The electrostatic plasma response to a small pulsed antenna in a magnetic field is analyzed. The ringing of the plasma at three discrete frequencies--the upper-hybrid frequency and two resonance cone branch frequencies--is evidenced, and the amplitudes of these frequency responses is determined as a function of the characteristic plasma frequencies, the angle of observation with respect to the magnetic field, and the pulse length. Applications to plasma diagnostics are discussed. It is shown that the upper hybrid response and the response at either of the resonance cone branch frequencies is adequate information to determine the plasma density, and the magnetic field magnitude and angle

Grabbe, Crockett L.

English

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r 
APPENDIX 5 
(NASA-CB-185454) THEOBY OF THE PULSE N89-26726 
BESPONSX PROH A SElALL A N T E N N A  I N  A 
HAGNETTZED PLASMA jlowa Uriiv.) 1 9  p 
CSCL L U f  Unclas 
G3/75 0219578 
Theory of the Pulse Response from a Small 
Antenna in a Magnetized Plasma 
CROCKETT L. GRABBE 
Department of physics and Astronomy. 
The University of Iowa 
Iowa City, Iowa 52242 
The electrostatic plasma response to a small pulsed antenna in a 
magnetic field is analyzed. The ringing of the plasma at three dis- 
crete frequencies--the upper-hybrid frequency and two resonance cone 
branch frequencies--is evidenced, and the amplitudes of these frequency 
responses is determined as a function of the characteristic plamsa fre- 
quencies, the angle of observation with respect to the magnetic field, 
and the pulse length. Applications to plasma diagnostics are dis- 
cussed. It is show that the upper hybrid response and the response at 
either of the resonance 
to determine the plasma 
angle . 
cone branch frequencies is adequate information 
density, and the magnetic field magnitude and 
https://ntrs.nasa.gov/search.jsp?R=19890017355 2020-03-20T02:17:25+00:00Z
t 
I 
8 
2 
I. INTROUDCTION 
The problem of the radiated wave fields excited by small sources 
in magnetized plasmas has been of considerable interest in recent years 
because of the highly directional resonance cone nature of the waves 
under certain conditions [1,21. In addition to being an interesting 
physical phenomena, resonance cones have been used as a diagnostic for 
density and temperature and form the channels along which power flows 
in certain radio frequency plasma heating experiments [3-51. Most of 
the work on this subject has concentrated on continuous source plasma 
antennas. 
The case of pulsed antennas is also of interest for a couple of 
reasons. 
ous source, and this may be of importance in possible applications to 
plasma diagnostics, such as that of space plasmas observed on satel- 
lites and space vehicles. Second, for the continuous source case 
observation can only be made along the resonance cone angle, which is a 
function of the density and magnetic field, so that the angles must be 
swept to find the resonance cones in diagnostic applications. 
it may be of advantage to place the transmitting and receiving probes 
along a fixed direction. 
First, the pulsed source can use less power than the continu- 
However, 
A n  experiment has been performed on the wave response of a small 
pulsed antenna [ 6 ] .  It was found that the response was a ringing at 
3 
three characteristic frequencies for any given angle 8 of observation: 
the upper-hybrid frequency and two frequencies which correspond to the 
frequencies that would create the upper and middle branch of the reso- 
nance cones at a cone angle 8. 
It is the purpose of this paper to develop the theory of the 
electrostatic response to a small pulsed antenna, which is done in 
Section 11. The possible applications to diagnostics is discussed in 
Section 111, and the results summarized in Section IV. 
. 
4 
11. PULSE RESPONSE THEORY 
We would l i ke  t o  solve the electrostat ic  Poisson's equations for 
a pulsed source. 
a Gaussian t i m e  dependence centered about t = 0: 
We will assume a cold plasma and a point source with 
where K is the plasma dielectr ic  tensor and T is the pulse length. We 
Fourier analyze these equations i n  space and Laplace transform in time 
t o  obtain 
where 
5 
3 s )  = 
I 
I 
I -  
Kl i% O 
. 
0 - i K H  Kl 9 
II 0 0 
The integral  over t can be converted into the standard form for the 
conjugate error function ( e r f c )  [71. For the magnetic f i e ld  in  the 
z direction: 
where 
and KH is the Hall term. 
form for  s -* - i w .  Choosing k = (k,,O,k,), we can evaluate the inverse 
integral  over k by well-known integral  forms from Gradstein and Rhyzik 
[ 81 (see also Ref. [l] 1 ,  or by rescaling of variables k, and k,: 
These components reduce t o  the well-known 
+ 
6 .  
I .  
Here 8 is defined to be the angle between the magnetic field and the 
observation point (the group velocity angle). The inverse Laplace 
transform is then 
where we have factored the denominator in Eq. (81, and 
The first two frequencies are the middle and upper branch resonance 
cone frequencies, and the upper-hybrid frequency. (Ion contribu- 
tions are omitted.) These frequencies are plotted versus w2 /w2 for 
various 8 in Fig. 1. The conjugate error function erfc contains the 
pulse length dependence. 
Pe uh 
To evaluate the integral in Eq. (91, we pick the branch cuts and 
the Laplace integration contour as shown in Fig. 2. Now the quantity 
7 
2 erfc y 
argument and asymptotic form 
is a monotonically decreasing function of y with the small 
Thus it is well behaved and contributes no singularities to the inte- 
gral in Eq. (9). 
the integral along the branch cuts. 
circle around the branch points are zero, while the sign change across 
each branch cut cause the integrals on each side of the branch cuts to 
be identical and additive. 
Hence we may use asymptotic techniques to evaluate 
The integral over CO and the 
Thus we may parameterize each integration 
path along the branch cuts by s = -+iwr - x, where x is real: 
3 
At asymptotic times such that e-wit e< 1 for each w, only the values of 
x such that x e< q will contribute significantly to the integral. 
This condition is satisfied for all three values of wi if t >> l/wl(e). 
Then by expanding to lowest order in x 
, 
8 
where 
Here w(y) is the w error function. (See Ref. 7, Chapter 7.) Also, 
The form of the solution in Eq. (15) clearly shows Ai(-Wi) = Ai(ui). * 
the ringing of the plasma at the two resonance cone frequencies w1 and 
w2 and the upper-hybrid frequency wuh, as observed by Simmonutti 161. 
The response dies away as t-'I2. The magnitude of the amplitudes %, 
A2, and A1 are plotted in Figs. 3 and 4 as a function of ( w 2  /w2 ) for 
three angles of observation 0 ,  and long and short pulse times T. 
Pe uh 
9 
111. PLASMA DIAGNOSTICS APPLICATION 
gY the use of transmitting and receiving probes in a plasma, one 
may measure through the use of a spectrum analyzer the three frequen- 
cies w 1 ,  w2 and w and their corresponding amplitudes Al, A2 and %. 
By comparing with the theoretical predictions given in a s .  (10) - (12) 
and (16) - (181, one may deduce information on the plasma characteris- 
tics. We w i l l  show that this ty-pe of measurement is adequate to deter- 
mine the plasma density, magnetic field strength, and magnetic field 
angle 8 relative to the transmitting-to-receiving-antenna line in a 
plasma for  which none of these is known. 
uh 
Let us define the ratios 
1 m E- 
uh 
w 
- w  
2 m, : - 
uh 
0 
w (19) 
which are obtained from the primary measurements. 
into the form 
Then put Eq. (12) 
10 
Divide Eqs. (10) and (11) by w& and substitute Eq. (21) into them to 
obtain the two equations 
w2 w2  
w2  
(T) (1 - x, sin2 0 = 0.2511 - (1 - 2m2)211/2 f . 
uh uh w 
(22 1 
Finally, evaluate the Mk in Eq. ( 2 0 )  through Eqs. (16) - (18) to obtain 
the two equations 
Eqs. (22) and (23) form two independent equations for the two 
unknowns ( w 2  /w2  ) and 8, and Eq. (21) gives (wge/w&) in terms of 
( w 2  /w2 ). 
of the frequencies and their amplitudes, this is sufficient information 
Pe uh 
Since all other quantities are hown through a measurement 
Pe uh 
to determine the plasma density (through (%e), magnetic magnitude 
(through 1, and magnetic field angle 8. (With multiple receiving 
antennas we could narrow the magnetic field down to a unique direc- 
, tion.) But we have two sets of measurements, either one of which is 
adequate to determine these three quantities: 
11 
(1) M,; 02, and their ratio m,. 
( 2 )  M ; 01, wUh and their ratio m . - - 
Thus we may use both sets of measurements in conjunction with each 
