Abstract

Consider a time-harmonic electromagnetic plane wave incident on a perfectly conducting biperiodic surface (crossed grating). The diffraction is modeled as a boundary value problem for the three-dimensional Maxwell equation. The surface is assumed to be a small and smooth deformation of a planar surface. In this paper, a novel approach is developed to solve the inverse diffraction grating problem in the near-field regime, which is to reconstruct the surface with resolution beyond Rayleigh’s criterion. The method requires only a single incident field with one polarization, one frequency, and one incident direction, and is realized by using the fast Fourier transform. Numerical results show that the method is simple, efficient, and stable to reconstruct biperiodic surfaces with subwavelength resolution.

© 2014 Optical Society of America

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2013 (8)

G. Bao, P. Li, and J. Lv, “Numerical solution of an inverse diffraction grating problem from phasless data,” J. Opt. Soc. Am. A 30, 293–299 (2013).
[Crossref]

A. Lechleiter and D. L. Nguyen, “On uniqueness in electromagnetic scattering from biperiodic structures,” ESAIM Math. Modell. Numer. Anal. 47, 1167–1184 (2013).
[Crossref]

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces,” SIAM J. Appl. Math. 73, 2162–2187 (2013).
[Crossref]

T. Cheng, P. Li, and Y. Wang, “Near-field imaging of perfectly conducting grating surfaces,” J. Opt. Soc. Am. A 30, 2473–2481 (2013).
[Crossref]

G. Bao and J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. 7, 377–396 (2013).
[Crossref]

A. Lechleiter and D. L. Nguyen, “Factorization method for electromagnetic inverse scattering from biperiodic structures,” SIAM J. Imaging Sci. 6, 1111–1139 (2013).
[Crossref]

H. Ammari, J. Garnier, and K. Solna, “Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,” Proc. Am. Math. Soc. 141, 3431–3446 (2013).
[Crossref]

H. Ammari, J. Garnier, and K. Solna, “Partial data resolving power of conductivity imaging from boundary measurements,” SIAM J. Math. Anal. 45, 1704–1722 (2013).
[Crossref]

2012 (2)

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic struture,” J. Comput. Phys. 231, 3007–3022 (2012).
[Crossref]

G. Bao, P. Li, and H. Wu, “A computational inverse diffraction grating problem,” J. Opt. Soc. Am. A 29, 394–399 (2012).
[Crossref]

2011 (6)

A. Malcolm and D. P. Nicholls, “A field expansions method for scattering by periodic multilayered media,” J. Acoust. Soc. Am. 129, 1783–1793 (2011).
[Crossref] [PubMed]

A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
[Crossref]

G. Bao, H. Zhang, and J. Zou, “Unique determination of periodic polyhedral structures by scattered electromagnetic fields,” Trans. Am. Math. Soc. 363, 4527–4551 (2011).
[Crossref]

G. Hu, J. Yang, and B. Zhang, “An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate,” Appl. Anal. 90, 317–333 (2011).
[Crossref]

G. Hu and B. Zhang, “The linear sampling method for inverse electromagnetic scattering by a partially coated bi-periodic structures,” Math. Methods Appl. Sci. 34, 509–519 (2011).
[Crossref]

J. Yang and B. Zhang, “Inverse electromagnetic scattering problems by a doubly periodic structure,” Math. Appl. Anal. 18, 111–126 (2011).

2009 (2)

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 79, 1–34 (2009).
[Crossref]

Y. Wu and Y. Y. Lu, “Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method,” J. Opt. Soc. Am. A 26, 2444–2451 (2009).
[Crossref]

2006 (2)

I. Akduman, R. Kress, and A. Yapar, “Iterative reconstruction of dielectric rough surface profiles at fixed frequency,” Inverse Probl. 22, 939–954 (2006).
[Crossref]

P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. 32, 136–156 (2006).
[Crossref]

2004 (2)

2003 (5)

Z. Chen and H. Wu, “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).
[Crossref]

T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[Crossref]

G. Bruckner and J. Elschner, “A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles,” Inverse Probl. 19, 315–329 (2003).
[Crossref]

J. Elschner, G. Hsiao, and A. Rathsfeld, “Grating profile reconstruction based on finite elements and optimization techniques,” SIAM J. Appl. Math. 64, 525–545 (2003).

S. Carney and J. Schotland, “Near-field tomography,” MSRI Ser. Math. Appl. 47, 133–168 (2003).

2002 (2)

F. Hettlich, “Iterative regularization schemes in inverse scattering by periodic structures,” Inverse Probl. 18, 701–714 (2002).
[Crossref]

G. Bruckner, J. Cheng, and M. Yamamoto, “An inverse problem in diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
[Crossref]

2001 (1)

P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, “A fully asynchronous multifrontal solver using distributed dynamic scheduling,” SIAM J. Matrix Anal. Appl. 23, 15–41 (2001).
[Crossref]

2000 (2)

S. Carney and J. Schotland, “Inverse scattering for near-field microscopy,” App. Phys. Lett. 77, 2798–2800 (2000).
[Crossref]

R. Kress and T. Tran, “Inverse scattering for a locally perturbed half-plane,” Inverse Probl. 16, 1541–1559 (2000).
[Crossref]

1999 (3)

R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, “An improved operator expansion algorithm for direct and inverse scattering computations,” Waves Random Media 9, 441–457 (1999).
[Crossref]

K. Ito and F. Reitich, “A high-order perturbation approach to profile reconstruction: I. Perfectly conducting gratings,” Inverse Probl. 15, 1067–1085 (1999).
[Crossref]

D. Dobson, “Optimal shape design of blazed diffraction grating,” J. Appl. Math. Optim. 40, 61–78 (1999).
[Crossref]

1998 (3)

J. Elschner and G. Schmidt, “Diffraction in periodic structures and optimal design of binary gratings: I. Direct problems and gradient formulas,” Math. Methods Appl. Sci. 21, 1297–1342 (1998).
[Crossref]

J. Elschner and G. Schmidt, “Numerical solution of optimal design problems for binary gratings,” J. Comput. Phys. 146, 603–626 (1998).
[Crossref]

