  We study a Helmholtz-type spectral problem related to the propagation of
electromagnetic waves in photonic crystal waveguides. The waveguide is created
by introducing a linear defect into a two-dimensional periodic medium. The
defect is infinitely extended and aligned with one of the coordinate axes. The
perturbation is expected to introduce guided mode spectrum inside the band gaps
of the fully periodic, unperturbed spectral problem. In the first part of the
paper, we prove that, somewhat unexpectedly, guided mode spectrum can be
created by arbitrarily "small" perturbations. Secondly we show that, after
performing a Floquet decomposition in the axial direction of the waveguide, for
any fixed value of the quasi-momentum $k_x$ the perturbation generates at most
finitely many new eigenvalues inside the gap.
