Polarized ultrashort brilliant multi-GeV $\gamma$-rays via single-shot laser-electron interaction

TL;DR

This paper demonstrates the generation of highly polarized multi-GeV gamma rays through single-shot laser-electron interactions, employing a novel Monte Carlo simulation to optimize polarization transfer in quantum radiation regimes.

Contribution

It introduces a Monte Carlo simulation method with electron-spin-resolved probabilities and shows efficient polarization transfer for high-energy gamma-ray production.

Findings

Generation of up to 95% polarized gamma rays

Production of multi-GeV gamma rays suitable for vacuum birefringence experiments

High-brilliance gamma-ray beams beneficial for physics applications

Abstract

Generation of circularly-polarized (CP) and linearly-polarized (LP) $\gamma$-rays via the single-shot interaction of an ultraintense laser pulse with a spin-polarized counterpropagating ultrarelativistic electron beam has been investigated in nonlinear Compton scattering in the quantum radiation-dominated regime. For the process simulation a Monte Carlo method is developed which employs the electron-spin-resolved probabilities for polarized photon emissions. We show efficient ways for the transfer of the electron polarization to the high-energy photon polarization. In particular, multi-GeV CP (LP) $\gamma$-rays with polarization of up to about 95\% can be generated by a longitudinally (transversely) spin-polarized electron beam, with a photon flux at a single shot meeting the requirements of recent proposals for the vacuum birefringence measurement in ultrastrong laser fields. Such high-energy, high-brilliance, high-polarization $\gamma$-rays are also beneficial for other applications in high-energy physics, nuclear physics, and laboratory astrophysics.

Equations

$$\frac{{\mathrm{d}}^2{W}_{fi}}{\mathrm{d}u\mathrm{d}\eta }=\frac{W_R}{2}\left(F_0+{\xi }_1{F}_1+{\xi }_2{F}_2+{\xi }_3{F}_3\right),$$

$$F_0$$

$$F_1$$

$$F_2$$

$$F_3$$

$${\xi }_1^{mix}=F_1/F_0,{\xi }_2^{mix}=F_2/F_0,{\xi }_3^{mix}=F_3/F_0.$$

Full text

Polarization is a crucial intrinsic property of a $\gamma$-photon. In astrophysics the $\gamma$-photon polarization provides detailed insight into the $\gamma$-ray emission mechanism and on properties of dark matter Laurent et al. (2011); Bœhm et al. (2017). Highly-polarized high-energy $\gamma$-rays are a versatile tool in high-energy Moortgat-Pick et al. (2008) and nuclear physics Uggerhøj (2005). For instance, polarized $\gamma$-rays of tens of MeV can be used to excite polarization-dependent photofission of the nucleus in the giant dipole resonance Speth and van der Woude (1981), while polarized $\gamma$-rays of up to GeV play crucial roles for the meson-photoproduction Akbar et al. (2017).

Recently, several proposals have been put forward to detect vacuum birefringence in ultrastrong laser fields, probing it with circularly-polarized (CP) or lineraly-polarized (LP) $\gamma$-photons of high-energies (larger than MeV and up to several GeV), see King and Elkina (2016); Ilderton and Marklund (2016); Ataman et al. (2017); Nakamiya and Homma (2017); Bragin et al. (2017), taking advantage of the fact that the QED vacuum nonlinearity is significantly enhanced for high-energy photons. As proved in Bragin et al. (2017), the use of CP rather than LP probe photons reduces the measurement time of vacuum birefringence and vacuum dichroism by two orders of magnitude.

