1R.E TRASSACTTOXS OAV dNTES;VAS A N D PROPAGATIOX

A Coaxial Low-Density Plasma Experiment” A. OLTEf., ~ M B E R , IRE, J. S. KINGt, AVD E. K. MILLERt

January

Summary-The design of an inverted coaxial plasma diode is discussed in considerable detail. The RF transmission and reflection measurements of a 15-cm-long diode are presented and compared with a preliminary theory. An attempt is made to account for the electron collision with the walls.

I. IXTRODUCTION

H E scattering of electromagnetic radiation b>- ionized gases with densities below 1013 ions per cm3 is of general interest in the communication

field. Detailed experimental data in the vicinity of the plasma frequency is of particular interest in revealing the importance of possible interaction mechanisms, such as inelastic scattering.

Measurements of the interaction of microwaves with gas discharges have been numerous since the early ex- periments of Tonks.’ Romell,2 in 1951, examined the differential scattering from a 3.2-cm diameter plasma column placed between a pair of Yagi antennas. Her~hberger ,~ in 1959, extending the experiments of Romell and Dattner,4 used a 0.5 cm diameter discharge tube placed across a 3.0-in X 1.5-in rectangular wave- guide, and between open waveguide ends. The response to variable plasma densities was observed at a fixed signal frequency of 2800 Mc. Golant, et al.,5 have ob- tained data from the positive column of a hot cathode gas discharge in glass bulbs axially inserted in a cylin- drical waveguide. Interpretation of these and similar experiments are complicated by the boundary condi- tions around the plasma and by the sensitivity of the results on the microwave structure near the plasma. Furthermore, they are fixed frequency measurements in which the functional dependence on the microwave fre- quency cannot be directly observed.

The present experiment was undertaken with three purposes in mind: 1) to produce an interaction in a simpler geometry, ie., one approaching the ideal of a free-space plane wave incident on a plane, uniform ion-

* Received by the PGAP, October 11, 1961. The work described in this paper was sponsored by the Advanced Research Projects Agency (Project Defender) and the Air Force Cambridge Research Labs. under Contract .IF 19(604)-8032.

Mich. t Radiation Laboratory, The University of Michigan, Ann Arbor,

Reo., vol. 37, pp. 1458-1483; June 1, 1931. L. Tonks, “The high frequency behavior of a plasma,” Pbys.

i’fature, vol. 167, p. 243; February 10, 1951. D. Romell, “Radio reflexions from a column of ionized gas,“

plasma,” J . A p p l . Pkys., vol. 31, pp. 417422; February, 1960. W. D. Hershberger, “Absorption and reflection spectrum of a

no. 2, pp. 309-350: 1957. A. Dattner, “The plasma resonator,” Ericsson Tech., vol. 13,

waveguides filled with plasma in the positive column of a discharge,” 5 V. E. Golant, et al., “Propagation of microwaves through

Smiet Phys.-Tech. Phys., vol. 6, pp. 38, 44; July, 1961.

ized gas barrier; 2) to observe the interaction as a con- tinuous function of signal frequency; and 3) to initially test the interpretability of this arrangement with the simplest plasma, ie., a collision-free electron cloud whose density and distribution should be accurately predicted. Extension of the measurements to ionized gases is intended as a logical second step and is planned for the near future.

I t is believed that the first two purposes largely have been achieved by the apparatus described below. The use of an electron cloud, however, has not necessarily produced an ideal plasma, because electron-conductor wall collisions have replaced gas collisions.

The apparatus is described in some detail in Section 11. Some initial measurements with a trial cathode are presented in Section 111. Preliminary interpretation of these data are also given.

