Radiation friction force effects on electron dynamics in ultra-intensity laser pulse
aa r X i v : . [ phy s i c s . p l a s m - ph ] O c t Radiation friction force effects on electron dynamics in ultra-intensity laser pulse
Yanzeng Zhang and Sergei Krasheninnikov
Mechanical and Aerospace Engineering Department,University of California San Diego, La Jolla, CA 92093, USA
The electron dynamics in the ultra-high intensity laser pulse with radiation friction force in theLandau-Lifshitz form are studied. It is demonstrated that widely used approximation, where onlythe term dominating the dissipation of electron kinetic energy is retained in the expression for theradiation friction, is incorrect for the case of diverging electron trajectories. As a matter of fact, forlarge friction force effects, all components of the radiation friction force in the Landau-Lifshitz formhave the same order in the equation of electron motion, being equally important for both electrontrajectory and thus energy gain in the case of diverging electron trajectories (e.g. determined bythe superposition of few electromagnetic waves).
It is well known that electron dynamics in a stronglyrelativistic electromagnetic field could be significantly al-tered by so called radiation damping effects [1, 2]. Insome contemporary literature these effects are also calledthe radiation friction effects. In particular, radiation fric-tion becomes important for the case of plasma interac-tions with ultra-intense laser pulse (e.g. see [3–13] andthe references therein).Recently it was found that for some cases the radia-tion friction results in electron trapping in the vicinity ofsome spatially localized points (attractors) [9–13]. How-ever, these studies were performed by using just a partof the Landau-Lifshitz expression [1] for the radiationfriction, which dominates in the dissipation of electronkinetic energy. We will see that this approximation isincorrect for the case of diverging electron trajectories corresponding to electron dynamics near such attractors.In this study we consider an impact of the radiationfriction on electron dynamics exposed to the combina-tion of monochromatic waves with the same frequencybut opposite propagation directions, which is widely usedin theoretical studies (e.g. see [6–12]). We use standardnormalized variables of ˆ t = tω , ˆ x = k x , ˆ v = v /c , and( ˆ E , ˆ B ) = e ( E , B ) /mωc , where e is the elementary charge, m is electron mass, c is the light speed, and ω = kc and k are, respectively, the wave frequency and wavenumber.Then the equation of electron motion in the electromag-netic field with radiation friction force in the Landau-Lifshitz form of f RF = ρ f ( γ f + γ f + f ) , (1)where γ is the relativistic factor, ρ f = 2 r e k/ ≪ r e = e /mc is the classical electron radius, and f = − v [( E + v × B ) − ( v · E ) ] , f = [( ∂ t + v · ▽ ) E + v × ( ∂ t + v · ▽ ) B ] , f = [ E × B + B × ( B × v ) + E ( v · E )] , (2)can be written as follows d P dt = − ( E + v × B ) + Q ( q ) f RF , (3) dγdt = − E · v + Q ( q ) f RF · v , (4)where P = γm v and the function Q ( q ) of q = γa s p ( E + v × B ) − ( v · E ) , (5)characterizes the quantum effects with a s = mc/ ~ k beingthe normalized Schwinger field, and ~ the Planck con-stant (e.g. see [8] and the references therein). In [11] itwas shown that for q ≤ Q ( q ) could be approximatedas Q ( q ) ≈ (1 + 18 q + 69 q + 73 q + 5 . q ) − / . (6)We notice that in Eqs. (2-5) we removed hats over thenormalized variables to simplify the expressions. For the discussion of the applicability of the Landau-Lifshitz ex-pression for the radiation friction see [5] and the refer-ences therein.For modest quantum effects, Q ( q ) ∼ γ ≈ a , where a is the normalized amplitude of laserwave vector potential, from Eq. (4) we find that the con-tributions of different components of the friction force, f i (i=1, 2, 3), to the Eq. (4) for electron kinetic energycould be estimated as follows: f ∼ a and f , f ∼ a (we use f = | f | to simplify the expression when estimat-ing forces magnitude). As a result, in super-relativisticregime where γ ≈ a ≫ f component of the frictionforce dominates in Eq. (4) and it starts to compete withthe Lorentz force for η f = a ρ f ≥ f component is often taken into account.However, electron energy gain/loss depends also on themagnitude of laser wave, which is a function of the spa-tial coordinate. Therefore, it’s necessary to evaluate animpact of the radiation friction on electron trajectory.From Eqs. (1-4) we find γ d v dt = − ( E + v × B ) + ( E · v ) v (7)+ Q ( q ) ρ f { f + γ [ f − ( f · v ) v ] + [ f − ( f · v ) v ] } , where the factor of γ = 1 − v in front of f compo-nent is eliminated as f is aligned with electron velocityas seen from Eq. (2), but not for f and f components.As a result, one could expect that for γ ≈ a , the con-tributions of all components of the radiation friction toelectron trajectory are comparable. This is in contrastto the equation for kinetic electron energy (4), where f component dominates.We notice that Eqs. (3, 4, 7) neglect the force f s ,related to the interaction of electrons spin with elec-tromagnetic field (e.g. see [14–16] and the referencestherein), the magnitude of which compared with those ofthe radiation friction force could be estimated as [7, 14–16]: f s /f ∼ /αγ , f s /f ∼ f s /f ∼ /αa , where α = e / ~ c = 1 /
