vary within 10%, but the shapes of the curve remain very similar. This means that for some combinations of shapes 𝒜 and ℬ, the creep compensation will perform very well, while for other combinations it may over-correct or under-correct the creep. If high accuracy is required and the DM shapes used are known in advance, we recommend adding a parameter Γ to Eq. (6): $B_{N} (t) = B_{N} (t_{N - 1}) - Γ \cdot \sum_{i = 0}^{N - 1} Δ x_{i, N} (t) \cdot [B_{N} \cdot B_{i}]$ . The user can tune Γ to achieve the best result for the particular DM shapes used. The default value of Γ will be 1.0 and the typical range will be between 0.9 and 1.1.

4.3. Usage and implementation considerations

Before the activation of the creep compensation, the DM is recommended to stabilize at B₀ for 2 hours or more; this time depends on the stability required and can be estimated from the x(t) curve, Fig. 6. The creep compensation can be activated without waiting after the DM has been turned on. In this case, the commands B₀ to be used in Eq. (6) are the commands −offsets(), provided by the manufacturer. These commands reproduce the DM shape in its rest state; the DM acquires this shape when switched off, or if switched on and no commands are applied.

The proof-of-concept creep compensation reported in [3], performed well only for times larger than 6 minutes after the flipping point; in the first 6 minutes the behavior was not understood. With the current study we confirmed that this limitation was entirely due to the thermal effect described in Sec. 3. After mitigating or avoiding the thermal effect, the creep compensation performs well for any t, including immediately after the flipping point.

An algorithm that would implement both compensations together, would have to first compensate for creep. The creep-compensated commands would be fed to the thermal compensation which would produce the commands to be sent to the DM. For even higher stability, a third step could be added to compensate for creep resulting from the thermal effect.

The implementation of the creep compensation algorithm is most practical if at the flipping points the user provides the difference between the current and the next DM shape, i.e. B_j+1 − B_j, rather than the DM shapes B_j+1, B_j.

4.4. Results

In the following we present the tests and the performance of the creep compensation algorithm. When testing the creep compensation algorithm, the thermal compensation was not activated. However, choosing commands B_j such that they all have a similar p, the impact of thermal effect was minimized. In the following plots, the contribution from the thermal effect is 3 nm RMS at the most.

For commands B_j in the following tests we used defocus, coma (both orientations) and spherical aberration with different amplitudes. The absolute amplitudes of the modes were between 0.05 and 0.07 of the maximum actuator stroke. For larger amplitudes, the interferometer could not reliably resolve the fringes. The differences between the shapes, ℬ_i − ℬ_j, yielded values between 332 nm RMS and 1058 nm RMS.

We performed tests with a varying number of flipping points. The most exhaustive test involved seven flipping points: ℬ₀ → ℬ₁ → ... → ℬ₆ → ℬ₇. To test repeatability, ℬ₁ and ℬ₆ were chosen to be the same in this test.

The DM was left to stabilize at ℬ₀ for 4 hours. Then each of the shapes ℬ₁, ℬ₂, ..., ℬ₆ was set to the DM for an equal amount of time, denoted as t_B. In the end, the shape of ℬ₇ was monitored for 60 minutes. The test was repeated several times, varying the t_B values in the range between 0.5 min and 20 min. Figure 8 shows the stability of this final shape, ℬ₇. Without creep compensation, the deviation in shape ℬ₇ reaches 110 nm RMS in 60 minutes. With the creep compensation, it stays withing 10 nm RMS.

Fig. 8 The stability of the final shape, ℬ₇, without creep compensation (upper) and with it (lower).

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Figure 9 demonstrates the effect of compensation in a similar way as Fig. 8, but for all the intermediate shapes, ℬ₁, ℬ₂, ..., ℬ₆ and only at times t_B. Without the creep compensation, these shapes creep between 90 and 200 nm RMS in 20 minutes. With the compensation, this is reduced to 8 – 13 nm RMS.

Fig. 9 The stability of the intermediate shapes in the test. Upper: without creep compensation and with it. Lower: with the creep compensation, i.e. the black curves from the upper plot are zoomed in.

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Figure 10 demonstrates the effect of creep on the repeatability of DM shapes. In our test we chose commands B₁ and B₆ to be the same. Without the creep compensation, the shapes ℬ₁ and ℬ₆ differ by 127 nm RMS in 100 minutes (red curve). With the compensation activated, this goes down to 12 nm RMS (blue curve). This and other test have demonstrated that the creep compensation guarantees a repeatable DM shape within 15 nm RMS over time periods of hours.

Fig. 10 The repeatability of DM shapes with and without the creep compensation.

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The results of these test demonstrate that the creep compensation guarantees a high stability of DM surface for a wide variety of use-cases. We conclude that the creep compensation normally corrects for 93% – 95% of the creep. In the worst of our tests it corrected for 85% of creep. This means that, for a 1000 nm RMS change in shape, the remaining instability will normally be between 12–17 nm RMS in the first 60 minutes, and in the worst case 36 nm RMS. Without creep compensation, the corresponding instability would be 240 nm RMS for a 1000 nm RMS change in shape.

5. Conclusions

We have tested the long-term stability of an Alpao deformable mirror, using an interferometer. We have observed two sources of surface instability. First, the thermal effect deforms the surface shape because the DM structure warms up due to the currents running through the actuator coils; the amplitude of the deformation is proportional to the sum of squared actuator commands, averaged over the last few minutes. Second, the DM surface creeps because the material has remembered the shapes that the DM surface had over the past few hours; we reported this in more detail in [3].

