Nonlinear Breit-Wheeler pair production in collisions of bremsstrahlung $\gamma$ quanta and a tightly focused laser pulse

Experimental efforts toward the detection of the nonperturbative strong-field regime of the Breit-Wheeler pair creation process plan to combine incoherent sources of GeV $\gamma$ quanta and the coherent fields of tightly focused optical laser pulses. This endeavor calls for a theoretical understanding of how the pair yields depend on the applied laser field profile. We provide estimates for the number of produced pairs in a setup where the high-energy radiation is generated via bremsstrahlung. Attention is paid to the role of the transversal and longitudinal focusing of the laser field, along with the incorporation of a Gaussian pulse envelope. We compare our corresponding results with predictions from plane-wave models and determine the parameters of focused laser pulses which maximize the pair yield at fixed pulse energy. Besides, the impact of various super-Gaussian profiles for the laser pulse envelope and its transverse shape is discussed.

Figure 1

Sketch of an experimental setup put forward to create $e^{-}{e}^{+}$-pairs from the collision of bremsstrahlung $\gamma$ quanta and a tightly focused laser pulse in the nonperturbative strong-field regime ($\xi \gg 1$, $\kappa \sim 1$) of the Breit-Wheeler process. The high-$Z$ target responsible for the generation of the $\gamma$ radiation is supposed to be thin. A magnet is located right after the target to deflect primarily the electron flux that traverses it. The spreading beam of bremsstrahlung photons forms a cone colored in blue. For more details on the planned experiment we refer the reader to Ref. [15].

Figure 2

Bremsstrahlung spectra according to Eqs. (1) (red dashed) and (2) (blue solid).

Figure 3

Plane-wave pulses ($w_0\to \infty$) with Gaussian and super-Gaussian-profiles with $n=4$ and $n=8$ are shown in blue, green and red, respectively. While the pure Gaussian envelope is dotted, the corresponding modulated functions linked to $n=4$ and $n=8$ are dashed and dot-dashed. These envelopes are given as references. This picture has been generated by setting $\tau =5\mathrm{fs}$ and $\omega =1.55\mathrm{eV}$.

Figure 4

Behavior of the laser intensity $I(\mathit{x},t)=E^2(\mathit{x},t)$ of a paraxial pulse [see Eq. (4)] in $t$ and $z$ (upper panel) and in $x$ and $z$ (lower panel). Here, the dashed lines represent wavefronts of bremsstrahlung radiation separated between each other by a distance ${\sigma }_z=2z_R$. The inset of the upper panel reveals the oscillatory feature of the strong field along $t$ and $z$ axes. We have used the same benchmark values and notation as in Table 1. For these values and $I_0\approx 2\times {10}^{22}\mathrm{W}/{\mathrm{cm}}^2$, the strong field condition $\xi (\mathit{x},t)\gg 1$ translates into $I(\mathit{x},t)/I_0\gg 2\times {10}^{-4}$ and $z_R=15.7\mu \mathrm{m}$.

Figure 5

Comparison of the numerically evaluated rate as given in Eq. (16) (black solid) with the analytical $\kappa \approx 1$ asymptote from Eq. (23) (red dashed) and the limiting case $R_{\kappa \ll 1}\approx \frac{\alpha m^2}{8{\omega }^{\prime }{V}_{\gamma }}{(\frac{3}{2})}^{3/2}\kappa {\mathrm{e}}^{-\frac{8}{3\kappa }}$ that is valid for $\kappa \ll 1$ (blue dotted) [4].

Figure 6

Differential number of pairs in dependence on the scaled energy of bremsstrahlung photons for $\xi =70$. We use the same benchmark values and notation as in Table 1.

Figure 7

Pair yield per radiating electron for constant crossed fields (blue dashed), pulsed plane wave (black) and pulsed Gaussian profile (red). The same benchmark parameters and notation of Table 1 have been used.

Figure 8

Dependence of the pair yield on the thickness of the bremsstrahlung bunch ${\sigma }_z$ for parameters given in Table 1. Moreover, the number of gamma photons in the pulse is kept constant.

Figure 9

Differential number of pairs for different branches of focal regions for a focused Gaussian pulse for $\xi =70$. The other parameters are given in Table 1.

Figure 10

Percentage of created particles from different focal regions for a focused Gaussian pulse. The other parameters are given in Table 1.

Figure 11

Distribution of created pairs in the longitudinal direction. We use the same benchmark values and notation as in Table 1.

Figure 12

Dependence of the ratio between the number of produced pairs with ($N$) and without ($N_{z=0}$) longitudinal focusing on the intensity parameter $\xi$ (upper panel) and the pulse length $\tau$ (lower panel). The other parameters are given in Table 1.

Figure 13

Impact of focusing when keeping the laser pulse energy constant. Here, the pair yield is maximal for all curves at the smallest considered value $x=0.5$, which corresponds to the largest intensity parameter $\xi =2{\xi }_0$, minimal beam waist $\tilde{w}=w_0/2$ (black) and minimal pulse duration $\tau ={\tau }_0/4$ (red dotted). The comparison is made for the reference values ${\xi }_0=70$, $w_0=2\mu \mathrm{m}$ and ${\tau }_0=30\mathrm{fs}$ corresponding to $x=1$.

Figure 14

Number of produced pairs as a function of the laser intensity and the beam waist while keeping the pulse energy constant. The optimal intensity point is found at $x\approx 0.9$, corresponding to $I\approx 3\times {10}^{22}\mathrm{W}/{\mathrm{cm}}^2$ ($\xi =120$) at $\tilde{w}=0.9\mu \mathrm{m}$. The comparison is made for the reference values ${\xi }_0=110$, $w_0=1\mu \mathrm{m}$ corresponding to $x=1$ and $E_0=10\mathrm{GeV}$.

Figure 15

Deviation in the number of created pairs when considering Gaussian pulse in paraxial approximation (black solid) and beyond paraxial pulses (dotted red and dashed blue) for different diffraction angles $\epsilon$ and $\xi =70$. Here, the energy of incident photons is fixed to ${\omega }^{\prime }=500\mathrm{MeV}$ (left panel) and ${\omega }^{\prime }=2.357\mathrm{GeV}$ (right panel).

Figure 16

Number of produced pairs for laser pulses with super-Gaussian time envelopes (red, blue dashed) compared to standard Gaussian with intensity parameter $\xi =70$ (black solid) when keeping the laser energy at the standard Gaussian level. Dotted curves result when the value of $\xi$ is kept the same for all pulse shapes.

Figure 17

Number of produced pairs for laser pulses with super-Gaussian time envelopes (red, blue dashed) compared to standard Gaussian (black solid) when keeping the laser energy at the standard Gaussian level. Hence, the intensity parameters read $\xi =70$ for $n=2$, $\xi =63.5$ for $n=4$ and $\xi =59.6$ for $n=8$.

Figure 18

Number of produced pairs for laser pulses with super-Gaussian spatial envelopes (red, blue dashed) in the transverse plane compared to standard Gaussian with intensity parameter $\xi$ (black solid) when keeping the laser energy at the standard Gaussian level. Dotted curves result when the value of $\xi$ is kept the same for all pulse shapes.

Figure 19

Local values of the quantum nonlinearity parameter $\kappa$ for Gaussian pulses with (red dotted) and without (black solid) longitudinal focusing at $t=0$, $r=0$.