Undulator Models and theUse of Long Straight Sectionsin the SPring-8 Storage Ring

K.Soutome, H.Tanaka, M.Takao, J.Schimizu, H.YoneharaAccelerator Group, SPring-8 (Japan)

Contents

Motivation

Model of Insertion Devices

Use of 30m-LSS

Low-Beta Insertion for 10T SCW

Summary

ID Model• For simulation with small amplitudes

=> Halbach-type ID model will be enough.

• For simulation with large amplitudes, e.g. for simulation ofbeam injection

=> ID model with nonlinear fields adequate in a wide range ofaperture is needed as an input of simulation code.

=> Also useful in analysis of some special (complicated) IDs

• SPring-8 storage ring has four 30m-LSS in addition to normalstraight sections. A long undulator is already installed and thereremain three sections for innovative light sources for future use.

• Independent local tuning of lattice functions is required.

=> Dynamic aperture and momentum acceptance must be keptlarge.

Use of 30m-LSS

Halbach-Type Model

Example:

Magnetic Field of ID15

by H.Tanaka

Amplitude of injected beam is about 10mm. Halbach-type

model fits only locally and we need to extend the model.

cf. E.Forest and K.Ohmi, KEK Report 92-14

By = Acos(kx)

on median plane (y=0, z=z0)

0.8

0.81

0.82

0.83

0.84

0.85

0.86

-15 -10 -5 0 5 10 15

3D CalculationFit: 0.845 cos(0.0148 x)Fit: 0.854 cos(0.0021 x)

By [

T]

x [mm]

Our Method• Put multipole thin lens at both ends of ID.

• Prepare 3D magnetic field data on appropriate mesh points.

(The field data is generated by our code or given as an input

file and interpolated.)

• Trace the elctron trajectory from the entrance to the exit of ID

by solving the equation of motion.

(We used the Runge-Kutta of 4th order with 10mm step.)

• Change the initial position and repeat to get mapping data.

• Fit the strengths of multipoles so that (x, x’, y, y’)exit is

reproduced.

We checked that the results are almost independ of the number

of thin lens positions. So we put multipole thin lens at the

entrance and exit of ID.

Example: ID17

Multi-Operation Mode Undulator: Figure-8 Mode H / V Polarization

Helical Mode Circ. Polarization Asymmetric Figure-8 Mode

Fast Helicity Switching (>10Hz)

by K.Shirasawa ID17 is now under tuning!

Magnetic Field of ID17

For 3D field calculations we used:

(1) our own code for IDs w/o iron yoke

(2) RADIA developed at the ESRF

by P.Elleaume, et.al.

⇒ Calculate for 3 cells and repeat the

data in the middle.

(example field data along off-axis at x=4mm, y=2mm)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 1000 2000 3000 4000

Bx

[T]

z [mm]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 1000 2000 3000 4000

ID17 Figure-8 Mode (Phase127mm/100A)Calculation by RADIA Measured with Hall Probe

By

[T]

z [mm]

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 1000 2000 3000 4000

Bz

[T]

z [mm]

by K.Shirasawa

Example of Multipole Fitting

Fitting with up to 16-poles

Agreement is good but not perfect.

=> ... to be discussed later

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8

tracefit

∆y[

mm

]

∆x[mm]

-0.1

-0.05

0

0.05

0.1

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

tracefit

∆y'

[mra

d]

∆x'[mrad]

-6

-4

-2

0

2

4

6

-10 -5 0 5 10

Multipole Fitting up to 16-Poles

trace fit

x[mm]

ID17(mod), Phase 127mm, +100A

Example of Multipole Fitting (cont.)

Fitting with up to 10-poles will be enough in this case.

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14 16

ID17(mod), Phase 127mm, +100A

dx[mm]dx'[mrad]dy[mm]dy'[mrad]

Order of Multipole

Multipole Fitting

Strength of MultipolesPM Only (w/o Yoke)

• Quadrupole and decapole components increase by iron yoke.

• Sextupole component increases by the electromagnets.

• Tune shift is about 0.02 and beta distortion is about 15%.

=> Correction by quadrupole (and sextupole) magnets are planned.

N4 -0.00011116 -0.0018663 0.0037541 -0.0052573S4 9.9597e-06 0.00015046 4.4570e-05 -4.6742e-05N6 0.00010288 0.0087676 0.51206 -0.50293S6 -0.42487 0.29724 0.29036 0.28689N8 -0.074613 2.9241 -5.3136 4.7213S8 -0.024056 -1.1979 0.12922 -1.7631N10 -1.9483 -66.899 -2250.7 2392.3S10 -195.82 392.23 238.69 191.85

Entrance

Exit N4 -0.00010579 -0.0017337 0.0037167 -0.0054727S4 -4.1264e-06 -6.5888e-05 -0.00012031 0.00019378N6 0.00016917 0.0071736 -0.41479 0.41441S6 0.42489 0.29397 0.28957 0.28844N8 -0.10868 1.6554 -5.5162 5.8290S8 -0.024460 -0.25619 0.12395 -1.3835N10 2.5984 -60.071 1581.5 -1283.7S10 200.36 343.45 197.75 311.90

“Default” I=+100A I=-100A

Simulation of Dynamic Aperture

0

2

4

6

8

10

-25 -20 -15 -10 -5 0 5 10 15

IdealLOMC-error

ID17 with 6polesID17 with 16poles

y(m

m)

x(mm)

+100A@Phase=127mm

0

2

4

6

8

10

-25 -20 -15 -10 -5 0 5 10 15

IdealLOMC-error

ID17 with 6polesID17 with 16poles

y(m

m)

x(mm)

-100A@Phase=127mm

Injection efficiency etc. will be affected.

Effects on Beam Lifetime

ID17 is of out-vacuum type and the above results indicate that byclosing the gap the momentum acceptance is reduced due to nonlinearfields or H-V coupling changed.

Lifetime determined only by the RF voltage

Normalize dataat low voltage.

by M.Takao

0

5

10

15

20

25

30

10 11 12 13 14 15 16

04/07/16 ID17 close04/07/16 ID17 open04/12/25 ID17 close04/12/25 ID17 open

RF Voltage [MV]

0

0.02

0.04

0.06

0.08

0.1

0.12

10 11 12 13 14 15 16

04/07/16 ID17 close04/07/16 ID17 open04/12/25 ID17 close04/12/25 ID17 open

RF Voltage [MV]

open

close

Improvement of the ModelUse polynomials instead of multipoles.

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8

tracefit

∆y[

mm

]

∆x[mm]

-0.1

-0.05

0

0.05

0.1

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

tracefit

∆y'

[mra

d]

∆x'[mrad]

Multipole Fitting Polynomial Fitting

-0.1

-0.05

0

0.05

0.1

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

tracefit

∆y'

[mra

d]∆x'[mrad]

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8

tracefit

∆y[

mm

]

∆x[mm]

SX: a6x2 + b6xy + c6y2e.g. SX: a6(x2- y2) + b6xy

Dependence on the Order of Polynomial

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8

Up to 4-Poles

tracefit

∆y[

mm

]

∆x[mm]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8

Up to 6-Poles

tracefit

∆y[

mm

]

∆x[mm]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8

Up to 8-Poles

tracefit

∆y[

mm

]

∆x[mm]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8

Up to 10-Poles

tracefit

∆y[

mm

]

∆x[mm]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8

Up to 12-Poles

tracefit

∆y[

mm

]

∆x[mm]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.2 0 0.2 0.4 0.6 0.8

Up to 16-Poles

tracefit

∆y[

mm

]

∆x[mm]

Dependence on the Order of Polynomial (cont.)

Polynomial fitting is better and the results can be used as an

input of a simulation code, though they do not represent real

fields of multipole correctors.

=> We are planning to use the new scheme in our simulation code.

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14 16

ID17(mod), Phase 127mm, +100A

dx[mm]dx'[mrad]dy[mm]dy'[mrad]

Order of Polynomial

Polynomial Fitting

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14 16

ID17(mod), Phase 127mm, +100A

dx[mm]dx'[mrad]dy[mm]dy'[mrad]

Order of Multipole

Multipole Fitting

Use of 30m-LSS

SPring-8 will come to the next phase of using “insertion devices”:

There remain three sections of 30m-long straight sections for

innovative light sources.

For the most efficient use of these sections independent local

tuning of lattice functions (beta, phase, dispersion) is required.

=> Symmetry of the ring is lowered.

=> Dynamic aperture and momentum acceptance become small.

How do we manage?

Possible Solution

(1) Keep Betatron Phase Matching (for on-momentum particle)

(2) Make Local Chromaticity Correction (for off-momentum particle)

(3) Add Counter-Sextupole (for cancellation of nonlinear kick)

0

1 0

2 0

3 0

4 0

5 0

6 0

0

0 . 5

1

1 . 5

2

2 . 5

3

0 5 0 100 150

Bet

atro

n F

un

ctio

n β

[m

]D

ispersion

F

un

ction η

[m]

Path Length [m]

βx

βy

ηx

Matching Section

∆νx=4π, ∆νy=2π, Sextupoles are weakly excited.

Present Condition of Matching Section

For independent local tuning of lattice functions:

10T SCW as a Test Case

Dipole Field

xmax = 7.3mm, x’max = ± 25mrad

Orbit at 10T

Basic ParametersNumber of Poles: 3S/C Wire: Nb3Sn and NbTiMaximum Field: 10TStored Energy at 10T: 400kJWeight: 1000kgMagnet Length: 1mPole Gap: 42mmBeam Chamber: 65mm(H), 20mm(V)

-2

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1 1.2

Exp.Cal.

Length [m]

-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1 1.2

x

x'

Length [m]

10T SCW as a Test Case (cont.)

Sextupole Field: 45T/m (focusing)Quadrupole Field: 0.50T (defocusing)

Fabricated at Budker INP by N.Mezentsev, et.al.

cf. M.Fedurin, et.al. NIM A448(2000)51, A470(2001)34.

-1

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1 1.2

Length [m]

-600

-400

-200

0

200

0 0.2 0.4 0.6 0.8 1 1.2

Length [m]

Emittance and SCW FieldAchromat Optics Non-Achromat Optics

Minimum shifts due to self-dispersion.

To keep the emittance small even at 10T

• Horizontal beta must be small.

• Horizontal dispersion must be vanished

or it must be controlled to cancel self-

dispersion by the wiggler.

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 2 4 6 8 10

β* = 25m20m15m10m5.0m4.0m3.0m2.5m2.0m1.5m0.7m

BSCW

* [T]

1

1.2

1.4

1.6

1.8

2

2.2

2.4

-0.1 -0.05 0 0.05 0.1

10T

β* = 10.0m 5.0m 4.0m 3.0m 2.5m 2.0m 1.5m 1.0m 0.7m

η0 [m]

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 2 4 6 8 10

β* = 25m 20m 15m 10m 5.0m 4.0m 3.0m 2.5m 2.0m 1.5m 1.0m 0.7m

BSCW

* [T]

η0 = 0

Low-Beta Insertion at LSS (tentative)

Achromat Optics Non-Achromat Optics

before

after

βx=2.5mηx=0

0

10

20

30

40

50

60

0.0

0.10

0.20

0.30

0.40

0.50

0 30 60 90

β [m

] η [m

]

s [m]

βy

βx

ηx

0

10

20

30

40

50

60

0.0

0.10

0.20

0.30

0.40

0.50

0 30 60 90

β [m

] η [m

]

s [m]

βy

βx

ηx

0

10

20

30

40

50

60

0.0

0.10

0.20

0.30

0.40

0.50

0 30 60 90

β [m

] η [m

]

s [m]

βy

βxη

x

0

10

20

30

40

50

60

0.0

0.10

0.20

0.30

0.40

0.50

0 30 60 90

β [m

] η [m

]

s [m]

βy β

x

ηx

Quadrupoles

Ring Parameters

ε εεδ ηβeff

ID

ID

= +22 2

Effective Emittance at Normal ID:

Achromat Optics Non-Achromat Optics

bef. aft. aft. bef. aft. aft. SCW 0T 10T SCW 0T 10T

βx[m] 23.5 2.5 - 21.7 2.5 -βy[m] 14.4 13.0 - 14.0 12.5 -ηx[m] 0 0 - 0.103 0 -σE/E 0.11 0.11 0.15 0.11 0.11 0.15ε[nmrad] 6.59 6.98 7.15 3.43 3.80 4.32εeff[nmrad] 6.59 6.98 7.15 3.71 4.08 4.84

Local Chromaticity CorrectionAchromat Optics (Half of the Matching Section)

before after

SFLOCAL

Horizontal chromaticity (or betatron phase jump for off-momentum particles)should be corrected locally in consideration of beam injection and beam lifetime.

0

10

20

30

40

50

60

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 10 20 30 40s[m]

ξx

ξy

βy

βx

ηx

0

10

20

30

40

50

60

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 10 20 30 40s[m]

ξx

ξyβ

y

βx

ηx

Counter-Sextupole

Similar to “noninterleaved sextupoles” scheme: K.L.Brown, IEEE Trans. Nucl. Sci. NS-26 (1979) 3490. L.Emery, in Proc. 1989 IEEE PAC (1989) p.1225. K.Oide and H.Koiso, PR E47(1993) 2010.

SXCOUNTER ∆νx = π

SFLOCAL

0

10

20

30

40

50

60

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40s[m]

νx

νy

βy

βx

ηx

Dynamic Aperture

After tuning the strength of SFLOCAL, SXCOUNTER and other harmonic sextupoles...

The work is still in progress to get a better solution ...

before (w/o Low-Beta Insertion)Achromat Optics

after (with Low-Beta Insertion)

We checked that momentum acceptance is enlarged by using counter-sextupoles.

with error fields

0

1

2

3

4

5

6

7

8

-20 -15 -10 -5 0 5 10

∆p/p = +1%∆p/p = 0∆p/p = -1%

x [mm]

0

1

2

3

4

5

6

7

8

-20 -15 -10 -5 0 5 10

∆p/p = +1%∆p/p = 0∆p/p = -1%

x [mm]

Summary• ID model we presented is valid for a wide range of

aperture. Obtained multipole/polynomial strengths can beused directly as an input of simulation code.

• We showed the effectiveness of this scheme by taking a“multi-operation mode undulator” ID17 as an example.

• For the most efficient use of LSSs independent local tuningof lattice functions (beta, phase, dispersion) is required.

• To keep the dynamic aperture and momentum acceptancewe proposed the following scheme: “Betatron PhaseMatching” & “Local Chromaticity Correction” &“Counter-Sextupole”.

• Beam test of lattice modification is planned using one LSS(w/o insertion devices).