Abstract

Many optical design programs use various forms of the damped least-squares method for local optimization. In this paper, we show that damped least-squares algorithms, with maximized computational speed, can create sensitivity with respect to changes in initial conditions. In such cases, starting points, which are very close to each other, lead to different local minima after optimization. Computations of the fractal capacity dimension show that sets of these starting points, which lead to the same minimum (the basins of attraction for that minimum), have a fractal structure. Introducing more damping makes the optimization process stable.

© 2009 Optical Society of America

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References

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  1. D. C. Sinclair, "Optical design software," in Handbook of Optics, Fundamentals, Techniques, and Design, Vol. 1,2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), 34.1-34.26.
  2. F. Bociort, "Optical system optimization," in Encyclopedia of optical engineering, R. G. Driggers, ed. (Marcel Dekker, New York, 2003), 1843-1850.Q1
  3. D. P. Feder, "Automatic optical design," Appl. Opt. 2, 1209-1226 (1963).
    [CrossRef]
  4. E. Ott, Chaos in dynamical systems, 2nd ed. (Cambridge University Press, Cambridge, 2002).
  5. H. Gross, H. Z¨ugge, M. Peschka, and F. Blechinger, "Principles of optimization," in Handbook of Optical Systems, Vol. 3 (Wiley-VCH, Weinheim, 2007), 291-370.
  6. C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, "Final state sensitivity: an obstruction to predictability," Phys. Lett. A 99, 415-418 (1983).
    [CrossRef]
  7. S.W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, "Fractal basin boundaries," Physica D 17, 125-153 (1985).
  8. C. Grebogi, E. Kostelich, E. Ott, and J. A. Yorke, "Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor," Physica D 25, 347-360 (1987).
  9. 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]
  10. H. E. Nusse, and J. A. Yorke, "Basins of attraction," Science 271, 1376-1380 (1996).
    [CrossRef]
  11. M. van Turnhout, and F. Bociort, "Predictability and unpredictability in optical system optimization," in Current Developments in Lens Design and Optical Engineering VIII, P. Z. Mouroulis, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE 6667, 666709 (2007).
  12. Optical Research Associates, CODE V, Pasadena, CA.
  13. ZEMAX Development Corporation, ZEMAX, Bellevue, WA.
  14. F. Bociort, E. van Driel, and A. Serebriakov, "Networks of local minima in optical system optimization," Opt. Lett. 29, 189-191 (2004).
    [CrossRef] [PubMed]
  15. F. Bociort and M. van Turnhout, "Generating saddle points in the merit function landscape of optical systems," in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59620S (2005).
    [CrossRef]
  16. S. N. Rasband, Chaotic dynamics of nonlinear systems (Wiley, New York, 1990).
  17. H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and fractals: New frontiers of science, 2nd ed. (Springer-Verlag, New York, 2004).

2004

1996

H. E. Nusse, and J. A. Yorke, "Basins of attraction," Science 271, 1376-1380 (1996).
[CrossRef]

1987

C. Grebogi, E. Kostelich, E. Ott, and J. A. Yorke, "Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor," Physica D 25, 347-360 (1987).

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]

1985

S.W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, "Fractal basin boundaries," Physica D 17, 125-153 (1985).

1983

C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, "Final state sensitivity: an obstruction to predictability," Phys. Lett. A 99, 415-418 (1983).
[CrossRef]

1963

Bociort, F.

Feder, D. P.

Grebogi, C.

C. Grebogi, E. Kostelich, E. Ott, and J. A. Yorke, "Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor," Physica D 25, 347-360 (1987).

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]

S.W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, "Fractal basin boundaries," Physica D 17, 125-153 (1985).

C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, "Final state sensitivity: an obstruction to predictability," Phys. Lett. A 99, 415-418 (1983).
[CrossRef]

Kostelich, E.

C. Grebogi, E. Kostelich, E. Ott, and J. A. Yorke, "Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor," Physica D 25, 347-360 (1987).

McDonald, S. W.

