Journal of the Korean Physical Society, Vol. 50, No. 5, May 2007, pp. 1243∼1251

Optical Zoom System Design for Compact Digital Camera Using Lens Modules

Sung-Chan Park,∗ Yong-Joo Jo, Byoung-Taek You and Sang-Hun Lee Department of Physics, Dankook University, Cheonan 330-714

(Received 5 February 2007)

By use of lens modules and third-order aberration theory, a new design approach can be applied to the three-group inner-focus zoom system. The optimum initial design satisfying the specific requirements and its real lens design from the lens modules are presented. An initial design with a focal length range of 4.3 to 12.9 mm is derived by assigning appropriate first-order quantities and third-order aberrations to each module along with the constraints required for the optimum solutions. By using an automatic design method rather than analytic approaches, we separately designed a real lens for each group at given conjugates and then combined them to establish an actual zoom system. The combination of the separately designed groups results in a system that satisfies the basic properties of the zoom system consisting of the original lens modules. When the aberrations are balanced, the finally designed three-group zoom lens is expected to fulfill all the requirements of a compact digital zoom camera.

PACS numbers: 42.15.Eq, 42.15.Fr Keywords: Lens module, Aberrations, Digital zoom camera

I. INTRODUCTION without detailed prescriptions. Zoom lens design using lens modules has the following advantages: modules can be used for each of the moving groups, and the param- The zoom lens design is usually divided into two tasks. eters defining the specifications, the third-order aberra- One is paraxial studies based on thin-lens theory, which tion characteristics, and the positions of the groups for give the first-order parameters, such as the focal length zooming can be varied to obtain the optimum design sat- of each group, the zoom ratio, the focal length range, the isfying the requirements. zooming locus, etc. The other is to set up the zoom lens In this paper, lens modules and aberration theory are system from the paraxial studies and balance aberrations used to discuss the optimum initial design of three-group [1–5]. These approaches, however, have several disadvan- inner-focus zoom lenses. This initial zoom system is de- tages. It is difficult to determine if the solutions obtained signed to satisfy specific requirements, and the real lens from paraxial studies satisfy all the requirements for the designs are obtained from the lens modules by using an zoom lens, such as packaging constraints, specifications, automatic design method. In this process, the real lens overall length, and so on. Since the aberrations of this for each group is quickly designed to match the first- starting zoom lens are not corrected, aberration balanc- and third-order aberrations of the module. Compared to ing at all zoom positions requires much more effort in an analytic design, this approach can dramatically save the design of multi-group zoom systems. time and effort. Thus, the separately designed groups The difficulties due to paraxial analyses can be over- are then combined to form an actual zoom lens. Finally, come by using the lens module design reported by residual aberration balancing results in a zoom lens that Stavroudis and Mercado [6], Kuper and Rimmer [7], and has enough performance over a range of f-number from Park and Lee [8]. Lens modules are the mathemati- 3.2 at the wide-field extreme to 4.5 at the narrow-field cal constructs that can model a complex optical sys- extreme positions. This zoom lens is expected to fulfill tem without actually doing the detailed design. The lens all requirements of a compact digital zoom camera. modules discussed by Kuper and Rimmer are based on mock ray tracing, which consists of tracing rays through a lens specified by one of its eikonal function rather than II. LENS MODULE DESIGN FOR THE its curvature, thickness, and indices. Lens modules can THREE-GROUP INNER-FOCUS ZOOM be used as a starting point for the design of a real lens and SYSTEM to model an arbitrary lens from measurable quantities The layout of the three-group inner-focus zoom system ∗E-mail: [email protected]; Fax: +82-41-550-3429 is shown in Figure 1. From the object to the image side, -1243- -1244- Journal of the Korean Physical Society, Vol. 50, No. 5, May 2007

Fig. 1. Layout of the three-group inner-focus zoom system: (a) wide-field position, (b) zooming position, and (c) narrow-field position.

