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A new approach to XUV: the ϱ cooling technique
Volume 81, number 5
OPTICS COMMUNICATIONS
1 March 1991
A new approach to XUV: the p cooling technique R. Bonifacio
INFN and Universit.~di Milano, Via Celoria, 16, 20133 Milano, Italy Received 31 July 1990; revised manuscript received 8 October 1990
We demonstrate that o n resonance and for a fixed energy and wavelength, there is a maximum value of the fundamental FEL parameter p when the wiggler parameter is one. This circumstance maximizes the efficiency and minimizes the high gain "effective energy spread" Ay/pT.Furthermore we show the possibilityof"effective" beam cooling in the proposed of two wigglerscheme for coherent harmonic generation in the exponential gain regime. This consists of tuning the second wiggler to the nth harmonics while changing simultaneously Bw and 2,, so that p increases and leads to a decrease of Ay/py in the second wiggler. This "pcooling" technique can be relevant in many applications and, in particular, (i) it allows, in principle, the tapering of the second wiggler so that one can reach an efficiency larger than 10% for the harmonic generation; (ii) it allows one to relax the beam emittanee requirement with a proper generalization of the Kim-Pellegrini criterion.
1. p-cooling
F r o m the resonance condition ( 2 ) we also see that the correspondent value o f aw is
In a high gain system both efficiency and energy spread are ruled by the f u n d a m e n t a l F E L p a r a m e t e r p [ 1 ] which scales as
a~=l,
(5)
where J is the current density, Bw is the wiggler field,
which in turn fixes the value o f Bw. If, for example, 2 = 80 n m a n d E = 300 MeV, it follows that 2w= 3 cm yields the m a x i m u m value for p. Let us note that the ratio between the generic value o f p a n d the optim u m value Pop, is given by
2w is the wiggler period, an y is the electron energy.
p/popt= [ Y ( 2 - Y) ]3/2,
B2/3,~4/311/3 p-~0.217
w
"~w
v
(SI),
(1)
Y
On the other hand, the resonance condition has the form 2 = 2 w ( )l +_a 2_ 272
--
3 2 2 w + K 22wBw 272 '
(2)
where K = e / 2 m n c 2 a n d a,,,=K2wBw is the wiggler parameter. Hence, at a given 2 a n d y we can write B2w= (2722-2w)/K223w,
(2a)
which substituted into ( 1 ), with J, 2 and 7 held constant, gives p3oC2w(2y22-2w)
•
(3)
This shows that p has a m a x i m u m when 2w = 2 y 2 .
(4)
(6)
where Y=2w/2y 2. The limitation Y~< 2, i.e. 2w,,< 2722, means that for 2 w> 2722 it is impossible to satisfy the resonance condition ( 2 ) . The possibility o f o p t i m izing p while keeping the resonance condition gives rise to a kind o f tapering which cools down the beam. In fact, we r e m e m b e r that in a high gain kystem the relevant energy spread is not given by Ay/y b u t by the "effective energy spread" Ay/py [ 1 ], since p is the linewidth in the high gain regime. Hence, by increasing p, one obtains an effective cooling o f the b e a m following an increase in the linewidth p. This would not be possible in a small gain system when the effective energy spread scales as NwAy/7, since the natural linewidth is 1/Nw, where Nw is the number o f wiggler periods. F o r example, one can imagine to reach the exponential gain regime with a given
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
31 1
Volume 81, number 5
OPTICS COMMUNICATIONS
2w~ y2t which gives the same 1. Hence, upon changing the wiggler period to 2w=2y 2, one cools the beam, since p increases so that A7/P7 decreases and the amount of "cooling" is given by (6).
