The 13th Symposium onAccelerator Science and Technology, Suita, Osaka, Japan, October 2001

CORRECTION OF MIRROR DISTORTION EFFECTS IN BEAM-SIZE
MEASUREMENT USING SR INTERFEROMETER
J.W. Flanagan, 8. Hiramatsu, H. Ikeda, T. Mitsuhashi, KEK, Tsukuba, Ibaraki, Japan

Interferometer
Slits

Abstract
The effective use of the synchrotron radiation interferometer to measure beam sizes depends on theability to measure and correct for systematic effects
introduced by the distortion of the surface of the extraction mirror. At KEKB it has been observed thatchanging beam currents cause large changes in the apparent beam size due to the heating of the mirror surface. In this paper we discuss methods developed tomonitor and correct for this distortion in the operationof the SR interferometer.

Extraction
MirrorD

Mask Plane

1 INTRODUCTION
Beam sizes are measured at KEKB using SR interferometers, one for the 8 GeV High Energy Ring
(HER) and one for the 3.5 GeV Low Energy Ring
(LER).[1] At either ring, when the beam current increases the temperature of the beryllium extraction
mirror rises, causing a deformation of the mirror surface as the mirror block expands. The resulting curvature changes the propagation angles of the light rays
which pass the interferometer slits, causing a change
in the apparent beam size which can be quite severe(30%).
The situation is illustrated in Figure 1. The heatdeformed mirror surface is shown as a solid black line,
as are the light paths from the SR source point to theinterferometer slits. The ideal ( undeformed) mirror
surface and its corresponding light paths are shown
as dashed lines. Immediately downstream of the extraction mirror is a dotted line representing a planewhere the pinhole mask (discussed below) for measuring mirror distortion is located. The light rays whichreach the interferometer slits depart from the source atan angular divergence of B. The distance between thelight rays when they reach the slits (at a distance Lfrom the light source) is D; the corresponding distanceat the mask plane (at distance L!) is Dl. If the distancefrom the extraction mirror to the mask plane is muchsmaller than the distance from the source to the mask
plane, and for small angular divergences, () ~ Dl/ L1.
If the mirror is undeformed, then Dl = D 0 1, whereDol/ Ll = D / L = Bo. Thus, the equation giving beam
size abeam as a function of interference pattern visibility 'Y, assuming a Gaussian beam, can be expressed
asSR Source Point

Figure 1: Effect of mirror distortion on ray propagation.

G"beam=~nBo

J~ ~2ln'Y= ),LI1r

Dol

J~ ~,
2ln'Y

where >.. is the wavelength under observation.
However, in the case of a deformed mirror, () =J 80
and Dl/ L1 =J D / L. In this case, it is necessary to
determine D1 in order to make the correct estimate of
the beam size:
abeam= :()J~ ln ~ =;r;, J~ ~·ln

(1)

One method of determining Dl is to use a Hartmannpinhole mask.[2] By moving a pinhole mask across themask plane, the spots on the mask plane from whichlight rays propagate to the interferometer slit positions
can be easily located (they are the locations at which
one or the other slit is illuminated by the SR lightwhich has passed through the pinhole), and D1 can bemeasured. D1 as a function of beam current is shownin Figure 2.
However, this method is not applicable in normal
operation, since the interferometer cannot be used
when the mask is in place due to one slit being always
blocked. For this reason we have developed a method
for measuring and correcting for this distortion during real-time operation of the interferometer based on-381-

The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001hER vs effective slit separation at mask Dt

Synch Slitsr otronLens
L ightInterferencePatternI

6 r-~---,--~--~--~---r--•.....!J--A-I D~

54hDt(mm) 3

IV/:'··.

--(:•.

('F

Figure 3: Interference pattern formation.

2
1

0 ~~--~--~--~--~--~--~
200 300 400 500 600 700 800 900
LER beam current hER (rnA)
Figure 2: Change in Dt as a function of beam current
due to mirror heating. This corresponds to a change
in apparent beam size of ~ 30% as the LER beam
current goes from 200 to 900 rnA.

modifying the fitting procedure used to analyze theinterference pattern visible after the slits.

