Laser pulse reflection by anisotropic plasma

Physics Letters A 376 (2012) 2306–2308

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Physics Letters A www.elsevier.com/locate/pla

Laser pulse reflection by anisotropic plasma K.Yu. Vagin, S.A. Uryupin ∗ P.N. Lebedev Physics Institute, Russian Academy of Sciences, Leninsky pr. 53, Moscow 119991, Russia

a r t i c l e

i n f o

Article history: Received 26 April 2012 Accepted 24 May 2012 Available online 28 May 2012 Communicated by V.M. Agranovich

a b s t r a c t The effect of anomalous amplification of test laser pulse reflected by a nonequilibrium plasma formed under atom ionization by a strong pulse of circularly polarized wave is described. It is shown that the gain in radiation intensity may reach ten orders of magnitude. The most effective amplification takes place for frequencies comparable with the Weibel instability growth rate. © 2012 Elsevier B.V. All rights reserved.

Keywords: Pulse amplification Weibel instability Radiation reflection

1. Introduction



2

4E i

cnE i  I 

3h¯ ω L At the ionization of substance atoms in the field of strong short laser pulse a nonequilibrium plasma with anisotropic electron velocity distribution is produced (see, for instance, [1–5]). Nonequilibrium plasma has unusual optical properties. In particular, the new properties of absorption and reflection of a high-frequency radiation by such plasma arise due to the influence of an alternating magnetic field on the electron kinetics in the skin layer [6–8]. In anisotropic plasma the development of Weibel instability is possible [9]. Therefore, the unusual optical properties established in the papers [6–8] are realized in the time interval with duration less than the inverse growth rate of Weibel instability. The possibility to overcome such theory limit was demonstrated in the paper [10]. In order to develop the approach proposed in [10] in this report reflection of test laser pulse by plasma formed under atom ionization in the field of intense pulse of circularly polarized wave is studied. It is shown that the field strength of the reflected radiation increases anomalously during the time interval of Weibel instability development. The effect reason is test pulse field amplification in the skin layer caused by Weibel instability development. 2. Ground state Assume that half-space z > 0 is occupied by plasma formed under ionization of matter in the strong laser pulse of circularly polarized wave. The ionizing pulse duration is considered small compared to the time of nonequilibrium photoelectron distribution variation. If ionizing radiation flux density I satisfies inequalities

*

Corresponding author. E-mail address: [email protected] (S.A. Uryupin).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.05.048

ω2pump cnE i , ω2L

(1)

where E i is the ionization potential of substance atoms, n is the photoelectron density, c is the speed of light, h¯ is the Planck constant, ωpump is the carrier frequency of ionizing radiation, ω L is the plasma frequency, then anisotropic distribution of photoelectrons over velocities is formed. One can approximate this distribution by the function [1,4]

F (v) 



n 4π 2 v E v 2T

exp −

v 2z 2v 2T

−

( v ⊥ − v E )2 2v 2T

 .

(2)

Here, velocities v E and v T have the form

v 2E

ω2L h¯ ω L =  v 2T = 2 nmc ωpump 2m 2I



I cnE i

,

(3)

where m is the electron mass. In such  nonequilibrium state momentum flux density tensor P i j = m dv v i v j F (v) is diagonal

P xx = P y y 

1 2

nmv 2E  P zz  nmv 2T .

(4)

Plasma described by the relations (2)–(4) is unstable against Weibel instability development [9]. 3. Test pulse reflection Consider the interaction between test pulse and nonequilibrium plasma. Assume that at the time t = 0 electromagnetic pulse









E L ( z, t ) = E L (t − z/c ), 0, 0 , B L ( z, t ) = 0, E L (t − z/c ), 0 ,

(5)

K.Yu. Vagin, S.A. Uryupin / Physics Letters A 376 (2012) 2306–2308

2307

falls on plasma surface. Here E L (t − z/c ) = E L η(t − z/c ) sin[ω0 (t − z/c )], ω0 < ω L , η(τ ) is the Heaviside function. In order to describe the response of a nonequilibrium plasma on the action of pulse (5), we use a system of equations for mean electron velocity u( z, t ), momentum flux density tensor P i j ( z, t ), vortex electric E( z, t ) and magnetic B( z, t ) fields (see, for example [11]). We assume that field (5) action on plasma is weak. We describe its influence in linear approximation on the field strength. The pulse of the form (5) leads to excitation in plasma electromagnetic field with components E( z, t ) = { E x ( z, t ), 0, 0} and B( z, t ) = {0, B y ( z, t ), 0}. At the same time one velocity component δ u( z, t ) = {δ u x ( z, t ), 0, 0} and two identical momentum flux density tensor components δ P xz (z, t ) = δ P zx (z, t ) are nontrivial. For these quantities we have the linearized system of equations Fig. 1. Path of integration in the complex plane of variable

∂δ u x 1 ∂δ P xz e + = E x, ∂t mn ∂ z m eB y ∂δ u x ∂δ P xz + P zz = ( P xx − P zz ) , ∂t ∂z mc ∂By ∂By ∂ Ex ∂ Ex +c = 0, +c = −4π en δ u x , ∂t ∂z ∂t ∂z

 Er t +

 = −E L t + +

(6)

 +∞

f (ω) = 0 dt f (t ) exp(i ωt ), where  > γ > 0 and γ is the exponential growth rate of the function f (t ). We assume that in plasma the perturbations of E x , B y , δ u x and δ P xz due to the action of pulse (5) much higher than their values due to thermal fluctuations. This allows us to assume that at time t = 0, the initial values of perturbations of these quantities are approximately equal zero. In the conditions of inequality (4) the system of Eqs. (6) permits significant simplification if the characteristic electron velocity along the anisotropy axis satisfies the inequality v T  L |ω|, where L is minimal characteristic length of variation in physical quantities along z-axis. Then from the system of Eqs. (6) we obtain second-order differential equation for the Laplace image of the electric field E x ( z, ω) in plasma

 ∂2 2 − k (ω) E x ( z, ω) = 0, ∂ z2

(7)

−∞+i 

ω ω −ω k (ω) = 2 c γ + ω2 2

2 L 2 0

2

( grow )

(8)

2π

Er

γ0 (τ ) = −η[τ ] E S −γ 0

√

and γ0 = ω L v E / 2c  ω L is the maximum possible growth rate of Weibel instability in spatially unlimited plasma with electron distribution of the form (2) [12,13]. The solution of Eq. (7) that does not grow up deep into the plasma is E x ( z = 0, ω) exp[−k(ω) z], z > 0, where for the function k(ω) defined by (8) we chose the branch in the plane of the complex variable ω satisfying the condition Re[k(ω)]  0 on the straight line Im[ω] =  along which the integration is performed in the inverse Laplace transform (see Fig. 1). In the case of axially-symmetrical with respect to z-axis electron distribution (2) the reflected field, that satisfies the Maxwell equations at z < 0, can be written as



dω



Er ( z, t ) = E r (t + z/c ), 0, 0 ,





Br ( z, t ) = 0, − E r (t + z/c ), 0 . (9)

Using the continuity of the tangential electric and magnetic field components at the plasma boundary z = 0, for the electric field outgoing from the plasma we have

E L (ω)

2ω

ω + ik(ω)c

z

e −i ω(t + c ) ,

(10)

where E L (ω) = E L ω0 /(ω02 − ω2 ) is the Laplace image of electric field (5) incident on plasma. In order to calculate the integral in (10) we use the integration contour in the plane of the complex variable ω shown in Fig. 1. According to (10) for t > 0 the front of the reflected field reaches z f = −ct. At the front, where z = z f , the field (10) is zero. The reflected field is localized in the domain z f < z < 0 and for time t > 0 in the point z < 0 is specified by the field strength on the plasma surface in an earlier time τ = t + z/c < t. Departing from the plasma surface field (10) consists of two parts. The first part corresponds to the sum of the field − E L (t + z/c ) and the contributions to the integral term in (10) from the contour segments in Fig. 1 passing along the real axis. This part describes non-increasing contribution to the reflected field, that is similar to the radiation field reflected by equilibrium plasma. As plasma is in nonequilibrium state there is an additional contribution to the outgoing field due to the Weibel instability development. This contribution to the integral part of (10) comes from the sides of the cut along the imaginary axis on the interval (0, i γ0 ) and has the form

where 2

z

ω.



c

+∞+ i

where P xx and P zz are described by expressions (4). To solve the system of Eqs. (6) for t > 0 we use the Laplace transform where the original function f (t ) and its image f (ω) are re +∞+i  lated by the relations f (t ) = (2π )−1 −∞+i  dω f (ω) exp(−i ωt ),



z c



γ02 − μ2 μ exp(μτ ), π ω02 + μ2 |μ|

dμ

(11)

where E S = (2ω0 /ω L ) E L is the amplitude of the electric field in plasma near the surface under the conditions of high-frequency skin-effect. Far from the front of the outgoing radiation, where τ = t + z/c  (ω0−1 , γ0−1 ), the growing field (11) may be approximated by the expression (grow)

Er

γ02

exp(γ0 τ ) . 2 2 (γ τ )3/2 γ + ω 2π 0 0 0

ES

(τ )  − √

(12) (grow)

We see from (12) that the amplitude of the function E r maximum when the frequency ω0 coincides with γ0 .

(τ ) has

4. Discussion The consideration given above is valid during limited time interval. On the one hand, the time must be greater than the inverse growth rate of Weibel instability in order to amplify the reflected field:

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K.Yu. Vagin, S.A. Uryupin / Physics Letters A 376 (2012) 2306–2308

−1

t > γ0

 ≈ 0.5 [ps]



1019 n [cm−3 ]

100 mv 2E [eV]

 .

(13)

On the other hand, the time t must be less than nonlinear saturation time of Weibel instability t N L . According to numerical studies of the Weibel instability [14], the instability linear stage, in which the field grows exponentially rapidly with time, takes place until the energy density of the quasistationary magnetic field, which has maximum near plasma boundary z = 0, reaches ten percent of the kinetic energy density of electrons B 2y ( z ≈ 0, t N L )/8π  0.05nmv 2E . Assuming γ0 ≈ ω0 , we have estimate:

γ t NL ≈ 14 +

1 2

 ln

n [cm−3 ] 1019



106



I L [W/cm2 ]

,

(14)

where I L = c E 2L /8π is the test radiation flux density incident on the plasma surface. Also the time t must be less than inverse photoelectron collision frequency as these collisions make the distribution function (2) isotropic. To estimate the appropriate time enough to consider the collisions of electrons with ions

νei−1 =

3m2 v 3E

16

√

π Z e 4 nΛ

 2 3/2    mv E [eV] 5 1019 ≈ 2 [ps] , ZΛ 100 n [cm−3 ]

(15)

where Z is the ion multiplicity, Λ is the Coulomb logarithm. According to (14) the time of nonlinear saturation of the Weibel instability t NL increasing with I L decreasing. However, the flux density of the test pulse should not be too small. The point is that, apart from the magnetic field produced by the test pulse, there is a spontaneous magnetic field B sp generated in thermal fluctuations of the current density in plasma. The spontaneous magnetic field is also amplified when the Weibel instability develops. Its amplification may be disregarded if the strength B sp is less than 2E L , the magnetic field strength in plasma produced by the incident test pulse at time less than γ0−1 . Neglecting the difference between the fluctuations in isotropic and anisotropic plasmas, let us roughly estimate the field strength B sp . According to [15] B 2sp /(4π nmv 2E ) ∼ ( v 2E /c 2 ) (ω3L /nc 3 )  1. As a result, we have a qualitative estimate for the minimum flux density of the amplified pulse:



2

I L  10 W/cm

     n [cm−3 ] 3/2 mv 2E [eV] 2 1019

100

.

(16)

The flux density of the test pulse should not be too big. In accordance with the condition of the nonlinear stabilization of the instability, we obtain the upper limit for I L :

I L < 1.25 · 10−2nmcv 2E

  2    n [cm−3 ] mv E [eV] .  6 · 1010 W /cm2 19 10

100

(17)

Note that the inequality (17) ensures that the amplitude of the electron oscillation velocity in the electric field of the test pulse is low compared to v E and justifies the use of linearized Eqs. (6). Since, according to (12) characteristic frequency ω ∼ γ0 and the external field penetrates into plasma on the skin depth, then simplifying assumption of smallness of the electron velocity along z-axis, which we have used in obtaining Eq. (7), is equivalent to vT  vE. Concluding the discussion, let us estimate the efficiency η of the reflected field amplification due to Weibel instability development, meaning by η the ratio of the reflected field flux density to the incident one. Thus it is clear that the maximum value of this ratio is obtained for times t ∼ t NL and equal η  1.25cnmv 2E /102 I L (see (17)). Comparing the inequalities (16) and (17) we see that the gain can reach ten orders of magnitude. 5. Conclusions From the consideration given above it is clear that the nonequilibrium anisotropic plasma can be an amplifier of weak pulses of electromagnetic radiation. The effect of amplification is realized during the limited time interval when the Weibel instability development occurs. The characteristic values of the gain may reach ten orders of magnitude. The spectrum of the reflected radiation is wide and has maximum at the frequencies of the order of the inverse instability growth rate. Acknowledgements The work has been performed under support of the RFBR, project No. 12-02-00744. References [1] N.B. Delone, V.P. Krainov, J. Opt. Soc. Am. B 8 (1991) 1207. [2] S.J. McNaught, J.P. Knauer, D.D. Meyerhofer, Phys. Rev. Lett. 78 (1997) 626. [3] W.P. Leemans, C.E. Clayton, W.B. Mori, K.A. Marsh, A. Dyson, C. Joshi, Phys. Rev. Lett. 68 (1992) 321. [4] N.B. Delone, V.P. Krainov, Phys.-Usp. 41 (1998) 469. [5] V.D. Mur, S.V. Popruzhenko, V.S. Popov, JETP 92 (2001) 777. [6] G. Ferrante, M. Zarcone, S.A. Uryupin, Europ. Phys. J. D 19 (2002) 349. [7] G. Ferrante, M. Zarcone, S.A. Uryupin, Europ. Phys. J. D 22 (2003) 109. [8] S.G. Bezhanov, S.A. Uryupin, Plasma Phys. Rep. 32 (2006) 423. [9] E.S. Weibel, Phys. Rev. Lett. 2 (1959) 83. [10] K.Yu. Vagin, S.A. Uryupin, JETP 111 (2010) 669. [11] V.Yu. Bychenkov, V.P. Silin, V.T. Tikhonchuk, Theor. Math. Phys. 82 (1990) 11. [12] V.P. Krainov, JETP 96 (2003) 430. [13] A.Yu. Romanov, V.P. Silin, S.A. Uryupin, JETP 99 (2004) 727. [14] R.C. Davidson, D.A. Hammer, I. Haber, C.E. Wagner, Phys. Fluids 15 (1972) 317. [15] A.I. Akhiezer, I.A. Akhiezer, R.V. Polovin, A.G. Sitenko, K.N. Stepanov, Plasma Electrodynamics, Pergamon, New York, 1975.