Betatron coupling : Merging Hamiltonian and matrix approaches

1098-4402= Betatron coupling is usually analyzed using either matrix formalism or Hamiltonian perturbation theory. The latter is less exact but provides a better physical insight. In this paper direct relations are derived between the two formalisms. This makes it possible to interpret the matrix approach in terms of resonances, as well as use results of both formalisms indistinctly. An approach to measure the complete coupling matrix and its determinant from turn-by-turn data is presented. Simulations using methodical accelerator design MAD-X, an accelerator design and tracking program, were performed to validate the relations and understand the scope of their application to real accelerators such as the Relativistic Heavy Ion Collider.


I. INTRODUCTION
Betatron coupling in circular accelerators has been widely studied using both matrix formalism and Hamiltonian perturbation theory.In the matrix formalism the transverse beam motion is parametrized by a 4 4 one-turn matrix factorized into a block diagonal normal mode form consisting of the respective Twiss functions and a 2 2 coupling matrix (C matrix).In the Hamiltonian approach the coupling elements are considered to be a perturbation to the uncoupled lattice.The equations of motion are expanded up to the first order in those perturbations.In this paper, direct relations between the two approaches are derived and discussed in detail.
Numerical methods such as Fourier transform and singular value decomposition (SVD) have been demonstrated to be effective in measuring relevant quantities in both formalisms [1][2][3][4].However, in the matrix formalism, measurements have been constrained to three of the four elements in the coupling matrix [2,3].The individual elements of the coupling matrix usually have complicated behavior around the ring.One often has to rely on fitting techniques to identify sources of local coupling [2].
An approach to extend the existing methods to measure the complete C matrix and hence the determinant is presented.The behavior of the determinant in the presence of coupling sources is discussed in detail and its advantage of localizing these coupling sources is transparent.The applicability of the expression relating closest tune approach (Q min ) and C matrix given in [5,6] is also studied.

II. HAMILTONIAN TERMS AND COUPLING MATRIX
To relate the coupling matrix to the Hamiltonian terms the expressions describing the turn-by-turn motion from both formalisms are compared.This is achieved by intro-ducing the canonical momentum in the matrix framework and constructing a complex variable.Prior to this the two approaches are briefly introduced.

A. Resonance driving terms
Using Hamiltonian and normal form theory a weakly coupled lattice is treated as a perturbation to the uncoupled lattice.The turn-by-turn normalized particle positions and momenta at a location s are described as [7] x ÿ i ph x 2I x p e i x ÿ 2if 1001 2I y q e i y ÿ 2if 1010 2I y q e ÿi y ; (1) where I x;y are the horizontal and the vertical invariants.The phases of the oscillations x;y are expressed as a function of the tunes Q x;y , the turn number N and the initial phases x0;y0 as x;y 2Q x;y N x0;y0 .The resonance driving terms (RDTs) f 1001 and f 1010 are proportional to the Hamiltonian terms [7] and drive the difference and the sum resonances, respectively.These terms are functions of the uncoupled lattice parameters at the location of both the coupling elements and the observation point s given by x sl y ; ( where k l is the lth integrated skew quadrupole strength,

B. Matrix formalism
In the matrix approach, the coupled motion [8] is parametrized by factoring the one-turn matrix into block diagonal normal mode form by means of the similarity transformation given by the symplectic 4 4 matrix V, of the form where I is the 2 2 identity matrix, C is the 2 2 coupling matrix, and C is its symplectic conjugate requiring jCj are the normalization matrices for the a and b modes [9].Note that jCj jCj.The normalized motion in the horizontal and vertical planes is given by [9] x px ŷ py Using the above expressions for normalized positions and momenta, the complex variables are given by Note that the convention for momenta used in Hamiltonian theory described in [7] is the negative of that used in matrix formalism described in [10] ( ph x ÿ px ).

C. Relating the C matrix to RDTs
The relation to the Hamiltonian formalism can now be established by directly comparing Eqs. ( 1) and (2) to Eqs. (7) where Re and Im stand for real and imaginary parts, respectively.The determinant of C can also be related to the RDTs as jCj 4 2 jf 1001 j 2 ÿ jf 1010 j 2 ; (13) and using jCj 2 1 yields These expressions have a direct interpretation: If jCj is positive the difference resonance (f 1001 ) dominates; and if it is negative the sum resonance (f 1010 ) dominates.From these expressions it is also observed that a null jCj does not imply null coupling, but jf 1001 j jf 1010 j.If jf 1010 j 2 > 1 4 jf 1001 j, then 2 < 0 and the particle motion is unstable [see discussion in [9] after Eq. ( 12)].
In [4] it was demonstrated that the amplitude of the RDTs remains constant along sections free of coupling sources and experiences abrupt jumps at locations with couplers.In [4] it is also shown that the relative longitudinal variations of the RDTs become smaller as the tunes approach the resonance.On the resonance, the amplitude of the RDTs becomes invariant around the ring.Thus, by virtue of Eq. ( 13), the determinant of C also tends to be invariant around the ring as the tunes approach the resonance.

