Abstract

This paper is concerned with the numerical solution of an inverse diffraction grating problem, which is to reconstruct a periodic grating profile from measurements of the phaseless diffracted field at a constant height above the grating structure. An efficient continuation method is developed to recover the Fourier coefficients of the periodic grating profile. The continuation proceeds along the wavenumber and updates are obtained from the Landweber iteration at each step. Numerical results are presented to show that the method can effectively reconstruct the shape of the grating profile.

© 2013 Optical Society of America

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References

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  1. G. Bao, L. Cowsar, and W. Masters, eds., Mathematical Modeling in Optical Science (SIAM, 2001).
  2. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
  3. G. Bao, D. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
    [CrossRef]
  4. O. Bruno and F. Reitich, “Numerical solution of diffraction probems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
    [CrossRef]
  5. Z. Chen and H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM J. Numer. Anal. 41, 799–826 (2003).
    [CrossRef]
  6. 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]
  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. H. Ammari, “Uniqueness theorems for an inverse problem in a doubly periodic structure,” Inverse Probl. 11, 823–833 (1995).
    [CrossRef]
  12. G. Bao, “A uniqueness theorem for an inverse problem in periodic diffractive optics,” Inverse Probl. 10, 335–340 (1994).
    [CrossRef]
  13. G. Bao, and A. Friedman, “Inverse problems for scattering by periodic structure,” Arch. Ration. Mech. Anal. 132, 49–72 (1995).
    [CrossRef]
  14. 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]
  15. G. Bao and Z. Zhou, “An inverse problem for scattering by a doubly periodic structure,” Trans. Am. Math. Soc. 350, 4089–4103 (1998).
    [CrossRef]
  16. G. Bruckner, J. Cheng, and M. Yamamoto, “Inverse problems of diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
    [CrossRef]
  17. F. Hettlich and A. Kirsch, “Schiffer’s theorem in inverse scattering for periodic structures,” Inverse Probl. 13, 351–361 (1997).
    [CrossRef]
  18. A. Kirsch, “Uniqueness theorems in inverse scattering theory for periodic structures,” Inverse Probl. 10, 145–152 (1994).
    [CrossRef]
  19. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).
  20. N. Garca and M. Nieto-Vesperinas, “Near-field optics inversescattering reconstruction of reflective surfaces,” Opt. Lett. 18, 2090–2092 (1993).
    [CrossRef]
  21. K. Ito and F. Reitich, “A high-order perturbation approach to profile reconstruction: I. perfectly conducting gratings,” Inverse Probl. 15, 1067–1085 (1999).
    [CrossRef]
  22. T. Arens and A. Kirsch, “The factorization method in inverse scattering from periodic structures,” Inverse Probl. 19, 1195–1211 (2003).
    [CrossRef]
  23. F. Hettlich, “Iterative regularization schemes in inverse scattering by periodic structures,” Inverse Probl. 18, 701–714 (2002).
    [CrossRef]
  24. G. Bruckner and J. Elschner, “A two-step algorithm for the reconstruction of perfectly reflecting periodic profiles,” Inverse Probl. 19, 315–329 (2003).
    [CrossRef]
  25. 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).
  26. G. Bao, P. Li, and H. Wu, “A computational inverse diffraction grating problem,” J. Opt. Soc. Am. A 29, 394–399 (2012).
    [CrossRef]
  27. G. Bao, J. Lin, and F. Triki, “Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data,” in Mathematical and Statistical Methods for Imaging, Vol. 548 of Contemporary Mathematics (AMS, 2011), pp. 45–60.
  28. A. Kirsch, “Diffraction by periodic structure,” in Inverse Problems in Mathematical Physics (Saariselkä, 1992), Vol. 422 of Lecture Notes in Physics (Springer, 1993), pp. 87–102.
  29. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).
  30. J. Desanto, G. Erdmann, W. Herman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surface,” Waves Random Media 8, 385–414 (1998).
    [CrossRef]

2012 (1)

2011 (1)

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]

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, “Inverse problems of diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
[CrossRef]

1999 (2)

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

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

1998 (4)

J. Desanto, G. Erdmann, W. Herman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surface,” Waves Random Media 8, 385–414 (1998).
[CrossRef]

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

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 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)

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

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

1993 (3)

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]

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

G. Bao, J. Lin, and F. Triki, “Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data,” in Mathematical and Statistical Methods for Imaging, Vol. 548 of Contemporary Mathematics (AMS, 2011), pp. 45–60.

