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Aberration analysis of electron mirrors by a differential algebraic method
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Optik
Optics
Optik 119 (2008) 90–96 www.elsevier.de/ijleo
Aberration analysis of electron mirrors by a differential algebraic method Liping Wang, John Rouse, Eric Munro, Haoning Liu, Xieqing Zhu Munro’s Electron Beam Software Ltd., 14 Cornwall Gardens, London SW7 4AN, England, UK Received 23 March 2006; accepted 22 June 2006
Abstract A differential algebraic (DA) method has been developed for the aberration analysis of electron mirrors. Since large ray slopes occur near the turning points in mirrors, the axial position is no longer suitable as the independent variable and the electron trajectory equation used in conventional lens theory is no longer feasible. A DA solution of the electron motion equation, wherein a single DA ray trace is performed on a non-standard extension of real number space called nDv, enables the aberrations of a mirror system to be obtained, in principle up to arbitrary order n, and with very high accuracy, due to the remarkable algebraic properties of nDv. With the DA method, the enormous effort to derive explicit formulae for the aberration coefficients of electron mirrors is avoided. A software package MIRROR_DA has been developed for the aberration analysis of electron mirrors, based on the DA method. Two examples of electron mirrors are presented. For the first example, for which the electrostatic and magnetic fields are represented by analytical models, the results computed with MIRROR_DA were shown to be in good agreement with those extracted by direct ray tracing, with relative deviations of less than 0.065% for all the primary aberration coefficients. The second example consists of a real magnetic lens and electrostatic mirror, with numerically computed fields, and from the results of MIRROR_DA, the spherical aberration coefficient Cs3 is almost cancelled out because of the correction effect of the mirror. The MIRROR_DA software is a novel, effective and precise tool for the aberration analysis of electron mirrors, capable of handling realistic and complicated systems of electron lenses and electron mirrors. r 2006 Elsevier GmbH. All rights reserved. Keywords: Differential algebra; Electron optics; Electron mirror; Aberration correction
1. Introduction In addition to traditional electrostatic and magnetic round lenses, many designers of modern charged particle optical systems have to consider more complicated optical elements, in the quest for greater resolution, higher speed and wide applicability. The electron mirror is one of such sought-after elements in recent years [1]. As mirrors can generate aberrations of opposite sign to those of round lenses, they can be used Corresponding author.
E-mail address: [email protected] (L. Wang). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.06.013
as aberration correctors in systems such as electron microscopes. LEEM systems [2] can also be used in a mirror mode, where the electrons are reflected from the sample surface, and the variation of potential across the sample yields a contrast variation in the reflected beam. It has therefore become important for designers to be able to compute the optical properties of mirrors and to accurately extract the individual aberration coefficients, so that the mirror can be designed to correct the existing column aberrations. The difficulty in simulating mirrors lies in the fact that the electron’s axial velocity is reduced to zero at the turning point and then reverses, relative to its initial
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value. This has traditionally caused a problem for conventional aberration theories, which usually assume that the axial velocity is finite and the beam slopes are small, and express the ray paths as functions of axial position. In a mirror, the axial velocity goes to zero, the beam slopes can be infinite at the turning point and the axial position is no longer a unique coordinate for the ray, since the ray reversed its direction. Electron mirrors were studied extensively by Kelman and coworkers in the 1970s [3]. Their investigations, however, were not complete and not consistently extensible. A time-dependent perturbation formalism was developed by Rose and Preikszas in the late 1990s, which allows one to calculate the aberration of a rotationally symmetric electron mirror [4,5]. In their work, analytical formulae are given for the coefficients of the spherical and the axial chromatic aberration, while the position of a particle is referred to that of an axial reference particle; the Lorentz equation is transformed into a set of inhomogeneous integral equations, which are then solved by an iterative procedure. However, the derivation of the explicit expressions for these coefficients is very arduous, and it is mandatory to employ an algebraic computer program, such as MOPS, which they developed for this purpose [5]. Recently, Wan et al. [6], during the development of PEEM3, developed a model of a mirror corrector, using a charged ring method to compute the potential distribution and computing the aberrations with the code COSY INFINITY developed by Berz, who first introduced the differential algebraic (DA) method into accelerator physics with great success [7]. Previously, we have applied the DA method to compute the aberrations of electron optical systems, such as realistic electrostatic and magnetic lenses [8]. Our previous work was focused on the DA solution of the electron trajectory equation, with the use of the axial position z as independent variable. We demonstrated that with a single DA ray trace, all the aberration coefficients are obtained simultaneously, in principle up to arbitrary order, and with very high accuracy. In our further research, we have found that the DA method also provides a powerful technique for the analysis of electron mirrors, if we use time instead of axial position as our independent variable. By using the DA method, the enormous effort usually required for deriving explicit formulae for the aberration coefficients of electron mirrors is eliminated. We have developed a software package called MIRROR_DA for the aberration analysis of electron mirrors, based on the DA method. This software is written in ANSI Standard C++. No special language or extensions are needed, since the DA quantities can be represented by a C++ Class. In this paper, our method is described in detail. Two examples are presented to show the accuracy and effectiveness of MIRROR_DA.
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2. DA method for electron mirrors In electron mirrors, the z-position cannot be used as independent variable, so it is mandatory to use time t, or an equivalent length, as the independent variable [4,5]. Following other researchers [4,5], we also make use of a reference ray, which is one that enters the mirror travelling along the optical axis with nominal kinetic energy F0. Its position (x(t), y(t), z(t)) at any time t, is (0,0,z(t)), expressed in Cartesian coordinates, where z denotes the axial position of the reference ray. The position of an arbitrary ray, at time t, is measured with respect to the reference ray, thus: (x(t), y(t), z(t)+h(t)), where h(t) is the axial separation of the arbitrary ray from the reference ray. Now consider the Lorentz equation of motion d ðmvÞ ¼ q ðE þ v BÞ, (1) dt where E is the electric field, B is the magnetic flux density, and m, q and v are the mass, charge and velocity of the particle. When linearized for the reference ray, this gives dz _ dz_ q ¼ z; ¼ E z ð0; 0; zÞ, (2) dt dt m as the reference ray is locatable solely by its axial position z. For an arbitrary ray, the Lorentz equation can be transformed into a set of equations in x, y and h as dx _ ¼ x; dt
dx_ q _ z z_By Þ, ¼ ðE x þ yB dt m
dy _ ¼ y; dt
dy_ q _ z Þ, ¼ ðE y þ z_Bx xB dt m
_ dh _ dh ¼ q ðE z þ xB € _ y yB _ x Þ z, ¼ h; (3) dt dt m _ y, _ z_ as well as z_ are components of where in Eq. (3), x, velocities, i.e. derivatives with respect to time, and _ We can trace the rays from the object plane h_ ¼ z_ z. by solving the above equations. At the object plane z ¼ z0, from the reference ray with nominal potential F0 (measured relative to where the electron is at rest), conservation of energy gives 1 _2 2 m z0
¼ qF0 ,
and so the initial velocity of the reference ray is rffiffiffiffiffiffiffiffiffiffiffi _z0 ¼ 2qF0 , m
(4)
(5)
This, along with z0 ¼ z0
(6)
is the initial condition for the reference ray. As for an arbitrary ray, suppose its position at the object plane is
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(x0, y0, z0), apparently h0 ¼ z0 z0 ¼ 0.
