Abstract

We present a formulation to analyze photonic periodic structures from viewpoints of sources and gain. The approach is based on a generalized eigenvalue problem and mode expansions of sources which sustain optical fields with phase boundary conditions. Using this scheme, we calculate power spectra, dispersion relations, and quality factors of Bloch modes in one-dimensional periodic structures consisting of dielectrics or metals. We also compare the results calculated from this scheme with those from the complex-frequency method. The outcomes of these two approaches generally agree well and only deviate slightly in the regime of low quality factors.

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    [CrossRef]
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    [CrossRef]
  37. A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron.39, 1253–1272 (2007).
    [CrossRef]
  38. S. W. Chang, “Full frequency-domain approach to reciprocal microlasers and nanolasers-perspective from Lorentz reciprocity,” Opt. Express19, 21116–21134 (2011).
    [CrossRef] [PubMed]
  39. S. W. Chang, “Confinement factors and modal volumes of micro- and nanocavities invariant to integration regions,” IEEE J. Sel. Top. Quantum. Electron.18, 1771–1780 (2012).
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  43. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley and Sons, 1989).
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    [CrossRef]

2012

A. G. Hanif, T. Arima, and T. Uno, “Finite-difference frequency-domain algorithm for band-diagram calculation of 2-D photonic crystals composed of Debye-type dispersive materials,” IEEE Antennas Wireless Propag. Lett.11, 41–44 (2012).
[CrossRef]

V. Siahpoush, T. Søndergaard, and J. Jung, “Green’s function approach to investigate the excitation of surface plasmon polaritons in a nanometer-thin metal film,” Phys. Rev. B85, 075305 (2012).
[CrossRef]

S. W. Chang, “Confinement factors and modal volumes of micro- and nanocavities invariant to integration regions,” IEEE J. Sel. Top. Quantum. Electron.18, 1771–1780 (2012).
[CrossRef]

2011

2010

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B81, 125431 (2010).
[CrossRef]

2009

W. Cai, R. Sainidou, J. Xu, A. Polman, and F. J. G. de Abajo, “Efficient generation of propagating plasmons by electron beams,” Nano Lett.9, 1176–1181 (2009).
[CrossRef] [PubMed]

2007

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron.39, 1253–1272 (2007).
[CrossRef]

A. D. Yaghjian, “Bidirectionality of reciprocal, lossy or lossless, uniform or periodic waveguides,” IEEE Microw. Wireless Compon. Lett.17, 480–482 (2007).
[CrossRef]

2006

R. L. Chern, C. C. Chang, and C. C. Chang, “Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions,” Phys. Rev. E73, 036605 (2006).
[CrossRef]

S. I. Bozhevolnyi and V. M. Shalaev, “Nanophotonics with surface plasmons–part I,” Photonics Spectra, Jan. Issue, 58–66 (2006).

V. M. Shalaev and S. I. Bozhevolnyi, “Nanophotonics with surface plasmons–part II,” Photonics Spectra, Feb. Issue, 66–73 (2006).

2005

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

C. C. Chang, R. L. Chern, C. C. Chang, and R. R. Hwang, “Interfacial operator approach to computing modes of surface plasmon polaritons for periodic structures,” Phys. Rev. B72, 205112 (2005).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-d modeling with accurate 2-d analysis,” IEEE J. Sel. Top. Quantum Electron.11, 1135–1142 (2005).
[CrossRef]

2004

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron.36, 213–221 (2004).
[CrossRef]

D. Pissoort and F. Olyslager, “Study of eigenmodes in periodic waveguides using the lorentz reciprocity theorem,” IEEE Trans. Microw. Theory Techn.52, 542–553 (2004).
[CrossRef]

Y. Cao, Z. Hou, and Y. Liu, “Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals,” Solid State Commun.132, 539–543 (2004).
[CrossRef]

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

2003

2002

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B65, 155120 (2002).
[CrossRef]

2001

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett.86, 3008–3011 (2001).
[CrossRef] [PubMed]

T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A64, 033812 (2001).
[CrossRef]

2000

Y. Tanaka, Y. Tomoyasu, and S. Tamura, “Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch,” Phys. Rev. B62, 7387–7392 (2000).
[CrossRef]

1999

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999).
[CrossRef] [PubMed]

1998

1997

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B55, 7427–7444 (1997).
[CrossRef]

1996

F. Olyslager, “Properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides,” IEEE Trans. Microw. Theory Techn.44, 2064–2075 (1996).
[CrossRef]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

1994

P. G. Petropoulos, “Phase error control for FD-TD methods of second and fourth order accuracy,” IEEE Trans. Antennas Propag.42, 859–862 (1994).
[CrossRef]

1987

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

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

1977

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys.22, 17–23 (1977).

1951

A. Kocabas, S. S. Senlik, and A. Aydinli, “Plasmonic band gap cavities on biharmonic gratings,” Phys. Rev. B77, 195130 (2008).

1924

J. R. Carson, “A generalization of reciprocal theorem,” Bell System Technical Journal3, 393–399 (1924).

1896

H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd.4, 176–187 (1896).

1850

F. Wu, D. Han, X. Hu, X. Liu, and J. Zi, “Complete surface plasmon-polariton band gap and gap-governed waveguiding, bending and splitting,” J. Phys.: Condens. Matter21, 185010 (2009).

Abbas, M. N.

Albin, S.

Arima, T.

A. G. Hanif, T. Arima, and T. Uno, “Finite-difference frequency-domain algorithm for band-diagram calculation of 2-D photonic crystals composed of Debye-type dispersive materials,” IEEE Antennas Wireless Propag. Lett.11, 41–44 (2012).
[CrossRef]

Aydinli, A.

A. Kocabas, S. S. Senlik, and A. Aydinli, “Plasmonic band gap cavities on biharmonic gratings,” Phys. Rev. B77, 195130 (2008).

Baek, J. H.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley and Sons, 1989).

Barnes, W. L.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

Benson, R. M.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron.39, 1253–1272 (2007).
[CrossRef]

Benson, T. M.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-d modeling with accurate 2-d analysis,” IEEE J. Sel. Top. Quantum Electron.11, 1135–1142 (2005).
[CrossRef]

Boriskina, S. V.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron.39, 1253–1272 (2007).
[CrossRef]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi and V. M. Shalaev, “Nanophotonics with surface plasmons–part I,” Photonics Spectra, Jan. Issue, 58–66 (2006).

V. M. Shalaev and S. I. Bozhevolnyi, “Nanophotonics with surface plasmons–part II,” Photonics Spectra, Feb. Issue, 66–73 (2006).

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett.86, 3008–3011 (2001).
[CrossRef] [PubMed]

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).

Cai, W.

W. Cai, R. Sainidou, J. Xu, A. Polman, and F. J. G. de Abajo, “Efficient generation of propagating plasmons by electron beams,” Nano Lett.9, 1176–1181 (2009).
[CrossRef] [PubMed]

Cao, Y.

Y. Cao, Z. Hou, and Y. Liu, “Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals,” Solid State Commun.132, 539–543 (2004).
[CrossRef]

Carson, J. R.

J. R. Carson, “A generalization of reciprocal theorem,” Bell System Technical Journal3, 393–399 (1924).

Chang, C. C.

R. L. Chern, C. C. Chang, and C. C. Chang, “Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions,” Phys. Rev. E73, 036605 (2006).
[CrossRef]

R. L. Chern, C. C. Chang, and C. C. Chang, “Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions,” Phys. Rev. E73, 036605 (2006).
[CrossRef]

C. C. Chang, R. L. Chern, C. C. Chang, and R. R. Hwang, “Interfacial operator approach to computing modes of surface plasmon polaritons for periodic structures,” Phys. Rev. B72, 205112 (2005).
[CrossRef]

C. C. Chang, R. L. Chern, C. C. Chang, and R. R. Hwang, “Interfacial operator approach to computing modes of surface plasmon polaritons for periodic structures,” Phys. Rev. B72, 205112 (2005).
[CrossRef]

Chang, S. W.

S. W. Chang, “Confinement factors and modal volumes of micro- and nanocavities invariant to integration regions,” IEEE J. Sel. Top. Quantum. Electron.18, 1771–1780 (2012).
[CrossRef]

S. W. Chang, “Full frequency-domain approach to reciprocal microlasers and nanolasers-perspective from Lorentz reciprocity,” Opt. Express19, 21116–21134 (2011).
[CrossRef] [PubMed]

Chang, Y. C.

