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SLAC - PUB - 3957
May 1986 (A) OPTICS MODULES FOR CIRCULAR ACCELERATOR DESIGN*
Karl L. Brown Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94%~ and Roger V. Servranckx University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N-0 W O
1 INTRODUCTION AND SUMMARY
This paper is intended as a companion paper to ‘Circular Machine Design Techniques and Tools’ presented at this conference by Roger Servranckx. The intent here is to provide a tutorial discussion on the basic optics of circular particle accelerators for the benefit of those readers who have a fundamental knowledge of charged particle optics but do not make it their profession to design particle accelerators.
We begin the tutorial by presenting the solutions of the first-order differential equations of motion for a single particle in a closed circular machine introducing the concepts of phase shift, beta functions, and the Courant-Snyder invariant. From these solutions we derive the transfer matrix between two points in the machine as a function of the phase shift and the parameters contained in the Courant-Snyder invariant. We then introduce typical optical building blocks (modules) used in circular machine designs and relate them to their characteristic transfer matrix elements, the phase shift through them, and the Courant-Snyder-Twiss parameters, /3, QI, and 7. Next we discuss the systematics of some elementary phase ellipse matching problems between optical modules.
- Work supported in part by the Department of Energy, contract DEAC03-76SF00515 and by the National Sciences and Engineering Research Council of Canada. Presented at the Second International Conference on Charged Optics, Albuquerque, New Mexico, May 19-23, 1986
The report ends with a discussion of second-order optical modules and how they are used to provide the momentum bandwidth needed for the design of a typical circular machine.
2 FIRST-ORDER OPTICS
2.1 NOTATIONS AND DEFINITIONS
As in TRANSPORT,[‘][” I31we represent the position and direction of travel of a particle via a six-dimensional vector:
x=
The coordinates x and y represent, respectively, the horizontal and vertical displacements at the position of the particle, and x’ and y’ represent the slopes of the projection of the trajectory in the same planes. The quantity 1 repre- sents the longitudinal position of the particle relative to a particle travelling on the reference trajectory with the reference momentum. The last coordinate 6 = (p - po)/po gives the fractional deviation of the momentum of the particle from the central design momentum of the system. In first-order optics, the motion is described by the following matrix equa- tion: 6 xi = (^) c Rjxoj 3 i= 1,2,...,6 (^) (2.1) j=l Equation (2.1) can also be rewritten in compact matrix notation as
X=RXo.
In optics studies it is customary first to study the properties of a set of optical elements by restricting the momentum of the test particles to one value (called the reference momentum), and then to study the properties as the momentum is changed. The elements Rij of the matrix R that contain one subscript with the value 6 are called chromatic terms. The elements Rij for which no subscript is equal to 6 are referred to as geometric terms.
of x(s) with respect to s yields
x’(s)= d-
-Lp,O COS($J(S)+ 4) - &$J(sin(ti(s) + 4)&) P(s) 2
sin(llr(s)+ 4) )
where we now define the function o(s) by
P’(s) = -24s)
Alternatively x’(s) can be written in the form ,. ~. ~. x’(s) = +Ertfq cos(x(s) + 4)
where x(s) satisfies the relation
ta+f+> - x(s)) = -&
or equivalently
sin($(s) - x(s)) = - x47&m and the function 7 (s) is defined by
1+ o(s) 7(s) = p(s) -
The functions ,0(s), o(s), and 7(s) are all periodic with the period L, where L is the length of the closed machine. Consider the values of the solution for x and its derivative at successive revolutions at a fixed point s. We can describe the motion at position s by plotting the values of x and x’ in the “z-phase plane”. Eliminating the trigonometric functions from the expressions of x(s) and x’(s) yields, after some manipulation, the ‘Courant-Snyder’ invariant15’ (the equation of the machine ellipse). The result is:
7(s)x2 + 24s)xx’ + P(s)x t2 = E ,
which shows that the positions (x,x’) of a particle at the coordinate s upon successive turns lie on an ellipse. The parameters QI, ,B, and 7 are sometimes referred to, in the literature, as the Twiss parameters.[” Similar equations may be derived for the y, y’ phase plane. We assume, in this discussion, that midplane symmetry is applicable and therefore there is no linear coupling between the x and y phase planes.
2.2.1 The Machine Ellipse
This ‘machine ellipse’ may also be represented in a matrix form as follows:
T = P(s) (
-44 7(s)^ )
(2.2)
where T has a determinant equal to 1. The equation of the ellipse characteristic of the machine may then be written in the matrix form
XtT-lX = E where (^) P-3)
The area of the ellipse is KITE.We^ can^ compute^ the^ maximum^ x excursion^ xmax and the maximum x ’ excursion x ‘max. They^ are given^ by^ the^ expressions:
Xmax = x6 E^ , X’ma=fi.
From the explicit equation of the ellipse one can also obtain the coordinates of the intercepts with the axes:
E Xinter = r ’
X ‘inter = ;.
and from these expressions one can deduce alternative expressions for the area of the ellipse:
Area = XE = XX~~X ‘inter = zzinterz lrnax.
This result can be generalized to dimension n. For^ n^ dimensions^ c is the product of one intercept, one maximum and (n - 2) maxima of subspace intercepts. Figure 1 illustrates these points in two dimensions.
2.3 THE RELATIONSHIP BETWEEN THE BEAM ELLIPSE AND THE
MACHINE ELLIPSE
Consider a closed machine that is characterized by the ellipse El with emit- tance e and area Al, as shown in Fig. 2. Let Pr denote a point on that ellipse and let 0 denote the origin of the axes. After successive turns around the machine the point Pr will reappear at Pz, P3, etc.
5-84 (^) I 4809All Fig. 2. The Superposition of Beam Ellipses Ez and E3 with a Machine Ellipse El.
Consider now an ellipse Es inscribed in El with a contact point at Pr. Let the ellipse E2 represent a beam of particles circulating in the machine. Ellipse E becomes, after one turn, ellipse E3 with contact point Pz. Ellipses E2 and E have the same area.
When the beam ellipse E2 is concentric and similar to the machine ellipse El, the beam is said to be matched to the machine. In this instance the beam reappears on successive turns as the same ellipse, but the individual particles in the beam rotate around the ellipse as did the points Pr etc.
Let us find the transfer matrix which transforms the machine ellipse defined by the input values pr and CY~at position sr into an ellipse with the values p and 02 at position sz.
Consider again the solutions as given by the Floquet theorem:
4s) =amos(ti(s) + 4) ,
44 = - P;.) \r(
- a(s)cos($(s)+ 4)+ sin($(s)+ 4) 1
.
Expanding the trigonometric functions and simplifying the notation gives
5 =@(cos+cos4- sin$sin+) ,
6’ crcos$cos~$-osin+sin~+sin$cos~+cos$sin$).
The point having $ = 0 is assumed to be associated with the values pr and o and x1 and xl’; these values then satisfy the following relations:
Xl’= - d
$’ 1 arcos4+sincj).
Denoting by /32, cy2, x2, and x ‘2 the values associated with $ nonzero, and eliminating cos 4 and sin 4 from the previous four equations, one gets
x2 =x \i
$cosd + w sin$,>+ xl’&Zsin ti , 1
x2’ =x
-a2 cos + - sin 1c,- o301 sin $ + or cos + m
- Xl’ F(cos$^ -^ ozsin$)^. 2
From the above equations we deduce the transfer matrix between position 1 and position 2 to be
R=
A$ + or sin A+)
- (1 +^ aroz)^ sin A+^ +^ ((~12-^ al)^ cos A$ m d-
$(cos A$ - (~2sin A$) 2 (
where A$ is the phase shift between position sr and 82.
a) A thin lens is characterized by sr = sz so that A$ = 0.
b) If Rrz = 0 (point to point imaging) then A+ = nr.
c) If Rir = 0 (parallel to point imaging) then tan A+ = -l/al.
d) For a drift of length L, Rrz = L and sin A$ = .L/@&.
It is perhaps worthwhile to comment on the meaning of ‘phase shift’ in a circular machine.
-M
I--
R= ~ 1 F 5-
.O
4
1 M
1
5399A
Fig. 3 Phase Shift for Point to Point Imaging
Consider Fig. 3 where we show a single lens imaging from point 1 to point 2. This corresponds to the matrix element Riz = 0. From the figure and Eq. (2.5) we can conclude that the phase shift is z. In this case we only need to know that R12 = 0 in order to conclude that the phase shift is zero or n?r. With the further information contained in Fig. 3 we know that the answer is z. No additional information about the incoming phase ellipse is necessary.
Now, in contrast, consider Fig. 4 where again we have a single lens but with the matrix element Rri = 0. This corresponds to parallel to point imaging.
Comparing again with Eq. (2.5), we^ discover^ that^ we^ need^ to^ know^ the orientation of the incoming phase ellipse, or, at the entrance of the module in order to evaluate the phase shift through the module.
Fig. 4 Phase Shift for Parallel to Point Imaging If w =^ 0, corresponding^ to^ an upright^ ellipse,^ then^ the^ phase^ shift^ is
otherwise
tan(A$) = -&
/--*+=n/2 + a2=
Fig. 5 Phase Shift for Point to Parallel Imaging As a third example consider Fig. 5 where R22 = 0, corresponding to point to parallel imaging. In this case we readily conclude that we must have a knowledge
(2.8) gives
tanA+ = R12^ L RI& - R12w = PI - La1 ’
Consider the extreme point on the beam ellipse shown in Fig. 6. As the beam travels through the drift space, this point will be displaced by Ax given by
where ,Bw is the /3 value achieved at the point where the beam has a waist,
xl= constant (^) y =constant
5-W Ax=Lx’^ Aa=-Ly^ 39WA
Fig. 6. The Transformation of an Ellipse through a Drift (Field-free) Space.