Fig. 2. First-order parameters for specifying a lens system: (a) thick-lens elements and (b) the lens module. the zoom system is composed of a fixed front lens group, a second lens group for compensation, and a third lens group for zooming. Their powers are denoted by k1, k2, and k3, respectively. The first group is always fixed. While the third group moves to the object side to have a longer focal length, the second group should move to keep the image position stationary. When the displacement of the groups for zooming is zero, i.e., at position 1, the distances between the preced- ing group and the succeeding group are specified by d11, d21, d31, as shown in Figure 1. When the displacement of the third lens group is maximum, i.e., at position 3, the zoom system has its longest focal length, and posi- tion 2 is located halfway between position 1 and position 3. The displacement is positive if a group moves from Fig. 3. Optimized zoom system consisting of three lens left to right. The object is set at infinity [9]. modules. In a zoom system, each group is generally composed of several thick lens elements, as shown in Figure 2(a). In that figure, when the higher-order aberrations are ne- each group of the zoom system could be replaced by the glected, the thick lens system could be specified by its thick-lens module by specifying its focal length (FLM ), first-order quantities and the third-order Seidel image front focal length (FFM ), back focal length (BFM ), mag- aberrations at given conjugate points. In other words, if nification (MGM ), entrance pupil position (EPM ), en- we assign the first-order quantities and the third-order trance pupil diameter (EDM ), field angle (β), and the aberrations of lens modules to the thick-lens system, then third-order aberrations, as shown in Figure 2. both lenses are equivalent to each other within the limits We have set up the zoom camera system shown in of the first- and the third-order properties [7,9]. Hence, Figure 1 with three thick-lens modules, for which initial Optical Zoom System Design for Compact Digital··· – Sung-Chan Park et al. -1245-

Fig. 4. Aberrations of an optimized zoom system consisting of three lens modules: (a) position 1 and (b) position 3.

first-order inputs are appropriately given to work as a all zoom positions. Collisions between modules must be zoom system. It is based on the first and second modules avoided during zooming, and enough mounting space is having negative power, but the third module having pos- required. We next selected the overall length to be as itive power. The aperture stop of the zoom system is lo- short as possible for a compact zoom system. In this cated in front of the third module so that the system has case, we required the overall length to be less than 18 symmetrical configuration with respect to the stop. This mm. layout is good for aberration balancing. The air space In order to get an optimum zoom system, we opti- between each module should be ensured for the mount- mized the lens module prescriptions so that the specific ing space. Since lens modules do not reflect higher-order constraints were satisfied. The design variables are the aberrations, it is desirable to reduce the aperture and focal length, the front and back focal lengths, the conju- field size of the system so that the third-order aberra- gate points, the spacings, and the aberration coefficients tions are dominant. We have taken the zoom system of each module. Figure 3 shows the initial design of the with a half image size of 1 mm and an f-number of F/5 zoom system obtained from this process. Focal lengths at position 1 to F/7 at position 3. The distances between range from 4.3 to 12.9 mm, and aberrations are corrected modules are constrained to be longer than 0.5 mm over quite well, as shown in Figure 4. Table 1 shows the data -1246- Journal of the Korean Physical Society, Vol. 50, No. 5, May 2007

Table 1. Design data (in mm) for the lens modules in the optimized lens module zoom system.

Module I Module II Module III

FLM −36.8400 −7.9621 3.9705

FFMM (b) 38.2414 7.8676 −5.6551

BFM −36.0248 −8.4947 −1.6746

MGM 0.0 0.0 0.0

EDM 1.0 1.0 1.0 Fig. 5. Schematic diagram of the thick-lens system with Field (β) 1.0◦ 1.0◦ 1.0◦ five elements. Thickness 0.8472 0.6972 7.4074

W040 0.0890 0.0100 −0.0506

W131 −0.0034 −0.0147 0.0175 (SIII ), Petzval curves(SIV ), distortion(SV ), longitudi- W 0.0750 0.0018 0.0028 222 nal chromatic aberration (SL), and transverse chromatic W220 −0.0043 0.0010 0.0024 aberration (ST ) of this system are expressed in terms of W311 −0.0015 −0.0005 −0.0011 Gaussian brackets, as [11–14]

F ocal length : f = 1/[k1, −d1/n1, k2,

Table 2. First-order specifications and zooming locus of ··· , k9, −d9/n9, k10], (1) the zoom system consisting of three lens modules (in mm). Back focal length : bfl = f · [k1, −d1/n1, k2,

