Abstract

A critical assessment of the finite element (FE) method for studying two-dimensional dielectric photonic crystals is made. Photonic band structures, transmission coefficients, and quality factors of various two-dimensional, periodic and aperiodic, dielectric photonic crystals are calculated by using the FE (real-space) method and the plane wave expansion or the finite difference time domain (FDTD) methods and a comparison is established between those results. It is found that, contrarily to popular belief, the FE method (FEM) not only reproduces extremely well the results obtained with the standard plane wave method with regards to the eigenvalue analysis (photonic band structure and density of states calculations) but it also allows to study very easily the time-harmonic propagation of electromagnetic fields in finite clusters of arbitrary complexity and, thus, to calculate their transmission coefficients in a simple way. Moreover, the advantages of using this real space method in the context of point defect cluster quality factor calculations are also stressed by comparing the results obtained with this method with those obtained with the FDTD one. As a result of this study, FEM comes out as an stable, robust, rigorous, and reliable tool to study light propagation and confinement in both periodic and aperiodic dielectric photonic crystals and clusters.

© 2013 OSA

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2011 (1)

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011).
[CrossRef]

2010 (4)

Huang-Ming Lee and Jong-Ching Wua, “Transmittance spectra in one-dimensional superconductor-dielectric photonic crystal,” J. Appl. Phys.107, 09E1491–3, (2010).

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702, (2010).
[CrossRef]

A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett.97, 1811061–3, (2010).
[CrossRef]

R. V. Nair and R. Vijaya, “Photonic crystal sensors: an overview,” Prog. Quant. Electron.34, 89–134, (2010).
[CrossRef]

2009 (2)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82, (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555, (2009).
[CrossRef] [PubMed]

2008 (2)

T. Baba, T. Kawasaki, H. Sasaki, J. Adachi, and D. Mori, “Large delay-bandwidth product and tuning of slow light pulse in photonic crystal coupled waveguide,” Opt. Express16, 9245–9253, (2008).
[CrossRef] [PubMed]

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. Express16, 113761 (2008).
[CrossRef]

2007 (2)

Takasumi Tanabe, Masaya Notomi, Eiichi Kuramochi, Akihiko Shinya, and Hideaki Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics1, 49–52, (2007).
[CrossRef]

M. -C. Lin and R. -F. Jao, “Finite element analysis of photon density of states for two-dimensional photonic crystals with in-plane light propagation,” Opt. Express15, 207–218 (2007).
[CrossRef] [PubMed]

2006 (3)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures,” Rev. Mod. Phys.78, 455–481, (2006).
[CrossRef]

A. J. Garcia-Adeva, “Band gap atlas for photonic crystals having the symmetry of the kagome and pyrochlore lattices,” New J. Phys.8, 86/1–14 (2006).
[CrossRef]

A. J. Garcia-Adeva, “Band structure of photonic crystals with the symmetry of a pyrochlore lattice,” Phys. Rev. B.73, 0731071 (2006).
[CrossRef]

2005 (2)

2004 (2)

W. R. Frei and H. T. Johnson, “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B70, 1651161 (2004).
[CrossRef]

J. L. Garcia-Pomar and M. Nieto-Vesperinas, “Transmission study of prisms and slabs of lossy negative index media,” Opt. Express12, 2081–2095, (2004).
[CrossRef] [PubMed]

2003 (3)

C. Lopez, “Material aspects of photonic crystals,” Adv. Mater.15, 1679–1704, (2003).
[CrossRef]

S. -H. Kim and Y. -H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron.39, 1081–1085, (2003).
[CrossRef]

L. F. Shen, Z. Ye, and S. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B68, 035109 (2003).
[CrossRef]

2002 (2)

J. M. Geremia, J. Williams, and H. Mabuchi, “Inverse-problem approach to designing photonic crystals for cavity QED experiments,” Phys. Rev. E66, 066606 (2002).
[CrossRef]

D. -S. Song, S. -H. Kim, H. -G. Park, C. -K. Kim, and Y. -H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett.80, 3901–3903, (2002).
[CrossRef]

2001 (1)

2000 (1)

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000).
[CrossRef]

1999 (1)

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math.59, 2108–2120, (1999).
[CrossRef]

1998 (1)

1997 (1)

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys.107, 6756–6769, (1997). Erratum, ibid. 109, 4128 (1998).
[CrossRef]

1996 (1)

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency”, Phys. Rev. B547837–7842, (1996).
[CrossRef]

1995 (1)

Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop.43, 1460–1463, (1995).
[CrossRef]

1994 (1)

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

1993 (1)

1991 (2)

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two dimensional dielectric lattices,” Phys. Rev. Lett.67, 2017–2020, (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett.67, 2295–2299, (1991).
[CrossRef] [PubMed]

Adachi, J.

