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Problem geometry of the time-domain scattering by a biperiodic structure
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2020, 25(1): 259-286.
doi: 10.3934/dcdsb.2019181
Analysis of time-domain Maxwell's equations in biperiodic structures
| 1. | School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China |
| 2. | Department of Mathematics, Purdue University, West Lafayette, IN47907, USA |
| 3. | School of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China |
This paper is devoted to the mathematical analysis of the diffraction of an electromagnetic plane wave by a biperiodic structure. The wave propagation is governed by the time-domain Maxwell equations in three dimensions. The method of a compressed coordinate transformation is proposed to reduce equivalently the diffraction problem into an initial-boundary value problem formulated in a bounded domain over a finite time interval. The reduced problem is shown to have a unique weak solution by using the constructive Galerkin method. The stability and a priori estimates with explicit time dependence are established for the weak solution.
Keywords:
Time-domain Maxwell's equations,
biperiodic structures,
diffraction gratings,
well-posedness and stability,
a priori estimates.
Mathematics Subject Classification:
78A45, 35A15, 35Q60.
Citation:
Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete and Continuous Dynamical Systems - B,
2020, 25
(1)
: 259-286.
doi: 10.3934/dcdsb.2019181
![]()
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Nonreflecting boundary conditions for the time-dependent wave equation, J. Comput. Phys., 180 (2002), 270-296.
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Maxwell's equations in periodic chiral structures, Math.Nachr., 251 (2003), 3-18.
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Numerical analysis of diffraction by periodic structures: TM polarization, Numer. Math., 75 (1996), 1-16.
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Variational approximation of Maxwell's equations in biperiodic structures, SIAM J. Appl. Math., 57 (1997), 364-381.
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G. Bao, D. Dobson and J. A. Cox,
Mathematical studies in rigorous grating theory, J. Opt. Soc. Am. A, 12 (1995), 1029-1042.
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G. Bao, Z. Chen and and H. Wu,
Adaptive finite-element method for diffraction gratings, J. Opt. Soc. Amer. A, 22 (2005), 1106-1114.
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G. Bao, L. Cowsar and W. Masters, Eds., Mathematical Modeling in Optical Science, Frontiers in Applied Mathematics, vol. 22, SIAM, Philadelphia, PA, 2001.
doi: 10.1137/1.9780898717594. |
| [10] |
G. Bao, P. Li and H. Wu,
An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures, Math. Comp., 79 (2010), 1-34.
doi: 10.1090/S0025-5718-09-02257-1. |
| [11] |
G. Bao and H. Yang,
A least-squares finite element analysis for diffraction problems, SIAM J. Numer. Anal., 37 (2000), 665-682.
doi: 10.1137/s0036142998342380. |
| [12] |
G. Bao, T. Cui and and P. Li,
Inverse diffraction grating of Maxwell's equations in biperiodic structures, Opt. Express, 22 (2014), 4799-4816.
doi: 10.1364/OE.22.004799. |
| [13] |
G. Bao and D. Dobson,
On the scattering by a biperiodic structure, Proc. Amer. Math. Soc., 128 (2000), 2715-2723.
doi: 10.1090/S0002-9939-00-05509-X. |
| [14] |
G. Bao and A. Friedman,
Inverse problems for scattering by periodic structure, Arch. Rational Mech. Anal., 132 (1995), 49-72.
doi: 10.1007/BF00390349. |
| [15] |
G. Bao, Y. Gao and P. Li,
Time-domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal., 292 (2018), 835-884.
doi: 10.1007/s00205-018-1228-2. |
| [16] |
Q. Chen and P. Monk,
Discretization of the time domain CFIE for acoustic scattering problems using convolution quadrature, SIAM J. Math. Anal., 46 (2014), 3107-3130.
doi: 10.1137/110833555. |
| [17] |
X. Chen and A. Friedman,
Maxwell's equations in a periodic structure, Trans. Amer. Math. Soc., 323 (1991), 465-507.
