  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 nonlinear and non-selfadjoint interior
transmission eigenvalue problems.
