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Resolution limit of probe-forming systems with magnetic quadrupole lens triplets and quadruplets
Nuclear Instruments and Methods in Physics Research B 201 (2003) 637–644 www.elsevier.com/locate/nimb
Resolution limit of probe-forming systems with magnetic quadrupole lens triplets and quadruplets A.G. Ponomarev a,*, K.I. Melnik a, V.I. Miroshnichenko a, V.E. Storizhko a, B. Sulkio-Cleff b a
Institute of Applied Physics, National Academy of Sciences of the Ukraine, Petropavlovskaja Street 58, Sumy 244030, Ukraine b Institute of Nuclear Physics, University of M€unster, M€unster, Germany Received 22 August 2002; received in revised form 10 December 2002
Abstract Over the past decade, in MeV ion beam microanalysis efforts to achieve a spatial resolution better than 0.1 lm with a beam current of 100 pA have been connected with microprobes of new generation where the probe is formed by means of separated magnetic quadrupole lens structures [1]. However, as was pointed out in [2], no dramatic improvements in spatial resolution have been produced so far. For better understanding of the situation the authors carried out theoretical studies of multiparameter sets of probe-forming systems based on separated triplets and quadruplets of magnetic quadrupole lenses. Comparisons were made between the highest current values attained at different systems for a given beam spot size. The maximum parasitic sextupole and octupole field components were found whose contributions to spot broadening are tolerable. It is shown that the use of modern electrostatic accelerators [3] and precision magnetic quadrupole lenses [4,5] makes it possible to eliminate the effect of chromatic aberrations and second- and third-order parasitic aberrations resulting from distortions of the quadrupole lens symmetry. Therefore probe-forming systems with triplets and quadruplets of magnetic quadrupole lenses have a lower theoretical spatial resolution limit which is restricted mainly by intrinsic spherical third-order aberrations in state-of-the-art microprobes. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 41.85.p; 41.85.Gy; 41.85.Lc; 41.85.Ne Keywords: Magnetic quadrupole lens; Probe-forming system; Optimization; Emittance
1. Introduction The beam spot size is determined by the total contribution coming from the linear image of the
*
Corresponding author. Fax: 380-542-223-760. E-mail address: [email protected] (A.G. Ponomarev).
object slit and the aberration-induced beam broadening that can be limited by an angular slit. So the same beam spot size can be obtained by various combinations of an angular and an object slits. As is known, the slits dimensions influence the emittance and consequently (considering brightness) the beam current at the entrance to the magnetic quadrupole lens structure. The procedure of selecting most suitable probe-forming
0168-583X/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-583X(02)02229-2
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systems using a two-parameter set for the separated Russian quadruplet of magnetic quadrupole lenses was discussed in detail in [6] where the aberration-induced beam spot broadening included contributions from chromatic and intrinsic spherical third-order aberrations. It was demonstrated that there are probe-forming systems in which demagnifications and aberrations are balanced out for a given beam spot size. A proper selection of the object and the angular slits dimensions permits maximum current to be obtained for these systems which we call ‘‘perfect’’. In this paper we take the next step to elucidate the effect of the sextupole and octupole parasitic aberrations in the probe-forming systems. The analysis procedures developed in [6–8] allow a correct comparison between systems with varying number of lenses. Here we continue the discussion presented in [8] by comparing the performance parameters for the separated high excitation triplet and the separated Russian quadruplet.
2. Calculations of parasitic aberrations in magnetic quadrupole lens structures using the matrizant method For purposes of clarity we will first dwell upon some aspects of our approach to the inclusion of parasitic aberrations into the optimization problem to be solved for the probe-forming systems. The calculations of the magnetic quadrupole lens aberrations including contributions from the parasitic sextupole and octupole components involve the axial field model. It assumes that in the lens there is a straight-line optical axis along which the field j~ Bj ¼ 0. Naturally, this is an approximation corresponding to the averaged flight path of a particle that enters the lens at zero angles retaining at the exit the position in the transverse plane and the direction of flight. In the arbitrary Cartesian _ _ coordinates ðx ; y ; zÞ where the z-axis is aligned with the optical lens axis, the magnetic quadrupole field can be described by the scalar potential as _ _
wð x ; y ; zÞ _
_2
_2
_
__
¼ U 2 ðzÞð x y Þ þ 2W 2 ðzÞ x y
_
_3
_
__2
_2 _
_3
þ U 3 ðzÞð x 3 x y Þ þ W 3 ðzÞð3 x y y Þ _ 00
_
_4
þ ðU 4 ðzÞ U 2 ðzÞ=12Þ x _
_ 00
_3 _
_
_ 00
__3
_
_2 _2
þ ð4W 4 ðzÞ W 2 ðzÞ=6Þ x y 6U 4 ðzÞ x y ð4W 4 ðzÞ þ W 2 ðzÞ=6Þ x y _ 00
_
_4
þ ðU 4 ðzÞ þ U 2 ðzÞ=12Þy þ ; _
ð1Þ
_
where U i ðzÞ and W i ðzÞ are the field components of the magnetic quadrupole lens. _ On the _assumption that the major W 2 ðzÞ and the skew U 2 ðzÞ quadrupole components are governed by the same distribution law, in the ðx; y; zÞ _ _ coordinates rotated about the ð x ; y ; zÞ coordinates through an angle _
1 a ¼ arc tg 2
U 2 ð0Þ _
! ;
ð2Þ
W 2 ð0Þ
(the rotation is about the z-axis) the scalar potential finally takes the form wðx; y; zÞ ¼ 2W2 ðzÞxy þ U3 ðzÞx3 þ 3W3 ðzÞx2 y 3U3 ðzÞxy 2 W3 ðzÞy 3 þ U4 ðzÞx4 þ ð4W4 ðzÞ W200 ðzÞ=6Þx3 y 6U4 ðzÞx2 y 2 ð4W4 ðzÞ þ W200 ðzÞ=6Þxy 3 þ U4 ðzÞy 4 þ ; ð3Þ where W2 ðzÞ is the major quadrupole field component, W3 ðzÞ and U3 ðzÞ are the major and the skew sextupole parasitic components, W4 ðzÞ and U4 ðzÞ are the major and the skew octupole parasitic components. The ðx; y; zÞ coordinates connected with the optical lens axis and antisymmetry planes of the major quadrupole component W2 ðzÞ are referred to as eigencoordinates and the components W3 ðzÞ, U3 ðzÞ, W4 ðzÞ and U4 ðzÞ parasitic eigencomponents of the magnetic quadrupole. Any minor transformation of a lens as an integral rigid body would lead to the redistribution of the multipole components and production of additional parasitic dipole and/or skew quadrupole components.
