  High fidelity microwave photon counting is an important tool for various
areas from background radiation analysis in astronomy to the implementation of
circuit QED architectures for the realization of a scalable quantum information
processor. In this work we describe a microwave photon counter coupled to a
semi-infinite transmission line. We employ input-output theory to examine a
continuously driven transmission line as well as traveling photon wave packets.
Using analytic and numerical methods, we calculate the conditions on the system
parameters necessary to optimize measurement and achieve high detection
efficiency.
