January  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

* Corresponding author: Gang Bao

Received  January 2019 Revised  March 2019 Published  January 2020 Early access  July 2019

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.

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
References:
[1]

B. AlpertL. 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. BaoD. 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. BaoZ. 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. BaoP. 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. BaoT. 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. BaoY. 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. GaoP. 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. GaoP. 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. LiL.-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. VeysogluR. 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. WangB. 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. WangG. BaoJ. LiP. 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. AlpertL. 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. BaoD. 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. BaoZ. 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. BaoP. 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. BaoT. 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. BaoY. 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. GaoP. 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. GaoP. 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. LiL.-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. VeysogluR. 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. WangB. 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. WangG. BaoJ. LiP. 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.

Figure 1.  Problem geometry of the time-domain scattering by a biperiodic structure
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