Abstract

We present detailed results of analytical and numerical investigation of the Shockley surface states in photonic crystals doped with a chain of alternating s- and p-type defects. Conditions for the existence and control of such surface states are studied using the empirical tight-binding model and verified by the finite-difference time-domain technique. We show for the first time, to our knowledge, that, in contrast to the case of solids, the Shockley states in photonic crystals with complete unit cells of defects do not appear simultaneously with the opening of the inverted bandgap between s and p bands. Rather, the width of the inverted bandgap must reach a critical value equal to the separation between the discrete levels. This results in a system size effect of the Shockley states in photonic crystals. We show how such system size effect can be used to control the surface states. We also demonstrate the control of the Shockley states by controlling the overlap of s and p bands, achieved through change in either the radius of one of the defects or the refractive index via electro-optical effects.

© 2007 Optical Society of America

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References

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  1. P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977).
    [CrossRef]
  2. P. Yeh and A. Yariv, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
    [CrossRef]
  3. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
    [CrossRef]
  4. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Observation of surface photons on periodic dielectric arrays," Opt. Lett. 18, 528-530 (1993).
    [CrossRef] [PubMed]
  5. J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715 (1996).
    [CrossRef]
  6. F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 59, 15112-15120 (1999).
    [CrossRef]
  7. Y. A. Vlasov, N. Moll, and S. J. McNab, "Observation of surface states in a truncated photonic crystal slab," Opt. Lett. 29, 2175-2177 (2004).
    [CrossRef] [PubMed]
  8. P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
    [CrossRef] [PubMed]
  9. E. Moreno, F. J. Garcia-Vidal, and L. Martin-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402(R) (2004).
    [CrossRef]
  10. A. Many, Y. Goldstein, and N. B. Grover, Semiconductor Surfaces (North-Holland, 1965).
  11. S. G. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992).
  12. I. E. Tamm, "On the possible bound states of electrons on a crystal surface," Phys. Z. Sowjetunion 1, 733-735 (1932).
  13. W. Shockley, "On surface states associated with a periodic potential," Phys. Rev. 56, 317-323 (1939).
    [CrossRef]
  14. N. Malkova and C. Z. Ning, "Shockley and Tamm surface states in photonic crystals," Phys. Rev. B 73, 113113 (2006).
    [CrossRef]
  15. N. Malkova and C. Z. Ning, "Tamm surface states in a finite chain of defects in a photonic crystal," J. Phys. Condens. Matter (to be published).
  16. E. Lidorokis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, "Tight-binding parametrization for photonic band gap materials," Phys. Rev. Lett. 81, 1405-1408 (1998).
    [CrossRef]
  17. A. L. Reynolds, U. Peschel, F. Lederer, P. J. Robertson, T. F. Krauss, and P. J. I. de Maagt, "Coupled defects in photonic crystals," IEEE Trans. Microwave Theory Tech. 49, 1860-1866 (2001).
    [CrossRef]
  18. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  19. N. Malkova, S. Kim, and V. Gopalan, "Jahn-Teller effect in two-dimensional photonic crystals," Phys. Rev. B 68, 045105 (2003).
    [CrossRef]

2006 (1)

N. Malkova and C. Z. Ning, "Shockley and Tamm surface states in photonic crystals," Phys. Rev. B 73, 113113 (2006).
[CrossRef]

2004 (3)

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

E. Moreno, F. J. Garcia-Vidal, and L. Martin-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402(R) (2004).
[CrossRef]

Y. A. Vlasov, N. Moll, and S. J. McNab, "Observation of surface states in a truncated photonic crystal slab," Opt. Lett. 29, 2175-2177 (2004).
[CrossRef] [PubMed]

2003 (1)

N. Malkova, S. Kim, and V. Gopalan, "Jahn-Teller effect in two-dimensional photonic crystals," Phys. Rev. B 68, 045105 (2003).
[CrossRef]

2001 (1)

A. L. Reynolds, U. Peschel, F. Lederer, P. J. Robertson, T. F. Krauss, and P. J. I. de Maagt, "Coupled defects in photonic crystals," IEEE Trans. Microwave Theory Tech. 49, 1860-1866 (2001).
[CrossRef]

1999 (1)

F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 59, 15112-15120 (1999).
[CrossRef]

1998 (1)

E. Lidorokis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, "Tight-binding parametrization for photonic band gap materials," Phys. Rev. Lett. 81, 1405-1408 (1998).
[CrossRef]

1996 (1)

J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715 (1996).
[CrossRef]

1993 (1)

1991 (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

1978 (1)

P. Yeh and A. Yariv, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
[CrossRef]

1977 (1)

1939 (1)

W. Shockley, "On surface states associated with a periodic potential," Phys. Rev. 56, 317-323 (1939).
[CrossRef]

1932 (1)

I. E. Tamm, "On the possible bound states of electrons on a crystal surface," Phys. Z. Sowjetunion 1, 733-735 (1932).

Agio, M.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Arjavalingam, G.

