Abstract

In a strongly scattering medium where Anderson localization takes place, constructive interference of local non-propagating waves dominate over the incoherent addition of propagating waves. This results in the disappearance of propagating waves within the medium, which significantly attenuates energy transmission. In this numerical study performed in the optical regime, we systematically found resonance modes, called eigenchannels, of a 2-D Anderson localized system that allow for the near-perfect energy transmission. We observed that the internal field distribution of these eigenchannels exhibit dense clustering of localized modes. This strongly suggests that the clustered resonance modes facilitate long-range energy flow of local waves. Our study explicitly elucidates the interplay between wave localization and transmission enhancement in the Anderson localization regime.

© 2012 OSA

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References

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  1. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958).
    [CrossRef]
  2. S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett.53(22), 2169–2172 (1984).
    [CrossRef]
  3. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature390(6661), 671–673 (1997).
    [CrossRef]
  4. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
    [CrossRef] [PubMed]
  5. O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun.51(6), 381–384 (1984).
    [CrossRef]
  6. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett.101(12), 120601 (2008).
    [CrossRef] [PubMed]
  7. M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
    [CrossRef] [PubMed]
  8. C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys.69(3), 731–808 (1997).
    [CrossRef]
  9. Z. Shi and A. Z. Genack, “Transmission Eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
    [CrossRef] [PubMed]
  10. W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B83(13), 134207 (2011).
    [CrossRef]
  11. J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett.94(11), 113903 (2005).
    [CrossRef] [PubMed]
  12. J. B. Pendry, “Quasi-extended electron-states in strongly disordered-systems,” J. Phys. C. Solid State20(5), 733–742 (1987).
    [CrossRef]
  13. C. Katherine, “V Phase-Measurement Interferometry Techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), 349–393.
  14. S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson Localization in Open 3D Media,” Phys. Rev. Lett.96(4), 043902 (2006).
    [CrossRef] [PubMed]
  15. M. A. Noginov, Tutorials in Complex Photonic Media (SPIE Press, 2009), 25, 696 p., 696 p. of plates.
  16. S. Zhang and A. Z. Genack, “Statistics of diffusive and localized fields in the vortex core,” Phys. Rev. Lett.99(20), 203901 (2007).
    [CrossRef] [PubMed]
  17. R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett.104(9), 093901 (2010).
    [CrossRef] [PubMed]
  18. R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007).
    [CrossRef]
  19. M. B. Kevin Vynck, F. Riboli, D. S. Wiersma, “Disordered optical modes for photon management.” http://arxiv.org/abs/1202.4601 .
  20. J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. Lond., A Contain. Pap. Math. Phys. Character205(387-401), 237–288 (1906).
    [CrossRef]
  21. N. Cherroret, S. E. Skipetrov, and B. A. van Tiggelen, “Transverse confinement of waves in three-dimensional random media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 056603 (2010).
    [CrossRef] [PubMed]

2012

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
[CrossRef] [PubMed]

Z. Shi and A. Z. Genack, “Transmission Eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
[CrossRef] [PubMed]

2011

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B83(13), 134207 (2011).
[CrossRef]

2010

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett.104(9), 093901 (2010).
[CrossRef] [PubMed]

N. Cherroret, S. E. Skipetrov, and B. A. van Tiggelen, “Transverse confinement of waves in three-dimensional random media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 056603 (2010).
[CrossRef] [PubMed]

2008

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett.101(12), 120601 (2008).
[CrossRef] [PubMed]

2007

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007).
[CrossRef]

S. Zhang and A. Z. Genack, “Statistics of diffusive and localized fields in the vortex core,” Phys. Rev. Lett.99(20), 203901 (2007).
[CrossRef] [PubMed]

2006

S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson Localization in Open 3D Media,” Phys. Rev. Lett.96(4), 043902 (2006).
[CrossRef] [PubMed]

2005

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett.94(11), 113903 (2005).
[CrossRef] [PubMed]

1997

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature390(6661), 671–673 (1997).
[CrossRef]

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys.69(3), 731–808 (1997).
[CrossRef]

1987

J. B. Pendry, “Quasi-extended electron-states in strongly disordered-systems,” J. Phys. C. Solid State20(5), 733–742 (1987).
[CrossRef]

1984

O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun.51(6), 381–384 (1984).
[CrossRef]

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett.53(22), 2169–2172 (1984).
[CrossRef]

1958

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958).
[CrossRef]

1906

J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. Lond., A Contain. Pap. Math. Phys. Character205(387-401), 237–288 (1906).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958).
[CrossRef]

Asakawa, K.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007).
[CrossRef]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature390(6661), 671–673 (1997).
[CrossRef]

Beenakker, C. W. J.

