  Superconducting parametric amplifiers have great promise for quantum-limited
readout of superconducting qubits and detectors. Until recently, most
superconducting parametric amplifiers had been based on resonant structures,
limiting their bandwidth and dynamic range. Broadband traveling-wave parametric
amplifiers based both on the nonlinear kinetic inductance of superconducting
thin films and on Josephson junctions are in development. By modifying the
dispersion property of the amplifier circuit, referred to as dispersion
engineering, the gain can be greatly enhanced and the size can be reduced. We
present two theoretical frameworks for analyzing and understanding such
parametric amplifiers: (1) generalized coupled-mode equations and (2) a finite
difference time domain (FDTD) model combined with a small signal analysis. We
show how these analytical and numerical tools may be used to understand device
performance.
