The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

Absolute Calibration of SR Interferometer by theMethod of Virtual Beam Broadening
J.W. Flanagan, T. Mitsuhashi, S. Hiramatsu, KEK, Tsukuba, Ibaraki 305-0801 Japan

Abstract
A method of performing absolute calibration of anSR interferometer has been developed which uses aweighted superposition of beam interferograms obtained over a range of beam positions at the SR source
point. This method permits the absolute calibrationof magniﬁcation factors in an SR interferometer beamsize measurement system using the precision of theBPM calibrations. Principles and preliminary test results from the KEKB LER are presented.

INTRODUCTION
At KEKB, synchrotron radiation (SR) interferometers are used to measure the transverse beam sizes inboth the Low Energy Ring (LER) and the High Energy Ring (HER) [1]. Deformations of the surfaces in
the optical path between the beam and the interferometer slits introduce magniﬁcation errors which need tobe corrected in order to measure the proper beam size.
In particular, the beryllium extraction mirror surface
curvature changes as a function of beam current. A
means of measuring and correcting for this distortionin real-time has been developed and put into use [2][3],but an independent method of checking the absolutecalibration of the SR interferometer system is needed
to verify the accuracy of this correction.
One method of carrying out this calibration is to use
beam size measurements taken at two diﬀerent beam
sizes, and solving for the calibration factor. If a beamof initial size σ0 is smeared over a gaussian distribution of locations of size σsmear to make a broadened
beam size σb , then the broadened beam size can berepresented by adding the initial beam size and thesmearing size in quadrature:σb =2σ02 + σsmear.

If it is assumed that the reported beam size from theinterferometer diﬀers from the true beam size by a factor A, due to magniﬁcation introduced into the system
by mirror distortions, then σ0 = Aσ0 , and σb = Aσb ,where σ0 is the (uncorrected) reported beam size fromthe interferometer for the initial beam size and σb isthat for the broadened beam size. Inserting these into
the above equations gives:

1σb =A

1σ0A

22+ σsmear.

From this, the calibration factor A, which is the ratio between the reported beam size and the true beam
size becomes:√A=σb 2 − σ0 2.
σsmear

One method of creating a smearing of the beam thathas been employed is to increase the beam emittanceby a known amount by using a dispersion bump [4].
Using this method, σsmear is calculated from the machine optics model, and thus has some model dependency in it. For the purposes of calibrating themonitor itself, it is desirable to have a method which
does not depend on the optics model. Such a modelindependent method of generating a precise σsmear isdescribed below.

MODEL INDEPENDENT METHOD:
VIRTUAL BROADENING
Instead of broadening the beam itself, we consider
the creation of a “virtually broadened” beam, created by taking the superposition of multiple interferograms, each one taken with the orbit of the beam atthe SR source point shifted by a diﬀerent transverseoﬀset. The justiﬁcation for this comes directly fromthe van Cittert-Zernike theory upon which the operation of the interferometer for beam-size measurement
is based. The visibility of the interference fringes froma ﬁnite source as a function of slit separation is givenby the Fourier transform of the source distribution; theinterference pattern observed can be treated as the superposition of many interference patterns (of visibility1) from a distribution of point sources. The virtually broadened beam interferogram is similarly constructed from the superposition of a distribution of(ﬁnite-sized) sources.
Explicitly, we move the beam at the SR source point
via transverse local bumps to a series of oﬀset locations
around the normal, unperturbed orbit. At each bumpheight (oﬀset from the normal orbit) i, we record theinterferogram Ii (see Fig. 1). Then we create theThe 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

Slits

BeamLens

Interference Pattern
I1

yy2
1

I2

Figure 1: Beam positions and interference patterns at two diﬀerent bump heights.

Ib =nNi Ii

i=1

By selecting the appropriate weighting factors Ni wecan set the virtual smearing width σsmear to be any
arbitrary size. We set the Ni as follows:

Ni = e−(yi −y0 )222σsmear,where yi − y0 is the height of the ith bump. Notethat the precision of each bump height determination
is determined by the precision of the BPM relative
calibrations, which is 2 − 3µm at KEKB [3], or about
2 − 3% of the vertical beam size at the SR monitor
source positions.
By ﬁtting the resulting weighted average interferogram with the same method we use for usual beam-size
monitoring, we can determine σb , and hence A.

were made in 50-micron steps, from −600 microns to+600 microns around the regular orbit position. The
raw data are shown in Figures 4 through 5. Figure 4
shows a scatterplot of reported beam sizes for several
measurements at each bump height. In Figure 5, theeﬀective slit separations at the extraction mirror areplotted for the same data. This is a measure of themagniﬁcation of the beam image due to distortions of
the mirror surface [2]. In order to limit our calibration
data set to raw beam images that diﬀer systematically
only in transverse oﬀset, data taken at bump heights
below −175 microns and above 380 microns were cut.
The beam sizes and mirror magniﬁcations within theuncut regions are seen to be relatively ﬂat. Figure 6
shows the weighting function used to create the composite virtual interferogram from the individual interferograms.
Un-smeared beam interferogram
Intensity (arb)virtual interferogram Ib as the weighted sum of theinterferograms:

1.6
1.4
1.2
10.8
0.6
0.4
0.2
0-400

-200

0microns

200400SIMULATIONSPRELIMINARY RESULTS
Test data were taken on 16 May 2003, using theLER vertical interferometer. Vertical position bumps

Figure 2: Simulated interferogram of real, unbroadened beam (σy = 100µm).

Smeared beam interferogram
Intensity (arb)

We did software simulations of this method, using
the KEKB LER vertical interferometer spacings, using
10 bumps spaced 50µm apart, with 1 interferograms
taken at each bump height, and with σ0 = 100µm andσsmear = 75µm. (These parameters match those of
the real calibration data, described in the next section.) In this case, σb = 125µm in the ideal case. The
result of ﬁtting the simulated virtual interferogram,σb (fit) = 124.881µm, shows that the virtual beam
size has been created with an accuracy of ≈ 0.1%. The
simulated interferograms are shown in Figs. 2 and 3.

0.5
0.4
0.3
0.2
0.1
0-400-200

0microns

200400Figure 3: Simulated interferogram of virtually broadened beam (σsmear = 75µm). Note decreased visibility as a result of beam smearing, evident as a reduced
peak-vally depth in the fringes.

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

Weighting factor
10.8
dimensionlessFitting the composite weighted sum of the histograms with the same ﬁtting procedure used on theindividual interferograms gives the result, A = 1.04 ±0.03. Errors include statistical errors, plus eﬀects ofchanging light asymmetry between diﬀerent images( 1%), and orbit ﬂuctuations and BPM scatter whichlead to errors in the smearing width ( 1.3%). The correction of the interferometer system magniﬁcation isthus good to within the measurement accuracy of thecalibration data.

0.60.40.2
Measured beam size
1350130<-CUT-600

CUT->-400-2000200
400
Bump height (microns)

600

125

Figure 6: Weighting factors used at each bump height.

microns

120115
110
105
10095
-600

-400-2000200
400
Bump height (microns)

600

Figure 4: Bump height versus raw beam size measurements.
Effective slit separation at extraction mirror
8.8mm

8.6

<-CUT

8.4

8.2

is as follows: The width of the smearing functionused, σsmeary , is 75µm. The corresponding momen√
2tum space width, σsmeary = 1+ασsmeary , is about
β
6.7µrad. At the wavelength of 500 nm, the verticalangular divergence of the SR beam, σψ , is 1.3 mrad.
Since σsmeary  σψ , any deviation from symmetry
in the light balance between two slits is completely
negligible.

ACKNOWLEDGMENTS
The authors wish to thank N. Iida and Y. Funakoshi
for valuable discussions, particularly on their dispersion bump (“iBump”) method for cross-calibration between the interferometer and the optics model. While
the principle of using measurements at two diﬀerent
beam sizes had been used for cross-calibrating thegated camera from the interferometer [6], they suggested its applicability to the calibration of the interferometer itself. The authors also wish to thank Prof.
Kamada for discussions on the phase-space issue.

CUT->

REFERENCES

8

7.8
-600

-400

-200
0200
400
Bump height (microns)

600

Figure 5: Bump height versus apparent slit separationon the extraction mirror surface, which is proportional
to the magniﬁcation factor.
It has been pointed out that smearing in the beam
position axis neglects the smearing in the transversemomentum axis that would occur if a beam werebroadened physically by increasing the emittance [5].
To fully account for this eﬀect, a two-dimensional
bump scan in, and superposition over, both position
and momentum axes would be needed. However, asimple way to evaluate the magnitude of the eﬀect[1] T. Mitsuhashi, J.W. Flanagan, S. Hiramatsu, “OpticalDiagnostic System for the KEK B-Factory,” Proceedings of EPAC 2000, Vienna, Austria (2000) p. 1783.
[2] J.W. Flanagan, S. Hiramatsu, H. Ikeda, T. Mitsuhashi,“Correction of Mirror Distortion Eﬀects in Beam-SizeMeasurement Using SR Interferometer,” Proc. 13thSymposium on Accelerator Science and Technology,Osaka, Japan (2001), p. 381.
[3] M. Arinaga, et al., “KEKB Beam Instrumentation Systems,” NIM A 499 (2003) pp. 100-137.
[4] N. Iida, Y. Funakoshi, private communication.
[5] S. Kamada, private communication.
[6] J.W. Flanagan, et al., “High-Speed Gated Camera Observations of Transverse Beam Size along Bunch Trainat the KEKB B-Factory,” Proceedings of EPAC 2000,Vienna, Austria (2000) p. 1119.