  We calculate the time-of-flight patterns of strongly interacting bosons
confined in two-dimensional square lattice in the presence of an artificial
magnetic field using quantum rotor model that is inherently combined with the
Bogolyubov approach. We consider various geometries of the magnetic flux, which
are expected to be realizable, or have already been implemented in experimental
settings. The flexibility of the method let us to study cases of the artificial
magnetic field being uniform, staggered or forming a checkerboard
configuration. Effects of additional temporal modulation of the optical
potential that results from application of Raman lasers driving particle
transitions between lattice sites are also included. The presented
time-of-flight patterns may serve as a verification of chosen gauge in
experiments, but also provide important hints on unconventional, non-zero
momentum condensates, or possibility of observing graphene-like physics
resulting from occurrence of Dirac cones in artificial magnetic fields in
systems of ultra-cold bosons in optical lattices. Also, we elucidate on
differences between effects of magnetic field in solids and the artificial
magnetic field in optical lattices, which can be controlled on much higher
level leading to effects not possible in condensed matter physics.
