Adaptive optics: A liquid formable mirror for high-order wavefront correction

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A Liquid Deformable Mirror

for High-Order Wavefront Correction

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus, prof. dr. ir. J. T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 18 december 2006 te 10.00 uur

door

Edgar Moreno VUELBAN

Master of Science

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Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. J.J.M. Braat.

Samenstelling promotiecommissie: Rector Magnificus, voorzitter

Prof. dr. ir. J.J.M. Braat, Technische Universiteit Delft, promotor Prof. dr. P. M. Sarro, Technische Universiteit Delft

Prof. dr. ir. A. Gisolf, Technische Universiteit Delft

Prof. dr. C. Dainty, National University of Ireland, Galway Ireland Dr. N. Bhattacharya, Technische Universiteit Delft

Dr. G. Vdovin, Flexible Optical BV, Netherlands

Dr. ir. S. Kuiper, Philips Research Laboratories, Netherlands Prof. dr H.P. Urbach, Technische Universiteit Delft, reservelid

Dr. N. Bhattacharya heeft als begeleidster in belangrijke mate aan de tot-standkoming van het proefschrift bijgedragen.

This work was supported by TNO Science and Industry, Delft. ISBN-10: 90-78314-06-0

ISBN-13: 978-90-78314-06-6

Cover photo credit: J. Spronck

Copyright c 2006 by Edgar M. Vuelban

A free electronic version of this thesis can be downloaded from: http://www.library.tudelft.nl/dissertations

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Clarita M. Vuelban,

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1

Introduction

Astronomers, since the time of Isaac Newton and Christian Huygens in the 17th century, have already realized that the cause of the twinkling behav-ior of stars is due to atmospheric turbulence. Twinkling is the result of the random variation of the intensity of light, emanating from a star, due to the random interference between waves from the same star passing through slightly different atmospheric paths [1]. Aside from the twinkling effect, the turbulent atmosphere causes image “quiver” and image spreading or blur-ring. Image “quiver”happens because the angle-of-arrival of light, coming from the star, is affected by the variation of the index of refraction along its path through the atmosphere. Image spreading is the result of the random high-order aberrations produced by the turbulent atmosphere. These three effects, caused by turbulence, greatly limit the resolving power of ground-based telescopes.

Though the effects of atmospheric turbulence on the images of astronom-ical point-like objects has long been known, it took three centuries before a scheme for compensating the effects of atmospheric turbulence has been proposed. It was Horace Babcock [2] in 1953 who first proposed a scheme for an adaptive compensation of atmospheric turbulence using a deform-ing element, called the Eidophor, with a feedback from a wavefront sensor. Review articles on adaptive optics (AO) [3, 4, 5] often refer to the Babcock proposal as the impetus in the beginning of the modern era of the field of Adaptive Optics. As discussed by Dainty [3], the principal purpose of an adaptive optics system in astronomy is to remove the turbulence-induced wavefront distortions and provide a high resolution image, if possible a diffraction-limited image of astronomical objects.

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dis-turbed.

The removal of wavefront distortions using an adaptive optics system is achieved by employing three different components; 1) wavefront sensor, 2) control unit, and 3) deformable mirror. A wavefront sensor senses and measures the incoming aberrated wavefront. The computation of the re-quired correction and the control of the shape of the deformable mirror is performed by the control unit. The deformable mirror does the actual cor-rection of the distorted wavefront. Most often, the measurement and correc-tion are carried out in a closed loop, in which several iteracorrec-tions of measure-ment and correction are performed until an almost perfect compensation is established.

Though all three components play an important role in the correction pro-cess, it is the deformable mirror that is often referred to as the “heart” of an AO system. The ability of an adaptive optics system to perform the compen-sation relies heavily on how good a deformable mirror can adapt its shape with respect to the incoming distorted wavefront, in order to minimize the distortion. Deformable mirrors are already in existence for a few decades now. Conventional deformable mirrors are based on a thin reflective glass plate with the actuators attached to the backside of this plate.

Deformable mirrors based on liquid deformation are quite rare. So far, there are only two liquid-based deformable mirror designs reported and demon-strated, aside from the one that is being proposed and discussed in this the-sis. It is interesting to recall that the original idea of Babcock for a deforming element is actually based on liquid surface deformation. As mentioned ear-lier, Babcock proposed a correcting device, based on the Eidophor system. The Eidophor projection system [6] consists, along with other important op-tical components, of a thin oil film covering a reflecting mirror. An electron gun bombards electric charges onto the surface of the oil and by means of an electrostatic force a controlled surface corrugation is achieved. Surprisingly, there was no reported concrete implementation of this attractive proposal. Dainty [3] pointed out that the technology at that time was not that devel-oped to meet the challenging task, and during that period it was suspected that the principle would only work on bright stars.

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actua-Control System Wavefront sensor high-resolution image Corrected Wavefront beamspliter distorted wavefront Deformable Mirror closed-loop system

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1.1. SURVEY OF STATE-OF-THE-ART DEFORMABLE MIRRORS

tors, very narrow actuator pitch, low power requirement, and large stroke dynamic range. This demand is triggered mainly by the planned develop-ment of ”giant” ground-based telescopes, which incorporate AO systems to correct for atmospheric turbulence-induced wavefront distortions. The pro-posed “giant” ground-based telescopes (with aperture diameters in excess of 10m), for example, need an estimated number of actuators in the range of 100x103 - 500x103 with an inter-actuator pitch of < 200 µm to 1 mm [7]. Aside from the needs of the astronomical community, deformable mirrors are quite indispensable in the field of vision science [8, 9, 10, 11, 12], free-space laser communication [13], and high-energy lasers [14, 15, 16].

1.1 Survey of state-of-the-art deformable mirrors

Deformable mirrors have been in existence for over three decades now. As mentioned earlier, early types of deformable mirror consist of a reflective thin glass plate, which is deformed by an external force such as electro-static, piezoelectric, magnetorestrictive, and electromagnetic, among others. Aside from thin glass plates, conventional DMs also employ clamped re-flective membranes [17, 18] . These first generation DMs suffer from limited spatial resolution due to a) high stiffness of the material, b) large cross-talks between neighboring actuators, and c) large interactuator spacings. Aside from these, early generation actuators have high energy dissipation and hys-teresis, especially for piezoelectric and electromagnetic actuators.

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a high voltage, ± 3.0 kV , in order to achieve a surface displacement of ± 1 µm. Primarily, the mirror was intended to correct for wavefront errors in the visible region.

Another conventional deformable mirror is the bimorph DM [20], depicted in Figure 1.3. The mirror consists of a thin reflective glass plate and a ce-ramic material (PZT), which are glued together. The PZT material is po-larized normal to its surfaces. A continuous conducting electrode is sand-wiched between the glass plate and the PZT material. The backside of the piezoelectric material is covered with a number of independent electrodes. When a voltage is applied between the continuous electrode and one of the actuators below, the lateral dimensions of the piezoelectric material change by an amount depending on the electric field, Ez, and the coefficient, d13, of

the piezoelectric tensor. The mirror thus bends, like a bimetallic strip. The voltage required to create a height difference of one wavelength, λ, between center and edge of an electrode is given by

Vλ 2t2λ/D2d13 (1.1)

where t is the thickness of the plate and D is the diameter of back electrode. Unlike the MPM, the surface deformation of a bimorph mirror is highly nonlocalized.

