Efficiency of radiation friction losses in laser-driven"hole boring"of dense targets

In the interaction of laser pulses of extreme intensity ($>10^{23}~{\rm W cm}^{-2}$) with high-density, thick plasma targets, simulations show significant radiation friction losses, in contrast to thin targets for which such losses are negligible. We present an analytical calculation, based on classical radiation friction modeling, of the conversion efficiency of the laser energy into incoherent radiation in the case when a circularly polarized pulse interacts with a thick plasma slab of overcritical initial density. By accounting for three effects including the influence of radiation losses on the single electron trajectory, the global `hole boring' motion of the laser-plasma interaction region under the action of radiation pressure, and the inhomogeneity of the laser field in both longitudinal and transverse direction, we find a good agreement with the results of three-dimensional particle-in-cell simulations. Overall, the collective effects greatly reduce radiation losses with respect to electrons driven by the same laser pulse in vacuum, which also shift the reliability of classical calculations up to higher intensities.


Introduction
The continuous progress of laser techniques making higher and higher electromagnetic intensities accessible for experiments has stimulated the growth of research areas such as relativistic dynamics and nonlinear optics in classical plasmas [1] and quantum electrodynamics in extremely strong fields [2,3]. Radiation friction (RF) is a problem of central interest in both the above mentioned areas. In the classical context, a modification of the Newton-Lorentz equation of motion for an electron by adding a new force term, named the RF force (RFF) or radiation reaction force, is necessary to make the electron dynamics self-consistent with the emission of radiation. Although the correct form of the RFF has been the subject of intense debate for decades and until recently [4,5], it now appears that in the classical limit the Landau-Lifshitz (LL) expression [6] gives a correct and consistent description [7,8]. The LL expression of the RFF has become the basis of classical simulations of superintense laser-plasma interaction [9,10] where RF losses (corresponding to the escape of high-frequency, incoherent radiation from the plasma) are important enough to affect the plasma dynamics.
When the frequency of the emitted radiation becomes sufficiently high that the energy and momentum of single photons are not negligible with respect to those of the radiating electron, a quantum electrodynamics (QED) description becomes necessary. However, a correct and effective description of "quantum RF" is an open issue. The first two experiments claiming for evidence of quantum RF signatures in nonlinear Thomson scattering of superintense laser pulses by ultrarelativistic electrons [11,12] came to somewhat different conclusions about which model better described the experimental results (see [13] for a discussion). Notice that these experiments involved laser-plasma physics in the generation via wakefield acceleration of a dense, short duration bunch of relativistic electrons in order to increase the luminosity in the gamma-ray region; however, the dynamics of the laser-bunch interaction was of single particle nature.
An alternate approach to investigate RF in the laboratory is to search for regimes where collective effects in the laser-plasma interaction boost radiation losses, so that RF signatures may become strong and unambigous. Several simulation works have shown highly efficient radiation losses (a few tens per cent of the laser pulse energy) in the interaction of circularly polarized (CP) pulses with dense thick targets [14,15,16,17,18,19]. In our previous work [19] we have made a first attempt of a classical model to estimate the conversion efficiency η rad of the laser energy into uncoherent radiation. In turn, the efficient absorption of CP light causes a strong transfer of angular momentum to the target, with the generation of ultrahigh magnetic fields (inverse Faraday effect) with strength achieving several Giga-Gauss which can provide a macroscopic signature of RF [19].
In Ref. [19] the scaling of η rad with the laser intensity agreed reasonably with the results of three-dimensional particle-in-cell (PIC) simulations of the laser-plasma interaction, up to intensities approaching 10 24 Wcm −2 . Beyond this limit, however, the model predicts unphysical values of η rad > 1 because neither the modification of the radiating electron trajectories due to RF nor the depletion of the laser pulse are taken into account. In addition, and more in general, at those intensities the classical description becomes questionable and quantum effects are expected to become relevant.
The aim of this paper is to provide an accurate estimate of η rad via analytical modeling. First, we use the solution by Zeldovich [20] to take self-consistently into account the RF losses into the electron trajectory. Then, we show that the amount of RF losses is strongly affected by the average motion of the plasma surface, the finite evanescence length of the electromagnetic (EM) field in the plasma, and the radially inhomogeneous distribution of the laser intensity. By accounting for these effects, analytical estimates in good agreement with the results of three-dimensional (3D) simulations are obtained. We also provide an estimate of the intensity limit under which, in the present context, RF can still be treated within classical electrodynamics.

Review of previous modeling and its limitations
In the regime of interest here, an ultraintense laser pulse of frequency ω and dimensionless field amplitude a 0 = eE L /m e ωc (with E L the electric field amplitude) interacts with a strongly overdense (electron density n e n c = m e ω 2 /4πe 2 , the cut-off density) plasma target which remains opaque to the laser light. The radiation pressure of the laser light is high enough to produce "hole boring " (HB) in the target, i.e. the plasma surface is driven at an average velocity where I L = cE 2 L /4π = m e c 3 n c a 2 0 is the laser intensity. Eq.(1) can be obtained by balancing the mass and momentum flows at the surface [21] and is valid for total reflection of the laser light in the frame co-moving with the surface, i.e. in the absence of dissipative effects. If a fraction η of the laser intensity is dissipated, for example due to RF losses, Eq.(1) may be modified by replacing I L with I L (1 − η/2). In the case of our simulations this would lead at most by a 5% decrease in v HB at the highest intensity considered (a 0 = 800).
In order for the interaction to remain in the HB regime during the whole duration of the laser pulse, the target must be "thick" enough that v HB τ L < D, where τ L is the laser pulse duration and D is the target thickness. In the opposite "thin" target limit v HB τ L D, the target can be accelerated as a whole and enter the "light sail" (LS) regime [22,23], where the scaling of the velocity v LS with intensity becomes much faster than (1). Thus, the same laser pulse parameters may enable to reach velocities v LS c in an ultrathin target while yielding v HB to be a fraction of c in a thick target. In particular, for the parameters of calculations presented below v HB ≈ (0.3 ÷ 0.6)c. The different acceleration regime may therefore explain the huge difference in the radiation efficiency between thick and thin targets. In fact, assuming that the electrons radiate in the field of a plane electromagnetic CP wave propagating along x, the radiated power is [24] where γ = 1/ 1 − v 2 /c 2 and v x is the velocity component parallel to the wavevector. Assuming that most of the radiating electrons co-move with the target ions, the factor (1 − v x /c) 2 leads to strong suppression of radiation emission for thin targets moving at v LS c, while the suppression is much less severe for thick targets as far as v HB is significantly smaller than c.
In the thick target case, the laser field penetrates into the skin layer where the electrons pile up under the action of the radiation pressure. The areal density of electrons in the skin layer can be estimated as [19,25] where r 0 = e 2 /mc 2 is the classical electron radius. For a 0 1, by estimating γ a 0 we obtain the radiated power per unit surface as I rad = N x P rad ∝ a 5 0 , which implies a ∝ a 3 0 scaling for the radiation loss efficiency, in good agreement with the simulation results.
For v x = 0, and assuming that the duration of the uncoherent high-energy emission is the same as the laser pulse, the conversion efficiency defined as a ratio of the energy emitted by radiating electrons U rad to that of the laser pulse U L is thus given by where the parameter is introduced, and λ is the laser wavelength. In Ref. [19] it was suggested that for thick targets an enhancement of radiation losses may originate from the non-steady dynamics of HB acceleration [25]. In particular, ion acceleration by the space-charge field causes a pulsed "collapse" of the electron density with the excess electrons returning towards the laser with negative velocity v x < 0, enhancing the RF losses by a sequence of radiation bursts. However, it is not straightforward to provide analytical estimates for either the rate of the bursts or the value of v x for the returning electrons. In particular, estimating v x would require to find the motion of the returning electrons in an inhomogeneous electric field with the radiation friction force included. For an order-of-magnitude estimate, we simply assumed (1 − v x /c) 2 1 and the number of the returning electrons to be N x [19] (i.e. most of the electrons in the skin layer to collapse). This leads again to an expression like (4) for the conversion efficiency, apart from a reduction factor < 1 accounting for the fact that the returning electrons radiate only for a fraction of the interaction time.
Apparently, the η rad ∼ a 3 0 scaling fairly agrees with the results of 3D simulations which give η rad ∼ a 3.2 0 up to intensities a 0 500, but the absolute value predictions of (4) are much higher than those observed in the simulations. This is not surprising since obviously (4) becomes invalid when approaching a critical value of the laser field amplitude where η rad 1, which is unphysical. In the simulation [19], η rad ≈ 0.08 for a 0 400. The quantitative disagreement makes also not possible, on the basis of the predicted scaling only, to understand whether the radiation is mostly due to electrons either remaining in the skin layer or returning towards the laser. The very limited nature of the estimate (4) for the conversion efficiency is due to several underlying shortcomings, such as the neglect of self-consistent RF effects on the electron motion, the absence of a more precise estimate of v x , and the inhomogeneity of the laser field in both the longitudinal and transverse directions. In the following we show that accounting for these effects, even if still in an approximated way, leads to a considerably smaller growth of the conversion efficiency at high intensities than that given by (4) and therefore substantially improves the agreement with the simulations.

Self-consistent electron motion
The model first introduced by Zeldovich [20] describes a stationary electron motion in the field of a strong circularly polarized plane wave with RF effects included self-consistently. Since RF allows absorption of momentum from the plane wave, a drag force is exerted on the electron along the direction of wave propagation (x for definiteness). Thus, in order to obtain a stationary solution an electric field E d along x is introduced in the model balancing the radiation drag. The complete EM fields are thus given by In the stationary regime an electron moves along a circle in the (y, z) plane and drifts along the x axis with a constant velocity: The phase shift θ is generated by the RFF. Neglecting the latter in the equations of monition gives θ = 0. For ultra-relativistic particles, the RFF is given by [6] F with the radiation power which differs from (2) by the replacement a 2 0 → γ 2 reflecting the fact that the circular motion of the electron is now determined jointly by the Lorentz and the RF forces. In the stationary regime the total force (with the centrifugal component included) vanishes. Projecting this condition on the axes of cylindrical coordinates and assuming that the value of the longitudinal electric field E d is known we obtain three equations which determine the values of γ, θ and v x : In principle the system of Eqs. (11-12-13) might be applied to study the motion of electrons in the space-charge field created by the ponderomotive force action, see examples e.g. in [26] where an approximate analytic description for the case of standing waves was developed. However, such space-charge field is highly inhomogeneous, which would already make an analytical estimate difficult. In addition, in the case under investigation the electron density is high enough for screening effects to be non-negligible: considering as an example the contribution of returning electrons, as those located exactly at the plasma-vacuum boundary return towards the incoming laser, the spacecharge field is partially canceled so that the electrons filling in inner layers will experience a lower force. A complete description of this scenario would require to resolve the electron plasma dynamics with RFF included. Since our primary aim is to relate the radiation losses to an average value of v x determined by the laser-plasma dynamics, we take v x as a parameter in the system, and following Zeldovich [20] we solve Eqs.(11)- (13) in the reference frame moving with the instant velocity v x of the radiating electron. In the following, we use the notations γ , v 0 , ξ , etc. for values measured in this reference frame. Setting v x = 0 and taking into account that γ 1, one may safely put v 0 ≈ c. Eliminating the angle θ from Eqs. (12), (13) we obtain an equation determining γ (ξ , a 0 ) (note that a 0 is relativistically invariant) [20]: For low intensities, a 0 a cr it gives γ = a 0 , as was used in [19]. In the opposite limit, a 0 a cr , the gamma-factor grows much slower with a 0 : Eq.(15), previously obtained in Ref. [27], corresponds to the limit in which the oscillation energy of the electron in the EM field equals the energy radiated per cycle. Remarkably, this single particle result corresponds, in our model where collective effects enter via (3) for the number of radiating electrons, to a total conversion of the laser pulse energy into radiation from the target. In fact it follows from Eqs. (14), (10) and (3) that so that η rad → 1 for a 0 a cr . Notice that ξ is determined by the laser wavelength measured in the laboratory frame.
We compare predictions of our model to the results of 3D PIC simulations (see [19] for the numerical set-up) which describe the interaction of a laser pulse with a plasma of thickness D > 10λ and initial density n 0 = 90n c . The supergaussian laser pulse is introduced via the time-dependent boundary condition at the plasma surface, x = 0, as described in [19] a(r, x = 0, t) = a 0 (y cos(ωt) + z sin(ωt))e −(r/r 0 with r = y 2 + z 2 , r 0 = 3.8λ and r L = 3.0λ. In our PIC calculations we varied the laser amplitude in the interval a 0 = 300 ÷ 750 which corresponds to the peak intensities (3.8 ÷ 23.7) · 10 23 W/cm 2 . As is seen on Fig.1, the values of η rad obtained from (16) for v x = 0 qualitatively reproduce the behavior of conversion efficiency extracted from the PIC simulation (shown by diamonds) in the whole interval of a 0 , although the absolute values appear considerably overestimated. Below we identify the sources of these differences and improve the model by accounting the respective effects for.