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| [18] |
Z. Chen and J.-C. Nédélec,
On Maxwell equations with the transparent boundary condition, J. Comput. Math., 26 (2008), 284-296.
|
| [19] |
Z. Chen and H. Wu,
An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 41 (2003), 799-826.
doi: 10.1137/S0036142902400901. |
| [20] |
D. Dobson,
A variational method for electromagnetic diffraction in biperiodic structures, Math. Modelling Numer. Anal., 28 (1994), 419-439.
doi: 10.1051/m2an/1994280404191. |
| [21] |
D. Dobson and A. Friedman,
The time-harmonic Maxwell equations in a doubly periodic structure, J. Math. Anal. Appl., 166 (1992), 507-528.
doi: 10.1016/0022-247X(92)90312-2. |
| [22] |
B. Engquist and A. Majda,
Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.
doi: 10.1090/S0025-5718-1977-0436612-4. |
| [23] |
L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, vol. 19, Graduate Studies in Mathematics, AMS, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
| [24] |
L. Fan and P. Monk,
Time dependent scattering from a grating, J. Comput. Phys., 302 (2015), 97-113.
doi: 10.1016/j.jcp.2015.07.067. |
| [25] |
Y. Gao and P. Li,
Analysis of time-domain scattering by periodic structures, J. Differential Equations, 261 (2016), 5094-5118.
doi: 10.1016/j.jde.2016.07.020. |
| [26] |
Y. Gao and P. Li,
Electromagnetic scattering for time-domain Maxwell's equations in an unbounded structure, Math. Models Methods Appl. Sci., 27 (2017), 1843-1870.
doi: 10.1142/S0218202517500336. |
| [27] |
Y. Gao, P. Li and Y. Li,
Analysis of time-domain elastic scattering by an unbounded structure, Math. Meth. Appl. Sci., 41 (2018), 7032-7054.
doi: 10.1002/mma.5214. |
| [28] |
Y. Gao, P. Li and B. Zhang,
Analysis of transient acoustic-elastic interaction in an unbounded structure, SIAM J. Math. Anal., 49 (2017), 3951-3972.
doi: 10.1137/16M1090326. |
| [29] |
M. J. Grote and J. B. Keller,
Exact nonreflecting boundary conditions for the time dependent wave equation, SIAM J. Appl. Math., 55 (1995), 280-297.
doi: 10.1137/S0036139993269266. |
| [30] |
T. Hagstrom,
Radiation boundary conditions for the numerical simulation of waves, Acta Numer., 8 (1999), 47-106.
doi: 10.1017/s0962492900002890. |
| [31] |
X. Jiang and P. Li, Inverse electromagnetic diffraction by biperiodic dielectric gratings, Inverse Probl., 33 (2017), 085004, 29pp.
doi: 10.1088/1361-6420/aa76b9. |
| [32] |
A. Lechleiter and D. L. Nguyen.,
On uniqueness in electromagnetic scattering from biperiodic structures, ESAIM: M2AN, 47 (2013), 1167-1184.
doi: 10.1051/m2an/2012063. |
| [33] |
P. Li, L.-L. Wang and A. Wood,
Analysis of transient electromagentic scattering from a three-dimensional open cavity, SIAM J. Appl. Math., 75 (2015), 1675-1699.
doi: 10.1137/140989637. |
| [34] |
J.-C. Nedelec and F. Starling,
Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equations, SIAM J. Math. Anal., 22 (1991), 1679-1701.
doi: 10.1137/0522104. |
| [35] |
R. Petit, ed., Electromagnetic Theory of Gratings, Springer, 1980.
doi: 10.1007/978-3-642-81500-3. |
| [36] |
P. Rayleigh,
On the dynamical theory of gratings, R. Soc. London Ser. A, 79 (1907), 399-416.
|
| [37] |
D. J. Riley and J.-M. Jin,
Finite-element time-domain analysis of electrically and magnetically dispersive periodic structures, IEEE Trans. Antennas and Propagation, 56 (2008), 3501-3509.
doi: 10.1109/TAP.2008.2005454. |
| [38] |
M. Veysoglu, R. Shin and J. A. Kong,
A finite-difference time-domain analysis of wave scattering from periodic surfaces: oblique incidence case, J. Electromagn. Waves Appl., 7 (1993), 1595-1607.
doi: 10.1163/156939393X00020. |
| [39] |
B. Wang and L.-L. Wang,
On L$^2$-stability analysis of time-domain acoustic scattering problems with exact nonreflecting boundary conditions, J. Math. Study, 47 (2014), 65-84.
doi: 10.4208/jms.v47n1.14.04. |
| [40] |
L.-L. Wang, B. Wang and X. Zhao,
Fast and accurate computation of time-domain acoustic scattering problems with exact nonreflecting boundary conditions, SIAM J. Appl. Math., 72 (2012), 1869-1898.
doi: 10.1137/110849146. |
| [41] |
Z. Wang, G. Bao, J. Li, P. Li and H. Wu,
An adaptive finite element method for the diffraction grating problem with transparent boundary conditions, SIAM J. Numer. Anal., 53 (2015), 1585-1607.
doi: 10.1137/140969907. |
| [42] |
Y. Wu and Y. Y. Lu,
Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method, J. Opt. Soc. Am. A, 26 (2009), 2444-2451.