C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, "Final state sensitivity: an obstruction to predictability," Phys. Lett. A 99, 415-418 (1983).
[CrossRef]

McDonald, S.W.

S.W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, "Fractal basin boundaries," Physica D 17, 125-153 (1985).

Nusse, H. E.

H. E. Nusse, and J. A. Yorke, "Basins of attraction," Science 271, 1376-1380 (1996).
[CrossRef]

Ott, E.

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]

C. Grebogi, E. Kostelich, E. Ott, and J. A. Yorke, "Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor," Physica D 25, 347-360 (1987).

S.W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, "Fractal basin boundaries," Physica D 17, 125-153 (1985).

C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, "Final state sensitivity: an obstruction to predictability," Phys. Lett. A 99, 415-418 (1983).
[CrossRef]

Serebriakov, A.

van Driel, E.

Yorke, J. A.

H. E. Nusse, and J. A. Yorke, "Basins of attraction," Science 271, 1376-1380 (1996).
[CrossRef]

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]

C. Grebogi, E. Kostelich, E. Ott, and J. A. Yorke, "Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor," Physica D 25, 347-360 (1987).

S.W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, "Fractal basin boundaries," Physica D 17, 125-153 (1985).

C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, "Final state sensitivity: an obstruction to predictability," Phys. Lett. A 99, 415-418 (1983).
[CrossRef]

Appl. Opt.

Opt. Lett.

Phys. Lett. A

C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, "Final state sensitivity: an obstruction to predictability," Phys. Lett. A 99, 415-418 (1983).
[CrossRef]

Physica D

S.W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, "Fractal basin boundaries," Physica D 17, 125-153 (1985).

C. Grebogi, E. Kostelich, E. Ott, and J. A. Yorke, "Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor," Physica D 25, 347-360 (1987).

Science

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]

H. E. Nusse, and J. A. Yorke, "Basins of attraction," Science 271, 1376-1380 (1996).
[CrossRef]

Other

M. van Turnhout, and F. Bociort, "Predictability and unpredictability in optical system optimization," in Current Developments in Lens Design and Optical Engineering VIII, P. Z. Mouroulis, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE 6667, 666709 (2007).

Optical Research Associates, CODE V, Pasadena, CA.

ZEMAX Development Corporation, ZEMAX, Bellevue, WA.

F. Bociort and M. van Turnhout, "Generating saddle points in the merit function landscape of optical systems," in Optical Design and Engineering II, L. Mazuray and R. Wartmann, eds., Proc. SPIE 5962, 59620S (2005).
[CrossRef]

S. N. Rasband, Chaotic dynamics of nonlinear systems (Wiley, New York, 1990).

H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and fractals: New frontiers of science, 2nd ed. (Springer-Verlag, New York, 2004).

E. Ott, Chaos in dynamical systems, 2nd ed. (Cambridge University Press, Cambridge, 2002).

H. Gross, H. Z¨ugge, M. Peschka, and F. Blechinger, "Principles of optimization," in Handbook of Optical Systems, Vol. 3 (Wiley-VCH, Weinheim, 2007), 291-370.

D. C. Sinclair, "Optical design software," in Handbook of Optics, Fundamentals, Techniques, and Design, Vol. 1,2nd ed., M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), 34.1-34.26.

F. Bociort, "Optical system optimization," in Encyclopedia of optical engineering, R. G. Driggers, ed. (Marcel Dekker, New York, 2003), 1843-1850.Q1

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Figures (12)

Fig. 1.
Fig. 1.

Two neighboring basins of attraction. After local optimization, all initial points in one basin lead to the same local minimum (LM1 or LM2).

Fig. 2.
Fig. 2.

Five local minima for a two-dimensional monochromatic doublet. The colors correspond to the colors used for the basins of attraction in subsequent figures.

Fig. 3.
Fig. 3.

