Experimental demonstrations of alignment and mode matching in optical cavities with higher-order Hermite-Gauss modes

Higher-order spatial laser modes have recently been investigated as candidates for reducing test-mass thermal noise in ground-based gravitational-wave detectors such as advanced LIGO. In particular, higher-order Hermite-Gauss (HG) modes have gained attention within the community for their more robust behaviors against random test-mass surface deformations and stronger sensing and control capacities. In this letter we offer experimental investigations on various aspects of HG mode interferometry. We have generated purified HG modes up to the 12-th order $\mathrm{HG}_{6,6}$ mode, with a power conversion efficiency of 38.8% and 27.7% for the $\mathrm{HG}_{3,3}$ and $\mathrm{HG}_{6,6}$ modes respectively. We demonstrated for the first time the misalignment and mode mismatch-induced power coupling loss measurements for HG modes up to the $\mathrm{HG}_{6,6}$. We report an excellent agreement with the extended numerical power loss factors that in the ``small power loss'' region converge to $2n+1$ or $n^2+n+1$ for a misaligned or mode mismatched $\mathrm{HG}_{n,n}$ mode. We also demonstrated the wavefront sensing (WFS) signal measurement for HG modes up to the $\mathrm{HG}_{6,6}$. The measurement result is accurately in accordance with theoretical WFS gain $\beta_{n,n-1}\sqrt{n} + \beta_{n,n+1}\sqrt{n+1}$ for an $\mathrm{HG}_{n,n}$ mode, with $\beta_{n,n-1}$ being the beat coefficient of the adjacent $\mathrm{HG}_{n,n}$ and $\mathrm{HG}_{n-1,n}$ modes on a split photodetector.