Abstract

We present numerical studies of two photonic crystal membrane microcavities, a short line-defect cavity with a relatively low quality (Q) factor and a longer cavity with a high Q. We use five state-of-the-art numerical simulation techniques to compute the cavity Q factor and the resonance wavelength λ for the fundamental cavity mode in both structures. For each method, the relevant computational parameters are systematically varied to estimate the computational uncertainty. We show that some methods are more suitable than others for treating these challenging geometries.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (1)

2016 (2)

W. Xue, Y. Yu, L. Ottaviano, Y. Chen, E. Semenova, K. Yvind, and J. Mørk, “Threshold characteristics of slow-light photonic crystal lasers,” Phys. Rev. Lett. 116, 063901 (2016).
[Crossref] [PubMed]

A. Taghizadeh, J. Mørk, and I.-S. Chung, “Numerical investigation of vertical cavity lasers with high-contrast gratings using the Fourier modal method,” J. Light. Technol. 34, 4240–4251 (2016).
[Crossref]

2015 (1)

P. Lodahl, S. Mahmoodian, and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures,” Rev. Mod. Phys. 87, 347–400 (2015).
[Crossref]

2014 (2)

2013 (3)

2012 (1)

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]

2011 (3)

D. M. Shyroki, “Modeling of sloped interfaces on a Yee grid,” IEEE Trans. Antennas Propag. 59, 3290–3295 (2011).
[Crossref]

A. M. Ivinskaya, A. V. Lavrinenko, and D. M. Shyroki, “Modeling of nanophotonic resonators with the finite-difference frequency-domain method,” IEEE Trans. Antennas Propag. 59, 4155–4161 (2011).
[Crossref]

M. Notomi, A. Shinya, K. Nozaki, T. Tanabe, S. Matsuo, E. Kuramochi, T. Sato, H. Taniyama, and H. Sumikura, “Low-power nanophotonic devices based on photonic crystals towards dense photonic network on chip,” IET Circuits, Devices Syst. 5, 84–93 (2011).
[Crossref]

2010 (4)

M. Okano, T. Yamada, J. Sugisaka, N. Yamamoto, M. Itoh, T. Sugaya, K. Komori, and M. Mori, “Analysis of two-dimensional photonic crystal L-type cavities with low-refractive-index material cladding,” J. Opt. 12, 075101 (2010).
[Crossref]

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
[Crossref]

P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photon. Rev. 4, 499–516 (2010).
[Crossref]

D. M. Shyroki, A. M. Ivinskaya, and A. V. Lavrinenko, “Free-space squeezing assists perfectly matched layers in simulations on a tight domain,” IEEE Antennas Wirel. Propag. Lett. 9, 389–392 (2010).
[Crossref]

2008 (3)

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[Crossref]

P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser Photon. Rev. 2, 514–526 (2008).
[Crossref]

A. V. Boriskin, S. V. Boriskina, A. Rolland, R. Sauleau, and A. I. Nosich, “Test of the FDTD accuracy in the analysis of the scattering resonances associated with high-Q whispering-gallery modes of a circular cylinder,” J. Opt. Soc. Am. A 25, 1169–1173 (2008).
[Crossref]

2007 (8)

F. Bordas, M. J. Steel, C. Seassal, and A. Rahmani, “Confinement of band-edge modes in a photonic crystal slab,” Opt. Express 15, 10890–10902 (2007).
[Crossref] [PubMed]

G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express 15, 11042–11060 (2007).
[Crossref] [PubMed]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref] [PubMed]

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[Crossref]

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99, 023902 (2007).
[Crossref]

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” phys. status solidi (b) 244, 3419–3434 (2007).
[Crossref]

D. M. Shyroki and A. V. Lavrinenko, “Perfectly matched layer method in the finite-difference time-domain and frequency-domain calculations,” Phys. Status Solidi (b) 244, 3506–3514 (2007).
[Crossref]

2006 (1)

