  Here we report a theoretical model based on Green's functions and averaging
techniques that gives ana- lytical estimates to the signal to noise ratio (SNR)
near the first parametric instability zone in parametrically- driven
oscillators in the presence of added ac drive and added thermal noise. The
signal term is given by the response of the parametrically-driven oscillator to
the added ac drive, while the noise term has two dif- ferent measures: one is
dc and the other is ac. The dc measure of noise is given by a time-average of
the statistically-averaged fluctuations of the position of the parametric
oscillator due to thermal noise. The ac measure of noise is given by the
amplitude of the statistically-averaged fluctuations at the frequency of the
parametric pump. We observe a strong dependence of the SNR on the phase between
the external drive and the parametric pump, for some range of the phase there
is a high SNR, while for other values of phase the SNR remains flat or
decreases with increasing pump amplitude. Very good agreement between
analytical estimates and numerical results is achieved.