A.G. Ponomarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 637–644
All the components are considered in the normalized form for two representations of the longitudinal distribution Wi ðzÞ ¼ Wi fj ðzÞ;
Ui ðzÞ ¼ Ui fj ðzÞ;
i ¼ 2; 3; 4; j ¼ 1; 2; with f1 ðzÞ ¼
8 < 1; :
1
2 3 1þeðC0 þC1 sþc2 s þC3 s Þ
0;
ð4Þ
;
jzj 6 z2 ; z2 < jzj 6 z0 ; s ¼ z=r0 ; jzj > z2 ;
where r0 is the lens bore radius, f2 ðzÞ ¼ hðz z1 Þ hðz þ z1 Þ with 0; z < 0; hðzÞ ¼ 1; z P 0; Rz z1 ¼ Leff =2 with Leff ¼ z0 0 f1 ðsÞ ds, ðz0 ; z2 Þ and ðz2 ; z0 Þ are the coordinates of the input and output fringe field region. f1 ðzÞ is used in approximation with allowance for the real fringe field and f2 ðzÞ corresponds to the conventional rectangular representation. To determine the transformation matrix for the magnetic quadrupole lens in which the field is described by the scalar potential (3) and expressions (4) matrizant method is employed [9–11]. In our case the phase moment space is represented by the vector ~ ¼ ðx; x0 ; y; y 0 ; xd; x0 d; yd; y 0 d; x2 ; xx0 ; x02 ; y 2 ; yy 0 ; y 02 ; X xy; x0 y; xy 0 ; x0 y 0 ; x3 ; x2 x0 ; xx02 ; x03 ; xy 2 ; xyy 0 ; xy 02 ; x0 y 2 ; x0 yy 0 ; x0 y 02 ; y 3 ; y 2 y 0 ; yy 02 ; y 03 ; yx2 ; yxx0 ; yx02 ; y 0 x2 ; y 0 xx0 ; y 0 x02 ÞT ;
ð5Þ
where x and y are the beam deviations from the optical lens axis (z), x0 and y 0 are angles dictating the beam direction, and d is the momentum spread of the beam particles. ~ vector along the z-axis is The evolution of the X determined by the relation ~ðzÞ ¼ Rðz=z0 ÞX ~ðz0 Þ; X
ð6Þ
where Rðz=z0 Þ is the 38 38 matrizant (transformation matrix) for the z0 to z-plane transformations, the first and the third matrizant rows containing a complete set of aberrations including
639
chromatic, parasitic second- and third-order aberrations due to the major and skew sextupole and octupole components and intrinsic third-order aberrations. For the rectangular representation of the field components analytical expressions were derived for all matrizant elements. This was done using present-day programs of analytical calculations and in part the distribution theory. The possibility of calculating the matrizant on the basis of the analytical expressions is important, permitting analysis of a large number (several thousand) of probe-forming systems to be performed within a reasonable time. In order to estimate errors in the matrizant computations for the rectangular representation of the field components calculations were carried out for a distribution approximating a real fringe field by the shuttle-sum method [12]. Fig. 1 shows some aberrations in a single magnetic quadrupole lens versus the relative width of the fringe field, m ¼ ðz0 z2 Þ=Leff . If m ! 0 for Leff ¼ const, aberrations approach corresponding values obtained for the rectangular representation. Note that aberrations for a zero width of the fringe field (rectangular model) are minimum for the magnetic quadrupole.
3. Calculations of parasitic field components in magnetic quadrupole lenses Parasitic field components, Wi and Ui , produced in a magnetic quadrupole lens by distortions of the quadrupole symmetry were found by solving numerically the Laplace equation for the scalar magnetic potential in the working lens area using the charge-density method [13]. It is known to be very accurate and ideally suited to determination of the field structure in ion-optic elements, permitting higher derivatives of the scalar magnetic potential to be calculated analytically by differentiating the integral operator nucleus [14]. Unlike [15], where an approximated analytical model is used, our calculations were performed for real pole tip shapes. The quadrupole symmetry distortions were simulated by a planar shift of one or two poles and the excitation error by that in prescribed
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analitical numerical
U3/W2 = W3/W2 [1/cm], r=0.635cm,
3
2
U4/W2 [1/cm ], r=0.635cm, U3/W2 = W3/W2 [1/cm], r=0.375cm,
2
2
U4/W2 [1/cm ], r=0.375cm.
1.0
0.006
Field components
0.5 0.0 -0.5 -1.0
2
/10
3
2
3
/10
-1.5 10
20
30
40
-40
-20
0
3
/10
20
40
∆r [µm]
(a)
80
6
60 40 20 2
/10
6
0
0.002 0.001 0.000 -0.001 -0.002
10
20
30
40
50
ν / L [%]
Fig. 1. Spherical aberrations of a single magnetic quadrupole lens as function of the fringing field relative width m ¼ ðz0 z2 Þ=Leff .
0.0
0.5
1.0
1.5
∆ [%]
Fig. 2. Relative parasitic sextupole (W3 =W2 ¼ U3 =W2 ) and octupole (W4 =W2 ¼ U4 =W2 ) field components as function of (a) variations in the pole gap fitting and (b) pole-tip excitation error for two aperture radii.
SEPARATED QUADRUPLET λ L1 s L2 L4 L3 s K1
a
g
SEPARATED TRIPLET a
L1
λ
K1
OBJECT
The choice of optimum probe-forming systems for a nuclear microprobe is based on a comparison between the highest emittance values that can be provided by these systems for a given beam spot size. To perform the comparative analysis twoparameter sets were constructed for perfect probeforming systems where independent parameters are the system length, l, and the object distance, a
-0.5
(b)
boundary conditions at one of the poles. Fig. 2 illustrates parasitic sextupole and octupole field components relative to the major quadrupole component versus the pole shifts and the excitation error, for two values of the aperture radius, ra ¼ ð0:635; 0:375 cmÞ.
