Is $n\sin\theta$ conserved along light path?

TL;DR

The paper investigates whether the quantity $n heta$ remains conserved along light paths in media with continuously varying speed, concluding that it generally does not, contrary to the classical Snell's law for discrete media.

Contribution

It clarifies the applicability of Snell's law in continuous media and demonstrates the importance of calculus of variations in understanding light propagation.

Findings

$n heta$ is not conserved in general for continuous media

Provides pedagogical insights into Fermat's principle and calculus of variations

Highlights limitations of classical Snell's law in variable-speed media

Abstract

Snell's law states that the quantity $n\sin\theta$ is unchanged in refraction of light passing from one medium to another. We inquire whether this is true in the general case where the speed of light varies continuously within a medium. It turns out to be an instructive exercise in application of Snell's law and Fermat's principle. It also provides good pedagogical problems in calculus of variations to deal with the subtleties of a variable domain of integration and inclusion of constraints. The final result of these exercises is that, contrary to an initial expectation, the answer to the question in the title is negative.