• Previous Article
    Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations
  • DCDS-S Home
  • This Issue
  • Next Article
    Radar cross section reduction of a cavity in the ground plane: TE polarization
June  2015, 8(3): 435-473. doi: 10.3934/dcdss.2015.8.435

Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures

1. 

Berlin Mathematical School, Technical University Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

2. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany

Received  November 2013 Revised  March 2014 Published  October 2014

This paper is concerned with the study of a new integral equation formulation for electromagnetic scattering by a $2\pi$-biperiodic polyhedral Lipschitz profile. Using a combined potential ansatz, we derive a singular integral equation with Fredholm operator of index zero from time-harmonic Maxwell's equations and prove its equivalence to the electromagnetic scattering problem. Moreover, under certain assumptions on the electric permittivity and the magnetic permeability, we obtain existence and uniqueness results in the special case that the grating is smooth and, under more restrictive assumptions, in the case that the grating is of polyhedral Lipschitz regularity.
Citation: Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435
References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (1995). Google Scholar

[2]

H. Ammari, Uniqueness theorems for an inverse problem in a doubly periodic structure,, Inverse Problems, 11 (1995), 823. doi: 10.1088/0266-5611/11/4/013. Google Scholar

[3]

T. Arens, Scattering by Biperiodic Layered Media: The Integral Equation Approach,, habilitation thesis, (2010). Google Scholar

[4]

G. Bao and D. C. Dobson, On the scattering by a biperiodic structure,, Proc. AMS, 128 (2000), 2715. doi: 10.1090/S0002-9939-00-05509-X. Google Scholar

[5]

A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations I. An integration by parts formula in Lipschitz polyhedra,, Math. Methods Appl. Sci., 24 (2001), 9. doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2. Google Scholar

[6]

A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications,, Math. Methods Appl. Sci., 24 (2001), 31. doi: 10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X. Google Scholar

[7]

A. Buffa, M. Costabel and C. Schwab, Boundary element methods for Maxwell's equations on non-smooth domains,, Numer. Math., 92 (2002), 679. doi: 10.1007/s002110100372. Google Scholar

[8]

A. Buffa, M. Costabel and D. Sheen, On traces for H(curl, Ω) in Lipschitz domains,, J. Math. Anal. Appl., 276 (2002), 845. doi: 10.1016/S0022-247X(02)00455-9. Google Scholar

[9]

A. Buffa, R. Hiptmair, T. von Petersdorff and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains,, Numer. Math., 95 (2003), 459. doi: 10.1007/s00211-002-0407-z. Google Scholar

[10]

B. Bugert, On Integral Equation Methods for Electromagnetic Scattering by Biperiodic Structures,, PhD thesis, (2014). Google Scholar

[11]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications,, SIAM, (2013). Google Scholar

[12]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results,, SIAM J. Math. Anal., 19 (1988), 613. doi: 10.1137/0519043. Google Scholar

[13]

M. Costabel and F. Le Louër, On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body,, SIAM J. Appl. Math., 71 (2011), 635. doi: 10.1137/090779462. Google Scholar

[14]

D. C. Dobson and A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure,, Anal. Appl., 166 (1992), 507. doi: 10.1016/0022-247X(92)90312-2. Google Scholar

[15]

D. C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures,, Math. Anal. Numer., 28 (1994), 419. Google Scholar

[16]

J. Elschner, R. Hinder, F. Penzel and G. Schmidt, Existence, uniqueness and regularity for solutions of the conical diffraction problem,, Math. Models Methods Appl. Sci., 10 (2000), 317. doi: 10.1142/S0218202500000197. Google Scholar

[17]

V. Yu. Gotlib, Solutions of the Helmholtz equation, concentrated near a plane periodic boundary,, J. Math. Sci., 102 (2000), 4188. doi: 10.1007/BF02673850. Google Scholar

[18]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations,, Springer, (2008). doi: 10.1007/978-3-540-68545-6. Google Scholar

[19]

