  While fundamental-mode discrete solitons have been demonstrated with both
self-focusing and defocusing nonlinearity, high-order-mode localized states in
waveguide lattices have been studied thus far only for the self-focusing case.
In this paper, the existence and stability regimes of dipole, quadrupole and
vortex soliton structures in two-dimensional lattices induced with a defocusing
nonlinearity are examined by the theoretical and numerical analysis of a
generic envelope nonlinear lattice model. In particular, we find that the
stability of such high-order-mode solitons is quite different from that with
self-focusing nonlinearity. As a simple example, a dipole (``twisted'') mode
soliton which may be stable in the focusing case becomes unstable in the
defocusing regime. Our results may be relevant to other two-dimensional
defocusing periodic nonlinear systems such as Bose-Einstein condensates with a
positive scattering length trapped in optical lattices.
