Phasefronts as surfaces of constant vertex time

¹ Complex Photonic Systems (COPS), University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
² JMO GmbH, Zugspitzstr. 66, 82131 Gauting, Germany

^* e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it. ,This email address is being protected from spambots. You need JavaScript enabled to view it.

Published online: 22 September 2025

Abstract

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.