Extremely Dense Gamma-Ray Pulses in Electron Beam-Multifoil Collisions

Sources of high-energy photons have important applications in almost all areas of research. However, the photon flux and intensity of existing sources is strongly limited for photon energies above a few hundred keV. Here we show that a high-current ultrarelativistic electron beam interacting with multiple submicrometer-thick conducting foils can undergo strong self-focusing accompanied by efficient emission of gamma-ray synchrotron photons. Physically, self-focusing and high-energy photon emission originate from the beam interaction with the near-field transition radiation accompanying the beam-foil collision. This near field radiation is of amplitude comparable with the beam self-field, and can be strong enough that a single emitted photon can carry away a significant fraction of the emitting electron energy. After beam collision with multiple foils, femtosecond collimated electron and photon beams with number density exceeding that of a solid are obtained. The relative simplicity, unique properties, and high efficiency of this gamma-ray source open up new opportunities both for applied and fundamental research including laserless investigations of strong-field QED processes with a single electron beam.

Intense sources of high-energy photons have broad applications in industry, medicine, materials science [1][2][3][4] as well as in nuclear physics, particle physics and laboratory astrophysics [5][6][7]. This large variety of applications motivated the construction of numerous facilities worldwide from synchrotrons and FELs [4,8], which provide bright sources of photons with energies up to some hundred keV, to Compton-based facilities aiming at photon energies up to 20 MeV [7].
The investigation of exotic phenomena such as lightby-light scattering, birefringence and dichroism of the quantum vacuum, catalytic generation of electronpositron cascades by high-energy photons, quark-gluon physics as well as the creation of dense electron-positron FIG. 1. Schematic setup. An ultrarelativistic electron beam sequentially collides with aluminum foils. At each beam-foil collision, a strong transverse force which focuses the electron beam and leads to copious gamma-ray emission is induced.
To this goal, here we introduce a novel concept for an ultraintense gamma-ray source based on the interaction of a single high-current ultrarelativistic electron beam with multiple submicrometer-thick conducting foils (see Fig. 1). By using fully 3D particle-in-cell (PIC) simulations, we show that: (i) An ultrarelativistic (10 GeV), dense (4.7 × 10 27 m −3 ) electron beam can be radially focused up to 3.8 × 10 29 m −3 , i.e., beyond FIG. 2. (a) Electron beam density, (b) transverse magnetic field, and (c) transverse electric field in the collision with a 0.5µm-thick aluminum foil. For comparison, the magnetic and electric beam self-fields are 3.1 × 10 4 T, and 9.4 × 10 12 V/m, respectively. (d) Electron beam to radiation energy conversion efficiency η as a function of σ ⊥ in the collision with one foil. The electron beam has 2 nC charge, 10 GeV energy, and σ = 0.55 µm. Black circles: 3D PIC simulations results; blue circles: reflected-beam model predictions. (e) Same as in panel (d) but for σ ⊥ = 0.55 µm and as a function of σ . the 1.8 × 10 29 m −3 electron density of solid aluminum; (ii) Electron beam focusing is accompanied by intense synchrotron photon emission with more than 30% of the electron beam energy converted into a 4.1 × 10 29 m −3 peak density collimated gamma-ray beam; (iii) When the electron beam density exceeds the foil electron density, the beam self-fields expel the target electrons and create an electron-depleted channel through the foil. The fields experienced by the beam electrons inside the plasma channel are so high that the quantum parameter χ ≈ γ|f ⊥ |/eF cr exceeds unity [41,42]. Here f ⊥ = q(E ⊥ + β × B) is the Lorentz force transverse to the beam velocity, γ is the beam relativistic factor, q = −e is the electron charge, F cr = m 2 e c 3 /e ≈ 1.3 × 10 18 V/m is the QED critical field [6,41,42]. This opens up the possibility of laserless strong-field QED investigations with only one ultrarelativistic electron beam [43].
We start by considering the free propagation of an electron beam in vacuum. The electric E and magnetic B self-fields of a cold electron beam in vacuum are related by B = β × E [44], where β = v/c is the normalized beam velocity (Gaussian units are employed for equations). Thus, f ⊥ = qE ⊥ /γ 2 is strongly suppressed for large γ, and the beam propagates almost ballistically over relatively long distances in vacuum.
