Ferrofluidic Deformable Mirrors for Adaptive Optics
Ermanno F. Borra (left) and Denis Brousseau (right)
[This is an invited article. The authors have built the first deformable liquid mirror from a magnetic liquid or “ferrofluid”, which is set to find wide range of applications including correction of aberration in the images of telescopes and many other optical devices. -- 2Physics.com]
Authors: Denis Brousseau and Ermanno F. Borra
Affiliation: Université Laval, Département de physique, de génie physique et d’optique and Centre d’Optique, Photonique et Laser (COPL), Québec, Canada.
For roughly the last 25 years, adaptive optic (AO) systems were primarily used for astronomical applications. In the last 10 years, the range of scientific applications of AO has soared and now includes vision science, medical imaging and free space optical communications to name only a few. These new applications have stimulated research for low-cost, high-stroke deformable mirrors with a large number of actuators. Most actual deformable mirrors are expensive, costing about $1000 US per actuator. Current high-stroke deformable mirrors like the Imagine Optics 52-actuator MIRAO DM can produce large deformations (50 µm peak-to-valley tilt) but having a larger number of actuators would greatly increase its cost. Micro-Electro-Mechanical Systems (MEMS) deformable mirrors having large numbers of actuators (over 1000) and fabricated by a technique similar to surface micromachining have a great potential for low cost, but are currently limited to strokes of only a few microns.
It is well known that a liquid follows an equipotential surface to a high degree of precision. For example, the surface of a rotating pool of mercury takes a parabolic shape which can be used as a the primary mirror of a low cost telescope (http://wood.phy.ulaval.ca/what.html and http://www.astro.ubc.ca/lmt/lm/index.html). During the last few years, we have developed a new type of deformable mirror made of a ferrofluid whose surface is shaped by an array of magnetic coils. A ferrofluid is a liquid that contains a suspension of small ferromagnetic particles (Ø ~ 10 nm) within a water- or oil- based carrier liquid. In the presence of an external magnetic field, the magnetic particles react with the field and the fluid surface takes a shape that is determined by the equilibrium between the magnetic, gravitational and surface tension forces. The equation that describes the shape of the surface can be derived using equations found in 
where µr and ρ are the relative permeability and density of the ferrofluid respectively, n is a unit vector perpendicular to the liquid surface and B is the external magnetic field vector at the liquid-air interface. The external magnetic field B can be produced by an array of small current carrying coils located just under the surface of the liquid. Based on this principle, we built a 37-channel deformable mirror prototype, made of a ferrofluid whose surface is actuated by a hexagonal array of small current carrying coils.
In standard modal control of deformable mirrors, the mirror surface is shaped by the linear addition of the individual response function of the actuators. We see, from the preceding equation, that in the case of a ferrofluid deformable mirror (FDM), the liquid surface deformation is non-linear with respect to the external magnetic field, and also depends on the individual orientation of the external magnetic field components. Consequently, conventional modal control of a FDM is impossible; however we have successfully developed a custom algorithm that is able to compute the currents that must be assigned to the coils for a given mirror surface shape.
Using a commonly available ferrofluid we found that a maximum deformation of over a millimeter can be achieved before reaching instability . In theory, much larger deformations (several mm) could be obtained with magnetic fields having components mostly parallel to the liquid surface and/or using ferrofluids having different physical properties.
Fig. 1. A custom ferrofluid developed in our labs is shown coated with a reflective layer of MeLLF and under the influence of a magnetic field from a permanent magnet located under the container. Picture clearly shows the very large deformation amplitudes that can be obtained.
Ferrofluids have a low reflectivity similar to motor oil and for many applications must be coated with a reflective layer. This can be done using reflective liquids based on interfacial films of silver particles known as Metal Liquid-Like Films or MeLLFs . MeLLFs combine the properties of metals and liquids, can be deformed and are therefore well adapted to applications in the field of liquid optics. MeLLFs are not compatible with currently available commercial ferrofluids, which are hydrophobic, and for compatibility with MeLLFs we had to developed a custom hydrophilic ferrofluid (see Fig. 1) . Our team is also considering the deposition of a chemical membrane on the ferrofluid.
Our prototype consists of 37 custom made coils (actuators) closely packed in a hexagonal array 35 mm in diameter (see Fig. 2). Each coil is made of about 200 loops of AWG28 magnet wire and has an external diameter of 5 mm. A small ferrite core is placed at the center of each coil to lower the current requirement of the device. An aluminum container (not seen) filled with a one-millimetre-thick layer of ferrofluid is placed on top.
Fig. 2. Our 37-channel prototype showing the hexagonal array of 37 coils of 5-mm diameter.
Total cost of the FDM was estimated at about $100 per actuator, including materials, electronics and shop time. Costs can certainly be reduced further with improved technology.
Using our algorithm, we have computed the required currents to produce standard Zernike polynomials (http://en.wikipedia.org/wiki/Zernike_polynomials). Those currents were then fed to the FDM and the resulting wavefronts were measured using a wavefront sensor (see Fig. 3).
Fig. 3. Experimental wavefronts representing Zernike polynomials reproduced by the FDM and measured using a Shack-Hartmann wavefront sensor. Each wavefront has a PV wavefront amplitude of about 5 μm.
