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Jump-over method of the FEL interaction among electron beam bunches for increasing the radiation power
&*H
Nuclear Instruments
and Methods in Physics Research A 362 (1995) 581-585
_-
ma l!!i!J ELSEVIER
NUCLEAR INSTRUMENTS 8 METHODS IN PHVSICS RESEARCH Secbon A
Jump-over method of the FEL interaction among electron beam bunches for increasing the radiation power Yoshikazu Miyahara SPring-8, Kamigori, Ako-gun, Hyogo 678-12, Japan
Received 25 August 1994; revised form received 7 March 1995 Abstract A new method to substantially increase the FEL radiation power in an electron storage ring is presented. In this method, the storage ring is operated with many bunches, and the FEL interaction of a light pulse with the beam bunches is jumped over for several bunches by modulating the closed orbit distortion of the stored beam in synchronism with the FEL interaction. The interaction interval for each bunch is lengthened considerably, and the radiation damping proceeds for a longer time. The energy spread is much reduced and the radiation power can be increased more than a factor of 10.
1. Introduction
2. Jump-over
In recent years, the FEL oscillation has been realized in several storage rings [l-6]. However, the radiation power is weak and unstable and far beyond practical uses. Accordingly, it is necessary to investigate experimentally and theoretically the methods of increasing and stabilizing the radiation power. Previously, we have reinvestigated the saturation mechanism of the radiation power [7] and proposed a method of stabilizing the power [8]. In the present paper, we discuss a new method of increasing the radiation power. As has been believed, the radiation power saturates because the FEL gain is decreased as the increase of the energy spread of electron beam which is caused by the FEL interaction between the electron beam and the radiation field or the light pulse. Meanwhile, the energy spread is decreased by the radiation damping, and increased by the quantum excitation. These effects can be described by FEL power equations, with which the equilibrium energy spread and the saturation power are determined 171. Because of this mechanism, the radiation power can be increased by strengthening the radiation damping, or more straightfowardly by decreasing the energy spread and thereby increasing the gain. As has been recognized, this can be done by increasing the beam energy or installing a damping wiggler. In the present paper, an alternative method of strengthening the radiation damping is presented. The method is analyzed with the FEL power equations given in the previous investigation.
So far, an FEL experiment in a storage ring has been performed with a single or a few bunched beam, in which the bunch distance is just the twice of the mirror distance. In this case, the light pulse in the mirror cavity interacts with the electron beam bunch in each revolution. Accordingly, the beam bunch or the energy spread is cooled by the radiation damping during only one revolution period before the next interaction. In the present scheme, a number of beam bunches are stored, and different beam bunches are managed to interact with the light pulse successively, but each bunch interacts after many revolutions. Consequently, each bunch is cooled much more before the next interaction, so that the gain is increased much, and the radiation power also increased substantially. This scheme can be obtained by modulating the closed orbit distortion (COD) in synchronism with the FEL interaction. The interaction is managed to act only when the COD is zero in the interaction region. Suppose there are ten bunches in a storage ring with a revolution period of Z’,, and mirrors are placed at a distance with a round trip time 0.9T, ( = qT,) of the light pulse. And suppose that the COD is modulated sinusoidally with a period of l.U, (see Fig. 1). In this scheme, we expect that the COD of the 9th bunch is zero at the place of wiggler in synchronism with the FEL interaction. The light pulse interacts with every ninth bunch succesively, and eight bunches are jumped over without the FEL interaction. Accordingly, each bunch makes the interaction at a period of 9T,, (= hT,), and the radiation damping
016%9002/95/$09.50 0 1995 Elsevier Science B.V. AI1 rights reserved SSDI 0168-9002(95)00345-2
method
Y. Miyahara / Nucl. Instr. and Me&
582
in Phys. Res. A 362 (1995) 581-585
E,/e: resonant beam energy in [VI, ‘yr: resonant beam energy divided by the rest mass energy of the electron. In addition, the energy spread is decreased by the radiation damping, and increased by the quantum excitation, which is given by Fig. 1. Schematic representation of the jump-over method of the FEL interaction among electron beam bunches. Closed circles indicate the bunches which interact with light pulse, and open circles the bunches jumped over without the interaction. The sinusoidal line represents the amplitude of COD in the FEL interaction region.
proceeds for nine times longer period compared with a single bunched beam operation. Such a COD can be produced with two steering magnets installed in the place with a betatron phase advance of rr as shown in Fig. 2.
da,2 dn
*To -(o,2 rE
--
- a,,z)
where n is the number of revolutions, To the damping time and a,, the energy spread in the steady state without the FEL interaction. Combining Eqs. (2) and (4), and together with Eq. (l), we get the following equations, dP
G-6 - -P,
dn-
q
(5)
(6) 3. FEL power equations We assume that a helical wiggler is installed in a dispersion free straight section, and treat the FEL interaction in the small signal regime. The radiation power density P in the optical cavity increases every time of the FEL interaction with the light pulse, which is determined by
$=(G-S)P,
Meanwhile, the gain averaged over the energy spread is decreased with the increase of the energy spread, and given by [7]
(7)
G=G,,&-)
a,
(1)
where / is the number of interactions with a period of and G is the gain per the interaction and 8 the radiation loss per the round trip. The energy spread of each bunch is increased every time of the interaction with a period of hT,,, which is expressed as [7]
qT,,,
da,*-p, dm
(F)=
-O.O675exp[-(N,a,)*/Zu~],
(u,=O.15), (9)
where us: energy spread in rms, N,: number of wiggler periods, A,,: wiggler period in [m], K,: K-parameter,
where A: FEL radiation wavelength, I,: beam current per bunch, I+.,:Alfen current ( = 17 000 A), 2: cross section of electron beam, os: angular synchrotron oscillation frequency, a+ momentum compaction factor, (F): average gain function. In Eq. (8), os/op appears because the peak current is inversely proportional to the bunch length, which is given by a; = (a,c/o,)o,. Accordingly, if numerical values of above parameters are given, we can solve Eqs. (S), (6), (7) and (9) on P, us and G as a function of n. In the steady state, we have dP/dn = due2/dn = 0, SO that
G=S
or
(10)
(F)=$IJ-, 0
Mirror Wiggler
Slit
Fig. 2. The COD modulation in the FEL interaction region. The wiggler radiation emitted off axis by the nonzero COD bunches are absorbed in the slit.
which determines tion power as
the energy spread as Us,, and the radia-
(11)
583
Y. Miyahara / Nucl. Instr. and Meth. in Phys. Res. A 362 (1995) 581-585
In consequence, the radiation power is increased by h times with the jump-over method in the many bunch operation compared with a single bunch operation. Numerical analysis of Eq. (10) indicates that uss depends very weakly on the beam current, and given as uss 5 1/2N, = SW/W, which is the half bandwidth of the wiggler radiation spectrum [7]. Noting that Q/E, = To/~E and a,, X= a-,, we find the maximum radiation power available by the storage ring FEL as follows,
(12)
where (Y, is the radiation damping time in energy, and & = AL/L,. Adding the quantum excitation effects to Eq. (13), we get the following differential equation, d% dt2
+2aEz
+ w,‘E=
- hw,’
sin(Rt)
+Q.E.
(YP
(15) In case of no Q.E. term, we get the following solution for the forced oscillation, e=Cexp(-(~,t)cos(~t+S)--DD,
The maximum power extracted from the optical cavity is ~~,SPlllax with S,, the loss coefficient of power extraction and S the cross section of the extracted power.
