Application of a global optimization process to the design of pickup heads for compact and digital versatile disks

Application of a global optimization process

to the design of pickup heads for compact and

digital versatile disks

Chao-Hsi Tsao Jyh-Long Chern

National Chiao Tung University Institute of Electro-Optical Engineering Microelectronics and Information System Research Center

Department of Photonics Hsinchu 300, Taiwan

E-mail: jlchern@faculty.nctu.edu.tw

Abstract. A global optimization process is developed for designing an objective lens for multiple-configuration applications. As a demonstra-tion, it is shown that diffraction-limit performance can be achieved for a pickup head used for both compact disks共wavelength 780 nm兲 and digi-tal versatile disks共wavelength range 635 to 650 nm兲 with different work-ing distances and focal lengths, at a numerical aperture of 0.6. © 2006 Society of Photo-Optical Instrumentation Engineers. 关DOI: 10.1117/1.2361281兴

Subject terms: pickup head; optical design; optimization.

Paper 050600R received Jul. 26, 2005; revised manuscript received Jan. 16, 2006; accepted for publication Mar. 22, 2006; published online Oct. 17, 2006.

Numerical optimization applied to complex problems has been an important field of science and technology. Ap-plication of various algorithms to a variety of topics can be found in the literature.1System and performance optimiza-tion of optical design to meet particular requirements has been one of the classical applications of numerical optimi-zation, and it is nontrivial in that its adjustable parameters are numerous and the response of the merit function to variation of the system parameters is very nonlinear. How-ever, a well-designed algorithm can provide a useful method to find the best solution for a sophisticated system even with a large number of variables. Studies of optimi-zation in optical system design can be traced back to early 1960, when computers were starting to be used.2 Among many released algorithms, the method of damped least squares共DLS兲 has been widely used and become a standard scheme for approaching a minimum.3 However, the DLS approach is only a method of local optimization, because of its search strategy. To handle the dramatically emergent fields of technology, new searching strategies are strongly demanded, while, on the other hand, better theoretical un-derstanding of the nature of optimization is desired for fur-ther exploration. For these purposes, recently, Koshel uti-lized the simplex algorithm to optimize illumination design, which is a large and difficult area in which more research is strongly required,4and Bociort, van Driel, and Serebriakov tried to realize the connection of local minima over the whole variable space.5 Generally, it can be acknowledged that the performance of local optimization depends greatly on the choice of initial design and the definition of merit function. For real applications, even with an available paraxial solution, it may not be easy to achieve a proper initial design, which can lead to the best solution or even one with diffraction-limited performance in imaging optics. Hence, a nonlocal, or even global, optimization algorithm is demanded for seeking a better solution within a con-strained domain of variables. Evidently, the situation

be-comes much more complicated for zoom lenses or optical systems with multiple configurations.6

As a real illustration of multiple-configuration applica-tions, we consider the pickup-head lens design for compact disks共CDs兲 and digital versatile disks 共DVDs兲, which has attracted much attention in recent years.7–11In those stud-ies, holographic optical elements 共HOEs兲8 and complex-surface designs7,9–11were proposed to solve the compatibil-ity problem of applying a single lens for two different wavelengths. With the progress of manufacturing technol-ogy, aspheric lenses are no longer impossible to fabricate. Naturally, a singlet lens with aspheric surfaces will be taken as a solution for the design of two-wavelength optical pickup heads. However, the need for different ranges of wavelengths, thicknesses of cover materials, and other op-tical and mechanical specifications raises difficulties in finding suitable coefficients for aspheric surfaces with the desired performance, which generally means diffraction-limit performance for all configurations. Nevertheless, the solution may be obtained with the aid of a well-developed global optimization process as shown below. In this paper, we design an objective lens with CD/DVD double

cation as a demonstration of the application of a global optimization process.

A higher numerical aperture 共NA兲 of 0.6 and shorter wavelengths共635 to 650 nm兲 than for the CD 共NA=0.45, ␭=750 nm兲 are necessary for the DVD in order to increase the data density. However, a higher NA will also increase the coma, which is proportional to共NA兲,3and the thickness of the cover materials. Thus, the substrate thickness adopted for the DVD was half that of the cover materials used for the CD. Table 1 shows the specifications of the CD and DVD pickup lenses, which constitute our design target for the single aspheric lens. A schematic diagram of lens layout is shown in Fig. 1. In this optimization, the NA for a CD is chosen as 0.6 instead of 0.45, the regular require-ment. The higher NA reflects the larger lens area that is needed for a single objective lens operated at two distinct wavelengths; to increase the NA for a CD means to take the outer region of the lens into consideration for both DVD and CD configurations, and this will tighten the specifica-tions and increase the difficulties in optimization.