4. Comparison between probe forming systems
-1.0
L2
s L3
g
TARGET
0
-1.5
K4
-40
6
K3
/10 3 6 /10
K3
2
-20
K2
3
-0.003 -0.006
ν / L [%]
Aber ra tions [ µm / mrad ]
0.000
50
Field components
0
0.003
K2
Aber ra tions [ µm / mrad ]
1.5
/10
l Fig. 3. Lens arrangement along the ion-optical line for separated triplet and quadruplet structures.
A.G. Ponomarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 637–644
(Fig. 3). Thus, we have a parameter set that includes various suitable lens positions along the optical line for separated quadrupole structures. Fig. 4 represents the demagnification and aberration coefficients for the two-parameter set of a high excitation triplet, while similar simulations but for the Russian quadruplet are given in our previous paper [6] where we describe at greater length how the expressions relating the demagnification and aberration coefficients to a and l were derived. The lens parameters and dimensions of the systems were selected such as to cover most microprobes available: Dx 1800 P1
|Dy|
effective lens lengths Li ¼ L ¼ 6:4 cm; lens power supply for the triplet k1 ¼ k2 and for the Russian quadruplet k1 ¼ k4 , k2 ¼ k3 , each even-number lens being rotated about its basic position through 90°; working distance g ¼ ð10 cm; 15 cmÞ; fixed distance in paired doublets s ¼ 3:3 cm (Fig. 3); system length was varied in the range 1:5 m 6 l 6 8:5 m; object distance, a: 1:0 m 6 a 6 l nL ðn 1Þ
s g where n ¼ 3; 4 is the number of lenses in the system; the triangular region of a values where a > l nL ðn 1Þs g is ignored as having no sense (Figs. 4–6).
400
1500 1200
R2.2
R2.3
900 600 300
P2.1
l
800 600 [cm] 400
0
200
(a)
600 400 200 a [cm] P2.3
E[µm2*mrad 2 ] 0
crest 1.5
P2.2
1.0 |Cpy|[µm/mrad/%] 400000 |Cpx|
200
260000
P1
l [cm]
200000 P2.1
100000 0
800 600 400
l [cm]
400 200
200
(b)
0.0 200
600 800
(a)
400 600 a [cm] R1
0
600
2*rx[ µm] 2*r [µm]
R1
P2.3
a [cm]
R2.1
0.5
400
300000
y
35
P2.2
100 80
[µm/mrad3] /104 3 /104 /104
P1
-76
1200000 55
-250
900000
60 40 R2.1
20
600000 P2.1
0 600 400
600
l [cm] 400
200
200
0 600
800
300000 800
(c)
641
0
0
600
0
P2.3
P2.2
Fig. 4. Demagnifications and aberration coefficients for a twoparameter set of a high-excitation triplet (g ¼ 10 cm). Varied parameters: l is the system length and a is the object distance.
400
l [cm] 400 200
a [cm]
(b)
R2.2
0
200
R2.3
a [cm]
Fig. 5. Plots of (a) emittance and (b) object slit dimensions for a two-parameter set of perfect probe-forming systems based on a high-excitation triplet (g ¼ 10 cm).
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A.G. Ponomarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 637–644
R2.2
R2.3
E[µm2*mrad 2] crest 1.5 1.0 R2.1
0.5
200 400
0.0 200
l [cm] 600
400 600 a [cm] R1
800
(a)
2*rx[µm] 2*r [µm]
R1
y
60
85
0:000538 [1/cm2 ]) field contributions, which corresponds to the þ10 lm displacement of one pole in the radial direction from the center for the aperture radius ra ¼ 0:635 cm. The 100% spot is the one formed by all particles that passed through the object and the angular slit. In order to estimate the effect of the parasitic field components on the resolution of probeforming systems based on triplets and quadruplets of magnetic quadrupole lenses the following calculations were carried out. Two-parameter sets were constructed for the triplet and the Russian quadruplet with different working distances (g ¼ 10 cm and 15 cm). For each set, by varying the magnitude of parasitic field components in the lenses and the beam spot size the highest emittance
50 40 W3=W4=0 W3/W2=0.00113, W4/W2=0.00159 W3/W2=0.00384, W4/W2=0.00514 W3/W2=0.01269, W4/W2=0.01557
30 20 10 0 600 400
600
l [cm] (b)
400 200
2
0 R2.3
200 a [cm]
R2.2
Fig. 6. Plots of (a) emittance and (b) object slit dimensions for a two-parameter set of perfect probe-forming systems based on a Russian quadruplet (g ¼ 10 cm).
20
2
800
30 Emittance [ µm *mrad ]
R2.1
10
0
(a)
0
1
2
3
4
0
1
2
3
4
2
Emittance [ µm *mrad ]
30
20
2
The two-parameter sets of probe-forming systems in Figs. 4–6 are subdivided into four regions like in [6], where they are discussed in more detail. The first region, P1, comprises systems with high demagnifications and large aberrations. Regions P2.i (i ¼ 1; 2; 3) include systems with moderate demagnifications and acceptable aberrations, P2.1 long systems with a small object distance, P2.2 short systems and P2.3 long systems with a large object distance. Figs. 5 and 6 show beam emittances and corresponding object slit dimensions for two-parameter sets of perfect probe-forming systems for a fixed 100% beam spot diameter of 0.3 lm with each lens having parasitic sextupole (W3 =W2 ¼ U3 =W2 ¼ 0:000375 [1/cm]) and octupole (W4 =W2 ¼ U4 =W2 ¼
(b)
10
0
d [ µm]
Fig. 7. Maximum emittance as function of the beam spot size for parameter sets of perfect probe-forming systems based on a high excitation triplet (g ¼ 10; 15 cm) for different parasitic lens field components.
