# American Institute of Mathematical Sciences

April  2021, 15(2): 285-314. doi: 10.3934/ipi.2020065

## Imaging junctions of waveguides

 1 Laboratoire POEMS, ENSTA Paris, 828 Boulevard des Maréchaux, 91120 Palaiseau, France 2 Université Paris-Saclay, CEA, LIST, F-91120 Palaiseau, France

* Corresponding author: Laurent Bourgeois

Received  April 2020 Revised  September 2020 Published  November 2020

In this paper we address the identification of defects by the Linear Sampling Method in half-waveguides which are related to each other by junctions. Firstly a waveguide which is characterized by an abrupt change of properties is considered, secondly the more difficult case of several half-waveguides related to each other by a junction of complex geometry. Our approach is illustrated by some two-dimensional numerical experiments.

Citation: Laurent Bourgeois, Jean-François Fritsch, Arnaud Recoquillay. Imaging junctions of waveguides. Inverse Problems & Imaging, 2021, 15 (2) : 285-314. doi: 10.3934/ipi.2020065
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##### References:
Waveguide with an abrupt change of properties
Obstacles within the waveguide
Exact data on the section $\Sigma^0$. Left: $R^0 = 1.1$. Right: $R^0 = 3$
A waveguide with a transition zone (the domain $B_R$ is hatched)
A junction of three half-waveguides (the domain $B$ is hatched)
Full-scattering, $\kappa = 40$ ($P = 13$) and $\tilde{\kappa} = 60$ ($\tilde{P} = 20$). Top left: obstacle 3 and exact data. Top right: obstacle 3 and noisy data. Bottom left: obstacle 4 and exact data. Bottom right: obstacle 4 and noisy data
Back-scattering for obstacle 1. Top left: $\kappa = 40$ ($P = 13$) and $\tilde{\kappa} = 20$ ($\tilde{P} = 7$), exact data. Top right: $\kappa = 40$ and $\tilde{\kappa} = 20$, noisy data. Middle left: $\kappa = \tilde{\kappa} = 40$ ($P = \tilde{P} = 13$), exact data. Middle right: $\kappa = \tilde{\kappa} = 40$, noisy data. Bottom left: $\kappa = 40$ ($P = 13$) and $\tilde{\kappa} = 60$ ($\tilde{P} = 20$), exact data. Bottom right: $\kappa = 40$ and $\tilde{\kappa} = 60$, noisy data
Back-scattering for obstacle 2. Top left: $\kappa = 40$ ($P = 13$) and $\tilde{\kappa} = 20$ ($P = 7$), exact data. Top right: $\kappa = 40$ and $\tilde{\kappa} = 20$, noisy data. Middle left: $\kappa = \tilde{\kappa} = 40$ ($P = \tilde{P} = 13$), exact data. Middle right: $\kappa = \tilde{\kappa} = 40$, noisy data. Bottom left: $\kappa = 40$ ($P = 13$) and $\tilde{\kappa} = 60$ ($\tilde{P} = 20$), exact data. Bottom right: $\kappa = 40$ and $\tilde{\kappa} = 60$, noisy data
Back-scattering for obstacle 4. Top left: $\kappa = 40$ ($P = 13$) and $\tilde{\kappa} = 20$ ($\tilde{P} = 7$), exact data. Top right: $\kappa = 40$ and $\tilde{\kappa} = 20$, noisy data. Middle left: $\kappa = \tilde{\kappa} = 40$ ($P = \tilde{P} = 13$), exact data. Middle right: $\kappa = \tilde{\kappa} = 40$, noisy data. Bottom left: $\kappa = 40$ ($P = 13$) and $\tilde{\kappa} = 60$ ($\tilde{P} = 20$), exact data. Bottom right: $\kappa = 40$ and $\tilde{\kappa} = 60$, noisy data
Full-scattering, obstacle 3, $\kappa = \tilde{\kappa} = 40$, $h = 0.65$ ($P = 9$) and $\tilde{h} = 1$ ($\tilde{P} = 13$). Left: exact data. Right: noisy data
Back-scattering for obstacle 1, $\kappa = \tilde{\kappa} = 30$ and $h>\tilde{h}$. Top left: $h = 1$ ($P = 10$) and $\tilde{h} = 0.5$ ($\tilde{P} = 5$), exact data. Top right: $h = 1$ and $\tilde{h} = 0.5$, noisy data. Bottom left: $h = 1$ ($P = 10$) and $\tilde{h} = 0.75$ ($\tilde{P} = 8$), exact data. Bottom right: $h = 1$ and $\tilde{h} = 0.75$, noisy data
Back-scattering for obstacle 1, $\kappa = \tilde{\kappa} = 30$ and $h<\tilde{h}$. Top left: $h = 0.5$ ($P = 5$) and $\tilde{h} = 1$ ($\tilde{P} = 10$), exact data. Top right: $h = 0.5$ and $\tilde{h} = 1$, noisy data. Bottom left: $h = 0.75$ ($P = 8$) and $\tilde{h} = 1$ ($\tilde{P} = 10$), exact data. Bottom right: $h = 0.75$ and $\tilde{h} = 1$, noisy data
Back-scattering for obstacle 2, $\kappa = \tilde{\kappa} = 30$ and $h>\tilde{h}$. Top left: $h = 1$ ($P = 10$) and $\tilde{h} = 0.5$ ($\tilde{P} = 5$), exact data. Top right: $h = 1$ and $\tilde{h} = 0.5$, noisy data. Bottom left: $h = 1$ ($P = 10$) and $\tilde{h} = 0.75$ ($\tilde{P} = 8$), exact data. Bottom right: $h = 1$ and $\tilde{h} = 0.75$, noisy data
Back-scattering for obstacle 2, $\kappa = \tilde{\kappa} = 30$ and $h<\tilde{h}$. Top left: $h = 0.5$ ($P = 5$) and $\tilde{h} = 1$ ($\tilde{P} = 10$), exact data. Top right: $h = 0.5$ and $\tilde{h} = 1$, noisy data. Bottom left: $h = 0.75$ ($P = 8$) and $\tilde{h} = 1$ ($\tilde{P} = 10$), exact data. Bottom right: $h = 0.75$ and $\tilde{h} = 1$, noisy data
Data on a single half-waveguide. Top left: exact data on section $\Sigma^0$. Top right: noisy data on section $\Sigma^0$. Middle left: exact data on section $\Sigma^1$. Middle right: noisy data on section $\Sigma^1$. Bottom left: exact data on section $\Sigma^2$. Bottom right: noisy data on section $\Sigma^2$
Top: data on two half-waveguides. Left: exact data on sections $\Sigma^0$ and $\Sigma^1$. Right: noisy data on sections $\Sigma^0$ and $\Sigma^1$. Bottom: data on three half-waveguides. Left: exact data on sections $\Sigma^0$, $\Sigma^1$ and $\Sigma^2$. Right: noisy data on sections $\Sigma^0$, $\Sigma^1$ and $\Sigma^2$
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