A well-defined merit function is able to reflect the re-quired performance and to drive the algorithm to find the optimal results efficiently and effectively. Such a merit function may have to be quite nontrivial and special.

How-ever, for a global optimization, the merit function should be of general purpose. For the design of an optical pickup lens, the most crucial criterion, beyond the specifications listed

Table 2 List of the variables used in global optimization and their boundaries.

Variable Lower limit Upper limit Definition

CV1 −1 1 Curvature of surface 1

TH 1 5 Thickness of lens

CV2 −1 1 Curvature of surface 2

CC1 0a 0a Conic constant of surface 1

CC2 0a 0a Conic constant of surface 2

AD1, AE1, AF1, AG1 0a 0a Aspheric coefficients of surface 1

AD2, AE2, AF2, AG2 0a 0a Aspheric coefficients of surface 2

WD1 1.65b 1.75 Working distance for configuration 1

WD2 1.3 1.4 Working distance for configuration 2

a_{Lower limit= upper limit= 0 means no bound for variable.}

b_{Limits of working distances are roughly decided by specification: ±3.7%.}

Fig. 2 Flow chart of optimization process. Table 1 Specifications of the DVD/CD pickup.

Specification DVD CD

Wavelength_共nm兲 635–650 780

Focal length共mm兲 3.36 3.39

Numerical aperture 0.6 0.6

Working distance共mm兲 1.71 1.35

in Table 1, is the spot size, which indicates the radius of the distribution of energy focused on the focal plane by the objective lens. Thus, the root-mean-square共rms兲 spot sizes produced by the two configurations on the focal plane were taken as a part of our merit function for the judgment cri-terion of optimization. Besides the spot sizes, the focal lengths of two configurations are also included in our merit function to ensure that the final design is able to meet the mechanical constraints. Thus, the merit function we took had the form

M =

兺

i no. of configurations

兵关wf共EFLi− fi兲兴2+共wsSi兲2其, 共1兲 where wfand wsare the specified weight; EFL, f, and S are the determined focal length of the system, the target focal length, and the determined rms spot size, respectively; and the summation index i labels the configurations. The weights for effective focal length and spot size are different—in the current demonstration, the former is 0.1 and the latter is 1—in order to balance the contributions of the terms in the optimization. It should be noticed that the definition of the merit function reflects the deviation be-tween the current performance共effective focal length and rms spot size in our case兲 and the expected performance. Consequently, our target is to minimize the value of the merit function. From this point of view, we can recognize it as an error function instead of a merit function.

To make the spot sizes at the two wavelengths small enough, especially for large numerical apertures, an as-pheric surface is introduced, whose parameters are vari-ables in addition to the curvatures and thickness of the lens. The most common aspheric surface is of the form

z = CV r

2

1 +关

冑

4_{+ AE r}6_{+ AF r}8

+ AG r10, 共2兲

where z is the surface sag, CV is the curvature of the sur-face, CC is the conic constant, and AD, AE, AF, and AG are the aspheric coefficients of higher-order terms of radius r. Table 2 shows the list of variables and their bounds for global optimization. A total of 15 degrees of freedom char-acterize the space of variables. It is noticed that the values chosen for the bounds on the working distances for the two configurations are necessary and could be very critical, since we have to consider both the specification and the budget for optimization. Practically, the working distance is

Fig. 3 Best six solutions of global optimization for the DVD configuration, with the simulated spot sizes

and values of the merit function. S1 and S2 denote the average spot sizes for the CD configuration and for the DVD configuration, respectively.

Fig. 4 Parameter variations of the six preliminary solutions and the

one of the mechanical specifications, and its limits are de-termined by the budget. Including the working distance among the system variables enables us to assure that the final design will not deviate from the desired specification and also provides new degrees of freedom to optimize the design of the CD/DVD objective lens.

For a complex optical design problem, even if the best solution can be found by a global search algorithm, it still takes large computational power. To make a useful but quick search, a method combining proximate and stochastic search, so that a topographic map with adjustable resolution 共topographic attempt兲 can be achieved, has been developed as illustrated in Fig. 2. To perform such an exploration, an

efficient check of multiple trial solutions is demanded. In this respect, the simplex method contributes superior capa-bility of exploring solutions, because randomly sampling

N + 1 points in the N-dimensional design space provides a

chance of hitting another local minimum in the neighbor-hood of the initial location; in contrast, a gradient-dependent local optimization algorithm can easily be trapped at a particular local minimum if the initial point is fixed.12 Our global optimization algorithm utilizes a modi-fied downhill simplex method for local minimum search and a hybrid algorithm combining topographic attempts and stochastic searching to randomly explore local minima in the value domain of the merit function. Among the com-ponents of the merit function, the deviations of effective focal length and rms spot size could be extremely large when an unreasonable design has been generated, which requires bounds on the space of reasonable designs. Con-sequently, it is possible that the optimizer will be trapped in a limited proximity region. Once this happens, a random

Table 3 The optimized parameters for the DVD/CD dual-purpose

lens design.

