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Kick and phase errors in spontaneous and amplified radiation
Nuclear Instruments and Methods in Physics Research A 445 (2000) 24}27
Kick and phase errors in spontaneous and ampli"ed radiation夽 Kwang-Je Kim* Accelerator Systems Division, Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439-4800, USA
Abstract Two types of magnet errors are considered } the random phase error (RPE), in which the phase errors are evenly distributed along the magnet, and the random kick error (RKE), in which the errors in the derivative of the phase are evenly distributed. We compute the reduction in performance of both spontaneous radiation and high-gain free-electron lasers for both types of errors within the framework of 1-D free-electron laser theory. 2000 Elsevier Science B.V. All rights reserved. PACS: 41.60.Cr Keywords: Free electron lasers; Magnet errors
1. Introduction E!ects of undulator errors on radiation performance have been studied by several authors [1}10]. The errors are most conveniently characterized by the phase error [6,8]
d (q)"k
dz [t !t ]. X XM t XM
(1)
Here k"2p/j, j is the radiation wavelength, z the distance along the undulator axis, t is the average XM z-velocity in the ideal undulator, t is the average X z-velocity in the presence of errors, q is the scaled longitudinal coordinate, q"2ok z"N/N %
(2)
夽 Work supported by the US Department of Energy, O$ce of Basic Energy Sciences, under Contract No. W-31-109-ENG-38. * Tel.: #630-252-4647; Fax: #630-252-7369. E-mail address: [email protected] (K.-J. Kim).
where o is the Pierce parameter [11], k "2p/j , j is the undulator period length, N"z/j (the number of the undulator periods), and N "1/4po % is the number of undulator periods in one nominal gain length. For cases considered here, N <1. % For well-optimized undulators corrected by suitably placed shims, the phase errors are distributed uniformly along the undulator with a constant rms value p [8}10]. This type of error will be referred ( to as the random phase error (RPE). In the presence of RPE, the intensity of the spontaneous radiation in the forward direction is reduced by the factor [8] R"1!p . (
(3)
For long undulators, the errors could be controlled by steering corrections at regular intervals [3,5}7]. In these cases, the errors in the phase derivative, d d /dq, rather than the phase errors themselves, are uniformly distributed along the undulator. For this type of error, which will be
0168-9002/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 1 0 7 - 8
K.-J. Kim / Nuclear Instruments and Methods in Physics Research A 445 (2000) 24}27
The function g(q) is an even function of q with a width p , normalized as O
Table 1 Reduction due to magnet errors Random phase error (RPE)
Random kick error (RKE)
Spontaneous radiation intensity in the forward direction
1!p (
1!=q
We will assume
High-gain FEL growth rate
p 1! ( q (3
g(q) dq"1.
1!=q
referred to as the random kick error (RKE), the reduction in the power growth of high-gain FELs was found to be [7] R"1! =q where
(4)
2 dd dq . (5) dq Here the angular brackets denote ensemble average. In this paper we generalize the above results to others cases } the reduction in spontaneous radiation for RKE and the reduction in high-gain growth for RPE. The results are summarized in Table 1. 1 =" ¹
2. Error types In this section, we introduce mathematical models for the two types of errors. We suppose that the errors originate from discrete points (such as the magnet poles), q "(Dq)j, j"0,1,2,2 . H 2.1. RPE The mathematical model for RPE is as follows:
(q)" e g(q!q ) (6) H H H where +e , is a set of independent random numbers, H and we have 1e e 2"ed . G H GH
25
(7)
(8)
p )Dq)1/N ;1. (9) O % The rms phase error p is given by ( 1 2 e p " 1 (q)2 dq" . (10) ( ¹ Dq For the calculation in Section 3, we will be dealing with the phase derivative
dd
" e f (q!q ), f (q)"dg/dq D(q), H H dq H and will need the following integrals:
\
dq
OY
\
(11)
dq f (q )f (q )[1, q , q #q ]"[0, , 0]. (12)
2.2. RKE With a set of random numbers +e , satisfying G Eq. (7), the mathematical model for RKE can be de"ned as dd (q) ,D(q)" e f (q!q ). (13) H H dq H However, the function f (q) here is quite di!erent from the case of RPE. It is an even function with normalization
f (q) dq"1.
