Small-Scale Self-Focusing Effects in Tapered Optical Beams

This memo presents a simple analysis of small-scale self focusing or filamentation effects in strongly diverging or converging optical beams (“tapered beams”) propagating in nonlinear or saturable optical media. In contrast to the collimated beam case where the nonlinear perturbations can be modelled as uniformly spaced periodic ripples, the nonlinear phase and amplitude ripples in the tapered-beam case can be modeled as nonlinear Newton’s rings or Fresnel zone plates. These perturbations, although they preserve their shape with distance, do not grow exponentially with distance as in the collimated beam case because they lack the Talbot interaction that is necessary for exponential growth. Instead they tend to grow either logarithmically or as a power of z with distance.