Spherical aberration correction with threefold symmetric line currents

Ultramicroscopy 161 (2016) 74–82

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Spherical aberration correction with threefold symmetric line currents Shahedul Hoque a,b,n, Hiroyuki Ito b, Ryuji Nishi a, Akio Takaoka a, Eric Munro c a b c

Research Center for Ultra-High Voltage Electron Microscopy, Osaka University, 7-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan Hitachi High-Technologies Corporation, 882, Ichige, Hitachinaka, Ibaraki 312‐8504, Japan Munro's Electron Beam Software Ltd., 14 Cornwall Gardens, London SW7 4AN, UK

art ic l e i nf o

a b s t r a c t

Article history: Received 6 August 2015 Received in revised form 3 November 2015 Accepted 6 November 2015 Available online 10 November 2015

It has been shown that N-fold symmetric line current (henceforth denoted as N-SYLC) produces 2N-pole magnetic fields. In this paper, a threefold symmetric line current (N3-SYLC in short) is proposed for correcting 3rd order spherical aberration of round lenses. N3-SYLC can be realized without using magnetic materials, which makes it free of the problems of hysteresis, inhomogeneity and saturation. We investigate theoretically the basic properties of an N3-SYLC configuration which can in principle be realized by simple wires. By optimizing the parameters of a system with beam energy of 5.5 keV, the required excitation current for correcting 3rd order spherical aberration coefficient of 400 mm is less than 1AT, and the residual higher order aberrations can be kept sufficiently small to obtain beam size of less than 1 nm for initial slopes up to 5 mrad. & 2015 Elsevier B.V. All rights reserved.

Keywords: Spherical aberration Aberration correction N-fold symmetric line current Sextupole corrector Hexapole corrector Magnetic multipole

1. Introduction Correction of spherical aberration of round lenses by means of sextupole field has been studied extensively [1–7]. Conventional aberration correctors consist of multipoles made of magnetic materials. The manufacturing of such multipoles requires high degree of mechanical precision, and the power supply unit must also be highly stable. Other major difficulties are the hysteresis and inhomogeneity of magnetic materials, which makes it extremely difficult to control and have sufficient repeatability of the aberration correction conditions. However, hysteresis and inhomogeneity can be overcome by eliminating magnetic materials. It has been shown that a combination of N parallel line currents with N-fold symmetry can produce 2N-pole magnetic fields [8]. We call such an arrangement “N-fold symmetric line current”, or in short N-SYLC. Since the line currents of N-SYLC can be realized by simple wires, it is in principle free of magnetic materials, thus of hysteresis and inhomogeneity. In this paper, a threefold symmetric line current (henceforth, N3SYLC in short) for 3rd order spherical aberration correction is proposed. The setup is similar to the hexapole magnetic corrector by Rose et al. [5–7], with the conventional magnetic multipoles replaced by N3-SYLC. The N3-SYLC in the present model can be realized by rectangular-shaped coils. The paraxial trajectories are controlled by rotationally symmetric round lenses, while N3-SYLC produces hexapole fields which result in negative spherical aberration. n

Corresponding author at: Research Center for Ultra-High Voltage Electron Microscopy, Osaka University, 7-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. http://dx.doi.org/10.1016/j.ultramic.2015.11.005 0304-3991/& 2015 Elsevier B.V. All rights reserved.

As a first step toward constructing an aberration correction system utilizing N3-SYLC, we investigate theoretically the basic electron optical properties of the N3-SYLC, assuming that it is free of defects in size, shape and configurations, and the power supply units are also perfectly stable. We show that it can indeed produce negative spherical aberration in combination with rotationally symmetric lenses. The effect of higher order aberrations of N3SYLC is also discussed. We also propose several versions of N3SYLC with different characteristics in terms of higher order aberrations and sensitivity.

2. Structure of N3-SYLC N3-SYLC consists of three line currents parallel to one another with threefold symmetry about an axis. We consider currents I through infinitely long wires parallel to and at distance R from the z-axis. The angle between the currents is 2π as shown is Fig. 1. The 3

magnetic scalar potential is given as follows (see Appendix A for derivation)

Ψ= − = −

⎞ μ0 I ⎛ r 3 1 r6 1 r9 ⎜ 3 sin 3θ + sin 9θ + ⋯⎟ sin 6θ + 6 9 2π ⎝ R 2R 3R ⎠ μ0 I 2π

∞

∑ n= 1

r 3n sin (3nθ ). nR3n

(1)

The above expression is a linear combination of 3n-pole fields, where n ¼1, 2, 3 and so on. The strength of each multipole component gets weaker by a factor of 13n as n increases. Hence, the nR

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the finite-size coils of Fig. 2(a) using the Biot and Savart Law, without making any approximations.

