Spatiotemporal control of laser intensity using differentiable programming

TL;DR

This work addresses the challenge of designing structured laser pulses that optimally exploit spatiotemporal degrees of freedom for nonlinear optics and plasma physics. It introduces a differentiable UPPE framework (SUPER-JAX) that couples a near-field parameterization of amplitude and phase with far-field propagation, enabling gradient-based inverse design via automatic differentiation. Through three case studies—longitudinally uniform intensity, a moving, constant-shaped intensity peak, and a uniform plasma column—the approach demonstrates substantial performance gains, with losses $\mathcal{L}$ reduced by at least a factor of $15$ and even $93\%$ in the nonlinear case when full spatiotemporal control is used. The results show the practical potential of differentiable pulse design for extending interaction lengths, controlling ionization dynamics, and guiding experimental realizations using diffractive optics or metasurfaces.

Abstract

Optical techniques for spatiotemporal control can produce laser pulses with custom amplitude, phase, or polarization structure. In nonlinear optics and plasma physics, the use of structured pulses typically follows a forward design approach, in which the efficacy of a known structure is analyzed for a particular application. Inverse approaches, in contrast, enable the discovery of new structures with the potential for superior performance. Here, an implementation of the unidirectional pulse propagation equation that supports automatic differentiation is combined with gradient-based optimization to design structured pulses with features that are advantageous for a range of nonlinear optical and plasma-based applications: (1) a longitudinally uniform intensity over an extended region, (2) a superluminal intensity peak that travels many Rayleigh ranges with constant duration, spot size, and amplitude, and (3) a laser pulse that ionizes a gas to form a uniform column of plasma. In the final case, optimizing the full spatiotemporal structure improves the performance by a factor of 15 compared to optimizing only spatial or only temporal structure, highlighting the advantage of spatiotemporal control.

Figures (5)

• Figure 1: Diagram of the optimization loop in the differentiable program SUPER-JAX. The spatiospectral structure of the near-field electric field $\hat{E}e^{i\hat{\phi}}$ is iteratively tuned via gradients of a loss function $\mathcal{L}$ to find an optimized pulse that produces the desired far-field behavior. Far-field propagation is simulated with the unidirectional pulse propagation equation, and gradients of $\mathcal{L}$ are calculated with automatic differentiation for each tunable parameter $\hat{x}$.
• Figure 2: Optimization for case 1: producing an extended, uniform focal region with a monochromatic laser beam. (a) The simulated (solid blue) and target (dashed orange) on-axis intensity of the laser beam. For comparison, the on-axis intensity produced by an axiparabola with a 2-cm focal region is also shown (dashed--dotted green). (b) The near-field coefficient $\hat{A}$ as a function of radius, parameterized with 20 B-splines.
• Figure 3: Optimization for case 2: creating a uniform, superluminal ($1.005c$) intensity peak with constant duration (40 fs) and spot size (10 $\mu$m). (a) The optimized intensity profile as a function of $z$ and $t-z/c$, which exhibits a focal velocity of $1.005c$ (dashed blue) over 8 mm. (b) Radial profile at the temporal location of maximum intensity along $z$. The fitted spot size (dashed blue) varies by only 6% over the target region. (c) Near-field electric field spectral amplitude. The individual near-field functions are shown for (d) $\hat{A}$, (e) $\hat{\omega}$, and (f) $\hat{\tau}$.
• Figure 4: Optimization for case 3: producing a uniform-density plasma column. (a) Lineouts of the electron density at the final simulation time for the four innermost radial cells (solid blue), compared with the target density (dashed orange). (b) The on-axis far-field intensity profile, which propagates at a superluminal velocity of 1.14$c$. (c) Near-field electric field spectral amplitude of the optimized pulse, which exhibits a central frequency that increases quadratically with radius. The individual near-field functions for (d) $\hat{A}$, (e) $\hat{\omega}$, and (f) $\hat{\tau}$ are also shown, each implemented with 20 B-splines.
• Figure 5: On-axis density profiles as a function of $z$ and $t-z/c$ for the labeled cases, along with the ratio of final to initial loss values $\mathcal{L}/\mathcal{L}_0$. (a) A laser pulse focused to the nominal focal point with no optimization. Cases optimized with (b) only spatial and (c) only temporal structuring are also shown, for which there is no appreciable reduction in the loss value. (d) Full optimization (same simulation shown in Fig. \ref{['fig:nonlinear-2']}), for which the loss value is reduced by 93%.