Rays and the ray equation: behaviour in optical systems, valid solutions to Maxwell's equations

Consider a packet composed of a roughly uniform train of waves spread out over a region that is substantially longer and wider than their mean wavelength. Any particular point of space and time, ${bf x}$ and $t$, it has a definite phase $Theta({bf x},t)$. Once we know this phase, we can define the local frequency $omega$ and wave-vector ${bf k}$ by $$ omega= -left(frac{partialTheta}{partial t}right)_x,qquad k_i = left(frac{partialTheta}{partial x_i }right)_t. $$ These definitions are motivated by the idea that $$ Theta({bf x},t)sim {bf k}cdot {bf x} -omega t, $$ at least locally.

We wish to understand how ${bf k}$ changes as the wave propagates through a slowly varying medium. Assume that the dispersion equation $omega=omega ({bf k})$, which is initially derived for a uniform medium, can be extended to $omega =omega({bf k}, {bf x})$, where the $bf x$ dependence arises, for example, as a result of a position-dependent refractive index. This assumption is only an approximation, but it is a good approximation when the distance over which the medium changes is much larger than the distance between wavecrests.

Applying the equality of mixed partials to the definitions of ${bf k}$ and $omega$ gives us $$ left(frac{partial omega}{partial x_i }right)_t= -left(frac{partial k_i}{partial t }right)_{bf x},quad left(frac{partial k_i}{partial x_j }right)_{x_i}= left(frac{partial k_j}{partial x_i }right)_{x_j}. quad (star) $$ The subscripts indicate what is being left fixed when we differentiate. We must be careful about this, because we want to use the dispersion equation to express $omega$ as a function of ${bf k}$ and ${bf x}$, and the wave-vector ${bf k}$ will itself be a function of ${bf x}$ and $t$.

Taking this dependence into account, we write $$ left(frac{partial omega}{partial x_i }right)_t= left(frac{partial omega}{partial x_i }right)_{bf k} + left(frac{partial omega}{partial k_j }right)_{bf x}left(frac{partial k_j}{partial x_i }right)_t. $$ We now use ($star$) to rewrite this as $$ left(frac{partial k_i}{partial t }right)_{bf x}+ left(frac{partial omega}{partial k_j }right)_{bf x}left(frac{partial k_i}{partial x_j }right)_t = -left(frac{partial omega}{partial x_i }right)_{bf k}. $$ Interpreting the left hand side as a convective derivative $$ frac{dk_i}{dt}=left(frac{partial k_i}{partial t }right)_{bf x}+({bf V}_gcdotnabla) k_i, $$ we read off that
$$ frac {dk_i}{dt}= -left(frac{partial omega}{partial x_i }right)_{bf k} $$ provided we are moving at velocity $$ frac {dx_i}{dt}=({bf V}_g)_i= left(frac{partial omega}{partial k_i }right)_{bf x}. $$ Since this is the group velocity, the packet of waves is actually travelling at this speed. The last two equations therefore tell us how the orientation and wavelength of the wave train evolve if we ride along with the packet as it is refracted by the inhomogeneity.

The formulae $$ dot {bf k} = -frac{partial omega}{partial {bf x} }, dot {bf x}= phantom -frac{partial omega}{partial {bf k} }, $$ are Hamilton's ray equations and are the basis for geometric optics. These Hamilton equations are identical in form to Hamilton's equations for classical mechanics except that ${bf k}$ is playing the role of the canonical momentum, ${bf p}$, and $omega({bf k}, {bf x})$ replaces the Hamiltonian, $H({bf p}, {bf x})$.