OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 25 — Dec. 16, 2013
  • pp: 30812–30841

Formulation for scalable optimization of microcavities via the frequency-averaged local density of states

Xiangdong Liang and Steven G. Johnson  »View Author Affiliations

Optics Express, Vol. 21, Issue 25, pp. 30812-30841 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (3314 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We present a technique for large-scale optimization of optical microcavities based on the frequency-averaged local density of states (LDOS), which circumvents computational difficulties posed by previous eigenproblem-based formulations and allows us to perform full topology optimization of three-dimensional (3d) leaky cavity modes. We present theoretical results for both 2d and fully 3d computations in which every pixel of the design pattern is a degree of freedom (“topology optimization”), e.g. for lithographic patterning of dielectric slabs in 3d. More importantly, we argue that such optimization techniques can be applied to design cavities for which (unlike silicon-slab single-mode cavities) hand designs are difficult or unavailable, and in particular we design minimal-volume multi-mode cavities (e.g. for nonlinear frequency-conversion applications).

© 2013 Optical Society of America

OCIS Codes
(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments
(230.5750) Optical devices : Resonators
(140.3945) Lasers and laser optics : Microcavities

ToC Category:
Optical Devices

Original Manuscript: September 16, 2013
Revised Manuscript: November 17, 2013
Manuscript Accepted: November 19, 2013
Published: December 6, 2013

