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Exact 1D model for coherent synchrotron radiation with shielding and bunch compression

Christopher Mayes and Georg Hoffstaetter

Cornell University, Ithaca, New York 14853, USA(Received 28 October 2008; published 19 February 2009)

Coherent synchrotron radiation has been studied effectively using a one-dimensional model for the

charge distribution in the realm of small angle approximations and high energies. Here we use Jefimenko’s

form of Maxwell’s equations, without such approximations, to calculate the exact wakefields due to this

effect in multiple bends and drifts. It has been shown before that the influence of a drift can propagate well

into a subsequent bend. We show, for reasonable parameters, that the influence of a previous bend can also

propagate well into a subsequent bend, and that this is especially important at the beginning of a bend.

Shielding by conducting parallel plates is simulated using the image charge method. We extend the

formalism to situations with compressing and decompressing distributions, and conclude that simpler

approximations to bunch compression usually overestimate the effect. Additionally, an exact formula for

the coherent power radiated by a Gaussian bunch is derived by considering the coherent synchrotron

radiation spectrum, and is used to check the accuracy of wakefield calculations.

DOI: 10.1103/PhysRevSTAB.12.024401 PACS numbers: 41.60.Ap, 41.75.�i, 41.85.Ew

I. INTRODUCTION

Coherent synchrotron radiation (CSR) is an importantdetrimental effect in modern particle accelerators with highbunch charges and short bunch lengths. It is a collectivephenomenon where the energy radiated at wavelengths onthe order of the bunch length is enhanced by the number ofcharges in the bunch. This CSR field can subsequentlyaffect particle motion. A comprehensive collection of pa-pers and references regarding this subject is found inRef. [1].

CSR is difficult to model using discrete particles exactlybecause the problem scales with the number of particles Nas N2. To simplify this, the one-dimensional modelprojects the transverse particle density onto the longitudi-nal dimension. Formulas for the CSR field from this linecharge are then used to calculate forces on each particleand then propagate the full bunch distribution. While thismakes the calculation tractable, the electromagnetic fieldson the world sheet of this charged line are singular.Pioneering efforts described in Refs. [2,3] circumventthis problem by examining the nonsingular terms only. Inmore detail, Ref. [4] regularizes the longitudinal forcebetween two charges traveling on the arc of a circle bysubtracting off the Coulomb force, calculated as if thecharges traveled on a straight line, from the force calcu-lated using Lienard-Wiechert fields. The result is an alwaysfinite CSR force.

This paper uses the less widely known Jefimenko formsof Maxwell’s equations [5], which allow one to calculateelectromagnetic fields by directly using the evolvingcharge and current densities, and which internally incor-porate all retardation effects. These equations are related toforms used in Refs. [2,6]. This is in contrast to the usualLienard-Wiechert approach, which gives fields due to

charges at their retarded times t0 and positions xðt0Þ, andone must invert equations of the form t� t0 ¼jxðt0Þ � xoj=c for the retarded time t0, where xo is anobservation point at a later time t and c is the speed oflight. While this latter method has proven useful in deriv-ing equations for (incoherent) synchrotron radiation ofsingle particles, the former is found to be useful for thecoherent fields of particle distributions.

II. EXACT 1D APPROACHES TO STEADY-STATECSR

In general, for given charge and current densities �ðx; tÞand Jðx; tÞ at position x and time t, the electric fieldEðx; tÞcan be calculated using Jefimenko’s form of Maxwell’sequations [5],

E ðx; tÞ ¼ 1

4��0

Zd3x0

�r

r3�ðx0; t0Þ þ r

cr2@t0�ðx0; t0Þ

� 1

c2r@t0Jðx0; t0Þ

�t0¼t�r=c

; (1)

in which r � x� x0, r � krk, �0 is the vacuum permittiv-ity and t0 is the retarded time. In this formulation, theretarded points x0 and times t0 are independent variables,so there are no functions that need to be inverted.Therefore, if one knows �, _�, and _J at all points in spacex0 and times t0 � t, with a dot denoting the time derivative,then this formula gives the electric field by directintegration.Now consider a line charge distribution, which follows a

path XðsÞ parametrized by distance s, has a unit tangentuðsÞ ¼ dXðsÞ=ds, and moves with constant speed �calong this path. A bunch with total charge Q and normal-ized line density � therefore has one-dimensional chargedensity and current

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 12, 024401 (2009)

1098-4402=09=12(2)=024401(16) 024401-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevSTAB.12.024401

�ðs; tÞ ¼ Q�ðs� sb � �ctÞ;Jðs; tÞ ¼ Q�cuðsÞ�ðs� sb � �ctÞ;

(2)

where sb is the location of the bunch center at time t ¼ 0.The rate of energy change per unit length of an elemen-

tary charge q at position s is dE=ds ¼ quðsÞ �Eðs; tÞ.Functions of this type are called wakefields. UsingEq. (1) with the one-dimensional bunch in Eq. (2) gives

dEds

ðs; tÞ ¼ Nrcmc2Z 1

�1ds0

�uðsÞ � rðs; s0Þ

rðs; s0Þ3 �ðsrÞ

� �uðsÞ � rðs; s0Þ

rðs; s0Þ2 �0ðsrÞ

þ �2 uðsÞ � uðs0Þrðs; s0Þ �0ðsrÞ

�; (3)

with the definitions

sr � s0 � s0 þ �rðs; s0Þ; (4)

s0 � sb þ �ct; (5)

r ðs; s0Þ � XðsÞ �Xðs0Þ; (6)

rðs; s0Þ � krðs; s0Þk; (7)

whereN ¼ Q=q is the number of elementary particles withmass m and classical radius rc ¼ q2=ð4��0mc2Þ, and theprime on � indicates a derivative of this function withrespect to its argument, i.e. �0ðxÞ ¼ d�=dx. Additionally,s0 is the center of the bunch at time t, and this is the onlyplace where the time dependence appears. The integrand isthus the contribution to the wakefield due to particlesbetween the retarded positions s0 and s0 þ ds0, withN�ðsrÞ being the charge density at retarded position s0and retarded time t0.

Unfortunately the integral in Eq. (3) diverges as s�s0 ! 0, which is a consequence of the one-dimensionalline charge model. This problem can be alleviated by usingthe regularization procedure originating in Saldin et al. [4],where the electric field E is split into two parts:

E ¼ ECSR þESC: (8)

The space charge (SC) part is the electric field of a linecharge moving on a straight path,

ESCðs; tÞ ¼ Q

4��0uðsÞ

Z 1

�1d~s

�s� ~s

js� ~sj3 �ðslÞ

� �s� ~s

js� ~sj2 �0ðslÞ þ �2 1

js� ~sj�0ðslÞ

�; (9)

with sl � ~s� s0 þ �js� ~sj, which can be integrated byparts, simplifying to

E SCðs; tÞ ¼ � QuðsÞ4��0�

2

Z 1

�1d~s

�0ð~s� s0 þ �js� ~sjÞjs� ~sj :

(10)

It will turn out to be useful to change variables in thisexpression, so that when combined with Eq. (3) the func-tion �0 can be factored. This can be done by setting ~s�s0 þ �js� ~sj ¼ s0 � s0 þ �rðs; s0Þ, with the conventionthat sgnðs� ~sÞ ¼ sgnðs� s0Þ. Noting that @rðs; s0Þ=@s0 ¼�rðs; s0Þ � uðs0Þ=rðs; s0Þ, this leads to

�1

js� ~sj ¼ sgnðs0 � sÞ 1þ �sgnðs0 � sÞs� s0 � �rðs; s0Þ ; (11)

d~s ¼ 1� �rðs; s0Þ � uðs0Þ=rðs; s0Þ1þ �sgnðs0 � sÞ ds0; (12)

so that

ESCðs; tÞ ¼ Q

4��0

uðsÞ�2

Z 1

�1ds0sgnðs0 � sÞ�0ðsrÞ

� 1� �rðs; s0Þ � uðs0Þ=rðs; s0Þs� s0 � �rðs; s0Þ : (13)

The resulting wakefield due to ECSR, called the CSR wake,is �

dECSR

ds

�¼ quðsÞ � ½Eðs; tÞ �ESCðs; tÞ�: (14)

This expression is finite, and shown in Ref. [4] to correctlyaccount for the coherent energy loss due to synchrotronradiation.The approach here is to be contrasted with the conven-

tional one taken in the literature using Lienard-Wiechertformulas. In terms of the quantities above, the electric fieldat position s due to a charge q at retarded time t0 ¼ t�rðs; s0Þ=c and retarded position s0 is

ELWðs; s0Þ ¼ q

4��0

�r� �ruðs0Þ

�2½r� �r � uðs0Þ�3

þ r� f½r� �ruðs0Þ� � �2u0ðs0Þg½r� �r � uðs0Þ�3

�; (15)

with r as in Eq. (3) suppressing the arguments. Therefore,the electric field at s due to a charge �ðst; tÞdst between stand st þ dst, as in Eq. (2), is found by inverting st ¼ s0 þ�rðs; s0Þ for s0 and using Eq. (15). This is often impossibleto do analytically, but fortunately for a distribution ofcharges the inversion can be circumvented by changingvariables. Because @rðs; s0Þ=@s0 ¼ �r � uðs0Þ=r from be-fore, the charge is

�ðst; tÞdst ¼Q�ðs0 � sb��ctþ�rÞ�1��

r �uðs0Þr

�ds0;

(16)

and the total electric field is

MAYES AND HOFFSTAETTER Phys. Rev. ST Accel. Beams 12, 024401 (2009)

024401-2

Eðs; tÞ ¼Z 1

�1dstELW½s; s0ðstÞ��ðst; tÞ (17)

¼ QZ 1

�1ds0

�1� �

r � uðs0Þr

�ELWðs; s0Þ�ðsrÞ:

(18)

We can use this to verify that ESC computed this wayagrees with the result using the Jefimenko approach. Forthe SC field one has rðsÞ ¼ ðs� s0ÞuðsÞ, uðs0Þ ¼ uðsÞ, andu0ðsÞ ¼ 0, giving

ESCðs; tÞ ¼ QuðsÞ4��0

Z 1

�1ds0

�þ sgnðs� s0Þðs� s0Þ2

� �ðs0 � s0 þ �js� s0jÞ: (19)

Equation (19) agrees with Eq. (10) when integratedby parts because, for s0 ¼ 0,

R½� þ sgnðs � s0Þ� �ðs � s0Þ�2ds0 ¼ ½� þ sgnðs � s0Þ�ðs � s0Þ�1, and� @

@s0�½s0 þ�ðs� s0Þsgnðs� s0Þ� ¼�½1��sgnðs� s0Þ� ��0ðs0 þ�js� s0jÞ, and similarly for all s0.

III. SINGLE BENDING MAGNET

Now we apply Eq. (3) to the geometry of an arc of acircle of curvature � and length B, shown in Fig. 1. Sets ¼ 0 at the entrance of the bend so that � ¼ �s is theangle into the bend. In terms of fixed Cartesian unit vectorsea and eb, the path coordinates and tangent vector are

X ðsÞ ¼ ��1 sinð�sÞea � ��1½1� cosð�sÞ�eb; (20)

u ðsÞ ¼ cosð�sÞea � sinð�sÞeb: (21)

Consider a bunch with its center at angle �0 ¼ �s0, anda test particle at angle �. The contribution to Eq. (3) of thisfinite arc is

dEds

ðsÞ��B

¼ Nrcmc2Z �B

0d�0

�sin

ð�rÞ3��ðsÞ

� �sin

ð�rÞ2�0ðsÞ þ �2 cos

�r�0ðsÞ

�; (22)

with s ¼ ��1� and the following definitions:

� �� 0; (23)

s � 1

�ð�� 0 � Þ þ �r; (24)

r � 1

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos

p: (25)

Thus is the angle between the test particle and theretarded source particle, and is positive when the formeris ahead of the latter. The first term of Eq. (22) can be

integrated by parts because @ð2� 2 cosÞ�1=2=@�0 ¼sinðÞð2� 2 cosÞ�3=2, and the wake greatly simplifies to

dEds

ðsÞ��B

¼ Nrcmc2� ��ðsÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2� 2 cosp

��¼�

¼�ð�B��Þ

þZ �

�ð�B��Þd

�2 cosðÞ � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos

p �0ðsÞ�: (26)

In terms of the variable, the space charge term in Eq. (13)can be split as

dESC

dsðsÞ ¼ �Nrcmc2

�Z �ð�B��Þ

�1dISCðÞ

þZ �

�ð�B��ÞdISCðÞ þ

Z 1

�dISCðÞ

�;

(27)

with the integrand

ISCðÞ � � sgn

�2

1� � sinðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�2 cos

p

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos

p �0ðsÞ; (28)

so that the contribution of the bend to the CSR wake is

dECSR

dsðsÞ

��B¼ Nrcmc2

� ��ðsÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos

p��¼�

¼�ð�B��Þ

þZ �

�ð�B��Þd�0ðsÞ

��2 cosðÞ � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos

p

þ sgnðÞ�2


p


p�

�Z �ð�B��Þ

�1dISCðÞ �

Z 1

�dISCðÞ

�:

(29)FIG. 1. Geometry for a single bend. The variable s parame-trizes the curve with radius 1=�. The coordinates are XðsÞ, andthe unit tangent vector is uðsÞ.

