emittance coupling Hong Qin12 1,2, Ronald C. Davidson 1, Moses Chung 3, John J. Barnard 4, Tai-Sen F. Wang 5 PAC11, New York, March 29, 2011

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1 Theoretical study of transverse-longitudinal emittance coupling Hong Qin1 1,, Ronald C. Davidson 1, Moses Chung 3, John J. Barnard 4, Tai-Sen F. Wang 5 1. PPPL, Princeton University. Univ. of Science and Technology of China 3. Handong Global University, Korean 4. LLNL 5. LANL PAC11, New York, March 9, 011

2 Transverse-longitudinal emittance coupling gives better beams FEL (LCLS, Emma 06, Kim 03 ) Heavy ion fusion

3 Emittance dynamics from covariance matrix Two issues with coupled emittance dynamics s æ ö x xz xx xz ç xz z zx zz º xx xz x xz ç zx zz xz z çè ø º ò f dxdzdp dp b x z s T () = s0 s M() s M() s Transfer matrix for coupled dynamics Initial covariance matrix

4 Previous work and present study Emittance exchange in one pass through the coupling component. Emma et al 06, Cornacchia et al 0, Kim 03 Eigen-emittance Williamson s theorem 36. Dragt s book, Yampolsky et al 11. Kishek et al 99. How about the transfer matrix M(s)? How about a coupled lattice? Emittance exchange? Present study

5 D coupled transverse dynamics 1 T T H = q Aq, q= ( xzxz,,, ) 10 free parameters æ k 0 ö æ kx kxz ö A =, k = ç è0 I ø èçk k ø xz z skew-quadrupole

6 Transfer matrix Original Courant-Snyder theory for uncoupled dynamics: SO( ) scalar Non-commutative generalization for coupled dynamics: p y SO( ( 4 )

7 Envelope equation Original Courant-Snyder theory for uncoupled dynamics: Envelope scalar Non-commutative generalization for coupled dynamics: envelope matrix

8 Courant-Snyder Invariant Original Courant-Snyder theory for uncoupled dynamics: w: envelope scalar Non-commutative generalization for coupled dynamics: w: envelope matrix

9 Phase advance Original Courant-Snyder theory for uncoupled dynamics: Non-commutative generalization for coupled dynamics:

10 Phase advance Original Courant-Snyder theory for uncoupled dynamics: Non-commutative generalization for coupled dynamics:

11 Twiss functions Original Courant-Snyder theory for uncoupled dynamics: b a g = w = - ww - = w + w Non-commutative generalization for coupled dynamics: T b = w w T a =-w w - T T g = w w + w w ( ) 1

12 How did we do it? General problem n n Hamiltonian Eq. æ 0 I ö q = J H, J = ç çè- I 0 ø

13 Time-dependent canonical transformation S(t) q H = S( t) q 1 T = q A( t) q T Symplectic group: SJS T = J Target Hamiltonian Two transformations to At ( ) = 0

14 Dynamics of physical emittance s T () = s0 s M() s M() s Eignen-emittances are invariants. But physical emittance measures beam qualities e = Det [ s ] = e ( s = 0) 4D 4D æ x xx ö e º Det( s ), s º x x x ç çè xx x ø æ z zz ö º º ç ç çè zz z ø e Det ( s ), s z z z ç Dt Det ( ), xz xz xz æ x xz ö e º s s º ç xz z çè ø

15 Example Matched solution of the envelope matrix w Coupling skew-quad

16 Interesting dynamics of physical emittance Interesting dynamics Emittance does not match the lattice. No emittance exchange. Classical uncertain principle and minimum emittance theorem. But can be explored for beam better property at the focal point.

17 An interesting case e = e e = e = x0 z0 x z But, e = e ¹ const. x z

18 Conclusions Coupled focusing lattice generates interesting emittance dynamics. Generalized Courant-Snyder theory for coupled dynamics is the tool to understand the coupled emittance dynamics. Coupled emittance dynamics can be explored for better beam properties at the focal point.

19 Courant-Snyder theory for uncoupled dynamics Phase advance j ( t ) = ò t 0 dt w ( t) qs () = Iws ()cos[ j() s + j ] 0 Courant (1958) CS invariant envelope æ ö b æqö æq0 ö [ cosj a0sinj] bb0sinj + = M( t) b 0 ç èq ø çèq M( t) = 0 ø ç 1 + aa0 a0- a b 0 ç- sinj+ cosj [ cosj- asinj] èç çè bb bb b ø 0 0 b

20 Courant-Snyder theory for uncoupled dynamics q I = + ( wq - w q) = const. w -3 w () s + k() s w() s = w () s Courant-Snyder invarianti Courant (1958) Envelope eq.