G. Bao and Z. Zhou, “An inverse problem for scattering by a doubly periodic structure,” Trans. Am. Math. Soc. 350, 4089–4103 (1998).
[Crossref]

1997 (2)

G. Bao, “Variational approximation of Maxwell’s equations in biperiodic structures,” SIAM J. Appl. Math. 57, 364–381 (1997).
[Crossref]

F. Hettlich and A. Kirsch, “Schiffer’s theorem in inverse scattering theory for periodic structures,” Inverse Probl. 13, 351–361 (1997).
[Crossref]

1995 (3)

G. Bao, D. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
[Crossref]

G. Bao and A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Rational Mech. Anal. 132, 49–72 (1995).
[Crossref]

H. Ammari, “Uniqueness theorems for an inverse problem in a doubly periodic structure,” Inverse Probl. 11, 823–833 (1995).
[Crossref]

1994 (3)

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

A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
[Crossref]

G. Bao, “A unique theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
[Crossref]

1993 (3)

1991 (2)

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

J. A. DeSanto and R. J. Wombell, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[Crossref]

Akduman, I.

I. Akduman, R. Kress, and A. Yapar, “Iterative reconstruction of dielectric rough surface profiles at fixed frequency,” Inverse Probl. 22, 939–954 (2006).
[Crossref]

Amestoy, P. R.

P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. 32, 136–156 (2006).
[Crossref]

P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, “A fully asynchronous multifrontal solver using distributed dynamic scheduling,” SIAM J. Matrix Anal. Appl. 23, 15–41 (2001).
[Crossref]

Ammari, H.

H. Ammari, J. Garnier, and K. Solna, “Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,” Proc. Am. Math. Soc. 141, 3431–3446 (2013).
[Crossref]

H. Ammari, J. Garnier, and K. Solna, “Partial data resolving power of conductivity imaging from boundary measurements,” SIAM J. Math. Anal. 45, 1704–1722 (2013).
[Crossref]

H. Ammari, “Uniqueness theorems for an inverse problem in a doubly periodic structure,” Inverse Probl. 11, 823–833 (1995).
[Crossref]

Arens, T.

T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[Crossref]

Bao, G.

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces,” SIAM J. Appl. Math. 73, 2162–2187 (2013).
[Crossref]

G. Bao and J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. 7, 377–396 (2013).
[Crossref]

G. Bao, P. Li, and J. Lv, “Numerical solution of an inverse diffraction grating problem from phasless data,” J. Opt. Soc. Am. A 30, 293–299 (2013).
[Crossref]

G. Bao, P. Li, and H. Wu, “A computational inverse diffraction grating problem,” J. Opt. Soc. Am. A 29, 394–399 (2012).
[Crossref]

G. Bao, H. Zhang, and J. Zou, “Unique determination of periodic polyhedral structures by scattered electromagnetic fields,” Trans. Am. Math. Soc. 363, 4527–4551 (2011).
[Crossref]

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 79, 1–34 (2009).
[Crossref]

G. Bao and Z. Zhou, “An inverse problem for scattering by a doubly periodic structure,” Trans. Am. Math. Soc. 350, 4089–4103 (1998).
[Crossref]

G. Bao, “Variational approximation of Maxwell’s equations in biperiodic structures,” SIAM J. Appl. Math. 57, 364–381 (1997).
[Crossref]

G. Bao and A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Rational Mech. Anal. 132, 49–72 (1995).
[Crossref]

G. Bao, D. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
[Crossref]

G. Bao, “A unique theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
[Crossref]

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces in dielectric media,” SIAJ J. Imaging Sci., to appear.

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001).
[Crossref]

Bruckner, G.

G. Bruckner and J. Elschner, “A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles,” Inverse Probl. 19, 315–329 (2003).
[Crossref]

G. Bruckner, J. Cheng, and M. Yamamoto, “An inverse problem in diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
[Crossref]

Bruno, O.

Carney, S.

S. Carney and J. Schotland, “Near-field tomography,” MSRI Ser. Math. Appl. 47, 133–168 (2003).

S. Carney and J. Schotland, “Inverse scattering for near-field microscopy,” App. Phys. Lett. 77, 2798–2800 (2000).
[Crossref]

Chen, Z.

Z. Chen and H. Wu, “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).
[Crossref]

Cheng, J.

G. Bruckner, J. Cheng, and M. Yamamoto, “An inverse problem in diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
[Crossref]

Cheng, T.

Coifman, R.

R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, “An improved operator expansion algorithm for direct and inverse scattering computations,” Waves Random Media 9, 441–457 (1999).
[Crossref]

Cowsar, L.

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001).
[Crossref]

Cox, J. A.

DeSanto, J. A.

J. A. DeSanto and R. J. Wombell, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[Crossref]

Dobson, D.

D. Dobson, “Optimal shape design of blazed diffraction grating,” J. Appl. Math. Optim. 40, 61–78 (1999).
[Crossref]

G. Bao, D. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
[Crossref]

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

D. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–340 (1993).
[Crossref]

Duff, I. S.

P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, “A fully asynchronous multifrontal solver using distributed dynamic scheduling,” SIAM J. Matrix Anal. Appl. 23, 15–41 (2001).
[Crossref]

Elschner, J.

G. Bruckner and J. Elschner, “A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles,” Inverse Probl. 19, 315–329 (2003).
[Crossref]

J. Elschner, G. Hsiao, and A. Rathsfeld, “Grating profile reconstruction based on finite elements and optimization techniques,” SIAM J. Appl. Math. 64, 525–545 (2003).

J. Elschner and G. Schmidt, “Diffraction in periodic structures and optimal design of binary gratings: I. Direct problems and gradient formulas,” Math. Methods Appl. Sci. 21, 1297–1342 (1998).
[Crossref]

J. Elschner and G. Schmidt, “Numerical solution of optimal design problems for binary gratings,” J. Comput. Phys. 146, 603–626 (1998).
[Crossref]

Friedman, A.

G. Bao and A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Rational Mech. Anal. 132, 49–72 (1995).
[Crossref]

Garnier, J.

H. Ammari, J. Garnier, and K. Solna, “Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,” Proc. Am. Math. Soc. 141, 3431–3446 (2013).
[Crossref]

H. Ammari, J. Garnier, and K. Solna, “Partial data resolving power of conductivity imaging from boundary measurements,” SIAM J. Math. Anal. 45, 1704–1722 (2013).
[Crossref]

Goldberg, M.

