Abstract

We present a semianalytic theory for the properties of two-dimensional photonic crystal waveguides of finite length. For single-mode guides, the transmission spectrum and field intensity can be accurately described by a simple two-parameter model. Analogies are drawn with Fabry–Perot interferometers, and generalized Fresnel coefficients for the interfaces are calculated.

© 2003 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Maede, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
  2. C. M. Soukoulis, ed., Photonic Crystals and Light Localization in the 21st Century, Vol. 563 of NATO ASI Series C: Mathematical and Physics Sciences (Kluwer Academic, Dordrecht, The Netherlands, 2001).
    [CrossRef]
  3. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
    [CrossRef] [PubMed]
  4. S.-Y. Lin, E. Chow, V. Hietala, P. P. Villeneuve, and J. D. Joannopoulos, Science 282, 274 (1998).
    [CrossRef] [PubMed]
  5. J. Yonekura, M. Ikeda, and T. Baba, J. Lightwave Technol. 17, 1500 (1999).
    [CrossRef]
  6. J. D. Joannopoulos, S. Fan, A. Mekis, and S. G. Johnson, in Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis, ed., Vol. 563 of NATO ASI Series C: Mathematical and Physics Sciences (Kluwer Academic, Dordrecht, The Netherlands, 2001), pp. 1–24.
    [CrossRef]
  7. S. Fan, P. R. Villeneuve, J. D. Joannopoulous, and H. A. Haus, Phys. Rev. Lett. 80, 960 (1998).
    [CrossRef]
  8. A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, Phys. Rev. E 63, 046612 (2001).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, 1993), Sec. 1.6.4.
  10. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. Martijn de Sterke, and P. A. Robinson, J. Opt. Soc. Am. A 17, 2165 (2000).
    [CrossRef]
  11. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, Phys. Rev. E 64, 046603 (2001).
    [CrossRef]
  12. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge U. Press, Cambridge, 1997).

2001

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, Phys. Rev. E 63, 046612 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, Phys. Rev. E 64, 046603 (2001).
[CrossRef]

2000

1999

1998

S.-Y. Lin, E. Chow, V. Hietala, P. P. Villeneuve, and J. D. Joannopoulos, Science 282, 274 (1998).
[CrossRef] [PubMed]

S. Fan, P. R. Villeneuve, J. D. Joannopoulous, and H. A. Haus, Phys. Rev. Lett. 80, 960 (1998).
[CrossRef]

1996

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Asatryan, A. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, Phys. Rev. E 63, 046612 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, Phys. Rev. E 64, 046603 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. Martijn de Sterke, and P. A. Robinson, J. Opt. Soc. Am. A 17, 2165 (2000).
[CrossRef]

Baba, T.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, 1993), Sec. 1.6.4.

Botten, L. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, Phys. Rev. E 64, 046603 (2001).
[CrossRef]

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, Phys. Rev. E 63, 046612 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. Martijn de Sterke, and P. A. Robinson, J. Opt. Soc. Am. A 17, 2165 (2000).
[CrossRef]

Busch, K.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, Phys. Rev. E 63, 046612 (2001).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Chow, E.

S.-Y. Lin, E. Chow, V. Hietala, P. P. Villeneuve, and J. D. Joannopoulos, Science 282, 274 (1998).
[CrossRef] [PubMed]

Fan, S.

S. Fan, P. R. Villeneuve, J. D. Joannopoulous, and H. A. Haus, Phys. Rev. Lett. 80, 960 (1998).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

J. D. Joannopoulos, S. Fan, A. Mekis, and S. G. Johnson, in Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis, ed., Vol. 563 of NATO ASI Series C: Mathematical and Physics Sciences (Kluwer Academic, Dordrecht, The Netherlands, 2001), pp. 1–24.
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge U. Press, Cambridge, 1997).

Haus, H. A.

S. Fan, P. R. Villeneuve, J. D. Joannopoulous, and H. A. Haus, Phys. Rev. Lett. 80, 960 (1998).
[CrossRef]

Hietala, V.

S.-Y. Lin, E. Chow, V. Hietala, P. P. Villeneuve, and J. D. Joannopoulos, Science 282, 274 (1998).
[CrossRef] [PubMed]

Ikeda, M.

Joannopoulos, J. D.

S.-Y. Lin, E. Chow, V. Hietala, P. P. Villeneuve, and J. D. Joannopoulos, Science 282, 274 (1998).
[CrossRef] [PubMed]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

J. D. Joannopoulos, R. D. Maede, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

J. D. Joannopoulos, S. Fan, A. Mekis, and S. G. Johnson, in Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis, ed., Vol. 563 of NATO ASI Series C: Mathematical and Physics Sciences (Kluwer Academic, Dordrecht, The Netherlands, 2001), pp. 1–24.
[CrossRef]

Joannopoulous, J. D.

S. Fan, P. R. Villeneuve, J. D. Joannopoulous, and H. A. Haus, Phys. Rev. Lett. 80, 960 (1998).
[CrossRef]

Johnson, S. G.

J. D. Joannopoulos, S. Fan, A. Mekis, and S. G. Johnson, in Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis, ed., Vol. 563 of NATO ASI Series C: Mathematical and Physics Sciences (Kluwer Academic, Dordrecht, The Netherlands, 2001), pp. 1–24.
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Lin, S.-Y.

S.-Y. Lin, E. Chow, V. Hietala, P. P. Villeneuve, and J. D. Joannopoulos, Science 282, 274 (1998).
[CrossRef] [PubMed]

Maede, R. D.

J. D. Joannopoulos, R. D. Maede, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Martijn de Sterke, C.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, Phys. Rev. E 63, 046612 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, Phys. Rev. E 64, 046603 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. Martijn de Sterke, and P. A. Robinson, J. Opt. Soc. Am. A 17, 2165 (2000).
[CrossRef]

McPhedran, R. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, Phys. Rev. E 64, 046603 (2001).
[CrossRef]

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, Phys. Rev. E 63, 046612 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. Martijn de Sterke, and P. A. Robinson, J. Opt. Soc. Am. A 17, 2165 (2000).
[CrossRef]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

J. D. Joannopoulos, S. Fan, A. Mekis, and S. G. Johnson, in Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis, ed., Vol. 563 of NATO ASI Series C: Mathematical and Physics Sciences (Kluwer Academic, Dordrecht, The Netherlands, 2001), pp. 1–24.
[CrossRef]

Nicorovici, N. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, Phys. Rev. E 63, 046612 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, Phys. Rev. E 64, 046603 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. Martijn de Sterke, and P. A. Robinson, J. Opt. Soc. Am. A 17, 2165 (2000).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge U. Press, Cambridge, 1997).

Robinson, P. A.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge U. Press, Cambridge, 1997).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge U. Press, Cambridge, 1997).

Villeneuve, P. P.

S.-Y. Lin, E. Chow, V. Hietala, P. P. Villeneuve, and J. D. Joannopoulos, Science 282, 274 (1998).
[CrossRef] [PubMed]

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, J. D. Joannopoulous, and H. A. Haus, Phys. Rev. Lett. 80, 960 (1998).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Winn, J. N.

J. D. Joannopoulos, R. D. Maede, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, 1993), Sec. 1.6.4.

Yonekura, J.

J. Lightwave Technol.

J. Opt. Soc. Am. A

NATO ASI Series C: Mathematical and Physics Sciences

C. M. Soukoulis, ed., Photonic Crystals and Light Localization in the 21st Century, Vol. 563 of NATO ASI Series C: Mathematical and Physics Sciences (Kluwer Academic, Dordrecht, The Netherlands, 2001).
[CrossRef]

NATO ASI Series C: Mathematical and Physics Sciences

J. D. Joannopoulos, S. Fan, A. Mekis, and S. G. Johnson, in Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis, ed., Vol. 563 of NATO ASI Series C: Mathematical and Physics Sciences (Kluwer Academic, Dordrecht, The Netherlands, 2001), pp. 1–24.
[CrossRef]

Phys. Rev. E

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, Phys. Rev. E 63, 046612 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Phys. Rev. Lett.

S. Fan, P. R. Villeneuve, J. D. Joannopoulous, and H. A. Haus, Phys. Rev. Lett. 80, 960 (1998).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Science

S.-Y. Lin, E. Chow, V. Hietala, P. P. Villeneuve, and J. D. Joannopoulos, Science 282, 274 (1998).
[CrossRef] [PubMed]

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77 (Cambridge U. Press, Cambridge, 1997).

M. Born and E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, 1993), Sec. 1.6.4.

J. D. Joannopoulos, R. D. Maede, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

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Figures (3)

Fig. 1
Fig. 1

(a) Field intensity (decimal log scale) in a finite cluster due to a point source (black dot). The waveguide corresponds to the removal of a line of cylinders of radius a/d=0.3, refractive index νc=3 (in air νb=1), and wavelength λ/d=3.19. (b) Supercell model approximating geometry (a), with guides spaced by D=11d.

Fig. 2
Fig. 2

(a) Transmission spectrum calculated with multipole (dashed curve) and asymptotic (solid curve) models. (b) ρ (solid curve), arg ρ (short-dashed curve), and arg μ (long-dashed curve) versus wavelength. The left (right) axis displays the magnitude (argument).

Fig. 3
Fig. 3

Comparison of the field intensity [Fig. 1(a)] along the waveguide axis, calculated with the multipole method (dashed curve) and the asymptotic model (solid curve).

Equations (3)

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

T=T23ΛLI-R21ΛLR23ΛL-1T12.
T=F+I-ρ2ΛLI-ρΛLρΛL-1F+-1.
T=1-ρ21-ρ2μ2L2v+δ2.

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