Position 1 Position 2 Position 3 ··· , k9, −d9/n9], (2) efl 4.3000 8.8000 12.9000 F ront focal length : ffl = −f · [−d1/n1, k2, bfl 1.1000 4.0050 6.3607 −d2/n2, ··· , k9, −d9/n9, k10], (3) ffl 3.1059 −3.8577 −11.9613 Magnification : M = 1/[−d0, k1, −d1/n1, d1i 3.7758 3.7877 2.1244 ··· , k9, −d9/n9, k10], (4) d2i 4.1721 1.2657 0.5000 4 X 2 d3i 1.1000 4.0050 6.3607 SI = u0 aj g2j−1wj, (5) 3 X SII = u0β ajbjg2j−1wj, (6) X S = u2β2 b2g w , (7) for each module, and Table 2 gives the zooming locus III 0 j 2j−1 j for each position. In Table 1, the values of W , W , 2 X 040 131 SIV = H kj/(njnj−1), (8) W222, W220, and W311 denote the third-order wave aber- 3 X 2 2 2 rations calculated at the edge of the field and at the exit SV = u0β bj/aj(bj g2j−1wj + d0kj /njnj−1), (9) pupil in units of waves at the d-line. Therefore, they cor- 2 X respond to the wave aberration coefficients for spherical SL = −u0 ajg2j−1∆(δn/n)j, (10) aberration, coma, astigmatism, Petzval curves, and dis- X ST = −u0β bjg2j−1∆(δn/n)j, (11) tortion, respectively [9,10]. In Table 2, dji (j = 1, 2, 3) are the air spaces between the lens modules at the zoom Where positions. The subscripts i denote a zoom position for zooming. aj = [−do, k1, −d1/n1, ··· , −dj−1/nj−1, −cj · nj−1],

