Accelerator Physics Linear Optics

A. Bogacz, G. A. Krafft, and T. Zolkin Jefferson Lab Colorado State University Lecture 3

USPAS Accelerator Physics June 2013 Linear Beam Optics Outline

• Particle Motion in the Linear Approximation • Some Geometry of Ellipses • Ellipse Dimensions in the β-function 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 Hill’s Equation – General Formula for Phase Advance – Transfer Matrix in Terms of β-function – Periodic Solutions • Non-periodic Solutions –Formulas for β-function and Phase Advance • Beam Matching

USPAS Accelerator Physics June 2013 Linear Particle Motion

Fundamental Notion: The Design Orbit is a path in an Earth- 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. σ :[0,1]→ R3 r σ →= XXYZ()σσσσ() () ,, () () Fundamental Notion: Path Length

222 ⎛⎞⎛⎞⎛⎞dX dY dZ ds=++⎜⎟⎜⎟⎜⎟ dσ ⎝⎠⎝⎠⎝⎠dddσσσ

USPAS Accelerator Physics June 2013 The Design Trajectory is the path specified in terms of the path length in the Earth-fixed reference frame. For a relativistic accelerator where the particles move at the velocity of light, Ltot=cttot. 3 sL :[0,tot ]→ R r s →= Xs()() XsYs () ,, Zs () ()

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 2013 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 14.3-14.5. Most vector analysis books have a similar, and more advanced discussion under the subject of “Frenet-Serret 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 2013 4-Fold Symmetric Synchrotron xˆ s s0 = 0 zˆ 1 yˆ vertical ρ sL= + ρπ /2 s7 2 xˆ zˆ

s = 3s 62 s3 L

s5 s42= 2s

USPAS Accelerator Physics June 2013 Its Design Trajectory

() 0,0,s 0 <

()()()−−LLss 2 ,0,ρ + −4 0,0, − 1 s45< ss<

()()−−L 2ρρ ,0,0 +() 1 − ρ cos() ρss −55 / ,0, − sin()() ss − / s 56 << ss

()()()−−L ,0, − +ss −667 1,0,0 s << ss ()()ρρρρρ ρ −−+,0,() sin()ss −7 / ,0,1 − cos()() s −s772/ρ ss<< 4 s

USPAS Accelerator Physics June 2013 Betatron Design Trajectory

sR :[0,2π ]→ R3 r sXsRsRRsR →= ()() cos / ( , sin ) / ,0 ( )

Use path length s as independent variable instead of t in the dynamical equations.

dd1 = dsΩc R dt

USPAS Accelerator Physics June 2013 Betatron Motion in s dr2δ Δ p +−()1 nrR Ω22 =Ω dt2 cc p dz2 δ +Ωnz2 = 0 dt 2 δ c ⇓ δ

dr2δ (1− n) 1 Δ p +=r ds22 R R p δ dz2 n +=z 0 ds22δ R δ USPAS Accelerator Physics June 2013 Bend Magnet Geometry

yˆ r B xˆ

Rectangular Magnet of Length L Sector Magnet xˆ zˆ

ρ ρ θ/2 θ

USPAS Accelerator Physics June 2013 Bend Magnet Trajectory

For a uniform magnetic field r dmV()γ r rr = ⎡ EV+× B⎤ dt ⎣ ⎦ dmV() γ x =−qV B dt zy dmV() γz = qV B dt xy dV2 dV2 x + Ω=2V 0 z +Ω2V =0 dt 2 cx dt 2 cz For the solution satisfying boundary conditions: rr X (0) == 0 VVz( 0) 0 z ˆ p X ()tt=Ω−=Ω−Ω=()cos1cos1()cccyρ ()() tqBm /γ qBy p Z ()ttt= sin()Ω=ccρ sin () Ω qBy

USPAS Accelerator Physics June 2013 Magnetic Rigidity

The magnetic rigidity is:

p BBρρ= = y 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= Bρθ(2sin( /2)) BL= Bρ sin (θ )

USPAS Accelerator Physics June 2013 Natural Focusing in Bend Plane

Perturbed Trajectory

Design Trajectory

Can show that for either a displacement perturbation or angular perturbation from the design trajectory

dx2 x dy2 y 22=− 22=− dsρ x () s dsρ y () s

USPAS Accelerator Physics June 2013 Quadrupole Focusing

r Bx( , y) =+Bs′( )( xyyˆˆx)

dv dv γγmqBsxmqBsx =−′′() y = ()y ds ds dx22Bs′′( ) dy Bs( ) +=xy0 −= 0 ds22 Bρρ ds B

Combining with the previous slide dx22⎡⎤11Bs′′( ) dy ⎡⎤ Bs( ) 22++⎢⎥xy =0 22 +−⎢⎥ = 0 ds⎣⎦⎢⎥ρρxy() s B ρρ ds⎣⎦⎢⎥() s B

USPAS Accelerator Physics June 2013 Hill’s Equation

Define focusing strengths (with units of m-2)

11Bs′( ) Bs′( ) ksxy()=+22 k =− ρ xys ()BsBρρρ()

dx22 dy +=ksx() 0 += ksy() 0 ds22xy 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 2013 Energy Effects

