Modal dynamics of magnetic-liquid deformable mirrors Denis Brousseau,* Simon Thibault, Ermanno F. Borra, and Simon F. Boivin Center for Optics, Photonics and Lasers (COPL), Laval University, Quebec G1V 0A6, Canada *Corresponding author: [email protected] Received 3 April 2014; revised 20 June 2014; accepted 20 June 2014; posted 23 June 2014 (Doc. ID 209482); published 23 July 2014
Magnetic-liquid deformable mirrors (MLDMs) were introduced by our group in 2004 and numerous developments have been made since then. The usefulness of this type of mirror in various applications has already been shown, but experimental data on their dynamics are still lacking. A complete theoretical modeling of MLDM dynamics is a complex task because it requires an approach based on magnetohydrodynamics. A purpose of this paper is to present and analyze new experimental data of the dynamics of these mirrors from open-loop step response measurements and show that a basic transfer function modeling is adequate to achieve closed-loop control. Also, experimental data on the eigenmodes dynamic is presented and a modal-based control approach is suggested. © 2014 Optical Society of America OCIS codes: (220.1080) Active or adaptive optics; (230.4040) Mirrors. http://dx.doi.org/10.1364/AO.53.004903
1. Introduction
Magnetic-liquid deformable mirrors (MLDMs) are deformable mirrors based on the actuation of the surface of a magnetic liquid (ferrofluid) by small current carrying coils, also referred to as actuators. MLDMs are not the only type of deformable mirror based on the actuation of a liquid. A deformable mirror based on the principle of total internal reflection of light from an electrostatically deformed liquid–air interface is presented in [1]. In the case of a MLDM, the shape that the surface of the liquid takes is determined by the addition of the component vectors of the magnetic field produced by each actuator [2]. Up until recently, this vectorial attribute proscribed the use of standard control algorithms in which the surface of the deformable mirror is predicted through the linear addition of the actuators’ influence functions. A way to counter this problem was introduced in [3] and experimentally validated in [4]. The linear superposition behavior of the influence functions is achieved by using a large and uniform magnetic field that is superimposed to the magnetic field produced by the MLDM actuators. Predicting the surface 1559-128X/14/224903-07$15.00/0 © 2014 Optical Society of America
produced by MLDMs is thus simpler and it was possible to demonstrate their usefulness in various applications where the high stroke they can achieve is put to practice. For example, using a MLDM to produce tunable reflective axicons is presented in [5] and a MLDM as a telescope simulator for optical subcomponents testing is presented in [6]. However, our experiments with MLDMs have shown that, while keeping a static shape on their surface was easy to implement, changing the surface at an optimized frequency while maintaining stability was challenging. A simple theotretical approach to the dynamics of an MLDM based on the harmonic oscillator was discussed in [7], and a more detailed theoretical development appears in [3] where the MLDM is considered as a multiple-input multipleoutput (MIMO) system with the currents applied to the actuators as inputs and the surface deflections as outputs. However, experimental data on MLDM dynamics are still lacking and are needed to investigate optimal control approaches. Furthermore, it appears that an extensive mathematical description, while useful, is not as crucial as gathering experimental data to improve MLDM control approaches in a simple manner. In this paper, we present new data on the dynamical behavior of MLDMs using actuator open-loop step response measurements. A basic PID 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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control approach based on a simple transfer function model of the MLDM is presented. Also, the relation between MLDM spatial frequencies and dynamical response is considered through open-loop step response measurements of the MLDM eigenmodes. We then introduce the idea that, using a basic transfer function model for each MLDM mode, a modal-based controller could be implemented with the result of an overall improved performance while using a simple dynamical description of the mirror. 2. Experimental Setup
The MLDM experimental setup is shown in Fig. 1 and is composed of the following principal components: a light source (SRC), a MLDM, and a wavefront sensor (WFS). Other optical elements, such as lenses (L1 to L5), beam splitters (BS1 and BS2), diaphragms, and fold mirrors (FM1 to FM3), are used to magnify, conjugate, and direct the light. The setup was designed to be customizable. For example, the FM2 location and pupil size have been chosen so that it could be replaced by a 97-actuator ALPAOSAS DM to perform dual deformable mirror experiments. A description of each of the main components is given below: Source: A Blue Sky Research fiber coupled laser source operating at a wavelength of 635 nm and having a maximum output power of 25 mW. MLDM: The one used here has 216 1.5-mm diameter coils arranged on a 16 × 16 square grid with a pitch of 2.5 mm (see Fig. 2). The coils are smaller than the ones used in previous studies and, apart from the modification to a square layout instead of a hexagonal one, the distance between the coils is about the
Fig. 1. MLDM setup showing the coupled laser source (SRC) operating at 635 nm, the MLDM, the wavefront sensor (WFS), and the relay optics. 4904
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Fig. 2. MLDM actuator array shown with the 120 active actuators and the 27.5-mm diameter pupil overlaid in red.
