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
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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 Earthfixed 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] → R 3
r
σ → X (σ ) = ( X (σ ) , Y (σ ) , Z (σ ) )
Fundamental Notion: Path Length
2
2
2
⎛ dX ⎞ ⎛ dY ⎞ ⎛ dZ ⎞
ds = ⎜
⎟ +⎜
⎟ +⎜
⎟ dσ
⎝ dσ ⎠ ⎝ dσ ⎠ ⎝ dσ ⎠
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.
s :[0, Ltot ] → R 3
r
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 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
s0 = 0
x̂
ẑ
s1
ŷ vertical
ρ
s2 = L + ρπ / 2
s7
ẑ
s6 = 3s2
s3
L
s5
s4 = 2 s2
USPAS Accelerator Physics June 2013
x̂
Its Design Trajectory
( 0, 0, s )
( 0, 0, L ) + ρ ( cos ( ( s − s1 ) / ρ ) − 1, 0,sin ( ( s − s1 ) / ρ ) )
( − ρ , 0, L + ρ ) + ( s − s2 )( −1,0,0 )
( − L − ρ , 0, L + ρ ) + ρ ( − sin ( ( s − s3 ) / ρ ) , 0, cos ( ( s − s3 ) / ρ ) − 1)
( − L − 2 ρ , 0, L ) + ( s − s4 )( 0,0, − 1)
( − L − 2 ρ , 0, 0 ) + ρ (1 − cos ( ( s − s5 ) / ρ ) , 0, − sin ( ( s − s5 ) / ρ ) )
( − L − ρ , 0, − ρ ) + ( s − s6 )(1,0,0 )
( − ρ , 0, − ρ ) + ρ ( sin ( ( s − s7 ) / ρ ) , 0,1 − cos ( ( s − s7 ) / ρ ) )
USPAS Accelerator Physics June 2013
0 < s < L = s1
s1 < s < s2
s2 < s < s3
s3 < s < s4
s4 < s < s5
s5 < s < s6
s6 < s < s7
s7 < s < 4s2
Betatron Design Trajectory
s :[0, 2π R ] → R
3
r
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 Ωc R dt
USPAS Accelerator Physics June 2013
Betatron Motion in s
d 2δ r
Δp
2
2
+ (1 − n ) Ω c δ r = Ω c R
2
dt
p
d 2δ z
2
+ nΩ c δ z = 0
2
dt
⇓
d 2δ r (1 − n )
1 Δp
δr =
+
2
2
ds
R
R p
d δz n
+ 2δz =0
2
ds
R
2
USPAS Accelerator Physics June 2013
Bend Magnet Geometry
r
B
Rectangular Magnet of Length L
x̂
ẑ
ρ
θ
ρ
θ/2
USPAS Accelerator Physics June 2013
ŷ
x̂
Sector Magnet
Bend Magnet Trajectory
For a uniform magnetic field
r
r r r
d (γ m V )
= ⎡⎣ E + V × B ⎤⎦
dt
d (γ m V x )
= − qV z B y
dt
d (γ m V z )
= qV x B y
dt
2
d 2Vx
d
Vz
2
2
+
Ω
=
0
V
+
Ω
Vz = 0
c
c
x
2
2
dt
dt
r
For the solution satisfying boundary conditions:
X (0) = 0
X (t ) =
p
cos ( Ω c t ) − 1) = ρ ( cos ( Ω c t ) − 1)
(
qB y
Z (t ) =
Ω c = qB y / γ m
p
s in ( Ω c t ) = ρ s in ( Ω c t )
qB y
USPAS Accelerator Physics June 2013
r
V ( 0 ) = V0 z zˆ
Magnetic Rigidity
The magnetic rigidity is:
B ρ = By ρ =
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 ρ ( 2 sin (θ / 2 ) )
BL = B ρ 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
d 2x
x
=
−
ρ x2 ( s )
ds 2
USPAS Accelerator Physics June 2013
d2y
y
=
−
ρ y2 ( s )
ds 2
Quadrupole Focusing
r
B ( x , y ) = B ′ ( s )( xyˆ + yxˆ )
γm
dv x
= − qB ′ ( s ) x
ds
d 2 x B′ ( s )
+
x=0
2
ds
Bρ
γm