11. EXPERIMENTAL DESIGN

The experimental apparatus is essentially a rigid 50- ohm transmission line with a wide-band R F generator a t one end, and suitable detector and load termination at the other end. A central 15-in section of this line is a removable coaxial diode which is operated in a high vacuum by looping the line through a 36-in deep by 18-in diameter vacuum manifold. The diode produces an electron plasma region with a length d to annular width rc-y , ratio of about 100. The complete line is shown in the scaled drawing of Fig. 1. The physical ar- rangement of components around the vacuum console is shown in Fig. 2. In the diode section the cathode forms the outer conductor and the anode the inner conductor of the coaxial line, in contrast to the usual arrangement. Thermionic emission is produced by radiant heating of the outer conductor surface by a tungsten wire heater surrounded by a platinum reflector. Anode heat from the electron bombardment is carried away by water flow inside the anode conductor. The water flow is dead- ended and thus requires only one exit path from the transmission line. As shown in Fig. 1, this exit path is through an adjustable shorted quarter-wave line to eliminate reflections.

The coaxial line was designed to minimize reflections and losses everywhere, in the absence of the ion cloud. Except for short external connecting cables the line is a rigid structure, mostly of nickel tubing, with central conductor supported by a minimum number of insulat- ing beads. Gradually tapered ends are used at the two external terminals and at the diode ends. DC isolation of the negative potential cathode is achieved by two large series capacitors of high dielectric constant.

1963 Olte. et a/.: ,I Coaxial Low-Density Plasma Experim.en.? 25

The long and smooth coaxial geometrv offers several keJ- advantages to the analysis of the experiment:

1) the waveguide can be made to propagate only the simplest wave, the transverse electromagnetic mode (TElI) ;

2) “unclean’’ boundary conditions at the extremities of the plasma region become insignificant with a large length-to-1%-idth ratio; and

3) the waveguide has a very broad-band response which permits variation of the transmitted RF fre- quency f around any fixed plasma frequency f,. While the reverse operation is also possible, enough uncer- tainty exists i n the exact determination of the plasma frequency that f, is better used as a parameter than a variable.

Fig. 1-Schematic of the experiment. -1) Concentric coolant tube en-

stub split tee block. Dj DC isolation capacitors. E) Teflon trance and exit. B ) Quarter-wave shorting stub. C) Shorting

x-acuum tight support beads. F) Support bead cooling coil. G) Tapered transitions to input and output coaxial cables. H) Vacuum tank. I ) 1,’acoum tank cover plate. J ) DC insulated vacu- um flanges. I<) Diode centering and tension plate. L) Tension plate assembly. kIj Tapered transitions to coaxial diode. Nj Coaxial diode, heater windings and heat reflector. 0) Return pat: coaxial line. P) Quartz support bead housing. Q) “Vacsorb forepumps. R ) “I-ac-ion“ high vacuum pump.

Fig. ?--Equipment assembly showing plumbing. v a c u ~ ~ m system, signal generators. and detectors.

There are, also, practical difficulties i n the design:

1) the primary difficulty is the construction of the long and narrow diode whose anode must be coaxially symmetric yet unsupported, small in diameter yet capa- ble of high power dissipation, and whose cathode re- quires a long internal oxide coated surface, not obtain- able from commercial tube manufacturers; and

3 ) the cathode is a “one-shot” device and the diode is unavailable for adjustment once the cathode is acti- vated inside the vacuum system.

The Coaxial Diode The details of t h e diode design are shown in Fig. 3.

The anode passes through a teflon vacuum seal above the vacuum cover plate, tapers to a small radius in the active cathode region, and joins the central conductor below a second taper. The bottom tip is threaded into a retention nut inside the inner conductor. Tightening th is nut applies the centering and tensioning action of the anode stretcher assembly shown in Fig. 1. The cathode is a nickel tube 15.75 in long, with an inside diameter of 0.356 in and a wall thickness of 0.012 in, which makes a sliding fit into the ends of the two heavy nickel taper sections. A 12-in longitudinal tungsten fila- ment heater is mounted on ceramic insulators which are positioned inside the cylindrical platinum heat shield. A difference in thermal expansion of the cathode over the anode of up to 0.15 in is allolved by the sliding joints, which, however, do not impair electrical continuity.

