Numerical third-order transfer map for solenoid

Nuclear Instruments and Methods in Physics Research A298 (1990) 441-459 North-Holland

441

Numerical third-order transfer map for solenoid Alex J. Drag Center for Theoretical Physics, University of Maryland College Park, MD 20742, USA

A method is described for computing numerically the Lie algebraic transfer map for a solenoid, including fringe-field effects, through third order . The method can easily be extended to fifth and even higher order if desired . Examples are given for a simple imaging system and a sextupole corrected spot forming system .

1 . Introduction The purpose of this paper is to describe the computation of the transfer map for the solenoid through third order and its implementation in the charged particle beam transport code MARYLIE 3 .0 . These computations are done using Lie algebraic methods, and a familiarity with these methods is assumed [1-4] . Identical results could also be obtained using differential algebraic or other methods [5] . Our treatment is purely classical ; for a quantum treatment of the paraxial problem, see ref . [6] . Our goal is to write in Hamiltonian form the equations of motion for trajectories in a solenoid, to express and expand this Hamiltonian in appropriate dimensionless coordinates, and to compute the third-order transfer map from this expansion. This is done in sections 7 and 8 . Sections 2-6 lay the ground work required for this effort, and sections 9 and 10 give simple examples of the use of a solenoid . A concluding section summarizes the results of this paper .

2. Lagrangian and Hamiltonian The Lagrangian for the relativistic motion of a particle with charge q and rest mass m in a magnetic field described by vector potential A is given by the familiar relation

be found by the usual procedure . One finds for the conjugate momenta the results

aL

PX= ôz =qAX+

mz 2 1-(z +y 2 ±22) /c 2 '

Py=

1-(X2+y2+z2)/c2 ,

PZ=

Y

= qAY A+

ôLZZ _

mi

1 _ (x2+y2+i2)/c2

Consequently, the Hamiltonian is given by the relation H=zpx +ypy + 1pz

=

-L

m2C4+c2[(PX-gAX)2+(Py_gAy)2+p~2] .

3. Introduction of z as an independent variable To proceed further, it is useful to replace the time by z as an independent variable . This results in a new Hamiltonian that will be denoted by the symbol K. Correspondingly, the time becomes a dependent varia-

L= -mc 2 J1-v 2 /c 2 +qv-A . We assume that the magnetic field is such that it can be derived from a vector potential for which only A X and A y are nonzero. Then, using the Cartesian coordinates shown in fig. 1, the Lagrangian can also be written in the form

L = -mc 2

1-( .z 2 +y 2 +i 2 )/c 2

+q*A X

+OA Y .

(2)

The Hamiltonian corresponding to eq . (2) can now

Fig. 1 . Coordinate system for the solenoid.

0168-9002/90/$03 .50 C 1990 - Elsevier Science Publishers B .V . (North-Holland)

V. THEORETICAL OPTICS

442

A J Dragt / Numerical third-order transfer map for solenoid

ble, and a momentum p, conjugate to the time is introduced. Following standard procedures [1], one does this by first writing the relation then solving this relation for pz to find the relation Pz=(P1/c2 - m2c2-(Px-qAx)2

-(

(Py

qAY)2)

1~2

,

(6)

and finally defining K by the relation

g = - Pi =- (Pilc 2 - m2c 2- (Px - gAx) 2 - (Py - qAy)

2)i/2.

(7)

Note that in finding eq. (6) it is necessary to select the sign of a square root . This is done in such a way that 2 is positive, as is obviously the case for orbits of interest .

Eventually we will need to expand the Hamiltonian in a power series about the design orbit (the design orbit for a solenoid is a straight line down the axis). With this end in mind, we devote this section to the expansion of A x and Ay, the vector-potential part of the Hamiltonian . Since we are interested in third-order effects, we will need to carry out an expansion through terms of fourth degree . The specification of the magnetic field of a solenoid, including fringe fields, requires fairly complicated mathematical expressions for all three components of B(x, y, z). However, it can be shown that if BJ0, 0, z) (the on-axis longitudinal component of B) is known, then Maxwell's equations plus the requirement of axial symmetry are sufficient to determine all components of B everywhere [7,8]. In particular, B can be obtained from a vector potential A (B = V x A) given by the expressions p2)>

Aj, =xU(z, p2 ),

(8)

Az = 0 .

n=o

where b2n p2

b2n

( -1 ) n (p 2 ) n b2n f 22n +ini(n + 1)t}

and

p2

az2n

(9)

>

(12)

Here L is the length of the body of the solenoid, and X is a characteristic length that describes the leading and trailing fringe-field falloff. As described below, this model has physically reasonable properties that should be adequate for at least preliminary calculations . It is the model currently available in MARYLIE [4]. However, other models could be employed with equal ease if one wished to make detailed calculations for a particular solenoid [9]. The soft-edge bump function has the properties bump(z, A,

L) = 1

bump(z, X, L)=0

f

.0

bump(z,

for

z E [0, L],

elsewhere,

(13)

a,

L)

dz

= L.

(14)

In particular, from the results above one has the relation

f ~ B, (0, 0, z) dz =BL .

(15)

Specifically, the soft-edge bump function is defined in terms of a soft-edge "signum" function by the relation bump(z, X, L)=2(sgn(z, A)-sgn(z-L, A)) .

(16)

The soft-edge signum function is in turn defined by the relation sgn(z, X)=tanh(z/Ä) .

(17)

The soft-edge signum function becomes the true (hard-edge) signum function in the limit that the characteristic length X goes to zero, lim sgn(z, X) = sgn(z) . a-o

sgn(z) = 1 if

z > 0,

sgn(z) = 0 if

z = 0,

sgn(z) = -1

denote the quantities

a2n Bz() , O+ Z)

=x 2 +y 2.

,

B, (0, 0, z) =B bump (z, X, L) .

(18)

Recall that the true signum function has the definition

Here U is defined by the relation U-

For simplicity, we will use for the on-axis longitudinal field a soft-edge "bump" function model of the form

bump( 12L+w, X, L)=bump(ZL-w, A, L),

4. Expansion of vector potential

Ax= -yU(z,

5. Choice of on-axis field

if

(19)

z < 0.

(10)

In this same limit the soft-edge bump function becomes the hard-edge bump function,

(11)

lim bump(z, À, L) =bump(z, L) . ~, ->o

(20)

443

A.J. Dragt / Numerical third-order transfer mapfor solenoid

The hard-edge bump function has the properties bump(z, L) = 1 for

z E (0, L),

bump(0, L) =bump(L, L)

(21)

= 2,

bump(z, L)=0 -elsewhere . It follows that the characteristic length X controls the rate of falloff of the fringe fields . The fringe-field region is large if X is large, and vanishes as X goes to zero .