ρ (1/+Δpp)

ppΔ Δ=xs() ()1cos/ −() sρ eB p y ρ

This solution is not a solution to Hill’s equation directly, but is a solution to the inhomogeneous Hill’s Equations

dx2 ⎡⎤11Bs′( ) Δ p 22++⎢⎥x = ds⎣⎦⎢⎥ρρρxx() s B() s p d 2 y ⎡⎤11Bs′() Δp +−⎢⎥y = ds22 s B s p ⎣⎦⎢⎥yyρρρ() ()

USPAS Accelerator Physics June 2013 Inhomogeneous Hill’s Equations

Fundamental transverse equations of motion in particle accelerators for small deviations from design trajectory

dx2 ⎡⎤11Bs′( ) Δ p 22++⎢⎥x = ds⎣⎦⎢⎥ρρρxx() s B() s p dy2 ⎡⎤11Bs′() Δ p +−⎢⎥y = ds22 s B s p ⎣⎦⎢⎥yyρρρ() () ρ radius of curvature for bends, B' transverse field gradient for magnets that focus (positive corresponds to horizontal focusing), Δp/p momentum deviation from design momentum. Homogeneous equation is 2nd order linear ordinary differential equation.

USPAS Accelerator Physics June 2013 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.

xs( )= xpx( s)++ Axs12( ) Bx x( s) y (s) =y py(sA) ++y 1(sB) yy 2(s) Because the inhomogeneous terms are proportional to Δp/p, the particular solution can generally be written as Δp Δp xsDs()= () ysDs() = () pxp pyp where the dispersion functions satisfy 2 ⎡⎤′′2 ⎡⎤ dDx 1111Bs() dDy Bs( ) 22++⎢⎥DDxy = 22 +−⎢⎥ = ds⎣⎦⎢⎥ρρρxxyy() s B() s ρρρ ds⎣⎦⎢⎥() s B() s

USPAS Accelerator Physics June 2013 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-of- arrival of the off-momentum particle compared to the on-momentum particle which traverses the design trajectory.

ds⎛⎞Δ p ΔzDsds=+ρ⎜⎟ρ () − ⎝⎠p Δp ds ds dzDs()Δ = () Δp p ρ ()s ρ (+) Ds() p Design Trajectory Dispersed Trajectory

s2 ⎪⎧ Dsx ( ) Dsy ( )⎪⎫ M 56 =+⎨ ⎬ds ∫ ρρ()ss() s1 ⎪⎩⎭xy⎪

USPAS Accelerator Physics June 2013 Solutions Homogeneous Eqn.

Dipole ⎛⎞xs ⎛ xs ⎞ () ⎛⎞cos()ss−− /ρρ sin ρ()() ss / ( i ) ⎜⎟= ⎜⎟()ii ⎜ ⎟ ⎜⎟dx⎜⎟ ⎜ dx ⎟ ⎜⎟()ss−−sin()()ssii /ρρ / ρ cos()() ss − / ⎜()i ⎟ ⎝⎠ds⎝⎠ ⎝ ds ⎠

Drift ⎛⎞x (sxs) ⎛( ) ⎞ ⎛⎞1 ss− i ⎜⎟= i ⎜ ⎟ ⎜⎟dx⎜⎟ ⎜ dx ⎟ ⎜⎟()s ⎝⎠01 ⎜()si ⎟ ⎝⎠ds ⎝ ds ⎠

USPAS Accelerator Physics June 2013 Quadrupole in the focusing direction kBB= ′ / ρ

⎛⎞xs( ) ⎛⎞ ⎛ xs( ) ⎞ cos( ks()−− sii) sin( ks() s) / k i ⎜⎟= ⎜⎟ ⎜ ⎟ ⎜⎟dx⎜⎟ ⎜ dx ⎟ ⎜⎟()ss⎜⎟−−kksssin cos kss − ⎜()i ⎟ ⎝⎠ds⎝⎠()()ii()() ⎝ ds ⎠

Thin Focusing Lens (limiting case when argument goes to zero!) ⎛⎞xs( +−ε ) ⎛⎞ xs( ε ) ⎜⎟⎛⎞10 ⎜⎟ dx= ⎜⎟ dx ⎜⎟()ss+−ε ⎝⎠−1/f 1 ⎜⎟()ε ⎝⎠ds ⎝⎠ ds Thin Defocusing Lens: change sign of f

USPAS Accelerator Physics June 2013 Solutions Homogeneous Eqn.

Drift ⎛⎞x (sxs) ⎛( ) ⎞ ⎛⎞1 ss− i ⎜⎟= i ⎜ ⎟ ⎜⎟dx⎜⎟ ⎜ dx ⎟ ⎜⎟()s ⎝⎠01 ⎜()si ⎟ ⎝⎠ds ⎝ ds ⎠

USPAS Accelerator Physics June 2013 Quadrupole in the focusing direction kBB= ′ / ρ

Quadrupole in the defocusing direction kBB= ′ / ρ

⎛⎞xs() ⎛⎞ ⎛ xs( ) ⎞ cosh()−−ks() sii sinh( −− ks() s) / − k i ⎜⎟= ⎜⎟ ⎜ ⎟ ⎜⎟dx⎜⎟ ⎜ dx ⎟ ⎜⎟()ss⎜⎟−−−kksssinh cosh −− kss ⎜()i ⎟ ⎝⎠ds⎝⎠()()ii()() ⎝ ds ⎠

