- Email: [email protected]
Stigmatic condition for electrostatic core-lens by using diagram of axis intercept
Optik 123 (2012) 1492–1496
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Stigmatic condition for electrostatic core-lens by using diagram of axis intercept R. Nishi ∗ , A. Takaoka Research Center for Ultra-High Voltage Electron Microscopy, Osaka University, 7-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan
a r t i c l e
i n f o
Article history: Received 23 March 2011 Accepted 14 August 2011
Keywords: Core-lens Curved optic axis Large tilt beam Stigmatic condition Aperture aberration
a b s t r a c t This paper clarifies a method for finding the stigmatic operating condition of electrostatic core-lenses applied to a large tilting optical system. The core-lens has many operating parameters such as configurations of electrodes and their voltage and it is important that how to search the suitable parameters systematically. For this purpose, the diagram of axis intercept is newly defined and applied. It is shown that the core-lens can realize the negative aperture aberration impossible for usual round lenses. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Gabor proposed electron optical system which has electrodes on the axis for aberration correction of the objective lens of electron microscopes [1]. This idea was expanded to usage a large tilting beam to an electron microscopy for 3D observation by Hoppe in 1970s [2]. In its electron optics, electrodes were placed on the rotationally symmetrical axis, and this arrangement was named ‘core-lens’. Typke analyzed the aberration terms of off-axial imaging near to the rotationally symmetrical axis. It was shown that the axial symmetry yielded the relation between the aberration constants considerably reducing the number of independent constants. For rays neighboring a pencil axis, expansions of the image coordinate and of the mixed eikonal were given in terms of invariants about the rotational axis [3]. Plies reported possibility of multitubular core-lenses as an objective lens of 3D electron microscopy. Models of core-lens using cylindrical mirror analyzers were numerically calculated considering curved optical axis and rotational symmetry. The coefficient of third-order aperture aberration and the coefficient of axial chromatic aberration of the first-order were minimized and both were obtained about 20 cm for 50 keV electrons and a focal length of 1 cm [4]. These researches aimed at the 3D electron microscopy (electron tomography) were pioneering analyses. However, subsequent advances of tomographic observation are performed by tilting the sample instead of tilting the beam. Therefore, the optical systems
∗ Corresponding author. Tel.: +81 6 6879 7941; fax: +81 6 6879 7942. E-mail address: [email protected] (R. Nishi). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.09.017
using core-lenses have not been studied after 1980s. Recently, a scanning electron microscopy in large tilting beam is required for 3D observation of LSI devices [5]. The core-lens has been interesting again in applications for a large tilting beam electron optics. Above researches discussed general features of image aberrations in off-axial imaging and numerical calculation for specific multitubular type core-lenses. Applying core-lenses to large tilting beam electron microscopes requires a decision-method of stigmatic condition for core-lens system. In this paper, we discuss how to find an operating condition of core-lenses useful for large tilting beam. Under the stigmatic condition, many parameters such as geometric arrangement of core-lenses and initial conditions of trajectories have to be decided. Here, we consider the trajectories that start at an object point on the axis of rotational symmetry and focus to the axis again at an image point. We assume the optic axis is symmetry with respect to the center plane between the object point and the image point. In order to clear a method to find suitable parameters, we use some simplified models whose electric field is given by the analytical function. The requirement of stigmatic and the amount of an aberration coefficient are analyzed by using the newly defined diagram, that is, the crossing point with the symmetrical axis is drawn as a function of beam tilt angles. 2. Diagram of axis intercept The rotationally symmetric axis (Z coordinate) of core-lenses and the optic axis (z coordinate) of beam flux are defined as shown in Fig. 1. The X-axis and the x-axis are taken perpendicular to the Z-axis and the z-axis, respectively. The Y and y axes are taken perpendicular to Z, X and z, x axes. The angles between the z-axis and
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aberration, when we select angle o or i as the horizontal axis. The curve has maximum value at o = 0 or i = 0. The curves II and III are the cases of core-lenses. The curves of axis intercept can be approximated by quadratic curves which have extremum Zi at an angle m . Then, the optic axis can be fixed. After decision of Zi , we can use the relation Z = Zc − Zi = xi /sin i for calculation of aberration using characteristic functions. Considering the second-order small value for the aperture aberration, xi = 3(0030)(xo )2 (see Appendix A), where (pqrs) stands for the coefficients of the characteristic which are the same meaning function, p q (pqrs)xo yo (xo )r (yo )s . Therefore, we obtain as usage in Ref. [6]; S = relation of Eq. (1) for the second-order aperture aberration, Z = − Fig. 1. A curved optic axis z of a core-lens is shown by dashed line and is on XZsection. Illustration at upper right is enlarged near the image plane Zi . A trajectory starting at object point Zo with an angle o crosses Z-axis of rotational symmetry at a point Zc with an angle i .
