Asymptotic electron motion in strong radiation-dominated regime
aa r X i v : . [ phy s i c s . p l a s m - ph ] J u l Asymptotic electron motion in strong radiation-dominated regime
A. S. Samsonov, E. N. Nerush, ∗ and I. Yu. Kostyukov Institute of Applied Physics of the Russian Academy of Sciences,46 Ulyanov St., Nizhny Novgorod 603950, Russia (Dated: July 12, 2018)We study electron motion in electromagnetic (EM) fields in the radiation-dominated regime. It isshown that the electron trajectories become close to some asymptotic trajectories in the strong fieldlimit. The description of the electron dynamics by this asymptotic trajectories significantly differsfrom the ponderomotive description that is barely applicable in the radiation-dominated regime. Theparticle velocity on the asymptotic trajectory is completely determined by the local and instant EMfield. The general properties of the asymptotic trajectories are discussed. In most of standing EMwaves (including identical tightly-focused counter-propagating beams) the asymptotic trajectoriesare periodic with the period of the wave field. Furthermore, for a certain model of the laser beamwe show that the asymptotic trajectories are periodic in the reference frame moving along the beamwith its group velocity that may explain the effect of the radiation-reaction trapping.
I. INTRODUCTION
If the amplitude of an optical field is such that an elec-tron gains in it energy of hundreds of its rest-mass en-ergy, the electron starts to emit synchrotron radiationand can lose its energy efficiently [1]. This phenomenon— radiation reaction — is highly important for theoreti-cal physics and astrophysics, therefore the motion of elec-trons in strong laser field nowadays is a topic of numeroustheoretical investigations [2–7], and it has been studiedrecently in the experiments [8, 9]. Also, the emission ofhard photons by electrons in a strong laser field lets one tomake a femtosecond broadband source of MeV photons,based on either laser pulse – electron beam collision [10–12], laser-plasma interaction [13–17] or electromagneticcascades [18, 19].In the interaction of a strong laser pulse with a plasma,radiation losses can significantly affect the plasma dy-namics, and, for instance, lead to less-efficient ion ac-celeration [20–24], the enhancement of the laser-drivenplasma wakefield [25, 26], highly efficient laser pulse ab-sorption [27], relativistic transparency reduction [28], andto the inverse Faraday effect [29].Despite of high importance of the radiation losses forlaser-plasma physics at high intensity, there is no generalconcept of the losses impact on the electron motion, andthis impact is considered mostly by ad hoc hypothesesand particle-in-cell (PIC) simulations. Only for a fewfield configurations the analytical solutions for motion ofemitting electron are present [30–32], whereas for the mo-tion of the non-emitting electron the Miller’s ponderomo-tive concept [33] is applicable in a vast number of cases.In the high-intensity field, the energy gained by theelectron can be significantly limited by the radiationlosses. In this case, in contrast to the low-intensity limit,the electron Lorentz factor becomes small in comparisonwith the field amplitude: γ/a ≪ ; here γ is the electron ∗ [email protected] Lorentz factor and a = eE /mcω is the normalized am-plitude of the electric field, E , ω is the typical angularfrequency of the field, c is the speed of light, m and e > are the electron mass and the magnitude of the electroncharge, respectively. The smallness of γ/a allows one tosimplify the analytical treatment of the electron motionin the strong radiation-dominated regime. This can beillustrated by a stationary Zel’dowich’s solution [30] forthe electron motion in the rotating electric field E ( t ) . Atmoderate field intensity the angle ϕ (between the particlevelocity and the vector − E ) is connected with the elec-tron Lorentz factor γ . However, in the strong radiationdominated regime ϕ and γ/a tend to zero (see Fig. 1),and the particle velocity coincides with the direction ofthe electric field ( v k E ), thus γ is not needed in order tocompute the particle trajectory.In Refs. [34, 35] the concept of the electron motion,that in the radiation-dominated regime can supersedethe ponderomotive concept, is discussed. It have beenshown in Ref. [34] that in the regime of dominated ra-diation friction the number of degrees of freedom, whichgovern the electron motion, is reduced. Namely, it isshown for the rotating electric field with the Gaussianenvelope, that on the time scales larger than the rotationperiod, the electron position is described by a first-orderdifferential equation that does not contain the electronmomentum. For this, the electron motion with Landau–Lifshitz radiation reaction have been considered. It isalso shown that in the radiation-dominated regime, elec-trons are not expelled from but are captured for a longtime by the strong-field region.In Ref. [35] it is shown for almost arbitrary field config-uration, that in the strong field limit in a timescale, muchsmaller than the timescale of the field variation, the di-rection of the electron velocity approaches some certaindirection that is determined only by the values of the lo-cal electric and magnetic fields. Then, as the electronvelocity is known, the electron trajectory can be recon-structed. This approach, called the “low-energy limit”,was used (but not described) for the fields of the linearlypolarized standing waves earlier [36]. / π / a ₀ a ₀ ₀ Figure 1. Electrons moving in the rotating electric field andexperiencing quantum radiation reaction for many periods ofthe rotation: (circles) the ratio of the mean Lorentz factor γ m to a and (triangles) the mean angle ϕ m between theparticle velocity and the vector opposite to the electric field,for different values of the field amplitude a . Bars depict thestandard deviations ± σ . Results of PIC-MC simulations forthe field angular frequency ω = 2 πc/λ , λ = 1 µ m. Let us emphasize that if the electron velocity is deter-mined not by the electron momentum but by the localfields, one can describe the plasma dynamics with hydro-dynamical equations. Indeed, in this case the currentsin the Maxwell’s equations depend only on the particledensity and particle velocity (i.e. on the particle densityand local EM fields), therefore the first-order equation forthe electron position together with the Maxwell’s equa-tions and the continuity equation form a closed systemof equations.In this paper we present the first step toward sucha hydrodynamical approach to the plasma dynamics inthe radiation-dominated regime. Namely, in Sec. II weestimate γ/a ratio and the threshold of the radiation-dominated regime. In Sec. III for arbitrary field con-figuration we find the