The Effect of Ultrastrong Magnetic Fields on Laser-Produced Gamma-Ray Flashes

Laser produced $gamma$-photons can make an important impact on applied and fundamental physics that require high $gamma$-photon yield and strong collimation. We propose addition of a constant magnetic field to the laser-solid interaction to obtain the aforementioned desired $gamma$-photon properties. The $gamma$-ray flash spatial and spectral characteristics are obtained via quantum electrodynamics particle-in-cell simulations. When the constant magnetic field aligns with the laser magnetic field then the $gamma$-ray emission is significantly enhanced. Moreover, the $gamma$a-photon spatial distribution becomes collimated, approximately in the form of a disk.

The 1980's witnessed the invention of the Chirped Pulse Amplification technique [1] resulting in the rapid growth of laser power, exceeding the petawatt (PW) level by the end of the 20th century [2].The 10 PW lasers [3] became reality with a femtosecond-class laser in ELI-NP, another laser of ten times higher energy being near completion in ELI-Beamlines and the upgrade of the Apollon laser.
When a PW-class laser interacts with matter energetic charged particles are generated, being the norm over the past years.Once multi-PW laser facilities came forward, generation and usage of γ-photons became one of the main tasks of those facilities.A high laser to γ-photon energy conversion efficiency, κ γ , is predicted [4][5][6].The γ-ray flashes suit, among others, photonuclear reactions [7], positron sources [8], neutron sources [9], extreme energy density materials science [10] and studies on fundamental processes [11].
The γ-ray flash is a result of the multiphoton Compton scattering process [12], which occurs when an electron collides with the laser, emitting a scattered γ-photon.The scattering process reads e − + N l ω l → e − + ω γ , where e − represents an electron, N l 1 is the number of laser photons, ω l is the laser frequency, and ω γ is the scattered γ-photon frequency.The γ-photon yield is quantified by the quantum nonlinearity parameter, χ e = γ e E −1 Here, E S = m 2 e c 3 /(e ) ≈ 1.3 × 10 18 Vm −1 is the Schwinger field (m e is the electron rest mass, c is the vacuum speed of light, e is the elementary charge and is the reduced Planck constant), γ e is the Lorentz factor of an electron with velocity v before scattering, E is the electric field and B is the magnetic field.High κ γ values require χ e 1 [4,5].The purpose of this paper is to demonstrate that a constant magnetic field (CMF) added to a laser field can enhance the γ-photon yield and directionality from lasersolid interactions.We initially study the single-electron dynamics under the effect of an ultraintense laser in addition to a CMF, with radiation reaction force taken into account.We then extend our study to particle-in-cell (PIC) simulations with the target being a lithium slab, as shown in Fig. 1.It has recently been shown that lithium combined with multi-PW lasers results in high κ γ [13]; although generation of a small electron-positron pair population through the multiphoton Breit-Wheeler process is possible under the conditions examined, it does not affects the γ-photon emission and is therefore ignored.We demonstrate that a strong CMF orthogonal to the laser propagation direction increases κ γ by several times.Moreover, if the CMF aligns with the laser magnetic field then the γ-ray flash distribution resembles a narrow disk along the laser electric field oscillating plane, increasing the emitted γ-photon fluence.
Let us measure velocity in c, momentum in m e c, distance in λ = λ/(2π) (where λ = 0.815 µm is the laser wavelength) and time in ω −1 l (where ω l = c/λ).We use a linearly polarized electromagnetic wave with an elec- These fields are in the megatesla scale for the wavelength under consideration [14].
A radiation reaction force, e G e /(3 γ e ), is considered [15], where p = [p x , p y , p z ] is the electron momentum and G e = (1 + 8.93χ e + 2.41χ 2 e ) −2/3 is the Gaunt factor which accounts for quantum electrodynamics correction of the radiation reaction effect [6].The equations of motion are and Here, r = [x, y, z], α = e 2 /(4πε 0 c) is the fine structure constant and ε 0 is the vacuum permittivity.The obtained single-electron trajectories are shown in Fig. 2(a-d).Fig. 2(a) corresponds to the reference case where only a plane electromagnetic wave is considered [16], without CMF.The electron drifts along the x-axis with a drift velocity, u d c; the electron trajectory in the electron average rest frame is a figure-eight.The radiation reaction force is maximized where the trajectory curvature is high.Projection of electron trajectories at those locations indicates that the radiation is emitted at two symmetric lobes [4,13].
