# Accelerator Physics Linear Optics S. A. Bogacz, G. Accelerator Physics Linear Optics S. A. Bogacz, G. A. Krafft, S. DeSilva, R. Gamage Jefferson Lab Old Dominion University Lecture 4 USPAS Accelerator Physics June 2016 Linear Beam Optics Outline Particle Motion in the Linear Approximation Some Geometry of Ellipses Ellipse Dimensions in the -function Descriptionfunction Description Area Theorem for Linear Transformations Phase Advance for a Unimodular Matrix Formula for Phase Advance Matrix Twiss Representation Invariant Ellipses Generated by a Unimodular Linear Transformation Detailed Solution of Hills Equation General Formula for Phase Advance Transfer Matrix in Terms of -function Descriptionfunction Periodic Solutions Non-function Descriptionperiodic Solutions Formulas for -function Descriptionfunction and Phase Advance Beam Matching USPAS Accelerator Physics June 2016 Linear Particle Motion Fundamental Notion: The Design Orbit is a path in an Earth-function Description

fixed reference frame, i.e., a differentiable mapping from [0,1] to points within the frame. As we shall see as we go on, it generally consists of arcs of circles and straight lines. :[0,1] R 3 X X , Y , Z Fundamental Notion: Path Length 2 2 2 dX dY dZ ds d d d d USPAS Accelerator Physics June 2016 The Design Trajectory is the path specified in terms of the path length in the Earth-function Descriptionfixed reference frame. For a relativistic accelerator where the particles move at the velocity of light, Ltot=cttot. s :[0, Ltot ] R 3 s X s X s , Y s , Z s The first step in designing any accelerator, is to specify bending magnet locations that are consistent with the arc portions of the Design Trajectory.

USPAS Accelerator Physics June 2016 Comment on Design Trajectory The notion of specifying curves in terms of their path length is standard in courses on the vector analysis of curves. A good discussion in a Calculus book is Thomas, Calculus and Analytic Geometry, 4th Edition, Articles 14.3-function Description14.5. Most vector analysis books have a similar, and more advanced discussion under the subject of Frenet-function DescriptionSerret Equations. Because all of our design trajectories involve only arcs of circles and straight lines (dipole magnets and the drift regions between them define the orbit), we can concentrate on a simplified set of equations that only involve the radius of curvature of the design orbit. It may be worthwhile giving a simple example. USPAS Accelerator Physics June 2016 4-Fold Symmetric Synchrotron s0 0 x z s1 y vertical s2 L / 2 s7 z

s6 3s2 s3 L s5 s4 2 s2 USPAS Accelerator Physics June 2016 x Its Design Trajectory 0, 0, s 0, 0, L cos s s1 / 1, 0,sin s s1 / , 0, L s s2 1,0,0 L , 0, L sin s s3 / , 0, cos s s3 / 1 L 2 , 0, L s s4 0,0, 1 L 2 , 0, 0 1 cos s s5 / , 0, sin s s5 / L , 0, s s6 1,0,0 , 0, sin s s7 / , 0,1 cos s s7 / USPAS Accelerator Physics June 2016 0 s L s1 s1 s s2 s2 s s3 s3 s s4 s4 s s5 s5 s s6 s6 s s7 s7 s 4 s2 Betatron Design Trajectory s :[0, 2 R] R

3 s X s R cos s / R , R sin s / R , 0 Use path length s as independent variable instead of t in the dynamical equations. d 1 d ds c R dt USPAS Accelerator Physics June 2016 Betatron Motion in s d 2 r p 2 2 1 n c r c R 2 dt p d 2 z 2 n c z 0 2 dt d 2 r 1 n 1 p

r 2 2 ds R R p d 2 z n 2 z 0 2 ds R USPAS Accelerator Physics June 2016 Bend Magnet Geometry B Rectangular Magnet of Length L x z /2 USPAS Accelerator Physics June 2016 y x Sector Magnet

Bend Magnet Trajectory For a uniform magnetic field d (mV ) E V B dt d (mVx ) qVz By dt d (mVz ) qVx By dt d 2Vx d 2Vz 2 2 V 0 c x cVz 0 2 2 dt dt For the solution satisfying boundary conditions: X 0 0 X t

p cos c t 1 cos ct 1 qBy Z t c qBy / m p sin c t sin c t qBy USPAS Accelerator Physics June 2016 V 0 V0 z z Magnetic Rigidity The magnetic rigidity is: B By p q It depends only on the particle momentum and charge, and is a convenient way to characterize the magnetic field. Given magnetic rigidity and the required bend radius, the required bend field is a simple ratio. Note particles of momentum 100 MeV/c have a rigidity of 0.334 T m. Normal Incidence (or exit) Long Dipole Magnet Dipole Magnet BL B 2sin / 2

BL B sin USPAS Accelerator Physics June 2016 Natural Focusing in Bend Plane Perturbed Trajectory Design Trajectory Can show that for either a displacement perturbation or angular perturbation from the design trajectory d 2x x ds 2 x2 s USPAS Accelerator Physics June 2016 d2y y ds 2 y2 s Quadrupole Focusing B x, y B s xy yx m dv x qB s x ds d 2 x B s

x 0 2 ds B m dv y ds qB s y d 2 y B s y 0 2 ds B Combining with the previous slide B s d 2x 1 2 x 0 2 ds B x s B s d2y 1 2

y 0 2 ds B y s USPAS Accelerator Physics June 2016 Hills Equation Define focusing strengths (with units of m-function Description2) kx s 1 2 x s B s d 2x k x s x 0 2 ds B ky 1

2 y s B s B d2y k y s y 0 2 ds Note that this is like the harmonic oscillator, or exponential for constant K, but more general in that the focusing strength, and hence oscillation frequency depends on s USPAS Accelerator Physics June 2016 Energy Effects 1 p / p x s p p 1 cos s / eBy p This solution is not a solution to Hills equation directly, but is a solution to the inhomogeneous Hills Equations

B s d 2x 1 1 p x ds 2 x2 s B x s p B s d2y 1 1 p y ds 2 y2 s B y s p USPAS Accelerator Physics June 2016 Inhomogeneous Hills Equations Fundamental transverse equations of motion in particle accelerators for small deviations from design trajectory B s d 2x 1 1 p 2

x 2 ds B x s p x s B s d2y 1 1 p 2 y 2 ds B y s p y s radius of curvature for bends, B' transverse field gradient for magnets that focus (positive corresponds to horizontal focusing), p/p p/p momentum deviation from design momentum. Homogeneous equation is 2nd order linear ordinary differential equation. USPAS Accelerator Physics June 2016 Dispersion From theory of linear ordinary differential equations, the general solution to the inhomogeneous equation is the sum of any solution to the inhomogeneous equation, called the particular integral, plus two linearly independent solutions to the homogeneous equation, whose amplitudes may be adjusted to account for boundary conditions on the problem. x s =x p s Ax x1 s Bx x2 s y s =y p s Ay y1 s B y y2 s

Because the inhomogeneous terms are proportional to p/p p/p, the particular solution can generally be written as x p s =Dx s p p y p s =D y s p p where the dispersion functions satisfy B s d 2 Dx 1 1 D 2 x ds 2 s B x s x d 2 Dy

ds 2 USPAS Accelerator Physics June 2016 1 B s 1 2 D y s B y s y M56 In addition to the transverse effects of the dispersion, there are important effects of the dispersion along the direction of motion. The primary effect is to change the time-function Descriptionof-function Description arrival of the off-function Descriptionmomentum particle compared to the on-function Descriptionmomentum particle which traverses the design trajectory. ds p D s

ds p ds p D s p z d z =D s p ds p s Design Trajectory s2 M 56 Dispersed Trajectory Dx s Dy s ds s1 x s y s USPAS Accelerator Physics June 2016

Solutions Homogeneous Eqn. Dipole x s cos s s / i dx s sin s s / / i ds sin s si / x si dx cos s si / si ds Drift x s x si 1 s s i dx dx s 0 1 si ds

ds USPAS Accelerator Physics June 2016 Quadrupole in the focusing direction k B / B x s cos k s si dx s k sin k s si ds k s si / k x si dx s cos k s si i ds sin

Quadrupole in the defocusing direction k B / B x s cosh k s si dx s k sinh k s s i ds k s si / k x si dx s cosh k s si i ds sinh

USPAS Accelerator Physics June 2016 Transfer Matrices Dipole with bend (put coordinate of final position in solution) x safter cos dx sin / safter ds x sbefore sin dx cos s before ds Drift x safter

1 dx 0 safter ds x sbefore Ldrift dx 1 s before ds USPAS Accelerator Physics June 2016 Thin Lenses f f Thin Focusing Lens (limiting case when argument goes to zero!) x slens 1 dx s 1/ f lens ds

x s 0 lens dx 1 slens ds Thin Defocusing Lens: change sign of f USPAS Accelerator Physics June 2016 Composition Rule: Matrix Multiplication! Element 1 Element 2 s0 s1 x s1 x s0 M 1 x s1 x s0 s2 x s2 x s1 M 2

x s2 x s1 x s2 x s0 M 2 M 1 x s2 x s0 More generally M tot M N M N 1...M 2 M 1 Remember: First element farthest RIGHT USPAS Accelerator Physics June 2016 Some Geometry of Ellipses y Equation for an upright ellipse 2 2 x y 1 a b b a x

