Aberrations of asymmetrized quadrupole lenses

NUCLEAR INSTRUMENTS AND METHODS

91 (1971) 407-411; ©

NORTH-HOLLAND PUBLISHING CO.

ABERRATIONS OF ASYMMETRIZED QUADRUPOLE LENSES GY. SZABO Institllte of Nllclear Research of tlte HI/ngarian Academy of Sciences, Debrecen, HilI/gary

and L. P. OVSYANNIKOVA

A. F. Ioffe Pltysicu- Technical Institllte, USSR Academy of Sciences, Leningrad, USSR

Received 14 July 1970 and in revised form 2 October 1970 The trajectory equation of combined asymmetrized quadrupole lenses is given in third order approximation. To il1ustrate the general features aberration curves have been computed for the Gaussian plane of image in the case of an electrostatic singlet.

1. Introduction A first order description of the field and the ionoptical behaviour of the asymmetrically fed electrostatic quadrupole lenses has been given in our previous papers 1.2). Asymmetrizing the quadrupole lenses one is furnished with the possibility of shifting continously the optical axis from the geometrical one i.e. correcting the position of the beam entering the transport system without changing the focusing properties of the lenses. One can expect, however, that a price is to be paid, namely accessory aberration may appear because of the asymmetrization. This effect has been investigated in third order approximation for the two-dimensional cases 3-5). In the present paper the three-dimensional case is treated for superimposed electrostatic and magnetic quadrupole fed asymmetrically. 2. General description Using the notations introduced in ref. 4 the potential function in first order approximation reads (1) where X

o = - PIt.,

Yo == - q 1 t.

suitable notations the potential function for the magnetic lenses is as follows

where Xo

=-

fIl'm

=-

(1/J2) (Pt -ql) r m ,

(3)

Yo=-5 1 1'm=-(1/.j2)(Pl+Ql)rm

are the displacements of the centre, while W is the magnetic pole strength, I'm denotes the radius of the aperture, and N is a constant depending on the form of the pole-pieces. In the electrostatical case the parameters P3 and q3 appear beside the asymmetry parameters PI and ql; these can be given by P3 = - PI'

q3 = q I

•

In magnetic case, similar to the relation (3), the parameters 53 and t 3 are the following: t 3 = (1/../2) (P3-Q3),

= (1/J2) (P3+Q3)'

53

The potential distribution for superimposed electrostatic and magnetic quadrupoles fed asYmmetrically in three-dimensional case and third order approximation can be written as follows:

are the displacements of the centre, PI and q 1 are the so-called asymmetry parameters. Their values dependbeside the form of the electrodes-oll the magnitude of the asymmetrization, namely on the asymmetry of the voltages of the electrodes opposite to each other. PI and q 1 are given for several electrode configurations of practical interest in refs. 3 and 4. r e denotes the radius of the aperture. It is known, that a parallel treatment can be given for the ion-optical behaviour of the magnetic quadrupoles in a co-ordinate system rotated with 45°. With 407

¢(x, y, z)

=

2 VK

[~f.(z) (x 2 _y2) 21'.

1

1

1'.

6 r.

+-[Pl(Z)X-Ql(Z)Y]- -

1

3

2

[P'{(Z)X3_Q~(Z)y3J

1

:1.

+-3 P3 (Z)(X -3xy )+-3 Q3 (Z)(y3_3YX) 3re 31'.

r:

I 4 - 24 f :'(Z)(x - y4)J + 2

WN[r~fm(z) xy

408

GY. SZAB6 AND L. P. OVSYANNIKOVA

-&

T;tz)(x 3+3 xy l)

- -b: S';(z)(l+3yx 2) +~

T3(z)(3x 2 y_y 3)

+J. [T,(z)x+S,(Z)y] rm

-

3 rm

--\-S3(Z)(X 3 -3X y2 )].

