Abstract

Optical absorption is omnipresent and often distributed non-uniformly in space. We present a numerical study on the effects of inhomogeneous absorption on transmission eigenchannels of light in highly scattering media. In the weak absorption regime, the spatial profile of a transmission channel remains similar to that without absorption, and the effect of inhomogeneous absorption can be stronger or weaker than homogeneous absorption depending on the spatial overlap of the localized absorbing region with the field intensity maximum of the channel. In the strong absorption regime, the high transmission channels redirect the energy flows to circumvent the absorbing regions to minimize loss. The attenuation of high transmission channels by inhomogeneous absorption is lower than that by homogeneous absorption, regardless of the location of the absorbing region. The statistical distribution of transmission eigenvalues in the former becomes broader than that in the latter, due to a longer tail at high transmission. The maximum enhancement factor of total transmission increases with absorption, eventually exceeds that without absorption.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Statistics and control of waves in disordered media

Zhou Shi, Matthieu Davy, and Azriel Z. Genack
Opt. Express 23(9) 12293-12320 (2015)

Time-resolved optoacoustic measurement of absorption of light by inhomogeneous media

A. A. Karabutov, N. B. Podymova, and V. S. Letokhov
Appl. Opt. 34(9) 1484-1487 (1995)

Frequency width of open channels in multiple scattering media

Jeroen Bosch, Sebastianus A. Goorden, and Allard P. Mosk
Opt. Express 24(23) 26472-26478 (2016)

References

  • View by:
  • |
  • |
  • |

  1. I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photon. 4, 320–322 (2010).
    [Crossref]
  2. S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
    [Crossref]
  3. W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white light-emitting diode lighting,” Appl. Opt. 52, 2602–2609 (2013).
    [Crossref] [PubMed]
  4. M. Grätzel, “Dye-sensitized solar cells,” J. Photochem. Photobiol. C. 4, 145–153 (2003).
    [Crossref]
  5. A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110, 6595–6663 (2010).
    [Crossref] [PubMed]
  6. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
    [Crossref] [PubMed]
  7. J. B. Pendry, “Light finds a way through the maze,” Physics 1, 20 (2008).
    [Crossref]
  8. Z. Shi and A. Z. Genack, “Transmission eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett. 108, 043901 (2012).
    [Crossref] [PubMed]
  9. M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).
  10. M. Davy, Z. Shi, J. Wang, and A. Z. Genack, “Transmission statistics and focusing in single disordered samples,” Opt. Express 21, 10367–10375 (2013).
    [Crossref] [PubMed]
  11. M. Kim, W. Choi, C. Yoon, G. H. Kim, and W. Choi, “Relation between transmission eigenchannels and single-channel optimizing modes in a disordered medium,” Opt. Lett. 38, 2994–2996 (2013).
    [Crossref] [PubMed]
  12. W. Choi, M. Kim, D. Kim, C. Yoon, C. Fang-Yen, Q. Park, and W. Choi, “Enrichment of deeply penetrating waves in disordered media,” arXiv:1308.6558 (2013).
  13. S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
    [Crossref] [PubMed]
  14. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
    [Crossref] [PubMed]
  15. Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
    [Crossref]
  16. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
    [Crossref] [PubMed]
  17. O. N. Dorokhov, “Localization and transmission coefficient for two coupled metal chains with disorder,” Solid State Commun. 44, 915–919 (1982).
    [Crossref]
  18. O. N. Dorokhov, “On the coexistence of localized and extended electronic states in the metallic phase,” Solid State Commun. 51, 381–384 (1984).
    [Crossref]
  19. P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Annals of Physics 181, 290–317 (1988).
    [Crossref]
  20. P. A. Mello and J.-L. Pichard, “Maximum-entropy approaches to quantum electronic transport,” Phys. Rev. B 40, 5276–5278 (1989)
    [Crossref]
  21. Y. V. Nazarov, “Limits of universality in disordered conductors,” Phys. Rev. Lett. 73, 134–137 (1994).
    [Crossref] [PubMed]
  22. A. Lamacraft, B. D. Simons, and M. R. Zirnbauer, “Localization from σ-model geodesics,” Phys. Rev. B 70, 075412 (2004).
    [Crossref]
  23. A. Altland, A. Kamenev, and C. Tian, “Anderson localization from the replica formalism,” Phys. Rev. Lett. 95, 206601 (2005).
    [Crossref] [PubMed]
  24. P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526 (1998).
    [Crossref]
  25. A. G. Yamilov and B. Payne, “Interplay between localization and absorption in disordered waveguides,” Opt. Express 21, 11688–11697 (2013).
    [Crossref] [PubMed]
  26. S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89, 224202 (2014).
    [Crossref]
  27. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
    [Crossref] [PubMed]
  28. W. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
    [Crossref] [PubMed]
  29. Y. D. Chong and A. D. Stone, “Hidden black: coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett. 107, 163901 (2011).
    [Crossref] [PubMed]
  30. H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
    [Crossref] [PubMed]
  31. H. Noh, S. M. Popoff, and H. Cao, “Broadband subwavelength focusing of light using a passive sink,” Optics Express 21, 17435–17446 (2013).
    [Crossref] [PubMed]
  32. I. M. Vellekoop, E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Demixing light paths inside disordered metamaterials,” Opt. Express 16, 67–80 (2008).
    [Crossref] [PubMed]
  33. W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
    [Crossref]
  34. P. A. Lee and D. S. Fisher, “Anderson localization in two dimensions,” Phys. Rev. Lett. 47, 882–885 (1981).
    [Crossref]
  35. C. W. J. Beenakker, J. C. J. Paasschens, and P. W. Brouwer, “Probability of reflection by a random laser,” Phys. Rev. Lett. 76, 1368–1371 (1996).
    [Crossref] [PubMed]
  36. M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
    [Crossref]
  37. M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Getting beneath the surface of opaque media: universal structure of transmission eigenchannel,” arXiv:1502.03642.
  38. M. Fink, W. A. Kuperman, J.-P Montagner, and A. Tourin, Imaging of Complex Media with Acoustic and Seismic Waves (Springer, 2002).
    [Crossref]
  39. L. V. Wang, Photoacoustic Imaging and Spectroscopy (CRC Press, 2009).
    [Crossref]

