Skip to main content Accesibility Help
×
×
Home

Near-Field Imaging of Interior Cavities

  • Peijun Li (a1) and Yuliang Wang (a1)
Abstract

A novel method is developed for solving the inverse problem of reconstructing the shape of an interior cavity. The boundary of the cavity is assumed to be a small and smooth perturbation of a circle. The incident field is generated by a point source inside the cavity. The scattering data is taken on a circle centered at the source. The method requires only a single incident wave at one frequency. Using a transformed field expansion, the original boundary value problem is reduced to a successive sequence of two-point boundary value problems and is solved in a closed form. By dropping higher order terms in the power series expansion, the inverse problem is linearized and an explicit relation is established between the Fourier coefficients of the cavity surface function and the total field. A nonlinear correction algorithm is devised to improve the accuracy of the reconstruction. Numerical results are presented to show the effectiveness of the method and its ability to obtain subwavelength resolution.

Copyright
Corresponding author
*Email addresses: lipeijun@math.purdue.edu (P. Li), wang2049@math.purdue.edu (Y.Wang)
References
Hide All
[1]Ammari, H., Gamier, J., and Solna, K., Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging, Proc. Amer. Math. Soc., 141 (2013), 34313446.
[2]Ammari, H., Garnier, J., and Solna, K., Limited view resolving power of linearized conductivity imaging from boundary measurements, SIAM J. Math. Anal., 45 (2013), 17041722.
[3]Bao, G., Cui, T. and Li, P., Inverse diffraction grating of Maxwells equations in biperiodic structures, Optics Express, 22 (2014), 47994816.
[4]Bao, G. and Li, P., Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 21622187.
[5]Bao, G. and Li, P., Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imaging Sci., 7 (2014), 867899.
[6]Bao, G. and Li, P., Convergence analysis in near-field imaging, Inverse Problems, to appear.
[7]Bao, G. and Lin, J., Near-field imaging of the surface displacement on an infinite ground plane, Inverse Problems and Imaging, 7 (2013), 377396.
[8]Bruno, O. and Reitich, F., Numerical solution of diffraction problems: a method of variation of boundaries, J. Opt. Soc. Am. A, 10 (1993), 11681175.
[9]Cakoni, F. and Colton, D., Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer, 2006.
[10]Cheng, T., Li, P. and Wang, Y., Near-field imaging of perfectly conducting grating surfaces, J. Opt. Soc. Am. A, 30 (2013), 24732481.
[11]Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.
[12]Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998.
[13]Courjon, D., Near-field Microscopy and Near-field Optics, Imperial College Press, London, 2003.
[14]NIST Digital Library of Mathematical Functions. dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.
[15]Derveaux, G., Papanicolaou, G., and Tsogka, C., Resolution and denoising in near-field imaging, Inverse Problems, 22 (2006), 14371456.
[16]Li, P. and Shen, J., Analysis of the scattering by an unbounded rough surface, Math. Meth. Appl. Sci., 35 (2012), 21662184.
[17]Li, P. and Wang, Y., Near-field imaging of obstacles, preprint.
[18]Malcolm, A. and Nicholls, D.P., A field expansions method for scattering by periodic multi-layered media, J. Acout. Soc. Am., 129 (2011), 17831793.
[19]Malcolm, A. and Nicholls, D.P., A boundary perturbation method for recovering interface shapes in layered media, Inverse Problems, 27 (2011), 095009.
[20]Nicholls, D.P. and Reitich, F., Shape deformations in rough surface scattering: cancellations, conditioning, and convergence, J. Opt. Soc. Am. A, 21 (2004), 590605.
[21]Nicholls, D.P. and Reitich, F., Shape deformations in rough surface scattering: improved algorithms, J. Opt. Soc. Am. A, 21 (2004), 606621.
[22]Nicholls, D.P. and Shen, J., A stable high-order method for two-dimensional bounded-obstacle scattering, SIAM J. Sci. Comput., 28 (2006), 13981419
[23]Qin, H. and Cakoni, F., Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005.
[24]Qin, H. and Colton, D., The inverse scattering problem for cavities, Appl. Numer. Math., 62 (2012), 699708.
[25]Qin, H. and Colton, D., The inverse scattering problem for cavities with impedance boundary condition, Adv. Comput. Math., 36 (2012), 157174.
[26]Zeng, F., Cakoni, F. and Sun, J., An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002.
[27]Zeng, F., Suarez, P., and Sun, J., A decomposition method for an interior inverse scattering problem, Inverse Problems and Imaging, 7 (2013), 291303.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed