Abstract

Topology-optimized designs of multiple-disk resonators are presented using level-set expression that incorporates surface effects. Effects from total internal reflection at the surfaces of the dielectric disks are precisely simulated by modeling clearly defined dielectric boundaries during topology optimization. The electric field intensity in optimal resonators increases to more than four and a half times the initial intensity in a resonant state, whereas in some cases the Q factor increases by three and a half times that for the initial state. Wavelength-scale link structures between neighboring disks improve the performance of the multiple-disk resonators.

© 2015 Optical Society of America

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    [Crossref]

2014 (2)

K. Yaji, T. Yamada, M. Yoshino, T. Matsumoto, K. Izui, and S. Nishiwaki, “Topology optimization using the lattice boltzmann method incorporating level set boundary expressions,” J. Comput. Phys. 274, 158–181 (2014).
[Crossref]

G. Fujii, T. Ueta, and M. Mizuno, “Level set-based topology optimization for anti-reflection surface,” Appl. Phys. A 116, 921–927 (2014).
[Crossref]

2013 (4)

G. Fujii, H. Watanabe, T. Yamada, T. Ueta, and M. Mizuno, “Level set based topology optimization for optical cloaks,” Appl. Phys. Lett. 102, 251106 (2013).
[Crossref]

L. Lan, F. Sun, Y. Liu, C. K. Ong, and Y. Ma, “Experimentally demonstrated a unidirectional electromagnetic cloak designed by topology optimization,” Appl. Phys. Lett. 103, 121113 (2013).
[Crossref]

J. Andkjær and O. Sigmund, “Topology optimized cloak for airborne sound,” J. Vib. Acoust. 135, 0410111 (2013).
[Crossref]

D. S. Wiersma, “Disordered photonics,” Nat. Photonics 7, 188–196 (2013).
[Crossref]

2012 (6)

G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “Study on transition from photonic-crystal laser to random laser,” Opt. Express 20, 7300–7315 (2012).
[Crossref] [PubMed]

G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “Finite element analysis of laser modes within photonic random media,” J. Phys. B: At. Mol. Phys. 45, 085404 (2012).
[Crossref]

G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “A study on the effect of filling factor for laser action in dielectric random media,” Appl. Phys. A 107, 35–42 (2012).
[Crossref]

Q. Zhang, J. Qi, X. Li, F. Yi, Z. Wang, and Y. Zhang, “Electrically pumped lasing from single ZnO micro/nanowire and poly(3,4- ethylenedioxythiophene):poly(styrenexulfonate) hybrid heterostructures,” Appl. Phys. Lett. 101, 043119 (2012).
[Crossref]

J. Andkjær, N. A. Mortensen, and O. Sigmund, “Towards all-dielectric, polarization-independent optical cloaks,” Appl. Phys. Lett. 100, 101106 (2012).
[Crossref]

M. Otomori, T. Yamada, K. Izui, S. Nishiwaki, and J. Andkjær, “A topology optimization method based on the level set method for the design of negative permeability dielectric metamaterials,” Comput. Methods Appl. Mech. Engrg. 237–240, 192–211 (2012).
[Crossref]

2011 (7)

S. Zhou, W. Li, Y. Chen, G. Sun, and Q. Li, “Topology optimization for negative permeability metamaterials using level-set algorithm,” Acta Mater. 59, 2624–2636 (2011).
[Crossref]

A. Kawamoto, T. Matsumori, S. Yamasaki, T. Nomura, T. Kondoh, and S. Nishiwaki, “Heaviside projection based topology optimization by a pde-filtered scalar function,” Struct. Multidiscip. Optim. 44, 19–24 (2011).
[Crossref]

J. Andkjær and O. Sigmund, “Topology optimized low-contrast all-dielectric optical cloak,” Appl. Phys. Lett. 98, 021112 (2011).
[Crossref]

T. Yamada, K. Izui, and S. Nishiwaki, “A level set-based topology optimization method for maximizing thermal diffusivity in problems including design-dependent effects,” J. Mech. Des. 133, 031011 (2011).
[Crossref]

T. Yamada, S. Yamasaki, K. I. S. Nishiwaki, and M. Yoshimura, “Design of compliant thermal actuators using structural optimization based on the level set method,” J. Comput. Inf. Sci. Eng. 11, 011005 (2011).
[Crossref]

