Abstract

We investigate statistical properties of collective optical excitations in disordered chains of microspheres using transfer-matrix method based on nearest-neighbors approximation. Radiative losses together with transmission and reflection coefficients of optical excitations are studied numerically. We found that for the macroscopically long chain, the transmission coefficient demonstrates properties typical for a one dimensional strongly localized system: log-normal distribution with parameters obeying standard scaling relation. At the same time, we show that the distribution function of the radiative losses behaves very differently from other lossy optical systems. We also studied statistical properties of the optical transport in short chains of resonators and demonstrated that even small disorder results in significant drop of transmission coefficient acompanied by strong enhancement of the radiative losses.

© 2011 OSA

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  1. A. B. Matsko and V. S. Ilchenko, “Optical Resonators With Whispering-Gallery Modes—Part I: Basics,” IEEE J. OF Sel. Topics in Q. El. 12, 3–14 (2006).
    [CrossRef]
  2. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [CrossRef]
  3. V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
    [CrossRef]
  4. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94, 203905 (2005).
    [CrossRef] [PubMed]
  5. B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B 75, 245327 (2007).
    [CrossRef]
  6. K. Grujic and O. G. Hellesø, “Dielectric microsphere manipulation and chain assembly by counter-propagating waves in a channel waveguide,” Opt. Express 15, 6470–6477 (2007).
    [CrossRef] [PubMed]
  7. J. Goeckeritz and S. Blair, “Optical characterization of coupled resonator slow-light rib waveguides,” Opt. Express 18, 18190–18199 (2010).
    [CrossRef] [PubMed]
  8. S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 db per sphere,” Appl. Phys. Lett. 92, 261111 (2008).
    [CrossRef]
  9. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18, 26505–26516 (2010).
    [CrossRef] [PubMed]
  10. V. N. Astratov, “Fundamentals and Applications of Microsphere Resonator Circuits,” in Photonic Microresonator Research and Applications , I. Chremmos, O. Schwelb, and N. Uzunoglu, eds., (Springer Series in Optical Sciences156, 2010), pp. 423–457.
    [CrossRef]
  11. L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. E 73, 036606 (2006).
    [CrossRef]
  12. Z. Chen, A. Taflove, and V. Backman, “Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres,” Opt. Lett. 31, 389–391 (2006).
    [CrossRef] [PubMed]
  13. G. S. Blaustein, M. I. Gozman, O. Samoylova, I. Y. Polishchuk, and A. L. Burin, “Guiding optical modes in chains of dielectric particles,” Opt. Express 15, 17380–17391 (2007).
    [CrossRef] [PubMed]
  14. M. Gozman, I. Polishchuk, and A. Burin, “Light propagation in linear arrays of spherical particles,” Phys. Lett. A 372, 5250 – 5253 (2008).
    [CrossRef]
  15. A. Petrov, M. Krause, and M. Eich, “Backscattering and disorder limits in slow light photonic crystal waveguides,” Opt. Express 17, 8676–8684 (2009).
    [CrossRef] [PubMed]
  16. D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
    [CrossRef]
  17. S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett. 32, 289–291 (2007).
    [CrossRef] [PubMed]
  18. S. Mookherjea, “Spectral characteristics of coupled resonators,” J. Opt. Soc. Am. B 23, 1137–1145 (2006).
    [CrossRef]
  19. P. Pradhan and N. Kumar, “Localization of light in coherently amplifying random media,” Phys. Rev. B 50, 9644–9647 (1994).
    [CrossRef]
  20. V. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. 73, 810–813 (1994).
    [CrossRef] [PubMed]
  21. J. Heinrichs, “Transmission, reflection and localization in a random medium with absorption or gain,” J. Phys.: Condens. Matter 18, 4781 (2006).
    [CrossRef]
  22. J. C. J. Paasschens, T. S. Misirpashaev, and C. W. J. Beenakker, “Localization of light: Dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996).
    [CrossRef]
  23. V. Freilikher and M. Pustilnik, “Phase randomness in a one-dimensional disordered absorbing medium,” Phys. Rev. B 55, R653–R655 (1997).
    [CrossRef]
  24. S. K. Joshi, D. Sahoo, and A. M. Jayannavar, “Modeling of stochastic absorption in a random medium,” Phys. Rev. B 62, 880–885 (2000).
    [CrossRef]
  25. D. V. Savin and H.-J. Sommers, “Distribution of reflection eigenvalues in many-channel chaotic cavities with absorption,” Phys. Rev. E 69, 035201 (2004).
    [CrossRef]
  26. C.-S. Deng, H. Xu, and L. Deych, “Optical transport and statistics of radiative losses in disordered chains of microspheres,” Phys. Rev. A 82, 041803 (2010).
    [CrossRef]
  27. M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).
  28. H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bi-sphere,” Phys. Rev. B 62, 7976–7997 (2000).
    [CrossRef]
  29. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Collective emission and absorption in a linear resonator chain,” Opt. Express 17, 15210 (2009).
    [CrossRef] [PubMed]
  30. A. D. Stone, J. D. Joannopoulos, and D. J. Chadi, “Scaling studies of the resistance of the one-dimensional anderson model with general disorder,” Phys. Rev. B 24, 5583–5596 (1981).
    [CrossRef]
  31. Z.-Q. Zhang, “Light amplification and localization in randomly layered media with gain,” Phys. Rev. B 52, 7960–7964 (1995).
    [CrossRef]
  32. X. Jiang and C. M. Soukoulis, “Transmission and reflection studies of periodic and random systems with gain,” Phys. Rev. B 59, 6159–6166 (1999).
    [CrossRef]
  33. G. Czycholl, B. Kramer, and A. MacKinnon, “Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results,” Z. Phys. B 43, 5–11 (1981).
    [CrossRef]
  34. B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys. 56, 1469 (1993).
    [CrossRef]
  35. B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions,” J. Phys. France 45, 1283–1295 (1984).
    [CrossRef]
  36. F. M. Izrailev, S. Ruffo, and L. Tessieri, “Classical representation of the one-dimensional Anderson model,” J. Phys. A: Math. Gen. 31, 5263 (1998).
    [CrossRef]
  37. A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006).
    [CrossRef]
  38. C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
    [CrossRef]

