Abstract

We present a general, rigorous, modal formalism for modeling light propagation and light emission in three-dimensional (3D) periodic waveguides and in aggregates of them. In essence, the formalism is a generalization of well-known modal concepts for translation-invariant waveguides to situations involving stacks of periodic waveguides. By surrounding the actual stack by perfectly-matched layers (PMLs) in the transverse directions, reciprocity considerations lead to the derivation of Bloch-mode orthogonality relations in the sense of E×H products, to the normalization of these modes, and to the proof of the symmetrical property of the scattering matrix linking the Bloch modes. The general formalism, which rigorously takes into account radiation losses resulting from the excitation of radiation Bloch modes, is implemented with a Fourier numerical approach. Basic examples of light scattering like reflection, transmission and emission in periodic-waveguides are accurately resolved.

© 2007 Optical Society of America

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  29. D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photon. Nanostruct. Fundam. Appl. 3, 120–128 (2005).
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  32. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  37. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
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    [CrossRef]
  40. S. F. Helfert, “Numerical stable determination of Floquet-modes and the application to the computation of band structures,” Opt. Quantum Electron. 36, 87–107 (2004).
    [CrossRef]
  41. C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  44. G. Lecamp, J. P. Hugonin, and P. Lalanne, “Remarkably large spontaneous emission β-factor in photonic crystal waveguides,” Phys. Rev. Lett. 99, 023902 (2007).
    [CrossRef] [PubMed]
  45. A. Baudrion, J. Weeber, A. Dereux, G. Lecamp, P. Lalanne, and S. Bozhevolnyi, “Influence of the filling factor on the spectral properties of plasmonic crystals,” Phys. Rev. B. 74, 125406 (2006).
    [CrossRef]
  46. J. C. Chen and K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
    [CrossRef]

2007 (1)

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

2006 (4)

A. Baudrion, J. Weeber, A. Dereux, G. Lecamp, P. Lalanne, and S. Bozhevolnyi, “Influence of the filling factor on the spectral properties of plasmonic crystals,” Phys. Rev. B. 74, 125406 (2006).
[CrossRef]

K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
[CrossRef]

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[CrossRef]

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

2005 (8)

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]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photon. Nanostruct. Fundam. Appl. 3, 120–128 (2005).
[CrossRef]

S. F. Helfert, “Determination of Floquet modes in asymmetric periodic structures,” Opt. Quantum Electron. 37, 185–197 (2005).
[CrossRef]

C. Ciminelli, F. Peluso, and M. N. Armenise, “Modeling and design of two-dimensional guided-wave photonic band-gap devices,” J. Lightwave Technol. 23, 886–901 (2005).
[CrossRef]

G. Lecamp, P. Lalanne, J. P. Hugonin, and J. M. Gerard, “Energy transfer through laterally confined Bragg mirrors and its impact on pillar microcavities,” IEEE J. Quantum Electron. 41, 1323–1329 (2005).
[CrossRef]

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

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]

2004 (7)

S. F. Helfert, “Numerical stable determination of Floquet-modes and the application to the computation of band structures,” Opt. Quantum Electron. 36, 87–107 (2004).
[CrossRef]

P. Bienstman, “Two-stage mode finder for waveguides with a 2D cross-section,” Opt. Quantum Electron. 36, 5–14 (2004).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mat. 3, 211–219 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

S. Hughes, “Enhanced single-photon emission from quantum dots in photonic crystal waveguides and nanocavities,” Opt. Lett. 29, 2659–2661 (2004).
[CrossRef] [PubMed]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

2003 (3)

P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high-Q, small-V microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
[CrossRef]

J. M. Elson, “Scattering losses from planar waveguides with material inhomogeneity,” Waves Random Media 13, 95–105 (2003).
[CrossRef]

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

2002 (5)

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

B. Gralak, S. Enoch, and G. Tayeb, “From scattering or impedance matrices to Bloch modes of photonic crystals,” J. Opt. Soc. Am. A 19, 1547–1554 (2002).
[CrossRef]

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Q. Cao, P. Lalanne, and J. P. Hugonin, “Stable and efficient Bloch-mode computational method for onedimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002).
[CrossRef]

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

2001 (3)

2000 (3)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[CrossRef]

1999 (1)

S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

1996 (1)

1995 (3)

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

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

J. C. Chen and K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
[CrossRef]

1994 (1)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

1991 (1)

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[CrossRef]

Absil, P. P.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Andreani, L. C.