Birner, A.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Observation of surface photons on periodic dielectric arrays," Opt. Lett. 18, 528-530 (1993).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Davison, S. G.

S. G. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992).

de Maagt, P. J. I.

A. L. Reynolds, U. Peschel, F. Lederer, P. J. Robertson, T. F. Krauss, and P. J. I. de Maagt, "Coupled defects in photonic crystals," IEEE Trans. Microwave Theory Tech. 49, 1860-1866 (2001).
[CrossRef]

Economou, E. N.

E. Lidorokis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, "Tight-binding parametrization for photonic band gap materials," Phys. Rev. Lett. 81, 1405-1408 (1998).
[CrossRef]

Elson, J. M.

J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715 (1996).
[CrossRef]

Garcia-Vidal, F. J.

E. Moreno, F. J. Garcia-Vidal, and L. Martin-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402(R) (2004).
[CrossRef]

Goldstein, Y.

A. Many, Y. Goldstein, and N. B. Grover, Semiconductor Surfaces (North-Holland, 1965).

Gopalan, V.

N. Malkova, S. Kim, and V. Gopalan, "Jahn-Teller effect in two-dimensional photonic crystals," Phys. Rev. B 68, 045105 (2003).
[CrossRef]

Gosele, U.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Grover, N. B.

A. Many, Y. Goldstein, and N. B. Grover, Semiconductor Surfaces (North-Holland, 1965).

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Halevi, P.

F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 59, 15112-15120 (1999).
[CrossRef]

Hong, C.-S.

Joannopoulos, J. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Observation of surface photons on periodic dielectric arrays," Opt. Lett. 18, 528-530 (1993).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Kim, S.

N. Malkova, S. Kim, and V. Gopalan, "Jahn-Teller effect in two-dimensional photonic crystals," Phys. Rev. B 68, 045105 (2003).
[CrossRef]

Kramper, P.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Krauss, T. F.

A. L. Reynolds, U. Peschel, F. Lederer, P. J. Robertson, T. F. Krauss, and P. J. I. de Maagt, "Coupled defects in photonic crystals," IEEE Trans. Microwave Theory Tech. 49, 1860-1866 (2001).
[CrossRef]

Lederer, F.

A. L. Reynolds, U. Peschel, F. Lederer, P. J. Robertson, T. F. Krauss, and P. J. I. de Maagt, "Coupled defects in photonic crystals," IEEE Trans. Microwave Theory Tech. 49, 1860-1866 (2001).
[CrossRef]

Lidorokis, E.

E. Lidorokis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, "Tight-binding parametrization for photonic band gap materials," Phys. Rev. Lett. 81, 1405-1408 (1998).
[CrossRef]

Malkova, N.

N. Malkova and C. Z. Ning, "Shockley and Tamm surface states in photonic crystals," Phys. Rev. B 73, 113113 (2006).
[CrossRef]

N. Malkova, S. Kim, and V. Gopalan, "Jahn-Teller effect in two-dimensional photonic crystals," Phys. Rev. B 68, 045105 (2003).
[CrossRef]

N. Malkova and C. Z. Ning, "Tamm surface states in a finite chain of defects in a photonic crystal," J. Phys. Condens. Matter (to be published).

Many, A.

A. Many, Y. Goldstein, and N. B. Grover, Semiconductor Surfaces (North-Holland, 1965).

Martin-Moreno, L.

E. Moreno, F. J. Garcia-Vidal, and L. Martin-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402(R) (2004).
[CrossRef]

McNab, S. J.

Meade, R. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Observation of surface photons on periodic dielectric arrays," Opt. Lett. 18, 528-530 (1993).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Moll, N.

Moreno, E.

E. Moreno, F. J. Garcia-Vidal, and L. Martin-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402(R) (2004).
[CrossRef]

Muller, F.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Ning, C. Z.

N. Malkova and C. Z. Ning, "Shockley and Tamm surface states in photonic crystals," Phys. Rev. B 73, 113113 (2006).
[CrossRef]

N. Malkova and C. Z. Ning, "Tamm surface states in a finite chain of defects in a photonic crystal," J. Phys. Condens. Matter (to be published).

Peschel, U.