first-order equation for the elec-tron position, by a method different from Ref. [35] andwith B-case (see below) considered separately. The right-hand-side of this equation is the velocity field that is fullydetermined by the local field vectors. It is shown that γ ≪ a is enough for this first-order equation to be validin the laser field. In Sec. IV we compare the solution ofthis first-order equation with the solution of the exactequations of the electron motion for a number of fieldconfigurations. In Sec. V we discuss the relation betweenthe velocity field and the Poynting vector. In Sec. VI thesymmetry of the velocity field induced by the symmetryof the Maxwell’s equations, is considered, and the dra-matic difference between the ponderomotive descriptionand the description by the velocity field in the radiation-dominated regime is demonstrated. Thus, in the subsec-tion VI A, in the limit of strong fields, the electron motionin a wide class of periodic standing waves is shown to beperiodic. From this, in the subsection VI B we show witha certain model of the laser beam that the beam can cap- ture the electrons and carries them along itself with thebeam group velocity. Sec. VII is the conclusion. II. STRONG RADIATION-DOMINATEDREGIME
In order to estimate the threshold value of the nor-malized field amplitude a for the radiation-dominatedregime, let us start from the equations of the electronmotion with the Landau–Lifshitz radiation reaction forceincorporated: d p dt = − E − v × B − F rr v , (1) dγdt = − vE − F rr v , (2)where time is normalized to /ω , v is the electron ve-locity normalized to the speed of light c , p = γ v is theelectron momentum normalized to mc , E and B are theelectric and the magnetic fields respectively (normalizedto mcω/e ), and F rr v is the main term of the radiationreaction force [37]: F rr = αγ ~ ωmc (cid:8) ( E + v × B ) − ( Ev ) (cid:9) . (3)Here α = e / ~ c ≈ / is the fine-structure constantand ω is the frequency characterizing the time-scale orthe space-scale of the field (e.g. angular frequency of thelaser field).The radiation losses increase sharply with the increaseof γ , therefore for some electron Lorentz factor γ = ¯ γ ,the further energy gain stops due to the losses. The cor-responding value ¯ γ can be found from Eq. (2) assumingthat the transverse and the longitudinal to v componentsof the Lorentz force are of the order of a : ¯ γ ≈ s αa mc ~ ω , (4)where a is the characteristic electric field strength. Inthe absence of the radiation reaction the electron en-ergy in the field can be estimated as γ ∼ a , thus theradiation-dominated regime corresponds to ¯ γ ≪ a hence a ≫ a ∗ = (cid:18) α mc ~ ω (cid:19) / . (5)Note for the laser wavelength λ = 1 µ m the ampli-tude a ∗ ≈ that corresponds to the intensity I ≈ × W cm − . This level of intensity is expectedto be reached in the near future with such facilities asELI-beamlines [38], ELI-NP [39], Apollon [40], Vulcan2020 [41], or XCELS [42].In the case of strong radiation losses the angle betweenthe Lorentz force and the electron velocity can be small,and the transverse to v component of the Lorentz forcebecomes much lower than the longitudinal one. How-ever, this doesn’t affect much the given estimates. Forinstance, for the electron motion in the rotating elec-tric field from the stationary Zel’dowich’s solution [30]we get ϕ ≈ γ/a and γ ≈ ( a /µ ) / ≪ a (where µ = 2 α ~ ω/ mc ) at a ≫ a ∗ , with the same estimatefor a ∗ (except the factor / in the parentheses, seeRef. [30]). Note also that the quantum consideration ofthe radiation reaction gives results that are close to theZel’dovich’s solution: in the Monte Carlo (MC) simula-tions the mean ϕ value is about π/ times larger than γ/a , and γ/a drops with the increase of the field am-plitude (Fig. 1).In what follows we assume that the field is far beyondthe threshold of the radiation-dominated regime, a ≫ a ∗ . III. VELOCITY FIELD AND ASYMPTOTICTRAJECTORIES
The reduced equations of the electron motion for arbi-trary field configuration can be derived as follows. Theequation for the electron velocity can be obtained fromEqs. (2) and (1), and is the following: d v dt = − γ (cid:26) E − v ( vE ) + v × B + F rr v γ (cid:27) , (6)where the first three terms in the parentheses approxi-mately correspond to the transverse to v component ofthe Lorentz force.If the angle ψ between the Lorentz force and theelectron velocity is noticeable ( ψ ∼ ), then the termwith F rr in Eq. (6) is negligible, because F rr /γ ∼ a α ~ ω/mc ≪ a for reasonable field amplitudes, E . E S /α , where E S = m c /e ~ is the Sauter–Schwingercritical field. Thus, as far as γ ≪ a , we have | d v /dt | ≫ . It means that the characteristic timescale of the veloc-ity vector variation is small, τ v ∼ γ/a ≪ . Therefore,on small time scales it can be assumed that the fields E and B in Eq. (6) are constant. In the constant EM fieldthe electron velocity v in a time of some τ v approachessome asymptotic direction. This direction correspondsto ψ → hence d v /dt = 0 , and can be found as follows. A. B-case
In the case E · B = 0 and B > E there is a refer-ence frame K ′ in which the field is purely magnetic, and B ′ k B (here strokes denote quantities in K ′ ). In K ′ theelectron goes along the helical path with its axis parallelto the direction of B ′ . The corresponding drift velocityof the electron in the laboratory reference frame K is thespeed of K ′ in K and can be found from the followingequation: E + v × B = 0 . (7) Let us note that Eq. (7) does not depend on the compo-nent of the velocity parallel to the magnetic field, so onecan choose this component arbitrarily (implying v < ).One can choose, for example, the solution with v · B = 0 ,i.e.: v = E × B B . (8)As shown in Sec. VI B the ambiguity of v in this case canbe resolved by additional physical considerations. B. E-case If E · B = 0 or E > B there is a reference frame K ′ , inwhich E ′ k B ′ or B ′ = 0 . The electron trajectory in K ′ asymptotically approaches the straight line parallel to E ′ ,and v approaches . Note that for the resulting electrontrajectory v · E < as far as the electron is acceleratingby the field.As v ≈ and the electron moves along the straightline, in the laboratory reference frame K the resulting v can be found from the equation d v /dt = 0 , that yields E − v ( vE ) + v × B = 0 , (9)Scalar multiplication of Eq. (9) by B , E and E × B leadsto the following solution: vB = EBvE , (10) v · E × B = E − ( vE ) , (11) vE = − s E − B + p ( E − B ) + 4( EB ) , (12) v · E × [ E × B ] = ( vE )( EB ) − ( vB ) E . (13)The right-hand-side of Eq. (12) is relativistic invariant,and we choose the sign “ − ” in order to obtain the stabletrajectory in K ′ . For the opposite sign, “ + ”, the electronin K ′ is decelerating and its velocity is reversed quicklyif initially v is not exactly parallel to the direction givenby Eq. (9). Note that vectors E , E × B , E × [ E × B ] form an orthogonal basis thus Eqs. (11)–(13) are enoughto determine v unambiguously. C. Asymptotic trajectory
