  In optical metrological protocols to measure physical quantities, it is, in
principle, always beneficial to increase photon number to improve measurement
precision. However, practical constraints prevent arbitrary increase of n due
to the imperfections of a practical detector, especially when the detector
response is dominated by saturation effect. In this work, we show that a
modified weak measurement protocol, namely, biased weak measurement
significantly improves the precision of optical metrology in the presence of
saturation effect. This method detects an ultra-small fraction of photons while
maintains considerable amount of metrological information. The biased
pre-coupling leads to an additional reduction of photons in the post-selection
and generates an extinction point in the spectrum distribution, which is
extremely sensitive to the estimated parameter and difficult to be saturated.
Therefore, the Fisher information can be persistently enhanced by increasing
the photon number. In our magnetic-sensing experiment, biased weak measurement
achieves precision approximately one order of magnitude better than those of
previously used methods. The proposed method can be applied in various optical
measurement schemes to circumvent detector saturation effect with low-cost
apparatuses.