3. Spherical aberration correction with N3-SYLC 3.1. A system for aberration correction

Fig. 1. Schematic diagram of N3-SYLC.

hexapole component r 3 sin 3θ is the most dominant one, which results in 2nd and higher order combination aberrations. The next dominant term in the potential is the dodecapole component, which results in 5th and higher order combination aberrations. The N3-SYLC described above can be realized by rectangular coils shown in Fig. 2(a). The electron trajectories are along the zaxis (optic axis). In each rectangular coil, current through the line AB parallel to the z-axis plays the role of the line current I in Fig. 1. Now, we make the following approximations for simplicity while performing analytical calculation:

The schematic diagram of a system using N3-SYLC for spherical aberration correction is shown in Fig. 3. It consists of four rotationally symmetric lenses (henceforth, round lenses) RL1, RL2, RL3 and RL 4 and two N3-SYLCs (S1 and S2), and is similar to the hexapole corrector by Rose et al. [5–7], with the conventional magnetic multipoles replaced by N3-SYLCs. The excitations of RL1 and RL3 are positive, while RL2 and RL 4 are negative. RL2 and RL3 will have the same value of excitation. The focal length of RL1 is f1, while RL2 and RL3 have the same focal length f2 . The N3-SYLCs, each having length L , are placed with their centers at distance f2 from RL2 and RL3. The characteristic function of the system of Fig. 3 consists of contributions from the round lenses and the N3-SYLCs, and can be expressed as follows.

F = F2RL + F3N 3 + F4RL + F6RL + F6N 3 + ⋯,

(2)

where the superscript N3 stands for N3-SYLC and RL for round lenses, and the numerical subscripts denote the total power of x and y in each term. The explicit expressions of the characteristic functions F3N 3 and F6N 3 are given in Appendix B (Eq. (B.1)). The paraxial trajectories are determined by F2RL . G (z ) and H (z ) trajectories are shown in Fig. 3. At the object plane z ¼ zo , they satisfy the following conditions: G (zo )=1, G′(zo )=0; H (zo )=0, H′(zo )=1. 3.2. Aberration correction

(1) Only the line current I through line AB in each coil is taken into account. Other parts of the coils are discarded. (2) The magnetic scalar potentials and magnetic flux-densities are confined only in the region of the coils. (3) In that region, the magnetic scalar potential is that generated by currents through infinitely long wires, i.e. it is independent of z (see Fig. 2(b)). Under these approximations, Eq. (1) can be applied to the coils of Fig. 2(a). However, while performing numerical calculations, for higher accuracy, we will determine the magnetic-flux densities by

In this section we derive analytical expressions for the aberrations of N3-SYLCs in the setup of Fig. 3. We assume thin lenses for RL1, RL2, RL3 and RL 4 , so that they only change the slopes of electron trajectories, and their fields do not overlap with the fields of the N3-SYLCs. The F3N 3 term results in second order aberration (ΔX2, ΔY2 ), which cancels out in the image plane due to the symmetry properties of the trajectories in the present arrangement. The details of the cancellation of the second order aberration are given in Appendix C.1. F4RL results in a 3rd order positive spherical aberration which is

Fig. 2. Realization of N3-SYLC (a) a realistic N3-SYLC and (b) infinite-wire potential approximation.

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Fig. 3. Schematic diagram of a system for spherical aberration correction using N3-SYLC.

the characteristic of rotationally symmetric lenses. However, the combination aberration of

(

∂ N3 ∂ N3 F , ∂Y F3 ∂X 3

)

and (ΔX2, ΔY2 ) results in

negative spherical aberration as follows.

⎧ ⎛ ημ 0 I ⎞2 4 3⎫ ⎪ ⎪ ΔX3N 3 = − M ⎨ 6⎜ X ′ X o′ 2 + Yo′ 2 ⎟ f1 L ⎬ 3 ⎪ ⎪ o ⎝ ⎠ 2 π R Φ ⎩ ⎭

(

⎧ ⎛ ημ 0 I ⎞2 4 3⎫ ⎪ ⎪ ΔY3N 3 = − M ⎨ 6⎜ Y ′ X o′ 2 + Yo′ 2 ⎟ f1 L ⎬ 3 ⎪ ⎪ o ⎝ ⎠ 2 π R Φ ⎩ ⎭

(

⎫

)⎪⎪

)

⎪ ⎬ ⎪ ⎪ ⎪ ⎭

(3)

Here, Xo′ and Yo′ are initial slopes at the object pane, M is the total magnification, μ0 is magnetic permeability in vacuum, and Φ is the electric potential which is a constant in this case. η is given by

η=

e , 2m 0

where, e is the absolute value of electron charge, and m0 is electron rest-mass. The derivation of Eq. (3) is given in Appendix C.2. Thus, the N3-SYLC system has a negative spherical aberration coefficient given by

⎛ ημ 0 I ⎞2 4 3 CsN3 = − 6 ⎜ ⎟f LM ⎝ 2π R 3 Φ ⎠ 1

(4)

If the spherical aberration coefficient of the round lenses is CsRL , then the overall spherical aberration coefficient of the system is

Cstotal = CsRL + CsN3.

