Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T03:23:37.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Near-Field Imaging of Interior Cavities

Published online by Cambridge University Press:  22 January 2015

Peijun Li  and
Yuliang Wang
Peijun Li*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Yuliang Wang
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
*
*Email addresses: lipeijun@math.purdue.edu (P. Li), wang2049@math.purdue.edu (Y.Wang)
Get access

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.

Keywords

65N2178A46Inverse scatteringnear-field imagingHelmholtz equation
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 2 , February 2015 , pp. 542 - 563
Copyright
Copyright © Global Science Press Limited 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[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), 3431–3446.Google Scholar
[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), 1704–1722.Google Scholar
[3]Bao, G., Cui, T. and Li, P., Inverse diffraction grating of Maxwells equations in biperiodic structures, Optics Express, 22 (2014), 4799–4816.Google Scholar
[4]Bao, G. and Li, P., Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162–2187.Google Scholar
[5]Bao, G. and Li, P., Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imaging Sci., 7 (2014), 867–899.Google Scholar
[6]Bao, G. and Li, P., Convergence analysis in near-field imaging, Inverse Problems, to appear.Google Scholar
[7]Bao, G. and Lin, J., Near-field imaging of the surface displacement on an infinite ground plane, Inverse Problems and Imaging, 7 (2013), 377–396.Google Scholar
[8]Bruno, O. and Reitich, F., Numerical solution of diffraction problems: a method of variation of boundaries, J. Opt. Soc. Am. A, 10 (1993), 1168–1175.Google Scholar
[9]Cakoni, F. and Colton, D., Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer, 2006.Google Scholar
[10]Cheng, T., Li, P. and Wang, Y., Near-field imaging of perfectly conducting grating surfaces, J. Opt. Soc. Am. A, 30 (2013), 2473–2481.Google Scholar
[11]Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.Google Scholar
[12]Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1998.Google Scholar
[13]Courjon, D., Near-field Microscopy and Near-field Optics, Imperial College Press, London, 2003.Google Scholar
[14]NIST Digital Library of Mathematical Functions. dlmf.nist.gov/, Release 1.0.6 of 2013-05-06.Google Scholar
[15]Derveaux, G., Papanicolaou, G., and Tsogka, C., Resolution and denoising in near-field imaging, Inverse Problems, 22 (2006), 1437–1456.Google Scholar
[16]Li, P. and Shen, J., Analysis of the scattering by an unbounded rough surface, Math. Meth. Appl. Sci., 35 (2012), 2166–2184.Google Scholar
[17]Li, P. and Wang, Y., Near-field imaging of obstacles, preprint.Google Scholar
[18]Malcolm, A. and Nicholls, D.P., A field expansions method for scattering by periodic multi-layered media, J. Acout. Soc. Am., 129 (2011), 1783–1793.Google Scholar
[19]Malcolm, A. and Nicholls, D.P., A boundary perturbation method for recovering interface shapes in layered media, Inverse Problems, 27 (2011), 095009.Google Scholar
[20]Nicholls, D.P. and Reitich, F., Shape deformations in rough surface scattering: cancellations, conditioning, and convergence, J. Opt. Soc. Am. A, 21 (2004), 590–605.Google Scholar
[21]Nicholls, D.P. and Reitich, F., Shape deformations in rough surface scattering: improved algorithms, J. Opt. Soc. Am. A, 21 (2004), 606–621.Google Scholar
[22]Nicholls, D.P. and Shen, J., A stable high-order method for two-dimensional bounded-obstacle scattering, SIAM J. Sci. Comput., 28 (2006), 1398–1419Google Scholar
[23]Qin, H. and Cakoni, F., Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005.Google Scholar
[24]Qin, H. and Colton, D., The inverse scattering problem for cavities, Appl. Numer. Math., 62 (2012), 699–708.Google Scholar
[25]Qin, H. and Colton, D., The inverse scattering problem for cavities with impedance boundary condition, Adv. Comput. Math., 36 (2012), 157–174.Google Scholar
[26]Zeng, F., Cakoni, F. and Sun, J., An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002.Google Scholar
[27]Zeng, F., Suarez, P., and Sun, J., A decomposition method for an interior inverse scattering problem, Inverse Problems and Imaging, 7 (2013), 291–303.Google Scholar