This paper discusses an innovative low-cost deformable mirror made of a magnetic liquid (ferrofluid) whose surface is actuated by an hexagonal array of small current carrying coils. Predicted and experimental performances of a 37-actuator ferrofluid deformable mirror are presented along with wavefront correction examples. We show the validity of the model used to compute the actuators currents to obtain a desired wavefront shape. We demonstrate that the ferrofluid deformable mirror can correct a 11 µm low order aberrated wavefront to a residual RMS error of 0.05 µm corresponding to a Strehl ratio of 0.82.
©2007 Optical Society of America
Babcock first introduced adaptive optics in 1953 by proposing the use of a deformable mirror to correct atmospheric seeing . Interestingly, his concept was based on the actuation of a liquid surface. The idea consisted of using electrodes to actuate an electrostatic charged oil film that covers the surface of a reflecting mirror. Technology not being ready at the time, his concept was never put in application. Since then, many adaptive optics systems have been developed and are now in operation for astronomical purposes. The last decade has seen adaptive optics applications soar and demonstrated that new low-cost and high stroke deformable mirrors with a large number of actuators were needed.
Liquid Mirror Telescopes (LMTs) are well-known examples of the use of a liquid to make optical components. LMTs have been built and used to obtain astronomical, space science and atmospheric sciences data [2–6]. The major advantages of liquid optics are their low-cost and the fact that the surface of a liquid follows an equipotential surface to a very high precision.
Using liquids to build a deformable mirror was discussed by Ragazzoni and Marchetti in 1994 . They proposed to use a current flowing through a conducting liquid (mercury) and actuate its surface with a coil lying underneath the liquid. However, mercury is poorly suited to make liquid magnetic mirrors since ferromagnetic mercury is unstable , and the high density of mercury necessitates strong magnetic fields and thus high electrical currents [7,10]. Liquid adaptive mirrors shaped by electro-capillary effects have also been discussed .
The drawbacks of using mercury for liquid deformable mirrors can be solved by using a magnetic liquid (ferrofluid) coated with a metallic layer and actuated by magnetic fields . Ferrofluid deformable mirrors that use arrays of actuators have been previously discussed . Other concepts and proposals of using ferrofluids for deformable mirrors have been discussed by Borra et al. in recent papers [14,15]. In this article, we present the first experimental demonstration of a 37-actuator ferrofluid deformable mirror.
2. Description of the ferrofluid deformable mirror
The ferrofluid deformable mirror consists of 37 custom made coils (actuators) arranged in a hexagonal array 35 mm in diameter (see Fig. 1). Aluminum and brass were selected as material for the structure of the mirror because they have very low ferromagnetic properties. Each actuator is made of about 200 loops of resin-coated copper wire 0.3 mm in diameter. The copper wire is wound in 4 layers around a hollow brass core 2 mm in diameter and thus covers a full length of about 15 mm. The effective external diameter of an actuator is 5 mm. The windings are maintained in place by an aluminum ring that fits the brass tubing. Each coil has an electrical resistivity of about 1 ohm. A small ferrite core 1.5 mm in diameter is inserted in the brass core to amplify the magnetic field produced by each actuator. Assembling the actuators is a delicate task but each one costs only a few dollars in material. The actuators are then inserted in an aluminum support and securely attached from below with a small nut. An aluminum container filled with a one millimeter thick layer of EFH1 ferrofluid from Ferrotec (USA) Corporation is placed on top of the actuators. EFH1 is an oil-based ferrofluid and has a relative magnetic permeability of 2.7, a density of 1.21x103 kg.m-3, a magnetization saturation of 400 gauss and a concentration of magnetic particles of about 5% per volume.
Currents are controlled by a LabView interface and three Adlink 6216V 16 bits and 16-channels analog output cards. A simple amplification stage permits a current range of 0 to 200 mA in each actuator. The direction of current flow in each actuator is determined by individually driving the actuators and using a simple compass. Including materials, driving electronics and shop time, the cost of the whole deformable mirror was estimated at about $100 per actuator. Most of this cost is due to the Adlink cards and could be decreased by using newer cards available with a higher number of outputs.
