Abstract

The localization of electromagnetic waves in lossless inhomogeneous dielectric media is studied. We consider a three-dimensional lossless periodic medium (photonic crystal) having a gap in the frequency spectrum (photonic bandgap). If such a medium is perturbed by either a single defect or a random array of defects, exponentially localized electromagnetic waves arise with frequencies in the gap. For a single defect, we derive equations for these midgap frequencies and estimate their number. For a random medium, we show the occurrence of Anderson localization for electromagnetic waves.

© 1998 Optical Society of America

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  1. P. W. Anderson, “A question of classical localization. A theory of white paint,” Philos. Mag. B 53, 505–509 (1985).
    [CrossRef]
  2. P. Sheng, ed., Scattering and Localization of Classical Waves (World Scientific, Singapore, 1990).
  3. S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
    [CrossRef]
  4. S. John, “The localization of light,” in Photonic Band Gaps and Localization, C. M. Soukoulis, ed., Vol. 308 of NATO ASI Ser. B. (Plenum, New York, 1993), pp. 1–22.
    [CrossRef]
  5. R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Photonic bound states in periodic materials,” Phys. Rev. Lett. 67, 3380–3384 (1991).
  6. J. Rarity, C. Weisbuch, eds., Microcavities and Photonic Bandgaps: Physics and Applications (Kluwer Academic, Dordrecht, The Netherlands, 1995).
  7. C. Soukoulis, ed., Photonic Band Gap Materials (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  8. A. Figotin, A. Klein, “Localization of electromagnetic and acoustic waves in random media. Lattice model,” J. Stat. Phys. 76, 985–1003 (1994).
    [CrossRef]
  9. A. Figotin, A. Klein, “Localization of classical waves II: Electromagnetic waves,” Commun. Math. Phys. 184, 411–441 (1997).
    [CrossRef]
  10. A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
    [CrossRef]
  11. A. Figotin, A. Klein, “Midgap defect modes in dielectric and acoustic media,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (to be published).
  12. Special issue on development and applications of materials exhibiting photonic bandgaps, J. Opt. Soc. Am. B 10, 279–413 (1993).
  13. P. R. Villeneuve, M. Piché, “Photonic band gaps in periodic dielectric structures,” Prog. Quantum Electron. 18, 153–200 (1994).
    [CrossRef]
  14. P. M. Hui, N. F. Johnson, “Photonic band-gap materials,” in Solid State Physics, H. Ehrenreich, F. Spaepen, eds. (Academic, New York, 1995), Vol. 49, pp. 151–203.
  15. P. R. Villeneuve, J. Joannopoulos, “Working at the speed of light,” Sci. Spectra 9, 18–24 (1997).
  16. J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
  17. C. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993).
  18. E. Yablonovitch, T. Gmitter, R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Donor and acceptor modes in photonic periodic structure,” Phys. Rev. B 44, 13772–13774 (1991).
  19. A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 68–88 (1996).
    [CrossRef]
  20. A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. II. 2D photonic crystals,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 1561–1620 (1996).
    [CrossRef]
  21. P. W. Anderson, “Absence of diffusion in certain random lattice,” Phys. Rev. 109, 1492–1505 (1958).
    [CrossRef]
  22. B. Shklovskii, A. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Heidelberg, 1984).
  23. I. M. Lifshits, S. A. Greduskul, L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).
  24. J. Fröhlich, T. Spencer, “Absence of diffusion in the Anderson tight binding model for large disorder or low energy,” Commun. Math. Phys. 88, 151–184 (1983).
    [CrossRef]
  25. J. Fröhlich, F. Martinelli, E. Scoppola, T. Spencer, “Constructive proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 101, 21–46 (1985).
    [CrossRef]
  26. H. Holden, F. Martinelli, “On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on L2(ℝν),” Commun. Math. Phys. 93, 197–217 (1984).
    [CrossRef]
  27. H. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators (Springer-Verlag, Heidelberg, 1987).
  28. H. Dreifus, A. Klein, “A new proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 124, 285–299 (1989).
    [CrossRef]
  29. R. Carmona, J. Lacroix, Spectral Theory of Random Schrödinger Operators (Birkhäuser, Boston, Mass., 1990).
  30. L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators (Springer-Verlag, Heidelberg, 1991).
  31. J. Combes, P. Hislop, “Localization for some continuous, random Hamiltonians in d-dimensions,” J. Funct. Anal. 124, 149–180 (1994).
    [CrossRef]
  32. A. Figotin, A. Klein, “Localization phenomenon in gaps of the spectrum of random lattice operators,” J. Stat. Phys. 75, 997–1021 (1994).
    [CrossRef]
  33. A. Figotin, A. Klein, “Localization of classical waves I: Acoustic waves,” Commun. Math. Phys. 180, 439–482 (1996).
    [CrossRef]
  34. J. Maynard, “Acoustic Anderson localization,” in Random Media and Composites, B. V. Kohn, G. W. Milton, eds., (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1988), pp. 206–207.
  35. W. Kohler, G. Papanicolaou, B. White, “Localization and mode conversion for elastic waves in randomly layered media,” Wave Motion 23, 1–22 and 181–201 (1996).
    [CrossRef]
  36. A. Maradudin, E. Montroll, G. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation (Academic, New York, 1963).
  37. L. Deych, A. Lisyansky, “Impurity localization of electromagnetic waves in polariton region,” Phys. Rev. Lett. (to be published).
  38. Research done by F. Klopp on Internal Lifshits tails for random perturbations of periodic Schrödinger operators.
  39. L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).
  40. M. Reed, B. Simon, Analysis of Operators, Vol. 4 of Methods of Modern Mathematical Physics (Academic, New York, 1978).
  41. M. Klaus, “Some applications of the Birman–Schwinger principle,” Helv. Phys. Acta 55, 49–68 (1982).

