CALCULATION, DESIGN, AND PRODUCTION OF OPTICAL SYSTEMS

Choosing an optical setup and designing compact objectives for mobile telephones

I. G. Bronshte n,a� V. A. Zverev, and I. L. Livshits

St. Petersburg State University of Information Technologies, Mechanics, and Optics, St. Petersburg

Kim Young-Gi, Kim Tae-Young, and Jung Phil-Ho

Samsung Electro-Mechanics Co. Ltd., Suwon, Republic of Korea�Submitted December 25, 2008�Opticheski� Zhurnal 76, 25–31 �May 2009�

This paper discusses fundamental questions of constructing the optical setup of a compact objec-tive and choosing the initial design for the system, based on the theory of the synthesis and com-position of optical systems. The planning of the optical systems of compact objectives for mobiletelephones is analyzed. An example of a design for a version of an objective is given.© 2009 Optical Society of America.

INTRODUCTION

It has become necessary in recent years to develop newtypes of objectives for the cameras of mobile telephones.This is caused, first of all, by the appearance of new radiationdetectors with higher resolving power, as well as a continu-ally growing demand for mobile telephones with built-invideo cameras. Thus, for example, there is a ubiquitous tran-sition from objectives for 2M �two-megapixel� video cam-eras to objectives for video cameras with a resolution of 3Mor more.

Such optical systems are currently being developed in allcountries that produce mobile telephones, the leaders amongwhich are South Korea and Japan.1–3 All the optical systemspresented in the patents described here consist of three orfour plastic lenses with surfaces of nonspherical shape. Therecontinues to be conventional competition among optical-systems developers to create the shortest, fairly fast objec-tive.

The objectives that are currently required must possessthe following properties and parameters:

• The relative aperture of the objective must be at least1:2.8.

• The angular field must be at least 2�=60°.• The image size is determined by the 1 /4 in. sensitive sur-

face of the CCD array, which corresponds to an imagediagonal of 2y�=4.5 mm.

• The size of the circle of confusion in the point imageformed by the objective must correspond to the pixel sizeof the CCD array, not exceeding 2 �m, within the entirefield.

• The structural parameters of the optical system of the ob-jective must satisfy the technological conditions of massproduction.

• The distance from the first surface of the optical system tothe image plane must not exceed 5 mm.

• For the CCD array to function normally, the angle that theprincipal ray makes with the optical axis in image spacemust not exceed �=23°.

268 J. Opt. Technol. 76 �5�, May 2009 1070-9762/2009/050

KNOWN DESIGNS OF OPTICAL SYSTEMS OF COMPACTOBJECTIVES

Let us consider the best-known designs of compact op-tical systems for mobile telephones.1–3 When examining thepatent sources, there was special interest in analyzing theconfiguration of the optical elements used in the optical set-ups of the objectives. The results of the analysis are shown inTable I. Here O is an afocal component whose focal power isclose to zero, P is a positive component having positive focalpower, and N is a negative component having negative focalpower.

The number of possible combinations of three elementstaken three at a time equals 33=27. Excluding combinationsthat are meaningless in practice, we get nineteen possibleversions of layouts of the objective. The information given inTable I makes it possible to form the following conclusions:

• Almost all the optical setups of compact objectives knownfrom patent sources consist of three optical elements�lenses�.

• Each of the optical systems contains at least one opticalelement with nonspherical surfaces.

• The elements with nonspherical surfaces are made fromplastic.

• The images formed by all the known optical systems havesmall distortion, not exceeding 3%.

• By no means do all objectives form an image whose qual-ity is determined by the diffraction of light.

Note: Versions of optical systems that possess negativefocal power were not considered in this analysis.

Our analysis shows that the largest number of solutionson the creation of compact objectives for the cameras ofmobile telephones are based on the P–N–P sequence—twopositive components with a negative component betweenthem. An aperture stop is usually placed after the first com-ponent, fairly close to it.1–3 Figure 1 shows the optical layoutof a compact objective according to U.S. Patent 6927925.4

268268-06$15.00 © 2009 Optical Society of America

CHOOSING THE INITIAL DESIGN OF THE OPTICALSYSTEM OF THE OBJECTIVE

It is well-known that, at moderate values of the relativeaperture and the angular field, the quality of the imageformed by an optical system is determined by the first terms

TABLE I. Results of a patent analysis of the optical systems of c

FIG. 1. Example of an optical layout of a compact objective.

