Abstract

We suggest a novel conceptual approach to describing the properties of waveguides and circuits in photonic crystals, based on effective discrete equations that include long-range interaction effects. We demonstrate, through the example of sharp waveguide bends, that our approach is very effective and accurate for the study of bound states and transmission spectra of photonic-crystal circuits and disclose the importance of evanescent modes in these phenomena.

© 2002 Optical Society of America

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References

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  1. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).
    [CrossRef]
  2. T. F. Krauss and R. M. De la Rue, Prog. Quantum Electron. 23, 51 (1999), and references therein.
    [CrossRef]
  3. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  4. A. Mekis, S. H. Fan, and J. D. Joannopoulos, Phys. Rev. B 58, 4809 (1998).
  5. A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannapoulos, Phys. Rev. Lett. 77, 3787 (1996).
    [PubMed]
  6. S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, Science 282, 274 (1998).
    [PubMed]
  7. S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, J. Opt. Soc. Am. B 18, 162 (2001).
  8. S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 80, 960 (1998).
  9. S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, Phys. Rev. E 62, 5777 (2000).
  10. S. F. Mingaleev and Yu. S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).
    [PubMed]
  11. T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, J. Vac. Sci. Technol. B 17, 2734 (1999).
    [CrossRef]
  12. A. R. McGurn, Phys. Rev. B 53, 7059 (1996).
    [CrossRef]

2001 (3)

2000 (1)

S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, Phys. Rev. E 62, 5777 (2000).

1999 (2)

T. F. Krauss and R. M. De la Rue, Prog. Quantum Electron. 23, 51 (1999), and references therein.
[CrossRef]

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, J. Vac. Sci. Technol. B 17, 2734 (1999).
[CrossRef]

1998 (3)

A. Mekis, S. H. Fan, and J. D. Joannopoulos, Phys. Rev. B 58, 4809 (1998).

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

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 80, 960 (1998).

1996 (2)

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

A. R. McGurn, Phys. Rev. B 53, 7059 (1996).
[CrossRef]

Chen, J. C.

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

Chow, E.

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

de Dood, M. J. A.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, J. Vac. Sci. Technol. B 17, 2734 (1999).
[CrossRef]

De la Rue, R. M.

T. F. Krauss and R. M. De la Rue, Prog. Quantum Electron. 23, 51 (1999), and references therein.
[CrossRef]

Fan, S.

Fan, S. H.

A. Mekis, S. H. Fan, and J. D. Joannopoulos, Phys. Rev. B 58, 4809 (1998).

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 80, 960 (1998).

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

Haus, H. A.

Hietala, V.

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

Joannapoulos, J. D.

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

Joannopoulos, J. D.

S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, J. Opt. Soc. Am. B 18, 162 (2001).

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001), http://www.opticsexpress.org .
[CrossRef] [PubMed]

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

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 80, 960 (1998).

A. Mekis, S. H. Fan, and J. D. Joannopoulos, Phys. Rev. B 58, 4809 (1998).

Johnson, S. G.

Kivshar, Yu. S.

S. F. Mingaleev and Yu. S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).
[PubMed]

S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, Phys. Rev. E 62, 5777 (2000).

Krauss, T. F.

T. F. Krauss and R. M. De la Rue, Prog. Quantum Electron. 23, 51 (1999), and references therein.
[CrossRef]

Kurland, I.

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

Lin, S. Y.

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

Manolatou, C.

McGurn, A. R.

A. R. McGurn, Phys. Rev. B 53, 7059 (1996).
[CrossRef]

Mekis, A.

A. Mekis, S. H. Fan, and J. D. Joannopoulos, Phys. Rev. B 58, 4809 (1998).

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

Mingaleev, S. F.

S. F. Mingaleev and Yu. S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).
[PubMed]

S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, Phys. Rev. E 62, 5777 (2000).

Polman, A.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, J. Vac. Sci. Technol. B 17, 2734 (1999).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).
[CrossRef]

Sammut, R. A.

S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, Phys. Rev. E 62, 5777 (2000).

Snoeks, E.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, J. Vac. Sci. Technol. B 17, 2734 (1999).
[CrossRef]

van der Drift, E.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, J. Vac. Sci. Technol. B 17, 2734 (1999).
[CrossRef]

Villeneuve, P. R.

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 80, 960 (1998).

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

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

Zijlstra, T.

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, J. Vac. Sci. Technol. B 17, 2734 (1999).
[CrossRef]

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

J. Vac. Sci. Technol. B (1)

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, J. Vac. Sci. Technol. B 17, 2734 (1999).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (2)

A. R. McGurn, Phys. Rev. B 53, 7059 (1996).
[CrossRef]

A. Mekis, S. H. Fan, and J. D. Joannopoulos, Phys. Rev. B 58, 4809 (1998).

Phys. Rev. E (1)

S. F. Mingaleev, Yu. S. Kivshar, and R. A. Sammut, Phys. Rev. E 62, 5777 (2000).

Phys. Rev. Lett. (3)

S. F. Mingaleev and Yu. S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).
[PubMed]

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

S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 80, 960 (1998).

Prog. Quantum Electron. (1)

T. F. Krauss and R. M. De la Rue, Prog. Quantum Electron. 23, 51 (1999), and references therein.
[CrossRef]

Science (1)

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

Other (1)

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Dispersion relation for the 2D photonic-crystal waveguide (shown in the inset) calculated by the supercell method3 (dashed curve) and from approximate equations (5) and (6) for L=7 (solid curve) and L=1 (dotted curve). The hatched areas are the projected band structure of a perfect 2D crystal.

Fig. 2
Fig. 2

Electric field En for two bound states supported by a 90° waveguide bend (shown at the top). The center of the bend is located at n=0.

Fig. 3
Fig. 3

Reflection coefficients calculated by the finite-difference time-domain method (dashed curves, from Ref. 5) and from Eqs. (3)–(6) with L=7 (solid curves) and L=1 (dotted curve in the top plot), for four different bend geometries.

Equations (8)

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

2+ω/c2xE(x|ω)=0,
E(x|ω)=ω/c2d2yG(x,y|ω)dyE(y|ω),
mMn,mωEm=0,  Mn,mω=dJn,mω-δn,m,
Jn,mω=ω/c2rdd2yG(xn,xm+y|ω)
T1,jω=-ML-jωMLω, j=1,2,,2L,  Tj,j+1=1, j=1,2,,2L-1.
TˆωΦp=expikpωΦp,
Emin=Φa-m1+rΦa-m2+p=3L+1λpinΦa-mp,
Emout=tΦm-b2+p=3L+1λpoutΦm-bp

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