- Email: [email protected]
Experimental ray-tracing – a precise and accurate method for the determination of the properties of optical systems
Nuclear Instruments and Methods in Physics Research B 158 (1999) 90±96
www.elsevier.nl/locate/nimb
Experimental ray-tracing ± a precise and accurate method for the determination of the properties of optical systems M. Maetz
a,*
~ez-Ruiz a, T. Schneider a, S. Scheloske a, A. Wallianos a, C. Wies a, , A. N un K. Traxel b a
b
Max-Planck-Institut f ur Kernphysik, P.O. Box 103980, 69029 Heidelberg, Germany Physikalisches Institut, Universit at Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany
Abstract We developed a new technique to determine the impact position of single pencil beams in the image plane with an accuracy of 0.2 lm. This technique bases on the overlapping of secondary electron images from a 1000 mesh grid. The measurements themselves can be obtained without any change in the routine setup if a secondary electron counter is installed. By choosing dierent object positions (x0 , y0 ) and divergence angles (h0 , u0 ) for the pencil beam the characteristics of the used lens system can be analysed and the lens errors can be calculated with high precision. Results for the Heidelberg quadrupole doublet will be reported. Ó 1999 Elsevier Science B.V. All rights reserved. PACS: 29.90; 41.75; 41.85; 61.16 Keywords: Ray-tracing; Nuclear microprobe; Secondary electrons; Lens errors; Ion optics
1. Introduction One of the most laborious tasks during the daily work of a nuclear microscopist is the challenge to get as much current as possible for a beam size as small as possible whereby the brightness of the used ion source represents the theoretical limit. Common lens systems, for example magnetic quadrupole multiplets, normally suer from parasitic aberrations which worsen the reachable reso-
* Corresponding author: Tel.: +49-6221-516-210; fax: +496221-516-540; e-mail: [email protected]
lution in a nuclear microprobe. Of course other in¯uences like ripple of the power supplies, mechanical vibrations or stray magnetic ®elds work in the same direction. A possible increase of the daily performance bases on the exact knowledge of the optical properties of the used lens system. With numerical ray-tracing or matrix methods the optical properties can be calculated for a given geometry. Parasitic eects which are present in practice can however only be modelled poorly. So there is the need to determine the optical properties experimentally. Up to now two methods that measure the optical properties of lens systems in nuclear
0168-583X/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 0 3 3 2 - 8
M. Maetz et al. / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 90±96
microprobes have been used. In the grid shadow method [1] the shadow pattern of a lm-sized grid as summary eect of all optical in¯uences is analysed. Experimental ray-tracing, the second method, bases on the determination of the impact coordinates (xi , yi ) of pencil beams in the image plane [3]. The higher eort involved to do experimental ray-tracing is counterbalanced by the fact that pencil beams can be chosen to display mainly or sometimes even only one of the ionoptical properties like demagni®cation, dependence on the divergence angles h and u separately, etc. In this paper we present the development of experimental ray-tracing with high accuracy and precision as a method which can be realized without any change in the routine set-up if a secondary electron counter is installed. The method is sensitive to the ion-optical properties only and ignores the above mentioned additional in¯uences on the beam size like for instance mechanical vibrations.
2. The method 2.1. The Heidelberg set-up The complete set-up of the Heidelberg proton microprobe including the lens system is described in detail in Ref. [4]. Only relevant information is given here for better understanding. The lens system consists of a magnetic quadrupole doublet, totally located inside the vacuum. The distance between the object slits and the aperture slits is 1957 mm, the distance between the object slits and the image plane is 4740 mm. The beam size has been measured to be 0.3 lm in STIM experiments. To get 100 pA of beam current the slits have to be opened to result in a 3 lm spot. This is due to the very low brightness of our ion source [5]. The secondary electron imaging system consists of two electrostatic de¯ectors to allow fast beam scanning. Only small regions (about 100 lm2 ) may be scanned to enhance repetition rates to a few seconds per image. To average out beam ¯uctuations the pixel contents from a number of loops
91
can be added and the image can be stored on hard disk. With secondary electron images from 1000 mesh grids the scanning system can be calibrated to give the PIXEL distance in lm. This secondary electron imaging plays a central role in our experimental ray-tracing method. 2.2. The concept of lens errors The mathematical formalism that is used to describe optical properties of a microprobe is based on the following assumptions: In a cartesian coordinate system the z-axis is de®ned by the beam axis with (x, y) (0, 0). The position of the beam in the object plane is denoted with the index o, the position in the image plane with index i. The divergence is measured by the angles h and u which represent the slopes of the particle path projected onto the xz- and yz-plane, respectively. The principle idea of the concept of lens errors is that the image coordinates of the beam (xi , yi ) may be written as polynomials in the system variables with the lens errors as coecients h. . . j . . .i (for details see Refs. [2,6]): X hxjak iak xi hxjx0 ix0
X
k
hxjsk isk hxjx0 DEix0 DE
k
hxjhi hxjhDEiDEh hxjh2 ih2 hxjh3 ih3 hxjh5 ih5 ! X hxjuqk iqk hxjui u k
hxju2 iu2 hxju3 iu3 hxjhuihu hxjhu2 ihu2 hxjh2 uih2 u hxjh3 u2 ih3 u2 hxjhu4 ihu4 X yi hyjy0 iy0 hyjak iak
X
k
hyjsk isk hyjy0 DEiy0 DE
k
hyjui hyjuDEiDEu hyju2 iu2 hyju3 iu3 hyju5 iu5
92
M. Maetz et al. / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 90±96
! X hyjhqk iqk hyjhi h
can be chosen to highlight certain ion-optical properties.