(5)

The condition for vanishing 3rd order spherical aberration is

3.3. Higher order aberrations of N3-SYLC We will not derive the analytical forms of the 4th and 5th order aberrations, rather rely on numerical calculations to evaluate their effects. However, we can estimate the dependence of the higher order aberrations on various parameters of the system without deriving the full analytical expressions. Since the characteristic function of N3-SYLC does not contain 4th order terms after the 3rd order terms (Eq. (2)), but only 6th order terms and higher, the 4th order aberration arises solely from the combination aberrations involving 2nd and 3rd order aberrations. On the other hand, 5th order aberrations arises from the 6th order term ( F6N 3) in the characteristic function (Eq. (2)), as well as combination aberrations of the lower order terms. The 4th and 5th order aberrations due to N3-SYLC can be expressed as follows (see Appendix C.3 for details).

ΔX 4 =

∑ n = 1,3

ΔX5 = ≡

ημo I 4π R

⎛ ημo I ⎞n ⎟ Un ⎜ ⎝ 2π R 3 ⎠

V+ 6

⎛ ημo I ⎞n ⎟ Vn 3⎠ ⎝ n = 1 2π R 4

∑⎜

⎛ ημ I ⎞ ⎛ ημ I ⎞ P⎜ o 6⎟ + Q ⎜ o 3⎟ ⎝ 2π R ⎠ ⎝ 4π R ⎠

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

(8)

Here, Un , Vn and V are some integrations. Expressions for the y components are similar. In the 5th order aberration, term P is caused by the 6th order term F6N 3 in the characteristic function of Eq. (2), while term Q is combination aberration involving lower order aberrations. Under the correction condition of Eq. (6), 1

I = Ic =

2π 6 ημ 0

3⎛ 1 R3Φ 2 f1−2 L− 2 ⎜

1 CsRL ⎞ 2

⎟ . ⎝ M ⎠

(6)

If RL 4 in Fig. 3 is used as the final objective lens, then I = Ic is the required excitation of the N3-SYLC for zero-spherical aberration. However, if there is a separate objective lens after RL 4 , then I >Ic such that the resultant negative aberration cancels the positive aberration of the final objective lens. From Eq. (6), the dependence of correction sensitivity on various parameters of the system is given by 3⎛ 1 Ic ∝R3Φ 2 f1−2 L− 2 ⎜

(9)

Therefore, ΔX 4 and Q in ΔX5 are independent of R when spherical aberration is corrected. Thus, R can be manipulated for suitable sensitivity without affecting these aberrations. However, term P in Eq. (7) is proportional to 13 under the correction conR

dition of Eq. (8). Thus, if the radius R is decreased for higher sensitivity, this aberration is increased, the effect of which is also discussed in §4. 3.4. Comparison between analytical and numerical results

1

CsRL ⎞ 2 ⎟ . ⎝ M ⎠

⎛ C RL ⎞ 2 3 ημo I ∝f1−2 L− 2 Φ ⎜ s ⎟ . 3 2π R ⎝ M ⎠

(7)

This shows how the sensitivity of the system can be controlled by manipulating different parameters of N3-SYLC. For example, we see that the required excitation is most sensitive to a change in the inner radius R of the N3-SYLC coils.

The approximations assumed in §2 and §3.2 for obtaining the condition for spherical aberration correction (Eqs. (3) and (6)) do not hold exactly in a real system. For more accurate results we resort to numerical calculations. However, as is shown below, under certain conditions, simplified analytical results hold with more than 90% accuracy. This simplicity is one of the key