USPAS Accelerator Physics June 2013 Transfer Matrices

Dipole with bend Θ (put coordinate of final position in solution) ⎛⎞xs() ⎛ xs( ) ⎞ ⎜⎟after⎛⎞cos()ΘΘρ sin () ⎜ before ⎟ = ⎜⎟ ⎜⎟dx−Θsin() /ρ cos Θ () ⎜ dx ⎟ ⎜⎟()ssafter⎝⎠ ⎜() before ⎟ ⎝⎠ds ⎝ ds ⎠

Drift

⎛⎞ ⎛ ⎞ xs( after) xs( before ) ⎜⎟⎛⎞1 Ldrift ⎜ ⎟ = ⎜⎟ ⎜⎟dx01 ⎜ dx ⎟ ⎜⎟()ssafter⎝⎠ ⎜() before ⎟ ⎝⎠ds ⎝ ds ⎠

USPAS Accelerator Physics June 2013 Quadrupole in the focusing direction length L

⎛⎞ ⎛ ⎞ xs( after ) ⎛⎞coskL sin kL / k xs( before ) ⎜⎟( ) ( ) ⎜ ⎟ = ⎜⎟ ⎜⎟dx⎜⎟ ⎜ dx ⎟ ⎜⎟()ssafter ⎜⎟− kkLkLsin() cos() ⎜()before ⎟ ⎝⎠ds⎝⎠ ⎝ ds ⎠

Quadrupole in the defocusing direction length L

⎛⎞ ⎛ ⎞ xs( after ) ⎛⎞cosh−−−kL sinh kL / k xs( before ) ⎜⎟( ) ( ) ⎜ ⎟ = ⎜⎟ ⎜⎟dx⎜⎟ ⎜ dx ⎟ ⎜⎟()ssafter ⎜⎟−−kkLkLsinh() cos() − ⎜()before ⎟ ⎝⎠ds⎝⎠ ⎝ ds ⎠ Wille: pg. 71

USPAS Accelerator Physics June 2013 Thin Lenses

f –f

Thin Focusing Lens (limiting case when argument goes to zero!) ⎛⎞xs( +−ε ) ⎛⎞ xs( ε ) lens⎛⎞10 lens ⎜⎟= ⎜⎟ ⎜⎟dx⎜⎟ ⎜⎟ dx ⎜⎟()sslens+−ε ⎝⎠−1/f 1 ⎜⎟() lens ε ⎝⎠ds ⎝⎠ ds Thin Defocusing Lens: change sign of f

USPAS Accelerator Physics June 2013 Composition Rule: Matrix Multiplication!

Element 1 Element 2

s0 s1 s2

⎛⎞⎛⎞x (sxs10) ( ) ⎛⎞⎛⎞x (sxs21) ( ) ⎜⎟⎜⎟= M ⎜⎟⎜⎟= M ′′1 ′′2 ⎝⎠⎝⎠x ()sxs10() ⎝⎠⎝⎠x ()sxs21()

⎛⎞x (sxs20) ⎛⎞) ⎜⎟= MM ⎜⎟( ′′21 ⎝⎠x ()sxs20 ⎝⎠()

More generally

M tot= MM N N −121... MM Remember: First element farthest RIGHT

USPAS Accelerator Physics June 2013 Some Geometry of Ellipses y Equation for an upright ellipse

2 2 ⎛ x ⎞ ⎛ y ⎞ b ⎜ ⎟ + ⎜ ⎟ =1 a x ⎝ a ⎠ ⎝ b ⎠

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 π appears on the RHS of the defining equation. For a general ellipse

Ax2 + 2Bxy + Cy 2 = D

USPAS Accelerator Physics June 2013 The area is easily computed to be Area D π ≡ ε = Eqn. (1) AC − B2 γ γ So the equation is equivalentlyα β xα2 + 2 xy + y 2 = ε

A B C = , = , and β = AC − B2 AC − B2 AC − B2

USPAS Accelerator Physics June 2013 When normalized in this manner, the equation coefficients clearly satisfy βγ −α 2 =1

Example: the defining equation for the upright ellipse may be rewritten in following suggestive way b a x2 + y 2 = ab = ε a b

β = a/b and γ = b/a, note xmax = a = βε , ymax = b = γε

USPAS Accelerator Physics June 2013 General Tilted Ellipse y Needs 3 parameters for a complete y=sx description. One way b a b x2 + ()y − sx 2 = ab = ε a b x a where s is a slope parameter, a is the maximum extent in the x-direction, and the y-intercept occurs at b, and again ε is the area of the ellipse divided by π b ⎛ a2 ⎞ a a ⎜1+ s2 ⎟x2 − 2s xy + y 2 = ab = ε ⎜ 2 ⎟ a ⎝ b ⎠ b b

USPAS Accelerator Physics June 2013 γ

Identify α b ⎛ a2 ⎞ a a = ⎜1+ s2 ⎟, = − s, β = ⎜ 2 ⎟ a ⎝ b ⎠ b b

Note that βγ – α2 = 1 automatically, and that the equation for ellipse becomes β α x2 + ( y + x)2 = βε

by eliminating the (redundant!) parameter γ

USPAS Accelerator Physics June 2013 Ellipse Dimensions in the β-function Description ε ⎛ α ⎞ ⎜ ⎟ ⎜− γ, γε ⎟ ⎝ ⎠ y y=sx=– α x / β βε ⎛ α ε ⎞ γε ⎜ ⎟ b = ε / β ⎜ ,− ⎟ ⎝ β ⎠ x ε γ a = βε