2 FITTING PROCEDURE
We fit the interference pattern to an equation of theform
y(x) =a+ bx +!!!:. :~:::>ix2.

•{A~+ A~+ 2A1A2'Y(~) 2 cos [,ec (~) (x- </>c)]},whereThis represents two single-slit diffraction terms ofamplitude A1 and A2 brought into overlap by the lensbehind the slits, modulated by a cosine double-slit
term with visibility 'Y determined by the spatial coherence of the SR light. The cos term has a period
determined by ,Be, with offset <l>c· The resulting pattern is of magnitude m, with a linear background terma+bx.

To account for bandpass effects which would otherwise artificially lower the visibility, we write the fitting function as a sum over a sample of wavelengths>.i passed by the bandpass filter with central frequency>.0 , each term weighted by the transmission ti at wavelength A£. The incident SR spectrum changes by only afew percent over the bandwidth used (10 nm, centeredaround 500 nm), and is taken as flat. The exponent
appearing on 'Y represents the variation of visibilityobserved at different wavelengths relative to that observed at the central wavelength, assuming a Gaussian
beam with the same apparent beam size at all wavelengths (i.e., using Equation 1).
In A1 and A2, a1 represents the asymmetry ~~+~~of the light intensities It and !2 impinging on the twoslits. ,Bs (1 ±as) represent the widths of the singleslit sine terms from each slit, with as representing the(usually very small) asymmetry between the widthsof the patterns from each slit. <l>s ± Ef represent thecenters of the single-slit terms, separated by 08 fromeach other.
Changes in 08 are correlated with deformations inthe mirror. The reason for this is that as Dt after themirror changes, the angle between the rays reaching
the slits changes. This in turn causes the angles between the rays after the lens to change. Since the surface of the camera CCD remains at a fixed distance (F)from the slits/lens, the centers of the single-slit terms
A 1 and A2 move relative to each other on the CCDsurface. By measuring the change in 08 the deformation can be monitored and corrected for in real-time
during interferometer usage.
The most important feature of the new fitting procedure which directly corrects for the effect of mirror
distortion on apparent beam size is the inclusion of
the 08 term. The fit procedure previously used[3] canbe derived from the current one by holding Os, a1 anda 8 at zero. The light intensity asymmetry parametera 1 corrects for the secondary effects of changes in thelight distribution due to mirror distortion.

3 CALIBRATION
For small values of 08 relative to the longitudinalcoherence depth, the relationship between Dt and 08
should be approximately linear. To measure this, wetook calibration data over the full range of beam cur--382-

The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

Figure 4: Screenshot of image-processing computer in LER optics hut during image acquisition and analysis.
are shown the single-slit components, measured with aremote-controlled shutter which can be used to block
the light from one slit at a time. Below that is thedouble-slit pattern, fit using the width of the singleslit patterns as measured from the shutter data. Preliminary tests of the system in the Spring of 2001 at
the KEKB LER indicate that the method works well.
It should be noted that another approach to thisproblem would be to put the slits in the tunnel justafter the extraction mirror. This would greatly reducethe magnification changes. However, because the angleof incidence on the slits/lens determines the separationof the single-slit terms on the camera CCD plane, itwould still be necessary to take 8s into account in orderto make accurate fits.

Dt vs separation of single-slit peaks 08
1W120

1~11Idata <>fit ...... -110-100-Os(arb) 90-80-70

<>~

60

050

I

5.5
5
4.5
43.5
3
Effective slit separation at mask Dt (mm)The authors wish to thank Professors S. Kurokawa
and K. Oide for their encouragement of this work. Theauthors also thank the KEKB commissioning team fortheir support and feedback.

Figure 5: Dt vs 08

rents in use, and plotted the effective slit separation
at the mask plane Dt versus 88 • The data are shownin Figure 5. The standard deviation to a linear fit isless than one percent, verifying that 88 can be used toestimate Dt without needing to use the pinhole mask
after initial calibration.

45 ACKNOWLEDGMENTS

OPERATION

Figure 4 shows the screen of the image processing
computer in the optics hut. In the upper left graph

6 REFERENCES[1] T. Mitsuhashi, J.W. Flanagan, S. Hiramatsu, "Optical
Diagnostic System for the KEK B-Factory," Proceedings of EPAC 2000, Vienna, Austria, p. 1783
[2] D. Malacara, ed., Optical Shop Testing, Second edition,Wiley (New York) 1992, pp. 367-396.
[3] J.W. Flanagan, S. Hiramatsu, T. Mitsuhashi, "Automatic Continuous Transverse Beam-size Measurement
System for KEKB," Proceedings of EPAC 2000, Vienna, Austria, p. 1714

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