(a) Merit function equimagnitude contours of the two-dimensional monochromatic doublet example (specifications are given in Tabs. 1 and 2). The larger blue points A, B, C, and D correspond to the local minima shown in Fig. 2. The smaller purple points are saddle points. In all figures of the same kind, the compact black regions contain configurations that suffer from ray failure. (b) Basins of attraction for the same doublet example (on a grid of 101 × 101 points), obtained by using differential equation (1) in a first stage of the optimization process in CODE V. The colors correspond to the local minima as shown in Fig. 2.

Fig. 4.
Fig. 4.

Basins of attraction of the two-dimensional monochromatic doublet, obtained with OPTSYS, for different values of the maximum damping factor on a grid of 401 × 401 points. The specifications of the doublet are given in Tabs. 1 and 2. The colors correspond to the local minima as shown in Fig. 2. (a) Large damping, (b) 10 times smaller damping.

Fig. 5.
Fig. 5.

Magnified regions of basin boundaries shown in Fig. 4(b). Figure 5(a) shows the positions of these regions within the parameter domain shown in Fig. 4(b).

Fig. 6.
Fig. 6.

Grid boxes that contain at least one point of the orange basin shown in Fig. 5(b), for five different grids with (a) 400 ×400, (b) 200×200, (c) 100×100, (d) 50 ×50, and (e) 25×25 points. The values of N(ε) are 23981, 8734, 3328, 1149, and 375, respectively.

Fig. 7.
Fig. 7.

Calculation of the capacity dimension D for the orange basin shown in Fig. 5(b). The slope of the straight line, which fits the data, gives D = 1.48 [see Eq. (4)].

Fig. 8.
Fig. 8.

Basins of attraction for the two-dimensional monochromatic doublet obtained with CODE V (a–d) and Zemax (e). (The gray contours are the equimagnitude contours of the merit function.) (a) Doublet with settings as given in Tabs. 1 and 2 and default optimization of CODE V. (b) Magnification of the small white rectangle in a). (c) Magnification of the small white rectangle in b). (d) Same as a), but with the thickness of the second lens equal to 0.3 mm. (e) Same as a), but with the damped-least-squares optimization of ZEMAX.

Fig. 9.
Fig. 9.

(a) Double Gauss system (without vignetting). (b) Another local minimum which is found after optimizing a two-dimensional set of starting points (see text).

Fig. 10.
Fig. 10.

Basins of attraction for a seven-dimensional Double Gauss optimization problem. The initial curvature of the first surface is plotted vertically, and the initial curvature of the second surface is plotted horizontally.

Fig. 11.
Fig. 11.

Optimization paths (dark colored lines) in the variable space of the doublet, obtained with (a) default optimization of CODE V, (b) OPTSYS, (c) damped least-squares optimization of ZEMAX. Four starting points for optimization are chosen very close to each other at the position ‘START’, and the iterations are shown as dots. Note that the sequence of iterations always remains inside the same basin as for the starting point, independent of the basin shape.

Fig. 12.
Fig. 12.

(a) Illustration of the external damping procedure. The starting point is shown in black, and the result obtained after one optimization step with CODE V is shown in white. The result after external damping is shown in gray. The external damping factor damps the optimization in such a way that it does not change the direction, but only shortens the step size. (b) Basins of attraction for the doublet after applying our external damping to the default optimization method of CODE V.

Tables (4)

Tables Icon

Table 1. Specifications for the monochromatic doublet example.

Tables Icon

Table 2. Curvatures, thicknesses, and refractive indices of the starting points for the monochromatic doublet example. For curvatures c 2 and c 3 the variation domain is given in brackets.

Tables Icon

Table 3. Capacity dimensions of the basins of attraction shown in Figs. 5(a)–(e). The letters A, C, D, and E correspond to the basins of the corresponding local minima, and ‘RF’ refers to starting configurations for which ray failure occurs during optimization. Recall the basin of minimum B has disappeared in Fig. 5.

Tables Icon

Table 4. Capacity dimensions of the basins of attraction shown in Figs. 8(a)–(e). The letters A–E correspond to the basins of the local minima.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

dxds=MFMF ,
N(ε)=kεD ,
N(εm)N(ε0)=(εmε0)D=(12)mD ,
mD=log2 N(ε0)N(εm) .

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