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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).
    [CrossRef]
  2. G. Bao, D. Dobson, J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
    [CrossRef]
  3. Z. Chen, 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]
  4. J. C. Nédélec, 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]
  5. Y. Wu, 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]
  6. G. Bao, L. Cowsar, W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001).
    [CrossRef]
  7. G. Bao, “A unique theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
    [CrossRef]
  8. G. Bao, A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Rational Mech. Anal. 132, 49–72 (1995).
    [CrossRef]
  9. G. Bruckner, J. Cheng, M. Yamamoto, “An inverse problem in diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
    [CrossRef]
  10. F. Hettlich, A. Kirsch, “Schiffer’s theorem in inverse scattering theory for periodic structures,” Inverse Probl. 13, 351–361 (1997).
    [CrossRef]
  11. A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
    [CrossRef]
  12. T. Arens, A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
    [CrossRef]
  13. G. Bao, P. Li, H. Wu, “A computational inverse diffraction grating problem,” J. Opt. Soc. Am. A 29, 394–399 (2012).
    [CrossRef]
  14. G. Bao, P. Li, J. Lv, “Numerical solution of an inverse diffraction grating problem from phasless data,” J. Opt. Soc. Am. A 30, 293–299 (2013).
    [CrossRef]
  15. G. Bruckner, J. Elschner, “A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles,” Inverse Probl. 19, 315–329 (2003).
    [CrossRef]
  16. J. Elschner, G. Hsiao, A. Rathsfeld, “Grating profile reconstruction based on finite elements and optimization techniques,” SIAM J. Appl. Math. 64, 525–545 (2003).
  17. F. Hettlich, “Iterative regularization schemes in inverse scattering by periodic structures,” Inverse Probl. 18, 701–714 (2002).
    [CrossRef]
  18. K. Ito, F. Reitich, “A high-order perturbation approach to profile reconstruction: I. Perfectly conducting gratings,” Inverse Probl. 15, 1067–1085 (1999).
    [CrossRef]
  19. A. Malcolm, D. P. Nicholls, “Operator expansions and constrained quadratic optimization for interface reconstruction: Impenetrable periodic acoustic media,” Wave Motion, to appear.
  20. D. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–340 (1993).
    [CrossRef]
  21. D. Dobson, “Optimal shape design of blazed diffraction grating,” J. Appl. Math. Optim. 40, 61–78 (1999).
    [CrossRef]
  22. J. Elschner, 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]
  23. J. Elschner, G. Schmidt, “Numerical solution of optimal design problems for binary gratings,” J. Comput. Phys. 146, 603–626 (1998).
    [CrossRef]
  24. I. Akduman, R. Kress, A. Yapar, “Iterative reconstruction of dielectric rough surface profiles at fixed frequency,” Inverse Probl. 22, 939–954 (2006).
    [CrossRef]
  25. R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, V. Rokhlin, “An improved operator expansion algorithm for direct and inverse scattering computations,” Waves Random Media 9, 441–457 (1999).
    [CrossRef]
  26. J. A. DeSanto, R. J. Wombell, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
    [CrossRef]
  27. R. Kress, T. Tran, “Inverse scattering for a locally perturbed half-plane,” Inverse Probl. 16, 1541–1559 (2000).
    [CrossRef]
  28. G. Bao, “Variational approximation of Maxwell’s equations in biperiodic structures,” SIAM J. Appl. Math. 57, 364–381 (1997).
    [CrossRef]
  29. G. Bao, P. Li, 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]
  30. D. Dobson, “A variational method for electromagnetic diffraction in biperiodic structures,” Math. Model. Numer. Anal. 28, 419–439 (1994).
  31. A. Lechleiter, D. L. Nguyen, “On uniqueness in electromagnetic scattering from biperiodic structures,” ESAIM Math. Modell. Numer. Anal. 47, 1167–1184 (2013).
    [CrossRef]
  32. H. Ammari, “Uniqueness theorems for an inverse problem in a doubly periodic structure,” Inverse Probl. 11, 823–833 (1995).
    [CrossRef]
  33. G. Bao, H. Zhang, J. Zou, “Unique determination of periodic polyhedral structures by scattered electromagnetic fields,” Trans. Am. Math. Soc. 363, 4527–4551 (2011).
    [CrossRef]
  34. G. Bao, Z. Zhou, “An inverse problem for scattering by a doubly periodic structure,” Trans. Am. Math. Soc. 350, 4089–4103 (1998).
    [CrossRef]
  35. G. Hu, J. Yang, 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]
  36. G. Hu, 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]
  37. J. Yang, B. Zhang, “Inverse electromagnetic scattering problems by a doubly periodic structure,” Math. Appl. Anal. 18, 111–126 (2011).
  38. A. Lechleiter, D. L. Nguyen, “Factorization method for electromagnetic inverse scattering from biperiodic structures,” SIAM J. Imaging Sci. 6, 1111–1139 (2013).
    [CrossRef]
  39. D. L. Nguyen, Spectral Methods for Direct and Inverse Scattering from Periodic Structures (PhD thesis, Ecole Polytechnique, Palaiseau, France, 2012).
  40. K. Sandfort, The factorization method for inverse scattering from periodic inhomogeneous media (PhD thesis, Karlsruher Institut für Technologie, 2010).
  41. O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
    [CrossRef]
  42. O. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
    [CrossRef]
  43. Y. He, D. P. Nicholls, J. Shen, “An efficient and stable spectral method for electromagnetic scattering from a layered periodic struture,” J. Comput. Phys. 231, 3007–3022 (2012).
    [CrossRef]
  44. A. Malcolm, D. P. Nicholls, “A field expansions method for scattering by periodic multilayered media,” J. Acoust. Soc. Am. 129, 1783–1793 (2011).
    [CrossRef] [PubMed]
  45. A. Malcolm, D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
    [CrossRef]
  46. D. P. Nicholls, F. Reitich, “Shape deformations in rough surface scattering: cancellations, conditioning, and convergence,” J. Opt. Soc. Am. A 21, 590–605 (2004).
    [CrossRef]
  47. D. P. Nicholls, F. Reitich, “Shape deformations in rough surface scattering: improved algorithms,” J. Opt. Soc. Am. A 21, 606–621 (2004).
    [CrossRef]
  48. G. Bao, P. Li, “Near-field imaging of infinite rough surfaces,” SIAM J. Appl. Math. 73, 2162–2187 (2013).
    [CrossRef]
  49. G. Bao, P. Li, “Near-field imaging of infinite rough surfaces in dielectric media,” SIAJ J. Imaging Sci., to appear.
  50. T. Cheng, P. Li, Y. Wang, “Near-field imaging of perfectly conducting grating surfaces,” J. Opt. Soc. Am. A 30, 2473–2481 (2013).
    [CrossRef]
  51. G. Bao, J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. 7, 377–396 (2013).
    [CrossRef]
  52. S. Carney, J. Schotland, “Inverse scattering for near-field microscopy,” App. Phys. Lett. 77, 2798–2800 (2000).
    [CrossRef]
  53. S. Carney, J. Schotland, “Near-field tomography,” MSRI Ser. Math. Appl. 47, 133–168 (2003).
  54. H. Ammari, J. Garnier, K. Solna, “Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,” Proc. Am. Math. Soc. 141, 3431–3446 (2013).
    [CrossRef]
  55. H. Ammari, J. Garnier, K. Solna, “Partial data resolving power of conductivity imaging from boundary measurements,” SIAM J. Math. Anal. 45, 1704–1722 (2013).
    [CrossRef]
  56. PHG (Parallel Hierarchical Grid), http://lsec.cc.ac.cn/phg/ .
  57. P. R. Amestoy, I. S. Duff, J. Koster, J.-Y. L’Excellent, “A fully asynchronous multifrontal solver using distributed dynamic scheduling,” SIAM J. Matrix Anal. Appl. 23, 15–41 (2001).
    [CrossRef]
  58. P. R. Amestoy, A. Guermouche, J.-Y. L’Excellent, S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. 32, 136–156 (2006).
    [CrossRef]

