Near fields in the vicinity of pulsed electron beams in space

P&net. Space Sci., Vol. 35, No. I, pp. I i-19, 1987 Printed in Great Britain.

0

00324633/87 $3.00+0.00 1987 Pergamon Journals Ltd.

NEAR FIELDS IN THE VICINITY OF PULSED ELECTRON BEAMS IN SPACE K. J. HARKER and P. M. BANKS Space, Telecommunications and Radioscience Laboratory, Stanford University, Stanford, CA 94305, U.S.A. (Received 20 h4ay 1986) Abstract-Most measurements of the radio frequency emissions from artificial electron beams in space have been made in close proximity to the beam. This paper presents theory and results for the near-field radiation associated with the operation of a pulsed electron beam, taking into account the effects of the ionospheric plasma. The beam is modeled as an infinite train of square-wave pulses and the radiation is obtained by adding coherently the radiation from each individual electron in the idealized helical trajectory assumed by the beam as it traverses the magnetized plasma. The mathematical solution is obtained by taking the Fourier transform in space and time of Maxwell’s equations and the driving modulated beam current, and then taking the inverse transform of the resulting electric field.Typical electric field strengths are presented for a range of modulation frequencies extending from below the ion cyclotron to the electron cyclotron frequency.

There has been considerable recent interest in electromagnetic wave emissions from electron beams launched in space from sources mounted on space vehicles, such as rockets and the Space Shuttle.

In previous work (Harker and Banks, 1984; Harker and Banks, 1985) we presented theory and results for the far-field radiation from a pulse-modulated electron beam. The beam consisted of either a finite (Harker and Banks, 1984) or infinite train (Harker and Banks, 1985) of square wave modulated electron bunches. The radiation spectrum studied was coherent spontaneous emission obtained by adding coherently the radiation from individual electrons moving in the segmented helical path assumed by the beam as it traversed the enveloping space plasma. Most of the wave measurements made to date with such beams have been made in the near field region. This includes experiments on the STS-3 (Banks et al., 1984; Shawhan et al., 1984), in the Spacelab- SEPAC experiment (Obayashi et al., 1982 ; Taylor and Obayashi, 1984) and on Spacelab- (Gurnett et ai., 1986). In STS-3 the fields emanating from the beam were made with the Plasma Diagnostic Package (PDP) (Shawhan and Murphy, 1983) mounted on the Remote Manipulator Arm (RMS), thereby confining the measurement distance to some tens of meters from its beam. Later experiments with a free-flying PDP on Z-2 have allowed wave measurements to greater distances of approximately 100 m from the beam (Gurnett et al., 1986). The purpose of this paper is to extend our previous

work to the near field case. By this we mean, specifically, that we will not introduce the usua1 far field assumption that the observation point is far enough from the beam for the radiating fields to behave as exp (ikr/r), but rather keep the expression for the fields in their most general form. Since the radiating and non-radiating fields are intermixed near the beam, the fields strengths are calculated instead of the power flux as in our previous work. Following the method of Harker and Banks (1985), we again assume an infinite train of pulses extending from z = 0 to co, and we assume that the ability of the beam to produce coherent fields falls off exponentially with distance along the beam. It should be further pointed out that the primary intent of this paper is to present the fundamental theory for the generation of near fields in pulsed electron beams. We will also give representative numerical solutions, which should give the reader a feeling for the general character of the solutions, including the variation of the electric fields with frequency and azimuth and cold plasma wave modes. A detailed quantitative comparison with results from experiments on the Space Shuttle and rockets will be covered in a sequel to this paper, since these are still in a preliminary phase. Other authors have recently discussed emission from pulsed electron beams (Lavergnat and Lehner, 1984; Oknuki and Adachi, 1984). Their work, however, considered only beams with zero pitch angle and did not consider near fields, their studies being solely confined to far field radiation. Most recently, Lavergnat et al. (1984) have presented results for the

12

K. I.

J. HARKER

and P. M.

BANKS

transform

B,

in space given in the form

E(r, t) exp [ - i(w,t - k - r)] drdt Wm k) = ~(2wxm)4

(4)

where wy = yw,, w,[ = 27ru,,/(il is the modulation frequency, d is the distance between pulses and y is the integer harmonic number. The inverse Fourier transform is then given by

E(r, t) =

E(w,, k) exp [i(w,t - k - r)] dk.

f Y=-rn s

If we use rotating

coordinates

then the electric field and current FIG. 1. ELECTRON BEAM COORDINATESYSTEM.

power flow from injected at non-zero

sinusoidally pitch angles.

modulated

ELECTRIC FIELD IN ROTATING

M,, =

In this section we begin our development for the theory of near fields resulting from the helical motion of a pulsed electron beam in a space plasma. Specifically, we derive the inverse transform for the electric field components in rotating coordinates. The electric beam is emitted from an electron source and spirals around a magnetic field line with parallel and perpendicular velocities u,, and uI. Figure 1 shows the coordinate system used in our analysis. The z-axis is the axis of the helix, and the x and y axes are chosen so that the electron source is located at the point (a, 0,O) where

(1)

(

as

a cos -,

VII

WXZ

a sin -,

VII

frequency. The

Z

%2

VII

$Pa,exp i(ct-

v)&

(8)

and the matrix elements P,” are given in Appendix and D is the plasma dispersion function

A,

D = -[E,~L:+E~~~+(E~+E,)~‘~:

-_(EOE,+E,E_ ,),u:-2E,a~Z+E&,E~

,I.

(9)

The quantities E,, Ed and E_ 1 appearing in these formulas are the relative dielectric tensor components in rotating coordinates defined by &,=l-

Z 2 %I wpe w(w + VW,,) - w(w - VW,)

(10)

where wpe and wpi are the ion and electron angular plasma frequencies and o,, is the ion cyclotron angular frequency. The refractive index p = (pL, cp,n) is related to the propagation vector k = (k,, rp, k,,) by the relation p

WJ

VI cos P) VII

(7)

(2)

>

with velocity - vI sin -,

are related by

where

COORDINATES

where w, is the electron cyclotron beam thereby follows a trajectory

defined as

E(w, k) = - & M(w, k) *J(w, k)

beams

(5)

VII

(3)

>

Since the wave is unbounded in time, we use a Fourier series transform in time and a Fourier integral

=

kc. w

(11)

The coordinate systems for the position vector r = (p, $, z) and the propagation and refractive index vectors are shown in Fig. 2. The dispersion function D can be written in the form

Near fields of pulsed electron beams in space

4

If we substitute the formal result

I

13 (7), (8) and (15) into (5), we obtain

2Nei exp i w,t-k,,z ’ (27~)~w,~Dy -k,ws(4-ti)-s(4

- 4) +xb]

- dk,,k,dk,d&

x sinq

(18)

% FIG.

2. REFRACTIVE INDEX AND POSITION VFCMR DINATES

D =

-eo(nZ-n,2)(n*-n;)

COORINTEGRATIONS OVER k,,, # and k,

(12)

where

& =

&{-(Eo+E.)P:+2~o~.~

~~(E,--O)*~:+4EO(E,E_,-E~)~:+4E~(E,2--E,E_,)},

In this section we carry out the integrations in (18) over k,,, k, and 4. The integrations over k,, and k, are carried out by contour integration, while the integration over 4 is carried out using Bessel function identities. The contour integration with respect to k,, has residues at the poles of the integral in (18). These occur at n2 = nz, n2 = ni, and n = n,, where

(13)

n, = bc ~ WY

and Es =

;(E,+E-,).

This contour

2Ne 4

$v,, (19)

reduces (18) to the form

s 2n

0

d’

Nek$.,, (k)bEE,y

o,t-k,pcos(c$-$)

xexpi

L

VlllY - 9> +&$]a.

