Variable-focus lens with 1-kHzbandwidth

Hiromasa OkuPRESTO, Japan Science and Technology Agency,

4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan

HiromasaOku@ipc.i.u-tokyo.ac.jp

Koichi HashimotoDepartment of System Information Sciences, Graduate School of Information Sciences,

Tohoku University, Aramaki aza Aoba 09, Aoba-ku Sendai-city Miyagi-pref. 980-8579, Japan

Koichi Hashimoto@ipc.i.u-tokyo.ac.jp

Masatoshi IshikawaDepartment of Information Physics and Computing, Graduate School of Information Science

and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

MasatoshiIshikawa@ipc.i.u-tokyo.ac.jp

http://www.k2.t.u-tokyo.ac.jp/

Abstract: This paper proposes a variable-focus lens with 1-kHz band-width. The lens transforms its shape rapidly using the liquid pressuregenerated by a piezo stack actuator. This mechanism also includes a built-inmotion amplifier with high bandwidth to compensate for the short workingrange of the piezo stack actuator. Prototypes have been developed to validatethe proposed design. A 1-kHz bandwidth of the lenses was confirmed bymeasuring the frequency responses. Refractive power ranging from−1/167to 1/129 mm−1 and a maximum resolution of 12.3 cycles/mm were attained.

© 2004 Optical Society of AmericaOCIS codes: (230.0230) Optical Devices; (230.3990) Microstructure Devices.

References and links1. J. Wals, J. Dovic, A. J. Niessen, M. Rieck, and R. M. G. Rijs, “Fast-Access Optical Drive,” Japanese J. Appl.

Phys. Part 139, 862–866 (2000).2. N. Sugiura and S. Morita, “Variable-focus liquid-filled optical lens,” Appl. Opt.32, 4181–4186 (1993).3. D.-Y. Zhang, V. Lien, Y. Berdichevsky, J. Choi, and Y.-H. Lo, “Fluidic adaptive lens with high focal length

tunability,” Appl. Phys. Lett.82, 3171–3172 (2003).4. N. Chronis, G. L. Liu, K.-H. Jeong, and L. P. Lee, “Tunable liquid-filled microlens array integrated with mi-

crofluidic network,” Opt. Express11, 2370–2378 (2003),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-19-2370.

5. L. G. Commander, S. E. Day, and D. R. Selviah, “Variable focal length microlenses,” Opt. Commun.177,157–170 (2000).

6. R. Day, E. Lacot, F. Stoeckel, and B. Berge, “Three-dimensional sensing based on a dynamically focused laseroptical feedback imaging technique.,” Appl. Opt.40, 1921–1924 (2001).

7. T. Kaneko, T. Ohmi, N. Ohya, N. Kawahara, and T. Hattori, “A New, Compact and Quick-Response DynamicFocusing Lens,” inTRANSDUCERS’971, 63–66 (1997).

8. Leonard Meirovitch, “Natural modes of vibration,” inAnalytical Methods in Vibrations(Macmillan, New York,1967).

9. National Astronomical Observatory, ed., inRika nenpyo (Chronological Scientific Tables),(Maruzen, Tokyo,1995).

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2138#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

10. Japan Institute of Metals, ed., inKinzoku Binran (Metals Handbook),(Maruzen, Tokyo, 1990).11. William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, “Modeling of data,” inNU-

MERICAL RECIPES in C, Second Edition(Cambridge University Press, 1992).

1. Introduction

High-speed focusing is required in many application fields as a result of the recent developmentand widespread use of optical devices. For example, rapid axial scanning of the focal planeis important for confocal scanning microscopes to acquire three-dimensional information ofobjects at high speed. High-speed focusing can also realize three-dimensional displays usingaxial scanning of the image plane position. In these application fields, since millisecond-orderscanning is required, we assume that 1-kHz bandwidth is adequate for our proposed high-speedfocusing device.

