Misalignment and mode mismatch error signals for higher-order Hermite-Gauss modes from two sensing schemes

The locking of lasers to optical cavities is ubiquitously required in the field of precision interferometry such as Advanced LIGO to yield optimal sensitivity. Using higher-order Hermite-Gauss (HG) modes for the main interferometer beam has been a topic of recent study, due to their potential for reducing thermal noise of the test masses. It has been shown however that higher-order HG modes are more susceptible to coupling losses into optical cavities: the misalignment and mode mismatch induced power losses scale as $2n+1$ and $n^{2}+n+1$ respectively with $n$ being the mode index. In this paper we calculate analytically for the first time the alignment and mode mismatch sensing signals for arbitrary higher-order HG modes with both the traditional sensing schemes (using Gouy phase telescopes and quadrant photodetectors) and the more recently proposed radio-frequency jitter-based sensing schemes (using only single element photodiodes). We show that the sensing signals and also the signal-to-shot noise ratios for higher-order HG modes are larger than for the fundamental mode. In particular, the alignment and mode mismatch sensing signals in the traditional sensing schemes scale approximately as $\sqrt{n}$ and $n$ respectively, whereas in the jitter-based sensing schemes they scale exactly as $2n+1$ and $n^{2}+n+1$, respectively, which exactly matches the decrease in their respective tolerances. This potentially mitigates the downside of higher-order HG modes for their suffering from excessive misalignment and mode-mismatch induced power losses.