Question

I have been reading a book about electrodynamics and I have stumbled upon the following matter which is, to me , contradictory. When the electromagnetic wave changes medium, it is subject to certain boundary conditions, such as Snell's law. Another condition is that the phase factors are the same:

(k⃗ i⋅x⃗ )=(k⃗ r⋅x⃗ )=(k⃗ t⋅x⃗ )

where the subscripts indicate the incident, reflected and transmitted wave, respectively, and are understood to be at the border between the media. On the other hand, we have the Fresnel coefficients R and T, indicating the reflective and transmissive coefficients for a particular polarization, which are both functions of both indices of refraction n1 (first medium) and n₂ (second medium). They can be defined such that:

Et=Ei⋅T(n1,n2)

and

Er=Ei⋅R(n1,n2)

If at least one index of refraction is complex, though, the Fresnel coefficients become complex themselves, which indicates a phase jump between the incident field and the reflected/transmitted field, contradicting the condition laid out above that the phases have to be the same. So which is it, what am I missing here?

Answer 1

The phase factors in the electromagnetic wave are not always the same when changing medium due to the complex Fresnel coefficients. If at least one index of refraction is complex, the coefficients become complex themselves, indicating a phase jump between incident and reflected/transmitted waves.

Step-by-step explanation:

The phase factors in the electromagnetic wave are not always the same when it changes medium. While the phase factors need to be the same for the incident, reflected, and transmitted waves at the boundary between the media, the Fresnel coefficients can cause a phase jump between the incident field and the reflected/transmitted field. This is because the Fresnel coefficients depend on the indices of refraction of the two media, and if at least one index of refraction is complex, the coefficients become complex themselves. This indicates a phase jump between the incident and reflected/transmitted waves, contradicting the condition that the phases have to be the same.