Abstract

Localized states in one-dimensional, open, disordered systems and their connection to the internal structure of random samples have been studied. It is shown that the localization of energy and anomalously high transmission associated with these states are due to the existence inside the sample of a transparent (for a given resonant frequency) segment with minimum size of the order of the localization length. An analogy between the stochastic scattering problem at hand and a deterministic quantum problem permits one to describe analytically some statistical properties of localized states with high transmission. It is shown that there is no one-to-one correspondence between the localization and high transparency: only a small fraction of the localized modes exhibit a transmission coefficient close to one. The maximum transmission is provided by the modes that are localized in the center, while the highest energy concentration occurs in cavities shifted towards the input. An algorithm is proposed to estimate the position of an effective resonant cavity and its pumping rate by measuring the resonant transmission coefficient. The validity of the analytical results is checked by extensive numerical simulations and wavelet analysis.

© 2004 Optical Society of America

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References

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  1. I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).
  2. M. V. Berry and S. Klein, “Transparent mirrors: rays, waves and localization,” Eur. J. Phys. 18, 222–228 (1997).
    [CrossRef]
  3. U. Frisch, C. Froeschle, J.-P. Scheidecker, and P.-L. Sulem, “Stochastic resonance in one-dimensional random media,” Phys. Rev. A 8, 1416–1421 (1973).
    [CrossRef]
  4. M. Ya. Azbel and P. Soven, “Transmission resonances and the localization length in one-dimensional disordered systems,” Phys. Rev. B 27, 831–836 (1983).
    [CrossRef]
  5. M. Ya. Azbel, “Eigenstates and properties of random systems in one dimension at zero temperature,” Phys. Rev. B 28, 4106–4125 (1983).
    [CrossRef]
  6. D. S. Wiersma, “The smallest random laser,” Nature 406, 132–133 (2000).
    [CrossRef] [PubMed]
  7. D. S. Wiersma and S. Cavalieri, “Light emission: A temperature-tunable random laser,” Nature 414, 708–709 (2001).
    [CrossRef] [PubMed]
  8. Xunyia Jiang and C. M. Soukoulis, “Time-dependent theory for random laser,” Phys. Rev. Lett. 85, 70–73 (2000).
    [CrossRef] [PubMed]
  9. H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
    [CrossRef]
  10. H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
    [CrossRef] [PubMed]
  11. E. N. Economou and C. M. Soukoulis, “Connection of localization with the problem of the bound state in a potential well,” Phys. Rev. B 28, 1093–1095 (1983).
    [CrossRef]
  12. D. Bohm, Quantum Theory (Prentice-Hall, New York, 1952).
  13. J. B. Pendry, “Quasi-extended electron states in strongly disordered systems,” J. Phys. C 20, 733–742 (1987).
    [CrossRef]
  14. S. Rytov, Yu. Kravtsov, and V. Tatarskii, Principles of Statistical Radiophysics IV: Wave Propagation Through Random Media (Springer-Verlag, Berlin, 1989).
  15. A. Grossman and J. Morlet, “Decomposition of Hardy function into square integrable wavelets of constant shape,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 723–736 (1984).
    [CrossRef]

2001 (2)

D. S. Wiersma and S. Cavalieri, “Light emission: A temperature-tunable random laser,” Nature 414, 708–709 (2001).
[CrossRef] [PubMed]

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
[CrossRef] [PubMed]

2000 (2)

Xunyia Jiang and C. M. Soukoulis, “Time-dependent theory for random laser,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

D. S. Wiersma, “The smallest random laser,” Nature 406, 132–133 (2000).
[CrossRef] [PubMed]

1999 (1)

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

1997 (1)

M. V. Berry and S. Klein, “Transparent mirrors: rays, waves and localization,” Eur. J. Phys. 18, 222–228 (1997).
[CrossRef]

1987 (1)

J. B. Pendry, “Quasi-extended electron states in strongly disordered systems,” J. Phys. C 20, 733–742 (1987).
[CrossRef]

1984 (1)

A. Grossman and J. Morlet, “Decomposition of Hardy function into square integrable wavelets of constant shape,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

1983 (3)

M. Ya. Azbel and P. Soven, “Transmission resonances and the localization length in one-dimensional disordered systems,” Phys. Rev. B 27, 831–836 (1983).
[CrossRef]

M. Ya. Azbel, “Eigenstates and properties of random systems in one dimension at zero temperature,” Phys. Rev. B 28, 4106–4125 (1983).
[CrossRef]

E. N. Economou and C. M. Soukoulis, “Connection of localization with the problem of the bound state in a potential well,” Phys. Rev. B 28, 1093–1095 (1983).
[CrossRef]

1973 (1)

U. Frisch, C. Froeschle, J.-P. Scheidecker, and P.-L. Sulem, “Stochastic resonance in one-dimensional random media,” Phys. Rev. A 8, 1416–1421 (1973).
[CrossRef]

Azbel, M. Ya.