A.G. Ponomarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 637–644
values were determined and surfaces were constructed similar to those shown in Figs. 5(a) and 6(a). Then for each surface the maximum was found corresponding to the highest emittance in the emittance set. Figs. 7 and 8 represent the maximum emittance values for various parameter sets of probe-forming systems and parasitic field components of the lenses versus the beam spot size. The subject of our discussion here is the beam emittance rather than the beam current since for different accelerator types and beam transport systems authors often report differing beam brightnesses, b, at the entrance to the microprobe line [16]. Hence, in order to determine required emittance, E, for prescribed current, I, one has to do a simple calculation: E ¼ I=b; then from Figs. 7 and 8, knowing the parasitic field components, the minimum spot size can be estimated. W3 =W4=0 W3 /W2=0.00113, W4/W2=0.00159 W3 /W2=0.00384, W4/W2=0.00514 W3 /W2=0.01269, W4/W2=0.01557
2
2
Emittance [ µm *mrad ]
40 30 20 10 0
(a)
0
1
2
3
4
2
2
Emittance [ µm *mrad ]
20 1.73
0 0 0.3
(b)
A qualitative analysis of contributions to the beam spot size from the object slit image and aberration-produced spot broadening suggests that probe-forming systems are feasible in which the demagnification and the aberration coefficients are balanced. These are represented in Figs. 5(a) and 6(a) by the crest line; differing values of the parasitic components do not cause this line to shift considerably for fixed beam spot dimensions, changing only the maximum emittance value. Figs. 5(b) and 6(b) show object slit dimensions for two-parameter sets of probe-forming systems which provide the corresponding fixed beam spot size. The surfaces exhibit typical valleys related to the effect of aberrations on the object slit dimensions comparable to those of the angular slit, i.e. the aberration-produced spot broadening is limited not only by the angular but also by the object slit, resulting in the reduced maximum emittance value (Figs. 5(a) and 6(a)). As follows from a comparison between the long and the short systems corresponding to regions P2.3 and P2.2 in Figs. 5 and 6, for practically similar emittance values long systems allow essentially larger object slit dimensions owing to higher demagnification coefficients. In real microprobe systems there are transverse oscillations of amplitude reaching 60 lm [17]. Thus the effective collimator dimensions, deff , are defined as ð7Þ
In the above relation the vibration term, Ddvib , leads to a significant increase in the beam spot size in short systems compared to long ones, as if parasitic vibrational beam scanning over the sample took place. It can be seen in Figs. 7 and 8 that the Russian quadruplet and the high excitation triplet have roughly equal spatial resolution over a wide range of beam spot sizes.
30
10
5. Results and discussion
deff ¼ dcol þ Ddvib ¼ dcol ð1 þ DdÞ:
40
643
1
2
3
4
d [ µm]
Fig. 8. Maximum emittance as function of the beam spot size for parameter sets of perfect probe forming systems based on Russian quadruplet (g ¼ 10; 15 cm) for different parasitic lens field components.
6. Conclusions The effect of lens parasitic fields on the microprobe spatial resolution was simulated for
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conventional magnetic quadrupole probe-forming systems. The calculations of the field structure in the lenses give the dependence of the parasitic components on the quadrupole asymmetries due to manufacture inaccuracies and pole-tip excitation errors. In this case lens parasitic fields W3 =W2 ¼ U3 =W2 ¼ 0:000375 [1/cm] and W4 =W2 ¼ U4 =W2 ¼ 0:000538 [1/cm2 ] have minimum permissible values above which the related parasitic aberrations become dominant. This imposes limits upon the lens manufacture requirements (<10 lm for ra ¼ 0:635 cm and <4 lm for ra ¼ 0:375 cm) and the choice of material that has to provide the magnetic induction imbalance at one of the poles produced by residual magnetization below 0.32% (for ra ¼ 0:635 cm) and 0.19% (for ra ¼ 0:375 cm) of maximum magnetic induction under operating conditions. From the above discussion it is clear that the use of conventional lenses [4,5] solves the problem of parasitic sextupole and octupole aberrations. Moreover, with present-day precision accelerators of new type [3] that have beam energy spread DE=E 105 , the contributions in the beam spot size broadening from chromatic aberrations can be made insignificant. Thus, the spatial resolution of magnetic quadrupole probe-forming systems based on triplets and quadruplets is mainly limited by intrinsic third-order aberrations. With the maximum beam brightness b ¼ 60 pA/(lm2 mrad2 ) for a 3 MeV proton beam [16] and beam current at the target I ¼ 100 pA, a sufficient emittance is E ¼ I=b ¼ 1:67 lm2 mrad2 . Fig. 7(b) shows a 100% beam diameter of 0.3 lm. This seems to be the spatial resolution limit for probeforming systems of interest. Their spatial resolution can only be increased by increasing the beam brightness from the accelerator. So at the present stage of development of accelerator-based microanalysis instrumentation, design and construction of high-brightness hydrogen/helium ion sources is a first priority.
Acknowledgements The authors acknowledge the assistance of Dr. S.M. Yudina with the preparation of this paper for publication. This work is supported by Ministry of Education and Science of the Ukraine, Project N2M712001 and BMBF/Berlin (Germany), Project UKR 00/003.
