doi: 10.3934/eect.2022020
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Stability properties for a problem of light scattering in a dispersive metallic domain

1. 

Université polytechnique Hauts-de-France, LAMAV, FR CNRS 2037, F-59313 - Valenciennes Cedex 9, France

2. 

Université Côte d'Azur, LJAD, CNRS, INRIA, 06108 Nice, France

Received  September 2021 Revised  January 2022 Early access April 2022

In this work, we study the well-posedness and some stability properties of a PDE system that models the propagation of light in a metallic domain with a hole. This model takes into account the dispersive properties of the metal. It consists of a linear coupling between Maxwell's equations and a wave type system. We prove that the problem is well posed for several types of boundary conditions. Furthermore, we show that it is polynomially stable and that the exponential stability is conditional on the exponential stability of the Maxwell system.

Citation: Serge Nicaise, Claire Scheid. Stability properties for a problem of light scattering in a dispersive metallic domain. Evolution Equations and Control Theory, doi: 10.3934/eect.2022020
References:
[1]

L. AlouiS. Ibrahim and M. Khenissi, Energy decay for linear dissipative wave equations in exterior domains, J. Differential Equations, 259 (2015), 2061-2079.  doi: 10.1016/j.jde.2015.03.018.

[2]

C. AmroucheC. BernardiM. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci., 21 (1998), 823-864.  doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.

[3]

H. Barucq and B. Hanouzet, Étude asymptotique du système de Maxwell avec la condition aux limites absorbante de Silver-Müller II, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1019–1024.

[4]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.

[5]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[6]

A. Boardman, Electromagnetic Surface Modes, John Wiley & Sons, 1972.

[7]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[9]

A. BuffaM. Costabel and D. Sheen, On traces for H(curl, $\Omega$) in Lipschitz domains, J. Math. Anal. Appl., 276 (2002), 845-867.  doi: 10.1016/S0022-247X(02)00455-9.

[10]

C. Carle, Numerische Verfahren für Plasmonische Nanostrukturen (in German), Master's thesis, Karlsruher Institut für Technologie, 2017.

[11]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.

[12]

M. Costabel, M. Dauge and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains, 2010.

[13]

M. Daoulatli, Energy decay rates for solutions of the wave equation with linear damping in exterior domain, Evol. Equ. Control Theory, 5 (2016), 37-59.  doi: 10.3934/eect.2016.5.37.

[14]

M. Eller, J. E. Lagnese and S. Nicaise, Stabilization of heterogeneous Maxwell's equations by linear or nonlinear boundary feedback, Electron. J. Differential Equations, (2002), No. 21, 26 pp.

[15]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, vol. 5 of Springer Series in Computational Mathematics, Springer, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[16]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[17]

Y. HuangJ. Li and W. Yang, Theoretical and numerical analysis of a non-local dispersion model for light interaction with metallic nanostructures, Comput. Math. Appl., 72 (2016), 921-932.  doi: 10.1016/j.camwa.2016.06.003.

[18]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[19] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford Univ. Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[20]

S. Nicaise, Stabilization of a Drude/vacuum model, Z. Anal. Anwend., 37 (2018), 349-375.  doi: 10.4171/ZAA/1618.

[21]

S. Nicaise and C. Scheid, Stability and asymptotic properties of a linearized hydrodynamic medium model for dispersive media in nanophotonics, Comput. Math. Appl., 79 (2020), 3462-3494.  doi: 10.1016/j.camwa.2020.02.006.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5 (2000), 87-137.  doi: 10.1051/cocv:2000103.

[24]

A. PiteletN. SchmittD. LoukresisC. ScheidH. D. GersemC. CiraciE. Centeno and A. Moreau, Influence of spatial dispersion on surface plasmons and grating couplers, J. Optical Society of America A, 36 (2019), 2989-2999.  doi: 10.1364/JOSAB.36.002989.

[25]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[26]

N. Schmitt, High-Order Simulation and Calibration Strategies for Spatially Dispersive Metals in Nanophotonics, PhD thesis, Côte d'Azur University, 2018.

[27]

Ch. Weber, A local compactness theorem for Maxwell's equations, Math. Meth. Appl. Sci., 2 (1980), 12-25.  doi: 10.1002/mma.1670020103.

show all references

References:
[1]

L. AlouiS. Ibrahim and M. Khenissi, Energy decay for linear dissipative wave equations in exterior domains, J. Differential Equations, 259 (2015), 2061-2079.  doi: 10.1016/j.jde.2015.03.018.

[2]

C. AmroucheC. BernardiM. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci., 21 (1998), 823-864.  doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.

[3]

H. Barucq and B. Hanouzet, Étude asymptotique du système de Maxwell avec la condition aux limites absorbante de Silver-Müller II, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1019–1024.

[4]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.

[5]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[6]

A. Boardman, Electromagnetic Surface Modes, John Wiley & Sons, 1972.

[7]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[9]

A. BuffaM. Costabel and D. Sheen, On traces for H(curl, $\Omega$) in Lipschitz domains, J. Math. Anal. Appl., 276 (2002), 845-867.  doi: 10.1016/S0022-247X(02)00455-9.

[10]

C. Carle, Numerische Verfahren für Plasmonische Nanostrukturen (in German), Master's thesis, Karlsruher Institut für Technologie, 2017.

[11]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.

[12]

M. Costabel, M. Dauge and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains, 2010.

[13]

M. Daoulatli, Energy decay rates for solutions of the wave equation with linear damping in exterior domain, Evol. Equ. Control Theory, 5 (2016), 37-59.  doi: 10.3934/eect.2016.5.37.

[14]

M. Eller, J. E. Lagnese and S. Nicaise, Stabilization of heterogeneous Maxwell's equations by linear or nonlinear boundary feedback, Electron. J. Differential Equations, (2002), No. 21, 26 pp.

[15]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, vol. 5 of Springer Series in Computational Mathematics, Springer, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[16]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[17]

Y. HuangJ. Li and W. Yang, Theoretical and numerical analysis of a non-local dispersion model for light interaction with metallic nanostructures, Comput. Math. Appl., 72 (2016), 921-932.  doi: 10.1016/j.camwa.2016.06.003.

[18]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[19] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford Univ. Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[20]

S. Nicaise, Stabilization of a Drude/vacuum model, Z. Anal. Anwend., 37 (2018), 349-375.  doi: 10.4171/ZAA/1618.

[21]

S. Nicaise and C. Scheid, Stability and asymptotic properties of a linearized hydrodynamic medium model for dispersive media in nanophotonics, Comput. Math. Appl., 79 (2020), 3462-3494.  doi: 10.1016/j.camwa.2020.02.006.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Math. Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5 (2000), 87-137.  doi: 10.1051/cocv:2000103.

[24]

A. PiteletN. SchmittD. LoukresisC. ScheidH. D. GersemC. CiraciE. Centeno and A. Moreau, Influence of spatial dispersion on surface plasmons and grating couplers, J. Optical Society of America A, 36 (2019), 2989-2999.  doi: 10.1364/JOSAB.36.002989.

[25]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[26]

N. Schmitt, High-Order Simulation and Calibration Strategies for Spatially Dispersive Metals in Nanophotonics, PhD thesis, Côte d'Azur University, 2018.

[27]

Ch. Weber, A local compactness theorem for Maxwell's equations, Math. Meth. Appl. Sci., 2 (1980), 12-25.  doi: 10.1002/mma.1670020103.

Figure 1.  Type of considered domain
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