DATA ACQUISITION AND ERROR ANALYSIS FOR PEPPERPOTEMITTANCE MEASUREMENTS

S. Jolly, Imperial College, London, UKJ. Pozimski, STFC/RAL, Chilton, Didcot, Oxon, UK/Imperial College, London, UK

J. Pfister, IAP, Frankfurt am Main, GermanyO. Kester, NSCL, East Lansing, Michigan, USA

D. Faircloth, C. Gabor, A. Letchford, S. Lawrie, STFC/RAL, Chilton, Didcot, Oxon, UK

Abstract

The pepperpot provides a unique and fast method ofmeasuring emittance, providing four dimensional corre-lated beam measurements for both transverse planes. Inorder to make such a correlated measurement, the pepper-pot must sample the beam at specific intervals. Such dis-continuous data, and the unique characteristics of the pep-perpot assembly, requires special attention be paid to boththe data acquisition and the error analysis techniques. Afirst-principles derivation of the error contribution to therms emittance is presented, leading to a general formulafor emittance error calculation. Two distinct pepperpot sys-tems, currently in use at GSI in Germany and RAL in theUK, are described. The data acquisition process for eachsystem is detailed, covering the reconstruction of the beamprofile and the transverse emittances. Error analysis forboth systems is presented, using a number of methods toestimate the emittance and associated errors.

INTRODUCTION

The use of pepperpots in measuring transverse emittanceis widespread. The pepperpot is unique in providing an in-stantaneous measurement of the 4 dimensional emittanceof a beam in a single shot. To do so the pepperpot sacrificesposition resolution by measuring the beam only at discreteintervals through an intercepting screen. With suitably fastanalysing software, this provides the opportunity of mea-suring and visualising the emittance of the beam in realtime. The disadvantage of using a pepperpot is that they arehighly destructive to the beam, primarily due to the inter-cepting screen, and the discontinuous nature of the positionmeasurement that results from segmenting the beam.

To fully categorise emittance measurement error, a firstprinciples analysis of the propagation of errors through thecalculation of rms emittance has been carried out. This re-sults in a general formula for the calculation of errors fromany method of emittance measurement. This error analysisprocedure is demonstrated for two contrasting pepperpotdesigns.

PEPPERPOT SYSTEMS

Error analysis has been carried out for two pepperpotsystems: from the HITRAP project at GSI [1] and the FrontEnd Test Stand (FETS) at RAL [2].

Figure 1: 3-D model of the FETS pepperpot assembly [3].

A CAD model of the FETS pepperpot assembly is shownin Fig. 1: full description of the FETS pepperpot deviceis given in [3]. The intercepting screen is a 100 μm thicktungsten foil with a square array of 41×41 holes, each50± 5 μm in diameter, on a 3± 0.01 mm pitch, giving atotal imaging area of 120×120 mm2. The beam is imagedwith a quartz scintillator, 10 mm from the tungsten screen,and a 2048×2048 pixel PCO 2000 high speed camera: thecamera-to-screen distance of 1100 mm gives a resolution of65 μm per pixel and an angular resolution of 6.5 mrad. Datais recorded from the camera direct to a multi-image TIFFfile and analysed with Matlab. Calibration is carried outusing a series of calibration marks on the rear copper platefacing the camera: 4 lines, forming a 125 mm×125 mmsquare around the intercepting screen, provide the neces-sary calibration information on the size, location and rota-tion of the pepperpot holes.

Figure 2: The HITRAP pepperpot setup (cf. [5]).

The setup of the GSI pepperpot system for the HITRAP

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experiment is outlined in [4], with recent modificationsdetailed in [5]: the complete system is shown in Fig. 2.The pepperpot can be equipped with various interceptingscreens: for recent measurements a 100 μm thick tung-sten foil was used, with an array of 19×19 holes, each100± 10 μm in diameter, on a 1.6± 0.02 mm pitch, toimage an area of 29×29 mm2. An Al2O3 scintillator ismounted 150 mm from the screen: pepperpot images arerecorded with a cooled fast shutter CCD camera with a1280×1024 pixel resolution, giving an angular resolutionof 0.3 mrad. A laser is used to calibrate the system: an im-age is recorded of the resulting light spots on the imagingscreen, these calibration spots are projected to horizontaland vertical axes and the maxima are defined as calibrationpositions for each individual row and column.

EMITTANCE ERROR ANALYSIS

In order to calculate an error on the emittance, it is nec-essary to derive an emittance error formula by propagatingthe errors on each measured quantity through the formulafor emittance. The rms emittance is used as it is mathemati-cally well defined, allowing such a first-principles approachto be used. Such an approach is valid only if the errors oneach variable are also well defined: this is addressed in thenext section. In the x-plane, the definition of εrms is:

εrms =√〈x2〉〈x′2〉 − 〈xx′〉2 (1)

=

√√√√√√

(∑Ni=1 ρix2

i

)(∑Nj=1 ρjx′2

j

)−(∑N

i=1 ρixix′i

)2

(∑Ni=1 ρi

)2

(2)

To calculate an error on the emittance, a variance, σ2εrms

,and hence a standard deviation, σεrms , must be derived.The variance on one term of the

∑Ni=1 ρix

2i series is given

by:

σ2ρx2 = 4ρ2x2σ2

x + x4σ2ρ (3)

Since, for addition, variances also add, the variance forthe complete series, σ2P

ρx2 is:

σ2Pρx2 =

N∑

i=1

(4ρ2

i x2i σ

2xi

+ x4i σ

2ρi

)(4)

. . . since every position measurement, xi, has its own er-ror, σxi , and each intensity measurement, ρi, has its ownerror σρi . The same calculation holds true for

∑Nj=1 ρjx

′2j

and its associated variance, σ2Pρx′2 :

σ2ρx′2 = 4ρ2x′2σ2

x′ + x′4σ2ρ (5)

σ2Pρx′2 =

N∑

j=1

(4ρ2

jx′2j σ2

x′j+ x′4

j σ2ρj

)(6)

The variance for the product of these two terms,σ2P

ρx2P

ρx′2 , is given by:

σ2Pρx2

Pρx′2 =

(N∑

i=1

4ρ2i x

2i σ

2xi

+ x4i σ

2ρi

)⎛

⎝N∑

j=1

ρjx′2j

⎞

⎠

2

+

(N∑

i=1

4ρ2i x

′2i σ2

x′i+ x′4

i σ2ρi

)⎛

⎝N∑

j=1

ρjx2j

⎞

⎠

2

(7)

Two more variances are needed: for the xx′ and ρ2 termsin Eqn. 2. Following the same procedure:

σ2ρxx′ = x2x′2σ2

ρ + ρ2x′2σ2x + ρ2x2σ2

x′ (8)

σ2Pρxx′ =

N∑

i=1

(x2

i x′2i σ2

ρi+ ρ2

i x′2i σ2

xi+ ρ2

i x2i σ

2x′

i

)(9)

σ2(P

ρxx′)2 = 4

(N∑

i=1

x2i x

′2i σ2

ρi+ ρ2

i x′2i σ2

xi+ ρ2

i x2i σ

2x′

i

)

×⎛

⎝N∑

j=1

ρjxjx′j

⎞

⎠

2

(10)

And:

σ2Pρ =

N∑

i=1

σ2ρi

(11)

σ2(P

ρ)2 = 4

(N∑

i=1

σ2ρi

)⎛

⎝N∑

j=1

ρj

⎞

⎠

2

(12)

As such, the variance for the numerator in Eqn. 2 is:

σ2Pρx2

Pρx′2−(

Pρxx′)2 =

(N∑

i=1

4ρ2i x

2i σ

2xi

+ x4i σ

2ρi

)⎛

⎝N∑

j=1

ρjx′2j

⎞

⎠

2

+

(N∑

i=1

4ρ2i x

′2i σ2

x′i+ x′4

i σ2ρi

)⎛

⎝N∑

j=1

ρjx2j

⎞

⎠

2

+

4

(N∑

i=1

x2i x

′2i σ2

ρi+ ρ2

i x′2i σ2

xi+ ρ2

i x2i σ

2x′

i

)⎛

⎝N∑

j=1

ρjxjx′j

⎞

⎠

2

− 8

(N∑

i=1

ρ2i x

4i x

′4i σ2

ρi+ ρ4

i x2i x

′4i σ2

xi+ ρ4

i x4i x

′2i σ2

x′i

)

(13)

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07 Hadron Accelerator Instrumentation

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The subtraction comes about through the cancellation ofterms in Eqn. 2. Propagating the errors through the divisionand square root gives a variance on εrms, σ2

εrms:

σ2ε =

(∑N

i=1 ρ2i x

2i σ

2xi

+x4

i σ2ρi

4

)(∑N

j=1 ρjx′2j

)2

ε2(∑N

k=1 ρk

)4

+

(∑N

i=1 ρ2i x

′2i σ2

x′i+

x′4i σ2

ρi

4

)(∑N

j=1 ρjx2j

)2

ε2(∑N

k=1 ρk

)4 +

(N∑

i=1

x2i x

′2i σ2

ρi+ ρ2

i x′2i σ2

xi+ ρ2

i x2i σ

2x′

i

)(N∑

j=1

ρjxjx′j

)2

ε2(∑N

k=1 ρk

)4

−2(∑N

i=1 ρ2i x

4i x

′4i σ2

ρi+ ρ4

i x2i x

′4i σ2

xi+ ρ4

i x4i x

′2i σ2

x′i

)

ε2(∑N

k=1 ρk

)4

+ε2(∑N

i=1 σ2ρi

)

(∑Nk=1 ρk

)2

(14)

The error on the rms emittance, σεrms , is the square rootof this value.

PEPPERPOT EMITTANCE ERRORS

The next stage is to identify the errors on each mea-sured quantity and use these to calculate σεrms . For pep-perpot measurements, the dominant errors are: the spacingof the holes for σx; the camera resolution and pepperpot-to-scintillator distance – defining the angular resolution – forσx′ ; and the inherent beam variation and signal noise in themeasurement apparatus for σρ. Certain errors, such as theshape and diameter of the holes, contribute to both σx andσρ: however, the dominant contribution to σx is clearly thehole spacing, and careful analysis of the hole size wouldallow this error to be removed. It is assumed that the holesize is smaller than the relative pixel size of the camera, al-lowing simple angles to be calculated through ray tracing,and that the camera orientation defines the beam orienta-tion. As such, calibration errors contribute only to errorson σx′ .

The emittance values derived from measurements for thetwo systems, along with the corresponding error estimatefor each contributing error, is shown in Table 1. Whereerrors are negligible, or error analysis has not been carriedout, no figure is shown. For the FETS system, an additionalangle error of ∼1 mrad per mm is included due to the in-accuracy of the calibration. Two sources of intensity errorare considered: beam noise, corresponding to the stochas-tic pulse-to-pulse variation in the beam, and the noise floor,

FETS HITRAPValue σ (%) Value σ (%)

Beam radius (mm) 45 – 17 –εx (π mm mrad) 0.61 – 0.24 –

Hole spacing (mm) 3 1.8 1.6 2.2Angle res. (mrad) 6.5 1.6 0.3 0.2Beam noise (%) 10 1.3 10 0.3Noise floor (%) 2 ∼ 0 10 1.2σε (π mm mrad) 0.029 4.8 0.010 3.9

Table 1: Percentage Emittance Error Contributions forPepperpot Measurements

a pessimistic figure representing the constant level of back-ground noise (quoted as a percentage of the maximum sig-nal). For the FETS system, each source of error contributesapproximately equally to the final error figure of ± 4.8%:For the GSI system, the dominant error is clearly the holespacing, with a contribution from the background noise:the angle resolution is considerably better than the FETSsystem and this is reflected in the error values. An inter-esting effect is that the beam noise contributes significantlymore for the FETS system but is dominated by the noisefloor in the HITRAP system: this is a result of the smallerbeam and lower light intensity producing a less intense pep-perpot image for the HITRAP pepperpot. This also con-tributes to the larger position error.

CONCLUSIONS

The formula for calculating rms emittance errors hasbeen applied successfully to 2 different pepperpot setups,with promising results. Further work is required to cate-gorise errors not included in this analysis, since these affectthe accuracy of the emittance measurement while not con-tributing to the error estimate. This has particular impor-tance when dealing with cut selection, something dealt within considerable detail by the SCUBEEx algorithm (see [6]and Refs therein). As such, this method constitutes a mini-mum estimate of the emittance error.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the contributions ofRoger Barlow, Louis Lyons and Martin Jolly in the analysisand derivation of emittance errors.

REFERENCES

[1] J Pfister et al., EPAC’08, THPP037, p. 3449.

[2] A Letchford et al., EPAC’08, THPP029, p. 3437.

[3] S Jolly et al., DIPAC’07, WEO2A01, p. 218.

[4] T Hoffmann et al., BIW 2000, AIP Conf. Proc. 546, p. 432.

[5] J Pfister et al., LINAC’08, TUP074, p. 555.

[6] M Stockli et al., Rev. Sci. Instrum. 75 (2004) 1646.

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