Rotating system for four-dimensional transverse rms-emittance measurements

Knowledge of the transverse four-dimensional beam rms-parameters is essential for applications that involve lattice elements that couple the two transverse degrees of freedom (planes). Of special interest is the removal of inter-plane correlations to reduce the projected emittances. A dedicated ROtating System for Emittance measurements (ROSE) has been proposed, developed, and successfully commissioned to fully determine the four-dimensional beam matrix. This device has been used at the High Charge injector (HLI) at GSI using a beam line which is composed of a skew quadrupole triplet, a normal quadrupole doublet, and ROSE. Mathematical algorithms, measurements, and results for ion beams of 83Kr13+ at 1.4 MeV/u are reported in this paper.


I. INTRODUCTION
Emittance is an important figure of merit for propagation of charged particle beams. It is defined as the amount of phase-space being occupied by the particle distribution to quantify the beam quality and to match the following optics. Precise knowledge from measurements of particle distribution parameters is important for accelerator design and for phase-space manipulation. However, most of the published work is just on separated measurements of two-dimensional x-x ′ and y-y ′ sub-phasespaces (planes) [1][2][3]. For simplicity correlations between the two planes, i.e. x-y, x-y ′ , x ′ -y, and x ′ -y ′ are often assumed as zero. However, such inter-plane correlations may be produced by inter-plane coupling fields such as dipole fringes, solenoids, and titled magnets or just by beam losses [4].
Ion beams extracted from Electron Cyclotron Resonance (ECR) ion sources have a complex structure in the four-dimensional phase-space [5]. Distributions with equal projected rms-emittances are strongly correlated after extraction [6][7][8]. Correlations increase the projected rms-emittances. Removing correlations reduces the effective emittances without introduction of beam loss through scraping for instance. In order to remove unknown correlations, they must be quantified by measurements. Accordingly, four-dimensional diagnostics has the major task of allowing for elimination of inter-plane coupling.
For beam lines comprising just non-coupling elements as upright quadrupoles, dipoles, and accelerating gaps, the horizontal and vertical beam dynamics are decoupled. The design of such beam lines can be accomplished ignoring eventual inter-plane correlations in the beam as long as the projected distribution parameters are known at the beam line entrance. This convenience of ignoring correlations cannot be further afforded, if the beam line includes coupling elements as solenoids or skew quadrupoles for instance. In that cases even the projected distribution parameters along the beam line will depend on inter-plane correlations at the entrance of the beam line [9].
Using standard slit/grid emittance measurement devices [10,11] and multi-slit/screen devices [12] as the pepper-pot technique the projected phase-space distributions, i.e. the horizontal and vertical rms-emittances, can be measured. The slit determines the location of the phase-space element. A subsequent grid measures the angular distribution of the ions that passed the slit. By moving the slit and recording the angular distribution at each slit position, the projected phases-space distribution is measured. The direction of movements of the slit and grid determines the plane onto which the fourdimensional distribution is projected. Inter-plane correlation matrix-elements cannot be measured directly using a slit/grid configuration.
There is considerable work on measuring fourdimensional distributions using pepper-pots [13][14][15] for ion beams at energies below 150 keV/u, where the beam can be stopped by the pepper-pot mask. However, this method is not applicable at energies beyond 150 keV/u due to technical reasons, i.e. doubtful readout by temperature-dependent screens and fixed resolutions by holes and screens [16]. Four-dimensional emittance measurements were proposed and conducted for instance in [17][18][19][20][21] at electron machines. Other options based on phase space tomography technique have been developed to reconstruct the two-dimensional phase space in [22] and the full four-dimensional phase space in [23,24]. The combination of skew quadrupoles with a slit/grid emittance measurement devices has been applied successfully for high intensity uranium ions at an energy of 11.4 MeV/u at GSI [25]. This paper reports on a method without skew quadrupoles that features reduced time (about a factor of three) needed to perform the measurements within about one hour.
The paper starts with an introduction of the parameters that quantify four-dimensional particle distributions. The third section is on the method of ROSE: analytical calculation and numerical analysis are elaborated comprehensively. The subsequent section is on showing the commissioning method and software for measuring and evaluating the full four-dimensional beam matrix. The fifth section shows the capability of ROSE to provide the input for successful elimination of inter-plane coupling.

II. FOUR-DIMENSIONAL RMS-QUANTITIES
Four-dimensional beam rms-emittance measurements require determination of ten unique elements of the second moments beam matrix. The 4×4 symmetric second moments beam matrix C can be expressed as [26] where x and y are the horizontal and vertical coordinates, respectively, and x ′ and y ′ are their derivatives with respect to the longitudinal coordinate. Four of the matrix elements quantify the coupling. If at least one of the elements of the off-diagonal sub-matrix is non-zero, the beam is transversely coupled. Projected rms-emittances ε x and ε y are quantities which are used to characterize the transverse beam quality in the laboratory coordinate system and are invariant under linear uncoupled (with respect to the laboratory coordinate system) symplectic transformations. Projected rmsemittances are the rms phase-space areas from projections of the particle distribution onto the planes, and their values are equal to the square roots of the determinants of the on-diagonal 2×2 sub-matrices, i.e. phasespace area divided by π: where µ refers to either x or y. The dimensionless parameter α relates to the µ-µ ′ correlation and the β-function refers to the beam width. They are defined as The eigen-emittances ε 1 and ε 2 are invariant under coupled linear symplectic transformations provided by solenoids or skew quadrupoles for instance [27]. None of the projected emittances can be smaller than the smaller of the two eigen-emittances. The eigen-emittances can be expressed as [28] The square matrix J is the skew-symmetric matrix with non-zero entries in the block diagonal off form and J is defined as The eigen-emittances are equal to projected rmsemittances if and only if all inter-plane correlations are zero. If the second moments beam matrix has correlations between horizontal and vertical phase-spaces (see Equ. 1), the eigen-emittances and projected rmsemittances are different. The product of the eigenemittances can not be larger than the product of projected rms-emittances. The four-dimensional beam rmsemittance is calculated as The coupling parameter t is introduced to quantify inter-plane coupling as and if t is equal to zero, there are no inter-plane correlations and the projected rms-emittances are equal to the eigen-emittances.

III. ROSE METHOD
ROSE has been developed to measure the full fourdimensional transverse beam matrix of ion beams as shown in Fig. 1. It is a slit/grid combination being installed inside a rotatable vacuum chamber. In the slit/grid system of ROSE the slit has an opening width of d slit =0.2 mm and the step width is typically δµ=0.5 mm. Slit and grid are separated by d=300 mm and the wire distance is d wire =1.0 mm and the intermediate step number for moving the grid is n=4. The spacial/angular resolution of the emittance measurements is accordingly and the spacial/angular resolution of the emittance measurements is determined to be 0.5 mm/0.9 mrad. The emittance measurement unit can be rotated around the beam axis by a total of 270 • , and the rotation can be done within two minutes. One emittance measurement takes about 15 minutes. For rotation no shutters need to be closed. The vacuum pressure increased from few 10 −8 mbar to few 10 −7 mbar during rotation. Afterward the pressure recovered within about three minutes. A detail description of the mechanical set-up can be found in [29]. ROSE has been installed at the HLI section [30], as it is fed by an ECR source that provides correlated beams. It is installed as a mobile set-up, i.e. the corresponding chamber may be installed at many locations along the versatile GSI beam lines. The complete beam line consists of one skew quadrupole triplet and one regular quadrupole doublet followed by the ROSE unit as shown in Fig. 2.

A. Mathematical algorithms
The transport of the beam matrix from location i to location f can be calculated as (see Fig. 3) As the ROSE beam line is without coupling elements if the skew quadrupole triplet is switched off, the offdiagonal sub-matrices of the transport matrices vanish, i.e. M xy =M yx =0. The transports M a or M b of single particle coordinates from location i to location f using magnet setting a or b are described by transfer matrices through from Equ. 12, the correlated beam second moments of the off-diagonal sub-matrices at location f using magnet settings a and b can be written as x Rotating clockwise the emittance measurement unit by θ is equivalent to rotating the beam by -θ around the beam axis. After rotation, the new particle coordinates using magnet settings a or b are transported by a simple rotation matrix  According to Equ. 16 horizontal second moments after rotation using magnet settings a and b are expressed as All elements of the transport matrices M a,b xx and M a,b yy are known. The second moments θ after rotation can be measured. Combining Equ. 13 to Equ. 19 the solution of the searched coupled matrix elements at location i can be summarized to a set of linear equations and with the same procedure for setting b and finally Since there are k=4 unknown coupling parameters and n=6 linear equations, at least k=4 of them are selected to solve the unknown parameters. There are possibilities to do so. From Equ. 20 two arbitrary equations can be removed and doing so in total 15 algorithms are generated. For instance algorithm #1 reads and algorithm #15 reads The algorithms offer the possibility to reconstruct the full beam matrix at location i from emittance measurements of different rotation angles through B. Measurement procedure Projected rms-emittance measurements can be performed at various angles, i.e. 0 • , 90 • , and Θ • (any angle which is not equivalent to 0 • or 90 • ) to reconstruct the full four-dimensional beam matrix C i . Rotation by 0 • /90 • will just measure the usual uncoupled second moments. Rotation by Θ • provides access to the coupled beam second moments. As will be shown below, this optics comprises just non-coupling linear elements. Using a rotatable emittance measurement device, a minimum of four, but more reliable six measurements is sufficient to measure the complete four-dimensional beam matrix: (I) measurements at θ=0 • with optics a (and b).
(III) measurements at θ=Θ • with optics a and b.
If just four measurements are applied to evaluate the full beam matrix at location i, the uncoupled second moments for setting f are calculated from the final uncoupled second moments for setting a, i.e. xx a f , xx ′ a f , and x ′ x ′ a f using the transport matrices M a xx and M b xx . The four measurements (projected rms-emittances and Twiss parameters) and their deliverables are: (I) θ=0 • , magnet setting a delivers parameters xx a f , xx ′ a f , and x ′ x ′ a f . (II) θ=90 • , magnet setting a delivers parameters yy a f , yy ′ a f , and y ′ y ′ a f . (III) θ=Θ • , magnet setting a delivers parameters xx a θ , xx ′ a θ , and From step (I) the uncorrelated second moments xx i , xx ′ i , and x ′ x ′ i are obtained at location i by simple back transformation through inversion of Equ. 12. From step (II) the uncorrelated beam-moments yy i , yy ′ i , and y ′ y ′ i are obtained at location i in the same way. Steps (III) and (IV) deliver xy a,b f , xy ′ a,b f + x ′ y a,b f , and x ′ y ′ a,b f at location f (Equ. 13 to Equ. 15). Finally, Equ. 32 determines the correlated second moments xy i , xy ′ i , x ′ y i , and x ′ y ′ i at location i. The fourdimensional second moments beam matrix is then finally reconstructed at location i from four measurements.
If six measurements are applied, the uncoupled second moments for setting b at location f can be measured directly.

