Abstract

In a weakly disordered sample, light waves (or electrons) propagate, on average, by diffusion. However, with some small probability, random high-quality cavities can be formed within the sample. Such cavities are due to rare events, i.e., to some rare disorder configurations which can support “almost localized” eigenstates and thus can trap the wave for a long time in a small region of space of sub-mean-free-path size. The almost localized states are nonuniversal in the sense that their character and likelihood are determined not only by the average strength of the disorder (the dimensionless conductance) but also by microscopic details of the system. In particular, they are extremely sensitive to the correlation radius Rc of the disordered potential. Moreover, on a lattice, a new type of almost localized state becomes possible that has no analog in the continuum. The likelihood of these lattice-specific states decreases with the increase of Rc, in sharp contrast to the situation in the continuum. We review the earlier work on the almost localized states in the continuum and develop a theory of those states on a lattice. We emphasize that extreme care must be taken in trying to simulate on a lattice the rare events in a continuous, random medium.

© 2004 Optical Society of America

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2002 (7)

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802–016805 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601–126604 (2002).
[CrossRef] [PubMed]

B. K. Nikolic, “Quest for rare events in mesoscopic disordered metals,” Phys. Rev. B 65, 012201–012204 (2002).
[CrossRef]

B. K. Nikolic and V. Z. Cerovski, “Structure of quantum disordered wave functions: weak localization, far tails, and mesoscopic transport,” Eur. Phys. J. B 30, 227–238 (2002).
[CrossRef]

A. Ossipov, T. Kottos, and T. Geisel, “Signatures of prelocalized states in classically chaotic systems,” Phys. Rev. E 65, 055209–055212 (2002).
[CrossRef]

V. Uski, B. Mehlig, and M. Schreiber, “Spatial structure of anomalously localized states in disordered conductors,” Phys. Rev. B 66, 233104–233107 (2002).
[CrossRef]

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807–013812 (2002).
[CrossRef]

2001 (4)

B. K. Nikolic and P. B. Allen, “Resistivity of a metal between the Boltzmann transport regime and the Anderson transition,” Phys. Rev. B 63, 020201–020204 (2001).
[CrossRef]

B. K. Nikolic, “Statistical properties of eigenstates in three-dimensional mesoscopic systems with off-diagonal or diagonal disorder,” Phys. Rev. B 64, 014203–014210 (2001).
[CrossRef]

The opposite case of light trapping by strongly non-Gaussian disorder was addressed numerically in C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903–183906 (2001).
[CrossRef]

V. Uski, B. Mehlig, and M. Schreiber, “Signature of ballistic effects in disordered conductors,” Phys. Rev. B 63, R241101–241104 (2001).
[CrossRef]

2000 (3)

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

V. Uski, B. Mehlig, R. A. Römer, and M. Schreiber, “Exact diagonalization study of rare events in disordered conductors,” Phys. Rev. B 62, R7699–R7702 (2000).
[CrossRef]

A. D. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Phys. Rep. 326, 259–382 (2000).
[CrossRef]

1999 (2)

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]

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
[CrossRef]

1997 (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

1995 (2)

B. A. Muzykantskii and D. E. Khmelnitskii, “Nearly localized states in weakly disordered conductors,” Phys. Rev. B 51, 5480–5483 (1995).
[CrossRef]

V. I. Fal’ko and K. B. Efetov, “Statistics of prelocalized states in disordered conductors,” Phys. Rev. B 52, 17413–17429 (1995).
[CrossRef]

1993 (2)

V. G. Karpov, “Resonant electronic states in disordered systems,” Phys. Rev. B 48, 4325–4328 (1993).
[CrossRef]

V. G. Karpov, “Quasilocalized sound excitations in disordered systems,” Phys. Rev. B 48, 12539–12549 (1993).
[CrossRef]

1966 (2)

B. I. Halperin and M. Lax, “Impurity-band tails in the high-density limit. I. Minimum counting methods,” Phys. Rev. 148, 722–740 (1966).
[CrossRef]

J. Zittartz and J. S. Langer, “Theory of bound states in a random potential,” Phys. Rev. 148, 741–747 (1966).
[CrossRef]

Allen, P. B.

B. K. Nikolic and P. B. Allen, “Resistivity of a metal between the Boltzmann transport regime and the Anderson transition,” Phys. Rev. B 63, 020201–020204 (2001).
[CrossRef]

Apalkov, V. M.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601–126604 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802–016805 (2002).
[CrossRef] [PubMed]

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Baughman, R. H.

