Effectiveness and Hydrodynamic Stability of Low-Entropy Laser-Driven Acceleration of Thin Foil.

Transcription

1 Effectiveness and Hydrodynamic Stability of Low-Entropy Laser-Driven Acceleration of Thin Foil. Sergey Gus kov and Masakatsu Murakami INTRODUCTION The importance to understand of effectiveness limits of laser-driven acceleration of a projectile in a dense state due to elaboration of fast ignition concept [1,], promising approach of inertial confinement fusion (ICF). The fast ignition approach based on impact of preliminary compressed thermonuclear fuel of ICF target by a laser-accelerated projectile was developed for impact fast ignition [3,4], actively studied now. To be able ignite DT-fuel the projectile must be accelerated up to the velocities corresponding to the plasma temperature of 5-1 kev. One can easily show that such a velocity attain V im = (3k B T i /m i ) 1/ (.7-1) 1 8 сm/s (where k B is the Boltzmann constant, T i, the plasma ion temperature, and m i, the plasma ion mass equal to.5 mass of neutron for an equimolar deuterium-tritium mixture). Thermonuclear plasma is created as a result of matter heating and compression by an impact-exited shock waves. The higher projectile density, the higher ignition efficiency. Therefore the impacting projectile should be not only accelerated up to above noted thermonuclear velocity but also it should have the maximal possible density. On this reason so called low entropy regime acceleration when the unablated part of a projectile is heated on the minimal level should be used. Additionally, to increase the final projectile density, the impact fast ignition scheme assumes the acceleration of a spherical layer along a conical channel [3]. The record velocity of laser-driven projectile was achieved in Gekko/HIPER experiments [4] destined to study impact fast ignition. Here the 14- µm thickness CH (5%C, 5%H) or CHBr (5%C, 47%H, 3%Br) foils were irradiated by third harmonic of Nd-laser radiation (λ=.35 µm) at the pulse intensity, I = W/cm and duration,.5 ns. The foil velocities of 6-7 km/s were measured. The given paper is devoted to a theoretical investigation of flat foil acceleration up to ultrahigh velocities of the order of 1 km/s. The research was done in one-dimensional approximation to understand the ultimate possibilities of low-entropy acceleration of projectile by laser pulse as well as to investigate the energetic efficiency and hydrodynamic stability of such an acceleration. The matched laser pulse and foil parameters which are able to provide the ultimate foil velocity were defined. The comparison with the results of numerical simulations and Gekko/HIPER experiments is presented. ULTIMATE VELOCITIES OF LOW-ENTROPY ACCELERATION A general solution of a problem on ablation-driven acceleration of a flat solid layer with initial density ρ when the applied constant-intensity energy pulse evaporates the matter with density ρ a [5] will be used. The solution shows that finial velocity of the layer (at the moment of cessation of the energy pulse) is determined by the sound velocity of ablated matter and the ratio of initial and final masses of condensed layer: $ "+ 1% u( #) = c ln & ' ( " ) a ( # ) (1) where γ is the adiabatic exponent of ablated matter; Δ and Δ(τ) are respectively the initial thickness of a layer and thickness of the unablated part of layer at the moment τ of the energy pulse cessation; sound velocity of the ablated matter c a is expressed through the pulse intensity I and the density of ablated matter (ablation density): 1/ 3 $ & " 1' I % ñ = a ( * + ) (, + 1-# a ). / () Velocity of the ablation wave front D is connected with the sound velocity c a of ablated matter by a relation coming from the requirement of equality of matter flows on the both sides of ablation surface D = a c (3) a According to Eq. (3) the final thickness of unablated layer is % c " & #(") = # 1 a a $ ' # ( ) * (4) The energetic efficiency of acceleration is characterized by the value of hydrodynamic efficiency, that is the ratio of the final kinetic energy of unablated part of the layer to the deposited energy: 1 # 1 & " ' " $ = 1 ln ( ) # * "(%) + "( t) (5) Maximum hydrodynamic efficiency is achieved if

2 Δ /Δ(τ) 5. For γ=5/3 the maximum value of η is.41. If a laser pulse is the external energy source and the target made from light elements then low-entropy acceleration occurs when Iλ -parameter is not higher than W cm - µm. Under these conditions, the laser light is effectively absorbed by inverse Bremsstrahlung mechanism, the ablation density is close to the critical plasma density 3 A " # " $ 1.83 % 1 a cr Z& (where A and Z are the atomic number and average ion charge, respectively), the hydrodynamics and electron heat conductivity make a major contribution to the energy flow at the ablation boundary, and the most effective laser coupling with a blow-off plasma and minimal heating of unablated part of the target are provided. The above solution (1)-(5) is a base of well-known rocket model of ablation-driven acceleration. By use the different approximations for ablation density ρ a this model is extensively and successfully applied to the analysis of laser-driven acceleration under the different conditions of target irradiation, including laser energy conversion to the energy of thermal radiation [6] and fast electrons [7]. The model applied the critical density as ablation density describes very well the results of experiments on a laser-driven acceleration of light-element targets under the conditions of the inverse Bremsstrahlung absorption (see, for example, the books [8], [9]). So, the solution (1)-(5) together with expression (6) is used here. The maximum ablation pressure at the interaction of laser beam with a flat target occurs when a lateral expansion of blow-off plasma is minimal. To provide conditions for a plane expansion of the blow-off plasma it is necessary to make a thermal plasma expansion length during a laser pulse less than radius of the laser beam focused onto the target surface: (6) c " R (7) a L By using Eqs. (1)-(6), requirement (7), and assuming that the ablated matter had been fully ionized, i.e. A/Z, and γ =5/3, we obtain the following relations for the final velocity (normalized for thermonuclear velocity of u f = 1 km/s), for thickness of the layer, and requirement for a plane expansion of the blow-off plasma: ( ) 1/ 3 I u( " ) =.83 # $ ln u ( ", (8) ) f ( ) 1/ 3 I" # $ (#) = 1 % 1.9 &, (9) $ $ ' " ( ) 1/ I" # $ R (1) L where I, τ, ρ, are measured in the units of 1 14 W сm -, ns, and g/сm 3, respectively; λ and Δ - in µm. The higher the sound velocity in the region of laser light absorption, the higher the final velocity of the target. Therefore, the final velocity increases with Iλ -parameter growth (see Eqs. (1) and (6)) and also increases weakly (by a logarithmic law) with increase in the ratio of initial thickness and final thickness of the foil. Let define the condition for plane expansion of the blow-off plasma in respect to the maximum Iλ -parameter, i.e. define the sign of equality in requirement (1). The ratio Δ /Δ(τ) determines unambiguously the hydrodynamic efficiency η (see Eq.(5)). To achieve simultaneously thermonuclear velocities and high hydrodynamic efficiency is of interest. For that reason let define, for certainty, ratio Δ /Δ(τ) as equal to 5, which corresponds to the maximum hydrodynamic efficiency, as noted above. Then Eq. (8) shows that the scale of the ultimate velocity achievable at low-entropy acceleration of a flat projectile actually corresponds to the thermonuclear velocity, and lies in the range of ( ) 1 3 km/s (Iλ =(1 5) 1 14 W сm - µm ). Such a velocity corresponds to impact-produced temperature of deuterium-tritium fuel equal to 1 kev, which is responsible for the thermonuclear reaction rate closed to the maximal value. THE MATCHED LASER PULSE AND FOIL PARAMETERS. The matched laser pulse and foil parameters corresponding to certain final foil velocity at the maximum hydrodynamic efficiency comes from Eqs. (8) (1) at the application of above defined value for Δ /Δ(τ) and the sign in Eq. (1) and introduction the connection of laser intensity with energy, pulse duration, and beam radius: / 3 ( 1/ 3 4 / 3 u( ") ) # $ %.45 & E ' * + L u f * +, - 5 / 3 & 1/ 3 / 3 u( ") ' " #. $ E % ( ) L u f ( ) * + (11) (1)

3 / 3 & 1/ 3 / 3 u( ") ' R # 48 $ E % ( ) (13) L L ( u f ) * + The foil thickness, pulse duration, and laser beam radius are increasing with the laser pulse energy by the same law as E L 1/3. For the fixed energy, each of those parameters decreases with the final velocity increasing, and the pulse duration decreases most strongly. The laser parameters, i.e. the pulse duration and the beam radius, increase with the wavelength, while the target thickness, on the contrary, decreases. The influence of the wavelength on the final state of the accelerated foil is ambiguous. On the one hand, final velocity of the foil increases with growth of wavelength, according to Eq.(8), because plasma temperature near the ablation surface increases with decrease in critical density, and hence, the sound velocity increases too (see Eq.()). On the other hand, with the wavelength increasing more evident become the processes responsible for reduction of the final projectile density due to a hydrodynamic expansion through a free surface of the layer. One of those processes is connected with the electron heat conductivity from blow-off plasma to unablated part of the target. Temperature enhancement near the ablation surface due to an increase in the wavelength leads to an enhancement of the energy flux provided by electron heat conductivity, and thereby, to more intensive heating of the accelerated part of a projectile. The second process represents the heating by a shock wave. The effect of thermal expansion is the greater the lesser the ratio of thermal expansion time t ex to laser pulse duration τ. The time t ex is scaled as the ratio of foil thickness to sound velocity in unablated part of the target, t ex Δ /c s. FINAL DENSITY OF ACCELERATED FOIL Let estimate the reduction of final projectile density stimulated by shock wave heating. The sound velocity с s behind the shock wave front propagating over a unablated part of the foil is connected with the sound velocity in blow-off plasma c a through the ratio of the ablation density to the density of unablated part of the target [1]: 1/ # $ % ( 1) & a ) * ' ( +, ñ = " "+ s s 1/ c a (14) where γ s is the adiabatic exponent of matter behind the shock wave front. By substituting Ex. () into Eq. (14), with account for Ex. (6), and using Eq. (1) for the pulse duration we get for the ratio t ex /τ (γ s =γ =5/3): t ex 1 1/ " 3.5 # 1 # $ % & (15) The ratio t ex / τ decreases quite rapidly with increase in the laser radiation wavelength, in accordance with the hyperbolic law. For a solid-state target and Nd-laser harmonic radiation wavelengths we have t ex < τ. So, using the adiabatic law of plane expansion for unablated layer one can easily estimate the final density of the accelerated projectile expressed as /( + 1) # ( " 1) /( + 1) t s 4 s s % = ex & ' ( # $ ) * (16) By substituting Eq. (15) in that expression we get (for γ s =5/3) " 1 " 3 / 4 " 3 / 8 # 1.1 $ (17) Thus, when a solid foil, for example, layer of DT-ice (ρ.16 g/сm 3 ) is accelerated by the action of a time-non-profiled laser pulse, which provides necessary energy for a projectile acceleration, i.e. the first three harmonics of the Nd-laser (λ=1.5,.5 and. 35 µm) and the KrF-laser (λ=.5 µm), the final density of accelerated foil will be less than the initial when the given by Eqs. (11)-(13) laser and projectile parameters are optimal to achieve thermonuclear velocity. According to relation (17), at the acceleration of a plastic layer (ρ 1 g/сm 3 ) the final density ρ will be about 4 times less than initial for the third Nd-laser harmonic, and about 8 for the first harmonic. To conserve the final density of accelerated foil higher than initial one the laser pulse duration must be less than optimal value to accelerate foil, which is given by Eq. (1), and must be less than thermal expansion time. According to Eqs. (1) and () the last requirement in the approximation of shock wave heating is written as 1/ 1/ 3 1/ 3 " # 41 $ % & I (18) The laser pulse duration, which provides the conservation of high foil density, decreases with the reduction of initial foil thickness, with radiation of wavelength and with growth of laser intensity. It should be noted that to increase the final density one should use a time-profiled laser pulse. To achieve a greater final density of accelerated projectile the choice of smaller laser radiation wavelength must be combined with the necessity to use the higher laser radiation intensity to provide the thermonuclear velocity of projectile. According to Eq. (8), to achieve a thermonuclear velocity at the third Nd-laser harmonic, the radiation intensity must be of W/сm, whereas for the first harmonic, it must be only W/сm.

4 Figure 1 shows the dependence of foil areal density (product of initial projectile thickness by density) versus laser pulse duration for two values of Iλ - parameter and two wavelengths, namely, the first (dashed lines) and the third (solid lines) Nd-laser harmonics. The lines with index 1 correspond to Iλ -parameter of W cm - µm and final foil velocity of 14 km/s, the lines with index correspond to Iλ parameter of W cm - µm and the final foil velocity of 1 km/s. For the first harmonic, value Iλ = W cm - µm, corresponds to the laser intensity, I= W/cm, Iλ = W cm - µm corresponds to I= W/cm. For the third harmonic the respective intensities are I=1 15 W/cm and I= W/cm b & "& #$'()* 3 1a b 1 a " #$$% 1b 3a #$µ* ##$, r, µm g/cm ", ns Fig. 1. The dependence of foil areal density versus pulse duration for two values of Iλ -parameter ( W cm - µm lines with index 1, and W cm - µm lines with index ) and for first (dash lines) and third (solid lines) harmonics of Nd-laser radiation. Figure illustrates foil areal density (dashed curves with index 1), pulse duration (pointed curves with index ), and laser beam radius (solid curves with index 3) versus the laser pulse energy for the thermonuclear final velocity (curves with a index), and half its value (b index). According to the data of Fig. 1 (solid line with index ), to achieve the thermonuclear velocity of the foil with areal density Δ ρ = µm g/cm 3 accelerated under the action of the third Nd-laser harmonic radiation of intensity I= W/cm the matched value of laser pulse duration should be close to 1.5 ns. The matched value of laser pulse duration for the foil with areal density 15 µm g/cm 3 is close to 1 ns. According to the data of Fig. (curves a and 3a), in the first case the matched values of laser beam radius and energy are 4 µm and 13 J, respectively, in the second case, 18 µm and 55 J. 1 1 Fig.. The dependencies of foil areal density (dash curves with index 1), pulse duration (pointed curves with index ), and laser beam radius (solid curves with index 3), versus the laser pulse energy for the final thermonuclear velocity of 1 km/s (curves with the index a), and half of its value (b index). The present analysis shows that Gekko/HIPER experiment conditions (I= W/cm, λ=.35µm, τ=-.5 ns, Δ ρ =14- µm g/cm 3 ) are close to optimal to achieve the thermonuclear velocities. But, it should be noted, that according to Eq. (18), the laser pulse duration, which could provide the conservation of high foil density under the above experimental conditions, is less than one used in the experiment, namely, τ d =(.8-1) ns. For this reason it should be supposed that accelerated foil density in the experiment was smaller than initial one. According to relation (17), that reduction is larger than twice. The presented here theoretical model predicts for Gekko/HIPER experiments conditions the final foil velocity of approximately 7-8 km/s that is closed to measured values. The model also is in a good agreement with final foil density results of the numerical simulations of Gekko/HIPER experiments which was published in [11]. The simulations were performed by means of the 1D hydrodynamic codes Diana and Rapid. In both codes the main physical processes typical for the inertial thermonuclear fusion were into account, namely, the electron and ion heat conductivity, the two-temperature state, the laser radiation absorption by the inverse Bremsstrahlung mechanism, the ionizaton and real equations of state. In addition, the code Rapid takes into account the resonance mechanism of laser light absorption. Numerical simulation shown the value of final foil velocity practically same as measured in Gekko/HIPER experiments. Especially, to discuss the problem of final foil density let consider the density profile evolution obtained in numerical simulation performed by Diana code for the polystyrene foil of the thickness of µm

5 and I= W/cm [11]. The equations of state was accounted for a cold pressure and an equilibrium ionization in the Sacha-Raizer approximation. Figure 3 illustrates the density profiles for the time moments t =., 1.,.,.4, and.8 ns in respect to the spatial co-ordinate. The initial position of the foil (ρ =1 g/cm 3 ) corresponds to the co-ordinate range of x 1-3 cm., g/cm 3 1,1,1 1E-3 1E-4 -,6 -,4 -,,, X, cm ns 1. ns. ns.4 ns.8 ns Fig. 3 The taken from [11] space-distribution of the foil density in the different moments of time for the case when the initial foil thickness is µm. One can see that under such conditions (I= W/cm and Δ = µm) the foil becomes loose by, approximately, the end of laser pulse, and an area with density higher than the normal (initial) is absent. At the moment of t=.4 ns the maximum density equals.1 g/cm 3, which is less than that predicted by the estimation based on a shock wave heating. According to the numerical simulation results, the electron heat conductivity gives a significant contribution to the foil heating. For thinner foils and greater intensities of the laser radiation the reduction of the final foil density is larger, and for thicker foils it is smaller. The absorption efficiency calculated by Rapid code in the conditions of Gekko/HIPER experiments varies in region of 7-8 % [11]. HYDRODYNAMIC STABILITY OF LASER-DRIVEN ACCELERATION OF THIN FOIL The problem of a hydrodynamic instability of the laser-driven acceleration pays much attention in literature. In a general statement this topic is a wide one, and needs particular consideration. Here we restrict ourselves by a simple evaluation of Rayleigh-Taylor instability effect as applied to the specific foil acceleration conditions considered in this work. For this purpose let use the well-known formula for an increment of a perturbation amplitude growth at a linear stage of instability evolution /1/ = " # k D,where m " # $ % & ' Ak g = m 1+ k L m a 1/ (19) In these expressions, k m is the unstable wavenumber, g is the acceleration, L a is the density gradient scale length at the ablation front, and A is