s __ __ ~

ELSEVIER Nuclear Instruments and Methods in Physics Research B 130 (1997) 90-96

Beam interactions with Materials A Atoms

Field reconstruction technique for testing magnetic quadru~o~e lenses

S. Lebed *, A. Ponomarev Applied Physics Institute, National Academy uf Sciences qf Ukraine, 244030 Sumy, Ukraine

Abstract

A number of analytical instruments make use of strong magnetic quadruple lenses. Performance properties of such instruments are determined by the presence of lens aberrations that arise from the structure of the lens magnetic field. This paper describes a technique for reconst~~tion and analysis of the 3-D field of the magnetic quad~pole lens. Measurements of the radial 8, or longitudinal B, component of the field are used as boundary conditions for the Laplace equation for magnetic scalar potential. Application of the charged density method for the solution of the boundary-value problem allows

the field and its multipole components to be determined to a high accuracy. The field reconstruction technique can be used for determination of both intrinsic and parasitic multipole components of the field in a magnetic quadrupole lens. This technique is suitable either for a single magnetic multipole lens or for a magnetic multipole lens system. 0 1997 Elsevier

Science B.V.

1. Introduction

A number of analytical inst~ments make use of

magnetic quadrupole lenses for focusing high-energy electron and ion beams. Improvement of such instru- ments is connected with the correction of lens aber- rations. Aberrations of a quadrupole lens can be

intrinsic and parasitic. Parasitic aberrations arise from mechanical imperfections of the lens, iron inhomo- geneity of the pole tips and lens misalignment in the focusing system. Intrinsic aberrations exist even in ideally manufactured lenses. Both parasitic and in- trinsic aberrations are a certain integral value of

* Corresponding author

interaction of electron or ion beam with multipole components of the lens magnetic field. Hence, know- ing the muitipole components of the field, we can

determine the relevant lens aberrations. Alterna- tively, knowing lens aberrations, we can indirectly estimate the relevant multipole components of the

magnetic field of quadrupole lens. In this connection there are two approaches in testing magnetic quadrupole lenses, based either on measurements of aberrations or on measurements of multipole compo- nents of the magnetic field. In the first case, the grid shadow method [l-3] has been widely used. This method assumes that spherical aberrations can be calculated with the grid shadow patterns formed by the beam passing through the grid, intrinsic and parasitic aberrations can be separated and multipole components of the field can be indirectly determined.

0168-583X/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved

PII SO168-583X(97)00302-9

S. Lebed, A. P~~#rnarev/N~~~. Instr. und Meth. in P&s. Res. B 130 (1997) W-96 91

In the second case, multipole components are often

measured using the rotating coil method [4,5]. This method assumes that multipole components of the

magnetic quadrupole field can be determined with the signal from a coil rotating in the lens magnetic field. But both methods allow the value of multipole components to be determined only inside the lens.

Quantitative measurement of magnetic quadrupole lens fringe fields have been conducted in [6,7] to

determine the axial profile of the multipole field components. Direct measurements of radial compo-

nent B, in the working area of the lens by a Hall

probe were used.

This paper presents a technique for field recon-

struction based on the solution of the 3-D Laplace

equation for magnetic scalar potential. Measurements of radial B, or longitudinal BZ component of the lens magnetic field are used as boundary conditions.

The boundary is a cylinder of rc < ra radius (ra is the lens aperture radius). Cylinder length is chosen

such that field on the cylinder ends is negligible. Component B, or B, are measured only on the side surface of the cylinder near the lens pole tips. Recon-

structed field is analyzed for the presence of multi- pole components.

2. Field reconst~ctio~ technique

Magnetic field in the air gap of the magnetic

quadrupole lens can be described by the magnetic scalar potential w( X, y, z) which satisfies the Laplace equation

Aw=O. (1)

Consider a cylindrical surface G limiting the work-

ing area of the lens, i.e. the area where the beam of charged particles is passing. This cylinder axis can be misaligned with the lens axis as it is shown in Fig. 1. If we know the surface G distribution of radial component B, we have

dw( XYYJ) aw( x,y,z) BrhYJ)l, = ar = an ?

G G G

(2)

\ I Fig, 1. Position of the boundary cyiindrical surface inside the magnetic quadrupole lens.

where nc is the external normal to the surface G. Boundary condition (2) determines the Neumann problem for the Laplace equation 1. On the other hand, if we know the surface G distribution of

longitudinal component BZ, then B,(x, Y, Z)/C fully determines the distribution of magnetic scalar poten-

tial w( X, y,z&. We can choose such cylinder length t that IBI on the cylinder ends is close to zero. Then

w( x,y,z)I~ = l_iL/lB,(+,~~t)l&

Boundary condition (3) determines the Dirichlet problem for the Laplace equation. Boundary-value problems (l), (2) and (I), (3) are nume~calIy solved by the charged density method [S]. This method has the following advantages: 1. surface limiting the region of solution can be

open; 2. high accuracy of solution; 3. higher derivatives from the solution can be analyt-

ically calculated with high accuracy; 4. solution and its higher derivatives can be calcu-

lated in any points of the determination region.

In the charged density method, the magnetic scalar potential w in any point of the dete~ination region (x,y,z>, including the boundary surface G, is de- scribed by

where p( xP, y,, , z,) is the determination point; q(xy, yy.z,,) is the point on the cylinder surface G;

R,, = (xp -x,>2+(Y~-Y,>2+(zp-zq)2;

11. MICROBEAM LENSES AND OPTICS

93 S. Lobed. A. P~~~[~~ure~, /Nwl. Insrr. und Me& in Phw. Rex B 130 (19971 90-96

d.S, is the infinitely small element of the surface G; a(q) is the value of surface density of ‘magnetic charges’.

Here it is not worthwhile to plunge deep into the physical meaning of ‘magnetic charge’. The problem

of finding magnetic fields for description of which the magnetic scalar potential is used is fully equiva-

lent to the electrostatic problem where surface den- sity of electric charges has a clear physical meaning.

Distribution of surface density of ‘magnetic

charges’ a(q) on the surface G is determined from

boundary conditions (3)

w( dlc=j&W&,dSCI. P>qEG, (5) G

or (2)

p,q E G. (6) Numerical solution in the charged density method

is based on the discretization of the surface G boundary and representation of cr(p> as a bilinear

approximation for each element e (Fig. 2). After the approximation of (T(P) has been substi-

tuted into (5) or (6) an integration is performed for each discretization point. This results in a set of linear equations with respect to unknown values of surface density of magnetic charges in the points

44 = w It is obvious that the kind of square matrix vector of the right part {f} will be different

(7) A and for (5)

Fig. 2. Diagram of discretization of boundary cylindrical surface.

and (6). A set of linear equations (7) is solved using the LU-factorization method [9]. When the vector of surface density of ‘magnetic charges’ (P) is found, the magnetic scalar potential w in any point of the working region can be calculated using (4). But we

are preferably interested in higher derivatives of scalar magnetic potential w. The main advantage of

the charged density method is that higher derivatives from w are found analytically by differentiation of

integral equation core 14)

a(l+j+k)N,( p> =:

ax;a,;a$ I() aq G

i, j, k = 0, 1,2,. . . (8)

It is obvious from (8) that, knowing {(T] values, we can calculate values of higher derivatives from mag- netic scalar potential in any points of the lens work-

ing region. Dete~ination of multipole components of the magnetic field, both parasitic and intrinsic, in

the quadrupole lens is based on the analysis of

magnetic scalar potential w( x, y, Z> obtained after boundary-value problems (l), (2) or (l), (3) have been solved.

Consider an example of the magnetic field of a quadrupole lens which has intrinsic multipole com- ponents and parasitic sextupole and octupole compo- nents. According to Szilagyi [8] the magnetic scalar potential of such a lens can be expressed in the rectangul~ coordinate system connected with the

lens axis as a series:

- ;w;(Z) +4w4(2) xy3, i

(9)

where W&z>, Wz(.d, W&z), W&z) are the dipole,

quadrupole, sextupole and octupole components, re-

spectively.

