An Analytic Approach for Symplectic Particle Tracking

in Complex 3-Dimensional Magnetic Structures Johannes Bahrdt, HZB / BESSY II, APS-ASD, October, 2011

Outline

• Need for accurate tracking in APPLE II devices

• Symplectic tracking based on generating functions

• Analytic description of arbitrary undulator fields

• Potential applications to other magnet structures

• Analytic equations for dynamic field integrals

• Benefits and limitations of shimming techniques

device design operational λ0 / mm periods Gap / mm By / Bz / T

U49-1 Hybrid 1998 - 49,4 83 15 0,799

U49-2 Hybrid 2000 - 49,4 83 15 0,788

U125-1 Hybrid 1998 – 2005 125 31 20 1,162

U125-2 Hybrid QPU 2000 - 125 31 15 1,360

U41 Hybrid 1999 - 41,2 79 15 0,659

U139 Hybrid 2004 - 139 10 15 1,471

UE56-1 APPLE II 1999 - 2003 56 2 x 30 16 0,771 / 0,529

UE56k APPLE II 2003 - 56 1 x 30 16 0,772 / 0,529

UE56-2 APPLE II 1999 - 56 2 x 30 16 0,772 / 0,529

UE46 APPLE II 2001 - 46,3 70 16 0,680 / 0,435

UE52 APPLE II 2002 - 52 77 16 0,742 / 0,505

UE49 APPLE II 2003 - 49 63 16 0,709 / 0,477

UE112 APPLE II 2006 - 112 32 20 0,994 / 0,765

Permanent Magnet Undulators of BESSY II

UE112 APPLE II Undulator at BESSY II

PETRA III APPLE II Undulator as built at HZB

J. Bahrdt et al., Proc. EPAC 2008, Genua, Italy; J. Bahrdt et al., SRI 2009, Melbourne Australia;

J. Bahrdt et al., IPAC 2010 Kyoto, Jap; J. Bahrdt et al., IPAC 2011 San Sebastian, Spain

L=5m

gap=11mm

The BESSY Soft X-ray polarimeter

Agreement between theory and measurement 0 undulator phase π

elliptical

mode

inclined

mode

Polarization Characterization

Universal Mode of APPLE Undulator

Modification of undulator radiation polarization by beamline components

J. Bahrdt et al, SRI 2009,

Melbourne, Australia

Müller-Matrix describes beamline effects

Parameters of beamline Müller matrix

and of analyzer vs. photon energy.

Gap and shift settings for normalized

Stokes parameters at the sample.

transmissions of field amplitudes

phase difference of electric field components

8

Need for symplectic tracking code in APPLE II structures

• symplectic and fast particle tracker dynamic kicks:

• extended undulator structure instead of thin lens approach

• simple interface (30 Fourier coefficients per undulator)

• full parametrization of undulator field in all operation modes

(linear superposition of magnet fields)

• simple implementation of shims

Numeric approach

• fast and symplectic, full turn FFAG orbit tracking

H. Lustfeld, Ph. F. Meads, G. Wüstefeld et al., LINAC 1984

• tracking of superconducting wave length shifter (strong field devices)

M. Scheer, G. Wüstefeld, EPAC 1992

Analytic approach

• tracking of undulators (nonlinear, weak fields)

J. Bahrdt, G. Wüstefeld, PAC 1991…

J. Bahrdt, G. Wüstefeld, Phys. Rev. Special Topics,

A & B 14, 040703 (2011)

History

𝜃𝑥/𝑦~𝜆02 ∙ 𝐵0

2/𝛾2

9

Hamilton‟s Principle of Stationary Action

A

B

𝛿𝑆 = 0 = 𝐿 𝑞 + 휀, 𝑞 + 휀 − 𝐿(𝑞, 𝑞 ) 𝑑𝑡𝐵

𝐴

= 휀𝜕𝐿

𝜕𝑞+ 휀

𝜕𝐿

𝜕𝑞

𝐵

𝐴

𝑑𝑡

휀 𝑡𝐴 = 휀 𝑡𝐵 = 0

𝛿𝑆 = 휀(𝑡)𝜕𝐿

𝜕𝑞−

𝑑

𝑑𝑡

𝜕𝐿

𝜕𝑞 𝑑𝑡

𝐵

𝐴

integration by parts using

theorem about the

calculus of variation

𝜕𝐿

𝜕𝑞−

𝑑

𝑑𝑡

𝜕𝐿

𝜕𝑞 = 0

Hamilton’s principle:

