Accelerator Phyic Linear Optic A. Bogacz, G. A. Krafft, and T. Zolkin Jefferon Lab Colorado State Univerity Lecture 3 USPAS Accelerator Phyic June 03
Linear Beam Optic Outline Particle Motion in the Linear Approximation Some Geometry of Ellipe Ellipe Dimenion in the β-function Decription Area Theorem for Linear Tranformation Phae Advance for a Unimodular Matrix Formula for Phae Advance Matrix Twi Repreentation 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 Beam Matching USPAS Accelerator Phyic June 03
Linear Particle Motion Fundamental Notion: The Deign Orbit i a path in an Earthfixed reference frame, i.e., a differentiable mapping from [0,] to point within the frame. A we hall ee a we go on, it generally conit of arc of circle and traight line. 3 σ :[0,] R r σ X σ = X σ Y σ Z σ Fundamental Notion: Path Length (,, ) d dx dy dz = + + dσ dσ dσ dσ USPAS Accelerator Phyic June 03
The Deign Trajectory i the path pecified in term of the path length in the Earth-fixed reference frame. For a relativitic accelerator where the particle move at the velocity of light, L tot =ct tot. 3 :[0, L tot ] R r X = X Y Z (,, ) The firt tep in deigning any accelerator, i to pecify bending magnet location that are conitent with the arc portion of the Deign Trajectory. USPAS Accelerator Phyic June 03
Comment on Deign Trajectory The notion of pecifying curve in term of their path length i tandard in coure on the vector analyi of curve. A good dicuion in a Calculu book i Thoma, Calculu and Analytic Geometry, 4 th Edition, Article 4.3-4.5. Mot vector analyi book have a imilar, and more advanced dicuion under the ubject of Frenet-Serret Equation. Becaue all of our deign trajectorie involve only arc of circle and traight line (dipole magnet and the drift region between them define the orbit), we can concentrate on a implified et of equation that only involve the radiu of curvature of the deign orbit. It may be worthwhile giving a imple example. USPAS Accelerator Phyic June 03
4-Fold Symmetric Synchrotron 7 ˆx 0 = 0 ẑ ŷ vertical ρ = L+ ρπ / ẑ ˆx = 3 6 3 5 L = 4 USPAS Accelerator Phyic June 03
It Deign Trajectory ( 0,0, ) ρ co (/ ρ),0,in (/ ρ) ( ρ,0, ρ ) (,0,0 ) ( L ρ,0, L) + ( 4 )( 0,0, ) ( ρ,0, ρ) (,0,0 ) ( ρ,0, ρ) + ρ in ( )/ ρ,0, co ( ) 0,0, 0 < < L= L + < < L + + < < 3 ( ( ) ) L ρ,0, L+ ρ + ρ in / ρ,0,co / ρ < < 3 3 3 4 ( ( )) ( ( 7 ) ) < < 4 5 L ρ,0,0 + ρ co / ρ,0, in / ρ < < 5 5 5 6 L + < < 6 6 7 / ρ < < 4 7 7 USPAS Accelerator Phyic June 03
Betatron Deign Trajectory 3 :[0, π R] R r X = R R R R ( co /, in /,0) Ue path length a independent variable intead of t in the dynamical equation. d d d = Ω R dt c USPAS Accelerator Phyic June 03
Betatron Motion in d δ r Δp + n Ω cδ r =ΩcR dt d δ z dt + nω δ z = d δ r Δp + δ r = d R R p c d z n δ d 0 n + δ z = R 0 p USPAS Accelerator Phyic June 03
Bend Magnet Geometry B r ˆx ŷ ˆx ẑ Rectangular Magnet of Length L Sector Magnet ρ θ ρ θ/ USPAS Accelerator Phyic June 03
For a uniform magnetic field Bend Magnet Trajectory dvx + Ω 0 cvx = r d( γ mv ) r r r = E V B dt + d( γ mvx ) = qvzby dt d( γ mvz ) = qvxby dt dvz +Ω 0 cvz = dt dt r For the olution atifying boundary condition: ( 0 ) 0 r X = V ( 0) = V ˆ 0 z z p X () t = ( co( Ωct) ) = ρ ( co( Ωct) ) Ω c = qby / γ m qb y p Z t ct ct qb () = in ( Ω ) = ρ in ( Ω ) y USPAS Accelerator Phyic June 03
Magnetic Rigidity The magnetic rigidity i: Bρ = B ρ = y p q It depend only on the particle momentum and charge, and i a convenient way to characterize the magnetic field. Given magnetic rigidity and the required bend radiu, the required bend field i a imple ratio. Note particle of momentum 00 MeV/c have a rigidity of 0.334 T m. Normal Incidence (or exit) Long Dipole Magnet Dipole Magnet = ( in ( /)) BL = Bρ in ( θ ) BL Bρ θ USPAS Accelerator Phyic June 03
Natural Focuing in Bend Plane Perturbed Trajectory Deign Trajectory Can how that for either a diplacement perturbation or angular perturbation from the deign trajectory d x d = x ρ x d y d y = ρ y USPAS Accelerator Phyic June 03
r B x y B xy yx (, ) = ( ˆ + ˆ) Quadrupole Focuing dv dv x y γm = qb x γm = qb y d d d y d x B B + x = 0 y = 0 ρ d Bρ d B Combining with the previou lide d x B d y B + 0 0 + x y = + = d ρx Bρ d ρy Bρ USPAS Accelerator Phyic June 03
Hill Equation Define focuing trength (with unit of m - ) B B kx = + k = ρ Bρ ρ Bρ y x y d x d y + k x x = 0 + k y y = 0 d d Note that thi i like the harmonic ocillator, or exponential for contant K, but more general in that the focuing trength, and hence ocillation frequency depend on USPAS Accelerator Phyic June 03
Energy Effect ρ ( +Δp/ p) p Δp Δ x = eb p co ( / ρ ) y ρ Thi olution i not a olution to Hill equation directly, but i a olution to the inhomogeneou Hill Equation d x x d y y B Δp + + x = d ρ Bρ ρx p B Δp + y = d ρ Bρ ρy p USPAS Accelerator Phyic June 03
Inhomogeneou Hill Equation Fundamental tranvere equation of motion in particle accelerator for mall deviation from deign trajectory d x x d y y B Δp + + x = d ρ Bρ ρx p B Δp + y = d ρ Bρ ρy p ρ radiu of curvature for bend, B' tranvere field gradient for magnet that focu (poitive correpond to horizontal focuing), Δp/p momentum deviation from deign momentum. Homogeneou equation i nd order linear ordinary differential equation. USPAS Accelerator Phyic June 03
Diperion From theory of linear ordinary differential equation, the general olution to the inhomogeneou equation i the um of any olution to the inhomogeneou equation, called the particular integral, plu two linearly independent olution to the homogeneou equation, whoe amplitude may be adjuted to account for boundary condition on the problem. = + + = + + x x A x B x y y A y B y p x x p y y Becaue the inhomogeneou term are proportional to Δp/p, the particular olution can generally be written a Δ xp D p x yp Dy p where the diperion function atify = = d B d B d D B d Dy B x + + Dx = + Dy = ρx ρ ρx ρy ρ ρy Δp p USPAS Accelerator Phyic June 03
M 56 In addition to the tranvere effect of the diperion, there are important effect of the diperion along the direction of motion. The primary effect i to change the time-ofarrival of the off-momentum particle compared to the on-momentum particle which travere the deign trajectory. ( Δ ) = d z D Δp p d ρ ρ ( + ) d Δp Δ z = ρ + D d ρ p d D Δp p Deign Trajectory Dipered Trajectory M 56 x = + D ρx D ρ y y d USPAS Accelerator Phyic June 03
Dipole Solution Homogeneou Eqn. x co (( )/ ) in (/ ) i i ρ ρ i ρ x dx = dx in (( )/ )/ co (/ ) i ρ ρ i ρ ( i ) d d Drift x x i i dx = dx 0 ( i ) d d USPAS Accelerator Phyic June 03
Quadrupole in the focuing direction k = B / Bρ ( ( )) i ( i) x co k in k / k x i dx = dx k in ( k ( )) co( ) ( i ) i k i d d Thin Focuing Len (limiting cae when argument goe to zero!) ( ε ) ( ε ) x + x 0 dx = dx ( + ε ) / f ( ε ) d d Thin Defocuing Len: change ign of f USPAS Accelerator Phyic June 03
Dipole Solution Homogeneou Eqn. x co (( )/ ) in (/ ) i i ρ ρ i ρ x dx = dx in (( )/ )/ co (/ ) i ρ ρ i ρ ( i ) d d Drift x x i i dx = dx 0 ( i ) d d USPAS Accelerator Phyic June 03
Quadrupole in the focuing direction k = B / Bρ ( ( )) i ( i) x co k in k / k x i dx = dx k in ( k ( )) co( ) ( i ) i k i d d Quadrupole in the defocuing direction k = B / Bρ ( ) i ( i) x coh k inh k / k x i dx = dx k inh ( k ( )) coh ( ) ( i ) i k i d d USPAS Accelerator Phyic June 03
