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Ray Optics

An approximation of wave optics that uses rays to model light propagation, simplifying calculations and providing a useful framework for many optical systems.
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The statement of the theorem

Let E(r,t)\mathbf{E}(\mathbf{r}, t) be the electric field satisfying the wave equation ∇2E−μϵ∂2E∂t2=0\nabla^2 \mathbf{E} - \mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 in a medium characterized by ϵ(r)\epsilon(\mathbf{r}) and μ(r)\mu(\mathbf{r}). Assume a monochromatic wave solution of the form E(r,t)=A(r)eikS(r)/ℏ\mathbf{E}(\mathbf{r}, t) = \mathbf{A}(\mathbf{r}) e^{i k S(\mathbf{r}) / \hbar}, where k=ω/ck = \omega/c and S(r)S(\mathbf{r}) is the eikonal function. The ray optics approximation is defined by the Eikonal equation, which mandates that the phase function S(r)S(\mathbf{r}) satisfies:\n(∇S(r))2=n2(r)c2ℏ2k2\left(\nabla S(\mathbf{r})\right)^2 = \frac{n^2(\mathbf{r})}{c^2} \hbar^2 k^2 \nwhere n(r)=ϵ(r)μ(r)/ϵ0μ0n(\mathbf{r}) = \sqrt{\epsilon(\mathbf{r})\mu(\mathbf{r})}/\sqrt{\epsilon_0 \mu_0} is the refractive index. The ray trajectory r(s)\mathbf{r}(s) parameterized by arc length ss is then governed by the Hamiltonian system derived from the phase function S(r)S(\mathbf{r}):\ndrds=∇S(r)\frac{d \mathbf{r}}{d s} = \nabla S(\mathbf{r})\ndds(12∣∇S(r)∣2)=Fext(r,s)\frac{d}{d s} \left( \frac{1}{2} \left| \nabla S(\mathbf{r}) \right|^2 \right) = \mathbf{F}_{ext}(\mathbf{r}, s) \nwhere Fext\mathbf{F}_{ext} represents any external forces or material inhomogeneities not accounted for by the background medium parameters.