  We report two new classes of face-centered photonic square lattices with flat
bands which we call the Lieb-I and the Lieb-II lattices. There are 5 and 7
sites in the corresponding unit cells of the simplest Lieb-I and Lieb-II
lattices, respectively. The number of flat bands $m$ in the new Lieb lattices
is related to the number of sites $N$ in the unit cell by $m=(N-1)/2$. Physical
properties of the lattices with even and odd number of flat bands are
different. We also consider localization of light in such Lieb lattices. If the
input beam excites the flat-band mode, it will not diffract during propagation,
owing to the strong localization in the flat-band mode. For the Lieb-II
lattice, we also find that the beam will oscillate and still not diffract
during propagation, because of the intrinsic oscillating properties of certain
flat-band modes. The period of oscillation is determined by the energy
difference between the two flat bands. This study provides a new platform for
the investigation of flat-band modes.
