Abstract

An optimization scheme based on topology optimization for transient response of photonic crystal structures is developed. The system response is obtained by a finite-element time-domain analysis employing perfectly matched layers as an absorbing boundary condition. As an example a waveguide-side-coupled microcavity is designed. The gradient-based optimization technique is applied to redistribute the material inside the microcavity such that the Q factors of a monopole and a dipole mode are improved by 375% and 285%, respectively, while maintaining strong coupling. This is obtained by maximizing the stored energy inside the microcavity in the decaying regime of the transient response. Manufacturable designs are achieved by filtering techniques capable of controlling minimum length scales of the design features.

© 2010 Optical Society of America

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  6. J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022–2024 (2004).
    [CrossRef]
  7. J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
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  9. P. I. Borel, A. Harpoth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
    [CrossRef] [PubMed]
  10. L. H. Frandsen, A. Harpoth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 degrees bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
    [CrossRef] [PubMed]
  11. P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71 (2005).
    [CrossRef]
  12. J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90(circle) photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
    [CrossRef]
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    [CrossRef]
  14. J. Dahl, J. S. Jensen, and O. Sigmund, “Topology optimization for transient wave propagation problems in one dimension design of filters and pulse modulators,” Struct. Multidiscip. Optim. 36, 585–595 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2009 (3)

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

J. S. Jensen, “Space-time topology optimization for one-dimensional wave propagation,” Int. J. Numer. Methods Eng. 198, 705–715 (2009).

J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
[CrossRef]

2008 (3)

A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, “The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers,” Opt. Express 16, 11376–11392 (2008).
[CrossRef] [PubMed]

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103, 033102 (2008).
[CrossRef]

J. Dahl, J. S. Jensen, and O. Sigmund, “Topology optimization for transient wave propagation problems in one dimension design of filters and pulse modulators,” Struct. Multidiscip. Optim. 36, 585–595 (2008).
[CrossRef]

2007 (2)

T. Nomura, K. Sato, K. Taguchi, T. Kashiwa, and S. Nishiwaki, “Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique,” Int. J. Numer. Methods Eng. 71, 1261–1296 (2007).
[CrossRef]

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
[CrossRef]

2005 (6)

B. Yue and M. N. Guddati, “Dispersion-reducing finite elements for transient acoustics,” J. Acoust. Soc. Am. 118, 2132–2141 (2005).
[CrossRef]

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71 (2005).
[CrossRef]

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90(circle) photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005).
[CrossRef] [PubMed]

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett. 86, 111114 (2005).
[CrossRef]

2004 (5)

2003 (1)

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 82, 1341–1343 (2003).
[CrossRef]

2002 (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

2001 (2)

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. Lett. 11, 152–154 (2001).
[CrossRef]

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

2000 (1)

Y. S. Chung, C. Cheon, I. H. Park, and S. Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[CrossRef]

1994 (1)

D. A. Tortorelli and P. Michaleris, “Design sensitivity analysis: overview and review,” Inverse Probl. Eng. 1, 71–105 (1994).
[CrossRef]

1989 (1)

D. A. Tortorelli and R. B. Haber, “First-order design sensitivities for transient conduction problems by an adjoint method,” Int. J. Numer. Methods Eng. 28, 733–752 (1989).
[CrossRef]

1987 (2)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

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

Akahane, Y.

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 82, 1341–1343 (2003).
[CrossRef]

Asano, T.

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 82, 1341–1343 (2003).
[CrossRef]

Avniel, Y.

Bendsøe, M. P.

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications, 2nd ed. (Springer Verlag, 2004).

Benisty, H.

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Borel, P. I.

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90(circle) photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71 (2005).
[CrossRef]

L. H. Frandsen, A. Harpoth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 degrees bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[CrossRef] [PubMed]

P. I. Borel, A. Harpoth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

Burger, M.

M. Burger, S. J. Osher, and E. Yablonovitch, “Inverse problem techniques for the design of photonic crystals,” IEICE Trans. Electron. E87C, 258–265 (2004).

Cheon, C.

Y. S. Chung, C. Cheon, I. H. Park, and S. Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[CrossRef]

Choquette, K. D.

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103, 033102 (2008).
[CrossRef]

Chung, Y. S.

Y. S. Chung, C. Cheon, I. H. Park, and S. Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[CrossRef]

Dahl, J.

J. Dahl, J. S. Jensen, and O. Sigmund, “Topology optimization for transient wave propagation problems in one dimension design of filters and pulse modulators,” Struct. Multidiscip. Optim. 36, 585–595 (2008).
[CrossRef]

De la Rue, R. M.