2014 (2)

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
[Crossref] [PubMed]

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89, 224202 (2014).
[Crossref]

2013 (5)

2012 (3)

H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (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, 043901 (2012).
[Crossref] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).

2011 (3)

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[Crossref]

W. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Y. D. Chong and A. D. Stone, “Hidden black: coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett. 107, 163901 (2011).
[Crossref] [PubMed]

2010 (5)

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110, 6595–6663 (2010).
[Crossref] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photon. 4, 320–322 (2010).
[Crossref]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
[Crossref]

2008 (4)

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

J. B. Pendry, “Light finds a way through the maze,” Physics 1, 20 (2008).
[Crossref]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[Crossref]

I. M. Vellekoop, E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Demixing light paths inside disordered metamaterials,” Opt. Express 16, 67–80 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (1)

A. Altland, A. Kamenev, and C. Tian, “Anderson localization from the replica formalism,” Phys. Rev. Lett. 95, 206601 (2005).
[Crossref] [PubMed]

2004 (1)

A. Lamacraft, B. D. Simons, and M. R. Zirnbauer, “Localization from σ-model geodesics,” Phys. Rev. B 70, 075412 (2004).
[Crossref]

2003 (1)

M. Grätzel, “Dye-sensitized solar cells,” J. Photochem. Photobiol. C. 4, 145–153 (2003).
[Crossref]

1999 (1)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

1998 (1)

P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526 (1998).
[Crossref]

1996 (1)

C. W. J. Beenakker, J. C. J. Paasschens, and P. W. Brouwer, “Probability of reflection by a random laser,” Phys. Rev. Lett. 76, 1368–1371 (1996).
[Crossref] [PubMed]

1994 (1)

Y. V. Nazarov, “Limits of universality in disordered conductors,” Phys. Rev. Lett. 73, 134–137 (1994).
[Crossref] [PubMed]

1989 (1)

P. A. Mello and J.-L. Pichard, “Maximum-entropy approaches to quantum electronic transport,” Phys. Rev. B 40, 5276–5278 (1989)
[Crossref]

1988 (1)

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Annals of Physics 181, 290–317 (1988).
[Crossref]

1984 (1)

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

1982 (1)

O. N. Dorokhov, “Localization and transmission coefficient for two coupled metal chains with disorder,” Solid State Commun. 44, 915–919 (1982).
[Crossref]

1981 (1)

P. A. Lee and D. S. Fisher, “Anderson localization in two dimensions,” Phys. Rev. Lett. 47, 882–885 (1981).
[Crossref]

Altland, A.