D. Simulations
To confirm the relations derived above, simulations were carried out with the aid of MAD-X [11].A simple ring consisting of 80 FODO cells is constructed using parameters shown in Table I.Three skew quadrupoles of different strengths were placed at arbitrary locations to introduce transverse coupling in the lattice.

f 1001 and C matrix
Twiss functions and coupling matrix elements in the form of R matrix (Edwards-Teng parametrization [8]) are directly available from MAD-X.The C matrix is determined by a simple transformation of the R matrix given by where C is obtained from Eq. ( 5) by normalizing out the beta dependence from C .To calculate the RDTs f 1001 and f 1010 from MAD-X, the first order approximation given by Eq. ( 3) is used.Figure 1 shows a comparison between the RDTs and C matrix elements as derived in Eqs. ( 9) and ( 13).The rms of the differences between the compared quantities are smaller than 10 ÿ6 , thus numerically validating the relations.

Dependence on skew quadrupole strengths
The relations between jCj and the RDTs are first order approximations.To investigate the accuracy of these relations, the mean of the ratio of the quantities compared in Fig. 1 is computed for increasing skew quadrupole strengths.The horizontal and vertical tunes are fixed at Q x 18:226 and Q y 17:232, respectively.Figure 2 shows a plot of this mean ratio along with the standard deviation of the ratio as a function of closest tune approach (Q min ) produced by the three skew quadrupoles.For this particular case, the quantities compared agree in the percent level for a Q min lower than 3 10 ÿ3 .

Stop-band limits
To explore the behavior of Eq. ( 13), a scan of horizontal tune (Q x ) is performed with the vertical tune fixed at Q y 0:228.Figure 3 shows the driving terms jf 1001 j 2 and jf 1010 j 2 as well as jCj= 2 plotted as a function Q x for the FODO lattice described above.The dominance of f 1001 or f 1010 is seen in Fig. 3 depending on the proximity to either the difference or sum resonance, respectively.This transition is also evident from the behavior of jCj= 2 as it switches sign when we move from difference to the sum resonance in accordance with Eq. ( 13).There are missing data points at Q x 0:5 due to the 1=2 integer resonance.

III. DETERMINANT OF C
It has been demonstrated that C 12 =, C 11 =, C 22 = are measured using turn-by-turn data from beam posi-  However, the jCj like the RDTs is a constant in coupler free regions and exhibit abrupt jumps at the location of a coupler.These discontinuities are intuitive and can be identified simply by visual inspection to the nearest BPM, as seen in Fig. 1.The jCj also allows one to easily estimate global quantities like Q min and local quantities like the strength and polarity of the coupler which are useful during machine operation.However, it will be seen that C 12 = has a better signal-to-noise ratio.
Additionally, it has better resolution to locate an isolated coupler more accurately using appropriate fitting techniques, while jCj has a resolution to within a region between the two nearest BPMs.where a and b are the phase advances between the two observation locations of the normal modes.We will demonstrate the effectiveness of such a calculation in a simple FODO lattice as well as in a complicated lattice such as RHIC where the coupling sources are mainly localized in the interaction regions with little or no coupling in the arcs.
It is also important to note that RDTs calculated using fast-Fourier-transform techniques in [4] can be extended to compute C matrix and hence the determinant according to Eqs. ( 11)- (15).The approach of N-turn map presented in [6] could also be extended to calculate both and jCj.