Parameter Value Parameter Value

CV1 0.390281 AG1 −0.000115 CV2 −0.213145 AD2 0.484489 TH 3.499182 AE2 −0.379871 CC1 −0.781059 AF2 0.032930 CC2 −9.9047e + 07 AG2 0.040510 AD1 0.023906 WD1 1.733523 AE1 −0.004754 WD2 1.400000 AF1 0.001182

Table 4 Comparison of specification and optimized results.

DVD CD

Parameter Spec. Optim. Diff. 共%兲 Spec. Optim. Diff. 共%兲 Focal length共mm兲 3.36 3.36378 +0.125 3.39 3.3862 −0.112 Working distance共mm兲 1.71 1.7335 +1.37 1.35 1.4 +3.7

Fig. 5 The layout of the final design, where _{共a兲 is for the DVD and 共b兲 is for the CD pickup. The}

casting mechanism is necessary to throw the optimizer to a new start position. There is inevitable risk that the new start position will be an unreasonable design; thus either pre-checking the value of the merit function for the new start or an enforcement operation13 will be useful to move the op-timizer to a new reasonable and effective initial design.

We adopted OSLO, commercial software for optical simulation with flexible programming capability,14 as the platform on which all algorithms are developed. A global search algorithm was employed to look for the local minima within the design space of 15 dimensions, which is limited by bounds on each variable. It is found that with only one trial run, the desired designs can be deduced. Nev-ertheless, as a detailed evaluation, extensive results corre-sponding to local minima in the space of the merit function were obtained by the optimization algorithm, and the best six sets with the lowest value of merit function are shown in Fig. 3. For simplification, only the results of configura-tion 1共designed for the DVD specification兲 are presented. In effect, this quick search is to identify the dominant

com-plex where optimized solutions can be found. Indeed, the

spot-size analysis of those six sets of variables shows that the complexes indicated by these solutions include good initial designs for further improvement.

As shown in Fig. 4, the major differences among these designs are in the curvatures and the thickness of the lens, which determine the optical power of the lens. In other words, the proposed optimization scheme is superior in lo-calizing the design to the specified optical power. Gener-ally, in influencing the optical performance, the conic con-stants are in competition with all the other aspheric coefficients. It is worthwhile to notice that the conic con-stants are also different in different designs, and this means that the proposed optimization is directly involved in tuning the conic constant, and hence the aspheric coefficients are less different for the six designs. It is also interesting to note that the working distances are nearly the same.

In the next stage, the simplex method was replaced by the usual DLS method for approaching the local minimum, because the solutions have been confirmed to a complex. Doing DLS approaches for these six new initial designs will reach our design goal if the curvatures of surfaces 1 and 2, the thickness of the lens, and the working distances are removed from the list of variables of optimization. The reason for removing the first three variables is physical: the curvatures and thickness determine the optical power of a lens. Either the focal length or the spot size will be strongly affected by changing these three parameters. Removing them cannot only decrease the complexity of variable do-main but also return the emphasis to other aspheric coeffi-cients during optimization, which leads to higher perfor-mance.

The best result, which is deduced from the preliminary design shown in Fig. 3共c兲, can be obtained by the same further optimization process among the six preliminary re-sults is shown in Fig. 5, and the optimized parameters of the lens design are listed in Table 3. We have a set of parameters, which can achieve focusing of the collimated incident beam with wavelengths of 635 to 650 nm and 780 nm on the image plane with minimum rms spot size and meet the specifications. The geometrical-optical esti-mate of the rms spot size at wavelengths of 635 to 650 nm

is 0.491␮m and that at a wavelength of 780 nm is 0.84␮m. They are both less than the theoretical diffraction limits, which are 0.918 and 1.094␮m, respectively. In this case, a diffraction-based verification is necessary. Figures 5共c兲 and 5共d兲 are graphs of the point spread functions of the corresponding cases. From them it is obvious that the en-ergy is almost all concentrated in a very small range with diffraction-limited performance, because both the Strehl ra-tios are larger than 0.8.

Table 4 shows the comparison of the required specifica-tions and the results of the final design. It can be seen that the design can reach the specifications within the range of allowance.

