Abstract

Seven different stochastic binary optimizers—based on the concepts of genetic algorithms and evolutionary strategies—are developed, applied to determine defect locations in several photonic crystal structures that serve as test cases, and compared by extensive statistical analysis. In addition to the stochastic optimizers, a quasi-deterministic optimizer based on an algorithm inspired by hill-climbing algorithms was implemented. The test cases include the prominent 90° photonic crystal waveguide bend and a photonic crystal power divider. The analysis of the results shows that many different photonic crystal structures with high transmission may be found for any operating frequency. All of the eight optimizers outperform standard codes—because they maintain an incomplete fitness table—and find the global optima with a high probability even when the number of fitness evaluations is much smaller than the number of potential solutions contained in the discrete search space. Based on the incomplete fitness table, an algorithm to estimate bit-fitness values is presented. The bit-fitness values are then used to improve the performance of some algorithms. The four best algorithms—an extended microgenetic algorithm, two mutation-based algorithms, and the quasi-deterministic algorithm inspired by hill-climbing algorithms—are considered to be of high value for the optimization of defects in photonic crystals and for similar binary optimization problems.

© 2007 Optical Society of America

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References

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  1. J. D. Jannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals—Molding the Flow of Light (Princeton, 1995).
  2. K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, Taylor & Francis, 2005).
    [CrossRef]
  3. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
    [CrossRef] [PubMed]
  4. J. Smajic, Ch. Hafner, and D. Erni, "Design and optimization of an achromatic photonic crystal bend," Opt. Express 11, 1378-1384 (2003).
    [CrossRef] [PubMed]
  5. J. Smajic, Ch. Hafner, and D. Erni, "Optimization of photonic crystal structures," J. Opt. Soc. Am. A 21, 2223-2232 (2004).
    [CrossRef]
  6. G. Winter, J. Périaux, M. Galan, and P. Cuesta, Genetic Algorithms in Engineering and Computer Science (Wiley, 1995).
  7. D. Quagliarella, J. Périaux, C. Poloni, and G. Winter, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science (Wiley, 1998).
  8. J. Smajic, Ch. Hafner, Cui Xudong, and R. Vahldieck, "Numerical optimization of photonic crystal structures," J. Comput. Theor. Nanosci. (to be published).
  9. J. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. A. B. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
    [CrossRef]
  10. Ch. Hafner, MaX-1, A Visual Electromagnetics Platform for PCs (Wiley, 1999).
  11. See http://metamaterial.ethz.ch/PowerDivider/divider.htm.

2005 (2)

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, Taylor & Francis, 2005).
[CrossRef]

J. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. A. B. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

2004 (1)

2003 (1)

1999 (1)

Ch. Hafner, MaX-1, A Visual Electromagnetics Platform for PCs (Wiley, 1999).

1998 (1)

D. Quagliarella, J. Périaux, C. Poloni, and G. Winter, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science (Wiley, 1998).

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

1995 (2)

G. Winter, J. Périaux, M. Galan, and P. Cuesta, Genetic Algorithms in Engineering and Computer Science (Wiley, 1995).

J. D. Jannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals—Molding the Flow of Light (Princeton, 1995).

Busch, K.

J. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. A. B. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Cuesta, P.

G. Winter, J. Périaux, M. Galan, and P. Cuesta, Genetic Algorithms in Engineering and Computer Science (Wiley, 1995).

Erni, D.

Fan, S.

J. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. A. B. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Galan, M.

G. Winter, J. Périaux, M. Galan, and P. Cuesta, Genetic Algorithms in Engineering and Computer Science (Wiley, 1995).

Hafner, Ch.

J. Smajic, Ch. Hafner, and D. Erni, "Optimization of photonic crystal structures," J. Opt. Soc. Am. A 21, 2223-2232 (2004).
[CrossRef]

J. Smajic, Ch. Hafner, and D. Erni, "Design and optimization of an achromatic photonic crystal bend," Opt. Express 11, 1378-1384 (2003).
[CrossRef] [PubMed]

Ch. Hafner, MaX-1, A Visual Electromagnetics Platform for PCs (Wiley, 1999).

J. Smajic, Ch. Hafner, Cui Xudong, and R. Vahldieck, "Numerical optimization of photonic crystal structures," J. Comput. Theor. Nanosci. (to be published).

Jannopoulos, J. D.

J. D. Jannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals—Molding the Flow of Light (Princeton, 1995).

Jiao, J. Y.

J. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. A. B. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Meade, R. D.

J. D. Jannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals—Molding the Flow of Light (Princeton, 1995).

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Miller, D. A. B.

J. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. A. B. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

Mingaleev, S. F.

J. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. A. B. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

Périaux, J.

D. Quagliarella, J. Périaux, C. Poloni, and G. Winter, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science (Wiley, 1998).

G. Winter, J. Périaux, M. Galan, and P. Cuesta, Genetic Algorithms in Engineering and Computer Science (Wiley, 1995).

Poloni, C.

D. Quagliarella, J. Périaux, C. Poloni, and G. Winter, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science (Wiley, 1998).

Quagliarella, D.

D. Quagliarella, J. Périaux, C. Poloni, and G. Winter, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science (Wiley, 1998).

Schillinger, M.

J. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. A. B. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

Smajic, J.

Vahldieck, R.

J. Smajic, Ch. Hafner, Cui Xudong, and R. Vahldieck, "Numerical optimization of photonic crystal structures," J. Comput. Theor. Nanosci. (to be published).

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Winn, J. N.

J. D. Jannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals—Molding the Flow of Light (Princeton, 1995).

Winter, G.

D. Quagliarella, J. Périaux, C. Poloni, and G. Winter, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science (Wiley, 1998).

