One of the boundary conditions of an EM wave crossing a boundary (dielectric materials, wave is TE polarized), where part of the wave is reflected and part is refracted, is
where E is the amplitude of the oscillating electric field of the incident wave, E' is that of the reflected wave, and E'' is that of the refracted or transmitted wave.
My concern is, how does this comply with conservation of energy? Especially in the case of going from high index of refraction to low index of refraction, so E' is in the same direction as E (no phase shift), it seems that an incident wave with some E is resulting in two new waves with E' and E'', where E'>0, and E''>E.
Doesn't this violate conservation of energy?
(Note: I have seen the proof of this boundary condition from Faraday's law with the shrinking loop and it makes sense to me, but I haven't been able to reconcile it with this problem.)
(I also considered that the transmitted wave might be moving slower than the incident wave, which would mean it has a smaller energy flux despite the higher E, but this is not the case when going from high to low index of refraction).