On the Properties of Quasi-periodic Boundary Integral Operators for the Helmholtz Equation

Rubén Aylwin1, Carlos Jerez-Hanckes2, José Pinto1
1Department of Electrical Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile
2Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Santiago, Chile

Tóm tắt

We study the mapping properties of boundary integral operators arising when solving two-dimensional, time-harmonic waves scattered by periodic domains. For domains assumed to be at least Lipschitz regular, we propose a novel explicit representation of Sobolev spaces for quasi-periodic functions that allows for an analysis analogous to that of Helmholtz scattering by bounded objects. Except for Rayleigh-Wood frequencies, continuity and coercivity results are derived to prove wellposedness of the associated first kind boundary integral equations.

Từ khóa


Tài liệu tham khảo

Alber, H.D.: A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent. Proc. R. Soc. Edinb. 82(3–4), 251–272 (1979)

Ammari, H.: Scattering of waves by thin periodic layers at high frequencies using the on-surface radiation condition method. IMA J. Appl. Math. 60(2), 199–214 (1998)

Ammari, H., Bao, G.: Coupling of finite element and boundary element methods for the scattering by periodic chiral structures. J. Comput. Math. 26(3), 261–283 (2008)

Ammari, H., He, S.: Homogenization and scattering for gratings. J. Electromagn. Waves Appl. 11(12), 1669–1683 (1997)

Ammari, H., Nédélec, J.-C.: Analysis of the diffraction from chiral gratings. In: Mathematical Modeling in Optical Science. SIAM, pp. 179–206 (2001)

Bao, G.: Finite element approximation of time harmonic waves in periodic structures. SIAM J. Numer. Anal. 32(4), 1155–1169 (1995)

Bao, G.: Variational approximation of Maxwell’s equations in biperiodic structures. SIAM J. Appl. Math. 57(2), 364–381 (1997)

Bao, G.: Recent mathematical studies in the modeling of optics and electromagnetics. J. Comput. Math. 22(2), 148–155 (2004)

Bao, G., Dobson, D.C.: On the scattering by a biperiodic structure. Proc. Am. Math. Soc. 128(9), 2715–2723 (2000)

Bao, G., Dobson, D.C., Cox, J.A.: Mathematical studies in rigorous grating theory. J. Opt. Soc. Am. A 12(5), 1029–1042 (1995)

Barnett, A., Greengard, L.: A new integral representation for quasi-periodic scattering problems in two dimensions. BIT Numer. Math. 51(1), 67–90 (2011)

Bruno, O.P., Fernandez-Lado, A.G.: Rapidly convergent quasi-periodic Green functions for scattering by arrays of cylinders–including Wood anomalies. Proc. R. Soc. A 473(2199), 20160802 (2017)

Bruno, O.P., Shipman, S.P., Turc, C., Stephanos, V.: Three-dimensional quasi-periodic shifted Green function throughout the spectrum, including Wood anomalies. Proc. R. Soc. A 473(2207), 20170242 (2017)

Chen, X., Friedman, A.: Maxwell’s equations in a periodic structure. Trans. Am. Math. Soc. 323(2), 465–507 (1991)

Cho, M.H., Barnett, A.H.: Robust fast direct integral equation solver for quasi-periodic scattering problems with a large number of layers. Opt. Express 23(2), 1775–1799 (2015)

Costabel, M.: Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)

Dobson, D.C.: A variational method for electromagnetic diffraction in biperiodic structures. ESAIM: Math. Model. Numer. Anal. 28(4), 419–439 (1994)

Dobson, D.C., Cox, J.A.: An integral equation method for biperiodic diffraction structures. Int. Conf. Appl. Theory Period. Struct. 1545, 106–114 (1991)

Dobson, D.C., Friedman, A.: The time-harmonic Maxwell equations in a doubly periodic structure. J. Math. Anal. Appl. 166(2), 507–528 (1992)

Elschner, J., Schmidt, G.: Diffraction in periodic structures and optimal design of binary gratings. Part I: direct problems and gradient formulas. Math. Methods Appl. Sci. 21(14), 1297–1342 (1998)

Jerez-Hanckes, C.: Modeling elastic and electromagnetic surface waves in piezoelectric tranducers and optical waveguides. Ph.D. thesis, École Polytechnique, Palaiseau, France (2008)

Kirsch, A.: Diffraction by periodic structures. In: Inverse Problems in Mathematical Physics. Springer, Berlin, pp. 87–102 (1993)

Kirsch, A.: Uniqueness theorems in inverse scattering theory for periodic structures. Inverse Prob. 10, 145–152 (1994)

Kress, R.: Linear Integral Equations, vol. 82, 3rd edn. Applied Mathematical Sciences (2014)

Lai, J., Kobayashi, M., Barnett, A.: A fast and robust solver for the scattering from a layered periodic structure containing multi-particle inclusions. J. Comput. Phys. 298, 194–208 (2015)

Lechleiter, A., Nguyen, D.-L.: Volume integral equations for scattering from anisotropic diffraction gratings. Math. Methods Appl. Sci. 36(3), 262–274 (2013)

Lechleiter, A., Zhang, R.: A floquet-bloch transform based numerical method for scattering from locally perturbed periodic surfaces. SIAM J. Sci. Comput. 39(5), B819–B839 (2017)

Linton, C.M.: The green’s function for the two-dimensional Helmholtz equation in periodic domains. J. Eng. Math. 33(4), 377–401 (1998)

Liu, Y., Barnett, A.: Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects. J. Comput. Phys. 324, 226–245 (2016)

McLean, W.C.H.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

Nédélec, J.C., Starling, F.: Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations. SIAM J. Math. Anal. 22(6), 1679–1701 (1991)

Nečas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, Berlin (2011)

Pestourie, R., Pérez-Arancibia, C., Lin, Z., Shin, W., Capasso, F., Johnson, S.G.: Inverse design of large-area metasurfaces. Opt. Express 26(26), 33732–33747 (2018)

Petit, R. (ed.): Electromagnetic Theory of Gratings. Springer, Berlin (1980)

Saranen, J., Vainikko, G.: Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer, Berlin (2002)

Schmidt, G.: Boundary integral methods for periodic scattering problems. In: Around the Research of Vladimir Maz’ya II. Springer, Berlin, pp. 337–363 (2010)

Schmidt, G.: Integral equations for conical diffraction by coated grating. J. Integral Equ. Appl. 23(1), 71–112 (2011)

Shiraishi, K., Higuchi, S., Muraki, K., Yoda, H.: Silver-film subwavelength gratings for polarizers in the terahertz and mid-infrared regions. Opt. Express 24(18), 20177–20186 (2016)

Silva-Oelker, G., Aylwin, R., Jerez-Hanckes, C., Fay, P.: Quantifying the impact of random surface perturbations on reflective gratings. IEEE Trans. Antennas Propag. 66(2), 838–847 (2018)

Silva-Oelker, G., Jerez-Hanckes, C., Fay, P.: Study of W/HfO\(_2\) grating selective thermal emitters for thermophotovoltaic applications. Opt. Express 26(22), A929–A936 (2018)

Starling, F., Bonnet-Bendhia, A.-S.: Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Methods Appl. Sci. 17, 305–338 (1994)

Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer, Berlin (2007)

Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces, vol. 3. Springer, Berlin (2007)

Zhang, B., Chandler-Wilde, S.N.: A uniqueness result for scattering by infinite rough surfaces. SIAM J. Appl. Math. 58(6), 1774–1790 (1998)