k
hyjh2 ih2 hyjh3 ih3 hyjhuihu
3. Results
hyjuh2 iuh2 hyju2 hiu2 h hyju3 h2 iu3 h2 hyjuh4 iuh4 :
1
The misalignment of the single lenses is fully represented by the rotation q, tilt a and translation s, DE is the energy shift of the particles compared to the ideal energy. 2.3. Experimental ray-tracing The ®rst choice to measure the optical properties of a lens system is the determination of the absolute values for the image coordinates (xi , yi ) with high accuracy to get their dependence on the various system variables. But in practice this is very dicult. Our method uses pencil beams with well de®ned parameters x0 , y0 , u, and h to produce secondary electron images of 1000 mesh grids. To produce the pencil beams (rays) the slit openings were 6 ´ 30 lm2 for the object and 40 ´ 40 lm2 for the aperture. We choose the origin of our coordinate system in the image plane by setting x0 y0 u h 0, i.e., we choose the pencil beam on the geometrical axis. The resulting image (Fig. 1) is then stored in the memory. Then another image is accumulated using a ray propagating in a dierent direction, i.e., one or more parameters x0 , y0 , u, and h being dierent from zero. Due to the lens errors this image is shifted by a certain amount dx, dy (Fig. 1). From the two images the exact number for the shift dx, dy is deduced in the following way (Fig. 1): To get a uniform grey scale all pixel contents less than about 15% of the maximum are set to zero, all others to 1. Then the second image is shifted pixel by pixel until complete overlap is obtained. The precision of the method can also be seen from Fig. 1 where the shift is only d 0.4 lm o the optimum. The experimental procedure makes sure that the measured coordinates (xi , yi ) are neither in¯uenced by time varying external stray ®elds or by noise from power supplies nor by mechanical vibrations. Another advantage is that pencil beams
3.1. Demagni®cation factors By varying the object coordinates x0 , y0 and choosing u h 0 only the demagni®cation factors áx | x0 ñ and áy | y0 ñ are directly measured: dx hxjx0 ix0 ;
dy hyjy0 iy0 :
The demagni®cation factors of our set-up as deduced from the measurements are: hxjx0 i ÿ0:1694 0:0016; hyjy0 i ÿ0:02447 0:00027: It should be stressed that these are the real demagni®cation factors because no lens aberrations dependent on u or h are involved in the measurements. This is for instance not true if áx | x0 ñ and áy | y0 ñ are deduced from beam size measurements. 3.2. Chromatic aberrations Varying the energy E of the beam the chromatic coecients (áx|hDE ñ and áy|uDE ñ) can be determined. By choosing symmetric o axis rays additional information about the alignment of the aperture slits can be achieved. Eight rays were used to measure the chromatic coecients, four points with h equal to zero (u 0.10, 0.20 mrad) and four points with u equal to zero (h 0.13, 0.26 mrad). For the points with h equal to zero the dierences dx, dy can be expressed as: dx 0 and dy áy|uDE ñuDE, whereas for the other four points (u 0): dx áx|hDE ñhDE and dy 0. The quality of the measurements can be seen from Fig. 2. Ideal alignment of the aperture slits leads to (áy|uDE ñu) ÿ(áy|uDE ñÿu) and (áx|hDE ñh) ÿ(áx|hDE ñÿh). Our results show ideal alignment within 3%. From the slopes and the values for u0 , and h0 the coecients of the chromatic aberration can be calculated as mean values áx|hDE ñ (2.162 0.024) lm/(keV mrad) and áy|uDE ñ (2.193 0.050) lm/(keV mrad).
M. Maetz et al. / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 90±96
93
Fig. 1. In a ®rst step all pixel contents less than about 15% of the maximum are set to 0, all others are set to 1 to get a uniform grey scale (1). Second, both images are overlapped where the pixel signal of the resulting image is given by the absolute value of the difference of the pixel contents of the images (2). Third, the second image is shifted pixel by pixel until ``complete'' overlap is obtained giving dx and dy (3). ``Complete'' overlap is de®ned by the minimum value of the subtracted overlap image. For comparison with the ``complete'' overlap also neighboured overlap images are given with a shift of only d 0.42 lm o the optimum (3). The pixel distance is 0.2 lm in both directions (®eld of view 27 ´ 27 lm2 ).