X. Zhang, H. Li, X. Tu, X. Wu, L. Liu, and L. Xu, “Suppression and hopping of whispering gallery modes in multiple-ring-coupled microcavity lasers,” J. Opt. Soc. Am. B 28, 483–488 (2011).
[Crossref]

K. Scholten, X. Fan, and E. T. Zellers, “Microfabricated optofluidic ring resonator structures,” Appl. Phys. Lett. 99, 141108 (2011).
[Crossref] [PubMed]

2010 (4)

S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidisc. Optim. 41, 495–505 (2010).
[Crossref]

A. R. Diaz and O. Sigmund, “A topology optimization method for design of negative permeability metamaterials,” Struct. Multidisc. Optim. 41, 163–177 (2010).
[Crossref]

T. Yamada, K. Izui, S. Nishiwaki, and A. Takezawa, “A topology optimization method based on the level set method incorporating a fictitious interface energy,” Comput. Methods Appl. Mech. Engrg. 199, 2876–2891 (2010).
[Crossref]

T. J. Kippenberg, “Particle sizing by mode splitting,” Nature 4, 9–10 (2010).

2009 (3)

Z. Luo, L. Tong, J. Luo, P. Wei, and M. Y. Wang, “Design of piezoelectric actuators using a multiphase level set method of piecewise constants,” J. Comput. Phys. 228, 2643–2659 (2009).
[Crossref]

A. Iga, S. Nishiwaki, K. Izui, and M. Yoshimura, “Topology optimization for thermal conductors considering design-dependent effects, including heat conduction and convection,” Int. J. Heat Mass Transfer 52, 2721–2731 (2009).
[Crossref]

D. C. Dobson and L. B. Simeonova, “Optimization of periodic composite structures for sub-wavelength focusing,” Appl. Math. Optim. 60, 133–150 (2009).
[Crossref]

2008 (4)

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

W. R. Frei, H. T. Johnsona, and D. A. Tortorelli, “Optimization of photonic nanostructures,” Comput. Methods Appl. Mech. Engrg. 197, 3410–3416 (2008).
[Crossref]

S. Zhou and Q. Li, “A variational level set method for the topology optimization of steady-state navier-stokes flow,” J. Comput. Phys. 227, 10178–10195 (2008).
[Crossref]

P. Wei and M. Y. Wang, “Piecewise constant level set method for structural topology optimization,” Int. J. Numer. Methods Engrg. 78, 379–402 (2008).
[Crossref]

2007 (8)

S. Y. Wang, K. M. Lim, B. Khoo, and M. Y. Wang, “An extended level set method for shape and topology optimization,” J. Comput. Phys. 221, 395–421 (2007).
[Crossref]

A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems,” J. Comput. Phys. 223, 469 (2007).
[Crossref]

L. He, C.-Y. Kao, and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comput. Phys. 225, 891–909 (2007).
[Crossref]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[Crossref]

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

P. I. Borel, B. Bilenberg, L. H. Frandsen, T. Nielsen, J. Fage-Pedersen, A. V. Lavrinenko, J. S. Jensen, O. Sigmund, and A. Kristensen, “Imprinted silicon-based nanophotonics,” Opt. Express 15, 1261–1266 (2007).
[Crossref] [PubMed]

G. H. Yoon, J. S. Jensen, and O. Sigmund, “Topology optimization of acoustic-structure interaction problems using a mixed finite element formulation,” Int. J. Numer. Methods Engrg. 70, 1049–1075 (2007).
[Crossref]

S. V. Boriskina, “Spectral engineering of bends and branches in microdisk coupled-resonator optical waveguides,” Opt. Express 15, 17371–17379 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (2)

D. Geuzebroek, E. Klein, H. Kelderman, N. Baker, and A. Driessen, “Compact wavelength-selective switch for gigabit filtering in access networks,” IEEE Photonics Technol. Lett. 17, 1041–1135 (2005).
[Crossref]

G. Allaire, F. Gournay, F. Jouve, and A. Toader, “Structural optimization using topological and shape sensitivity via a level set method,” Control and Cybernetics 34, 59–80 (2005).

