Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a...

22
1)llc FILE COPY Naval Research Laboratory Wuhton, DC 20S.80oo NRL Memorandum Report 6295 AD-A200 399 Relativistic Focusing and Beat Wave Phase Velocity Control in the Plasma Beat Wave Accelerator E. ESAREY AND A. TING Berkeley Research Associates P.O. Box 852 Springfield, VA 22150 P. SPRANGLE Plasma Theory Branch Plasma Physics Division DTIC - ELECTE OCT 25 19881 September 22, 1988 DcJ Approved for public release; distribution unlimited. J.../

Transcript of Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a...

Page 1: Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a plasma with the resulting beat wave phase velocity equal to the speed of light. Focusing

1)llc FILE COPYNaval Research LaboratoryWuhton, DC 20S.80oo

NRL Memorandum Report 6295

AD-A200 399

Relativistic Focusing and Beat Wave Phase VelocityControl in the Plasma Beat Wave Accelerator

E. ESAREY AND A. TING

Berkeley Research AssociatesP.O. Box 852

Springfield, VA 22150

P. SPRANGLE

Plasma Theory BranchPlasma Physics Division

DTIC -

ELECTE

OCT 25 19881 September 22, 1988DcJ

Approved for public release; distribution unlimited.

J.../

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SECURIty CLASSIFICATION OF THIS PAGE

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la REPORT SECURITY CLASSIFICATION 1b RESTRICTIVE MARKINGS

UNCLASSIFIED ______________________2a. SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION/ AVAILABILITY OF REPORT

Approved for public release; distribution2b. DECLASSIFICATION / DOWNGRADING SCHEDULE unlimited.

4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5 MONITORING ORGANIZATION REPORT NLV-BERIS,

NRL Memorandum Report 62956a. NAME OF PERFORMING ORGANIZATION 6b OFFICE SYMBOL 7a NAME OF MONITORING ORGANiZA' ONI (if applicable)Naval Research Laboratory Code 4790

6c. ADDRESS (City, State, and ZIP Code) 7b ADDRtESS (City, State arid ZIP Code)

Washington, DC 20375-5000

8a. NAME OF FUNDING/ISPONSORING J8b OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT DENTIF-CAT ON N ;,MBE-'ORGANIZATION (if applicablIe)

U.S._Department ofEnergy _______ _________ ____________

Sc. ADDRESS (City. State, and ZIP Code) 10 SOURCE OF FUNDING NUMBERS

*PROGRAM IPROET 31TASK VVORK uNITWsigoDC 20545 ELEMENT NO -983 NO IACCESSiON NO

WasingonDOE I A00411. TITLE (include Socursty Clasification)Relativiatic Focusing and Beat Wave Phase Velocity Control in the PlasmaBeat Wave Accelerator

12. PERSONAL AUTHOR(S)Esarey,* E., Ting,* A. and Sprangle, P.

13a. TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year. Month. Day) isAC ~Interim FROM TO ___ 1988 September 22 2 4

16 SUPPLEMENTARY NOTATION

*Berkeley Research Assoc., P.O. Box 852, Springfield, VA 22150

17. COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block nufller)

FIELD GROUP SUB-GROUP

19 ABSTRACT (Continue on reverse if necessary and identify by block number)

7 Relativistic focusing allows two colinear short pulse radiation beams, provided they are of sufficiently

high power, to propagate through a plasma without diffracting. By further accounting for finite radial beamgeometry, it is possible for the phase velocity of the radiation beat (ponderomotive) wave to equal the speedof light. This removes one of the limiting factors, phase detuning between the accelerated electrons and thebeat wave, in determining the maximum energy gain in the plasma beat wave accelerator.

