Abstract

A novel approach is presented to solving the inverse diffractive grating problem in near-field optical imaging, which is to reconstruct perfectly conducting grating surfaces with resolution beyond the diffraction limit. The grating surface is assumed to be a small and smooth deformation of a plane surface. An analytical solution of the direct grating problems is derived by using the method of transformed field expansion. Based on the analytic solution, an explicit reconstruction formula is deduced for the inverse grating problem. The method requires only a single incident field and is realized efficiently by using the fast Fourier transform. Numerical results show that the method is simple, stable, and effective in reconstructing grating surfaces with super-resolved resolution.

© 2013 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
  2. G. Bao, D. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
    [CrossRef]
  3. 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]
  4. 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]
  5. G. Bao, Z. Chen, and H. Wu, “Adaptive finite element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
    [CrossRef]
  6. G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001).
  7. A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
    [CrossRef]
  8. G. Bao, “A unique theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
    [CrossRef]
  9. H. Ammari, “Uniqueness theorems for an inverse problem in a doubly periodic structure,” Inverse Probl. 11, 823–833 (1995).
    [CrossRef]
  10. F. Hettlich and A. Kirsch, “Schiffer’s theorem in inverse scattering theory for periodic structures,” Inverse Probl. 13, 351–361 (1997).
    [CrossRef]
  11. G. Bao and A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Ration. Mech. Anal. 132, 49–72 (1995).
    [CrossRef]
  12. G. Bao and Z. Zhou, “An inverse problem for scattering by a doubly periodic structure,” Trans. Am. Math. Soc. 350, 4089–4103 (1998).
    [CrossRef]
  13. 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]
  14. G. Bruckner, J. Cheng, and M. Yamamoto, “An inverse problem in diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
    [CrossRef]
  15. N. García and M. Nieto-Vesperinas, “Near-field optics inverse-scattering reconstruction of reflective surfaces,” Opt. Lett. 24, 2090–2092 (1993).
    [CrossRef]
  16. K. Ito and F. Reitich, “A high-order perturbation approach to profile reconstruction: I. Perfectly conducting gratings,” Inverse Probl. 15, 1067–1085 (1999).
    [CrossRef]
  17. T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
    [CrossRef]
  18. F. Hettlich, “Iterative regularization schemes in inverse scattering by periodic structures,” Inverse Probl. 18, 701–714 (2002).
    [CrossRef]
  19. G. Bruckner and J. Elschner, “A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles,” Inverse Probl. 19, 315–329 (2003).
    [CrossRef]
  20. 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).
  21. G. Bao, P. Li, and H. Wu, “A computational inverse diffraction grating problem,” J. Opt. Soc. Am. A 29, 394–399 (2012).
    [CrossRef]
  22. G. Bao, P. Li, and J. Lv, “Numerical solution of an inverse diffraction grating problem from phaseless data,” J. Opt. Soc. Am. A 30, 293–299 (2013).
    [CrossRef]
  23. D. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Euro. J. Appl. Math. 4, 321–340 (1993).
  24. D. Dobson, “Optimal shape design of blazed diffraction grating,” Appl. Math. Optim. 40, 61–78 (1999).
  25. 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).
  26. J. Elschner and G. Schmidt, “Numerical solution of optimal design problems for binary gratings,” J. Comput. Phys. 146, 603–626 (1998).
    [CrossRef]
  27. D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College, 2003).
  28. 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]
  29. D. P. Nicholls and F. Reitich, “Shape deformations in rough surface scattering: improved algorithms,” J. Opt. Soc. Am. A 21, 606–621 (2004).
    [CrossRef]
  30. O. Bruno and F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
    [CrossRef]
  31. 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]
  32. P. Li and J. Shen, “Analysis of the scattering by an unbounded rough surface,” Math. Methods Appl. Sci. 35, 2166–2184 (2012).
  33. A. Malcolm and D. P. Nicholls, “A boundary perturbation method for recovering interface shapes in layered media,” Inverse Probl. 27, 095009 (2011).
    [CrossRef]
  34. G. Bao and J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. 2, 377–396 (2013).
  35. G. Bao and P. Li, “Near-field imaging of infinite rough surfaces,” SIAM J. Appl. Math. (to be published).
  36. G.-Q. Zhang, “Integrated solutions of ordinary differential equation system and two-point boundary value problems. I. Integrated solution method,” J. Comp. Math. 3, 245–254 (1981).

2013 (2)

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

G. Bao and J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. 2, 377–396 (2013).

2012 (2)

P. Li and J. Shen, “Analysis of the scattering by an unbounded rough surface,” Math. Methods Appl. Sci. 35, 2166–2184 (2012).

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

2011 (3)

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]

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]

2005 (1)

2004 (2)

2003 (4)

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).

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]

1999 (2)

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,” Appl. Math. Optim. 40, 61–78 (1999).

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).

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 (1)

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 and A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Ration. 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]

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

1994 (2)

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)

N. García and M. Nieto-Vesperinas, “Near-field optics inverse-scattering reconstruction of reflective surfaces,” Opt. Lett. 24, 2090–2092 (1993).
[CrossRef]

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

O. Bruno and F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
[CrossRef]

1991 (1)

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]

1981 (1)

G.-Q. Zhang, “Integrated solutions of ordinary differential equation system and two-point boundary value problems. I. Integrated solution method,” J. Comp. Math. 3, 245–254 (1981).

Ammari, H.

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

G. Bao and J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. 2, 377–396 (2013).

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, Z. Chen, and H. Wu, “Adaptive finite element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
[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, 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. Ration. 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, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001).

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces,” SIAM J. Appl. Math. (to be published).

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.

Chen, Z.

G. Bao, Z. Chen, and H. Wu, “Adaptive finite element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
[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]

Cheng, J.

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

Courjon, D.

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College, 2003).

Cowsar, L.

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

Cox, J. A.

Dobson, D.

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

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, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Euro. J. Appl. Math. 4, 321–340 (1993).

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).

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. Ration. Mech. Anal. 132, 49–72 (1995).
[CrossRef]

García, N.

N. García and M. Nieto-Vesperinas, “Near-field optics inverse-scattering reconstruction of reflective surfaces,” Opt. Lett. 24, 2090–2092 (1993).
[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]

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).

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]

Li, P.

G. Bao, P. Li, and J. Lv, “Numerical solution of an inverse diffraction grating problem from phaseless 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]

P. Li and J. Shen, “Analysis of the scattering by an unbounded rough surface,” Math. Methods Appl. Sci. 35, 2166–2184 (2012).

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces,” SIAM J. Appl. Math. (to be published).

Lin, J.

G. Bao and J. Lin, “Near-field imaging of the surface displacement on an infinite ground plane,” Inverse Probl. Imag. 2, 377–396 (2013).

Lv, J.

Malcolm, A.

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]

Masters, W.

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

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]

Nicholls, D. P.

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]

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

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]

Nieto-Vesperinas, M.

N. García and M. Nieto-Vesperinas, “Near-field optics inverse-scattering reconstruction of reflective surfaces,” Opt. Lett. 24, 2090–2092 (1993).
[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.

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).

Shen, J.

P. Li and J. Shen, “Analysis of the scattering by an unbounded rough surface,” Math. Methods Appl. Sci. 35, 2166–2184 (2012).

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]

Wu, H.

Yamamoto, M.

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

Zhang, G.-Q.

G.-Q. Zhang, “Integrated solutions of ordinary differential equation system and two-point boundary value problems. I. Integrated solution method,” J. Comp. Math. 3, 245–254 (1981).

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]

Appl. Math. Optim. (1)

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

Arch. Ration. Mech. Anal. (1)

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

Euro. J. Appl. Math. (1)

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

Inverse Probl. (10)

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

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

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

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

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]

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

F. Hettlich and A. Kirsch, “Schiffer’s theorem in inverse scattering theory for periodic structures,” Inverse Probl. 13, 351–361 (1997).
[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. 2, 377–396 (2013).

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]

J. Comp. Math. (1)

G.-Q. Zhang, “Integrated solutions of ordinary differential equation system and two-point boundary value problems. I. Integrated solution method,” J. Comp. Math. 3, 245–254 (1981).

J. Comput. Phys. (1)

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

J. Opt. Soc. Am. A (7)

Math. Methods Appl. Sci. (2)

P. Li and J. Shen, “Analysis of the scattering by an unbounded rough surface,” Math. Methods Appl. Sci. 35, 2166–2184 (2012).

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).

Opt. Lett. (1)

N. García and M. Nieto-Vesperinas, “Near-field optics inverse-scattering reconstruction of reflective surfaces,” Opt. Lett. 24, 2090–2092 (1993).
[CrossRef]

SIAM J. Appl. Math. (1)

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. Math. Anal. (1)

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]

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

Other (4)

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

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College, 2003).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

G. Bao and P. Li, “Near-field imaging of infinite rough surfaces,” SIAM J. Appl. Math. (to be published).

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Figures (3)

Fig. 1.
Fig. 1.

Problem geometry.

Fig. 2.
Fig. 2.

Example 1: finite Fourier grating; the exact surface (solid line) is plotted against the reconstructed surface (dashed line) using different h, ε, δ. (a) h=0.1λ, ε=0.01, δ=0.01; (b) h=0.2λ, ε=0.01, δ=0.01; (c) h=0.1λ, ε=0.1, δ=0.01; and (d) h=0.1λ, ε=0.01, δ=0.05.

Fig. 3.
Fig. 3.

Example 2: infinite Fourier grating; the exact surface (solid line) is plotted against the reconstructed surface (dashed line) using different h, ε, δ. (a) h=0.02λ, ε=0.01, δ=0.01; (b) h=0.1λ, ε=0.01, δ=0.01; (c) h=0.11λ, ε=0.1, δ=0.01; and (d) h=0.02λ, ε=0.01, δ=0.05.

Equations (101)

