Linear coupling parameterization in the betatron phase rotation frame

Transcription

1 C-A/AP/#165 September 004 Linear coupling parameterization in the betatron phase rotation frame Yun Luo Collider-Accelerator Department Brookhaven National Laboratory Upton, NY 11973

2 CAD/AP/165 September 1, 004 Linear coupling parameterization in the betatron phase rotation frame Yun Luo Brookhaven National Laboratory, Upton, NY 11973, USA Following Ripken s linear coupling parameterization procedure, the linear coupling parameterization in the betatron phase rotation frame is introduced. n the parameterization, the two eigenmodes betatron oscillation phases and phase advances are well defined. The general epression of the coordinates (,, y, in the laboratory frame are given independent of different beam ecitations. The transfer matri P from the betatron phase rotation frame to the laboratory frame are discussed in detail. t can be obtained from the eigenvectors of the one-turn transfer map T. t also has tight connections to Twiss and coupling parameters defined in Edwards-Teng s parameterization. t can be constructed from BPM turn-by-turn data with the precisely measured eigentunes and betatron initial phases. The phase ellipses and beam sizes in different projection planes are easily obtained in the betatron phase rotation parameterization. The physical meanings of the coupling parameter r and the coupling matri C defined in Edwards-Teng s parameterization become clear. The two eigenmode betatron phase advances, the matri P, therefore Twiss and coupling parameters at point s are described in terms of the section transfer map T 1 and Twiss and coupling parameters at the starting point s 1. The linear coupling betatron phase rotation parameterization is useful for the turn-by-turn BPM data interpretation. The three dimensional linear coupling parameterization in the betatron phase rotation frame can be obtained in the same way as the two-dimensional parameterization. And the parameterization using (, p, y, py canonical coordinates is similar to that shown here with (,, y, coordinates. The betatron phase rotation parameterization can be adjusted for the transport line linear coupling research purpose, too. 1 ntroduction The Courant-Snyder parameterization [1] of the uncoupled one-dimensional particle s betatron oscillation in the circular accelerators has been proved to be a great success. The pursuit to cast the coupled twodimensional transverse betatron motion into an elegant way as the one-dimensional parameterization is continuing. Here we give another approach in its sort. There are two main directions to parameterize the two-dimensional linearly coupled betatron motions. One is to start from the eigenvectors of the one-turn 4 4 transfer matri to construct the particle s four coordinates in the laboratory frame. The eigenvectors transfer from one point to another in the ring so that the coordinates in the eigenvector frame are constant. Comparing to the one-dimensional Courant-Snyder parameterization, new Twiss parameters in the two-dimensional situation are defined accordingly. One wellknown parameterization in this sort was achieved by G. Ripken [, 3, 4]. There, four β and four α functions are defined. Knowing the epression of the four coordinates in the laboratory frame, the phase ellipses and beam sizes into different observation planes are easily obtained. Another approach is to directly decouple the one-turn transfer map in the laboratory frame into an uncoupled 4 4 one-turn map through the matri similarity transformation. One successful approach in this direction had been obtained by Edwards and Teng [5, 6] and improved by others [7, 8, 9, 10]. The parameterization is proved successful. There two sets of Twiss parameters from two eigenmodes are defined and a coupling matri is introduced. t has been widely used in coupling measurements, where the normalized C is used [11, 9]. Comparing the two successful parameterizations by Ripken, Edwards and Teng, we find that Ripken s parameterization give straight-forward epression of the particle s coordinates in the laboratory frame. There the particle motion s phases and phase advances between two points are both well defined. The shortcoming 1

3 of this approach is that it is a little complicated. The inverse transfer from its defined Twiss parameters to the one-turn transfer map hasn t been found [1]. Edwards-Teng s parameterization is based on the matri manipulations. The transfers between its defined Twiss, coupling parameters and the one-turn map are easily obtained. The unsolved problem of the parameterization is its ambiguous definitions of betatron phases and phase advances. The general epression of one particle s motion in the laboratory frame is not given. So it is not used to calculate the phase ellipses, phase areas and beam sizes in different projection planes. The relation between the one-turn transfer map s eigenvectors and its defined Twiss and coupling parameters are not given. With high performance digital BPMs and the modern analytical techniques [11, 13, 14, 15, 16, 17, 18], the betatron phase advances can be very precisely measured in the beam eperiments. The merit of the phase measurement is that it is independent from the BPM position offsets and their gains. t has attracted much attention in how to use them in the past few years [19]. For a specific betatron oscillation, the two eigenmodes phases and phase advances are all well defined. To include them into the linear coupling parameterizations are demanded. This article will give an approach to tackle those mysteries. Following the procedure of Ripken s parameterization, we transfer the four coordinates in the laboratory frame to the betatron phase rotation frame. Each laboratory coordinate will have four terms. Each eigenmode contributes two terms. t will be found that the transfer matri from the laboratory frame to the betatron phase rotation frame can be obtained simply from the eigenvectors of the one-turn map in the laboratory frame. The projection phase ellipses and their areas, the coupling tilt angle in the y plane are also easily obtained. The betatron phase rotation parameterization is self-consistent formalism. t is found that the transfer matri from the laboratory frame to the betatron phase rotation frame have very tight connections to Twiss and coupling parameters defined in Edwards-Teng s parameterization. So they are directly borrowed into the betatron phase rotation parameterization. The article is organized in the following way: first we introduce the phase rotation frame starting from Ripken s parameterization but with different epression of the four laboratory coordinates. Then we discuss how to construct the matri P analytically and eperimentally. The phase ellipses and their areas, and the beam sizes are given. The propagation of the matri P in terms of the section transfer matri T 1 are obtained. Phase rotation frame n the general linearly coupled situation, it is known that the displacements and y include the contributions from both eigenmodes. we assume that mode is more linked to the horizontal plane, and mode is more linked to the vertical plane. The nth turn s and y can be casted as: { n = A 1,1 cos(πµ 1 (n 1 + φ 1,1 + A 1, cos(πµ (n 1 + φ 1,, (1 y n = A,1 cos(πµ 1 (n 1 + φ,1 + A, cos(πµ (n 1 + φ, where the amplitude coefficients A and initial phases φ both have subscripts (i, j. The first subscript represents the horizontal or vertical plane. The second subscript represents mode or mode P matri Eq. (1 has a very clear physical meaning. The tunes µ 1, and the initial phase φ i,j can be precisely obtained from BPM turn-by-turn data in beam eperiments. They are all well defined quantities for a specific betatron motion. The angles and also can be casted into two modes contributions like Eq. (1. However, the shortcoming of this parameterization is that there are two many A s and φ s and no clear relations between them. One simple alternative is to cast the coordinates with the action-angle variables as the uncoupled onedimensional situation [1], where and y are: { = J β cos Φ y = J y β y cos Φ y, ( where J,y are the horizontal and vertical actions, Φ and Φ y are the phases of the betatron oscillations. By taking differentials of the above two equations, we obtain two terms for each or if we still keep the betatron phases Φ,y.