In beam optics, the equations for ellipses are normalized (by multiplication of the ellipse equation by ab) so that the area of the ellipse divided by appears on the RHS of the defining equation. For a general ellipse Ax 2 2 Bxy Cy 2 D USPAS Accelerator Physics June 2016 The area is easily computed to be Area D AC B 2 Eqn. (1) So the equation is equivalently x 2 2xy y 2 A AC B 2 , B AC B 2 , and USPAS Accelerator Physics June 2016

C AC B 2 When normalized in this manner, the equation coefficients clearly satisfy 2 1 Example: the defining equation for the upright ellipse may be rewritten in following suggestive way b 2 a 2 x y ab a b = a/b and = b/a, note xmax a , ymax b USPAS Accelerator Physics June 2016 General Tilted Ellipse y Needs 3 parameters for a complete description. One way b 2 a 2 x y sx ab a b y=sx b x

a where s is a slope parameter, a is the maximum extent in the x-function Descriptiondirection, and the y-function Descriptionintercept occurs at b, and again is the area of the ellipse divided by 2 2 b a a a 2 2 1 s 2 x 2 s xy y ab a b b b USPAS Accelerator Physics June 2016 Identify 2 b 2 a 1 s 2 , a b a s, b a b

Note that 2 = 1 automatically, and that the equation for ellipse becomes 2 x 2 y x by eliminating the (redundant!) parameter USPAS Accelerator Physics June 2016 Ellipse in the -function Description , y y=sx= x / , b / x

a As for the upright ellipse xmax , USPAS Accelerator Physics June 2016 ymax Area Theorem for Linear Optics Under a general linear transformation x' M 11 y ' M 21 M 12 x M 22 y an ellipse is transformed into another ellipse. Furthermore, if det (M) = 1, the area of the ellipse after the transformation is the same as that before the transformation. Pf: Let the initial ellipse, normalized as above, be 0 x 2 2 0 xy 0 y 2 0 USPAS Accelerator Physics June 2016 Effect of Transformation 2

2 Let the final ellipse be x 2 xy y The transformed coordinates must solve this equation. x2 2 xy y2 M x, y x, y 0 x 2 2 0 xy 0 y 2 0 M 1 The transformed coordinates must also solve the initial equation transformed. x ' M 11 M 12 x M M y ' 21 22 y 1 1 x M 11 M 12 x 1 1

y y M 21 M 22 USPAS Accelerator Physics June 2016 Because 1 M x M 1 y The transformed ellipse is 11 M 1 M 1 21 x' 12

y ' 22 x2 2xy y2 0 1 M M 1 11 M 1 2 1 1 2 M M M M M M M M M 2 M M M 11 0 2 M 1 1

12 0 11 1 11 1 2 12 0 22 1 12 1 12 0 21 1 1

0 21 1 21 1 22 USPAS Accelerator Physics June 2016 0 0 21 2 22 1 0 22 0 Because (verify!) 2 00 02

M 1 2 1 2 M 21 12 M 1 2 1 2 M 11 00 2 0 22 2 M 1

1 1 1 M M M 11 22 12 1 2 det M the area of the transformed ellipse (divided by ) is, by Eqn. (1) 0 Area 0 | det M | 00 02 det M 1 USPAS Accelerator Physics June 2016 21 Tilted ellipse from the upright ellipse In the tilted ellipse the y-function Descriptioncoordinate is raised by the slope with respect to the un-function Descriptiontilted ellipse

x' 1 0 x y' s 1 y b 0 , a a 0 , b 0 0, b a 2 s , a b a s, b 1 M s 21 a b Because det (M)=1, the tilted ellipse has the same area as the upright ellipse, i.e., = 0.

USPAS Accelerator Physics June 2016 Phase Advance of a Unimodular Matrix Any two-function Descriptionby-function Descriptiontwo unimodular (Det (M) = 1) matrix with | Tr M| < 2 can be written in the form 1 0 cos sin M 0 1 The phase advance of the matrix, , gives the eigenvalues of the matrix = ei, and cos = (Tr M)/2. Furthermore 2=1 Pf: The equation for the eigenvalues of M is 2 M 11 M 22 1 0 USPAS Accelerator Physics June 2016 Because M is real, both and * are solutions of the quadratic. Because Tr M 2 i 1 Tr M / 2 2 For |Tr M| < 2, * =1 and so 1,2 = ei. Consequently cos = (Tr M)/2. Now the following matrix is trace-function Descriptionfree. M 11 M 22 1 0

2 cos M 0 1 M 21 USPAS Accelerator Physics June 2016 M 12 M 22 M 11 2 Simply choose M 11 M 22 , 2 sin M 12 , sin M 21 sin and the sign of to properly match the individual matrix elements with > 0. It is easily verified that 2 = 1. Now

1 0 cos 2 sin 2 M 0 1 2 and more generally 1 0 cos n sin n M 0 1 n USPAS Accelerator Physics June 2016 Therefore, because sin and cos are both bounded functions, the matrix elements of any power of M remain bounded as long as |Tr (M)| < 2. NB, in some beam dynamics literature it is (incorrectly!) stated that the less stringent |Tr (M)| 2 ensures boundedness and/or stability. That equality cannot be allowed can be immediately demonstrated by counterexample. The upper triangular or lower triangular subgroups of the two-function Descriptionby-function Descriptiontwo unimodular matrices, i.e., matrices of the form 1 x

1 0 or 0 1 x 1 clearly have unbounded powers if |x| is not equal to 0. USPAS Accelerator Physics June 2016 Significance of matrix parameters Another way to interpret the parameters , , and , which represent the unimodular matrix M (these parameters are sometimes called the Twiss parameters or Twiss representation for the matrix) is as the coordinates of that specific set of ellipses that are mapped onto each other, or are invariant, under the linear action of the matrix. This result is demonstrated in Thm: For the unimodular linear transformation 1 0 cos sin M 0 1 with |Tr (M)| < 2, the ellipses USPAS Accelerator Physics June 2016 x 2 2xy y 2 c are invariant under the linear action of M, where c is any constant. Furthermore, these are the only invariant ellipses. Note that the theorem does not apply to I, because |Tr (I)| = 2.

Pf: The inverse to M is clearly M 1 1 0 cos 0 1 sin By the ellipse transformation formulas, for example ' sin 2 sin cos sin cos sin sin 1 2 sin cos sin sin cos 2 2 2 2 2 2 2

2 USPAS Accelerator Physics June 2016 2 2 2 2 Similar calculations demonstrate that ' = and ' = . As det (M) = 1, c' = c, and therefore the ellipse is invariant. Conversely, suppose that an ellipse is invariant. By the ellipse transformation formula, the specific ellipse i x 2 2 i xy i y 2 is invariant under the transformation by M only if cos sin 2 2 cos sin sin i 1 2 sin 2 i cos sin sin 2 sin 2 cos sin sin

i i TM i TM v , i USPAS Accelerator Physics June 2016 i sin 2 cos sin sin i cos sin 2 i i.e., if the vector v is ANY eigenvector of TM with eigenvalue 1. All possible solutions may be obtained by investigating the eigenvalues and eigenvectors of TM. Now TM v v has a solution w hen Det TM I 0 i.e., 2 2 4 cos 2 1 1 0 Therefore, M generates a transformation matrix TM with at least

one eigenvalue equal to 1. For there to be more than one solution with = 1, 2 2 1 2 4 cos 1 0, cos 1, or M I USPAS Accelerator Physics June 2016 and we note that all ellipses are invariant when M = I. But, these two cases are excluded by hypothesis. Therefore, M generates a transformation matrix TM which always possesses a single nondegenerate eigenvalue 1; the set of eigenvectors corresponding to the eigenvalue 1, all proportional to each other, are the only vectors whose components (i, i, i) yield equations for the invariant ellipses. For concreteness, compute that eigenvector with eigenvalue 1 normalized so ii i2 = 1 M 21 / M 12 i v1,i i M 11 M 22 / 2 M 12 1 i

All other eigenvectors with eigenvalue 1 have v1 v1,i / c, for some value c. USPAS Accelerator Physics June 2016 Because Det (M) =1, the eigenvector v1,i clearly yields the invariant ellipse x 2 2xy y 2 . Likewise, the proportional eigenvector v1 generates the similar ellipse 2 x 2xy y 2 c Because we have enumerated all possible eigenvectors with eigenvalue 1, all ellipses invariant under the action of M, are of the form x 2 2xy y 2 c USPAS Accelerator Physics June 2016 To summarize, this theorem gives a way to tie the mathematical representation of a unimodular matrix in terms of its , , and , and its phase advance, to the equations of the ellipses invariant under the matrix transformation. The equations of the invariant ellipses when properly normalized have precisely the same , , and as in the Twiss representation of the matrix, but varying c. Finally note that throughout this calculation c acts merely as a scale parameter for the ellipse. All ellipses similar to the starting

ellipse, i.e., ellipses whose equations have the same , , and , but with different c, are also invariant under the action of M. Later, it will be shown that more generally 2 x 2 2xx' x'2 x 2 x'x / is an invariant of the equations of transverse motion. USPAS Accelerator Physics June 2016 Applications to transverse beam optics When the motion of particles in transverse phase space is considered, linear optics provides a good first approximation of the transverse particle motion. Beams of particles are represented by ellipses in phase space (i.e. in the (x, x') space). To the extent that the transverse forces are linear in the deviation of the particles from some pre-function Description defined central orbit, the motion may analyzed by applying ellipse transformation techniques. Transverse Optics Conventions: positions are measured in terms of length and angles are measured by radian measure. The area in phase space divided by , , measured in m-function Descriptionrad, is called the emittance. In such applications, has no units, has units m/radian. Codes that calculate , by widely accepted convention, drop the per radian when reporting results, it is implicit that the units for x' are radians. USPAS Accelerator Physics June 2016 Linear Transport Matrix Within a linear optics description of transverse particle motion, the particle transverse coordinates at a location s along the beam line are described by a vector x s dx

s ds If the differential equation giving the evolution of x is linear, one may define a linear transport matrix Ms',s relating the coordinates at s' to those at s by x s ' x s dx M dx s ', s s ' s ds ds USPAS Accelerator Physics June 2016 From the definitions, the concatenation rule Ms'',s = Ms'',s' Ms',s must apply for all s' such that s < s'< s'' where the multiplication is the usual matrix multiplication. Pf: The equations of motion, linear in x and dx/ds, generate a motion with x s x s ' ' x s ' x s M dx M M dx M s '', s dx dx s '', s ' s '', s ' s ', s s s ' ' s ' s ds ds

ds ds for all initial conditions (x(s), dx/ds(s)), thus Ms'',s = Ms'',s' Ms',s. Clearly Ms,s = I. As is shown next, the matrix Ms',s is in general a member of the unimodular subgroup of the general linear group. USPAS Accelerator Physics June 2016 Ellipse Transformations Generated by Hills Equation The equation governing the linear transverse dynamics in a particle accelerator, without acceleration, is Hills equation* d 2x K s x 0 2 ds Eqn. (2) The transformation matrix taking a solution through an infinitesimal distance ds is x s ds 1 dx s ds ds K s ds rad ds x s x s dx M dx

rad s s ds , s s 1 ds ds * Strictly speaking, Hill studied Eqn. (2) with periodic K. It was first applied to circular accelerators which had a periodicity given by the circumference of the machine. It is a now standard in the field of beam optics, to still refer to Eqn. 2 as Hills equation, even in cases, as in linear accelerators, where there is no periodicity. USPAS Accelerator Physics June 2016 Suppose we are given the phase space ellipse s x 2 2 s xx' s x'2 at location s, and we wish to calculate the ellipse parameters, after the motion generated by Hills equation, at the location s + ds s ds x 2 2 s ds xx' s ds x'2 ' Because, to order linear in ds, Det Ms+ds,s = 1, at all locations s, ' = , and thus the phase space area of the ellipse after an infinitesimal displacement must equal the phase space area before the displacement. Because the transformation through a finite interval in s can be written as a series of infinitesimal displacement transformations, all of which preserve the phase space area of the transformed ellipse, we come to two important conclusions: USPAS Accelerator Physics June 2016 1. The phase space area is preserved after a finite integration of Hills equation to obtain Ms',s, the transport matrix which can be used to take an ellipse at s to an ellipse at s'. This conclusion holds generally for all s' and s. 2. Therefore Det Ms',s = 1 for all s' and s, independent of the details of the functional form K(s). (If desired, these two conclusions may be verified more analytically by showing

that d 2 0 s s 2 s 1, s ds may be derived directly from Hills equation.) USPAS Accelerator Physics June 2016 Evolution equations for , functions The ellipse transformation formulas give, to order linear in ds So ds s ds 2 s rad ds s ds s s s Kds rad rad d 2 s s ds rad d s s s K rad ds rad USPAS Accelerator Physics June 2016 Note that these two formulas are independent of the scale of the