We are now able to introduce a full set of deviation variables (J, 11, T; Pt, Pr,, PT) according to the prescription x = e, Y = r1, t = (z/v°) + T, PX = P£, Py = P >

(28)

Pt = Pr + PT

6. Deviation variables and their Hamiltonians The purpose of this section is to define regular and dimensionless deviation variables, and to find the Hamiltonians describing the motion in these variables. We begin by the introduction of regular (not yet dimensionless) deviation variables. Since the design orbit is a straight line down the axis of the solenoid, we may introduce transverse deviation variables ~, rl, by the definitions

We must still verify that eq . (28) is a canonical transformation, and find the transformed Hamiltonian . It is easily verified that the transformation of eq. (28) can be obtained from the transformation function F2 defined by the expression F2 = xPg+YP17 +~t - (zlVo)~(PT+Pl) .

(29 )

Specifically, following the standard rules [101 gives the results

(22) To complete this set, we need to find the time as a function of z along the design orbit. According to Hamilton's equations we have the result _dt__8K __ dz ôp,

i

c2

8F2

(P2lc2)_tn2c2_(PX+Py)

In particular, evaluating eq . (23) on the design orbit gives the result ° _dt __ P (24) dZ design orbit C 2 ( p o2/C 2) - m2C2 Here p° is the value of pt on the design orbit. Specifically, p° is given by the relation =

H

-

design orbi - -

2 2

m2Ca +poC

(25)

,

where pa is the design momentum. It follows from eq . (24) that v°, the velocity on the design orbit, is given by the relation o vz

dz dt

_ design

_ _dt orbit - ( dz

1 design orbit)

__

_

C2po

Pt

(26)

(Note that v0 > 0 as desired and expected.) Consequently, the time along the design orbit is given by the relation t ( Z ) = Z/v°.

apn =Y,

Pt (23)

A0

âF2

(27)

vo >

(30)

Px= ax =PE,

Pl

8F2 ôt =Pt + PT'

Evidently eq . (28) agrees with eq . (30) . Consequently, the transformation to deviation variables is indeed canonical. Let H (not to be confused with the H of section 2) be the "new" Hamiltonian associated with the new (deviation) variables . It is connected to the "old" Hamiltonian K given in eq . (7) by the standard relation [101 :

For many purposes it is useful to describe trajectories in terms of dimensionless variables. The last task of this section is to define these variables and their associated Hamiltonian . Let t° be some convenient scale length. We have found it useful to introduce dimensionless variables (X, Y, T; P., Py , P,) defined in terms of the deviation V. THEORETICAL OPTICS

444

A .J. Dragt / Numerical third-order transfer map for solenoid B02 XPxYPy

variables of the previous paragraphs, and the scale length, by the rules

3Bô X ZPy + Y ZPy ) + 16e

X = ~/e, Y= q/e, T = cT/C, Px = P¬ /P., Py = P,,/Pa,

+ (BZ- B3)( X2+Y2)(XPy -YPx) 16C 4 + (Bô -4BoB2 )(X + 2X2 Y 2 + Y4 ) 128e - P 2 P (p2+Py ) 2 + (3 ) We

PT

(32)

= PTl(Poc)

These are the standard MARYLIE variables [4] . The transformation of eq . (32) is not canonical since it does not preserve Poisson brackets. However, it is easily verified from eq. (32) that the new and old Poisson brackets all differ from each other only by the common factor (poe). In this case it can be shown that the evolution of the new dimensionless variables can still be described by a new Hamiltonian [111 . Call this "new" Hamiltonian K (not to be confused with the Hamiltonian K of section 3) . Then K is connected to the "old" Hamiltonian H given in eq. (31) by the relation (33)

K = H1(Poe) .

7. Expansion of Hamiltonian The purpose of this section is to expand the Hamiltonian K given by eq. (33) as a power series in the dimensionless deviation variables of eq . (32), and to comment on various properties of this expansion [12] . Upon expanding K, one finds the results K=Ko+K,+K2+K3+K4 +''',

(34)

where the homogeneous polynomials Kn of degree n through terms of degree 4 are given by the relations Ko = 1/(P2Y2e),

K2 =

21

+

y _

P2

21(XPy - YPx) + 8e( X2+Y2 )

2P2Y2e,

PT (Px + Py ) Bo P,(XPY -YPx ) K3 2P?' 2,ße -

+

Bp(X o T 2 +Y 2 ) P* + 283Y21, 8PC

Px + 2 PXPy + Py K4 -

81

Bo (Pz + Py) (XPy - YPx ) 4e

Bo (X2PX + Y 2 Py ) + 16e

4e

(3 - ,ß 2 )Bo pT (XPy - YPx ) 4P 2e Z + (3 - PZ )B0P2(X +Y Z ) 16p 2é

+ (5

-PZ) . 8P4Y2e

(3s)

Here the quantities Ben are dimensionless strengths and dimensionless longitudinal gradients defined in terms of the b2n by the relations e2n+ 1 B2n = b2n1G,

(36)

where G is the magnetic rigidity Bp of particles on the design orbit . The rigidity G is given by the relation G = Po/4,

( 37 )

and has units of tesla meters . Also, P and y are the standard relativistic factors for the design orbit . They are related to po and p°, the design momentum and negative design energy, respectively, by the equations Po = PYmc, p°= -ymc 2 .

(38)

As is evident from eq. (35), the Hamiltonian K has dimensions of inverse length . Note also that K, vanishes as it should. If it did not, the phase-space path obtained by setting all deviation variables equal to zero (the design trajectory) would not be a solution of the equations of motion . We next remark that the constant piece Ko is irrelevant to the actual motion, and need not be computed . It is presented here only as an aid for those who may wish to carry out the expansion of eq . (35) independently . Finally, we observe that the expansion of eq . (35) can be written in the more compact form 2 __ P Z _ Bo(Q X P) - e~ BôQ P2 K2 21 2C + 8e + 2P2Y2e, K3 = PTK2/P, K =

BO(P . Q) 2 - B"P 2 (QXP) - ez 8e 4e 8e 2 (BZ - Bô)Q 2 (Q X P) - ez 3B.P 2Q + 16e + 161 (p2)2

(3-P2)p2P2 (Bô - 4BaB2)(Q2)2 4P21 + 128e +

A .J. Dragt / Numerical third-order transfer map for solenoid

(3 - ß 2)B0*(QXP) . ez 4ß 2e + (3 - ß 2 )B0P,Q 2 + (5- ß2)P° . 16ß 2C 8ß'YZe

(39)

Here the quantities P and Q denote the two-dimensional vectors given by the relations P = PXex + Pyey, Q=Xe.+Yey.

(40)

This form makes it manifest that K is rotationally symmetric about the z-axis, as should be expected based on the underlying axial symmetry of a solenoidal magnetic field. It also shows, since K3 is proportional to Pr, that all second-order aberrations of the solenoid transfer map are purely chromatic .

The transfer map M for the solenoid is found by following the Hamiltonian flow generated by K from some initial z-coordinate z' to some final z-coordinate z f. It can be shown that the transfer map obeys the equation of motion [2]: M=

M : - K: .

