CC BY
11Annual Moscow Workshop «Physics of Nonideal Plasmas» (Moscow, 3-4 December 2002)
Weibel instability in plasmas produced by a super-intense femtosecond laser pulse
Krainov V.P. (krainov@online.ru) Moscow Institute of Physics and Technology,
1. Introduction
Erich Weibel [1] (see also the textbook [2]) was the first who predicted spontaneously growing transverse quasi-static electromagnetic waves which appear in a plasma due to an anisotropic velocity distribution of electrons. The maximum increment of this instability for the wave frequency ro is (in non-relativistic approximation)
Imro = (u/c)rop, Rero = 0, where rop is the plasma frequency and u is the average velocity of electrons in the (longitudinal or transverse) direction where this velocity has a maximum. This simple result is valid under the condition of a strong anisotropy of the velocity distribution in longitudinal and transverse directions. It corresponds to the electromagnetic waves with small wavelengths (kc >> rop, where k is the wave number).
This approach has been successfully applied in Ref. [3] for electrons produced in the tunneling ionization of atoms by a strong low-frequency linearly polarized laser field. The corresponding average velocities of electrons along the field strength polarization u¡¡ and in the transverse plane u± differ strongly each from other. Their ratio was found in Ref. [4]:
where the quantity
u±/u|| = y/31/2,
Y = ю (2Ei)1/2/F
is the Keldysh parameter (atomic system of units e=m=ft=l is used in this paper). Here F is the field strength amplitude, and ro is the laser frequency. The quantity Ei >> ro is the ionization potential of the atom (or atomic ion). In the case of tunneling ionization the condition y << l is fulfilled. It was found in Ref. [3] that the maximum instability increment is (as we have said above)
Imro = (u||/c)rop.
Here the average longitudinal velocity u|| is given in Ref. [4]:
u|| = (3F3/2)1/2/(ro (2Ei)3/4).
In this work we consider the Weibel effect for the barrier-suppression ionization that occurs at the irradiation of atoms and atomic ions by the field of a super-intense laser pulse with the peak intensity larger than 101 W/cm2 The corresponding anisotropic distribution of ejected electrons was obtained in Ref. [5] (both for linearly and for circularly polarized fields). First we neglect the collisions of strongly heated ejected electrons with the electrons and atomic ions (having in mind, for example, the cluster plasma [6]) since Weibel instability is developed during very short time of order of rop These collisions are taken into account in the final part of the paper.
The Weibel instability is the most remarkable at the peak of the super-intense femtosecond laser pulse. We solve the problem in the linear regime only when the perturbation of the kinetic distribution function is less than the unperturbed distribution function. It will be found that the
real part of the frequency ro of Weibel electromagnetic field is much less compared to the laser frequency ro . Therefore we can consider Vlasov-Maxwell equations for Weibel field independent on the Maxwell equation for external laser field.
2. Linearly polarized field 2.1. Dispersion relation
First the case of linearly polarized laser pulse is considered. The Boltzmann transport equation for Weibel electromagnetic field, retaining only linear terms of perturbation, is of the known form [1,2]:
df/dt + v(f/dr) = (E + (1/c)[v,B])(df /dv). Here f (v) is an anisotropic stationary distribution of electrons, and f is the perturbation of the distribution function, while E and B represent the perturbation of the electric field strength and of the magnetic induction in the electromagnetic wave, respectively (i.e. the Weibel field). Collisions will be considered below.
Assuming that the first order quantities f(v,r,t), E(r,t), and B(r,t) depend on r and t only through the Fourier factor exp(irot + ikr), one obtains the equation for Weibel field with the fixed frequency ro and the wave vector k:
i(ro + kv)f = E(df /dv) - (1/c)[v,(df /dv)]-B. (1)
It follows from the Maxwell equation
rot E = -(1/c)(dB/dt), the connection between the electric and magnetic generated fields:
B = -(c/ro)[k,E].