00 Form 1473. JUN 86 P'revious editions ire obsoiete . -.

S/N I)I)T.LF..Iti.6or)I

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CONTENTS

INTRODUCTION ................................................. I

ANALYSIS OF RADIATION FOCUSING AND BEAT WAVE PHASEVELOCITY CONTROL ....................................................................... 3

DISCUSSION.................................................................................. 6

ACKNOWLEDGEMENTS ................................. I................................ 6

REFERENCES ................................................................................ 7

DISTRIBU77ON LISTS.......................................................................13

ACC~On, tny,-

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RELATIVISTIC FOCUSING AND BEAT WAVE PHASE VELOCITYCONTROL IN THE PLASMA BEAT WAVE ACCELERATOR

Introduction

Recently there has been much interest in plasma based accelerator schemes, such as

the plasma beat wave accelerator (PBWA),' - 3 for producing ultra-high energy electrons.

This has led to a renewed interest in the study oi the propagation of intense radiation beams

through a plasma.4 - " In the PBWA two colinear radiation beams of frequencies w1 ,W2are incident on a umiorm plasma. By appropriately choosing the difference il lhe i a.e

frequencies to be equal to the electron plasma frequency wp, Aw = wl - ,2 = wp, where

wP/w? << 1, it is possible for the radiation beat wave to resonantly drive large amplitude

electron plasma waves. In the ideal wave breaking limit,' 4 the maximum accelerating

electric field Em is given by E, (mtc 2 /e)wp/c " .97. ,yi eV/cm where np is the plasma

density in cm - . For example, np = 1.6 x 1016 cm - 3 gives Em _ 120 MeV/cm which

implies that an electron could be accelerated to 1.2 TeV in 100 meters.

To realize such an acceleration scheme it is necessary that i) the radiation beallis

propagate at high intensity over distances large compared to the Rayleigh length ZR =

r /2c, where r, is the radiation spot size, and that ii) phase resonance between the

accelerating electrons and the plasma wave be maintained over an equally large distance.

In vacuum, radiation diffracts over distances on the order of ZR, which can be relatively

short. Hence, in order to maintain high intensity beams it is necessary to rely on focusing

enhancement from the plasma. In the PBWA the phase velocity of the plasma wave isequal to the phase velocity of the radiation beat wave which, in the 1-D limit, is given by

vp/c = Aw/Ak = 1 - w'/2w', where Ak = k, - k2 is the difference in th*. wave numbers

of the two beams. Since the velocity of an ultra-relativistic electron is approximately the

speed of light, the electrons out run the plasma wave and become "detuned" in a length"5

Ld A W A /w., where A1, = 21rc/wp. For wl/wp = 25 and np = 1.6 x 1016 cm - 3 , this gives

L," 16 cm and a maximum electron energy gain of AE - ELd - 2 GeV. In order to

increase the energy gain beyond this detuning limit, it is necessary to increase the phase

velocity of the plasma beat wave.

This paper addresses the two points mentioned above concerning the realization of

the PBWA. As is shown below, matched beam solutions are possible in which the two

radiation beams propagate with constant spot sizes provided the radiation is of stuffici ntlv

high power. This allows the radiation beams to propagate over distances larger than

the Rayleigh length while maintaining their high intensities. For example, two radiation

beams with equal spot sizes are matched provided t.h, power in each beam is P = P, 3.

Manuscrpt approved July 7, 1988.

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IEEEI !U EE 11 I I i I - i -- I

where P, : 17 x 10'w 2/wP W is the power threshold for relativistic focusing of a single

radiation beam in a plasma. 4" In addition, by including finite radial beam profiles along

with relativistic focusing, the phase velocity of the beat wave can be tuned to the speed

of light. This is accomplished by appropriately choosing the initial spot sizes and powersof the radiation beams. Hence, phase resonance between the electron and beat wave can

be maintained beyond the 1-D detuning length Ld and, consequently, substantially higher

electron energies can be achieved. Figure 1 shows schematically the propagation of two

matched radiation beams through a plasma with the resulting beat wave phase velocityequal to the speed of light.

Focusing of radiation beams in a plasma occurs through the combined effects of rela-

tivistic, ponderomotive and thermal self-focusing.4 -13 Typically these processes occur on

widely separate time scales. Relativistic focusing 4- 8 occurs on the shortest time scale,rR - 1/w, which is the time scale at which the electrons respond to the radiation field.