A. Calculation of C 12 = using SVD
For the purpose of this paper, we use an SVD-based method to construct the physical modes of beam [3,12].We will briefly introduce this method to show how the C matrix and its determinant are computed.A data matrix B tm consisting of t columns of turn-by-turn data at the mth BPM is decomposed using SVD [13] given by where is a diagonal matrix of singular values i with 1 2 n 0. U tt and V mm are normalized eigenvectors of AA T and A T A representing the temporal and spatial patterns of the leading singular values.
In the presence of coupling, the physical modes of beam motion are related to the SVD modes by a 4 4 rotation matrix O [12].The normal modes are reconstructed from BPM data of either horizontal or vertical planes with the knowledge of the O matrix.From the horizontal plane, C 12 = is given by [ 17) and ( 19)-( 21), jCj= 2 can now be calculated from turn-by-turn BPM data.Single-particle tracking is performed using MAD-X on the FODO lattice described earlier to generate turn-by-turn data at a large number of BPM locations.For all simulations, we assume dual-plane BPMs capable of measuring transverse positions in both planes, unless otherwise mentioned.The location of BPMs ( 1) and (2) in Eq. ( 17) to calculate C 21 = is arbitrary and only requires that there are no coupling sources present between them.For the purpose of the simulations BPM ( 2) is chosen to be the next upstream detector from location (1). Figure 4 shows a comparison of MAD-X model and SVD computed values of jCj= 2 .The overall discrepancy is smaller than 0.01% and increases up to 1% in the coupler regions.To minimize the discrepancy in the coupler region, C 21 = at the BPM location just before the coupler can be calculated using the previous BPM (downstream) instead of the BPM upstream.This maintains the region between the two BPMs to be coupler free and the calculation of C 21 = exact.
To investigate the effect of noise in BPMs, different levels of Gaussian noise were introduced into turn-byturn tracking data used to compute C matrix. Figure 5 shows the normalized rms of the difference for jCj= 2 and C matrix elements between model and calculated values as a function of noise =signal amplitude.It is clear that noise deteriorates the measurements.We observe that C 12 = is more robust against noise than the other elements of the C matrix.The discrepancy of the effect of noise between C 12 = and C11 22 can be attributed to the calculation of b;a and is under study.C 21 = inherently has large errors since Eq. ( 17) is exact only in a coupler free region.This can be improved by the choice of appropriate BPMs for the calculation as explained above.It was shown that the choice of number of turns in tracking affects the computation of C matrix due to the periodicity effect caused by the number of significant digits in the tune [3].In real data C11 22 unlike C 12 = were shown to be susceptible to large errors due to signal ''leaks'' in BPMs and their corresponding electronics [1,10].

C. Calculation of jCj= 2 for RHIC lattice
Single-particle tracking using RHIC lattice was performed to verify the applicability of this approach for a more realistic operating accelerator with several coupling sources.RHIC consists of two threefold symmetric rings with six interaction regions.Each arc is made of 11 FODO cells with 80 phase advance, and interaction regions consist of almost the same FODO cells without the dipoles [14].A model of RHIC containing realistic but uncorrected errors is used to track a single particle for 2000 turns with Q x 28:266 and Q y 29:212.The Q min for this lattice is 4:37 10 ÿ2 : Figure 6 shows a calculation of jCj= 2 from turn-by-turn data compared to model values from MAD-X.One can clearly see that the coupling sources are quite strong and mainly located in interaction regions.The agreement between model and calculated values of jCj= 2 is better in the arcs than in the interaction regions (IRs).
The larger discrepancies arising in the IRs are due to the presence of strong coupling sources.However, even in a complicated lattice like RHIC, the sources are clearly identified and all errors are below 5%.
For the above simulation, we assumed dual-plane BPMs in the lattice.However, this is not true for RHIC and most operating hadron colliders.RHIC consists of 160 BPMs per plane per ring: 72 dual-plane BPMs distributed through the IRs and 176 single-plane BPMs distributed in the arcs capable of acquiring 1024 turns.The single-plane BPMs in RHIC are confined to the arc regions where coupling sources are minimal.A more sophisticated algorithm to estimate jCj= 2 with lattices consisting of both single and double plane BPMs is under investigation.In real accelerators, many BPMs routinely fail resulting in unreliable data.It has been shown that preprocessing of BPM data is usually effective in removing faulty BPMs and maintaining data integrity [15] to obtain reliable measurements.

D. Calculation of skew quadrupole strengths
In [16,17] a method to obtain multipolar strengths from the measurement of RDTs was proposed.Skew quadrupole strengths are equivalently obtained from the measurement of the C matrix by use of the above relations.According to Fig. 7 we assume that only one skew quadrupole of integrated strength k exists between the two BPMs where the C matrices have been measured.From [16,17] the integrated strength k is given by k ÿ4e ÿi skew Here k is the strength of coupler, skew x;y and skew x;y are the Twiss functions at the location of the skew quadrupole, l x;y and lÿ1 x;y are the betatron phases at the lth and l ÿ 1th BPMs, respectively, and f l 1001 and f lÿ1 1001 are the corresponding RDTs.These terms are given by Eq. ( 9) as a function of the measured C matrix.The RDT f 1010 can also be used leading to a similar equation.It is also interesting to relate the change of the determinant of where x;y skew x;y ÿ l x;y are the phase advances between the skew quadrupole and the second location of observation.Using Eq. (A4), Eq. ( 24) can also be expressed as To determine the applicability of the above expressions, a simulation using strongly coupled RHIC lattice is performed.The Q min for this lattice is 4:37 10 ÿ2 .Figure 8 shows skew quadrupole strengths determined from Eq. ( 23) for the RHIC lattice.The presence of large coupling sources leads to relative errors of calculated strengths in the 20% level.This is due to the fact that these expressions are first order approximations and deviate with large coupling as illustrated in Sec.II D 2.

