Harmonic generation from laser-driven vacuum

We investigate the feasibility that, in the field of a superstrong standing laser wave, high-order harmonics of the pumping laser are generated from vacuum. Analytical calculations employing adiabatic perturbation theory show that, for laser electric fields larger than $E_{\mathrm{cr}}=m^2{c}^3/\hslash e=1.3\times {10}^{16}\mathrm{V}/\mathrm{cm}$, high-order harmonics are generated. The harmonic spectrum shows a wide plateau followed by a cutoff. The cutoff starts approximately at photon energies $\hslash {\omega }_M\sim \sqrt{\hslash ec{E}_L}$, with $E_L$ being the amplitude of the laser field. In the opposite limit $E_L\ll E_{\mathrm{cr}}$, the emission of high harmonics is very unlikely. In this case, a feasibility analysis for the experimental observation of the photon-photon scattering process using x-ray free electron lasers shows that the requirements are much less restrictive than those required to observe electron-positron pair creation.

Figure 1

Harmonic spectrum as given by Eq. (4.38) in $\mathrm{c}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ for ${\omega }_L=2.4\times {10}^{-6}m=1.2\mathrm{eV}$ and $E_L={10}^{-4}{E}_{\mathrm{cr}}=1.3\times {10}^{12}\mathrm{V}/\mathrm{cm}$.

Figure 2

Function $f_q({\beta }_L)$ [see Eq. (5.6)] with ${\beta }_L=0.025$ corresponding to ${\omega }_L=12.5\mathrm{KeV}$.

Figure 3

VHHG spectrum with ${\rho }_L=10$ corresponding to $E_L=1.3\times {10}^{17}\mathrm{V}/\mathrm{cm}$, ${\beta }_L=0.025$ corresponding to ${\omega }_L=12.5\mathrm{KeV}$ and $\vartheta =\pi /4$. We used Eq. (5.24) for the first $80$ harmonics, Eq. (5.22) for the harmonics $550$–$800$, and a $5$-order polynomial interpolating function in the remaining part (dotted part of the curve). The dotted vertical line corresponds to the cutoff formula (5.23) and the solid one to the corrected value $q_M\sim 553$ obtained from Eq. (5.25).

Figure 4

Feynman diagram of the photon plus electron-positron pair production in the presence of a constant and uniform electric field. The thick fermion lines indicate that the electron and positron states have to be calculated in the presence of the external field.

Figure 5

Feynman diagram corresponding to the amputated one-loop $n$-point function (a3) with $n=8$.

Figure 6

Feynman diagram representation of Eq. (a7). The thick fermion line in the last Feynman diagram indicates that the corresponding propagators are calculated in the presence of the external field $A_{\mu }(x)$ [see Eq. (a7)].