137 is the fine structure constant. Inour simulations we choose such a that f s < f , f andthus we neglect the force f s . On the other hand, for thecase of a strong friction, η f >
1, from the energy balancewe have γ ∼ ( ρ f a ) − / < a . As a result, for η f > f on electron motion would be small in com-parison with that produced by f and f . However, wewill keep the force f for completeness, which can also tosome extent mimic the impact of f s as we found that f s and f could be comparable in our simulations.Nonetheless, since ρ f ≪
1, an impact of radiation fric-tion on electron trajectory is small unless we are dealingwith the situation where electron trajectories are stronglydiverging. Such strongly diverging electron trajectoriesare typical in the vicinity of electron trapping sites (at-tractors) considered in [9–12]. Therefore, one could ex-pect that for such case the impact of all the componentsof the radiation friction force on electron trajectory and,therefore, electron energy gain will be comparable.To illustrate the impact of f and f components of theradiation friction force on the electron dynamics, we willconsider the electromagnetic field in the form of a stand-ing wave [6–12], which is characterized by the followingvector potential A = a cos ( z ) cos ( t ) e x , (8)which will be employed to numerically solve Eqs. (1-4)with and without f and f components.We notice that parameters q and η f depend on thelaser wave amplitude and wavenumber differently. Toreveal an impact of f and f components more clearlyfor large friction force case, in our simulations we take λ = 1 µm and a = 1000 (corresponding to laser inten-sity of I = 1 . × W cm − ) such that η f ≈
12 and q in Eq. (5) will be of order unity and thus only modest t z without f and f with f and f (a) t γ without f and f with f and f (b) FIG. 1. Electron dynamics for λ = 1 µm , a = 1000 (laserintensity of I = 1 . × W cm − ), and initial conditions z (0) = 0, P x (0) = 0 and P z (0) = 100. The solid blue curve(red diamond marker) corresponds to the simulation resultswithout (with) f and f components. variation of the function Q ( q ) appears. As indicated byEqs. (1-3) the y-component of the electron equation ofmotion has only dissipated term originated from f forthe vector potential in Eq. (8) and thus it is reasonableto set the y-component of electron momentum as zerowhich will be conserved and so is the y coordinate. There-fore, we need to only deal with 4-dimensional equations.Under such set up, electron dynamics with different ini-tial coordinates of z (the x coordinate doesn’t affect theelectron motion and thus is chosen as zero initially) andmomenta ( P = P x e x + P z e z ) are investigated.Shown in Fig. 1 is the results for electron with z (0) = 0, P x (0) = 0 and P z (0) = 100, where electron performsperiodic motion. The solid blue curve and red diamondmarker denote, respectively, the results without and with f and f components. As we can see for this non-diverging electron trajectory, the impact of f and f com-ponents on such highly relativistic electron dynamics iscompletely negligible.However, this is not the case for electron with stronglydiverging trajectories. For example, the electron motionsfor z (0) = 0 .
01, and P x (0) = P z (0) = 0 have been shownin Fig. 2 where the results without and with f and f components are displayed, respectively, by the solid blueand dash-dot red curves (we omit initial stage of elec-tron motion for z coordinates and gamma-factors ver-sus time where the gamma-factors are settling down).As one can see, f and f components of the radiationfriction force make significant impact on all parametersshown in Fig. 2. They adjust the electron trajectories[Fig. 2(a)] and thus the energy gains [Fig. 2(b)], where γ < a in this region due to strong radiation friction ef-fects. Fig. 2(a) shows that, in accordance with the resultsof [9–12], due to an impact of radiation friction electronstarts to be trapped in the vicinities of zero electric fieldat z = π/ ± nπ with n = 0 , , ... (attractors). Whetherincluding f and f components or not leads to electronending up in different attractors (we found that the sameattractor of z = − π/ f and f compo-nents is obtained for electron motion with f but not f ).The divergence of electron trajectory under the impactof f and f is clearly seen in Fig. 2(c) where electron tra-jectories are shown from t = 0. However, such divergingeffects depend on initial conditions (e.g., for initial coor-
30 60 90-2-1012 30 60 900200400600 -2 -1 0 1 2-30-20-100
FIG. 2. Electron dynamics for the same parameters withFig. 1 but different initial conditions of z (0) = 0 . P x (0) = P z (0) = 0. The solid blue and dash-dot red curves denotethe simulation results without (with) f and f components,respectively. dinate z (0) = 0 .
02 and the same momentum with Fig. 2the electrons with and without f and f components fallinto the same attractor near z = π/ f and f components onthe electron dynamics in the same attractor, we initiallyput the electron into the attractor z = π/ P x (0) = 0, P z (0) = 20. The results are shown in Fig. 3,which demonstrates that f and f components could sig-nificantly affect the electron trajectory and thus energy t γ without f and f with f and f (b) FIG. 3. Electron dynamics for the same conditions with Fig. 2but z (0) = π/ P z (0) = 20. gain even when they are around the same attractor. Wenotice that it is hard to obtain reliable long time simula-tion results of electron motion for such strongly divergingelectron trajectories in Figs. 2 and 3, but the relativelyshort time simulations already exhibit the significant im-pacts of f and f components.We see fast oscillations of electron trajectory at trap-ping sites [e.g., see Fig. 3(a)], which were also observedin [10, 11]. These oscillations are just the gyro-motionof relativistic electron in magnetic field in the vicin-ity of electric node as seen from Eq. (7). In our nor-malized units the gyro-frequency and gyro-radius canbe expressed, respectively, as Ω gyro = | B ( t ) | /γ and ρ gyro ≈ γ/ | B ( t ) | , where the magnitude of the mag-netic field directed in y-direction depends on time andz-coordinate as B ( t ) = − a sin ( z ) cos ( t ) and the electronmotion speed is approximately taken as unity. For thecase of large radiation friction where γ ≪ a [e.g, seeFig. 3(b)], for most of the period of electromagnetic wavewe have Ω gyro ≫ ρ gyro ≪ µ ∝ (cid:2) γ v (cid:3) / | B ( t ) | [1] is not conserved due to radiation friction, which con-tinuously dissipates electron energy. However, for rela-tively short period of time around B ( t ) = 0, electrongyro-radius becomes large ( ρ gyro ≥
1) and electron expe-riences large departure from trapping site [see Fig. 3(a)],gaining kinetic energy from the laser [see Fig. 3(b)]. Moreclearly one could see it from Fig. 4 where electron z-coordinate (solid red), gamma-factor (dash-dot green)and local gyro-radius ρ localgyro (blue diamond) are displayedas the functions of time for the case of Fig. 3 without f and f components (notice that ρ localgyro in Fig. 4 is notgoing to infinity when B ( t ) = 0 because of discrete sam-pling). It shows that the electron begins to effectivelyexchange energy with laser when it performs large excur-sion away from the electric node and into the region withstrong electric field.In conclusion, we demonstrate that widely used as-sumption that for strong radiation friction η f >
1, theleading role in dynamics of highly relativistic electron isplayed by f component of the Landau-Lifshitz radiationfriction force [see Eq. (2)] is incorrect for the case of di-verging electron trajectory. For latter case f and f com-ponents of the radiation friction force, having the sameorder as f component in equation of electron motion fortrajectories (7), are equally important for electron bothtrajectory and energy gain.This work has been supported by the University ofCalifornia Office of the President Lab Fee grant number LFR-17-449059. FIG. 4. The logarithms of electron gamma-factor (dashedgreen) and local gyro-radius ρ localgyro (blue diamond) (quanti-fied by the left y axis) and electron coordinate z (solid red)(corresponding to the right y axis) versus time for case ofFig. 2 without f and f components.[1] L. D. Landau and E. M. Lifshitz, The Classical Theoryof Fields, V. 2, Course of Theoretical Physics (Elsevier,2009).[2] Y. B. Zel’dovich, Sov. Phys. Uspekhi , 79 (1975).[3] E. Sarachik and G. Schappert, Phys. Rev. D , 2738(1970).[4] A. Zhidkov, J. Koga, A. Sasaki, and M. Uesaka, Phys.Rev. Lett. , 185002 (2002).[5] S. V. Bulanov, T. Z. Esirkepov, M. Kando, J. K. Koga,and S. S. Bulanov, Physical Review E , 056605 (2011).[6] A. R. Bell and J. G. Kirk, Phys. Rev. Lett. , 200403(2008).[7] G. Lehmann and K. H. Spatschek, Phys. Rev. E. ,056412 (2012).[8] S. V. Bulanov, T. Z. Esirkepov, J. Koga, and T. Tajima,Plasma Phys. Rep. , 196 (2004).[9] J. G. Kirk, A. R. Bell, and I. Arka, Plasma Phys. Contr. Fusion , 085008 (2009).[10] A. Gonoskov, A. Bashinov, I. Gonoskov, C. Harvey,A. Ilderton, A. Kim, M. Marklund, G. Mourou, andA. Sergeev, Phys. Rev. Lett. , 014801 (2014).[11] T. Z. Esirkepov, S. S. Bulanov, J. K. Koga, M. Kando,K. Kondo, N. N. Rosanov, G. Korn, and S. V. Bulanov,Phys. Lett. A , 2044 (2015).[12] S. V. Bulanov, T. Z. Esirkepov, J. K. Koga, S. S. Bu-lanov, Z. Gong, X. Q. Yan, and M. Kando, J. PlasmaPhys. , 905830202 (2017).[13] L. L. Ji, A. Pukhov, I. Y. Kostyukov, B. F. Shen, andK. Akli, Phys. Rev. Lett. , 145003 (2014).[14] M. Tamburini, F. Pegoraro, A. D. Piazza, C. H. Keitel,and A. Macchi, New J. Phys. , 123005 (2010).[15] S. M. Mahajan, F. A. Asenjo, and R. D. Hazeltine, Mon.Not. R. Astron. Soc. , 4112 (2015).[16] M. Wen, H. Bauke, and C. H. Keite, Sci. Rep.6