TUBES(U) NAVAL SURFACE - i/i UNCLASSIFIED NSC/TR-82-530 ...

AD-A126 116 THEORY OF CUSPTRON MICROWRVE TUBES(U) NAVAL SURFACE - i/iWEAPONS CENTER SILVER SPRING ND H S UHN ET AL. OCT 82

UNCLASSIFIED NSC/TR-82-530 SBI-AD-F588 147

FEF/ 9/ EhhhhhhhhhhiIEIIIIIIIIIIIIEEllIhlllIIhllEEhlllhllllllhEEllllllllllllE

r..-.

-:4-0atl

lowo

• j MICROCOPY RESOLUTION TEST CHART!. NATIONAL BUREAU OF STANDARDS-1963-A

.f II

io"M

111 .1qI.oIII

IqlJ~ fJ=~

. . .... .......

-4 AMEOR

BY HLAN& ACHUNGM. KI

RESEARCH~~~' AN TEHWG DEATMN

OCTBE 1982

AP~~~rovad~ fo -kd~wb d tC

Ago

NAVA SURACEWEAPNS CNTE

Dabigm, ~ ~ ~ ~ ~ ~ -Vigna2480Sle SrnMrln 01

*,.= . - -.. -. , ;-,. . j_:- - - • - -. _ . u . - ;.;, - ... '- i'%_~ "- ""J , -.

UNCLASSIFIED

'ECUflITY CLASSIFICATION OF THIS PAGE (W7ie. Date Entered)

REPORT OCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM

I. REPORT NMBER 2. GOVT ACCESSION NO. 3. RECIPIENTVS CATALOG NUMBER

NSWC TR 82-530 -A ZA 116)4. TITLE (and Subtitle) S. TYpe Or REPORT & PERIOD COVERED

THEORY OF CUSPTRON MICROWAVE TUBES

6. PERFORMING ORG. REPORT NUMBER

7. AuTOR(s) III. CONTRACT OR GRANT NUMMERWa)

Han S. Uhm and Chung M. Kim

9. PERFORMING ORGANIZATION NAME AND ADDRESS1 10. PROGRAM ELEMENT. PROJECT, TASKAREA A WORK UNIT NUMI1MI 'Naval Surface Weapons Center (Code R41) 6 WOK NI NUMBERSWhie.ak61152N; ZRO0001; ZROll09;:.-.White Oak R01AA400

Silver Spring, MD 20910I1. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

October 198213. NUMBER Of PAGES

7014. MONITORING AGENCY NAME S ADDRESS(I dllerent freg, Controlling Office) IS. SECURITY CLASS. (of this eetoft)

UNCLASSIFIEDISa. OCkASSIFICATIONi DOWNGRADING

SCHEDU L E

14. DISTRIBUTION STATEMENT (of this Repot)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract eltered In Block 20, it different Item Report)

ISl. SUPPLEMENTARY NOTES

19. KEY WORDS (Contlnue on reverse aide It neceeaary and Identity by block num bor) / . I

Cusptron Microwave Tubes

20. ABSTRACT (Continue on reverse side If necesarty and Identitfy by block rnumber)Stability properties of the negative-mass instability in a rotating annular

electron beam (E-layer) is investigated, in connection with applications onthe cusptron microwave tubes. Analysis is carried out for an infinitelylong E-layer propagating through a magnetron-type conducting wall and propa-

4 gating parallel to an applied axial magnetic field. We assume that the

layer is thin and very tenous. A closed algebraic dispersion relation of thenegative-mass instability is obtained, including the important influence of

DO FjAN7 1473 EDITION 00 1 NOV 65 IS 0BSOLETE UNCLASSIFIEDS-N 01 02.LF1 4.6i04 SECURITY CLASSIFICATION Of THIS PAGE (hen Dots Etered)

UNCLASSIFIEDSECURITY CLASSIFICATION COP T4S PAG (When Dat Ent'eod

conducting boundaries on the mode control in microwave generation and amplification. It is showtn that for typical persent experimental bram parameters, gainand efficiency of the cusptron can be more than five times those of the gyro-tron. Moreover, with an appropriate geometric configuration, perturbations

.-.- with azimuthal mode number N can be dominantly unstable, thereby optimizing,. the microwave power output for radiation with high frequency w N=c, where- w is the electron cyclotron frequency and N is the resonator number in thec

conducting wall.

IiI

XO.699ofl For.

WTIC !AE

% ,,

.. U.Smour ed

* Distribution/

Availabi11t Cod'9S. aInd/or

Dist Pe alt

* L"NCLASS:FIEDSECURITY CLASSIFICATION OF1!MIS AGE(mIRn 009a ltoet.)

NSWC TR 82-530

FOREWORD

Stability properties of the negative-mass instability in a rotating

annular electron beam (E-layer) is investigated, in connection with applica-

tions on the cusptron microwave tubes. Analysis is carried out for an

infinitely long E-layer propagating through a magnetron-type conducting

wall and propagating parallel to an applied axial magnetic field. We

assume that the E-layer is thin and very tenous, A closed algebraic

dispersion relation of the negative-mass instability is obtained, including

the important influence of conducting boundaries on the mode control in

microwave generation and amplification. It is shown that for typical

present experimental beam parameters, gain and efficiency of the cusptron

can be more than five times those of the gyrotron. Moreover, with an

appropriate geometric configuration, perturbations with azimuthal mode

number N can be dominantly unstable, thereby optimizing the microwave

power output for radiation with high frequency w Nwc, where w is the

electron cyclotron frequency and N is the resonator number in the conducting

wall.

Approved by:

IRA M. BLATSTEIN, HeadRadiation Division

i

NSWC TR 82-530

CONTENTS

Section Pae

I INTRODUCTION. .. .................... .. . . . 1

*II VLASOV-MAXWELL THEORY ...................... 4

III VACUUM DISPERSION RELATION .. ................... 15

IV NEGATIVE-MASS INSTABILITY IN MAGNETRON-TYPE CONDUCTOR. ...... 18

V CONCLUSIONS..... ............ .. . . .27

VI ACKNOWLEDGEMENTS.................... .... 29

APPENDIX A--MAGNETIC WAVE ADMITTANCES .............. A-

NSWC TR 82-530

ILLUSTRATIONS

Figure Page

I CROSS SECTIONAL VIEW OF CUSPTRON ...... .................. .. 30

2 PLOTS OF THE PARAMETERS n (SOLID CURVES) and , (DASHED CURVES)VERSUS RATIO R c/Ra [OBTAINED FROM EQ. (45)] FOR N - 6, s - 0,

a - 4/12 AND THREE LOWEST RADIAL MODES .... ............... .. 31

3 PLOTS OF PARAMETER n VERSUS RATIO R /R OBTAINED FROM EQ. (45)c a

FOR s - 0, a - /2N, LOWEST RADIAL MODE NUMBER, AND SEVERALVALUES OF N ........... ............................ . 32

4 PLOTS OF PARAMETER n VERSUS RATIO R /Ra OBTAINED FROM EQ. (45)

FOR N - 6, a - 4/12, LOWEST RADIAL MODE NUMBER AND DIFFERENT

VALUES OF s ....... .. .. ............................ .. 33

5 PLOTS OF PARAMETER n VERSUS RATIO R c/Ra OBTAINED FROM EQ. (45)

FOR N - 8, s - 0, LOWEST RADIAL MODE NUMBER AND SEVERAL VALUESOF a ........... .. ................................ ... 34

6 THE STRAIGHT LINES w - k8 c + L c AND w - kc/8z INTERSECT AT2 2C

(w0 ,k 0 ) = (wcy z , w cazy z/c). THE VACUUM WAVEGUIDE MODE""22 2 22 /

w - (k2c2 + ni c /, 2) PASSES THROUGH (we0 k0 ) PROVIDEDa0R w /c " n/zyz ...... .... ........................... .... 35a cz

7 PLOT OF THE GEOMETRIC FACTOR r(w,kb) VERSUS NORMALIZED fREQUENCY

w/w c IN EQ. (28) FOR N - 6, Z " 6, s - 0, a - 4/12, R c/Ra = 1.4

AND R w /- 0.4433 CORRESPONDING TO THE GRAZING CONDITION FOR

= - 0.2 .......... .............................. 36

8 PLOTS OF (a) NORMALIZED GROWTH RATE 1w AND DOPPLER-SHIFTED

REAL OSCILLATION FREQUENCY r/ w VERSUS w/w OBTAINED FROMr c c

EQ. (47) FOR ELECTRON BEAM PARAMETERS v = 0.002, .5 - 0.04,

Be w 0.4, Bz m 0.2, FIELD INDEX n = 0 AND THE GRAZING CONDITIONS

CORRESPONDING TO EQ. (53), AND PARAMETERS OTHERWISE IDENTICAL

TO FIG. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

iii-

NSWC TR 82-530

ILLUSTRATIONS (Cont.)

Figure Page

9 PLOTS OF (a) NORMALIZED GROWTH RATE 0 1/w AND (b) DOPPLER-

SHIFTED REAL OSCILLATION FREQUENCY nr/wc VERSUS w/w OBTAINED

FROM EQ. (47) FOR R c/R a - 1.4, Ra w c/c - 0.4433, SEVERAL

DIFFERENT VALUES OF THE FIELD INDEX n, AND PARAMETERS OTHERWISEIDENTICAL TO FIG. 8 ......... ........................ .. 39

10 PLOTS OF (a) NORMALIZED GROWTH RATE a ±/we AND (b) DOPPLER-

SHIFTED REAL FREQUENCY r /wc VERSUS NORMALIZED AXIAL WAVE

NUMBER kc/ c OBTAINED FROM EQ. (48) FOR THE PARAMETERSC- 41IDENTICAL TO FIG. 8................ ........

11 PLOTS OF (a) NORMALIZED GROWTH RATE 0 1/w AND (b) DOPPLER-

SHIFTED REAL FREQUENCY ar/wc VERSUS w/wc FOR Be = 0.96, THE

GRAZING CONDITIONS CORRESPONDING TO EQ. (53), AND PARAMETERS

OTHERWISE IDENTICAL TO FIG. 8 ..... ................. .... 43

12 PLOT OF THE NORMALIZED GROWTH RATE 0i/wc VERSUS w/wc OBTAINED

FROM EQS. (47) AND (58) FOR 8e , 0.96, N - 24, k - 25,

SRc /Ra = 1.1, Ra w c/c - 1.069, a - 7/2N, AND PARAMETERS OTHERWISE

IDENTICAL TO FIG. 8 ......... ........................ .. 45

A-1 CONFIGURATION OF A RESONATOR ....... ................... ... A-6

iv

.0l

NSWC TR 82-530

I. INTRODUCTION

[' b

In recent years, there have been numerous experimental and theoretical

investigations on high power and high frequency microwave devices such as

gyrotrons I- 3 relativistic magnetrons4 and free electron lasers.5 ,6 High

power microwaves have been also produced by the negative-mass instability7 9

of a relativistic rotating electron layer (E-layer) in a conducting

waveguide. This relatively new scheme of microwave generation by a

rotating E-layer has many attractive features in practical application.

Using the mode control scheme currently under investigation,I0 for example,

this device could be a tunable, high-frequency microwave tube with low

magnetic fields. One of the simple ways to produce a rotating E-layer is

use of a magnetic cusp field through which an annular electron beam be-11

comes the same radius rotating beam. After passing through a cusped

magnetic field, a nonrelativistic E-layer is propagating through a

convential magnetron-type conducting wall called the cusptron. The name

"'cusptron" is originated from the cusp and the magnetron. In this paper,

we investigate the negative-mass stability properties of the E-layer in

a magnetron-type conductor, in connection with the application on the

cusptron microwave tube.

The stability analysis of the negative-mass instability is carried

out for anE-layer with radius R0 propagating through a mangetron-type

conductor with its inner-and outer-most radii R and Rc, respectively.

Equilibrium and stability properties are calculated for the

electron distribution function (Eq. (1)] in which all electrons have the

same energy and the same canonical angular momentum but a Lorentzian

' "" ' / " " . . , . o . i , . : iI

* . .. . . . . _ , . ., . .. .-, . _ .'*'-. ___ . _._.. . . . .. .. .

NSWC.TR 82-530

distribution in the axial canonical momentum. The stability analysis is

calculated within the framework of the linearized Vlasov-Maxwell equations

for an infinitely long E-layer propagating parallel to an applied magnetic

field Boe with an axial velocity 0 ce. We assume that the E-layer is thin

and very tenuous. The formal dispersion relation [Eq. (29)] of the negative-

mass instability is obtained in Section II, including the important

.. influence of conducting boundaries on the mode control in microwave

amplification and generation.

In Section III, properties of the vacuum dispersion relation in a

magnetron-type conductor are investigated without including the influence

of beam electrons. It is shown that the vacuum dispersion relation reduces

to three distructive modes; the transverse electric (TE), the transverse

magnetic (TM), and the magnetron TE modes. From numerical calculation, the

cut-off frequency wct Mc/Ra is obtained in terms of the ratio Rc /Ra* In

Section IV, the negative-mass stability properties of an E-layer is numerically

investigated. Several points are noteworthy from the numerical calculation for

a nonrelativistic cusptron. First, optimum coupling occurs between the beam

and the fundamental 2Tr modes. It is shown for the fundamental 2Tr mode that for

typical present experimental beam parameters, gain and efficiency of the cusptron

can be more than five times those of the gyrotron. Second, for the

applied magnetic field satisfying the grazing condition of the I = N

perturbations, other azimuthal perturbations with Z 0 N are suppressed.

Here t is the azimuthal harmonic number of perturbations and N is the

number of resonator in the magnetron-type conductor. Under this grazing

condition, the I - N mode perturbation is the dominant unstable mode,

optimizing the microwave power output for radiation with frequency

W = c . Therefore, even for relatively low magnetic field, high frequency

2

NSWC TR 82-530

microwave can be amplified by making use of the cusptron with N 2.

Finally, the growth rate and Doppler-shifted real oscillation frequency

are substantially increased by changing the applied field index from

zero to a small positive value. Preliminary investigation of a rela-

tivistic cusptron amplifier is also carried out in Section IV.

I.

i-3

NSWC TR 82-530

II. VLASOV-MAXWELL THEORY

As illustrated in Fig. 1, the equilibrium configuration consists

of a nonneutral electron layer (E-layer) that is infinite in axial extent

and aligned parallel to an applied magnetic field B0 (r)iZ. The electron

layer is accomplished by passing of a hollow electron beam through an

ideal cusp magnetic field. Therefore, the electron layer is in a fast

rotational equilibrium, where all electrons have positive canonical

angular momentum. The mean radius of the E-layer is denoted by R0 . We

also assume that the radial thickness of the E-layer is 24 which is much

less than the equilibrium radius R0 , i.e., a <<R0 . The mean motion

of the E-layer is in the azimuthal direction, and the applied magnetic

field provides radial confinement of the electrons. As shown in Fig. 1,

the outer conductor is a magnetron-type configuration with its inner-

and outer-most radii R and R ,respectively. The angle of the open spaces

in the magnetron-type conductor is denoted by 2a. In the theoretical

analysis, cylindrical polar coordinates (r, 8, z) are employed. In the

present analysis, we assume that v/1 << 1, where v - Nbe2/mc2 is Budker's

parameter and -?mc2 is the electron energy. Here Nb is the total number

of electrons per unit axial length, -e and m are the charge and rest

mass of electrons, respectively. Consistent with the low-density assump-

tion, we neglect the influence of equilibrium self-fields.

In the present analysis, we investigate the equilibrium and stability

properties for the choice of equilibrium distribution function

w N 4 A 6(y- 61)Sfb (H, Pe, 8 z ) 2 - 2 2

41 mc2 P z ) 2 + P )

14

NSWC TR 82-530

2 (m 2c4 22)where H - ymc c + c p is he total energy, Pz is the axial

. - canonical momentum, P0 - r(p0 - (e/c)Ao(r)] is the canonical angular

momentum, wc - eB0 (Ro)/imc is the electron cyclotron frequency, A0 (r)

is the axial component of the equilibrium vector potential,

P0 - (e/c)RoA0 (R0 ) - (e/2c)Ro2B0 (R0) and 9, z and A constants.

In the subsequent stability analysis, we examine the linearized

Vlasov-Maxwell equations for perturbations about a thin E-layer

equilibrium described by Eq. (1). To calculate stability quantities

of the E-layer, we adopt a normal-mode approach in which all E-layer

perturbations are assumed to vary with time and space according to

6rP(z,t) = '(r)exp {i(Z9 + kz - wt)} ,

where Imw > 0. Here, w is the complex eigenfrequency, k is the axial

wavenumber, and Z is the azimuthal harmonic number. The Maxwell

equations for the perturbed electric and magnetic field amplitudes can

be expressed as

" x E(x) -i )(x)--V X- -- X

V x A(x) -(4ir/c)5(x) -i(w/c)t(x) (2)

where 2(x) and 9(x) are the perturbed electric and magnetic fields,

()=-ef d 3p v tb(x'?) (3)

is the perturbed current density.

0VIx (X')1S) - e aTexp(-i) ( + f (4)

tb(hxe fn -@. (

is the perturbed distribution function, and T t' t.

5

i , . . '. " . .. - .. _ . . .. .o L o~- . -. - . - " , * -* *. ,- . ,, :-. ~ - . .- . - . .

NSWC TR 82-530

From Eq. (2), it is straightforward to show that

21"~ ~ E-gzr (r +- - t Cr) . (r) (5)ar Zz W ie r jz j c LB

where

p 2 2 /2C k2 (6)

For present purpose, we assume that

"ll - 1W - - k$cl <<z g'

la << R0 (7)

where 6z - z/Imc and c is the speed of light in vacuo. The perturbedz

axial and azimuthal electric fields 1iJ(r) and Ez(0) are continuous

across the beam boundaries (r --R and r = R2). Here R - R - a and2 1 0

R2 = R0 + a are the inner and outer boundaries of the E-layer, respec-

tively. Moreover, within the context of Eq. (7), it is valid to approximate

Eke (Rl) .= EL(RO) Eje(R 2 )

(8)

Elz(Rl) EzCR0 ) EXz(R 2).

Integrating Eq. (5) from r = - to r - R2 + 6 and taking the limit

6 .* 0+, we obtain

+ 4 r R2B z(R2 ) - BLz(R1-) = -- I drJ (r) (9)

1

where d0 (R + ). Similarly, the discontinuity of

the azimuthal component of the perturbed magnetic field is given by

6

. I

77 7-75 . 7 ... ~

NSWC TR E -530

B toe(R 2+ ) 4 toe(Rl.) f dr3Iz (r), (10)

fRa

For convenience in the subsequent Analysis, ye introduce the

normalized electric and magnetic wave admittances, 12.d and b+~, defined

at the inner and outer surfaces of the E-layer by

d+- r 0 2 () d 2)

and

b+ LB,, (R 2 10jPr OB tir)Bt R

(12)

b- Ii CR1 -)/r /a r) 2 z(r)] R,

From Eq. (2), it is -traightforward to show that

:-Btz Cr) i- rEz (r) + E r)]

)b +t i-w - Ez(r) (13)

p r cp

outside the E-layer and the radial coordinate r satisfying r Ra

Substituting Eq. (13) into Eqs. (9) and (10) and making use of Eqs. (11)

and (12) we obtain

7

4

NSWC TR 82-530

,,R,

";!%,, ,:' " ( b .+ b ) 2 R 0i ( R ' ) k i ( 1 )1 - - i 4 ,_

. (14)

(d_+ d+)A (R)m i 4w 2dr3B (r) +i 4rp2 O ]2dri~ (r)

where use has been made of Eq. (8). Evidently, an evaluation of the

azimuthal and axial components of the perturbed current density

(Jr.t and J z) is required for a detailed stability analysis.

