Abstract

We report on the numerical structural optimization of two-dimensional photonic crystal (PhC) power dividers by using two different classes of optimization algorithms, namely, a modified truncated Newton (TN) gradient search as deterministic local optimization scheme and an evolutionary optimization representing the probabilistic global search strategies. Because of the severe accuracy requirements during optimization, the proper PhC device has been simulated by using the multiple-multipole program that is contained in the MaX-1 software package. With both optimizer classes, we found reliable and promising solutions that provide vanishing power reflection and perfect power balance at any specified frequency within the photonic bandgap. This outcome is astonishing in light of the discrete nature inherent in the underlying PhC structure, especially when the optimizer is allowed to intervene only within a very small volume of the device. Even under such limiting constraints structural optimization is not only feasible but has proven to be highly successful.

© 2004 Optical Society of America

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    [CrossRef]
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2003 (4)

2002 (2)

A. Chutinan, M. Okano, S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 80, 1698–1700 (2002).
[CrossRef]

E. Moreno, D. Erni, Ch. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618/1–12 (2002).
[CrossRef]

2001 (1)

2000 (2)

D. Erni, D. Wiesmann, M. Spühler, S. Hunziker, E. Moreno, B. Oswald, J. Fröhlich, Ch. Hafner, “Applications of evolutionary optimization algorithms in computational optics,” Appl. Comput. Electromagn. Soc. J. 15, 43–60 (2000).

S. G. Nash, “A survey of truncated-Newton methods,” J. Comput. Appl. Math. 124, 1–2, 45–49 (2000).
[CrossRef]

1999 (3)

E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
[CrossRef]

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villenueve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

1996 (2)

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

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic bandgap boundaries,” J. Appl. Phys. 79, 7483–7492 (1996).
[CrossRef]

1994 (1)

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, K. Kash, “Novel applications of photonic bandgap materials: Low-loss bends and high-Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1983 (2)

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

R. S. Dembo, T. Steihaug, “Truncated-Newton algorithms for large-scale unconstrained optimization,” Math. Program. 26, 190–212 (1983).
[CrossRef]

1965 (1)

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

1964 (1)

R. Fletcher, C. M. Reeves, “Function minimisation by conjugate gradients,” Comput. J. 7, 147–154 (1964).
[CrossRef]

1962 (1)

W. Spendley, G. R. Hext, F. R. Himsworth, “Sequential application of simplex designs in optimization and evolutionary operation,” Technometrics 4, 441–461 (1962).
[CrossRef]

1953 (1)

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculation by fast computing machines,” J. Chem. Phys. 21, 1087–1091 (1953).
[CrossRef]

Alerhand, O. L.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, K. Kash, “Novel applications of photonic bandgap materials: Low-loss bends and high-Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Bäck, Th.

Th. Bäck, F. Hoffmeister, H.-P. Schwefel, “A survey of evolution strategies,” Proceedings of the Fourth International Conference on Genetic Algorithms, R. K. Belew, B. A. Norman, eds., (Morgan Kaufman, San Mateo, Calif., 1991), pp. 2–9.

Benisty, H.

Centeno, E.

E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
[CrossRef]

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

Chen, J. C.

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

Chutinan, A.

A. Chutinan, M. Okano, S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 80, 1698–1700 (2002).
[CrossRef]

Dembo, R. S.

R. S. Dembo, T. Steihaug, “Truncated-Newton algorithms for large-scale unconstrained optimization,” Math. Program. 26, 190–212 (1983).
[CrossRef]

Devenyi, A.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, K. Kash, “Novel applications of photonic bandgap materials: Low-loss bends and high-Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Eberhart, R. C.

R. C. Eberhart, J. A. Kennedy, “New optimizer using particle swarm theory,” in Proceedings of the Sixth International Symposium on Micro Machine and Human Science (IEEE Press, Piscataway, N.J., 1995), pp. 39–43.

J. Kennedy, R. C. Eberhart, “Particle swarm optimization,” in Proceedings of IEEE International Conference on Neural Networks (IEEE Press, Piscataway, N.J., 1995), pp. 1942–1948.

Erni, D.

J. Smajic, Ch. Hafner, D. Erni, “Automatic calculation of band diagrams of photonic crystals using the multiple multipole program,” Appl. Comput. Electromagn. Soc. J. 18, 172–180 (2003).

J. Smajic, Ch. Hafner, D. Erni, “On the design of photonic crystal multiplexers,” Opt. Express 11, 566–571 (2003).
[CrossRef] [PubMed]

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

E. Moreno, D. Erni, Ch. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618/1–12 (2002).
[CrossRef]

D. Erni, D. Wiesmann, M. Spühler, S. Hunziker, E. Moreno, B. Oswald, J. Fröhlich, Ch. Hafner, “Applications of evolutionary optimization algorithms in computational optics,” Appl. Comput. Electromagn. Soc. J. 15, 43–60 (2000).

J. Smajic, Ch. Hafner, K. Rauscher, D. Erni, “Analysis of photonic crystal waveguides using the open supercell approach,” in European Optical Society Topical Meeting on Optics in Computing (European Optical Society, Hannover, Germany, 2004), pp. 49–50.

Fan, S.

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villenueve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

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

Felbacq, D.

E. Centeno, D. Felbacq, “Guiding waves with photonic crystals,” Opt. Commun. 160, 57–60 (1999).
[CrossRef]

E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
[CrossRef]

Ferrini, R.

Fletcher, R.

R. Fletcher, C. M. Reeves, “Function minimisation by conjugate gradients,” Comput. J. 7, 147–154 (1964).
[CrossRef]

Fröhlich, J.

D. Erni, D. Wiesmann, M. Spühler, S. Hunziker, E. Moreno, B. Oswald, J. Fröhlich, Ch. Hafner, “Applications of evolutionary optimization algorithms in computational optics,” Appl. Comput. Electromagn. Soc. J. 15, 43–60 (2000).

Gelatt, C. D.

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

Goldberg, D. E.

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

Guizal, B.

E. Centeno, B. Guizal, D. Felbacq, “Multiplexing and demultiplexing with photonic crystals,” J. Opt. A Pure Appl. Opt. 1, L10–L13 (1999).
[CrossRef]

Hafner, Ch.

J. Smajic, Ch. Hafner, D. Erni, “Automatic calculation of band diagrams of photonic crystals using the multiple multipole program,” Appl. Comput. Electromagn. Soc. J. 18, 172–180 (2003).

J. Smajic, Ch. Hafner, D. Erni, “On the design of photonic crystal multiplexers,” Opt. Express 11, 566–571 (2003).
[CrossRef] [PubMed]

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

E. Moreno, D. Erni, Ch. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618/1–12 (2002).
[CrossRef]

D. Erni, D. Wiesmann, M. Spühler, S. Hunziker, E. Moreno, B. Oswald, J. Fröhlich, Ch. Hafner, “Applications of evolutionary optimization algorithms in computational optics,” Appl. Comput. Electromagn. Soc. J. 15, 43–60 (2000).

Ch. Hafner, Post-Modern Electromagnetics Using Intelligent MaXwell Solvers (Wiley, Chichester, UK, 1999).

Ch. Hafner, MaX-1: A Visual Electromagnetics Platform (WileyChichester, UK, 1998).

J. Smajic, Ch. Hafner, K. Rauscher, D. Erni, “Analysis of photonic crystal waveguides using the open supercell approach,” in European Optical Society Topical Meeting on Optics in Computing (European Optical Society, Hannover, Germany, 2004), pp. 49–50.

Haus, H. A.

Hext, G. R.

W. Spendley, G. R. Hext, F. R. Himsworth, “Sequential application of simplex designs in optimization and evolutionary operation,” Technometrics 4, 441–461 (1962).
[CrossRef]

Himsworth, F. R.

W. Spendley, G. R. Hext, F. R. Himsworth, “Sequential application of simplex designs in optimization and evolutionary operation,” Technometrics 4, 441–461 (1962).
[CrossRef]

Hoffmeister, F.

Th. Bäck, F. Hoffmeister, H.-P. Schwefel, “A survey of evolution strategies,” Proceedings of the Fourth International Conference on Genetic Algorithms, R. K. Belew, B. A. Norman, eds., (Morgan Kaufman, San Mateo, Calif., 1991), pp. 2–9.

Holland, J. H.

J. H. Holland, Adaptation in Natural and Artificial Systems (MIT Press, Cambridge, Mass., 1975).

Houdré, R.

Hunziker, S.

D. Erni, D. Wiesmann, M. Spühler, S. Hunziker, E. Moreno, B. Oswald, J. Fröhlich, Ch. Hafner, “Applications of evolutionary optimization algorithms in computational optics,” Appl. Comput. Electromagn. Soc. J. 15, 43–60 (2000).

Joannopoulos, J. D.

C. Manolatou, S. G. Johnson, S. Fan, P. R. Villenueve, H. A. Haus, J. D. Joannopoulos, “High-density integrated optics,” J. Lightwave Technol. 17, 1682–1692 (1999).
[CrossRef]

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

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, K. Kash, “Novel applications of photonic bandgap materials: Low-loss bends and high-Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

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

Johnson, S. G.