2010 (3)

2009 (3)

2008 (3)

M. Gozman, I. Polishchuk, and A. Burin, “Light propagation in linear arrays of spherical particles,” Phys. Lett. A 372, 5250 – 5253 (2008).
[CrossRef]

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[CrossRef]

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 db per sphere,” Appl. Phys. Lett. 92, 261111 (2008).
[CrossRef]

2007 (4)

2006 (6)

Z. Chen, A. Taflove, and V. Backman, “Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres,” Opt. Lett. 31, 389–391 (2006).
[CrossRef] [PubMed]

S. Mookherjea, “Spectral characteristics of coupled resonators,” J. Opt. Soc. Am. B 23, 1137–1145 (2006).
[CrossRef]

L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. E 73, 036606 (2006).
[CrossRef]

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical Resonators With Whispering-Gallery Modes—Part I: Basics,” IEEE J. OF Sel. Topics in Q. El. 12, 3–14 (2006).
[CrossRef]

J. Heinrichs, “Transmission, reflection and localization in a random medium with absorption or gain,” J. Phys.: Condens. Matter 18, 4781 (2006).
[CrossRef]

2005 (1)

Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94, 203905 (2005).
[CrossRef] [PubMed]

2004 (2)

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

D. V. Savin and H.-J. Sommers, “Distribution of reflection eigenvalues in many-channel chaotic cavities with absorption,” Phys. Rev. E 69, 035201 (2004).
[CrossRef]

2000 (2)

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bi-sphere,” Phys. Rev. B 62, 7976–7997 (2000).
[CrossRef]