R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, “An improved operator expansion algorithm for direct and inverse scattering computations,” Waves Random Media 9, 441–457 (1999).
[Crossref]

Guermouche, A.

P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. 32, 136–156 (2006).
[Crossref]

He, Y.

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic struture,” J. Comput. Phys. 231, 3007–3022 (2012).
[Crossref]

Hettlich, F.

F. Hettlich, “Iterative regularization schemes in inverse scattering by periodic structures,” Inverse Probl. 18, 701–714 (2002).
[Crossref]

F. Hettlich and A. Kirsch, “Schiffer’s theorem in inverse scattering theory for periodic structures,” Inverse Probl. 13, 351–361 (1997).
[Crossref]

Hrycak, T.

R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, “An improved operator expansion algorithm for direct and inverse scattering computations,” Waves Random Media 9, 441–457 (1999).
[Crossref]

Hsiao, G.

J. Elschner, G. Hsiao, and A. Rathsfeld, “Grating profile reconstruction based on finite elements and optimization techniques,” SIAM J. Appl. Math. 64, 525–545 (2003).

Hu, G.

G. Hu, J. Yang, and B. Zhang, “An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate,” Appl. Anal. 90, 317–333 (2011).
[Crossref]

G. Hu and B. Zhang, “The linear sampling method for inverse electromagnetic scattering by a partially coated bi-periodic structures,” Math. Methods Appl. Sci. 34, 509–519 (2011).
[Crossref]

Israeli, M.

R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, “An improved operator expansion algorithm for direct and inverse scattering computations,” Waves Random Media 9, 441–457 (1999).
[Crossref]

Ito, K.

K. Ito and F. Reitich, “A high-order perturbation approach to profile reconstruction: I. Perfectly conducting gratings,” Inverse Probl. 15, 1067–1085 (1999).
[Crossref]

Kirsch, A.

T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[Crossref]

F. Hettlich and A. Kirsch, “Schiffer’s theorem in inverse scattering theory for periodic structures,” Inverse Probl. 13, 351–361 (1997).
[Crossref]

A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
[Crossref]

Koster, J.

P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, “A fully asynchronous multifrontal solver using distributed dynamic scheduling,” SIAM J. Matrix Anal. Appl. 23, 15–41 (2001).
[Crossref]

Kress, R.

I. Akduman, R. Kress, and A. Yapar, “Iterative reconstruction of dielectric rough surface profiles at fixed frequency,” Inverse Probl. 22, 939–954 (2006).
[Crossref]

R. Kress and T. Tran, “Inverse scattering for a locally perturbed half-plane,” Inverse Probl. 16, 1541–1559 (2000).
[Crossref]

L’Excellent, J.-Y.

P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. 32, 136–156 (2006).
[Crossref]

P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, “A fully asynchronous multifrontal solver using distributed dynamic scheduling,” SIAM J. Matrix Anal. Appl. 23, 15–41 (2001).
[Crossref]

Lechleiter, A.

A. Lechleiter and D. L. Nguyen, “On uniqueness in electromagnetic scattering from biperiodic structures,” ESAIM Math. Modell. Numer. Anal. 47, 1167–1184 (2013).
[Crossref]

A. Lechleiter and D. L. Nguyen, “Factorization method for electromagnetic inverse scattering from biperiodic structures,” SIAM J. Imaging Sci. 6, 1111–1139 (2013).
[Crossref]

Li, P.

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces,” SIAM J. Appl. Math. 73, 2162–2187 (2013).
[Crossref]

T. Cheng, P. Li, and Y. Wang, “Near-field imaging of perfectly conducting grating surfaces,” J. Opt. Soc. Am. A 30, 2473–2481 (2013).
[Crossref]

G. Bao, P. Li, and J. Lv, “Numerical solution of an inverse diffraction grating problem from phasless data,” J. Opt. Soc. Am. A 30, 293–299 (2013).
[Crossref]

G. Bao, P. Li, and H. Wu, “A computational inverse diffraction grating problem,” J. Opt. Soc. Am. A 29, 394–399 (2012).
[Crossref]

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 79, 1–34 (2009).
[Crossref]

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces in dielectric media,” SIAJ J. Imaging Sci., to appear.

Lin, J.

G. Bao and J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. 7, 377–396 (2013).
[Crossref]

Lu, Y. Y.

Lv, J.

Malcolm, A.

A. Malcolm and D. P. Nicholls, “A field expansions method for scattering by periodic multilayered media,” J. Acoust. Soc. Am. 129, 1783–1793 (2011).
[Crossref] [PubMed]

A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
[Crossref]

A. Malcolm and D. P. Nicholls, “Operator expansions and constrained quadratic optimization for interface reconstruction: Impenetrable periodic acoustic media,” Wave Motion, to appear.

Masters, W.

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001).
[Crossref]

Nédélec, J. C.

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

Nguyen, D. L.

A. Lechleiter and D. L. Nguyen, “Factorization method for electromagnetic inverse scattering from biperiodic structures,” SIAM J. Imaging Sci. 6, 1111–1139 (2013).
[Crossref]

A. Lechleiter and D. L. Nguyen, “On uniqueness in electromagnetic scattering from biperiodic structures,” ESAIM Math. Modell. Numer. Anal. 47, 1167–1184 (2013).
[Crossref]

D. L. Nguyen, Spectral Methods for Direct and Inverse Scattering from Periodic Structures (PhD thesis, Ecole Polytechnique, Palaiseau, France, 2012).

Nicholls, D. P.

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic struture,” J. Comput. Phys. 231, 3007–3022 (2012).
[Crossref]

A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
[Crossref]

A. Malcolm and D. P. Nicholls, “A field expansions method for scattering by periodic multilayered media,” J. Acoust. Soc. Am. 129, 1783–1793 (2011).
[Crossref] [PubMed]

D. P. Nicholls and F. Reitich, “Shape deformations in rough surface scattering: cancellations, conditioning, and convergence,” J. Opt. Soc. Am. A 21, 590–605 (2004).
[Crossref]

D. P. Nicholls and F. Reitich, “Shape deformations in rough surface scattering: improved algorithms,” J. Opt. Soc. Am. A 21, 606–621 (2004).
[Crossref]

A. Malcolm and D. P. Nicholls, “Operator expansions and constrained quadratic optimization for interface reconstruction: Impenetrable periodic acoustic media,” Wave Motion, to appear.

