Uniqueness and numerical method for phaseless inverse diffraction grating problem with known superposition of incident point sources. (English) Zbl 07878750
Summary: In this paper, we establish the uniqueness of identifying a smooth grating profile with a mixed boundary condition (MBC) or transmission boundary conditions (TBCs) from phaseless data. The existing uniqueness result requires the measured data to be in a bounded domain. To break this restriction, we design an incident system consisting of the superposition of point sources to reduce the measurement data from a bounded domain to a line above the grating profile. We derive reciprocity relations for point sources, diffracted fields, and total fields, respectively. Based on Rayleigh’s expansion and reciprocity relation of the total field, a grating profile with a MBC or TBCs can be uniquely determined from the phaseless total field data. An iterative algorithm is proposed to recover the Fourier modes of grating profiles at a fixed wavenumber. To implement this algorithm, we derive the Fréchet derivative of the total field operator and its adjoint operator. Some numerical examples are presented to verify the correctness of theoretical results and to show the effectiveness of our numerical algorithm.
{© 2024 IOP Publishing Ltd}
{© 2024 IOP Publishing Ltd}
MSC:
| 78Axx | General topics in optics and electromagnetic theory |
| 35Rxx | Miscellaneous topics in partial differential equations |
| 35Jxx | Elliptic equations and elliptic systems |
References:
| [1] | Arens, T.; Kirsch, A., The factorization method in inverse scattering from periodic structures, Inverse Problems, 19, 1195-211, 2003 · Zbl 1330.35519 · doi:10.1088/0266-5611/19/5/311 |
| [2] | Bao, G., A uniqueness theorem for an inverse problem in periodic diffractive optics, Inverse Problems, 10, 335, 1994 · Zbl 0805.35144 · doi:10.1088/0266-5611/10/2/009 |
| [3] | Bao, G., Variational approximation of Maxwell’s equations in biperiodic structures, SIAM J. Appl. Math., 57, 364-81, 1997 · Zbl 0872.65108 · doi:10.1137/S0036139995279408 |
| [4] | Bao, G.; Li, P.; Lv, J., Numerical solution of an inverse diffraction grating problem from phaseless data, J. Opt. Soc. Am. A, 30, 293-9, 2013 · doi:10.1364/JOSAA.30.000293 |
| [5] | Bao, G.; Cowsar, L.; Masters, W., Mathematical Modeling in Optical Science, 2001, SIAM · Zbl 0964.00050 |
| [6] | Bao, G.; Lin, Y., Determination of random periodic structures in transverse magnetic polarization, Commun. Math. Res., 37, 271-96, 2021 · Zbl 1513.78003 · doi:10.4208/cmr.2021-0003 |
| [7] | Bao, G.; Li, P., Maxwell’s Equations in Periodic Structures, 2022, Springer · Zbl 1484.35001 |
| [8] | Bonnet Bendhia, A-S; Starling, F., Guided waves by electromagnetic gratings and non uniqueness examples for the diffraction problem, Math. Methods Appl. Sci., 17, 305-38, 1994 · Zbl 0817.35109 · doi:10.1002/mma.1670170502 |
| [9] | Bruckner, G.; Elschner, J., The numerical solution of an inverse periodic transmission problem, Math. Methods Appl. Sci., 28, 757-78, 2005 · Zbl 1070.35119 · doi:10.1002/mma.588 |
| [10] | Chen, X.; Friedman, A., Maxwell’s equations in a periodic structure, Trans. Am. Math. Soc., 323, 465-507, 1991 · Zbl 0727.35131 |
| [11] | Chen, Z.; Wu, H., An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 41, 799-826, 2003 · Zbl 1049.78018 · doi:10.1137/S0036142902400901 |
| [12] | Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, 2019, Springer · Zbl 1425.35001 |
| [13] | DeSanto, J.; Erdmann, G.; Hereman, W.; Misra, M., Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces, Waves Random Media, 8, 385-414, 1998 · Zbl 0922.35112 · doi:10.1088/0959-7174/8/4/001 |
| [14] | Deuflhard, P.; Engl, H.; Scherzer, O., A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions, Inverse Problems, 14, 1081-106, 1998 · Zbl 0915.65053 · doi:10.1088/0266-5611/14/5/002 |
| [15] | Dobson, D. C., A variational method for electromagnetic diffraction in biperiodic structures, ESAIM Math. Model. Numer. Anal., 28, 419-39, 1994 · Zbl 0820.65087 · doi:10.1051/m2an/1994280404191 |
