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AFOSR-TR- L1

Final Technical Reportto the

Air Force Office of Scientific Research

Grant Number: AFOSR-89-0463Grantee: Jackson State UniversityDates: 1 September 1989 to 30 September 1990

Research Title: Mathematical Analysis of ThreeFree-Electron-Laser Issues

Principal Investigator: Professor Shayne Johnston

REPO T D CUM NTATON AGEForm Approved

REPOT D CUM NTAION PAG 0MB No. 0704-0188

Putsc 'ew"'¶m' ouraer, or tm- co~le¶cmc ~i ftor,navo, set' -t~ !3 io eraqe ' lor Oer,'eionse. nctuding the tfme'C or 'e- -nghtrucvcmei. searcmt.-q evsr nq :iara sourcesgathevog a I'd 'al.l AIM-nq the data reeded. and ConOWe¶ing aro 'ev-e-g me : ol~e c li oftO' O m formationl Sena cotini,.@vtS regarding thish b.rd@9 astimate oram )tr i ter isoC~t o3 ,,

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1. AGENCY USE ONLY (Lidlv* blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

II Final Tech Report, 01 Sep 89 - 30 Sep9C4. TITLE AND SUBTMTE S. FUNDING NUMBERS

MATHEMATICAL ANALYSIS OF THREE AFOSR-89-0463FREE-ELECTRON-LASER ISSUES 612 34A

6. AUTHOR(S)

Shayne Johnston

7. PERFORMING ORGANIZATION NAME(S) ANO ADORESS(ES) 8. PERFORMING ORGANIZATIONREPORT NUMBER

Dept of Pntyscis and Atmospheric SciencesJackson State UniversityJackson, MS 39217-0460

9. SPONSORINGý MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING, MONITORING

AFOSR/N14 AGENCY REPORT NUMBER

Boiling AFB DC 20332-6448 AFOSR-89-0463

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION iAVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The Principal investigator has concentrated mainly on Project 2 duringthis initial support period. in late August 1989, the preliminaryresults were presented in a paper given at the Eleventh internationalConference on Free Electron Lasers (see Digest of paper, Appendix E).Further results including a guide magnetic field were presented in asecond paper given at the Thirty-First Annual Plasma Divisionalmeeting of the American Physical Society held in November 1989 (seeabstract, Appendix F). Two important developments have been (1) therecognition that the conditions for-minimal axial degradation and forimmunity to saturation by trapping (Research objective 6) can besatisfied simultaneously, and (2) a numerical example of a 33 TWsubmicron radiation source (see Appendix D, Submitted paper).

14. SUBJECT TERMS IS. NUMBER OF PAGES

16. PRiCE CODE

17. SECURITY CLASSIFICATION 1 8. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACTirT %QTWTrn ~ iT~rT A(TFVTrT I NCLASSIFIED SAR

Table of Contents

Participation of Principal Investigator ....................... 1

Participation of Students ........................................ I

Progress towards Research Objectives .......................... I

Establishment of Computer Research Laboratory ................... 2

Future Work .... .................................................. 3

APPENDICES

A. Grant Documentation ... ....................................... 4B. Summary of Research Objectives .............................. 6C. Paper Published in Nuclear Instruments

and Methods in Physics Research ...................... 8D. Paper Submitted for Publication ............................ 11

E. Digest of Paper Presented at the Eleventh InternationalConference on Free Electron Lasers ................... 22

F. Abstract of Paper Presented at the Plasma DivisionalMeeting of the American Physical Society .............. 25

Participation of Principal Investigator

The Principal Investigator for this grant, Professor Shayne Johnston,devoted 50% of his time during the academic year 1989-90 and 2.0 summermonths during Summer 1990 to this grant. The 50% release time from teachingduties was honored by the University and contributed to the hiring of anadditional physics faculty member. On the Departmental level, a separateroom was made available to the PI for the establishment of a computerresearch laboratory.

Participation of Students1. Two undergraduate physics majors were active participants in the

research. In particular, they each received academic credit (SpecialTopics in Physics, 3.0 credit hours) in the Fall Semester 1989 forsupervised readings in the field of free-electron lasers. One of thesestudents, Mr.Quinton L. Williams, was a senior who graduated with a B.S.degree in physics in Spring 1990 and has now gone on to graduate studyin physics at Georgia Tech. The other student, Mr. John E. Foster, wasselected by the University in Fall 1990 as its sole HEADWAE StudentHonoree, i.e., its Student-of-the-Year, in recognition of hisoutstanding scholarly and extra curricular record. They both did a finejob rendering computer support.

2. A graduate student in comput2r science, Mr. Vijaykanth R. Tummalapally,also participated in the research. In particular, he was very helpfulin planning and implementing the acquisition of computer hardware andsoftware. Mr. Tummalapally will complete the requirements for the M.S.degree during Fall 1990.

Progress Towards Research Objectives

The research objectives of the three component projcts of thisgrant are summarized in Appendix B. The progress towards theseobjectives is summarized as follows:

Project 1: Sideband Control by Optical Guiding

During the Spring Semester 1990, Mr. Quinton Williams and Mr. JohnFoster performed numerical studies of parametric and forcing excitationof nonlinear oscillators (Research Objective 6). The results of theseinvestigations are interesting but incomplete, i.e., not yetpublishable.

Project 2: A Path to Ultra-High-Power Free-Electron Lasers

The Principal Investigator has concentrated mainly on Project 2during this initial support period. In late August 1989, thepreliminary results were presented in a paper given at the EleventhInternational Conference on Free Electron Lasers (see Digest of paper,Appendix E). Further results including a guide magnetic field werepresented in a second paper given at the Thirty-First Annual PlasmaDivisional Meeting of the American Physical Society held in November1989 (see abstract, Appendix F). In addition, two papers were prepared

for publication. The first, entitled "Higher-Power Free-Electron

Lasers", was published in Nuclear Instruments and Methods in PhysicsResearch A296, 532 (1990) (see Appendix C). The second, entitled "APath to Ultra-High-Power Free-Electron Lasers", was submitted toPhysical Review A in April 1990 and is currently being revised (seeAppendix D).