Chen, Y.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B81, 125431 (2010).
[CrossRef]

Cheng, C. W.

Chern, R. L.

R. L. Chern, C. C. Chang, and C. C. Chang, “Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions,” Phys. Rev. E73, 036605 (2006).
[CrossRef]

C. C. Chang, R. L. Chern, C. C. Chang, and R. R. Hwang, “Interfacial operator approach to computing modes of surface plasmon polaritons for periodic structures,” Phys. Rev. B72, 205112 (2005).
[CrossRef]

Chutinan, A.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999).
[CrossRef] [PubMed]

de Abajo, F. J. G.

W. Cai, R. Sainidou, J. Xu, A. Polman, and F. J. G. de Abajo, “Efficient generation of propagating plasmons by electron beams,” Nano Lett.9, 1176–1181 (2009).
[CrossRef] [PubMed]

Erland, J.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett.86, 3008–3011 (2001).
[CrossRef] [PubMed]

Erni, D.

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B65, 155120 (2002).
[CrossRef]

Evans, B. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

Giovannini, H.

Gregersen, N.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B81, 125431 (2010).
[CrossRef]

Guo, S.

Hafner, C.

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B65, 155120 (2002).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Arctech House, 2005), 3rd ed.

Han, D.

F. Wu, D. Han, X. Hu, X. Liu, and J. Zi, “Complete surface plasmon-polariton band gap and gap-governed waveguiding, bending and splitting,” J. Phys.: Condens. Matter21, 185010 (2009).

Hanif, A. G.

A. G. Hanif, T. Arima, and T. Uno, “Finite-difference frequency-domain algorithm for band-diagram calculation of 2-D photonic crystals composed of Debye-type dispersive materials,” IEEE Antennas Wireless Propag. Lett.11, 41–44 (2012).
[CrossRef]

Hibbins, A. P.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

Hou, Z.

Y. Cao, Z. Hou, and Y. Liu, “Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals,” Solid State Commun.132, 539–543 (2004).
[CrossRef]

Hu, X.

F. Wu, D. Han, X. Hu, X. Liu, and J. Zi, “Complete surface plasmon-polariton band gap and gap-governed waveguiding, bending and splitting,” J. Phys.: Condens. Matter21, 185010 (2009).

Hvam, J. M.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett.86, 3008–3011 (2001).
[CrossRef] [PubMed]

Hwang, R. R.

C. C. Chang, R. L. Chern, C. C. Chang, and R. R. Hwang, “Interfacial operator approach to computing modes of surface plasmon polaritons for periodic structures,” Phys. Rev. B72, 205112 (2005).
[CrossRef]

Imada, M.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008), 2nd ed.

John, S.

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

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008), 2nd ed.

Ju, Y. G.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

Jung, J.

V. Siahpoush, T. Søndergaard, and J. Jung, “Green’s function approach to investigate the excitation of surface plasmon polaritons in a nanometer-thin metal film,” Phys. Rev. B85, 075305 (2012).
[CrossRef]

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999).
[CrossRef] [PubMed]

Kim, S. B.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

Kim, S. H.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

Kitson, S. C.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

Kocabas, A.

A. Kocabas, S. S. Senlik, and A. Aydinli, “Plasmonic band gap cavities on biharmonic gratings,” Phys. Rev. B77, 195130 (2008).

Kuzmiak, V.

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B55, 7427–7444 (1997).
[CrossRef]

Kwon, S. H.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999).
[CrossRef] [PubMed]

Lee, Y. H.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

Lemarchand, F.

Leosson, K.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett.86, 3008–3011 (2001).
[CrossRef] [PubMed]

Liu, X.

F. Wu, D. Han, X. Hu, X. Liu, and J. Zi, “Complete surface plasmon-polariton band gap and gap-governed waveguiding, bending and splitting,” J. Phys.: Condens. Matter21, 185010 (2009).

Liu, Y.

Y. Cao, Z. Hou, and Y. Liu, “Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals,” Solid State Commun.132, 539–543 (2004).
[CrossRef]

Lodahl, P.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B81, 125431 (2010).
[CrossRef]

Lorentz, H. A.

