7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Observing the birth of relativistic Coulomb fields
M. Ota1, K. Kan2, T. Matsui3, and M. Nakajima4
1 National Institute for Fusion Science (NIFS),
2
National Institutes for Quantum Science and Technology (QST),
3 Department of Electrical and Electronic Engineering, Mie University,
4 Institute of Laser Engineering (ILE), Osaka University
e-mail (speaker): ota.masato@nifs.ac.jp
Relativity changed the concept of time and space. The
theory showed that scales of time and space are notabsolute but relative: time and space can expand andcontract. Special relativity describes the difference
between physics in a static system and in an inertialsystem, especially when the relative speed between thesystems is as high as the speed of light. A. Einsteinpresented a paper about special relativity in 1905 titled“On the electrodynamics of moving bodies” [1].
Relativistic phenomena predicted in the paper such as
time dilation and rest mass energy were demonstrated
and applied to GPS and nuclear power plants,
respectively. However, the contraction of the Coulomb
field, which is the phenomenon referred to by the title ofthe paper, has not yet been demonstrated. This is because
ultrafast electric-field measurement with a temporal
resolution of sub-picosecond was necessary. We haveconducted electro-optic sampling[2] on the Coulomb fieldaround a high-energy electron beam with the hightemporal resolution; we have succeeded in visualizing
the electric-filed contraction and observing the processof the relativistic Coulomb-field generation[3].
Figure 1 shows the simplified schematic of theexperimental setup, which is conducted by using aphotocathode linac at SANKEN, Osaka University.
Ti:sapphire laser with a pulse width of ≤130 fs (FWHM)
and a central wave length of 800 nm is divided into two:
one is for the generation of the electron beam and theother is for the probe laser. The electron beam generated
by the linac propagates in the air after passing throughthe acceleration tube. The Coulomb field around thebeam contracts in the propagation direction due to therelativistic effect. Therefore, the Coulomb field can beregarded as a half-cycle terahertz pulse. When theelectron beam passed by the ZnTe crystal, which has theFig.1 Schematic of the experimental setup.

Pockels effect, birefringence was induced inside thecrystal depending on the strength and the temporal
evolution of the Coulomb field. Measuring themodulation of the polarization of the probe laser enables
us to evaluate the information of the Coulomb field.
In the experiment, we put an aluminum foil in frontof the ZnTe crystal with a distance D and made theCoulomb field around the electron beam canceled. In thissituation, after the passage of the foil, the Coulomb field
propagates in the air with a spherical shape whose radiuscorresponds to the propagation distance of the beam (D),as predicted by the Liénard-Wiechert potentials (LWP).
Figures 2 (a-c) show the experimental results. Here, thespherical Coulomb field asymptotically coincides withthe relativistically contracted planar electric field with
the increase of the propagation distance. The
experimental results are quantitatively verified by thenumerical simulation, as shown in Figs. 2 (e-f). This
result corresponds to verifying the theory that theLorentz transformation of the electromagnetic potentials
corresponds to the solution of the LWP in the far fieldfrom a boundary. Therefore, we have succeeded indemonstrating the relativistic Coulomb fields. In addition,
our result is one of the most direct and intuitive
experimental results of relativity.
References[1] A. Einstein: Ann. Phys. 17, 891 (1905).
[2] Wu, Q. & Zhang, X. C. Appl. Phys. Lett. 67, 3523–3525 (1995).
[3] M. Ota et al., Nat. Phys. 18, 1436, (2022).

Fig.2
(a-c)
Experimentalresults.
(e-f)
Particle-in-cell simulation results. The D on thelower left in each figure denotes the distancebetween the Al foil and the ZnTe crystal.