Effects of Drift and Diffusion of Excited States on Spatial Hole Burning and Laser Oscillation

Abstract

The rate equation for the population inversion of homogeneously broadened 4‐level lasers is solved for the cases of small and large mobility of excited states. Considered are drift and diffusion, i.e., collective and random motions along the laser axis. Both drift and diffusion smooth spatial holes which are burned into the population inversion by standing‐wave fields. A drift results also in a drag effect, i.e., a phase lag between the population inversion and the standing‐wave field. For homogeneously broadened lasers, sufficient smoothing of spatial holes will result in spontaneous full‐power single‐frequency operation. This is examined by calculating the unsaturated gains of secondary modes as a function of the excitation, drift, and diffusion parameters. Spatial hole burning is insignificant for gas lasers because of their large carrier diffusion. Furthermore, ionic drift in gas‐ion lasers alone could effectively quench spatial hole burning. This is in contrast to dye lasers where drifts, i.e., longitudinal flow rates of the order of the sound velocity, would be required to obtain a single‐frequency output. In solid‐state lasers like ruby and Nd: YAG, energy diffusion between lattice sites is shown to significantly reduce spatial hole depths although the matrix elements for the exchange are quite small. An experiment is described which confirms the theory and allows the measurement of small diffusion constants of excited laser states. For the ⁴F_3/2 level in Nd: YAG a value of 5×10⁻⁷ cm²/sec is obtained. This energy exchange among Nd³⁺ sites contributes 0.3 GHz to the homogeneous linewidth. Implications for axial mode selection and single‐frequency oscillation of various lasers are discussed.