Effects of the longitudinal velocity
An analysis of the 3D distribution functions of the radiation power density P(x, r, v x ) (calculated as P = −n e v·F rad ) and of the electron and ion density n e,i (x, r, v x ) extracted from the PIC simulation shows that most of the emitted radiation comes from electrons having velocities v x > 0, and located close to the receding front of the ion density. This is illustrated for the a 0 = 500 case in Fig.2 where space-time plots in the (x, t) plane are shown for the radiation power and the particle densities at r = 1λ, where the former has its radial maximum. The density fronts move in the forward direction with average velocity 0.41c, in fair agreement with the value v HB = 0.47c given by Eq.(1). Small oscillations in the front position are visible in correspondence of the generation of plasma bunches in the forward direction, as discussed in Ref. [25]. The power density plot shows that most of the emission originates close to the hole boring front. Emission due to returning electrons with velocity −c is visible after t = 11T L , but its contribution to the total emitted power is small, presumably because of the low density in the returning jets (as seen the n e (x, t) plot). Consistently with these observations, we assume that on the average the radiating electrons move with velocity v x = v HB given by (1). In this way, we obtain a result shown on Fig.1 by a

Effects of field inhomogeneity
Finally, we account the attenuation of the laser field in the plasma and the dependence of the laser intensity on time and its radial distribution in the focal spot. The laser field amplitude a 0 is not constant within the evanescence length s , but dropping down, leading to a considerable decrease of the "efficient" value of a 0 entering Eqs. (15) and (16). Fig.3 based on the HB model of [25] sketches the electron and the ion density distributions along the propagation direction at the initial stage of the interaction when the electrons are pushed forward by light pressure, while the ions still remain immobile and homogeneously distributed inside the plasma layer. Taking the electron density for x > d in the form Efficiency of radiation friction losses in laser-driven "hole boring" of dense targets 9 we replace a step distribution employed in Ref. [19] by a decaying exponent. Assuming that n p0 n 0 with n p0 being the maximal density of electrons and n 0 is the initial density equal to that of ions, we obtain for the electric field inside the layer with the maximal value achieved at the electron surface. Taking into account that n p0 s = N x a 0 /r 0 λ (3) we obtain for the maximal longitudinal field where E cl = e/r 2 0 = m 2 c 4 /e 3 = 1.81 · 10 18 V/cm is the critical field of classical electrodynamics which is 1/α = 137 times greater than that of quantum electrodynamics E cr = m 2 c 3 /e . Note that for λ 1µm, E d E cr at a 0 (400ξ) −1 ≈ 1.6·10 5 , according to (21), so that in this case E d E L E cr .
Within the same approximation the local equilibrium condition for the electrons inside the layer requires that the laser field amplitude drops accordingly, a(x) = a 0 exp(−(x − d)/ s ). Then the global equilibrium condition for the whole layer reads Here we take into account that the intensity of reflected radiation is (1 − η rad )I L . The arial radiation power (intensity) is where the power P (x, v x = 0) is given by (10) in the reference frame co-moving with the electrons and γ (ξ , a 0 ) is expressed from Eq. (14), so that This gives the following equation for the conversion efficiency with an approximate solution which employs the fact that η rad /2 1 up to very high values of a 0 ; in particular, at a 0 = 750 which was at the limit of our numerical calculation, η rad /2 ≈ 0.11. In the limiting cases of weak and strong fields the integral in (24) can be solved analytically giving and η rad → 0.78 , a 0 → ∞ .
