Space-time singularities in spatially-chirped Laguerre-Gaussian beams of any order

TL;DR

This paper addresses how spatial chirp modifies space-time singularities in ultrashort Laguerre-Gaussian beams by deriving a paraxial frequency-domain description and a single integral $\mathcal{I}_{l,p}$ that governs the time-domain field. It presents analytic solutions for low-order cases (notably $p=0$ and select $l,p$ combinations) and provides a numerical recipe to handle higher orders, revealing 3D space-time singularity curves with rich topology that depend on the radial index $p$, azimuthal index $l$, and chirp parameter $B$. The study also analyzes propagation effects, noting a transition from spatial chirp to pulse-front tilt and how singularities warp, merge, or detach in time as the beam propagates, with notable differences between purely radial, vortical, and mixed cases. Overall, the work demonstrates that simple spatial chirp can generate highly nontrivial space-time topologies, with potential implications for attosecond optics, particle manipulation, and ultrafast information transfer.

Abstract

The electric field distributions and space-time singularity curves are computed for ultrashort pulsed Laguerre-Gaussian laser beams having spatial chirp. Due to the breaking of cylindrical symmetry by the spatial chirp, the singularities trace complicated curves in space-time, which also vary for different combinations of radial and vortical orders. Analytical solutions are mostly presented along with a recipe for numerically calculating higher orders. The behavior of the singularities upon propagation is also shown, along with a discussion of the extension towards few-cycle pulses. These results are an example of how a simple physical scenario can result in highly complicated singular behavior in space-time.

Figures (8)

• Figure 1: Sketch of the physical scenario under consideration. An ultrashort Laguerre-Gauss beam (pictured here with $\{l,p\}=\{0,2\}$) is sent through a dispersive element (here a prism) to acquire pulse-front tilt (and angular dispersion). When focused, such a pulse-beam will have spatial chirp. We study in this work how the field behaves near such a focus, i.e. an ultrashort Laguerre-Gaussian beam with spatial chirp.
• Figure 2: Amplitude and phase profiles of the spatially-chirped LG beam with $l=0$ and $p=1$, at the focus ($z=0$) and with $B=1$. The amplitude is shown for cuts at $t=0$ (a) and $x=0$ (b), where the hyperbola shape is traced in a solid white line. Amplitude and phase are shown for different $y$ positions (c--e) denoted by the dashed white lines in (a), where the phase is shown on the right in each case. These cuts paint a picture of the 3D singularity curve, which is shown in (f). All amplitude profile plots are normalized to their own maximum, and all phase plots span $[\pi,\pi]$, which will be true for all figures.
• Figure 3: Amplitude and phase profiles of the spatially-chirped LG beam with $l=0$ and $p=2$, at the focus ($z=0$) and with $B=1$. The amplitude is shown for cuts at $t=0$ (a) and $x=0$ (b), where the two hyperbola-like curves are traced in solid white lines. Amplitude and phase are shown for different $y$ positions (c--e) denoted by the dashed white lines in (a), where the phase is shown on the right in each case.
• Figure 4: Amplitude profiles of the spatially-chirped LG beam with $l=0$ and $p=2$, at the focus ($z=0$). The three plane cuts (a--c) are shown for a number of $B$ values (i--v). Isosurfaces at $1/5000$ of the peak intensity are shown for $B=1$ (d) and $B=\sqrt{2}$ (e) tracing the singularities where they leave the cardinal planes.
• Figure 5: Amplitude profiles of the spatially-chirped LG beam with $l=1$ and $p=1$, at the focus ($z=0$). The three cuts (a--c) are shown for a number of $B$ values (i--iv).