Monte Carlo simulation of radiation emitted from

the BM ports of the new ESRF lattice

M. Sanchez del Rio, B. Nash

June 20, 2014

Abstract

This document explains a model to Monte Carlo sample wiggler sources.The model follows the implementation in the SHADOW ray tracing code.The results are applied to the simulation of the emission of the wigglersto be used at the bending magnet beamlines for the new ESRF lattice.

1 Introduction

This paper includes the equations needed to simulate the emission from wig-glers. The model is based on the fact that there is no interference of the photonsemitted from different points along the electron trajectory, usually a good ap-proximation for insertion deviced with deviation parameter K >> 1. The com-putation goest though different phases: calculation of the trajectory, velocityand curvature for an electron moving in an arbitrary magnetic field, computa-tion of the spectrum of the emitted radiation, sampling mechanisms to createrays from the radiation distribution, and lately the effect of the electron beamemittance. The effects of the electron beam emittance have been treated indepth. The SHADOW model has been corrected and a the new implementationis described.

2 Electron in a magnetic field

Let us define our reference system with coordinate s along the eletron orbit, ytangent to the orbit at a given point, x in the horizontal plane, and z in vertical.

In usual wigglers the magnetic field is directed along the z direction to makethe electron moving in the horizontal plane. See Fig. 2 for an example of wigglermagnetic field.

An electron of energy E will move with velocity:

vxc

≈ (0.3/E)

∫Bz(y)dy (1)

where vx is the velocity along x, c is the speed of the light, E is the electronenergy in Gev, Bz the magnetic field in T and coordinates and distances are inm.

1

Figure 1: Magnetic field for the wigglers to be placed at the ESRF new lattice.

The trajectory is given by the integral of the velocity:

x ≈ (0.3/E)

∫ ∫Bz(y)dy

2 (2)

To keep the electron in orbit, the electron must have the same position anddirection at the entrance and exit of the wiggler. This is guaranteed if:∫

Bz(y)dy = 0 (3)

In most practical cases we can write the magnetic field along the axis y farfrom its ends as a sinusoidal function with period λw and B0 the peak magneticfield. An arbitrary magnetic field Bz(y) can be defined numerically, but it maybe convinient to describe it as a sum of different harmonic components:

Bz(y) =N∑

n=1

Bn sin(2π

λwny + θn) (4)

A general expression of harmonic decomposition is in Ref. [4]. The interestof the harmonic decomposition is that trajectories and velocities are calculatedanalytically thus with less errors than using numerical integration. For thestandard wiggler:

vxc

=K

γcos(

2π

λwy) (5)

where K is the deflection parameter:

K = 93.4B0[T ]λw[m] (6)

2

(K >> 1 for wiggler, so the emission on different points on the trajactory doesnot interfere) and γ is the electron energy in units of electron energy at rest(mec

2 = 0.511MeV ).The trajectory is given by:

x ≈ 0.3

E

∫ ∫Bz(y)dy

2 =λwK

2πγsin(

2π

λwy) (7)

where .In SHADOW [2] the computation of the electron trajectories for standard

wigglers are done by a preprocessor called ”epath”. The calculations for anarbitrary magnetic field (numerically defined or using harmonic decomposition)is done by the preprocessor epath b which defines:

Bz(y) =N∑

n=1

Bn cos(2π

λwny)

vx(y)

c=

−0.3

E

λw

2πn

N∑n=1

Bn[sin(2π

λwny)− sin(−πn)]

x(y) =−0.3

E(λw

2πn)2

N∑n=1

Bn[cos(2π

λwny)− cos(−πn)] (8)

and:

vy(y)

c=

√(v0c)2 − (

vxc)2

γ =E

mec2= (1− (

vxc)2)−1/2

x(y) =−0.3

E(λw

2πn)2

N∑n=1

Bn[cos(2π

λwny)− cos(−πn)]

1

R(y)= − e

γmec

c

v0Bz(y) (9)

3

Table 1: Harmonic decomposition n,Bn for the magnetic field in Fig. 2.

1 0.3931932 0.7354933 0.3135394 −0.1639425 −0.1685666 −0.0356707 −0.0203428 −0.0060859 0.03461310 0.02241711 −0.00986212 −0.00518913 0.00760614 0.00129615 −0.00482216 −0.00156517 −0.00022318 −0.00131319 0.00034420 0.00192821 0.00068422 −0.00053123 −0.00030024 −0.00000625 −0.00013326 −0.00005127 0.00007628 0.00002629 −0.00004630 0.00002131 0.00004732 0.00002033 −0.00002134 −0.00001135 −0.00001436 0.00000037 −0.00000238 0.00001139 −0.00000040 0.000003

3 wiggler radiation

The full emission spectrum produced by the 3-pole wiggler with magnetic fieldin Fig. 2 (BMHard) is compared with a the 0.4 T bending magnet (integratedover 2 mrad horizontal divergence) in Fig. 3.

4

Figure 2: Calculated (XOP) spectra produced by the 3-pole wiggler at BMHardlocation (solid line) and the 0.4T bending magnet at BMSoft (dashed line).

4 Electron beam size and divergence character-isation. Evolution in a straight section.

Let us suppose we are given the Twiss parameters β(s0), α(s0), γ(s0) at positions0, and the emittance is ϵ. We consider the horizontal phase space here, withcoordinates (x, x′) An electron beam with non-zero emittance ϵx,z is distributedabout the orbit with individual electrons executing betatron oscillations. Inthe case of a Gaussian electron beam (expected in electron storage ring), theemittance is related to the so-called Courant Snyder invariant Jx,z via

ϵ = ⟨J⟩ (10)

The Courant-Snyder invariant at s is given by

J =1

2(γx2 + 2αxx′ + βx′2) (11)

We note that for a constant value of J , this represents the equation of an ellipse.As the Twiss parameters change, the shape will change, but the area remains thesame. Now, we recall that the Twiss parameters must satisfy the relationships[3]:

βγ = 1 + α2

α = −1

2

dβ

ds(12)

and one may compute their value at another position (s) (assuming no focussingelement between s and s0) via

β(s) = β0 − 2α0(s− s0) + γ0(s− s0)2

5

α(s) = α0 − γ0(s− s0)γ(s) = γ0 (13)

A waist (minimum in beam size) is given when α = 0. Thus, for a given(β0, α0, γ0), we find that the waist occurs at a distance

sw =α0

γ0(14)

with the value of β at the waist,

βw =1

γ0(15)

Now, the normalized electron beam distribution is given by

f(x, x′) =1

2πϵe−

J(x,x′)ϵ (16)

This Gaussian distribution may be characterized by the matrix of second mo-ments

Σ =

(⟨x2⟩ ⟨xx′⟩⟨xx′⟩ ⟨x′2⟩

)(17)

with, for example

⟨x2⟩ =∫

dxdx′x2f(x, x) (18)

and likewise for the other second moments. Notationally, we may also write

σx =√

⟨x2⟩σ′x =

√⟨x′2⟩ (19)

Computing these quantities, using eqns. (11) and (16), we find that

⟨x2⟩ = βϵ⟨xx′⟩ = −αϵ⟨x′2⟩ = γϵ (20)

One may also confirm that⟨J⟩ = ϵ (21)

It is sometimes useful to introduce a set of normalized coordinates (u, u′)in which the motion in phase space is simply circular. In the uncoupled linearcontext here, we may represent this transformation as(

uu′

)=

( 1√β

0

α√β

√β

)(xx′

)(22)

and the inverse transformation is given by(xx′

)=

( √β 0

− α√β

1√β

)(uu′

)(23)

In terms of the normalized coordinates, the Courant-Snyder invariant may berepresented simply as

J =1

2(u2 + u′2) (24)

6

5 Inclusion of dispersion

There is an additional effect that increases the horizontal beam size, even inthe ideal ring with no errors. This is the effect of orbit changes with energy,and is captured in the concept of the dispersion function η(s) and its derivativeη′ = dη/ds. For a particle with a given energy deviation δ = ∆E/E, the particlewill oscillate about an orbit displaced by in the horizontal by ηδ. Each particlein the bunch will have a slightly different energy with a Gaussian distributiongiven by

ρ(δ) =1√2πσδ

e− δ2

2σ2δ (25)

Now, one applies the analysis of the previous section, but to the coordinate

xβ = x− ηδ (26)

One then sees that the horizontal beam size and divergence are increased via

⟨x2⟩ = ⟨x2β⟩+ η2σ2

δ (27)

⟨x′2⟩ = ⟨x′2β ⟩+ η′2σ2

δ (28)

In case one is given η and η′ at one position, one can compute them at anotherposition, assuming there is not dipole or quadrupole field in between:

η2 = η1 + (s2 − s1)η′1

η′2 = η′1 (29)

6 Electron beam sampling with emittance

The electron coordinates (position and divergence or angle) in both horizontalx and vertical z directions follow a bivariate normal distribution with moments:⟨x2⟩, ⟨x′2⟩, ⟨xx′⟩. At the beam waist ⟨xx′⟩ = 0, ⟨x2⟩ = σ2

x, ⟨x′2⟩ = σ′2x, and theemittance is: ϵx = σxσ′x. The same equations hold for the vertical plane byreplacing x by z.

We are interested in computing these moments at any point y of the wigglerknowing their values at the waist. The optical functions at a y position in thewiggler (s = y−sw from waist) we can calculated using Eqs. 13. The sw is theposition of the waist from the center of the wiggler (y = 0) calculated usingEq. 14. The moments can be expressed from the electron orbit functions usingEq. 20.

The moments at any point of the wiggler axis distant s from the horizontalwaist are1:

⟨x2⟩ = s2σx′2 + σ2x

1

they ones implemented in SHADOW (routine gauss) were different (bug fixed, using newroutine binormal): ( √

d2σx′2 + σ2x

dσx′2√d2σx′2+σ2

xdσx′2√

d2σx′2+σ2x

σxσx′√d2σx′2+σ2

x

)(30)

7

⟨xx′⟩ = sσx′2⟨x′2⟩ = σx′2 (31)

Again, the same equations hold for thevertical plane replacing subindices xby z.

To do the sampling from a general covariance matrix, we assume that onecan find a matrix L such that

Σ = LLT (32)

For any 2x2 covariance matrix

Σ =

(σ21 σ1σ2ρ

σ1σ2ρ σ22

)(33)

The matrix L is found by the Cholesky decomposition method to be

L =

(σ1 0σ2ρ σ2

√1− ρ2

)(34)

which, when the Σ matrix is expressed in terms of the Twiss parameters repro-duces equation (23). If you get two random numbers (xu

1 , xu2 ) from the computer

generator in U(0, 1) we can create two numbers following a Normal distributionin N(0, 1) using the Box and Muller decomposition:

xn1 =

√−2 ln(xu

1 ) cos 2πxu2

xn2 =

√−2 ln(xu

1 ) sin 2πxu2 (35)

the result of applying L to these two numbers gives two new numbers followinga binormal correlated distribution with covariance Σ.

7 Hard and Soft bend locations for upgrade lat-tice

The center of the soft bend source location (BMSoft, in the lattice) is locatedat a distance s = 13.33 meters from the start of the cell, and the center ofthe hard bend source location (BMHard) is located at s = 13.717 meters. A3-pole wiggler (3PW) is only used for the current Hard bend dipoles (s=13.717m). The soft bend dipoles, located at s=13.33 m will come from the dipoleitself which has a field of 0.41 T (radius R = 50.02 m). The full BM length issend−sstart = 13.565−12.807 = 0.76m covering an emission angle of 15.7mrad,separated in the Hard and Soft BM (at s = 13.133) thus in 10.7 + 4.8 mrad.The 3PW will be between the dipole (DQ2C) and the quadrupole QF8D. TheTwiss parameters for these positions on the S28 lattice are in Table 7.

Table 2: Twiss parameters at the wiggler center for the Hard and Soft locations.

location s [m] αx βx [m] γx [m−1] αz βz [m] γz [m−1]BMSoft 13.33 -0.49 0.46 2.72 1.38 5.10 0.57BMHard 13.717 -2.03 1.44 3.56 2.68 3.16 2.59

8

Table 3: Second moments at the wiggler center for the Hard and Soft locations.

location√⟨x2⟩ ⟨xx′⟩

√⟨x′2⟩

√⟨z2⟩ ⟨zz′⟩

√⟨z′2⟩

[µm] [pm] [µrad] [µm] [pm] [µm]BMSoft 8.306624 73.5 20.199 5.049752 -6.9 1.688194BMHard 14.697 304.50 23.108 3.975 -13.400 3.599

Table 4: Moments at three positions (center and edges) of the wiggler [1].

moment unit y = −0.1 m y = 0 y = 0.1 m⟨x2⟩ µm2 12.67 15.70 16.8⟨xx′⟩ pm 251.10 304.5 357.9⟨x′2⟩ µrad2 23.11 23.11 23.11⟨z2⟩ µm2 4.31 3.97 3.64⟨zz′⟩ pm -14.69 -13.40 -12.11⟨z′2⟩ µrad2 3.60 3.60 3.60

The equilibrium horizontal emittance is 150 pm. The vertical emittance mayvary depending on the conditions. Let us suppose it is 5 pm. We then find forthe second moments at BMSoft and BMHard (see Table 7)).