S. K. Joshi, D. Sahoo, and A. M. Jayannavar, “Modeling of stochastic absorption in a random medium,” Phys. Rev. B 62, 880–885 (2000).
[CrossRef]

1999 (2)

X. Jiang and C. M. Soukoulis, “Transmission and reflection studies of periodic and random systems with gain,” Phys. Rev. B 59, 6159–6166 (1999).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[CrossRef]

1998 (1)

F. M. Izrailev, S. Ruffo, and L. Tessieri, “Classical representation of the one-dimensional Anderson model,” J. Phys. A: Math. Gen. 31, 5263 (1998).
[CrossRef]

1997 (1)

V. Freilikher and M. Pustilnik, “Phase randomness in a one-dimensional disordered absorbing medium,” Phys. Rev. B 55, R653–R655 (1997).
[CrossRef]

1996 (1)

J. C. J. Paasschens, T. S. Misirpashaev, and C. W. J. Beenakker, “Localization of light: Dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996).
[CrossRef]

1995 (1)

Z.-Q. Zhang, “Light amplification and localization in randomly layered media with gain,” Phys. Rev. B 52, 7960–7964 (1995).
[CrossRef]

1994 (2)

P. Pradhan and N. Kumar, “Localization of light in coherently amplifying random media,” Phys. Rev. B 50, 9644–9647 (1994).
[CrossRef]

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. 73, 810–813 (1994).
[CrossRef] [PubMed]

1993 (1)

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys. 56, 1469 (1993).
[CrossRef]

1984 (1)

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions,” J. Phys. France 45, 1283–1295 (1984).
[CrossRef]

1981 (2)

G. Czycholl, B. Kramer, and A. MacKinnon, “Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results,” Z. Phys. B 43, 5–11 (1981).
[CrossRef]

A. D. Stone, J. D. Joannopoulos, and D. J. Chadi, “Scaling studies of the resistance of the one-dimensional anderson model with general disorder,” Phys. Rev. B 24, 5583–5596 (1981).
[CrossRef]

Artemyev, M. V.

B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B 75, 245327 (2007).
[CrossRef]

Ashili, S. P.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

Assefa, S.

Astratov, V. N.

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 db per sphere,” Appl. Phys. Lett. 92, 261111 (2008).
[CrossRef]

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006).
[CrossRef]

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

V. N. Astratov, “Fundamentals and Applications of Microsphere Resonator Circuits,” in Photonic Microresonator Research and Applications , I. Chremmos, O. Schwelb, and N. Uzunoglu, eds., (Springer Series in Optical Sciences156, 2010), pp. 423–457.
[CrossRef]

Backman, V.

Beenakker, C. W. J.

J. C. J. Paasschens, T. S. Misirpashaev, and C. W. J. Beenakker, “Localization of light: Dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996).
[CrossRef]

Blair, S.

Blaustein, G. S.

Burin, A.

M. Gozman, I. Polishchuk, and A. Burin, “Light propagation in linear arrays of spherical particles,” Phys. Lett. A 372, 5250 – 5253 (2008).
[CrossRef]

Burin, A. L.

Cai, W.

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006).
[CrossRef]

Chadi, D. J.

A. D. Stone, J. D. Joannopoulos, and D. J. Chadi, “Scaling studies of the resistance of the one-dimensional anderson model with general disorder,” Phys. Rev. B 24, 5583–5596 (1981).
[CrossRef]

Chen, Z.

Chipouline, A.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Cooper, M. L.

Czycholl, G.

G. Czycholl, B. Kramer, and A. MacKinnon, “Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results,” Z. Phys. B 43, 5–11 (1981).
[CrossRef]

Deng, C.-S.

C.-S. Deng, H. Xu, and L. Deych, “Optical transport and statistics of radiative losses in disordered chains of microspheres,” Phys. Rev. A 82, 041803 (2010).
[CrossRef]

Derrida, B.

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions,” J. Phys. France 45, 1283–1295 (1984).
[CrossRef]

Deych, L.