A. L. Reynolds, U. Peschel, F. Lederer, P. J. Robertson, T. F. Krauss, and P. J. I. de Maagt, "Coupled defects in photonic crystals," IEEE Trans. Microwave Theory Tech. 49, 1860-1866 (2001).
[CrossRef]

Ramos-Mendieta, F.

F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 59, 15112-15120 (1999).
[CrossRef]

Rappe, A. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Observation of surface photons on periodic dielectric arrays," Opt. Lett. 18, 528-530 (1993).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Reynolds, A. L.

A. L. Reynolds, U. Peschel, F. Lederer, P. J. Robertson, T. F. Krauss, and P. J. I. de Maagt, "Coupled defects in photonic crystals," IEEE Trans. Microwave Theory Tech. 49, 1860-1866 (2001).
[CrossRef]

Robertson, P. J.

A. L. Reynolds, U. Peschel, F. Lederer, P. J. Robertson, T. F. Krauss, and P. J. I. de Maagt, "Coupled defects in photonic crystals," IEEE Trans. Microwave Theory Tech. 49, 1860-1866 (2001).
[CrossRef]

Robertson, W. M.

Sandoghdar, V.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Shockley, W.

W. Shockley, "On surface states associated with a periodic potential," Phys. Rev. 56, 317-323 (1939).
[CrossRef]

Sigalas, M. M.

E. Lidorokis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, "Tight-binding parametrization for photonic band gap materials," Phys. Rev. Lett. 81, 1405-1408 (1998).
[CrossRef]

Soukoulis, C. M.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

E. Lidorokis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, "Tight-binding parametrization for photonic band gap materials," Phys. Rev. Lett. 81, 1405-1408 (1998).
[CrossRef]

Stesliska, M.

S. G. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992).

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Tamm, I. E.

I. E. Tamm, "On the possible bound states of electrons on a crystal surface," Phys. Z. Sowjetunion 1, 733-735 (1932).

Tran, P.

J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715 (1996).
[CrossRef]

Vlasov, Y. A.

Wehrspohn, R. B.

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

Yariv, A.

P. Yeh and A. Yariv, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
[CrossRef]

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977).
[CrossRef]

Yeh, P.

P. Yeh and A. Yariv, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
[CrossRef]

P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977).
[CrossRef]

Appl. Phys. Lett. (1)

P. Yeh and A. Yariv, "Optical surface waves in periodic layered media," Appl. Phys. Lett. 32, 104-105 (1978).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. L. Reynolds, U. Peschel, F. Lederer, P. J. Robertson, T. F. Krauss, and P. J. I. de Maagt, "Coupled defects in photonic crystals," IEEE Trans. Microwave Theory Tech. 49, 1860-1866 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (2)

Phys. Rev. (1)

W. Shockley, "On surface states associated with a periodic potential," Phys. Rev. 56, 317-323 (1939).
[CrossRef]

Phys. Rev. B (6)

N. Malkova and C. Z. Ning, "Shockley and Tamm surface states in photonic crystals," Phys. Rev. B 73, 113113 (2006).
[CrossRef]

N. Malkova, S. Kim, and V. Gopalan, "Jahn-Teller effect in two-dimensional photonic crystals," Phys. Rev. B 68, 045105 (2003).
[CrossRef]

E. Moreno, F. J. Garcia-Vidal, and L. Martin-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402(R) (2004).
[CrossRef]

J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715 (1996).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, "Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane," Phys. Rev. B 59, 15112-15120 (1999).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Electromagnetic Bloch waves at the surface of a photonic crystal," Phys. Rev. B 44, 10961-10964 (1991).
[CrossRef]

Phys. Rev. Lett. (2)

P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Muller, R. B. Wehrspohn, U. Gosele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113903 (2004).
[CrossRef] [PubMed]

E. Lidorokis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, "Tight-binding parametrization for photonic band gap materials," Phys. Rev. Lett. 81, 1405-1408 (1998).
[CrossRef]

Phys. Z. Sowjetunion (1)

I. E. Tamm, "On the possible bound states of electrons on a crystal surface," Phys. Z. Sowjetunion 1, 733-735 (1932).

Other (4)

N. Malkova and C. Z. Ning, "Tamm surface states in a finite chain of defects in a photonic crystal," J. Phys. Condens. Matter (to be published).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

A. Many, Y. Goldstein, and N. B. Grover, Semiconductor Surfaces (North-Holland, 1965).

S. G. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992).

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Figures (8)

Fig. 1
Fig. 1

Coupled defect structures with unit cells including one s defect (small filled circles) and one p defect (large open circles) embedded in a perfect host PC (gray circles): (a) structure 1 of six complete unit cells; (b) structure 2 of seven incomplete unit cells. The source (S), port (P), perfect matched layers (gray boxes), and reference system are shown.

