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Symmetric single-quadrupole-magnet scan method to measure the 2D transverse beam parameters
Nuclear Instruments and Methods in Physics Research A 743 (2014) 103–108
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Symmetric single-quadrupole-magnet scan method to measure the 2D transverse beam parameters Eduard Prat Paul Scherrer Institut, 5232 Villigen, Switzerland
art ic l e i nf o
a b s t r a c t
Article history: Received 14 October 2013 Received in revised form 9 January 2014 Accepted 9 January 2014 Available online 21 January 2014
Precise measurements of the transverse beam parameters are essential to control and optimize all types of charged particle beams. In this paper we present a novel method that uses one quadrupole magnet and one profile monitor to measure the transverse beam emittance and optics. In comparison to a conventional single-quadrupole scan measurement, this new technique measures the two transverse planes simultaneously. This novel procedure is faster, more intuitive and allows keeping under control the required quadrupole gradient and the beam sizes at the profile monitor. The application of the method is illustrated with the SwissFEL Injector Test Facility. & 2014 Elsevier B.V. All rights reserved.
Keywords: Linear accelerators Free-electron lasers Electron optics Beam characteristics
1. Introduction
The transport of the 2D beam matrix from s0 to s can be described by a matrix formalism [1,2]:
An accurate knowledge of the transverse beam properties is fundamental in all types of charged particle accelerators. For instance, in Free-Electron Lasers (FEL) a high quality electron beam with low emittance and matched optics is required for the optimal lasing process. The 2-dimensional (2D) transverse beam matrix composed of the second-order moments of the beam distribution describes the statistical properties of the beam: ! 〈u2 〉 〈uu0 〉 ð1Þ suu ¼ 〈uu0 〉 〈u02 〉
suu ðsÞ ¼ R suu ðs0 Þ RT
where u refers to either x (horizontal coordinate) or y (vertical coordinate), and u0 is the derivative of u with respect to the longitudinal coordinate. The beam emittance and optical functions (α, β and γ, also called Twiss parameters) can be derived as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εu ¼ detðsuu Þ ð2Þ
αu ¼ 〈uu0 〉=εu ;
βu ¼ 〈u2 〉=εu ;
γ u ¼ 〈u02 〉=εu :
ð3Þ
When we multiply the emittance by the beam total momentum we get the so-called normalized emittance:
εN u ¼
p εu mc
ð4Þ
where p is the central momentum of the beam, m is the particle mass and c is the speed of light. 0168-9002/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2014.01.021
where R is the transverse transfer matrix from s0 to s: ! Ruu Ruu0 R¼ : Ru0 u Ru0 u0
ð5Þ
ð6Þ
According to Eqs. (1), (5) and (6), the beam size at a position s can be expressed as follows: 〈u2 〉s ¼ R2uu 〈u2 〉s0 þ R2uu0 〈u02 〉s0 þ 2Ruu Ruu0 〈uu0 〉s0 :
ð7Þ
The elements of suu can be reconstructed at s0 by measuring the beam size at s for different optics transformations between s0 and s. At least three measurements (i.e. three equations) are needed to reconstruct the three second moments of the beam, but more measurements will allow improving the robustness of the reconstruction. Ideally, the betatron phase-advance μu between the observation and reconstruction points should be scanned progressively between 0 and π for the best reconstruction of the 2D parameters. In order to obtain the 2D parameters the beam sizes can be measured at different positions along a fixed lattice (multipleposition approach like FODO measurements) or alternatively the optics between the reconstruction and the observation points can be varied with one or more quadrupole magnets (multiple-optics approach or quadrupole scans). The multiple-optics approach has the inconvenience of modifying the optics during a measurement, but it is more compact, that requires less equipment and is more flexible than the multiple-position strategy. Because of that in
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quadrupole scans there is no need for a long diagnostic section, therefore the overall length of a typical accelerator is reduced. Refs. [3–8] give more detailed information about definitions and measurement procedures concerning the 2D transverse beam parameters. In a conventional single-quadrupole scan measurement the phase-advance is controlled only in one plane at a time. Moreover, the beam size can go through a very tight focus at the measurement position during the scan, which makes the reconstruction of the parameters very difficult and sensitive to beam size measurement errors. In addition, the beam size variations can be very large along the scan, which may give rise to dynamic range problems for the profile monitor. This document presents a special single-quadrupole scan measurement procedure with the following features:
40 focusing plane defocusing plane
β [m]
30 20
Convergent beam 10 0
0
5
focusing plane defocusing plane
β [m]
The horizontal and vertical phase-advances are scanned simulta
2. Description and formulas The measurement set-up consists simply of a quadrupole, a drift space, and a profile monitor, which measures the beam sizes in both planes simultaneously. The quadrupole entrance is the reference point where the beam parameters (emittance and optical functions) are reconstructed. All the equations presented in this section are obtained assuming thin-lens approximation for the quadrupole magnet. The optics at the reconstruction point need to be identical in both planes if we want to simultaneously scan x and y with a single quadrupole. We define the optics at the reconstruction position as follows:
αx ðs0 Þ ¼ αy ðs0 Þ ¼ α0 βx ðs0 Þ ¼ βy ðs0 Þ ¼ β0 γ x ðs0 Þ ¼ γ y ðs0 Þ ¼ γ 0 :
ð8Þ
As exemplified in Fig. 1, the propagated β-functions downstream of the quadrupole cross each other at a given point only if the beam is initially convergent. This happens for any quadrupole gradient at a distance L from the quadrupole such that:
α0 L ¼ β 0 :
ð9Þ
If this condition is fulfilled the sum of the phase-advances in x and y (between the quadrupole and the profile monitor) will be
15
150
100
neously. This improves the time of the measurement by a factor of about two with respect to conventional measurements. The β-function at the measurement point is the same in x and y for any quadrupole gradient. Consequently the ratio between the beam size in x and y should be constant for the whole scan (i.e. round beam if the emittances are equal in both planes), but varies for mismatched beam parameters at the reconstruction point. It will be easy and intuitive to measure whether the beam is matched or not. The scan produces a gentle waist. The β-function at the waist is relatively large (with typical values of 10 m) and the beam size variations at the measurement location are kept within moderate limits. Therefore, dynamic problems of the profile monitors and space charge effects are avoided. The range of the quadrupole gradient required for the measurement stays within reasonable limits, which minimizes possible chromatic effects and trajectory deviations during the scan. The β-function at the measurement position and the required quadrupole strengths can be tuned by changing some parameters, such as the covered phase-advance or the initial optics.
10
s [m]
Divergent beam
50
0
0
5
10
15
s [m] Fig. 1. β-function along a drift after a quadrupole with a certain strength. The upper plot shows a case where the initial beam is convergent (β0 ¼ 10 m, α0 ¼ 1) and the lower plot corresponds to a divergent initial beam (β0 ¼ 10 m, α0 ¼ 1). Only if the beam is initially convergent there is a position where the β-functions are the same in both planes. This point is at a distance L ¼ β0 =α0 from the quadrupole (in our example L ¼ 10 m). This condition holds for any gradient of the quadrupole magnet and also satisfies μx þ μy ¼ π.
exactly 1801:
μx þ μy ¼ π :
ð10Þ
This means that the scan will be equivalent in x and y, since for the reconstruction of the 2D parameters a certain phase-advance μ provides the same information as π μ. The integrated focusing of the quadrupole can be obtained as a function of the phase-advance and the initial β-function using the following equation: 1 1 1 ¼ kl ¼ ¼ : f tan μx β 0 tan μy β 0
ð11Þ
One can see that for no quadrupole gradient the phase-advance in x and y is identical and equal to π =2. The range of the quadrupole gradients to perform the scan will depend on the phase-advance that we want to cover. For example, for β0 ¼ 10 m we will need kl ¼ 7 0:10 m 1 to cover π =2 or kl ¼ 70:24 m 1 to scan 3π =4. The β-function at the measurement point can be obtained as follows:
β¼
L2
β0 sin 2 μ
ð12Þ
where μ is the phase-advance in x or y. The observed minimum βfunction at the measurement location occurs for no quadrupole gradient, when μx ¼ μy ¼ π =2:
βmin ¼
L2
β0
:
ð13Þ
This quantity can be controlled by changing the initial optics and the distance between the quadrupole and the profile monitor. For our example where β0 ¼ 10 m and L ¼10 m, β min will also be 10 m. From Eqs. (12) and (13) it can be seen that the ratio between the β-function and the βmin function only depends on the covered
E. Prat / Nuclear Instruments and Methods in Physics Research A 743 (2014) 103–108
x 10−5
4
2
50 10
x 10−5
2 y’N [m1/2]
100
4
x y x’N [m1/2]
β [m]
150
105
0 −2
0 −2
1 −4 −4
0 xN
0.5 0.25
x y
4
0 0.4 σx [m]
0
2
[m1/2]
−4 −4
4 x 10−5
−2
0
4 reconstruction
1
4 x 10−5
x 10−4
reconstruction
3
virtual meas.
2
2
yN [m1/2]
x 10−4
3
0.2
kl [1/m]
−2
σy [m]
μ [π]
0.75
virtual meas.
2 1
−0.2 0
−0.4
5
10 scan index
scan index
Fig. 2. β-function, phase-advance and quadrupole strength as a function of the scan index when the phase-advance is progressively covered from 151 to 1651 (for example α0 ¼ 1, β0 ¼ 10 m, L ¼ 10 m).
phase-advance:
β 1 ¼ : βmin sin 2 μ
0
ð14Þ
For instance, if the phase-advance is scanned from 151 to 1651 the
β-function will increase by a factor of 15 with respect to the waist (i. e. around a factor of 4 in beam size). To implement this measurement the lattice and the initial optics have to be chosen in accordance with Eqs. (8) and (9). Four quadrupoles upstream of the measurement set-up should be sufficient to set the optics at the entrance of the measurement quadrupole. Then the covered phase-advance has to be chosen, and the quadrupole gradients required for the scan will be obtained from Eq. (11). Since all the above equations are obtained assuming thin-lens approximation for the quadrupole, the real phase-advances will slightly differ from the assumed ones and the real β-functions will not be exactly the same as the ones obtained from Eq. (12). These differences are, however, negligible in practice (as long as L is significantly longer than the quadrupole length). Fig. 2 shows the scan parameters for our example (α0 ¼1, β0 ¼ 10 m, L ¼10 m) when the phase-advance is progressively covered from 151 to 1651. The symmetry of the method can be clearly seen: for any scan index the β-functions are the same and the phase-advance is equivalent in x and y. More details about how to obtain the different formulas presented in this section can be found in Appendix A. Like in any other controlled optics-based emittance measurement, the proposed method is optimized for certain design incoming optics but it also works for initially mismatched beams. In practice, the incoming optics can differ from that expected. In this case the phase-advances and the β-functions at the measurement location will differ from the calculated ones and the scan will not be symmetric. In most of the situations, however, it will still be possible to determine the emittance and Twiss parameters to a first approximation. Then, based on the reconstructed optical functions, some quadrupoles upstream of the measurement setup should be varied to match the incoming optics to the design.
15
0
0
5
10
15
scan index
Fig. 3. Simulation of a measurement at the SITF for a matched beam of 250 MeV and a projected emittance of 500 nm in both planes. Upper plots show the normalized phase-space for x (left) and y (right)—the blue circles are the reconstructed normalized ellipses and the red lines correspond to the beampsize ffiffiffiffiffiffi 0 measurements.1 p u ffiffiffiffiffi and ffi u are normalized to the measured values: uN ¼ u= β u , 0 0 uN ¼ ðuαu þ u β u Þ= βu . In this way the reconstructed ellipses are always circles. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
This procedure (measurement-matching) should be repeated until the beam is properly matched and the scan is symmetric as described above.