Tranfer Matrice Dipole with bend Θ (put coordinate of final poition in olution) after before x x co( Θ) ρ in ( Θ) dx = in / co dx ρ after Θ Θ before d d Drift after before x x Ldrift dx = 0 dx after before d d USPAS Accelerator Phyic June 03
Quadrupole in the focuing direction length L Quadrupole in the defocuing direction length L after after before x co( kl) in ( kl) / k x dx = in co dx after k kl kl before d d before x coh ( kl) inh ( kl) / k x dx = inh co dx after k kl kl before d d Wille: pg. 7 USPAS Accelerator Phyic June 03
Thin Lene f f Thin Focuing Len (limiting cae when argument goe to zero!) ( ε ) ( ε ) x len + x len 0 dx = dx ( ) / f len + ε ( len ε ) d d Thin Defocuing Len: change ign of f USPAS Accelerator Phyic June 03
Compoition Rule: Matrix Multiplication! Element Element 0 x x 0 = M x x 0 More generally Remember: Firt element farthet RIGHT x x = M x x x x 0 = M M x x 0 M = M M... M M tot N N USPAS Accelerator Phyic June 03
Some Geometry of Ellipe Equation for an upright ellipe y x a + y b = b a x In beam optic, the equation for ellipe are normalized (by multiplication of the ellipe equation by ab) o that the area of the ellipe divided by π appear on the RHS of the defining equation. For a general ellipe Ax + Bxy + Cy = D USPAS Accelerator Phyic June 03
The area i eaily computed to be Area π ε = So the equation i equivalently AC D B Eqn. () γ x + αxy + βy = ε A B γ =, α =, and β = AC B AC B AC C B USPAS Accelerator Phyic June 03
When normalized in thi manner, the equation coefficient clearly atify βγ α = Example: the defining equation for the upright ellipe may be rewritten in following uggetive way a x a + y = ab b b = ε β = a/b and γ = b/a, note x = a = βε, max y max = b = γε USPAS Accelerator Phyic June 03
General Tilted Ellipe Need 3 parameter for a complete decription. One way y y=x a x a + ab b b ( y x) = = ε b a x where i a lope parameter, a i the maximum extent in the x-direction, and the y-intercept occur at b, and again ε i the area of the ellipe divided by π + a a b x b xy + y a b a b = ab = ε USPAS Accelerator Phyic June 03
Identify b a a γ = = = a + b, α, β b a b Note that βγ α = automatically, and that the equation for ellipe become x + β y + αx = βε by eliminating the (redundant!) parameter γ USPAS Accelerator Phyic June 03
Ellipe Dimenion in the β-function Decription y ε α, γ γε y=x= α x / β γε b = ε / β x βε, α ε β ε γ a = βε A for the upright ellipe xmax = βε, y max = γε Wille: page 8 USPAS Accelerator Phyic June 03
Area Theorem for Linear Optic Under a general linear tranformation x' M = y' M x y an ellipe i tranformed into another ellipe. Furthermore, if det (M) =, the area of the ellipe after the tranformation i the ame a that before the tranformation. M M Pf: Let the initial ellipe, normalized a above, be γ = 0x + α 0xy + β0 y ε 0 USPAS Accelerator Phyic June 03
Becaue ( ) ( M M ) x x' = y ( ) ( M M ) y' The tranformed ellipe i γ x + αxy + βy = ε0 ( M ) ( M ) ( M ) ( M ) γ = γ + α + β 0 0 0 α = M M γ + M M + M M α + M M β 0 0 0 ( M ) ( M ) ( M ) ( M ) β = γ + α + β 0 0 0 USPAS Accelerator Phyic June 03
Becaue (verify!) ( βγ α = β ) 0γ 0 α0 (( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) M M M M M M M M + ( )( βγ ) 0 0 α0 det M = the area of the tranformed ellipe (divided by π) i, by Eqn. () Area π = ε = β 0 γ 0 ε 0 = ε 0 det M α det M 0 USPAS Accelerator Phyic June 03
USPAS Accelerator Phyic June 03 Tilted ellipe from the upright ellipe In the tilted ellipe the y-coordinate i raied by the lope with repect to the un-tilted ellipe = y x y x 0 ' ' b a b a b a a b = = + = β α γ,, 0 0 0, 0,, b a M a b γ α β = = = = Becaue det (M)=, the tilted ellipe ha the ame area a the upright ellipe, i.e., ε = ε 0.
