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Nonlinear propagation and ponderomotive force of electromagnetic plane waves of arbitrary polarisation in a magnetoplasma
Volume 106A, number 9
PHYSICS LETTERS
31 December 1984
NONLINEAR PROPAGATION AND PONDEROMOTIVE FORCE OF ELECTROMAGNETIC PLANE WAVES OF ARBITRARY POLARISATION IN A MAGNETOPLASMA R. UMA and D. SUBBARAO Fusion Cell, Centre of Energy Studies, Indian Institute of Technology, New Delhi 110 O16, India Received 12 December 1983 Revised manuscript received 4 September 1984
Ponderomotive self-effects of plane uniform electromagnetic modes of a magnetoplasma introduce stop- and pass-band structures in the dispersion characteristics of the waves resulting in Bragg reflection, etc.
We present in this letter some results on the nonlinear propagation of arbitrarily polarised uniform plane electromagnetic waves in a magnetoplasma. This forms a sequel to the studies on ponderomotive effects in a magnetoplasrna discussed recently by Subbarao [1], Statham and ter Haar [2]. An understanding of the nonlinear propagation and reflection of electromagnetic waves is of interest in wave heating of magnetically confined thermonuclear plasmas [3], effects of megagauss fields on laser absorption in laser-driven fusion [4], soliton propagation [5] and astrophysical plasmas [5] and a host of nonlinear wave phenomena in plasma physics including ionosphere modification [6]. Ponderomotive effects of waves in a plasma are usually considered when the waves are inhomogeneous in their intensity profiles [7,8]. However, in a magnetoplasma, in the presence of a single linear mode, even a uniform plane wave has been shown to give rise to significant effects on propagation characteristics for sufficiently large amplitudes [1 ]. Since it is in general not possible to launch a wave in one pure polarisation in the weird geometries involved in the wave-launching schemes mentioned above, we consider the presence of both the characteristic linear modes in the magnetoplasma; a study of wave accessibility and dispersion characteristics then reveals, even in a weakly nonlinear case, significant features such as 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
total non-penetration of the wave due to Bragg reflection. It is found convenient to calculate, following refs. [ 1,2], the steady-state ponderomotive force density from the time independent part of the first moment of the Vlasov equation viz. the fluid momentum equation: Fpc, = (pc~E + c - l J a X B ) - (4rr/COp2a)(V'JcJc~), (1)
where o~refers to the charge species, pa and Ja are the charge and current densities respectively, and ¢Opa is the characteristic plasma frequency. Expressing the field E in terms of the unit eigenvectors ~1 and #2 corresponding to the characteristic modes, i.e. E = E 1 + E 2 where E 1,2 = E1,2~1,2 exp [i(k 1,2 .r - cot)] and using the orthonormal relations for the displacement vector D a n d g : ki.D j = 0 and D i . E ] = n25i] (i = 1, 2; ] = 1, 2), where n i is the refractive index for the ith linear mode, eq. (1) may be simplified to Fp = (i/16rr) × {(k 1 - k2)" [(1 - oj2/o~ 2) ( E I E ~ - E2E~) + (oa2/oa 2) (EID ~ - E2D]) ] + (k I - k2)(E 1 "E~)
+ c.c.},
(2)
423
Volume 106A, number 9
PHYSICS LETTERS
