Linear Beam Optics and Particle Motion in Accelerator Physics
Accelerator Physics
Linear Optics
S. A. Bogacz, G. A. Krafft, S. DeSilva, R. Gamage
Jefferson Lab
Old Dominion University
Lecture 4
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
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
.
Fundamental Notion:
Path Length
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,
L
tot
=ct
tot
.
The first step in designing any accelerator, is to specify
bending magnet locations that are consistent with the arc
portions of the Design Trajectory.
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
, 4
th
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.
4-Fold Symmetric Synchrotron
L
ρ
vertical
Its Design Trajectory
Betatron Design Trajectory
Use path length
s
as independent variable instead of
t
in the
dynamical equations.
Betatron Motion in
s
Bend Magnet Geometry
ρ
Rectangular Magnet of Length
L
Sector Magnet
θ
Bend Magnet Trajectory
For a uniform magnetic field
For the solution satisfying boundary conditions:
Magnetic Rigidity
The magnetic rigidity is:
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.
Long Dipole Magnet
Normal Incidence (or exit)
Dipole Magnet
Natural Focusing in Bend Plane
Perturbed Trajectory
Design Trajectory
Can show that for either a displacement perturbation or angular perturbation
from the design trajectory
Quadrupole Focusing
Combining with the previous slide
Hill’s Equation
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
Define focusing strengths (with units of m
-2
)
Energy Effects
This solution is not a solution to Hill’s equation directly, but
is
a solution to the
inhomogeneous Hill’s Equations
Inhomogeneous Hill’s Equations
Fundamental transverse equations of motion in particle
accelerators for small deviations from design trajectory
ρ
radius of curvature for bends,
B'
transverse field gradient
for magnets that focus (positive corresponds to horizontal
focusing),
Δ
p/p
momentum deviation from design
momentum. Homogeneous equation is 2
nd
order
linear
ordinary differential equation.
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.
Because the inhomogeneous terms are proportional to
Δ
p/p
, the particular
solution can generally be written as
where the dispersion functions satisfy
M
56
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.
Design Trajectory
Dispersed Trajectory
Solutions Homogeneous Eqn.
Dipole
Drift
Quadrupole in the focusing direction
Quadrupole in the defocusing direction
Transfer Matrices
Dipole with bend
Θ
(put coordinate of final position in solution)
Drift
Thin Lenses
Thin Focusing Lens (limiting case when argument goes to
zero!)
Thin Defocusing Lens: change sign of
f
f
–f
Composition Rule: Matrix Multiplication!
Remember: First element farthest RIGHT
Element 1
Element 2
More generally
Some Geometry of Ellipses
y
Equation for an upright ellipse
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
The area is easily computed to be
So the equation is equivalently
Eqn. (1)
Example: the defining equation for the upright ellipse may be
rewritten in following suggestive way
β
= a/b
and
γ = b/a,
note
When normalized in this manner, the equation coefficients
clearly satisfy
General Tilted Ellipse
x
y
b
a
Needs 3 parameters for a complete
description. One way
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
y=sx
Identify
Note that
βγ – α
2
=
1 automatically, and that the e
quation for
ellipse becomes
by eliminating the (redundant!) parameter
γ
Ellipse in the
β-
function Description
As for the upright ellipse
Area Theorem for Linear Optics
Under a general linear transformation
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
Effect of Transformation
Let the final ellipse be
The transformed coordinates must solve this equation.
The transformed coordinates must also
solve the initial equation transformed.
Because
The transformed ellipse is
Because (verify!)
the area of the transformed ellipse (divided by
π
) is,
by Eqn. (1)
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
Because det (
M
)=1, the tilted ellipse has the same area as the
upright ellipse, i.e.,
ε
=
ε
0
.
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
Pf: The equation for the eigenvalues of
M
is
The
phase advance
of the matrix,
μ
,
gives the eigenvalues of the
matrix
λ
=
e
±
iμ
, and cos
μ =
(Tr
M
)/2. Furthermore
βγ
–
α
2
=1
Because
M
is real, both
λ
and
λ
* are solutions of the
quadratic.
Because
For |Tr
M|
< 2,
λ
λ
* =1 and so
λ
1,2
=
e
±
iμ
. Consequently cos
μ
=
(Tr
M
)/2. Now the following matrix is trace-free.
Simply choose
and the sign of
μ
to properly match the individual matrix
elements with
β >
0
. It is easily verified that
βγ – α
2
=
1. Now
and more generally
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
clearly have unbounded powers if |
x
| is not equal to 0.
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
with |Tr (
M
)| < 2, the ellipses
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
By the ellipse transformation formulas, for example
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
is invariant under the transformation by
M
only if
i.e., if the vector is
ANY
eigenvector of
T
M
with eigenvalue 1.
All possible solutions may be obtained by investigating the
eigenvalues and eigenvectors of
T
M
. Now
i.e.,
Therefore,
M
generates a transformation matrix
T
M
with at least
one eigenvalue equal to 1. For there to be more than one solution
with
λ
= 1,
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
T
M
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 (
γ
i
,
α
i
,
β
i
) yield equations for the
invariant ellipses. For concreteness, compute that eigenvector with
eigenvalue 1 normalized so
β
i
γ
i
– α
i
2
=
1
All other eigenvectors with eigenvalue 1 have , for
some value
c.
Because we have enumerated all possible eigenvectors with
eigenvalue 1, all ellipses invariant under the action of
M
, are of the
form
Because Det (
M
) =1, the eigenvector
clearly yields the
invariant ellipse
Likewise, the proportional eigenvector generates the similar
ellipse
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
is an invariant of the equations of transverse motion.
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.
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
If the differential equation giving the evolution of
x
is linear, one
may define a linear transport matrix
M
s
'
,
s
relating the coordinates
at
s
'
to those at
s
by
From the definitions, the concatenation rule
M
s
'
'
,
s
=
M
s
'
'
,
s
'
M
s
'
,
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
for all initial conditions (
x
(
s
),
dx/ds
(
s
)), thus
M
s
'
'
,
s
=
M
s
'
'
,
s
'
M
s
'
,
s
.
Clearly
M
s
,
s
=
I
. As is shown next, the matrix
M
s
'
,
s
is in general a
member of the unimodular subgroup of the general linear group.
Ellipse Transformations Generated by
Hill’s Equation
The equation governing the linear transverse dynamics in a
particle accelerator, without acceleration, is
Hill’s equation*
*
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.
Eqn. (2)
The transformation matrix taking a solution through an
infinitesimal distance
ds
is
Suppose we are given the phase space ellipse
at location
s
, and we wish to calculate the ellipse parameters, after
the motion generated by Hill’s equation, at the location
s + ds
Because, to order linear in
ds
, Det
M
s+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:
1.
The phase space area is preserved after a finite integration of
Hill’s equation to obtain
M
s
'
,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
M
s
'
,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
may be derived directly from Hill’s equation.)
Evolution equations for
α
,
β
functions
The ellipse transformation formulas give, to order linear in
ds
So
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,
x
max
, 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
x
max
(
s
) =
w
(
s
) = (
εβ
)
1/2
(
s
)
Such a differential equation describing the evolution of the
maximum extent of an ellipse being transformed is known as an
envelope equation
.
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.
It should be noted, for consistency, that the same
β
(
s
) =
w
2
(
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.
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
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
where
are two particular solutions to Hill’s equation, provided that
and where
A
,
B
, and
c
are constants (in
s
)
That specific solution with boundary conditions
x
(
s
1
) =
x
1
and
dx
/
ds
(
s
1
) =
x'
1
has
Eqns. (3)
Therefore, the unimodular transfer matrix taking the solution at
s
=
s
1
to its coordinates at
s
=
s
2
is
where
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
is also periodic by Eqn. (3b), and
μ
L
is independent of
s
.
The transfer matrix for a single period reduces to
where the (now periodic!) matrix functions are
By Thm. (2), these are the ellipse parameters of the periodically
repeating, i.e.,
matched
ellipses.
General formula for phase advance
In terms of the
β
-function, the phase advance for the period is
and more generally the phase advance between any two
longitudinal locations
s
and
s
'
is
Transfer Matrix in terms of
α
and
β
Also, the unimodular transfer matrix taking the solution from
s
to
s
' is
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.
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
M
s+L,s
(
s
). From this find
μ
L
(which is independent
of the location chosen!) from cos
μ
L
= (
M
11
+
M
22
) / 2, and by
choosing the sign of
μ
L
so that
β
(
s
) =
M
12
(
s
) / sin
μ
L
is positive.
Likewise,
α
(
s
) = (
M
11
-
M
22
) / 2 sin
μ
L
. 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.
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
,
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.
Pseudoharmonic Solution
gives
Using the
x
(
s
) equation above and the definition of
ε
, the
solution may be written in the standard “pseudoharmonic” form
The the origin of the terminology “phase advance” is now obvious.
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.
Definition of
β
function
where
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
α
0
,
β
0
, and
γ
0
. A function
β
(
s
)
is simply defined by the
ellipse transformation rule
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,
where
β
(
s
) and
α
(
s
) are the ellipse functions at the entrance of
the region described by transport matrix
M
s
'
,s
.
Applied to the
situation at hand yields
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
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.
Dispersion Calculation
Begin with the inhomogeneous Hill’s equation for the
dispersion.
Write the general solution to the inhomogeneous equation for
the dispersion as before.
Here
D
p
can be any particular solution, and we suppose that the
dispersion and it’s derivative are known at the location
s
1
, and
we wish to determine their values at
s
.
x
1
and
x
2
are linearly
independent solutions to the homogeneous differential equation
because they are transported by the transfer matrix solution
M
s
,
s
1
already found.
To build up the general solution, choose that particular solution
of the inhomogeneous equation with homogeneous boundary
conditions
Evaluate
A
and
B
by the requirement that the dispersion and it’s
derivative have the proper value at
s
1
(
x
1
and
x
2
need to be
linearly independent!)
3 by 3 Matrices for Dispersion Tracking
Particular solutions to inhomogeneous equation for constant
K
and constant
ρ
and vanishing dispersion and derivative at
s
= 0
M
56
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.
Design Trajectory
Dispersed Trajectory
Solenoid Focussing
Can also have continuous focusing in both transverse directions by applying solenoid
magnets:
Busch’s Theorem
For cylindrical symmetry magnetic field described by a vector potential:
Conservation of Canonical Momentum gives Busch’s Theorem:
Beam rotates at the Larmor frequency which implies coupling
Radial Equation
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!
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
G. A. Krafft
Jefferson Lab
Explore the fundamental concepts of linear beam optics and particle motion in accelerator physics, covering topics such as design trajectory, path length, phase advance, transfer matrix, and more. Understand the intricacies of designing accelerators and the mathematical representations involved in optimizing particle trajectories. Delve into the geometry of ellipses, Twiss representation, and the significance of path length in Earth-fixed reference frames. Gain insights into beam matching and the importance of curvature in accelerator design.
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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 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 -function Periodic Solutions Non-periodic Solutions Formulas for -function and Phase Advance Beam Matching USPAS Accelerator Physics June 2016
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. :[0,1] = 3 R ( ) ( ) ( ) ( ) ( ) , , X X Y Z Fundamental Notion: Path Length 2 2 2 dX d dY d dZ d = + + ds d USPAS Accelerator Physics June 2016
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 :[0, ] R s L tot ( ) ( ) ( ) ( ) ( ) = , , 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 14.3-14.5. 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 2016
4-Fold Symmetric Synchrotron x 1s s = 0 z 0 y vertical = + /2 s L 7s 2 x z = 3 s s 3s 6 2 L = 2 5s s s 4 2 USPAS Accelerator Physics June 2016
Its Design Trajectory ( ) = 0,0, 0 s s L s 1 ( ) ( ) ( ) ( ) ( ) + ( ) + ( 0,0, cos / 1,0,sin )( 2 s / L s s s s s s s 1 ) ( 1 1 2 ) + ( L ,0 , 1,0,0 L s s s s 2 3 ( L ) ( ) ( ) ) ( ) ( ) + + ,0, sin / ,0,cos / 1 L s s s s s s s 3 3 3 4 ( ) ( + )( ) )( 6 s ) ( L 2 ,0, 0,0, 1 ) ( ( ) 1,0,0 ( ) ( ) 7 / s L s ( s s s s 4 4 5 ( ) ( ) ) ) 2 ,0,0 + 1 cos / ,0, sin / s s s s s s s 5 5 5 6 ( ) ( + ( ,0, L s s s s 6 7 ) ( ) ( ) ( ) + ,0, sin / ,0,1 cos 4 s s s s s s 7 7 2 USPAS Accelerator Physics June 2016
Betatron Design Trajectory 3 :[0,2 ] R s R ( ) ( ) ( ) ( ) = cos / , sin R / ,0 s X s R s R s R Use path length s as independent variable instead of t in the dynamical equations. 1 d ds d = R dt c USPAS Accelerator Physics June 2016
Betatron Motion in s 2 d r p p ( ) + = 2 c 2 c 1 n r R 2 dt 2 d z + = 2 c 0 n z 2 dt ( ) 1 n 2 1 R p d r p + = r 2 2 ds R z 2 d n + = 0 z 2 2 ds R USPAS Accelerator Physics June 2016
Bend Magnet Geometry y B x Rectangular Magnet of Length L Sector Magnet x z /2 USPAS Accelerator Physics June 2016
Bend Magnet Trajectory For a uniform magnetic field ( ) d mV dt mV dt mV dt = + E V B ( ) d = x qV B z y ( ) d = qV B z x y 2 2 d V dt d V dt + = + = 2 c 2 c 0 0 x V V z x z 2 2 ( ) 0 ( ) 0 = = For the solution satisfying boundary conditions: 0 z 0 X V V z p ( ) ( ) ( ) ( ) ( ) = = = cos 1 cos 1 / X t t t qB m c c c y qB y p ( ) ( ) ( ) = = sin sin Z t t t c c qB y USPAS Accelerator Physics June 2016
Magnetic Rigidity The magnetic rigidity is: p q = = B B y 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) Dipole Magnet Long Dipole Magnet ( ) ( ) ( ) = B = B sin 2sin /2 BL BL 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 2 2 d x ds x d y ds y = = ( ) s ( ) s 2 2 y 2 2 x USPAS Accelerator Physics June 2016
Quadrupole Focusing ( ) ( )( ) B s = + , B x y xy yx v ds d v ds d ( ) ( ) qB s x qB s y y = = x m m ( ) B ( ) B B s B s 2 2 d x ds d y ds + = = 0 0 x y 2 2 Combining with the previous slide ( ) B ( ) B B s B s 2 2 1 1 d x ds d y ds + + = + = 0 0 x y ( ) s ( ) s 2 2 x 2 2 y USPAS Accelerator Physics June 2016
Hills Equation Define focusing strengths (with units of m-2) ( ) B ( ) B B s B s 1 1 ( ) s = + = k k ( ) s ( ) s x y 2 x 2 y 2 2 d x ds d y ds ( ) s x ( ) s y + = + = 0 0 k k x y 2 2 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 p p p ( ) ( ) x s ( ) = 1 cos / s eB y This solution is not a solution to Hill s equation directly, but is a solution to the inhomogeneous Hill s Equations ( ) B B s 2 1 1 d x ds p p + + = x ( ) s ( ) s 2 2 x x ( ) B B s 2 1 1 d y ds p p + = y ( ) s ( ) s 2 2 y y USPAS Accelerator Physics June 2016
Inhomogeneous Hills Equations Fundamental transverse equations of motion in particle accelerators for small deviations from design trajectory ( ) B B s 2 1 1 d x ds p p + + = x ( ) s ( ) s 2 2 x x ( ) B B s 2 1 1 d y ds p p + = y ( ) s ( ) s 2 2 y y radius of curvature for bends, B' transverse field gradient for magnets that focus (positive corresponds to horizontal focusing), 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. ( ) ( ) ( ) ( ) 1 2 = p x x x s x s A x s B x s + + ( ) y s ( ) s ( ) ( ) s + + = y A y s B y 1 2 p y y Because the inhomogeneous terms are proportional to p/p, the particular solution can generally be written as p x s D s p where the dispersion functions satisfy ( ) ( ) x x s B s p p ( ) ( ) ( ) s ( ) s = = y D p x p y ( ) B B s B s 2 d D ds 2 1 1 1 1 d D ds y + + = + = x D D ( ) ( ) s ( ) s x y 2 2 2 2 y y 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-of- arrival of the off-momentum particle compared to the on-momentum particle which traverses the design trajectory. ds p p ( ) = + z D s ds ds p ds p ( ) ( ) = d z D s ( ) s p p ( ) ( ) + D s Design Trajectory Dispersed Trajectory ( ) ( ) s ( ) ( ) s s D s D s 2 y = + x M ds 56 x y s 1 USPAS Accelerator Physics June 2016
Solutions Homogeneous Eqn. Dipole ( ) ( ) s ds ( ) ( ) s ds ( ( ) / ( ( ) x s dx x s dx ( s ) / ( s ) / cos / sin / s s s s i i i = ( ) ( ) ) ) sin cos s s i i i Drift ( ) ( ) s ( ) ( ) s ds x s dx ds x s dx 1 0 s s i = i 1 i USPAS Accelerator Physics June 2016
B B = Quadrupole in the focusing direction / k ( ) ( ) ( ) ( ) s ( ) ( ) s ds ( ) ( ) x s dx ds x s dx cos sin / k s s k s s k i i i = ( ) ( ) ( ) ( ) sin cos k k s s k s s i i i B B = Quadrupole in the defocusing direction / k ( ) ( ) ( ) ( ) s ( ) ( ) s ds ( ) ( ) x s dx ds x s dx cosh sinh / k s s k s s k i i i = ( ) ( ) ( ) ( ) sinh cosh k k s s k s s i i i USPAS Accelerator Physics June 2016
Transfer Matrices Dipole with bend (put coordinate of final position in solution) ( ( after s ds ) ( ) x s x s ( ) ( ) ( ) ( ) cos sin sin cos after before = dx dx ds ) ( ) / before s Drift ( ) ( ) x s x s 1 0 L after before = drift dx ds dx ds ( ) ( ) 1 after s before s USPAS Accelerator Physics June 2016
Thin Lenses f f Thin Focusing Lens (limiting case when argument goes to zero!) ( ) ( ) 1/ lens s ds ( ) + x s dx x s dx ds 1 0 1 lens lens = ( ) + f lens s Thin Defocusing Lens: change sign of f USPAS Accelerator Physics June 2016
Composition Rule: Matrix Multiplication! Element 1 Element 2 1s 0s 2s ( ) ( ) x s ( ) ( ) x s ( ) ( ) 2 x s ( ) ( ) x s ( ) ( ) x s x s x s x s x s 1 0 2 1 = = M M 1 2 1 0 2 1 ( ) ( ) x s x s x s 2 0 = M M 2 1 0 More generally = ... M M M M M 1 2 1 tot N N Remember: First element farthest RIGHT USPAS Accelerator Physics June 2016
Some Geometry of Ellipses y Equation for an upright ellipse 2 2 b x y + = 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 appears on the RHS of the defining equation. For a general ellipse + + = 2 2 2 Ax Bxy Cy D USPAS Accelerator Physics June 2016
The area is easily computed to be Area D = Eqn. (1) 2 AC B So the equation is equivalently + 2 + = 2 2 x xy y A B C = = = , , and 2 2 2 AC B AC B AC B USPAS Accelerator Physics June 2016
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 a + = = 2 2 x y ab a b = a = =b = , x ymax = a/b and = b/a, note max USPAS Accelerator Physics June 2016
General Tilted Ellipse y Needs 3 parameters for a complete description. One way y=sx b a b ( ) 2 + = = 2 x y sx ab x a b 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 is the area of the ellipse divided by + 1 2 b a a a + = = 2 2 2 2 s x s xy y ab 2 a b b b USPAS Accelerator Physics June 2016
Identify + 1 2 b a a a = = = 2 , , s s 2 a b b b Note that 2 = 1 automatically, and that the equation for ellipse becomes ( ) 2 + + = 2 x y x by eliminating the (redundant!) parameter USPAS Accelerator Physics June 2016
Ellipse in the -function Description , y y=sx= x / , = / b x = a = = , x y As for the upright ellipse max max USPAS Accelerator Physics June 2016
Area Theorem for Linear Optics Under a general linear transformation ' M M x x 11 12 = ' M M y y 21 22 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 + x + = 2 2 2 xy y 0 0 0 0 USPAS Accelerator Physics June 2016
Effect of Transformation + + = 2 2 2 x xy y Let the final ellipse be The transformed coordinates must solve this equation. x y + + = 2 2 2 x y ( ) , x y M + + = 2 2 2 x xy y ( ) 0 0 0 0 x y , M 1 ' ' M M M M x y x y 11 12 = 21 2 2 1 1 x y x y M M M M = 11 12 The transformed coordinates must also solve the initial equation transformed. 1 1 21 22 USPAS Accelerator Physics June 2016
Because ( ( ) ) ( ( ) ) 1 1 M M ' ' x y x y 11 12 = 1 1 M M 21 22 The transformed ellipse is + + = 2 2 2 x x y y 0 ( ) ( ) ( ) ) ( 12 ) ( ) 2 2 = + + 1 1 1 1 2 M M M M 0 0 0 11 11 21 21 ( ) ) ) ( ) ( 11 ) ( ) ( 11 ( ) ( 12 ) ( ) ( 21 ) = + + + 1 1 1 1 1 1 1 1 M M M M M M M M 0 0 0 12 22 21 22 ( ( ) ( 2 2 = + + 1 1 1 1 2 M M M M 0 0 0 12 22 22 USPAS Accelerator Physics June 2016
Because (verify!) ( ) = 2 2 0 0 0 ) ( ( ) ( 21 ) ( ) ( 11 ) )( ( ) ( 11 ) ( 22 ) ( 12 ) 2 2 2 2 + 1 1 1 1 1 1 1 1 2 M M M M M M M M 12 22 21 ( ) 2 = 2 0 1 det M 0 0 the area of the transformed ellipse (divided by ) is, by Eqn. (1) Area = = = 0 | det | M 0 2 0 1 det M 0 0 USPAS Accelerator Physics June 2016
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 = s y ' ' 1 0 x x 1 y b a a b ( ) = = = = 1 , 0, , M s 0 0 0 21 b a a a = + = = 2 , , s s a b b 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-by-two 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 = e i , and cos = (Tr M)/2. Furthermore 2=1 Pf: The equation for the eigenvalues of M is ( ) + + = 2 1 0 M M 11 22 USPAS Accelerator Physics June 2016
Because M is real, both and * are solutions of the quadratic. Because ( ) 2 Tr M ( ( ) )2 = 1 Tr / 2 i M For |Tr M| < 2, * =1 and so 1,2 = e i . Consequently cos = (Tr M)/2. Now the following matrix is trace-free. = 1 0 2 M M 11 22 M 1 0 12 ( ) cos M 2 M M 22 11 M 21 USPAS Accelerator Physics June 2016
Simply choose M M M M = = = , , 11 22 12 21 2 sin sin sin and the sign of to properly match the individual matrix elements with > 0. It is easily verified that 2 = 1. Now and more generally 1 0 ( ) ( ) = 2 + 2 2 cos sin M 0 1 1 0 ( ) ( ) = n + n Mn cos sin 0 1 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-by-two unimodular matrices, i.e., matrices of the form 1 0 1 1 x 0 x or 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
+ + = 2 2 2 x xy y 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 1 0 ( ) ( ) = 1 cos sin M 0 1 By the ellipse transformation formulas, for example ( ( ( ( = + = cos sin ) + )( ) + ( ) 2 = 1 + sin + + cos + + 2 2 ' sin 2 sin cos sin cos sin ) ) = 2 2 2 2 2 2 2 2 sin sin 2 2 USPAS Accelerator Physics June 2016
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 + 2 xy x i i + = 2 2 y i is invariant under the transformation by M only if ( ) ( )( 2 ) ( ) )( 2 2 cos sin 2 cos sin sin sin i i ( )( 2 ) ( ) = + cos sin sin 1 2 + sin cos sin + sin 2 i i ( ) ( )( ) ( ) sin 2 cos sin sin cos sin i i i , T T v M i M i USPAS Accelerator Physics June 2016
v i.e., if the vector is ANY eigenvector of TM with eigenvalue 1. All possible solutions may be obtained by investigating the eigenvalues and eigenvectors of TM. Now solution w a has = v v T M i.e., ( ) = hen Det 0 T I M ( )( ) + 2 4cos + = 2 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 4cos 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 i i i2 = 1 = 1 i / M M 21 12 i ( ) = = / 2 v M M M , 1 11 22 12 i i = i/ v v c All other eigenvectors with eigenvalue 1 have , for some value c. 1 , 1 USPAS Accelerator Physics June 2016
v, 1 Because Det (M) =1, the eigenvector invariant ellipse Likewise, the proportional eigenvector generates the similar ellipse ( + 2 xy x c clearly yields the i + + = 1v 2 2 2 . x xy y ) = + 2 2 y Because we have enumerated all possible eigenvectors with eigenvalue 1, all ellipses invariant under the action of M, are of the form xy x + + 2 = 2 2 y 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 xx x = + + = ( ) / ( ) 2 + + 2 2 2 ' x 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- 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-rad, 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 ( ) ( ) ds x s dx s If the differential equation giving the evolution of x is linear, one may define a linear transport matrix Ms',srelating the coordinates at s' to those at s by ( ) ( ) ds ( ) ( ) s ds ' x s x s = dx dx M ' s s , ' s USPAS Accelerator Physics June 2016
From the definitions, the concatenation rule Ms'',s= Ms'',s'Ms',smust 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 ( ) ( ) ( ) ds ds for all initial conditions (x(s), dx/ds(s)), thus Ms'',s= Ms'',s'Ms',s. ( ) s ' ' ( ) ( ) s ds ( ) ( ) s ds ' x s x x s x s = = = dx dx dx dx M M M M ' ' ' s s , ' ' , ' ' ' , ' ' ' , ' s s s s s s s 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 Hill s Equation The equation governing the linear transverse dynamics in a particle accelerator, without acceleration, is Hill s equation* 2 d x ( ) s + = Eqn. (2) 0 K x 2 ds The transformation matrix taking a solution through an infinitesimal distance ds is ( ) ( ) ( ) s ( ) ( ) s ds + ds rad x s ds x s x s 1 = dx dx dx M ( ) + + s ds , s ds s ( ) s rad 1 K ds 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 2016