As for the upright ellipse xmax = βε , ymax = γε Wille: page 81

USPAS Accelerator Physics June 2013 Area Theorem for Linear Optics

Under a general linear transformation

⎛ x'⎞ ⎛ M11 M12 ⎞⎛ x⎞ ⎜ ⎟ = ⎜ ⎟⎜ ⎟ ⎝ y'⎠ ⎝ M 21 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. γ Pf: Let the initial ellipse, normalizedα as above, be β 2 2 0 x + 2 0 xy + 0 y = ε0

USPAS Accelerator Physics June 2013 Because −−11 xx⎛⎞(MM) ( ) ' ⎛⎞⎜⎟11 12 ⎛ ⎞ ()()⎜⎟= ⎜ ⎟ ⎝⎠yy⎜⎟MM−−11 ⎝' ⎠ γ⎝⎠21 22 The transformed ellipse isα 2 β 2 x + 2 xy + y = ε0

22 γγαβ=++ (MMMM−−−−1111) 2( ) ( ) ( ) 11000 11 21 21 α βγαβ=+++()()()()()()MM−−11γαβ MM −− 11 MM −− 11()() MM −− 11 11 12000() 11 22 12 21 21 22 () ()()()22 =++ MMMM−−−−1111 2 12000 12 22 22

USPAS Accelerator Physics June 2013 Because (verify!)

22 βγ β α α γ −=( 00 − 0) ()()()()()()()()22 22 ×+−MM−−11 MM −− 112 MMMM −−−− 1111 ( 21 12 11 22 11 22 12 21 )

2 21− =−()()00 0 det M

the area of the transformedβγ αellipse (divided by π) is, by Eqn. (1) β γ Area εα π = ε = 0 = ε | det M | 2 −1 0 0 0 − 0 det M

USPAS Accelerator Physics June 2013 Tilted ellipse from the upright ellipse In the tilted ellipse the y-coordinate is raised by the slope with respect to the un-tilted ellipse ⎛ x'⎞ ⎛1 0⎞⎛ x⎞ ⎜ ⎟ = ⎜ ⎟⎜ ⎟ ⎝ y'⎠ ⎝ s 1⎠⎝ y⎠ baγ γαβ===, 0, , ()M −1 =−s 000ab21 α b a a a ∴ = + s2 , = − s, β = a b b b Because det (M)=1, the tilted ellipse has the same area as the

upright ellipse, i.e., ε = ε0.

USPAS Accelerator Physics June 2013 Phase Advance of a Unimodular Matrix

Any two-by-two unimodular (Det (M) = 1) matrix with |Tr M| < 2 can be written inμ the formα β ⎛1 0⎞ ⎛ γ ⎞ M = ⎜ ⎟cos()+ ⎜ α⎟sin μ () ⎝0 1⎠ ⎝− − ⎠

The phase advance of the matrix, μ, gives the eigenvalues of the matrix λ = e iμ, and cos μ = (Tr M)/2. Furthermore βγ–α2=1

Pf: The equation λfor the eigenvalues of M is

2 − (M11 + M 22 )λ +1 = 0

USPAS Accelerator Physics June 2013 Because M is real, both λ and λ* are solutions of the quadratic. Because Tr(M ) λ = ± i 1−()Tr()M / 2 2 2 iμ For |Tr M| < 2, λλ* =1 and so λ1,2 = e . Consequently cos μ = (Tr M)/2. Now the following matrix is trace-free. ⎛ M − M ⎞ ⎜ 11 22 M ⎟ ⎛1 0⎞ 2 12 M − ⎜ ⎟cos()μ = ⎜ ⎟ 0 1 M 22 − M11 ⎝ ⎠ ⎜ M ⎟ ⎝ 21 2 ⎠

USPAS Accelerator Physics June 2013 α

μ β Simply choose μ γ M − M M M = 11 22 , = 12 , = − 21 2sin sin sin μ and the sign of μ to properly μmatch αthe individual matrix elements with β > 0. It is easily verified thatβ βγ – α2 = 1. Now ⎛1 0⎞ ⎛ γ ⎞ M 2 = ⎜ ⎟cos()2 + ⎜ α⎟sin 2μ () ⎜0 1⎟ ⎜− − ⎟ ⎝ ⎠ μ ⎝ α ⎠ and more generally β 1 0 γ n ⎛ ⎞ ⎛ α⎞ M = ⎜ ⎟cos()n + ⎜ ⎟sin nμ () ⎝0 1⎠ ⎝− − ⎠

USPAS Accelerator Physics June 2013 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-by-two 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 2013 Significance of matrix parameters

Another way to interpret the parameters α, β, and γ, 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 linearμ transformationα β ⎛1 0⎞ ⎛ γ ⎞ M = ⎜ ⎟cos()+ ⎜ α⎟sin μ () ⎝0 1⎠ ⎝− − ⎠ with |Tr (M)| < 2, the ellipses