3. Modeling the DM
Ferrofluids are liquids that contain a suspension of about 10 nm diameter ferromagnetic particles within a liquid carrier. In the presence of an external magnetic field, the magnetic particles align themselves with the field and the liquid becomes magnetized . For a ferrofluid surface at rest in the plane z=0, the equation that describes the ferrofluid surface amplitude h under the applied magnetic field is given by 
where ρ is the density of the ferrofluid, µr the relative magnetic permeability of the ferrofluid and Bn and Bt are the normal and tangential components (relative to the xy plane) of the external magnetic field. The relative magnetic permeability µr of the ferrofluid is assumed to be independent of the external magnetic field since in most cases it is extremely small compared to the saturation magnetization of the ferrofluid. The maximum surface amplitudes achievable with ferrofluids are limited by the so-called Rosensweig instability . For this application it is not a limitation since it corresponds to an amplitude of over a millimeter for EFH1 ferrofluid . Particle aggregation effects have been observed on ferrofluids in the presence of magnetic fields . The low concentration of particles in EFH1 combined to the fact that magnetic field produced by our actuators is about 1/100 of the magnetization saturation of EFH1 cause this phenomenon to be negligible . Ferrofluids have a reflectivity of about 4% and must be coated with a layer of colloidal silver nanoparticles called MELLF (Metal-Like Liquid Film) to render them reflective . Although MELLFs are not compatible with commercial ferrofluids, we have successfully developed our own ferrofluid having characteristics that makes it compatible with the reflective colloidal film. However, we should notice that several applications might not need a high reflectivity. In this work we used a non-coated ferrofluid as high reflectivity was not needed for our testing purposes.
3.2 Wavefront calculation
The magnetic field produced by the actuators at the liquid surface plane is calculated assuming that each actuator is made of multiple single current loops. The on-axis magnetic field produced by a single current loop is given by a simple equation. However, elsewhere, the magnetic field must be evaluated from
where a is the loop radius, r is the distance measured from the axis of the loop, z is the vertical distance measured from the plane of the loop, and Ft and Fn are dimensionless functions depending only on the loop radius and the coordinates. These two functions contain elliptical integrals and have to be numerically calculated. Once they are calculated for a given loop radius and an array of sample points, they remain constant and the magnetic field is just scaled by the current I flowing through the coil. Consequently, the total magnetic field shape produced by one actuator is obtained by adding the contributions from each of its 200 loops and does not need to be evaluated each time the current is changed. Assuming that the ferrite core becomes uniformly magnetized when current flows in the actuator, its impact is as though a uniform current density is circulating on its surface. The presence of the ferrite core is thus equivalent to having additional current loops at the innermost radii values of the windings and therefore does not appreciably change the functions Ft and Fn but simply gives a higher magnetic field value for a given current. This assumption is confirmed by finite element analysis of an actuator having a ferrite core. Finally, all simulations are made neglecting the presence of ferrite cores and merely lowering the calculated currents by a scaling factor of about 6.
Using Eq. (1) and (2), we computed the profile of the influence function of a single actuator (see black curve in Fig. 2). The surface of the ferrofluid lies 4 mm above the actuators. The influence function can be fitted with a Gaussian having a FWHM (Full Width at Half Maximum) of about 9 mm. The FWHM of the influence function implies that the coupling between the actuators is negligible starting at the 3rd neighboring actuator. We also found that the FWHM of the influence function increases linearly with the distance between the ferrofluid surface and the actuators.
Because the surface of the ferrofluid deformable mirror is shaped by adding the individual vector components of the magnetic field from each actuator, we cannot use scalar addition of the influence functions of the actuators to determine the profile of the mirror surface. At first, we thought that this would render the control algorithm of such a deformable mirror more complex, but we find that the additional steps involved do not noticeably increase computer time. This vectorial behavior could also be used to advantage: by inverting the current flow in some actuators, new combinations of magnetic field components would give a larger number of available wavefront surfaces. Simulations showed that using both inverted and noninverted currents has a limited impact on the generated surfaces. However, it might still be useful when driving the mirror in close-loop but it has not been investigated yet.
3.3 Simulation results
To find the assigned currents to the 37 actuators for producing a given wavefront, we use the Newton-Raphson method and minimize the wavefront residual RMS error between the desired and the produced wavefront. A MATLAB algorithm was written and is used to calculate the required driving currents for the 37 actuators to generate the first 27 Zernike polynomials using OSA convention . The surface of the ferrofluid is assumed to be 4 mm above the actuators for all simulations. Several calculations have demonstrated that this value provides the lowest residuals for a broad range of Zernike polynomials.