4 So in the linearly coupled case, if we keep the two eigenmodes phases, there will probably be four terms for each coordinate of (,, y, at one point in the ring, then we get: cos Φ 1 y = P sin Φ 1 cos Φ, (3 sin Φ P = p 11 p 1 p 13 p 14 p 1 p p 3 p 4 p 31 p 3 p 33 p 34 p 41 p 4 p 43 p 44, (4 where Φ 1, are the two eigenmodes betatron oscillation phases. The one turn s betatron phase advances are πµ 1 for mode, πµ for mode. Eq. (3 looks like that from Ripken s epression of (,, y, coordinates. However, here we include four terms for each coordinate, and only two betatron phases are introduced.. Phase rotations Here we give the frequently used phase rotation matri in accelerator physics: ( cos( Φi sin( Φ R( Φ i = i sin( Φ i cos( Φ i ( R( Φ1 0 R( Φ 1, Φ = 0 R( Φ, (5. (6 Careful readers will notice that both of them are clockwise rotation matri. n order to get the phase advance propagation like Φ 1, s = Φ 1, s1 + Φ 1,, we change the signs of the sin functions in Eq. (3: p 11 p 1 p 13 p 14 cos Φ 1 y = p 1 p p 3 p 4 p 31 p 3 p 33 p 34 sin Φ 1 cos Φ (7 p 41 p 4 p 43 p 44 sin Φ t is easily to prove: cos(φ 1 + Φ 1 sin(φ 1 + Φ 1 cos(φ + Φ sin(φ + Φ = R( Φ 1, Φ cos(φ 1 sin(φ 1 cos(φ sin(φ 1, (8 So the coordinates in the betatron phase rotation frame are rotating with the two eigenmodes phase advances when the particle travels along the ring. The amplitude of coordinates in the rotation frame are constant..3 Actions J 1 and J By now the elements of P are related with the particle s oscillation amplitudes or energy. n order to make p ij independent of individual particles, here we introduce actions into. n the uncoupled situation, the action J 1 of mode will appear in the left upper block of P. Action J of mode will appear in the right lower block of P. Then we modify Eq. (7 as: y = p 11 p 1 p 13 p 14 p 1 p p 3 p 4 p 31 p 3 p 33 p 34 p 41 p 4 p 43 p 44 J1 cos Φ 1 J J 1 sin Φ 1 cos Φ J sin Φ The propagation of the coordinates in the betatron phase rotation frame is : J1 cos(φ 1 + Φ 1 J 1 sin(φ 1 + Φ 1 J cos(φ + Φ = R( Φ 1, Φ J sin(φ + Φ J1 cos(φ 1 J 1 sin(φ 1 J cos(φ J sin(φ. (9 1. (10 3

