  This paper is concerned with a nonlinear imaging problem, which aims to
reconstruct a locally perturbed, perfectly reflecting, infinite plane from
intensity-only (or phaseless) far-field or near-field data. A recursive Newton
iteration algorithm in frequencies is developed to reconstruct the locally
rough surface from multi-frequency intensity-only far-field or near-field data,
where the fast integral equation solver developed in [39] is used to solve the
direct scattering problem in each iteration. For the case with far-field data,
a main feature of our work is that the incident field is taken as a
superposition of two plane waves with different directions rather than one
plane wave, so the location and shape of the local perturbation of the infinite
plane can be reconstructed simultaneously from intensity-only far-field data
with multiple wave numbers. This is different from previous work on inverse
scattering from phaseless far-field data, where only the shape reconstruction
was considered due to the translation invariance property of the phaseless
far-field pattern corresponding to one plane wave as the incident field.
Finally, numerical examples are carried out to demonstrate that our
reconstruction algorithm is stable and accurate even for the case of
multiple-scale profiles.
