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High-gain free-electron laser pulse amplifiers with variable wigglers
Volume 41, number 4
OPTICS COMMUNICATIONS
15 April 1982
HIGH-GAIN FREE-ELECTRON LASER PULSE AMPLIFIERS WITH VARIABLE WIGGLERS
A.T. GEORGES and W.H. LOUISELL Department of Physics, University of Southern California, Los Angeles, CA 90007, USA Received 28 September 1981 Revised manuscript received 27 November 1981
Calculations are presented for pulse amplification in high-gain free-electron lasers with variable wiggler magnets. Better than 90% peak electron trapping and overall peak energy extraction efficiencies of over 60% are demonstrated for 1 ~m radiation. Pulse reshaping and phase modulation effects are investigated.
Recently, the efficient transfer, through stimulated emission, of the kinetic energy of a highly relativistic electron beam to a co-propagating laser beam has attracted great interest in single-pass free-electron laser (FEL) amplifiers [ 1 - 3 ] . A 500 MeV electron beam with a peak current density of 105 A/cm 2 has a peak power density of 5 X 1013 W/cm 2. Such high electron beam power densities can be obtained in electron storage rings [4] which produce high quality beams with fractional energy spread of less than 10 -3. High power FEL amplifiers would be very useful in boosting the. power of conventional laser oscillators to the levels required for laser fusion. Theuse of damage-free and renewable electrons as amplifying medium is an advantage over conventional glass laser amplifiers which can be damaged by intense radiation. The key to high energy extraction efficiency in a FEL amplifier is the design of the wiggler magnet which imparts a small transverse motion to the electrons. It is through this transverse motion that the electrons couple to a transverse laser field (As) and exchange energy [5]. The variation of the wiggler period (kw) and magnetic vector potential (34) with distance (z) must be chosen such that .the electrons are adiabatically trapped in the ponderomotive potential and maintained in resonance (Xw = 2~,2Xs/(1 + 2M2), where Xs is the wavelength of the laser field and 3, the electron energy in rest energy units) as they decelerate. Since the rate of decrease of the electron energy depends on 282
the growing laser field, the optimum kw(z ) and M(z) values can be determined only by dynamic design or from trial and error solutions of the self-consistent equa. tions for the electron motion and the growth of the laser field. Computer simulation studies reported in ref. [1 ] have shown that wigglers with constant magnetic vector potential and period variation of the form Xw(Z) = Xw(O) exp [-(Z/Zo)4 ] give 100% electron trapping under conditions of high gain (input electron beam power density >> input laser intensity). These calculations, however, involve several approximations and are not self-consistent. For example, the laser field is not calculated from Maxwell's equation, but from an energy conservation relation which does not give information about the phase of the field. Moreover, they assume continuous (steady-state) electron beam and laser field, while high current electron beam sources and high power lasers operate in the pulsed mode and, therefore, it is important to account for the slippage of the electron and laser pulses as they move with different speeds. In this paper, we report the results of self-consistent calculations on the performance of high gain FEL pulse amplifiers with exponentially decreasing wiggler period and constant magnetic vector potential. In the one-dimensional approximation, the basic equations for a FEL amplifier with a variable helical wiggler can be written in the form [6,7]
Volume 41, number 4
OPTICS COMMUNICATIONS 1.0
dT(z, z e, 0 o)/dz = --i(ks/7/3z)
X (M(Z)As(z,r)exp[iO(z,re,O0) ] - c.c.},
(1)
dO(z,re,Oo)/dz = kw(Z ) + ks(1 - 1//3z) ,
(2)
0.8
~ 0.6 3.0.4
(d/dZ)As(z,r) = - i uOe M(z) ~ p ( r e ) 2ksm Oo
15 April 1982
0.2
exp [-iO(z,re,O0) ]
N3'(Z,re,O0)
(3)
'
where the transverse potentials for the wiggler, M(z), and the laser field, As(z,r ), are measured in units of mc/e, with e,m being the electron charge and mass, and c the speed of light, k s = 21r/Xs and kw(Z) = 27r/ Xw(Z) are the wave vectors for the laser field and the wiggler. ~z ~ [1 - (1 + 2/142)/72 ] 1/2 is the axial electron velocity normalized to the velocity of light, r = t - z/c is the proper time of the laser field and re(Z ) = t - f~c-l~l(z',re,OO)dZ ' the proper time of an electron. The angle 0 = f~kw(z')dz' - kscr + 00 measures the electron phase in the ponderomotive potential and 00 is its value at the entrance of the wiggler which is assumed to be uniformly distributed between zero and 27r. N is the number of representative electrons in the distribution for 00. P(%) is the electron charge density. In the computer simulation we consider a gaussian electron pulse with full width at half maximum (FWHM) of 25 ps and an overlapping gaussian incident laser pulse with FWHM of 50 ps. The trajectories of 900 representative electrons and the complex amplitude (As(z,r)) of the laser field are calculated self-consistently from eqs. (1)-(3). These coupled nonlinear differential equations are solved using a second-order predictor-corrector method [8] with interpolation to account for the slippage of the electron and laser pulses. As in ref. [ 1], we consider the case of monoenergetic electrons with 7 = 141.42 (72.47 MeV) interacting with a laser field of 1/am wavelength. The wiggler vector potential is assumed to be constant and equal to M = 1/V~ (initial magnetic field strength ~5.37 kG), and the initial wiggler period is determined from the resonance condition to be Xw = 272Xs/(1 + 2M 2) = 2 cm. An exponential variation, Xw(Z) = Xw(0) exp [-(Z/Zo)n ], was assumed for the wiggler period and the parameters z 0 and n were varied to maximize the fraction of electrons trapped. For a peak
O
I
5
~ _ _
I
I
6 9 DISTANCE IN METERS
12
Fig. 1. Normalized e l e c t ron energy versus distance along the wiggler for 36 electrons w i t h uni form phase d i s t r i b u t i o n at the center of the electron pulse.
laser intensity of 108 W/cm 2 and a peak current density of 3 X 104 A/cm 2 (electron beam power density = 2.17 X 1012 W/cm 2) it was determined that the values n = 4 and z 0 = 10.2 m give the highest electron trapping and energy extraction for the shortest amplifier length of 12 m. Ref. [1] uses the same values as above except for z 0 which is taken to be equal to 12.73 m. Fig. 1 shows the normalized electron energy as a function of distance inside the wiggler for 36 representative electrons at the center of the electron pulse. The initial phase (00) of the electrons is uniformly distributed between zero and 2n. Note that only three of these electrons are untrapped ("-92% trapping efficiency) and at z = 12 meters an average 62% of the energy is extracted from the trapped electrons. The overall energy extraction efficiency is 57% and could be made higher by going to slightly greater wiggler lengths. The final energy spread of the trapped electrons normalized to their initial energy is 1.5%. Fig. 2 shows the relative electron phase for the same group of electrons as a function of distance. The phases of the trapped electrons oscillate about either of two different average values separated by 2n and are localized in the decelerating half-cycles (buckets) of the periodic ponderomotive potential [9]. The period of the oscillations and their average value decrease as the wiggler period decreases and the electrons lose their energy to the growing laser field. The decrease in the average value of the oscillations is absent in the approximate results of ref. [1] (fig. 5) which incidentally show 100% electron trapping. Our calculations show that even for a 283
Volume 41, number 4 3~
OPTICS COMMUNICATIONS
/
2~ :
: /
,
,.
- 2-,r I ~ - ~ 1 - - - . 0 3
, , , .
,
I 6 DISTANCE
I 9 IN
k 12
METERS
Fig. 2. Relative phase versus distance for the same electron group as in fig. 1. more slowly decreasing wiggler period (z 0 > 12.73 m) some electrons whose phase falls between two adjacent trapping wells are not trapped. As will be explained below, the reason for the less than perfect electron trapping lies in the dynamics of a high gain FEL amplifier with decreasing wiggler period. Fig. 3 shows the amplification of the peak laser intensity along the wiggler. The gain per unit length is very high during the first two meters of the wiggler, where the laser intensity is much weaker than the electron beam power density. The initial high gain is followed by oscillations in the laser intensity which die out as the laser intensity becomes higher than the electron beam power density and the gain saturates. The oscillations in the laser intensity arise as follows: The growing laser field makes the electrons lose their energy too fast and they are driven out of resonances, absorbing energy back from the field. However, as the elec-
15 April 1982
trons move down the wiggler where the wiggler period decreases, they are brought again into resonance and lose energy to the field. This cycle repeats, but the amplitude of the oscillation decreases as the laser field grows more slowly. Comparing figs. 2 and 3 we see that it is during the dips in the laser intensity that weakly trapped electrons become detrapped. Therefore, a small degree of electron detrapping appears to be inevitable in high gain FEL amplifiers. The normalized input and output laser (solid line) and electron (dashed line) pulses are shown in fig. 4 as functions of the proper time of the laser field. The output laser pulse is asymmetric with the trailing edge amplified more than the leading edge, while its peak is delayed relative to the peak of the incident pulse by about 2.5 ps. These effects are due to the slippage between the electron and laser pulses which can be seen in fig. 4. Note that the FWHM of the output laser pulse in this case is only 11 ps, but increases with higher peak currents. The electron output pulse shape is calculated from the exit times re(Z = 12m) of the 900 representative electrons and the corresponding initial instantaneous charge densities. As can be seen, the leading and trailing edge of the electron pulse remain essentially undistorted and lag relative to the incident field by (1 - f3z(O))z/c = 2 ps. The center part of the electron pulse, however, is severely distorted. Trapped electrons at the center of the pulse lose about 62% of their initial energy and at z = 12 m they lag relative to incident field by ~3.5 ps. These slower electrons pile up on the trailing side of the pulse and distort its shape. The break in the leading side of the electron pulse cor-
1013
fiE "} 10'2
1.0-
j
~electron
m z tO 11 u3
0.6
10 m
<
0.4
electron
i~p.t ~ , , e / /
J
t /
]
/
\\
loser
o~tp~,
\ \
,~ 10 9 Ld 0.. 10
_ 0
3 DISTANCE
1 6 IN
I METERS
Fig. 3. Peak laser intensity versus distance. 284
output pulse
d o.a
J
f
/ /
t
I 9
I
I 12
O
15
I 30 TIME
©.~.% 45
60
(psec)
Fig. 4. Normalized laser and electron flux as functions of the proper time of the laser field.
Volume 41, number 4
OPTICS COMMUNICATIONS
responds to the threshold current for efficient electron trapping. Finally, fig. 5 shows the phase of the output laser pulse. The phase of the input laser pulse is assumed to be constant and equal to zero. As can be seen, the amplification gives rise to strong phase modulation. The instantaneous frequency of the leading side of the pulse is ~ 5 0 GHz lower than the center frequency of the pulse, while the trailing side is "~60 GHz higher. The phase modulation increases the effective spectral bandwidth of the pulse beyond the inverse of the pulsewidth. Calculations with the initial electron energy being 1% higher or lower than the nominal design value and with all other parameters being the same showed the following behavior. When the electron energy was increased by l% the electron trapping efficiency and the amplifier gain fell off dramatically. On the other hand, when the electron energy was decreased by the same amount the amplifier gain remained essentially the same. Tb.is asymmetry in the performance of a highgain FEL amplifier with variable wiggler period can be explained as follows. Lower energy electrons are injected into the wiggler below resonance, but are brought into resonance as the wiggler period decreases and become trapped. Higher energy electrons are injected above resonance and are driven farther above resonance as they move in the wiggler, never being trapped. The less than 0.1% electron energy spread in storage rings is well within the tolerance limits of the high gain FEL amplifiers considered here. Increasing or decreasing
15 April 1982
the incident laser intensity by a factor of two had no significant effect on the performance of the amplifier. Using higher peak electron currents caused the laser output pulse to broaden and develop several peaks, Similar results have been obtained for FEL amplifiers designed for wavelengths at 10.6/am and 0.5 #m with electron energies 3, = 100 and 3' = 1000, respectively. In the work reported in ref. [3] for FEL amplifier at 0.5/Jm with electron energy 3' = 1000 and magnetic field strength of 1 kG, it was found that unrealistic wiggler lengths of 2 0 0 - 3 0 0 m are required for adiabatic trapping of the electron and high energy extraction efficiency. Our calculations show that by using a 35 kG initial magnetic field strength, the required wiggler length is reduced by a factor of ten. In general, the higher the electron energy is, the stronger the wiggler magnet should be, in order to keep the ratio of transverse to axial electron velocity constant (vt/u z ~ ]14/')" ~ 1/200). We should point out that magnetic field strengths of 35 kG are not unrealistic and that 50 kG superconducting wiggler magnets are in operation [10]. It would be more practical, however, to amplify 0.5/am radiation using less energetic electrons (7 ~ 200) and weaker wiggler fields (M ~ 1/x/~). In conclusion, we have shown that peak energy extraction efficiencies of over 60% are possible in FEL amplifiers with exponentially decreasing wiggler period. The power gain and output power level for this type of amplifiers are determined by the electron beam power density and for realistic current densities they can exceed 104 and 1 TW/cm 2, respectively. This work was partially supported by the National Science Foundation under grant No. ENG-7623704.
77"
References a.
o
-
I
0
I
~5 TIME
50 (psec)
I
45
I 6O
Fig. 5. Laser output phase as a function the proper time of the field.
[1] S.A. Mani, in: Free-electron generators of coherent radiation, Physics of Quantum Electronics, Vol. 7, eds. S.F. Jacobs, H.S. Pilloff, M. Sargent III, M.O. Scully and R. Spitzer (Addison-Wesley, Reading, Mass. 1990) p. 589. [2] D. Prosnitz, A. Szoke and V.K. Neff, ibid, p. 571. [3] H.R. Hiddleston, S.B. Segall and G.C. Catella, ibid, p. 729. [4] C. Pellegrini, ibid, p. 415. [5] F.A. Hopf, P. Meystre, M.O. Scully and W.H. Louisell, Optics Comm. 18 (1976) 413; Phys. Rev. Lett. 37 (1976) 1342.
285
Volume 41, number 4
OPTICS COMMUNICATIONS
[6] T.G. Kuper, G.T. Moore and M.O. ScuUy, Optics Comm. 34 (1980) 117; H. A1-Abawi, J.K. Mc Iver, G.T. Moore and M.O. Scully, in: Free-electron generators of coherent radiation, Physics of Quantum Electronics, Vol. 8, eds. S.F. Jacobs, G.T. Moore, H.S. Pilloff, M. Sargent, M.O. ScuUy and R. Spitzer (Addison-Wesley, Reasing, Mass., 1982) p. 415.
286
15 April 1982
[7] W.H. Louisell, C.D. Cantrell and W.A. Wegener, article in ref. [1], p. 623. [8] A. Icsevgi and W.E. Lamb, Jr,, Rev. Phys. 185 (1969) 517. [9] N.M. KroU, P.L. Morton and M.N. Rosenbluth, article inref. [1],p. 89. [10] H. Winick, G. Brown, K. Halbach and J. Harris, Physics Today 34 (1981) 51.
OPTICS COMMUNICATIONS
15 April 1982
HIGH-GAIN FREE-ELECTRON LASER PULSE AMPLIFIERS WITH VARIABLE WIGGLERS
A.T. GEORGES and W.H. LOUISELL Department of Physics, University of Southern California, Los Angeles, CA 90007, USA Received 28 September 1981 Revised manuscript received 27 November 1981
Calculations are presented for pulse amplification in high-gain free-electron lasers with variable wiggler magnets. Better than 90% peak electron trapping and overall peak energy extraction efficiencies of over 60% are demonstrated for 1 ~m radiation. Pulse reshaping and phase modulation effects are investigated.