the Z-axis at the object point Zo and the image point Zi are defined as o and i , respectively. The angle between the trajectory x(z) and z-axis at the object and image planes are denoted by ˛o and ˛i , respectively. In Appendix A, these values are rewritten by the slope x (z) of trajectory; xo = tan ˛o and xi = − tan ˛i . Similar relations are also rewritten for Y component as shown in Fig. 1; yo = tan ˇo and yi = − tan ˇi . Rotationally symmetrical optical system shows the feature that the trajectories starting from the object point on the axis focus on the image point on the axis independently of the rotational angle ϕ. When trajectories on XZ-section focuses on the axis, trajectories rotated by ϕ with respect to the axis also focuses on the axis. The focusing in the azimuthally section is always at least one order better than in the radial section [4]. Therefore, it is enough that the trajectories are taken only on the XZ- or xz-section into account. The operating condition of core-lenses has to be satisfied stigmatic. The curved optic axis is decided after clarification of stigmatic. Then, the trajectory x(z) is defined. On the other hand, the trajectory X(Z) can be defined regardless of the optic axis and trajectories started on Z-axis always cross the Z-axis when the lens is convex. This crossing point is named as axis intercept Zc and is shown by the inserted illustration in Fig. 1. Diagram in which Zc is plotted as a function of trajectory parameters, we call it diagram of axis intercept and use it for decision of operating condition. As shown in Fig. 2, in the case of a round lens (curve I), the curve of axis intercept is quadratic due to the 3rd-order aperture
II III
Zi
0
(1)
A coefficient of the quadratic curve of axis intercept around om gives the aperture aberration coefficient (0030) using Eq. (1). If the curve of axis intercept is convex upward such as a round lens, a coefficient of the aperture aberration is positive. If the curve is convex downward, the coefficient is an inverse sign contrary to the round lens, that is, negative. 3. Stigmatic condition In order to analyze the stigmatic condition of core-lenses, we adopt a ring charge and point charges whose fields are well known analytically. Furthermore, we assume that the arrangement of corelenses is symmetry with respect to the center plane between the object and the image points. 3.1. A model of ring and point charges The most simplified arrangement of core-lens consists of single ring and single point charge as shown in Fig. 3(a). A ring charge works as a round lens whose aperture aberration is positive. On the other hand, a positive point charge has opposite property against the ring charge, that is, the closer trajectories to the axis are more strongly converged. Combining both elements, it is expected to work as a core-lens in large tilting beam. At first, variables such as charge amount Qring , Qpoint , distance X, Z and kinetic energy W when an electron goes to infinite distance are normalized. For normalization, we use notations: Q0 = Xˆ =
40 aW , e
X , a
Zˆ =
−q =
−Qring Q0
,
k=
Qpoint Qring
,
(2)
Z , a
(3)
where e is the elementary charge and 0 is the permittivity in vacuum, a is a radius of the ring charge. Then, the potential distribution ˆ X, ˆ Z) ˆ is given as follows: (
Zc I
3(0030) 2 (xo ) . sin o
θm
K(g) −2q ˆ X, ˆ Z) ˆ = ( + ˆ + 1)2 + Zˆ 2 (|X| g =
ˆ 4|X|
ˆ + 1)2 + Zˆ 2 (|X|
kq Xˆ 2 + Zˆ 2
, (4)
,
where K(g) is a complete elliptic integral of the first kind. In the following, the symbol ‘∧ ’ in the normalized coordinate variables will be omitted for simplicity.
θo or θi
Fig. 2. Diagram of axis intercept shows Zc as a function of some parameter of beam flux; angles o , i . The curve I corresponds to a round lens and the curves II and III correspond to core-lenses. The curves have extremum Zi at a point m , which represents om or im .
3.2. Searching procedure of stigmatic condition Let us discuss the procedure to determine the stigmatic condition of the core-lenses. In the case of the central symmetry
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Fig. 3. (a) The model is alignment of a negative ring charge −q and a positive point charge kq. (b) The trajectories starting from symmetry plane (Z = 0) at Xs parallel to Z-axis (only shown in k = 0.05). (c) The axis intercept Zc plotted as a function of Xs from (b). (d) The Zc plotted as a function of i from (b). (e) The trajectories starting at object point Zo = − Zi on the axis with starting angle o ; neighborhood of om = im (only shown in k = 0.05). (f) Diagram of axis intercept as a function of o . Parameters of model are q = 0.9, k = 0.05, 0.03, 0.01.
trajectory, the trajectory is parallel to the axis at the symmetry plane (Z = 0). Calculation is enough on XZ-section as described before. The procedure is as follows:
1. The trajectory starts parallel to the axis at Xs on the symmetry plane Z = 0, and the starting position Xs changes to calculate some trajectories (Fig. 3(b)). 2. The axis intercept Zc is plotted as a function of the Xs (Fig. 3(c)). 3. If this curve does not have an extremum, change the lens parameters and calculate again from the step 1. If there is an extremum Zi , beam flux is focused by the first-order at the image plane Z = Zi . 4. From Fig. 3(b), the Zc is plotted again as a function of i . The curve is fitted to quadratic function around the extremum Zi at im ; the diagram of axis intercept (Fig. 3(d)). 5. Next, trajectories start from the axis of rotational symmetry. The starting position is Zo = − Zi and starting angle of the optic axis is om = im . The starting angles o of the trajectories change near the initial angle om as a parameter (Fig. 3(e)). 6. From Fig. 3(e), the relation between the starting angle o and the axis intercept Zc is known and this relation is plotted; the diagram of axis intercept. The plotted points are fitted by quadratic function as shown in Fig. 3(f).
Fig. 4. (a) A model of cascade arrangement of core-lenses consisting of triple point charges. (b) Axis intercept Zc calculated from symmetry plane at the starting position Xs . (c) and (d) Diagrams of axis intercept as a function of i or o calculated from the symmetry plane or the object point, respectively. The coefficient of the aperture aberration is negative because the quadratic curves in (c) and (d) are convex downward (q = 1, k = 1, and k = 10 only (d)).
