Prove that in the case where there are more than two...

Prove that in the case where there are more than two optically different media sandwiched together, with air on the left and air on the right, the angle at which light returns to air is independent of the indices of refraction of the interior media (Eigure 23-53). In other words, the refraction angle, $\theta_{\mathrm{n}}$, depends only on $n_{1}$, $\theta_{1}$, and $n_{i}$ (not $n_{2}$, $\left.n_{3}, n_{4}, \ldots\right)$. Example 23-1

Key Concepts

Snell's Law

Snell's law relates the angles of incidence and refraction at an interface between two media with different refractive indices. It establishes that the product of the refractive index and the sine of the angle of incidence is constant across the interface. This fundamental law is applied repeatedly at each boundary in a multilayer system and forms the basis for analyzing how light propagates and refracts through complex structures.

Conservation of the Parallel Component of the Wave Vector

When light traverses different media, the component of its wave vector that is parallel to the interfaces remains unchanged. This conservation arises from the translational symmetry along the interfaces, ensuring that the phase matching condition is satisfied at each boundary. It is this invariance that leads to the overall cancellation of the interior media's indices of refraction in determining the final emerging angle in the configuration described.

Optical Path Invariance in Multilayer Systems

In a system comprising several layers with different refractive indices, the overall deflection of light depends only on the boundary conditions at the first and last media (air in this case) because the cumulative effect of the layered interiors cancels out. This invariance is a consequence of the consistent application of Snell’s law and the conservation of the parallel component, ensuring that the exit angle is determined solely by the external media and the initial angle of incidence.

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