Pralet, S.

P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. 32, 136–156 (2006).
[Crossref]

Rathsfeld, A.

J. Elschner, G. Hsiao, and A. Rathsfeld, “Grating profile reconstruction based on finite elements and optimization techniques,” SIAM J. Appl. Math. 64, 525–545 (2003).

Reitich, F.

Rokhlin, V.

R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, “An improved operator expansion algorithm for direct and inverse scattering computations,” Waves Random Media 9, 441–457 (1999).
[Crossref]

Sandfort, K.

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media (PhD thesis, Karlsruher Institut für Technologie, 2010).

Schmidt, G.

J. Elschner and G. Schmidt, “Numerical solution of optimal design problems for binary gratings,” J. Comput. Phys. 146, 603–626 (1998).
[Crossref]

J. Elschner and G. Schmidt, “Diffraction in periodic structures and optimal design of binary gratings: I. Direct problems and gradient formulas,” Math. Methods Appl. Sci. 21, 1297–1342 (1998).
[Crossref]

Schotland, J.

S. Carney and J. Schotland, “Near-field tomography,” MSRI Ser. Math. Appl. 47, 133–168 (2003).

S. Carney and J. Schotland, “Inverse scattering for near-field microscopy,” App. Phys. Lett. 77, 2798–2800 (2000).
[Crossref]

Shen, J.

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic struture,” J. Comput. Phys. 231, 3007–3022 (2012).
[Crossref]

Solna, K.

H. Ammari, J. Garnier, and K. Solna, “Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,” Proc. Am. Math. Soc. 141, 3431–3446 (2013).
[Crossref]

H. Ammari, J. Garnier, and K. Solna, “Partial data resolving power of conductivity imaging from boundary measurements,” SIAM J. Math. Anal. 45, 1704–1722 (2013).
[Crossref]

Starling, F.

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

Tran, T.

R. Kress and T. Tran, “Inverse scattering for a locally perturbed half-plane,” Inverse Probl. 16, 1541–1559 (2000).
[Crossref]

Wang, Y.

Wombell, R. J.

J. A. DeSanto and R. J. Wombell, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[Crossref]

Wu, H.

G. Bao, P. Li, and H. Wu, “A computational inverse diffraction grating problem,” J. Opt. Soc. Am. A 29, 394–399 (2012).
[Crossref]

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 79, 1–34 (2009).
[Crossref]

Z. Chen and H. Wu, “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).
[Crossref]

Wu, Y.

Yamamoto, M.

G. Bruckner, J. Cheng, and M. Yamamoto, “An inverse problem in diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
[Crossref]

Yang, J.

J. Yang and B. Zhang, “Inverse electromagnetic scattering problems by a doubly periodic structure,” Math. Appl. Anal. 18, 111–126 (2011).

G. Hu, J. Yang, and B. Zhang, “An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate,” Appl. Anal. 90, 317–333 (2011).
[Crossref]

Yapar, A.

I. Akduman, R. Kress, and A. Yapar, “Iterative reconstruction of dielectric rough surface profiles at fixed frequency,” Inverse Probl. 22, 939–954 (2006).
[Crossref]

Zhang, B.

G. Hu, J. Yang, and B. Zhang, “An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate,” Appl. Anal. 90, 317–333 (2011).
[Crossref]

J. Yang and B. Zhang, “Inverse electromagnetic scattering problems by a doubly periodic structure,” Math. Appl. Anal. 18, 111–126 (2011).

G. Hu and B. Zhang, “The linear sampling method for inverse electromagnetic scattering by a partially coated bi-periodic structures,” Math. Methods Appl. Sci. 34, 509–519 (2011).
[Crossref]

Zhang, H.

G. Bao, H. Zhang, and J. Zou, “Unique determination of periodic polyhedral structures by scattered electromagnetic fields,” Trans. Am. Math. Soc. 363, 4527–4551 (2011).
[Crossref]

Zhou, Z.

G. Bao and Z. Zhou, “An inverse problem for scattering by a doubly periodic structure,” Trans. Am. Math. Soc. 350, 4089–4103 (1998).
[Crossref]

Zou, J.

G. Bao, H. Zhang, and J. Zou, “Unique determination of periodic polyhedral structures by scattered electromagnetic fields,” Trans. Am. Math. Soc. 363, 4527–4551 (2011).
[Crossref]

App. Phys. Lett. (1)

S. Carney and J. Schotland, “Inverse scattering for near-field microscopy,” App. Phys. Lett. 77, 2798–2800 (2000).
[Crossref]

Appl. Anal. (1)

G. Hu, J. Yang, and B. Zhang, “An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate,” Appl. Anal. 90, 317–333 (2011).
[Crossref]

Arch. Rational Mech. Anal. (1)

G. Bao and A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Rational Mech. Anal. 132, 49–72 (1995).
[Crossref]

ESAIM Math. Modell. Numer. Anal. (1)

A. Lechleiter and D. L. Nguyen, “On uniqueness in electromagnetic scattering from biperiodic structures,” ESAIM Math. Modell. Numer. Anal. 47, 1167–1184 (2013).
[Crossref]

Eur. J. Appl. Math. (1)

D. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–340 (1993).
[Crossref]

Inverse Probl. (13)

F. Hettlich, “Iterative regularization schemes in inverse scattering by periodic structures,” Inverse Probl. 18, 701–714 (2002).
[Crossref]

K. Ito and F. Reitich, “A high-order perturbation approach to profile reconstruction: I. Perfectly conducting gratings,” Inverse Probl. 15, 1067–1085 (1999).
[Crossref]

G. Bruckner and J. Elschner, “A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles,” Inverse Probl. 19, 315–329 (2003).
[Crossref]

G. Bruckner, J. Cheng, and M. Yamamoto, “An inverse problem in diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
[Crossref]

