Inverse Faraday Effect driven by Radiation Friction

A collective, macroscopic signature to detect radiation friction in laser-plasma experiments is proposed. In the interaction of superintense circularly polarized laser pulses with high density targets, the effective dissipation due to radiative losses allows the absorption of electromagnetic angular momentum, which in turn leads to the generation of a quasistatic axial magnetic field. This peculiar"inverse Faraday effect"is investigated by analytical modeling and three-dimensional simulations, showing that multi-gigagauss magnetic fields may be generated at laser intensities $>10^{23}~\mbox{W cm}^{-2}$.

A collective, macroscopic signature to detect radiation friction in laser-plasma experiments is proposed. In the interaction of superintense circularly polarized laser pulses with high density targets, the effective dissipation due to radiative losses allows the absorption of electromagnetic angular momentum, which in turn leads to the generation of a quasistatic axial magnetic field. This peculiar "inverse Faraday effect" is investigated by analytical modeling and three-dimensional simulations, showing that multi-gigagauss magnetic fields may be generated at laser intensities > 10 23 W cm −2 .
The development of ultrashort pulse lasers with petawatt power has opened new perspectives for the study of high field physics and ultra-relativistic plasmas [1,2]. In this context, the longstanding problem of radiation friction (RF) or radiation reaction has attracted new interest. RF arises from the back-action on the electron of the electromagnetic (EM) field generated by the electron itself and plays a dominant role in the dynamics of ultra-relativistic electrons in strong fields. A considerable amount of work has been devoted both to revisiting the RF theory [3] and to its implementation in laser-plasma simulations [4][5][6][7][8], as well as to the study of radiationdominated plasmas in high energy astrophysics, see e.g. Refs. [9].
While RF is still an open matter both for classical and quantum electrodynamics [2], RF models have not been discriminated experimentally yet. This circumstance led to several proposals of devoted experiments providing clear signatures of RF, e.g. in nonlinear Thomson scattering [10], compton scattering [11], modification of Raman spectra [12], electron acceleration in vacuum [13], radiative trapping [14] or γ-ray emission from plasma targets [15]. Most of these studies are based on single particle effects, and RF signatures are found in modifications of observables such as emission patterns and spectra when RF is included in the modeling. Detecting such modifications may require substantial improvements in reducing typical uncertainities in laser-plasma experiments. Instead, in this Letter we propose to use a collective, macroscopic effect induced by RF, namely the generation of multi-gigagauss, quasi-steady, axial magnetic fields in the interaction of a circularly polarized (CP) laser pulse with a dense plasma. This is a peculiar form of the Inverse Faraday effect (IFE) [16] and may be more accessibile experimentally than single-particle effects. In fact, the IFE has been previously studied in different regimes of laser-plasma interactions [17-19, and references therein]. By using three-dimensional (3D) particle-in-cell (PIC) simulations, we find that at laser intensities foreseeable with next generation facilities pro-ducing multi-petawatt [20] or even exawatt pulses [21], the magnetic field created by the RF-driven IFE in dense plasma targets reaches multi-gigagauss values with a direction dependent on the laser polarization, which confirms its origin from the "photon spin". The magnetic field is slowly varying on times longer than the pulse duration and may be detected via optical polarimetry techniques [22], providing an unambiguous signature of the dominance of RF effects. The effect might also be exploited to create strongly magnetized laboratory plasmas in so far unexplored regimes (see e.g. [23]).
The IFE is due to absorption of EM angular momentum [24], which in general is not proportional to energy absorption. As an example of direct relevance to the present work, let us consider a mirror boosted by the radiation pressure of a CP (with positive helicity, for definiteness) laser pulse. From a quantum-mechanical point of view, the laser pulse of frequency ω propagating alongx corresponds to N incident photons with total energy Nhω and angular momentum Nhx. If the mirror is perfect, N is conserved in any frame. If the mirror moves alongx, the reflected photons are red-shifted leading to EM energy conversion into mechanical energy (up to 100% if the mirror velocity ∼ c) but there is no spin flip for the reflected photons, hence no absorption of angular momentum. However, if the electrons in the mirror emit high-frequency photons, a greater number of incident low-frequency photons must be absorbed with their angular momentum. From a classical point of view, absorption of angular momentum requires some dissipation mechanism [19] which, in our example, implies a nonvanishing absorption in the rest frame of the mirror.
