  Weak cross-Kerr nonlinearities between single photons and coherent states are
the basis for many applications in quantum information processing. These
nonlinearities have so far mainly been discussed in terms of highly idealized
single-mode models. We develop a general theory of the interaction between
continuous-mode photonic pulses and apply it to the case of a single photon
interacting with a coherent state. We quantitatively study the validity of the
usual single-mode approximation using the concepts of fidelity and conditional
phase. We show that high fidelities, non-zero conditional phases and high
photon numbers are compatible, under conditions where the pulses fully pass
through each other and where unwanted transverse-mode effects are suppressed.
