  We derive a sufficient condition for the existence of index-guided modes in a
very general class of dielectric waveguides, including photonic-crystal fibers
(arbitrary periodic claddings, such as ``holey fibers''), anisotropic
materials, and waveguides with periodicity along the propagation direction.
This condition provides a rigorous guarantee of cutoff-free index-guided modes
in any such structure where the core is formed by increasing the index of
refraction (e.g. removing a hole). It also provides a weaker guarantee of
guidance in cases where the refractive index is increased ``on average''
(precisely defined). The proof is based on a simple variational method,
inspired by analogous proofs of localization for two-dimensional attractive
potentials in quantum mechanics.