In the above procedure, the quadratic coefficient of the quadratic curve in Fig. 3(f) are roughly proportional to that in Fig. 3(d). Namely, calculation of trajectories from the symmetry plane gives approximately proportional coefficient of the aperture aberration. This feature decreases amount of calculation and parameters to search. According to the above procedure shown in Fig. 3, we obtained the operating conditions and the aperture aberration coefficient. We use two normalized parameters; the ring charge −q and the
R. Nishi, A. Takaoka / Optik 123 (2012) 1492–1496
a
Second model of core-lens system is the cascade arrangement of core-lenses and it is the simplest alignment of three point charges with different polarities. Electric field of a point charge is equal to that of the sphere electrodes. The positive charge gives the effect as a convex lens, while the negative charge gives the effect as a concave lens. The combination of convex-concave-convex core-lenses has some invaluable characteristics such as negative aperture aberration like that of multipole lenses. Here, we examine the feature of beam flux with a large tilting angle to Z-axis. We also assume that the alignment of positive–negative–positive charges is symmetry with respect to the symmetry plane as shown in Fig. 4(a). The variables are normalized like the previous model. The coordinates are normalized by the distance b between the pointcharges. The center charge −qQ0 and the both side charges kqQ0 are normalized by standard charge Q0 = 40 bW/e, then normalized charges are kq, − q, kq. For this model, the operating condition (q, k, Xsm , im , Zi , or om ) of the first-order focusing is determined by the above mentioned procedure. Then, small quadratic coefficient approximated near the extremum gives suitable operating condition for the model. Fig. 4(b)–(d) shows the diagrams of Zc in the case that the sign of the center charge is negative and the sign of both side charges are positive. Fig. 4(b) shows the curve as a function of Xs by calculation of trajectories from the symmetry plane. This curve has the local minimum at Zi = 3.83. Fig. 4(c) shows the curve of axis intercept as a function of i . This curve also has the minimum. This means that the sign of the coefficient of the aperture aberration is negative for this model. A solid line is fitted by a quadratic function near the extremum. Fig. 4(d) shows the diagram of axis intercept as a function of o for the trajectory starting from the axis. They are calculated under the condition of starting point of Zo = − Zi = − 3.83 on the Z-axis and the starting angle around om = 0.361 radian whose value are determined in Fig. 4(b) and (c). The coefficient of the aperture aberration (0030) is −115 from the quadratic coefficient in Fig. 4(d) by using Eq. (1) when q = 1, k = 1. If the ratio k becomes larger, the coefficient becomes smaller. For example, the coefficient (0030) is −15 at k = 10. This shows that the aperture aberration coefficient is negative and the amount of (0030) is −15 × 10 = − 150 mm for the distance between the charges of 10 mm.
Xo on ϕ-secti
ratio k, that is, the point charge is kq. When the ratio k = 0.05, the first-order focusing point Zi is formed at the position of Z = 1.43 as shown in Fig. 3(b). We change the ratio k and plot Zc as a function of crossing angle i , which is the diagram of axis intercept shown in Fig. 3(d). Solid lines show fitted curves of a quadratic function. An increase in k causes an decrease in the quadratic coefficient of the diagram and an increase in i . Fig. 3(e) shows trajectories of the ratio k = 0.05 which are calculated from the object point on the axis, and they are focused with first-order on the point of Z = 1.43, which is almost same as Fig. 3(b). The difference between them is negligible small. Fig. 3(f) shows the diagram of axis intercept as a function of starting angle o . Solid lines in Fig. 3(f) are fitted well more wider range than that in Fig. 3(d). A coefficient of the aperture aberration of core-lenses can be obtained from the quadratic coefficient of fitted curves in Fig. 3(f) using Eq. (1). The smaller the curvature around the extremum becomes, the smaller the coefficient of the aperture aberration becomes. From this approximated quadratic curve, the quadratic coefficient is −150 when k = 0.05. The coefficient of the aperture aberration (0030) is calculated at −(− 150/3) sin 0.30 ∼ 15 by using Eq. (1). This normalized value is corresponding to the aperture aberration of 150 mm for the radius of the ring charge of 10 mm.
3.3. A model of triple point charges
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ϕ
xocos θo
ξocosθo Yo
b
yo
ηo D
Xo′
C tan θo
Yo′
B A
ϕ
xo′ cos 2 θo
yo′ cosθo
O
Fig. A.1. Coordinates of position and gradient by rotating XZ-section with small angle ϕ with respect to Z-axis. (a) The position (Xo , Yo ) at the object plane. Due to x-axis is tilted by the angle o , xo is transferred to xo cos o in X coordinate. (b) The gradient (Xo , Yo ) at the object plane.
4. Conclusion To consider the stigmatic condition and the properties of the electrostatic core-lenses, the ”diagram of axis intercept” and the ”curve of axis intercept” are newly proposed. The diagram is shown as crossing point of trajectories to the axis of core-lens as a function of beam tilt angle . The extremum of the curve corresponds to the stigmatic condition on the Z-axis. The procedure to search the extremum leads to the operating condition of core-lens. Then, if the curve has a small quadratic coefficient, the aperture aberration is also small. It is shown that the model combined single ring and single point charge has the positive aperture aberration, but the model aligned triple point charges with positive and negative sign has invaluable negative aperture aberration. This procedure will be extended to search the important condition of dispersion free for core-lenses. Acknowledgments We wish to express our gratitude to Dr. Katsumi Ura, professor emeritus at Osaka University, for helpful suggestions and comments. We would like to thank to Prof. Hirotaro Mori and Prof. Toshiaki Suhara at Research Center for Ultra-High Voltage Electron Microscopy in Osaka University, for the support of this study. Appendix A. Second-order aberrations of core-lenses The field of the core-lenses is rotational symmetry and the corelens-systems are pseudo stigmatic. This is because the paraxial trajectories from the axis of rotational symmetry go on the plane including the rotationally symmetric axis and cross the axis again. Furthermore, if the optical system has symmetry with respect to the center plane (Z = 0) between the object and the image, the system is stigmatic. This is accountable as follows. Due to om = im for the optic axis, Fig. A.1(b), which shows gradient at the object plane, also shows gradient at the image plane by substitution of suffixes o → i. Because the lengths AB and AC are same at the object plane and the image plane, xi = xo and yi = yo , that is, angular magnifications of x- and y-directions are also same. Magnification
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of x- and y-directions are therefore same, and the system is stigmatic. The core-lenses can be viewed as a plane symmetrical system about the XZ-section. Number of the aberration terms of secondorder is reduced to 10 compared to non-symmetrical system. Assuming the magnification to be one, the second-order aberrations xi and yi are given by differentiating the characteristic function with respect to xo or yo . xi
yi
⎫ = ⎪ ⎪ ⎪ ⎪ 2 ⎬ 2 +(0111)yo yo + (0210)yo + (0012)(yo ) , = (0111)yo xo + (1101)xo yo + 2(1002)xo yo ⎪ ⎪ ⎪ ⎪ ⎭ 3(0030)(xo )2
+ 2(1020)xo xo
+ (2010)xo2
(A.1)
(A.2)
(A.3)
(A.4)
This result is consistent with Typke’s result [3]. When above relation is valid, next relations are obtained 3(0030) = 2(0022) cos o , sin o
(0004) = 0.
(A.7)
yo = xo cos o tan ϕ = xo ϕ cos o + O(4). cos ϕ =
(2010)xo2 / cos
yi = i cos o sin ϕ = (2010)xo2 ϕ cos o = (2010)xo yo .
xi = (2010)xo2 + (0210)yo2 , yi = (1101)xo yo .
(A.8) ϕ
(A.9) (A.10)
(A.5)
Next, we discuss non-aperture aberrations. As shown in Fig. A.1(a), the trajectory with small deviation of (xo , yo ) shifted from the optic axis is considered. We call the plane ϕ-section, where the trajectory exists. The section is rotated with respect to Z-axis by the angle ϕ from XZ-section. The coordinate system on the ϕsection is called (, ), where we select = 0. The initial position is selected as ( o , o ). Because the trajectory stays on the same ϕ-section, o equals to zero. For the coordinate
(A.11)
Comparing the coefficients of these equations and Eqs. (A.9) and (A.10), (0210) = 0,
(2010) = (1101),
(A.12)
are obtained about distortion aberrations. We assumed the system is symmetry at the plane of the center, following aberration coefficients are also zero. (1020) = (0111) = (1002) = 0.
Considering aperture aberrations, Zx = Zy because the trajectory starting from the axis crosses to the axis again. Therefore, Eq. (A.2) is put to be equal to Eq. (A.3). To be valid this relation independently of xo and yo , it is required to be (0012) = 0.
i = 0.
Considering xo = yo = 0, from Eq. (A.1),
Zy = − cos o {2(0012)xo + 2(0022)(xo )2 + 4(0004)(yo )2 }{1 + O(1)}.
(A.6)
= (2010)xo2 /cos2 ϕ + O(3),
xi = i cos ϕ = = (2010)xo2 + O(4).
At first, the aperture aberrations of the second-order are considered by the diagram of axis intercept. Considering up to 4th-order characteristic function for aperture aberrations, i is replaced by i = o + O(1), where O(1) means first-order small value. The secondorder small values are taken into account. The differences Zx and Zy between the image point Zi and the crossing points Zc of x and y components by aberrations are 1 {3(0030)(xo )2 + (0012)(yo )2 }{1 + O(1)}, sin o
i = (0030)o2 + 2(1020)o o + (2010)o2
(2010)o2
+2(0012)xo yo .
Zx = −
(, ), it holds the same format as Eq. (A.1) because the core-lenses have rotation symmetry about the Z-axis.
(A.13)
Finally, if we consider the second-order of (xo , yo ), remaining aberrations are the aperture aberration and distortion aberrations. Using the diagram of axis intercept, we can obtain the coefficient of the aperture aberration as follows; (same as Eq. (1)). Zx = Zy = −
3(0030) 2 (xo ) . sin o
(A.14)
References [1] D. Gabor, A zonally corrected electron lens, Nature 158 (1946) 198. [2] H. Hoppe, Dreidimensional abbildende Elektronenmikroskope, I. Prinzip der geräte, Z. Naturforsch. 27a (1972) 919–929. [3] D. Typke, Image aberrations of axially symmetric imaging systems with off-axial rays: three-dimensionally imaging electron microscopes (part III), Nucl. Instrum. Methods 187 (1981) 217–226. [4] E. Plies, Calculations of electrostatic multitubular core-lenses for a 3D imaging electron microscope: three-dimensionally imaging electron microscopes, (part IV), Nucl. Instrum. Methods 187 (1981) 227–235. [5] K. Abe, Y. Tsuruga, S. Okada, T. Nomura, H. Aoki, H. Fuji, H. Koike, A. Hamaguchi, Y. Yamazaki, Three-dimensional measurement by tilting and moving objective lens in CD-SEM(III), in: Proc. SPIE, vol. 5752, 2005, pp. 1200–1208. [6] P. Hawkes, E. Kasper, Basic geometrical optics, in: Principles of Electron Optics, vol. 1, Academic Press, London, 1989.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Stigmatic condition for electrostatic core-lens by using diagram of axis intercept R. Nishi ∗ , A. Takaoka Research Center for Ultra-High Voltage Electron Microscopy, Osaka University, 7-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan
a r t i c l e
i n f o
Article history: Received 23 March 2011 Accepted 14 August 2011
Keywords: Core-lens Curved optic axis Large tilt beam Stigmatic condition Aperture aberration
a b s t r a c t This paper clarifies a method for finding the stigmatic operating condition of electrostatic core-lenses applied to a large tilting optical system. The core-lens has many operating parameters such as configurations of electrodes and their voltage and it is important that how to search the suitable parameters systematically. For this purpose, the diagram of axis intercept is newly defined and applied. It is shown that the core-lens can realize the negative aperture aberration impossible for usual round lenses. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Gabor proposed electron optical system which has electrodes on the axis for aberration correction of the objective lens of electron microscopes [1]. This idea was expanded to usage a large tilting beam to an electron microscopy for 3D observation by Hoppe in 1970s [2]. In its electron optics, electrodes were placed on the rotationally symmetrical axis, and this arrangement was named ‘core-lens’. Typke analyzed the aberration terms of off-axial imaging near to the rotationally symmetrical axis. It was shown that the axial symmetry yielded the relation between the aberration constants considerably reducing the number of independent constants. For rays neighboring a pencil axis, expansions of the image coordinate and of the mixed eikonal were given in terms of invariants about the rotational axis [3]. Plies reported possibility of multitubular core-lenses as an objective lens of 3D electron microscopy. Models of core-lens using cylindrical mirror analyzers were numerically calculated considering curved optical axis and rotational symmetry. The coefficient of third-order aperture aberration and the coefficient of axial chromatic aberration of the first-order were minimized and both were obtained about 20 cm for 50 keV electrons and a focal length of 1 cm [4]. These researches aimed at the 3D electron microscopy (electron tomography) were pioneering analyses. However, subsequent advances of tomographic observation are performed by tilting the sample instead of tilting the beam. Therefore, the optical systems
∗ Corresponding author. Tel.: +81 6 6879 7941; fax: +81 6 6879 7942. E-mail address: [email protected] (R. Nishi). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.09.017
using core-lenses have not been studied after 1980s. Recently, a scanning electron microscopy in large tilting beam is required for 3D observation of LSI devices [5]. The core-lens has been interesting again in applications for a large tilting beam electron optics. Above researches discussed general features of image aberrations in off-axial imaging and numerical calculation for specific multitubular type core-lenses. Applying core-lenses to large tilting beam electron microscopes requires a decision-method of stigmatic condition for core-lens system. In this paper, we discuss how to find an operating condition of core-lenses useful for large tilting beam. Under the stigmatic condition, many parameters such as geometric arrangement of core-lenses and initial conditions of trajectories have to be decided. Here, we consider the trajectories that start at an object point on the axis of rotational symmetry and focus to the axis again at an image point. We assume the optic axis is symmetry with respect to the center plane between the object point and the image point. In order to clear a method to find suitable parameters, we use some simplified models whose electric field is given by the analytical function. The requirement of stigmatic and the amount of an aberration coefficient are analyzed by using the newly defined diagram, that is, the crossing point with the symmetrical axis is drawn as a function of beam tilt angles. 2. Diagram of axis intercept The rotationally symmetric axis (Z coordinate) of core-lenses and the optic axis (z coordinate) of beam flux are defined as shown in Fig. 1. The X-axis and the x-axis are taken perpendicular to the Z-axis and the z-axis, respectively. The Y and y axes are taken perpendicular to Z, X and z, x axes. The angles between the z-axis and
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aberration, when we select angle o or i as the horizontal axis. The curve has maximum value at o = 0 or i = 0. The curves II and III are the cases of core-lenses. The curves of axis intercept can be approximated by quadratic curves which have extremum Zi at an angle m . Then, the optic axis can be fixed. After decision of Zi , we can use the relation Z = Zc − Zi = xi /sin i for calculation of aberration using characteristic functions. Considering the second-order small value for the aperture aberration, xi = 3(0030)(xo )2 (see Appendix A), where (pqrs) stands for the coefficients of the characteristic which are the same meaning function, p q (pqrs)xo yo (xo )r (yo )s . Therefore, we obtain as usage in Ref. [6]; S = relation of Eq. (1) for the second-order aperture aberration, Z = − Fig. 1. A curved optic axis z of a core-lens is shown by dashed line and is on XZsection. Illustration at upper right is enlarged near the image plane Zi . A trajectory starting at object point Zo with an angle o crosses Z-axis of rotational symmetry at a point Zc with an angle i .
