Recovering piecewise constant refractive indices by a single far-field pattern

We are concerned with the inverse scattering problem of recovering an inhomogeneous medium by the associated acoustic wave measurement. We prove that under certain assumptions, a single far-field pattern determines the values of a perturbation to the refractive index on the corners of its support. These assumptions are satisfied, for example, in the low acoustic frequency regime. As a consequence if the perturbation is piecewise constant with either a polyhedral nest geometry or a known polyhedral cell geometry, such as a pixel or voxel array, we establish the injectivity of the perturbation to far-field map given a fixed incident wave. This is the first unique determinancy result of its type in the literature, and all of the existing results essentially make use of infinitely many measurements.Peer reviewe

Unique Determination of the Shape of a Scattering Screen from a Passive Measurement

We consider the problem of fixed frequency acoustic scattering from a sound-soft flat screen. More precisely, the obstacle is restricted to a two-dimensional plane and interacting with an arbitrary incident wave, it scatters acoustic waves to three-dimensional space. The model is particularly relevant in the study and design of reflecting sonars and antennas, cases where one cannot assume that the incident wave is a plane wave. Our main result is that given the plane where the screen is located, the far-field pattern produced by any single arbitrary incident wave determines the exact shape of the screen, as long as it is not antisymmetric with respect to the plane. This holds even for screens whose shape is an arbitrary simply connected smooth domain. This is in contrast to earlier work where the incident wave had to be a plane wave, or more recent work where only polygonal scatterers are determined.Peer reviewe