Weak-value amplification in a shot-noise-limited interferometer

We study the weak-value amplification (WVA) in a phase measurement with an optical interferometer in which shot noise limits the sensitivity. We compute the signal and the shot noise including the full-order interaction terms of the WVA, and show that the shot-noise contribution to a phase shift in a pointer variable is always larger than the final variance of the pointer variable. This yields the difference in estimating the noise level up to a factor of 1.5. To clarify an advantage for practical uses of the WVA, we discuss the signal-to-noise ratio and its optimization in the presence of the shot noise.