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[CrossRef]

D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photon. Nanostruct. Fundam. Appl. 3, 120–128 (2005).
[CrossRef]

Armenise, M. N.

Asatryan, A. A.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Baba, T.

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[CrossRef]

Baets, R.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Baudrion, A.

A. Baudrion, J. Weeber, A. Dereux, G. Lecamp, P. Lalanne, and S. Bozhevolnyi, “Influence of the filling factor on the spectral properties of plasmonic crystals,” Phys. Rev. B. 74, 125406 (2006).
[CrossRef]

Bienstman, P.

P. Bienstman, “Two-stage mode finder for waveguides with a 2D cross-section,” Opt. Quantum Electron. 36, 5–14 (2004).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

P. Bienstman, Rigorous and efficient modeling of wavelength scale photonic components (Gent University PhD thesis in English, 2001).

Bogaerts, W.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Botten, L. C.

K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Bozhevolnyi, S.

A. Baudrion, J. Weeber, A. Dereux, G. Lecamp, P. Lalanne, and S. Bozhevolnyi, “Influence of the filling factor on the spectral properties of plasmonic crystals,” Phys. Rev. B. 74, 125406 (2006).
[CrossRef]

Byrne, M. A.

K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
[CrossRef]

Campillo, A. J.

R. K. Chang and A. J. Campillo, Optical processes in microcavities (World Scientific London, 1996).
[CrossRef]

Cao, Q.

Chang, R. K.

R. K. Chang and A. J. Campillo, Optical processes in microcavities (World Scientific London, 1996).
[CrossRef]

Chen, J. C.

J. C. Chen and K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
[CrossRef]

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Chu, S. T.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Chutinan, A.

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[CrossRef]

Ciminelli, C.

De Mesel, K.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

de Sterke, C. M.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Dereux, A.

A. Baudrion, J. Weeber, A. Dereux, G. Lecamp, P. Lalanne, and S. Bozhevolnyi, “Influence of the filling factor on the spectral properties of plasmonic crystals,” Phys. Rev. B. 74, 125406 (2006).
[CrossRef]

Dossou, K.

K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
[CrossRef]

Elson, J. M.

J. M. Elson, “Scattering losses from planar waveguides with material inhomogeneity,” Waves Random Media 13, 95–105 (2003).
[CrossRef]

Enoch, S.

Fan, S. H.

S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Gaylord, T. K.

Gerace, D.

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[CrossRef]

D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photon. Nanostruct. Fundam. Appl. 3, 120–128 (2005).
[CrossRef]

Gerard, J. M.

G. Lecamp, P. Lalanne, J. P. Hugonin, and J. M. Gerard, “Energy transfer through laterally confined Bragg mirrors and its impact on pillar microcavities,” IEEE J. Quantum Electron. 41, 1323–1329 (2005).
[CrossRef]

Gill, D.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Gralak, B.

Grann, E. B.

Hamann, H. F.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

Hamano, T.

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[CrossRef]

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Helfert, S. F.

S. F. Helfert, “Determination of Floquet modes in asymmetric periodic structures,” Opt. Quantum Electron. 37, 185–197 (2005).
[CrossRef]

S. F. Helfert, “Numerical stable determination of Floquet-modes and the application to the computation of band structures,” Opt. Quantum Electron. 36, 87–107 (2004).
[CrossRef]

Hryniewicz, J. V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Hughes, S.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

S. Hughes, “Enhanced single-photon emission from quantum dots in photonic crystal waveguides and nanocavities,” Opt. Lett. 29, 2659–2661 (2004).
[CrossRef] [PubMed]

Hugonin, J. P.