In the case here investigated, effective dissipation is provided by the RF force which makes the electron dynamics consistent with the radiative losses. In order to demonstrate IFE induced by RF, we consider a regime of ultra-high laser intensity I L > 10 23 W cm −2 and thick plasma targets (i.e. with thickness much greater than the evanescence length of the laser field) where the radiative energy loss is a large fraction of the laser energy as shown by simulations with RF included [25][26][27][28]. We use a simple model to account for such losses and provide a scaling law with the laser intensity. The power radiated by an electron moving with velocity v x along the propagation axis of a CP pulse of amplitude E L = (m e ωc/e)a 0 ≡ B 0 a 0 (with ω the laser frequency) is Since at the relevant frequencies ω rad ≃ a 2 0 ω the radiation is incoherent, the total radiated power by N comoving electrons will be N P rad . For thin targets accelerated by the CP laser pulse ("light sail" regime), all electrons move with the foil at v x ≃ c, and there is no high-frequency oscillation driven by the v × B force. Thus the radiation is strongly suppressed by the factor (1 − v x /c) 2 ≪ 1, as observed in simulations [6,29]. In contrast, RF losses become very important for thick targets [25,27,28] ("hole boring" regime) because the acceleration of the plasma surface has a pulsed nature [26,30,31] with a dense bunch of electrons being periodically dragged towards the incident laser pulse, i.e. in a counterpropagating configuration (v x < 0).
In order to estimate the number of radiating electrons per unit surface we consider the dynamic picture of hole boring [30,32]. As illustrated in Fig.1, at the surface of the plasma the radiation pressure generates a positively charged layer of electron depletion (of thickness d) and a related pile-up of electrons in the skin layer (of thickness ℓ s ), i.e. the evanescent laser field region. Ions are accelerated in the skin layer leaving it at a time τ i at which an ion bunch neutralized by accompanying electrons is formed. At this instant, the equi-librium between ponderomotive and electrostatic forces is lost and the excess electrons in the skin layer will quickly return back towards the charge depletion region. The number per unit surface of returning electrons is N x = (n p0 − n 0 )ℓ s where n p0 is the electron density in the skin layer at the beginning of the acceleration stage. Using the model of Refs. [30,32] N x may be estimated from the balance of electrostatic and radiation pressures: eE d n p0 ℓ s /2 = 2I L /c where E d = 4πen 0 d is the peak field in the depletion region, and n p0 ℓ s = n 0 (d + ℓ s ) because of charge conservation. Solving these equations in the limit n p0 ≫ n 0 we obtain N x ≃ a 0 /r c λ where λ = 2πc/ω is the laser wavelength, r c = e 2 /m e c 2 and we used I L = m e cω 2 a 2 0 /(4πr c ). Thus, the total radiated intensity is I rad = P rad N x . In order to compare with the laser intensity I L we take into account that the radiation is emitted as bursts corresponding to the periodic return of electrons towards the laser, i.e. for a fraction f τ ≃ τ e /(τ e + τ i ) of the interaction stage where τ e is the time interval during which the electrons move backwards. Analysis of laser piston oscillations in Ref. [26] suggests that τ e ≃ τ i so we take f τ ≃ 1/2 for our rough estimate. Assuming (1 − v x /c) 2 ∼ 1 we obtain for the fraction of radiated energy to the laser pulse energy If the energy of electrons is mainly due to the motion in the laser field, then γ ≃ (1 + a 2 0 ) 1/2 ∼ a 0 for a 0 ≫ 1 and η rad ∝ a 3 0 . For λ = 0.8 µm, η rad ∼ 1 for a 0 ∼ 400, corresponding to I L ∼ 7 × 10 23 W cm −2 . This orderof-magnitude estimate implies that for such intensities a significant part of the laser energy is lost as radiation, strongly affecting the interaction dynamics. A more precise estimate would require to account both for the energy depletion of the laser and for the trajectory modification of the electrons due to the RF force.
A 3D approach is essential to model the phenomena of angular momentum absorption and magnetic field generation, thus we rely on massively parallel PIC simulations in which RF is implemented following the approach described in Ref. [6] (see Ref. [8] for a benchmark with other approaches).
The laser pulse is initialized in a way that at the waist plane x = 0 (coincident with the target boundary) the normalized amplitude of the vector potential a = eA/m e c 2 would be a(x = 0, r, t) = a 0 (ŷ cos(ωt) ±ẑ sin(ωt)) where r = (y 2 + z 2 ) 1/2 . Both radial profiles with n = 2 (Gaussian, G) and n = 4 (super-Gaussian, SG) have been used in the simulations. For all the results shown below, we take r l = 3λ and radius r 0 = 3.8λ. The plus and minus sign in the expression for a correspond to positive and negative helicity, respectively. The pulse energy is given  microns and a polarity inverting with the pulse helicity is generated. The comparison of Fig.2 d) and e) shows that B x has similar values and extension for a Gaussian pulse. The field is slowly varying over more than a ten laser cycles (∼30 fs) time, with no sign of rapid decay at the end of the simulation, as shown in Fig.2 f).
