  In atomic and molecular phase measurements using laser-induced fluorescence
detection, optical cycling can enhance the effective photon detection
efficiency and hence improve sensitivity. We show that detecting many photons
per atom or molecule, while necessary, is not a sufficient condition to
approach the quantum projection limit for detection of the phase in a two-level
system. In particular, detecting the maximum number of photons from an
imperfectly closed optical cycle reduces the signal-to-noise ratio (SNR) by a
factor of $\sqrt{2}$, compared to the ideal case in which leakage from the
optical cycle is sufficiently small. We derive a general result for the SNR in
a system in terms of the photon detection efficiency, probability for leakage
out of the optical cycle per scattered photon, and the product of the average
photon scattering rate and total scattering time per atom or molecule.