Xiangdong Liang and Steven G. Johnson, "Formulation for scalable optimization of microcavities via the frequency-averaged local density of states," Opt. Express 21, 30812-30841 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Nomura, K. Tanabe, S. Iwamoto, and Y. Arakawa, “High-Q design of semiconductor-based ultrasmall photonic crystal nanocavity,” Opt. Express18, 8144–8150 (2010). [CrossRef] [PubMed]
  2. B. Bourdin, “Filters in topology optimization,” Int. J. Numer. Methods Eng.50, 2143–2158 (2001). [CrossRef]
  3. M. P. Bendsoe and O. Sigmund, Topology Optimization: Theory, Methods and Applications, 2nd ed. (Springer, 2003).
  4. M. Y. Wang, X. Wang, and D. Guo, “A level set method for structural topology optimization,” Comput. Methods Appl. Mech. Eng.192, 227–246 (2003). [CrossRef]
  5. G. Allaire, F. Jouve, and A.-M. Toader, “Structural optimization using sensitivity analysis and a level-set method,” J. Comput. Phys.194, 363–393 (2004). [CrossRef]
  6. O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sinica25, 227–239 (2009). [CrossRef]
  7. E. Andreassen, A. Clausen, M. Schevenels, B. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidiscip. Optim.43, 1–16 (2011). [CrossRef]
  8. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev.5, 308–321 (2011). [CrossRef]
  9. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express15, 16161–16176 (2007). [CrossRef] [PubMed]
  10. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
  11. H. Hashemi, C. W. Qiu, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “A diameter–bandwidth product limitation of isolated-object cloaking,” Phys. Rev. A86,013804 (2012). [CrossRef]
  12. D. P. Bertsekas, Nonlinear Programming, 2nd ed. (Athena Scientific, 1999).
  13. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Series in Pure and Applied Optics), 2nd ed. (Wiley-Interscience, 2007).
  14. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  15. D. G. Rabus, Integrated Ring Resonators, Vol. 127 of Springer Series in Optical Sciences(Springer, 2007).
  16. S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett.10, 2408–2411 (2010). [CrossRef] [PubMed]
  17. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. Quantum Electron.38, 850–856 (2002). [CrossRef]
  18. C. W. Wong, P. T. Rakich, S. G. Johnson, M. Qi, H. I. Smith, E. P. Ippen, L. C. Kimerling, Y. Jeon, G. Barbastathis, and S. G. Kim, “Strain-tunable silicon photonic band gap microcavities in optical waveguides,” Appl. Phys. Lett.84, 1242–1244 (2004). [CrossRef]
  19. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425, 944–947 (2003). [CrossRef] [PubMed]
  20. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express13, 1202–1214 (2005). [CrossRef] [PubMed]
  21. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater.4, 207–210 (2005). [CrossRef]
  22. D. C. Dobson and F. Santosa, “Optimal Localization of Eigenfunctions in an Inhomogeneous Medium,” SIAM J. Appl. Math.64, 762–774 (2004). [CrossRef]
  23. C.-Y. Kao and F. Santosa, “Maximization of the quality factor of an optical resonator,” Wave Motion45, 412–427 (2008). [CrossRef]
  24. 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]
  25. J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express18, 3793–3804 (2010). [CrossRef] [PubMed]
  26. A. W. Snyder and J. Love, Optical Waveguide Theory, Science Paperbacks, 190 (Springer, 1983).
  27. R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, “Smallest possible electromagnetic mode volume in a dielectric cavity,” IEE Proceedings - Optoelectronics145, 391–397 (1998). [CrossRef]
  28. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69,674 (1946).
  29. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006). [CrossRef]
  30. A. F. Koenderink, “On the use of Purcell factors for plasmon antennas,” Opt. Lett.35, 4208–4210 (2010). [CrossRef] [PubMed]
  31. B. Zhen, S.-L. Chua, J. Lee, A. W. Rodriguez, X. Liang, S. G. Johnson, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Enabling enhanced emission and low-threshold lasing of organic molecules using special Fano resonances of macroscopic photonic crystals,” Proc. Natl. Acad. Sci. U. S. A.110, 13711–13716 (2013). [CrossRef] [PubMed]
  32. E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J.48, 2103–2132 (1969). [CrossRef]
  33. F. Ladouceur, “Roughness, inhomogeneity, and integrated optics,” J. Lightwave Technol.15, 1020–1025 (1997). [CrossRef]
  34. V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Strain-tunable high-Q optical microsphere resonator,” Opt. Commun.145, 86–90 (1998). [CrossRef]
  35. M. Soltani, S. Yegnanarayanan, and A. Adibi, “Ultra-high Q planar silicon microdisk resonators for chip-scale silicon photonics,” Opt. Express15, 4694–4704 (2007). [CrossRef] [PubMed]