Let us set b = 150, except where otherwise mentioned.For the b x case (see supplementary figure) the electron trajectory forms periodic spiral-like patterns, with the period decreasing for increasing b x .The ratio of its zaxis to y-axis extent is b x .The radiation reaction force initially obtains low values of 0.0018m e cω l and slowly decreasing thereafter.The particle trajectory is approximately the same whether or whether not the radiation reaction is considered, with γ ≈ 12.Note that the b x case combined with a circularly polarized laser in the absence of radiation reaction has been studied six decades ago, referred to as the autoresonance effect where an electron can be accelerated to high energies [17,18].However, inclusion of radiation reaction suppresses the autoresonance scheme [19,20].
For the b y case (see supplementary figure) the electron propagation direction rotates approximately −12 • (for the parameters defined) on the xz-plane.By shifting the electron trajectory along the new propagation axis the trajectories form figure-eight patterns, with the trajectory period decreasing for increasing b y .Thus, the highest γ-photon energy yield is expected at two lobes located at negative z-values.The Lorentz factor obtains a peak value of γ ≈ 390, whilst the radiation reaction force reaches values of ≈ 55m e cω l , revealing strong γ-photon emission.
If the radiation reaction is ignored, the b z case gives circular trajectories along the xy-plane as shown in Fig. 2(b), passing through z = 0 and symmetrically around the y-axis; no drifting of the electron trajectory occurs [21].In this case, the Lorentz factor is O(10 3 ).The picture changes by the inclusion of radiation reaction, where the electron trajectory drifts along an axis (with u d ≈ 0.518) forming an angle θ ≈ arccos(1 − b z /a) ≈ 55 • with the laser propagation axis.The electron trajectories form periodic high-curvature patterns, as shown in Fig. 2(c).By considering the electron average rest frame then a heart-figure is revealed, as seen in Fig. 2(d).The Lorentz factor peaks at γ ≈ 1440, with a period matching that of the high-curvature patterns appearance.The b z case corresponds to the highest radiation reaction force values, at ≈ 320m e cω l , and strongest γ-photon emission is expected for this case.Due to the electron trajectory restricted on the xy-plane, all γ-photons are emitted as a γ-ray disk.Now we realize the laser-target interaction through three-dimensional relativistic quantum electrodynamic PIC simulations.We use the EPOCH [22,23] code, enabling the Photons (acknowledging the radiation reaction effect) and the Higuera-Cary (obtaining accurate high-energy electron trajectories) preprocessor directives.For similar interaction parameters, in the ab- sence of the CMF, Compton γ-photons dominate over Bremsstrahlung γ-photons within the simulation time [13].Therefore, the Bremsstrahlung preprocessor directive is not enabled.Although a small population of electron-positron pairs (orders of magnitude smaller than the target electrons) is generated during the interaction [13], they are ignored as we are only interested in the γ-ray flash characterization.
The laser and CMF parameters used in the PIC simulations match those used in the single-electron model, where now the pulsed nature of the laser is considered with 17λ focused diameter, 40ω −1 l pulse duration and a pulse temporal offset of two standard deviations.Three simulation sets are performed, each having the CMF oriented along one of the three Cartesian axes.The laser focuses in the center of the simulation box, with box open boundaries at ±15.36 µm in each direction.Normal laser incidence on the target (yz-plane) requires cells of 10 nm along the x-axis and 40 nm on the other two, where eight macroelectrons and eight macroions are assigned per cell.The laser peak reaches the focal spot at 65 fs, while it requires 110 fs for γ-photon emission to increase asymptotically, indicating that Compton γ-photon emission occurs in timescales comparable to the laser duration.
The γ-photon radiant intensity, I Ω , being the total γphoton energy emitted per solid angle per pulse duration maps the γ-photon directionality.The b = 0 case corresponds to a peak I Ω reference value of I Ω0 .Fig. 3(a-d  The γ-ray disk reappears in Fig. 3(d), corresponding to the b z case.Here, the γ-photons pile-up on a narrow distribution on the xy-plane.Projection of the electron trajectories (see Fig. 2(c)) suggests that the γ-ray disk is not uniform along the xy-plane.For the b z ≈ 300 case, a lobe of I Ω /I Ω0 > 3 occurs, squeezed within an angle of < 10 • at full-width-at-half-maximum along z-axis.
