Abstract

Fundamental concepts in the quasi-one-dimensional geometry of disordered wires and random waveguides in which ideas of scaling and the transmission matrix were first introduced are reviewed. We discuss the use of the transmission matrix to describe the scaling, fluctuations, delay time, density of states, and control of waves propagating through and within disordered systems. Microwave measurements, random matrix theory calculations, and computer simulations are employed to study the statistics of transmission and focusing in single samples and the scaling of the probability distribution of transmission and transmittance in random ensembles. Finally, we explore the disposition of the energy density of transmission eigenchannels inside random media.

© 2015 Optical Society of America

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2015 (2)

M. Davy, Z. Shi, J. Wang, X. Cheng, and A. Z. Genack, “Transmission eigenchannels and the densities of states of random media,” Phys. Rev. Lett. 114, 033901 (2015).
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M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, “Universal structure of transmission eigenchannels inside opaque media,” Nat. Commun. 6, 6893 (2015).
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2014 (10)

A. Pẽna, A. Girschik, F. Libisch, S. Rotter, and A. A. Chabanov, “The single-channel regime of transport through random media,” Nat. Commun. 5, 3488 (2014).
[Crossref] [PubMed]

S. E. Skipetrov and I. M. Sokolov, “Absence of Anderson localization of light in a random ensemble of point scatterers,” Phys. Rev. Lett. 112, 023905 (2014).
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B. Gérardin, J. Laurent, A. Derode, C. Prada, and A. Aubry, “Full transmission and reflection of waves propagating through a maze of disorder,” Phys. Rev. Lett. 113, 173901 (2014).
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X. Cheng and A. Z. Genack, “Focusing and energy deposition inside random media,” Opt. Lett. 39, 6324 (2014).
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S. Popoff, A. Goetschy, S. Liew, A. D. Stone, and H. Cao, “Coherent control of total transmission of light through disordered media,” Phys. Rev. Lett. 112, 133903 (2014).
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O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive real-time imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
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X. Hao, L. Martin-Rouault, and M. Cui, “A self-adaptive method for creating high efficiency communication channels through random scattering media,” Sci. Rep. 4, 5874 (2014).
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Y. Bromberg and H. Cao, “Generating non-Rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
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Z. Shi, J. Wang, and A. Z. Genack, “Microwave conductance in random waveguides in the cross-over to Anderson localization and single-parameter scaling,” Proceedings of the National Academy of Sciences (PNAS) 111, 2926–2930 (2014).
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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,” Phys. Rev. Lett. 112, 023904 (2014).
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2013 (10)

T. Sperling, W. Bührer, C. Aegerter, and G. Maret, “Direct determination of the transition to localization of light in three dimensions,” Nat. Photonics 7, 48 (2013).
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B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, and C. Yang, “Speckle-scale focusing in the diffusive regime with time-reversal of variance-encoded light (TROVE),” Nat. Photonics 7, 300 (2013).
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Y. Choi, T. R. Hillman, W. Choi, N. Lue, R. R. Dasari, P. T. C. So, W. Choi, and Z. Yaqoob, “Measurement of the time-resolved reflection matrix for enhancing light energy delivery into a scattering medium,” Phys. Rev. Lett. 111, 243901 (2013).
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T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media non-invasively using the photoacoustic transmission matrix,” Nat. Photonics 8, 58 (2013).
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H. P. Paudel, C. Stockbridge, J. Mertz, and T. Bifano, “Focusing polychromatic light through strongly scattering media,” Opt. Express 21, 17299 (2013).
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H. Yu, T. R. Hillman, W. Choi, J. Lee, M. S. Feld, R. R. Dasari, and Y. Park, “Measuring large optical transmission matrices of disordered media,” Phys. Rev. Lett. 111, 153902 (2013).
[Crossref] [PubMed]

A. Goetschy and A. Stone, “Filtering random matrices: the effect of incomplete channel control in multiple scattering,” Phys. Rev. Lett. 111, 063901 (2013).
[Crossref] [PubMed]

Z. Shi, M. Davy, J. Wang, and A. Z. Genack, “Focusing through random media in space and time: a transmission matrix approach,” Opt. Lett. 38, 2714–2716 (2013).
[Crossref] [PubMed]