C.-S. Deng, H. Xu, and L. Deych, “Optical transport and statistics of radiative losses in disordered chains of microspheres,” Phys. Rev. A 82, 041803 (2010).
[CrossRef]

Deych, L. I.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. E 73, 036606 (2006).
[CrossRef]

Dignam, M. M.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[CrossRef]

Eich, M.

Franchak, J. P.

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

Freilikher, V.

V. Freilikher and M. Pustilnik, “Phase randomness in a one-dimensional disordered absorbing medium,” Phys. Rev. B 55, R653–R655 (1997).
[CrossRef]

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. 73, 810–813 (1994).
[CrossRef] [PubMed]

Fussell, D. P.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[CrossRef]

Gardner, E.

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions,” J. Phys. France 45, 1283–1295 (1984).
[CrossRef]

Goeckeritz, J.

Gozman, M.

M. Gozman, I. Polishchuk, and A. Burin, “Light propagation in linear arrays of spherical particles,” Phys. Lett. A 372, 5250 – 5253 (2008).
[CrossRef]

Gozman, M. I.

Green, W. M. J.

Grujic, K.

Gupta, G.

Hara, Y.

Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94, 203905 (2005).
[CrossRef] [PubMed]

Heinrichs, J.

J. Heinrichs, “Transmission, reflection and localization in a random medium with absorption or gain,” J. Phys.: Condens. Matter 18, 4781 (2006).
[CrossRef]

Hellesø, O. G.

Hughes, S.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[CrossRef]

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Collective emission and absorption in a linear resonator chain,” Opt. Express 17, 15210 (2009).
[CrossRef] [PubMed]

A. B. Matsko and V. S. Ilchenko, “Optical Resonators With Whispering-Gallery Modes—Part I: Basics,” IEEE J. OF Sel. Topics in Q. El. 12, 3–14 (2006).
[CrossRef]

Izrailev, F. M.

F. M. Izrailev, S. Ruffo, and L. Tessieri, “Classical representation of the one-dimensional Anderson model,” J. Phys. A: Math. Gen. 31, 5263 (1998).
[CrossRef]

Jayannavar, A. M.

S. K. Joshi, D. Sahoo, and A. M. Jayannavar, “Modeling of stochastic absorption in a random medium,” Phys. Rev. B 62, 880–885 (2000).
[CrossRef]

Jiang, X.

X. Jiang and C. M. Soukoulis, “Transmission and reflection studies of periodic and random systems with gain,” Phys. Rev. B 59, 6159–6166 (1999).
[CrossRef]

Jimba, Y.

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bi-sphere,” Phys. Rev. B 62, 7976–7997 (2000).
[CrossRef]

Joannopoulos, J. D.

A. D. Stone, J. D. Joannopoulos, and D. J. Chadi, “Scaling studies of the resistance of the one-dimensional anderson model with general disorder,” Phys. Rev. B 24, 5583–5596 (1981).
[CrossRef]

Joshi, S. K.

S. K. Joshi, D. Sahoo, and A. M. Jayannavar, “Modeling of stochastic absorption in a random medium,” Phys. Rev. B 62, 880–885 (2000).
[CrossRef]

Kanaev, A. V.

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006).
[CrossRef]

Käsebier, T.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Kley, E.-B.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Kramer, B.

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys. 56, 1469 (1993).
[CrossRef]

G. Czycholl, B. Kramer, and A. MacKinnon, “Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results,” Z. Phys. B 43, 5–11 (1981).
[CrossRef]

Krause, M.

Kumar, N.

P. Pradhan and N. Kumar, “Localization of light in coherently amplifying random media,” Phys. Rev. B 50, 9644–9647 (1994).
[CrossRef]

Kuwata-Gonokami, M.

Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94, 203905 (2005).
[CrossRef] [PubMed]

Lacis, A.

M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

Lee, R. K.

Liang, W.

MacKinnon, A.