The common ways of producing high-energy polarized $\gamma$-rays are linear Compton scattering Bocquet et al. (1997); Nakano et al. (1998); Omori et al. (2006); Alexander et al. (2008); Blanpied et al. (1999) and bremsstrahlung Olsen and Maximon (1959); Kuraev et al. (2010); Abbott et al. (2016); Lohmann et al. (1994). The advantage of the former is that it employs unpolarized electron beams, and the emitted $\gamma$-photon polarization is determined by the driving laser polarization, while in the latter the spin of the scattering electron determines the $\gamma$-photon polarization Berestetskii et al. (1982). However, in linear Compton scattering the electron-photon collision luminosity is rather low. The collision luminosity can be increased by using high-intensity lasers, but in this case the interaction regime moves into the nonlinear regime, when the radiation formation length is much smaller than the laser wavelength. Then during the photon formation the laser field does not vary much and the emission process acquires similarity to bremsstrahlung. Namely, in the nonlinear regime the circular polarization of the emitted $\gamma$-photon requires longitudinally spin-polarized (LSP) electrons. Nevertheless, the nonlinear regime of Compton scattering is beneficial for the generation of polarized $\gamma$-photons, because the polarization is enhanced at high $\gamma$-photon energies Bocquet et al. (1997), and the typical emitted photon energy is increased in the quantum nonlinear regime, becoming comparable with the electron energy Ritus (1985); Di Piazza et al. (2012). In the nonlinear regime the relative bandwidth of emission is increased Boca and Florescu (2011), which is not suitable for photonic applications involving narrow resonances, and stimulated investigations for the bandwidth reduction, see e.g. Ghebregziabher et al. (2013); Terzić et al. (2014). However, vacuum birefringence is not a resonant effect and its measurement does not require a small $\gamma$-photon bandwidth, but mostly high flux of highly-polarized high-energy photons.

Regarding deficiencies connected with the bremsstrahlung mechanism, the incoherent bremsstrahlung cannot generate LP $\gamma$-photons, and the scattering angle and emission divergence are both relatively large Baier et al. (1998). Furthermore, for coherent bremsstrahlung Ter-Mikaelian (1972); Uggerhøj (2005), the current density of the impinging electrons and the radiation flux is limited by the damage threshold of the crystal material Lohmann et al. (1994); Carrigan and Ellison (1987); V. M. Biryukov and Kotov (1997).

With rapid developments of strong laser techniques, stable (energy fluctuation $\sim$ 1%) ultrashort ultraintense laser pulses can reach peak intensities of the scale of $10^{22}$ W/cm2 with a duration of about tens of femtoseconds Danson et al. (2015); Gales et al. (2018); ELI ; Vul ; Exa ; for Relativistic Laser Science (CoReLS), opening new ways to generate high-energy high-flux $\gamma$-rays Sarri et al. (2014); Cole et al. (2018); Poder et al. (2018); Li et al. (2015); Magnusson et al. (2019); Xie et al. (2017) in the nonlinear regime of Compton scattering Goldman (1964); Nikishov and Ritus (1964); Ritus (1985); Bula et al. (1996). Moderately polarized $\gamma$-photons have been predicted in strong fields in electron-spin-averaged treatment King et al. (2013); King and Elkina (2016); Sinha et al. (2019). However, the polarization properties of radiation are essentially spin-dependent in the nonlinear regime Baier et al. (1998); Ivanov et al. (2004), differing from those in the linear regime CAI ; Sun and Wu (2011); Petrillo et al. (2015); An et al. (2018)), which calls for comprehensive spin-resolved studies, especially in the most attractive high-energy regime.

In this Letter, the feasibility of generation of polarized ultrashort multi-GeV brilliant $\gamma$-rays via nonlinear Compton scattering with spin-polarized electrons is investigated theoretically (see Fig. 1). High-flux $\gamma$-photons with polarization beyond 95% are shown to be feasible in a single-shot interaction, along with new applications in high-energy, astro- and strong laser physics. The investigation is based on the developed Monte Carlo method for simulation of polarization-resolved radiative processes in the interaction of an ultrastrong laser beam with a relativistic spin-polarized electron beam. While the scheme for CP $\gamma$-photons includes an arbitrarily-polarized (AP) laser pulse colliding with a LSP electron bunch (see Fig. 1(a)), the scheme for LP $\gamma$-photons employs an elliptically-polarized (EP) laser pulse with a small ellipticity colliding with a transversely spin-polarized (TSP) electron bunch (see Fig. 1(b)). The spin-dependent radiation reaction in a laser field with a small ellipticity yields the separation of $\gamma$-photons with respect to the polarization and the enhancement of the polarization rate.