Selection of the outer conductor as the cathode was made to optimize the radial uniformity of the electron density distribution. If p is the electron density, r, the cathode radius, and lo the diode current per u n i t length,

Fig. 3-Coaxial diode and the heater. -Yi ‘Tapered transition to coaxial diode. B) ;\node and outer coolant tube: O.156-in outside diame- ter, 0.136-it1 inside diamctcr. C! Inncr coolant tube: 0.093-in outside diameter. 0.085-in inside diameter. D ) Ceramic guide and spacers for heater rods. E,! 0.030-in diameter tungsten heater rods. 12 rods equally spaced around cathode on 0.600-in diameter rod circle. Fj Platinum heater reflector: 0,800-in outside diame- ter. 0.i90-in inside diameter. 12.43i in long. G) Sickel cathode: 0.380 in-outside diameter, 0.356-,11 inside diameter, 15.750 i n long.

26 IRE TRANSACTIOhTS ON ANTEXNAS A1VD PROPAGATION January

the cylindrical form of Child’s law for space charge limited operation can be written as an expansion in 7

where the function /3 both for the regular and the in- verted diode was obtained by Langmuir and Blodgett;6 EO, m and e are the permittivity of free space (MKS), the electronic mass, and charge. The density function is quite different, depending on whether T is greater or less than yC. Fig. 4 is a plot of p ( r ) vs r / rc and Y J r for the normal and inverted diodes, respectively. Both func- tions go to infinity a t r = rc but the inverted diode has a minimum point and reverses slope. Hence, selection of the anode radius such that y c / r is about 3.0 or less, results in a reasonably uniform density over all but the outer radii, in contrast with the normal case. The ratio used in the present experiment is 2.28. Since the plasma frequency depends on the 1/2 power of p ( r ) we conclude that the variation off, around its average is less than 8 per cent over 60 per cent of the plasma area, as com- pared to 50 per cent variation over this same area in the normal diode. Furthermore, the influence of the spike in p ( r ) near the cathode is less important, in the inverted case, in determining the effective average dielectric constant for the plasma.

The radial dimensions should be as large as possible for adequate coolant area and for precision of annular spacing. They are, however, severely limited by power requirements. The companion equation to (1) relating voltage and charge density is

Fig. 4-The diode electron density vs radius.

charge between coaxial cylinders,” Phys. Rev., vol. 22, pp. 34i-356; 6 I. Langmuir and K. B. Blodgett, “Currents limited by space

October, 1923.

Hence the plate power per unit length is, from (1) and (21,

The ratio rO/yc is approximately fixed by the need for density uniformity, and the value of p a t r = r . is typ- ical of the annular region average. Hence, (3) displays the strong dependence of power on r, and p average. There is evidently a need t o minimize Y, to reach the highest possible p . The first diode was constructed with the \-alues

Y c / Y a = 2.28

r, = 0.078 in.

A safe upper bound for heat flux across the anode wall is about 100 w/cm2. For this limit and the above radii, (3) gives the electron density at the anode p(r,) = 1.6X 109/cm3.

I t is not reasonable to push the electron density much above 2.0 X l o9 because the radii become unusably small and high pressure cooling is necessary. Electron densi- ties near or above 1010/cm3 can only be reached by pulsed operation of the diode plate voltage. This has not been attempted as yet.