1997 (3)

A. Figotin, A. Klein, “Localization of classical waves II: Electromagnetic waves,” Commun. Math. Phys. 184, 411–441 (1997).
[CrossRef]

A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
[CrossRef]

P. R. Villeneuve, J. Joannopoulos, “Working at the speed of light,” Sci. Spectra 9, 18–24 (1997).

1996 (4)

A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 68–88 (1996).
[CrossRef]

A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. II. 2D photonic crystals,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 1561–1620 (1996).
[CrossRef]

A. Figotin, A. Klein, “Localization of classical waves I: Acoustic waves,” Commun. Math. Phys. 180, 439–482 (1996).
[CrossRef]

W. Kohler, G. Papanicolaou, B. White, “Localization and mode conversion for elastic waves in randomly layered media,” Wave Motion 23, 1–22 and 181–201 (1996).
[CrossRef]

1994 (4)

P. R. Villeneuve, M. Piché, “Photonic band gaps in periodic dielectric structures,” Prog. Quantum Electron. 18, 153–200 (1994).
[CrossRef]

J. Combes, P. Hislop, “Localization for some continuous, random Hamiltonians in d-dimensions,” J. Funct. Anal. 124, 149–180 (1994).
[CrossRef]

A. Figotin, A. Klein, “Localization phenomenon in gaps of the spectrum of random lattice operators,” J. Stat. Phys. 75, 997–1021 (1994).
[CrossRef]

A. Figotin, A. Klein, “Localization of electromagnetic and acoustic waves in random media. Lattice model,” J. Stat. Phys. 76, 985–1003 (1994).
[CrossRef]

1993 (1)

Special issue on development and applications of materials exhibiting photonic bandgaps, J. Opt. Soc. Am. B 10, 279–413 (1993).

1991 (3)

E. Yablonovitch, T. Gmitter, R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Donor and acceptor modes in photonic periodic structure,” Phys. Rev. B 44, 13772–13774 (1991).

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Photonic bound states in periodic materials,” Phys. Rev. Lett. 67, 3380–3384 (1991).