C. Minimizing the measurement errors
The vector Λ j (see Equ. 32) is sensitive to the emittance measurements at location f . During emittance measurements, finite grid bin results in finite resolution and background noise have influence on the measured second moments. The typical error of directly measured second moments is about 10%. These errors enter into the inversion of Equ. 32.
In the following the minimization of the measurement error of the coupled second moments by making use of the so-called condition number of a matrix is described. If the inverse Γ j −1 exists, the condition number of a square matrix Γ j is defined as This quantity is always bigger than or equal to 1.0. Let Γ T j be the transpose of the square matrix Γ j , then the spectral norm, or two-norm, of a matrix is defined as the square root of the maximum eigenvalue of Γ T j Γ j where ρ is the function that computes the spectral radius of Γ j . Since the matrix Γ T j Γ j is symmetric and all of its eigenvalues are real-valued and non-negative then ρ(Γ T j Γ j ) is the largest of these eigenvalues. The square matrix Γ j , related to the transfer matrixelements from location i to location f , is invertible but ill-conditioned if its condition number is too large. The condition number associated with the linear equations (see Equ. 32) is a measure for how ill-conditioned the matrix is. If the condition number is large, even a small error in emittance measurements may lead to radically different results for the beam coupling parameter evaluations. On the other hand, if the condition number is small the error in evaluation will not exceed notably the error in emittance measurements. The numerical stability (degeneration of the system) is better if the condition number is small. Well-conditioned matrices have condition numbers which are closed to 1.0.
We summarize that in order to obtain reasonable evaluation results it needs: (I) one reference emittance measurement with 100% transmission efficiency between location i and location f to obtain projected beam parameters at location i (ondiagonal section of beam matrix of C i ).
(II) that all quadrupoles are varied numerically in a brute-force method to check each setting for full transmission efficiency from location i to location f , and for reasonable beam sizes on slit/grid ( 2 mm< σ rms <5 mm in our case). In the plane spanned by the two quadrupole gradients these settings form finite areas. We refer to these areas as safety islands in the following.
(III) that all settings from safety islands are combined to determine combinations of two settings a and b corresponding to a low condition number.

IV. MEASUREMENTS AND EVALUATIONS
In a first measurement the projected rms-emittances and Twiss parameters at the exit of the ROSE beam line were measured as listed in Tab. I. A beam of 83 Kr 13+ at 1.4 MeV/u has been used, the beam intensity through the ROSE was about 20 eµA, and space-charge effects can be neglected in this case. The skew triplet and normal doublet were switched off and the transmission through the set-up was 100%. Uncoupled second moments at location i were obtained from 0 • /90 • measurements at location f to reconstruct the on-diagonal section of C i . In order to match reasonable beam sizes on the slit/gird locations and to assure full transmission, the strengths of Q 1 and Q 2 were varied numerically to check all available doublet settings, i.e. the safety islands including all reasonable doublet settings were defined. Combining two settings of the doublet (Q 1 and Q 2 , Q ′ 1 and Q ′ 2 ) from the safety islands, the corresponding condition number of the matrix Γ j was calculated using Equ. 33. Since there are N doublet settings inside the safety islands, N 2 combinations of two doublet settings were obtained. Finally, the combination of doublet settings a and b with minimum condition number of the matrix Γ j is applied.
The doublet setting a Q1/Q2=13.2/-12.6 T/m and setting b Q1 ′ /Q2 ′ =9.4/-10.2 T/m were selected as they provide low condition numbers for the majority of the algorithms. The safety islands and the selected doublet settings are plotted in Fig. 4.