However, no previous focusing mechanisms have yet attained a 1-kHz bandwidth. For exam-ple, the axial tracking mechanism of optical disks, which is one known fast-focusing system,has a first resonant frequency of around 100 Hz [1].

A lens that realizes variable focusing by changing its shape is considered a key device toachieve high bandwidth. Although several studies have reported such variable-focus lens struc-tures [2, 3] and micro variable-focus lens arrays [4, 5], few have aimed at achieving high band-width. Day et al. have developed a variable-focus lens based on the modification of the wetta-bility of a surface [6]; this lens took 0.1 s to scan the entire focal-length range from minimum tomaximum. Kaneko et al. have developed variable-focus lenses with deformable glass surfaces[7] using a PZT bimorph actuator; however, their prototype was limited to a bandwidth of 150Hz.

In this paper, we propose a new structure of a variable-focus lens that can realize 1-kHzbandwidth.

2. Focusing mechanism

Fig. 1. (a) Schematic diagram of the proposed lens structure, and (b) its focusing mecha-nism.

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2139#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

The proposed lens focuses rapidly by changing the lens shape. Figure 1 shows its structureand focusing mechanism. The lens surfaces are composed of two parallel transparent disks,at least one of which is deformable. The space between the disks is filled with a transparentliquid. One disk is deformed by the pressure difference between the liquid and the atmosphere.Convex and concave lenses can be realized by increasing and decreasing the pressure to pushout or pull in the disk, respectively. The pressure of the liquid is generated by an actuator whichthrusts against an elastic plate called acylinder. By deforming the cylinder, pressure is appliedto the liquid and transmitted to the lens surfaces.

The elastic lens surface and the cylinder must be designed to have a first resonance frequencyof transverse vibrations that is sufficiently higher than 1 kHz. This is because a lower resonancefrequency causes resonance vibrations, which makes it difficult to rapidly control the focallength.

A bandwidth of more than 1 kHz is also required for the actuator. One suitable actuator isa piezo stack actuator, which has a high bandwidth (up to∼ 10 kHz). However, the workingrange of the piezo stack actuator is too short (∼ 10 µm) for actuating the lens surface directly.To solve this problem, the lens structure includes a built-in motion amplifier making use ofthe incompressibility of the liquid: The cross-sectional area of the cylinderS is designed to beseveral tens of times larger than that of the lens surfaces, and therefore the distortion of thelens surface is approximatelyS/s times that of the cylinder.

3. Theoretical analysis of the dynamic and static deformation of the elastic surfaces

Analysis of the static and dynamic behavior of deformable parts, such as the lens surfacesand the cylinder, is essential to properly design the lens. Thus, a theoretical analysis of thedeformation will be explained here to prepare for further discussion on the design optimizationin the next section.

a

d

z

r

O

d/2

Fig. 2. Disk model.

Elastic circular plates whose edges were rigidly fixed to the container of liquid were adoptedas the lens surface and the cylinder. Hence, the lens surface is modeled as a disk of radiusa andthicknessd, formed of a material of Young’s modulusE, densityρ, and Poisson’s ratioν ; theperiphery of the disk is completely fixed. A schematic diagram of this model is shown in Fig. 2.

According to the theory of vibration [8], the natural frequencyω of such a disk is

ωm,n = k2m,nd

√1

12(1−ν2)Eρ

(m,n = 0,1,2, · · ·). (1)

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2140#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

Here, m and n indicate the vibration mode. The parameterkm,n is the solution of

Im(km,na)Jm−1(km,na)−Jm(km,na)Im−1(km,na) = 0. (2)

where Im is the m-th modified Bessel function of the first kind, andJm is the m-th Besselfunction of the first kind. A vibration mode withm= 0,n = 1 corresponds to the first naturalvibration. Note thatkm,n is proportional to 1/a because it is a solution of Eq. (2).

