interaction between a plasma and a strong electromagnetic wave

Physica 98C (1980) 313-324 © North-Holland Publishing Company

INTERACTION BETWEEN A PLASMA AND A STRONG ELECTROMAGNETIC WAVE

HIROSHI TAKAHASHI* Tokyo Institute of Technology, Ookayma, Meguro ku, Tokyo, Japan

Received 14 August 1978

The interaction between a high density plasma and a strong electromagnetic wave are studied by using the quantum- mechanical two times Green's function method. The equations of motion for Green's function of a many component plasma in an electromagnetic wave with several frequencies and polarization vectors are obtained as a set of simultaneous equations. The equations of motion are solved in the case of a plasma which is composed of electrons and heavy ions, and the dispersion relation and the density correlation functions for electrons and ions are formulated in closed form. The quantum-mechanical dispersion relation is similar to the type of express/on obtained by Silin for the cla~cal plasma. The effects of exchange Coulomb interactions, which become appreciable in high density plasmas, on the formulation are studied for the case of Fermion particles by using Hubbard's approximation.

I. Introduction

Interaction between a plasma and a strong electromagnetic wave produces a parametric coupling. This parametric coupling has been discussed by using the electromagnetic fluid theory [I, 2] and the classical plasma theory [3, 4] ; the phenomena of the oscillating two stream instability [5, 6] and stimulated Brlllouin and Raman scattering [I, 5] have also been formulated.

In order to give a more quantitative basis for the high density plasma, a quantum theory should be developed. The first application of the quantum-mechanical Green's function method to this problem was performed by Goldman [7]. He treated the parametric couple of external radiation to a pair of collective longitudinal mode in a plasma with two components by examining the periodic Green's function for a non-linear Poisson's equation. However, his formulation [7, 8] is rather complicated, so that the higher order coupling (in terms of in eq. (2.28)) is difficult to be taken into account.

In this paper, we will use the two times Green's function method formulated by creation and annihilation operators. The equation of motion for the Green's function is obtained as the hierarchy equation. The two creation and two annihilation operator Green's function is decoupled, and a closed form equation in the form of a Hartree-Fock type equation can be formulated.

In section 2, a Hamiltonian for the coupled system of the plasma with many components and the electromagnetic wave is formulated by using the second quantization method. The simultaneous equations for the Green's functions of the plasma particles are obtained by using the two times Green's function method. Section 3 treats the case when the plasma is composed of the electron and the heavy ions. The same t y p e o f dispersion relation corresponding to Silin's classical formula [3, 4] is obtained by solving'the simultaneous equations. The density correlation functions for electrons and ions are also formulated. By using the approximation obtained by Hubbard, the simple evaluation of the exchange coulomb interaction was carried out.

*Present address: Brookhaven National Laboratory, Upton, Long Island, New York, USA.

313

314 H. Takahashi/Interaction between a plasma and a strong electromagnetic wave

2. Formalism

Let us consider a plasma composed of electrons and ions which is irradiated by a strong laser. The wave function • (i)(r, t) of the ith particle with charge q(O in the very strong electromagnetic wave is expressed as

t-

, , . : ,ox, ( : , I , q ,A,t 2mO)

(2.1)

where C(i),k(i) and m are respectively normalization constant, canonical momentum and mass of the particle. The electromagnetic wave is assumed to be very strong so that it is treated as a classical vector potential A. The interaction between the plasma and the electromagnetic wave is treated by the dipole approximation. Throughout this paper, the unit of~ = C - 1 will be used. When the electromagnetic wave is composed of the several modes pro- pagating along the z direction, it is expressed by

A(t) = ~ [,a-.xs cos (COst + Os) + Ay s sin (COst + Os)], S

(2.2)

where Axs and Ay s are respectively x and y components of polarization vector A s. The total Hamiltonian of the plasma composed of I kinds of particles under the laser irradiation is expressed by

i{[1 ] • s,t i ] ( i¢/) s,t

(2.3)

where E F is the Fermi potentialwhen the particle is a Fermion. ~dii)(r s - rt) is the Coulomb interaction potential between the ith'and the flh particles.

To obtain the second quantization formalism, the wave function of the particle ~q)(r, t) and its adjoint function are expressed as

• O)(r, t) = ~_,. ag~oi(t ) exp (-ik~)" r), k I o I

(2.4)

• *(0(r, t) = ~ a~l ( t ) exp (ik~). r) (2.5) kl Ol

and the total'Hamiltonian is expressed by the creation and annihilation operators a~lot, aklOl as follows

i,/(i~ /) Oklozl~a2 (2.6)

where

H. Takahash i / I n t e rac t i on b e t w e e n a p l a s m a a n d a s t rong e l e c t r o m a g n e t i c wave 315

:1~ 9 ¢ D = (k~i) + q(t)A(t))2 - E~ ), (2.7) 101 ~. ,' 2m(O

:~/)(Q) 4,q(%e) 1__ = ~ " Q2, (2.8)

here v(i)(v if)) is the volume per i kind (] kind) particle in the configuration space. And the prime over the summation in eq. (6) means that the term of Q = 0 is omitted in the summation.

On the other hand, from the following second quantization formalism for the density operator of the i particle

p(O(r, t) = ~*(O(r, t)~I~O)(r,t) = ~ a~ l exp ( ik~O..; ja~o,(t) exp ( - Jk 2 • r ) (2.9)

k lk2OlO2

we obtain the Fourier transform of the density operator

,(O(k,t)= J'p(O(r, OeiX'rdr = ~. a~ol(t)a~+kOl(t). kl ° l

(2.10)

The Fourier transform of the density correlation function (p(O(r, t), p(0(r', t')) can be expressed by

jdr dr'(p(i)(r, t ), p ff)(r', t'))e -ik" (r- : ) = F(i/)(k, t - t')

= O(O(k, t), p(/Yf(k, t)) = ~ . (O'f t (0 a (i)t ¢t'~a (i) :t'~> (aklal()akl+kal(t), k2+ko2~ J k2o2~ s. kzozk2~

(2.11)

To calculate this, let us consider the following two times retarded Green's function

- <<a~,o,(t): ,kx+ko,(t); , ,~)+*ko.(t') , ,~)o.(r)>> XlOl~ , . . . .

k2~2

= ((a~l~(t)a~l+k(t); p(/Yf(k, t)>)

= -iO(t - t') ( [a~z( t )a~+k( t ) ; pfJ)f(k, t)] ), (2.12)

where

¢o> = Tr (e-a/To)/rr (e-H/r), [,4, B] = AB - aA,

and Tis the temperature. The Fourier transform of the Green's function

1 I O(t) = J

"0

for t ~ 0

for t < 0 (2.13)


G(iJ~k, E) = ~. G~ j) (/¢, E~ (2.14) ~.1o'1. - -

kl Ol

and the Fourier transform 4~?g) of the correlation function F(ii)(k, t) are related by the following equation

/i/)(k, ~) -- -[2/(e - e / r - 1)] Im G(ii)(k, E + ie), (2.15)

where e -~ 0 is assumed. To obtain the hierarchy equation for the Green's function eq. (14), we first construct the quantum-mechanical

of motion for the operator a~3~oi and a ~ i (From now on we omit the time argument t of the operator a equation and a t unless it is necessary):

at i Ok2~ (2.16)

~)t (2.17)

The equation of motion for the Green's function eq. (14)]s then obtained as

ac~q) (k, t - t') i - K l ° l '~ ~t = ( -7~a l ( t ) + T~ ÷kal(t))G~)al(k, t - t') + 6(t " t ' )6 i~n~al - n~ +kal)

(q) Ot ;'>t , a '), a 0 . v

i)t "), "') , a i) • ff)t k . - ( ( a ~ l a l a ~ e z a ~ - o ~ z ~ 1 + k + O a 1 , P ( , t ' ) ) ) ]

(2.18)

To obtain the Green's function G~,/)a (k, t - t ') we have to know the higher order of Green's function expressed • . . 1~1 ~ . ~ . .

