Abstract

We extend the multipole method to allow for rod-type defects in woodpiles composed of infinitely long cylinders. A coupled-resonator optical waveguide and a linear waveguide are considered, where each waveguide is embedded in a woodpile cladding. For both structures, low-loss waveguiding is observed (Q1×1043×104). Decreasing the radius of the defect rod shifts the transmission resonances to shorter wavelengths. The reflection and transmission coefficients of the woodpile are derived for the case of normal incidence in the long-wavelength limit, and it is shown that both the individual layers and the entire assemblage of layers homogenize to one-dimensional dielectric slabs. Expressions for the effective permittivities are given.

© 2010 Optical Society of America

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2008 (2)

2007 (1)

F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic bandgap,” Adv. Mater. 19, 1567-1570 (2007).
[CrossRef]

2006 (3)

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. Wu, J. Serbin, and M. Gu, “Two-photon polymerization for three-dimensional microfabrication,” J. Photochem. Photobiol., A 181, 1-11 (2006).
[CrossRef]

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,” Microwave Opt. Technol. Lett. 1, 133-145 (2006).

2005 (1)

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]

2003 (5)

A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses,” Appl. Phys. Lett. 83, 4480-4482 (2003).
[CrossRef]

G. A. Ozin, “The photonic opal: the jewel in the crown of optical information processing,” Chem. Commun. (Cambridge) 21, 2639-2643 (2003).

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]

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic bandgaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[CrossRef]

2002 (3)

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]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (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]

2001 (3)

2000 (2)

1999 (1)

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118-6127 (1999).
[CrossRef]

1998 (1)

K. Busch and S. John, “Photonic bandgap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

1996 (1)

R. C. McPhedran, C. G. Poulton, N. A. Nicorovici, and A. B. Movchan, “Low frequency corrections to the static effective dielectric constant of a two-dimensional composite material,” Proc. R. Soc. London, Ser. A 452, 2231-2245 (1996).
[CrossRef]

1995 (1)

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic bandgaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135-1145 (1995).
[CrossRef]

1994 (1)

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

1987 (1)

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

1904 (1)

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[CrossRef]

Arsh, A.

A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses,” Appl. Phys. Lett. 83, 4480-4482 (2003).
[CrossRef]

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Interference lithography for 3D photonic band gap crystal layer by layer fabrication,” in Materials Research Society Symposium Proceedings, 2001, E.D.Jones, O.Manasreh, K.D.Choquette, D.J.Friedman, and D.K.Johnstone, eds.,Vol. 692, K2.9.1.
[CrossRef]

Asatryan, A. 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,” Microwave Opt. Technol. Lett. 1, 133-145 (2006).

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 I. Method,” J. Opt. Soc. Am. A 17, 2165-2177 (2000).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118-6127 (1999).
[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]

Botten, L. 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,” Microwave Opt. Technol. Lett. 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]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[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 I. Method,” J. Opt. Soc. Am. A 17, 2165-2177 (2000).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118-6127 (1999).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic bandgaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135-1145 (1995).
[CrossRef]

Braun, P. V.

F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic bandgap,” Adv. Mater. 19, 1567-1570 (2007).
[CrossRef]

Bulla, D.

Busch, K.

K. Busch and S. John, “Photonic bandgap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Centeno, E.

Cheng, J.

J. Cheng, 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]

de Dood, M.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic bandgaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[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,” Microwave Opt. Technol. Lett. 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 I. Method,” J. Opt. Soc. Am. A 17, 2165-2177 (2000).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118-6127 (1999).
[CrossRef]

Deubel, M.

G. von Freymann, S. Wong, G. A. Ozin, S. John, F. Pérez-Willard, M. Deubel, and M. Wegener, in Conference on Lasers and Electro-Optics, 2005, CTuU5, pp. 1002-1004.

Dossou, K. B.

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,” Microwave Opt. Technol. Lett. 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]

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]

Enoch, S.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic bandgaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[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]

Fan, S.

F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic bandgap,” Adv. Mater. 19, 1567-1570 (2007).
[CrossRef]

Feigel, A.