S. Hoque et al. / Ultramicroscopy 161 (2016) 74–82

advantages of N-SYLC; various parameters can be estimated analytically without performing detailed numerical calculations. In our model, while performing numerical calculation, ring shaped coils (henceforth called ring coils) are used as RL1, RL2, RL3 and RL 4 , and thin lens approximation is not imposed. Magneticflux densities by ring coils are analytically solvable, so are the flux densities by N3-SYLC. That all the fields involved in the model are analytically solvable makes the results highly accurate. The values of different parameters used in the numerical calculations are as follow. (1) All four ring coils have the same radius Rc ¼10 mm and same focal length f ¼25 mm. Then the total length of the system is 25 × 8 ¼ 200 mm. (2) The N3-SYLCs have length L , inner radius R and outer radius 55 mm each. (3) The electric potential is 5500 V. In order to estimate the effects of fringe fields, and check the validity of Eq. (4), numerical calculations are performed under the following two conditions. (1) Discard the fringe effects of N3-SYLC by applying the approximations of §2. We will denote CsN3 obtained in this manner as “ Num. CsN3 (fringeless)”. Aberration coefficients are obtained by means of a differential algebraic method [9]. (2) The magnetic-flux densities by the finite-size coils of Fig. 2 (a) are determined using the Biot and Savart Law, without assuming the approximations of §2. We will denote CsN3 obtained in this manner as “ Num. CsN3 (full)”. Calculations are performed using a software called the LANTERN developed by MEBS [10] (see Appendix D for a short description of the software). The electron trajectories are solved by the RungeKutta method and the aberration coefficients are calculated from the aberration diagrams at the Gaussian image plane. We will denote CsN3 obtained analytically by Eq. (4) as “ Analytic CsN3”. We have determined the negative spherical aberration coefficient of N3-SYLC as a function of length L , for two values of the inner radius, namely, R ¼2 mm and 10 mm. The results are shown in Fig. 4. In Fig. 4, the ratio ( Analytic CsN3)/( Num. CsN3 (full)) is plotted by white spots, while the ratio ( Num. CsN3 (fringless))/( Num. CsN3 (full)) by black spots. The results for R ¼2 mm is shown by round-shaped markers, whereas for R ¼10 mm by triangular markers. It can be seen that the fringe field effects become less significant for larger values of L . However, when the edges of N3-SYLC approach the R

77

ring coils (in the present case, for L ¼50 mm, the edges of N3-SYLC coils coincide with the ring coils), the fringe field effects become significant. The difference between ( Analytic CsN3)/( Num.CsN3 (full)) and ( Num. CsN3 (fringeless))/( Num. CsN3 (full)) is due to the thin lens approximations for the round lenses in the analytical calculation. It can be seen from Fig. 4 that simplified analytical results hold with more than 90% accuracy for 20 mm
approach the round lenses. The 3rd order positive spherical aberration coefficient of the ring coils in the present model (Fig. 3) is found to be 400 mm by numerical calculation. The value of N3-SYLC excitation current Ic to correct this aberration depends on the system parameters. For example, for L¼ 35 mm and R ¼10 mm, Ic ≅82 AT. It is necessary to keep the current of the N3-SYLC under a few Ampere-Turns in a real system considering the heating of the coils and the capability of commercially available current supply units. From Eq. (6), it is clear that Ic can be lowered by making the inner radius R smaller. At the same time higher order residual aberrations are also needed to be taken into account, which is discussed in §4.

4. Variations of N3-SYLC system The N3-SYLC system discussed in the previous section will be called N3-SYLC3-3. In order to obtain better characteristics in terms of higher order aberrations, we propose two variations of the N3SYLC system, which will be called N3-SYLC3-3’ and N3-SYLC3x2. The construction of each type is shown in Fig. 5. In N3-SYLC3-3’, S2 is rotated with respect to S1 by an angle of π about the z-axis, with opposite sign for the current. In N3-SYLC3x2, two N3-SYLCs with a relative rotation π about the z-axis and opposite-sign currents are combined in both S1 and S2. The magnetic scalar potential of each case under the approximations of §2 is given in Table 1. It is to be noted that in N3-SYLC3-3’, the dodecapole component of the potential has opposite signs in S1 and S2, whereas it cancels out in N3-SYLC3x2. Aberrations in N3-SYLC3-3, N3-SYLC3-3’ and N3-SYLC3x2 are calculated using the LANTERN software. We have considered two values for radius R , namely 10 mm and 2 mm. Shorter radius is considered for higher sensitivity. The excitation current Ic for canceling the 3rd order spherical aberrations of the round lenses is the same for N3-SYLC3-3 and N3-SYLC3-3’, which is expected since the hexapole component in the potential is the same in both. N3-SYLC3x2 needs half the excitation, because, as can be seen form the potentials given in Table 1, the hexapole component is doubled

Fig. 4. Comparison between negative spherical aberration coefficients obtained by simplified analytical calculation and accurate numerical calculations.

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S. Hoque et al. / Ultramicroscopy 161 (2016) 74–82

Fig. 5. Construction of N3-SYLC3-3, N3-SYLC3-3’ and N3-SYLC3x2.

