Abstract

Consider the diffraction of a time-harmonic plane wave incident on a perfectly reflecting periodic surface. A continuation method on the wavenumber is developed for the inverse diffraction grating problem, which reconstructs the grating profile from measured reflected waves a constant distance away from the structure. Numerical examples are presented to show the validity and efficiency of the proposed method.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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]
  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. 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]
  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. R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980).
  6. G. Bao, L. Cowsar, and W. Masters, eds., Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics (SIAM, 2001).
  7. D. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–340 (1993).
    [CrossRef]
  8. D. Dobson, “Optimal shape design of blazed diffraction grating,” Appl. Math. Optim. 40, 61–78 (1999).
    [CrossRef]
  9. 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]
  10. J. Elschner and G. Schmidt, “Numerical solution of optimal design problems for binary gratings, J. Comput. Phys. 146, 603–626 (1998).
    [CrossRef]
  11. A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
    [CrossRef]
  12. G. Bao, “A unique theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
    [CrossRef]
  13. H. Ammari, “Uniqueness theorems for an inverse problem in a doubly periodic structure,” Inverse Probl. 11, 823–833 (1995).
    [CrossRef]
  14. F. Hettlich and A. Kirsch, “Schiffer’s theorem in inverse scattering theory for periodic structures,” Inverse Probl. 13, 351–361 (1997).
    [CrossRef]
  15. G. Bao and A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Ration. Mech. Anal. 132, 49–72 (1995).
    [CrossRef]
  16. G. Bao and Z. Zhou, “An inverse problem for scattering by a doubly periodic structure,” Trans. Am. Math. Soc. 350, 4089–4103 (1998).
    [CrossRef]
  17. 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]
  18. G. Bruckner, J. Cheng, and M. Yamamoto, “An inverse problem in diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
    [CrossRef]
  19. D. Colton and R. Kress, “Inverse Acoustic and Electromagnetic Scattering Theory,” 2nd ed., Vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1998).
  20. N. García and M. Nieto-Vesperinas, “Near-field optics inverse-scattering reconstruction of reflective surfaces,” Opt. Lett. 18, 2090–2092 (1993).
    [CrossRef]
  21. J. B. Keller, “Singularities and Rayleigh’s hypothesis for diffraction gratings,” J. Opt. Soc. Am. A 17, 456–457 (2000).
    [CrossRef]
  22. R. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
    [CrossRef]
  23. K. Ito and F. Reitich, “A high-order perturbation approach to profile reconstruction: I. perfectly conducting gratings,” Inverse Probl. 15, 1067–1085 (1999).
    [CrossRef]
  24. T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
    [CrossRef]
  25. F. Hettlich, “Iterative regularization schemes in inverse scattering by periodic structures,” Inverse Probl. 18, 701–714 (2002).
    [CrossRef]
  26. G. Bruckner and J. Elschner, “A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles,” Inverse Probl. 19, 315–329 (2003).
    [CrossRef]
  27. 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).
    [CrossRef]
  28. Y. Chen, “Inverse scattering via Heisenberg uncertainty principle,” Inverse Probl. 13, 253–282 (1997).
    [CrossRef]
  29. G. Bao and P. Li, “Inverse medium scattering problems for electromagnetic waves, SIAM J. Appl. Math. 65, 2049–2066 (2005).
    [CrossRef]
  30. G. Bao and P. Li, “Inverse medium scattering for the Helmholtz equation at fixed frequency,” Inverse Probl. 21, 1621–1641 (2005).
    [CrossRef]
  31. G. Bao and P. Li, “Numerical solution of an inverse medium scattering problem for Maxwell’s equations at fixed frequency,” J. Comput. Phys. 228, 4638–4648 (2009).
    [CrossRef]
  32. 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]
  33. H. Ammari, J. Garnier, H. Kang, M. Lim, and K. Solna, “Multistatic imaging of extended targets,” SIAM J. Imaging Sci., to be published.
  34. H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
  35. G. Bao, Z. Chen, and H. Wu, “Adaptive finite element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
    [CrossRef]

2011

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]

2009

G. Bao and P. Li, “Numerical solution of an inverse medium scattering problem for Maxwell’s equations at fixed frequency,” J. Comput. Phys. 228, 4638–4648 (2009).
[CrossRef]

2005

G. Bao and P. Li, “Inverse medium scattering problems for electromagnetic waves, SIAM J. Appl. Math. 65, 2049–2066 (2005).
[CrossRef]

G. Bao and P. Li, “Inverse medium scattering for the Helmholtz equation at fixed frequency,” Inverse Probl. 21, 1621–1641 (2005).
[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]

2003

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).
[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]

2002

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]

2000

1999

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

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]

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

1998

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

Y. Chen, “Inverse scattering via Heisenberg uncertainty principle,” Inverse Probl. 13, 253–282 (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

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

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

1991

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]

1973

R. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

Ammari, H.

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

H. Ammari, J. Garnier, H. Kang, M. Lim, and K. Solna, “Multistatic imaging of extended targets,” SIAM J. Imaging Sci., to be published.

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, 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 P. Li, “Numerical solution of an inverse medium scattering problem for Maxwell’s equations at fixed frequency,” J. Comput. Phys. 228, 4638–4648 (2009).
[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 P. Li, “Inverse medium scattering problems for electromagnetic waves, SIAM J. Appl. Math. 65, 2049–2066 (2005).
[CrossRef]

G. Bao and P. Li, “Inverse medium scattering for the Helmholtz equation at fixed frequency,” Inverse Probl. 21, 1621–1641 (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 and A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Ration. 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]

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, Y.

Y. Chen, “Inverse scattering via Heisenberg uncertainty principle,” Inverse Probl. 13, 253–282 (1997).
[CrossRef]

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]

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]

Colton, D.

D. Colton and R. Kress, “Inverse Acoustic and Electromagnetic Scattering Theory,” 2nd ed., Vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1998).

Cox, J. A.

Dobson, D.

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

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

Engl, H.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

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.

Garnier, J.

H. Ammari, J. Garnier, H. Kang, M. Lim, and K. Solna, “Multistatic imaging of extended targets,” SIAM J. Imaging Sci., to be published.

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]

Hanke, M.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

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).
[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]

Kang, H.

H. Ammari, J. Garnier, H. Kang, M. Lim, and K. Solna, “Multistatic imaging of extended targets,” SIAM J. Imaging Sci., to be published.

Keller, J. B.

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]

Kress, R.

D. Colton and R. Kress, “Inverse Acoustic and Electromagnetic Scattering Theory,” 2nd ed., Vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1998).

Li, P.

G. Bao and P. Li, “Numerical solution of an inverse medium scattering problem for Maxwell’s equations at fixed frequency,” J. Comput. Phys. 228, 4638–4648 (2009).
[CrossRef]

G. Bao and P. Li, “Inverse medium scattering problems for electromagnetic waves, SIAM J. Appl. Math. 65, 2049–2066 (2005).
[CrossRef]

G. Bao and P. Li, “Inverse medium scattering for the Helmholtz equation at fixed frequency,” Inverse Probl. 21, 1621–1641 (2005).
[CrossRef]

Lim, M.

H. Ammari, J. Garnier, H. Kang, M. Lim, and K. Solna, “Multistatic imaging of extended targets,” SIAM J. Imaging Sci., to be published.

Millar, R.

R. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[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]

Neubauer, A.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

Nieto-Vesperinas, M.

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).
[CrossRef]

Reitich, F.

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

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]

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]

Schmidt, G.

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]

Solna, K.

H. Ammari, J. Garnier, H. Kang, M. Lim, and K. Solna, “Multistatic imaging of extended targets,” SIAM J. Imaging Sci., to be published.

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.

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]

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, 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.

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

Arch. Ration. Mech. Anal.

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

Eur. J. Appl. Math.

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

Inverse Probl.

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]

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

Y. Chen, “Inverse scattering via Heisenberg uncertainty principle,” Inverse Probl. 13, 253–282 (1997).
[CrossRef]

G. Bao and P. Li, “Inverse medium scattering for the Helmholtz equation at fixed frequency,” Inverse Probl. 21, 1621–1641 (2005).
[CrossRef]

J. Comput. Phys.

G. Bao and P. Li, “Numerical solution of an inverse medium scattering problem for Maxwell’s equations at fixed frequency,” J. Comput. Phys. 228, 4638–4648 (2009).
[CrossRef]

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

Math. Methods Appl. Sci.

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]

Opt. Lett.

Radio Sci.

R. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

SIAM J. Appl. Math.

G. Bao and P. Li, “Inverse medium scattering problems for electromagnetic waves, SIAM J. Appl. Math. 65, 2049–2066 (2005).
[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).
[CrossRef]

SIAM J. Imaging Sci.

H. Ammari, J. Garnier, H. Kang, M. Lim, and K. Solna, “Multistatic imaging of extended targets,” SIAM J. Imaging Sci., to be published.

SIAM J. Math. Anal.

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.

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.

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]

Waves Random Media

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

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

D. Colton and R. Kress, “Inverse Acoustic and Electromagnetic Scattering Theory,” 2nd ed., Vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1998).

R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, 1980).

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

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.

Problem geometry.

Fig. 2.
Fig. 2.

Evolution of the reconstructions in Example 1. Solid curve, test profile; dotted curve, reconstructed profile. Left: reconstruction at κ=1. Right: reconstruction at κ=2.

Fig. 3.
Fig. 3.

Evolution of the reconstructions in Example 2. Solid curve, test profile; dotted curve, reconstructed profile. Left column from top to bottom: reconstruction at κ=1, reconstruction at κ=2, reconstruction at κ=3. Right column from top to bottom: reconstruction at κ=4, reconstruction at κ=5, reconstruction at κ=6.

Fig. 4.
Fig. 4.

Evolution of the reconstructions in Example 3. Solid curve, test profile; dotted curve, reconstructed profile. Left column from top to bottom: reconstruction at κ=1, reconstruction at κ=3. Right column from top to bottom: reconstruction at κ=5, reconstruction at k=7.

Equations (33)

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

S={(x,y)R2:y=f(x),0<x<Λ},
βn={κ2αn2,forκ>|αn|,iαn2κ2,forκ<|αn|.
Δu+κ2u=0,inΩ,
u+uinc=0,onS.
u=nZAneiαnx+iβny,y>maxx(0,Λ)f(x),
Γ={(x,y0)R2:x(0,Λ),y0>maxx(0,Λ)f(x)},
u(x,y)=0Λϕ(s)G(x,y;s,0)ds,
G(x,y;s,t)=i2πnZ1βneiαn(xs)+iβn|yt|,(x,y)(s,t).
u(x,y0)=0Λϕ(s)G(x,y0;s,0)ds.
ϕ(s)=nZϕneiαns,
u(x,y0)=nZuneiαnx,
un=1Λ0Λu(x,y0)eiαnxdx.
ϕn=iβnuneiβny0.
(Tfϕ)(x)=0Λϕ(s)G(x,f(x);s,0)ds.
(Tfϕ)(x)=nZψneiαnx+iβnf(x),
ψn={uneik2αn2y0,fork>|αn|,uneαn2k2y0e2αn2k2y0+γfork<|αn|,
(Tfϕ)(x)+uinc(x,f(x))L2(0,Λ)2=0,
nZψneiαnx+iβnf(x)+ei(αxβf(x))L2(0,Λ)2=0.
|n|Nψneiαnx+iβnf(x)+ei(αxβf(x))L2(0,Λ)2=0.
f(x)=c0+m=1[c2m1cos(mx)+c2msin(mx)],
f(x)c0+m=1M[c2m1cos(mx)+c2msin(mx)].
fk(x)=c0+m=1k[c2m1cos(mx)+c2msin(mx)],
gl(Ck,x)=|n|Nψneiαnx+iβnf(x)+ei(αxβf(x)),
Hl(Ck)=02π|gl(Ck,x)|2dx.
H(Ck)=0,
Ck(i+1)=Ck(i)τkDH(Ck(i))H(Ck(i)),i=1,2,,
DH=(Hlcj)l=1,,L,j=0,1,,2k
c˜0+m=1k˜[c˜2m1cos(mx)+c˜2msin(mx)]
c˜j={cj,forj2k,0,forj>2k,
u(x,y0)u(x,y0)(1+σrand),
f(x)=1.5+0.2cosx+0.2cos2x.
f(x)=1.7+0.05ecos2x+0.04ecos3x.
f(x)={1.5,forx(π2,3π2),1.0,otherwise.

Metrics