Two important developments have been (1) the recognition that theconditions for minimal axial degradation and for immunity to saturationby trapping (Research Objective 6) can be satisfied simultaneously, and(2) a numerical example of a 33 TW submicron radiation source (seeAppendix D, Submitted paper).

Finally, the recent acquisition by the PI of an IBM RS60OO0/Model530 workstation (through a different grant) has expanded the futurescope of this work to include particle simulations. Such simulationsare needed to study properly the principal constraint in the proposedscheme, namely, the potential degradation of the microbunches due toboth the initial electron energy spread and the energy spread inducedsubsequently by electrostatic repulsion.

Project 3: The Orbital-Instability Operating Point

Mr. Vijaykanth Tummalapally performed some useful numerical studiesof single-particle orbits in combined helical wiggler and axial guidemagnetic fields during the spring and summer of 1990. Again, therecently-acquired IBM R6000 workstation will be valuable in thecontinuation of this project. Particle simulations of a non-coldhelical electron beam near orbital instability are needed to resolve thecomplicated and disparate orbital dynamics and the concomitant emittedradiation.

Establishment of Computer Research Laboratory

The Principal Investigator has installed the computer hardwarepurchased through this grant in his newly-established computer researchlaboratory located in Room 301A of Just Science Hall, adjacent to hisoffice, Room 301B. The following equipment was purchased:

1. An Everex 386/33 Mhz with 8 MB RAM and 330 MB hard disk.2. An Intel 80387 math coprocessor.3. A Weitek 3167 coprocessor.4. A dual-coprocessor board.5. A NEC Multisync 5D color monitor.6. A NEC graphics card.7. A NEC 890XL PostScript laser printer.8. An IBM PS/2 Model 70 with 2 MB RAM and VGA graphics.9. A Panasonic KX-1124 dot-matrix printer.10. A Hewlett-Packard 7550A graphicsplotter.

In addition, a variety of mathematical and programming software wasacquired.

The Everex 386/33 and the IBM PS/2 will be connected via Ethernetto an IBM R6000/Model 530 workstation which was recently obtained by thePI through a separate DOE grant and which has also been installed in thecomputer lab. As noted earlier, the availability of the IBM RCSCmachine makes feasible on expansion in scope of the present work toinclude large particle simulations.

Future Work:

The research objectives of this grant (see Appendix B) were

proposed in the context of a three-year proposal and many of them remainto be addressed. Continuing support of this work by AFOSR has beenrequested. In any event, the PI considers the computer capabilityprovided by this grant to be vital to his continued researchproductivity.

Appendix B: Summary of Research Objectives

Project 1: Sideband Control by Optical Guiding

1. A determination of group velocity in the nonlinear saturated regime.The analytic model of this regime due to Antonsen and Levush isappealing in its mathematical tractability but it is not yet clearhow to accommodate the concept of group velocity within thatframework.

2. A mathematical investigation of the limitations of thegroup-velocity concept in the presence of gain.

3. An extension of the analysis to include space-charge effects andwaveguide boundary conditions.

4. An analytic and numerical study of the sensitivity of the controlcondition to the distribution of trapped orbits.

5. An investigation of sideband seeding at saturation and theconditions for the validity of the conventional linear theory. Thisissue will involve the analysis of a certain Mathieu equation.

6. An investigation of multiple sideband generation by parametriccoupling. Although there has been some limited study in theengineering literature of combined parametric and forcing excitationof nonlinear systems, this past work is restricted to the case ofsteady-state oscillations and weak nonlinearity.

Project 2: A Path to Ultra-High-Power Free-Electron Lasers

i. Inclusion of the radiation field E in the Lorentz-Dirac equationrof motion. The neglect Er is inappropriate if the ponderomotiveforce becomes competitive with the constant field E , thecondition for this being E > V E 1/Kw.

r ^.A 0, 0 0

2. A careful analytic and numerical study of the bunch degLadationissue. Derivation of an upper bound on the permissible interactionlength.

3. Relaxation of the assumption P/W <K( and inclusion of theeffects of transverse interference across the face of the disk. Aproper treatment of a radiating charged structure would bereminiscent of the old extended-electron theories. Is such adisk stable or subject to clumping on smaller scales?

4. Investigation of the prospects for optical guiding in the class oflaboratory devices considered here.

5. Further analysis of the scheme of frozen microbunches proposed by Yuas a means of preserving the integrity of the macroparticle model.

6. Inclusion of a guide magnetic field B in the Lorentz-Diracequation of motion. Further mathematical analysis of the conditionsfor minimum axial degradation and for immunity to saturation bytrapping.

Project 3: The Orbital-Instability Operating Point

1. A thorough analytic investigation of the emission of radiation by anelectron beam with a nonzero energy spread near the point ofexponential instability of the helical orbits.

2. A clarification and exploitation of mathematical analogies with thetheory of radio-frequency heating of nonuniform plasmas by phasemixing.

3. The use of a helical-ribbon model for the electron beam in a gaincalculation which includes both radial gradients and a finite energyspread.

4. The =•,alysis of velocity-shear instabilities associated with theribbon model.

5. Exploration of a new concept: A dual-beam free-electron laserconsisting of a primary electron beam coupled to a concentrictenuous control beam near orbital instability. It has been shown byStenflo for a one-dimensional wiggler field that the presence of atenuous secondary electron beam near orbital instability candestabilize the electrostatic plasma mode suppoted by the primaryelectron beam. The implication is that such a tenuous control beamcould thereby enhance the gain in a Raman free-electron laser. Toassess this possibility, it is necessary to perform athree-dimensional analysis which takes account of the radialvari'tion of the wiggler field. The primary and secondary electronbeams then no longer physically overlap in space, but instead havedifferent radial locations according to their energies. However, ina cylindrical waveguide, the beams remain coupled by the boundaryconditions on the fields. Does the instability persist in thesecircumstances? The appropriate dispersion relation has been derivpdby the PI but has not yet been solved numerically.

Nudlear Instrumtents and Mietlhods in Ph_%sics Research A246 (199) 512-514

Na'rth-Faalland

11iGI-JER-POWER FREE-ELECTRON LASERS

Shia'. ne JOHINSTON!X'1'.rtopian w '/ Ilia i m aii, I .otia.piaa,yp S, temaacp. ha, A. %,,a Siate t Lti erwa'ra. 11 0. Bhv I 'th)W. Jcwk win. WS fX I'.' L S.4

I'lie m konina lt pr. '~cesý s spon'at aneousL emiassio, 't en haniced h% prehu nchang oil a le ngth scale shoart comtpared %%,ith the Aaselength.Mind ILAaNtIraCd I. ti'ro'ng axiaai electric: field. (ienera~lk speAkaag. thle potential foar verv high powmer levels is achieved ,at the epense ofrhiasc cohIerence rclkatisc to tile ca'aaentional free-clectroal liser,

This paper concern,. a radical variant p1.21 of the The new proposed scheme has two unconventionalfree-electron laser hased on enhanced and driven spaln- ingredient% which serve. respectively, to enhance and itotaneous emlissionf. The essential idea is founded upon an sustain the spontaneous emission. The enhancemente~act nJlathcIMatcal solution [21 of thle Lorelt,- Dirac occurs hecause the electron beam is prehunched on aequationf and on a spontaneous radiative-reactiotl effect length scale..wbILh is L/tort compared with the radiatedwhich is completely omnitted from the usual theoretical waveleng:h. In contrast, the bunching which occurs indescrtptioat of free-electron lasers. 'Thus, it is proposed an ordinarv free-elet-tron laser is oin the sante lengthto stud% a new, donmain iii paranmeter space w here thlis -scale as the wavelength and is due to the ponderomotivetlorma~lk ne-ligtble effect becotme., dominant, force. Here, the microbuinches behave as macroparticles

Ihle prinlcipal advatltage of this ncxs class, of labtara- of charge Ve and mass Ni,, which radiate coherentlv.tor-ý de.iceS ai. the poatenttal for very high povxer levels. Althloughl thle ponderomotive effect, which--va-ries--,Gieneraul]% ýpeakinlg. this high power is achieved at the ts ot. is unaffected, the scattering cross section and theexpense of phlas coherence relative Ill the conventional wAiggler-raia~tio~n-pressure effect vary as e-/in and -sofree-clectron laser, although the radiation spectrum still are enhanced by the large factor N, Consequently [6].consi is aaf halrp emissiion lines with smnall contributions spontaneous emission can dominate stimulated emis-from wkell-separated harmonics provided the wiggler sion. and the relevant dynanmical equation then becomespump pararimeter K,~ =2,1,"kc s-atisfies K,,~ < 1. and the Lorenti.-Dirac equation including enhanced radia-tile enmissioan is coinfitned to a foarwaird cone of angular live reaction rather than the customary penduluim equa-width I -,' tar highly relatiistic electrons with y~ - 1. lion.

rfile utlttimate Power linmitatbons inherent in rree-dele- The -second key ingredient. sustainment of this en-trim laser devices are an important consideration for hanced emission, is achieved simpl ' by applying a strongsuch praoposed applications asN laser propulsion of axial electric field Fab inifio and sal pre-establishingspacecraft 1 31, removal (if chlorofluorocarbons from the at a very high v.alue, the. level of the radiation field atearth*-, atmosphere [4[ and, of course, antimissile de- which ponderomotive buckets can even formi. In thefcnse s,.stcms. The saturation mechanism itl eansen- meantime, a balance is atutonmatically struck [71 in whichtaaonal free-electron lasers is electron trapping in the all of the energy gained from E,, is immediately shed ascoherent ponrderomotive potential wells. iFfficiency-en- enhanced spontaneous radiation at the free-electron-hanceanent schemes have been devised to prashibi~t the laser wavelength. The electrons maintain a constanto nset of saturation by tapering the wiggler tllagnetic- energy y, in this asymptotic state of balance I1I1 and actfield strength air winclenglh [51, Such tapering sceminmply as at catalytic internmediary %%ith 1007 efficiency..ire conceptualh, equivalent to providing a longitudinal It is asumned that the preliminary bunching has beenaccelerating electric field E,, ito restore the energv trails- accomplished by utiliz'ing it conventional saturatedferred front the electrons to thie radiation. the strength free-electron-laser stage which results in a train of disk-of L ,being ex~ternally programmed to balance the shaped structures having an axial thickness 8 and atpoilderotnotive force. rThe rough idea underh11ing thuis transverse raditus p. In order to treat the electron hunchPaper is to increase the radiated power by muaking E,. is as at tiacral particle without extended structure, tilelorge as possibile, Toi achieve this goal. a very differenut inequalities 6 -14ý X and I p/'y., )-oý X mlust he satisfied.kind oif bKil,,te should he airrangedl. \% here X denotes the radiated waiselength. X - A ý( +

I Ises aer -. acrut l'ahli ,herN I1N, (Nai-rt ih- iiiil~anal

S Johnston Higher-power FELN

K').---. The motion of the microbunches is then More remarkable than the mere emtencc oM thi,

governed bv the fullb relativistic Lorentz--Dirac equa- steadv-state solution is the fact [I1 that it i- an attraictor

tion the energy -y('r) alwa,,s becomes, aomptoticallh Con-

du ,stant and equal to -f. the value determined bs the

. (at2t-y -u× ) balance condition (6). regardless of the initial ener -,or the field strengths E,, and B, involved. The tran-.ent

.. ,dy l dength scale for the asymptotic state of balance to beSd - d 7 j achieved becomes accessible in the lahorator,, f r