-

vi,

integration

X

-s(o-;)+us]

(2~) Bv,,+ i(q-kkllvll --swd

x singexp[is(g

-SW,

(14)

and n,’ and ni are associated with the positive and negative values of the radical. The equation for the Fourier transform of the current has been derived by Harker and Banks (1985). If we transform their current to rotating coordinates, then this current is given by J,(w, k) = -

w,

k,,, =

%b

(15)

x P,,a, sin 2v B

/I

(20)

where

where

B=

(17) b is the pulse length, d is the distance between pulses, fl = l/Z is the inverse coherence length of the beam defined in Harker and Banks (1985), and J,(k,a) are Bessel functions.

exp [- ik,,,(k&l k,,a(kJGz(kd - k;&Jl x [Bv,, + @, - k,,,(k&,,-wJl expI- ik,,&&l + k,,dkJk$ikJ -k;AkJ x WV,, + i(w,- k,,dkJq -mJl (21)

K.

J.

HARKER

and P. M. BANKS 2

0

-2 log

E

W/ml -4

-6

. EP E* _______^. Ez -

-8

-10

-4.0

-6.0

-2.0

-3.0

.O

-1.0

log Wr&,

FIG. 3. LOGARITHM OFTHEPOLAR

COORDINATE

OF THE

BEAM GYRORADIUS

WHEN

AS A FUNCTION

OF THE

ELECTRIC

FIELD

ALL BEAM ENERGY LOGARITHM

AT THE

IS TRANSVERSE

OF W/WCC FOR S =

FIG. 4. SAMEASFIG. 3, BUTFOR

~M~~N~

ROOT

2 OFpI.

0 AND ROOT 1 OF /iI.

Beam parameters are lOOOV,100 mA and 30” pitch angle. Plasma parameters are an electron plasma to cyclotron frequency ratio, 6, of 8, an ion to electron mass ratio of 29,300 (O+), and w,/c = 0.02. The plot assumes that /l = 0, that the harmonic number y and the duty cycle factor sin (r&/d) are unity, and ignores the factor exp (it). In this figure log (wLH/c& corresponds to -2.23, and log (w&e,_j) to -4.47.

- co to cc so that we could use a contour integration. Second, we would prefer a Hankel function of the second kind of preference to Jy_,(k,p), since this type of Hankel function falls off as exp (- ik,p) in accordance with the radiation condition. It turns out that we can cure both of these defects by a transfo~ation of the type first introduced by Sommerfeld (1949). We first substitute the identity

We next integrate

.L,@,P)

over 4, using the formula

2n exp i(z cos B- n0) do = 27ci-“J,(z), s0

- exp i7c(s- a)Hj!,(k,

p exp ( - zi))]

(24)

(22) into (23). We can also show that

to obtain K(p,II/,z) =

= ![@%,p)

L-k,.) f CCexprlqst-((s--cr)ll/lia y=-m Y 9

s m

X

0 k,dkL

a,( -kJ

~ek~v,, (27r)%J&&,ypavav x sin

2

.t_.(k,p)B, I/

(23)

Equation (23) suffers from two defects. First we would prefer the limits for the kL integration be from

= (- l>“-‘p&,)

(25)

= (- I)“-“aV(kL).

(26)

The integral resulting from (23) consists of two terms, the first containing the first term in (24), and the second containing the second term in (24). If we replace k I by -k, in the second term of the integral, and use (25) and (26), we find that it is identical to the first except that the integration is from --m to 0. Therefore, we have achieved our above stated goal, having reduced (23) to the form

15

Near fields of pulsed electron beams in space

s m

-(s-cc)lJh]i=

-cc

Nek$,,v,

k,dk,

2(27c)*o&&,y

P&7”

x sin~J,~.(k,a)H!“.(k,p)B.

(27) log E

II

(V/m)

‘.