  36. L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997). [CrossRef]
  37. K. Inoue and K. Ohtaka, Photonic Crystals: Physics, Fabrication and Applications, Springer Series in Optical Sciences (Springer, 2010).
  38. A. Oskooi and S. G. Johnson, “Electromagnetic wave source conditions,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, and S. G. Johnson, eds. (Artech, 2013), Chap. 4, pp. 65–100.
  39. J.-M. Gerard and B. Gayral, “Strong Purcell effect for InAs quantum boxes in three-dimensional solid-state microcavities,” J. Lightwave Technol.17, 2089–2095 (1999). [CrossRef]
  40. O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E58, 3909–3915 (1998). [CrossRef]
  41. G. D’Aguanno, N. Mattiucci, M. Centini, M. Scalora, and M. J. Bloemer, “Electromagnetic density of modes for a finite-size three-dimensional structure,” Phys. Rev. E69,057601 (2004). [CrossRef]
  42. J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley, 1975).
  43. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).
  44. M. E. Peskin and D. V. Schroeder, An Introduction To Quantum Field Theory (Frontiers in Physics) (Westview, 1995).
  45. S. G. Johnson, “Numerical methods for computing Casimir interactions,” in Casimir Physics, Vol. 834 of Lecture Notes in Physics, D. Dalvit, P. Milonni, D. Roberts, and F. da Rosa, eds. (Springer, 2011), Chap. 6. [CrossRef]
  46. L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinemann, 1984).
  47. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).
  48. N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Physica B405, 2915–2919 (2010). [CrossRef]
  49. L. V. Ahlfors, Complex Analysis, 3rd ed. (McGraw-Hill, 1978).
  50. A. W. Rodriguez, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “Theoretical ingredients of a Casimir analog computer,” Proc. Natl. Acad. Sci. U. S. A.107, 9531–9536 (2010). [CrossRef] [PubMed]
  51. X. Liang, Ph.D. thesis, Massachusetts Institute of Technology, 2013.
  52. S. Scheel, L. Knöll, and D. G. Welsch, “Spontaneous decay of an excited atom in an absorbing dielectric,” Phys. Rev. A60, 4094–4104 (1999). [CrossRef]
  53. C. Van Vlack and S. Hughes, “Finite-difference time-domain technique as an efficient tool for calculating the regularized Green function: applications to the local-field problem in quantum optics for inhomogeneous lossy materials,” Opt. Lett.37, 2880–2882 (2012). [CrossRef] [PubMed]
  54. G. Strang, Computational Science and Engineering (Wellesley-Cambridge, 2007).
  55. A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microwave Theory35, 688–696 (1987). [CrossRef]
  56. K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals, Optical Science and Engineering (CRC, 2005). [CrossRef]
  57. W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” J. Comput. Phys.231, 3406–3431 (2012). [CrossRef]
  58. T. A. Davis, Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms) (SIAM, 2006). [CrossRef]
  59. S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, “Efficient Management of Parallelism in Object Oriented Numerical Software Libraries,” In Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds. (Birkhäuser, 1997). [CrossRef]
  60. S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc Users Manual,” Technical Report No. ANL-95/11 - Revision 3.3, Argonne National Laboratory (2012).
  61. S. Balay, J. Brown, K. Buschelman, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, “PETSc Web page,” http://www.mcs.anl.gov/petsc , 2012.
  62. P. Hénon, P. Ramet, and J. Roman, “PaStiX: a high-performance parallel direct solver for sparse symmetric positive definite systems,” Parallel Computing28, 301–321 (2002). [CrossRef]
  63. S. G. Johnson, “The NLopt nonlinear-optimization package,” http://ab-initio.mit.edu/nlopt .
  64. D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program.45, 503–528 (1989). [CrossRef]
  65. K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optimiz.12, 555–573 (2002). [CrossRef]
  66. J. Nocedal and S. J. Wright, Numerical Optimization (Springer, 2000).
  67. A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optimiz.41, 365–384 (2009). [CrossRef]
  68. A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20, 21558–21575 (2012). [CrossRef] [PubMed]
  69. Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S.-Y. Lin, “High-quality-factor Bragg onion resonators with omnidirectional reflector cladding,” Opt. Lett.28, 2144–2146 (2003). [CrossRef] [PubMed]
  70. Z. Artstein, “Discrete and continuous bang-bang and facial spaces or: Look for the extreme points,” SIAM Rev.22, 172–185 (1980). [CrossRef]
  71. A. F. Oskooi, Ph.D. thesis, Massachusetts Institute of Technology, 2010.