A. Altland, A. Kamenev, and C. Tian, “Anderson localization from the replica formalism,” Phys. Rev. Lett. 95, 206601 (2005).
[Crossref] [PubMed]

Beenakker, C. W. J.

C. W. J. Beenakker, J. C. J. Paasschens, and P. W. Brouwer, “Probability of reflection by a random laser,” Phys. Rev. Lett. 76, 1368–1371 (1996).
[Crossref] [PubMed]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
[Crossref]

Boschloo, G.

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110, 6595–6663 (2010).
[Crossref] [PubMed]

Brouwer, P. W.

P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526 (1998).
[Crossref]

C. W. J. Beenakker, J. C. J. Paasschens, and P. W. Brouwer, “Probability of reflection by a random laser,” Phys. Rev. Lett. 76, 1368–1371 (1996).
[Crossref] [PubMed]

Cao, H.

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89, 224202 (2014).
[Crossref]

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
[Crossref] [PubMed]

H. Noh, S. M. Popoff, and H. Cao, “Broadband subwavelength focusing of light using a passive sink,” Optics Express 21, 17435–17446 (2013).
[Crossref] [PubMed]

H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

W. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

Choi, W.

M. Kim, W. Choi, C. Yoon, G. H. Kim, and W. Choi, “Relation between transmission eigenchannels and single-channel optimizing modes in a disordered medium,” Opt. Lett. 38, 2994–2996 (2013).
[Crossref] [PubMed]

M. Kim, W. Choi, C. Yoon, G. H. Kim, and W. Choi, “Relation between transmission eigenchannels and single-channel optimizing modes in a disordered medium,” Opt. Lett. 38, 2994–2996 (2013).
[Crossref] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[Crossref]

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[Crossref]

W. Choi, M. Kim, D. Kim, C. Yoon, C. Fang-Yen, Q. Park, and W. Choi, “Enrichment of deeply penetrating waves in disordered media,” arXiv:1308.6558 (2013).

W. Choi, M. Kim, D. Kim, C. Yoon, C. Fang-Yen, Q. Park, and W. Choi, “Enrichment of deeply penetrating waves in disordered media,” arXiv:1308.6558 (2013).

Choi, Y.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).

Chong, Y.

H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

Chong, Y. D.

Y. D. Chong and A. D. Stone, “Hidden black: coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett. 107, 163901 (2011).
[Crossref] [PubMed]

W. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Davy, M.

M. Davy, Z. Shi, J. Wang, and A. Z. Genack, “Transmission statistics and focusing in single disordered samples,” Opt. Express 21, 10367–10375 (2013).
[Crossref] [PubMed]

M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Getting beneath the surface of opaque media: universal structure of transmission eigenchannel,” arXiv:1502.03642.

Dorokhov, O. N.

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

O. N. Dorokhov, “Localization and transmission coefficient for two coupled metal chains with disorder,” Solid State Commun. 44, 915–919 (1982).
[Crossref]

Fang-Yen, C.

W. Choi, M. Kim, D. Kim, C. Yoon, C. Fang-Yen, Q. Park, and W. Choi, “Enrichment of deeply penetrating waves in disordered media,” arXiv:1308.6558 (2013).

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[Crossref]

Fink, M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
[Crossref]

M. Fink, W. A. Kuperman, J.-P Montagner, and A. Tourin, Imaging of Complex Media with Acoustic and Seismic Waves (Springer, 2002).
[Crossref]

Fisher, D. S.

P. A. Lee and D. S. Fisher, “Anderson localization in two dimensions,” Phys. Rev. Lett. 47, 882–885 (1981).
[Crossref]

Ge, L.

W. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Genack, A. Z.

M. Davy, Z. Shi, J. Wang, and A. Z. Genack, “Transmission statistics and focusing in single disordered samples,” Opt. Express 21, 10367–10375 (2013).
[Crossref] [PubMed]

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

M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Getting beneath the surface of opaque media: universal structure of transmission eigenchannel,” arXiv:1502.03642.

