Generation of High Order Harmonics in Vacuum for Various Configurations of Interacting Electromagnetic Field

TL;DR

The paper tackles high-order harmonic generation in the QED vacuum using the Heisenberg–Euler Lagrangian, focusing on the perturbative regime where $\eta \ll 1$ and on two geometries: a theoretical $4\pi$-dipole in-coming field and two crossing Gaussian beams. It derives explicit, leading-order HOH intensities and angular distributions, expressed through matrix elements $H_n$ and $K_n$, and provides detailed results for both the dipole and crossing-beam configurations, including comparisons to plane-wave limits. The study reveals that odd harmonics $m=2n+1$ are produced with calculable rates, but practical observability is limited by Schwinger plasma screening, implying that only the lower harmonics (e.g., $m=3$ and $m=5$) are feasible in optical regimes; it also clarifies momentum-conservation cancellations that afflicted earlier approximations. The results deliver analytic guidance for experimental design in nonlinear QED and reconcile them with prior plane-wave analyses, while mapping the impact of focusing and beam geometry on HOH efficiency.

Abstract

High order harmonic (HOH) generation by interacting extremely intense electromagnetic waves in the quantum vacuum is investigated within the framework of the Heisenberg-Euler formalism. We consider here the process in the lowest order of a perturbation theory relative to the electromagnetic (EM) beam intensity, giving contribution to the HOH generation. The main expressions are obtained for a general geometry, whyle polarizations of different sub-beams forming the EM beam focus are almost the same. Nevertheless, explicit expressions for the HOH generation are derived for the $4π$-dipole in-coming waves and for the two crossing Gaussian beams. The former geometry of the EM beam is optimal at a given EM wave power, whereas the latter one is more realistic from the experimental point of view. We consider also a relationship of our present general results with the results, obtained earlier for the HOH generation during of collision of two plane electromagnetic waves.

Figures (7)

• Figure 1: Diagram representation of the definition (\ref{['eq:HELagr']}) for $\mathcal{L}_{HE}$ as a power series of $F_{\mu\nu}$ presented below in Eq. (\ref{['Qser']}).
• Figure 2: Dependencies of $\bar{H}_1$ (blue) and $\bar{K}_1$ (red) on the angle $\Theta$. They determine generation of the 3rd harmonics. The lines show real parts of these coefficients. For our normalization of the field $\tilde{\pmb{A}}^{in}$, the imaginary parts vanish.
• Figure 3: Dependencies of $\bar{H}_2$ (blue) and $\bar{K}_2$ (red) on the angle $\Theta$. They determine generation of the 5th harmonics. The lines show real parts of these coefficients. For our normalization of the field $\tilde{\pmb{A}}^{in}$ the imaginary parts vanish.
• Figure 4: Angular distribution of the HOH quanta. Shown is $|\bar{H}_n+\bar{K}_n/(2n+2)|^2$, normalized on $|\bar{H}_n+\bar{K}_n/(2n+2)|^2$, integrated over the whole solid angle, $4\pi$, as functions of the angle $\Theta$ for $n=1$, 2, 3, and 4. They are proportional to angular distribution of the harmonics of the number $m=2n+1=3,$ 5, 7, and 9. The dashed gray line shows angular distribution of the incident wave ($n=0$), normalized in the same way.
• Figure 5: Sketch of the two intersection Gaussian beams considered in Sec \ref{['TwoBeams']}. The coordinate system used for the definitions of the incident waves structure is shown in the upper right corner.