The axial length of the diode must be long to provide enough plasma length to produce clearly measurable changes in the reflection and transmission coefficients, I’ and T, in the vicinity of the plasma frequency. The anode cannot be bead supported in and near the active cathode region, however, because of sputtering from the cathode. Hence, the anode length between closest bead supports should be small for mechanical rigidity. A compromise is therefore necessary in the choice of length. The effect of the plasma length, d, on T and I? can be demonstrated by considering the simplest pos- sible model of interaction, namely, the passage of the principal coaxial mode (TEM) through a uniform elec- tron cloud in which no electron collisions occur, either with gas molecules or container walls. The equation of motion for the electrons gives the well-known dispersion relation

where the propagation vectors inside and outside the electron cloud are, respectively, k =271-f&< and K O = 271-f 4G. The plasma frequency

where N is the number of electrons per cm3. The elec- tron cloud is in effect a dielectric region with specific capacity R = ( k / k , ) 2 = € / E O . Use of the appropriate boundary conditions for incident, reflected, and trans-

mitted plane Ix-aves leads to expressions for 7 and I? in terms of the plasma length d. These are

and

Yt'hen the medium is lossless, we must have ! r = 4 1 - 1 ~j i n order to conserve energy. If the effect of elastic collisions is taken into account bq7 including the usual Lorentz damping term in the equation of 1110-

tion, the dispersion relation is modified by the inclusion of a collision frequency, v, and (4) becomes

Fig. 5 is a plot of from (621) for several plasma lengths d expressed i n fractions of the plasn~a wave- length (X, = c.;fp). The solid curves correspond to v = 0 and the dotted curves show the effect of including a par- ticular collision frequency. The effect of the actual non- uniformity i n the electron distribution is discussed i n Section 111. Since some frequencqr dependent fluctua- tion will occur i n the measurements due to line reflec- tions, the mininlum desirable length should evidently not be less than about $ X p . The heater length used for the present diode is 31.5 cm and, due to the axial tem- perature profile, this restricts the active region to about 23 cm. This corresponds to a wavelength of 92 cm or a plasma frequent!, of 326 >IC and a density p of 1.3 X lo9 electrons,l'cm3. Since this is near the density limit noted above, we conclude that the span of electron plasma fre- quencies available for good data is restricted to the re- gion 100-400 >IC.

The composition and activation of the cathode oxide surface follon-ed normal procedures recommended b). tube manufacturers. An unusual problem was encoun-

f If,

Fig. 5-Power transmission coefiicient vs frequency.

tered in coating the long, inside surface. BaC02 and S rC02 (Sylvania C10 triple carbonate) i n a nitrocellu- lose lacquer and a111>71 acetate binder was applied as a liquid coating on this surface by the cathophoretic proc- ess. X technique of spin-dr\.ing was developed to pro- mote a uniform depth of coating. The coated cathode was then mounted in the vacuum system and activated b,, heating from the diode heater. Difficulties in deter- mining the actual pressure and temperatures a t the cathode surface, as well as an axial temperature gradient were encountered; the technique is still i n a formative stage.

The A node Coolitzg System

The single-ended water cooling channel was designed to operate at an available tap pressure of 50 psi. For a heat flux of 100 n;/cm2 and the diode geometry already described, this corresponds to a bulk temperature rise of 30.3"C and a heat dissipation of 3160 w. Local boiling temperatures should not \-et be reached according to conventional film drop calculation. Operation of elec- trically heated test models suggests, however, that local boiling does co~nmence near 1500 11' input and the region of instable film boiling is reached above 2500 w. Conse- quent]>-, an upper limit of 2400 w has so far been im- posed on the diode operation. This corresponds, by (3), to an electron density of 1.43X10g rather than the an- ticipated densit). of 1.6X109. The discrepancy in heat transfer properties is not yet understood and, because of inadequate test models, may not actually exist in the electron diode. The exceedingl>v small dimensions for the stainless steel tubing, shown i n Fig. 3 , make sup- port brackets for the inner tube impractical and conse- quentl~r lack of concentricity is unavoidable. This ma); provide an explanation for the discrepancy.

Coaxial Support Beads

The center conductor of the coaxial loop is supported by ten dielectric beads, each of which presents a dielec- tric discontinuit!. and hence a possible reflection. These reflections have been effectively compensated in the low frequency region below 3000 >;IC by adjustment of the bead radial dimensions. The six beads, shown in Fig. 1, are subjected to temperatures up to 200°C inside the vacuum system. Consequently, quartz ( K =4.1) was used in this region. The rest of the beads are made of teflon.