A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses,” Appl. Phys. Lett. 83, 4480-4482 (2003).
[CrossRef]

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Interference lithography for 3D photonic band gap crystal layer by layer fabrication,” in Materials Research Society Symposium Proceedings, 2001, E.D.Jones, O.Manasreh, K.D.Choquette, D.J.Friedman, and D.K.Johnstone, eds.,Vol. 692, K2.9.1.
[CrossRef]

Felbacq, D.

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]

García-Santamaría, F.

F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic bandgap,” Adv. Mater. 19, 1567-1570 (2007).
[CrossRef]

Gralak, B.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic bandgaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[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]

Gu, M.

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]

Hong, R.

J. Cheng, 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]

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]

Jia, B.

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 Univ. Press, 2008).

John, S.

K. Busch and S. John, “Photonic bandgap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

G. von Freymann, S. Wong, G. A. Ozin, S. John, F. Pérez-Willard, M. Deubel, and M. Wegener, in Conference on Lasers and Electro-Optics, 2005, CTuU5, pp. 1002-1004.

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 Univ. Press, 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]

Kawashima, S.

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]

Klebanov, M.

A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses,” Appl. Phys. Lett. 83, 4480-4482 (2003).
[CrossRef]

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Interference lithography for 3D photonic band gap crystal layer by layer fabrication,” in Materials Research Society Symposium Proceedings, 2001, E.D.Jones, O.Manasreh, K.D.Choquette, D.J.Friedman, and D.K.Johnstone, eds.,Vol. 692, K2.9.1.
[CrossRef]

Kotler, Z.

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Interference lithography for 3D photonic band gap crystal layer by layer fabrication,” in Materials Research Society Symposium Proceedings, 2001, E.D.Jones, O.Manasreh, K.D.Choquette, D.J.Friedman, and D.K.Johnstone, eds.,Vol. 692, K2.9.1.
[CrossRef]

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]

Lewis, J. A.

F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic bandgap,” Adv. Mater. 19, 1567-1570 (2007).
[CrossRef]

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]

Lousse, V.

F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic bandgap,” Adv. Mater. 19, 1567-1570 (2007).
[CrossRef]

Luther-Davies, B.

Lyubin, V.

A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses,” Appl. Phys. Lett. 83, 4480-4482 (2003).
[CrossRef]

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Interference lithography for 3D photonic band gap crystal layer by layer fabrication,” in Materials Research Society Symposium Proceedings, 2001, E.D.Jones, O.Manasreh, K.D.Choquette, D.J.Friedman, and D.K.Johnstone, eds.,Vol. 692, K2.9.1.
[CrossRef]

Maxwell Garnett, J. C.

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[CrossRef]

Maystre, D.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic bandgaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[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,” Microwave Opt. Technol. Lett. 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]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[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 I. Method,” J. Opt. Soc. Am. A 17, 2165-2177 (2000).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118-6127 (1999).
[CrossRef]

R. C. McPhedran, C. G. Poulton, N. A. Nicorovici, and A. B. Movchan, “Low frequency corrections to the static effective dielectric constant of a two-dimensional composite material,” Proc. R. Soc. London, Ser. A 452, 2231-2245 (1996).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic bandgaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135-1145 (1995).
[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 Univ. Press, 2008).

Modinos, A.

Moroz, A.

Movchan, A. B.

R. C. McPhedran, C. G. Poulton, N. A. Nicorovici, and A. B. Movchan, “Low frequency corrections to the static effective dielectric constant of a two-dimensional composite material,” Proc. R. Soc. London, Ser. A 452, 2231-2245 (1996).
[CrossRef]

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,” Microwave Opt. Technol. Lett. 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]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[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 I. Method,” J. Opt. Soc. Am. A 17, 2165-2177 (2000).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118-6127 (1999).
[CrossRef]

R. C. McPhedran, C. G. Poulton, N. A. Nicorovici, and A. B. Movchan, “Low frequency corrections to the static effective dielectric constant of a two-dimensional composite material,” Proc. R. Soc. London, Ser. A 452, 2231-2245 (1996).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic bandgaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135-1145 (1995).
[CrossRef]

Noda, S.

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]

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]

Okano, 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]

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]

Ozin, G. A.

G. A. Ozin, “The photonic opal: the jewel in the crown of optical information processing,” Chem. Commun. (Cambridge) 21, 2639-2643 (2003).

G. von Freymann, S. Wong, G. A. Ozin, S. John, F. Pérez-Willard, M. Deubel, and M. Wegener, in Conference on Lasers and Electro-Optics, 2005, CTuU5, pp. 1002-1004.

Pérez-Willard, F.

G. von Freymann, S. Wong, G. A. Ozin, S. John, F. Pérez-Willard, M. Deubel, and M. Wegener, in Conference on Lasers and Electro-Optics, 2005, CTuU5, pp. 1002-1004.

Poulton, C. G.

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]

R. C. McPhedran, C. G. Poulton, N. A. Nicorovici, and A. B. Movchan, “Low frequency corrections to the static effective dielectric constant of a two-dimensional composite material,” Proc. R. Soc. London, Ser. A 452, 2231-2245 (1996).
[CrossRef]

Robinson, P. A.

Serbin, J.

S. Wu, J. Serbin, and M. Gu, “Two-photon polymerization for three-dimensional microfabrication,” J. Photochem. Photobiol., A 181, 1-11 (2006).
[CrossRef]

Sfez, B.

A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses,” Appl. Phys. Lett. 83, 4480-4482 (2003).
[CrossRef]

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Interference lithography for 3D photonic band gap crystal layer by layer fabrication,” in Materials Research Society Symposium Proceedings, 2001, E.D.Jones, O.Manasreh, K.D.Choquette, D.J.Friedman, and D.K.Johnstone, eds.,Vol. 692, K2.9.1.
[CrossRef]

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]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[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]

Stefanou, N.

Taflove, A.

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

Tayeb, G.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic bandgaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[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]

Veinger, M.

A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses,” Appl. Phys. Lett. 83, 4480-4482 (2003).
[CrossRef]

Ventura, M. J.

von Freymann, G.

G. von Freymann, S. Wong, G. A. Ozin, S. John, F. Pérez-Willard, M. Deubel, and M. Wegener, in Conference on Lasers and Electro-Optics, 2005, CTuU5, pp. 1002-1004.

Wegener, M.

G. von Freymann, S. Wong, G. A. Ozin, S. John, F. Pérez-Willard, M. Deubel, and M. Wegener, in Conference on Lasers and Electro-Optics, 2005, CTuU5, pp. 1002-1004.

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,” Microwave Opt. Technol. Lett. 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 Univ. Press, 2008).

Wong, S.

G. von Freymann, S. Wong, G. A. Ozin, S. John, F. Pérez-Willard, M. Deubel, and M. Wegener, in Conference on Lasers and Electro-Optics, 2005, CTuU5, pp. 1002-1004.

Wu, S.

S. Wu, J. Serbin, and M. Gu, “Two-photon polymerization for three-dimensional microfabrication,” J. Photochem. Photobiol., A 181, 1-11 (2006).
[CrossRef]

Xu, M.

F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic bandgap,” Adv. Mater. 19, 1567-1570 (2007).
[CrossRef]

Yablonovitch, E.

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

Yang, J.

J. Cheng, 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]

Yannopapas, V.

Zhou, G.

Adv. Mater. (1)

F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic bandgap,” Adv. Mater. 19, 1567-1570 (2007).
[CrossRef]

Appl. Phys. Lett. (2)

A. Feigel, M. Veinger, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Three-dimensional simple cubic woodpile photonic crystals made from chalcogenide glasses,” Appl. Phys. Lett. 83, 4480-4482 (2003).
[CrossRef]

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]

Chem. Commun. (Cambridge) (1)

G. A. Ozin, “The photonic opal: the jewel in the crown of optical information processing,” Chem. Commun. (Cambridge) 21, 2639-2643 (2003).