2013 (8)

G. Bao, P. Li, 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, D. L. Nguyen, “On uniqueness in electromagnetic scattering from biperiodic structures,” ESAIM Math. Modell. Numer. Anal. 47, 1167–1184 (2013).
[CrossRef]

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

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

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

A. Lechleiter, 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, 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, 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, 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, H. Wu, “A computational inverse diffraction grating problem,” J. Opt. Soc. Am. A 29, 394–399 (2012).
[CrossRef]

2011 (6)

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

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

2009 (2)

G. Bao, P. Li, 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, 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, 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, S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. 32, 136–156 (2006).
[CrossRef]

2004 (2)

2003 (5)

Z. Chen, 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, A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[CrossRef]

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

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

R. Kress, 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, V. Rokhlin, “An improved operator expansion algorithm for direct and inverse scattering computations,” Waves Random Media 9, 441–457 (1999).
[CrossRef]

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

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

G. Bao, 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, 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, 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, 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, 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, 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, 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, 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, A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[CrossRef]

Bao, G.

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

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

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

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

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

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

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

Bruckner, G.

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

Bruno, O.

Carney, S.

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

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

Chen, Z.

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

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

Friedman, A.

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

Garnier, J.

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

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

Goldberg, M.

R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
[CrossRef]

F. Hettlich, 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, 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, A. Yapar, “Iterative reconstruction of dielectric rough surface profiles at fixed frequency,” Inverse Probl. 22, 939–954 (2006).
[CrossRef]

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

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

Li, P.

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

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

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

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

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

Lin, J.

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

A. Malcolm, 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, 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, W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001).
[CrossRef]

Nédélec, J. C.

J. C. Nédélec, 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, D. L. Nguyen, “On uniqueness in electromagnetic scattering from biperiodic structures,” ESAIM Math. Modell. Numer. Anal. 47, 1167–1184 (2013).
[CrossRef]

A. Lechleiter, D. L. Nguyen, “Factorization method for electromagnetic inverse scattering from biperiodic structures,” SIAM J. Imaging Sci. 6, 1111–1139 (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, 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, 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, D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
[CrossRef]

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

A. Malcolm, 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, S. Pralet, “Hybrid scheduling for the parallel solution of linear systems,” Parallel Comput. 32, 136–156 (2006).
[CrossRef]

Rathsfeld, A.

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

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

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

Shen, J.

Y. He, D. P. Nicholls, 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, 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, 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, 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, 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, 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, H. Wu, “A computational inverse diffraction grating problem,” J. Opt. Soc. Am. A 29, 394–399 (2012).
[CrossRef]

G. Bao, P. Li, 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, 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, M. Yamamoto, “An inverse problem in diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
[CrossRef]

Yang, J.

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

Yapar, A.

I. Akduman, R. Kress, 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, 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, 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, B. Zhang, “Inverse electromagnetic scattering problems by a doubly periodic structure,” Math. Appl. Anal. 18, 111–126 (2011).

Zhang, H.

G. Bao, H. Zhang, 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, 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, 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, J. Schotland, “Inverse scattering for near-field microscopy,” App. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

Appl. Anal. (1)

G. Hu, J. Yang, 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, 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, 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, F. Reitich, “A high-order perturbation approach to profile reconstruction: I. Perfectly conducting gratings,” Inverse Probl. 15, 1067–1085 (1999).
[CrossRef]

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

F. Hettlich, 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, 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, A. Yapar, “Iterative reconstruction of dielectric rough surface profiles at fixed frequency,” Inverse Probl. 22, 939–954 (2006).
[CrossRef]

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

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

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

Y. He, D. P. Nicholls, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, J. Zou, “Unique determination of periodic polyhedral structures by scattered electromagnetic fields,” Trans. Am. Math. Soc. 363, 4527–4551 (2011).
[CrossRef]

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

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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 ) | .

Metrics