  This paper is concerned with inverse scattering of plane waves by a locally
perturbed infinite plane (which is called a locally rough surface) with the
modulus of the total-field data (also called the phaseless near-field data) at
a fixed frequency in two dimensions. We consider the case where a Dirichlet
boundary condition is imposed on the locally rough surface. This problem models
inverse scattering of plane acoustic waves by a one-dimensional sound-soft,
locally rough surface; it also models inverse scattering of plane
electromagnetic waves by a locally perturbed, perfectly reflecting, infinite
plane in the TE polarization case. We prove that the locally rough surface is
uniquely determined by the phaseless near-field data generated by a countably
infinite number of plane waves and measured on an open domain above the locally
rough surface. Further, a direct imaging method is proposed to reconstruct the
locally rough surface from the phaseless near-field data generated by plane
waves and measured on the upper part of the circle with a sufficiently large
radius. Theoretical analysis of the imaging algorithm is derived by making use
of properties of the scattering solution and results from the theory of
oscillatory integrals (especially the method of stationary phase). Moreover, as
a by-product of the theoretical analysis, a similar direct imaging method with
full far-field data is also proposed to reconstruct the locally rough surface.
Finally, numerical experiments are carried out to demonstrate that the imaging
algorithm with phaseless near-field data and full far-field data are fast,
accurate and very robust with respect to noise in the data.
