Josephson Parametric Amplifiers (JPA) are nonlinear devices that are used for quantum sensing and qubit readout in the microwave regime. While JPAs regularly operate in the quantum limit, their gain saturates for very small (few photons) input power. In a previous work, we showed that the saturation power of JPAs is not limited by pump depletion, but instead by the high-order nonlinearity of Josephson junctions, the nonlinear circuit elements that enable amplification in JPAs. Here, we present a systematic study of the nonlinearities in JPAs, we show which nonlinearities limit the saturation power, and present a strategy for optimizing the circuit parameters for achieving the best possible JPA. For concreteness, we focus on JPAs that are constructed around a Josephson Ring Modulator (JRM). We show that by tuning the external and shunt inductors, we should be able to take the best experimentally available JPAs and improve their saturation power by $sim 15$ dB. Finally, we argue that our methods and qualitative results are applicable to a broad range of cavity based JPAs.
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