Abstract

We extend the fictitious source superposition method in order to model linear defects in photonic woodpiles, and we use the method to model a waveguide that is created by changing either the radius or refractive index of a single rod of an infinite woodpile composed of chalcogenide glass cylinders. In one instance, a nearly constant dispersion was observed over a sizable kx interval, where kx is the Bloch vector in the waveguiding direction, making this a compelling geometry for slow-light waveguides. The principal advantage of the method is that it does not rely on a supercell, thus avoiding what is possibly the greatest source of inefficiency present in most of the other methods that are used for modeling these structures. Instead, the method proceeds by placing an artificial source inside each rod of the defect layer and then subsequently taking an appropriate field superposition to remove all but one of these sources. The remaining source can then be used to mimic the fields that would be produced by a defect rod.

© 2011 Optical Society of America

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).
  2. H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
    [CrossRef]
  3. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
    [CrossRef]
  4. N. Yamamoto, S. Noda, and A. Chutinan, “Development of one period of a three-dimensional photonic crystal in the 510 μmwavelength region by wafer fusion and laser beam diffraction pattern observation techniques,” Jpn. J. Appl. Phys. 37, L1052–L1054 (1998).
    [CrossRef]
  5. J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
    [CrossRef] [PubMed]
  6. E. Nicoletti, G. Zhou, B. Jia, M. J. Ventura, D. Bulla, B. Luther-Davies, and M. Gu, “Observation of multiple higher-order stopgaps from three-dimensional chalcogenide glass photonic crystals,” Opt. Lett. 33, 2311–2313 (2008).
    [CrossRef] [PubMed]
  7. T. F. Krauss, “Why do we need slow light?” Nat. Photon. 2, 448–450 (2008).
    [CrossRef]
  8. 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]
  9. S. Johnson, M. Povinelli, M. Soljačić, A. Kralis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
    [CrossRef]
  10. S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227–229 (2004).
    [CrossRef] [PubMed]
  11. M. Imada, L. H. Lee, M. Okano, S. Kawashima, and S. Noda, “Development of three-dimensional photonic-crystal waveguides at optical-communication wavelengths,” Appl. Phys. Lett. 88, 171107 (2006).
    [CrossRef]
  12. S. Kawashima, K. Ishizaki, and S. Noda, “Light propagation in three-dimensional photonic crystals,” Opt. Express 18, 386–392(2010).
    [CrossRef] [PubMed]
  13. D. J. Kan, A. A. Asatryan, C. G. Poulton, and L. C. Botten, “Multipole method for modeling linear defects in photonic woodpiles,” J. Opt. Soc. Am. B 27, 246–258 (2010).
    [CrossRef]
  14. J. Chen, R. Hong, and J. Yang, “Analysis of planar defect structures in three-dimensional layer-by-layer photonic crystals,” J. Appl. Phys. 104, 063111 (2008).
    [CrossRef]
  15. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef] [PubMed]
  16. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  17. A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystals,” Appl. Phys. Lett. 75, 3739–3741 (1999).
    [CrossRef]
  18. M. Okano, S. Kako, and S. Noda, “Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,” Phys. Rev. B 68, 235110 (2003).
    [CrossRef]
  19. S. Kawashima, L. H. Lee, M. Okano, M. Imada, and S. Noda, “Design of donor-type line-defect waveguides in three-dimensional photonic crystals,” Opt. Express 13, 9774–9781 (2005).
    [CrossRef] [PubMed]
  20. S. Kawashima, M. Okano, M. Imada, and S. Noda, “Design of compound-defect waveguides in three-dimensional photonic crystals,” Opt. Express 14, 6303–6307 (2006).
    [CrossRef] [PubMed]
  21. S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
    [CrossRef]
  22. L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol. 1, 133–145 (2006).
  23. F. Zolla, R. Petit, and M. Cadilhac, “Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources,” J. Opt. Soc. Am. A 11, 1087–1096 (1994).
    [CrossRef]
  24. G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
    [CrossRef]
  25. A. Figotin and V. Goren, “Resolvent method for computations of localized defect modes of H-polarization in two-dimensional photonic crystals,” Phys. Rev. E 64, 056623 (2001).
    [CrossRef]
  26. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. part II. properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2109 (2000).
    [CrossRef]
  27. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
    [CrossRef] [PubMed]
  28. S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
    [CrossRef]