3. Implementation at the SwissFEL Injector Test Facility The SwissFEL project [9], planned at the Paul Scherrer Institute, foresees the realization of a SASE X-ray FEL with a photon wavelength down to 0.1 nm. The SwissFEL Injector Test Facility (SITF) [10] is a 250 MeV accelerator with the main goal to demonstrate the high-brightness electron beam required to drive the SwissFEL facility. The SITF further serves as a platform for the development and validation of accelerator components needed for the SwissFEL project. At the SITF the distance between the quadrupole and the profile monitor is 10.83 m. The optics at the quadrupole entrance that fulfill Eq. (9) are chosen to be: β0 ¼ 15.00 m and α0 ¼1.385. The quadrupole gradients used in the measurements are chosen to cover 1501 of phase-advance (from 151 to 1651): according to Eq. (11) the integrated focusing of the quadrupole has to be scanned in the range 70.249 m 1. From Eq. (12) we get that the β-function for a matched beam will vary between 7.8 m and 116.4 m. In reality, taking into account the non-zero length of the quadrupole, these numbers are slightly different but practically equivalent: for a matched beam the β-functions will vary between 8.0 m and 117.5 m and the phase-advance will be covered from 15.51 to 165.01. We have simulated this measurement method using a Gaussian beam with a normalized emittance of 500 nm in both planes and 1 Lower plots show the reconstructed and virtually measured beam sizes as a function of the scan index. The emittance can be reconstructed with a statistical error of 2.6% (for 5% beam-size measurement error).
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E. Prat / Nuclear Instruments and Methods in Physics Research A 743 (2014) 103–108
Emittance error [%] 20
5
18
4
16
3
14
2 α
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
12
1 0
10
−1
8
−2
6 4
−3 0
20
40
60
80
100
β [m] 20
3
Fail rate [%]
Emittance error [%]
2.5 15
10
2
y [mm]
1.5 1
5 0.5 0
1
2
3 M
4
0
5
1
2
3 M
4
5
Fig. 4. Upper plot shows the relative emittance error for different incoming optics. The color code is limited to 20% to make the details around the matched solution visible, which is indicated by the white dot. A relative beam size measurement error of 5% is assumed. The lower plots show the emittance error (left) and the failed fits (right) as a function of the mismatch parameter for the cases for which M r 5. Per each point we performed simulations for 1000 different random seeds.
x 10−5
x 10−5
2
y’N [m1/2]
x’N [m1/2]
2
0
−2
−2
−2
0
xN [m1/2] 4
0
−2
2 x 10−5
0
yN [m1/2]
x 10−4
8
x 10−4
reconstruction
2
1
0
measurement
6
σy [m]
σx [m]
reconstruction
measurement
3
2 x 10−5
4
2
0
5
10
scan index
15
0
0
5
10
15
scan index
Fig. 5. Measurement example at the SITF for a mismatched beam. Upper plots show the normalized phase-space for x (left) and y (right). Lower plots show the measured and reconstructed beam size measurements. The reconstructed normalized emittances are 275 7 7 nm in x and 397 7 15 nm in y. The errors correspond to the statistical uncertainties calculated from the statistical errors of the beam size measurements—the beam sizes were measured 10 times per every scan index. See also Fig. 6.
an energy of 250 MeV. The phase-advance is scanned from 151 to 1651 in steps of 101. We have assumed 5% of beam-size measurement error. Fig. 3 shows an example of a virtual measurement for a
x [mm]
Fig. 6. Measurement example at the SITF for a mismatched beam: beam images as a function of the scan index. See also Fig. 5.
matched beam. The calculation of the emittance and the optics is done by finding the values that best fit the measured beam sizes in a least-squares procedure. We performed simulations for 50,000 different seeds to evaluate the errors of the reconstructed parameters. The actual values of the emittance and the optics can be reconstructed accurately—the statistical error of the reconstructed emittance is 2.6%. The profile monitor used for the measurements is a YAG screen with a measured resolution of about 15 μm. This corresponds to a normalized emittance resolution of about 14 nm for an energy of 250 MeV—taking β ¼ βmin 8 m. The calibration error of the profile monitor is of the order of 1% [11], which causes a systematic error of the reconstructed emittances of around 2%. The calibration uncertainty is dominant over other systematic errors like energy uncertainties or quadrupole field errors—the magnitude of these two error sources is around 0.1% according to the accuracy of beam momentum and magnetic field measurements done at the SITF. Fig. 4 shows how the statistical error of the reconstructed emittance varies with the mismatch parameter,2 assuming that the relative beam size measurement error is 5%. The figure shows how the emittance error increases with the mismatch, therefore the beam should always be matched at the entrance of the quadrupole to avoid a systematic increase of the errors of the reconstructed parameters. Even for strongly mismatched beams (M r 5), however, the fit works most of the times and the emittance error is below 20%. Figs. 5 and 6 show an example of a measurement done at the SITF for an initially mismatched beam. Every measurement takes between 90 and 120 s. This time is dominated by the data acquisition and saving. The beam energy was 210 MeV and the charge was 200 pC. As in the virtual measurement, the quadrupole strengths were chosen to scan the phase-advance from 151 to 1651
2 The mismatch parameter is defined as in Ref. [5]: M ¼ 12ðβγ d 2ααd þ γβd Þ, where the subscript d denotes the design values. For a perfectly matched beam M¼ 1.
E. Prat / Nuclear Instruments and Methods in Physics Research A 743 (2014) 103–108
x 10−5
x 10−5
2
2
y’N [m1/2]
x’N [m1/2]
107
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
−2
−2
−2
0
xN [m1/2]
2 x 10−5
−2
0
2
yN [m1/2]
x 10−5
x 10−4
x 10−4 4
4
reconstruction measurement
3
reconstruction measurement
3
σy [m]
σx [m]
y [mm]
2
1
1
0
2
0
5
10
scan index
15
0
0
5
10
15
scan index
x [mm]
Fig. 7. Measurement example at the SITF for a well-matched beam. The reconstructed normalized emittances are 288 74 nm in x and 329 7 4 nm in y. See also Fig. 8.
Fig. 8. Measurement example at the SITF for a well-matched beam: beam images as a function of the scan index. See also Fig. 7.
in steps of 101. The reconstructed mismatch parameter was 1.76 in x and 1.90 in y. Due to the optics mismatch the scan was not symmetric. The beam sizes in the profile monitor varied between 55 μm and 630 μm. After two matching iterations using five quadrupoles placed upstream of the measurement section the beam could be properly matched. Figs. 7 and 8 show the results for this measurement. The mismatch parameter was 1.01 in x and 1.00 in y and the normalized emittances were around 300 nm in both planes (288 74 nm in x and 329 7 4 nm in y). These values slightly differ from the first emittances most likely due to the larger systematic errors in the initial mismatched case (see Fig. 4). The scan was symmetric and the beam size in the profile monitor varied between 80 μm and 320 μm—less variation than in the case of the mismatched beam. As it can be seen from Figs. 6 and 8, by just looking at the images during the scan it is possible to check whether the beam is matched or not, which is a practical advantage during operation. The initial optics are typically close to the design in routine operation and it requires hardly more than two iterations to match the optics. Therefore the proposed technique can extensively be used as described in the manuscript.
We have demonstrated with the SITF that by applying this method the 2D beam parameters can be reconstructed with small errors.
Acknowledgments Many thanks go to Bolko Beutner, Sven Reiche, and Thomas Schietinger for fruitful discussions and careful proof-reading of the manuscript.
Appendix A. Demonstration of formulas in Section 2 This appendix is based on the matrix formalism described in Refs. [1,2]. The horizontal and vertical transfer matrices of a lattice consisting of a quadrupole magnet (thin-lens approximation) and a drift space are Rx ¼ Rdrift Rquad x ¼ Ry ¼ Rdrift Rquad y ¼
4. Conclusion We have presented a novel method to measure the 2D emittance and optics. It consists of a symmetric single quadrupole-scan that measures the two transverse planes simultaneously and equivalently. This new and intuitive procedure is about two times faster than the conventional quadrupole scans. The beam sizes in the measurement position are kept in a range than can easily be measured with standard profile monitors. The method requires relatively small quadrupole fields, which is convenient to control possible chromatic effects or transverse kicks during the measurement. The parameters of the scan (quadrupole strengths, β-functions covered phase-advance, initial optics and drift length) are related to each other by a few simple formulas, which allows designing the measurement set-up in an easy and convenient way.
1
L
0
1
1 0
L 1
1
0
1f
1
1
0
þ 1f
1
0
!
¼@ 0
!
¼@
1 Lf
L
1f
1
1 þ Lf
L
þ 1f
1
1 A 1 A:
ðA:1Þ
The β-function at the observation position s can be obtained from the optical functions at the reconstruction point s0 and the transfer matrix elements using the following equations:
βx ðsÞ ¼ R211 βx ðs0 Þ 2R11 R12 αx ðs0 Þ þ R212 γ x ðs0 Þ βy ðsÞ ¼ R233 βy ðs0 Þ 2R33 R34 αy ðs0 Þ þ R234 γ y ðs0 Þ:
ðA:2Þ
Considering that the initial optical functions are the same in x and y (Eq. (8)) and the expressions of the transfer elements for our lattice (Eq. (A.1)) we get the following: ! ! L2 L L2 βx ðsÞ ¼ 1 þ 2 2 β0 2L 2 α0 þ L2 γ 0 f f f ! ! 2 L L L2 βy ðsÞ ¼ 1 þ 2 þ 2 β0 2L þ 2 α0 þL2 γ 0 : ðA:3Þ f f f
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E. Prat / Nuclear Instruments and Methods in Physics Research A 743 (2014) 103–108
Subtracting the previous two expressions with βx ðsÞ ¼ βy ðsÞ ¼ β for any quadrupole gradient, we immediately arrive at Eq. (9), α0 L ¼ β 0 . We can now write some elements of Rx and Ry as a function of the optical functions and the phase-advances, and equate them to the terms written in Eq. (A.1): qffiffiffiffiffiffiffiffiffi R12 ¼ ββ0 sin μx ¼ L ðA:4Þ R34 ¼
qffiffiffiffiffiffiffiffiffi
ββ0 sin μy ¼ L
ðA:5Þ
sffiffiffiffiffiffi R11 ¼
β L ð cos μx þ α0 sin μx Þ ¼ 1 f β0 β L ð cos μy þ α0 sin μy Þ ¼ 1 þ : f β0
ðA:7Þ
ðA:8Þ
On the other hand, if we sum the expressions of Eqs. (A.6) and (A.7) we obtain the following: sffiffiffiffiffiffi
β ð cos μx þ cos μy þ 2α0 sin μx Þ ¼ 2: β0
ðA:10Þ
From Eqs. (A.8) and (A.10) we get the result of Eq. (10):
μx þ μy ¼ π :
ðA:11Þ
Finally, Eqs. (11) and (12) can be obtained by combining Eqs. (A.4), (A.5), (A.6) and (A.7). References
Eqs. (A.4) and (A.5) can only be fulfilled if: sin μx ¼ sin μy :
cos μx ¼ cos μy :
ðA:6Þ
sffiffiffiffiffiffi R33 ¼
pffiffiffiffiffiffiffiffiffiffiffiffi If now one replaces sin μx by L= β β 0 (Eq. (A.4)) and uses the condition from Eq. (9) it follows that:
ðA:9Þ
[1] K.L. Brown, SLAC Report No. 75, 1982. [2] J. Rossbach, P. Schmüser, in: S. Turner (Ed.) Proceedings of the 5th CAS General Accelerator Physics Course, Jyväskylä, 1992, p. 17. [3] C. Lejeune, J. Aubert, in: A. Septier (Ed.) Applied Charged Particle Optics, Part A. Advances in Electronics and Electron Physics, Supplement 13A, Academic Press, New York, 1980. [4] H. Wiedemann, Particle Accelerator Physics I, Springer, Berlin, 1998. [5] M. Minty, F. Zimmermann, Measurement and Control of Charged Particle Beams, Springer, Berlin, 2003. [6] F. Löhl, Master's thesis, University of Hamburg, 2005. [7] J. Bengtsson, W. Leemans, T. Byrne, in: Proceedings of the 15th Particle Accelerator Conference, Washington, 1993, p. 567. [8] P.G. Tenenbaum, CLIC-Note No. 326, 1997. [9] R. Ganter, et al., PSI Report No. 10-04, 2012. [10] M. Pedrozzi, et al., PSI Report No. 10-05, 2010. [11] R. Ischebeck, private communication.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Symmetric single-quadrupole-magnet scan method to measure the 2D transverse beam parameters Eduard Prat Paul Scherrer Institut, 5232 Villigen, Switzerland
art ic l e i nf o
a b s t r a c t
Article history: Received 14 October 2013 Received in revised form 9 January 2014 Accepted 9 January 2014 Available online 21 January 2014
Precise measurements of the transverse beam parameters are essential to control and optimize all types of charged particle beams. In this paper we present a novel method that uses one quadrupole magnet and one profile monitor to measure the transverse beam emittance and optics. In comparison to a conventional single-quadrupole scan measurement, this new technique measures the two transverse planes simultaneously. This novel procedure is faster, more intuitive and allows keeping under control the required quadrupole gradient and the beam sizes at the profile monitor. The application of the method is illustrated with the SwissFEL Injector Test Facility. & 2014 Elsevier B.V. All rights reserved.
Keywords: Linear accelerators Free-electron lasers Electron optics Beam characteristics
1. Introduction
The transport of the 2D beam matrix from s0 to s can be described by a matrix formalism [1,2]:
An accurate knowledge of the transverse beam properties is fundamental in all types of charged particle accelerators. For instance, in Free-Electron Lasers (FEL) a high quality electron beam with low emittance and matched optics is required for the optimal lasing process. The 2-dimensional (2D) transverse beam matrix composed of the second-order moments of the beam distribution describes the statistical properties of the beam: ! 〈u2 〉 〈uu0 〉 ð1Þ suu ¼ 〈uu0 〉 〈u02 〉
suu ðsÞ ¼ R suu ðs0 Þ RT
where u refers to either x (horizontal coordinate) or y (vertical coordinate), and u0 is the derivative of u with respect to the longitudinal coordinate. The beam emittance and optical functions (α, β and γ, also called Twiss parameters) can be derived as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εu ¼ detðsuu Þ ð2Þ
αu ¼ 〈uu0 〉=εu ;
βu ¼ 〈u2 〉=εu ;
γ u ¼ 〈u02 〉=εu :
ð3Þ
When we multiply the emittance by the beam total momentum we get the so-called normalized emittance:
εN u ¼
p εu mc
ð4Þ
where p is the central momentum of the beam, m is the particle mass and c is the speed of light. 0168-9002/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2014.01.021
where R is the transverse transfer matrix from s0 to s: ! Ruu Ruu0 R¼ : Ru0 u Ru0 u0
ð5Þ
ð6Þ
According to Eqs. (1), (5) and (6), the beam size at a position s can be expressed as follows: 〈u2 〉s ¼ R2uu 〈u2 〉s0 þ R2uu0 〈u02 〉s0 þ 2Ruu Ruu0 〈uu0 〉s0 :
ð7Þ
The elements of suu can be reconstructed at s0 by measuring the beam size at s for different optics transformations between s0 and s. At least three measurements (i.e. three equations) are needed to reconstruct the three second moments of the beam, but more measurements will allow improving the robustness of the reconstruction. Ideally, the betatron phase-advance μu between the observation and reconstruction points should be scanned progressively between 0 and π for the best reconstruction of the 2D parameters. In order to obtain the 2D parameters the beam sizes can be measured at different positions along a fixed lattice (multipleposition approach like FODO measurements) or alternatively the optics between the reconstruction and the observation points can be varied with one or more quadrupole magnets (multiple-optics approach or quadrupole scans). The multiple-optics approach has the inconvenience of modifying the optics during a measurement, but it is more compact, that requires less equipment and is more flexible than the multiple-position strategy. Because of that in
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quadrupole scans there is no need for a long diagnostic section, therefore the overall length of a typical accelerator is reduced. Refs. [3–8] give more detailed information about definitions and measurement procedures concerning the 2D transverse beam parameters. In a conventional single-quadrupole scan measurement the phase-advance is controlled only in one plane at a time. Moreover, the beam size can go through a very tight focus at the measurement position during the scan, which makes the reconstruction of the parameters very difficult and sensitive to beam size measurement errors. In addition, the beam size variations can be very large along the scan, which may give rise to dynamic range problems for the profile monitor. This document presents a special single-quadrupole scan measurement procedure with the following features:
40 focusing plane defocusing plane
β [m]
30 20
Convergent beam 10 0
0
5
focusing plane defocusing plane
β [m]
The horizontal and vertical phase-advances are scanned simulta
2. Description and formulas The measurement set-up consists simply of a quadrupole, a drift space, and a profile monitor, which measures the beam sizes in both planes simultaneously. The quadrupole entrance is the reference point where the beam parameters (emittance and optical functions) are reconstructed. All the equations presented in this section are obtained assuming thin-lens approximation for the quadrupole magnet. The optics at the reconstruction point need to be identical in both planes if we want to simultaneously scan x and y with a single quadrupole. We define the optics at the reconstruction position as follows:
αx ðs0 Þ ¼ αy ðs0 Þ ¼ α0 βx ðs0 Þ ¼ βy ðs0 Þ ¼ β0 γ x ðs0 Þ ¼ γ y ðs0 Þ ¼ γ 0 :
ð8Þ
As exemplified in Fig. 1, the propagated β-functions downstream of the quadrupole cross each other at a given point only if the beam is initially convergent. This happens for any quadrupole gradient at a distance L from the quadrupole such that:
α0 L ¼ β 0 :
ð9Þ
If this condition is fulfilled the sum of the phase-advances in x and y (between the quadrupole and the profile monitor) will be
15
150
100
neously. This improves the time of the measurement by a factor of about two with respect to conventional measurements. The β-function at the measurement point is the same in x and y for any quadrupole gradient. Consequently the ratio between the beam size in x and y should be constant for the whole scan (i.e. round beam if the emittances are equal in both planes), but varies for mismatched beam parameters at the reconstruction point. It will be easy and intuitive to measure whether the beam is matched or not. The scan produces a gentle waist. The β-function at the waist is relatively large (with typical values of 10 m) and the beam size variations at the measurement location are kept within moderate limits. Therefore, dynamic problems of the profile monitors and space charge effects are avoided. The range of the quadrupole gradient required for the measurement stays within reasonable limits, which minimizes possible chromatic effects and trajectory deviations during the scan. The β-function at the measurement position and the required quadrupole strengths can be tuned by changing some parameters, such as the covered phase-advance or the initial optics.
10
s [m]
Divergent beam
50
0
0
5
10
15
s [m] Fig. 1. β-function along a drift after a quadrupole with a certain strength. The upper plot shows a case where the initial beam is convergent (β0 ¼ 10 m, α0 ¼ 1) and the lower plot corresponds to a divergent initial beam (β0 ¼ 10 m, α0 ¼ 1). Only if the beam is initially convergent there is a position where the β-functions are the same in both planes. This point is at a distance L ¼ β0 =α0 from the quadrupole (in our example L ¼ 10 m). This condition holds for any gradient of the quadrupole magnet and also satisfies μx þ μy ¼ π.
exactly 1801:
μx þ μy ¼ π :
ð10Þ
This means that the scan will be equivalent in x and y, since for the reconstruction of the 2D parameters a certain phase-advance μ provides the same information as π μ. The integrated focusing of the quadrupole can be obtained as a function of the phase-advance and the initial β-function using the following equation: 1 1 1 ¼ kl ¼ ¼ : f tan μx β 0 tan μy β 0
ð11Þ
One can see that for no quadrupole gradient the phase-advance in x and y is identical and equal to π =2. The range of the quadrupole gradients to perform the scan will depend on the phase-advance that we want to cover. For example, for β0 ¼ 10 m we will need kl ¼ 7 0:10 m 1 to cover π =2 or kl ¼ 70:24 m 1 to scan 3π =4. The β-function at the measurement point can be obtained as follows:
β¼
L2
β0 sin 2 μ
ð12Þ
where μ is the phase-advance in x or y. The observed minimum βfunction at the measurement location occurs for no quadrupole gradient, when μx ¼ μy ¼ π =2:
βmin ¼
L2
β0
:
ð13Þ
This quantity can be controlled by changing the initial optics and the distance between the quadrupole and the profile monitor. For our example where β0 ¼ 10 m and L ¼10 m, β min will also be 10 m. From Eqs. (12) and (13) it can be seen that the ratio between the β-function and the βmin function only depends on the covered
E. Prat / Nuclear Instruments and Methods in Physics Research A 743 (2014) 103–108
x 10−5
4
2
50 10
x 10−5
2 y’N [m1/2]
100
4
x y x’N [m1/2]
β [m]
150
105
0 −2
0 −2
1 −4 −4
0 xN
0.5 0.25
x y
4
0 0.4 σx [m]
0
2
[m1/2]
−4 −4
4 x 10−5
−2
0
4 reconstruction
1
4 x 10−5
x 10−4
reconstruction
3
virtual meas.
2
2
yN [m1/2]
x 10−4
3
0.2
kl [1/m]
−2
σy [m]
μ [π]
0.75
virtual meas.
2 1
−0.2 0
−0.4
5
10 scan index
scan index
Fig. 2. β-function, phase-advance and quadrupole strength as a function of the scan index when the phase-advance is progressively covered from 151 to 1651 (for example α0 ¼ 1, β0 ¼ 10 m, L ¼ 10 m).
phase-advance:
β 1 ¼ : βmin sin 2 μ
0
ð14Þ
For instance, if the phase-advance is scanned from 151 to 1651 the
β-function will increase by a factor of 15 with respect to the waist (i. e. around a factor of 4 in beam size). To implement this measurement the lattice and the initial optics have to be chosen in accordance with Eqs. (8) and (9). Four quadrupoles upstream of the measurement set-up should be sufficient to set the optics at the entrance of the measurement quadrupole. Then the covered phase-advance has to be chosen, and the quadrupole gradients required for the scan will be obtained from Eq. (11). Since all the above equations are obtained assuming thin-lens approximation for the quadrupole, the real phase-advances will slightly differ from the assumed ones and the real β-functions will not be exactly the same as the ones obtained from Eq. (12). These differences are, however, negligible in practice (as long as L is significantly longer than the quadrupole length). Fig. 2 shows the scan parameters for our example (α0 ¼1, β0 ¼ 10 m, L ¼10 m) when the phase-advance is progressively covered from 151 to 1651. The symmetry of the method can be clearly seen: for any scan index the β-functions are the same and the phase-advance is equivalent in x and y. More details about how to obtain the different formulas presented in this section can be found in Appendix A. Like in any other controlled optics-based emittance measurement, the proposed method is optimized for certain design incoming optics but it also works for initially mismatched beams. In practice, the incoming optics can differ from that expected. In this case the phase-advances and the β-functions at the measurement location will differ from the calculated ones and the scan will not be symmetric. In most of the situations, however, it will still be possible to determine the emittance and Twiss parameters to a first approximation. Then, based on the reconstructed optical functions, some quadrupoles upstream of the measurement setup should be varied to match the incoming optics to the design.
15
0
0
5
10
15
scan index
Fig. 3. Simulation of a measurement at the SITF for a matched beam of 250 MeV and a projected emittance of 500 nm in both planes. Upper plots show the normalized phase-space for x (left) and y (right)—the blue circles are the reconstructed normalized ellipses and the red lines correspond to the beampsize ffiffiffiffiffiffi 0 measurements.1 p u ffiffiffiffiffi and ffi u are normalized to the measured values: uN ¼ u= β u , 0 0 uN ¼ ðuαu þ u β u Þ= βu . In this way the reconstructed ellipses are always circles. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
This procedure (measurement-matching) should be repeated until the beam is properly matched and the scan is symmetric as described above.
3. Implementation at the SwissFEL Injector Test Facility The SwissFEL project [9], planned at the Paul Scherrer Institute, foresees the realization of a SASE X-ray FEL with a photon wavelength down to 0.1 nm. The SwissFEL Injector Test Facility (SITF) [10] is a 250 MeV accelerator with the main goal to demonstrate the high-brightness electron beam required to drive the SwissFEL facility. The SITF further serves as a platform for the development and validation of accelerator components needed for the SwissFEL project. At the SITF the distance between the quadrupole and the profile monitor is 10.83 m. The optics at the quadrupole entrance that fulfill Eq. (9) are chosen to be: β0 ¼ 15.00 m and α0 ¼1.385. The quadrupole gradients used in the measurements are chosen to cover 1501 of phase-advance (from 151 to 1651): according to Eq. (11) the integrated focusing of the quadrupole has to be scanned in the range 70.249 m 1. From Eq. (12) we get that the β-function for a matched beam will vary between 7.8 m and 116.4 m. In reality, taking into account the non-zero length of the quadrupole, these numbers are slightly different but practically equivalent: for a matched beam the β-functions will vary between 8.0 m and 117.5 m and the phase-advance will be covered from 15.51 to 165.01. We have simulated this measurement method using a Gaussian beam with a normalized emittance of 500 nm in both planes and 1 Lower plots show the reconstructed and virtually measured beam sizes as a function of the scan index. The emittance can be reconstructed with a statistical error of 2.6% (for 5% beam-size measurement error).
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Emittance error [%] 20
5
18
4
16
3
14
2 α
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
12
1 0
10
−1
8
−2
6 4
−3 0
20
40
60
80
100
β [m] 20
3
Fail rate [%]
Emittance error [%]
2.5 15
10
2
y [mm]
1.5 1
5 0.5 0
1
2
3 M
4
0
5
1
2
3 M
4
5
Fig. 4. Upper plot shows the relative emittance error for different incoming optics. The color code is limited to 20% to make the details around the matched solution visible, which is indicated by the white dot. A relative beam size measurement error of 5% is assumed. The lower plots show the emittance error (left) and the failed fits (right) as a function of the mismatch parameter for the cases for which M r 5. Per each point we performed simulations for 1000 different random seeds.
x 10−5
x 10−5
2
y’N [m1/2]
x’N [m1/2]
2
0
−2
−2
−2
0
xN [m1/2] 4
0
−2
2 x 10−5
0
yN [m1/2]
x 10−4
8
x 10−4
reconstruction
2
1
0
measurement
6
σy [m]
σx [m]
reconstruction
measurement
3
2 x 10−5
4
2
0
5
10
scan index
15
0
0
5
10
15
scan index
Fig. 5. Measurement example at the SITF for a mismatched beam. Upper plots show the normalized phase-space for x (left) and y (right). Lower plots show the measured and reconstructed beam size measurements. The reconstructed normalized emittances are 275 7 7 nm in x and 397 7 15 nm in y. The errors correspond to the statistical uncertainties calculated from the statistical errors of the beam size measurements—the beam sizes were measured 10 times per every scan index. See also Fig. 6.
an energy of 250 MeV. The phase-advance is scanned from 151 to 1651 in steps of 101. We have assumed 5% of beam-size measurement error. Fig. 3 shows an example of a virtual measurement for a
x [mm]
Fig. 6. Measurement example at the SITF for a mismatched beam: beam images as a function of the scan index. See also Fig. 5.
matched beam. The calculation of the emittance and the optics is done by finding the values that best fit the measured beam sizes in a least-squares procedure. We performed simulations for 50,000 different seeds to evaluate the errors of the reconstructed parameters. The actual values of the emittance and the optics can be reconstructed accurately—the statistical error of the reconstructed emittance is 2.6%. The profile monitor used for the measurements is a YAG screen with a measured resolution of about 15 μm. This corresponds to a normalized emittance resolution of about 14 nm for an energy of 250 MeV—taking β ¼ βmin 8 m. The calibration error of the profile monitor is of the order of 1% [11], which causes a systematic error of the reconstructed emittances of around 2%. The calibration uncertainty is dominant over other systematic errors like energy uncertainties or quadrupole field errors—the magnitude of these two error sources is around 0.1% according to the accuracy of beam momentum and magnetic field measurements done at the SITF. Fig. 4 shows how the statistical error of the reconstructed emittance varies with the mismatch parameter,2 assuming that the relative beam size measurement error is 5%. The figure shows how the emittance error increases with the mismatch, therefore the beam should always be matched at the entrance of the quadrupole to avoid a systematic increase of the errors of the reconstructed parameters. Even for strongly mismatched beams (M r 5), however, the fit works most of the times and the emittance error is below 20%. Figs. 5 and 6 show an example of a measurement done at the SITF for an initially mismatched beam. Every measurement takes between 90 and 120 s. This time is dominated by the data acquisition and saving. The beam energy was 210 MeV and the charge was 200 pC. As in the virtual measurement, the quadrupole strengths were chosen to scan the phase-advance from 151 to 1651
2 The mismatch parameter is defined as in Ref. [5]: M ¼ 12ðβγ d 2ααd þ γβd Þ, where the subscript d denotes the design values. For a perfectly matched beam M¼ 1.
E. Prat / Nuclear Instruments and Methods in Physics Research A 743 (2014) 103–108
x 10−5
x 10−5
2
2
y’N [m1/2]
x’N [m1/2]
107
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
−2
−2
−2
0
xN [m1/2]
2 x 10−5
−2
0
2
yN [m1/2]
x 10−5
x 10−4
x 10−4 4
4
reconstruction measurement
3
reconstruction measurement
3
σy [m]
σx [m]
y [mm]
2
1
1
0
2
0
5
10
scan index
15
0
0
5
10
15
scan index
x [mm]
Fig. 7. Measurement example at the SITF for a well-matched beam. The reconstructed normalized emittances are 288 74 nm in x and 329 7 4 nm in y. See also Fig. 8.