"where u yr ,iIs the proper time. Q e sufficiently large. The attractive character of the ,olu-tion ensures that the balance is self-regulating and in-

12, e e B ni,c and T, = 2 e 3m,cý. Let the electro- sensitive to small field errors.

magnetic fields comprise a helical-w.iggler magnetic fieldC C An important constraint in the proposed scheme ,,

2, = Qý ( i cos k,: -- sin k , : ). (2) the potential degradation of the microhunches due toboth the initial electron energy spread and the ener,.\

thand ah ufrmadialte le field and hencet - .o . Note spread induced subsequently by, electrostatic repulsion

that the radiated field and hence the ponderomotive Initial estimates [21 suggest that the upper bound on the

force have been omitted completely from eq. (1) in permissible interaction length is quite stringent. Thereaccordance with the preceding discussion.

are at least two possible ways to circumvent his limita-In order to solve eq. (1) explicitly, transform to a

helical coordinate system which rotates with the wiggler tion. One way is to adapt the idea of -frozen" mucro-

field: bunches as proposed by Yu [8]. i.e., to sattsfN imula-neouslv the resonance conditions for bunching and for

I -i sin k,: +f cos A,: radiation. Thus. the electrons would continue to see thebunching lasers while traversing the wiggler region and

. -i cos k: -• sin k:). one would have y,, = y_-, .A second possibility is to introduce a strong axial

and seek a solution for constant y. iu•. u.. u,. One is guide magnetic field B, into the wiggler region. The

thus led 12] to the following three conditions which relation between the axial velocity r_ and the energ, -

define an exact stead,-state solution: for a helical orbit then becomes [9]

i,= -V,,k.cu,u'( 1u2-u ). (3) _ =1. -1-C 'Y: I" ( - 12,, -yk ,r v: "

1' •2_ A, + .-VT,,k,•cuuj( I - u1 u! ). (4)where Q,, e I B, ic. Implicit differentiation of eq

f2).-f2o u, = \'m~k~ui4( u?+ u• ). (5) (7) then yields a condition for di:, d-y to wariish. vi?..

For given u, and upon elimination of u1, eqs. (3) and -Q,;'y = k A:( I ( K1 ). (8)

14) yield a cubic equation for u, which always has one

real positive root. Condition (5) then determines the When the guide magnetic field satisfies condition iS).

corresponding axial electric field. This exact solution the axial degradation of the microhunches is thus mini-

reduces to the customary helical orbit when radiative- mized. The condition corresponds to a stable Type II

reaction corrections and E,) are ignored. orbit 191 on the strong-field side of magnetoresonance.

In the limiting case -y >> I and .VAr,,c << 1. condi- The use of an axial guide field also raises another

lions (3)-I5) reduce to a, = 0. u. = S2,'kc. and interesting possibility. If. instead of condition 18). the

guide magnetic field were chosen to sattsf,,

The steady-state condition (6) has a simple interpreta- y [ (), _ ')1

tion in the rest frame of the electron microbunch whereit expresses a balance between the dc electric force and then the ponderomotive potenial can be show.n ito

the rate at which momentum is removed from the vanish [9]. Under such circumstances a conventional

incident wiggler field (cf. F= (OT/c). This latter force free-electron laser would have zero gain whereas the

is independent of the radiated field level, unlike the devices described in this paper ,%ould not only. stillponderomotive force which is proportional to E,. The radiate but also w.ould be totall, immune to saturation

new balance condition (6) sustains the helical motion of by trapping and to sideband instabilities.the electrons and the concomitant spontaneous emis- In conclusion, we note some directions for further

sion. The static electric field is balanced not against the research as follows:

coherent ponderomotive force but instead against the (I ) Inclusion of the radiation field E, in the

hunch-enhanced radiation pressure force. Lorentz-Dirac equation of motion. The neglect of E, i,

III TH)oiRN

534 S. Johnston , Higher-power FELI

inappropriate if the ponderomotive force becemes corn- Acknowledgementspetitive with the constant field E,. the condition for This work was supported in part bx the LiS. Depart-this being E, > 2 yý -'0 / K,. ment of Energ, under the aupice, of th,

(2 ) A ca re fu l a n a ly tic a n d n u m e rica l stu d , o f th e JS o B L ' G E Cnd or tiu Pro r m I t peJSU,'LBL/'AGMNEF Consortium Program. It is, pre-

bunch degradation issue. Derivation of an upper boundon te prmisibl intracion engh. ently supported by. the LU.S..Air Force Office of Scien-on the permissible interaction length. Uific Research.

(3) Relaxation of the assumption p/-,., << A and

inclusion of the effects of transverse interference acrossthe face of the disk. A proper treatment of a radiating Referencescharge structure would be reminiscent of the old ex- Ill S. Johnton. Bull. Ani. Phx., Soc. 3211Q , j 1,SIs27tended-electron theories [10]. Is such a disk stable or [21 S. Johnston. Bull. Am. Phýs Soc. 33 (19,,) 1884,

subject to clumping on smaller scales? [31 J.T. Kare (ed.|. Proc. SDIO DARP- Workshop on Laser

(4) Investigation of the prospects for optical guiding Propulsion. CONF-860788 (Lawrence Liermore Nationalin the class of laboratory devices considered here. Laborator-.. November 1986).

(5) Further analysis of the scheme of frozen micro- [41 T.H. Stix. Bull. Am. Phy•. Soc. 33 (19•8) 1993. 5R22.

bunches proposed by Yu [81 as a means of preserving [51 N.M. Kroll. P.L. Morton and M.N. Ro.enbluth. IEEE J.

the integrity of the macroparticle model. Quantum Electron. QE-17 (1981) 1436.field B,, in the [61 S. Johnston and R.M. Kulsrud. Ph\,. Fluids 20 09lTh

(6) Inclusion of a guide magnetic 1674.