F. Hettlich and A. Kirsch, “Schiffer’s theorem in inverse scattering theory for periodic structures,” Inverse Probl. 13, 351–361 (1997).
[Crossref]

A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
[Crossref]

T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[Crossref]

H. Ammari, “Uniqueness theorems for an inverse problem in a doubly periodic structure,” Inverse Probl. 11, 823–833 (1995).
[Crossref]

G. Bao, “A unique theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
[Crossref]

I. Akduman, R. Kress, and A. Yapar, “Iterative reconstruction of dielectric rough surface profiles at fixed frequency,” Inverse Probl. 22, 939–954 (2006).
[Crossref]

J. A. DeSanto and R. J. Wombell, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[Crossref]

R. Kress and T. Tran, “Inverse scattering for a locally perturbed half-plane,” Inverse Probl. 16, 1541–1559 (2000).
[Crossref]

A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
[Crossref]

Inverse Probl. Imag. (1)

G. Bao and J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. 7, 377–396 (2013).
[Crossref]

J. Acoust. Soc. Am. (1)

A. Malcolm and D. P. Nicholls, “A field expansions method for scattering by periodic multilayered media,” J. Acoust. Soc. Am. 129, 1783–1793 (2011).
[Crossref] [PubMed]

J. Appl. Math. Optim. (1)

D. Dobson, “Optimal shape design of blazed diffraction grating,” J. Appl. Math. Optim. 40, 61–78 (1999).
[Crossref]

J. Comput. Phys. (2)

J. Elschner and G. Schmidt, “Numerical solution of optimal design problems for binary gratings,” J. Comput. Phys. 146, 603–626 (1998).
[Crossref]

Y. He, D. P. Nicholls, and J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic struture,” J. Comput. Phys. 231, 3007–3022 (2012).
[Crossref]

J. Opt. Soc. Am. A (9)

Math. Appl. Anal. (1)

J. Yang and B. Zhang, “Inverse electromagnetic scattering problems by a doubly periodic structure,” Math. Appl. Anal. 18, 111–126 (2011).

Math. Comput. (1)

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 79, 1–34 (2009).
[Crossref]

Math. Methods Appl. Sci. (2)

J. Elschner and G. Schmidt, “Diffraction in periodic structures and optimal design of binary gratings: I. Direct problems and gradient formulas,” Math. Methods Appl. Sci. 21, 1297–1342 (1998).
[Crossref]

G. Hu and B. Zhang, “The linear sampling method for inverse electromagnetic scattering by a partially coated bi-periodic structures,” Math. Methods Appl. Sci. 34, 509–519 (2011).
[Crossref]

Math. Model. Numer. Anal. (1)

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

MSRI Ser. Math. Appl. (1)

S. Carney and J. Schotland, “Near-field tomography,” MSRI Ser. Math. Appl. 47, 133–168 (2003).

Parallel Comput. (1)

P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, and S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. 32, 136–156 (2006).
[Crossref]

Proc. Am. Math. Soc. (1)

H. Ammari, J. Garnier, and K. Solna, “Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,” Proc. Am. Math. Soc. 141, 3431–3446 (2013).
[Crossref]

SIAM J. Appl. Math. (3)

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces,” SIAM J. Appl. Math. 73, 2162–2187 (2013).
[Crossref]

G. Bao, “Variational approximation of Maxwell’s equations in biperiodic structures,” SIAM J. Appl. Math. 57, 364–381 (1997).
[Crossref]

J. Elschner, G. Hsiao, and A. Rathsfeld, “Grating profile reconstruction based on finite elements and optimization techniques,” SIAM J. Appl. Math. 64, 525–545 (2003).

SIAM J. Imaging Sci. (1)

A. Lechleiter and D. L. Nguyen, “Factorization method for electromagnetic inverse scattering from biperiodic structures,” SIAM J. Imaging Sci. 6, 1111–1139 (2013).
[Crossref]

SIAM J. Math. Anal. (2)

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

H. Ammari, J. Garnier, and K. Solna, “Partial data resolving power of conductivity imaging from boundary measurements,” SIAM J. Math. Anal. 45, 1704–1722 (2013).
[Crossref]

SIAM J. Matrix Anal. Appl. (1)

P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, “A fully asynchronous multifrontal solver using distributed dynamic scheduling,” SIAM J. Matrix Anal. Appl. 23, 15–41 (2001).
[Crossref]

SIAM J. Numer. Anal. (1)

Z. Chen and H. Wu, “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).
[Crossref]

Trans. Am. Math. Soc. (2)

G. Bao, H. Zhang, and J. Zou, “Unique determination of periodic polyhedral structures by scattered electromagnetic fields,” Trans. Am. Math. Soc. 363, 4527–4551 (2011).
[Crossref]

G. Bao and Z. Zhou, “An inverse problem for scattering by a doubly periodic structure,” Trans. Am. Math. Soc. 350, 4089–4103 (1998).
[Crossref]

Waves Random Media (1)

R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, “An improved operator expansion algorithm for direct and inverse scattering computations,” Waves Random Media 9, 441–457 (1999).
[Crossref]

Other (7)

D. L. Nguyen, Spectral Methods for Direct and Inverse Scattering from Periodic Structures (PhD thesis, Ecole Polytechnique, Palaiseau, France, 2012).

K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media (PhD thesis, Karlsruher Institut für Technologie, 2010).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).
[Crossref]

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001).
[Crossref]

A. Malcolm and D. P. Nicholls, “Operator expansions and constrained quadratic optimization for interface reconstruction: Impenetrable periodic acoustic media,” Wave Motion, to appear.

PHG (Parallel Hierarchical Grid), http://lsec.cc.ac.cn/phg/ .

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces in dielectric media,” SIAJ J. Imaging Sci., to appear.

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Figures (4)

Fig. 1
Fig. 1

Geometry of the diffraction grating problem.

Fig. 2
Fig. 2

The exact grating profile ψ. (a) Example 1: smooth grating profile with finite Fourier modes; (b) Example 2: non-smooth grating profile with infinite Fourier modes.

Fig. 3
Fig. 3

Example 1: Reconstructed grating surfaces by using different h with ε = 0.025λ and δ = 5%. (a) h = 0.4λ; (b) h = 0.3λ; (c) h = 0.2λ; (d) h = 0.1λ.