2004 (7)

J. K. Guest, J. H. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Engrg. 61, 238–254 (2004).
[Crossref]

J. Liu, J. Yao, J. Yao, and T. H. Yeap, “Single-longitudinal-mode multiwavelength fiber ring laser,” IEEE Photonics Technol. Lett. 16, 1020–1022 (2004).
[Crossref]

Y. Li, K. Saitou, and N. Kikuchi, “Topology optimization of thermally actuated compliant mechanisms considering time-transient effect,” Finite Elem. Anal. Des. 40, 1317–1331 (2004).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.-Y. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express 12, 1551–1561 (2004).
[Crossref] [PubMed]

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. Harpøth, 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. Harpøth, P. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[Crossref] [PubMed]

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

2001 (1)

S. Osher and F. Santosa, “Level-set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. 171, 272–288 (2001).
[Crossref]

2000 (1)

J. A. Sethian and A. Wiegmann, “A structural boundary design via level-set and immersed interface methods,” J. Comput. Phys. 163, 489–529 (2000).
[Crossref]

1999 (3)

J. Sokolowski and A. Zochowski, “On the topological derivatives in shape optimization,” SIAM J. Control Optim. 37, 1251–1272 (1999).
[Crossref]

J. Sokolowski and A. Zochowski, “On the topological derivative in shape optimization,” SIAM J. Control and Optim. 37, 1251–1272 (1999).
[Crossref]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[Crossref]

1998 (1)

O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima,” Struct. Optim. 16, 68–75 (1998).
[Crossref]

1995 (2)

D. S. Wiersma, M. P. van Albada, and A. Lagendijk, “Random laser?” Nature 373, 203 (1995).
[Crossref]

Z. D. Ma and N. Kikuchi, “Topological design for vibrating structures,” Comput. Methods Appl. Mech. Engrg. 121, 259–280 (1995).
[Crossref]

1994 (2)

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J. K. Guest, J. H. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Engrg. 61, 238–254 (2004).
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M. Otomori, T. Yamada, K. Izui, S. Nishiwaki, and J. Andkjær, “A topology optimization method based on the level set method for the design of negative permeability dielectric metamaterials,” Comput. Methods Appl. Mech. Engrg. 237–240, 192–211 (2012).
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T. Yamada, K. Izui, and S. Nishiwaki, “A level set-based topology optimization method for maximizing thermal diffusivity in problems including design-dependent effects,” J. Mech. Des. 133, 031011 (2011).
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T. Yamada, K. Izui, S. Nishiwaki, and A. Takezawa, “A topology optimization method based on the level set method incorporating a fictitious interface energy,” Comput. Methods Appl. Mech. Engrg. 199, 2876–2891 (2010).
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G. Allaire, F. Gournay, F. Jouve, and A. Toader, “Structural optimization using topological and shape sensitivity via a level set method,” Control and Cybernetics 34, 59–80 (2005).

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A. Kawamoto, T. Matsumori, S. Yamasaki, T. Nomura, T. Kondoh, and S. Nishiwaki, “Heaviside projection based topology optimization by a pde-filtered scalar function,” Struct. Multidiscip. Optim. 44, 19–24 (2011).
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D. Geuzebroek, E. Klein, H. Kelderman, N. Baker, and A. Driessen, “Compact wavelength-selective switch for gigabit filtering in access networks,” IEEE Photonics Technol. Lett. 17, 1041–1135 (2005).
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S. Y. Wang, K. M. Lim, B. Khoo, and M. Y. Wang, “An extended level set method for shape and topology optimization,” J. Comput. Phys. 221, 395–421 (2007).
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Y. Li, K. Saitou, and N. Kikuchi, “Topology optimization of thermally actuated compliant mechanisms considering time-transient effect,” Finite Elem. Anal. Des. 40, 1317–1331 (2004).
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Z. D. Ma and N. Kikuchi, “Topological design for vibrating structures,” Comput. Methods Appl. Mech. Engrg. 121, 259–280 (1995).
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T. J. Kippenberg, “Particle sizing by mode splitting,” Nature 4, 9–10 (2010).

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
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Kitagawa, Y.

Klein, E.

D. Geuzebroek, E. Klein, H. Kelderman, N. Baker, and A. Driessen, “Compact wavelength-selective switch for gigabit filtering in access networks,” IEEE Photonics Technol. Lett. 17, 1041–1135 (2005).
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H. Eschenauer, V. Kobelev, and A. Schumacher, “Bubble method for topology and shape optimization of structures,” Struct. Optim. 8, 42–51 (1994).
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A. Kawamoto, T. Matsumori, S. Yamasaki, T. Nomura, T. Kondoh, and S. Nishiwaki, “Heaviside projection based topology optimization by a pde-filtered scalar function,” Struct. Multidiscip. Optim. 44, 19–24 (2011).
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Q. Zhang, J. Qi, X. Li, F. Yi, Z. Wang, and Y. Zhang, “Electrically pumped lasing from single ZnO micro/nanowire and poly(3,4- ethylenedioxythiophene):poly(styrenexulfonate) hybrid heterostructures,” Appl. Phys. Lett. 101, 043119 (2012).
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Y. Li, K. Saitou, and N. Kikuchi, “Topology optimization of thermally actuated compliant mechanisms considering time-transient effect,” Finite Elem. Anal. Des. 40, 1317–1331 (2004).
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S. Y. Wang, K. M. Lim, B. Khoo, and M. Y. Wang, “An extended level set method for shape and topology optimization,” J. Comput. Phys. 221, 395–421 (2007).
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S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
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L. Lan, F. Sun, Y. Liu, C. K. Ong, and Y. Ma, “Experimentally demonstrated a unidirectional electromagnetic cloak designed by topology optimization,” Appl. Phys. Lett. 103, 121113 (2013).
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Z. D. Ma and N. Kikuchi, “Topological design for vibrating structures,” Comput. Methods Appl. Mech. Engrg. 121, 259–280 (1995).
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A. Kawamoto, T. Matsumori, S. Yamasaki, T. Nomura, T. Kondoh, and S. Nishiwaki, “Heaviside projection based topology optimization by a pde-filtered scalar function,” Struct. Multidiscip. Optim. 44, 19–24 (2011).
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K. Yaji, T. Yamada, M. Yoshino, T. Matsumoto, K. Izui, and S. Nishiwaki, “Topology optimization using the lattice boltzmann method incorporating level set boundary expressions,” J. Comput. Phys. 274, 158–181 (2014).
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[Crossref] [PubMed]

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S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
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G. Fujii, T. Ueta, and M. Mizuno, “Level set-based topology optimization for anti-reflection surface,” Appl. Phys. A 116, 921–927 (2014).
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J. Andkjær, N. A. Mortensen, and O. Sigmund, “Towards all-dielectric, polarization-independent optical cloaks,” Appl. Phys. Lett. 100, 101106 (2012).
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K. Yaji, T. Yamada, M. Yoshino, T. Matsumoto, K. Izui, and S. Nishiwaki, “Topology optimization using the lattice boltzmann method incorporating level set boundary expressions,” J. Comput. Phys. 274, 158–181 (2014).
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A. Kawamoto, T. Matsumori, S. Yamasaki, T. Nomura, T. Kondoh, and S. Nishiwaki, “Heaviside projection based topology optimization by a pde-filtered scalar function,” Struct. Multidiscip. Optim. 44, 19–24 (2011).
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[Crossref]

A. Iga, S. Nishiwaki, K. Izui, and M. Yoshimura, “Topology optimization for thermal conductors considering design-dependent effects, including heat conduction and convection,” Int. J. Heat Mass Transfer 52, 2721–2731 (2009).
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A. Kawamoto, T. Matsumori, S. Yamasaki, T. Nomura, T. Kondoh, and S. Nishiwaki, “Heaviside projection based topology optimization by a pde-filtered scalar function,” Struct. Multidiscip. Optim. 44, 19–24 (2011).
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L. He, C.-Y. Kao, and S. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comput. Phys. 225, 891–909 (2007).
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S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
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A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems,” J. Comput. Phys. 223, 469 (2007).
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Q. Zhang, J. Qi, X. Li, F. Yi, Z. Wang, and Y. Zhang, “Electrically pumped lasing from single ZnO micro/nanowire and poly(3,4- ethylenedioxythiophene):poly(styrenexulfonate) hybrid heterostructures,” Appl. Phys. Lett. 101, 043119 (2012).
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A. Bermúdez, L. Hervella-Nieto, A. Prieto, and R. Rodríguez, “An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems,” J. Comput. Phys. 223, 469 (2007).
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Q. Zhang, J. Qi, X. Li, F. Yi, Z. Wang, and Y. Zhang, “Electrically pumped lasing from single ZnO micro/nanowire and poly(3,4- ethylenedioxythiophene):poly(styrenexulfonate) hybrid heterostructures,” Appl. Phys. Lett. 101, 043119 (2012).
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S. Zhou, W. Li, Y. Chen, G. Sun, and Q. Li, “Topology optimization for negative permeability metamaterials using level-set algorithm,” Acta Mater. 59, 2624–2636 (2011).
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Acta Mater. (1)

S. Zhou, W. Li, Y. Chen, G. Sun, and Q. Li, “Topology optimization for negative permeability metamaterials using level-set algorithm,” Acta Mater. 59, 2624–2636 (2011).
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Appl. Math. Optim. (1)

D. C. Dobson and L. B. Simeonova, “Optimization of periodic composite structures for sub-wavelength focusing,” Appl. Math. Optim. 60, 133–150 (2009).
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Appl. Phys. A (2)

G. Fujii, T. Ueta, and M. Mizuno, “Level set-based topology optimization for anti-reflection surface,” Appl. Phys. A 116, 921–927 (2014).
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G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “A study on the effect of filling factor for laser action in dielectric random media,” Appl. Phys. A 107, 35–42 (2012).
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Appl. Phys. Lett. (8)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
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K. Scholten, X. Fan, and E. T. Zellers, “Microfabricated optofluidic ring resonator structures,” Appl. Phys. Lett. 99, 141108 (2011).
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Q. Zhang, J. Qi, X. Li, F. Yi, Z. Wang, and Y. Zhang, “Electrically pumped lasing from single ZnO micro/nanowire and poly(3,4- ethylenedioxythiophene):poly(styrenexulfonate) hybrid heterostructures,” Appl. Phys. Lett. 101, 043119 (2012).
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G. Fujii, H. Watanabe, T. Yamada, T. Ueta, and M. Mizuno, “Level set based topology optimization for optical cloaks,” Appl. Phys. Lett. 102, 251106 (2013).
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L. Lan, F. Sun, Y. Liu, C. K. Ong, and Y. Ma, “Experimentally demonstrated a unidirectional electromagnetic cloak designed by topology optimization,” Appl. Phys. Lett. 103, 121113 (2013).
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J. Andkjær, N. A. Mortensen, and O. Sigmund, “Towards all-dielectric, polarization-independent optical cloaks,” Appl. Phys. Lett. 100, 101106 (2012).
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Comput. Methods Appl. Mech. Engrg. (6)

W. R. Frei, H. T. Johnsona, and D. A. Tortorelli, “Optimization of photonic nanostructures,” Comput. Methods Appl. Mech. Engrg. 197, 3410–3416 (2008).
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Control and Cybernetics (1)

G. Allaire, F. Gournay, F. Jouve, and A. Toader, “Structural optimization using topological and shape sensitivity via a level set method,” Control and Cybernetics 34, 59–80 (2005).

Finite Elem. Anal. Des. (1)

Y. Li, K. Saitou, and N. Kikuchi, “Topology optimization of thermally actuated compliant mechanisms considering time-transient effect,” Finite Elem. Anal. Des. 40, 1317–1331 (2004).
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IEEE Photonics Technol. Lett. (2)

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J. Liu, J. Yao, J. Yao, and T. H. Yeap, “Single-longitudinal-mode multiwavelength fiber ring laser,” IEEE Photonics Technol. Lett. 16, 1020–1022 (2004).
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Int. J. Heat Mass Transfer (1)

A. Iga, S. Nishiwaki, K. Izui, and M. Yoshimura, “Topology optimization for thermal conductors considering design-dependent effects, including heat conduction and convection,” Int. J. Heat Mass Transfer 52, 2721–2731 (2009).
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Int. J. Numer. Methods Engrg. (4)

G. H. Yoon, J. S. Jensen, and O. Sigmund, “Topology optimization of acoustic-structure interaction problems using a mixed finite element formulation,” Int. J. Numer. Methods Engrg. 70, 1049–1075 (2007).
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J. Comput. Inf. Sci. Eng. (1)

T. Yamada, S. Yamasaki, K. I. S. Nishiwaki, and M. Yoshimura, “Design of compliant thermal actuators using structural optimization based on the level set method,” J. Comput. Inf. Sci. Eng. 11, 011005 (2011).
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J. Comput. Phys. (10)

