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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

$$E+E'=E''$$

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).

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The boundary conditions for electromagnetic field of constant frequency $\omega>0$ at an interface of two media are that the tangential components (the components in the plane of the interface) of electric and magnetic fields $\overrightarrow{E}$ and $\overrightarrow{H}$ are continuous. If these conditions are satisfied, the normal component of the Poynting vector (which vector equals $\overrightarrow{E}\times\overrightarrow{H}$ up to a constant factor) is continuous at the interface, so the energy flux is the same from both sides of the interface.

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  • $\begingroup$ @Benitok: I am not sure (I suspect some caveats may be necessary, for example, in the case of total internal reflection), as the incident and the reflected waves interfere, but the normal components of the Poynting vector (of total field) are continuous everywhere on the interface. $\endgroup$ – akhmeteli Dec 11 '16 at 17:17
  • $\begingroup$ This is my intuition of energy conservation - what goes into some area (the energy of the incident wave), must come out (the sum of the energies of the reflected and transmitted waves). So is it correct to say $|S|sin\theta=|S'|sin\theta+|S''|sin\phi$ at the interface ($\theta$ and $\phi$ are the angles of incidence and refraction respectively)? If so, is there a way to show this is consistent with the boundary conditions? I used Snell's law to cancel every $sin\theta$ and substituted each S with its respective EXH, but couldn't get a clear result. If not, what's wrong with my intuition? $\endgroup$ – Benitok Dec 11 '16 at 17:38
  • $\begingroup$ @Benitok: Sorry, I have nothing to add at the moment. $\endgroup$ – akhmeteli Dec 11 '16 at 17:44

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