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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
SEMEOE
... WC R...1 -13
SBANDWIDTH ANDEREQUENCY ENHANCEMENTOF FREE ELECTRON JASER IN A MILDLYRELATIVISTIC ELECTRON BEAM -
BV
RESEARCH AND TECHNOLOGY DEPARTMENT ELECTE
Ml -. , JUIL 29 19810MA I ,1
Approved for public release, distribution unlimited
SW
NAVAL SURFACE WEAPONS CENTER
.LKS Dahigren, Virginia 22448 0 Silver Spring, Maryland 20910
,L *' (
/
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (When Date Entered)
READ INSTRUCTIONSREPORT DOCUMENTATION PAGE REOMTINGTORBEFORE COMPLETING FORMI. REPORT NUMBER j2oGVTACCESS|ON N. 3. RECIPIENT'S CATALOG NUMBER
NSWC TR 81-103__ _ _ _ _ _ _
4. TITLE (nci Subtitle) S. TYPE OF REPORT & PERIOD COVERED
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
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WORK UNIT NUMBERS
Naval Surface Weapons CenterCode R41White Oak. Silver Spring- MD 20910
11. CONTROLLING OFFICE NAME AND ADDRESS 12, REPORT DATE
Anril 198113. NUMBER OF PAGES
22IC4 MONITORING AGENCY NAME & AODRESS(II different trom Controlling Olflice) IS. SECURITY CLASS. (of this report)
UNCLASSIFIEDISo. DECLASSIFICATION/DOWNGRADING
SCHEDULE
IS. DISTRIBUTION STATEMENT (of this Report)
Approved for public release, distribution unlimited
17. DISTRIBUTION STATEMENT (of the ebstract entierd in Block 20, If different from Report)
I&. SUPPLEMENTARY NOTES
IS. KEY WORDS (Continu on reverse side if necessary aid Identify by block number)
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
SE C'RIT"€
C'.. ASSIFICATION OPL "I41I PAGE r
37hen Dete Nnree.1
. ..... . ... .. .. ... " .. .. ... .." " " -" .° " ".. .. ,,,h, ;.--. ... . m -" ."'V ,f .-,. ....
UNCLASSIFIED. 4... ITy ' ASIFtICATiOh 1 THIS PAGEfWhon Data Entered)
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGEfwhen Date Entered;
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
By-- - ----- -Distribution!
vailability CodesAvail and/or
Dist Special
1/2
I'.!
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.
15/16
NSWC TR 81-103
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