Because of the vector-dependent response of our device, we suspected that trying to fit real wavefronts made from combining several Zernike polynomials would result in lower wavefront residual errors than by adding the residuals of each Zernike that made up the original wavefront. We performed experiments to test this assumption. We purposely introduced optical aberrations of 0.58 µm RMS wavefront amplitude and 2.42 µm PV wavefront amplitude in our wavefront measurement setup. PSFs before and after correction can be seen in Fig. 4. The achieved Strehl ratio of the corrected wavefront is 0.84 at a wavelength of 659.5 nm.
We also introduced much greater amplitude aberrations with PV and RMS wavefront amplitudes of 11.43 and 2.58 µm respectively. The RMS residual error of the corrected wavefront was measured to be 0.15 µm. We found that this error drops to 0.05 µm if we consider only the low order aberration terms. Correction for high spatial frequency Zernike polynomials would improve if the FDM had a greater number of actuators.
Fig. 4. Experimental result showing the PSF (log scale) of an aberrated wavefront (left) corrected by using our deformable mirror (right). Strehl ratio of corrected wavefront is 0.84 (659.5 nm).
Although we got promising results, some drawbacks remain. We need to bias the surface of the liquid to allow for a push-pull effect as the amplitude varies as the square of the current applied to a given actuator (deformations can only be positive). This reduces the available stroke of the mirror and also adds a surface residual error.
A novel way to control those liquid mirrors has recently been introduced by Iqbal and Amara, and solves most of these drawbacks . The technique consists of adding a constant and uniform magnetic field whose orientation is along the direction perpendicular to the surface of the liquid. The amplitude of this constant magnetic field is about 10 times greater than what is produced by the coils (~ 2.5 gauss). The magnetic field of the actuators acts as a small perturbation of the uniform field and this linearizes the response of the liquid (as shown in Fig. 5). This also has the effect of amplifying the stroke produced by the coils, reducing the required currents, so that ferrite cores in the actuators are no longer necessary, and making negative deformations possible.
Fig. 5. Measured amplitudes of the deformations produced by a single actuator in the presence of an external uniform magnetic field, as a function of current in the coil. The red and blue curves correspond to external magnetic fields of 25 and 30 gauss respectively. Negative deformations can be produced by inverting the current flow. The actuator used in the experiment has no ferrite core.
Until recently, we thought that those liquid mirrors were limited to a time response of only a few tens of hertz, because when driven at frequencies higher than about 20 Hz, we saw a rapid loss in amplitude response of the liquid and a phase lag of over 90 degrees appeared between the driving signal and the resulting liquid deformation, quite similar to the response of a low-pass RLC filter.
We demonstrated that the amplitude loss can be overcome by overdriving the coils with a very short and high amplitude current pulse launched at the beginning of each driving signal. By using this technique, a desired surface deformation is reached faster and the remaining signal stabilize the liquid shape.
We also demonstrated that the phase lag can be countered by increasing the viscosity of the ferrofluid. The critical frequency (a 90 degrees phase lag) was improved from 20 to 450 Hz by increasing the viscosity of the ferrofluid from 6 cP to 450 cP (viscosity of water is 1 cP and SAE 50 motor viscosity is about 500 cP). Since a single square wave signal sent to the liquid corresponds to two corrections (rise and fall), this actually implies a frequency response of 900 Hz. But increasing the viscosity also increases the time required for the liquid to stabilize. However, a solution to this problem is to use overdriving pulses that give an initial velocity to the liquid as discussed in the preceding paragraph.
To conclude, we have demonstrated a liquid deformable mirror prototype that can produce standard aberration terms, and we successfully corrected a 11 µm PV amplitude aberrated wavefront, yielding a residual RMS wavefront error of 0.05 µm. A new technique linearizes the response of these new deformable mirrors and allows the use of regular control algorithms. This will simplify our goal to demonstrate closed-loop operation of these new mirrors.
We have also shown the counterintuitive result that using a liquid having a sufficiently high viscosity improves their frequency response up to 900 Hz. By using both overdriving pulses and a higher viscosity ferrofluid, utilizing these mirrors in closed-loop at a running frequency of hundreds of hertz, appears to be possible. This will enable these mirrors to be used in many more applications than we previously thought. Ongoing tests on the chemical deposition of thin chemical membranes on ferrofluids could also improve the response of FDMs. We are now building a new prototype having 91 actuators of 2-mm diameter, thus reducing the footprint and allowing a higher density of actuators.
 "Interaction of a magnetic liquid with a conductor containing current and a permanent magnet"
V. V. Kiryushin and A. V. Nazarenko, Fluid Dynamics, 23, 306–311 (1988). Abstract.
 R. E. Rosensweig, "Ferrohydrodynamics". (Dover, 1997).
 "Nanoengineered astronomical optics",
E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre, Astron. Astrophys. 419, 777-782 (2004). Abstract.
 "Ethylene Glycol Based Ferrofluid for the Fabrication of Magnetically Deformable Liquid Mirrors"
J. -P. Déry, E. F. Borra, and A. M. Ritcey, Chem. Mater. 20 (2008). Abstract.
 "Modeling of a Magnetic-Fluid Deformable Mirror for Retinal Imaging Adaptive Optics Systems"
A. Iqbal and F. B. Amara, International Journal of Optomechatronics 1, 180-208 (2007). Abstract.