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys. 56, 1469 (1993).
[CrossRef]

G. Czycholl, B. Kramer, and A. MacKinnon, “Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results,” Z. Phys. B 43, 5–11 (1981).
[CrossRef]

Maleki, L.

Matsko, A. B.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Collective emission and absorption in a linear resonator chain,” Opt. Express 17, 15210 (2009).
[CrossRef] [PubMed]

A. B. Matsko and V. S. Ilchenko, “Optical Resonators With Whispering-Gallery Modes—Part I: Basics,” IEEE J. OF Sel. Topics in Q. El. 12, 3–14 (2006).
[CrossRef]

Mishchenko, M.

M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

Misirpashaev, T. S.

J. C. J. Paasschens, T. S. Misirpashaev, and C. W. J. Beenakker, “Localization of light: Dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996).
[CrossRef]

Miyazaki, H.

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bi-sphere,” Phys. Rev. B 62, 7976–7997 (2000).
[CrossRef]

Möller, B. M.

B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B 75, 245327 (2007).
[CrossRef]

Mookherjea, S.

Mukaiyama, T.

Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94, 203905 (2005).
[CrossRef] [PubMed]

Oh, A.

Paasschens, J. C. J.

J. C. J. Paasschens, T. S. Misirpashaev, and C. W. J. Beenakker, “Localization of light: Dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996).
[CrossRef]

Pertsch, T.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Petrov, A.

Polishchuk, I.

M. Gozman, I. Polishchuk, and A. Burin, “Light propagation in linear arrays of spherical particles,” Phys. Lett. A 372, 5250 – 5253 (2008).
[CrossRef]

Polishchuk, I. Y.

Pradhan, P.

P. Pradhan and N. Kumar, “Localization of light in coherently amplifying random media,” Phys. Rev. B 50, 9644–9647 (1994).
[CrossRef]

Pustilnik, M.

V. Freilikher and M. Pustilnik, “Phase randomness in a one-dimensional disordered absorbing medium,” Phys. Rev. B 55, R653–R655 (1997).
[CrossRef]

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. 73, 810–813 (1994).
[CrossRef] [PubMed]

Roslyak, O.

L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. E 73, 036606 (2006).
[CrossRef]

Ruffo, S.

F. M. Izrailev, S. Ruffo, and L. Tessieri, “Classical representation of the one-dimensional Anderson model,” J. Phys. A: Math. Gen. 31, 5263 (1998).
[CrossRef]

Sahoo, D.

S. K. Joshi, D. Sahoo, and A. M. Jayannavar, “Modeling of stochastic absorption in a random medium,” Phys. Rev. B 62, 880–885 (2000).
[CrossRef]

Samoylova, O.

Savchenkov, A. A.

Savin, D. V.

D. V. Savin and H.-J. Sommers, “Distribution of reflection eigenvalues in many-channel chaotic cavities with absorption,” Phys. Rev. E 69, 035201 (2004).
[CrossRef]

Scherer, A.

Schmidt, C.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Schneider, M. A.

Seidel, D.

Shuvayev, V.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Sommers, H.-J.

D. V. Savin and H.-J. Sommers, “Distribution of reflection eigenvalues in many-channel chaotic cavities with absorption,” Phys. Rev. E 69, 035201 (2004).
[CrossRef]

Soukoulis, C. M.

X. Jiang and C. M. Soukoulis, “Transmission and reflection studies of periodic and random systems with gain,” Phys. Rev. B 59, 6159–6166 (1999).
[CrossRef]

Stone, A. D.

A. D. Stone, J. D. Joannopoulos, and D. J. Chadi, “Scaling studies of the resistance of the one-dimensional anderson model with general disorder,” Phys. Rev. B 24, 5583–5596 (1981).
[CrossRef]

Taflove, A.

Takeda, K.

Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94, 203905 (2005).
[CrossRef] [PubMed]

Tessieri, L.

F. M. Izrailev, S. Ruffo, and L. Tessieri, “Classical representation of the one-dimensional Anderson model,” J. Phys. A: Math. Gen. 31, 5263 (1998).
[CrossRef]

Travis, L.

