optical design and multiobjective optimization of miniature zoom optics with liquid lens element

Optical design and multiobjective optimization ofminiature zoom optics with liquid lens element

Jung-Hung Sun,1 Bo-Ren Hsueh,2 Yi-Chin Fang,3,*John MacDonald,4 and Chao-Chang Hu5

1Department of Mechanical and Automation Engineering, National Kaohsiung First University ofScience and Technology, 1 University Road, Yenchau, Kaohsiung 824, Taiwan

2Institute of Engineering Science and Technology National Kaohsiung First University ofScience and Technology, 1 University Road, Yenchau, Kaohsiung 824, Taiwan

3Graduate of Electro- Optical Engineering, National Kaohsiung First University of Science andTechnology, 1 University Road, Yenchau, Kaohsiung 824, Taiwan, ROC

4J. J. Thomson Physical Laboratory, The University of Reading, P.O. Box 220, Whiteknights, Reading RG6 6AF, UK5ITRI-South, BuildingYanjiou 1st, No. 31, Gongye 2nd Road, Annan District, Tainan City 709, Taiwan

*Corresponding author: [email protected]

Received 10 November 2008; revised 3 February 2009; accepted 4 February 2009;posted 10 February 2009 (Doc. ID 103921); published 16 March 2009

We propose an optical design for miniature 2:5× zoom fold optics with liquid elements. First, we reducethe volumetric size of the system. Second, this newly developed design significantly reduces the numberof moving groups for this 2:5× miniature zoom optics (with only two moving groups compared with thefour or five groups of the traditional zoom lens system), thanks to the assistance of liquid lens elements inparticular. With regard to the extended optimization of this zoom optics, relative illuminance (RI) and themodulation transfer function (MTF) are considered because the more rays passing through the edge ofthe image, the lower will be the MTF, at high spatial frequencies in particular. Extended optimizationemploys the integration of the Taguchi method and the robust multiple criterion optimization (RMCO)approach. In this approach, a Pareto optimal robust design solution is set with the aid of a certain designof the experimental set, which uses analysis of variance results to quantify the relative dominanceand significance of the design factors. It is concluded that the Taguchi method and RMCO approachis successful in optimizing the RI and MTF values of the fold 2:5× zoom lens system and yields betterand more balanced performance, which is very difficult for the traditional least damping square methodto achieve. © 2009 Optical Society of America

OCIS codes: 110.0110, 120.3620, 220.1000, 220.0220.

1. Introduction

Zoom lenses were developed a long time ago. Thanksto the rapid development of optical materials andcomputers, miniature zooms have been widely usedfor many advanced optical systems, such as photo-graphy for mobile phones, projection systems, andmicroscopes. Generally speaking, the design of min-

iature zoom optics is complicated, not only by the pro-blems of assembling the optics and components of azoom mechanical system [1], but also because ofthe multiconfiguration method [2] and complicateddesign of the mechanical cams [3].

Several years ago, the Philips Company an-nounced the liquid lens as a varifocal component thattakes advantage of compact lens size without the me-chanical cam and curve movement. The liquid lenswas expected to play a significant role in minimizingthe overall length for an optical zoom lens [3–6].

0003-6935/09/091741-17$15.00/0© 2009 Optical Society of America

20 March 2009 / Vol. 48, No. 9 / APPLIED OPTICS 1741

However, optics with a liquid lens has been studiedfor some time, unfortunately without success so far[4–7]. In the present study, we demonstrate a specialfold optical 2:5× zoom lens optical design with onlytwo moving groups for zoom, focus, and compensa-tion, compared with the four or five groups with com-plicated cam of the traditional zooms. In addition,the design comprises a fold optical zoom lens systemwith one prism assembled in a noncoaxial opticalsystem, which has the advantage of further reducingvolumetric size without any sacrifice of system per-formance. Generally speaking, optical design witha liquid lens element is difficult to optimize, mainlybecause of the extreme spherical and coma aberra-tion, but also because of the axial chromatic aberra-tions. These severe axial aberrations are generatedmainly from oblique incident rays close to the liquidlens element. In the present research, first, we de-monstrate a methodology for the optical design offold zoom optics, which may not only solve the severeaberration problem but also further reduce the volu-metric size of the system. Second, in this newlydeveloped design, the number of moving groupsfor this 2:5× miniature zoom optics is significantlyreduced to two, thanks to the assistance of a newdesign methodology for liquid lens elements inparticular.With regard to the issue of extended optimization,

