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Optics Communications 275 (2007) 22–26

Aberration analyses for liquid zooming lenses without moving parts

Zhuo Wang *, Yong Xu, Yang Zhao

Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202, United States

Received 19 October 2006; received in revised form 13 March 2007; accepted 15 March 2007

Abstract

Liquid lenses with an adjustable focal length are of particular interest for their potential applications in compact imaging or zoomingsystems. We have analyzed the aberrations for liquid zooming lenses and compared them with those of traditional lenses. Based on the3rd order aberration theory, possible structures for the liquid zooming lens without moving mechanism were determined. We showedthat a two-group liquid lens system is the simplest one that contains no moving mechanical parts and has enough parameters to correctall chromatic aberrations for any focal length and most monochromatic aberrations for certain focal lengths.� 2007 Elsevier B.V. All rights reserved.

1. Introduction

Liquid lenses with an adjustable focal length have manypotential applications [1]. The idea of liquid lenses and theoperating principles are similar to some lenses found innature (e.g. human eyes). Liquid lenses are ideal for com-pact imaging modules where focusing and zooming with-out moving parts are desired. With the progress of solidstate imaging devices, a commercially available 1/300 CMOSdetector now contains mega-pixels (the pixel size is around3 lm · 3 lm), and the corresponding lens module has theoverall length and diameter of several millimeters [2]. Suchsystems have been incorporated into portable devices suchas mobile phones, PDAs and laptops. For compact cameraphones, it is difficult to package the moving mechanism forauto-focus and cam mechanism for zooming, which areboth standard for digital cameras. Liquid lenses can beused to solve these problems.

Liquid lenses show a promising future for both auto-focus [3] and zooming [1]. For focusing purposes, becausethe object distance is much larger than focal length, liquidlenses typically make a minor adjustment in the focallength by changing the curvature of one surface of a lens.Usually the adjustment has little effect on the aberrations,

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.03.032

* Corresponding author. Tel.: +1 313 577 4933.E-mail address: wangzhs@wayne.edu (Z. Wang).

especially when the varying surface is carefully selected.For example, one may want to select the surface whoseaberration is not sensitive to curvature variation, i.e. sur-faces with small incident angle of light. This situation issimilar to the case where after test plate fitting designersneed to change one or more curvatures to maintain ele-ment’s or system’s power. The other application of liquidlenses is zooming. Conventional optical systems achievezooming by precisely adjusting the relative positions of dif-ferent lens groups. The moving parts in a conventionalzooming system make it difficult to miniaturize the system.On the contrary, liquid lenses contain no moving mechan-ical parts and can be used for miniature zooming systems.

Though liquid lenses with an adjustable focal length areof special interest, most imaging analyses are based on firstorder optics (Gaussian optics) (e.g. [1,4,5]). When the focallength is really short (e.g. micro-lens array [5]), first orderoptics analysis may be enough. However, to improve theimaging quality, aberration analyses for liquid lenses areneeded. In this paper we study the aberration characteris-tics for zooming systems without moving parts. Specialemphasis is on elucidating their differences from those oftraditional lenses. We also want to point out that the cur-rent liquid lens systems do not have enough degree of free-dom to correct aberrations. We present a two-groupstructure that can lead to liquid lens systems with moredegrees of freedom for aberration correction.

Z. Wang et al. / Optics Communications 275 (2007) 22–26 23

Two methods are currently employed to change the cur-vature of the liquid lens: one is by electric wetting [3,6] andthe other is by injecting fluids in a deformable transparentchamber [1,7]. Our analysis applies to both methods. Weassume that the surfaces of liquid lenses are spherical,which is a reasonable approximation [1,4,5].

2. Approach

Our approach in this study is based on design and anal-ysis method developed by Slyusarev [8] in the pre-computerera. It is well known that one of main purposes of aberra-tion analyses is to provide guidelines for proper structureselection. Our goal is to do such analyses for liquid lens sys-tems. It should be noted that with the advance of comput-ing, the speed of ray tracing has increased enormously andthus optimization becomes possible. For a specific liquidlens, we may easily get all 3rd order and 5th order aberra-tions from ray-tracing results. However, even though thedesign methodology has changed considerably, we stillneed to use aberration analyses whenever the optimizationprogram gets lost at a local minimum. Our analyses aretheoretically to tell whether the structure is capable of cor-recting aberrations, which cannot be done with optimiza-tion programs.