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett.97, 1811061–3, (2010).
[CrossRef]

Andonegui, I.

I. Andonegui and A. J. Garcia-Adeva, (unpublished).

Arcizet, O.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011).
[CrossRef]

Avniel, Y.

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. Express16, 113761 (2008).
[CrossRef]

Baba, T.

Bayindir, M.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000).
[CrossRef]

Benson, T. M.

C. Sibilia, T. M. Benson, M. Marciniak, and T. Szoplik, Photonic Crystals: Physics and Technology (Springer, Milano, 2008).
[CrossRef]

Berenger, J.

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

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702, (2010).
[CrossRef]

Beveratos, A.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011).
[CrossRef]

Biswas, R.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000).
[CrossRef]

Braive, R.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011).
[CrossRef]

Brent, R. P.

R. P. Brent, Algorithms for Minimization without Derivatives (Courier Dover Publications, 1973).

Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82, (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555, (2009).
[CrossRef] [PubMed]

Chan, J.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555, (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82, (2009).
[CrossRef] [PubMed]

Cox, S. J.

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math.59, 2108–2120, (1999).
[CrossRef]

Dalichaouch, R.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B10, 314–321, (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two dimensional dielectric lattices,” Phys. Rev. Lett.67, 2017–2020, (1991).
[CrossRef] [PubMed]

Dobson, D. C.

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math.59, 2108–2120, (1999).
[CrossRef]

Eichenfield, M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82, (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555, (2009).
[CrossRef] [PubMed]

Fan, S.

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency”, Phys. Rev. B547837–7842, (1996).
[CrossRef]

Frei, W. R.

W. R. Frei and H. T. Johnson, “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B70, 1651161 (2004).
[CrossRef]

Garcia-Adeva, A. J.

A. J. Garcia-Adeva, “Band gap atlas for photonic crystals having the symmetry of the kagome and pyrochlore lattices,” New J. Phys.8, 86/1–14 (2006).
[CrossRef]

A. J. Garcia-Adeva, “Band structure of photonic crystals with the symmetry of a pyrochlore lattice,” Phys. Rev. B.73, 0731071 (2006).
[CrossRef]

I. Andonegui and A. J. Garcia-Adeva, (unpublished).

Garcia-Pomar, J. L.

Gavartin, E.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011).
[CrossRef]

Geremia, J. M.

J. M. Geremia, J. Williams, and H. Mabuchi, “Inverse-problem approach to designing photonic crystals for cavity QED experiments,” Phys. Rev. E66, 066606 (2002).
[CrossRef]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett.67, 2295–2299, (1991).
[CrossRef] [PubMed]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

He, S.

L. F. Shen, Z. Ye, and S. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B68, 035109 (2003).
[CrossRef]

Hernandez-Figueroa, H. E.

Ho, K. M.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000).
[CrossRef]

Hwang, J. K.

Hyun, S. B.

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702, (2010).
[CrossRef]

Istrate, E.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures,” Rev. Mod. Phys.78, 455–481, (2006).
[CrossRef]

Jao, R. -F.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetism (Wiley–IEEE press, New York, 2002).

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702, (2010).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8, 173–190 (2001).
[CrossRef] [PubMed]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency”, Phys. Rev. B547837–7842, (1996).
[CrossRef]

Joannopoulos, J. J.

J. J. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, New Jersey, 1995).

Johnson, H. T.

W. R. Frei and H. T. Johnson, “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B70, 1651161 (2004).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702, (2010).
[CrossRef]

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. Express16, 113761 (2008).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8, 173–190 (2001).
[CrossRef] [PubMed]

Kawasaki, T.

Kim, C. -K.

D. -S. Song, S. -H. Kim, H. -G. Park, C. -K. Kim, and Y. -H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett.80, 3901–3903, (2002).
[CrossRef]

Kim, S. -H.

S. -H. Kim and Y. -H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron.39, 1081–1085, (2003).
[CrossRef]

D. -S. Song, S. -H. Kim, H. -G. Park, C. -K. Kim, and Y. -H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett.80, 3901–3903, (2002).
[CrossRef]

Kingsland, D. M.

Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop.43, 1460–1463, (1995).
[CrossRef]

Kippenberg, T. J.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011).
[CrossRef]

Koshiba, M.

Kroll, N.

Kuramochi, Eiichi

Takasumi Tanabe, Masaya Notomi, Eiichi Kuramochi, Akihiko Shinya, and Hideaki Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics1, 49–52, (2007).
[CrossRef]

Lee, Huang-Ming

Huang-Ming Lee and Jong-Ching Wua, “Transmittance spectra in one-dimensional superconductor-dielectric photonic crystal,” J. Appl. Phys.107, 09E1491–3, (2010).

Lee, J. F.

Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop.43, 1460–1463, (1995).
[CrossRef]

Lee, R.

Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop.43, 1460–1463, (1995).
[CrossRef]

Lee, Y. H.

Lee, Y. -H.

S. -H. Kim and Y. -H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron.39, 1081–1085, (2003).
[CrossRef]

D. -S. Song, S. -H. Kim, H. -G. Park, C. -K. Kim, and Y. -H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett.80, 3901–3903, (2002).
[CrossRef]

Leung, K. M.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett.67, 2295–2299, (1991).
[CrossRef] [PubMed]

Lin, M. -C.

Lopez, C.

C. Lopez, “Material aspects of photonic crystals,” Adv. Mater.15, 1679–1704, (2003).
[CrossRef]

Mabuchi, H.

J. M. Geremia, J. Williams, and H. Mabuchi, “Inverse-problem approach to designing photonic crystals for cavity QED experiments,” Phys. Rev. E66, 066606 (2002).
[CrossRef]

Mandelshtam, V. A.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys.107, 6756–6769, (1997). Erratum, ibid. 109, 4128 (1998).
[CrossRef]

Marciniak, M.

C. Sibilia, T. M. Benson, M. Marciniak, and T. Szoplik, Photonic Crystals: Physics and Technology (Springer, Milano, 2008).
[CrossRef]

McCall, S. L.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B10, 314–321, (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two dimensional dielectric lattices,” Phys. Rev. Lett.67, 2017–2020, (1991).
[CrossRef] [PubMed]

Meade, R. D.

J. J. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, New Jersey, 1995).

Mori, D.

Nair, R. V.

R. V. Nair and R. Vijaya, “Photonic crystal sensors: an overview,” Prog. Quant. Electron.34, 89–134, (2010).
[CrossRef]

Nieto-Vesperinas, M.

Notomi, Masaya

Takasumi Tanabe, Masaya Notomi, Eiichi Kuramochi, Akihiko Shinya, and Hideaki Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics1, 49–52, (2007).
[CrossRef]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702, (2010).
[CrossRef]

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. Express16, 113761 (2008).
[CrossRef]

Ozbay, E.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000).
[CrossRef]

Painter, O.

A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett.97, 1811061–3, (2010).
[CrossRef]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82, (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555, (2009).
[CrossRef] [PubMed]

Park, H. -G.

D. -S. Song, S. -H. Kim, H. -G. Park, C. -K. Kim, and Y. -H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett.80, 3901–3903, (2002).
[CrossRef]

Platzman, P. M.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B10, 314–321, (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two dimensional dielectric lattices,” Phys. Rev. Lett.67, 2017–2020, (1991).
[CrossRef] [PubMed]

Robert-Philip, I.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011).
[CrossRef]

Rodriguez-Esquerre, V. F.

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702, (2010).
[CrossRef]

Ryu, H. Y.

Sacks, Z.

Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop.43, 1460–1463, (1995).
[CrossRef]

Safavi-Naeini, A. H.

A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett.97, 1811061–3, (2010).
[CrossRef]

Sagnes, I.

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011).
[CrossRef]

Sakoda, K.

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

Sargent, E. H.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures,” Rev. Mod. Phys.78, 455–481, (2006).
[CrossRef]

Sasaki, H.

Schultz, S.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B10, 314–321, (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two dimensional dielectric lattices,” Phys. Rev. Lett.67, 2017–2020, (1991).
[CrossRef] [PubMed]

Shen, L. F.