Considering the electron motion on a timescale of thefield variation timescale, t ∼ ≫ τ v , one can neglectthe dynamics of the electron while it is approaching theconstant-field-approximation asymptotic solution, andassume that in every time instant the electron velocityis determined by Eq. (7) or Eq. (9) which depend onlyon the instant (and local) fields. Thus, the electron tra-jectory is governed by the following reduced-order equa-tions: d r dt = v , (14) E − v ( vE ) + v × B = 0 , (15)where the last equation determines the velocity field v and can be used in both B- and E-cases (in B-case ityields Eq. (7)). From here on we call the solution ofEqs. (14)–(15) “asymptotic trajectory” because, first, lo-cally it corresponds to the asymptotic ( t → ∞ ) electrontrajectory in the constant-field-approximation, and, sec-ond, it describes the electron trajectory in asymptoticallystrong field ( a ≫ a ∗ ).Note that the reasoning about the electron trajectoryin the radiation-dominated regime is also valid if the pa-rameter χ is large ( χ ≈ γF ⊥ /eE s , see Ref. [43, 44], where F ⊥ is the component of the Lorentz force perpendicularto the particle velocity). In this case ( χ ≫ ) the syn-chrotron emission is described by the quantum formulaeand Eq. (3) is not valid, however, it is still possible todescribe the electron trajectory classically between thephoton emission events [43, 45] because ℓ f ≪ ℓ W . Here ℓ f ∼ mc /F ⊥ is the radiation formation length, i.e. thedistance within which the emission of a single photonoccurs, and ℓ W ∼ c/W is the mean distance that theelectron passes without the photon emission; W is thefull probability rate of the photon emission. Estimating W ∼ mce ~ χ / γ , (16)we obtain ℓ f /ℓ W ∼ α/χ / < / ≪ . Therefore, theelectron moves classically between the short events of thephoton emission. Note also that for optical frequencies ℓ W /λ ∼ ~ ω/ ( αχ / mc ) ≪ . IV. SIMPLE EXAMPLES
In order to test the asymptotic description of the elec-tron trajectory (Eqs. (14) and (15)) we compare numer-ical solutions of them with numerical solutions of theclassical equations of the electron motion with the radi-ation reaction taken into account by the inclusion of theLandau–Lifshitz force [37] or by the recoil of the emit-ted photons described in the quasiclassical framework ofBaier–Katkov [43, 45]. Numerical solution of the fullequations of the electron motion is based on the Vay’spusher [46] where the Landau–Lifshitz force is taken intoaccount with the Euler’s method or, alternatively, thequantum recoil is taken into account by the Monte Carlo(MC) technique similarly to the QUILL [47, 48] code (seealso Appendix A). In order to solve Eqs. (14)–(15) we usethe classical Runge–Kutta method. The test results forvarious field configurations are present below. y x v(x, y) a = 500 a = 2 ∙
10³ a = 10 ⁴ MCLL
Figure 2. Velocity field Eq. (15) (arrows) and the full elec-tron trajectories in the fields Eq. (17) for different values ofthe field magnitude: a = 500 (dashed lines), a = 2 × (dash-dotted lines) and a = 1 × (solid lines). The elec-trons start from x = ± . with its Lorentz factor γ = 100 and the momentum along y axis. The trajectories at x < arecomputed with the Landau–Lifshitz radiation reaction takeninto account, while the trajectories at x > are computedwith radiation reaction taken into account by Monte Carlotechnique and quantum formulae Eq. (A12). Bars depict thestandard deviation ( ± σ ) of the final electron position com-puted with 400 trajectories. A. Rotating electric field
In the rotating electric field of the amplitude a Eq. (15) gives v = − E /E , that coincides with the high-field limit ( a ≫ a ∗ ) of the Zel’dovich’s stationary solu-tion [30] utilizing the main term of the Landau–Lifshitzforce. This stationary solution can be updated by tak-ing into account quantum corrections to the radiation-reaction force [49], that also yields v → − E /E in thehigh-field limit. MC simulations demonstrate the samebehavior, however, high dispersion of the angle between v and E is evident, see Fig. 1. B. Static B-node
Let us start from the following simple field configura-tion: E y = a , B z = a x, (17)and the other components of the fields are zero.In Fig. 2 the velocity field Eq. (8) ( | x | > ) andEqs. (11) and (12) ( | x | ≤ ) is depicted by the arrows. Inthe left half of Fig. 2 the electron trajectories computedwith the Landau–Lifshitz force are shown, and in theright half of Fig. 2 the electron trajectories are computedwith Monte Carlo technique and quantum synchrotronformulae. Obviously, the shape of the electron trajecto-ries computed with Monte Carlo approach is slightly dif-ferent for different runs, so the bars depict the standarddeviation of the electron final position. The trajectoriesare computed for different a values, namely a = 500 , × , × which correspond to dashed, dash-dottedand solid lines, respectively. First, it is seen that athigher a values the real electron velocity coincides betterwith the velocity field that induces the asymptotic tra-jectories. Second, the Landau–Lifshitz approach demon-strate slightly better coincidence, because in the Landau–Lifshitz approach the mean electron energy generally lessthan in the quantum approach.Note that the fields Eq. (17) resemble the B node ofa standing linearly polarized wave, however, in the lin-early polarized standing wave the sign of E × B | x variesin time, and the node attracts the asymptotic electrontrajectories during a half of a period, and repels themduring the other half. C. Linearly polarized standing wave
In the linearly polarized standing wave asymptoticelectron trajectories can be found analytically. The fieldsof the linearly polarized standing wave read as follows: E = y a cos( t ) cos( x ) , (18) B = z a sin( t ) sin( x ) , (19)where y and z are the unit vectors along the y and z axes, respectively. Then from Eqs. (8), (11) and (12) weget: v = ( x tg( t ) tg( x ) ± y p − tg ( t ) tg ( x ) , E > B x ctg( t ) ctg( x ) , E < B. (20)Since the fields are homogeneous along the y axis elec-tron’s motion along it is not of any interest. Then x ( t ) of the asymptotic trajectory is found from the followingalgebraic equations: ( sin( x ) cos( t ) = sin( x ) cos( t ) , E > B cos( x ) sin( t ) = cos( x ) sin( t ) , E < B. (21)where the starting point x = x ( t ) also belongs to theregion E > B or E < B . For instance, the electrontrajectory initially is determined by the first of Eqs. (21),then it reaches the point with E = B ; after that thetrajectory is determined by the second of Eqs. (21) upto the moment when the electron reaches another pointwith E = B and so on. For E = B Eqs. (18) and (19)yields | tg( x ) tg( t ) | = 1 (22)with the following solution: x = ± t + π πn, n = 0 , ± , ± , ... (23) For the electron starting from the point x at the mo-ment t = 0 the chain of points ( x , t ) , ( x , t ) , ... atwhich E = B is the following. First, from Eqs. (21) and(22) under the assumption that initially E > B , we have: ctg x = tg t = s x ) − (24)The coordinate x can be found from the observationthat x = x and t = π − t obey the second of Eqs. (21)with x , t replaced by x , t . Also, x = x and t = π − t obey Eq. (22) (because x and t obey them).Analogously, x = x and t = π + t . Then the electrontrajectory periodically repeat itself (see Fig. 3 (a)).Note that the electron trajectory in the linearly polar-ized standing wave is periodic in the framework of thepresented asymptotic theory. As shown in Sec. VI A,this is just an example of the general behaviour of theelectron trajectories in standing waves in the radiation-dominated regime. However, it can seem that this be-havior contradicts the anomalous radiative trapping [36](ART). Really, ART is caused by a drift of the electronbetween the asymptotic trajectories given by Eqs. (21).This drift takes many periods of the field [36] and cannot be described by the presented asymptotic theory.The asymptotic electron trajectories computed withEqs. (14) and (15) are shown in Fig. 3 (a) with paleblue and beige lines (the computed trajectories coincideexactly with the analytical solutions Eqs. (21)). Sixelectron trajectories computed with Vay’s pusher andMonte Carlo technique for the photon emission are alsodepicted: for a = 1 × by green lines, and for a = 1 × by red lines. Fig. 3 (b) shows the en-ergy of the electrons on the trajectories A and B fromFig. 3 (a). The coincidence of the electron trajectoriescomputed for a = 1 × with the asymptotic trajecto-ries are evident, opposite to a = 1 × case in that thecondition γ ≪ a is not fulfilled. Note that in the case of a = 1 × the electrons moving according to the MCapproach to the radiation reaction, become closer to theB-nodes for each subsequent period, that is the effect ofART. V. ABSORPTION-INDUCED TRAPPING
It follows from Eqs. (8) and (11) that the angle be-tween the asymptotic velocity v and the Poynting vector S ∝ E × B is always less than π/ , i.e. v · S > . Thishints that the electron motion in the radiation-dominatedregime can be connected with the energy flow of the elec-tromagnetic fields. Let us consider the region containingcurrents which (partially) absorb the incoming electro-magnetic wave. In the average, the Poynting vector isdirected into the region of the currents, and we suggestthat in the radiation-dominated regime this region at-tracts the electron trajectories. In this section we verifythis suggestion in a couple of examples. c t / x / (cid:1) B(a) / a ₀ ct / (cid:0) B c t / x / (cid:2) theory, E-case theory, B-caseVay + MC, a = Vay + MC, a = ⁴(c) Figure 3. The electron motion (a), (b) in the field of the lin-early polarized standing electromagnetic wave Eqs. (18)–(19)and (c) in the field of two counter-propagating linearly polar-ized waves with a plane at x = 0 absorbing 70% of the incom-ing energy (see Eqs. (25)–(26), R = 0 . ). Thin lines depictasymptotic trajectories obtained by the numerical integrationof Eqs. (14) for the E- and B-cases (beige and pale blue, re-spectively). Thick lines correspond to the numerical integra-tion of the classical electron motion equations with quantumradiation reaction (A12) taken into account by Monte Carlotechnique. It is worth to mention that the thin lines in (a)coincides with the analytical solution Eq. (21) and with thethin green lines in Fig. 2 from Ref. [36]. A plane wave pushes initially immobile electrons ap-proximately in the direction of the Poynting vector, soit can seem that the absorption-induced trapping can berealized without strong radiation reaction. However, asseen from the examples below, in the absence of strongradiation losses if the electrons have been accelerated bya wave, then they can not be turned back by a counter-propagating wave. Thus the radiation reaction may causeelectron trapping in the region with strong absorption ofthe electromagnetic energy.
A. Counter-propagating linearly polarized wavespartially absorbing by a plane
The field of two counter-propagating linearly polarized(along the y axis) waves, that is partially absorbing at theplane x = 0 , can be written as follows: E = y a { cos( t ) cos( x ) − . − R ) cos( x ∓ t ) } , (25) B = z a { sin( t ) sin( x ) ∓ . − R ) cos( x ∓ t ) } , (26)where in ∓ the upper sign corresponds to x > and thelower one corresponds to x < , and R is the reflectioncoefficient. The asymptotic and Vay+MC electron tra-jectories in this field are shown in Fig. 3 (c) with thesame color codes as in Fig. 3 (a). Here R = 0 . thatmeans absorption of of the wave energy in the plane x = 0 . As seen from the figure, the electron trajectoriesare attracted by the plane x = 0 in the strong radiation-dominated regime, whereas at moderate intensity of thewaves the electrons easily pass the plane. The mean stan-dard deviation of x computed for the Vay+MC electrontrajectories for ten periods of the wave and x = 0 . λ is about . λ for a × and . λ for a = 1 × . B. Multipole wave absorbing by a current loop
The field of a multipole harmonic wave that is com-pletely absorbing by a current loop can be obtained bytime reversal of the field emitting by a current loop (seeApp. B). The electron motion in the absorbing multi-pole wave with the angular frequency ω = 2 πc/λ for aloop radius r ℓ = λ = 1 µ m is shown in Fig. 4, wherein the cylindrical coordinate system the “wire” positionis marked with the black cross. The z axis is the axisof the loop. Fig. 4 (a) demonstrates the magnetic fieldof the multipole wave at t = 0 . Fig. 4 (b) shows theasymptotic electron trajectories, Figs. 4 (c) and (d) showthe electron trajectories computed by Vay and MC algo-rithm for the loop current magnitude I = 1 × andfor I = 5 × , respectively. The trajectories start at t = 0 from the sphere shown by a thick dashed line andare computed up to t = 5 λ/c .It is seen from Fig. 4 that the current loop attractsthe asymptotic electron trajectories. However, it is seenthat the absorption-induced trapping is not really a strict z / r / (cid:3) B(a) r / (cid:4)
E-case B-case(b) z / r / (cid:5) = (c) r / (cid:6) = 5 ∙ Figure 4. (a) The magnetic field of a multipole wave that isentirely absorbing by a current loop (see App. B), the loopradius is r ℓ = λ = 1 µ m, t = 0 . The axis of the loop coincideswith the z axis and the position of a “wire” is shown by theblack cross. (b) Asymptotic electron trajectories for E- andB-case (orange and blue, respectively) in the multipole wave.The electrons start to move at t = 0 from the points on thecircle ( r + z ) / = 1 . r ℓ (thick black dashed line). Thebottom plots show the Vay+MC electron trajectories in thefield of the multipole wave for (c) I = 1 × and (d) I =5 × . All trajectories are computed for t ∈ [0 , λ/c ] . trapping but just means that electrons in the radiation-dominated regime stay for a long time in the region withthe currents absorbing the incoming waves. VI. EMISSION-ABSORPTION SYMMETRYAND GENERAL PROPERTIES OFASYMPTOTIC TRAJECTORIES
In this section we consider the properties of the elec-tron trajectories described by Eqs. (14) and (15). Forthis purpose let us consider the well-known symmetry ofthe Maxwell’s equations, namely, the following transform t ∗ = − t, (27) E ∗ = − E , (28) B ∗ = B , (29) ρ ∗ = − ρ, j ∗ = j . (30)does not change them, i.e. they leads to the Maxwell’sequations for the starred variables; here ρ is the chargedensity and j is the current density. From here on wedenote E , B , j evolving in time t as initial system and E ∗ , B ∗ , j ∗ evolving in time t ∗ as starred system. Thissymmetry is the relation between a system of currentsemitting some fields and the system of currents absorbingthe fields: namely, the Poynting vector, the j · E productand the time direction in the starred system is oppositeto that in the initial system.According to Eq. (15), in the starred system the ve-locity field v ∗ relates to the velocity field of the initialsystem v as follows: v ∗ ( r , t ∗ ) = − v ( r , − t ∗ ) , (31)that obeys the stability condition v ∗ · E ∗ < . Thus, inthe starred system the velocity field and the time direc-tion are opposite to that in the initial system, that leadsto the same trajectories in the starred system r ∗ ( t ∗ ) as inthe initial system, passed by the electrons in the oppositedirection: d r ∗ /dt ∗ = v ∗ ( r ∗ , t ∗ ) = − v ( r ∗ , − t ∗ ) .Let us note a fundamental difference between theasymptotic trajectories described by Eqs. (15), (14), andby the ponderomotive description. The ponderomotiveforce is determined by the distribution of E and B andis indifferent to the transform Eqs. (27)–(30), whereasthis transform reverses the direction of the electron mo-tion in the case when Eqs. (15) and (14) are applicable,namely, when radiation reaction is strong.In order to illustrate the difference between the pon-deromotive description and the description by the veloc-ity field Eq. (15) the following toy example can be consid-ered. The first laser pulse propagates along the direction x and scatters an electron aside. Then the second pulseis formed from the first one with the substitution (28)–(29), and according to the Maxwell’s equations this pulsetravels in the direction − x . In the framework of theponderomotive description the second laser pulse is notimportant because it will never meet the electron scat-tered by the first pulse. At the same time, consideringthe first laser pulse as the initial system of fields, we seethat the second laser pulse is equivalent to the starredsystem of fields. In the case of strong radiation reac-tion the asymptotic approach is valid, and the electronaccording to Eq. (31) will pass along its preceding trajec-tory in the opposite direction in the field of the secondpulse, i.e. the electron will be brought back to its initialposition by the second laser pulse.Therefore, the asymptotic description of the electronmotion Eqs. (14) and (15) implies that the electrons arenot scattered by, but stay for a long time in the fieldof a laser pulse or in a laser beam. This conclusion isin a good agreement with the results of theoretical con-siderations and numerical simulations showing that theponderomotive force can be significantly suppressed bythe radiation reaction [34, 50]. A. Asymptotic trajectories in standing waves
We see in Sec. IV that the reduced equations lead to pe-riodic electron trajectories in the linearly polarized stand-ing electromagnetic wave. Here we show, that Eqs. (14)and (15) always lead to a periodic electron trajectories ina wide class of fields, namely in the periodic fields whichcan be represented in the following form: E = f ( r , t ) − f ( r , − t ) , (32) B = g ( r , t ) + g ( r , − t ) , (33)where E = f ( r , t ) , B = g ( r , t ) is the solution of Maxwell’sequations for some charge density ρ and current density j (for the sake of simplicity let us consider ρ = 0 and j = 0 ). This representation means that the fields are thesum of the fields of some system and the fields of thecorresponding starred system. In this case the symmetry(27)–(30) leads to the same fields of the starred system asin the initial system, i.e. E ∗ ( r , t ∗ ) = E ( r , t ∗ ) , B ∗ ( r , t ∗ ) = B ( r , t ∗ ) , hence it should lead to the same velocity field v ∗ ( r , t ∗ ) = v ( r , t ∗ ) , that together with Eq. (31) yields v ( r , − t ) = − v ( r , t ) . (34)Thus, the velocity field in the electromagnetic fields (32)–(33) is an odd function of time. Consequently, the timereversal conserves the equation for the electron position, d r d ( − t ) = v ( r , ( − t )) , (35)and the electron position r ( t ) is an even function of time.Therefore, r ( t ) − r ( − t ) = Z t − t v ( r ( t ) , t ) dt = Z t v ( r ( t ) , t ) dt + Z t v ( r ( − t ) , − t ) dt = 0 . (36)As far as the velocity field governed by Eqs. (8) and(10)–(13) is a single-valued function of the electromag-netic fields, and the fields are periodic in time, the veloc-ity field is also periodic with the same period, T . Thus,the velocity field is an odd function relative to any time instant t = nT , where n is an integer. Let the electronstarts to move at t = nT − T / , then it comes to thestarting point a period later, r ( nT + T /
2) = r ( nT − T / ,then due to the periodicity of v at t = ( n + 1) T , we have r (( n + 1) T + T /
2) = r ( nT + T / . Therefore, in theframework of the asymptotic approach, the electron ismoving periodically back and forth along the same pathin the periodic fields Eqs. (32)–(33). B. Asymptotic trajectories in a laser beam of finitediameter