(41)

Here a dot indicates differentiation with respect to the independent variable z. What is needed is the solution to eq. (41) evaluated at z = zf subject to the initial condition M = I at z = z' . The solution to eq. (41) for the Hamiltonian K given by eq. (35) or (39) can not be written in analytic form, but instead requires numerical integration . We will turn to that task shortly . First, however, we will discuss an aspect of M that can be derived analytically . We begin by observing from eq . (39) that K can be written in the form (42) K = - (Bo/2e) JZ + K rest where Jz denotes the quantity Jz =(QXP) -ez =XPy -YPx,

Moreover, it is easy to check that the Lie operators Jz : and : K: commute. This is a simple consequence of the rotational invariance of K as is manifest in the form of eq. (39). It follows that M can be written as the product of two commuting maps in the form M = MrotM rest, (45) where the rotational part Mro' and the remaining part M rest obey the equations of motion M rot = M rot :(B./2C)Jz : , (46) Mrest =

(43)

and K rest denotes the remaining terms in the Hamiltonian. It is easily verified that Jz has the properties :J, :X = [XPy -YPX , X] =Y, :J :Y= -X, :Jz:PX =Py , :Jz:Py = -Px . Consequently, as the notation is meant to indicate, the Lie operator :J, : is the generator of rotations about the z-axis .

M rest : -

K rest :

.

(47)

The solution to eq. (46) can be written in the closed analytic form Mrot = exp(9 : (48) J, :), where B is the angle given by the relation

0=f zr (B./2e) dz . Z

8. Computation of transfer map

445

(49)

In particular, if the interval (z', z f ) contains all of the solenoid including (to the accuracy desired) all of the fringe-field region, then it follows from eqs. (14), (49), and the definitions (10) and (36) that eq. (49) for 0 can also be written in the form 0 = (BL/2G ). (50) Let ~ be a six-dimensional vector whose entries are the dimensionless phase-space variables, e= (X, Y, -r ; P., Py , Pr ) . (51)

Also, consider initial and final phase-space coordinates ~' and ~f connected by the relation yf = Mrotyt, (52) Here the function J. appearing in eq. (48) is to be viewed as a function of the variables ~'. Then, using the properties in eq. (44), one finds the results X f = X' cos 0 + Y' sin B, (53) Y f =- X' sin g+Y'cos0. From these results, and from looking at fig. 1, it follows that Mrot rotates phase-space points counterclockwise about the z-axis by angle 0. Next we examine the effect of Mrest . From eqs. (39), (42), and (43) one finds that Kzest is given by the relation 2 B2Q2 PT2 Krest = P (54) + 2ß272e . 2e + 8C It follows that, in paraxial approximation, M rest has identical effects on the dynamical pairs (X P.) and (Y,PY ). In particular, Mrest in paraxial approximation does not produce any coupling between the X, Y-planes. We conclude from the above discussion and the factorization (45) that, in paraxial approximation, the coupling between planes produced by a solenoid consists of a V . THEORETICAL OPTICS

446

A .J. Dragi / Numerical third-order transfer map for solenoid

sol Fig. 2. Schematic of simple solenoidal imaging system .

simple rotation described by eqs. (48) and (50) . This is consistent with the spiraling motion expected for a positively charged particle in a uniform magnetic field directed along the z-axis. We now turn to the full determination of M by means of numerical integration . What is desired is a factorized representation for M in the standard MARYLIE form e M(55) = e fz : f3 e f4 .

What is needed is the solution to eqs. (57), (60), and (61) evaluated at z = z f subject to the initial conditions R = I, g3 = g4 = 0 at z = z' . This solution is computed numerically by MARYLIE using special-purpose builtin GENMAP routines [13-15] . GENMAP, when working through third order, integrates numerically 224 simultaneous first-order differential equations [16] . The equation count is as follows : 6 for the design orbit, 36 for eq . (57) for the matrix R, 56 for eqs . (60) for the coefficients of 93, and 126 for eqs. (61) for the coefficients of g4 . Strictly speaking, the numerical calculation of the design orbit is not necessary for the case of a solenoid since this orbit is known analytically . However, GENMAP is designed to handle the general case where the design orbit also needs to be determined numerically. The numerical integration is begun using a seventhorder Runge-Kutta routine, and continued using an eleventh-order Adams routine [16,17] . The number of integration steps can be specified by the user, and results that are essentially accurate to machine precision can be found if enough steps are employed .

It is convenient to begin with the reverse order factorization M=e 94= e- 93 e sz .

(56)

Once the terms in the factorization (56) have been found, the passage to the factorization (55) is a simple matter of the Lie-algebraic concatenation of maps . The virtue of the factorization (56) is that is leads to a simple standard set of differential equations for the transfer matrix R (corresponding to the factor e92 ), and for the polynomials g3 and g4 . It can be shown from eq . (41) that R obeys the differential equation Here S is a symmetric matrix defined in terms of K 2 by the relation K2

I - 2 Y-' Sab, a , b , a,b

(58)

and J is the standard symplectic matrix J=( _0

1 ).

(59)

Also, it can be shown that g3 and 94 obey the equations 93= -K 3 nt , 94 -_

-K ,nt 3 K4nt + . 93 : 2

(60) 61

Here the interaction Hamiltonian Kant is defined by the relation Kint(y' (62) z)=Ka(R(z)~, z) . For further detail, see ref. [2] .

9 . Simple example of imaging system Consider the simple imaging system depicted schematically in fig . 2 . It consists of a solenoid preceded and followed by suitably chosen drift spaces of equal length . Appendix A shows a MARYLIE run set up to model this simple system . This MARYLIE run computes field profiles, writes out the transfer map for the system, and traces rays using this map . Figs. 3 and 4 show the profiles of the on-axis longitudinal field bo and its second derivative b 2 as computed by MARYLIE for thus system . How well does such a system perform when secondand third-order aberrations are taken into account? Inspection of the transfer map displayed in appendix A shows that the system is imaging . Also f3 is purely chromatic (as is consistent with the form of K 3 as given in eq . (39)), and several terms in f4 are sizable . Correspondingly, all second-order aberrations are purely chromatic, and there may be important geometric third-order aberrations [18] . Suppose an object consisting of the word MARYLIE as shown in fig . 5 is imaged by this system . Fig. 6 shows the result as given by MARYLIE ray tracing through third order. At least three features of the image are worthy of comment. First, the image is essentially inverted as would be expected from the light optics analog of fig . 2 . Second, the image is rotated, and careful numerical computation shows that the amount of rotation agrees with that predicted by eqs. (50) and (53) .

447

A.J. Dragt / Numerical third-order transfer mapforsolenoid 0 020

0015-, bo(z),

Tesla 0 010 -

0 005 -

z, meters

Fig . 3 . Profile of

bo (z) = BJ0,

0, z) (tesla), for the solenoid used in the example of appendix A. In thus example, L = 0 .10 m, X = 0 .01 m, and B = 0 .02 T . The solenoid is centered at z = 0 .10 m .