1989 (1)

H. Dreifus, A. Klein, “A new proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 124, 285–299 (1989).
[CrossRef]

1985 (2)

P. W. Anderson, “A question of classical localization. A theory of white paint,” Philos. Mag. B 53, 505–509 (1985).
[CrossRef]

J. Fröhlich, F. Martinelli, E. Scoppola, T. Spencer, “Constructive proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 101, 21–46 (1985).
[CrossRef]

1984 (1)

H. Holden, F. Martinelli, “On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on L2(ℝν),” Commun. Math. Phys. 93, 197–217 (1984).
[CrossRef]

1983 (1)

J. Fröhlich, T. Spencer, “Absence of diffusion in the Anderson tight binding model for large disorder or low energy,” Commun. Math. Phys. 88, 151–184 (1983).
[CrossRef]

1982 (1)

M. Klaus, “Some applications of the Birman–Schwinger principle,” Helv. Phys. Acta 55, 49–68 (1982).

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattice,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “A question of classical localization. A theory of white paint,” Philos. Mag. B 53, 505–509 (1985).
[CrossRef]

P. W. Anderson, “Absence of diffusion in certain random lattice,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Brommer, D.

E. Yablonovitch, T. Gmitter, R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Donor and acceptor modes in photonic periodic structure,” Phys. Rev. B 44, 13772–13774 (1991).

R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Photonic bound states in periodic materials,” Phys. Rev. Lett. 67, 3380–3384 (1991).

Carmona, R.

R. Carmona, J. Lacroix, Spectral Theory of Random Schrödinger Operators (Birkhäuser, Boston, Mass., 1990).

Combes, J.

J. Combes, P. Hislop, “Localization for some continuous, random Hamiltonians in d-dimensions,” J. Funct. Anal. 124, 149–180 (1994).
[CrossRef]

Cycon, H.

H. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators (Springer-Verlag, Heidelberg, 1987).

Deych, L.

L. Deych, A. Lisyansky, “Impurity localization of electromagnetic waves in polariton region,” Phys. Rev. Lett. (to be published).

Dreifus, H.

H. Dreifus, A. Klein, “A new proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 124, 285–299 (1989).
[CrossRef]

Efros, A.

B. Shklovskii, A. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Heidelberg, 1984).

Figotin, A.

A. Figotin, A. Klein, “Localization of classical waves II: Electromagnetic waves,” Commun. Math. Phys. 184, 411–441 (1997).
[CrossRef]

A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
[CrossRef]

A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 68–88 (1996).
[CrossRef]

A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. II. 2D photonic crystals,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 1561–1620 (1996).
[CrossRef]

A. Figotin, A. Klein, “Localization of classical waves I: Acoustic waves,” Commun. Math. Phys. 180, 439–482 (1996).
[CrossRef]

A. Figotin, A. Klein, “Localization phenomenon in gaps of the spectrum of random lattice operators,” J. Stat. Phys. 75, 997–1021 (1994).
[CrossRef]

A. Figotin, A. Klein, “Localization of electromagnetic and acoustic waves in random media. Lattice model,” J. Stat. Phys. 76, 985–1003 (1994).
[CrossRef]

A. Figotin, A. Klein, “Midgap defect modes in dielectric and acoustic media,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (to be published).

L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators (Springer-Verlag, Heidelberg, 1991).

Froese, R.

H. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators (Springer-Verlag, Heidelberg, 1987).

Fröhlich, J.

J. Fröhlich, F. Martinelli, E. Scoppola, T. Spencer, “Constructive proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 101, 21–46 (1985).
[CrossRef]

J. Fröhlich, T. Spencer, “Absence of diffusion in the Anderson tight binding model for large disorder or low energy,” Commun. Math. Phys. 88, 151–184 (1983).
[CrossRef]

Gmitter, T.

E. Yablonovitch, T. Gmitter, R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Donor and acceptor modes in photonic periodic structure,” Phys. Rev. B 44, 13772–13774 (1991).

Greduskul, S. A.

I. M. Lifshits, S. A. Greduskul, L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).

Hislop, P.

J. Combes, P. Hislop, “Localization for some continuous, random Hamiltonians in d-dimensions,” J. Funct. Anal. 124, 149–180 (1994).
[CrossRef]

Holden, H.

