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A powerful and efficient multibeam microwave FEL amplifier
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH
Nuclear Instruments and Methods in Physics Research A 341 (1994) 300-304 North-Holland
Seoon A
A powerful and efficient multibeam microwave FEL amplifier 1. Boscolo a °*, G. Jiantning
° University and INFN of Milan, Italy n University and INFN of Ban, Italy
a,l,
P. Radaelli °, V. Variale b
The design of a 400 MW-20 GHz multibeam FEL amplifier, aimed to be a source for a high gradient linac is presented. A 120 A, 5 MV electron beam from an electrostatic accelerator is simulated. FEL simulation confirms an efficiency as high as 70/. The space-charge effect is strongly reduced by the cavity walls.
1. Introduction The FEL as a powerful high frequency microwave source is a goal to pursue after the Livermore experiment [1]. In fact, a GW power level (at 35 GHz) and an efficiency of 45% [2] were reached. A theoretical efficiency up to 70%, for an FEL operating at a frequency lower than 20 GHz, seems feasible as long as the electron beam (eb) energy is higher than 5 MeV (y > 10) [3]. The encouraging results of the study on powerful electrostatic accelerators [4] and on the microwave FEL efficiency, give momentum to the proposal of an RF source based on an FEL driven by an electrostatic accelerator . We call this source FELTRON. In this paper we discuss a multibeam design of FELTRON, tailored to the TeV high gradient structure (HGS). In a previous paper [5] we saw that efficiency, power, repetition rate, dimension, capability to produce a set of parallel rf beams and costs meet the requirements of HGS. A HGS operating at 20 GHz with an accelerating gradient of 100 MV/m, asks for a power of 200 MW, a set of 6-10 rf beams of 50 ns long, and a rep rate of 1 kHz . The linac must be fed at each meter due to high losses . Considerations on the recovery of the spent eb (that is on the overall efficiency) and on the number of sources per unitary length lead to a machine design with 3-4 separated electron beams. In fact, to push further the overall efficiency, it is likely to recover the set of spent electron beams. This can be done only by
* Corresponding author . On leave from Southwest Jiaotong University, Chengdu, P.R . China.
realigning the electron beams after the FEL interaction . The alignment of a set of beams seems possible for a small number of beams. Simple assessments indicate that three separate beams, having an average energy spread of a few percent relative to one-another, are realignable . The eb power must be 600 MW in order to get 400 MW or RF beam power (with 70% extraction efficiency). We choose a current I = 120 A and a voltage V= 5 MV . In this paper we draw our attention to the effect of the space charge . It is shown that the conducting walls have an important role in strongly reducing the space charge force. Because of this reduction we can state that the Compton regime is shifted to lower frequencies than those expected with the usual theory . A powerful electrostatic accelerator provides a long cb, 150 ns for a three beam FEL (500 ns for a ten beam FEL) . The long eb is composed of pieces with a stepwise energy shape (Figs. I and 2) . Two adjacent beams have an energy difference of l% : Ay,,/y=
l%o .
It
Fig. 1 Voltage drop of the high voltage terminal with the superposition of the sawtooth generated by a small ramp induction lmac .
0168-9002/94/$07.00 © 1994 - Elsevier Science B V. All rights reserved SSDI0168-9002(93)E1096-G
1. Boscolo et al. I Nucl Instr. and Meth to Phys Res. A 341 (1994) 300-304 5 20 WV
5 14 61eV
i
5 26 MV
Table 1 FEL parameters
50 ns
1
526 MeV
3 cm
AY1 J 1Y=1%
301
514 MeV DISPERSIVE SECTION
Fig. 2 . Sketch of the electron beam before and after the magnetic dispersive section ; the pieces of the beam, scaled in energy, are spatially separated ; the dispersive section is composed of a doublet of thin magnets and a drift space in between . A dispersive system divides the long eb into a set of parallel beams, 50 ns long each (Fig . 2).