Lorentz-Dirac equation of motion. Further mathemati- [7i F,A. Jackson. J. Math. Phys. 25 (1984) 1584.

cal analysis of the conditions for rmnimum axial de- [81 L.H. Yu. Ph,,ys. Rev. Lett. 53 01)84) 254,gradation [eq. (8)) and for immunity to saturation by [9[ HP. Freund et al.. Ph•s. Re%. A26 1•982) 21•)4.trapping [eq. (9)1. [101 T. Erber. Fortschr. Phys. 9 (19611 343.

April 1990

A Path to Ultra-High-Power Free-Electron Lasers

Shayne JohnstonDepartment of Physics and Atmospheric Sciences

Jackson State UniversityJackson, Mississippi 39217

Abstract

A radical variant of the free-electron laser is proposed and analyzed.

The radiation is generated by spontaneous emission, enhanced by prebunching

on a length scale short compared with the wavelength, and is sustained by

a driving axial electric field. In contrast with conventional efficiency-

enhancement schemes which modify the pendulum equation, the electric field

is here balanced against the rate at which momentum is removed from the pump

field alone. An explicit solution of the Lorentz-Dirac equation governing

the microbunches demonstrates that this balance is self-regulating and

nonlinearly stable. A numerical example is presented of a submicron radiation

source with a peak powerof 33 TW. Generally speaking, the potential for very

high power levels is achieved at the expense of phase coherence relative to

the conventional free-electron laser.

I. Introduction

The ultimate oower limitations inherent in free-electron laser devices

are an important consideration for such proposed applications as laser

propuision of spacecrafti, removal of chlorofluorocarbons from the earth's

atmosphere 2 and, of course, antimissile defense systems. 3 The saturation

mechanism in conventional free-electron lasers is electron trapping in the

coherent ponderomotive potential produced bv the beating of the wiggler and

radiation fields. Efficiency-enhancement schemes have been devised to pro-

hibit the onset of saturation by tapering the wiggler magnetic field strength

or wavelength. 4 Such tapering schemes are conceptually equivalent to providing

a longitudinal accelerating electric field Eo to restore the energy transferrea

from the electrons to the radiation, the strength of Eo being externally pro-

grammed to balance the ponderomotive force. The rough idea underlying this

paper is to increase the radiated power by making Eo as large as possible. To

achieve this goal, a very different kind of balance should be arranged.

The scheme proposed and analyzed in this paper has two unconventional

ingredients which serve respectively to enhance and to sustain the spontaneous

emission. The enhancement occurs because the electron beam is prebunched on

a length scale which is short compared with the radiated wavelength. In

contrast, the bunching which occurs in an ordinary free-electron laser is on

the same length scale as the wavelength and is due to the ponderomotive force.

Here, the microbunches behave as macroparticles of charge Ne and mass Nm which

radiate coherently. Although the ponderomotive effect, which varies as elm, is

unaffected, the scattering cross-section and the wiggler-radiation-pressure

effect vary as e 2 /m and so are enhanced by the large factor N. Consequently 5 ,

spontaneous emission can dominate stimulated emission, and the relevant dynami-

cal ecuation then becomes the Lorentz-Dirac eauation including enhanced radiativ:e

reaction rather than the customary pendulum equation.

The second key ingredient, sustainment of this enhanced emission, is

achieved simply by applying a strong axial electric field Eo ab initio and

so ore-establishing at a very high value the level of the radiation field at

which ponderomotive buckets can even form. in the meantime, a balance is

autcmatically struck6 in which all of the energy gained from Eo is immediately

she. as enhanced spontaneous radiation at the free-electron-laser wavelength.

The eiectronsmaintain a constant energy in this asymptotic state of balance7

and act simply as a catalytic intermediary with 100% efficiency.

The paper is organized as follows. In Section II, the relativistic

Lorentz-Dirac equation governing the microbunches is solved exactly to justify

the preceding claims. The radiated power to be expected in such a device is

discussed in Section III. The critical issue of degradation of the micro-

bunches is 2xamined in Section IV. A numerical example of an intense source

of submicror radiation is presented in Section V. Finally, a summary of

conclusions is given in Section VI.

II. Solution of the Lorentz-Dirac Equation

Consider the fully relativistic Lorentz-Dirac equation

_--alpS~(1)

AZ

+ N T.-

where u= v/c' C denotes proper time}-l-t = _ let •moC;

"= e 0 B/mC, and to = 2e 2 /3moc 3 = 6.24 x 10- 2 4 s. Let the electro-

magnetic fields comprise a helical wiggler magnetic field

-P Aand a uniform axial electric field J.E = - .A . Note that the

radiated field and hence the ponderomotive force have been completely omitted

from Ec. (1) in -.ccordance with the preceding discussion. We shall return to

this point in Section III.

In order to solve Eq. (1), transform to a helical coordinate system which

rotates with the wiggler field, i.e.,

A A Ael = -x sin kwz + v cos kw;,

A Ae2 = -x cos kwz - y sin kwz

^ Ae3----z,

and seek a solution for constant , Ul, u2, u 3 . One is thus led to the followinz

three conditions ,.hich define an exact steady-state solution:

t.X ~~ ~ f4 Z . A 4 C2..U 1 + I 31 N -uU, U, LA + U + U% (3)

..1 ULL1 N NT. C LA 3

For given u3 and upon elimination of ul, Eqs. (3) and (4) yield a cubic

equation for u2 which always has one real positive root. Condition (5) then

determines the corresponding axial electric field. This exact solution reduces

to the customary helical orbit when radiative-reaction corrections and E. are

ignored.

In the limiting case >»M and NR'Cc C << I conditions

(3), (4) and (5) reduce to u 1l 0, u 2 9Odd•kwC, and

(6)

The steady-state condition (6) admits a simple interpretation in the rest frame

of the electron microbunch where it expresses a balance between the dc electric

force and the rate at which momentum is removed from the incident wiggler field

(cf. F = I 6/c). This balance sustains the helical motion and its concomitant

spontaneous emission.

