Strong-Field Saturation Effects in Laser Media

Using Lamb's model, we analyze the effects of gain saturation by strong traveling-wave fields in dilute laser media. Using approximate solutions of an integral equation for the population inversion density (PID), the index of refraction and incremental gain are studied for arbitrarily strong fields. Effects of atomic motion are included for a Maxwellian velocity distribution, but pressure effects are neglected. The case of a monochromatic field leads to the saturation results of Gordon, White, and Rigden, which are studied as a function of frequency. For a small ratio of natural to Doppler linewidths, there is a transition from inhomogeneous to homogeneous broadening for sufficiently strong fields. An effect of particular interest is the generation of waves at $2{\omega }_1-{\omega }_2$, $2{\omega }_2-{\omega }_1$, and higher order sidebands by two strong input signals at ${\omega }_1$ and ${\omega }_2$. The source of the parametric gain at these frequencies is the time-dependent gain saturation due to the presence of multiple strong fields. For ${\omega }_1-{\omega }_2$ small compared to the decay rates ${\gamma }_a$ and ${\gamma }_b$ of the laser levels, the gain at these intermodulation sidebands is computed as a function of the field strengths. The limiting cases of homogeneous and inhomogeneous (due to atomic motion) broadening are studied in detail. These two cases give essentially the same results, for a given unsaturated gain. Numerical results indicate that a first-order side-band intensity at least 10% as large as that of the inducing fields can be easily observed in practice. The integral equation for the PID is converted into an infinite set of linear algebraic equations for a typical solid-state laser. The conditions under which this set of equations can be limited to a finite number is discussed, and the dependence of the sideband gain on ${\omega }_1-{\omega }_2$ is calculated. For large ${\omega }_1-{\omega }_2$, the gain at the $m\mathrm{th}$ sideband decreases as ${({\omega }_1-{\omega }_2)}^{-m}$.