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys.69(3), 731–808 (1997).
[CrossRef]

Bertolotti, J.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett.94(11), 113903 (2005).
[CrossRef] [PubMed]

Cherroret, N.

N. Cherroret, S. E. Skipetrov, and B. A. van Tiggelen, “Transverse confinement of waves in three-dimensional random media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 056603 (2010).
[CrossRef] [PubMed]

Choi, W.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
[CrossRef] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
[CrossRef] [PubMed]

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B83(13), 134207 (2011).
[CrossRef]

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B83(13), 134207 (2011).
[CrossRef]

Choi, Y.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
[CrossRef] [PubMed]

Dorokhov, O. N.

O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun.51(6), 381–384 (1984).
[CrossRef]

Engelen, R. J. P.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Garnett, J. C. M.

J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. Lond., A Contain. Pap. Math. Phys. Character205(387-401), 237–288 (1906).
[CrossRef]

Genack, A. Z.

Z. Shi and A. Z. Genack, “Transmission Eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
[CrossRef] [PubMed]

S. Zhang and A. Z. Genack, “Statistics of diffusive and localized fields in the vortex core,” Phys. Rev. Lett.99(20), 203901 (2007).
[CrossRef] [PubMed]

Gersen, H.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007).
[CrossRef]

Ghulinyan, M.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett.94(11), 113903 (2005).
[CrossRef] [PubMed]

Gottardo, S.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett.94(11), 113903 (2005).
[CrossRef] [PubMed]

Heller, E. J.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett.104(9), 093901 (2010).
[CrossRef] [PubMed]

Höhmann, R.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett.104(9), 093901 (2010).
[CrossRef] [PubMed]

Ikeda, N.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007).
[CrossRef]

John, S.

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett.53(22), 2169–2172 (1984).
[CrossRef]

Kaplan, L.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett.104(9), 093901 (2010).
[CrossRef] [PubMed]

Kim, J.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
[CrossRef] [PubMed]

Kim, M.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
[CrossRef] [PubMed]

Kuhl, U.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett.104(9), 093901 (2010).
[CrossRef] [PubMed]

Kuipers, L.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007).
[CrossRef]

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature390(6661), 671–673 (1997).
[CrossRef]

Mosk, A. P.

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B83(13), 134207 (2011).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett.101(12), 120601 (2008).
[CrossRef] [PubMed]

Park, Q. H.

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B83(13), 134207 (2011).
[CrossRef]

Park, Q.-H.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
[CrossRef] [PubMed]

Pavesi, L.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett.94(11), 113903 (2005).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Quasi-extended electron-states in strongly disordered-systems,” J. Phys. C. Solid State20(5), 733–742 (1987).
[CrossRef]

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature390(6661), 671–673 (1997).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Shi, Z.

Z. Shi and A. Z. Genack, “Transmission Eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
[CrossRef] [PubMed]

Skipetrov, S. E.

N. Cherroret, S. E. Skipetrov, and B. A. van Tiggelen, “Transverse confinement of waves in three-dimensional random media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 056603 (2010).
[CrossRef] [PubMed]

S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson Localization in Open 3D Media,” Phys. Rev. Lett.96(4), 043902 (2006).
[CrossRef] [PubMed]

Stöckmann, H. J.

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett.104(9), 093901 (2010).
[CrossRef] [PubMed]

Sugimoto, Y.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007).
[CrossRef]

van Tiggelen, B. A.

N. Cherroret, S. E. Skipetrov, and B. A. van Tiggelen, “Transverse confinement of waves in three-dimensional random media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 056603 (2010).
[CrossRef] [PubMed]

S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson Localization in Open 3D Media,” Phys. Rev. Lett.96(4), 043902 (2006).
[CrossRef] [PubMed]

Vellekoop, I. M.

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett.101(12), 120601 (2008).
[CrossRef] [PubMed]

Wiersma, D. S.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett.94(11), 113903 (2005).
[CrossRef] [PubMed]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature390(6661), 671–673 (1997).
[CrossRef]

Yoon, C.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
[CrossRef] [PubMed]

Zhang, S.

S. Zhang and A. Z. Genack, “Statistics of diffusive and localized fields in the vortex core,” Phys. Rev. Lett.99(20), 203901 (2007).
[CrossRef] [PubMed]

J. Phys. C. Solid State

J. B. Pendry, “Quasi-extended electron-states in strongly disordered-systems,” J. Phys. C. Solid State20(5), 733–742 (1987).
[CrossRef]

Nat. Photonics,

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics, in press. (2012).
[CrossRef] [PubMed]

Nat. Phys.