G. Granet and L. Li, “Convincingly converged results for highly conducting periodically perforated thin films with square symmetry,” J. Opt. A Pure Appl. Opt. 8, 546–549 (2006).
[Crossref]

2005 (2)

C. Sauvan, P. Lalanne, and J. P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev. B 71, 165118 (2005).
[Crossref]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[Crossref]

2004 (2)

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[Crossref] [PubMed]

E. Jørgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higher order hierarchical Legendre basis functions for electromagnetic modeling,” IEEE Trans. Antennas Propag. 52, 2985–2995 (2004).
[Crossref]

2003 (3)

Z.-Y. Li and K.-M. Ho, “Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides,” Phys. Rev. B 68, 245117 (2003).
[Crossref]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

T. F. Krauss, “Planar photonic crystal waveguide devices for integrated optics,” Phys. Status Solidi (a) 197, 688–702 (2003).
[Crossref]

2001 (2)

2000 (1)

1997 (1)

1996 (1)

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

1992 (1)

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[Crossref]

1975 (1)

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 23, 623–630 (1975).
[Crossref]

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.  69, 681 (1946).

1928 (1)

R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen differenzengleichungen der mathematischen physik,” Math. Ann. 100, 32–74 (1928).
[Crossref]

Akahane, Y.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

Asano, T.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

Atatüre, M.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref] [PubMed]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[Crossref]

Badolato, A.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref] [PubMed]

Bai, Q.

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

Bordas, F.

Boriskin, A. V.

Boriskina, S. V.

Breinbjerg, O.

E. Jørgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higher order hierarchical Legendre basis functions for electromagnetic modeling,” IEEE Trans. Antennas Propag. 52, 2985–2995 (2004).
[Crossref]

O. S. Kim, E. Jørgensen, P. Meincke, and O. Breinbjerg, “Higher-Order Hierarchical Legendre Basis Functions in Applications,” “The Fourth Swedish Conference on Computational Electromagnetics,” (Lund University, 2007), pp. 239–246.

Brodwin, M. E.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 23, 623–630 (1975).
[Crossref]

Burger, S.

B. Maes, J. Petráček, S. Burger, P. Kwiecien, J. Luksch, and I. Richter, “Simulations of high-Q optical nanocavities with a gradual 1D bandgap,” Opt. Express 21, 6794–6806 (2013).
[Crossref] [PubMed]

J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” phys. status solidi (b) 244, 3419–3434 (2007).
[Crossref]

S. Burger, J. Pomplun, F. Schmidt, and L. Zschiedrich, “Finite-element method simulations of high-Q nanocavities with 1D photonic bandgap,” Proc. SPIE7933, 79330T (2011).
[Crossref]

S. Burger, P. Gutsche, M. Hammerschmidt, S. Herrmann, J. Pomplun, F. Schmidt, B. Wohlfeil, and L. Zschiedrich, “Hp-finite-elements for simulating electromagnetic fields in optical devices with rough textures,” Proc. SPIE9630, 96300S (2015).
[Crossref]

S. Burger and L. Zschiedrich, “Numerical investigation of photonic crystal microcavities in silicon-on-insulator waveguides,” Proc. SPIE7609, 76091Q (2010).
[Crossref]

Busch, K.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
[Crossref]

Cao, Q.

Chen, Y.

W. Xue, Y. Yu, L. Ottaviano, Y. Chen, E. Semenova, K. Yvind, and J. Mørk, “Threshold characteristics of slow-light photonic crystal lasers,” Phys. Rev. Lett. 116, 063901 (2016).
[Crossref] [PubMed]

Chung, I.-S.

A. Taghizadeh, J. Mørk, and I.-S. Chung, “Numerical investigation of vertical cavity lasers with high-contrast gratings using the Fourier modal method,” J. Light. Technol. 34, 4240–4251 (2016).
[Crossref]

Courant, R.

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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]

Shinya, A.

M. Notomi, A. Shinya, K. Nozaki, T. Tanabe, S. Matsuo, E. Kuramochi, T. Sato, H. Taniyama, and H. Sumikura, “Low-power nanophotonic devices based on photonic crystals towards dense photonic network on chip,” IET Circuits, Devices Syst. 5, 84–93 (2011).
[Crossref]

Shyroki, D. M.