The evolution of the Twiss parameters along the 3PW axis is determined byEqs. 13. In terms of the moments, the values at a given point y are calculatedfrom the values at the given location in Table 7 using:

⟨x2⟩y = ⟨x2⟩+ 2⟨xx′⟩y + ⟨x′2⟩y2

⟨xx′⟩y = ⟨xx′⟩+ ⟨x′2⟩y⟨x′2⟩y = ⟨x′2⟩ (36)

The change of the electron beam moments change along the wiggler is sum-marized in Table 7 and a plot of the X space phase at the widget edges is inFig. 7.

9

Figure 3: Monte Carlo sampling of the electron phase space x′[rad] vs. x[m] atthe wiggler edges: y = −0.1 (red) and y = 0.1 (black).

It is also possible to calculate the moments at a given point knowing thedistance to the waist. From Eq. 14 we calculate the positions of the horizontal(x) and vertical (z) with respect to the BMSoft and BMHard positions (Ta-bles 7 and 7). These tables also contain the moments at these waist positions(parameters used by SHADOW). Note that at the waist position the relationϵ = σσ′ holds, because correlations are zero.

Note that due to the quadrupole, the waist is actually not where we calculatessupposing no focusing. But for the purpose of beam size evolution just within thewiggler it is correct, as parameters are calculated back and forth from the wigglerposition always supposing free space. The correct values and the calculated onesare shown in Fig. 7.

The plots of phase spaces (x, x′), (z, z′) and real (x, z) and divergence (x′, z′)space is given in Fig. 8 for the photons emitted by the 3-pole wiggler neglectingelectron emittances. Fig. 8 contains the plots of the electron phase space withthe same plot limits. Last, Fig. 8 contains the plots for the photon beam includ-ing all effects (wiggler emission and electron dispersion) which is approximatelythe convolution of the two previous models.

10

Table 5: Horizontal waist parameters and positions. Relative positions meansfrom ID to waist. Note that SHADOW input requites the distance from thewaist to the ID center, thus the opposite of the values given here.

location X waist [m] X waist [m] σx σ′x ϵx

(relative) (absolute) [µm] [µrad] [pm]BMSoft -0.180 13.150 7.46 20.20 150BMHard -0.570 13.147 6.51 23.11 150

Table 6: Vertical waist parameters and positions. Relative positions means fromID to waist. Note that SHADOW input requites the distance from the waist tothe ID center, thus the opposite of the values given here.

location Z waist [m] Z waist [m] σz σ′z ϵz

(relative) (absolute) [µm] [µrad] [pm]BMSoft 2.421 15.751 2.97 1.69 5BMHard 1.035 14.752 1.39 3.60 5

BMSoft s=13.33 BMHard s=13.717

5.10

3.16

Approx waist

14.75

DQ2C QF8D

1.44

0.46

Approx waist

13.15

Figure 4: Twiss parameters in the vecinity of BMSoft and BMHard positions[1].

8 Future work

1. inclusion of orbit offset errors and recompute velocities and curvaturesfrom them

2. unify shadow definitions with standard one in literature

3. 3D case: helicoidal wiggler: unify models

4. include dispersion in SHADOW models

11

Figure

5:Photon

phasespacewithou

tem

ittances(len

gthin

cm,an

gles

inra

d.

12

Figure

6:Electronphasespace(len

gthin

cm,an

gles

inra

d.

13

Figure

7:Photonphasespace(len

gthin

cm,an

gles

inra

d.Fullsimulation

includingelectron

emittance

effects.

14

A Python code for calculating Twiss parame-ters and moments at BMHard and BMSoftlocations

from numpy import sq r t

#S28 ESRF l a t t i c e parameters f o r w i gg l e r s to be used at the BMbeamnlimes

l o c a t i o n = 0 # 0=s o f t bend , 1=hard bend# emittancesex = 150e−12 # m. radez = 5e−12 # m. rad

i f l o c a t i o n == 0 :#s o f ts = 13.33 # m

ax = −0.49bx = 0.46 # mgx = 2.72 # mˆ−1

az = 1.38bz = 5.10gz = 0.57

else :#hards = 13.717 # m

ax = −2.03bx = 1.44 # mgx = 3.56 # mˆ−1

az = 2.68bz = 3.16gz = 2.59

print ( ”−−−−−−−−−−− inputs −−−−−−−−−−−−−−−−\n” )print ( ” emittances [m. rad ] : ex , ez : %e , %e \n”%(ex , ez ) )i f l o c a t i o n == 0 :