  72. J. Lu, S. Boyd, and J. Vuckovic, “Inverse design of a three-dimensional nanophotonic resonator,” Opt. Express19, 10563–10570 (2011). [CrossRef] [PubMed]
  73. B. Osting and M. I. Weinstein, “Long-lived scattering resonances and Bragg structures,” SIAM J. Appl. Math.73, 827–852 (2013). [CrossRef]
  74. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1996).
  75. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004). [CrossRef]
  76. P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta27, 321–335 (1980). [CrossRef]
  77. L.-A. Wu, M. Xiao, and H. J. Kimble, “Squeezed states of light from an optical parametric oscillator,” J. Opt. Soc. Am. B4, 1465–1475 (1987). [CrossRef]
  78. Z. Y. Ou and H. J. Kimble, “Enhanced conversion efficiency for harmonic generation with double resonance,” Opt. Lett.18, 1053–1055 (1993). [CrossRef] [PubMed]
  79. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2)and χ(3)harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express15, 7303–7318 (2007). [CrossRef] [PubMed]
  80. I. B. Burgess, Y. Zhang, M. W. McCutcheon, A. W. Rodriguez, J. Bravo-Abad, S. G. Johnson, and M. Loncar, “Design of an efficient terahertz source using triply resonant nonlinear photonic crystal cavities,” Opt. Express17, 20099–20108 (2009). [CrossRef] [PubMed]
  81. Z.-F. Bi, A. W. Rodriguez, H. Hashemi, D. Duchesne, M. Loncar, K.-M. Wang, and S. G. Johnson, “High-efficiency second-harmonic generation in doubly-resonant χ(2)microring resonators,” Opt. Express20, 7526–7543 (2012). [CrossRef] [PubMed]
  82. M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal nanocavity with a Quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express16, 19136–19145 (2008). [CrossRef]
  83. A. R. Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express16, 12084–12089 (2008). [CrossRef]
  84. Z. M. Meng, F. Qin, Y. Liu, and Z. Y. Li, “High-Q microcavities in low-index one-dimensional photonic crystal slabs based on modal gap confinement,” J. Appl. Phys.109,043107 (2011). [CrossRef]
  85. J. T. Robinson, C. Manolatou, L. Chen, and M. Lipson, “Ultrasmall mode volumes in dielectric optical micro-cavities,” Phys. Rev. Lett.95,143901 (2005). [CrossRef]
  86. M. Nomura, “GaAs-based air-slot photonic crystal nanocavity for optomechanical oscillators,” Opt. Express20, 5204–5212 (2012). [CrossRef] [PubMed]
  87. S. Kita, K. Nozaki, S. Hachuda, H. Watanabe, Y. Saito, S. Otsuka, T. Nakada, Y. Arita, and T. Baba, “Photonic Crystal Point-Shift Nanolasers With and Without Nanoslots Design, Fabrication, Lasing, and Sensing Characteristics,” IEEE J. Sel. Top. Quantum Electron.17, 1632–1647 (2011). [CrossRef]
  88. Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express18, 23844–23856 (2010). [CrossRef] [PubMed]
  89. O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima,” Struct. Multidiscip. Optim.16, 68–75 (1998).
  90. S. J. Osher and R. P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2002).
  91. C. Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods, Appl. Phys. B81, 235–244 (2005). [CrossRef]
  92. J. K. Guest, J. H. Prévost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Eng.61, 238–254 (2004). [CrossRef]
  93. O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim.33, 401–424 (2007). [CrossRef]
  94. S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidiscip. Optim.41, 495–505 (2010). [CrossRef]
  95. F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim.43, 767–784 (2011). [CrossRef]
  96. A. Ben-Tal, L. El Ghaoui, and A. S. Nemirovski, Robust Optimization (Princeton University, 2009).
  97. D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Res.58, 161–178 (2010). [CrossRef]
  98. F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B28, 387–397 (2011). [CrossRef]
  99. H. Men, R. M. Freund, N. C. Nguyen, J. Saa-Seoane, and J. Peraire, “Fabrication-adaptive optimization with an application to photonic crystal design,” arXiv:1307.5571.
  100. A. Faraon, C. Santori, Z. Huang, V. M. Acosta, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers to photonic crystal cavities in monocrystalline diamond,” Phys. Rev. Lett.109,033604 (2012). [CrossRef] [PubMed]
  101. L. Li, M. Trusheim, O. Gaathon, K. Kisslinger, C.-J. Cheng, M. Lu, D. Su, X. Yao, H.-C. Huang, I. Bayn, A. Wolcott, R. M. Osgood, and D. Englund, “Reactive ion etching: Optimized diamond membrane fabrication for transmission electron microscopy,” J. Vac. Sci. Technol. B31,06FF01 (2013). [CrossRef]
  102. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express18, A366–A380 (2010). [CrossRef] [PubMed]
  103. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U. S. A.107, 17491–17496 (2010). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited