Cylindrical Geometry

The combination of volumetric heating of the laser material by the absorbed pump radiation and surface cooling required for heat extraction leads to a nonuniform temperature distribution in the rod. This results in a distortion of the laser beam due to a temperature- and stress-dependent variation of the index of refraction. The thermal effects which occur in the laser material are thermal lensing and thermal stress-induced birefringence.
An additional issue associated with thermal loading is stress fracture of the laser material. Stress fracture occurs when the stress induced by temperature gradients in the laser material exceeds the tensile strength of the material. The stressfracture limit is given in terms of the maximum power per unit length dissipated as heat in the laser medium.
The particular temperature profile which exists in the laser material depends, to a large degree, on the mode of operation, that is, cw- pumped, single-shot, or repetitively pulse-pumped. In the case of cw operation, a long cylindrical laser rod with uniform internal heat generation and constant surface temperature assumes a quadratic radial temperature dependence. This leads to a similar dependence in both the index of refraction and the thermal-strain distributions.
In a pulse-pumped system, laser action occurs only during the pump pulse or shortly thereafter in the case of Q-switching. Theoretical and experimental investigations have shown that heat transport during the pump pulse, which usually has a duration between 0.2 and 5ms, can be neglected. Therefore, in single-shot operation, optical distortions are the result of thermal gradients generated by nonuniform pump-light absorption.
In repetitively pulse-pumped systems, distortions will occur from the cumulative effects of nonuniform pump processes and thermal gradients due to cooling. Which effects dominate depend on the ratio of the pulse interval time to the thermal relaxation time constant of the rod. At repetition intervals, which are short compared to the thermal relaxation time of the laser rod, a quasi-thermal steady state will be reached where the distortions from pumping become secondary to the distortions produced by the removal of heat from the laser material.