Equation (27) is now in a form suitable for evaluation by contour integration. The residues correspond to the poles in B, which is defined in (21). Since the Hankel function of the second kind goes to zero as the imaginary component of its argument goes to - co, the path of the contour integral is closed by a semicircular path lying in the lower half of the k, plane. The result of this contour integration, after simplification, is the formula

D

FIG. 5. SAME AS FIG. 3, BUT FOR s = 1 AND ROOT 1 OF pI.

where R = 377 Cl is the impedance of free space and ZB= - Nez,, is the beam current. It is readily demonstrated from the cold plasma dispersion relation that

‘& _ dn* -

(Eg-E,)~L:+2EOn2-2&O&, 2E,~L:+(&,+EO)I12-E,&_,-EO&,’

(29)

In order to understand the summation over q and to define the factor Q(pl,) in (28), let us summarize our above analysis by noting first that the roots (n, pL) appearing in this equation correspond to the simultaneous satisfaction of (19) and the dispersion relation (9). The solutions for p,, obtained by solving (9) for ZA:in terms of n*, are given by

These roots may be complex, and only two of the four possible roots are valid. The summation over q in (28) is the summation over these two valid roots. Since the integration path for k, (and hence pr) in (27) encircled the lower half plane, the roots pLq (represented by the summation q in the formula) must have a negative imaginary component. If the imaginary component of the root vanishes, then the radiation condition dictates that the real component of the root must be positive. The factor Q is related to whether the root n, associated with pLIObelongs to the a or b branch of the dispersion relation. If one substitutes pLr for p, in (13) and observes that the positive radical gives n,2, then Q(ZA~,) = 1. If, on the other hand, the negative radical in (13) gives nz, then Q(p*,) = - 1.

TRANSFORMATION

P :, =

+

[8E,E.f-2(E,

from rotating

to

E, = cos $Ex + sin $E, Eti = -sin$E,+cos+E,,

+h)(w,+~I~-l)l~: +(E~E,--E,E_,)2]1’2}.

COORDINATES

We now consider the conversion polar coordinates. Since

~~-I(E.+EO)~~-(~DE,+EIE-l)l s T!I[(E,--E&Z:

TO POLAR

(30)

we readily determine

(31)

from (31) and (6) that the com-

K. J. HARKBRand

P. M. BANKS

c

=

SW,

-

a$

w,t-t ------Z-S& VII

These equations represent our final results for the near fields in the vicinity of a rectangular wave modulated electron beam. RESULTS

FIG. 6. SAMEAS&X 3, BUTFOR

S =

1 AND ROOT 2 OF #A.

punents of the dectric field Xnpolar ~oor~nates given by the equation

are

E, = --jl: [E, e-j+ + E_ , e’*]

J2

(32)

E,=z&-LE_,e@] 4 E; = Eo. substituting f&C%

wlxre

(LB)into these equations and simplifying

Results for a r~~resen~tive case have been plotted in Figs 3 and 4 for Cerenkov radiation (S = 0), in Figs 5 and 5 for normal cyclotron radiation (S = i), and in Figs 7 and 8 for anomabus cyclotron radiation (s = - 1). These figures present as a function of frequency the variation of the electric field as determined by (33)+X). The electric field is generated by a 1000 V, 100 mA electron beam transversing a plasma at a 30” pitch angle with up&~ = 8 and in which oxygen is the positive ionic component. The plot assumes radiation at the fundamental modulation frequency, a duty cycle b/d such that the factor sin (~~~1~ is unity, and an inverse coherence length, Iy, equal to zero. In all six figures the beam is sampled at the helix radius @ = avjal) at which ah the beam energy is ~~sve~, and the phase term exp& which has no effect on amplitude, is ignored. As mentioned earlier, there are two roots which are included in the summation for the electric field, We shall designate these as roots 1 and 2, generally corresponding, respectively, to the upper (+) and lower (-) sign in (30). The only exception is for the Cerenkov solution below the ion-cyclotron frequency. Here, for continuity with the solutions above the ion-, cyclotron frequency, it is necessary to associate roots 1 and 2 with the lower (-) and upper (+) roots, respectively, of (30). With these de~nit~ons in mind, Figs 3,5 and 7 i&&rate the fields associated with root I, and Figs 4,5 and 8 ihustrate those associated with root 2. Table 1 gives the behavior of the roots for each of the curves in Figs 3-8. Regions of propagation of course correspond to positive roots of pi. The wave branches associated with each root are also included in the table. The general character ofthe propagation region for cold plasma waves in the frequency range given in Figs 3-g is shown in Fig. 2 of Harker and Banks (19X5). Figures 5 and 7 are not plotted below an abscissa of - 1.7 for the fottuwing reasons. As the curves approach the lower hybrid frequency, ,s~ remains real

Near fields of pulsed electron beams in space 2

a

log

E

(V/m)

-4log

E

(V/m)

-8-

I

-0.6

FIG. 7. SAMEAS FIG. 3, BUT FOR s = - 1 AND ROOT 1 OF PL.

and becomes increasingly larger. Because of the Bessel and Hankel functions in (33~(35), this causes the curves to oscillate so rapidly with respect to the abscissa that they can no longer be plotted. Also, the criterion for cold plasma theory that the wavelength be greater than the Debye wavelength is violated. The same situation holds true below the ioncyclotron frequency. In the region between the lower hybrid and the ion cyclotron frequency, pI is imaginary and so large that the field magnitudes become negligible. SUMMARY

The results presented here are of general applicability to the generation of fields by modulated electron beams. However, because much simpler expansions prevail in the far field region, the main applicability of the theory presented here will be to regions near to the beam. This is a region of premier importance, since here is where most of the experimental measurements of beam-generated fields have been recorded. In our previous paper on the far field radiation (Harker and Banks, 1985) in the long field approxi-

-10-r -6.0

FIG. 8. SAMEAS FIG. 3, BUT FOR s = - 1 AND ROOT 2 OF P1.

mation there were three variables, CO,k, and k,,, and also three constraints, the first two of which are the dispersion relation and the wave-particle interaction condition. In the case of the assumed infinite pulse train, we know from the theory of the Fourier series that the frequency of the emissions must lie at multiples of the fundamental modulation frequency. This represents the third condition. This implies that all three variables are determined, and that therefore all the radiation at a given frequency is concentrated into radiation at a single polar angle, 4. In the case of the long-beam theory summarized in (33~(35), the fields are a function of the spatial coordinates (I, 4, z) and again of w, k,, and k,. However, w, k,, and k, are again bound by the same three constraints as above, and are, therefore, determined. Therefore, as expected, only the spatial dependance on (r, 1(1, z) remains, and the fields are distributed throughout space, except perhaps for surfaces in the region where certain field components may vanish. Another interesting result of this work is the nature of the dependence on beam length. This dependence is exponential [exp (-z/L)], the same variation assumed

18

K. J. HARKER

and

P. M. BANKS FIGS 3-8

TABLE I.~~AVIOROFR~~OF~,.IN

Figure

s

Real

Root

3

0

1

-0.13 to -2.23 whistler

4

0

2

-0.60 to -5.00 whistler and slow Alfven

5

1

1

-0.27 to -2.23 whistler -4.47 to -5.00 fast Alfvtn

1

2

6 7

-1

I

8

-1

2

-0.02 to -2.23 whistler -4.47 to - 5.00 fast AlfvCn

where 1 = w,/c, multiplied by a factor of order unity determined by the characteristics of the surrounding plasma. For the physical configuration considered in this paper and a modulation frequency of 10 kHz, this gives a field of around 1 mV m- ‘. Since instrumentation on the shuttle should generally have a sensitivity of around 10i5Vm-‘, these fields should be detectable. Bush et al. (1986) reported a tentative field strength of 0.3 nT for the emissions observed during the SpaceZab-2 mission. For a typical parallel refractive index of 100, this corresponds to an electric field of 1 m Vm- ‘, a figure which is in rough agreement with the results in Figs 3--S. Acknowledgements-This research was supported through contract No. F-I9628-84-K-0014 from the Rome Air DevelCenter and through

NASA grant NAGW

-2.23

to -5.00

Complex 0.00 to -0.13 0.00 to -0.60

-2.23

to -4.47

0.00 to -0.27

-0.27 to -5.00

0.00 to -0.27

-2.23 to -4.47

0.00 to -0.02

0.00 to - 5.00

for the driving current. There is no longer any linear dependence on the beam length, as in the case of the far field. This difference results from the fact that the near field is a local effect, whereas the far fields are an integrated effect of the whole beam. Examination of equations (36) and (37) and the numerical results show that the near fields are of the form

opment

Type of root imaginary

235.