doi: 10.1364/JOSAA.26.002444. |
show all references
References:
| [1] |
B. Alpert, L. Greengard and T. Hagstrom,
Nonreflecting boundary conditions for the time-dependent wave equation, J. Comput. Phys., 180 (2002), 270-296.
doi: 10.1006/jcph.2002.7093. |
| [2] |
H. Ammari,
Uniqueness theorems for an inverse problem in a doubly periodic structure, Inverse Problems, 11 (1995), 823-833.
doi: 10.1088/0266-5611/11/4/013. |
| [3] |
H. Ammari and G. Bao,
Maxwell's equations in periodic chiral structures, Math.Nachr., 251 (2003), 3-18.
doi: 10.1002/mana.200310026. |
| [4] |
G. Bao,
Finite element approximation of time harmonic waves in periodic structures, SIAM J. Numer. Anal., 32 (1995), 1155-1169.
doi: 10.1137/0732053. |
| [5] |
G. Bao,
Numerical analysis of diffraction by periodic structures: TM polarization, Numer. Math., 75 (1996), 1-16.
doi: 10.1007/s002110050227. |
| [6] |
G. Bao,
Variational approximation of Maxwell's equations in biperiodic structures, SIAM J. Appl. Math., 57 (1997), 364-381.
doi: 10.1137/S0036139995279408. |
| [7] |
G. Bao, D. Dobson and J. A. Cox,
Mathematical studies in rigorous grating theory, J. Opt. Soc. Am. A, 12 (1995), 1029-1042.
doi: 10.1364/JOSAA.12.001029. |
| [8] |
G. Bao, Z. Chen and and H. Wu,
Adaptive finite-element method for diffraction gratings, J. Opt. Soc. Amer. A, 22 (2005), 1106-1114.
doi: 10.1364/JOSAA.22.001106. |
| [9] |
G. Bao, L. Cowsar and W. Masters, Eds., Mathematical Modeling in Optical Science, Frontiers in Applied Mathematics, vol. 22, SIAM, Philadelphia, PA, 2001.
doi: 10.1137/1.9780898717594. |
| [10] |
G. Bao, P. Li and H. Wu,
An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures, Math. Comp., 79 (2010), 1-34.
doi: 10.1090/S0025-5718-09-02257-1. |
| [11] |
G. Bao and H. Yang,
A least-squares finite element analysis for diffraction problems, SIAM J. Numer. Anal., 37 (2000), 665-682.
doi: 10.1137/s0036142998342380. |
| [12] |
G. Bao, T. Cui and and P. Li,
Inverse diffraction grating of Maxwell's equations in biperiodic structures, Opt. Express, 22 (2014), 4799-4816.
doi: 10.1364/OE.22.004799. |
| [13] |
G. Bao and D. Dobson,
On the scattering by a biperiodic structure, Proc. Amer. Math. Soc., 128 (2000), 2715-2723.
doi: 10.1090/S0002-9939-00-05509-X. |
| [14] |
G. Bao and A. Friedman,
Inverse problems for scattering by periodic structure, Arch. Rational Mech. Anal., 132 (1995), 49-72.
doi: 10.1007/BF00390349. |
| [15] |
G. Bao, Y. Gao and P. Li,
Time-domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal., 292 (2018), 835-884.
doi: 10.1007/s00205-018-1228-2. |
| [16] |
Q. Chen and P. Monk,
Discretization of the time domain CFIE for acoustic scattering problems using convolution quadrature, SIAM J. Math. Anal., 46 (2014), 3107-3130.
doi: 10.1137/110833555. |
| [17] |
X. Chen and A. Friedman,
Maxwell's equations in a periodic structure, Trans. Amer. Math. Soc., 323 (1991), 465-507.
doi: 10.2307/2001542. |
| [18] |
Z. Chen and J.-C. Nédélec,
On Maxwell equations with the transparent boundary condition, J. Comput. Math., 26 (2008), 284-296.
|
| [19] |
Z. Chen and H. Wu,
An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 41 (2003), 799-826.
doi: 10.1137/S0036142902400901. |
| [20] |
D. Dobson,
A variational method for electromagnetic diffraction in biperiodic structures, Math. Modelling Numer. Anal., 28 (1994), 419-439.
doi: 10.1051/m2an/1994280404191. |
| [21] |
D. Dobson and A. Friedman,
The time-harmonic Maxwell equations in a doubly periodic structure, J. Math. Anal. Appl., 166 (1992), 507-528.
doi: 10.1016/0022-247X(92)90312-2. |
| [22] |
B. Engquist and A. Majda,
Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.
