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Measurement, analysis and modification of the fifth-order fringe field components of magnetic quadrupole lenses
24
Nuclear Instruments and Methods in Physics Research B54 (1991) 24-27 North-Holland
Measurement, analysis and modification of the fifth-order fringe field components of magnetic quadrupole lenses G.R. Moloney, D.N. Jamieson and G.J.F. Legge School of Physics, U~iversify of Me~~~ne,
~a$kville 3052, Australia
Extensive, quantitative measurements of magnetic quadrupole lens fringe fields have been conducted. The fringe field region of magnetic quadrupole lenses has been shown to contain significant contamination by higher-order multipole fields. These multipole components will cont~bute to the aberration coefficients of the lens. One of the largest components is the duodecapde ~mponent,
which contributes to the fifth-order geometric aberrations of the lens. The measured multipole profiles in the fringe field region of the Melbourne quadrupoles are presented. We also present the results of an investigation into the effect of modifying the quadmpole pole tip profile at the pole ends.
1. Induction
tial, II, may be expressed in the cylindrical
Previous measurements have shown the presence of higher-order multipole components in magnetic quadrupole lenses. Mechanical misalignments in lens construction may lead to cont~nation of the quadrupole field by sextupole and octupole fields. Such contamination has been commonly observed in magnetic quadrupole lenses. Jamieson showed, using the grid-shadow method, multipole contamination in the Melbourne magnetic quadrupoles (11. Magnetic field measurements have also shown the presence of sextupole contamination in the Melbourne lenses [2]. Such contamination has also been observed in other quadrupole lenses by Martin and Goloskie [3], Breese et al. [4], and Jamieson et al. 15-71. A lens with no such ~s~~rnents will still possess some duodecapole and 20-pole contamination. Parzen [8] showed this contamination could be minimised by choosing a suitable ratio of pole tip radius to bore radius. However, the duodecapole and ZO-pole components become significant in the fringe field region of conventional lenses. The new two-stage ion microprobe [9], nearing completion at Melbourne, will be very sensitive to lens aberrations. We are interested in minimising the contribution of the duodecapole field components to the lens aberrations. Measurements of the field profile of magnetic quadrupole lenses enable us to determine the effect of lens design changes on the lens optics. As will be seen, the axial form of the quadrupole fields is complex. Analysis of the lenses by the grid-shadow method yields no information on the axial dependence of the multipole fields. The small sampling area (= 1 mm2) of the Hall effect probe used for the measurements gives more detail than the conventional rotating coil methods. Following Szilagyi [lo], the magnetic scalar poten0168-583X/91/$03.50
coordinate
system (r, o, z) as a Fourier series in cy, M u(r, a, z) = [(Y&, z) cos(ma)
2
m=O
+b,(r,
z) sin(mcu>],
(1)
where a,(r,
2) = f Am,*(Z)r2k+m, k=O k-0
and t -l)km!
A m,k=
#2k”(
4Q!(m+k)!
z)
’
In
(3)
where the superscripts, (2k), on U and W denote the 2k th derivative with respect to z. The U,(z) and W,(z) represent the profiles of the mth-order multipole in the multipole expansion. For a magnetic quadrupole lens, assuming no constructional or excitational asymmetries, u satisfies the following symmetry conditions: u(r,
(Y, z) =u(-r, = --u(r,
(Y, z) =u(r, -ix,
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
2)
2).
The only terms in u satisfying tions are those such that m=2,6,10,14;~~.
fn-ff,
(4)
these symmetry
condi-
(5)
25
G.R. Maloney et al. / Fifth-orderfringefieid components
4
=
vu =
cL)RR/Eje /4-+X
CUWT
VOLTAGIS:
Fig. 1. Block diagram of the Meibourne magnetic field mapping ins~ment.
Thus, to sixth order quad~pole lens, u(r,
(Y, 2) = [ W,(.z)r”-
in r,
for an ideal magnetic
&WF(t)r4
x sin(2cr) + We( t)r6
C &+Wp(z)P] sin(6a).
(6)
In the central region of a quadrupole, u(r, a, z) is
independent of z, and the W,(t) are constant. Thus the derivatives in z vanish and we obtain the familiar two-dimensional multipole expansion for u(r, a);
zf(r, a) = mf or m[Umcos(ma)
+ W, sin(ma)].
