Abstract

We propose a novel, to our knowledge, algorithm of iterative simulated quenching with temperature rescaling for designing diffractive optical elements, based on an analogy between simulated annealing and statistical thermodynamics. The temperature is iteratively rescaled at the end of each quenching process according to ensemble statistics to bring the system back from a frozen imperfect state with a local minimum of energy to a dynamic state in a Boltzmann heat bath in thermal equilibrium at the rescaled temperature. The new algorithm achieves much lower cost function and reconstruction error and higher diffraction efficiency than conventional simulated annealing with a fast exponential cooling schedule and is easy to program. The algorithm is used to design binary-phase generators of large irregular spot arrays. The diffractive phase elements have trapezoidal apertures of varying heights, which fit ideal arbitrary-shaped apertures better than do trapezoidal apertures of fixed heights.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [CrossRef]
  2. J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
    [CrossRef]
  3. U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
    [CrossRef] [PubMed]
  4. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  5. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
    [CrossRef] [PubMed]
  6. J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
    [CrossRef]
  7. S. Kirckpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef]
  8. S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 721–141 (1984).
    [CrossRef] [PubMed]
  9. P. J. M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).
  10. L. Ingber, “Simulated annealing: practice versus theory,” J. Math. Comput. Model. 18, 29–57 (1993).
    [CrossRef]
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, Cambridge, UK, 1992), pp. 444–455.
  12. H. Szu, R. Hartley, “Fast simulated annealing,” Phys. Lett. 122, 157–162 (1987).
    [CrossRef]
  13. K. H. Hoffmann, P. Salomon, “The optimal simulated annealing schedule for a simple model,” J. Phys. A 23, 3511–3523 (1990).
    [CrossRef]
  14. G. Ruppeiner, J. M. Pedersen, P. Salamon, “Ensemble approach to simulated annealing,” J. Phys. I 1, 455–470 (1991).
  15. R. Frost, P. Heineman, “Simulated annealing a heuristic for parallel stochastic optimization,” in Proceedings of the 1997 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA’97) (Computer Science Research, Education, and Applications, Las Vegas, Nev., 1997).
  16. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  17. I. M. Barton, P. Blair, M. R. Taghizadeh, “Diffractive phase elements for pattern formation: phase-encoding geometry considerations,” Appl. Opt. 36, 9132–9137 (1997).
    [CrossRef]
  18. J. W. Goodman, A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
    [CrossRef]

1999 (1)

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

1997 (1)

1993 (1)

L. Ingber, “Simulated annealing: practice versus theory,” J. Math. Comput. Model. 18, 29–57 (1993).
[CrossRef]

1992 (2)

1991 (1)

G. Ruppeiner, J. M. Pedersen, P. Salamon, “Ensemble approach to simulated annealing,” J. Phys. I 1, 455–470 (1991).

1990 (1)

K. H. Hoffmann, P. Salomon, “The optimal simulated annealing schedule for a simple model,” J. Phys. A 23, 3511–3523 (1990).
[CrossRef]

1989 (1)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

1987 (1)

H. Szu, R. Hartley, “Fast simulated annealing,” Phys. Lett. 122, 157–162 (1987).
[CrossRef]

1984 (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 721–141 (1984).
[CrossRef] [PubMed]

1983 (1)

S. Kirckpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

1977 (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1970 (1)

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

1953 (1)

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Aarts, E. H. L.

P. J. M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).

Barton, I. M.

Blair, P.

Dammann, H.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, Cambridge, UK, 1992), pp. 444–455.

Frost, R.

R. Frost, P. Heineman, “Simulated annealing a heuristic for parallel stochastic optimization,” in Proceedings of the 1997 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA’97) (Computer Science Research, Education, and Applications, Las Vegas, Nev., 1997).

Gelatt, C. D.

S. Kirckpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 721–141 (1984).
[CrossRef] [PubMed]

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 721–141 (1984).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gillet, J.-N.

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Hartley, R.

H. Szu, R. Hartley, “Fast simulated annealing,” Phys. Lett. 122, 157–162 (1987).
[CrossRef]

Heineman, P.

R. Frost, P. Heineman, “Simulated annealing a heuristic for parallel stochastic optimization,” in Proceedings of the 1997 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA’97) (Computer Science Research, Education, and Applications, Las Vegas, Nev., 1997).

Hoffmann, K. H.

K. H. Hoffmann, P. Salomon, “The optimal simulated annealing schedule for a simple model,” J. Phys. A 23, 3511–3523 (1990).
[CrossRef]

Ichikawa, H.

Ingber, L.

L. Ingber, “Simulated annealing: practice versus theory,” J. Math. Comput. Model. 18, 29–57 (1993).
[CrossRef]

Jaakkola, T.

Jahns, J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Kirckpatrick, S.

S. Kirckpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Klotz, E.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Krackhardt, U.

Kuisma, S.

Mait, J. N.

Metropolis, N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Miller, J. M.

Noponen, E.

Pedersen, J. M.

G. Ruppeiner, J. M. Pedersen, P. Salamon, “Ensemble approach to simulated annealing,” J. Phys. I 1, 455–470 (1991).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, Cambridge, UK, 1992), pp. 444–455.

