  We propose a protocol for the second-order nonlinear phase estimation with a
coherent state as input and balanced homodyne detection as measurement
strategy. The sensitivity is sub-Heisenberg limit, which scales as $N^{-3/2}$
for $N$ photons on average. By ruling out hidden resources in quantum Fisher
information, the fundamental sensitivity limit is recalculated and compared to
the optimal sensitivity of our protocol. In addition, we investigate the effect
of photon loss on sensitivity, and discuss the robustness of measurement
strategy. The results indicate that our protocol is nearly optimal and robust.