EXACT 1D MODEL FOR COHERENT . . . Phys. Rev. ST Accel. Beams 12, 024401 (2009)

024401-3

A. Steady state

In the practical environment of a particle acceleratorwith a bunched beam, one is typically only concernedwith electric fields around the bunch center. Because ofthe rotational symmetry, there will be an angle into abending magnet beyond which the CSR wake, relative tothe bunch center, does not change. Note that in Eq. (24) thequantity z ¼ ��1ð�� 0Þ is the distance along the pathahead of the bunch center, and define the extent of thebunch lb � zþ � z�, where zþ is the head particle coor-dinate, and z� is the tail particle coordinate. Henceforth thesymbol z will refer to the longitudinal coordinate relativeto the bunch center: z ¼ s� s0. The particle at zþ isaffected by a particle at z� at retarded angle max foundby inverting

�lb ¼ max � �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cosmax

p: (30)

Similarly, a particle at z� is affected by a particle at zþ atretarded angle min found by inverting

� �lb ¼ min � �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cosmin

p: (31)

When the bunch center is at an angle �0 >max � �zþ,only particles within the bend affect the wakefield. The‘‘steady state’’ (s.s.) CSR wake is then

WCSRs:s:ðzÞ ¼ Nrcmc2

Z max

min

d

��2 cosðÞ � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos

p

þ sgnðÞ�2


p


p��0½z� �ðÞ�;

(32)

where

�ðÞ ¼ ��1ð� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos

p Þ (33)

is the distance behind the test particle at z. The notation

WCSRðzÞ � dECSR

dsðs0 þ zÞ (34)

is used to refer to the CSR wake immediately surroundingthe bunch center at s0.

In the ultrarelativistic approximation (� ! 1) with asmall normalized bunch length �lb � 1, and thus �1, the steady-state formula in Eq. (32) greatly simplifies.The 1=�2 term in Eq. (28) puts ISC ! 0, and the term in theintegrand ½�2 cosðÞ � 1�=½2j sinð=2Þj� � �jj=2. Thefunction �ðÞ for � ! 1 is approximately

�ðÞ ��3=ð24�Þ for > 02=� for < 0:

(35)

Figure 2 plots the inverse of Eq. (33) for positive and various energies. One sees that the approximation inEq. (35) is increasingly good for higher energies, butgreatly overestimates at the smallest distances.Changing variables using Eq. (35), the ultrarelativistic

steady-state CSR wake is

W CSR�!1

ðzÞ ¼ �Nrcmc2�Z lb

0d�

��2�0ðz� �Þð3��Þ1=3 þ ��

8�0ðzþ �Þ

�: (36)

The first term in this integral is derived by an alternatemethod in Ref. [4]. The scaling here is apparent by writingthe distribution in the normalized form

�ðz��Þ � 1

~�

�z��

�; (37)

�0ðz� �Þ � 1

2~�0�z��

�; (38)

where 2 is the variance of �, so that ~� has unit variance.

Also using normalized ~z � z= and ~� � �= gives

W CSR�!1

ð~zÞ ¼�Nrcmc2ð�Þ2=32

Z lb=

0d~�

��2~�0ð~z� ~�Þð3~�Þ1=3 þð�Þ4=3

~�

8~�0ð~zþ ~�Þ

�: (39)

Now one can see that the particles in front of the testparticle, represented in the last term in the integrand,

influence the wake by roughly a factor of ð�Þ4=3 lessthan particles behind, and that the primary contributionto the CSR wake scales with the factor in front of theintegral in Eq. (39). However, it is interesting to note thateven as � ! 1, where a charge radiates infinitely morepower in the forward direction than the backward direc-tion, there is still a finite CSR force from particles ahead ofthe test particle. In light of the primary scaling, we define acharacteristic CSR energy change per unit length as

W0 � Nrcmc2ð�Þ2=32

: (40)

10 7 10 6 10 5 10 4 0.001 0.01

10 5

10 4

0.001

0.01

0.1

α

2561286432168γ

FIG. 2. (Color) The inverse of Eq. (33) for positive at variousenergies. The dashed green curve (� ! 1) is ¼ ð24��Þ1=3,the inverse of Eq. (35).


024401-4

The ultrarelativistic approximation in Eq. (36) is com-pared to the exact formula Eq. (32) in Fig. 3 for variousenergies and a particular value of �. One sees thatEq. (36) represents the largest possible effect. The CSRwake due only to particles in front of the test particle isshown in Fig. 4, emphasizing again that these forwardparticles contribute only a small amount to the total CSRwake.

Neglecting the contribution due to forward particles, theultrarelativistic stead-state CSR wake in Eq. (39) scales

withW0 and depends only on the shape of ~�. Factoring outW0 from the exact steady-state CSR wake in Eq. (32), theexact result additionally depends on � and �. Therefore,to quantify the appropriateness of the ultrarelativistic ap-

proximation, the ratio of the average energy lost (per unitlength) of a Gaussian bunch using the exact Eq. (32) to thatusing approximate Eq. (36) is shown in Fig. 5 for a prac-tical range of these parameters. At a given energy, one seesthat Eq. (36) is a good approximation for the relatively longbunches. This can be understood from Fig. 2, because theapproximation in Eq. (35) has a relative error for a finiteenergy that diverges for small .A systematic method for calculating the CSR wake

using Lienard-Wiechert formulas in the small angle, rela-tivistic approximations has been developed in Ref. [7] forarbitrary combinations of drifts and bends. Using the cor-responding equation in Ref. [7] for the geometry of a bend,and the appropriate Jacobian factor, the steady-state CSRwake to second order in and 1=� is

W CSRSHMS;s:s:

ðzÞ ¼ �Nrcmc2Z max

0d

��1

2�2þ 2

8

�

��

2þ �22

þ �23=4� 1

=2þ �23=24

�

� �0�z� ��1

�

2�2þ 3

24

��: (41)

Compared to Eq. (36), this expression is a significantlybetter approximation of Eq. (32) for low � and a practicalrange of �, shown in Fig. 6.

B. Shielding by parallel plates

The presence of a conducting beam chamber canstrongly change CSR wakefields. If particle trajectoriesare planar within a chamber of finite height and infinite

4 2 0 2 40.6

0.4

0.2

0.0

0.2

z σ

WC

SRz

W0

2561286432168γ

FIG. 3. (Color) The steady-state CSR wake for various relativ-istic � using Eq. (32), compared to Eq. (36) plotted as green.Here � ¼ 3� 10�5 for a Gaussian bunch, represented in lightblue.

4 2 0 2 40.00

0.02

0.04

0.06

0.08

0.10

0.12

z σ

κ σ

43

WC

SRz

W0

842γ

FIG. 4. (Color) The steady-state CSR wake for various relativ-istic � due only to particles ahead of the test particle, i.e.negative in Eq. (32), and the second term in the integrand ofEq. (36). A Gaussian bunch is used, with � ¼ 3� 10�5, andthe wake has been scaled by ð�Þ�4=3. Compared with Fig. 3 thisdemonstrates that the contribution to the CSR wake of particlesahead of the test particle is insignificant compared to thosebehind.

0.99

0.9

0.7

0.5

0.3

0.1

10 7 10 6 10 5 10 4 10 3

101

102

103

104

κ σ

γ

FIG. 5. The ratio of the average energy of a Gaussian bunchusing the exact Eq. (32) to that using approximate Eq. (36) in apractical range of the parameters � and �.


024401-5

width, then the CSR wakefields can be calculated by usingthe image charge method for infinite parallel plates. Herewe will formulate an exact solution to the 1D CSR problemfor such chamber walls.

For more realistic chamber geometries, CSR fields canchange in more complex ways, and it is therefore importantto know when the CSR forces deviate significantly fromthose of the parallel plate case. Effects that can change thewakefield include reflections of CSR waves on these wallsand shadowing effects from bends in the chamber wall. Fora rectangular cross section, it has been observed thatreflected waves are only important when the sidewallsare closer together than the height of the chamber (see,for example, Ref. [7]). By shadowing, we mean that someretarded positions have a straight line connection to theobservation point that interferes with the vacuum chamber,e.g. in a bend.

The kick due to a single image bunch at height h is easilyadapted from Eq. (3) as

dEds

ðs; t; hÞ ¼ Nrcmc2Z 1

�1ds0

�uðsÞ � r

ðr2 þ h2Þ3=2 �ðshÞ

þ��2 uðsÞ � uðs0Þ

ðr2 þ h2Þ1=2 � �uðsÞ � rr2 þ h2

��0ðshÞ

�;

(42)

with the argument

sh � s0 � sb � �ctþ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ h2

p; (43)

and with r and u retaining their meaning from Eq. (3).Parallel plates require an image bunch for each plate, and

an image bunch for each of those, ad infinitum. For the realbunch with orbit midway between plates separated by adistance H, symmetry gives the total image kick

dEimages

dsðs; tÞ ¼ X1

n¼�1n�0

ð�1Þn dEds

ðs; t; nHÞ (44)

¼ 2X1n¼1

ð�1Þn dEds

ðs; t; nHÞ: (45)

If the real bunch has a vertical offset V, the total image kickis modified to

dEimages

dsðs; tÞ ¼ X1

n¼�1;n�0even

dEds

ðs; t; nHÞ

� X1n¼�1odd

dEds

ðs; t; nH� 2VÞ: (46)

In a bend, the contribution of the image bunches to the CSRwake within the bend, following Eq. (26), is

dEimages

dsðsÞ

��B¼ Nrcmc2

X1n¼1

2ð�1Þn

��ðs;nÞ

r;n

��¼�

¼�ð�Lm��Þ

þZ �

�ð�Lm��Þd

�2 cosðÞ � 1

r;n�0ðs;nÞ

�;

(47)


r;n �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cosþ ðn�HÞ2

q; (48)

s;n � ��1ð�� 0 � þ �r;nÞ: (49)

Notice that the integrands do not need to be regularized bythe SC term, because they are always finite due to thealways positive factor ðn�HÞ2.Because of the infinite number of image layers needed, a

finite bend can never be exactly in the steady state.However, due to their increased distances and angles, therelevant contribution image number n will be negligiblebeyond some maximum image number. This point is illus-trated in Fig. 7, where the contributions to the CSR wake offive individual images are shown along with their sum withthe free space wake, to give the total shielded wake.

C. Retarded bunch visualization

For a given particle at time t within the bunch, it isevident that the retarded bunch density can be very dis-torted relative to the actual bunch density. From Eq. (16),the retarded bunch density at position s0 as seen by aparticle at position s is

0.99

0.95

0.9

10 7 10 6 10 5 10 4 10 3 10 2

5

10

15

20

κ σ

γ

FIG. 6. Similar to Fig. 5, but with the ratio of the averageenergy of Eq. (32) to Eq. (41), showing that the latter is anexcellent approximation at relatively low energies.


024401-6

�retðs0; sÞ ¼ �ðs0 � sb � �ctþ �rÞ�1� �

r � uðs0Þr

�:

(50)

In the steady state, the geometry of a bend can be used inEq. (50). Moving to coordinates relative to the bunchcenter, the steady-state density seen by a test particle atzt within the bunch as a function of z0 is

�rets:s:ðz0; ztÞ ¼

�1� �

sin½�ðzt � z0Þ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos½�ðzt � z0Þ�p �

� �fz0 þ ��1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos½�ðzt � z0Þ�

qg: (51)

This retarded density is illustrated in Fig. 8 for a Gaussianbunch distribution for various test particles. There one seesthat the density in front of the test particle is compressed toroughly =ð1þ �Þ � =2, concentrated in an apparent

spike at the right of the plot. The density behind the testparticle occupies the majority of the plot. While it mayseem that the curves shown are Gaussian in form, this isonly true for the left sides of the curves; the right sides havebeen extended and diluted due to the Jacobian factor inEq. (50). Similarly, the retarded density of an image bunchat height h is

�retðs0; h; sÞ ¼�1� �

r � uðs0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ h2

p�

� �ðs0 � sb � �ctþ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ h2

pÞ: (52)

Figure 9 shows the retarded densities for a Gaussianbunch and several image bunches within a bend. In thisexample, the first and second image bunches as seen byparticles in the rear of the bunch are actually closer than thereal retarded bunch.

IV. MULTIPLE BENDS AND DRIFTS

A. CSR in multiple bends

In this section the general formula Eq. (3), regularizedby Eq. (10), is applied to the geometry of multiply con-nected bends and drifts. Shielding by conducting parallelplates is added as in Eq. (45). It has been seen in Eq. (39)that the primary contribution to the CSR wake in a bend isdue to particles behind the test particle, so for brevity thepath is given behind the test particle only.Let the bunch center be at length s0 inside bend 1 of

length B1 and positive curvature �1, preceded by drift 1 oflength D1, preceded by bend 2 of length B2 and curvature�2 � 0, as shown in Fig. 10. A drift follows bend 1,referred to as D0. A negative curvature �2 signifies abend in the opposite direction of bend 1. With s ¼ 0located at the beginning of bend 1, the path coordinates are

X ðsÞ ¼

8>>><>>>:XD0

ðsÞ for s > B1

XB1ðsÞ for 0< s � B1

sea for �D1 < s � 0XB2

ðsÞ for s � �D1;

(53)

where the paths in the individual elements are

XD0ðsÞ �

�sinð�1B1Þ

�1

þðs�B1Þcosð�1B1Þ�ea

þ�1

�1

½cosð�1B1Þ� 1�� ðs�B1Þ sinð�1B1Þ�eb;

(54)

XB1ðsÞ � sinð�1sÞ

�1

ea � ½1� cosð�1sÞ��1

eb; (55)

zt 3σ

2σ

1σ

0

1σ

6000 5000 4000 3000 2000 1000 0z' σ

FIG. 8. (Color) The steady-state retarded distribution�ret;s:s:ðz0; ztÞ for various test particles zt in Eq. (51) using a

Gaussian bunch with standard deviation ¼ 0:3 mm and en-ergy 1 GeV, in a magnet of bending radius ��1 ¼ 10:0 m.

5 4 3 2 1 0 1 2 3 4 50.6

0.4

0.2

0.0

0.2

0.4

0.6

z σ

WC

SRz

W0

Positive ImagesNegative ImagesParallel PlatesFree Space

FIG. 7. (Color) The free space and shielded CSR wakes in abend. The contributions to the shielded wake of individual imagebunches are shown in red and blue. A 1 GeV Gaussian bunchwith ¼ 0:3 mm is used in a bend of radius ��1 ¼ 10:0 m. Theshielding height H ¼ 2 cm.


024401-7

XB2ðsÞ �

�sinð�2D1 þ �2sÞ

�2

�D1

�ea � 1� cosð�2D1 þ �2sÞ

�2

eb: (56)

The tangent vector is then

u ðsÞ ¼

8>>><>>>:cosð�1B1Þea � sinð�1B1Þeb for s > B1

cosð�1sÞea � sinð�1sÞeb for 0< s � B1

1ea for �D1 < s � 0cosð�2D1 þ �2sÞea � sinð�2D1 þ �2sÞeb for s � �D1:

(57)

Straightforward calculation gives the total CSR wake atposition s in the bend (0< s < B1) due to these differentsections of the path

dECSR

ds

��totð0< s < B1Þ ¼ dECSR

ds

��B1

þdECSR

ds

��D1

þ dECSR

ds

��B2

þ� � � (58)

with B1,D1, and B2 signifying the contributions from bend1, drift 1, and bend 2, respectively. Because of their length,these terms are written out in the Appendix.

A visualization of the retarded bunch and images of thisgeometry, similar to Fig. 9, is shown in Fig. 11. Eventhough the bunch has progressed 50 cm into bend 1, itsees much of the retarded bunch inside bend 2, especiallyfor test particles zt in the front of the bunch.

Two principal effects of CSR on the bunch distributionare a loss of energy and an increase in energy spread. Theseare calculated using the CSR wake WCSRðzÞ and the bunchdistribution �ðzÞ, where the average energy change per unitlength hWCSRi and the standard deviation WðWCSRÞ overthe distribution are

hWCSRi �Z zþ

z�WCSRðzÞ�ðzÞdz; (59)

W ��Z zþ

z�W2

CSRðzÞ�ðzÞdz� hWCSRi2�1=2

: (60)

FIG. 10. Geometry for bends and drifts. The variable s pa-rametrizes the path XðsÞ, with s ¼ 0 at the beginning of elementB1. The names B1, D1, etc. also serve to indicate the elementlength. The dashed line is for a prior bend with negativecurvature.

zt 3σ , 2σ , 1σ , 0, 1σ , 2σ , 3σ

7000 6000 5000 4000 3000 2000 1000 0z σ

FIG. 9. (Color) The same as Fig. 8, along with image charges at heights nH ¼ n� 2 cm (not to scale), which are approximately atheights n� 67, and calculated using Eq. (52).


024401-8

The term W is important because it contributes to thecorrelated energy spread in a bunch.

To show how these quantities change as a bunch pro-gresses through a bend, Fig. 12 plots hWCSRi and W ,normalized by W0, versus different bunch center coordi-nates s0 in bend 1 using Eq. (58) with D1 ¼ 1 m and �1 ¼�2 ¼ 1=10 m. In the literature, the wake near the begin-ning of bend 1 is often calculated as if the prior drift lengthD1 ! 1 [4,8], so such calculations are plotted in dottedlines for comparison. From the difference between the twoapproaches, one sees the effect of bend 2, where the CSRwake at s0 ¼ 0 is nonzero. In this example, they coincide

after about 1.4 and 1.8 m for the free space and shieldedcases, respectively.It has been mentioned at the start of Sec. III B that

sidewalls can change the CSR field. The shadowing effectcan be especially relevant in the case of two bends that areseparated by a drift. In this example, the vector from asource particle at z ¼ �8000 to the center of the bunch(z ¼ 0) correspondingly requires that the vacuum chamberhalf-width must be greater than approximately 3 cm.

B. CSR in a drift between bends

The nonzero CSR wake at the beginning of bend 1 inFig. 12 is evidence that the wake in a drift region after abend also needs to be considered. This exit wake in theregionD0 following bend 1 is calculated using Eq. (3) withEqs. (53) and (57) around the center of a bunch at s0 >B1.Because the bunch is moving in a straight line, the regu-larization procedure simply removes the need to integrateany s0 > B1 for the real bunch. Therefore we can useEq. (3) for bend 1, drift 1, and earlier elements, andsubtract the space charge terms for s0 < B1. Image charges,however, still require terms for s0 >B1. The total exit wakeis then

dECSR

ds

��totðs > B1Þ ¼

dEimages

ds

��D0

þdECSR

ds

��B1

þ dECSR

ds

��D1

þ� � � ; (61)

where the individual terms due to element elementsD0,B1,D1 are written out in the Appendix. For a magnet of lengthB1 ¼ 3 m, the exit wakes in the following drift D0 areshown in Fig. 13 for bunch centers in the following three

0.0 0.5 1.0 1.5 2.0

0.4

0.2

0.0

0.2

0.4

s0 m

Wcs

rW

0W

W0

Parallel PlatesFree Space

FIG. 12. (Color) The average energy loss and energy spreadinduced, per unit length, of the CSR wake for a Gaussian bunchthrough the length of a bend in free space as well as betweenparallel plates withH ¼ 2 cm. Solid lines haveD1 ¼ 1 m, whiledashed lines have D1 ! 1. Parameters used are ��1

1 ¼ ��12 ¼

10 m, z ¼ 0:3 mm, with an energy of 1 GeV.

9000 8000 7000 6000 5000 4000 3000 2000 1000 0z σ

FIG. 11. (Color) Similar to Fig. 9, but with a 1 m drift (shaded in gray) between two magnets of curvature �1 ¼ �2 ¼ 1=10 m. Thecenter of the bunch is 50 cm into the bend. A Gaussian bunch distribution is used with z ¼ 0:3 mm, an energy of 1 GeV, and ashielding height H ¼ 2 cm.


024401-9

meters between parallel plates and in free space. Theaverage and standard deviation of the wakes through thisregion are shown in Fig. 14. In the shielded situation, onesees that the bunch actually gains some energy in a shortlength following the bend, and that the total energy lossbetween parallel plates is negligible compared to the freespace losses. Energy spread, however, is qualitatively thesame in both cases.

V. BUNCH COMPRESSION

Bunch compression or decompression can be achievedin a bending magnet if there is a correlation betweenenergy and longitudinal position of particles in the bunch.To exactly calculate CSR for this, however, requires at leasta two-dimensional model, because particles of differentenergies travel on different orbits. In the framework of theone-dimensional model described by Eq. (3), this effectcan be approximately modeled by allowing the bunchlength to be time dependent, and neglecting variations in

the velocity �c. The density and current are then

�ðs; tÞ ¼ Q1

ðtÞ~�

�s� sb � �ct

ðtÞ�;

Jðs; tÞ ¼ Q�c1

ðtÞ~�

�s� sb � �ct

ðtÞ�;

(62)

where ~� has unit norm and variance with respect to s, as inEq. (37). The time derivative of �ðs; tÞ is

@

@t�ðs; tÞ ¼ ��c

Q~�0ðstÞ2

� _

�Q~�ðstÞ2

þ st

Q~�0ðstÞ2

�:

(63)

with st � s� sb � �ct. Note that _=ð�cÞ is on the orderof =B in a magnet of length B, and (s� sb � �ct) is onthe order of for all relevant ðs; tÞ, and therefore the termin brackets is on the order of =B � 1 relative to the firstterm, and will be neglected. With such an approximation,the CSR wake in a bunch compression system can be

3.0 3.5 4.0 4.5 5.0 5.5 6.00.4

0.3

0.2

0.1

0.0

0.1

s0 m

Wcs

rW

0

Free SpaceParallel Plates

3.0 3.5 4.0 4.5 5.0 5.5 6.00.00

0.05

0.10

0.15

0.20

0.25

0.30

s0 m

σW

W0

Free SpaceParallel Plates

FIG. 14. (Color) The average energy loss and energy spread per unit length of the exit wakes in Fig. 13. In this example, shielding byparallel plates drastically reduces the energy loss, but only marginally reduces the energy spread, when compared to free spacecalculations.

4 2 0 2 4

0.2

0.1

0.0

0.1

0.2

0.3

z σ

Wcs

rW

0

6.00

5.00

4.00

3.01s0 m

4 2 0 2 40.6

0.4

0.2

0.0

z σ

Wcs

rW

0

6.00

5.00

4.00

3.01s0 m

FIG. 13. (Color) CSR wakes for various bunch centers s0 >B1 calculated using Eq. (61). The left graph uses parallel plates separatedby a distance H ¼ 2 cm, while the right graph is for free space [n ¼ 0 terms only in Eqs. (A6) and (A7), and without Eq. (A5)]. Thebending radius ��1

1 ¼ 10 m, and the bunch has a Gaussian profile with ¼ 0:3 mm and an energy of 1 GeV.


024401-10

modeled by simply making the substitutions

�ðsrÞ ! 1

ðtretÞ~�

�sr

ðtretÞ�

(64)

�0ðsrÞ ! 1

½ðtretÞ�2~�0�

srðtretÞ

�(65)

tret ¼ t�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ðnHÞ2

q=c (66)

in all of the previous formulas, with r ¼ kXðsÞ �Xðs0Þk,as in Eq. (3). This accounts for the real charges (n ¼ 0) andimage charges (n � 0) at the appropriate retarded times.

Calculations for the average and standard deviation ofthe CSR wake with a linearly compressing bunch throughthe length of bend 1 in free space are shown in Fig. 15(a).The above approximation is referred to as method 1.Method 2 calculates the instantaneous CSR wake of acompressing bunch at each point in the bend as if it alwayshad its instantaneous length. Such a scheme is essentiallywhat particle tracking codes (e.g. elegant [9], Bmad[10]) use for CSR simulation. For reference, method 3

calculates the CSR wake for a noncompressing bunchthat maintains the same length as the final compressedlength in methods 1 and 2. In this example, method 2overestimates the CSR effect compared to the more real-istic method 1, and both exhibit a much smaller effect thanmethod 3. At the end of the magnet (s0 ¼ 3 m), the CSRwake, according to method 1, has yet to reach its corre-sponding steady-state strength.Figure 15(b) shows these same calculations but between

parallel plates with H ¼ 2 cm. One sees that the energyloss in method 2 is similar to that in method 1, butthe energy spread induced is overestimated. Free spaceand shielded calculations are repeated with D1 ! 1in Figs. 15(c) and 15(d), which when compared withFigs. 15(a) and 15(b) one can see the effect of the previousbend B2.

VI. COHERENT POWER SPECTRUM

Some of the first CSR calculations are found in anoriginally unpublished report by Schwinger [11]. Herewe use one of his methods to derive an exact expression

0.0 0.5 1.0 1.5 2.0 2.5 3.0− 0.4

− 0.2

0.0

0.2

0.4

s0 (m)

⟨Wcs

r⟩/W

0σ

W/W

0

Method 3Method 2Method 1

0.0 0.5 1.0 1.5 2.0 2.5 3.0− 0.4

− 0.2

0.0

0.2

0.4

s0 (m)

⟨Wcs

r⟩/W

0σ

W/W

0


0.0 0.5 1.0 1.5 2.0 2.5 3.0− 0.4

− 0.2

0.0

0.2

0.4

s0 (m)

⟨Wcs

r/W0⟩

σW

/W0


0.0 0.5 1.0 1.5 2.0 2.5 3.0− 0.4

− 0.2

0.0

0.2

0.4

s0 (m)

⟨Wcs

r/W0⟩

σW

/W0


FIG. 15. (Color) The average and standard deviation of the CSR wake in free space (a) and between parallel plates with H ¼ 2 cm (b)over a Gaussian bunch, compressing from ¼ 0:9 mm to ¼ 0:6 mm linearly through bend 2, and from ¼ 0:6 mm to ¼0:3 mm linearly through bend 1, using methods described in the text. Parts (a) and (b) have D1 ¼ 1 m, while parts (c) and (d) haveD1 ! 1. The lengths B1 ¼ B2 ¼ 3 m, the bending radii are ��1

1 ¼ ��12 ¼ 10 m, and the energy is 1 GeV.


024401-11

for the coherent energy loss by a Gaussian beam, which isthen used to verify our earlier calculations. Consider thepower spectrum due to a single particle moving on a circlewith velocity �c, which is proportional to the absolute

square of the Fourier transform electric field Eð1Þð�; tÞ,integrated over solid angle �, as in

dPð1Þ

d!/Z

d�

��Z 1

�1dtei!tEð1Þð�; tÞ

��2

: (67)

For N particles moving on this circle with positions s ¼sn þ �ct, the total electric field can be written in terms ofthe single particle’s electric field (sn ¼ 0), as in

E ðNÞð�; tÞ ¼ XNn¼1

Eð1Þð�; t� tnÞ; (68)

where the time deviations tn ¼ sn=ð�cÞ. By changingvariables, this means that the N particle power spectrumis simply

dPðNÞ

d!¼

��XNn¼1

ei!tn

��2dPð1Þ

d!: (69)

These phase factors can be separated into terms with m ¼n and m � n,

dPðNÞ

d!¼

�XNm¼1

ei!tmXNn¼1

e�i!tn

�dPð1Þ

d!

¼ NdPð1Þ

d!þ dPð1Þ

d!

XNm¼1

exp

�i!

sm�c

�

� XNn¼1n�m

exp

��i!

sn�c

�; (70)

so that the second term can be written as a correlationbetween different particlesXm�n

exp

�i!

sm � sn�c

�’ NðN � 1Þ

Zds�ðsÞ exp

�i!

s

�c

�

�Z

ds0�ðs0Þ exp��i!

s0

�c

�(71)

using the normalized particle distribution �ðsÞ along thecircle. The N particle power spectrum is then

dPðNÞ

d!ð!Þ ’ N

dPð1Þ

d!|fflfflffl{zfflfflffl}incoherent

þ NðN � 1Þ��Z

ds�ðsÞ exp�i!s

�c

��2dPð1Þ

d!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}coherent

:

(72)

The first term in Eq. (72) is the incoherent power spectrum,

while the second is the coherent power spectrum. Thesquared integral is called the form factor.In free space, the well-known single particle power

spectrum is

dPð1Þ

d!ð!Þ ¼ Pð1Þ

!c

S

�!

!c

�; (73)

where !c � 32�

3c� is the critical frequency [5,12]. The

function S is defined as

Sð�Þ � 9ffiffiffi3

p8�

�Z 1

�dxK5=3ðxÞ; (74)

in which K is a modified Bessel function. The integralR10 SðxÞdx ¼ 1, and the total power lost by a single particle

is

Pð1Þ � 23rcmc3�4�4�2: (75)

For a Gaussian distribution with variance 2 and � � 1,the form factor is, extending the integration limits toinfinity,

��Z 1

�1ds

expði !s�c � s2

22Þffiffiffiffiffiffiffi2�

p

��2¼ exp

��2!2

�2c2

�

¼ exp

��ac

!

!c

�2�; (76)

defining the coherence factor

ac � 3

2��3� ¼

�c!c: (77)

The total power spectrum per particle for an N-particleGaussian distribution with various values of ac is shown inFig. 16. One sees from the exponential that the lowerfrequencies, up to a cutoff frequency around ! ¼ �c=,are enhanced by a factor of N by the coherent part of

p

0123456789

10 5 10 4 0.001 0.01 0.1 1 1010 4

0.1

100

105

108

ω ωc

1N

dPN

d ω

FIG. 16. The power spectrum in Eq. (72), per particle, using aGaussian form factor with various values of the coherenceparameter ac ¼ 2p, defined in Eq. (77). The lower frequenciesare enhanced by a factor of N, and in this example N ¼ 109.


024401-12

Eq. (72). The spectrum at higher frequencies agrees withthe familiar single particle spectrum in Eq. (73).

It turns out that Eq. (72) can be integrated exactly for aGaussian distribution. Explicitly, the total power radiatedby N particles is

PðNÞ ¼ NPð1Þ Z 1

0SðxÞdxþ NðN � 1ÞPð1Þ 9

ffiffiffi3

p8�

�Z 1

0xe�a2cx

2

�Z 1

xK5=3ðyÞdy

�dx

¼ NPð1Þ þ NðN � 1ÞPð1Þ 9ffiffiffi3

p8�

Z 1

0K5=3ðyÞ

��Z y

0xe�a2cx

2dx

�dy

¼ NPð1Þ þ NðN � 1ÞPð1ÞTc

�3

2��3�

�; (78)

in which the final integral yields the coherence functiondefined as

TcðacÞ � 9

32ffiffiffiffi�

pa3c

exp

�1

8a2c

�K5=6

�1

8a2c

�� 9

16a2c: (79)

The limit lima!0þTcðaÞ ¼ 1, which is to say that aninfinitely narrow bunch radiates as one charge. In practicalsituations ac 1, so an asymptotic expansion of Tc givesthe useful approximation

TcðacÞ 9�ð56Þ

162=3ffiffiffiffi�

p�1

ac

�4=3 � 9

16

�1

ac

�2

þ 9�ð56Þ32 � 22=3 ffiffiffiffi

�p

�1

ac

�10=3 þ � � � : (80)

The first term in Eq. (80) is given in Ref. [13]. Figure 17compares this first term to the exact expression in Eq. (79)and to all three terms in Eq. (80). One sees an excellent

approximation for ac * 50 using the first term and forac * 1 using all three terms in Eq. (80). Also, the averagecoherent energy lost per particle per unit length is

PðNÞ

N�c

�coh

¼ � 2

3ðN � 1Þrcmc2�4�3�2Tc

�2

3�3�

�(81)

� �ð56Þ61=3

ffiffiffiffi�

p W0 þ � � � ; (82)

using W0 defined in Eq. (40). The numerical coefficient

�ð5=6Þ6�1=3��1=2 ’ 0:350. The same procedure inEqs. (78) and (80) can be carried out for a uniform distri-bution of length �L with the same variance 2, implying

that �L ¼ 2ffiffiffi3

p. The result yields the same form as

Eq. (82), except with the numerical coefficient 2�4=3 ’0:397. This term was originally derived in Ref. [11].To verify that the CSR wake does indeed represent the

coherent energy lost, the relative difference of the averageenergy loss using the steady-state wake of a Gaussianbunch in Eq. (32) to the result in Eq. (81) is plotted inFig. 18. One sees that the relative difference is at most 1%in this practical parameter range, and that occurs withrelatively long bunches. We speculate that this error iscaused by the regularization procedure that subtracts thespace charge term from the longitudinal electric field.The relevance of the coherence function depends on

the number of particles N � 1 ’ N. The coherent powerradiated equals the incoherent power radiated whenN � TcðacÞ ¼ 1, illustrated in Fig. 19. Using Eq. (80), the

FIG. 18. (Color) The relative difference jðb� aÞj=b for a theaverage energy lost using Eq. (32), and b being the result inEq. (81).

1 10 100ac

0.001

0.01

0.1

1

10

100

Tc ac

FIG. 17. (Color) The coherence function TcðacÞ of Eq. (79) isplotted in red. The green curve is the first term in the asymptoticexpansion in Eq. (80) [13], and the blue curve uses all threeterms in Eq. (80).


024401-13

coherent power dominates the total power when

� &N3=4

�3: (83)

VII. CONCLUSION

The wakefield due to CSR of a one-dimensional bunchtraveling on a curve without small angle or high energyapproximations has been derived using Jefimenko’s formsof Maxwell’s equations. This exact solution allowed us toquantify the accuracy of the approximations of the steady-state CSR wake in a bend given in Refs. [4,7] showing thatthe former is inaccurate at low energies and long bunchlengths, and that the latter is much more accurate down tolow energies. All approximations tend to overestimate theCSR wake. For planar orbits the equations are extended to

include shielding by perfectly conducting parallel platesusing the image charge method.The formulas have been applied to the geometry of a

bend preceded by a drift, preceded by another bend, andshow that the CSR wake well inside the downstream bendis influenced by the upstream bend for the parameters used.In fact, a bunch near the entrance of a bend is influenced bythe CSR wake due to the previous bend much more than bythat due to the previous drift. Shielding by parallel platesreduces the energy loss rate significantly, but the effect onreducing energy spread increase is far less dramatic, inboth the drift and bend regions.Bunch compression has been added to this model by

allowing the bunch length to be time dependent, so that theretarded charge density seen by a test particle is appropri-ately taken into account. This method has been comparedto simple methods used by particle simulation codes Bmadand elegant, and it is shown that these tend to over-estimate the effect [9,10].Additionally, an exact expression for the coherent power

lost by a one-dimensional Gaussian bunch moving in acircle has been derived by integrating the power spectrum,following the method of Schwinger [11]. When comparedto the energy loss rate by the CSR wake, the two showslight deviations. This could be due to the regularizationprocedure for the one-dimensional CSR wake that sub-tracts off the space charge term.

APPENDIX A: FORMULAS FOR MULTIPLEBENDS AND DRIFTS

For ease of reading, the individual terms in Eqs. (58) and(61) have been deferred to here. They are calculated byapplying Eq. (3), regularized by Eq. (10), including imagecharges as in Eq. (45), to the geometry in Eq. (53).In Eq. (58), the first term dEcsr=dsjB1

is the sum of

Eqs. (29) and (47) with � ! �1 and � ! �1s, explicitly:

dECSR

dsðsÞ

��B1

¼ Nrcmc2�Z b

a

d

��2 cosðÞ � 1

2j sinð=2Þj þ 1

�2

sgnðÞ � � cosð=2Þ� 2�j sinð=2Þj

��0ðsÞ � �1�ðsÞ

2j sinð=2Þj��b

a

þZ 1

�a

d�1

�2

�0ðz��Þ�

þZ 1

�b

d�1

�2

�0ðzþ�Þ�

þ X1n¼1

2ð�1Þn��1�ðs;nÞ

r;n

��b

a

þZ b

a

d�2 cosðÞ � 1

r;n�0ðs;nÞ

��(A1)


a � �1ðs� B1Þ; b � �1s; �a � s� 2�1

�1

sin

��1s

2

�; �b � B1 � sþ 2�

1

�1

sin

��1ðB1 � sÞ

2

�;

r;n �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cosþ ðn�1HÞ2

q; s � s� s0 � 1

�1

ð� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos

p Þ; s;n � s� s0 � 1

�1

ð� �r;nÞ:(A2)

Some trigonometric functions have been simplified, and the space charge integrals have changed variables to

Incoherent

Coherent

105 106 107 108 109 1010N100

104

106

108

ac

FIG. 19. (Color) The dividing line where the coherent powerequals the incoherent power, i.e. the total power is twice theincoherent power. Below this line, the coherent power dominatesthe total power.


024401-14

� ¼ ð� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cos

p Þ=�1. These terms account for the regularized CSR wake and image charges in bend 1. The nextterms are

dECSR

ds

��D1

¼ Nrcmc2Z D1

0dL

X1n¼0

ð2� �n;0Þð�1Þn�TL

R3L;n

�ðsL;nÞ þ��2 cosð�1sÞ

RL;n

� �TL

R2L;n

��0ðsL;nÞ

�

RL;n � 1

�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cosð�1sÞ þ 2�1L sinð�1sÞ þ ð�1LÞ2 þ ð�1nHÞ2

q

TL � L cosð�1sÞ þ 1

�1

sinð�1sÞsL;n � �L� s0 þ �RL;n

(A3)

dECSR

ds

��B2

¼ Nrcmc2Z B2

0dL

X1n¼0

ð2� �n;0Þð�1Þn�TL

R3L;n

�ðsL;nÞ þ��2 cosð�1sþ �2LÞ

RL;n

� �TL

R2L;n

��0ðsL;nÞ

�

RL;n �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�cosð�1sÞ � 1

�1

þ 1� cosð�2LÞ�2

�2 þ

�D1 þ sinð�1sÞ

�1

þ sinð�2LÞ�2

�2 þ ðnHÞ2

s

TL � D1 cosð�1sÞ þ �2 � �1

�1�2

sinð�1sÞ þ 1

�2

sinð�1sþ �2LÞsL;n � �L�D1 � s0 þ �RL;n:

(A4)

Note that the lower limit of the sums has been set to n ¼ 0 to account for the real charges as well as image charges,necessitating the use of Kronecker’s delta. Alternatively, if only free space terms are desired, the above formulas can beused with the n ¼ 0 term only. The dummy variable s0 has been rescaled to Lwhich integrates backwards over the length ofthe appropriate element. The terms RL;n, TL, and sL;n are redefined after each equation in order to keep the naming sane.

Similarly, the wake at s > B1 after bend, as in Eq. (61), contains the terms

dECSR

ds

��D0

¼ �Nrcmc2�Z B1

�1ds0

�1

ðs� s0Þ2 �½s0 � s0 þ �ðs� s0Þ� þ �

�� 1

s� s0�0½s0 � s0 þ �ðs� s0Þ�

�

þ X1n¼1

2ð�1Þn �½B1 � s0 þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs� B1Þ2 þ ðnHÞ2p �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs� B1Þ2 þ ðnHÞ2p

þ X1n¼1

2ð�1ÞnZ 1

0dL

�0½Lþ B1 � s0 þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs� B1 � LÞ2 þ ðnHÞ2p �

�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs� B1 � LÞ2 þ ðnHÞ2p �

; (A5)

dECSR

ds

��B1

¼ Nrcmc2X1n¼0

ð2� �n;0Þð�1ÞnZ B1

0dL

�TL

R3L;n

�ðsL;nÞ þ��2 cosð�1LÞ

RL;n

� �TL

R2L;n

��0ðsL;nÞ

�

RL;n �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� 2 cosð�1LÞ

�21

þ 2ðs� B1Þ sinð�1LÞ

�1

þ ðs� B1Þ2 þ ðnHÞ2s

TL � s� B1 þ 1

�1

sinð�1LÞsL;n � �Lþ B1 � s0 þ �RL;n;

(A6)

dECSR

ds

��D1

¼ Nrcmc2X1n¼0

ð2� �n;0Þð�1ÞnZ D1

0dL

�TL

R3L;n

�ðsL;nÞ þ��2 cosð�1B1Þ

RL;n

� �TL

R2L;n

��0ðsL;nÞ

�

RL;n �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Lþ ðs� B1Þ cosð�1B1Þ þ sinð�1B1Þ

�1

�2 þ

�cosð�1B1Þ � 1

�1

� ðs� B1Þ sinð�1B1Þ�2 þ ðnHÞ2

s

TL � s� B1 þ L cosð�1B1Þ þ 1

�1

sinð�1B1Þ

sL;n � �L� s0 þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2L þ ðnHÞ2

q:

(A7)


024401-15

[1] J. B. Murphy, ‘‘An Introduction to Coherent SynchrotronRadiation in Storage Rings,’’ ICFA Beam DynamicsNewsletter No. 35 (2004).

[2] Y. S. Derbenev, J. Rossback, and E. L. Saldin, ReportNo. TESLA-FEL 1995-05, 1995.

[3] J. B. Murphy, S. Krinsky, and R. L. Gluckstern, inProceedings of the Particle Accelerator Conference,Dallas, TX, 1995 (IEEE, New York, 1995), Vol. 5,pp. 2980–2982.

[4] E. L. Saldin, E. A. Schneidmiller, and M.V. Yurkov, Nucl.Instrum. Methods Phys. Res., Sect. A 398, 373 (1997).

[5] J. D. Jackson, Classical Electrodynamics (Wiley, NewYork, 1999), 3rd ed.

[6] R. Warnock, G. Bassi, and J. A. Ellison, Nucl. Instrum.Methods Phys. Res., Sect. A 558, 85 (2006).

[7] D. C. Sagan, G. H. Hoffstaetter, C. E. Mayes, and U. Sae-Ueng, arXiv:0806.2893v1.

[8] T. Agoh and K. Yokoya, Phys. Rev. ST Accel. Beams 7,054403 (2004).

[9] M. Borland, Phys. Rev. ST Accel. Beams 4, 070701(2001).

[10] D. Sagan, Nucl. Instrum. Methods Phys. Res., Sect. A 558,356 (2006).

[11] J. Schwinger, Report No. LBNL-39088; A QuantumLegacy: Seminal Papers of Julian Schwinger, edited byKimball A. Milton (World Scientific, Singapore, 2000).

[12] A.W. Chao and M. Tigner, Handbook of AcceleratorPhysics and Engineering (World Scientific, Singapore,2006).

[13] J. S. Nodvick and D. S. Saxon, Phys. Rev. 96, 180 (1954).


024401-16

exact 1d model for coherent synchrotron radiation with ...hoff/papers/08csr_mayes.pdf · exact 1d...

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