Gigan, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

Goetschy, A.

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
[Crossref] [PubMed]

Grätzel, M.

M. Grätzel, “Dye-sensitized solar cells,” J. Photochem. Photobiol. C. 4, 145–153 (2003).
[Crossref]

Hagfeldt, A.

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110, 6595–6663 (2010).
[Crossref] [PubMed]

IJzerman, W. L.

Kamenev, A.

A. Altland, A. Kamenev, and C. Tian, “Anderson localization from the replica formalism,” Phys. Rev. Lett. 95, 206601 (2005).
[Crossref] [PubMed]

Kim, D.

W. Choi, M. Kim, D. Kim, C. Yoon, C. Fang-Yen, Q. Park, and W. Choi, “Enrichment of deeply penetrating waves in disordered media,” arXiv:1308.6558 (2013).

Kim, G. H.

Kim, J.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).

Kim, M.

M. Kim, W. Choi, C. Yoon, G. H. Kim, and W. Choi, “Relation between transmission eigenchannels and single-channel optimizing modes in a disordered medium,” Opt. Lett. 38, 2994–2996 (2013).
[Crossref] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).

W. Choi, M. Kim, D. Kim, C. Yoon, C. Fang-Yen, Q. Park, and W. Choi, “Enrichment of deeply penetrating waves in disordered media,” arXiv:1308.6558 (2013).

Kloo, L.

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110, 6595–6663 (2010).
[Crossref] [PubMed]

Kumar, N.

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Annals of Physics 181, 290–317 (1988).
[Crossref]

Kuperman, W. A.

M. Fink, W. A. Kuperman, J.-P Montagner, and A. Tourin, Imaging of Complex Media with Acoustic and Seismic Waves (Springer, 2002).
[Crossref]

Lagendijk, A.

Lamacraft, A.

A. Lamacraft, B. D. Simons, and M. R. Zirnbauer, “Localization from σ-model geodesics,” Phys. Rev. B 70, 075412 (2004).
[Crossref]

Lee, P. A.

P. A. Lee and D. S. Fisher, “Anderson localization in two dimensions,” Phys. Rev. Lett. 47, 882–885 (1981).
[Crossref]

Lerosey, G.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
[Crossref]

Liew, S. F.

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
[Crossref] [PubMed]

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89, 224202 (2014).
[Crossref]

Mello, P. A.

P. A. Mello and J.-L. Pichard, “Maximum-entropy approaches to quantum electronic transport,” Phys. Rev. B 40, 5276–5278 (1989)
[Crossref]

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Annals of Physics 181, 290–317 (1988).
[Crossref]

Montagner, J.-P

M. Fink, W. A. Kuperman, J.-P Montagner, and A. Tourin, Imaging of Complex Media with Acoustic and Seismic Waves (Springer, 2002).
[Crossref]

Mosk, A. P.

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89, 224202 (2014).
[Crossref]

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white light-emitting diode lighting,” Appl. Opt. 52, 2602–2609 (2013).
[Crossref] [PubMed]

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[Crossref]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photon. 4, 320–322 (2010).
[Crossref]

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

I. M. Vellekoop, E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Demixing light paths inside disordered metamaterials,” Opt. Express 16, 67–80 (2008).
[Crossref] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[Crossref] [PubMed]

Nazarov, Y. V.

Y. V. Nazarov, “Limits of universality in disordered conductors,” Phys. Rev. Lett. 73, 134–137 (1994).
[Crossref] [PubMed]

Nieuwenhuizen, T. M.

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

Noh, H.

H. Noh, S. M. Popoff, and H. Cao, “Broadband subwavelength focusing of light using a passive sink,” Optics Express 21, 17435–17446 (2013).
[Crossref] [PubMed]

H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

W. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Paasschens, J. C. J.

C. W. J. Beenakker, J. C. J. Paasschens, and P. W. Brouwer, “Probability of reflection by a random laser,” Phys. Rev. Lett. 76, 1368–1371 (1996).
[Crossref] [PubMed]

Park, J.

M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Getting beneath the surface of opaque media: universal structure of transmission eigenchannel,” arXiv:1502.03642.

Park, Q.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).