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S={(x,y)R2:y=f(x),0<x<Λ},
f(x)=εg(x),gC2(R).
Ω={(x,y)R2:f(x)<y<h,0<x<Λ}
Γ={(x,y)R2:y=h,0<x<Λ},
Δu+κ2u=0inΩ.
u=0onS.
u=uinc+ud=eiκy+nZAneiαnx+iβny,
αn=n(2πΛ),βn={(κ2αn2)1/2forκ>|αn|,i(αn2κ2)1/2forκ<|αn|.
q(x)=nZqneiαnx,qn=1Λ0Λq(x)eiαnxdx.
(Tq)(x)=nZiβnqneiαnx.
yu=Tu+ρonΓ,
x˜=x,y˜=h(yfhf),
D={(x˜,y˜)R2:0<y˜<h,0<x<Λ}.
c12wx2+c22wy2+c32wxy+c4wy+c1κ2w=0,
c1=(hf)2,c2=[f(hy)]2+h2,c3=2f(hy)(hf),c4=(hy)[f(hf)+2(f)2].
w(x,y)=0ony=0.
yw=(1fh)(Tw+ρ)ony=h.
w(x,y;ε)=k=0wk(x,y)εk.
2wkx2+2wky2+κ2wk=vkinD,
vk=2gh2wk1x2+2g(hy)h2wk1xy+g(hy)hwk1y+2κ2ghwk1g2h22wk2x2(g)2(hy)2h22wk2y22gg(hy)h22wk2xy+[2(g)2gg](hy)h2wk2yκ2g2h2wk2,
wk(x,y)=0ony=0
ywkTwk=ρkony=h,
ρ0=ρ,
ρ1=(gh)(Tw0+ρ),
ρk=(gh)Twk1.
wk(x,y)=nZwk(n)(y)eiαnx,vk(x,y)=nZvk(n)(y)eiαnx,ρk(x)=nZρk(n)eiαnx.
d2wk(n)dy2+βn2wk(n)=vk(n),0<y<h,
wk(n)=0aty=0,
dwk(n)dyiβnwk(n)=ρk(n)aty=h.
wk(n)(y)=K1(n)(y)ρk(n)0hK2(n)(y,z)vk(n)(z)dz,
K1(n)(y)=eiβnh2iβn(eiβnyeiβny)
K2(n)(y,z)={eiβny2iβn(eiβnzeiβnz),z<y,eiβnz2iβn(eiβnyeiβny),z>y.
uδ(x,h)=u(x,h)+O(δ),
w(x,y)=w0(x,y)+εw1(x,y)+O(ε2).
wδ(x,h)=w0(x,h)+εw1(x,h)+O(ε2)+O(δ).
εw1(x,h)=(wδ(x,h)w0(x,h))+O(ε2)+O(δ),
εw1(x,h)=wδ(x,h)w0(x,h),
v0=0,ρ0=2iκeiκh,
v0(n)=0,ρ0(n)={2iκeiκh,n=0,0,n0.
w0(n)(y)=K1(n)(y)ρ0(n)=eiκh2iκ(eiκyeiκy)ρ0(n)={eiκyeiκy,n=0,0,n0,
w0(x,y)=nZw0(n)(y)eiαnx=eiκyeiκy.
ρ1(x)=(gh)yw0(x,h)=2iκh1cos(κh)g(x),
ρ1(n)=2iκh1cos(κh)gn,
v1(x,y)=2gh2w0x2+2g(hy)h2w0xy+g(hy)hw0y+2κ2ghw0.
v1=2iκh1(hy)cos(κy)g(x)4iκ2h1sin(κy)g(x),
v1(n)=[4iκ2h1sin(κy)+2iκh1(hy)αn2cos(κy)]gn.
w1(n)(y)=K1(n)(y)ρ1(n)0hK2(n)(y,z)v1(n)(z)dz.
w1(n)(h)=K1(n)(h)ρ1(n)0hK2(n)(h,z)v1(n)(z)dz.
w1(n)(h)=κeiβnhβnh(eiβnheiβnh)cos(κh)gn+κeiβnhβnh0h(eiβnzeiβnz)×[2κsin(κz)αn2(hz)cos(κz)]gndz.
0h(eiβnzeiβnz)sin(κz)dz=iβnαn2(eiβnh+eiβnh)sin(κh)καn2(eiβnheiβnh)cos(κh)
0h(eiβnzeiβnz)(hz)cos(κz)dz=(κ2+βn2)αn4(eiβnheiβnh)cos(κh)+2iκβnαn4×(eiβnh+eiβnh)sin(κh)2iβnhαn2.
w1(n)(h)=2iκeiβnhgn.
fn=i2κ(wδ(n)(h)w0(n)(h))eiβnh,
w0(n)(h)={eiκheiκh,n=0,0,n0.
SNR=min{ε2,δ1}.
e(ω2κ2)1/2h=SNR,
ωκ=[1+(logSNRκh)2]1/2,
fn=i2κ(wδ(n)(h)w0(n)(h))eiβnhχ[ω,ω],
χ[ω,ω](αn)={1for|αn|ω,0for|αn|>ω.
f(x)|αn|ωfneiαnx.
uδ(x,h)=u(x,h)(1+δrand),
g(x)=0.5sin(2πx)+0.5sin(6πx).
g(x)=0.5ecos(4πx)+0.4ecos(6πx)1.5.
u(y)+M(y)u(y)=f(y),
A0u(y)|y=0=r0,
B1u(y)|y=h=s1,
Φ(y)+M(y)Φ(y)=0,
Φ(0)=Im,
det[A0B1Φ(h)]0.
A=AM+D0A,A(0)=A0,
r=Af+D0r,r(0)=r0,
B=BM+D1B,B(h)=B1,
s=Bf+D1s,s(h)=s1.
[A(y)B(y)]Cm×m
[A(y)B(y)]u(y)=[r(y)s(y)].
u+η2u=f,0<y<h,
u=0aty=0,
uiηu=gaty=h.
v+Mv=f,
A0v(0)=0,
B1v(h)=g,
v=[v1v2],f=[0f],M=[01η20],A0=[10],B1=[iη1].
u(y)=K1(y)g0hK2(y,z)f(z)dz,
K1(y)=eiηh2iη(eiηyeiηy)
K2(y,z)={eiηy2iη(eiηzeiηz),z<y,eiηz2iη(eiηyeiηy),z>y.
Q1MQ=N,
N=[iη00iη],Q=[11iηiη],Q1=12iη[iη1iη1].
Φ(y)=Q[eiηyeiηy]Q1,
det[A0B1Φ(h)]=|10iηeiηheiηh|=eiηh0.
D0=iη,D1=iη,
A=AM+iηA,A(0)=A0,
r=Af+iηr,r(0)=0,
B=BMiηB,B(h)=B1,
s=Bfiηs,s(h)=g.
A=[A1,A2]=12iη[iη(1+e2iηy),1e2iηy],
B=[B1,B2]=[iη,1],
r=0yeiη(yz)A2(z)f(z)dz,
s=eiη(hy)gyheiη(zy)f(z)dz.
[A1A2B1B2][uu]=[rs].
u=rB2sA2A1B2B1A2.
A1B2B1A2=1.

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