Fig. 1: Liénard–Wiechert potentials.
From: Ultrafast visualization of an electric field under the Lorentz transformation
a, The generation mechanism of the spherical electromagnetic wavefront is explained by the superposition of the multiple spherical electromagnetic potentials (the blue-green circles) generated from the moving electron beam (the yellow ellipsoid). Here, the electromagnetic potentials are emitted consecutively in all directions (solid angle of 4π) at the position through which the beam propagated; they overlap in the propagation direction of the beam (forward side). This mechanism requires the beam to travel near the speed of light in a vacuum. b,c, Schematics of the LWP in 3D (b) and 2D (c) space, respectively, in an inertial system. Both schematics in 3D and 2D show the evolution of the spherical electromagnetic potentials around a relativistic electron beam at times t = t1, t2 and t3 (t1 < t2 < t3) after the beam has passed through a metallic boundary (grey plate). The wavefront can be considered to be flat at t → ∞, as shown in c. Here, the beam propagates along the z direction at nearly the speed of light. In c, vectors E and B represent the electric field and magnetic field of the electromagnetic wave derived by the LWP, respectively, at t = t2. These vectors have axial symmetry around the z axis.