| [16] | Dong, H.; Zhang, D.; Chi, Y., An iterative scheme for imaging acoustic obstacle from phaseless total-field data, Inverse Probl. Imaging, 16, 925-42, 2022 · Zbl 1494.65093 · doi:10.3934/ipi.2022005 |
| [17] | 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, 1297-342, 1998 · Zbl 0913.65121 · doi:10.1002/(SICI)1099-1476(19980925)21:143.0.CO;2-C |
| [18] | Elschner, J.; Yamamoto, M., An inverse problem in periodic diffractive optics: reconstruction of Lipschitz grating profiles, Appl. Anal., 81, 1307-28, 2002 · Zbl 1028.78008 · doi:10.1080/0003681021000035551 |
| [19] | Elschner, J.; Schmidt, G.; Yamamoto, M., An inverse problem in periodic diffractive optics: global uniqueness with a single wavenumber, Inverse Problems, 19, 779-87, 2003 · Zbl 1121.78325 · doi:10.1088/0266-5611/19/3/318 |
| [20] | Elschner, J.; Schmidt, G.; Yamamoto, M., Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number, J. Inverse Ill-Posed Probl., 11, 235-44, 2003 · Zbl 1038.78014 · doi:10.1515/156939403769237024 |
| [21] | Elschner, J.; Yamamoto, M., Uniqueness results for an inverse periodic transmission problem, Inverse Problems, 20, 1841-52, 2004 · Zbl 1077.78009 · doi:10.1088/0266-5611/20/6/009 |
| [22] | Elschner, J.; Hu, G., Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves, Inverse Problems, 26, 2010 · Zbl 1210.35285 · doi:10.1088/0266-5611/26/11/115002 |
| [23] | Hanke, N.; Neubauer, A.; Scherzer, O., A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72, 21-37, 1995 · Zbl 0840.65049 · doi:10.1007/s002110050158 |
| [24] | Hanke, M.; Hettlich, F.; Scherzer, O., The Landweber iteration for an inverse scattering problem, pp 909-15, 1995, American Society of Mechanical Engineers |
| [25] | Hettlich, F.; Kirsch, A., Schiffer’s theorem in inverse scattering theory for periodic structures, Inverse Problems, 13, 351-61, 1997 · Zbl 0873.35108 · doi:10.1088/0266-5611/13/2/010 |
| [26] | Hettlich, F., The Landweber iteration applied to inverse conductive scattering problems, Inverse Problems, 4, 931-47, 1998 · Zbl 0917.35160 · doi:10.1088/0266-5611/14/4/011 |
| [27] | Hettlich, F., Iterative regularization schemes in inverse scattering by periodic structures, Inverse Problems, 18, 701-14, 2002 · Zbl 1002.35124 · doi:10.1088/0266-5611/18/3/311 |
| [28] | Hu, G.; Qu, F.; Zhang, B., A linear sampling method for inverse problems of diffraction gratings of mixed type, Math. Methods Appl. Sci., 35, 1047-66, 2012 · Zbl 1251.35188 · doi:10.1002/mma.2511 |
| [29] | Ji, X.; Liu, X., Inverse elastic scattering problems with phaseless far field data, Inverse Problems, 11, 2019 · Zbl 1423.35449 · doi:10.1088/1361-6420/ab2a35 |
| [30] | Jiang, X.; Li, P.; Lv, J.; Zheng, W., An adaptive finite element PML method for the elastic wave scattering problem in periodic structure, ESAIM Math. Model. Numer. Anal., 51, 2017-47, 2017 · Zbl 1408.74048 · doi:10.1051/m2an/2017018 |
| [31] | Jiang, X.; Li, P.; Lv, J.; Zheng, W., Convergence of the PML solution for elastic wave scattering by biperiodic structures, Commun. Math. Sci., 4, 987-1016, 2018 · Zbl 1402.74051 · doi:10.4310/CMS.2018.v16.n4.a4 |
| [32] | Jiang, X.; Li, P.; Lv, J.; Wang, Z.; Wu, H.; Zheng, W., An adaptive edge finite element DtN method for Maxwell’s equations in biperiodic structures, IMA J. Numer. Anal., 42, 2794-828, 2022 · Zbl 1517.65113 · doi:10.1093/imanum/drab052 |
| [33] | Jin, Q.; Amato, U., A discrete scheme of Landweber iteration for solving nonlinear ill-posed problems, J. Math. Anal. Appl., 1, 187-203, 2001 · Zbl 0992.47032 · doi:10.1006/jmaa.2000.7090 |
| [34] | Kirsch, A.; Pävarinta, L.; Somersalo, E., Diffraction by periodic structures, pp 87-102, 1993, Springer |
| [35] | Kirsch, A., Uniqueness theorems in inverse scattering theory for periodic structures, Inverse Problems, 10, 145, 1994 · Zbl 0805.35155 · doi:10.1088/0266-5611/10/1/011 |
| [36] | Kirsch, A.; Kleinman, R. E., An inverse problem for periodic structures, Methoden und Verfahren der Mathematischen Physik, pp 75-93, 1995, Peter Lang |
| [37] | Klibanov, M. V., Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74, 392-410, 2014 · Zbl 1293.35188 · doi:10.1137/130926250 |
