Unique determination of the index of refraction from phase-less near-field data in two dimensions

Motivated by non-destructive testing of optical fiber, we consider the problem of determining the index of refraction of a two-dimensional medium from magnitude of the total field resulting from known incident plane waves at a fixed frequency. The finiteness of the fiber cross section allows us to assume that the index of refraction is variable over a compact set. The measurement set is a closed curve, e.g., a circle of fixed radius, containing the fiber cross section. We use complex geometric optics (CGO) solutions to show that the inverse problem for the unknown index of refraction is unique in $C^{k}$ for large enough $k$. The key tool in this work is the multi-term expansion of the CGO solutions which we rely on to address the uniqueness question.