12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

IBIC2023, Saskatoon, CanadaJACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 4 5I. Pinayev†, Brookhaven National Laboratory, Upton, NY, USA
Abstract
The quadrupole scan is commonly used for measurementof beam emittance. The found dependence of the beam sizevs. quadrupole strength is fitted with parabola, which coefficients are used for emittance calculations. The measurement errors can cause substantial variations in the emittance value. Sometimes the fitted parabola has negative
minimum value, making impossible emittance calculation.
We propose more robust data processing modifying thequadrupole scan procedure using or weighted fit for parabola. The experimental results are presented.

INTRODUCTION
Emittance serves as measure of the transverse phasespace occupied by the beam as well as a measure of beam
quality in many applications. By definition [1] emittance is""𝜀!" = 𝜎!" 𝜎!#− 𝜎!!#monitor is 3.63 m. Emittance for the blue curve is 0.19 mm
mrad, and for the red curve it is 0.11 mm mrad, differenceis factor of 2.
Moreover, due to the statistical variations the value ofthe fitted parabola minimum can be less than zero making
the data set unusable for obtaining value of the emittance.

MODIFICATION OF SCAN
We are going to modify the procedure to avoid such subtracting. We can present beam angular distribution as sumof correlated and uncorrelated parts, so that for each particle 𝑥 # = 𝛼𝑥 + 𝑥(′, where 𝛼 is correlation factor and 𝑥(′ isuncorrelated angular spread. Correlation factor can befound from the covariance:
𝛼 = 𝜎!!# ⁄𝜎!"
Beam emittance can be found from Eq. (1)(1)"where 𝜎!" and 𝜎!#are variances, and 𝜎!!# is covariance ofbeam distribution.
Commonly used technique is scanning of a focusing element such as a quadrupole or a solenoid and measuring
dependence of the transverse beam size on a profile monitor [2, 3]. The obtained data are fit with a parabola which
coefficients are used for calculating emittance. In somecases, even slight changes in the data can cause substantial
variation of the emittance value. This is due to that we are
subtracting two large numbers. Example of such measurement is shown in Fig. 1.

(2)" "𝜀!" = 𝜎!$𝜎!%#$(3)"where 𝜎!$is r.m.s beam size at focusing element, and 𝜎!"%#$

is uncorrelated angular spread.
After passing through a focusing element with focal
length 𝐹 and drift with length 𝐿 the beam size at the observation point can be found&""𝜎!" = .1 − '0 𝜎!$+""")
2𝛼𝐿21 − "!3𝜎!$+ 𝐿" (𝛼 " 𝜎!$+ 𝜎!%#$(4)The minimum size is observed when 1 − 𝐿⁄𝐹 = −𝛼𝐿and minimal size is:"𝜎!()*= 𝐿" 𝜎!"%#$(5)

So, from 1 − 𝐿⁄𝐹()* corresponding to the minimal
beam size one can find the first order correlation factor α
between transverse positions and angles of the particles inthe bunch, and the uncorrelated angular spread can be calculated using the minimal beam size.
Now we need to find beam size at focusing element tocalculate beam emittance. For this purpose, we can measure the beam size when focusing element is off (1⁄𝐹 = 0):""""𝜎!" = 𝜎!$+ 2𝛼𝐿𝜎!$+ 𝛼 " 𝐿" 𝜎!$+ 𝜎!()*Figure 1: The horizontal axis shows ratio of distance fromthe solenoid center to the profile monitor and solenoid’s
focal length. Vertical axis shows the square of the r.m.s.
beam size in mm2.
The maximal difference between the parabolic fit coefficients is 0.2%. Distance from the solenoid to the profile____________________________________________* Work supported by Brookhaven Science Associates, LLC under Contract No. DE-SC0012704 with the U.S. Department of Energy† pinayev@bnl.gov.

07 Machine Parameter Measurements(6)

and+ $ ,+ $$+ $ ,+#%&'##"#%&'
𝜎!$= (./0&)$ = (&⁄'%&' )$(7)

Emittance can be found using formula below:+ $ ,+ $+$#%&'𝜀 " = (&#⁄' #%&')$ &$%&'(8)

If the beam is converging to the minimal size with focusing element off or close to such condition, then there canbe significant error in determination of the beam size at theMOP045
127Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

ROBUST EMITTANCE MEASUREMENTS*

Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

IBIC2023, Saskatoon, CanadaJACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0
d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 4 5

focusing element. In this case we can use the setting of thefocusing element providing for substantially different
beam size and use the formula below for calculation of thebeam size$+#$ ,+#%&'"
𝜎!3= (.,&⁄' )$/"0&(.,&⁄' )/&$0$(9)

If we set 1⁄𝐹 = 𝛼 (passing through the focusing element removes linear correlation) and measured beam size
will be""𝜎!" = 𝜎!$+ 𝐿" 𝜎!%#$that relative error in the beam size measurement is relative
constant, then we can modify the error function in the following way:+$

𝜒 " = ∑ 91 − +$+&, :%()*"(13)

Figure 2 shows the fitting curves for the same set of experimental data points with regur and weighted fits.

(10)

Because the beam is converging (𝛼 = −1⁄𝐿), this will
require defocusing element which cannot be realized withsolenoids. But we can set focusing so that waist will beachieved in the middle between focusing element and theprofile monitor (1⁄𝐹 = 𝛼 + 2⁄𝐿). In this case beam size
at profile monitor will be close to beam size in the focusing
element and Eq. (10) can be used as well.

WEIGHTED FIT
Another approach for solving the problem is to utilize
the weighted fit. In most cases well-known formulas or
standard functions for the polynomial fit are used. Such approach implies that errors in the measurement of beam size
are independent of beam size and are constant. The following error function is used for fit evaluation:"""𝜒 = ∑2𝜎(456− 𝜎7)83"(11)

It is possible to measure few data points at the fixed focusing element setting to find the variation of the measured
beam size.
"𝜒 =∑

$$
9+%()*,++&,:$+(--.-$(12)

Implementing multiple measurements at each setting of
focusing element slows down the process. If we assume

Figure 2: Experimental data of quadrupole scan with regular fit (light blue) and weighted fit (dark blue).

CONCLUSIONS
Two methods are proposed and experimentally tested forincreasing robustness of the emittance measurements.

REFERENCES[1] S.Y. Lee, “Transverse Motion”, in Accelerator Physics,
World Scientific, pp. 53-55, 1999.
[2] M.G. Minty and F. Zimmermann, “Transverse Beam Emittance Measurement and Control”, in Measurement and Control of Charged Particle Beams, Springer, pp. 101-104, 2003.
[3] M.C. Ross, N. Phinney, G. Quickfall, J.C. Sheppard,
H. Shoaee, “Automated Emittance Measurement in theSLC”, in Proc. PAC’87, Washington, DC, USA, pp. 725-728.

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07 Machine Parameter Measurements