American Institute of Mathematical Sciences

November  2013, 7(4): 1393-1407. doi: 10.3934/ipi.2013.7.1393

Reconstruction of penetrable grating profiles

 1 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2 LSEC and Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, China

Received  September 2012 Revised  June 2013 Published  November 2013

This paper is concerned with the inverse problem of recovering a penetrable grating profile in the TM-polarization case from the scattered field measured only above the structure, corresponding to a countably infinite number of incident quasi-periodic waves. A sampling method is proposed to reconstruct the penetrable grating profile based on a near field linear operator equation in $l^2$. The mathematical justification of the sampling method is established and numerical results are presented to show the validity of the inversion algorithm.
Citation: Jiaqing Yang, Bo Zhang, Ruming Zhang. Reconstruction of penetrable grating profiles. Inverse Problems and Imaging, 2013, 7 (4) : 1393-1407. doi: 10.3934/ipi.2013.7.1393
References:
 [1] T. Arens and N. Grinberg, A complete factorization method for scattering by periodic surfaces, Computing, 50 (2005), 111-132. doi: 10.1007/s00607-004-0092-0. [2] T. Arens and A. Kirsch, The factorization method in inverse scattering from periodic structures, Inverse Problems, 519 (2003), 1195-1211. doi: 10.1088/0266-5611/19/5/311. [3] G. Bruckner and J. Elschner, A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles, Inverse Problems, 19 (2003), 315-329. doi: 10.1088/0266-5611/19/2/305. [4] G. Bruckner and J. Elschner, The numerical solution of an inverse periodic transmission problem, Math. Meth. Appl. Sci., 28 (2005), 757-778. doi: 10.1002/mma.588. [5] A-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem, Math. Meth. Appl. Sci., 17 (1994), 305-338. doi: 10.1002/mma.1670170502. [6] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. [7] D. Colton, R. Kress and P. Monk, Inverse scattering from an orthotropic medium, J. Comput. Appl. Math., 81 (2007), 269-298. doi: 10.1016/S0377-0427(97)00065-4. [8] J. Elschner, G. C. Hsiao and A. Rathsfeld, Grating profile reconstruction based on finite elements and optimization techniques, SIAM J. Appl. Math., 64 (2003), 525-545. doi: 10.1137/S0036139902420018. [9] F. Hettlich, Iterative regularization schemes in inverse scattering by periodic structures, Inverse Problems, 18 (2002), 701-714. doi: 10.1088/0266-5611/18/3/311. [10] G. Hu, F. Qu and B. Zhang, A linear sampling method for inverse problems of diffraction gratings of mixed type, Math. Methods Appl. Sci., 35 (2012), 1047-1066. doi: 10.1002/mma.2511. [11] G. Hu and B. Zhang, The linear sampling method for the inverse electromagnetic scattering by a partially coated bi-periodic structure, Math. Methods Appl. Sci., 34 (2011), 509-519. doi: 10.1002/mma.1375. [12] K. Ito and F. Reitich, A high-order perturbation approach to profile reconstruction: I. Perfectly conducting grating, Inverse Problems, 15 (1999), 1067-1085. doi: 10.1088/0266-5611/15/4/315. [13] A. Lechleiter, Imaging of periodic dielectrics, BIT Numer. Math., 50 (2010), 59-83. doi: 10.1007/s10543-010-0255-7. [14] A. Malcolm and D. P. Nicholls, A boundary perturbation method for recovering interface shape in layered media, Inverse Problems, 27 (2011), 095009 (18pp). doi: 10.1088/0266-5611/27/9/095009. [15] G. Schmidt, Integral equations for conical diffraction by coated gratings, J. Integral Equat. Appl., 23 (2011), 71-112. doi: 10.1216/JIE-2011-23-1-71. [16] J. Yang and B. Zhang, Uniqueness results in the inverse scattering problem for periodic structures, Math. Methods Appl. Sci., 35 (2012), 828-838. doi: 10.1002/mma.1609. [17] J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 035004 (17pp). doi: 10.1088/0266-5611/28/3/035004.

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References:
 [1] T. Arens and N. Grinberg, A complete factorization method for scattering by periodic surfaces, Computing, 50 (2005), 111-132. doi: 10.1007/s00607-004-0092-0. [2] T. Arens and A. Kirsch, The factorization method in inverse scattering from periodic structures, Inverse Problems, 519 (2003), 1195-1211. doi: 10.1088/0266-5611/19/5/311. [3] G. Bruckner and J. Elschner, A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles, Inverse Problems, 19 (2003), 315-329. doi: 10.1088/0266-5611/19/2/305. [4] G. Bruckner and J. Elschner, The numerical solution of an inverse periodic transmission problem, Math. Meth. Appl. Sci., 28 (2005), 757-778. doi: 10.1002/mma.588. [5] A-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem, Math. Meth. Appl. Sci., 17 (1994), 305-338. doi: 10.1002/mma.1670170502. [6] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. [7] D. Colton, R. Kress and P. Monk, Inverse scattering from an orthotropic medium, J. Comput. Appl. Math., 81 (2007), 269-298. doi: 10.1016/S0377-0427(97)00065-4. [8] J. Elschner, G. C. Hsiao and A. Rathsfeld, Grating profile reconstruction based on finite elements and optimization techniques, SIAM J. Appl. Math., 64 (2003), 525-545. doi: 10.1137/S0036139902420018. [9] F. Hettlich, Iterative regularization schemes in inverse scattering by periodic structures, Inverse Problems, 18 (2002), 701-714. doi: 10.1088/0266-5611/18/3/311. [10] G. Hu, F. Qu and B. Zhang, A linear sampling method for inverse problems of diffraction gratings of mixed type, Math. Methods Appl. Sci., 35 (2012), 1047-1066. doi: 10.1002/mma.2511. [11] G. Hu and B. Zhang, The linear sampling method for the inverse electromagnetic scattering by a partially coated bi-periodic structure, Math. Methods Appl. Sci., 34 (2011), 509-519. doi: 10.1002/mma.1375. [12] K. Ito and F. Reitich, A high-order perturbation approach to profile reconstruction: I. Perfectly conducting grating, Inverse Problems, 15 (1999), 1067-1085. doi: 10.1088/0266-5611/15/4/315. [13] A. Lechleiter, Imaging of periodic dielectrics, BIT Numer. Math., 50 (2010), 59-83. doi: 10.1007/s10543-010-0255-7. [14] A. Malcolm and D. P. Nicholls, A boundary perturbation method for recovering interface shape in layered media, Inverse Problems, 27 (2011), 095009 (18pp). doi: 10.1088/0266-5611/27/9/095009. [15] G. Schmidt, Integral equations for conical diffraction by coated gratings, J. Integral Equat. Appl., 23 (2011), 71-112. doi: 10.1216/JIE-2011-23-1-71. [16] J. Yang and B. Zhang, Uniqueness results in the inverse scattering problem for periodic structures, Math. Methods Appl. Sci., 35 (2012), 828-838. doi: 10.1002/mma.1609. [17] J. Yang, B. Zhang and R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28 (2012), 035004 (17pp). doi: 10.1088/0266-5611/28/3/035004.
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