Phae Advance of a Unimodular Matrix Any two-by-two unimodular (Det (M) = ) matrix with Tr M < can be written in the form M = 0 0 co α ( μ ) + in( μ ) γ β α The phae advance of the matrix, μ, give the eigenvalue of the matrix λ = e iμ, and co μ = (Tr M)/. Furthermore βγ α = Pf: The equation for the eigenvalue of M i λ ( M + M ) λ + = 0 USPAS Accelerator Phyic June 03
Becaue M i real, both λ and λ* are olution of the quadratic. Becaue λ = Tr ( M ) ± i ( Tr( M )/ ) For Tr M <, λλ* = and o λ, = e iμ. Conequently co μ = (Tr M)/. Now the following matrix i trace-free. M 0 0 co ( μ ) M = M M M M M USPAS Accelerator Phyic June 03
Simply chooe α = M M in μ, β = M, in μ γ = M in μ and the ign of μ to properly match the individual matrix element with β > 0. It i eaily verified that βγ α =. Now M M n = 0 and more generally = 0 0 co 0 co α ( μ ) + in( μ ) γ α β α ( nμ ) + in( nμ ) γ β α USPAS Accelerator Phyic June 03
Therefore, becaue in and co are both bounded function, the matrix element of any power of M remain bounded a long a Tr (M) <. NB, in ome beam dynamic literature it i (incorrectly!) tated that the le tringent Tr (M) enure boundedne and/or tability. That equality cannot be allowed can be immediately demontrated by counterexample. The upper triangular or lower triangular ubgroup of the two-by-two unimodular matrice, i.e., matrice of the form 0 x x 0 clearly have unbounded power if x i not equal to 0. or USPAS Accelerator Phyic June 03
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 USPAS Accelerator Phyic June 03
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 USPAS Accelerator Phyic June 03
USPAS Accelerator Phyic June 03 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 i invariant under the tranformation by M only if, in co in in co in in in co in in in co in in in co in co v T T M i i i M i i i i i i r + + + = β α γ β α γ μ α μ μ β μ α μ μ β μ γ μ α μ μ βγ μ β μ α μ μ γ μ γ μ α μ μ α μ β α γ ε β α γ = + + y xy x i i i
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 T i.e., M v rλ λ r λ v r = 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 =, + + = = =± 4 co μ 0, co μ, or M I M USPAS Accelerator Phyic June 03
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 r v = α = β ( ), i i M M / M βi γ = α β r r v = εv / i c All other eigenvector with eigenvalue have,, for ome value c. USPAS Accelerator Phyic June 03
Becaue Det (M) =, the eigenvector invariant ellipe γ x + αxy + βy = ε. v r clearly yield the Likewie, the proportional eigenvector generate the imilar ellipe ε c ( 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 r v,i γx + αxy + βy = c USPAS Accelerator Phyic June 03
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. USPAS Accelerator Phyic June 03
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. USPAS Accelerator Phyic June 03
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 () USPAS Accelerator Phyic June 03
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. USPAS Accelerator Phyic June 03
Ellipe Tranformation Generated by Hill Equation The equation governing the linear tranvere dynamic in a particle accelerator, without acceleration, i Hill equation* ( + d) ( + d) = K d x d () d () = 0 + K x The tranformation matrix taking a olution through an infiniteimal ditance d i x dx 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. USPAS Accelerator Phyic June 03
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: USPAS Accelerator Phyic June 03
. 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.) USPAS Accelerator Phyic June 03
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 USPAS Accelerator Phyic June 03
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) / () ( ε ) dw + K w = 3 d /rad. w Such a differential equation decribing the evolution of the maximum extent of an ellipe being tranformed i known a an envelope equation. USPAS Accelerator Phyic June 03
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. USPAS Accelerator Phyic June 03
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 ± = + + USPAS Accelerator Phyic June 03
USPAS Accelerator Phyic June 03 are two particular olution to Hill equation, provided that () () and (), 3 w c d d w c w K d w d = = + μ and where A, B, and c are contant (in ) That pecific olution with boundary condition x( ) = x and dx/d ( ) = x' ha + = ' ' ' x x e w ic w e w ic w e w e w B A i i i i μ μ μ μ Eqn. (3)
USPAS Accelerator Phyic June 03 Therefore, the unimodular tranfer matrix taking the olution at = to it coordinate at = i Δ + Δ Δ Δ + Δ Δ Δ =,,,,,,, ' in ' co co ' ' in ' ' in in ' co ' x x c w w w w w w w w c w w w w w w c c w w c w w w w x x μ μ μ μ μ μ μ where () d w c = = Δ, μ μ μ