same as in previous MLDMs. Simulations have shown that both the hexagonal and square layout of actuators results in similar performance but, from an electromechanical standpoint, the square layout is easier to fabricate. Because the setup is used for other research work with already matched pupil relays, only the first 120 coils (identified in red in Fig. 2) can be fitted within the existing pupil that corresponds to a 27.5 mm diameter at the MLDM. The remaining coils of the MLDM are all inactive and set to a default current value of zero. Each resin-coated coil consists of roughly 300 turns of 0.13-mm diameter magnet wire wound on a 1-mm diameter brass core. The coils are supplied in a current by an ALPAO SAS electronic box. The current in the MLDM actuators is controlled using the electronic box through the ALPAO Core Engine (ACE) software. The external magnetic field that linearizes the response of the actuators is produced by a Maxwell bobbin [8]. The actuator array is placed within the Maxwell bobbin with the top of the actuators lying near the middle plane of the Maxwell bobbin form, a position where the magnetic field produced by the bobbin is uniform. A container filled with a layer of commercial ferrofluid is put down on top of the coils. The current within the Maxwell bobbin can be increased from 0 to 2.5 A using an external constant current source and the typical driving current is 500 mA. A fold mirror is placed on top of the Maxwell bobbin to redirect the optical axis of the setup to the ferrofluid surface. Ferrofluid: A commercially available EFH1 lowviscosity ferrofluid from Ferrotec (USA) Corp. is used. EFH1 ferrofluid has a density ρ of 1210 kg∕m3, a viscosity υ of 6 mPa · s, a surface tension σ of 29 mN/m, and a relative magnetic permeability μ of 2.6. Wavefront sensor: The WFS is a Shack–Hartmann made by Optocraft GmbH. The lenslet array consists
of 35 × 35 microlenses having a 150-μm pitch and the pupil fills 17 × 17 subapertures at the MLM. The maximum frame rate of the WFS is close to 400 Hz and a default value of 100 Hz is used (see Subsection 3.B). The ferrofluid container has a radius of 37.5 mm resulting, when using EFH1 ferrofluid, in a Bond number (B0 ρgR2 ∕σ) of 575, which indicates that the mirror operates in a regime dominated by gravity waves. The applied magnetic field also has an influence on the ferrofluid dispersion relation but the magnetic field values at which we operate the MLDM are lower than what has been identified to produce an impact to the ferrofluid dispersion relation [9]. Moreover, a 1-mm thickness layer of ferrofluid is used and intensity measurements made with a highspeed photodiode, when driving a coil at increasing frequencies, have shown no resonant peaks at natural sloshing frequencies of a liquid confined to a cylindrical container. Resonant peaks were only observed when the ferrofluid layer thickness was equal or larger than 2 mm. The fact that the action of the actuators is confined to a diameter of 27.5 mm while the container has a diameter of 75 mm, together with the fact that we did not observe resonant peaks at ferrofluid layer thicknesses below 2 mm, makes us conclude that there is no direct link between vibration modes and the MLDM eigenmodes. 3. Results A.
MLDM Eigenmodes
The influence function of each of the 120 actuators was recorded using the ALPAO ACE software. Like in previous studies [2], the influence function of an actuator can still be well represented by a Gaussian function although the exact FWHM of the profile can vary according to the diameter of the actuator and the ferrofluid layer thickness. Figure 3 shows the experimental eigenvalues of the singular value decomposition (SVD) for the MLDM when the 120 actuators identified in Fig. 2 are used. It can be seen
Fig. 3. MLDM eigenvalues. Mode numbers superior to 30 are discarded due to their low gain relative to the maximum gain.
that mode numbers higher than 30 have gains lower than 1% and to prevent saturating the actuators and amplify noise, mode numbers superior to 30 were discarded. Having to discard a large number of modes is a consequence of the tighter actuator spacing on this particular MLDM resulting in a higher overlap of the influence functions. Figure 4 shows modes 1–30 in WFS space. B. Actuator Open-Loop Step Response
A theoretical approach to the dynamics of an MLDM based on the simple harmonic oscillator is discussed in [7], and a more detailed theoretical development appears in [3]. The experimental results presented in [7] were obtained by measuring the displacement on a position-sensitive device (PSD) of a laser beam reflected from the surface of a MLDM. Although it provides valuable experimental data on the dynamical response of MLDMs at high recording frequency, the PSD method has its drawbacks: since the measurement is made on a single point of the mirror surface, the recorded response signal is related only to the dynamics of the slope at this very specific position of the MLDM surface. Also, considering that it is very difficult to determine exactly the location where the laser beam hits the surface of the MLDM, and the large lever arm on the PSD setup, the calibration and interpretation of the PSD data in terms of amplitude are rendered highly difficult. Although well theoretically detailed, the analytical treatment presented in [3] where the Navier–Stokes equations are used to describe the MLDM dynamics lacks in terms of experimental results. In the subsequent text, we present the experimental open-loop response of the MLDM to different step input commands to complement our knowledge of the MLDM dynamics. Figure 5 displays the measured open-loop step response of the MLDM when a command is sent to the central actuator. The blue curve shows the measured wavefront peak-to-valley (PV), while the red curve
Fig. 4. MLDM eigenmodes in WFS space. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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Fig. 5. Open-loop step response of the MLDM when a command is sent to the central actuator. The blue curve is the measured wavefront PV and the red curve is the mean of the x and y slope maximums as a function of time. Amplitude is given in normalized arbitrary units. The sampling frequency is 100 Hz.
