# resulting refractive index of multiple layers with different indexes

I am wondering whether my chain of thought is right because this seems very counterintuitive to me.

But I calculated that if I have multiple layers with different refractive indexes stacked onto each other, the only indexes that matter are the one of the layer where the beams are emerging from and the one where the beam is finally immersing into.

E.g. if my layers are:

Light source -> Epoxy -> Silicone -> Water

the resulting relation is

sin(a_water) = (n_epoxy / n_water) sin(a_epoxy)

Does it really not matter what material is in between? I searched the internet but I could not find an answer. As many times as I calculate this, the refractive index of the silicone layer cancel out.

Thank you very much.

• – Frobenius Jul 29 '18 at 18:37

Snell's Law $$n_1\sin\theta_1=n_2\sin\theta_2=n_3\sin\theta_3=...$$ applies continuously as the ray crosses multiple layers of refractive material, provided that the refractive index varies in only one direction, and that the angles of incidence $\theta_1,\theta_2$ etc are measured relative to that direction. The variation can be smooth and continuous, it need not be abrupt changes between distinct layers. It does not need to vary monotonically (constant increase or decrease), it can vary anyhow.