other to give two independent determinations of the plasma density and 
magnetic field angle and magnitude, and this w i l l  help improve the 
accuracy of the diagnostics. 
Our model puts two conditions on the antenna that must be satis- 
fied to ensure the validity of the results here for plasma diagnostic 
applications. First, the electrostatic approximation requires the 
transmitting antenna be of a monopole type (spherically symmetric) to 
guard against unwanted transverse wave components. Secondly, the 
antennas must be sufficiently small to be approximated by a delta func- 
tion. 
%x > 
that approximation. Both of these approximation requirements can 
easily be met for most applications. 
Since kin - 2n/L where L is the antenna size and we require 
and (w/k) << c, that means L << 2 n c / m  is the condition for 
12 
IV. SUMMARY AND CONCLUSIONS 
The plasma response to a pulsed plasma exhibits a discrete fre- 
quency response at the upper-hybrid frequency and the two resonance 
cone branch frequencies for the given angle of observation. 
tude of these responses was calculated as a function of the various 
plasma and response frequencies, the observation angle, and the pulse 
length. These are summarized in Eqs. (16) - (18) and Figs. 2 and 3. A 
possible application of these results is to plasma diagnostics for the 
plasma density and magnetic field magnitude and angle. The determina- 
tion of these quantities from pulse response measurements is carried 
out through Eqs. (21) - (23). 
The ampli- 
Some restrictions and limitations of the model used here were 
mentioned. 
eliminate unwanted transverse wave components and ensure an electro- 
static response. 
required to make the delta function antenna configuration valid. 
Finally, thermal effects will modify the results in two ways. 
The first is that a spherical antenna should be used to 
The second is that antenna sizes L << 2nc/y, are 
The 
first is that the cold plasma amplitudes blow up near 8 = 71/2 and thus 
are not valid there. The 
second is that other plasma frequency responses w i l l  occur, such as the 
Bernstein and cyclotron harmonic wave resonances. 
Thermal effects W i l l  limit those amplitudes. 
Investigations of 
thermal effects and finite antenna-size effects are planned for a 
future paper. 
This research was sponsored by NASA grant NAG364 and grant 
ATM-8312514 with the National Science Foundation. 
14 
REFERENCES 
c 
[ll R. Fisher and R. Gould, "Resonance cones in the field pattern of a 
radio frequency probe in a warm anisotropic plasma,lt Phys. Fluids, 
vol. 14, p. 857, 1971. 
[2] H. Kuehl, "Interference structure near the resonance cone,I1 Phys. 
Fluids, vol. 16, p. 1311, 1973. 
[31 R. Briggs and R. Parker, "Transport of RF energy to the lower 
hybrid resonance in an inhomogeneous plasma," Phys. Rev. Lett., 
vol. 29, p. 852, 1972. 
[41 C. Grabbe, I%nergy flow and absorption along resonance cones near 
the lower hybrid," Phys. Fluids, vol. 22, p. 1323, 1979. 
[51 P. Bellan and M. Porkolab, "Propagation and mode conversion of 
lower hybrid waves generated by a finite source," Phys. Fluids, 
vol. 17, p. 1592, 1974. 
[ 6 ]  M. Simmonutti, "Transient response from small antenna in an 
anisotropic plasma,lI Phys. Fluids, vol. 19, p. 608, 1976. 
[7] M. Abromowitz and L. Step, Handbook of Mathematical Functions. 
Washington, D.C.: National Bureau of Standards, 1964. 
[8] I. Gradsteyn and I. Fthyzik, Table of Integrals, Series and 
Products. New York: Academic, 1965. 
15 
FIGURE CAPTIONS 
Fig. 1. Integration contour for the inverse Laplace transform in 
Eq. ( 9 ) .  Branch cuts and branch points are indicated. 
Fig. 2. Plots of the two resonance cone branch frequencies for  three 
angles 0 .  
Fig. 3. Plots of the amplitudes of the upper-hybrid and resonance cone 
responses for three angles 0 and a short pulse time. 
Fig. 4. Plots of the amplitudes of the upper-hybrid and resonance 
cone responses fo r  three angles 0 and a long pulse time. 
Re s 
Fig. 1 
4 
I 
. '  ' 
Fig. 3 
C - G84-628 
I I I I I I I I I I 
I 1 1 I I 1 I I I I 
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.001 ’ 
w 2 h U h  2 
Pe 
Fig. 4 


Theory of the pulse response from a small antenna in a magnetized plasma

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