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

A nearly optimal and robust protocol for nonlinear phase estimation using coherent states