21 +K.I~B ~ --
n7 2
,
(7)
where 2 and 1 refer the second and first wiggler, respectively. After, imposing the consistency condition (7) with a given ratio P2/Pl, for fixed values of J and 7, we obtain easily: (~-~)
312
3=
(l+a2 -)0 X n ~212'
(~_2)3 a~z aZ~=
--
-X5 =
(l+a~
1)
nX
(9) "
) ( = 2 d 2 , = ( 1+ a Z ) / 2 n ,
In a previous paper [ 2 ] we proposed the following technique to produce coherent spontaneous emission (C.S.E.) at a harmonics frequency. One should use a first wiggler to drive almost to saturation a small signal at the same fundamental wavelength 2. This gives strong bunching at the harmonics due to the coherent driving of the exponentially growing fundamental. There will be C.S.E. from the bunching on the nth harmonics. However the efficiency of the method, as originally proposed, is greatly reduced by the energy spread produced by the first wiggler. The effective energy spread is increased further since the second wiggler is tuned by simple reducing the wiggler field Bw as in ref. [2]. In fact in this way one reduces p, as discussed in ref. [ 2 ], and thus increase the effective energy spread Ay/p7. The large initial energy spread of the second wiggler causes a small efficiency and the practical impossibility of tapering because a large fraction of the electrons remain out of the bucket even after the C.S.E. radiation process. We now demonstrate that the second wiggler can be tuned to the nth harmonics by changing Bw and 2w so that the p parameter can be increased, in principle arbitrarily, and yields an effective cooling in the second wiggler. This should increase the efficiency of the FEL process and open the possibility of tapering at the nth harmonics. In order to tune the second wiggler on the nth harmonics one must have 72
where X = 1 2 / 2 1 , and a~ = K ~ I B I is the parameter of wiggler 1. Furthermore by using ( 1 ) and (7) one has
From (8) we see that P2/P~ is optimized by choosing
2. The two wiggler technique for harmonic generation
22 +K2z3B~z
1 March 1991
(10)
which from (9) implies (11)
az=l
as it must be according to our previous discussion. The corresponding value ofp2/p~ is (P2/ P~ ) 3 = (l + a~ ) / 2nal .
(12)
If in view of tapering one wants a higher value of a2 one must choose a value of X < X a s one can see easily from (9). Because we also need P2/P~ > 1 we find immediately from (12) that the inequality
l
al ~ nWx/-n-2-~ or al
~>n+~l
must be satisfied. Hence the effective cooling in the second wiggler can be done only for large or small values of the first wiggler parameter according to the inequality (13). Let us note that also even value of the nth harmonics can be considered because the first wiggler produces bunching also at even harmonics even i f no field is produced [ 3 ]. However, as already noticed [ 2 ], this bunching radiates coherently once if is brought into resonance in the second wiggler: Explicit expressions of the various parameters which satisfy the previous conditions can be given as follows: The inequality (13) is satisfied by the choice al=al=(np+~l)
+-~ (p>~l),
(14)
which, for nZpZ>>l, can be approximated by a~ - (2np) -+i. Upon inserting the identity 1 + a 2 = 2npal into (8) and (9) one obtains easily X - 2 2 / 2 ~ = 2 p a d ( l +a2,) ,
(15)
and (P2/Pl )3=4p2a2 / (1 + a2) 2 .
(8)
(13)
We note that P2/P~ > 1 implies
(16)
Volume 81, number 5
p+
OPTICS COMMUNICATIONS
1
px/~-~_l~ a z ~ p + x / p 2 - - 1 ,
(17)
and that P2/Pl is the maximum for a2= 1, which gives
P21Pl =p2/3.
(18)
Finally, since a2/al = B23'2/BI3' 1, and after using ( 15 ) we have
B2/ BI =a2( 1 + aE) /2pa 2 .
(19)
Hence the p-cooling technique, for fixed p > 1 can be obtained, in principle, up to an arbitrary value of p2/ p l = p 3/2 provided a2 = 1 and 22/3.1 =pal, where d~ must be chosen according to (14) and BE/BI= 1/ pd 2. If, for easier tapering one wants a2 > 1 one must use the more general expressions ( 15)-(17). Explicit examples will be discussed elsewhere. We now discuss the relevance of p-cooling to respect to emittance limitations in high gain regime. The Kim-Pellegrini emittance criterion has the form:
~ <...73'12n .
(20)
This limitation, however, is valid for spontaneous emission or for a low gain system. A more general criterion reads 7 3.p (Av~
~ ~\T)'
(21)
where 3.~=3.wxf2/aw is the betatron wavelength and Av/v is the FEL full width. Eq. (21) reduces to (20) when
Ap/v=I/N,,,
(a)
as for spontaneous emission and assuming 3'p=NwAw.
(b)
Inequality (21) to within factors 2 can be derived with the same calculations reported in ref. [ 4 ], without assuming (a) or (b). For the high-gain system A v / v = 2 p and eq. (21) becomes
73. ( 23'pp~
¢~< ~--~\~-~--/.
(22)
A precise derivation ofeq. (22) will be given elsewhere [ 5 ]. In view of this inequality the advantage of p-cooling is evident. We now show that limitation
! March 199i
(22) is less restrictive than (20) because in the exponential gain regime it must be: 22pp/Aw >/x/~.
(23)
In fact in a high-gain FEL the Raylength range Zr = ztr2/2 must be larger than the gain length 2w/4rrp [6] i.e.,
ltr2 /2 >~2w/ 4Ztp .
(24)
Since r 2 = 2 # / 2 r t y , we can write (24) as
73. 3.,~ e >/2--np3.---~"
(25)
If we combine (22) and (25) we obtain
Y3' 3.__~__~<.%e<.%y3.3.pp 2np3.p x 3'w "
(26)
The necessary condition for the validity of eq. (26) is given by eq. (23). After, inserting 3.p into ( 23 ), we have 3.pp 7P 3'w aw
-
Po aw
,
(27)
where po = (topaw2w/8Ztc) 2/3, aw=eBw/m and top_2 _ eEne/%me. Thus, we can write explicitly 3.pp ~2.2X 10_4j (Amp/mm2) 3.w(cm) 3.w Bw(T) "
(28)
This condition, equivalent to (23), sets a limitation on the current density, 3'w and Bw, which, independently from electron energy, must be satisfied in a high-gain FEL to match the condition on the emittance and on the Raylength range. In conclusion, we have shown the possibility of a p-cooling technique for reducing the effective energy spread AT/P7 by increasing p in a high-gain FEL. In particular, we have given explicit expression for p cooling in a two wiggler scheme for harmonic generation which should facilitate tapering in the second wiggler. Finally we have introduced a emittance limitation which is valid for a saturated beam. This limitation is shown to be less restrictive than the KimPellegrini limitation and can be further relaxed by the p cooling technique. K_rinsky and Yu [ 7 ] have already shown that the linear gain in the exponential regime is not reduced by emittance if the inequality of eq. (22) is satisfied 313
Volume 81, number 5
OPTICS COMMUNICATIONS
i n a strong sense, i.e., ~ , ~ rhs. It c a n b e s h o w n [ 5] that b o t h the l i n e a r g a i n a n d the s a t u r a t i o n i n t e n s i t y are n o t affected i f the i n e q u a l i t y o f eq. ( 2 2 ) holds.
Acknowledgments I a m m o s t l y grateful to L.M. N a r d u c c i for carefully reading, criticizing a n d i m p r o v i n g the manuscript.
References [ 1 ] R. Bonifaeio, C. Pellegrini and L.M. Narducci, Optics Comm. 50 (1984) 373.
314
1 March 1991
[2] R. Bonifacio, L. De Salvo Souza, P. Pierini and E.T. Seharlemann, Nuel. Instr. Meth. A 296 (1990) 787. [ 3 ] R. Bonifaeio, L. De Salvo and P. Pierini, Nuel. Instr. Meth. A 193 (1990) 627. [4] C. Pellegrini and J.B. Murphy, in: Proe. Joint US-CERN Particle Accelerator School, South Padre Islands, Texas, (1986), eds. M. Month and S. Turner (Springer, Berlin, 1988). [ 5 ] R.Bonifaeio and B.W.J. McNeil, in preparaUon. [ 6 ] See for example W.A. Barletta and A.M. Sessler, in: High gain, high power FEL, eds. R. Bonifacio, L. De Salvo Souza and C. Pellegrini (Nonh-HoUand, Amsterdam, 1989) p. 211. [ 7 ] Li-Hua Yu and S. Krinsky, Phys. Lett. A 129 (1988) 463.
OPTICS COMMUNICATIONS
1 March 1991
A new approach to XUV: the p cooling technique R. Bonifacio
INFN and Universit.~di Milano, Via Celoria, 16, 20133 Milano, Italy Received 31 July 1990; revised manuscript received 8 October 1990
We demonstrate that o n resonance and for a fixed energy and wavelength, there is a maximum value of the fundamental FEL parameter p when the wiggler parameter is one. This circumstance maximizes the efficiency and minimizes the high gain "effective energy spread" Ay/pT.Furthermore we show the possibilityof"effective" beam cooling in the proposed of two wigglerscheme for coherent harmonic generation in the exponential gain regime. This consists of tuning the second wiggler to the nth harmonics while changing simultaneously Bw and 2,, so that p increases and leads to a decrease of Ay/py in the second wiggler. This "pcooling" technique can be relevant in many applications and, in particular, (i) it allows, in principle, the tapering of the second wiggler so that one can reach an efficiency larger than 10% for the harmonic generation; (ii) it allows one to relax the beam emittanee requirement with a proper generalization of the Kim-Pellegrini criterion.
1. p-cooling
F r o m the resonance condition ( 2 ) we also see that the correspondent value o f aw is
In a high gain system both efficiency and energy spread are ruled by the f u n d a m e n t a l F E L p a r a m e t e r p [ 1 ] which scales as
a~=l,
(5)
where J is the current density, Bw is the wiggler field,
which in turn fixes the value o f Bw. If, for example, 2 = 80 n m a n d E = 300 MeV, it follows that 2w= 3 cm yields the m a x i m u m value for p. Let us note that the ratio between the generic value o f p a n d the optim u m value Pop, is given by
2w is the wiggler period, an y is the electron energy.
p/popt= [ Y ( 2 - Y) ]3/2,
B2/3,~4/311/3 p-~0.217
w
"~w
v
(SI),
(1)
Y
On the other hand, the resonance condition has the form 2 = 2 w ( )l +_a 2_ 272
--
3 2 2 w + K 22wBw 272 '
(2)
where K = e / 2 m n c 2 a n d a,,,=K2wBw is the wiggler parameter. Hence, at a given 2 a n d y we can write B2w= (2722-2w)/K223w,
(2a)
which substituted into ( 1 ), with J, 2 and 7 held constant, gives p3oC2w(2y22-2w)
•
(3)
This shows that p has a m a x i m u m when 2w = 2 y 2 .
(4)
(6)
where Y=2w/2y 2. The limitation Y~< 2, i.e. 2w,,< 2722, means that for 2 w> 2722 it is impossible to satisfy the resonance condition ( 2 ) . The possibility o f o p t i m izing p while keeping the resonance condition gives rise to a kind o f tapering which cools down the beam. In fact, we r e m e m b e r that in a high gain kystem the relevant energy spread is not given by Ay/y b u t by the "effective energy spread" Ay/py [ 1 ], since p is the linewidth in the high gain regime. Hence, by increasing p, one obtains an effective cooling o f the b e a m following an increase in the linewidth p. This would not be possible in a small gain system when the effective energy spread scales as NwAy/7, since the natural linewidth is 1/Nw, where Nw is the number o f wiggler periods. F o r example, one can imagine to reach the exponential gain regime with a given
0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
31 1
Volume 81, number 5
OPTICS COMMUNICATIONS
2w~ y2t which gives the same 1. Hence, upon changing the wiggler period to 2w=2y 2, one cools the beam, since p increases so that A7/P7 decreases and the amount of "cooling" is given by (6).
