In the present paper, we will first present a boundary-type solution for total internal reflection at an interface between two nonabsorbing media, in which the instantaneous, time varying and time averaged radiant fluxes have been determined at all points in the two media. Solutions for the s and p polarizations were found for which the instantaneous tangential E and H components and the normal components of the radiant flux were continuous in crossing the interface. From these radiant fluxes, it was possible to derive equations for the flow lines, to determine the instantaneous radiant fluxes along these flow lines, and to see how the methods of propagation differed in the two media and for the two polarizations. These ideas have also been extended to long wavelength reflection at ocean-air interfaces, and similar phenomena appear there.
A. I. Mahan, A. I. Mahan,
"Deeper Look At Total Internal Reflection And Some Of Its Implications", Proc. SPIE 0190, Los Alamos Conference on Optics 1979, (13 November 1980); doi: 10.1117/12.957801; https://doi.org/10.1117/12.957801