Optical Zoom System Design for Compact Digital Camera Using Lens Modules

Transcription

1 Journal of the Korean Physical Society, Vol. 50, No. 5, May 2007, pp 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 (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: Eq, Fr Keywords: Lens module, Aberrations, Digital zoom camera I. INTRODUCTION The zoom lens design is usually divided into two tasks. One is paraxial studies based on thin-lens theory, which give the first-order parameters, such as the focal length of each group, the zoom ratio, the focal length range, the zooming locus, etc. The other is to set up the zoom lens system from the paraxial studies and balance aberrations [1 5]. These approaches, however, have several disadvantages. It is difficult to determine if the solutions obtained from paraxial studies satisfy all the requirements for the zoom lens, such as packaging constraints, specifications, overall length, and so on. Since the aberrations of this starting zoom lens are not corrected, aberration balancing at all zoom positions requires much more effort in the design of multi-group zoom systems. The difficulties due to paraxial analyses can be overcome by using the lens module design reported by Stavroudis and Mercado [6], Kuper and Rimmer [7], and Park and Lee [8]. Lens modules are the mathematical constructs that can model a complex optical system without actually doing the detailed design. The lens 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 its curvature, thickness, and indices. Lens modules can be used as a starting point for the design of a real lens and to model an arbitrary lens from measurable quantities scpark@dankook.ac.kr; Fax: 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 parameters defining the specifications, the third-order aberration characteristics, and the positions of the groups for zooming can be varied to obtain the optimum design satisfying the requirements. In this paper, lens modules and aberration theory are used to discuss the optimum initial design of three-group inner-focus zoom lenses. This initial zoom system is designed to satisfy specific requirements, and the real lens designs are obtained from the lens modules by using an automatic design method. In this process, the real lens for each group is quickly designed to match the firstand third-order aberrations of the module. Compared to an analytic design, this approach can dramatically save time and effort. Thus, the separately designed groups are then combined to form an actual zoom lens. Finally, residual aberration balancing results in a zoom lens that has enough performance over a range of f-number from 3.2 at the wide-field extreme to 4.5 at the narrow-field extreme positions. This zoom lens is expected to fulfill all requirements of a compact digital zoom camera. II. LENS MODULE DESIGN FOR THE THREE-GROUP INNER-FOCUS ZOOM SYSTEM The layout of the three-group inner-focus zoom system is shown in Figure 1. From the object to the image side,

2 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 k 1, k 2, and k 3, 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 preceding group and the succeeding group are specified by d 11, d 21, d 31, 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 position 2 is located halfway between position 1 and position 3. The displacement is positive if a group moves from left to right. The object is set at infinity [9]. 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 neglected, the thick lens system could be specified by its first-order quantities and the third-order Seidel image aberrations at given conjugate points. In other words, if we assign the first-order quantities and the third-order aberrations of lens modules to the thick-lens system, then both lenses are equivalent to each other within the limits of the first- and the third-order properties [7,9]. Hence, Fig. 3. modules. Optimized zoom system consisting of three lens each group of the zoom system could be replaced by the thick-lens module by specifying its focal length (FL M ), front focal length (FF M ), back focal length (BF M ), magnification (MG M ), entrance pupil position (EP M ), entrance pupil diameter (ED M ), field angle (β), and the third-order aberrations, as shown in Figure 2. We have set up the zoom camera system shown in Figure 1 with three thick-lens modules, for which initial

3 Optical Zoom System Design for Compact Digital Sung-Chan Park et al 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 zoom system. It is based on the first and second modules having negative power, but the third module having positive power. The aperture stop of the zoom system is located in front of the third module so that the system has symmetrical configuration with respect to the stop. This layout is good for aberration balancing. The air space between each module should be ensured for the mounting space. Since lens modules do not reflect higher-order aberrations, it is desirable to reduce the aperture and field size of the system so that the third-order aberrations are dominant. We have taken the zoom system with a half image size of 1 mm and an f-number of F/5 at position 1 to F/7 at position 3. The distances between modules are constrained to be longer than 0.5 mm over all zoom positions. Collisions between modules must be avoided during zooming, and enough mounting space is required. We next selected the overall length to be as short as possible for a compact zoom system. In this case, we required the overall length to be less than 18 mm. In order to get an optimum zoom system, we optimized the lens module prescriptions so that the specific constraints were satisfied. The design variables are the focal length, the front and back focal lengths, the conjugate points, the spacings, and the aberration coefficients of each module. Figure 3 shows the initial design of the zoom system obtained from this process. Focal lengths range from 4.3 to 12.9 mm, and aberrations are corrected quite well, as shown in Figure 4. Table 1 shows the data