starting ellipse , and in theory may be integrated directly for (s) and (s) given the focusing function K(s). A somewhat easier approach to obtain (s) is to recall that the maximum extent of an ellipse, xmax, is ()1/2(s), and to solve the differential equation describing its evolution. The above equations may be combined to give the following non-function Descriptionlinear equation for xmax(s) = w(s) = ()1/2(s) 2 2 / rad d w K s w . 2 3 ds w Such a differential equation describing the evolution of the maximum extent of an ellipse being transformed is known as an envelope equation. USPAS Accelerator Physics June 2016 It should be noted, for consistency, that the same (s) = w2(s)/ is obtained if one starts integrating the ellipse evolution equation from a different, but similar, starting ellipse. That this is so is an exercise. The envelope equation may be solved with the correct boundary conditions, to obtain the -function Descriptionfunction. may then be obtained from the derivative of , and by the usual normalization formula. Types of boundary conditions: Class I periodic boundary conditions suitable for circular machines or periodic focusing lattices, Class IIinitial condition boundary conditions suitable for linacs or recirculating

machines. USPAS Accelerator Physics June 2016 Solution to Hills Equation in Amplitude-Phase form To get a more general expression for the phase advance, consider in more detail the single particle solutions to Hills equation 2 d x K s x 0 2 ds From the theory of linear ODEs, the general solution of Hills equation can be written as the sum of the two linearly independent pseudo-function Descriptionharmonic functions x s Ax s Bx s where x s w s e i s USPAS Accelerator Physics June 2016 are two particular solutions to Hills equation, provided that d 2w c2 K sw 3 2 ds w and d

c s 2 , Eqns. (3) ds w s and where A, B, and c are constants (in s) That specific solution with boundary conditions x(s1) = x1 and dx/ds (s1) = x'1 has i s1 w s1 e A ic i s1 B w' s1 w s e 1 i s1 w s1 e ic i s1 w' s1 e w s 1

USPAS Accelerator Physics June 2016 1 x1 x'1 Therefore, the unimodular transfer matrix taking the solution at s = s1 to its coordinates at s = s2 is w s2 w s2 w' s1 cos s2 , s1 sin s2 , s1 w s c 1 c w s2 w' s2 w s1 w' s1 x2 sin s2 , s1 w s w s 1 2 c 2 1 x '2

w' s1 w' s2 cos s2 ,s1 w s w s 2 1 w s2 w s1 sin s2 , s1 c w s1 cos s2 , s1 w s2 where s2 c s2 , s1 s2 s1 2 ds w s s1 USPAS Accelerator Physics June 2016

x 1 w' s2 w s1 x' sin s2 , s1 1 c Case I: K(s) periodic in s Such boundary conditions, which may be used to describe circular or ring-function Descriptionlike accelerators, or periodic focusing lattices, have K(s + L) = K(s). L is either the machine circumference or period length of the focusing lattice. It is natural to assume that there exists a unique periodic solution w(s) to Eqn. (3a) when K(s) is periodic. Here, we will assume this to be the case. Later, it will be shown how to construct the function explicitly. Clearly for w periodic sL s s L s with L s c 2 w s

ds is also periodic by Eqn. (3b), and L is independent of s. USPAS Accelerator Physics June 2016 The transfer matrix for a single period reduces to w s w' s w2 s cos L sin L sin L c c c w s w' s w s w' s w' s w s 1 sin cos sin L

L L 2 w2 s c c 1 0 cos L sin L 0 1 where the (now periodic!) matrix functions are w s w' s s , c w2 s s , c 1 2 s s s By Thm. (2), these are the ellipse parameters of the periodically

repeating, i.e., matched ellipses. USPAS Accelerator Physics June 2016 General formula for phase advance In terms of the -function Descriptionfunction, the phase advance for the period is L ds L s 0 and more generally the phase advance between any two longitudinal locations s and s' is s' s ', s ds s s USPAS Accelerator Physics June 2016 Transfer Matrix in terms of and Also, the unimodular transfer matrix taking the solution from s to s' is M s ', s

s ' cos s ',s s sin s ',s s 1 s ' s sin s ', s 1 s ' s cos s' s s ', s s' s sin s ', s

s cos s ',s s' sin s ',s s ' Note that this final transfer matrix and the final expression for the phase advance do not depend on the constant c. This conclusion might have been anticipated because different particular solutions to Hills equation exist for all values of c, but from the theory of linear ordinary differential equations, the final motion is unique once x and dx/ds are specified somewhere. USPAS Accelerator Physics June 2016 Method to compute the -function Our previous work has indicated a method to compute the -function Description function (and thus w) directly, i.e., without solving the differential equation Eqn. (3). At a given location s, determine the one-function Descriptionperiod transfer map Ms+L,s (s). From this find L (which is independent of the location chosen!) from cos L = (M11+M22) / 2, and by choosing the sign of L so that (s) = M12(s) / sin L is positive. Likewise, (s) = (M11-function DescriptionM22) / 2 sin L. Repeat this exercise at every location the -function is desired. By construction, the beta-function Descriptionfunction and the alpha-function Descriptionfunction, and hence w, are periodic because the single-function Descriptionperiod transfer map is periodic. It is straightforward to show w=(c(s))1/2 satisfies the envelope equation. USPAS Accelerator Physics June 2016 Courant-Snyder Invariant Consider now a single particular solution of the equations of motion generated by Hills equation. Weve seen that once a particle is on an invariant ellipse for a period, it must stay on that

ellipse throughout its motion. Because the phase space area of the single period invariant ellipse is preserved by the motion, the quantity that gives the phase space area of the invariant ellipse in terms of the single particle orbit must also be an invariant. This phase space area/, 2 x 2 2xx' x'2 x 2 x'x / is called the Courant-function DescriptionSnyder invariant. It may be verified to be a constant by showing its derivative with respect to s is zero by Hills equation, or by explicit substitution of the transfer matrix solution which begins at some initial value s = 0. USPAS Accelerator Physics June 2016 Pseudoharmonic Solution x s dx s ds s cos s,0 0 sin s,0 0 1 s 0 sin s , 0 1

s 0 s 0 cos s , 0 gives x s 0 sin s ,0 0 dx 0 cos s,0 s sin s,0 ds 0 s x s s x' s s x s / s x 2 2 2 0 2 0 x'0 0 x0 / 0 Using the x(s) equation above and the definition of , the solution may be written in the standard pseudoharmonic form 0 x'0 0 x0 x s s cos s , 0 where tan x0

The the origin of the terminology phase advance is now obvious. 1 USPAS Accelerator Physics June 2016 Case II: K(s) not periodic In a linac or a recirculating linac there is no closed orbit or natural machine periodicity. Designing the transverse optics consists of arranging a focusing lattice that assures the beam particles coming into the front end of the accelerator are accelerated (and sometimes decelerated!) with as small beam loss as is possible. Therefore, it is imperative to know the initial beam phase space injected into the accelerator, in addition to the transfer matrices of all the elements making up the focusing lattice of the machine. An initial ellipse, or a set of initial conditions that somehow bound the phase space of the injected beam, are tracked through the acceleration system element by element to determine the transmission of the beam through the accelerator. The designs are usually made up of well-function Description understood modules that yield known and understood transverse beam optical properties. USPAS Accelerator Physics June 2016 Definition of function Now the pseudoharmonic solution applies even when K(s) is not periodic. Suppose there is an ellipse, the design injected ellipse, which tightly includes the phase space of the beam at injection to the accelerator. Let the ellipse parameters for this ellipse be 0, 0, and 0. A function (s) is simply defined by the ellipse transformation rule 2 2 s M 12 s 0 2M 12 s M 11 s 0 M 11 s 0

2 2 M 12 s 0 M 11 s 0 M 12 s / 0 where M s,0 M 11 s M 12 s M 21 s M 22 s USPAS Accelerator Physics June 2016 One might think to evaluate the phase advance by integrating the beta-function Descriptionfunction. Generally, it is far easier to evaluate the phase advance using the general formula, tan s ', s M s M s M s ', s s ', s 12 11 s ', s 12

where (s) and (s) are the ellipse functions at the entrance of the region described by transport matrix Ms',s. Applied to the situation at hand yields tan s , 0 M 12 s 0 M 11 s 0 M 12 s USPAS Accelerator Physics June 2016 Beam Matching Fundamentally, in circular accelerators beam matching is applied in order to guarantee that the beam envelope of the real accelerator beam does not depend on time. This requirement is one part of the definition of having a stable beam. With periodic boundary conditions, this means making beam density contours in phase space align with the invariant ellipses (in particular at the injection location!) given by the ellipse functions. Once the particles are on the invariant ellipses they stay there (in the linear approximation!), and the density is preserved because the single particle motion is around the invariant ellipses. In linacs and recirculating linacs, usually different purposes are to be achieved. If there are regions with periodic focusing lattices within the linacs, matching as above ensures that the beam USPAS Accelerator Physics June 2016 envelope does not grow going down the lattice. Sometimes it is advantageous to have specific values of the ellipse functions at specific longitudinal locations. Other times, re/matching is done to preserve the beam envelopes of a good beam solution as changes in the lattice are made to achieve other purposes, e.g. changing the dispersion function or changing the chromaticity of regions where there are bends (see the next chapter for definitions). At a minimum, there is usually a matching done in