(7)
Fig. 2. Side and end views of the lens pole tips. The upper diagram shows the unmodified pole tip, with the sharply terminated ends. The lower diagram shows the modified pole tips with the rounded pole ends. 1. MICROPROBE TECHNOLOGY
G.R. Maloney et al. / Fifth-order fringe field components
26
In the fringe field region of a quadrupole lens these derivatives do not vanish, and must be considered when calculating the higher-order ion-optical properties of the system. The IV;’ term in u will contribute to the intrinsic spherical aberration coefficients of the lens. The Wp and W, terms will contribute to the fifth-order aberrations of the lens.
- a~-. Q2 (modified)
2. Experimental To measure the field profiles of magnetic quadrupole lenses, a computer-controlled magnetic field mapping instrument has been constructed. The instrument is shown schematically in fig. 1. This instrument enables us to map the radial component of the magnetic field, B, = Vp, as a function of r, (Y and z. The measurements are analysed by the computer to produce the field profiles presented. At a given value of r and z, the computer rotates the Hall probe about the magnetic axis. The data are Fourier transformed in a to yield the b,( r, z) values in eq. (1). In each z-plane, a polynomial in r is fit to each of the a,(r, z) and b,(r, z) by the least-squares method. This yields values for the Am,k(~) and B,,,k(~) coefficients in eq. (3). The multipole field profiles, U,(z) = A,,c( z) and W,(z) = B,,,(z), are thus obtained. As part of the ion-optics program conducted at Melbourne, detailed measurements have been made of the magnetic field profile of a quadrupole lens, hereafter called Ql. The lens, constructed at Melbourne, is similar to those presently in use on the Melbourne proton microprobe, described by Legge et al. [ll]. Measurements have also been made of the field profile of a second lens, hereafter called 42. The pole tips of 42 have been modified in an attempt to determine the influence of pole-tip termination on the quadrupole, m = 2, and duodecapole, m = 6, profiles. Fig. 2 shows a cross section of the pole tips of both quadrupoles, showing the modified design.
2.OE-05 l.OE-05 -
-l.OE-05 2.OE-05 -
1.OE-05
0 25
and results
The measured magnetic field profiles of the original and modified quadrupole lenses are shown in fig. 3. The data points shown represent the measured values of the B,++(z) in eq. (3). The profiles have been normalised so that B,,, = 1.0 in the central region of the quadrupole. An empirical field profile function has been fit to the
35
40
Fig. 3. The measured multipole profiles of two magnetic quadrupole lenses. Data for both the modified and unmodified lenses are shown. All profiles have been normalised so that E2., =1 in the centre of the lens. The pole ends are indicated by the dotted line.
quadrupole profile data by the least-squares The function fit is 1, f(Z)
=
method.
OIZ
l/(1 + eCO+CIS+C2S2+C3S3),zi
3. Measurements
30
Distance from centre of lens, z (mm)
(8)
z2
where s = z/r,, and r, = 6.35 mtn is the lens bore radius. The fitted curves are shown in fig. 3. The curves shown in the plots of B,,,(z) and B,,,(z) are those expected from the function fitted to the quadrupole, B,,(z), profile. From eq. (3), we expect B,,,(z) = - &L&(z) and B2,*(z) = &B&(z). The agreement is quite good for both quadrupole lenses. This indicates the accuracy of the method for measuring the profiles. The curve drawn in the plot of B6,,Jz) is a cubic spline through the data points of Ql, and is intended only to guide the eye.
G. R. Maloney et al. / Fifth-order fringe field components
4. Discussion It is evident from the data that the modified pole tip has had no significant effect on the magnetic field profile of the lens, and would have a negligible effect on the aberrations of the lens. This is not surprising, as the modification made is quite small. This is, however, only the start of a program to analyse the effect of progressively greater “rounding” of the pole-tip ends. We aim to reduce the duodecapole field in the fringe field region, thus reducing the fifth-order aberrations introduced to the lens. Also, by changing the quadrupole profile we may be able to produce a form with lower second and fourth derivatives. This will reduce the B,,,(z) and B2,2(z) profiles, hence reducing the intrinsic spherical aberration of the lens. The numerical raytracing program OXRAY [12] will be used to calculate aberration coefficients for the profiles measured.