H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd.4, 176–187 (1896).

Maradudin, A. A.

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B55, 7427–7444 (1997).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008), 2nd ed.

Merzbacher, E.

E. Merzbacher, Quantum Mechanics (Wiley and Sons, New York, 1998), 3rd ed.

Mochizuki, M.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

Moreno, E.

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B65, 155120 (2002).
[CrossRef]

Mørk, J.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B81, 125431 (2010).
[CrossRef]

Nielsen, T. R.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B81, 125431 (2010).
[CrossRef]

Noda, S.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

Nosich, A. I.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron.39, 1253–1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-d modeling with accurate 2-d analysis,” IEEE J. Sel. Top. Quantum Electron.11, 1135–1142 (2005).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron.36, 213–221 (2004).
[CrossRef]

O’Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999).
[CrossRef] [PubMed]

Olyslager, F.

D. Pissoort and F. Olyslager, “Study of eigenmodes in periodic waveguides using the lorentz reciprocity theorem,” IEEE Trans. Microw. Theory Techn.52, 542–553 (2004).
[CrossRef]

F. Olyslager, “Properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides,” IEEE Trans. Microw. Theory Techn.44, 2064–2075 (1996).
[CrossRef]

Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999).
[CrossRef] [PubMed]

Park, H. G.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

Petropoulos, P. G.

P. G. Petropoulos, “Phase error control for FD-TD methods of second and fourth order accuracy,” IEEE Trans. Antennas Propag.42, 859–862 (1994).
[CrossRef]

Pissoort, D.

D. Pissoort and F. Olyslager, “Study of eigenmodes in periodic waveguides using the lorentz reciprocity theorem,” IEEE Trans. Microw. Theory Techn.52, 542–553 (2004).
[CrossRef]

Polman, A.

W. Cai, R. Sainidou, J. Xu, A. Polman, and F. J. G. de Abajo, “Efficient generation of propagating plasmons by electron beams,” Nano Lett.9, 1176–1181 (2009).
[CrossRef] [PubMed]

Preist, T. W.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

Russell, P.

P. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003).
[CrossRef] [PubMed]

Sainidou, R.

W. Cai, R. Sainidou, J. Xu, A. Polman, and F. J. G. de Abajo, “Efficient generation of propagating plasmons by electron beams,” Nano Lett.9, 1176–1181 (2009).
[CrossRef] [PubMed]

Sambles, J. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

Scherer, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999).
[CrossRef] [PubMed]

Senlik, S. S.

A. Kocabas, S. S. Senlik, and A. Aydinli, “Plasmonic band gap cavities on biharmonic gratings,” Phys. Rev. B77, 195130 (2008).

Sentenac, A.

Sewell, P.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron.39, 1253–1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-d modeling with accurate 2-d analysis,” IEEE J. Sel. Top. Quantum Electron.11, 1135–1142 (2005).
[CrossRef]

Shalaev, V. M.

V. M. Shalaev and S. I. Bozhevolnyi, “Nanophotonics with surface plasmons–part II,” Photonics Spectra, Feb. Issue, 66–73 (2006).

S. I. Bozhevolnyi and V. M. Shalaev, “Nanophotonics with surface plasmons–part I,” Photonics Spectra, Jan. Issue, 58–66 (2006).

Shih, M. H.

Siahpoush, V.

V. Siahpoush, T. Søndergaard, and J. Jung, “Green’s function approach to investigate the excitation of surface plasmon polaritons in a nanometer-thin metal film,” Phys. Rev. B85, 075305 (2012).
[CrossRef]

Skliarov, O. P.

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys.22, 17–23 (1977).

Skovgaard, P. M. W.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett.86, 3008–3011 (2001).
[CrossRef] [PubMed]

Smotrova, E. I.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron.39, 1253–1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-d modeling with accurate 2-d analysis,” IEEE J. Sel. Top. Quantum Electron.11, 1135–1142 (2005).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron.36, 213–221 (2004).
[CrossRef]

Søndergaard, T.