dv y
ds
= qB ′ ( s ) y
d 2 y B′ ( s )
−
y=0
2
ds
Bρ
Combining with the previous slide
B′ ( s ) ⎤
d 2x ⎡ 1
+⎢ 2
+
⎥x=0
2
ds
B ρ ⎥⎦
⎢⎣ ρ x ( s )
B′ ( s ) ⎤
d2y ⎡ 1
+⎢ 2
−
⎥y=0
2
ds
B ρ ⎥⎦
⎢⎣ ρ y ( s )
USPAS Accelerator Physics June 2013
Hill’s Equation
Define focusing strengths (with units of m-2)
kx ( s ) =
1
ρ
2
x
(s)
+
B′ ( s )
d 2x
+ kx ( s ) x = 0
2
ds
Bρ
ky =
1
ρ
2
y
(s)
−
B′ ( s )
Bρ
d2y
+ ky (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 2013
Energy Effects
ρ (1 + Δ p / p )
Δx ( s ) =
p Δp
1 − cos ( s / ρ ) )
(
eB y p
ρ
This solution is not a solution to Hill’s equation directly, but is a solution to the
inhomogeneous Hill’s Equations
B′ ( s ) ⎤
1 Δp
d 2x ⎡ 1
+
+
=
x
⎢
⎥
ρx (s) p
ds 2 ⎢⎣ ρ x2 ( s )
B ρ ⎥⎦
B′ ( s ) ⎤
1 Δp
d2y ⎡ 1
+⎢
−
⎥y=
ρy (s) p
ds 2 ⎣⎢ ρ y2 ( s )
B ρ ⎥⎦
USPAS Accelerator Physics June 2013
Inhomogeneous Hill’s Equations
Fundamental transverse equations of motion in particle
accelerators for small deviations from design trajectory
B′ ( s ) ⎤
1 Δp
d 2x ⎡ 1
+⎢ 2
+
⎥x=
2
ds
B ρ ⎥⎦
ρx (s) p
⎢⎣ ρ x ( s )
B′ ( s ) ⎤
d2y ⎡ 1
1 Δp
+⎢ 2
−
⎥y=
2
ds
B ρ ⎥⎦
ρy (s) p
⎢⎣ ρ y ( s )
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.
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 Δp/p, the particular
solution can generally be written as
Δp
x p ( s ) =Dx ( s )
p
Δp
y p ( s ) =D y ( s )
p
where the dispersion functions satisfy
B′ ( s ) ⎤
d 2 Dx ⎡ 1
1
+
+
=
D
⎢ 2
⎥ x
ρ
ρ
ρx (s)
ds 2
s
B
⎢⎣ x ( )
⎥⎦
d 2 Dy
ds 2
USPAS Accelerator Physics June 2013
⎡ 1
B′ ( s ) ⎤
1
+⎢ 2
−
=
D
⎥ y
ρ
ρ
ρy (s)
s
B
⎢⎣ 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-ofarrival of the off-momentum particle compared to the on-momentum particle which
traverses the design trajectory.
Δp ⎞
ds ⎛
D
s
ρ
+
(
)
⎟ − ds
p ⎠
ρ ⎜⎝
ds
Δp
D (s)
p
Δz =
d ( Δz ) =D ( s )
Δ p ds
p ρ (s)
ρ (+)
Design Trajectory
⎧⎪ D x ( s ) D y ( s ) ⎫⎪
= ∫⎨
+
⎬ds
s1 ⎪
⎩ ρ x ( s ) ρ y ( s ) ⎭⎪
s2
M 56
Dispersed Trajectory
USPAS Accelerator Physics June 2013
Solutions Homogeneous Eqn.
Dipole
⎛ x ( s ) ⎞ ⎛ cos s − s / ρ
(( i ) )
⎜
⎟=⎜
⎜⎜ dx ( s ) ⎟⎟ ⎜ − sin ( ( s − s ) / ρ ) / ρ
i
⎝
⎝ ds
⎠
ρ sin ( ( s − si ) / ρ ) ⎞ ⎛⎜ x ( si ) ⎞⎟
⎟ dx
cos ( ( s − si ) / ρ ) ⎟⎠ ⎜⎜ ( si ) ⎟⎟
⎝ ds
⎠
Drift
⎛ x (s) ⎞
⎜
⎟ = ⎛1
⎜⎜ dx ( s ) ⎟⎟ ⎜⎝ 0
⎝ ds
⎠
⎛ x s ⎞
s − si ⎞ ⎜ ( i ) ⎟
⎟ dx
1 ⎠ ⎜⎜ ( si ) ⎟⎟
⎝ ds
⎠
USPAS Accelerator Physics June 2013
Quadrupole in the focusing direction k = B ′ / B ρ
(
)
⎛ x ( s ) ⎞ ⎛ cos k ( s − si )
⎜
⎟=⎜
dx
⎜⎜ ( s ) ⎟⎟ ⎜⎜ k sin k s s
−
− i)
(
⎝ ds
⎠ ⎝
(
)
(
)
k ( s − si ) / k ⎞ ⎛ x ( si ) ⎞
⎟⎜
⎟
dx
⎟
cos k ( s − si ) ⎟ ⎜⎜ ( si ) ⎟⎟
⎠
⎠ ⎝ ds
sin
(
)
Thin Focusing Lens (limiting case when argument goes to
zero!)
⎛ x (s + ε ) ⎞
⎜
⎟=⎛ 1
⎜⎜ dx ( s + ε ) ⎟⎟ ⎜⎝ −1/ f
⎝ ds
⎠
⎛ x s −ε ) ⎞
0⎞⎜ (
⎟
⎟ dx
1 ⎠ ⎜⎜ ( s − ε ) ⎟⎟
⎝ ds
⎠
Thin Defocusing Lens: change sign of f
USPAS Accelerator Physics June 2013
Solutions Homogeneous Eqn.
Dipole
⎛ x ( s ) ⎞ ⎛ cos s − s / ρ
(( i ) )
⎜
⎟=⎜
⎜⎜ dx ( s ) ⎟⎟ ⎜ − sin ( ( s − s ) / ρ ) / ρ
i
⎝
⎝ ds
⎠
ρ sin ( ( s − si ) / ρ ) ⎞ ⎛⎜ x ( si ) ⎞⎟
⎟ dx
cos ( ( s − si ) / ρ ) ⎟⎠ ⎜⎜ ( si ) ⎟⎟
⎝ ds
⎠
Drift
⎛ x (s) ⎞
⎜
⎟ = ⎛1
⎜⎜ dx ( s ) ⎟⎟ ⎜⎝ 0
⎝ ds
⎠
⎛ x s ⎞
s − si ⎞ ⎜ ( i ) ⎟
⎟ dx
1 ⎠ ⎜⎜ ( si ) ⎟⎟
⎝ ds
⎠
USPAS Accelerator Physics June 2013
Quadrupole in the focusing direction k = B ′ / B ρ
(
)
⎛ x ( s ) ⎞ ⎛ cos k ( s − si )
⎜
⎟=⎜
dx
⎜⎜ ( s ) ⎟⎟ ⎜⎜ − k sin k s − s
( i)
⎝ ds
⎠ ⎝
(
)
(
)
k ( s − si ) / k ⎞ ⎛ x ( si ) ⎞
⎟⎜
⎟
dx
⎟
cos k ( s − si ) ⎟ ⎜⎜ ( si ) ⎟⎟
⎠
⎠ ⎝ ds
sin
(
)
Quadrupole in the defocusing direction k = B ′ / B ρ
(
)
⎛ x ( s ) ⎞ ⎛ cosh − k ( s − si )
⎜
⎟=⎜
dx
⎜⎜ ( s ) ⎟⎟ ⎜⎜
− k sinh − k ( s − si )
⎝ ds
⎠ ⎝
(
)
(
)
− k ( s − si ) / − k ⎞ ⎛ x ( si ) ⎞
⎟⎜
⎟
dx
⎟⎟ ⎜⎜ ( s ) ⎟⎟
cosh − k ( s − si )
i
⎠
⎠ ⎝ ds
sinh
USPAS Accelerator Physics June 2013
(
)
Transfer Matrices
Dipole with bend Θ (put coordinate of final position in solution)
⎛ x ( safter ) ⎞
⎜
⎟ ⎛ cos ( Θ )
⎜ dx
⎟ = ⎜ − sin ( Θ ) / ρ
⎜ ( safter ) ⎟ ⎝
⎝ ds
⎠
⎛ x ( sbefore ) ⎞
ρ sin ( Θ ) ⎞ ⎜
⎟
⎟ ⎜ dx
⎟
cos ( Θ ) ⎠ ⎜ ( s
before ) ⎟
⎝ ds
⎠
Drift
⎛ x ( safter ) ⎞
⎜
⎟ ⎛1
⎜ dx
⎟ = ⎜0
⎜ ( safter ) ⎟ ⎝
⎝ ds
⎠
⎛ x ( sbefore ) ⎞
Ldrift ⎞ ⎜
⎟
⎟
⎟
1 ⎠ ⎜⎜ dx ( s
before ) ⎟
⎝ ds
⎠
USPAS Accelerator Physics June 2013
Quadrupole in the focusing direction length L
⎛ x ( safter ) ⎞ ⎛ cos k L
⎜
⎟ ⎜
⎜ dx
⎟=⎜
⎜ ( safter ) ⎟ ⎜⎝ − k sin k L
⎝ ds
⎠
(
)
(
k L / k ⎞ ⎛ x ( sbefore ) ⎞
⎟
⎟⎜
⎟
⎟⎟ ⎜ dx s
cos k L
( )
⎠ ⎜⎝ ds before ⎟⎠
sin
)
(
(
)
)
Quadrupole in the defocusing direction length L
⎛ x ( safter ) ⎞ ⎛ cosh − k L
⎜
⎟ ⎜
⎜ dx
⎟=⎜
⎜ ( safter ) ⎟ ⎜⎝ − k sinh − k L
⎝ ds
⎠
(
(
)
− k L / − k ⎞ ⎛ x ( sbefore ) ⎞
⎟
⎟⎜
⎟
⎟⎟ ⎜ dx s
cos − k L
( )
⎠ ⎜⎝ ds before ⎟⎠
sinh
)
(
Wille: pg. 71
USPAS Accelerator Physics June 2013
(
)
)
Thin Lenses
–f
f
Thin Focusing Lens (limiting case when argument goes to
zero!)
