Transient response in ultra-high speed liquid lenses

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2013 J. Phys. D: Appl. Phys. 46 075102

(http://iopscience.iop.org/0022-3727/46/7/075102)

Download details:

IP Address: 128.112.36.203

The article was downloaded on 24/01/2013 at 14:30

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 46 (2013) 075102 (7pp) doi:10.1088/0022-3727/46/7/075102

Transient response in ultra-high speedliquid lensesM Duocastella and C B Arnold

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 08544, USA

E-mail: cbarnold@princeton.edu

Received 15 September 2012, in final form 29 November 2012Published 23 January 2013Online at stacks.iop.org/JPhysD/46/075102

AbstractLiquid lenses are appealing for applications requiring adaptive control of the focal length, butcurrent methods depend on factors such as liquid inertia that limit their response time to tens ofmilliseconds. A tunable acoustic gradient index (TAG) lens uses sound energy to radiallyexcite a fluid-filled cylindrical cavity and produce a continuous change in refractive powerthat, at steady state, enables rapid selection of the focal length on time scales shorter than 1 µs.However, the time to reach steady state is a crucial parameter that is not fully understood. Herewe characterize the dynamics of the TAG lens at the initial moments of operation as a functionof frequency. Based on this understanding, we develop a model of the lens transients whichincorporates driving frequency, fluid speed of sound and viscosity, and we show that is in goodagreement with the experimental results providing a method to predict the lens behaviour atany given time.

(Some figures may appear in colour only in the online journal)

1. Introduction

In several fields, including high-speed three-dimensionalmicroscopy [1], flow cytometry [2] or laser materialsprocessing [3], there exists a demand for adaptive lensescapable of changing focus at high speeds. The conventionalapproach of mechanically moving an ensemble of opticalelements to achieve a user tunable focal length is inherentlylimited by the slow response time of the physical actuation.This has spurred research for faster and more robustalternatives, particularly liquid lenses [4], which can takeadvantage of a number of mechanisms to change the focus.Successful implementations have demonstrated the changingcurvature of an interface between two immiscible fluids usingelectrowetting [5, 6], acoustic radiation force [7, 8], stimuli-responsive hydrogels [9], or adjusting the flow conditions [10];the modulation of the surface profile of an encapsulated liquiddroplet [11] by means of electrostatic [12] or pneumatic [13]forces; or the generation of a refractive index change withina liquid through the controlled diffusion of solutes [14]. Inmany conditions, such approaches provide ample performancewith response times of tens of milliseconds and focal-scanningrates below the kHz range. A much faster focusing mechanismcan be achieved using oscillating liquid lenses [15]. Insteadof waiting for the lens to reach a fixed focus condition, the

lens is in a continuous state of changing focus, and a fast-speed camera or synchronized pulsed illumination enablesone to capture images at different focal planes [15–18]. Anexample of this type of device is the tunable acoustic gradientindex of refraction (TAG) lens, an acoustically driven opticalelement [19]. In the TAG lens, a fluid-filled cylindrical cavityis acoustically excited causing a periodic standing wave inthe fluid, and consequently, a periodic change in the indexof refraction due to density modulations [20]. Since theexcitation signal can have frequencies up to the MHz range,the focal-scanning rate at steady state can also be of that order,significantly faster than other methods [18].

In previous work, the theory behind the TAG lensoperation was developed for the steady-state regime [20], andthe feasibility for using the device for imaging [18, 21, 22]and materials processing [23] was demonstrated. Here, weanalyse transients in the lens behaviour as it approaches thesteady state as a function of frequency. Based on these results,we find the time to reach steady state primarily depends onthe mechanical properties of the filling fluid as well as thefrequency of the driving signal. We present experimentalcharacterization of this effect and develop a model to describethe temporal evolution of standing waves within the TAGlens, which is in good agreement with the experimentalresults.

0022-3727/13/075102+07$33.00 1 © 2013 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 46 (2013) 075102 M Duocastella and C B Arnold

functiongenerator

beam expander

LaserWavefrontsignal (S1)

RF signal(S2)

Laser pulse(S3)

wavefrontresponse

pulsegenerator controllable

delay time 15 ns

time

10 ms

(a) (b)

RF signal (S2)

(S1)

(S3)

TAGlens

wavefrontsensor

Figure 1. Experimental setup. (a) Scheme of the setup used to characterize the transient behaviour of the TAG lens. (b) Temporal diagramof the different signals involved in the synchronization system.

2. Experimental setup

We use a TAG lens (TAG Optics Inc., TL2-B-355V) with aninner diameter of 16 mm and a length of 20 mm. We drivethe lens with a function generator (Syscomp, WGM-201) thatproduces a sinusoidal signal between 500 kHz and 1 MHz andan ac voltage of 20 V peak to peak. The lens filling fluidis silicone oil with a viscosity of 100 centistokes, a staticrefractive index of 1.403 and a speed of sound of 1000 m s−1.The characterization setup for the TAG lens is schematizedin figure 1(a). We use a pulse generator (Stanford Research,DG-535) triggered by a computer to control the delay timebetween laser pulses from a Nd : YVO4 laser (15 ns duration,355 nm wavelength, M2 = 1.3) and the RF signal that drivesthe TAG. We analyse the laser beam 1 cm after passing throughthe lens using a Shack–Hartmann wavefront sensor (ThorlabsWFS 150 C). The beam is expanded prior to the lens entranceto ensure a flat wavefront and we set an electronic aperture forthe wavefront analysis of 1 mm. In this set of experiments, thedelay times used are presented in figure 1(b). We increment thetime separation between the RF signal and the laser pulses insteps of 0.1 and 0.5 µs enabling data acquisition with sufficienttemporal resolution.

3. Study of the lens transient behaviour

The optical performance of the TAG lens critically depends onthe driving frequency selected [20]. Two different scenarioscan be distinguished: when the driving frequency correspondsto a resonance of the lens, and the off-resonance case. Ingeneral, working on resonance is preferred since it enablesrefractive powers about 2 orders of magnitude higher thanoff-resonance [20]. In this experiment, we measure the TAGresonances using impedance spectroscopy (Solartron 1260A).The plot of impedance versus frequency is presented in figure 2.The well-defined peaks in the plot, with approximatelyconstant spacing between each other, indicate the presenceof resonances in piezoelectrics, and consequently, in the TAGlens.

We first characterize the transient behaviour of the TAGwhen driven on resonance. In particular, we select the 582 kHzresonance. The evolution of the lens refractive power over

Figure 2. Determination of the TAG lens resonances. Impedancespectroscopy measurement of the lens where each peak correspondsto a lens resonant frequency.

the first moments of operation presented in figure 3(a) revealsa first change in optical power at 8 µs. We define this timeas τ , which corresponds to the time an acoustic wave takesto travel from the piezoelectric to the lens centre. After thistime, the lens optical power oscillates at the same frequencyas the driving signal, with the TAG continuously changingbetween a converging and a diverging lens [18]. Notably, theamplitude of the oscillation periodically increases every 2 τ

until, at 1.6 ms, the optical power variation is below 5%, asshown in figure 3(b). At this point, we consider that a standingwave is formed and that steady state is reached. The analysis ofthe frequency components of TAG transient computed with afast Fourier transform (FFT) algorithm presented in figure 3(c)reveals a single frequency component at 582 kHz.

For off-resonance conditions, we present two frequencies,586.5 and 595.5 kHz, separated, respectively, by 3.5 and12.5 kHz from the closest resonance at 582 kHz. We canobserve in figures 4(a) and (d) common trends with the TAGbehaviour on resonance. In particular, in both cases the lensrefractive power oscillates with varying amplitude until atabout 1.6 ms it stabilizes reaching steady state. In addition,the first change in refractive power also occurs at 1 τ and isfollowed by periodic changes in amplitude every 2 τ . However,

2

J. Phys. D: Appl. Phys. 46 (2013) 075102 M Duocastella and C B Arnold

582

normalized time

time (ms)

10

5

0

-5

-100 0.40.2 0.6 1.00.8 1.2 1.61.4 1.8

(b) transient behavior steady-state

time (ms)

-100 0.40.2 0.6 1.00.8 1.2 1.61.4 1.8

10

5

0

-5

normalized time

(d)(c)

frequency (kHz)

500 600 700550 650

Amplitude(a.u.)