(14)
For q )q(q , L L> O L d (q)" f (q ) dq " e . (15) H H In RKE, the error is conveniently characterized by the average value of the square of the phase error per gain length = given by Eq. (5) [7]:
1 e =" 1(d (¹))2" . ¹ Dq
(16)
SECTION I.
26
K.-J. Kim / Nuclear Instruments and Methods in Physics Research A 445 (2000) 24}27
3. Calculation of reduction factors
Therefore,
We follow closely the method in Ref. [7]. The 1-D free-electron laser (FEL) equation in the presence of the phase error can be written as dH dP da "!iH#iD(q)a, "P, "!a. (17) dq dq dq Here a is the scaled "eld amplitude, and H and P are the collective variables representing the phase and momentum, respectively [11]. Note that the phase derivative D(q)"dd /dq appears in the FEL equation as a q-dependent detuning. In the absence of errors, D(q)"0, the solution of Eq. (17) with initial conditions a"0, H "H, and P"0 at q"0, is given by H e HH O a (q)" . 3 j H H
(18)
Here
(j , j , j )"
1!(3i 1#(3i , , !1 2 2
(19)
are the eigenvalues. This solution describes both the spontaneous emission for q;1 and the high gain behavior for q*1 with the leading growth rate j . Solving Eq. (17) in perturbation theory to second order in D, a(q)"a (q)[1#iF (q)!F (q)]. M
(20)
Here
O 1 F (q)" U(q!q )D(q )a (q ) dq , M a (q) M
(21)
O OY 1 F (q)" dq dq U(q!q )U(q !q ) a (q) M ;D(q )D(q )a (q ). M
1"F (q)"2"e U(q!q !q ) H \ H a (q #q ) H dq (26) ;f (q ) M a (q) M OY 1F (q)2"e dq dq U(q!q !q )U H \ H \ ;(q !q ) f (q ) f (q )a (q #q )a\(q). (27) M H M Since the function f (q) vanishes outside a small region "q")p ;1, the integrands in the above can O be expanded in q and q . Let us consider various cases. For spontaneous emission q, q , q , and q are all H much smaller than unity. Since a (q)"iHq#0(q) M and U(q)"1#0(q) for q;1, we obtain
(q #q ) H R"1#e dq f (q ) q H OY (q #q ) !2 dq dq f (q ) f (q ) H . (28) q \ \ For RPE, the "rst term in the curly bracket of Eq. (28) gives a negligible contribution, and the second term can be evaluated making use of Eq. (12). The result is
, 1 eN R"1!e "1! "1!p . (29) ( q q H For RKE, f (q) is an even function normalized by Eq. (14). Therefore,
(22) R"1#e H
The Green's function U is given by 1 U(q)" eGHH O. 3 H
"a(q)""R"a (q)" (24) M R"1#1"F (q)"2!1(F #FH)2. (25) This is the general expression for the reduction factor. For the error models discussed in the previous section, we insert Eq. (11) (or Eq. (13)) to obtain
q q H ! H q q
eN =q "1! "1! . 6 6 (30)
(23)
Eqs. (29) and (30) give the "rst row of Table 1.
K.-J. Kim / Nuclear Instruments and Methods in Physics Research A 445 (2000) 24}27
In the exponential gain regime, the leading behavior is a (q)P(H /3j )e H O, UP(1/3)e H O. M M Eq. (25) is then easily evaluated
Ne "F (q)"" f (q ) dq , 9 "0 for RPE, Ne " 9
for RKE.
(31)
In calculating F from Eq. (27), we can replace the functions U(q!q !q ), a (q #q ), and a (q) in the H M H M integrand by the leading exponential behavior. However, the function U(q !q ) must be evaluated exactly since "q !q ";1. We obtain
Ne 1F (q)2" dq 9 L \ OY dq e H \HL OY\O f (q ) f (q ). (32) ; \ The function exp i(j !j )(q !q ) can be ex L panded to 1#i(j !j )(q !q ), and the integral L can then be evaluated using the properties of f (q) discussed in Section 2. In this way one obtains
Ne 1F (q)2" 2(3 Ne " 6
for RPE, for RKE.
