Animated Solution for Physics - Optics: A ray of light passes through four transparent media with refractive indices ${\mu }_1$, ${\mu }_2$, ${\mu }_3$ and ${\mu }_4$ as shown in the figure. The surfaces of all media are parallel. If the emergent ray $CD$ is parallel to the incident ray $AB$, we must have

A Journey Through Media

Imagine a light ray as a car driving through different terrains. In this problem, our light ray embarks on a journey through four distinct transparent media, each characterized by its own refractive index: ${\mu }_1$, ${\mu }_2$, ${\mu }_3$, and ${\mu }_4$. The boundaries separating these media are perfectly parallel to one another.

The ray enters the first medium along the path $AB$ and, after a series of refractions, emerges from the fourth medium along the path $CD$. The crucial piece of information provided is that the emergent ray $CD$ is perfectly parallel to the incident ray $AB$.

Generalized Snell's Law

To unlock this problem, we need to look beyond the standard application of Snell's Law at a single boundary. When dealing with a stack of parallel media, Snell's Law reveals a beautiful, generalized symmetry.

At any given interface, the product of the refractive index of the medium and the sine of the angle the ray makes with the normal remains constant. Mathematically, this is expressed as:

This means we can directly relate the conditions in the first medium to the conditions in the final medium, completely bypassing the complex mathematics of the intermediate layers. Let the angle of incidence in the first medium be $i$, and the angle of emergence in the fourth medium be $e$. Applying our generalized law, we get:

Now, let's bring in the critical constraint: the incident ray $AB$ is parallel to the emergent ray $CD$.

Because the interfaces are parallel, the normal lines drawn at any point on these interfaces are also parallel to each other. If two parallel rays intersect a set of parallel normal lines, basic geometry dictates that the angles they form must be identical. Therefore, the angle of incidence $i$ is exactly equal to the angle of emergence $e$:

Armed with this geometric truth, we return to our generalized Snell's Law equation and substitute $e$ with $i$:

Assuming the angle of incidence is not zero (the ray is not hitting the surface head-on), $\sin i$ is a non-zero value. We can safely divide both sides of the equation by $\sin i$, which elegantly simplifies to:

This result carries a profound physical insight: when light passes through a series of parallel transparent slabs, the intermediate layers (${\mu }_2$ and ${\mu }_3$) only serve to shift the ray laterally. They have absolutely no effect on the final angle of emergence. For the emergent ray to exit parallel to its original path, the refractive index of the final medium must be identical to that of the initial medium.