J. Appl. Phys. (1)

J. Cheng, 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 (3)

J. Photochem. Photobiol., A (1)

S. Wu, J. Serbin, and M. Gu, “Two-photon polymerization for three-dimensional microfabrication,” J. Photochem. Photobiol., A 181, 1-11 (2006).
[CrossRef]

Microwave Opt. Technol. Lett. (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,” Microwave Opt. Technol. Lett. 1, 133-145 (2006).

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 (2)

Opt. Lett. (2)

Philos. Trans. R. Soc. London, Ser. A (1)

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[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 (7)

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]

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic bandgaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118-6127 (1999).
[CrossRef]

K. Busch and S. John, “Photonic bandgap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, “Photonic bandgaps for arrays of perfectly conducting cylinders,” Phys. Rev. E 52, 1135-1145 (1995).
[CrossRef]

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers,” Phys. Rev. E 66, 056604 (2002).
[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]

Phys. Rev. Lett. (1)

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

Proc. R. Soc. London, Ser. A (1)

R. C. McPhedran, C. G. Poulton, N. A. Nicorovici, and A. B. Movchan, “Low frequency corrections to the static effective dielectric constant of a two-dimensional composite material,” Proc. R. Soc. London, Ser. A 452, 2231-2245 (1996).
[CrossRef]

Other (4)

G. von Freymann, S. Wong, G. A. Ozin, S. John, F. Pérez-Willard, M. Deubel, and M. Wegener, in Conference on Lasers and Electro-Optics, 2005, CTuU5, pp. 1002-1004.

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

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

A. Feigel, Z. Kotler, B. Sfez, A. Arsh, M. Klebanov, and V. Lyubin, “Interference lithography for 3D photonic band gap crystal layer by layer fabrication,” in Materials Research Society Symposium Proceedings, 2001, E.D.Jones, O.Manasreh, K.D.Choquette, D.J.Friedman, and D.K.Johnstone, eds.,Vol. 692, K2.9.1.
[CrossRef]

Supplementary Material (1)

» Media 1: AVI (394 KB)     

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Figures (12)

Fig. 1
Fig. 1

Photonic woodpile. A linear waveguide can be created by altering the properties of a single cylinder (red/dark cylinder).

Fig. 2
Fig. 2

(a) Up-down symmetric unit cell consisting of multiple cylinders that are aligned parallel to the x axis and whose centers lie along the y axis. The surfaces U + and U must be chosen so that the cylinders are completely contained inside region A. (b) Configuration of the incident field with wave vector k. The polarization angle δ is defined as the angle between vector ν = k × z ̂ ( k sin θ ) and direction of the electric field, with δ = 0 , π 2 corresponding to T E and T M polarization, respectively.

Fig. 3
Fig. 3

(a) Incoming and diffracted fields, with phase origin at P, as well as the reflection and transmission matrices associated with the fields above and below the unit cell. (b) The unit cell with phase origins P 1 and P 2 adjusted to give the grating a total thickness of h.

Fig. 4
Fig. 4

(a) For the Bloch analysis, a pair of orthogonal rods constitutes a single point of a body-centered tetragonal (BCT) lattice for which the primitive vectors are a 1 = d x ̂ (not shown), a 2 = d y ̂ , and a 3 = ( d 2 ) x ̂ + ( d 2 ) y ̂ + 2 h z ̂ . Offsetting the phase origins P 1 and P 2 laterally, as indicated above, has the effect of interleaving the layers of an infinite stack. (b) The boundaries of the first Brillouin zone (thin lines) and the SBZ (thick lines) of a prolate BCT lattice, such as the lattice shown in part (a).

Fig. 5
Fig. 5

Plot of the number of propagating Bloch modes for each (normalized) k as the in-plane Bloch vector k t = ( k x , k y ) traverses the boundary of the SBZ. White indicates the absence of propagating states. The structural parameters of the woodpile are given in Section 3.

Fig. 6
Fig. 6

Location of transmission maxima (top) for T M incidence and the corresponding Q-factors (bottom) for the CROW using defect sizes of (a) r w = 0 , (b) r w = 0.5 r , and (c) r w = 0.8 r as k x varies. The transmittance is negligible except for the resonances. As r w increases, the resonances move to longer wavelengths. Fields at the point indicated (arrow) in part (b) are shown in Fig. 7.