3 I'm

jection of the trajectory onto the xoz focusing plane can be derived

(4)

Here F
The asymmetry parameters P,(z), Qi(Z), Sj(z), T,(z) depend on the values of asymrnetrization and on the shape of electrodes and poles. They can be obtained by multiplying the corresponding P" qt, S" " parameters of two-dimensional lenses by the normalized functions fc(z) and fm(z) describing the z-dependence of the electrostatic and magnetic fields, respectively. Using the expression (4) the equation for the pro-

=-

fJ~ [tim x(3 X,2+ y'2)_ ImYx'y' -/~xyy'

- -& f':: x(x 1 + 3l)] + fJ~ [I. X(X,l + y'2) - f3~f.2 x(x 2 _

),2) -

tl: x'(x2.- y2) - t I:' x 3J (6)

F d denotes third order terms appearing as a consequence of the asymmetrization. Here it was assumed

30

x 10 (mmj

fO

-o'+-----,....-----r---14.:,.----'l----f-'--...,...--"-2

-fO

-20

Fig. 1. Aberration curves for electrostatic quadrupole singlet asymmetrized in both directions to 40%. ( The x-lIxis is extended by n factor 10.) The true-to-scale shape of the envelope is to be seen in the inset.

409

ABERRATIONS OF ASYMMETRIZED QUADRUPOLE LENSES

that Plo Qb S" T, are of the same order of magnitude as the coordinates and their derivatives.

The equation for the projection of trajectory onto the yoz defocusing plane can be obtained by the replacements

F d = f3:[-rcx'(P~x-Q~ y)+r.y'(P l l+Qlx')

+ (l/r.)P 3(x 2 -

-lP~ I"cX2

y2) - (2/r.)Q3 x y

in the eqs. (5)-(7). In the central part of the lens-far from the edgesthe trajectory equations of the two-dimensional lens come back. Their form for the pure electrostatic case is given in refs. 3 and 4. The eq. (5) was solved by the method of variations of parameters. For the sake of simplicity it was assumed that the z-dependence of all the field parameters can be described by the same function.ttz) i.e.

- f3: Ie r. (3 PI x 2 - PI / - 2 Q1 X .1') - 2f3;r;Pl(P 1 X-QIY)] + + f3~ [y' rm(S~ X+ T{ y)

+

*

T;'l'm(x

2

+/) -

+ -!-S'{ rmxy

(I/"m) S3(X

2

_ )12)

- (2/rn,) T3 xy - ·~;rm T 1 (x"+ y")J

+ P; P; [H. I'm T

j

(x

2

_

P j = pJ(z) , QI = qd(z) , S/ = sJ(z) ,

y2)

The strength parameters

P. and Pm are

y{mm) [10.'10)

~j':MO~. 2°FG.20)

-20

20~a10)

~

x (mm)

r: -W)Ht~~:~) tF +FtalO) i+2o.tOJ{4:3o.lo4(*O,IO) +F Ho.°H o. +r2Q20)H:3o.H2U

t 20 )

("420)

x (mm)

-0

-20

20j

-20

(mm)(20.1,0) HJ(mm) ~(mm)(4({I,O) (30.40)

x (mm)

-0

-0

td(z) .

This assumption is almost valid e.g. in the case of concave cylindrical quadrupole lenses as it was shown in ref. 7. The first order solution is obtained as follows

(7)

+ r.(P l X-QI .1') (fmx+ T j I'm)]'

1/ =

(0.0) I

2 4' I

lao )

0

2

'"

W2o.OJ 0

2

'"

r

3

/40)

0

2

0

X (mm)

Fig. 2. Envelope of aberration curves for electrostatic quadrupole singlet using entry aperture with radius rl = r/2, and the asymmetl'ization changes from 0 to 40% in both directions. (The x-axis is extended by n factor 10.)

GY. SZAB6 AND L. P. OVSYANNIKOVA

410

where X3 denoted the trajectory shift caused by the asymmetrization, The contribution of the terms of third order is as shown in: 8 • 9) Xq

= X2(Z)

with coefficients

Expressions for the aberration coefficients in the Gaussian plane of image can be deduced by quadra~ ture in the case of a point object. Derivatives of the function describing the z-dependence of the field have been eliminated by particular integration of (8)8.9). In the resulting formulae different models can be used for the z-dependence of the field.

f:o xt(z)[Fq(z)+Fd(z)]dz

- x 1 (z)

to

(8)

x 2 (z)[Fq(z)+F d(z)]dz.

The trajectories calculated in first approximation are to be substituted into the expressions for F q and F d • The equation for the projection of trajectory onto the xoz plane in third order approximation is X (Z)

~

= LJ

I)

I

aUkl Xo Xo

3. Aberration curves for the electrostatic case To illustrate the results of the analytical calculations (for a more detailed description see ref. 10) aberration curves have been computed for the case of an electro-

(9)

k " YoYo ,

3

• (mm) !J

20

30

to x

-to 20

'.. . .........t!l'r \

\

'" \

\I ( /

,.