1

0

2

4

3

5

6

modelexperiment

(a)

time ( s)µ

0 2015105 40353025

1 53

normalized time

1.0

0.5

0

-0.5

-1.0

Figure 3. Transient behaviour of the TAG on resonance conditions, at a frequency of 582 kHz. (a) Experimental values of the TAG lensoptical power at the first moments of operation. In the upper horizontal axis, time has been normalized by τ , being the time an acoustic wavetakes to propagate from the piezoelectric to the lens centre. (b) Measured temporal evolution of the TAG lens optical power until reachingsteady state. The inset corresponds to a detail of the first moments of the lens operation where time has been normalized by τ . (c) Frequencycomponents of the signals corresponding to (b) and (d). (d) Model that predicts the TAG optical power evolution over time.

in this case the amplitude does not monotonically increase, asobserved on resonance, but it is periodically modulated. Theperiod of the amplitude modulation at 586.5 kHz is 0.28 mswhereas at 595.5 kHz the period is 0.08 ms. Such periodicity(T ) follows the simple relationship:

T = 1

|�f | , (1)

where �f is the separation of the driving frequency from itsclosest resonance. Another relevant aspect that is worth notingis the decrease in the TAG refractive power as one movesfurther from resonance, in good agreement with experimentsat steady state [20]. Concerning the signal analysis in thefrequency domain, we can observe in figure 4(c) that for the586.5 kHz case there are two frequency components. Themost intense one corresponds to the driving frequency, whereasthe other one corresponds to the closest resonance frequency.For the 595.5 kHz case, the signal presents several frequencypeaks (figure 4(f )). The most intense one corresponds to thedriving frequency, wheres the other peaks can be related to theremaining resonance frequencies (see figure 2).

4. Modelling the lens transients

We can describe the TAG lens transients using the linearacoustic model that successfully predicted the lens steady-state

behaviour [20]. Accordingly, the density fluctuations in thelens fluid obey the damped wave equation:

∇2(c2s ρ + ν

∂ρ

∂t) − ∂2ρ

∂t2= 0, (2)

where cs is the fluid speed of sound and ν = 7/4µ, withµ being the kinematic viscosity. The force imparted by thepiezoelectric walls to the fluid can be written as the followingNeumann boundary condition:

∂ρ

∂r

∣∣∣∣r=r0

= ρ0vAωc2s

ν2ω2 + c4s

sin(ωt) − ρ0vAω2ν

ν2ω2 + c4s

cos(ωt), (3)

where r0 is the lens radius, ρ0 is the static density, vA is thevelocity of the piezoelectric wall and ω is the driving frequency.We can expand the solution to the above equation as a sum ofeigenfunctions:

ρ(r, t) = r(Asin(ωt) + Bcos(ωt))

+∞∑

m=1

J0(kmr)

[e−νk2

mt/2

(Cmsin

(kmt

2

√4c2 − ν2k2

m

)

+Dmcos

(kmt

2

√4c2 − ν2k2

m

))

+Emsin(ωt) + Fmcos(ωt)

](4)

3

J. Phys. D: Appl. Phys. 46 (2013) 075102 M Duocastella and C B Arnold

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

time (ms )

0 0.40.2 0.6 1.00.8 1.2 1.61.4 1.8

transient behaviour steady-sate

frequency (kHz)

500 600 700550 650

Am

plit

ud

e(a

.u.)

1.0

0

0.5

model

experiment

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

time (ms )

0 0.40.2 0.6 1.00.8 1.2 1.61.4 1.8

transient behaviour steady-sate

time (ms )

0 0.40.2 0.6 1.00.8 1.2 1.61.4 1.8

0

0.2

0.4

0.6

0.8

1.0

-0.4

-0.6

-0.8

-1.0

-0.2

transient behaviour steady-sate

frequency (kHz)

500 600 700550 650

Am

plit

ud

e(a

.u.)