5 .4 Betatron phases Φ 1 and Φ n Eq. (10 there is still an ambiguity in the definitions of eigenmodes phases Φ 1,. n principle, one can define arbitrary Φ 1, in Eq. (10, there still will be a constant P at one point in the ring. We notice that in the uncoupled one-dimensional situation, Courant and Snyder defined the displacement s phase as the eigenmode s phase. Therefore there is only one term for the displacement coordinate. Then or has two terms if we keep the displacement s phase. Among them, one term is in-phase to the displacement, another is out-of-phase to the displacement. Here we forcedly define Φ 1 as the phase of mode s contribution to displacement, and Φ as the phase of mode s contribution to y displacement. Therefore, the element p 1 and p 34 in matri P equal to zero: y = p 11 0 p 13 p 14 p 1 p p 3 p 4 p 31 p 3 p 33 0 p 41 p 4 p 43 p 44 J1 cos Φ 1 J J 1 sin Φ 1 cos Φ J sin Φ. (11 The eigenmode phases at one point in the ring are: { Φ1 = πµ 1 (n 1 + φ 1,0 Φ = πµ (n 1 + φ,0, (1 φ 1,0 and φ,0 are the initial betatron phases for the two eigenmodes at that point. They can be precisely measured eperimentally from the turn-by-turn BPM data. Each coordinate in the laboratory frame is then given : = p 11 J1 cos Φ 1 + p 13 J cos Φ p 14 J sin Φ = p 1 J1 cos Φ 1 p J1 sin Φ 1 + p 3 J cos Φ p 4 J sin Φ (13 y = p 31 J1 cos Φ 1 p 3 J1 sin Φ 1 + p 33 J cos Φ = p 41 J1 cos Φ 1 p 4 J1 sin Φ 1 + p 43 J sin Φ p 44 J sin Φ Eq. (13 is a general epression to (,, y, coordinates of one particle s motion. t has very clear physical meanings, and it is convenient for the eperimental BPM turn-by-turn data interpretations. And in the betatron phase rotation frame, the eigentunes, the eigenmodes initial phases and the eigenmodes phase advances are clearly defined. The particle oscillation energy and two eigenmodes action are also well defined. n the following, we will see that Eq. (13 is useful for the analytical calculations..5 Reduce to uncoupled situation n the uncoupled situation, all elements in the two off-diagonal blocks of matri P will disappear: p J1 cos Φ 1 y = p 1 p p 33 0 J 1 sin Φ 1 J cos Φ 0 0 p 43 p 44. (14 J sin Φ The four coordinates in the laboratory frame are: { = p11 J1 cos Φ 1 = p 1 J1 cos Φ 1 p J1 sin Φ 1 (15 { y = p33 J cos Φ = p 43 J cos Φ p 44 J sin Φ (16 Comparing to the uncoupled one-dimensional Courant-Snyder parameterization: { = J β cos( 1 β ds + φ,0 = α J /β cos( 1 β ds + φ,0 J /β sin( 1 β ds + φ,0, (17 { y = Jy β y cos( 1 β y ds + φ y,0 = α y Jy /β y cos( 1 β y ds + φ y,0 J y /β y sin( 1 β y ds + φ y,0, (18 4

6 we find the connections of p ij s to Twiss parameters defined in Courant-Synder s one-dimensional parameterization as: p 11 = β p 1 = α β p = 1/ β. (19 p 33 = βy p 43 = α y βy p 44 = 1/ β y And the betatron phase connections are: { Φ1 = 1 β ds + φ,0 Φ = 1 β y ds + φ y,0, (0 which coincides with our previous definitions..6 One eigenmode ecitation n order to measure the betatron oscillation optics, we coherently shake the beam with one eigenmode s frequency [11, 0]. f only mode is activated, the oscillation of the center of the bunch is: p 11 0 p 13 p 14 J1 cos Φ 1 y = p 1 p p 3 p 4 p 31 p 3 p 33 0 J 1 sin Φ 1 0, (1 p 41 p 4 p 43 p 44 0 each coordinate in the laboratory frame is given by: = p 11 J1 cos Φ 1 = p 1 J1 cos Φ 1 p J1 sin Φ 1 y = p 31 J1 cos Φ 1 p 3 J1 sin Φ 1 = p 41 J1 cos Φ 1 p 4 J1 sin Φ 1. ( f only mode is activated, the oscillation of the center of the bunch is given by: p 11 0 p 13 p 14 0 y = p 1 p p 3 p 4 p 31 p 3 p J cos Φ p 41 p 4 p 43 p 44, (3 J sin Φ each coordinate in the laboratory frame is given by: = p 13 J cos Φ p 14 J sin Φ = p 3 J cos Φ p 4 J sin Φ (4 y = p 33 J cos Φ = p 43 J cos Φ p 44 J sin Φ So mode s betatron phase is reflected in coordinate when the beam is shaked with mode s frequency, the mode s betatron phase is reflected in y coordinate when the beam is shaked with mode s frequency. The phase in y displacement with mode frequency shaking, or the phase in displacement with mode frequency shaking reflects the off-diagonal terms of matri P. P. Bagley [11], R. Talman [1], G. Bourianoff [] also derived and y coordinates under one eigenmode ecitation. Twiss parameters and/or normalized coupling matri C are used. Comparing to their epressions, Eq. ( and Eq. (4 are simple, which is only one specification to the general epression Eq. (13..7 Matri P and T 1 The coordinates in the laboratory frame at s can be obtained from those at s 1 by: y = T 1 y, (5 5 1