REFERENCE Banks, P, M., Fraser-Smith, A. C., Inan, U. S., Reeves, G., Raitt, W. J., Kurth, W. S., Murphy, G., Anderson, R. and

Shawhan, S. D. (1984) Observations of VLF emissions from a pulsed gun on the Space Shuttle. EOS 65, 1054. Bush, R. I., Banks, P. M., Neubert, T., Harker, K. J., Raitt, W. J., Gurnett, D. A. and Kurth, W. S. (1986) Observation of wave stimulation by pulsed electron beam experiments on Spacelab-2. Abstracts of the ~ati~~nal Rudio Science Meeting, 13-16 January 1986, Boulder, CO, p. 157. Gumett, D. A., Kurth, W. S., Steinberg, J. T., Banks, P. M., Bush, R. 1. and Raitt, W. J. (1986) Whistler-mode radiation from the Spacelabelectron beam. Geophys. Res. Lett. 13, 225. Harker, K. J. and Banks, P. M. (1984) Radiation from pulsed electron beams in space plasmas. Radio Sci. 19,454 . Harker, K. J. and Banks, P. M. (1985) Radiation from long pulse train electron beams in space plasmas. Planet. Space Sci. 33, 953. Kuo, Y. Y., Harker, K. J. and Crawford, F. W. (1976) Radiation by whistler by helical electron and proton beams. J. geophys. Res. 81,2356.

Lavergnat, J. and Lehner, T. (1984) Low frequency radiation characteristics of a modulated electron beam immersed in a magnetized plasma. IEEE Trans. Antennas Prapagat. AP-32, 171. Lavergnat, J., Lehner, T. and Matthieussent, G. (1984) Coherent spontaneous emission from a modulated beam injected in a magnetized plasma. Physics Fluids 27, 1632. Obayashi, T., Kawashima, N., Kuriki, K., Nagatomo, M., Ninomiya, K., Sasaki, S., Ushirokawa, A., Kudo, I., Ejiri, M., Roberts, W. T., Chappell, R., Burch, J. and Banks, P. (1982) Space experiments with particle accelerators (SEPAC), in Artijicial Particle Beams in Space Plasma Studies (Edited by Grandal, B.). Plenum Press, New York. Ohnuki, S. and Adachi, S. (1984) Radiation of electromagnetic waves from an electron beam antenna in an ionosphere. Radio Sci. 19,925. Shawhan, S. D., Murphy, G. B., Banks, P. M., Williamson, P. R. and Raitt, W. J. (1984) Wave emissions from DC and modulated electron beams on STS 3. Radio Sci. 19, 471. Shawhan, S. D. and Murphy, G. B. (1983) Plasma Diagnostics Package Assessment of the STS-3 Orbiter Environ-

Near fields of pulsed electron beams in space ment and Systems fir Science. Paper presented at AIAA

21st Aerospace Sciences Meeting, Reno, Nevada (Jan~lary 10-13, 1983). Sommerfeld, A. (1949) Partial L%&zrentid Equations in Physics, Vol. VI, Chapter VI. Academic Press, New York. Taylor, W. W. t. and Obayashi, T. (1984) Wave-particle &teractions induced by SEPAC on Spacelab I. P&c. 1984 Symposium on the Eflects of the kmosphere on C31 Systems, l-3 May 1984, Alexandria, VA. APPENDIX A : MATRIX

COMPONEN’IS

P,,

The matrix elements P,* are obtained from those givert

19

by Kuo ef of. (1976) by factoring exp i (ix- v)d, factors :

out the k: and

P,,$= ‘z&i-

f&_i-+ fs,)p~-c@z2-t+..,s, Pi),, = P,,o= fjI--F-*f#iin/2 ii* Pt.-l = P-.,,, = P:@*-Qd/2 PO.0=~~n2-(E,fE_,)(n*+:~:)+E,E_, PO._,= P_1.0= (/.?-e,)~lnj2”2 P ._,,._, =$&L-

(E~+JjC&i:--8$i2+EIE0