The geometrical effect of an active element cross-section on the laser gain

An improved method for calculating the dependence of a laser emission gain on the tube cross-section’s geometry has been developed. In this connection the general solution of the Helmholtz equation was considered. But the solution in the form of an infinite series holds the potential for errors. In practice, a researcher has to replace the infinite series by a finite one. Some measures for solving the problems arising in this case were proposed. We have obtained an approximate solution of the Helmholtz equation that is convenient for practice, and a modified method for finding the coefficients of expansion has been developed. The method was tested for some cross-sectional forms that allowed independent theoretical calculation. As a result, the calculations accuracy was demonstrated to improve.