Finally, the image clearly suffers from third-order geometric aberrations as expected .

scope . There is also a sextupole placed at the crossover between the two solenoids. For the moment, this sextupole should be ignored . Appendix B shows a MARYLIE run set up to study the performance of this system . This run computes various transfer maps and traces rays by applying these maps to an initial conditions file to produce a final conditions file . As indicated in fig . 7, the incoming rays are selected to be parallel to the optical axis (PX = Py = 0) and to lie

10. Simple example of spot-forming system As a second simple example, consider the spot-forming system shown in fig . 7 . It consists of two solenoids and suitably chosen drifts. Such a system might serve as the basis for a scanning transmission electron micro-

100

50

0

-50

-100 000

Fig. 4 . Profile of

005 b2(Z) = B~'(0, 0, z)

010

015

020

(T/mz ), for the solenoid used in the example of appendix A . V . THEORETICAL OPTICS

A.J. Dragt / Numerical third-order transfer mapfor solenoid

448

40-

20 -

y,

meters -20-

-40- I -005

000

1 0 10

005

Fig. 5. Object to be imaged by the simple solenoidal imaging system example of appendix A.

06~ y, meters

04502-

00-

ô

meters

0 -1

02 -

04~ x, meters

-06 J -06

x, meters -04

00

-02

,T, . . 1 02

04

.

-10 06

Fig. 6. Image of the word MARYLIE produced by the simple solenoidal imaging system example of appendix A.

t

i

tt~2

i

I 0 x70-s

-10

I

5

Fig. 8. Plot of initial conditions for inconung rays of spot-forming system.

3

i

I

Fig. 7. A simple spot-forming system consisting of two solenoids and suitably chosen drifts .

A.J. Dragt / Numerical third-order transfer map for solenoid

449

As explained earlier, f is purely chromatic, and has no effect on rays having the design energy . Moreover, the first factor in Ml describes the linear (paraxial) part of M. It follows that the intermediate variables describe paraxial results . In particular, for the rays shown in fig . 8, one has the results .

y, meters

Xtnt = lrint = 0

(65)

since the system is designed to produce a perfect spot in the paraxial approximation . Now consider the second step in which the map M2 given by M2 = e f4

(66)

acts on the intermediate variables to produce the final results x, meters -6

-4

-2

0 X10,

2

4

6

Fig . 9 . Final focal spot patterns produced by the simple spot-forming system of fig . 7 .

~afm = M2 yânt .

(67)

Expanding eq. (66) and use of eq . (67) gives, through third order, the relation y ftn= (1+ :f4 :) ~mt =y ant ] . a +[f4 > Jant

(68)

In particular, in view of eq . (65), one has the results on two cylinders having radii of 50 and 100 [ m, respectively. They are also selected to be monoenergetic. Fig . 8 shows the result of plotting the X and Y entries of the initial conditions file for these rays . Altogether, 200 rays are shown . As advertized, the incoming rays do indeed lie on two cylinders . As an indication of the performance of this spot-forming system, fig. 9 displays the X and Y entries of the final conditions file for rays traced to the paraxial final focus image plane . Evidently, the rays are not brought to a point focus, but instead again form two circles . Moreover, while the ratio of the radii of the initial circles of fig . 8 is 2 to 1, the ratio of the radii of the two final circles of fig . 9 is 8 to 1 . This is what is to be expected from third-order geometrical aberrations since 2 3 =8 . Not all aberrations are detrimental to the performance of a spot-forming system. Suppose the transfer map for the system is written in the form of eq . (55) . Then, as has been seen earlier, fs has only chromatic terms and thus has no effect on monoenergetic rays . Moreover, most terms in f4 have no effect . To see that this is so, suppose that the map (55) is viewed as a two-step process [19] . In the first step the map Ml given by Ml = e -f2 e f,

(63)

acts on the initial variables to produce a set of intermediate variables, mt

m

= My l, a

(64)

Xfin = [ f 4 lrfm = [ f4

Xmt~

(69)

lrmt ] .

Evidently, X ftn and Y ft n fail to form a spot only if the Poisson brackets in eq . (69) are nonvanishing when evaluated for the intermediate paraxial results . It follows from the nature of the Poisson bracket and eq. (65) that in order to produce a nonvanishing result in eq . (69), f4 must have entries that contain powers of Px and Py., and no terms in X or Y. Also, by reflection symmetry, such entries in f4 can contain only even powers of P and Py . Consequently, in the MARYLIE monomial labelling scheme, the only possible detrimental terms in f4 for a solenoidal spot-forming system are f (140) = f (040000) =coefficient of Px , f (149) = f (020200) = coefficient of PxPy ,

(70)

f (195) = f (000400) = coefficient of P4 . Finally, by the axial symmetry of a system consisting only of solenoids and drifts, the terms Px, P2Py, and PY can occur for such a system only in the combination 2 2 (P ) . Inspection of the transfer map exhibited in appendix B shows that this is indeed the case . That is, one has the relations f (140) = f (195), f(149)=2f(140) .

(71)

The facts that for a solenoid spot-forming system there are only the three detrimental terms (70), and that V. THEORETICAL OPTICS

450

A.J . Dragt / Numerical third-order transfer map for solenoid

these three terms are related by the two conditions (71), suggest that it may be possible to correct such a system with a single corrector . This is indeed the case [20] . Suppose, as illustrated in fig . 7, a sextupole is placed at the (paraxial) crossover point between the two solenoids. Then it can be shown that such a sextupole can be used to correct the detrimental terms (70) in f4. Moreover, by placing the sextupole at the crossover, it can be shown that no detrimental geometric terms in f3 are produced [21] . This is verified explicitly in the second part of the MARYLIE run shown in appendix B. Observe that when the sextupole is properly powered, the entries f (140), f (149) and f(195) in the transfer map all vanish . Moreover, all the geometric entries in f produced by the sextupole contain terms in X and Y, and thus are not detrimental . If the incoming rays of fig. 8 are traced through the corrected spot-forming system, one should find (through third order) a perfect spot . This has been verified to be the case . However, now fourth- and fifth-order aberrations come into play . Their treatment is beyond the scope of this paper . Among other things, a proper treatment would require a fifth-order treatment of solenoids, drifts, and sextupoles, and at least a hard-edge model of sextupole fringe fields .

Although fourth- and fifth-order effects cannot be handled with a third-order code, the fourth- and fifthorder effects arising from "cross-couplings" of secondand third-order effects can be estimated numerically by tracing rays through a system element by element rather than simply using the total transfer map through third order . This is done in the last part of the MARYLIE run in appendix B . Fig. 10 shows the spot pattern resulting from this calculation . Note that the pattern is now considerably smaller than that of fig. 9, thus indicating that third-order geometric aberrations have indeed been corrected . Moreover, the ratio of the sizes of the two patterns is now about 16 to 1 . This is consistent with domination by fourth-order aberrations, since 24 = 16 . Such aberrations are produced by an effective fs . Terms m fs are to be expected as the result of map concatenation since the Poisson bracket of a third-order polynomial (f arising from the sextupole) with a fourth-order polynomial (f4 arising from a solenoid) is a fifth-order polynomial . Finally, terms in fb are expected as the result of Poisson bracketing two f4 polynomials . 11 . Concluding discussion A description has been given of the numerical calculation of the third-order transfer map for a solenoid . This calculation is based on the expansion of the Hamiltonian, eq. (35), and the differential eqs . (57), (60), and (61) . Fringe-field effects were modeled using a soft-edge bump profile. Other profiles could have been used with equal ease. If desired, these results could be extended easily to fifth and even higher order [22] . Indeed, a fifth-order GENMAP already exists [23] . Finally, as an application, it was shown how a sextupole can be used in conjunction with solenoids to improve the performance of a spot-forming system .

y, meters

4--~

2--~

2~

Acknowledgements

4,

x, meters

x10`

Fig . 10 . Estimated final focal spot patterns for sextupole-corrected system .