M. Davy, Z. Shi, J. Wang, and A. Z. Genack, “Transmission statistics and focusing in single disordered samples,” Opt. Express,  21, 10367–10375 (2013).
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M. A. Castellanos-Beltran, D. Q. Ngo, W. E. Shanks, A. B. Jayich, and J. G. E. Harris, “Measurement of the full distribution of persistent current in normal-metal rings,” Phys. Rev. Lett. 110, 156801 (2013).
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2012 (8)

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. Photonics 6, 583–587 (2012).
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S. Tripathi, R. Paxman, T. Bifano, and K. C. Toussaint, “Vector transmission matrix for the polarization behavior of light propagation in highly scattering media,” Opt. Express 20, 16067–16076 (2012).
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A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling wave in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
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Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109, 203901 (2012).
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M. Davy, Z. Shi, and A. Z. Genack, “Focusing through random media: eigenchannel participation number and intensity correlation,” Phys. Rev. B 85, 035105 (2012).
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K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound-pulse-guided digital phase conjugation,” Nat. Photonics 6, 657 (2012).
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D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20, 1733–1740 (2012).
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Z. Shi and A. Z. Genack, “Transmission eigenvalues and the bare conductance in the crossover to Anderson localization,” Phys. Rev. Lett. 108, 043901 (2012).
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2011 (7)

X. Xu, H. Liu, and L. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5, 154 (2011).
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E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106, 193905 (2011).
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J. Wang and A. Z. Genack, “Transport through modes in random media,” Nature,  471, 345 (2011).
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W. Choi, A. P. Mosk, Q. Park, and W. Choi, “Transmission eigenchannels in a disordered medium,” Phys. Rev. B 83, 134207 (2011).
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J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, “Control of light transmission through opaque scattering media in space and time,” Phys. Rev. Lett. 106, 103901 (2011).
[Crossref] [PubMed]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[Crossref]

D. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, “Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium,” Nat. Commun. 2, 447 (2011).
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2010 (8)

E.G. van Putten and A.P. Mosk, “The information age in optics: measuring the transmission matrix,” Physics 3, 22 (2010).
[Crossref]

S. Zhang, Y. Lockerman, and A. Z. Genack, “Mesoscopic speckle,” Phys. Rev. E 82, 051114 (2010).
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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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B. Payne, A. Yamilov, and S. E. Skipetrov, “Anderson localization as position-dependent diffusion in disordered waveguides,” Phys. Rev. B 82, 024205 (2010).
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I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
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I. M. Vellekoop and C. M. Aegerter, “Scattered light fluorescence microscopy: imaging through turbid layers,” Opt. Lett. 35, 1245 (2010).
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S. Popoff, G. Lerosey, M. Fink, A. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 1–5 (2010).
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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).
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2009 (2)

Z. 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. B 79, 144203 (2009).
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A. Aubry and A. Derode, “Random matrix theory applied to acoustic backscattering and imaging in complex media,” Phys. Rev. Lett. 102, 084301 (2009).
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2008 (6)

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
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Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
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Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
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C. Tian, “Supersymmetric field theory of local light diffusion in semi-infinite media,” Phys. Rev. B 77, 064205 (2008).
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N. Cherroret and S. E. Skipetrov, “Microscopic derivation of self-consistent equations of Anderson localization in a disordered medium of finite size,” Phys. Rev. E 77, 046608 (2008).
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H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. van Tiggelen, “Localization of ultrasound in a three dimensional elastic network,” Nat. Phys. 4, 945–948 (2008).
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2007 (2)

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
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T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two dimensional photonic lattices,” Nature 446, 52–55 (2007).
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2006 (3)

P. Sebbah, B. Hu, J. Klosner, and A. Z. Genack, “Quasimodes of spatially extended field distributions within nominally localized random waveguides,” Phys. Rev. Lett. 96, 183902 (2006).
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S. E. Skipetrov and B. A. van Tiggelen, “Dynamics of Anderson localization in open 3D media,” Phys. Rev. Lett. 96, 043902 (2006).
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M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
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2005 (1)

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson localized 1D Systems,” Phys. Rev. Lett. 94, 113903 (2005).
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2003 (1)

A. A. Chabanov, A. Z. Genack, and Z.-Q. Zhang, “Breakdown of diffusion in dynamics of extended waves in mesoscopic media,” Phys. Rev. Lett. 90, 203903 (2003).
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2002 (4)

