  We demonstrate that crossed arrays of optical fibers support the
double-discrete linear and nonlinear propagation of light beams, in which not
only the transverse coordinate (the fiber's number) is discrete, but also the
longitudinal (propagation) coordinate, i.e., the number of the fiber-crossing
site, is effectively discrete too. In the linear limit, this transmission
regime features double-discrete self-collimation. The nonlinear fishnet arrays
with both focusing and defocusing nonlinearities give rise to double-discrete
spatial solitons. Solitons bifurcating from two different branches of the
linear dispersion relation feature strong interactions and form composite
states. In the continuum limit, the model of the nonlinear fishnet reduces to a
system of coupled-mode equations similar to those describing Bragg gratings,
but without the cross-phase-modulation terms.