Fig. 2. Measured shifts dx, dy by varying the energy of the proton beam for two arbitrarily chosen pencil rays out of the eight (h 0; u 0.20, left and h 0.26; u 0, right). From the slopes of the ®tted lines and the values for h and u the chromatic aberrations can be calculated.
3.3. Higher order aberrations 3.3.1. The coecients To determine the divergence dependent aberrations the image coordinates xi and yi for a set of 100 rays were measured. All those rays originated in the origin (x0 y0 0), the angles h and u, which cover the range of 2 mrad, were chosen to
display symmetries and simple geometric con®gurations. Eight points were measured twice to check the precision. The values were reproduced with an uncertainty of Ddx 0.18 lm and Ddy 0.16 lm. Fitting of the suitable polynomials (Eq. (1)) to all datapoints results in the aberration coecients of Table. 1 (upper part). The values agree within the given errors with the results of a measurement done about one year ago [7] when the same rays (phase space points) had been analysed. Only the astigmatic coecients áx|hñ and áy|uñ dier which is a result of dierent focusing quality. This means that the experimental values are reproducible to a high degree. As will be seen later, some of the rays with h, u > 0.5 mrad are only badly described by the lens error concept (Eq. (1)). As these rays are far o the acceptance in routine use, a second ®t was done, but only to the data resulting from the rays inside a cone with SQRT(h2 + u2 ) 6 0.5 mrad (72 rays), which is still bigger than the routine acceptance of the aperture. The resulting aberration coecients can also be seen in Table. 1 (lower part). Surprisingly many of the coecients are completely dierent.
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M. Maetz et al. / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 90±96
Table 1 Coecient
Value (lm/mradn )
Data set: h 6 1 mrad, u 6 1.5 mrad áx|hñ (ÿ5.81 0.64) (2.42 0.14) áx|h2 ñ (ÿ3.5 1.0) áx|h3 ñ 5 (1.10 0.36) áx|h P ñ (ÿ7.54 0.86) ( k áx|uqñqk áx|uñ) (ÿ6.12 0.47) áx|u2 ñ (ÿ6.3 1.1) áx|u3 ñ áx|huñ (ÿ2.26 2.34) (ÿ55 23) áx|hu2 ñ (8.6 3.4) áx|h2 uñ (21 26) áx|h3 u2 ñ (5 24) áx|hu4 ñ
áy|uñ áy|u2 ñ áy|u3 ñ 5 áy|u P ñ ( k áy|hqñqk áy|hñ) áy|h2 ñ áy|h3 ñ áy|huñ áy|uh2 ñ áy|u2 hñ áy|u3 h2 ñ áy|uh4 ñ
(18.1 1.1) (3.98 0.38) (ÿ35.9 4.3) (26.3 3.4) (ÿ1.65 0.13) (ÿ0.159 0.044) (0.268 0.075) (ÿ2.00 0.58) (8.9 5.8) (ÿ2.43 0.84) (ÿ39.9 6.4) (ÿ6.7 5.9)
Data set: SQRT(h2 u2 ) 6 0.5 mrad áx|hñ (ÿ8.61 0.41) (2.24 0.27) áx|h2 ñ (17.0 6.5) áx|h3 ñ 5 (ÿ45 20) áx|h P ñ (ÿ6.32 0.28) ( k áx|uqñqk + áx|uñ) (ÿ9.83 0.31) áx|u2 ñ (ÿ11.0 1.4) áx|u3 ñ áx|huñ (ÿ3.92 0.60) (ÿ65 12) áx|hu2 ñ (ÿ0.6 1.7) áx|h2 uñ (96 53) áx|h3 u2 ñ (50 49) áx|hu4 ñ
áy|uñ áy|u2 ñ áy|u3 ñ 5 áy|u P ñ ( k áy|hqñqk + áy|hñ) áy|h2 ñ áy|h3 ñ áy|huñ áy|uh2 ñ áy|u2 hñ áy|u3 h2 ñ áy|uh4 ñ
(16.49 0.62) (ÿ0.54 0.41) (ÿ12 10) (ÿ39 31) (ÿ1.79 0.28) (ÿ1.69 0.31) (0.52 1.43) (ÿ3.84 0.71) (0.12 14.1) (ÿ3.3 2.1) (22 62) (ÿ42 58)
Coecient
Value (lm/mradn )
In the upper part of the table the coecients are listed resulting from the ®t of all datapoints. The coecients in the lower part are the ®t results from the inner cone (SQRT(h2 u2 ) 6 0.5 mrad). Extreme
With the knowledge of the optical properties of the lens system the ``ideal'' phase volume can be calculated. Ideal phase volume means the openings of object and aperture slits to get maximum beam current into a given beam size. The displacement dr SQRT(dx2 + dy2 ) as function of the divergence angles h and u is shown in Fig. 3 for the inner cone < 0:5 mrad. The calculations have been done with the parameters deduced from the ®t to the inner 72 points. 3.3.2. Selected data sets To get a visual impression of the imaging properties of our lens system the hitting points of rays in the image plane are shown for cones (x0 y0 0, SQRT(h2 + u2 ) constant) in Fig. 4. and for the cross (h or u 0) as well as for the diagonals h u and h ÿu in Fig. 5. The full lines represent the calculations with the help of the
Fig. 3. The displacement dr SQRT(dx2 dy2 ) as function of the divergence angles h and u for the inner cone is shown as contour plot. The given numbers represent the displacement in lm.
