  This article discusses self-organization in cold atoms via light-mediated
interactions induced by feedback from a single retro-reflecting mirror.
Diffractive dephasing between the pump beam and the spontaneous sidebands
selects the lattice period. Spontaneous breaking of the rotational and
translational symmetry occur in the 2D plane transverse to the pump. We
elucidate how diffractive ripples couple sites on the self-induced atomic
lattice. The nonlinear phase shift of the atomic cloud imprinted onto the
optical beam is the parameter determining coupling strength. The interaction
can be tailored to operate either on external degrees of freedom leading to
atomic crystallization for thermal atoms and supersolids for a quantum
degenerate gas, or on internal degrees of freedom like populations of the
excited state or Zeeman sublevels. Using the light polarization degrees of
freedom on the Poincar{\'e} sphere (helicity and polarization direction),
specific irreducible tensor components of the atomic Zeeman states can be
coupled leading to spontaneous magnetic ordering of states of dipolar and
quadrupolar nature. The requirements for critical interaction strength are
compared for the different situations. Connections and extensions to
longitudinally pumped cavities, counterpropagating beam schemes and the CARL
instability are discussed.