w=(o,Z-a,2)1’z,
4. Effects of orbit modulation
D,=
sin(ot++), (16) (17)
P, d/a,
(18)
[(
ws2- f12)2 + (2a,R)2]1’2
Here we consider crudely the effects of the orbit modulation on the synchrotron oscillation of the electron beam or on the energy spread and bunch length, which affects the FEL again. Suppose that the orbit length around the circumference is modulated sinusoidally as L = L, + AL sin( .f&) with 0 = w,/2q, where w0 is the revolution frequency. As for the relative energy E of an electron and the time difference T from the synchronous electron, we get the following relations, d& (Y - = -w,%’ dt z
the other hand, in case of no forced oscillation, the electron energy is balanced by the quantum excitation and the radiation damping, resulting in the Gaussian distribution of electron energy. The mean number and the mean energy of the photons emitted from an electron per revolution are given respectively by [9] On
2ff,c$E,
= (YpE+ p, cos( at),
5 27r (n,> = zJ5FY’
(14)
Table 1 Examples
of parameters
of a storage ring. The values in the left column are of UVSOR [ll]
Beam energy Bending radius Radiation energy Momentum compaction factor Circumference Orbit modulation length Orbit modulation frequency (q = 0.9) Revolution frequency RF frequency RF voltage Synchrotron oscillation freq. Synchrotron oscillation tune Natural energy spread Natural bunch length Forced oscillation amplitude energy time
E
750 2.2 12.7 0.032 53.2 30 3.13
P ua ffP
LO AL R/2a %/2T %E/27F
in Japan MeV m keV m mm MHz
5.64
VaF %/2T “s ab OI
90.2 50 13.0 0.0023 4.4 x lo- 4 170
496 100 43.4 0.0077 4.4 x 1o-4 52
4 0,
3.0 x 1o-7 29
3.4 x 1o-6 29
MHZ MHZ kV kH2
Ps
Ps
584
Y.Miyahara / Nucl. Instr. and Meth. in Phys. Res. A 362 (1995) 581-585
where y is the relativistic electron energy and U, the critical energy of the radiation. At an electron energy of 1 GeV, for instance, we get (n,) = 132 and (u) = 135 eV. The relative change in the electron energy is (u)/E = 1.3 X lo-‘, being very small compared with the energy spread about 10e3. The number of photon emission per synchrotron oscillation period amounts to lo5 for a tune lJ5= 0.001. This successive reduction of the small amount of energy or the excitation in the longitudinal phase space is counter-balanced by the radiation damping. Therefore, we can imagine that each electron rotates the phase space at the synchrotron oscillation frequency with nearly a constant amplitude given by the probability of the Gaussian distribution. The electrons really migrate in the phase space at random because of the stochastic photon emission, while rotating the phase space between the successive photon emission. The electrons always fill the phase space with that distribution, and are not distinguished each other, so that we can suppose that there is no damping of the oscillation amplitude. Then from Eqs. (16)-(B) we can express the energy of an electron as follows, E=Q
cos(w,t+S)-DD,
sin(Rt+4),
sin(o,t+S)+D,cos(Rt++),
5. Discussion We have shown that the FEL radiation power can be increased drastically with the jump-over method by modulating the COD in a many bunched beam operation. Since the radiation power is very sensitive to the fluctuation of the gain, the COD modulation should be highly stabilized. Especially, the modulation frequency should avoid the Fourier component of the resonant frequency given below
~~,m
(25) (21)
Meanwhile, we get the following expression for the time difference from Eqs. (13) and (21) with (Y, = 0, r=IpcO %
considerable in the shorter bunch case. If we can reduce the orbit modulation length to about a half of the above value, the gain reduction in the latter case is not serious either. There is a possibility, however, that the stochastic process also increases the energy spread in response to the increase of the bunch length due to the forced oscillation. Even in this situation, the gain is not reduced much if the forced oscillation amplitude D, is reduced to the half.
(23)
(24) Now, we consider numerically how much the energy spread and the bunch length are affected by the modulation. Table 1 represents two examples of parameters of a storage ring composed of the double bend achromatic lattice structure operated at an RF frequency of 90 and 496 MHz. The orbit modulation length AL = 30 mm will be enough to produce a COD modulation amplitude about 5-10 mm for the jump-over method in the FEL interaction region. As shown in the Table, the modulation or forced oscillation frequency is much faster than the synchrotron oscillation frequency. The forced oscillation amplitude in energy is very small as D, = 3.0 X lo-’ or 3.4 X 10m6 compared with the natural energy spread a, = 4.4 X 10m4, while the amplitude in time is D, = 29 ps, being small or comparable in reference to the natural bunch length a, = 170 ps and 52 ps, respectively. Thus, in both cases, the increase in energy spread is negligible. The increase in the bunch length or the decrease in the peak current and the gain is still negligible in the longer bunch case, but
In addition, a longer bunch length is more preferable to avoid the increase of the energy spread and bunch length in the COD modulation, as discussed in Section 4. To get a sufficient gain, the total beam current in the many bunch operation needs be considerably higher than that in the single bunch operation. In such a situation the wiggler radiation itself might induce radiation damage on the mirrors. The damage is, however, can be reduced considerably by installing an absorber slit on the mirror axis, since most of the wiggler radiation is emitted off axis by the nonzero COD bunches. In addition, it is noticed that the beam current is only necessary to be so high as to provide the initial gain higher than the radiation loss in the optical cavity. This is because the FEL radiation power is nearly independent of the beam current once the FEL oscillation is obtained as discussed in detail in Ref. [7].
Acknowledgement The author is grateful to Dr. A. Ando for the discussion on Section 4.
References [l] M. Billardon, P. Elleaume, J.M. Ortega, C. Bazin, M. Bergher, M. Velghe, Y. Petroff, D.A.G. Deacon, K.E. Robinson and J.M.J. Madey, Phys. Rev. L&t. 51 (1983) 1652. [2] V.N. Litvinenko, Synchrotron Radiation News 1 (5) (1988) 18.
Y. Miyahura / Nucl. lnstr. and Meth. in Phys. Res. A 362 (199.5) 581-585 [3] M.E. Couprie, M, Billardon, M. Velghe, C. Bazin, J.M. Ortega, R. Prazeres and Y. Petroff, Nucl. Instr. and Meth. A 296 (1990) 13. [4] T. Yamazaki, K. Yamada, S. Sugiyama, H. Ohgaki, T. Tomimasu, T. Noguchi, T. Mikado, M. Chiwaki, and R. Suzuki, Nucl. Instr. and Meth. A 309 (1991) 343. [5] S. Takano, H. Hama and G. Isoyama, Nucl. Instr. and Meth. A 331 (1993) 20. 161 T. Yamazaki, K. Yamada, S. Sugiyama, H. Ohgaki, N. Sei,
[7] [8] [9] [lo] [ll]
585
T. Mikado, T. Noguchi, M. Chiwaki, R. Suzuki, M. Kawai, M. Yokoyama, K. Owaki, S. Hamada, K. Aizawa, Y. Oku, A. Iwata, and M. Yoshiwa, Nucl. Instr. and Meth. A 331 (1993) 27. Y. Miyahara, Nucl. Instr. and Meth. A 349 (1994) 247. Y. Miyahara, Nucl. Instr. and Meth. A 351 (1994) 568. M. Sands, SLAC-121, UC-28 (1970) Chap. V. M. Billardon, Phys. Rev. Lett. 65 (1990) 713. UVSOR Activity Report 1993, UVSOR-21 (1994) 193.
Nuclear Instruments
and Methods in Physics Research A 362 (1995) 581-585
_-
ma l!!i!J ELSEVIER
NUCLEAR INSTRUMENTS 8 METHODS IN PHVSICS RESEARCH Secbon A
Jump-over method of the FEL interaction among electron beam bunches for increasing the radiation power Yoshikazu Miyahara SPring-8, Kamigori, Ako-gun, Hyogo 678-12, Japan
Received 25 August 1994; revised form received 7 March 1995 Abstract A new method to substantially increase the FEL radiation power in an electron storage ring is presented. In this method, the storage ring is operated with many bunches, and the FEL interaction of a light pulse with the beam bunches is jumped over for several bunches by modulating the closed orbit distortion of the stored beam in synchronism with the FEL interaction. The interaction interval for each bunch is lengthened considerably, and the radiation damping proceeds for a longer time. The energy spread is much reduced and the radiation power can be increased more than a factor of 10.
1. Introduction
2. Jump-over
In recent years, the FEL oscillation has been realized in several storage rings [l-6]. However, the radiation power is weak and unstable and far beyond practical uses. Accordingly, it is necessary to investigate experimentally and theoretically the methods of increasing and stabilizing the radiation power. Previously, we have reinvestigated the saturation mechanism of the radiation power [7] and proposed a method of stabilizing the power [8]. In the present paper, we discuss a new method of increasing the radiation power. As has been believed, the radiation power saturates because the FEL gain is decreased as the increase of the energy spread of electron beam which is caused by the FEL interaction between the electron beam and the radiation field or the light pulse. Meanwhile, the energy spread is decreased by the radiation damping, and increased by the quantum excitation. These effects can be described by FEL power equations, with which the equilibrium energy spread and the saturation power are determined 171. Because of this mechanism, the radiation power can be increased by strengthening the radiation damping, or more straightfowardly by decreasing the energy spread and thereby increasing the gain. As has been recognized, this can be done by increasing the beam energy or installing a damping wiggler. In the present paper, an alternative method of strengthening the radiation damping is presented. The method is analyzed with the FEL power equations given in the previous investigation.
So far, an FEL experiment in a storage ring has been performed with a single or a few bunched beam, in which the bunch distance is just the twice of the mirror distance. In this case, the light pulse in the mirror cavity interacts with the electron beam bunch in each revolution. Accordingly, the beam bunch or the energy spread is cooled by the radiation damping during only one revolution period before the next interaction. In the present scheme, a number of beam bunches are stored, and different beam bunches are managed to interact with the light pulse successively, but each bunch interacts after many revolutions. Consequently, each bunch is cooled much more before the next interaction, so that the gain is increased much, and the radiation power also increased substantially. This scheme can be obtained by modulating the closed orbit distortion (COD) in synchronism with the FEL interaction. The interaction is managed to act only when the COD is zero in the interaction region. Suppose there are ten bunches in a storage ring with a revolution period of Z’,, and mirrors are placed at a distance with a round trip time 0.9T, ( = qT,) of the light pulse. And suppose that the COD is modulated sinusoidally with a period of l.U, (see Fig. 1). In this scheme, we expect that the COD of the 9th bunch is zero at the place of wiggler in synchronism with the FEL interaction. The light pulse interacts with every ninth bunch succesively, and eight bunches are jumped over without the FEL interaction. Accordingly, each bunch makes the interaction at a period of 9T,, (= hT,), and the radiation damping
016%9002/95/$09.50 0 1995 Elsevier Science B.V. AI1 rights reserved SSDI 0168-9002(95)00345-2
method
Y. Miyahara / Nucl. Instr. and Me&
582
in Phys. Res. A 362 (1995) 581-585
E,/e: resonant beam energy in [VI, ‘yr: resonant beam energy divided by the rest mass energy of the electron. In addition, the energy spread is decreased by the radiation damping, and increased by the quantum excitation, which is given by Fig. 1. Schematic representation of the jump-over method of the FEL interaction among electron beam bunches. Closed circles indicate the bunches which interact with light pulse, and open circles the bunches jumped over without the interaction. The sinusoidal line represents the amplitude of COD in the FEL interaction region.
proceeds for nine times longer period compared with a single bunched beam operation. Such a COD can be produced with two steering magnets installed in the place with a betatron phase advance of rr as shown in Fig. 2.
da,2 dn
*To -(o,2 rE
--
- a,,z)
where n is the number of revolutions, To the damping time and a,, the energy spread in the steady state without the FEL interaction. Combining Eqs. (2) and (4), and together with Eq. (l), we get the following equations, dP
G-6 - -P,
dn-
q
(5)
(6) 3. FEL power equations We assume that a helical wiggler is installed in a dispersion free straight section, and treat the FEL interaction in the small signal regime. The radiation power density P in the optical cavity increases every time of the FEL interaction with the light pulse, which is determined by
$=(G-S)P,
Meanwhile, the gain averaged over the energy spread is decreased with the increase of the energy spread, and given by [7]
(7)
G=G,,&-)
a,
(1)
where / is the number of interactions with a period of and G is the gain per the interaction and 8 the radiation loss per the round trip. The energy spread of each bunch is increased every time of the interaction with a period of hT,,, which is expressed as [7]
qT,,,
da,*-p, dm
(F)=
-O.O675exp[-(N,a,)*/Zu~],
(u,=O.15), (9)
where us: energy spread in rms, N,: number of wiggler periods, A,,: wiggler period in [m], K,: K-parameter,
where A: FEL radiation wavelength, I,: beam current per bunch, I+.,:Alfen current ( = 17 000 A), 2: cross section of electron beam, os: angular synchrotron oscillation frequency, a+ momentum compaction factor, (F): average gain function. In Eq. (8), os/op appears because the peak current is inversely proportional to the bunch length, which is given by a; = (a,c/o,)o,. Accordingly, if numerical values of above parameters are given, we can solve Eqs. (S), (6), (7) and (9) on P, us and G as a function of n. In the steady state, we have dP/dn = due2/dn = 0, SO that
G=S
or
(10)
(F)=$IJ-, 0
Mirror Wiggler
Slit
Fig. 2. The COD modulation in the FEL interaction region. The wiggler radiation emitted off axis by the nonzero COD bunches are absorbed in the slit.