The large teflon beads have a three-fold design re- quirement and were selected after considerable explora- tion. They must provide good R F impedance matching, mechanical strength against accidental vibrations and a tension force of up to 100 pounds, and they must in- sure a high vacuum seal. The beads were surface etched to provide a bonding surface. A vacuum tight bond to the nickel coaxial walls was achieved with 0.010-in thick coating of epoxy resin. 3Ietal and teflon walls were scored 0.002 in deep to promote bonding and the epoxy was built up on successive applications to seal against

28 IRE TRANSACTIONS O N A~1.TE~VL\7AS AND PROPAGATION Janua y

bubble leaks. External cooling coils insure that the teflon temperatures remain below 100°C. Operating ex- perience with these beads has been quite satisfactory.

Vacuum System

The vacuum system was designed to provide a vac- uum below mm Hg which is sufficient for the pro- tection of the activated cathode. The vacuum console was engineered by Varian Associates; most of the com- ponents are visible in Fig. 2. The 150-liter test chamber is evacuated through a 5-in manifold by two valved “Vacsorb” adsorption pumps and a “Vac-ion” ion pump. A Philips-Granville controlled leak valve also connects to the manifold. All test penetrations into the system are made through the removable cover plate by means of several removable port flanges. These pene- trations, all insulated, include the two coaxial seals, six thermocouple leads, and five heater and diode current leads. The test chamber walls can be preheated to 90‘C for outgassing, and are water cooled during operation. Starting pressures are indicated by a thermocouple vac- uum gauge and high vacua by the ion pump current directly. The ion pump has a flat pump rate character- istic of 140 litersjsec for air; the pump rate varies with the kind of gas being pumped.

Operating experience has shown that when the cham- ber is outgassed and without the diode heater on, indi- cated pressures of 4x lo-* mm Hg are normally ob- tained, and this can be accomplished from open tank conditions in about 12 hours. With the heater on, the level rises to as high as 4 X lo-’ mm Hg. These pressures are believed to be quite satisfactory for a replaceable cathode. Longitudinal slits in the connecting tapered sections provide the pumping orifice. Thus, there is some uncertainty, due to the closed nature of the diode design, of the actual pressure in the diode during activa- tion outgassing. This has hampered the activation oper- ations.

R F System The microwave detection is accomplished with con-

ventional apparatus. Fig. 6 is a typical arrangement used for measuring both transmitted and reflected power relative to input power. If the reflections from the coaxial line without electrons is small, the measure- ment of the transmission coefficient is especially simple:

Reflected Transmitted

H.P 4158

Fig. 6-Typical detection system. ‘

1.1 is just the ratio of transmitted voltage amplitudes for the heated electron tube, with plate-voltage-on rela- tive to plate-voltage-off. An alternative method for measuring reflected power utilizes a slotted line in place of the directional coupler and the VSWR is found with plate-voltage-on and off. In the reflection case, losses in the cable and the tube must be accounted for. The per- formance of the line and necessary corrections to the data are discussed in Section 111.

111. EXPERIMENTAL DATA AND PRELININARV ANALYSIS

In the schematic of the experiment (Fig. l), the RF input and output lines are shown entering the vacuum console through the cover plate. The block diagram of the microwave circuit is shown in Fig. 6. The power reflection and transmission coefficients of the circuit be- tween the input and the output reference planes are shown in Fig. 7. This measurement is carried out with the cathode hot, but with no dc voltage on the diode, The reflections in the loop are small from 100-1000 Mc, moderate from 1000-3000 >:IC, and unsatisfactorily large from 3000-4000 Mc. The power transmission co- efficient varies from 0.9 at 200 Mc to 0.6 at 2000 Mc; in general i t is decreasing with increasing frequency.

