  This paper is concerned with uniqueness in inverse acoustic scattering with
phaseless far-field data at a fixed frequency. The main difficulty of this
problem is the so-called translation invariance property of the modulus of the
far-field pattern generated by one plane wave as the incident field. Based on
our previous work (J. Comput. Phys. 345 (2017), 58-73), the translation
invariance property of the phaseless far-field pattern can be broken by using
infinitely many sets of superpositions of two plane waves as the incident
fields at a fixed frequency. In this paper, we prove 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 condition that the obstacle is a priori 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 $c_1$.
To the best of our knowledge, this is the first uniqueness result in inverse
scattering with phaseless far-field data. Our proofs are based essentially on
the limit of the normalized eigenvalues of the far-field operators which is
also established in this paper by using a factorization of the far-field
operators.