shows the mean of the x and y slope maximums as a function of time. The WFS frame rate is set to 10 ms following a rule of thumb of 10% of the ∼100 ms typical response time of the MLDM. As noted in [2], the influence function of a MLDM actuator can be well described by a Gaussian function. Therefore, measuring the wavefront slope maximum as a function of time provides information about the evolution of the standard deviation of the influence function while the wavefront PV measure provides information about the influence function amplitude with time. In the figure, the overshoot peak of the wavefront slope maximum exceeds by about 1% the steadystate value while the overshoot peak of the wavefront PV is less than the steady-state value by about 3.5%. Also, the overshoot peak of the wavefront slope maximum is reached near 60 ms while the overshoot peak of the wavefront PV is reached 40 ms later at about 100 ms. A careful examination of the wavefront shape evolution during the first 100 ms confirms that the influence function amplitude continues to increase after its width has reached a minimum and then starts back to increase. This suggests that the hydrodynamic flow is still moving ferrofluid from the area surrounding the actuator to the volume of the influence function just above the driven actuator. Figure 6 displays the measured open-loop step response of the MLDM when a command is sent to the central actuator and its two closest neighbors are driven with an inverse command (reverse current) at 0.5× the command of the central actuator. The blue curve on the graph shows the measured wavefront PV while the red shows the mean of the x and y slope maximums as a function of time. Here it can be seen that, although the overshoot has increased, the steady-state value is reached faster and the 4906
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Fig. 6. Open-loop step response of the MLDM when a command is sent to the central actuator and its closest two neighbors are driven with an inverse command of 0.5× the command of the central actuator. The blue curve shows the measured wavefront PV while the red marker shows the mean of the x and y slope maximums as a function of time. Amplitude is given in normalized arbitrary units. The sampling frequency is 100 Hz.
wavefront PV progression matches the one of the maximum slope. The difference with the preceding measure implies that there is dynamical coupling taking place between the actuators. This effect would need to be taken into account in MLDM control approaches. C.
MLDM Transfer Function Model
A basic transfer function model for the MLDM was established using Matlab fminsearch unconstrained nonlinear optimization function and the experimental data of the wavefront PV open-loop step response of a single actuator. The overshoot peak amplitude being less than the steady-state value and, considering it takes about 1 s to reach it, we assumed that the model is a sum of a first- and a second-order transfer function: H MLDM s g1
ω20 1 ; (1) g2 2 1 sT s 2ζω0 s ω20
where g1 and g2 are the gains of each transfer function, T is the time constant of the first-order transfer function, and ω0 and ζ are, respectively, the natural frequency and the damping constant of the secondorder transfer function. The best fit parameters Table 1.
Best Fit Parameters of MLDM Transfer Function Model
Parameter g1 T g2 ω0 ζ
Value 0.135 0.3136 s−1 0.865 37.792 rad/s 0.5749
Fig. 7. Modeled open-loop step response of the MLDM central actuator based on transfer function of Eq. (1).
found are given in Table 1, and Fig. 7 shows the simulated step response based on this modeled transfer function in comparison to the actual recorded step response. The transfer function has one real pole at −3.19, two conjugate complex poles at −21.7 30.9j and −21.7 − 30.9j, and two zeroes at −3.63 and −2918. All poles have negative real parts assuring system stability. D.
Basic Closed-Loop PID Control
The basic closed-loop control of the MLDM is done using the velocity form of the standard PID controller [10]: ut ut − 1 K c et − et − 1
K c Δt et Ti
K cTd et − 2et − 1 et − 2; Δt
(2)
where ut is the actuator command vector, et is the actuator command error vector, Δt is the WFS frame rate, and K c , T i , and T d are the controller gain, integral time, and derivative time, respectively. The open-loop transfer function of the setup is then H open H wfs H delay H PID H DAC H MLDM , where H wfs is the WFS transfer function modeled as an integrator, H delay is the WFS computational delay, H PID is the PID controller, and H DAC is the digital-toanalog converter modeled as a zero-order hold. A first-order low-pass filter having a 7 Hz cutoff frequency is applied to the command error of the derivative term in the PID controller to prevent noise amplification. The final command output has a simple antiwindup scheme to prevent actuator saturation where ut is set to the maximum absolute possible command in case the computed command exceeds this maximum. The values of K c 0.6, T i 0.024, and T d 0.02 were found using Matlab and entered to the MLDM controller code but it was apparent that the gain K c of 0.6 was too large and made the loop become unstable right at the start. After fine-tuning the controller parameters directly
Fig. 8. Wavefront RMS amplitude on the MLDM as a function of time at start of closing the loop using the basic PID controller. The closed-loop frequency is 97 Hz.
on the experimental setup, optimized values of K c 0.36, T i 0.028, and T d 0.022 were found to give the best results. Figure 8 shows the wavefront rms amplitude when the loop is closed with these parameters on the setup static aberrations and room turbulence. The closed-loop frequency is 97 Hz. E. Modal Open-Loop Step Response
The results presented in Subsections 3.B and 3.D point to the fact that the dynamical response of the MLDM is closely related to the spatial frequency of the surface produced. The overshoot seen in Fig. 6 where three neighboring actuators are driven at the same time in push-pull mode caused the need to lower the loop gain from 0.6 to 0.36 in Subsection 3.D. The open-loop step responses of all the MLDM modes were recorded to better understand the link between its spatial and dynamic behavior. For this, the command vector of each eigenmode (eigenvectors) was sent as a step input to the MLDM and the response recorded using the WFS at a sampling frequency of
Fig. 9. Wavefront PV open-loop step responses of MLDM for modes 1–12. Amplitude is given in normalized arbitrary units. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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Fig. 10. Open-loop step response of MLDM mode 30. The blue curve is the measured wavefront PV and the red curve is the mean of the x and y slope maximums as a function of time. Amplitude is given in normalized arbitrary units.