7 or through the betatron phase rotation frame: y = P R( Φ 1, Φ P 1 So we obtain: y 1. (6 T 1 = P R( Φ 1, Φ P 1. (7 f the phase advances between the two points are known, then we can calculate matri P with the section transfer map T 1 and matri P 1 : For the one-turn transfer map, we obtain: 3 Construction of matri P P = T 1 P 1 R ( Φ 1, Φ. (8 T = P R(πµ 1, πµ P. (9 n this section, we discuss how to construct the matri P analytically and eperimentally. t will be found that the matri P can be obtained from the eigenvectors of the one-turn transfer map T in the laboratory frame, and has tight connections with Twiss and coupling parameters defined in Edwards-Teng s parameterization. 3.1 Matri P and eigenvectors We assume the four eigenvectors of the one-turn transfer matri T are: v 1, v1, v, v. Their eigenvalues λ i, i = 1,, 3, 4, are e iπµ1, e iπµ1, e iπµ, e iπµ, respectively. T v 1 = e iπµ1 v 1 T v 1 = e iπµ1 v 1 T v = e iπµ v T v = e iπµ v. (30 Substituting Eq. (9 into the above equations, after some matri manipulations, we get: R(πµ 1, πµ (P v 1 = e iπµ1 (P v 1 R(πµ 1, πµ (P v1 = (P v e iπµ1 1 R(πµ 1, πµ (P v = e iπµ (P. (31 v R(πµ 1, πµ (P v = (P v e iπµ We normalize the eigenvectors of the one-turn transfer map T according to: v1, T v 1, = 1 (3 However, after the normalization, v 1, and v 1, e iφ both meet Eq. (3, and have the same eigenvalues. n order to assure p 1 and p 34 in the matri P equal to zero, here we will force v 1 and v1 s first elements to be purely imaginary numbers. Their amplitudes will not be changed. The reason for that will be clear in the following when we connect matri P to the eigenvectors of the one-turn transfer map T. f the phase of the first element of v 1 is e iθ1, we multiply v 1 by e i( π/ θ. Then the phase of the first element of v1 is, we multiply v e iθ1 1 by ei(π/+θ1. The same procedure is needed for the eigenvectors v and v. f the phase of the third element of v is e iθ, we multiply v by e i( π/ θ. Then the phase of the third element of v is e iθ, we multiply v by e i(π/+θ. After that the first elements of v 1 and v1, the third elements of v and v are all purely imaginary numbers. For the rotation matri R(πµ 1, πµ, there are also four normalized eigenvectors. Their eigenvalues λ i, i = 1,, 3, 4, are e iπµ1, e iπµ1, e iπµ, e iπµ, respectively. The normalized eigenvectors of the matri R(πµ 1, πµ are ṽ 1, ṽ1, ṽ, ṽ : ṽ 1 = 1 (-i T ṽ1 = 1 (i T ṽ = 1 (0 0 -i 1 T, (33 ṽ 1 = (0 0 i 1 T 6

8 R(πµ 1, πµ ṽ 1 = e iπµ1 ṽ 1 R(πµ 1, πµ ṽ 1 = e iπµ1 ṽ 1 R(πµ 1, πµ ṽ = e iπµ ṽ R(πµ 1, πµ ṽ = e iπµ ṽ. (34 Comparing Eq. (31 and Eq. (34, we obtain: P v 1 = 1 (-i T P v1 = 1 (i T P v = 1 (0 0 -i 1 T P v 1 = (0 0 i 1 T, (35 The eigenvectors of matri T and R(πµ 1, πµ are all comple. Here we recombine the above four equations: P i (v 1 v1 = ( T P 1 (v 1 + v1 = ( T P i (v v = ( T, (36 P 1 (v + v = ( T Defining four new vectors from the eigenvectors of matri T: e 1 = i (v 1 v1 e = 1 (v 1 + v1 e 3 = i (v v e 4 = 1 (v + v, (37 then Eq. (36 can be re-written as: P e 1 = ( T P e = ( T P e 3 = ( T P e 4 = ( T, (38 or in a compact way: P e =, (39 where matri e is constructed as: e = (e 1 e e 3 e 4. (40 is 4 4 identity unit matri. From Eq. (39, we obtain: P = e. (41 Since P s determinant is 1, so we should normalize matri e to make its determinant to be 1 in the end of the procedure. Then we find that the matri P can be directly constructed from the eigenvectors of the one-turn transfer map T in a sample way. t is easy to check that p 1 and p 34 in the matri P are both zero from the above procedure. 3. Matri P and Twiss, coupling parameters Edwards-Teng s parameterization decouples the coupled one-turn transfer map like: A 0 T = V V 0 B, (4 r C V = C +. r (43 7

9 n order to easily distinguish Twiss parameter γ, the author uses r in the matri V instead of γ in other literatures. Each matri A and matri B are parameterized like one-dimensional Courant-Snyder parameterization [1]. Matri A and matri B can be further parameterized as U i R(πµ i U i : ( βi 0 U i = α i / β i 1/, (44 β i Then the one-turn map T is: Comparing to Eq. (9 and Eq. (45, we obtain: T = V U R(πµ 1, πµ U V, (45 U1 0 U =. (46 0 U P = V U, (47 P = V U = 1. (48 t is interesting that the matri P is the same as the matri V U defined in Edwards-Teng s parameterization. Epanding V U into Twiss and coupling parameters defined in Edwards-Teng s parameterization, we obtain: P11 P1 ru1 CU P = = P1 P C +. (49 U 1 ru r β 1 0 c 11 β c 1 α / β c 1 / β P = α 1 r/ β 1 r/ β 1 c 1 β c α / β c / β c 1 α 1 / β 1 c β1 c 1 / β 1 r β 0 c 11 α 1 / β 1 + c 1 β1 c 11 / β 1 α r/ β r/. (50 β The matri P is independent of Edwards-Teng s parameterization. However, since Edwards-Teng s parameterization gives the well-defined Twiss parameters and the coupling matri C, we would like directly borrow them into the betatron phase rotation parameterization. On the other hand, we can calculate Twiss parameters and coupling parameters from matri P. For eample, the two eigenmodes Twiss parameters and r can be obtained in the following equations: r = p 11 p = p 33 p 44 β 1 = p 11 /p α 1 = p 1 /p γ 1 = (1 + α 1 /β 1 = (p 1 + p /p 11p. (51 β = p 33 /p 44 α = p 43 /p 44 γ = (1 + α /β = (p 43 + p 44/p 33 p 44 The coupling matri C can be obtained, for eample, through P1 and U : C = P1 U = P1 P /r. (5 For the one-turn map there are ten independents. For the matri P there are eight independents since two independents µ 1 and µ have been split ed out. There are different ways to define those eight independents of the matri P. 3.3 Construct matri P eperimentally To fully construct matri P, we need to know the turn-by-turn (,, y, data at one point in the ring. n each interaction region of the Relativistic Heavy on Collider ( RHC, there are two DXBPMs which are close to the last magnet DX and face to the P. There is no other magnet between the two BPMs if we ignore or switch off the detector s solenoid field, so we obtain the turn-by-turn angle informations of the particle s motion at the two BPMs. Here we give an eample of the matri P construction through the simulation data. The two DXBPMs in the R8 in the Blue ring are used to get the angle information. The design distance from the two BPMs to the design P is 8.33 m. We designate the DXBPM in the upstream of P8 as DXBPM81, and the DXBPM 8