Ponderomotive focusing -" depends on the expulsion of ions from the radiation chan-

nel and thus occurs on a time scale given roughly by 'rp r./C,, where C, is the ion

acoustic speed. Thermal focusing1'2 ,3 relies on heating of the plasma by the radiation

beam and typically occurs on an even longer scale. This paper is concerned with rela-

tivistic focusing, and hence the analysis is applicable to lasers with pulse lengths TL in

the range rR < -rL < -rp, which is the region of interest for the PBWA. Physically, rela-tivistic focusing arises solely from the relativistic electron quiver velocity, Vq = ca, /-r-L,

in the combined radiation field. Here aj = eA±/mc2 is the normalized radiation vector

potential and 'yj = /T + a2_ is the relativistic gamma factor for an electron in a helically

polarized radiation field. The focusing mechanism for a single beam is that a radiation

profile peaked on axis leads to an index of refraction profile, n 1 - (,p/w)2 /2-y±, which

has a minimum on axis. The radiation beam, therefore, focuses along the axis.' When

the radiation power is greater than the critical power P, for relativistic self-focusing, it is

possible for the envelope of a single radiation beam to propagate at a constant spot size.T

For two colinear beams, however, the situation is more complicated due to the coupling

of one beam to the other through the relativistic gamma factor. The analysis presented

below indicates that matched beam propagation for two beams is only possible for a finite

range of the parameter R, such that 1/(v/8 - 1) < R < 0 - 1. where R = (r,l /r,l )2 is

the square of the ratio of the spot sizes of the two beams. The radiation power required

to obtain matched beam propagation for two beams is near that for a single beani. P.

Control of the beat wave phase velocity is most easily understood by the following

heuristic argument. The finite radial extent of the radiation beams gives rise to a small

2

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effective perpendicular wave number k1 . Here k1 is a function not only of the spot size

but also of the power due to the relativistic focusing effects. The existence of k1 gives

rise to an effective parallel wave number given by kll !:_ (1 - wp/2w2 - c2 k±/2w2 )w/c.

Hence, the parallel phase velocity of the beat wave is now given by vp/c 1_ 1 - w/2w2 +

(k21 - k2 2 )c2 /2wwp. By appropriately choosing the initial spot sizes and powers of the

two radiation beams, it is possible to have the last term in the expression for vp/c cancel

the second term thus providing vp = c.

Analysis of Radiation Focusing and Beat Wave Phase Velocity Control

The analysis starts with the wave equation for the vector potential of the conibin'-d

radiation field, (V2 - c-8 2 /&tI)A± = -(4ir/c)J±, where J± is the transverse current

density. In order to study the effects of relativistic focusing alone, only the current resulting

from the electron quiver motion is needed, J± = -enpVq. The wave equation is then given

by(P 122 --1/2,

( _a±= a. (1 + alI' + ja 2 -L 21aIa 21cosA1/) (1)

where a± = a, + a2 . Throughout the following, a subscript 1 refers to the radiation beam

of frequency wl, and a subscript 2 refers to the radiation beam of frequency w2. The factor

within the square root is the relativistic gamma factor 7±, assuming helically polarized

radiation. Here, A4 = Pj -- t2 is the phase of the beat wave and ('1,2 = kl,-z-W l.t4 2 ,

where 01,2 is the slowly evolving phase of the radiation field.

In order to examine the qualitative diffractive properties of the radiation beams, it is

helpful to consider the index of refraction nl, 2 of each beam. The approximate index of

refraction associated with each beam is obtained in the following manner. First, tile I-D

limit of the left hand side of Eq. (1) is taken, assuming a - exp(i$I). Next, Eq. (1) is

divided by the phase factor exp(iZ) of either beam 1 or 2 and then averaged over a period

of the beat phase Al. In the mildly relativistic limit 1al,212 < 1, the index of refraction

for each beam is given by

n- klc/w = I -(w /2w)(1 - jai 12/2 - lal 2"), (2a)

n2= k 2 c/w 2 = 1 - (w/2w2)(1 - lai 1- a2 12 /2). (2b)

