  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.
