  We present a new theoretical framework to analyze microwave amplifiers based
on the dc SQUID. Our analysis applies input-output theory generalized for
Josephson junction devices biased in the running state. Using this approach we
express the high frequency dynamics of the SQUID as a scattering between the
participating modes. This enables us to elucidate the inherently nonreciprocal
nature of gain as a function of bias current and input frequency. This method
can, in principle, accommodate an arbitrary number of Josephson harmonics
generated in the running state of the junction. We report detailed calculations
taking into account the first few harmonics that provide simple
semi-quantitative results showing a degradation of gain, directionality and
noise of the device as a function of increasing signal frequency. We also
discuss the fundamental limits on device performance and applications of this
formalism to real devices.
