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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 and 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), 731-808. arXiv:cond-mat/9612179">arXiv:cond-mat/9612179" target="_blank">arXiv:cond-mat/9612179

[2]

R. E. Collins, "Field Theory of Guided Waves," Mac Graw-Hill, New York, 1960.

[3]

O. N. Dorokhov, On the coexistence of localized and extended electronic states in the metallic phase, Solid State Commun., 51 (1984), 381-384. doi: 10.1016/0038-1098(84)90117-0.

[4]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media," Springer, New York, 2007.

[5]

J. Garnier, Multi-scaled diffusion-approximation Applications to wave propagation in random media, ESAIM Probab. Statist., 1 (1997), 183-206. doi: 10.1051/ps:1997107.

[6]

J. Garnier and G. Papanicolaou, Pulse propagation and time reversal in random waveguides, SIAM J. Appl. Math., 67 (2007), 1718-1739.

[7]

P. Gérard and E. Leichtnam, Ergodic properties of the eigenfunctions for the Dirichlet problem, Duke Math. Journal, 71 (1993), 559-607.

[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), 391-398. doi: 10.1137/1104038.

[9]

Y. Imry, Active transmission channels and universal conductance fluctuations, Europhys. Lett., 1 (1986), 249. doi: 10.1209/0295-5075/1/5/008.

[10]

P. A. Mello, P. Pereyra and N. Kumar, Macroscopic approach to multichannel disordered conductors, Ann. Phys., 181 (1988), 290-317. doi: 10.1016/0003-4916(88)90169-8.

[11]

K. A. Muttalib, Random matrix theory and the scaling theory of localization, Phys. Rev. Lett., 64 (1990), 745-747. doi: 10.1103/PhysRevLett.65.745.

[12]

Y. V. Nazarov, Limits of universality in disordered conductors, Phys. Rev. Lett., 73 (1994), 134-137. doi: 10.1103/PhysRevLett.73.134.

[13]

G. C. Papanicolaou, Wave propagation in a one-dimensional random medium, SIAM J. Appl. Math., 21 (1971), 13-18. doi: 10.1137/0121002.

[14]

G. Papanicolaou, L. Ryzhik and K. Sølna, Statistical stability in time reversal, SIAM J. Appl. Math., 64 (2004), 1133-1155. doi: 10.1137/S0036139902411107.

[15]

G. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Itô-Schrödinger equation, SIAM Multiscale Model. Simul., 6 (2007), 468-492 (electronic). doi: 10.1137/060668882.

[16]

J. B. Pendry, A. MacKinnon and A. B. Pretre, Maximal fluctuations - a new phenomenon in disordered systems, Physica A, 168 (1990), 400-407. doi: 10.1016/0378-4371(90)90391-5.

[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), 587-609.

[18]

I. M. Vellekoop and A. P. Mosk, Universal optimal transmission of light through disordered materials, Phys. Rev. Lett., 101 (2008), 120601. arXiv:0804.2412v2

[19]

I. M. Vellekoop and A. P. Mosk, Focusing coherent light through opaque strongly scattering media, Opt. Lett., 32 (2007), 2309-2311. doi: 10.1364/OL.32.002309.

[20]

S. Zelditch and M. Zworski, Ergodicity of eigenfunctions for ergodic billiards, Commun. Math. Phys., 175 (1996), 673-682. doi: 10.1007/BF02099513.

show all references

References:
[1]

C. W. J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys., 69 (1997), 731-808. arXiv:cond-mat/9612179">arXiv:cond-mat/9612179" target="_blank">arXiv:cond-mat/9612179

[2]

R. E. Collins, "Field Theory of Guided Waves," Mac Graw-Hill, New York, 1960.

[3]

O. N. Dorokhov, On the coexistence of localized and extended electronic states in the metallic phase, Solid State Commun., 51 (1984), 381-384. doi: 10.1016/0038-1098(84)90117-0.

[4]

J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, "Wave Propagation and Time Reversal in Randomly Layered Media," Springer, New York, 2007.

[5]

J. Garnier, Multi-scaled diffusion-approximation Applications to wave propagation in random media, ESAIM Probab. Statist., 1 (1997), 183-206. doi: 10.1051/ps:1997107.

[6]

J. Garnier and G. Papanicolaou, Pulse propagation and time reversal in random waveguides, SIAM J. Appl. Math., 67 (2007), 1718-1739.

[7]

P. Gérard and E. Leichtnam, Ergodic properties of the eigenfunctions for the Dirichlet problem, Duke Math. Journal, 71 (1993), 559-607.

[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), 391-398. doi: 10.1137/1104038.

[9]

Y. Imry, Active transmission channels and universal conductance fluctuations, Europhys. Lett., 1 (1986), 249. doi: 10.1209/0295-5075/1/5/008.

[10]

P. A. Mello, P. Pereyra and N. Kumar, Macroscopic approach to multichannel disordered conductors, Ann. Phys., 181 (1988), 290-317. doi: 10.1016/0003-4916(88)90169-8.

[11]

K. A. Muttalib, Random matrix theory and the scaling theory of localization, Phys. Rev. Lett., 64 (1990), 745-747. doi: 10.1103/PhysRevLett.65.745.

[12]

Y. V. Nazarov, Limits of universality in disordered conductors, Phys. Rev. Lett., 73 (1994), 134-137. doi: 10.1103/PhysRevLett.73.134.

[13]

G. C. Papanicolaou, Wave propagation in a one-dimensional random medium, SIAM J. Appl. Math., 21 (1971), 13-18. doi: 10.1137/0121002.

[14]

G. Papanicolaou, L. Ryzhik and K. Sølna, Statistical stability in time reversal, SIAM J. Appl. Math., 64 (2004), 1133-1155. doi: 10.1137/S0036139902411107.

[15]

G. Papanicolaou, L. Ryzhik and K. Sølna, Self-averaging from lateral diversity in the Itô-Schrödinger equation, SIAM Multiscale Model. Simul., 6 (2007), 468-492 (electronic). doi: 10.1137/060668882.

[16]

J. B. Pendry, A. MacKinnon and A. B. Pretre, Maximal fluctuations - a new phenomenon in disordered systems, Physica A, 168 (1990), 400-407. doi: 10.1016/0378-4371(90)90391-5.

[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), 587-609.

[18]

I. M. Vellekoop and A. P. Mosk, Universal optimal transmission of light through disordered materials, Phys. Rev. Lett., 101 (2008), 120601. arXiv:0804.2412v2

[19]

I. M. Vellekoop and A. P. Mosk, Focusing coherent light through opaque strongly scattering media, Opt. Lett., 32 (2007), 2309-2311. doi: 10.1364/OL.32.002309.

[20]

S. Zelditch and M. Zworski, Ergodicity of eigenfunctions for ergodic billiards, Commun. Math. Phys., 175 (1996), 673-682. doi: 10.1007/BF02099513.

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