Skip to main content
SpringerLink
Log in

Imaging of periodic dielectrics

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

We consider imaging of periodic penetrable structures from measurements of scattered electromagnetic waves. The importance of this problem stems from the decreasing size of periodic structures in photonic devices, together with an increasing demand in fast non-destructive testing. This demand makes qualitative inverse scattering techniques particularly attractive since they do not use time consuming optimization techniques for reconstruction but rather directly transform measured data into a picture of the scattering object. We present the Factorization method as an algorithm for imaging of a special class of periodic dielectric structures known as diffraction gratings. Our sampling method computes a picture of the shape of the periodic structure from measured near-field data in a rapid way. We provide numerical examples illustrating this imaging technique.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arens, T.: Scattering by biperiodic layered media: the integral equation approach. Habilitation Thesis, Universität Karlsruhe (2010)

  2. Arens, T., Grinberg, N.: A complete factorization method for scattering by periodic structures. Computing 75, 111–132 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arens, T., Kirsch, A.: The factorization method in inverse scattering from periodic structures. Inverse Probl. 19, 1195–1211 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bao, G., Chen, Z., Wu, H.: Adaptive finite-element method for diffraction gratings. J. Opt. Soc. Am. A 22, 1106–1114 (2005)

    Article  MathSciNet  Google Scholar 

  5. Bonnet-Bendhia, A.-S., Starling, F.: Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Methods Appl. Sci. 17, 305–338 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carney, P.S., Schotland, J.C.: Near-field tomography. In: Inside Out: Inverse Problems and Applications. Math. Sci. Res. Inst. Publ., vol. 47, pp. 133–168. Cambridge Univ. Press, Cambridge (2003)

    Google Scholar 

  7. Charalambopoulus, A., Kirsch, A., Anagnostopoulus, K.A., Gintides, D., Kiriaki, K.: The factorization method in inverse elastic scattering from penetrable bodies. Inverse Probl. 23, 27–51 (2007)

    Article  Google Scholar 

  8. Elschner, J., Schmidt, G.: Diffraction of periodic structures and optimal design problems of binary gratings. Part I: Direct problems and gradient formulas. Math. Meth. Appl. Sci. 21, 1297–1342 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  10. Gohberg, I.C., Kreĭn, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Transl. Math. Monographs, vol. 18. American Mathematical Society, Providence (1969)

    MATH  Google Scholar 

  11. Groß, H., Rathsfeld, A.: Mathematical aspects of scatterometry—an optical metrology technique. In: Intelligent Solutions for Complex Problems—Annual Research Report 2007. Weierstrass Institute for Applied Analysis and Stochastics, Berlin (2007)

  12. Groß, H., Model, R., Bär, M., Wurm, M., Bodermann, B., Rathsfeld, A.: Mathematical modelling of indirect measurements in scatterometry. Measurement 39, 782–794 (2006)

    Article  Google Scholar 

  13. Hatit, S.B., Foldyna, M., Martino, A.D., Drévillon, B.: Angle-resolved Mueller polarimeter using a microscope objective. Phys. Status Solidi (a) 205, 743–747 (2008)

    Article  Google Scholar 

  14. Kirsch, A.: Diffraction by periodic structures. In: Proc. Lapland Conf. on Inverse Problems, pp. 87–102. Springer, Berlin (1993)

    Google Scholar 

  15. Kirsch, A.: The factorization method for a class of inverse elliptic problems. Math. Nachr. 278, 258–277 (2004)

    Article  MathSciNet  Google Scholar 

  16. Lechleiter, A.: Factorization methods for photonics and rough surface scattering. PhD thesis, Universität Karlsruhe, Karlsruhe, Germany (2008)

  17. Lechleiter, A.: The Factorization method is independent of transmission eigenvalues. Inverse Probl. Imag. 3, 123–138 (2009)

    Article  MathSciNet  Google Scholar 

  18. McLean, W.: Strongly Elliptic Systems and Boundary Integral Operators. Cambridge University Press, Cambridge (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA Saclay Ile de France and Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Armin Lechleiter

Authors
  1. Armin Lechleiter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Armin Lechleiter.

Additional information

Communicated by Erkki Somersalo.

This work was supported by the Deutsche Forschungsgemeinschaft DFG through the Graduiertenkolleg GRK 1294/1: Analysis, Simulation und Design nanotechnologischer Prozesse.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lechleiter, A. Imaging of periodic dielectrics. Bit Numer Math 50, 59–83 (2010). https://doi.org/10.1007/s10543-010-0255-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-010-0255-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation