Mode-matching sensing through RF Higher-Order Modulation for Gravitational Wave detectors

The current generation of ground-based gravitational wave detectors, including LIGO, Virgo, and KAGRA, has achieved unprecedented sensitivity, detecting strains on the order of 10^-21. To further improve sensitivity and broaden the reach of gravitational wave astronomy, ongoing efforts focus on refining interferometric techniques, enhancing detector technologies, and developing innovative noise reduction strategies. In this context, it has become critical to develop new techniques for online percent-level sensing of mode-mismatch errors between the circulating laser beam and the many optical cavities constituting the interferometer; this thesis presents a proof-of-principle demonstration of a novel mode-matching sensing technique with reduced hardware requirements. One of the primary challenges in achieving higher sensitivity is addressing quantum noise, which arises from fundamental limits imposed by quantum mechanics. This noise is due to the fluctuations of the vacuum states entering the interferometer outport and is made up of two contributions: shot noise dominating at high frequencies and radiation pressure noise at low frequencies. To mitigate its impact and improve the sensitivity of the instruments, the current GW detectors involve replacing the vacuum state with squeezed vacuum states. These states obey the Heisenberg Uncertainty Principle, for which reducing the phase fluctuations inevitably leads to an increase in the amplitude fluctuations, and vice versa. However, during the latest scientific run O3, Virgo and LIGO successfully implemented squeezing to reduce shot noise as it was the dominant noise source at high frequency. In the current upgrades, GW interferometers are also limited at low frequencies by radiation pressure noise. This necessitates a Frequency-Dependent Squeezing technique, implemented using a 300m Filter Cavity (FC) slightly detuned with respect to the squeezed vacuum frequency. This detuning introduces a frequency-dependent phase shift in the squeezed vacuum state, which allows to obtain broadband reduction of quantum noise. However, adding the FC leads inevitably to losses which can degrade the squeezing level and, therefore, the overall quantum noise reduction. One of the primary factors contributing to the reduction in performance is the imperfection in mode-matching (mode-mismatch) between the fundamental mode of the optical cavity and that of the incident squeezed vacuum beam. This mismatch is characterized by differences in dimension and position of the waist, which can be represented by an additional spurious higher order mode (HOM), namely a Laguerre-Gaussian LG01 mode. When the cavity is locked on the fundamental mode, this HOM is reflected and carries important information on the origin of the mismatch. To address this issue in the current and future GW detector, like Advanced Virgo+, we introduce a novel method for sensing mode-mismatch through RF Higher Order Mode Modulation. This technique is based on a custom electro-optical lens (EOL) made up of a lithium-niobate crystal equipped with a suitable set of electrodes. The shape of the electrodes determines the ability of the object to act as a lens by exploiting the electro-optical effect. With the EOL, the laser beam is modulated generating sidebands on the LG01 mode. The modulation frequency is chosen to be twice the HOM spacing. In this way, when the cavity is locked, one of the sidebands resonates inside the cavity while the other is reflected together with the LG01 carrier. By measuring the beat signal between the carrier and sideband on a single-element photodiode, we can extract two error signals using I/Q demodulation at the sideband frequency. These signals are proportional to the real and imaginary parts of the LG01 mode which correspond to the waist size and waist position mismatch, respectively.