MOP045

Robust Emittance Measurement
Igor Pinayev, Brookhaven National Laboratory, Upton, NY, USA
Emittance serves as measure of the transverse phase space occupied by the beam as well as ameasure of beam quality in many applications. By definition emittance is:

The horizontal axis shows ratio of
distance from the solenoid center to theprofile 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 thesolenoid to the profile 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, difference is factor of 2.

""𝜀!" = 𝜎!" 𝜎!#− 𝜎!!#"where 𝜎!" and 𝜎!#are variances, and 𝜎!!# is covariance of beam distribution.
Commonly used technique is scanning of a focusing element such as a quadrupole or asolenoid and measuring dependence of the transverse beam size on a profile monitor. The
obtained data are fit with a parabola which coefficients are used for calculating emittance. In
some cases, even slight changes in the data can cause substantial variation of the emittancevalue. This is due to that we are subtracting two large numbers.

MODIFICATION OF SCAN
We can present beam angular distribution as sum of correlated and uncorrelated parts, so thatfor each particle 𝑥 # = 𝛼𝑥 + 𝑥′
( , where 𝛼 is correlation factor and 𝑥′( is uncorrelated angularspread. Correlation factor can be found from the covariance:
𝛼 = 𝜎!!# ⁄𝜎!"

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
𝜒Beam emittance can be found"" ""= 1 𝜎(,-. − 𝜎/)0

" "𝜀!" = 𝜎!$𝜎!#$%""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"𝐿""""𝜎!" = 1 −𝜎!$+ 2𝛼𝐿 1 − '& 𝜎!$+ 𝐿" 𝛼 " 𝜎!$+ 𝜎!#$%𝐹The minimum size is observed when 1 − 𝐿⁄𝐹 = −𝛼𝐿 and minimal size is:""𝜎!()*= 𝐿" 𝜎!#$%Now we need to find beam size at focusing element to calculate beam emittance. For thispurpose, we can measure the beam size when focusing element is off (1⁄𝐹 = 0)""""
𝜎!" = 𝜎!$+ 2𝛼𝐿𝜎!$+ 𝛼 " 𝐿" 𝜎!$+ 𝜎!()*""""𝜎−𝜎𝜎−𝜎!
!
!()*!()*"𝜎!$ =="1 + 𝛼𝐿𝐿⁄𝐹()* "Emittance can be found using formula"""𝜎−𝜎𝜎!
!()* !()*"
𝜀 =𝐿⁄𝐹()* " 𝐿"

If the beam is converging to the minimal size with focusing element off or close to suchcondition, then there can be significant error in determination of the beam size at the focusing
element. In this case we can use the setting of the focusing element providing for substantiallydifferent beam size and use the formula below for calculation of the beam size""𝜎−𝜎!
!()*"𝜎!+ =1 − 𝐿⁄𝐹 " + 2𝛼𝐿 1 − 𝐿⁄𝐹 + 𝐿" 𝛼 "If we set 1⁄𝐹 = 𝛼 (passing through the focusing element removes linear correlation) andmeasured beam size will be""𝜎!" = 𝜎!$+ 𝐿" 𝜎!#$%Because the beam is converging (𝛼 = −1⁄𝐿), this will require defocusing element whichcannot be realized with solenoids. But we can set focusing so that waist will be achieved in themiddle between focusing element and the profile monitor (1⁄𝐹 = 𝛼 + 2⁄𝐿). In this case beam
size at profile monitor will be close to beam size in the focusing element.

Implementing multiple measurements at each setting of focusing element
slows down the process. If we assume that relative error in the beam size
measurement is relative constant, then we can modify the error function in
the following way:𝜒" = 1

""
𝜎/)0
1− "
𝜎(,-.

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