R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys.3(6), 401–405 (2007).
[CrossRef]

Nature

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature390(6661), 671–673 (1997).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Philos. Trans. R. Soc. Lond., A Contain. Pap. Math. Phys. Character

J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. Lond., A Contain. Pap. Math. Phys. Character205(387-401), 237–288 (1906).
[CrossRef]

Phys. Rev.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958).
[CrossRef]

Phys. Rev. B

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B83(13), 134207 (2011).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

N. Cherroret, S. E. Skipetrov, and B. A. van Tiggelen, “Transverse confinement of waves in three-dimensional random media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(5), 056603 (2010).
[CrossRef] [PubMed]

Phys. Rev. Lett.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson Localized 1D systems,” Phys. Rev. Lett.94(11), 113903 (2005).
[CrossRef] [PubMed]

S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson Localization in Open 3D Media,” Phys. Rev. Lett.96(4), 043902 (2006).
[CrossRef] [PubMed]

S. Zhang and A. Z. Genack, “Statistics of diffusive and localized fields in the vortex core,” Phys. Rev. Lett.99(20), 203901 (2007).
[CrossRef] [PubMed]

R. Höhmann, U. Kuhl, H. J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: A microwave study,” Phys. Rev. Lett.104(9), 093901 (2010).
[CrossRef] [PubMed]

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett.53(22), 2169–2172 (1984).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett.101(12), 120601 (2008).
[CrossRef] [PubMed]

Z. Shi and A. Z. Genack, “Transmission Eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett.108(4), 043901 (2012).
[CrossRef] [PubMed]

Rev. Mod. Phys.

C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys.69(3), 731–808 (1997).
[CrossRef]

Solid State Commun.

O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun.51(6), 381–384 (1984).
[CrossRef]

Other

M. A. Noginov, Tutorials in Complex Photonic Media (SPIE Press, 2009), 25, 696 p., 696 p. of plates.

C. Katherine, “V Phase-Measurement Interferometry Techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1988), 349–393.

M. B. Kevin Vynck, F. Riboli, D. S. Wiersma, “Disordered optical modes for photon management.” http://arxiv.org/abs/1202.4601 .

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Figures (5)

Fig. 1
Fig. 1

(a) Geometry of a disordered medium and (b) the determination of localization length. (a) A medium composed of randomly distributed square particles is prepared in the x-z plane. The width of the medium in the x-direction is 130 μm and the thickness in the z-direction is varied. The incident wave covers the entire width while the output is sampled along the black dashed line whose width is W = 90 μm. (b) Transmittance values of random media as a function of thickness. Data represented by squares are acquired for np = 1.6 media and circles for np = 2.5 media. The error bar is obtained from the statistical distribution of the transmittances taken at various incident angles. Dashed and dotted lines are linear fits to the squares and circles, respectively.

Fig. 2
Fig. 2

Field distribution within the disordered media. (a), (b) Intensity maps inside random media with np = 1.6 and 2.5, respectively. The incident wave is a plane wave propagating along the z-direction. The color bar shows the amplitude in arbitrary units. Scale bar, 10 μm. (c), (d) Angular spectrum maps for (a) and (b), respectively. Horizontal and vertical axes are the wave numbers along the x and z directions, respectively, in units of the free space wave number, k0. The color bar indicates the amplitude in arbitrary units.

Fig. 3
Fig. 3

(a) Transmission matrix and (b) its singular value distribution. The kxi and kxo are the wave numbers along the x-direction for the input and output channels, respectively. The color bar indicates the amplitude in arbitrary units. Singular values are sorted in descending order for the eigenchannel index. Singular value distribution for np = 1.6 medium is shown in blue for comparison.

Fig. 4
Fig. 4

Internal field distribution of eigenchannels for the disordered medium of np = 2.5. For (a)-(e), eigenchannel indices are 1, 6, 11, 31, and 291, respectively, and their transmittance values are 80, 37, 12, 0,39, and 2.0 × 10-7%, respectively. The color bar indicates the base-10 logarithm of the amplitude. The aspect ratio is set to have different scales for the horizontal and vertical axes for better visibility. Both scale bars indicates 5 μm.

Fig. 5
Fig. 5

Effective width, Weff , of the internal field. Effective width, which is as a function of eigenchannel index, is defined through Eq. (2).

Equations (2)

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H=UτV,
Δx(z)= x 2 I(x,z)dx ( xI(x,z)dx ) 2 ( I(x,z)dx ) 2

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