⎛ x ( slens + ε ) ⎞
⎜
⎟=⎛ 1
⎜⎜ dx ( s + ε ) ⎟⎟ ⎜⎝ −1/ f
lens
⎝ ds
⎠
⎛ x s −ε ) ⎞
0 ⎞ ⎜ ( lens
⎟
⎟ dx
1 ⎠ ⎜⎜ ( slens − ε ) ⎟⎟
⎝ ds
⎠
Thin Defocusing Lens: change sign of f
USPAS Accelerator Physics June 2013
Composition Rule: Matrix Multiplication!
Element 1
Element 2
s0
s1
⎛ x ( s1 ) ⎞
⎛ x ( s0 ) ⎞
⎜
⎟ = M1 ⎜
⎟
⎝ x ′ ( s1 ) ⎠
⎝ x ′ ( s0 ) ⎠
s2
⎛ x ( s2 ) ⎞
⎛ x ( s1 ) ⎞
⎜
⎟ = M2 ⎜
⎟
⎝ x ′ ( s2 ) ⎠
⎝ x ′ ( s1 ) ⎠
⎛ x ( s2 ) ⎞
⎛ x ( s0 ) ⎞
⎜
⎟ = M 2M1 ⎜
⎟
⎝ 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 2013
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 π appears on the RHS of the defining
equation. For a general ellipse
Ax 2 + 2 Bxy + Cy 2 = D
USPAS Accelerator Physics June 2013
The area is easily computed to be
Area
π
D
≡ε =
Eqn. (1)
AC − B 2
So the equation is equivalently
γx 2 + 2αxy + βy 2 = ε
γ=
A
AC − B 2
, α=
B
AC − B 2
, and β =
USPAS Accelerator Physics June 2013
C
AC − B 2
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 2 a 2
x + y = ab = ε
b
a
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
description. One way
b 2 a
2
x + ( y − sx ) = ab = ε
b
a
y=sx
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
2
⎞ 2
b⎛
a
a
a 2
2
⎜⎜1 + s 2 ⎟⎟ x − 2 s xy + y = ab = ε
a⎝
b ⎠
b
b
USPAS Accelerator Physics June 2013
Identify
2
⎛
b
2 a ⎞
γ = ⎜⎜1 + s 2 ⎟⎟,
b ⎠
a⎝
a
α = − s,
b
a
β=
b
Note that βγ – α2 = 1 automatically, and that the equation for
ellipse becomes
x 2 + (βy + αx ) = βε
2
by eliminating the (redundant!) parameter γ
USPAS Accelerator Physics June 2013
Ellipse Dimensions in the β-function
Description
⎛
⎞
ε
⎜ −α
, γε ⎟⎟
⎜
γ
⎝
⎠
y
γε
y=sx=– α x / β
⎛
ε ⎞⎟
⎜ βε ,−α
⎜
β ⎟⎠
⎝
b= ε/β
x
ε
γ
a = βε
As for the upright ellipse
xmax = βε ,
Wille: page 81
USPAS Accelerator Physics June 2013
ymax = γε
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
γ 0 x 2 + 2α 0 xy + β 0 y 2 = ε 0
USPAS Accelerator Physics June 2013
Because
(
(
−1
⎛
M
⎛ x⎞ ⎜
⎜ ⎟=⎜
M −1
⎝ y⎠
⎝
) ( )
) (M )
⎞ x'
⎞
12 ⎟ ⎛
⎟ ⎜⎝ y ' ⎟⎠
22 ⎠
M −1
11
−1
21
The transformed ellipse is
γx 2 + 2αxy + βy 2 = ε 0
(
γ=
α = ( M −1 )
11
β=
(M )
M −1
−1
12
)
2
11
(
γ 0 + 2 ( M −1 )
11
(M )
2
12
−1
11
γ0
)
α 0 + ( M −1 ) β 0
2
21
21
( M ) + ( M ) ( M ) )α + ( M ) ( M )
+ 2(M ) (M ) α + (M ) β
γ 0 + ( M −1 )
−1
(
M −1
−1
22
−1
−1
12
−1
12
−1
−1
22
USPAS Accelerator Physics June 2013
0
0
21
21
2
22
−1
0
22
β0
Because (verify!)
((
× M
βγ − α 2 = ( β 0γ 0 − α 02 )
−1
) (M ) + (M ) (M )
2
−1
21
2
−1
12
2
−1
11
(
= β 0γ 0 − α
2
0
2
22
(
− 2 M −1
)( det M )
−1
) (M ) (M ) (M )
−1
11
−1
22
−1
12
2
the area of the transformed ellipse (divided by π) is, by Eqn. (1)
Area
π
=ε =
ε0
β 0γ 0 − α 02 det M −1
USPAS Accelerator Physics June 2013
= ε 0 | det M |
21
)
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 ⎠
b
γ0 = ,
a
a
β0 = ,
b
α 0 = 0,
b a 2
∴ γ = + s ,
a b
a
α = − s,
b
−1
M
( ) = −s
21
a
β=
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 in the form
⎛α
⎛ 1 0⎞
⎟⎟ cos(μ ) + ⎜⎜
M = ⎜⎜
⎝−γ
⎝ 0 1⎠
β ⎞
⎟⎟ sin (μ )
−α ⎠
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 for the eigenvalues of M is
λ2 − (M 11 + 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 )
2
λ=
± i 1 − (Tr (M ) / 2 )
2
For |Tr M| < 2, λ λ* =1 and so λ1,2 = e iμ. Consequently cos μ
= (Tr M)/2. Now the following matrix is trace-free.
⎛ M 11 − M 22
⎜
⎛ 1 0⎞
2
⎟⎟ cos(μ ) = ⎜
M − ⎜⎜
⎜⎜ M
⎝ 0 1⎠
21
⎝
USPAS Accelerator Physics June 2013
⎞
M 12
⎟
⎟
M 22 − M 11 ⎟
⎟
2
⎠
Simply choose
M 11 − M 22
α=
,
2 sin μ
M 12
β=
,
sin μ
M 21
γ =−
sin μ
and the sign of μ to properly match the individual matrix
elements with β > 0. It is easily verified that βγ – α2 = 1. Now
⎛ 1 0⎞
⎛α
⎟⎟ cos(2 μ ) + ⎜⎜
M = ⎜⎜
⎝ 0 1⎠
⎝−γ
2
and more generally
⎛ 1 0⎞
⎛α
⎟⎟ cos(nμ ) + ⎜⎜
M = ⎜⎜
⎝ 0 1⎠
⎝−γ
n
USPAS Accelerator Physics June 2013
β ⎞
⎟⎟ sin (2 μ )
−α ⎠
β ⎞
⎟⎟ sin (nμ )
−α ⎠
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 linear transformation
⎛ 1 0⎞
⎛α
⎟⎟ cos(μ ) + ⎜⎜
M = ⎜⎜
⎝ 0 1⎠
⎝−γ
with |Tr (M)| < 2, the ellipses
USPAS Accelerator Physics June 2013
β ⎞
⎟⎟ sin (μ )
−α ⎠
γx 2 + 2αxy + 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.
Pf: The inverse to M is clearly
M
−1
⎛ 1 0⎞
⎛α
⎟⎟ cos(μ ) − ⎜⎜
= ⎜⎜
⎝ 0 1⎠
⎝−γ
β ⎞
⎟⎟ sin (μ )
−α ⎠
By the ellipse transformation formulas, for example
β ' = β 2 sin 2 μ γ + 2(− β sin μ )(cos μ + α sin μ )α + (cos μ + α sin μ )2 β
(
(
)
)
= (sin μ + cos μ )β = β
= β sin 2 μ 1 + α 2 − 2 βα 2 sin 2 μ + β cos 2 μ + βα 2 sin 2 μ
2
2
USPAS Accelerator Physics June 2013
Similar calculations demonstrate that α' = α and γ' = γ. 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
γ i x 2 + 2α i xy + β i y 2 = ε
is invariant under the transformation by M only if
(cos μ − α sin μ )
2(cos μ − α sin μ )(γ sin μ )
⎛ γ i ⎞ ⎛⎜
⎜ ⎟
1 − 2 βγ sin 2 μ
⎜α i ⎟ = ⎜ − (cos μ − α sin μ )(β sin μ )
2
⎜ β ⎟ ⎜⎜
(
)
sin
β
μ
− 2(cos μ + α sin μ )(β sin μ )
⎝ i⎠ ⎝
⎛γ i ⎞
⎜ ⎟
r
≡ TM ⎜α i ⎟ ≡ TM v ,
⎜β ⎟
⎝ i⎠
2
USPAS Accelerator Physics June 2013
⎞⎛ γ i ⎞
(γ sin μ )2
⎟⎜ ⎟
(cos μ + α sin μ )(γ sin μ )⎟⎜α i ⎟
(cos μ + α sin μ )2 ⎟⎟⎠⎜⎝ β i ⎟⎠
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
r
r
TM vλ = λvλ
i.e.,
(
has a solution when Det (TM − λI ) = 0
)
λ 2 + ⎡⎣ 2 − 4 cos 2 μ ⎤⎦ λ + 1 (1 − λ ) = 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,
1 + ⎡⎣ 2 − 4 cos 2 μ ⎤⎦ + 1 = 0, cos 2 μ = 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 eigenvalue 1, all proportional to each other, are the only
vectors whose components (γi, αi, βi) yield equations for the
invariant ellipses. For concreteness, compute that eigenvector with
eigenvalue 1 normalized so βiγi – αi2 = 1
− M 21 / M 12
⎛γi ⎞
⎛
⎞ ⎛γ ⎞
⎜ ⎟
⎜
⎟ ⎜ ⎟
r
v1,i = ⎜ α i ⎟ = β ⎜ (M 11 − M 22 ) / 2 M 12 ⎟ = ⎜ α ⎟
⎜β ⎟
⎜
⎟ ⎜β ⎟
1
⎝ i⎠
⎝
⎠ ⎝ ⎠
r
r
All other eigenvectors with eigenvalue 1 have v1 = εv1,i / c, for
some value c.
USPAS Accelerator Physics June 2013
r
Because Det (M) =1, the eigenvector v1,i clearly yields the
invariant ellipse
γx 2 + 2αxy + βy 2 = ε .
r
Likewise, the proportional eigenvector v1 generates the similar
ellipse
ε
(
γx
c
2
)
+ 2αxy + βy 2 = ε
Because we have enumerated all possible eigenvectors with
eigenvalue 1, all ellipses invariant under the action of M, are of the
form
γx 2 + 2αxy + 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 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 α, β, and γ,
but with different c, are also invariant under the action of M.
Later, it will be shown that more generally
(
)
ε = γx 2 + 2αxx'+ βx'2 = x 2 + (βx'+αx )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 predefined 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 ⎟ = 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 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 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 ⎟
s + ds , s ⎜
rad ⎟⎜ (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 Hill’s equation, even in cases, as in linear accelerators, where there is no periodicity.
USPAS Accelerator Physics June 2013
Suppose we are given the phase space ellipse
γ (s )x 2 + 2α (s )xx'+ β (s )x'2 = ε
at location s, and we wish to calculate the ellipse parameters, after
the motion generated by Hill’s equation, at the location s + ds
2
2
(
)
(
)
(
)
γ s + ds x + 2α s + ds xx'+ β s + ds x' = 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
details of the functional form K(s). (If desired, these two
conclusions may be verified more analytically by showing
that
(
)
d
βγ − α 2 = 0 → β (s )γ (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,
β functions
The ellipse transformation formulas give, to order linear in ds
So
ds
β (s + ds ) = −2α
+ β (s )
rad
ds
α (s + ds ) = −γ (s )
+ α (s ) + β (s )Kds rad
rad
dβ
2α (s )
(s ) = −
ds
rad
γ (s )
dα
(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) and α(s) given the focusing function K(s). A somewhat
easier approach to obtain β(s) is to recall that the maximum
extent of an ellipse, xmax, is (εβ)1/2(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
ε / rad )
(
d 2w
.
+ 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 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
2
d x
+ K (s )x = 0
2
ds
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
x± (s ) = w(s )e ± iμ ( s )
USPAS Accelerator Physics June 2013
are two particular solutions to Hill’s equation, provided that
2
2
d w
c
+ K (s )w = 3
2
ds
w
and
dμ
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
⎛
w(s1 )e
⎛ A⎞ ⎜
⎜⎜ ⎟⎟ = ⎜ ⎡
ic ⎤ iμ ( s1 )
⎝ B ⎠ ⎜ ⎢ w' (s1 ) + w(s ) ⎥ e
1 ⎦
⎝⎣
iμ ( s1 )
⎞
w(s1 )e
⎟
⎡
ic ⎤ −iμ ( s1 ) ⎟
⎢ w' (s1 ) −
⎥e
⎟
(
)
w
s
1 ⎦
⎣
⎠
USPAS Accelerator Physics June 2013
−iμ ( s1 )
−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 Δμ s2 , s1 −
sin Δμ s2 , s1
⎜
(
)
w
s
c
1
⎜
c
⎡ w(s2 )w' (s2 )w(s1 )w' (s1 ) ⎤
⎛ x2 ⎞ ⎜ −
⎜
⎜⎜ ⎟⎟ =
2
⎢1 +
⎥ sin Δμ s2 , s1
(
)
(
)
w
s
w
s
c
⎦
2
1 ⎣
⎝ x '2 ⎠ ⎜
⎜
⎡ w' (s1 ) w' (s2 ) ⎤
−⎢
−
⎜⎜
⎥ cos Δμ s2 , s1
(
)
(
)
w
s
w
s
2
1 ⎦
⎣
⎝
w(s2 )w(s1 )
sin Δμ s2 , s1
c
w(s1 )
cos Δμ s2 , s1
w(s2 )
where
s2
Δμ s2 , s1 = μ (s2 ) − μ (s1 ) = ∫
s1
USPAS Accelerator Physics June 2013
c
ds
w (s )
2
⎞
⎟
⎟
⎟⎛ x ⎞
⎟⎜⎜ 1 ⎟⎟
w' (s2 )w(s1 )
x'
+
sin Δμ s2 , s1 ⎟⎝ 1 ⎠
⎟
c
⎟⎟
⎠
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
assume this to be the case. Later, it will be shown how to
construct the function explicitly. Clearly for w periodic
φ (s ) = μ (s ) − μ L s
with
μL =
s+ L
∫
s
c
ds
w (s )
2
is also periodic by Eqn. (3b), and μL is independent of s.
USPAS Accelerator Physics June 2013
The transfer matrix for a single period reduces to
⎛
⎞
w(s )w' (s )
w2 (s )
⎜
⎟
cos μ L −
sin μ L
sin μ L
c
c
⎜
⎟
c ⎡ w(s )w' (s )w(s )w' (s ) ⎤
w' (s )w(s )
⎜
⎟
−
+
+
1
sin
μ
cos
μ
sin
μ
L
L
L⎟
2
⎜ w2 (s ) ⎢
⎥
c
c
⎣
⎦
⎝
⎠
β ⎞
⎛ 1 0⎞
⎛α
⎟⎟ cos(μ L ) + ⎜⎜
⎟⎟ sin (μ L )
= ⎜⎜
⎝ 0 1⎠
⎝ − γ −α ⎠
where the (now periodic!) matrix functions are
w(s )w' (s )
α (s ) = −
,
c
w 2 (s )
β (s ) =
,
c
1 + α 2 (s )
γ (s ) =
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
L
ds
μL = ∫
β (s )
0
and more generally the phase advance between any two
longitudinal locations s and s' is
Δμ s ', s
s'
ds
=∫
β (s )
s
USPAS Accelerator Physics June 2013
Transfer Matrix in terms of α and β
Also, the unimodular transfer matrix taking the solution from s
to s' is
M s ', s
⎛
β (s ')
⎜
(cos Δμ s ',s + α (s )sin Δμ s ',s )
(
)
β
s
⎜
=⎜
⎡(1 + α (s ')α (s ))sin Δμ s ', s ⎤
1
⎜−
⎢
⎥
⎜
(
(
)
(
)
)
α
s
'
α
s
cos
μ
+
−
Δ
(
)
(
)
β s' β s ⎣
s ', s ⎦
⎝
⎞
⎟
β (s ')β (s ) sin Δμ s ', s
⎟
⎟
β (s )
(cos Δμ s ',s − α (s')sin Δμ s ',s )⎟⎟
β (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 βfunction (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 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/π,
(
)
ε = γx 2 + 2αxx'+ βx'2 = x 2 + (βx'+αx )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
Pseudoharmonic Solution
⎛
β (s )
⎜
(cos Δμ s,0 + α 0 sin Δμ s,0 )
⎛ x(s ) ⎞ ⎜
β0
⎜ dx ⎟ =
⎜ (s )⎟ ⎜
⎡(1 + α (s )α 0 )sin Δμ s , 0 ⎤
1
⎜
⎝ ds ⎠ −
⎢
⎥
⎜
(
(
)
)
α
s
α
μ
cos
+
−
Δ
(
)
β
s
β
0
s ,0 ⎦
0 ⎣
⎝
gives
⎞
⎟⎛ x ⎞
⎟⎜ 0 ⎟
⎟⎜ dx ⎟
β0
(cos Δμ s,0 − α (s )sin Δμ s,0 )⎟⎟⎜⎝ ds 0 ⎟⎠
β (s )
⎠
β (s )β 0 sin Δμ s ,0
(x (s ) + (β (s )x' (s ) + α (s )x(s )) )/ β (s ) = (x
2
2
2
0
)
+ (β 0 x'0 +α 0 x0 ) / β 0 ≡ ε
2
Using the x(s) equation above and the definition of ε, the
solution may be written in the standard “pseudoharmonic” form
⎛ β 0 x'0 +α 0 x0 ⎞
⎟⎟
x(s ) = εβ (s ) cos(Δμ s ,0 − δ ) where δ = tan ⎜⎜
x0
⎝
⎠
The the origin of the terminology “phase advance” is now obvious.
−1
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 wellunderstood “modules” that yield known and understood transverse
beam optical properties.
USPAS Accelerator Physics June 2013
Definition of β 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 α0, β0, and γ0. A function β(s) is simply defined by the
ellipse transformation rule
β (s ) = (M 12 (s ))2 γ 0 − 2M 12 (s )M 11 (s )α 0 + (M 11 (s ))2 β 0
[
]
= (M 12 (s )) + (β 0 M 11 (s ) − α 0 M 12 (s )) / β 0
2
where
M s ,0
2
⎛ M 11 (s ) M 12 (s )⎞
⎟⎟
≡ ⎜⎜
⎝ M 21 (s ) M 22 (s )⎠
USPAS Accelerator Physics June 2013
One might think to evaluate the phase advance by integrating
the beta-function. Generally, it is far easier to evaluate the phase
advance using the general formula,
tan Δμ s ', s =
(M )
) − α (s )(M )
β (s )(M s ', s
s ', s 12
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 hand yields
tan Δμ s , 0
M 12 (s )
=
β 0 M 11 (s ) − α 0 M 12 (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.
d 2D
1
+
=
K
s
D
( )
ρ (s)
ds 2
Write the general solution to the inhomogeneous equation for
the dispersion as before.