Fig. 4
Fig. 4

Example 2: Reconstructed grating surfaces by using different h with ε = 0.0125λ and δ = 5%. (a) h = 0.2λ; (b) h = 0.1λ; (c) h = 0.05λ; (d) h = 0.025λ.

Tables (4)

Tables Icon

Table 1 Example 1: Relative error of the reconstructions by using different ε with h = 0.4λ and δ = 0.0.

Tables Icon

Table 2 Example 1: Relative error of the reconstructions by using different h with ε = 0.025λ and δ = 5%.

Tables Icon

Table 3 Example 2: Relative error of the reconstructions by using different ε with h = 0.2λ and δ = 0.0.

Tables Icon

Table 4 Example 2: Relative error of the reconstructions by using different h with ε = 0.0125λ and δ = 5%.

Equations (87)

Equations on this page are rendered with MathJax. Learn more.

ϕ ( ρ ) = ε ψ ( ρ ) ,
E inc = t exp ( i κ ( α ρ β z ) ) , H inc = s exp ( i κ ( α ρ β z ) ) .
s 1 = α 2 t 3 + β t 2 , s 2 = ( α 1 t 3 + β t 1 ) , s 3 = α 1 t 2 α 2 t 1 .
E j inc = t j exp ( i κ z ) , H j inc = s j exp ( i κ z ) .
× E inc i κ H inc = 0 , × H inc + i κ E inc = 0 , in 3 .
× E i κ H = 0 , × H + i κ E = 0 , in Ω S ,
E = 0 and H = 0 in Ω S .
E × ν S = 0 on S ,
ν 1 = ϕ x ( 1 + ϕ x 2 + ϕ y 2 ) 1 / 2 , ν 2 = ϕ y ( 1 + ϕ x 2 + ϕ y 2 ) 1 / 2 , ν 3 = 1 ( 1 + ϕ x 2 + ϕ y 2 ) 1 / 2 .
u ( ρ ) = n 2 u n exp ( i α n ρ ) , u n = Λ 1 1 Λ 2 1 0 Λ 1 0 Λ 2 u ( ρ ) exp ( i α n ρ ) d ρ .
u Γ = ν Γ × ( u × ν Γ ) = ( u 1 ( ρ , h ) , u 2 ( ρ , h ) , 0 ) ,
T u = ( v 1 ( ρ , h ) , v 2 ( ρ , h ) , 0 ) ,
u j ( ρ , h ) = n 2 u j n ( h ) exp ( i α n ρ ) , v j ( ρ , h ) = n 2 v j n ( h ) exp ( i α n ρ ) ,
{ v 1 n ( h ) = 1 κ β n [ ( κ 2 α 2 n 2 ) u 1 n ( h ) + α 1 n α 2 n u 2 n ( h ) ] , v 2 n ( h ) = 1 κ β n [ ( κ 2 α 1 n 2 ) u 2 n ( h ) + α 1 n α 2 n u 1 n ( h ) ) ] .
( × E ) × ν Γ = i κ T E Γ + f , on Γ ,
f = i κ ( H inc × ν Γ T E Γ inc ) = ( f 1 , f 2 , 0 ) .
f 1 = 2 i κ t 1 exp ( i κ h ) and f 2 = 2 i κ t 2 exp ( i κ h ) .
{ × ( × E ) κ 2 E = 0 in Ω , E × ν S = 0 on S , ( × E ) × ν Γ i κ T E Γ = f on Γ .
Δ E j + κ 2 E j = 0 in Ω .
x E 1 + y E 2 + z E 3 = 0 in Ω .
E 2 + ϕ y E 3 = 0 , E 1 + ϕ x E 3 = 0 , ϕ y E 1 ϕ x E 2 = 0 , on z = ϕ ( ρ ) .
z E 1 x E 3 = i κ H 1 + f 1 , z E 2 y E 3 = i κ H 2 + f 2 , on z = h ,
{ H 1 n ( h ) = 1 κ β n [ ( κ 2 α 2 n 2 ) E 1 n ( h ) + α 1 n α 2 n E 2 n ( h ) ] , H 2 n ( h ) = 1 κ β n [ ( κ 2 α 1 n 2 ) E 2 n ( h ) + α 1 n α 2 n E 1 n ( h ) ] .
x ˜ = x , y ˜ = y , z ˜ = h ( z ϕ h ϕ ) ,
D = { ( x ˜ , y ˜ , z ˜ ) 3 : 0 < z ˜ < h } = 2 × ( 0 , h ) .
c 1 2 E j x 2 + c 1 2 E j y 2 + c 2 2 E j z 2 c 3 2 E j x z c 4 2 E j y z c 5 E j z + κ 2 c 1 E j = 0 in D ,
c 1 = ( h ϕ ) 2 , c 2 = ( ϕ x 2 + ϕ y 2 ) ( h z ) 2 + h 2 , c 3 = 2 ϕ x ( h z ) ( h ϕ ) , c 4 = 2 ϕ y ( h z ) ( h ϕ ) , c 5 = ( h z ) [ ( ϕ x x + ϕ y y ) ( h ϕ ) + 2 ( ϕ x 2 + ϕ y 2 ) ] .
x E 1 + y E 2 ( h z h ϕ ) ( ϕ x z E 1 + ϕ y z E 2 ) + ( h h ϕ ) z E 3 = 0 in D .
E 2 + ϕ y E 3 = 0 , E 1 + ϕ x E 3 = 0 , ϕ y E 1 ϕ x E 2 = 0 , z = 0 .
( h h ϕ ) z E 1 x E 3 = i κ H 1 + f 1 , ( h h ϕ ) z E 2 y E 3 = i κ H 2 + f 2 .
