Abstract

This paper reports on two important issues that arise in the context of the global optimization of photonic components where large problem spaces must be investigated. The first is the implementation of a fast simulation method and associated matrix solver for assessing particular designs and the second, the strategies that a designer can adopt to control the size of the problem design space to reduce runtimes without compromising the convergence of the global optimization tool. For this study an analytical simulation method based on Mie scattering and a fast matrix solver exploiting the fast multipole method are combined with genetic algorithms (GAs). The impact of the approximations of the simulation method on the accuracy and runtime of individual design assessments and the consequent effects on the GA are also examined. An investigation of optimization strategies for controlling the design space size is conducted on two illustrative examples, namely, 60° and 90° waveguide bends based on photonic microstructures, and their effectiveness is analyzed in terms of a GA’s ability to converge to the best solution within an acceptable timeframe. Finally, the paper describes some particular optimized solutions found in the course of this work.

© 2010 Optical Society of America

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  1. F. Monifi, M. Djavid, A. Ghaffari, and M. S. Abrishamian, “Design of efficient photonic crystal bend and power splitter using super defects,” J. Opt. Soc. Am. B 25, 1805–1810 (2008).
    [CrossRef]
  2. J. Smajic, C. Hafner, and D. Erni, “Design and optimisation of an achromatic photonic crystal bend,” Opt. Express 11, 1378–1384 (2003).
    [CrossRef] [PubMed]
  3. C. Hafner, C. Xudong, J. Smajic, and R. Vahldieck, “Efficient procedures for the optimization of defects in photonic crystal structures,” J. Opt. Soc. Am. A 24, 1177–1188 (2007).
    [CrossRef]
  4. J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
    [CrossRef]
  5. P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimisation of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
    [CrossRef]
  6. C. Styan, A. Vukovic, P. Sewell, and T. M. Benson, “An adaptive synthesis tool for rib waveguide design,” J. Lightwave Technol. 22, 2793–2800 (2004).
    [CrossRef]
  7. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC Press, 1993).
  8. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  9. M. P. Ioannidou, N. C. Skaropoulos, and D. P. Chrissoulidis, “Study of interactive scattering by clusters of spheres,” J. Opt. Soc. Am. A 12, 1782–1789 (1995).
    [CrossRef]
  10. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
    [CrossRef]
  11. L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (MIT Press, 1988).
  12. N. A. Gumerov, R. Duraiswami, and E. A. Borovikov, “Data structures, optimal choice of parameters, complexity results for generalized multilevel fast multipole methods in d dimensions,” University of Maryland Institute for Advanced Computer Studies Tech. Report UMIACS-TR-≠2003-28 (also CS-TR-≠2258), Available at https://www.cs.umd.edu/Library/TRs/CS-TR-4458/CS-TR-4458.pdf.
  13. N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassilou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
    [CrossRef]
  14. N. A. Gumerov and R. Duraiswami, “Computation of scattering from clusters of spheres using the fast multipole method,” J. Acoust. Soc. Am. 117, 1744–1761 (2005).
    [CrossRef] [PubMed]
  15. R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
    [CrossRef]
  16. G. Winter, J. Périaux, M. Galan, and P. Cuesta, Genetic Algorithms in Engineering and Computer Science (Wiley, 1995).
  17. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, 1989).
  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).
  19. J. G. Kori, Numerical Recipes in C, 2nd ed. (Laxmi, 1992).
  20. J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
    [CrossRef]
  21. Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms (Wiley, 1999).
  22. M. K. Moghaddam, M. M. Mirsalehi, and A. R. Attari, “A 60° photonic crystal waveguide bend with improved transmission characteristics,” Opt. Appl. XXXIX, 307–317 (2009).

2009 (1)

M. K. Moghaddam, M. M. Mirsalehi, and A. R. Attari, “A 60° photonic crystal waveguide bend with improved transmission characteristics,” Opt. Appl. XXXIX, 307–317 (2009).

2008 (1)

2007 (1)

2005 (2)

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimisation of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

N. A. Gumerov and R. Duraiswami, “Computation of scattering from clusters of spheres using the fast multipole method,” J. Acoust. Soc. Am. 117, 1744–1761 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

2001 (1)

J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
[CrossRef]

1999 (1)

Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms (Wiley, 1999).

1995 (3)

M. P. Ioannidou, N. C. Skaropoulos, and D. P. Chrissoulidis, “Study of interactive scattering by clusters of spheres,” J. Opt. Soc. Am. A 12, 1782–1789 (1995).
[CrossRef]

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

J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

1994 (2)

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

1993 (2)

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC Press, 1993).

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

1992 (2)

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassilou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

J. G. Kori, Numerical Recipes in C, 2nd ed. (Laxmi, 1992).

1989 (1)

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, 1989).

1988 (1)

L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (MIT Press, 1988).

1965 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Abrishamian, M. S.

Attari, A. R.

M. K. Moghaddam, M. M. Mirsalehi, and A. R. Attari, “A 60° photonic crystal waveguide bend with improved transmission characteristics,” Opt. Appl. XXXIX, 307–317 (2009).