L. F. Shen, Z. Ye, and S. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B68, 035109 (2003).
[CrossRef]

Shinya, Akihiko

Takasumi Tanabe, Masaya Notomi, Eiichi Kuramochi, Akihiko Shinya, and Hideaki Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics1, 49–52, (2007).
[CrossRef]

Sibilia, C.

C. Sibilia, T. M. Benson, M. Marciniak, and T. Szoplik, Photonic Crystals: Physics and Technology (Springer, Milano, 2008).
[CrossRef]

Sigalas, M. M.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000).
[CrossRef]

Smith, D.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two dimensional dielectric lattices,” Phys. Rev. Lett.67, 2017–2020, (1991).
[CrossRef] [PubMed]

Smith, D. R.

Song, D. -S.

D. -S. Song, S. -H. Kim, H. -G. Park, C. -K. Kim, and Y. -H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett.80, 3901–3903, (2002).
[CrossRef]

Sopaheluwakan, A.

A. Sopaheluwakan, Defect States and Defect Modes in 1D Photonic Crystals (MSc Thesis, University of Twente, 2003).

Szoplik, T.

C. Sibilia, T. M. Benson, M. Marciniak, and T. Szoplik, Photonic Crystals: Physics and Technology (Springer, Milano, 2008).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Tanabe, Takasumi

Takasumi Tanabe, Masaya Notomi, Eiichi Kuramochi, Akihiko Shinya, and Hideaki Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics1, 49–52, (2007).
[CrossRef]

Taniyama, Hideaki

Takasumi Tanabe, Masaya Notomi, Eiichi Kuramochi, Akihiko Shinya, and Hideaki Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics1, 49–52, (2007).
[CrossRef]

Taylor, H. S.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys.107, 6756–6769, (1997). Erratum, ibid. 109, 4128 (1998).
[CrossRef]

Temelkuran, B.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000).
[CrossRef]

Tuttle, G.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000).
[CrossRef]

Vahala, K. J.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555, (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82, (2009).
[CrossRef] [PubMed]

Vijaya, R.

R. V. Nair and R. Vijaya, “Photonic crystal sensors: an overview,” Prog. Quant. Electron.34, 89–134, (2010).
[CrossRef]

Villeneuve, P. R.

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency”, Phys. Rev. B547837–7842, (1996).
[CrossRef]

Vuckovic, J.

Waks, E.

Williams, J.

J. M. Geremia, J. Williams, and H. Mabuchi, “Inverse-problem approach to designing photonic crystals for cavity QED experiments,” Phys. Rev. E66, 066606 (2002).
[CrossRef]

Winger, M.

A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett.97, 1811061–3, (2010).
[CrossRef]

Winn, J. N.

J. J. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, New Jersey, 1995).

Wua, Jong-Ching

Huang-Ming Lee and Jong-Ching Wua, “Transmittance spectra in one-dimensional superconductor-dielectric photonic crystal,” J. Appl. Phys.107, 09E1491–3, (2010).

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett.67, 2295–2299, (1991).
[CrossRef] [PubMed]

Ye, Z.

L. F. Shen, Z. Ye, and S. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B68, 035109 (2003).
[CrossRef]

Zhang, L.

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. Express16, 113761 (2008).
[CrossRef]

Adv. Mater. (1)

C. Lopez, “Material aspects of photonic crystals,” Adv. Mater.15, 1679–1704, (2003).
[CrossRef]

Appl. Phys. Lett. (2)

A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett.97, 1811061–3, (2010).
[CrossRef]

D. -S. Song, S. -H. Kim, H. -G. Park, C. -K. Kim, and Y. -H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett.80, 3901–3903, (2002).
[CrossRef]

Comp. Phys. Comm. (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702, (2010).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. -H. Kim and Y. -H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron.39, 1081–1085, (2003).
[CrossRef]

IEEE Trans. Ant. Prop. (1)

Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop.43, 1460–1463, (1995).
[CrossRef]

J. Appl. Phys. (2)

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000).
[CrossRef]

Huang-Ming Lee and Jong-Ching Wua, “Transmittance spectra in one-dimensional superconductor-dielectric photonic crystal,” J. Appl. Phys.107, 09E1491–3, (2010).