H. Holden, F. Martinelli, “On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on L2(ℝν),” Commun. Math. Phys. 93, 197–217 (1984).
[CrossRef]

Hui, P. M.

P. M. Hui, N. F. Johnson, “Photonic band-gap materials,” in Solid State Physics, H. Ehrenreich, F. Spaepen, eds. (Academic, New York, 1995), Vol. 49, pp. 151–203.

Joannopoulos, J.

P. R. Villeneuve, J. Joannopoulos, “Working at the speed of light,” Sci. Spectra 9, 18–24 (1997).

R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Photonic bound states in periodic materials,” Phys. Rev. Lett. 67, 3380–3384 (1991).

E. Yablonovitch, T. Gmitter, R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Donor and acceptor modes in photonic periodic structure,” Phys. Rev. B 44, 13772–13774 (1991).

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

John, S.

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

S. John, “The localization of light,” in Photonic Band Gaps and Localization, C. M. Soukoulis, ed., Vol. 308 of NATO ASI Ser. B. (Plenum, New York, 1993), pp. 1–22.
[CrossRef]

Johnson, N. F.

P. M. Hui, N. F. Johnson, “Photonic band-gap materials,” in Solid State Physics, H. Ehrenreich, F. Spaepen, eds. (Academic, New York, 1995), Vol. 49, pp. 151–203.

Kirsch, W.

H. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators (Springer-Verlag, Heidelberg, 1987).

Klaus, M.

M. Klaus, “Some applications of the Birman–Schwinger principle,” Helv. Phys. Acta 55, 49–68 (1982).

Klein, A.

A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
[CrossRef]

A. Figotin, A. Klein, “Localization of classical waves II: Electromagnetic waves,” Commun. Math. Phys. 184, 411–441 (1997).
[CrossRef]

A. Figotin, A. Klein, “Localization of classical waves I: Acoustic waves,” Commun. Math. Phys. 180, 439–482 (1996).
[CrossRef]

A. Figotin, A. Klein, “Localization phenomenon in gaps of the spectrum of random lattice operators,” J. Stat. Phys. 75, 997–1021 (1994).
[CrossRef]

A. Figotin, A. Klein, “Localization of electromagnetic and acoustic waves in random media. Lattice model,” J. Stat. Phys. 76, 985–1003 (1994).
[CrossRef]

H. Dreifus, A. Klein, “A new proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 124, 285–299 (1989).
[CrossRef]

A. Figotin, A. Klein, “Midgap defect modes in dielectric and acoustic media,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (to be published).

Kohler, W.

W. Kohler, G. Papanicolaou, B. White, “Localization and mode conversion for elastic waves in randomly layered media,” Wave Motion 23, 1–22 and 181–201 (1996).
[CrossRef]

Kuchment, P.

A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. II. 2D photonic crystals,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 1561–1620 (1996).
[CrossRef]

A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 68–88 (1996).
[CrossRef]

Lacroix, J.

R. Carmona, J. Lacroix, Spectral Theory of Random Schrödinger Operators (Birkhäuser, Boston, Mass., 1990).

Lifshits, I. M.

I. M. Lifshits, S. A. Greduskul, L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).

Lisyansky, A.

L. Deych, A. Lisyansky, “Impurity localization of electromagnetic waves in polariton region,” Phys. Rev. Lett. (to be published).

Maradudin, A.

A. Maradudin, E. Montroll, G. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation (Academic, New York, 1963).

Martinelli, F.

J. Fröhlich, F. Martinelli, E. Scoppola, T. Spencer, “Constructive proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 101, 21–46 (1985).
[CrossRef]

H. Holden, F. Martinelli, “On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on L2(ℝν),” Commun. Math. Phys. 93, 197–217 (1984).
[CrossRef]

Maynard, J.

J. Maynard, “Acoustic Anderson localization,” in Random Media and Composites, B. V. Kohn, G. W. Milton, eds., (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1988), pp. 206–207.

Meade, R.

E. Yablonovitch, T. Gmitter, R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Donor and acceptor modes in photonic periodic structure,” Phys. Rev. B 44, 13772–13774 (1991).