most optical design work so far has been done use ofthe least damping square (LDS) as a quick meritfunction. Some genetic algorithms have been studiedbefore, but have not been successful in global optimi-zation [8–10]. Traditionally, optical design faces thedilemma of a trade-off between the modulationtransfer function (MTF) and relative illuminance(RI). The higher the RI is, the more rays will passthrough the edge of the image. These deleteriousrays will seriously degrade the systematic MTF, inparticular at the edge of the image. Generally speak-ing, it is very difficult to overcome this dilemma bythe LDS method.In this study, in order to achieve maximum perfor-

mance in the design process by extended optimiza-tion, the Taguchi method is employed in theoptimization procedures with orthogonal arrays ofstatistically designed experiments, in the hope of

obtaining the best results with the fewest possibleexperiments [6]. In the history of the industry, theTaguchi method has been successfully developed tooptimize and analyze design processes with staticand dynamic characteristics [11–14]; however, withthis method it is difficult to optimize systems withintercorrelated multiple performance characteris-tics, such as the fold zoom optics in the presentresearch. In this paper, the Taguchi method has beenfurther studied and developed to handle multiple ob-jective product design problems, using generativetechniques and a rational approach to treat the con-straints in such problems.

The present paper addresses this problem andproposes a methodology that further develops theTaguchi method to incorporate multiple objectivesand constraints in the product design. In this meth-odology, statistical analysis (analysis of variance,ANOVA) concepts are used to obtain a well-diversi-fied, though not necessarily exhaustive, set of Paretooptimal solutions. Experiments are simulated, andthe results are evaluated by using the optical soft-ware tool CODE V (Optical Research Associates).

In Sections 2 and 3, we demonstrate the physicalinsights of liquid lenses and the methodology of op-tical design for a fold optical zoom lens system. InSection 4, MTF and RI are defined and their actionexplained. The Taguchi method and the multiple-criteria optimization approach are described inSections 5 and 6. The problem’s formulation andresults are presented in Section 7, while the conclu-sions can be found in Section 8.

2. Physical Insights into Liquid Lenses

The liquid lens included two different refractiveindexes. The radii of curvature and liquid thicknessvary with driving voltages; for an example, seeFig. 1(a), which shows the schematic cross sectionof an electrowetting optics. The refractive index ofthe insulating liquid is higher than that of the con-ducting liquid, which gives the lens a negative opticalpower. Figure 1(b) shows the adjusted lens shape onapplication of a voltage. When the voltage differenceis applied, it pulls the conducting liquid toward thewall. The insulating liquid does not react to theelectrical field [3].

Fig. 1. (a) Schematic cross section of an electrowetting lens. (b) Lens shape adjusted by application of a voltage to achieve ray conver-gence [3].

1742 APPLIED OPTICS / Vol. 48, No. 9 / 20 March 2009

This research used the Arctic 320 liquid lens [15].The liquid lens as shown in Fig. 2 is described. Thetwo windows of the liquid lens are made of the sameglass material (Glass 1 ¼ Glass 2). Table 1 gives theindex of refraction versus wavelength for the glassmaterial. Table 2 gives the liquids (at 20°C) indexof refraction versus wavelength. Table 3 gives the ra-dius of curvature and liquid thickness at severaldriving voltages (at 25°C) and shows that the radiusof curvature was limited to −17:896 and 3:965mm.The constructional drawing of the lens is as shownin Fig. 3.

3. Methodology of Optical Design for a Fold OpticalZoom Lens System

Optical design for a liquid lens presents difficulties inoptimization work, not only because of the extremespherical and coma aberration introduced by the li-quid lens element but also because of the axial chro-matic aberrations. These severe axial aberrationsare generated mainly from oblique incident raysclose to the liquid lens element [16,17]. Exampleswere given to show that a miniature 3× zoom lenscannot have an overall length of less than 20mm be-cause, first, oblique rays through the aperture stopand liquid lens element will introduce enormous

spherical and coma aberrations, which cannot be

eliminated because of the low index of the optical ma-terials of the liquid lens element. The higher the in-dex of the optical materials, the fewer aberrationswill be introduced. Second, the requirement of over-all length sharpens the chief ray angle, so that comabecomes very severe. According to [16,17], it is verydifficult to design 3× zoom optics with a liquid lens

Fig. 2. Schematic drawing of liquid lens [15].