USPAS Accelerator Physics June 2013 γ α x2 + 2 xy + βy 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. μ α Pf: The inverse to M is clearly β ⎛1 0⎞ ⎛ γ ⎞ β M −1 = ⎜ ⎟cos()− ⎜ α⎟sin μ () β ⎜0 1⎟ ⎜− − ⎟ μ ⎝ ⎠ ⎝ ⎠ βBy the ellipseγ transformation formulas, for example 2 μ2 β 2 '= ()sin α+ 2(− sinμ )(cos + sin ) + (cos + sin ) μ = sinμ2 ()1+ 2 − βα2 2 sin 2 +α cos2 + 2 sin 2 ()μ β μ μ α = sin 2 + cos2 = β β μ μ α βα μ μ β USPAS Accelerator Physics June 2013 Similar calculations demonstrate that α' = α and γ' = γ. As det (M) = 1, c' = c, and thereforeγ the ellipse is invariant. Conversely, suppose that an ellipse is invariant. Byα the ellipse transformation formula, the specific ellipse β 2 2 i x + 2 i xy + i y = ε is invariant under the transformation by M only if γ

α 2 2 ⎛ ⎞ ⎛ μ ()cos − sin 2(cos − sin )( sin ) ( sin ) ⎞⎛ ⎞ ⎜ i ⎟ ⎜ α ⎟⎜ i ⎟ ()()β μ μ 2 ()() ⎜ i ⎟ = ⎜− cosα − sin sin 1− 2 sin cos + sin sin ⎟⎜ i ⎟ ()⎜ 2 ( )()(2 )⎟ ⎜ ⎟ ⎜ μsinβ − 2 cos + sin sin cos + sin ⎟⎜ ⎟ ⎝ i ⎠ ⎝ β μ μ ⎠⎝ i ⎠ γ μ α ⎛ i ⎞ μ γ ⎜ ⎟ r βγ μ ≡ T α⎜ ⎟ ≡ T v, M i M μ μ ⎜ ⎟ α ⎝ βi ⎠ μ γ β μ μ μ α μ γ μ γ α μ μ α

USPAS Accelerator Physics June 2013 β r 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 rλ r TM v = λvλ has a solution when Det (TM − λI )= 0 i.e.,

22 ()λμλλ+ ⎡⎤⎣⎦24cos−+−= 11( ) 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 λ = 1, 22 1+−⎣⎦⎡⎤ 2 4cosμμ += 1 0, cos = 1, or M =±I

USPAS Accelerator Physics June 2013 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 eigenvalueγ 1, all proportional to each other, are the only vectors whoseα components (γi, αi, βi) yield equations for the invariant ellipses. Forβ concreteness, compute that eigenvector with 2 eigenvalue 1 normalizedβ so βiγi – αi = 1 ⎛ ⎞ ⎛ − M / M ⎞ γ⎛ ⎞ ⎜ i ⎟ ⎜ 21 12 ⎟ ⎜ ⎟ r α v1,i = ⎜ i ⎟ = ⎜()M11 − M 22 / 2M12 ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ i ⎠ ⎝ 1 ⎠ ⎝ β ⎠ r r All other eigenvectors with eigenvalue 1 have v 1 = ε v 1 , i / c , for some value c.

USPAS Accelerator Physics June 2013 Because Det (M) =1,γ the eigenvector vr clearly yields the α 1,i invariant ellipse β x2 + 2 xy + y 2 = ε. γ r Likewise, the proportional eigenvector v 1 generates the similar ellipse α ε β (x2 + 2 xy + y 2 )= ε c Because we have enumerated all possible eigenvectors with eigenvalue 1, all ellipses invariant under the action of M, are of the form γ α x2 + 2 xy + βy 2 = c

USPAS Accelerator Physics June 2013 To summarize, this theorem gives a way to tie the mathematical representation of a unimodular matrix in terms of its α, β, and γ, 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 α, β, and γ as in the Twiss representation of the matrix, but varying c. Finally εnote that throughout this calculation c acts merely as a scale parameterγ for the ellipse. All ellipses similar to the starting α ellipse, i.e., ellipses whose βequations have the same α, β, and γ, but with different c, are also invariant under the action of M. Later, it will be shown that more generallyβ α = x2 + 2 xx'+ x'2 = (x2 + ()x'+ x 2 )/ β is an invariant of the equations of transverse motion.

USPAS Accelerator Physics June 2013 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- 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 π, ε, measured in m-rad, is called the emittance. In such applications, α has no units, β has units m/radian. Codes that calculate β, 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 2013 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 2013 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 ⎟ = ⎜ dx ⎟ = M ⎜ dx ⎟ = M M ⎜ dx ⎟ s'',s ⎜ ()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 2013 Ellipse Transformations Generated by Hill’s Equation The equation governing the linear transverse dynamics in a particle accelerator, without acceleration, is Hill’s equation* d 2 x + K()s x = 0 Eqn. (2) ds2 The transformation matrix taking a solution through an infinitesimal distance ds is ⎛ x()s + ds ⎞ ⎛ ds ⎞⎛ x(s) ⎞ ⎛ x(s) ⎞ 1 ⎜ dx ⎟ = ⎜ ⎟⎜ dx ⎟ ≡ M ⎜ dx ⎟ ⎜ ()s + ds ⎟ ⎜ rad ⎟⎜ ()s ⎟ s+ds,s ⎜ ()s ⎟ ⎝ ds ⎠ ⎝− K()s ds rad 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 Hill’s equation, even in cases, as in linear accelerators, where there is no periodicity.