D. M. Shyroki, “Modeling of sloped interfaces on a Yee grid,” IEEE Trans. Antennas Propag. 59, 3290–3295 (2011).
[Crossref]

A. M. Ivinskaya, A. V. Lavrinenko, and D. M. Shyroki, “Modeling of nanophotonic resonators with the finite-difference frequency-domain method,” IEEE Trans. Antennas Propag. 59, 4155–4161 (2011).
[Crossref]

D. M. Shyroki, A. M. Ivinskaya, and A. V. Lavrinenko, “Free-space squeezing assists perfectly matched layers in simulations on a tight domain,” IEEE Antennas Wirel. Propag. Lett. 9, 389–392 (2010).
[Crossref]

D. M. Shyroki and A. V. Lavrinenko, “Perfectly matched layer method in the finite-difference time-domain and frequency-domain calculations,” Phys. Status Solidi (b) 244, 3506–3514 (2007).
[Crossref]

Silberstein, E.

Søndergaard, T.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC, 2014).

Song, B.-S.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
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Sorensen, D. C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, Philadelphia, 1998).
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Sözüer, H. S.

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[Crossref]

Steel, M. J.

Stobbe, S.

P. Lodahl, S. Mahmoodian, and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures,” Rev. Mod. Phys. 87, 347–400 (2015).
[Crossref]

Strauss, M.

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
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Sugaya, T.

M. Okano, T. Yamada, J. Sugisaka, N. Yamamoto, M. Itoh, T. Sugaya, K. Komori, and M. Mori, “Analysis of two-dimensional photonic crystal L-type cavities with low-refractive-index material cladding,” J. Opt. 12, 075101 (2010).
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M. Okano, T. Yamada, J. Sugisaka, N. Yamamoto, M. Itoh, T. Sugaya, K. Komori, and M. Mori, “Analysis of two-dimensional photonic crystal L-type cavities with low-refractive-index material cladding,” J. Opt. 12, 075101 (2010).
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M. Notomi, A. Shinya, K. Nozaki, T. Tanabe, S. Matsuo, E. Kuramochi, T. Sato, H. Taniyama, and H. Sumikura, “Low-power nanophotonic devices based on photonic crystals towards dense photonic network on chip,” IET Circuits, Devices Syst. 5, 84–93 (2011).
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A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method(Artech House, 2004), 3rd ed.

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A. Taghizadeh, J. Mørk, and I.-S. Chung, “Numerical investigation of vertical cavity lasers with high-contrast gratings using the Fourier modal method,” J. Light. Technol. 34, 4240–4251 (2016).
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Tanabe, T.

M. Notomi, A. Shinya, K. Nozaki, T. Tanabe, S. Matsuo, E. Kuramochi, T. Sato, H. Taniyama, and H. Sumikura, “Low-power nanophotonic devices based on photonic crystals towards dense photonic network on chip,” IET Circuits, Devices Syst. 5, 84–93 (2011).
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Taniyama, H.

M. Notomi, A. Shinya, K. Nozaki, T. Tanabe, S. Matsuo, E. Kuramochi, T. Sato, H. Taniyama, and H. Sumikura, “Low-power nanophotonic devices based on photonic crystals towards dense photonic network on chip,” IET Circuits, Devices Syst. 5, 84–93 (2011).
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P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
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P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
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E. Jørgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higher order hierarchical Legendre basis functions for electromagnetic modeling,” IEEE Trans. Antennas Propag. 52, 2985–2995 (2004).
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K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
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P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
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M. Okano, T. Yamada, J. Sugisaka, N. Yamamoto, M. Itoh, T. Sugaya, K. Komori, and M. Mori, “Analysis of two-dimensional photonic crystal L-type cavities with low-refractive-index material cladding,” J. Opt. 12, 075101 (2010).
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Yamamoto, N.