The latter number is obtained directly from (25), as the approximation η rad /2 1 is no longer valid in this limit. As is clearly seen from (27) in the limit of low intensities (which on practice means the laser field amplitude up to a 0 400), the field attenuation inside the plasma layer leads to further suppression of the radiation losses by a factor 0.4. A similar suppression effect emerges due to the laser amplitude dependence on the transverse coordinate and time. Assuming that the dimensionless laser amplitude in the focal waist possess axial symmetry a(r, t) = a 0 g(r/r 0 , ct/r L ) and integrating the radiation power over the transverse coordinate and time we obtain that the function f (ξ , a 0 ) in (26) is replaced by the factor S(ξ , a 0 ) = g 5 (ρ, τ )f (ξ , a 0 g(ρ, τ ))dρdτ where ρ = (r/r 0 ) 2 and τ = ct/r L . Finally apparently leading to additional suppression of the convergence efficiency. For the supergaussian pulse (17)  (− ln(y 2 /y 1 )) 3/4 f (ξ , a 0 y 2 ) .(32) In the strong field limit f (a) ∼ 1/a 3 so that the integrands in (30) are proportional one to another, leaving the limit (28) unchanged. Instead, in the weak-field limit f ≈ 1/5, which gives for (32) S(a 0 a cr ) ≈ 2 3/4 /5 7/4 , and consequently η rad ≈ 0.20ξ a 3 0 . The resulting dependence η rad (a 0 ) calculated for a supergaussian pulse (17) along (31) and (32) is shown on Fig. 1 by a solid red line and demonstrates an impressive improvement of (16): in the interval of intensities a 0 = 400 ÷ 800 the calculated values do not deviate from the PIC result by more than 20%. Residual discrepancies may largely be ascribed to the fact that (1) tends to overestimate the actual recession velocity since complete reflection is assumed. Notice that radiation losses also contribute to decrease the reflectivity R = 1 − η rad and hence reduce the recession velocity, which is principle may create a positive feedback for the enhancement of radiation emission. However, since η rad is quite smaller than unity, these effects appear not to play a significant role.

Extension of the classical regime of interaction towards higher intensities
Although the radiation losses appear high compared to those in the "light sail" regime, their significant relative suppression caused by the RFF leads to a specific freezing of the electron lateral motion, so that the relativistic γ-factor grows much slower (15) than in the perturbative domain a 0 a cr where the RFF is negligible. This in turn shifts the boarder between the classical and the quantum regime of interaction to considerably higher intensities. Quantum effects become dominant when the characteristic energy of emitted photons approaches the electron kinetic energy. For electrons moving along the circle with the period T = 2π/ω this condition reads: restricting the classical domain by: For λ = 800nm Eq.(34) gives γ q ≈ 600. When the effect of the RFF is not accounted for, γ = a 0 , so that the classical regime is restricted by a 0 < 600, corresponding to intensity I L = 2 · 10 24 W/cm 2 . This intensity is only 2.5 times higher than that corresponding to a = a cr ≈ 400 (6) when the RFF starts playing the dominant role in the electron dynamics. However, as follows from Eq. (14), the effect of the the RFF itself alters the function γ(a 0 ) in the domain a 0 > a cr . As a result, the condition (34) reads a 0 ≤ a q ≡ a 0 (γ q ) ≈ 2 · 10 3 , shifting the boundary of the quantum domain to considerably higher intensities, i.e., ∼ 2 · 10 25 W/cm 2 in our case. Moreover, the effect of the longitudinal velocity on ξ as considered in Section 4, will lead to a further slowdown in the increase of γ with a 0 , shifting the classical boundary towards even higher intensities. Note that a cr ∼ λ 1/3 and a q ∼ λ, so that the domain of intensities where the RFF dominates the plasma dynamics and remains classical expands approximately ∼ λ 2 with increasing of the laser wavelength.

Conclusions
In conclusion, we have presented a selfconsistent analytic model for the interaction of superintense laser pulses with thick plasma in the hole boring regime. The inclusion of the RFF along the lines of Zeldovich's work [20] allowed calculating the conversion efficiency of the laser energy into high frequency radiation in the wide range of intensities.
After accounting the effects of (a) the global hole boring motion of the plasma and (b) of the laser field inhomogeneity in space and time, our result demonstrated a good quantitative agreement with the outcome of the PIC simulation. The effect of the RFF, in combination with the factors (a) and (b), results in a much slower (compared to predictions made in [19]) increase of the conversion efficiency with the laser intensity, so that η ≈ 0.25 at I L = 3 · 10 24 W/cm 2 . Our analysis also predicts that, within the setup which provides the hole boring regime of interaction, quantum corrections to the RFF will hardly be significant at intensities I L < 10 25 W/cm 2 .