269 J. Opt. Technol. 76 �5�, May 2009

of an expansion in series of the aberration as a function ofthe aperture and field angles in a point image—i.e., by theso-called primary aberrations of the image. Consequently, forany optical system whose parameters have been optimizedusing the quality of the image formed by it as a criterion, theprimary aberrations must be compensated first of all.

The primary aberrations are determined by the followingcoefficients:

• spherical aberration

SI = �i=1

i=k

hiQi, �1�

• coma

SII = �i=1

i=k

HiQi − J�i=1

i=k

Wi, �2�

• astigmatism

SIII = �i=1

i=kHi

2

hiQi − 2J�

i=1

i=kHi

2

hiWi + J2 � �i+1�i+1 − �i�i

hi, �3�

ct objectives.

ompa

269Bronshte n et al.

• Petzval curvature of the image surface

SIV = � �i+1�i+1 − �i�i

hi, �4�

• distortion

�5�

�i is the deformation coefficient of a spherical surface in theequation x2+y2−2riz− �1+�i�z2, �i is the angle formed be-tween the axial virtual ray and the optical axis, �i=1 /ni, hi

and Hi are the distances from the optical axis to the pointswhere the axial and principal virtual rays intersect the prin-cipal planes of the optical surfaces, and J is the Lagrange–Helmholtz invariant.

When the optical system is a thin component,5 i.e., whenthe thickness of the lenses that compose it can be neglected,the heights of the virtual rays on the principal planes of thesurfaces are

h1 = h2 = . . . = hk = h ,

H1 = H2 = . . . = Hk = H , �6�

with

�i=1

i=k

hiQi = h�i=1

i=k

Qi, �i=1

i=k

HiQi = H�i=1

i=k

Qi.

If �i=1i=kQi=0 in the optical system thus developed, S1

=0. In this case, if �i=1i=kWi=0, SII=0 when the other coeffi-

cients are nonzero.Thus, only aplanatic correction of the primary aberra-

tions can be obtained in the image formed by the thin com-ponent, regardless of the number of nonspherical surfaces init.6

To compensate all the primary aberrations of the image,the surfaces of the system must be dispersed along the opti-cal axis in such a way that the heights are hi�hj and h�

�h�. The inequality conditions of the heights are naturallysatisfied in reflective optical systems. Thus, for example, it is

270 J. Opt. Technol. 76 �5�, May 2009

quite possible to achieve plananastigmatic correction of theaberrations in a three-mirror system with a finite distancebetween the reflecting surfaces.7,8

In traditional systems with small but finite thickness ofthe lenses, inequality of the heights can be obtained by in-creasing the surface curvature. However, this results in theappearance of large, hard-to-remove high-order aberrations.This is explained by the fact that replacing spherical withnonspherical surfaces in traditional optical systems does notalways have the desired �or expected� effect.

Let the parameters be Q1=Q2= . . . =Qk=0 when thepoints where the virtual rays intersect the principal planes ofthe surfaces are at unequal heights. Each of the surfaces ofsuch a system in fact forms an image in which there is nospherical aberration. That is, in general, when the other co-efficients are nonzero for each surface, the coefficient SI=0.Professor V. N. Churilovski� called such surfaces anaberra-tional surfaces.9 Classical Newtonian, Cassegrainian, andGregorian reflective systems, which were fairly widely usedin the last century because it is comparatively easy to moni-tor the surface shape during fabrication, can serve as an ex-ample of such systems, consisting of anaberrational surfaces.The image that they form has acceptable quality only withina very small angular field, which can be broadened by usinglens compensators of the field aberrations.

In the optical systems of compact objectives, the thick-ness of the lenses cannot be considered small by comparisonwith the length of the system because of technological prob-lems of fabrication. Therefore, the difference in the heightsof the points where the virtual rays intersect the principalplanes of the surfaces of the system, as on reflective systems,is obtained naturally. In this case, the lens surfaces of suchobjectives �at least, a fair number of them� need not be anab-errational. On the other hand, because of the technologicalproblems of fabrication and to avoid the appearance of largehigh-order aberrations, the coefficients of surface deforma-tion need not be great; consequently, the surface parametersPi must be fairly small even in the initial system.

The successful choice of the initial optical system deter-mines the final success in creating it. This problem can besuccessfully solved by using the theory of the synthesis andcomposition of optical systems, proposed and developed byProfessor M. M. Rusinov10 and developed further in Refs. 11and 12.