Table 1. Index of Refraction versusWavelength λ for Glass Material [15]

λ (mm) Index

700.0 1.5150650.0 1.5164590.0 1.5187550.0 1.5206480.0 1.5252430.0 1.5300

Table 2. Index of Refraction versus Wavelength for Liquids [15]

Wavelength PC100 (Liquid A) H100 (Liquid B)

400 1.41178 1.51297448 1.40729 1.50340489 1.40451 1.49772541 1.40180 1.49250589 1.39988 1.48894654 1.39791 1.48535703 1.39671 1.48332

Table 3. Radius of Curvature and Liquid Thickness at DrivingVoltages [15]

Voltage (V) R (mm) 1=f (diopters) Φ=2 (mm) e (mm) u (mm)

0 to 32 −17:896 −5:03 2.150 0.000 0.72033 −21:242 −4:24 2.138 0.012 0.70934 −26:402 −3:41 2.127 0.023 0.69935 −35:082 −2:57 2.115 0.035 0.68836 −52:731 −1:71 2.104 0.046 0.67837 −108:020 −0:83 2.093 0.057 0.66738 1635.997 0.06 2.082 0.068 0.65639 93.995 0.96 2.072 0.078 0.64540 48.026 1.87 2.061 0.089 0.63541 32.095 2.80 2.051 0.099 0.62442 24.014 3.75 2.040 0.110 0.61343 19.130 4.70 2.030 0.120 0.60244 15.862 5.67 2.020 0.130 0.59045 13.521 6.66 2.010 0.140 0.57946 11.764 7.65 2.001 0.149 0.56847 10.396 8.66 1.991 0.159 0.55748 9.302 9.68 1.981 0.169 0.54549 8.408 10.70 1.972 0.178 0.53450 7.663 11.75 1.962 0.188 0.52251 7.033 12.80 1.953 0.197 0.51152 6.494 13.86 1.943 0.207 0.49953 6.028 14.93 1.934 0.216 0.48754 5.620 16.01 1.925 0.225 0.47555 5.262 17.10 1.915 0.235 0.46356 4.943 18.21 1.906 0.244 0.45157 4.659 19.32 1.897 0.253 0.43958 4.404 20.43 1.888 0.262 0.42759 4.174 21.56 1.879 0.271 0.41560 3.965 22.70 1.869 0.281 0.402

Fig. 3. Constructional drawing of liquid lens [15].


element if the overall length is required to be lessthan 20mm.In contrast, fold optics comprising a fold optical

zoom lens system with one prism and two liquidlenses assembled in a noncoaxial optical systemhas the advantage of better performance and reason-able volumetric size, which may be well accommo-dated in a mobile phone. The employment of a

prism not only reduces the volumetric size of thezoom optics, for the front diameter of the first ele-ment, in particular, but the prism accommodatesthe chief rays and the marginal rays and makesthe rays somewhat closer to telecentric before pas-sing through the aperture stop and the liquid lenselement, with the result that spherical and comaaberration is no longer so severe. The smootherthe rays that can pass through the optical surface,the less aberration will be introduced. The diameterof the front lens is 8:46mm and of the rear lens is5:5mm. The initial design specification is as shownin Table 4.

This design has another advantage of involving agroup with less motion, thanks to the liquid lens ele-ment. The methodology introduced in this researchrequires the power of the liquid lens to be mainly re-sponsible for the zoom function; from the other pointof view, the two moving groups are basically intendedas compensation. The optical design is completed bythe following steps: first, the goals and performancesare established for the demands of the fold opticalzoom lens system. Second, the fold optical zoom lenssystem is set up by the optical design software CODEV, including the radius of curvature and liquid thick-ness of the liquid lens element, according to Table 3.After setting up the liquid lens element, two or threegroups are chosen for compensation within a con-strained moving space. Third, the fold optical zoomlens system is optimized by the constraining condi-tion, such as overall length in the x direction andthe y direction, the front element of a semi-aperture,and the glass material of the prism. After optimiza-tion by CODE V, the initial condition design resultcan be obtained. The flow chart for the design ofthe fold optical zoom lens system is shown in Fig. 4.

The initial design goals and optical performancerequirements are shown in Table 5. The fold opticalzoom system is presented in Figs. 5–7. The perfor-mance of the zoom lens in the three zoom positionsis presented in Figs. 8–16, shown as the MTF re-sponse, field curvature, distortion characteristics,and rays fan. The MTF at maximum field is equalto or more than 38%; the field curvature, distortioncharacteristic and rays fan are less than or equal to0:03mm, 4%, and 0:01018mm, respectively.

4. Modulation Transfer Function and RelativeIllumination

The MTF is an important value of image quality. Theoptical transfer function is defined as

OPTðvÞ ¼ MTFðvÞ exp½iPTFðvÞ�; ð1Þ

Table 4. Fold Optical Zoom Lens DesignSpecification

Specification Value (mm)

Overall lengthx Direction 10.10y Direction 23.80DiameterFront lens 8.68Rear lens 5.76

Fig. 4. Flow chart for the design of a fold optical zoom lenssystem.