CLOSEST TUNE APPROACH
The following expression relating the closest tune approach and the determinant of C is given in [5,6], This equation cannot hold true in general since its left-hand side is invariant around the ring but the right-hand side is not, as explained in Sec.II C.Only close to the difference resonance the determinant of C tends to be invariant and Eq. ( 26) is considered to be a good approximation.Under this assumption the closest tune approach can also be related to the resonance terms, and since we assume the tunes to be close to the difference resonance, the approximation jf 1001 j jf 1010 j might also be used [5,6].A computer simulation is performed to investigate the validity of Eqs. ( 26) and ( 27).The same FODO lattice with three skew quadrupoles were used with horizontal and vertical tunes Q x 18:226 and Q y 17:239, respectively.Since jCj varies around the ring,  9) and (10).Middle: jCj= 2 is plotted as a function longitudinal position.A representation of the lattice (dipoles in black and quadrupoles in red) is shown in the bottom graph.The horizontal and vertical tunes are Q x 28:266 and Q y 29:212, respectively, and Q min 4:37 10 ÿ2 .
Q min is calculated with the three values of jCj at the three locations of skew quadrupoles and plotted as a function of increasing skew quadrupole strengths as shown in Fig. 9.We observe a dispersion in the Q min curves depending on the choice of the jCj value used to calculate Q min .If the tunes are closer to the coupling resonance, the relative longitudinal variation in jCj around the ring is smaller, hence reducing this dispersion.

V. CONCLUSIONS
Direct relations are established between the coupling matrix and the RDTs.This allows reinterpretation of the coupling matrix in terms of resonances and using results from both formalisms indistinctly.Numerical simulations are carried out to confirm these relations and explore their scope of application to real accelerators.The determinant of C and the RDTs (f1001 1010 ) have been demonstrated to exhibit distinct behavior that unambiguously reveals the region of the coupling sources.
A new approach to compute the full C matrix and hence the determinant from turn-by-turn data is presented and comparison to model shows excellent agreement.An approach to extract the skew quadrupole strengths previously using RDTs is also extended to C matrix.The applicability of the expression for Q min from the jCj has been discussed.

2 1 .
By means of another similarity transformation, the dependence is normalized out of C,

2 | − C|/γ 2 FIG. 3 .
FIG.3.(Color) jf 1001 j 2 , jf 1010 j 2 , and jCj= 2 plotted as a function of Q x (Q y 0:228).The stop bands at the sum and the difference resonance show the dramatic increase in these functions.The 1=2 integer resonance line is plotted as a reference.

C|/γ 2 FIG. 5 .FIG. 4 .
FIG. 5. (Color) Normalized rms of the difference of jCj= 2 between MAD-X model and SVD computed values for increasing amount of Gaussian noise in turn-by-turn data.Horizontal and vertical tunes are Q x 18:226 and Q y 17:257, respectively.

FIG. 6 .FIG. 7 .
FIG. 6. (Color) Top: Comparison of jCj= 2 between MAD-X model and SVD computed values from tracking data.Middle: Difference between model and calculated values of jCj= 2 .Bottom: A representation of the lattice (dipoles in black and quadrupoles in red) is shown in the bottom graph.The horizontal and vertical tunes were Q x 28:266 and Q y 29:212, respectively, and Q min 4:37 10 ÿ2 .

FIG. 8 .
FIG. 8. (Color) Top: Skew quadrupole strengths calculated from RDTs and C matrix are compared to MAD-X model values.Note that RDTs are calculated from C matrix using Eqs.(9) and(10).Middle: jCj= 2 is plotted as a function longitudinal position.A representation of the lattice (dipoles in black and quadrupoles in red) is shown in the bottom graph.The horizontal and vertical tunes are Q x 28:266 and Q y 29:212, respectively, and Q min 4:37 10 ÿ2 .

TABLE I .
FODO lattice parameters.
FIG. 2. (Color) Top: Mean of the ratio of jC 12 ÿ C 21 iC 11 C 22 =j and 4jf 1001 j.Bottom: Mean of the ratio of jCj= 2 and 4jf 1001 j 2 ÿ jf 1010 j 2 .The error bars represent the standard deviation of the ratio of the respective quantities.The horizontal and vertical tunes are Q x 18:226 and Q y 17:232.To calculate the determinant, we propose a method to calculate C 21 given C 12 , C 11 , and C 22 at two locations with an arbitrary phase advance in both normal modes.Given two locations (1) and (2) in a coupler free region, C 1 21 is given (see Sec. 1 in the appendix) by C 1 21 ÿC 1 11 cos a sin b C 1 12 cos a sin b C 1 22 sin a cos b ÿ C 2 12 =sin a sin b ; (17) BETATRON COUPLING: MERGING HAMILTONIAN AND . . .