When a beam collides with a conductor, it can be subject to strong near-field coherent transition radiation (CTR), which alters the nearly perfect cancellation of the electric and magnetic terms in the Lorentz force. Electromagnetic boundary conditions require that the electric field component tangential to the surface of a perfect conductor must be continuous and zero at the conductor surface, whereas the tangential magnetic field can be discontinuous and remains large [44]. Thus, when an electron beam encounters a conductor, the magnetic term of the Lorentz force, which drives beam focusing, can overcome the electric term, which drives beam expansion. Effectively, when the beam length is smaller than its transverse size, this process can be visualized as a beam colliding with its image charge (see below and Supplemental Material). Notice that a large f ⊥ naturally results in intense emission of radiation. For instance, in the classical regime the radiated power (mean photon energy) is proportional to γ 2 f 2 ⊥ (γ 2 f ⊥ ) [42,44]. For modeling, we consider an ultrarelativistic cold electron beam with cylindrical symmetry around its propagation axis x. The description is simplified by employing cylindrical coordinates with r = y 2 + z 2 , θ = arctan(z/y), and x being the radial, azimuthal and vertical components, respectively. We assume that cylindrical symmetry is preserved throughout the interaction. Hence, fields are independent of θ, the azimuthal electric field E θ and the radial B r and vertical B x components of the magnetic field are zero. Here beam and conductor fields are denoted by the superscript b and c, respectively. For an ultrarelativistic charge distribution ρ(x, r, t) = ρ 0 e −r 2 /2σ 2 ⊥ e −(x−x0−vt) 2 /2σ 2 with N e electrons, initial position x 0 , velocity v along x, and peak charge density and B b θ (x, r, t) = βE b r (x, r, t), provide an approximate solution to Maxwell equations up to terms of order 1/γ 2 around the beam [43]. To evaluate E c r (x, r, t) and B c θ (x, r, t), we consider a flat perfectly conducting foil with front surface at x = 0. When the electron beam is outside the conductor, the method of images can be employed for determining E c r (x, r, t) and B c θ (x, r, t) in x < 0 [45]. This method cannot be applied when the beam enters the foil, because the image would be located in x < 0, where conductor fields must satisfy sourcefree Maxwell equations. However, when σ ⊥ σ one can approximate E c r (x, r, t) and B c θ (x, r, t) with the image charge fields. This "reflected' beam" approximation holds because CTR, which is emitted with transverse size σ ⊥ and typical wavelength σ , undergoes weak diffraction over a Rayleigh length of approximately σ 2 ⊥ /σ σ from the boundary. The opposite limit σ ⊥ σ , cor- 3. Beam evolution. First column, initial electron beam density (a1), its magnetic (b1) and electric (c1) field, and the initial photon density (d1). Second to sixth column, same quantities as in the first column but at the 3rd (a2)-(d2), the 6th (a3)-(d3), the 7th (a4)-(d4), the 12th (a5)-(d5), and the 16th (a6)-(d6) beam-foil interaction, respectively.
responds to the magnetostatic approximation, yielding a vanishing B c θ and a surface-localized E c r (see Supplemental Material). Note that beam focusing in the σ ⊥ σ limit has been demonstrated in accelerators [46][47][48][49][50].
The electron beam to radiated energy conversion efficiency η can be calculated from (1) is employed for the beam and image charge fields. The average energy radiated per particle per unit time is conveniently approximated as [42] γm e c 2 qN e .
(2) In Eq. (2) we have assumed that all electrons have the same initial momentum and energy γm e c 2 . Furthermore, we have neglected the change of γ during the beam-foil interaction. The triple integral in Eq. (2) can be carried out numerically. Figure 2 shows the results of 3D PIC simulations of a cold electron beam colliding with one 0.5µm-thick aluminum foil. The electron beam has 2 nC charge, 10 GeV energy, and Gaussian spatial distribution with σ = 0.55 µm, σ ⊥ = 1.25 µm, and 9.2 × 10 26 m −3 den-sity. Figure 2(a) displays a snapshot of the electron beam density when the beam center has reached the front surface of the foil. Figures 2(b) and 2(c) show the transverse magnetic B ⊥ and electric field E ⊥ , respectively. Whilst B ⊥ is amplified and its peak value nearly doubles with respect to the beam self-field (3.1 × 10 4 T), E ⊥ is suppressed and much smaller than the beam self-field (9.4 × 10 12 V/m). ] plots η in electron beam-single foil collision with the same parameters as above but for σ = 0.55 µm (σ ⊥ = 0.55 µm ) and σ ⊥ (σ ) ranging from 0.275 µm to 1.25 µm. Black circles and blue circles correspond to 3D PIC simulation and reflected model results, respectively. These simulations confirmed that the mechanism of beam focusing and photon emission is robust and effective in a range of parameters that could become within reach of existing accelerator facilities such as FACET II [51].