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
[CrossRef]

Bayram, S. B.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807–013812 (2002).
[CrossRef]

Cao, 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]

Cerovski, V. Z.

B. K. Nikolic and V. Z. Cerovski, “Structure of quantum disordered wave functions: weak localization, far tails, and mesoscopic transport,” Eur. Phys. J. B 30, 227–238 (2002).
[CrossRef]

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]

Efetov, K. B.

V. I. Fal’ko and K. B. Efetov, “Statistics of prelocalized states in disordered conductors,” Phys. Rev. B 52, 17413–17429 (1995).
[CrossRef]

Fal’ko, V. I.

V. I. Fal’ko and K. B. Efetov, “Statistics of prelocalized states in disordered conductors,” Phys. Rev. B 52, 17413–17429 (1995).
[CrossRef]

Frolov, S. V.

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
[CrossRef]

Geisel, T.

A. Ossipov, T. Kottos, and T. Geisel, “Signatures of prelocalized states in classically chaotic systems,” Phys. Rev. E 65, 055209–055212 (2002).
[CrossRef]

Halperin, B. I.

B. I. Halperin and M. Lax, “Impurity-band tails in the high-density limit. I. Minimum counting methods,” Phys. Rev. 148, 722–740 (1966).
[CrossRef]

Hinklin, T.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807–013812 (2002).
[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, X.

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

Karpov, V. G.

V. G. Karpov, “Quasilocalized sound excitations in disordered systems,” Phys. Rev. B 48, 12539–12549 (1993).
[CrossRef]

V. G. Karpov, “Resonant electronic states in disordered systems,” Phys. Rev. B 48, 4325–4328 (1993).
[CrossRef]

Khmelnitskii, D. E.

B. A. Muzykantskii and D. E. Khmelnitskii, “Nearly localized states in weakly disordered conductors,” Phys. Rev. B 51, 5480–5483 (1995).
[CrossRef]

Kottos, T.

A. Ossipov, T. Kottos, and T. Geisel, “Signatures of prelocalized states in classically chaotic systems,” Phys. Rev. E 65, 055209–055212 (2002).
[CrossRef]

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Laine, R. M.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807–013812 (2002).
[CrossRef]

Langer, J. S.

J. Zittartz and J. S. Langer, “Theory of bound states in a random potential,” Phys. Rev. 148, 741–747 (1966).
[CrossRef]

Lax, M.

B. I. Halperin and M. Lax, “Impurity-band tails in the high-density limit. I. Minimum counting methods,” Phys. Rev. 148, 722–740 (1966).
[CrossRef]

Mehlig, B.

V. Uski, B. Mehlig, and M. Schreiber, “Spatial structure of anomalously localized states in disordered conductors,” Phys. Rev. B 66, 233104–233107 (2002).
[CrossRef]

V. Uski, B. Mehlig, and M. Schreiber, “Signature of ballistic effects in disordered conductors,” Phys. Rev. B 63, R241101–241104 (2001).
[CrossRef]

V. Uski, B. Mehlig, R. A. Römer, and M. Schreiber, “Exact diagonalization study of rare events in disordered conductors,” Phys. Rev. B 62, R7699–R7702 (2000).
[CrossRef]

Mirlin, A. D.

A. D. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Phys. Rep. 326, 259–382 (2000).
[CrossRef]

Muzykantskii, B. A.

B. A. Muzykantskii and D. E. Khmelnitskii, “Nearly localized states in weakly disordered conductors,” Phys. Rev. B 51, 5480–5483 (1995).
[CrossRef]

Nikolic, B. K.

B. K. Nikolic, “Quest for rare events in mesoscopic disordered metals,” Phys. Rev. B 65, 012201–012204 (2002).
[CrossRef]

B. K. Nikolic and V. Z. Cerovski, “Structure of quantum disordered wave functions: weak localization, far tails, and mesoscopic transport,” Eur. Phys. J. B 30, 227–238 (2002).
[CrossRef]

B. K. Nikolic, “Statistical properties of eigenstates in three-dimensional mesoscopic systems with off-diagonal or diagonal disorder,” Phys. Rev. B 64, 014203–014210 (2001).
[CrossRef]

B. K. Nikolic and P. B. Allen, “Resistivity of a metal between the Boltzmann transport regime and the Anderson transition,” Phys. Rev. B 63, 020201–020204 (2001).
[CrossRef]

Ossipov, A.

