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whenever I come to the derivation of the Fresnel equations I get stuck on the boundary condition for the component of the E-Field that is parallel to the surface.

I know for the parallel components Maxwell dictates that:

[itex]E_{1t}[/itex] = [itex]E_{2t}[/itex].

For the parallel incoming light field component [itex]E_{it}[/itex], the reflected component [itex]E_{rt}[/itex] and the refracted one [itex]E_{tt}[/itex] it holds that:

[itex]E_{it}[/itex] + [itex]E_{rt}[/itex] = [itex]E_{tt}[/itex].

I always think about time though. I have the sequence in my head: ray coming in and then we have the refracted and reflected beam. Does that not apply because we just assume, that everything is happening at once?

Would be very nice if something could shed some light on this. Thank you very much in advance :) and have a good one,

spookyfw