Abstract

A rigorous semi-analytic approach to the modelling of coupling, guiding and propagation in complex microstructures embedded in two-dimensional photonic crystals is presented. The method, which is based on Bloch mode expansions and generalized Fresnel coefficients, is shown to be able to treat photonic crystal devices in ways which are analogous to those used in thin film optics with uniform media. Asymptotic methods are developed and exemplified through the study of a serpentine waveguide, a potential slow wave device.

© 2004 Optical Society of America

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  5. T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B 66, 041303(R) (2002).
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  8. S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunnelling through Localized States,” Phys. Rev. Lett. 80, 960–963 (1998).
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    [Crossref]
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    [Crossref]
  18. 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]
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    [Crossref]
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    [Crossref]
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2003 (8)

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066616 (2003).
[Crossref]

A. Martinez, A. Griol, P. Sanchis, and J. Marti, “Mach-Zehnder interferometer employing coupled-resonator optical waveguides,” Opt. Lett. 28, 405–407 (2003).
[Crossref] [PubMed]

T.P. White, L.C. Botten, R.C. McPhedran, and C.M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. 28, 2452–2454 (2003).
[Crossref] [PubMed]

L.C. Botten, A.A. Asatryan, T.N. Langtry, T.P. White, C.M. de Sterke, and R.C. McPhedran, “Semi-analytic treatment for propagation in finite photonic crystal waveguides,” Opt. Lett. 28, 854–856 (2003).
[Crossref] [PubMed]

S.F. Mingaleev and K. Busch, “Scattering matrix approach to large-scale photonic crystal circuits,” Opt. Lett. 28, 619–621 (2003).
[Crossref] [PubMed]

Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. E 68, 155101 (2003).

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]

K. Busch, S.F. . Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys.: Condens. Matter 15, 1233–1254 (2003).
[Crossref]

2002 (4)

2001 (2)

A. Chutinan, M. Mochizuki, M.. Imada, and S. Noda, “Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 79, 2690–2692 (2001).
[Crossref]

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[Crossref]

2000 (3)

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. 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.2, Method,” J. Opt. Soc. Am. A. 17, 2177–2190 (2000).
[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.1, Method,” J. Opt. Soc. Am. A. 17, 2165–2176 (2000).
[Crossref]

1999 (2)

J. Yonekura, M. Ikeda, and T. Baba, “Analysis of Finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method,” J. Lightwave Technol. 17, 1500–1508 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[Crossref]

1998 (1)

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunnelling through Localized States,” Phys. Rev. Lett. 80, 960–963 (1998).
[Crossref]

1996 (1)

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[Crossref] [PubMed]

Asatryan, A. A.

L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures,” in preparation.

Asatryan, A.A.

L.C. Botten, A.A. Asatryan, T.N. Langtry, T.P. White, C.M. de Sterke, and R.C. McPhedran, “Semi-analytic treatment for propagation in finite photonic crystal waveguides,” Opt. Lett. 28, 854–856 (2003).
[Crossref] [PubMed]

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[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.1, Method,” J. Opt. Soc. Am. A. 17, 2165–2176 (2000).
[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.2, Method,” J. Opt. Soc. Am. A. 17, 2177–2190 (2000).
[Crossref]

Baba, T.

Bayindir, M.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

Botten, L. C.

L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures,” in preparation.

S. Chen, R. C. McPhedran, C. M. De Sterke, and L. C. Botten, “Optimal tapers in photonic crystal waveguides,” submitted to Appl. Phys. Lett.

Botten, L.C.

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, A.A. Asatryan, T.N. Langtry, T.P. White, C.M. de Sterke, and R.C. McPhedran, “Semi-analytic treatment for propagation in finite photonic crystal waveguides,” Opt. Lett. 28, 854–856 (2003).
[Crossref] [PubMed]

T.P. White, L.C. Botten, R.C. McPhedran, and C.M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. 28, 2452–2454 (2003).
[Crossref] [PubMed]

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[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.1, Method,” J. Opt. Soc. Am. A. 17, 2165–2176 (2000).
[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.2, Method,” J. Opt. Soc. Am. A. 17, 2177–2190 (2000).
[Crossref]

Busch, K.

K. Busch, S.F. . Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys.: Condens. Matter 15, 1233–1254 (2003).
[Crossref]

S.F. Mingaleev and K. Busch, “Scattering matrix approach to large-scale photonic crystal circuits,” Opt. Lett. 28, 619–621 (2003).
[Crossref] [PubMed]

Chak, P.

Chen, J.C.

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[Crossref] [PubMed]

Chen, S.

S. Chen, R. C. McPhedran, C. M. De Sterke, and L. C. Botten, “Optimal tapers in photonic crystal waveguides,” submitted to Appl. Phys. Lett.

Chutinan, A.

A. Chutinan, M. Mochizuki, M.. Imada, and S. Noda, “Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 79, 2690–2692 (2001).
[Crossref]

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

De Sterke, C. M.

S. Chen, R. C. McPhedran, C. M. De Sterke, and L. C. Botten, “Optimal tapers in photonic crystal waveguides,” submitted to Appl. Phys. Lett.

L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures,” in preparation.

de Sterke, C.M.

T.P. White, L.C. Botten, R.C. McPhedran, and C.M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. 28, 2452–2454 (2003).
[Crossref] [PubMed]

L.C. Botten, A.A. Asatryan, T.N. Langtry, T.P. White, C.M. de Sterke, and R.C. McPhedran, “Semi-analytic treatment for propagation in finite photonic crystal waveguides,” Opt. Lett. 28, 854–856 (2003).
[Crossref] [PubMed]

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[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.1, Method,” J. Opt. Soc. Am. A. 17, 2165–2176 (2000).
[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.2, Method,” J. Opt. Soc. Am. A. 17, 2177–2190 (2000).
[Crossref]

Enoch, S.

Fan, S.

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066616 (2003).
[Crossref]

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunnelling through Localized States,” Phys. Rev. Lett. 80, 960–963 (1998).
[Crossref]

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[Crossref] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teulolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1988).

Forchel, A.

T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B 66, 041303(R) (2002).
[Crossref]

Garcia-Martin, A.

K. Busch, S.F. . Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys.: Condens. Matter 15, 1233–1254 (2003).
[Crossref]

Gralak, B.

Griol, A.

Hamermesh, M.

M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, 1962).

Happ, T. D.

T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B 66, 041303(R) (2002).
[Crossref]

Hermann, D.

K. Busch, S.F. . Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys.: Condens. Matter 15, 1233–1254 (2003).
[Crossref]

Ho, K.-M.

Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. E 68, 155101 (2003).

Ikeda, M.

Imada, M..

A. Chutinan, M. Mochizuki, M.. Imada, and S. Noda, “Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 79, 2690–2692 (2001).
[Crossref]

Joannopoulos, J.D.

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunnelling through Localized States,” Phys. Rev. Lett. 80, 960–963 (1998).
[Crossref]

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[Crossref] [PubMed]

Kamp, M.

T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B 66, 041303(R) (2002).
[Crossref]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[Crossref]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[Crossref]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[Crossref]

Kulakovskii, V.D.

T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B 66, 041303(R) (2002).
[Crossref]

Kurland, I.

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[Crossref] [PubMed]

Langtry, T. N.

L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures,” in preparation.

Langtry, T.N.

Li, Z.-Y.

Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. E 68, 155101 (2003).

Marti, J.

Martinez, A.

McPhedran, R. C.

S. Chen, R. C. McPhedran, C. M. De Sterke, and L. C. Botten, “Optimal tapers in photonic crystal waveguides,” submitted to Appl. Phys. Lett.

L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures,” in preparation.

McPhedran, R.C.

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]

T.P. White, L.C. Botten, R.C. McPhedran, and C.M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. 28, 2452–2454 (2003).
[Crossref] [PubMed]

L.C. Botten, A.A. Asatryan, T.N. Langtry, T.P. White, C.M. de Sterke, and R.C. McPhedran, “Semi-analytic treatment for propagation in finite photonic crystal waveguides,” Opt. Lett. 28, 854–856 (2003).
[Crossref] [PubMed]

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[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.1, Method,” J. Opt. Soc. Am. A. 17, 2165–2176 (2000).
[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.2, Method,” J. Opt. Soc. Am. A. 17, 2177–2190 (2000).
[Crossref]

Mekis, A.

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[Crossref] [PubMed]

Mingaleev, S.F.

Mingaleev, S.F. .

K. Busch, S.F. . Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys.: Condens. Matter 15, 1233–1254 (2003).
[Crossref]

Mochizuki, M.

A. Chutinan, M. Mochizuki, M.. Imada, and S. Noda, “Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 79, 2690–2692 (2001).
[Crossref]

Nicorovici, N.A.

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, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[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.1, Method,” J. Opt. Soc. Am. A. 17, 2165–2176 (2000).
[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.2, Method,” J. Opt. Soc. Am. A. 17, 2177–2190 (2000).
[Crossref]

Noda, S.

A. Chutinan, M. Mochizuki, M.. Imada, and S. Noda, “Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 79, 2690–2692 (2001).
[Crossref]

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

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[Crossref]

Ozbay, E.

Pereira, S.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teulolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1988).

Reithmaier, J.-P.

T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B 66, 041303(R) (2002).
[Crossref]

Robinson, P.A.

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.1, Method,” J. Opt. Soc. Am. A. 17, 2165–2176 (2000).
[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.2, Method,” J. Opt. Soc. Am. A. 17, 2177–2190 (2000).
[Crossref]

Sanchis, P.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[Crossref]

Schillinger, M.

K. Busch, S.F. . Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys.: Condens. Matter 15, 1233–1254 (2003).
[Crossref]

Sipe, J.E.

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]

Soukoulis, C. M.

C. M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, (Kluwer, Dordrecht2001).

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[Crossref]

Tartakovskii, I.I.

T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B 66, 041303(R) (2002).
[Crossref]

Tayeb, G.

Teulolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teulolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1988).

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[Crossref]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teulolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1988).

Villeneuve, P.R.

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunnelling through Localized States,” Phys. Rev. Lett. 80, 960–963 (1998).
[Crossref]

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[Crossref] [PubMed]

Wang, Z.

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066616 (2003).
[Crossref]

White, T. P.

L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures,” in preparation.

White, T.P.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

Wolfram, S.

S. Wolfram, The Mathematica Book, 3rd Ed., (Wolfram Media/Cambridge University Press, 1996).

Yeh, P.

P. Yeh, Optical waves in layered media, (Wiley, New York, 1988), Ch. 6.

Yonekura, J.

Appl. Phys. Lett. (2)

A. Chutinan, M. Mochizuki, M.. Imada, and S. Noda, “Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 79, 2690–2692 (2001).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. 74, 1370–1372 (1999).
[Crossref]

J. Lightwave Technol. (1)

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

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

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.2, Method,” J. Opt. Soc. Am. A. 17, 2177–2190 (2000).
[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.1, Method,” J. Opt. Soc. Am. A. 17, 2165–2176 (2000).
[Crossref]

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

J. Phys.: Condens. Matter (1)

K. Busch, S.F. . Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys.: Condens. Matter 15, 1233–1254 (2003).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. B (2)

T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B 66, 041303(R) (2002).
[Crossref]

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

Phys. Rev. E (4)

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066616 (2003).
[Crossref]

Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. E 68, 155101 (2003).

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[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]

Phys. Rev. Lett. (2)

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[Crossref] [PubMed]

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunnelling through Localized States,” Phys. Rev. Lett. 80, 960–963 (1998).
[Crossref]

Other (8)

C. M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, (Kluwer, Dordrecht2001).

S. Chen, R. C. McPhedran, C. M. De Sterke, and L. C. Botten, “Optimal tapers in photonic crystal waveguides,” submitted to Appl. Phys. Lett.

W. H. Press, B. P. Flannery, S. A. Teulolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1988).

M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, 1962).

L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures,” in preparation.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

P. Yeh, Optical waves in layered media, (Wiley, New York, 1988), Ch. 6.

S. Wolfram, The Mathematica Book, 3rd Ed., (Wolfram Media/Cambridge University Press, 1996).

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Figures (4)

Fig. 1.
Fig. 1.

A typical three segment photonic crystal device showing the component regions M1, M2 and M3, three lateral supercells of the model and a constituent grating bounded by dashed lines.

Fig. 2.
Fig. 2.

Two equivalent serpentine waveguide geometries (assuming no tunneling through the guide ends). Both are characterized by the period Dy and the double and single guide lengths, L 1 and L 2.

Fig. 3.
Fig. 3.