USPAS Accelerator Physics June 2013 γ α Supposeγ we are given the phase spaceβ ellipse 2 2 ()s x +α2 (s)xx'+ (s)x' = ε at location s, and we wish to calculate the ellipse parameters, after the motion generated by Hill’s equation,β at the location s + ds ()s + ds x2 + 2 (s + ds)()xx'+ s + ds x'2 = ε'

Because, to order linear in ds, Det Ms+ds,s = 1, at all locations s, ε' = ε, 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 2013 1. The phase space area is preserved after a finite integration of

Hill’s 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 detailsβγ of the functional form K(s). (If desired, these two conclusionsα may be verified more analytically by showing that β d γ ()− 2 = 0 → ()()s s −α 2 s = ()1, ∀s ds may be derived directly from Hill’s equation.)

USPAS Accelerator Physics June 2013 Evolution equations for the α, ββ functions The ellipseα transformation formulas give, to order linear in ds α ds ()s + γds = −2 + β ()s rad ds α ()()s + ds = − s + ()s + β s ()Kds rad rad So dβ 2α s ()s = − () ds rad dα γ (s) ()s = β s ()K rad − ds rad

USPAS Accelerator Physics June 2013 Note that these two formulas are independent of the scale of the starting ellipse ε, and in theory may be integrated directly for β(s)andα(s) given the focusing function K(s). A somewhat easier approach to obtain β(s) is to recall that the maximum 1/2 extent of an ellipse, xmax, is (εβ) (s), and to solve the differential equation describing its evolution. The above equations may be combined to give the following non-linear equation for xmax(s) = w(s) = (εβ)1/2(s) 2 dw2 ε ()/rad +=Ksw() . ds23 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 2013 It should be noted, for consistency, that the same β(s) = w2(s)/ε 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 β-function. α may then be obtained from the derivative of β, and γ by the usual normalization formula. Types of boundary conditions: Class I— periodic boundary conditions suitable for circular machines or periodic focusing lattices, Class II—initial condition boundary conditions suitable for linacs or recirculating machines.

USPAS Accelerator Physics June 2013 Solution to Hill’s Equation in Amplitude-Phase form To get a more general expression for the phase advance, consider in more detail the single particle solutions to Hill’s equation d 2 x + K()s x = 0 ds2 From the theory of linear ODEs, the general solution of Hill’s equation can be written as the sum of the two linearly independent pseudo-harmonic functions

x(s)= Ax+ (s)+ Bx− (s) where

±iμ ()s x± (s)= w(s)e

USPAS Accelerator Physics June 2013 are two particular solutions to Hill’s equation, provided that d 2w c2 dμ c + K()s w = and ()s = , Eqns. (3) ds2 w3 ds w2 s ()

and where A, B, andμ c are constants (in s)

That specific solution with boundary conditions x(s1) = x1 and dx/ds (s ) = x' has 1 1 μ μ −1 i ()s1 −i ()s1 ⎛ w()s1 e w()s1 e ⎞ ⎛ A⎞ ⎜ ⎟ ⎛ x1 ⎞ ⎜ ⎟ = ⎜ ⎡ ic ⎤ i ()s ⎡ ic ⎤ −iμ ()s ⎟ ⎜ ⎟ ⎜ B⎟ w'()s + e 1 w'()s − e 1 ⎜ x' ⎟ ⎝ ⎠ ⎜ ⎢ 1 ()⎥ ⎢ 1 ()⎥ ⎟ ⎝ 1 ⎠ ⎝ ⎣ w s1 ⎦ ⎣ w s1 ⎦ ⎠

USPAS Accelerator Physics June 2013 μ

μ Therefore, the unimodular transfer matrix taking the solution at

s = s1 to its coordinates at s = s2 isμ μ μ ⎛ w()s w(s )w'(s ) w(s )w(s ) ⎞ ⎜ 2 cos Δ − 2 1 sin Δ 2 1 sin Δ ⎟ () s2 ,s1 s2 ,s1 s2 ,s1 ⎜ w s1 c μ c ⎟ ⎜ c ⎡ w()()()()s w' s w s w' s ⎤ ⎟ ⎛ x2 ⎞ − 1+ 2 2 1 1 sin Δ ⎛ x1 ⎞ ⎜ ⎟ = ⎜ ⎢ 2 ⎥ s2 ,s1 ⎟⎜ ⎟ ⎜ ⎟ w()()s2 w s1 ⎣ c ⎦ w()s w'()()s w s ⎜ ⎟ ⎝ x'2 ⎠ ⎜ 1 2 1 ⎟⎝ x'1 ⎠ cos Δ s ,s + sin Δμs ,s ⎜ ⎡w'()s w'()s ⎤ w()s 2 1 c 2 1 ⎟ ⎜ − 1 − 2 cos Δ 2 ⎟ ⎜ ⎢ () ()⎥ s2 ,s1 ⎟ ⎝ ⎣ w s2 w s1 ⎦ ⎠

where μ μ s2 c Δ = ()s − μ s ()= ds s2 ,s1 2 1 ∫ w2 s () s1

USPAS Accelerator Physics June 2013 Case I: K(s) periodic in s

Such boundary conditions, which may be used to describe circular or ring-like 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 φ assume this to beμ the case. Later, it will be shown how to construct the function μexplicitly. Clearly for w periodic s+L c ()s = s ()− s with μ = ds L L ∫ 2 () s w s is also periodic by Eqn. (3b), and μL is independent of s.

