American Institute of Mathematical Sciences

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January  2020, 25(1): 259-286. doi: 10.3934/dcdsb.2019181

Analysis of time-domain Maxwell's equations in biperiodic structures

Gang Bao 1,, , Bin Hu 1, , Peijun Li 2, and  Jue Wang 3,

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  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.

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 & Continuous Dynamical Systems - B, 2020, 25 (1) : 259-286. doi: 10.3934/dcdsb.2019181
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.  Google Scholar

[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.  Google Scholar

[3]

H. Ammari and G. Bao, Maxwell's equations in periodic chiral structures, Math.Nachr., 251 (2003), 3-18.  doi: 10.1002/mana.200310026.  Google Scholar

[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.  Google Scholar

[5]

G. Bao, Numerical analysis of diffraction by periodic structures: TM polarization, Numer. Math., 75 (1996), 1-16.  doi: 10.1007/s002110050227.  Google Scholar

[6]

G. Bao, Variational approximation of Maxwell's equations in biperiodic structures, SIAM J. Appl. Math., 57 (1997), 364-381.  doi: 10.1137/S0036139995279408.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Z. Chen and J.-C. Nédélec, On Maxwell equations with the transparent boundary condition, J. Comput. Math., 26 (2008), 284-296.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numer., 8 (1999), 47-106.  doi: 10.1017/s0962492900002890.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

R. Petit, ed., Electromagnetic Theory of Gratings, Springer, 1980. doi: 10.1007/978-3-642-81500-3.  Google Scholar

[36]

P. Rayleigh, On the dynamical theory of gratings, R. Soc. London Ser. A, 79 (1907), 399-416.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

H. Ammari and G. Bao, Maxwell's equations in periodic chiral structures, Math.Nachr., 251 (2003), 3-18.  doi: 10.1002/mana.200310026.  Google Scholar

[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.  Google Scholar

[5]

G. Bao, Numerical analysis of diffraction by periodic structures: TM polarization, Numer. Math., 75 (1996), 1-16.  doi: 10.1007/s002110050227.  Google Scholar

[6]

G. Bao, Variational approximation of Maxwell's equations in biperiodic structures, SIAM J. Appl. Math., 57 (1997), 364-381.  doi: 10.1137/S0036139995279408.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Z. Chen and J.-C. Nédélec, On Maxwell equations with the transparent boundary condition, J. Comput. Math., 26 (2008), 284-296.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numer., 8 (1999), 47-106.  doi: 10.1017/s0962492900002890.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

R. Petit, ed., Electromagnetic Theory of Gratings, Springer, 1980. doi: 10.1007/978-3-642-81500-3.  Google Scholar

[36]

P. Rayleigh, On the dynamical theory of gratings, R. Soc. London Ser. A, 79 (1907), 399-416.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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