R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Photonic bound states in periodic materials,” Phys. Rev. Lett. 67, 3380–3384 (1991).

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Montroll, E.

A. Maradudin, E. Montroll, G. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation (Academic, New York, 1963).

Papanicolaou, G.

W. Kohler, G. Papanicolaou, B. White, “Localization and mode conversion for elastic waves in randomly layered media,” Wave Motion 23, 1–22 and 181–201 (1996).
[CrossRef]

Pastur, L.

L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators (Springer-Verlag, Heidelberg, 1991).

Pastur, L. A.

I. M. Lifshits, S. A. Greduskul, L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).

Piché, M.

P. R. Villeneuve, M. Piché, “Photonic band gaps in periodic dielectric structures,” Prog. Quantum Electron. 18, 153–200 (1994).
[CrossRef]

Rappe, A.

R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Photonic bound states in periodic materials,” Phys. Rev. Lett. 67, 3380–3384 (1991).

E. Yablonovitch, T. Gmitter, R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Donor and acceptor modes in photonic periodic structure,” Phys. Rev. B 44, 13772–13774 (1991).

Reed, M.

M. Reed, B. Simon, Analysis of Operators, Vol. 4 of Methods of Modern Mathematical Physics (Academic, New York, 1978).

Schiff, L.

L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).

Scoppola, E.

J. Fröhlich, F. Martinelli, E. Scoppola, T. Spencer, “Constructive proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 101, 21–46 (1985).
[CrossRef]

Shklovskii, B.

B. Shklovskii, A. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Heidelberg, 1984).

Simon, B.

H. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators (Springer-Verlag, Heidelberg, 1987).

M. Reed, B. Simon, Analysis of Operators, Vol. 4 of Methods of Modern Mathematical Physics (Academic, New York, 1978).

Spencer, T.

J. Fröhlich, F. Martinelli, E. Scoppola, T. Spencer, “Constructive proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 101, 21–46 (1985).
[CrossRef]

J. Fröhlich, T. Spencer, “Absence of diffusion in the Anderson tight binding model for large disorder or low energy,” Commun. Math. Phys. 88, 151–184 (1983).
[CrossRef]

Villeneuve, P. R.

P. R. Villeneuve, J. Joannopoulos, “Working at the speed of light,” Sci. Spectra 9, 18–24 (1997).

P. R. Villeneuve, M. Piché, “Photonic band gaps in periodic dielectric structures,” Prog. Quantum Electron. 18, 153–200 (1994).
[CrossRef]

Weiss, G.

A. Maradudin, E. Montroll, G. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation (Academic, New York, 1963).

White, B.

W. Kohler, G. Papanicolaou, B. White, “Localization and mode conversion for elastic waves in randomly layered media,” Wave Motion 23, 1–22 and 181–201 (1996).
[CrossRef]

Winn, J.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Yablonovitch, E.

E. Yablonovitch, T. Gmitter, R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Donor and acceptor modes in photonic periodic structure,” Phys. Rev. B 44, 13772–13774 (1991).

Commun. Math. Phys. (6)

A. Figotin, A. Klein, “Localization of classical waves II: Electromagnetic waves,” Commun. Math. Phys. 184, 411–441 (1997).
[CrossRef]

J. Fröhlich, T. Spencer, “Absence of diffusion in the Anderson tight binding model for large disorder or low energy,” Commun. Math. Phys. 88, 151–184 (1983).
[CrossRef]

J. Fröhlich, F. Martinelli, E. Scoppola, T. Spencer, “Constructive proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 101, 21–46 (1985).
[CrossRef]

H. Holden, F. Martinelli, “On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on L2(ℝν),” Commun. Math. Phys. 93, 197–217 (1984).
[CrossRef]

H. Dreifus, A. Klein, “A new proof of localization in the Anderson tight binding model,” Commun. Math. Phys. 124, 285–299 (1989).
[CrossRef]

A. Figotin, A. Klein, “Localization of classical waves I: Acoustic waves,” Commun. Math. Phys. 180, 439–482 (1996).
[CrossRef]

Helv. Phys. Acta (1)

M. Klaus, “Some applications of the Birman–Schwinger principle,” Helv. Phys. Acta 55, 49–68 (1982).

J. Funct. Anal. (1)

J. Combes, P. Hislop, “Localization for some continuous, random Hamiltonians in d-dimensions,” J. Funct. Anal. 124, 149–180 (1994).
[CrossRef]

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

Special issue on development and applications of materials exhibiting photonic bandgaps, J. Opt. Soc. Am. B 10, 279–413 (1993).

