Theory of third-harmonic generation using Bessel beams, and self-phase-matching

Taking Bessel beams (J(0) beam) as a representation of a conical beam, and a slowly varying envelope approximation (SVEA) we obtain the results for the theory of third-harmonic generation from an atomic medium, We demonstrate how the phenomenon of self-phase-matching is contained in the transverse-phase-matching integral of the theory. A method to calculate the transverse-phase-matching integral containing four Bessel functions is described which avoids the computer calculations of the Bessel functions. In order to consolidate the SVEA result an alternate method is used to obtain the exact result for the third-harmonic generation. The conditions are identified in which the exact result gees over to thr result of the SVEA. The theory for multiple Bessel trams is also discussed which has been shown to be the source of the wide width of the efficiency curve of the third-harmonic generation observed in experiments.