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Active mode mismatch sensing and control can facilitate optimal coupling in optical cavity experiments such as interferometric gravitational wave detectors. In this paper, we demonstrate a radio-frequency (RF) beam wavefront curvature modulation-based mode mismatch sensing scheme inspired by the previously proposed RF beam jitter alignment sensing scheme. The proposed mode mismatch sensing scheme uses an electro-optic lens (EOL) device that is designed to provide the required beam wavefront curvature actuation, as well as a mode converting telescope that rephases the RF second-order modes and generates a non-vanishing mode mismatch sensing signal. We carefully investigate the total second-order mode generation from the wavefront actuation both analytically and numerically, taking the effects of Gaussian beam size evolution and the second-order mode phase mismatch cancellation into consideration. We demonstrate the second-order mode generation as a function of the incident beam waist size and the electro-optic crystal size which, along with a “trade-off” consideration of the beam size at the edges of the crystal and the clipping loss, provides us with guidance for designing the beam profile that interacts with the crystal to improve the EOL modulation efficiency. 

Abstract Odd-indexed higher-order Hermite–Gauss (HG) modes are compatible with four-quadrant segmented mirrors due to their intensity nulls along the principal axes, which guarantees minimum beam intensity illuminating the bond lines between the segments thus leading to low power loss. However, a misplaced HG beam can cause extra power loss due to the bright intensity spots probing the bond lines. This paper analytically and numerically studies the beam displacement tolerances on a segmented mirror for the $\mathrm{H}{\mathrm{G}}_{3,3}$mode. We conclude that for ‘effective’ bond lines with 6 µm width, and the $\mathrm{H}{\mathrm{G}}_{3,3}$beam size chosen to guarantee 1 ppm clipping loss when centered, the beam can be rotated by roughly 1^∘or laterally displaced by 4% of its beam size while keeping the total power on the bond lines under 1 ppm. We also demonstrate that the constrained beam displacement parameter region that guarantees a given power loss limit, or the beam displacement tolerance, is inversely proportional to the bond line thickness. 

This paper analytically and numerically investigates misalignment and mode-mismatch-induced power coupling coefficients and losses as a function of Hermite–Gauss (HG) mode order. We show that higher-order HG modes are more susceptible to beam perturbations when, for example, coupling into optical cavities: the misalignment and mode-mismatch-induced power coupling losses scale linearly and quadratically with respect to the mode indices, respectively. As a result, the mode-mismatch tolerance for the ${\mathrm{H}\mathrm{G}}_{3,3}$mode is reduced to a factor of 0.28 relative to the currently used ${\mathrm{H}\mathrm{G}}_{0,0}$mode. This is a potential hurdle to using higher-order modes to reduce thermal noise in future gravitational-wave detectors.