aberration coecients deduced from the full data set, the dotted lines represent the ``inner'' data set.
M. Maetz et al. / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 90±96
95
Fig. 4. The hitting points of rays in the image plane are shown for cones (x0 y0 0, SQRT(h2 u2 )2[0.25, 0.5, 0.75, 1.0] mrad). The full lines represent the calculations with the help of the aberration coecients deduced from the full data set, the dotted lines represent the data set with SQRT(h2 + u2 ) 6 0.5 mrad. The given letters show the measured points. Dierent scaling for the left and for the right plots have to be taken into account. If only the inner regions are ®tted the resulting polynomial describes this regions very well, but the outer points (h, u > 0.5 mrad) are described very badly and vice versa.
Fig. 5. Measured shifts dx, dy in the image plane are shown for the cross (h 0) as well as for the diagonal h u (the data for the rays with u 0 and with h u give the same results). The full lines represent the calculations with the help of the aberration coecients deduced from the full data set, the dotted lines represent the data set with SQRT(h2 u2 ) 6 0.5 mrad. If only the inner regions are ®tted the resulting polynomial describes this regions very well, but the outer points (h, u > 0.5 mrad) are described very badly and vice versa.
Fig. 4 shows clearly that the region h > 0, u > 0.25 mrad (points M, N, O) is the worst. Another aspect is the fact that the higher aberration coecients depend strongly on the ®tted phase space (radii in Fig. 4). If only the inner
regions are ®tted the resulting polynomial describes this regions very well, but the outer points (h, u > 0.5 mrad) are described very badly and vice versa. Fig. 5 reveals clearly the poorer focusing quality in x than in y for setup. Even for
96
M. Maetz et al. / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 90±96
h 0 y-divergences u P 1 mrad in¯uence xi stronger than yi (Fig. 5, right side). From beam size measurements we suspected this behaviour since long. We thought however that ¯uctuating stray ®elds and noise in the power supplies were responsible. These in¯uences can now be excluded (see 2.3). 4. Conclusions A new method of experimental ray-tracing, based on secondary electron images produced by selected pencil beams has been developed. The method allows to identify the image coordinates (xi , yi ) of any ray (x0 , y0 , h, u) with a precision of Ddx 0.18 lm and Ddy 0.16 lm. An optimization to still reduce this values is straightforward. The ionoptical properties of the Heidelberg setup have been measured, giving precise values for demagni®cation and the chromatic errors. The latter have been measured for 8 rays with h, u < 0.3 mrad only. The higher order aberrations have been analysed for two aperture openings h 6 1 mrad, u 6 1.5 mrad and SQRT(h2 u2 ) 6 0.5 mrad. The resulting aberration coecients dier signi®cantly, the smaller phase space coecients cannot describe the measured results for the bigger phase
space and vice versa. In routine use only small phase spaces (h, u) are used to produce the microbeam. Therefore our results contradict the demand as formulated in [8] to analyse the ionoptical properties for an aperture setting as big as possible. Even more, the question about the signi®cance of the aberration concept to improve the lens properties must be thought over. We propose to do direct ray-tracing to observe the results of technical modi®cations.
References [1] D.N. Jamieson, G.J.F. Legge, Nucl. Instr. and Meth. B 29 (1987) 544. [2] M.B.H. Breese, D.N. Jamieson, P.J.C. King, Materials Analysis using a Nuclear Microprobe, Wiley, New York, 1996. [3] F.W. Martin, Nucl. Instr. and Meth. B 54 (1991) 17. [4] K. Traxel, P. Arndt, J. Bohsung, K.U. Braun-Dullaeus, M. Maetz, D. Reimold, H. Schiebler, A. Wallianos, Nucl. Instr. and Meth. B 104 (1995) 19. [5] R.A. Szymanski, D.N. Jamieson, Nucl. Instr. and Meth. B 130 (1997) 80. [6] G.W. Grime, F. Watt, Beam Optics of Quadrupole forming Systems, Adam Hilger, Bristol, 1984. [7] M. Maetz, Ph.D. thesis, University of Heidelberg, 1997, unpublished. [8] D.N. Jamieson, Nucl. Instr. and Meth. B 104 (1995) 650.