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Englund, D.

Frandsen, L. H.

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90(circle) photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71 (2005).
[CrossRef]

P. I. Borel, A. Harpoth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

L. H. Frandsen, A. Harpoth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 degrees bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[CrossRef] [PubMed]

Frei, W. R.

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103, 033102 (2008).
[CrossRef]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett. 86, 111114 (2005).
[CrossRef]

Fushman, I.

Guddati, M. N.

B. Yue and M. N. Guddati, “Dispersion-reducing finite elements for transient acoustics,” J. Acoust. Soc. Am. 118, 2132–2141 (2005).
[CrossRef]

Guest, J. K.

J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
[CrossRef]

Haber, R. B.

D. A. Tortorelli and R. B. Haber, “First-order design sensitivities for transient conduction problems by an adjoint method,” Int. J. Numer. Methods Eng. 28, 733–752 (1989).
[CrossRef]

Hahn, S. Y.

Y. S. Chung, C. Cheon, I. H. Park, and S. Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[CrossRef]

Harpoth, A.

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90(circle) photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71 (2005).
[CrossRef]

L. H. Frandsen, A. Harpoth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 degrees bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[CrossRef] [PubMed]

P. I. Borel, A. Harpoth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Houdre, R.

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Hvam, J. M.

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Jensen, J. S.

J. S. Jensen, “Space-time topology optimization for one-dimensional wave propagation,” Int. J. Numer. Methods Eng. 198, 705–715 (2009).

J. Dahl, J. S. Jensen, and O. Sigmund, “Topology optimization for transient wave propagation problems in one dimension design of filters and pulse modulators,” Struct. Multidiscip. Optim. 36, 585–595 (2008).
[CrossRef]

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71 (2005).
[CrossRef]

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90(circle) photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022–2024 (2004).
[CrossRef]

P. I. Borel, A. Harpoth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

L. H. Frandsen, A. Harpoth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 degrees bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[CrossRef] [PubMed]

J. S. Jensen and O. Sigmund have prepared a manuscript to be called “Topology optimization for nano-photonics—a review”.

Jin, J.

J. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays (Wiley, 2007).

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, H. T.

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103, 033102 (2008).
[CrossRef]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett. 86, 111114 (2005).
[CrossRef]

Johnson, S. G.

Kashiwa, T.

T. Nomura, K. Sato, K. Taguchi, T. Kashiwa, and S. Nishiwaki, “Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique,” Int. J. Numer. Methods Eng. 71, 1261–1296 (2007).
[CrossRef]

Kim, G. H.

Koshiba, M.

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. Lett. 11, 152–154 (2001).
[CrossRef]

Krauss, T. F.

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Kristensen, M.

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90(circle) photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71 (2005).
[CrossRef]

L. H. Frandsen, A. Harpoth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 degrees bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[CrossRef] [PubMed]

P. I. Borel, A. Harpoth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

Lavrinenko, A. V.

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

Lee, Y. H.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

Michaleris, P.

D. A. Tortorelli and P. Michaleris, “Design sensitivity analysis: overview and review,” Inverse Probl. Eng. 1, 71–105 (1994).
[CrossRef]

Mochizuki, M.

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 82, 1341–1343 (2003).
[CrossRef]

Nishiwaki, S.

T. Nomura, K. Sato, K. Taguchi, T. Kashiwa, and S. Nishiwaki, “Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique,” Int. J. Numer. Methods Eng. 71, 1261–1296 (2007).
[CrossRef]

Noda, S.

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 82, 1341–1343 (2003).
[CrossRef]

Nomura, T.

T. Nomura, K. Sato, K. Taguchi, T. Kashiwa, and S. Nishiwaki, “Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique,” Int. J. Numer. Methods Eng. 71, 1261–1296 (2007).
[CrossRef]

Notomi, M.

Oesterle, U.

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Olivier, S.

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Osher, S. J.

M. Burger, S. J. Osher, and E. Yablonovitch, “Inverse problem techniques for the design of photonic crystals,” IEICE Trans. Electron. E87C, 258–265 (2004).

Oskooi, A. F.

Park, I. H.

Y. S. Chung, C. Cheon, I. H. Park, and S. Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[CrossRef]

Rattier, M.

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Riley, D. J.

J. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays (Wiley, 2007).

Sasaki, S.

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. Lett. 11, 152–154 (2001).
[CrossRef]

Sato, K.