Barrett, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Benisty, H.

J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
[CrossRef]

Benson, T. M.

Berry, M.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Borel, P. I.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimisation of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

Borovikov, E. A.

N. A. Gumerov, R. Duraiswami, and E. A. Borovikov, “Data structures, optimal choice of parameters, complexity results for generalized multilevel fast multipole methods in d dimensions,” University of Maryland Institute for Advanced Computer Studies Tech. Report UMIACS-TR-≠2003-28 (also CS-TR-≠2258), Available at https://www.cs.umd.edu/Library/TRs/CS-TR-4458/CS-TR-4458.pdf.

Chan, T. F.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Chew, W. C.

J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

Chrissoulidis, D. P.

Coifman, R.

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

Cuesta, P.

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

Demmel, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Djavid, M.

Donato, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Dongarra, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Duraiswami, R.

N. A. Gumerov and R. Duraiswami, “Computation of scattering from clusters of spheres using the fast multipole method,” J. Acoust. Soc. Am. 117, 1744–1761 (2005).
[CrossRef] [PubMed]

N. A. Gumerov, R. Duraiswami, and E. A. Borovikov, “Data structures, optimal choice of parameters, complexity results for generalized multilevel fast multipole methods in d dimensions,” University of Maryland Institute for Advanced Computer Studies Tech. Report UMIACS-TR-≠2003-28 (also CS-TR-≠2258), Available at https://www.cs.umd.edu/Library/TRs/CS-TR-4458/CS-TR-4458.pdf.

Eijkhout, V.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Engheta, N.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassilou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Erni, D.

Felbacq, D.

Forchel, A.

J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
[CrossRef]

Frandsen, L. H.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimisation of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

Galan, M.

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

Ghaffari, A.

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, 1989).

Greengard, L.

L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (MIT Press, 1988).

Gumerov, N. A.

N. A. Gumerov and R. Duraiswami, “Computation of scattering from clusters of spheres using the fast multipole method,” J. Acoust. Soc. Am. 117, 1744–1761 (2005).
[CrossRef] [PubMed]

N. A. Gumerov, R. Duraiswami, and E. A. Borovikov, “Data structures, optimal choice of parameters, complexity results for generalized multilevel fast multipole methods in d dimensions,” University of Maryland Institute for Advanced Computer Studies Tech. Report UMIACS-TR-≠2003-28 (also CS-TR-≠2258), Available at https://www.cs.umd.edu/Library/TRs/CS-TR-4458/CS-TR-4458.pdf.

Hafner, C.

Harpøth, A.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimisation of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

Ioannidou, M. P.

Kamp, M.

J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
[CrossRef]

Kori, J. G.

J. G. Kori, Numerical Recipes in C, 2nd ed. (Laxmi, 1992).

Kristensen, M.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimisation of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC Press, 1993).

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC Press, 1993).

Maystre, D.

Michielssen, E.

Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms (Wiley, 1999).

Mirsalehi, M. M.

M. K. Moghaddam, M. M. Mirsalehi, and A. R. Attari, “A 60° photonic crystal waveguide bend with improved transmission characteristics,” Opt. Appl. XXXIX, 307–317 (2009).

Moghaddam, M. K.

M. K. Moghaddam, M. M. Mirsalehi, and A. R. Attari, “A 60° photonic crystal waveguide bend with improved transmission characteristics,” Opt. Appl. XXXIX, 307–317 (2009).

Monifi, F.

Moosburger, J.

J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
[CrossRef]

Murphy, W. D.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassilou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Oesterle, U.

J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
[CrossRef]

Olivier, S.

J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
[CrossRef]

Périaux, J.

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

Pozo, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Rahmat-Samii, Y.

Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms (Wiley, 1999).

Rokhlin, V.

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassilou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Romine, C.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Sewell, P.

Skaropoulos, N. C.

Smajic, J.

Song, J. M.

J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Styan, C.

Tayeb, G.

Vahldieck, R.

Van der Vorst, H.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Vassilou, M. S.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassilou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Vukovic, A.

Wandzura, S.

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

Weisbuch, C.

J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
[CrossRef]

Winter, G.

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

Xing, P. F.

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimisation of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

Xudong, C.

Appl. Phys. Lett. (1)

J. Moosburger, M. Kamp, A. Forchel, S. Olivier, H. Benisty, C. Weisbuch, and U. Oesterle, “Enhanced transmission through photonic-crystal-based bent waveguides by bend engineering,” Appl. Phys. Lett. 79, 3579–3581 (2001).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: A pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassilou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

J. Acoust. Soc. Am. (1)

N. A. Gumerov and R. Duraiswami, “Computation of scattering from clusters of spheres using the fast multipole method,” J. Acoust. Soc. Am. 117, 1744–1761 (2005).
[CrossRef] [PubMed]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (3)

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

Microwave Opt. Technol. Lett. (1)

J. M. Song and W. C. Chew, “Multilevel fast multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microwave Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

Opt. Appl. (1)

M. K. Moghaddam, M. M. Mirsalehi, and A. R. Attari, “A 60° photonic crystal waveguide bend with improved transmission characteristics,” Opt. Appl. XXXIX, 307–317 (2009).