Let us first introduce our new Monte Carlo method for simulation of polarized $\gamma$-photon emissions during the interaction of polarized relativistic electrons with ultrastrong laser fields. Photon emissions are treated quantum mechanically, while the electron dynamics semiclassically. At each simulation step the photon emission is determined by the total emission probability, and the photon energy by the spectral probability, using the common algorithms  Ridgers et al. (2014); Elkina et al. (2011); Green and Harvey (2015). The spin of the electron after the emission is determined by the spin-resolved emission probabilities according to the algorithm of Ref. Li et al. (2019); See more details in sup . For determination of the photon polarization, we employ the polarized photon emission probabilities by polarized electrons in the local constant field approximation Ritus (1985); Baier et al. (1998); Di Piazza et al. (2018); Ilderton et al. (2019); Podszus and Di Piazza (2019); Ilderton (2019); Di Piazza et al. (2019), which are derived in the QED operator method of Baier-Katkov Baier et al. (1973). This approximation is valid in ultrastrong laser field, when the invariant field parameter is large $a_{0}\equiv|e|E_{0}/(m\omega_{0})\gg 1$ Ritus (1985); Baier et al. (1998), with laser field amplitude $E_{0}$, frequency $\omega_{0}$, and the electron charge $e$ and mass $m$. Relativistic units with $c=\hbar=1$ are used throughout. The radiation probabilities are characterized by the quantum strong field parameter $\chi\equiv|e|\sqrt{-(F_{\mu\nu}p^{\nu})^{2}}/m^{3}$ Ritus (1985), where $F_{\mu\nu}$ is the field tensor, and $p^{\nu}$ the four-vector of the electron momentum. The angle-integrated radiation probability of a polarized photon with a polarized electron reads:

[TABLE]

where $W_{R}={\alpha m}/\left[{8\sqrt{3}\pi\lambdabar_{c}\left(k\cdot p_{i}\right)}{\left(1+u\right)^{3}}\right]$, $\alpha$ is the fine structure constant, $u=\varepsilon_{\gamma}/\left(\varepsilon_{i}-\varepsilon_{\gamma}\right)$, $\lambdabar_{c}$ the Compton wavelength, $\varepsilon_{\gamma}$ the emitted photon energy, $\varepsilon_{i}$ the electron energy before radiation, $\eta=k\cdot r$ the laser phase, while $p_{i}$, $k$, and $r$ are four-vectors of the electron momentum before radiation, laser wave-vector, and coordinate, respectively. The photon polarization is represented by the Stokes parameters ($\xi_{1}$, $\xi_{2}$, $\xi_{3}$), defined with respect to the axes $\hat{{\bf e}}_{1}=\hat{\bf a}-\hat{\bf v}(\hat{\bf v}\hat{\bf a})$ and $\hat{{\bf e}}_{2}=\hat{\bf v}\times\hat{\bf a}$ McMaster (1961), with the photon emission direction $\hat{\bf n}$ along the electron velocity ${\bf v}$ for the ultrarelativistic electron, $\hat{\bf v}={\bf v}/|{\bf v}|$, and the unit vector $\hat{{\bf a}}={\bf a}/|{\bf a}|$ along the electron acceleration ${\bf a}$. The variables introduced in Eq. (1) read:

[TABLE]

where $u^{\prime}=2u/3\chi$, ${\rm IntK}_{\frac{1}{3}}(u^{\prime})\equiv\int_{u^{\prime}}^{\infty}{\rm d}z{\rm K}_{\frac{1}{3}}(z)$, ${\rm K}_{n}$ is the $n$-order modified Bessel function of the second kind, ${\bf S}_{i}$ and ${\bf S}_{f}$ are the electron spin-polarization vector before and after radiation, respectively, $|{\bf S}_{i,f}|=1$, and ${\bf S}_{if}\equiv{\bf S}_{i}\cdot{\bf S}_{f}$. Our Monte Carlo algorithm to determine the photon polarization yields the following, taking into account that the averaged polarization of the emitted photon is in a mixed state:

[TABLE]

Here we choose as the basis for the emitted photon Lipps and Tolhoek (1954); McMaster (1961) the two orthogonal pure states with the Stokes parameters $\hat{\bm{\xi}}^{\pm}\equiv\pm$($\xi_{1}^{mix}$, $\xi_{2}^{mix}$, $\xi_{3}^{mix}$)/$\xi_{0}^{mix}$ with $\xi_{0}^{mix}\equiv\sqrt{(\xi_{1}^{mix})^{2}+(\xi_{2}^{mix})^{2}+(\xi_{3}^{mix})^{2}}$.