Figure 1.4, shows a schematic of a commercially available magnetic de-formable mirror from Imagine Eyes [21]. The DM consists of a thin silver-coated flexible membrane, which is deformed by a set of 52 magnetic ac-tuators. Permanent magnets are attached to the backside of the reflective membrane while the corresponding actuator coils are situated underneath. When a voltage is applied to the individual coils, a magnetic force is created between the coil and the permanent magnet. The resulting magnetic force is proportional to the electrical current in the actuator coil. The mirror has an initial surface flatness below 0.1 µm RMS over the entire pupil. The actuator pitch of this DM is 2.5 mm.

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1.1. SURVEY OF STATE-OF-THE-ART DEFORMABLE MIRRORS

Top view

Elevation Aluminized glass mirror

Piezoelectric ceramic Addressing electrodes Common electrode Electrical addressing leads

Figure 1.2: The monolithic piezoelectric mirror ( adapted from [19]).

Reﬂecting surface Glass PZT material Back electrodes Inner electrode incident beam

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coil F B I

I

magnet reflective membrane

Figure 1.4: Schematic diagram of a magnetic deformable mirror from Imag-ine Eyes (Adapted from [22]).

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electro-1.1. SURVEY OF STATE-OF-THE-ART DEFORMABLE MIRRORS U1 U2 U3 U4 Al mirror Mirror die Nitride membrane Oxide insulation Bo!om die Glue Chip holder Al electrodes Control voltages

Figure 1.5: The micromachined DM. An application of a voltage Ui results

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static attraction, which causes the cantilever to deflect toward the substrate. Silicon nitride Si wafer substrate Address pad (Polysilicon) Reflective layer (Polysilicon) Actuator (Polysilicon) Sacrificial layer (Oxide)

Figure 1.6: The Boston MEMS-based DM (Adapted from [25]).

Table 1.1: Typical DM specifications

DM parameter Value (Boston MEMs-based DM)

Number of actuators 1024

Array geometry hexagonal, square array Actuator pitch ≤ 500 microns

Maximum Stroke 3 microns Response time ≤ 1 ms Voltage requirement ≤100 V

Some of the common deformable mirror specifications [26] include a) num-ber of actuators, b) array geometry, c) actuator spacing, d) minimum allow-able stroke, e) response time, and f) actuator power requirement. Tallow-able 1.1 lists the values of the parameters of MEMs-based DM. The values presented are based on the Boston DM and is indicative of the state-of-the-art MEMs-based deformable mirrors.

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1.2. RESEARCH ON LIQUID-BASED DEFORMABLE MIRROR

and the flexible faceplate DMs provide similar correction performance. The residual correction error (rms) for a piston-type DM is in the range 22-24%, while modal-type DMs have a rms error of 9-10 %. In a similar study, Dal-imier and Dainty [28] have evaluated the ability of three commercially avail-able DMs to compensate the aberrations of the human eye. They compared the performance of a 37 actuator membrane mirror, a 19 actuator piezo mir-ror, and a 35 actuator bimorph mirror. For each mirmir-ror, the author fitted the measured mirror modes with Zernike polynomials and other typical oc-ular aberrated wavefronts. According to the authors, the bimorph mirror showed the lowest root mean square error. The performance of the 19 ac-tuator piezo mirror could be promising if the number of acac-tuators can be increased.

1.2 Research on liquid-based deformable mirror

l

I

2

I1

w liquid mercury container

Figure 1.7: An illustration of the mercury-based liquid deformable mirror proposed and demonstrated by Ragazzoni and colleagues [29]. An elec-trical current I2passes through an actuator, situated underneath the liquid

mercury container, while another current I1 passes through mercury. A

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1994 [29]. Figure 1.7 shows the design concept of the mercury-based DM. An electrical current I2 passes through the conducting liquid and another

current I1 runs through an actuator coil situated underneath. The principle

of the device is based on a well known fact that two long parallel wires car-rying currents exert a force on each other [30]. When the two currents flow in the same direction then an attractive force is generated, while a repul-sive force is created when the two currents flow in opposite direction. In a similar manner, an attractive or repulsive force is created between mercury and the actuator underneath. A repulsive force between the coil and mer-cury induces a deformation of the liquid surface. The amount of the surface deformation can be approximated by [29]

h = 2x10−7NI1I2

ρgld (1.2)

where I2 and I1 are the currents passing through the actuator coil and the

liquid respectively, N is the number of turns of the coil, ρ is the density of mercury, g is the acceleration due to gravity, d is the distance between the actuator and the barycenter of the fluid volume, and l is the dimension of the container. The authors have demonstrated a deformation height of 475 ± 195nm. This design concept suffers a lot of drawbacks such as a) high energy dissipation, b) heating of the liquid, c) health and safety concern with regards to the use of mercury, d) large actuator cross-talk, and e) huge overall power requirement.

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1.3. MOTIVATION OF THE RESEARCH

1.3 Motivation of the research

The planned giant ground-based telescopes will require highly sophisti-cated AO subsystems, particularly on deformable mirrors. DMs for this giant telescopes require a high number of actuators (> 10000), aside from other demands mentioned earlier. Meeting these demands using tional deformable mirrors pose a big challenge and is pushing the conven-tional DM technology to the edge. Thus, a lot of research efforts are geared toward finding innovative solutions to these demands. A good innovation would be the use of liquid as a deformable medium.

Only a limited number of research and development efforts on LDM were reported in the literature during the last three decades. This limited research efforts might have stemmed from the obvious fact that only a few liquids (e.g., mercury, liquid gallium, and some eutectic alloys of gallium) have high reflectivity. Another possible reason for the lack of research and devel-opment effort on liquid DM might be the realization that a liquid-based de-formable mirror is limited by its orientation. One can only surmise the real reasons for such lack of research and development effort on liquid-based de-formable mirrors. There is not much reported knowledge on liquid-based DM, thus there is a need to further study this subject matter. Liquids have some advantageous properties compared to thin plates and stretched mem-branes. Liquid is a highly deformable medium compared to a thin glass plate or a stretched membrane. This fact makes the use of a liquid as a deformable surface an attractive solution to the problem of low spatial res-olution inherent in most conventional DMs. Also, a liquid does not suffer from mechanical resonance and it is optically flat at its initial undisturbed state. Another motivation of this research work is to find a way to reduce the fabrication cost of a DM.

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1.4 Scope of this thesis

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2

High-Order Wavefront

Correction

This chapter presents a brief overview of turbulence effects in the atmo-sphere and describe some concepts in wavefront correction in astronomy. The modal representation and correction of distorted wavefront caused by atmospheric turbulence is discussed.