Here we stress that many field configurations couldbe reduced to the form of periodic fields that obey theemission-absorption symmetry Eqs. (27)–(30). In theprevious subsection we also assumed that the velocityfield is a single-valued function of the fields. This is notstrictly true in the B-case, because one can add to v from Eq. (8) a vector parallel to B . The effect of thisambiguity is also discussed in this section.Let us consider the fields of TE11 mode of a rectangu-lar metallic waveguide: E x = 0 , (37) E y = a cos( k y y ) sin( k z z ) cos( t − k x x ) , (38) E z = − a k y k z sin( k y y ) cos( k z z ) cos( t − k x x ) , (39) B x = a ( k z + k y ) k z cos( k y y ) cos( k z z ) sin( t − k x x ) , (40) B y = − k x E z , (41) B z = k x E y , (42)where we assume that the wave angular frequency Ω =( k x + k y + k z ) / = 1 (here we use the normalizationfrequency ω equal to the frequency of the wave, and,as before, the time is normalized to /ω , coordinatesare normalized to c/ω , k is the wavenumber normal-ized to ω/c ). These fields obey the metallic boundaryconditions at y = 0 , ± ℓ y , ± ℓ y , ... ( E z = 0 ) and at z = 0 , ± ℓ z , ± ℓ z , ... ( E y = 0 ). Here ℓ y = π/k y and ℓ z = π/k z are the sizes of the waveguide along the y - and z -axes, respectively.The fields Eqs. (37)–(42) are the solution of theMaxwell’s equations not only inside the waveguide butin the open space as well because this fields can be repre-sented as a sum of plane waves. Particularly, we considerthese fields in the region y ∈ [ − ℓ y / , ℓ y / and z ∈ [0 , ℓ z ] as the model of the laser beam of finite diameter. If ℓ y ≫ ℓ z , the electric field is mainly directed along the y -axis and reaches its maximum in the center of the beam.The fields Eqs. (37)–(42) are shown in Fig. 5 (a) for ℓ y = 4 π , ℓ z = 2 π and t = x = 0 . The asymptotic electrontrajectory computed for these fields is shown in Fig. 5 (b),where ξ = x − v g t , v g = k x ≈ . is the group velocityof the electromagnetic wave, and the trajectory starts at t = 0 , x = 0 , y = 0 . and z = 0 . and is computed up z / (cid:7) y (cid:8) (cid:9) B E(a) (cid:10) (cid:11),(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18) t (cid:19) τ theory, (t) theory, y(t)(b) z(cid:20) λ (cid:21) (cid:22) λ Vay + MC, a = 700 theory(c) (cid:23)(cid:24) λ (cid:25) (cid:26) λ Vay + MC, a = theory
Figure 5. (a) The electric and magnetic fields (red and blue arrows, respectively) of the continued TE11 mode of a waveguide,Eqs. (38)–(39) and (41)–(42), at t = 0 . (b) The asymptotic trajectory of an electron computed with Eqs. (14), (8) and (10)–(13)in the laboratory reference frame; ξ = x − v g t , where v g is the group velocity of the TE11 mode. (c), (d) The same asymptotictrajectory (thick line) and five full electron trajectories (thin lines) starting at the point x = 0 , y = 0 . λ , z = 0 . λ ( λ = 1 µ m, t ∈ [0 , τ ] ), for (c) a = 700 and (d) a = 4 × . to t = 2 τ , where τ = 2 πk x ( v φ − v g ) = 2 π − k x (43)is the intrinsic timescale of the task, v φ = 1 /v g is thephase velocity of the wave. For Fig. 5 cτ /λ ≈ . . InFigs. 5 (c) and (d) the same asymptotic trajectory isshown as the thick blue line. Five electron trajectories inthe fields given by Eqs. (37)-(42) are shown in Figs. 5 (c)and (d). These trajectories start at the same point as theasymptotic trajectory, but they are computed by Vay’salgorithm with quantum radiation reaction incorporatedby Monte Carlo technique. For Figs. 5 (c) and (d) we use a = 700 and a = 4 × , respectively.It is seen from Figs. 5 (b), (c) and (d) that the asymp-totic trajectory is quasiperiodic, that is in a qualitativeagreement with the fact that for a = 4 × the electronsstay for a long time in the high-field region. However,as we see below, the asymptotic trajectories being com-puted in the laboratory reference frame yield the valuesof ξ and the values of the trajectory period which do notcoincide well with that for real electron trajectories. Thereason for that is that Eq. (8) is not Lorentz invariant,namely if one compute v from it in some reference frame,in another reference frame he obtain v ′ = E ′ × B ′ + aB ′ ,where a is a constant. Let us transform the fields Eqs. (37)–(42) to the ref-erence frame K ′ moving along the x axis with the groupvelocity of the fields v g . This lead to the following fields: E ′ y = a k ⊥ cos( k y y ) sin( k z z ) cos( k ⊥ t ′ ) , (44) E ′ z = − a k ⊥ k y k z sin( k y y ) cos( k z z ) cos( k ⊥ t ′ ) , (45) B ′ x = a ( k z + k y ) k z cos( k y y ) cos( k z z ) sin( k ⊥ t ′ ) , (46) E ′ x = B ′ x = B ′ y = B ′ z = 0 , (47)where k ⊥ = p − k x . These fields do not depend on x ′ and for all the electrons in these fields the compo-nent of the Lorentz force along the x ′ axis is absent.Furthermore, the electrons due to the radiation reaction“forget” their initial direction of motion, hence we con-clude that the average velocity of the electrons in thefields Eqs. (44)–(47) is v ′ x = 0 . Therefore, in K the av-erage electron velocity is v x = v g hence ξ = const that isin good agreement with results of Vay+MC simulations.Note that ξ = const does not coincide with the result ofthe asymptotic consideration in the laboratory referenceframe (see Fig. 5 (b)). Also, a wrong value of v x leads toa wrong value of the period of y and z coordinates of theelectron in the framework of the asymptotic approach.0The substitution t ′ → t ′ + π/ k ⊥ yields that the elec-tric field given by Eqs. (44)–(45) are odd functions oftime and the magnetic field Eq. (46) is the even functionof time in K ′ . As follows from Sec. VI A in this casethe electron trajectories are periodic in the radiation-dominated regime and their period is equal to π/k ⊥ in K ′ . Therefore, in the laboratory reference frame in theradiation-dominated regime the electrons move along the x -axis with the group velocity of the laser beam, and, as y ′ = y and z ′ = z , the electron trajectories are periodicin the yz plane with the period πk ⊥ q − v g = τ. (48)Thus, the ambiguity of the velocity field in the asymp-totic approach can be resolved by appropriate choose ofthe reference frame.Therefore, we show that the asymptotic description,Eqs. (14) and (15), leads to periodic trajectories in awide class of standing waves (e.g. formed by laser beamsof finite diameter), and to electron motion along the laserbeam with its group velocity with periodic transverse mo-tion. The latter may explain the effect of the radiation-reaction trapping [50]. VII. CONCLUSION
To conclude, here we show that in the radiation-dominated regime the electrons tend to move with ve-locity that is determined by the fields only, see Eq. (15).This means that the electron trajectory can be foundfrom the first-order equation, Eq. (14). We call this veloc-ity asymptotic because it can be found as the asymptoticelectron velocity ( t → ∞ ) in the constant field approxi-mation. The reason for reduction of the equation orderis that the electron energy in the radiation-dominatedregime is small ( γ ≪ a ), the electrons are “light” andare easily turned by the laser field to the asymptotic di-rection in a time much smaller than the characteristicvariation time of the electromagnetic fields. The velocityfield v ( r , t ) corresponds to the absence of the componentof the Lorentz force transverse to the electron velocity,so v is also called the radiation-free direction [35].In a number of the electromagnetic field configura-tions we found the numerical solutions of the reduced-order equations and the full equations of electron mo-tion with the radiation