independent variation

of coordinates qi

Modified Hamilton’s principle:

independent variation of

coordinates qi and momenta pi

𝛿 𝐿 𝑞, 𝑞 𝑑𝑡 = 0 𝐵

𝐴

𝛿 (𝑝𝑞 − 𝐻)𝑑𝑡 = 0 𝐵

𝐴

≡

Newton‟s Laws

Lagrange Euler equation

2nd order DEs in N variables:

Hamilton‟s equations of motion

1st order DEs in 2N variables:

Differential equations: complete description of system but:

- small step sizes required (pretty nasty for undulators)

- symplectic integrator needed

Alternative method: Integration of Hamilton Jacobi Equation

- permits step sizes as long as undulator length

- symplectic even for long step sizes

10

Equivalent Descriptions for System Evolution

𝜕𝐿

𝜕𝑞−

𝑑

𝑑𝑡

𝜕𝐿

𝜕𝑞 = 0

𝜕𝐻

𝜕𝑥= −𝑝 𝑥

𝜕𝐻

𝜕𝑦= −𝑝 𝑦

𝜕𝐻

𝜕𝑝𝑥= 𝑥

𝜕𝐻

𝜕𝑝𝑦= 𝑦

Hamiltonian of relativistic particle in magnetic field:

𝐻 = (𝑝 − 𝑒𝐴 )2𝑐2 +𝑚02𝑐4

Canonical variables: 𝑥, 𝑦, 𝑝𝑥, 𝑝𝑦, and independent variable t

Hamiltonian is independent upon t.

Change independent variable from time t to z to enable

further transformation to cyclic coordinates; new Hamiltonian:

𝐻 = −𝑝 𝑧 = −𝐻 2

𝑐2−𝑚0

2𝑐2 − 𝑝 𝑥 − 𝑒𝐴 𝑥2− 𝑝 𝑦 − 𝑒𝐴 𝑦

2− 𝑒𝐴 𝑧

Normalization and 2nd order expansion in 𝑝𝑥, 𝑝𝑦, 𝑥3

𝑥3~1/𝑘𝐵𝜚 small quantity in undulators; well suited as expansion variable

11

Analytic Approach for Symplectic Mapping I

3

2

3

2

3 2/)(2/)(1 xAxApxAppH zyyxxz

Goal:

Change of canonical variables to cyclic variables using generating functions

Usual procedure of canonical transformation (keeps stationarity of action):

𝑝𝑖𝑞 𝑖 − 𝐻 = 𝑃𝑖𝑄 𝑖 − 𝐾 +𝑑𝐹

𝑑𝑧

with various options for 𝐹

We choose 𝐹 = 𝐹3 𝑄, 𝑝, 𝑧 + 𝑞𝑖𝑝𝑖 getting the new Hamiltonian:

𝐾 𝑄𝑥𝑖 , 𝑄𝑦𝑖 , 𝑃𝑥𝑖 , 𝑃𝑦𝑖 = 0 = 𝐻 𝑥𝑓 , 𝑦𝑓 , 𝑝𝑥𝑓, 𝑝𝑦𝑓 +𝜕𝐹3(𝑄𝑥𝑖 , 𝑄𝑦𝑖 , 𝑝𝑥𝑓, 𝑝𝑦𝑓, 𝑧)

𝜕𝑧

I

and the differential equations:

𝑃 = −𝜕𝐹3𝜕𝑄

𝑞 = −𝜕𝐹3𝜕𝑝

Note:

final coordinates and momenta 𝑥𝑓 , 𝑦𝑓 , 𝑝𝑥𝑓, 𝑝𝑦𝑓 depend upon z

initial coordinates and momenta 𝑄𝑥𝑖 , 𝑄𝑦𝑖 , 𝑃𝑥𝑖 , 𝑃𝑦𝑖 are constants of motion 12

Analytic Approach for Symplectic Mapping II

With substitutions

get HJE

Insert Taylor series expansion of generating function in HJE

expansion variables: 𝑝𝑥𝑓, 𝑝𝑦𝑓, 𝑥3

Each individual expansion term must be zero.

Iterative solution and determination of z-derivatives of 𝑓𝑖𝑗𝑘

Integration of 𝜕𝑓𝑖𝑗𝑘/𝜕𝑧 along z yields generating function.

implicit formulation

of transformation:

02/)(2/)(1 33

2

33

2

33 zzyyxx FxAxAFxAF

𝑝𝑥𝑓 = −𝜕𝐹3𝜕𝑥𝑓

= −𝐹3𝑥 𝑝𝑦𝑓 = −𝜕𝐹3𝜕𝑦𝑓

= −𝐹3𝑦

ijk

kj

yf

i

xijk xppfF 33

yFp

xFp

pFy

pFx

y

x

yff

xff

/

/

/

/

3

3

3

3

Analytic Approach for Symplectic Mapping III

Generating Function in 2nd Order

A 2nd order expansion of GF includes the following terms

Note: the term 𝑥3 always appears (last index ≥ 1)

thus, GF is linear in momenta in 2nd order expansion

Integration of expansion coefficients with respect to z:

From the generating function the transformation map is evaluated

.2/)(2/)( 2

001

2

001002

001101

001011

001

yyxxz

xxz

yyz

zz

AfAff

Aff

Aff

Af

.)')/(()')/(()2/1(

)')/((

)')/((

22

002

011

101

001

dzdzyAAdzxAAf

dzdzyAAf

dzdzxAAf

dzAf

zyzx

zy

zx

z

Transformation Map

2nd order means 1st order in momenta:

In this case the implicit form can be converted into an explicit form:

With pn = (1−f011y)(1−f101x)−f011xf101y

Note: Transformation is not limited to 2nd order, however, the

Newton Raphson Method has to be used for higher order

expansions to solve the implicit equations

.001002011101

011

001002011101

101

yyyfyxfyyfy

yff

xxyfxxfxxfx

xff

ffpfpfpp

fzpyy

ffpfpfpp

fzpxx

.

/))())(1((

/))())(1((

011

001002101001002101

101

001002011001002011

fyff

nxxxyyyyxyf

fxff

nyyyxxxxyxf

zpfyy

pffpfffpfp

zpfxx

pffpfffpfp

Example I: Simple Planar Undulator Structure

Scalar potential of a Halbach type undulator:

Derivation of vector potential from scalar potential

which is

From this expressions the particle dynamics in undulators can be derived

Note:

Analytic expressions of the vector potential are required since GF is

derived from analytic integration over z

).cos()sinh()cos()/( 0 kzykxkkBV yxy

.0

)/(

)/(

2

1

z

y

x

A

CdzxVA

CdzyVA

.0

)sin()sinh()sin())/()((

)sin()cosh()cos()/(

0

0

z

yxzyxy

yxzx

A

kzykxkkkkBA

kzykxkkBA

Example II: Arbitrary Periodic Structure in x and z

The following functions cos(kxnx)cos(kzmz), sin(kxnx)cos(kzmz)

cos(kxnx)sin(kzmz), sin(kxnx)sin(kzmz)

form a basis on the interval S=[-λx0/2, λx0/2]x[-λz0/2, λz0/2]

for all periodic functions in x and z direction with periodicity λx0 and λz0

Complete description of arbitrary periodic magnet fields in a plane:

Fourier coefficients:

Ansatz for 3D field:

with 𝛻𝐵 = 𝛻 × 𝐵 = 0:

solving the differential equation:

with midplane symmetry:

))sin()sin()sin()cos(

)cos()sin()cos()cos((),(

,,

,

0 0

,

zkxksszkxkcs

zkxksczkxkcczxB

zjxijizjxiji

zjxiji

n

i

m

j

zjxijiy

01

01

/2

/2

xxxi

zzzj

iikk

jjkk

jijijiji sssccscc ,,,, ,,,

)(),(),,( ,, yHzxBzyxB ijijyijy

222~

2 /)( yHHkk ijjxi

)exp()exp( ~,2~

,1 ykcykcHjyijyiij

)cosh( ~,

ykcHjyiij

Skip sin-terms for planar undulator (symmetric in x)

and get usual Halbach fields:

Derivation of Fourier coeffifficients

- undulator:

single trans. field distribution

- wiggler:

seveveral trans. field distributions

)cos(

...

...

~

1

~,,

1

~,1,

1

~,01,0

mj

m

jjnmn

m

jjnn

m

jj

zkcC

cC

cC

)cos()cosh()cos(),,( ~~,

0 1

~,

zkykxkczyxBjjyi

n

i

m

j

xijiy

...5,3,1~

22~~

,

j

kkk xijjyi

BESSY U125 Wiggler I

solve system of

linear equations

(1,5,9… for ppm structure)

Field reconstruction using different

numbers of harmonics J. Bahrdt et al., IPAC 2011

The 3rd field harmonic has to be

included for a a field reconstruction

on the percent level

For undulators usually 1st field harmonic

is sufficient

Shimmed device

U125-2 Magic fingers

BESSY U125 Wiggler II

Theoretic dynamic

field integrals

22

0

0

0

04

03

02

01

1

4321

)2/cos(

))(sin(

))(cos(

)/)()/((

)/)()/((

)/)()/((

)/)()/((

)(

zxiyi

xxi

zz

xixi

xixi

zz

yk

xisyixicyiyi

yk

i

zz

yk

xisyixicyiyi

yk

i

zz

yk

xisyixicyiyi

yk

i

zz

yk

xisyixicyiyi

yk

i

n

i

iiii

kkk

kik

zkc

xxks

xxkc

cnkeBsBcBkeV

cnkeBsBcBkeV

cnkeBsBcBkeV

cnkeBsBcBkeV

VVVVV

zyi

zyi

zyi

zyi

)(VB

Scalar potential of APPLE II field

Example III: APPLE II Undulator

.0

)/(

)/(

2

1

z

y

x

A

CdzxVA

CdzyVA

Accuracy of APPLE II Field Description

analytic representation (black)

simulation with RADIA (red)

Parametrization of magnet field distribution with high accuracy

By at phase = 0 Bx at phase = pi

in midplane

4mm vertical off-axis

8mm vertical off-axis

22

Comparison with ELEGANT

Aimin Xiao

Planar undulator

Period length 28mm

Analytic approach and

symplectic tracker

cwigg of ELEGANT

Acceleration of undulator tracking

By more than an order of magnitude

23

First Results for APPLE II

Aimin„s implementation of

analytic method into ELEGANT

Comparison of thick lens

and thin lens approach

phase = 0

phase = pi

phase = pi/2

Aimin Xiao

• Other complicated undulator structures such as

Figure 8 undulator for lin. pol. SPRING 8 helical ID

and reduced on axis power density

• Dipole magnet:

Curved coordinate system along the reference orbit is used

particle displacements and momenta with resect to this reference orbit

starting from a modified Hamiltonian 𝑝𝑠 the same method is applicable

• Fringe fields in quadrupoles and sextupoles

• …

Other Interesting Applications

Courtesy of H. Kitamura Courtesy of B. Diviacco

𝑞𝑖

𝑞 𝑖

25

Analytic Kick Maps (Thin Lens Approximation)

Kick

phase space volume is preserved with kick maps

𝑞𝑖

𝑞 𝑖

𝑞 𝑖𝑛𝑒𝑤=𝑞 𝑖

𝑜𝑙𝑑 +𝑊(𝑞𝑖)

Dynamic Kicks in a Planar Undulator

))cosh()cosh()sinh()sinh(1

(

)cos()sin()(2

~,2

~,1

~,2

~,1

~,2

2

2

~,1

01 02 1

212~

1~

,2~

,12

ykykykykk

k

k

xkxkk

kcc

B

z

jijijiji

jyi

xi

jyi

n

i

n

i

m

j

xixi

j

xi

jiji

f

x

The integrated dynamic kicks due to the wiggling motion in undulators

follow directly from generating function:

These expressions can be used in a thin lens approximation (kick map)

which assumes a localized kick of the strength of the integrated quantity

))sin()sin()cos()cos((

)sinh()cosh(1

)(2

21~

,2

2121~

,2

01 02 1

~,2

~,12

~

~,2

~,12

xkxkk

kkxkxkk

ykykk

ccB

z

xixi

jyi

xixixixijyi

n

i

n

i

m

jjyijyi

j

jiji

f

y

x/f002x y/f002y

Elliptical mode (φ2=0)

𝑐𝑝𝑝, 𝑠𝑝𝑝 analytic functions of φ1

𝑒𝑖 , 𝑒𝑗 include gap dependence

For general expressions in universal mode and for off-midplane expressions see

J. Bahrdt, G. Wüstefeld, Phys. Rev. Special Topics, A & B 14, 040703 (2011)

Dynamic Kicks of APPLE II Undulators

.)(

8

)(

8

0

0 0

00

2

22

0 0

00220

22

0

n

i

n

j

xjxjxixi

yj

xj

yi

xijiji

n

i

n

j

xjxjxixixjjiji

y

ppppx

sscsk

k

k

keecc

Bk

Lf

sccckeeccBk

Lf

sfcf

)).(

)(()(

8

)()(

8

0000

0000

0 0222/

0000

2

0 0222/

22

2/

22

2/

44

0

xjxjxixixjxjxixiyj

xjxjxixixjxjxixixj

yi

xin

i

n

j

jiji

xjxjxixixjxjxjxixi

yj

xj

yi

xin

i

n

j

jiji

papay

papapapax

ccsssscck

csscsccskk

keecc

Bk

Lg

cssskccsck

k

k

keecc

Bk

Lf

scg

scfsfcf

Inclined mode (φ1=0):

𝑐𝑝𝑝 , 𝑠𝑝𝑝 analytic functions of φ2

f0

fπ

fπ/2

4 generic

functions

22

2/ papa scg

L-Shims for Dynamic Kick Reduction

Field integrals of L-Shim

analytic representation (black)

simulation with RADIA (red)

Parametrization of shim field integral with high accuracy

J. Chavanne et al., Proceedings of

the EPAC 2000, Vienna, Austria

.

]2/exp[

)cosh()sin()sinh()cos(

]2/exp[

)sinh()cos()cosh()sin(

0

0

yixi

yi

n

i

yixiiyixiiy

yi

n

i

yixiiyixii

yi

xix

kk

gk

ykxksykxkcB

gk

ykxksykxkck

kB

Implementation in our tracking scheme:

Active Compensation of Dynamic Kicks

with Flat Wires: BESSY II UE112 APPLE II

current settings for gaps of 20mm

24mm, 30mm and 40mm

J. Bahrdt et. al., EPAC 2008, Genoa, Italy

water cooled

undulator

vacuum chamber

30

,1

))(

(

1)2,mod()12/(2

02

0

0

0

)2),1mod(()2/)1((2

0

)2,mod(2

0

11)2,mod()12/(2

02

00

1

m

istarti

mmim

iij

m

m

x

mmimm

mistartii

ij

m

istarti

mmim

i

ij

m

m

y

xr

cydxB

xr

cmh

xr

cdxB

.1)(

2)1(1

1

2)(2)2,mod(

1)(2)2,mod(

12/

0,

12

)1(

1

2

0

2

0

2

0

)1()1(

)1(

m

jji

ij

mh

j

ic

istartimj

istartimj

yxr

mistart

Multipole Decomposition of Flat Wire Field Integrals

Accuracy of Multipole Expansion

Accuracy of field integral description (y=0)

with z0=0,, 10, 20, 30mm and y0=10mm.

Thick lines: exact fields integrals, dotted lines:

approximation with m=20.

y

z y0

z0 position of wire

multipole expansions is senseless

outside radius of convergence

(defined by magnet field sources)

use instead: multipole specs at various hor. trans. positions

UE112 shimming effect

x-kick per ID passage (vertical linear mode) particles distributed on horizontal phase

space ellipse, semi axes: 30mm / 1.87mrad

red: no shims

blue: wire shims powered

BESSY II: 1000-turns tracking (x-x‟-plane)

frequency

map

EPAC 2006 Bahrdt, Scheer& Wüstefeld

Particle Tracking for BESSY UE112 APPLE II

30mm

400ura

d

UE112: Tune Shift and Beam Size Variation

dynamic multipoles with and

without active compensation

horizontal and vertical tunes

vs horizontal displacement:

black: tune correction off, wires off

blue: tune correction on, wires off

red: no tune correction, wires on

source size variation with row phase of the

UE112 at gap = 24mm in the elliptical mode

(top) and the inclined mode (bottom). Black,

blue: currents switched off; red, magenta:

currents switched on.

UE112: Recovery of Injection Efficiency

Active shimming of the UE112 is essential for a top-up mode.

- inclined mode: no alternative to active compensation

- elliptical mode: improvement of passive shimming results

The next step: Compensation for universal mode

Static multipoles:

Complete description of two dimensional straight line field integral distributions

on a source free circular disc

is an analytic function; the bar indicates a straight line integration

Cauchy Riemann relations: are equivalent to the

2D-Maxwell equations

“Dynamic multipoles” (DM):

is not an analytic function; the tilde indicates an integration

along a wiggling trajectory

Cauchy Riemann relations

are not fullfilled:

Note: By principle “dynamic multipoles” can not be compensated

with shims which are usually described by static field integrals

Static Multipoles and so-called „Dynamic Multipoles“

0/~

/~

)~

,~

( 002002 xyyxyxyx ffyBxBBB

0/~

/~

)~

,~

( 002002 yyxxxyzyx ffyBxBBB

yx BiBF

y

B

x

B

y

B

x

B

xy

yx

)(

)(

yx BiBF~~~

36

„Dynamic Multipoles“ of Planar Undulator

,)1(

)1(

1)1(22/

1

21)(

)1(22/

1

211)(

in

i

in

xni

in

x

in

i

in

yni

in

y

yxaB

yxaBStatic regular multipoles

are usually described by:

In contrast the

“Dynamic multipoles” of a

planar undulator with only one

transverse and one long.

Fourier component:

),)2()2()!2(

1

)!12(

1)1()(

)2()!1(

1)1)(((

8

~

22/

1

122121

2

3

1112/

2

3

2)2,mod(,0

)(

inn

i

iin

y

i

x

i

x

y

x

nn

x

n

x

y

xn

n

y

yxkkini

kk

k

xkn

kk

k

kB

LB

).)2()!1(

1)(

)2()!21(

1)2(

)!22(

1)1(

)2()!12(

1)2(

)!22(

1)1(

(8

~

11

2

2

21212/

1

22221

12122/

1

22221

2

2

2)2,mod(,0

)(

nn

yyy

y

x

inin

y

n

i

ii

x

i

y

inin

y

n

i

ii

x

i

y

y

x

n

n

x

ykn

kkk

k

ykin

xki

k

ykin

xki

kk

k

kB

LB

37

„Dynamic Multipoles“ of Planar Undulator

Special Cases

.~~

~~

1)1(22/

1

2)(

)1(22/

1

21)(

in

i

in

xni

n

x

in

i

in

yni

n

y

yxaB

yxaBCase 1:

The undulator focusses horizontally

with the same strength as vertically

Case 2:

The undulator defocusses horizontally

And the period length is long:

Case 3:

The undulator has infinitely wide poles

Note: In all cases the “dynamic

multipoles (DM)” are principally

different from static multipoles

22

yx kk .~~

)1(~~

1)(

112/)(

n

xn

n

x

nn

yn

n

y

ybB

xbB

.~~

0~

1)(

)(

n

xn

n

x

n

y

ycB

B

,)1(

)1(

1)1(22/

1

21)(

)1(22/

1

211)(

in

i

in

xni

in

x

in

i

in

yni

in

y

yxaB

yxaB

38

Shimming of „Dynamic Multipoles“

Why does shimming of DM work at all?

Shimming of DM in the midplane has no principle limitations, but

vertical off-axis effects are enhanced; this is acceptable because:

• usually, vertical beta-function smaller than horizontal beta functions

• usually vertical emittance smaller than horizontal emittance

• large particle amplitudes occur during horizontal injection

What about gap dependency?

DM are expected to drop off much faster

than shim field integrals due 𝐵2 dependency, but:

• detailed considerations show similar gap dependence of

dynamic multipoles and static multipoles for long period lengths

• DM scale with square of period length; Murphy‟s Law does not apply

39

Summary and Outlook

Fast analytic, symplectic GF-based tracking scheme

one step per undulator is possible

Analytic description of undulator fields and shim field integrals

simplifies interface

Analytic expressions for analytic kicks

Extension of method to other undulator structures

and accelerator magnets