Abstract
The paper is concerned with the reconstruction of a defect in the core of a two-dimensional open waveguide from the scattering data. Since only a finite numbers of modes can propagate without attenuation inside the core, the problem is similar to the one-dimensional inverse medium problem. In particular, the inverse problem suffers from a lack of uniqueness and is known to be severely ill-posed. To overcome these difficulties, we consider multi-frequency scattering data. The uniqueness of solution to the inverse problem is established from the far field scattering information over an interval of low frequencies.
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References
Ammari H, Iakovleva E, Kang H. Reconstruction of a small inclusion in a two-dimensional open waveguide. SIAM J Appl Math, 2005, 65: 2107–2127
Ammari H, Triki F. Resonances for microstrip transmission lines. SIAM J Appl Math, 2004, 64: 601–636
Bao G, Li P. Inverse medium scattering problems for electromagnetic waves. Inverse Probl, 2005, 21: 1621–1641
Bao G, Li P. Inverse medium scattering problems for electromagnetic waves. SIAM J Appl Math, 2005, 65: 2049–2066
Bao G, Triki F. Error estimates for recursive linearization of inverse medium problems. J Comput Math, 2010, 28: 725–744
Bao G, Lin J, Triki F. A multi-frequency inverse source problem. J Differential Equations, 2010, 249: 3443–3465
Bao G, Lin J, Triki F. An inverse source problem with multiple frequency data. Comptes rendus Mathematiques, 2011, 349: 855–859
Bartashevskii E L, Drobakhin O O, Slavin I V. Interpretation of results of multifrequency measurements in cheking multilayer dielectric structures. Measurement Techniques, 1984, 27: 511–514
Bonnet-Ben Dhia A S, Chorfi L, Dakhia G, et al. Diffraction by a defect in an open waveguide: a mathematical analysis based on a modal radiation condition. SIAM J Appl Math, 2009, 70: 677–693
Chen Y, Rokhlin V. On the inverse scattering problem for the Helmholtz equation in one dimension. Inverse Probl, 1992, 8: 365–391
Ciraolo G, Magnanini R. A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides. Math Methods Appl Sci, 2009, 32: 1183–1206
Coddington E A, Levinson N. Theory of Differential Equations. New York: McGraw-Hill, 1955
Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. In: Applied Mathematical Sciences, vol. 93. Berlin: Springer-Verlag, 1992
Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. New York: Springer-Verlag, 1977
Magnanini R, Santosa F. Wave propagation in a 2-D optical waveguide. SIAM J Appl Math, 2000, 61: 1237–1252
Snyder A W, Love D. Optical Waveguide Theory. London: Chapman & Hall, 1974
Wilcox C. Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water. Arch Rat Mech Anal, 1975, 60: 259–300
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Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday
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Bao, G., Triki, F. Reconstruction of a defect in an open waveguide. Sci. China Math. 56, 2539–2548 (2013). https://doi.org/10.1007/s11425-013-4696-8
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DOI: https://doi.org/10.1007/s11425-013-4696-8