  This paper considers the inverse problem of scattering of time-harmonic
acoustic and electromagnetic plane waves by a bounded, inhomogeneous,
penetrable obstacle with embedded objects inside. A new method is proposed to
prove that the inhomogeneous penetrable obstacle can be uniquely determined
from the far-field pattern at a fixed frequency, disregarding its contents. Our
method is based on constructing a well-posed interior transmission problem in a
small domain associated with the Helmholtz or modified Helmholtz equation and
the Maxwell or modified Maxwell equations. A key role is played by the
smallness of the domain which ensures that the lowest transmission eigenvalue
is large so that a given wave number k is not an eigenvalue of the interior
transmission problem. Another ingredient in our proofs is a priori estimates of
solutions to the transmission scattering problems with data in Lp (1<p<2),
which are established in this paper by using the integral equation method. A
main feature of the new method is that it can deal with the acoustic and
electromagnetic cases in a unified way and can be easily applied to deal with
inverse scattering by unbounded rough interfaces.
