American Institute of Mathematical Sciences

Advanced
December  2011, 15(3): 597-621. doi: 10.3934/dcdsb.2011.15.597

Optimal transmission through a randomly perturbed waveguide in the localization regime

Josselin Garnier 1,

1. 

Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, site Chevaleret, case 7012, 75205 Paris Cedex 13, France

Received  January 2010 Revised  July 2010 Published  February 2011

We demonstrate that increased power transmission through a random single-mode or multi-mode channel can be obtained in the localization regime by optimizing the spatial wave front or the time pulse profile of the source. The idea is to select and excite the few modes or the few frequencies whose transmission coefficients are anomalously large compared to the typical exponentially small value. We prove that time reversal is optimal for maximizing the transmitted intensity at a given time or space, while iterated time reversal is optimal for maximizing the total transmitted energy. The statistical stability of the optimal transmitted intensity and energy is also obtained.
Keywords: random media, Wave propagation, asymptotic analysis..
Mathematics Subject Classification: Primary: 35L05, 35R60; Secondary: 60F0.
Citation: Josselin Garnier. Optimal transmission through a randomly perturbed waveguide in the localization regime. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 597-621. doi: 10.3934/dcdsb.2011.15.597
References:
[1]

C. W. J. Beenakker, Random-matrix theory of quantum transport,, Rev. Mod. Phys., 69 (1997), 731.   Google Scholar

[2]

R. E. Collins, "Field Theory of Guided Waves,", Mac Graw-Hill, (1960).   Google Scholar

[3]

O. N. Dorokhov, On the coexistence of localized and extended electronic states in the metallic phase,, Solid State Commun., 51 (1984), 381.  doi: 10.1016/0038-1098(84)90117-0.  Google Scholar

[4]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media,", Springer, (2007).   Google Scholar

[5]

J. Garnier, Multi-scaled diffusion-approximation Applications to wave propagation in random media,, ESAIM Probab. Statist., 1 (1997), 183.  doi: 10.1051/ps:1997107.  Google Scholar

[6]

J. Garnier and G. Papanicolaou, Pulse propagation and time reversal in random waveguides,, SIAM J. Appl. Math., 67 (2007), 1718.   Google Scholar

[7]

P. Gérard and E. Leichtnam, Ergodic properties of the eigenfunctions for the Dirichlet problem,, Duke Math. Journal, 71 (1993), 559.   Google Scholar

[8]

M. E. Gertsenshtein and V. B. Vasil'ev, Waveguides with random inhomogeneities and Brownian motion in the Lobachevsky plane,, Theory Probab. Appl., 4 (1959), 391.  doi: 10.1137/1104038.  Google Scholar

[9]

Y. Imry, Active transmission channels and universal conductance fluctuations,, Europhys. Lett., 1 (1986).  doi: 10.1209/0295-5075/1/5/008.  Google Scholar

[10]

P. A. Mello, P. Pereyra and N. Kumar, Macroscopic approach to multichannel disordered conductors,, Ann. Phys., 181 (1988), 290.  doi: 10.1016/0003-4916(88)90169-8.  Google Scholar

[11]

K. A. Muttalib, Random matrix theory and the scaling theory of localization,, Phys. Rev. Lett., 64 (1990), 745.  doi: 10.1103/PhysRevLett.65.745.  Google Scholar

[12]

Y. V. Nazarov, Limits of universality in disordered conductors,, Phys. Rev. Lett., 73 (1994), 134.  doi: 10.1103/PhysRevLett.73.134.  Google Scholar

[13]

G. C. Papanicolaou, Wave propagation in a one-dimensional random medium,, SIAM J. Appl. Math., 21 (1971), 13.  doi: 10.1137/0121002.  Google Scholar

[14]

G. Papanicolaou, L. Ryzhik and K. Sølna, Statistical stability in time reversal,, SIAM J. Appl. Math., 64 (2004), 1133.  doi: 10.1137/S0036139902411107.  Google Scholar

[15]

G. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Itô-Schrödinger equation,, SIAM Multiscale Model. Simul., 6 (2007), 468.  doi: 10.1137/060668882.  Google Scholar

[16]

J. B. Pendry, A. MacKinnon and A. B. Pretre, Maximal fluctuations - a new phenomenon in disordered systems,, Physica A, 168 (1990), 400.  doi: 10.1016/0378-4371(90)90391-5.  Google Scholar

[17]

J.-L. Pichard, N. Zanon, Y. Imry and A. D. Stone, Theory of random multiplicative transfer matrices and its implications for quantum transport,, J. Phys., 51 (1990), 587.   Google Scholar

[18]

I. M. Vellekoop and A. P. Mosk, Universal optimal transmission of light through disordered materials,, Phys. Rev. Lett., 101 (2008).   Google Scholar

[19]

I. M. Vellekoop and A. P. Mosk, Focusing coherent light through opaque strongly scattering media,, Opt. Lett., 32 (2007), 2309.  doi: 10.1364/OL.32.002309.  Google Scholar

[20]

S. Zelditch and M. Zworski, Ergodicity of eigenfunctions for ergodic billiards,, Commun. Math. Phys., 175 (1996), 673.  doi: 10.1007/BF02099513.  Google Scholar

show all references

References:
[1]

C. W. J. Beenakker, Random-matrix theory of quantum transport,, Rev. Mod. Phys., 69 (1997), 731.   Google Scholar

[2]

R. E. Collins, "Field Theory of Guided Waves,", Mac Graw-Hill, (1960).   Google Scholar

[3]