In the next section, we will give a short review of theaberration theory by Slyusarev and apply it to the liquidlens systems.

3. Lens aberration theory

Aberration theories for traditional lenses have beenstudied in detail [8–10]. To study the aberration for liquidlenses, we should compare the case with aberration analy-ses for traditional lenses. In the traditional lens design, it isoften convenient to normalize focal length of the designedlens to 1. After the design is finished, we ‘scale’ the lens sys-tem to the desired focal length, while the ‘‘FOV’’ (field ofview) and ‘‘F/#’’ are held constant. In scaling the system,we need to vary not only the curvature but also the spacebetween lenses and their surfaces and the location of theimaging plane. If there are no moving parts in the system,we need at least two groups of lenses [4], and a strictrequirement on the power distribution as discussed later.

hp1y

Liquid lenswith varyin

-u

Fig. 1. Schematic plot for Seidel sum ca

What will happen to each sub-group if its focal length var-ies? We consider two cases here.

Case A: each sub-group corrects aberration separately.We still need to scale the sub-group to get the desired focallength. Similarly, not only the curvature, but the spacebetween surfaces needs to be scaled. However, even if thespace between surfaces is fixed, such scaling is feasible forthe thin lens group because its aberration is not sensitiveto the thickness of the lens.

Case B: each sub-group does not correct aberration sep-arately. There must be complex aberration compensationamong groups.

Using the method below, we can explore both cases.The following analyses are based on 3rd order aberra-

tion theory. It is valid for systems with small incidentangles (<20�) and small aperture (NA < 0.1). For a typicalzooming lens, the incident angle is 32–12� (f 0 = 35 mm–105 mm for 135 camera), and the typical aperture is 0.18(F/# 2.8). Thus we need to consider higher order aberra-tions. However, 3rd order optics is always important andis chosen as a good design principle since low primary aber-ration will lead to lower high order aberrations due tolower induced higher order aberration. Thus if the primaryaberrations are small, we are balancing smaller amounts ofaberration. This leads to a better corrected and less sensi-tive system.

The Seidel sum calculation is based upon tracing theaxial and the principal rays [8,10] (i.e. marginal and chiefrays in many places), as shown in Fig. 1. Two lens-groupswith power u1 and u2 are separated by a space d. The dis-tance between the last optical surface and object plane (alsoknown as back focal length) is denoted as bfl. If the inci-dent heights of the axial ray and the principal ray on thelens elements of a lens group are almost the same, such lensgroup may be regarded as thin lens group. The correspond-ing projection heights are denoted as h and hp respectively.We are interested in such system for the following reasons:practical lens system is a combination of such groups; theaberration for such groups has very neat expression; a thicklens can be treated as the combination of two thin lenseswith a plane parallel plate whose Seidel sums take particu-larly simple forms [8]; and the aberration of such a thin lensgroup is not sensitive to the spacing, which is desired asdiscussed.

principal ray

aperture stop

O-hp2

h1 h2

1 g 1

d bfl

Liquid lens 2 with varying 2

axial ray

u

ϕϕ

lculation of liquid thin lens groups.

Object Image

variablesurfaces

Fig. 2. Liquid lens with two varying surfaces.

24 Z. Wang et al. / Optics Communications 275 (2007) 22–26

For thin lens groups, the Seidel sums can be written as[8,10]

SI ¼X

h4u3P ð1ÞSII ¼

Xh3hpu

3P þ JX

h2u2W ð2Þ

SIII ¼X

h2h2pu

3P þ 2JX

hhpu2W þ J 2

Xu ð3Þ

SIV ¼ J 2X

pu ð4Þ

SV ¼X

hh3pu

3P þ 3JX

h2pu

2W þ J 2X hp

h2u 3þ pð Þ ð5Þ

where every sum is over all thin lens groups; J is the La-grange–Helmholtz invariant of the system; the Petzvalsum is given by p ¼

P uini=u, u is overall power of one thin

lens group, ui and ni correspond to the power of ith thinlens and its refraction index, respectively. Since for com-mon glass materials and liquids ni ranges from 1.3 to 1.7,p is almost a constant that is not related to the lens struc-ture. For each thin lens group, P and W are the normalizedparameters corresponding to h ¼ 1 and u ¼ 1.