* A. Perturbed Current Density

" "" In this section, we evaluate the perturbed distribution function

and subsequently the perturbed current density. As indicated in Eq. (4),

the particle trajectories, r'(O), 8'(-) and z'((T), in the equilibrium

fields are required in order to evaluate the perturbed distribution

function. The applied axial magnetic field B (r)jz is approximated by

B0(r) - B0 (R0) (1 - np/R0) (15)

" where p r 0 and

n B (R) B0(r) (16)

B(0 ) . r I R

0: is the magnetic field index. Therefore, the azimuthal component of the

vector potential is expressed as

rA 0 (r) - RoA 0 (R0 ) + B0 (%) f dr r 1 L ) " (17)

*'8. °

8

-~ -- iK1 11 ~x S

NSWC TR 82-530

?Making use of the Hamiltonian H - cmc2 . (m2c 4 + c p2) , the canonical

angular momentum P8 a r [Pa - (e/c)%(r)I and Eq. (17), we obtain the

electron trajectories13

P r'- R0 = wc6Pl mr 2N + Arn(wrl +)

z = z + p z/im , (18)

!clW c JPe R

where

2

= - - (19)W 2!Wr2 n)wc2 (20)

is the radial betatron frequency-squared of electrons, y8 - (1 - %2)-

is the relativistic mass ratio associated with the aximuthal electron

velocity a8 c, Ar is the amplitude for radial betatron oscillations,P - P0Pe = 0 and T t' - t. Within the context of Eq. (7), it is

valid to approximate

. 6Pe

e= c - 2 (21)YmR%

and to neglect the small oscillatory term cp'/R 0.

After a simple algebraic manipulation that makes use of Eq. (7),

the perturbed distribution function in Eq. (4) is approximated by12'13

P 9

NSWC TR 82-530

SkV V a

fb { I(,- z) ROE e(R) + W 2.(R 0) fbo

+:e. . (22)

[+ (I v (R0 ) -

where the orbit integral I is defined by

T m d exp (i [z£' -9) + k(z' -Z) WT

(23)

kp iSpe -1- + -

and use has been made of Eq. (8). Substituting Eq. (23) into Eq. (22)

and carrying out a tedious but straightforward algebra, we obtain the

perturbed distribution function

2 2 2ec I +( 2

b = im 2 (w _ k, _ kVa) o(% ) + Izitz(%) (24)

Subsequently, integrals of the perturbed current density in the left-hand

side of Eq. (14) are given by

•f RfRidr . (r) - ( / B) dr J, (r)

R R

.r - (RO) + 1z ,) (25)

where the effective susceptibility a is defined by

10

-- -. -, . . ! . 1.- ,,- -. ._- ,.- -

o - +. .._ -' . --- . -. - .. '_--. , _' , , _ _. -- - -

NSWC TR 82-530

2 2 k2R2ck)2 - L kR (26)R 2 (W - L ~c - kSz + ikl zcIA/YZ3)2

and v - Ne2/mc2 is Budker's parameter.

Substituting Eq. (25) into Eq. (14), we obtain the matrix equation

relating ile(R.) and ilz(%). The condition for a nontrivial solution

to this matrix equation is that the determinent of the matrix vanishes.

This gives the general dispersion relation

F tw2 (kc-2 b + b+ d -d+] + 1 0 (27)

p2c2 +d+

where use has been made of the approximation w 1 wc + k8z c, which is

consistent with Eq. (7). Defining the geometric factor r(w,k)

rp2c2 2 Wc2 (kc- $..)2 1r(wk" 2 [d + z (28)

and substituting Eq. (26) into Eq. (27), the dispersion relation in

Eq. (27) is expressed as

v c2 X2 - k2RO2-Jr-- (29)

r(w,k) - - 2 ( 1W k c (z)cA/z3)2(w - ~w- k ,c + ilktI ch/

which can be used to determine the complex eigenfrequency Q w - Lwc - cc z

in terms of various physical parameters.

B. Geometric Factor r(wk)

The evaluation of the wave admittances at the boundaries of the

E-layer is required in order to complete the dispersion relation. To

11

NSWC TR 82-530

make the theoretical analysis tractable, in the subsequent analysis, we

concentrate the lowest modes of the electromagnetic waves inside the

14,1resonator where R < r < R • Moreover, the previous study 1 ,15 have".-:a c

exhibited that the lowest modes in the resonator dominates the wave

interaction. In this regard, for the transverse magnetic (TM) mode, we

select (Appendix)

z() 0, Ra 4 r R Rc (30)

which is lowest mode properly satisfying all the necessary boundary

conditions. The Maxwell equations of the TM mode in the region

0 r 4 R can be expressed as

at a

Sp E (r)-0 (31)~rrr ~2 )

2 2 2 2except the inside of the E-layer (i.e., R < r < R). Here p w /c - k

is defined in Eq. (6). The physically acceptable solutions to Eq. (31)

is given by

A J Jr(pr) 0 R

B 1z (r) - B[JI(pr) - Ji(n)NI(pr)/N1(n)] (32)

R2 r<R a

* where JI(x) and N (x) are Bessel functions of the first and second kind,

respectively, the parameter n is defined by

2 ( c k 2)R 2 (33)

12

NSWC TR 82-530

and A and B are constants. Substituting Eq. (32) into Eq. (11) and

carrying out a straightforward algebra with a << %, we obtain a sum

of the electric wave admittance

i 2J (n)/irt

d + d+ - ( ( T - (34)

*where t - R/a

Similarly, for the transverse electric (TE) mode, the sume of the

magnetic wave admittances is expressed as (Appendix)

b + b+ - (35)" b-b -

, ()+ G(w,k)Nt (M

where

-. I (n)D(w,k). .~ G(w,k) -- ,(36)

I-i N (n)F(wk)

- n (n) /in \2D(w,k) j n, W ( nn

"n

,J 10 (n)Nl( ) - J1 (C)N0 (n)

Not J1 ( )N(n0) - Jl(n)N 1

is the vaccuum dispersion function,

F(wk) D(wk) - ["-n N'() 1 -( --n ) (38)

I I

the integer n in Eq. (37) is defined by

n s + mN (39)

13

NSWC TR 32-530

N is the number of the resonators, a is the half angle of the open

spaces as shown in Fig. 1, a is an integer satisfying 0 < s 4 N - 1,

the prime(') denotes (d/dx)J (x), and finally the parameter is defined

by

2 2 2 2 2 (40)R C R /R 22W/ _k) c(0

Pbr detailed informatiou in deriving Eq. (35), we urge the reader to

read the Appendix. Substituting Eqs. (34) and (35) into Eq. (28)

completes the dispersion relation of the cusptron microwave tube.

4

14

NSWC TR 82-530

III. VACUUM DISPERSION RELATION

In the absence of the beam IV + 0 in Eq. (29)], the dispersion

relation in Eq. (27) reduces to the TM dispersion relation

J t(r) 0O (41)

and to the TE dispersion relation

G I (w,k) - 0 , (42)

where use has been made of Eqs. (28), (34) and (35). The TM dispersion

relation is equivalently expressed as

2W - 2 in (43)

- a

a

where 8in is the nth root of J1($2n) - 0. Similarly, it is shown from

Eq. (36) that the TE dispersion relation can be equivalently expressed

as the ordinary TE mode

- k --in (44)

c R 2

a

and the magnetron TE mode

D(w,k) n (sin na)=- 3 '(n) \ I

n

J0 (n)N1 () - J1 ()No()

N- I"j.(;)N 1 (n) - jI(iiN 1W 0 (45)

1.5

nor

NSWC TR 82-530

where an is the nth root of JI (an) - 0. In Eq. (45), N is the

-€ number of resonators, a is the half angle of the open space in resonator,

the integer n in Eq. (45) is defined by n - s + iN, s - 0, 1, 2, ... N - 1,

and the parameter n and C are defined by n2 2 Ra2/R c2 22 p2Ra2

(W2/c - k2)Ra2 in Eq. (33).

In the remainder of this section, we investigate properties of the

magnetron TE mode in Eq. (45) which relates the parameter n to the

parameter C. In other words, the value of the parameter n is evaluated

in term of Rc/Ra - c/n. Once we determine value of the parameter n, the

dispersion curve in (wk) parameter space is obtained from the relation

W2/C2 _ k2 2/R 2 (46)/ck i /a

Shown in Fig. 2 are plots of the parameters n (solid curves) and C

(dabhed curves) versus the radius ratio Rc/Ra for N - 6, s - 0, a - w/12

and the three lowest radial modes. Several points are noteworthy in

Fig. 2. First, as expected, value of the parameter n decreases

monotonically as the ratio Rc/Ra increases. However, value of the

parameter C stays relatively steady. Second, we note that values of

parameters n and c at Rc/R a - 1 are given by n - C 01 - 3.83 for the

lowest radial mode (the fundamental 2w mode), a02 - 7.02 for the second

and a61 - 7.50 for the third mode. Finally, it is shown for the third

radial mode that the value of the parameter T has a plateau for

1.2 <R /R 1.4.

Figure 3 is plots of the parameter n versus ratio R /R for s - 0,c a

a - w/2N, the lowest radial mode number, and several values of N.i4Remarkably, the value of the parameter n is relatively independent of

16

NSWC TR 82-530

the number N for N ;0 3 and for the lowest radial mode number. On the

A, other hand, for N - 2 and Rc/R a - 1, the parameter n is given

n a " 21 - 3.05. Dependence of the parameter n on the integer s is

presented in Fig. 4 where parameter n is plotted versus the ratio

R /R. for N - 6, a - w/12, lowest radial mode number and differentca

values of s. After a careful examination of Eq. (45), we note that the

dispersion relation for s - 1 is identical to that for s - N - 1, and

so on. In this regard, plots in Fig. 4 are presented only for the

integer s satisfying 0 4 s 4 N/2. Obviously from Fig. 4, we conclude

that the parameter n depends sensitively on the integer s. For R a 1,

the value of the parameter n is given by in - asl for each s. Of course,

the vacuum dispersion relation in Eq. (45) is also investigated for

different values of the half angle a. Shown in Fig. 5 are plots of

the parameter n versus the ratio Rc/R a for N - 8, s 0 0, lowest radial

mode number and several values of a. Parameter n reduces with increasing

value of the half angle a for the range 1 < R c/Ra < 1.4, which is a typical

parameter range of present experiments.

