Comparative study of focusing with scalar and vector beams in an active Raman gain system

We investigate the focusing characteristics of scalar and vector beams within an atomic medium. An active-Raman-gain configuration is employed to achieve significant Kerr nonlinearity in a four-state atomic system. The probe beams can attain focusing within the medium through careful selection of input beam intensities and the spatial profile of the control field. We analytically derive the linear and third-order nonlinear susceptibilities for both scalar and vector probe beams. Our observations indicate that, in addition to the energy transfer from the control beam to the probe beam, the giant cross-Kerr nonlinearity facilitates the focusing of the scalar probe beam into a significantly smaller spot size. Conversely, the vector probe beams exhibit gain-induced narrowing. Furthermore, we evaluate the state of polarization for the vector beam at the minimum beam waist, observing a polarization rotation and a change in ellipticity during propagation. Through the mechanism of focusing, we achieve a reduced spot size for the probe beam, which may have substantial implications for resolution enhancement in microscopy applications.