Opt. Commun. (1)

P. F. Xing, P. I. Borel, L. H. Frandsen, A. Harpøth, and M. Kristensen, “Optimisation of bandwidth in 60° photonic crystal waveguide bends,” Opt. Commun. 248, 179–184 (2005).
[CrossRef]

Opt. Express (1)

Other (9)

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (MIT Press, 1988).

N. A. Gumerov, R. Duraiswami, and E. A. Borovikov, “Data structures, optimal choice of parameters, complexity results for generalized multilevel fast multipole methods in d dimensions,” University of Maryland Institute for Advanced Computer Studies Tech. Report UMIACS-TR-≠2003-28 (also CS-TR-≠2258), Available at https://www.cs.umd.edu/Library/TRs/CS-TR-4458/CS-TR-4458.pdf.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC Press, 1993).

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

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, 1989).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

J. G. Kori, Numerical Recipes in C, 2nd ed. (Laxmi, 1992).

Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms (Wiley, 1999).

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Figures (23)

Fig. 1
Fig. 1

Schematic presentation of the scattering structure where N scatterers are placed on a square lattice.

Fig. 2
Fig. 2

Schematic description of scattering between two distant holes p and q via clusters and super-clusters.

Fig. 3
Fig. 3

Strategies for re-defining design spaces: (a) physical separation of design domains, (b) projection of input and output beams to identify important scatterers.

Fig. 4
Fig. 4

PBG 90° bend waveguide based on a rods-in-air platform: (a) field profile, (b) comparison of the output field with the target Gaussian output.

Fig. 5
Fig. 5

PBG 60° bend waveguide on the holes-in-substrate platform: (a) field profile, (b) comparison of the output field with the target Gaussian output.

Fig. 6
Fig. 6

Absolute error between the FMM and LU solver obtained for different numbers of Bessel functions n b and FMM growth factors n g .

Fig. 7
Fig. 7

Relative runtime compared against that of LU solver with n b = 6 . Runtime is given for both the LU and FMM solvers for different numbers of Bessel functions n b and FMM growth factors n g .

Fig. 8
Fig. 8

Convergence of the overlap and residual error of the iterative GMRES/FMM solver as a function of the number of GMRES iterations for 90° and 60° PBG bends.

Fig. 9
Fig. 9

Different 90° PBG structures defined by the ratio of the lattice pitch to scatterer diameter analyzed using the iterative GMRES/FMM solver for different residual errors R T and compared against the LU solver in terms of (a) absolute fitness error, (b) relative runtime.

Fig. 10
Fig. 10

Different 60° PBG structures defined by the ratio of the lattice constant to hole radius are analyzed using the iterative GMRES/FMM solver for different residual errors R T and compared against the LU solver in terms of (a) absolute fitness error, (b) relative computational runtime.

Fig. 11
Fig. 11

Convergence of the best solution in each GA generation for different residual errors R T of the iterative FMM solver during optimization of (a), (b) 60° and (c), (d) 90° photonic bends.

Fig. 12
Fig. 12

GA convergence for different levels of the FMM solver n g for optimization of (a) 60° and (b) 90° photonic bend. Inset shows normalized runtime.

Fig. 13
Fig. 13

Convergence of the best solution for different Bessel functions n b for optimization of (a) 60° and (b) 90° photonic bend. Inset shows normalized runtime.

Fig. 14
Fig. 14

Average and best GA fitness and normalized time as a function of number of scatterers in region I for the 90° bend.

Fig. 15
Fig. 15

Best fitness as a function of different switching criteria g s for the 90° bend.

Fig. 16
Fig. 16

Impact of the thresholds on the runtime and on the optimized GA fitness for the fixed GA seed.

Fig. 17
Fig. 17

Design space size as a function of number of generations for both 60° and 90° bend and for the threshold T U = 0.1 .

Fig. 18
Fig. 18

Impact of T U and the frequency of the resizing of the problem space on the best and average GA fitness and normalized runtime.

Fig. 19
Fig. 19

Convergence of the best solution for the 90° bend optimization and for thresholds of 0, 0.1, and 0.4.

Fig. 20
Fig. 20

Optimized 60° photonic bend: (a) field profile, (b) output field cross section with fitness of 0.8188.

Fig. 21
Fig. 21

Two optimized 90° photonic bends: (a) field profile and (b) output field cross section with fitness of 0.9498; (c) field profile and (d) output field cross section of fitness 0.9345.

Fig. 22
Fig. 22

Optimized 90° photonic bend within an embedded PBG structure: (a) field profile, (b) output field cross section.

Fig. 23
Fig. 23

Selectivity of optimized designs compared with the PBG solutions: (a) 60° bend, (b) 90° bend.

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