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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  15. 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]
  16. 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]
  17. P. Lalanne and J. P. Hugonin, "Bloch-wave engineering for high-Q, small-V microcavities," IEEE J. Quantum Electron. 39, 1430-1438 (2003).
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  18. 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]
  19. 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]
  20. 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]
  21. 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).
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  24. S. F. Helfert, "Determination of Floquet modes in asymmetric periodic structures," Opt. Quantum Electron. 37, 185-197 (2005).
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  25. A. Chutinan and S. Noda, "Waveguides and waveguide bends in two-dimensional photonic crystal slabs," Phys. Rev. B 62, 4488-4492 (2000).
    [CrossRef]
  26. 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]
  27. S. Hughes, "Enhanced single-photon emission from quantum dots in photonic crystal waveguides and nanocavities," Opt. Lett. 29, 2659-2661 (2004).
    [CrossRef] [PubMed]
  28. 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]
  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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  31. 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]
  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).
    [CrossRef]
  33. 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]
  34. 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]
  35. 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]
  36. 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]
  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).
    [CrossRef]
  38. L. F. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  39. Q. Cao, P. Lalanne and J. P. Hugonin, "Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides," J. Opt. Soc. Am. A 19, 335-338 (2002).
    [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]
  42. 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]
  43. 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]
  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, 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, 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 one-dimensional 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]

IEEE J. Quantum Electron. (5)

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]

P. Lalanne and J. P. Hugonin, "Bloch-wave engineering for high-Q, small-V microcavities," IEEE J. Quantum Electron. 39, 1430-1438 (2003).
[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. Lalanne, "Electromagnetic analysis of photonic crystal waveguides operating above the light cone," IEEE J. Quantum Electron. 38, 800-804 (2002).
[CrossRef]

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]

IEEE J. Sel. Top. Quantum Electron. (2)

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]

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]

IEEE Photon. Technol. Lett. (2)

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]

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]

IEEE Trans. Antennas Propag. (1)

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. Comput. Phys. (1)

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]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (6)

Microwave Opt. Technol. Lett. (2)

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

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]

Nat. Mat. (1)

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

Nature (1)

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]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (3)

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

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

Photon. Nanostruct. Fundam. Appl. (1)

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]

Phys. Rev. A (1)

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]

Phys. Rev. B (4)

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

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]

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]

Phys. Rev. B. (1)

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

Phys. Rev. E (3)

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]

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]

Phys. Rev. Lett. (3)

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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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