Geometric optics

Definition

Let $\Omega \subseteq \R^n$ be open and let $c \colon \Omega \to (0, \infty)$ be smooth. Consider the reduced wave equation $\Delta u + \frac{\omega^2}{c(x)^2} u = 0$ at a high frequency $\omega = 1/\varepsilon$. The geometric optics ansatz is a formal series $$u(x) = e^{i\varphi(x)/\varepsilon} \sum_{k=0}^{\infty} \varepsilon^k a_k(x),$$ with a real phase $\varphi$ and amplitudes $a_k$. Inserting it into the equation and setting the coefficient of each power of $\varepsilon$ to zero gives:

1. the eikonal equation $\abs{\nabla \varphi}^2 = c(x)^{-2}$, from the power $\varepsilon^{-2}$;
2. the transport equation $2\nabla\varphi \cdot \nabla a_0 + (\Delta\varphi)\, a_0 = 0$, from the power $\varepsilon^{-1}$;
3. the transport equations $2\nabla\varphi \cdot \nabla a_k + (\Delta\varphi)\, a_k = i\, \Delta a_{k-1}$ for $k \ge 1$.

Geometric optics is the approximation of $u$ by such a series, with $\varphi$ and the $a_k$ solving these equations. The rays are the integral curves of the gradient $\nabla\varphi$.