All-optical signatures of strong-field QED in the vacuum emission picture

We study all-optical signatures of the effective nonlinear couplings among electromagnetic fields in the quantum vacuum, using the collision of two focused high-intensity laser pulses as an example. The experimental signatures of quantum vacuum nonlinearities are encoded in signal photons, whose kinematic and polarization properties differ from the photons constituting the macroscopic laser fields. We implement an efficient numerical algorithm allowing for the theoretical investigation of such signatures in realistic field configurations accessible in experiment. This algorithm is based on a vacuum emission scheme and can readily be adapted to the collision of more laser beams or further involved field configurations. We solve the case of two colliding pulses in full $3+1$-dimensional spacetime and identify experimental geometries and parameter regimes with improved signal-to-noise ratios.

Figure 1

Diagrammatic representation of the single photon vacuum emission process (2). The double line denotes the dressed fermion propagator accounting for arbitrarily many couplings to the external field $\overline{A}$, represented by the wiggly lines ending at crosses.

Figure 2

Leading contribution to the single photon vacuum emission process in the limit of weak external fields.

Figure 3

Sketch of the transverse field amplitude profile of a generic Gaussian beam (left) and the special case of a Gaussian beam with infinite Rayleigh range, but finite beam waist (right) as a function of the longitudinal coordinate (measured along the propagation axis). Here, $\mathrm{\Theta }$ is the total angular spread, $w_0$ is the beam waist and ${\mathrm{z}}_R$ is the Rayleigh range over which the beam diameter increases by a factor of $\sqrt{2}$.

Figure 4

Top: Sketch of the mapping from a regular grid ($k_x$, $k_y$) to a polar grid with fixed radius ($\phi$). Light gray (dark blue) nodes represent the discretization in Cartesian coordinates (polar coordinates) in momentum space. As gray and blue nodes generally do not overlap, we apply cubic interpolation. Bottom: Sketch of the coordinate systems used. Spatial as well as momentum coordinates are originally given in Cartesian coordinate systems. In spherical coordinates the angles $\phi$ and $\vartheta$ give the longitude and latitude ($\vartheta \in [0,\pi ]$), respectively. In our numerical calculation only regular grids were used.

Figure 5

Sketch of the collision geometry considered in Sec. 5a. Two Gaussian laser pulses collide under an angle ${\vartheta }_2$ with respect to their beam axes; the offset between the beam foci is ${\overrightarrow{x}}_0=0$. Note that an angle of ${\vartheta }_2=0°(180)°$ corresponds to co(counter)propagating laser beams.

Figure 6

Total number of signal photons $N$ attainable per shot in the collision of two identical high-intensity laser pulses ($w_{0,1}=w_{0,2}=\lambda =800\mathrm{nm}$, $W=25\mathrm{J}$, $\tau =25\mathrm{fs}$) plotted as a function of the collision angle ${\vartheta }_2$. The dashed line shows the results for the advanced description of the colliding laser fields in terms of pulsed Gaussian beams, evaluated numerically with our algorithm. In addition, we present results for the toy-model benchmark scenario of keeping $w_{0,1}=w_{0,2}=\lambda$ finite but formally sending ${\mathrm{z}}_{R,b}\to \infty$. The latter scenario is analyzed in two different ways: by means of a fully numerical calculation with our algorithm (solid line), and by performing the Fourier transform from position to momentum space analytically, and numerically integrating over the outgoing signal photon momenta with Maple™ (cross symbols).

Figure 7

Directional emission characteristics of signal photons for two identical laser pulses colliding under an angle of ${\vartheta }_2=135°$. Top: Three-dimensional plot of the total number density $\rho (\phi ,\vartheta )$. For illustration, we also include a projection of the emission characteristics onto the xy-plane (gray). Bottom: Projection of the directional emission characteristics (top) onto the collision plane of the laser pulses (xz-plane). For comparison, the forward cones of the colliding Gaussian laser beams with $f^{\#}=1$ and delimited by the beams’ divergences ${\theta }_b=\frac{1}{\pi }$ representing the background are highlighted in gray.

