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doi: 10.3934/dcdsb.2022106
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## Time reversal of surface plasmons

 Department of Mathematics, Colorado State University, Fort Collins CO, 80523, USA

* Corresponding author: Olivier Pinaud

Received  November 2021 Revised  April 2022 Early access June 2022

Fund Project: The author is supported by NSF grant DMS-2006416

We study in this work the so-called "instantaneous time mirrors" in the context of surface plasmons. The latter are associated with high frequency waves at the surface of a conducting sheet. Instantaneous time mirrors were introduced in [3], with the idea that singular perturbations in the time variable in a wave-type equation create a time-reversed focusing wave. We consider the time-dependent three-dimensional Maxwell's equations, coupled to Drude's model for the description of the surface current. The time mirror is modeled by a sudden, strong, change in the Drude weight of the electrons on the sheet. Our goal is to characterize the time-reversed wave, in particular to quantify the quality of refocusing. We establish that the latter depends on the distance of the source to the sheet, and on some physical parameters such as the relaxation time of the electrons. We also show that, in addition to the plasmonic wave, the time mirror generates a free propagating wave that offers, contrary to the surface wave, some resolution in the direction orthogonal to the sheet. Blurring effects due to non-instantaneous mirrors are finally investigated.

Citation: Olivier Pinaud. Time reversal of surface plasmons. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022106
##### References:
 [1] 2D Materials: Properties and Devices. Cambridge University Press, 2017. [2] H. Ammari and E. O. Hiltunen, Time-dependent high-contrast subwavelength resonators, J. Comput. Phys., 445 (2021), Paper No. 110594, 18 pp. doi: 10.1016/j.jcp.2021.110594. [3] V. Bacot, M. Labousse, A. Eddi, M. Fink and E. Fort, Time reversal and holography with spacetime transformations, Nature Physics, 12 (2016), 972-977.  doi: 10.1038/nphys3810. [4] G. Bal, M. Fink and O. Pinaud, Time reversal by time-dependent perturbations, SIAM J. Appl. Math., 79 (2019), 754-780.  doi: 10.1137/18M1216894. [5] Yu. V. Bludov, A. Ferreira, N. M. R. Peres and M. I. Vasilevskiy, A primer on surface plasmons-polaritons in graphene, Internat. J. Modern Phys. B, 27 (2013), 1341001, 74 pp. doi: 10.1142/S0217979213410014. [6] L. Borcea, J. Garnier and K. Solna, Wave propagation and imaging in moving random media, Multiscale Model. Simul., 17 (2019), 31-67.  doi: 10.1137/18M119505X. [7] M. Fink, Time reversed acoustics, Physics Today, 50 (1997), 34-40. [8] J. Garnier, Wave propagation in periodic and random time-dependent media, Multiscale Model. Simul., 19 (2021), 1190-1211.  doi: 10.1137/20M1377734. [9] K. A. Lurie, An Introduction to the Mathematical Theory of Dynamic Materials, volume 15. Springer, 2007. [10] D. Margetis and M. Luskin, On solutions of maxwell's equations with dipole sources over a thin conducting film, J. Math. Phys., 57 (2016), 042903, 32 pp. doi: 10.1063/1.4945083. [11] P. A. Martin, Acoustics and dynamic materials, Mechanics Research Communications, 105 (2020), 103502.  doi: 10.1016/j.mechrescom.2020.103502. [12] G. W. Milton and O. Mattei, Field patterns: A new mathematical object, Proc. A., 473 (2017), 20160819, 26 pp. doi: 10.1098/rspa.2016.0819. [13] O. Pinaud, Instantaneous time mirrors and wave equations with time-singular coefficients, SIAM J. Math. Anal., 53 (2021), 4401-4416.  doi: 10.1137/20M1384361. [14] P. Reck, C. Gorini, A. Goussev, V. Krueckl, M. Fink and K. Richter, Dirac quantum time mirror, Phys. Rev. B, 95 (2017), 165421.  doi: 10.1103/PhysRevB.95.165421. [15] P. Reck, C. Gorini, A. Goussev, V. Krueckl, M. Fink and K. Richter, Towards a quantum time mirror for non-relativistic wave packets, New J. Phys., 20 (2018), 033013, 9 pp. doi: 10.1088/1367-2630/aaae98. [16] J. Wilson, F. Santosa and P. A. Martin, Temporally manipulated plasmons on graphene, SIAM J. Appl. Math., 79 (2019), 1051-1074.  doi: 10.1137/18M1226889. [17] J. Wilson, F. Santosa, M. Min and T. Low, Temporal control of graphene plasmons, Phys. Rev. B, 98 (2018), 081411.  doi: 10.1103/PhysRevB.98.081411.

