I'm having trouble understanding the reflectivity and transmissivity (the ratios of energy reflected and transmitted through a boundary separating different media). Since energy is proportional to the square of the field, if you have a superposition of two fields at a point, to obtain the energy density at that place you cannot just sum the energies of the separate fields. You have to sum the fields and then square the total field. So my problem is that to calculate the total energy that enters a patch of boundary, on the media of the incident wave, there are two fields, the incident and the reflected. Yet somehow everywhere I look they separate the energies, as the incident energy and the reflected energies, and then assume that the total energy entering the patch is the sum (incident energy - reflected energy) (with a minus sign because the reflected energy goes away from the patch). I don't understand this. Can someone help me here?
Because the waves are going in opposite directions, they don't interfere, so taking the difference of the energies is appropriate.
As far as I know, the reflected and incident waves do interfere. In fact, at normal incidence, with light polarized perpendicular to the plane of incidence, if the index of refraction of the second media is large enough, the reflected wave will almost annihilate the incident wave.
So back to my question, what is the logic of neglecting this interference in deriving reflectivity and transmissivity coefficients? Shouldn't one take into account that there are two fields in the first media, the incident and the reflected? I think one should use Poynting formula on the sum of the two waves, instead of calculating the Poynting vector of each field separately, which is what I see is done to calculate the reflectivity.