# Refraction across two interfaces: Snell's laws for a light ray and the law of sines for a triangle

I got intrigued by a paradoxical thought when thinking of light rays that are refracted twice.

A light ray travelling across a medium impinges a transparent slab at a certain angle of incidence. This slab has a finite thickness and is made of a medium having other optical properties. The ray gets refracted into this second medium. The ray is further refracted into the outer medium after traversing that intermediate medium.

The three ray angles with respect to the normal of the interface are $$\theta_1$$, $$\theta_2$$, $$\theta_3$$. The speeds of light in the three media are $$c_1, c_2, c_3$$. The Snell's laws at the two interfaces result from the application of Fermat's principle:

$$\frac{\sin \theta_1}{c_1} = \frac{\sin \theta_2}{c_2}$$ and $$\frac{\sin \theta_2}{c_2} = \frac{\sin \theta_3}{c_3}$$.

Trivially this implies:

$$\frac{\sin \theta_1}{c_1} = \frac{\sin \theta_2}{c_2} = \frac{\sin \theta_3}{c_3} \tag{1}\label{1}$$

Typically it is observed that, when $$c_3=c_1$$, the last angle of refraction is the same as the first angle of incidence ($$\theta_3 =\theta_1$$) as if there was no slab. Only an offset occurs because of the slab in the middle.

So far, plain sailing.

Turning to trigonometry, the last equality has a striking similarity with the law of sines (Wikipedia). Given a triangle with sides $$a,b,c$$ and opposite angles $$\alpha, \beta, \gamma$$, it holds:

$$\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \tag{2}\label{2}$$

Trivially, if I draw a triangle with side lengths $$c_1, c_2, c_3$$, or a scaled measure of that, the angles $$\theta_1, \theta_2, \theta_3$$ are set.
This would imply that the optical properties of the media completely determine the angles of the ray path, no matter what. The first angle of incidence cannot be a free parameter either.

This conclusion looks like plain nonsense from a physical viewpoint.

Where is the fallacy/blunder in the reasoning? Which physical consideration breaks the parallel between the chained Snell's laws \eqref{1} and the law of sines \eqref{2}?

• Therefore, the fallacy would have consisted in hastening oneself into constructing a triangle. Three quantities $c_1, c_2, c_3$ can easily be seen as measures of three sides, but the sheer act of construction imposes silently a constraint $\theta_1 + \theta_2 + \theta_3 = \pi$, which is implied naturally in the law of sines. This constraint may be unwarranted elsewhere, as is for Snell's laws here. Likewise, the constraint $c_1^2 = c_2^2 + c_3^2 - 2 c_2c_3 \cos \theta_1$, the law of cosines, would fit the geometry of a triangle but would have no bearing on the physics of refraction. Jan 4 at 17:02