Theory of two-photon resonant up-conversion with ultrashort pulses. I. Third-harmonic generation

The theory of third-harmonic generation under perfect two-photon resonant conditions is analysed for an ultrashort fundamental pulse. When the conversion efficiency is small, the fundamental pulse propagates under the two-photon self-induced transparency conditions and analytic solutions governing its evolution can be obtained. It is shown that for exact index-matching, analytic solutions can also be found for the third-harmonic signal at any propagation depth in the non-linear medium. In the general case, the energy conversion efficiency can be expressed in terms of elliptic integrals, although simpler formulae which apply in various limits are also presented. The solutions are compared with experimental results in the literature, as well as with other theoretical treatments.