  We consider two-dimensional superconductor/ferromagnet/superconductor
junctions and investigate the subgap modes along the junction interface. The
subgap modes exhibit characteristics similar to the Yu-Shiba-Rusinov states
that originate form the interplay between superconductivity and ferromagnetism
in the magnetic junction. The dispersion relation of the subgap modes shows
qualitatively different profiles depending on the transport state (metallic,
half-metallic, or insulating) of the ferromagnet. As the spin splitting in the
ferromagnet is increased, the subgap modes bring about a $0$-$\pi$ transition
in the Josephson current across the junction, with the Josephson current
density depending strongly on the momentum along the junction interface (i.e.,
the direction of the incident current). For clean superconductor-ferromagnet
interfaces (i.e., strong coupling between superconductors and ferromagnet), the
subgap modes develop flat quasi-particle bands that allow to engineer the wave
functions of the subgap modes along an inhomogeneous magnetic junction.