2010 (2)

2008 (5)

J. Chen, R. Hong, and J. Yang, “Analysis of planar defect structures in three-dimensional layer-by-layer photonic crystals,” J. Appl. Phys. 104, 063111 (2008).
[CrossRef]

E. Nicoletti, G. Zhou, B. Jia, M. J. Ventura, D. Bulla, B. Luther-Davies, and M. Gu, “Observation of multiple higher-order stopgaps from three-dimensional chalcogenide glass photonic crystals,” Opt. Lett. 33, 2311–2313 (2008).
[CrossRef] [PubMed]

T. F. Krauss, “Why do we need slow light?” Nat. Photon. 2, 448–450 (2008).
[CrossRef]

S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
[CrossRef] [PubMed]

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

2006 (3)

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol. 1, 133–145 (2006).

M. Imada, L. H. Lee, M. Okano, S. Kawashima, and S. Noda, “Development of three-dimensional photonic-crystal waveguides at optical-communication wavelengths,” Appl. Phys. Lett. 88, 171107 (2006).
[CrossRef]

S. Kawashima, M. Okano, M. Imada, and S. Noda, “Design of compound-defect waveguides in three-dimensional photonic crystals,” Opt. Express 14, 6303–6307 (2006).
[CrossRef] [PubMed]

2005 (4)

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (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]

S. Johnson, M. Povinelli, M. Soljačić, A. Kralis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

S. Kawashima, L. H. Lee, M. Okano, M. Imada, and S. Noda, “Design of donor-type line-defect waveguides in three-dimensional photonic crystals,” Opt. Express 13, 9774–9781 (2005).
[CrossRef] [PubMed]

2004 (1)

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227–229 (2004).
[CrossRef] [PubMed]

2003 (2)

M. Okano, S. Kako, and S. Noda, “Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,” Phys. Rev. B 68, 235110 (2003).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

2002 (1)

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

2001 (2)

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

A. Figotin and V. Goren, “Resolvent method for computations of localized defect modes of H-polarization in two-dimensional photonic crystals,” Phys. Rev. E 64, 056623 (2001).
[CrossRef]

2000 (1)

1999 (1)

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystals,” Appl. Phys. Lett. 75, 3739–3741 (1999).
[CrossRef]

1998 (1)

N. Yamamoto, S. Noda, and A. Chutinan, “Development of one period of a three-dimensional photonic crystal in the 510 μmwavelength region by wafer fusion and laser beam diffraction pattern observation techniques,” Jpn. J. Appl. Phys. 37, L1052–L1054 (1998).
[CrossRef]

1994 (3)

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

F. Zolla, R. Petit, and M. Cadilhac, “Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources,” J. Opt. Soc. Am. A 11, 1087–1096 (1994).
[CrossRef]

Asatryan, A. A.

Baker, N. J.

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

Biswas, R.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Botten, L. C.

D. J. Kan, A. A. Asatryan, C. G. Poulton, and L. C. Botten, “Multipole method for modeling linear defects in photonic woodpiles,” J. Opt. Soc. Am. B 27, 246–258 (2010).
[CrossRef]

S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
[CrossRef] [PubMed]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol. 1, 133–145 (2006).

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. part II. properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2109 (2000).
[CrossRef]

Bulla, D.

Cadilhac, M.

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Chen, J.

J. Chen, R. Hong, and J. Yang, “Analysis of planar defect structures in three-dimensional layer-by-layer photonic crystals,” J. Appl. Phys. 104, 063111 (2008).
[CrossRef]

Chigrin, D. N.

Choi, D.

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

Chutinan, A.

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystals,” Appl. Phys. Lett. 75, 3739–3741 (1999).
[CrossRef]

N. Yamamoto, S. Noda, and A. Chutinan, “Development of one period of a three-dimensional photonic crystal in the 510 μmwavelength region by wafer fusion and laser beam diffraction pattern observation techniques,” Jpn. J. Appl. Phys. 37, L1052–L1054 (1998).
[CrossRef]

de Sterke, C. M.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol. 1, 133–145 (2006).

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. part II. properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2109 (2000).
[CrossRef]

Dossou, K. B.

S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008).
[CrossRef] [PubMed]

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol. 1, 133–145 (2006).

Dowling, J. P.

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

Eggleton, B. J.

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

El-Kady, I.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Figotin, A.

A. Figotin and V. Goren, “Resolvent method for computations of localized defect modes of H-polarization in two-dimensional photonic crystals,” Phys. Rev. E 64, 056623 (2001).
[CrossRef]

Fleming, J. G.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Goren, V.

A. Figotin and V. Goren, “Resolvent method for computations of localized defect modes of H-polarization in two-dimensional photonic crystals,” Phys. Rev. E 64, 056623 (2001).
[CrossRef]

Gu, M.

Ha, S.

Hagness, S. C.

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

Ho, K. M.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Hong, R.

J. Chen, R. Hong, and J. Yang, “Analysis of planar defect structures in three-dimensional layer-by-layer photonic crystals,” J. Appl. Phys. 104, 063111 (2008).
[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]

Imada, M.

M. Imada, L. H. Lee, M. Okano, S. Kawashima, and S. Noda, “Development of three-dimensional photonic-crystal waveguides at optical-communication wavelengths,” Appl. Phys. Lett. 88, 171107 (2006).
[CrossRef]

S. Kawashima, M. Okano, M. Imada, and S. Noda, “Design of compound-defect waveguides in three-dimensional photonic crystals,” Opt. Express 14, 6303–6307 (2006).
[CrossRef] [PubMed]

S. Kawashima, L. H. Lee, M. Okano, M. Imada, and S. Noda, “Design of donor-type line-defect waveguides in three-dimensional photonic crystals,” Opt. Express 13, 9774–9781 (2005).
[CrossRef] [PubMed]

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227–229 (2004).
[CrossRef] [PubMed]

Ishizaki, K.

Jacobs, S.

S. Johnson, M. Povinelli, M. Soljačić, A. Kralis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Jia, B.

Joannopoulos, J.

S. Johnson, M. Povinelli, M. Soljačić, A. Kralis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Joannopoulos, J. D.

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

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

Johnson, S.

S. Johnson, M. Povinelli, M. Soljačić, A. Kralis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Johnson, S. G.

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

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

Kako, S.

M. Okano, S. Kako, and S. Noda, “Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,” Phys. Rev. B 68, 235110 (2003).
[CrossRef]

Kan, D. J.

Kawashima, S.

Kivshar, Y. S.

Kralis, A.

S. Johnson, M. Povinelli, M. Soljačić, A. Kralis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Krauss, T. F.

T. F. Krauss, “Why do we need slow light?” Nat. Photon. 2, 448–450 (2008).
[CrossRef]

Lamont, M. R.

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

Lavrinenko, A. V.

Lee, L. H.

M. Imada, L. H. Lee, M. Okano, S. Kawashima, and S. Noda, “Development of three-dimensional photonic-crystal waveguides at optical-communication wavelengths,” Appl. Phys. Lett. 88, 171107 (2006).
[CrossRef]

S. Kawashima, L. H. Lee, M. Okano, M. Imada, and S. Noda, “Design of donor-type line-defect waveguides in three-dimensional photonic crystals,” Opt. Express 13, 9774–9781 (2005).
[CrossRef] [PubMed]

Lin, S. Y.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Luther-Davies, B.

E. Nicoletti, G. Zhou, B. Jia, M. J. Ventura, D. Bulla, B. Luther-Davies, and M. Gu, “Observation of multiple higher-order stopgaps from three-dimensional chalcogenide glass photonic crystals,” Opt. Lett. 33, 2311–2313 (2008).
[CrossRef] [PubMed]

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

Madden, S. J.