| [38] | Klibanov, M. V., A phaseless inverse scattering problem for the 3-D Helmholtz equation, Inverse Probl. Imaging, 11, 3263-276, 2017 · Zbl 1359.35227 · doi:10.3934/ipi.2017013 |
| [39] | Klibanov, M. V.; Romanov, V. G., Uniqueness of a 3-D coefficient inverse scattering problem without the phase information, Inverse Problems, 33, 2017 · Zbl 1516.35519 · doi:10.1088/1361-6420/aa7a18 |
| [40] | Li, S.; Lv, J.; Wang, Y., Numerical method for the inverse interior scattering problem from phaseless data, Inverse Probl. Imaging, 18, 776-96, 2024 · Zbl 07896891 · doi:10.3934/ipi.2023054 |
| [41] | Meier, A.; Arens, T.; Chandler-Wilde, S.; Kirsch, A., A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces, J. Int. Equ. Appl., 3, 281-321, 2000 · Zbl 1170.78365 |
| [42] | Novikov, R. G., Formulas for phase recovering from phaseless scattering data at fixed frequency, Bull. Sci. Math., 139, 923-36, 2015 · Zbl 1330.35277 · doi:10.1016/j.bulsci.2015.04.005 |
| [43] | Niu, T.; Lv, J.; Wu, D., Uniqueness in phaseless inverse electromagnetic scattering problem with known superposition of incident electric dipoles, Math. Methods Appl. Sci., 46, 17692-703, 2023 · Zbl 1534.78001 · doi:10.1002/mma.9526 |
| [44] | Petit, R., Electromagnetic Theory of Gratings, 1980, Springer |
| [45] | Rathsfeld, A.; Hsiao, G. C.; Elschner, J., Grating profile reconstruction based on finite elements and optimization techniques, SIAM J. Appl. Math., 64, 525-45, 2004 · Zbl 1059.35165 · doi:10.1137/S0036139902420018 |
| [46] | Rathsfeld, A.; Schmidt, G.; Kleemann, B., On a fast integral equation method for diffraction gratings, Commun. Comput. Phys., 1, 984-1009, 2006 · Zbl 1116.78032 · doi:10.34657/2786 |
| [47] | Strycharz, B., An acoustic scattering problem for periodic, inhomogeneous media, Math. Methods Appl. Sci, 21, 969-83, 1998 · Zbl 0907.35095 · doi:10.1002/(SICI)1099-1476(19980710)21:103.0.CO;2-Y |
| [48] | Strycharz, B., Uniqueness in the inverse transmission scattering problem for periodic media, Math. Methods Appl. Sci, 22, 753-72, 1999 · Zbl 0931.35120 · doi:10.1002/(SICI)1099-1476(199906)22:93.0.CO;2-U |
| [49] | Sun, F.; Zhang, D.; Guo, Y., Uniqueness in phaseless inverse scattering problems with known superposition of incident point sources, Inverse Problems, 35, 2019 · Zbl 1423.35456 · doi:10.1088/1361-6420/ab3373 |
| [50] | Wood, R., On a remarkable case of uneven distribution of light in a diffraction grating spectrum, Proc. Phys. Soc., 18, 269-75, 1902 · doi:10.1088/1478-7814/18/1/325 |
| [51] | Xu, X.; Hu, G.; Zhang, B.; Zhang, H., Uniqueness in inverse diffraction grating problems with infinitely many plane waves at a fixed frequency, SIAM J. Appl. Math., 83, 302-26, 2023 · Zbl 1512.35681 · doi:10.1137/22M1480963 |
| [52] | Xu, X.; Zhang, B.; Zhang, H., Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency. II, SIAM J. Appl. Math., 78, 3024-39, 2018 · Zbl 1403.78010 · doi:10.1137/18M1196820 |
| [53] | Yang, J.; Zhang, B., Uniqueness results in the inverse scattering problem for periodic structures, Math. Methods Appl. Sci., 35, 828-38, 2012 · Zbl 1236.35207 · doi:10.1002/mma.1609 |
| [54] | Zhang, D.; Guo, Y., Uniqueness results on phaseless inverse acoustic scattering with a reference ball, Inverse Problems, 34, 2018 · Zbl 1442.35553 · doi:10.1088/1361-6420/aac53c |
| [55] | Zhang, D.; Guo, Y.; Sun, F.; Liu, H., Unique determinations in inverse scattering problems with phaseless near-field measurements, Inverse Problems Imaging, 14, 569-82, 2020 · Zbl 1441.78017 · doi:10.3934/ipi.2020026 |
| [56] | Zhang, M.; Lv, J., Numerical method of profile reconstruction for a periodic transmission problem from single-sided data, Commun. Comput. Phys, 24, 435-53, 2018 · Zbl 1488.35630 · doi:10.4208/cicp.OA-2017-0169 |
| [57] | Zhang, R.; Sun, J., Efficient finite element method for grating profile reconstruction, J. Comput. Phys., 32, 405-19, 2015 · Zbl 1349.65594 · doi:10.1016/j.jcp.2015.09.016 |
| [58] | Zheng, J.; Cheng, J.; Li, P.; Lu, S., Periodic surface identification with phase or phaseless near-field data, Inverse Problems, 33, 2017 · Zbl 1516.35547 · doi:10.1088/1361-6420/aa8cb3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
