  A thin atomic gas in an optical resonator exhibits a phase transition from a
homogeneous density to crystalline order when laser illuminated orthogonal to
the resonator axis. We study this well-known self-organization phenomenon for a
generalized pumping scheme using a femtosecond pulse train with a frequency
spectrum spanning a large bandwidth covering many cavity modes. We show that
due to simultaneous scattering into adjacent longitudinal cavity modes the
induced light forces and the atomic dynamics becomes nearly
translation-invariant along the cavity axis. In addition the laser bandwidth
introduces a new correlation length scale within which clustering of the atoms
is energetically favorable. Numerical simulations allow us to determine the
self-consistent ordering threshold power as function of bandwidth and atomic
cloud size. We find strong evidence for a change from a second order to a first
order self-ordering phase transition with growing laser bandwidth when the size
of the atomic cloud gets bigger than the clustering length. An analysis of the
cavity output reveals a corresponding transition from a single to a double
pulse traveling within the cavity. This doubles the output pulse repetition
rate and creates a new time crystal structure in the cavity output. Simulations
also show that multi-mode operation significantly improves cavity cooling
generating lower kinetic temperatures at a much faster cooling rate.