Our results, expanded over a CMF amplitude range, are summarized in Fig. 4. The b x , b y and b z cases are represented by the blue, green and red lines respectively.The continuous lines show κ γ as a function of b.If no magnetic field is applied, the laser interaction with the lithium target results in κ γ ≈ 20 %.For the b x case, applying a CMF up to b x ≈ 150 increases κ γ , but further increase of b x results in rapid decrease of κ γ .This effect is due to magnetically induced transparency of the target [24], where a highly transparent target results in weak laser-target interaction, therefore low γ-photon emission.The dashed blue line shows I Ω /I Ω0 steadily decreasing for increasing b x .
Our PIC simulations reveal that magnetically induced transparency does not occur if the CMF is along yaxis, within our parameters of interest.Although I Ω /I Ω0 remains approximately constant, the extent of the γphoton lobes increases.As a result, κ γ gradually increases, reaching κ γ ≈ 40 % for b y = 300.
Target transparency also occurs for the b z case [25][26][27], where b z > 150 results in decrease of κ γ .However, at b z = 150 we obtain κ γ ≈ 60 %, three times higher than the reference case.The b z case is of particular interest also due to its high I Ω , as seen by the red dashed line in Fig. 4(a).Although optimal κ γ occurs at b z = 150, optimal I Ω /I Ω0 occurs at b z = 300 due to a narrower (along z-axis) γ-photon lobe.
The γ-photon energy spectrum depends on the γphoton detection angle.This is seen in Fig. 5(a), where the normalized fluence is shown, for the three b z cases examined as labeled in the figure .For the b = 0 case two symmetric lobes are obtained.However, by increasing the CMF value the first lobe shifts closer to the laser propagation axis while the second lobe shifts far from the axis.In addition, the figure depicts with the blue shadowed region the case where only γ-photons of energy larger than 500m e c 2 are considered.There, a dominant lobe exists on the laser propagation axis with a divergence of approximately 10 • at full-width-at-halfmaximum, and another less prominent lobe at angles of −45 • , −65 • and −110 • for the b z = 75, b z = 150 and b z = 300 cases respectively.
For applications, one needs to know how the γ-photon number changes per energy interval.Therefore, we calculate the ratio of the γ-photon spectrum at the peak I Ω location for the b z case, to the reference case.For b z = 150, I Ω peaks at 20 • to the laser propagation axis, and for b z = 300 it peaks at 15 • ; for b = 0, the reference I Ω peaks at 45 • .The spectra ratios for the b z = 150 and b z = 300 cases are shown in Fig. 5(b) with red and blue lines respectively.In both cases, the γ-photon number, N γ , is increased three (for higher γ-photon energies) to five (for lower γ-photon energies) times compared to the reference γ-photon number, N γ0 .However, considering peak I Ω only of higher energy γ-photons, results in spectra ratios as shown in Fig. 5(c).The lobe on the laser propagation axis contains approximately twice as high energy γ-photons compared to the reference case, taken at 50 • .The second lobe contains approximately same number of high energy γ-photons as in the reference case.
In conclusion, the spatial and spectral distributions of the emitted γ-photons are obtained via PIC simulations, where an a = 350 laser interacts with a lithium foil.The radiation reaction force alters the trajectory of a singleelectrons moving under the influence of an ultraintense laser in addition to a CMF.The radiation reaction force obtains high values when the CMF is transverse to the laser propagation axis.If the CMF aligns with the laser electric field then the γ-photons are emitted in two broad lobes, whilst if it coincides with the laser magnetic field then the electron moves strictly on the xy-plane and the γ-photons are emitted mostly along that plane.Specifically, the radiant intensity for the b z = 300 case is increased by a factor of more than three.Highest κ γ is obtained for the b z = 150 case, reaching up to 60 %, three times higher than the reference case.Moreover, the amplitude of the lower and higher part of the γphoton energy spectrum is increased by a factor of five and two respectively.The enhancement of the low energy γ-photon number at optimal emission angles suits photonuclear reactions [7].The higher energy γ-photon number is doubled along the laser propagation axis, necessary for electron-positron pair generation the nonlinear Breit-Wheeler process [11].
This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRA CZ (ID:90254).The EPOCH code is in part funded by the UK EPSRC grants EP/G054950/1, EP/G056803/1, EP/G055165/1 and EP/M022463/1.

