Laval University

Liquid mirrors and adaptive optics group

# Ferrofluid Derfomable Mirrors

## Introduction

AA deformable mirror (DM) is the central element of an adaptive optics system. The DM changes shape to correct for a phenomenon known as atmosphering seeing. Atmospheric seeing is the result of air masses of different densities (and thus of different refractive indices) causing uneven deformations in incoming wavefronts. The same phenomenon causes the stars to twinkle and is responsible for the "mirage" effect that can be seen over hot pavement. Atmospheric seeing is the main limiting factor when it comes to image quality for earth-based observatories. The use of adaptive optics allows the observatory to compensate for these wave deformations by changing the shape of the primary mirror in real time.

Traditional DMs are made with glass or metal faceplates deformed by a number of actuators. These mirrors are very precise, but generally very expensive. The advantage of liquid mirrors is that they provide a high quality, seamless optical surface while absolutely no mechanical polishing is required. The aim of this project is to take advantage of these properties to make a low cost DM.

In previous attempts to deform liquid mirrors, the goal was generally to modify the shape of the mirror as a whole and not make the kind of rapidly changing wavefront corrections needed for adaptive optics. These attempts focused on mercury, as was used in the first generation of rotating liquid mirrors at Laval.

The problem with using mercury as a DM is that because of its high density, a substantial force is needed to deform it. Using magnetic liquids (ferrofluids) represents a solution to this drawback. deformable mirror (DM) is the central element of an adaptive optics system. The DM changes shape to correct for a phenomenon known as atmosphering seeing. Atmospheric seeing is the result of air masses of different densities (and thus of different refractive indices) causing uneven deformations in incoming wavefronts. The same phenomenon causes the stars to twinkle and is responsible for the "mirage" effect that can be seen over hot pavement. Atmospheric seeing is the main limiting factor when it comes to image quality for earth-based observatories. The use of adaptive optics allows the observatory to compensate for these wave deformations by changing the shape of the primary mirror in real time.

Traditional DMs are made with glass or metal faceplates deformed by a number of actuators. These mirrors are very precise, but generally very expensive. The advantage of liquid mirrors is that they provide a high quality, seamless optical surface while absolutely no mechanical polishing is required. The aim of this project is to take advantage of these properties to make a low cost DM.

In previous attempts to deform liquid mirrors, the goal was generally to modify the shape of the mirror as a whole and not make the kind of rapidly changing wavefront corrections needed for adaptive optics. These attempts focused on mercury, as was used in the first generation of rotating liquid mirrors at Laval.

The problem with using mercury as a DM is that because of its high density, a substantial force is needed to deform it. Using magnetic liquids (ferrofluids) represents a solution to this drawback.

## What is a ferrofluid

A ferrofluid is a colloidal suspension of magnetic (ferrimagnetic or ferromagnetic) nanoparticles with sizes in the range between 3 and 20 nanometers in a carrier liquid. Particles have no natural affinities for liquids. It is thus essential to introduce a stabilising agent (surfactant) to increase the solubility of particles in the chosen carrier liquid and also to ensure repulsive forces between particles to prevent agglomeration. In the presence of an external magnetic field, these magnetic particles align themselves with the field and the bulk of the liquid becomes magnetized. The surface of the liquid can thus be shaped according to the magnetic field geometry. We use this property to shape the liquid surface as we would do with conventionnal deformable mirrors.

Common ferrofluids look like motor oil and their reflectivity of ferrofluids is quite low (about 4%). This is not a problem for testing purpose of our deformable mirrors and for certain optical applications, but can be troublesome for applications that require a highly reflective surface. This is why ferrofluids need to be coated with a reflective liquid layer called a MeLLF. Unfortunately, MeLLFs are not compatible with commercial ferrofluids and we had to develop our own ferrofluids. This has been achived by a team of chemists working in our group under the supervision of Prof. Anna M. Ritcey.