More remarkable than the mere existence of this steady-state solution is

the fact 7 that it is an attractor: the energy VIT) always becomes asymptotically

constant and equal to . the value determined by the balance condition (6),

Vj (7)

regardless of the initial energy V. or the field strengths E. and Bw involved.

For Y >>l, the transient length scale L, for this asymptotic state of balance

to be achieved is given by the formula 7

Sc+ / (8)

which is accessible in the laboratory for N sufficiently large. The attractive

character of the solution ensures that the balance is rugged, i.e., that it is

self-regulating and insensitive to small field errors.

UII. Radiated Power

Consider next the radiated power level that is attained. We assume here

that the preliminary bunching has been accomplished by utilizing a conventional

saturated free-electron-laser stage which results in a train of disc-shaped

structures having an axial thickness I , a transverse radius ? and spatial

separation & . The density of the microbunches is then limited by electro-

static repulsion to the upper bound given by Antonsen 8 . Since we have treated

the electron bunch as a macroparticle without extended structure, the inequalities

, 4< k and ( e/•ea' « must be satisfied, where •

denotes the radiated wavelength Xw beyo nd - /e i . Let Lta= Nwiew

be the length of the magnetic wiggler beyond the transient

distance (8) and let N' denote the number of macroparticles whose radiation

becomes superimposed during a wiggler transit time. Since N' =

it follows that

N/ W (9)

( VW/Vo)

In the absence of saturation, the power radiated by the train is then sustained

for a time equal to the length of the remainder of the train at the level

NN ae.I E.O '00

The factor N' in Eq. (10) ignores the regular spacing of the microbunches; it

is correct in the limit in which the parameter (V-O (A/ tends to

zero, being otherwise an underestimate. The large factor ( I - Vac/ce )-l

is associated with time dilation 9 and has been noted previously by the author. 5

The radiated power (10) emanates from a single train of macroparticles.

If the laser bunching stage is pulsed repetitively in coordination with the

electron source, then a sequence of radiating trains will traverse the wiggler,

one after another. At this point, we have a superradiant source with the potential

for high average power. Although the emission process is spontaneous rather than

stimulated, nevertheless the radiation spectrum will consist of sharp emission

lines with small contributions from well-separated harmonics provided the wiggler

pump parameter I z -.fw/A•c satifies Kw<l. For highly relativistic electrons

with Y> I , the emission is confined to a forward cone of angular width

1/ . Compared with a conventional free-electron-laser amplifier, the basic

trade-off here is to gain power at the expense of phase coherence.

Alternatively, for higher peak power, an oscillator configuration is possible

in which the radiation from successive trains is stored between mirrors in a cavity

and the intracavity power allowed to grow. It is then natural to enquire about

the power level at which the neglect of the radiation field Er in the equation

of motion (1) becomes questionable. The condition that the ponderomotive force

be:ome competitive with the constant field Eo can be written

E?. lac 4Eo /k w E 11

witi the corresponding ambient power level being C ( E, / sr)"('iw')

whe:e w denotes the radiation waist. If the intracavity power reached this level,

the device would then operate as a conventional free-electron laser, bunching the

macroparticles and saturating shortly thereafter. In order to generate coherent

radiation by stimulated emission, i.e., to extract net gain from the exactly

resonant macrcparticles, one would now increase the electric field Eo with a

programmed time dependence EO(t) in the spirit of conventional tapering.

Note tiat in an ordinary high-gain free-electron laser, one seeks to main-

tain w- 0 (e.g., by optical guiding) since the gain mechanism requires the

presence of the radiation field. In the present scheme, however, it is desir-

able (and irevitable via diffraction) to have w>> ? to reduce Er. The critical

power level corresponding to Eq. (11) exceeds the saturated power in an untapered

high-gain free-eiectron laser 1 O when the large factor (w/ ? )2 is taken into

account.

IV. Integrity of the Microbunches

An important constraint in the present scheme is the potential degradation

of the microbunches due both to the initial electron energy spread A1 and to

the energy spread induced subsequently by electrostatic repulsion. The axial

velocity V depends on • according to the relation(V 1 /G): [ --(i4I4)/•i ,

and thus an energy spread ( lTb implies a corresponding velocity spread

+ (12

which translates to axial spreading in space. The macroparticle model breaks

down when the axial spreading & becomes on the order of the spacing A and

the distance LA for this to happen can be determined from the formula 1 1

4 3 V 1. (13)

2 • let 'EW) AS'

Awhere E denotes the longitudinal self electric field of the

microbunch. 12

Formula (13) represents a stringent upper bound on the permissible inter-

action length ( Leo + Lw ). However, there are at least two possible ways

to circumvent this limitation. One way is to adapt the idea of "frozen" micro-

bunches as proposed by Yu13, i.e., to satisfy simultaneously the resonance

conditions for prebunching and for radiation. Thus, the electrons would

continue to see the bunching lasers while traversing the wiggler region and

radiating, and one would have 4e = Voo with La.-- 0 •

A second possibility is to introduce a strong axial guide magnetic field

30 into the wiggler region. The relation between the axial velocity V and

the energy • for a helical orbit then becomes15

% ~ I I

where ao lei BO / Mac Implicit differentiation of Eq. (14) then

yields a condition for A to vanish, viz.,

-fl-H V, Itv (+ ki)(15)

When the guide magnetic field satisfies condition (15), the axial degradation

of the microbunches is thus minimized. The condition corresponds to a stable

Type 17 orbitl 5 c1 the strong-field side of magnetoresonance.

V. Numerical Example of an Intense Submicron Source

The following numerical example assumes the coexistence of state-of-the

art tecinologies without any consideration of the details of the experimental

configuration. 7he purpose of the example is simply to emphasize the potential

for hig:i power in the class of devices considered.