W. Choi, M. Kim, D. Kim, C. Yoon, C. Fang-Yen, Q. Park, and W. Choi, “Enrichment of deeply penetrating waves in disordered media,” arXiv:1308.6558 (2013).

Park, Q. H.

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[Crossref]

Payne, B.

Pendry, J. B.

J. B. Pendry, “Light finds a way through the maze,” Physics 1, 20 (2008).
[Crossref]

Pereyra, P.

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Annals of Physics 181, 290–317 (1988).
[Crossref]

Pettersson, H.

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110, 6595–6663 (2010).
[Crossref] [PubMed]

Pichard, J.-L.

P. A. Mello and J.-L. Pichard, “Maximum-entropy approaches to quantum electronic transport,” Phys. Rev. B 40, 5276–5278 (1989)
[Crossref]

Popoff, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
[Crossref]

Popoff, S. M.

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
[Crossref] [PubMed]

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89, 224202 (2014).
[Crossref]

H. Noh, S. M. Popoff, and H. Cao, “Broadband subwavelength focusing of light using a passive sink,” Optics Express 21, 17435–17446 (2013).
[Crossref] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

Psaltis, D.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[Crossref]

Shi, Z.

M. Davy, Z. Shi, J. Wang, and A. Z. Genack, “Transmission statistics and focusing in single disordered samples,” Opt. Express 21, 10367–10375 (2013).
[Crossref] [PubMed]

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

M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Getting beneath the surface of opaque media: universal structure of transmission eigenchannel,” arXiv:1502.03642.

Simons, B. D.

A. Lamacraft, B. D. Simons, and M. R. Zirnbauer, “Localization from σ-model geodesics,” Phys. Rev. B 70, 075412 (2004).
[Crossref]

Stone, A. D.

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
[Crossref] [PubMed]

H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

Y. D. Chong and A. D. Stone, “Hidden black: coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett. 107, 163901 (2011).
[Crossref] [PubMed]

W. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

Sun, L.

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110, 6595–6663 (2010).
[Crossref] [PubMed]

Tian, C.

A. Altland, A. Kamenev, and C. Tian, “Anderson localization from the replica formalism,” Phys. Rev. Lett. 95, 206601 (2005).
[Crossref] [PubMed]

M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Getting beneath the surface of opaque media: universal structure of transmission eigenchannel,” arXiv:1502.03642.

Tourin, A.

M. Fink, W. A. Kuperman, J.-P Montagner, and A. Tourin, Imaging of Complex Media with Acoustic and Seismic Waves (Springer, 2002).
[Crossref]

Tukker, T. W.

van Putten, E. G.

van Rossum, M. C. W.

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photon. 4, 320–322 (2010).
[Crossref]

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

I. M. Vellekoop, E. G. van Putten, A. Lagendijk, and A. P. Mosk, “Demixing light paths inside disordered metamaterials,” Opt. Express 16, 67–80 (2008).
[Crossref] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[Crossref] [PubMed]

Vos, W. L.

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89, 224202 (2014).
[Crossref]

W. L. Vos, T. W. Tukker, A. P. Mosk, A. Lagendijk, and W. L. IJzerman, “Broadband mean free path of diffuse light in polydisperse ensembles of scatterers for white light-emitting diode lighting,” Appl. Opt. 52, 2602–2609 (2013).
[Crossref] [PubMed]

Wan, W.

W. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Wang, J.

Wang, L. V.

L. V. Wang, Photoacoustic Imaging and Spectroscopy (CRC Press, 2009).
[Crossref]

Yamilov, A. G.

Yang, C.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[Crossref]

Yaqoob, Z.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[Crossref]

Yoon, C.

M. Kim, W. Choi, C. Yoon, G. H. Kim, and W. Choi, “Relation between transmission eigenchannels and single-channel optimizing modes in a disordered medium,” Opt. Lett. 38, 2994–2996 (2013).
[Crossref] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).

W. Choi, M. Kim, D. Kim, C. Yoon, C. Fang-Yen, Q. Park, and W. Choi, “Enrichment of deeply penetrating waves in disordered media,” arXiv:1308.6558 (2013).

Zirnbauer, M. R.

