Synchrotron Radiation Outputfrom the ILC Undulator

David Newton

University of Liverpool, Cockcroft Institute

Positron Workshop

Durham

28/10/2009

Synchrotron Radiation Output from the ILC Undulator – p.1/21

Outline• The analytic tracking code

• Extension of the code to calculate synchrotron radiation

• Characterising the helical undulators from a field map

• Characterising the helical undulators from prototype measurements

• Synchrotron emmission from the helical undulator

• Efficiency of the code

Synchrotron Radiation Output from the ILC Undulator – p.2/21

Analytic Tracking Code

• Characterise an arbrtrary magnetic field in terms of it’s multipole

expansion and generalised gradients to produce an analytical description

of field as a fuction of the longitudinal coordinatea

• Use the analytical expression in differential algebra or Lie algebra code to

generate a Taylor or Lie (symplectic) map for the dynamics inthe magnet.

• Evaluate the analytical expressions to perform a numericalintegration

giving a fast particle tracking code to describe the evolution of the

canonical coordinates within the magnet.

• The C++ code that has been has been written has a modular structure

which facilitates extending the code

• A Synchrotron Radiation Module is being implemented which calculates

the synchrotron emission from a particle into an arbitrary observation

point

• eg ILC Helical undulator

aVenturini and DragtSynchrotron Radiation Output from the ILC Undulator – p.3/21

Synchrotron Radiation Calculation

Accelerated charges radiate energy. The observed electricfield a of the

emitted radiation is:

~E(t) =e

4πǫ0c

(

c(1 − |~β|2)(~n − ~β)

|~R|2(1 − ~n~β)3+

~n × [(~n − ~β) × ~β]

|~R|(1 − ~n~β)3

)

RET

d2U

dΩdt= ǫ0cE

2r2 = ~G(t)2

~G(ω) =1√2π

∫

∞

−∞

~G(t) exp(ıωt)dt

d2I

dΩdω= | ~G(ω)|2 + | ~G(−ω)|2

aJackson

Synchrotron Radiation Output from the ILC Undulator – p.4/21

Implementation

At each step of the tracking code, the sychrotron variables,X , X, X

(X = x, y, z), are calculated by estimating the radius of curvature of the

particle (using the two adjacent integration steps)

These variables are saved to a file and subsequently used to calculate the

electric field at an arbitrary observation point.

At the end of the integration the differential intensity is found by performing aDFT on the electric field components.

Synchrotron Radiation Output from the ILC Undulator – p.5/21

Benchmarking

• Starting with a constant magnetic field (By = 1 T), model the field in

terms of it’s generalised gradients, produce analytical descriptions of the

field and numerically integrate the motion of the particle through the field.

• E = 100 MeV

• R=33.3333 cm• L=10 cm• 10,000 integration steps

• At each integration step calculate the electric field observed at an

observation point, perform a fourier transform on the field to produce the

frequency distribution and compare with the analytical description

Synchrotron Radiation Output from the ILC Undulator – p.6/21

Benchmarking the synchrotron calculation

/ Hzν1210 1310 1410 1510

ω d

Ωd,n

)ω

I(2 d

-3410

-3310

Analytical

Numerical

Accurate to∼ 10−4

Synchrotron Radiation Output from the ILC Undulator – p.7/21

Field Map of the Helical Undulator a

/ degreesφ0 50 100 150 200 250 300 350

z / m

-0.004

-0.002

0

0.002

0.004

-1

-0.5

0

0.5

1

Synchrotron Radiation Output from the ILC Undulator – p.8/21

Measurements of the prototype undulator a

z / m0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Bx

/ T

-1

-0.5

0

0.5

1

Magnet 2: Bx

z / m0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

By

/ T

-1

-0.5

0

0.5

1

Magnet 2: By

Synchrotron Radiation Output from the ILC Undulator – p.9/21

Particle Tracking Comparison

Tracking the dynamical variables x and y with a simulated field map and a

measured fieldmap.

z / m0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x / m

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

-610×x, measured

x, computed

z / m0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

y / m

0

1

2

3

4

5

6

-610×

y, measured

y, computed

Synchrotron Radiation Output from the ILC Undulator – p.10/21

Particle Tracking Comparison

Tracking the dynamical variables Px and Py with a simulated field map and a

measured fieldmap.

z / m0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Px

-5

-4

-3

-2

-1

0

1

2

3

4

-610×Px, measured

Px, computed

z / m0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Py

-4

-2

0

2

4

6

8

10

12

-610×Py, measured

Py, computed

Synchrotron Radiation Output from the ILC Undulator – p.11/21

Electric Field

The electric field (Ex) observed on axis, 10 m from the end of the measured

undulator fieldmap.