References [1] V. Brazhnik, A. Dymnikov, R. Hellborg, S. Lebed, J. Pallon, V. Storizhko, Nucl. Instr. and Meth. B 77 (1993) 29. [2] D.N. Jamieson, Nucl. Instr. and Meth. B 181 (2001) 1. [3] D.J.W. Mous, R.G. Haitsma, T. Butz, R.-H. Flagmeyer, D. Lehmann, J. Vogt, Nucl. Instr. and Meth. B 130 (1997) 31. [4] M.B.H. Breese, G.W. Grime, W. Linford, M. Harold, Nucl. Instr. and Meth. B 158 (1999) 48. [5] C.G. Ryan, D.N. Jamieson, Nucl. Instr. and Meth. B 158 (1999) 97. [6] V.A. Brazhnik, V.I. Miroshnichenko, A.G. Ponomarev, V.E. Storizhko, Nucl. Instr. and Meth. B 174 (2001) 385. [7] A.G. Ponomarev, Ph.D. Thesis, Sumy 1999, unpublished. [8] V.A. Brazhnik, S.A. Lebed, V.I. Miroshnichenko, A.G. Ponomarev, V.E. Storizhko, Nucl. Instr. and Meth. B 171 (2000) 558. [9] A. Dymnikov, R. Hellborg, Nucl. Instr. and Meth. A 330 (1993) 323. [10] A.D. Dymnikov, G.M. Osetinskij, Fiz. Elem. Chastits At. Yadra 20 (1989) 694; Sov. J. Part. Nucl. 20 (1989) 293. [11] A.D. Dymnikov, G. Martinez, A.H. Azbaid, Nucl. Instr. and Meth. A 403 (1998) 195. [12] A.D. Dymnikov, Nucl. Instr. and Meth. A 363 (1995) 435. [13] M. Szilagyi, in: I. Brodie, J.J. Murray (Eds.), Microdevices: Physics and Fabrication Technologies, Plenum, 1988. [14] S. Lebed, A. Ponomarev, Nucl. Instr. and Meth. B 130 (1997) 90. [15] K. Halbach, Nucl. Instr. and Meth. 74 (1969) 147. [16] R. Szymanski, D.N. Jamieson, Nucl. Instr. and Meth. B 130 (1997) 80. [17] O. Schmelmer, G. Dollinger, G. Datzmann, C. Goeden, H.-J. K€ orner, Nucl. Instr. and Meth. B 158 (1999) 107.
Resolution limit of probe-forming systems with magnetic quadrupole lens triplets and quadruplets A.G. Ponomarev a,*, K.I. Melnik a, V.I. Miroshnichenko a, V.E. Storizhko a, B. Sulkio-Cleff b a
Institute of Applied Physics, National Academy of Sciences of the Ukraine, Petropavlovskaja Street 58, Sumy 244030, Ukraine b Institute of Nuclear Physics, University of M€unster, M€unster, Germany Received 22 August 2002; received in revised form 10 December 2002
Abstract Over the past decade, in MeV ion beam microanalysis efforts to achieve a spatial resolution better than 0.1 lm with a beam current of 100 pA have been connected with microprobes of new generation where the probe is formed by means of separated magnetic quadrupole lens structures [1]. However, as was pointed out in [2], no dramatic improvements in spatial resolution have been produced so far. For better understanding of the situation the authors carried out theoretical studies of multiparameter sets of probe-forming systems based on separated triplets and quadruplets of magnetic quadrupole lenses. Comparisons were made between the highest current values attained at different systems for a given beam spot size. The maximum parasitic sextupole and octupole field components were found whose contributions to spot broadening are tolerable. It is shown that the use of modern electrostatic accelerators [3] and precision magnetic quadrupole lenses [4,5] makes it possible to eliminate the effect of chromatic aberrations and second- and third-order parasitic aberrations resulting from distortions of the quadrupole lens symmetry. Therefore probe-forming systems with triplets and quadruplets of magnetic quadrupole lenses have a lower theoretical spatial resolution limit which is restricted mainly by intrinsic spherical third-order aberrations in state-of-the-art microprobes. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 41.85.p; 41.85.Gy; 41.85.Lc; 41.85.Ne Keywords: Magnetic quadrupole lens; Probe-forming system; Optimization; Emittance
1. Introduction The beam spot size is determined by the total contribution coming from the linear image of the
*
Corresponding author. Fax: 380-542-223-760. E-mail address: [email protected] (A.G. Ponomarev).
object slit and the aberration-induced beam broadening that can be limited by an angular slit. So the same beam spot size can be obtained by various combinations of an angular and an object slits. As is known, the slits dimensions influence the emittance and consequently (considering brightness) the beam current at the entrance to the magnetic quadrupole lens structure. The procedure of selecting most suitable probe-forming
0168-583X/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-583X(02)02229-2
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systems using a two-parameter set for the separated Russian quadruplet of magnetic quadrupole lenses was discussed in detail in [6] where the aberration-induced beam spot broadening included contributions from chromatic and intrinsic spherical third-order aberrations. It was demonstrated that there are probe-forming systems in which demagnifications and aberrations are balanced out for a given beam spot size. A proper selection of the object and the angular slits dimensions permits maximum current to be obtained for these systems which we call ‘‘perfect’’. In this paper we take the next step to elucidate the effect of the sextupole and octupole parasitic aberrations in the probe-forming systems. The analysis procedures developed in [6–8] allow a correct comparison between systems with varying number of lenses. Here we continue the discussion presented in [8] by comparing the performance parameters for the separated high excitation triplet and the separated Russian quadruplet.
