  The Cram\'er-Rao bound and the quantum Fisher information have been tools
used extensively for interferometric phase sensitivity. Most scenarios
considering a Mach-Zehnder interferometer with two input sources focused on the
phase-matched case, when the Fisher information is maximal. Under this
constraint, the best sensitivity is achieved for a balanced (50/50) input beam
splitter. In this paper, we take a different approach: we allow the beam
splitter transmission coefficient as well as the input phase mismatch to be
variable parameters. We then search for a pair of these parameters that
maximizes the Fisher information. We find that for the double coherent input
the maximum Fisher information can always be reached in the unbalanced case for
a carefully chosen input phase mismatch. For the coherent plus squeezed vacuum
case we find that under certain circumstances a threshold phase mismatch
exists, beyond which the optimum Fisher information is found for the degenerate
case. For the squeezed-coherent plus squeezed vacuum case we find that the
optimum actually occurs when the squeezing angles of the two inputs are in
anti-phase.