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

McPhedran, R. C.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol. 1, 133–145 (2006).

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. part II. properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2109 (2000).
[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, 2008).

Nicoletti, E.

Nicorovici, N. A.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol. 1, 133–145 (2006).

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. part II. properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2109 (2000).
[CrossRef]

Noda, S.

S. Kawashima, K. Ishizaki, and S. Noda, “Light propagation in three-dimensional photonic crystals,” Opt. Express 18, 386–392(2010).
[CrossRef] [PubMed]

M. Imada, L. H. Lee, M. Okano, S. Kawashima, and S. Noda, “Development of three-dimensional photonic-crystal waveguides at optical-communication wavelengths,” Appl. Phys. Lett. 88, 171107 (2006).
[CrossRef]

S. Kawashima, M. Okano, M. Imada, and S. Noda, “Design of compound-defect waveguides in three-dimensional photonic crystals,” Opt. Express 14, 6303–6307 (2006).
[CrossRef] [PubMed]

S. Kawashima, L. H. Lee, M. Okano, M. Imada, and S. Noda, “Design of donor-type line-defect waveguides in three-dimensional photonic crystals,” Opt. Express 13, 9774–9781 (2005).
[CrossRef] [PubMed]

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227–229 (2004).
[CrossRef] [PubMed]

M. Okano, S. Kako, and S. Noda, “Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,” Phys. Rev. B 68, 235110 (2003).
[CrossRef]

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystals,” Appl. Phys. Lett. 75, 3739–3741 (1999).
[CrossRef]

N. Yamamoto, S. Noda, and A. Chutinan, “Development of one period of a three-dimensional photonic crystal in the 510 μmwavelength region by wafer fusion and laser beam diffraction pattern observation techniques,” Jpn. J. Appl. Phys. 37, L1052–L1054 (1998).
[CrossRef]

Ogawa, S.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227–229 (2004).
[CrossRef] [PubMed]

Okano, M.

S. Kawashima, M. Okano, M. Imada, and S. Noda, “Design of compound-defect waveguides in three-dimensional photonic crystals,” Opt. Express 14, 6303–6307 (2006).
[CrossRef] [PubMed]

M. Imada, L. H. Lee, M. Okano, S. Kawashima, and S. Noda, “Development of three-dimensional photonic-crystal waveguides at optical-communication wavelengths,” Appl. Phys. Lett. 88, 171107 (2006).
[CrossRef]

S. Kawashima, L. H. Lee, M. Okano, M. Imada, and S. Noda, “Design of donor-type line-defect waveguides in three-dimensional photonic crystals,” Opt. Express 13, 9774–9781 (2005).
[CrossRef] [PubMed]

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227–229 (2004).
[CrossRef] [PubMed]

M. Okano, S. Kako, and S. Noda, “Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,” Phys. Rev. B 68, 235110 (2003).
[CrossRef]

Pelusi, M. D.

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

Petit, R.

Poulton, C. G.

D. J. Kan, A. A. Asatryan, C. G. Poulton, and L. C. Botten, “Multipole method for modeling linear defects in photonic woodpiles,” J. Opt. Soc. Am. B 27, 246–258 (2010).
[CrossRef]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

Povinelli, M.

S. Johnson, M. Povinelli, M. Soljačić, A. Kralis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[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]

Robinson, P. A.

Sigalas, M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

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]

Smith, G. H.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Soljacic, M.

S. Johnson, M. Povinelli, M. Soljačić, A. Kralis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Sözüer, H. S.

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

Sukhorukov, A. A.

Taeed, V. G.

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

Taflove, A.

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

Ventura, M. J.

Wilcox, S.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol. 1, 133–145 (2006).

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

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, 2008).

Yamamoto, N.

N. Yamamoto, S. Noda, and A. Chutinan, “Development of one period of a three-dimensional photonic crystal in the 510 μmwavelength region by wafer fusion and laser beam diffraction pattern observation techniques,” Jpn. J. Appl. Phys. 37, L1052–L1054 (1998).
[CrossRef]

Yang, J.

