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Simulation of higher-harmonics generation in a FEL
Nuclear Instruments and Methods in Physics Research A304 (1991) 638-640 North-Holland
638
Simulation of higher-harmonics generation in a FEL Sin-ichiro Kuruma
a,
Kumoki Mima
b,
Katsuhiro Ohi °, Sadao Nakai
b
Institute for Laser Technology, 2-6 Yamadaoka, Suita Osaka 565, Japan h Institute of Laser Engineering (ILE), Osaka University, 2-6 Yamadaoka, Suua Osaka 565, Japan ` Faculty of Engineering, Kansai University, 3-3-35 Yamate, Suita Osaka 564, Japan
and Chiyoe Yamanaka "
Nonlinear evolution of higher-harmonics radiation in a free electron laser is studied by a 1D simulation code . The parameters of the induction lmac FEL at ILE are: electron beam energy Eb = 6 MeV, current Ib =100 A, radius r b = 3 mm, wiggler wavelength X,, = 6 cm, magnetic field B w = 3.2 kG, wiggler length Lw = 3 m, input power P,1 =1 kW at X,1 = 515 win, we calculated that the output power at the fundamental, 3rd ("'~53=172 gym) and 5th (X55=103 lim) harmonic is 30, 1 and 0.15 MW, respectively . The electron beam current dependence of the radiation output power has been studied. It has been found that the radiation power Pn (n =1, 3, 5) is approximately proportional to Ib 17 . Furthermore, we found that the fundamental mode is efficiently converted to the higher harmonics when a high-intensity fundamental mode is injected and the wiggler is inversely tapered. 1. Introduction In order to have large gain in a FEL, it is necessary to enlarge the strength of the wiggler magnetic field [1]. But if the K parameter that depends on the wiggler magnetic field and period is larger than 1, then the magnitude of higher harmonics becomes larger [2]. So, in the region of short-wavelength radiation, the damage of the mirror is larger . But if we can pick up the higher-harmonics radiation effectively, then we can get shorter-wavelength radiation by using a low-energy electron beam . So it is important to analyze the characteristics of higher-harmonics generation for a compact FEL system . Hence we develop a 1D multifrequency simulation code that can describe the nonlinear evolution of the fundamental and higher-harmonics radiation. With this simulation code, it is possible to analyze the coupling of the fundamental to higher-harmonics radiation. In section 2, the model equations are described. In section 3, a numerical example is described and discussed, and section 4 is a summary. 2. Model equation The electron beam is assumed to propagate along the z-direction which is parallel to the axis of the plane wiggler. The vector potential of the wiggler magnetic field is given by A, (z)
=K(z) cos k,,zex ,
where A,(z) is normalized by mc 2/e and K(z) is the K parameter of the wiggler magnetic field ; ex is the unit vector in the x-direction . The vector potential of
the electromagnetic radiation field normalized by mc 2/e is assumed to have the form A, t, z)=Ea,,(Z)cos{kSnZ-Wsnt+Bsn(Z)1ex+
where a sn (z) is the amplitude function of the radiation field and BS (z) is the phase shift function . The microscopic current density is given by the summation over individual particles, j(t, z) =- (en b LINT ) ~v,(z, t,o)8[t-T,(z, t,o) ]
(3) where is the total number of beam electrons in the interaction region of length L, n b is the average electron density, v,(z, t, o) is the velocity of the ith electron at the position z, where the ith particle is assumed to enter the interaction region (i.e . cross the z = 0 plane) at time t,o . Substituting the current density of eq. (3) into the Maxwell equations with eq . (2) and carrying out a Fourier transformation with respect to t, we obtain /U'0(z, t,o),
NT
asn( Z / - - (wPKNz0/2ksn) X
) t j(- 1)/2( a ) -J(n+i)/2(a)1
sin (4)
$nr~+
Bsn(z) _ -(wPK,8z0/2ksnatn) X((14ziYi ) {J(n-1)/2(a)-J(n+1)/2(a)J cosq,n,), (5)
n¢,+ 2 2 a = ksnK/8Y,,
0168-9002/91/$03 .50 © 1991 - Elsevier Science Publishers B.V . (North-Holland)
( 7)
63 9
S Kuruma et al. / Higher-harmonics generation
where cop is the plasma frequency of the electron beam normalized by k,,c, and ßro is the initial average velocity of the electron normalized by the speed of light. A prime denotes a derivative with respect to zP = kwz, and we neglect the terms of second derivatives of zP. In eqs. (4) and (5), ( . . . ) represents a time average over a period of the fundamental wave, namely 21T/w,, ; and 4,, is the ponderomotive phase of the i th electron for the fundamental radiation, (8) V~l(z)=I-kso(I-/ßs,)//ß,,, _ { 1 - (1 + K2/2)/Y,2 )1/2
(9)
The equation of motion of the i th particle is
1000
100 Pi
a 0
D 0
n
0
z
v
P3
100
10
M
D 1 0 z v 0 m
3
Ps 10
y'(z) = (K12,8,,Y,)Yksna,,(z)(J(n-1),12(a) n
In eqs. (4)-(10), subscript i is the particle number and subscript n is the radiation mode number, n = 1, 3, 5, 7, - - - representing the fundamental, 3rd, 5th, 7th, harmonics mode, respectively. Both the linear and nonlinear evolution of a FEL amplifier can be investigated by eqs. (4)-(7) with eqs. (8)-(10) for the orbits of an ensemble of electrons having initial phases -m < ¢,o <
3. Numerical example and discussion In this section, we show two numerical examples . One corresponds to the FEL experiment at ILE, and another corresponds to the accelerator design of an inverse free electron laser (IFEL) . The parameters of the FEL are chosen as follows. The wiggler period X w = 6 cm, wiggler magnetic field strength B,, = 3.2 kG (K = 1.8), electron beam energy Eb = 6 MeV (y = 12 .74), electron beam current Ib = 100 A, electron beam radius rb = 3 mm . In this case, the radiation wavelength of the fundamental mode is approximately A, = 515 I.Lm. We neglect the effect of the space charge wave, because the parameter wp /yo y~ = 0.02 is much smaller than 1. In fig. 1, we show the temporal evolution of the radiation power strength for
100
1000 BEAM CURRENT (A)
0.1
Fig. 2. Beam current dependence on saturation power. the fundamental, 3rd, 5th, 7th and 9th harmonics modes. The initial power of the fundamental mode is about 1 kW and that of higher-harmonics modes is zero . In this figure, saturation occurs at z = 2 m and the saturation power of the fundamental (X,i = 515 gym), 3rd (X53 172 Wm) and 5th 055 = 103 wm) harmonics is P51 = 30 MW, P, = 1 MW and P55 = 0.15 MW, respectively . In fig. 2, we show the beam current dependence on the saturation power of the fundamental, 3rd and 5th harmonics modes. In this figure, we can see that the output power is proportional to the electron beam current to the power 1 .17.
s
D 0
D .1 0 z
v O m
10
Pi 106
P3 Ps P7
Ps
3
10
1 2 AXIAL LENGTH Z (m)
Fig. 1 . Temporal evolution of radiation power.
3
-rr/2
0
4% (rad)
+Tr/2
Fig. 3. Bunching and acceleration characteristics vs tapering rate of magnetic field . VIII . NUMERICAL SIMULATIONS
640
S. Kuruma et al / Higher-harmonics generation
Table 1 Output power distribution at z = 6 m (input : P = 60 GW, e-KE
P = 0 W, E,, =110
[GWJ
MeV, 1,, =100 A, tapering rate is 800 G/m)
Wave energy [GWJ
Fundamental Yymc z Ace. Detrapped Without higher harmonics With higher harmonics
30 .68 30 .45
7.98 8.24
38,65 38 .69
The parameters of the IFEL are chosen as follows: The wiggler period  w = 12 cm, initial wiggler magnetic field strength B.,o =1,15 kG (K = 1.32), wiggler length L w = 6 m, input electron beam energy E,, =I 10 MeV (y = 216), electron beam current I,, = 100 A, electron beam radius r,, = 0.1 mm, input radiation power Po = 60 GW and wavelength X, = 2.33 pin. We simulate the bunching and acceleration characteristics of the electron beam, which depend on the tapering rate of the wiggler magnetic field. Fig. 3 shows the bunching of the electron beam energy in the phase space of the fundamental radiation. In this case, we can see that the bunched electrons are accelerated to an energy of about 700 MeV, for the case of tapering the wiggler magnetic field at 1 .6 kG/m and the trapping rate is 43 .5°x . For this system, we studied the effects of higher-harmonics radiation. The results for the output power distribution (at an axial length z = 6 m and the tapering rate is 800 G/m) as shown m table 1 . In this case, the total input power is 71 GW (60 GW of the fundamental radiation mode and 11 GW of the electron beam). The output power of the electron beam, the fundamental, 3rd, 5th and 7th harmonics is 38 .69 GW, 31 .40 GW, 875 MW, 6.8 MW and 9.9 MW, respectively . From table 1 we can see that the effect of higher-harmonics radiation on the bunched electrons is small but fundamental radiation is converted to higher-harmonics radiation effectively.
Higher harmonics
Po
P3
PS
P7
P9
32 .34 31 .40
0 0 875
0 0.0068
0 0.0099
0 0.0222
YPn 0 091
4. Summary A one-dimensional multifrequency simulation code for analysis of higher-harmonics radiation of a FEL is developed. From simulations with this code, the power ratios of the 3rd and 5th harmonics to the fundamental radiation are clarified. The electron beam dependence on the radiation output power has also been found. We also found that the fundamental mode is effectively converted to higher harmonics when a high-intensity fundamental mode is injected and the wiggler is inversely tapered. So we can use this system as a higherharmonics converter.
References [11 P. Sprangle, R.A . Smith and V. Granatstein, Infrared and Millimeter Waves Sources of Radiation, vol. l, ed . K.J . Button (Acadenuc Press, 1979) p. 279 [2) C.A . Brau, Free Electron Lasers (Academic Press, 1990) p. 73 .
638
Simulation of higher-harmonics generation in a FEL Sin-ichiro Kuruma
a,
Kumoki Mima
b,
Katsuhiro Ohi °, Sadao Nakai
b
Institute for Laser Technology, 2-6 Yamadaoka, Suita Osaka 565, Japan h Institute of Laser Engineering (ILE), Osaka University, 2-6 Yamadaoka, Suua Osaka 565, Japan ` Faculty of Engineering, Kansai University, 3-3-35 Yamate, Suita Osaka 564, Japan
and Chiyoe Yamanaka "
Nonlinear evolution of higher-harmonics radiation in a free electron laser is studied by a 1D simulation code . The parameters of the induction lmac FEL at ILE are: electron beam energy Eb = 6 MeV, current Ib =100 A, radius r b = 3 mm, wiggler wavelength X,, = 6 cm, magnetic field B w = 3.2 kG, wiggler length Lw = 3 m, input power P,1 =1 kW at X,1 = 515 win, we calculated that the output power at the fundamental, 3rd ("'~53=172 gym) and 5th (X55=103 lim) harmonic is 30, 1 and 0.15 MW, respectively . The electron beam current dependence of the radiation output power has been studied. It has been found that the radiation power Pn (n =1, 3, 5) is approximately proportional to Ib 17 . Furthermore, we found that the fundamental mode is efficiently converted to the higher harmonics when a high-intensity fundamental mode is injected and the wiggler is inversely tapered. 1. Introduction In order to have large gain in a FEL, it is necessary to enlarge the strength of the wiggler magnetic field [1]. But if the K parameter that depends on the wiggler magnetic field and period is larger than 1, then the magnitude of higher harmonics becomes larger [2]. So, in the region of short-wavelength radiation, the damage of the mirror is larger . But if we can pick up the higher-harmonics radiation effectively, then we can get shorter-wavelength radiation by using a low-energy electron beam . So it is important to analyze the characteristics of higher-harmonics generation for a compact FEL system . Hence we develop a 1D multifrequency simulation code that can describe the nonlinear evolution of the fundamental and higher-harmonics radiation. With this simulation code, it is possible to analyze the coupling of the fundamental to higher-harmonics radiation. In section 2, the model equations are described. In section 3, a numerical example is described and discussed, and section 4 is a summary. 2. Model equation The electron beam is assumed to propagate along the z-direction which is parallel to the axis of the plane wiggler. The vector potential of the wiggler magnetic field is given by A, (z)
=K(z) cos k,,zex ,
where A,(z) is normalized by mc 2/e and K(z) is the K parameter of the wiggler magnetic field ; ex is the unit vector in the x-direction . The vector potential of
the electromagnetic radiation field normalized by mc 2/e is assumed to have the form A, t, z)=Ea,,(Z)cos{kSnZ-Wsnt+Bsn(Z)1ex+
where a sn (z) is the amplitude function of the radiation field and BS (z) is the phase shift function . The microscopic current density is given by the summation over individual particles, j(t, z) =- (en b LINT ) ~v,(z, t,o)8[t-T,(z, t,o) ]
(3) where is the total number of beam electrons in the interaction region of length L, n b is the average electron density, v,(z, t, o) is the velocity of the ith electron at the position z, where the ith particle is assumed to enter the interaction region (i.e . cross the z = 0 plane) at time t,o . Substituting the current density of eq. (3) into the Maxwell equations with eq . (2) and carrying out a Fourier transformation with respect to t, we obtain /U'0(z, t,o),
NT
asn( Z / - - (wPKNz0/2ksn) X
) t j(- 1)/2( a ) -J(n+i)/2(a)1
sin (4)
$nr~+
Bsn(z) _ -(wPK,8z0/2ksnatn) X((14ziYi ) {J(n-1)/2(a)-J(n+1)/2(a)J cosq,n,), (5)
n¢,+ 2 2 a = ksnK/8Y,,
0168-9002/91/$03 .50 © 1991 - Elsevier Science Publishers B.V . (North-Holland)
( 7)
63 9
S Kuruma et al. / Higher-harmonics generation
where cop is the plasma frequency of the electron beam normalized by k,,c, and ßro is the initial average velocity of the electron normalized by the speed of light. A prime denotes a derivative with respect to zP = kwz, and we neglect the terms of second derivatives of zP. In eqs. (4) and (5), ( . . . ) represents a time average over a period of the fundamental wave, namely 21T/w,, ; and 4,, is the ponderomotive phase of the i th electron for the fundamental radiation, (8) V~l(z)=I-kso(I-/ßs,)//ß,,, _ { 1 - (1 + K2/2)/Y,2 )1/2
(9)
The equation of motion of the i th particle is
1000
100 Pi
a 0
D 0
n
0
z
v
P3
100
10
M
D 1 0 z v 0 m
3
Ps 10
y'(z) = (K12,8,,Y,)Yksna,,(z)(J(n-1),12(a) n
In eqs. (4)-(10), subscript i is the particle number and subscript n is the radiation mode number, n = 1, 3, 5, 7, - - - representing the fundamental, 3rd, 5th, 7th, harmonics mode, respectively. Both the linear and nonlinear evolution of a FEL amplifier can be investigated by eqs. (4)-(7) with eqs. (8)-(10) for the orbits of an ensemble of electrons having initial phases -m < ¢,o <
3. Numerical example and discussion In this section, we show two numerical examples . One corresponds to the FEL experiment at ILE, and another corresponds to the accelerator design of an inverse free electron laser (IFEL) . The parameters of the FEL are chosen as follows. The wiggler period X w = 6 cm, wiggler magnetic field strength B,, = 3.2 kG (K = 1.8), electron beam energy Eb = 6 MeV (y = 12 .74), electron beam current Ib = 100 A, electron beam radius rb = 3 mm . In this case, the radiation wavelength of the fundamental mode is approximately A, = 515 I.Lm. We neglect the effect of the space charge wave, because the parameter wp /yo y~ = 0.02 is much smaller than 1. In fig. 1, we show the temporal evolution of the radiation power strength for
100
1000 BEAM CURRENT (A)
0.1
Fig. 2. Beam current dependence on saturation power. the fundamental, 3rd, 5th, 7th and 9th harmonics modes. The initial power of the fundamental mode is about 1 kW and that of higher-harmonics modes is zero . In this figure, saturation occurs at z = 2 m and the saturation power of the fundamental (X,i = 515 gym), 3rd (X53 172 Wm) and 5th 055 = 103 wm) harmonics is P51 = 30 MW, P, = 1 MW and P55 = 0.15 MW, respectively . In fig. 2, we show the beam current dependence on the saturation power of the fundamental, 3rd and 5th harmonics modes. In this figure, we can see that the output power is proportional to the electron beam current to the power 1 .17.