Co-isider an electron beam as designed by Barletta16 for use in a laboratory

x-ray laser. We take Y = 688 (i.e., approximately one-third the design energy),

but otherwise adot.t the remaining design parameters as follows: bunch length

1.2 ps, bunch spacing 0.26 ns, number of bunches 5, repetition rate 200 Hz,

number of particles per bunch 7.2xi09 , normalized emittance 0.001 mm-rad, focused

transverse radius 24.1 Am and energy spread (t6f1) = 0.I%.

Let the formation of microbunches now be accomplished bi beating on intense

KrF laser (wave.length 0.248 um, power 2GW, pulse length 12 ns) against a helical

magnetic wiggler (Ký, =iXw = 11.7 cm). The electrons are resonant with the beat

potential and will bunch to form macroparticles with N = 4.5 x 10u and

S< 02 0.5 . Next, we hold these microbunches frozen by allowing

the prebunching fields to extend into the primary wiggler region. The primary

wiggler (co-wound with the bunching wiggler) is taken to have Aw = 20 cm,

Kw = 1 and Lw = 16.0 m. Intense radiation is then emitted at the wavelength

= 0.42 /Am.

The accelerating electric field Eo required to sustain this radiation is

found from Eq. (6) to be E0 = 2 x 108 V/m. The scheme considered here converts

all of the work done by this state-of-the art 1 7 accelerating gradient to submicron

radiation. The corresponding peak power is, from Eq. (10), an immense 33 TW,

20

emitted in 1.2 ps pulses at a repetition rate of 1000 pulses per second.

VI. Conclusion

This paper has addressed a general question about free-electron lasers,

viz., what are the ultimate power limitations inherent in this emerging new

technology? It is clear that the optimum arrangement would be to apply the

maximum possible accelerating gradient to the electrons and then to convert

all of this work to radiation with 100% efficiency. A convcn:ional tapered

free-electron laser can't reach this optimum state because it is governed bv

the physics of saturation by trapping which leads to the result that E(taper)

is much smaller that E (state of the art). This paper has analyzed an alternative

scheme by which the optimum state can indeed be attained.

As illustrated in the numerical example in Section V, the frozen-microbunch

version of the scheme requires the existence of a powerful laser with a wavelength

"*shorter than that which one desires to generate. The principal advantage of this

new class of devices is high power. Generally speaking, the potential for very

high power levels is achieved at the expense of phase coherence relative to the

conventional free-electron laser.

Acknowledgment

This work was supported in part by the U. S. Department of Energy under the

auspices of the LBL/JSU/AGMEF Science Consortium. it is presently supported by

the U. S. Air Force Office of Scientific Research under Grant No. AFOSR-89-0463.

REFERENCES

1. Proceecings of the SDIO/DARPA Workshop on Laser Propulsion, J. T. Kare,Ed., CONF-860788, Lawrence Livermore National Laboratory (November1986).

2. T. J. Scix, Bull. Am. Phys. Soc. 33, 1993, 5R22 (1988).

3. Report -:o the American Physical Society of the Study Group on Scienceand Technology of Directed Energy Weapons (American Physical Society,New YorK, 1987).

4. N. M. Kroll, P. L. Morton and M. N. Rosenbluth, IEEE J. Quant.Electronics- E-W7_ 1436 (1981).

5. S. Johnston and R. M. Kulsrud, Phys. Fluids 20, 1674 (1977).

6. E. A. Jackson, J. Math. Phys. 25, 1584 (1984).

7. S. Johnston, dull. Am. Phys. Soc. 32, 1827 (1987).

8. T. M. Antonsen, Phys. Rev. Lett. 58, 211 (1987).

9. W. Rindler, Special Relativity (Oliver and Boyd, Edinburgh, 1960), 2nded., p. 53, problem 1.

10. J. C. ;arrison and J. Wong, Optics Comm. 62, 119 (1987).

11. S. Johnston, Bull. Am. Phys. Soc. 33, 1884 (1988).

12. C. M. Tang, H. Freund, P. Sprangle and W. Colson, in Free ElectronGenerators of Coherent Radiation, Physics of Quantum Electronics,Volume 8, ed. S. F. Jacobs et al. (Addison-Wesley, Reading, Mass., 1982), p.503.

13. L. H. Yu, Phys. Rev. Lett. 53, 254 (1984).

14. S. Johnston, Bull. Am. Phys. Soc. 34, 1984 (1989).

15. H. P. Freund, P. Sprangle, D. Dillenburg, E. H. da Journada, R. S.Schneider, and B. Liberman, Phys. Rev A 26, 2004 (1982).

16. W. A. Barletta, Lawrence Livermore National Laboratory, Livermore, CA,Report UCRL-99661 (1988).

17. A. M. Sessler, Physics Today 41(1), 26 (1988).

:22

FEL '89CONFERENCE DIGEST

11th International Conference on

FREE ELECTRONLASERS

FEL

'89

August 28 - September 1, 1989Ritz Cariton Hotei

Naples, Florida

LEOSI

23

P2.10 Higher-Power Free-Electron Lasers

Shayne Johnston

Jackson State UniversityJackson, Mississippi

The dominant process is spontaneous emission, enhanced by prebunching on alength scale short compared with the wavelength, and sustained by a s:rong axialelectric field. Generally speaking, the potential for very high power levelsis achieved at the expense of phase coherence relative to the conventional free-electron laser.

