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Magnetic-field generation by a finite-radius electromagnetic beam
Volume 95A, number 5
PHYSICS LETTERS
2 May 1983
MAGNETIC-FIELD GENERATION BY A FINITE-RADIUS ELECTROMAGNETIC BEAM O.M. GRADOV P.N. Lebedev Physical Institute, Moscow, USSR and L. STENFLO Institute of Physics, Umea University, S-901 87 Umea, Sweden Received 27 September 1982 Revised manuscript received 23 February 1983
We point out that magnetic fields can be generated in a plasma because of the inhomogeneity in the radial intensity distribution of a linearly polarized electromagnetic beam.
Almost all of the mechanisms which previously have been proposed as explanations for magnetic-field generation in plasmas (see e.g. refs. [1,2]) suppose that the large radiation intensity changes the plasma parameters significantly. In the present lettea we shall consider a new mechanism which indicates that magnetic fields can be generated in a nonlinear medium, even if its density and temperature are not changed. This turns out to be possible if the intensity distribution of the incident electromagnetic beam is radially inhomogeneous. Let us thus consider a uniform plasma in which a linearly polarized electromagnetic wave propagates along the z axis
E(r, t) = YcEo(r ) exp(-icot + ikz),
(1)
where .~ is the unit vector along the x axis and r (X 2 +y2)1/2. The ponderomotive force associated with VE02, which is radial, induces obviously changes in the plasma parameters which, for simplicity, are considered to be independent of the z and 0 coordinates. A convenient subdivision of the fields and velocities into rapidly (characteristic times co-1) oscillating and slowly (times of the order of the inverse ion plasma frequency) changing quantities yields a system of selfconsistent equations (see e.g. refs. [3,4]) in which the averaged slow electric field can be neglected as the =
0 031-9163/83/0000-0000] $ 03.00 © 1983 North-Holland
ions will maintain quasi-neutrality (see e.g. ref. [4] ). It will now turn out that the electron density n o is essentially constant, that the generated magnetic field B ~ Bo(r)O is azimuthal and that the slow velocity u ~ Uz$ is in the z direction. Focusing our attention on the average equation of m o m e n t u m we then further assume that the electron thermal velocity Ore is sufficiently small (i.e. (mco/ec)Vte is much smaller than Eo/c as well as the initial magnetic-field magnitude) and the pressure term is accordingly neglected [4]. After a certain transient period, a final state is thus given by a balance between the ponderomotive and Lorentz forces, i.e.
~a(~2)/ar ~ - uza,
(2)
where [2 = eBo/m, "~ = (elm) ft E(r, t l ) d t 1 and the bracket ( ) represents the mean value over the time 27rico. By means of the Maxwell equation
VX B ~, laonoeUz$
(3)
and eq. (2) we then obtain
O(rI2) 2~Or = -(co 2p/C2)r2 0 (0 2 ) / a r ,
(4)
i.e. r
~2=coPFfdr 1 2
rc L
-11/2 rl(-aoZ/arl>J ,
(s)
0 233
Volume 95A, number 5
PHYSICS LETTERS
where 6Op is the electron plasma frequency. Considering an electromagnetic beam with gaussian intensity distribution, i.e.
(u 2) = w 2 exp(_r2/r2),
(6)
where r 0 is the effective beam radius, we rewrite (5) in the form n = t.Op7ro~ -W- [1 -- (1 + r2/r20)exp(_r2/r20) ] 1/2
(7)
The generated magnetic field is thus proportional to r for r ~ r 0, and to r - I for r ~ r O, whereas in the region r ~ r O, ~2 is o f the order of ~pW/C. Supposing
234
2 May 1983
that w/c is larger than 0.01, we then note that the magnitude ofB o [as defined by (7)] seems to be comparable to the dc magnetic field of any other suggested mechanism (see e.g. ref. [2] ).
References [ 1] M. Kono, M.M. ~ori~ and D. ter Haar, J. Plasma Phys. 26 (1981) 123. [ 2] R. Balian and J.-C. Adam, eds., Laser-plasma interaction (North-Holland, Amsterdam, 1982). [3] O.M. Gradov and L. Stenflo, Z. Naturforsch. 35a (1980) 461. [4] L.M. Gorbunov, Soy. Phys. Usp. 16 (1973) 217.
PHYSICS LETTERS
2 May 1983
MAGNETIC-FIELD GENERATION BY A FINITE-RADIUS ELECTROMAGNETIC BEAM O.M. GRADOV P.N. Lebedev Physical Institute, Moscow, USSR and L. STENFLO Institute of Physics, Umea University, S-901 87 Umea, Sweden Received 27 September 1982 Revised manuscript received 23 February 1983
We point out that magnetic fields can be generated in a plasma because of the inhomogeneity in the radial intensity distribution of a linearly polarized electromagnetic beam.
Almost all of the mechanisms which previously have been proposed as explanations for magnetic-field generation in plasmas (see e.g. refs. [1,2]) suppose that the large radiation intensity changes the plasma parameters significantly. In the present lettea we shall consider a new mechanism which indicates that magnetic fields can be generated in a nonlinear medium, even if its density and temperature are not changed. This turns out to be possible if the intensity distribution of the incident electromagnetic beam is radially inhomogeneous. Let us thus consider a uniform plasma in which a linearly polarized electromagnetic wave propagates along the z axis
E(r, t) = YcEo(r ) exp(-icot + ikz),
(1)
where .~ is the unit vector along the x axis and r (X 2 +y2)1/2. The ponderomotive force associated with VE02, which is radial, induces obviously changes in the plasma parameters which, for simplicity, are considered to be independent of the z and 0 coordinates. A convenient subdivision of the fields and velocities into rapidly (characteristic times co-1) oscillating and slowly (times of the order of the inverse ion plasma frequency) changing quantities yields a system of selfconsistent equations (see e.g. refs. [3,4]) in which the averaged slow electric field can be neglected as the =
0 031-9163/83/0000-0000] $ 03.00 © 1983 North-Holland
ions will maintain quasi-neutrality (see e.g. ref. [4] ). It will now turn out that the electron density n o is essentially constant, that the generated magnetic field B ~ Bo(r)O is azimuthal and that the slow velocity u ~ Uz$ is in the z direction. Focusing our attention on the average equation of m o m e n t u m we then further assume that the electron thermal velocity Ore is sufficiently small (i.e. (mco/ec)Vte is much smaller than Eo/c as well as the initial magnetic-field magnitude) and the pressure term is accordingly neglected [4]. After a certain transient period, a final state is thus given by a balance between the ponderomotive and Lorentz forces, i.e.
~a(~2)/ar ~ - uza,
(2)
where [2 = eBo/m, "~ = (elm) ft E(r, t l ) d t 1 and the bracket ( ) represents the mean value over the time 27rico. By means of the Maxwell equation
VX B ~, laonoeUz$
(3)
and eq. (2) we then obtain
O(rI2) 2~Or = -(co 2p/C2)r2 0 (0 2 ) / a r ,
(4)
i.e. r
~2=coPFfdr 1 2
rc L
-11/2 rl(-aoZ/arl>J ,
(s)
0 233
Volume 95A, number 5
PHYSICS LETTERS
where 6Op is the electron plasma frequency. Considering an electromagnetic beam with gaussian intensity distribution, i.e.
(u 2) = w 2 exp(_r2/r2),
(6)
where r 0 is the effective beam radius, we rewrite (5) in the form n = t.Op7ro~ -W- [1 -- (1 + r2/r20)exp(_r2/r20) ] 1/2
(7)
The generated magnetic field is thus proportional to r for r ~ r 0, and to r - I for r ~ r O, whereas in the region r ~ r O, ~2 is o f the order of ~pW/C. Supposing
234
2 May 1983
that w/c is larger than 0.01, we then note that the magnitude ofB o [as defined by (7)] seems to be comparable to the dc magnetic field of any other suggested mechanism (see e.g. ref. [2] ).
References [ 1] M. Kono, M.M. ~ori~ and D. ter Haar, J. Plasma Phys. 26 (1981) 123. [ 2] R. Balian and J.-C. Adam, eds., Laser-plasma interaction (North-Holland, Amsterdam, 1982). [3] O.M. Gradov and L. Stenflo, Z. Naturforsch. 35a (1980) 461. [4] L.M. Gorbunov, Soy. Phys. Usp. 16 (1973) 217.