Ultra-Short flying-focus

Abstract

Achromatic flying-focus enables programmable control of intensity peak velocity, with applications in ultrafast optics. However, spatiotemporal coupling inherently elongates ultrashort pulses by introducing frequency-dependent focusing and arrival-time dispersion. We present a theoretical model identifying this pulse-lengthening effect and propose a radially-dependent spectral chirp to compensate for chromatic timing mismatches. Numerical simulations confirm that this approach preserves both pulse duration and programmed flying-focus velocity over extended focal regions. Additionally, dispersive media such as plasmas can naturally mitigate elongation. These results extend achromatic flying-focus techniques to ultrashort pulses, enabling new opportunities in laser--plasma interactions and high-field nonlinear optics.

Figures (7)

• Figure 1: Mitigation of pulse broadening at the focus of an axiparabola for different wavelengths. Without any radial delay or chirp (top), the pulse propagates along the z-axis at a superluminal speed, deviating from the $z=ct$ trajectory. Adding a quartic radial delay (middle) compensates this temporal shift at the central frequency, chromatic effects lead to pulse stretching along the focal line. Adding both a radial delay and a radial chirp (bottom) aligns all spectral components along the $z=ct$ trajectory, preserving the pulse duration while maintaining the programmed flying-focus velocity.
• Figure 2: Simulated on-axis intensity in the frame of a particle moving at speed $c$. Without radial chirp (top), the imposed radial delay yields a luminal apparent propagation velocity but leads to progressive pulse stretching and reduced peak intensity. With radial chirp (bottom), the pulse duration is preserved along the entire focal line while maintaining a luminal apparent velocity.
• Figure 3: Simulated on-axis intensity in the frame of a particle moving at speed $c$. Quartic chirp alone (top) leads to alternating temporal stretching and partial compression along the focal line, with a compression point near mid-propagation. Quartic chirp combined with additional global and quadratic chirp (bottom) maintains temporal compression along the full focal line.
• Figure 4: Illustration for different wavelengths of a stepped transmissive doublet for generating radially-dependent spectral chirp.
• Figure 5: Simulated on-axis intensity in the frame of a particle moving at speed $c$, using a stepped doublet with radial delay $\tau_d(r)=\beta_0r^4/3$ and $\gamma=2$. The pulse duration is preserved along the entire focal line while maintaining a luminal apparent velocity.