V. Siahpoush, T. Søndergaard, and J. Jung, “Green’s function approach to investigate the excitation of surface plasmon polaritons in a nanometer-thin metal film,” Phys. Rev. B85, 075305 (2012).
[CrossRef]

T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A64, 033812 (2001).
[CrossRef]

Stockman, M. I.

M. I. Stockman, “Electromagnetic theory of SERS,” in Surface-Enhanced Raman Scattering, Topics in Applied Physics, K. Kneipp, M. Moskovits, and H. Kneipp, Eds. (Springer-Verlag, 2006). 47–65.
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Arctech House, 2005), 3rd ed.

Tamura, S.

Y. Tanaka, Y. Tomoyasu, and S. Tamura, “Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch,” Phys. Rev. B62, 7387–7392 (2000).
[CrossRef]

Tanaka, Y.

Y. Tanaka, Y. Tomoyasu, and S. Tamura, “Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch,” Phys. Rev. B62, 7387–7392 (2000).
[CrossRef]

Tomoyasu, Y.

Y. Tanaka, Y. Tomoyasu, and S. Tamura, “Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch,” Phys. Rev. B62, 7387–7392 (2000).
[CrossRef]

Tromborg, B.

T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A64, 033812 (2001).
[CrossRef]

Uno, T.

A. G. Hanif, T. Arima, and T. Uno, “Finite-difference frequency-domain algorithm for band-diagram calculation of 2-D photonic crystals composed of Debye-type dispersive materials,” IEEE Antennas Wireless Propag. Lett.11, 41–44 (2012).
[CrossRef]

Vlasov, A. G.

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys.22, 17–23 (1977).

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008), 2nd ed.

Wu, F.

F. Wu, D. Han, X. Hu, X. Liu, and J. Zi, “Complete surface plasmon-polariton band gap and gap-governed waveguiding, bending and splitting,” J. Phys.: Condens. Matter21, 185010 (2009).

Xu, J.

W. Cai, R. Sainidou, J. Xu, A. Polman, and F. J. G. de Abajo, “Efficient generation of propagating plasmons by electron beams,” Nano Lett.9, 1176–1181 (2009).
[CrossRef] [PubMed]

Yablonovitch, E.

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

Yaghjian, A. D.

A. D. Yaghjian, “Bidirectionality of reciprocal, lossy or lossless, uniform or periodic waveguides,” IEEE Microw. Wireless Compon. Lett.17, 480–482 (2007).
[CrossRef]

Yang, J. K.

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999).
[CrossRef] [PubMed]

Yokoyama, M.

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

Zi, J.

F. Wu, D. Han, X. Hu, X. Liu, and J. Zi, “Complete surface plasmon-polariton band gap and gap-governed waveguiding, bending and splitting,” J. Phys.: Condens. Matter21, 185010 (2009).

Bell System Technical Journal

J. R. Carson, “A generalization of reciprocal theorem,” Bell System Technical Journal3, 393–399 (1924).

IEEE Antennas Wireless Propag. Lett.

A. G. Hanif, T. Arima, and T. Uno, “Finite-difference frequency-domain algorithm for band-diagram calculation of 2-D photonic crystals composed of Debye-type dispersive materials,” IEEE Antennas Wireless Propag. Lett.11, 41–44 (2012).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-d modeling with accurate 2-d analysis,” IEEE J. Sel. Top. Quantum Electron.11, 1135–1142 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum. Electron.

S. W. Chang, “Confinement factors and modal volumes of micro- and nanocavities invariant to integration regions,” IEEE J. Sel. Top. Quantum. Electron.18, 1771–1780 (2012).
[CrossRef]

IEEE Microw. Wireless Compon. Lett.

A. D. Yaghjian, “Bidirectionality of reciprocal, lossy or lossless, uniform or periodic waveguides,” IEEE Microw. Wireless Compon. Lett.17, 480–482 (2007).
[CrossRef]

IEEE Trans. Antennas Propag.

P. G. Petropoulos, “Phase error control for FD-TD methods of second and fourth order accuracy,” IEEE Trans. Antennas Propag.42, 859–862 (1994).
[CrossRef]

IEEE Trans. Microw. Theory Techn.

F. Olyslager, “Properties of and generalized full-wave transmission line models for hybrid (bi)(an)isotropic waveguides,” IEEE Trans. Microw. Theory Techn.44, 2064–2075 (1996).
[CrossRef]

D. Pissoort and F. Olyslager, “Study of eigenmodes in periodic waveguides using the lorentz reciprocity theorem,” IEEE Trans. Microw. Theory Techn.52, 542–553 (2004).
[CrossRef]

J. Phys.: Condens. Matter

F. Wu, D. Han, X. Hu, X. Liu, and J. Zi, “Complete surface plasmon-polariton band gap and gap-governed waveguiding, bending and splitting,” J. Phys.: Condens. Matter21, 185010 (2009).

Nano Lett.

W. Cai, R. Sainidou, J. Xu, A. Polman, and F. J. G. de Abajo, “Efficient generation of propagating plasmons by electron beams,” Nano Lett.9, 1176–1181 (2009).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron.36, 213–221 (2004).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron.39, 1253–1272 (2007).
[CrossRef]

Photonics Spectra

S. I. Bozhevolnyi and V. M. Shalaev, “Nanophotonics with surface plasmons–part I,” Photonics Spectra, Jan. Issue, 58–66 (2006).

V. M. Shalaev and S. I. Bozhevolnyi, “Nanophotonics with surface plasmons–part II,” Photonics Spectra, Feb. Issue, 66–73 (2006).

Phys. Rev. A

T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A64, 033812 (2001).
[CrossRef]

Phys. Rev. B

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B81, 125431 (2010).
[CrossRef]

V. Siahpoush, T. Søndergaard, and J. Jung, “Green’s function approach to investigate the excitation of surface plasmon polaritons in a nanometer-thin metal film,” Phys. Rev. B85, 075305 (2012).
[CrossRef]

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B65, 155120 (2002).
[CrossRef]

Y. Tanaka, Y. Tomoyasu, and S. Tamura, “Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch,” Phys. Rev. B62, 7387–7392 (2000).
[CrossRef]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B54, 6227–6244 (1996).
[CrossRef]

A. Kocabas, S. S. Senlik, and A. Aydinli, “Plasmonic band gap cavities on biharmonic gratings,” Phys. Rev. B77, 195130 (2008).

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B55, 7427–7444 (1997).
[CrossRef]

C. C. Chang, R. L. Chern, C. C. Chang, and R. R. Hwang, “Interfacial operator approach to computing modes of surface plasmon polaritons for periodic structures,” Phys. Rev. B72, 205112 (2005).
[CrossRef]

Phys. Rev. E

R. L. Chern, C. C. Chang, and C. C. Chang, “Analysis of surface plasmon modes and band structures for plasmonic crystals in one and two dimensions,” Phys. Rev. E73, 036605 (2006).
[CrossRef]

Phys. Rev. Lett.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett.86, 3008–3011 (2001).
[CrossRef] [PubMed]

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

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

Radio. Eng. Electron. Phys.

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys.22, 17–23 (1977).

Science

P. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003).
[CrossRef] [PubMed]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284, 1819–1821 (1999).
[CrossRef] [PubMed]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science293, 1123–1125 (2001).
[CrossRef] [PubMed]

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science305, 1444–1447 (2004).
[CrossRef] [PubMed]

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science308, 670–672 (2005).
[CrossRef] [PubMed]

Solid State Commun.

Y. Cao, Z. Hou, and Y. Liu, “Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals,” Solid State Commun.132, 539–543 (2004).
[CrossRef]

Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd.

H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd.4, 176–187 (1896).

Other

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley and Sons, 1989).

E. Merzbacher, Quantum Mechanics (Wiley and Sons, New York, 1998), 3rd ed.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Arctech House, 2005), 3rd ed.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

M. I. Stockman, “Electromagnetic theory of SERS,” in Surface-Enhanced Raman Scattering, Topics in Applied Physics, K. Kneipp, M. Moskovits, and H. Kneipp, Eds. (Springer-Verlag, 2006). 47–65.
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008), 2nd ed.

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Figures (7)

Fig. 1
Fig. 1

The schematic diagram of the unit cell Ωuc in a generic periodic structure. The active region and its complement are denoted as Ωa and Ωb, respectively.

