WAVEFRONT CONTROL FOCUS of the light ' s own group velocity at any chosen speed, including superluminal velocities [ 2 ]. This flying focus decouples where and when a pulse is focused from how the beam physically propagates, thus addressing all three limitations at once. This article introduces the optical principles behind these techniques and their potential for compact laser-plasma accelerators.
LASER-PLASMA ACCELERATION AND DEPHASING In the nonlinear regime of laser-wakefield acceleration( LWFA), the laser is intense enough to expel all electrons from its path, leaving in its wake a spherical cavity composed only of ions( the“ bubble”), surrounded by a dense layer of electrons( Fig. 1). An electron bunch trapped at the rear of this cavity is both accelerated longitudinally and focused transversely, with its total energy gain primarily determined by the distance over which it remains in the accelerating phase of the wakefield.
Since the wake is created by the laser, the accelerating structure propagates at the laser ' s group velocity vg in the plasma, which is slightly lower than the speed of light in vacuum c. The accelerated electrons, on the other hand, quickly reach a relativistic energy and therefore travel at a velocity very close to c. Thus, trapped electrons slowly advance through the bubble and, after a distance known as the dephasing length, they overtake the accelerating region of the plasma wave and enter a decelerating phase, at which point they reach their maximum energy.
This limit does not depend on how the laser is focused but follows directly from the fact that the accelerating structure, bound to the laser, propagates at v g < c. Therefore, overcoming dephasing requires controlling the velocity of the wakefield, which is set by the laser ' s own velocity in the plasma. Structured light makes this velocity directly and arbitrarily tunable, including superluminal values.
TUNABLE SUPERLUMINAL VELOCITIES WITH FLYING FOCUS While superluminal laser propagation might sound counterintuitive, spatiotemporal pulse shaping makes it achievable without violating special relativity, as no information or energy travels faster than c. The flying focus concept relies on focusing different part of the beam to distinct longitudinal positions along an extended focal line. By introducing a delay between these different parts, their relative arrival times at focus can be arbitrarily set to tune the effective propagation velocity of the peak laser field along the line.
The first implementations of flying focus were chromatic: a chirped pulse passes through a strongly chromatic lens that focuses each spectral component at a different longitudinal position [ 2 ]. Since the frequencies arrive in sequence, the focal point moves along the propagation axis over time, with its velocity tunable via the chirp. However, at any position along the focal line, only a narrow spectral slice is in focus. This fundamentally limits laser-plasma acceleration, which requires broad bandwidths to
Figure 1. Principle of LWFA and dephasing. The electron density is represented in grayscale.
Photoniques 140 I www. photoniques. com 57