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

Optimal transmission through a randomly perturbed waveguide in the localization regime

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.
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

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