17

NSWC TR 82-530

IV. NEGATIVE-MASS INSTABILITY IN MAGNETRON-TYPE CONDUCTOR

In this section, we investigate stability properties of the

negative-mass instability in a E-layer propagating through a magnetron-

type conductor, by making use of the dispersion relation in Eq. (29).

The growth rate and bandwidth of the negative-mass instability are

directly related to the gain and bandwidth of the cusptron amplifier

or oscillator. Making use of the fact that the "Doppler-shifted"

eigenfrequency 9 in Eq. (7) is well removed from the electron cyclotron

resonance, i.e., IlI < < Wc, and evaluating the function r(w,k) at

k - (w - lwc)/Bzc for the amplifier and at w - - 1w + kB c for

the oscillator, the dispersion relation in Eq. (29) can be approximated

:I by

(wk c g-kr) ki 2z3

(47)

. 2"Z2k 2. 2 - 2

2 YR 0 2 bED

for the amplifier and

[:.r(%,k) +(~r r + i Ik8c3

*O (48)2

2'R02 (t2U- k20)I %

18

NSWC TR 82-530

for the oscillator. In the remainder of this section, the growth rate

S- InMS and the Doppler-shifted real oscillation frequency ar - ReQ arer

numerically calculated from Eq. (47) for the amplifier and from Eq. (48)

for the oscillator. Numerical calculation is carried out for the

nonrelativistic electron beam parameters v - 0.002, A - 0.04, O8 - 0.4

and 8z " 0.2 corresponding to j - 1.118. For a relativistic beam,

8e " 0.96 and 8z " 0.2 corresponding to y - 5.1.

To the lowest order, the eigenfrequency w and axial wavenumber k

are obtained from the simultaneous solution of the vacuum waveguide mode

dispersion relation,

2 2k 2 (49)

c 2R2a

and the condition for cyclotron resonance

w= k + k5 c. (50)

Moreover, to maximize the growth rate and efficiency of microwave

generation and amplification, it is required that the group velocity of

the vacuum waveguide mode in Eq. (49) be approximately equal to the

beam velocity, i.e.,

2V - dw/dk - kc2/W 8 c . (51)g 2

Solving Eqs. (50) and (51) for the characteristic frequency and axial

wavenumber (w,k) - 'w0,k0), we find (Fig. 6)

W Z YZ2 k0 - Z czYz 2 /C . (52)

19

NSWC TR 82-530

For maximum growth, it is also required that (w0 k0) solve Eq. (49) in

• .leading order. Therefore, for maximum efficiency, we find that R should* a

satisfy

R - nc/w cy . (53)a cz

Under the grazing condition in Eq. (53), the cyclotron resonance mode

in Eq. (50) is a tangetial line of the vacuum waveguide mode in Eq. (49),

as shown in Fig. 6. Noting 8 % Wc/C, it is found from Eq. (53) that

the beam radius is determined from

R0/Ra ffi c/Rawc = eyz/n , (54)

and that the parameter n should satisfy the inequality

- nlty > o(55)

for a physically acceptable cusptron microwave tube.

A. Nonrelativistic Cusptron Amplifier

In this section, we summarize results of numerical calculation

from Eq. (47) for the amplifier in a nonrelativistic electron beam6

09 . 0.4) propagating through a unifrom applied magnetic field with

the field index n 0 0. Shown in Fig. 7 is a plot of the geometric

factor r(w,k b) versus normalized frequency w/wc in Eq. (28) for N =.6,gc

Z - 6, s - 0, a - n/12, R c/Ra = 1.4 and Ra w c 0.4433 corresponding

to the grazing condition in Eq. (53) for the parameter n = 2.715 in

Fig. 2. Here the axial wavenumber k is substituted by kb ( - Zc)/5 c

consistent with the dispersion relation in Eq. (47) for amplifiers. As

20

NSWC TR 82-530

expected from the condition Ra w c - 0.4433, the curve of the geometric

tactor r grazes the horizontal line. Values of the geometric factor r

are very close to zero in a considerable range (i.e., 6 <w/w < 6.5 in

Fig. 7) of w space, thereby exhibiting possibility of broad unstable

frequency range.

Figure 8 shows plots of (a) normalized growth rate Oi/wc and

(b) Doppler-shifted real frequency Qr/W c versus w/w obtained from

Eq. (47) for the electron beam parameters v - 0.002, A - 0.04,

ae M 0.4, Oz - 0.2, the field index n - 0 and the grazing conditions

(i.e., R w /c - 0.4433 for R /R 1.4, 0.4934 for R /R - 1.3,

0.5453 for R /R = 1.2, 0.5905 for R /R - 1.1 and 0.6257 for R /R 1),

corresponding to fundamental 21T mode, and parameters otherwise identical

to Fig. 7. For each value of R c/R a, the growth rate curve consists of

two parts; solid line corresponding to relatively large Doppler-shifted

real frequency and dashed line corresponding to very small Doppler-shifted

real frequency [Figs. 8(a) and (b)]. The efficiency of microwave tube is

directly proportional to the Doppler-shifted real frequency.16 In this

regard, even though instability of the dashed curve exhibits large growth

rate, there is no sifnificant amplification in this frequency range

4i corresponding to small Doppler-shifted real frequency. Obviously from

Fig. 8(a), the amplification growth rate [solid curve in Fig. 8(a)]

reduces as R c/Ra approaches to unity. In particular, for R c/Ra f 1,

the amplification growth rate vanishes, thereby smoothly connecting

two dashed curves in Fig. 8(a). In this case (R c/Ra = i), the

Doppler-shifted real frequency Qr/w c is less than 0.003. We also

'I

4 21

NSWC TR 82-530

note from Figure 8(a) that the maximum growth rate of the cusptron is more

[aathan five times that for conventional gyrotron amplifier [Curve for Rcl/R a

1.4 in Figure 8(a)]. Also the Doppler-shifted real frequency for R /R -c a

1.4 in Figure 8(b) exhibits a strong possibility of very high efficiency in

microwaye amplification.

Numerical investigation of Equation (47) has been also carried out for

a broad range of physical parameters s, 1, a, N and various radial mode

number. From this numerical calculation, we maki several conclusions.

First, under the grazing condition Ra = nc/Nw yz corresponding to =,

there is no amplification growth rate of instability for perturbations with

+ N azimuthal harmonic number. Second, optimizing value of the parameter

SRaw /c according to Equation (53), the £ + N perturbations also have substantial

amount of the amplification growth rate with relatively large Doppler-shifited

real frequency. However, comparing with Figure 8, we conclude that the

amplification growth rate of the I = N perturbation is largest and is most

effective means of the microwave amplification. Third, after optimizing the

parameter Rawc/c according to Equation (53), the s = 0 perturbation is the

best in the microwave amplification. Fourth, it is found that the lowest

radial mode perturbation is dominant unstable mode. Fifth, we also found

from the numerical calculation that after optimizing R aw c, the growth rate and

Doppler-shifted real frequency are almost independent of a in the range

i/4N < a < v/2N for s - 0, £ = N and the lowest radial mode number. However,

increasing a from n/2N to 7/N reduces drastically the growth rate and Doppler-

shifted real frequency. Finally, stability properties have been investigated

I

I

NSWC TR 82-530

also for N = 4 and parameters otherwise identical to Figure 8. It has been

shown from numerical calculation that the maximum growth rate and Doppler-

shifted real frequency of N = 4 are comparable to those of N - 6 case. However, the

optimum value of the growth rate and real frequency occurs at Rc/Ra - 1.8

and R aw c/c - 0.4494. After consideration of all of these properties, we

conclude for the nonrelativistic electron beam with - 0.4 that the optimum

physical parameters for microwave amplification are N - 6, £ - 6, s - 0,

- g/12, R c/R -1.4 and R w /c - 0.4433.

B. Effect of Non-Zero Field Index

As shown in Equations (47) and (48), the coupling coefficient is directly

proportional to the parameter

1 1 2. + n (56)1-n - 2 1 - n

where n is the field index defined in Equation (16). Evidently from Equation

(56), for nonrelativistic beam with 682 << 1, small increase of the field

index from zero makes a big difference in the coupling coefficient, thereby

enhancing the gain and efficiency of the microwave amplification. Shown in

Figure 9 is plots of (a) normalized growth rate 1w c and (b) Doppler-shifted

real oscillation frequency 2/wc obtained from Equation (47) for R c/Ra = 1.4,

R aw c/c - 0.4433, several different values of the field index n, and parameters

otherwise indentical to Figure 8. Obviously, the growth rate and real

frequency increase substantially by increasing the field index from zero to

a small positive value. However, since the applied magnetic field is not

anymore uniform along the axial direction for non-zero field index,

23

NSWC TR 82-530

tapering of the conducting wall radius Ra is required in order to match

the grazing condition in Equation (53) (e.g., Rawc/c - 0.4433 in Figure 9).

C. Nonrelativistic Cusptron Oscillator

The growth rate and Doppler-shifted real oscillation frequency are

numerically obtained from Equation (48) for the oscillations in a nonrelativistic

electron beam. Figure 10 is plots of (a) normalized growth rate and (b)

Doppler-shifted real frequency versus normalized axial wavenumber kc/w obtainedC

from Equation (48) for the parameters identical to Figure 8. Normalized

lowest order eigenfrequency wo/wc = t + k8z/c/w is also shown in the horizon-

tal scale in Figure 10. Comparing Figure 10(a) with Figure 8(a), we note

that unstable range in frequency space for the oscillation is broader than

that for the amplifier. Perfurbations in the amplifier are unstable only

for the positive k-space. Moreover, the real frequency in the oscillator

is considerably different from that in the amplifier [Figures 8(b) and 10(b)].

D. Relativistic Cusptron Amplifier

Preliminary investigation of Equation (47) has been carried out for the

amplifiers in a relativistic electron beam with 80 - 0.96 and 8z - 0.2.

After a careful examination of Equation (55) and Figure 2, we note for the

relativistic electron beam (0, - 1) that the first available coupling occurs

at the third lowest radial mode number where the parameter n is larger than

0 N for a reasonable range of R c/R . Figure 11 shows plots of (a) normalized

growth rate i 1w and (b) Doppler-shifted real frequency 0r /w versus /Wic rc c

for 8 - 0.96, the grazing conditions (R wc - 1.1101 for R/ aR m 1.5,a cc a

6 1.1679 for R c/R - 1.25 and 1.2249 for R c/Ra m 1), and parameters otherwise

21

NSWC TR 82-530

identical to Figure 8. Contrary to a nonrelativistic cusptron amplifiers

(in Figure 8), the growth rate and bandwidth of instability in Figure 11,: incrase drastically as Rc /Ra approaches to unity. However, for Rc/R a m 1.5

and % C /c - 1.1101, the Doppler-shifted real frequency of perturbations with

+ . N azimuthal mode number vanishes, while the f - N perturbation has

* considerably large Doppler-shifted real frequency. In this regard, by

selecting Rc/Ra - 1.5 a, Ra c /c - 1.1101, microwaves with I - N mode

perturbations are dominantly amplified, thereby optimizing the microwave

power output for radiation with frequency w = w . On the other hand, forC

_ RR " 1, various other modes compete with the £ - N mode, leading to multi-

mode amplification. Therefore, even though the growth rate for Rc /Ra - 1.5

. is less than that for R /R - 1, geometric configuration with R /R - 1.5 isc a c a

more effective to amplify microwaves. Numerical investigation of Equation

(47) for N - 12 and N - 24 also exhibits very similar properties.

After a careful examination of the geometric factor r(w, k) for a broad

range of various physical parameters, it can be found

r(w, kb) 0,b (57)

,- k)/9k] k - kb = 0,

for particular values of R w /c and frequency w. In this case, in order toa c

correctly evaluate the gain of the cusptron, we approximate Equation (29) by

r Ia1 (2, z kb 21z2C2 (k )k1 r(., - --28 2 2 2~

3 -2

a kJc (.2u - ~2R 2 ). (58)2R02

z 0

25

NSWC TR 82-530

Of course, the dispersion relation in Equation (47) is used to obtain the

gain for a broad range of physical parameters except w satisfying Equation

(57). Obviously, Equation (47) fails to estimate the gain for this

frequency range. We therefore make use of Equation (58) to obtain the gain

at the frequency satisfying Equation (57). We also emphasize the reader

that the gain of the cusptron amplifier at the frequency w corresponding

to Equation (57) is significantly greater than that at other frequencies.

In order to illustrate a high gain cusptron amplification, shown in

Figure 12 are plot of the normalized growth rate S1 1w versus w/w obtainedi c c

from Equations (47) and (58) for -0 = 0.96, N - 24, ti 25, Rc/Ra - 1.1,

R aw c/c = 1.069 and parameters otherwise identical to Figure 8. The dashed

curves in Figure 12 are plot of the gain obtained from Equation (47) in the

frequency range satisfying Equation (57). Obviously Equation (47) fails in

this frequency range. However, Equation (58) correctly evaluates the growth

rate in this region. As expected, the maximum gain in Figure 12 is considerably

enhanced in comparison with that of ordinary cusptron amplifier (Figures 8

and 11). However, the Doppler-shifted real frequency in this frequency range

is comparable or smaller than that for other frequency ranges. Other detailed

properties of the relativistic cusptron amplifiers and oscillators are currently

under investigation by authors for a broad range of physical parameters and

will be published elsewhere.

26I

NSWC TR 82-530

V. CONCLUSIONS

In this paper, we have investigated the negative-mass stability

properties of an E-layer propagating through a magnetron-type conductor,

in connection with application on the cusptron amplifier and oscillator.

Stability analysis has been carried out within the framework of the

linearized Vlasov-Maxwell equations, assuming that the E-layer is thin

and in a fast rotational equilibrium. The formal dispersion relation

of the negative-mass instability was obtained in Section V, including the

important influence of the magnetron-type conductcr which has periodic

resonators. Properties of the vacuum waveguide mode in a magnetron-type

conductor have been briefly investigated in Section III, without including

the influence of beam electrons. Making use of the vacuum dispersion

relation, the cut-off frequency w = nc/R has been calculated in termsct a

of the radius ratio Rc/Ra. For the number N of the resonator more than three,

it has been shown for the fundamental 2w mode with s - 0 that wR a/c aO 0

3.83 when R /R - 1. In Section IV, the negative-mass stability propertiesc a

of an E-layer were numerically investigated, including the important

influence of the magnetron-type conductor. Several points are noteworthy

from this numerical calculation for a nonrelativistic cusptron amplifier

or oscillator. First, optimum coupling occurs between the beam and the

fundamental 2r modes. Utilizing the fundamental 2w mode, it was shown

that for typical present experimental beam parameters, gain and efficiency

of the cusptron can be more than five times those of the gyrotron. Second,

for the applied magnetic field satisfying the grazing condition Ra /c

n/NY of the Z = N perturbations, other azimuthal perturbations with Z # N

z

27

.4I

NSWC TR 82-530

are suppressed. Therefore, under this grazing condition, the I = N mode

perturbation is the dominant unstable mode, optimizing the microwave

power output for radiation with frequency w = Lw c In this regard, even

for relatively small applied field, high frequency microwaves can be

amplified by making use of the cusptron with N > 2. Finally, the growth

rate and Doppler-shifted real oscillation frequency are substaatially

increased by changing the applied field index from zero to a small positive

n value. Preliminary investigation of a relativistic cusptron amplifier has

been carried out, and it has been shown that the stability trend of the

relativistic cusptron is similar to the nonrelativistic case.

28

0o

.* NSWC TR 82-530

VI. ACKNOWLEDGEMENTS

This research was supported by the Independent Research Fund at

the Naval Surface Weapons Center.

:I

'" 29

4-

PJSWC TR 82-530

MAGNETRON-TYPE CONDUCTOR

RESONATOR

0R

0c

0R

SR

E-LAYER

FIGURE 1. CROSS SECTIONAL VIEW OF CUSPTRON

30

NSWC TR 82-530

II N=68' s-=0

77 a lr/1 2

01I1 2 345 6

Rc/Ra

FIGURE 2. PLOTS OF THE PARAMETERS 77 (SOLID CURVES) AND (DASHED CURVES) VERSUSRATIO RcIRa [OBTAINED FROM EQ. (45)] FOR N 6, s =0, a 7r/12 AND THREE LOWESTRADIAL MODES

31

. . . . . . . . . . . .

NSWC TR 82-530

4- s =0a r/2N

N=877 N=6

2 N=4

N=2

13 5

FIGURE 3. PLOTS OF PARAMETER 77 VERSUS RATF) R C/R a OBTAIINED FROM EQ. (45) FOR s=0,airf2N, LOWEST RADIAL MODE NUMBER, AND SEVERAL VALUES OF N

32

-

NSWC TR 82-530

4 N=6a=7r/12

s=2

b77

2

s= 3 =O

01 3 5

L'L"L.:RC/R

a

FIGURE 4. PLOTS OF PARAMETER r VERSUS RATIO RcIRa OBTAINED FROM EQ. (45) FOR N-6,a - /12, LOWEST RADIAL MODE NUMBER AND DIFFERENT VALUES OF s

33

I

NSWC TR 82.530

4 7r/32 N=87r/l16 s=0

3 7r/327r/8

12

01 35

Rc/Ra

FIGURE 5. PLOTS OF PARAMETER 17VERSUS RATIO R /R1 OBTAINED FROM EQ. (45) FOR N =8,

s 0, LOWEST RADIAL MODE NUMBER AND SEVERAL VALUES OF c

34

o ,, . . .. ; ., . ° . ,., - . . . - ,. *, . -'--' J,-. .. _ ._. -........ .

NSWC TR 82-530

W 2/C2 k2 = 72/R2a

k

,.

FIGURE 6. THE STRAIGHT LINES w k1zc + 2w AND wA~ kc/O3z INTERSECT AT (wk) =

(Qw-t Qw2C2 2 172R 2 2 /

Z, O ' 0.THE VACUUM WAVEGUIDE MODE W~ (k2 /RaPASSES THROUGH (wo, k) PROVIDED RawcIC

35

NSWC TR 82-530

N=6 Rc/Ra =1.4"':" s =040...R c/c= 0.4433

a c20 a=ir/12

6.4-1 0 6.2 6.6

-20

-40

-60

FIGURE 7. PLOT OF THE GEOMETRIC FACTOR F(w,kb) VERSUS NORMALIZED FREQUENCYO w/c IN EO. (28) FOR N - 6, Q = 6, s - 0, a - ir/12, Rc/Ra - 1.4 AND Raw = 0.4433

CORRESPONDING TO THE GRAZING CONDITION FOR ,' " 3.2

3

36

NSWC TR 82-530

*1"

(a) v=0.002 A =0.04,

::: 'iR 8 /Ra 1.4

0-

x ~ ~1.3I/

li \ 11.2 1 I

6.0 6.2 6.4 6.6

FIGURE 8. PLOTS OF (a) NORMALIZED GROWTH RATE ni/wc AND DOPPLER-SHIFTED REALOSCILLATION FREQUENCY Sr/w VERSUS w/w OBTAINED FROM EQ. (47) FORELECTRON BEAM PARAMETERS v = 0.002. L = 0.04, 00 = 0.4, Oz = 0.2, FIELD INDEXn = 0 AND THE GRAZING CONDITIONS CORRESPONDING TO EQ. (53), ANDPARAMETERS OTHERWISE IDENTICAL TO FIGURE 7

37I

a . * -. .S

NSWC TR 82-530

(b) V=0.002, A=0.04,g°= 0 "4 ' f3z =0.2,

12-

x -

3.

oc 6

06.0 6.2 6.4 6.6

FIGURE 8. (CONTINUED)

38

- -

NSWC TR 82-530

(a) Rc/Ra= 1.4, V =0.002,Rawc/C=0.4433, A=0.04,

n=0.2 e 0.4 , /3z =0.216

n n= 0. 1

0 n-O-i:8

6.0 6.2 6.4 6.6wcC

FIGURE 9. PLOTS OF (a) NORMALIZED GROWTH RATE 1i/wc AND (b) DOPPLER-SHIFTED REAL

OSCILLATION FREQUENCY 2r/wc VERSUS w/w/c OBTAINED FROM EQ. (47) FOR

Rc/Ra = 1.4, Ra w/C = 0.4433, SEVERAL DIFFERENT VALUES OF THE FIELD INDEX

n, AND PARAMETERS OTHERWISE IDENTICAL TO FIGURE 8

39

NSWC TR 82-530

24- (b) Rc/Ra =1.4, P=0.002,Ra w /c=O.4433, A=0.041

0.~ 4,'~ I = 0.2

* 016

01

04

NSWC TR 82-530

(a) =0.002,, A=0.04,0 =0.4, fPZ=0.2, n=O

00I!8-

x

c4 0

0 c/i a

-0.5 0 1.25 2.5kc/ c

5.9 6.0 6.25 6.5Wb/cOc

FIGURE 10. PLOTS OF (a) NORMALIZED GROWTH RATE fi/wc AND (b) DOPPLER-SHIFTED

REAL FREQUENCY Qr/wc VERSUS NORMALIZED AXIAL WAVE NUMBER kc/w c

OBTAINED FROM EQ. (48) FOR THE PARAMETERS IDENTICAL TO FIGURE 8

41

NSWC TR 82-530

6

4 f 0 0 0 4 , 0,=0.2, n=o

0

-0.5 0 1.25 2.5kc/cuw

59 6.0 6.25 6.5

b0c

FIGURE 10. (CONTINUED)

42

"Wa

NSWC TR 82-530

(a) v=0.002, A=0.04,10- go =0.96, 8z=0.2

* I

" a

5- Rc/Ra=1

Ii1.25

0 1.5

6 6.4 6.8

FIGURE 11. PLOTS OF (a) NORMALIZED GROWTH RATE 2i/c AND (b) DOPPLER-SHIFTED REALFREQUENCY *r/wc VERSUS w/w c FOR go = 0.96, THE GRAZING CONDITIONS

CORRESPONDING TO EQ. (53). AND PARAMETERS OTHERWISE IDENTICALTO FIGURE 8

.4

NSWC TR 82-530

* . 0 () V 0.002, A 0.04,0 =0.96, #3 =0.2

* . 0

: . x

5 3 5a-

1.25_ 1.5

06

6 6.4 6.8- /Icwoc

FIGURE 11. (CONTINUED)

44i4

I

NSWC TR 82-530

" N=24, 1=25,s =O, Rc/Ra = 1.1,

3 RaCwc/C = 1.069 '/ \ v 0.002,

o A=0.04,0 0=0.96~20-

025

25 26 27

S.4

FIGURE 12. PLOT OF THE NORMALIZED GROWTH RATE Qi/w c VERSUS wl/c OBTAINED FROM

EQS. (47) AND (58) FOR 0 = 0.96, N - 24, Q - 25, Rc/Ra . 1.1, Rawc/c = 1.069, a = /2 ,

AND PARAMETERS OTHERWISE IDENTICAL TO FIGURE 8

45

I

I.

NSWC TR 82-530

REFERENCES

1. V. A. Flyagin, A. V. Gaponov, M. I. Petelin, and V. K. Yulpatov,IEEE Trans. Microwave Theory Tech. MTT-25, 514 (1977).

2. J. L. Hirschfield and V. L. Granatstein, IEEE Trans. MicrowaveTheory Tech. MTT-25, 528 (1977).

3. H. S. Uhm, R. C. Davidson and K. R. Chu, Phys. Fluids 21, 1877(1978).

4. A. Palevsky and G. Bekefi, Phys. Fluids 22, 986 (1979).

5. D. A. G. Deacon, L. R. Elias, J. M. M. Madey, G. J. Ramian,H. A. Schwettman and T. I. Smith, Phys. Rev. Lett. 38, 897 (1977)

6. H. S. Uhm and R. C. Davidson, Phys. Fluids 24, 1541 (1981).

7. H. S. Uhm and R. C. Davidson, J. Appl. Phys. 49, 593 (1978).

8. W. W. Destler, D. W. Hudgings, M. J. Rhee, S. Kawasaki, andV. L. Granatstein, J. Appl. Phys. 48, 3291 (1977).

9. P. Sprangle, J. Appl. Phys. 47, 2935 (1978).

10. W. W. Destler, R. L. Weiler, and C. D. Strittler, Appl. Phys. Lett.38, 570 (1981).

11. M. J. Rhee and W. W. Destler, Phys. Fluids 17, 1574 (1974).

12. H. S. Uhm and R. C. Davidson, Phys. Fluids 21, 265 (1978).

13. H. S. Uhm and R. C. Davidson, Phys. Fluids 20, 771 (1977).

14. N. M. Kroll and W. E. Lamb, Jr., J. Appl. Phys. 19, 166 (1948).

15. G. B. Collins, Microwave Magnetrons, (New York, McGraw-Hill Co.,1948) Chap. 2.

16. P. Sprangle and W. M. Manheimer, Phys. Fluids 18, 224 (1975).

46

NSWC TR 82-530

APPENDIX A

Magnetic Wave Admittances

In this section, we obtain expressions for the wave admittances at

the boundaries of an E-layer in a magnetron-type conductor. Shown in

Fig. A-I is one of the resonators in a magnetron-type conductor. Obviously,

the electromagnetic field in the resonator is required to satisfy the

boundary conditions

r (r R) z(r R) = 0(r R)= 0,

(A.l)

B(e= ±a) E (e ±a) = (e ±M) -0e z r

The previous study1 5'16 has shown that the lowest azimuthal mode in the

resonator dominates the wave and beam interaction. Moveover, the

theoretical analysis is considerably simplified when the electromagnetic

field in the resonator is represented by the lowest azimuthal mode.

Therefore, after a careful examination of the boundary conditions in

Eq. (A.1), it is shown that the electric and magnetic fields in the.4

resonator are expressed as

B b [Jo(pr) - Jl(C)N0 (pr)/Nl(4)] exp {ikz} , (A.2)

E= ib(W/cp)[Jl(pr) - JI( )NI(pr)/NI( )] exp {ikz} , (A.3)

B = - (kc/w)E, (A.4)

A-I

NSWC TR 82-530

for the TE mode and

E B -E 0 (A.5)r e z

for the TM mode. In Eq. (A.2), J (x) and N (x) are Bessel functions of2. 2.the first and second kind, respectively, C - (W2 /C 2 - k2 R and b is

a constant. It is also emphasized that Eq. (A.5) is valid provided

W2/c2 _ k2 # 0. Making use of Eq. (A.5), the electric wave admittances

have been obtained in Eq. (34).

In the interaction region 0 4 r < Ra, the perturbed axial mangetic

field is expressed as

Sbin Jn(pr) exp { i(n8 + kz) }

on

- .- O <r <R 0B 4 r (A.6)

z

pbn Zn(pr) exp {i(ne + kz)}n nZ

R <r R0 a

where the function Zn (pr) is defined by

Jn(Pr) ,n

Z (pr) - (A.7)n

J2Z(pr) + G2IN2I(pr) , n .

and constant GZ is determined from the proper boundary condition. From

Eq. (A.3), we obtain the azimuthal electric field

A-20e

,I I _* I ., , ,.1*, ,!. . .. ,I il jZ . .... ! -. ,!

NSWC TR 82-530

-E(Ra + ) -i(w/cp) [Jl10) - J1 (r)Nl(n)/Nl( )1 exp { ikz } , (A.8)

for

Lr a < < 2-- + .+N N

and Ee(R ) = 0, otherwise, where N is the number of the resonators,

q0, 1, .. N -1, and *( )denotes 60 * (R.j + 6). It is also shownb +

from Eq. (A.6) that

e(Ra-) - -i(w/cp) 1 b Z (n) exp {i(ne + kz)} (A.9)a n nn

where the prime (') denotes (d/dx)Z n(x).

The constants bn are determined by the average field matching15 16

f dE(Ra) exp {-in}': f o

N-1 ((2nq/N)+ a

-E C R a+) d I d expf-mnej (A.10)q a 0 fJ(2irqf/N) -a

which gives

• 1

for n - s + inN, and b - 0 otherwise, where m is any integer and s = 0,n

1, ... N - 1. Substituting Eq. (A.l1)into Eq. (A.6), the axial magnetic

field in R0 < r < R is expressed as0i a

A-3

• 4- '

NSWC TR 82-530

Nc [F- J ( ") - 1l(

Br -b-I-i-- Nl(n) 1L.P (A.12)

CO Z(pr) (snnm-x Z n / exp {i(ne + kz)}MEi -- z W(n) n I

Resonance is determined by the requirement that the average value of

the axial magnetic field in the interaction region match that in the

resonator, i.e.,

(2wq/N) + a

deB z(R ) - 2B z(R ) . (A.13)

(2irq/N) - a

Substituting Eqs. (A.2) and (A.12) into Eq. (A.13), and carrying out a

tedious but straightforward algebra, we determine the constant G

J£ (n)D(w,k)

G (D,k) - (A.14)'..N£ ()F(,k)

where the vacuum dispersion function D(w,k) is defined by

' n(n) in na) 2

D(wk)m = -J (n )

(A. 15)

" J(n)NI() - Jl(C)N0W: Na JI1()NIl() - Jl (n)N l()

and the function F(w,k) is given by

A

bI,- A-4

.I

NSWC TR 82-530

F(w,k) aD(w,k) -___ ic J(A.16)(ni) N2 W>

The perturbed axial magnetic field with azimuthal harmonic number

2 is given by

{.~r bl 2 J 2I(pr) , 0 4 r < RO ,

]r b, [i. (pr) + G2IN 2I(pr)] , < r -4 R (

from Eq. (A.6). Making use of Eq. (A.17) and the definitions of the

magnetic wave admittances in Eq. (12), it can be shown that sum of the

magnetic wave admittances at r - R0 is expressed as

2b_ +. = I) (A.18)I I

b +b+ J- ()+GN {

where R0/R c =nRR = (w2/c2 - k2) R0.