2.1 Turbulence effects

Turbulence is characterized by unstable fluid flow, where the particles of the fluid move in irregular path resulting in an exchange of momentum in one portion of the fluid into the other. This type of fluid flow is rotational, and nonlinear [33]. In the atmosphere, turbulence is manifested by the random fluctuations in wind velocity. The onset of turbulent flow can be predicted by a single parameter - the Reynolds number, Re. The Reynolds number is given by

Re = vH_ν (2.1)

where v, and H are the characteristic velocity and characteristic size of the flow, ν is the kinematic viscosity. For atmospheric turbulence, v is the mean wind velocity vw, while H is the thickness of the turbulence layer and ν is

the kinematic viscosity of air (ν = 1.5 × 10−5_m2_{/s). Values of Re above 4000,}

characterize turbulent flow. If we assume that the mean wind velocity vwis

1 m/s and the characteristic turbulent layer in the atmosphere is 1 m, then we get Re = 6667.

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2.2. KOLMOGOROV MODEL OF TURBULENCE

phase [1]. These turbulence effects cause substantial degradation of the re-sulting images of astronomical objects seen through a telescope.

2.2 Kolmogorov model of turbulence

The Kolmogorov model assumes that energy is added to the fluid medium in the form of large-scale disturbances or eddies (designated as the ”outer scale”, L0), which then breaks into smaller and smaller structures (the “inner

scale”, l0) [34]. The outer scale corresponds to the largest scale size for which

the eddies may be considered to be isotropic. The inner scale, on the other hand, corresponds to the eddy size below which dissipation of energy in the eddy through viscous effects becomes important. In terms of the Reynolds number, the two scales of turbulence are related by the simple relation,

lo=

Lo

Re3/4 (2.2)

Eq. 2.2 basically shows that as the wind velocity vwincreases, the Reynolds

number also increases and in effect the inner scale of turbulence becomes smaller. At the inner scale, the kinetic energy is dissipated into heat by molecular (viscous) friction. The inner scale can be as small as 1 mm near the ground and about 1 cm near the tropopause [35]. The value for the outer scale is still not well established, as measurements by various groups showed great disagreements [36].

In studying turbulent flows, Kolmogorov [37] introduced the concept of a structure tensor, Di j, to describe the non-stationary random fluctuations of

the flow. This structure tensor is given by [37]

Di j(r) =

D

[vi(r1+ r) − vi(r1)][vj(r1+ r) − vj(r1)]

E

(2.3) where vi and vj refer to the different components of the velocity, r is the

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turbulent flow is incompressible. With these three assumptions, the struc-ture tensor Di jbecomes a single structure function,

Dv(r) =

D

[vr(r1+ r) − vr(r1)]2

E

(2.4) Kolmogorov found that Dvhas a simpler universal form

Dv(r) = Cv2r2/3 (2.5)

where C2

v is the velocity structure constant. Eq. 2.5 is valid in the “inertial

range” , in which the separation r lies in the region lo < r < Lo, where lo

and Lo are the inner and outer scale of turbulence described earlier. The

justifications of using structure functions to study atmospheric turbulence effects are well-elucidated by Tatarskii [38] and will not be discussed further in this section.

The Kolmogorov model is only valid in the inertial range specified by the inner and outer scales of turbulence. The model also does not account for the intermittency of the turbulence, in which small-scale structures of tur-bulence occur in bursts, with intervening quiescent periods [34].

2.3 Temperature and refractive index fluctuations

The presence of turbulence in the atmosphere results in the mixing of air from different altitudes, causing temperature variations [34]. Since the index of refraction of air n depends on the temperature (aside from the pressure and the concentration of the water vapor), it also fluctuates as a consequence of these temperature fluctuations. Based on the Kolmogorov 2/3 law depen-dence on r, the structure functions for the temperature and refractive index can be written as [34]

DT(r) = C2_Tr2/3 (2.6)

Dn(r) = C2nr2/3 (2.7)

where C2

T and Cn2 are the structure parameters for temperature and

refrac-tive index respecrefrac-tively. Various models for the dependence of the refracrefrac-tive index structure constant, Cn2 on the altitude have been proposed. Among

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2.4. TEMPORAL DEPENDENCE OF AMPLITUDE AND PHASE VARIATIONS

Hufnagel-Valley model, includes an additional term for the surface layer turbulence which can be strong, especially at desert sites during the day.

2.4 Temporal dependence of amplitude and phase

vari-ations

The turbulence in the atmosphere is primarily driven by winds and local eddies. When the wind velocity, vw, is high the movement of eddies across

an optical aperture (e.g., telescope) becomes very fast. The results of this fast movement of eddies are high frequency amplitude and phase fluctuations [1]. The frequency spectrum of the amplitude variation, FA( f )is constant for

low frequencies and varies as f−8/3_{for high frequencies. It has been shown} [39] that this frequency variation can be written as

FA( f )f →0= Sg C2_N vw k2/3L7/3 FA( f )f →∞= 2.192 C2_N vw k2/3L7/3 f fg !−8/3 (2.8) where L is the propagation path length, Sg is a constant (for plane waves,

Sg is equal to 0.851, while for spherical waves, Sg = 0.191), and fg is the

characteristic frequency. Greenwood [40] has shown that the characteristic frequency, fg, can be expressed as

fg=

0.4vw

√

λL (2.9)

where λ is the wavelength of the light beam. Similarly, the power frequency spectrum for phase variation Fφ( f ) shows a f−8/3 _{power law dependence} [1],

Fφ( f )_{f →∞} = 0.0326k2f−8/3

Z L

0

C2_N(z)vw5/3(z)dz (2.10)

2.5 Modal correction of turbulence-induced aberrated

wavefront

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respectively, as it propagates through the atmosphere. Most often, it is the phase fluctuation that is being compensated by an adaptive optics systems [35].

The compensation of atmospheric turbulence-induced phase variations is implemented either by modal and zonal compensation. The zonal method compensates this phase distortion by controlling the individual segment of the aperture [41]. In contrast, the modal compensation corrects for several modes of the spatial expansion of the total phase distortion function. In several published articles [42, 43, 44], it has been shown that a modal com-pensation is much better than the zonal comcom-pensation in terms of the small error propagation introduced by the presence of noise.

In the modal compensation, the phase, φ (x), can be decomposed into a set of basis functions, φ(x) = ∞ X j = 1 ajFj(x) (2.11)

where x is the spatial coordinate, aj is the jth expansion coefficient

associ-ated with the jth basis function Fj(x). The most commonly used basis

func-tions are the Zernike polynomials [45, 41, 46, 47] and Karhunen-Loeve (KL) functions [48, 41, 49].

Zernike polynomials

In terms of the basis functions, the Zernike polynomials are commonly used because of their simple analytical expressions and because of the fact that the lower-order Zernike polynomials represent the classical aberrations such astigmatism, coma, tilt, spherical aberration, etc. Zernike polynomials are a set of polynomials defined on a unit circle [45]. These polynomials are a product of angular functions and radial polynomials. The Zernike polyno-mials can be expressed as

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2.5. MODAL CORRECTION OF TURBULENCE-INDUCED ABERRATED WAVEFRONT where Rm_n (r) = (n−m)/2 X s = 0 (−1)s(n − s)! s! [(n + m) /2 − s]! [(n − m) /2 − s]!r n−2s _(2.13)

The radial degree n and the azimuthal frequency m should satisfy m ≤ n and

n − m even. The index j orders the polynomials. For a given N, modes with

lower values of m are ordered first.

