Fisher-Information Efficient Metagratings for Transverse Displacement Metrology at the Measurement Independent Shot-Noise Limit

We derive the measurement independent precision bound for transverse displacement metrology using a classical source. Using tools from quantum estimation theory and metamaterial designs, we prove that the bound is achieved efficiently using optimized metagratings by detecting only a very small amount of photons used to probe the displacement, and we also reveal the direct link between the resonant property in the unit cell and the conditions to achieve the bound: one with the 0th dipole resonance and the other one with the anapole condition of the 1st dipole.