show that simulation results approach the prediction of the reflected field model with increasing (decreasing) σ ⊥ (σ ). For beam density smaller than the foil electron density, simulations indicate that foil thickness is irrelevant provided that collisions and plasma instabilities remain negligible. By contrast, foil thickness is important when the electron beam density exceeds the conductor density [43]. Note that synchrotron photon emission also occurs when the beam exits the foil, as E r is suppressed at the rear foil surface and B θ grows gradually during the beam exit [52]. However, for σ ⊥ σ , the rear surface contribution to the radiated energy is subdominant, and is neglected in our model.
The above considerations suggest that the effect can be substantially enhanced by colliding the self-focused beam with further foils. In fact, the increased beam density due to self-focusing results in stronger beam self-fields at the subsequent beam-foil collisions. Consequently, one expects that both self-focusing and photon emission should grow with increasing number of beam-foil collisions. This expectation is corroborated by our simulation results (see Figs. [3][4]. Note that, for efficient selffocusing, the distance between two consecutive foils needs to be sufficiently large to allow beam self-field restoration around its propagation axis (r σ ⊥ ). This requires that the travelled distance is much larger than σ ⊥ . Furthermore, the interfoil distance needs to be short enough to prevent beam expansion. This can be estimated by considering the effect of |f ⊥ | ≈ |2qB θ | calculated at x ≈ x 0 + vt and r ≈ σ ⊥ , i.e., where focusing is stronger. For σ ⊥ σ , CTR extends approximately over a distance σ 2 ⊥ /σ , which is larger than the beam length σ . Thus, |f ⊥ | lasts for approximately σ /c, and the deflection angle is ϑ ≈ |f ⊥ |σ /γm e c 2 . Hence, to prevent defocusing the interfoil distance must be much smaller than σ ⊥ /ϑ.
In our multifoil 3D PIC simulations, the electron beam has 2 nC charge, Gaussian spatial and momentum dis-tribution with σ = σ ⊥ = 0.55 µm, 10 GeV mean energy, 212 MeV full width at half maximum (FWHM) energy spread, and 3 mm-mrad normalized emittance (ϑ ≈ 2 mrad with these parameters). The beam collides with 20 consecutive aluminum foils with 0.5 µm thickness, 10 µm interfoil distance, and 1.8 × 10 29 m −3 initial electron density. The computational box size is 6.6 µm(x) × 8.8 µm(y) × 8.8 µm(z) with 528(x) × 352(y) × 352(z) gridpoints, 4 particles-per-cell (ppc) for beam electrons and 8 ppc for foil electrons and ions were used. Simulations were independently performed with Smilei [53,54] and CALDER [55] PIC codes with good agreement. The initial self-consistent beam fields, the effect of field and collisional ionization and binary Coulomb collisions were included. Synchrotron and bremsstrahlung emission, and multiphoton Breit-Wheeler and Bethe-Heitler pair production were implemented with state-of-the-art Monte-Carlo methods [43,54,56,57]. Consistently with the submicrometer foil thickness, simulations showed that collisional processes are negligible. Figure 3 displays snapshots of the electron and gamma beam evolution (see Supplemental Material for a movie). Until the 6 th foil, the beam interacts with the field "reflected" by each foil. This leads to the self-focusing of the beam which gradually increases its density (see the first to third column of Fig. 3). The electron beam density rises from its initial value of 4.7 × 10 27 m −3 to 8.2 × 10 28 m −3 after the 6 th foil, while the maximum photon density and χ are 2.9 × 10 28 m −3 and 0.8, respectively [see Fig. 3(a3)-(d3)]. In the interaction with the 7 th foil, the electron beam density reaches 3.8 × 10 29 m −3 , which exceeds the foil density of 1.8 × 10 29 m −3 . Hence, the foil is unable to reflect the fields of the beam, and a channel where foil electrons are expelled is created [see Fig. 3(b4)-(c4)]. Here χ and the photon beam density rise up to 3 and 4.1 × 10 29 m −3 , respectively. Moreover, a fraction of approximately 10 −4 photons with energies > 2m e c 2 convert into e − e + pairs via the multiphoton Breit-Wheeler process. Electron beam density stops increasing when it becomes larger than the foil electron density. In the following beam-foil collisions, the electron density beam profile undergoes longitudinal modulations [see Fig. 3(a5)-(a6)]. These modulations arise because the reflected field strength is dependent on the longitudinal position and stronger around the rear part of the beam, which yields a longitudinally inhomogeneous focusing force. Figure 4(a) plots the initial (black dashed line) and final (blue line) electron beam energy distribution after the interaction with 20 consecutive foils. The broad distribution around approximately 5 GeV results from intense synchrotron emission occurring in the central and rear part of the electron beam. The residual peak around the initial electron beam energy is indicative of the small synchrotron and collisional energy losses of the front part of the beam, which experiences only weak amplitude CTR. Figure 4(b) reports the final photon spectrum and the conversion efficiency η (inset) as a function of the number of crossed foils. The increase of η at the 7 th foil is due to the extremely high beam density and, consequently, to the ultrastrong fields induced inside the foil. After colliding with 20 foils, more than 30% of the electron beam energy is converted into a collimated (5 mrad rms photon energy angular distribution), 4 fs FWHM duration, 2.8 × 10 29 m −3 peak density gamma-ray pulse.
In summary, we have introduced a new scheme to efficiently produce extremely dense gamma-ray beams from the interaction of a high-current ultrarelativistic electron beam with a sequence of thin foils. This scheme also provides a promising route to producing solid-density ultrarelativistic electron beams and to exploring strong-field QED processes with a single electron beam without the need of an external powerful laser drive.
Acknowledgments -This work was performed in the framework of the E-305 collaboration. E-305 is a SLAC experiment whose aims include the generation of bright gamma rays, in particular in electron beam-solid interaction. This article comprises parts of the PhD thesis work of Archana Sampath to be submitted to the Heidelberg University, Germany.
The Let us consider an ultrarelativistic particle beam with cylindrical symmetry around its propagation axis and moving along x > 0 at a velocity v in the laboratory frame. In its rest frame, the beam density distribution is chosen as Here primed quantities refer to their values in the beam rest frame, n 0 = N e /(2π) 3/2 σ 2 ⊥ σ , and N e is the number of beam particles of individual charge q. We assume that the longitudinal and transverse (rms) beam dimensions fulfill σ = γσ σ ⊥ = σ ⊥ , where γ = (1 − v 2 /c 2 ) −1/2 is the beam relativistic factor. Hence, the beam self-electric field can be evaluated from Gauss' law in the long-beam limit which is valid for r σ , i.e., r γσ . An inverse Lorentz transformation yields the beam self-field in the laboratory frame Here we have introduced the beam density in the laboratory n 0 = γn 0 and, for simplicity, the time origin is chosen such that at t = 0 the beam center is located at x = 0. By using the Weizsäcker-Williams approximation [1] the beam electric self-field can be decomposed into plane waves (pseudophotons) whose electric components are
