Photon Acceleration in a Flying Focus

A high-intensity laser pulse propagating through a medium triggers an ionization front that can accelerate and frequency upshift the photons of a second pulse. The maximum upshift is ultimately limited by the accelerated photons outpacing the ionization front or the ionizing pulse refracting from the plasma. Here, we apply the flying focus—a moving focal point resulting from a chirped laser pulse focused by a chromatic lens—to overcome these limitations. Theory and simulations demonstrate that the ionization front produced by a flying focus can frequency upshift an ultrashort optical pulse to the extreme ultraviolet over a centimeter of propagation. An analytic model of the upshift predicts that this scheme could be scaled to a novel tabletop source of spatially coherent x rays.

Figure 1

A schematic demonstrating photon acceleration in the ionization front of a flying focus. A negatively chirped drive pulse propagating at its group velocity $v_d<0$ forms a focus that counterpropagates at the velocity $v_f=c$, triggering an ionization front traveling at $c$. The resulting electron density is indicated by shading. A witness pulse (drawn in purple) copropagates with the ionization front at velocity $v_g$ and continually upshifts in frequency.

Figure 2

Profiles of the plasma guiding structure created when the flying focus drive pulse ionizes the shaped gas target. (a) The transverse profile as a function of moving frame coordinate $-\xi =z-ct$ at $z=0.7\mathrm{cm}$. (b) A line out corresponding to the dashed line in (a). (c) The guiding profile experienced by a typical photon within the witness pulse along the length of the accelerator (i.e., at each $z$, the transverse profile was extracted at the $\xi$ location of the photon).

Figure 3

A series of snapshots of an 87 fs witness pulse with an initial wavelength $\lambda =400\mathrm{nm}$ copropagating with a temporal gradient in the electron density. The snapshots are taken at propagation distances of (a) 0.1, (b) 0.3, (c) 0.8, and (d) 1.05 cm and plotted in the moving frame $-\xi =z-ct$. The pulse is modeled by photons initially spaced evenly in time over 87 fs. Each photon is represented by a circle colored to correspond to its vacuum wavelength (color bar). The electron density and scale length, $L=n_e/{\partial }_{\xi }{n}_e$, are shown as solid black and dashed gray lines, respectively.

Figure 4

(a) A plot of vacuum wavelength $\lambda$ over propagation distance $z$ for a pulse modeled by a series of photons evenly distributed over an input duration of 87 fs, each with a color corresponding to the initial temporal location within the pulse $\mathrm{\Delta }{t}_i$ (color bar). For comparison, the analytic case is plotted for the highest-frequency photon (dashed black line). (b) The distribution of wavelength over the final duration of the witness pulse $\mathrm{\Delta }{t}_f$ for the simulated case shown in (a).

Figure 5

A plot of minimum vacuum wavelength over propagation distance, displayed for four separate ionization front velocities. Drawn in a green dotted line is a reproduction of the simulated case in Fig. 4, with a flying focus focal velocity of $v_f=c$; in a red dashed line is a superluminal case in which $v_f=1.1c$; in a blue dash-dotted line is a subluminal case in which $v_f=0.9c$; in a black solid line is a case in which no flying focus is formed, and instead, the ionization front travels at the group velocity of the drive pulse $v_d=0.96c$.