P. Sebbah, B. Hu, A. Z. Genack, R. Pnini, and B. Shapiro, “Spatial field correlation: the building block of mesoscopic fluctuations,” Phys. Rev. Lett. 88, 123901 (2002).
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V. Gopar, K. A. Muttalib, and P. Wölfle, “Conductance distribution in disordered quantum wires: crossover between the metallic and insulating regimes,” Phys. Rev. B 66, 174204 (2002).
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L. S. Froufe-Pérez, P. García-Mochales, P. A. Serena, P. A. Mello, and J. J. Sáenz, “Conductance distributions in quasi-one-dimensional Disordered Wires,” Phys. Rev. Lett. 89, 246403 (2002).
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V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and pre-localized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
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2001 (4)

A. A. Chabanov and A. Z. Genack, Photon localization in resonant media,” Phys. Rev. Lett. 87, 153901 (2001).
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M. Ruhlander and C. M. Soukoulis, “The probability distribution of the conductance at the mobility edge,” Physica B 296, 32–35 (2001).
[Crossref]

K. Slevin, P. Markoš, and T. Ohtsuki, “Reconciling conductance fluctuations and the scaling theory of localization,” Phys. Rev. Lett. 86, 3594–3597 (2001).
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A. García-Martín and J. J. Sáenz, “Universal conductance distributions in the crossover between diffusive and localization regimes,” Phys. Rev. Lett. 87, 116603 (2001).
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2000 (2)

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
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A. D. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Phys. Rep. 326, 259 (2000).
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1999 (7)

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).
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F. Scheffold, R. Lenke, R. Tweer, and G. Maret, “Localization or classical diffusion of light?” Nature,  389, 206 (1999).
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K. A. Muttalib and P. Wölfle, “One-sided” log-normal distribution of conductances for a disordered quantum wire,” Phys. Rev. Lett. 83, 3013–3016 (1999).
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C. M. Soukoulis, X. Wang, Q. Li, and M. M. Sigalas, “What is the right form of the probability distribution of the conductance at the mobility edge?” Phys. Rev. Lett. 82, 668 (1999).
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P. Markoš, “Probability distribution of the conductance at the mobility edge,” Phys. Rev. Lett. 83, 588–591 (1999).
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A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of wave dynamics in random media,” Phys. Rev. Lett. 82, 715 (1999).
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H. Cao, Y. Zhao, S. Ho, E. Seelig, Q. Wang, and R. Chang, “Random laser action in semiconductor power,” Phys. Rev. Lett. 82, 2278 (1999).
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1998 (4)

M. Brandbyge and M. Tsukada, “Local density of states from transmission amplitudes in multichannel systems,” Phys. Rev. B 57, 15088 (1998).
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V. Plerou and Z. Wang, “Conductances, conductance fluctuations, and level statistics on the surface of multilayer quantum Hall states,” Phys. Rev. B 58:1967–1979 (1998).
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E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545 (1998).
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F. Scheffold and G. Maret, “Universal conductance fluctuations of light,” Phys. Rev. Lett. 81, 5800 (1998).
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1997 (4)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671 (1997).
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M. Stoytchev and A. Z. Genack, “Measurement of the probability distribution of total transmission in random waveguides,” Phys. Rev. Lett. 79, 309–312 (1997).
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C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys. 69, 731–808 (1997).
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K. Slevin and T. Ohtsuki, “The Anderson transition: time reversal symmetry and universality,” Phys. Rev. Lett. 78, 4083–4086 (1997).
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1996 (2)

A. Lagendijk and B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143 (1996).
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S. A. van Langen, P. W. Brouwer, and C. W. J. Beenakker, “Non-perturbative calculation of the probability distribution of plane-wave transmission through disordered waveguide,” Phys. Rev. E 53, 1344 (1996).
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1995 (5)

Th. M. Nieuwenhuizen and M. C. van Rossum, “Intensity distribution of waves transmitted through a multiple scattering medium,” Phys. Rev. Lett. 74, 2674 (1995).
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E. Kogan and M. Kaveh, “Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium,” Phys. Rev. B 52, R3813 (1995).
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K. Frahm, “Equivalence of the Fokker-Planck approach and the nonlinear sigma model for disordered wires in the unitary symmetry class,” Phys. Rev. Lett. bf  74, 4706 (1995).
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B. A. Muzyhantskii and D. E. Khmelnitskii, “Nearly localized states in weakly disordered conductors,” Phys. Rev. B 51, 5480–5484 (1995).
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G. Iannacone, “General relation between density of states and dwell times in mesoscopic systems,” Phys. Rev. B 51, 4727 (1995).
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1994 (5)

N. Lawandy, R. Balachandran, A. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature,  368, 436 (1994).
[Crossref]

A. Z. Genack and J. M. Drake, “Scattering for super-radiation,” Nature 368, 400 (1994).
[Crossref]

J. B. Pendry, “Symmetry and transport of waves in 1D disordered systems,” Adv. Phys. 43, 461–542 (1994).
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Y. V. Nazarov, “Limits of universality in disordered conductors,” Phys. Rev. Lett. 73, 134–137 (1994).
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J. de Boer, M. van Rossum, M. van Albada, T. M. Nieuwenhuizen, and A. Lagendijk, “Probability distribution of multiple scattered light measured in total transmission,” Phys. Rev. Lett. 73, 2567–2570 (1994).
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1993 (1)

R. Weaver, “Anomalous diffusivity and localization of classical waves in disordered media: the effect of dissipation,” Phys. Rev. B 47, 1077 (1993).
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1992 (2)

M. Fink, “Time-reversal of ultrasonic fields part I: basic principles,” IEEE Trans. Sonics Ultrason. 39, 555–567 (1992).
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M. S. Birman and D. R. Yafaev, “Spectral properties of the scattering matrix,” Algebra i Analiz 4, 1 (1992).

1991 (3)

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, and A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132 (1991).
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H. U. Baranger, D. P. DiVincenzo, R. A. Jalabert, and A. D. Stone, “Classical and quantum ballistic-transport anomalies in microjunctions,” Phys. Rev. B 44, 10637 (1991).
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A. Z. Genack and N. Garcia, “Observation of photon localization in a three-dimensional disordered system,” Phys. Rev. Lett. 66, 2064 (1991).
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1990 (6)

A. Z. Genack and J. M. Drake, “Relationship between optical intensity fluctuations and pulse propagation in random media,” Europhys. Lett. 11, 331 (1990).
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M. P. van Albada, J. F. de Boer, and A. Lagendijk, “Observation of long-range intensity correlation in the transport of coherent light through a random medium,” Phys. Rev. Lett. 64, 2787 (1990).
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Proceedings of the National Academy of Sciences (PNAS) (1)

Z. Shi, J. Wang, and A. Z. Genack, “Microwave conductance in random waveguides in the cross-over to Anderson localization and single-parameter scaling,” Proceedings of the National Academy of Sciences (PNAS) 111, 2926–2930 (2014).
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Figures (18)

