The propagation of a laser pulse through a gas amplifier is an important practical problem. In particular, in discharge-pumped gas amplifiers it is known that the amplification coefficient of the lasing medium is a function of the longitudinal and transversal coordinates inside the cavity. This is a consequence of practical aspects such as the position an shape of the electrodes inside the cavity . Also, for long enough laser pulses, diffusion of the active centers take place during the amplification of the pulse.
G. E. Torres-Cisneros,
"Pulse Propagation In A Gaseous Laser Amplifier", Proc. SPIE 0813, Optics and the Information Age, (1 January 1987); doi: 10.1117/12.967230; https://doi.org/10.1117/12.967230