J. Chen, R. Hong, and J. Yang, “Analysis of planar defect structures in three-dimensional layer-by-layer photonic crystals,” J. Appl. Phys. 104, 063111 (2008).
[CrossRef]

Yoshimoto, S.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227–229 (2004).
[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).
[CrossRef] [PubMed]

Zhou, G.

Zolla, F.

Appl. Phys. B (1)

S. Johnson, M. Povinelli, M. Soljačić, A. Kralis, S. Jacobs, and J. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293(2005).
[CrossRef]

Appl. Phys. Lett. (2)

M. Imada, L. H. Lee, M. Okano, S. Kawashima, and S. Noda, “Development of three-dimensional photonic-crystal waveguides at optical-communication wavelengths,” Appl. Phys. Lett. 88, 171107 (2006).
[CrossRef]

A. Chutinan and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystals,” Appl. Phys. Lett. 75, 3739–3741 (1999).
[CrossRef]

Int. J. Microw. Opt. Technol. (1)

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the fictitious source superposition method,” Int. J. Microw. Opt. Technol. 1, 133–145 (2006).

J. Appl. Phys. (1)

J. Chen, R. Hong, and J. Yang, “Analysis of planar defect structures in three-dimensional layer-by-layer photonic crystals,” J. Appl. Phys. 104, 063111 (2008).
[CrossRef]

J. Mod. Opt. (1)

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

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

J. Opt. Soc. Am. B (1)

Jpn. J. Appl. Phys. (1)

N. Yamamoto, S. Noda, and A. Chutinan, “Development of one period of a three-dimensional photonic crystal in the 510 μmwavelength region by wafer fusion and laser beam diffraction pattern observation techniques,” Jpn. J. Appl. Phys. 37, L1052–L1054 (1998).
[CrossRef]

Nat. Photon. (1)

T. F. Krauss, “Why do we need slow light?” Nat. Photon. 2, 448–450 (2008).
[CrossRef]

Nature (1)

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, “All-metallic three-dimensional photonic crystals with a large infrared bandgap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (1)

Opt. Photon. News (1)

S. J. Madden, D. Choi, M. R. Lamont, V. G. Taeed, N. J. Baker, M. D. Pelusi, B. Luther-Davies, and B. J. Eggleton, “Chalcogenide glass photonic chips,” Opt. Photon. News 19, 19–23 (2008).
[CrossRef]

Phys. Rev. B (1)

M. Okano, S. Kako, and S. Noda, “Coupling between a point-defect cavity and a line-defect waveguide in three-dimensional photonic crystal,” Phys. Rev. B 68, 235110 (2003).
[CrossRef]

Phys. Rev. E (3)

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, “Modeling of defect modes in photonic crystals using the fictitious source superposition method,” Phys. Rev. E 71, 056606 (2005).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

A. Figotin and V. Goren, “Resolvent method for computations of localized defect modes of H-polarization in two-dimensional photonic crystals,” Phys. Rev. E 64, 056623 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

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]

Science (1)

S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227–229 (2004).
[CrossRef] [PubMed]

Solid State Commun. (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Other (2)

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

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

Supplementary Material (3)

» Media 1: AVI (471 KB)     
» Media 2: AVI (441 KB)     
» Media 3: AVI (446 KB)     

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Figures (8)

Fig. 1
Fig. 1

Photonic woodpile: a linear waveguide can be created by changing the properties of a single cylinder (red).

Fig. 2
Fig. 2

(a) Spherical coordinates of the wave vector k = ( k x , k y , k z ) . (b) Successive unit cells of a grating whose cylinders are parallel to the x axis, with j indexing the cylinders in order of increasing displacement along the y axis.

