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In a material with a Kerr nonlinearity such as glass it is possible to generate the third harmonic of an incident wave, however in general, due to material dispersion such a process is not phase-matched and so rarely occurs with any useful efficiency. The theory of third harmonic generation thus remained a theoretical curiousity which meant that the work of Armstrong et al. [1] who found an analytic solution to the problem (which was further extended by Puell and Vidal [2]) has been forgotten. Thus for example Grubsky and Savchenko who derived formally identical equations for third harmonic generation in optical nanowires found similar solutions to Armstrong et al using a different approach [3]. The later work was influenced by the fact that efficient third harmonic generation is possible in waveguides using modal dispersion to ensure phase matching between light at frequency co in the fundamental mode and light at frequency Sco in a higher order mode. Such phase matching is only possible in nanowires where the diameter of the waveguide is less than the wavelength of light and typically the refractive index contrast is large since the cladding is usually assumed to be air. It is worth noting that all of these authors although they solved the general problem only concentrated on the solutions for which the third harmonic was initially zero. However given the greater availability of high power

Analytic Theory of Two Wave Interactions in a waveguide with a χ^(3)nonlinearity.