References [1] D.N. Jamieson and G.J.F. Legge, Nucl. Instr. and Meth. B29 (1987) 544. [2] G.R. Maloney and G.J.F. Legge, Proc. 5th Aust. Conf. on Nuclear Techniques of Analysis, AINSE, 1987, unpublished.
27
[3] F.W. Martin and R. Goloskie, these Proceedings (2nd Int. Conf. on Nuclear Microprobe Technology and AppIications, Melbourne, Australia, 1990) Nucl. Instr. and Meth. B54 (1991) 64. [4] M.B.H. Breese, D.N. Jamieson, and J.A. Cookson, Nucl. Instr. and Meth. B47 (1990) 443. [5] D.N. Jamieson, G.W. Grime and F. Watt, Nucl. Instr. and Meth. B40/41 (1989) 669. Fl D.N. Jamieson, C.G. Ryan and S.H. Sie, these Proceedings (2nd Int. Conf. on Nuclear Microprobe Technology and Applications, Melbourne, Australia, 1990) Nucl. Instr. and Meth. B54 (1991) 33. 171 D.N. Jamieson and U.A.S. Tapper, Nucl. Instr. and Meth. B44 (1989) 227. charged beams, PI G. Parzen, Magnetic fields for transporting BNL 50536, ISA 76-13 (1976). D.N. Jamieson and G.J.F. Legge, these 191 G.R. Maloney, Proceedings (2nd Int. Conf. on Nuclear Microprobe Technology and Applications, Melbourne, Australia, 1990) Nucl. Instr. and Meth. B54 (1991) 68. [lOI M. Szilagyi, Electron and Ion optics, Microdevices: Physics and Fabrication Technologies, eds. I. Brodie and J.J. Murray (Plenum, 1988). [ill G.J.F. Legge, D.N. Jamieson, P.M.J. O’Brien, and A.P. Mazzohni, Nucl. Instr. and Meth. 197 (1982) 563. 1121 G.W. Grime, F. Watt, G.D. Blower, J. Takacs and D.N. Jamieson, Nucl. Instr. and Meth. 197 (1982) 97.
I. MICROPROBE
TECHNOLOGY
Nuclear Instruments and Methods in Physics Research B54 (1991) 24-27 North-Holland
Measurement, analysis and modification of the fifth-order fringe field components of magnetic quadrupole lenses G.R. Moloney, D.N. Jamieson and G.J.F. Legge School of Physics, U~iversify of Me~~~ne,
~a$kville 3052, Australia
Extensive, quantitative measurements of magnetic quadrupole lens fringe fields have been conducted. The fringe field region of magnetic quadrupole lenses has been shown to contain significant contamination by higher-order multipole fields. These multipole components will cont~bute to the aberration coefficients of the lens. One of the largest components is the duodecapde ~mponent,
which contributes to the fifth-order geometric aberrations of the lens. The measured multipole profiles in the fringe field region of the Melbourne quadrupoles are presented. We also present the results of an investigation into the effect of modifying the quadmpole pole tip profile at the pole ends.
1. Induction
tial, II, may be expressed in the cylindrical
Previous measurements have shown the presence of higher-order multipole components in magnetic quadrupole lenses. Mechanical misalignments in lens construction may lead to cont~nation of the quadrupole field by sextupole and octupole fields. Such contamination has been commonly observed in magnetic quadrupole lenses. Jamieson showed, using the grid-shadow method, multipole contamination in the Melbourne magnetic quadrupoles (11. Magnetic field measurements have also shown the presence of sextupole contamination in the Melbourne lenses [2]. Such contamination has also been observed in other quadrupole lenses by Martin and Goloskie [3], Breese et al. [4], and Jamieson et al. 15-71. A lens with no such ~s~~rnents will still possess some duodecapole and 20-pole contamination. Parzen [8] showed this contamination could be minimised by choosing a suitable ratio of pole tip radius to bore radius. However, the duodecapole and ZO-pole components become significant in the fringe field region of conventional lenses. The new two-stage ion microprobe [9], nearing completion at Melbourne, will be very sensitive to lens aberrations. We are interested in minimising the contribution of the duodecapole field components to the lens aberrations. Measurements of the field profile of magnetic quadrupole lenses enable us to determine the effect of lens design changes on the lens optics. As will be seen, the axial form of the quadrupole fields is complex. Analysis of the lenses by the grid-shadow method yields no information on the axial dependence of the multipole fields. The small sampling area (= 1 mm2) of the Hall effect probe used for the measurements gives more detail than the conventional rotating coil methods. Following Szilagyi [lo], the magnetic scalar poten0168-583X/91/$03.50
coordinate
system (r, o, z) as a Fourier series in cy, M u(r, a, z) = [(Y&, z) cos(ma)
2
m=O
+b,(r,
z) sin(mcu>],
(1)
where a,(r,
2) = f Am,*(Z)r2k+m, k=O k-0
and t -l)km!
