Advertisement

Science in China Series A: Mathematics

, Volume 51, Issue 6, pp 1059–1070 | Cite as

Reconstruction of the shape of object with near field measurements in a half-plane

  • HePing DongEmail author
  • FuMing Ma
Article

Abstract

We consider a mathematical problem modelling some characteristics of near field optical microscope. We take a monofrequency line source to illuminate a sample with constant index of refraction and use the scattered field data measured near the sample to reconstruct the shape of it. Mixed reciprocity relation and factorization method are applied to solve our problem. Some numerical examples to show the feasibility of the method are presented.

Keywords

Helmholtz equation inverse problem mixed reciprocity relation factorization method 

MSC(2000)

35J05 78A46 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Courjon D, Bainier C. Near field microscopy and near field optics. Rep Progr Phys, 57: 989–1028 (1994)CrossRefGoogle Scholar
  2. 2.
    Girard C, Dereux A. Near-field optics theories. Rep Progr Phys, 59: 657–699 (1996)CrossRefGoogle Scholar
  3. 3.
    Carney P S, Schotland J C. Near-field tomography. Inside Out: Inverse Problems and Applications, 47: 133–168 (2003)MathSciNetGoogle Scholar
  4. 4.
    Bao G, Li P J. Inverse medium scattering problems in near-field optics. J Comput Math, 25: 252–265 (2007)MathSciNetGoogle Scholar
  5. 5.
    Potthast R. Point Sources and Multipoles in Inverse Scattering Theory. Boca Raton, FL: Chapman and Hall/CRC Press, 2001zbMATHGoogle Scholar
  6. 6.
    Potthast R. A point source method for inverse acoustic electromagnetic obstacle scattering problems. IMA J Appl Math, 61: 119–140 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kirsch A. Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory. Inverse Problems, 16: 413–429 (1999)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. 2nd ed. Berlin: Springer, 1998zbMATHGoogle Scholar
  9. 9.
    Kirsch A. An Introduction to the Mathematical Theory of Inverse Problems. Berlin: Springer, 1996zbMATHGoogle Scholar
  10. 10.
    Kress R. Linear Integral Equations. 2nd ed. Berlin: Springer, 1999zbMATHGoogle Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Institute of MathematicsJilin UniversityChangchunChina

Personalised recommendations