Fig. 1
Fig. 1 Illustration of a wave returning to a coherent volume of (λ/2)3 in three dimensions and its time-reversed partner.
Fig. 2
Fig. 2 Fluctuations of conductance of a gold wire at 10 mK as a function of magnetic field applied perpendicular to the wire. The wire is of length 310 nm and width 25 nm. (From Ref. [37])
Fig. 3
Fig. 3 Spectra of the transmission coefficients normalized to ensemble average values. In- and out-of-phase field transmission coefficients, field E ba / T ba / , transmission coefficients of intensity sba = Tba/〈Tba〉, total transmission sa = Ta/〈Ta〉 and transmittance s = T/〈T〉 for microwave radiation propagating through a random waveguides for a diffusive sample (left column) and a localized sample (right column). For diffusive waves, δ > 1, many modes contribute to transmission at all frequencies and for all source and detector positions. In contrast, for localized waves δ < 1, distinct peaks appear when the incident radiation is on resonance with a quasi-normal mode. The resonance condition holds for all source and detector positions and therefore sharp peaks remain even when transmission is integrated over space.
Fig. 4
Fig. 4 Scaling of optical transmission through a wedged sample consisting of TiO2 embedded in polystyrene. (a,b) Log-log and semi-log plots of the scaling of the transmission coefficient, T (L). The inverse variation of T (L) with L is seen in the log-log plot and the exponentially falloff of transmission beyond the absorption length La is seen in the semi-log plot. The absorption length La = 112 ± 5μm of the sample is indicated by the dashed line in both figures. The solid line is a fit of the expression T (L) = 5αD/vsinh(αL) to the data. Here D is the diffusion coefficient, v is the speed of light in the medium and α is the inverse of La. (c) Spectra of transmission for various sample lengths. The measured transmission in these experiments never reaches zero because the polarized intensity only vanishes at point singularities while measurements are made with a finite aperture placed before the photomultiplier tube. (From Ref. [41])
Fig. 5
Fig. 5 Absolute value of the wave field in the scattering medium at f = 0.36 MHz associated with (a), an incident plane wave, (b), a closed eigenchannel, (c), an open eigenchannel deduced from the measured S matrix, and (d), an open eigenchannel deduced from the normalized matrix Ŝ. The corresponding intensities averaged over the wave guide section (y axis) are shown versus depth x in lower panels (e)(h). They are all normalized by the intensity at the plane of sources (x = 0). (From Ref. [91])
Fig. 6
Fig. 6 The variance of the normalized total transmission vs. the inverse of the eigenchannel participation number M is obtained from measurements by grouping single samples with the same value of M−1 from ensembles with a wide range of values of g. The solid line is var(NTa/T) = M−1. (From Ref. [119])
Fig. 7
Fig. 7 Probability distributions of normalized total transmission and intensity in sub-ensembles of random samples with similar values of M−1. (a) The probability distributions of normalized total transmission P(NTa/T) for samples with M−1 = 0.17 ± 0.01 from two random ensembles with g = 3.9 (Green open circles) and g = 0.17 (red filled circles) are seen to overlap, confirming that P(NTa/T) depends only on the value of M. The solid curve is a calculation of P(Ta/Ta〉) with var(Ta/Ta〉) replaced with M−1 in the expression from Ref. [59, 60]. (b) P(NTa/T) for M−1 in the range of for two ensembles with g = 0.37 and 0.17. The solid line is an exponential distribution, exp(−NTa/T). (c,d) The normalized intensity distributions, P(N2Tba/T) for the same as in (a,b). The solid lines are the calculations of the distribution based upon P(NTa/T) in (a,b) and the exponential falloff of the distribution of the normalized intensity P(NTba/Ta) = exp(−NTba/Ta). (From Ref. [119])
Fig. 8
Fig. 8 Probability distribution of microwave conductance. (a) P(lnT) for g = 0.37 (red dots) and 0.045 (green asterisk). The solid black line is a Gaussian fit to the data. For g = 0.045, all of the data points are included, whereas for g = 0.37, only data to the left of the peak is used in the fit. (b) P(T) for g = 0.37 in a semi-log plot exhibits an exponential tail. (From Ref. [30])
Fig. 9
Fig. 9 Coulomb gas model of transmission eigenvalues and conductance. (a) Average positions of charges and their images as the position of the first charge x1 change in the random ensemble with g = 0.37. The dashed lines show the average positions of the charges for this ensemble. The curly brackets in this case represents the average over a subset of transmission matrices with the specified value of x1. (b) Average positions of charges vs. ln T in the same ensemble. Here, the curly brackets indicate the averaging is over a subset of transmission matrices with the specified value of lnT. (From Ref. [30])
Fig. 10