DEVELOPING THE DESIGN OF THE INITIAL OPTICALSYSTEM OF A COMPACT OBJECTIVE

Three steps can be distinguished in the process of devel-oping the optical system of a compact objective:

• choosing the initial optical system,• calculating the size of the optical system simultaneously

with its parametric synthesis,• optimizing the parameters of the optical system according

to the criterion of image quality. This stage of the devel-opment of an optical system in fact reduces to automati-cally correcting the aberrations of the image formed by thesystem thus designed.

270Bronshte n et al.

Let us consider each of the design stages:Stage I. According to the classification of objectives pro-

posed in Refs. 11 and 12, the optical system of the requiredobjective has a complexity index equal to 7. The structural-synthesis formula of such a system has the form

B�PA� + K�PP� + K�II� ,

where B is conventionally designated as the base element, Pas a surface concentric with the center of the entrance pupil,A as an aplanatic surface, K as a correction element, and I asa near-focus surface.

A detailed description of the properties of the surfacesand their use is given in Ref. 11.

Stage 2. The quality of the image formed by the opticalsystem must correspond to the structure of the multielementoptical-radiation detector. For a radiation detector whosesensitive surface has a discrete structure, it is customary todetermine the requirement on the image quality by the con-trast required for a specific spatial frequency of the object.As a rule, the Nyquist frequency is adopted as such a fre-quency.

When the diagonal of the CCD array used in the caseunder consideration is 1 /4 in., equal to 4.5 mm, and the ratioof the sides of its surface equals 4:3, the sides of the sensitivesurface are a=3.6 mm along the horizontal and b=2.7 mmalong the vertical. The number of pixels in a high-qualityarray is 2048 along the horizontal and 1536 along the verti-cal. From this, we find the distance between the pixels as

� = a/2048 = b/1536 = 0.00175 mm or 1.75 �m.

In this case, the number of pixels per millimeter is

p = 1/� = 2048/a = 1536/b = 569 pixel/mm.

The Nyquist frequency is N= �1 /2p�=284 line /mm.Note that the limiting frequency of the spatial-frequency

spectrum allowed by the optical system with diffraction qual-ity of the image equals N0�D / ��f��. When the relative ap-erture is 1:2.8 and the wavelength is 0.587 �m, the fre-quency is N0�600 line /mm. When the linear image fieldequals 4.5 mm and the angular object field is assumed to be65°, the focal length of the objective is f�=3.57. With thespecified relative aperture of the developed optical system,with a compromise choice of the angular field of the objectand the correspondingly calculated focal length of the sys-tem, and with the structural-synthesis formula chosen fromthe condition for the required correction of aberrations, wefind the radii of curvature of the spherical surfaces and thethicknesses of the lenses and the air gaps; i.e., in parallelwith solving the problem of size limitations, we solve theproblem of parametric synthesis of the initial optical system.The plane-parallel plate should be taken into account in thiscase as a necessary element of the radiation detector, since itis located in a convergent pencil of rays and introduces ab-errations into the image. Diffraction quality of the imagecannot be achieved if its influence is neglected. Note that it isconvenient to choose the lens thicknesses by means of thegraphic editor of the program used here.

271 J. Opt. Technol. 76 �5�, May 2009

Stage 3. The problem of correcting the image aberrationsassumes that the multiparametric problem of mutual com-pensation of highly nonlinear functions has been solved. As arule, there are no such problems with a rigorously analyticalsolution. Therefore, the procedure for automatically correct-ing the aberrations of the image formed by the optical systembeing designed includes in fact two interconnected proce-dures: automatically correcting the image aberrations bymeans of the corresponding software, and a heuristic proce-

FIG. 2. Fundamental optical layout of the initial version of an objective.

FIG. 3. Optical layout of a version of an objective for a mobile telephoneafter optimization.

271Bronshte n et al.

dure of controlling the process of automatically correctingthe aberrations and accordingly the process of automaticallydesigning the optical system.

At this stage, the intelligent contribution of the optical-system developer consists of a competent or, one could al-most say, a lucky construction of an estimating function, fol-lowed by controlling the process of optimizing the systemparameters by varying both the estimating function by as-signing various weighting characteristics and by introducingand varying the weighting characteristics on the parametersof the optical system. The developer’s qualifications and ex-perience are just as important when carrying out this stage ofplanning the optical system as when choosing the initial sys-tem.