Figure 8

Total number of signal photons polarized perpendicularly to the high-intensity laser beams $N_{\perp }$ plotted as a function of the collision angle ${\vartheta }_2$. Both laser pulses ($w_{0,1}=w_{0,2}=\lambda =800\mathrm{nm}$, $W=25\mathrm{J}$, $\tau =25\mathrm{fs}$) are polarized perpendicularly to the collision plane. The dashed (solid) curve shows the result obtained from a numerical calculation for pulsed Gaussian beams (the benchmark scenario with $w_{0,b}=\lambda$ finite, but ${\mathrm{z}}_{R,b}\to \infty$). The cross symbols display data for the benchmark scenario obtained by performing the Fourier transform analytically and evaluating the momentum integral numerically with Maple™.

Figure 9

Directional emission characteristics of perpendicularly polarized signal photons for two identical laser pulses colliding under an angle of ${\vartheta }_2=135°$. Top: Three-dimensional plot of the number density ${\rho }_{\perp }(\phi ,\vartheta )$. Bottom: Projection of the directional emission characteristics (top) onto the collision plane of the laser pulses (xz-plane). For comparison, the forward cones of the colliding Gaussian laser beams with $f^{\#}=1$ and delimited by the beams’ divergences ${\theta }_b=\frac{1}{\pi }$ are highlighted in gray.

Figure 10

Directional emission characteristics of perpendicularly polarized signal photons for two identical laser pulses colliding under an angle of ${\vartheta }_2=175.8°$. Top: Three-dimensional plot of the number density ${\rho }_{\perp }(\phi ,\vartheta )$. For better visibility of the directional emission characteristics, we adopt a perspective different from the other plots. Bottom: Projection of the directional emission characteristics (top) onto the collision plane of the laser pulses (xz-plane). The forward cones of the colliding Gaussian laser beams focused down to $f^{\#}=1$ and delimited by the beams’ divergences ${\theta }_b=\frac{1}{\pi }$ are highlighted in gray.

Figure 11

Impact of a relative shift between the laser foci on the integrated numbers of signal photons $N$ (blue solid line, left scale) and $N_{\perp }$ (orange dashed line, right scale) for two identical laser pulses colliding in a counterpropagation geometry, i.e., ${\vartheta }_2=180°$. Both laser pulses are polarized perpendicularly to the collision plane. Top: Transverse shift with ${\overrightarrow{x}}_0=({\mathrm{x}}_0,0,0)$ in units of the waist size $w_{0,1}=w_{0,2}=\lambda$. Bottom: Longitudinal shift along the common beam axis with ${\overrightarrow{x}}_0=(0,0,{\mathrm{z}}_0)$ in units of the Rayleigh range ${\mathrm{z}}_{R,1}={\mathrm{z}}_{R,2}=\pi \lambda$.

Figure 12

Scenario (i): The two Gaussian beams of fundamental and doubled frequency are focused to a beam waist of $w_{0,1}=w_{0,2}=\lambda$. In this scenario the beam divergences fulfill ${\theta }_2={\theta }_1/2$. We depict the case of zero offset, ${\overrightarrow{x}}_0=0$, and ${\vartheta }_2\in \{0°,180°\}$.

Figure 13

Scenario (ii): The Gaussian beam of fundamental (doubled) frequency is focused to a waist size of $w_{0,1}=\lambda$ ($w_{0,2}=\lambda /2$). In this scenario the beam divergences fulfill ${\theta }_1={\theta }_2$. We depict the case of zero offset, ${\overrightarrow{x}}_0=0$, and ${\vartheta }_2\in \{0°,180°\}$.

Figure 14

Integrated numbers of signal photons attainable in the various scenarios (o)-(ii) plotted as a function of the collision angle ${\vartheta }_2$. We depict results for (o) the collision of two fundamental-frequency laser pulses focused to $w_{0,1}=w_{0,2}=\lambda$, and the collision of fundamental and doubled frequency laser pulses focused to waist sizes (i) $w_{0,2}=w_{0,1}=\lambda$ and (ii) $w_{0,2}=w_{0,1}/2=\lambda /2$. Top: Total number of signal photons $N$. Bottom: Number of signal photons emitted outside the forward cones of the colliding Gaussian laser beams $N_{>\theta }$, delimited by the beams’ radial divergences ${\theta }_b$.