M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

Tünnermann, A.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Vlasov, Y. A.

Woggon, U.

B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B 75, 245327 (2007).
[CrossRef]

Xia, F.

Xu, H.

C.-S. Deng, H. Xu, and L. Deych, “Optical transport and statistics of radiative losses in disordered chains of microspheres,” Phys. Rev. A 82, 041803 (2010).
[CrossRef]

Xu, Y.

Yang, S.

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 db per sphere,” Appl. Phys. Lett. 92, 261111 (2008).
[CrossRef]

Yariv, A.

Yurkevich, I.

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. 73, 810–813 (1994).
[CrossRef] [PubMed]

Zhang, Z.-Q.

Z.-Q. Zhang, “Light amplification and localization in randomly layered media with gain,” Phys. Rev. B 52, 7960–7964 (1995).
[CrossRef]

Appl. Phys. Lett. (3)

V. N. Astratov, J. P. Franchak, and S. P. Ashili, “Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder,” Appl. Phys. Lett. 85, 5508–5510 (2004).
[CrossRef]

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 db per sphere,” Appl. Phys. Lett. 92, 261111 (2008).
[CrossRef]

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006).
[CrossRef]

IEEE J. OF Sel. Topics in Q. El. (1)

A. B. Matsko and V. S. Ilchenko, “Optical Resonators With Whispering-Gallery Modes—Part I: Basics,” IEEE J. OF Sel. Topics in Q. El. 12, 3–14 (2006).
[CrossRef]

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

J. Phys. A: Math. Gen. (1)

F. M. Izrailev, S. Ruffo, and L. Tessieri, “Classical representation of the one-dimensional Anderson model,” J. Phys. A: Math. Gen. 31, 5263 (1998).
[CrossRef]

J. Phys. France (1)

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions,” J. Phys. France 45, 1283–1295 (1984).
[CrossRef]

J. Phys.: Condens. Matter (1)

J. Heinrichs, “Transmission, reflection and localization in a random medium with absorption or gain,” J. Phys.: Condens. Matter 18, 4781 (2006).
[CrossRef]

Opt. Express (6)

Opt. Lett. (3)

Phys. Lett. A (1)

M. Gozman, I. Polishchuk, and A. Burin, “Light propagation in linear arrays of spherical particles,” Phys. Lett. A 372, 5250 – 5253 (2008).
[CrossRef]

Phys. Rev. A (2)

C.-S. Deng, H. Xu, and L. Deych, “Optical transport and statistics of radiative losses in disordered chains of microspheres,” Phys. Rev. A 82, 041803 (2010).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Phys. Rev. B (10)

P. Pradhan and N. Kumar, “Localization of light in coherently amplifying random media,” Phys. Rev. B 50, 9644–9647 (1994).
[CrossRef]

J. C. J. Paasschens, T. S. Misirpashaev, and C. W. J. Beenakker, “Localization of light: Dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996).
[CrossRef]

V. Freilikher and M. Pustilnik, “Phase randomness in a one-dimensional disordered absorbing medium,” Phys. Rev. B 55, R653–R655 (1997).
[CrossRef]

S. K. Joshi, D. Sahoo, and A. M. Jayannavar, “Modeling of stochastic absorption in a random medium,” Phys. Rev. B 62, 880–885 (2000).
[CrossRef]

A. D. Stone, J. D. Joannopoulos, and D. J. Chadi, “Scaling studies of the resistance of the one-dimensional anderson model with general disorder,” Phys. Rev. B 24, 5583–5596 (1981).
[CrossRef]

Z.-Q. Zhang, “Light amplification and localization in randomly layered media with gain,” Phys. Rev. B 52, 7960–7964 (1995).
[CrossRef]

X. Jiang and C. M. Soukoulis, “Transmission and reflection studies of periodic and random systems with gain,” Phys. Rev. B 59, 6159–6166 (1999).
[CrossRef]