When a static pressureP is applied to one surface of the disk, the deformationz= R(r) isdenoted by the following polynomial:

R(r) =a4P64D

{( ra

)4−2

( ra

)2+1

}, (3)

whereD denotes the flexural rigidity, given by:

D = Ed3/12(1−ν2). (4)

4. Optimizing the design of the proposed lens

A high natural frequency of transverse vibrations is preferable for the elastic plates forming thelens surface and the cylinder. This is because, resonance vibrations easily occur in a disk witha low natural frequency. Consequently, complicated feedback control is needed to control thelens at high-speed, which increases the cost and complexity of the system. At the same time, thelens is also required to have a focal length of sufficient range. Hence, the lens surfaces and thecylinder need to have not only a high first natural frequency but also high elasticity. However, ahigh elasticity is not consistent with a high natural frequency. Therefore, the trade-off betweenthese properties should be considered.

To evaluate the elasticity of the disk, we define a new parameterS, which is the ratio of thecurvature at the center of the disk to the static pressure of the liquid. The refractive power of thelens, which is the inverse of the focal length, is related to the curvature of the disk. Assuming aplano-convex or plano-concave lens, the relation is denoted by 1/ f = (nr−1)c, wherenr is therefractive index of the material that forms the lens. Thus,S represents the refractive power perunit pressure; moreover, largeS indicates large elasticity.

According to Eq. (3), the curvature at the center of the diskc is

c =a2

16DP. (5)

We can obtainSusing the relations forc andD in Eq. (5) and Eq. (4), as follows:

S=cP

=a2

16D=

34

a2(1−ν2)Ed3 . (6)

Table 1. Relations between parameters

radiusa thicknessd materialE,ρ,ν

ω0,1 ∝ 1/a2 ∝ d ∝√

Eρ(1−ν2)

S ∝ a2 ∝ 1/d3 ∝√

1−ν2

E

Table 1 summarizes how the first natural frequencyω0,1 andSdepend on each parameter ofthe disk. The disk can be properly designed by considering these relations.

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2141#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

As for the disk dimensions, that is, radiusa and thicknessd, a thin, small disk is betterthan a thick, large disk. To explain the reason, consider how to attain a natural frequency fourtimes higher for a given disk of radiusa0, thicknessd0, first natural frequencyω0 and elasticityS0. Assuming a change in radius to new radiusa, a is required to bea0/2 to attain naturalfrequency four times higher. Insertinga = a0/2 in each factor in table 1 yieldsω = 22ω0 andSr = S0/22 = S0/4. In contrast, if we modify the thickness to new thicknessd, d is requiredto be 4d0. Inserting this in the relations in table 1 yieldsω = 4ω0 andSt = S0/43 = S0/64.According to the result,Sr is 16 times larger thanSt. This indicates that modifying the radiuscan attain a much higher elasticity than modifying the thickness. Hence, to attain a desired firstnatural frequency, modifying the radius is better than modifying the thickness. Therefore, a lenswith smaller lens radius can attain a given refractive power with less liquid pressure.

10

10

10

10

10

10

10

10 100 1000 10000 100000

Zn

Al

Cd

Glass (potash soda zinc)

Glass (potash soda lead)

Glass (alumino-silicate)

Glass (borosilicate)

Glass (high lead)

Glass (fused silica)

Au

Ag

Si N

Fe

Cu

Ni

Polyethylene

Polystylene

Polydimethylsiloxane

(PDMS)

0

1

2

3

4

5

6

3 4

First natural frequency ω0,1 [arbitrary unit]Low freq. High freq.

Co

effic

ien

t o

f th

e e

lasticity [

arb

itra

ry u

nit]

Rig

idF

lexib

le

Fig. 3. Materials [4][9][10]: Glasses were arbitrarily selected from a thin sheet glass cat-alogue (Corning Inc.). Metals are plotted because they can be used as a material for thecylinder.