In the above expressions, the first term (the unity) represents vacuum diffraction while the

second term is the contribution from the ambient plasma. The remaining terms represent

focusing from the radiation fields. More specifically, a term proportional to (lai 2 2 la, 2) '2

3

Page 7: Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a plasma with the resulting beat wave phase velocity equal to the speed of light. Focusing

results from the individual contributions of beam 1 and 2 to the relativistic factor 7±,

while the remaining term proportional to jai, 2 12 /2 results from the contribution of the

beat wave to -y. Radiation focusing occurs when cn/or < 0. Hence, the contribution

from the radiation terms to n1 ,2 provide focusing for radiation profiles peaked on axis. For

sufficiently high power, these focusing terms dominate the vacuum diffraction and provide

overall focusing of the radiation beams.

Envelope equations describing the evolution of the spot size r,(z) of each beam are

derived by applying the "source-dependent expansion" (SDE)1'6 17 to Eq. (1). This is

accomplished by expanding the normalized vector potential al,2 for each beam into a

series of Gaussian-Laguerre polynomials and using orthogonality properties to determinetheir coefficients. The SDE differs from the typical vacuum modal expansion in that the

parameters characterizing the Gaussian-Laguerre polynomials, such as the width of the

Gaussian, are functions of z which depend on the "source", i.e. the right hand side of

Eq. (1). Assuming that each beam is adequately described by the lowest order Gaussian

mode a = laoo Iexp[i/3 -(1 - ia)r 2 /rf], then the parameters lao0, 13, a and r, are given

by Iaoo(z)l = aor.o/r,(z), a(z) = (w/4c)dr2/dz along with the following equations:

4r 2 1 -8W 1\+ (3a)t wr~WI(1 + R)

d2 4c 2 [ w( 8W1 1 )]bdz-- s2 _ - 1- W, 1 W+ IR) (3b)Z2 ;22 r32 22(1+ R)2

d 2c 1 + - , 3 4W 2,R(l + 2R)0 11 = - -2- 2' " ' -+ ( 4 a )

dz 2, r., a 4c (2 W (1 +R) 2 /J 4.

2c 1 (3 4rj- wr~ [2 + -p, -, +_ -W 1W2 2 4c 2 (2 W1+ R 2)] (4b)

where the mildly relativistic limit was taken, Iai, 212 < 1. Physically, W = (Wpaoro/4c)2 =

P/Pc where P is the power in one of the beams and P, is the critical power necessary

for relativistic focusing of a single beam.7 Here aq and ro are the initial amplitude ofthe vector potential on axis and the initial 3pot size of each beam. The parameter a is

related to the curvature of the radiation wavefront and the parameter 3 is important in

that it represents a correction to the parallel wave number on axis k1 = ;/c + d3/dz. This

relation is used below to determine the beat wave phase velocity on axis.

Equations (3a) and (3b) describe the envelope evolution for each beam as it propagates

through the plasma. Setting the right-hand sides of Eqs. (3a) and (3b) equal to zero gives

4

Page 8: Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a plasma with the resulting beat wave phase velocity equal to the speed of light. Focusing

matched beam solutions for which the beams propagate without diffracting. Matched beamsolutions are obtained for values of R in the range 1/(vi - 1) < R < v/g - 1 provided tilenormalized power W of each beam is given by

W, = [8R 2 (1 + R) 2 - (1 + R)4][64R 2 - (1 + R) 4 ]', (5a)

W2 = [8(l + R)2 - (1 + R)4 )[64R 2 - (1 + R) 4]- '. (5b)

This is illustrated in the following limits: For R = 1, then W, = W 2 = 1/3. As R -1/(v/8--1), then W, - 0 and W 2 --- 1. As R -- v/8- -1, then W, -. 1 and W2 - 0.Hence, it is possible for a beam close to the critical power to confine a second beam which

has a smaller spot size and a smaller power.

Matched beam propagation occurs when R, W, and W2 are specified as indicated

above. For example, once R is chosen in the range 1/(v/8 - 1) < R < v/8 - 1, then W1

and W2 are given by Eqs. (5a) and (5b). The actual magnitudes of the spot sizes r,1 andr, 2 are undetermined and only their ratio has been specified. Specifying a value for r,gives a value for the radiation beat wave phase velocity on axis according to the relationc/vP = 1 + cAfI /Aw, where A,8' = do3/dz- d/32/dz. Alternatively, requiring vp = C

for a given set of matched beam parameters R, W, and W2 specifies r, 1 . For example,

as R - v/8 -1, requiring vp= c gives k2r, - 5w,/Aw, where k , = wp/c. For R = 1,requiring v, = c gives k r, :-- 2. As R -- 1/(v/8- 1), it is not possible to have t, = c. For

applications in the PBWA, it may be desirable to have kprl >> 1. This implies that itmay be desirable to choose a matched beam case with R > 1. For example, R = 1.5 gives

W, _ 0.7, W 2 : 0.1 andk~r, = 2.6w,/Aw.

As a final illustration, the above results are applied to parameters similar to those inthe UCLA beat wave excitation experiment," where wl _ 2.0 x 1014 sec - 1 and Aw w _9.7 x 10- 2 (which implies n, 10"7 cm- 3 ). A test electron with intial energy given by

70 = 50 is accelerated by plasma waves generated in the following two special cases: i) Amatched beam case with the beat wave phase velocity tuned to the speed of light, v, = c.where R = 1.5, rsl = 8.3 x 10- 3 cm, P, = 1.3 x 1012 W and P, = 1.6 < 1011 W; and

ii) the same parameters as case i) only now the beat wave phase velocity is given by the

1-D limit, vp/c = 1 - w/2w2, and the radiation beams are assumed to undergo vacuum

Rayleigh diffraction, r8 = r0(1 + :2/4 )1/a. The results of case ii) are shown in Fig. 2and the results of case i) are shown in Fig. 3. Figure 2 indicates that the test electron

outruns the plasma wave and begins to be deaccelerated after approximately 0.5 cm witha maximum energy gain of A-y = 210. In Fig. 3, however, phase resonance between the

5

Page 9: Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a plasma with the resulting beat wave phase velocity equal to the speed of light. Focusing

electron and beat wave is maintained which allows an energy gain of AY a- 6500 in 8 cm.

This energy gain continues in a linear fashion until it becomes limited by some non-ideal

effect such as pump depletion.19

Discussion

In summary, it has been shown that two colinear, short-pulse, Gaussian radiation

beams can propagate through a uniform plasma without diffracting due to relativistic

focusing. This occurs for values of R in the range 1/(v/8 - 1) < R < v'8 - 1, provided W,

and W 2 are specified according to Eqs. (5a) and (5b). In addition, it is possible to tune

the phase velocity of the radiation beat wave to the speed of light for cases where R > 1.

This is accomplished by appropriately choosing r,,. In an actual PBWA, Aw = WP and the

envelope behavior of the radiation beams becomes more complicated due to the presence

of large amplitude resonantly driven plasma waves.2" However, the analysis presented here

remains valid for the front of the radiation pulse (the first several plasma wavelengths)

where the amplitude of the plasma wave remains small. In the small amplitude linit, the

phase velocity of the plasma wave is equal to that of the radiation beat wave.21 Assuming

that the phase velocity of the plasma wave remains fixed to its initial value, then the above

analysis indicates that it is possible to tune this phase velocity to the speed of light. This

implies that phase detuning between the plasma wave and the electrons can, in principle,

he avoided which results i. a substantiallv higher energy gain in the PBWA.

Acknowledgements

The authors would like to acknowledge useful discussions with C.-M. Tang. This work

was supported by the U.S. Department of Encrgy.

I

4,

6I

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References

1) T. Tajima and J.M. Dawson, Phys. Rev. Lett. 43, 267 (1979).

2) C. Joshi and T. Katsouleas, eds., Laser Acceleration of Particles, AlP Coulf. Proc.

