The Eikonal and Ray-Tracing Equations

The short-wavelength approximation that turns wave propagation into the trajectories of rays through a plasma.

The Eikonal Ansatz

When the wavelength is much shorter than the scale over which the plasma varies, a wave can be written as an amplitude times exp(i S), where S is a rapidly varying phase, the eikonal. The local wavevector is k = grad S and the frequency is omega = -dS/dt. Substituting into the wave equation and keeping the leading order recovers the local dispersion relation D(omega, k, x, t) = 0 at each point.

Ray Equations

Treating the dispersion relation as a Hamiltonian D in the phase space of position and wavevector yields Hamilton-like ray equations: dx/dt = -(dD/dk)/(dD/domega) is the group velocity, and dk/dt = (dD/dx)/(dD/domega) describes how the wavevector refracts as the ray traverses inhomogeneous plasma. The rays trace the path of energy flow, bending toward resonances and reflecting at cutoffs.

Absorption Along the Ray

Ray tracing gives geometry; the power deposited is computed by carrying an imaginary part of the dispersion relation (from the warm dielectric tensor) along the ray, so the wave energy is attenuated where kinetic damping is strong, typically at cyclotron-resonance layers. This separates the fast phase evolution from the slow amplitude evolution and makes heating calculations tractable.

Relevance

Ray-tracing and its extension, beam tracing, are standard tools for designing electron-cyclotron heating and current drive, predicting where a launched beam deposits power and drives current. For the Hyperion breeder concept, ray tracing through the strongly varying field, from about 8 T on axis to 16.84 T peak, positions resonance layers for candidate heating schemes at the design stage; these are computations for a machine not yet built.