From these results, we can also derive adequate materials for the disk. Figure 3 shows a scat-ter plot of various materials depending on their first natural frequency and elasticity. Materialsplotted at the upper right, such as glasses and plastics commonly used for lenses, are adequatefor the disk. This is one advantage of the proposed lens, because common lens fabricationtechniques and materials, such as optical coatings can be used.

5. Experimental results

5.1. Fabrication of the prototypes

Based on the above discussion, two kinds of prototypes were fabricated: prototypes #1 and #2.Prototype #1 was fabricated simply to determine whether the proposed structure could achievea 1-kHz response. Thus, its structure was designed to be simple and easy to fabricate, butnot durable. After confirming the response speed of prototype #1, prototype #2 was designed

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2142#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

Table 2. Mechanical specifications

Prototype number #1 #2Lens diameter [mm] 5.0 7.0Lens plate thickness [µm] 20 40Lens plate material Silica glass Silica glassFirst natural frequency of the lens plate [kHz] 7.2 7.2Cylinder diameter [mm] 27 42Lens thickness* [mm] ∼ 34 3.5

*Assuming the lens surface is flat.

to measure more detailed characteristics, such as surface profile and resolution. Mechanicalspecifications of these prototypes are shown in Table 2.

Prototype #1 had a body composed of a silica glass tube as a rigid container, and it hadtwo 5-mm diameter lens surfaces placed facing each other at the opposite sides of the tube. Aschematic diagram of the structure is shown in Fig. 4(b). The inner diameter of the glass tubewas 27 mm and the lens thickness was about 34 mm. One lens surface was a thin elastic silicaglass disk with a natural frequency of 7.2 kHz, which was sufficiently higher than the required1 kHz. The other, a rigid surface, was a 1.0-mm thick silica glass plate. The elastic lens surfaceconsisted of a 20-µm thick and 5.5-mm diameter glass plate bonded on a 1.0-mm thick glassplate having a 5-mm diameter hole at the center, as shown in the inset of Fig. 4(b).

5.1.1. Prototype #1

5.5φ

5.0φ

7.5φ

20µm

10x10x0.5

5

φ27

0.5 Elastic lens surfacefabrication process

Cylinder (SUS304)

Unit: mm

Optical axis

The piezostack actuatorpresses here

Tubes with a stopcock(glass)

Optical axisLens surface

10mm

(a) (b)

4

Body(glass)

glue

glue

glue

glue

Fig. 4. Prototype #1 :(a) photograph, and (b) schematic figure of the structure and fabrica-tion process

One end of the glass tube was covered with a 5-mm thick acrylic plate with two inlets for thetransparent liquid, and the other end was covered with a 0.5-mm thick stainless steel (SUS304)deformable plate with a diameter of 27 mm, which functioned as the cylinder. The liquid inletswere connected to conventional stopcocks of the type normally used in chemical experiments.

The container was filled with a fused silica index-matching oil (Cargille laboratories) whoserefractive index matched that of fused silica (1.4587 at a wavelength of 5893A).

Prototype #2 was designed to overcome a drawback encountered in prototype #1: Through

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2143#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

frequency response measurements of prototype #1, it was found that the structure of the elasticlens surface was very fragile. In fact, prototype #1 was broken just after the frequency responsemeasurement; it lacked sufficient durability to measure various characteristics, such as the lenssurface deformation shape, resolution, etc.

5.1.2. Prototype #2

Fig. 5. Prototype #2:(a) photograph, and (b) schematic diagram of the structure and fabri-cation process.

To solve this problem, the elastic lens surface of prototype #2 was designed to be much moredurable than that of prototype #1; its structure was as follows. The elastic lens surface consistedof one 40-µm-thick silica glass plate sandwiched between two identical 480-µm-thick silicaglass substrates. The two substrates had a circular hole 7-mm in diameter at identical positionsso that the 40-µm-thick glass plate could function as an elastic lens surface with 7-mm diameterand 7.2-kHz natural frequency. A schematic diagram of the fabrication process is shown in theinset of Fig. 5(b). This sandwich structure provided sufficient durability for the elastic lenssurface. A silica glass plate of thickness 1 mm was adopted as the other lens surface.