100 Hz. Figure 9 presents the step responses for modes 1–12 while Fig. 10 presents the 30th mode. The overshoot is seen to increase with mode number for lower-order modes while the slow first-order transfer function component slowly decreases within the first 10 modes. It appears from these results that the basic PID controller based on the open-loop step response of a single actuator, introduced and tested in Subsection 3.D, is underestimating the impact of the overshoot seen in some of the higher-order modes of the MLDM. That is why it was necessary to lower the gain of the controller on the experimental setup. 4. Conclusion
It was shown that a basic PID controller based on the open-loop step response of a single actuator was adequate to close the loop on the MLDM. A first set of optimal parameters for the PID controller was found by first using a simple transfer function model for the MLDM. The parameters were then fine-tuned on the setup because the PID controller gain K c was too large and made the loop to become unstable. A careful examination of the open-loop step response of each of the MLDM modes made apparent that some higher-order modes exhibit a larger overshoot than what is seen on the open-loop step response of a single actuator. When the controller gain K c is lowered from 0.6 to 0.36 and the changes on the other parameters remain minor, the loop can be successfully closed on the MLDM. The impact of the overshoot seen in the higher-order modes is removed when the gain is lowered. The optimal parameter set found on the experimental setup forms a compromise that takes care of the differences between the step responses of the modes. The modes presented in Fig. 4 form a base to describe the wavefronts that the MLDM can produce. When we consider also using the dynamical response 4908
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of these modes, the surface of the MLDM could be described within the controller as composed of dynamic eigenmodes [11]. Each mode could then be changed independently using its individual PID controller based on the transfer function of that mode. The computational load of having a controller based on the modes should not pose a problem considering the speed of the MLDM response. Such a controller could also be optimized to track specific modes. We plan to implement such a controller on a MLDM. This will require to model the transfer function of each mode and also to optimize the controller parameters for each of these modes as well. It is obvious that modes that are symmetrical (e.g., modes 1 and 2) will share the same transfer function and controller parameters. The principal task will be to evaluate if such a modal-based controller will significantly improve or not the performance of the MLDM compared to the set of parameters we found as a compromise. A feed-forward controller could also be used to improve the open-loop transfer function of the MLDM modes. This idea will be put forward when a proper model of the modes is developed and we learn more about the repeatability of the MLDM response. The present paper concentrates on obtaining new and important data on MLDM dynamics using a commercial ferrofluid without considering how the ferrofluid choice and other implements could change the MLDM dynamics. For example, unpublished initial results on laying a thin membrane on a custom ferrofluid, both chemically made by chemists within our group, seem to indicate that the open-loop step response steady state could be reached about 5 times faster without significantly altering the shape of the influence function although requiring a somewhat increased command signal. Another approach being investigated is using overdriving techniques, similar to what is being used in LCD technology, to lower the time required to reach the open-loop steady-state value. Initial results indicate that combining the use of a membrane and overdriving could allow a tenfold improvement on the time required to reach the steady-state value of the open-loop step response. The authors would like to thank Raytheon ELCAN Optical Technologies for manufacturing the MLDM used in this paper and ALPAO SAS which graciously made modifications to their Core Engine software for connecting to the MLDM. This work has been supported by grants from Natural Sciences and Engineering Research Council (NSERC). References 1. E. S. ten Have and G. Vdovin, “Characterization and closedloop performance of a liquid mirror adaptive optical system,” Appl. Opt. 51, 2155–2163 (2012). 2. D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express 15, 18190–18199 (2007). 3. A. Iqbal and F. B. Amara, “Modeling of a magnetic-fluid deformable mirror for retinal imaging adaptive optics systems,” Int. J. Optomechatron. 1, 180–208 (2007).
4. D. Brousseau, E. F. Borra, M. Rochette, and D. BouffardLandry, “Linearization of the response of a 91-actuator magnetic liquid deformable mirror,” Opt. Express 18, 8239–8250 (2010). 5. D. Brousseau, J. Drapeau, M. Piché, and E. F. Borra, “Generation of Bessel beams using a magnetic liquid deformable mirror,” Appl. Opt. 50, 4005–4010 (2011). 6. S. Thibault, D. Brousseau, and E. F. Borra, “Liquid deformable mirror for advanced sub-optical system testing,” Proc. SPIE 7739, 773910 (2010).
7. J. Parent, E. F. Borra, D. Brousseau, A. M. Ritcey, J.-P. Déry, and S. Thibault, “Dynamic response of ferrofluidic deformable mirrors,” Appl. Opt. 48, 1–6 (2009). 8. J. Clerk-Maxwell, Treatise on Electricity and Magnetism (Clarendon, 1873). 9. F. Boyer and E. Falcon, “Wave turbulence on the surface of a ferrofluid in a magnetic field,” Phys. Rev. Lett. 101, 244502 (2008). 10. A. Visioli, Practical PID Control (Springer, 2006). 11. T. Ruppel, “Model-based feed forward control of large deformable mirrors,” Eur. J. Control 3, 261–272 (2011).
