Uniqueness in determining refractive indices by formally determined far-field data

Abstract

We present two uniqueness results for the inverse problem of determining an index of refraction by the corresponding acoustic far-field measurement encoded into the scattering amplitude. The first one is a local uniqueness in determining a variable index of refraction by the fixed incident-direction scattering amplitude. The inverse problem is formally posed with such measurement data. The second one is a global uniqueness in determining a constant refractive index by a single far-field measurement. The arguments are based on the study of certain non-self-adjoint interior transmission eigenvalue problems.

Keywords:

• inverse scattering
• refractive index
• uniqueness
• formally-determined
• interior transmission eigenvalues

AMS Subject Classifications:

• 35A02
• 35R30

Additional information

Funding

This work was supported by the German Research Foundation (DFG), [grant number HU 2111/1-1]; the NSF of China [grant number 11371115], [grant number 11201453], [grant number 91130022] and the NSF grant, DMS 1207784.