quadrupole scan and tomography studies on...

Quadrupole scan and tomography studies on VELA

C.S. Edmonds, B.L. Militsyn, D.J. Scott, C.P. Topping and A. Wolski

25 July 2016

Overall goals

The overall goals of the quad scan and tomography studies are:

• to develop and demonstrate novel techniques for detailed characterisation of beam properties (in particular, the transverse phase space distribution) including coupling and space charge effects;

• to apply these techniques to understand the machine behaviour under various conditions;

• to implement the techniques in tools that can be used routinely in the control room for machine characterisation and tuning.

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Drift length 𝐿𝐷

Screen

Quadrupole, strength 𝑘1𝐿

s0 s1

For any individual particle: 𝑥1𝑝𝑥1=1 𝐿𝐷0 1

1 0−𝑘1𝐿 1

𝑥0𝑝𝑥0

This leads to: 𝑥12 = 𝑎(𝑘1𝐿)

2+𝑏 𝑘1𝐿 + 𝑐

where 𝑎, 𝑏 and 𝑐 are functions of 𝑥02 , 𝑥0𝑝𝑥0 , 𝑝𝑥0

2 (and 𝐿𝐷).

If we fit a quadratic curve to a plot of 𝑥12 vs 𝑘1𝐿, then from the fit

coefficients 𝑎, 𝑏 and 𝑐 we can determine 𝑥02 , 𝑥0𝑝𝑥0 and 𝑝𝑥0

2 , and hence obtain the emittance and Courant-Snyder parameters.

Conventional quad scan

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𝛽𝑥0 =2𝑎

𝑞

𝛼𝑥0 =2𝑎 + 𝑏𝐿𝐷𝑞𝐿𝐷

𝜀𝑥 =𝑞

2𝐿𝐷2

where 𝑞 = 4𝑎𝑐 − 𝑏2

𝑥12 = 𝑎(𝑘1𝐿)

2+𝑏 𝑘1𝐿 + 𝑐

𝑥02 = 𝛽𝑥0𝜀𝑥

𝑥0𝑝𝑥0 = −𝛼𝑥0𝜀𝑥𝑝𝑥02 = 𝛾𝑥0𝜀𝑥

𝛽𝑥0𝛾𝑥0 − 𝛼𝑥02 = 1

ALICE data


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The “conventional” quad scan method has some challenges and limitations:

• Accurate beam size estimates are best obtained for a well-defined image with 𝑥2 ≈ 𝑦2 … but this is not always what we are provided with!

• The horizontal and vertical degrees of freedom are treated separately: no account is taken of coupling.

• Space-charge effects may not be negligible (depending on beam energy and charge density).

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Quad scans: a more general method

To address some of the limitations of the conventional quad scan method, consider a more general transformation from 𝑠0 to 𝑠1:

in which case:

where:

and 𝑀𝑖,𝑗 are elements of the transfer matrix from 𝑠0 to 𝑠1.

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If we vary the transfer matrix from 𝑠0 to 𝑠1 (by changing one or more quad strengths), we obtain a set of beam sizes:

where 𝐶 is a matrix with elements expressed in terms of elements of the different transfer matrices from 𝑠0 to 𝑠1:

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To determine the beam distribution at 𝑠0, we just have to invert the matrix 𝐶.

In the case of three settings of the beamline from 𝑠0 to 𝑠1, 𝐶 is a square (3 × 3) matrix:

If we use 𝑁 settings of the beamline, 𝐶 is an 𝑁 × 3 matrix.

In this case, we can still “invert” 𝐶 (using, for example, singular value decomposition) to find a best fit of the distribution parameters to the data.

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Quad scans: two degrees of freedom

The “general” method described on the previous slides can be extended to two (transverse) degrees of freedom:

The matrix 𝐷 (3𝑁 rows by 10 columns) is more complicated than the matrix 𝐶… but the method works in just the same way.

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Given the (4 × 4) matrix Σ with elements Σ𝑖,𝑗 = 𝑥𝑖𝑥𝑗 , where 𝑥𝑖

are the elements of the vector 𝑥, 𝑝𝑥 , 𝑦, 𝑝𝑦 , we can obtain the

(generalised) Courant-Snyder parameters 𝛽𝑖,𝑗𝑘 and (normal mode)

beam emittances 𝜀𝑘:

Σ𝑖,𝑗 = 𝛽𝑖,𝑗𝑘 𝜀𝑘

𝑘=I,II

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Using the more general method:

• we can keep 𝑥2 ≈ 𝑦2 at the observation screen (by varying a set of quadrupoles as necessary);

• transverse coupling can be included by measuring all the second-order moments of the beam distribution on the observation screen;

• space-charge can be included in a linear approximation in the transfer matrix from 𝑠0 to 𝑠1.