S. Zhou and Q. Li, “A variational level set method for the topology optimization of steady-state navier-stokes flow,” J. Comput. Phys. 227, 10178–10195 (2008).
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K. Yaji, T. Yamada, M. Yoshino, T. Matsumoto, K. Izui, and S. Nishiwaki, “Topology optimization using the lattice boltzmann method incorporating level set boundary expressions,” J. Comput. Phys. 274, 158–181 (2014).
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S. Osher and J. A. Sethian, “Fronts propagating with curvature dependent speed algorithms based on hamilton-jacobi formulations,” J. Comput. Phys. 78, 12–49 (1988).
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J. A. Sethian and A. Wiegmann, “A structural boundary design via level-set and immersed interface methods,” J. Comput. Phys. 163, 489–529 (2000).
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S. Y. Wang, K. M. Lim, B. Khoo, and M. Y. Wang, “An extended level set method for shape and topology optimization,” J. Comput. Phys. 221, 395–421 (2007).
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J. Mech. Des. (1)

T. Yamada, K. Izui, and S. Nishiwaki, “A level set-based topology optimization method for maximizing thermal diffusivity in problems including design-dependent effects,” J. Mech. Des. 133, 031011 (2011).
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J. Opt. Soc. Am. B (1)

J. Phys. B: At. Mol. Phys. (1)

G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “Finite element analysis of laser modes within photonic random media,” J. Phys. B: At. Mol. Phys. 45, 085404 (2012).
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J. Vib. Acoust. (1)

J. Andkjær and O. Sigmund, “Topology optimized cloak for airborne sound,” J. Vib. Acoust. 135, 0410111 (2013).
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Nat. Photonics (1)

D. S. Wiersma, “Disordered photonics,” Nat. Photonics 7, 188–196 (2013).
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Nature (4)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
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T. J. Kippenberg, “Particle sizing by mode splitting,” Nature 4, 9–10 (2010).

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
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D. S. Wiersma, M. P. van Albada, and A. Lagendijk, “Random laser?” Nature 373, 203 (1995).
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Opt. Express (7)

G. Fujii, T. Matsumoto, T. Takahashi, and T. Ueta, “Study on transition from photonic-crystal laser to random laser,” Opt. Express 20, 7300–7315 (2012).
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Y. Watanabe, Y. Sugimoto, N. Ikeda, N. Ozaki, A. Mizutani, Y. Takata, Y. Kitagawa, and K. Asakawa, “Broadband waveguide intersection with low- crosstalk in two-dimensional photonic crystal circuits by using topology optimization,” Opt. Express 14, 9502–9507 (2006).
[Crossref] [PubMed]

P. I. Borel, B. Bilenberg, L. H. Frandsen, T. Nielsen, J. Fage-Pedersen, A. V. Lavrinenko, J. S. Jensen, O. Sigmund, and A. Kristensen, “Imprinted silicon-based nanophotonics,” Opt. Express 15, 1261–1266 (2007).
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S. V. Boriskina, “Spectral engineering of bends and branches in microdisk coupled-resonator optical waveguides,” Opt. Express 15, 17371–17379 (2007).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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

Opt. Lett. (2)

SIAM J. Control and Optim. (1)

J. Sokolowski and A. Zochowski, “On the topological derivative in shape optimization,” SIAM J. Control and Optim. 37, 1251–1272 (1999).
[Crossref]

SIAM J. Control Optim. (1)

J. Sokolowski and A. Zochowski, “On the topological derivatives in shape optimization,” SIAM J. Control Optim. 37, 1251–1272 (1999).
[Crossref]

Struct. Multidisc. Optim. (3)

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

S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidisc. Optim. 41, 495–505 (2010).
[Crossref]

A. R. Diaz and O. Sigmund, “A topology optimization method for design of negative permeability metamaterials,” Struct. Multidisc. Optim. 41, 163–177 (2010).
[Crossref]

Struct. Multidiscip. Optim. (1)

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[Crossref]

Struct. Optim. (3)

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[Crossref]

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Other (2)

A. A. Novotny and J. Sokolowski, Topological derivatives in shape optimization (Springer, 2013).
[Crossref]

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Figures (15)