M. Okano, T. Yamada, J. Sugisaka, N. Yamamoto, M. Itoh, T. Sugaya, K. Komori, and M. Mori, “Analysis of two-dimensional photonic crystal L-type cavities with low-refractive-index material cladding,” J. Opt. 12, 075101 (2010).
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Yvind, K.

W. Xue, Y. Yu, L. Ottaviano, Y. Chen, E. Semenova, K. Yvind, and J. Mørk, “Threshold characteristics of slow-light photonic crystal lasers,” Phys. Rev. Lett. 116, 063901 (2016).
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J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive finite element method for simulation of optical nano structures,” phys. status solidi (b) 244, 3419–3434 (2007).
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S. Burger, J. Pomplun, F. Schmidt, and L. Zschiedrich, “Finite-element method simulations of high-Q nanocavities with 1D photonic bandgap,” Proc. SPIE7933, 79330T (2011).
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S. Burger, P. Gutsche, M. Hammerschmidt, S. Herrmann, J. Pomplun, F. Schmidt, B. Wohlfeil, and L. Zschiedrich, “Hp-finite-elements for simulating electromagnetic fields in optical devices with rough textures,” Proc. SPIE9630, 96300S (2015).
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S. Burger and L. Zschiedrich, “Numerical investigation of photonic crystal microcavities in silicon-on-insulator waveguides,” Proc. SPIE7609, 76091Q (2010).
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IEEE J. Quantum Electron. (1)

N. Gregersen, S. Reitzenstein, C. Kistner, M. Strauss, C. Schneider, S. Höfling, L. Worschech, A. Forchel, T. R. Nielsen, J. Mørk, and J.-M. Gérard, “Numerical and experimental study of the Q factor of high-Q micropillar cavities,” IEEE J. Quantum Electron. 46, 1470–1483 (2010).
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IEEE Trans. Antennas Propag. (4)

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
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D. M. Shyroki, “Modeling of sloped interfaces on a Yee grid,” IEEE Trans. Antennas Propag. 59, 3290–3295 (2011).
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E. Jørgensen, J. L. Volakis, P. Meincke, and O. Breinbjerg, “Higher order hierarchical Legendre basis functions for electromagnetic modeling,” IEEE Trans. Antennas Propag. 52, 2985–2995 (2004).
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M. Notomi, A. Shinya, K. Nozaki, T. Tanabe, S. Matsuo, E. Kuramochi, T. Sato, H. Taniyama, and H. Sumikura, “Low-power nanophotonic devices based on photonic crystals towards dense photonic network on chip,” IET Circuits, Devices Syst. 5, 84–93 (2011).
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A. Taghizadeh, J. Mørk, and I.-S. Chung, “Numerical investigation of vertical cavity lasers with high-contrast gratings using the Fourier modal method,” J. Light. Technol. 34, 4240–4251 (2016).
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J. Opt. (1)

M. Okano, T. Yamada, J. Sugisaka, N. Yamamoto, M. Itoh, T. Sugaya, K. Komori, and M. Mori, “Analysis of two-dimensional photonic crystal L-type cavities with low-refractive-index material cladding,” J. Opt. 12, 075101 (2010).
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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
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Opt. Express (6)

Opt. Lett. (1)

Phys. Rep. (1)

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007).
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V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
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G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99, 023902 (2007).
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W. Xue, Y. Yu, L. Ottaviano, Y. Chen, E. Semenova, K. Yvind, and J. Mørk, “Threshold characteristics of slow-light photonic crystal lasers,” Phys. Rev. Lett. 116, 063901 (2016).
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D. M. Shyroki and A. V. Lavrinenko, “Perfectly matched layer method in the finite-difference time-domain and frequency-domain calculations,” Phys. Status Solidi (b) 244, 3506–3514 (2007).
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Rev. Mod. Phys. (1)

P. Lodahl, S. Mahmoodian, and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures,” Rev. Mod. Phys. 87, 347–400 (2015).
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[Crossref]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method(Artech House, 2004), 3rd ed.

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L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012), chap. 8, pp. 224–281.
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A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC, 2014).

