  We show how off-resonant light scattering can provide quantitative
information on antiferromagnetic ordering of a two-species fermionic atomic gas
in a tightly-confined two-dimensional optical lattice. We analyze the emerging
magnetic ordering of atoms in the mean-field and in random phase approximations
and show how the many-body static and dynamic correlations, evaluated in the
standard Feynman-Dyson perturbation series, can be detected in the scattered
light signal. The staggered magnetization reveals itself in the magnetic Bragg
peaks of the individual spin components. These magnetic peaks, however, can be
considerably suppressed in the absence of a true long-range antiferromagnetic
order. The light scattered outside the diffraction orders can be collected by a
lens a with highly improved signal-to-shot-noise ratio when the diffraction
maxima are blocked. The collective and single-particle excitations are
identified in the spectrum of the scattered light. We find that the
spin-conserving and spin-exchanging atomic transitions convey information on
density, longitudinal spin, and transverse spin correlations. The different
correlations and scattering processes exhibit characteristic angular
distribution profiles for the scattered light and, e.g., the diagnostic signal
of transverse spin correlations could be separated from the signal by the
scattering direction, frequency, or polarization. We also analyze the detection
accuracy by estimating the number of required measurements, constrained by the
heating rate that is determined by inelastic light scattering events. The
imaging technique could be extended to the two-species fermionic states in
other regions of the phase diagram where the ground state properties are still
not fully understood.