The chromatic aberration of the system is

SIC ¼X

h2C ð6ÞSIIC ¼

XhhpC ð7Þ

where C ¼P u

t, t is the Abbe number and the sum is overevery thin lens.

In Eqs. (1)–(7), two kinds of parameters are involved: P

and W are aberration parameters decided in the stage ofaberration analysis (based on 3rd order aberration theory);h, hp and u are other parameters decided in the stage of lay-out (based on Gaussian optics).

4. Liquid lens first order analysis

Layout stage is critical to the success of a design. Inorder to do aberration analysis, we need a first order anal-ysis (layout) first. Here we suggest the object is located atinfinity. For two-group structure sketched in Fig. 1, iflenses and image plane do not move, we have

u2 þu1

1� du1

¼ 1

bfl¼ constant ð8Þ

The corresponding overall power will be

u ¼ 1� du1

bflð9Þ

Now it is obvious that ‘‘no moving part’’ places arestraint on the power distribution. If we have three groups(e.g. like a Cooke triplet), ‘‘no moving part’’ imposes a sim-ilar restraint.

5. Liquid lens third order analysis

At the layout stage, the aberrations that are related onlyto h, hp and u also need to be considered. In the primary aber-ration realm, they are SIV, SIC, and SIIC. Though distortion

SV is related to P and W, because in many practical designseach group corrects aberration separately (i.e. P and W aresmall) or the hp is quite small in the group where P and W

are large, SV is simplified as J 2P hp

h2 uð3þ pÞ and then con-sidered at the layout stage. Such a treatment is of courseapproximate.

For the two-group structure shown in Fig. 1, to correctfield curvature we have u1 þ u2 ¼ 0. Thus the positivepower and negative power must be separated since theoverall power is u ¼ u1 þ u2 � du1u2. If u1 < 0 andu2 > 0, we have retro-focus structure, favoring the aberra-tion correction for large FOV. If u1 > 0 and u2 < 0, wehave telephoto structure. If u1 > 0 and u2 > 0, such astructure is called Petzval lens and it exhibits a large Petz-val sum [11]. Thus the system is only usable for small FOV(e.g. half FOV = 7�). However, to satisfy Eq. (8), thereexists a residual field curvature. Only when blf ¼f1ðf1=d � 1Þ, the field curvature is corrected. In otherwords, during zooming we can only correct SIV at a certainfocal length. It is preferred to correct SIV for large FOV(short focal length end).

The situation for distortion is the same. Because of Eq.(8), there is residual distortion. Since distortion is propor-tional to the third power of FOV, it is desirable to correctit for large FOV.

Considering the two chromatic aberrations, if we havejust one thin lens in one group, SIC can only be correctedfor one certain focal length and SIIC cannot be correctedsimultaneously [10]. Thus the simplest way to correct chro-matic aberration is to replace the singlet in each lens groupwith a doublet, and to correct color aberrations separately.It is worth noticing that for imaging system with very shortfocal length (e.g. 3 mm in [2]), to reduce cost the same plas-tic material (e.g. PMMA or ZEONEX) is used for alllenses. Such a system does not consider color aberrationsat all. Of course, the imaging quality can only be called fair(suitable for low end applications, e.g. VGA (640 · 480),not SXGA (1280 · 1024)). Designers need to balancebetween quality and cost.

Now we move to the stage of aberration design. For theremaining three monochromatic aberrations, we first con-sider Case A (see Section 3) where aberrations for twogroups are corrected separately, i.e. P and W for everygroup are 0. This is also the case for a traditional zooming

Liquid lens 1 with varying 1

Liquid lens 2 with varying 2

Object

Image

ϕ ϕ

Fig. 3. Liquid zooming lens consisting of two tunable lenses; the shape of the lens can be either biconvex or biconcave.

