The detectors use resonant transducers (and therefore there are two resonance modes coupled to the gravitational field) in order to obtain high coupling and high Q.
For the discussion on the detector sensitivity
and frequency bandwidth it is sufficient to
consider the simplest resonant antenna, a cylinder of high Q
material, strongly coupled to a non
resonant transducer followed by a very low noise electronic amplifier.
The equation for the end bar displacement is
We consider here only the noise which can be easily modeled, the
sum of two terms: the thermal (Brownian) noise
and the electronic noise.
The power spectrum due to the thermal noise is
By referring the noise to the displacement of the bar ends, we obtain
the power spectrum of the displacement due to the Brownian noise:
To this noise we must add the wide-band noise due to the electronic amplifier (the contribution to the narrow-band noise due to the amplifier has been already included in ).
For sake of simplicity we consider an electromechanical transducer that
converts the vibration of the detector in a voltage signal
The GW spectrum that can be detected with SNR=1 is:
-(or strain sensitivity) is the square root
We remark that the best spectral sensitivity, obtained at the
resonance frequency of the detector, only depends on the temperature
T, on the mass M and on the
quality factor Q of the detector, provided .
Note that this condition is rather different from that
required for optimum pulse sensitivity.
The bandwidth of the detector is found by imposing that
be equal to twice the value .
We obtain , in terms of the frequency
An example of the Nautilus strain sensitivity is given
in this picture