  It is well known that the modulus of the far-field pattern (or phaseless
far-field pattern) is invariant under translations of the scattering obstacle
if only one plane wave is used as the incident field, so the shape but not the
location of the obstacle can be recovered from the phaseless far-field data. In
this paper, it is proved that the translation invariance property of the
phaseless far-field pattern can be broken if superpositions of two plane waves
are used as the incident fields for all wave numbers in a finite interval.
Based on this, a recursive Newton-type iteration algorithm in frequencies is
then developed to recover both the location and the shape of the obstacle
simultaneously from multi-frequency phaseless far-field data. Numerical
examples are also carried out to illustrate the validity of the approach and
the effectiveness of the inversion algorithm.
