linearity of the electron optics

1

Linearity of the Electron Optics

Alexey Burov

RR Talks

2

Examples of YAG images

In many cases, YAG images of e-beam had distinct triangular shape, pointing to a sextupole nonlinearity (Sasha, Lionel).

This nonlinearity has to be seen in differential orbit measurements.

The differential orbit measurements were done by Mary:• A5 & A6 correctors Oct 29 (not clean in CS) , Nov 6 (OK for CS);•S2 & S5 - Nov 1;•S3 – Nov 26; S4 – Nov 29.

Nonlinearity Coefficient

• For every mentioned corrector, 10 equidistant kicks were applied, from the highest possible negative, to the highest positive. For every corrector j, the BPM data were saved in P163 format.

• To see nonlinearity in these data, I prepared a MathCad file, which makes a linear fit for every given BPM i , and subtract this fit from the data

• For every BPM i, the nonlinearity is characterized by

• For every corrector, the measure of nonlinearity of its trajectories can be presented as

3

;)(max

)~

(max2

10, jiCSiL

jiCSij rr

Xdk

)LinearFit(~

jijiji XXX

)~

(min)~

(max~

1010 jijiji XXXd

m6.1/ BBrL

222jijiji yxr

),( jijiji yxX

4

Example for CXA05

1 0.5 0 0.50.3

0.2

0.1

0

0.1

0.2

Corrector Current, Amp

BP

M O

ffse

t, c

m

1 0.5 0 0.51

0.5

0

0.5

1


BP

M O

ffse

t, c

m

BXS05, cm BYS05, cm

1 0.5 0 0.50.3

0.2

0.1

0

0.1

0.2


X n

onlin

eari

ty, m

m

XBPM_Name "ipR:BXS05"

1 0.5 0 0.50.2

0.1

0

0.1

0.2

0.3


Y n

onlin

eari

ty, m

m

YBPM_Name "ipR:BYS05"

E-Cool Line

5

Results

6

Corr BPM K [μrad/mm2]

CXA5 XC4 12

CYA5 XC7 3.5

CXA6 XC4 14

CYA6 YC8 13

CXS2 YC3 4.4

CYS2 YC8 13

CXS3 YC8 2.5

CYS3 YC8 4.0

CXS4 YC3 1.6

CYS4 YC3 3.0

CXS5 XC7 1.0

CYS5 YC9 0.8

1. The numbers are not negligible. Currently, the angle is estimated ~ 100μrad on the axis.

2. Nonlinearities for below CS2 are small.

3. Also: nonlinearities for BS2, BS3 were always small (not shown here).

4. Max nonlinearity are always at XC4, YC8 ( ≈ π/2 of the Larmor phase).

5. Hypothesis: The main source of nonlinearity is located at the lens S3

Accuracy (CYS2 kicks)

7

subDirA "07nov011505,CYS02I kicks from -.4a to +0a\" subDirB "07nov011515,CYS02I kicks from +.1a to +.5a\"

0.4 0.2 0 0.2 0.4 0.62

1

0

1

2

3


Y B

PM O

ffse

t, m

m

YBPM_Name "ipR:BYC80"

0.4 0.2 0 0.2 0.4 0.60.05

0

0.05

0.1


X n

onlin

eari

ty, m

m

0.4 0.2 0 0.2 0.4 0.60.2

0.1

0

0.1

0.2

0.3

0.4


Y n

onli

neari

ty, m

m YBPM_Name "ipR:BYC80"

0.4 0.2 0 0.2 0.4 0.60.2

0.1

0

0.1

0.2

0.3


Y n

on

lin

eari

ty,

mm YBPM_Name "ipR:BYC30"

Error of CYS2 current (or is it MI perturbation?) is equivalent to 100-200 microns of the bpm error. And this case is not unique… How can we avoid this?

How the error was evolved downstream the line…

8

0.4 0.2 0 0.2 0.4 0.60.1

0

0.1

0.2


X n

onlin

earit

y, m

m


0.4 0.2 0 0.2 0.4 0.60.05

0

0.05

0.1


X n

onlin

earit

y, m

m


0.4 0.2 0 0.2 0.4 0.60.02

0

0.02

0.04

0.06

0.08


Y no

nlin

earit

y, m

m


0.4 0.2 0 0.2 0.4 0.60.1

0.05

0

0.05

0.1

Corrector Current, AmpX

non

linea

rity,

mm


0.4 0.2 0 0.2 0.4 0.60.1

0.05

0

0.05

0.1


Y n

onlin

earit

y, m

m


0.4 0.2 0 0.2 0.4 0.60.1

0

0.1

0.2

Corrector Current, AmpY

non

linea

rity,

mm


How to check the hypothesis

• To check the SS3 hypothesis, a local bump CYS2-CYS3-CYS4 can be done.

• Then, for several bump mults, bpm data can be taken and compared with the nonlinear parts of Nov 1 CYS2 measurements.

• Nov 1 CYS2 & CXS2 measurements should be redone, due to their high errors for some points.

9

Cylindrical Aberrations of Ideal Lens

• Focusing strength of a solenoid can be derived as:

• Assuming

• In local bumps, this nonlinearity has to be seen as a cubic parabola in a plane of the bump, and ~nothing in the other plane, as soon as the lens is optically thin.

• For

10

)1()(

21

4)( 2

04

2

2

22

2

rFrOdzB

dzBr

B

dzBrF

100 ,4/)/)tanh((1)/tanh(1)( FlarazlazBzB

)3/(1 al

rrF

)(

%5.0)3/(:cm30,cm4,cm4.1 22 alrrlar

Bump nonlinearity

• For a linear local 3-bump around a doublet

• A nonlinear offset in the bpm BS4 , for

11

+ -

F F

sCS2 CS3 CS4

BS4

srFrx 22/

%5.2yields,m2m5,G400 1 x/rFsB

BS3

SS3

Cx/yS2 Bump Data

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CYS2 bump

NameX "R:BXS04S" NameY "R:BYS04S"

2 1 0 1 23

2

1

0

1

2

Xtjj

Ytjj

Itjj


2 1 0 10.5

0

0.5

1

Lens current, A

next

bpm

off

set,

mm

Xtjj

Ytjj

Itjj

CXS2 bump

%3.4%3.6 y/rx/r %5.1%14 y/rx/r

Huge nonlinearity of SS3! 14%/2.5%=6 times of the ideal level

CS2 bumps show:

• Nonlinearities seen in these bumps:– None of them has symmetry of the solenoidal lens

– Y-bump is ~ 3 times more nonlinear as X-bump (as in P163, p.5)

– Absolute value of Y-bump nonlinearity is 5 times higher the ideal level (also agrees with the P163 data in p.5).

– Dependences of BS3(CS2) are all linear at the noise level.

– Thus, these bump data confirm my hypothesis about extremely nonlinear S3 lens

13

CS3 bump

14


20 10 0 10 201.5

1

0.5

0

0.5

Xtjj

Ytjj

Itjj

CYS3 bump


20 10 0 10 200.8

0.6

0.4

0.2

0

0.2

Xtjj

Ytjj

Itjj

CXS3 bump

%1%5.2 y/rx/r %3%5 y/rx/r

Nonlinearity of SS4 is 5%/2.5% = 2 times of the ideal level.

Lens T4

• The same way of measurements with the 3-bump around ST4 (Mary, 08Feb21_linearity measurement) yields non-linearity as 4 times of the ideal level.

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NameX "R:BXT05S" NameY "R:BYT05S"

15 10 5 0 5 100.3

0.2

0.1

0

0.1

0.2

0.3

offset at the lens, mm

nonl

inea

rity

at t

he n

ext b

pm, m

m

Xnljj

Ynljj

Itjj

max Xnl( ) min Xnl( )0.5 max It( ) min It( )( )

0.041

max Ynl( ) min Ynl( )1 max It( ) min It( )( )

0.024

max It( ) min It( )2

9.845

Conclusions

• Bump measurements give the same ratios of nonlinearities as P163.

• The hypothesis of extreme SS3 nonlinearity (6 times more than natural , 3 times more than SS4) is confirmed.

• Other measured lenses have following nonlinearities: – SS4: 2 times of the natural level

– ST4: 4 times of the natural level

• It is reasonable to explore the most linear part for the beam position in SS3 (~-5 mm), and to optimize optics, reducing the field in SS3.

16

linearity of the electron optics

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