www.elsevier.nl/locate/nimb
Experimental ray-tracing ± a precise and accurate method for the determination of the properties of optical systems M. Maetz
a,*
~ez-Ruiz a, T. Schneider a, S. Scheloske a, A. Wallianos a, C. Wies a, , A. N un K. Traxel b a
b
Max-Planck-Institut f ur Kernphysik, P.O. Box 103980, 69029 Heidelberg, Germany Physikalisches Institut, Universit at Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany
Abstract We developed a new technique to determine the impact position of single pencil beams in the image plane with an accuracy of 0.2 lm. This technique bases on the overlapping of secondary electron images from a 1000 mesh grid. The measurements themselves can be obtained without any change in the routine setup if a secondary electron counter is installed. By choosing dierent object positions (x0 , y0 ) and divergence angles (h0 , u0 ) for the pencil beam the characteristics of the used lens system can be analysed and the lens errors can be calculated with high precision. Results for the Heidelberg quadrupole doublet will be reported. Ó 1999 Elsevier Science B.V. All rights reserved. PACS: 29.90; 41.75; 41.85; 61.16 Keywords: Ray-tracing; Nuclear microprobe; Secondary electrons; Lens errors; Ion optics
1. Introduction One of the most laborious tasks during the daily work of a nuclear microscopist is the challenge to get as much current as possible for a beam size as small as possible whereby the brightness of the used ion source represents the theoretical limit. Common lens systems, for example magnetic quadrupole multiplets, normally suer from parasitic aberrations which worsen the reachable reso-
* Corresponding author: Tel.: +49-6221-516-210; fax: +496221-516-540; e-mail: [email protected]
lution in a nuclear microprobe. Of course other in¯uences like ripple of the power supplies, mechanical vibrations or stray magnetic ®elds work in the same direction. A possible increase of the daily performance bases on the exact knowledge of the optical properties of the used lens system. With numerical ray-tracing or matrix methods the optical properties can be calculated for a given geometry. Parasitic eects which are present in practice can however only be modelled poorly. So there is the need to determine the optical properties experimentally. Up to now two methods that measure the optical properties of lens systems in nuclear
0168-583X/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 0 3 3 2 - 8
M. Maetz et al. / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 90±96
microprobes have been used. In the grid shadow method [1] the shadow pattern of a lm-sized grid as summary eect of all optical in¯uences is analysed. Experimental ray-tracing, the second method, bases on the determination of the impact coordinates (xi , yi ) of pencil beams in the image plane [3]. The higher eort involved to do experimental ray-tracing is counterbalanced by the fact that pencil beams can be chosen to display mainly or sometimes even only one of the ionoptical properties like demagni®cation, dependence on the divergence angles h and u separately, etc. In this paper we present the development of experimental ray-tracing with high accuracy and precision as a method which can be realized without any change in the routine set-up if a secondary electron counter is installed. The method is sensitive to the ion-optical properties only and ignores the above mentioned additional in¯uences on the beam size like for instance mechanical vibrations.
2. The method 2.1. The Heidelberg set-up The complete set-up of the Heidelberg proton microprobe including the lens system is described in detail in Ref. [4]. Only relevant information is given here for better understanding. The lens system consists of a magnetic quadrupole doublet, totally located inside the vacuum. The distance between the object slits and the aperture slits is 1957 mm, the distance between the object slits and the image plane is 4740 mm. The beam size has been measured to be 0.3 lm in STIM experiments. To get 100 pA of beam current the slits have to be opened to result in a 3 lm spot. This is due to the very low brightness of our ion source [5]. The secondary electron imaging system consists of two electrostatic de¯ectors to allow fast beam scanning. Only small regions (about 100 lm2 ) may be scanned to enhance repetition rates to a few seconds per image. To average out beam ¯uctuations the pixel contents from a number of loops
91
can be added and the image can be stored on hard disk. With secondary electron images from 1000 mesh grids the scanning system can be calibrated to give the PIXEL distance in lm. This secondary electron imaging plays a central role in our experimental ray-tracing method. 2.2. The concept of lens errors The mathematical formalism that is used to describe optical properties of a microprobe is based on the following assumptions: In a cartesian coordinate system the z-axis is de®ned by the beam axis with (x, y) (0, 0). The position of the beam in the object plane is denoted with the index o, the position in the image plane with index i. The divergence is measured by the angles h and u which represent the slopes of the particle path projected onto the xz- and yz-plane, respectively. The principle idea of the concept of lens errors is that the image coordinates of the beam (xi , yi ) may be written as polynomials in the system variables with the lens errors as coecients h. . . j . . .i (for details see Refs. [2,6]): X hxjak iak xi hxjx0 ix0
X
k
hxjsk isk hxjx0 DEix0 DE
k
hxjhi hxjhDEiDEh hxjh2 ih2 hxjh3 ih3 hxjh5 ih5 ! X hxjuqk iqk hxjui u k
hxju2 iu2 hxju3 iu3 hxjhuihu hxjhu2 ihu2 hxjh2 uih2 u hxjh3 u2 ih3 u2 hxjhu4 ihu4 X yi hyjy0 iy0 hyjak iak
X
k
hyjsk isk hyjy0 DEiy0 DE
k
hyjui hyjuDEiDEu hyju2 iu2 hyju3 iu3 hyju5 iu5
92
M. Maetz et al. / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 90±96
! X hyjhqk iqk hyjhi h
can be chosen to highlight certain ion-optical properties.
