  We analyze the amplification by the Aharonov-Albert-Vaidman weak quantum
measurement on a Sagnac interferometer [P. B. Dixon et al., Phys. Rev. Lett.
102, 173601 (2009)] up to all orders of the coupling strength between the
measured system and the measuring device. The amplifier transforms a small tilt
of a mirror into a large transverse displacement of the laser beam. The
conventional analysis has shown that the measured value is proportional to the
weak value, so that the amplification can be made arbitrarily large in the cost
of decreasing output laser intensity. It is shown that the measured
displacement and the amplification factor are in fact not proportional to the
weak value and rather vanish in the limit of infinitesimal output intensity. We
derive the optimal overlap of the pre- and post-selected states with which the
amplification become maximum. We also show that the nonlinear effects begin to
arise in the performed experiments so that any improvements in the experiment,
typically with an amplification greater than 100, should require the nonlinear
theory in translating the observed value to the original displacement.
