Efficient field calculation of 3D bunches on general trajectories

Nuclear Instruments and Methods in Physics Research A 445 (2000) 338}342

E$cient "eld calculation of 3D bunches on general trajectories M. Dohlus*, A. Kabel, T. Limberg Deutsches Elektronen Synchrotron DESY, MPY, Notkestr. 85, D-22607 Hamburg, Germany

Abstract The program TRAFIC4 calculates the e!ects of space charge forces and coherent synchrotron radiation on bunches moving on curved trajectories. It calculates the "elds acting on the particles, as they travel along the beamline, from "rst principles. The bunch is modeled by small overlapping 1d, 2d or 3d continuous Gaussian sub-bunches. Previously, a 1d or 2d integration of the retarded source distribution was used to calculate the "elds of 1d and 2d sub-bunches. A new approximation is proposed to avoid the need for multi-dimensional integration: a 2d or 3d sub-bunch can be interpreted as the convolution of a 1d sub-bunch with a transverse density function. In the same way, the "eld can be obtained by a convolution of the 1d-"eld with the transverse density function. The near "eld of a 1d sub-bunch can be split into a singular part which is dominated by local e!ects and a residual part, which depends essentially on long-range interactions. As the singular part can be described by analytical functions, the convolution can be performed e$ciently. The residual part depends weakly on the transverse o!set. Therefore, only one or few sampling points are needed for the convolution, which signi"cantly reduces the numerical e!ort.  2000 Elsevier Science B.V. All rights reserved. Keywords: Coherent synchrotron radiation

1. Introduction Dispersive bunch compressors are used to achieve the required bunch properties for some FEL applications, e.g., &50 lm bunch length, normalized emittance of few 10\ rad for the TESLATTF SASE FEL [1]. For this type of compressor a linear, length-correlated energy spread is superimposed by a phase shift of the accelerating RF and transformed into a length reduction by a dispersive magnet arrangement. Due to the curved trajectories and the shape variations during the compression processes transient EM "elds are generated [2] which have to be taken into account for an appropriate analysis of the beam dynamic. The

* Corresponding author.

program TRAFIC4 [3}5] splits the real bunch } with time dependent shape } into a set of subbunches with "xed shapes, but with individual energies and paths. The "elds of these sub-bunches are calculated by a retarded potential approach and considered together with the external "elds in the tracking calculation for an ensemble of test particles. To obtain "nite "elds, the sub-bunches have to be at least two-dimensional and smooth "elds are generated only by three dimensional charge distributions. The numerical e!ort for a two- or even three-dimensional source integration limits the capability to simulate many arrangements under various operating conditions. To solve this problem, a new approximation is proposed which avoids multi-dimensional integration: for a particular type of two- and three-dimensional charge distributions, which can be described

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 3 9 - X

M. Dohlus et al. / Nuclear Instruments and Methods in Physics Research A 445 (2000) 338}342

by a convolution of a 1D distribution with a transverse density function, the "eld quantities can be obtained by a convolution of the 1D "eld quantities with the density function. The near "eld of a 1d bunch can be split into a singular part that is dominated by local e!ects and the residual part, which depends essentially on long-range interactions. As the singular part can be described by analytical functions, the convolution can be performed e$ciently. The residual part depends weakly on the transverse o!set, so that only one or few sampling points are needed for the convolution.

2. Theory The 1d "eld solver of TRAFIC4 calculates the magnetic #ux density B and the normalized Lorentz force W"E#* ;B for a line charge denR sity j(s, t) traveling along the path r (s) by a numerQ ical integration of



4pe c B"  



\



4pe W(r, t)" 

e ;Gb ds Q Q

 \



#

(1a)

(e !e );e ;Gb b ds R Q Q Q R 

\

G(1!b b ) ds Q R

 

G"!

 jQ (s, t) (b !b )e ds Q Q c R R \ 




j(s, t) j(s, t) jQ (s, t) "R # R R c R 

butions can be obtained by the convolution of the one-dimensional density with a transverse density function g(x , x ) which is de"ned for an arbitrary   plane r (x , x )"x e #x e with e ) e "0 E         and g(x , x ) dx dx "1. Therefore, the 3d     charge density is described by g(x , x )   o(r"r (s)#x e #x e , t)"j(s, t) . Q     " (e ;e ) ) * r "   Q Q (2) Of course, this class is not su$cient to model more general shape variations as they happen in a bunch compressor, but it is adequate for an e$cient subbunch approach. In the present version of TRAFIC4 [3], the Lorentz force and the magnetic #ux density are computed by the convolution of the corresponding 1d quantities (X"W or B) with the transverse density:



X (r, t)" X(r!r (x , x ), t)g(x , x ) dx dx . (3) E       E This convolution is very time consuming as two (three) dimensional integrations have to be carried out for the calculation of two (three) dimensional source distributions. 2.1. One-dimensional integration with pole extraction

 j(s, t) ! b b * e ds R Q R Q Q \ #

339



(1b)

(1c)

with R"r!r (s), R"""R"", t"t!R/c , Q  * "b c e and e "* r . The line charge density is R R  R Q Q Q derived from the current #ow i(s, t):"c j (t!t (s))  G A by j(s, t):" j (t!t (s))/b (s) with b "(c * t )\. G A Q Q  Q A A class of two- and three-dimensional source distri-

As the integrands and integrals of Eq. (1) are singular for observer positions on the path, r (s), Q a pole extraction technique is used for the numerical integration. Therefore, the path point r "r (s ) with minimal distance to the observer is  Q  searched for. In the following, we calculate the normalized Lorentz force for the case * "* :"c b (s )e (s ) of an observer velocity R   Q  Q  identical to the source velocity at the closest path point. A di!erent velocity can be taken into account by adding (* !* );B. The normalized R  Lorentz force and the magnetic #ux density are split into a singular and a non-singular part W"W #W , B"B #B where the non1. ,1 1. ,1 singular part (index NS) can be calculated numerically for any observation point. The singular part

SECTION VI.

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M. Dohlus et al. / Nuclear Instruments and Methods in Physics Research A 445 (2000) 338}342

(index SP) is analytically known:






2b 4pe B "  (e j );w (*r)#  d ;e j         1. c 



;w (*r)!(d j );w (*r)    

 (4a)

4pe W "2b d ;(e j #d j );w (*r)  1.        1 2 j w (*r)# e j # d j w (*r) #   2    c    #2b d j w (*r) (4b)     with










w (*r)"Dr/ ""*r"" (5a)  w (*r)"ln( ""*r "" /R ) (5b)   w (*r)"*r ln( ""*r "" /R ) (5c)   *r"r!r , e "e (s ), d "* e (s ), b "b (s ),   Q   Q Q   Q  j "j(s , t), j "* j(s , t) and R an arbitrary    Q   normalisation length. 2.2. Convolution method For the calculation of the "elds W (r, t) and E B (r, t) of a bunch with the transverse density funcE tion, g(x , x ), the convolution integral Eq. (3) has   to be solved. As the non-singular parts of W and B are smooth functions of r and the singular parts are analytically given, the numerical e!ort for the convolution can be signi"cantly reduced. This is especially true if the 2d plane r (x , x ) of the E   transverse density function is perpendicular to the path at the point r "r (s ). For this case, and for  Q  transverse bunch dimensions which are of the same order or smaller than the longitudinal dimension, the non-singular parts are almost sampled by the convolution: B +B , = += . For ,1E ,1 ,1E ,1 bunches with larger transverse dimensions only few points of B and W have to be evaluated and ,1 ,1 "tted to compute the convolution. The condition that the 2d plane r (x , x ) is perpendicular to the E   path at the closest point r has some further  advantages: the observation point lies in the plane r !r (x , x ) and all other observation  E   points in this plane have the same closest path point r "r (s ). Therefore, the coe$cients  Q 

s , e , d , b , j , j in Eqs. (4a) and (4b) are con      stant for all arguments (r!r (x , x ), t) of B and E   1. W in the convolution integral Eq. (3). The convol1. uted singular parts B , W are obtained from 1.E 1.E Eqs. (4a) and (4b) by replacing the functions w , w , w by   



w (*r)" w (*r!r (x , x ))g(x , x ) dx dx E  E       (6a)



w (*r)" w (*r!r (x , x ))g(x , x ) dx dx E  E       (6b)



w (*r)" w (*r!r (x , x ))g(x , x ) dx dx .  E       E (6c)

2.2.1. 2D bunches For 2d bunches the transverse density function g(x ) depends only on one parameter and the con volution integrals are reduced to a 1d integration over x with r (x )"x e . Therefore, only few  E    scalar integrals have to be computed to obtain the functions a 1 w (*r)" I #e I , w (*r)" I , E   E a  2a  1a 1 w (*r)" I # e I E 2a  2  

(7)

with

   

I (a, b)"  "



\  \ 

a g (b!x) dx, I (a, b)  a#x x g(b!x) dx a#x





 

a#x g(b!x) dx, R \   a#x I (a, b)" x ln g(b!x) dx (8)  R \  and b"e ) *r, a"*r!be , a"""a"". The substi  tution of these functions into Eqs. (4a) and (4b) leads to the contributions B , W and together 1.E 1.E

I (a, b)" 

a ln

M. Dohlus et al. / Nuclear Instruments and Methods in Physics Research A 445 (2000) 338}342

with B +B , W +W a numerically e$,1E ,1 ,1E ,1 cient approximation is obtained.

with

2.2.2. Round 3d Gaussian bunches The 2d convolutions, Eqs. (6a)}(6c) for threedimensional beams with a round Gaussian transverse density function g(x , x )"g ((x #x /     p )/((2np) can be reduced to the computation of P P three scalar functions of one parameter:

H (x)"(2px 

e w (*r"r e )" H (rD /p ) E P r 

(9a)





r w (*r"r e )"H (r /p )#ln E  P R 



r w (*r"r e )"*r H (r /p )#ln  P E R 

(9b) (9c)

341



 I (ox) exp(!ox) g(x!o)do   (10a)



H (x)"(2p 





o ln




o I (ox) x 

;exp(!ox) g(x!o) do



(10b)




(2p  o o ln I (x) H (x)"  x x   ;exp(!ox)g(x!o) do

(10c)

Fig. 1. Longitudinal and radial component of W"E#* ;B for a bunch in circular motion (parameters: curvature radius 10 m, bunch R charge 1 nC, bunch length p"50 lm, c"100). All "gures compare the results of a direct multi-dimensional integration (solid lines) and the singularity convolution method (dotted lines). The "elds of a 2d disc-shaped bunch are shown in (a) and (c) and of a 3d spherical bunch in (b) and (d). The disc-shaped bunch lies in the plane of curvature.

SECTION VI.

342

M. Dohlus et al. / Nuclear Instruments and Methods in Physics Research A 445 (2000) 338}342

and g(x)"exp(!x/2)/(2p. The functions H (x),  H (x), H (x) have been calculated numerically and   are approximated by Chebychev polynomials.

Acknowledgements We wish to thank E.L. Saldin, E.A. Schneidmiller and J. Rossbach for many useful discussions.

3. Calculations References To verify the presented singularity convolution method, W +W #W , we compare this E ,1 ,1E method with a direct multi-dimensional integration for a bunch in circular motion. Fig. 1 shows the longitudinal and radial component of the normalized Lorentz force for 2d and 3d bunches with q"1 nC, c"100 and the radius of curvature R "10 m. The 2d bunch is a round Gaussian disc  in the longitudinal-radial plane with p"50 lm, the 3d bunch is a spherical Gaussian distribution with the same p. The ordinate in Fig. 1 is the longitudinal bunch coordinate (from head to tail for increasing s). All curves are in good agreement, while the numerical e!ort for the singularity convolution method is about two orders smaller than for the multi-dimensional integration.

[1] R. Brinkmann, Conceptual design of a 500 GeV e#elinear collider with integrated X-ray laser facility, DESY 1997-048, ECFA 1997-192. [2] E. Saldin, E. Schneidmiller, M. Yurkov, On the coherent radiation of an electron bunch moving in an arc of a circle, TESLA-FEL 96-14. [3] M. Dohlus, A. Kabel, T. Limberg, Wake "elds of a bunch on a general trajectory due to coherent synchrotron radiation, PAC97 [4] M. Dohlus, A. Kabel, T. Limberg, Uncorrelated emittance growth in the TTF-FEL bunch compression sections due to coherent synchrotron radiation and space charge e!ects, EPAC98. [5] K. Rothemund, M. Dohlus, U.V. Rienen, Calculation of coherent "elds of charged particle bunches on general trajectories, FEL99, same Conference.