The dynamics of non-linear optical absorption

On traversing materials with absorbing dyes, weak optical beams develop a Beer (exponential) profile, while intense beams develop a spatially initially linear and then finally an exponential profile. This anomalous, deep penetration due to photo-bleaching of surface layers is important for heavy dye-loading and intense beams, for instance in photo-actuation. We address the problem of the evolution in time from initial Beer's Law to deeply penetrating optical profiles in dyes. Our solution of the coupled, non-linear, partial differential equations governing the spatio-temporal decay of the Poynting flux and the non-linear dynamics of the \textit{trans-cis} conversion is applicable to general systems of photo-active molecules under intense irradiation, for instance in biology, in spectroscopy and in opto-mechanical devices.