Using the probabilities for the photon emission in these states $W_{fi}^{\pm}$ given by Eq. (1), we define the stochastic procedure with a random number $N_{r}\in[0,1]$: if $W_{fi}^{+}/\overline{W}_{fi}\geq N_{r}$, the $\hat{\bm{\xi}}^{+}$ photon state is chosen, otherwise the photon state is set to $\hat{\bm{\xi}}^{-}$, see sup . The Stokes parameters of each emitted photon are rotated from the instantaneous frame $(\xi_{1},\xi_{2},\xi_{3})$ defined with respect to the basis vectors $\hat{\textbf{e}}_{1}$, $\hat{\textbf{e}}_{2}$ and $\hat{\textbf{n}}$ to the observation frame $(\xi^{(o)}_{1},\xi^{(o)}_{2},\xi^{(o)}_{3})$ defined with respect to the basis vectors $\hat{\textbf{o}}_{1}$, $\hat{\textbf{o}}_{2}$ and $\hat{\textbf{o}}_{3}$, see sup ; McMaster (1961). Here, $\overline{W}_{fi}\equiv W_{R}F_{0}$ is the electron-spin-resolved radiation probability averaged by the photon polarization, cf. Li et al. (2019). Averaging over the electron spin in Eq. (1) yields $\xi_{2}=0$, indicating that only LP $\gamma$-photons can be generated with unpolarized electrons, in accordance with King et al. (2013).

The simulation results for the generation of CP $\gamma$-rays are shown in Fig. 2. A realistic tightly-focused Gaussian LP laser pulse is used Salamin and Keitel (2002); Salamin et al. (2002); sup , with peak intensity $I_{0}\approx 3.45\times 10^{21}$ W/cm2 ($a_{0}=50$), wavelength $\lambda_{0}=1$ $\mu$m, pulse duration $\tau=10T_{0}$ with period $T_{0}$, and focal radius $w_{0}=5$ $\mu$m. The electron beam counterpropagating with the laser pulse (polar angle $\theta_{e}=180^{\circ}$ and azimuthal angle $\phi_{e}=0^{\circ}$) is fully LSP with $(\overline{S}_{x},\overline{S}_{y},\overline{S}_{z})=(0,0,-1)$. It has a cylindrical form with radius $w_{e}=2\lambda_{0}$, length $L_{e}=3\lambda_{0}$ and electron number $N_{e}=5\times 10^{6}$ (density $n_{e}\approx 1.33\times 10^{17}$ cm*-3*). The electron density has a transversely Gaussian and longitudinally uniform distribution. The electron initial kinetic energy is $\varepsilon_{0}=10$ GeV, the angular divergence $\Delta\theta=0.2$ mrad, and the energy spread $\Delta\varepsilon_{0}/\varepsilon_{0}=0.06$. The emittance of the electron beam is estimated $\epsilon_{e}\approx 4\times 10^{-4}$ mm$\cdot$mrad. In our simulations, the electron-positron pair creations and their further radiations are taken into account. Such electron bunches are achievable via laser wakefield acceleration Leemans et al. (2014); Gonsalves et al. (2019) with further radiative polarization Li et al. (2019), or alternatively, via directly wakefield acceleration of LSP electrons Wen et al. (2019).