(a) Band diagram for a serpentine tine waveguide with L 1=L 2=5d, where q is the Bloch coefficient along the waveguide, and Dy =L 1+L 2. (b) Transmission through an FDC with same parameters (dashed), one period of the serpentine waveguide (dotted) and two periods (solid). Note that for frequencies below ωd/(2πc)=0.3064, the double guide cavity only supports a single, odd mode, and thus the analytic result of (19) does not apply.

Fig. 4.
Fig. 4.

(a) Band diagram for a serpentine waveguide with L 1=L 2=7d. The solid curve is calculated with the full numerical simulation while the dashed curve is calculated using the approximation (19) L 1=7.5d, L 2=6.7d (b) Transmission through a FDC with L=7d (dashed), 2 periods of the serpentine guide (dotted) and 3 periods of the serpentine guide.

Equations (22)

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𝓣 𝓯 = μ 𝓯 where 𝓣 = ( T R T 1 R R T 1 T 1 R T 1 ) , 𝓯 = ( f f + ) ,
𝓣 T 𝓠 𝓣 = 𝓠 , where 𝓠 = [ 0 Q Q 0 ] .
𝓣 = 𝓕 𝓕 1 with 𝓕 = [ F F + F + F ] , = [ Λ 0 0 Λ 1 ] , Λ = diag ( μ i ) .
𝓕 T 𝓠 𝓕 = where = ( 0 I I 0 )
𝓕 H 𝓣 p 𝓕 = 𝓣 m where 𝓣 p = ( I r i I e i I e I r ) and 𝓣 m = ( I m i I m ¯ i I m ¯ I m ) ,
R ij = ( F i ) 1 ( I R j R i ) 1 ( R j R i ) F i ,
T ij = ( F j ) 1 ( I R i R j ) 1 ( I R i 2 ) F i ,
r = R 12 δ + T 21 Λ L c + , c = T 12 δ + R 21 Λ L c + ,
c + = R 23 Λ L c , t = T 23 Λ L c ,
R = R 13 = R 12 + T 21 Λ L R 23 Λ L ( I R 21 Λ L R 23 Λ L ) 1 T 12 ,
T = T 13 = T 23 Λ L ( I R 21 Λ L R 23 Λ L ) 1 T 12 .
S 12 2 = I , where S 12 = ( R 12 T 21 T 12 R 21 )
R 13 = T 12 1 ( R 21 + Λ L R 23 Λ L ) ( I R 21 Λ L R 23 Λ L ) 1 T 12 ,
= T 21 1 ( I Λ L R 23 Λ L R 21 ) 1 ( R 21 + Λ L R 23 Λ L ) T 21 .
S 13 H S 13 = I 13 , where S 13 = ( R 13 T 31 T 13 R 31 ) , I 13 = ( I 1 0 0 I 3 ) ,
w 1 T = w 3 T = [ 1 0 0 0 ] , w 2 T = [ 1 0 0 0 0 1 0 0 ] .
T ˜ 13 = T ˜ 23 Λ ˜ L ( I R ˜ 21 Λ ˜ L R ˜ 23 Λ ˜ L ) 1 T ˜ 12 ,
R 13 = ρ f = cos 2 ( Δ β L ) exp ( 2 i β ¯ L ) 1 + sin 2 ( Δ β L ) exp ( 2 i β ¯ L ) , T 13 = τ f = i exp ( i β ¯ L ) sin ( Δ β L ) ( 1 + exp ( 2 i β ¯ L ) ) 1 + sin 2 ( Δ β L ) exp ( 2 i β ¯ L ) ,
𝓣 s = ( τ s ρ s 2 τ s ρ s τ s ρ s τ s 1 τ s ) ,
𝓣 s = 𝓣 f 2 , where 𝓣 f = ( τ f ρ f 2 τ f ρ f τ f ρ f τ f 1 τ f )
μ + 1 μ 1 2 = τ f 2 ρ f 2 + 1 τ f , i. e . cos ( q D y 2 ) = ( 1 τ f ) ,
cos ( q D 2 ) = sin ( Δ β L 1 ) sin ( β ¯ L 1 + β L 2 ) + cos 2 ( Δ β L 1 ) sin ( 2 β ¯ L 1 + β L 2 ) 2 sin ( Δ β L 1 ) cos ( β ¯ L 1 ) .

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