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Magnetic-liquid deformable mirrors (MLDMs) were introduced by our group in 2004 and numerous developments have been made since then. The usefulness of this type of mirror in various applications has already been shown, but experimental data on their dynamics are still lacking. A complete theoretical modeling of MLDM dynamics is a complex task because it requires an approach based on magnetohydrodynamics. A purpose of this paper is to present and analyze new experimental data of the dynamics of these mirrors from open-loop step response measurements and show that a basic transfer function modeling is adequate to achieve closed-loop control. Also, experimental data on the eigenmodes dynamic is presented and a modal-based control approach is suggested. © 2014 Optical Society of America OCIS codes: (220.1080) Active or adaptive optics; (230.4040) Mirrors. http://dx.doi.org/10.1364/AO.53.004903
1. Introduction
Magnetic-liquid deformable mirrors (MLDMs) are deformable mirrors based on the actuation of the surface of a magnetic liquid (ferrofluid) by small current carrying coils, also referred to as actuators. MLDMs are not the only type of deformable mirror based on the actuation of a liquid. A deformable mirror based on the principle of total internal reflection of light from an electrostatically deformed liquid–air interface is presented in [1]. In the case of a MLDM, the shape that the surface of the liquid takes is determined by the addition of the component vectors of the magnetic field produced by each actuator [2]. Up until recently, this vectorial attribute proscribed the use of standard control algorithms in which the surface of the deformable mirror is predicted through the linear addition of the actuators’ influence functions. A way to counter this problem was introduced in [3] and experimentally validated in [4]. The linear superposition behavior of the influence functions is achieved by using a large and uniform magnetic field that is superimposed to the magnetic field produced by the MLDM actuators. Predicting the surface 1559-128X/14/224903-07$15.00/0 © 2014 Optical Society of America
produced by MLDMs is thus simpler and it was possible to demonstrate their usefulness in various applications where the high stroke they can achieve is put to practice. For example, using a MLDM to produce tunable reflective axicons is presented in [5] and a MLDM as a telescope simulator for optical subcomponents testing is presented in [6]. However, our experiments with MLDMs have shown that, while keeping a static shape on their surface was easy to implement, changing the surface at an optimized frequency while maintaining stability was challenging. A simple theotretical approach to the dynamics of an MLDM based on the harmonic oscillator was discussed in [7], and a more detailed theoretical development appears in [3] where the MLDM is considered as a multiple-input multipleoutput (MIMO) system with the currents applied to the actuators as inputs and the surface deflections as outputs. However, experimental data on MLDM dynamics are still lacking and are needed to investigate optimal control approaches. Furthermore, it appears that an extensive mathematical description, while useful, is not as crucial as gathering experimental data to improve MLDM control approaches in a simple manner. In this paper, we present new data on the dynamical behavior of MLDMs using actuator open-loop step response measurements. A basic PID 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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control approach based on a simple transfer function model of the MLDM is presented. Also, the relation between MLDM spatial frequencies and dynamical response is considered through open-loop step response measurements of the MLDM eigenmodes. We then introduce the idea that, using a basic transfer function model for each MLDM mode, a modal-based controller could be implemented with the result of an overall improved performance while using a simple dynamical description of the mirror. 2. Experimental Setup
The MLDM experimental setup is shown in Fig. 1 and is composed of the following principal components: a light source (SRC), a MLDM, and a wavefront sensor (WFS). Other optical elements, such as lenses (L1 to L5), beam splitters (BS1 and BS2), diaphragms, and fold mirrors (FM1 to FM3), are used to magnify, conjugate, and direct the light. The setup was designed to be customizable. For example, the FM2 location and pupil size have been chosen so that it could be replaced by a 97-actuator ALPAOSAS DM to perform dual deformable mirror experiments. A description of each of the main components is given below: Source: A Blue Sky Research fiber coupled laser source operating at a wavelength of 635 nm and having a maximum output power of 25 mW. MLDM: The one used here has 216 1.5-mm diameter coils arranged on a 16 × 16 square grid with a pitch of 2.5 mm (see Fig. 2). The coils are smaller than the ones used in previous studies and, apart from the modification to a square layout instead of a hexagonal one, the distance between the coils is about the
Fig. 1. MLDM setup showing the coupled laser source (SRC) operating at 635 nm, the MLDM, the wavefront sensor (WFS), and the relay optics. 4904
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Fig. 2. MLDM actuator array shown with the 120 active actuators and the 27.5-mm diameter pupil overlaid in red.
