USPAS Course on Recirculated and Energy Recovered Linear Accelerators

Transcription

1 USPAS Coure on Recirculated and Energy Recovered Linear Accelerator G. A. Krafft and L. Merminga Jefferon Lab I. Bazarov Cornell Lecture 6 7 March 005

2 Lecture Outline. Invariant Ellipe Generated by a Unimodular Linear Tranformation. Detailed Solution of Hill Equation General Formula for Phae Advance Tranfer Matrix in Term of β-function Periodic Solution. Non-periodic Solution Formula for β-function and Phae Advance. Diperion Calculation. Beam Matching 7 March 005

3 Significance of matrix parameter Another way to interpret the parameter α, β, and γ, which repreent the unimodular matrix M (thee parameter are ometime called the Twi parameter or Twi repreentation for the matrix) i a the coordinate of that pecific et of ellipe that are mapped onto each other, or are invariant, under the linear action of the matrix. Thi reult i demontrated in Thm: For the unimodular linear tranformation M = 0 0 co with Tr (M) <, the ellipe α ( µ ) + in( µ ) γ β α 7 March 005

4 β γx + αxy + βy = are invariant under the linear action of M, where c i any contant. Furthermore, thee are the only invariant ellipe. Note that the theorem doe not apply to ±I, becaue Tr (±I) =. Pf: The invere to M i clearly M = 0 0 co α c ( µ ) in( µ ) γ β α By the ellipe tranformation formula, for example ' = β in µ γ + β in µ co µ + α in µ α + co = = ( ) ( )( ) ( µ + α in µ ) ( ) β in µ + α βα in µ + β co µ + βα in µ ( in µ + co µ ) β = β β 7 March 005

5 Similar calculation demontrate that α' = α and γ' = γ. A det (M) =, c' = c, and therefore the ellipe i invariant. Converely, uppoe that an ellipe i invariant. By the ellipe tranformation formula, the pecific ellipe γ x + α xy + β y = ε i i invariant under the tranformation by M only if γ i αi = βi T M i ( co µ α in µ ) ( co µ α in µ )( γ in µ ) ( γ in µ ) ( co µ α in µ )( β in µ ) βγ in µ ( co µ + α in µ )( γ in µ ) ( β in µ ) ( co µ + α in µ )( β in µ ) ( co µ + α in µ ) γ i αi T βi M v, i γ i α i βi 7 March 005

6 v i.e., if the vector i ANY eigenvector of T M with eigenvalue. All poible olution may be obtained by invetigating the eigenvalue and eigenvector of T M. Now i.e., T M v λ λ λ = v ha I = a olution when Det ( T λ ) 0 ( λ µ λ )( λ) + 4co + = 0 Therefore, M generate a tranformation matrix T M with at leat one eigenvalue equal to. For there to be more than one olution with λ =, + + = = M =± I 4co µ 0, co µ, or M 7 March 005

7 and we note that all ellipe are invariant when M = ±I. But, thee two cae are excluded by hypothei. Therefore, M generate a tranformation matrix T M which alway poee a ingle nondegenerate eigenvalue ; the et of eigenvector correponding to the eigenvalue, all proportional to each other, are the only vector whoe component (γ i, α i, β i ) yield equation for the invariant ellipe. For concretene, compute that eigenvector with eigenvalue normalized o β i γ i α i = γ i M / M v = α = β ( ), i i M M / M βi γ = α β v = εv i / c All other eigenvector with eigenvalue have,, for ome value c. 7 March 005

8 Becaue Det (M) =, the eigenvector invariant ellipe γ x + αxy + βy = ε. v clearly yield the Likewie, the proportional eigenvector generate the imilar ellipe ε c v,i ( γx + αxy + βy ) = ε Becaue we have enumerated all poible eigenvector with eigenvalue, all ellipe invariant under the action of M, are of the form γx + αxy + βy = c 7 March 005

