OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 8 — Apr. 13, 2009
  • pp: 6436–6450

Chaotic behavior in an algorithm to escape from poor local minima in lens design

Maarten van Turnhout and Florian Bociort  »View Author Affiliations


Optics Express, Vol. 17, Issue 8, pp. 6436-6450 (2009)
http://dx.doi.org/10.1364/OE.17.006436


View Full Text Article

Enhanced HTML    Acrobat PDF (880 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

In lens design, damped least-squares methods are typically used to find the nearest local minimum to a starting configuration in the merit function landscape. In this paper, we explore the use of such a method for a purpose that goes beyond local optimization. The merit function barrier, which separates an unsatisfactory solution from a neighboring one that is better, can be overcome by using low damping and by allowing the merit function to temporarily increase. However, such an algorithm displays chaos, chaotic transients and other types of complex behavior. A successful escape of the iteration trajectory from a poor local minimum to a better one is associated with a crisis phenomenon that transforms a chaotic attractor into a chaotic saddle. The present analysis also enables a better understanding of peculiarities encountered with damped least-squares algorithms in conventional local optimization tasks.

© 2009 Optical Society of America

OCIS Codes
(080.2720) Geometric optics : Mathematical methods (general)
(220.2740) Optical design and fabrication : Geometric optical design
(220.3620) Optical design and fabrication : Lens system design
(080.1753) Geometric optics : Computation methods

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: February 17, 2009
Revised Manuscript: March 24, 2009
Manuscript Accepted: April 1, 2009
Published: April 2, 2009

Citation
Maarten van Turnhout and Florian Bociort, "Chaotic behavior in an algorithm to escape from poor local minima in lens design," Opt. Express 17, 6436-6450 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6436


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. G. W. Forbes and A. E. Jones, "Towards global optimization with adaptive simulated annealing," Proc. SPIE 1334, 144-153 (1991). [CrossRef]
  2. T. G. Kuper and T. I. Harris, "Global optimization for lens design - an emerging technology," Proc. SPIE 1780,14-28 (1992).
  3. M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, "Lens design: Global optimization with Escape Function," Opt. Rev.  6, 463-470 (1995).
  4. K. E. Moore, "Algorithm for global optimization of optical systems based on genetic competition," Proc. SPIE 3780, 40-47 (1999). [CrossRef]
  5. L. W. Jones, S. H. Al-Sakran, and J. R. Koza, "Automated synthesis of both the topology and numerical parameters for seven patented optical lens systems using genetic programming," Proc. SPIE 5874, 587403 (2005). [CrossRef]
  6. J. P. McGuire, Jr., "Designing easily manufactured lenses using a global method," Proc SPIE 6342, 63420O (2006). [CrossRef]
  7. J. R. Rogers, "Using global synthesis to find tolerance-insensitive design forms," Proc SPIE 6342, 63420M (2006). [CrossRef]
  8. O. Marinescu and F. Bociort, "Network search method in the design of EUV lithographic objectives," Appl. Opt. 46, 8385-8393 (2007). [CrossRef] [PubMed]
  9. D. C. Sinclair, "Optical design software," in Handbook of Optics, Fundamentals, Techniques, and Design, M. Bass, E. W. Van Stryland, D. R. Williams, and Wolfe W. L., eds., 2nd ed., (McGraw-Hill, New York, 1995) Vol. 1, 34.1-34.26.
  10. M. Laikin, The Method of Lens Design, Lens design, 4th ed. (CRC Press, Boca Raton, FL, 1996).
  11. A. Serebriakov, F. Bociort, and J. Braat, "Finding new local minima by switching merit functions in optical system optimization," Opt. Eng. 44, 100,501 (2005). [CrossRef]
  12. H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Principles of optimization, in Handbook of Optical Systems, (Wiley-VCH, Weinheim, 2007) Vol. 3, 291-370.
  13. D. Shafer, "How to optimize complex lens designs," Laser Focus World 29, 135-138 (1993).
  14. D. Shafer, "Global optimization in optical design," Comput. Phys. 8, 188-195 (1994).
  15. M. van Turnhout and F. Bociort, "Instabilities and fractal basins of attraction in optical system optimization," Opt. Express 17, 314-328 (2009). URL http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-1-314. [CrossRef] [PubMed]
  16. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974).
  17. E. Ott, Chaos in dynamical systems, 2nd ed. (Cambridge University Press, Cambridge, 2002).
  18. H. E. Nusse and J. A. Yorke, "Basins of attraction," Science 271, 1376-1380 (1996). [CrossRef]
  19. S. N. Rasband, Chaotic dynamics of nonlinear systems (Wiley, New York, 1990).
  20. I. Castillo and E. R. Barnes, "Chaotic behavior of the affine scaling algorithm for linear programming," SIAM J. Optim. 11, 781-795 (2000). [CrossRef]
  21. B. Davies, "Period Doubling," in Encyclopedia of Nonlinear Science, A. Scott, ed., (Routledge, New York, 2004).
  22. C. Grebogi, E. Ott, and J. A. Yorke, "Chaos, strange attractors, and fractal basin boundaries in non-linear dynamics," Science 238, 632-638 (1987). [CrossRef] [PubMed]
  23. Y.-C. Lai and C. Grebogi, "Converting transient chaos into sustained chaos by feedback control," Phys. Rev. E 49, 1094-1098 (1994). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Multimedia

Multimedia FilesRecommended Software
» Media 1: AVI (87 KB)      QuickTime

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited