AO-AI02 164 NAVAL SURFACE WEAPONS CENTER SILVER SPRING MD F/6 20/S

11"r ~BANDWIDTH AND FREQUENCY ENHANCEMENT OF FREE ELECTRON LASIER IN 7A--CTCIU)APR 81 H S U HM

UNCLASSIFIED NSWC/TR-81-103 NLmh EEEIIEIIE/II/I/imo

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SBANDWIDTH ANDEREQUENCY ENHANCEMENTOF FREE ELECTRON JASER IN A MILDLYRELATIVISTIC ELECTRON BEAM -

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RESEARCH AND TECHNOLOGY DEPARTMENT ELECTE

Ml -. , JUIL 29 19810MA I ,1

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BANDWIDTH AND FREQUENCY ENHANCEMENT OF FREE Final - April 1981

ELECTRON LASER IN A MILDLY RELATIVISTIC ELECTRON 6. PERFORMNGORG. REPORT NUMER

BEAM7. AUTHOR(s) S, CONTRACT OR GRANT NUMBER(I)

Han S. Uhm

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Naval Surface Weapons CenterCode R41White Oak. Silver Spring- MD 20910

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Microwave RadiationElectron BeamFree Electron LaserIWideband Amplified 01 n.. " ( ', -

2V O, lSTRACT (Continue on reverse side 'if ne'cessary andt Identiy' by ?hoc n ber) ''

The bandwidth and frequency enhancemntf the free electron laserinstability in a mildly relativistic (: 1.15) electron beam propagatingthrough a dielectric loaded waveguide is presented. For an appropriate

choice of the dielectric constanta i nd the thickness of the dielectricmaterial, it is shown that the instability bandwidth and frequency can begreatly enhanced for specified values of the beam energy and the wigglerwavelength.

DO IO, 1473\ OND JAN !N or 1 NOV 65 ,:. oSsoLE'E UNCLASSIFIEDC N ) 1020 G14- De0 t

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NSWC TR 81-103

FOREWORD

The bandwidth and frequency enhancement of the free electron laser

instability in a mildly relativistic (y< 1.15) electron beam propagating

through a dielectric loaded waveguide is presented. For an appropriate choice

of the dielectric constant C and the thickness of the dielectric material, it

is shown that the instability bandwith and frequency can be greatly enhanced

for specified values of the beam energy and the wiggler wavelength. This

research was supported by the Independent Research Fund at the Naval

Surface Weapons Center.

H. R. RIEDL

By direction

Accession For

NTIS RAIDTIC TAB 5

U1nannounced 09 Justificatio

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vailability CodesAvail and/or

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NSWC TR 81-103

CONTENTS

Chapter Page

TEXT ..... .... .... ................................. 5

ACKNOWLEDGEMENTS ..... .... ........................... .11

BIBLIOGRAPHY .......... ............................. .15

ILLUSTRATIONS

Figure Page

1 PLOT OF THE DIELECTRIC DISPERSION RELATION IN THE PARAMETER SPACE(w,k) FOR f = 1, Rw/Rc = 0.8 AND SEVERAL VALUES OF THE DIELECTRICCONSTANT 2. THE STRAIGHT LINES REPRESENT w (k + ko)Bc FOR Y 1.107AND SEVERAL VALUES OF k0Rc ..................... 12

2 PLOT OF NORMALIZED RADIATION FREQUENCY w/koc VERSUS i FOR (a) THESHW MODE, y = 1.1, (b) THE LHW MODE, y = 1.15, SEVERAL VALUES OFkoRc, AND PARAMETERS OTHERWISE IDENTICAL TO FIG. 1 ... ..... .. 13

3/4

NSWC TR 81-103

One of the most basic instabilities that characterize a relativistic

electron beam propagating through a helical (or undulator) wiggler1-7

magnetic field is the free electron laser instability. In recent

years, the free electron laser instability has been extensively inves-

tigated with particular emphasis on applications to high power microwave

generation. In the previous theoretical studies of this instability,

it appears that as a result of the relativistic Doppler effect, the

frequency w of the microwave radiation from the electron beam passing

through a wiggler field with the axial wavelength X0 = 21r/k 0 is given

by w - (k + ko)Bc - 2lU + 0)$k 0c. Here k is the axial wavenumber

of the radiation, y = (1 - 82) -1/2 is the relativistic factor of the

beam electrons, and c is the speed of light in vacuo. In this regard,

in order to generate high frequency microwave radiation, high energy

beams (y - 1) are required. However, it is very undesirable to have a

high y value in a practical, compact microwave tube. I, therefore,

develop a new idea to enhance the frequency upshift without making use

of a high y value. Moreover, I also present a new promising scheme

for a broad bandwidth microwave amplifier.

The previous analysis 6 '7 by the author for an electron beam in a

perfectly conducting waveguide shows that the free electron laser

instability is essentially a mode coupling between the electromagnetic

and electrostatic modes, whose dispersion relations are expressed as

NSWC TR 81-103

2 22 k2 in/R c , TE mode,

2 - 2 R2 ()

c i2 /R TM modeLBIn c

and

w= (k + k , (2)

respectively, for a tenuous beam. Note that for a tenuous beam, the

electrostatic mode can be approximated by the free streaming mode in

Eq. (2). In Eq. (1), 6 n and aIn are the nth roots of the Bessel function

J (B) = 0 and its derivative J'(an) = 0, respectively, of order Z.2 in I i

Rc is the radius of a grounded conducting wall, and TE and TM represent

the transverse electric and transverse magnetic modes, respectively.

However, for present purposes, I assume a tenuous electron beam

propagating through a cylindrical waveguide loaded with a dielectric

material in the range R w< r < R c . Therefore, the radial profile of the

dielectric constant is given by c(r) - 1, for 0 < r < Rw , and e(r) = e

for Rw< r < Rc . The permeability 4 of the dielectric material

differs from unity by only a few parts in 105, thereby approximating u = 1

in the subsequent analysis. Cylindrical polar coordinates (r,O,z)

are introduced. In the remainder of this article, properties of the

mode coupling of the free electron laser in a dielectric loaded waveguide

are investigated, in connection with enhancement of the frequency and

bandwidth of the microwave radiation from a mildly relativistic

electron beam.

It 1s, therefore, required to derive the dispersion relation of the

transverse electromagnetic mode in a dielectric loaded waveguide.

In the analysis, a normal-mode approach is adopted in which all components

of the electromagnetic field are assumed to vary according to 6 (k,t)

6 .

NSWC TR 81-103

(r)exp{i(H + kz - wt)}, where X is the azimuthal harmonic number.

The Maxwell equations for the electromagnetic field amplitudes can be

expressed as

X ;(~x) i(W/c)j(k)

(3)

X x (x) -i(w/c)e-(r)E(x)

without including the influence of the beam presence. In Eq. (3), ( )

and (x) are the electric and magnetic fields. Making use of Eq. (3),

it is straightforward to show that

B(r) i2 - rE (r) - -- B (r) , (4)cp p r

E(r) -- i-- B (r) - - Ez(r) , (5)e 2 ar z 2 z

cp p r

and

r2 f+ p B ) 0, (6)r 2 (r

where p 2 w 2(r)/c - k , E8(r) and E(r) are the azimuthal and axial

components of the electric field and Be(r) and B z(r) are the azimuthal

and axial components of the magnetic field.