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

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]

G. Lecamp, P. Lalanne, J. P. Hugonin, and J. M. Gerard, “Energy transfer through laterally confined Bragg mirrors and its impact on pillar microcavities,” IEEE J. Quantum Electron. 41, 1323–1329 (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]

P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high-Q, small-V microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
[CrossRef]

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Q. Cao, P. Lalanne, and J. P. Hugonin, “Stable and efficient Bloch-mode computational method for onedimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002).
[CrossRef]

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

Ibanescu, M.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Iga, K.

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[CrossRef]

Joannopoulos, J. D.

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mat. 3, 211–219 (2004).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

J. D. Joannopoulos, R. Meade, and J. Winn, Photonic crystals: molding the flow of light (Princeton University Press NJ, 1995).

Johnson, F. G.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Johnson, S. G.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Kim, W. J.

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

King, O.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Kingsland, D. M.

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

Kolodziejski, L. A.

S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Koyama, F.

T. Baba, T. Hamano, F. Koyama, and K. Iga, “Spontaneous emission factor of a microcavity DBR surface-emitting laser,” IEEE J. Quantum Electron. 27, 1347–1358 (1991).
[CrossRef]

Krauss, T. F.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Kuang, W.

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

Lalanne, P.

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

A. Baudrion, J. Weeber, A. Dereux, G. Lecamp, P. Lalanne, and S. Bozhevolnyi, “Influence of the filling factor on the spectral properties of plasmonic crystals,” Phys. Rev. B. 74, 125406 (2006).
[CrossRef]

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]

G. Lecamp, P. Lalanne, J. P. Hugonin, and J. M. Gerard, “Energy transfer through laterally confined Bragg mirrors and its impact on pillar microcavities,” IEEE J. Quantum Electron. 41, 1323–1329 (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]

P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high-Q, small-V microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
[CrossRef]

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

Q. Cao, P. Lalanne, and J. P. Hugonin, “Stable and efficient Bloch-mode computational method for onedimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002).
[CrossRef]

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

Langtry, T. N.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Lecamp, G.

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

A. Baudrion, J. Weeber, A. Dereux, G. Lecamp, P. Lalanne, and S. Bozhevolnyi, “Influence of the filling factor on the spectral properties of plasmonic crystals,” Phys. Rev. B. 74, 125406 (2006).
[CrossRef]

G. Lecamp, P. Lalanne, J. P. Hugonin, and J. M. Gerard, “Energy transfer through laterally confined Bragg mirrors and its impact on pillar microcavities,” IEEE J. Quantum Electron. 41, 1323–1329 (2005).
[CrossRef]

Lee, J. F.

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

Lee, R.

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

Lee, R. K.

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[CrossRef]

Li, K.

J. C. Chen and K. Li, “Quartic perfectly matched layers for dielectric waveguides and gratings,” Microwave Opt. Technol. Lett. 10, 319–323 (1995).
[CrossRef]

Li, L. F.

Lidorikis, E.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall NY, 1983).

McNab, S. J.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

McOrist, J.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Meade, R.

J. D. Joannopoulos, R. Meade, and J. Winn, Photonic crystals: molding the flow of light (Princeton University Press NJ, 1995).

Mock, A.

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

Moerman, I.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Moharam, M. G.

Nicorovici, N. A.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Noda, S.

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[CrossRef]

Notomi, M.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

O’Boyle, M.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

O’Brien, J.

W. Kuang, W. J. Kim, A. Mock, and J. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron. 12, 1183–1195 (2006).
[CrossRef]

Peluso, F.

Pommet, D. A.

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Ramunno, L.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Rodier, J. C.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Sacks, Z. S.

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

Sauvan, C.

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]

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[CrossRef]

Seiferth, E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Shinya, A.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Silberstein, E.

Sipe, J. E.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Skorobogatiy, M. A.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall NY, 1983).

Soljacic, M.

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mat. 3, 211–219 (2004).
[CrossRef]

Taillaert, D.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Takahashi, C.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Takahashi, J.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Talneau, A.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Tayeb, G.

Trakalo, M.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Van, V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Van Daele, P.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Vassallo, C.

C. Vassallo, Optical waveguide concepts (Elsevier Amsterdam, 1991).

Verstuyft, S.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

Weeber, J.

A. Baudrion, J. Weeber, A. Dereux, G. Lecamp, P. Lalanne, and S. Bozhevolnyi, “Influence of the filling factor on the spectral properties of plasmonic crystals,” Phys. Rev. B. 74, 125406 (2006).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

White, T. P.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Winn, J.

J. D. Joannopoulos, R. Meade, and J. Winn, Photonic crystals: molding the flow of light (Princeton University Press NJ, 1995).

Xu, Y.

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[CrossRef]

Yamada, K.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Yariv, A.

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[CrossRef]

Yokohama, I.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Young, J. F.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
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IEEE J. Quantum Electron. (5)

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Figures (7)

Fig. 1.
Fig. 1.

Actual and computational geometries considered in this work. (a) Sketch of a geometry composed of aggregate of different periodic-waveguide sections. The geometry is assumed to be surrounded by infinite uniform claddings in the x- and y-transverse directions. (b) Associated computational system obtained by bounding the actual waveguide with PMLs (in blue) in the transverse directions. (c) A periodic-waveguide section of (a). (d) Associated periodic waveguide bound with PMLs. In (b) and (d), the PMLs have a finite thickness that is not represented for the sake of clarity.

Fig. 2.
Fig. 2.

The two solutions used for deriving the S-matrix reciprocity relation. At a given frequency ω, the QNBMs of the left periodic-waveguide section (z<L1), labelled by “L” for “left”, are denoted by Φ (p,L) and those on the right side (z>L2), labelled by “R” for “right”, by Φ (p,R), p being a relative integer. The geometries are arbitrary and may contain sources for the left and right ends of the structure and for L1<z<L2. The whole system is assumed to be surrounded by PMLs (not shown) everywhere in the transverse directions.

Fig. 3.
Fig. 3.

Some implications of Eq. 21. (a) The complex modal transmission-coefficients do not depend on the propagation sense. (b) Same property for the cross modal reflection-coefficients. (c) The excitation amplitude of a QNBM by a dipole source J δ(r-r0) located at point r=r0 is equal to the scalar product between the source J and the field E(r0) scattered at the dipole location by exciting the same geometry with the reciprocal QNBM.

Fig. 4.
Fig. 4.

Excitation of QNBMs by a Dirac dipole source J δ(r-r 0) located at point r=r 0. The D(p,R) and D(p,L) coefficients represent the modal amplitude coefficients of the excited forward- and backward-QNBMs, respectively. The periodic waveguide is not necessarily symmetric for the study, as shown by the échelette profile.

Fig. 5.
Fig. 5.

QNBM calculation of a PhC waveguide. (a) Schematic view of the PhC waveguide formed by removing a line defect in the ΓK direction of a 2D PhC structure composed of a triangular lattice of air holes (lattice constant a=0.24 µm) etched into a silicon slab (n=3.55). The slab thickness is 0.6a and the air holes radii 0.29a. The inset shows the dispersion relation of the fundamental guided QNBM Φ(-1). (b) Display of the 300-first normalized propagation constants of the QNBMs for a frequency a/λ=0.255, point A in the inset. Blue dots and red squares are obtained for (fPML)-1=(1+i) and (fPML)-1=5(1+i), respectively.

Fig. 6.
Fig. 6.

Scattering at the interface between two periodic sections. (a) Schematic top view of the 3D scattering problem. The PhC parameters are the same as in the caption of Fig. 5. (b) Convergence of the a-FMM for the modal reflectivity R of the fundamental guided QNBM Φ(-1). The calculation is performed for a/λ=0.255, point A in the inset of Fig. 5(a).

Fig. 7.
Fig. 7.

