Self-absorption of synchrotron radiation in a laser-irradiated plasma

Electrons at the surface of a plasma that is irradiated by a laser with intensity in excess of 1023 Wcm−2 are accelerated so strongly that they emit bursts of synchrotron radiation. Although the combination of high photon and electron density and electromagnetic field strength at the plasma surface makes particleparticle interactions possible, these interactions are usually neglected in simulations of the high-intensity regime. Here we demonstrate an implementation of two such processes: photon absorption and stimulated emission. We show that, for plasmas that are opaque to the laser light, photon absorption would cause complete depletion of the multi-keV region of the synchrotron photon spectrum, unless compensated by stimulated emission. Our results motivate further study of the density dependence of QED phenomena in strong electromagnetic fields. ∗ tom.blackburn@physics.gu.se 1 ar X iv :2 00 5. 00 30 2v 1 [ ph ys ic s. pl as m -p h] 1 M ay 2 02 0


I. INTRODUCTION
Radiation emission from accelerated electrons is a ubiquitous feature of regions of strong electromagnetic field. In astrophysical environments [1], or in laser-matter interactions at the highintensity frontier [2], the fields can be so strong that the interactions must be described within the framework of quantum electrodynamics (QED) [3][4][5]. Experiments at the next generation of high-intensity laser facilities [6][7][8] will produce high-energy γ rays via quantum synchrotron emission (also called nonlinear Compton scattering) in a variety of geometries [9][10][11][12][13]. Particle-in-cell (PIC) simulations, extended to include the one-particle to two-particle ('1 to 2') strong-field QED processes of photon emission and electron-positron pair creation [14,15], play an essential role in modelling these interactions. However, for every emission process, there is a corresponding absorption process. To date, the inverse ('2 to 1') processes of one-photon absorption [16] and pair annihilation to one photon [17,18] have been neglected in PIC simulations.
Here we consider the effect of one-photon absorption in a scenario where the photons are absorbed by the same population of relativistic electrons that emitted them. In an astrophysical context, this phenomenon is known as synchrotron self-absorption [19]. It leads to a steep cutoff at low frequency in the emission spectra from, e.g., supernovae [20], gamma-ray burst afterglows [21,22], and black hole X-ray binaries [23]. In principle, the irradiation of a solid target by a laser of intensity 10 23 Wcm −2 is a platform for studying self-absorption, because of the combination of strong electromagnetic field, high electron density, and high photon density at the plasma surface. A consistent treatment of photon absorption must include stimulated emission, which is the competing, induced process. To do so, we construct a cross section for stimulated emission in QED that is valid within the locally constant, crossed fields approximation; to the best of our knowledge, a cross section from QED has not previously been reported. We present an implementation of both processes as binary interactions between macroparticles in a PIC code.
Simulating a laser-plasma interaction, we find that while photon absorption suppresses the multi-keV region of the synchrotron spectrum, this suppression is countered by stimulated emission.
Our results demonstrate that it is feasible to include particle-particle interactions in studies of laser-driven plasmas.

II. INDUCED PROCESSES
The following master equation determines the evolution of the number of photons, N(k), with momentum k [24]: Here w(p, k) is the rate at which an electron with momentum p emits photons with momentum k and f (p) is the electron distribution function, defined by dN e = f (p) d 3 p/(2π) 3 . (We use units such thath = c = 1 throughout). The first term in square brackets on the RHS of eq. (1) describes 'spontaneous emission', which is the quantum synchrotron emission already included in laserplasma simulations [14,15]. The following two terms correspond, respectively, to the induced processes of stimulated emission and photon absorption. Unlike spontaneous emission, they depend on the density of photons already present. All three processes depend on the electron and photon momenta, p µ and k µ , and the strength of the electromagnetic field F µν , which is implicit in w(p, k).
Conservation of momentum means that an electron in vacuum cannot absorb radiation without some associated emission of radiation. Absorption can occur, however, for an electron in a background electromagnetic field F µν (where the required emissions appear as 'absorption' of negative frequency modes from the background [16]). If the field is weak compared to the critical field of QED, E cr = m 2 /e [25,26], and if it varies sufficiently slowly such that quantum processes can be considered to be instantaneously constant, the interaction is controlled by the quantum parameters χ e = F µν p ν /(mE cr ) and χ γ = F µν k ν /(mE cr ), where p and k are the electron and photon momenta, e is the elementary charge and m is the electron mass.
The rates of absorption and stimulated emission, the number of events per unit volume and time, may be expressed in terms of the invariant flux F = n e n γ k.p/(k 0 p 0 ) and the relevant cross section σ , where n e and n γ are the electron and photon number densities. The cross sections can be obtained by substituting into eq. (1) w(p, k) = (2π) 3 2V dW d 3 k , where dW d 3 k is the triple-differential rate of photon emission (as given in [27] for a constant, crossed field) and V is a volume factor, and dividing through by the flux F. We find where s = χ γ /χ e andz = (2z/s)(k.p/m 2 ) for both processes. In the remaining two auxiliary variables, g = 1/2 + s 2 /[4(1 ± s)] and z = {s/[χ e (1 ± s)]} 2/3 , choosing the positive (negative) sign yields the cross section for absorption (stimulated emission). The cross section for absorption, obtained in this way, agrees with the result of a direct calculation from strong-field QED [16]. To the best of our knowledge, a QED cross section for stimulated emission has not previously been reported.
This result is obtained in the locally constant, crossed field approximation (LCFA), under which the rate for a QED process in an arbitrary background field may be replaced with its equivalent in a constant, crossed field [28]. The validity of this approximation depends on the normalized field amplitude a 0 = eE 0 /(mω 0 ), where E 0 is the electric field strength and ω 0 is the field's frequency of oscillation. The LCFA holds for the '1 to 2' processes of Compton scattering [29][30][31][32][33] and nonlinear Breit-Wheeler pair creation [34] if a 0 satisfies a 0 1 and a 3 0 /χ e,γ 1, as under these conditions the formation length is much smaller than the scale of variation of the background field.
In a pulsed background, however, there are always temporal regions where the local value of a 0 is small, and hence the assumptions of the LCFA are automatically violated. Compton scattering and Breit-Wheeler pair creation 'self-regulate' in this situation [33]; while the fractional error in the rate is large in such regions, the rate itself is small (in fact, vanishing) due to the behaviour of the Airy functions appearing there, and thus the absolute error is small. The question arises as to what extent these statements apply also to induced processes, which depend on additional kinematic variables.
A comparison of the LCFA for one-photon absorption eq. (2) with the full QED result [16] in a monochromatic plane-wave background shows good agreement for s χ e /a 3 0 . Absorption is, though, more likely in regions where a 0 is not large. In very short pulses, these regions can contribute a significant proportion of the total probability [16]. However, note that [16] benchmarked absorption using externally injected photons, which overlap with the electrons even in free space.
Here we consider photons that are emitted by the electron population itself, so that overlap takes place only in the high-field region, a 0 1, where emission is most likely. As the LCFA is satisfied for the emission process in this regime, and emission and absorption take place in the same region of space, it should also be satisfied for the absorption process.
Emission of a photon by an electron, followed by absorption of that photon by another electron, may be viewed as the component of Møller scattering (ee → ee, in a strong field) in which the intermediate photon is real. A complete treatment of electron-electron scattering in a background field would include off-shell and interference contributions; this has been done for monochromatic [35][36][37][38] and pulsed electromagnetic waves [39] at low intensity a 0 1, with particular focus on resonances in the transition amplitude. These resonances occur when the intermediate photon goes on shell, which significantly enhances the interaction probability over its value in vacuum. This is precisely the interaction under consideration here. It should dominate the virtual component, i.e. direct electron-electron scattering, which is, in its usual classical description [40], negligible for laser-plasmas.

A. Analytical estimates
Let us first determine the laser and plasma parameters for which one-photon absorption becomes important. Consider a population of electrons, with number density n e , performing a circular orbit with Lorentz factor γ, quantum parameter χ e and gyroradius R c = γ 2 /(mχ e ). Let the space be filled by photons with number density n γ , quantum parameter χ γ and energy ω, all propagating in the same direction and in the plane of the electron orbit.
Defining θ to be the angle between the electron and photon momenta and assuming γ 1 and θ 1, the argument of the Airy function in eq. (2) may be cast asz θ 2 /θ 2 c , for θ c = [mχ e /(γ 2 ω)] 1/3 . This shows that the cross section is suppressed for θ > θ c , i.e. unless the electron and photon are almost collinear, so it occurs once per orbit. In general, both absorption and stimulated emission are likeliest for low-energy photons propagating at small angles to the electron trajectory.
The number of events per unit volume n abs = Fσ (t)dt, where F = n e n γ k.p/(k 0 p 0 ) is the invariant flux, σ (t) the instantaneous cross section, and the integral is taken over the interval where p is close to parallel with k. Assuming that s = χ γ /χ e 1 and the angle between electron and photon θ (t) = t/R c 1, we obtain We integrate eq. (3) using the fact that for ξ 1. The fraction of photons absorbed by the electrons is given by: In the case that the electrons are in a plasma that is driven by a circularly polarized laser with angular frequency ω 0 , we can set χ e = γ 2 ω 0 /m and express the density n e in terms of the critical density n cr = mω 2 0 /(4πα). We define the self-absorption frequency ω abs as the largest frequency for which the absorption fraction f abs 1: Photons with energies smaller than ω abs , which lies in the multi-keV range for overdense plasmas, should be efficiently absorbed. Note that there is no dependence on the laser intensity.
