  This paper is concerned with uniqueness in inverse acoustic scattering with
phaseless far-field data at a fixed frequency. In our previous work ({\em SIAM
J. Appl. Math. \bf78} (2018), 1737-1753), by utilizing spectral properties of
the far-field operator we proved for the first time that the obstacle and the
index of refraction of an inhomogeneous medium can be uniquely determined by
the phaseless far-field patterns generated by infinitely many sets of
superpositions of two plane waves with different directions at a fixed
frequency under the a priori assumption that the obstacle is known to be a
sound-soft or non-absorbing impedance obstacle and the index of refraction $n$
of the inhomogeneous medium is real-valued and satisfies that either $n-1\ge
c_1$ or $n-1\le-c_1$ in the support of $n-1$ for some positive constant $c_1$.
In this paper, we remove the a priori assumption on the obstacle and the index
of refraction of the inhomogeneous medium by adding a reference ball to the
scattering system together with a simpler method of using Rellich's lemma and
Green's representation formula for the scattering solutions. Further, our new
method is also used to prove uniqueness in determining a locally rough surface
from the phaseless far-field patterns corresponding to infinitely many sets of
superpositions of two plane waves with different directions as the incident
fields at a fixed frequency.