same as in previous MLDMs. Simulations have shown that both the hexagonal and square layout of actuators results in similar performance but, from an electromechanical standpoint, the square layout is easier to fabricate. Because the setup is used for other research work with already matched pupil relays, only the first 120 coils (identified in red in Fig. 2) can be fitted within the existing pupil that corresponds to a 27.5 mm diameter at the MLDM. The remaining coils of the MLDM are all inactive and set to a default current value of zero. Each resin-coated coil consists of roughly 300 turns of 0.13-mm diameter magnet wire wound on a 1-mm diameter brass core. The coils are supplied in a current by an ALPAO SAS electronic box. The current in the MLDM actuators is controlled using the electronic box through the ALPAO Core Engine (ACE) software. The external magnetic field that linearizes the response of the actuators is produced by a Maxwell bobbin [8]. The actuator array is placed within the Maxwell bobbin with the top of the actuators lying near the middle plane of the Maxwell bobbin form, a position where the magnetic field produced by the bobbin is uniform. A container filled with a layer of commercial ferrofluid is put down on top of the coils. The current within the Maxwell bobbin can be increased from 0 to 2.5 A using an external constant current source and the typical driving current is 500 mA. A fold mirror is placed on top of the Maxwell bobbin to redirect the optical axis of the setup to the ferrofluid surface. Ferrofluid: A commercially available EFH1 lowviscosity ferrofluid from Ferrotec (USA) Corp. is used. EFH1 ferrofluid has a density ρ of 1210 kg∕m3, a viscosity υ of 6 mPa · s, a surface tension σ of 29 mN/m, and a relative magnetic permeability μ of 2.6. Wavefront sensor: The WFS is a Shack–Hartmann made by Optocraft GmbH. The lenslet array consists
of 35 × 35 microlenses having a 150-μm pitch and the pupil fills 17 × 17 subapertures at the MLM. The maximum frame rate of the WFS is close to 400 Hz and a default value of 100 Hz is used (see Subsection 3.B). The ferrofluid container has a radius of 37.5 mm resulting, when using EFH1 ferrofluid, in a Bond number (B0 ρgR2 ∕σ) of 575, which indicates that the mirror operates in a regime dominated by gravity waves. The applied magnetic field also has an influence on the ferrofluid dispersion relation but the magnetic field values at which we operate the MLDM are lower than what has been identified to produce an impact to the ferrofluid dispersion relation [9]. Moreover, a 1-mm thickness layer of ferrofluid is used and intensity measurements made with a highspeed photodiode, when driving a coil at increasing frequencies, have shown no resonant peaks at natural sloshing frequencies of a liquid confined to a cylindrical container. Resonant peaks were only observed when the ferrofluid layer thickness was equal or larger than 2 mm. The fact that the action of the actuators is confined to a diameter of 27.5 mm while the container has a diameter of 75 mm, together with the fact that we did not observe resonant peaks at ferrofluid layer thicknesses below 2 mm, makes us conclude that there is no direct link between vibration modes and the MLDM eigenmodes. 3. Results A.
MLDM Eigenmodes
The influence function of each of the 120 actuators was recorded using the ALPAO ACE software. Like in previous studies [2], the influence function of an actuator can still be well represented by a Gaussian function although the exact FWHM of the profile can vary according to the diameter of the actuator and the ferrofluid layer thickness. Figure 3 shows the experimental eigenvalues of the singular value decomposition (SVD) for the MLDM when the 120 actuators identified in Fig. 2 are used. It can be seen
Fig. 3. MLDM eigenvalues. Mode numbers superior to 30 are discarded due to their low gain relative to the maximum gain.
that mode numbers higher than 30 have gains lower than 1% and to prevent saturating the actuators and amplify noise, mode numbers superior to 30 were discarded. Having to discard a large number of modes is a consequence of the tighter actuator spacing on this particular MLDM resulting in a higher overlap of the influence functions. Figure 4 shows modes 1–30 in WFS space. B. Actuator Open-Loop Step Response
A theoretical approach to the dynamics of an MLDM based on the simple harmonic oscillator is discussed in [7], and a more detailed theoretical development appears in [3]. The experimental results presented in [7] were obtained by measuring the displacement on a position-sensitive device (PSD) of a laser beam reflected from the surface of a MLDM. Although it provides valuable experimental data on the dynamical response of MLDMs at high recording frequency, the PSD method has its drawbacks: since the measurement is made on a single point of the mirror surface, the recorded response signal is related only to the dynamics of the slope at this very specific position of the MLDM surface. Also, considering that it is very difficult to determine exactly the location where the laser beam hits the surface of the MLDM, and the large lever arm on the PSD setup, the calibration and interpretation of the PSD data in terms of amplitude are rendered highly difficult. Although well theoretically detailed, the analytical treatment presented in [3] where the Navier–Stokes equations are used to describe the MLDM dynamics lacks in terms of experimental results. In the subsequent text, we present the experimental open-loop response of the MLDM to different step input commands to complement our knowledge of the MLDM dynamics. Figure 5 displays the measured open-loop step response of the MLDM when a command is sent to the central actuator. The blue curve shows the measured wavefront peak-to-valley (PV), while the red curve
Fig. 4. MLDM eigenmodes in WFS space. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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Fig. 5. Open-loop step response of the MLDM when a command is sent to the central actuator. The blue curve is the measured wavefront PV and the red curve is the mean of the x and y slope maximums as a function of time. Amplitude is given in normalized arbitrary units. The sampling frequency is 100 Hz.