(33)
Collecting these results, we obtain p R"1! ( q for RPE, (3 "1!Ne"1!=q for RKE. (34) Eq. (33) reproduces the second row of Table 1.
27
4. Discussion We have computed the reduction in the spontaneous intensity and in the high-gain growth for two types of errors, RPE and RKE. In the case of RKE, the average value of the square of the phase error per gain length = was used as the measure of the error magnitude rather than the rms phase error p that was used in the case of RPE. It is possible to ( introduce p for RKE by, for example, ( 1 p " (d !mn) . (35) ( L N L Here, d is the phase error for q )q)q given L L L> by (15). The parameter n is determined by minimizing Eq. (34). One then obtains p "=q/10. (
References [1] B.M. Kincaid, J. Opt. Soc. Am. B 2 (1985) 1294. [2] C.J. Elliott, B. McVey, Proceedings of Adriatico Research Conference, Undulator Magnets for Synchrotron Radiation and FELs, Trieste, June 1987, World Scienti"c, Singapore, 1988, p. 143. [3] H.D. Shay, E.T. Scharlemann, Nucl. Instr. and Meth. A 272 (1988) 601. [4] B. Diviacco, R.P. Walker, Proceedings of the 1989 US Particle Accelerator Conference, IEEE Conference Record 89CH2669-0, 1989, p. 1259. [5] E. Esarey et al., Nucl. Instr. and Meth. A 296 (1990) 423. [6] B.L. Bobbs et al., Nucl. Instr. and Meth. A 296 (1990) 574. [7] L.H. Yu et al., Phys. Rev. A 45 (1992) 1163. [8] R.P. Walker, Nucl. Instr. and Meth. A 335 (1993) 328. [9] B. Diviacco, R.P. Walker, Nucl. Instr. and Meth. A 386 (1996) 522. [10] R. Dejus, Rev. Sci. Instr. 66 (1995) 1875. [11] R. Bonifacio, C. Pellegrini, L.N. Narducci, Opt. Commun. 50 (1984) 373.
SECTION I.
Kick and phase errors in spontaneous and ampli"ed radiation夽 Kwang-Je Kim* Accelerator Systems Division, Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439-4800, USA
Abstract Two types of magnet errors are considered } the random phase error (RPE), in which the phase errors are evenly distributed along the magnet, and the random kick error (RKE), in which the errors in the derivative of the phase are evenly distributed. We compute the reduction in performance of both spontaneous radiation and high-gain free-electron lasers for both types of errors within the framework of 1-D free-electron laser theory. 2000 Elsevier Science B.V. All rights reserved. PACS: 41.60.Cr Keywords: Free electron lasers; Magnet errors
1. Introduction E!ects of undulator errors on radiation performance have been studied by several authors [1}10]. The errors are most conveniently characterized by the phase error [6,8]
d (q)"k
dz [t !t ]. X XM t XM
(1)
Here k"2p/j, j is the radiation wavelength, z the distance along the undulator axis, t is the average XM z-velocity in the ideal undulator, t is the average X z-velocity in the presence of errors, q is the scaled longitudinal coordinate, q"2ok z"N/N %
(2)
夽 Work supported by the US Department of Energy, O$ce of Basic Energy Sciences, under Contract No. W-31-109-ENG-38. * Tel.: #630-252-4647; Fax: #630-252-7369. E-mail address: [email protected] (K.-J. Kim).
where o is the Pierce parameter [11], k "2p/j , j is the undulator period length, N"z/j (the number of the undulator periods), and N "1/4po % is the number of undulator periods in one nominal gain length. For cases considered here, N <1. % For well-optimized undulators corrected by suitably placed shims, the phase errors are distributed uniformly along the undulator with a constant rms value p [8}10]. This type of error will be referred ( to as the random phase error (RPE). In the presence of RPE, the intensity of the spontaneous radiation in the forward direction is reduced by the factor [8] R"1!p . (
(3)
For long undulators, the errors could be controlled by steering corrections at regular intervals [3,5}7]. In these cases, the errors in the phase derivative, d d /dq, rather than the phase errors themselves, are uniformly distributed along the undulator. For this type of error, which will be
0168-9002/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 1 0 7 - 8
K.-J. Kim / Nuclear Instruments and Methods in Physics Research A 445 (2000) 24}27
The function g(q) is an even function of q with a width p , normalized as O
Table 1 Reduction due to magnet errors Random phase error (RPE)
Random kick error (RKE)
Spontaneous radiation intensity in the forward direction
1!p (
1!=q
We will assume
High-gain FEL growth rate
p 1! ( q (3
g(q) dq"1.