The angle-resolved number density and average circular polarization of all emitted photons are given in Figs. 2(a)  and (b). The $\gamma$-ray pulse duration is determined by the electron bunch length: $\tau_{\gamma}\approx\tau_{e}\approx 10$ fs Li et al. (2015). The total number of emitted $\gamma$-photons $N_{\gamma}$ at these parameters is about $8.04\times 10^{7}$ for $\varepsilon_{\gamma}\geq 1$ MeV and about $1.23\times 10^{7}$ for $\varepsilon_{\gamma}\geq 1$ GeV, i.e., $N_{\gamma}$ is approximately one order of magnitude larger than $N_{e}$, which is in accordance with the analytical estimation $N_{\gamma}\sim\alpha a_{0}N_{e}\tau/T_{0}$ Ritus (1985). For comparison the polarized $\gamma$-photon number in the linear Compton scattering is $N_{\gamma}/N_{e}\approx 10^{-3}$ at $\varepsilon_{\gamma}\approx 56$ MeV Omori et al. (2006). The total flux of the $\gamma$-rays with $\varepsilon_{\gamma}\geq 1$ MeV is large due to the shortness of the pulse: ${\cal F}_{\gamma}\approx 8.04\times 10^{21}$ s*-1*. The circular polarization of $\gamma$-photon $P_{\gamma}^{CP}$ is proportional to the $\gamma$-photon energy $\varepsilon_{\gamma}$ (derived from Eqs. (2) and (4)), and significantly higher for higher photon energies, see Fig. 2(c). Intuitively speaking, the polarization of emitted $\gamma$-photons (photon helicity in the case of CP) is transfered from the angular momentum (helicity) of electrons. The larger the average energy of emitted $\gamma$-photons, the smaller the number of emitted photons per electron will be. As the electron carries a certain helicity, the transferred average helicity per photon will be larger in the case of smaller photon number, i.e., in the case of high photon energies. For instance, multi-GeV $\gamma$-rays of about 10-GeV, 8-GeV and 6-GeV can be emitted with circular polarization of about 99%, 94% and 81%, respectively, and with quite significant brilliances of about $6.25\times 10^{18}$, $3.94\times 10^{20}$ and $1.36\times 10^{21}$ photons/(s mm2 mrad2 0.1% BW), respectively, which are comparable with the brilliance of unpolarized multi-MeV $\gamma$-rays obtained in a recent experiment Sarri et al. (2014). The detection of vacuum birefringence demands $N_{\gamma}\sim 10^{7}-10^{9}$ CP GeV $\gamma$-photons Bragin et al. (2017). Taking into account that LSP electron bunches with $N_{e}\approx 5\times 10^{8}$ are feasible Wen et al. (2019), we estimate the number of CP $\gamma$-photons at 4 GeV from Fig. 2(c) for such bunches (using $\Delta\varepsilon_{\gamma}/\varepsilon_{\gamma}\sim 0.01$) to reach the required target number for vacuum birefringence $N_{\gamma}\approx 10^{6}$ in a single-shot interaction.

Figure 2 indicates that the photon polarization is mostly determined by the incoming electron spin vector $\bm{S}_{i}$, where the Monte Carlo and the average methods provide identical results; See Fig. 2(c). During multiple photon emissions the average spin of the electron beam from the initial longitudinal direction is gradually oriented along the laser magnetic field in the transverse direction. This effect reduces the circular polarization in the case of multiple photon emissions; See Fig. 2(d).

Generation of LP $\gamma$-rays is analyzed in Fig. 3. As an unpolarized (or TSP) electron beam head-on collides with a LP laser pulse, an average polarization of about 55% can be obtained sup , pointed out already in King et al. (2013); King and Elkina (2016). However, by harnessing the scheme with an EP laser pulse Li et al. (2019), much higher polarization can be achieved (see Fig. 1(b)). We focus on the characteristics of the LP $\gamma$-photons in the high-density region of -10 mrad $<\theta_{x,y}<$ 10 mrad (in this region the total radiation flux ${\cal F}_{\gamma}\approx 2.82\times 10^{21}$ s*-1* and the average linear polarization $\overline{P^{LP}}\approx$ 55.2%).