Fig. 2
Fig. 2

The unit cell of a 1D periodic bilayer structure. The widths of Ωa, Ωb, and the whole unit cell are d1, d2 and a = d1 + d2, respectively.

Fig. 3
Fig. 3

(a) The behaviors of |ωΔεr,n0(ω)| as a function of the normalized frequency for TE modes with l = 2 and 3. The periodic structure is composed of nondispersive and lossless dielectric/dielectric bilayers. (b) The band diagrams from the loci of minima of |ωΔεr,nk(ω)| at ky = 0 (sold lines) and counterparts from the complex-ω method (red circles). For clear illustrations, the magnitudes |ωΔεr,nk(ω)| increase as the curves bend into the figure.

Fig. 4
Fig. 4

Comparisons between band diagrams of the two lowest-order TE and TM modes using the proposed formulation (solid lines) and complex-ω method (red circles) at ky = π/(2a) in the periodic structure of nondispersive and lossless dielectric/dielectric bilayers.

Fig. 5
Fig. 5

(a) Band diagrams, (b) Q factors Qnk versus kxa/π, and (c) quasi-3D views of Pnk(ω) as a function of kxa/π. The periodic structure is composed of nondispersive but lossy dielectric/metal bilayers. The propagation constant ky is set to zero.

Fig. 6
Fig. 6

(a) The band diagrams and (b) quasi-3D views of Pnk(ω) of the three lowest TE modes in the periodic structure of dispersive and lossy dielectric/metal bilayers. The propagation constant ky is set to zero.

Fig. 7
Fig. 7

(a) The band diagrams for the three lowest-order TM modes and (b) lateral 3D views of Pnk(ω) for TM modes with l = 2 and 3. The periodic structure is composed of dispersive and lossy dielectric/metal bilayers. The x component of the electric-field magnitude at the BZ center and boundary are also shown in (a). The propagation constant ky is set to 0.9π/a.

Tables (1)

Tables Icon

Table 1 Comparisons between resonance frequencies as well as Q factors, and a list of Δεr,n0(ωn0) for the four lowest TE modes at the BZ center in Fig. 5.

Equations (33)

