Stato dei lavori

Ottimizzazione dei wiggler di DANE

Simona Bettoni

Outline Method to reduce the integrated octupole in the wiggler of DANE

Analysis tools at disposal:

→ Multipolar analysis: In (also vs x shift at the entrance)

→ Tracking: x (y) and x’ (y’) vs x (y) shift at the entrance (tools

Tosca+Matlab)

Shifted poles & cut poles models

Axis optimization

Analysis of the results:

→ Multipolar analysis

→ Tracking

→ Comparison with the experimental data at disposal

In the future

Other methods to reduce the integrated octupole

CURVED POLE

MOVING MAGNETIC AXIS

New method

Reduction of the octupole around the beam trajectory in the region of the poles

Compensation of the integrated octupole in each semiperiod

Proposed by Pantaleo

Multipolar expansion of the field with respect to the beam trajectory

1. Determination of the beam trajectory starting from the measured data

2. Fit of By between -3 cm and +3 cm by a 4º order polynomial in x

centered in xT(z) = xT

-1.5 -1 -0.5 0 0.5 1 1.5-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

z (m)

x (m

)

443

32

210 TTTTTy xxbxxbxxbxxbbxxB

xT +3 cm

xT -3 cm

Beam trajectory (xT)

)(zbb nn

The integrated multipoles in periodic magnets

44

33

2210 ''''' xbxbxbxbbxB TTTTT

yT

44

33

2210 xbxbxbxbbxB AAAAA

yA

Even

multipoles

→

Odd

multipoles

→

...34242

232322222121212 Tj

AjTj

AjTj

Ajj

Ajj

T xbcxbcxbcbcb

...33232

222221212222 Tj

AjTj

AjTj

Ajj

Ajj

T xbcxbcxbcbcb

In a displaced system of reference:

bAk → defined in the reference centered in OA (wiggler axis)

bTk → defined in the reference centered in OT

(beam trajectory)x’

y’

x

y

OA O T

xT

Left-right symmetry of the

magnet Multipoles change sign from a pole to

the next one Sum from a pole to the next one

,...2,1,0j

Txxx '

Method to reduce the integrated octupole: displacement of the magnetic field

WITHOUT POLE MODIFICATION

In each semiperiod the particle trajectory is always on one side with respect the magnetic axis

In each semiperiod the particle travels on both sides with respect to the magnetic axis

Opportunely choosing the B axis is in principle possible to make zero the integrated octupole in each semiperiod

WITH THE POLE MODIFICATION

...366443 TA

TAT xbcxbcb Octupo

le

↑

Optimization of the pole of the wiggler

FC1-like FC2-like

Goals Reduce as less as possible the magnetic

field in the gap

Maintain the left-right symmetry

-9

-7

-5

-3

-1

1

3

5

7

9

-112 -96 -80 -64 -48 -32 -16 0 16 32 48 64 80 96 112

z (cm)

x (c

m)

Positron trajectory

By (T)FC 2FC 1

Analysis

For each z fit of By vs x in the system of reference perpendicular to the beam trajectory

-1.5 -1 -0.5 0 0.5 1 1.5-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

z (m)

x (m

)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

z (m)

x (m

)

Cut poles model: analysis perpendicular to s

I3 calculated over the entire wiggler varies of more than a factor 2 if the

analysis is performed perpendicular to s and not to z!

Cut poles model

-200

-150

-100

-50

0

50

100

150

200

250

300

-0.16 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0.16

z (m)

b3 (T

/m̂3)

Perpendicular to z

Perpendicular to s (Matlab)

Perpendicular to s (Tosca)

IFC = 693 A

Sector poles wiggler

Cut the poles in z to have sector poles

I3 calculated over the entire wiggler perpendicular to z is 9.09 T/m3 with respect to 4.13 T/m3 of the analysis perpendicular to z

-200

-150

-100

-50

0

50

100

150

200

250

300

-0.16 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0.16

z (m)

b3 (T

/m̂3)

Alfa = 0

Straight poles

Sector poles

IFC = 693 A

Shifted poles solution

$ and field roll-off

Shifted poles model

For the moment shifted the coils with the poles

Cut-shifted poles: the comparison of the field (at the same current = 550 A in FC)

SHIFTED POLES

CUT POLES

Cut-shifted poles: the comparison of the field (at the same current = 550 A in FC)

0.8

1

1.2

1.4

1.6

1.8

2

-0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07x (m)

By

(T)

Cut poles (550 A)Cut poles (693 A)Shifted poles (550 A)

With the shifted poles solution, the field roll-off is improved, therefore the shims can be eliminated maintaining more or less the same dependence of the solution on the x-shift at the entrance.

Shim thick in cut poles solution = 1.15 mm x 2 = 2.3 mm/37 mm = 6 % gap

By(z = 0, x = 0)SHIFTED POLES = By(z = 0, x = 0)CUT POLES+7.6%

Trajectory optimization

Determined the best value of the current in HC to minimize the integral of By over z

430

415

460

452

429

y = -158.43x + 67936

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

400 410 420 430 440 450 460 470

IHC (A)

Inte

gral

of B

y dz

(G

.cm

)

Trajectory optimization

Exit angle = 8 x 10-2 mrad

x-shift exit-entrance = 0.13 mm

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-1.2 -0.7 -0.2 0.3 0.8

z (m)

x (m

)

By integrated over z = 2 G.m

Tools analysis: multipoles with Tosca & Matlab

TOSCA

1. Determination of the best beam trajectory (tracking Tosca)

2. For each z found By in the points on a line of ±3 cm around (xTR, 0, zTR,) and perpendicular to the trajectory

3. Fit of the By at each point of the line (Tosca) at steps of 1 mm (fit Matlab)

MATLAB

1. Determination of the best beam trajectory (tracking Tosca/0 the integral of By over z)

2. For each found z points on a line of ±3 cm around (xTR, 0, zTR,) and perpendicular to the trajectory

3. Fit of the By at each point of the line at steps of 1 mm interpolated by Matlab

Tools analysis: tracking

1. Beam enters at several x

2. Tosca tracks the trajectory of each beam

3. Calculated the x exit-xTR NOM and x’exit in function of the x-shift at the entrance

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-10

-8

-6

-4

-2

0

2

4

6

8x 10

-3

x enter (m)

angl

e ex

it (r

ad)

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

x enter (m)

x ex

it (m

)

The curves are only to show the tool

-400

-300

-200

-100

0

100

200

-0.16 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0.16

z (m)

b3

(T/m

3)

Axis 1.15 cmAxis 1 cmAxis 0.85 cmAxis 0.5 cm

Axis optimization

For the moment used these codes to optimize the position of the axis

Multipoles

Presence of spikes in my analysis

Multipoles

Beam trajectory at fixed z and parabolic interpolation in z

-150

-100

-50

0

50

100

150

-1.2 -0.7 -0.2 0.3 0.8

z (m)

b3 (T/m

3)Miro passo 2 mm

Miro passo 1 cm

Spikes

Spikes: solved problem

-150

-125

-100

-75

-50

-25

0

25

50

75

100

125

150

175

-1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9 -0.85 -0.8 -0.75

z (m)

b3 (T

/m3)

Miro passo 2 mm

Miro passo 1 cm

Mio con passo 2 mm

Integrato

-150

-125

-100

-75

-50

-25

0

25

50

75

100

125

150

175

-1.05 -1 -0.95 -0.9 -0.85 -0.8 -0.75

z (m)

b3 (T

/m3)

Miro passo 2 mm

Miro passo 1 cm

Mio con passo 2 mm

Integrato

Axis optimization

Minimized I3 calculated in the entire wiggler

y = -294.07x + 216.19

-150

-100

-50

0

50

100

0.2 0.4 0.6 0.8 1 1.2

Posizione asse (-) (cm)

I3 s

u T

UT

TO

il

wig

gle

r (T

/m2)

0.73 cm

Multipolar analysis: to summarize

Multipolar analysis (entire wiggler)I0 (T.m) I1 (T) I2 (T/m) I3 (T/m2) I4 (T/m3)

Tosca (2 mm step)

-1.17E-04 2.09 -1.13 0.13 87.8

Tosca (1 cm step)

4.6E-05 2.10 -1.25 -0.98 211

Miro (2 mm step)

1.87E-04 2.09 -1.13 -1.01 95.0

Miro (1 cm step)

1.08E-04 2.08 -1.14 -1.32 101

To do the first optimization I used this technique

Analysis of the results: tracking (±3 cm)

Beam enters from x = xTR NOM-3 cm to x = xTR NOM+3 cm at steps of 1 mm, where xTR NOM is the position of entrance of the nominal trajectory

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

z (m)

x (m

)

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

x enter (m)

x ex

it (m

)

y = - 0.76*x2 + 0.93*x - 8e-005

y = 13*x3 - 0.76*x2 + 0.92*x - 8e-005

y = 53*x4 + 13*x3 - 0.8*x2 + 0.92*x - 7.6e-005

y = 1.5e+004*x5 + 53*x4 - 2.7*x3 - 0.8*x2 + 0.92*x - 7.6e-005

data 1

quadratic

cubic 4th degree

5th degree

Analysis of the results: tracking: the x exit (±3 cm)

The fit is satisfactory already for the 3rth-4rth order

Coefficient of the 3rd order term = 13 m-2

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3

x enter (m)

exit

angl

e (r

ad)

y = 10*x3 - 0.64*x2 - 0.065*x - 2.4e-005

y = 19*x4 + 10*x3 - 0.65*x2 - 0.065*x - 2.3e-005

y = 1.2e+004*x5 + 19*x4 - 2.3*x3 - 0.65*x2 - 0.062*x - 2.3e-005

data 1

cubic 4th degree

5th degree

Analysis of the results: tracking: the x’ exit (±3 cm)

The fit is satisfactory for the 3th-4th order

Coefficient of the 3rd order term = 10 rad/m3

Analysis of the results: comparison with the experimental data

I could compare the results only with the results of the experimental map at about 700 A

Ho riscalato curva di Miro x_exit = x_exitMIRO-x_exitMIRO(xENTR = 0)

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05

x-x0 (m)

Exi

t x (m

)Experimental (Miro)

Asse073CorrHC430

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05

x-x0 (m)

Exi

t x (m

)Experimental (Miro)Asse073CorrHC430Asse075CorrHC430

Analysis of the results: comparison with the experimental data

I could compare the results only with the results of the experimental map at about 700 A

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05

x-x0 (m)

Exi

t ang

le (ra

d)Experimental (Miro)

Asse073CorrHC430

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05

x-x0 (m)

Exi

t ang

le (ra

d)Experimental (Miro)Asse073CorrHC430Asse075CorrHC430

Analysis of the results: tracking: the y exit

-6.00E-04

-4.00E-04

-2.00E-04

0.00E+00

2.00E-04

4.00E-04

6.00E-04

-3.E-03 -2.E-03 -1.E-03 0.E+00 1.E-03 2.E-03 3.E-03

Entrance y (m)

Exi

t y (m

)

x = x_nom-0.01

x = x_nom

x = x_nom+0.01

Analysis of the results: tracking: the y’ exit

-3.00E-03

-2.00E-03

-1.00E-03

0.00E+00

1.00E-03

2.00E-03

3.00E-03

-3.E-03 -2.E-03 -1.E-03 0.E+00 1.E-03 2.E-03 3.E-03

Entrance y (m)

Exi

t y' (

m)

x = x_nom-0.01

x = x_nom

x = x_nom+0.01

Conclusions

Shifted poles - cut poles solution comparison:

The field roll-off is improved no shim increased BPEAK

Cheaper

At present:

improved the linearity zone of x and x’ with respect to the field map at

dipsosal

In the future:

Shifted poles solution analysis:

Analysis of the field maps by Dragt, Mitchell and Venturini (the map considered the

best one by us, one with the poles more centered and one with the poles more

shifted)

Measurement of the field map of the wiggler at I = 550 A to have a real

comparison with the results of the simulation (at LNF, at ENEA?)

Di scorta…

Situation in the present configuration (I = 693 A): x exit

y = 727.49x4 + 307.4x3 + 6.1081x2 + 1.0132x + 0.0175

y = 2E+06x6 + 357542x5 + 3602x4 - 290.86x3 + 1.028x2 + 1.2127x + 0.0182

y = 316785x5 + 7063.2x4 - 231.14x3 - 0.7598x2 + 1.1965x + 0.0183

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04

x-x0 (m)

Exi

t x

(m)

Sperimentali (conti Miro)

Poly. (Sperimentali (conti Miro))

Poly. (Sperimentali (conti Miro))

Poly. (Sperimentali (conti Miro))

The fit is satisfactory for the 5rth-6rth order

│Coefficient of the 3rd order term │ >200 m-2

Situation in the present configuration (I = 693 A): x’ exit

The fit is satisfactory for the 6rth order

│Coefficient of the 3rd order term │ ~600 rad/m3

y = 7E+06x6 + 669739x5 + 2028.2x4 - 592.4x3 - 1.6938x2 + 0.2157x + 0.0004

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04

x-x0 (m)

Exi

t ang

le (ra

d)

Experimental (Miro)

Poly. (Experimental (Miro))

Trajectory optimization

To determine the best value of I in HC for the several axis displacements

0.75

0

1.15

y = 52.323x + 390.2

380

390

400

410

420

430

440

450

460

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Axis position with respect to the geometric axis (m)

Opt

imiz

ed I in

HC (A)

Fine!