Fig. 10 Approach to single-parameter scaling in multi-channel random systems. ℛ ≡ −var(lnT)/〈lnT〉, 〈MT〉/〈T〉 and 〈M−1T〉/〈T〉 are plotted vs. L/ξ. The dashed line is the prediction of SPS in the limit of deeply localized samples in 1D disordered materials. (From Ref. [30])
Fig. 11
Fig. 11 Contrast in optimal focusing vs. eigenchannel participation number M. The open circles and squares represent measurements from transmission matrices N = 30 and 66 channels, respectively. The filled triangles give results for N′ × N′ matrices with N′ = 30 for points selected from a larger matrix of size N = 66. Phase conjugation is applied within the reduced matrix to achieve maximal focusing. Eq. 4 is represented by the solid red and dashed blue curves for N = 30 and 66, respectively. (From Ref. [119])
Fig. 12
Fig. 12 The ensemble average of normalized intensity for focused radiation (blue circles) is compared to Eq. 5 (blue solid line) for L = 61 cm for diffusive and localized waves. The transmitted field for a single polarization is measured along a line with a spacing of 2 mm and the source antenna is translated along 49 points on a square grid with 9 mm spacing. Fr) (blue dots) is fit with the theoretical expression obtained from the Fourier transform of the specific intensity (dashed blue line). The black dashed line is proportional to 〈Tba〉/〈Ifocr = 0)〉 = 1/N. Equation 5 is not valid for localized waves, but good agreement is obtained when κ is replaced by 1/(μ − 1) in Eq. 5 with μ determined experimentally. (From Ref. [101])
Fig. 13
Fig. 13 Optimal focusing inside a random sample. (a) Intensity distribution in the transverse dimension for optimal focusing at (L/4,0) and (L/2,0). (b) Spatial variation of the contrast μ(x) and eigenchannel participation number M(x). (From Ref. [108])
Fig. 14
Fig. 14 Controlling pulse transmission through a random medium with a time dependent TM. Spectra of the TM for microwave radiation propagating through a random waveguide with a length L = 61 cm are measured from 14.7 to 15.7 GHz in 3200 steps. The TM is determined with a single polarization between pairs of 45 points on the incident and output surfaces. The time dependent transmission matrix is obtained from spectra of the transmitted field between all points a and b, tba(ν). These spectra are multiplied by a Gaussian pulse centered in the measured spectrum at ν0 =15.2 GHz with bandwidth σν =150 MHz and then Fourier transformed into the time domain. This gives the time response at the detector to an incident Gaussian pulse launched by a source antenna with bandwidth σt =1/2π/σν. (a) The time variation Iba(t′) of an incident pulse launched at the center of the input surface and detected at the center of the output surface in a single realization of the random sample and the average of the time-of-flight distribution 〈I(t′)〉. (b) Phase conjugation is applied numerically to the same configuration as in (a) to focus at t′= 33 ns at the center of the output surface. The Whittaker-Shannon sampling theorem is used to obtain high-resolution spatial intensity patterns shown in the inset of (b). (From Ref. [146])
Fig. 15
Fig. 15 Time evolution of the maximal focusing contrast 〈μ〉. μ is well described by Eq. 7 after the time of the ballistic arrival, t′ ∼ 21 ns. At early times, the signal-to-noise ratio is too low to analyze the transmission matrix. (From Ref. [146])
Fig. 16
Fig. 16 Spectra of transmission eigenvalues τn and the eigenchannel dwell times n/dω for n = 1, 5, 15, 25 for a diffusive sample with g = 6.9. The dwell time for the eigenchannels is the contribution of the eigenchannel to the DOS. (From Ref. [115])
Fig. 17
Fig. 17 Comparison of two approaches for finding the DOS of a disordered material. (a) Contributions of the individual modes in Eq. 9 to the DOS. The integral of each mode over the angular frequency is unity. (b) DOS determined from the TM (red curve) by summing spectra of n/dω and modes in (a). (From Ref. [115])
Fig. 18
Fig. 18 Spatial profile of the energy distribution of transmission eigenchannel inside the sample. (a) Ensemble averages of the eigenchannel energy density profiles Wτ(x) for eigenvalues τ = 1, 0.5, 0.1, and 0.001 for a diffusive sample with L/ξ = 0.05. Wτ(x) is the energy integrated over the transverse dimension and is normalized to equal τ at the output surface. (b) Ensemble averages of energy density profiles for all transmission eigenchannels with the eigenchannel indices n from 1 to N. The linear falloff of the average of the energy density over all eigenchannels is in accord with the diffusion theory. (From Ref. [118])

Equations (9)

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τ n = 1 / cosh 2 ( L / ξ n ) .
var ( N T a / T ) = n N τ n 2 ( n N τ n ) 2 = 1 / M .
C ba , b a = δ aa δ bb + M 1 ( δ aa + δ bb ) .
μ = 1 1 / M 1 / N .
I foc ( Δ r ) β T β = F ( Δ r ) + κ 1 + κ .
t ba ( ω ) = m t ba m Γ m / 2 Γ m / 2 + i ( ω ω m ) .
μ = 1 / ( 1 / M ( t ) 1 / N ) .
ρ ( ω ) = 1 π n N d θ n d ω .
ρ ( ω ) = m ρ m ( ω ) = 1 π Γ m / 2 π ( Γ m / 2 ) 2 + ( ω ω m ) 2 .

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