With the use of the Zernike polynomials, an arbitrary phase function, φ (r, θ), over a unit circular aperture can be expanded as

φ(r, θ) = ∞ X

j = 1

ajZj(r, θ) (2.14)

where the amplitudes ajof the Zernike components are given by

aj =

Z

r φ(r, θ) Zj(r, θ)drdθ (2.15)

Noll [45] has shown that a Zernike representation of the spectrum of the phase fluctuations (the so-called Wiener spectrum) can be obtained from Eq. 2.14 by evaluating the covariance of the expansion coefficients. Usually, in representing a turbulence-induced wavefront distortions, the Zernike com-ponents are considered as Gaussian random variables with zero mean. The analytic expression for the covariance between two Zernike polynomials Zj

and Zj′ having amplitudes a_jand a_j′ is given by [45]

D ajaj′ E = c0Γ[(n + n ′_{+ 5/3)/2] (D/r}₀₎5/3 Γ[(n − n′₊ 17 3)/2] Γ[(n′− n + 17 3)/2 i Γ[(n + n′₊ 23 3)/2] ,(2.16) f or j − j′ even D ajaj′ E = 0, f or j − j′ odd

where c0= 2.2698(−1)(n+n′−2m)/2δmm′[(n + 1)(n′+ 1)]1/2, D is the diameter of the

aperture, rois the so-called Fried parameter [50], Γ[•] is the Gamma function.

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and aj′. With Eq. 2.16, the residual error ∆Z_N, after a certain degree of phase

correction, can be easily calculated as

∆ZN = ∆1− N X j=2 D aj2 E , [rad2] (2.17)

where ∆1 = 1.03242(D/ro)5/3rad2is the total atmospheric wavefront residual

error after piston correction.

Table 2.1: Correlations of some high order Zernike coefficients ajand aj′.

Correlated Correlation

coefficient pairs (ajaj′) magnitude a2a16, a3a17 7.54 x10−4 a50a50 1.70 x10−4 a2a30, a3a29 9.52 x10−6 a200a240 6.99 x10−6 a500a500 2.30 x10−6 a2649a2505 7.57 x10−8 a6216a6216 2.40 x10−8 a2a380 5.74 x10−11 a2a498 1.74 x10−11

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2.5. MODAL CORRECTION OF TURBULENCE-INDUCED ABERRATED WAVEFRONT gain as G(N) = q ∆ZN− q ∆KLN q ∆ZN (2.18) where ∆Z

N and ∆KLN are the residual errors using Zernike basis functions and

Karhunen-Loeve basis functions respectively. It is important to note that the modal compensation technique presented here is in the context of Kol-mogorov model of turbulence. The effects of a Zernike expansion for non-Kolmogorov turbulence have been studied in [51].

An extension of this modal approach using the Zernike polynomials and the KL function to high order (N =14x103_{) was carried out. Since the KL}

functions do not have analytic expressions, a technique proposed by Rod-dier [49] to create KL functions from Zernike polynomials was used in our study. Let Czbe the covariance matrix formed from the Zernike polynomials

Zjand Zj′. The matrix C_zcan be diagonalized with a unitary matrix U such

that the unitary transformation of the matrix Czyields a completely

diago-nal matrix consisting of the KL eigenvalues of atmospheric wavefront phase expansion. This procedure can be numerically implemented by performing a singular value decomposition (SVD) of the covariance matrix Cz.

The resulting mean-square residual error as a function of the mode num-ber is shown in Figure 2.1. It can be observed from the graph that at a very high mode number N ≈ 14000 the residual errors computed using the KL functions and the Zernike polynomials are the same, indicating that the KL basis functions do not have any significant advantage over the Zernike polynomials. Figure 2.1 revealed that in fact, the use of KL functions is only advantageous up to N=7500. Dai [42] calculated the mean-square residual error only up to 500 modes. Correcting beyond 7500 modes, the relative gain of using KL functions decreases sharply. Another insight that can be drawn from the figure is that in the context of Kolmogorov model of turbu-lence, the correction of the “very high order modes” would not result in a dramatic reduction of the mean-square residual error. Using KL functions in the modal compensation revealed that the residual error curve flattens out at N > 1x104_{. This behavior might be due to the accuracy of the singular}

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0 2500 5000 7500 10000 12500 15000 10−5 10−4 10−3 10−2 10−1 100 101 Mode number σ 2 [(D/r o ) (5 /3 ) units] Zernike polynomials KL functions

(a) mean-square residual phase error after correcting N number of modes using 1) Zernike polynomial and 2) Karhunen-Loeve functions . D is the aperture diameter and rois the Fried parameter.

0 2500 5000 7500 10000 12500 15000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Mode number Gain

(b) Relative gain of using KL functions over the Zernike polynomials. The gain is defined as G(N) =

√ ∆Z N− √ ∆KL N √ ∆ZN .

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2.6. SUMMARY

quite small as the number of modes is increased. Thus, the residual error for the KL curve tends to flatten out at high mode number.

2.6 Summary

Atmospheric turbulence distorts the wavefront coming from distant astro-nomical objects. The distortion is basically caused by the fluctuation of the index of refraction of air. Most often, the behavior of atmospheric turbu-lence is studied using the Kolmogorov model which views the turbuturbu-lence as a cascade system. A lot of efforts have already been done on establishing the statistics of the turbulence-induced wavefront phase variations.

A modal expansion of the wavefront phase distortion can be established us-ing orthogonal polynomials like Zernike polynomials and Karhunen-Loeve functions. An extension of the modal approach of correcting wavefront phase distortion at a very high order (number of modes up to 14×103_),

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3

Device Principle

The principle of operation of the proposed liquid deformable mirror (LDM) is presented in this chapter. The advantages and major drawbacks of this proposed LDM are also elucidated. Conventional electrocapillary actuation and the electrocapillary actuation used in this device are discussed.The de-sign concept was initially validated using a test device and the results of this validation are presented in this chapter. Lastly, the implementation of the design concept onto different platforms is also presented.

3.1 Design concept of an electrostatic LDM

Figure 3.1(a) illustrates the concept of the proposed LDM. The device basi-cally consists of an array of vertibasi-cally-oriented capillaries. These capillaries are filled with two immiscible liquids, a conducting liquid and a viscous dielectric liquid, where the dielectric liquid overfills the top end of the capi-laries and forms a thin layer on top. A free-floating reflective membrane is utilized to remedy the problem of the low reflectivity of a pure liquid surface. Another solution to the problem of low reflectivity is to use a self-assembling reflective colloidal film spread at the surface of the dielectric liquid [32, 52].