In conclusion, we have successfully demonstrated a pro-cess combining global and local optimizations, mainly with a modified downhill-simplex algorithm, by which a dual-purpose CD/DVD pickup head was designed with diffraction-limited performance in both configurations. The optimization is achieved through aspheric coefficients. This study also shows that the computer-aided design with ap-propriate algorithms and optimization methods will be very helpful in advancing the development of emergent technol-ogy and applications for optical system design.

Acknowledgment

This work was supported, in part, by the National Science Council, Taiwan, R.O.C., under project No. 93-2215-E-009-057. We thank Lambda Research Corp. for educational support of OSLO.

References

1. See http://www.optimization-online.org/for Web resources and more information.

2. M. J. Hayford, “Optimization methodology,” in Geometrical Optics,

Proc. SPIE 531, 68–81共1985兲; G. K. Hearn, “The evolution of

op-timization algorithms,” in Lens Design, SPIE Crit. Rev. CR41, 54–70 共1992兲.

3. D. C. Sinclair, “Optical design software,” Chap. 34 in Handbook of

Optics, M. Bass, E. W. Van Stryland, D. R. Williams, and W. L.

Wolfe, Eds., Vol. I, pp. 18–20, OSA共1995兲.

4. R. J. Koshel, “Simplex optimization method for illumination design,”

Opt. Lett. 30, 649–651共2005兲.

5. F. Bociort, E. van Driel, and A. Serebriakov, “Networks of local minima in optical system optimization,” Opt. Lett. 29, 189–191 共2004兲.

6. A. Mann, Ed., Selected Papers on Zoom Lenses, SPIE共1993兲. 7. P.-Y. Liu, H.-P. D. Shieh, J.-J. Ju, S.-T. Tsai, T.-M. Yang, T.-K.

Chang, and J.-S. Liu, “A novel dual focus objective lens for DVD/CD pick-up head,” IEEE Trans. Magn. 34, 462–464共1998兲.

8. R. Katayama, Y. Komatsu, and Y. Ono, “Dual wavelength optical head for 0.6 mm and 1.2 mm substrate thicknesses,” Jpn. J. Appl.

Phys., Part 1 36, 460–466共1997兲.

9. Y. Tanaka, M. Yamagata, Y. Komma, S. Mizuno, and K. Nagashima, “Lens design for optical head compatible with compact disk and digi-tal versatile disk,” Jpn. J. Appl. Phys., Part 1 37, 2179–2183共1998兲. 10. C. W. Lee, C. S. Chung, J. H. Yoo, Y. H. Lee, T. K. Kim, Y. K. Son, S. J. Kim, P. Y. Seong, and K. S. Kim, “A compact disk-recordable compatible digital versatile disk pickup with a lens for 780 nm and 650 nm wavelengths,” Jpn. J. Appl. Phys., Part 1 37, 2197–2200 共1998兲.

11. J.-H. Yoo, C.-W. Lee, K.-H. Cho, H.-S. Choi, and J.-W. Lee, “An optical head with special annular lens for laser disc-compatible digital versatile disc pickup,” Jpn. J. Appl. Phys., Part 1 37, 2184–2188 共1998兲.

12. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,

Numerical Recipes in C, The Art of Scientific Computing, Chap. 10,

Cambridge Univ. Press共1992兲.

13. I. Ono, S. Kobayashi, and K. Yoshida, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” in Proc. Int. Lens Design Conf. 1990, Proc. SPIE 1354共1990兲. 14. Lambda Research Inc., OSLO, http://www.lambdares.com.

Chao-Hsi Tsao received the MS degree

from the Department of Physics of National Cheng Kung University共NCKU兲 of Taiwan in 1998. From 2000 to 2003, he worked in a semiconductor company as a technical de-velopment engineer. Currently, he is pursu-ing his PhD degree in the Department of Photonics, Institute of Electro-Optical Engi-neering of National Chiao Tung University 共NCTU兲. His research interests include opti-cal design and global optimization algo-rithms.

Jyh-Long Chern received his BS and MS

degrees in physics from the National Tsing Hua University, Hsinchu, Taiwan, in 1984 and 1986, respectively. He received his PhD in optical science from the University of New Mexico, Albuquerque, New Mexico, USA, in 1991. He was a postdoctoral fellow at the Basic Research Laboratories, NTT 共Nippon Telegram and Telephone Corpora-tion兲, Japan, from April 1991 to August 1992. After his postdoctoral career, he joined the National Sun Yat-sen University, Kaohsiung, Taiwan, in 1992, as an associate professor of physics. In 1995, he became a full professor. In 1996, he moved to the National Cheng Kung Uni-versity, Tainan, Taiwan, as a full professor of physics. In August 2002, he joined the faculty of National Chiao-Tung University, where he is currently a professor of electro-optical engineering.