4 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 FL M FFM M (b) BF M MG M ED M Field (β) Thickness W W W W W Table 2. First-order specifications and zooming locus of the zoom system consisting of three lens modules (in mm). Position 1 Position 2 Position 3 efl bfl ffl d 1i d 2i d 3i Fig. 5. Schematic diagram of the thick-lens system with five elements. (S III ), Petzval curves(s IV ), distortion(s V ), longitudinal chromatic aberration (S L ), and transverse chromatic aberration (S T ) of this system are expressed in terms of Gaussian brackets, as [11 14] F ocal length : f = 1/[k 1, d 1 /n 1, k 2,, k 9, d 9 /n 9, k 10 ], (1) Back focal length : bfl = f [k 1, d 1 /n 1, k 2,, k 9, d 9 /n 9 ], (2) F ront focal length : ffl = f [ d 1 /n 1, k 2, d 2 /n 2,, k 9, d 9 /n 9, k 10 ], (3) Magnification : M = 1/[ d 0, k 1, d 1 /n 1,, k 9, d 9 /n 9, k 10 ], (4) S I = u 4 0 a 2 j g 2j 1 w j, (5) S II = u 3 0β a j b j g 2j 1 w j, (6) for each module, and Table 2 gives the zooming locus for each position. In Table 1, the values of W 040, W 131, W 222, W 220, and W 311 denote the third-order wave aberrations calculated at the edge of the field and at the exit pupil in units of waves at the d-line. Therefore, they correspond to the wave aberration coefficients for spherical aberration, coma, astigmatism, Petzval curves, and distortion, respectively [9,10]. In Table 2, d ji (j = 1, 2, 3) are the air spaces between the lens modules at the zoom positions. The subscripts i denote a zoom position for zooming. III. REAL LENS DESIGN FOR EACH GROUP A thick-lens system composed of real lens elements is used to design each group, which is equivalent to the thick-lens module given in Table 1. The schematic diagram of this lens system is depicted in Figure 5. The aperture stop lies on the first surface, and the chief ray makes an angle β with the optical axis at the stop. The focal length (f), the front focal length (ffl), the back focal length (bfl), magnification (M) at a given conjugate, and the third-order Seidel aberration coefficients for spherical aberration (S I ), coma (S II ), astigmatism Where and S III = u 2 0β 2 b 2 jg 2j 1 w j, (7) S IV = H 2 k j /(n j n j 1 ), (8) S V = u 0 β 3 b j /a j (b 2 jg 2j 1 w j + d 2 0k 2 j /n j n j 1 ), (9) S L = u 2 0 aj g 2j 1 (δn/n) j, (10) S T = u 0 β b j g 2j 1 (δn/n) j, (11) a j = [ d o, k 1, d 1 /n 1,, d j 1 /n j 1, c j n j 1 ], b j = 1 for j = 1, = [ d 1 /n 1, k 2,, d j 1 /n j 1, c j n j 1 ] for j > 1, w j = g 2j /n 2 j g 2j 2 /n 2 j 1, g 2j = [ d 0, k 1, d 1 /n 1,, d j 1 /n j 1, k j ], g 2j 1 = [ d 0, k 1, d 1 /n 1,, k j 1, d j 1 /n j 1 ], g 2j 1 = [ d 0, k 1, d 1 /n 1,, d j 2 /n j 2, k j 1 ], (δn/n) j = {(n F n C )/n d } j {(n F n C )/n d } j 1. In these equations, k j (j = 1, 2,, 10) is the optical power of each surface, d j (j = 0, 1, 2,, 10) is the distance between surfaces, and u j (j = 0, 1, 2,, 10) is the