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Tuesuday Morning4

fte.0acni Low is scde denrcally and ezperimnuaily. Two supprimeg toe growth of the low bvquency solutin The thaxoycomplimeemtry dtliettical approaches wo the problect ot optical predicts tue ratio of cavity Q's required to insure stable opm-

wedngR to beC3d Cin* goo AgreemntWit CWA onue inteIIon of the high frequency mode in an oscillator as well as the

giesnmand pishcvdunsfroua uiigo n useful gain for the high frequency mode in amplifier.wavlegt -6idn ln the Plcm beamu in fth Coliumbia FE1L isobtained experimentally by analysis of special "ringdwn- dama and *This work was supported by -ONR and DOE.

v V11e wfth numerical aluiscasms. Data is FreseIted for both the

slight enhacmnt in the sidebu shift. 3T 11 7Grant No. N0014-796-0769 and the National Science Foundanon. Short-Period Wigglers for a Free Electron Laser,* DI.J

GrmN.RS-731-Padack. .l.H. Boosk. Y. CA-Ael. W.W. Deader, TM. AtL - j$ Pmensaan addrew physics Department. Weimiann inanute of men. Jr.. V.L. Granmasteln.J.D- Mayergom R H.ackm and -

Science. 76100 Rehovot. Lsvael Hf. Bluem. Univ. of Maryland - We presn the results of recent

* paimenW studies of short-period (t, 1 cm) clectruncagnetwigglers and the iproparation of relativistic sheet electron beamsthrough thes wigglers. We have studied wiggler field uniformity

3T 8 NtIllmetor Wveelength metal Grattan rree as a function of mechanical tolerances and alternate mnascot con-Electron baser, Y. Vieber. A.Ctsbet. E. "arate. figuw~Aosw. Two critical issues for high average power ('- I MW)University of Californaza Imaws. -Preliminary wiggler-fo~cused sheet beam. EEL& awe beam. stability and beamezpetimomtal results at a metal grating tree electron utewvgiewls epeeteprmnalager deas&"me to operate I* the loner millimeter (2- itreto ytewvgiewls epeeteprmna3 ) -0wenthmt retume and mwimn verailed sraff measurements on both of th enisue as well as analytic theoryref lecttors will he presented. The device consists of rltn necpe urn oba aaees xfrnnatwo oppoezag plasmr gratings which fore a slow wavereaigntcpedurntobamaaets.Ep iml

suppetsm structure. latmeatlag With a mildly verifictioni of the theory is reported. Finally, the first obse- 4relativlsti~c electron beas. The electron besmi is vation oi wiggler-induced radiation was obtained from the sheet

generated us&"g a rectilinear Pierce geometry diode bea FEL in an oscillator configuration.and a tharmisezc cathode capable at pradsucai ame@I"ru beem with a current density of '611ms. The *Supported by SDIO/IST/ONR through a contract administeredelectron acceleratmig voltage cam be varied up to 251 yHryDaon aoaom ndteUT o.and--bas a pulse duration of 5sm. A theoretical byHryDaod Tiaaisad h .. DE"disesaxee at the go&& and outpat radiation wavelemgth ?Nava Research Laboratorydo$edse am the grating Parameters will1 also bepresented.

3T 12A Comparative Scaling Study of Elarmonsic and Fun-

3T 9 Enhanced Sustained spontaneous Emission with dlamental Free Electron Lasers,* J.H. BOOSKE and B. LE-Guide Magetsic Field. S. Johnston, Jackson State VUSH, University of Maryland-For many of the myriad possa-University -- The potential role of a guide magnetic be ~ o reEeto ees(~eapii ifield in an unconventional freeeolectron-lamer schesme 1 i plctos fFe lcrnL4m(E ,aFi

is studied. The dominant process in this schem is pae nraiyii h riec tfxdvlaeo iinz

intense spontaneous emission, enhanced by prebunching ing the beam voltage at fixed frequency. With - -- i it nor-.on a Length scale short compared with the wavelength. Minaliaiona. some simple scnalng relations and a geral calcul-and sustained by a strong axial electric field. It is tion of coupling coeffcienrta. wadiscusthe quantltaaive tradooffsshow that the guide maignetic field can be tunia-either between umg short period wiggler for fundamental interacti~onto minimize a&&Ia degradation of the microbunches(dvz/dVi-*O) or to cause the ponderonotive potential to vesu5s hronicoperation with more conventional wigglers.

vanish. in the latter case, a conventional FEL wouldhave zero gain whereas the devices considered hers Work supported by ONR and U.S. DoE.would not oniv still radiate but would be totallyimee--o saturation by trapping mand to sideband

Lostabiltites.

IS. Johnston. Bull. As. Phys. Soc. 33, 1884, 2F6 (1988) 3T13Designs and Experiments for High Average Power

FEL. Using Sheet Electront Beacco and Short PerioidWlgglo.*s S.W. Bidwell. J.H. Dooske. Y. Carnel. MZ.X hang.D.J. Radack, T.M. Antonsen. Jr.. V.L. Grana~sasan. B. Levusra.

3T10o Interaction of High and Low Frequency Waves in a W.W. Dentier. P.E. Latham. and LD. Mayargoy, UnvFree Flectron [jus.* N. METZLEIL P. E. LATHAM. T. IM. Maya and H.P. Freuind., Science Applications sinteruauonaiANTONSEN. and B. LEV USH, LPR. Univervity of Maryland . a We will disu 1ddgn wavdeasblt studies icCollege Park. MD.-Under certain circumstances a free electron ahigh average power f(-1M ) nliet-wv10OOG )miner can operate at two widely spaced frequencies. For exm FEL using a short-period wiggler (4.- I cia) and a sliest aeic-ple, with a planar wiggler ampification will occur mat odd har- tron beam (V6 - 0.5 - 1.0 MV)TADA1YMe include 001111deratiomsmonies of the fundamental frequency. A second example is the of cavity wall heating, high voltage electro gun desin. od

case in which the interaction occurs in a waveguide with a su control. etc. We will also present preliinmary results of expert-ciently high cut-off frequency giving rise to two interesetions oi menu aon a sheet beama FEL oscillnator utilizang a 100 ons. 300400

the beam and waveguide dispersion curves. We have studied the WN pulse-iine accelerator.

competition of the low and high frequezncy waves in both amn- *Supported by the U.S. Departmeont of Eciergy-.andSDIO/IST Iplifier and oscillator configurations. We haue found parameter throuigh a contract administered by the Harry Diamond Labors-regumes wnere the presence of a high frequency wave nounnariw .onad.

VoL .34,- No. 9(1989) 1984