The problem area is geometric optics, namely refraction across homogeneous media with constant speed of light.
I explain the three steps of a methodological doubt that popped up.
Polite request: please use answers for answering and comments for commenting or asking.
Step 1: Where Snell's law comes from
I am aware that Snell's law is derived from Fermat's principle of least time.
Snell's law is derived upon applying Fermat's principle to an idealized ray between two points, say $A$ and $B$, at either side of a single refractive surface, $\sigma$, a plane for simplicity.
Once you determine the traversal time $T$ and minimize it with respect to the position of the point of incidence $R$ on the refractive surface, say with coordinate $r$, the statement $\delta T(r)=0$ tells us that the quantity $\sin \theta/c$ has to be the same across the refractive interface $\sigma$.
There is no need to develop the algebra further and determine the position of $R$, neither do the positions of $A$ and $B$ need to be specified. Snell's law is in fact a necessary and sufficient condition for a point of incidence $R$ to be found, so it applies to any ray and any point on the refractive interface $\sigma$. (Leaving aside the rays resulting into reflection, for simplicity.)
Broadly speaking, if I phrase it right, Snell's law is a corollary of Fermat's principle.
So far, plain sailing.
Step 2: A similar refraction problem with a single interface
Let's turn to drawing a real ray between an illuminated point $A$ and an eye point $B$ at either side of a single refractive surface $\sigma$. The positions of $A$ and $B$ are now meaningful for this problem. The objective is to determine the position of the point of incidence $R$.
Fermat's principle can be used for this problem of course. You need to move on from where Snell's reasoning stopped. The minimization problem $\delta T(r)=0$ is one algebraic equation in one unknown. You solve it and find the coordinate of $R$ on the refracting interface that is physically meaningful. Of course, Snell's law is implicit in this result.
This too is plain sailing.
Step 3: Refraction across two interfaces
Now, suppose that between $A$ and $B$ (known) there are two refractive interfaces, $\sigma_1$ and $\sigma_2$ (planes, for simplicity). The ray between $A$ and $B$ crosses three media with a different speed of light. I want to determine the ray path between $A$ and $B$ as before: the challenge is to determine the two points of incidence of the ray on the interfaces, $R_1$ and $R_2$.
Fermat's principle applies here too. The traversal time $T$ is the sum of the times to traverse the three media across the segments $AR_1$, $R_1 R_2$, $R_2 B$. The traversal time has to be minimized. Here, this statement gives one equation with two unknowns, $\delta T(r_1, r_2) = 0$. I then looked for one more geometric condition to link $r_1$ and $r_2$ and use the same algebra as the system with a single refractive interface.
Here, I thought of calling in Snell's law. Using the prior knowledge of the relationship between the angles of incidence and refraction at each interface, I could express the traversal time as $T(r_1)$ or $T(r_2)$, minimize it, and fix either $R_1$ or $R_2$ first.
This is where the methodological doubt kicked in.
- Snell's law is in itself a corollary of Fermat's principle, that is the result of a minimization problem, $\delta T = 0$.
- If I use Snell's law to anticipate the ray behaviour across two interfaces, I am basically feeding into the new $T$ the result of a prior $\delta T = 0$. The new $T$ has to be minimized yet, though. The new minimization problem should bring about, as a corollary, that the Snell's law applies on the same interfaces where I had imposed it.
Basically, I would be sneaking into the problem's givens a part of the solution, with a potential to corrupt the method that guarantees the correctness of the solution.
So, the main question is whether this concern is well posed in the first place.
- If so, which other non-Snell condition should be imposed to solve for $r_1$ and $r_2$?
- If not, what makes the trick or rather what debunks the concern?