Recently, the efficient transfer, through stimulated emission, of the kinetic energy of a highly relativistic electron beam to a co-propagating laser beam has attracted great interest in single-pass free-electron laser (FEL) amplifiers [ 1 - 3 ] . A 500 MeV electron beam with a peak current density of 105 A/cm 2 has a peak power density of 5 X 1013 W/cm 2. Such high electron beam power densities can be obtained in electron storage rings [4] which produce high quality beams with fractional energy spread of less than 10 -3. High power FEL amplifiers would be very useful in boosting the. power of conventional laser oscillators to the levels required for laser fusion. Theuse of damage-free and renewable electrons as amplifying medium is an advantage over conventional glass laser amplifiers which can be damaged by intense radiation. The key to high energy extraction efficiency in a FEL amplifier is the design of the wiggler magnet which imparts a small transverse motion to the electrons. It is through this transverse motion that the electrons couple to a transverse laser field (As) and exchange energy [5]. The variation of the wiggler period (kw) and magnetic vector potential (34) with distance (z) must be chosen such that .the electrons are adiabatically trapped in the ponderomotive potential and maintained in resonance (Xw = 2~,2Xs/(1 + 2M2), where Xs is the wavelength of the laser field and 3, the electron energy in rest energy units) as they decelerate. Since the rate of decrease of the electron energy depends on 282
the growing laser field, the optimum kw(z ) and M(z) values can be determined only by dynamic design or from trial and error solutions of the self-consistent equa. tions for the electron motion and the growth of the laser field. Computer simulation studies reported in ref. [1 ] have shown that wigglers with constant magnetic vector potential and period variation of the form Xw(Z) = Xw(O) exp [-(Z/Zo)4 ] give 100% electron trapping under conditions of high gain (input electron beam power density >> input laser intensity). These calculations, however, involve several approximations and are not self-consistent. For example, the laser field is not calculated from Maxwell's equation, but from an energy conservation relation which does not give information about the phase of the field. Moreover, they assume continuous (steady-state) electron beam and laser field, while high current electron beam sources and high power lasers operate in the pulsed mode and, therefore, it is important to account for the slippage of the electron and laser pulses as they move with different speeds. In this paper, we report the results of self-consistent calculations on the performance of high gain FEL pulse amplifiers with exponentially decreasing wiggler period and constant magnetic vector potential. In the one-dimensional approximation, the basic equations for a FEL amplifier with a variable helical wiggler can be written in the form [6,7]
Volume 41, number 4
OPTICS COMMUNICATIONS 1.0
dT(z, z e, 0 o)/dz = --i(ks/7/3z)
X (M(Z)As(z,r)exp[iO(z,re,O0) ] - c.c.},
(1)
dO(z,re,Oo)/dz = kw(Z ) + ks(1 - 1//3z) ,
(2)
0.8
~ 0.6 3.0.4
(d/dZ)As(z,r) = - i uOe M(z) ~ p ( r e ) 2ksm Oo
15 April 1982
0.2
exp [-iO(z,re,O0) ]
N3'(Z,re,O0)
(3)
'
where the transverse potentials for the wiggler, M(z), and the laser field, As(z,r ), are measured in units of mc/e, with e,m being the electron charge and mass, and c the speed of light, k s = 21r/Xs and kw(Z) = 27r/ Xw(Z) are the wave vectors for the laser field and the wiggler. ~z ~ [1 - (1 + 2/142)/72 ] 1/2 is the axial electron velocity normalized to the velocity of light, r = t - z/c is the proper time of the laser field and re(Z ) = t - f~c-l~l(z',re,OO)dZ ' the proper time of an electron. The angle 0 = f~kw(z')dz' - kscr + 00 measures the electron phase in the ponderomotive potential and 00 is its value at the entrance of the wiggler which is assumed to be uniformly distributed between zero and 27r. N is the number of representative electrons in the distribution for 00. P(%) is the electron charge density. In the computer simulation we consider a gaussian electron pulse with full width at half maximum (FWHM) of 25 ps and an overlapping gaussian incident laser pulse with FWHM of 50 ps. The trajectories of 900 representative electrons and the complex amplitude (As(z,r)) of the laser field are calculated self-consistently from eqs. (1)-(3). These coupled nonlinear differential equations are solved using a second-order predictor-corrector method [8] with interpolation to account for the slippage of the electron and laser pulses. As in ref. [ 1], we consider the case of monoenergetic electrons with 7 = 141.42 (72.47 MeV) interacting with a laser field of 1/am wavelength. The wiggler vector potential is assumed to be constant and equal to M = 1/V~ (initial magnetic field strength ~5.37 kG), and the initial wiggler period is determined from the resonance condition to be Xw = 272Xs/(1 + 2M 2) = 2 cm. An exponential variation, Xw(Z) = Xw(0) exp [-(Z/Zo)n ], was assumed for the wiggler period and the parameters z 0 and n were varied to maximize the fraction of electrons trapped. For a peak
O
I
5
~ _ _
I
I
6 9 DISTANCE IN METERS
12
Fig. 1. Normalized e l e c t ron energy versus distance along the wiggler for 36 electrons w i t h uni form phase d i s t r i b u t i o n at the center of the electron pulse.
laser intensity of 108 W/cm 2 and a peak current density of 3 X 104 A/cm 2 (electron beam power density = 2.17 X 1012 W/cm 2) it was determined that the values n = 4 and z 0 = 10.2 m give the highest electron trapping and energy extraction for the shortest amplifier length of 12 m. Ref. [1] uses the same values as above except for z 0 which is taken to be equal to 12.73 m. Fig. 1 shows the normalized electron energy as a function of distance inside the wiggler for 36 representative electrons at the center of the electron pulse. The initial phase (00) of the electrons is uniformly distributed between zero and 2n. Note that only three of these electrons are untrapped ("-92% trapping efficiency) and at z = 12 meters an average 62% of the energy is extracted from the trapped electrons. The overall energy extraction efficiency is 57% and could be made higher by going to slightly greater wiggler lengths. The final energy spread of the trapped electrons normalized to their initial energy is 1.5%. Fig. 2 shows the relative electron phase for the same group of electrons as a function of distance. The phases of the trapped electrons oscillate about either of two different average values separated by 2n and are localized in the decelerating half-cycles (buckets) of the periodic ponderomotive potential [9]. The period of the oscillations and their average value decrease as the wiggler period decreases and the electrons lose their energy to the growing laser field. The decrease in the average value of the oscillations is absent in the approximate results of ref. [1] (fig. 5) which incidentally show 100% electron trapping. Our calculations show that even for a 283
Volume 41, number 4 3~
OPTICS COMMUNICATIONS
/
2~ :
: /
,
,.
- 2-,r I ~ - ~ 1 - - - . 0 3
, , , .
,
I 6 DISTANCE
I 9 IN
k 12
METERS
Fig. 2. Relative phase versus distance for the same electron group as in fig. 1. more slowly decreasing wiggler period (z 0 > 12.73 m) some electrons whose phase falls between two adjacent trapping wells are not trapped. As will be explained below, the reason for the less than perfect electron trapping lies in the dynamics of a high gain FEL amplifier with decreasing wiggler period. Fig. 3 shows the amplification of the peak laser intensity along the wiggler. The gain per unit length is very high during the first two meters of the wiggler, where the laser intensity is much weaker than the electron beam power density. The initial high gain is followed by oscillations in the laser intensity which die out as the laser intensity becomes higher than the electron beam power density and the gain saturates. The oscillations in the laser intensity arise as follows: The growing laser field makes the electrons lose their energy too fast and they are driven out of resonances, absorbing energy back from the field. However, as the elec-
15 April 1982
trons move down the wiggler where the wiggler period decreases, they are brought again into resonance and lose energy to the field. This cycle repeats, but the amplitude of the oscillation decreases as the laser field grows more slowly. Comparing figs. 2 and 3 we see that it is during the dips in the laser intensity that weakly trapped electrons become detrapped. Therefore, a small degree of electron detrapping appears to be inevitable in high gain FEL amplifiers. The normalized input and output laser (solid line) and electron (dashed line) pulses are shown in fig. 4 as functions of the proper time of the laser field. The output laser pulse is asymmetric with the trailing edge amplified more than the leading edge, while its peak is delayed relative to the peak of the incident pulse by about 2.5 ps. These effects are due to the slippage between the electron and laser pulses which can be seen in fig. 4. Note that the FWHM of the output laser pulse in this case is only 11 ps, but increases with higher peak currents. The electron output pulse shape is calculated from the exit times re(Z = 12m) of the 900 representative electrons and the corresponding initial instantaneous charge densities. As can be seen, the leading and trailing edge of the electron pulse remain essentially undistorted and lag relative to the incident field by (1 - f3z(O))z/c = 2 ps. The center part of the electron pulse, however, is severely distorted. Trapped electrons at the center of the pulse lose about 62% of their initial energy and at z = 12 m they lag relative to incident field by ~3.5 ps. These slower electrons pile up on the trailing side of the pulse and distort its shape. The break in the leading side of the electron pulse cor-
1013
fiE "} 10'2
1.0-
j
~electron
m z tO 11 u3
0.6
10 m
<
0.4
electron
i~p.t ~ , , e / /
J
t /
]
/
\\
loser
o~tp~,
\ \
,~ 10 9 Ld 0.. 10
_ 0
3 DISTANCE
1 6 IN
I METERS
Fig. 3. Peak laser intensity versus distance. 284
output pulse
d o.a
J
f
/ /
t
I 9
I
I 12
O
15
I 30 TIME
©.~.% 45
60
(psec)
Fig. 4. Normalized laser and electron flux as functions of the proper time of the laser field.
Volume 41, number 4
OPTICS COMMUNICATIONS
responds to the threshold current for efficient electron trapping. Finally, fig. 5 shows the phase of the output laser pulse. The phase of the input laser pulse is assumed to be constant and equal to zero. As can be seen, the amplification gives rise to strong phase modulation. The instantaneous frequency of the leading side of the pulse is ~ 5 0 GHz lower than the center frequency of the pulse, while the trailing side is "~60 GHz higher. The phase modulation increases the effective spectral bandwidth of the pulse beyond the inverse of the pulsewidth. Calculations with the initial electron energy being 1% higher or lower than the nominal design value and with all other parameters being the same showed the following behavior. When the electron energy was increased by l% the electron trapping efficiency and the amplifier gain fell off dramatically. On the other hand, when the electron energy was decreased by the same amount the amplifier gain remained essentially the same. Tb.is asymmetry in the performance of a highgain FEL amplifier with variable wiggler period can be explained as follows. Lower energy electrons are injected into the wiggler below resonance, but are brought into resonance as the wiggler period decreases and become trapped. Higher energy electrons are injected above resonance and are driven farther above resonance as they move in the wiggler, never being trapped. The less than 0.1% electron energy spread in storage rings is well within the tolerance limits of the high gain FEL amplifiers considered here. Increasing or decreasing
15 April 1982
the incident laser intensity by a factor of two had no significant effect on the performance of the amplifier. Using higher peak electron currents caused the laser output pulse to broaden and develop several peaks, Similar results have been obtained for FEL amplifiers designed for wavelengths at 10.6/am and 0.5 #m with electron energies 3, = 100 and 3' = 1000, respectively. In the work reported in ref. [3] for FEL amplifier at 0.5/Jm with electron energy 3' = 1000 and magnetic field strength of 1 kG, it was found that unrealistic wiggler lengths of 2 0 0 - 3 0 0 m are required for adiabatic trapping of the electron and high energy extraction efficiency. Our calculations show that by using a 35 kG initial magnetic field strength, the required wiggler length is reduced by a factor of ten. In general, the higher the electron energy is, the stronger the wiggler magnet should be, in order to keep the ratio of transverse to axial electron velocity constant (vt/u z ~ ]14/')" ~ 1/200). We should point out that magnetic field strengths of 35 kG are not unrealistic and that 50 kG superconducting wiggler magnets are in operation [10]. It would be more practical, however, to amplify 0.5/am radiation using less energetic electrons (7 ~ 200) and weaker wiggler fields (M ~ 1/x/~). In conclusion, we have shown that peak energy extraction efficiencies of over 60% are possible in FEL amplifiers with exponentially decreasing wiggler period. The power gain and output power level for this type of amplifiers are determined by the electron beam power density and for realistic current densities they can exceed 104 and 1 TW/cm 2, respectively. This work was partially supported by the National Science Foundation under grant No. ENG-7623704.
77"
References a.
o
-
I
0
I
~5 TIME
50 (psec)
I
45
I 6O
Fig. 5. Laser output phase as a function the proper time of the field.
[1] S.A. Mani, in: Free-electron generators of coherent radiation, Physics of Quantum Electronics, Vol. 7, eds. S.F. Jacobs, H.S. Pilloff, M. Sargent III, M.O. Scully and R. Spitzer (Addison-Wesley, Reading, Mass. 1990) p. 589. [2] D. Prosnitz, A. Szoke and V.K. Neff, ibid, p. 571. [3] H.R. Hiddleston, S.B. Segall and G.C. Catella, ibid, p. 729. [4] C. Pellegrini, ibid, p. 415. [5] F.A. Hopf, P. Meystre, M.O. Scully and W.H. Louisell, Optics Comm. 18 (1976) 413; Phys. Rev. Lett. 37 (1976) 1342.
285
Volume 41, number 4
OPTICS COMMUNICATIONS
[6] T.G. Kuper, G.T. Moore and M.O. ScuUy, Optics Comm. 34 (1980) 117; H. A1-Abawi, J.K. Mc Iver, G.T. Moore and M.O. Scully, in: Free-electron generators of coherent radiation, Physics of Quantum Electronics, Vol. 8, eds. S.F. Jacobs, G.T. Moore, H.S. Pilloff, M. Sargent, M.O. ScuUy and R. Spitzer (Addison-Wesley, Reasing, Mass., 1982) p. 415.
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