Effects of spatial modes on third-harmonic generation

Abstract

The theory of third-harmonic generation is extended to the treatment of a generating electric field having a cylindrical spatial-mode structure. In the tight-focusing approximation it is proved that third-harmonic generation is only possible in negatively dispersive media ($\Delta k<\mathrm{}0\mathrm{}$). For a laser beam which consists of a pure $\mathrm{TE}{\mathrm{M}}_{\mathrm{pl}}$ mode, the generated field is a superposition of modes with a fixed angular mode number $3l$ and a radial mode number in the range from 0 to $3p$ with the exception of $3p-1$, which is always 0. The generated power has a functional dependence on $b\Delta k$ which is single peaked for $p=0\mathrm{}$ and multipeaked for $p\ne 0$. For a phase-matched situation, the conversion efficiency decreases as either $p$ or $l$ increases. However, for a system where the number density of the negatively dispersive medium is proportional to the mismatch wave vector $\Delta k$, the TE${\mathrm{M}}_{00}$ mode will not give the maximum conversion efficiency. It is noted that reliable determinations of refractive indices from phase-matching curves and third-order susceptibilities from absolute efficiencies of third-harmonic generation depend critically upon knowledge of the spatial structure of the generating field.