T. Nomura, K. Sato, K. Taguchi, T. Kashiwa, and S. Nishiwaki, “Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique,” Int. J. Numer. Methods Eng. 71, 1261–1296 (2007).
[CrossRef]

Shi, P.

Shinya, A.

Sigmund, O.

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

J. Dahl, J. S. Jensen, and O. Sigmund, “Topology optimization for transient wave propagation problems in one dimension design of filters and pulse modulators,” Struct. Multidiscip. Optim. 36, 585–595 (2008).
[CrossRef]

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
[CrossRef]

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90(circle) photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71 (2005).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022–2024 (2004).
[CrossRef]

P. I. Borel, A. Harpoth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

L. H. Frandsen, A. Harpoth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 degrees bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[CrossRef] [PubMed]

J. S. Jensen and O. Sigmund have prepared a manuscript to be called “Topology optimization for nano-photonics—a review”.

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications, 2nd ed. (Springer Verlag, 2004).

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, 1949).

Smith, C. J. M.

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Svanberg, K.

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

Taguchi, K.

T. Nomura, K. Sato, K. Taguchi, T. Kashiwa, and S. Nishiwaki, “Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique,” Int. J. Numer. Methods Eng. 71, 1261–1296 (2007).
[CrossRef]

Tanaka, Y.

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 82, 1341–1343 (2003).
[CrossRef]

Tortorelli, D. A.

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett. 86, 111114 (2005).
[CrossRef]

D. A. Tortorelli and P. Michaleris, “Design sensitivity analysis: overview and review,” Inverse Probl. Eng. 1, 71–105 (1994).
[CrossRef]

D. A. Tortorelli and R. B. Haber, “First-order design sensitivities for transient conduction problems by an adjoint method,” Int. J. Numer. Methods Eng. 28, 733–752 (1989).
[CrossRef]

Tsuji, Y.

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. Lett. 11, 152–154 (2001).
[CrossRef]

Vuckovic, J.

Weisbuch, C.

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

Yablonovitch, E.

M. Burger, S. J. Osher, and E. Yablonovitch, “Inverse problem techniques for the design of photonic crystals,” IEICE Trans. Electron. E87C, 258–265 (2004).

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

Yang, L. R.

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

Yue, B.

B. Yue and M. N. Guddati, “Dispersion-reducing finite elements for transient acoustics,” J. Acoust. Soc. Am. 118, 2132–2141 (2005).
[CrossRef]

Zhang, L.

Appl. Phys. Lett. (5)

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022–2024 (2004).
[CrossRef]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett. 86, 111114 (2005).
[CrossRef]

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 82, 1341–1343 (2003).
[CrossRef]

C. J. M. Smith, R. M. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001).
[CrossRef]

Comput. Methods Appl. Mech. Eng. (1)

J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
[CrossRef]

Electron. Lett. (1)

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71 (2005).
[CrossRef]

IEEE Microw. Wirel. Compon. Lett. (1)

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. Lett. 11, 152–154 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90(circle) photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

Y. S. Chung, C. Cheon, I. H. Park, and S. Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[CrossRef]

IEICE Trans. Electron. (1)

M. Burger, S. J. Osher, and E. Yablonovitch, “Inverse problem techniques for the design of photonic crystals,” IEICE Trans. Electron. E87C, 258–265 (2004).

Int. J. Numer. Methods Eng. (3)

T. Nomura, K. Sato, K. Taguchi, T. Kashiwa, and S. Nishiwaki, “Structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique,” Int. J. Numer. Methods Eng. 71, 1261–1296 (2007).
[CrossRef]

J. S. Jensen, “Space-time topology optimization for one-dimensional wave propagation,” Int. J. Numer. Methods Eng. 198, 705–715 (2009).

D. A. Tortorelli and R. B. Haber, “First-order design sensitivities for transient conduction problems by an adjoint method,” Int. J. Numer. Methods Eng. 28, 733–752 (1989).
[CrossRef]

Inverse Probl. Eng. (1)

D. A. Tortorelli and P. Michaleris, “Design sensitivity analysis: overview and review,” Inverse Probl. Eng. 1, 71–105 (1994).
[CrossRef]

J. Acoust. Soc. Am. (1)

B. Yue and M. N. Guddati, “Dispersion-reducing finite elements for transient acoustics,” J. Acoust. Soc. Am. 118, 2132–2141 (2005).
[CrossRef]

J. Appl. Phys. (1)

W. R. Frei, H. T. Johnson, and K. D. Choquette, “Optimization of a single defect photonic crystal laser cavity,” J. Appl. Phys. 103, 033102 (2008).
[CrossRef]

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

Opt. Express (5)

Phys. Rev. Lett. (2)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

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

SIAM J. Optim. (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

Struct. Multidiscip. Optim. (2)

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
[CrossRef]

J. Dahl, J. S. Jensen, and O. Sigmund, “Topology optimization for transient wave propagation problems in one dimension design of filters and pulse modulators,” Struct. Multidiscip. Optim. 36, 585–595 (2008).
[CrossRef]

Other (9)

J. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays (Wiley, 2007).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 2008).