A. Ossipov, T. Kottos, and T. Geisel, “Signatures of prelocalized states in classically chaotic systems,” Phys. Rev. E 65, 055209–055212 (2002).
[CrossRef]

Raikh, M. E.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802–016805 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601–126604 (2002).
[CrossRef] [PubMed]

Rand, S. C.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807–013812 (2002).
[CrossRef]

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Römer, R. A.

V. Uski, B. Mehlig, R. A. Römer, and M. Schreiber, “Exact diagonalization study of rare events in disordered conductors,” Phys. Rev. B 62, R7699–R7702 (2000).
[CrossRef]

Schreiber, M.

V. Uski, B. Mehlig, and M. Schreiber, “Spatial structure of anomalously localized states in disordered conductors,” Phys. Rev. B 66, 233104–233107 (2002).
[CrossRef]

V. Uski, B. Mehlig, and M. Schreiber, “Signature of ballistic effects in disordered conductors,” Phys. Rev. B 63, R241101–241104 (2001).
[CrossRef]

V. Uski, B. Mehlig, R. A. Römer, and M. Schreiber, “Exact diagonalization study of rare events in disordered conductors,” Phys. Rev. B 62, R7699–R7702 (2000).
[CrossRef]

Sebbah, P.

The opposite case of light trapping by strongly non-Gaussian disorder was addressed numerically in C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903–183906 (2001).
[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]

Shapiro, B.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601–126604 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802–016805 (2002).
[CrossRef] [PubMed]

Soukoulis, C. M.

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

Uski, V.

V. Uski, B. Mehlig, and M. Schreiber, “Spatial structure of anomalously localized states in disordered conductors,” Phys. Rev. B 66, 233104–233107 (2002).
[CrossRef]

V. Uski, B. Mehlig, and M. Schreiber, “Signature of ballistic effects in disordered conductors,” Phys. Rev. B 63, R241101–241104 (2001).
[CrossRef]

V. Uski, B. Mehlig, R. A. Römer, and M. Schreiber, “Exact diagonalization study of rare events in disordered conductors,” Phys. Rev. B 62, R7699–R7702 (2000).
[CrossRef]

Vanneste, C.

The opposite case of light trapping by strongly non-Gaussian disorder was addressed numerically in C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903–183906 (2001).
[CrossRef]

Vardeny, Z. V.

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
[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, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Williams, G. R.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807–013812 (2002).
[CrossRef]

Yoshino, K.

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
[CrossRef]

Zakhidov, A.

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
[CrossRef]

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]

Zittartz, J.

J. Zittartz and J. S. Langer, “Theory of bound states in a random potential,” Phys. Rev. 148, 741–747 (1966).
[CrossRef]

Eur. Phys. J. B (1)

B. K. Nikolic and V. Z. Cerovski, “Structure of quantum disordered wave functions: weak localization, far tails, and mesoscopic transport,” Eur. Phys. J. B 30, 227–238 (2002).
[CrossRef]

Nature (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Phys. Rep. (1)

A. D. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Phys. Rep. 326, 259–382 (2000).
[CrossRef]

Phys. Rev. (2)

B. I. Halperin and M. Lax, “Impurity-band tails in the high-density limit. I. Minimum counting methods,” Phys. Rev. 148, 722–740 (1966).
[CrossRef]

J. Zittartz and J. S. Langer, “Theory of bound states in a random potential,” Phys. Rev. 148, 741–747 (1966).
[CrossRef]

Phys. Rev. A (1)

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807–013812 (2002).
[CrossRef]

Phys. Rev. B (11)

V. G. Karpov, “Resonant electronic states in disordered systems,” Phys. Rev. B 48, 4325–4328 (1993).
[CrossRef]

V. G. Karpov, “Quasilocalized sound excitations in disordered systems,” Phys. Rev. B 48, 12539–12549 (1993).
[CrossRef]

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999).
[CrossRef]

B. A. Muzykantskii and D. E. Khmelnitskii, “Nearly localized states in weakly disordered conductors,” Phys. Rev. B 51, 5480–5483 (1995).
[CrossRef]

V. I. Fal’ko and K. B. Efetov, “Statistics of prelocalized states in disordered conductors,” Phys. Rev. B 52, 17413–17429 (1995).
[CrossRef]