Figure 15

Partitioning of the attainable numbers of signal photons into the energy regimes $\mathrm{\Delta }({\omega }_1)$ and $\mathrm{\Delta }(2{\omega }_1)$ for the various scenarios (o)-(ii). Both high-intensity laser pulses are polarized perpendicularly to the collision plane. The segment with center frequency $\mathrm{k}={\omega }_1$ ($2{\omega }_1$) is depicted by • ($+$) symbols. Naturally, there is no $\mathrm{k}\approx 2{\omega }_1$ signal for the collision of two fundamental-frequency beams. Top: Total number of signal photons $N$. Bottom: Integrated number of signal photons emitted outside the forward cones of the colliding Gaussian laser beams $N_{>\theta }$.

Figure 16

Partitioning of the perpendicularly polarized signal photons into the frequency regimes $\mathrm{\Delta }({\omega }_1)$ and $\mathrm{\Delta }(2{\omega }_1)$ for the various scenarios (o)-(ii). Both high-intensity laser pulses are polarized perpendicularly to the collision plane. The segment with center frequency $\mathrm{k}={\omega }_1$ ($2{\omega }_1$) is depicted by • ($+$) symbols. Naturally, no $\mathrm{k}\approx 2{\omega }_1$ signal is induced in the collision of two fundamental-frequency beams. Top: Total number of perpendicularly polarized signal photons $N_{\perp }$. Bottom: Integrated number of perpendicularly polarized signal photons emitted outside the forward cones of the colliding Gaussian laser beams $N_{\perp ,>\theta }$.

Figure 17

Differential number of signal photons $\frac{{\mathrm{d}}^{2}N}{\mathrm{d}\phi \mathrm{d}\cos \vartheta }$ for $\phi =0$, i.e., in the collision plane, for a collision angle of ${\vartheta }_2=135°$, plotted as a function of the polar angle $\vartheta$ for the scenarios (o)-(ii). The dashed lines at $\vartheta =0°$ ($\vartheta =135°$) indicate the propagation direction of the high-intensity laser beam of frequency ${\omega }_1$ (${\omega }_2$). The different peak widths at $\vartheta \approx 135°$ for scenarios (i) and (ii) can be traced back to the different focusing of the frequency-doubled laser. Generically, a harder focusing results in a wider far-field divergence.

Figure 18

Comparison of the normalized differential number of signal photons $\frac{{\mathrm{d}}^{2}N}{\mathrm{d}\phi \mathrm{d}\cos \vartheta }$ for $\phi =0$ with the far-field photon distributions of the high-intensity laser beams. Here, we exemplarily limit ourselves to scenario (i) and a collision angle of ${\vartheta }_2=135°$. As the frequency-doubled Gaussian beam is only focused to $w_{0,2}=\lambda$, its divergence fulfills ${\theta }_2={\theta }_1/2$.

Figure 19

Convergence of the total number of signal photons $N$ as a function of the number of radial momentum grid points in $N_{\mathrm{\Delta }(\mathrm{k})}$ for various sizes of the sampling region. The sampling interval in $\mathrm{k}$ has a total length of $2L_{\mathrm{k}}$ and is centered around $\omega \approx 1.54\mathrm{eV}$. Two identical laser pulses ($\lambda =800\mathrm{nm}$, $W=25\mathrm{J}$, $\tau =25\mathrm{fs}$) are focused to $w_{0,1}=w_{0,2}=\lambda$. They are assumed to be polarized perpendicularly to the collision plane and collide under an angle of ${\vartheta }_2=135°$. The benchmark toy model is used here to allow for a comparison with the semianalytical result (black line).

Figure 20

Convergence of the total number of signal photons $N$ as a function of the interval length $[-L_{\mathrm{z}},L_{\mathrm{z}}]$ for two sets of grid points in the $\mathrm{z}$ direction. Two identical laser pulses ($\lambda =800\mathrm{nm}$, $W=25\mathrm{J}$, $\tau =25\mathrm{fs}$) are focused to $w_{0,1}=w_{0,2}=\lambda$ and collide in a counterpropagation geometry. Both pulses are polarized perpendicularly to the collision plane. The benchmark toy model is used here to allow for a comparison with the semianalytical result (black line).