M. Ya. Azbel and P. Soven, “Transmission resonances and the localization length in one-dimensional disordered systems,” Phys. Rev. B 27, 831–836 (1983).
[CrossRef]

M. Ya. Azbel, “Eigenstates and properties of random systems in one dimension at zero temperature,” Phys. Rev. B 28, 4106–4125 (1983).
[CrossRef]

Berry, M. V.

M. V. Berry and S. Klein, “Transparent mirrors: rays, waves and localization,” Eur. J. Phys. 18, 222–228 (1997).
[CrossRef]

Cao, C. Q.

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
[CrossRef] [PubMed]

Cao, H.

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
[CrossRef] [PubMed]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Cavalieri, S.

D. S. Wiersma and S. Cavalieri, “Light emission: A temperature-tunable random laser,” Nature 414, 708–709 (2001).
[CrossRef] [PubMed]

Chang, R. P. H.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Economou, E. N.

E. N. Economou and C. M. Soukoulis, “Connection of localization with the problem of the bound state in a potential well,” Phys. Rev. B 28, 1093–1095 (1983).
[CrossRef]

Frisch, U.

U. Frisch, C. Froeschle, J.-P. Scheidecker, and P.-L. Sulem, “Stochastic resonance in one-dimensional random media,” Phys. Rev. A 8, 1416–1421 (1973).
[CrossRef]

Froeschle, C.

U. Frisch, C. Froeschle, J.-P. Scheidecker, and P.-L. Sulem, “Stochastic resonance in one-dimensional random media,” Phys. Rev. A 8, 1416–1421 (1973).
[CrossRef]

Grossman, A.

A. Grossman and J. Morlet, “Decomposition of Hardy function into square integrable wavelets of constant shape,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

Ho, S. T.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Jiang, Xunyia

Xunyia Jiang and C. M. Soukoulis, “Time-dependent theory for random laser,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

Klein, S.

M. V. Berry and S. Klein, “Transparent mirrors: rays, waves and localization,” Eur. J. Phys. 18, 222–228 (1997).
[CrossRef]

Kumar, P.

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
[CrossRef] [PubMed]

Ling, Y.

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
[CrossRef] [PubMed]

Morlet, J.

A. Grossman and J. Morlet, “Decomposition of Hardy function into square integrable wavelets of constant shape,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Quasi-extended electron states in strongly disordered systems,” J. Phys. C 20, 733–742 (1987).
[CrossRef]

Scheidecker, J.-P.

U. Frisch, C. Froeschle, J.-P. Scheidecker, and P.-L. Sulem, “Stochastic resonance in one-dimensional random media,” Phys. Rev. A 8, 1416–1421 (1973).
[CrossRef]

Seelig, E. W.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Soukoulis, C. M.

Xunyia Jiang and C. M. Soukoulis, “Time-dependent theory for random laser,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

E. N. Economou and C. M. Soukoulis, “Connection of localization with the problem of the bound state in a potential well,” Phys. Rev. B 28, 1093–1095 (1983).
[CrossRef]

Soven, P.

M. Ya. Azbel and P. Soven, “Transmission resonances and the localization length in one-dimensional disordered systems,” Phys. Rev. B 27, 831–836 (1983).
[CrossRef]

Sulem, P.-L.

U. Frisch, C. Froeschle, J.-P. Scheidecker, and P.-L. Sulem, “Stochastic resonance in one-dimensional random media,” Phys. Rev. A 8, 1416–1421 (1973).
[CrossRef]

Wang, Q. H.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma and S. Cavalieri, “Light emission: A temperature-tunable random laser,” Nature 414, 708–709 (2001).
[CrossRef] [PubMed]

D. S. Wiersma, “The smallest random laser,” Nature 406, 132–133 (2000).
[CrossRef] [PubMed]

Xu, J. Y.