in N3-SYLC3x2. For smaller R , the required excitation Ic decreases according to Eq. (6)’. These results along with the aberration diagrams are shown in Fig. 6. It is to be noted that for R ¼2 mm, Ic is less than 1AT, which is small enough to be suitable for designing a working system. The aberration diagrams reveal the residual higher order aberrations, which restricts the beam size after the correction of the 3rd order spherical aberration. We would like to increase the sensitivity of the corrector by making R smaller. However, the residual aberration of N3-SYLC3-3 increases significantly as R decreases, while it remains unchanged in N3-SYLC3x2. In N3-SYLC3-3’ the residual aberration increases, but the increase is significantly suppressed compared to N3-SYLC3-3. These results can be understood in terms of Eq. (7), and the potentials given in Table 1. Under the condition I = Ic , in case of N3-SYLC3-3, the 4th order aberration is unaffected but the 5th order aberration increases as discussed in §3.3. In N3-SYLC3-3’, this increase in S1 is suppressed by the opposite sign increase in S2. This is because the dodecapole component of the potential, which results in the P term in Eq. (7), has opposite signs in S1 and S2. As for N3-SYLC3x2, the dodecapole component is absent in the potential, which means that the P term

is absent in the 5th order aberration. Considering the value of excitation current for aberration correction and the size of the residual aberrations, N3-SYLC3x2 is the most attractive of the three. However, the higher sensitivity may lead to the requirements of higher precision of size, position and excitation current of the N3SYLCs. In that case, N3-SYLC3-3’ with less sensitivity but still with small residual aberrations may be more practical.

5. Conclusion A combination of three parallel line currents with threefold symmetry (N3-SYLC) produces hexapole magnetic fields, which can be used to correct the 3rd order spherical aberration of rotationally symmetric lenses. With suitable arrangement and dimensions, theoretically the required excitation for aberration correction can be made less than 1AT, while the residual higher order aberrations can be kept small enough to obtain beam size of less than 1 nm for initial slopes up to 5 mrad (without considering diffraction effect). The structure of N-SYLC in general is simple and has the

Table 1 Scalar magnetic potentials in three N3-SYLC setup. Setup N3-SYLC3-3 N3-SYLC3-3’ N3-SYLC3x2

S1

S2

−

μ0 I ⎛ r 3 ⎜ 2π ⎝ R3

sin 3θ +

−

μ0 I ⎛ r 3 ⎜ 2π ⎝ R3

sin 3θ +

−

μ0 I ⎛ r 3 ⎜ π ⎝ R3

⎞ sin 3θ + O r 9 ⎟ ⎠

r6 2R6 r6 2R6

⎞ sin 6θ +⋯⋯⋯⎟ ⎠

−

μ0 I ⎛ r 3 ⎜ 2π ⎝ R3

sin 3θ +

⎞ sin 6θ +⋯⋯⋯⎟ ⎠

−

μ0 I ⎛ r 3 ⎜ 2π ⎝ R3

sin 3θ −

−

μ0 I ⎛ r 3 ⎜ π ⎝ R3

⎞ sin 3θ + O r 9 ⎟ ⎠

( )

r6 2R6 r6 2R6

⎞ sin 6θ +⋯⋯⋯⎟ ⎠ ⎞ sin 6θ +⋯⋯⋯⎟ ⎠

( )

S. Hoque et al. / Ultramicroscopy 161 (2016) 74–82

79

Fig. 6. Aberration diagrams at the Gaussian image plane under zero 3rd order spherical aberration condition. In each case, diagrams are shown for initial slopes 10, 8, 6, and 4 mrad.

potential to be a versatile electron optical component, realizable with lower cost than conventional magnetic multipoles. The magnetic fields induced by N-SYLC are analytically solvable, which makes analytical calculations of aberration coefficients possible. This should greatly facilitate designing a working system. Since the fields by N-SYLC are linearly proportional to the excitation current, it is in principle easy to control the system by means of simple addition of line currents or combination of different types of NSYLCs. This should be useful in controlling higher order residual aberrations. N-SYLC does not use magnetic materials and are hence free of the problems of hysteresis and inhomogeneity. Absence of magnetic materials also implies that magnetic interference is less likely to enter the region near the electron trajectories. Because the fields in N-SYLC are directly induced by line currents, it is free of the problems of saturation unlike magnetic materials. Since the fields are almost perpendicular to the electron trajectories, it should have high sensitivity. Thus the N3-SYLC proposed here has

the potential to be suitable is for aberration correction of high energy beams as well. In principle, N3-SYLC can replace the conventional magnetic multipoles of spherical aberration correctors. However, effects of errors in shapes and positions of the wires and other parameters, and fluctuation of excitation current are needed to be considered while designing a working system. The aberrations due to these imperfections will be investigated in the next step. The effect of the fields induced by N-SYLC on adjacent magnetic materials in the electron microscopes are also needed to be taken into account. Shielding of such undesired effects on magnetic materials will be essential while designing a full system. These issues will be addressed in future works.

Acknowledgment We thank Prof. Toshiaki Suhara of Osaka University and Momoyo Enyama of Hitachi Ltd. for helpful discussions and suggestions. We

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S. Hoque et al. / Ultramicroscopy 161 (2016) 74–82

also thank Fatema Tuz Zohora of Bangladesh University of Engineering and Technology for helping with writing a computer program for calculating aberration coefficients using DA method.

Then, the contribution of the N3-SYLC to the characteristic function of the system of Fig. 3 is given by

( )

F N3 = F3N3 + F6N3 + O r 9 =

Appendix A

ημ 0 I 3 x − 3xy2 2π R 3 ημ 0 I 6 + x − 15x 4 y2 + 15x2y 4 − y6 + O r 9 . 4π R 6

(

)

(

Magnetic scalar potential of N-SYLC We consider current I through an infinite line parallel to and at a distance R from the z -axis as shown in Fig. A.1. The orientation is characterized by angle α .

)

( )

(B.1)

Appendix C Aberration calculation In this section we will calculate the aberrations in the system shown in Fig. 3, which is a combination of four round lenses and two N3-SYLCs. The electron trajectories can be expressed as follows

→ ⎯ ⎯→ ⎯ ⎯⎯⎯⎯→ ⎯⎯⎯⎯→ ⎯⎯⎯⎯→ X =XG +ΔX2 +ΔX3 +ΔX 4 +⋯, ⎯→ ⎯ where, X⃗ is vector (X , Y ) and so on. XG are the paraxial trajectories nd determined by the 2 order term in the characteristic function ⎯⎯⎯⎯→ (Eq. 2), and ΔXn are n-th order aberrations (n is integer). Here, as is the standard practice, we have used the rotating coordinate system given as follows.

X = x cos φ + y sin φ

Fig. A.1. Current I through infinite line.

Y = − x sin φ + y cos φ The magnetic-flux densities at point (r, θ ) are given by

Br =

μ0 I sin (θ − α) , 2πR 1 + r¯ 2 − 2r¯ cos (θ − α)

Bθ =

μ0 I cos (θ − α) − r¯ , 2πR 1 + r¯ 2 − 2r¯ cos (θ − α)

φ′ =

r

where, r ̅ = R . These expressions can be expanded as follows.

Br =

μ0 I 2πR

∑n = 0 r¯ n sin {(n + 1)(θ − α )} ,

Bθ =

μ0 I 2πR

∑n = 0 r¯ n cos {(n + 1)(θ − α )} .

where, B is the on-axis value of the rotationally symmetric field by the round lenses. The paraxial trajectories are linear combinations of G (z ) and H (z ) trajectories, which are the two independent solutions of the paraxial ray equations with the following boundary conditions

( z o ) = 1, G′( z o ) = 0; H ( zGo ) = 0, H′( z o ) = 1.

∞

μ0I 2π

∞

∑ n= 1

rn sin {n (θ −α )} . nR n

(A.1) 2π 3

Adding the potentials for yields the potential for − the three-fold symmetric line current as follows.

μ0I 2π

∞

∑ n= 1

where, z =zo is the object plane. We consider the following G (z ) and H (z ) trajectories under thin lens approximation for the ring coils (Fig. 3).

⎫ ⎪ ⎪ ⎪ 1 ⎬ G = + ( z − 2f2 ) ( z3 ≤ z ≤ z 4 ); f1 > 0. ⎪ f1 ⎪ H = + f1 ( z1 ≤ z ≤ z2 ), H = − f1 ( z3 ≤ z ≤ z 4 ); f1 > 0.⎪ ⎭ G= −

2π α=0, 3 ,

Ψ= −

,

∞

The magnetic scalar potential Ψ is obtained by integrating the magnetic-flux densities. The result is as follows.

Ψ= −

ηB 2 Φ

r 3n sin (3nθ ). nR3n

(A.2)

1 ( z + 2f2 ) ( z1 ≤ z ≤ z2 ), f1

The aberrations approximations.

can

be

determined

by

(C.1) successive

C.1. 2nd order aberration

⎯⎯⎯⎯→ The differential equations for the 2nd order aberrations ΔX2 are as follows.

Appendix B Characteristic function The vector potentials of the N3-SYLC in orthogonal coordinates are as follows.

ΔX2′ ′ +

η2B 2 ΔX2 4Φ

ΔY2′ ′ +

η2B 2 ΔY2 4Φ

Ax = 0, Ay = 0, μ0 I

(x + O ( r ).

Az = −

2πR 3

9

3

)

− 3xy2 −

μ0 I 4πR 6

(x

6

− 15x 4 y2 + 15x2y 4 − y6

)

= =

(

F2x XG, YG, XG′, YG′, z Φ

(

F2y XG, YG, XG′, YG′, z Φ

),

),

where,

⎛ ⎛ d ∂ ∂ ⎞ N3 d ∂ ∂ ⎞ N3 F2x ≡ ⎜ − + + ⎟ F3 . ⎟ F3 , F2y ≡ ⎜ − ⎝ ⎝ dz ∂Y ′ ∂Y ⎠ dz ∂X′ ∂X ⎠

S. Hoque et al. / Ultramicroscopy 161 (2016) 74–82

The equations can be solved by the method of variation of parameters. We will consider electrons emitting from the z-axis at the object plane z =zo , i.e., with initial position Xo ¼ Yo ¼0 and initial slopes Xo′ , Yo′ . The solutions are

