Optimal design of resonant-mass gravitational wave antennas

A new generation of resonant-mass gravitational wave antennas, to be operated at ultralow temperatures, is under development by several research groups. This paper presents a theory for the optimal design of the new antennas. First, a general sensitivity limit is derived, which may be applied to any linear instrument for which the design figure of merit is the signal-to-noise ratio (SNR). By replacing the amplifier by its noise resistance and considering the energy dissipated in the noise resistance when a signal is applied, it is possible to show that the optimally filtered SNR is less than or equal to $E_r$/(${\mathrm{kT}}_n$), the energy dissipated in the noise resistance divided by Boltzmann’s constant times the amplifier noise temperature. This sensitivity limit will be achieved if the instrument is lossless, in which case the energy dissipated in the noise resistance is equal to the energy deposited in the system by the signal. For resonant-mass gravitational wave antennas, if the amplifier is identified as the mechanical amplifier (transducer and electronic amplifier together), then the lossless limit is accessible in practice. A useful point of view is that optimal antenna designs are those that are most loss tolerant—those that achieve the limiting SNR with the lowest possible mechanical Q values. The techniques of network synthesis may be used to design mechanical networks for matching the main antenna mass to the mechanical amplifier that are optimal in this sense. A class of loss-tolerant networks has been synthesized; their properties are summarized in a set of design charts that give the Q requirements and bandwidth as a function of the number of modes, the temperature, and the amplifier noise resistance and noise temperature.