Table 5. Fold Optical Zoom Lens Design Goals and Performance

Position EFL (mm) Field of View Image High f-number MTF 100 lp=mm Field Curvature Distortion

1 5.8 28:88° 2:64mm 3.3 y ¼ 0mm, 60% y ¼ 2:64mm, 38% �0:02mm �4%2 10 17:74° 2:64mm 5.6 y ¼ 0mm, 76% y ¼ 2:64mm, 68% �0:01mm �4%3 15 12:04° 2:64mm 8.2 y ¼ 0mm, 77% y ¼ 2:64mm, 67% �0:03mm �4%


where MTF is the real part of the spatial frequencyfor the optical transfer function and PTF is the phasetransfer function. If the resolution of the object is Moand the resolution of the image is Mi, then

MTF ¼ Mi=Mo: ð2ÞThe MTF of the optical visual system is calculated

as the square root of the absolute value of the Fouriertransformation of Iðx; yÞ [18]:

MTFðu; vÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi‖FFT½Iðx; yÞ�‖

q; ð3Þ

where u; v are the coordinates in the spatial frequen-cies plane and FFT is the fast Fourier transform.

The MTF represents the image performance ofmost optical image systems and components. An ex-ample is given such that the performance evaluationof the human eye is the definition of a merit function(MF) [19,20],

MF ¼Z2π

0

Zρ2ρ1

MTFðρ; θÞρdρdθ; ð4Þ

where ðρ; θÞ are the polar coordinates.RI is employed to evaluate the brightness of the

light from corner to center. To compute the valuesof RI, we first read out the pixel values for each R,G, and B (red, green, and blue) channel in the centerof the image. These values will be used as baselinenumbers. We then read out the R, G, and B pixelvalues at each of the four corners. RI can becomputed as the ratio of corner luminance Y(¼0:3 ×Rþ 0:59 ×Gþ 0:11 × BÞ) to the center lumi-nance [19]. The RI of each field point is computed,including all effects of vignette, pupil expansion,and cos 4 (but not variations of transmission orthe angular sensitivity of the coatings).

Traditionally, optical design faces the dilemma of atrade-off between MTF and RI. The higher the RI,the more rays will pass through the edge of theimage. These deleterious rays will seriously degradethe systematic MTF, at the edge of the image inparticular. Generally speaking, it is very difficultto overcome this dilemma by the LDS method.

5. Taguchi Method

A. Engineering System and Parameter Design

In modern and planned engineering systems, theTaguchi method plays the role of an optimizer. Asshown in Fig. 17, an engineering system generallyconsists of four sections: the signal factor, control fac-tor, noise factor, and output response. The signal fac-tor is the input from the user to the system for aspecified output response. If the system’s output re-sponse changes with the input signal, the system isconsidered to possess dynamic characteristics, ac-cording to the Taguchi method. Parameters that

Fig. 5. 2:5× zoom lens layout (zoom 1).




Fig. 8. Performance with MTF for (zoom 1).



are easy or inexpensive to control are chosen as thecontrol factors. In contrast, parameters that are dif-ficult or expensive to control are designated as noisefactors.The Taguchi method uses an orthogonal array to

execute experiments and to analyze results. Usingan orthogonal array can substantially reduce thetime and cost of developing a new product or techni-que and thereby increase the competitiveness of theproduct in the open market. Orthogonal arrays con-sist of inner and outer columns, with the former de-signated the control factors and the latter the input

signal and noise factors. The principle of the Taguchimethod is to allow the design factors to be subjectedto the tests of the noise factors located at the outercolumns of the array, such that the optimized controlfactors will be effective in combating the influence ofthe noise factors acting on the product quality.

B. Control and Noise Factors and Their Levels

The eight dominant design parameters for develop-ing the zoom lens are identified as the control factors,which are listed in Table 6 and Fig. 18, together withtheir alternative levels. It is noted that most of the


Fig. 11. Performance of field curves (zoom 1). Fig. 12. Performance of field curves (zoom 2).


factors have three levels, except for factors A and B,which are the overall length dimensions of the zoomlens; level 2 of factors A and B is the overall length ofthe optics, and level 1 is an 0:1mm reduction overall;at level 3 there is an increase of overall length of0:1mm. Factor C is the front element of the semi-aperture of the zoom lens, where level 2 is the lenssemi-aperture, level 1 is the lens semi-aperture withan 0:1mm reduction and level 3 is the lens overalllength increased by 0:1mm. Factor D is the materialof the prism, where level 1 is BK4, level 2 is NBK7,and level 3 is BAK4. For the sake of considering thetolerance and misalignment of system design, noisefactors are brought in. The noise factors are listed inTable 7, together with their alternative levels. Thetolerance of the overall length dimensions and thefront element of the semi-aperture, level 1, is the0:05mm reduction; level 2 is increased by 0:05mm.

Fig. 13. Performance of field curves (zoom 3).