Substituting this equation into Eq. (1), we find the equation containing only the electric field:
i(ro + kv)f = E(df /dv) + (1/ro)[v,(df /dv)]-[k,E]. (2)
Let us assume that the wave vector k is directed along the X axis, and the electric field strength E is directed along the Z axis. Then one finds the function f from Eq. (2):
f = (iE/(ro + kvx)){-df /(dvz + (k/ro)(vz(df )/dvx - vx(df )/dvz)}. (3)
The second Maxwell equation is of the form:
rot B = (4n/c)j +(1/c)(dE)/dt). (4)
Here the electric current density j is determined by the distribution function:
j(r,t) = -Jvf(v,r,t)d3v.
Substituting this expression and Eq. (3) into Eq. (4), we obtain the Vlasov equation in the form:
i[k,B] = (ick2/ro)E =
= (4rc/c)Jv[(iE)/(ro + kvx)]{df /dvz - (k/ro)(vz(df )/dvx - v^df )/dvz)}d3v + (iro/c)E. (5) Projection of this vector equation to Z axis gives the dispersion relation between the frequency ro and the wave number k of the Weibel field:
k2c2 - ro2 = 4nH"-»vz dvx dvy dvz {df /(dvz - [(kvz)/(ro + kvx)](df /dvx)}. (6)
We simplify the first term in the right side of this equation taking into account the normalization condition for the unperturbed distribution function
J/Jf d3v = n,
where n is the number density of free electrons:
k2c2 - ro2 + rop2 = -4nkdvx dvy dvz [(vz2)/(ro + kvx)](Sf /(dvx). (7)
Here we define the plasma frequency
rop = (4nn)1/2.
We assume that the inequality ro>>kvx is valid in the tunneling and barrier-suppression ionization regimes. The corresponding restriction for the wave number k will be given below. Then we can expand the denominator in Eq. (7):
1/(ro+kvx) = (1/ro) - (kvx)/ro2 +...
Substituting this expansion into Eq. (7), we integrate by parts:
k2c2 - ro2 + rop2 = (4nk/ro2)JM-M dvy J«.«, v^dvj«.«, vx(3f /(3vx)dvx =
= -(4nk2/ro2)J«-« dvy J«-« vz2dv z J f dvx,
or finally
k2c2 - ro2 + rop2 = -rop2(k2/ro2)<vz2>. (8)
Here <vz2> is the average square of the longitudinal electron velocity.
The most interesting range of wave numbers k in the dispersion relation (8) is kvx << ro << kc; then we can neglect the term -ro2 in the left side of the dispersion equation (8). Hence, one obtains from Eq. (8):
ro2 = - [(k2<vz2>)/(rop2 + k2c2)]rop2. (9)
Thus, the frequency is a pure imaginary quantity that produces the Weibel plasma instability. The maximum value of this instability increment is achieved at kc >> rop > ro (short wavelength limit):
ro2 = -(<vz2>/c2)rop2. (10)
2.2. Tunneling ionization
In the case of the tunneling ionization the distribution function f is of a Gaussian form [4,7]. Hence, <vz2> = u||2, where
u||2 = 3ro /(2y3).
Here ro is the laser frequency and y is the Keldysh parameter (see Introduction). Thus, it follows from Eq. (10) the result of Ref. [3]:
ro2 = -(u||2/c2)rop2 = -[(3ro )/(2y3c2)]rop2 = -{(3F3)/[(2ro 2(2Ei)3/V]}rop2. The above condition ro >> kvx produces the range of the wave numbers
rop << kc << rop/y.
The spatial scale of the Weibel linearly polarized field is
X > yc/rop.
The upper limit for the spatial scale is y 1 times larger than this estimate. 2.3. Barrier-suppression ionization
Now we consider the case of barrier-suppression ionization by a linearly polarized field. According to Ref. [5] we have the anisotropic distribution of ejected electrons in the form:
f (vz,v±) - n-Ai2{(2Ei + vz2y2/3+v±2)/(2Ff3}. (11)
Here Ai(x) is the Airy function. This distribution reduced to the tunneling limit [4] under the condition of a weak field (compared to the atomic field):
F << (2Ei)3/2.
Thus, the deviation from the tunneling limit for the square of the instability increment, Eq. (10), is determined by the ratio of squares of average longitudinal velocities:
<vz2>/uf = G(s), and ro2 = -G(s)[(3F3)/(2ro 2(2Ei)3/V)]rop2,
(12)
where the dimensionless field parameter s is determined by
s = (2Ei)/(2F)2/3. The universal function G(s) is given by the expression
G(s) = 4s1/2{J «dt J Vdz Ai2(s + t + z2)/{J «dt J «dz Ai2(s + t + z2)}. (13)
We have G(s) ^ 1 at s >> 1 (tunneling limit) as it should do.
It is seen that in the case of barrier-suppression ionization the increment increases more slowly with the increase of the laser field F than in the case of the tunneling ionization. Thus, we can conclude that an electromagnetic field is generated in a plasma with the same linear polarization as the initial laser radiation field which produces these anisotropic plasma electrons. The frequency (see Eq. (10)) of this field does not contain the real component, so that this is a
quasi-stationary field, but with exponentially growing amplitude of electric and magnetic strengths (the electric field is small compared to the magnetic field).
2.4. Relativistic effects
We discuss in brief the relativistic generalization of the tunneling results. According to Ref. [8] the velocity distribution of ejected electrons along the field polarization is of the form (in the case of a weak relativistic effect):
f exp{-(vzY/3ro )[1 + (3vz2/4y2c2)]}. Here the second term in the exponent is responsible for the relativistic effect. It diminishes the average longitudinal velocity:
u||2(rel) = u||2/[1 + 3u||2/(2yc)2] where (see the Introduction) u||2 = 3ro /(2y3). This decreasing of the instability increment is in agreement with the relativistic results of Ref. [9]. Thus, one obtains instead of Eq. (10)
Imro = (u||(rel)/c)rop.
3. Circularly polarized field
3.1. Tunneling ionization
Let us consider in this Section the Weibel instability produced in a plasma during the tunneling and barrier-suppression ionization of atoms (or atomic ions) by a circularly polarized laser femtosecond pulse. We direct again the wave vector k of the laser field and of the produced electromagnetic perturbation Weibel field along X axis. The perturbation electric field strength E is also circularly polarized and rotates in the plane YZ. Hence,
E = E(iz +i-iy)exp(irot + ikr). Here iy, iz are unit vectors along the axis Y and Z, respectively. Eq. (2) takes the form:
i(ro + kvx)f = (E/v||)(vz+ivy)(df /dv||) + (kE/ro)(vz+ivy)[(vx/v||)(df /(dvz)-df /(dvx].
Here v|| is the electron velocity in the polarization plane YZ. Hence, one obtains the perturbation distribution function
f = i(E/ro)(vz+ivy){-(1/v||)(df /dv||) + (k/(ro + kvx))(Sf /dvx)}. Instead of Eq. (6) we find the dispersion relation in the form:
k2c2- ro2 = 4u2j "v||3dv||J".„dvx{(1/v||)(df /dv||) - (k/(ro + kvx))(df /dvx)}. (14)
The unperturbed anisotropic electron distribution function for the tunneling ionization is of the form (see, for example, Ref. [10]):
f = (n/(4n2u2v ))exp{-(vx2+(vrv )2)/(2u2)}. (15)
Here v = F/ro is the ponderomotive electron velocity, and the average velocity along the wave vector is
u = (F/(2(2Ei)1/2)1/2
(again F is the laser field strength amplitude, and Ei is the atomic ionization potential).
It should be noted that u << v , i.e. the width of the distribution is small compared to its shift in longitudinal direction. The first term in the right side of Eq. (14) vanishes since the integrand is the odd function of the argument (v||-v ). Then the dispersion relation (14) takes the form:
k2c2 - ro2 = -4rc2j " vfdvn L " dvx (k/(ro + kvx))(df /dvx).
We assume again that ro >> kvx, i.e., ro >> ku and expand the denominator in Taylor series:
1/(ro+kvx) = 1/ro - kvx/ro2+... Integrating by parts, we simplify the dispersion relation:
k2c2 - ro2 = -(4rc2k2/ro2)J " vfdvn L " dvx f . (16)
Substituting Eq. (15) into Eq. (16), one obtains
ra4 - (kcro)2 - (ropkv )2/2 = 0.
The solution of this equation is
ro2 = -((kc)2)/2){[1+2(ropv )/(kc2)2]1/2-1} < 0. Thus, the frequency ro is a pure imaginary quantity that produces the circularly polarized exponentially increasing electromagnetic wave. Its real part is zero, so that this is a quasi-stationary wave. In the short wave limit
k >> ropv /c2 (17)
we simplify this solution having in mind that v = F/ro :
ro2 = -(1/2)(v /c)2rop2 = -(1/2)(F/ro c)2rop2. (18)
The condition ro >> ku restricts the wave number k from above:
k2 << Frop2(2Ei)1/2/(ro 2c2). (19)
1/2
Inequalities (17) and (18) do not contradict each to other under the condition F << c2(2Ei) which is satisfied up to very high values of the laser field intensities (c = 137 a.u.). The condition (19) restricts the spatial scale of the Weibel circularly polarized field
X > ro c/[rop(F(2Ei)1/2)1/2].
The upper limit for spatial scale is
X < ro c2)/(ropF).
3.2. Barrier-suppression ionization
In the case of barrier-suppression ionization by a circularly polarized field the unperturbed distribution function was found in Ref. [5]:
f (vz,v±) - n-Ai2{(2Ei + (v|| - v )2 + vx2)/(2F)2/3}. Here Ai(x) is the Airy function. Substituting this expression into the general Eq. (16), it is easy to check that the dispersion relation is the same as in the case of the tunneling ionization. Thus the maximum increment of Weibel instability is determined again by simple Eq. (18) also for barrier-suppression ionization. The spatial scale of the Weibel circularly polarized field is (at F -(2Ei)3/2:
X > ro c/(ropEi).
3.3. Relativistic effects
In the non-relativistic limit we have v = F/ro << c. Hence, the Weibel increment is small compared to the plasma frequency rop. The anisotropic relativistic distribution of ejected electrons was obtained in Ref. [10]. The most of electrons are ejected not in the polarization plane of circularly polarized laser radiation, but at the definite angle 9^0 with respect to this polarization plane. This angle is determined from the relation (see, for example, Ref. [11]):
tan9 = F/(2ro c).
The normalized unperturbed relativistic distribution function is of the form [10]
f (px,p||) = (n/(4n2ur 2v ))exp{-[(px - v tan9)2 + (pM - v )2]/(2u^)}. Here px, p|| are the momentum components of the ejected electron along the wave vector and in the polarization plane, respectively. The relativistic width ur of the distribution is given by expression [10]:
ur2 = u2[1+(F/ro c)1/2]2/(1+(F/ro c)2)
where (see above)
u2 = F/(2(2Ei)1/2)
is the non-relativistic width.
Substituting Eq. (20) into Eq. (16), we find instead of Eq. (18):
ro2 = -(1/2)(<v||2>/c2)rop2.
1/2 " Here (<v||2>) is the average relativistic velocity in the polarization plane (compare with Eq.
(10) for linearly polarized field). This quantity can be expressed via the corresponding
1/2
relativistic momentum p|| = F/ro and the relativistic energy Erel = c(p||2+px2+c2) due to narrow peaks in the unperturbed distribution (20):
(<v||2>)1/2/c =(p||c)/Erel = (F/ro )/(F2/(2ro 2c)+c).
Thus, one finally obtains
ro = -{(i/V2)(F/ro )/[F2/(2ro 2c)+c]}rop. (11)
It is seen that relativistic effects diminish the increasing of the Weibel increment for circularly polarized field analogously to the case of linear polarization (see above). The maximum value of this increment is achieved at the field strength F = ro cV2:
romax = -(i/2)rop.
4. Electron collisions
In this Section we discuss how collisions of electrons with other electrons and atomic ions restrict the development of Weibel instability in plasma. Let us consider, for example, the irradiation of solid matter by a linearly polarized femtosecond laser pulse with the peak intensity of I = 101 W/cm2 and the laser frequency ro = 1.17 eV = 0.043 a.u.. The peak laser field strength is then F=16.6 a.u. According to H. Bethe rule for barrier-supression ionization F = Ei2/(4Z) (see Ref. [7]) where Z is the charge of atomic ion and Ei is its ionization potential, we find that this field produces atomic ions with the average multiplicity Z ~ 30 having the ionization potential of the order Ei ~ 1.2 keV. We use further the number density of atoms in the
irradiated solid target na = 5-1022 cm 3. Of course, during the laser pulse plasma expands, and its expansion at the peak of laser pulse depends on the pulse duration. We assume femtosecond pulse durations, and therefore such an expansion can be neglected (see estimates below).
At the leading edge of the laser pulse first electrons ejected due to inner ionization have small velocities. Velocities of produced electrons increase with time when multicharged ions are produced. We consider the final stage of the inner ionization when ions with maximum charge multiplicity Z are produced, and the corresponding electrons have maximum velocities. Their number density can be estimated as equal to the number density of atomic ions na. Other electrons are heated at the leading edge of the laser pulse, but uniformly due to inverse induced bremsstrahlung (see, for example, [12]). The typical kinetic electron energies are of order of 1 keV, i.e. the electron velocities of these slow electrons are of order of u¡¡ ~ u_l ~ 8 a.u. These velocities are small compared to the velocity of electrons produced at the final stage of inner ionization (see estimates below). Therefore slow electrons do not destroy Weibel effect.
The value of Keldysh parameter is y = 0.024<<1. Thus, the multicharged ionization of atoms
is the quasi-static above-barrier process. The longitudinal (along the field strength direction)
1/2
velocity is u|| = (3ro /2y3) = 67.0 a.u. (c = 137 a.u.), while the transverse velocity uL = [F/(2(2Ei)1/2]1/2 = 0.94 a.u. The average energy of these fast electrons is u||2/2 = 60 keV, that is, it is much larger than the velocity of slow electrons even after their heating at the leading edge of
the laser pulse. It is seen that u|| >> uL, i.e. the velocity distribution of ejected electrons is
1/2
strongly anisotropic one. The plasma frequency of fast electrons is rop = (4nna) = 0.30 a.u. = 8.2 eV.
Now we derive the rate of non-relativistic electron-electron collisions:
Vee = (4uZnaAe)/(u||3) = 0.0040 fs >. Here A ~ 10 is the Coulomb logarithm for electron-electron collisions. The laser-driven electron
momentum is p = F/ro = 386 a.u. It corresponds to electron velocity v = cp /(p 2 +
1/2
c2) = 0.94c, i.e. an electron is an ultra-relativistic particle. The velocity v should not be taken into account in electron-electron collision rate, since all electrons oscillate coherently. But, of course, we should substitute the laser-driven electron momentum p into the expression for the rate of relativistic (Mott) electron-ion collisions (Ai ~ 10 is the Coulomb logarithm for electron-ion collisions):
Vei = v (4nZ2naAi)/(p 2c2) = 0.0016 fs >. Thus we can neglect the electron-ion collisions at the alignment of longitudinal and transverse velocities (see also the analogous statement in Ref. [3]). Under these conditions the time for
alignment of velocities and, hence, for the end of growing of Weibel field, is determined according to Ref. [3]:
T = 1/(LVee).
Here L = 2ln(2v||/v±) - 3. Substituting the above values of velocities and the rate of electron-electron collisions, we find L = 6.9 and t = 36 fs.
Now we can derive the maximum increment of Weibel instability in the case of linearly polarized field:
Imro = (u||(rel)/c)rop = 0.030 a.u. and Imro-T = 45. Thus, the strong Weibel magnetic field
B(t) = B exp(Imro-t)
can be produced before electron-electron collisions will align the longitudinal and transverse velocities. The spatial scale of this field is larger than X - 6A. The upper limit for this scale is 240 A. We cannot determine the initial spontaneous Weibel magnetic field B which appears in plasma, since we solve the linear Maxwell equations (see Conclusion for nonlinear effects).
These results refer to the peak intensity of 101 W/cm2. However, if the peak intensity is 101 W/cm2 or less, it follows from analogous derivations that no Weibel instability is developing during the time of existence of strong anisotropic velocity distribution. But, if we diminish the number density of a matter up to gaseous values, the Weibel instability develops even at relatively low intensities (see also [3]).
In the case of a circularly polarized field with the same parameters as above, we find according to Eq. (21):
Imro = 0.4rop = 0.12 a.u.
Hence, the Weibel field is much greater in the case of circular polarization compared to the case
of linear polarization of the laser radiation since Imro-T = 180 (we used qualitatively the same estimate for T as we did above for linearly polarized field). The lower limit for spatial scale of this field is
X = (ro c)/(ropEi) - 0.8 A.
The upper limit for this scale is
X = (ro c2)/(ropF) - 85 A.
5. Expansion of plasma cloud
Above estimates refer to the solid density of a matter irradiated by a laser pulse. Let us consider here restriction for duration of laser pulse in order to neglect the expansion of produced plasma. The first mechanism is the Coulomb explosion. The Newton equation for model plasma ball is of the form
M(d2R)/(dt2) = QZ/R2.
Here M is the mass of the atomic ion, R is the current radius of produced plasma, Z is the charge multiplicity of the atomic ion, and Q is the charge of the plasma cloud. First we assume that all electrons are removed from this plasma ball due to outer ionization. Then Q = (4n/3)naZR 3. Here R is the initial radius of the plasma ball which is of order of laser focal radius. na is the initial number density of atoms in the irradiated solid matter.
Integrating the Newton equation we obtain the energy conservation law
(M/2)(dR/dt)2 = QZ/R -QZ/R. Let us estimate the time At for doubling the expanding cloud. Substituting R = 2R into this equation, one obtains
Thus,
dR/dt = (QZ/MR )12 = AR)/At = R /At. At = (MR 3/QZ)1/2 ~ (1/2Z)(M/na)1/2.
It is seen that the value of the initial radius R does not contain in this expression for doubling time At. Substituting M=100 a.m., Z = 30 and na = 0.0074 a.u. we find At = 2 fs.
Of course, the much more typical case is realized when only small part of electrons produced due to inner ionization are removed from the plasma cloud (outer ionization). If 1% of electrons is ejected from this ball, then according to the last equation we obtain the time for doubling of the ball radius At = 20 fs.
It should be noted that the excursion length of oscillation of ejected electrons in the laser field is F/ro 2 = 4700 A. Though this is very large quantity, but it does not change above estimates, since electrons oscillate in the plane of metal surface (at the normal direction of the laser ray) and return periodically back to the plasma ball.
Now we estimate the velocity of hydrodynamic expansion which occurs with the velocity of ionic sound:
Vi = (ZTe/M)1/2.
Here Te is the average electron temperature which is determined mainly by slow electrons produced at the early stage of the laser pulse. According to Ref. [12] we have Te ~ 1 keV. Hence, Vi ~ 0.077 a.u. If we consider the time interval At = 20 fs, then the surface of the plasma cloud shifts to the distance AR = 35 A. Thus, in the case of super-intense femtosecond laser pulses hydrodynamic expansion can be neglected compared to the Coulomb explosion. The typical times for Coulomb explosion allow to state that at the duration of laser pulse less than 30-50 fs the solid density of a matter does not change significantly in the peak of laser pulse. Real expansion begins after this peak.
6. Conclusion
Thus, we find that the plasma instability produces quasi-static (the frequency does not contain the real part) magnetic field B. The corresponding quasi-static electric field E is much less if we have in mind the short wave limit kc >> ro:
E = (ro/kc)B << B.
Let us estimate the maximum value of this field for the case of circularly polarized field. Our derivations are valid in the linear approximation when f << f . According to above results we rewrite this inequality for a circularly polarized field in the form:
(Ek2v )/ro3 << 1, or (Bkv )/(ro2c) << 1. Substituting the minimum value of the wave number k ~ ropF/(ro c2) and ro ~ rop(v /c), we find the maximum magnetic field (for circularly polarized field):
Bmax - ropc[1/(1+(F/ro c)2)1/2]. It is seen that relativistic effects strongly diminishes the Weibel magnetic field. This field is produced during the time t of order of the atomic time: t ~ (Imro) 1 ~ (Imrop) 1 << t where t is the time for alignment of velocities (see previous Section).
The atomic unit of magnetic induction is Bat = 17.13 MGs. Substituting F/(ro c) = 2.8 and rop = 0.30 a.u., one obtains Bmax ~ 230 MGs.
According to the energy conservation law the magnetic energy of the unit volume of plasma B2/4n should be less than the kinetic energy nv 2/2 of electrons in the same volume (see also Refs. [3, 13]). Thus, one obtains the inequality
Bmax < ropv .
It does not contradict to the above estimate of Bmax.
Now we consider the analogous estimates for the linearly polarized field. The condition f << f gives the inequalities
(Ek2u||)/ro3 << 1, or (Bku||)/(ro2c) << 1.
Substituting the minimum value of the wave number k - rop/c (see Section 2.2) and ro - uprop/c, we find
Bmax - ropu||(rel) - (ro /^^pc. Substituting y = 0.024, ro = 0.043 a.u. and rop = 0.30 a.u., we obtain Bmax = 70 MGs. Thus, circular polarization of laser radiation is more preferable for obtaining of high steady magnetic fields than linear polarization.
This material has been presented at Annual Moscow workshop "Physics of Non-ideal Plasmas" (PNP-2002) 3-4 December 2002.
References
[1] E.S. Weibel, Phys. Rev. Lett. 2, 83 (1959)
[2] A.F. Alexandrov, L.S. Bogdankevich, A.A. Rukhadze, Principles of Plasma Electrodynamics, Springer-Verlag, Berlin, 1984, p. 162.
[3] V.I. Arefyev, V.P. Silin, S.A. Uryupin, Phys. Lett. A 255, 307 (1999).
[4] N.B. Delone, V.P. Krainov, J. Opt. Soc. Am. B 8, 1207 (1991).
[5] V.P. Krainov, J. Opt. Soc. Am. B 14, 425 (1997).
[6] V.P. Krainov, M B. Smirnov, Phys. Rep. 370, 237 (2002).
[7] N.B. Delone, V.P. Krainov, Multiphoton Processes in Atoms, Springer-Verlag, Berlin, 2000.
[8] V.P. Krainov, Optics Express, 2, 268 (1997).
[9] P.A. Polyakov, Izv. RAN, Ser. Phys., 63, 2374 (1999) (in Russian).
[10] V.P. Krainov, J. Phys. B: At. Mol. Opt. Phys. 32, 1607 (1999).
[11] H.R. Reiss, J. Opt. Soc. Am. B 7, 574 (1990).
[12] A.Yu. Romanov, V.P. Silin, S.A. Uryupin, JETP, 84, 687 (1997).
