  This manuscript is the second in a series presenting fast direct solution
techniques for solving two-dimensional wave scattering problems from
quasi-periodic multilayered structures. The fast direct solvers presented in
the series are for the linear system that results from the discretization of a
robust integral formulation. The fast direct solver presented in this
manuscript has a computational cost that scales linearly with respect to the
number of discretization points on the interfaces and the number of layers. The
latter is an improvement over the previous solver and makes the new solver more
efficient especially for problems involving multiple incident angles and
changes to the layered media. Numerical results illustrate the improved
performance of the new solver over the previous one.