Kash, K.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, K. Kash, “Novel applications of photonic bandgap materials: Low-loss bends and high-Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Kennedy, J.

J. Kennedy, R. C. Eberhart, “Particle swarm optimization,” in Proceedings of IEEE International Conference on Neural Networks (IEEE Press, Piscataway, N.J., 1995), pp. 1942–1948.

Kennedy, J. A.

R. C. Eberhart, J. A. Kennedy, “New optimizer using particle swarm theory,” in Proceedings of the Sixth International Symposium on Micro Machine and Human Science (IEEE Press, Piscataway, N.J., 1995), pp. 39–43.

Kirkpatrick, S.

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

Kurland, I.

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

Loncar, M.

Manolatou, C.

Mead, R.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Meade, R. D.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, K. Kash, “Novel applications of photonic bandgap materials: Low-loss bends and high-Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

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

Mekis, A.

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

Metropolis, N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculation by fast computing machines,” J. Chem. Phys. 21, 1087–1091 (1953).
[CrossRef]

Moosburger, J.

Moreno, E.

E. Moreno, D. Erni, Ch. Hafner, “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method,” Phys. Rev. E 66, 036618/1–12 (2002).
[CrossRef]

D. Erni, D. Wiesmann, M. Spühler, S. Hunziker, E. Moreno, B. Oswald, J. Fröhlich, Ch. Hafner, “Applications of evolutionary optimization algorithms in computational optics,” Appl. Comput. Electromagn. Soc. J. 15, 43–60 (2000).

Nash, S. G.

S. G. Nash, “A survey of truncated-Newton methods,” J. Comput. Appl. Math. 124, 1–2, 45–49 (2000).
[CrossRef]

Nelder, J. A.

J. A. Nelder, R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
[CrossRef]

Noda, S.

A. Chutinan, M. Okano, S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 80, 1698–1700 (2002).
[CrossRef]

Okano, M.

A. Chutinan, M. Okano, S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 80, 1698–1700 (2002).
[CrossRef]

Oswald, B.

D. Erni, D. Wiesmann, M. Spühler, S. Hunziker, E. Moreno, B. Oswald, J. Fröhlich, Ch. Hafner, “Applications of evolutionary optimization algorithms in computational optics,” Appl. Comput. Electromagn. Soc. J. 15, 43–60 (2000).

Qiu, M.

Rauscher, K.

J. Smajic, Ch. Hafner, K. Rauscher, D. Erni, “Analysis of photonic crystal waveguides using the open supercell approach,” in European Optical Society Topical Meeting on Optics in Computing (European Optical Society, Hannover, Germany, 2004), pp. 49–50.

Rechenberg, I.

I. Rechenberg, “Cybernetic solution path of an experimental problem,” (Ministry of Aviation, Royal Aircraft Establishment, London, 1965).

Reeves, C. M.

R. Fletcher, C. M. Reeves, “Function minimisation by conjugate gradients,” Comput. J. 7, 147–154 (1964).
[CrossRef]

Rosenbluth, A. W.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculation by fast computing machines,” J. Chem. Phys. 21, 1087–1091 (1953).
[CrossRef]

Rosenbluth, M. N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculation by fast computing machines,” J. Chem. Phys. 21, 1087–1091 (1953).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).

Scherer, A.

Schwefel, H.-P.

Th. Bäck, F. Hoffmeister, H.-P. Schwefel, “A survey of evolution strategies,” Proceedings of the Fourth International Conference on Genetic Algorithms, R. K. Belew, B. A. Norman, eds., (Morgan Kaufman, San Mateo, Calif., 1991), pp. 2–9.

Smajic, J.

J. Smajic, Ch. Hafner, D. Erni, “Automatic calculation of band diagrams of photonic crystals using the multiple multipole program,” Appl. Comput. Electromagn. Soc. J. 18, 172–180 (2003).

J. Smajic, Ch. Hafner, D. Erni, “On the design of photonic crystal multiplexers,” Opt. Express 11, 566–571 (2003).
[CrossRef] [PubMed]

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

J. Smajic, Ch. Hafner, K. Rauscher, D. Erni, “Analysis of photonic crystal waveguides using the open supercell approach,” in European Optical Society Topical Meeting on Optics in Computing (European Optical Society, Hannover, Germany, 2004), pp. 49–50.

Smith, D. A.

R. D. Meade, A. Devenyi, J. D. Joannopoulos, O. L. Alerhand, D. A. Smith, K. Kash, “Novel applications of photonic bandgap materials: Low-loss bends and high-Q cavities,” J. Appl. Phys. 75, 4753–4755 (1994).
[CrossRef]

Spendley, W.

