Carrier mode selective working point and side band imbalance in LIGO I

Abstract

In gravitational wave interferometers, the input laser beam is phase modulated to generate radio-frequency side bands that are used to lock the cavities. The mechanism is the following: the frequency of the side bands and the carrier is chosen in such a way that their response to small changes of the longitudinal degrees of freedom is different. This difference is therefore monitored and it serves as an error signal for controlling the optical cavity lengths, as they are linearly related to the set of observed phases between carrier and side bands. Among the others, one longitudinal degree of freedom is optimally sensitive to the space-time distortions propagating through the cosmos, as predicted by the general theory of relativity. The observation of the astrophysical signal relies on the measurement of that specific degree of freedom. The entire problem is more complex when the transverse degrees of freedom are taken into account, because the relative phase between the fields also depends on their overlap. In order to establish an unambiguous relation between length changes and phase measurements, there must be one circulating optical mode and the only difference between carrier and side bands must be their amplitude. We will show that the variability of the transverse degrees of freedom and their different actions on carrier and side band fields puts a severe limit on this assumption. Unless the system is made of perfect and perfectly matched optical cavities, it is never governed by one unique coherent state and any adjustment of the optical lengths results from a compromise between the lengths that are optimal for the carrier field and the side band ones. Such a compromise alters the correspondence between error signals and cavity lengths, calculated in the one-dimensional treatment. We assess the strength of this effect and relate it to the sensitivity of the instrument (which relies on the reconstruction of that correspondence) in realistic circumstances.