The laser intensity does, however, play a role, in that the origin of the photons that are to be absorbed is electron synchrotron radiation, which only becomes substantial if the laser intensity is sufficiently high [10][11][12]. We now estimate the properties of this emission for the scenario of a laser-irradiated, overdense plasma. Only electrons within the skin layer are exposed to strong electromagnetic fields; the effective value of the laser amplitude is reduced by screening from a 0 , its value in vacuum, to a eff a 0 n cr /n e [10]. (This result strictly applies only in the nonrelativistic limit [41], but it is consistent with the simulation results to be presented.) Electrons are accelerated on segments of circular trajectories, with Lorentz factor γ a eff , and emit synchrotron radiation with a characteristic frequency of ω cr γ 3 ω 0 . We expect the LCFA to be valid for the emission and absorption of photons that satisfy s > χ e /a 3 0 , which is equivalent to ω > ω 0 . This is satisfied for both the self-absorption frequency ω abs and the characteristic frequency of emission ω cr : with n e = 100n cr and a 0 = 400, for example, γ a eff 40, χ e = γ 2 ω 0 /m 5 × 10 −3 and ω cr 100 keV. The treatment of synchrotron radiation as incoherent requires that the frequencies of interest ω > ω coh , where ω coh = n 1/3 e is an upper limit for the onset of coherence effects [15]. Both ω abs and ω cr meet this requirement by at least a factor of two.
The cross sections for stimulated emission and absorption are similar in magnitude for s 1 [42]. The balance between the two is determined by the gradient in momentum space of the electron distribution function: net absorption occurs when this is negative, i.e. there are more electrons at lower energy than at higher energy [24]. This dependence on the electron distribution function means that we turn to numerical methods, i.e. particle-in-cell simulations.
Particle-in-cell simulations now incorporate both the quantum emission and absorption of synchrotron radiation, in addition to classical, relativistic plasma dynamics. In this work, emission is modelled in the usual Monte Carlo approach [14,15] by integrating the LCFA rate [3,28] along the electron trajectory and sampling the quantum synchrotron spectrum. We use a spectrum that is differential in both energy and scattering angle [27,43]. Absorption and stimulated emission are incorporated as a binary interaction between macroparticles. Each macrophoton (index i) is initialized on creation with optical depths against absorption and stimulated emission τ i ∼ exp(−τ i ), where = abs, stim. At every timestep, the interaction probability P i j is calculated for all pairwise combinations of macroelectrons j and macrophotons i that are located in the same grid cell, using the cross sections given in eq. (2): P i j = w j (c∆t/V )(k i .p j /k 0 i p 0 j )σ , where w j is the macroelectron weight, ∆t is the timestep, V is the volume of a grid cell, k is the four-momentum of the photon, and p is the four-momentum of the electron.
While the cross sections eq. (2) are derived for a plane electromagnetic wave in the constant field limit, it is applied to arbitrary background fields in our code. To do so, we replace s → k 0 /p 0 in the factor ofz/z appearing in the prefactor. (Elsewhere it remains s = χ γ /χ e .) The purpose of this change is to guarantee that the cross section is positive. We have verified that it does not change the final results of our simulations, as eq. (2) is strongly suppressed unless the electron and photon are almost collinear.
The macrophoton's optical depths are updated as τ i → τ i − P i j for each electron (index j), until one of τ i falls below zero. If absorption occurs (τ abs j < 0), the macroelectron momentum is updated as p j → p j + w i k i /w j , where w i is the weight of the macrophoton, and the macrophoton is deleted from the simulation. If stimulated emission occurs (τ stim j < 0), the macroelectron momentum is updated as p j → p j − k i and a new macrophoton with momentum k i and weight w j is added to the simulation. Should both optical depths fall below zero simultaneously, a pseudorandom number U is drawn on the unit interval and absorption selected if U < P abs i j /(P abs i j + P stim i j ); otherwise stimulated emission is selected. Benchmarking against analytical results are given in appendix A.

C. Results
As an example, we simulate the interaction of a 10-fs (fwhm duration), circularly polarized laser pulse with a slab of fully ionized carbon plasma, density n e = 100n cr and thickness 5.0 µm, at normal incidence. The laser amplitude is a 0 = 400 and its wavelength λ = 800 nm, which yields an electron density of 1.7 × 10 23 cm −3 . The simulation is performed in 1D, with 1000 cells per micron and 200 particles per cell for each species.