W. Spendley, G. R. Hext, F. R. Himsworth, “Sequential application of simplex designs in optimization and evolutionary operation,” Technometrics 4, 441–461 (1962).
[CrossRef]

Spühler, M.

D. Erni, D. Wiesmann, M. Spühler, S. Hunziker, E. Moreno, B. Oswald, J. Fröhlich, Ch. Hafner, “Applications of evolutionary optimization algorithms in computational optics,” Appl. Comput. Electromagn. Soc. J. 15, 43–60 (2000).

Steihaug, T.

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Figures (10)

Fig. 1
Fig. 1

Left, top view of the initial PhC power divider topology with input and output ports. The crystal area (i.e., the proper branching region) that is subject to optimization is contained within the indicated rectangle near the center. Right, the performance of the initial device is given by the spectral response of the power reflection (R) and the power transmission with respect to the upper (Tu) and lower (Td) output port. The spectral constraints relevant to the various optimization schemes are indicated by labels 1, 2 and 3.

Fig. 2
Fig. 2

Outcome of the sensitivity analysis: The movement toward an improved power divider performance is depicted for every single rod by a corresponding displacement vector. This vector is also assigned to the gradient vector in 2D real space. The rod’s radius variations for a better performance are encoded according to the corresponding filling color (see details in the text).

Fig. 3
Fig. 3

Left, top view of an optimal PhC power divider topology, which was achieved after a single optimization step. The step size (as a result of the 1D search procedure inherent in the modified TN scheme) in the direction of the gradient is 250 nm=0.25a, i.e., 25% of the lattice constant. Right, the frequency response of the PhC power divider already yields vanishing power reflection around the specified normalized frequency of ωa/(2πc)=0.36.

Fig. 4
Fig. 4

Left, top view of the resulting PhC power divider for the optimization procedure that minimizes power reflection at a normalized operation frequency of ωa/(2πc)=0.38. Three iteration steps were needed to reach the optimum, whereas for the last one a 1D search step size of 50 nm=0.05a, i.e., 5% of the lattice constant has been applied. Right, the frequency response of the PhC power divider yields a residual power reflection of R=0.06% at the specified frequency.

Fig. 5
Fig. 5

Left, top view of the resulting PhC power divider for the optimization procedure that minimizes power reflection at a normalized operation frequency of ωa/(2πc)=0.40. For the 1D search in the last iteration step a step size of 12 nm=0.012a, i.e., 1.2% of the lattice constant was required. Right, the frequency response of the PhC power divider yields a residual power reflection of R=0.25% at the specified frequency.

Fig. 6
Fig. 6

Frequency responses of the PhC power divider after several consecutive optimization schemes associated with the different normalized operation frequencies ωa/(2πc)={0.36, 0.37, 0.38, 0.39, 0.40} labeled 1, 2,…5. Nearly vanishing power reflection was achieved for each of the specified operation frequencies.

Fig. 7
Fig. 7

Poynting vector field distributions within two optimized PhC power dividers. Structure with minimized power reflection at a normalized operation frequency of (left) ωa/(2πc)=0.36, (right) ωa/(2πc)=0.40.

Fig. 8
Fig. 8

PhC power divider structure adapted for the application of a standard GA. For symmetry reasons the resulting binary representation of the proper branching area (i.e., the rectangle including the most significant lattice sites) is encoded into 12 bits spanning a search space that consists of 4096 different solutions.

Fig. 9
Fig. 9

Spectral response and device topology of two optimal PhC power dividers: left, globally optimal solution and right, second-best solution with respect to the discrete search space. These solutions have been used to evaluate the performance of various proposed evolutionary optimization schemes as elucidated in the text.

Fig. 10
Fig. 10

Poynting vector field pattern of the second-best power splitter at the critical normalized operation frequency ωa/(2πc)=0.416 where strong reflection and relatively low accuracy are obtained. This frequency coincides with the upper resonance peak in the spectral response presented in Fig. 9.

Equations (10)

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

H(xk)pk=-g(xk),
xk+1=xk+pk,
F=F(x1,, xN; y1,, yN; r1,, rN),
FxiF(xi+hx/2)-F(xi-hx/2)hx,
FyiF(yi+hy/2)-F(yi-hy/2)hy,
FriF(ri+hr/2)-F(ri-hr/2)hr.
FxiF(xi+hx)-F(xi)hx,
FyiF(yi+hy)-F(yi)hy,
FriF(ri+hr)-F(ri)hr.
maxα:FXk+α Fk|Fk|,Xk+1=Xk+α Fk|Fk|,

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