Fig. 3
Fig. 3

The incoming ( f I ± ) and diffracted fields ( f D ± ), as well as the reflection (R) and transmission (T) matrices associated with the waveguide layer. Each cylinder of the layer contains a fictitious source ( Q n j , V ). The phases of f ± must be adjusted in order to give the waveguide layer an artificial thickness h equal to the distance between adjacent layers. The phase-adjusted fields are given by F I ± and F D ± . This procedure also necessitates an additional pair of scattering matrices, so that there are two such matrices associated with the top surface of the waveguide ( R ˜ a and T ˜ a ) and another two associated with the bottom surface ( R ˜ b and T ˜ b ). The semi-infinite cladding regions below and above the waveguide layer are characterized by the R and R reflection matrices.

Fig. 4
Fig. 4

Comparison of the dispersion curves, computed using the FSS method with N = 4 , 10, and 40 integration intervals, for the linear waveguide. The radius of the defect rod is 0.075 d (i.e., half that of the cladding rods). The frequency at the point labeled A, which corresponds to k x = 2.075 / d , is listed in Table 1 for different truncation parameters and for different values of N. Mode frequencies computed using Eq. (38) with a supercell containing N s = 4 cylinders are shown for comparison with the N = 4 curve computed using the FSS method. Note that if the trapezoidal rule is used to carry out the integration (as it is here), then the FSS method is equivalent to the supercell method when N s = N .

Fig. 5
Fig. 5

Dispersion curves for the linear waveguide for different values of r d , which specifies the radius of the defect rod. Here, r is the radius of the cladding rods. Note that the complete bandgap of the cladding spans the normalized frequencies 0.4990 d / λ 0.5245 .

Fig. 6
Fig. 6

Plots of the energy density, ε E 2 , of the electric field [(a) and (b)], and of the energy density, μ H 2 , of the magnetic field [(c) and (d)] for the linear waveguide for the “slow-light” point, i.e., point B ( k x d / π = 1.0 , d / λ 0.5091 ), indicated in Fig. 5, where ε ( y , z ) and μ ( y , z ) are the relative permittivity and relative permeability. The view looks down the defect rod—the radius of which is r d = 0.5 r . Plots (a) and (c) are for the plane x / d = 0.0 , where x is the direction parallel to the defect rod. Plots (b) and (d) are for the plane x / d = 0.25 .

Fig. 7
Fig. 7

Plot of Re ( S x ) for points B [parts (a) and (b)] (Media 1), C [parts (c) and (d)] (Media 2), and D [parts (e) and (f)] (Media 3) of Fig. 5, where S x is the x component of the Poynting vector. The view looks down the defect rod, the radius of which is r d = 0.5 r . Plots (a), (c), and (e) are for the plane x / d = 0.0 , where x is the direction parallel to the defect rod. Plots (b), (d), and (f) are for the plane x / d = 0.25 . Animations that show how Re ( S x ) varies as x moves along the defect rod are available online.

Fig. 8
Fig. 8

Dispersion curves for the linear waveguide for different values of n d , which specifies the refractive index of the defect rod. The refractive index of the cladding rods is n c = 2.68 (chalcogenide glass).

Tables (1)

Tables Icon

Table 1 Convergence of the (Normalized) Frequencies, d / λ , Near Point A of Fig. 4, with k x = 2.075 / d Being Fixed

Equations (61)

Equations on this page are rendered with MathJax. Learn more.