2 . General considerations on the microwave FEL of FELTRON project For the high power requirement, the FEL amplifier must operate in high gain + tapered regime. This means that the Pierce parameter p has to be greater than 0 .01, i .e . (for the definition of p and its ranges in the different regimes see for example ref. [6]) 113 2 3 2l3 I QO A~ / p = 0 .00833fB 3 > 0 .01, (2) / ( a h12 h/2 ) Yr a n =0 .6681,[T]A [cm] is the wiggler parameter, B o and A n are the field and period of the wiggler, yr is the beam resonant energy, I is the beam current, and a and h are the waveguide dimensions . In order to get that high value for p, a tradeoff between the wiggler parameter a o , the beam energy yr and the current density on the waveguide mode (2I/ah) must be worked out . The voltage V and the current I depend on the technological possibilities of the electrostatic accelerator . The voltage and current tradeoff is fixed at V = 5 MV and I = 120 A. The eb power results then in en VI = 600 MW and in turn the rf power results in RF = 420 MW . This rf output is divided (77 into two rf beams as shown in Fig . 3 . The tradeoff between the height h of the waveguide, and so in turn of the wiggler gap, the wiggler field intensity B and
P = = 70%) P
Energy (MeV) (y =11 .28) Current (A) Energy spread Normalized emittance (mrad) Electron beam radius (mm) Pierce parameter yll Undulator length (m) Undulatortype Polarization w-wave wavelength (cm) Wiggler period (cm) Peak field (KG) Wiggler parameter Gap (cm) Waveguide (cm 2 ) Small signal gain Mode Frequency (GHz)
5 .25 120 < 1 .5% E < 10-3 4 p-3% 2 .2 9 hybrid with canted poles planar A, =1 .5 A o = 10 B D = 7 .5 ao = 5 g = 3 ah = 3 * 3 G o = 11 TE O , 20
period A o (in order to match the synchronism condition and the efficiency requirement) is discussed in ref. [3] . The FEL parameters are listed in Table 1 .
70%
3 . Multibeam FEL design The multibeam FEL is obtained by separating a unique long eb into several pieces . The FEL is composed of a unique wiggler with a set of parallel waveguides attached to one another as shown in Fig. 4 . The matching of the ponderomotive wave velocity cu/k,, and the longitudinal eb velocity upm = 6,,c, for the different energies, is obtained by varying the wavenumber k ll = (k 2 - k 2 ) 1 / 2 = (k 2 - (,r/h)2)1/2 (see ref . [3]). From the synchronism condition As= A i,(1 + a2) y2 A o ljl +ao 1 -
~2
2hy,6ii
Electron Beams
Fig . 3 . Sketch of the FEL amplifier with three input electrons beams and the six output RF beams .
Fig . 4. Schematic of the three beam FEL with a unique wiggler . The three waveguide heights are different in order to match the requirement of the synchronism condition with different y. V. PERFORMANCE STUDIES
L Boscolo et al. I Nucl. Instr. and Meth. in Phys. Res. A 341 (1994) 300-304
302
we obtain the value of h at the three different y. The relation between the waveguide height and the beam energy is approximately given by 8h = 8y/3 . Choosing the three waveguide vertical dimensions h, = 0 .03 m, h 2 = 0 .0317 m, h 3 = 0 .0341 m we obtain y, = 11 .28, y2 = 11 .16, and y3 = 11 .04, respectively . In order to find the quantities of our interest, that is bunching b, length of the regular and tapered wiggler sections, rf power per beam and efficiency, we have to solve the set of motion equations, including space charge and without the usual approximation y2 >> 1 (since yll = 2.2). We have used the equations of ref. [7] . We remark that the normalized space-charge force
F,
e
p(r, z, t) =p(r)p(z, t) = ( I - - ) z Y_ (P .» ri; "t
e ""H
+ c .c . ) ,
(6)
where r,, is the beam radius and P",
=
I
2 b", > ßc,rr r.2
0= (kll+ko)z -
E,
mc z
would be two times higher than the ponderomotive force FP. = aoe,fu/yßll if the longitudinal electric field E, is calculated from the equation V -E = ple a . To perform the calculation of the electric field due to the space charge when a bunched electron beam travels within a pipe we have to solve the equation 2
A parabolic transverse beam profile p(r) _ (1 r = lrh) [8] and a (z, t) dependence with three harmonics of the ponderomotive potential are assumed
1 _a2
C2
E ôt 2sc
_
J
Solving Eq . (5) we find for the field E,, -i
with the waveguide boundary condition. In Eq . (5) the charge density p is related to the current density by J,, = pv . We notice that J, as well as p, must refer to the wiggling trajectory . In our problem (with high magnetic field and relatively small y) the coordinates along the trajectory cannot be approximated with the coordinate z. To solve Eq . (5) we must define the eb distribution function p(r, z, t) . This problem has been treated in ref. [8] in the case of cylindrical symmetry, i.e . a cylindrical waveguide . We can follow the same framework even if we have a rectangular waveguide, because the wall effect does not depend on the symmetry. In fact, the effect of a metallic wall is to set the image charges, whose action does not depend on the geometric form but on the distances. The metallic wall of the waveguide creates an image eb of positive charge which runs coupled with the real eb . The reduction of the interaction between two adjacent eb bunches comes from the fact that each eb bunch also sees a bunch of opposite sign . When the relative distance between the bunches is the same or almost the same as the distance equal to between the real and image beam (that is, when the transverse waveguide dimension is about the radiation wavelength), the two symmetric positive and negative charge trains are tightly coupled (the field lines go from positive to negative charges) . The cylindrical geometry is chosen because the solution of Eq . (5) is relatively simple to find .
m
e (ku+kll~ m iC
ail.
( P at +v e 0
1
YPm e
1-
4 2 g 2 ,,2 Mr
r-
( rh, - fmI,(mgr)(,
where g = (ko+kll ) /y, I,,(mghl 2 )Kz(mgrn) - K,,(mghl 2 )Iz(mgr,,)
The coupling of the longitudinal electron field to the electrons is given by the equation of motion for the energy of an electron d yJ
e
dt
mc2vJ
E" .
Combining Eq . (8) with Eq . (7) we get: d yJ
e
dZ
me- ,,
"e
e mc -
E, c' H' ,
where J,(m~), with ~---a /[2(1 +a2)], arises from an average over a wiggler period . The number of harmonics to consider in our calculations has been estimated by calculating the harmonic bunching b , with the FEL code and the reduction factor f =J F,"/m, where F -_
I -
4
r2 / - - - fmlolmgr) rh m z g rn z
I.
In Fig. 5 the curves of b", are plotted for m = 1, 3, 5. The bunching of the first harmonic is very tight, but the others are three times less (within the tapered
I. Boscolo et al. / Nucl. Instr. and Meth . in Phys. Res. A 341 (1994) 300-304
303
wiggler section) . The product b.J0F ,/rn reduces itself at higher harmonics to a negligible value compared
with the first one. The space-charge reduction factor f for the FELTRON project comes out to be about 0.05. With this number the space-charge force becomes very small with respect to the ponderomotive force. We point out that the space-charge term calculated in this way has the expected dependences on y, wavelength
and waveguide dimensions relative to the radiation wavelength. We expect a reduction of the space charge with y, with frequency and with h . Incidentally, we notice that the bunching at higher harmonics could be higher starting the tapering at the proper z for each
harmonic, but we start at the maximum of the first harmonic . In support to our arguments, we remark
that, at the Livermore experiment [1], although the parameters gave a ratio between the space charge
force and the ponderomotive force of about two, the power and bunching where not affected by the space
0 1
1
2
3
4
5
67 . . 8 .. WIGGLER LENGTH (m)
Fig. 6. Output power increases with interaction distance z in the tapered regime
charge, even if the cavity was 10 cm wide (the other dimension was 3 em).
We have taken this wall effect into consideration in our computer code by rewriting the space charge field with an ad hoc value of the beam area Asp to match
around z - 5 m comes from the fact that we have used a linear tapering, while the variation of
y and ao
the real field as written in Eq . (9).
(which has a value around 2.5
at that point) is no
solving numerically the equations of motion . The dia-
1 (we do not have the auto-tapering option in our
reaches a value around 0.8 and that it remains substantially stable up to the end of the process. Down the
Fig. 6, is PRF = 420 MW, as expected with a wiggler
the electrons in the bucket
FELs with three equal and matched waveguides . The difference in power in the first case results in about
We report in Figs . 5 and 6 the results obtained by
longer linear, since ao2 is not negligible with respect to
gram of the bunching factor b, Fig. 5, shows that b
code). The extracted power, illustrated by the plot of
tapered wiggler the decrease is due to smearing out of due to
the non-linear
effects. The change in the slope of the bunching curve
field reduction up to 0.5 kG .
We have compared the output power of the parallel
5%, i.e . 20 MW, while in the second case the difference decreases to 2-3% .
5. Conclusions A 20 GHz microwave FEL set up with a set of 3 parallel electron beams is studied . The voltage and current are respectively 5 MV and 120 A for a peak power of 600 MW. The Pierce gun has a gradient of 5 MV/m
and a cathode area of about 10 CM 2 . The electron beam has been simulated from the gun up to
the exit of the accelerating column . The linear voltage
drop of the high voltage terminal due to the 120 A current is transformed into a stepwise ladder by the superposition of a sawtooth voltage. The electron beam exits the accelerator with a succession of sections scaled down in energy of 1% from one to another. A disper-
Fig. 5 . Bunching versus wiggler length for the first, third and fifth harmonics .
sive magnetic system separates those sections into a set of parallel electron beam pulses, each 50 ns long . The FEL has a hybrid wiggler, 20 cm wide, with a set of parallel rectangular waveguides attached to each other. The horizontal dimension is the same for all the V. PERFORMANCE STUDIES
304
I Boscolo et al. /Nucl Instr. and Meth . in Phys . Res. A 341 (1994) 300-304
waveguides, 30 mm, while the dimension is increased (starting from 30 mm) with the decreasing energy of the beams. This variation of the vertical size is because we have to fulfil the synchronism condition. The small variation in the output power that comes from the different power of the electron beams is unimportant . Each FEL channel should deliver 400 MW RF power with an extraction efficiency of 70% . That power is then divided into two parts of 200 MW each, because this is the power level required by the high gradient structure of the TeV collider . The FEL has been simulated with a 1-D code . The space-charge effect is reduced by more than 90% with the parameters of the project. This reduction is due to the image beam created by the metallic wall of waveguide.
References [1] T.J . Orzechowski et al ., Phys . Rev Lett . 57 (1986) 2172 [2] A.M . Sessler, Proc . CAS/CERN 9003, Chester College, Chester, UK, 6-13 April 1989, ed S. Turner, p 335 [3] I. Boscolo, Opt. Commun . 98 (1993) 193. [4] I. Boscolo, F. Giuliani, M. Valentim and M. Roche, IEEE Trans Nucl Sci NS-39 (1992) 308; 1. Boscolo, F. Giuliani and M. Roche, Nucl . Instr. and Meth A 318 (1992) 465 [5] I. Boscolo and L. Elias, Nucl . In-,tr . and Meth A 309 (1991) 585 . [6] R. Bonifacio, C. Pellegrini and L. Narducci, Opt. Commun. 50 (1984) 373 . [7] T.J . Orzechoswki, E.T. Scharleman and D.B . Hopkms, Phys . Rev. A 35 (1987) 2184 . [8] E.T . Scharlemann, W.M Fawley, B .R . Anderson and T.J . Orzechowski, Nucl . Instr. and Meth . A 250 (1986) 150.
Nuclear Instruments and Methods in Physics Research A 341 (1994) 300-304 North-Holland
Seoon A
A powerful and efficient multibeam microwave FEL amplifier 1. Boscolo a °*, G. Jiantning
° University and INFN of Milan, Italy n University and INFN of Ban, Italy
a,l,
P. Radaelli °, V. Variale b
The design of a 400 MW-20 GHz multibeam FEL amplifier, aimed to be a source for a high gradient linac is presented. A 120 A, 5 MV electron beam from an electrostatic accelerator is simulated. FEL simulation confirms an efficiency as high as 70/. The space-charge effect is strongly reduced by the cavity walls.
1. Introduction The FEL as a powerful high frequency microwave source is a goal to pursue after the Livermore experiment [1]. In fact, a GW power level (at 35 GHz) and an efficiency of 45% [2] were reached. A theoretical efficiency up to 70%, for an FEL operating at a frequency lower than 20 GHz, seems feasible as long as the electron beam (eb) energy is higher than 5 MeV (y > 10) [3]. The encouraging results of the study on powerful electrostatic accelerators [4] and on the microwave FEL efficiency, give momentum to the proposal of an RF source based on an FEL driven by an electrostatic accelerator . We call this source FELTRON. In this paper we discuss a multibeam design of FELTRON, tailored to the TeV high gradient structure (HGS). In a previous paper [5] we saw that efficiency, power, repetition rate, dimension, capability to produce a set of parallel rf beams and costs meet the requirements of HGS. A HGS operating at 20 GHz with an accelerating gradient of 100 MV/m, asks for a power of 200 MW, a set of 6-10 rf beams of 50 ns long, and a rep rate of 1 kHz . The linac must be fed at each meter due to high losses . Considerations on the recovery of the spent eb (that is on the overall efficiency) and on the number of sources per unitary length lead to a machine design with 3-4 separated electron beams. In fact, to push further the overall efficiency, it is likely to recover the set of spent electron beams. This can be done only by
* Corresponding author . On leave from Southwest Jiaotong University, Chengdu, P.R . China.
realigning the electron beams after the FEL interaction . The alignment of a set of beams seems possible for a small number of beams. Simple assessments indicate that three separate beams, having an average energy spread of a few percent relative to one-another, are realignable . The eb power must be 600 MW in order to get 400 MW or RF beam power (with 70% extraction efficiency). We choose a current I = 120 A and a voltage V= 5 MV . In this paper we draw our attention to the effect of the space charge . It is shown that the conducting walls have an important role in strongly reducing the space charge force. Because of this reduction we can state that the Compton regime is shifted to lower frequencies than those expected with the usual theory . A powerful electrostatic accelerator provides a long cb, 150 ns for a three beam FEL (500 ns for a ten beam FEL) . The long eb is composed of pieces with a stepwise energy shape (Figs. I and 2) . Two adjacent beams have an energy difference of l% : Ay,,/y=
l%o .
It
Fig. 1 Voltage drop of the high voltage terminal with the superposition of the sawtooth generated by a small ramp induction lmac .
0168-9002/94/$07.00 © 1994 - Elsevier Science B V. All rights reserved SSDI0168-9002(93)E1096-G
1. Boscolo et al. I Nucl Instr. and Meth to Phys Res. A 341 (1994) 300-304 5 20 WV
5 14 61eV
i
5 26 MV
Table 1 FEL parameters
50 ns
1
526 MeV
3 cm
AY1 J 1Y=1%
301
514 MeV DISPERSIVE SECTION
Fig. 2 . Sketch of the electron beam before and after the magnetic dispersive section ; the pieces of the beam, scaled in energy, are spatially separated ; the dispersive section is composed of a doublet of thin magnets and a drift space in between . A dispersive system divides the long eb into a set of parallel beams, 50 ns long each (Fig . 2).
2 . General considerations on the microwave FEL of FELTRON project For the high power requirement, the FEL amplifier must operate in high gain + tapered regime. This means that the Pierce parameter p has to be greater than 0 .01, i .e . (for the definition of p and its ranges in the different regimes see for example ref. [6]) 113 2 3 2l3 I QO A~ / p = 0 .00833fB 3 > 0 .01, (2) / ( a h12 h/2 ) Yr a n =0 .6681,[T]A [cm] is the wiggler parameter, B o and A n are the field and period of the wiggler, yr is the beam resonant energy, I is the beam current, and a and h are the waveguide dimensions . In order to get that high value for p, a tradeoff between the wiggler parameter a o , the beam energy yr and the current density on the waveguide mode (2I/ah) must be worked out . The voltage V and the current I depend on the technological possibilities of the electrostatic accelerator . The voltage and current tradeoff is fixed at V = 5 MV and I = 120 A. The eb power results then in en VI = 600 MW and in turn the rf power results in RF = 420 MW . This rf output is divided (77 into two rf beams as shown in Fig . 3 . The tradeoff between the height h of the waveguide, and so in turn of the wiggler gap, the wiggler field intensity B and
P = = 70%) P
Energy (MeV) (y =11 .28) Current (A) Energy spread Normalized emittance (mrad) Electron beam radius (mm) Pierce parameter yll Undulator length (m) Undulatortype Polarization w-wave wavelength (cm) Wiggler period (cm) Peak field (KG) Wiggler parameter Gap (cm) Waveguide (cm 2 ) Small signal gain Mode Frequency (GHz)
5 .25 120 < 1 .5% E < 10-3 4 p-3% 2 .2 9 hybrid with canted poles planar A, =1 .5 A o = 10 B D = 7 .5 ao = 5 g = 3 ah = 3 * 3 G o = 11 TE O , 20
period A o (in order to match the synchronism condition and the efficiency requirement) is discussed in ref. [3] . The FEL parameters are listed in Table 1 .