reaction taken into account bythe Monte Carlo technique and the Baier–Katkov syn-chrotron formulae [43]. The comparison between thesesolutions demonstrates that the reduced-order equationscan be used for a qualitative description of the electrontrajectories for a greater or of the order of thousand foroptical wavelengths. In order to stress these high valuesof a we call the solutions of the reduced equations ofmotion as asymptotic trajectories ( a → ∞ ).Also we demonstrate that the reduced-order equationsfor the electron trajectories in the radiation-dominated regime are the useful analytical tool. First, they pre-dict the electron trapping in the regions where the wavefield is absorbed, see Sec. V. This result can be impor-tant for the theoretical consideration of the field absorp-tion by the QED cascade in the counter-propagatinglaser waves [18]. Second, contrary to the concept ofthe ponderomotive force, the asymptotic theory leads toperiodic electron trajectories in a wide class of stand-ing electromagnetic fields (including the case of counter-propagating tightly focused laser beams, see Sec. VI A).This result is in a good agreement with Ref. [34] thatdemonstrates the reduction of the ponderomotive forcein the radiation-dominated regime. Furthermore, us-ing a certain configuration of the laser beam we demon-strate that the beam in the radiation-dominated regimedoes not push the electrons aside, but captures and car-ries them with the group velocity of the beam. Thisresult probably explains the radiation-reaction trappingobserved in the numerical simulation of Ref. [50].Therefore, the concept of the ponderomotive force isnot applicable in the radiation-dominated regime and canbe replaced by the description of the asymptotic electrontrajectories. This concept implies that velocities of theelectrons in a given point are the same hence the elec-trons (positrons) in the radiation-dominated regime canbe described in the framework of the hydrodynamical ap-proach. The Maxwell’s equations, in which the electroncurrent is determined only by the plasma density andby the local field values (see Eq. (15)), together with thecontinuity equation for the plasma density are formed theclosed system of equations. Note that the reduced-orderequations gives the positive field work on the electrons( v · E < ) hence the plasma in the framework of theasymptotic theory is always an absorbing medium. Inmore details this hydrodynamical approach will be con-sidered elsewhere. ACKNOWLEDGMENTS
We thank A. V. Bashinov and V. A. Kostin for fruitfuldiscussions. We are grateful to E. V. Frenkel who broughtour attention to the symmetries of the Maxwell’s equa-tions, and to T. Docker for his help with haskell-chart library.This research was supported by the Grants Council un-der the President of the Russian Federation (Grant No.MK-2218.2017.2). The study of the absorption-inducedtrapping was supported by the Russian Science Founda-tion through Grant No. 16–12-10383
Appendix A: Tests of numerical instruments1. Radiation reaction: classical limit
In order to test the Vay’s solver for the equations ofmotion [46] coupled with Landau–Lifshitz radiation re-1action force (taken into account by Euler method) let usconsider electron motion in constant crossed electric andmagnetic fields: E y = a / , B z = a , (A1) E x = E z = B x = B y = 0 . (A2)In the reference frame K ′ moving along x axis withthe speed V = 0 . the electric field vanishes and theonly z component of the magnetic field remains: B ′ z = B z √ − V , where the stroke marks quantities in K ′ .Taking into account the Landau–Lifshitz radiation re-action, for relativistic electron motion in K ′ we obtain(assuming γ ≫ ): dγ ′ /dt ′ = − Cγ ′ , (A3) dw ′ /dt ′ = iB ′ z w ′ /γ ′ , (A4) dv ′ z /dt ′ = 0 , (A5)where C = 23 e ~ c ~ ωmc B ′ z v ⊥ , (A6) w ′ = v ′ x + iv ′ y , v ⊥ = v ′ x + v ′ y and ω is just some frequencyused for normalization of time. The solution of Eqs. (A3)-(A5) is the following: γ ′ = γ ′ γ ′ Ct ′ , (A7) w ′ = w ′ exp (cid:18) iB ′ z γ ′ ( t ′ + γ Ct ′ (cid:19) , (A8)and x ′ + iy ′ = x ′ + iy ′ + w ′ s iπ B ′ z C exp (cid:18) − iB ′ z γ ′ C (cid:19) × ( erf r B ′ z C i ( t ′ + 1 γ ′ C ) ! − erf r B ′ z C i γ ′ C !) , (A9)where subscript denotes t ′ = 0 and erf( x ) = 2 √ π Z x exp( − t ) dt (A10)is the error function.Figure 6 (a) demonstrates the electron trajectory in the xy plane obtained with the numerical integration of theequation of motion taking into account radiation reactionin Landau–Lifshitz form (solid blue line) for a = 50 . , ω = 2 πc/λ , λ = 1 . nm, x = y = 0 , v x ( t = 0) ≃ . , v z ( t = 0) ≃ . and γ = 63 . Note that these parametersensure v ′⊥ ( t ) = v ′ x ( t = 0) = 0 . V , leading in thelaboratory reference frame to the cycloid-like trajectorywith points of dy/dx → ∞ . x Vay + LL LL, y(t →∞ ) Vay + MC(a) 0125 0 10 20 50 70 (cid:27) A: Boltzmann, t = 22 B: Vay + MC, t = 22C: Boltzmann, t = 49.5 D: Vay + MC, t = 49.5D C BA(b)
Figure 6. (a) The electron trajectory in the crossed electricand magnetic fields (A1)-(A2) ( a = 50 . , λ = 1 . nm) com-puted by numerical integration of the classical equations ofthe electron motion with radiation reaction taken into accountby means of the main term of the Landau–Lifshitz radiationreaction force (solid line) and by means of Monte Carlo tech-nique and quantum emission probability (A12) (dashed line).The dotted line depicts y ( t → ∞ ) found from Eq. (A9). Theelectrons initially have v x ≃ . , v z ≃ . and γ = 63 . (b) Inthe same fields, the energy distribution of the electrons withthe same initial momentum, computed by the same methodas for the dashed line in the subplot (a) (lines B and D) andwith numerical integration of the Boltzmann equation (A11),for different time instants. For the given parameters we obtain B ′ z = 43 . , γ ′ = γ ′ ( t ′ = 0) = 43 . and C = 4 . × − , and from Eq. (A7)at the time instant t ′ = 6 π we get γ ′ ( t ′ ) = γ ′ / . Ne-glecting displacement of the particle, x ′ , and assuming v ′ x ( t ′ ) = V , we finally get for the laboratory referenceframe: t = (1 − V ) − / t ′ ≈ and γ ( t ) ≈ . . Itshould be mentioned that at the time instance at which v ′ x = V , in the laboratory reference frame the Lorentzfactor reaches its local maximum. Numerical solver usingLandau–Lifshitz force demonstrates that the local maxi-2mum of γ closest to t = 22 is reached at t ≈ . and is γ ≈ . that is quite close to the predicted value.Equation (A9) yields y ′ ( t ′ → ∞ ) ≈ . for theabove-mentioned parameters. This value ( y ( t → ∞ ) = y ′ ( t ′ → ∞ ) ) is depicted as gray dotted line in Fig. 6, andin a good agreement with the value obtained with the nu-merical solver. The dashed orange line is got by means ofparticle pusher that takes into account with the quantumformulae and is described in the next subsection.