This variation is due primarily to the wall losses of the coaxial line. From the behavior of the transmission and the reflection parameters one may conclude that the mi- crowave structure of Fig. 1 may be represented by lossy transmission lines connecting the diode to the input and the output reference planes, as shown in Fig. 6. This is a sufficiently good approximation for frequencies from 100-3000 Mc. For such a circuit the power transmission coefficient of the diode is readily obtained by monitor- ing the power at the output terminal with the cathode at the proper temperature and the dc voltage on the diode alternately on and off. The ratio of the power out- put with the dc voltage on, to that with the voltage off, is then the power transmission coefficient for tha t sec- tion of the coaxial line which contains the electrons.

The measurement of the power reflection coefficient of the diode is not as simple. The power reflection co- efficient is first measured at the input terminal, and then

Fig. ’I-Loop power transmission and reflection coefficients vs frequency a t zero amperes.

1562 Olte, ef al. : -4 Coaxial Low-Density Plasma Experiment 29

transferred to the input of the coaxial diode. Obviously for this operation one must know the losses in the con- necting line. The measured results of the attenuation (n-ith the plate voltage off) between the input and the output terminals (;.e., for the entire circuit) are shown i n Fig. 8. The loner curve is computed from the skin depth loss formula by assuming reasonable wall con- ductivities and permeabilities. I t exhibits the expected dependence on the square root of the frequency. .I curve m:hich is a reasonable fit to the measured points shows the same frequency dependence, but the actual losses are about 1.5 times higher. This is not unusual in xliew of some uncertainty of the wall conductivity and especially of the permittivity of the nickel walls. The appropriate fraction of this loss was used to advance the reflection coefficient from the input terminal to the input of the diode.

The power reflection and transmissisn coefficients, as a function of frequency for three specific cases of anode currents (0.5, 1 and 2 amperes) in a 15-cm long diode, are shown i n Figs. 9, 10, and 11. Sote tha t good repeat- ability is obtained between the three or four runs shown. The accuracy of the measurements is of the order of i -5 per cent. The interesting features are: 1) that the power transmission coefficient is flat over an appreci- able frequency interval a t lower frequencies; 2) that there is an absorption peak a t a high frequency; and 3) that the power reflection coefficient remains small a t all times. For a lossless medium the power reflection and transmission coefficients should add up to unity at all frequencies. Since this does not happen, the electron stream appears lossy to the microwaves up to fairly high frequencies.

In this configuration the interaction of the space charge limited electron stream with the incident electro- n~agnetic field is a reasonably well-defined boundary value problem. From the solution of Maxwell's equa- tions, subject to the appropriate boundary conditions, one can predict the reflection and the transmission co- efficients for the diode. However, the complete solution is of considerable difficulty and it still remains to be worked out. Only an approximate solution will be at- tempted here. The metal walls of the diode appear to

0 Experimental Theoretical

* 3/2 Theoretical 2 0 251 I O D y

I .5 c D D

OlO' I

' ' ' 1'0" Frequency (mc/s)

Fig. 8-Loop attenuation loss vs frequency.

the electrons as large molecules, and hence we may talk of electron-molecule collisions. The collisions between the electrons and the h>-pothetical molecules are in- elastic, whereby the total momentum of the electron is reduced to a low value. The dc field intensity is suffi- cientll- strong so that the drift velocit>T of the electrons is large compared to the random velocit~. component.

Fig. 9-lliode power reflection and transmission coefficients L-S frequenr)- at 0.5-ampere anode current.

Theoretical 0 A I + Transmltied

Frequency (mc/s)'

Fig. 10-Diode power reflection and transmission coefficients vs frequency a t 1-ampere anode current.

I

I Theoretlca I = - 0 I t Transmltted

3 -

Frequency (mc/s)

Fig. 11-Diode power reflection and transmission coefficients vs frequency a t 2-ampere anode current.