Z. Wang et al. / Optics Communications 275 (2007) 22–26 25

lens where the aberration is corrected separately for differ-ent moving groups. If the spherical aberration and comaare corrected, for any power and incident height, the resid-ual astigmatism is 1/p times the curvature, which cannot becorrected for any focal length as we explained. The resulthere is similar to that of fixed power system [8,10]. To bal-ance aberrations, depending on designer’s taste and experi-ence, some amount of P and W is left for every group tocorrect astigmatism and the whole system is still free ofspherical aberration and coma.

For Case B (see Section 3) where aberration compensa-tions exist between the two groups, we have four variablesP1, P2, W1 and W2. In this case SV cannot be simplified andconsidered at layout stage. If we can find suitable combina-tion of these variables, it is possible to correct SI, SII, SIII

and SV to 0 simultaneously for every focal length. Thequestion now is what values of P and W are realizable.

P and W are related not only to the structure of lenses,but also to the conjugate distance. In case the object islocated at infinity, we denote them as P1 and W1. P

and W for finite conjugate system are functions of P1

and W1 [8]. For a singlet, P1 and W1 are not independentvariables [8]:

P1 � P10 þ 0:85ðW 1 � 0:14Þ2 ð10Þwhere P10 ¼

nð4n�1Þ4ðnþ2Þðn�1Þ2. Most optically transparent liquids

have refractive index between 1.33 and 1.65 [12]. Thusfor typical liquid lens P10 ranges from 3.96 (for n = 1.33)to 1.50 (for n = 1.65).

There are different types of liquid lenses. For liquid lensusing electric wetting, only a few types of liquids may beused. For liquid lens using injecting fluids, almost all opti-cally transparent liquids can be used. In this case, higherrefractive index liquid will lead to lower value of P10 .

In case of a doublet, there exist similar relations betweenP1 and W1, except now P10 can vary in a large range frompositive to negative. For traditional design, it is possible tofind an appropriate glass pair to satisfy the requirements ofP1, W1 and C. One difficulty for liquid lenses is the lim-ited number of materials, which limits the achievable P1,W1 and C. The other difficulty is P10 is fixed for each glass(liquid) pair. P10 may satisfy the requirements of P1 andW1 for a certain focal length. Because the required P1

and W1 vary with focal length, the system may be unableto satisfy the requirements for any focal length because ofEq. (10). Whether such a design strategy is successfuldepends on the particulars of the design.

As comparisons, we want to point out that the liquid lenswith one or two surface changes are not good enough foraberration correction. For liquid lens with one surfacechange [3], it is impossible to realize zooming withoutmoving parts. For the structure reported in [1], shown inFig. 2, where the system is composite with a convex-planarliquid lens, plane liquid and planar-concave liquid lens, P1,W1 are fixed (for convex-planar liquid lens, P1 ¼ 13=6;W 1 ¼ 1=3; for planar-convex liquid lens, P1 ¼ 9;W 1 ¼ �3). Thus we can only correct the primary aberra-tions at a certain power. The situation is similar in theconfiguration reported in [4], shown in Fig. 3, where the1st lens group and the 2nd lens group only contain a singleliquid lens, and the shape of the lens may be biconvex orbiconcave. All these structures do not consider the colorcorrection.

6. Conclusions

We have analyzed the aberration characteristics for liquidlenses. We show that a separate two-group structure is suffi-cient for aberration correction. Because there are no movingmechanical parts, it constrains the power distribution in thelayout stage, and thus the aberration sums SIV and SV

remain large. Since the space between the surfaces of liquidlenses is not a free variable for current liquid lenses, a thinlens group is a suitable model for such systems because itsaberrations are not sensitive to spacing. Depending on thedesigner’s preference and experience, one can choose to cor-rect aberrations separately for every group or to compensateaberrations between groups. We have analyzed both possi-bilities. It is impossible to compensate the aberrationbetween groups for all focal lengths. But the configurationof two air spaced doublets is the simplest one that containsno moving mechanical parts and has enough parametersto correct all chromatic aberrations for any focal lengthand all monochromatic aberrations for certain focal lengths.

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