1!=q
referred to as the random kick error (RKE), the reduction in the power growth of high-gain FELs was found to be [7] R"1! =q where
(4)
2 dd dq . (5) dq Here the angular brackets denote ensemble average. In this paper we generalize the above results to others cases } the reduction in spontaneous radiation for RKE and the reduction in high-gain growth for RPE. The results are summarized in Table 1. 1 =" ¹
2. Error types In this section, we introduce mathematical models for the two types of errors. We suppose that the errors originate from discrete points (such as the magnet poles), q "(Dq)j, j"0,1,2,2 . H 2.1. RPE The mathematical model for RPE is as follows:
(q)" e g(q!q ) (6) H H H where +e , is a set of independent random numbers, H and we have 1e e 2"ed . G H GH
25
(7)
(8)
p )Dq)1/N ;1. (9) O % The rms phase error p is given by ( 1 2 e p " 1 (q)2 dq" . (10) ( ¹ Dq For the calculation in Section 3, we will be dealing with the phase derivative
dd
" e f (q!q ), f (q)"dg/dq D(q), H H dq H and will need the following integrals:
\
dq
OY
\
(11)
dq f (q )f (q )[1, q , q #q ]"[0, , 0]. (12)
2.2. RKE With a set of random numbers +e , satisfying G Eq. (7), the mathematical model for RKE can be de"ned as dd (q) ,D(q)" e f (q!q ). (13) H H dq H However, the function f (q) here is quite di!erent from the case of RPE. It is an even function with normalization
f (q) dq"1.
(14)
For q )q(q , L L> O L d (q)" f (q ) dq " e . (15) H H In RKE, the error is conveniently characterized by the average value of the square of the phase error per gain length = given by Eq. (5) [7]:
1 e =" 1(d (¹))2" . ¹ Dq
(16)
SECTION I.
26
K.-J. Kim / Nuclear Instruments and Methods in Physics Research A 445 (2000) 24}27
3. Calculation of reduction factors
Therefore,
We follow closely the method in Ref. [7]. The 1-D free-electron laser (FEL) equation in the presence of the phase error can be written as dH dP da "!iH#iD(q)a, "P, "!a. (17) dq dq dq Here a is the scaled "eld amplitude, and H and P are the collective variables representing the phase and momentum, respectively [11]. Note that the phase derivative D(q)"dd /dq appears in the FEL equation as a q-dependent detuning. In the absence of errors, D(q)"0, the solution of Eq. (17) with initial conditions a"0, H "H, and P"0 at q"0, is given by H e HH O a (q)" . 3 j H H
(18)
Here
(j , j , j )"
1!(3i 1#(3i , , !1 2 2
(19)
are the eigenvalues. This solution describes both the spontaneous emission for q;1 and the high gain behavior for q*1 with the leading growth rate j . Solving Eq. (17) in perturbation theory to second order in D, a(q)"a (q)[1#iF (q)!F (q)]. M
(20)
Here
O 1 F (q)" U(q!q )D(q )a (q ) dq , M a (q) M
(21)
O OY 1 F (q)" dq dq U(q!q )U(q !q ) a (q) M ;D(q )D(q )a (q ). M
1"F (q)"2"e U(q!q !q ) H \ H a (q #q ) H dq (26) ;f (q ) M a (q) M OY 1F (q)2"e dq dq U(q!q !q )U H \ H \ ;(q !q ) f (q ) f (q )a (q #q )a\(q). (27) M H M Since the function f (q) vanishes outside a small region "q")p ;1, the integrands in the above can O be expanded in q and q . Let us consider various cases. For spontaneous emission q, q , q , and q are all H much smaller than unity. Since a (q)"iHq#0(q) M and U(q)"1#0(q) for q;1, we obtain
(q #q ) H R"1#e dq f (q ) q H OY (q #q ) !2 dq dq f (q ) f (q ) H . (28) q \ \ For RPE, the "rst term in the curly bracket of Eq. (28) gives a negligible contribution, and the second term can be evaluated making use of Eq. (12). The result is
, 1 eN R"1!e "1! "1!p . (29) ( q q H For RKE, f (q) is an even function normalized by Eq. (14). Therefore,
(22) R"1#e H
The Green's function U is given by 1 U(q)" eGHH O. 3 H
"a(q)""R"a (q)" (24) M R"1#1"F (q)"2!1(F #FH)2. (25) This is the general expression for the reduction factor. For the error models discussed in the previous section, we insert Eq. (11) (or Eq. (13)) to obtain
q q H ! H q q
eN =q "1! "1! . 6 6 (30)
(23)
Eqs. (29) and (30) give the "rst row of Table 1.
