Abstract

This work presents results of ab-initio simulations of continuous wave transport in disordered absorbing waveguides. Wave interference effects cause deviations from diffusive picture of wave transport and make the diffusion coefficient position- and absorption-dependent. As a consequence, the true limit of a zero diffusion coefficient is never reached in an absorbing random medium of infinite size, instead, the diffusion coefficient saturates at some finite constant value. Transition to this absorption-limited diffusion exhibits a universality which can be captured within the framework of the self-consistent theory (SCT) of localization. The results of this work (i) justify use of SCT in analyses of experiments in localized regime, provided that absorption is not weak; (ii) open the possibility of diffusive description of wave transport in the saturation regime even when localization effects are strong.

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    [CrossRef]
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2013

C. Tian, “Hydrodynamic and field-theoretic approaches to light localization in open media,” Physica E49, 124–153 (2013).
[CrossRef]

T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Phot.7, 48–52 (2013).
[CrossRef]

B. Payne, T. Mahler, and A. Yamilov, “Effect of evanescent channels on position-dependent diffusion in disordered waveguides,” Waves in Random and Complex Media23, 43–55 (2013).
[CrossRef]

A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

2011

J. Wang and A. Z. Genack, “Transport through modes in random media,” Nature471, 345–348 (2011).
[CrossRef] [PubMed]

2010

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science327, 1352–1355 (2010).
[CrossRef] [PubMed]

B. Payne, J. Andreasen, H. Cao, and A. Yamilov, “Relation between transmission and energy stored in random media with gain,” Phys. Rev. B82, 104204 (2010).
[CrossRef]

A. Yamilov and B. Payne, “Classification of regimes of wave transport in quasi-one-dimensional nonconservative random media,” J. Mod. Opt.57, 1916–1921 (2010).
[CrossRef]

C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett.105, 263905 (2010).
[CrossRef]

B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B82, 024205 (2010).
[CrossRef]

2009

Z. Q. Zhang, A. A. Chabanov, S. K. Cheung, C. H. Wong, and A. Z. Genack, “Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media,” Phys. Rev. B79, 144203 (2009).
[CrossRef]

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of anderson localization,” Phys. Today62, 24–29 (2009).
[CrossRef]

2008

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4, 945–948 (2008).
[CrossRef]

N. Cherroret and S. E. Skipetrov, “Microscopic derivation of self-consistent equations of anderson localization in a disordered medium of finite size,” Phys. Rev. E77, 046608 (2008).
[CrossRef]

C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B77, 064205 (2008).
[CrossRef]

2006

M. Störzer, P. Gross, C. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett.96, 063904 (2006).
[CrossRef] [PubMed]

2001

L. I. Deych, A. Yamilov, and A. A. Lisyansky, “Scaling in one-dimensional localized absorbing systems,” Phys. Rev. B64, 024201 (2001).
[CrossRef]

2000

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

B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett.84, 4333–4336 (2000).
[CrossRef] [PubMed]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404, 850–853 (2000).
[CrossRef] [PubMed]

1999

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

1998

P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B57, 10526–10536 (1998).
[CrossRef]

1997

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

1994

V. D. 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]

J. B. Pendry, “Symmetry and transport of waves in one-dimensional disordered systems,” Adv. Phys.43, 461–542 (1994).
[CrossRef]

1993

J. Kroha, C. M. Soukoulis, and P. Wölfle, “Localization of classical waves in a random medium: A self-consistent theory,” Phys. Rev. B47, 11093–11096 (1993).
[CrossRef]

1987

P. A. Lee, D. A. Stone, and H. Fukuyamak, “Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field,” Phys. Rev. B35, 1039–1070 (1987).
[CrossRef]

1984

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

1980

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22, 3519–3526 (1980).
[CrossRef]

D. Vollhardt and P. Wölfle, “Diagrammatic, self-consistent treatment of the anderson localization problem in d≤ 2 dimensions,” Phys. Rev. B22, 4666–4679 (1980).
[CrossRef]

1979

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions,” Phys. Rev. Lett.42, 673–676 (1979).
[CrossRef]

1958

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

Abrahams, E.

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22, 3519–3526 (1980).
[CrossRef]

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions,” Phys. Rev. Lett.42, 673–676 (1979).
[CrossRef]

Aegerter, C.

M. Störzer, P. Gross, C. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett.96, 063904 (2006).
[CrossRef] [PubMed]

Aegerter, C. M.

T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Phot.7, 48–52 (2013).
[CrossRef]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, Cambridge, UK, 2007).
[CrossRef]

Anderson, P. W.

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22, 3519–3526 (1980).
[CrossRef]

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions,” Phys. Rev. Lett.42, 673–676 (1979).
[CrossRef]

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

Andreasen, J.

B. Payne, J. Andreasen, H. Cao, and A. Yamilov, “Relation between transmission and energy stored in random media with gain,” Phys. Rev. B82, 104204 (2010).
[CrossRef]

Beenakker, C. W.

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

Brouwer, P. W.

P. W. Brouwer, “Transmission through a many-channel random waveguide with absorption,” Phys. Rev. B57, 10526–10536 (1998).
[CrossRef]

Bührer, W.

T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Phot.7, 48–52 (2013).
[CrossRef]

Cao, H.

A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

B. Payne, J. Andreasen, H. Cao, and A. Yamilov, “Relation between transmission and energy stored in random media with gain,” Phys. Rev. B82, 104204 (2010).
[CrossRef]

Chabanov, A. A.

Z. Q. Zhang, A. A. Chabanov, S. K. Cheung, C. H. Wong, and A. Z. Genack, “Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media,” Phys. Rev. B79, 144203 (2009).
[CrossRef]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404, 850–853 (2000).
[CrossRef] [PubMed]

Chandresekhar, S.

S. Chandresekhar, Radiative Transfer (Dover, New York, 1960).

Cherroret, N.

N. Cherroret and S. E. Skipetrov, “Microscopic derivation of self-consistent equations of anderson localization in a disordered medium of finite size,” Phys. Rev. E77, 046608 (2008).
[CrossRef]

Cheung, S.

C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett.105, 263905 (2010).
[CrossRef]

Cheung, S. K.

Z. Q. Zhang, A. A. Chabanov, S. K. Cheung, C. H. Wong, and A. Z. Genack, “Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media,” Phys. Rev. B79, 144203 (2009).
[CrossRef]

Deych, L. I.

L. I. Deych, A. Yamilov, and A. A. Lisyansky, “Scaling in one-dimensional localized absorbing systems,” Phys. Rev. B64, 024201 (2001).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fisher, D. S.

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22, 3519–3526 (1980).
[CrossRef]

Freilikher, V. D.

V. D. 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]

Fukuyamak, H.

P. A. Lee, D. A. Stone, and H. Fukuyamak, “Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field,” Phys. Rev. B35, 1039–1070 (1987).
[CrossRef]

Garcia, P. D.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science327, 1352–1355 (2010).
[CrossRef] [PubMed]

Genack, A. Z.

J. Wang and A. Z. Genack, “Transport through modes in random media,” Nature471, 345–348 (2011).
[CrossRef] [PubMed]

Z. Q. Zhang, A. A. Chabanov, S. K. Cheung, C. H. Wong, and A. Z. Genack, “Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media,” Phys. Rev. B79, 144203 (2009).
[CrossRef]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature404, 850–853 (2000).
[CrossRef] [PubMed]

Gross, P.

M. Störzer, P. Gross, C. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett.96, 063904 (2006).
[CrossRef] [PubMed]

Hu, H.

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4, 945–948 (2008).
[CrossRef]

John, S.

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

Kroha, J.

J. Kroha, C. M. Soukoulis, and P. Wölfle, “Localization of classical waves in a random medium: A self-consistent theory,” Phys. Rev. B47, 11093–11096 (1993).
[CrossRef]

Lagendijk, A.

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of anderson localization,” Phys. Today62, 24–29 (2009).
[CrossRef]

B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett.84, 4333–4336 (2000).
[CrossRef] [PubMed]

Lee, P. A.

P. A. Lee, D. A. Stone, and H. Fukuyamak, “Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field,” Phys. Rev. B35, 1039–1070 (1987).
[CrossRef]

Licciardello, D. C.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions,” Phys. Rev. Lett.42, 673–676 (1979).
[CrossRef]

Lisyansky, A. A.

L. I. Deych, A. Yamilov, and A. A. Lisyansky, “Scaling in one-dimensional localized absorbing systems,” Phys. Rev. B64, 024201 (2001).
[CrossRef]

Lodahl, P.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science327, 1352–1355 (2010).
[CrossRef] [PubMed]

Mahler, T.

B. Payne, T. Mahler, and A. Yamilov, “Effect of evanescent channels on position-dependent diffusion in disordered waveguides,” Waves in Random and Complex Media23, 43–55 (2013).
[CrossRef]

Maret, G.

T. Sperling, W. Bührer, C. M. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Phot.7, 48–52 (2013).
[CrossRef]

M. Störzer, P. Gross, C. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett.96, 063904 (2006).
[CrossRef] [PubMed]

Mirlin, A.

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

Montambaux, G.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, Cambridge, UK, 2007).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nieuwenhuizen, T. M.

M. C. 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.

A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

Page, J. H.

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4, 945–948 (2008).
[CrossRef]

Payne, B.

B. Payne, T. Mahler, and A. Yamilov, “Effect of evanescent channels on position-dependent diffusion in disordered waveguides,” Waves in Random and Complex Media23, 43–55 (2013).
[CrossRef]

A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

B. Payne, J. Andreasen, H. Cao, and A. Yamilov, “Relation between transmission and energy stored in random media with gain,” Phys. Rev. B82, 104204 (2010).
[CrossRef]

A. Yamilov and B. Payne, “Classification of regimes of wave transport in quasi-one-dimensional nonconservative random media,” J. Mod. Opt.57, 1916–1921 (2010).
[CrossRef]

B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B82, 024205 (2010).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Symmetry and transport of waves in one-dimensional disordered systems,” Adv. Phys.43, 461–542 (1994).
[CrossRef]

Pustilnik, M.

V. D. 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]

Ramakrishnan, T. V.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions,” Phys. Rev. Lett.42, 673–676 (1979).
[CrossRef]

Redding, B.

A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

Sapienza, L.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science327, 1352–1355 (2010).
[CrossRef] [PubMed]

Sarma, R.

A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

Skipetrov, S. E.

B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B82, 024205 (2010).
[CrossRef]

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4, 945–948 (2008).
[CrossRef]

N. Cherroret and S. E. Skipetrov, “Microscopic derivation of self-consistent equations of anderson localization in a disordered medium of finite size,” Phys. Rev. E77, 046608 (2008).
[CrossRef]

Smolka, S.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science327, 1352–1355 (2010).
[CrossRef] [PubMed]

Soukoulis, C. M.

J. Kroha, C. M. Soukoulis, and P. Wölfle, “Localization of classical waves in a random medium: A self-consistent theory,” Phys. Rev. B47, 11093–11096 (1993).
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C. Tian, “Hydrodynamic and field-theoretic approaches to light localization in open media,” Physica E49, 124–153 (2013).
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C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett.105, 263905 (2010).
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C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B77, 064205 (2008).
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H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4, 945–948 (2008).
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L. Y. Zhao, C. Tian, Z. Q. Zhang, and X. D. Zhang, “Unusual Brownian motion of photons in open absorbing media,” arXiv:1304.0516 (2013).

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C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett.105, 263905 (2010).
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Z. Q. Zhang, A. A. Chabanov, S. K. Cheung, C. H. Wong, and A. Z. Genack, “Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media,” Phys. Rev. B79, 144203 (2009).
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L. Y. Zhao, C. Tian, Z. Q. Zhang, and X. D. Zhang, “Unusual Brownian motion of photons in open absorbing media,” arXiv:1304.0516 (2013).

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Nat. Phys.

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, “Localization of ultrasound in a three-dimensional elastic network,” Nat. Phys.4, 945–948 (2008).
[CrossRef]

Nature

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[CrossRef]

J. Kroha, C. M. Soukoulis, and P. Wölfle, “Localization of classical waves in a random medium: A self-consistent theory,” Phys. Rev. B47, 11093–11096 (1993).
[CrossRef]

Z. Q. Zhang, A. A. Chabanov, S. K. Cheung, C. H. Wong, and A. Z. Genack, “Dynamics of localized waves: Pulsed microwave transmissions in quasi-one-dimensional media,” Phys. Rev. B79, 144203 (2009).
[CrossRef]

C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B77, 064205 (2008).
[CrossRef]

B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B82, 024205 (2010).
[CrossRef]

B. Payne, J. Andreasen, H. Cao, and A. Yamilov, “Relation between transmission and energy stored in random media with gain,” Phys. Rev. B82, 104204 (2010).
[CrossRef]

P. A. Lee, D. A. Stone, and H. Fukuyamak, “Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field,” Phys. Rev. B35, 1039–1070 (1987).
[CrossRef]

P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, “New method for a scaling theory of localization,” Phys. Rev. B22, 3519–3526 (1980).
[CrossRef]

L. I. Deych, A. Yamilov, and A. A. Lisyansky, “Scaling in one-dimensional localized absorbing systems,” Phys. Rev. B64, 024201 (2001).
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B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett.84, 4333–4336 (2000).
[CrossRef] [PubMed]

M. Störzer, P. Gross, C. Aegerter, and G. Maret, “Observation of the critical regime near anderson localization of light,” Phys. Rev. Lett.96, 063904 (2006).
[CrossRef] [PubMed]

C. Tian, S. Cheung, and Z. Zhang, “Local diffusion theory for localized waves in open media,” Phys. Rev. Lett.105, 263905 (2010).
[CrossRef]

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[CrossRef] [PubMed]

Phys. Today

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of anderson localization,” Phys. Today62, 24–29 (2009).
[CrossRef]

Physica E

C. Tian, “Hydrodynamic and field-theoretic approaches to light localization in open media,” Physica E49, 124–153 (2013).
[CrossRef]

Rev. Mod. Phys.