2. Calculations of parasitic aberrations in magnetic quadrupole lens structures using the matrizant method For purposes of clarity we will first dwell upon some aspects of our approach to the inclusion of parasitic aberrations into the optimization problem to be solved for the probe-forming systems. The calculations of the magnetic quadrupole lens aberrations including contributions from the parasitic sextupole and octupole components involve the axial field model. It assumes that in the lens there is a straight-line optical axis along which the field j~ Bj ¼ 0. Naturally, this is an approximation corresponding to the averaged flight path of a particle that enters the lens at zero angles retaining at the exit the position in the transverse plane and the direction of flight. In the arbitrary Cartesian _ _ coordinates ðx ; y ; zÞ where the z-axis is aligned with the optical lens axis, the magnetic quadrupole field can be described by the scalar potential as _ _
wð x ; y ; zÞ _
_2
_2
_
__
¼ U 2 ðzÞð x y Þ þ 2W 2 ðzÞ x y
_
_3
_
__2
_2 _
_3
þ U 3 ðzÞð x 3 x y Þ þ W 3 ðzÞð3 x y y Þ _ 00
_
_4
þ ðU 4 ðzÞ U 2 ðzÞ=12Þ x _
_ 00
_3 _
_
_ 00
__3
_
_2 _2
þ ð4W 4 ðzÞ W 2 ðzÞ=6Þ x y 6U 4 ðzÞ x y ð4W 4 ðzÞ þ W 2 ðzÞ=6Þ x y _ 00
_
_4
þ ðU 4 ðzÞ þ U 2 ðzÞ=12Þy þ ; _
ð1Þ
_
where U i ðzÞ and W i ðzÞ are the field components of the magnetic quadrupole lens. _ On the _assumption that the major W 2 ðzÞ and the skew U 2 ðzÞ quadrupole components are governed by the same distribution law, in the ðx; y; zÞ _ _ coordinates rotated about the ð x ; y ; zÞ coordinates through an angle _
1 a ¼ arc tg 2
U 2 ð0Þ _
! ;
ð2Þ
W 2 ð0Þ
(the rotation is about the z-axis) the scalar potential finally takes the form wðx; y; zÞ ¼ 2W2 ðzÞxy þ U3 ðzÞx3 þ 3W3 ðzÞx2 y 3U3 ðzÞxy 2 W3 ðzÞy 3 þ U4 ðzÞx4 þ ð4W4 ðzÞ W200 ðzÞ=6Þx3 y 6U4 ðzÞx2 y 2 ð4W4 ðzÞ þ W200 ðzÞ=6Þxy 3 þ U4 ðzÞy 4 þ ; ð3Þ where W2 ðzÞ is the major quadrupole field component, W3 ðzÞ and U3 ðzÞ are the major and the skew sextupole parasitic components, W4 ðzÞ and U4 ðzÞ are the major and the skew octupole parasitic components. The ðx; y; zÞ coordinates connected with the optical lens axis and antisymmetry planes of the major quadrupole component W2 ðzÞ are referred to as eigencoordinates and the components W3 ðzÞ, U3 ðzÞ, W4 ðzÞ and U4 ðzÞ parasitic eigencomponents of the magnetic quadrupole. Any minor transformation of a lens as an integral rigid body would lead to the redistribution of the multipole components and production of additional parasitic dipole and/or skew quadrupole components.
A.G. Ponomarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 637–644
All the components are considered in the normalized form for two representations of the longitudinal distribution Wi ðzÞ ¼ Wi fj ðzÞ;
Ui ðzÞ ¼ Ui fj ðzÞ;
i ¼ 2; 3; 4; j ¼ 1; 2; with f1 ðzÞ ¼
8 < 1; :
1
2 3 1þeðC0 þC1 sþc2 s þC3 s Þ
0;
ð4Þ
;
jzj 6 z2 ; z2 < jzj 6 z0 ; s ¼ z=r0 ; jzj > z2 ;
where r0 is the lens bore radius, f2 ðzÞ ¼ hðz z1 Þ hðz þ z1 Þ with 0; z < 0; hðzÞ ¼ 1; z P 0; Rz z1 ¼ Leff =2 with Leff ¼ z0 0 f1 ðsÞ ds, ðz0 ; z2 Þ and ðz2 ; z0 Þ are the coordinates of the input and output fringe field region. f1 ðzÞ is used in approximation with allowance for the real fringe field and f2 ðzÞ corresponds to the conventional rectangular representation. To determine the transformation matrix for the magnetic quadrupole lens in which the field is described by the scalar potential (3) and expressions (4) matrizant method is employed [9–11]. In our case the phase moment space is represented by the vector ~ ¼ ðx; x0 ; y; y 0 ; xd; x0 d; yd; y 0 d; x2 ; xx0 ; x02 ; y 2 ; yy 0 ; y 02 ; X xy; x0 y; xy 0 ; x0 y 0 ; x3 ; x2 x0 ; xx02 ; x03 ; xy 2 ; xyy 0 ; xy 02 ; x0 y 2 ; x0 yy 0 ; x0 y 02 ; y 3 ; y 2 y 0 ; yy 02 ; y 03 ; yx2 ; yxx0 ; yx02 ; y 0 x2 ; y 0 xx0 ; y 0 x02 ÞT ;
ð5Þ
where x and y are the beam deviations from the optical lens axis (z), x0 and y 0 are angles dictating the beam direction, and d is the momentum spread of the beam particles. ~ vector along the z-axis is The evolution of the X determined by the relation ~ðzÞ ¼ Rðz=z0 ÞX ~ðz0 Þ; X
ð6Þ
where Rðz=z0 Þ is the 38 38 matrizant (transformation matrix) for the z0 to z-plane transformations, the first and the third matrizant rows containing a complete set of aberrations including
639
chromatic, parasitic second- and third-order aberrations due to the major and skew sextupole and octupole components and intrinsic third-order aberrations. For the rectangular representation of the field components analytical expressions were derived for all matrizant elements. This was done using present-day programs of analytical calculations and in part the distribution theory. The possibility of calculating the matrizant on the basis of the analytical expressions is important, permitting analysis of a large number (several thousand) of probe-forming systems to be performed within a reasonable time. In order to estimate errors in the matrizant computations for the rectangular representation of the field components calculations were carried out for a distribution approximating a real fringe field by the shuttle-sum method [12]. Fig. 1 shows some aberrations in a single magnetic quadrupole lens versus the relative width of the fringe field, m ¼ ðz0 z2 Þ=Leff . If m ! 0 for Leff ¼ const, aberrations approach corresponding values obtained for the rectangular representation. Note that aberrations for a zero width of the fringe field (rectangular model) are minimum for the magnetic quadrupole.
3. Calculations of parasitic field components in magnetic quadrupole lenses Parasitic field components, Wi and Ui , produced in a magnetic quadrupole lens by distortions of the quadrupole symmetry were found by solving numerically the Laplace equation for the scalar magnetic potential in the working lens area using the charge-density method [13]. It is known to be very accurate and ideally suited to determination of the field structure in ion-optic elements, permitting higher derivatives of the scalar magnetic potential to be calculated analytically by differentiating the integral operator nucleus [14]. Unlike [15], where an approximated analytical model is used, our calculations were performed for real pole tip shapes. The quadrupole symmetry distortions were simulated by a planar shift of one or two poles and the excitation error by that in prescribed
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analitical numerical
U3/W2 = W3/W2 [1/cm], r=0.635cm,
3
2
U4/W2 [1/cm ], r=0.635cm, U3/W2 = W3/W2 [1/cm], r=0.375cm,
2
2
U4/W2 [1/cm ], r=0.375cm.
