Structured coupling beam control of six-wave mixing: from Helical Ince-Gaussian to Laguerre-Gaussian modes

Abstract We theoretically investigate six-wave mixing (SWM) in a coherently driven six-level atomic system using a structured coupling field. The probe and generated-field responses are obtained using the density-matrix formalism combined with the Maxwell Bloch equations. A Helical Ince–Gaussian (HIG) beam is introduced as the coupling field, and its ellipticity is varied to study the influence of its transverse structure on the nonlinear mixing process. In the zero ellipticity limit, the HIG modes reduce to the corresponding Laguerre–Gaussian (LG) modes. The effect of the LG modes is further examined for radial indices p = 0,1 and azimuthal indices l = 1,2. The normalized generated-field intensity increases with increasing l for both radial orders, whereas increasing p reduces the intensity due to the additional radial node and associated spatial redistribution of the coupling field. Among the modes considered, the LG mode (p,l) = (0,2) produces the largest response, with an approximately 1.06-fold enhancement over the Gaussian coupling case. These results demonstrate that the spatial structure of the coupling field provides an additional means of controlling higher-order nonlinear wave-mixing processes.