Factorization method for imaging a local perturbation in inhomogeneous periodic layers from far field measurements

Houssem Haddar; Alexander Konschin

doi:10.3934/ipi.2019067

Article Contents

2020, Volume 14, Issue 1: 133-152. Doi: 10.3934/ipi.2019067

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Factorization method for imaging a local perturbation in inhomogeneous periodic layers from far field measurements

Houssem Haddar^1, and
Alexander Konschin^2, ,

1.
CMAP, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
2.
Center for Industrial Mathematics, University of Bremen, Bibliothekstraẞe 5, 28359 Bremen, Germany

^* Corresponding author: Alexander Konschin
^* Corresponding author: Alexander Konschin

Received: March 31, 2019

Revised: July 31, 2019

Published: November 2019

A. Konschin was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 281474342/GRK2224/1

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Abstract

We analyze the Factorization method to reconstruct the geometry of a local defect in a periodic absorbing layer using almost only incident plane waves at a fixed frequency. A crucial part of our analysis relies on the consideration of the range of a carefully designed far field operator, which characterizes the geometry of the defect. We further provide some validating numerical results in a two dimensional setting.

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Mathematics Subject Classification: Primary: 78A46, 35R30; Secondary: 35Q60.

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Figure 1. Example for the refractive index $ n = n_p + q $

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Figure 2. Three periods of the parameters and results for Example $ 1 $

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Figure 3. Three periods of main parameters for the numerical experiments

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Figure 4. Results for Example $ 2 $ with $ k^2 = 0.4 $, $ q = q_1 $, $ \nu = 10^{-3} $ and $ \nu = 10^{-6} $

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Figure 5. Results for Example $ 3 $ with $ k^2 = 1.8 $, $ \nu = 1 $ and $ \nu = 10^{-3} $

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Figure 6. Results for Example $ 4 $ with $ k^2 = 0.09 $, $ \nu = 10^{-3} $ and with $ k^2 = 3 $, $ \nu = 1 $

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Figure 7. Results for FM and LSM with $ k^2 = 3 $ and $ 1\% $ relative noise

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