k
hyjh2 ih2 hyjh3 ih3 hyjhuihu
3. Results
hyjuh2 iuh2 hyju2 hiu2 h hyju3 h2 iu3 h2 hyjuh4 iuh4 :
1
The misalignment of the single lenses is fully represented by the rotation q, tilt a and translation s, DE is the energy shift of the particles compared to the ideal energy. 2.3. Experimental ray-tracing The ®rst choice to measure the optical properties of a lens system is the determination of the absolute values for the image coordinates (xi , yi ) with high accuracy to get their dependence on the various system variables. But in practice this is very dicult. Our method uses pencil beams with well de®ned parameters x0 , y0 , u, and h to produce secondary electron images of 1000 mesh grids. To produce the pencil beams (rays) the slit openings were 6 ´ 30 lm2 for the object and 40 ´ 40 lm2 for the aperture. We choose the origin of our coordinate system in the image plane by setting x0 y0 u h 0, i.e., we choose the pencil beam on the geometrical axis. The resulting image (Fig. 1) is then stored in the memory. Then another image is accumulated using a ray propagating in a dierent direction, i.e., one or more parameters x0 , y0 , u, and h being dierent from zero. Due to the lens errors this image is shifted by a certain amount dx, dy (Fig. 1). From the two images the exact number for the shift dx, dy is deduced in the following way (Fig. 1): To get a uniform grey scale all pixel contents less than about 15% of the maximum are set to zero, all others to 1. Then the second image is shifted pixel by pixel until complete overlap is obtained. The precision of the method can also be seen from Fig. 1 where the shift is only d 0.4 lm o the optimum. The experimental procedure makes sure that the measured coordinates (xi , yi ) are neither in¯uenced by time varying external stray ®elds or by noise from power supplies nor by mechanical vibrations. Another advantage is that pencil beams
3.1. Demagni®cation factors By varying the object coordinates x0 , y0 and choosing u h 0 only the demagni®cation factors áx | x0 ñ and áy | y0 ñ are directly measured: dx hxjx0 ix0 ;
dy hyjy0 iy0 :
The demagni®cation factors of our set-up as deduced from the measurements are: hxjx0 i ÿ0:1694 0:0016; hyjy0 i ÿ0:02447 0:00027: It should be stressed that these are the real demagni®cation factors because no lens aberrations dependent on u or h are involved in the measurements. This is for instance not true if áx | x0 ñ and áy | y0 ñ are deduced from beam size measurements. 3.2. Chromatic aberrations Varying the energy E of the beam the chromatic coecients (áx|hDE ñ and áy|uDE ñ) can be determined. By choosing symmetric o axis rays additional information about the alignment of the aperture slits can be achieved. Eight rays were used to measure the chromatic coecients, four points with h equal to zero (u 0.10, 0.20 mrad) and four points with u equal to zero (h 0.13, 0.26 mrad). For the points with h equal to zero the dierences dx, dy can be expressed as: dx 0 and dy áy|uDE ñuDE, whereas for the other four points (u 0): dx áx|hDE ñhDE and dy 0. The quality of the measurements can be seen from Fig. 2. Ideal alignment of the aperture slits leads to (áy|uDE ñu) ÿ(áy|uDE ñÿu) and (áx|hDE ñh) ÿ(áx|hDE ñÿh). Our results show ideal alignment within 3%. From the slopes and the values for u0 , and h0 the coecients of the chromatic aberration can be calculated as mean values áx|hDE ñ (2.162 0.024) lm/(keV mrad) and áy|uDE ñ (2.193 0.050) lm/(keV mrad).
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Fig. 1. In a ®rst step all pixel contents less than about 15% of the maximum are set to 0, all others are set to 1 to get a uniform grey scale (1). Second, both images are overlapped where the pixel signal of the resulting image is given by the absolute value of the difference of the pixel contents of the images (2). Third, the second image is shifted pixel by pixel until ``complete'' overlap is obtained giving dx and dy (3). ``Complete'' overlap is de®ned by the minimum value of the subtracted overlap image. For comparison with the ``complete'' overlap also neighboured overlap images are given with a shift of only d 0.42 lm o the optimum (3). The pixel distance is 0.2 lm in both directions (®eld of view 27 ´ 27 lm2 ).