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. USPAS Accelerator Phyic June 03
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. USPAS Accelerator Phyic June 03
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 Δμ ', ' = d β () USPAS Accelerator Phyic June 03
M Tranfer Matrix in term of α and β ', Alo, the unimodular tranfer matrix taking the olution from to ' i = β β ( ' ) ( co Δμ () ) ', + α in Δμ', β ' β in Δμ', β () ( ) () ( ) + α ' α in Δμ', β co Δμ', α ' in Δμ ', ( ' ) β + ( α( ' ) α ) co Δμ 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. ', USPAS Accelerator Phyic June 03
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. USPAS Accelerator Phyic June 03
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. USPAS Accelerator Phyic June 03
x dx d () () = give Peudoharmonic Solution β β β () ( co Δμ + α in Δμ ) β () 0 () β 0 x dx ( + α() α ) ( () ) 0 in Δμ,0 β0 co Δμ Δ,0 in,0 d α μ 0 + ( α() α ) co Δμ β (),0 0 0,0,0 β in Δμ ( () ( () () ) ) x β x' + α x / β = x + ( β x' + α x ) 0,0 / β ε + 0 0 0 0 0 0 Uing the x() equation above and the definition of ε, the olution may be written in the tandard peudoharmonic form x β x' + α x () () 0 0 0 0 = εβ co Δμ =,0 δ where δ tan x0 The the origin of the terminology phae advance i now obviou. 0 USPAS Accelerator Phyic June 03
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. USPAS Accelerator Phyic June 03
β where 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 () ( ()) = M ( ) γ 0 M M α0 + M [( ()) ( () ()) ] = M + β0m α0m / β0 M M M M,0 () M () β 0 USPAS Accelerator Phyic June 03
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 Δμ ', = 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 Δμ,0 = β M 0 M () α M () 0 USPAS Accelerator Phyic June 03
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 USPAS Accelerator Phyic June 03
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. USPAS Accelerator Phyic June 03
Diperion Calculation Begin with the inhomogeneou Hill equation for the diperion. d D d + K D = Write the general olution to the inhomogeneou equation for the diperion a before. 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. ρ + + D D Ax Bx = p USPAS Accelerator Phyic June 03
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,, USPAS Accelerator Phyic June 03
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 () K ρ ( coh ( K ) ) ρ K ρ ( co( K) ) D' p,0 () inh K ρ ( K) ρ in K ρ ( K) USPAS Accelerator Phyic June 03
M 56 In addition to the tranvere effect of the diperion, there are important effect of the diperion along the direction of motion. The primary effect i to change the time-ofarrival of the off-momentum particle compared to the on-momentum particle which travere the deign trajectory. ( Δ ) = d z D Δp p d ρ ρ ( + ) d Δp Δ z = ρ + D d ρ p d D Δp p Deign Trajectory Dipered Trajectory M 56 x = + D ρx D ρ y y d USPAS Accelerator Phyic June 03
Solenoid Focuing Can alo have continuou focuing in both tranvere direction by applying olenoid magnet: B( z) z USPAS Accelerator Phyic June 03
Buch Theorem For cylindrical ymmetry magnetic field decribed by a vector potential: r ˆ A= Aθ z, r θ Bz = ( raθ ( z, r) ) i nearly contant r r Bz( r = 0, z) r B z( r = 0, z) r Aθ ( z, r) Br = Conervation of Canonical Momentum give Buch Theorem: Pθ = γmr θ + qraθ = cont for particle with & θ = 0 where B = 0, P = 0 & & qb Ω γ m z c γmr θ = = = ωlarmor Beam rotate at the Larmor frequency which implie coupling z θ USPAS Accelerator Phyic June 03
d dt k = Radial Equation ( γmr& ) γmrω = qr & θb = γmrω L z c thin len focal length = f ω β e 4β Bz dz z γ m c L z L weak compared to quadrupole for high γ If go to full ¼ ocillation inide the magnetic field in the thick len cae, all particle end up at r = 0! Non-zero emittance pread out perfect focuing! y x USPAS Accelerator Phyic June 03
Larmor Theorem Thi reult i a pecial cae of a more general reult. If go to frame that rotate with the local value of Larmor frequency, then the tranvere dynamic including the magnetic field are imply thoe of a harmonic ocillator with frequency equal to the Larmor frequency. Any force from the magnetic field linear in the field trength i tranformed away in the Larmor frame. And the motion in the two tranvere degree of freedom are now decoupled. Pf: The equation of motion are d ( γmr& ) γmr & θ = qr & θbz dt γmr & θ + qaθ = con = Pθ d ( γmr& ) γmr & θ dt + γmrθ ω γmrω = qr & θ B qrω B γmr & θ = P d dt γmr γmr & θ γmrω ( &) θ L L z L z = L γmr & θ = Pθ -D Harmonic Ocillator USPAS Accelerator Phyic June 03