USPAS Accelerator Physics June 2013 μ

μ The transfer matrix for a single period reduces to ⎛ w(s)w'(s) w2 (s) ⎞ ⎜ cos L − sin L sin L ⎟ ⎜ c μ c μ ⎟ ⎜ c ⎡ w()s w' s ()w s w ()' s ⎤ () μ w'()s w s () ⎟ − 1+ sin cos + sin α⎜ 2 ⎢ 2 μ ⎥ α L L L ⎟ ⎝ w ()s ⎣ c ⎦ c ⎠ β μ ⎛1 0⎞ ⎛ γ ⎞ α = ⎜ ⎟cos()L + ⎜ ⎟sin μ L () ⎝0 1⎠ β ⎝− − ⎠ where the (now periodic!) matrix functions are γ w s w'(s) w2 (s) 1+α 2 (s) ()s = − (), ()s = , ()s = c c β()s By Thm. (2), these are the ellipse parameters of the periodically repeating, i.e., matched ellipses.

USPAS Accelerator Physics June 2013 General formula for phase advance

In terms of the β-function, the phase advance for the period is

μ L ds = L ∫ 0 β ()s and more generally the phase advance between any two longitudinal locations s and s'is

μ s' ds Δ = s',s ∫ s β ()s

USPAS Accelerator Physics June 2013 β β μ Transfer Matrixα in terms of α and β β α μ Also, the unimodularβ transferα matrix taking the solution from s α μ β to s' is α β μ β μ ⎛ ()s' ⎞ ⎜ ()cos Δ + ()s sin Δ β ()()s' s sin Δ ⎟ ⎜ ()s s',s s',s μ s',s ⎟ M = α s',s ⎜ ()⎡ 1+ ()()s' s sin Δ ⎤ () ⎟ ⎜ 1 s',s s ⎟ − ⎢ ⎥ ()cos Δ s',s − ()s' sin Δμs',s ⎜ +()()s' − s ()cos Δ () ⎟ ⎝ ()()s' s ⎣ 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 Hill’s 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 2013 Method to compute the β-function Our previous work has indicated a method to compute the β- function (and thus w) directly, i.e., without solving the differential equation Eqn. (3). At a given location s, determine the one-period

transfer map Ms+L,s (s). From this find μL (which is independent of the location chosen!) from cos μL = (M11+M22) / 2, and by choosing the sign of μL so that β(s) = M12(s) / sin μL is positive. Likewise, α(s) = (M11-M22) / 2 sin μL. Repeat this exercise at every location the β-function is desired.

By construction, the beta-function and the alpha-function, and hence w, are periodic because the single-period transfer map is periodic. It is straightforward to show w=(cβ(s))1/2 satisfies the envelope equation.

USPAS Accelerator Physics June 2013 Courant-Snyder Invariant Consider now a single particular solution of the equations of motion generated by Hill’s equation. We’ve seen that once a particle is on an invariant ellipse for a period, it must stay on that ellipse εthroughout its motion. Because the phase space area of the γ single period invariantα ellipse is preserved by the motion, the quantity that gives the phaseβ space area of the invariant ellipse in terms of the single particle orbit must also be an invariant. This β phase space area/π, α = x2 + 2 xx'+ x'2 = (x2 + ()x'+ x 2 )/ β is called the Courant-Snyder invariant. It may be verified to be a constant by showing its derivative with respect to s is zero by Hill’s equation, or by explicit substitution of the transfer matrix solution which begins at some initial value s = 0.

USPAS Accelerator Physics June 2013 β

β μ α μ β α Pseudoharmonicβ α Solution α μ β β α β β μ ⎛ s μ ⎞ ⎜ ()cos()Δα + sin Δ ()s sin Δ ⎟ ⎛ x()s ⎞ s,0 0 s,0 β 0 s,0 ⎛ x0 ⎞ ⎜ ⎟ ⎜ 0 μ ⎟⎜ ⎟ dx = α dx ⎜ ()s ⎟ ⎜ ()⎡ 1+ ()s sin Δ β ⎤ ⎟⎜ ⎟ ⎜ 1 0 s,0 0 ⎟⎜ ⎟ ⎝ ds ⎠ − ⎢ ⎥ ()cos Δ s,0 − ()s sin Δμs,0 ⎝ ds 0 ⎠ ⎜ +()()s − cos Δ s () ⎟ ⎝ ()s 0 ⎣ 0 s,0 ⎦ β ⎠ gives α εβ β (x()2 ()s + s ()x' s + ()(s)x(s) 2 )/ (s)= (x2 + ( x' + x )2 )/ ≡ ε μ 0 0 0 0 0 0 Using the x(s) equation aboveδ and the definition of ε, the solution may be written in the standardδ “pseudoharmonic” form β ⎛ x' +α x ⎞ −1⎜ 0 0 0 0 ⎟ x()s = s cos ()()Δ s,0 − where = tan ⎜ ⎟ ⎝ x0 ⎠ The the origin of the terminology “phase advance” is now obvious.

USPAS Accelerator Physics June 2013 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- understood “modules” that yield known and understood transverse beam optical properties.