Fig. 14. Performance of rays fan (zoom 1).


Regarding the noise factors, orthogonal arrays con-sist of the inner (control factors) and outer columns(noise factors).

C. System Performance Evaluation usingSignal/Noise Ratio

The signal-to-noise (S/N) ratio originates in the com-munication field. The Taguchi method expands itsfunction into the quality engineering area. For anyengineered system, the input signal, control factors,and noise factors come together to perform the designfunction of the system. Some measurable output re-sponses, generally referred to as the performancecharacteristics, are used to express how well the sys-tem performs its functions. The so-called S/N ratio isused to evaluate the output response. As statedabove, the performance characteristics to be mea-sured are the MTF and RI of the system. In anystudy, it is better to have a larger MTF and RI.

The larger-the-better (LTB) S/N ratio formula is

LTBS=N ¼ −10 log�1n

Xni¼1

1

y2i

�; ð5Þ

where yi denotes the values of the MTF and RI.The S/N ratio measures the level of system perfor-

mance and the effects of noise factors on perfor-mance. The higher this ratio, the more the systemis doing what it was intended to do, regardless ofnoise factors; the system is more robust againstnoise. ANOVA is performed on this S/N ratio to de-termine the effects of different factors on perfor-mance. Specifically, the sum of squares, meansquares, F values, and percentage contribution arecomputed to determine which factors contribute sig-nificantly to the variance and mean response. This



information is then used to predict the optimumsettings for the various design factors.

6. Design for a Robust Multiple-Criteria OptimizationApproach (RMCO)

The Taguchi method, in its basic form, does not han-dle multiple objective formulations or constraints.

This study, following [21,22], extends the Taguchimethod to address such cases. The problem set-up for the multiple responses case is similar tothat for the single response case except that eachobjective and constraint function will result in a se-parate ANOVA table. A two-step procedure is thenfollowed to identify the noninferior set:


Fig. 17. Engineering system using the Taguchi method.

Table 6. Control Factors and Their Levels (L9)

Control Factors

Level

1 2 3

A Surface 1 to surface 5 length 4:9mm 5:0mm 5:1mmB Surface 7 to image length 19:3mm 19:4mm 19:5mmC Semi-aperture of front element 4:3mm 4:4mm 4:5mmD Prism glass material BK4 NBK7 BAK4


1. Eliminate all factor levels that cause any of theconstraints to be violated.

2. Identify those factor levels, in the reducedspace, that significantly affect any of the objectives.

A factor is assumed to have a dominant effect on anobjective function if its percentage contribution (or Fvalue) is considerably higher than that of all theother factors for this objective. It has a significant ef-fect if the percentage contribution is greater than adesigner specified cutoff value. Details of this cutoffvalue and its selection are addressed below in thissection.

The S/N ratio for the constraint functions are setup for the larger-is-better case if it is a lesser-thanconstraint (and also the reverse), so that the factorsthat violate the constraint can be easily identified.Using the ANOVA table, the effect of the various fac-tor levels on the constraint functions is consideredfirst. If the ANOVA of a constraint function indicatesthat the function is significantly affected by a factor,the level of the factor causing the violation is identi-fied. If the mean effect of this level on the constraintfunction is beyond the constraint limit, then this le-vel is eliminated from further consideration. Notethat, before constraint mean effects are comparedwith the constraint limit, the limit has to be trans-formed by the same transformation used to obtainthe S/N ratio from the constraint equation. In thismanner, the infeasible designs are filtered out fromthe initial design space.

The ANOVA of each of the objective functions arethen considered, and the design space is furtherreduced according to the following rules:

1. If a factor has a significant effect on all objec-tive functions, then all the levels that optimize atleast one objective are selected.

2. If a factor has a dominant effect on a single ob-jective, the factor level that optimizes this objective isselected, regardless of its significance for other ob-jectives.

3. If a factor has an insignificant effect on all theobjectives, then the designer can use his or her dis-cretion to determine the objective most affected bythis factor before identifying the best level for thefactor.

Table 7. Noise Factors (Tolerance) and Their Levels (L4)

Noise Factors

Levels

1 (mm) 2 (mm)

A Surface 1 to surface 5 length 0.05 −0:05B Surface 7 to image length 0.05 −0:05C Semi-aperture of front element 0.05 −0:05

Fig. 19. Flow chart of the robust multiple criterion optimizationprocedure [21,22].