G. Hu and A. Rathsfeld, Scattering of time-harmonic electromagnetic plane waves by perfectly conducting diffraction gratings,, IMA Journal of Applied Mathematics, (2014), 1. doi: 10.1093/imamat/hxt054. Google Scholar

[20]

I. V. Kamotski and S. A. Nazarov, The augmented scattering matrix and exponentially decaying solutions of an elliptic problem in a cylindrical domain,, Journal of Mathematical Sciences, 111 (1988), 3657. doi: 10.1023/A:1016377707919. Google Scholar

[21]

R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics,, SIAM J. Appl. Math., 48 (1988), 307. doi: 10.1137/0148016. Google Scholar

[22]

A. Lechleiter and D.-L. Nguyen, On uniqueness in electromagnetic scattering from biperiodic structures,, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 1167. doi: 10.1051/m2an/2012063. Google Scholar

[23]

D. Maystre, Integral methods,, in Electromagnetic Theory of Gratings, (1980), 63. Google Scholar

[24]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar

[25]

J.-C. Nedelec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equations,, SIAM J. Appl. Math., 22 (1991), 1679. doi: 10.1137/0522104. Google Scholar

[26]

G. Schmidt, Boundary integral methods for periodic scattering problems,, in Around the Research of Vladimir Maz'ya II, 12 (2010), 337. doi: 10.1007/978-1-4419-1343-2_16. Google Scholar

[27]

G. Schmidt, Conical diffraction by multilayer gratings: A recursive integral equations approach,, Applications of Mathematics, 58 (2013), 279. doi: 10.1007/s10492-013-0014-6. Google Scholar

[28]

O. Steinbach and M. Windisch, Modified combined field integral equations for electromagnetic scattering,, SIAM J. Numer. Anal., 47 (2009), 1149. doi: 10.1137/070698063. Google Scholar

[29]

K. Yosida, Functional Analysis,, Springer, (1980). Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Academic Press, (1995). Google Scholar

[2]

H. Ammari, Uniqueness theorems for an inverse problem in a doubly periodic structure,, Inverse Problems, 11 (1995), 823. doi: 10.1088/0266-5611/11/4/013. Google Scholar

[3]

T. Arens, Scattering by Biperiodic Layered Media: The Integral Equation Approach,, habilitation thesis, (2010). Google Scholar

[4]

G. Bao and D. C. Dobson, On the scattering by a biperiodic structure,, Proc. AMS, 128 (2000), 2715. doi: 10.1090/S0002-9939-00-05509-X. Google Scholar

[5]

A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations I. An integration by parts formula in Lipschitz polyhedra,, Math. Methods Appl. Sci., 24 (2001), 9. doi: 10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2. Google Scholar

[6]

A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell's equations II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications,, Math. Methods Appl. Sci., 24 (2001), 31. doi: 10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X. Google Scholar

[7]

A. Buffa, M. Costabel and C. Schwab, Boundary element methods for Maxwell's equations on non-smooth domains,, Numer. Math., 92 (2002), 679. doi: 10.1007/s002110100372. Google Scholar

[8]

A. Buffa, M. Costabel and D. Sheen, On traces for H(curl, Ω) in Lipschitz domains,, J. Math. Anal. Appl., 276 (2002), 845. doi: 10.1016/S0022-247X(02)00455-9. Google Scholar

[9]

A. Buffa, R. Hiptmair, T. von Petersdorff and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains,, Numer. Math., 95 (2003), 459. doi: 10.1007/s00211-002-0407-z. Google Scholar

[10]

B. Bugert, On Integral Equation Methods for Electromagnetic Scattering by Biperiodic Structures,, PhD thesis, (2014). Google Scholar

[11]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications,, SIAM, (2013). Google Scholar

[12]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results,, SIAM J. Math. Anal., 19 (1988), 613. doi: 10.1137/0519043. Google Scholar

[13]