H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bi-sphere,” Phys. Rev. B 62, 7976–7997 (2000).
[CrossRef]

B. M. Möller, U. Woggon, and M. V. Artemyev, “Bloch modes and disorder phenomena in coupled resonator chains,” Phys. Rev. B 75, 245327 (2007).
[CrossRef]

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[CrossRef]

Phys. Rev. E (2)

L. I. Deych and O. Roslyak, “Photonic band mixing in linear chains of optically coupled microspheres,” Phys. Rev. E 73, 036606 (2006).
[CrossRef]

D. V. Savin and H.-J. Sommers, “Distribution of reflection eigenvalues in many-channel chaotic cavities with absorption,” Phys. Rev. E 69, 035201 (2004).
[CrossRef]

Phys. Rev. Lett. (2)

V. Freilikher, M. Pustilnik, and I. Yurkevich, “Effect of absorption on the wave transport in the strong localization regime,” Phys. Rev. Lett. 73, 810–813 (1994).
[CrossRef] [PubMed]

Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94, 203905 (2005).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys. 56, 1469 (1993).
[CrossRef]

Z. Phys. B (1)

G. Czycholl, B. Kramer, and A. MacKinnon, “Conductivity and localization of electron states in one dimensional disordered systems: Further numerical results,” Z. Phys. B 43, 5–11 (1981).
[CrossRef]

Other (2)

M. Mishchenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University Press, Cambridge, 2002).

V. N. Astratov, “Fundamentals and Applications of Microsphere Resonator Circuits,” in Photonic Microresonator Research and Applications , I. Chremmos, O. Schwelb, and N. Uzunoglu, eds., (Springer Series in Optical Sciences156, 2010), pp. 423–457.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic of a disordered chain composed of N coupled microspheres with a finite length L = Nd. R and T are the reflection and transmission coefficient of the fundamental Bloch mode.

Fig. 2
Fig. 2

The localization length ξ as a function of frequency for a chain of N = 1000 spheres and for disorder strength δ = 10 3.

Fig. 3
Fig. 3

Main frame: ld /ξ as a function of δ 2 for different frequencies around the band center. Inset: ld /ξ as a function of δ 2/3 for different frequencies around the band edge. The straight lines show the linear relationships between ld /ξ and δ 2, ld /ξ and δ 2/3.

Fig. 4
Fig. 4

Scaling relation between τ and ρ. The inset shows τ as a function of ξ.

Fig. 5
Fig. 5

(a) Linear relationship between A and ρ for a single realization and for a wide range of frequencies. (b) Distribution function of A: numerical histogram and its fit by the distribution of ξ.

Fig. 6
Fig. 6

Scaling of mean value and variance of the losses A for several values of frequencies in the vicinity of band center.

Fig. 7
Fig. 7

Evolution of 〈A〉 for short chain N = 6. (a) 〈A〉 as a function of frequency for various values of disorder strength δ. (b) 〈A〉 as a function of disorder strength δ for various values of frequency.

Fig. 8
Fig. 8

Evolution of 〈T〉 and 〈R〉 for short chain N = 6. (a) 〈T〉 and 〈R〉 as functions of frequency for δ = 10−3. The frequencies are all around upper band edge frequency. (b) 〈T〉 as a function of δ for various values of frequency.

Fig. 9
Fig. 9

(a) Dependence of 〈T〉 and 〈R〉 on the length of the chain for N = 6 and δ = 10−3 at band edge frequency x = 21.838. (b) Dependence of 〈A〉 on the length of the chain for N = 6 and for several values of disorder strength at band edge frequency x = 21.838.

Fig. 10
Fig. 10

Probability distribution f (A) of losses A for (a) different chain length but fixed disorder strength δ = 10−3, (b) different disorder strength but fixed chain length N = 6. The frequency was kept as x = 21.954 for all of these calculations.