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, “Inverse problems of diffractive optics: conditional stability,” Inverse Probl. 18, 415–433 (2002).
[CrossRef]

Bruno, O.

Chen, Z.

Z. Chen and H. Wu, “An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures,” SIAM J. Numer. Anal. 41, 799–826 (2003).
[CrossRef]

Cheng, J.

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

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

Cox, J. A.

Desanto, J.

J. Desanto, G. Erdmann, W. Herman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surface,” Waves Random Media 8, 385–414 (1998).
[CrossRef]

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).

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]

Erdmann, G.

J. Desanto, G. Erdmann, W. Herman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surface,” Waves Random Media 8, 385–414 (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]

Garca, N.

Herman, W.

J. Desanto, G. Erdmann, W. Herman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surface,” Waves Random Media 8, 385–414 (1998).
[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 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 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]

A. Kirsch, “Diffraction by periodic structure,” in Inverse Problems in Mathematical Physics (Saariselkä, 1992), Vol. 422 of Lecture Notes in Physics (Springer, 1993), pp. 87–102.

Kress, R.

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

Li, P.

Lin, J.

G. Bao, J. Lin, and F. Triki, “Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data,” in Mathematical and Statistical Methods for Imaging, Vol. 548 of Contemporary Mathematics (AMS, 2011), pp. 45–60.

Misra, M.

J. Desanto, G. Erdmann, W. Herman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surface,” Waves Random Media 8, 385–414 (1998).
[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]

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).

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 probems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
[CrossRef]

Schmidt, G.

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

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

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]

Triki, F.

G. Bao, J. Lin, and F. Triki, “Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data,” in Mathematical and Statistical Methods for Imaging, Vol. 548 of Contemporary Mathematics (AMS, 2011), pp. 45–60.

Wu, H.

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

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, “Inverse problems of 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. (1)

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

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]

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. (9)

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

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

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

F. Hettlich and A. Kirsch, “Schiffer’s theorem in inverse scattering 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]

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]

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 (3)

Math. Methods Appl. Sci. (1)

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. (1)

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

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

Waves Random Media (1)

J. Desanto, G. Erdmann, W. Herman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surface,” Waves Random Media 8, 385–414 (1998).
[CrossRef]

Other (6)

G. Bao, J. Lin, and F. Triki, “Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data,” in Mathematical and Statistical Methods for Imaging, Vol. 548 of Contemporary Mathematics (AMS, 2011), pp. 45–60.

A. Kirsch, “Diffraction by periodic structure,” in Inverse Problems in Mathematical Physics (Saariselkä, 1992), Vol. 422 of Lecture Notes in Physics (Springer, 1993), pp. 87–102.

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

G. Bao, L. Cowsar, and W. Masters, eds., Mathematical Modeling in Optical Science (SIAM, 2001).

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

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Figures (3)

Fig. 1.
Fig. 1.

Evolution of the reconstructions in Example 1. Left column from top to bottom: reconstructions at κ=1, reconstructions at κ=3. Right column from top to bottom: reconstructions at κ=2, reconstructions at κ=4.

Fig. 2.
Fig. 2.

Evolution of the reconstructions in Example 2. Left column from top to bottom: reconstructions at κ=1, reconstructions at κ=2. Right column from top to bottom: reconstructions at κ=3, reconstructions at κ=4.

Fig. 3.
Fig. 3.

Evolution of the reconstructions in Example 3. Left column from top to bottom: reconstructions at κ=1, reconstructions at κ=2, and reconstructions at κ=3. Right column from top to bottom: reconstructions at κ=4, reconstructions at κ=5, and reconstructions at κ=6.