The main correction parameters in the optical systemunder consideration are the coefficients of the equations thatdescribe the nonspherical surfaces. It is important in this caseto keep in mind that the form of the equation that describes anonspherical surface has a substantial effect on the characterof the aberration correction. When the optical system whoseparameters are given in this paper as an example was beingdesigned, the nonspherical surfaces were defined by an equa-tion of the form

z = G1R2 + G3R4 + G6R6 + G10R8 + G16R

10. �7�

TABLE II. Frequency-contrast characteristics of the objective sho

272 J. Opt. Technol. 76 �5�, May 2009

The following coefficients of the equations of the non-spherical surfaces were used as correction parameters: G1,G3, G6, G10, and G16.

The authors faced the following serious problems in cal-culating and then optimizing the parameters of the opticalsystem under consideration:

• the need to correct chromatic aberrations without beingable to choose the lens material;

• the serious conflicts that arise when it is necessary to cor-rect distortion while simultaneously achieving the neces-sary size of the optical system �the length of the objectivemust not exceed its focal length by more than a factor of1.2�;

• ensuring the required angles of incidence of the principalrays on the image detector, which also come into conflictwith correction of the distortion;

• the requirements on the allowable deviations of the surfaceshape and the thickness of the optical items when they arefabricated, due to mass-production conditions.

The extremely nonlinear character of the interaction ofthe parameters that determine the aberrations of the imageformed by such systems should be especially emphasized.This has the effect that the values of the coefficients of theequations that define the nonspherical surfaces assume great

n Fig. 3.

wn i

272Bronshte n et al.

significance when the parameters are being optimized, andthis hampers the convergence of the process of automaticallycorrecting the aberrations. The need to develop complex op-tical systems of this type increases the requirements both onthe level of software being used and on the developer’squalifications.

Figure 2 shows the optical layout of the initial system,obtained as a result of its parametric synthesis from surfaceswith known properties. The initial aberrations of the systemare small.

Since parametric synthesis gives us an optical systemthat contains only spherical surfaces, we carry out the pro-cess of synthesizing the initial system with a lower relativeaperture. For example, let the initial limiting spatial fre-quency of the imaged object equal N0=200 line /mm for anaberration-free image. According to the formula N0

�D / ��f��, we then get that, when �=0.587 �m, the relativeaperture of the optical system must equal D / f�=1:8.5. In thecase under consideration, the process of synthesizing the ini-tial system was carried out with D / f�=1:8.5.

As a result of automatically correcting the aberrations,the optical system whose layout is shown in Fig. 3 was ob-tained for the objective. Table II gives the frequency-contrastcharacteristics �FCCs� for the compact objective developedhere. A comparison of the values given here with the valuesof ideal FCCs, also shown in Table II, makes it possible toconclude that the optical system developed by the techniqueexplained here forms a high-quality image, close to the dif-fraction limit of resolution over the entire image field.

273 J. Opt. Technol. 76 �5�, May 2009

The extremely crucial nature of the problem of finding amore perfect design of an optical system that possesseshigher optical characteristics while reducing the size and thenumber of components is determined by the massive demandfor mobile telephones with built-in video cameras.

a�Email: irina@jupiter.spb.ru

1Y. Shinohara, “Imaging lens,” U.S. Patent 6,795,253 B2 �2004�.2M. Isono, “Imaging lens,” U.S. Patent 2004/0021957 A1 �2004�.3K. Sato, “Imaging lens,” U.S. Patent 6,961,191 B2 �2005�.4V. A. Zverev, Principles of Geometrical Optics. A Textbook �SPbGITMO�TU�, 2002�.

5G. G. Slyusarev, Methods of Designing Optical Systems �Mashinostroenie,Leningrad, 1969�.

6V. A. Zverev and A. N. Shepelevich, “The concept of a thin component ina system of reflective surfaces,” Opt. Zh. 73, No. 12, 21 �2006� �J. Opt.Technol. 73, 840 �2006��.

7V. A. Zverev and A. N. Shepelevich, “Parametric model of a three-component system of reflective surfaces,” Opt. Zh. 74, No. 4 47 �2007��J. Opt. Technol. 74, 263 �2007��.

8V. N. Churilovski�, Theory of Optical Devices �Mashinostroenie, Lenin-grad, 1966�.

9M. M. Rusinov, Composite of Optical Systems �Mashinostroenie, Lenin-grad, 1989�.

10I. L. Anitropova, “Formalizing the heuristic synthesis procedure in lensdesign,” Opt. Soc. Am. Proc. 22, 6 �1994�.

11I. L. Livshits and A. V. Salnikov, “CAD based on developed algorithm andexpert rules in proposed in automate lens,” Tech. Digest ICO 4228, 157�2004�.

12H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE 47,

617 �1962�.

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