A. Lamacraft, B. D. Simons, and M. R. Zirnbauer, “Localization from σ-model geodesics,” Phys. Rev. B 70, 075412 (2004).
[Crossref]

Annals of Physics (1)

P. A. Mello, P. Pereyra, and N. Kumar, “Macroscopic approach to multichannel disordered conductors,” Annals of Physics 181, 290–317 (1988).
[Crossref]

Appl. Opt. (1)

Chem. Rev. (1)

A. Hagfeldt, G. Boschloo, L. Sun, L. Kloo, and H. Pettersson, “Dye-sensitized solar cells,” Chem. Rev. 110, 6595–6663 (2010).
[Crossref] [PubMed]

J. Photochem. Photobiol. C. (1)

M. Grätzel, “Dye-sensitized solar cells,” J. Photochem. Photobiol. C. 4, 145–153 (2003).
[Crossref]

Nat. Commun. (1)

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
[Crossref]

Nat. Photon. (3)

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photon. 4, 320–322 (2010).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photon. 6, 581–585 (2012).

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photon. 2, 110–115 (2008).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Optics Express (1)

H. Noh, S. M. Popoff, and H. Cao, “Broadband subwavelength focusing of light using a passive sink,” Optics Express 21, 17435–17446 (2013).
[Crossref] [PubMed]

Phys. Rev. B (5)

P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B 57, 10526 (1998).
[Crossref]

W. Choi, A. P. Mosk, Q. H. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
[Crossref]

S. F. Liew, S. M. Popoff, A. P. Mosk, W. L. Vos, and H. Cao, “Transmission channels for light in absorbing random media: from diffusive to ballistic-like transport,” Phys. Rev. B 89, 224202 (2014).
[Crossref]

P. A. Mello and J.-L. Pichard, “Maximum-entropy approaches to quantum electronic transport,” Phys. Rev. B 40, 5276–5278 (1989)
[Crossref]

A. Lamacraft, B. D. Simons, and M. R. Zirnbauer, “Localization from σ-model geodesics,” Phys. Rev. B 70, 075412 (2004).
[Crossref]

Phys. Rev. Lett. (11)

A. Altland, A. Kamenev, and C. Tian, “Anderson localization from the replica formalism,” Phys. Rev. Lett. 95, 206601 (2005).
[Crossref] [PubMed]

Y. V. Nazarov, “Limits of universality in disordered conductors,” Phys. Rev. Lett. 73, 134–137 (1994).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

P. A. Lee and D. S. Fisher, “Anderson localization in two dimensions,” Phys. Rev. Lett. 47, 882–885 (1981).
[Crossref]

C. W. J. Beenakker, J. C. J. Paasschens, and P. W. Brouwer, “Probability of reflection by a random laser,” Phys. Rev. Lett. 76, 1368–1371 (1996).
[Crossref] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 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, 043901 (2012).
[Crossref] [PubMed]

S. M. Popoff, A. Goetschy, S. F. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
[Crossref] [PubMed]

Y. D. Chong and A. D. Stone, “Hidden black: coherent enhancement of absorption in strongly scattering media,” Phys. Rev. Lett. 107, 163901 (2011).
[Crossref] [PubMed]

H. Noh, Y. Chong, A. D. Stone, and H. Cao, “Perfect coupling of light to surface plasmons by coherent absorption,” Phys. Rev. Lett. 108, 186805 (2012).
[Crossref] [PubMed]

Physics (1)

J. B. Pendry, “Light finds a way through the maze,” Physics 1, 20 (2008).
[Crossref]

Rev. Mod. Phys. (1)

M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
[Crossref]

Science (1)

W. Wan, Y. D. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889–892 (2011).
[Crossref] [PubMed]

Solid State Commun. (2)

O. N. Dorokhov, “Localization and transmission coefficient for two coupled metal chains with disorder,” Solid State Commun. 44, 915–919 (1982).
[Crossref]

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

Other (4)

W. Choi, M. Kim, D. Kim, C. Yoon, C. Fang-Yen, Q. Park, and W. Choi, “Enrichment of deeply penetrating waves in disordered media,” arXiv:1308.6558 (2013).

M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Getting beneath the surface of opaque media: universal structure of transmission eigenchannel,” arXiv:1502.03642.

