Diagonalization of the length sensing matrix of a dual recycled laser interferometer gravitational wave antenna

Next generation gravitational wave antennas employ resonant sideband extraction (RSE) interferometers with Fabry-Perot cavities in the arms as an optical configuration. In order to realize stable, robust control of the detector system, it is a key issue to extract appropriate control signals for longitudinal degrees of freedom of the complex coupled-cavity system. In this paper, a novel length sensing and control scheme is proposed for the tuned RSE interferometer that is both simple and efficient. The sensing matrix can be well diagonalized, owing to a simple allocation of two rf modulations and to a macroscopic displacement of the cavity mirrors, which cause a detuning of the rf modulation sidebands.

Figure 1

The schematic of the RSE interferometer optical configuration. The power-recycling mirror (PRM) is placed between the beam splitter and the laser source to enhance the circulating laser power inside the interferometer, whereas the signal-extraction mirror (SEM) is placed between the beam splitter and the dark port to enhance the detector response, which makes the whole interferometer system more complex. The length degrees of freedom and the control signal-extraction ports are shown in addition to the optical arrangement.

Figure 2

Resonant conditions for optical fields, carrier, and modulation sidebands. Optical fields are color coded. The legend in the upper left indicates the order of the fields, entering the interferometer from the left (laser). The carrier field resonates inside both arm cavities and the power-recycling cavity, which gives enhanced response to gravitational wave signals. The phase modulation is designed to circulate inside the power-recycling and the signal-extraction cavity, whereas the amplitude modulation circulates only inside the power-recycling cavity. The central part of RSE, the dual-recycled Michelson interferometer, can be viewed as a coupled cavity, as is shown in the lower figure. The coupling connecting the two cavities is indicated by a plane mirror representing the Michelson interferometer.

Figure 3

The schematic of the idea “Delocation.” The same color coding and legend as in Fig. 2 are used. The goal is to make (only) one of the modulations slightly off-resonant inside the cavity. By giving the same amount of macroscopic displacements to PRM and SEM, only AM sidebands will be detuned inside the power-recycling cavity, without affecting the PM sidebands and of course the carrier. The translation to the coupled cavity is also shown.