10 in the downstream of P8 as DXBPM8. n the following we use SAD code [3] to produce the BPM turn-by-turn data. n the simulation, only one free oscillation particle circulates in the Blue ring. The two eigentunes can be precisely obtained from the turn-by-turn and/or y data with the fast Fourier transformation( FFT technique. The uncoupled tunes are µ,0 = 8., µ y,0 = 9.3. We set the skew quadrupole family 1 s integrated strength to be m 1 to introduce the coupling. From FFT of ( + y turn-by-turn data at DXBPM81, the two tunes are obtained 8.16, 9.375, respectively nitial phases Knowing the two eigentunes µ 1 and µ, we first obtain the initial phases of two eigenmodes. Multiplying cos(πµ 1 (n 1, sin(πµ 1 (n 1 to turn-by-turn data, and taking summations to the maimum Nth turn, we get: φ 1,0 is given by: C 1 = N i=1 i cos(πµ 1 (n 1 S 1 = N i=1 i sin(πµ 1 (n 1 sin φ 1,0 = S 1 / S 1 + C 1 cos φ 1,0 = C 1 / S 1 + C 1 φ 1,0 = arctan ( S 1 /C 1. The same procedure to y turn-by-turn BPM data, we get: C = N i=1 y i cos(πµ (n 1 the φ,0 is given by: S = N i=1 y i sin(πµ (n 1 sin φ,0 = S / S + C cos φ,0 = C / S + C φ,0 = arctan ( S /C., (53. (54, (55. (56 From turn-by-turn data at DXBPM81, we get mode s betatron oscillation initial phase From y turn-by-turn data from DXBPM81, we get mode s betatron oscillation initial phase At DXBPM8, we similarly get mode and mode s initial phases and So the eigenmodes phase advances between the two BPMs are and for mode and mode, respectively. From the analytical calculation with SAD code, they are and Here we see that the phase advances are very precisely reproduced from the simulation turn-by-turn BPM data Matri A Eq. (11 can be rewritten as: J1 p 11 J 1 p 1 J p 13 J1 J p 14 y = p 1 J 1 p J p 3 J1 J p 4 p 31 J 1 p 3 J p 33 J1 J p 34 p 41 J 1 p 4 J p 43 J p 44 cos Φ 1 sin Φ 1 cos Φ sin Φ. (57 n order to facilitate narration, we define an interim matri A as : J1 p 11 J 1 p 1 J p 13 J1 J p 14 A11 A1 A = = p 1 J 1 p J p 3 J1 J p 4 A1 A p 31 J 1 p 3 J p 33 J1 J p 34 p 41 J 1 p 4 J p 43. (58 J p 44 we still use harmonic analysis technique to get the coefficients A ij. For eample, from turn-by-turn data we get: A 11 = ( N i=1 i cos(πµ 1 (n 1 + φ 1,0 /N A 1 = ( N i=1 i sin(πµ 1 (n 1 + φ 1,0 /N A 13 = ( N i=1. (59 i cos(πµ (n 1 + φ,0 /N A 14 = ( N i=1 i sin(πµ (n 1 + φ,0 /N 9

11 Similarly from, y and turn-by-turn data, we obtain A j, A 3j and A 4j, j = 1,, 3, 4. From the above (,, y, simulation data at DXBPM81, we obtain matri A: A = ( Normalize matri P Matri A includes the action informations. From the determinant of the left upper block and the right lower block of matri A, we get the ratio of the two actions: k = J 1 /J = A11 / A. (61 According to the definition of matri A, we first invert the signs of the elements of matri A, then divide the first two column elements by k. Normalizing the new matri s determinant to 1, then we get the matri P. Please be advised the order of the procedures. Based on the above obtained A, we obtain: P = (6 From matri P, we can continue to calculate Twiss and coupling parameters, the one-turn map. f we know P 1 and P at two points in the ring, the transfer matri T 1 between them can be obtained. The section transfer map is hopefully useful for the field error diagnostics, on-line optics modeling and so on Construct P from eigenvectors Here we want to take a little time to check the method of the matri P construction from the eigenvectors from the eigenvectors of one-turn transfer matri T. According to Eq. (9, the one-turn transfer map at DXBPM81 is obtained: T = (63 Solve out the eigenvectors and normalize them according to the above given procedures in section 3.1, we re-construct the matri P from the eigenvectors of one-turn transfer map T: P = , ( which is close to the original one Eq. (6. 4 Phase ellipses and Σ matri n the uncoupled one-dimensional situation, it is well-known that the particle traces out an ellipse in the or y plane. For the linearly coupled situation, there are two modes contributions in each laboratory coordinate shown in Eq. (13. There are several ellipses in different projection planes. 10