Table 8. Experimental Setup and S/N Ratio

Control Factors Performance Evaluation of Quality

Test A B C D MTF S/N (db) RI S/N (db)

1 4.9 19.3 4.3 BK4 −6:22831 −2:170392 4.9 19.4 4.4 NBK7 −6:32192 −1:993383 4.9 19.5 4.5 BAK4 −6:11848 −1:950454 5.0 19.3 4.4 BAK4 −6:02679 −1:955675 5.0 19.4 4.5 BK4 −6:03110 −1:955676 5.0 19.5 4.3 NBK7 −6:15196 −2:203027 5.1 19.3 4.5 NBK7 −5:95288 −1:986498 5.1 19.4 4.3 BAK4 −6:04195 −2:122369 5.1 19.5 4.4 BK4 −5:97418 −2:06618

Fig. 18. Control factors of zoom optics.


This procedure thus attempts to identify superiordesigns, but note that it does not ensure theelimination of all nonsuperior designs from the finalset. The proportion of nonsuperior designs in thefinal set depends to a great extent on what the de-signer determines as the cutoff value for identifyingsignificant factors. A high value for this cutoff indi-cates that the designer does not want any nonsuper-ior designs in the final set. In this case, the designeris willing to eliminate some superior designs for thesake of reducing the cost of processing the final set.Alternatively, a low cutoff value will result in a largefinal design set and consequently a sizeable numberof nonsuperior designs, which then have to be filteredout by further processing. Depending on the choice ofthe designer, further processing would mean eitheranother iteration of the reduced design space or fil-tering out the nonsuperior designs by using a stan-dard procedure, such as the technique of dominatedapproximations. Thus the cutoff value is a trade-offbetween the number of noninferior designs that thedesigner requires and the amount of effort he or sheis willing to put in to identify them.The procedure outlined above does not guarantee

an exhaustive noninferior set, but it should be notedthat the enumeration of such a set may not bepreferred in most practical design situations, sinceit will only compound the complexity of the finalselection process. At the same time, providing a re-presentative set in a systematic manner as shownhere will greatly facilitate the identification of appro-priate designs as either final or capable of further re-finement. The flow chart describing this technique ofrobust multiple criterion optimization is shownin Fig. 19.

7. Problem Formulation and Results

The application of this methodology to multicriteriaoptimization problems is demonstrated by a fold op-tical 2:5× zoom lens system design problem involvingtwo objectives and two constraints. The objective ofthis problem is to raise the performance of the zoom

lens, whichmaximizes the RI andMTF under a givenspecification of lens. The problem can be expressed asfollows:

1. Maximize the RI at position 1 of the maxi-mum field.

2. Maximize the MTF at position 1 of the maxi-mum field.

Maximization is subject to the constraintsRI ≥ 60% at position 1 of the maximum field,MTF ≥ 40% at position 1 of the maximum field.

To capture the nonlinear effects of the various re-sponses, in the present study three levels within thespecified range were initially chosen for each of thefour design variables. Thus, neglecting interactioneffects, the total of the degrees of freedom of the sys-tem is 8. The L9 orthogonal array was used to designthe control factors for this problem. The L4 orthogo-nal array was used to design the noise factors for thisproblem. The constraint equation was used only toeliminate invalid designs. The experiments are setup as shown in Table 8.

The objective functions are computed for each com-bination of design variables and for each combinationof noise values. The S/N ratios for the various experi-ments are shown in Table 8. The mean response ofeach factor level is then computed for each objectivefunction. These are listed in Table 9.

The ANOVA of the objective functions (Table 9)shows that factor A has a significant effect on theMTF (objective 1), whereas only factor C significantlyinfluences the RI of the zoom lens (objective 2). Ad-ditionally, factor A and factor C are dominant factorsfor the performance of the zoom lens, since their per-centage contributions for MTF (70.55%) and RI(86.73%) are much greater than those of any otherfactor. The levels of factor A and factor C that opti-mize the MTF (level 3) and RI (level 3) are thussingled out for further consideration.

Table 9. ANOVA for All Objectives

Measure Level 1 Level 2 Level 3 Sum of Squares Degrees of Freedom Mean of Squares Contribution (%)

Objective 1 (MTF)Factor A −6:2229 −6:06995 −5:98967 0.084237 2 0.042118 70.551Factor B −6:06933 −6:13166 −6:08154 0.006547 2 0.003273 5.482Factor C −6:14074 −6:10763 −6:03416 0.017854 2 0.008927 14.953Factor D −6:07786 −6:14225 −6:06241 0.01076 2 0.005380 9.011Residual −5:684341 × 10−14 0Error −1:136868 × 10−13 0Total 0.119398 8

Objective 2 (RI)Factor A −2:03807 −2:03812 −2:05834 0.000819 2 0.00041 1.049524Factor B −2:03752 −2:02381 −2:07321 0.003903 2 0.001952 4.997543Factor C −2:16526 −2:00508 −1:9642 0.067749 2 0.033875 86.734887Factor D −2:06408 −2:06096 −2:00949 0.005638 2 0.002819 7.218044Residual 7:10543 × 10−15 0Error 7:10543 × 10−15 0Total 0.078111 8


