Transverse Distance Estimation with Higher-Order Hermite-Gauss modes

We explore the use of higher-order Hermite-Gauss modes for sensing optically induced transverse displacements. In the small-displacement regime, we show that projective measurements onto the two neighboring spatial modes yield optimal Fisher information, linearly scaling with the mode order $m$. We further extend the analysis to arbitrary displacement values and derive general expressions for the Fisher information, demonstrating that higher-order modes continue to outperform the fundamental Gaussian mode even at larger separations. This approach enables enhanced displacement sensitivity with only a minimal number of measurements, offering a simple and scalable alternative to conventional Spatial Mode Demultiplexing schemes. We provide a proof-of-principle experimental demonstration using spatial light modulators, showing an order-of-magnitude reduction in estimation variance when employing Hermite-Gauss modes of order $m = 8$ and $m = 17$. These results highlight the potential of structured light for ultrasensitive displacement sensing and may enable new applications in birefringence measurements with broadband or low-coherence light sources.