Dynamical symmetries in laser harmonic generation via a cubic nonlinearity

TL;DR

This work addresses the problem of deriving robust selection rules for high-harmonic generation in nonlinear media under complex, multi-mode laser configurations. It develops a symmetry-centered framework that unifies the beat-wave picture with the full set of spatiotemporal symmetries, and connects HHG to classical diffraction theorems via Noether's theorem and Floquet-Bloch theory through fundamental modes and synthetic dimensions. The authors derive explicit 1D/2D/3D harmonic-progression equations in terms of $K$-vectors, establish photon-counting rules for the participation of each mode, and demonstrate how DC modes and symmetry breaking generate higher-dimensional progressions, reconciling photon-counting and symmetry approaches across literature examples. The framework offers a standardized algorithmic path to predict and engineer harmonic spectra for structured targets and chiral media, effectively bridging HHG with established diffraction theory.

Abstract

In our earlier work on harmonic generation with complex light [Nature Communications 15, 6878 (2024)], we demonstrated how the harmonic spectrum of a complex laser beam in a nonlinear medium can be obtained through the judicious application of the ``beat wave'' concept. In this paper, we show how the same results can be obtained via the full set of symmetries of the initial laser-target configuration, and how this can be reconciled with the ``beat wave'' approach. We also highlight the connections between our work and existing theory for diffraction of EM waves from crystals: Laue equations, Mathieu equation, and theorems by Noether, Floquet and Bloch. The specific nature of our approach to harmonic spectra allows these connections to be revealed. We illustrate this with numerous examples taken from existing literature to show the wide applicability of our approach.

Figures (3)

• Figure 1: Hierarchy of harmonic progressions in 2-D $(\omega/\sigma, \ell/\sigma)$ space. (a) A single CP mode, $A^2$ obeys two independent continuous symmetries, given by $\epsilon X_1$ and $\epsilon X_2$, leading to two independent Noether-conserved quantities, indicated by two independent lines, given by $\Delta K \cdot X_1 = \Delta K \cdot X_2 = 0$. There is only one crossing, so no harmonics possible. (b) Two independent CP modes, one continuous symmetry of $A^2$ becomes discrete, "is broken", e.g. $\epsilon X_1$ becomes $X_1$. The "broken" symmetry leads to a collection of equally spaced parallel lines $\Delta K \cdot X_1 = 2\pi n_1$ (Laue), while the remaining continuous symmetry leads to a single line $\Delta K \cdot X_2 = 0$ (Noether). The line crossings lead to a 1-D harmonic progression. (c) Three independent CP modes, breaking the other continuous symmetry of $A^2$ as well. This leads to two sets of parallel lines $\Delta K \cdot X_1 = 2\pi n_1$ and $\Delta K \cdot X_2 = 2\pi n_2$; the crossings form a 2-D regular grid (Laue).
• Figure 2: A simple illustration of the connection between Noether's Theorem (vertical direction) and the Laue equations (horizontal direction) via light diffraction off a grating. See text for a full description.
• Figure 3: (a) The $(\omega/\sigma, \ell/\sigma)$ harmonic spectrum based on results by E. Pisanty et al.torusknot. Unlike in the $(\omega, \ell)$ spectrum, the harmonic peaks (open circles) in this spectrum are all on a single line, while this line and the spacing between peaks are fully determined by the two pump modes (filled circles). The equation for the line, $2(\omega/\sigma)/3 - \ell/\sigma = -1/3 = \mathrm{Const,}$ represents a Noether-conserved quantity, whose value is the same for all harmonics. The positions of the harmonics on the line is given by $(\omega/\sigma-1)/3 = n \in \mathbb{Z}$. (b) The $(\omega/\sigma, \ell/\sigma)$ harmonic spectrum based on results by G. Lerner et al.symmetry3. The three independent pump modes (filled circles) are given by $(\omega/\sigma, \ell/\sigma) = (2,1)$, $(-2,0)$ and $(-3,1)$, driving a 2-D harmonic grid. Harmonics (open circles) occur at the crossings of the lines $\ell/\sigma = m\in\mathbb{Z}$ and $(\omega/\sigma + \ell/\sigma +2)/5 = n\in\mathbb{Z}$.