shows the mean of the x and y slope maximums as a function of time. The WFS frame rate is set to 10 ms following a rule of thumb of 10% of the ∼100 ms typical response time of the MLDM. As noted in [2], the influence function of a MLDM actuator can be well described by a Gaussian function. Therefore, measuring the wavefront slope maximum as a function of time provides information about the evolution of the standard deviation of the influence function while the wavefront PV measure provides information about the influence function amplitude with time. In the figure, the overshoot peak of the wavefront slope maximum exceeds by about 1% the steadystate value while the overshoot peak of the wavefront PV is less than the steady-state value by about 3.5%. Also, the overshoot peak of the wavefront slope maximum is reached near 60 ms while the overshoot peak of the wavefront PV is reached 40 ms later at about 100 ms. A careful examination of the wavefront shape evolution during the first 100 ms confirms that the influence function amplitude continues to increase after its width has reached a minimum and then starts back to increase. This suggests that the hydrodynamic flow is still moving ferrofluid from the area surrounding the actuator to the volume of the influence function just above the driven actuator. Figure 6 displays the measured open-loop step response of the MLDM when a command is sent to the central actuator and its two closest neighbors are driven with an inverse command (reverse current) at 0.5× the command of the central actuator. The blue curve on the graph shows the measured wavefront PV while the red shows the mean of the x and y slope maximums as a function of time. Here it can be seen that, although the overshoot has increased, the steady-state value is reached faster and the 4906
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Fig. 6. Open-loop step response of the MLDM when a command is sent to the central actuator and its closest two neighbors are driven with an inverse command of 0.5× the command of the central actuator. The blue curve shows the measured wavefront PV while the red marker shows the mean of the x and y slope maximums as a function of time. Amplitude is given in normalized arbitrary units. The sampling frequency is 100 Hz.
wavefront PV progression matches the one of the maximum slope. The difference with the preceding measure implies that there is dynamical coupling taking place between the actuators. This effect would need to be taken into account in MLDM control approaches. C.
MLDM Transfer Function Model
A basic transfer function model for the MLDM was established using Matlab fminsearch unconstrained nonlinear optimization function and the experimental data of the wavefront PV open-loop step response of a single actuator. The overshoot peak amplitude being less than the steady-state value and, considering it takes about 1 s to reach it, we assumed that the model is a sum of a first- and a second-order transfer function: H MLDM s g1
ω20 1 ; (1) g2 2 1 sT s 2ζω0 s ω20
where g1 and g2 are the gains of each transfer function, T is the time constant of the first-order transfer function, and ω0 and ζ are, respectively, the natural frequency and the damping constant of the secondorder transfer function. The best fit parameters Table 1.
Best Fit Parameters of MLDM Transfer Function Model
Parameter g1 T g2 ω0 ζ
Value 0.135 0.3136 s−1 0.865 37.792 rad/s 0.5749
Fig. 7. Modeled open-loop step response of the MLDM central actuator based on transfer function of Eq. (1).
found are given in Table 1, and Fig. 7 shows the simulated step response based on this modeled transfer function in comparison to the actual recorded step response. The transfer function has one real pole at −3.19, two conjugate complex poles at −21.7 30.9j and −21.7 − 30.9j, and two zeroes at −3.63 and −2918. All poles have negative real parts assuring system stability. D.
Basic Closed-Loop PID Control
The basic closed-loop control of the MLDM is done using the velocity form of the standard PID controller [10]: ut ut − 1 K c et − et − 1
K c Δt et Ti
K cTd et − 2et − 1 et − 2; Δt
(2)
where ut is the actuator command vector, et is the actuator command error vector, Δt is the WFS frame rate, and K c , T i , and T d are the controller gain, integral time, and derivative time, respectively. The open-loop transfer function of the setup is then H open H wfs H delay H PID H DAC H MLDM , where H wfs is the WFS transfer function modeled as an integrator, H delay is the WFS computational delay, H PID is the PID controller, and H DAC is the digital-toanalog converter modeled as a zero-order hold. A first-order low-pass filter having a 7 Hz cutoff frequency is applied to the command error of the derivative term in the PID controller to prevent noise amplification. The final command output has a simple antiwindup scheme to prevent actuator saturation where ut is set to the maximum absolute possible command in case the computed command exceeds this maximum. The values of K c 0.6, T i 0.024, and T d 0.02 were found using Matlab and entered to the MLDM controller code but it was apparent that the gain K c of 0.6 was too large and made the loop become unstable right at the start. After fine-tuning the controller parameters directly
Fig. 8. Wavefront RMS amplitude on the MLDM as a function of time at start of closing the loop using the basic PID controller. The closed-loop frequency is 97 Hz.
on the experimental setup, optimized values of K c 0.36, T i 0.028, and T d 0.022 were found to give the best results. Figure 8 shows the wavefront rms amplitude when the loop is closed with these parameters on the setup static aberrations and room turbulence. The closed-loop frequency is 97 Hz. E. Modal Open-Loop Step Response
The results presented in Subsections 3.B and 3.D point to the fact that the dynamical response of the MLDM is closely related to the spatial frequency of the surface produced. The overshoot seen in Fig. 6 where three neighboring actuators are driven at the same time in push-pull mode caused the need to lower the loop gain from 0.6 to 0.36 in Subsection 3.D. The open-loop step responses of all the MLDM modes were recorded to better understand the link between its spatial and dynamic behavior. For this, the command vector of each eigenmode (eigenvectors) was sent as a step input to the MLDM and the response recorded using the WFS at a sampling frequency of
Fig. 9. Wavefront PV open-loop step responses of MLDM for modes 1–12. Amplitude is given in normalized arbitrary units. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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Fig. 10. Open-loop step response of MLDM mode 30. The blue curve is the measured wavefront PV and the red curve is the mean of the x and y slope maximums as a function of time. Amplitude is given in normalized arbitrary units.