doi: 10.1090/S0025-5718-1977-0436612-4. |
| [23] |
L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, vol. 19, Graduate Studies in Mathematics, AMS, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
| [24] |
L. Fan and P. Monk,
Time dependent scattering from a grating, J. Comput. Phys., 302 (2015), 97-113.
doi: 10.1016/j.jcp.2015.07.067. |
| [25] |
Y. Gao and P. Li,
Analysis of time-domain scattering by periodic structures, J. Differential Equations, 261 (2016), 5094-5118.
doi: 10.1016/j.jde.2016.07.020. |
| [26] |
Y. Gao and P. Li,
Electromagnetic scattering for time-domain Maxwell's equations in an unbounded structure, Math. Models Methods Appl. Sci., 27 (2017), 1843-1870.
doi: 10.1142/S0218202517500336. |
| [27] |
Y. Gao, P. Li and Y. Li,
Analysis of time-domain elastic scattering by an unbounded structure, Math. Meth. Appl. Sci., 41 (2018), 7032-7054.
doi: 10.1002/mma.5214. |
| [28] |
Y. Gao, P. Li and B. Zhang,
Analysis of transient acoustic-elastic interaction in an unbounded structure, SIAM J. Math. Anal., 49 (2017), 3951-3972.
doi: 10.1137/16M1090326. |
| [29] |
M. J. Grote and J. B. Keller,
Exact nonreflecting boundary conditions for the time dependent wave equation, SIAM J. Appl. Math., 55 (1995), 280-297.
doi: 10.1137/S0036139993269266. |
| [30] |
T. Hagstrom,
Radiation boundary conditions for the numerical simulation of waves, Acta Numer., 8 (1999), 47-106.
doi: 10.1017/s0962492900002890. |
| [31] |
X. Jiang and P. Li, Inverse electromagnetic diffraction by biperiodic dielectric gratings, Inverse Probl., 33 (2017), 085004, 29pp.
doi: 10.1088/1361-6420/aa76b9. |
| [32] |
A. Lechleiter and D. L. Nguyen.,
On uniqueness in electromagnetic scattering from biperiodic structures, ESAIM: M2AN, 47 (2013), 1167-1184.
doi: 10.1051/m2an/2012063. |
| [33] |
P. Li, L.-L. Wang and A. Wood,
Analysis of transient electromagentic scattering from a three-dimensional open cavity, SIAM J. Appl. Math., 75 (2015), 1675-1699.
doi: 10.1137/140989637. |
| [34] |
J.-C. Nedelec and F. Starling,
Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equations, SIAM J. Math. Anal., 22 (1991), 1679-1701.
doi: 10.1137/0522104. |
| [35] |
R. Petit, ed., Electromagnetic Theory of Gratings, Springer, 1980.
doi: 10.1007/978-3-642-81500-3. |
| [36] |
P. Rayleigh,
On the dynamical theory of gratings, R. Soc. London Ser. A, 79 (1907), 399-416.
|
| [37] |
D. J. Riley and J.-M. Jin,
Finite-element time-domain analysis of electrically and magnetically dispersive periodic structures, IEEE Trans. Antennas and Propagation, 56 (2008), 3501-3509.
doi: 10.1109/TAP.2008.2005454. |
| [38] |
M. Veysoglu, R. Shin and J. A. Kong,
A finite-difference time-domain analysis of wave scattering from periodic surfaces: oblique incidence case, J. Electromagn. Waves Appl., 7 (1993), 1595-1607.
doi: 10.1163/156939393X00020. |
| [39] |
B. Wang and L.-L. Wang,
On L$^2$-stability analysis of time-domain acoustic scattering problems with exact nonreflecting boundary conditions, J. Math. Study, 47 (2014), 65-84.
doi: 10.4208/jms.v47n1.14.04. |
| [40] |
L.-L. Wang, B. Wang and X. Zhao,
Fast and accurate computation of time-domain acoustic scattering problems with exact nonreflecting boundary conditions, SIAM J. Appl. Math., 72 (2012), 1869-1898.
doi: 10.1137/110849146. |
| [41] |
Z. Wang, G. Bao, J. Li, P. Li and H. Wu,
An adaptive finite element method for the diffraction grating problem with transparent boundary conditions, SIAM J. Numer. Anal., 53 (2015), 1585-1607.
doi: 10.1137/140969907. |
| [42] |
Y. Wu and Y. Y. Lu,
Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method, J. Opt. Soc. Am. A, 26 (2009), 2444-2451.
doi: 10.1364/JOSAA.26.002444. |
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