  Polarized light microscopy using path-entangled $N$-photon states (i.e., the
N00N states) has been demonstrated to surpass the shot-noise limit at very low
light illumination. However, the microscopy images suffer from divergence of
phase sensitivity, which inevitably reduces the image quality. Here, we show
that due to experimental imperfections, such a singularity also takes place in
the microscopy that uses twin-Fock states of light for illumination. We propose
two schemes to completely eliminate this singularity: (i) locking the phase
shift sensed by the beams at the optimal working point, by using a spatially
dependent offset phase; (ii) a combination of two binary-outcome photon
counting measurements, one with a fixed offset phase and the other without any
offset phase. Our observations remain valid for any kind of binary-outcome
measurement and may open the way for quantum-enhanced microscopy with high $N$
photon states.
