Studies in non-linear optics

Abstract

A general method, of approach to resonant non-linear optical phenomena involving travelling waves has been developed. Maxwell's equations are solved for the electric field in a spherical Fabry-Perot type optical resonator, which encloses uniaxial arisotropic media. The specific case of propagation perpendicular to the optic axis is considered but the theory can be extended to cover the general case including double refraction. In the presence of more than one optical field of this form, when the medium enclosed in the resonator is non-linear, by expanding the polarisation in terms of the electric field in the normal way coupled mode equations are obtained for amplitudes of the eigen modes concerned. This general formalism is then used to examine resonant second harmonic generation in the small conversion approximation from a fundamental beam in the lowest order 0-0 mode. Analytical solutions are obtained in three limiting cases, weak focussing, strong focussing and the focus removed infinitely from the non-linear medium. The general case is solved numerically. From the results the values of the variable parameters can he obtained which give the maximum output in any given mode. Graphs are presented giving the output variation in 0-0 and 0-2 modes with focus position, phase matching, focussing and spot size. From there it can be seen that the optimum focussing for the 0-0 mode occurs at 1/20 = 5.65 (1 crystal length, zo one half the confocal parameter). Secondly degenerate parametric amplification between two lowest order inodes is examined under the approximation that the pump beam is undepleted. An analytic solution is obtained for the single pass amplification in the weak focussing limit and preliminary results of numerical computations for the general case are given. From these results a value for the optimum threshold condition is calculated.