J. Stat. Phys. (3)

A. Figotin, A. Klein, “Localized classical waves created by defects,” J. Stat. Phys. 86, 165–177 (1997).
[CrossRef]

A. Figotin, A. Klein, “Localization phenomenon in gaps of the spectrum of random lattice operators,” J. Stat. Phys. 75, 997–1021 (1994).
[CrossRef]

A. Figotin, A. Klein, “Localization of electromagnetic and acoustic waves in random media. Lattice model,” J. Stat. Phys. 76, 985–1003 (1994).
[CrossRef]

Philos. Mag. B (1)

P. W. Anderson, “A question of classical localization. A theory of white paint,” Philos. Mag. B 53, 505–509 (1985).
[CrossRef]

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattice,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Phys. Rev. B (1)

E. Yablonovitch, T. Gmitter, R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Donor and acceptor modes in photonic periodic structure,” Phys. Rev. B 44, 13772–13774 (1991).

Phys. Rev. Lett. (1)

R. Meade, D. Brommer, A. Rappe, J. Joannopoulos, “Photonic bound states in periodic materials,” Phys. Rev. Lett. 67, 3380–3384 (1991).

Phys. Today (1)

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

Prog. Quantum Electron. (1)

P. R. Villeneuve, M. Piché, “Photonic band gaps in periodic dielectric structures,” Prog. Quantum Electron. 18, 153–200 (1994).
[CrossRef]

Sci. Spectra (1)

P. R. Villeneuve, J. Joannopoulos, “Working at the speed of light,” Sci. Spectra 9, 18–24 (1997).

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (2)

A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 68–88 (1996).
[CrossRef]

A. Figotin, P. Kuchment, “Band-gap structure of spectra of periodic dielectric and acoustic media. II. 2D photonic crystals,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 1561–1620 (1996).
[CrossRef]

Wave Motion (1)

W. Kohler, G. Papanicolaou, B. White, “Localization and mode conversion for elastic waves in randomly layered media,” Wave Motion 23, 1–22 and 181–201 (1996).
[CrossRef]

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A. Maradudin, E. Montroll, G. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation (Academic, New York, 1963).

L. Deych, A. Lisyansky, “Impurity localization of electromagnetic waves in polariton region,” Phys. Rev. Lett. (to be published).

Research done by F. Klopp on Internal Lifshits tails for random perturbations of periodic Schrödinger operators.

L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).

M. Reed, B. Simon, Analysis of Operators, Vol. 4 of Methods of Modern Mathematical Physics (Academic, New York, 1978).

J. Maynard, “Acoustic Anderson localization,” in Random Media and Composites, B. V. Kohn, G. W. Milton, eds., (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1988), pp. 206–207.

R. Carmona, J. Lacroix, Spectral Theory of Random Schrödinger Operators (Birkhäuser, Boston, Mass., 1990).

L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators (Springer-Verlag, Heidelberg, 1991).

H. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators (Springer-Verlag, Heidelberg, 1987).

B. Shklovskii, A. Efros, Electronic Properties of Doped Semiconductors (Springer-Verlag, Heidelberg, 1984).

I. M. Lifshits, S. A. Greduskul, L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

C. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993).

P. M. Hui, N. F. Johnson, “Photonic band-gap materials,” in Solid State Physics, H. Ehrenreich, F. Spaepen, eds. (Academic, New York, 1995), Vol. 49, pp. 151–203.

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[CrossRef]

P. Sheng, ed., Scattering and Localization of Classical Waves (World Scientific, Singapore, 1990).

J. Rarity, C. Weisbuch, eds., Microcavities and Photonic Bandgaps: Physics and Applications (Kluwer Academic, Dordrecht, The Netherlands, 1995).