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Rosenbluth, A. W.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M. N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Ruppeiner, G.

G. Ruppeiner, J. M. Pedersen, P. Salamon, “Ensemble approach to simulated annealing,” J. Phys. I 1, 455–470 (1991).

Salamon, P.

G. Ruppeiner, J. M. Pedersen, P. Salamon, “Ensemble approach to simulated annealing,” J. Phys. I 1, 455–470 (1991).

Salomon, P.

K. H. Hoffmann, P. Salomon, “The optimal simulated annealing schedule for a simple model,” J. Phys. A 23, 3511–3523 (1990).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sheng, Y.

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Streibl, N.

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
[CrossRef] [PubMed]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Szu, H.

H. Szu, R. Hartley, “Fast simulated annealing,” Phys. Lett. 122, 157–162 (1987).
[CrossRef]

Taghizadeh, M. R.

Teller, A. H.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, Cambridge, UK, 1992), pp. 444–455.

Turunen, J.

van Laarhoven, P. J. M.

P. J. M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).

Vasara, A.

Vecchi, M. P.

S. Kirckpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, Cambridge, UK, 1992), pp. 444–455.

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Westerholm, J.

Appl. Opt. (3)

IBM J. Res. Dev. (1)

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell. (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. 6, 721–141 (1984).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

J. Math. Comput. Model. (1)

L. Ingber, “Simulated annealing: practice versus theory,” J. Math. Comput. Model. 18, 29–57 (1993).
[CrossRef]

J. Phys. A (1)

K. H. Hoffmann, P. Salomon, “The optimal simulated annealing schedule for a simple model,” J. Phys. A 23, 3511–3523 (1990).
[CrossRef]

J. Phys. I (1)

G. Ruppeiner, J. M. Pedersen, P. Salamon, “Ensemble approach to simulated annealing,” J. Phys. I 1, 455–470 (1991).

Opt. Acta (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Opt. Commun. (1)

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

Opt. Eng. (1)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Lett. (1)

H. Szu, R. Hartley, “Fast simulated annealing,” Phys. Lett. 122, 157–162 (1987).
[CrossRef]

Science (1)

S. Kirckpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Other (3)

P. J. M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).

R. Frost, P. Heineman, “Simulated annealing a heuristic for parallel stochastic optimization,” in Proceedings of the 1997 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA’97) (Computer Science Research, Education, and Applications, Las Vegas, Nev., 1997).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University, Cambridge, UK, 1992), pp. 444–455.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Iterative simulated quenching algorithm flowchart.

Fig. 2
Fig. 2

Grating period of trapezoidal apertures with varying heights where R = 11 and J = 11.

Fig. 3
Fig. 3

Binary-amplitude trapezoidal-shaped aperture.

Fig. 4
Fig. 4

(a) Designed hologram period plotted as a 256 × 256 pixel array, (b) irregular 21 × 21 spot array with 9 × 9 central orders missing obtained from hologram shown in (a).

Fig. 5
Fig. 5

(a) Designed hologram period plotted as 256 × 256 pixel array, (b) irregular 15 × 15 spot array obtained from hologram shown in (a).

Fig. 6
Fig. 6

(a) Average system energy versus time step (the optimized hologram is shown in Fig. 5), (b) acceptance probability after temperature rescaling versus quenching iteration number.

Fig. 7
Fig. 7

(a) Average system energy versus time step for both iterative simulated quenching and conventional single simulated quenching (irregular 9 × 9 spot-array generators are designed), (b) close-up of (a).

Tables (2)

Tables Icon

Table 1 Designs of Binary-Phase Diffractive Optical Elements by Iterative Simulated Quenchinga

Tables Icon

Table 2 Designs by Iterative Simulated Quenching versus Conventional Single Simulated Quenchinga

Equations (21)

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

Gω, T=exp-Eω/TωΩexp-Eω/T,
hΔEk=exp-ΔEk/T.
Gωk, ThΔEk=Gωk+1, T.
Tk=c/ln1+k,
Tk+1=bTk,  where 0<b<1 so that Tk=T0 expb-1k,
Tk=T0 exp-ckQ/D,
E=Ω EωGω, T=1Ωexp-Eω/TΩ Eωexp-Eω/T.
CT=dEdT=σ2T2,
σ2=E2-E2.
CEeq-EeTeq-Teσeq2Teq2,
Teq=12σeq2Eeq-Ee1±1-4TeEeq-Ee/σeq21/2.
12σeq2Eeq-EeTeqσeq2Eeq-Ee.
tk=T0 exp-ck.
D=4J-1/2R+R-1=2J-1R-1.
Tmn=expiϕ-1r=0r-1j=1joddJ-2 Umnrj+δm0δn0,
Umn=iΔy/2πmexp-iπnyu+yd×exp-iπnb+dsincmd-b+nΔy-exp-iπma+csincmc-a+nΔy,
U0n=i/2πnexp-iπnyu+yd×d-cexp-iπnΔy-b-aexpiπnΔy-d-b-c+asincnΔy.
U00=Δy/2d-c+b-a.
xp=xp±2up-1fEif xp-1<xp<xp+1xp=xpotherwise,
E=mn|Tmn|2-βSmn2,
β=β0N2-Me-1,

Metrics