Fig. 2
Fig. 2

Energy spectrum of the defect chain with four complete unit cells for Δ = 0 (a), (b); 0.4 (c), (d); 0.8 (e), (f); and 3 (g), (h). (a), (c), (e), (g) Solid and dotted–dashed curves show dispersion relations of infinite coupled and uncoupled chains, respectively; the stars show the discrete spectrum of the finite chain with coupling. (b), (d), (f), (h) Wave functions for each eigenvalue of the discrete chain presented in ascending order from the bottom to the top of the figure.

Fig. 3
Fig. 3

Energy spectrum of structure 2 with four incomplete unit cells for Δ = 0 (a), (b); 0.1 (c), (d); 0.8 (e), (f); and 3 (g), (h). (a), (c), (e), (g) Solid and dotted–dashed curves show dispersion relations of infinite coupled and uncoupled chains, respectively, while stars show the discrete spectrum of the finite chain. (b), (d), (f), (h) Wave functions for each eigenvalue of the discrete chain in ascending order from the bottom to the top of the figure.

Fig. 4
Fig. 4

Critical value Δ cr as (a) a function of the number of defects and (b) as a function of ( α s α p ) β for the chain of four unit cells. The data calculated from Eq. (4) are shown by solid curves. The approximate value of the critical coupling 2 N is shown by a dashed curve in (a).

Fig. 5
Fig. 5

Defect chain containing s and p modes with six complete unit cells. (a) Calculated transmission coefficient, where the large arrows indicate the two surface modes. The structure is shown in the inset. (b) Theoretical dispersion relationship: the two crossed dashed–dotted curves show the spectrum of the uncoupled chains of the s and p defects ( β s p , p s = 0 ) . The solid and dashed curves represent the supercell plane-wave calculation and the fitted dispersion relation ω 1 , 2 ( k ) of Eq. (3) for the infinite chain, respectively. The stars show the calculated spectrum of the finite chain. (c), (d), (e), (f) The E z distributions for the modes pointed out by arrows in (a) are presented in descending order from top to bottom. (g), (h), (i), (j) The theoretical wave functions (dashed curves) in comparison with the amplitudes of the fields along the defect chain found from the FDTD simulations (solid curves) for the chains of (g), (h) six and (i), (j) ten unit cells.

Fig. 6
Fig. 6

Defect chain of s and p modes with seven incomplete unit cells. (a) Calculated transmission coefficient. The arrows indicate the two surface modes. The structure is shown in the inset. (b) Theoretical dispersion relationship: all the notations are similar to those in Fig. 5b. (c), (d) The E z distributions for the two surface modes indicated by arrows in (a).

Fig. 7
Fig. 7

Surface states calculated from the FDTD simulations (star markers) and the band edges calculated using the supercell plane-wave technique (dashed line and curve) for structure 1 of six unit cells as a function of R s .

Fig. 8
Fig. 8

(a) Calculated surface states (stars) and band edges of the hybridized bands (dashed lines) for structure 1 with R s = 0.075 a as a function of the number of unit cells. The dashed areas at the band edges illustrate the uncertainty in defect radii. The E z distributions for the modes of (b), (c) high and (d), (e) low frequency for the structure of (b), (d) four and (c), (e) ten unit cells.

Equations (6)

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

i d d t a l = α s a l + β s ( a l 1 + a l + 1 ) + β s p b l + β p s b l 1 ,
i d d t b l = α p b l + β p ( b l 1 + b n + 1 ) + β s p a l + β p s a l + 1 ,
a n ( t ) = A n exp ( i ω t ) = A exp ( i k d n i ω t ) ,
b n ( t ) = B n exp ( i ω t ) = B exp ( i k d n i ω t ) .
ω 1 , 2 ( k ) = 1 2 { α s + α p + 2 ( β s + β p ) c ± [ α s α p + 2 c ( β s β p ) ] 2 + 4 β ̃ s p 2 } ,
[ α s ω β s p β s 0 0 0 0 0 0 0 β s p α p ω β p s β p 0 0 0 0 0 0 β s β p s α s ω β s p β s 0 0 0 0 0 0 β p β s p α p ω β p s β p 0 0 0 0 0 0 0 0 0 0 β s β p s α s ω β s p 0 0 0 0 0 0 0 β p β s p α p ω ] ( A 1 B 1 A 2 B 2 A N B N ) = 0 ,

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