G. Winter, J. Périaux, M. Galan, and P. Cuesta, Genetic Algorithms in Engineering and Computer Science (Wiley, 1995).

Xudong, Cui

J. Smajic, Ch. Hafner, Cui Xudong, and R. Vahldieck, "Numerical optimization of photonic crystal structures," J. Comput. Theor. Nanosci. (to be published).

Yasumoto, K.

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, Taylor & Francis, 2005).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. A. B. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Phys. Rev. Lett. (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Other (7)

G. Winter, J. Périaux, M. Galan, and P. Cuesta, Genetic Algorithms in Engineering and Computer Science (Wiley, 1995).

D. Quagliarella, J. Périaux, C. Poloni, and G. Winter, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science (Wiley, 1998).

J. Smajic, Ch. Hafner, Cui Xudong, and R. Vahldieck, "Numerical optimization of photonic crystal structures," J. Comput. Theor. Nanosci. (to be published).

Ch. Hafner, MaX-1, A Visual Electromagnetics Platform for PCs (Wiley, 1999).

See http://metamaterial.ethz.ch/PowerDivider/divider.htm.

J. D. Jannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals—Molding the Flow of Light (Princeton, 1995).

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals (CRC Press, Taylor & Francis, 2005).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Sharp 90° bend in a 2D PhC with the shortest possible 45° waveguide section. The rectangle shows the binary optimization area for 10 bit optimizations. The numbers indicate the cells that correspond to a certain bit. The configuration shown is represented by the bit string 1110110001.

Fig. 2
Fig. 2

Same bend structure as in Fig. 1 with a slightly longer 45° waveguide section and larger optimization area for 15 bit optimizations. The configuration shown is represented by the bit string 111010001111111. Note that five cells (bit numbers 11–15) are added to the cells shown in Fig. 1. For the 14 bit optimizations, bit 4 is fixed to 0, and only 14 bits, 1–3 and 5–15, are optimized.

Fig. 3
Fig. 3

Configuration and optimization area of the PhC power divider. As in Figs. 1, 2, the numbers indicate the bit numbers. The configuration shown is represented by the bit string 110110010011.

Fig. 4
Fig. 4

Fitness values for all of the 16,386 individuals of the 14 bit optimization problem. The bit string that characterizes an individual is obtained by binary representation of the corresponding integer number.

Fig. 5
Fig. 5

Numbers of individuals with fitness above a certain value for the 14 bit (90° bend with large optimization area), 12 bit (power splitter), and three 10 bit test cases (90° bend with large optimization area: squares, HF; curve without symbols, CF; crosses, LF).

Fig. 6
Fig. 6

Time-average Poynting vector field for the best LF solution of the 10 bit test case (90° bend).

Fig. 7
Fig. 7

Time-average Poynting vector field for the best CF solution of the 10 bit test case (90° bend).

Fig. 8
Fig. 8

Time-average Poynting vector field for the best HF solution of the 10 bit test case (90° bend).

Fig. 9
Fig. 9

Time-average Poynting vector field for the best (HF) solution of the 14 bit test case (90° bend).

Fig. 10
Fig. 10

Time-average Poynting vector field for the second-best solution of the 14 bit test case (90° bend).

Fig. 11
Fig. 11

Transmission response for the three 10 bit bend optimization cases: LF (maximum transmission at normalized frequency 0.33), CF (maximum transmission at normalized frequency 0.3867), and HF (maximum transmission at normalized frequency 0.42). Only the two best solutions are shown for each case. The model numbers (183, 37, etc.) are the integer representations of the corresponding 10 bit strings (0010110111, 0000100101, etc.), where 1 stands for a rod and 0 stands for a defect at the corresponding position (least-significant bit corresponds to position 1 in Fig. 1).

Fig. 12
Fig. 12

Transmission response for the 14 bit bend optimization case. Only the two best solutions are shown for the HF case (maximum transmission at normalized frequency 0.42).

Tables (10)

Tables Icon

Table 1 Population Size for the Eight Different Runs for the Seven Stochastic Optimizers (STAT, MGA0, MGA1, MGA2, MUT0, MUT1, MUT2) and for the HC Algorithm RHC1 a

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Table 2 Probabilities of Finding Global Optimum in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizers a

Tables Icon

Table 3 Average Relative Fitness (Fitness Found by the Algorithm or Fitness of the Global Optimum) in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizers a

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Table 4 Average Number of Fitness Calls When an IFT Is Used and the Algorithm Is Stopped As Soon As It Finds the Global Optimum, Averaged over All Test Case 100, 200, and 400 Fitness Evaluations, for All Eight Optimizers a

Tables Icon

Table 5 Same as in Table 4, When No IFT Is Used a

Tables Icon

Table 6 Same as in Table 2, When All Algorithms Are Stopped After 100 Fitness Evaluations

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Table 7 Same as in Table 2, When All Algorithms Are Stopped After 200 Fitness Evaluations

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Table 8 Same as in Table 2, When All Algorithms Are Stopped After 400 Fitness Evaluations

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Table 9 Probabilities of Finding the Global Optimum in Percent for the RHC1 Algorithm, for the 14 Bit Bend Optimization Stopped after 100, 200, 400 Fitness Evaluations (Columns B1, B2, B4, Respectively) and for the Power Divider Optimization Stopped After 100, 200, 400 Fitness Evaluations (Columns D1, D2, D4, Respectively) a

Tables Icon

Table 10 Bit-Fitness Values for the Ten Bits of the HF Solution of the 90° Bend Structure a

Equations (3)

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

Fitness = T E ,
bit fitness = ( f 1 f 0 ) e ,
bit fitness = 0.5 ( π + arctan ( ( f 1 f 0 ) e ) ) π ,

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