0.15

0

0.10

0.05

model

experiment

time (ms )

0 0.40.2 0.6 1.00.8 1.2 1.61.4 1.8

0

0.2

0.4

0.6

0.8

1.0

-0.4

-0.6

-0.8

-1.0

-0.2

transient behaviour steady-sate

(a) (b)

(d)

(c)

(e)(f)

Figure 4. Transient behaviour of the TAG off-resonance. Experimental (a) and modelled (b) evolution of the lens optical power at afrequency of 586.5 kHz (3.5 kHz away from resonance). (c) Representation of both experimental data and model in the frequency domain.The signals present two frequency components, which correspond to the driving frequency and the closest resonance at 582 kHz.Experimental (d) and modelled (e) evolution of the lens optical power at a frequency of 595.5 kHz (12.5 kHz away from resonance). Thecorresponding representation of the signals in the frequency domain reveal a higher peak at the driving frequency and a series of smallerpeaks at the closest resonant frequencies.

where km = xm/r0 is the location of the mth zero of J1(x) = 0,and the coefficients are given by

A = ρ0vAωc2s

ν2ω2 + c4s

(5)

B = − ρ0vAω2ν

ν2ω2 + c4s

(6)

Em =(

2ρ0vAω

r20 J 2

0 (kmr0)

)

×[k2m(c4

s − ω2ν2) − c2s ω

2

ν2ω2 + c4s

ω2∫ r0

0J0(kmr)r2 dr

− (ω2 − c2s k

2m)

∫ r0

0J0(kmr) dr

]

×[ω2(ω2 − 2c2

s k2m) + k4

m(ν2ω2 + c4s )

]−1

(7)

Fm =(

2ρ0vAω2ν

r20 J 2

0 (kmr0)

)

ω2−2c2s k

2m

ν2ω2+c4s

ω2∫ r0

0 J0(kmr)r2 dr − k2m

∫ r0

0 J0(kmr) dr

ω2(ω2 − 2c2s k

2m) + k4

m(ν2ω2 + c4s )

. (8)

Note the comparison with [20]. The only undeterminedcoefficients are Cm and Dm, which depend on the initialboundary conditions. Considering ρ(r, 0) = 0 and

∂ρ(r,t)

∂r|t=0 = 0, we obtain

Dm = −2B∫ r0

0 J0(kmr)r2 dr

r20 J 2

0 (kmr0)− Fm (9)

Cm = 2

km

√4c2

s − ν2k2m

×(−2ωA

∫ r0

0 J0(kmr)r2 dr

r20 J 2

0 (kmr0)+

νk2mDm

2− ωEm

).

(10)

Since the experimental measurements are taken at thevicinity of the lens centre r ≈ 0, and for low viscosityfluids where ν � c2

s /ω, the only terms that will contributesignificantly to the final solution are Dm and Fm, whichyields to

ρ(0, t) =∞∑

m=1

(e−νk2

mt/2Dmcos(kmcst) + Fmcos(ωt))

. (11)

The relative small changes in the fluid density comparedwith the static density enable one to use a linearized versionof the Lorentz–Lorenz equation that relates fluid density withrefractive index [24]. Therefore, the temporal evolution of thefluid refractive index (n), and consequently, the lens opticalpower (P ), will be proportional to (11)

P ∝ n ∝ ρ �⇒ P = Cρ (12)

4

J. Phys. D: Appl. Phys. 46 (2013) 075102 M Duocastella and C B Arnold

9 2 9211511 251 3 5 7 13 17 19 23 27 31

normalized time

-1.0

-0.5

0

0.5

1.0

time (ms )

0 0.05 0.10 0.15 0.20 0.25

-1.0

-0.5

0

0.5

1.0

time (ms )

0 0.05 0.10 0.15 0.20 0.25

9 2 9211511 251 3 5 7 13 17 19 23 27 31

normalized time(a)

frequency (kHz)

900600 700 800

Am

plit

ud

e(a

.u.)