V. Uski, B. Mehlig, R. A. Römer, and M. Schreiber, “Exact diagonalization study of rare events in disordered conductors,” Phys. Rev. B 62, R7699–R7702 (2000).
[CrossRef]

V. Uski, B. Mehlig, and M. Schreiber, “Signature of ballistic effects in disordered conductors,” Phys. Rev. B 63, R241101–241104 (2001).
[CrossRef]

V. Uski, B. Mehlig, and M. Schreiber, “Spatial structure of anomalously localized states in disordered conductors,” Phys. Rev. B 66, 233104–233107 (2002).
[CrossRef]

B. K. Nikolic and P. B. Allen, “Resistivity of a metal between the Boltzmann transport regime and the Anderson transition,” Phys. Rev. B 63, 020201–020204 (2001).
[CrossRef]

B. K. Nikolic, “Statistical properties of eigenstates in three-dimensional mesoscopic systems with off-diagonal or diagonal disorder,” Phys. Rev. B 64, 014203–014210 (2001).
[CrossRef]

B. K. Nikolic, “Quest for rare events in mesoscopic disordered metals,” Phys. Rev. B 65, 012201–012204 (2002).
[CrossRef]

Phys. Rev. E (1)

A. Ossipov, T. Kottos, and T. Geisel, “Signatures of prelocalized states in classically chaotic systems,” Phys. Rev. E 65, 055209–055212 (2002).
[CrossRef]

Phys. Rev. Lett. (5)

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802–016805 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601–126604 (2002).
[CrossRef] [PubMed]

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

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[CrossRef]

The opposite case of light trapping by strongly non-Gaussian disorder was addressed numerically in C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903–183906 (2001).
[CrossRef]

Other (5)

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Coherent random lasing and ‘almost localized’ photon modes,” http://arxiv.org/abs/cond-mat/0212231.

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, “Breakdown of diffusion in dynamics of extended waves in mesoscopic media,” http://arxiv.org/abs/cond-mat/0211651.

B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, “Distribution of mesoscopic fluctuations and relaxation processes in disordered conductors,” in Mesoscopic Phenomena in Solids, B. L. Altshuler, P. A. Lee, and R. A. Webb, eds. (North-Holland, Amsterdam, 1991).

M. Patra, “Influence of spatial correlations on the lasing threshold of random lasers,” http://arxiv.org/abs/cond-mat/0302506.

M. L. Mehta, Random Matrices and the Statistical Theory of Energy Levels (Academic, New York, 1991).

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Figures (3)

Fig. 1
Fig. 1

(a) Structure of a two-dimensional resonator illustrated schematically; only half of the ring-shaped waveguide (blank region) is shown. (b) Optimal fluctuation of the dielectric constant δ(ρ) (solid curve) and the corresponding field distribution (dotted curve). Dashed curve outside the shaded region of width d illustrates evanescent leakage.

Fig. 2
Fig. 2

Rationale for the structure of the resonator. Upon wrapping a stripe with enhanced dielectric constant into a ring, a wave-guided mode is transformed into a whispering-gallery mode.

Fig. 3
Fig. 3

Dimensionless function Φ(u) defined in Eq. (17) is plotted. Inset: Normalized modulus of the log-density of random resonators S˜m=Sm(k0Rc)/Sm(0) calculated from Eq. (24) for =4, Q=50, and m=15, is plotted versus the dimensionless correlation radius k0Rc.

Equations (71)