The best carrier liquid for this specific application appeared to be ethylene glycol due to its relatively high surface tension and its low vapour pressure limiting the evaporation. The metallic silver particles are concentrated and sprayed on the surface of a few millimetres thickness of ferrofluid. The liquid mirror spread on the surface of the ferrofluid exhibits excellent reflectivity properties comparable to those previously reported for silver nanoparticles spread on water (see the Reflective liquids section). Furthermore, interferometry measurements indicate that the reflective film forms a smooth surface with a root mean square roughness of approximately λ / 20.

Illustration showing the structure of the ferrofluid-mellf interface.

Photographs of a homemade ferrofluid coated with a MeLLF.
The image at right shows the liquid deformed by the presence of a magnetic field of a permanent magnet located under the container.

More information on ferrofluids can be found here and a video on how a ferrofluid works here (QuickTime required).

## Basic theory

The equations that govern a ferrofluidâs reaction to an external magnetic field have been known for nearly half a century (principaly developed by R. E. Rosensweig, the theory surrounding ferrofluids and their application is explained in Rosensweigâs excellent book Ferrohydrodynamics). The most interesting relationship for the purposes of our work is the equation that gives the amplitude of a deformation as a function of the external magnetic field:

$h=\frac{\mu_r-1}{2\mu_r\mu_0\rho g}\left[\vert \mathbf{B}.\mathbf{n}\vert^2+\mu_r\vert \mathbf{B}\times \mathbf{n}\vert^2\right]$

where ρ is the liquid density, μr is the liquid magnetic permeability and $\mathbf{n}$ is a unit vector perpendicular to the surface of the liquid. Essentially this states that the height of the fluid at each point is proportional to the square of the applied magnetic field. Hence, by creating the proper magnetic field configuration, any shape can be produced onto the surface of the liquid. Furthermore, the maximum deformation produced on the surface of the liquid is only limited in amplitude by the magnetization saturation of the ferrofluid and/or the Rosensweig instability. For commercial ferrofluids, this amplitude can be as high as one millimeter.

Permanent magnets were first considered but they usually create irregular magnetic fields and their strength cannot be modified without moving them. Another solution is to use coils of copper wire to which an electrical current is applied, thus inducing a magnetic field. We therefore simulated a variety of possibilities to create a magnetic field that could produce typical know optical aberrations.

## Earlier experiments

The first prototype of a ferrofluid mirror was designed by Laird in 2004. The mirror consisted of several small magnetic coils (actuators) of about 5 mm in diameter. A ferrite core was inserted at the center of each actuator to increase the magnetic field produced. The actuators were aligned in a square geometry as illustrated in the following photograph.

While Laird's ferrofluid DM succeded in demonstrating the low-cost and achievable high strokes of such deformable mirrors, several drawbacks remained. The influence function of the actuators was too broad, greatly reducing the high spatial frequencies aberrations the FDM could produce. It took a while to realize that Laird's actuators had two major problems. The ferrite cores diameter was too large, reducing the minimum diameter of the copper windings. We found that the copper windings that contribute the most to the influence function profile are the ones nearest to the center of the actuators. By reducing the size of the ferrite cores, we could make copper windings nearer to the center of the actuators, while still benefit from their presence to amplify the magnetic field. The other problem of Laird's actuators was found to be their square geometry. Rioux demontrated in 2005 that an hexagonal geometry of the actuators was better suitable for a ferrofluid DM. The hexagonal geometry, while not changing the influence function profile, produces lower surface fitting errors. These conclusions from Laird's work allowed us to construct a new prototype that is presented in the section The 37-actuator prototype.

The initials of the COPL (Centre for Optics, Photonics and Lasers), where this research was being carried out, spelled using Laird's DM.

## The 37-actuators prototype

In the meantime, you can check the poster and/or the paper we produced for the June 2008 SPIE Astronomical Telescopes and Instrumentation conference in Marseille.

## Bandwidth of a ferrofluid DMs

In the meantime, you can check the poster and/or the paper we produced for the June 2008 SPIE Astronomical Telescopes and Instrumentation conference in Marseille.