Energy Density Distribution in a Polished Cylinder of Laser Material

A calculation of the flux distribution inside a smooth-walled laser cylinder is made for two limiting assumptions about the nature of the incident pumping flux. The idealized case of a line source at one focus of an elliptical reflector producing a convergent cylindrical wavefront at the other focus leads in the expected way to a singularity at the center of the crystal, and for sufficiently large values of the product of the doping concentration and diameter, the flux density goes through a minimum before increasing. This corresponds to the relatively rare physical situation in which the laser crystal is much larger than the pump source and both are very small compared to the reflector. A much more realistic assumption of isotropic flux distribution cannot be integrated in closed form and required a machine calculation. This reveals that the distribution is strongly dependent on doping, and that for light doping the flux density in the central region is approximately three-fold higher than in the peripheral regions. This is used as a hypothesis to explain low observed laser efficiencies by postulating that for typical operation the central region is undergoing laser action while the more voluminous peripheral region is absorbing pump energy and converting it to fluorescence. Some experimental evidence to support this hypothesis is obtained by measurements of the inversion, averaged over the whole crystal, by means of absorptivity measurements. It has been shown that laser action commences while the crystal is still, on the average, absorbing.