M. Costabel and F. Le Louër, On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body,, SIAM J. Appl. Math., 71 (2011), 635. doi: 10.1137/090779462. Google Scholar

[14]

D. C. Dobson and A. Friedman, The time-harmonic Maxwell equations in a doubly periodic structure,, Anal. Appl., 166 (1992), 507. doi: 10.1016/0022-247X(92)90312-2. Google Scholar

[15]

D. C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures,, Math. Anal. Numer., 28 (1994), 419. Google Scholar

[16]

J. Elschner, R. Hinder, F. Penzel and G. Schmidt, Existence, uniqueness and regularity for solutions of the conical diffraction problem,, Math. Models Methods Appl. Sci., 10 (2000), 317. doi: 10.1142/S0218202500000197. Google Scholar

[17]

V. Yu. Gotlib, Solutions of the Helmholtz equation, concentrated near a plane periodic boundary,, J. Math. Sci., 102 (2000), 4188. doi: 10.1007/BF02673850. Google Scholar

[18]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations,, Springer, (2008). doi: 10.1007/978-3-540-68545-6. Google Scholar

[19]

G. Hu and A. Rathsfeld, Scattering of time-harmonic electromagnetic plane waves by perfectly conducting diffraction gratings,, IMA Journal of Applied Mathematics, (2014), 1. doi: 10.1093/imamat/hxt054. Google Scholar

[20]

I. V. Kamotski and S. A. Nazarov, The augmented scattering matrix and exponentially decaying solutions of an elliptic problem in a cylindrical domain,, Journal of Mathematical Sciences, 111 (1988), 3657. doi: 10.1023/A:1016377707919. Google Scholar

[21]

R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics,, SIAM J. Appl. Math., 48 (1988), 307. doi: 10.1137/0148016. Google Scholar

[22]

A. Lechleiter and D.-L. Nguyen, On uniqueness in electromagnetic scattering from biperiodic structures,, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 1167. doi: 10.1051/m2an/2012063. Google Scholar

[23]

D. Maystre, Integral methods,, in Electromagnetic Theory of Gratings, (1980), 63. Google Scholar

[24]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000). Google Scholar

[25]

J.-C. Nedelec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equations,, SIAM J. Appl. Math., 22 (1991), 1679. doi: 10.1137/0522104. Google Scholar

[26]

G. Schmidt, Boundary integral methods for periodic scattering problems,, in Around the Research of Vladimir Maz'ya II, 12 (2010), 337. doi: 10.1007/978-1-4419-1343-2_16. Google Scholar

[27]

G. Schmidt, Conical diffraction by multilayer gratings: A recursive integral equations approach,, Applications of Mathematics, 58 (2013), 279. doi: 10.1007/s10492-013-0014-6. Google Scholar

[28]

O. Steinbach and M. Windisch, Modified combined field integral equations for electromagnetic scattering,, SIAM J. Numer. Anal., 47 (2009), 1149. doi: 10.1137/070698063. Google Scholar

[29]

K. Yosida, Functional Analysis,, Springer, (1980). Google Scholar

[1]

Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2019181

[2]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355

[3]

Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.

[4]

Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1881-1905. doi: 10.3934/cpaa.2013.12.1881

[5]

Paola Loreti, Daniela Sforza. Inverse observability inequalities for integrodifferential equations in square domains. Evolution Equations & Control Theory, 2018, 7 (1) : 61-77. doi: 10.3934/eect.2018004

[6]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159

[7]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164

[8]

Kunquan Lan, Wei Lin. Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1943-1960. doi: 10.3934/dcdsb.2018256

[9]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

[10]

Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006

[11]

Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547

[12]

Xiaotao Huang, Lihe Wang. Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054

[13]

Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229

[14]

Saroj Panigrahi. Liapunov-type integral inequalities for higher order dynamic equations on time scales. Conference Publications, 2013, 2013 (special) : 629-641. doi: 10.3934/proc.2013.2013.629

[15]

Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155

[16]

Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301

[17]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613

[18]

Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems & Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229

[19]

José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327

[20]

Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]