E j ( ρ , z ; ε ) = k = 0 E j ( k ) ( ρ , z ) ε k .
Δ E j ( k ) + κ 2 E j ( k ) = F j ( k ) in D ,
F j ( k ) = 2 ψ h 2 E j ( k 1 ) x 2 + 2 ψ h 2 E j ( k 1 ) y 2 + 2 ( h z ) ψ x h 2 E j ( k 1 ) x x + 2 ( h z ) ψ y h 2 E j ( k 1 ) y z + ( h z ) ( ψ x x + ψ y y ) h E j ( k 1 ) z + 2 κ 2 ψ h E j ( k 1 ) ψ 2 h 2 2 E j ( k 2 ) x 2 ψ 2 h 2 2 E j ( k 2 ) y 2 ( h z ) 2 ( ψ x 2 + ψ y 2 ) h 2 2 E j ( k 2 ) z 2 2 ψ ψ x ( h z ) h 2 2 E j ( k 2 ) x z 2 ψ ψ y ( h z ) h 2 2 E j ( k 2 ) y z + ( h z ) [ 2 ( ψ x 2 + ψ y 2 ) ψ ( ψ x x + ψ y y ) ] h 2 E j ( k 2 ) z κ 2 ψ 2 h 2 E j ( k 2 ) .
x E 1 ( k ) + y E 2 ( k ) + z E 3 ( k ) = w ( k ) in D ,
w ( k ) = ψ h ( x E 1 ( k 1 ) + y E 2 ( k 1 ) ) + ( h z h ) ( ψ x z E 1 ( k 1 ) + ψ y z E 2 ( k 1 ) ) .
E 1 ( k ) = u 1 ( k ) , E 2 ( k ) = u 2 ( k ) ,
u 1 ( k ) ( ρ ) = ψ x E 3 ( k 1 ) , u 2 ( k ) = ψ y E 3 ( k 1 ) .
x E 1 ( k ) + y E 2 ( k ) + z E 3 ( k ) = u 3 ( k ) , z = 0 ,
u 3 ( k ) ( ρ ) = ψ h ( x E 1 ( k 1 ) + y E 2 ( k 1 ) ) + ( ψ x z E 1 ( k 1 ) + ψ y z E 2 ( k 1 ) ) .
{ z E 1 ( k ) x E 3 ( k ) = i κ H 1 ( k ) + v 1 ( k ) , z E 2 ( k ) y E 3 ( k ) = i κ H 2 ( k ) + v 2 ( k ) ,
v 1 ( 0 ) = f 1 , v 1 ( 1 ) = ψ h z E 1 ( 0 ) , v 1 ( k ) = ψ h ( x E 3 ( k 1 ) + i κ H 1 ( k 1 ) ) , v 2 ( 0 ) = f 2 , v 2 ( 1 ) = ψ h z E 2 ( 0 ) , v 2 ( k ) = ψ h ( y E 3 ( k 1 ) + i κ H 2 ( k 1 ) ) ,
{ H 1 n ( k ) ( h ) = 1 κ β n [ ( κ 2 α 2 n 2 ) E 1 n ( k ) ( h ) + α 1 n α 2 n E 2 n ( k ) ( h ) ] , H 2 n ( k ) ( h ) = 1 κ β n [ ( κ 2 α 1 n 2 ) E 2 n ( k ) ( h ) + α 1 n α 2 n E 1 n ( k ) ( h ) ] .
x E 1 ( k ) + y E 2 ( k ) + z E 3 ( k ) = v 3 ( k ) ,
v 3 ( k ) ( ρ ) = ψ h ( x E 1 ( k 1 ) + y E 2 ( k 1 ) ) .
Δ E j ( 0 ) + κ 2 E j ( 0 ) = 0 in D .
x E 1 ( 0 ) + y E 2 ( 0 ) + z E 3 ( 0 ) = 0 in D .
E 1 ( 0 ) ( ρ , 0 ) = 0 , E 2 ( 0 ) ( ρ , 0 ) = 0 .
z E 3 ( 0 ) ( ρ , 0 ) = x E 1 ( 0 ) ( ρ , 0 ) y E 2 ( 0 ) ( ρ , 0 ) = 0 .
{ z E 1 ( 0 ) ( ρ , h ) x E 3 ( 0 ) ( ρ , h ) = i κ H 1 ( 0 ) ( ρ , h ) + f 1 ( ρ ) , z E 2 ( 0 ) ( ρ , h ) y E 3 ( 0 ) ( ρ , h ) = i κ H 2 ( 0 ) ( ρ , h ) + f 2 ( ρ ) .
x E 1 ( 0 ) ( ρ , h ) + y E 2 ( 0 ) ( ρ , h ) + z E 3 ( 0 ) ( ρ , h ) = 0 .
E j ( 0 ) ( ρ , z ) = n 2 E j n ( 0 ) ( z ) exp ( i α n ρ ) , f j = n 2 f j n exp ( i α n ρ ) ,
d 2 E j n ( 0 ) ( z ) d z 2 + ( κ 2 | α n | 2 ) E j n ( 0 ) ( z ) = 0 , 0 < z < h .
E 1 n ( 0 ) = 0 , E 2 n ( 0 ) = 0 , E 3 n ( 0 ) = 0 ,
{ E 1 n ( 0 ) i α 1 n E 3 n ( 0 ) = i β n [ ( κ 2 α 2 n 2 ) E 1 n ( 0 ) + α 1 n α 2 n E 2 n ( 0 ) ] + f 1 n , E 2 n ( 0 ) i α 2 n E 3 n ( 0 ) = i β n [ ( κ 2 α 1 n 2 ) E 2 n ( 0 ) + α 1 n α 2 n E 1 n ( 0 ) ] + f 2 n , E 3 n ( 0 ) + i α 1 n E 1 n ( 0 ) + i α 2 n E 2 n ( 0 ) = 0 .
E j 0 ( ρ , z ) = t j ( exp ( i κ z ) exp ( i κ z ) ) .
Δ E j ( 1 ) + κ 2 E j ( 1 ) = F j ( 1 ) in D ,
F j ( 1 ) = 2 ψ h 2 E j ( 0 ) x 2 + 2 ψ h 2 E j ( 0 ) y 2 + 2 ( h z ) ψ x h 2 E j ( 0 ) x z + 2 ( h z ) ψ y h 2 E j ( 0 ) y z + ( h z ) ( ψ x x + ψ y y ) h E j ( 0 ) z + 2 κ 2 ψ h E j ( 0 ) .
