Interaction of photons traversing a slowly varying electromagnetic background

When two electromagnetic fields counterpropagate, they are modified due to mutual interaction via the polarized virtual electron-positron states of the vacuum. By studying how photon-photon scattering effects such as birefringence and four-wave mixing evolve as the fields pass through one another, we find a significant increase during overlap when both electromagnetic variants can be nonzero. The results have particular relevance for calculations based on a constant field background.

Figure 1

The Heisenberg-Euler Lagrangian contains effective vertices for classical electromagnetic fields interacting via the quantum effects described by the four-photon scattering box diagram and six-photon scattering hexagon diagram.

Figure 2

The smaller panels plot snapshots of the total electric field above the larger panels showing the corresponding state of the scattered overlap (red dashed) and asymptotic (blue solid) second harmonic field at times $t_4>t_3>t_2>t_1$. The electric fields are in units of the probe field amplitude, ${\mathcal{E}}_p$.

Figure 3

The second harmonic overlap field (dashed line) generated in four-photon scattering can dominate the asymptotic field (dot-dashed line) generated in six-photon scattering. Agreement is also shown between simulation (points) and theory (solid line) for ${\mathcal{E}}_s=0.02$, ${\mathcal{E}}_p=0.005$, ${\lambda }_p=2000\mathrm{nm}$, ${\tau }_s=6.4{\lambda }_p$ and ${\tau }_p=5{\lambda }_p$, where the field is in units of the probe amplitude, ${\mathcal{E}}_p$.

Figure 4

Both asymptotic (dot-dashed line) and overlap (dashed line) signals for frequency down-conversion originate from four-photon scattering. The leading-order asymptotic signal is due to the change in background due to interaction with the probe. Theory (solid line) and simulation (points) agree and show a backwards-propagating signal. The parameters are the same as in Fig. 3 and fields in units of the probe amplitude, ${\mathcal{E}}_p$.