A. Beam with low coupling
Measurements at 0 • , 90 • , and -30 • using settings a and b with the skew triplet being switched off were done. The measured Twiss parameters together with the projected rms-emittances are listed in Tab. II. Two evaluations using four or six measurements were done independently for comparison.
Evaluating six measurements (setting a and b at 0 • /90 • /-30 • ) the correlated second moments at location i, their corresponding eigen-emittances, and condition number for each algorithm are shown in Fig. 6. The averaged beam second moments matrix C(6) applying six measurements at location i is calculated as (in units of mm and mrad) Evaluation of the two eigen-emittances of C(4)/ C(6) reveals ε 1 =2.7/2.6 mm mrad and ε 2 =1.6/1.6 mm mrad. The corresponding coupling parameters t are 0.1/0.1. Both evaluations produce similar eigen-emittances and coupling parameters. The rms-ellipses of the matrices C(4)/ C(6) in the projections are shown in Fig. 7.
Evaluation of the two eigen-emittances of C ( 4)/ C(6) reveals ε 1 =2.4/2.5 mm mrad and ε 2 =2.1/1.5 mm mrad. The corresponding coupling parameters t are 1.2/1.8. The beam is significantly coupled. Comparing the beam matrices C(4) and C(6), the difference between their larger eigen-emittances is small but the smaller eigenemittances are different. The projected rms-ellipses of these matrices are shown in Fig. 10. According to the rms-ellipses in the projections, the rms-ellipses look very similar but feature different eigen-emittances and coupling parameters.
The larger uncertainty on the measured eigenemittances, especially ε 2 , for a beam inhabiting considerable correlations is already known from conventional emittance measurements in one single-plane (see appendix). The uncertainty of the measured single plane emittance is larger if the beam is strongly convergent or divergent, i.e. if it is correlated. This is just from the fact that the final observable, i.e. the emittance, is calculated from a difference (see Equ. 2) between measured quantities. Differences are much more prone to errors from their constituents as sums or products. This sensitivity known from single-plane emittance measurements occurs in four-dimensional measurements as well, as the eigen-emittances, especially ε 2 , are also calculated from differences of measured quantities (Equ. 4). Therefore a way to reduce the error on the measured emittance is to reduce the beam correlations prior to the emittance measurements. This method has been applied successfully for single-plane measurements. In order to apply it to four-dimensional emittance measurements it needs to be demonstrated that the measured data are sufficiently accurate to perform this reduction of correlations.

V. DECOUPLING PROSPECT
Any arbitrary beam line including at least three interplane coupling elements may serve to remove all interplane correlations. Here a beam line composed of a skew quadruplet enclosed by two normal quadruplets is chosen. If this beam line is set to decouple the beam matrix C(4) calculated from four measurements of the large coupling case, the corresponding decoupling transport matrix R(4) is determined from the required gradients as  algorithms C(6) d j = R(4) C(6) j R(4) T , j = 1, 2, · · · 15 (41) in order to test its decoupling capability. The coupling parameters before and after decoupling for each algorithm are plotted in Fig. 11. The coupling parameters of matrices C(6) d j are lower than 0.1 for each algorithm, i.e. the beam is practically decoupled. The decoupling transfer matrix R(4), constructed from four measurements will decouple all cases from six measurements with reasonable condition numbers. Accordingly, even for beams being considerably coupled, just four measurements are required to determine the four-dimensional beam parameters with sufficient precision to allow for elimination of all inter-plane correlations by an appropriate beam line.

VI. CONCLUSION
A new method using an rotatable slit/grid emittance measurement device called ROSE has been developed and commissioned to measure the four-dimensional second order beam matrix. It will allow precise and mobile four-dimensional emittance measurements without additional elements. This unique set-up works with high reliability. During ROSE commissioning, it was found that three of the parameters extracted from the fourdimensional beam matrix (eigen-emittance ε 1,2 and the t-parameter) are sensitive even to small errors in the measurements. Despite careful choice of the optics reducing this sensitivity, fluctuations in ε 1,2 and t were observed for a beam with considerable correlation. This observation confirms results from earlier single-plane emittance measurements, that featured larger errors in case the beam was correlated. However, ROSE can provide as major deliverable the optics to fully decouple a correlated beam. This optics is quite insensitive to the exact value of ε 1,2 and t as it just depends on the second moments. The latter were measured with sufficient precision. ROSE therefore can provide the input for advanced coupled beam dynamics methods as the four-dimensional beam envelope modell [31][32][33][34][35][36].

ACKNOWLEDGMENTS
One of the authors, Chen Xiao, would like to express his sincere thanks to Peter Forck at GSI for fruitful discussions.