by the two creation and two annihilation operators shown m the last term of eq. (18). The equation of motion for these higher order Green's function can be obtained in the same way as the above by differentiating these in terms of time t. From this procedure, we get the next higher order of Green's function expressed by the three creation and three annihilation operators. In this way, we will obtain the hierarchy equation which is very hard to solve. To avoid this complexity, we will use the approximate expression for the above higher order Green's function expressed by the two creation and two annihilation operators as follows. That is, the higher order of Green's function in eq. (18) is decoupled as

a i ) t "')t a i') i ) t • q ) + k t " ' "

,i t "~ t

(2.19)

H. Takahasht/lnteraction between a plasma and a strong dectromagnetic wave 317 ((a i)f a " ) *a ,') ,a t9 • (l)f - ( (a~a~)+ko~( t ) ;p( / ) f (k , t )> > ~I0'1 J~iO~ ~--00'2 ~l-l-k+Ool(t),P (k, t)))- ~kl,Qn~lO , ....

-T- 6i, f ~o1, ~ (Sk 1, ~-on~?,l((a~ol ak[ +kOl(t); P(J)t(k, t)))

"t- ak~ ' (k I +k + Q)n~o 1 ((a~lO 1 a~? +kOl(t); pq)(k, f)))} "F g(klOl)(l)(k~o~)(j')(k2_Oo~)(f)(k I +k +0'o0(0( t - t').

(2.20)

In these equations, the upper or lower sign is applied to Fermion or Boson particle. The last terms, g are the terms which corresponds to the higher correlated Green's functions. We assume that these terms are smali enough that we can neglect them.

This approximation corresponds to the Hartree-Fock approximation, and the first terms in RHSof eqs. (19) and (20) contribute to the direct Coulomb interaction and the terms in the parentheses { } contribute to the exchange Coulomb interaction. And the second terms in the parentheses { } contribute to the change of particle in kinetic energies Tk(~o. (t) and T~)+ ko- (t) are shown in the following.

Substituting eq. (119) and (20)linto eq. (18), and arranging it, we obtain the equation of motion for the Green's function as follows

i)ex a : l a l , ~ t (i~t -I- T(kO1al~t)-- T~1+kOl(t))G~iJ) (k, -t')ffi~i,j~(t-t')(n~?.l - n~+kOl )

(n (i) nff) ", ' ~ g1°1"-" gl+k°lYG~ ij) ( k , t t ' ) 4" 4'/rq(0/~,(i)k2 V.klOl - n~+ko,) ]~ ~ q(~)G~2]~,(k, ' - t ' )~ 4ffq (~2 ~ (k 2 _k~) 2 . /f.2Ol ~

= Q ( t - t ' ) , (2.21)

where

T(k Oolex(t) = T ( t~ (t) • 4,q (02 ~, n~o,/ (k 1 - k2) 2. (2.22)

To solve eq. (21), let us divide the time dependent energy difference into the time independent part and dependent part as follows

T(ki)leOlX(t)_ T~i~e~k(f)= ~i)ex ~i).x q(O x lo I " k I + ko 1 - " ~ 3 k " A ( t ) , (2.23)

where

k 2 ~0ex = [ ~ "'101 ], 2m(O -- EI~) ) • 4ffq (02 ~ n~o,1/(~1 -k2) 2.

k2 (2.24)

The integration of eq. (21)is obtained as

-~gex ,gex . q(t9 __1 G(kilDal(k, t - t ' ) = - i ~ e x p [ + i ( ~ l O l - ~ l + k ¢ l ) ( t - t # ) - l - m - ~ s~ ( {k. Ax s

318 H. Takahashi/lnteraction between ¢ plasma and a strong electromagnetic wave

X [sin (cost + as) - sin (Wst"+ as) ] - k . Ay s [cos (cost + as) - cos (Wst"+ Os)] } ] Q(t" - t ')dt",

where Q(t - t') is the RHS of eq. (21). By using the following formula,

(2.25)

exp (-izSin ~O) = ~ JL(Z) exp (-iL~b) L -- _ o o

(2.26)

we obtain the Fourier transform of eq. (21) as follows

r ~ l j ) ( k , . e) = U 7. exp [-,,.,(0,- "s)] Y. S,,,.(4"b exp , ',)] s Ls= -.0. Ms =--**

X Q(Y's(Ms - Ls) ~°s + E - ie) ~( / ) ex ~r(i)e~

E + .,t k lO . 1 - - .,t k l + . g o 1 - ~,sLs6os

(2.27)

where

z s = qk" A s / m w s, 1

A s is a difference vector ofAsx -AsY' I

A s = tan -1 (AsylAsx).

Substituting the Fourier transform of RHS of eq. (21) to eq. ~z/) and rearranging it, we get

(2.28)

JLs_Nst. s ) ~ JLs-Ms(zli))exp [--i(Os-- As)(Ns- Ms)] s s L s Ms

0 (n~'~lOl - - n~l +ko 1)

+

- g 1 ¢ 7 1 k2o 1 s

i) ni)~ s)] 4nq(02~(0~x - 0ex ~ ( n ~ l a , - ~l+k*l).,/~zfr-(#) k , E + ~ M s w . (2.29)

(E + T, sLsW s + , klal - s .~/'/~l+kOl) - (k 2 - k l ) 2

If the last term of RHS, which is due to the exchange Coulomb interaction, is approximated in the same way as in the works of Bailyn [12] and Hove [13], we get

S k 1 s

It. Takahashi/Interaction between a plasma and a strong electromagnetic wave

s L~ M~ s

k 2 'f al s j'~ s

ff/) k E k2 ~ ~°1 ~ " $ s '

where

_ n~to 1 - n~3 l+ko l L k l o l ( k , E ) _ ~ ( 0 + ~i)ex ~i)ex '

• -~ A £ 1 O l - - - £ 1 + k O l

/-# o.i ,~

kl

319

(2.30)

(2.31)

(2.32)

al

1 _oil ff)extir,,.., E')=I'~kl 1]-lk~llr~(k2_kl)2L~02ol (k'E)" By operating the {H s ~NsJks_Ns(z (i)) exp 0(0 s -- As)Ns)} to eq. (30) and rearranging it, we get

(il)(k,E-I- ~ Kd'°s)= { " [ - [ ~ Jks_Ns(Z~i))exp (i(Os -- As)N s) ~GUo~(k,E'I" ~ N'sc%) S 8 JV S S

(2.33)

(2.34)

={n ~, JKs-Ma(z~ i))exp(i(Os-As)Ms)} ~(k,E+ ~Ksc%)~i, j s Ms s

k2"f ol ~ Ksc° " ' re2 s

where

L<~)¢k,~= L~'~(k,e)/[l 4~q ('32

(2.35)

(2.36)

Furthermore, after similar manipulation, we get


s s K s M s

+--~ -.(i)L(o (I,,E + . . . . , s / ' * i s

(2.37)

where

L(i)(k, E) = ~. La(~)(k, E), Ol

- -~ q(i)2LO)(k, E ) ~ L(i)(k, E), ~(i)(k,E ) = I1 4"a . . 1

The dmpersion relatiot| and the correlation function are obtakned by solving eq. (37).

(2.38)

'(2.39)

3. Electron and very heavy ion plasma

In general it is not easy to solve eq. (2.37), but the case of the plasma composed of electron and very heavy ions can be simplified as follows. If we take superficies i = 1 and i = 2 . . . I for electron and ions respectively, and the ions are so heavy that from eq. (2.28) we can make z = 0 for i ~ 1. Then the JKt_ Mt(Z, 0))_ is replaced by ~IQ, Mr" Thus eq. (2.37) for the Green's function G (i' D(k, E +Ntl, s~o s) becomes

$ $

+ ~ q(i)L(i)(k,E + ~ Lsc%)~f qO')G(f'/)(k,E + ~ Lscos), (3.1)

o°'"(,.~ + ~,.~--In ~ _,.,.~1~ ~ ~_~.,.~i~ °xp E-i,o.-~.,,,.- ~,i} s s Ks

X 4~"(1)L(1)(k,E.L2 '4 ~ + X Ks°Js)[1 - ~ X q(f)2L(/') -1 s z~*l

i ~1 s i '~1 s

×~,1,o,1~ ~ + ~ , ) ] ,32, 8

H. Takahashi/Interaetion between a plasma and a strong electromagnetic wave 321

Furthermore, for the oscillator with energy E much smaller than the frequency Us O f the external field, we 03 can make L (k, E + ZsMs6os) = 0 (M s ~ 0 and i 4~ 1). Then the Green s function G(I,D(k, E) becomes the

following simple form,

+""". ++": l ' -,.++'r ",W" p'("' +' +:+

4 n i l 411" q(i,)2L(i3(k ' q(i32Z(i)(k,E) ] i ' l l

~ - - 1 '

x l4rt"(1)J'~X J2Ks(z~l))tg(l'(k,E+ X Kso°s) [1- ~ X q(i')2LU')( k'E)] '~.~' [sK s s i ' • 1

The dispersion relation is obtained by putting the term inside the 1st parenthesis, [ ] equals zero. Similarly, we can get the other Green's functions as follows.

(3.4)

G(l'k)(k,E)ffi~2~Ks J2K~g(1)~q(1)L(l~k,E+~s Ks6~s)q(J)L(/)(k,E) A-l, (3.5)

G'(U)(k. E) = -~ q(O ~.(i)(k. E) q(J> L(J)(k. E) [1 + e(l>(k.E')] A -I, (3.6)

G(iO(k, E)= Z(i)(k, E')[1- (1 + e(1)(k,E')) ~2 X q q')2g ffS( k, E)]A -1 , /' ~iand l

(3.7)

6'Oa)(k. E) = G(X/>(k. ~ . (3.8)

G(m(k, E) = C(~i)(k. e), (3.9)

where i, j ~ 1 and

A = I - - e(1)(k, E) e(2)(k, E~) ,

_+n" q(m|rT X" j2 (z(1))~Z(1)~t., e(1)(k'E)- 2 I 1 ~ , K, s E+XKsws)' k | , ~ : . J T" ,

-1

i@1

(3.1o)

(3.11)

(3.1'2)


[ ~]-1 L(O(k,~=L(O(k,~ 1 - ~ f~,x q~2L~')(k, , (3.13)

~ = L(~)(k, ~ [ 1 - ~ , , (mia ) - - ~]-1 (3.14) Z(l)(k,

Dispersion relation in this case is expressed as,

e(l)(k, E) e(2)(k, E) = 1. (3.15)

By taking S = 1 and I = 2, we can get the similar dispersion relation obtained by Silin for the classical plasma:

4nq(l)2 Z y2(z(1))[1 4#r 2- Kw)I-IL(1)(k,E + Kw) --k2 K - ~'] q(1) L(1)(k, E +

X 4~ ,.,(2)2[ 1 4n q(2)2 L(2)(k, .-]-1L(2)( k, E) = 1. (3.16) k 2 ~ L - k 2 "~'J

That is, the 6e e and 6e i in Silin's formula [7, 8] are respectively replaced by quantum-mechanical longitudinal di- electric permittivities (4n/k2)q (1)2 L(1)(k, E)and (4n/k2)q (2)2L(2Xk, E)which include the Coulomb exchange interaction. When we take electro-mgnetic intensity A = 0, the formula for the Green's functions becomes the familiar Green's function of the non electromagnetic wave as follows:

GOD(k, E) = L(1)(k, .E') AT"I ,

O(t/)(k, E) = ~ q(O iY)(k, ~ q~0 L~O(k, E) ~i 1,

Z G(H)(k'E) ffi Z(O(k'E)[1-- ~ ]' . iand l q(f)2L(fXk,~{1-{- 4'/r Z(1)(k, E)} ] ~ r - 1 , / 6 2

(3.17)

(3.18)

(3.19)

where

4n.,(1)2L(1)(k ~.4~r Z q(i)2L(0(k,E) • (3.20) Af= l -k2 q ~ "~J k2 i¢1

The dispersion relation in the non electromagnetic wave becomes

4~ Zq(02Lq)(k, E) = 1. (3.21) k2 i

As shown in the previous section, the correlation functions are obtained by using eq. (2.15)from these Green's functions.

H. Takahashi/Interaction between a plasma and a strong electromagnetic wave 323

So far, we have derived the dispersion relation and the correlation function for the plasma which is under laser irradiation. By using the second quantization formalism, we could include the exchange Coulomb interaction in the formula. The closed form for the permittivity of Fermi particles has been formulated in the references [11 ], but this formalism for the exchange Coulomb interaction is valid for only small momentum changes k. Therefore, let us study the exchange Coulomb interaction using the simple approximate formula obtained by Hubbard. According to his theory , the exchange Coulomb interaction appeared in the formula as the screening of the direct Coulomb interaction in the following formula

41re2 [1 - k2 (3.22)

k 2 ~ 2(k 2 + k2F '

where e is the electron charge, and k F is the Fermi momentum, which is expressed by

k F = (37t2n0) 1/3, (3.23)

n o in.eq. (23) is the electron density. When the momentum becomes larger than kF, this screening factor becomes appreciable. As shown in eq. (23)

the Fermi momentum k F is a function of electron density no, the plasma frequency Wp corresponding to this electron density is

(3.24)

When an electromagnetic wave propagates in this plasma, the wave interacts strongly with the plasma when the plasma frequency resonates with the electromagnetic frequency. Furthermore, reflection or scattering will occur. Therefore, we will estimate the exchange Coulomb interaction effect inJ this case. The momentum 0f the electromagnetic wave is expressed by

27r ~P- -=~41re2n0 (3.25) k = " ~ = c

and the momentum change is also of the same order. Thus, by substituting eqs. (23) and (25) into eq. (22), we get the screening factor as shown as a function of electron density (see table I). The screening effect becomes

Table I

The screening factor due to the exchange Coulomb interaction (7, = electro magnetic wavelength; kf = Fermi momentum and n o = electron number density)

n o 7` k f k 2 (cm -a) (cm) 1 - 2(k2 + k~

10 Is 1.056 × 10 _~. 3.094 x 10 s 0.9656 1016 3.342 × 10 - s 6.665 × 10 s 0.9313 1017 1.056 × 10 -s 1.436 x 106 0.8722 1018 3.342 x 10 --6 3.094 x 106 0.7875 1019 1.056 × 10 -6 6.665 x 106 0.6929 102o 3.342 × 10 -7 1.436 x 107 0.6129 1021 1.056 × lO -7 3.094 × 107 0.5596 1022 3.342 X 10-8 6.665 × 107 0.5296 1023 1,056 X 10 -8 1.436 × 108 0.5142 1024 3.342 X 10 -9 3.094 × 108 0.5067


appreciable when the electron density is more than 1017 . The electron densities of 1019 and 1021 produce approximately the plasma frequencies which correspond to the CO 2 laser and Nd ruby laser frequencies. The screening factors for these cases are 0.6929 and 0.5596, respectively. For much higher density this screening factor approaches 0.5 which corresponds to the value due t~ Fermi's exclusion principle.

References

[1] S. Jorna and K. Bruckner, Rev. Mod. Phys. 46 (1974) 325. [2] D. F. Dubois and M. V. Goldman, Phys. Rev. 164 (1967) 207. [3] V. P. Silin, Soy. Phys. JETP 30 (1969) 105. [4] V. P. Silin, Soy. Phys. Usp. 15 (1973) 742. [5] K. Nishikawa, J. Phys. Soc. Japan 24 G968) 916, 1152. [6] P. K. Kaw and J. M. Dawson, Phys. Fluids 12 (1969) 2586. [7] M. V. Goldman, Ann. Phys. 38 (1966) 95; 38 (1966) 117. [ 8] I~ Kadanoff and G. Bayra, Quantum Statistical Mechanics (Benjamin, New York, 1962). [9] T. Matsubua, l~og. Theor. Phys. 14 (1955) 351.

[10] D. N. Zubaxev, Soy. Phys. Usp. 3 (1960) 320. [11] H. Takahaslti, Physiea 51 (197I) 333. [12] 1~ Batlyn, PhyL Rev. 136 (1961) A1321. [13] B. Dayal and P. Srlvastava, Proc. Roy. Soc. A283 (1965) 394.

interaction between a plasma and a strong electromagnetic wave

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