  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.