D ( s ) =D p ( s ) + Ax1 ( s ) + Bx2 ( s )
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
D p ,0 ( s1 ) = D ′p ,0 ( s1 ) = 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!)
⎛ A ⎞ ⎛ x1 ( s1 )
⎜ ⎟ = ⎜ x′ s
⎝ B⎠ ⎝ 1 ( 1)
−1
x2 ( s1 ) ⎞ ⎛ D ( s1 ) ⎞
⎟ ⎜
⎟
x2′ ( s1 ) ⎠ ⎝ D ′ ( s1 ) ⎠
(
) D (s ) + (M )
(s − s ) + (M ) D (s ) + (M )
D ( s2 ) = D p ,0 ( s2 − s1 ) + M s2 , s1
D ′ ( s2 ) = D ′p ,0
2
1
s2 , s1
11
21
USPAS Accelerator Physics June 2013
1
s2 , s1
1
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 )
11
s2 , s1
12
21
s2 , s1
22
0
D p ,0 ( s2 − s1 ) ⎞
⎟ ⎛ D ( s1 ) ⎞
⎜
⎟
D ′p ,0 ( s2 − s1 ) ⎟ ⎜ D ′ ( s1 ) ⎟
⎟
⎟⎟ ⎜⎝ 1 ⎟⎠
1
⎠
Particular solutions to inhomogeneous equation for constant K
and constant ρ and vanishing dispersion and derivative at s = 0
K<0
K=0
) )
Dp,0(s)
1
cosh
K ρ
( (
K s −1
D'p,0(s)
1
(
Kρ
sinh
Ks
)
s2
2ρ
s
ρ
USPAS Accelerator Physics June 2013
K>0
(
(
Ks
1
sin
Kρ
(
Ks
1
1 − cos
Kρ
))
)
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-ofarrival of the off-momentum particle compared to the on-momentum particle which
traverses the design trajectory.
Δp ⎞
ds ⎛
D
s
ρ
+
(
)
⎟ − ds
p ⎠
ρ ⎜⎝
ds
Δp
D (s)
p
Δz =
d ( Δz ) =D ( s )
Δ p ds
p ρ (s)
ρ (+)
Design Trajectory
⎧⎪ Dx ( s ) D y ( s ) ⎫⎪
= ∫⎨
+
⎬ds
s1 ⎪
⎩ ρ x ( s ) ρ y ( s ) ⎭⎪
s2
M 56
Dispersed Trajectory
USPAS Accelerator Physics June 2013
Solenoid Focussing
Can also have continuous focusing in both transverse directions by applying solenoid
magnets:
B(z)
z
USPAS Accelerator Physics June 2013
Busch’s Theorem
For cylindrical symmetry magnetic field described by a vector potential:
r
A = Aθ ( z , r ) θˆ
∴ Aθ ( z , r )
1 ∂
rAθ ( z , r ) ) is nearly constant
(
r ∂r
Bz ( r = 0, z ) r
Bz′ ( r = 0, z ) r
Br =
2
2
Bz =
Conservation of Canonical Momentum gives Busch’s Theorem:
Pθ = γ mr 2θ& + qrAθ = const
for particle with θ& = 0 where Bz = 0, Pθ = 0
Ωc
qBz
&
γ mr θ = −
=−
= −ω Larmor
2γ m
2
2
Beam rotates at the Larmor frequency which implies coupling
USPAS Accelerator Physics June 2013
Radial Equation
d
(γ mr& ) − γ mrω L2 = qrθ& Bz = −2γ mrω L2
dt
ω L2
∴k = 2 2
βz c
thin lens focal length
∞
∫−∞
1
=
f 4 β z2γ 2 m 2 c 2
e2
Bz2 dz
weak compared to quadrupole for high 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
x
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
(γ mr& ) − γ mrθ& 2 = qrθ& Bz
dt
γ mr 2θ& + qAθ = cons = Pθ
d
(γ mr& ) − γ mrθ&′2 + 2γ mrθ ′ω L − γ mrω L2 = qrθ&′Bz − qrω L Bz
dt
γ mr 2θ&′ = Pθ
d
(γ mr& ) − γ mrθ&′2 = −γ mrω L2 ⎪⎫ 2-D Harmonic Oscillator
dt
⎬
⎪⎭
γ mr 2θ&′ = Pθ
USPAS Accelerator Physics June 2013