Due to the electron-spin-dependence of radiation, namely, that the radiation probability is larger when the electron spin is anti-parallel to the rest frame magnetic field Li et al. (2019), the TSP electrons more probably emit photons in the half cycles with $B_{y}<0$. In the given EP laser field, the $y$-component of the electron momentum in those half-cycles with $B_{y}<0$ is positive, $p_{y}>0$, and the corresponding $\theta_{y}=p_{y}/p_{z}<0$, since $p_{z}<0$. Then, the $\gamma$-photons are more emitted with $\theta_{y}<0$, see Fig. 3(c). The relative asymmetry of the photon emission $\Delta\mathcal{N_{\gamma}}$ corresponds to the relative difference of the radiation probabilities for ${\bm{S}_{i}}$ being parallel and anti-parallel to the magnetic field Li et al. (2019) and is most significant around $\varepsilon_{\gamma}\approx 1.2$ GeV ($\varepsilon_{\gamma}/\varepsilon_{0}\approx 0.6$). Due to radiative electron-spin effects, electrons and $\gamma$-photons are split into two parts along the minor axis ($y$ axis) of the polarization ellipse Li et al. (2019). In contrast to the case with unpolarized electrons King et al. (2013); King and Elkina (2016), here at emission angles $\theta_{y}<0$ ($\theta_{y}>0$) the $\gamma$-photon polarization is proportional (inversely proportional) to its energy, see Fig. 3(d). Thus, specially highly LP $\gamma$-rays can be obtained in the high-energy region.

For $\theta_{y}<0$ in Fig. 3(d), although the average linear polarization of all emitted photons is $\overline{P^{LP}}\approx$ 58.3%, the high-energy photons achieve even higher linear polarization: at photon energies of about 2 GeV, 1 GeV, 0.4 GeV and 0.2 GeV, the polarizations are of about 95%, 82%, 70% and 64%, respectively, and the brilliances of about $8.0\times 10^{16}$, $1.2\times 10^{20}$, $2.9\times 10^{20}$ and $3.5\times 10^{20}$ photons/(s mm2 mrad2 0.1% BW), respectively. For $\theta_{y}>0$, the sign of polarization is energy dependent sup . Moreover, the depolarization effect due to multiple photon emissions in this case is not significant, because subsequent photon emissions generate LP $\gamma$-photons as well, see the black-dotted and green-dashed curves in Fig. 3(d). Furthermore, using $N_{e}\approx 3\times 10^{8}$ Wen et al. (2019), LP $\gamma$-photons of $N_{\gamma}\sim 10^{5}-10^{6}$ within 1.015 GeV $\leq\varepsilon_{\gamma}\leq 1.021$ GeV ($\Delta\varepsilon_{\gamma}/\varepsilon_{\gamma}\approx 0.0059$) can be obtained, which satisfy the requirement ($N_{\gamma}\gtrsim 6.4\times 10^{4}$) of the vacuum birefringence measurement with LP photons Nakamiya and Homma (2017).

Finally, we have investigated the impact of the laser and electron beam parameters on the polarization of $\gamma$-rays sup . Generally, larger energy spread $\Delta\varepsilon_{0}/\varepsilon_{0}=0.1$, larger angular divergence of 1 mrad, and different collision angles $\theta_{e}=179^{\circ}$ and $\phi_{e}=90^{\circ}$ do not disturb significantly the quality of $\gamma$-ray polarization. In generation of CP and LP $\gamma$-rays, the polarization is robust with respect to the variation of parameters, e.g., $\tau$, $\varepsilon_{0}$, $a_{0}$ and $\epsilon$. As the initial polarization of the electron beam decreases, the polarization of emitted $\gamma$-photons declines as well.

In conclusion, brilliant multi-GeV CP (LP) $\gamma$-rays with polarization up to about 95% are shown to be feasible in the nonlinear regime of Compton scattering with ultrarelativistic longitudinally (transversely) spin-polarized electrons. A photon number applicable for vacuum birefringence measurement in ultrastrong laser fields is achievable in a single-shot interaction. High degree of linear polarization is obtained due to the spin-dependent radiation reaction in a laser field with a small ellipticity, which induces separation of $\gamma$-photons with respect to the polarization.

Acknowledgement: We thank J. Evers, T. Wistisen, Y.-S. Huang, R.-T. Guo and Y. Wang for helpful discussion. Y. Li, F. Wan and J. Li thank Prof. C. Keitel for hospitality. This work is supported by the National Natural Science Foundation of China (Grants Nos. 11874295, 11804269, 11905169), the National Key R&D Program of China (Grant No. 2018YFA0404801) and the Science Challenge Project of China (No. TZ2016099).