ΔX2 = ΔY2 =

3ημ0 2π Φ R

3

3ημ0

( −G ∫ IT H dz + H ∫ IT GH dz), ( −G ∫ IT H dz + H ∫ IT GH dz), x

y

2π Φ R 3

3

x

3

y

2

2

where,

Tx ≡ X o′ 2 cos 3φ − 2X o′ Yo′ sin 3φ − Yo′ 2 cos 3φ, Ty ≡ − X o′ 2 sin 3φ − 2X o′ Yo′ cos 3φ + Yo′ 2 sin 3φ. Under the thin lens approximation for the ring coils, φ is a constant in the region of the N3-SYLCs. Moreover, the rotation by the ring coil (RL2) at the exit of the first N3-SYLC is canceled by the opposite rotation by the ring coil ( RL3) at the entrance of the second N3-SYLC. Then, ΔX2 becomes ⎛ ⎞⎞ ⎛ z2 ⎞ ⎛ z2 3 z4 z4 Tx ⎜ − G ⎜ ∫ H dz + ∫ H 3dz ⎟ + H ⎜ ∫ GH 2dz + ∫ GH 2dz ⎟ ⎟ z z z z ⎠⎠ ⎝ ⎠ ⎝ ⎝ 3 1 3 1 2π Φ R ⎛ ⎞⎞ ⎛ z2 ⎛ z2 3 z4 z4 3ημ 0 3 ⎞ = Tx ⎜ − G ⎜ ∫ f1 dz + ∫ ( −f1 ) dz ⎟⎠ + H ⎜⎝ ∫z Gf12 dz + ∫z Gf12 dz ⎟⎠ ⎟. z3 ⎝ z1 ⎠ 3 1 2π Φ R 3 ⎝

ΔX2 =

3ημ 0

3

At the image plane z = zi , H ¼0, and hence,

ΔX2 = −

3ημ 0 2π Φ R

⎛

T G (zi ) ⎜ 3 x ⎝

∫z

z2

1

f13 dz +

3

C.2. 3rd order aberration

⎯⎯⎯⎯→ The 3rd order aberrations ΔX3 are caused by the rotationally symmetric round lenses (ring coils in Fig. 3) and the N3-SYLCs. Here we will consider the contributions from the N3-SYLCs only, ⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯→ which will be denoted as ΔX3N 3 . The differential equations for ΔX3N 3 are η2B 2 ΔX3N 3 4 Φ

=

(

∂ΔX2 F ∂X 2x

φ ΔY3N 3′′ + =

(

+

∂F ΔX2 ∂X2x

+

∂ΔY2 F ∂X 2y

+ ΔY2

∂F2y ∂X

)( X , Y ) ≡ W , G

G

x

η2B 2 ΔY3N 3 4 Φ

∂ΔX2 F2x ∂Y

+ ΔX2

∂F2x ∂Y

T ≡ − GH2 ∫ Q 3 H2dz + H3 ∫ Q 3 GHdz − GH ∫ Q 3 H3dz + H2 ∫ Q 3 GH2dz, Q3 =

ημ0 I 2πR 3

.

Performing the integrations using Eq. (C.1) yields the following expressions for the 3rd order aberration at the Gaussian image plane.

⎧ ΔX3N 3 = − M ⎨ 6 ⎩

(

2πR 3 Φ

⎧ ΔY3N 3 = − M ⎨ 6 ⎩

(

2πR 3 Φ

ημ0 I

ημ0 I

2

)f 2

⎫ ⎭

( Xo′2 + Yo′2 ),

⎫ ⎭

( Xo′2 + Yo′2 ),

4 3 1 L ⎬ X o′

)f

4 3 1 L ⎬ Yo′

where, M is the total magnification. C.3. Higher order aberrations Aberrations up to 5th order by both the round lenses and the N3-SYLCs are shown in Table C.1. In this table, x and y components of aberrations are denoted compactly as ΔUn . ∂ n stands for n-th order partial derivative with respect to x and y , and term like ∂ 2F3N 3 ∙(ΔU2 )2 means that it originates from the combination of

∂ 2F3N 3 and (ΔU2 )2. As is shown in Appendix A, the characteristic functions of N3-

⎞

z4

∫z ( − f1 )3dz⎟⎠ = 0.

Similar is the result for ΔY2. Thus, the second order aberrations cancel out because H is an odd function. The physical meaning of the cancellation mechanism is that, the N3-SYLC has a deflection effect proportional to the coil length L , and in the system of Fig. 3 the deflection effect by S2 cancels the opposite-sign deflection by S1.