Tab

le10

.NoninferiorDes

ignSet

Test

Con

trol

factors

AB

CD

MTF

100lp=mm

(%)

RI(%

)

Field

(x,y)

(deg

)(0.00,

0.00

)(0.00,

7.85

)(0.00,

15.42)

(0.00,

22.48)

(0.00,

28.88)

(0.00,

0.00

)(0.00,

7.85

)(0.00,

15.42)

(0.00,

22.48)

(0.00,

28.88)

15.1

19.3

4.5

BK4

60.0

60.0

49.0

47.0

43.0

100

94.3

85.3

74.6

61.9

25.1

19.4

4.5

NBK7

60.0

60.0

49.0

46.0

43.0

100

94.2

85.3

74.6

62.1

35.1

19.5

4.5

BAK4

60.0

59.0

48.0

46.0

43.0

100

94.4

85.4

74.8

62.0

45.1

19.3

4.5

BAK4

60.0

59.0

48.0

46.0

43.0

100

94.4

85.4

74.8

62.0

55.1

19.4

4.5

BK4

60.0

59.0

48.0

46.0

43.0

100

94.4

85.4

74.8

62.0

65.1

19.5

4.5

NBK7

60.0

60.0

48.0

47.0

43.0

100

94.2

85.3

74.6

62.1

75.1

19.3

4.5

NBK7

60.0

60.0

48.0

47.0

43.0

100

94.2

85.3

74.6

62.1

85.1

19.4

4.5

BAK4

60.0

59.0

48.0

46.0

43.0

100

94.4

85.4

74.8

62.0

95.1

19.5

4.5

BK4

60.0

60.0

48.0

47.0

43.0

100

94.3

85.3

74.6

61.9


The factor levels are chosen that optimize the re-spective objectives, the design space is narroweddown, and the new levels (bounds) for the design fac-tors are identified as follows:

A¼5:1mm; 19:3≤B≤19:5mm; C¼4:5mm;

and D is the material of the prism, where level 1 isBK4, level 2 is NBK7, and level 3 is BAK4.At this point, the designer has the option of either

(1) performing another iteration on the reducedspace to further narrow down the search or (2) iden-tifying the noninferior set by combining the variouslevels of each factor. The predicted levels can be usedto formulate new design levels. Alternatively, thecombination of the various levels in the reduceddesign space results in a design set as shown inTable 10. The graphical representation of the result-ing objective functions for all the superior designs isshown in Fig. 20. All the designs in this set, for thisparticular example, are Pareto optimal designs.To test the effect of the initial conditions on the ro-

bustness of the noninferior design set [23], we eitherreduced or increased the length of the contribution

maximum factor C 0:1mm. The results are shownin cases 1 and 2 of Table 11. Cases 3 and 4 of Table 11show the dimensions of factors A, B, and C, which arereduced and increased by 1%. In cases 1 and case themaximum change in MTF of the initial conditionsand that of the noninferior design set are 9.864%and 1.364%, and the RI maximum changes are1.492% and 0.434%. In cases 3 and 4, the maximumchange in MTF of the initial conditions and that ofthe noninferior design set are 8.926% and 1.748%,and the RI maximum changes are 0.398% and0.214%. From the results shown above, the MTFand RI could not change as the dimensions of the fac-tors were reduced and increased. The noninferior de-sign set was therefore robust.

8. Conclusions

The research concludes that the optical design for2:5× zoom optics with a liquid lens element andthe proposed method, optimized by the Taguchimethod, coupled with a multiple-criteria optimiza-tion approach (MCOA), is effective and efficientwhen applied to the MTF and RI of 2:5× zoom opticswith a liquid lens element. It meets the demands of

Fig. 20. Graph of MTF against RI in position 1.


Tab

le11

.Robustnes

sExp

erim

entforInitialConditionsan

dtheNoninferiorDes

ignSet

MTF

100lp=mm

(%)

RI(%

)

Field

(x,y)(deg

)(0.00,

0.00

)(0.00,

7.85

)(0.00,

15.42)

(0.00,

22.48)

(0.00,

28.88)

(0.00,

0.00

)(0.00,

7.85

)(0.00,

15.42)

(0.00,

22.48)

(0.00,

28.88)

Initialcond

ition

60.5

48.0

47.0

41.0

38.0

100

94.6

85.3

75.7

61.8

Cas

e1

60.0

58.0

48.0

46.0

42.0

100

94.4

85.4

74.9

54.2

Cas

e2

60.0

58.0

48.0

46.0

44.0

100

94.4

85.4

74.9

61.9

Increa

sedPercent

(%)