0.15

0.10

0.05

0.20

0

mode section 4

experiment

(b) (c)

Figure 5. Prediction with the TAG lens model. (a) Temporal evolution of the TAG at a frequency of 680 kHz as computed using the modeldescribed in section 4. (b) Experimental data obtained at 680 kHz, in good agreement with the predicted behaviour. (c) The correspondingrepresentation of (a) and (b) in the frequency domain confirms the good agreement between model and experiment.

with C being the proportionality constant between density andoptical power.

The numerical computation of (11) for the resonant andoff-resonant cases experimentally analysed are presented infigures 3(d), 4(b) and 4(e). We only use C as a fittingparameter. Note, though, that it is possible to derive thefundamental terms in C as done in prior work [20].

We can observe in all cases a very good agreement betweenmodel and experiment. Notably, the model describes thecontinuous oscillation of the TAG optical power, the changes inthe oscillation amplitude every 2 τ as well as the time to reachsteady state. In addition, the plots of the computed transientsand experimental data in the frequency domain presented infigures 2(d), 3(c) and 3(f ) reveal that both signals have thesame frequency components. Therefore, the model not onlyprovides good visual agreement but quantitatively reproducesthe experiment.

The capability of the model to predict the lens transientsis analysed in figure 5. We first compute the temporalresponse of the lens at a frequency of 680 kHz (figure 5(a)).Theapparently random signal obtained presents the same trends aspreviously described, including the change in amplitude every2 τ . By collecting the experimental data at this frequency weshow in figure 5(b) that the experiment follows the behaviourpredicted by the model. Model and experiment also have thesame frequency components, as presented in figure 5(c). Weconclude that the model can be used to predict transients giventhe dimensions of the lens and the properties of the fillingfluid. The capability to predict the lens transient enables oneto operate the lens prior to reaching steady state. In fact, themodel opens the door to the selection of user-defined transientswhich can be used to shape light in order to induce uniqueeffects on materials.

5. Additional interpretation of the lens transients

The solution found in (11) represents the temporal evolutionof the TAG lens as a Fourier–Bessel expansion with time-dependent coefficients. A more intuitive interpretation ofthe observed transient behaviour consists in considering theTAG lens as a cylindrical resonator in which waves with the

frequency of the driving signal travel back and forth interferingwith each other (figure 6). In this way, the first change in theTAG optical power occurs when the first wave, generated atthe piezoelectric, reaches our detector placed at the centre ofthe lens (time 1 τ ). This wave continues propagating towardsthe piezoelectric, reaching it at 2 τ . At this moment, the waveinterferes with an incoming new wave. The resulting wavepropagates to the centre of the lens, arriving at this point at3 τ . Since this resulting wave has a different amplitude thanthe first wave, we observe a change in refractive power. Thisprocess is repeated successively, producing variations in theamplitude of the lens power every 2 τ . On resonance, thesuccessive interferences are constructive, and the waves addup to produce a monotonic increase in the amplitude. Off-resonance, interferences are partially constructive (or partiallydestructive), and the additive process between waves is lessefficient and is time dependent. As a result, a periodicmodulation in the amplitude is observed and the final refractivepower is lower than on resonance. Since the fluid has a certainviscosity, waves suffer losses in successive roundtrips due toviscous damping. Such damping ultimately balances out thegain injected in the cavity through the driving signal, leadingto steady state.

According to this interpretation, we can write the equationcorresponding to the additive process of cylindrical waves atthe vicinity of the lens centre as

ρ(0, t) ∝i(t)∑m=0

γ msin (−ωt + mφ) ,

i(t) =

0, for 1 � t < 3τ

1, for 3τ � t < 5τ

...

(13)

where φ is the phase shift between waves and γ is thedamping factor. On resonance, waves will add up and weassume a 0 phase shift. Off resonance, the phase shiftis 2τω. By rewriting ω = ωresonance + �ω, where �ω

expresses the proximity to resonance, we obtain φ = 2τ�ω.Notably, (13) with this value of φ corresponds to a wave withperiodicity T = 2π/�ω = 1/�f , in good agreement with

5

J. Phys. D: Appl. Phys. 46 (2013) 075102 M Duocastella and C B Arnold

t=0

t=3τ

piezoelectri c

fluid

t=2τ

t=1τ

sound waves

Figure 6. Scheme of the model based on considering the TAG as a resonator. Cylindrical waves travel back and forth in the TAG, interferingwith each other and producing periodic changes in the lens optical power every 2 τ .

model section 4

model section 5 model section 5

frequency (kHz)

900600 700 800

Am

plit

ud

e(a

.u.)

0.15

0.10

0.05

0.20

0

model section 4

frequency (kHz)500 600 700550 650

Am

plit

ud

e(a

.u.)

1

0

2

4

3

5

6

(b)(a)

Figure 7. Comparison between the two models that described the TAG lens temporal evolution. Plot of the frequency components of thelens transient at (a) resonance conditions, at 582 kHz; and (b) off-resonance conditions, at 680 kHz. Note in both cases the good agreementbetween models.

the empirical relationship described in (1). Concerning thedamping factor, we assume the classical attenuation coefficientfor acoustic waves [25], which in this particular case can bewritten as

γ = e−2τν ω2

c2s (14)

with 2 τ being the time delay between successive a wave.The calculation of (13) is in good agreement with

experiment and with the model described in section 4, as canbe observed in figure 7 for both resonance and off-resonanceconditions. In addition, the model allows further insights aboutthe TAG transient behaviour and in particular about the time toreach steady state. Using (13), the amplitude (A) of a travellingwave after l roundtrips can be rewritten as

Al = A0

l∑i=0

γ i. (15)

Assuming that steady state is reached when amplitudechanges between consecutive waves are below 5%, and usingelementary algebra, yields to

Al+1 − Al

A0= γ l < 0.05 �⇒ tsteady state ≈ −c2

s ln(0.05)

ω2ν.

(16)

According to this equation, the time to reach steady state for theexperimental conditions analysed in figure 3 is 1.3 ms, which isin close agreement with the measured value. In addition, (16)also unveils strategies to reduce the lens transient time. Thisincludes driving the lens at higher frequencies, or using fluidswith higher viscosities or lower speeds of sound. In any of thesemethods the number of cavity roundtrips that acoustic wavesexperience is reduced, which can result in loss of optical power.In order to compensate for this effect, one can increase thegain per roundtrip by increasing the power of the lens drivingsignal.

6

J. Phys. D: Appl. Phys. 46 (2013) 075102 M Duocastella and C B Arnold

6. Conclusion

Under the conditions studied in this paper we havedemonstrated that the TAG lens can reach steady state in a 1 mstime-scale. Once in steady state, using pulsed illuminationit is possible to select between different focal lengths intime scales below 1 µs. The duration of the lens transientresponse depends on the driving frequency as well as onthe properties of the filling fluid including speed of soundand viscosity. The lens temporal behaviour also dependsstrongly on the driving frequency. On resonance, the lensoptical power monotonically increases, whereas off resonanceit can periodically vary or even present apparently randomforms. Two robust models have been used to describe the lenstransients that are in good agreement with all the experimentalconditions analysed. The models enable one to predict thetime to reach steady state as well as to determine the lensoptical power at any given time prior to reaching steady state.This makes possible to operate the lens during transients. Byappropriately selecting a particular temporal evolution of thelens optical power one could shape light in a fast and controlledway, with important applications in relevant areas such as laserprocessing or microscopy.

Acknowledgments

The authors acknowledge financial support from AFOSR.In addition, we acknowledge Christian Theriault from TAGOptics Inc. for providing the lens.