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3.1. DESIGN CONCEPT OF AN ELECTROSTATIC LDM

The upward or downward movement of liquid inside the capillary is facil-itated by an electrocapillary pressure. This electrocapillary pressure is the result of the reduction of the interfacial energy between the wall and the conducting liquid when a potential difference, between the conducting liq-uid and the solid electrode, exists. Any movement of the liqliq-uid inside the capillary is transmitted to the top end of the channel, thereby inducing a deformation of the free liquid surface. This deformation can be tuned to physically correct for an incoming aberrated wavefront.

liquid 2 (dielectric) liquid 1 (conducting) inlet floating reflective membrane capillaries hydrophobic coating dielectric layer solid electrode V substrate

Figure 3.1: Design concept of an electrocapillary-actuated liquid deformable mirror. Inset is the detailed structure of the capillary wall showing several layers of materials.

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forms a convex meniscus, due to the nonwetting property of the hydropho-bic material. Generally, the conducting liquid has a higher surface tension value compared to the dielectric viscous liquid. This initial convex meniscus shape has a consequence on the regime of liquid movement inside the chan-nel. First, at low input voltage the liquid-liquid meniscus inside will flip from a convex to a concave shape, without triggering bulk transport of liq-uid inside the channel. When the applied voltage exceeds a certain thresh-old voltage, bulk liquid transport follows the meniscus flipping regime. These two regimes can be utilized for wavefront correction depending on the required amount of deformable mirror surface vertical displacement. The fundamental issue of surface waves is addressed in the proposed de-vice. Surface waves would not pose any serious impediment on the opera-tion of the device when the viscosity of the liquid is well chosen. In Chapter 4 of this thesis, an analysis of the surface wave propagation and damping is presented.

3.2 Advantages and disadvantages of the device

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down-3.3. ELECTROCAPILLARY ACTUATION

ward). However, such limitation can be remedied with the use of beam directing optical components. Another limitation of the device is that it is only suitable for dynamic correction since the liquid surface will not assume a static position after actuation (it will go back to its initial flat state, after some time, after actuation).

3.3 Electrocapillary actuation

The electrostatic actuation of the liquid inside the capillary is implemented using the electrocapillary effect. This electrocapillary effect has been already demonstrated in 1895 by Pellat [53] (see Figure 3.2). Pellat’s original set-up consists of two plane parallel uncoated electrodes, which are oriented vertically. These plane electrodes are spaced at a distance d and are partially filled with a dielectric liquid of mass density ρ and electric permittivity ǫr.

Gas, with permeability equal to free space, ǫo= 8.854 x10−12Fm−1, fills the

other half of the electrodes. With this configuration, the equilibrium height of rise Hpis given as [53]

Hp≈

(ǫr− ǫo)E2

2ρg (3.1)

where E is the uniform electric field between the plates, and g is the acceler-ation due to gravity. Similarly, the pressure gradient (for infinite electrodes) is given by [54] ∆Pp= " ǫoE2 2 (κ − 1)(κ − 2) 3 # (3.2) where κ is the specific inductive capacity of the dielectric liquid.

The equilibrium height of rise using the electrocapillary actuation mentioned in section 3.1 and that is used in the proposed device can be expressed as

He = ho+

dǫV2

ρgτ (3.3)

where hois the initial height of the liquid, τ is the thickness of the dielectric

coating on the electrode, ǫ = ǫrǫo(ǫris the dielectric constant of the dielectric

coating, rather than the dielectric liquid), and V is the applied voltage. The generated electrocapillary pressure is given by [55],

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Figure 3.2: Pellat’s electrocapillary set-up.

The difference between the conventional electrocapillary actuation and the actuation presented here is that in the former, as soon as the final height is reached after the actuation, it is not possible to pull back the liquid down-wards because this new height is already the preferred equilibrium height. In contrast, in the electrocapillary actuation presented in section 3.1 the downward flow of liquid is possible due to the hydrophobic layer that pvents the liquid to adhere to its raised height after the applied field is re-moved. Another difference is that the flow of the liquid on Pellat’s set-up is due to a bulk force that pushes the liquid upward. In the set-up presented in this chapter, the effect is more confined to the inner wall surface (interfacial tension reduction). The electric field existing inside the conducting liquid is almost negligible compared to Pellat’s set-up. In terms of the flow regime, the conventional electrocapillary set-up only exhibits a sudden upward flow as soon as the voltage exceeds the threshold voltage. This mechanism is slower, as shown by [56], compared to the flow regime mentioned in section 3.1.

3.4 Proof of concept

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liq-3.4. PROOF OF CONCEPT

uids (NaCl solution and n-Hexadecane), without the floating membrane. The device consists of a 64x64 array of capillaries arranged in a hexagonal

Figure 3.3: Test device with a 64x64 array of capillaries, a) the device (inside the red circle) covered with a thin layer of liquid, b) Gold membrane (inside the red circle) freely floating on top of the device.

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Figure 3.4: Top view of the hexagonal structure of the capillaries.

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3.4. PROOF OF CONCEPT

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(a) Results of the response time measurement on the test device. 1) The input signal (pulse with am-plitude of 196 V and duration of 100 ms) to the ac-tuators, b) Detected photodector (PD) signal. The equivalent liquid displacement is approximately 1 mm. The insulating liquid used is n-Hexadecane (viscosity = 3 cSt)

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(b) Results of the interferometric measurement on the test device. a) One of the frames of the recorded interferogram, b) Reconstructed shape of the liquid deformation. The liquid used in this ex-periment is silicone oil (viscosity of 500 cSt).

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a) b) c) d) e) f) g)

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3.5. PROTOTYPE

(a) PCB-based substrate (b) Si-based substrate

Figure 3.7: Different substrates for the liquid deformable mirror.

3.5 Prototype

In realizing a prototype, two different substrates were employed. These substrates are based on printed circuit board (PCB), shown in Figure 3.7a, and silicon wafer, Figure 3.7b . Figure 3.8 shows the working PCB-based prototype device attached to the holder.

Fabricating the device on a silicon wafer has some advantages. First, the surface of the wafer is optically smooth and flat. Second, the sub-millimeter pitch of the actuator can be easily realized without putting too much con-straints on the interconnects. Third, it is lightweight and hence it can be used in applications that demands lightweight components. Fourth, the de-vice can be scaled up to a few tens of centimeters. Fifth, the integration of control electronics on the wafer could be a possibility. Bulk production of such a device is possible since the processing technology is compatible with the standard MEMS processes. Such bulk production could substantially reduce the fabrication cost.

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3.6. SUMMARY

Table 3.1: Specifications of the prototype LDM

Substrate Capillary Pitch[mm] Dielectric

diameter[µm] coating

Si wafer 200 0.4 silicon dioxide

PCB 500 1.15 parylene C

rication capability of the manufacturer.

3.6 Summary

A new design of a liquid deformable mirror is proposed and presented in this chapter. The proposed device consist of an array of capillaries, filled with two immiscible liquids. The actuation mechanism used in this device is based on the electrocapillary effect. A comparison between the differences of the electrocapillary effect employed in the device and the conventional electrocapillary effect is also elucidated in this chapter. The advantages of this proposed device include 1) minimal power dissipation, 2) large stroke dynamic range, 3) initial flat surface, 4) high number of actuators, among others. However, the device is limited by its orientation and is mainly suit-able for dynamic correction.