t / ns0 1 2 3 4 5 6

-910×

-1E

x / V

m

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

610×

t / ns1 1.2 1.4 1.6 1.8

-910×-1

Ex

/ Vm

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

610×

Synchrotron Radiation Output from the ILC Undulator – p.12/21

Synchrotron Energy Spectrum Comparison

The energy spectrum, calculated on axis, 10 m from the end of the simulated

undulator field map (right) and the measured field map (left)

Energy / MeV0 5 10 15 20 25 30 35

0

10

20

30

40

50

60

70

80

90-2410×

ω dΩd,n)ωI(2d

Energy / MeV0 5 10 15 20 25 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-2110×

ω dΩd,n)ωI(2d

Synchrotron Radiation Output from the ILC Undulator – p.13/21

Synchrotron Energy Spectrum (simulated field map)

Energy / MeV0 5 10 15 20 25 30

-3110

-3010

-2910

-2810

-2710

-2610

-2510

-2410

-2310

-2210

-2110

-2010 ω dΩd,n)ωI(2d

Synchrotron Radiation Output from the ILC Undulator – p.14/21

Energy emitted at the end of the undulator

Energy emitted into 1689 points at the end of the undulator

Problem with scaling (ignore the values)

x / m-0.03 -0.02 -0.01 0 0.01 0.02 0.03

-310×

y / m

-0.03

-0.02

-0.01

0

0.01

0.02

0.03-310×

-1110

-1010

-910

Synchrotron Radiation Output from the ILC Undulator – p.15/21

Undulator 6 cell latticeGAP2 0.19 DRIFUND 1.79 DRIFGAP3 0.16 DRIFUND 1.79 DRIFGAP2 0.19 DRIFGAP4 0.2 DRIFGAP2 0.19 DRIFUND 1.79 DRIFGAP3 0.16 DRIFUND 1.79 DRIFGAP2 0.19 DRIFGAP4 0.2 DRIFGAP2 0.19 DRIFUND 1.79 DRIFGAP3 0.16 DRIFUND 1.79 DRIFGAP2 0.19 DRIFGAP1 0.65 DRIFQDU 0.25 QUADQDU 0.25 QUAD

GAP1 0.65 DRIF

Synchrotron Radiation Output from the ILC Undulator – p.16/21

Tracking through the lattice

Evolution of the position and momenta variables in the lattice

(note the systematic error sums)

z / m0 2 4 6 8 10 12 14

x,y

/ m

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-310×

x, measured

y, computed

z / m0 2 4 6 8 10 12 14

Px,

Py

0

0.01

0.02

0.03

0.04

0.05

0.06

-310×

Px, measured

Py, computed

Synchrotron Radiation Output from the ILC Undulator – p.17/21

Electric fields and energy spectrum from the lattice

The electric field, Ex, and the energy spectrum calculated on-axis, at the end

of the final undulator

t / ns0 2 4 6 8 10 12 14 16 18 20 22

-910×

-1E

/ V

m

0

200

400

600

800

1000

1200

14001210×

Energy / MeV0 10 20 30 40 50

-2110

-2010

-1910

-1810

-1710

-1610

-1510

-1410

-1310

-1210

-1110

-1010

-910

-810

-710

-610 ω dΩd,n)ωI(2d

Synchrotron Radiation Output from the ILC Undulator – p.18/21

Power output at the lattice end

Power output at the end of the final undulator in the lattice (1,965 points)

x / m-0.001 0 0.001

y / m

-0.001

0

0.001

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-310×

Synchrotron Radiation Output from the ILC Undulator – p.19/21

Tracking through the lattice

Evolution of the position and momenta variables in the lattice

(note the systematic error cancels)

z / m0 2 4 6 8 10 12 14

x,y

/ m

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06-310×

x, measured

y, computed

z / m0 2 4 6 8 10 12 14

Px,

Py

-5

0

5

10

15

-610×Px, measured

Py, computed

Synchrotron Radiation Output from the ILC Undulator – p.20/21

Code Efficiency

• Analytic tracking code has been heavily optimised for speed

• Synchrotron extension has not been optimised at all so far

• Typical run times (2.66 GHz processor, 4Mb RAM)• Calculate analytic transfer matrices (100,000 steps per undulator) 1 hr

0” 9’

• Numerical evaluation of the matrices for a particular trajectory and

calculation of the synchrotron variables 9’ 11”• Calculating fields and power absorbed at an observation point ∼ 3”• Field interpolation and FFT - not benchmarked (∼ O seconds)

• Comparisons with other codea

• SPUR: (2m field map, 1225 points, 45 processors, (integration steps ???) 4

hrs

• SPECTRA: (2m field map, accuracy level 1 (???)) 68 hrs

• This work: (10,000 integration steps, 2500 points) 2 hrs 5’

aSynchrotron Radiation Output from the ILC Undulator – p.21/21