S. Lebed, A. Ponomareu /Nucl. Instr. and Meth. in Phys. Res. B 130 (1997) 90-96 93

In order to simplify the description and ignore the conversion of the coordinate system we assume that the lens axis is aligned with the cylinder axis where one of the field components B, or B, is measured.

The multipole field components will be deter- mined by

1 = -X).( X>Y,Z) - kw;‘,,( X,Y,Z)

8 II x.y=o

= $(w;:&.Y.Z) -wx”;.,y x, ,z I ( y )I] 1 . x,y=o

(10)

To determine multipole components of the mag-

netic quadrupole field of higher order, it is necessary to save members of the series describing these com- ponents in the expression of magnetic scalar poten-

tial w (9) and repeat the procedure described above.

3. a-Testing

a-Testing of field reconstruction technique means a primary testing in order to 1. check the technique ideology;

2. check the adequacy of the numerical code to the technique;

3. check the capabilities of the technique.

For this purpose an idealized model has been chosen which has an analytical solution. As has been

mentioned above, the magnetostatic problem, which

allows magnetic field to be described using the

magnetic scalar potential, is fully equivalent to the electrostatic problem. Therefore an electrostatic

quadrupole lens has been taken as a test model which has pole tips in the form of infinitely thin conductors of finite length with given distribution of

Table 1 influence of discretization parameters on the accuracy of calculation of tield gradient on the lens axis

Z w;,.(z)(A) A$,lB,l Aw;,(B,)

IO*20 10*28 l5*60 IO*20 IO*28 15*60

50. 1.665 le-4

40. 1.0547e-3

30. 1.8408e-2

28. 3.5505e-2

26. 63295e-2

25. 7.9988e-2

24. 9.668Oe-2

22. 1.2447e- 1

20. 1.4156c-1

16. 1.5488e- 1

12. 1.5826e-1

10. 1.5889e-1

6. 1.5946e-1

2. 1.59&k-l

7.Oe-5

9.Oe-4

2.7e-3

4.2e-3

I&-2

2.3e-2

2.9e-2

3.5e-2

3.1e-2

l.3e-2

7.1e-3

6.5e-3

6.01~3

5.9e-3

4.9e-5 6.6e-4 2.2e-2 2.oe-2 7.6e.-3 1.3e-3 1 .oe-2 2.2e-2 2.5e-2 l.2e-2 4.3e-3 3.3e-3 2.8e-3 2.7e-3

7.7e-5

2.2e-4 3.7e-3 4.Oe-3

I .9e-3

2.9e-4

2.4e3

4.6e-3

4.3e-3

2.6e-3

1 .Oe-3

S.le-4

6.4e-4

6.Oe-4

2.8e-4 2.4e-3 2.Oe-2

I .9e-2

8.le-3

2.3e-4

8.6e-3

2sk-2

2.le-2

6.7e-3

6.le-3

1.2e-3

2.le-5

7.2e-4

2.8e-4 2.4e-3 2.0e-2

I .9e-2

8.3e-3

6.6e-5

8.4e-3

1.9e-2

2.0e-2

6.3e-3

1.4e-3

1.5e-3

3.le-4

3.8e-4

l.3e-4

1.4e-5

5.9e-4

l.4e-3

1 Be-3

7.le-6

1 Be-3

1.4e-3

6.1e-4

3.Oe-4

3.le-5

2.le-5

2.le-5

2.1e-5

0. 1.59661~1 5.9e-3 2.7e-3 6.Oe-4 7.8e-4 4.4e-4 1.9e-5

(B,), (B,) is the component used as a boundary condition; (A) is the analytical solution; {N) is the numerical solution; IO* 20 means that

number of points on z axis is n, = 10, on 0 angle is n, = 20; r1 = 7 mm; rC = 5 mm; L = 100 mm; I = 50 mm; A$! = Iw:,( A) - w,“~{N)I.

II. MICROBEAM LENSES AND OPTICS

94 S. Lebed. A. Ponomure~ / Nucl. Inslr. and kfeth. in Phys. Res. B 130 (I 997) 90-96

electric charge density. Scalar potential w in such a lens is described by the analytical expression

w( x,y,z) = i (- l)‘y i=l -l/2

si( W Ijl(X-Xi)2+(y-y,)2+(Z-t)2 .