USPAS Accelerator Physics June 2013 Definition of β function

βNow 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.γ Let the ellipse parameters for this

ellipse be α0, β0, and γ0. A function β(s) is simply defined by the ellipse transformation ruleβ α ()s = M()s ()2 − 2M (s)()Mα s + (M (s))2 12 0 12 11 0 11 β 0 (() )()2 () () 2 = []M12 s + 0M11 s − 0M12 s / β0 where ⎛ M (s)()M s ⎞ ⎜ 11 12 ⎟ M s,0 ≡ ⎜ () ()⎟ ⎝ M 21 s M 22 s ⎠

USPAS Accelerator Physics June 2013 One might think to evaluate the phase advance by integrating the beta-function.μ Generally, it is far easier to evaluate the phase advance using the general formula, β (M ) tan Δ = s',s 12 s',s ()s ()M −α()s ()M s',s 11 s',s 12 where β(s) and α(s) are the ellipse functions at the entrance of the region described by transport matrix Ms',s. Applied to the situation at handμ yields

β M12 (s) tan Δ s,0 = () () 0M11 s −α0M12 s

USPAS Accelerator Physics June 2013 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 2013 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 2013 Dispersion Calculation Begin with the inhomogeneous Hill’s equation for the dispersion. dD2 1 +=KsD() ds2 ρ s () Write the general solution to the inhomogeneous equation for the dispersion as before.

D (sDs)= p ( )++ AxsBxs12( ) ( )

Here Dp can be any particular solution. Suppose that the dispersion and it’s derivative are known at the location s1, and we wish to determine their values at s2. x1 and x2, because they are solutions to the homogeneous equations, must be transported by the transfer matrix solution Ms2,s1 already found.

USPAS Accelerator Physics June 2013 To build up the general solution, choose that particular solution of the inhomogeneous equation with boundary conditions

Dspp,0( 1)= Ds′ ,0( 1 )= 0 Evaluate A and B by the requirement that the dispersion and it’s derivative have the proper value at s1 (x1 and x2 need to be linearly independent!) −1 ⎛⎞A ⎛⎞⎛⎞xs11() xs 21 () Ds 1 () = ⎜⎟⎜⎟ ⎜⎟ ′′() () ′() ⎝⎠B ⎝⎠⎝⎠xs11 xs 21 Ds 1

D (sDssM2,021,1,) =−+pssss( ) ( ) DsM( ) +( ) Ds′( 1) 2111 21 12

D′′()sDssM2,021,1,=−+pssss (() ) DsM() +() Ds ′() 1 2121 21 22

USPAS Accelerator Physics June 2013 3 by 3 Matrices for Dispersion Tracking

⎛⎞ (MMDssss,,,021) ( ss) p ()− ⎛⎞2111 21 12 ⎛⎞ D ()sDs2 ⎜⎟1 () ⎜⎟()⎜⎟ ⎜⎟ D′′′sM2,=−()()ss M ss ,,0211 DssDs p ()() ⎜⎟⎜⎟2121 21 22 ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠11⎜⎟00 1 ⎝⎠ ⎝⎠ Particular solutions to inhomogeneous equation for constant K and constant ρ and vanishing dispersion and derivative at s = 0

K < 0 K = 0 K > 0

1 s2 1 Dp,0(s) (cosh( Ks)− 1) (1cos− ( Ks)) K ρ 2ρ K ρ 1 s 1 D' (s) sinh ( Ks) sin ( Ks) p,0 K ρ ρ K ρ

ds⎛⎞Δ p ΔzDsds=+ρ⎜⎟ρ () − ⎝⎠p Δp ds ds dzDs()Δ = () Δp p ρ ()s ρ (+) Ds() p Design Trajectory Dispersed Trajectory

s2 ⎪⎧ Dsx ( ) Dsy ( )⎪⎫ M 56 =+⎨ ⎬ds ∫ ρρ()ss() s1 ⎪⎩⎭xy⎪

USPAS Accelerator Physics June 2013 Solenoid Focussing

Can also have continuous focusing in both transverse directions by applying solenoid magnets:

Bz( )

USPAS Accelerator Physics June 2013 Busch’s Theorem

For cylindrical symmetry magnetic field described by a vector potential:

r 1 ∂ AAzr==(),θˆ B() rAzr() , is nearly constant θθz rr∂

Brzz==()0, zr Br′ () 0, zr ∴ Azr(), B= θ 22r Conservation of Canonical Momentum gives Busch’s Theorem:

2 PmrqrAconstθθ=+=γθ&

for particle with & = 0 where BPz == 0, 0 qB Ω mrγθ2 & =− ω z =−θ c =− 22m Larmor

Beam rotates at the Larmor frequencyγ which implies coupling θ

USPAS Accelerator Physics June 2013 Radial Equation

d 22 ()γγωθmr& −==− mrLz qr& B2 mr γω L dt 2 ∴ k = ωL 22c z β thin lens focal length ∞ eBdz22 1 ∫ z = −∞ weak compared to quadrupole for high γ β 2222 fmc4 z γ

If go to full ¼ oscillation inside the magnetic field in the “thick” lens case, all particles end up at r = 0! Non-zero emittance spreads out perfect focusing! y

USPAS Accelerator Physics June 2013 Larmor’s Theorem

This result is a special case of a more general result. If go to frame that rotates with the local value of Larmor’s 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 2 ()γγθθmr& −= mr&& qr Bz dt γθ2 mr& += qAθθ cons = P

d 22 ()mr& −+ mr&&′′2 mrL −=− mrLzLz qr ′ B qr B dt γγθγθωγωθ ω 2 mrγθ&′ = P θ d 22⎫ ()mr& −=− mr&′ mr L ⎪ dt γγθγω⎬ 2-D Harmonic Oscillator 2 mr&′ = P ⎭⎪ γθ USPAS Acceleratorθ Physics June 2013