9 To ummarize, thi theorem give a way to tie the mathematical repreentation of a unimodular matrix in term of it α, β, and γ, and it phae advance, to the equation of the ellipe invariant under the matrix tranformation. The equation of the invariant ellipe when properly normalized have preciely the ame α, β, and γ a in the Twi repreentation of the matrix, but varying c. Finally note that throughout thi calculation c act merely a a cale parameter for the ellipe. All ellipe imilar to the tarting ellipe, i.e., ellipe whoe equation have the ame α, β, and γ, but with different c, are alo invariant under the action of M. Later, it will be hown that more generally ( ( ) ) x + βx' αx / β ε = γx + αxx' + βx' = + i an invariant of the equation of tranvere motion. 7 March 005

10 Application to tranvere beam optic When the motion of particle in tranvere phae pace i conidered, linear optic provide a good firt approximation of the tranvere particle motion. Beam of particle are repreented by ellipe in phae pace (i.e. in the (x, x') pace). To the extent that the tranvere force are linear in the deviation of the particle from ome predefined central orbit, the motion may analyzed by applying ellipe tranformation technique. Tranvere Optic Convention: poition are meaured in term of length and angle are meaured by radian meaure. The area in phae pace divided by π, ε, meaured in m-rad, i called the emittance. In uch application, α ha no unit, β ha unit m/radian. Code that calculate β, by widely accepted convention, drop the per radian when reporting reult, it i implicit that the unit for x' are radian. 7 March 005

11 Definition of the Linear Tranport Matrix Within a linear optic decription of tranvere particle motion, the particle tranvere coordinate at a location along the beam line are decribed by a vector x dx d ( ) () If the differential equation giving the evolution of x i linear, one may define a linear tranport matrix M ', relating the coordinate at ' to thoe at by x( ' ) x() dx d ( ' ) = M ', dx d () 7 March 005

12 From the definition, the concatenation rule M '', = M '',' M ', mut apply for all ' uch that < '< '' where the multiplication i the uual matrix multiplication. Pf: The equation of motion, linear in x and dx/d, generate a motion with x () x( '' ) x( ' ) x( ) M = = = ' ', dx dx M dx M M dx () ( ) '', ' ( ) '', ' ', '' ' d d d d for all initial condition (x(), dx/d()), thu M '', = M '',' M ',. () Clearly M, = I. A i hown next, the matrix M ', i in general a member of the unimodular ubgroup of the general linear group. 7 March 005

13 Ellipe Tranformation Generated by Hill Equation The equation governing the linear tranvere dynamic in a particle accelerator, without acceleration, i Hill equation* d x d () = 0 + K x The tranformation matrix taking a olution through an infiniteimal ditance d i x dx d ( + d) ( + d) = K () d rad d x rad dx d ( ) () M Eqn. () x dx d ( ) () + d, * Strictly peaking, Hill tudied Eqn. () with periodic K. It wa firt applied to circular accelerator which had a periodicity given by the circumference of the machine. It i a now tandard in the field of beam optic, to till refer to Eqn. a Hill equation, even in cae, a in linear accelerator, where there i no periodicity. 7 March 005

14 Suppoe we are given the phae pace ellipe () + α( ) + β ( ) x xx' x' ε γ = at location, and we wih to calculate the ellipe parameter, after the motion generated by Hill equation, at the location + d ( ) ( ) ( ) + d x + α + d xx' + β + d x' ' γ = ε Becaue, to order linear in d, Det M +d, =, at all location, ε' = ε, and thu the phae pace area of the ellipe after an infiniteimal diplacement mut equal the phae pace area before the diplacement. Becaue the tranformation through a finite interval in can be written a a erie of infiniteimal diplacement tranformation, all of which preerve the phae pace area of the tranformed ellipe, we come to two important concluion: 7 March 005

15 . The phae pace area i preerved after a finite integration of Hill equation to obtain M ',, the tranport matrix which can be ued to take an ellipe at to an ellipe at '. Thi concluion hold generally for all ' and.. Therefore Det M ', = for all ' and, independent of the detail of the functional form K(). (If deired, thee two concluion may be verified more analytically by howing that d d ( ) ()() βγ α = 0 β γ α () =, may be derived directly from Hill equation.) 7 March 005