FIG. 1 .
FIG. 1.An a = 350 laser (figure inset) interacting with a lithium foil, in the presence of a CMF (black dashed arrows), at 80 fs PIC simulation time.The case where the CMF corresponds to the bz = 150 case is depicted.The laser-foil interaction results in enhanced γ-photon emission, illustrated by blue spheres (energy larger than 500 mec 2 , with a ballistic offset of 10 µm).The green-red and the gray colors show the laser field and the target respectively.

FIG. 2 .
FIG. 2. Electron trajectory by solving equations of motion under the influence of a linearly polarized electromagnetic wave of a ≈ 350 plus a CMF of (a) b = 0, (b) bz = 150 ignoring radiation reaction, (c) bz = 150 including radiation reaction and (d) bz = 150 including radiation reaction, in electron average rest frame.

FIG. 3 .
FIG. 3. IΩ/IΩ0 as obtained by PIC simulations of an a = 350 laser interacting with a lithium target in the presence of a CMF, for the cases of (a) b = 0, (b) bx = 300, (c) by = 300 and (d) bz = 300.For the bz = 300 case the color-bar saturates (green color).
) depicts I Ω /I Ω0 , with the CMF orientation labeled in the figure.Fig.3(a) shows a γ-photon double lobe structure, as predicted by previous PIC simulations and in

FIG. 4 .
FIG. 4. left axis -solid lines: κγ as a function of b, due to the interaction of an a = 350 laser with a lithium foil in the presence of a CMF.bx, by and bz correspond to the blue, green and red lines respectively, reaching a κγ value of 60 % for the bz ≈ 150 case.right axis -dashed lines: Peak IΩ/IΩ0 as a function of b.

FIG. 5 .
FIG. 5. (a) Normalized fluence in radial directions on the xy-plane, within 1 • full angle.The cases b = 0, bz = 75, bz = 150, bz = 300 correspond to the black, red, blue and green lines respectively.The blue filled area corresponds to the bz = 150 case, by considering only γ-photons of energy larger than 500mec 2 .(b) The ratio of bz to b = 0 γ-photon energy spectra, measured at peak IΩ within an 1 • full angle.(c) The ratio of bz to b = 0 γ-photon energy spectra, measured at peak IΩ (where here for IΩ we consider only γ-photons of energy larger than 500mec 2 ), within a 10 • full angle.The bz ≈ 150 and bz ≈ 300 cases correspond to the red and blue lines respectively.