Note: to include space-charge, some iteration is needed, since the transfer matrix (and hence the matrix 𝐷) depends on the initial beam distribution.

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Quad scans: application in VELA

• Use four quadrupoles between 𝑠0 and 𝑠1 to keep the beam sizes (roughly) constant over the range of a scan.

• Vary horizontal and vertical phase advances (separately) from 10° to 170°.

• Investigate how beam distribution depends on various parameters: – BSOL setting;

– bunch charge;

– gun phase…

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𝑠0 𝑠1


Example results (20 pC bunch charge):

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• Some results (and preliminary analysis) to be contributed by Chris…

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Phase space tomography

• Quadrupole scans provide basic information on the optics; in particular we obtain the Courant-Snyder parameters and beam emittances describing the second-order moments of the beam distribution.

• In practice, the beam can have a distribution with more structure than (for example) a simple Gaussian: a full description needs more than just a set of values for the second-order moments.

• Phase space tomography provides a way to obtain a detailed image of the beam phase space distribution.

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Tomography: basic principles

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http://www.cmu.edu/me/xctf/xrayct

Tomography: basic principles A set of (usually 1D) projections taken over a range of rotation angles is known as a sinogram.

Various algorithms (with different pros and cons…) can be used to reconstruct a 2D distribution from a sinogram, for example:

• Filtered back-projection (FBP)

• Maximum entropy (MENT)

• Algebraic reconstruction (ART)

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By Kostmo - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=9928788

original (“phantom”) sinogram reconstruction

Phase space tomography: 1 d.o.f.

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𝑥

𝑝𝑥 𝑝𝑥

𝑥

𝑥 𝑥

image intensity (projected onto x-axis)

image intensity (projected onto x-axis)

rotation in horizontal phase space

Phase space tomography on VELA Beam phase space tomography has been applied at a number of facilities, including:

• CERN SPS (longitudinal phase space tomography);

• Fermilab (longitudinal – see Duncan’s talk \\CCROFTDELL4\Tomography\Documents);

• UMER (transverse, in strong space-charge regime);

• PITZ.

Tomography studies on VELA have a number of goals:

• to demonstrate tomography in normalised phase space;

• to demonstrate beam phase space tomography in two (transverse) degrees of freedom;

• to provide a detailed characterisation of the beam, including the effects of space charge.

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Tomography in normalised phase space By an appropriate normalisation of phase space, a “stretched” distribution can be made to appear “round”: this potentially improves the ability to distinguish detailed features.

• Choose some Courant-Snyder parameters at the reconstruction point (based on a “best guess” of the actual beam distribution).

• Calculate the phase advance from the reconstruction point to the observation point: this is tomographic rotation angle.

• Calculate the Courant-Snyder 𝛽 function at the observation point: scale the

measured co-ordinates by 1 𝛽 when constructing the sinogram.

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Reconstruction in real phase space (simulation) Reconstruction in normalised phase space (simulation)

K.M. Hock et al. / Nuclear Instruments and Methods in Physics Research A 642 (2011) 36–44

Tomography in two degrees of freedom • By using the full beam image from a YAG screen (rather than a projection

onto the 𝑥 or 𝑦 axis), we can determine the beam distribution in 4D phase space, 𝜌(𝑥, 𝑝𝑥, 𝑦, 𝑝𝑦).

• This allows characterisation of coupling between the degrees of freedom.

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Original distribution (simulation) Reconstruction (simulation)

K.M. Hock, A. Wolski / Nuclear Instruments and Methods in Physics Research A 726 (2013) 8–16

Tomography in two degrees of freedom

For VELA, we plan to use the ART algorithm.

The observed intensity 𝐼(𝑥, 𝑦) at a given point on the YAG screen can be related to the 4D phase space distribution by:

𝐼 𝑥𝑖 , 𝑦𝑗 = 𝑊𝑖𝑗𝑘𝑙 𝜌 𝑥𝑖 , 𝑝𝑥𝑘 , 𝑦𝑗 , 𝑝𝑦𝑙 .

𝑘,𝑙

This can be “flattened” to give a matrix equation:

𝐼 𝑖,𝑗,𝑛 = 𝑊 𝑖,𝑗,𝑛 , 𝑖′,𝑗′,𝑘,𝑙 𝜌 𝑖′,𝑗′,𝑘,𝑙𝑖′,𝑗′,𝑘,𝑙

where 𝑛 = 1…𝑁 is an index to indicate the rotation angle.

By inverting the matrix 𝑊, we can reconstruct the 4D phase space distribution 𝜌 from a set of observed 2D beam intensities 𝐼.

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Tomography in 2 d.o.f. in VELA

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Simulation Normalised phase space Real phase space

Tomography in 2 d.o.f. in VELA

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Preliminary experimental results

23 June 2015 10 pC bunch charge 4.5 MeV/c momentum