The appropriate boundary conditions of E z(r) and Bz(r) at r = Rc

are given by Ez(Rc) - [(D/r)Bz(r) ]r=R 0. Moreover, the fields

k(r), Ee(r), and Ei(r) are continuous across the boundary (r - Rw)

of the dielectric material. Evidently, the solutions to Eq. (6) are

given by a linear combination of the Bessel functions of the first kind

J (pr) and the second kind NI(pr) of order z. After a tedious but

straightforward algebra, it can be shown that the dispersion relation

of the electromagnetic mode in a dielectric loaded waveguide is

7

NSWC TR 81-103

expressed as

E 2(n2 -42)(n2 e2)

DT(w,k)DT(w,k) )22 22T 4 4 (7)

where the TE and TM dielectric functions are defined by

DE = 1 (8))~()-J(Oj) 1 ~ 4DT n J (n)Nj(C) J£C )NZ(n) (8)

and

DM = e Jj(n)NZ(C) JZ(;)Nj(n) 1 Jj(4) (9)

T ri J z (n)N9 ~ - J9 (;)N() (9).(4

respectively, and the parameters 4, , and n are defined by

2 2 _2 2 2W2/c -k - /R , (10)

2- 2 - 2 ~2 2w2/c /R , (ii)c

and n M w/R , and the prime denotes J'(x) = dJ /dx and N'(x)

dN /dx. Several points are noteworthy from Eqs. (7) - (11).

First, the dispersion relations of the TE and TM modes aredecoupled for I - 0. Second, in the limit of ; - 1 or R w/Rc -+ 1,

the dispersion relation in Eq. (7) can be simplified as Eq. (10) with

/R w -= n/Rc for the TE mode and with E/Rw = Ogn/R c for the TM mode.

Third, for a completely filled dielectric waveguide (Rw w 0), Eqs. (7) -

(9) can be also reduced to Eq. (11) with a = n for the TE mode and

with X - n for the TM mode.

For given values of the dielectric constant i and the ratio Pw/Rc,

the parameter & is determined from Eqs. (7) - (9) in terms of . The

oscillation frequency w and axial wavenumber k in a dielectric loaded

waveguide are obtained from the simultaneous solution of Eqs. (10)

and (11) for specified 4 and C. Figure 1 is a plot of the dielectric

NSWC TR 81-103

dispersion relation in the parameter space (w,k) for Z = 1, R /Rc = 0.8

and several values of the dielectric constant i. The straight lines in

Fig. 1 represent the free streaming mode in Eq. (2) for y = 1.107 and

several values of the normalized wiggler wavenumber k 0Rc . The dispersion

curves in Fig. I correspond to the lowest radial mode number. In a

range of physical parameters, the free streaming mode w = (k + k0 )6c

intersects the dielectric dispersion curve of the electromagnetic mode,

thereby indicating the free electron laser instability. The mode

coupling occurs at k - kp, distinguishing two cases; (a) the short

helical wavelength (SHW) mode corresponding to the normalized mode

coupling wavenumber k R = 9.3 for 6 = 2 and k 0Rc = 9 in Fig. 1 andpc O

(b) the long helical wavelength (LHW) mode corresponding to k R = 12.3pc

for = 6 and 0Rc W 1.5 in Fig. 1.

Short Helical Wavelength Mode. The SHW mode is the high frequency

operation of the free electron laser instability. The normalized

radiation frequency w/k0 Bc = (kp + k0)/k 0 versus the dielectric constant g

is plotted in Fig. 2(a) for the SHW mode, y 1.1, several values of k0Rc,

and parameters otherwise identical to Fig. 1. The electromagnetic

dispersion relation of a short axial wavelength mode satisfying kRc - 1

1/2can be approximated by w = kc/I /

, thereby giving the normalized

radiation frequency

w/koac = (1 - 5/2)- (12)

Shown also in Fig. 2(a) is plot of w/k0 8c in Eq. (12). Obviously from

Eq. (12) and Fig. 2(a), the normalized radiation frequency w/ko~c

increases rapidly as the dielectric constant i increases from unity

- /2to = In this regard, it is important to emphasize that the

9

- " ' - -. .. . - -i "~,t~~ . ." * .....