Equations (27)

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E inc ( n ) = l , m [ ζ l , m ( n ) N m , l ( 1 ) ( r r n ) + η l , m ( 1 ) M m , l ( 1 ) ( r r n ) ] ,
E int ( n ) = l , m [ c l , m ( n ) N m , l ( 1 ) ( r r n ) + d l , m ( 1 ) M m , l ( 1 ) ( r r n ) ] .
E s = n = 1 N l , m [ a l , m ( n ) N m , l ( 3 ) ( r r n ) + b l , m ( n ) M m , l ( 3 ) ( r r n ) ] ,
a l , m ( n ) = α l ( n ) { ζ l , m ( n ) + j n l , m [ a l , m ( j ) U l , m l , m ( r j r n ) + b l , m ( j ) V l , m l , m ( r j r n ) ] } .
α l ( n ) = i β l ( n ) g l ( n ) i β l ( n ) .
α l ( n ) = i γ l , s ( n ) ω ω l , s ( n ) + i γ l , s ( n ) ,
1 α n ( l ) a n ( l , m ) = U n , n 1 ( l , m ) a n 1 ( l , m ) + U n , n + 1 ( l , m ) a n + 1 ( l , m ) .
ω ω 0 + i γ = 2 γ U cos ( q d ) ,
n = 1 N | a n | 2 = J N + 1 + J 1 .
J n = i 2 U n , n 1 ( a n 1 a n * a n 1 * a n ) ,
W s = 1 4 π d φ d θ sin θ n · 1 2 Re ( E s × H s * ) ,
( a n + 1 a n ) = T n ( a n a n 1 ) , T n = ( 1 α n U n , n + 1 U n , n 1 U n , n + 1 1 0 ) ,
a n = a n + e iqnd + a n e iqnd a n 1 = a n + e iq ( n 1 ) d + a n e iq ( n 1 ) d ,
J n = U n , n 1 sin ( kd ) [ e ( 2 n 1 ) β d | a n | 2 e ( 2 n 1 ) β d | a n + | 2 ] + U n , n 1 i sinh ( β d ) [ ( a n + ) * a n e i ( 2 n 1 ) kd ( a n ) * a n + e i ( 2 n 1 ) kd ] ,
T = U N + 1 , N U 1 , 0 e 2 N β d | T N + 1 l | 2 ,
R = e 2 β d | R N + 1 l | 2 i sinh ( β d ) [ R N + 1 l * e ikd R N + 1 l e ikd ] e β d sin ( kd )
A + R + T = 1 ,
A = n = 1 N | a n | 2 U 1 , 0 e β d sin ( kd )
P n = 1 2 i sin ( q d ) ( 1 α n e iqd ( U n , n + 1 + U n , n 1 ) α n U n , n + 1 e 2 iqnd 1 α n ( e iqd U n , n + 1 + e iqd U n , n 1 ) α n U n , n + 1 e 2 iqnd 1 α n ( e iqd U n , n + 1 + e iqd U n , n 1 ) α n U n , n + 1 1 α n e iqd ( U n , n + 1 + U n , n 1 ) α n U n , n + 1 ) .
r n l = a n + a n , r n r = a n + 1 a n + 1 + , t n l = a n + 1 a n , t n r = a n + a n + 1 + .
P n = ( 1 / t n r r n l / t n r r n r / t n r t n l r n l r n r / t n r ) .
T N r = t N r T N 1 r 1 r N l R N 1 r , R N r = r N r + R N 1 r t N l t N r 1 r N l R N 1 r , T N l = t N l T N 1 l 1 r N l R N 1 r , R N l = R N 1 l + r N l T N 1 l T N 1 r 1 r N l R N 1 r .
1 ξ = 1 ξ 0 + 1 l d ,
l d ξ = l d ξ 0 + 1 { δ 2 + 1 band   center δ 2 / 3 + 1 band   edges .
f u ( A ) A 2 exp ( a u / A ) ,
f u ( A ) A 2 exp [ ( a A ) 2 / ( b 2 A 2 ) ] .
n | a n | 2 = 1 / ( 1 exp [ d / ξ ] ) ξ ,

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