2. Radiation reaction: general case
The quantum radiation reaction can be taken into ac-count in Vay’s pusher by means of Monte Carlo tech-nique. To do this we use the alternative event genera-tor [44] based on Baier–Katkov synchrotron formula [37,45]. The event generator checks at every time step if thephoton emission occurs, and if it does, the electron mo-mentum is decreased on the momentum of the emittedphoton. Using of classical description of the electron tra-jectory together with the quantum formula for the photonemission is valid because the radiation formation lengthin strong fields ( a ≫ ) is much smaller than the fieldcharacteristic scale [37, 45, 51].In order to test Vay’s pusher coupled with Monte Carloevent generator we compute the energy distribution ofthe electrons in the crossed fields Eqs. (A1)–(A2). Theresulting spectra are compared with the spectra obtainedfrom the Boltzmann equation in the reference frame K ′ .As mentioned above, in the reference frame K ′ movingalong x axis with velocity V = 0 . the electrons see thepure magnetic field directed along the z axis. Therefore,in K ′ the Boltzmann equation that describes the electronenergy distribution f ′ ( t ′ , γ ′ ) is the following: ∂f ′ ( t ′ , γ ′ ) ∂t ′ = Z ∞ γ ′ w ( ǫ, ǫ − γ ′ ) f ′ ( t ′ , ǫ ) dǫ − W ( γ ′ ) f ′ ( t ′ , γ ′ ) , (A11)where w ( ǫ, ǫ γ ) = − αǫ ℓ ǫ (cid:20)Z ∞ κ Ai( ξ ) dξ + (cid:18) κ + ǫ γ χ κ / ǫ (cid:19) Ai ′ ( κ ) (cid:21) , (A12)is the distribution of the photon emission probability bythe electron with the Lorentz factor ǫ over the photonenergy ε γ normalized to mc , i.e. over ǫ γ = ε γ /mc (seeRefs. [37, 45]), and χ = ǫ ℓ B ′ z v ⊥ ǫ, (A13) κ = (cid:20) ǫ γ ( ǫ − ǫ γ ) χ (cid:21) / , (A14) W ( γ ′ ) = Z γ ′ w ( γ ′ , ǫ γ ) dǫ γ (A15) is the overall emission probability for an electron withthe Lorentz factor γ ′ , ǫ ℓ = ~ ω/mc , ω is the frequencyused for normalization of time.The Boltzmann equation (A11) can be solved numer-ically as follows. In finite-difference method the distri-bution function f ′ ( γ ′ ) is represented as a vector, andthe right-hand-side of the Eq. (A11) is represented asthe product of a matrix and a vector. Then Eulermethod can be used, and the computation of f ′ ( t ′ , γ ′ ) from f ′ ( t ′ = 0 , γ ′ ) is reduced to a matrix exponentia-tion, that can be done with square-and-multiply algo-rithm that have logarithmic complexity on the numberof time steps. Then the distribution function in the initialreference frame can be found from f ′ ( t ′ , γ ′ ) with Lorentztransformation. For that one should neglect the electrondisplacement in K ′ (i.e., x ′ ( t ′ ) − x ′ (0) ) and assume thatin K ′ the angles ϕ ′ between x ′ axis and v ′⊥ are uniformlydistributed on the interval [0 , π ) : t = t ′ Γ , (A16) γ = γ ′ Γ(1 +
V v ⊥ cos ϕ ′ ) , (A17) f ( γ ) ∝ Z f ′ ( γ ′ ) dγ ′ dγ dϕ ′ = Z f ′ ( γ ′ )(1 + V v ⊥ cos ϕ ′ ) dϕ ′ , (A18)where the integration should be performed over the pathdetermined by the value of γ and Eq. (A17); Γ = (1 − V ) − / . It is worth noting that for correctness of themethod the step along γ ′ in the finite-difference schemeshould be much smaller than the width of the emissionspectrum. Thus, especially small step of γ ′ should byused in the classical regime.Figure 6 (b) demonstrates the electron spectra in thecrossed fields Eq. (A1)-(A2) with a = 50 . and λ =1 . nm used for the normalization. The electrons initially(at t = 0 ) move along x and z axes ( v x ≃ . , v z ≃ . )and have Lorentz factor γ = 63 . Curves A and C areobtained by Eq. (A11) for t = 22 and t = 49 . , respec-tively. Curves B and D represent the spectra of particles whose trajectory is computed by Vay’s pushercoupled with Monte Carlo event generator, for t = 22 and t = 49 . , respectively.In K ′ the parameters of the simulations yield the quan-tum parameter χ ′ ( t ′ = 0) = 2 , and if Landau–Lifshitzradiation reaction is used, χ ′ drops down to χ ′ ( t = 22) =0 . and χ ′ ( t = 49 .
5) = 0 . (see Eq. (A7)). However, ini-tially χ ′ & that leads to wide emission spectrum andwide resulting spectrum of the electrons. Moreover, theoverall emission probability is not very high and a sig-nificant fraction of electrons do not emit photons at all.These electron fractions form peaks clearly seen on thecurves A and B. The position of the peak on the curveA corresponds to non-emitting electrons with v ′ x = − . that according to Eq. (A17) gives γ ≈ . However,in Monte Carlo simulation at t = 22 the distribution ofelectrons over ϕ ′ is far from the uniform one, and most3of the non-emitting electrons moves with v x ≈ . lead-ing to the peak at γ = γ ( t = 0) . Thus, the difference ofcurves A and B comes from the assumption of uniformelectron distribution over the angle ϕ ′ . This assumptionbecomes more reliable at later times ( t = 49 . ), and thedifference between two methods of the spectra computa-tion vanishes (see curves C and D).Therefore, the results of the Vay’s pusher coupled withthe Landau–Lifshitz radiation reaction force or with theMonte Carlo event generator (that uses some approx-imate expression for fast computation of the emissionprobability) coincides well with the results obtained byother methods. Appendix B: Multipole wave