Preliminary validation of the design concept, using a test device, showed that it can indeed achieve a large stroke (≈ 1 mm) and the rise time of the liquid surface when actuated is relatively fast (few milliseconds).

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4

Numerical Modeling

This chapter presents the numerical modeling of the 1) influence function of the liquid DM, 2) electrocapillary rise of the liquid column inside the capil-lary, and 3) surface waves.

A brief overview of the governing equations relevant to the modeling prob-lems is presented. An approximate analytic solution of the surface deforma-tion is discussed, as well as the electrocapillary rise of the liquid inside the capillary. The problem of surface waves is elaborated in detail in this chap-ter. The numerical modeling of the influence function is implemented us-ing the finite element method (FEM) with the arbitrary Lagrangian-Eulerian (ALE) technique. The influence function is basically the profile of the mirror surface when an actuator deformed it [57]. In the modeling, the shapes of the surface of a bare liquid and a floating membrane are studied and com-pared. The influence of the membrane properties (e.g., thickness and mod-ulus of elasticity) on the deformation has been considered. A brief overview of the basic concepts of FEM and ALE is also presented in this chapter.

4.1 Governing Equations

The approach used here to model the movement of liquid inside the channel and the eventual deformation of the liquid surface is based on the contin-uum model, which treats the liquid as continuous and indefinitely divisible [58]. Basically, the equations of motion of the liquid involved the momen-tum balance and the continuity condition.

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4.1. GOVERNING EQUATIONS

The equation governing the deformation of a “thin plate” is used to model the membrane deformation. The rationale behind employing such equation is that the floating membrane still exhibits some stiffness even at a thickness of just a few microns.

4.1.1 Equations of motion

In modeling the motion of liquid, we consider only the case of an incom-pressible Newtonian fluid. For the general problem of modeling fluid flow, the interested reader is referred to [59, 60, 61]. By ‘Newtonian fluid’ we mean a fluid wherein the shear force per unit area is proportional to the negative of the local velocity gradient [59]. This condition can be expressed as

τ = −µ∇v (4.1)

where µ is the dynamic viscosity of the liquid, τ is the shear stress tensor,

vis the velocity vector. For a fluid to be considered as incompressible, two conditions should be met. First, in the steady state case, the velocity of the fluid vf should be much smaller than the velocity of sound (cs), (see Eq. 4.2).

Second, for the unsteady flow, Eq. 4.2 should be supplemented by the con-dition that the characteristic transient time tc should be much greater than

the ratio between a characteristic length l (e.g., length of the flow channel) and the velocity of sound (cs) [60], (see Eq. 4.3).

vf << cs (4.2)

tc >>

l

cs (4.3)

With the conditions stated above, the general equations of motion simplify to ρ" ∂v ∂t + (v · ∇) v # = −∇p + µ∇2v + F (4.4) and ∇ · v = 0 (4.5)

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the liquid (e.g., gravity). Eq. 4.4 is generally known as the Navier-Stokes (NS) equation. Eq. 4.5 is the continuity condition for fluid flow.

Axisymmetric case:

In axisymmetric form, the NS equation is separable according to ρ ∂vz ∂t + vr ∂vz ∂r + vz ∂vz ∂z ! = −∂p_∂z + µ       1 r ∂ ∂r r ∂vz ∂r ! + ∂v 2 z ∂z2       (4.6)

for the z-component, and ρ ∂vr ∂t + vr ∂vr ∂r + vz ∂vr ∂z ! = −∂p_∂r + µ" 1 r ∂ ∂r r ∂vr ∂r ! + ∂ 2_v r ∂z2 − vr r2 # (4.7) for the r-component. Similarly, the continuity condition is rewritten as

1 rvr+ ∂vr ∂r + ∂vz ∂z = 0 (4.8) Dimensionless form:

Most often, it is advantageous to write the above equations (Eqs. 4.6- 4.8) into their dimensionless forms by setting

v = v∗V , p = p∗ρV2, _{∇ =} 1 D ! ∇∗ (r, z) = (r∗D, z∗D) , t = _D V t∗, _∇2= 1 D2 ! ∇∗2

where V is the characteristic velocity (e.g., average velocity of flow), D is the characteristic scaling length (e.g.,the diameter of the channel), (x, y, z) are the spatial coordinates. The quantity Re is the Reynolds number, Re = (DVρ/µ). Then Eqs. 4.6, 4.7, and 4.8 become

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4.1. GOVERNING EQUATIONS

4.1.2 Electrostatics

The actuation used in the device is based on the electrocapillary effect, which is basically a consequence of electrostatic interaction between charged bod-ies. The electric field and the electrostatic force are the important electro-static parameters that are needed for the calculation. The electric field is defined as the negative of the electric potential gradient,

E = −∇V (4.12)

The corresponding force is given by [54],

Fv= ρeE (4.13)

where ρe is the electrostatic charge density.

4.1.3 Thin plate equations

The deformation of a free floating membrane is modeled using the thin-plate equation. The thin-plate equation is based on Kirchhoffs bending theory. In Kirchhoff’s bending theory, the components of the stresses can be written as [62], σx = − Ez 1 − ν2 ∂2_w ∂x2 + ν ∂2w ∂y2 ! (4.14) σy = − Ez 1 − ν2 ∂2_w ∂y2 + ν ∂2_w ∂x2 ! (4.15) τxy= − Ez 1 + ν ∂2_w ∂y∂x ! (4.16) where E is the modulus of elasticity, ν is Poisson’s ratio and w is the com-ponent of the displacement vector in the z-direction. The general governing equation for the deflections occuring in thin plate bending can be written as [62], ∂4w ∂x4 + 2 ∂4w ∂x2_∂y2 + ∂4w ∂y4 = p D (4.17) where D = Eh3_/12

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H d_c r z S h b substrate liquid 2 liquid 1 L

Figure 4.1: Model geometry for determining the influence function of the deformable mirror. H is the thickness of liquid 2 (oil), liquid 1 is the con-ducting liquid, dcis the diameter of the capillary, L is the length of the

cap-illary, S is the profile of the liquid surface, and hbis the height of the liquid

surface bump.

order to determine the deflection function, it is required to integrate Eq. 4.17 with the constants of integration dependent upon the imposed boundary conditions.

4.2 First order approximation of the influence function

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4.2. FIRST ORDER APPROXIMATION OF THE INFLUENCE FUNCTION

We let the pressure Pi (the subscript i is used just to denote the pressure

term due to the capillary outflow), due to the liquid outflow, be a function of the mean velocity Dvj

E

(again here, the subscript j is just to denote the mean velocity of the outflow and does not imply a component of the velocity vector) and the spatial coordinate r. This can be written as

Pi = 1 2(ρl1 + ρl2) _D vj E2 g (r) (4.18)

where ρl1, ρl2are the densities (assumed as constants) of liquid 1 and liquid 2 respectively,Dvj

E

is the mean velocity of the outflow. The exact spatial distri-bution of the pressure Piis highly nontrivial to determine analytically.