12 4.1 Projection in the plane from mode (,, y, coordinates from mode is : J1 cos Φ 1 y = P J 1 sin Φ ( P11 = P1 P1 P J1 cos Φ 1 J 1 sin Φ (65 The projection in the plane is given: = P11 ( J1 cos Φ 1 J 1 sin Φ 1. (66 Since P11 is matri, its determinant is r, so Multiply P11 + to both sides of Eq. (66, we obtain: P11 + = r J 1 p 0 p 1 p 11 P11 + = r P11. (67 ( We normalize P11 + to a unit determinant matri P11 + : P11 + = P11 + p 0 /r = = p 1 p 11 cos Φ1, (68 sin Φ 1 = r J cos Φ1 1. (69 sin Φ 1 ( p /r 0 p 1 /r p 11 /r. (70 Then we re-write Eq. (68 as: P11 + = r J cos Φ1 1, (71 sin Φ 1 P11 + is a matri, its determinant equals to 1, so after its transferring, the phase area will not change. X X = P11 +, (7 ( X X = r J 1 cos Φ1, (73 sin Φ 1 where (X, X are the new coordinates. The phase area in X X plane, or in the plane from mode s contribution is: ε, = r πj 1. (74 The projection in the plane from mode is an ellipse. From Eq. (69, we get: or in Twiss parameters, Under the uncoupled situation, the above ellipse reduced to: which is the same as Courant-Snyder s parameterization [1]. (p + ( p 1 + p = r 4 J 1, (75 γ 1 + α 1 + β 1 = J1 r, (76 γ 1 + α 1 + β 1 = J 1, (77 11

13 4. Projection in the y plane from mode Similarly, we calculate the projection in the y plane from the contribution of mode : ( y J1 cos Φ = P1 1. (78 J 1 sin Φ 1 P1 + y Normalize P1 + to a unit determinant matri P1 + : then we obtain: P1 + y ( Y Y = (1 r J cos Φ1 1. (79 sin Φ 1 P1 + = P1 + / 1 r, (80 = 1 r J cos Φ1 1. (81 sin Φ 1 = 1 r J cos Φ1 1. (8 sin Φ 1 Then the phase area in the Y Y plane or in the y plane contributed from mode is: The phase shape in the y plane is an ellipse. From Eq. (79: ε,y = (1 r πj 1. (83 (p 4 y p 3 + ( p 41 y + p 31 = (1 r J 1. (84 ts shape and orientation is decided by the elements of P1, or the coupling matri C defined in Edwards- Teng s parameterization. The total phase area from mode in the y and planes is: ε = ε, + ε,y = πj 1. (85 We define the phase area partition ratio κ between plane and y plane projection phase areas from mode as: κ = ε,y /ε, = (1 r /r. ( Projection in the y plane from mode We also can obtain the projection in the y plane from mode : ( p11 0 J1 cos Φ = 1 y p 31 p 3 J 1 sin Φ 1. (87 t is easy to prove that the projection phase shape in y plane from mode is still an ellipse. (p 3 + ( p 31 + p 11 y = (c 1 /r r 4 J 1. (88 ts shape and orientation are decided by the coupling matri C. ts phase area is: ε, y = p 11 p 3 πj 1 = c 1 /r r πj 1. (89 f c 1 = 0, no phase area into y plane. n the one eigenmode ecitation, and y turn-by-turn coordinates at the dual direction BPMs can be measured, then we can measure some coupling parameters in the y projection plane. 1

14 4.4 Projections from mode Similar to mode, mode also has projections in the y,, y planes. These coordinates from mode are given by: 0 0 y = P 0 P11 P1 J cos Φ = 0 P1 P J cos Φ J sin Φ, (90 J sin Φ ( y ( y ( = ( = P J cos Φ J sin Φ ( = P1 J cos Φ ( p13 p 14 p 33 0 J sin Φ ( J cos Φ, (91 J sin Φ (9. (93 These phase ellipses areas are obtained: ε,y = r πj ε, = (1 r πj. (94 ε, y = c 1 /r r πj And we also see the phase projection area of mode is conservative: ε = ε,y + ε, = πj (95 The phase area partition ratio κ between the and y plane projections from mode is defined as: κ = ε, /ε,y = (1 r /r, (96 then κ = κ = κ. 4.5 Σ matri and tilt angle t is interesting to investigate the projections of all particles in one bunch in different projection planes, therefore Σ matri is defined [4]: Σ = σ σ, σ,y σ, σ, σ σ,y σ, σ y, σ y, σ y σ y, σ y, σ y, σ,y σ = < > < > < y > < > < > < > < y > < > < y > < y > < y > < y > < > < > < y > < >. (97 Among them, the particles projection into the y planes is more interesting. Using Eq. (13, it is easy to obtain: < > = p 11 < J 1 > +p 13 < J > +p 14 < J > < y > = p 31 < J 1 > +p 3 < J 1 > +p 33 < J >. (98 < y > = p 11 p 31 < J 1 > +p 13 p 33 < J > Normally the projection of the beam in the y plane doesn t like an ellipse since each particle s projection is combination of two ellipses. However, here we still define the coupling tilt angle for one bunch as other literatures [5]: < y > tan Ψ = < > < y >, (99 which is valid for < > < y > 0. The tilt angle is eperimentally observable through the synchrotron light monitor for the electron beam when the coupling is weak. 13

