  Quantum noise in a model of singly resonant frequency doubling including
phase mismatch and driving in the harmonic mode is analyzed. The general
formulae about the fixed points and their stability as well as the squeezing
spectra calculated linearizing around such points are given. The use of a
nonlinear normalization allows to disentangle in the spectra the dynamic
response of the system from the contributions of the various noisy inputs. A
general ``reference'' model for one-mode systems is developed in which the
dynamic aspects of the problem are not contaminated by static contributions
from the noisy inputs. The physical insight gained permits the elaboration of
general criteria to optimize the noise suppression performance. With respect to
the squeezing in the fundamental mode the optimum working point is located near
the first turning point of the dispersive bistability induced by cascading of
the second order nonlinear response. The nonlinearities induced by conventional
crystals appear enough to reach it being the squeezing ultimately limited by
the escape efficiency of the cavity. In the case of the harmonic mode both,
finite phase mismatch and/or harmonic mode driving allow for an optimum dynamic
response of the system something not possible in the standard phase matched
Second Harmonic Generation. The squeezing is then limited by the losses in the
harmonic mode, allowing for very high degrees of squeezing because of the
non-resonant nature of the mode. This opens the possibility of very high
performances using artificial materials with resonantly enhanced
nonlinearities. It is also shown how it is possible to substantially increase
the noise reduction and at the same time to more than double the output power
for parameters corresponding to reported experiments.