F j ( 1 ) ( ρ , z ) = 2 κ 2 t j h ψ ( exp ( i κ z ) exp ( i κ z ) ) i κ t j ( h z ) h ( ψ x x + ψ y y ) ( exp ( i κ z ) + exp ( i κ z ) ) .
x E 1 ( 1 ) + y E 2 ( 1 ) + z E 3 ( 1 ) = w ( 1 ) in D ,
w ( 1 ) ( ρ , z ) = ψ h ( x E 1 ( 0 ) + y E 2 ( 0 ) ) + ( h z h ) ( ψ x z E 1 ( 0 ) + ψ y z E 2 ( 0 ) ) = i κ ( h z ) h ( t 1 ψ x + t 2 ψ y ) ( exp ( i κ z ) + exp ( i κ z ) ) .
E 1 ( 1 ) ( ρ , 0 ) = u 1 ( 1 ) ( ρ ) = ψ x ( ρ ) E 3 ( 0 ) ( ρ , 0 ) = 0 , E 2 ( 1 ) ( ρ , 0 ) = u 2 ( 1 ) ( ρ ) = ψ y ( ρ ) E 3 ( 0 ) ( ρ , 0 ) = 0 .
z E 3 ( 1 ) ( ρ , 0 ) = w ( 1 ) ( ρ , 0 ) x E 1 ( 1 ) ( ρ , 0 ) y E 2 ( 1 ) ( ρ , 0 ) = 2 i κ ( t 1 ψ x + t 2 ψ y ) .
z E 1 ( 1 ) x E 3 ( 1 ) = i κ H 1 ( 1 ) + v 1 ( 1 ) , z E 2 ( 1 ) y E 3 ( 1 ) = i κ H 2 ( 1 ) + v 2 ( 1 ) ,
v j ( 1 ) ( ρ ) = ψ h z E j ( 0 ) ( ρ , h ) = i κ t j h ( exp ( i κ h ) + exp ( i κ h ) ) ψ .
x E 1 ( 1 ) ( ρ , h ) + y E 2 ( 1 ) ( ρ , z ) + z E 3 ( 1 ) ( ρ , z ) = 0 .
ψ ( ρ ) = n 2 ψ n exp ( i α n ρ ) , E j ( 1 ) ( ρ , z ) = n 2 E j n ( 1 ) ( z ) exp ( i α n ρ ) , F j ( 1 ) ( ρ , z ) = n 2 F j n ( 1 ) ( z ) exp ( i α n ρ ) ,
F j n ( 1 ) ( z ) = [ 2 κ 2 t j h ( exp ( i κ z ) exp ( i κ z ) ) + i κ t j ( h z ) h ( α 1 n 2 + α 2 n 2 ) ( exp ( i κ z ) + exp ( i κ z ) ) ] ψ n .
d 2 E j n ( 1 ) ( z ) d z 2 + ( κ 2 | α n | 2 ) E j n ( 1 ) ( z ) = F j n ( 1 ) ( z ) , 0 < z < h ,
E 1 n ( 1 ) = 0 , E 2 n ( 1 ) = 0 , E 3 n ( 1 ) ( 0 ) = 2 κ ( α 1 n + α 2 n ) ψ n ,
{ E 1 n ( 1 ) i α 1 n E 3 n ( 1 ) = i β n [ ( κ 2 α 2 n 2 ) E 1 n ( 1 ) + α 1 n α 2 n E 2 n ( 1 ) ] + v 1 n ( 1 ) , E 2 n ( 1 ) i α 2 n E 3 n ( 1 ) = i β n [ ( κ 2 α 1 n 2 ) E 2 n ( 1 ) + α 1 n α 2 n E 1 n ( 1 ) ] + v 2 n ( 1 ) , E 3 n ( 1 ) + i α 1 n E 1 n ( 1 ) + i α 2 n E 2 n ( 1 ) = 0 ,
v j n ( 1 ) = i κ t j h ( exp ( i κ h ) + exp ( i κ h ) ) ψ n .
E 1 n ( 1 ) ( h ) = 2 i κ t 1 exp ( i β n h ) ψ n and E 2 n ( 1 ) ( h ) = 2 i κ t 2 exp ( i β n h ) ψ n ,
E j δ ( ρ , h ) = E j ( ρ , h ) + 𝒪 ( δ ) ,
E j ( ρ , h ) = E j ( 0 ) ( ρ , h ) + ε E j ( 1 ) ( ρ , h ) + 𝒪 ( ε 2 ) + 𝒪 ( δ ) .
ε E j ( 1 ) ( ρ , h ) = E j δ ( ρ , h ) E j ( 0 ) ( ρ , h )
ϕ n = ( 2 i κ t j ) 1 [ E j n δ ( h ) E j n ( 0 ) ( h ) ] exp ( i β n h ) ,
E j n ( 0 ) ( h ) = t j ( exp ( i κ h ) exp ( i κ h ) ) δ 0 n .
SNR = min { ε 2 , δ 1 } .
exp ( ( κ c 2 κ 2 ) 1 / 2 h ) = SNR ,
κ c κ = [ 1 + ( log SNR κ h ) 2 ] 1 / 2 ,
ϕ n = ( 2 i κ t j ) 1 [ E j n δ ( h ) E j n ( 0 ) ( h ) ] exp ( i β n h ) χ n ,
χ n = { 1 for | α n | κ c , 0 for | α n | > κ c .
ϕ ( ρ ) n ϕ n exp ( i α n ρ ) = | α n | κ c ( 2 i κ t j ) 1 [ E j n δ ( h ) E j n ( 0 ) ( h ) ] exp ( i ( α n ρ β n h ) ) = | α n | κ c ( 2 i κ t j ) 1 E j n δ ( h ) exp ( i ( α n ρ β n h ) ) + ( 2 i κ ) 1 ( 1 exp ( 2 i κ h ) ) .
E 1 δ ( ρ , h ) = E 1 ( ρ , h ) ( 1 + δ rand ) ,
e = ϕ ϕ δ , ε 0 , R ϕ 0 , R ,
ψ ( x , y ) = 0.6 sin ( 2 π x ) sin ( 2 π y ) + sin ( 4 π x ) sin ( 4 π y ) .
ψ ( x , y ) = | sin ( 2 π x ) sin ( 2 π y ) | | cos ( 2 π x ) cos ( 2 π y ) | .

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