Near-field linear sampling method for an inverse problem in an electromagnetic waveguide

Abstract

We consider the problem of determining the shape and location of an unknown penetrable object in a perfectly conducting electromagnetic waveguide. The inverse problem is posed in the frequency domain and uses multistatic data in the near field. In particular, we assume that we are given measurements of the electric scattered field due to point sources on a cross-section of the waveguide and measured on the same cross-section, which is away from the scatterer but not in the far field.The problem is solved by using the Linear Sampling Method (LSM) and we also discuss the generalized LSM. We start by giving a brief discussion of the direct problem and its associated interior transmission problem. Then, we adapt and analyze the LSM to deal with the inverse problem. This extends the work on the LSM for perfectly conducting scatterers in a waveguide by one of us (Yang) to the detection of penetrable objects. We provide several useful results concerning reciprocity and the density of fields due to single layer potentials. We also prove the standard results for the LSM in the waveguide context. Finally we give numerical results to show the performance of the method for simple shapes.

Imaging junctions of waveguides

“…With this convention, the transmission conditions hold in H 1{2 pΣ 0 q andH´1 {2 pΣ 0 q, respectively. By using a similar weak formulation as in Proposition 1 in the bounded domain W b R , we would prove that problem (19) has a unique solution G s p¨, yq such that pG s p¨, yq| W0 , G s p¨, yq|W 0 q P H 1 loc pW 0 qˆH 1 loc pW 0 q. Then Gp¨, yq " G 0 p¨, yq`G s p¨, yq is the solution to problem (17).…”

“…Then Gp¨, yq " G 0 p¨, yq`G s p¨, yq is the solution to problem (17). Next, we remark by using the decomposition u n " gǹ ,0`v s n for all n P N, where gǹ ,0 plays the role of an incident wave while v s n plays the role of a scattered wave, that the field v s n satisfies the transmission problem Comparing systems (19) and 20, by linearity we obtain that for y P W 0 and x P W , we have…”

“…Note that in [3], our method was successfully tested in the presence of real data coming from an ultrasonic NDT experiment on a steel plate. We here mention several other works based on sampling-type methods in acoustic waveguides [15,20,21,22,5,6] and electromagnetic waveguides [7,19].…”

A sampling type method in an electromagnetic waveguide

“…[22] developed the linear sampling method for an acoustic waveguide in the time domain. [23] investigated the linear sampling method in an electromagnetic waveguide. We also mention the inverse scattering in an acoustic waveguide [27], the time migration imaging method in an acoustic terminating waveguide [28], in an electromagnetic waveguide [13] and the time reversal imaging in an electromagnetic terminating waveguide [4].…”

“…In the electromagnetic waveguides, there have been few work [13,4,23]. In particular, [13] considered a reverse time migration imaging method, where the Helmholtz-Kirchhoff identity was used to show that the imaging function peaks in the scatterer.…”