  Guiding electronic waves in a manner similar to photon transmission in
optical fibers is key for developing the electron-optics toolbox. Here we
outline a `weak guiding' approach, in which efficient diffraction around
disorder results in low-loss, high-finesse electron guiding. We describe an
implementation of this scheme for guiding along a narrow-width line gate in
gapless and gapped graphene. A simple model for weak guiding, which relies on
the Jackiw-Rebbi midgap states, is introduced and solved. The weak-guiding
modes are shown to exist for confining potential of either sign and no matter
how strong or weak. Modes evolve in a cyclic manner upon varying gate
potential, repeatedly sweeping the Dirac gap and becoming dispersionless (flat
band) at certain magic values of gate potential. Large mode widths facilitate
diffraction around disorder in the core region, enabling exceptionally large
mean free paths and long-range ballistic propagation.