16 Evolution equation for the α, β function The ellipe tranformation formula give, to order linear in d So d β + rad d α + d = γ + α + β rad ( + d) = α β () ( ) ( ) () () Kd rad dα d dβ d () = () = β () K α () rad rad ( ) γ rad 7 March 005

17 Note that thee two formula are independent of the cale of the tarting ellipe ε, and in theory may be integrated directly for β()andα() given the focuing function K(). A omewhat eaier approach to obtain β() i to recall that the maximum extent of an ellipe, x max, i (εβ) / (), and to olve the differential equation decribing it evolution. The above equation may be combined to give the following non-linear equation for x max () = w() = (εβ) / () dw d + K w = ( ) ( ε ) 3 /rad. w Such a differential equation decribing the evolution of the maximum extent of an ellipe being tranformed i known a an envelope equation. 7 March 005

18 It hould be noted, for conitency, that the ame β() = w ()/ε i obtained if one tart integrating the ellipe evolution equation from a different, but imilar, tarting ellipe. That thi i o i an exercie. The envelope equation may be olved with the correct boundary condition, to obtain the β-function. α may then be obtained from the derivative of β, and γ by the uual normalization formula. Type of boundary condition: Cla I periodic boundary condition uitable for circular machine or periodic focuing lattice, Cla II initial condition boundary condition uitable for linac or recirculating machine. 7 March 005

19 Solution to Hill Equation in Amplitude-Phae form To get a more general expreion for the phae advance, conider in more detail the ingle particle olution to Hill equation d x d () = 0 + K x From the theory of linear ODE, the general olution of Hill equation can be written a the um of the two linearly independent peudo-harmonic function where x ( ) = Ax ( ) Bx ( ) x + + ( ) ( ) ± iµ ( ) w e ± = 7 March 005

20 are two particular olution to Hill equation, provided that d w + d K c w () w = and () = (), and where A, B, and c are contant (in ) 3 dµ d c w Eqn. (3) That pecific olution with boundary condition x( ) = x and dx/d ( ) = x' ha A B = w ( ) ( ) iµ ( ) ( ) iµ ( ) e w e w ic e ( ) '( ) w e iµ iµ ' + ( ) w ( ) w ic x ( ) x' 7 March 005

21 Therefore, the unimodular tranfer matrix taking the olution at = to it coordinate at = i x x' = ( ) ( ) w w c w ( ) w( ) co µ '( ) ( ) w w, w + w ( ) w' ( ) w( ) w( ) ( ) w' ( ) w( ) w' ( ) ( ) ( ) w' w c c co µ in µ,, in µ, ( ) ( ) w w co µ c, + w' in µ x ( ) ( ) w x' c, in µ, where µ ( ) ( ), = µ µ = w c () d 7 March 005

22 Cae I: K() periodic in Such boundary condition, which may be ued to decribe circular or ring-like accelerator, or periodic focuing lattice, have K( + L) = K(). L i either the machine circumference or period length of the focuing lattice. It i natural to aume that there exit a unique periodic olution w() to Eqn. (3a) when K() i periodic. Here, we will aume thi to be the cae. Later, it will be hown how to contruct the function explicitly. Clearly for w periodic φ + L = µ L with µ L = w () µ () c () d i alo periodic by Eqn. (3b), and µ L i independent of. 7 March 005

23 The tranfer matrix for a ingle period reduce to c w () co µ + L ( ) w' ( ) w ( ) () w' () w() w' () w' () w() w w = 0 c c 0 co in µ ( µ ) + in( µ ) L L in µ L α γ co µ + β α where the (now periodic!) matrix function are () w ( ) ( ) L c L in µ c L in µ L ( ) () w w α α() = ', β () =, γ () = + c c β By Thm. (), thee are the ellipe parameter of the periodically repeating, i.e., matched ellipe. 7 March 005

24 General formula for phae advance In term of the β-function, the phae advance for the period i µ L L = 0 β d () and more generally the phae advance between any two longitudinal location and 'i µ ', ' = d β () 7 March 005

