Nonlinear dynamics of high power radiation in a magnetized plasma

Volume 137, number 1,2

PHYSICS LETTERS A

1 May 1989

NONLINEAR DYNAMICS OF HIGH POWER RADIATION IN A MAGNETIZED PLASMA A. C A R D I N A L I a, A.V. K H I M I C H b, M. L O N T A N O c, E.I. R A K O V A b and A.M. SERGEEV b a CentroRicerchesull'Energia, EURATOM-ENEAAssociation, Frascati, ltaly b InstituteofAppliedPhysics, SovietAcademyofSciences, Gorky, USSR c lstitutodiFisica delPlasma, EURATOM-ENEA-CNRAssociation, Milan, Italy Received 31 October 1988; revised manuscript received 31 January 1989; accepted for publication 1 March 1989 Communicated by R.C. Davidson

The dynamics in time and space ofa Gaussian beam of high power free electron laser (FEL) radiation, polarized in the ordinary mode, propagating in a magnetized plasma, is presented within a paraxial approximation. The process of self-focusingof radiation and the subsequent formation of a focus point, running back towards the incoming high frequency power, are numerically computed. Some implications on the optimal FEL operations in a plasma heating scenario are discussed.

In the near future, new electron cyclotron resonant (ECR) heating scenarios will occur when high power free electron laser (FEL) sources will be available [ 1 ]. In these operational regimes a sequence o f short pulses o f electromagnetic ( E M ) radiation will be injected in a tokamak plasma, each pulse being characterized by a very high peak power (Po ~ 10 G W ) and short time length ( r ~ 50 ns). Electric fields larger than 105 V / c m [2] will be able to drive nonlinear effects which can substantially affect the propagation and absorption characteristics of the medium. In the foreseen experimental conditions [ 1 ] the most important nonlinear mechanism which can arise is the self-focusing o f the injected radiation, due to the dependence of the EM wave phase velocity on the local inhomogeneous electric field amplitude; if an effective focusing occurs, the structure of the high frequency ( H F ) electric field in the plasma would be essentially nonlinear, and we should reconsider the analysis of the absorption properties o f the EC waves at the resonant layer with respect to the standard linear theory. The nonlinear dynamics o f FEL radiation, propagating in the form o f an ordinary mode ( O M ) in a magnetized plasma, has been studied first in ref. [ 3 ], with reference to the range o f the experimental parameters expected in the M T X project [ 1 ]. It was shown that the set o f equations for the complex am-

plitude of the H F electric field and for the nonlinear plasma density perturbation has the following form:

OE O2E 0 ZE -i-~x + ~ + ~ - nE=O,

(1)

O2n 021El 2 Ot 2 Oz 2

(2)

In eqs. (1) and (2) the dimensionless variables are related to the physical ones through the following relations: --2 --

~z,

ZO

x(

2kz2

--

xl/2

1--

--,y

,

ZO

1- ~

--, x,

2 2 Zo~2p~ On c2( 1 -t2~o/t2 2 ) n

Ezo

~

--, E ,

---~n

f2peet ( 2P \Mmc3(l_i2~/f22)

),/2 -,t,

(3)

where 12 and s'2peare, respectively, the injected wave and electron plasma frequencies, e, m, M are the electron charge, the electron and the ion masses, P is the injected power, Zo is the transverse dimension o f the beam and k is the wavenumber. Furthermore, we refer to an EC wave beam, polarized in the OM,

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Volume 137, number 1,2

PHYSICS LETTERS A

which propagates along the x direction, almost perpendicularly to the external magnetic field Bo, which points in the z direction. In the subsequent analysis we will make use of the so-called aberrationless approach [4] which allows us to follow the dynamics, in space and time, of the axial part of the wave beam in the plasma. Within this approximation, we represent the field amplitude and the density perturbation by means of the following trial functions: E=

1

~/2a(x, t)b(x, t) ( z2 y2 ) × exp 2a 2 2b 2 +ia+iflz2+iTy 2 ,

n = - n o ( x , t) exp

c2(x, t)

d2(x, t) '

(4)

where a(x, t) and b(x, t), c(x, t) and d(x, t) are the characteristic transverse dimension of the wave beam and of the density perturbation, respectively; fl and 7 describe the curvature of the wavefront and a is the phase shift. The substitution of expressions (4) in eqs. (1) and (2) and the expansion in powers of y2 and z 2, give rise to the following equations:

02a 1 O~2 - a3 -fa,

02f 3 Ot2 - a s b,

02b 1 O~2 - b 3

02g 1 Ot2 - a 3 b 3 ,

gb,

(5)

where ~= 2x, f = no/c 2, g= no/d 2. Before proceeding to the solution of eqs. ( 5 ) let us comment the limits of validity of the approximation used here; the solution of self-focusing problems in nonlinear optics and plasma physics has demonstrated that the aberrationless approach is in excellent agreement, as far as the average characteristics of the beam (i.e. scales, point of focus) are concerned, with other methods such as variational approaches and numerical simulations. As we have already mentioned it is valid for the central part of the beam, where actually the most part of the EM energy is concentrated, while it fails in the peripheral region of the spot. In the steady state it is possible to write the exact Hamiltonian of the system and through a variational approach to derive the model equations for typical beam scales associated with Gaussian trial 48

1 May 1989

functions: these equations are the same as those derived within our approximation. In the dynamical case the comparison should be made with the results of numerical simulations. Let us go back to our problem. The set of equations (5) has the following asymptotic solution, for a-,0: a~ ~

-

~+

u dt'

b-aft,

f = l / a 4, g = l / b 4,

, (6)

corresponding to a field distribution in the form of a running focus. The focal point (which represents a singularity for the field) travels at an arbitrary velocity u (t) toward the high frequency source and the half-apex angle of the cone surface near the focus increases as the singularity slows down. In principle, the law of velocity variation can be determined only after the complete solution of the problem in the whole space, including the region close to the boundary, where the solution is not self-similar; numerically, however, we can determine the law u(t) whithin the aberrationless approximation. We have studied numerically the set of equations (5). Typical results of the computations are shown in fig. 1. An ordinary diffractive structure a=b= ( 1 +~2)J/2, in the absence of plasma perturbations, was taken as initial condition. Then, with increasing of the nonlinear density perturbation, the transition from defocusing to self-focusing takes place resulting inevitably in the singularity formation. The structure of the distribution near the singularity has obviously the self-similar form of (6). The superimposed small oscillations are caused by the matching of the self-similar solution with the field structure near the plasma boundary. The formation of running focuses can then prevent the direct transmission of EM energy to the region of EC resonance or decrease the efficiency of EC damping. To avoid this effect, the experimental parameters have to be chosen so that the duration of the single FEL pulse be smaller than the time of singularity formation:

zu
--

This expression can be written in the form

(7)

Volume 137, number 1,2

PHYSICS LETTERS A

t : 3224

t= O 0 0 0

1.00

1,00

0.80

0.80

0.60

0.60

0.40

0.40

0.20

0.20

(a)

"'-...b,,

(d) I

000

i

0.00

1 May 1989

t = 1131

I

I

I

I

t = 3394

b

100

1.00 a

0.80

0.80

0.60

060

0.40

0.40

0.20

0.20

a

(b)

0.00

I

I

I

i

~\

(e)

,.

0.00

I

t=2.828 t = 3 691

1.00 0,80

0.80

0.60

060

0.40

0.20 0.00 0.00

0.40

(C)

0.20 '

'

'

~

0.10.

0.20

0.30

0.40

0.00 0.00

0.50

(f) I 0.10

I 0.20

~_ I

0.30

~I

0.40

I 0.50

Fig. 1. The functions a(~) (full line) and b(~) (dashed line), solutions of eqs. (5), are plotted versus ~, for different times t. The evolution of the initial divergent pattern of the wave beam (a) towards the convergent structure (b, c, d ), and the subsequent formation of the focus (e, f ), receding against the incoming radiation are shown. All variables are normalized according to the transformation ( 3 ).

~Q2z~Mmc3( 1 -.Qpe/~Q 2) 1 ru< 2e212~e W ~ ~,

(8)

where W--Pzu is the energy of the FEL pulse. For representative values of M T X parameters (i.e., plasma m i n o r radius a = 15 cm, average plasma density ~ 2 X 1014 c m - 3 , average electron temperature ~re~ 1 keV, magnetic field Bo ~ 7 T, beam half-size Zo= 1-2 cm, injected frequencyf~. ~ 200 GHz, single pulse energy FV~ 200 J) we can estimate the value of the time of focus formation as tr 7 X 10-1, Zo2(cm) s, which turns to be very small with

respect to the pulse duration (5 × 1 0 - 8 s); furthermore, for the same parameters, we can evaluate the typical plasma d i m e n s i o n on the l-axis in fig. 1 with the formula ~ = 4 × lO-2a(cm)/z~ ( c m ) , which gives ~ 0.6 or 0.15 for Zo= 1 or 2 cm, respectively. It is clear that to prevent the appearance of the nonlinear effects discussed here the pulse of the EM radiation, for a fixed injected energy and beam size, has to be shortened.

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References [ 1 ] K.I. Thomassen, Free electron laser experiments in AlcatorC, Lawrence Livermore National Laboratory, Livermore, USA, LLL-PROP-00202 (1986). [2 ] W.M. Nevins, T.D. Rognlien and B.I. Cohen, Phys. Rev. Lett. 59 (1987) 60.

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1 May 1989

[3] A. Cardinali, M. Lontano and A.M. Sergeev, in: Proc. 15th Europ. Conf. on Controlled fusion and plasma heating, Vol. 3, eds. S. Pesic and J. Jacquinot (EPS, Dubrovnik, 1988) p. 976; and Phys. Fluids, accepted for publication. [4] A.V. Petrischev and V.I. Talanov, Kvantovaja Elektron. 6 (1971) 35.