15 5 Propagation of matri P Here we investigate the propagation of matri P. The phase advances, Twiss parameters and coupling matri at s are also given in the elements of the section transfer matri T 1. Propagation of matri P is the key to other parameters propagations. 5.1 Matri P s propagation n order to easily distinguish the elements of P at position s 1 and s, we denote the quantity at s has a tilde overhead. The transfer matri T 1 in the laboratory frame from s 1 to s is written into block matri: t 11 t 1 t 13 t 14 T11 T1 T 1 = = t 1 t t 3 t 4 T1 T t 31 t 3 t 33 t 34. (100 t 41 t 4 t 43 t 44 According to Eq. (8, the propagation of P from s 1 to s can be re-written as: ( P11 P1 T11 T1 P11 P1 R = ( Φ 1 0 P1 P T1 T P1 P 0 R ( Φ, (101 P11 = (T11 P11 + T1 P1 R ( Φ 1, (10 P1 = (T1 P11 + T P1 R ( Φ 1, (103 P1 = (T11 P1 + T1 P R ( Φ, (104 P = (T1 P1 + T P R ( Φ. (105 From the above equations, if we know the phase advance Φ 1, Φ, the matri P at s can be obtained with the matri P 1 and the transfer matri T The phase advances As we know the phase advances are independent from particle to particle. So here we investigate one particle which only has mode s coordinates at s 1 in the betatron phase rotation frame, just like one eigenmode ecitation. The coordinates in the laboratory frame at s 1 is: y 1 = J 1 The coordinates in the laboratory frame at s is: y = J 1 From Eq. (5, we obtain: p 11 cos(φ 1 + Φ 1 p 1 cos(φ 1 + Φ 1 p sin(φ 1 + Φ 1 p 31 cos(φ 1 + Φ 1 p 3 sin(φ 1 + Φ 1 p 41 cos(φ 1 + Φ 1 p 4 sin(φ 1 + Φ 1 For p 11, we obtain: p 11 cos Φ 1 p 1 cos Φ 1 p sin Φ 1 p 31 cos Φ 1 p 3 sin Φ 1 p 41 cos Φ 1 p 4 sin Φ 1 p 11 cos(φ 1 + Φ 1 p 1 cos(φ 1 + Φ 1 p (Φ 1 + Φ 1 p 31 cos(φ 1 + Φ 1 p 3 sin(φ 1 + Φ 1 p 41 cos(φ 1 + Φ 1 p 4 sin(φ 1 + Φ 1 = t 11 t 1 t 13 t 14 t 1 t t 3 t 4 t 31 t 3 t 33 t 34 t 41 t 4 t 43 t 44. (106. (107 p 11 cos Φ 1 p 1 cos Φ 1 p sin Φ 1 p 31 cos Φ 1 p 3 sin Φ 1 p 41 cos Φ 1 p 4 sin Φ 1 (108 p 11 cos(φ 1 + Φ 1 = (t 11 p 11 + t 1 p 1 + t 13 p 31 + t 14 p 41 cos Φ 1 (t 1 p + t 13 p 3 + t 14 p 4 sin Φ 1. (109 14

16 For simplicity, we define: So Φ 1 is given: And we also obtain: G = T 1 P 1, (110 G ij = Σt ik p kj, k = 1,, 3, 4. (111 cos Φ 1 = G 11 / G 11 + G 1 sin Φ 1 = G 1 / G 11 + G 1 Φ 1 = arctan(g 1 /G 11 p 11 =. (11 G 11 + G 1. (113 The phase advance Φ for mode also can be achieved in the above procedure ecept we assume the particle only have mode coordinates in the betatron phase rotation frame at s 1. Φ and p 33 are obtained: cos Φ = G 33 / G 33 + G 34 sin Φ = G 34 / G 33 + G 34 Φ = arctan(g 34 /G 33 p 33 =, (114 G 33 + G 34. ( p ij at s Knowing the eigenmodes betatron oscillation phase advances Φ 1 and Φ, according to Eq. (10 to Eq. (105, the matri P is given: P11 = 1 p G11 G 1 G11 G 1, ( G 1 G G 1 G 11 P1 = 1 p G31 G 3 11 G 41 G 4 P1 = 1 p G13 G G 3 G 4 P = 1 p G33 G G 43 G 44 t is easy to check that the elements p 1 = 0, p 34 = 0. G11 G 1, (117 G 1 G 11 ( G33 G 34 G 34 G 33, (118 G33 G 34. (119 G 34 G Twiss and coupling parameters at s Then we are able to describe Twiss parameters at s in these at s 1 and the transfer matri T 1 s elements. β 1 = p 11 / p = or G 11 +G 1 G 11G G 1G 1 α 1 = p 1 / p = G11G1+G1G G 11G +G 1G 1, (10 β = p 33 / p 34 = S 33 +S 34 G 33G 44 G 34G 43. (11 α = p 43 / p 44 = G33G43+G34G44 G 33G 44+G 34G 43 The coupling parameter r at s can be obtained: From Eq. (5, Eq. (103, Eq. (104, we get: r = p 11 p = G 11 G G 1 G 1, (1 r = p 33 p 44 = G 33 G 44 G 34 G 43. (13 r C = P1 P = (T11 P1 + T1 P (P1 T1 + T P = T11 T1 + T1 T + T11 (r C T + T1 (r C T1. (14 15