S. Burger, P. Gutsche, M. Hammerschmidt, S. Herrmann, J. Pomplun, F. Schmidt, B. Wohlfeil, and L. Zschiedrich, “Hp-finite-elements for simulating electromagnetic fields in optical devices with rough textures,” Proc. SPIE9630, 96300S (2015).
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Figures (10)

Fig. 1
Fig. 1 PhC membrane geometry (black) and electric field Ey profile for the M1 mode in the L9 cavity in the y = 0 (top) and the z = 0 (bottom) plane corresponding to the center of the cavity and membrane, respectively. The parameters Dx, Dy and Dz are the computational distances from the membrane to the perfectly matched layer (dark grey).
Fig. 2
Fig. 2 Resonance wavelength (red, left axis) and Q factor (black, right axis) as function of the cavity Ln computed using the SIE approach. In all cases, the cavity is surrounded by 6 holes as in our L9 configuration. The solid lines are curve fits to guide the eye.
Fig. 3
Fig. 3 Discretization schemes. (a) FDTD/FDFD: (Upper) Representative cutouts of the photonic crystal structures defined by air holes (white) in InP (black) material for different spatial mesh (red grid) resolutions. (Lower) The meshed material distributions for the corresponding cutouts. (b) FEM: Mesh of the PhC membrane L5 geometry used in the sFEM simulations with h = 130 nm. Due to symmetry planes, only one eighth of the full device is required with corresponding boundary conditions set on the grey faces. The InP PhC membrane (blue) is perforated with an in-plane triangular lattice of air cylinders (orange). The mesh consists solely of prisms in the interior and additional bricks in the exterior PML domain (not shown). Edge and face ansatz functions based on Jacobi polynomials are shown for degrees p = 1, 2, 3 and 4 above the mesh. (c) Staircase discretization scheme employed by the aFMM. 33 layers are used per unit cell. (d) SIE: Accurate representation of the geometry of the PhC membrane with higher-order quadrilateral elements.
Fig. 4
Fig. 4 Nonuniform discretization mesh in the FDFD simulations. The discretization step decreases towards the center of the cavity.
Fig. 5
Fig. 5 L5: (a) Resonance wavelength λ, (b) relative wavelength deviation, (c) Q factor, (d) relative Q factor deviation, (e) computation time and (f) memory requirement as function of resolution setup. Notice the disconnected y axis in (c).
Fig. 6
Fig. 6 L5: Relative (a) resonance wavelength and (b) Q factor deviations as function of domain setup.
Fig. 7
Fig. 7 L9: (a) Resonance wavelength λ, (b) relative wavelength deviation, (c) Q factor, (d) relative Q factor deviation, (e) computation time and (f) memory requirement as function of resolution setup. Notice the disconnected y axis in (a) and (c).
Fig. 8
Fig. 8 L9: Relative (a) resonance wavelength deviations and (b) Q factor deviations as function of domain setup.
Fig. 9
Fig. 9 Error bar plot of the resonance wavelength and the Q factor (a,b) for the L5 cavity and (c,d) for the L9 cavity. The green horizontal lines mark average values computed from the FDTD, FDFD, FEM and SIE results. Notice the disconnected y axis in (b) and (d).
Fig. 10
Fig. 10 L5, tFEM: Q and λ (a) as function of mesh size for p = 2 and (b) as function of polynomial order p for resolution setup 4. The (a) data is identical to those in Fig. 5.

Tables (7)

Tables Icon

Table 1 Parameters of PhC membranes.

Tables Icon

Table 2 Best estimates of M1 wavelengths and Q factors for each computational method.

Tables Icon

Table 3 Overview of typical computational resources for the L9 calculations.

Tables Icon

Table 4 Data for the numerical resolution study for the L5 cavity.

Tables Icon

Table 5 Data for the domain size study for the L5 cavity.

Tables Icon

Table 6 Data for the numerical resolution study for the L9 cavity.

Tables Icon

Table 7 Data for the domain size study for the L9 cavity.

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