the Z-axis at the object point Zo and the image point Zi are defined as o and i , respectively. The angle between the trajectory x(z) and z-axis at the object and image planes are denoted by ˛o and ˛i , respectively. In Appendix A, these values are rewritten by the slope x (z) of trajectory; xo = tan ˛o and xi = − tan ˛i . Similar relations are also rewritten for Y component as shown in Fig. 1; yo = tan ˇo and yi = − tan ˇi . Rotationally symmetrical optical system shows the feature that the trajectories starting from the object point on the axis focus on the image point on the axis independently of the rotational angle ϕ. When trajectories on XZ-section focuses on the axis, trajectories rotated by ϕ with respect to the axis also focuses on the axis. The focusing in the azimuthally section is always at least one order better than in the radial section [4]. Therefore, it is enough that the trajectories are taken only on the XZ- or xz-section into account. The operating condition of core-lenses has to be satisfied stigmatic. The curved optic axis is decided after clarification of stigmatic. Then, the trajectory x(z) is defined. On the other hand, the trajectory X(Z) can be defined regardless of the optic axis and trajectories started on Z-axis always cross the Z-axis when the lens is convex. This crossing point is named as axis intercept Zc and is shown by the inserted illustration in Fig. 1. Diagram in which Zc is plotted as a function of trajectory parameters, we call it diagram of axis intercept and use it for decision of operating condition. As shown in Fig. 2, in the case of a round lens (curve I), the curve of axis intercept is quadratic due to the 3rd-order aperture
II III
Zi
0
(1)
A coefficient of the quadratic curve of axis intercept around om gives the aperture aberration coefficient (0030) using Eq. (1). If the curve of axis intercept is convex upward such as a round lens, a coefficient of the aperture aberration is positive. If the curve is convex downward, the coefficient is an inverse sign contrary to the round lens, that is, negative. 3. Stigmatic condition In order to analyze the stigmatic condition of core-lenses, we adopt a ring charge and point charges whose fields are well known analytically. Furthermore, we assume that the arrangement of corelenses is symmetry with respect to the center plane between the object and the image points. 3.1. A model of ring and point charges The most simplified arrangement of core-lens consists of single ring and single point charge as shown in Fig. 3(a). A ring charge works as a round lens whose aperture aberration is positive. On the other hand, a positive point charge has opposite property against the ring charge, that is, the closer trajectories to the axis are more strongly converged. Combining both elements, it is expected to work as a core-lens in large tilting beam. At first, variables such as charge amount Qring , Qpoint , distance X, Z and kinetic energy W when an electron goes to infinite distance are normalized. For normalization, we use notations: Q0 = Xˆ =
40 aW , e
X , a
Zˆ =
−q =
−Qring Q0
,
k=
Qpoint Qring
,
(2)
Z , a
(3)
where e is the elementary charge and 0 is the permittivity in vacuum, a is a radius of the ring charge. Then, the potential distribution ˆ X, ˆ Z) ˆ is given as follows: (
Zc I
3(0030) 2 (xo ) . sin o
θm
K(g) −2q ˆ X, ˆ Z) ˆ = ( + ˆ + 1)2 + Zˆ 2 (|X| g =
ˆ 4|X|
ˆ + 1)2 + Zˆ 2 (|X|
kq Xˆ 2 + Zˆ 2
, (4)
,
where K(g) is a complete elliptic integral of the first kind. In the following, the symbol ‘∧ ’ in the normalized coordinate variables will be omitted for simplicity.
θo or θi
Fig. 2. Diagram of axis intercept shows Zc as a function of some parameter of beam flux; angles o , i . The curve I corresponds to a round lens and the curves II and III correspond to core-lenses. The curves have extremum Zi at a point m , which represents om or im .
3.2. Searching procedure of stigmatic condition Let us discuss the procedure to determine the stigmatic condition of the core-lenses. In the case of the central symmetry
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Fig. 3. (a) The model is alignment of a negative ring charge −q and a positive point charge kq. (b) The trajectories starting from symmetry plane (Z = 0) at Xs parallel to Z-axis (only shown in k = 0.05). (c) The axis intercept Zc plotted as a function of Xs from (b). (d) The Zc plotted as a function of i from (b). (e) The trajectories starting at object point Zo = − Zi on the axis with starting angle o ; neighborhood of om = im (only shown in k = 0.05). (f) Diagram of axis intercept as a function of o . Parameters of model are q = 0.9, k = 0.05, 0.03, 0.01.
trajectory, the trajectory is parallel to the axis at the symmetry plane (Z = 0). Calculation is enough on XZ-section as described before. The procedure is as follows:
1. The trajectory starts parallel to the axis at Xs on the symmetry plane Z = 0, and the starting position Xs changes to calculate some trajectories (Fig. 3(b)). 2. The axis intercept Zc is plotted as a function of the Xs (Fig. 3(c)). 3. If this curve does not have an extremum, change the lens parameters and calculate again from the step 1. If there is an extremum Zi , beam flux is focused by the first-order at the image plane Z = Zi . 4. From Fig. 3(b), the Zc is plotted again as a function of i . The curve is fitted to quadratic function around the extremum Zi at im ; the diagram of axis intercept (Fig. 3(d)). 5. Next, trajectories start from the axis of rotational symmetry. The starting position is Zo = − Zi and starting angle of the optic axis is om = im . The starting angles o of the trajectories change near the initial angle om as a parameter (Fig. 3(e)). 6. From Fig. 3(e), the relation between the starting angle o and the axis intercept Zc is known and this relation is plotted; the diagram of axis intercept. The plotted points are fitted by quadratic function as shown in Fig. 3(f).
Fig. 4. (a) A model of cascade arrangement of core-lenses consisting of triple point charges. (b) Axis intercept Zc calculated from symmetry plane at the starting position Xs . (c) and (d) Diagrams of axis intercept as a function of i or o calculated from the symmetry plane or the object point, respectively. The coefficient of the aperture aberration is negative because the quadratic curves in (c) and (d) are convex downward (q = 1, k = 1, and k = 10 only (d)).
In the above procedure, the quadratic coefficient of the quadratic curve in Fig. 3(f) are roughly proportional to that in Fig. 3(d). Namely, calculation of trajectories from the symmetry plane gives approximately proportional coefficient of the aperture aberration. This feature decreases amount of calculation and parameters to search. According to the above procedure shown in Fig. 3, we obtained the operating conditions and the aperture aberration coefficient. We use two normalized parameters; the ring charge −q and the
R. Nishi, A. Takaoka / Optik 123 (2012) 1492–1496
a
Second model of core-lens system is the cascade arrangement of core-lenses and it is the simplest alignment of three point charges with different polarities. Electric field of a point charge is equal to that of the sphere electrodes. The positive charge gives the effect as a convex lens, while the negative charge gives the effect as a concave lens. The combination of convex-concave-convex core-lenses has some invaluable characteristics such as negative aperture aberration like that of multipole lenses. Here, we examine the feature of beam flux with a large tilting angle to Z-axis. We also assume that the alignment of positive–negative–positive charges is symmetry with respect to the symmetry plane as shown in Fig. 4(a). The variables are normalized like the previous model. The coordinates are normalized by the distance b between the pointcharges. The center charge −qQ0 and the both side charges kqQ0 are normalized by standard charge Q0 = 40 bW/e, then normalized charges are kq, − q, kq. For this model, the operating condition (q, k, Xsm , im , Zi , or om ) of the first-order focusing is determined by the above mentioned procedure. Then, small quadratic coefficient approximated near the extremum gives suitable operating condition for the model. Fig. 4(b)–(d) shows the diagrams of Zc in the case that the sign of the center charge is negative and the sign of both side charges are positive. Fig. 4(b) shows the curve as a function of Xs by calculation of trajectories from the symmetry plane. This curve has the local minimum at Zi = 3.83. Fig. 4(c) shows the curve of axis intercept as a function of i . This curve also has the minimum. This means that the sign of the coefficient of the aperture aberration is negative for this model. A solid line is fitted by a quadratic function near the extremum. Fig. 4(d) shows the diagram of axis intercept as a function of o for the trajectory starting from the axis. They are calculated under the condition of starting point of Zo = − Zi = − 3.83 on the Z-axis and the starting angle around om = 0.361 radian whose value are determined in Fig. 4(b) and (c). The coefficient of the aperture aberration (0030) is −115 from the quadratic coefficient in Fig. 4(d) by using Eq. (1) when q = 1, k = 1. If the ratio k becomes larger, the coefficient becomes smaller. For example, the coefficient (0030) is −15 at k = 10. This shows that the aperture aberration coefficient is negative and the amount of (0030) is −15 × 10 = − 150 mm for the distance between the charges of 10 mm.
Xo on ϕ-secti
ratio k, that is, the point charge is kq. When the ratio k = 0.05, the first-order focusing point Zi is formed at the position of Z = 1.43 as shown in Fig. 3(b). We change the ratio k and plot Zc as a function of crossing angle i , which is the diagram of axis intercept shown in Fig. 3(d). Solid lines show fitted curves of a quadratic function. An increase in k causes an decrease in the quadratic coefficient of the diagram and an increase in i . Fig. 3(e) shows trajectories of the ratio k = 0.05 which are calculated from the object point on the axis, and they are focused with first-order on the point of Z = 1.43, which is almost same as Fig. 3(b). The difference between them is negligible small. Fig. 3(f) shows the diagram of axis intercept as a function of starting angle o . Solid lines in Fig. 3(f) are fitted well more wider range than that in Fig. 3(d). A coefficient of the aperture aberration of core-lenses can be obtained from the quadratic coefficient of fitted curves in Fig. 3(f) using Eq. (1). The smaller the curvature around the extremum becomes, the smaller the coefficient of the aperture aberration becomes. From this approximated quadratic curve, the quadratic coefficient is −150 when k = 0.05. The coefficient of the aperture aberration (0030) is calculated at −(− 150/3) sin 0.30 ∼ 15 by using Eq. (1). This normalized value is corresponding to the aperture aberration of 150 mm for the radius of the ring charge of 10 mm.
3.3. A model of triple point charges
1495
ϕ
xocos θo
ξocosθo Yo
b
yo
ηo D
Xo′
C tan θo
Yo′
B A
ϕ
xo′ cos 2 θo
yo′ cosθo
O
Fig. A.1. Coordinates of position and gradient by rotating XZ-section with small angle ϕ with respect to Z-axis. (a) The position (Xo , Yo ) at the object plane. Due to x-axis is tilted by the angle o , xo is transferred to xo cos o in X coordinate. (b) The gradient (Xo , Yo ) at the object plane.
4. Conclusion To consider the stigmatic condition and the properties of the electrostatic core-lenses, the ”diagram of axis intercept” and the ”curve of axis intercept” are newly proposed. The diagram is shown as crossing point of trajectories to the axis of core-lens as a function of beam tilt angle . The extremum of the curve corresponds to the stigmatic condition on the Z-axis. The procedure to search the extremum leads to the operating condition of core-lens. Then, if the curve has a small quadratic coefficient, the aperture aberration is also small. It is shown that the model combined single ring and single point charge has the positive aperture aberration, but the model aligned triple point charges with positive and negative sign has invaluable negative aperture aberration. This procedure will be extended to search the important condition of dispersion free for core-lenses. Acknowledgments We wish to express our gratitude to Dr. Katsumi Ura, professor emeritus at Osaka University, for helpful suggestions and comments. We would like to thank to Prof. Hirotaro Mori and Prof. Toshiaki Suhara at Research Center for Ultra-High Voltage Electron Microscopy in Osaka University, for the support of this study. Appendix A. Second-order aberrations of core-lenses The field of the core-lenses is rotational symmetry and the corelens-systems are pseudo stigmatic. This is because the paraxial trajectories from the axis of rotational symmetry go on the plane including the rotationally symmetric axis and cross the axis again. Furthermore, if the optical system has symmetry with respect to the center plane (Z = 0) between the object and the image, the system is stigmatic. This is accountable as follows. Due to om = im for the optic axis, Fig. A.1(b), which shows gradient at the object plane, also shows gradient at the image plane by substitution of suffixes o → i. Because the lengths AB and AC are same at the object plane and the image plane, xi = xo and yi = yo , that is, angular magnifications of x- and y-directions are also same. Magnification
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of x- and y-directions are therefore same, and the system is stigmatic. The core-lenses can be viewed as a plane symmetrical system about the XZ-section. Number of the aberration terms of secondorder is reduced to 10 compared to non-symmetrical system. Assuming the magnification to be one, the second-order aberrations xi and yi are given by differentiating the characteristic function with respect to xo or yo . xi
yi
⎫ = ⎪ ⎪ ⎪ ⎪ 2 ⎬ 2 +(0111)yo yo + (0210)yo + (0012)(yo ) , = (0111)yo xo + (1101)xo yo + 2(1002)xo yo ⎪ ⎪ ⎪ ⎪ ⎭ 3(0030)(xo )2
+ 2(1020)xo xo
+ (2010)xo2
(A.1)
(A.2)
(A.3)
(A.4)
This result is consistent with Typke’s result [3]. When above relation is valid, next relations are obtained 3(0030) = 2(0022) cos o , sin o
(0004) = 0.
(A.7)
yo = xo cos o tan ϕ = xo ϕ cos o + O(4). cos ϕ =
(2010)xo2 / cos
yi = i cos o sin ϕ = (2010)xo2 ϕ cos o = (2010)xo yo .
xi = (2010)xo2 + (0210)yo2 , yi = (1101)xo yo .
(A.8) ϕ
(A.9) (A.10)
(A.5)
Next, we discuss non-aperture aberrations. As shown in Fig. A.1(a), the trajectory with small deviation of (xo , yo ) shifted from the optic axis is considered. We call the plane ϕ-section, where the trajectory exists. The section is rotated with respect to Z-axis by the angle ϕ from XZ-section. The coordinate system on the ϕsection is called (, ), where we select = 0. The initial position is selected as ( o , o ). Because the trajectory stays on the same ϕ-section, o equals to zero. For the coordinate
(A.11)
Comparing the coefficients of these equations and Eqs. (A.9) and (A.10), (0210) = 0,
(2010) = (1101),
(A.12)
are obtained about distortion aberrations. We assumed the system is symmetry at the plane of the center, following aberration coefficients are also zero. (1020) = (0111) = (1002) = 0.
Considering aperture aberrations, Zx = Zy because the trajectory starting from the axis crosses to the axis again. Therefore, Eq. (A.2) is put to be equal to Eq. (A.3). To be valid this relation independently of xo and yo , it is required to be (0012) = 0.
i = 0.
Considering xo = yo = 0, from Eq. (A.1),
Zy = − cos o {2(0012)xo + 2(0022)(xo )2 + 4(0004)(yo )2 }{1 + O(1)}.
(A.6)
= (2010)xo2 /cos2 ϕ + O(3),
xi = i cos ϕ = = (2010)xo2 + O(4).
At first, the aperture aberrations of the second-order are considered by the diagram of axis intercept. Considering up to 4th-order characteristic function for aperture aberrations, i is replaced by i = o + O(1), where O(1) means first-order small value. The secondorder small values are taken into account. The differences Zx and Zy between the image point Zi and the crossing points Zc of x and y components by aberrations are 1 {3(0030)(xo )2 + (0012)(yo )2 }{1 + O(1)}, sin o
i = (0030)o2 + 2(1020)o o + (2010)o2
(2010)o2
+2(0012)xo yo .
Zx = −
(, ), it holds the same format as Eq. (A.1) because the core-lenses have rotation symmetry about the Z-axis.
(A.13)
Finally, if we consider the second-order of (xo , yo ), remaining aberrations are the aperture aberration and distortion aberrations. Using the diagram of axis intercept, we can obtain the coefficient of the aperture aberration as follows; (same as Eq. (1)). Zx = Zy = −
3(0030) 2 (xo ) . sin o
(A.14)
References [1] D. Gabor, A zonally corrected electron lens, Nature 158 (1946) 198. [2] H. Hoppe, Dreidimensional abbildende Elektronenmikroskope, I. Prinzip der geräte, Z. Naturforsch. 27a (1972) 919–929. [3] D. Typke, Image aberrations of axially symmetric imaging systems with off-axial rays: three-dimensionally imaging electron microscopes (part III), Nucl. Instrum. Methods 187 (1981) 217–226. [4] E. Plies, Calculations of electrostatic multitubular core-lenses for a 3D imaging electron microscope: three-dimensionally imaging electron microscopes, (part IV), Nucl. Instrum. Methods 187 (1981) 227–235. [5] K. Abe, Y. Tsuruga, S. Okada, T. Nomura, H. Aoki, H. Fuji, H. Koike, A. Hamaguchi, Y. Yamazaki, Three-dimensional measurement by tilting and moving objective lens in CD-SEM(III), in: Proc. SPIE, vol. 5752, 2005, pp. 1200–1208. [6] P. Hawkes, E. Kasper, Basic geometrical optics, in: Principles of Electron Optics, vol. 1, Academic Press, London, 1989.