M. Fink, W. A. Kuperman, J.-P Montagner, and A. Tourin, Imaging of Complex Media with Acoustic and Seismic Waves (Springer, 2002).
[Crossref]

L. V. Wang, Photoacoustic Imaging and Spectroscopy (CRC Press, 2009).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1 Schematic of the 2D disordered waveguide used in our numerical simulation. Dielectric cylinders are placed randomly in a waveguide with perfect-reflecting sidewalls. (a) Qu: homogeneous distribution of absorbers across the entire random arrays of cylinders. (b) Q3: absorbers are confined to the three circles with diameter Da. (c) Q1: absorbers are concentrated in a single circular region of diameter Da in the middle of the disordered waveguide.
Fig. 2
Fig. 2 Statistical distribution of transmission eigenvalues P(τ) for disordered waveguide with homogeneous and inhomogeneous absorption. (a) Without absorption, P(τ) (filled circles) exhibits the bimodal distribution given by Eq. (2) (dashed line). (b) Spatial distribution of the electric field amplitude |Ez(x, y)| of the highest transmission eigenchannel in one random realization of disordered waveguide. (c) Cross-section-averaged intensity along the x-direction, having the maximum at the center of the waveguide, which coincides with the absorbing region in Q1 (marked by the vertical dashed lines). (d) P(τ) for weak absorption αL/ξa = 0.4 [α = 1 for Qu, α = 0.46 for Q3 and α = 0.27 for Q1]. The peak near τ = 1 is diminished and shifted towards smaller τ. The shift in the case of inhomogeneous absorption is slightly larger than that of homogeneous absorption, as the open channels experience more attenuation due to better spatial overlap with the localized absorbing regions. (e) P(τ) for strong absorption αL/ξa = 1.8. Q1 has the highest transmission among the three cases. (f) The normalized width Wτ of P(τ) as a function of the absorption strength αL/ξa. (g, h, i) Spatial map of the normalized Poynting vector S⃗′(x, y) for the maximum transmission channel in the disordered waveguide with homogeneous absorption Qu (g), three absorbing regions Q3 (h) and one absorbing region Q1 (i) at αL/ξa = 1.8. Dashed circles in (h, i) mark the boundary of the absorbing regions. When absorption is strong and inhomogeneous, main energy flows bypass the absorbing regions.
Fig. 3
Fig. 3 Comparison of the maximum transmission eigenvalue τ1 and the average of all transmission eigenvalues τ̄ in the presence of inhomogeneous absorption to that of homogeneous absorption. (a) τ1 decreases with absorption in all three cases. The drop is the fastest for Q1 when absorption is weak [αL/ξa < 1] but switches to the slowest when absorption is strong [αL/ξa > 1]. (b) Similar trends are observed for the change of τ̄ with absorption. (c) The ensemble-averaged ratio 〈τ1/τ̄〉 shows the fastest reduction for Q1 compared to Qu and Q3 in the weak absorption regime. (d) 〈τ1/τ̄〉 starts to increase in the strong absorption regime, and eventually exceeds the ratio without absorption in Q1 and Q3.
Fig. 4
Fig. 4 Modification of transmission eigenchannels in disordered waveguide with a single absorbing region in the middle Q1. (a) Ensemble-averaged ratio τ n / τ n 0 for individual eigenchannels. The reduction of τ1 is the largest when absorption is weak D a / ξ a ( 1 ) = 0.32 but becomes less than other eigenchannels such as τ3 when absorption is strong D a / ξ a ( 1 ) = 1.45. (b) The absorption of individual eigenchannels An. The first eigenchannel is the one that experiences the most absorption at weak absorption but it is replaced by the third eigenchannel at higher absorption level. (c, d) Ensemble average of cross-section-averaged electric field intensity 〈I(x)〉 for the first and third eigenchannels. Both are normalized to their maximal values when there is no absorption. With weak absorption D a / ξ a ( 1 ) = 0.32 (brown dotted lines), the field intensity of the first eigenchannel decreases more at the waveguide center than the third eigenchannel. The vertical dashed lines mark the boundary of the absorbing region. When absorption is strong D a / ξ a ( 1 ) = 1.45, the reduction of field intensity behind the absorbing region becomes less for the first eigenchannel than the third eigenchannel as the former is modified to circumvent the absorbing region.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

t = U Σ V ,
P ( τ ) = τ ¯ 2 1 τ 1 τ ,

Metrics