No. 130 (Amer. Inst. Phys, New York, 1985).

3) T. Katsouleas, ed., IEEE Trans. Plasma Sci. PS-15 (1987).

4) C. Max, J. Arons and A.B. Langdon, Phys. Rev. Lett. 33, 209 (1974).

5) P. Sprangle and C.-M. Tang, in Laser Acceleration of Particles, ed. by C. Joshi and

T. Katsouleas, AlP Conf. Proc. No. 130 (Amer. lust. Phys, New York, 1985). p.

156.

6) G. Schmidt and W. Horton, Comments Plasma Phys. 9, 85 (1985).

7) P. Sprangle, C.-M. Tang and E. Esarey, IEEE Trans. Plasma Sci. PS-iS. 145 (1987).

8) G.Z. Sun, E. Ott, Y.C. Lee and P. Guzdar, Phys. Fluids 30, 526 (1987).

9) P.K. Kaw, G. Schmidt and T. Wilcox, Phys. Fluids 16, 1522 (1973).

10) C. Max, Phys. Fluids 19, 74 (1976).

11) F.S. Felber, Phys. Fluids 23, 1410 (1980).

12) D.A. Jones, E.L. Kane, P. Lalousis, P. Wiles and H. Hora, Phys. Fluids 25, 2295

(1982).

13) A. Schmitt and R.S.B. Ong, J. Appl. Phys. 54, 3003 (1983).

14) M.N. Rosenbiuth and C.Q Liu, Phys. Rev. Lett. 29, 701 (1972).

15) T. Katsouleas, C. Joshi, J.M. Dawson, F.F. Chen, C.E. Clayton, W.B. N'lori. C'.

Darrow, and D. Umstadter, in Laser Acceleration of Particles, ed. by C. Joslii and T.

Katsouleas, AIP Conf. Proc. No. 130 (Amer. Inst. Phys, New York, 1985), p. 63.

16) P. Sprangle, A. Ting and C.-M. Tang, Phys. Rev. Lett. 59, 202 (1987).

17) P. Sprangle, A. Ting and C.-M. Tang, Pliys. Rev. A 36, 2773 (1987).

18) C. Joshi, C.E. Clayton, C. Darrow and D. Unistadter, in Laser Acceleration of P tr-

tidles, ed. by C. Joshi and T. Katsouleas, AlP Conf. Proc. No. 130 (Amier. hist.

Phys, New York, 1985), p. 99.

19) W. Horton and T. Tajima, in Laser Acceleration of Particles, ed. by C. Joshi and T.

7

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Katsouleas, AlP Conf. Proc. No. 130 (Amer. Inst. Phys, New York, 1985), p. 179.

20) C. Joslii, C.E. Clayton and F.F. Chien, Phys. Rev. Lett. 48, 874 (1982).

21) C.-M. Tang, P. Sprangle and R.N. Sudan, Phys. Fluids 28, 1974 (1985).

Page 12: Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a plasma with the resulting beat wave phase velocity equal to the speed of light. Focusing

>N

EN

mo

06 oSr --

o-

0)0

0))

0 -0

0 <

CL -

Page 13: Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a plasma with the resulting beat wave phase velocity equal to the speed of light. Focusing

140

0 100-

4W

00

,w

60

iI

25 MeV

20 1 , 1 I I

0 2 4 6 8

z (cm)FIG. 2. Test electron acceleration with the same parameters as Fig. 3 except the phase

velocity is given by the 1-D limit and the radiation beams undergo vacuum Rayleigh

diffraction.

10

Page 14: Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a plasma with the resulting beat wave phase velocity equal to the speed of light. Focusing

>2

CP

WI

25 MeV

0 1 1

0 2 4 68

z (CM)FIG. 3. Test electron acceleration for a matched beam case with t',p C, Where R 15

rj= 8.3 X 10-3 CM, p1 = 1.3 x 1012 W and PA = 1.6 x< 10" W.