bj = 1 for j = 1,

= [−d1/n1, k2, ··· , −dj−1/nj−1, −cj · nj−1] for j > 1, III. REAL LENS DESIGN FOR EACH GROUP 2 2 wj = g2j/nj − g2j−2/nj−1, g = [−d , k , −d /n , ··· , −d /n , k ], A thick-lens system composed of real lens elements is 2j 0 1 1 1 j−1 j−1 j used to design each group, which is equivalent to the g2j−1 = [−d0, k1, −d1/n1, ··· , kj−1, −dj−1/nj−1], thick-lens module given in Table 1. The schematic di- g2j−1 = [−d0, k1, −d1/n1, ··· , −dj−2/nj−2, kj−1], agram of this lens system is depicted in Figure 5. The aperture stop lies on the first surface, and the chief ray and makes an angle β with the optical axis at the stop. The ∆(δn/n)j = {(nF − nC )/nd}j − {(nF − nC )/nd}j−1. focal length (f), the front focal length (ffl), the back focal length (bfl), magnification (M) at a given conju- In these equations, kj (j = 1, 2, ···, 10) is the optical gate, and the third-order Seidel aberration coefficients power of each surface, dj (j = 0, 1, 2, ···, 10) is the dis- for spherical aberration (SI ), coma (SII ), astigmatism tance between surfaces, and uj (j = 0, 1, 2, ···, 10) is the Optical Zoom System Design for Compact Digital··· – Sung-Chan Park et al. -1247- convergence angle of the ray from the axial object point, two additional equations, Eqs. (10) and (11), be zero and as shown in Figure 5. Therefore, the optical power kj is solved. In this research, general glass selections for the given by cj(nj − nj−1), where cj and nj are the curva- chromatic aberration correction are carried out instead ture and the refractive index of surface. The refractive of solving the equations. For the glass choices, flint glass indices in the object (n0) and image space (n10) are as- is used for the negative-power elements and crown glass sumed to be unity, and the square brackets denote the for the positive-power elements [15–17]. Gaussian brackets. In the initial zoom system design using lens modules, For the system to be equivalent to the thick-lens mod- the first and second groups are required to be as com- ules to within the limit of the first- and third-order prop- pact as possible to have a slim camera. Therefore, it erties, all the first-order quantities and all the third-order is desirable to design the first group as a single element aberrations of the real lens should be equal to those of with a negative-power meniscus lens. The meniscus lens the lens module: is convex to the object, and this configuration is useful to correct distortion at the wide-field zoom position. In FLM = 1/[k1, −d1/n1, k2, ··· , k9, −d9/n9, k10], (12) the single lens case, there are four design variables, i.e., BFM = f · [k1, −d1/n1, k2, ··· , k9, −d9/n9], (13) c1, c2, d0, and d1. Therefore, four constraints given by FFM = −f · [−d1/n1, k2, −d2/n2, Eqs. (12) to (15) can be satisfied by specifying the lens ··· , k , −d /n , k ], (14) design variables by using the automatic design method 9 9 9 10 in Code-V. After a few iterations, the real lens of the first MGM = 1/[−d0, k1, −d1/n1, group is obtained. Since this group is useful to correct ··· , k9, −d9/n9, k10], (15) the off-axis aberrations, we selected the E48R, which is S plastic material and easily aspherized. W = I , (16) 040 8 The design data of the first lens group with small chro- S matic aberrations are selected and evaluated. This real W = II , (17) 131 2 lens and lens module I of Table 1 exhibit the expected aberration properties; i.e., there are aberrations that are SIII W222 = , (18) not corrected, but the agreement for the first-order prop- 2 erties is complete. SIV W220 = , (19) The second group with strong negative power is mod- 4 eled into a single lens with BACD16. Using the same SV W311 = , (20) method as described for the first lens group design, the 2 solution for the second group was obtained. This group where W040, W131, W222, W220, and W311 are the third- is equivalent to lens module II of Table 1 within the first- order wave aberrations of the lens module given in Ta- order properties. ble 1. If Eqs. (12)∼(20) are satisfied simultaneously, The third group has a focal length of 3.9705 mm. This the real lens is equivalent to the lens module, except for strong power reduces the displacement amount of this chromatic aberrations. However, it is very complicated group to have a higher zoom ratio. Also, this group to handle all the first-order quantities and third-order is required to balance the aberrations generated by the aberrations at the same time. first and the second groups. Therefore, many lens ele- By extensive computer calculations, an analytic ap- ments are needed to have a lens system equivalent to the proach to obtain the real lens data has been reported lens modules. The third lens group has an IR-cut filter [9]. However, it is very hard work to handle all the (Infrared Ray cut filter). This filter cuts the infrared equations analytically. In this paper, an automatic de- ray and improves the image quality on the CCD image sign method is proposed to design a real lens equivalent plane. Since the filter is a plane parallel plate and its to the module of each group. The design variables of index is assumed to be that of BK7, it generates addi- real lenses are changed to obtain a lens system in which tional aberrations. It is known, however, that the Seidel the four first-order quantities and the third-order aber- aberrations induced by moving the parallel plate along rations are matched to those of the lens modules. Thus, the optical axis are unchanged. the constraints are composed of the four first-orders and The third lens group generally has a cemented dou- the third-order aberrations of each lens module given in blet, it is useful to correct the chromatic aberrations and Table 1. Therefore, the real lens that satisfies the con- coma. The design variables are the seven curvatures c1, straints for each group is equivalent to the lens module c2, c3, ···, c7, and the seven distances d0, d1, d2, ···, within the limit of the first- and the third-order proper- d6. There are nine constraints given by Eqs. (12) to ties. At the stage of initial lens system design, the groups (20), which can be satisfied by using the automatic de- are required to be as compact as possible to improve the sign method to specify the design variables. From this portability of the camera. Therefore, each group must process, the real lens data of the third group is obtained. be designed as a few elements. Therefore, this group is equivalent to lens module III. In a zoom system, it is desirable to have each group in- Table 3 lists the real lens design data of each group. dependently achromatized. However, that requires that -1248- Journal of the Korean Physical Society, Vol. 50, No. 5, May 2007