A silica glass disk of diameter 42 mm and thickness 1.41 mm was adopted as the cylinderpart; its natural frequency was also 7.2 kHz. As discussed previously, glasses are one suitablematerial for use as the elastic plates. However, because the center of the cylinder is thrust by apiezo stack actuator, this may cause the glass plate to crack. To prevent this, a thin metal sheetwas placed on the center of the glass plate.

The body of prototype #2 was made from stainless steel (SUS304). The lens thickness wasdesigned to be 3.5 mm when the elastic surface was flat. The body had two inlets for thetransparent liquid, which were connected to diaphragm valves (Fujikin). Fused-silica matchingoil was adopted as the transparent liquid, as in prototype #1.

5.2. Dynamic response

The frequency responses of the prototypes were measured. Both prototypes were actuated bya piezo stack actuator (P-841.10, Physik Instrumente) with a 15-µm working range and an 18-kHz natural frequency. This actuator includes a strain gauge position sensor with high resolution(¡ 1 nm). The input was the displacement of the piezo stack actuator obtained by the positionsensor, and the output was the displacement of the center of the deformed lens surface, asmeasured by a laser displacement meter (LC-2430, Keyence). A personal computer controlled

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2144#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

10 100 1000

10

1

0.1

Gai

n

0.60.40.2

0-0.2-0.4-0.6

Pha

se [r

ad]

Frequency [Hz]

0.1

1

10

10 100 1000Frequency [Hz]

Pha

se [r

ad]

Gai

n

-0.4

-0.2

0

0.2

0.4

(a) (b)

Fig. 6. Bode plot of frequency response for prototype #1 (a) and prototype #2 (b). The inputwas the displacement of the piezo stack actuator, and the output was the displacement ofthe center of the lens surface. The gain is the ratio of the maximum amplitude of the lens tothe maximum amplitude of the piezo stack actuator. In this plot, the gain was normalizedby dividing the entire set using a certain value. The phase plot shows the shift between theoutput and the input.

0 0.5 1 1.5 2 2.5 3�-15

�-10

�-5

0

5

10

15Piezo stack actuator

Lens surface

Dis

plac

emen

t [µm

]

Time [ms]0 0.5 1 1.5 2 2.5 3

�-15

�-10

�-5

0

5

10

15Piezo stack actuator

Lens surfaceD

ispl

acem

ent [

µm]

Time [ms]

(a) (b)

Fig. 7. Trajectories of the lens surface and the piezo stack actuator displacement during themeasurement of frequency response of 1 kHz for prototype #1 (a) and prototype #2 (b).

these two instruments. The Bode-plot of prototype #1 shown in Fig. 6(a) demonstrates that theproposed lens maintained a gain of approximately 1 and a phase shift of approximately 0-radup to a frequency of 1 kHz. The response of prototype #2 shown in Fig. 6(b) is almost the sameas that of prototype #1 except for one small reproducible resonance at 340 Hz. We suspectthat this is caused by a part supporting the lens and the piezo stack actuator. Figure 7 showstrajectories of the actuator and the lens surface displacements. According to these results, theproposed focusing mechanism has a bandwidth of at least 1 kHz.

The viscous resistance of the liquid causes pressure difference between the lens surface andthe cylinder. If this difference is larger than the maximum pressure that the actuator can gener-ate, then the actuator can not control the lens surface displacement. We estimated this pressure

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2145#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

difference using finite element method (FEM). The maximum pressure difference was esti-mated at 58 Pa, assuming that 1-kHz sinusoidal input was applied to prototype #2. This valueis sufficiently smaller than the maximum pressure of the piezo stack actuator, 7.2× 105 Pa.Therefore, the viscous resistance is not a problem for prototype #2.

