Propagation of Light Pulses in a Laser Amplifier

The problem of a light pulse propagating in a nonlinear laser medium is investigated. The electromagnetic field is treated classically and the active medium consists of thermally moving atoms which have two electronic states with independent decay constants ${\gamma }_a$ and ${\gamma }_b$ in addition to the decay constant ${\gamma }_{\mathrm{ab}}$ describing the phase memory. The self-consistency requirement that the field sustained by the polarized medium be equal to the field inducing the polarization leads to coupled equations of motion for the density matrix, and equations of propagation for the electromagnetic field. Although the theory is developed for a Doppler-broadened gaseous medium, it may also be applied to a solid medium with inhomogeneous broadening. A unified treatment is given encompassing a wide range of pulse durations from cw signals to psec pulses. Continuous pumping is allowed, as well as any amount of detuning of the carrier frequency of the pulse from the atomic resonance frequency. The three independent decay constants ${\gamma }_a$, ${\gamma }_b$, and ${\gamma }_{\mathrm{ab}}$ provide greater flexibility than that obtained by using $\frac{1}{T_1}$ and $\frac{1}{T_2}$. The equations are solved analytically in a few specialized cases and numerically in the general case. Flow charts for accomplishing the numerical integration are given. Among the special problems considered is the apparent paradox of pulses propagating faster than the velocity of light under circumstances described by Basov et al. It is shown that this contradiction with relativity arises from the use of an unphysical initial condition.