Equations (49)

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

Γf={(x,y)R2:y=f(x),0<x<2π},
Ωf={(x,y)R2:y>f(x),0<x<2π}
uinc=ei(αxβy)
βn={κ2αn2,forκ>|αn|,iαn2κ2,forκ<|αn|.
Δu+κ2u=0,inΩf,
u+uinc=0,onΓf.
u=nZAneiαnx+iβny,y>maxx(0,2π)f(x),
Γ0={(x,y0)R2:x(0,2π),y0>maxx(0,2π)f(x)},
K(f)=u(x,y0).
Kf(h)=u(x,y0)
Δu+κ2u=0inΩf,
u=h1+(f)2nuonΓf,
F(f)=|u(x,y0)|2=K(f)K¯(f),
F(f)=|u(x,y0)|2.
Ff(h)=2Re[K¯(f)Kf(h)]=2Re[u¯(x,y0)u(x,y0)]
K(f+h)=K(f)+Kf(h)+Rf(h),
F(f+h)=K(f+h)K¯(f+h)=[K(f)+Kf(h)+Rf(h)][K¯(f)+K¯f(h)+R¯f(h)]=F(f)+2Re[K¯(f)Kf(h)]+|Kf(h)|2+|Rf(h)|2+2Re[Rf(h)(K¯(f)+K¯f(h))].
K(f+h)=K(f)+2Re[K¯(f)Kf(h)]+o(h1,),
G(x,y;s,t)=i2πnZ1βnexp(iαn(xs)+iβn|yt|).
wgi(s,t)=02πG(x,y0;s,t)u¯(x,y0)g(x)dx,ty0,
(Ff)*g=2Re[nu(x,f(x))(nwgi(x,f(x))+nwg(x,f(x)))],
Δwg+κ2wg=0inΩf,
wg+wgi=0onΓf.
u(x,y0)=Γf[u(s,t)nG(x,y0;s,t)nu(s,t)G(x,y0;s,t)]dS.
Ff(h),g=02π2Re[u¯(x,y0)u(x,y0)]g(x)dx=2Re02πu¯(x,y0)g(x)Γf[u(s,t)nG(x,y0;s,t)nu(s,t)G(x,y0;s,t)]dSdx=2ReΓfu(s,t)02πu¯(x,y0)nG(x,y0;s,t)g(x)dxdS2ReΓfnu(s,t)02πu¯(x,y0)G(x,y0;s,t)g(x)dxdS=2ReΓfu(s,t)nwgi(s,t)dS2ReΓfnu(s,t)wgi(s,t)dS=2ReΓfu(s,t)nwgi(s,t)dS+2ReΓfnu(s,t)wg(s,t)dS,
Γf(unwgnuwg)dS=0.
h,(Ff)*g=Ffh,g=2ReΓfu(s,t)(nwgi(s,t)+nwg(s,t))dS=202πh(x)Re[nu(x,f(x))×(nwgi(x,f(x))+nwg(x,f(x)))]dx,
Fκ,f=2ReK¯κ(f)Kκ,f,
Fκm+1(f)Fκm+1(fm)+Fκm+1,fm(ffm).
Fκm+1,fmδf=Rm+1.
δf=τ(Fκm+1,fm)*Rm+1,
fm+1=fm+δf.
f(x)c0+m=1M[c2m1cos(mx)+c2msin(mx)].
f(x)c0+m=1N[c2m1cos(mx)+c2msin(mx)]+m=N+1M[c2m1cos(mx)+c2msin(mx)],
Δun+κ2un=0inΩfk(n),
un+ei(αxβfk(n)(x))=0onΓfk(n),
Φk=[1,cosx,sinx,,cos((N+k)x),sin((N+k)x)].
Kκ(fk(n))=un(x,y0),
Fκ(fk(n))=|un(x,y0)|2.
R(n)=Fκ(f)Fκ(fk(n)),
fk(n+1)=fk(n)+τκ(Fκ,fk(n))*R(n),
ΦkCk(n)=fk(n)
Ck(n+1)Ck(n)>Ck(n)Ck(n1).
fk˜(x)=c˜0+m=1N+k˜[c˜2m1cos(mx)+c˜2msin(mx)]
c˜j={cj,forj2(N+k),0,forj>2(N+k),
|u(x,y0)|2:=|u(x,y0)|2(1+σrand),
f(x)=1.0+0.2cosx+0.2cos2x,
f(x)=1.7+0.06ecos(2x)+0.05ecos(3x),
f(x)={1.5,forx(π2,3π2),1.0,otherwise.

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