\ I'

\

"'f~

d\ II

"

Ii

-f

III I

/,\ 1/ ' ..._ I

I

I

-3

"

"" ,

\

\

\

\\

111

I

-20

" (r;-2,5mm\

I I'

III II'

fO(mm

-to 5mm

,

,\

y (mm)

I

/

40

.-"

/

f

/

/

/

x

I

I

0

\

,"

/

/

/

,,

I

I

I

I I

2

(mm)

",/'

,.'",.'" -20

Fig, 3. Aberration curves for a beam entering with an axis parallel to the lens axis. The object point is 5.095 mm far from the axis, and the trajectory = 0 crosses the axis in the Gaussian image plane on the exit side.

x;

ABERRATIONS OF ASYMMETRIZED QUADRUPOLE LENSES

static quadrupole singlet with point objects located on the z axis and outside of it, respectively. In the entry plane circles of different radii were considered and the points have been determined where the trajectories through these circles cross the Gaussian image plane. The curves belonging to different entry apertures do not overlap completely because of the non-linear term appearing in (9), so the envelope giving the actual form of the beam has had be calculated, too. The aberration curves presented here are calculated for an electrostatic quadrupole singlet of 150 mm length with concave cylindrical electrodes (2e = 45°) and 15 mm aperture. Both object and image distances are 1000 mm measured from the entry and exit planes, respectively. In fig. I aberration curves are shown for the lens asymmetrized in both directions to 40 percent. The continuous curve demonstrates the envelope, the dotted lines are aberration curves for different entry apertures. The x-axis is extended by a factor 10. In the inset the envelope is demonstrated true-to-scale. In fig. 2 envelopes of aberration curves for entry aperture with radius 1'[ = 1'/2 can be seen for the above mentioned quadrupole singlet. In both planes the asymmetrization is changed from 0 to 40 percent. The figure shows the broadening of the beam because of asymmetrization. This effect, however, does not deeply influence the working conditions of a transport system using a small value of asymmetry and near to the paraxial beam. As was shown previously 1,2), systems composed of asymmetryzed quadrupole lenses are suitable for steering the beam in beam transport systems. For this

411

particular reason it was important to deal with beams which do not enter along the axis. Let a beam enter the lens with an axis parallel to the axis of the singlet in a given distance .dx. Let the beam axis intersect Gaussian image plane in the geometrical axis. This can be realised by the asymmetrization of the lens field. The maximum value of .dx depends on the actual parameters of the lens and on the value of the asymmetrization. Such a case is shown in fig. 3, where Llx = 5.095 mm and the asymmetrization equals to 40%. (The x-axis is extended by a factor 10, and the true-toscale envelope can be seen in the inset.) The authors wish to express their gratitude to Dr. E. Koltay for calling their attention to the subject as well as for the many helpful discussions. References 1) E. Koltay and Gy. Szab6, ATOMKI KBzl. 6 (1964) 105. 2) E. Koltay and Gy. Szab6, Nucl. Instr. and Meth. 35 (1965) 88. 3) L. P. Ovsyannikova and S. Va. Yavor, Zh. Tekhn. Fiz. 38

(1968) 1810.

4) L. P. Ovsyannikova, S. Va. Yavor, E. Koltay and Gy. Szab6,

Nucl. Instr. and Meth. 74 (1969) 185.

5) L. Bodnar, E. Kollay and Gy. Szab6, ATOMKI Kllzl. 11

(1969) 137.

6) A. D. Dymnikov, T. Va. Fishkova and S. Va. Xavor, Zh.

Tekhn. Fiz. 34 (1964) 1711.

7) A. Kiss, E. Kallay. L. P. Ovsyannikova and S. Va. Yavor, Nucl. Instr. and Meth. 78 (1970) 238. 8) A. D. Dymnikov, T. Va. Fishkova, L. P. Ovsyannikova and

S. Va. Yavor, Nucl. Instr. and Meth. 42 (1966) 293.

9) P. W. Hawkes, Quadrupole optics, Tracts in modern physics,

vol. 42 (Springer.Verlag, Berlin, 1966). P. Ovsyannikova and Gy. Szab6, ATOMKI Kozl. 11, no. 1 (1971).

10) L.