J. Chem. Phys. (1)

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys.107, 6756–6769, (1997). Erratum, ibid. 109, 4128 (1998).
[CrossRef]

J. Comp. Phys. (1)

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

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

Takasumi Tanabe, Masaya Notomi, Eiichi Kuramochi, Akihiko Shinya, and Hideaki Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics1, 49–52, (2007).
[CrossRef]

Nature (2)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82, (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555, (2009).
[CrossRef] [PubMed]

New J. Phys. (1)

A. J. Garcia-Adeva, “Band gap atlas for photonic crystals having the symmetry of the kagome and pyrochlore lattices,” New J. Phys.8, 86/1–14 (2006).
[CrossRef]

Opt. Express (6)

Phys. Rev. B (3)

L. F. Shen, Z. Ye, and S. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B68, 035109 (2003).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency”, Phys. Rev. B547837–7842, (1996).
[CrossRef]

W. R. Frei and H. T. Johnson, “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B70, 1651161 (2004).
[CrossRef]

Phys. Rev. B. (1)

A. J. Garcia-Adeva, “Band structure of photonic crystals with the symmetry of a pyrochlore lattice,” Phys. Rev. B.73, 0731071 (2006).
[CrossRef]

Phys. Rev. E (1)

J. M. Geremia, J. Williams, and H. Mabuchi, “Inverse-problem approach to designing photonic crystals for cavity QED experiments,” Phys. Rev. E66, 066606 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett.67, 2295–2299, (1991).
[CrossRef] [PubMed]

E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two dimensional dielectric lattices,” Phys. Rev. Lett.67, 2017–2020, (1991).
[CrossRef] [PubMed]

Prog. Quant. Electron. (1)

R. V. Nair and R. Vijaya, “Photonic crystal sensors: an overview,” Prog. Quant. Electron.34, 89–134, (2010).
[CrossRef]

Rev. Mod. Phys. (1)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures,” Rev. Mod. Phys.78, 455–481, (2006).
[CrossRef]

SIAM J. Appl. Math. (1)

S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math.59, 2108–2120, (1999).
[CrossRef]

Other (14)

I. Andonegui and A. J. Garcia-Adeva, (unpublished).

J. J. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, New Jersey, 1995).

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

MPB on-line manual, http://ab-initio.mit.edu/wiki/index.php/MPB\_manual .

J. Jin, The Finite Element Method in Electromagnetism (Wiley–IEEE press, New York, 2002).

A. Sopaheluwakan, Defect States and Defect Modes in 1D Photonic Crystals (MSc Thesis, University of Twente, 2003).

C. Sibilia, T. M. Benson, M. Marciniak, and T. Szoplik, Photonic Crystals: Physics and Technology (Springer, Milano, 2008).
[CrossRef]

R. P. Brent, Algorithms for Minimization without Derivatives (Courier Dover Publications, 1973).

Elmer – Finite Element Software for Multiphysical Problems, http://www.csc.fi/elmer/index.phtml .

The unofficial numerical electromagnetic code archives, http://www.si-list.org/swindex2.html .

The EMAP Finite Element Modeling Codes, http://www.emclab.umr.edu/emap.html .

Comsol multiphysics and Electromagnetics module, http://www.comsol.com .

T. A. Davis, UMFPACK 4.6: Unsymmetric MultiFrontal sparse LU factorization package, http://www.cise.ufl.edu/research/sparse/umfpack/ .

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

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Figures (9)

Fig. 1
Fig. 1

On the left: band structure calculated with the MPB (blue circles) and COMSOL® (red line) software packages for TE-polarized EM waves. On the right: Ez patterns in the unit cell calculated with MPB (upper row in each rectangle) and COMSOL® at the special symmetry points Γ, X, and M.

Fig. 2
Fig. 2

Left panel: Transmittance for TE-polarized waves propagating in the ΓM direction of the hexagonal photonic crystal cluster depicted in the inset. Central panel: Comparison between the band structure calculated with COMSOL® (red line) and MPB (blue circles) for TE-polarized EM waves along the boundary of the irreducible part of the 1BZ. The inset shows the unit cell used for the calculations. Right panel: Transmittance for TE-polarized waves propagating in the ΓK direction of the hexagonal photonic crystal cluster depicted in the inset. Inside the green rectangle: Comparison between the Ez field patterns at the M point calculated with MPB (on the left) and FEM (on the right) for the first eight bands (the band index increases from bottom to top). A portion of the photonic crystal that contains 3 ×3 unit cells is displayed in order to show the hexagonal symmetry of the modes.

Fig. 3
Fig. 3

(a) Evolution of the relative error with the number of elements in which the lattice is discretized. (b) Evolution of the simulation run time with the number of elements in which the lattice is discretized.