K.-J. Kim / Nuclear Instruments and Methods in Physics Research A 445 (2000) 24}27
In the exponential gain regime, the leading behavior is a (q)P(H /3j )e H O, UP(1/3)e H O. M M Eq. (25) is then easily evaluated
Ne "F (q)"" f (q ) dq , 9 "0 for RPE, Ne " 9
for RKE.
(31)
In calculating F from Eq. (27), we can replace the functions U(q!q !q ), a (q #q ), and a (q) in the H M H M integrand by the leading exponential behavior. However, the function U(q !q ) must be evaluated exactly since "q !q ";1. We obtain
Ne 1F (q)2" dq 9 L \ OY dq e H \HL OY\O f (q ) f (q ). (32) ; \ The function exp i(j !j )(q !q ) can be ex L panded to 1#i(j !j )(q !q ), and the integral L can then be evaluated using the properties of f (q) discussed in Section 2. In this way one obtains
Ne 1F (q)2" 2(3 Ne " 6
for RPE, for RKE.
(33)
Collecting these results, we obtain p R"1! ( q for RPE, (3 "1!Ne"1!=q for RKE. (34) Eq. (33) reproduces the second row of Table 1.
27
4. Discussion We have computed the reduction in the spontaneous intensity and in the high-gain growth for two types of errors, RPE and RKE. In the case of RKE, the average value of the square of the phase error per gain length = was used as the measure of the error magnitude rather than the rms phase error p that was used in the case of RPE. It is possible to ( introduce p for RKE by, for example, ( 1 p " (d !mn) . (35) ( L N L Here, d is the phase error for q )q)q given L L L> by (15). The parameter n is determined by minimizing Eq. (34). One then obtains p "=q/10. (
References [1] B.M. Kincaid, J. Opt. Soc. Am. B 2 (1985) 1294. [2] C.J. Elliott, B. McVey, Proceedings of Adriatico Research Conference, Undulator Magnets for Synchrotron Radiation and FELs, Trieste, June 1987, World Scienti"c, Singapore, 1988, p. 143. [3] H.D. Shay, E.T. Scharlemann, Nucl. Instr. and Meth. A 272 (1988) 601. [4] B. Diviacco, R.P. Walker, Proceedings of the 1989 US Particle Accelerator Conference, IEEE Conference Record 89CH2669-0, 1989, p. 1259. [5] E. Esarey et al., Nucl. Instr. and Meth. A 296 (1990) 423. [6] B.L. Bobbs et al., Nucl. Instr. and Meth. A 296 (1990) 574. [7] L.H. Yu et al., Phys. Rev. A 45 (1992) 1163. [8] R.P. Walker, Nucl. Instr. and Meth. A 335 (1993) 328. [9] B. Diviacco, R.P. Walker, Nucl. Instr. and Meth. A 386 (1996) 522. [10] R. Dejus, Rev. Sci. Instr. 66 (1995) 1875. [11] R. Bonifacio, C. Pellegrini, L.N. Narducci, Opt. Commun. 50 (1984) 373.
SECTION I.