J. S. Jensen and O. Sigmund have prepared a manuscript to be called “Topology optimization for nano-photonics—a review”.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

http://www.esf.org/euryi.

http://www.topopt.dtu.dk.

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications, 2nd ed. (Springer Verlag, 2004).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, 1949).

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Figures (11)

Fig. 1
Fig. 1

Illustration of a Γ K directional PhC-WG structure as the computational domain. It consists of the solution domain Ω S and the transition domain Ω T that are truncated by the PML region Ω PML . The circle encloses the scattering design region Ω D . In Ω E (black circular region) the energy is maximized. The PhC is built by blocks of size a / 2 × 3 a / 2 , where Ω T Ω S contains 30 × 14 building blocks, and Ω PML is extended with 24 and 4 blocks on both sides in directions Γ K and Γ M , respectively.

Fig. 2
Fig. 2

Monopole mode. (a) Initial MC geometry. (b) Optimized MC geometry.

Fig. 3
Fig. 3

(a) Logarithmic envelope of normalized stored energy for the monopole mode. (b),(c) H z -field distribution for the initial and optimized MC geometries, respectively. The material distribution is shown with x e t = 0.6 as threshold.

Fig. 4
Fig. 4

Dispersion curve of the Γ K directional PhC-WG. The bandgap exists between the dielectric (lower) and air (higher) bands in the normalized frequency range a / λ 0 = 0.21 0.33 . The horizontal lines represent the dipole mode frequency of 0.2480 and the monopole mode frequency of 0.3030.

Fig. 5
Fig. 5

Dipole mode. (a) Initial MC geometry. (b) Optimized MC geometry.

Fig. 6
Fig. 6

(a) Logarithmic envelope of normalized stored energy for the dipole mode. (b),(c) H z -field distribution for the initial and optimized MC geometries, respectively. The material distribution is shown with x e t = 0.6 as threshold.

Fig. 7
Fig. 7

Optimized designs for the monopole MC mode. (a) Isolated MC. Optimized coupled system geometries for (b) T 1 = 0 , (c) T 1 = T max , and (d) T 1 = 4 T max .

Fig. 8
Fig. 8

Monopole mode. (a) Logarithmic envelope of stored energy U ( t ) . Transmission spectrum for (b) initial design, (c) isolated MC, coupled system (d) T 1 = 0 , (e) T 1 = T max , and (f) T 1 = 4 T max .

Fig. 9
Fig. 9

Optimized designs for the dipole MC mode. (a) Isolated MC. Optimized coupled system geometries for (b) T 1 = 0 , (c) T 1 = T max , and (d) T 1 = 4 T max .

Fig. 10
Fig. 10

Dipole mode. (a) Logarithmic envelope of stored energy U ( t ) . Transmission spectrum for (b) initial design, (c) isolated MC, coupled system (d) T 1 = 0 , (e) T 1 = T max , and (f) T 1 = 4 T max .

Fig. 11
Fig. 11

H z -field distribution for the best optimized WG-side-coupled MC candidate for the dipole mode when T 1 = 4 T max . The material distribution is shown with x e t = 0.6 as threshold.

Tables (1)

Tables Icon

Table 1 Q [ 10 3 ] Factors for Coupled System Configurations

Equations (42)