Table 3. Design data of an initial real lens zoom system. Table 4. First-order specifications and zooming locus of an initial real lens zoom system (in mm). No. Radius Thickness Glass Object Infinity Infinity Position 1 Position 2 Position 3 1 13.3189 0.8500 E48R efl 4.3000 8.8003 12.9000 2 7.7469 ∗3.7758 Air bfl 1.1000 4.0052 6.3607 3 −4.0099 0.7000 BACD16 HOYA ffl 3.1057 −3.8582 −11.9613 4 −22.5989 ∗4.1721 Air d1i 3.7758 3.7877 2.1244 5(Stop) 3.1937 1.3267 E48R d2i 4.1721 1.2657 0.5000 6 −9.6633 0.1000 Air d3i 1.1000 4.0057 6.3607 7 5.4862 0.7000 FDS90 HOYA 8 2.9500 1.0287 FC5 HOYA 9 20.6236 2.9046 Air group designs; however, the mono-chromatic aberrations 10 −3.0000 0.7000 E48R are similar to each other at the zoom positions. In prin- 11 −6.9161 0.1000 Air ciple, each lens module can represent a very complex lens group consisting of an arbitrary number of elements. If 12 Infinity 0.5500 BK7 SCHOTT ∗ one had the freedom to change the number of elements in 13 Infinity 1.1000 Air the actual design; one would also expect its performance Image Infinity 0.0000 to be improved [18]. (∗ denotes the airspaces of moving groups at position 1.) Returning to the zoom system in Figure 6, since we reduced the aperture and field size so that the third- or- der aberrations were dominant, the f-number is too large, and the image size is too small. If current specifications for a compact digital zoom camera are to be met, the aperture and the field size should be increased to F/3.2 at position 1 and to F/4.5 at position 3. The half image size should be 2.2 mm for 1/4-inch CCD. In a large, ex- tended aperture and field system, however, higher-order aberrations that are not corrected in the previous design become significant. In order to improve the overall performance of the zoom system with an extended aperture and field, we easily balanced the aberrations of the starting data given in Table 3 by using the lens design program Code-V. In Fig. 6. Layout of an initial real lens zoom system. this process, the first-order layouts are fixed. To correct the residual aberrations, we used aspheric surfaces. The equation for the aspheric surface is given as IV. ACTUAL ZOOM SYSTEM DESIGN BY COMBINING REAL LENS GROUPS ch2 Z = 1 + p1 − (1 + K)c2h2 The groups separately designed in the previous sec- +Ah4 + Bh6 + Ch8 + Dh10 + ··· , tion, due to the zooming locus of Table 2, are then com- bined to establish a complete zoom system. If a zoom where c is the curvature around the axis, h is the ray system equivalent to the lens module zoom system is to height on the aspheric surface, K is a conic constant, and be achieved, the airspaces (dji) between groups should A, B, C, and D are aspheric coefficients. The aspheric be set according to the zooming locus of Table 2 at each surface has many design parameters, so aberrations can position. This procedure results in a zoom system equiv- be well corrected [19]. alent to the lens module zoom system, as shown in Fig- Finally, a zoom system having good performance is ure 6, and the design data are listed in Table 3. Ta- obtained. The layout of the system is shown in Figure ble 4 shows the first-order specifications of the combined 8, and its first-order properties are equal to those of the real lens zoom system. The agreement for the first-order starting lens. Figure 9 illustrates the field aberrations quantities between both zoom systems is complete. Fig- plot, and Figure 10 shows the modulation transfer func- ure 7 illustrates the aberrations of the system at two tion (MTF) characteristics of the system at two extreme extreme positions. From Figure 4 and Figure 7, com- positions. Aberrations are significantly reduced, and the parisons of the two cases show the expected aberration MTF at 200 lp/mm is more than 30 % at all zoom po- properties: there are color aberrations and residual aber- sitions over all fields. The relative illuminations are cal- rations that are not corrected in the first and second lens culated at the marginal field of all zoom positions. In Optical Zoom System Design for Compact Digital··· – Sung-Chan Park et al. -1249-

Fig. 7. Aberrations of an initial real lens zoom system: (a) position 1 and (b) position 3.

Fig. 8. Layout of an aberration-balanced zoom system with 1/4- inch CCD. -1250- Journal of the Korean Physical Society, Vol. 50, No. 5, May 2007

Fig. 9. Aberrations of an aberration-balanced zoom system: (a) position 1 and (b) position 3.