Fig. 2. Measured shifts dx, dy by varying the energy of the proton beam for two arbitrarily chosen pencil rays out of the eight (h 0; u 0.20, left and h 0.26; u 0, right). From the slopes of the ®tted lines and the values for h and u the chromatic aberrations can be calculated.
3.3. Higher order aberrations 3.3.1. The coecients To determine the divergence dependent aberrations the image coordinates xi and yi for a set of 100 rays were measured. All those rays originated in the origin (x0 y0 0), the angles h and u, which cover the range of 2 mrad, were chosen to
display symmetries and simple geometric con®gurations. Eight points were measured twice to check the precision. The values were reproduced with an uncertainty of Ddx 0.18 lm and Ddy 0.16 lm. Fitting of the suitable polynomials (Eq. (1)) to all datapoints results in the aberration coecients of Table. 1 (upper part). The values agree within the given errors with the results of a measurement done about one year ago [7] when the same rays (phase space points) had been analysed. Only the astigmatic coecients áx|hñ and áy|uñ dier which is a result of dierent focusing quality. This means that the experimental values are reproducible to a high degree. As will be seen later, some of the rays with h, u > 0.5 mrad are only badly described by the lens error concept (Eq. (1)). As these rays are far o the acceptance in routine use, a second ®t was done, but only to the data resulting from the rays inside a cone with SQRT(h2 + u2 ) 6 0.5 mrad (72 rays), which is still bigger than the routine acceptance of the aperture. The resulting aberration coecients can also be seen in Table. 1 (lower part). Surprisingly many of the coecients are completely dierent.
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Table 1 Coecient
Value (lm/mradn )
Data set: h 6 1 mrad, u 6 1.5 mrad áx|hñ (ÿ5.81 0.64) (2.42 0.14) áx|h2 ñ (ÿ3.5 1.0) áx|h3 ñ 5 (1.10 0.36) áx|h P ñ (ÿ7.54 0.86) ( k áx|uqñqk áx|uñ) (ÿ6.12 0.47) áx|u2 ñ (ÿ6.3 1.1) áx|u3 ñ áx|huñ (ÿ2.26 2.34) (ÿ55 23) áx|hu2 ñ (8.6 3.4) áx|h2 uñ (21 26) áx|h3 u2 ñ (5 24) áx|hu4 ñ
áy|uñ áy|u2 ñ áy|u3 ñ 5 áy|u P ñ ( k áy|hqñqk áy|hñ) áy|h2 ñ áy|h3 ñ áy|huñ áy|uh2 ñ áy|u2 hñ áy|u3 h2 ñ áy|uh4 ñ
(18.1 1.1) (3.98 0.38) (ÿ35.9 4.3) (26.3 3.4) (ÿ1.65 0.13) (ÿ0.159 0.044) (0.268 0.075) (ÿ2.00 0.58) (8.9 5.8) (ÿ2.43 0.84) (ÿ39.9 6.4) (ÿ6.7 5.9)
Data set: SQRT(h2 u2 ) 6 0.5 mrad áx|hñ (ÿ8.61 0.41) (2.24 0.27) áx|h2 ñ (17.0 6.5) áx|h3 ñ 5 (ÿ45 20) áx|h P ñ (ÿ6.32 0.28) ( k áx|uqñqk + áx|uñ) (ÿ9.83 0.31) áx|u2 ñ (ÿ11.0 1.4) áx|u3 ñ áx|huñ (ÿ3.92 0.60) (ÿ65 12) áx|hu2 ñ (ÿ0.6 1.7) áx|h2 uñ (96 53) áx|h3 u2 ñ (50 49) áx|hu4 ñ
áy|uñ áy|u2 ñ áy|u3 ñ 5 áy|u P ñ ( k áy|hqñqk + áy|hñ) áy|h2 ñ áy|h3 ñ áy|huñ áy|uh2 ñ áy|u2 hñ áy|u3 h2 ñ áy|uh4 ñ
(16.49 0.62) (ÿ0.54 0.41) (ÿ12 10) (ÿ39 31) (ÿ1.79 0.28) (ÿ1.69 0.31) (0.52 1.43) (ÿ3.84 0.71) (0.12 14.1) (ÿ3.3 2.1) (22 62) (ÿ42 58)
Coecient
Value (lm/mradn )
In the upper part of the table the coecients are listed resulting from the ®t of all datapoints. The coecients in the lower part are the ®t results from the inner cone (SQRT(h2 u2 ) 6 0.5 mrad). Extreme
With the knowledge of the optical properties of the lens system the ``ideal'' phase volume can be calculated. Ideal phase volume means the openings of object and aperture slits to get maximum beam current into a given beam size. The displacement dr SQRT(dx2 + dy2 ) as function of the divergence angles h and u is shown in Fig. 3 for the inner cone < 0:5 mrad. The calculations have been done with the parameters deduced from the ®t to the inner 72 points. 3.3.2. Selected data sets To get a visual impression of the imaging properties of our lens system the hitting points of rays in the image plane are shown for cones (x0 y0 0, SQRT(h2 + u2 ) constant) in Fig. 4. and for the cross (h or u 0) as well as for the diagonals h u and h ÿu in Fig. 5. The full lines represent the calculations with the help of the
Fig. 3. The displacement dr SQRT(dx2 dy2 ) as function of the divergence angles h and u for the inner cone is shown as contour plot. The given numbers represent the displacement in lm.