NSWC TR 81-103

submillimeter microwave radiation can be easily produced by this scheme

even for a moderate electron energy (Y " 1.15). The limitation of

the radiation frequency is the availability of the proper dielectric

material in the present time.

Long Helical Wavelength Mode. After a careful examination of

Fig. 1, it is noted that the LHW mode coupling can occur only for the

dielectric loaded waveguide. Figure 2(b) is plots of the normalized

radiation frequency w/k0Bc - (k + ko)/k 0 versus E for the LIH mode,

y - 1.15, several values of k0Rc, and parameters otherwise identical

to Fig. 1. Note that the normalized wiggler wavenumber k0Rc for the

LHW mode is much smaller than that for the SHW mode. However, by an

appropriate choice of the dielectric material, the radiation frequency W

for the LHW mode can be many times of the wiggler frequency k0 c.

Wide Bandwidth Amplifier. An outstanding microwave amplification

requires a broad instability bandwidth. As shown in Fig. 1, the

dispersion curves of the free streaming and dielectric waveguide modes

for kORc - 4.43 and i - 4 coincide practically in the range 4.5 < kRc <

thereby indicating possibilities of wide bandwidth amplifier. In

general, for a specified beam energy y, proper choice of the dielectric

constant c and the wiggler wavenumber k0 gives a wide band free

electron laser amplifier. The instability bandwidth can be easily

more than fifty percent.

Finally, I conclude this article by pointing out that the instability

growth rate for large wavenumber perturbations (kR >> 1) is substantiallyc

reduced by the axial momentum spread of the beam electrons,6'7

limiting the enhancement of the bandwidth ard radiation frequency.

However, the axial momentum spread of an electron beam for the free

10

NSWC TR 81-103

electron laser instability can be much less than that for other

microwave tubes such as the gyrotron. The growth rate and

bandwidth of the free electron laser instability are currently under

investigation by the author for a broad range of physical

parameters, including the influence of the axial momentum spread on

stability behavior.

Acknowledgments. This research was supported by the Independent

Research Fund at the Naval Surface Weapons Center.

NSWC TR 81-103

Rkc

FIGURE 'I PLOT OF THE DIELECTRIC DISPERSION RELATION IN THE PARAMETER SPACE(w~k) FORi -1, Rw/Rc -0.8 AND SEVERAL VALUES OF THE DIELECTRICCONSTANT Z. FOR oy -1.107 AND SEVERAL VALUES OF koRc. THE STRAIGHTLINES REPRESENT w -(k+k0 )O3c

12

NSWC TR 81-103

10 1

Rw/Rc = 0.8 Rw/RC =0.81

L=1wi koRClO0 cO

ko f c kogc

koRc 6

koRc8

0 €

03 5 7

FIGURE 2 PLOT OF NORMALIZED RADIATION FREQUENCY w/kn 3c VERSUS @FOR (a) THE SHW MODE,7 -1.1, (b) THE LHW MODE, / =1.15, SEVERALVALUES OF k0R AND PARAMETERS OTHERWISE IDENTICAL TO FIG. 1

1

t 13/14

NSWC TR 81-103

BIBLIOGRAPHY

1. 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).

2. P. Sprangle, C. Tang, and W. M. Manheimer, Phys. Rev. Lett. 43, 1932 (1979).

3. A. T. Lin and J. M. Dawson, Phys. Rev. Lett. 42, 1670 (1979).

4. I. B. Bernstein and J. L. Hirshfield, Phys. Rev. A20, 1661 (1979).

5. R. C. Davidson and H. S. Uhm, Phys. Fluids 23, 2076 (1980).

6. H. S. Uhm and R. C. Davidson, "Theory of Free Electron Laser Instability in

a Relativistic Annular Electron Beam", submitted for publication.

7. H. S. Uhm and R. C. Davidson, "Free Electron Laser Instability for a

Relativistic Solid Electron Beam in a Helical Wiggler Field", submitted for

publication.

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