Basic understanding of Snell law through three mediums, angles of incidence

Question

I've light traveling through three mediums, so the snell law would be
$$n_1sin{theta_1}=n_2sin{theta_2}=n_3sin{theta_3}$$
Suppose that I work with the first and the last equation
$$n_1sin{theta_1}=n_3sin{theta_3}$$
The point I'm missing here is that if I only work with $n_1$ and $n_3$, let's say in my problem, that means that my light at the end doesn't depend on $n_2$ !? I mean my final equation or formula wouldn't have $n_2$ in it, that seems incorrect!
My goal is to compute $Delta x$ wrt the angles and $n_i$, so if I work with $n_1$ and $n_3$, $Delta x$ wouldn't have $n_2$ in it!
My question is what I do miss about that?

Not_Einstein · Accepted Answer

"Is it equivalent as if I didn't have the intermediate medium at all?"
As @Math_Whiz had pointed out, the final angle would be the same if the intermediate layer weren't there. But the distance the ray travels in that layer affects the displacement that you want to calculate. Imagine that the intermediate layer were very thin. The ray inside the layer would have the direction you show but only travel a very short horizontal distance. It would then exit the layer a short horizontal distance from where it entered. The $Delta$x would then be much less than shown in your diagram.
By the way, the first equation you show should have $theta_2$ in the middle quantity.

Math_Whiz · Answer

Yes, the angle wouldn't depend upon the intermediate mediums through which the light ray passes through. But if you are talking about path differences, it would depend upon all mediums.
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Basic understanding of Snell law through three mediums, angles of incidence

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