References

[1] Yang L, Raighne A M, McCabe E M, Dunbar L A andScharf T 2005 Confocal microscopy usingvariable-focal-length microlenses and an optical fiberbundle Appl. Opt. 44 5928–36

[2] Gerstner A O, Laffers W and Tárnok A 2009 Clinicalapplications of slide-based cytometry: an updateJ. Biophoton. 2 463–9

[3] Kuwano R, Tokunaga T, Otani Y and Umeda N 2005 Liquidpressure varifocus lens Opt. Rev. 12 405–8

[4] Graham-Rowe D 2006 Liquid lenses make a splash NaturePhoton. sample 2–4

[5] Kuiper S and Hendriks B H W 2004 Variable-focus liquid lensfor miniature cameras Appl. Phys. Lett. 85 1128–30

[6] Miccio L, Finizio A, Grilli S, Vespini V, Paturzo M,De Nicola S and Pietro Ferraro 2009 Tunable liquidmicrolens arrays in electrode-less configuration and theiraccurate characterization by interference microscopy Opt.Express 17 2487–99

[7] Oku H and Ishikawa M 2009 High-speed liquid lens with 2 msresponse and 80.3 nm root-mean-square wavefront errorAppl. Phys. Lett. 94 221108

[8] Koyama D, Isago R and Nakamura K 2010 Compact,high-speed variable-focus liquid lens using acousticradiation force Opt. Express 18 25158–69

[9] Dong L, Agarwal A K, Beebe D J and Jiang H 2006 Adaptiveliquid microlenses activated by stimuli-responsivehydrogels Nature 442 551–4

[10] Mao X, Waldeisen J R, Juluri B K and Huang T J 2007Hydrodynamically tunable optofluidic cylindrical microlensLab Chip 7 1303–8

[11] Song W, Vasdekis A E and Psaltis D 2012 Elastomer basedtunable optofluidic devices Lab Chip 12 3590–7

[12] Binh-Khiem N, Matsumoto K and Shimoyama I 2008Polymer thin film deposited on liquid for varifocalencapsulated liquid lenses Appl. Phys. Lett.93 124101

[13] Chronis N, Liu G, Jeong K H and Lee L 2003 Tunableliquid-filled microlens array integrated with microfluidicnetwork Opt. Express 11 2370–8

[14] Mao X, Lin S S, Lapsley M I, Shi J, Juluri B K andHuang T J 2009 Tunable liquid gradient refractive index(l-grin) lens with two degrees of freedom Lab Chip9 2050–8

[15] McLeod E, Hopkins A B and Arnold C B 2006 Multiscalebessel beams generated by a tunable acoustic gradient indexof refraction lens Opt. Lett. 31 3155–7

[16] Lopez C A and Hirsa A H 2008 Fast focusing using apinned-contact oscillating liquid lens Nature Photon.2 610–3

[17] Stan C A 2008 Liquid optics: oscillating lenses focus fastNature Photon. 2 595–6

[18] Mermillod-Blondin A, McLeod E and Arnold C B 2008High-speed varifocal imaging with a tunable acousticgradient index of refraction lens Opt. Lett. 33 2146–8

[19] McLeod E and Arnold C B 2008 Optical analysis oftime-averaged multiscale bessel beams generated by atunable acoustic gradient index of refraction lens Appl. Opt.47 3609–18

[20] McLeod E and Arnold C B 2007 Mechanics and refractivepower optimization of tunable acoustic gradient lensesJ. Appl. Phys. 102 033104

[21] Olivier N, Mermillod-Blondin A, Arnold C B andBeaurepaire E 2009 Two-photon microscopy withsimultaneous standard and extended depth of fieldusing a tunable acoustic gradient-index lens Opt. Lett.34 1684–6

[22] Duocastella M, Sun B and Arnold C B 2012 Simultaneousimaging of multiple focal planes for three-dimensionalmicroscopy using ultra-high-speed adaptive opticsJ. Biomed. Opt. 17 050505

[23] Mermillod-Blondin A, McLeod E and Arnold C B 2008Dynamic pulsed-beam shaping using a tag lens in the nearUv Appl. Phys. A 93 231–4

[24] Born M and Wolf E 2003 Principles of Optics:Electromagnetic Theory of Propagation, Interference andDiffraction of Light (Cambridge: Cambridge UniversityPress)

[25] Pierce A D 1989 Acoustics an Introduction to its PhysicalPrinciples and Applications (Woodbury NY: AcousticalSociety of America)

7