Phasefronts as surfaces of constant vertex time

In the geometric optics approximation typical of far fields (vanishing wavelength), wavefronts emerge as constant eikonal surfaces. The electric (E) and magnetic (H) field vectors are mutually perpendicular and tangent to the wavefront. They follow planar elliptical trajectories that reach their elliptical vertices simultaneously, “in phase”. Near a dipole, scatterer, or edge, however, the vanishing wavelength assumption fails. Here, E and H behave differently. They still follow planar elliptical trajectories, albeit with different phases, ec-centricities, and planes. We define (i) “vertex time” as the position dependent time when E or H reach their respective ellipse vertex, and (ii) “phasefronts” as surfaces of constant electric or magnetic vertex time. The spatial gradients of the vertex time are perpendicular to the E and H phasefronts, defining separate E and H phase velocity vector fields. As we move into the far field, the E and H phasefronts converge to each other and to the wavefront, providing a quantitative measure of how much the local disturbance deviates from “far field”. Applying this concept to the field of a monochromatic point dipole, we find that, contrary to common assumptions, the dipole has no far field near its axis, no matter how far away.