where c.c. refers to the complex conjugate. This can be seen to vanish when k 1,2 [1B0 and in general oscillates in space with period 2rr[k t - k 2 [ - I . Consider an arbitrarily polarised electromagnetic wave launched obliquely at the plasma-vacuum boundary into the plasma. The eigenmodes propagating at angles 01 and 0 2 with respect to the normal :~ to the interface then satisfy Snell's law and also (k I k 2 ) ' r = O, r being a vector on the boundary taken alongy so that k 1 - k 2 is along ~; the magnetic field B 0 makes an angle 0 B with the normal. It is easily shown that the steady-state ponderomotive force in eq. (2) creates a density modulation along £ in the plasma. This may be determined selfconsistently from the electrostatic field generated in response to the ponderomotive force acting only on the electrons for co >> OOci. In particular, for normal incidence of the electromagnetic wave when the eigenmodes propagate collinearly, the density modulation -
-
6n_
1
1 - 2~2/eo 2
ElzE z cost(k1
n
- k 2 ) z - II (3a)
is created by the z component o f the force density (Fpe) z = (k 1 - k2)(ElzE~z/8rr ) (2oo2/6Op2 - 1) × sin[(k 1 - k2)z ] ,
where a} 2 = 4k2(0B -- Oi)/(k I - k2) 2 , k 2 = 4k 2 sin 20in/(k 1 - k2) 2 and
424
=_ qi
(On2/an)(Fpe)z (Te+Ti)(kl -k2)2
(4b)
n=no
w i t h y = (k 1 - k2)Y/2 and ~ = [(k 1 - k2)z + q~]/2, 4' being the phase of the complex ponderomotive force and n o the electron density. As dictated by the well-known stability diagram for the Mathieu equation [12], the unstable bands are situated for small q near ,
al,2 =al,2 - k
2
= m 2,
(5)
where m = 0, 1,2, .... The only physical solution for Zi(z) in the stable pass and unstable stop bands for example near a ~ 1 is then given by [11]
Zi(z ) "~ Z exp(-/az~ {exp(iz) + R 1 e x p ( - i S ) + A 3 [exp(3iz) + R 3 e x p ( - 3 i z ) ] + ...},
(6a)
where R 1 = exp(2io), R 2 = [(C 3 + iS3)/(C 3 - i S 3 ) ] exp(6io), A 3 = i(C 3 - iS3) exp(2io),
(3b)
where ire, T i are the electron and ion temperatures respectively in eV and E 1 z, E2z are as in ref. [9]. The other components of the force are associated with magnetic field generation [ 10]. The modulation in (3a) along with the force in (3b) vanishes for k 1,2 I1B0 and k 1,2 j- B0In the parametric approximation where the nonlinearity in eq. (3a) is calculated using unmodified linear modes, the wave equation of the eigenmodes E i = Eoi exp(ikyy)Zi(z-- ) reduces to the Mathieu equation [ 11 ] [d2/d2 -2 + (a] ,2 - k2) - 2ql,2 c°s 2z-] Z i = 0 ,
31 December 1984
(4a)
and o=~,arccos6,
# = ~1 q ( 1 - 62)1/2,
6 = (Z/q) [1 - (a + ~q2)1/2] , 1
s3 = - - g q + 6 ~ q2 cos 2 o + . . . , c3 _,_ a q2 sin 2 0 - - s ~ q 3 s i n 4 o + ....
(6b)
In the stop band, eq. (8) is a superposition of decaying (amplification is not possible in a passive medium) spatial harmonics and represents an evanescent mode of the plasma; accordingly at the plasmavacuum boundary the component of the electromagnetic wave with the polarisation of the normal mode is totally Bragg reflected. It may be recognised that Bragg reflection in the stop bands can be either total or partial depending on whether both the normal modes are in the stop bands or only one of them lies in the stop band. The former is possible in two cases: (1) Both the stop bands are of the same order and the bandwidths are sufficiently large for the bands to overlap. The band
Volume 106A, number 9
PHYSICS LETTERS
width for example, near a ~ 1 is q/a so that this case implies a I - a 2 ~ q/a; (2) The stop bands are of different orders (case (2) is elaborated later - fig. la). The validity of the solution in eq. (8) is restricted to the region near the boundary of the stop bands (deffmed by/.t = 0) where the coefficients IR 11, ~R2[ decay to zero.
0.3~
5
ta)
/
¢ B1
A2 ~,,,,,,,~FE
/
0.2
¢
~. 0.2:
In the pass bands, far from the stop bands, [R 1 [ = JR21 = 0 and/z is purely imaginary in eq. (8). The propagation constant may be approximately written in this region as [ a - 1 \1/2 k~(k I -k2)~aq2) , (7) ( a - 1) 2 _ q 2 provided a 4: m 2, m = 1,2 . . . . . The cascading to low wavelengths (corresponding to the lowest order wave number) should increase the collisionless absorption through nonlinear Landau damping as described for an unmagnetised plasma by Kaw et al. Further, for a < 0 which corresponds to the linear cut-off for the modes, k is real implying anomalous penetration of the wave into the plasma [10]. To investigate the propagation of any given wave in the magnetoplasma one may consider either eq. (7a) or the stability diagram depending on the initial parameters given. As an illustration, we consider the propagation of high frequency waves with 6o > 6o~1) > 6o(.ol) > 6o~2) when the two normal waves are the fast extraordinary (FE or right circularly polarised) wave with n2FE = 1 -- (6o2e/6o2) [1 -- (6oce/6o) cos 0FE ] and the ordinary wave (O or left circularly polarised) withn02 = 1 -(6op2e/6o2)[1 + (6oce/6o)cos 00] where w~l), 6o~2) are the linear cut-offs for FE and O respectively and 6o(**1)is the resonance of FE. Given 6oce =0.1 6ope, 6o = 1.5 6ope,0B = 60°, 0FE = 74°,00 90 °, eqs. (5) give the critical angle of incidence as 0in = 47 ° for which FE lies in the stop band near aFE ~ 1 and O lies in the stop band neara 0 ~ 4. The incident wave is expected to be totally Bragg reflected at this angle. Alternatively given 0in as for instance for the case of normal incidence, 0 B = 10 ° and Wee = 0.1 6ope, to determine the frequency band gaps, the stability diagram should be superimposed by the curves of the two normal modes by plotting a i versus qi, i = I, 2; along these curves 6o varies monotonically. From fig. la, the curve corresponding to FE may be seen to intersect the unstable shaded region near 1, 4 ...;
31 December 1984
/
0.11
/ J
0.1t
L
1.0
3.0
I
J
5.0
i
7.O
9.0
Q
~
1.12
1.10 \
3
(b)
FE
/7 7
1.08
/i 1.06
= 0.2
I 0.3
i
I O.t. KC/a~p
L
I
0.5
I
Fig. 1. (a) The stability diagram for the Mathieu equation superimposed by the curves a(to) and q(to) (to increadng towards the fight on the curves) corresponding to the fast extraordinary (FE) and the ordinary (O) waves for propagation at 0 = 10° with respect to the magnetic field and toe/top = 0.1 ; (b) dbpersion curves (Brillouin diagram) derived from (a) above for the quasilongitudinal case: continuous curve nonlinear and dotted curve the linear case.
while the O-wave intersects a ~ 4 and higher order stop bands. Correspondingly the dispersion curve (Brillouin diagram) of the FE curve shows a band gap (fig. lb) in 60 (and k); the a ~ 4 stop band is too narrow to be seen and is indicated by a square in fig. lb. At 6o = 1.076 6ope, FE lies in the a ~ 1 stop band while O lies in the a ~ 4 stop band, so that total Bragg reflection of the incident wave would occur. Finally it may be noted that the strength of the nonlinearity does not affect the location of the stop bands but it does affect the width. When cyclotron 425
31 December 1984
PHYSICS LETTERS
Volume 106A, number 9 Table 1 no (cm-3)
magnetically confined fusion plasma laser-plasma ionosphere
Power required for
:re (eV~
104
1010
109
1020 10s - 106
102 - 104 1
10_1015 10-4 10-3
1013 1014 10-3 10-2
The authors would like to thank Professor M.S. Sodha and the editor for their valuble comments. Financial support from the Indian Space Research Organisation and the Department of Science and Technology is acknowledged.
(8)
where h i is the normalised spatial damping rate, F = F ( w , Wpe, Wee ) and fiT1,2 = E 1,2~(noTe) 1/2 may be determined from eq. (6b). For normal incidence, the threshold powers are given in table l, where h i = 0.01, normalised power E2/87rTe ~ l, [Ell = otl/2lEI, IE21 = (I - ~)I/2[EI and the optimum value of a maximising E1E2, i.e. a = 1/2, has been used. In conclusion, we have shown that the propagation of finite-amplitude electromagnetic waves in a magnetoplasma is governed by the Mathieu equation in the parametric approximation. The incident angles and other conditions under which total or partial Bragg reflection of the incident wave occurs are discussed. It is proposed that this could be an important mechanism for reflection of incident light in laserplasma interaction occurring at threshold comparable to stimulated Brillouin scattering. The pass bands may be used to dump energy from time-modulated electromagnetic waves into the plasma [10] ; the stop bands may open up new channels for microwave and radio communication [10]. Although the ponderomotive force and density modulation considered here may be derived as in refs. [ 1 4 - 1 6 ] it is found that the derivation based on eq. (1) retains most of the information dictated by the Abraham model [17] for the stress tensor.
426
stimulated Brillouin scattering (W/cm2)
101 s
or collisional damping is strong, the stop bands are narrowed and more significantly, the nonlinearity parameter has to be above a certain threshold for Bragg reflection. The threshold condition is given by
E 1 E 2 = ki/F ,
nonlinear Bragg reflection (W/cm2)
References [ 1] D. Subbarao, Report liT DELHI/CES/FUS (1981). [2] G. Statham and D. ter Haar, Plasma Phys. 25 (1983) 681. [3] M. Porkolab, in: Fusion, ed. E. Teller (Academic Press, New York, 1981). [4] P.M. Bellan and J. Adam, Laser-plasma interaction (Stern, Dusseldorf, 1983). [5] S.G. Thornhill and D. ter Haar, Phys. Rep. 43 (1978) 43. [61 A.V. Gurevich, Nonlinear phenomena in ionosphere (Springer, Berlin, 1978). [7] H. Motz and C.J.H. Watson, Advances in electron and electronic physics. Vol. 23 (Academic Press, New York, 1967) p. 168. [8] J.A. Stamper, Phys. Fluids 15 (1975) 735. [9] K.C. Yeh and C.H. Liu, Theory of ionospheric waves (Academic Press, New York, 1972). [t0] R. Uma and D. Subbarao, Report lIT DELHI/CES/FUS (1983), in preparation. [ 11 ] N.W. Mclachlan,Theory and application of Mathicu functions (Dover, New York, 1964). [12] L. BriUouin,Wave propagation in periodic structures (McGraw-Hill, New York, 1946) pp. 174-176. [13] P.K. Kaw, A.T. Lin and J.M. Dawson, Phys. Fluids 16 (1973) 1967. [14] L.P. Pitaevskii, Soy. Phys. JETP 12 (1961) 1008. [15] V.I. Karpman and A.G. Shagalov, J. Plasma Phys. 27 (1982) 215. [16] A.N. Kaufman, J.R. Cary and N.R. Perira, Phys. Fluids 22 (1979) 790. [17] F.N.H. Robinson, Phys. Rep. 16 (1975) 313.
PHYSICS LETTERS
31 December 1984
NONLINEAR PROPAGATION AND PONDEROMOTIVE FORCE OF ELECTROMAGNETIC PLANE WAVES OF ARBITRARY POLARISATION IN A MAGNETOPLASMA R. UMA and D. SUBBARAO Fusion Cell, Centre of Energy Studies, Indian Institute of Technology, New Delhi 110 O16, India Received 12 December 1983 Revised manuscript received 4 September 1984
Ponderomotive self-effects of plane uniform electromagnetic modes of a magnetoplasma introduce stop- and pass-band structures in the dispersion characteristics of the waves resulting in Bragg reflection, etc.
We present in this letter some results on the nonlinear propagation of arbitrarily polarised uniform plane electromagnetic waves in a magnetoplasma. This forms a sequel to the studies on ponderomotive effects in a magnetoplasrna discussed recently by Subbarao [1], Statham and ter Haar [2]. An understanding of the nonlinear propagation and reflection of electromagnetic waves is of interest in wave heating of magnetically confined thermonuclear plasmas [3], effects of megagauss fields on laser absorption in laser-driven fusion [4], soliton propagation [5] and astrophysical plasmas [5] and a host of nonlinear wave phenomena in plasma physics including ionosphere modification [6]. Ponderomotive effects of waves in a plasma are usually considered when the waves are inhomogeneous in their intensity profiles [7,8]. However, in a magnetoplasma, in the presence of a single linear mode, even a uniform plane wave has been shown to give rise to significant effects on propagation characteristics for sufficiently large amplitudes [1 ]. Since it is in general not possible to launch a wave in one pure polarisation in the weird geometries involved in the wave-launching schemes mentioned above, we consider the presence of both the characteristic linear modes in the magnetoplasma; a study of wave accessibility and dispersion characteristics then reveals, even in a weakly nonlinear case, significant features such as 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
total non-penetration of the wave due to Bragg reflection. It is found convenient to calculate, following refs. [ 1,2], the steady-state ponderomotive force density from the time independent part of the first moment of the Vlasov equation viz. the fluid momentum equation: Fpc, = (pc~E + c - l J a X B ) - (4rr/COp2a)(V'JcJc~), (1)
where o~refers to the charge species, pa and Ja are the charge and current densities respectively, and ¢Opa is the characteristic plasma frequency. Expressing the field E in terms of the unit eigenvectors ~1 and #2 corresponding to the characteristic modes, i.e. E = E 1 + E 2 where E 1,2 = E1,2~1,2 exp [i(k 1,2 .r - cot)] and using the orthonormal relations for the displacement vector D a n d g : ki.D j = 0 and D i . E ] = n25i] (i = 1, 2; ] = 1, 2), where n i is the refractive index for the ith linear mode, eq. (1) may be simplified to Fp = (i/16rr) × {(k 1 - k2)" [(1 - oj2/o~ 2) ( E I E ~ - E2E~) + (oa2/oa 2) (EID ~ - E2D]) ] + (k I - k2)(E 1 "E~)
+ c.c.},
(2)
423
Volume 106A, number 9
PHYSICS LETTERS
where c.c. refers to the complex conjugate. This can be seen to vanish when k 1,2 [1B0 and in general oscillates in space with period 2rr[k t - k 2 [ - I . Consider an arbitrarily polarised electromagnetic wave launched obliquely at the plasma-vacuum boundary into the plasma. The eigenmodes propagating at angles 01 and 0 2 with respect to the normal :~ to the interface then satisfy Snell's law and also (k I k 2 ) ' r = O, r being a vector on the boundary taken alongy so that k 1 - k 2 is along ~; the magnetic field B 0 makes an angle 0 B with the normal. It is easily shown that the steady-state ponderomotive force in eq. (2) creates a density modulation along £ in the plasma. This may be determined selfconsistently from the electrostatic field generated in response to the ponderomotive force acting only on the electrons for co >> OOci. In particular, for normal incidence of the electromagnetic wave when the eigenmodes propagate collinearly, the density modulation -
-
6n_
1
1 - 2~2/eo 2
ElzE z cost(k1
n
- k 2 ) z - II (3a)
is created by the z component o f the force density (Fpe) z = (k 1 - k2)(ElzE~z/8rr ) (2oo2/6Op2 - 1) × sin[(k 1 - k2)z ] ,
where a} 2 = 4k2(0B -- Oi)/(k I - k2) 2 , k 2 = 4k 2 sin 20in/(k 1 - k2) 2 and
424
=_ qi
(On2/an)(Fpe)z (Te+Ti)(kl -k2)2
(4b)
n=no
w i t h y = (k 1 - k2)Y/2 and ~ = [(k 1 - k2)z + q~]/2, 4' being the phase of the complex ponderomotive force and n o the electron density. As dictated by the well-known stability diagram for the Mathieu equation [12], the unstable bands are situated for small q near ,
al,2 =al,2 - k
2
= m 2,
(5)
where m = 0, 1,2, .... The only physical solution for Zi(z) in the stable pass and unstable stop bands for example near a ~ 1 is then given by [11]
Zi(z ) "~ Z exp(-/az~ {exp(iz) + R 1 e x p ( - i S ) + A 3 [exp(3iz) + R 3 e x p ( - 3 i z ) ] + ...},
(6a)
where R 1 = exp(2io), R 2 = [(C 3 + iS3)/(C 3 - i S 3 ) ] exp(6io), A 3 = i(C 3 - iS3) exp(2io),
(3b)
where ire, T i are the electron and ion temperatures respectively in eV and E 1 z, E2z are as in ref. [9]. The other components of the force are associated with magnetic field generation [ 10]. The modulation in (3a) along with the force in (3b) vanishes for k 1,2 I1B0 and k 1,2 j- B0In the parametric approximation where the nonlinearity in eq. (3a) is calculated using unmodified linear modes, the wave equation of the eigenmodes E i = Eoi exp(ikyy)Zi(z-- ) reduces to the Mathieu equation [ 11 ] [d2/d2 -2 + (a] ,2 - k2) - 2ql,2 c°s 2z-] Z i = 0 ,
31 December 1984
(4a)
and o=~,arccos6,
# = ~1 q ( 1 - 62)1/2,
6 = (Z/q) [1 - (a + ~q2)1/2] , 1
s3 = - - g q + 6 ~ q2 cos 2 o + . . . , c3 _,_ a q2 sin 2 0 - - s ~ q 3 s i n 4 o + ....
(6b)
In the stop band, eq. (8) is a superposition of decaying (amplification is not possible in a passive medium) spatial harmonics and represents an evanescent mode of the plasma; accordingly at the plasmavacuum boundary the component of the electromagnetic wave with the polarisation of the normal mode is totally Bragg reflected. It may be recognised that Bragg reflection in the stop bands can be either total or partial depending on whether both the normal modes are in the stop bands or only one of them lies in the stop band. The former is possible in two cases: (1) Both the stop bands are of the same order and the bandwidths are sufficiently large for the bands to overlap. The band
Volume 106A, number 9
PHYSICS LETTERS
width for example, near a ~ 1 is q/a so that this case implies a I - a 2 ~ q/a; (2) The stop bands are of different orders (case (2) is elaborated later - fig. la). The validity of the solution in eq. (8) is restricted to the region near the boundary of the stop bands (deffmed by/.t = 0) where the coefficients IR 11, ~R2[ decay to zero.
0.3~
5
ta)
/
¢ B1
A2 ~,,,,,,,~FE
/
0.2
¢
~. 0.2:
In the pass bands, far from the stop bands, [R 1 [ = JR21 = 0 and/z is purely imaginary in eq. (8). The propagation constant may be approximately written in this region as [ a - 1 \1/2 k~(k I -k2)~aq2) , (7) ( a - 1) 2 _ q 2 provided a 4: m 2, m = 1,2 . . . . . The cascading to low wavelengths (corresponding to the lowest order wave number) should increase the collisionless absorption through nonlinear Landau damping as described for an unmagnetised plasma by Kaw et al. Further, for a < 0 which corresponds to the linear cut-off for the modes, k is real implying anomalous penetration of the wave into the plasma [10]. To investigate the propagation of any given wave in the magnetoplasma one may consider either eq. (7a) or the stability diagram depending on the initial parameters given. As an illustration, we consider the propagation of high frequency waves with 6o > 6o~1) > 6o(.ol) > 6o~2) when the two normal waves are the fast extraordinary (FE or right circularly polarised) wave with n2FE = 1 -- (6o2e/6o2) [1 -- (6oce/6o) cos 0FE ] and the ordinary wave (O or left circularly polarised) withn02 = 1 -(6op2e/6o2)[1 + (6oce/6o)cos 00] where w~l), 6o~2) are the linear cut-offs for FE and O respectively and 6o(**1)is the resonance of FE. Given 6oce =0.1 6ope, 6o = 1.5 6ope,0B = 60°, 0FE = 74°,00 90 °, eqs. (5) give the critical angle of incidence as 0in = 47 ° for which FE lies in the stop band near aFE ~ 1 and O lies in the stop band neara 0 ~ 4. The incident wave is expected to be totally Bragg reflected at this angle. Alternatively given 0in as for instance for the case of normal incidence, 0 B = 10 ° and Wee = 0.1 6ope, to determine the frequency band gaps, the stability diagram should be superimposed by the curves of the two normal modes by plotting a i versus qi, i = I, 2; along these curves 6o varies monotonically. From fig. la, the curve corresponding to FE may be seen to intersect the unstable shaded region near 1, 4 ...;
31 December 1984
/
0.11
/ J
0.1t
L
1.0
3.0
I
J
5.0
i
7.O
9.0
Q
~
1.12
1.10 \
3
(b)
FE
/7 7
1.08
/i 1.06
= 0.2
I 0.3
i
I O.t. KC/a~p
L
I
0.5
I
Fig. 1. (a) The stability diagram for the Mathieu equation superimposed by the curves a(to) and q(to) (to increadng towards the fight on the curves) corresponding to the fast extraordinary (FE) and the ordinary (O) waves for propagation at 0 = 10° with respect to the magnetic field and toe/top = 0.1 ; (b) dbpersion curves (Brillouin diagram) derived from (a) above for the quasilongitudinal case: continuous curve nonlinear and dotted curve the linear case.
while the O-wave intersects a ~ 4 and higher order stop bands. Correspondingly the dispersion curve (Brillouin diagram) of the FE curve shows a band gap (fig. lb) in 60 (and k); the a ~ 4 stop band is too narrow to be seen and is indicated by a square in fig. lb. At 6o = 1.076 6ope, FE lies in the a ~ 1 stop band while O lies in the a ~ 4 stop band, so that total Bragg reflection of the incident wave would occur. Finally it may be noted that the strength of the nonlinearity does not affect the location of the stop bands but it does affect the width. When cyclotron 425
31 December 1984
PHYSICS LETTERS
Volume 106A, number 9 Table 1 no (cm-3)
magnetically confined fusion plasma laser-plasma ionosphere
Power required for
:re (eV~
104
1010
109
1020 10s - 106
102 - 104 1
10_1015 10-4 10-3
1013 1014 10-3 10-2
The authors would like to thank Professor M.S. Sodha and the editor for their valuble comments. Financial support from the Indian Space Research Organisation and the Department of Science and Technology is acknowledged.
(8)
where h i is the normalised spatial damping rate, F = F ( w , Wpe, Wee ) and fiT1,2 = E 1,2~(noTe) 1/2 may be determined from eq. (6b). For normal incidence, the threshold powers are given in table l, where h i = 0.01, normalised power E2/87rTe ~ l, [Ell = otl/2lEI, IE21 = (I - ~)I/2[EI and the optimum value of a maximising E1E2, i.e. a = 1/2, has been used. In conclusion, we have shown that the propagation of finite-amplitude electromagnetic waves in a magnetoplasma is governed by the Mathieu equation in the parametric approximation. The incident angles and other conditions under which total or partial Bragg reflection of the incident wave occurs are discussed. It is proposed that this could be an important mechanism for reflection of incident light in laserplasma interaction occurring at threshold comparable to stimulated Brillouin scattering. The pass bands may be used to dump energy from time-modulated electromagnetic waves into the plasma [10] ; the stop bands may open up new channels for microwave and radio communication [10]. Although the ponderomotive force and density modulation considered here may be derived as in refs. [ 1 4 - 1 6 ] it is found that the derivation based on eq. (1) retains most of the information dictated by the Abraham model [17] for the stress tensor.
426
stimulated Brillouin scattering (W/cm2)
101 s
or collisional damping is strong, the stop bands are narrowed and more significantly, the nonlinearity parameter has to be above a certain threshold for Bragg reflection. The threshold condition is given by
E 1 E 2 = ki/F ,
nonlinear Bragg reflection (W/cm2)
References [ 1] D. Subbarao, Report liT DELHI/CES/FUS (1981). [2] G. Statham and D. ter Haar, Plasma Phys. 25 (1983) 681. [3] M. Porkolab, in: Fusion, ed. E. Teller (Academic Press, New York, 1981). [4] P.M. Bellan and J. Adam, Laser-plasma interaction (Stern, Dusseldorf, 1983). [5] S.G. Thornhill and D. ter Haar, Phys. Rep. 43 (1978) 43. [61 A.V. Gurevich, Nonlinear phenomena in ionosphere (Springer, Berlin, 1978). [7] H. Motz and C.J.H. Watson, Advances in electron and electronic physics. Vol. 23 (Academic Press, New York, 1967) p. 168. [8] J.A. Stamper, Phys. Fluids 15 (1975) 735. [9] K.C. Yeh and C.H. Liu, Theory of ionospheric waves (Academic Press, New York, 1972). [t0] R. Uma and D. Subbarao, Report lIT DELHI/CES/FUS (1983), in preparation. [ 11 ] N.W. Mclachlan,Theory and application of Mathicu functions (Dover, New York, 1964). [12] L. BriUouin,Wave propagation in periodic structures (McGraw-Hill, New York, 1946) pp. 174-176. [13] P.K. Kaw, A.T. Lin and J.M. Dawson, Phys. Fluids 16 (1973) 1967. [14] L.P. Pitaevskii, Soy. Phys. JETP 12 (1961) 1008. [15] V.I. Karpman and A.G. Shagalov, J. Plasma Phys. 27 (1982) 215. [16] A.N. Kaufman, J.R. Cary and N.R. Perira, Phys. Fluids 22 (1979) 790. [17] F.N.H. Robinson, Phys. Rep. 16 (1975) 313.