AD-AL49 686 BETATRON-SYNCHROTRON DETRRPPING IN A TAPERED WIGGLER i/iFREE ELECTRON LASER(U) NRVAL RESEARCH LAB ISSHINGTON DCP SPRRNGLE ET AL. 31 DEC 84 NRL-MR-5445
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Betatron-Synchrotron Detrapping in a Tapered Wiggler Free Electron Laser -12 P-ERSONAL AUTHOR(S)
* Sprangle, P. and Tang C.M.Ila TYPE Of REPORT i3b TIME COVERED 14 DATE OF REPORT (Yea, Month. Day) 5 PAGE COUNTInterim FROM 6_§70 _TO 1W874 1984 December 31 16
*16 SUPPLEMENTARY NOTATION This work was sponsored by the Defense Advanced Research Projects Agencyunder Contract 3817, and the Department of Energy under Contract DE-AI05-83ER40117.
IiCOSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)
L FIELD GROUP SUB-GROUP Free electron laseri,
*~ Betatron-synchrotron stability. -
19 ABSTRACT (Continue on reverse if necessary and identify by block number)
Betatron -synchrotron resonance detrapping is shown to take place when the wiggler magnetic field amplitudeis tapered. This resonance exists even if the radiation wavefronts are not curved and is only dependent on thetransverse gradient of the tapered wiggler field. 1. r - ..
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BETATRON-SYNCHROTRON DETRAPPING IN ATAPERED WIGGLER FREE ELECTRON LASER
It has been pointed out by M. Rosenbluth1 that electron detrapping in the--'S
ponderomotive wave can occur if a resonance between the electron's synchrotron
and betatron oscillations exist. Synchrotron oscillations are due to the
2-7-- ,trapped electrons oscillating longitudinally in the ponderomotive wave,
while the betatron oscillations are predominantly transverse electron
oscillations due to the transverse spatial gradients associated with the
wiggler field.8 - 14 If the radiation wavefronts are curved these two
oscillations can be resonantly coupled and can lead to electron detrapping in
the ponderomotive wave.1,15-18 When the radiation wavefronts are curved the
electrons which are undergoing transverse betatron oscillations will
experience periodically different phases in the ponderomotive wave. When the
periodically changing phase, due to the transverse betatron oscillations, is
resonant with the electron's synchrotron oscillations these later oscillations
can be amplified and result in detrapping.
A very qualitative model for this process can be seen by considering the
pendulum equation I- 3 '6 - 7'15- 18 for electrons in a wiggler field with
transverse spatial gradients and a radiation field with a curved wavefront.
For electrons deeply trapped in the ponderomotive wave the pendulum equation, .
roughly speaking, reduces to a driven harmonic oscillator equation. The
characteristic frequency of the electron oscillation is the sychrotron
frequency and is proportional to the fourth root of the radiation field
power. The amplitude of the periodic driving term in the oscillator equation
is proportional to the inverse of the radius of curvature associated with the
radiation field and the period of the driving term is proportional to the
electron betatron period. The electron phase, described by the pendulum
Manuscript approved October 22, 1984.
: :., . . : .. ,. ... .: -..:. .- : :.. :- :.- -a- . -- . . -" .., , . . .... .
S-.:equation, can be amplified if the frequency of the driving term is resonant
with the characteristic trapping frequency. This detrapping mechanism could
limit the level of radiation power generated by the FEL.
In this paper we suggest and analyze an alternative mechanism for
sychrotron-betatron resonant detrapping which does not depend on the curvature
of the radiation wavefronts. We sho that even for a one dimensional
radiation field sychrotron-betatron resonant detrapping can take place for
tapered wiggler fields, i.e., the wiggler magnetic field amplitude.
To analyze this synchrotron-betatron resonant detrapping mechanism we
choose a tapered linearly polarized wiggler field described by the vector
potential
zj0 A (y,z) A (z)coshfk (z)y)cos(( k(z')dz'e ,(1)
w w w - x
where Aw(z) and kw(z) are the spatially slowly varying amplitude and
wavenumber. It will be assumed later that kw(z)y is somewhat less than unity
2so that cosh(k y) I + (k Y) /2. The one dimensional radiation field is
w w
described by the vector potential .1AL(z,t) = AL sin(kz-wt+O)ex, (2)
where the amplitude AL, wavenumber k = w/c , frequency w and phase 0 are
assumed constant. Since (1) and (2) are independent of the x coordinate, the
electron's canonical momentum in the x direction is conserved, i.e.,
d(P - jel(A + AL)/c) • ex/dt 0 where P - Ym v = (P ,P ,P ) denotes the* Z 0' x y z
electron's mechanical momenta. The relativistic particle orbit equations
become
Py- 2 2 3A2(y,z,t) (3a)
y 2ym C .
2 a
S. . -. . - ._- - . . . - - - "
2 2- 1L. aA (y,z,t) (b
z 2 a2 ym c0
where A(y,z,t) (A (y,z) + tL(Z,t)), e x is the total x component of the .
2 2)!/2.vector potential and x = (I + P ° P/mc2) /. In obtaining (3) we used the
0
fact that
* 0
p= lel A(y,z,t) + P (4)x c ox' , .
and assumed that the injected momentum in the x direction is zero, i.e.,
P = 0. Using (3) and (4) the electron's axial velocity, vz, is given by
= 2l 2 + _) A2 (y,z,t). (5)z 2y 2 m2 c2 c0
In what follows it proves convenient to perform a transformation from the
independent time variable t to the independent position variable z. An
electron's phase with respect to the ponderomotive wave, in terms of the
position variable z, is defined as 0
z(Yo ',o Z ) =) + f ( WZ) + k - w/z(y ,*o,'o,z'))dz', (6)
0 0
where yo is the electron's y position at z = 0, * is the phase at z = 0,0
= al/z = k (0) + k - w/v and v is the axial velocity at z = 0SZ= w 0 0(assumed to be identical for all electrons). Quantities denoted with the
superscript - are functions of the initial condition variables yo,0 0 and
the independent variable z. Thus, for example, v is the axial velocity at
position z of an electron with initial condition (at z - 0) variables
•.-.- . . .- . . . . ..
.: ======= ======== .- . .- .:: .ii : -i: : . . . :: :" : : . . * • -"
yo,'o and €o" Differentiating (6) twice yields
Vk + ,-/-2 (7)z z 0
where - denotes the operator a/3z. Upon performing the operation
a/az + v C 2 a/3t in (5), transforming the resulting equation from the
independent variable t to the variable z and substituting the result into (7),
we arrive at the generalized pendulum equation for a tapered wiggler with .
transverse spatial gradients,
= le12ww -2 2 2-32ym c v
o z •os2(k 2 2 "
f(2AwA' cosh (ky) + A k sinh(2k y)cos 2 r k dz-)0
2 2 -z0
k Aw cosh (kwY) sin(2 f kdz')w w k wd.'"0
+ (A'A cosh(kwj) + AwALkw sinh(kwY))sinZ
+ (kw + k - z /c2)AwA.Lcosh(kC)cosp (8)
where v = w/(k + k - Z') and = (z). The last term in (8) represents thez w
usual ponderomotive potential wave.
Before going on to simplify (8) an equation describing the electrons
betatron motion (transverse oscillations in y) is needed. Using (3 a) and 0
assuming IkwyI<<1 and AL<<IAwj we find that
4
. .... _ *: *::.. . .. .. . . ..
S 2 lelA2k 2 zY - cos 2 k (z')dz')y. (9)
y m c 0
In arriving at (9) we have also assumed that 1yyI<<Yy, these approximations
can be shown to be well satisfied. Transforming (9) from the independent
variable t to z and setting v = c yields the following equation for the
betatron orbits,
2oc+ k(z ( +cos(2 rk(z')dz'))y 0, (0
where k (z) = ecA ) -kw/v!- and w -2eJAw(z)l moc is the1/ an 0w w eIw 0/Yoc
normalized wiggle velocity. Since the betatron wavenumber, k,, is much
greater than kw, the betatron motion can be separated into a slowly varying
part and a small rapidly varying part. Neglecting the rapid variations
in (z), the solution of (10) is 0
y(z) = ( ()/k(f k (z')dz + (O)), (11)
0
where y(O) and (0) are constants.
We can now proceed to simplify the pendulum equation in (8). Assuming
(ky)2<< 1, A- << k Aw, k- << k2 and keeping only the slowly varying terms,ww w w w w
(8) reduces to the form
_ d k d A 2
d2 + K (z)cos2 [ ddz 2 dz dz-
z
E(z)(I + cos(2 f k dz' + 2 (0)))I, (12)0
.- o.- , -.- --.--. . /.- -
62 A/-2 2c5)1/2where K (Z) ml Ae A(m is the synchrotron wavenumber,
a' a2 = lelw/(4y2m2c5) (z) - v2(0)(k (0)/2k (z))d(Awkw)2 /dz and (11) was used
to replace W(z). We can further simplify (12) by noting that
dk dA2
2w 2 wCL &(z) << adz ' dz
Therefore we may keep only the driving term associated with the betatron
oscillations which could amplify the synchrotron oscillations, i.e.,z
cos(2 f k dz' + 2o(0)), and (12) reduces to0 0
d 2' + K2(Z)sin= -d" w
dz2 dz
2 dA2 z
- w d- + &(z)cos(2 f k dz-)1, (13)
0
where for later convenience we have shifted the phase 4 by w/2,
i.e. 4 = 4 + 7r/2, and set (0) = 0. Note that the electron's energy,
neglecting transverse wiggler gradients, is determined by the relation
1 12 AwALk2yw 4-2-4sin*. (14)
2ym° c0O
Defining the resonant phase, *R' in the usual way, i.e.,
sin*R (dkw/dZ - c2 dA2 /dz)/K2, (15)
* 0
the pendulum equation (13) becomes
*'. 6 S
• .,;. ... -. ... [ ' -, . . - , - " l.. ;. -.... , -- .. ..' 6
*''°" .0"" ..j, "' " • . . , . " "" - ° - m . "', . " ,". '
- .-- ... .
2 z+ K2 sinW = K2 (1 + e cos(2 r k dz))sin*Rd 2 s s R- .'
dz2 02
dk z- c(1 + -2- cos(2f k dz'), (16)k- dz--cs2
w 0
where c = k2 y2(0)/2.w
We now consider the particularly simple illustration of a wiggler field
with a constant period (k /3z = 0) and linearly changing amplitude,w
Aw(z) - Aw(0) + 6Awz/L, where Aw(0) >> I6AwI is constant and L is the length of
the wiggler field. The pendulum equation in (13) for the case where
16A w/A w(0) << 1, Ks = 2k P( + 6) and 161 << 1, i.e., the synchrotron and
betatron oscillations are resonant, reduces to
2d24i + sin p = ( + € cos~l + 6)Z)sinPR, (17)
dZ2
2
where Z = K z and sin R = - y( 6A /A (0))/(2k L ). For the case where Aw is
constant and k w(z) k w(0) + 6k z/L varies linearly, the pendulum equation in
(13) reduces to
sini= (- y cosO + 6)Z)sin R, (18)
dZ2 w z
where sin R = (6w/k (0)/(2P2L k (0)).
R w w w w w
Rather than obtain an approximate solution to (17) or (18), using a multiple
time scale approach, we simply solve (17) numerically. Initially the phases are
distributed uniformly between 0 and 2w with aZ/Dz = 0. Figure (1) shows the
precentage of trapped particles as a function of normalized distance for
sin*R = 0.3 and E = 0.1 and 0.15. Initially approximately 65% of the particles
are trapped and for c = 0 this fraction is of cause maintained. After about 5
synchrotron oscillations 55% are trapped for c = 0.1 and 50% for c = 0.15.
Since we have assumed zero beam emtttance, these results apply only to those
7
. . . --
electrons initially on the outer edge of the beam, i.e., those having the largest
value of c. Figure (2) shows the percentage of particles detrapped after 10
synchrotron oscillations as a function of the mismatch parameter 6, for
siniP R = 0.3 and c = 0.1, 0.15. The percentage of detrapped particles maximize
when 6 < 0 since the more deeply trapped particles oscillate slightly faster than
those trapped nearer the phase space sepratrix. Here again these results apply
to only a small fraction of the total number of beam electrons, those initially
near the edge of the beam.