70%
3 . Multibeam FEL design The multibeam FEL is obtained by separating a unique long eb into several pieces . The FEL is composed of a unique wiggler with a set of parallel waveguides attached to one another as shown in Fig. 4 . The matching of the ponderomotive wave velocity cu/k,, and the longitudinal eb velocity upm = 6,,c, for the different energies, is obtained by varying the wavenumber k ll = (k 2 - k 2 ) 1 / 2 = (k 2 - (,r/h)2)1/2 (see ref . [3]). From the synchronism condition As= A i,(1 + a2) y2 A o ljl +ao 1 -
~2
2hy,6ii
Electron Beams
Fig . 3 . Sketch of the FEL amplifier with three input electrons beams and the six output RF beams .
Fig . 4. Schematic of the three beam FEL with a unique wiggler . The three waveguide heights are different in order to match the requirement of the synchronism condition with different y. V. PERFORMANCE STUDIES
L Boscolo et al. I Nucl. Instr. and Meth. in Phys. Res. A 341 (1994) 300-304
302
we obtain the value of h at the three different y. The relation between the waveguide height and the beam energy is approximately given by 8h = 8y/3 . Choosing the three waveguide vertical dimensions h, = 0 .03 m, h 2 = 0 .0317 m, h 3 = 0 .0341 m we obtain y, = 11 .28, y2 = 11 .16, and y3 = 11 .04, respectively . In order to find the quantities of our interest, that is bunching b, length of the regular and tapered wiggler sections, rf power per beam and efficiency, we have to solve the set of motion equations, including space charge and without the usual approximation y2 >> 1 (since yll = 2.2). We have used the equations of ref. [7] . We remark that the normalized space-charge force
F,
e
p(r, z, t) =p(r)p(z, t) = ( I - - ) z Y_ (P .» ri; "t
e ""H
+ c .c . ) ,
(6)
where r,, is the beam radius and P",
=
I
2 b", > ßc,rr r.2
0= (kll+ko)z -
E,
mc z
would be two times higher than the ponderomotive force FP. = aoe,fu/yßll if the longitudinal electric field E, is calculated from the equation V -E = ple a . To perform the calculation of the electric field due to the space charge when a bunched electron beam travels within a pipe we have to solve the equation 2
A parabolic transverse beam profile p(r) _ (1 r = lrh) [8] and a (z, t) dependence with three harmonics of the ponderomotive potential are assumed
1 _a2
C2
E ôt 2sc
_
J
Solving Eq . (5) we find for the field E,, -i
with the waveguide boundary condition. In Eq . (5) the charge density p is related to the current density by J,, = pv . We notice that J, as well as p, must refer to the wiggling trajectory . In our problem (with high magnetic field and relatively small y) the coordinates along the trajectory cannot be approximated with the coordinate z. To solve Eq . (5) we must define the eb distribution function p(r, z, t) . This problem has been treated in ref. [8] in the case of cylindrical symmetry, i.e . a cylindrical waveguide . We can follow the same framework even if we have a rectangular waveguide, because the wall effect does not depend on the symmetry. In fact, the effect of a metallic wall is to set the image charges, whose action does not depend on the geometric form but on the distances. The metallic wall of the waveguide creates an image eb of positive charge which runs coupled with the real eb . The reduction of the interaction between two adjacent eb bunches comes from the fact that each eb bunch also sees a bunch of opposite sign . When the relative distance between the bunches is the same or almost the same as the distance equal to between the real and image beam (that is, when the transverse waveguide dimension is about the radiation wavelength), the two symmetric positive and negative charge trains are tightly coupled (the field lines go from positive to negative charges) . The cylindrical geometry is chosen because the solution of Eq . (5) is relatively simple to find .
m
e (ku+kll~ m iC
ail.
( P at +v e 0
1
YPm e
1-
4 2 g 2 ,,2 Mr
r-
( rh, - fmI,(mgr)(,
where g = (ko+kll ) /y, I,,(mghl 2 )Kz(mgrn) - K,,(mghl 2 )Iz(mgr,,)
The coupling of the longitudinal electron field to the electrons is given by the equation of motion for the energy of an electron d yJ
e
dt
mc2vJ
E" .
Combining Eq . (8) with Eq . (7) we get: d yJ
e
dZ
me- ,,
"e
e mc -
E, c' H' ,
where J,(m~), with ~---a /[2(1 +a2)], arises from an average over a wiggler period . The number of harmonics to consider in our calculations has been estimated by calculating the harmonic bunching b , with the FEL code and the reduction factor f =J F,"/m, where F -_
I -
4
r2 / - - - fmlolmgr) rh m z g rn z
I.
In Fig. 5 the curves of b", are plotted for m = 1, 3, 5. The bunching of the first harmonic is very tight, but the others are three times less (within the tapered
I. Boscolo et al. / Nucl. Instr. and Meth . in Phys. Res. A 341 (1994) 300-304
303
wiggler section) . The product b.J0F ,/rn reduces itself at higher harmonics to a negligible value compared
with the first one. The space-charge reduction factor f for the FELTRON project comes out to be about 0.05. With this number the space-charge force becomes very small with respect to the ponderomotive force. We point out that the space-charge term calculated in this way has the expected dependences on y, wavelength
and waveguide dimensions relative to the radiation wavelength. We expect a reduction of the space charge with y, with frequency and with h . Incidentally, we notice that the bunching at higher harmonics could be higher starting the tapering at the proper z for each
harmonic, but we start at the maximum of the first harmonic . In support to our arguments, we remark
that, at the Livermore experiment [1], although the parameters gave a ratio between the space charge
force and the ponderomotive force of about two, the power and bunching where not affected by the space
0 1
1
2
3
4
5
67 . . 8 .. WIGGLER LENGTH (m)
Fig. 6. Output power increases with interaction distance z in the tapered regime
charge, even if the cavity was 10 cm wide (the other dimension was 3 em).
We have taken this wall effect into consideration in our computer code by rewriting the space charge field with an ad hoc value of the beam area Asp to match
around z - 5 m comes from the fact that we have used a linear tapering, while the variation of
y and ao
the real field as written in Eq . (9).
(which has a value around 2.5
at that point) is no
solving numerically the equations of motion . The dia-
1 (we do not have the auto-tapering option in our
reaches a value around 0.8 and that it remains substantially stable up to the end of the process. Down the
Fig. 6, is PRF = 420 MW, as expected with a wiggler
the electrons in the bucket
FELs with three equal and matched waveguides . The difference in power in the first case results in about
We report in Figs . 5 and 6 the results obtained by
longer linear, since ao2 is not negligible with respect to
gram of the bunching factor b, Fig. 5, shows that b
code). The extracted power, illustrated by the plot of
tapered wiggler the decrease is due to smearing out of due to
the non-linear
effects. The change in the slope of the bunching curve
field reduction up to 0.5 kG .
We have compared the output power of the parallel
5%, i.e . 20 MW, while in the second case the difference decreases to 2-3% .
5. Conclusions A 20 GHz microwave FEL set up with a set of 3 parallel electron beams is studied . The voltage and current are respectively 5 MV and 120 A for a peak power of 600 MW. The Pierce gun has a gradient of 5 MV/m
and a cathode area of about 10 CM 2 . The electron beam has been simulated from the gun up to
the exit of the accelerating column . The linear voltage
drop of the high voltage terminal due to the 120 A current is transformed into a stepwise ladder by the superposition of a sawtooth voltage. The electron beam exits the accelerator with a succession of sections scaled down in energy of 1% from one to another. A disper-
Fig. 5 . Bunching versus wiggler length for the first, third and fifth harmonics .
sive magnetic system separates those sections into a set of parallel electron beam pulses, each 50 ns long . The FEL has a hybrid wiggler, 20 cm wide, with a set of parallel rectangular waveguides attached to each other. The horizontal dimension is the same for all the V. PERFORMANCE STUDIES
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waveguides, 30 mm, while the dimension is increased (starting from 30 mm) with the decreasing energy of the beams. This variation of the vertical size is because we have to fulfil the synchronism condition. The small variation in the output power that comes from the different power of the electron beams is unimportant . Each FEL channel should deliver 400 MW RF power with an extraction efficiency of 70% . That power is then divided into two parts of 200 MW each, because this is the power level required by the high gradient structure of the TeV collider . The FEL has been simulated with a 1-D code . The space-charge effect is reduced by more than 90% with the parameters of the project. This reduction is due to the image beam created by the metallic wall of waveguide.
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