the Atwood number, given by (ρ - ρ a )/ (ρ + ρ a ), where ρ is the density of unablated foil, ρ a is the ablation density; the velocity of evaporation wave D is given by Eq. (3). The constant β in the ablation-driven stabilization term, according to numerical simulation, equals 3. In one-dimension foil acceleration problem the most dangerous (with respect to foil destruction) mode of perturbation corresponds to the wavelength equal to the foil thickness, and, thus, k m =π/δ. The gradient scale length estimation, L a =c a τ, gives, under the discussed conditions, the value of several hundred microns. So, for the most dangerous mode k m L a >> 1 and, thus, γ (Ag/L a ) 1/. Taking additionally into account that g=u/τ, and A=1 (since ρ a << ρ), Eq. () gives the following increment for the most dangerous mode 1/ ' u ( " 1 c # = 6 a a $ " % ) c * & a +, () Equation () shows that under acceleration of a plane foil up to the velocities not higher than the sound velocity of the ablated matter the relation τ<γ -1 is always valid. This means that in the above case the influence of hydrodynamic instability on the plane foil acceleration is small. The influence becomes stronger under "supersonic" acceleration of the foil, when the foil velocity higher than sound velocity on ablation surface is achieved (u > c a ). Introduce Eqs. (1), () and (6) into Eq. (), and assume (as was made earlier) that Δ/Δ(τ) = 5, γ = 5/3. Then from Eq.() one can see that the requirement τ<γ -1 is fulfilled if $ 7 " # % >.1 1/ 3 ( I# ) At Iλ =1 14 W cm - µm and λ=1.6 µm the pulse duration should exceed.7 ns for 3 µm thick СН-foil, and. ns for 1 µm foil thickness. For the third harmonic (λ=.35µm) the limit value of laser pulse duration becomes smaller: τ>.8 ns for foil thickness 3 µm, and τ>.3 ns for foil thickness 1 µm. So, for the third harmonic the condition of stable acceleration is valid for the laser and foil parameters, which correspond to the foil acceleration to thermonuclear velocity in a

6 dense state. CONCLUSION The thermonuclear velocity of a flat projectile can be reached under low-entropy laser-driven acceleration. The velocity of 1 km/s of dense plastic foil can be reached at the enough high laser light absorption efficiency of 7-8 % and enough high hydrodynamic efficiency of about %. The requirement of a conservation of high projectile's density at the end of acceleration process leads to overhead limitation for the laser pulse duration. The dominant dependence of the foil final velocity on laser parameters is the velocity growth with the laser intensity and wavelength growth according to the scaling relation u (Iλ) 1/3. Despite of the foil velocity growth with the wavelength growth, it is more preferable to make use of the third harmonic of Nd-laser radiation. This is stipulated by the fact that under the decrease of laser radiation wavelength the role of stimulated plasma processes becomes smaller and the absorption efficiency enhances, the hydrodynamic stability of foil acceleration grows, and the foil density decreases more slowly. The choice of shorter wavelength of the laser radiation should be combined with the necessity to use the higher intensity the smaller is the wavelength, but so that Iλ - parameter does not exceed the value of W сm - µm. The achievement of high density of a projectile accelerated to thermonuclear velocity is the most complicated problem in impact fast ignition. The solution of this problem would require an application of different techniques, such as the projectile acceleration in a conical channel, multi-layer design of the projectile, and a profiled pulse. [6] S.Yu. Gus'kov, S.V. Mesyats, V.B. Rozanov, JETP 16, 138 [7] S.Yu. Gus'kov, V.V. Zverev, V.B. Rozanov 1983 Quantum Electronics 1(4) 8 [8] J.J. Duderstadt and G.A. Moses, Inertial confinement fusion. John Wiley and Sons, New York (198) [9] S. Atzeni and J. Meyer-ter-Vehn, The physics of inertial fusion. Oxford University Press (4) [1] S.Yu. Gus'kov, S. Borodziuk, M. Kalal et al, Quantum Electronics 34(11), 989 (4) [11] S.Yu. Gus kov, H.Azechi, N.N. Demchenko et al, Plasma Physics and Controlled Fusion 49, 1689 (7) [1] H.Takabe, K. Mima, L. Montierth, R.L. Morse, Phys. Fluids 8, 3676 (1985) ACKNOWLEDGMENT The work was supported by the joint grant of Japan Society for the Promotion of Science and Russia Foundation of Basic Research (# JF). REFERENCES [1] N.G. Basov, S.Yu. Gus'kov, L.P. Feoktistov, J. Soviet Laser Research 13, 396 (199) [] M. Tabak et al, Phys. Plasmas 1(5), 166 (1994) [3] M. Murakami, H. Nagatomo, H. Azechi et al, Nuclear Fusion 46, 99 (6) [4] M. Murakami, H. Nagatomo, T. Sakaiya et al, Plasma Physics and Controlled Fusion 47, B815 (5) [5] Yu.V. Afanas'ev, E.G. Gamaly, O.N. Krokhin, V.B. Rozanov, PMM 39, 451 (1975)