The bandwidth is limited by the first resonance frequency of the disks, such as lens surfaceand cylinder. For example, the lens surface of prototype #2 had a natural frequency of 7.2 kHz.To prevent resonant vibration, the lens must be operated at lower frequency than 7.2 kHz.

Static lens surface profiles were measured using a laser confocal displacement meter (LT-8110, Keyence) that can measure the displacement of one point. To acquire a two-dimensionalsurface profile, this sensor was fixed on a computer-controlled xyz-stage and the profile wasmeasured by two-dimensional scanning. Eleven surface profiles were measured by varying thedisplacement of the piezo stack actuator from 0 to 10µm. The lens was set to be plano-concaveprofile at the initial displacement (0µm) of the actuator. The acquired profiles were subjectedto curve fitting using the fourth-order polynomial theoretically described in Eq. (3). Results areshown in Table 3.

5.3. Deformation profile of the lens surface

Table 3. Deflected lens surfaces were fitted with theoretical fourth-order polynomials. Thedisplacement of the piezo stack actuator (δ ), coefficients of the fitted curves (r4, r2), good-ness of the fit (Q), estimated focal length (f), and estimated Seidel spherical aberration (SA)are shown.

Fitted curve coefficientsδ [µm] r4 r2 Q f [mm] SA

0.00 (−1.8±0.1)×10−4 (6.5±0.1)×10−3 6.21×10−13 -167 -0.1000.99 (−1.6±0.1)×10−4 (5.5±0.1)×10−3 8.93×10−12 -195 -0.0891.99 (−1.1±0.1)×10−4 (4.0±0.1)×10−3 0.0171 -270 -0.0613.00 (−2.3±0.9)×10−5 (1.5±0.1)×10−3 0.944 -714 -0.0123.99 (4.9±0.7)×10−5 (−1.72±0.08)×10−3 1.00 632 0.0275.00 (7±1)×10−5 (−3.1±0.1)×10−3 1.00 345 0.0396.00 (1.2±0.1)×10−4 (−4.4±0.1)×10−3 0.999 244 0.0666.99 (1.4±0.1)×10−4 (−5.4±0.1)×10−3 0.479 199 0.0788.00 (1.7±0.1)×10−4 (−6.7±0.1)×10−3 4.53×10−8 162 0.0948.99 (1.6±0.1)×10−4 (−7.6±0.1)×10−3 2.78×10−26 144 0.0899.99 (1.6±0.1)×10−4 (−8.40±0.09)×10−3 6.78×10−151 129 0.089

To determine the validity of the fourth-order polynomial model, the goodness of fit[11] wascalculated by fitting the curve using a certain data set to obtain estimated curve parameters,and then estimating the probability that the actual curve for the measured data set matched theestimated curve. The goodness of fit is expressed by probabilityQ. The closer the probabilityQis to one, the more valid the model is. Table 3 shows the probabilityQ and focal lengthf . Thevalue ofQ is close to one when the focal length is less than about -200 mm or longer than about200 mm. In contrast, when the focal length is between these values, theQ value is significantlysmaller than one, which suggests that the adopted model is no longer valid.

The focal length and Seidel spherical aberration coefficient were also calculated based onthe fitted curves. The Seidel spherical aberration coefficient was estimated using optical CADsoftware (ZEMAX).

Assuming that the deformation of the cylinder is denoted by Eq. (3), the relation between the

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2146#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

0 2 4 6 8 10- 8

- 6

- 4

- 2

0

2

4

6

8

PZT displacement [µm]

Ref

ract

ive

pow

er (1

/f) [D

= 1

/m]

f = -167 [mm]

f = 129 [mm]

Plano-convex

Plano-concave

f = � [mm]

Fig. 8. Refractive power versus piezo stack actuator displacement

refractive power and the displacement of the actuator is

1f

=4(nr−1)aC

2

aL4 ∆C +K, (7)

where, f is the focal length,nr is refractive index,∆C is the actuator displacement, andK isa constant depending on the initial profile when∆C equals 0. Besides,aC andaL indicate theradius of the cylinder and the lens surface, respectively. According to Eq. (7), refractive power1/ f is a linear function of∆C with slope 4(nr−1)aC

2/aL4.