The first 12 computed Zernike polynomials can be seen in Fig. 3. Printthrough caused by the hexagonal geometry of the actuators is clearly visible on the wavefronts. This effect is artificial in great part because the computed wavefronts do not take into account liquid surface tensions which would smooth the ferrofluid surface.
For example, Fig. 4 shows the currents (without ferrite cores) needed to produce the trefoil and spherical aberration terms. Currents do not exceed 300 mA for all the computed Zernikes meaning that, for actuators with ferrite cores, the currents will be less than 50 mA to obtain the same wavefront amplitudes. The geometrical distribution of currents in Fig. 4 is not entirely symmetrical because the simulations include the presence of Earth’s magnetic field component. This 0.18 gauss horizontal component (chosen to be along the y-axis in the simulations) contributes to the overall magnetic field created by the actuators and creates a small asymmetry in the current distribution.
We calculated the RMS residual errors for the 27 Zernike polynomials and plotted a histogram of results in Fig. 5. The residuals are below 0.2 up to the 16th Zernike except for the spherical aberration term for which the residual is slightly under 0.3. Zernikes above 16 have higher residuals but still provides about 50% wavefront correction. Our computations predict that the residual errors would decrease by a factor of about 2 by increasing the number of actuators from 37 to 127.
4. Experimental setup
The computed currents for the first 20 Zernike polynomials were applied to the ferrofluid deformable mirror actuators and the produced wavefronts were measured using an Imagine Optics HASO HR44 wavefront sensor (see Fig. 6). The setup was designed to image a 25 mm diameter circular region of the ferrofluid surface upon the 5 mm pupil of the Shack-Hartmann. To compensate for the residual aberrations in the optical setup, the wavefront produced when the current source is turned off is subtracted from each measurement. The quality of the ferrofluid surface at rest was also measured using a Zygo interferometer and found to have better than 40 nm residual surface RMS error. Wavefronts are reconstructed using the first 36 Zernike polynomials coefficients calculated by the software provided by Imagine Optics. By comparing the reconstructed wavefronts with the added zonal residuals, we find that truncating the Zernike modes above 36 introduces a mean residual wavefront RMS error of about 8 nm on the reconstructed wavefront.
The central actuator was fed a current of 100 mA and the wavefront recorded with the HASO. The profile of the experimental response of this actuator is represented by the red curve in Fig. 2. The experimental influence function is as predicted: a Gaussian fit of theoretical data gives a FWHM of 8.4 mm and 8.9 mm for the experimental data. As stated before, the FWHM of the influence function increases linearly with distance between the liquid surface and the actuators. The difficulty to correctly evaluate this distance can result in a small discrepancy between the FWHM of the calculated and the experimental influence function.
The first 16 out of 20 of the produced Zernike polynomials are shown in Fig. 7. All these wavefronts were obtained by direct assignment of the properly scaled computed currents to the 37 actuators. Only a small number of actuators needed manual current adjustments, probably because of possible small breaks in the ferrite cores, difference in the exact number of turns or fabrication errors of the actuators. When data from the wavefront sensor will be coupled to our algorithm, the manual adjustment of currents in some of the actuators will not be necessary as they will be automatically taken care of by the control algorithm. Apart from these differences, the model that describes the wavefronts produced by the ferrofluid deformable mirror is thus excellent at calculating the required currents for a given wavefront.
A histogram of experimental residual wavefront RMS errors for the first 20 Zernike polynomials is presented in Fig. 8. Low order modes errors are close to predicted values, while higher orders have residuals 1.5 to 2 times larger than predicted.
Except for the 19th term, the general tendency between predicted and experimental residuals is similar. For higher terms, the residuals are greater than expected due to the difficulty of perfectly adjusting the currents for such high frequency terms.
4.2 Wavefront correction examples
The performance of the ferrofluid deformable mirror to compensate for complex wavefronts was demonstrated by two experiments. The first test consisted in correcting for a purposely introduced misalignment of lenses L2 and L3 in the setup of Fig. 6. The misalignment introduced spherical aberration and astigmatism in the system (see left of Fig. 9). The wavefront PV and RMS amplitudes of the aberrated wavefront were measured to be 2.42 and 0.58 µm respectively.
