Optimal optical measurement of small displacements

We derive the quantum noise limit for the optical beam displacement of a TEM₀₀ mode. Using a multimodal analysis, we show that the conventional split detection scheme for measuring beam displacement is non-optimal with ∼80% efficiency. We propose a new displacement measurement scheme that is optimal for small beam displacement. This scheme utilizes a homodyne detection set-up that has a TEM₁₀ mode local oscillator. We show that although the quantum noise limit to displacement measurement can be surpassed using squeezed light in appropriate spatial modes for both schemes, the TEM₁₀ homodyning scheme outperforms split detection for all values of squeezing.