Imaging of bi-anisotropic periodic structures from electromagnetic near-field data. (English) Zbl 1486.35469
Summary: This paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency. The factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35P25 | Scattering theory for PDEs |
| 35Q61 | Maxwell equations |
| 35R09 | Integro-partial differential equations |
| 65R20 | Numerical methods for integral equations |
| 78A45 | Diffraction, scattering |
Keywords:
factorization method; Maxwell’s equations; bi-anisotropic periodic structures; inverse electromagnetic scattering; sampling methodsReferences:
| [1] | T. Arens and N. Grinberg, A complete factorization method for scattering by periodic surfaces, Computing 75 (2005), no. 2-3, 111-132. · Zbl 1075.35103 |
| [2] | T. Arens and A. Kirsch, The factorization method in inverse scattering from periodic structures, Inverse Problems 19 (2003), no. 5, 1195-1211. · Zbl 1330.35519 |
| [3] | G. Bao, L. Cowsar and W. Masters, Mathematical Modeling in Optical Science, Front. Appl. Math. 22, Society for Industrial and Applied Mathematics, Philadelphia, 2001. · Zbl 0964.00050 |
| [4] | G. Bao, T. Cui and P. Li, Inverse diffraction grating of Maxwell’s equations in biperiodic structures, Optics Express 22 (2014), 4799-4816. |
| [5] | D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems 12 (1996), no. 4, 383-393. · Zbl 0859.35133 |
| [6] | W. Dorfler, A. Lechleiter, M. Plum, G. Schneider and C. Wieners, Photonic Crystals: Mathematical Analysis and Numerical Approximation, Springer, Basel, 2012. |
| [7] | J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles, Commun. Comput. Phys. 12 (2012), no. 5, 1434-1460. · Zbl 1373.74011 |
| [8] | H. Haddar and T.-P. Nguyen, Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers, Comput. Math. Appl. 74 (2017), no. 11, 2831-2855. · Zbl 1398.35225 |
| [9] | I. Harris, D.-L. Nguyen, J. Sands and T. Truong, On the inverse scattering from anisotropic periodic layers and transmission eigenvalues, preprint (2020), https://arxiv.org/abs/1908.05801. |
| [10] | X. Jiang and P. Li, Inverse electromagnetic diffraction by biperiodic dielectric gratings, Inverse Problems 33 (2017), no. 8, Article ID 085004. · Zbl 1378.78023 |
| [11] | A. Kirsch, Uniqueness theorems in inverse scattering theory for periodic structures, Inverse Problems 10 (1994), no. 1, 145-152. · Zbl 0805.35155 |
| [12] | A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems 14 (1998), no. 6, 1489-1512. · Zbl 0919.35147 |
| [13] | A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford Lecture Ser. Math. Appl. 36, Oxford University, Oxford, 2008. · Zbl 1222.35001 |
| [14] | A. Lechleiter, Imaging of periodic dielectrics, BIT 50 (2010), no. 1, 59-83. · Zbl 1194.78018 |
| [15] | A. Lechleiter and D.-L. Nguyen, Factorization method for electromagnetic inverse scattering from biperiodic structures, SIAM J. Imaging Sci. 6 (2013), no. 2, 1111-1139. · Zbl 1282.35266 |
| [16] | A. Lechleiter and R. Zhang, Reconstruction of local perturbations in periodic surfaces, Inverse Problems 34 (2018), no. 3, Article ID 035006. · Zbl 1457.65169 |
| [17] | T. G. Mackay and A. Lakhtakia, Electromagnetic Anisotropy and Bi-Anisotropy: A Field Guide, World Scientific, Singapore, 2010. |
| [18] | D.-L. Nguyen, Shape identification of anisotropic diffraction gratings for TM-polarized electromagnetic waves, Appl. Anal. 93 (2014), no. 7, 1458-1476. · Zbl 1295.78009 |
| [19] | D.-L. Nguyen, A volume integral equation method for periodic scattering problems for anisotropic Maxwell’s equations, Appl. Numer. Math. 98 (2015), 59-78. · Zbl 1329.65318 |
| [20] | D.-L. Nguyen, The factorization method for the Drude-Born-Fedorov model for periodic chiral structures, Inverse Probl. Imaging 10 (2016), no. 2, 519-547. · Zbl 1348.35326 |
| [21] | D.-L. Nguyen, Direct and inverse electromagnetic scattering problems for bi-anisotropic media, Inverse Problems 35 (2019), no. 12, Article ID 124001. · Zbl 1435.78013 |
| [22] | T.-P. Nguyen, Differential imaging of local perturbations in anisotropic periodic media, Inverse Problems 36 (2020), no. 3, Article ID 034004. · Zbl 1435.78014 |
| [23] | G. F. Roach, I. G. Stratis and A. N. Yannacopoulos, Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics, Princeton Ser. Appl. Math., Princeton University, Princeton, 2012. · Zbl 1250.78001 |
| [24] | K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media, PhD thesis, Karlsruher Institut für Technologie, 2010. |
| [25] | G. Schmidt, On the diffraction by biperiodic anisotropic structures, Appl. Anal. 82 (2003), no. 1, 75-92. · Zbl 1037.35086 |
| [26] | J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems 28 (2012), no. 3, Article ID 035004. · Zbl 1236.78021 |
| [27] | R. Zhang and B. Zhang, Near-field imaging of periodic inhomogeneous media, Inverse Problems 30 (2014), no. 4, Article ID 045004. · Zbl 1294.78008 |
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