A Class of Radiation Conditions for Open Periodic Waveguides

In this paper we consider the propagation of waves in an open waveguide in the half space R + 2 = { x ∈ R 2 : x 2 > 0 } under Dirichlet- or Neumann boundary condition for x 2 = 0 . The index of refraction n= n( x) is periodic along the axis of the waveguide (which we choose to be the x 1 − axis) and equal to one for x 2 > h 0 for some h 0 > 0 . We show first existence and uniqueness of a solution for the absorbing case, i.e. where the index of refraction is given by n( x)+ iεq( x) with ε>0 and some function q which is periodic with respect to x 1 , vanishes for x 2 > h 0 , and satisfies the angular spectral representation radiation condition. Then we prove convergence of the solution as ε tends to zero. We show that the limit solves the source problem for n( x) and satisfies a radiation condition which depends, first, on the choice of the absorption function q and, second, whether or not a cut-off value is critical with a non-evanescent mode. We show that the existence of non-evanescent modes is responsible for a slower decay of the solution along the axis of the waveguide. MSC: 35J05