show all references

##### References:
 [1] 2D Materials: Properties and Devices. Cambridge University Press, 2017. [2] H. Ammari and E. O. Hiltunen, Time-dependent high-contrast subwavelength resonators, J. Comput. Phys., 445 (2021), Paper No. 110594, 18 pp. doi: 10.1016/j.jcp.2021.110594. [3] V. Bacot, M. Labousse, A. Eddi, M. Fink and E. Fort, Time reversal and holography with spacetime transformations, Nature Physics, 12 (2016), 972-977.  doi: 10.1038/nphys3810. [4] G. Bal, M. Fink and O. Pinaud, Time reversal by time-dependent perturbations, SIAM J. Appl. Math., 79 (2019), 754-780.  doi: 10.1137/18M1216894. [5] Yu. V. Bludov, A. Ferreira, N. M. R. Peres and M. I. Vasilevskiy, A primer on surface plasmons-polaritons in graphene, Internat. J. Modern Phys. B, 27 (2013), 1341001, 74 pp. doi: 10.1142/S0217979213410014. [6] L. Borcea, J. Garnier and K. Solna, Wave propagation and imaging in moving random media, Multiscale Model. Simul., 17 (2019), 31-67.  doi: 10.1137/18M119505X. [7] M. Fink, Time reversed acoustics, Physics Today, 50 (1997), 34-40. [8] J. Garnier, Wave propagation in periodic and random time-dependent media, Multiscale Model. Simul., 19 (2021), 1190-1211.  doi: 10.1137/20M1377734. [9] K. A. Lurie, An Introduction to the Mathematical Theory of Dynamic Materials, volume 15. Springer, 2007. [10] D. Margetis and M. Luskin, On solutions of maxwell's equations with dipole sources over a thin conducting film, J. Math. Phys., 57 (2016), 042903, 32 pp. doi: 10.1063/1.4945083. [11] P. A. Martin, Acoustics and dynamic materials, Mechanics Research Communications, 105 (2020), 103502.  doi: 10.1016/j.mechrescom.2020.103502. [12] G. W. Milton and O. Mattei, Field patterns: A new mathematical object, Proc. A., 473 (2017), 20160819, 26 pp. doi: 10.1098/rspa.2016.0819. [13] O. Pinaud, Instantaneous time mirrors and wave equations with time-singular coefficients, SIAM J. Math. Anal., 53 (2021), 4401-4416.  doi: 10.1137/20M1384361. [14] P. Reck, C. Gorini, A. Goussev, V. Krueckl, M. Fink and K. Richter, Dirac quantum time mirror, Phys. Rev. B, 95 (2017), 165421.  doi: 10.1103/PhysRevB.95.165421. [15] P. Reck, C. Gorini, A. Goussev, V. Krueckl, M. Fink and K. Richter, Towards a quantum time mirror for non-relativistic wave packets, New J. Phys., 20 (2018), 033013, 9 pp. doi: 10.1088/1367-2630/aaae98. [16] J. Wilson, F. Santosa and P. A. Martin, Temporally manipulated plasmons on graphene, SIAM J. Appl. Math., 79 (2019), 1051-1074.  doi: 10.1137/18M1226889. [17] J. Wilson, F. Santosa, M. Min and T. Low, Temporal control of graphene plasmons, Phys. Rev. B, 98 (2018), 081411.  doi: 10.1103/PhysRevB.98.081411.
Left panel: Comparison exact/asymptotic expressions of $\mathcal J_P$. The lines represent the exact expression, and the stars the approximate one. Right panel: representation of $\mathcal J_P$ for several $\zeta$. Observe the peaked behavior around $r = 0$ for small $\zeta$
Absolute value of $\mathcal J_P$ (normalized to one). Left panel: $cT = 5z_0$, $z_0 = 10\ell_0$. Right panel: $cT = 15z_0$, $z_0 = 10\ell_0$. Observe the loss of resolution with the increased $T$
Comparison exact/asymptotic expressions for the complex root $s_+$
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