A Flying Focus with Arbitrary Directionality

TL;DR

This work addresses the limitation that conventional flying-focus pulses move only along the propagation direction. It introduces a two-dimensional flying focus realized by a chirped pulse passing through a diffractive lens and a diffraction grating, yielding a wavelength-dependent focus z_f(λ) = $\frac{\lambda_0 f_i f_{L0}}{\lambda f_i + \lambda_0 f_{L0}}$ and transverse location y_f(λ), with a trajectory described by θ_F(λ) = $\arctan\left[\frac{dY_F}{d\lambda}\left(\frac{dZ_F}{d\lambda}\right)^{-1}\right]$ and a focal range L_F = $\int \sqrt{\left(\frac{dY_F}{d\lambda}\right)^2 + \left(\frac{dZ_F}{d\lambda}\right)^2}\,d\lambda$ that can be steered by Bragg angle and focal-length ratios. Paraxial frequency-domain simulations and 2D PIC plasma simulations validate the analytic framework across parameter spaces, demonstrating arbitrary-angle, long-range focal motion and high-power performance (v_Z ≈ $-1.16c$, v_Y ≈ $-0.20c$, θ_F ≈ 10°) with peak intensities near $6.4\times10^{16}$ W/cm^2. The results establish holographic plasma optics as a viable route for high-damage-threshold, multi-dimensional control of focal trajectories, enabling new applications in laser wakefield acceleration, THz steering, and surface-harmonic generation.

Abstract

Flying focus techniques produce laser pulses whose focal points travel at arbitrary, controllable velocities. While this flexibility can enhance a broad range of laser-based applications, existing techniques constrain the motion of the focal point to the propagation direction of the pulse. Here, we introduce a flying focus configuration that decouples the motion of the focus from the propagation direction. A chirped laser pulse focused and diffracted by a diffractive lens and grating creates a focal point that can move both along and transverse to the propagation direction. The focal length of the lens, grating period, and chirp can be tuned to control the direction and velocity of the focus. Simulations demonstrate this control for a holographic configuration suited to high-power pulses, in which two off-axis pump beams with different focal lengths encode the equivalent phase of a chromatic lens and grating in a gas or plasma. For low-power pulses, conventional solid-state or adaptive optics can be used instead. Multi-dimensional control over the focal trajectory enables new configurations for applications, including laser wakefield acceleration of ions, steering of broadband THz radiation, and surface harmonic generation.

Figures (4)

• Figure 1: Schematic of a flying focus with arbitrary directionality. (a) A chirped pulse propagates through a chromatic lens and a grating, creating a flying focus at an angle $\theta_\mathrm{F}$ with respect to the pulse propagation direction with a focal range $L_\mathrm{F}$. (b) A diffractive lens disperses the frequencies longitudinally, producing (c) a focus that travels along the same direction as the pulse. (d) A diffraction grating disperses the frequencies transversely, producing (e) a focus that travels perpendicular to the propagation direction of the pulse. (f) A diffractive lens and grating combination disperses the frequencies both longitudinally and transversely, producing (g) a focus that travels at an angle $\theta_\mathrm{F}$ with respect to the propagation direction.
• Figure 2: Design space for an arbitrary-directionality flying focus. (a) The angle at the central wavelength with respect to its propagation direction and (b) normalized focal range as a function of Bragg angle $\theta_\mathrm{B}$ and focal length ratio $f_\mathrm{L0}/f_i$. The contours show the analytical results [Eqs. \ref{['eq:flying focus direction']} and \ref{['eq:longitudinal displacement']}] and the markers indicate the parameters used in the simulations.
• Figure 3: Focal location relative to the central wavelength $\lambda_0 = 800\ \mathrm{nm}$ for normalized focal ranges $L_\mathrm{F}\lambda_0/f_i\Delta \lambda$ equal to (a) 0.2 and (b) 0.25. The theory (dashed lines) agrees with the simulations (circles and triangles). The circles and triangles correspond to the same markers in Fig. \ref{['fig:analytic calculation of flying focus direction and distance']}, which indicate the focal length ratios $f_\mathrm{L0}/f_i$ and Bragg angles $\theta_\mathrm{B}$ used in the simulations.
• Figure 4: PIC simulation of a two-dimensional flying focus produced by focusing and diffracting a chirped laser pulse with a combined plasma zone plate and grating. (a) The pulse intensity at three different times, corresponding to the moments at which light at different wavelengths reach focus, showing that the focal spot moves at an angle with respect to the propagation direction of the central wavelength. The dots show the focal location as a function of time. (b) The longitudinal and transverse displacement with respect to the focal location of the central wavelength as a function of time and wavelength. (c) The focused intensity as a function of time and wavelength.