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× × E ( r ) = ( ω b c ) 2 ε ¯ ¯ ( r ) E ( r ) , E ( r ) = e i k r F ( r ) ,
ε ¯ ¯ r ( r + R m ) = ε ¯ ¯ r ( r ) ; F ( r + R m ) = F ( r ) , m = 1 3 ,
× × E k ( r ) ( ω c ) 2 ε ¯ ¯ r ( r , ω ) E k ( r ) = i ω μ 0 J s , k ( r ) = 0 , r Ω a ,
J s , k ( r + R m ) = e i k R m J s , k ( r ) , m = 1 3.
E k ( r ) n c n k ( ω ) f n k ( r , ω ) ,
J s , k ( r ) n c n k ( ω ) j s , n k ( r , ω ) .
j s , n k ( r , ω ) = i ω ε 0 Δ ε r , n k ( ω ) U ( r ) f n k ( r , ω ) ,
U ( r + R m ) = U ( r ) , m = 1 3 ,
× × f n k ( r , ω ) ( ω c ) 2 ε ¯ ¯ r ( r , ω ) f n k ( r , ω ) = ( ω c ) 2 Δ ε r , n k ( ω ) U ( r ) f n k ( r , ω ) ,
f n k ( r + R m , ω ) = e i k R m f n k ( r , ω ) , m = 1 3.
f n k ( r , ω ) = e i k r ψ n k ( r , ω ) , ψ n k ( r + R m , ω ) = ψ n k ( r , ω ) , m = 1 3 ,
j s , n k ( r , ω ) = e i k r ς s , n k ( r , ω ) , ς s , n k ( r + R m , ω ) = ς s , n k ( r , ω ) , m = 1 3 ,
ς s , n k ( r , ω ) = i ω ε 0 Δ ε r , n k ( ω ) U ( r ) ψ n k ( r , ω ) ,
J s , k ( r ) = a ( ω ) j s , n k ( r , ω ) ,
𝒥 2 V a = Ω a d r | J s , k ( r ) | 2 = | a ( ω ) 2 ε 0 2 ω Δ ε r , n k ( ω ) | 2 Ω a d r | f n k ( r , ω ) | 2 ,
P n k ( ω ) = 1 2 Ω a d r Re [ J s , k * ( r ) E k ( r ) ] = | a ( ω ) | 2 2 Ω a d r Re [ j s , n k * ( r , ω ) f n k ( r , ω ) ] = 𝒥 2 V a 2 ε 0 Im [ 1 ω Δ ε r , n k ( ω ) ] .
Q n k = i 2 Δ ε r , n k ( ω n k ) [ ω Δ ε r , n k ( ω ) ] ω | ω = ω n k .
2 ϕ n k ( x ) x 2 + { ( ω c ) 2 [ ε a ( ω ) + Δ ε r , n k ( ω ) ] k y 2 } ϕ n k ( x ) = 0 , x Ω a ,
2 ϕ n k ( x ) x 2 + [ ( ω c ) 2 ε b ( ω ) k y 2 ] ϕ n k ( x ) = 0 , x Ω b ,
ϕ n k ( x ) = { A e i M x + B e i M x , x Ω a , C e i N x + D e i N x , x Ω b , { M 2 = ( ω / c ) 2 [ ε a ( ω ) + Δ ε r , n k ( ω ) ] k y 2 , N 2 = ( ω / c ) 2 ε b ( ω ) k y 2 ,
( 0 0 0 0 ) = ( 1 1 1 1 M N M N τ τ e i ( k x a M d 1 ) e i ( k x a + M d 1 ) e i N d 2 e i N d 2 M N e i ( k x a M d 1 ) M N e i ( k x a + M d 1 ) τ e i N d 2 τ e i N d 2 ) ( A B C D ) ,
τ = { [ ε a ( ω ) + Δ ε r , n k ( ω ) ] / ε b ( ω ) ( TM ) , 1 ( TE ) .
cos ( k x a ) = cos ( M d 1 ) cos ( N d 2 ) + 1 2 ( M τ N + τ N M ) sin ( M d 1 ) sin ( N d 2 ) .
g n k ( r , ω ) = 1 i ω μ 0 × f n k ( r , ω ) e i k r φ n k ( r , ω ) ,
φ n k ( r + R m , ω ) = φ n k ( r ) , m = 1 3 ,
× ψ n k ( r , ω ) + i k × ψ n k ( r , ω ) = i ω μ 0 φ n k ( r , ω ) ,
× φ n k ( r , ω ) + i k × φ n k ( r , ω ) = i ω ε 0 ε ¯ ¯ r ( r , ω ) ψ n k ( r , ω ) + ς s , n k ( r , ω ) .
S uc d a [ ψ n k ( r , ω ) × φ n k ( r , ω ) ψ n k ( r , ω ) × φ n k ( r , ω ) ] = Ω a d r i ( k + k ) [ ψ n k ( r , ω ) × φ n k ( r , ω ) ψ n k ( r , ω ) × φ n k ( r , ω ) ] Ω uc d r [ ψ n k ( r , ω ) ς s , n k ( r , ω ) ψ n k ( r , ω ) ς s , n k ( r , ω ) ] .
0 = Ω uc d r [ ψ n k ( r , ω ) ς s , n k ( r , ω ) ψ n k ( r , ω ) ς s , n k ( r , ω ) ] = i ω ε 0 [ Δ ε r , n k ( ω ) Δ ε r , n k ( ω ) ] Ω a d r ψ n k ( r , ω ) ψ n k ( r , ω ) .
Ω a d r ψ n k ( r , ω ) ψ n k ( r , ω ) = Ω a d r f n k ( r , ω ) f n k ( r , ω ) δ n n Λ n k ( ω ) ,
Ω a d r ς s , n k ( r , ω ) ς s , n k ( r , ω ) = Ω a d r j s , n k ( r , ω ) j s , n k ( r , ω ) δ n n Θ n k ( r , ω ) ,
Θ n k ( ω ) = [ ω ε 0 Δ ε r , n k ( ω ) ] 2 Λ n k ( ω ) ,
c n k ( ω ) = 1 Θ n k ( ω ) Ω a d r e i k r J s , k ( r ) ς s , n k ( r , ω ) = 1 Θ n k ( ω ) Ω a d r J s , k ( r ) j s , n k ( r , ω ) .

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