This work was supported in part by US DOE Grant no . DE-AS05-80ER-10666 . I am grateful to Filippo Neri for carrying out independent calculations that verified the correctness of the MARYLIE solenoid GENMAP routines, and to Johannes van Zeijts for help in the preparation of the figures for this paper .

A.J. Dragt / Numerical third-order transfer mapfor solenoid

451

Appendix A #comment This is a MARYLIE run for a simple solenoidal imaging system consisting of a soleniod and two identical drifts . The beam parameters are those for 200 Kev electrons . #beam 1 .6490338242227880E-03 0 .3913901524593803 1 .000000000000000 1 .000000000000000 #menu fileout pmif 1 .00000000000000 12 .0000000000000 3 .00000000000000 Sol sol 0 .000000000000000E+00 0 .200000000000000 200 .000000000000 15 .0000000000000 1 .00000000000000 0 .000000000000000E+00 solpar psl 5 .000000000000000E-02 0 .100000000000000 1 .000000000000000E-02 2 .000000000000000E-02 3 .00000000000000 0 .000000000000000E+00 dr drft 0 .535356676117240 mapout ptm 3 .00000000000000 3 .00000000000000 0 .000000000000000E+00 0 .000000000000000E+00 1 .00000000000000 raysin rt 13 .0000000000000 14 .0000000000000 -1 .00000000000000 0 .000000000000000E+00 0 .000000000000000E+00 0 .000000000000000E+00 trace rt 0 .000000000000000E+00 14 .0000000000000 4 .00000000000000 0 .000000000000000E+00 1 .00000000000000 0 .000000000000000E+00 end end #lines lens 1*solpar 1*sol image 1*dr 1*lens 1*dr #lumps #loops #labor 1*fileout 1*image 1*mapout 1*raysin 1*trace 1*end zi= 0 .0000000000000000E+00 zf= 0 .2000000000000000 ns= 200 di= 5 .0000000000000000E-02 length= 0 .1000000000000000 cl= 1 .0000000000000000E-02 B= 2 .0000000000000000E-02 iopt= 0

V . THEORETICAL OPTICS

452

A.J. Dragt / Numerical third-order transfer map for solenoid

matrix for map is . -8 .21698E-01 1 .84973E-11 -5 .69924E-01 1 .25536E-11 -2 .58151E+00 -8 .21698E-01 -1 .79052E+00 -5 .69924E-01 5 .69924E-01 -1 .25536E-11 -8 .21698E-01 1 .84973E-11 1 .79052E+00 5 .69924E-01 -2 .58151E+00 -8 .21698E-01 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 nonzero elements in generating polynomial are f ( 33) =f ( f( 38)=f( f ( 42) =f ( f ( 45) =f ( f ( 53) =f ( f ( 57) =f ( f ( 60) =f ( f ( 67) =f ( f( 70)=f( f ( 76)=f ( f ( 83)=f ( f( 84)=f( f( 85)=f( f ( 86)=f ( f ( 87) =f ( f ( 90) =f ( f ( 91) =f ( f ( 92) =f ( f ( 95) =f ( f ( 96) =f ( f ( 99) =f ( f(104)=f( f(105)=f( f(106)=f( f(107)=f( f (110)=f ( f(111)=f( f(114)=f( f(l19)=f( f(120)=f( f(121)=f( f(124)=f( f(129)=f( f(130)=f( f(135)=f( f(140)=f( f(141)=f( f(142)=f( f(145)=f( f(146)=f( f(149)=f( f(154)=f( f(155)=f( f(156)=f(

20 11 10 10 02 01 01 00 00 00 00 40 31 30 30 22 21 21 20 20 20 20 13 12 12 11 11 11 il 10 10 10 10 10 10 04 03 03 02 02 02 02 01 01

00 00 10 01 00 10 01 20 il 02 00 00 00 10 01 00 10 01 20 11 02 00 00 10 01 20 il 02 00 30 21 12 10 03 01 00 10 01 20 il 02 00 30 21

01 01 01 01 01 01 01 01 01 01 03 00 00 00 00 00 00 00 00 00 00 02 00 00 00 00 00 00 02 00 00 00 02 00 02 00 00 00 00 00 00 02 00 00

=-0 .655ß0320280398D+01 )= 0 .54617177522132D+01 =-0 .11102230246252D-15 )= 0 .ß7214204611686D+00 =-0 .173ß4718667755D+O1 =-0 .ß7214204611686D+00 =-0 .32959746043559D-16 =-0 .655ß0320280398D+01 )= 0 .54617177522132D+01 =-0 .173ß4718667755D+01 =-0 .9762ß367345340D+00 =-0 .266092633ß0345D+03 )= 0 .65913133073316D+03 =-0 .91ß70955287732D-14 )= 0 .13422056761ß55D+02 =-0 .61925222623999D+03 =-0 .13422056761ß55D+02 =-0 .16557753197390D+02 =-0 .5321ß526760689D+03 )= 0 .65913133073316D+03 =-0 .20ß62384876004D+03 =-0 .17244726416ß08D+02 )= 0 .260656ß5021104D+03 )= 0 .16557753197390D+02 )= 0 .52703543ß59992D+01 )= 0 .65913133073316D+03 =-0 .ß2125675495990D+03 )= 0 .260656ß5021104D+03 )= 0 .20ß74041534917D+02 )= 0 .610622663543ß4D-14 )= 0 .13422056761ß55D+02 =-0 .16557753197390D+02 =-0 .31361096151744D-15 )= 0 .52703543ß59992D+01 )= 0 .157ß2632902935D+01 =-0 .414ß3707281162D+02 =-0 .52703543ß59992D+01 )= 0 .191ß1704835614D-14 =-0 .208623ß4876004D+03 )= 0 .260656ß5021104D+03 =-0 .ß2967414562323D+02 =-0 .66442345797430D+01 =-0 .13422056761ß55D+02 )= 0 .16557753197390D+02

0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 1 .00000E+00 0 .00000E+00

0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 1 .35765E+00 1 .00000E+00

A .J. Dragt / Numerical third-order transfer map for solenoid

f(159)=f( f(164)=f( f(165)=f( f(170)=f( f(175)=f( f(176)=f( f(179)=f( f(184)=f( f(185)=f( f(190)=f( f(195)=f( f(200)=f( f(209)=f(

01 01 01 01 00 00 00 00 00 00 00 00 00

12 10 03 01 40 31 22 20 13 11 04 02 00

00 02 00 02 00 00 00 02 00 02 00 02 04

453

)=-0 .52703543859992D+01 )=-0 .15782632902935D+01 )= 0 .38029475402102D-14 )= 0 .26020852139652D-16 )=-0 .26609263380345D+03 )= 0 .65913133073316D+03 )=-0 .61925222623999D+03 )=-0 .17244726416808D+02 )= 0 .26065685021104D+03 )= 0 .20874041534917D+02 )=-0 .41483707281162D+02 )=-0 .66442345797430D+01 )=-0 .15854058234226D+01