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
[CrossRef] [PubMed]

Zhao, Y. G.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Eur. J. Phys. (1)

M. V. Berry and S. Klein, “Transparent mirrors: rays, waves and localization,” Eur. J. Phys. 18, 222–228 (1997).
[CrossRef]

J. Phys. C (1)

J. B. Pendry, “Quasi-extended electron states in strongly disordered systems,” J. Phys. C 20, 733–742 (1987).
[CrossRef]

Nature (2)

D. S. Wiersma, “The smallest random laser,” Nature 406, 132–133 (2000).
[CrossRef] [PubMed]

D. S. Wiersma and S. Cavalieri, “Light emission: A temperature-tunable random laser,” Nature 414, 708–709 (2001).
[CrossRef] [PubMed]

Phys. Rev. A (1)

U. Frisch, C. Froeschle, J.-P. Scheidecker, and P.-L. Sulem, “Stochastic resonance in one-dimensional random media,” Phys. Rev. A 8, 1416–1421 (1973).
[CrossRef]

Phys. Rev. B (3)

M. Ya. Azbel and P. Soven, “Transmission resonances and the localization length in one-dimensional disordered systems,” Phys. Rev. B 27, 831–836 (1983).
[CrossRef]

M. Ya. Azbel, “Eigenstates and properties of random systems in one dimension at zero temperature,” Phys. Rev. B 28, 4106–4125 (1983).
[CrossRef]

E. N. Economou and C. M. Soukoulis, “Connection of localization with the problem of the bound state in a potential well,” Phys. Rev. B 28, 1093–1095 (1983).
[CrossRef]

Phys. Rev. Lett. (3)

Xunyia Jiang and C. M. Soukoulis, “Time-dependent theory for random laser,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
[CrossRef] [PubMed]

SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. (1)

A. Grossman and J. Morlet, “Decomposition of Hardy function into square integrable wavelets of constant shape,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

Other (3)

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

D. Bohm, Quantum Theory (Prentice-Hall, New York, 1952).

S. Rytov, Yu. Kravtsov, and V. Tatarskii, Principles of Statistical Radiophysics IV: Wave Propagation Through Random Media (Springer-Verlag, Berlin, 1989).

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Figures (9)

Fig. 1
Fig. 1

Transmission coefficient as a function of the wavelength.

Fig. 2
Fig. 2

Amplitude of the field inside the sample, normalized to the incident wave amplitude, as a function of the coordinate for the three wavelengths marked in Fig. 1.

Fig. 3
Fig. 3

Amplitude of the field, normalized to the incident wave amplitude, as a function of the coordinate inside the whole sample (thin black curves), in the left part (gray curves), middle part (thick black curves), and right part (light gray curves) taken as a separate sample each for λb (top panel) and λc (bottom panel).

Fig. 4
Fig. 4

Two-humped potential profile.

Fig. 5
Fig. 5

a, Spacing between resonantly transparent eigen modes T1, and b, inverse spacing between eigen modes, as functions of the length of the sample.

Fig. 6
Fig. 6

Logarithm of the half-width of the resonances as a function of the length of the sample.

Fig. 7
Fig. 7

Probability for a mode localized at a given point to provide the value of the transmission coefficient T, as a function of the dimensionless coordinate x/L.

Fig. 8
Fig. 8

Same probability as in Fig. 7 represented by colors. Black curve displays the transmission coefficient as a function of the coordinate of the point of localization calculated by Eq. (16).

Fig. 9
Fig. 9

Field amplitudes (light lines) and wavelet transformations (dark lines) as functions of the coordinate in two random realizations.

Equations (21)

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T=Θ12+Θ222Θ1Θ22 cos2 12(π-J)+14 4Θ1Θ2+14Θ1Θ22 sin2 12(π-J)-1.
J=2l0k(x)dx2kl0,
T=1+14 4Θ2-14Θ22 sin2 12(π-J)-1.
J=π(2n-1)n=0, 1, 2,.
Δkres1/l0.
T11+4l02(k-kres)2Θ4,
δk12l0Θ2.
Tres=2Θ1Θ2Θ12+Θ222.
|A|22Θ1Θ22Θ12+Θ22.
l0=lloc.
l1,2=L-lloc2±d,
Θ1,2=exp(l1,2/lloc)=expL-lloc2lloc±dlloc.
Δk1L,
Δkres1lloc.
δk1lloc exp-Llloc,
Tres(d)=4[exp(2d/lloc)+exp(-2d/lloc)]2,
|A(d)|2=8 exp(L/lloc-1-2d/lloc)[exp(2d/lloc)+exp(-2d/lloc)]2.
|A|2 exp(L/2lloc)1.
|A|22 exp(l1/lloc)1,if exp(l1/lloc)exp(l2/lloc)22 exp[(2l2-l1)/lloc]1,if exp(2l2/lloc)exp(l1/lloc)l1exp(l2/lloc),22 exp[(2l2-l1)/lloc]1,if exp(l1/lloc)exp(2l2/lloc)
d=1/4 ln 1/3lloc-0.27lloc.
n˜2(qres)=x1x2n2(x) exp(-i2kx)dx.

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