5 Optical Zoom System Design for Compact Digital Sung-Chan Park et al convergence angle of the ray from the axial object point, as shown in Figure 5. Therefore, the optical power k j is given by c j (n j n j 1 ), where c j and n j are the curvature and the refractive index of surface. The refractive indices in the object (n 0 ) and image space (n 10 ) are assumed to be unity, and the square brackets denote the Gaussian brackets. For the system to be equivalent to the thick-lens modules to within the limit of the first- and third-order properties, all the first-order quantities and all the third-order aberrations of the real lens should be equal to those of the lens module: F L M = 1/[k 1, d 1 /n 1, k 2,, k 9, d 9 /n 9, k 10 ], (12) BF M = f [k 1, d 1 /n 1, k 2,, k 9, d 9 /n 9 ], (13) F F M = f [ d 1 /n 1, k 2, d 2 /n 2,, k 9, d 9 /n 9, k 10 ], (14) MG M = 1/[ d 0, k 1, d 1 /n 1,, k 9, d 9 /n 9, k 10 ], (15) W 040 = S I 8, (16) W 131 = S II 2, (17) W 222 = S III 2, (18) W 220 = S IV 4, (19) W 311 = S V 2, (20) where W 040, W 131, W 222, W 220, and W 311 are the thirdorder wave aberrations of the lens module given in Table 1. If Eqs. (12) (20) are satisfied simultaneously, the real lens is equivalent to the lens module, except for chromatic aberrations. However, it is very complicated to handle all the first-order quantities and third-order aberrations at the same time. By extensive computer calculations, an analytic approach to obtain the real lens data has been reported [9]. However, it is very hard work to handle all the equations analytically. In this paper, an automatic design method is proposed to design a real lens equivalent to the module of each group. The design variables of real lenses are changed to obtain a lens system in which the four first-order quantities and the third-order aberrations are matched to those of the lens modules. Thus, the constraints are composed of the four first-orders and the third-order aberrations of each lens module given in Table 1. Therefore, the real lens that satisfies the constraints for each group is equivalent to the lens module within the limit of the first- and the third-order properties. At the stage of initial lens system design, the groups are required to be as compact as possible to improve the portability of the camera. Therefore, each group must be designed as a few elements. In a zoom system, it is desirable to have each group independently achromatized. However, that requires that two additional equations, Eqs. (10) and (11), be zero and solved. In this research, general glass selections for the chromatic aberration correction are carried out instead of solving the equations. For the glass choices, flint glass is used for the negative-power elements and crown glass for the positive-power elements [15 17]. In the initial zoom system design using lens modules, the first and second groups are required to be as compact as possible to have a slim camera. Therefore, it is desirable to design the first group as a single element with a negative-power meniscus lens. The meniscus lens is convex to the object, and this configuration is useful to correct distortion at the wide-field zoom position. In the single lens case, there are four design variables, i.e., c 1, c 2, d 0, and d 1. Therefore, four constraints given by Eqs. (12) to (15) can be satisfied by specifying the lens design variables by using the automatic design method in Code-V. After a few iterations, the real lens of the first group is obtained. Since this group is useful to correct the off-axis aberrations, we selected the E48R, which is plastic material and easily aspherized. The design data of the first lens group with small chromatic aberrations are selected and evaluated. This real lens and lens module I of Table 1 exhibit the expected aberration properties; i.e., there are aberrations that are not corrected, but the agreement for the first-order properties is complete. The second group with strong negative power is modeled into a single lens with BACD16. Using the same method as described for the first lens group design, the solution for the second group was obtained. This group is equivalent to lens module II of Table 1 within the firstorder properties. The third group has a focal length of mm. This strong power reduces the displacement amount of this group to have a higher zoom ratio. Also, this group is required to balance the aberrations generated by the first and the second groups. Therefore, many lens elements are needed to have a lens system equivalent to the lens modules. The third lens group has an IR-cut filter (Infrared Ray cut filter). This filter cuts the infrared ray and improves the image quality on the CCD image plane. Since the filter is a plane parallel plate and its index is assumed to be that of BK7, it generates additional aberrations. It is known, however, that the Seidel aberrations induced by moving the parallel plate along the optical axis are unchanged. The third lens group generally has a cemented doublet, it is useful to correct the chromatic aberrations and coma. The design variables are the seven curvatures c 1, c 2, c 3,, c 7, and the seven distances d 0, d 1, d 2,, d 6. There are nine constraints given by Eqs. (12) to (20), which can be satisfied by using the automatic design method to specify the design variables. From this process, the real lens data of the third group is obtained. Therefore, this group is equivalent to lens module III. Table 3 lists the real lens design data of each group.