How-ever, experimental data [63, 64] on jet studies suggest that the spatial distri-bution of the pressure can be approximated in the form g(r) = exp(−β(r/dc)2),

where dc is the diameter of the capillary and β is a fitting parameter (e.g.,

β = dc/H, ratio between the capillary diameter and the liquid thickness H).

The mean velocity of liquid outflow is the consequence of electrocapillary actuation. An electrocapillary pressure results when a potential difference between the solid electrode and the conducting liquid (see Figure 3.1) oc-curs and consequently enables the liquid to flow inside the capillary. In order to calculate the mean velocity of the liquid outflow, it is assumed that the flow inside the capillary follows Poiseuille flow [65]. The mean velocity can be expressed as,

< vj >=

dcǫoǫrV₀2

16 [η1zo+ η2(L − zo)] τ

(4.19) where ǫois the permittivity in vacuum, ǫr and τ are the dielectric constant

and thickness of the insulator respectively, V0is the applied voltage, η1and

η2are the dynamic viscosities of the lower and upper liquids respectively, zo

is the length of the lower liquid column and L is the length of the capillary. The pressure due to the outflow can now be written in the form

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−100 −80 −60 −40 −20 0 20 40 60 80 100 r* z* z* max z* FWHM

Figure 4.2: The computed influence function. The z∗and r∗are dimension-less quantities, given by z∗₌ z

H and r∗= r

dc. H is the liquid thickness and dc is the capillary diameter.

The profile of the surface deformation is expressed as

d2z dr2 = 1 2σ        1 2(ρl1+ ρl2) < vj > 2_exp        −β r_d c !2       − ρl2g(z − H)        n 1 + (dz/dr)2o3/2 (4.21) where σ is the surface tension value of the liquid 2 and g is the acceleration due to gravity. Equation 4.21 can be written in dimensionless form as

d2z∗ dr∗2 = " βEsexp(−βr∗ 2 ) − B₂o z∗_{− 1} # ( 1 + 1 β2 dz∗/dr∗ 2 )3/2 (4.22) where Bo is the Bond number given by Bo = (ρl2gd

2 c)/σ, β = dc/H, Es = (dc/4σ)(ρl1+ ρl2) D vj E2 , r = dcr∗, z = Hz∗.

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4.2. FIRST ORDER APPROXIMATION OF THE INFLUENCE FUNCTION 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 β FWH M Bo = 0.01 Bo = 0.05 Bo = 0.1 Es = 1 *

(a) Full-width-at-half-maximum (FWHM∗_{) as a function of β (β = d}

c/H)

(at different Bond numbers Boand Es = 1). The illustrated FWHM∗is a

dimensionless quantity. The actual FWHM is FWHM = dcFWHM∗.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 15 20 25 30 35 40 β R 20 Bo = 0.01 Bo = 0.05 Bo = 0.1 E_s = 1

(b) Spreading at 20 percent of the maximum deformation as a function of β (at different Bond numbers Boand Es= 1).

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Figure 4.2 shows the computed profile of the liquid-air interface based on Equation 4.22. Figures 4.3a and 4.3b show the dependence of the widths of the surface profile as a function of the parameter β at different values of the Bond number. Figure 4.3a is the full-width-at-half-maximum FWHM∗ while Figure 4.3b is the spread at 20 percent of the maximum value. The extent of spreading decreases as the parameter β increases. This means that as the liquid thickness H ≤ dc, the narrower the spreading becomes.

Simi-larly, increasing the Bond number (Bo = (ρ g (dc)2)/σ)results to a narrower

spreading. The parameter Eshas no influence on the spreading of the profile

of the interface.

4.3 Electrocapillary rise

Since the two liquids are incompressible, the electrocapillary rise time pro-vides a good estimate of how fast the rise time of the top liquid surface will be when liquid actuation starts. In calculating the capillary rise, we only consider bulk transport of liquid inside the capillary and neglect the effect of the meniscus inversion regime. Unlike the conventional capillary rise prob-lem, where the liquid rises when the capillary is in contact with the liquid bath, in our case the liquid column is initially at a certain height inside the capillary. It is only through the application of the electrocapillary force that liquid starts to rise. This situation happens because the inner wall of the capillary is hydrophobic, thereby preventing the liquid to rise by conven-tional capillary means. The rate of electrocapillary rise can be determined by ρr2 z + r 2 d2_z dt2 ! + 8ηz dz dt ! + r2ρgz − rǫ_τoǫrV2= 0 (4.23) where z = z(t) is the vertical displacement, η is the sum of the dynamic viscosities of the two liquids (η1 = ηH2O and η2 = ηgenericliquid), r is the ra-dius of the capillary, ρ is the total density of the two liquids (ρ1 = ρH2Oand

ρ2 = ρgenericliquid), and g is the acceleration due to gravity. Aside from the

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4.4. SURFACE WAVE PHENOMENA 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10 −5 Displacement [m] time [s] (a) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 −3 −2 −1 0 1 2 3 4x 10 −3 Ve locity [m/s] time [s] (b)

Figure 4.4: Electrocapillary rise of a two-liquid system (water and a generic liquid) in a capillary. The figure show the a) displacement and b) velocity as a function of time. The dynamic viscosity of the generic liquid is 2 mPa-s while the capillary diameter is 500 µm

From Figure 4.4 it can be observed that at a lower viscosity, an undamped oscillatory movement of the liquid happens. This situation is undesirable for the operation of the liquid deformable mirror. An increase of the vis-cosity value of the liquid results in a critically damped case, see Figure 4.5. The overdamped case is depicted in Figure 4.6 The electrocapillary rise be-havior of the liquid in a capillary diameter of 250 µm is depicted in Figures 4.7, 4.8, 4.9. At a lower viscosity value, see Figure 4.7, a critical oscillation is already observed. Increasing the viscosity at a smaller capillary radius immediately results in a damped oscillation of the liquid.