Kapchinskij-Vladimirskij (KV) distribution

k w = w, x x x x y y y y Kb Kb kx = kqx -,

21 Kapchinskij-Vladimirskij (KV) distribution (1959): Uncoupled N æi I ö = 1 + -, pee ç ø b x y f d KV pee x y è ex ey The only known solution x y I = + ( wx - xw ), I = + ( wy -yw ). x x x y y y w w x y -3-3 w + k w = w, w + k w = w, x x x x y y y y Kb Kb kx = kqx -, ky = kqy -, a( a + b) b( a + b) a º e w, b º e w. x x y y k qxy = k = qyx 0

physical meaning No b function uncoupled CS transfer matrix Have to

22 Many ways [Teng, 71] to parameterize the transfer matrix Symplectic rotation form [Edward-Teng, 73]: Lee Teng uncoupled No apparent physical meaning No b function uncoupled CS transfer matrix Have to define beta function from particle trajectories? [Ripken, 70], [Wiedemann, 99]

23 A hint from 1D C-S theory æ b ö [ cosj a0sinj] bb0sinj + b 0 M( t) = ç 1 + aa - 0 a0 a b 0 ç- ç sinj+ cosj [ cosj- asinj] çè bb bb b ø 0 0 t dt b( t) = w ( t), a( t) =- ww, j( t) =ò. 0 b( t )

24 Application: Strongly coupled system Stability completely l determined d by phase advance Theorem 1: Unstable P () t has an eigenvalue with l ¹ 1 c one turn generalized phase advance Theorem : T c Stable JP [ ( t )- P ( t )] is positive (negative)-definite c

25 Application: Weakly coupled system Stability determined by uncoupled phase advance Theorem 1: Unstable P ( t) has an eigenvalue with l ¹ 1 c Unstable uncoupled tune cos f = cos f p x y uncoupled tune x ny = n (sum and difference resonace) n

26 Application: Weakly coupled system Stability determined d by uncoupled phase advance Theorem : T c Stable JP [ ( t )- P ( t )] is positive (negative)-definite c stable sin f = sin f x y unstable tbl n + n = (sum resonace) x y n

27 Numerical example mis-aligned FODO lattice mis-alignment angle k æcos[ q( s) ] sin[ q( s) ] ö = k q çè sin [ q ( s ) ] - cos[ [ q ( s ) ] ø [Barnard, 96]

28 Envelope matrix w

29 Rotation matrix P 1

30 Rotation matrix P

31 Non-Commutative Courant-Snyder theory for coupled transverse dynamics Step I S = JAS -SJA c c æs 1 S ö æ 0 I öæbi 0 öæs1 Sö æs1 Söæ 0 I öæk 0ö = - ç ès S ø çè-i 0øè ç 0 b øèçs S ø èçs S øèç-i 0 øè ç0 I ø 3 4 I

32 S = 0 w º S 4 S4 = bi S4 - S4k 1 b º b - I w - = b w -wk S is symplectic: 1 T b - ww = I

33 Step II z = Pz H = c 0 P =-PJA c

34 Generalized KV distribution in coupled focusing lattice k qxy = k ¹ 0 qyx æksx ksxy ö - y = - ksx, k s = ç çèk k ø syx sy ( ) T ( ) w + wk = w w w k T -1-1T T T T T ICS = x w w x+ ( x w -x w )( wx -w x) T - y- k x kx, k = k + k q q s Nb w ICS f = æ - ö KV d 1 Aep èç e ø n( x, y, s ) = òdxdyf KV ì T -1-1T N b / A, 0 x w w x< e, = ï í ï T -1-1T ïî 0, e < x w w x.

Envelope matrix equation ( - 1 ) - 1 ( -

1T b * º w - w - 1 1 0 Q - l b * Q = æ ö

35 Envelope matrix equation ( - 1 ) - 1 ( - + k = 1 ) w w w w w T T k = k + k q s k s æksx ksxy ö = ç çè k k ø syx sy - Kb æ1/ a 0 ö = Q Q a + b ç è 0 1/ / b ø -1 ks 1 1T b * º w - w Q - l b * Q = æ ö çè 0 l ø Q = ( v, v ) 1 a / 1 = e l b = e/ l

36 T * 1 1T, * w - w - x b x= e b º Q Q l 0 b Q = ç çè 0 l ø - 1 * æ 1 ö ( v, v ) X a Y + = 1 = 1 X -1 b æ ö æxö Q = ç èy ø èçy ø a = e l / 1 b = / e l æ y / xö æxö - ks =- ç è y / y ø ç è y ø - Kb æ1/ a 0 ö = Q Q a + b çè 0 1/ b ø -1 ks æ y / Xö Kb æ1/ a 0 öæxö - = ç y / Y a b è ø + çè 0 1/ b øèçy ø

37 Generalized KV beam: pulsating & tumbling k æcos[ q( s) ] sin[ q( s) ] ö = k q ç èsin[ q( s) ] -cos[ q( s) ] ø Skew quad rotation angle

Beam Dynamics with Space- Charge

Beam Dynamics with Space- Charge Chris Prior, ASTeC Intense Beams Group, RAL and Trinity College, Oxford 1 1. Linear Transverse Review of particle equations of motion in 2D without space-charge - Courant-Snyder