φ ΔX3N 3 ′′ +

81

Table C.1 Structure of the aberrations of a system consisting of N3-SYLC and rotationally symmetric lenses. Aberration order (ΔUn ≡ ΔXn + iΔYn )

Aberrations

2nd order ΔU2

F3N 3

3rd order ΔU3

∂F3N 3 ∙ΔU2 , F4RL

4th order ΔU4

∂F3N 3 ∙ΔU3 , ∂ 2F3N 3 ∙(ΔU2 )2 , ∂F4RL ∙ΔU2

5th order ΔU5

∂F3N 3 ∙ΔU4 , ∂ 2F3N 3 ∙ΔU2 ∙ΔU3 , ∂F4RL ∙ΔU3 , ∂ 2F4RL ∙(ΔU2 )2F6N 3, F6RL

SYLC depend on the excitation current I and inner radius R of the coils in the following way.

F3N 3∝

ημo I ημ I ≡κ 3, F6N 3∝ o 6 ≡κ6. 2πR3 4π R

Thus, considering the aberration structures given in Table C.1, the 4th and 5th order aberrations can be expressed as follows.

ΔU4 = κ33 U3 + κ 3 U1 ΔU5 = κ6 VN 3 + κ34 V4 + κ33 V3 + κ32 V2 + κ 3 V1 + VRL

+

∂ΔY2 F ∂Y 2y

+ ΔY2

∂F2y ∂Y

)( X , Y ) ≡ W . G

G

y

Here, Un , Vn , VN3, VRL ( n = 1, 2, 3, and 4 ) are some integrations involving the terms given in Table C.1.

The solutions are

ΔX3N 3 =

1 Φ

ΔY3N 3 =

1 Φ

( −G ∫ HW ⎡⎣ X x

(

G,

YG, z⎤⎦ dz + H ∫ GWx ⎡⎣ XG , YG, z⎤⎦ dz ,

)

−G ∫ HWx ⎡⎣ XG , YG, z⎤⎦ dz + H ∫ GWx ⎡⎣ XG , YG, z⎤⎦ dz .

)

We will again consider electrons emitting from the z-axis at the object plane z =zo, i.e., with initial position Xo ¼ Yo ¼0. Under the thin lens approximation of the ring coils, the solutions become

ΔX3N 3 =

18 Φ

( −G ∫ Q

3 HTdz

+ H ∫ Q 3 GTdz X o′ X o′ 2 + Yo′ 2 ,

ΔY3N 3 =

18 Φ

( −G ∫ Q

3 HTdz

+ H∫Q3

where,

) ( GTdz) Y ′ ( X ′ o

o

)

2

)

+ Yo′ 2 ,

Appendix D LANTERN software LANTERN software is a program for simulating the properties of electrostatic and magnetic optical elements, namely, point or ringshaped charges, ring-shaped coils, toroidal coils and so on. The fields of these elements can be calculated analytically. The electron trajectories are computed with high accuracy using the full analytical fields, by implementing a 5th order Runge-Kutta formula with adaptive step size control. The error bound in the computations presented in this paper is 1.0e–12 m. Trajectories with

82

S. Hoque et al. / Ultramicroscopy 161 (2016) 74–82

different initial conditions (e.g. initial slopes, azimuth angles and energies) are calculated, and the geometrical and chromatic aberration coefficients are extracted by fitting the (x,y) positions of the computed trajectories at the image plane.

[5] [6]

References [1] V. Beck, A hexapole spherical aberration corrector, Optik 53 (1979) 241–255. [2] H. Rose, Correction of aperture aberrations in magnetic systems with threefold symmetry, Nucl. Instrum. Methods Phys. Res. 187 (1981) 187–199. [3] Z. Shao, On the fifth order aberration in a sextupole corrected probe forming system, Rev. Sci. Instrum. 59 (1988) 2429–2437. [4] E. Chen, C. Mu. New development in correction of spherical aberration of

[7]

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electro- magnetic round lens, in: K. Kuo, J. Yao (Eds.), Proceedings of International Symposium on Electron Microscopy 1990 World Scientific Singapore, pp. 28–35. H. Rose, Outline of a spherically corrected semi-aplanatic medium-voltage transmission electron microscope, Optik 85 (1990) 19–24. M. Haider, G. Braunshausen, E. Schwan, Correction of the spherical aberration of a 200 kV TEM by means of a hexapole corrector, Optik 99 (1995) 167–179. M. Haider, H. Rose, S. Uhlemann, B. Kabius, K. Urban, Towards 0.1 nm resolution with the first spherically corrected transmission electron microscope, J. Electron. Microsc. 47 (1998) 395–405. R. Nishi, H. Ito, S. Hoque. 18th International Microscopy Congress (IMC2014) 2014, Prague, Czech Republic, 200–201. M. Berz, Modern Map Methods in Particle Beam Physics, Academic Press, San Diego, 1999. LANTERN Software Version 3a, 2014, Munro's Electron Beam Software Ltd., London, U.K.