21 +K.I~B ~ --
n7 2
,
(7)
where 2 and 1 refer the second and first wiggler, respectively. After, imposing the consistency condition (7) with a given ratio P2/Pl, for fixed values of J and 7, we obtain easily: (~-~)
312
3=
(l+a2 -)0 X n ~212'
(~_2)3 a~z aZ~=
--
-X5 =
(l+a~
1)
nX
(9) "
) ( = 2 d 2 , = ( 1+ a Z ) / 2 n ,
In a previous paper [ 2 ] we proposed the following technique to produce coherent spontaneous emission (C.S.E.) at a harmonics frequency. One should use a first wiggler to drive almost to saturation a small signal at the same fundamental wavelength 2. This gives strong bunching at the harmonics due to the coherent driving of the exponentially growing fundamental. There will be C.S.E. from the bunching on the nth harmonics. However the efficiency of the method, as originally proposed, is greatly reduced by the energy spread produced by the first wiggler. The effective energy spread is increased further since the second wiggler is tuned by simple reducing the wiggler field Bw as in ref. [2]. In fact in this way one reduces p, as discussed in ref. [ 2 ], and thus increase the effective energy spread Ay/p7. The large initial energy spread of the second wiggler causes a small efficiency and the practical impossibility of tapering because a large fraction of the electrons remain out of the bucket even after the C.S.E. radiation process. We now demonstrate that the second wiggler can be tuned to the nth harmonics by changing Bw and 2w so that the p parameter can be increased, in principle arbitrarily, and yields an effective cooling in the second wiggler. This should increase the efficiency of the FEL process and open the possibility of tapering at the nth harmonics. In order to tune the second wiggler on the nth harmonics one must have 72
where X = 1 2 / 2 1 , and a~ = K ~ I B I is the parameter of wiggler 1. Furthermore by using ( 1 ) and (7) one has
From (8) we see that P2/P~ is optimized by choosing
2. The two wiggler technique for harmonic generation
22 +K2z3B~z
1 March 1991
(10)
which from (9) implies (11)
az=l
as it must be according to our previous discussion. The corresponding value ofp2/p~ is (P2/ P~ ) 3 = (l + a~ ) / 2nal .
(12)
If in view of tapering one wants a higher value of a2 one must choose a value of X < X a s one can see easily from (9). Because we also need P2/P~ > 1 we find immediately from (12) that the inequality
l
al ~ nWx/-n-2-~ or al
~>n+~l
must be satisfied. Hence the effective cooling in the second wiggler can be done only for large or small values of the first wiggler parameter according to the inequality (13). Let us note that also even value of the nth harmonics can be considered because the first wiggler produces bunching also at even harmonics even i f no field is produced [ 3 ]. However, as already noticed [ 2 ], this bunching radiates coherently once if is brought into resonance in the second wiggler: Explicit expressions of the various parameters which satisfy the previous conditions can be given as follows: The inequality (13) is satisfied by the choice al=al=(np+~l)
+-~ (p>~l),
(14)
which, for nZpZ>>l, can be approximated by a~ - (2np) -+i. Upon inserting the identity 1 + a 2 = 2npal into (8) and (9) one obtains easily X - 2 2 / 2 ~ = 2 p a d ( l +a2,) ,
(15)
and (P2/Pl )3=4p2a2 / (1 + a2) 2 .
(8)
(13)
We note that P2/P~ > 1 implies
(16)
Volume 81, number 5
p+
OPTICS COMMUNICATIONS
1
px/~-~_l~ a z ~ p + x / p 2 - - 1 ,
(17)
and that P2/Pl is the maximum for a2= 1, which gives
P21Pl =p2/3.
(18)
Finally, since a2/al = B23'2/BI3' 1, and after using ( 15 ) we have
B2/ BI =a2( 1 + aE) /2pa 2 .
(19)
Hence the p-cooling technique, for fixed p > 1 can be obtained, in principle, up to an arbitrary value of p2/ p l = p 3/2 provided a2 = 1 and 22/3.1 =pal, where d~ must be chosen according to (14) and BE/BI= 1/ pd 2. If, for easier tapering one wants a2 > 1 one must use the more general expressions ( 15)-(17). Explicit examples will be discussed elsewhere. We now discuss the relevance of p-cooling to respect to emittance limitations in high gain regime. The Kim-Pellegrini emittance criterion has the form:
~ <...73'12n .
(20)
This limitation, however, is valid for spontaneous emission or for a low gain system. A more general criterion reads 7 3.p (Av~
~ ~\T)'
(21)
where 3.~=3.wxf2/aw is the betatron wavelength and Av/v is the FEL full width. Eq. (21) reduces to (20) when
Ap/v=I/N,,,
(a)
as for spontaneous emission and assuming 3'p=NwAw.
(b)
Inequality (21) to within factors 2 can be derived with the same calculations reported in ref. [ 4 ], without assuming (a) or (b). For the high-gain system A v / v = 2 p and eq. (21) becomes
73. ( 23'pp~
¢~< ~--~\~-~--/.
(22)
A precise derivation ofeq. (22) will be given elsewhere [ 5 ]. In view of this inequality the advantage of p-cooling is evident. We now show that limitation
! March 199i
(22) is less restrictive than (20) because in the exponential gain regime it must be: 22pp/Aw >/x/~.
(23)
In fact in a high-gain FEL the Raylength range Zr = ztr2/2 must be larger than the gain length 2w/4rrp [6] i.e.,
ltr2 /2 >~2w/ 4Ztp .
(24)
Since r 2 = 2 # / 2 r t y , we can write (24) as
73. 3.,~ e >/2--np3.---~"
(25)
If we combine (22) and (25) we obtain
Y3' 3.__~__~<.%e<.%y3.3.pp 2np3.p x 3'w "
(26)
The necessary condition for the validity of eq. (26) is given by eq. (23). After, inserting 3.p into ( 23 ), we have 3.pp 7P 3'w aw
-
Po aw
,
(27)
where po = (topaw2w/8Ztc) 2/3, aw=eBw/m and top_2 _ eEne/%me. Thus, we can write explicitly 3.pp ~2.2X 10_4j (Amp/mm2) 3.w(cm) 3.w Bw(T) "
(28)
This condition, equivalent to (23), sets a limitation on the current density, 3'w and Bw, which, independently from electron energy, must be satisfied in a high-gain FEL to match the condition on the emittance and on the Raylength range. In conclusion, we have shown the possibility of a p-cooling technique for reducing the effective energy spread AT/P7 by increasing p in a high-gain FEL. In particular, we have given explicit expression for p cooling in a two wiggler scheme for harmonic generation which should facilitate tapering in the second wiggler. Finally we have introduced a emittance limitation which is valid for a saturated beam. This limitation is shown to be less restrictive than the KimPellegrini limitation and can be further relaxed by the p cooling technique. K_rinsky and Yu [ 7 ] have already shown that the linear gain in the exponential regime is not reduced by emittance if the inequality of eq. (22) is satisfied 313
Volume 81, number 5
OPTICS COMMUNICATIONS
i n a strong sense, i.e., ~ , ~ rhs. It c a n b e s h o w n [ 5] that b o t h the l i n e a r g a i n a n d the s a t u r a t i o n i n t e n s i t y are n o t affected i f the i n e q u a l i t y o f eq. ( 2 2 ) holds.
Acknowledgments I a m m o s t l y grateful to L.M. N a r d u c c i for carefully reading, criticizing a n d i m p r o v i n g the manuscript.
References [ 1 ] R. Bonifaeio, C. Pellegrini and L.M. Narducci, Optics Comm. 50 (1984) 373.
314
1 March 1991
[2] R. Bonifacio, L. De Salvo Souza, P. Pierini and E.T. Seharlemann, Nuel. Instr. Meth. A 296 (1990) 787. [ 3 ] R. Bonifaeio, L. De Salvo and P. Pierini, Nuel. Instr. Meth. A 193 (1990) 627. [4] C. Pellegrini and J.B. Murphy, in: Proe. Joint US-CERN Particle Accelerator School, South Padre Islands, Texas, (1986), eds. M. Month and S. Turner (Springer, Berlin, 1988). [ 5 ] R.Bonifaeio and B.W.J. McNeil, in preparaUon. [ 6 ] See for example W.A. Barletta and A.M. Sessler, in: High gain, high power FEL, eds. R. Bonifacio, L. De Salvo Souza and C. Pellegrini (Nonh-HoUand, Amsterdam, 1989) p. 211. [ 7 ] Li-Hua Yu and S. Krinsky, Phys. Lett. A 129 (1988) 463.