aberration coecients deduced from the full data set, the dotted lines represent the ``inner'' data set.
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Fig. 4. The hitting points of rays in the image plane are shown for cones (x0 y0 0, SQRT(h2 u2 )2[0.25, 0.5, 0.75, 1.0] mrad). The full lines represent the calculations with the help of the aberration coecients deduced from the full data set, the dotted lines represent the data set with SQRT(h2 + u2 ) 6 0.5 mrad. The given letters show the measured points. Dierent scaling for the left and for the right plots have to be taken into account. If only the inner regions are ®tted the resulting polynomial describes this regions very well, but the outer points (h, u > 0.5 mrad) are described very badly and vice versa.
Fig. 5. Measured shifts dx, dy in the image plane are shown for the cross (h 0) as well as for the diagonal h u (the data for the rays with u 0 and with h u give the same results). The full lines represent the calculations with the help of the aberration coecients deduced from the full data set, the dotted lines represent the data set with SQRT(h2 u2 ) 6 0.5 mrad. If only the inner regions are ®tted the resulting polynomial describes this regions very well, but the outer points (h, u > 0.5 mrad) are described very badly and vice versa.
Fig. 4 shows clearly that the region h > 0, u > 0.25 mrad (points M, N, O) is the worst. Another aspect is the fact that the higher aberration coecients depend strongly on the ®tted phase space (radii in Fig. 4). If only the inner
regions are ®tted the resulting polynomial describes this regions very well, but the outer points (h, u > 0.5 mrad) are described very badly and vice versa. Fig. 5 reveals clearly the poorer focusing quality in x than in y for setup. Even for
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M. Maetz et al. / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 90±96
h 0 y-divergences u P 1 mrad in¯uence xi stronger than yi (Fig. 5, right side). From beam size measurements we suspected this behaviour since long. We thought however that ¯uctuating stray ®elds and noise in the power supplies were responsible. These in¯uences can now be excluded (see 2.3). 4. Conclusions A new method of experimental ray-tracing, based on secondary electron images produced by selected pencil beams has been developed. The method allows to identify the image coordinates (xi , yi ) of any ray (x0 , y0 , h, u) with a precision of Ddx 0.18 lm and Ddy 0.16 lm. An optimization to still reduce this values is straightforward. The ionoptical properties of the Heidelberg setup have been measured, giving precise values for demagni®cation and the chromatic errors. The latter have been measured for 8 rays with h, u < 0.3 mrad only. The higher order aberrations have been analysed for two aperture openings h 6 1 mrad, u 6 1.5 mrad and SQRT(h2 u2 ) 6 0.5 mrad. The resulting aberration coecients dier signi®cantly, the smaller phase space coecients cannot describe the measured results for the bigger phase
space and vice versa. In routine use only small phase spaces (h, u) are used to produce the microbeam. Therefore our results contradict the demand as formulated in [8] to analyse the ionoptical properties for an aperture setting as big as possible. Even more, the question about the signi®cance of the aberration concept to improve the lens properties must be thought over. We propose to do direct ray-tracing to observe the results of technical modi®cations.
References [1] D.N. Jamieson, G.J.F. Legge, Nucl. Instr. and Meth. B 29 (1987) 544. [2] M.B.H. Breese, D.N. Jamieson, P.J.C. King, Materials Analysis using a Nuclear Microprobe, Wiley, New York, 1996. [3] F.W. Martin, Nucl. Instr. and Meth. B 54 (1991) 17. [4] K. Traxel, P. Arndt, J. Bohsung, K.U. Braun-Dullaeus, M. Maetz, D. Reimold, H. Schiebler, A. Wallianos, Nucl. Instr. and Meth. B 104 (1995) 19. [5] R.A. Szymanski, D.N. Jamieson, Nucl. Instr. and Meth. B 130 (1997) 80. [6] G.W. Grime, F. Watt, Beam Optics of Quadrupole forming Systems, Adam Hilger, Bristol, 1984. [7] M. Maetz, Ph.D. thesis, University of Heidelberg, 1997, unpublished. [8] D.N. Jamieson, Nucl. Instr. and Meth. B 104 (1995) 650.