Fig. 7
Fig. 7

Field intensity in the vicinity of the CROW (only one period in the horizontal direction is shown): (a) | E x | 2 , (b) | E y | 2 , (c) | E z | 2 , (d) | H x | 2 , (e) | H y | 2 , and (f) | H z | 2 . The radius of the defect rods (dashed circles) is r w = 0.5 r , and the parameters of the incoming field are k x d π = 0.59 , k y = 0 , and λ d = 1.958 , i.e., for the point indicated in Fig. 6b.

Fig. 8
Fig. 8

Transmittance and Q factors for a linear waveguide for TM incidence with r w = 0.5 r and k y = 0 fixed and k x d π = 0.86 (red/dotted), 0.92 (blue/dashed), and 1.0 (green/solid). The resonance shifts to shorter wavelengths as k x increases.

Fig. 9
Fig. 9

Transmittance and Q factors for a linear waveguide for T M incidence with k x d π = 1 and k y = 0 fixed. The defect size is r w = 0 (red/dotted), i.e., cylinder completely removed, r w = 0.5 r (blue/dashed), r w = 0.8 r (green/thin), and r w = r , i.e., no defect (black/thick). The resonances shift to longer wavelengths as r w increases.

Fig. 10
Fig. 10

Field intensity in the vicinity of the linear waveguide: (a) | E x | 2 (Media 1), (c) | E y | 2 , (e) | E z | 2 , (b) | H x | 2 , (d) | H y | 2 , and (f) | H z | 2 . The radius of the defect rod (dashed circle) is r w = 0.5 r , and the parameters of the incident field are k x π d = 1 , k y = 0 , and λ d = 1.936 , i.e., for the resonance indicated in Fig. 9. An animation showing | E x | 2 for the entire unit cell is available online.

Fig. 11
Fig. 11

Transmittance of a 16-layer woodpile for (a) T E and (b) T M incidence. The incident field is perpendicular to the grating plane. The exact values are shown in red/thick, while the blue/thin curve corresponds to a 16-layer stack of alternating homogeneous slabs, each with thickness h = d and permittivity given either by Eq. (44) or Eq. (45). The green/dashed curve corresponds to a single slab of thickness 8 d with permittivity given by Eq. (46).

Fig. 12
Fig. 12

(a) Four-layer photonic woodpile and successive approximations using (b) alternating slabs and (c) a single block of dielectric. For the woodpile parameters given in Section 3C, the approximations above can be used in place of a woodpile when λ d > 10 .

Equations (81)

