Geometrical optics

Geometrical optics

As a mathematical study, geometrical optics emerges as a short-wavelength limit for solutions to hyperbolic partial differential equations. For a less mathematical introduction, please see optics. In this short wavelength limit, it is possible to approximate the solution locally by

u(t,x) approx a(t,x)e^{i(kcdot x - omega t)}

where k, omega satisfy a dispersion relation, and the amplitude a(t,x) varies slowly. More precisely, the leading order solution takes the form

a_0(t,x) e^{ivarphi(t,x)/varepsilon}.
The phase varphi(t,x)/varepsilon can be linearized to recover large wavenumber k:= nabla_x varphi, and frequency omega := -partial_t varphi. The amplitude a_0 satisfies a transport equation. The small parameter varepsilon enters the scene due to highly oscillatory initial conditions. Thus, when initial conditions oscillate much faster than the coefficients of the differential equation, solutions will be highly oscillatory, and transported along rays. Assuming coefficients in the differential equation are smooth, the rays will be too. In other words, refraction does not take place. The motivation for this technique comes from studying the typical scenario of light propagation where short wavelength light travels along rays that minimize (more or less) its travel time. Its full application requires tools from microlocal analysis.

A Simple Example

Starting with the wave equation for (t,x) in mathbb{R}timesmathbb{R}^n

L(partial_t, nabla_x) u := left(frac{partial^2}{partial t^2} - c(x)^2 Delta right)u(t,x) = 0, ;; u(0,x) = u_0(x),;; u_t(0,x) = 0

one looks for an asymptotic series solution of the form

u(t,x) sim a_varepsilon(t,x)e^{ivarphi(t,x)/varepsilon} = sum_{j=0}^infty i^j varepsilon^j a_j(t,x) e^{ivarphi(t,x)/varepsilon}.
One may check that
L(partial_t,nabla_x)(e^{ivarphi(t,x)/varepsilon}) a_varepsilon(t,x) = e^{ivarphi(t,x)/varepsilon}
left(left(frac{i}{varepsilon} right)^2 L(varphi_t, nabla_xvarphi)a_varepsilon + frac{2i}{varepsilon} V(partial_t,nabla_x)a_varepsilon + frac{i}{varepsilon} (a_varepsilon L(partial_t,nabla_x)varphi) + L(partial_t,nabla_x)a_varepsilon right) with
V(partial_t,nabla_x) := frac{partial varphi}{partial t} frac{partial}{partial t} - c^2(x)sum_j frac{partial varphi}{partial x_j} frac{partial}{partial x_j}

Plugging the series into this equation, and equating powers of varepsilon, we find that the most singular term O(varepsilon^{-2}) satisfies the eikonal equation (in this case called a dispersion relation),

0 = L(varphi_t,nabla_xvarphi) = (varphi_t)^2 - c(x)^2(nabla_x varphi)^2.
To order varepsilon^{-1} we find that the leading order amplitude must satisfy a transport equation
2V a_0 + (Lvarphi)a_0 = 0

With the definition k : = nabla_x varphi, omega := -varphi_t, the eikonal equation is precisely the dispersion relation one would get by plugging the plane wave solution e^{i(kcdot x - omega t)} into the wave equation. The value of this more complicated expansion is that plane waves cannot be solutions when the wavespeed c is non-constant. However, one can show that the amplitude a_0 and phase varphi are smooth, so that on a local scale we have plane waves.

To justify this technique, one must show that the remaining terms are small in some sense. This can be done using energy estimates, and an assumption of rapidly oscillating initial conditions. It also must be shown that the series converges in some sense.

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