which determines tion power as
the energy spread as Us,, and the radia-
(11)
583
Y. Miyahara / Nucl. Instr. and Meth. in Phys. Res. A 362 (1995) 581-585
In consequence, the radiation power is increased by h times with the jump-over method in the many bunch operation compared with a single bunch operation. Numerical analysis of Eq. (10) indicates that uss depends very weakly on the beam current, and given as uss 5 1/2N, = SW/W, which is the half bandwidth of the wiggler radiation spectrum [7]. Noting that Q/E, = To/~E and a,, X= a-,, we find the maximum radiation power available by the storage ring FEL as follows,
(12)
where (Y, is the radiation damping time in energy, and & = AL/L,. Adding the quantum excitation effects to Eq. (13), we get the following differential equation, d% dt2
+2aEz
+ w,‘E=
- hw,’
sin(Rt)
+Q.E.
(YP
(15) In case of no Q.E. term, we get the following solution for the forced oscillation, e=Cexp(-(~,t)cos(~t+S)--DD,
The maximum power extracted from the optical cavity is ~~,SPlllax with S,, the loss coefficient of power extraction and S the cross section of the extracted power.
w=(o,Z-a,2)1’z,
4. Effects of orbit modulation
D,=
sin(ot++), (16) (17)
P, d/a,
(18)
[(
ws2- f12)2 + (2a,R)2]1’2
Here we consider crudely the effects of the orbit modulation on the synchrotron oscillation of the electron beam or on the energy spread and bunch length, which affects the FEL again. Suppose that the orbit length around the circumference is modulated sinusoidally as L = L, + AL sin( .f&) with 0 = w,/2q, where w0 is the revolution frequency. As for the relative energy E of an electron and the time difference T from the synchronous electron, we get the following relations, d& (Y - = -w,%’ dt z
the other hand, in case of no forced oscillation, the electron energy is balanced by the quantum excitation and the radiation damping, resulting in the Gaussian distribution of electron energy. The mean number and the mean energy of the photons emitted from an electron per revolution are given respectively by [9] On
2ff,c$E,
= (YpE+ p, cos( at),
5 27r (n,> = zJ5FY’
(14)
Table 1 Examples
of parameters
of a storage ring. The values in the left column are of UVSOR [ll]
Beam energy Bending radius Radiation energy Momentum compaction factor Circumference Orbit modulation length Orbit modulation frequency (q = 0.9) Revolution frequency RF frequency RF voltage Synchrotron oscillation freq. Synchrotron oscillation tune Natural energy spread Natural bunch length Forced oscillation amplitude energy time
E
750 2.2 12.7 0.032 53.2 30 3.13
P ua ffP
LO AL R/2a %/2T %E/27F
in Japan MeV m keV m mm MHz
5.64
VaF %/2T “s ab OI
90.2 50 13.0 0.0023 4.4 x lo- 4 170
496 100 43.4 0.0077 4.4 x 1o-4 52
4 0,
3.0 x 1o-7 29
3.4 x 1o-6 29
MHZ MHZ kV kH2
Ps
Ps
584
Y.Miyahara / Nucl. Instr. and Meth. in Phys. Res. A 362 (1995) 581-585
where y is the relativistic electron energy and U, the critical energy of the radiation. At an electron energy of 1 GeV, for instance, we get (n,) = 132 and (u) = 135 eV. The relative change in the electron energy is (u)/E = 1.3 X lo-‘, being very small compared with the energy spread about 10e3. The number of photon emission per synchrotron oscillation period amounts to lo5 for a tune lJ5= 0.001. This successive reduction of the small amount of energy or the excitation in the longitudinal phase space is counter-balanced by the radiation damping. Therefore, we can imagine that each electron rotates the phase space at the synchrotron oscillation frequency with nearly a constant amplitude given by the probability of the Gaussian distribution. The electrons really migrate in the phase space at random because of the stochastic photon emission, while rotating the phase space between the successive photon emission. The electrons always fill the phase space with that distribution, and are not distinguished each other, so that we can suppose that there is no damping of the oscillation amplitude. Then from Eqs. (16)-(B) we can express the energy of an electron as follows, E=Q
cos(w,t+S)-DD,
sin(Rt+4),
sin(o,t+S)+D,cos(Rt++),
5. Discussion We have shown that the FEL radiation power can be increased drastically with the jump-over method by modulating the COD in a many bunched beam operation. Since the radiation power is very sensitive to the fluctuation of the gain, the COD modulation should be highly stabilized. Especially, the modulation frequency should avoid the Fourier component of the resonant frequency given below
~~,m
(25) (21)
Meanwhile, we get the following expression for the time difference from Eqs. (13) and (21) with (Y, = 0, r=IpcO %
considerable in the shorter bunch case. If we can reduce the orbit modulation length to about a half of the above value, the gain reduction in the latter case is not serious either. There is a possibility, however, that the stochastic process also increases the energy spread in response to the increase of the bunch length due to the forced oscillation. Even in this situation, the gain is not reduced much if the forced oscillation amplitude D, is reduced to the half.
(23)
(24) Now, we consider numerically how much the energy spread and the bunch length are affected by the modulation. Table 1 represents two examples of parameters of a storage ring composed of the double bend achromatic lattice structure operated at an RF frequency of 90 and 496 MHz. The orbit modulation length AL = 30 mm will be enough to produce a COD modulation amplitude about 5-10 mm for the jump-over method in the FEL interaction region. As shown in the Table, the modulation or forced oscillation frequency is much faster than the synchrotron oscillation frequency. The forced oscillation amplitude in energy is very small as D, = 3.0 X lo-’ or 3.4 X 10m6 compared with the natural energy spread a, = 4.4 X 10m4, while the amplitude in time is D, = 29 ps, being small or comparable in reference to the natural bunch length a, = 170 ps and 52 ps, respectively. Thus, in both cases, the increase in energy spread is negligible. The increase in the bunch length or the decrease in the peak current and the gain is still negligible in the longer bunch case, but
In addition, a longer bunch length is more preferable to avoid the increase of the energy spread and bunch length in the COD modulation, as discussed in Section 4. To get a sufficient gain, the total beam current in the many bunch operation needs be considerably higher than that in the single bunch operation. In such a situation the wiggler radiation itself might induce radiation damage on the mirrors. The damage is, however, can be reduced considerably by installing an absorber slit on the mirror axis, since most of the wiggler radiation is emitted off axis by the nonzero COD bunches. In addition, it is noticed that the beam current is only necessary to be so high as to provide the initial gain higher than the radiation loss in the optical cavity. This is because the FEL radiation power is nearly independent of the beam current once the FEL oscillation is obtained as discussed in detail in Ref. [7].
Acknowledgement The author is grateful to Dr. A. Ando for the discussion on Section 4.
References [l] M. Billardon, P. Elleaume, J.M. Ortega, C. Bazin, M. Bergher, M. Velghe, Y. Petroff, D.A.G. Deacon, K.E. Robinson and J.M.J. Madey, Phys. Rev. L&t. 51 (1983) 1652. [2] V.N. Litvinenko, Synchrotron Radiation News 1 (5) (1988) 18.
Y. Miyahura / Nucl. lnstr. and Meth. in Phys. Res. A 362 (199.5) 581-585 [3] M.E. Couprie, M, Billardon, M. Velghe, C. Bazin, J.M. Ortega, R. Prazeres and Y. Petroff, Nucl. Instr. and Meth. A 296 (1990) 13. [4] T. Yamazaki, K. Yamada, S. Sugiyama, H. Ohgaki, T. Tomimasu, T. Noguchi, T. Mikado, M. Chiwaki, and R. Suzuki, Nucl. Instr. and Meth. A 309 (1991) 343. [5] S. Takano, H. Hama and G. Isoyama, Nucl. Instr. and Meth. A 331 (1993) 20. 161 T. Yamazaki, K. Yamada, S. Sugiyama, H. Ohgaki, N. Sei,
[7] [8] [9] [lo] [ll]
585
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