3.0.2 A Thin Lens
A focusing thin lens has the following transfer matrix:
R=
from which one derives
t :.
x2 = x1 = a constant and^ Az’=~~‘-z!‘=-$.
The Twiss parameters transform according to formula (2.7),
which gives
pz = /?I = a constant (^) and Acx=cY~-cY~=~. Pl
The relation (2.8) gives
tan A~,!J= R &lPl - R12a =
and so A$ = 0 because the integral
J
ds AlCl= P(,)=O
since the thin lens has a length .equal to zero. The^ transformation^ of an ellipse through a focusing thin lens is illustrated in Fig. 7.
r t
I XI ’ int
= (^) $ = a constant
XI =--.-+x1XI 2 I= I
= constant
x = constant 0 = constant
3989A
Fig. 7. The Transformation of an Ellipse through a Focusing Thin Lens.
which gives
/32 = pi = a constant and Ao=02-ol=
p1 sin CI! P - As for the thin lens, the relation (2.9) shows that A$ = 0 for the zero length dipole.
3.1 STUDY^ OF^ SIMPLE^ USEFUL^ COMPOSITE^ MODULES
Using the basic elements discussed in the previous section we shall now explore some typical composite modules.
3.1.1 Basic Focusing Module
If a focusing thin lens of focal length F is placed between two drifts of length F, the transfer matrix for the composite system is
From the matrix R we observe that angles are transformed to displacements and displacements to angles as follows:
x2 = FxI’ and 372 L-5- F’
From the relation (2.7) we have
from which P2 = F271 and^ 03.^ =^ -a^. Relations (2.8) and^ (2.9)^ yield
tan A+ = -$ and^ sinA$^ =^ &&
from which we can conclude the following result: If ~1 = cll2 = 0 then, since sinA$ > 0, we must have AII, = 7r/2 and F = dm.
This relation links the lens focal length F and the length L = 2F of the
. module^ to^ the^ magnitude^ of the^ /3 values. Practical tw+dimensional modules based on this concept are typically achieved by symmetric triplets or by quadruplets, as shown in Fig. 8. For the triplet, the focal length is different in the two phase planes (x,x’) and (y, y ‘) because of basic properties of triplets. If it is required that Fz =^ F,,^ then^ a symmetric^ quadruplet^ array^ of quadrupoles may be used as illustrated in Fig. 8.
-.
F,= Fy
6-84 fl^ f2^ f2^ fl^ 4809A
Fig. 8. A Triplet and a Quadruplet Lens System Possessing Parallel to Point and Point to Parallel Imaging in Both Planes.
3.1.2 The FODO Array
The FODO array is perhaps the most common building block used in the design of machine lattices and beam lines. Its structure is illustrated in Fig. 9 when it is composed entirely with quadrupoles. A FODO array with interspersed dipoles is discussed in Ref. 7. It is informative to study the FODO array at two different observation points in order to better understand its basic properties.
- First case: The cell begins and ends at the center of a lens, then the transfer matrix for the x and y planes is obtained by the following multiplication:
defocusing lenses is given by
Pmax 1 +^ sin(P/2)
- P (^) min = 1 - sin(j.k/2) *
Note that this ratio is independent of the length of the cell.
- Second case: If we now begin the FODO array in the middle of one of its drifts; the transfer matrix for one cell is given by
R=(i“i”)(*i,f:)(ii:) (,:,,Y)(i-“i”);
then
L
c+cYs ps
2L--
R= 4f --YS c -^ as
from which we obtain L cosp= l-- ( 2j2^ ) ’
which is the same as in case 1, but
&Y= & (2 -^ sin2(p/2))
and
ffz,y = F
2 sin(j.4/2) sinp.
The last two relations show that at this location we have the result
Pz = Py and^ a!, =^ -cxy^ ,
which is the same property possessed by a thin lens quadrupole.
A case of particular interest is obtained when ~1 = 7r/2. This corresponds to (L/f) = fi. This FODO cell is then often referred to as a ‘quarter-wave’ or X/4 transformer and is shown schematically in Fig. 10.
*--+L/2-j- L+L/2+
f f 6 - 84 4809A
Fig. 10. The X/4 Transformer.
The transfer matrix R of this quarter-wave transformer is
and we have the interesting property R11 = (^) -R22 and R11, and R22 both change signs between the x and y planes. This is a useful cell for phase space matching as will be discussed later.
3.1.3 A Telescopic System
The optical system illustrated in Fig. 11 is called telescopic.
Its transfer matrix is given by
R=(tl:)(-ll/F. Y)(i“:“)(-:, :)(i T)
= -&/FI 0 -F;,F2)^ =^ (-,”^ -P/M)^ ’^ (3.1)