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μ L 1 ( t ) H z + [ x ( L 2 , x ( t ) ε x ) + y ( L 2 , y ( t ) ε y ) ] H z = J B , z t ,
L 1 ( t ) = 2 t 2 + σ x + σ y ε t + σ x σ y ε 2 ,
L 2 , p ( t ) = 1 a p   exp [ b p t ] u ¯ ( t ) ,     p = x , y ,
σ ( r ) = σ max ( ρ d ) m ,
σ max = ( m + 1 ) log 10 ( R 0 ) 2 d Z 0 .
n ( ε 1 ̃ H z ) Y H z t = 0     on   Γ abs ,
n ( ε 1 ̃ H z ) Y ( n ρ 1 ) H 0 , z inc f t = 0     on   Γ inc ,
V [ μ T z L 1 ( t ) H z ( t ) + T z ̃ H z ( t ) + T z J B , z t ] d V + S [ Y c T z H z t + T z U z ] d S = 0 ,
H z ( r , t ) = i = 1 N N i ( r ) u i ( t ) ,
e = 1 M ( T e u ̈ + R e u ̇ + S e u + g e f e ) = 0 ,
T i j e = μ N i , N j Ω e ,
R i j e = μ ( σ x + σ y ) ε i 1 N i , N j Ω PML e + Y c N i , N j Γ ABS e ,
S p , i j e = ε 1 N i / p , N j / p Ω e ,     p = x , y ,
S i j e = μ σ x σ y ε i 2 N i , N j Ω PML e + S x , i j e + S y , i j e ,
g i e = j S x , i j e ψ x , j + S y , i j e ψ y , j ,     e Ω PML ,
f i e = N i , J B , z / t Ω S + N i , U z Γ inc e ,
ψ p , j = a p   exp [ b p t ] u ¯ ( t ) u j ( t ) ,     p = x , y .
T d u ̇ n + 1 / 2 = T δ v n 1 / 2 ,
T d v ̇ n = R u ̇ n + 1 / 2 S u n g n + f n ,
u ̈ n = u n + 1 2 u n + u n 1 Δ t 2 ,
u ̇ n = u n + 1 u n 1 2 Δ t .
ψ p , j n = exp [ b p Δ t ] ψ p , j n 1 + a p Δ t 2 ( u j n + exp [ b p Δ t ] u j n 1 ) .
ε r 1 ( x e ) = ( 1 x e ) ( ε r I ) 1 + x e ( ε r I I ) 1 .
Φ ( x ) = 0 T F ( u , u ̇ , u ̈ , x ) d t ,
r ( u , u ̇ , u ̈ , x ) = f ( T u ̈ + R u ̇ + S u + g ) = 0 .
F ̂ = F ( u , u ̇ , u ̈ , x ) + λ T r ( u , u ̇ , u ̈ , x ) ,
Φ x e = 0 T ( F u u x e + F u ̇ u ̇ x e + F u ̈ u ̈ x e + F x e + λ T x e r + λ T [ r u u x e + r u ̇ u ̇ x e + r u u ̈ x e + r x e ] ) d t .
Φ x e = [ λ T r u ̇ u x e + λ T r u ̈ u ̇ x e λ ̇ T r u ̈ u x e ] 0 T + 0 T ( F u ̇ u ̇ x e + F u ̈ u ̈ x e + F x e ) d t + 0 T ( r u ̈ λ ̈ r u ̇ λ ̇ + r u λ F u ) u x e d t + 0 T λ T r x e d t .
r u ̈ λ ̈ r u ̇ λ ̇ + r u λ = F u ,
r u ̈ λ ¯ ̈ + r u ̇ λ ¯ ̇ + r u λ ¯ = F u ,
Φ x e = 0 T ( F x e + λ T r x e ) d t .
Φ ( x ) = 0 T F ( u , u ̇ , u ̈ , x ) u ¯ ( t T 1 ) d t ,
max x R M Φ ( x ) = log 10 [ 0 T u T Q u u ¯ ( t T 1 ) d t ] ,
s .t . :     Governing   Eq .   ( 9 )
e v e x e < V f ,     0 x e 1 ,     e Ω D .
u ¯ ( t T 1 ) 1 2 tanh [ 2 β ( t T 1 ) / δ T ] tanh [ β ] + 1 2
1 / Q in = 1 / Q e + 1 / Q 0 ,
U ( t ) = U 0   exp [ ω 0 t / Q 0 ] ,
g ( t ) = exp [ ( t T 0 ) 2 / τ 2 ] sin [ 2 π ω 0 ( t T 0 ) ] ,
P ( ω ) = 1 2 Re [ n Γ out E ω × H ω d Γ ] ,
| T ( ω ) | 2 = 1 1 4 Q e 2 ( 1 + 2 Q e Q 0 ) ( ω ω 0 ω 0 ) 2 + 1 4 Q e ( 1 + Q e Q 0 ) 2 .
| T ( ω 0 ) | 2 = Q in 2 Q 0 2 Q e 2 Q 0 2 + O ( [ Q e / Q 0 ] 3 ) ,

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