(7)
Considering an arbitrary ray with deviation DF from the nominal potential, because 2 2 1 _ _2 (8) 2m x0 þ y0 þ z_0 ¼ qðF0 þ DFÞ, the axial component of its initial velocity is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qðF0 þ DFÞ ð1 þ DF=F0 Þ _ ¼ z0 , z_0 ¼ 2 2 0 0 m 1þx0þy0 1 þ x0 20 þ y0 20
_
(9)
thus h_0 ¼ z_0 z_ 0 ¼ z_ 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ð1 þ DF=F0 Þ 1 . 1 þ x0 20 þ y0 20
(10)
The other components of its initial velocity are x_ 0 ¼ x00 z_0 ; 0
y_ 0 ¼ y00 z_0 ,
(11)
0
where x 0 and y 0 are the initial slope components of the ray. Assume that, at time tf, the reference ray reaches the image plane z ¼ zi and its position and velocity are denoted by xi (apparently xi ¼ zi) and xi. At the same time, the position and velocity of the arbitrary ray can be written as xðtf Þ :¼ xf ; yðtf Þ :¼ yf ; hðtf Þ :¼ hf ; zðtf Þ :¼ zf ¼ zi þ hf , _ f Þ :¼ h_f ; z_ðtf Þ :¼ z_f ¼ z_ i þ h_f . _ f Þ :¼ x_ f ; yðt _ f Þ :¼ y_ f ; hðt xðt (12)
It should be noted that (xf, yf) do not represent the off-axial position of the arbitrary ray at the image plane z ¼ zi, which we denote as (xi, yi). This is because, in general, the off-axis rays reach the image plane later than the corresponding axial reference ray. In the case of a ‘‘field-free’’ region near the image plane, we can use a straight line to project the ray to its final position at the image plane, i.e. by linear extrapolation. However, to be more general, we use quadratic extrapolation: xi ¼ xf
hf x_ f h2f ðx€ f z_f z€f x_ f Þ þ , z_f 2_z3f
yi ¼ yf
hf y_ f h2f y€ f z_f z€f y_ f þ . z_f 2_z3f
which is called nDv. As already noted [8,9], nDv is the collection of DA-quantities, of order n, with v expansion variables. Remarkably, it has many properties similar to R, such as arithmetic operations and their laws, ‘‘zero’’ and ‘‘unit’’, ordering, ‘‘positive’’ and ‘‘negative’’, etc. However, nDv has many other algebraic properties that distinguish it from R, such as the existence of
(13)
Starting from the above equations, to derive the formulae for the aberration coefficients of electron mirrors requires enormous effort on formulae manipulations, inevitably involving the use of computer algebraic programs. To avoid this, we have developed the technique and software using DA to compute the optical properties and aberrations of mirrors. The DA method computes aberrations, theoretically to arbitrary order, in a single ray trace, using a non-standard analysis technique on an extension of the real space R,
infinitesimals d and their nilpotent property. Fundamental functions in R (such as exponent, logarithm, and trigonometric functions) can always be extended into nDv, and many familiar properties of these functions remain while extended. Series expansions of these functions in nDv are the same as in R. For the analysis of electron mirrors, the DA method uses all the equations stated in previous paragraphs. However, all variables for the arbitrary ray are transformed into DA quantities: _ y; _ h_ x; y; h; x; ðreal variablesÞ
!
_ _ _ _ _ _
x; y ; h ; x_ ; y_ ; h_ . ðDA quantitiesÞ
The potentials and fields of each optical element are represented by radial expansions of field harmonics. The electric field and the magnetic flux density are determined by gradients of scalar potentials. The potentials and fields are also transformed into DA quantities. In the DA ray tracing, the same numerical techniques that are used in traditional ray tracing can be employed, but all steps are transferred from the real space R to nDv. After a single DA ray trace in nDv, DA quantities are obtained at the image plane zi, which can be represented as 2 3 2 3 Aijklm x 6 7 6B 7 iþjþkþlþmpn X 6y7 6 ijklm 7 i j 0k 0l m6 6 07 ¼ 7, (14) x0 y0 x 0 y 0 d 6 6x 7 7 4 5 4 C ijklm 5 i;j;k;l;m¼0n Dijklm y0
n
zi
_ z, y0 ¼ y=_ _ z, and d denotes the relative where x0 ¼ x=_ energy deviation (DF/F0). The subscript n indicates that the DA raytrace is computed up to nth order. The summation indices i, j, k, l and m take all possible values from 0 to n, subject to the constraint that i+j+k+l+mpn. At the image plane zi, the coefficients Aijklm–Dijklm represent the optical properties and aberrations of the system. These coefficients are extracted from the corresponding DA quantities. After some simple manipulations, they provide the aberration coefficients in conventional form, theoretically up to any desired order and with extremely high accuracy. Additional great advantages of the DA method presented here are that it avoids painstaking derivation of aberration formulae and the ray-trace is valid for all values of the axial velocity and ray slope.
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Based on the above principle and formulae, we have developed the software MIRROR_DA for the aberration analysis of electron mirrors. In MIRROR_ DA, the DA ray trace is performed by a fifth-order Runge–Kutta technique, with adaptive step size control [10]. The electric and magnetic fields are reconstructed from Hermite series fitting of the axial distribution functions extracted from the field computation. This is very accurate because series of Hermite functions are continuously differentiable and are especially suitable for field functions that tend to zero at faraway axial distances [8,11].
magnetic lens, with 2 cylindrical polepieces, with 22 mm radius and 8 mm gap. Electrons were launched with an initial energy of 6996.2 eV, from an object plane at a point z0, on the left-hand side of the mirror, travelling to the right. They were reflected at about 6.5 mm to the right of the mid-plane and then refocused at the Gaussian image plane zi. The position of z0 was adjusted until z0 and zi were coincident, which occurred when z0 ¼ 107.681868 mm, measured relative to the mid-plane of the mirror. This example was simulated using both MIRROR_ DA and the analytic field model program. The complex coordinate at the Gaussian image plane, wi ¼ xi+iyi, can be expressed as
3. Examples
wi ¼ Mw0 þ cs3 s20 s~0 þ kR s20 w~ 0 þ kL s0 s~0 w0
One of the examples presented here is used to check the results of the aberration computation by the DA method. This example has a simple structure with analytic model fields. For electrostatic mirror fields we use an analytic model with multicylinder equi-radius electrodes, while for the magnetic fields, we use an analytic model consisting of a pair of polepieces of cylinder shape. In both cases, we assume that the potential varies linearly across the gaps, at the radius of the electrodes or polepieces. In these structures, the potential distribution, and the electric and magnetic fields, can be computed analytically using Fourier– Bessel series, similar to our previous study on bipotential electrostatic round lenses [10]. The derivation of analytic formulae for these models will be reported in a separate paper. A program has been written for direct ray tracing through these analytical model fields, which can compute selected rays starting from specified initial conditions, and output their positions on the screen. By computing well-chosen bunches of rays, we are able to extract the aberration coefficients (The extraction formulae will also be reported in a separate paper.) This program has been used to check the aberrations computed with our newly developed MIRROR_DA software. One of the examples analysed is shown in Fig. 1. It consists of an electrostatic mirror, with 2 cylindrical electrodes, with 20 mm radius and 10 mm gap, and a
þ fs0 w0 w~ 0 þ a~s0 w20 þ d 3 w20 w~ 0 þ cc s0 d þ ccm2 w0 d þ . . . ,
where w0 ¼ x0+iy0 and s0 ¼ sx0+isy0 is the complex coordinate and slope at the object plane, w~ 0 and s~0 are their conjugates, M is the complex magnification (which incorporates the image rotation angle), and the other coefficients are the primary aberration coefficients. Or equivalently, wi ¼ wg þ C s3 s2g s~g þ K R s2g w~ g þ K L sg s~g wg þ Fsg wg w~ g þ A~sg w2g þ D3 w2g w~ g þ C c sg dg þ C cm2 wg dg þ . . . ,
Electrode (V1= 10000 volts)
ð16Þ
where wg ¼ Mw0 is the complex Gaussian image point and dg ¼ DF/Fi is the fractional energy deviation at the Gaussian image plane. For this example, the complex magnification is (0.24451010.9696467i) as computed with MIRROR_DA, which is in good agreement with (0.2442750.969706i) extracted from the ray tracing in the analytic fields. The comparison of all primary aberration coefficients (as in Eq. (16)) between the results from MIRROR_DA and those extracted from direct ray-trace in analytic fields are shown in Table 1. As can be seen, the agreement is good between the DA results and the direct-ray-trace results. This demonstrates the great accuracy of the DA method in
Excitation = 750 AT Polepiece
ð15Þ
Polepiece Electrode (V2 = -50volts)
10mm scale bar
Fig. 1. Mirror system for aberration analysis, showing electrodes, polepieces and illustrative rays.
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Table 1. Comparison of primary aberration coefficients (as defined in Eq. (16)) between DA computation and direct ray-trace computation in analytic fields (values are in S.I. Units) Aberration coefficient Cs3 KR KL F A D3 Cc Ccm2
Spherical: Coma: Field curv.: Astig.: Distortion: Chromatic:
Extracted from raytrace in analytic fields
Computed with DA software
Difference in magnitude (%)
51.307 486.126+69.5835i 972.243138.921i 9376.19 4516.93+1315.79i 44 334.76341.0i 2.61757 24.7965+2.30714i
51.297687496 485.94870092+69.513773538i 971.89740184139.02754707i 9371.8318876 4514.1555599+1317.0234213i 44 325.5896996342.0968041i 2.6178132092 24.798835783+2.3088217965i
0.0181 0.0388 0.0368 0.0510 0.0645 0.0205 0.0093 0.0115
75 Magnetic lens
(mm)
Mirror
0
"Non-mirror case"
zi
z0 Vm "Mirror case"
-75
220
0 z (mm)
Fig. 2. Mirror system for aberration analysis, showing mirror electrodes and magnetic lens.
aberration analysis of electron mirrors using our software MIRROR_DA. The above example is only an illustrative one, to show the power and accuracy of the DA method for mirrors, as its electrostatic and magnetic fields use an analytic model so we can compare with the exact results. However, MIRROR_DA is not restricted to electron mirrors with analytic model fields. On the contrary, it can handle realistic electron mirrors with complicated configurations effectively and precisely. Here we present another example, which has a more complicated structure, as shown in Fig. 2. It consists of a magnetic lens and an electrostatic mirror corrector. The object plane is at z0 ¼ 60 mm, and the image plane is at zi ¼ 220 mm. The mid-plane of the gap of the magnetic lens is at z ¼ 210 mm, while the two electrode mirror is positioned so that its reflection electrode surface is at z ¼ 2.8 mm. This position was chosen because it was found that at this plane the combined spherical aberration of the lens and mirror is almost zero. This mirror system was analysed with our MIRROR_DA software. The axial potential function and the axial magnetic flux density distribution were obtained from our SOFEM software [12], which uses the second-order finite element method. Then these axial
functions were fitted by Hermite series [8,11]. These data were fed into MIRROR_DA, and the optical properties and aberrations were calculated by the DA method. Without the mirror, the magnetic lens can focus a 20 keV electron beam travelling directly to the right, from z0 to zi, with an excitation of about 1904.6 AT. In this case, the magnification is 0.1346059 and the rotation angle is 63.0981, and the spherical aberration coefficient Cs3 is +14.992 mm, as calculated by the DA method. According to Scherzer’s theorem, this spherical aberration coefficient Cs3 is always positive for ordinary round lenses. Now the mirror is added, and a 20 keV beam is launched from z0, travelling leftwards towards the mirror, and the mirror potential, Vm, is adjusted until the reflected beam is focused again at z0; this occurs when Vm ¼ 5948.3543 V. The beam then travels through the magnetic lens at 20 keV to the image plane zi, as before. From the MIRROR_DA software, the turning point is at around z ¼ 6.5631 mm, relative to the mirror electrode, and the spherical aberration coefficient Cs3 is 0.0001926 mm. Thus, with the correction effect of this mirror Cs3 is almost cancelled out. This cancellation of Cs3 was achieved by adjusting the axial position of the mirror. As calculated from the MIRROR_DA software, the traces of primary rays are plotted in Fig. 3, where ra is the primary focusing ray (i.e. launching from z-axis with unit initial slope), rb is the primary field ray (i.e. launching parallel to z-axis with unit initial radial coordinate and zero initial slope), as no aperture is present in this rotationally symmetrical system. From the DA quantities computed from MIRROR_ DA, we can easily plot the spot diagram at the final plane. For the example shown in Fig. 2, the spot on the screen is shown in Fig. 4 for a 0.6 0.6 mm2 squareshaped beam. Note that the image rotation introduced by the magnetic lens has been removed in Fig. 4. (One also has the other option to have the rotation shown during the plotting procedure of our software.) It can clearly be seen that the blurs are much smaller when the mirror is used.
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rb
60
220
(a)
ra 6.563
(c)
60
220 z (mm)
60
95
220
220
z (mm)
(b)
z (mm)
rb
ra
L. Wang et al. / Optik 119 (2008) 90–96
6.563
(d)
60
220
220
z (mm)
Fig. 3. Primary rays of the example as in Fig. 2, as calculated by MIRROR_DA software: (a) focusing ray without mirror; (b) field ray without mirror; (c) focusing ray with mirror and (d) field ray with mirror.
Fig. 4. Spot diagrams at the image plane, for the example shown in Fig. 2, for a square shaped beam: (a) without mirror and (b) with mirror.
In principle, a mirror with more electrodes can provide more adjustment of correcting properties (e.g. a tetrode mirror [5,6] enables the correction of both primary spherical aberration and axial chromatic aberration). Furthermore, in practice, the use of mirror in a system will inevitably be combined with bending elements to separate the incident beam and reflected beam and prevent the reflected beam from travelling back to the gun or sample [1,5,6]. These topics are not the scope of this paper, whose focus is reporting the method and results of our MIRROR_DA program.
4. Conclusions Aberration analysis of electron mirrors is more complicated than for conventional electron optical systems, because large ray slopes occur in the vicinity of turning points and the axial position is no longer suitable to be chosen as the independent variable. To derive the formulae of aberration coefficients of electron
mirrors, starting from the Lorentz equation of motion, normally requires immense effort of formulae manipulations, inevitably involving the use of computer algebraic programs. Based on our previous work of introducing the DA method to the aberration computation of conventional systems, we have developed the DA method for the analysis of electron mirrors, where the electron trajectory equation requiring the ray slopes to be small quantities all along the trajectory is no longer feasible. Focusing on the DA solution of the electron motion equation, we have found that, by a single DA ray tracing performed on non-standard nDv, all the aberration coefficients of a mirror system can be obtained simultaneously, in principle up to arbitrary order n, and with very high accuracy, due to the special algebraic properties of nDv. By using the DA method, the painstaking effort of deriving explicit formulae for the aberration coefficients of electron mirrors is eliminated. We have developed the software MIRROR_DA for the aberration analysis of electron mirrors, based on the DA method. Two examples of electron mirrors have been presented. For the first example, for which the electrostatic and magnetic fields have analytical models, the results computed with MIRROR_DA agreed excellently with those extracted from direct ray-tracing in the analytical model fields, with relative deviations of less than 0.065% for all the primary aberration coefficients. The second example, of a more complicated structure, has been computed by MIRROR_DA. In this example, the aberrations of a magnetic lens were calculated. And then a mirror with realistic electrode geometry was added and analysed, and from the results of MIRROR_DA, Cs3 is almost cancelled out, due to the correction effect of the mirror. It is concluded that this software is a novel, effective and precise tool for the aberration analysis of electron mirrors, capable of handling realistic and complicated electron optical configurations.
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References [1] J. Feng, E. Forest, A.A. MacDowell, et al., An X-ray photoemission electron microscope using electron mirror aberration corrector for the study of complex materials, J. Phys.: Condens Matter 17 (2005) S1339–S1350. [2] E. Bauer, LEEM basics, Surf. Rev. Lett. 5 (1998) 275–1286. [3] V.M. Kelman, L.M. Sekunova, E.M. Yakushev, Sov. Phys. Tech. Phys. 18 (1974) 1142, 1799, 1809. [4] H. Rose, D. Preikszas, Time-dependent perturbation formalism for calculating the aberrations of systems with large ray gradients, Nucl. Instrum. Methods—Phys. Res. A 363 (1995) 301–315. [5] D. Preikszas, H. Rose, Correction properties of electron mirrors, J. Electron Microsc. 46 (1) (1997) 1–9. [6] W. Wan, J. Feng, H.A. Padmore, et al., Simulation of a mirror corrector for PEEM3, Nucl. Instrum. Methods— Phys. Res. A 519 (2004) 222–229.
[7] M. Berz, Differential algebraic description of beam dynamics to very high orders, Particle Accelerators 24 (1989) 109–124. [8] L.-P. Wang, J. Rouse, H. Liu, E. Munro, X. Zhu, Simulation of electron optical systems by differential algebraic method combined with Hermite fitting for practical lens fields, Microelectron. Eng. 73–74 (2004) 90–96. [9] L.-P. Wang, T.-T. Tang, B.-J. Cheng, Differential algebraic theory and method for arbitrary high order aberrations of electron optics, Optik 111 (2000) 285–289. [10] E. Munro, X. Zhu, J. Rouse, H. Liu, A study of ways of improving the speed and accuracy of computing fields, trajectories and aberrations in electron optics, in: Proceedings of the Sixth Seminar on ‘‘Recent Trends in Charged Particle Optics and Surface Physics Instrumentation’’, Skalsky Dvur. ISI Brno, 1998. [11] H. Liu, E. Munro, J. Rouse, X. Zhu, Simulation methods for multipole imaging systems and aberration correctors, Ultramicroscopy 93 (2002) 271–291. [12] SOFEM User Manual v.1.6, MEBS Ltd., London, 2002.
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Optik
Optics
Optik 119 (2008) 90–96 www.elsevier.de/ijleo
Aberration analysis of electron mirrors by a differential algebraic method Liping Wang, John Rouse, Eric Munro, Haoning Liu, Xieqing Zhu Munro’s Electron Beam Software Ltd., 14 Cornwall Gardens, London SW7 4AN, England, UK Received 23 March 2006; accepted 22 June 2006
Abstract A differential algebraic (DA) method has been developed for the aberration analysis of electron mirrors. Since large ray slopes occur near the turning points in mirrors, the axial position is no longer suitable as the independent variable and the electron trajectory equation used in conventional lens theory is no longer feasible. A DA solution of the electron motion equation, wherein a single DA ray trace is performed on a non-standard extension of real number space called nDv, enables the aberrations of a mirror system to be obtained, in principle up to arbitrary order n, and with very high accuracy, due to the remarkable algebraic properties of nDv. With the DA method, the enormous effort to derive explicit formulae for the aberration coefficients of electron mirrors is avoided. A software package MIRROR_DA has been developed for the aberration analysis of electron mirrors, based on the DA method. Two examples of electron mirrors are presented. For the first example, for which the electrostatic and magnetic fields are represented by analytical models, the results computed with MIRROR_DA were shown to be in good agreement with those extracted by direct ray tracing, with relative deviations of less than 0.065% for all the primary aberration coefficients. The second example consists of a real magnetic lens and electrostatic mirror, with numerically computed fields, and from the results of MIRROR_DA, the spherical aberration coefficient Cs3 is almost cancelled out because of the correction effect of the mirror. The MIRROR_DA software is a novel, effective and precise tool for the aberration analysis of electron mirrors, capable of handling realistic and complicated systems of electron lenses and electron mirrors. r 2006 Elsevier GmbH. All rights reserved. Keywords: Differential algebra; Electron optics; Electron mirror; Aberration correction
1. Introduction In addition to traditional electrostatic and magnetic round lenses, many designers of modern charged particle optical systems have to consider more complicated optical elements, in the quest for greater resolution, higher speed and wide applicability. The electron mirror is one of such sought-after elements in recent years [1]. As mirrors can generate aberrations of opposite sign to those of round lenses, they can be used Corresponding author.
E-mail address: [email protected] (L. Wang). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.06.013
as aberration correctors in systems such as electron microscopes. LEEM systems [2] can also be used in a mirror mode, where the electrons are reflected from the sample surface, and the variation of potential across the sample yields a contrast variation in the reflected beam. It has therefore become important for designers to be able to compute the optical properties of mirrors and to accurately extract the individual aberration coefficients, so that the mirror can be designed to correct the existing column aberrations. The difficulty in simulating mirrors lies in the fact that the electron’s axial velocity is reduced to zero at the turning point and then reverses, relative to its initial
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value. This has traditionally caused a problem for conventional aberration theories, which usually assume that the axial velocity is finite and the beam slopes are small, and express the ray paths as functions of axial position. In a mirror, the axial velocity goes to zero, the beam slopes can be infinite at the turning point and the axial position is no longer a unique coordinate for the ray, since the ray reversed its direction. Electron mirrors were studied extensively by Kelman and coworkers in the 1970s [3]. Their investigations, however, were not complete and not consistently extensible. A time-dependent perturbation formalism was developed by Rose and Preikszas in the late 1990s, which allows one to calculate the aberration of a rotationally symmetric electron mirror [4,5]. In their work, analytical formulae are given for the coefficients of the spherical and the axial chromatic aberration, while the position of a particle is referred to that of an axial reference particle; the Lorentz equation is transformed into a set of inhomogeneous integral equations, which are then solved by an iterative procedure. However, the derivation of the explicit expressions for these coefficients is very arduous, and it is mandatory to employ an algebraic computer program, such as MOPS, which they developed for this purpose [5]. Recently, Wan et al. [6], during the development of PEEM3, developed a model of a mirror corrector, using a charged ring method to compute the potential distribution and computing the aberrations with the code COSY INFINITY developed by Berz, who first introduced the differential algebraic (DA) method into accelerator physics with great success [7]. Previously, we have applied the DA method to compute the aberrations of electron optical systems, such as realistic electrostatic and magnetic lenses [8]. Our previous work was focused on the DA solution of the electron trajectory equation, with the use of the axial position z as independent variable. We demonstrated that with a single DA ray trace, all the aberration coefficients are obtained simultaneously, in principle up to arbitrary order, and with very high accuracy. In our further research, we have found that the DA method also provides a powerful technique for the analysis of electron mirrors, if we use time instead of axial position as our independent variable. By using the DA method, the enormous effort usually required for deriving explicit formulae for the aberration coefficients of electron mirrors is eliminated. We have developed a software package called MIRROR_DA for the aberration analysis of electron mirrors, based on the DA method. This software is written in ANSI Standard C++. No special language or extensions are needed, since the DA quantities can be represented by a C++ Class. In this paper, our method is described in detail. Two examples are presented to show the accuracy and effectiveness of MIRROR_DA.
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2. DA method for electron mirrors In electron mirrors, the z-position cannot be used as independent variable, so it is mandatory to use time t, or an equivalent length, as the independent variable [4,5]. Following other researchers [4,5], we also make use of a reference ray, which is one that enters the mirror travelling along the optical axis with nominal kinetic energy F0. Its position (x(t), y(t), z(t)) at any time t, is (0,0,z(t)), expressed in Cartesian coordinates, where z denotes the axial position of the reference ray. The position of an arbitrary ray, at time t, is measured with respect to the reference ray, thus: (x(t), y(t), z(t)+h(t)), where h(t) is the axial separation of the arbitrary ray from the reference ray. Now consider the Lorentz equation of motion d ðmvÞ ¼ q ðE þ v BÞ, (1) dt where E is the electric field, B is the magnetic flux density, and m, q and v are the mass, charge and velocity of the particle. When linearized for the reference ray, this gives dz _ dz_ q ¼ z; ¼ E z ð0; 0; zÞ, (2) dt dt m as the reference ray is locatable solely by its axial position z. For an arbitrary ray, the Lorentz equation can be transformed into a set of equations in x, y and h as dx _ ¼ x; dt
dx_ q _ z z_By Þ, ¼ ðE x þ yB dt m
dy _ ¼ y; dt
dy_ q _ z Þ, ¼ ðE y þ z_Bx xB dt m
_ dh _ dh ¼ q ðE z þ xB € _ y yB _ x Þ z, ¼ h; (3) dt dt m _ y, _ z_ as well as z_ are components of where in Eq. (3), x, velocities, i.e. derivatives with respect to time, and _ We can trace the rays from the object plane h_ ¼ z_ z. by solving the above equations. At the object plane z ¼ z0, from the reference ray with nominal potential F0 (measured relative to where the electron is at rest), conservation of energy gives 1 _2 2 m z0
¼ qF0 ,
and so the initial velocity of the reference ray is rffiffiffiffiffiffiffiffiffiffiffi _z0 ¼ 2qF0 , m
(4)
(5)
This, along with z0 ¼ z0
(6)
is the initial condition for the reference ray. As for an arbitrary ray, suppose its position at the object plane is
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(x0, y0, z0), apparently h0 ¼ z0 z0 ¼ 0.