A-5

NSWC TR 82-530

I2.a

FIGURE A-1. CONFIGURATION OF A RESONATOR

A-6

NSWC TR 82-530

DISTRIBUTION

Copies Conies

Dr. Richard E. Aamodt 1 Dr. J. BaylessScience Application Inc. DARPA Attn: DEO934 Pearl St. Suite A 1400 Wilson Blvd.Boulder, CO 80302 Arlington, VA 22209

Dr. Saeyoung Ahn 1 Dr. Robert BehringerCode 5205 ONRNaval-Research Lab. 1030 E. GreenWashington, D.C. 20375 Pasadena, CA 91106

Dr. Wahab A. Ali 1 Prof. George BekefiNaval Research Lab. Code 4700 Bldg. 36-213, MIT4555 Overlook Ave., S.W. 77 Massachusetts Ave.Washington, D.C. 20375 Cambridge, MA 02139

Dr. Donald Arnush 1 Dr. Gregory BenfordTRW, Plasma Physics Dept. Physics DepartmentR1/107 University of California1 Space Park Irvine, CA 92717Rodondo Beach, CA 90278

Dr. Jim BenfordDr. J.M. Baird 1 Physics International Co.Code 4740 (B-K Dynamics) 2700 Merced St.Naval Research Lab. San Leandro, CA 94577

* 4555 Overlook Ave., S.W.Washington, D.C. 20375 Dr. Kenneth D. Bergeron

Plasma Theory Div. - 5241Dr. William A. Barletta 1 Sandia LaboratoriesLawrence Livermore National Lab. Albuquerque, NM 87115L-321University of California Dr. T. BerlincourtLivermore, CA 94550 Office of Naval Research

Department of the NavyDr. L. Barnett 1 Arlington, VA 22217Code 4740 (B-K Dynamics)Naval Research Lab.4555 Overlook Ave., S.W.Washington, D.C. 20375

I

~(1)

i

NSWC TR 82-530

DISTRIBUTION (Cont.)

Copies Copies

Dr. I. B. Bernstein 1 Dr. Neal CarronYale University Mission Research Corp.Mason Laboratory 735 State Street400 Temple Street Santa Barbara, CA 93102New Haven, CT 06520

Dr. Frank ChambersDr. 0. Book Lawrence Livermore National Lab.Code 4040 L-321Naval Research Laboratory University of California

* Washington, D.C. 20375 Livermore, CA 94550

Dr. Jay Boris Prof. F. ChenNaval Research Lab. Code 4040 Dept. of E. E.4555 Overlook Ave., S.W. UCLAWashington, D.C. 20375 Los Angeles, CA 90024

Dr. Howard E. Brandt Dr. M. CaponiHarry Diamond Labs TRW Advance Tech. Lab.2800 Powder Mill Road I Space ParkAdelph, MD 20783 Redondo Beach, CA 90278

Dr. R. BriggsLawrence Livermore Lab. Dr. K. R. ChuP.O. Box 808 Naval Research Lab. Code 4740Livermore, CA 94550 4555 Overlook Ave., S.W.

Washington, D.C. 20375Dr. K. BruecknerLa Jolla Institute Dr. Timothy CoffeyP.O. Box 1434 Naval Research Lab.La Jolla, CA 92038 4555 Overlook Ave., S.W.

Washington, D.C. 20375

Dr. Herbert L. BuchananLawrence Livermore National Lab Dr. W. J. CondellL-321 Office of Naval Research Code 421University of California Department of the NavyLivermore, CA 94550 Arlington, VA 22217

Dr. K. J. Button 1 Dr. G. CoopersteinMassachusetts Institute of Technology Naval Research Lab.Francis Bitter National Magnet Washington, D.C. 20375

LaboratoryCambridge, MA 02139 Dr. Edward Cornet

W.J. Schafer Associates, Inc.1901 North Fort Myer Dr.Arlington, VA 22209

(2)

a

NSWC TR 82-530

DISTRIBUTION (Cont.)

Copies Copies

- Prof. R. Davidson 1 Dr. D. EccleshallPlasma Fusion Center U.S. Army Ballistic Research Lab.Massachusetts Inst. of Technology Aberdeen Proving Ground, MD 21005Cambridge, MA 02139

Dr. Barbara EpsteinDr. J. Dawson Sandia LaboratoriesDept. of Physics Albuquerque, NM 87185UCLALos Angeles, CA 90024 Dr. A. Fisher

Physics Dept.Dr. W. Destlar University of CaliforniaDept. of Electrical Engineering Irvine, CA 92664

• .University of MarylandCollege Park, MD 20742 Dr. R. J. Faehl

Los Alamos Scientific Lab.Prof. P. Diament Los Alamos, NM 87544Columbia UniversityDept. of Electrical Engineering Dr. Leon FeinsteinNew York, NY 10027 Science Applications, Inc.

5 Palo Alto SquareProf. W. Doggett 1 Palo Alto, CA 94304NC State UniversityP.O. Box 5342 Dr. Franklin FelberRaleigh, NC 27650 Western Research Corporation

8616 Comierce Ave.

Dr. H. Dreicer 1 San Diego, CA 92121Plasma Physics DivisionLos Alamos Scientific Lab. Dr. Richard FernslorLos Alamos, NM 87544 Code 4770

Naval Research Lab.Dr. A. Drobot 4555 Overlook Ave., S.W.Naval Research Laboratory Washington, D.C. 20375Code 4790 (SAl)Washington, D.C. 30275 Dr. Thomas Fessenden

Lawrence Livermore National Lab.Prof. W. E. Drummond L-321Austin Research Associates University of California1901 Rutland Drive Livermore, CA 94550Austin, TX 78758

Prof. H. H. Fleischmann I1

Lab. for Plasma Studies andSchool of Applied andEngr. Physics

Cornell UniversityIthaca, N'Y 14850

(3)

*-

NSWC TR 82-530

DISTRIBUTICN (Cont.)

Copies Copies

Dr. Robert Fossum, Director 1 Dr. Robert GreigDARPA Naval Research Lab. (Code 4763)1400 Wilson Boulevard 4555 Overlook Ave., S.W.Arlington, VA 22209 Washington, D.C. 20375

Dr. H. Freund 1 Dr. J.U. GuilloryNaval Research Lab. Code 4790 Jaycor4555 Overlook Ave., S.W. 20050 Whiting St. Suite 500Washington, D.C. 20375 Alexandria, VA 22304

Dr. 1M. Friedman 1 Dr. Irving Haber* Code 4700.1 Code 4790

Naval Research Laboratory Naval Research Lab.Washington, D.C. 20375 Washington, D.C. 20375

Dr. B. Godfrey Prof. D. HammerMission Res. Corp. Lab. of Plasma Studies1400 San Mateo Blvd, S.E. Cornell UniversitySuite A Ithaca, NY 14850

• .Albuquerque, NM 87108Dr. Robert Hill

Dr. T. Godlove 1 Physics Division, #341Office of Inertial Fusion National Science FoundationU.S. Department of Energy Washington, D.C. 20550Washington, D.C. 20545

Dr. J. L. HirshfieldDr. Jeffry Golden Yale UniversityNaval Research Laboratory Mason Laboratory

* Washington, D.C. 20375 400 Terple StreetNew Haven, CT 06520

Dr. S. GoldsteinJaycor (Code 4770) Dr. Richard HubbardNaval Research Lab. Code 4790 (Jaycor)Washington, D.C. 20375 Naval Research Lab.

4555 Overlook Ave., S.W.Dr. Victor Granatstein Washington, D.C. 20375Naval Research Lab.Washington, D.C. 20375 Dr. Bertram Hui

Naval Research Lab. Code 4790Dr. S. Graybill 4555 Overlook Ave., S.W.

, Harry Diamond Lab. Washington, D.C. 203752800 Powder Mill Rd.Adelphi, MD 20783 Dr. S. Humphries

Sandia Laboratories

Dr. Michael Greenspan Albuquerque, NM 87115McDonnell Douglas Corp.St. Louis, MO 63166

(4)

NSWC TR 82-530

DISTRIBUTION (Cont.)

Copies Copies

Dr. Robert Johnston 1 Dr. Kwang Je KimScience Applications, Inc. Bldg. 64 #223

. 5 Palo Alto Square A & FR Div.Palo Alto, CA 94304 Lawrence Berkeley Lab.

D H d rBerkeley, CA 94720'" Dr. Howard Jory

Varian Associates, Bldg. 1 Dr. Jin Joong Kim611 Hansen Way North Carolina State UniversityPalo Alto, CA 94303 Raleigh, NC 27607

* Dr. Glenn Joyce 1 Prof. N. M. Kroll* Code 4790 La Jolla Institutes

Naval Research Lab. P.O. Box 14344555 Overlook Ave., S.W La Jolla, CA 92038

. Washington, D.C. 20375

Dr. M. Lampe* Dr. Sel-g Kainer 1 Naval Research Lab. Code 4790

Naval Research Lab. 4555 Overlook Ave., S.W.

4555 Overlook Ave., S.W.Washington, D.C. 203752075

Dr. L. J. Laslet1Dr. C.A. Kapetanakos Lawrence Berkeley Lab.Plasma Physics Division 1 Cyclotron Road

* Naval Research Laboratory Berkeley, CA 96720Washington, D.C. 20375

Dr. Y. Y. LauDr. Denis Keefe Naval Research Lab. Code 4740 (SAI)

• Lawrence Beskeley Lab. 4555 Overlook Ave., S.W.1 Cyclotron Road Washington, D.C. 20375Berkeley, CA 94720

Dr. Douglas Keeley Dr. Glen R. Lambertson

* Science Applications, Inc. Lawrence Berkeley Lab.