Our model is somewhat idealized since, among other things, beam emittance
has been neglected. The neglect of emittance implies that all the electrons have
the same initial betatron phase, 4(O), see Eq. (11). Hence, those electrons
injected near the axis will not experience the betatron synchrotron detrapping
since the value of c for these electrons is much smaller than those near the edge
of the beam.
ACKNOWLEDGMENTS0
This work is sponsored by DARPA under Contract 3817, and DOE under Contract
DE-AI05-83ER40117.0
8
.2 70-~4-
v 6
0-
60-
S50-
Sin %PR 0.3 60.15, 8 0.2o 40 -" 5o-
C 0
040
0 10 20 30 40 50 60 70a)n Z= KsZ
Fig. 1 - Percentage of trapped electrons as a function of normalized •distance for sinR = 0.3, and two values of E.
9
Aj -j.
4 00
30 After 10 SynchrotronOscillations -0
" 4-
7.
20 0
0 20 15
0.0
E 0. 100 10 -Sin *'R = 0.3
C
00
L3.. :
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2.
Fig. 2 - Percentage of trapped electrons after 10 synchrotron oscillations as a function ofthe mismatch parameter 6 for sin R =0.3 and e = 0.1, and 0.15.
4-q
.
a)
* 4 - 4 ° , . . .. , . . . . . . . .° .. . ' : - .."" ° "
C 10
00
References
1. M. N. Rosenbluth, "Two-Dimensional Effects in FEL's", paper No. I-ARA-83-U-
45 (ARA-488), Austin Research Asso., TX, 1983.
2. W. B. Colson, Phys. Lett. 64A, 190 (1977).
3. N. M. Kroll, P. L. Morton and M. N. Rosenbluth, IEEE J. Quantum Elec.
QE-17, 1436 (1981).
4. P. Sprangle, C.-M. Tang and W. M. Manheimer, Phys. Rev. Lett. 43, 1932
(1979).
5. P. Sprangle, C.-M. Tang and W. M. Manheimer, Phys. Rev. A21, 302 (1980).
6. C.-M. Tang and P. Sprangle, J. Applied Phys., 52, 3148 (1981).
7. C. M. Tang and P. Sprangle, Proc. of the 1981 IEEE Intl. Conf. on Infrared
and Millimeter Waves, Miami Beach, FL, 7-12 Dec 1981, pp. F-3-1.
8. T. I. Smith and J. M. J. Madey, Appl. Phys. B27, 195 (1982).
9. P. Diament, Phys. Rev. A23, 2537 (1981).
10. V. K. Nell, "Emittance and Transport of Electron Beams in a Free Electron
Laser", SRI Tech. Report JSR-79-10, SRI International, 1979. ADA081064
11. C. M. Tang, Proc. of the Intl. Conf. on Lasers "82, 164 (1983).
12. P. Sprangle and C.-M. Tang, Appl. Phys. Lett. 39, 677 (1981).
13. C. M. Tang and P. Sprangle, Free-Electron Generator of Coherent Radiation,
Phys. of Quantum Electronics, Vol. 9, (ed. by Jacobs, Moore, Pilloff,
Sargent, Scully and Spitzer), Addison-Wesley Publ. Co., Reading, MA, 627
(1982). p. 627
14. A. Gover, H. Freund, V. L. Granatstein, J. H. McAdoo, and C. M. Tang, "Basic
Design Considerations for Free Eelctron Laser Driven by Electron Beams from
RF Accelerators", Infrared and Millimeter Waves, Vol 12, ed. K. J. Button.
15. C.-M. Tang and P. Sprangle, Free-Electron Generators of Coherent Radiation,
SPIE Proc. 453, (ed. by C. A. Brau, S. F. Jacobs and M. 0. Scully),
Bellingham, WA, p. 11 (1983).11
. . * * *. . . . .
16. C. M. Tang, "Particle Dynamics Associated with a Free Electron Laser", to be
published in the Proc. of Lasers '83, held at San Francisco, CA, Dec 12-16,
1983.
17. W. M. Fawley, D. Prosnitz and E. T. Scharlemann, accepted for Phys. Rev. A.
18. C. M. Tang and P. Sprangle, "Resonance Between Betatron and Synchrotron
Oscillations in a Free Electron Laser: A 3-D Numerical Study", to be
published in Nuclear Instruments and Methods in Physics Research (Section
A).
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