  We investigate interference of optical fields by examining the probability
distribution of photon detection. The usual description of interference
patterns in terms of superposition of classical mean fields with definite
phases is elucidated in quantum fashion. Especially, for interference of two
independent mixtures of number states with Poissonian or sub-Poissonian
statistics, despite lack of intrinsic phases, it is found that the joint
probability has a distinct peak manifold in the multi-dimensional space of the
detector outcomes, which is along the trajectory of the mean-field values as
the relative phase varies on the unit circle. Then, an interference pattern
should mostly appear in each shot of measurement as a point in the peak
manifold with a randomly chosen relative phase. On the other hand, for
super-Poissonian sources the mean-field description is likely invalidated with
rather broad probability distributions.