s
D 0
D .1 0 z
v O m
10
Pi 106
P3 Ps P7
Ps
3
10
1 2 AXIAL LENGTH Z (m)
Fig. 1 . Temporal evolution of radiation power.
3
-rr/2
0
4% (rad)
+Tr/2
Fig. 3. Bunching and acceleration characteristics vs tapering rate of magnetic field . VIII . NUMERICAL SIMULATIONS
640
S. Kuruma et al / Higher-harmonics generation
Table 1 Output power distribution at z = 6 m (input : P = 60 GW, e-KE
P = 0 W, E,, =110
[GWJ
MeV, 1,, =100 A, tapering rate is 800 G/m)
Wave energy [GWJ
Fundamental Yymc z Ace. Detrapped Without higher harmonics With higher harmonics
30 .68 30 .45
7.98 8.24
38,65 38 .69
The parameters of the IFEL are chosen as follows: The wiggler period  w = 12 cm, initial wiggler magnetic field strength B.,o =1,15 kG (K = 1.32), wiggler length L w = 6 m, input electron beam energy E,, =I 10 MeV (y = 216), electron beam current I,, = 100 A, electron beam radius r,, = 0.1 mm, input radiation power Po = 60 GW and wavelength X, = 2.33 pin. We simulate the bunching and acceleration characteristics of the electron beam, which depend on the tapering rate of the wiggler magnetic field. Fig. 3 shows the bunching of the electron beam energy in the phase space of the fundamental radiation. In this case, we can see that the bunched electrons are accelerated to an energy of about 700 MeV, for the case of tapering the wiggler magnetic field at 1 .6 kG/m and the trapping rate is 43 .5°x . For this system, we studied the effects of higher-harmonics radiation. The results for the output power distribution (at an axial length z = 6 m and the tapering rate is 800 G/m) as shown m table 1 . In this case, the total input power is 71 GW (60 GW of the fundamental radiation mode and 11 GW of the electron beam). The output power of the electron beam, the fundamental, 3rd, 5th and 7th harmonics is 38 .69 GW, 31 .40 GW, 875 MW, 6.8 MW and 9.9 MW, respectively . From table 1 we can see that the effect of higher-harmonics radiation on the bunched electrons is small but fundamental radiation is converted to higher-harmonics radiation effectively.
Higher harmonics
Po
P3
PS
P7
P9
32 .34 31 .40
0 0 875
0 0.0068
0 0.0099
0 0.0222
YPn 0 091
4. Summary A one-dimensional multifrequency simulation code for analysis of higher-harmonics radiation of a FEL is developed. From simulations with this code, the power ratios of the 3rd and 5th harmonics to the fundamental radiation are clarified. The electron beam dependence on the radiation output power has also been found. We also found that the fundamental mode is effectively converted to higher harmonics when a high-intensity fundamental mode is injected and the wiggler is inversely tapered. So we can use this system as a higherharmonics converter.
References [11 P. Sprangle, R.A . Smith and V. Granatstein, Infrared and Millimeter Waves Sources of Radiation, vol. l, ed . K.J . Button (Acadenuc Press, 1979) p. 279 [2) C.A . Brau, Free Electron Lasers (Academic Press, 1990) p. 73 .