4.4 Surface wave phenomena

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3 3.5x 10 −5 Displacement [m] time [s] (a) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 −0.5 0 0.5 1 1.5 2 2.5 3x 10 −3 Ve locity [m/s] time [s] (b)

Figure 4.5: Electrocapillary rise of two liquid system in a capillary, a) dis-placement, b) velocity. The dynamic viscosity of the upper liquid is 10 mPa-s while the capillary diameter imPa-s 500 µm

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3x 10 −5 Displacement [m] time [s] (a) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.2 0.4 0.6 0.8 1x 10 −3 Ve locity [m/s] time [s] (b)

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4.4. SURFACE WAVE PHENOMENA 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 1 2 3 4 5 6x 10 −5 Displacement [m] time [s] (a) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 −1 0 1 2 3 4 5 6 7x 10 −3 Ve locity [m/s] time [s] (b)

Figure 4.7: Electrocapillary rise of a two-liquid system (water and a generic liquid) in a capillary, a) displacement, b) velocity. The dynamic viscosity of the upper liquid is 2 mPa-s while the capillary diameter is 250 µm

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 1 2 3 4 5 6x 10 −5 Displacement [m] time [s] (a) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3 3.5x 10 −3 Ve locity [m/s] time [s] (b)

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5x 10 −5 Displacement [m] time [s] (a) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.2 0.4 0.6 0.8 1x 10 −3 Ve locity [m/s] time [s] (b)

Figure 4.9: Electrocapillary rise of a two-liquid system in a capillary, a) dis-placement, b) velocity. The dynamic viscosity of the upper liquid is 50 mPa-s while the capillary diameter is 250 µm

the dispersion relation is different [68]. In general, for λ (wavelength) ≪ α (capillary length), surface waves can be considered as capillary waves (char-acterized by a very short wavelength and high frequency) while for λ ≫ α, gravity waves dominate. The intermediate case, where λ is close to the cap-illary length, is called ’capcap-illary-gravity waves’. Surface wave propagation is accompanied by damping. The amount of damping is dependent on the kinematic viscosity of the liquid and the wavelength.

4.4.1 Dispersion relation for an unbounded free liquid surface

Gravity waves:

To determine the dispersion relation of gravity waves, we use Eqs. 4.4 and 4.5, with µ = 0 for the inviscid case and F = ρg. Here, the term (v · ∇)v can be considered small. Hence, the pressure term reduces to the so-called Bernoulli’s equation, p = −ρgz − ρ∂φ/∂t. We express the velocity vector in terms of the velocity potential, v = ∇φ. In terms of the velocity potential, the equation of motion and the continuity condition can be rewritten as,

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4.4. SURFACE WAVE PHENOMENA ∂φ ∂z + 1 g ∂2φ ∂t2 ! = 0 (4.25)

For simplicity, we consider a two-dimensional wave propagation, along the x-axis and the vertical axis (z-axis). Along the y-axis, the motion is assumed to be uniform. The generalization of the propagation in two directions can be found in [66, 67]. Also, we consider the case λg ≪ h, where the liquid

thickness is much greater than the wavelength of the gravity waves. The expression for the velocity potential that satisfies the condition set forth in Eqs. 4.24 and 4.25 is

φ = Ae(kz)cos(kx − ωt) (4.26)

Using the expression for the potential given in Eq. 4.26 into Eq. 4.25, we obtain the dispersion relation for gravity waves at infinite depth,

ω2_o_{= gk} (4.27)

where k = 2π/λ, is the wavenumber. The corresponding velocity distribu-tion in the x and z direcdistribu-tions can be written as

vx = −A k ekzsin(kx − ωt) (4.28)

vz= A k ekz cos(kx − ωt) (4.29)

The subsequent group velocity,Vg, for the propagation of gravity waves, is

given by Vg = ∂ω/∂k = 1 2 r g k (4.30)

For the case of a finite liquid depth, h, Eqs. 4.27 and 4.30 result in

ω2_o = g k tanh(kh), (4.31) Vg = 1 2 s g k tanh(kh) tanh(kh) + kh cosh2(kh) ! (4.32) Capillary waves:

In the case of capillary waves, the pressure difference at the liquid surface follows the Laplace formula [60]. The equation of motion in terms of the velocity potential is given by

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for z = 0. Similarly, we use the form of the solution given in Eq. 4.26 and substitute this into Eq. 4.33. The resulting dispersion relation for capillary waves can be expressed as

ω2_o= σk

3

ρ (4.34)

Again, for a finite liquid depth, h, Eq. 4.34 is rewritten in the form, ω2_o= σk

3

ρ tanh(kh) (4.35)

Capillary-Gravity waves:

The dispersion relation for the capillary-gravity waves, at a finite liquid depth h, is given by ω2_o ₌ σk 3 ρ + gk ! tanh(kh) (4.36)

In this intermediate case, the gravity and surface tension both influence the frequency of oscillations of the surface waves.

4.4.2 Characteristic frequencies of oscillation in a bounded vessel

In the previous section we derived the dispersion relation of surface waves on a liquid surface, which extends indefinitely. Here, we examine the influ-ence of a bounded geometry on the surface waves.

The equation of motion (Eq. 4.25) of the liquid will still be the same. How-ever, we seek a solution of the velocity potential in the form,

φ = f (x, y)cosh(k[z + h])cos(ωt) (4.37) which satisfies the imposed boundary conditions on the sides of the vessel. Let us consider a rectangular vessel with sides x = [0, a], y = [0, b]. At the edges of the vessel, the velocity components of the velocity vector should obey the conditions,

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4.4. SURFACE WAVE PHENOMENA

In this case, not all values of the wavenumber will be permissible. The al-lowable values of the wavenumber are given by

k2 = π2 m 2 a2 + n2 b2 ! (4.40) where m and n are integers. Now, the dispersion relation for the capillary-gravity waves will be dependent on the allowed k values,

ω2 =         σ ρ π 2( m2 a2 + n2 b2 )!32 + g π2( m 2 a2 + n2 b2 )!12        tanh         h π2( m 2 a2 + n2 b2 )!12        (4.41)

4.4.3 Damping of surface waves

As presented in the previous sections, waves can be generated on the liquid surface. These waves are detrimental to the operation of the device. The presence of these unwanted waves can complicate the control of the device since there will be no quiescent surface at which optical wavefront correc-tion can take place. Here, we consider the case of using a viscous liquid and examine the effect of viscosity on the wave propagation. The disper-sion relation for a viscous liquid differs from the inviscid case by the fact that there is energy dissipation happening in a viscous liquid. A detailed derivation of the surface damping coefficient can be found in [66, 68]. The general dispersion relation for a viscous liquid can be expressed as

ω2 = ω2o− β2 (4.42)

where β2 _{= (2νk}2₎2 _{is the viscous damping coefficient, ν is the kinematic}

viscosity, ωois the dispersion relation for the inviscid case.

Adaptive optics: A liquid formable mirror for high-order wavefront correction

A Liquid Deformable Mirror

for High-Order Wavefront Correction

Clarita M. Vuelban,

Contents

1

Introduction

1.1

Survey of state-of-the-art deformable mirrors

1.2

Research on liquid-based deformable mirror

I

I1

I1

1.3

Motivation of the research

1.4

Scope of this thesis

2

High-Order Wavefront

Correction

2.1

Turbulence effects

2.2

Kolmogorov model of turbulence

2.3

Temperature and refractive index fluctuations

2.4

Temporal dependence of amplitude and phase

vari-ations

2.5

Modal correction of turbulence-induced aberrated

wavefront

2.6

Summary

3

Device Principle

3.1

Design concept of an electrostatic LDM

3.2

Advantages and disadvantages of the device

3.3

Electrocapillary actuation

3.4

Proof of concept

3.5

Prototype

3.6

Summary

4

Numerical Modeling

4.1

Governing Equations

4.2

First order approximation of the influence function

4.3

Electrocapillary rise

4.4

Surface wave phenomena