17 5.5 Reduce to uncoupled situation Under the uncoupled situation, G = p 11 t 11 + p 1 t 1 p t p 11 t 1 + p 1 t p t p 33 t 33 + p 43 t 34 p 44 p p 33 t 43 + p 43 t 44 p 44 p 44. (15 The phase advances then equal to: { Φ1 = arctan(p t 1 /(p 11 t 11 + p 1 t 1 Φ = arctan(p 44 t 34 /(p 33 t 33 + p 43 t 34, (16 or in Twiss parameters at s 1 { Φ 1 = arctan(t 1 /(β 1 t 11 α 1 t 1 Φ = arctan(t 34 /(β t 33 α t 34. (17 β 1 at s is: α 1 at s is: β 1 = ((p t 1 + (p 11 t 11 + p 1 t 1 /(p11p(t 11 t t 1 t 1 = β 1 t 11 α 1 t 11 t 1 + γ 1 t 1. α 1 = (p t 1 t + (p 11 t 11 + p 1 t 1 (p 11 t 1 + p 1 t /(p 11 p (t 1 t 1 t 11 t = t 11 t 1 β 1 + (t 11 t + t 1 t 1 α 1 t 1 t γ 1. (18 (19 All of them are the same to that from the uncoupled one-dimensional situation. For the mode, we also have similar conclusion. 5.6 Reduces to uncoupled section Under the uncoupled situation,t1 = 0, T1 = 0. From Eq. (10, we obtain: P11 = P11. (130 Since r = P11 and r = P11, so we obtain that r keeps constant in the uncoupled section. And the propagation of coupling matri propagation in Eq. (14 reduces to: C = T11 C T. ( n the interaction region n the drifts of interaction regions, from the β waist where α i = 0, we have: p 11 s p p 13 + s p 3 p 14 + s p 4 G = 0 p p 3 p 4 p 31 + s p 41 p 3 + s p 4 p 33 s p 44 p 41 p 4 0 p 44, (13 s is the distance from the β waist. Then the phase advances from the waist is given: { Φ1 (s = arctan(s p /p 11 = arctan(s/β 1. (133 Φ (s = arctan(s p 44 /p 33 = arctan(s/β Twiss parameters at s are given: { β1 = p 11 +(s p p 11p = β1 + s /β1 β = p 33 +(s p44 p 33p 44 = β + s /β, (134 16

18 { α1 = s β 1 α = s β. (135 These epressions are same to the uncoupled one-dimensional situation, no matter of the coupling parameters at the β waist. The phase advances between the two BPMs close to the P are used to measure β s at the P. 6 Acknowledgments The author thanks S. Peggs, R. Talman (Cornell for the simulating discussions on the local and global coupling observables. Thank D. Trbojevic and T. Roser introduced the author to the topic of the RHC global decoupling with skew quadrupole modulation technique. And thank S. Peggs and T. Roser for their careful readings and corrections of the draft. During the writing of the article, the author got encouragements from many people and the author would like to thank them. They are Y. Cai(SLAC, A. Chao(SLAC, W. Fischer, D. Trbojevic, F. Zimmerman(CERN. This work is supported by U.S. DOE under contract No. DE-AC0-98CH References [1] E.D. Courant, H.S. Snyder, Ann. Phys. 3 (1958. [] G. Ripken, DESY nternal Report: R1-70/04 (1970. [3] F. Willeke, G. Ripken, CERN Yellow Report [4] H. Wiedemann, Particle Accelerator Physics, Nonlinear and Higher-Order Beam Dynamics, Springer- Verlag, [5] L.C. Teng, NAL Report FN-9, [6] D. Edwards, L. Teng, PAC 1973, p.885. [7] S. Peggs, Particle Accelerator 1, p.19, 198. [8] M. Billing, Cornell Report CBN 85- ( [9] D. Sagan, D. Rubin, Phys. Rev. ST Accel. Beam, (1999. [10] R. Talman, in Lecture Notes in Physics, 343, M. Month and P. Bryant, Editors, 1988 [11] P. Bagley, D. Rubin, PAC 1987, p [1] F. C. selin, The MAD Program Version 8.13-Physicist Manual, p5, unpublished. [13] P.castro, et al., PAC [14] J. rwin, C.X. Wang, Y.T. Yan, Phys. Rev. Lett. 8, p.1684 (1999. [15] C.-. Wang, V. Sajaev, and C.-Y. Yao, Phys. Rev. ST Accel. Beams 6, (003 [16] Y.T. Yan, Y. Cai, et al., SLAC-Pub-10369, 004. [17] D. Sagan, Phys. Rev. ST Accel. Beam 4, 1801 (001. [18] D. Sagan, R. Meller, et al., Phys. Rev. ST Accel. Beam 3, (000. [19] Y. Cai, Phys. Lett. A. 150, 6 (003 [0] J. Borer, C. Bovet, et al., Harmonic Analysis of Cohenerent Bunch Oscillation in LEP, Proceedings of The Third EPAC, Berlin, Germanny, p108, 199. [1] Talman, Shaker Response in a Coupled Lattice, 1996, unpublished. 17

19 [] G. Borianoff, S. Hunt, D. Mathieson, F. Pilat, R. Talman, Determination of Coupled-Lattice Properties Using Turn-By-Turn Data. [3] SAD code homepage: [4] K. Brown, in CERN Yellow Report 80-04, [5] F. C. selin, The MAD Program Version 8.13-Physicist Manual, p49, unpublished. 18