−0:83

20.83

2.13

12.20

13.16

0−0:21

0.12

−1:06

−6:07

Cas

e3

60.0

56.0

48.0

45.0

43.0

100

94.8

85.6

75.1

62.7

Cas

e4

60.0

58.0

48.0

45.0

43.0

100

94.5

85.5

74.9

61.8

Increa

sedPercent

(%)

−0:83

18.75

2.13

9.76

13.16

00.05

0.29

−0:92

0.73

One

setof

noninferiorde

sign

60.0

60.0

49.0

46.0

43.0

100

94.2

85.3

74.6

62.1

Cas

e1

60.0

60.0

49.0

47.0

45.0

100

94.7

85.7

75.0

62.5

Cas

e2

60.0

60.0

49.0

47.0

45.0

100

94.7

85.7

75.0

62.5

Increa

sedPercent(%

)0

00

2.17

4.65

00.53

0.46

0.54

0.64

Cas

e3

60.0

59.0

49.0

47.0

45.0

100

94.4

85.4

74.6

62.1

Cas

e4

60.0

60.0

49.0

48.0

45.0

100

94.7

85.5

74.9

62.5

Increa

sedPercent(%

)0

−0:83

03.26

4.65

00.37

0.18

0.20

0.32


Tab

le12

.Perform

ance

Eva

luationoftheInitialConditionsan

dtheNoninferiorDes

ignSet

MTF

100lp=mm

(%)

RI(%

)

Field

(x,y)(deg

)(0.00,

0.00

)(0.00,

7.85

)(0.00,

15.42)

(0.00,

22.48)

(0.00,

28.89)

(0.00,

0.00

)(0.00,

7.85

)(0.00,

15.42)

(0.00,

22.48)

(0.00,

28.89)

Initialcondition

60.5

48.0

47.0

41.0

38.0

100

94.6

85.3

75.7

61.8

Non

inferior

design

set

160

.060

.049

.047

.043

.010

094

.385

.374

.661

.92

60.0

60.0

49.0

46.0

43.0

100

94.2

85.3

74.6

62.1

360

.059

.048

.046

.043

.010

094

.485

.474

.862

.04

60.0

59.0

48.0

46.0

43.0

100

94.4

85.4

74.8

62.0

560

.059

.048

.046

.043

.010

094

.485

.474

.862

.06

60.0

60.0

48.0

47.0

43.0

100

94.2

85.3

74.6

62.1

760

.060

.048

.047

.043

.010

094

.285

.374

.662

.18

60.0

59.0

48.0

46.0

43.0

100

94.4

85.4

74.8

62.0

960

.060

.048

.047

.043

.010

094

.385

.374

.661

.9In

crea

sedPercentage

(%)

−0:83

24.0

2.6

13.3

13.2

0−0:31

0.05

−1:33

0.32


the specifications, which the traditional least damp-ing square (LDS) method would do with great diffi-culty. In conclusion, the experimental results may besummarized as follows:

1. A miniature zoom with liquid lens elementscomplicates the work of optical design. Results showthat the optical design of fold zoom optics with aprism will perform reasonably well with the advan-tage of a liquid lens element as the zoom function,thanks to the fact that the telecentric chief raypasses through the liquid lens elements.2. After further extended optimization by the Ta-

guchi method, taking a multiple-criteria optimiza-tion approach which uses ANOVA results, RI forthe lens is still more than 60% in the extremely wideposition and more than 80% in telephoto. Conclu-sively, the MTF for zoom position 1 was further in-creased by 10.45% and the RI for zoom position 1was reduced by 0.254%, as shown in Table 12.3. An optical design of a 2:5× zoom lens optics as-

sembled with one prism and two liquid lens elementscan be successful, despite the minimization of volu-metric size. Compared with a traditional lens, whoseoverall length is often more than 15mm in the x di-rection and 25mm in the y direction, this opticsachieves 33% and 4.8% savings in volumetric sizein the x and y directions, respectively.4. With the optical software CODE V, only the

MTF or distortion is used to proceed with the toler-ance analysis. The tolerance of control factors is setto be the noise factors.5. This proposed fold optical zoom lens system

design could be fabricated because, during optimiza-tion, fabrication constrains such as the diameter ofthe front and the rear lens element was set andthe tolerance was analyzed.6. The MTF and RI could not change as the di-

mensions of the factors were reduced and increasedfor the noninferior design set, showing that the non-inferior design set was robust.

This work was supported by the Industrial Re-search Technology Institute, Taiwan, Republic ofChina, under grant number B200-96MB1.

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optical design and multiobjective optimization of miniature zoom optics with liquid lens element

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