For the case q,(t) = 1, i = 1, 2, 3, 4.

w(x,y,z)= 5 (-1)‘ln i= 1

(11)

z+1/2+yi(x-x,)2+(y-yi)‘+(z+1/2)2

z--1/2+ (X-X,)2+(y-y;)2+(z-1/2)2

The testing procedure was the same for all cases.

Cylindrical surface G of radius rC and length L was

located inside the lens. Surface G discretization is given in Fig. 2. The values of field components B, or B, were computed for discretization points using

(11).

Boundary-value problems were solved by the method Using a generator of random numbers, we have described in Section 2. Eq. (8) was used for calcula- varied values B, and BZ calculated in Test 1 within

tions of higher derivatives from the scalar potential for given points. For the same points the relevant derivatives from scalar potential set by (11) were

also calculated. Then numerical results were com- pared with the results of analytical solution. For all

tests, charge density q,(t) was taken as dimension-

less and all linear dimensions were measured in mm. It should be noted that a great number of calculations

were done, but only some of them demonstrating

general tendencies are listed in Tables l-3.

3.1. Test 1. Influence of discretization parameters on the solution accuracy

Table 1 gives the results of testing for the case when the cylinder axis is aligned with the lens axis

and q&t) = 1, i = 1, 2, 3, 4. Field gradients were

calculated on the lens axis. Analysis of data (Table 1) shows that a decreased discretization step leads to

more accurate numerical solutions.

3.2. Test 2. Influence of errors in B, and BZ mea- surements on the numerical solution of the relevant boundary-value problems

Table 2 Influence of error of field component B, or B, measurement on the accuracy of calculation of field gradient on the lens axis

z +(z)(A) Aw;,(B,) Aw,“,lB;)

0.1% 1% 5% 0.1% 1% 5%

50. 1.665 le-4 7.7e-5 7.7e-5 7.7e-5

40. l.O547e-3 1 .le-4 1.7e-4 1 .le-4

30. 1.8408e-2 5.5e-3 5.5e-3 5.3e-3

28. 3.5505e-2 7.3e-3 7.3e-3 7.Oe-3

26. 6.3295e-2 3.5e-3 3.5e-3 3.le-3

25. 7.9988e-2 3.3e-4 3.8e-4 8.4e-4

24. 9.668Oe-2 4.2e-3 4.2e-3 4.8e-3

22. 1.2447e- 1 S.Oe-3 8.Oe-3 86e-3

20. 1.4156e-1 6.2e-3 6.3e-3 6.8e-3

16. 1.5488e-1 2.2e-3 2.3e-3 2.7e-3 12. 1.5826e-1 9.3e-4 l.le-3 1.6e-3 10. 1.5889e-1 7.2e-4 9.6e-4 1.5e-3

6. 1.5946e- 1 5.3e-4 8.2e-4 1 k-3 2. 1.5964e- 1 4.8e-4 7.7e-4 1.7e-3 0. 1.5966e- 1 4.9e-4 l.le-4 1 k-3

0.1% means maximal error (max(A B)/max( B)) * 100% in prescribing B, or B,.

1.2e-4

1.7e-5

2.9e-4 7.Oe4

56e-4

2.8e-5

6.le-4

7.4e-4

3.2e-4 l.&-4 8.k5 5.le-5 4.3e-5 2.Oe-5 1.5e-5

1.2e-4 1.7e-5

2.8e-4 6.8e-4

5.2e-4

4.8e-5

6.Oe-4

6.5e-4

l.Se-4 3.5e-5 8.le-5

1 k-5 2.8e-4 2.2e-4 1.6e-4

1.2e-4

1.7e-5

2.5e-4 6.le-4

4.6e4

6.5e-5

5.5e-4

4.9e-4

3.&-5 3.8e-4 8.5e-4 6.le-4 1.7e-4 l_le-4 5.7e-4

S. Lebed, A. Ponomureu / Nuci. Insrr. and Meth. in Phys. Res. B 130 II 997) 90-96 95

Table 3

Accuracy of determination of sextupole and octupole field components

z W,(r) AW,fB,} AW&B;J wjw AW&) AWd{ B;1

50. - 2.87OC!e-8

40. - 44468e-7

30. - 2.9465e-5

28. - 7.4504-5

26. - 1.6294~4 25. - 2.1993e-4

24. - 2.7693e-4

22. - 3.6537e-4

20. -4,1041e-4

16. - 4.3533e-4

12. - 4.3897e-4

10. - 4.3942e-4

6. - 4.3973e-4

2. - 4.398 le-4

0. - 4.3982e-4

1.2e-8 I .7e-7 I .8e-5 30%5

1.5e-5

1.7e-6

1.9e-5

3.4e-5

2.2e-5 7.0e-6

4.0e-6

3.6e-6

3.9e-6

5.3e-6

5.5e-6

7.4e-8 2.4e-8

I .oe-5

2.4e-5

1.4e-5

3.8e-7

1.5e-5

2.5e-5

l.le-5

2.6e-6

3.0e-7

2.2e-6

4.7e-6

I .Se-6

2.2e-7

3.3048e-10 1.2593e-8

3.1354e-6

I.O298e-5

2.7244e-5

3.894le-5

5.0639e-5

6.7584e-5

7,4748e-5

7.761 le-5 7.7851e-5

7.7871e-5

7.7881e-5

7.7882e-5

7.7883e-5

7.2.e.11 4.5e-9 3.k6

6.7e-6

3.6e-6

7.7e-7

5.le-6

8.2e-6

4.9e-6

2.4e-6

2.5e-6

2.k6

2.2e-6

I .9e-6

I .8e-6

3.8e-9 3.5e-7

2.4e-6

5.9e-6

3.3e-6

7.h7

4.7e-6

6.9e-6

2.9e-6

I .5e-7

3.4e7

7.f%7

7Se-7

8.6e-7

1.2e-6

by = /W,(A) - W,(N)/, i = 3, 4.

the given limit. Table 2 illustrates field gradients

distribution along the axis depending on the error in given B, and B, values on the boundary.

3.3. Test 3. kzjluence of radius and length of cylinder sulfate on the accuracy of reconstructed field analy- sis

Variation of radius of cylindrical surface within 0.2 < rC/ra < 0.9 shows that radius should not be

chosen too small causing degradation of accuracy of higher derivatives. Measurements of field compo- nents B, or B, must be taken preferably near the

pole tips. Cylinder length L is to be chosen taking into account small field IBI on the cylinder ends. This allows a longitudinal distribution of higher

derivatives to be obtained with high accuracy.

3.4. Test 4. Accuracy of determination of sextupo~e and octupole field components of q~drupole lens

In this paper we have considered the field recon- struction technique and the analysis of magnetic field

applied to magnetic quadrupole lenses. However, this technique can also be used for investigation of

correcting magnetic octupole lenses and deflecting

magnets and also for any other magnetic system having air gap with straight axis in its structure.

A certain defect was introduced into the lens Testing of the numerical code for this technique causing an error in the given charge density of one shows that the error in the field gradient on the lens of electrodes q,(t) = 1 + 8; qi(t) = 1, i= 2, 3, 4. axis is of the order of 0.1%. The error of potential The lens was also turned with respect to the z axis sextupole and octupole field components is of the by an angle y. Field components B, and B,, the order of 1%. An increased field component order values of which included all possible errors of field gives rise to the error. For the field components measurements, were calculated on the cylinder sur- usually used for lens characterization, errors of these face. After boundary value problems cl), (2) and (11, orders are acceptable.

(3) had been solved the distributions of sextupole

and octupole field components along the lens axis

were calculated using relationships (10). Data given in Table 3 show high accuracy of the results inside

the lens. Some degradation of accuracy is observed on the lens ends, which is due to the small number of discretization points along the z axis on the lens

ends.

4. Discussion

II. MICROBEAM LENSES AND OPTICS

96 S. Lebed, A. Ponomarel’/ Nucl. Instr. and &leth. in Phys. Rex B 130 (I 997) 90-96

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