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ΔρΨ+[k2-U(ρ)]Ψ=0,
k2=ωc2=k02,U(ρ)=-δ(ρ)k02,
ln P=-12W2 dρdρδ(ρ)δ(ρ)κ(ρ-ρ),
dρκ(ρ-ρ)K(ρ-ρ1)=δ(ρ-ρ1).
δ(ρ)δ(ρ)=W2K(ρ-ρ)K(0)=1,
Nm(kl, ωT)exp[-Sm(kl, ωT)],
Ψ(ρ, θ)=(2πρ)-1/2χm(ρ)exp(imθ),
Lˆχm=d2χmdx2+δk02χm=1k02χm,
d2χmdx2=1k021-xdχm,
ln(ωT)=2m3 13/2.
Ξ{δ}=ln P-λ(χm|Lˆ|χm-1k02χm|χm),
δ(x)=dx1K0(x-x1)χm2(x1).
Sm=πρ01k02W2 dz1dz2χm2(z1)χm2(z2)K0(z1-z2),
d2χm(z)dz2+χm(z)13/2k0 dz1K0(z-z1)χm2(z1)=χm(z).
F {χ}=dzχm2(z)-1dzdχm(z)dz2-1213/2k0×dz1dz2χm2(z1)K0(z1-z2)χm2(z2).
Sm=243-3/2π1/2m131/2 Φ(11/2k0Rc)(Wk0Rc)2,
Φ(u)=33/226 (5+9+16u2)5/2(3+9+16u2)3/2.
(kl)-1=W24π2 dqdϕq3δ(q2+2kq cos ϕ)×d2ρK(ρ)exp(iqρ)=2W2k02 0π/2dα sin2 α0dρρK(ρ)J0(2kρ sin α),
(kl)-1=πWk0Rc21/22Fk2Rc22,
F(x)=exp(-x)[I0(x)-I1(x)].
kl=4π(k0RcW)2fork0Rc1;
kl=45/2k0Rcπ1/2W2fork0Rc1.
Sm(k0Rc1)=2π331/2kl|ln ωT|.
Sm(k0Rc>1)Sm(k0Rc1)=Φ(11/2k0Rc)π1/2(1/2k0Rc)3.
Hˆ=r,rcrcr+rVrcrcr,
P(Vr)=1WθW2-|Vr|,
P(Vr)=2π1/2W exp-4Vr2W2.
VrVr=βW2K(|r-r|),
Epx,py=2(cos px+cos py),
ψpx,py(nx, ny)=L-1 exp(ipxnx+ipyny),
ν(E)=-ππ dpx2π -ππ dpy2πδ(E-Epx,py)=4π2(4+|E|) K4-|E|4+|E|,
ν(E)=14π,
ν(E)=-12π2 ln|E|.
Ep=4-(px2+py2),
ψ(2n)=An,ψ(2n+1)=Bn,
(E-V0)An+1=Bn+1+Bn,
(E+V0)Bn=An+An+1.
(E+V0)An=Bn+Bn-1,
(E-V0)Bn-1=An-1+An.
An=A0 exp(iπn)exp(-2γ|n|),
Bn=B0 exp(iπn)exp(-2γ|n|),
(E-V0)A0=B0[1-exp(2γ)],
(E+V0)B0=A0[1-exp(-2γ)],
γ(V0)=arcsinh12(V02-E2)1/2.
Vnx,ny=V0exp(iπnx)+exp(iπny)fornx>0,ny>0,-exp(iπnx)+exp(iπny)fornx<0,ny>0,exp(iπnx)-exp(iπny)fornx>0,ny<0,-exp(iπnx)-exp(iπny)fornx<0,ny<0,
ψ(2nx, 2ny)=AnxCny,
ψ(2nx, 2ny+1)=AnxDny,
ψ(2nx+1, 2ny)=BnxCny,
ψ(2nx+1, 2ny+1)=BnxDny.
(E-2V0)Anx+1Cny+1=Anx+1(Dny+1+Dny)+Cny+1(Bnx+1+Bnx),
EAnx+1Dny=Anx+1(Cny+Cny+1)+Dny(Bnx+1+Bnx),
EBnxCny+1=Bnx(Dny+1+Dny)+Cny+1(Anx+Anx+1),
(E+2V0)BnxDnx=Bnx(Cny+Cny+1)+Dny(Anx+Anx+1).
γ(V0)=arcsinh12 V02-E241/2.
ln(ωT)=γN.
Nexp-16N2V02W2.
Nexp-16V02 ln2(ωT)W2 arcsinh2{2-1[V02-(E/2)2]1/2}.
F(E)=minV0V0arcsinh{2-1[V02-(E/2)2]1/2},
Nexp(-S),
S=16F2(E)ln2(ωT)W2.
P1=-W/2W/2 dV1W-W/2W/2 dVNWθi=1Nγ(Vi)-ln(ωT).
P1=N-ln(ωT)WNN!.
N(P12)4N1-ln(ωT)WN8N2,
S=8α0 ln2(ωT)W2,
α0=minZZ2 lnZZ-12.45.
P(Vnx,ny, Vmx,my)
exp-4Vnx,ny2W2-4Vmx,my2W2
-ξθ(Rc-|ρn-ρm|)(Vnx,ny-Vmx,my)2W2,
S(Rc1)Rc2F2(E)ln2(ωT)W2.
lRcW2.
S(Rc1)S(Rc=0)Rc1.

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