the first parts of the injector, to take the phase space that is generated by the particle source, and change this phase space in a way towards agreement with the nominal transverse focusing design of the rest of the accelerator. The ellipse transformation formulas, solved by computer, are essential for performing this process. USPAS Accelerator Physics June 2016 Dispersion Calculation Begin with the inhomogeneous Hills equation for the dispersion. d 2D 1 ds 2 K s D s Write the general solution to the inhomogeneous equation for the dispersion as before. D s =D p s Ax1 s Bx2 s x1 s M s , s1 ;1,1 D s1 M s , s1 ;1,1D s1 x2 s M s , s1 ;2,1 D s1 M s , s1 ;2,2 D s1 Here Dp can be any particular solution, and we suppose that the dispersion and its derivative are known at the location s1, and we wish to determine their values at s. x1 and x2 are linearly independent solutions to the homogeneous differential equation because they are transported by the transfer matrix solution Ms,s1 already found. USPAS Accelerator Physics June 2016

To build up the general solution, choose that particular solution of the inhomogeneous equation with homogeneous boundary conditions D s D s 0 p ,0 1 p ,0 1 Evaluate A and B by the requirement that the dispersion and its derivative have the proper value at s1 (x1 and x2 need to be linearly independent!) M s1, s1 1 0 A B 1 0 1 D s M s s M D s M D s2 D p ,0 s2 s1 M s2 , s1 D s2 Dp ,0 2 1

s2 , s1 11 21 USPAS Accelerator Physics June 2016 1 1 s2 , s1 s2 , s1 12 D s1 22 D s1 3 by 3 Matrices for Dispersion Tracking M s ,s 2 1 D s2 D s

M s2 , s1 2 1 0 M M s2 , s1 12 11 s2 , s1 21 0 22 D p ,0 s2 s1 D s1 Dp ,0 s2 s1 D s1

1 1 Particular solutions to inhomogeneous equation for constant K and constant and vanishing dispersion and derivative at s = 0 K<0 Dp,0(s) 1 cosh K D'p,0(s) 1 K sinh K=0 s2 2 1 1 cos K

Ks s 1 sin K Ks Ks 1 K>0 Ks USPAS Accelerator Physics June 2016 M56 In addition to the transverse effects of the dispersion, there are important effects of the dispersion along the direction of motion. The primary effect is to change the time-function Descriptionof-function Description

arrival of the off-function Descriptionmomentum particle compared to the on-function Descriptionmomentum particle which traverses the design trajectory. ds p D s ds p ds p D s p z d z =D s p ds p s Design Trajectory s2 M 56 Dispersed Trajectory

Dx s Dy s ds s1 x s y s USPAS Accelerator Physics June 2016 Solenoid Focussing Can also have continuous focusing in both transverse directions by applying solenoid magnets: B z z USPAS Accelerator Physics June 2016 Buschs Theorem For cylindrical symmetry magnetic field described by a vector potential: A A z , r 1 rA z , r is nearly constant r r Bz r 0, z r Bz r 0, z r A z , r Br 2

2 Bz Conservation of Canonical Momentum gives Buschs Theorem: P mr 2 qrA const for particle with 0 where Bz 0, P 0 mr 2 qBz c Larmor 2m 2 Beam rotates at the Larmor frequency which implies coupling USPAS Accelerator Physics June 2016 Radial Equation d mr mrL2 qr Bz 2mrL2 dt L2 k 2 2 z c thin lens focal length e2 2 B z dz

1 2 2 2 2 f 4 z m c weak compared to quadrupole for high If go to full oscillation inside the magnetic field in the thick lens case, all particles end up at r = 0! Non-function Descriptionzero emittance spreads out perfect focusing! y x USPAS Accelerator Physics June 2016 Larmors Theorem This result is a special case of a more general result. If go to frame that rotates with the local value of Larmors frequency, then the transverse dynamics including the magnetic field are simply those of a harmonic oscillator with frequency equal to the Larmor frequency. Any force from the magnetic field linear in the field strength is transformed away in the Larmor frame. And the motion in the two transverse degrees of freedom are now decoupled. Pf: The equations of motion are d mr mr 2 qr Bz dt mr 2 qA cons P d mr mr 2 2mr L mrL2 qr Bz qr L Bz dt mr 2 P d 2 2

mr mr mr L dt 2-function DescriptionD Harmonic Oscillator mr 2 P USPAS Accelerator Physics June 2016

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