  Scattering problems with locally perturbed periodic surfaces have been
studied both theoretically and numerically in recent years. In this paper, we
will discuss the regularity results of the Bloch transform of the total fields.
The idea is inspired by Theorem a in \cite{Kirsc1993}, which considered how the
total field depends on the wave numbers and the incident angles, with a family
of plain incident fields and a smooth enough periodic surface. We will show
that when the incident field satisfies some certain conditions, the Bloch
transform of the total field depends analytically on the quasi-periodicities
one the straight line $\R$ except for a countable number of points, while near
such points, a square-root like singularity exists. We also give some examples
to show that the conditions are satisfied by a large number of commonly used
incident fields. This result also provides a probability to improve the
numerical solution of this kind of problems, which is expected to be discussed
in our future papers.