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

γ s 2 = ( k n b ) 2 ( α p 2 + β q 2 )
F I ± = [ [ E I , s ± ] [ F I , s ± ] ] and F D ± = [ [ E D , s ± ] [ F D , s ± ] ]
E t = ( μ b ɛ b ) 1 4 s ξ s 1 2 [ E I , s e i γ s z + E D , s + e i γ s z ] e i ( α p x + β q y ) R s E + ξ s 1 2 [ F I , s e i γ s z + F D , s + e i γ s z ] e i ( α p x + β q y ) R s M ,
z ̂ × H t = ( ɛ b μ b ) 1 4 s ξ s 1 2 [ E I , s e i γ s z E D , s + e i γ s z ] e i ( α p x + β q y ) R s E + ξ s 1 2 [ F I , s e i γ s z F D , s + e i γ s z ] e i ( α p x + β q y ) R s M ,
[ F D F D + ] = [ T a R b R a T b ] [ F I F I + ] .
[ 2 + k 2 ] V ( r ) = 0
V ( r ) = A [ V ( r ) r 2 G ( r r ) G ( r r ) r 2 V ( r ) ] d r ,
= C [ V ( r ) n G ( r r ) G ( r r ) n V ( r ) ] d r ,
[ 2 + k 2 ] G ( r ) = n = δ ( r n D y ̂ ) exp ( i β 0 n D ) .
V l ( r l ) = n = [ A n l , V J n ( k r l ) + B n l , V H n ( k r l ) ] e i n θ r l e i α p x .
V l ( r l ) = n = B n l , V H n ( k r l ) e i n θ r l e i α p x + n = J n ( k r l ) e i n θ r l e i α p x [ j = 1 N c m = B m j , V S n m l j + q = ( J n q l , δ q , V + J n q l , + δ q + , V ) ] ,
A = S B + J D + J + D + .
A = [ [ A l , E ] [ A l , H ] ] ,
V l ( r l ) = n = C n l , V J n ( k , l r l ) e i n θ r l e i α p x ,
A = M B .
B = ( M + S ) 1 J D ,
F = D + 2 D G K B ,
[ F D F D + ] = [ I 2 D X ̂ 1 Z 1 G K ( M + S ) 1 J Z X ̂ ] [ F I F I + ] ,
[ F D F D + ] = [ F I F I + ] k n b k 2 D T 1 X Z 1 [ K s L J s K s L J a K a L J s K a L J a ] Z X T [ F I F I + ] ,
T = [ I I I I ] ,
R a = R b = k n b 2 k 2 D X Z 1 ( K ̃ s ε s L ̃ s J ̃ s K ̃ a ε a L ̃ a J ̃ a ) Z X ,
T a = T b = I k n b 2 k 2 D X Z 1 ( K ̃ s ε s L ̃ s J ̃ s + K ̃ a ε a L ̃ a J ̃ a ) Z X .
F = [ F 1 T F 0 T F 1 T ] T ,
R a s + 1 = R ̃ a + T ̃ b R a s ( I R ̃ b R a s ) 1 T ̃ a ,
T a s + 1 = T a s ( I R ̃ b R a s ) 1 T ̃ a ,
R b s + 1 = R b s + T a s R ̃ b ( I R a s R ̃ b ) 1 T b s ,
T b s + 1 = T ̃ b ( I R a s R ̃ b ) 1 T b s .
[ R ̃ a T ̃ b T ̃ a R ̃ b ] = [ Q P 0 0 Q P ] [ R a T b T a R b ] [ P Q 1 0 0 P Q 1 ] ,
Q = { diag [ e i ( α p + β q ) d 2 ] , for a shifted layer , I , otherwise . }
[ R ̃ a ( 2 ) T ̃ b ( 2 ) T ̃ a ( 2 ) R ̃ b ( 2 ) ] = [ Q 0 0 Q 1 ] [ R a ( 2 ) T b ( 2 ) T a ( 2 ) R b ( 2 ) ] [ Q 1 0 0 Q ] .
F 2 = μ F 1 ,
T F 1 = μ F 1 .
D s E , = δ s 0 cos δ
D s H , = δ s 0 sin δ ,
δ i j = { 1 , for i = j , 0 , otherwise . }
T = s | t s | 2 ,
R E E i k h 2 1 N c l = 1 N c f l ( ɛ l 1 ) ,
R H H i k h 2 1 N c l = 1 N c 2 f l ɛ l + 1 ɛ l 1 π h 3 d f l ,
T E E 1 + R E E ,
T H H 1 + R H H .
R pair E E = R H H + T E E T H H R E E ( 1 R E E R H H ) ,
