Vacuum high-harmonic generation in the shock regime

Electrodynamics becomes nonlinear and permits the self-interaction of fields when the quantized nature of vacuum states is taken into account. The effect on a plane probe pulse propagating through a stronger constant crossed background is calculated using numerical simulation and by analytically solving the corresponding wave equation. The electromagnetic shock resulting from vacuum high-harmonic generation is investigated and a nonlinear shock parameter identified.

Figure 1

An illustration of the weak-field expansion of the vacuum polarization diagram.

Figure 2

In the left-hand diagram, ${\omega }_j\in \{{\omega }_s,3{\omega }_s,2{\omega }_p\pm {\omega }_s,2{\omega }_p\pm 3{\omega }_s\}$. If the strong field is approximated as constant and the three strong-field photon legs are suppressed, in an effective approach, six-photon scattering of the probe can be represented as a triple interaction. The $\pm$ refer to incoming and outgoing photons respectively.

Figure 3

An example of the harmonics generated in the probe due to effective self-interaction in a slowly varying background.

Figure 4

A plot of how the functions describing how the occurrence of higher harmonics varies with probe propagation length.

Figure 5

The simulational setup for the collision of a Gaussian probe pulse with a constant pulse of various strengths (indicated by different line styles) and identical polarization. The size of the system is taken to be $3.2\times {10}^{-2}\text{cm}$.

Figure 6

The relative intensity of the second harmonic generated by single six-photon scattering for ${\mathcal{E}}_p={10}^{-3}$.

Figure 7

High-harmonic generation from multiple four-photon scattering for ${\nu }_1=3.3\times {10}^{-4}$.

Figure 8

Harmonic spectra in the parallel setup for different regimes of solution: (a) ${\nu }_2=0.05$, (b) ${\nu }_2=0.6$, and (c) ${\nu }_2=1$. The dots show the leading-order perturbative term, the dashed line is the all-order analytical solution, and the solid line is from numerical simulation.

Figure 9

After passing through the polarized vacuum in the parallel setup, the probe pulse wave fronts can steepen significantly.

Figure 10

Comparison of the power-law exponent $\gamma \left(\nu \right)$ for different values of the shock parameter $\nu$, as calculated using the fourth and tenth harmonics from the analytical (dashed line) and numerical (points) solutions.

Figure 11

Harmonic spectra from numerical simulation of the perpendicular setup for different regimes of solution: (a) ${\nu }_2=0.05$, (b) ${\nu }_2=0.6$, and (c) ${\nu }_2=1$. The thick blue (thin green) peaks are harmonics parallel to the probe (strong) pulse.

Figure 12

A probe that is initially polarized perpendicular to the background (blue dashed line) experiences different shocks in the ${\varepsilon }^{\perp }$ (dot-dashed red line) and ${\varepsilon }^{\parallel }$ (solid green line) modes.

Figure 13

High-harmonic generation for the perpendicular setup when four- and six-photon scattering are present and four-photon scattering is much more prevalent than six-photon scattering $({\upsilon }_1=100,{\nu }_2=1)$. The first panel (a) is for just four-photon scattering, the second panel (b) for just six-photon scattering, and the third panel (c) for when both are present. The thick blue (thin green) peaks are again harmonics parallel to the probe (strong) pulse.

Figure 14

The weakly dispersive case for the perpendicular setup. The initially ${\varepsilon }^{\parallel }$ polarized probe (blue dashed line) experiences the mixture of the probe-independent vacuum refractive index (here ${\upsilon }_1=1$) and the shock-inducing probe-dependent vacuum refractive index (here ${\nu }_2=1$). The ${\varepsilon }^{\parallel }$ mode (dot-dashed red line) and ${\varepsilon }^{\perp }$ mode (solid green line) behave differently.

Figure 15

When the dispersion is increased $({\upsilon }_1=5,{\nu }_2=1)$, a different type of shock is formed in the ${\varepsilon }^{\parallel }$ mode (dot-dashed red line) and the shock in the ${\varepsilon }^{\perp }$ mode (solid green line) is reduced, where the initially ${\varepsilon }^{\parallel }$ polarized probe is plotted by the blue dashed line.

Figure 16

The probe pulse after having scattered in the strong background when ${\upsilon }_1=100,{\nu }_2=1$.