V. THEORETICAL OPTICS

454

A .J . Dragt / Numerical third-order transfer map for solenoid

Appendix B #comment This is a MARYLIE run for a sextupole corrected solenoidal spot forming system . The system consists of two solenoids, a sextupole, and suitably chosen drifts . The performance of the system is studied first with the sextupole turned off . Then the performance is studied with the sextupole turned on . The beam parameters are those for 200 KeV electrons . #beam 1 .6490338242227880E-03 0 .3913901524593803 1 .000000000000000 1 .000000000000000 #menu fileout pmif 1 .00000000000000 12 .0000000000000 3 .00000000000000 sol sol 0 .000000000000000E+00 0 .200000000000000 200 .000000000000 15 .0000000000000 1 .00000000000000 0 .000000000000000E+00 solpar psl 5 .000000000000000E-02 0 .100000000000000 1 .000000000000000E-02 2 .000000000000000E-02 3 .00000000000000 0 .000000000000000E+00 drl drft 0 .117055375019450 dr2 drft 0 .435356676117240 dr3 drft 0 .535356676117240 nosex drft 0 .200000000000000 sex sext 0 .200000000000000 112 .308881352040 mapout ptm 3 .00000000000000 3 .00000000000000 0 .000000000000000E+00 0 .000000000000000E+00 1 .00000000000000 clear iden raysin rt 13 .0000000000000 14 .0000000000000 -1 .00000000000000 0 .000000000000000E+00 0 .000000000000000E+00 0 .000000000000000E+00 trace14 rt 0 .000000000000000E+00 14 .0000000000000 4 .00000000000000 1 .00000000000000 0 .000000000000000E+00 0 .000000000000000E+00 circ16 circ 0 .000000000000000E+00 16 .0000000000000 5 .00000000000000 1 .00000000000000 1 .00000000000000 3 .00000000000000 end end #lines lens 1*solpar l*sol sectl 1*liens 1*drl sect2 1*dr2 1*liens 1*dr3 spot 1*sectl 1*nosex 1*sect2

455

A .J. Dragt / Numerical third-order transfer map for solenoid

cspot 1*sectl #lumps liens 1*lens #loops onebyone 1 *cspof #labor 1*fileout 1*spot 1*mapout 1*raysin 1*tracel4 1*clear 1*cspot 1*mapout 1*clear 1*raysin 1*onebyone 1*circl6 1*end

1*sex

1*sect2

zi= 0 .0000000000000000E+00 zf= 0 .2000000000000000 ns= 200 di= 5 .0000000000000000E-02 length= 0 .1000000000000000 c1= 1 .0000000000000000E-02 B= 2 .0000000000000000E-02 iopt= 0 lump liens

constructed and stored .( 1)

matrix for map is -3 .14100E-11 -1 .11525E-01 -8 .14177E-11 1 .10076E+00 -5 .89300E-01 2 .94253E+00 8 .14177E-11 2 .98124E-01 -3 .14099E-11 -2 .94253E+00 1 .57530E+00 1 .10076E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00

-2 .98124E-01 -1 .57530E+00 -1 .11525E-01 -5 .89300E-01 0 .00000E+00 0 .00000E+00

0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 1 .00000E+00 0 .00000E+00

0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 1 .80324E+00 1 .00000E+00

nonzero elements in generating polynomial are f( f( f( f( f( f( f( f( f( f( f( f( f(

33)=f( 38)=f( 42)=f( 45)=f( 53)=f( 57) =f 60)=f 67)=f( 70)=f( 76)=f 83)=f 84)=f 85)=f(

20 11 10 10 02 01 01 00 00 00 00 40 31

00 00 10 01 00 10 01 20 il 02 00 00 00

01 01 01 01 01 01 01 01 01 01 03 00 00

=-0 .17658058518376D+02 )= 0 .94852372412875D+01 =-0 .11102230246252D-15 )= 0 .17442840922314D+01 =-0 .21740134030118D+01 =-0 .17442840922314D+01 )= 0 .64184768611142D-16 =-0 .17658058518376D+02 )= 0 .94852372412875D+01 =-0 .21740134030118D+01 =-0 .12967055473886D+01 =-0 .42970234492190D+04 )= 0 .32269150327953D+04

V. THEORETICAL OPTICS

456

f ( 86) =f ( f ( 87) =f ( f ( 90) =f ( f( 91)=f( f( 92)=f( f ( 95) =f ( f ( 96) =f ( f ( 99) =f ( f(104)=f( f(105)=f( f(106)=f( f(107)=f( f(110)=f( f(ill)=f( f(114)=f( f(l19)=f( f(120)=f( f '(121)=f ( f(124)=f( f(129)=f( f(130) = f( f(135)=f( f(140)=f( f(141)=f( f(142)=f( f(145)=f( f(146)=f( f(149)=f( f(154)=f( f(155)=f( f(156)=f( f(159)=f( f(164)=f( f(165)=f( f(170)=f( f(175)=f( f(176)=f( f(179)=f( f(184)=f( f(185)=f( f(190)=f( f(195)=f( f(200)=f( f(209)=f(

A.J. Dragt / Numerical third-order transfer map for solenoid

30 30 22 21 21 20 20 20 20 13 12 12 11 11 11 11 10 10 10 10 10 10 04 03 03 02 02 02 02 01 01 01 01 01 01 00 00 00 00 00 00 00 00 00

10 01 00 10 01 20 il 02 00 00 10 01 20 il 02 00 30 21 12 10 03 01 00 10 01 20 11 02 00 30 21 12 10 03 01 40 31 22 20 13 11 04 02 00

00 00 00 00 00 00 00 00 02 00 00 00 00 00 00 02 00 00 00 02 00 02 00 00 00 00 00 00 02 00 00 00 02 00 02 00 00 00 02 00 02 00 02 04

)= 0 .10990791610155D-11 )= 0 .65441175561517D+02 =-0 .12361481942807D+04 =-0 .65441175561518D+02 =-0 .33115506392027D+02 =-0 .ß5940468984381D+04 )= 0 .32269150327953D+04 =-0 .41646227812029D+03 =-0 .78471229268399D+02 )= 0 .32693721440358D+03 )= 0 .33115506392027D+02 )= 0 .66302197073150D+01 )= 0 .32269150327953D+04 =-0 .1639371ß323208D+04 )= 0 .3269372144035ßD+03 )= 0 .63ß64245024043D+02 =-0 .44ß64112425103D-12 )= 0 .6544117556151ßD+02 =-0 .3311550639202ßD+02 )= 0 .85981555641979D-15 )= 0 .6630219707314ßD+01 )= 0 .31565265805794D+01 =-0 .44175327180ß45D+02 =-0 .66302197073144D+01 =-0 .16072950262402D-12 =-0 .41646227ß12029D+03 )= 0 .32693721440358D+03 =-0 .ß8350654361689D+02 =-0 .1229860ß980533D+02 =-0 .65441175561517D+02 )= 0 .33115506392026D+02 =-0 .66302197073140D+01 =-0 .31565265ß05794D+01 =-0 .15992372356943D-12 )= 0 .13704315460217D-15 =-0 .42970234492190D+04 )= 0 .32269150327953D+04 =-0 .123614ß1942807D+04 =-0 .7847122926ß399D+02 )= 0 .3269372144035ßD+03 )= 0 .63864245024043D+02 =-0 .44175327180845D+02 =-0 .12298608980533D+02 =-0 .21057450636476D+01

matrix for map is . -3 .14100E-11 -1 .11525E-01 -8 .14177E-11 -2 .98124E-01 1 .10076E+00 -5 .89300E-01 2 .94253E+00 -1 .57530E+00 8 .14177E-11 2 .98124E-01 -3 .14099E-11 -1 .11525E-01 -2 .94253E+00 1 .57530E+00 1 .10076E+00 -5 .89300E-01 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00

0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 1 .00000E+00 0 .00000E+00

0 .00000E+00 0 .00000E+00 0 .00000E+00 0 .00000E+00 1 .80324E+00 1 .00000E+00

A .J. Dragt / Numerical third-order transfer map for solenoid

45 7

nonzero elements in generating polynomial are

f( 28)=f( f( 29)=f( f( 30)=f( f( 31)=f( f( 33)=f( f( 34)=f( f( 35)=f( f( 36)=f( f( 38)=f( f( 39)=f( f( 40)=f( f( 42)=f( f( 43)=f( f( 45)=f( f( 49)=f( f( 50)=f( f( 51)=f( f( 53)=f( f( 54)=f( f( 55)=f( f( 57)=f( f( 58)=f( f( 60)=f( f( 64)=f( f( 65)=f( f( 67)=f( f( 68)=f( f( 70)=f( f( 74)=f( f( 76)=f( f( 83)=f( f( 84)=f( f( 85)=f( f( 86)=f( f( 87)=f( f( 89)=f( f( 90)=f( f( 91)=f( f( 92)=f( f( 94)=f( f( 95)=f( f( 96)=f( f( 98)=f( f( 99)=f( f(101)=f( f(104)=f( f(105)=f( f(106)=f( f(107)=f( f(109)=f( f(110)=f( f(111)=f( f(113)=f(

30 21 20 20 20 12 11 11 11 10 10 10 10 10 03 02 02 02 01 01 01 01 01 00 00 00 00 00 00 00 00 40 31 30 30 30 22 21 21 21 20 20 20 20 20 20 13 12 12 12 11 11 11

00 00 10 01 00 00 10 01 00 20 11 10 02 01 00 10 01 00 20 11 10 02 01 30 21 20 12 11 03 02 00 00 00 10 01 00 00 10 01 00 20 11 10 02 01 00 00 10 01 00 20 11 10

00 00 00 00 01 00 00 00 01 00 00 01 00 01 00 00 00 01 00 00 01 00 01 00 00 01 00 01 00 01 03 00 00 00 00 01 00 00 00 01 00 00 01 00 01 02 00 00 00 01 00 00 01

)=-0 .12266535473644D+04 )= 0 .70150994979934D+02 )=-0 .14506107005751D+05 )= 0 .27652954437927D+03 )=-0 .17658058518376D+02 )=-0 .11164576497837D+02 )= 0 .55305908875855D+03 )=-0 .88019713851287D+02 )= 0 .94852372412875D+01 )= 0 .36799606420931D+04 )=-0 .14030198995987D+03 )=-0 .11102230246252D-15 )= 0 .11164576497837D+02 )= 0 .17442840922314D+01 )=-0 .23960922135302D-09 )=-0 .44009856925644D+02 )=-0 .29585294214485D-08 )=-0 .21740134030118D+01 )=-0 .70150994979934D+02 )= 0 .22329152995673D+02 )=-0 .17442840922314D+01 )= 0 .71878503149492D-09 )= 0 .64184768611142D-16 )= 0 .48353690019169D+04 )=-0 .27652954437927D+03 )=-0 .17658058518376D+02 )= 0 .44009856925643D+02 )= 0 .94852372412875D+01 )= 0 .98623331723502D-09 )=-0 .21740134030118D+01 )=-0 .12967055473886D+01 )= 0 .30922794328686D+07 )=-0 .22523777098737D+04 )=-0 .11059549248627D-08 )= 0 .65441178312748D+02 )=-0 .32818490291503D+04 )= 0 .13799509513192D+04 )=-0 .65441178308386D+02 )=-0 .33115506324463D+02 )= 0 .51483788230301D+04 )= 0 .61845588657372D+07 )=-0 .22523777098773D+04 )=-0 .38810349748714D+05 )=-0 .49020795853713D+05 )= 0 .20294492625488D+05 )=-0 .78471229268399D+02 )=-0 .22820129348399D+03 )= 0 .33115506319236D+02 )= 0 .66302196866374D+01 )=-0 .20688902993536D+03 )=-0 .22523777098759D+04 )= 0 .10080149361007D+06 )= 0 .40588985250975D+05

V. THEORETICAL OPTICS

458

A.J. Dragt / Numerical third-order transfer mapfor solenoid

f(114)=f( 11 02 00 =-0 .2282012934ß367D+03 f (116)=f ( 11 01 01 =-0 .1631079621941ßD+04 f (ll9)=f ( 11 00 02 )= 0 .63ß64245024043D+02 f (120) =f ( 10 30 00 )= 0 .6984980371ß824D-09 f(121)=f( 10 21 00 )= 0 .6544117830976ßD+02 f(123)=f( 10 20 01 )= 0 .98455470ß74510D+04 f(124)=f( 10 12 00 =-0 .33115506320365D+02 f(126)=f( 10 11 01 =-0 .10296757646060D+05 f(129)=f( 10 10 02 )= 0 .ß5981555641979D-15 f(130)=f( 10 03 00 )= 0 .66302196845396D+01 f (132)=f ( 10 02 01 >= 0 .206ß8902993536D+03 f(135)=f( 10 01 02 )= 0 .31565265ß05794D+01 f (140)=f ( 04 00 00 )= 0 .15596760756867D-06 f(141)=f( 03 10 00 =-0 .66302196842922D+01 f(142)=f( 03 01 00 =-0 .341693420519ß5D-09 f(144)=f( 03 00 01 )= 0 .14546665346466D+02 f(145)=f( 02 20 00 =-0 .49020795853713D+05 f (146)=f ( 02 11 00 =-0 .2282012934ß311D+03 f (148)=f ( 02 10 01 =-0 .815539ß1097089D+03 f (149)=f ( 02 02 00 )= 0 .311345ß6257059D-06 f(151)=f( 02 01 01 )= 0 .17202533229117D+03 f (154) =f ( 02 00 02 =-0 .1229ß608980533D+02 f(155)=f( 01 30 00 =-0 .65441178312079D+02 f(156)=f( 01 21 00 )= 0 .33115506323137D+02 f(158)=f( 01 20 01 =-0 .5148378ß230301D+04 f(159)=f( 01 12 00 =-0 .66302196856ß75D+01 f(161)=f( 01 11 01 )= 0 .41377ß05987073D+03 f(164)=f( 01 10 02 =-0 .31565265ß05794D+O1 f(165)=f( 01 03 00 )= 0 .10361883963708D-09 f (167)=f ( 01 02 01 =-0 .43639996039400D+02 f (170) =f ( 01 01 02 )= 0 .13704315460217D-15 f(175)=f( 00 40 00 )= 0 .3092279432ß686D+07 f(176)=f( 00 31 00 =-0 .2252377709ß740D+04 f(178)=f( 00 30 01 )= 0 .12936783249571D+05 f (179)=f ( 00 22 00 )= 0 .13799509513192D+04 f(181)=f( 00 21 01 =-0 .202944926254ß8D+05 f(184)=f( 00 20 02 =-0 .7ß471229268399D+02 f(185)=f( 00 13 00 =-0 .228201293483ß4D+03 f(187)=f( 00 12 01 )= 0 .81553981097089D+03 f (190)=f ( 00 11 02 )= 0 .63ß64245024043D+02 f (195)=f ( 00 04 00 )= 0 .15588757203491D-06 f (197)=f ( 00 03 01 =-0 .57341777430391D+02 f(200)=f( 00 02 02 =-0 .1229ß608980533D+02 f(209)=f( 00 00 04 =-0 .21057450636476D+01 circulating through onebyone liens liens drl sex dr2 dr3