Page 15: Relativistic Focusing and Beat Wave Phase Velocity Control ...matched radiation beams through a plasma with the resulting beat wave phase velocity equal to the speed of light. Focusing

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Major Bart Clare Dr. Adam DrobotUSASDC Science Applications Intl. Corp.P..O. Box 15280 1710 Goodridge Dr.Arlington, VA 22215-0500 Mail Stop G-B-1

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Brookhaven National LaboratoryDr. B. Cole Upton, NY 11973Univ. of WisconsinMadison, VI 53706 Dr. H. Figueroa

1-130 Knudsen HallIDr. Francis T. Cole U. C. L. A.Fermi National Accelerator Laboratory Los Angeles, CA 90024Physics SectionP. 0. Box 500 Dr. Jorge FontanaBatavia, IL 60510 Elec. and Computer Eng. Dept.

Univ. of Calif. at Santa BarbaraDr. Richard Cooper Santa Barbara, CA 93106Los Alamos National Laborat)ryP. 0. Box 1663 Dr. David ForslundLos Alamos, NM 87545 Los Alamos National Laboratorv

P. 0. Box 1663Dr. Ernest D. Courant Los Alamos, NM 87545Brookhaven National LaboratoryUpton, NY 11973

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Dr. P. GarabedianCourant Inst. of Math. Sciences Dr. G. L. JohnstonNov York University NY16-232New York, NY 10012 M. I. T.

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Jackson, MS 39217Dr. Dennis GillLos Alamos. National Laboratory Dr. Mike JonesP. 0. Box 1663 MS B259Los Alamos, NM 87545 Los Alamos National Laboratory

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Department of PhysicsDr. J. Hays Los Angeles, CA 90024TRWOne Space Park Dr. Rhon Keinigs MS-259Redondo Beach, CA 90278 Los Alamos National Labortory

P. 0. Box 1663Dr. Wendell Horton Los Alamos, NM 87545University of TexasPhysics Dept., RLM 11.320 Dr. Kvang-Je KimAustin, TX 78712 Lavrence Berkeley Laboratory

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Los Alamos Nationai LaboratoryDr. Robert A. Jameson P. 0. Box 1663. MS E531Los Alamos National Laboratory Los Alamos, NM .7{>5AT-Division, MS H811P.O. Box 1663 Dr. Ed KnaopLos Alamos, NM 87545 Los Alamos National Laboratory

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Dr. Peter Kneisel Prof. Kim MolvigCornell University Plasma Fusion CenterF. R. Nevan Lab. of Nucl. Studies Room NV16-240Ithaca, NY 14853 M.I.T.

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New York, NY 10012Dr. N. C. Luhmann, Jr.7702 Boelter Hall Dr. Robert J. NobleU. C. L. A. S.C1A.C., Bin 26Los Angeles, CA 90024 Stanford University

P.O. Box 4349Dr. Clare Max Stanford, CA 94305Institute of Geophysics& Planetary Physics Dr. J. NoremLawrence Livermore National Laboratory Argonne National LaboratoryP. 0. Box 808 Argonne, IL 60439 0Livermore, CA 94550

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Dr. Robert B. PalmerBrookhaven National Laboratory Dr. Harvey A. Rose, T-DOTUpton. NY 11973 Los Alanos National Laboratory

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Prof. Ravi SudanDr. Richard L. Sheffield Electrical Engineering DepartmentLos Alamos National Laboratory Cornell UniversityP.O. Box 1663, MS H825 Ithaca, NY 14853Los Alamos, NM 87545

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Dr. T. TajimaDr. J. D. Simpson Department of PhysicsArgonne National Laboratory and Institute for Fusion StudiesArgonne, IL 60439 University of Texas

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Dr. Perry B. WilsonStanford Linear Accelerator CenterStanford UniversityP.O. Box 4349Stanford, CA 94305

Dr. V. WooApplied Science DepartmentUniversity of California at DavisDavis, CA 95616

Dr. Jonathan VurteleN.I.T. PNV 16-234Plasma Fusion CenterCambridge, MA 02139

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