The currents needed to assign to the ferrofluid deformable mirror were calculated in the same manner as described above and applied to the actuators. As before, individual manual corrections were made to some of the actuators and the histogram of the residuals after correction is shown at right of Fig. 9.
The wavefront residual RMS error of the corrected wavefront is 0.05 µm, corresponding in our case to a correction RMS residual better than a tenth of a wave.
From these coefficients, the PSFs were computed and appear in Fig. 10. The PSF of the aberrated wavefront can be seen at left and the corrected PSF at right corresponds to a Strehl ratio of 0.82.
The second test consisted of placing two small Petri dishes directly in the beam between the ferrofluid deformable mirror and lens L3, producing high amplitude aberrations with a wavefront PV of 11.43 µm and a RMS of 2.58 µm (see Fig. 11). Required currents to compensate the aberrations were calculated and applied to the mirror, resulting in a residual wavefront having an RMS error of 0.15 µm. Considering only the 3rd order aberrations, the wavefront PV and RMS amplitude of the aberrated wavefront stays about the same, but the corrected wavefront has now a wavefront residual RMS error of 0.05 µm. This clearly shows the ability of the mirror to correct low order modes. Residuals for higher terms will be less for a ferrofluid deformable mirror having a higher number of actuators.
5. Dynamic behavior of the ferrofluid deformable mirror
Even though the major objective of this paper is to describe the static performance of a ferrofluid deformable mirror, some remarks of its dynamic behavior can be made.
5.1 Sensitivity to vibrations
It is well-known that the surface of a liquid is sensitive to vibrations. The fact that we are using a layer thickness of one millimeter greatly reduces this problem and, as demonstrated by Borra et al., the sensitivity to vibrations can be decreased significantly by using a layer of liquid thinner than a millimeter . Our experiments show that the sensitivity to vibrations is also significantly reduced when the ferrofluid deformable mirror is under use, the magnetic field from the actuators acting as a stabilizing effect. Notwithstanding a minimal damping installation and a machine shop in operation just above our lab, we found that the amplitude of disturbances induced by vibrations was less than 1/10 of a wave. In fact, we found that the air flows coming from the ceiling ventilation of the lab were a bigger problem than vibrations. These air flows were found to be easily minimized by placing a Mylar film of optical quality over the ferrofluid container.
5.2 Response time
Preliminary investigations of the response time of the ferrofluid deformable mirror demonstrated that the mirror can be driven at a frequency of 40 Hz. Ongoing tests using a bias magnetic field and/or by overdriving the actuators have shown that we can reach higher driving frequencies. Ferrofluids with different physical parameters, higher viscosity for example, are also under investigation and promise to provide driving frequencies of hundreds of hertz.
A low-cost ferrofluid deformable mirror using 37 magnetic actuators has been demonstrated. Surface deformation modeling of the mirror was carried out to produce Zernike polynomials. The performance of the mirror has been experimentally investigated by producing wavefronts of the first 20 Zernike polynomials and by correcting up to 11 µm static wavefront errors with a residual RMS error of 0.05 µm. We have been able to achieve a Strehl ratio of 0.82 with the mirror.
The Zernike polynomial experimental results indicate that the deformation obtained is predictable, thus validating the theoretical model. The ferrofluid deformable mirror is now ready for the development of a closed-loop operation mode. Preliminary tests demonstrate a response time of 40 Hz, but it is still under investigation and recent results show that the mirror could be driven at frequencies of hundreds of hertz.
The 37-actuator ferrofluid deformable mirror is composed of standard components having a total cost less than a 100$ per actuator, taking into account parts and labor in our mechanical shop. We can presume that costs will further decrease for a better engineered mirror. The exceptional low-cost of the device makes it particularly attractive for a large number of adaptive optics applications and experimental development. The results were obtained with 37 actuators but simulations have shown that a higher number of actuators would result in smaller residuals for high aberration terms. Ongoing research shows that the diameter of the actuators can be reduced by a factor of 2.5, allowing us to build smaller deformable mirrors (e.g. a 2-inches diameter ferrofluid deformable mirror having more than 450 actuators).
Considering the fact that this ferrofluid deformable mirror is a prototype, the results presented here are very encouraging. A better control of the fabrication process of actuators and mirror assembly will certainly lead to better performance.
This research was supported by the Natural Sciences and Engineering Research Council of Canada.
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