Dipole emission into PhC waveguides closed at one extremity by a PhC mirror. (a) Schematic top view of the 3D problem. The PhC parameters are the same as in the caption of Fig. 5. The dipole is parallel to the x-axis and is located in the central plane of the membrane at z0=0. (b) Convergence of the a-FMM for the β-factor defined as the power emitted into Φ(1) normalized to the total power emitted. The calculation is performed for a/λ=0.255, point A in the inset of Fig. 5(a).

Equations (49)

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× E = j ω μ ( r ) H and × H = j ω ε ( r ) E + J δ ( r r 0 ) ,
× E 1 = j ω 1 μ H 1 and × H 1 = j ω 1 ε E 1 + J 1 δ ( r r 1 ) ,
× E 2 = j ω 2 μ H 2 and × H 2 = j ω 2 ε E 2 + J 2 δ ( r r 2 ) .
S ( E 2 × H 1 ) dS = V j ( ω 1 E 2 T ε E 1 + ω 2 H 1 T μ H 2 ) d V E 2 ( r 1 ) J 1 .
S ( E 2 × H 1 E 1 × H 2 ) d S = V j [ ω 1 ( E 2 T ε E 1 H 2 T μ H 2 ) ω 2 ( E 1 T ε E 2 H 1 T μ H 2 ) ] d V
[ E 2 ( r 1 ) J 1 E 1 ( r 2 ) J 2 ] .
z = z 2 ( E 2 × H 1 E 1 × H 2 ) z dS z = z 1 ( E 2 × H 1 E 1 × H 2 ) z dS =
j ( ω 1 ω 2 ) V ( E 1 T ε E 2 H 1 T μ H 2 ) dV [ J 1 E 2 ( r 1 ) J 2 E 1 ( r 2 ) ] .
F z ( Φ 1 , Φ 2 ) = S ( E 2 × H 1 E 1 × H 2 ) z dS ,
E V ( Φ 1 , Φ 2 ) = V ( E 1 T ε E 2 H 1 T μ H 2 ) dV .
F z 2 ( Φ 1 , Φ 2 ) F z 1 ( Φ 1 , Φ 2 ) = j ( ω 1 ω 2 ) E V + J 2 E 1 ( r 2 ) J 1 E 2 ( r 1 ) .
E ( m ) ( r + a z ) , H ( m ) ( r + a z ) > = E ( m ) ( r ) , H ( m ) ( r ) > ,
F z ( Φ ( p , ω ) , Φ ( q , ω ) ) ( 1 exp { j [ k p ( ω ) k q ( ω ) ] a } ) = j ( ω ω ) E Cell ( z ) ( Φ ( p , ω ) , Φ ( q , ω ) ) .
F z ( Φ ( p , ω ) , Φ ( q , ω ) ) = z ( E ( q , ω ) × H ( p , ω ) E ( p , ω ) × H ( q , ω ) ) z dS = F ( p , ω ) δ p , q ,
F z ( Φ ( p , ω ) , Φ ( p , ω ) ) { 1 exp [ j { k p ( ω ) k p ( ω ) } a ] } = j ( ω ω ) E Cell ( z ) ( Φ ( p , ω ) , Φ ( p , ω ) ) .
F ( p , ω ) = E ( p , ω ) v g ( p ) a ,
E ( p , ω ) = 2 Cell E ( p ) T ε E ( p ) dV = 2 Cell H ( p ) T μ H ( p ) dV .
E ( 1 , r ) , H ( 1 , r ) > = E ( 1 , r ) * , H ( 1 , r ) * > ,
2 z Re ( E ( 1 , r ) × H ( 1 , r ) * ) z dS = F ( 1 , ω ) , and
Cell [ E ( 1 , r ) * ε E ( 1 , r ) + H ( 1 , r ) * μ H ( 1 , r ) ] dV = E ( 1 , ω ) ,