O. N. Dorokhov, On the coexistence of localized and extended electronic states in the metallic phase,, Solid State Commun., 51 (1984), 381.  doi: 10.1016/0038-1098(84)90117-0.  Google Scholar

[4]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media,", Springer, (2007).   Google Scholar

[5]

J. Garnier, Multi-scaled diffusion-approximation Applications to wave propagation in random media,, ESAIM Probab. Statist., 1 (1997), 183.  doi: 10.1051/ps:1997107.  Google Scholar

[6]

J. Garnier and G. Papanicolaou, Pulse propagation and time reversal in random waveguides,, SIAM J. Appl. Math., 67 (2007), 1718.   Google Scholar

[7]

P. Gérard and E. Leichtnam, Ergodic properties of the eigenfunctions for the Dirichlet problem,, Duke Math. Journal, 71 (1993), 559.   Google Scholar

[8]

M. E. Gertsenshtein and V. B. Vasil'ev, Waveguides with random inhomogeneities and Brownian motion in the Lobachevsky plane,, Theory Probab. Appl., 4 (1959), 391.  doi: 10.1137/1104038.  Google Scholar

[9]

Y. Imry, Active transmission channels and universal conductance fluctuations,, Europhys. Lett., 1 (1986).  doi: 10.1209/0295-5075/1/5/008.  Google Scholar

[10]

P. A. Mello, P. Pereyra and N. Kumar, Macroscopic approach to multichannel disordered conductors,, Ann. Phys., 181 (1988), 290.  doi: 10.1016/0003-4916(88)90169-8.  Google Scholar

[11]

K. A. Muttalib, Random matrix theory and the scaling theory of localization,, Phys. Rev. Lett., 64 (1990), 745.  doi: 10.1103/PhysRevLett.65.745.  Google Scholar

[12]

Y. V. Nazarov, Limits of universality in disordered conductors,, Phys. Rev. Lett., 73 (1994), 134.  doi: 10.1103/PhysRevLett.73.134.  Google Scholar

[13]

G. C. Papanicolaou, Wave propagation in a one-dimensional random medium,, SIAM J. Appl. Math., 21 (1971), 13.  doi: 10.1137/0121002.  Google Scholar

[14]

G. Papanicolaou, L. Ryzhik and K. Sølna, Statistical stability in time reversal,, SIAM J. Appl. Math., 64 (2004), 1133.  doi: 10.1137/S0036139902411107.  Google Scholar

[15]

G. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Itô-Schrödinger equation,, SIAM Multiscale Model. Simul., 6 (2007), 468.  doi: 10.1137/060668882.  Google Scholar

[16]

J. B. Pendry, A. MacKinnon and A. B. Pretre, Maximal fluctuations - a new phenomenon in disordered systems,, Physica A, 168 (1990), 400.  doi: 10.1016/0378-4371(90)90391-5.  Google Scholar

[17]

J.-L. Pichard, N. Zanon, Y. Imry and A. D. Stone, Theory of random multiplicative transfer matrices and its implications for quantum transport,, J. Phys., 51 (1990), 587.   Google Scholar

[18]

I. M. Vellekoop and A. P. Mosk, Universal optimal transmission of light through disordered materials,, Phys. Rev. Lett., 101 (2008).   Google Scholar

[19]

I. M. Vellekoop and A. P. Mosk, Focusing coherent light through opaque strongly scattering media,, Opt. Lett., 32 (2007), 2309.  doi: 10.1364/OL.32.002309.  Google Scholar

[20]

S. Zelditch and M. Zworski, Ergodicity of eigenfunctions for ergodic billiards,, Commun. Math. Phys., 175 (1996), 673.  doi: 10.1007/BF02099513.  Google Scholar

[1]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[2]

Chang-Yeol Jung, Alex Mahalov. Wave propagation in random waveguides. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 147-159. doi: 10.3934/dcds.2010.28.147
[3]

D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843
[4]

Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678-685. doi: 10.3934/proc.2015.0678
[5]

Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951
[6]

Andrey Shishkov, Laurent Véron. Propagation of singularities of nonlinear heat flow in fissured media. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1769-1782. doi: 10.3934/cpaa.2013.12.1769
[7]

Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048
[8]

Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1
[9]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[10]

S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305
[11]

Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks & Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001
[12]

Shi-Liang Wu, Cheng-Hsiung Hsu. Propagation of monostable traveling fronts in discrete periodic media with delay. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2987-3022. doi: 10.3934/dcds.2018128
[13]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239
[14]

Fathi Dkhil, Angela Stevens. Traveling wave speeds in rapidly oscillating media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 89-108. doi: 10.3934/dcds.2009.25.89
[15]

Andrew Homan. Multi-wave imaging in attenuating media. Inverse Problems & Imaging, 2013, 7 (4) : 1235-1250. doi: 10.3934/ipi.2013.7.1235
[16]

Patrick W. Dondl, Michael Scheutzow. Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients. Networks & Heterogeneous Media, 2012, 7 (1) : 137-150. doi: 10.3934/nhm.2012.7.137
[17]

Guillaume Bal, Olivier Pinaud. Self-averaging of kinetic models for waves in random media. Kinetic & Related Models, 2008, 1 (1) : 85-100. doi: 10.3934/krm.2008.1.85
[18]

Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473
[19]

Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867
[20]

Ralph Lteif, Samer Israwi, Raafat Talhouk. An improved result for the full justification of asymptotic models for the propagation of internal waves. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2203-2230. doi: 10.3934/cpaa.2015.14.2203

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences