  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.