30 IRE TR.AMSACTIO;Y’S OX AKTENNAS AXD PROPAGATIOS January

The cathode to anode spacing constitutes the mean-free path distance, and the inverse of the dc electron transit time may be taken as the collision frequency v. The col- u

lision frequency absorption mechanism is normally de- rived for elastic collisions. Here we are arbitrarily ex- tending the mechanism to a hypothetical gas with only E 5XlO8 . .

inelastic collisions allowed. 13 4 x IO8- The plasma frequency in the diode is shown in Fig.

12. These curves are computed from the electron densi- ties obtained from the Child’s Law potential distribu- tion in the diode. The plasma frequency variation radially in the interelectrode space is a microscopic be- havior. We have to obtain a macroscopic parameter, at Radius, r, in cm

least an effective average over the mean-free-path dis- tance for the electrons. If one approximates the plasma

2 amps 5 6 X 10’-

s: W

t

Fig. 12-Plasma frequency vs diode radius for different anode currents.

0.5 1 1 .62X108 I 12.25X108 I l . l l X 1 0 8 1 0.198 I 1.442 1 0.452

1.0 1 2 . 0 5 ~ 1 0 8 0.452 0.442 0.198 19.5 X108 I 1.65X108 2.0 I 2.6OX1O8

0.452 0.442 0.198 15.0 X108 1 1 .35X108

frequency in the diode by a step function, the lower value cf.) given by the flat portion of the curves, and the higher value (j,), describing the conditions near the cathode, given by the location of the absorption peak in the transmission curves, then one ma)- obtain an equivalent plasma frequency from the solution of the principal mode7 obtained for such a step function dis- tribution. Since the arguments of the characteristic equation for the principal mode in the region of inter- est are considerably less than unity, we may approxi- mate both solutions of the Bessel’s equation and of the modified Bessel’s equations by the first terms in the series. This gives the plasma frequency in a very simple form

I+L(

. .

where Y,, denotes the position of the step in the electron density, and f is the frequency of the incident microwave radiation. The values of the parameters used in the sub- sequent calculations are summarized in Table I.

The plasma frequency of (8) is regarded as the mac- roscopic description of the electron stream in the diode. Using this particular plasma frequency and the collision

lines of composite sections,” IRE TFMNS. ON ASTE~~;KAS AZD PROPA- ’I R. W. Klopfenstein, “Low frequency waves on transmission

GATION, t-01. AP-2, pp. 103-109; July, 1954.

frequency (the inverse of the dc transit time) in the dis- persion relation, (7) gives a reasonable description of the medium in the diode. The voltage transmission and reflection coefficients, respectively, of such a medium of length, d, are given by (6a) and (6b). The squared am- plitudes of (6a) and (6b) for the appropriate param- eters are shown either as solid or dashed curves in Figs. 9, 10, and 11. The computed power transmission coeffi- cient predicts absorption as the microwave frequency approaches the high plasma frequency close to the cathode. As f decreases sufficiently below fi the right- hand side of (8) becomes negative (;.e., the effective plasma frequency for the diode is imaginary), and this together with the collision frequency predicts a growing wave. Therefore, the power transmission coefficient swings above unity. The power reflection coefficient in this area shows a spike approaching unity. As the micro- wave frequency approaches the lower plasma frequency f2, the power reflection coefficient increases and the power transmission coefficient decreases. However, the changes are not rapid enough to give a good fit to the measurements. For frequencies considerably above the lower plasma frequency the absorption is higher than predicted by the theory. For these frequencies the trans- mission and reflection behavior is only slightly influ- enced by the high plasma frequency near the cathode. The calculated power transmission curves level out only below 100 kfc.