(7)
Considering an arbitrary ray with deviation DF from the nominal potential, because 2 2 1 _ _2 (8) 2m x0 þ y0 þ z_0 ¼ qðF0 þ DFÞ, the axial component of its initial velocity is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qðF0 þ DFÞ ð1 þ DF=F0 Þ _ ¼ z0 , z_0 ¼ 2 2 0 0 m 1þx0þy0 1 þ x0 20 þ y0 20
_
(9)
thus h_0 ¼ z_0 z_ 0 ¼ z_ 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ð1 þ DF=F0 Þ 1 . 1 þ x0 20 þ y0 20
(10)
The other components of its initial velocity are x_ 0 ¼ x00 z_0 ; 0
y_ 0 ¼ y00 z_0 ,
(11)
0
where x 0 and y 0 are the initial slope components of the ray. Assume that, at time tf, the reference ray reaches the image plane z ¼ zi and its position and velocity are denoted by xi (apparently xi ¼ zi) and xi. At the same time, the position and velocity of the arbitrary ray can be written as xðtf Þ :¼ xf ; yðtf Þ :¼ yf ; hðtf Þ :¼ hf ; zðtf Þ :¼ zf ¼ zi þ hf , _ f Þ :¼ h_f ; z_ðtf Þ :¼ z_f ¼ z_ i þ h_f . _ f Þ :¼ x_ f ; yðt _ f Þ :¼ y_ f ; hðt xðt (12)
It should be noted that (xf, yf) do not represent the off-axial position of the arbitrary ray at the image plane z ¼ zi, which we denote as (xi, yi). This is because, in general, the off-axis rays reach the image plane later than the corresponding axial reference ray. In the case of a ‘‘field-free’’ region near the image plane, we can use a straight line to project the ray to its final position at the image plane, i.e. by linear extrapolation. However, to be more general, we use quadratic extrapolation: xi ¼ xf
hf x_ f h2f ðx€ f z_f z€f x_ f Þ þ , z_f 2_z3f
yi ¼ yf
hf y_ f h2f y€ f z_f z€f y_ f þ . z_f 2_z3f
which is called nDv. As already noted [8,9], nDv is the collection of DA-quantities, of order n, with v expansion variables. Remarkably, it has many properties similar to R, such as arithmetic operations and their laws, ‘‘zero’’ and ‘‘unit’’, ordering, ‘‘positive’’ and ‘‘negative’’, etc. However, nDv has many other algebraic properties that distinguish it from R, such as the existence of
(13)
Starting from the above equations, to derive the formulae for the aberration coefficients of electron mirrors requires enormous effort on formulae manipulations, inevitably involving the use of computer algebraic programs. To avoid this, we have developed the technique and software using DA to compute the optical properties and aberrations of mirrors. The DA method computes aberrations, theoretically to arbitrary order, in a single ray trace, using a non-standard analysis technique on an extension of the real space R,
infinitesimals d and their nilpotent property. Fundamental functions in R (such as exponent, logarithm, and trigonometric functions) can always be extended into nDv, and many familiar properties of these functions remain while extended. Series expansions of these functions in nDv are the same as in R. For the analysis of electron mirrors, the DA method uses all the equations stated in previous paragraphs. However, all variables for the arbitrary ray are transformed into DA quantities: _ y; _ h_ x; y; h; x; ðreal variablesÞ
!
_ _ _ _ _ _
x; y ; h ; x_ ; y_ ; h_ . ðDA quantitiesÞ
The potentials and fields of each optical element are represented by radial expansions of field harmonics. The electric field and the magnetic flux density are determined by gradients of scalar potentials. The potentials and fields are also transformed into DA quantities. In the DA ray tracing, the same numerical techniques that are used in traditional ray tracing can be employed, but all steps are transferred from the real space R to nDv. After a single DA ray trace in nDv, DA quantities are obtained at the image plane zi, which can be represented as 2 3 2 3 Aijklm x 6 7 6B 7 iþjþkþlþmpn X 6y7 6 ijklm 7 i j 0k 0l m6 6 07 ¼ 7, (14) x0 y0 x 0 y 0 d 6 6x 7 7 4 5 4 C ijklm 5 i;j;k;l;m¼0n Dijklm y0
n
zi
_ z, y0 ¼ y=_ _ z, and d denotes the relative where x0 ¼ x=_ energy deviation (DF/F0). The subscript n indicates that the DA raytrace is computed up to nth order. The summation indices i, j, k, l and m take all possible values from 0 to n, subject to the constraint that i+j+k+l+mpn. At the image plane zi, the coefficients Aijklm–Dijklm represent the optical properties and aberrations of the system. These coefficients are extracted from the corresponding DA quantities. After some simple manipulations, they provide the aberration coefficients in conventional form, theoretically up to any desired order and with extremely high accuracy. Additional great advantages of the DA method presented here are that it avoids painstaking derivation of aberration formulae and the ray-trace is valid for all values of the axial velocity and ray slope.
