  We consider the propagation of broad optical beams through slab waveguides
with a purely quadratic nonlinearity and containing linear and nonlinear
long-period quasi-phase-matching gratings. An exact Floquet analysis on the
periodic, plane-wave solution shows that the periodicity can drastically alter
the growth rate of the modulational instability but that it never completely
removes the instability. The results are confirmed by direct numerical
simulation, as well as through a simpler, approximate theory for the averaged
fields that accurately predicts the low-frequency part of the spectrum.
