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Most electrodynamics textbooks derive the behavior of light on the boundary of two mediums with the starting hypothesis that there are only three waves: incident, transmitted and reflected. Why only these three?

Griffiths certainly does not talk about it; I can't find the explanation in Zangwill and Jackson either. Quote from Zangwill, "Everyday experience tells us that an incident plane wave which approaches medium 2 from medium 1 'splits' into a reflected wave confined to medium 1 and a refracted wave confined to medium 2." (pg.588)

I wonder how to derive this result theoretically. If we employ a left-right symmetry argument, naively from the ray picture, there could be another transmitted ray going off to the left, or even maybe a ray that goes straight up and down. From the wave picture, we get continuous translation symmetry on one axis and discrete translation symmetry on the other two, but these symmetries do not seem to help.

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  • $\begingroup$ It might make more sense if your question asked, "Why is there only ONE transmitted wave and ONE reflected wave?" $\endgroup$ – sammy gerbil Feb 17 at 18:04
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    $\begingroup$ There is only one transmitted and one reflected wave (assuming linear media) because there can be only one of each that satisfies conservation of momentum, also known as phase matching in this context. $\endgroup$ – garyp Feb 17 at 18:10
  • $\begingroup$ There is only one refracted wave path with least time. Also, only one wave with angle of reflection equal to angle of incidence. $\endgroup$ – Sam Feb 17 at 18:36
  • $\begingroup$ Many of the results such as "angle of reflection = angle of incidence", or "phase matching"(if I understand your comment correctly) are originally derived by assuming that there are only these 3 waves, at least in Griffiths. It wouldn't make much sense to use these results coming back and prove there can only be 3 such waves. $\endgroup$ – XYSquared Feb 18 at 1:11
  • $\begingroup$ I am not sure if momentum conservation or the principle of least time/action can help here. It's not clear which picture you are talking about (wave/ray/particle picture). Could either of you elaborate? $\endgroup$ – XYSquared Feb 18 at 1:13
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This isn't justified a priori. Instead, this is a useful heuristic that we use to build a solution of the Maxwell equations that we're interested in. Once that solution has been constructed, the method for finding it becomes irrelevant: it is a solution, and that is all we need to know.

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  • $\begingroup$ It seems to me that the OP is asking for a method (derivation) that leads to the solution. $\endgroup$ – garyp Feb 19 at 14:09
  • $\begingroup$ @garyp that is standard material found in any textbook on the subject, which OP had clearly already seen. I disagree with your reading of the question, but OP can clarify if I'm mistaken. $\endgroup$ – Emilio Pisanty Feb 19 at 14:39
  • $\begingroup$ I have worked through Griffiths so I know the derivations starting from this assumption. Some people mentioned that this was only a simple model comparing to the more complex like birefringent etc. Yet it seems a complete coincidence that the in this simplest model (linear and isotropic media with no birefringent properties), our everyday experience is correct. How can we be sure that there are only these three rays? I don't mind if people use advanced methods or look at microscopic pictures. I hope to know that our theory is powerful enough to justify our everyday observations. $\endgroup$ – XYSquared Feb 19 at 17:22
  • $\begingroup$ @XYSquared This is indeed a simplified model in that it ignores birefringence and other similar effects, but if you included all of those complications, your key problem would remain ─ when we address those situations, we also make a starting postulate of a finite number of plane waves, and that comes from our intuition about what the problem will look like. If that intuition is mistaken and we put in too many rays, some of the amplitudes will naturally resolve to zero; if we don't put in enough alternatives, the combination will not be a solution to the Maxwell equations. $\endgroup$ – Emilio Pisanty Feb 19 at 18:28
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    $\begingroup$ As explained in this answer, the justification for the three-rays postulate comes from the fact that it gives a solution. The existence-and-uniqueness theorem for the Maxwell equations tells us that there is one and only one solution with the characteristics we want (one incoming beam at the angle required, and no additional components transporting energy towards the boundary). Having found the solution (through intuition, fancy math, or anything in between) we know that it is the solution, and the details of how it was constructed become irrelevant. $\endgroup$ – Emilio Pisanty Feb 19 at 18:30

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