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Based on the above principle and formulae, we have developed the software MIRROR_DA for the aberration analysis of electron mirrors. In MIRROR_ DA, the DA ray trace is performed by a fifth-order Runge–Kutta technique, with adaptive step size control [10]. The electric and magnetic fields are reconstructed from Hermite series fitting of the axial distribution functions extracted from the field computation. This is very accurate because series of Hermite functions are continuously differentiable and are especially suitable for field functions that tend to zero at faraway axial distances [8,11].
magnetic lens, with 2 cylindrical polepieces, with 22 mm radius and 8 mm gap. Electrons were launched with an initial energy of 6996.2 eV, from an object plane at a point z0, on the left-hand side of the mirror, travelling to the right. They were reflected at about 6.5 mm to the right of the mid-plane and then refocused at the Gaussian image plane zi. The position of z0 was adjusted until z0 and zi were coincident, which occurred when z0 ¼ 107.681868 mm, measured relative to the mid-plane of the mirror. This example was simulated using both MIRROR_ DA and the analytic field model program. The complex coordinate at the Gaussian image plane, wi ¼ xi+iyi, can be expressed as
3. Examples
wi ¼ Mw0 þ cs3 s20 s~0 þ kR s20 w~ 0 þ kL s0 s~0 w0
One of the examples presented here is used to check the results of the aberration computation by the DA method. This example has a simple structure with analytic model fields. For electrostatic mirror fields we use an analytic model with multicylinder equi-radius electrodes, while for the magnetic fields, we use an analytic model consisting of a pair of polepieces of cylinder shape. In both cases, we assume that the potential varies linearly across the gaps, at the radius of the electrodes or polepieces. In these structures, the potential distribution, and the electric and magnetic fields, can be computed analytically using Fourier– Bessel series, similar to our previous study on bipotential electrostatic round lenses [10]. The derivation of analytic formulae for these models will be reported in a separate paper. A program has been written for direct ray tracing through these analytical model fields, which can compute selected rays starting from specified initial conditions, and output their positions on the screen. By computing well-chosen bunches of rays, we are able to extract the aberration coefficients (The extraction formulae will also be reported in a separate paper.) This program has been used to check the aberrations computed with our newly developed MIRROR_DA software. One of the examples analysed is shown in Fig. 1. It consists of an electrostatic mirror, with 2 cylindrical electrodes, with 20 mm radius and 10 mm gap, and a
þ fs0 w0 w~ 0 þ a~s0 w20 þ d 3 w20 w~ 0 þ cc s0 d þ ccm2 w0 d þ . . . ,
where w0 ¼ x0+iy0 and s0 ¼ sx0+isy0 is the complex coordinate and slope at the object plane, w~ 0 and s~0 are their conjugates, M is the complex magnification (which incorporates the image rotation angle), and the other coefficients are the primary aberration coefficients. Or equivalently, wi ¼ wg þ C s3 s2g s~g þ K R s2g w~ g þ K L sg s~g wg þ Fsg wg w~ g þ A~sg w2g þ D3 w2g w~ g þ C c sg dg þ C cm2 wg dg þ . . . ,
Electrode (V1= 10000 volts)
ð16Þ
where wg ¼ Mw0 is the complex Gaussian image point and dg ¼ DF/Fi is the fractional energy deviation at the Gaussian image plane. For this example, the complex magnification is (0.24451010.9696467i) as computed with MIRROR_DA, which is in good agreement with (0.2442750.969706i) extracted from the ray tracing in the analytic fields. The comparison of all primary aberration coefficients (as in Eq. (16)) between the results from MIRROR_DA and those extracted from direct ray-trace in analytic fields are shown in Table 1. As can be seen, the agreement is good between the DA results and the direct-ray-trace results. This demonstrates the great accuracy of the DA method in
Excitation = 750 AT Polepiece
ð15Þ
Polepiece Electrode (V2 = -50volts)
10mm scale bar
Fig. 1. Mirror system for aberration analysis, showing electrodes, polepieces and illustrative rays.
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Table 1. Comparison of primary aberration coefficients (as defined in Eq. (16)) between DA computation and direct ray-trace computation in analytic fields (values are in S.I. Units) Aberration coefficient Cs3 KR KL F A D3 Cc Ccm2
Spherical: Coma: Field curv.: Astig.: Distortion: Chromatic:
Extracted from raytrace in analytic fields
Computed with DA software
Difference in magnitude (%)
51.307 486.126+69.5835i 972.243138.921i 9376.19 4516.93+1315.79i 44 334.76341.0i 2.61757 24.7965+2.30714i
51.297687496 485.94870092+69.513773538i 971.89740184139.02754707i 9371.8318876 4514.1555599+1317.0234213i 44 325.5896996342.0968041i 2.6178132092 24.798835783+2.3088217965i
0.0181 0.0388 0.0368 0.0510 0.0645 0.0205 0.0093 0.0115
75 Magnetic lens
(mm)
Mirror
0
"Non-mirror case"
zi
z0 Vm "Mirror case"
-75
220
0 z (mm)
Fig. 2. Mirror system for aberration analysis, showing mirror electrodes and magnetic lens.
aberration analysis of electron mirrors using our software MIRROR_DA. The above example is only an illustrative one, to show the power and accuracy of the DA method for mirrors, as its electrostatic and magnetic fields use an analytic model so we can compare with the exact results. However, MIRROR_DA is not restricted to electron mirrors with analytic model fields. On the contrary, it can handle realistic electron mirrors with complicated configurations effectively and precisely. Here we present another example, which has a more complicated structure, as shown in Fig. 2. It consists of a magnetic lens and an electrostatic mirror corrector. The object plane is at z0 ¼ 60 mm, and the image plane is at zi ¼ 220 mm. The mid-plane of the gap of the magnetic lens is at z ¼ 210 mm, while the two electrode mirror is positioned so that its reflection electrode surface is at z ¼ 2.8 mm. This position was chosen because it was found that at this plane the combined spherical aberration of the lens and mirror is almost zero. This mirror system was analysed with our MIRROR_DA software. The axial potential function and the axial magnetic flux density distribution were obtained from our SOFEM software [12], which uses the second-order finite element method. Then these axial
functions were fitted by Hermite series [8,11]. These data were fed into MIRROR_DA, and the optical properties and aberrations were calculated by the DA method. Without the mirror, the magnetic lens can focus a 20 keV electron beam travelling directly to the right, from z0 to zi, with an excitation of about 1904.6 AT. In this case, the magnification is 0.1346059 and the rotation angle is 63.0981, and the spherical aberration coefficient Cs3 is +14.992 mm, as calculated by the DA method. According to Scherzer’s theorem, this spherical aberration coefficient Cs3 is always positive for ordinary round lenses. Now the mirror is added, and a 20 keV beam is launched from z0, travelling leftwards towards the mirror, and the mirror potential, Vm, is adjusted until the reflected beam is focused again at z0; this occurs when Vm ¼ 5948.3543 V. The beam then travels through the magnetic lens at 20 keV to the image plane zi, as before. From the MIRROR_DA software, the turning point is at around z ¼ 6.5631 mm, relative to the mirror electrode, and the spherical aberration coefficient Cs3 is 0.0001926 mm. Thus, with the correction effect of this mirror Cs3 is almost cancelled out. This cancellation of Cs3 was achieved by adjusting the axial position of the mirror. As calculated from the MIRROR_DA software, the traces of primary rays are plotted in Fig. 3, where ra is the primary focusing ray (i.e. launching from z-axis with unit initial slope), rb is the primary field ray (i.e. launching parallel to z-axis with unit initial radial coordinate and zero initial slope), as no aperture is present in this rotationally symmetrical system. From the DA quantities computed from MIRROR_ DA, we can easily plot the spot diagram at the final plane. For the example shown in Fig. 2, the spot on the screen is shown in Fig. 4 for a 0.6 0.6 mm2 squareshaped beam. Note that the image rotation introduced by the magnetic lens has been removed in Fig. 4. (One also has the other option to have the rotation shown during the plotting procedure of our software.) It can clearly be seen that the blurs are much smaller when the mirror is used.