" 5 Palo Alto Square 1 Cyclotron Road, Bldg. 47

Palo Alto, CA 94304 Berkeley, CA. 94720

Dr. Hogil Kim Dr. J. Carl LeaderA & FR Div. McDonnell Douglas Corp.Lawrence Berkeley Lab. Box 516Berkeley, CA 94720 St. Louis, MU 63166

Dr. Hong Chul KimA & FR Div.Laurence Berkeley Lab.Berkeley, CA 94720

(5)

NSWC TR 82-530

DISTRIBUTION (Cont.)

Copies Copies

Dr. W. M. Manheimer 1 Dr. James MarkNaval Research Lab. L-477Cd470Lawrence Livermore Lab,.:'.Code 4790

4555 Overlook Ave., S. W. Livermore, CA 94550Washington, D. C. 20375

Dr. Edward P. Lee Dr. Jon A. MasamitsuLawrence Livermore National Lab Lawrence Livermore National LabL-321 L-321University of California University of CaliforniaLivermore, CA 94550 Livermore, CA 94550

Dr. Ray Lemke Dr. Bruce R. MillerAir Force Weapons Lab Div. 5246

* Kirtland Air Force Base Sandia Laboratories- Albuquerque, NM 87117 Albuquerque, NF 87115

Dr. Anthony T. Lin 1 Dr. Melvin MonthUniversity of California Department of Energy

- Los Angeles, CA 90024 High Energy PhysicsWashington, D.C. 20545

Dr. C. S. Liu 1Dept. of Physics Dr. Philip MortonUniversity of Maryland Stanford Linear Accelerator Center

* College Park, MD 20742 P.O. Box 4349Stanford, CA 94305

Dr. Tom Lockner 1* Sandia Laboratories Dr. Don Murphy

Albuquerque, NM 87115 Naval Research Lab4555 Overlook Ave., S.W.

Dr. Conrad Longmire Washington, D.C. 20375Mission Research Corp.735 State Street Dr. Won NamkungSanta Barbara, CA 93102 E. E. Department

University of MarylandProf. R. V. Lovelace 1 College Park, MD 20742School of Applied and Eng. PhysicsCornell University Prof. J. Nation

. Ithaca, NY 14853 Lab of Plasma StudiesCornell University

Dr. Joseph A. Mangano Ithaca, NY 14850' DARPA

1400 Wilson Blvd. Dr. V. Kelvin NeilArlington, VA 22209 Lawrence Livermore Nat'l Lab

P.O. Box 608, L-321Livermore, CA 94550

(

(6)

NSWC TR 82-530

DISTRIBLTION (Cont.)

Copies Covies

Dr. Barry Newberger Dr. R. PostMail Stop 608 Lawrence Livermore LabLos Alamos National Lab University of CaliforniaLos Alamos, NM 87544 P.O. Box 808

Livermore, CA 94550Dr. C. L. Olson 1Sandia Lab Dr. D. S. PronoAlbuquerque, NM 87115 Lawrence Livermore Lab

P.O. Box 808

Livermore, CA 94550Dr. Edward OttDept. of Physics Dr. S. PutnamUniversity of Maryland Physics Internal Co.College Park, MD 20742 2700 Merced St.

San Leandro, CA 94577Dr. Peter PalmadessoBldg AO #07 Dr. Sid PutnamNaval Research LabWashington, D.C. 20375 Pulse Sciences, Inc.

1615 Broadway, Suite 610

Dr. Ron Parkinson Oakland, CA 94612

Science Applications, Inc. Dr. Michael1200 Prospect St.RaegP.O. Box 2351 Naval Research Lab Code 4763La Jolla, CA 92038 4555 Overlook Ave., S.W.a JWashington,

D.C. 20375

Dr. Richard Patrick 1AVCO - Everett Research Lab, Inc. Dr. M. e. Read2385Revee Bach kwyNaval Research Lab Code 47402385 Revere Beach Pkwy 4555 Overlook Ave., S.W.Everett, MA 02149 Washington, D.C. 20375

Dr. Robert PechacekNaval Research Lab Code 4763 Prof. M. Reiser

4555 Overlook Ave., S.W. Dept. of Physics & Astronomy

Washington, D.C. 20375 University of MarylandCollege Park, YD 20742

Dr. Sam PennerNational Bureau of Standards Dr. M. E. RensinkBldg. 245 Lawrence Livermore Lab

Washington, D.C. 20234 P.O. Box 808Livermore, CA 94550

Dr. Michael Picone 1Naval Research Lab Dr. Doon-thong hee&555 OvrokBeSW . E. Department4555Overlook A. , S.W. University of MarylandWashington, D.C. 20375 College Park, MD 20742

(7)

NSWC TR 82-530

DISTRIBUTION (Cont.)

Copies Copies

Dr. C. W. Roberson 1 Dr. William SharpNaval Research Lab Code 4740 Naval Research Lab Code 4790 (SAI)4555 Overlook Ave., S.W. 4555 Overlook Ave., S.W.Washington, D.C. 20375 Washington, D.C. 20375

Dr. J. A. Rome Dr. D. StrawOak Ridge National Lab AFnLOak Ridge, TN 37850 Kirtland AFB, NM 87117

Dr. Marshall N. RosenbluthUniversity of Texas at Austin Dr. John SiambisInst. for Fusion Studies Science Applications, Inc.RLM 11.218 5 Palo Alto SquareAustin, TX 78712 Palo Alto, CA 94304

Prof. Norman Rostoker 1 Dr. J. S. SilversteinDept. of Physics Code 4740 (HDL)University of California Naval Research LabIrvine, CA 92664 4555 Overlook Ave., S.W.

Washington, D.C. 20375

Dr. C. F. SharnNaval Sea Systems Command Dr. M. Lee $loan

Department of the Navy Austin Research Associates, Inc.

Washington, D. C. 20363 1901 Rutland DriveAustin, Texas 78758

Prof. S. T Schlesinger 1Columbia University Dr. L. SmithDept. of Electrical Engineering Lawrence Berkeley LabNew York, NY 10027 1 Cyclotron Road

Berkeley, CA 94770Prof. George SchmidtPhysics Dept. Dr. Joel A. SnowStevens Institute of Technology Senior Technical AdvisorHoboken, NJ 07030 Office of Energy Research

4U.S. Department of Energy, M.S. E084Dr. Andrew M. Sessler Washington, D.C. 20585Lawrence Berkeley LabBerkeley, CA 94720 Dr. Philip Sprangle

Naval Research Lab Code 4790Dr. J. D. Sethian 1 4555 Overlook Ave., S.W.Naval Research Lab Code 4762 Washington, D.C. 20375Washington, D.C. 20375

(8)

NSWC TR 82-530

DISTRIBUTION (Cont.)

Copies Copies

Dr. Doug Strickland Dr. John E. WalshNaval Research Lab Code 4790 (Beers) Department of Physics

. 4555 Overlook Ave., S W. Dartmouth CollegeWashington, D.C. 20375 Hanover, NH 03755

Dr. Charles D. Striffler Dr. WasneskiE. E. Dept. Naval Air Systems CommandUniv. of Maryland Department of the NavyCollege Park, MD 20742 Washington, D.C. 20350

Prof. R. SudanLab of Plasma Studies Dr. N. R. Vanderplaats

Cornell University Naval Research Laboratory

Ithaca, NY 14850 Code 68054555 Overlook Ave., S. W.

Dr. C. M. Tang Washington, D.C. 20375Naval -Research Lab Code 47904555 Overlook Ave., S.W.Washington, D.C. 20375 Lt. Col. W. Wnitaker

Defense Advanced Research ProjectsDr. R. Temkin 1 AgencyPlasma Fusion Center 1400 Wilson BoulevardMassachusetts Institute of Technology Arlington, VA 22209Cambridge, MA 02139

Dr. Mark WilsonDr. Lester E. Thode I National Bureau of StandardsMail Stop 608 Gaithersburg, MD 20760Los Alamos National LabLos Alamos, NM 87544 Dr. Gerold Yonas

Sandia Lab

Dr. James R. Thompson Albuquerque, NM 87115

Austin Research Associate, Inc.1901 Rutland Drive Dr. Simon S. YuAustin, Texas 78758 Lawrence Livermore National Lab

L-321.4 Dr. D. Tidman University of California

Jaycor Livermore, CA 94550205 S. Whiting St.Alexandria, VA 22304 Defense Technical Information

CenterDr. A. W. Trivelpiece Cameron Station

4 Science Applications, Inc. Alexandria, VA 22314 12- San Diego, CA 92123S e9Internal distribution:

RR04R40

(9)

.7 -1.

NSWC TR 82-530

DISTRIBUTION (Cont.)

Copies

Internal Distribution (Cont.)

R401 1R43 (C. W. Lufcy) 1R44 (T. F. Zien) IR45 (H. R. Riedl) 1R13 (J. Forbes)R41 (P. 0. Hesse) 1R41 (R. Cawley) 1R41 (M. H. Cha) 1R41 (H. C. Cben) 1R41 (J. Y. Choe) 1R41 (R. Fiorito) 1R41 (0. F. Goktepe) 1R41 (M. J. Rhee) 1R41 (D. W. Rule) 1R41 (Y. C. Song) 1R41 (H. S. Uhm) 1R43 (A. D. Krall) 1F 1F14 (H, C. Coward) 1F40 (,. F. Cavanagh) 1FIO (K. C. Baile) 1F46 (D. G. Kirkpatrick) 1F34 (R. A. Smith) 1F34 (E. Nolting) 1F34 (V. L. Kenyon) 1F04 (M. F. Rose) IF34 (F. Sazama) 1N14 (R. Biegalski) 1E431 9E432 3E35 1

(10)

I.

j

II