The above model predicts the behavior a t t he high frequencies reasonably well, including the absorption. If a better approximation could be obtained which in-

corporated the gradually changing character of the plasma frequency near the cathode, probablJ- one could account full!, for the absorption dip. The main diffi- culty is a t frequencies belouTLf1. The predicted attenua- tion effects do not fall off fast enough to match the es- perimental curves. There does not appear to be an>- col- lision frequency which would predict this behavior.

Since the electron density variation i n the inverted diode is relatively flat and since the first higher order niode occurs around 15 Gc, the solution of the trans- mission and reflection coefficients, on the basis of the principal modes onlJ7, is a reasonably good approxima- tion. If the electron-a-all interaction truly simulates in- elastic gas collisions, then the preliminary data indicate that absorption from such collisions is greater than from the normal elastic collisions for a substantial frequency interval.

nIeasurements for the pure electron case have been conlpleted for the 15-cm diode. In the next phase of the

experimental programs, a neutral gas, i.e., nitrogen or helium, n-ill be added to the s!-stem. The experiment n.ill then proceed with plasmas obtained hl- ionizing these gases. a W of the experiments ]vi11 be repeated n-ith 3 23- cnl cathode.

A~CKS\'O\VI,EDGMII.:NT

The authors would like to thank Prof. I<. 31. Siege1 and R. E. Hiatt for their advice and support i n the course of this investigation. -4 debt is owed to Prof. S. Silver, Dr. R. F. Goodrich, Dr. G. On-y-ang, and Dr. D. L. Sengupta for many suggestions. The advice of B. \;t'olk (Sylvania Research Laborator!;, Bayside, N.Y.), G. Becker (Electronics Research Laboratory, Gniversity of California, Berkele!.)? 1,'. R. Rurris and L. E. Paul (Electron Physics Laborator!,, University of Michigan), on the problems of technology was indis- pensable. G. McIlvain and E;. l'oung are thanked for their help in carrying out the experiment.

Interaction of a High-Intensity EM Field with a Low-Density Plasma*

KUN-MU CHENf

Summary-The interaction of a high-intensity EM field with a low-density plasma, including a collisionless plasma and a weakly ionized gas, is considered. The electron velocity distribution function is obtained exactly from a Boltzmann equation in each case without making any small perturbation approximation. The electron velocity distribution function obtained gives rise to significant results for some plasma parameters a s functions of the intensity of EM field. It is also shown that conventional results for the small field case can be obtained from the results presented in this paper.

INTRODUCTION

N THIS paper the behavior of a lon:-densit\; plasma when it interacts with a high-intensity EM field is investigated. It is u s u a l i ~ ~ assumed that a lolv-

densit). plasma can be considered as a medium having a permittivit!,,

* Received by the PGLIP, September 15, 1961. The work re- ported in this paper was supported by the -Advanced Research Projects .Agency and the AF Cambridge Research Labs., XF Systems Command under Contract KO. AF 19(601) i-128, ARPX Order

f Radiation Laboratory, The University of biichigan, Ann Arbor, lli-60.

Mich.

a conductivity,

lLge2 v

I f l , w? + v 2 and a permeabilit!- p =vu when it interacts with a small EAf field, where w is the angular frequent!. of the inci- dent E h I field, up is the plasma frequency of electron, e and me are the charge and the mass of electron! is the density of plasma, v is the collision frequency of electrons and neutral particles, and E, , and pr , are the per- mittivity and the permeability of free space. This model of a plasma, valid i u a small field case, is suspected to he inaccurate, ii not incorrect, when the incident Ell field is of high intensit!-. The purpose of the stud!. presented in th i s paper is to explore the basic properties of a plasma when it interacts with a high-intensit). E N field. The present study deals with a low-density plasma which includes two types: 1) a collisionless plasma and 2) a weaklq~ ionized plasma. The first case is a plasma in which the collision between particles is so rare that the collision effect can be entirel!; neglected. This model is applicable in some Iow-densitJr laboratory plasmas. The second case is a plasma of weakly- ionized gas type in which only the collision between electrons and neutral

C r = - - - - 1