1.0
0.006
Field components
0.5 0.0 -0.5 -1.0
2
3
2
3
-1.5 10
20
30
40
-40
-20
0
3
20
40
∆r [µm]
(a)
80
6
60 40 20 2
6
0
0.002 0.001 0.000 -0.001 -0.002
10
20
30
40
50
ν / L [%]
Fig. 1. Spherical aberrations of a single magnetic quadrupole lens as function of the fringing field relative width m ¼ ðz0 z2 Þ=Leff .
0.0
0.5
1.0
1.5
∆ [%]
Fig. 2. Relative parasitic sextupole (W3 =W2 ¼ U3 =W2 ) and octupole (W4 =W2 ¼ U4 =W2 ) field components as function of (a) variations in the pole gap fitting and (b) pole-tip excitation error for two aperture radii.
SEPARATED QUADRUPLET λ L1 s L2 L4 L3 s K1
a
g
SEPARATED TRIPLET a
L1
λ
K1
OBJECT
The choice of optimum probe-forming systems for a nuclear microprobe is based on a comparison between the highest emittance values that can be provided by these systems for a given beam spot size. To perform the comparative analysis twoparameter sets were constructed for perfect probeforming systems where independent parameters are the system length, l, and the object distance, a
-0.5
(b)
boundary conditions at one of the poles. Fig. 2 illustrates parasitic sextupole and octupole field components relative to the major quadrupole component versus the pole shifts and the excitation error, for two values of the aperture radius, ra ¼ ð0:635; 0:375 cmÞ.
4. Comparison between probe forming systems
-1.0
L2
s L3
g
TARGET
0
-1.5
K4
-40
6
K3
K3
2
-20
K2
3
-0.003 -0.006
ν / L [%]
Aber ra tions [ µm / mrad ]
0.000
50
Field components
0
0.003
K2
Aber ra tions [ µm / mrad ]
1.5
l Fig. 3. Lens arrangement along the ion-optical line for separated triplet and quadruplet structures.
A.G. Ponomarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 637–644
(Fig. 3). Thus, we have a parameter set that includes various suitable lens positions along the optical line for separated quadrupole structures. Fig. 4 represents the demagnification and aberration coefficients for the two-parameter set of a high excitation triplet, while similar simulations but for the Russian quadruplet are given in our previous paper [6] where we describe at greater length how the expressions relating the demagnification and aberration coefficients to a and l were derived. The lens parameters and dimensions of the systems were selected such as to cover most microprobes available: Dx 1800 P1
|Dy|
effective lens lengths Li ¼ L ¼ 6:4 cm; lens power supply for the triplet k1 ¼ k2 and for the Russian quadruplet k1 ¼ k4 , k2 ¼ k3 , each even-number lens being rotated about its basic position through 90°; working distance g ¼ ð10 cm; 15 cmÞ; fixed distance in paired doublets s ¼ 3:3 cm (Fig. 3); system length was varied in the range 1:5 m 6 l 6 8:5 m; object distance, a: 1:0 m 6 a 6 l nL ðn 1Þ
s g where n ¼ 3; 4 is the number of lenses in the system; the triangular region of a values where a > l nL ðn 1Þs g is ignored as having no sense (Figs. 4–6).
400
1500 1200
R2.2
R2.3
900 600 300
P2.1
l
800 600 [cm] 400
0
200
(a)
600 400 200 a [cm] P2.3
E[µm2*mrad 2 ] 0
crest 1.5
P2.2
1.0 |Cpy|[µm/mrad/%] 400000 |Cpx|
200
260000
P1
l [cm]
200000 P2.1
100000 0
800 600 400
l [cm]
400 200
200
(b)
0.0 200
600 800
(a)
400 600 a [cm] R1
0
600
2*rx[ µm] 2*r [µm]
R1
P2.3
a [cm]
R2.1
0.5
400
300000
y
35
P2.2
100 80
P1
-76
1200000 55
-250
900000
60 40 R2.1
20
600000 P2.1
0 600 400
600
l [cm] 400
200
200
0 600
800
300000 800
(c)
641
0
0
600
0
P2.3
P2.2
Fig. 4. Demagnifications and aberration coefficients for a twoparameter set of a high-excitation triplet (g ¼ 10 cm). Varied parameters: l is the system length and a is the object distance.
400
l [cm] 400 200
a [cm]
(b)
R2.2
0
200
R2.3
a [cm]
Fig. 5. Plots of (a) emittance and (b) object slit dimensions for a two-parameter set of perfect probe-forming systems based on a high-excitation triplet (g ¼ 10 cm).
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A.G. Ponomarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 637–644
R2.2
R2.3
E[µm2*mrad 2] crest 1.5 1.0 R2.1
0.5
200 400
0.0 200
l [cm] 600
400 600 a [cm] R1
800
(a)
2*rx[µm] 2*r [µm]
R1
y
60
85
0:000538 [1/cm2 ]) field contributions, which corresponds to the þ10 lm displacement of one pole in the radial direction from the center for the aperture radius ra ¼ 0:635 cm. The 100% spot is the one formed by all particles that passed through the object and the angular slit. In order to estimate the effect of the parasitic field components on the resolution of probeforming systems based on triplets and quadruplets of magnetic quadrupole lenses the following calculations were carried out. Two-parameter sets were constructed for the triplet and the Russian quadruplet with different working distances (g ¼ 10 cm and 15 cm). For each set, by varying the magnitude of parasitic field components in the lenses and the beam spot size the highest emittance
50 40 W3=W4=0 W3/W2=0.00113, W4/W2=0.00159 W3/W2=0.00384, W4/W2=0.00514 W3/W2=0.01269, W4/W2=0.01557
30 20 10 0 600 400
600
l [cm] (b)
400 200
2
0 R2.3
200 a [cm]
R2.2
Fig. 6. Plots of (a) emittance and (b) object slit dimensions for a two-parameter set of perfect probe-forming systems based on a Russian quadruplet (g ¼ 10 cm).