6 Journal of the Korean Physical Society, Vol. 50, No. 5, May 2007 Table 3. Design data of an initial real lens zoom system. No. Radius Thickness Glass Object Infinity Infinity E48R Air BACD16 HOYA Air 5(Stop) E48R Air FDS90 HOYA FC5 HOYA Air E48R Air 12 Infinity BK7 SCHOTT 13 Infinity Air Image Infinity ( denotes the airspaces of moving groups at position 1.) Fig. 6. Layout of an initial real lens zoom system. IV. ACTUAL ZOOM SYSTEM DESIGN BY COMBINING REAL LENS GROUPS The groups separately designed in the previous section, due to the zooming locus of Table 2, are then combined to establish a complete zoom system. If a zoom system equivalent to the lens module zoom system is to be achieved, the airspaces (d ji ) between groups should be set according to the zooming locus of Table 2 at each position. This procedure results in a zoom system equivalent to the lens module zoom system, as shown in Figure 6, and the design data are listed in Table 3. Table 4 shows the first-order specifications of the combined real lens zoom system. The agreement for the first-order quantities between both zoom systems is complete. Figure 7 illustrates the aberrations of the system at two extreme positions. From Figure 4 and Figure 7, comparisons of the two cases show the expected aberration properties: there are color aberrations and residual aberrations that are not corrected in the first and second lens Table 4. First-order specifications and zooming locus of an initial real lens zoom system (in mm). Position 1 Position 2 Position 3 efl bfl ffl d 1i d 2i d 3i group designs; however, the mono-chromatic aberrations are similar to each other at the zoom positions. In principle, each lens module can represent a very complex lens group consisting of an arbitrary number of elements. If one had the freedom to change the number of elements in the actual design; one would also expect its performance to be improved [18]. Returning to the zoom system in Figure 6, since we reduced the aperture and field size so that the third- order 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, extended 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 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 ch 2 Z = (1 + K)c 2 h 2 +Ah 4 + Bh 6 + Ch 8 + Dh 10 +, where c is the curvature around the axis, h is the ray height on the aspheric surface, K is a conic constant, and A, B, C, and D are aspheric coefficients. The aspheric surface has many design parameters, so aberrations can be well corrected [19]. Finally, a zoom system having good performance is obtained. The layout of the system is shown in Figure 8, and its first-order properties are equal to those of the starting lens. Figure 9 illustrates the field aberrations plot, and Figure 10 shows the modulation transfer function (MTF) characteristics of the system at two extreme positions. Aberrations are significantly reduced, and the MTF at 200 lp/mm is more than 30 % at all zoom positions over all fields. The relative illuminations are calculated at the marginal field of all zoom positions. In

7 Optical Zoom System Design for Compact Digital Sung-Chan Park et al 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.

8 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.

9 Optical Zoom System Design for Compact Digital Sung-Chan Park et al system was improved further, which keeping its firstorder layouts fixed. A compact system with a zoom ratio of 3X, whose aperture was F/3.2 at the wide field position 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 reasonably as a digital camera zoom system. As a result, the design of a zoom system using lens modules is broken down into the simple problem of designing individual groups separately and combining them, and quickly provides good solutions. Fig. 11. Chief ray angle of incidence on the image plane. 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. That is a small value, so that stable image quality for zooming can be realized. The overall length is less than 18 mm, so it is a compact zoom system. Consequently, this system has enough performance to satisfy the requirements of a current digital zoom camera system. V. CONCLUSION From the properties of lens modules, we set up an optimized a zoom system consisting of three lens modules with a reduced aperture and field. The optimum initial design with a zoom ratio of 3X was derived by assigning first-order quantities and third-order aberrations to each module along with the specific constraints. From an automatic design procedure, a good design for the real lens of each group was quickly obtained by matching the four first-order quantities and the thirdorder aberrations of the module at given conjugates. The separately designed groups were combined to establish an actual zoom system. This system was found to be almost equivalent to the zoom system consisting of the three lens modules with reduced aperture and field. The agreement between both lenses was good; however, the presence of higher-order aberrations made it difficult to achieve perfect agreement. Thus, it is desirable to design the initial system in the reduced region by using the firstand third-order inputs and then to extend the system to meet our goals. Through balancing of the higher-order aberrations in the extended aperture and field, the performance of zoom ACKNOWLEDGMENTS The present research was conducted by the research fund of Dankook University in REFERENCES [1] K. Yammji, in Progress in Optics VI, edited by E. Wolf (North-Holland, Amsterdam, 1967), p [2] M. S. Yeh, S. G. Shiue and M. H. Lu, Opt. Eng. 34, 1826 (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). [6] O. N. Stavroudis and R. I. Mercado, J. Opt. Soc. Am. 65, 509 (1975). [7] T. G. Kuper and M. P. Rimmer, Proc. SPIE 892, 140 (1988). [8] S. C. Park and J. U. Lee, J. Korean Phys. Soc. 32, 815 (1988). [9] S. C. Park and R. R. Shannon, Opt. Eng. 35, 1668 (1996). [10] W. T. Welford, Aberration of Optical Systems (Adams Hilger Ltd., Bristol, 1986). [11] M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958). [12] S. C. Park and K. B. Kim, Proc. SPIE 2539, 1192 (1995). [13] K. Tanaka, in Process in Optics XXIII, edited by E. Wolf (North-Holland, Amsterdam, 1986), p. 63. [14] M. Herzberger, J. Opt. Soc. Am. 33, 651 (1943). [15] S. C. Park and Y. S. Kim, J. Korean Phys. Soc. 41, 205 (2002). [16] Warren J. Smith, Modern Optical Engineering: The Design of Optical Systems (McGraw-Hill, New York, 1990). [17] Warren J. Smith, Modern Lens Design (McGraw-Hill, New York, 1992). [18] T. M. Jeong and G. Y. Yoon, J. Korean Phys. Soc. 49, 121 (2006). [19] J. Choi, T. H. Kim, H. J. Kong and J. U. Lee, J. Korean Phys. Soc. 47, 631 (2005).