The Existence of Eigenvalues for the Integral Equations of Laser Theory

In this paper the general integral equations governing the mode spectra of optical masers are investigated from a point-of-view based upon certain theoretical results for Hölder continuous kernels. Using an estimation originally performed by Fredholm, it is proved that the homogeneous integral equation (x) = ₀^bK (x, y) (y) dy has at least one eigenvalue for Hölder continuous kemels K with exponent a 1 2 and with nonvanishing trace. All the integral equations which have been treated in laser theory s0 far can be “factored” into one-dimensional equations with continuously differentiable kernels, to which this result applies directly. Although in practice the vanishing of the trace is the exception rather than the rule, the later sections of this paper are devoted to demonstrations of the nonvanishing character of the trace nf several of the common “laser kernels” associaled with practical reflector configurtions. These results provide in almost all cases the first rigorous proofs of the existence of eigenvalues and eigenfunctions for the integral equations of the optical maser.