We prove the existence of bound guided modes for the Helmholtz equation on lossless penetrable periodic slabs. We handle both robust modes, for which
no Bragg harmonics propagate away from slab, as well as nonrobust standing modes,
which exist in the presence of propagating Bragg harmonics. The latter are made
possible by symmetries of the slab structure, which prevent coupling of energy to the
propagating harmonics. These modes are isolated in wavevector-frequency space, as
they disappear under a perturbation of the wavevector. The main tool is a volumetric
integral equation of Lippmann-Schwinger type that has a self-adjoint kernel.