= R H H + R E E + O ( k 2 ) ,
R pair H H ,
ɛ T E 1 = 1 N c l = 1 N c f l ( ɛ l 1 )
ɛ T M 1 = 1 N c l = 1 N c 2 f l ɛ l + 1 ɛ l 1 π h 3 d f l
ɛ pair 1 = [ h 1 ( ɛ T E 1 ) + h 2 ( ɛ T M 1 ) ] ( h 1 + h 2 )
ɛ eff = 1 + 2 f 1 ɛ 1 + 1 ɛ 1 1 f 1 .
G ( r r ) = i 4 m = H m ( k r ) J m ( k r ) exp ( i m θ r ) exp ( i m θ r ) + n = J n ( k r ) exp ( i n θ r ) s = S n s J s ( k r ) exp ( i s θ r )
G ( r l r j c j + c l ) = i 4 m = J m ( k r l ) exp ( i m θ r l ) × s = S m s l j J s ( k r j ) exp ( i s θ r j )
S m = n 0 H m ( k | c n | ) e i β 0 n D exp [ i m θ c n ] ,
S m l j = n = H m ( k | c n l j | ) e i β 0 n D exp [ i m θ c n l j ] ,
K = [ ( K ) T ( K + ) T ] T ,
K ± = [ K 1 , ± K N c , ± 0 0 0 0 K 1 , ± K N c , ± ] ,
L ̃ s a = [ M ̃ E E + S ̃ s a ε s a M ̃ E H M ̃ H E M ̃ H H + S ̃ a s ε a s ] 1 ,
ε m = { 1 2 , for m = 0 , 1 , for m > 0 }
M = [ M E E M E H M H E M H H ] .
M l , n E E = η 2 ( I ) η 3 ( I ) H n ( k I r l ) J n ( k I r l ) Δ × { [ η 1 ( I ) η 1 ( II ) ] 2 η 2 ( I ) η 3 ( I ) J 2 H 3 } ,
M l , n H H = η 2 ( I ) η 3 ( I ) H n ( k I r l ) J n ( k I r l ) Δ × { [ η 1 ( I ) η 1 ( II ) ] 2 η 2 ( I ) η 3 ( I ) J 3 H 2 } ,
M l , n E H = [ η 1 ( I ) η 1 ( II ) ] η 2 ( I ) η 3 ( I ) Δ × 2 k μ I π k I 2 r l ,
M l , n H E = ( ɛ I μ I ) M l , n E H ,
Δ = J n ( k I r l ) 2 η 2 ( I ) η 3 ( I ) × { [ η 1 ( I ) η 1 ( II ) ] 2 η 2 ( I ) η 3 ( I ) J 2 J 3 } ,
J j = J n ( k I r l ) J n ( k I r l ) η j ( II ) J n ( k II r l ) η j ( I ) J n ( k II r l ) ,
H j = H n ( k I r l ) H n ( k I r l ) η j ( II ) J n ( k II r l ) η j ( II ) J n ( k II r l ) .
R E E R H H 1 2 k D × [ K ( M E E + S ) 1 J K ( M E E + S ) 1 J ] ,
R H H R E E 1 2 k D × [ K ( M H H + S ) 1 J K ( M H H + S ) 1 J ] ,
T E E T H H 1 1 2 k D × [ K ( M E E + S ) 1 J + K ( M E E + S ) 1 J ] ,
T H H T E E 1 1 2 k D × [ K ( M H H + S ) 1 J + K ( M H H + S ) 1 J ] ,
M l , 0 E E 4 i f l α 2 ( ɛ l 1 ) ,
M l , 1 E E M l , 0 H H = 32 π i f l 2 α 4 ( ɛ l 1 ) ,
M l , 1 H H M l , 0 E E ( ɛ l + 1 ) ,
K ( M E E H H + S ) 1 J 4 l = 1 N c 1 M l , 0 E E H H
K ( M E E H H + S ) 1 J = 0 ,
K ( M E E H H + S ) 1 J 8 l = 1 N c 1 M l , 1 E E H H 4 π i 3 α 2 .
R E E i k h 2 1 N c l = 1 N c f l ( ɛ l 1 ) = O ( α ) ,
T E E 1 + R 0 E E ,
R H H i k h 2 ( α 2 8 π N c ) l = 1 N c f l 2 ( ɛ l 1 ) = O ( α 3 ) ,
T H H 1 R 0 H H .
R E E R 0 E E + Δ E E R 0 E E ,
Δ E E i k h 2 ( 2 α 2 N c ) l = 1 N c f l 2 8 π ɛ l 1 π 3 ( k h f l ) 2 = O ( α 3 )
R H H R 0 H H + Δ H H ,
Δ H H i k h 2 ( 2 N c ) l = 1 N c f l ɛ l + 1 ɛ l 1 π h 3 d f l = O ( α ) .

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