100 Hz. Figure 9 presents the step responses for modes 1–12 while Fig. 10 presents the 30th mode. The overshoot is seen to increase with mode number for lower-order modes while the slow first-order transfer function component slowly decreases within the first 10 modes. It appears from these results that the basic PID controller based on the open-loop step response of a single actuator, introduced and tested in Subsection 3.D, is underestimating the impact of the overshoot seen in some of the higher-order modes of the MLDM. That is why it was necessary to lower the gain of the controller on the experimental setup. 4. Conclusion
It was shown that a basic PID controller based on the open-loop step response of a single actuator was adequate to close the loop on the MLDM. A first set of optimal parameters for the PID controller was found by first using a simple transfer function model for the MLDM. The parameters were then fine-tuned on the setup because the PID controller gain K c was too large and made the loop to become unstable. A careful examination of the open-loop step response of each of the MLDM modes made apparent that some higher-order modes exhibit a larger overshoot than what is seen on the open-loop step response of a single actuator. When the controller gain K c is lowered from 0.6 to 0.36 and the changes on the other parameters remain minor, the loop can be successfully closed on the MLDM. The impact of the overshoot seen in the higher-order modes is removed when the gain is lowered. The optimal parameter set found on the experimental setup forms a compromise that takes care of the differences between the step responses of the modes. The modes presented in Fig. 4 form a base to describe the wavefronts that the MLDM can produce. When we consider also using the dynamical response 4908
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of these modes, the surface of the MLDM could be described within the controller as composed of dynamic eigenmodes [11]. Each mode could then be changed independently using its individual PID controller based on the transfer function of that mode. The computational load of having a controller based on the modes should not pose a problem considering the speed of the MLDM response. Such a controller could also be optimized to track specific modes. We plan to implement such a controller on a MLDM. This will require to model the transfer function of each mode and also to optimize the controller parameters for each of these modes as well. It is obvious that modes that are symmetrical (e.g., modes 1 and 2) will share the same transfer function and controller parameters. The principal task will be to evaluate if such a modal-based controller will significantly improve or not the performance of the MLDM compared to the set of parameters we found as a compromise. A feed-forward controller could also be used to improve the open-loop transfer function of the MLDM modes. This idea will be put forward when a proper model of the modes is developed and we learn more about the repeatability of the MLDM response. The present paper concentrates on obtaining new and important data on MLDM dynamics using a commercial ferrofluid without considering how the ferrofluid choice and other implements could change the MLDM dynamics. For example, unpublished initial results on laying a thin membrane on a custom ferrofluid, both chemically made by chemists within our group, seem to indicate that the open-loop step response steady state could be reached about 5 times faster without significantly altering the shape of the influence function although requiring a somewhat increased command signal. Another approach being investigated is using overdriving techniques, similar to what is being used in LCD technology, to lower the time required to reach the open-loop steady-state value. Initial results indicate that combining the use of a membrane and overdriving could allow a tenfold improvement on the time required to reach the steady-state value of the open-loop step response. The authors would like to thank Raytheon ELCAN Optical Technologies for manufacturing the MLDM used in this paper and ALPAO SAS which graciously made modifications to their Core Engine software for connecting to the MLDM. This work has been supported by grants from Natural Sciences and Engineering Research Council (NSERC). References 1. E. S. ten Have and G. Vdovin, “Characterization and closedloop performance of a liquid mirror adaptive optical system,” Appl. Opt. 51, 2155–2163 (2012). 2. D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express 15, 18190–18199 (2007). 3. A. Iqbal and F. B. Amara, “Modeling of a magnetic-fluid deformable mirror for retinal imaging adaptive optics systems,” Int. J. Optomechatron. 1, 180–208 (2007).
4. D. Brousseau, E. F. Borra, M. Rochette, and D. BouffardLandry, “Linearization of the response of a 91-actuator magnetic liquid deformable mirror,” Opt. Express 18, 8239–8250 (2010). 5. D. Brousseau, J. Drapeau, M. Piché, and E. F. Borra, “Generation of Bessel beams using a magnetic liquid deformable mirror,” Appl. Opt. 50, 4005–4010 (2011). 6. S. Thibault, D. Brousseau, and E. F. Borra, “Liquid deformable mirror for advanced sub-optical system testing,” Proc. SPIE 7739, 773910 (2010).
7. J. Parent, E. F. Borra, D. Brousseau, A. M. Ritcey, J.-P. Déry, and S. Thibault, “Dynamic response of ferrofluidic deformable mirrors,” Appl. Opt. 48, 1–6 (2009). 8. J. Clerk-Maxwell, Treatise on Electricity and Magnetism (Clarendon, 1873). 9. F. Boyer and E. Falcon, “Wave turbulence on the surface of a ferrofluid in a magnetic field,” Phys. Rev. Lett. 101, 244502 (2008). 10. A. Visioli, Practical PID Control (Springer, 2006). 11. T. Ruppel, “Model-based feed forward control of large deformable mirrors,” Eur. J. Control 3, 261–272 (2011).
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