The fraction η rad of the laser energy dissipated by RF reaches values up to η rad ≃ 0.24 for a 0 = 600 as shown in Fig.3 a) . A fit to the data gives η rad ∝ a 3.1 0 , close to the η rad ∼ a 3 0 prediction of our model. Fig.3 a) also shows the peak magnetic field B max scaling as ∼ a 3.8 0 up to the highest value B max ≃ 28B 0 = 3.75 GG for a 0 = 500. The decrease down to B max ≃ 22B 0 for a 0 = 600 is related to the early interruption of the hole boring stage due to the breakthrough of the laser pulse through the target as observed in this case. Notice that we do not show simulations for a 0 < 200 since in such case the RF losses become too close to the percentage of energy which is lost due to numerical errors ( < ∼ 1%). However, the inferred scaling would predict B max ∼ 8 MG for a 0 = 100, which may be still detectable making an experimental test closer. To sketch an analytical model for IFE, let us first observe that the density of angular momentum of the laser pulse L z = r×(E×B)/4πc = −r∂ r I L (r)/(2cω), with I L (r) the radial profile of the intensity, vanishes on axis and has its maximum at the edge of the beam. We thus consider angular momentum absorption to occur in a thin cylindrical shell of radius R ≃ r 0 , thickness δ ≪ R, and length h. The temporal growth of the axial field B x induces an azimuthal electric field E φ , which in turn allows the absorbed angular momentum to be transfered from electrons to ions. Assuming that the electron and ion shells rotate with angular velocities Ω e,i , respectively, we may write for the angular momenta L e = I e Ω e and L i = I i Ω i where I e = 2πR 3 δhm e n e and I i = (Am p /Zm e )I e are the momenta of inertia for electrons and ions, respectively. The global evolution of the angular momenta of electrons and ions may be described by the equations where M abs is the torque due to angular momentum absorption (related to the absorbed power P abs by M abs = P abs /ω) and M E is the torque due to E φ : The rotation of the electrons induces a current density j eφ ≃ −en e Ω e R. Neglecting the displacement current, in the limiting case h ≫ R the field B x ≃ 4πj eφ δ/c and it is uniform as in a solenoid. In the opposite limit h ∼ δ ≪ R, the current distribution may be approximated by a thin wire of cross-section ∼ hδ, and E φ (R) can be obtained via the self-induction coefficient of a coil [33].
We thus obtain where ω 2 p = 4πn e e 2 /m e . The geometrical factor which shows that the electron rotation follows promptly the temporal profile of M abs (t), and that effect of the inductive field on electrons is equivalent to effective inertia. Since in our conditions I ′ e ∼ (ω 2 p /ω 2 )I e = (n e /n c )I e ≫ I e , the l.h.s. term in Eq.(4) can be neglected and M E ≃ M abs holds. Thus, from Eq.(4) we obtain i.e. the total angular momentum of ions is much larger that of electrons. This is in agreement with the simulation results [ Fig.3 b)]. In turn, posing M E ≃ M abs in Eq.(5) and using The total angular momentum absorbed L abs = U abs /ω where the absorbed energy is U abs ≃ η rad U L , assuming RF as the main source of dissipation. We thus estimate the final magnetic field as The product n e h is the surface density of the region where dissipation and angular momentum absorption occur. Thus, with reference to Fig.1 we may estimate n e h ≃ n p0 ℓ s ≃ (I L /πe 2 c) 1/2 = 2n c a 0 c/ω (for n p0 ≫ n 0 ). Noticing that B 0 /en c = 2λ we eventually obtain If η rad ∝ a 3 0 then B xm ∝ a 4 0 , in good agreement with the observed scaling in Fig.3 a). If we pose R ≃ r 0 , the laser initial beam radius, and δ ≃ λ, the radial width of the angular momentum density, for a 0 = 500, η rad = 0.16 and G = 1 Eq.(11) yields B xm ≃ 4.8B 0 . The discrepancy with the observed value of ≃ 28B 0 may be attributed to the nonlinear evolution and self-channeling of the laser pulse in the course of the hole boring process. For instance, Fig.2 shows that the magnetic field is generated in a region of radius ∼ 2λ. Further analysis of the simulation data shows both a slight increase (by a factor ∼ 1.2) of the laser amplitude on the axis and a localizaton of the densities of both EM and mechanical angular momenta in a narrow layer of ∼ 0.5λ width. Posing R ≃ 2λ, δ ≃ 0.5λ and an effective a 0 ≃ 600 in the above estimate yields B xm ≃ 23B 0 , which is in fair agreement with the simulation results considering the roughness of the model.
In conclusion, we showed in 3D simulations that in the interaction of superintense, circularly polarized laser pulses with thick, high density targets the strong radiation friction effects lead to angular momentum absorption and generation of multi-gigagauss magnetic fields via the Inverse Faraday effect. Simple models for the efficiency of radiative losses, the transfer of angular momentum to ions and the value of the magnetic field are in fair agreement with the simulation results for what concerns both the scaling with intensity and order-ofmagnitude estimates. With the advent of multi-petawatt laser systems, the investigated effect may provide a laboratory example of radiation-dominated, strongly magnetized plasmas and a macroscopic signature of radiation friction, providing a test bed for related theories.
Suggestions from D. Bauer are gratefully acknowledged. The simulations were performed using the computing resources granted by the John von Neumann-Institut für Computing (Research Center Jülich) under the project HRO01. T.V.L. acknowledges DFG within the SFB 652. S.P. acknowledges support of the excellence center for applied mathematics and theoretical physics within MEPhI Academic Excellence Project (contract No. 02.a03.21.0005, 27.08.2013).