f ± = [ [ f s TE ± ] [ f s TM ± ] ] ,
f D , p ± = [ [ f D , ( p , q ) TE ± ] [ f D , ( p , q ) TM ± ] ] ,
V p , x ( r ) = n = [ A p , n j , V J n ( k e ρ ) + B p , n j , V H n ( k e ρ ) ] e i n θ e i α p x ,
V p , x ( r ) = n = C p , n j , V J n ( k i ρ ) e i n θ e i α p x ,
A p = S p B p + J p Z X p f I , p + J p + Z X p + f I , p + ,
A p = M p B p ,
B p = L p J p Z X p f I , p L p J p + Z X p + f I , p + ,
f D , p = f I , p + ( 2 / d ) G p ( Z X p ) 1 K p B p ,
f D , p + = f I , p + + ( 2 / d ) G p ( Z X p + ) 1 K p + B p ,
G p = [ G p 0 0 G p ] ,
f D , p = T p , a f I , p + R p , b f I , p + ,
f D , p + = R p , a f I , p + T p , b f I , p + ,
V p , x ( r ) = n = [ C p , n j , V J n ( k i ρ ) + Q p , n j , V H n ( k i ρ ) ] e i n θ e i α p x ,
A p = ( M p B p + N p Q p ) ,
B p = Y p f I , p + Y p + f I , p + + Y p fs Q p ,
f D , p = T p , a f I , p + R p , b f I , p + + Q p Q p ,
f D , p + = R p , a f I , p + T p , b f I , p + + Q p + Q p ,
A p = M ^ p B p .
B p = H p Q p ,
H p = ( M ^ p M p ) 1 N p Q p .
f D = T a f I + R b f I + ,
f D + = R a f I + T b f I + ,
f D = T a f I + R b f I + + Q Q ,
f D + = R a f I + T b f I + + Q + Q ,
B = Y f I + Y + f I + + Y f s Q ,
B = HQ ,
Q = [ [ Q p E ] [ Q p H ] ] ,
F I ± = P 1 f I ± , F D ± = P f D ± ,
P = [ P 0 0 P ] ,
F D = T ˜ a F I + R ˜ b F I + + P Q Q ,
F D + = R ˜ a F I + T ˜ b F I + + P Q + Q ,
F I + = R F D , F I = R F D + .
F I ± = D ± Q ,
D ± = ( I G T ˜ G ± ) 1 ( G T ˜ G ± Q ± + G Q ) ,
B = ZQ ,
Z = ( Y P D + Y + P D + + Y f s ) Q .
( Z H ) Q = 0 .
· = d 2 π π d π d · d β 0 .
Q n j , V = Q n 0 , V e i β 0 j d = { Q n 0 , V , for     j = 0 0 , for     j 0 ,
Q = Q ,
B = HQ = HQ ,
B = ZQ = Z Q ,
Z ( k , k x ) H ( k , k x ) Q = 0 ;
( I E E ± ) F D ± = 0 ,
E = ( I R ˜ a R ) 1 T ˜ b R ,
E + = ( I R ˜ b R ) 1 T ˜ a R ,
R = S R S 1 ,
S = [ S 0 0 S ] ,
N = [ N E E N E H N H E N H H ] .
N n E E = η 3 ( II ) J 2 H / Δ ,
N n H H = η 2 ( II ) J 3 H / Δ ,
N n E H = η 2 ( II ) [ η 1 ( I ) η 1 ( II ) ] H / Δ ,
N n H E = η 3 ( II ) [ η 1 ( I ) η 1 ( II ) ] H / Δ ,
Δ = J n ( k I r ) 2 { [ η 1 ( I ) η 1 ( II ) ] 2 J 2 J 3 } ,
J j = η j ( I ) { J n ( k I r ) J n ( k I r ) η j ( II ) J n ( k II r ) η j ( I ) J n ( k II r ) } ,
H = J n ( k I r ) H n ( k II r ) × { H n ( k II r ) H n ( k II r ) J n ( k II r ) J n ( k II r ) } .
R p = [ R TE , TE R TE , TM R TM , TE R TM , TM ] ,
f D , ( p , q o ) V + = q i R q o q i V , TE f I , ( p , q i ) TE + R q o q i V , TM f I , ( p , q i ) TM + T q o q i V , TE f I , ( p , q i ) TE + + T q o q i V , TM f I , ( p , q i ) TM + .
R = [ R TE , TE R TE , TM R TM , TE R TM , TM ] ,
f D , s o V + = s i R s o s i V , TE f I , s i TE + R s o s i V , TM f I , s i TM + T s o s i V , TE f I , s i TE + + T s o s i V , TM f I , s i TM + ,
R TE , TE [ s o , s i ] = { R p o TE , TE [ q o , q i ] , for     p o = p i 0 , otherwise .

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