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rb
60
220
(a)
ra 6.563
(c)
60
220 z (mm)
60
95
220
220
z (mm)
(b)
z (mm)
rb
ra
L. Wang et al. / Optik 119 (2008) 90–96
6.563
(d)
60
220
220
z (mm)
Fig. 3. Primary rays of the example as in Fig. 2, as calculated by MIRROR_DA software: (a) focusing ray without mirror; (b) field ray without mirror; (c) focusing ray with mirror and (d) field ray with mirror.
Fig. 4. Spot diagrams at the image plane, for the example shown in Fig. 2, for a square shaped beam: (a) without mirror and (b) with mirror.
In principle, a mirror with more electrodes can provide more adjustment of correcting properties (e.g. a tetrode mirror [5,6] enables the correction of both primary spherical aberration and axial chromatic aberration). Furthermore, in practice, the use of mirror in a system will inevitably be combined with bending elements to separate the incident beam and reflected beam and prevent the reflected beam from travelling back to the gun or sample [1,5,6]. These topics are not the scope of this paper, whose focus is reporting the method and results of our MIRROR_DA program.
4. Conclusions Aberration analysis of electron mirrors is more complicated than for conventional electron optical systems, because large ray slopes occur in the vicinity of turning points and the axial position is no longer suitable to be chosen as the independent variable. To derive the formulae of aberration coefficients of electron
mirrors, starting from the Lorentz equation of motion, normally requires immense effort of formulae manipulations, inevitably involving the use of computer algebraic programs. Based on our previous work of introducing the DA method to the aberration computation of conventional systems, we have developed the DA method for the analysis of electron mirrors, where the electron trajectory equation requiring the ray slopes to be small quantities all along the trajectory is no longer feasible. Focusing on the DA solution of the electron motion equation, we have found that, by a single DA ray tracing performed on non-standard nDv, all the aberration coefficients of a mirror system can be obtained simultaneously, in principle up to arbitrary order n, and with very high accuracy, due to the special algebraic properties of nDv. By using the DA method, the painstaking effort of deriving explicit formulae for the aberration coefficients of electron mirrors is eliminated. We have developed the software MIRROR_DA for the aberration analysis of electron mirrors, based on the DA method. Two examples of electron mirrors have been presented. For the first example, for which the electrostatic and magnetic fields have analytical models, the results computed with MIRROR_DA agreed excellently with those extracted from direct ray-tracing in the analytical model fields, with relative deviations of less than 0.065% for all the primary aberration coefficients. The second example, of a more complicated structure, has been computed by MIRROR_DA. In this example, the aberrations of a magnetic lens were calculated. And then a mirror with realistic electrode geometry was added and analysed, and from the results of MIRROR_DA, Cs3 is almost cancelled out, due to the correction effect of the mirror. It is concluded that this software is a novel, effective and precise tool for the aberration analysis of electron mirrors, capable of handling realistic and complicated electron optical configurations.
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References [1] J. Feng, E. Forest, A.A. MacDowell, et al., An X-ray photoemission electron microscope using electron mirror aberration corrector for the study of complex materials, J. Phys.: Condens Matter 17 (2005) S1339–S1350. [2] E. Bauer, LEEM basics, Surf. Rev. Lett. 5 (1998) 275–1286. [3] V.M. Kelman, L.M. Sekunova, E.M. Yakushev, Sov. Phys. Tech. Phys. 18 (1974) 1142, 1799, 1809. [4] H. Rose, D. Preikszas, Time-dependent perturbation formalism for calculating the aberrations of systems with large ray gradients, Nucl. Instrum. Methods—Phys. Res. A 363 (1995) 301–315. [5] D. Preikszas, H. Rose, Correction properties of electron mirrors, J. Electron Microsc. 46 (1) (1997) 1–9. [6] W. Wan, J. Feng, H.A. Padmore, et al., Simulation of a mirror corrector for PEEM3, Nucl. Instrum. Methods— Phys. Res. A 519 (2004) 222–229.
[7] M. Berz, Differential algebraic description of beam dynamics to very high orders, Particle Accelerators 24 (1989) 109–124. [8] L.-P. Wang, J. Rouse, H. Liu, E. Munro, X. Zhu, Simulation of electron optical systems by differential algebraic method combined with Hermite fitting for practical lens fields, Microelectron. Eng. 73–74 (2004) 90–96. [9] L.-P. Wang, T.-T. Tang, B.-J. Cheng, Differential algebraic theory and method for arbitrary high order aberrations of electron optics, Optik 111 (2000) 285–289. [10] E. Munro, X. Zhu, J. Rouse, H. Liu, A study of ways of improving the speed and accuracy of computing fields, trajectories and aberrations in electron optics, in: Proceedings of the Sixth Seminar on ‘‘Recent Trends in Charged Particle Optics and Surface Physics Instrumentation’’, Skalsky Dvur. ISI Brno, 1998. [11] H. Liu, E. Munro, J. Rouse, X. Zhu, Simulation methods for multipole imaging systems and aberration correctors, Ultramicroscopy 93 (2002) 271–291. [12] SOFEM User Manual v.1.6, MEBS Ltd., London, 2002.
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