Fig. 10. MTF characteristics of an aberration-balanced zoom system: (a) position 1 and (b) position 3. Optical Zoom System Design for Compact Digital··· – Sung-Chan Park et al. -1251-

system was improved further, which keeping its first- order layouts fixed. A compact system with a zoom ratio of 3X, whose aperture was F/3.2 at the wide field posi- tion and F/4.5 at the narrow field position, and which had an image size of 1/4- inch on a CCD, was obtained. The zoom system developed in this work performs rea- sonably as a digital camera zoom system. As a result, the design of a zoom system using lens modules is bro- ken down into the simple problem of designing individual groups separately and combining them, and quickly pro- vides good solutions.

ACKNOWLEDGMENTS Fig. 11. Chief ray angle of incidence on the image plane. The present research was conducted by the research fund of Dankook University in 2005. this system, relative illuminations are more than 70 % over all positions. Figure 11 shows the chief ray angle of incidence (AOI) into image plane. The variation of AOI from a wide to a narrow field is less than 8.46 degree. REFERENCES That is a small value, so that stable image quality for zooming can be realized. The overall length is less than [1] K. Yammji, in Progress in Optics VI, edited by E. Wolf 18 mm, so it is a compact zoom system. Consequently, (North-Holland, Amsterdam, 1967), p. 105. this system has enough performance to satisfy the re- [2] M. S. Yeh, S. G. Shiue and M. H. Lu, Opt. Eng. 34, 1826 quirements of a current digital zoom camera system. (1995). [3] K. Tanaka, Appl. Opt. 21, 2174 (1982). [4] K. Tanaka, Appl. Opt. 21, 4075 (1982). [5] K. Tanaka, Appl. Opt. 22, 2174 (1983). V. CONCLUSION [6] O. N. Stavroudis and R. I. Mercado, J. Opt. Soc. Am. 65, 509 (1975). From the properties of lens modules, we set up an [7] T. G. Kuper and M. P. Rimmer, Proc. SPIE 892, 140 optimized a zoom system consisting of three lens modules (1988). with a reduced aperture and field. The optimum initial [8] S. C. Park and J. U. Lee, J. Korean Phys. Soc. 32, 815 design with a zoom ratio of 3X was derived by assigning (1988). [9] S. C. Park and R. R. Shannon, Opt. Eng. 35, 1668 first-order quantities and third-order aberrations to each (1996). module along with the specific constraints. [10] W. T. Welford, Aberration of Optical Systems (Adams From an automatic design procedure, a good design Hilger Ltd., Bristol, 1986). for the real lens of each group was quickly obtained by [11] M. Herzberger, Modern Geometrical Optics (Inter- matching the four first-order quantities and the third- science, New York, 1958). order aberrations of the module at given conjugates. The [12] S. C. Park and K. B. Kim, Proc. SPIE 2539, 1192 (1995). separately designed groups were combined to establish [13] K. Tanaka, in Process in Optics XXIII, edited by E. Wolf an actual zoom system. This system was found to be (North-Holland, Amsterdam, 1986), p. 63. almost equivalent to the zoom system consisting of the [14] M. Herzberger, J. Opt. Soc. Am. 33, 651 (1943). three lens modules with reduced aperture and field. The [15] S. C. Park and Y. S. Kim, J. Korean Phys. Soc. 41, 205 (2002). agreement between both lenses was good; however, the [16] Warren J. Smith, Modern Optical Engineering: The De- presence of higher-order aberrations made it difficult to sign of Optical Systems (McGraw-Hill, New York, 1990). achieve perfect agreement. Thus, it is desirable to design [17] Warren J. Smith, Modern Lens Design (McGraw-Hill, the initial system in the reduced region by using the first- New York, 1992). and third-order inputs and then to extend the system to [18] T. M. Jeong and G. Y. Yoon, J. Korean Phys. Soc. 49, meet our goals. 121 (2006). Through balancing of the higher-order aberrations in [19] J. Choi, T. H. Kim, H. J. Kong and J. U. Lee, J. Korean the extended aperture and field, the performance of zoom Phys. Soc. 47, 631 (2005).