print ( ” So f t bend l o c a t i o n \n” )print ( ” at l a t t i c e l o c a t i o n s : %f m”%(s ) )

else :print ( ”Hard bend l o c a t i o n \n” )

print ( ”X alpha , beta , gamma: %f %f %f \n”%(ax , bx , gx ) )print ( ”Z alpha , beta , gamma: %f %f %f \n”%(az , bz , gz ) )print ( ”−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n” )

xx2 = ex∗bxxxp = −ex∗axxp2 = ex∗gx

zz2 = ez∗bzzzp = −ez∗azzp2 = ez∗gz

print ( ”moments ( at the w igg l e r p o s i t i o n ) : \n” )print ( ” sq r t (<x2>) : %e”%(sq r t ( xx2 ) ) )print ( ” <x xp> : %e”%(xxp ) )print ( ” sq r t (<xp2>) : %e”%(sq r t ( xp2 ) ) )print ( ” ” )print ( ” sq r t (<z2>) : %e”%(sq r t ( zz2 ) ) )print ( ” <z zp> : %e”%(zzp ) )print ( ” sq r t (<zp2>) : %e”%(sq r t ( zp2 ) ) )

# waist p o s i t i o n sdx = ax/gxdz = az/gzprint ( ” ” )

15

print ( ”X waist at ( r e l a t i v e ) %f , Z waist at ( r e l a t i v e ) %f \n ”%(dx , dz ) )print ( ”X waist at ( abso lu te ) s=%f , Z waist at ( abso lu te ) s=%f \n ”%(s+dx

, s+dz ) )

# op t i c a l f unc t i on s at waistbx w = bx − 2∗ax∗dx + gx∗dx∗dxax w = ax − gx∗dxgx w = gx

bz w = bz − 2∗az∗dz + gz∗dz∗dzaz w = az − gz∗dzgz w = gz

print ( ” ” )print ( ” o p t i c a l f unc t i on s at X waist : ” )print ( ” alpha , beta , gamma: %f ,%f ,% f ”%(ax w , bx w , gx w ) )print ( ” ” )print ( ” o p t i c a l f unc t i on s at Z waist : ” )print ( ” alpha , beta , gamma: %f ,%f ,% f ”%(az w , bz w , gz w ) )

#moments at waistxx2 w = bx w∗exxxp w = −ax w∗exxp2 w = gx w∗ex

zz2 w = bz w∗ ezzzp w = −az w∗ ezzp2 w = gz w∗ ez

print ( ” ” )print ( ”moments at X waist : ” )print ( ” sq r t (<x2>) : %e”%(sq r t ( xx2 w ) ) )print ( ” <x xp> : %e”%(xxp w ) )print ( ” sq r t (<xp2>) : %e”%(sq r t ( xp2 w ) ) )print ( ” ” )print ( ”moments at Z waist : ” )print ( ” sq r t (<z2>) : %e”%(sq r t ( zz2 w ) ) )print ( ” <z zp> : %e”%(zzp w ) )print ( ” sq r t (<zp2>) : %e”%(sq r t ( zp2 w ) ) )

print ( ” ” )print ( ” emit tances s at waist ( double check ) : ” )print ( ” ex , ez : %e , %e”%(sq r t ( xx2 w∗xp2 w ) , sq r t ( zz2 w∗zp2 w ) ) )

B ShadowVUI workspace for BMHard

A ShadowVUI workspace containing the inputs for creating the 3-pole wigglersource is available in the SHADOW interface. To run it in the ESRF NICEcluster, start the SHADOW interface using XOP 2.4: /scisoft/xop2.4/xop

shadowvui, then load the workspace with the name: ESRFnewlattice-BMHard.wsfrom the examples directory (ShadowVUI → Load Example Workspace → From

Example Dir...).

References

[1] ESRF. ASD Upgrade: TDS Ch 2: Accelertor and source upgrade. TDSDraft version 1.3, 2014.

[2] Manuel Sanchez del Rio, Niccolo Canestrari, Fan Jiang, and Franco Cer-rina. SHADOW3: a new version of the synchrotron X-ray optics modellingpackage. Journal of Synchrotron Radiation, 18(5):708–716, Sep 2011.

[3] H. Wiedermann. Particle Accelerator Physics. Springer, 2007.

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[4] Y. K. Wu, E. Forest, and D. S. Robin. Explicit symplectic integrator fors-dependent static magnetic field. Phys. Rev. E, 68:046502, Oct 2003.

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