The y components of the incident and reflected electromagnetic field, as well as the electron, ion and photon number densities at t = 13.3 fs are shown in fig. 1(a). (Time t = 0 corresponds to the centre of the laser pulse crossing x = 0, the location of the unperturbed vacuum interface.) Electrons near the plasma surface are accelerated on circular orbits by the laser fields, with perpendicular momenta p ⊥ ma eff , as shown in fig. 1(b), and displaced by the radiation pressure in the x-direction, as shown in fig. 1(c). This establishes a charge-separation field that accelerates the ions in turn. In the steady state, the velocity of the hole-boring front is  are absorbed, 90% are absorbed before they have propagated a distance of 10 nm. If the radiation escapes the skin layer, it is highly unlikely to be absorbed thereafter.
The radiation spectrum at the end of the simulation, when the plasma is no longer driven by the laser, is shown in fig. 2. As emission takes place when the electron momentum is instantaneously perpendicular to the laser fields, in the rest frame of the plasma surface, we expect the synchrotron radiation to appear predominantly at polar angles θ satisfying cos θ β hb , where β hb is the holeboring velocity. This is confirmed by fig. 2(b) and (c), which show the radiation spectrum as a function of energy and polar angle. (θ = 0 corresponds to forward emission, i.e. parallel to the laser wavevector.) There is a significant reduction in the number of multi-keV photons when one-photon absorption is taken into account. The threshold energy at which the spectrum is suppressed is consistent with our theoretical estimate eq. (6), substituting n e /n cr = 100. However, this suppression is countered by stimulated emission, leading to a photon spectrum that is almost identical to the 'spontaneous emission only' result. (When both absorption and stimulated emission are included, the photon spectrum is effectively resampled at every timestep, leading to increased statistical noise.) In astrophysical scenarios, it is expected that net absorption causes the spectrum to be suppressed as ω 5/2 at ω ω cr [22], assuming that the electron population has a power-law distribution of energies dN e /dγ ∝ γ −p (p > 0) and that each electron emits and absorbs at a single frequency ω cr (γ). This is not observed here, as the electron perpendicular momentum distribution shown in fig. 1(b), while having negative gradient, is not sufficiently broad.

IV. CONCLUSIONS
In this paper we have considered the interplay between the standard strong-field QED process of nonlinear Compton scattering, or spontaneous photon emission, and the particle-particle processes of absorption and stimulated emission. By constructing cross sections for these processes within the same scheme (based on the locally constant field approximation) used for spontaneous emission, we have shown that it is feasible to include induced, particle-particle processes in simulations of laser-plasma interactions. This allows us to capture phenomena that are primarily dependent on density. While photon absorption occurs prolifically for multi-keV synchrotron photons in a laserplasma interaction, net absorption is weak because of stimulated emission. Our results motivate investigation into the density dependence of QED phenomena in strong fields, which adds a new axis to the standard parameter space of intensity (a 0 ) and energy (χ e,γ ).
eA(φ ) = ma 0 sin(φ ) cos 2 (πφ /L) for phases |φ | < L/2, where L = 4π. A beam of electrons, with initial energy γ 0 m and density n e , and a beam of photons, with energy ω and density n γ , are injected into this pulse: the electron beam counterpropagates into the laser pulse, and we vary the initial angle between the photon beam and laser wavevector θ 0 . (θ 0 = 0 corresponds to the electron and photon beams being initially parallel to one another, i.e. both counterpropagating to the laser.) A suitable observable is the fraction, f abs , of photons absorbed from the initial beam. Analytically, this is given by where σ int = 1 L L/2 −L/2 σ (φ ) dφ is the integrated cross section (Eq. 32 in [16]) and τ = L/ω 0 is the laser duration. In fig. 3 we compare the fraction of absorbed photons eq. (A1) using σ int calculated analytically from [16], with that obtained numerically by the PIC simulations outlined in section III B. To ensure a fair comparison, photon emission (both spontaneous and stimulated) and current deposition are disabled in the simulations.
The results of our PIC implementation (points) show excellent agreement with the analytical predictions (solid lines) over parameter scans in the field strength a 0 , initial photon beam angle θ 0 , and energy ω. In particular, the PIC implementation correctly resolves the peak structure seen in the dependence of the absorbed fraction f abs on the field strength a 0 . These peaks arise when the electrons and photons are brought into alignment at a local maximum of the field amplitude, i.e. when the instantaneous angle between the electron momentum and the laser wavevector, θ e (φ ) eA(φ )/(mγ 0 ), satisfies θ e (φ ) = θ 0 , at a phase φ where ∂ φ A(φ ) = 0. For the two-cycle pulse under consideration here, the matching condition is γ 0 θ 0 /a 0 = 0.211 and 0.870.
The densities employed to generate fig. 3, n e = n γ = 10 34 m −3 , are sufficiently high that ignoring current deposition is unphysical. However, as discussed in the main text above, one can alleviate this problem by considering the absorption of synchrotron photons generated in the hole-boring regime. The simulations discussed in the main text do include the fields generated by the plasma.