Refractive power (= 1/ f ) is plotted against the displacement of the piezo actuator in Fig.8. This figure shows that the linearity of the refractive power is not good. This bad linearitycaused by fabrication errors in the dimensions of the lens surface periphery. Theoretical slopeof prototype #2 is 5.4×106 m−2, and measured slope was∼ 1.4×106 m−2 that was almosta quarter of the theoretical value. We suspect that this is caused by the distortion of a partsupporting the lens and the piezo stack actuator.

The measured and fitted surface deflection profiles are shown in Fig. 9. As the focal lengthbecome short, the error between the measured and fitted surface profiles shows a saddle-likeshape; this indicates that the deflected surface profile is not point symmetric. This asymmetrycould be caused by fabrication errors in the dimensions of the lens surface periphery.

Prototype #2 was able to maintain a given surface profile while each surface profile measure-ment that took about 5 minutes. Furthermore, no fluid leakage was observed. However, wefound that the focal length of prototypes depended on its temperature, since the liquid volumeexpands with increasing temperature. One way to solve this problem is to regulate the tempera-ture to be constant.

5.4. Resolution

The resolution was also measured using prototype #2. A USAF 1951 test chart illuminated bya white light source (halogen lamp) was used to measure the resolution. A CMOS imager wasplaced at the image plane to acquire the images formed by prototype #2 with various positive

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2147#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

Fig. 9. Measured surface profiles, fitted curves, and error profiles. Surface profiles (a) ofprototype #2 were measured at various focal lengths. Each surface profile was fitted witha fourth-order polynomial. Fitted curves are shown in column (b). Column (c) shows theerror between the measured surface profile and the fitted curve.

focal lengths. The resolution was visually read from the chart image. The results are shown inTable 4. Based on these results, a maximum resolution of 12.3 cycles/mm was achieved. Figure10 shows the captured image of the standard chart by the CMOS imager when the focal lengthwas 174 mm.

Aberration correction could be achieved without loss of bandwidth using two or more of suchlenses. An initial numerical simulation has suggested that the aberrations could be correctedadequately.

Prototype #2 had small N.A., or large f-number, because the ratio of the lens diameter tothe focal length was small. The maximum radius of the proposed lens was estimated at 5 mmconsidering the maximum force of the actuator. Therefore, the lowest f-number of the proposedlens is about 13 at the shortest focal length. This f-number is not sufficient for practical appli-cations. In order to attain small f-number, it is necessary to develop an adequate optical systemincluding the proposed lens.

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2148#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004

Table 4. Resolution was measured using a USAF 1951 standard test target at various focallengthsf . Vertical and horizontal resolutions are shown.

Resolution [cycles / mm]f [mm] NA Vertical Horizontal

341 0.0103 2.78 2.21275 0.0127 3.19 3.19231 0.0151 6.03 4.79222 0.0158 9.54 9.54214 0.0164 12.3 12.3186 0.0188 11.5 10.2174 0.0201 9.75 8.69

Fig. 10. Image of the positive standard test chart using the proposed lens prototype whenthe focal length was 174 mm.

6. Conclusion

In this paper, a new variable-focus lens with 1-kHz bandwidth is proposed. Based on the dis-cussion of the mechanical characteristics, two kinds of prototypes were designed and fabri-cated. A bandwidth of 1 kHz of the proposed lens was confirmed by measuring the frequencyresponse. Also, a wide range of refractive power, from−1/167 to 1/129 mm−1, and a resolutionof 12.3 cycles/mm were attained.

(C) 2004 OSA 17 May 2004 / Vol. 12, No. 10 / OPTICS EXPRESS 2149#4158 - $15.00 US Received 4 April 2004; revised 30 April 2004; accepted 2 May 2004