Fig. 4
Fig. 4

(a) On the left: Band structure of a 5 ×5 square lattice periodic supercell calculated with MPB (blue circles) and FEM (red lines), wherein a defect state has been excited around ω a 2 π c = 0.466 ±0.002. On the right: density of photonic states (DOS), calculated by FEM. The band gap region is clearly distinguishable and a weakly localized defect mode merging from the upper band can be seen. In the inset, Ez patterns for a 5 × 5 supercell calculated by both methods are shown for the M symmetry point. (b) On the left: band structure of a 5×5 triangular lattice calculated with MPB (blue circles) and FEM (red line). A single defect state merges at ω a 2 π c = 0.285 ±0.003. On the right: density of photonic states, calculated by FEM. In the inset, Ez patterns for the Γ symmetry point. An hexagonal supercell has been used for the FEM calculation.

Fig. 5
Fig. 5

(a) On the left: band structure of a 5 ×5 square lattice periodic supercell calculated with MPB (blue circles) and FEM (red lines). The rod radii has been set to 0.17a, which ensures a better localization of the unique defect mode. On the right: density of photonic states, calculated by FEM. The defect mode clearly shows up in the gap region, since the defect state is strongly confined. In the inset, the Ez patterns for a 5 ×5 supercell calculated with both methods are shown for the M symmetry point. All these patterns match the ones presented in Fig. 4(a) but the modal volume decreases significantly for 0.17a rod lattices. (b) Triangular lattice made of 0.17a rods where the central rod has been removed. The band gap increases and so it does the defect mode localization around the point defect.

Fig. 6
Fig. 6

(a) Transmittance curves for different cluster sizes in a rectangular arrangement of dielectric rods of radius 0.38a with a central defect. (b) Analogous to (a) for radius 0.17a. The insets show a detailed view of the defect mode curves indicated by the red arrow. Logarithmic scale has been used for the vertical axis.

Fig. 7
Fig. 7

Analogous to figure 6 but for triangular arrangements of dielectric rods with a central defect. The radii of the rods are (a) 0.38a and (b) 0.17a.

Fig. 8
Fig. 8

Top: quality factor estimation as a function of the Lorentzian shaped resonance FWHM for a number of different sized square clusters of dielectric rods with radii (a) 0.38a and (b) 0.17a. The FWHM value is given in dimensionless normalized units of a/λ. Logarithmic scale has been used for both axes. Down: analogous computation results obtained for triangular arrangements of rods with radii (c) 0.38a and (d) 0.17a.

Fig. 9
Fig. 9

(a) evolution of the electromagnetic field inside the defect rod for a 0.38a arrangement and for different cluster dimensions. A point source is excited in the cavity but after some periods the source is extinguished and although the electromagnetic field still remains in the cavity it experiences an exponential decay. (b) analogous computation results obtained for triangular defect cluster.

Tables (1)

Tables Icon

Table 1 Quality factor calculations for different cluster sizes in square and triangular lattice rod-type PC single-defect cavities using FEM, FDTD, and harmonic inversion methods.

Equations (16)

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2 E z ( r ) + k 0 2 r ( r ) E z ( r ) = 0 ,
E ( r , t ) = E z ( r ) exp ( i ω t ) z ^
H ( r , t ) = i k 0 × r ( r ) E ( r , t ) .
( H z ( r ) r ( r ) ) k 0 2 H z ( r ) = 0 ,
H ( r , t ) = H z ( r ) exp ( i ω t ) z ^
E ( r , t ) = i k 0 ( r ) × H ( r , t ) .
n ^ × H = 0 ,
n ^ × E = 0 .
E z ( r + R ) = exp ( i k R ) E z ( r )
H z ( r + R ) = exp ( i k R ) H z ( r )
z ^ n ^ × ( × E z z ^ ) i β E z = 2 i β E 0 z
z ^ n ^ × ( × H z z ^ ) i β H z = 2 i β H 0 z ,
z n ^ × ( × E z z ^ ) i k 0 E z = i k 0 ( 1 k 0 n ^ ) E 0 z e i k 0 r
z n ^ × ( × H z z ^ ) i k 0 H z = i k 0 ( 1 k 0 n ^ ) H 0 z e i k 0 r ,
Q = f c FWHM ,
Q = 2 π | E ( t ) | 2 | E ( t ) | 2 | E ( t + T ) | 2 .

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