Fig. 8. Measurement example at the SITF for a well-matched beam: beam images as a function of the scan index. See also Fig. 7.
in steps of 101. The reconstructed mismatch parameter was 1.76 in x and 1.90 in y. Due to the optics mismatch the scan was not symmetric. The beam sizes in the profile monitor varied between 55 μm and 630 μm. After two matching iterations using five quadrupoles placed upstream of the measurement section the beam could be properly matched. Figs. 7 and 8 show the results for this measurement. The mismatch parameter was 1.01 in x and 1.00 in y and the normalized emittances were around 300 nm in both planes (288 74 nm in x and 329 7 4 nm in y). These values slightly differ from the first emittances most likely due to the larger systematic errors in the initial mismatched case (see Fig. 4). The scan was symmetric and the beam size in the profile monitor varied between 80 μm and 320 μm—less variation than in the case of the mismatched beam. As it can be seen from Figs. 6 and 8, by just looking at the images during the scan it is possible to check whether the beam is matched or not, which is a practical advantage during operation. The initial optics are typically close to the design in routine operation and it requires hardly more than two iterations to match the optics. Therefore the proposed technique can extensively be used as described in the manuscript.
We have demonstrated with the SITF that by applying this method the 2D beam parameters can be reconstructed with small errors.
Acknowledgments Many thanks go to Bolko Beutner, Sven Reiche, and Thomas Schietinger for fruitful discussions and careful proof-reading of the manuscript.
Appendix A. Demonstration of formulas in Section 2 This appendix is based on the matrix formalism described in Refs. [1,2]. The horizontal and vertical transfer matrices of a lattice consisting of a quadrupole magnet (thin-lens approximation) and a drift space are Rx ¼ Rdrift Rquad x ¼ Ry ¼ Rdrift Rquad y ¼
4. Conclusion We have presented a novel method to measure the 2D emittance and optics. It consists of a symmetric single quadrupole-scan that measures the two transverse planes simultaneously and equivalently. This new and intuitive procedure is about two times faster than the conventional quadrupole scans. The beam sizes in the measurement position are kept in a range than can easily be measured with standard profile monitors. The method requires relatively small quadrupole fields, which is convenient to control possible chromatic effects or transverse kicks during the measurement. The parameters of the scan (quadrupole strengths, β-functions covered phase-advance, initial optics and drift length) are related to each other by a few simple formulas, which allows designing the measurement set-up in an easy and convenient way.
1
L
0
1
1 0
L 1
1
0
1f
1
1
0
þ 1f
1
0
!
¼@ 0
!
¼@
1 Lf
L
1f
1
1 þ Lf
L
þ 1f
1
1 A 1 A:
ðA:1Þ
The β-function at the observation position s can be obtained from the optical functions at the reconstruction point s0 and the transfer matrix elements using the following equations:
βx ðsÞ ¼ R211 βx ðs0 Þ 2R11 R12 αx ðs0 Þ þ R212 γ x ðs0 Þ βy ðsÞ ¼ R233 βy ðs0 Þ 2R33 R34 αy ðs0 Þ þ R234 γ y ðs0 Þ:
ðA:2Þ
Considering that the initial optical functions are the same in x and y (Eq. (8)) and the expressions of the transfer elements for our lattice (Eq. (A.1)) we get the following: ! ! L2 L L2 βx ðsÞ ¼ 1 þ 2 2 β0 2L 2 α0 þ L2 γ 0 f f f ! ! 2 L L L2 βy ðsÞ ¼ 1 þ 2 þ 2 β0 2L þ 2 α0 þL2 γ 0 : ðA:3Þ f f f
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Subtracting the previous two expressions with βx ðsÞ ¼ βy ðsÞ ¼ β for any quadrupole gradient, we immediately arrive at Eq. (9), α0 L ¼ β 0 . We can now write some elements of Rx and Ry as a function of the optical functions and the phase-advances, and equate them to the terms written in Eq. (A.1): qffiffiffiffiffiffiffiffiffi R12 ¼ ββ0 sin μx ¼ L ðA:4Þ R34 ¼
qffiffiffiffiffiffiffiffiffi
ββ0 sin μy ¼ L
ðA:5Þ
sffiffiffiffiffiffi R11 ¼
β L ð cos μx þ α0 sin μx Þ ¼ 1 f β0 β L ð cos μy þ α0 sin μy Þ ¼ 1 þ : f β0
ðA:7Þ
ðA:8Þ
On the other hand, if we sum the expressions of Eqs. (A.6) and (A.7) we obtain the following: sffiffiffiffiffiffi
β ð cos μx þ cos μy þ 2α0 sin μx Þ ¼ 2: β0
ðA:10Þ
From Eqs. (A.8) and (A.10) we get the result of Eq. (10):
μx þ μy ¼ π :
ðA:11Þ
Finally, Eqs. (11) and (12) can be obtained by combining Eqs. (A.4), (A.5), (A.6) and (A.7). References
Eqs. (A.4) and (A.5) can only be fulfilled if: sin μx ¼ sin μy :
cos μx ¼ cos μy :
ðA:6Þ
sffiffiffiffiffiffi R33 ¼
pffiffiffiffiffiffiffiffiffiffiffiffi If now one replaces sin μx by L= β β 0 (Eq. (A.4)) and uses the condition from Eq. (9) it follows that:
ðA:9Þ
[1] K.L. Brown, SLAC Report No. 75, 1982. [2] J. Rossbach, P. Schmüser, in: S. Turner (Ed.) Proceedings of the 5th CAS General Accelerator Physics Course, Jyväskylä, 1992, p. 17. [3] C. Lejeune, J. Aubert, in: A. Septier (Ed.) Applied Charged Particle Optics, Part A. Advances in Electronics and Electron Physics, Supplement 13A, Academic Press, New York, 1980. [4] H. Wiedemann, Particle Accelerator Physics I, Springer, Berlin, 1998. [5] M. Minty, F. Zimmermann, Measurement and Control of Charged Particle Beams, Springer, Berlin, 2003. [6] F. Löhl, Master's thesis, University of Hamburg, 2005. [7] J. Bengtsson, W. Leemans, T. Byrne, in: Proceedings of the 15th Particle Accelerator Conference, Washington, 1993, p. 567. [8] P.G. Tenenbaum, CLIC-Note No. 326, 1997. [9] R. Ganter, et al., PSI Report No. 10-04, 2012. [10] M. Pedrozzi, et al., PSI Report No. 10-05, 2010. [11] R. Ischebeck, private communication.