20
2
800
30 Emittance [ µm *mrad ]
R2.1
10
0
(a)
0
1
2
3
4
0
1
2
3
4
2
Emittance [ µm *mrad ]
30
20
2
The two-parameter sets of probe-forming systems in Figs. 4–6 are subdivided into four regions like in [6], where they are discussed in more detail. The first region, P1, comprises systems with high demagnifications and large aberrations. Regions P2.i (i ¼ 1; 2; 3) include systems with moderate demagnifications and acceptable aberrations, P2.1 long systems with a small object distance, P2.2 short systems and P2.3 long systems with a large object distance. Figs. 5 and 6 show beam emittances and corresponding object slit dimensions for two-parameter sets of perfect probe-forming systems for a fixed 100% beam spot diameter of 0.3 lm with each lens having parasitic sextupole (W3 =W2 ¼ U3 =W2 ¼ 0:000375 [1/cm]) and octupole (W4 =W2 ¼ U4 =W2 ¼
(b)
10
0
d [ µm]
Fig. 7. Maximum emittance as function of the beam spot size for parameter sets of perfect probe-forming systems based on a high excitation triplet (g ¼ 10; 15 cm) for different parasitic lens field components.
A.G. Ponomarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 201 (2003) 637–644
values were determined and surfaces were constructed similar to those shown in Figs. 5(a) and 6(a). Then for each surface the maximum was found corresponding to the highest emittance in the emittance set. Figs. 7 and 8 represent the maximum emittance values for various parameter sets of probe-forming systems and parasitic field components of the lenses versus the beam spot size. The subject of our discussion here is the beam emittance rather than the beam current since for different accelerator types and beam transport systems authors often report differing beam brightnesses, b, at the entrance to the microprobe line [16]. Hence, in order to determine required emittance, E, for prescribed current, I, one has to do a simple calculation: E ¼ I=b; then from Figs. 7 and 8, knowing the parasitic field components, the minimum spot size can be estimated. W3 =W4=0 W3 /W2=0.00113, W4/W2=0.00159 W3 /W2=0.00384, W4/W2=0.00514 W3 /W2=0.01269, W4/W2=0.01557
2
2
Emittance [ µm *mrad ]
40 30 20 10 0
(a)
0
1
2
3
4
2
2
Emittance [ µm *mrad ]
20 1.73
0 0 0.3
(b)
A qualitative analysis of contributions to the beam spot size from the object slit image and aberration-produced spot broadening suggests that probe-forming systems are feasible in which the demagnification and the aberration coefficients are balanced. These are represented in Figs. 5(a) and 6(a) by the crest line; differing values of the parasitic components do not cause this line to shift considerably for fixed beam spot dimensions, changing only the maximum emittance value. Figs. 5(b) and 6(b) show object slit dimensions for two-parameter sets of probe-forming systems which provide the corresponding fixed beam spot size. The surfaces exhibit typical valleys related to the effect of aberrations on the object slit dimensions comparable to those of the angular slit, i.e. the aberration-produced spot broadening is limited not only by the angular but also by the object slit, resulting in the reduced maximum emittance value (Figs. 5(a) and 6(a)). As follows from a comparison between the long and the short systems corresponding to regions P2.3 and P2.2 in Figs. 5 and 6, for practically similar emittance values long systems allow essentially larger object slit dimensions owing to higher demagnification coefficients. In real microprobe systems there are transverse oscillations of amplitude reaching 60 lm [17]. Thus the effective collimator dimensions, deff , are defined as ð7Þ
In the above relation the vibration term, Ddvib , leads to a significant increase in the beam spot size in short systems compared to long ones, as if parasitic vibrational beam scanning over the sample took place. It can be seen in Figs. 7 and 8 that the Russian quadruplet and the high excitation triplet have roughly equal spatial resolution over a wide range of beam spot sizes.
30
10
5. Results and discussion
deff ¼ dcol þ Ddvib ¼ dcol ð1 þ DdÞ:
40
643
1
2
3
4
d [ µm]
Fig. 8. Maximum emittance as function of the beam spot size for parameter sets of perfect probe forming systems based on Russian quadruplet (g ¼ 10; 15 cm) for different parasitic lens field components.
6. Conclusions The effect of lens parasitic fields on the microprobe spatial resolution was simulated for
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conventional magnetic quadrupole probe-forming systems. The calculations of the field structure in the lenses give the dependence of the parasitic components on the quadrupole asymmetries due to manufacture inaccuracies and pole-tip excitation errors. In this case lens parasitic fields W3 =W2 ¼ U3 =W2 ¼ 0:000375 [1/cm] and W4 =W2 ¼ U4 =W2 ¼ 0:000538 [1/cm2 ] have minimum permissible values above which the related parasitic aberrations become dominant. This imposes limits upon the lens manufacture requirements (<10 lm for ra ¼ 0:635 cm and <4 lm for ra ¼ 0:375 cm) and the choice of material that has to provide the magnetic induction imbalance at one of the poles produced by residual magnetization below 0.32% (for ra ¼ 0:635 cm) and 0.19% (for ra ¼ 0:375 cm) of maximum magnetic induction under operating conditions. From the above discussion it is clear that the use of conventional lenses [4,5] solves the problem of parasitic sextupole and octupole aberrations. Moreover, with present-day precision accelerators of new type [3] that have beam energy spread DE=E 105 , the contributions in the beam spot size broadening from chromatic aberrations can be made insignificant. Thus, the spatial resolution of magnetic quadrupole probe-forming systems based on triplets and quadruplets is mainly limited by intrinsic third-order aberrations. With the maximum beam brightness b ¼ 60 pA/(lm2 mrad2 ) for a 3 MeV proton beam [16] and beam current at the target I ¼ 100 pA, a sufficient emittance is E ¼ I=b ¼ 1:67 lm2 mrad2 . Fig. 7(b) shows a 100% beam diameter of 0.3 lm. This seems to be the spatial resolution limit for probeforming systems of interest. Their spatial resolution can only be increased by increasing the beam brightness from the accelerator. So at the present stage of development of accelerator-based microanalysis instrumentation, design and construction of high-brightness hydrogen/helium ion sources is a first priority.
Acknowledgements The authors acknowledge the assistance of Dr. S.M. Yudina with the preparation of this paper for publication. This work is supported by Ministry of Education and Science of the Ukraine, Project N2M712001 and BMBF/Berlin (Germany), Project UKR 00/003.
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