In the case of a perfect conductor with front surface at x = 0, the sum of the transverse incident ( E b ⊥ ) and induced/scattered conductor ( E c ⊥ ) electric fields should vanish at the boundary, so that The scattered field can then be exactly computed at any time and position in vacuum (x < 0) by using the plane-wave decomposition [3] We have introduced the longitudinal wavenumber In Eq. (S9), Fourier components with ω 2 /c 2 > k 2 ⊥ describe propagating waves, while components with ω 2 /c 2 < k 2 ⊥ correspond to evanescent waves confined to the conductor's boundary [4]. The choice of the minus sign in the definition of k x is consistent with waves propagating/damped along x < 0. Equation (S9) can be simplified by expressing the transverse wavenumber as k ⊥ = k ⊥ (cos φ r + sin φ φ) and performing the integration over φ, which gives This equation is accurate in the domain of validity of the field approximation in Eq. (S3), and therefore holds in the near beam region that is of central interest here. The longitudinal electric field follows from Gauss' law in vacuum [5], yielding Finally, the induced magnetic field is given by Faraday's law and by using Eq. (S10) one obtains Figures S1 and S2 display 2D maps of the conductor electric (a)-(b) and magnetic (c) fields as computed numerically from Eqs. (S11), (S13), and (S15). We first consider an ultrarelativistic electron (q = −e) beam with γ = 2 × 10 4 , σ ω p /c = 2 and σ ⊥ ω p /c = 6, where ω p = 4πn 0 e 2 /m e is the beam plasma frequency. These fields are compared with the radial electric field of the beam's image charge E r r (d) (the "reflected field"), obtained by setting q → −q and v → −v in Eq. (S3). At t = 0 (see Fig. S1 and B c θ are very similar to the reflected field around the boundary (E c r ≈ −B c θ ≈ E r r ≈ 2.7 m e cω p /e). The induced longitudinal electric field has a maximum amplitude slightly lower (E c x ≈ −2.4 m e cω p /e), reached at r = 0 (in contrast to the transverse fields which peak at r ≈ 1.6σ ⊥ ≈ 9.5cω p ). At ω p t = 4 (see Fig. S2), the transverse induced fields detach from the boundary as is consistent for a propagating wavepacket. Yet diffraction causes these radiated fields to drop in amplitude (E c r ≈ 2.4 m e cω p /e and B c θ ≈ −2.3 m e cω p /e), and to reach their maxima at a larger radius (rω p /c ≈ 11) than the reflected field. The longitudinal induced field remains mainly confined to the boundary and has been intensified (E c x ≈ −3.8 m e cω p /e). Around the boundary and r ≈ σ ⊥ , the magnetic field is then mainly driven through ∂B c θ /c∂t ≈ ∂E c x /∂r > 0, and so becomes of positive polarity [see Fig. S2(c)] Figures S3(a)-(d) plot the same quantities as Figs. S1-S2 but for a spherical beam (σ = σ ⊥ = 2c/ω p ) and at t = 0. The scattered fields show a substantial divergence because the dominant wavenumber λ ∼ σ is close to the transverse beam size σ ⊥ . This translates into a longitudinal electric field E c x about twice stronger than E c r . The amplitude of the scattered magnetic field is close to, yet a bit lower than the beam self-field (B c θ ≈ −0.64 m e cω p /e vs. B b θ ≈ 0.90 m e cω p /e). The case of a longitudinally elongated beam (σ ω p /c = 6, σ ⊥ ω p /c = 1) is illustrated at two successive times in Figs. S4 and S5. The small σ ⊥ /σ ratio leads to most of the induced fields being evanescent, and thus strongly localized at the conductor's boundary. At both t = 0 (see Fig. S4) and ω p t = 5 (see Fig. S5), the longitudinal electric field exceeds its transverse counterpart. In parallel, the induced magnetic field gets considerably weaker than the magnetic self-field of the beam, and changes polarity along the boundary [see Fig. S5(c)].

B. Evaluation of the surface magnetic field in the radiating and nonradiating regimes
We now demonstrate that the magnetic field induced at the boundary equals the beam self-field when σ ⊥ σ and tends to vanish when σ ⊥ σ . To this purpose, we start by the expression of B c θ at x = 0 − B c θ (0 − , r, t) = − √ 8π n 0 q vc σ σ 2 When σ ⊥ σ , the dominant integration range over k ⊥ is such that k 2 ⊥ ω 2 /c 2 ≈ 2/σ 2 , so that the scattered field mainly consists of propagating waves. Approximating k x ≈ −ω/c yields It is easily seen that the above approximate expression of B c θ (0 − , r, t) exactly coincides with −E c r (0 − , r, t) from Eq. (S11). Since by construction E c r (0 − , r, t) = −E b r (0 − , r, t) = −B b θ (0 − , r, t), we conclude that B c θ (0 − , r, t) = B b θ (0 − , r, t) in the radiating regime. The calculation is more involved when σ ⊥ σ . By using Eq. (S10), we first decompose the integral over k ⊥ in Eq. (S16) as (S18)