C. Soukoulis, ed., Photonic Band Gap Materials (Kluwer Academic, Dordrecht, The Netherlands, 1996).

A. Figotin, A. Klein, “Midgap defect modes in dielectric and acoustic media,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (to be published).

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Figures (2)

Fig. 1
Fig. 1

(a) Slab of a photonic crystal with an interior defect of higher dielectric constant. (b) The defect is shown on the cross section of the slab as a darker square.

Fig. 2
Fig. 2

Equations for the eigenvalues for the defect eigenmodes take the form ρn+(ω)=1, where the functions ρn+(ω) are the eigenvalues of an auxiliary compact operator depending on the spectral parameter ω.

Equations (112)

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tB=-×E,·B=0,
tD=×H·D=0,
B(x, t)=μ(x)H(x, t),
D(x, t)=ε(x)E(x, t).
0<ε-ε(x)ε+<
E(x, t)=½[μ(x)|H(x, t)|2+ε(x)|E(x, t)|2],
E=EH,E=3E(x, t)dx,
-i tΨt=MΨt,
Ψt=HtEt,M=0iμ×-iε×0,
(H1, H2)μ=3H1(x)¯H2(x)μ(x)dx,
(E1, E2)=3E1(x)¯E2(x)ε(x)dx,
Hμ<,·μ(x)H(x)=0.
Eε<,·ε(x)E(x)=0.
(Ψ1, Ψ2)=½[(H1, H2)μ+(E1, E2)ε],
EΨ=Ψ2=(Ψ, Ψ)=½(Hμ2+Eε2).
Ψt=exp(itM)Ψ0,
EΨt=Ψt2=Ψ02=EΨ0.
2t2Ht=-1μ× 1ε×Ht,HtinSμ,
2t2Et=-1ε× 1μ×Et,EtinSε.
MH=1μ× 1ε×,ME=1ε× 1μ×,
ME=UMHU*,
UH=-iε× 1MHH,
σ(M)=σ(MH)[-σ(MH)].
Ψ±,t=[exp(±itMH)H0,±U exp(±itMH)H0],H0inSμ.
MHHω=ω2Hω,HωinSμ,
UHω=-iωε×Hω,
MΨ±ω=±ωΨ±ω,
Ψ±ω=(H±ω, E±ω),
E±ω=±UH±ω=±-iωε×H±ω.
μ(x)=1,
M=MH=×1ε×,
M=MH=×1ε×actingonL2.
limR inft 1E|x|RE(x, t)dx=1.
3½[μ(x)|Hω(x)|2+ε(x)|Eω(x)|2]dx<,
|Ψω(x)|const. exp-|x|Lω.
ε0(x+n)=ε0(x)foranynfromL.
M0Ψ(x)=×1ε0(x)×Ψ(x),
M0Ψωn(k)(x)=ωn2(k)Ψωn(k)(x),n=1, 2, ,
Ψωn(k)(x)=exp(ik·x)Φωn(k)(x),
Φωn(k)(x+n)=Φωn(k)(x),ninL,
ω1(k)ω2(k)ωn(k).
σ(M0)=n=1,2, In.
G0(ω; x, y)=(M0-ω2I)-1(x, y),
|G0(ω; x, y)|Cω exp-|x-y|Lω,|x-y|.