F L 2 ( Φ 1 , Φ 2 ) F L 1 ( Φ 1 , Φ 2 ) = E 1 ( r 2 ) J 2 E 2 ( r 1 ) J 1 .
Φ 1 = p > 0 I 1 ( p , L ) Φ ( p , L ) + D 1 ( p , L ) Φ ( p , L ) ,
Φ 2 = p > 0 I 2 ( p , L ) Φ ( p , L ) + D 2 ( p , L ) Φ ( p , L ) ,
F L 1 ( Φ 1 , Φ 2 ) = 4 p > 0 ( I 1 ( p , L ) D 2 ( p , L ) I 2 ( p , L ) D 1 ( p , L ) ) .
F L 2 ( Φ 1 , Φ 2 ) = 4 p > 0 ( I 2 ( p , R ) D 1 ( p , R ) I 1 ( p , R ) D 2 ( p , R ) ) .
4 p > 0 ( I 1 ( p , L ) D 2 ( p , L ) + I 1 ( p , R ) D 2 ( p , R ) ) E 2 ( r 1 ) J 1 = 4 p > 0 ( I 2 ( p , L ) D 1 ( p , L ) + I 2 ( p , R ) D 1 ( p , R ) ) E 1 ( r 2 ) J 2 .
( I 1 ) T S I 2 E 2 ( r 1 ) J 1 = ( I 2 ) T S I 1 E 1 ( r 2 ) J .
for z > z 0 , Φ = p > 0 D ( p , R ) Φ ( p ) ,
and for z < z 0 , Φ = p > 0 D ( p , L ) Φ ( −p ) ,
D ( m , L ) = E ( m ) ( r 0 ) J exp ( j k m z 0 ) 4 , and
D ( m , R ) = E ( −m ) ( r 0 ) J exp ( −j k m z 0 ) 4 ,
P 1 = P 1 = I 2 E ( 1 ) ( r 0 ) u 2 16 .
H ( r ) = p , q ( U xpq x + U ypq y + U xpq z ) exp ( jp G x x + jq G y y ) ,
E ( r ) = p , q ( S xpq x + S y pq y + S zpq z ) exp ( j p G x x + j q G y y ) ,
1 k 0 d [ Ψ ] d z = Ω ( z ) [ Ψ ] ,
Ψ ( p ) = n = 1 N b n ( p ) exp ( λ n ( p ) z ) W n ( p ) + f n ( p ) exp ( λ n ( p ) z ) W n ( p ) ,
[ b ( i ) f ( t ) ] = [ S 11 S 12 S 21 S 22 ] [ b ( t ) f ( i ) ] ,
[ I S 12 0 S 22 ] [ b ( i ) f ( i ) ] = ρ [ S 11 0 S 21 I ] [ b ( i ) f ( i ) ] .
[ b QNM f QNBM ] = S T [ b QNBM f QNM ] = [ S 11 S 12 S 21 S 22 ] [ b QNBM f QNM ] .
S 22 = ( F + ) 1 , S 12 = B + S 22 , S 21 = S 22 F and S 11 = B S 12 F ,
[ b M f W ] = S T [ b W f M ] .
[ b W + f W + ] = [ D ( L ) D ( R ) ] + [ b W f W ] ,
Φ ( r + a z , ω ) = Φ ( r , ω ) exp ( jk a ) .
F z 2 ( Φ , Φ m ) F z 1 ( Φ , Φ m ) = E ( m ) ( r 0 ) 2 exp ( j k m z 0 ) .
F z 1 ( Φ , Φ m ) = E ( m ) ( r 0 ) 2 exp ( j k m z 0 ) { exp [ j ( k m + k ) a ] 1 } 1
X ˆ = X ( x ) , Y ˆ = Y ( y ) , Z ˆ = Z ( z ) ,
μ ˆ = L μ L Det ( L ) , ε ˆ = L ε L Det ( L ) , J ˆ = L J ,
V ( E T ε E H T μ H ) dV = V ˆ ( E ˆ T ε ˆ E ˆ H ˆ T μ ˆ H ˆ ) d V ˆ .
S ( E × H ) z dS = S ˆ ( E ˆ × H ˆ ) z d S ˆ .

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