A m,k=
#2k”(
4Q!(m+k)!
z)
’
In
(3)
where the superscripts, (2k), on U and W denote the 2k th derivative with respect to z. The U,(z) and W,(z) represent the profiles of the mth-order multipole in the multipole expansion. For a magnetic quadrupole lens, assuming no constructional or excitational asymmetries, u satisfies the following symmetry conditions: u(r,
(Y, z) =u(-r, = --u(r,
(Y, z) =u(r, -ix,
0 1991 - Elsevier Science Publishers B.V. (North-Holland)
2)
2).
The only terms in u satisfying tions are those such that m=2,6,10,14;~~.
fn-ff,
(4)
these symmetry
condi-
(5)
25
G.R. Maloney et al. / Fifth-orderfringefieid components
4
=
vu =
cL)RR/Eje /4-+X
CUWT
VOLTAGIS:
Fig. 1. Block diagram of the Meibourne magnetic field mapping ins~ment.
Thus, to sixth order quad~pole lens, u(r,
(Y, 2) = [ W,(.z)r”-
in r,
for an ideal magnetic
&WF(t)r4
x sin(2cr) + We( t)r6
C &+Wp(z)P] sin(6a).
(6)
In the central region of a quadrupole, u(r, a, z) is
independent of z, and the W,(t) are constant. Thus the derivatives in z vanish and we obtain the familiar two-dimensional multipole expansion for u(r, a);
zf(r, a) = mf or m[Umcos(ma)
+ W, sin(ma)].
(7)
Fig. 2. Side and end views of the lens pole tips. The upper diagram shows the unmodified pole tip, with the sharply terminated ends. The lower diagram shows the modified pole tips with the rounded pole ends. 1. MICROPROBE TECHNOLOGY
G.R. Maloney et al. / Fifth-order fringe field components
26
In the fringe field region of a quadrupole lens these derivatives do not vanish, and must be considered when calculating the higher-order ion-optical properties of the system. The IV;’ term in u will contribute to the intrinsic spherical aberration coefficients of the lens. The Wp and W, terms will contribute to the fifth-order aberrations of the lens.
- a~-. Q2 (modified)
2. Experimental To measure the field profiles of magnetic quadrupole lenses, a computer-controlled magnetic field mapping instrument has been constructed. The instrument is shown schematically in fig. 1. This instrument enables us to map the radial component of the magnetic field, B, = Vp, as a function of r, (Y and z. The measurements are analysed by the computer to produce the field profiles presented. At a given value of r and z, the computer rotates the Hall probe about the magnetic axis. The data are Fourier transformed in a to yield the b,( r, z) values in eq. (1). In each z-plane, a polynomial in r is fit to each of the a,(r, z) and b,(r, z) by the least-squares method. This yields values for the Am,k(~) and B,,,k(~) coefficients in eq. (3). The multipole field profiles, U,(z) = A,,c( z) and W,(z) = B,,,(z), are thus obtained. As part of the ion-optics program conducted at Melbourne, detailed measurements have been made of the magnetic field profile of a quadrupole lens, hereafter called Ql. The lens, constructed at Melbourne, is similar to those presently in use on the Melbourne proton microprobe, described by Legge et al. [ll]. Measurements have also been made of the field profile of a second lens, hereafter called 42. The pole tips of 42 have been modified in an attempt to determine the influence of pole-tip termination on the quadrupole, m = 2, and duodecapole, m = 6, profiles. Fig. 2 shows a cross section of the pole tips of both quadrupoles, showing the modified design.
2.OE-05 l.OE-05 -
-l.OE-05 2.OE-05 -
1.OE-05
0 25
and results
The measured magnetic field profiles of the original and modified quadrupole lenses are shown in fig. 3. The data points shown represent the measured values of the B,++(z) in eq. (3). The profiles have been normalised so that B,,, = 1.0 in the central region of the quadrupole. An empirical field profile function has been fit to the
35
40
Fig. 3. The measured multipole profiles of two magnetic quadrupole lenses. Data for both the modified and unmodified lenses are shown. All profiles have been normalised so that E2., =1 in the centre of the lens. The pole ends are indicated by the dotted line.
quadrupole profile data by the least-squares The function fit is 1, f(Z)
=
method.
OIZ
l/(1 + eCO+CIS+C2S2+C3S3),zi
3. Measurements
30
Distance from centre of lens, z (mm)
(8)
z2
where s = z/r,, and r, = 6.35 mtn is the lens bore radius. The fitted curves are shown in fig. 3. The curves shown in the plots of B,,,(z) and B,,,(z) are those expected from the function fitted to the quadrupole, B,,(z), profile. From eq. (3), we expect B,,,(z) = - &L&(z) and B2,*(z) = &B&(z). The agreement is quite good for both quadrupole lenses. This indicates the accuracy of the method for measuring the profiles. The curve drawn in the plot of B6,,Jz) is a cubic spline through the data points of Ql, and is intended only to guide the eye.
G. R. Maloney et al. / Fifth-order fringe field components
4. Discussion It is evident from the data that the modified pole tip has had no significant effect on the magnetic field profile of the lens, and would have a negligible effect on the aberrations of the lens. This is not surprising, as the modification made is quite small. This is, however, only the start of a program to analyse the effect of progressively greater “rounding” of the pole-tip ends. We aim to reduce the duodecapole field in the fringe field region, thus reducing the fifth-order aberrations introduced to the lens. Also, by changing the quadrupole profile we may be able to produce a form with lower second and fourth derivatives. This will reduce the B,,,(z) and B2,2(z) profiles, hence reducing the intrinsic spherical aberration of the lens. The numerical raytracing program OXRAY [12] will be used to calculate aberration coefficients for the profiles measured.
References [1] D.N. Jamieson and G.J.F. Legge, Nucl. Instr. and Meth. B29 (1987) 544. [2] G.R. Maloney and G.J.F. Legge, Proc. 5th Aust. Conf. on Nuclear Techniques of Analysis, AINSE, 1987, unpublished.
27
[3] F.W. Martin and R. Goloskie, these Proceedings (2nd Int. Conf. on Nuclear Microprobe Technology and AppIications, Melbourne, Australia, 1990) Nucl. Instr. and Meth. B54 (1991) 64. [4] M.B.H. Breese, D.N. Jamieson, and J.A. Cookson, Nucl. Instr. and Meth. B47 (1990) 443. [5] D.N. Jamieson, G.W. Grime and F. Watt, Nucl. Instr. and Meth. B40/41 (1989) 669. Fl D.N. Jamieson, C.G. Ryan and S.H. Sie, these Proceedings (2nd Int. Conf. on Nuclear Microprobe Technology and Applications, Melbourne, Australia, 1990) Nucl. Instr. and Meth. B54 (1991) 33. 171 D.N. Jamieson and U.A.S. Tapper, Nucl. Instr. and Meth. B44 (1989) 227. charged beams, PI G. Parzen, Magnetic fields for transporting BNL 50536, ISA 76-13 (1976). D.N. Jamieson and G.J.F. Legge, these 191 G.R. Maloney, Proceedings (2nd Int. Conf. on Nuclear Microprobe Technology and Applications, Melbourne, Australia, 1990) Nucl. Instr. and Meth. B54 (1991) 68. [lOI M. Szilagyi, Electron and Ion optics, Microdevices: Physics and Fabrication Technologies, eds. I. Brodie and J.J. Murray (Plenum, 1988). [ill G.J.F. Legge, D.N. Jamieson, P.M.J. O’Brien, and A.P. Mazzohni, Nucl. Instr. and Meth. 197 (1982) 563. 1121 G.W. Grime, F. Watt, G.D. Blower, J. Takacs and D.N. Jamieson, Nucl. Instr. and Meth. 197 (1982) 97.
I. MICROPROBE
TECHNOLOGY