Fig. 1
Fig. 1 (a) Structure of a multiple-disk resonators. The domain sizes are L design y = 0.5 × L design x, L out x = 1.2 × L design x, L out y = 0.6 × L design x, L wg = 0.12 × L design x, L grid = L design x / 300, and R disk = 0.3 × L design x. The position of the posing source is ( x d , y d ) = ( L out x , R disk + L wg / 2 ). (b) The clearly defined dielectric boundary and level-set function. ”+/−” symbols represent the signs of the level-set functions defined on the grid points. The symbol ”0” represents the location at which the level-set functions interpolated linearly in each grid become zero. Dielectric boundaries are modeled by the lines connecting the zero points of level-set functions in each grid.
Fig. 2
Fig. 2 Dielectric boundary and the iso-surface of level-set function.
Fig. 3
Fig. 3 Flowchart of topology optimization process.
Fig. 4
Fig. 4 A resonance frequency ω L design x / 2 π c = 4.65362 and a WGM behavior in the initial configuration with εdm = 4 and εair = 1 at the resonance frequency.
Fig. 5
Fig. 5 A resonance mode of a single disk at the dip frequency ω L design x / 2 π c = 4.66074 with εdm = 4 and εair = 1.
Fig. 6
Fig. 6 The other two resonant modes, in the initial configuration, originated in the mode in the single disk.
Fig. 7
Fig. 7 Objective functional value F versus optimization step. Parameters are set to Δt = 1 × 10−8 and K(ϕ) = 1.
Fig. 8
Fig. 8 Obtained optimal configurations, the value of regularization parameter τ, optimization steps, and the perimeter of the optimal configurations L p / L design x.
Fig. 9
Fig. 9 Left and right link structures between neighboring disks for the obtained optimal configurations.
Fig. 10
Fig. 10 The distribution of total electric field, E z / | E i 0 |, where | E i 0 | is the amplitude of incident electric field at the center of design domain Ωdesign.
Fig. 11
Fig. 11 Amplitude distributions of total electric field, E z / | E i 0 |, where | E i 0 | is the amplitude of the incident electric field at the center of design domain Ωdesign.
Fig. 12
Fig. 12 Transmission spectrum near the resonance frequency ω L design x / 2 π c = 4.65362.
Fig. 13
Fig. 13 Dispersion relation in the waveguide. The resonance frequency ω L design x / 2 π c = 4.65362 is represented by black dashed line. Even and odd modes are plotted by red and blue dots, respectively.
Fig. 14
Fig. 14 Transmission spectrum in the waveguide T versus normalized frequency ω L design x / 2 π c.
Fig. 15
Fig. 15 Mode volume versus normalized frequency ω L design x / 2 π c.

Tables (1)

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Table 1 Q factor and mode volume.

Equations (12)

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2 E z + ω 2 c 2 ε ( x ) E z = ω 2 ε 0 c 2 D d δ ( x x d ) ,
ε ( x ) = { ε air + χ ( ε dm ε air ) x Ω design ε air x Ω out ε dm x Ω wg ,
χ ( ϕ ( x ) ) = { 1 if x Ω dm 0 if x Ω design \ Ω dm ,
{ 1 ϕ ( x ) < 0 for x Ω design \ Ω dm ϕ ( x ) = 0 for x Γ dm 0 < ϕ ( x ) 1 for x Ω dm \ Γ dm .
maximize ϕ F = 1 F 0 Ω dm E z E z * d Ω ,
minimize ϕ F r = F + Ω design 1 2 τ | ϕ | 2 d Ω ,
ϕ t = K ( ϕ ) ( F ¯ τ 2 ϕ ) ,
ϕ ( t + Δ t ) Δ t = ϕ ( t ) Δ t K ( ϕ ) [ F ¯ τ 2 ϕ ( t ) ] .
T = Γ out Re { E × H * 2 } n out d Γ Γ in Re { E × H * 2 } n in d Γ ,
L link / L design x = { 1.82 × 10 1 τ = 5 × 10 5 1.66 × 10 1 τ = 1 × 10 5 1.61 × 10 1 τ = 8 × 10 6 .
λ dm / L design x = 1 / [ ( ω L design x / 2 π c ) ε dm ] , = 1.00 × 10 1 .
L circuit x L design = 1 Δ ( ω L design x 2 π c ) ε dm .

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