A .J. Dragt / Numerical third-order transfer map for solenoid References

[2] [3] [4]

[6]

[8]

[10] [11] [121

[13]

A .J. Dragt, Lectures on Nonlinear Orbit Dynamics, in : Physics of High Energy Particle Accelerators, American Institute of Physics Conf. Proc ., vol . 87, eds. R.A . Carrigan et al . (1982) . A .J. Dragt and E. Forest, J . Math. Phys . 24 (1983) 2734. A .J. Dragt, F . Neri, G. Rangarajan, D.R. Douglas, L .M . Healy and R.D . Ryne, Ann . Rev. Nucl . Part. Sci. 3 8 (1988) 455 . A .J. Dragt et al ., MARYLIE 3 .0 User's Manual, University of Maryland Physics Department (1988), unpublished . For a description of differential algebraic methods, see M . Berz, Part . Accel. 24 (1989) 109 ; alternatively, as developed by Berz and others, one may directly integrate Hamiltonianlike differential equations for the aberration coefficients as they appear in a Taylor series expansion. An advantage of both these methods is that they facilitate implementation to arbitrary order . An advantage of the Lie algebraic approach, since it works with Lie generators rather than Taylor series, is that it requires the integration of only about a third as many differential equations . However, it requires somewhat different programming at each order . R . Jagannathan, Quantum Theory of Electron Lenses Based on the Dirac Equation, Institute of Mathematical Sciences preprint, Madras, India (1990) . A .B . El-Kareh and J.C . El-Kareh, Electron Beams, Lenses, and Optics, vol. 2 (Academic Press, New York, 1970) p . 20 ; see also H . Bateman, Partial Differential Equations of Mathematical Physics, (Cambridge University Press, London and New York, 1959) p . 406 . Note that A as defined by eqs . (8)-(11) vanishes in regions where B = 0 . Consequently, according to eq . (3), canonical and mechanical momenta are equal in field-free regions ; they generally differ elsewhere. For an extensive discussion of solenoidal fields and their treatment, see the remarkable two-volume set by P.W . Hawkes and E. Kasper, Principles of Electron Optics (Academic Press, 1989) . H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, 1980) p . 383 . A .J . Dragt, University of Maryland Lecture Notes on Classical Mechanics, unpublished . These calculations were done using MATHEMATICA ; for a description, see S . Wolfram, MATHEMATICA, A System for Doing Mathematics by Computer (AddisonWesley, 1988) . GENMAP refers to a series of programs that take as input real (calculated or measured) magnetic and/or electric field data for some beam-line element, and then generate the transfer map describing motion through that element . These programs have not all been fully developed and/or completely documented . References include refs. [14,15] below, and A .J . Dragt and R. Ryne, Proc. 1987 IEEE Particle Accelerator Conf. 2 (1987) 1081 .

45 9

[14] E. Forest, Ph.D . Thesis, University of Maryland Physics Department (1984) . [15] R.D . Ryne, Ph.D . Thesis, University of Maryland Physics Department (1987) . [16] Note that the word order is used with many different meanings : in the case of aberrations, third order means terms of degree three in the Taylor series expansion of final conditions in terms of initial conditions. In the case of differential equations, first order means that only first derivatives occur . In the case of a numerical integration method, nth order means that the solution is accurate through terms of degree n in the local time step . [17] These routines are more accurate than they need be for the present purpose. They were originally developed for code testing, and hence were readily available for other uses. A description of general Adams and low-order Runge-Kutta routines may be found in ref. [11]. For a description of high-order Runge-Kutta routines see F . Ceschino and J. Kuntzmann, Numerical Solution of Initial Value Problems (Prentice-Hall, 1966) . [18] For the purposes of MARYLIE output , the phase-space variables are arranged in the order (X, P, Y, Py , T, PT ) . In the MARYLIE monomial labelling scheme, entries in f3 are given by f (28) through f(83), and entries in f4 are given by f(84) through f(209) . In the expanded notation f( * * * * * * ), the entries * label the relevant powers of X, P, Y, Py , T, P respectively. Note that, as displayed in appendix A, the R32, Rio, R32, and R 34 components of the matrix part of the transfer map all vanish as is required for an imaging system. This imaging condition was achieved in a separate MARYLIE fitting run that is not shown . Note also that all the nonzero entries in f3 involve powers of PT . Consequently, f is purely chromatic. [191 Recall that Lie transformations act in the order in which they appear when equations are read from left to right . See refs . [1], [3], [14], or [15] . [201 AN . Crewe and D . Kopf, Optik (Stuttgart) 55 (1980) 1 ; 56 (1980) 391 ; 57 (1980) 313 ; 60 (1982) 271 ; AN . Crewe and D.B . Salzman, Ultramicroscopy 9 (1982) 373 ; AN . Crewe, Optik (Stuttgart) 69 (1984) 24 ; see also V.D . Beck, Optik (Stuttgart) 53 (1979) 241 ; Ji-Ye Ximen, Optik (Stuttgart) 65 (1983) 295 . [21] For a Lie algebraic treatment of the sextupole corrector, see ref . [14] and A.J . Dragt and E . Forest, Advances in Electronics and Electron Physics, ed. P . Hawkes, vol . 67 (Academic Press, New York, 1986) p. 65 . [22] The extension of the expansion (35) to higher order is easily done with MATHEMATICA . The extension of eqs . (57), (60), and (61) is given in ref. [2] . [231 F . Nen, Numerical Calculation of Third and Fifth-Order Transfer Maps for General Beam Line Elements, University of Maryland Physics Department, preprint, in preparation.

V . THEORETICAL OPTICS