In Fig. 33-71, two light rays pass from air through five...

In Fig. $33-71$, two light rays pass from air through five layers of transparent plastic and then back into air. The layers have parallel interfaces and unknown thicknesses; their indexes of refraction are $\overline{n_{1}}=1.7, n_{2}=1.6, n_{3}=1.5, n_{4}=1.4,$ and $n_{5}=1.6 .$ Ray $b$ is incident at angle $\theta_{b}=20^{\circ} .$ Relative to a normal at the last interface, at what angle do (a) ray $a$ and (b) ray $b$ emerge? (Hint: Solving the problem algebraically can save time.) If the air at the left and right sides in the figure were, instead, glass with index of refraction $1.5,$ at what angle would (c) ray $a$ and (d) ray $b$ emerge?

Key Concepts

Snell's Law

Snell's Law is the fundamental principle governing refraction, expressing the relationship between the sine of the angle of incidence and the sine of the angle of refraction at an interface between two media with different refractive indices. It is mathematically written as n? sin ?? = n? sin ??, which helps in calculating how a light ray bends when passing from one medium to another.

Refractive Index

The refractive index of a material quantifies how much the speed of light is reduced inside that material compared to a vacuum. It is a key parameter in studying light propagation since differences in refractive indices between media determine the degree of bending of light at interfaces according to Snell's Law.

Parallel Interfaces

When multiple layers with parallel interfaces are involved, while light rays may bend at each interface, the net angular deviation of a ray depends on the refractive indices of the layers. Specifically, if the initial and final media have the same refractive index, the emergent ray is parallel to the incident ray despite the intermediate refractions, making the analysis more straightforward.

Multi-Layer Optical Systems

A multi-layer optical system consists of several materials with different refractive indices arranged sequentially. In such systems, the light ray undergoes successive refractions at each interface, and the overall behavior can be predicted by applying Snell's Law repeatedly. This concept is important for understanding and designing optical devices where precise control of light paths is required.

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