πR2|G0(ω; x, y)|2||x-y|=RπR2Cω ×exp-2RLω R 0.
χ(y)=1ifyisinPC,0otherwise
χx(y)=χx(y-x).
G(ω)=(M-ω2I)-1,G0(ω)=(M0-ω2I)-1
A=supΨ0 AΨΨ.
χxG(ω)χyC0ηexp-|x-y|Lω,
Lω=4(2ε--1+ω2+η)η,
χx×G(ω)χyC(1+ω2)ηexp-|x-y|Lω,
maxkinBZ ωn2(k)=ωa2.
a-ωn2(k)ci|k-kn,i|2
ξ(ωa2)=lim supn0  Tr{[χIS|G0(ωa2+η)|PSχ]2},
ξ(ωb2)=lim supn0  Tr{[χIS|G0(ωb2-η)|PSχ]2},
a2U>π228m,
ε(x)=εε,l(x)=εifxisinΛl,ε0(x)otherwise.
σess(M)=σess(M0).
(M-ω2I)ΨδΨ,Ψ2=|Ψ(x)|2dx,
MΦω=ω2Φω
Φω(x)=-G0(ω; x, y)[(M-M0)Φω](y)dy,
Φω(x)=-ΛG0(ω; x, y)[(M-M0)Φω](y)dy.
|G0(ω; x, y)|C1 exp(-C2|x-y|).
l2>C(Λ, ωa, ωb),
C(Λ, ωa, ωb)=158(ωa2+ωb2)(ωb2-ωa2)2.
l2>79τγ2,
MΨ=ω2Ψ,ωisin(ωa, ωb).
HΨ=(M+I)-1Ψ=(ω2+1)-1Ψ,
ωisin(ωa, ωb).
HΨ=H0Ψ+VΨ=ξΨ,ξisnotinσ(H0),
Ψ=-R0(ξ)VΨ,R0(ξ)=(H0-ξI)-1.
R(ξ)=-VR0(ξ)V,
R(ξ)Φ=Φ,Φ=VΨ,
S(ω)Φ=Φ,Φ=VΨ,ωisin(ωa, ωb),
S(ω)=R[(ω2+1)-1]=(ω2+1)V M0+IM0-ω2IV.
S(ω)Φ=-Φ,Φ=-VΨ,ωisin(ωa, ωb),
S(ω)=(ω2+1)-V M0+IM0-ω2I-V.
ρn+(ω)=1,n=1, 2, ,
Ψi=M0+IM0-ωi2IVΦi
ρn-(ω)=-1,n=1, 2, ,
Ψi=-M0+IM0-ωi2I-VΦi
NH(α, β)=Tr χ(α,β)(H).
ε(x)=ε0(x)1+θ(x),
-1<θ-θ(x)θ+<,
NM(ωa2, ωb2)Cωa2,ε0,±,θ+,δθ+2(l+3)9+δξ(ωa2)<,
Cωa2,ε0,±,θ+,δ=(ωa2+1)2δC[1+ε--(6+δ)]C,
C=1+6 ε0,+ε0,-ε-+1ε-2,
NM(ωa2, ωb2)Cωb2,ε0,±,θ-,δθ-2(l+3)9+δξ(ωb2)<,
Cωb2,ε0,±,θ-,δ=(b+1)2δC[1+ε0,--(6+δ)]C,
C=1+6 ε+ε0,-ε0,-+1ε0,-2,
Px1,,xN=Px1+m,,xN+m.
σpp(A)={ω1, ω2, }.
σ(A)=σpp(A)¯σc(A),
εg,ζ(x)=ε0(x)γg,ζ(x),
γg,ζ(x)=1+giin3ζiui(x),
0<ε0,-ε0(x)ε0,+<
0<U-U(x)iin3ui(x)U+<
ε±=εg,±=ε0,±(1±gU+)
Mg=Mg,ζ=M(εg,ζ),
Mg=Mg,ζ=M(εg,ζ).
1U+1-ωaωb
g01U+min1, ωbωaU+/U--1,
g[ωa2, ωb2]=[ωa2, ωa2(g)][ωb2(g), ωb2].
ωa2ωa2(1+gU+)U-/U+ωa2(g)ωa21-gU+,
ωb2(1-gU+)ωb2(g)ωb2(1+gU+)U-/U+ωb2.
1-γ1ρ(t)dtKγηfor0γ1,
GΛ(ω)=(MΛ-ω2I)-1.
Gg,ζ,Λ(ω)1ηQωη|Λ|2,
Γx,LGg,ζ,x,L(ω)χxexp[-m(L/2)],
{ΛL(x)isregular}1-1Lp,
1-γ1ρ(t)dt,-1-1+γρ(t)dtKγηfor0γ1,

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