For realistic use cases the thermal effect can reach aberrations in the range 100 – 200 nm RMS. The creep effect can result in even higher aberrations, about 24% of the change in DM shapes, i.e. 240 nm RMS for each 1000 nm RMS.

We have developed and tested a software compensation for each of the effects. The compensation corrects for about 90% of the thermal effect. The creep compensation can correct for up to 95% of the creep, and in the worst case observed it corrected for 85% of the creep.

Given the excellent linearity and repeatability properties of the Alpao DM, together with its zero hysteresis and large actuator stroke, this gives an excellent candidate for open-loop (aka feed-forward) applications in adaptive optics. Compared to other DMs [15] it is perhaps the optimal choice currently available.

Funding

This work was financed in the United Kingdom by the Science and Technology Facilities Council through the Impact Acceleration Account and through grant ST/L00075X/1. It was also supported by FP7 OPTICON program, EC grant agreement No. 312430.

References and links

1. H. W. Babcock, “The possibility of compensating astronomical seeing,” PASP 65, 229 (1953). [CrossRef]

2. R. Davies and M. Kasper, “Adaptive optics for astronomy,” Annu. Rev. Astron. Astrophys. 50, 305–351 (2012). [CrossRef]

3. U. Bitenc, N. A. Bharmal, T. J. Morris, and R. M. Myers, “Assessing the stability of an ALPAO deformable mirror for feed-forward operation,” Opt. Express 22, Iss. 10, pp. 12438–12451 (2014). [CrossRef] [PubMed]

4. E. Gendron, F. Vidal, M. Brangier, T. Morris, Z. Hubert, A. Basden, G. Rousset, R. Myers, F. Chemla, A. Longmore, T. Butterley, N. Dipper, C. Dunlop, D. Geng, D. Gratadour, D. Henry, P. Laporte, N. Looker, D. Perret, A. Sevin, G. Talbot, and E. Younger, “MOAO first on-sky demonstration with CANARY,” Astron. Astrophys. 529, L2 (2011). [CrossRef]

5. O. Lardière, R. Nash, J.-P. Markes, D. Andersen, C. Bradley, C. Blain, R. Desmarais, D. Gamroth, M. Ito, K. Jackson, P. Lach, and L. Pham, “Final opto-mechanical design of Raven, a MOAO science demonstrator for Subaru,” Proc. SPIE 8447, 844753 (2012). [CrossRef]

6. B. L. Ellerbroek and C. R. Vogel, “Inverse problems in astronomical adaptive optics,” Inverse Probl. 25, 063001 (2009). [CrossRef]

7. C. Bourgenot, C. D. Saunter, G. D. Love, and J. M. Girkin, “Comparison of closed loop and sensorless adaptive optics in widefield optical microscopy,” J. Europ. Opt. Soc. Rap. Public. 8, 13027 (2013). [CrossRef]

8. K. S. K. Wong, Y. Jian, M. Cua, S. Bonora, R. J. Zawadzki, and M. V. Sarunic, “In vivo imaging of human photoreceptor mosaic with wavefront sensorless adaptive optics optical coherence tomography,” Biomed. Opt. Express 6(2), 580–590 (2015). [CrossRef] [PubMed]

9. D.J. Wahl, Y. Jian, S. Bonora, R.J. Zawadzki, and M.V. Sarunic, “Wavefront sensorless adaptive optics fluorescence biomicroscope for in vivo retinal imaging in mice,” Biomed. Opt. Express 7(1), 1–12 (2016). [CrossRef] [PubMed]

10. M. Booth., “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. express , 14(4), 1339–1352 (2006). [CrossRef] [PubMed]

11. Y. N. Sulai and A. Dubra, “Non-common path aberration correction in an adaptive optics scanning ophthalmoscope,” Biomed. Opt. Express 5(9), 3059–3073 (2014). [CrossRef] [PubMed]

12. M. Lamb, D. R. Andersen, J.-P. Véran, C. Correia, and O. Lardière, “Calibrating the non-common path aberrations on the MOAO system RAVEN and first science results using RAVEN,” AO4ELT4 - Conference Proceedings, https://doi.org/10.20353/K3T4CP1131535 (2016).

13. A. Zepp, S. Gładysz, and K. Stein, “Holographic wavefront sensor for fast defocus measurement,” Adv. Opt. Techn. 2(5–6), 433–437 (2013).

14. Private communication with Andreas Zepp, Fraunhofer IOSB, Germany.

15. A. Kellerer, F. Vidal, E. Gendron, Z. Hubert, D. Perret, and G. Rousset, “Deformable mirrors for open-loop adaptive optics,” Proc. SPIE 8447, 844765 (2012). [CrossRef]

B_N(t)	actuator commands that will at time t generate DM shape ℬ_N
B_N(t_N−1)	= B_N−1(t_N−1) + ΔB_N,N−1, the initial value of the actuator commands that at time t_N−1 give shape ℬ_N ; ΔB_N,N−1 is user input
B_N − B_i	$\sum_{j = i}^{N - 1} Δ B_{j + 1, j}$ , the commands that compensate for creep
ΔB_j+1,j	user input actuator commands, B_j+1 − B_j
Δx_i,N (t)	x(t − t_i, t_Bi) − x(t_N−1 − t_i, t_Bi)
x(t − t_i, t_Bi)	Correction factor determined with a calibration. It represents the amount of creep at time t − t_i if the DM had been at shape ℬ_i for t_Bi.
t_i	the point in time when the DM surface was changed from ℬ_i to ℬ_i+1
t_Bi = t_i − t_i−1	the length of time the DM surface had shape ℬ_i.

Software compensation method for achieving high stability of Alpao deformable mirrors

Abstract

1. Introduction