Wave Front Sensing demodulated at the difference frequency between two phase-modulation sidebands in a compound interferometer configuration for a gravitational-wave detector

TL;DR

This work tackles the limitation of conventional WFS in large-scale gravitational-wave detectors where arm-axis fluctuations dominate alignment signals. It introduces Phase-Modulated-sideband × Phase-Modulated-sideband Wave Front Sensing (PMPMWFS), which demodulates the beat at the difference frequency $f_{ ext{a}}-f_{ ext{b}}$ between two anti-resonant PM sidebands, enabling decoupled sensing of PRC and incident-beam axes from arm-cavity axes. The authors derive the PMPMWFS theory using a composite optical resonator model with reflection matrices and Gouy-phase effects, and validate it with KAGRA's PRXARM measurements, showing orthogonal end-mirror signals and reduced arm-axis coupling. Alignment control experiments demonstrate stable locking for over an hour and a net gain in transmission, indicating PMPMWFS as a viable and scalable approach for decoupled alignment sensing in future detectors, including RSE configurations. Overall, PMPMWFS provides a robust mechanism to separate multiple alignment degrees of freedom and enhance the stability of next-generation gravitational-wave observatories.

Abstract

Precise alignment sensing and control are essential for maintaining the stability of laser interferometric gravitational-wave detectors. Conventional Wave Front Sensing technique (WFS), which relies on the beat between the carrier and phase-modulated (PM) sidebands, is dominated by arm-axis signals when the carrier resonates in the full interferometer. This dominance limits the detection of other optical axes, such as the Power Recycling Cavity (PRC) and incident beam axes. To address this problem, we propose a novel sensing technique, "Phase-Modulated-sideband $\times$ Phase-Modulated-sideband Wave Front Sensing" (PMPMWFS), which demodulates the beat signal at the difference frequency between two anti-resonant PM sidebands. We derived the theoretical response of PMPMWFS and experimentally demonstrated it using the Power-Recycled X-arm (PRXARM) configuration of KAGRA. The results show that PMPMWFS effectively decouples angular fluctuation signals of the PRC and incident beam from those of the arm cavity and provides orthogonal signal components for the end mirror of the arm cavity. Furthermore, feedback control using PMPMWFS achieved stable interferometer locking for over one hour. These results demonstrate that PMPMWFS offers an effective sensing method for decoupling multiple alignment degrees of freedom in future gravitational-wave detectors.

Figures (9)

• Figure 1: RSE schematic diagram in KAGRA. The incident light is split by the BS into the X- and Y-directions. FPCs are installed in both arms of the MI, referred to as the X- and Y-arm cavity. The PRM forms the PRC with the two input mirrors of the FPCs (ITMX and ITMY). Similarly, the Signal Recycling Mirror (SRM) forms the Signal Recycling Cavity (SRC) with the ITMs. The gravitational wave signal is detected at the SRC's output port as a change in the optical path difference between the two arm cavities. Three PM sidebands are generated by the EOMs. The carrier beam resonates in both the PRC and the arm cavities. The PM sideband fields at $f_\mathrm{1}$, $f_\mathrm{2}$, and $f_\mathrm{3}$ exhibit distinct resonance states: the sideband field at $f_\mathrm{1}$ resonates with both the PRC and SRC, the sideband field at $f_\mathrm{2}$ resonates only with the PRC, and the sideband field at $f_\mathrm{3}$ is anti-resonant with all resonators. The reflected light from the interferometer is redirected by the Input Faraday Isolator (IFI) and detected by the QPDs.
• Figure 2: Schematic of the PRXARM configuration in KAGRA. Red, blue, orange, and purple lines indicate the optical paths for the carrier, $f_\mathrm{2}$ sideband, $f_\mathrm{3}$ sideband, and difference frequency $f_\mathrm{3}-f_\mathrm{2}$, respectively. The WFS and PMPMWFS signals are obtained by demodulating the QPD signals on the REFL port at frequencies $f_\mathrm{2}$ and $f_\mathrm{3}-f_\mathrm{2}$, respectively, and then passing them through low-pass filters. The demodulated WFS signal is separated into the I-phase signal (at the reference demodulation phase) and the Q-phase signal (phase-shifted by 90 degrees from the reference). The X-arm cavity transmission power is detected by a photodiode at its transmission port.
• Figure 3: I-Q plane plot calculated for the PMPMWFS signal and conventional WFS signal in PRXARM. The radial axis is plotted on a logarithmic scale. Figure \ref{['FIG:cal_PRXARM_a']} shows the beat component between the $f_\mathrm{2}$ and $f_\mathrm{3}$ sidebands. Figure \ref{['FIG:cal_PRXARM_b']} shows the beat component between the carrier and the difference frequency $f_\mathrm{3}-f_\mathrm{2}$ sideband, and Fig. \ref{['FIG:cal_PRXARM_c']} shows the PMPMWFS signal obtained by combining these components, while Fig. \ref{['FIG:cal_PRXARM_d']} shows the calculated WFS signal. The argument represents the optimal demodulation phase in degrees at which the signal from each mirror is maximized. The magnitude represents the signal normalized by the tilt of each mirror. Note that the Gouy phase was calculated assuming 0 degrees. Furthermore, the demodulation phase for all degrees of freedom was rotated so that the ETMX angle signal in the I-phase component was minimized for both PMPMWFS and conventional WFS results. The rotated phase applied in Fig. \ref{['FIG:cal_PRXARM_c']} was also applied in Fig. \ref{['FIG:cal_PRXARM_a']} and \ref{['FIG:cal_PRXARM_b']}.
• Figure 4: Gouy phase dependence of the magnitude of I-phase and Q-phase signals when the Gouy phase $\eta_{\mathrm{pd}}$ is swept from -180 to 180 degrees. The horizontal axis represents the Gouy phase $\eta_{\mathrm{pd}}$ degrees, and the vertical axis represents the signal magnitude normalized by the mirror angle. The Figures \ref{['FIG:cal_PRXARM_gouy_a']} and \ref{['FIG:cal_PRXARM_gouy_b']} show the graphs of the PMPMWFS and WFS signals, respectively. The upper and lower figures show the graphs of I-phase and Q-phase signals, respectively. The demodulation phase is set such that the ETMX angle signal is minimized in the I-phase at the Gouy phase of 0 degrees.
• Figure 5: Measured results and theoretical calculation results for the PRXARM. Figure \ref{['FIG:measure_PRXARM_a']} shows the PMPMWFS signals obtained by tilting each mirror in the pitch angle. The signals in Fig. \ref{['FIG:measure_PRXARM_a']} were obtained by normalizing the measured values to the local tilt of each mirror measured by the optical lever, when each mirror was periodically tilted at 1.3 Hz. Figure \ref{['FIG:measure_PRXARM_c']} presents the corresponding theoretical results. The upper row of each plot corresponds to the Gouy phase of -55 degrees, while the lower row corresponds to the Gouy phase of 50 degrees. Each plot shows the signals with the demodulation phase rotated to minimize the ETMX angle signal in the I-phase component.