25 M ', Tranfer Matrix in term of α and β Alo, the unimodular tranfer matrix taking the olution from to ' i = β β ( ' ) ( co µ () ) ( ) ( ) ', + α in µ ', β ' β in µ ', β () ( ( ) ( )) () ( ( ) ) + α ' α in µ ', β co µ ', α ' in µ ', ( ' ) β ( ) + ( α( ' ) α( ) ) co µ β ( ' ) Note that thi final tranfer matrix and the final expreion for the phae advance do not depend on the contant c. Thi concluion might have been anticipated becaue different particular olution to Hill equation exit for all value of c, but from the theory of linear ordinary differential equation, the final motion i unique once x and dx/d are pecified omewhere. ', 7 March 005

26 One method to compute the β-function Our previou work ha indicated a method to compute the β- function (and thu w) directly, i.e., without olving the differential equation Eqn. (3). At a given location, determine the one-period tranfer map M +L, (). From thi find µ L (which i independent of the location choen!) from co µ L = (M +M ) /, and by chooing the ign of µ L o that β() = M () / in µ L i poitive. Likewie, α() = (M -M ) / in µ L. Repeat thi exercie at every location the β-function i deired. By contruction, the beta-function and the alpha-function, and hence w, are periodic becaue the ingle-period tranfer map i periodic. It i traightforward to how w=(cβ()) / atifie the envelope equation. 7 March 005

27 Courant-Snyder Invariant Conider now a ingle particular olution of the equation of motion generated by Hill equation. We ve een that once a particle i on an invariant ellipe for a period, it mut tay on that ellipe throughout it motion. Becaue the phae pace area of the ingle period invariant ellipe i preerved by the motion, the quantity that give the phae pace area of the invariant ellipe in term of the ingle particle orbit mut alo be an invariant. Thi phae pace area/π, ( ( ) ) x + βx' αx / β ε = γx + αxx' + βx' = + i called the Courant-Snyder invariant. It may be verified to be a contant by howing it derivative with repect to i zero by Hill equation, or by explicit ubtitution of the tranfer matrix olution which begin at ome initial value = 0. 7 March 005

28 x dx d () () Peudoharmonic Solution = give β β β () ( co µ + α in µ ) β () 0 () β 0 x dx ( + α() α ) ( () ) 0 in µ,0 β0 co µ,0 in,0 d α µ 0 + ( α() α ) co µ β (),0 0 0,0,0 β in µ ( () ( () () ( ) ( )) ) x β x' + α x / β ( ) = x + ( β x' + α x ) 0,0 ( ) / β ε Uing the x() equation above and the definition of ε, the olution may be written in the tandard peudoharmonic form () () ( ) β 0x' 0 + α0x0 x = εβ co µ =,0 δ where δ tan x0 The the origin of the terminology phae advance i now obviou. 0 7 March 005

29 Cae II: K() not periodic In a linac or a recirculating linac there i no cloed orbit or natural machine periodicity. Deigning the tranvere optic conit of arranging a focuing lattice that aure the beam particle coming into the front end of the accelerator are accelerated (and ometime decelerated!) with a mall beam lo a i poible. Therefore, it i imperative to know the initial beam phae pace injected into the accelerator, in addition to the tranfer matrice of all the element making up the focuing lattice of the machine. An initial ellipe, or a et of initial condition that omehow bound the phae pace of the injected beam, are tracked through the acceleration ytem element by element to determine the tranmiion of the beam through the accelerator. The deign are uually made up of wellundertood module that yield known and undertood tranvere beam optical propertie. 7 March 005

30 Definition of β function Now the peudoharmonic olution applie even when K() i not periodic. Suppoe there i an ellipe, the deign injected ellipe, which tightly include the phae pace of the beam at injection to the accelerator. Let the ellipe parameter for thi ellipe be α 0, β 0, and γ 0. A function β() i imply defined by the ellipe tranformation rule β where () ( ()) = M ( ) ( ) ( ( )) γ 0 M M α0 + M [( ()) ( ( ) ( )) ] = M + β0m α0m / β0 M M ( ) M ( ) M,0 () M () β 0 7 March 005

31 One might think to evaluate the phae advance by integrating the beta-function. Generally, it i far eaier to evaluate the phae advance uing the general formula, tan µ ', = β ( M ) ', ()( M ) ()( ) ', α M ', where β() and α() are the ellipe function at the entrance of the region decribed by tranport matrix M ',. Applied to the ituation at hand yield tan µ,0 = β M 0 M ( ) () α M () 0 7 March 005

32 Diperion Calculation Begin with the inhomogeneou Hill equation for the diperion. d D d ( ) + K D = ρ ( ) Write the general olution to the inhomogeneou equation for the diperion a before. ( ) ( ) + ( ) + ( ) D D Ax Bx = p Here D p can be any particular olution. Suppoe that the diperion and it derivative are known at the location, and we wih to determine their value at. x and x, becaue they are olution to the homogeneou equation, mut be tranported by the tranfer matrix olution M, already found. 7 March 005

33 To build up the general olution, chooe that particular olution of the inhomogeneou equation with boundary condition ( ) ( ) D D p,0 = p,0 = 0 Evaluate A and B by the requirement that the diperion and it derivative have the proper value at (x and x need to be linearly independent!) ( ) ( ) ( ) ( ) ( ) ( ) A x x D = B x x D ( ) = ( ) + ( ) ( ) + ( ) ( ) D D M D M D p,0,, ( ) = ( ) + ( ) ( ) + ( ) ( ) D D M D M D p,0,, 7 March 005

34 3 by 3 Matrice for Diperion Tracking ( M ) ( ) ( ), M, D p,0 ( ) ( ) p ( ) D ( ) D( ) D ( ) M ( ), M, D,0 = D 0 0 Particular olution to inhomogeneou equation for contant K and contant ρ and vanihing diperion and derivative at = 0 K < 0 K = 0 K > 0 D p,0 () D' p,0 () K ρ ( coh( K) ) ( co ) ( K ) ( K) ρ inh in ( K) K ρ ρ K ρ K ρ 7 March 005

35 Longitudinal Stability in Detail For the microtron K i = / ρ i ( ( )) Dx, p,0= ρi co / ρi 0 πρi 56 πρ 0 i ( ( ρi) ) M = co / d = πρ i 7 March 005

36 M long M 56 0 = ce i = π frf Ginφ 0 4π ρi frf πρ i γ tanφ cγ i cei π frf Ginφ ( long ) γ = mc G co Tr M < 0 < νπ tanφ < Same reult for racetrack microtron if no optic or identity optic in the return traight φ 7 March 005

37 General Polytron 56 πρ / n i 0 ( ( ρi) ) πρi ( π) ( π ) { } M = co / d = n/ in / n / n Full one turn map, auming ynchronou phae ame for all linac i M n long M long 4π ρi frf πρ i { S( n/π) } γ tanφ { S( n/π) } cγ in cei = G π frf inφ n ( ) in / S x x x If ν denote the full one turn path length increment νπ φ π φ 0 < tan / n < 0 < l tan / n< 7 March 005

38 Beam Matching Fundamentally, in circular accelerator beam matching i applied in order to guarantee that the beam envelope of the real accelerator beam doe not depend on time. Thi requirement i one part of the definition of having a table beam. With periodic boundary condition, thi mean making beam denity contour in phae pace align with the invariant ellipe (in particular at the injection location!) given by the ellipe function. Once the particle are on the invariant ellipe they tay there (in the linear approximation!), and the denity i preerved becaue the ingle particle motion i around the invariant ellipe. In linac and recirculating linac, uually different purpoe are to be achieved. If there are region with periodic focuing lattice within the linac, matching a above enure that the beam 7 March 005

39 envelope doe not grow going down the lattice. Sometime it i advantageou to have pecific value of the ellipe function at pecific longitudinal location. Other time, re/matching i done to preerve the beam envelope of a good beam olution a change in the lattice are made to achieve other purpoe, e.g. changing the diperion function or changing the chromaticity of region where there are bend (ee the next chapter for definition). At a minimum, there i uually a matching done in the firt part of the injector, to take the phae pace that i generated by the particle ource, and change thi phae pace in a way toward agreement with the nominal tranvere focuing deign of the ret of the accelerator. The ellipe tranformation formula, olved by computer, are eential for performing thi proce. 7 March 005