M. C. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys.71, 313–371 (1999).
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L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with anderson-localized modes,” Science327, 1352–1355 (2010).
[CrossRef] [PubMed]

Waves in Random and Complex Media

B. Payne, T. Mahler, and A. Yamilov, “Effect of evanescent channels on position-dependent diffusion in disordered waveguides,” Waves in Random and Complex Media23, 43–55 (2013).
[CrossRef]

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S. Chandresekhar, Radiative Transfer (Dover, New York, 1960).

L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging(Wiley-Interscience, 2007).

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A. Yamilov, R. Sarma, B. Redding, B. Payne, H. Noh, and H. Cao, “Position-dependent diffusion of light in disordered waveguides,” arXiv:1303.3244 (2013).

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L. Y. Zhao, C. Tian, Z. Q. Zhang, and X. D. Zhang, “Unusual Brownian motion of photons in open absorbing media,” arXiv:1304.0516 (2013).

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Figures (3)

Fig. 1
Fig. 1

Comparison between position-dependent diffusion coefficient d(z) = D(z)/D0 found from ab-initio numerical simulations (c.f. Eq. (2), blue curves) and self-consistent theory (Eqs. (57), red curves). (a) In passive systems (L/ξ = 0.4, 0.7, 1.5, 3.0 and 4.4) the diffusion coefficient diminishes in the interior of the system due to the enhanced return probability and, hence, stronger localization correction caused by wave interference. (b) For fixed length (L/ξ = 4.4), an increase of absorption suppresses the localization corrections to the position-dependent diffusion coefficient. The five curves correspond to L/ξa0 = 0.0, 3.3, 5.7, 9.8 and 21. (c) When absorption (ξ/ξa0 = 1.3) is added to the five samples shown in (a), the position-dependent diffusion coefficient no longer decreases below its saturated plateau value Dp, see Eq. (8). In all cases SCT agrees well with the ab-initio simulation of wave transport in disordered waveguides. The cause of the small deviations are discussed in the text.

Fig. 2
Fig. 2

Existence of the minimum diffusion coefficient is seen from the evolution of d1/2D(z = L/2)/D0 with the increase of the system size. In the passive systems (ξ/ξa0 = 0, open circles) the limit is expected to be zero. In absorbing systems (ξ/ξa0 = 0.3, 0.7, 1.3, 2.9 shown as cross, diamond, upward and downward triangle symbols respectively) saturation corresponds to formation of the plateau region seen in Figs. 1(b,c). The saturation value Dp increases monotonically with an increase of ξ/ξa0. Five solid lines are obtained from the self-consistent theory Eqs. (57) for each value of the absorption strength. Qualitative prediction of the minimum value of position-dependent diffusion coefficient in SCT is supported by the numerical simulations. The agreement is also quantitative for Dp/D0 ≳ 0.2.

Fig. 3
Fig. 3

Universal scaling of the local diffusion coefficient in the middle of the sample d1/2D(z = L/2)/D0 is described by a single parameter. The ab-initio numerical simulations in disordered waveguides with absorption (symbols) agree well with Eq. (8), which we derived based on SCT.

Equations (8)

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

{ 2 + k 2 [ 1 + δ ε ( r ) ] } E ( r ) = 0 .
D ( z ) = J z ( z ) / [ d 𝒲 ( z ) / d z ] .
J z ( ± ) ( z ) = ( v / π ) 𝒲 ( z ) ( D ( z ) / 2 ) d 𝒲 ( z ) / d z ,
v = π 2 J z ( + ) ( z ) + J z ( ) ( z ) 𝒲 ( z ) .
[ ( L ξ a 0 ) 2 ζ d ( ζ ) ζ ] C ^ ( ζ , ζ ) = δ ( ζ ζ ) ,
1 d ( ζ ) = 1 + 2 L ξ C ^ ( ζ , ζ ) ,
C ^ ( ζ , ζ ) z 0 L d ( ζ ) ζ C ^ ( ζ , ζ ) = 0
d p 1 = 1 + ( ξ a 0 / ξ ) d p 1 / 2 ,

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