American Institute of Mathematical Sciences

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

Reconstruction of penetrable grating profiles

Jiaqing Yang 1, , Bo Zhang 2, and  Ruming Zhang 2,

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.
Keywords: grating profile, Sampling method, transmission condition, quasi-periodic boundary value problem, impedance Green's function..
Mathematics Subject Classification: Primary: 35R30, 35J05; Secondary: 78A46, 78M5.
Citation: Jiaqing Yang, Bo Zhang, Ruming Zhang. Reconstruction of penetrable grating profiles. Inverse Problems & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[6]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Lechleiter, Imaging of periodic dielectrics, BIT Numer. Math., 50 (2010), 59-83. doi: 10.1007/s10543-010-0255-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. Lechleiter, Imaging of periodic dielectrics, BIT Numer. Math., 50 (2010), 59-83. doi: 10.1007/s10543-010-0255-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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