- Email: [email protected]
Cs-corrector compensating for the chromatic aberration and the spherical aberration of electron lenses
Ultramicroscopy 203 (2019) 139–144
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Magnetic Cc/Cs-corrector compensating for the chromatic aberration and the spherical aberration of electron lenses
T
H. Rosea, A. Nejatib, H. Müllerb,
⁎
a b
Ulm University, Meyerhofstr. 27, D-89081 Ulm, Germany CEOS GmbH, Englerstr. 28, D-69126 Heidelberg, Germany
ABSTRACT
Aberration correction in transmission electron microscopy has proven feasible and useful over a large range of acceleration voltages. The spherical aberration has been corrected for beam energies from 15 kV [1] up to 1.2 MeV [2] while the correction of the chromatic aberration has been achieved for beam energies ranging from 20 kV[3] up to 300 kV[4]. Above this threshold the conventional correction principle based on mixed electric and magnetic focusing elements becomes infeasible with present technology [5]. For conventional electron sources at high voltages the relative energy width of the beam gets so small that chromatic correction becomes less important. Nevertheless, for new applications with pulsed electron sources with energy spreads in the order of 100 eV chromatic aberration will become a limiting factor even at high energies [6]. To enable chromatic aberration correction for such systems a novel type of a feasible, purely magnetic multipole aberration corrector with curved optic axis is proposed which is capable of compensating for the chromatic and spherical aberration up to several MeV.
1. Introduction The chromatic aberration of magnetic systems with a straight optic axis is unavoidable [4]. In order to correct for this aberration in systems with a straight optic axis one must introduce mixed electric and magnetic quadrupole elements. Unfortunately, the electric field strength is limited to values smaller than about 7 kV/mm. Owing to this limitation the application of electric-magnetic correctors is limited to electron energies smaller than about 300 kV provided that the length of the corrector cannot surpass 1 m. Correction of chromatic aberration is possible in magnetic systems with a curved optic axis which is produced by the dipole component of the magnetic field. In addition, we must incorporate quadrupole elements for focusing and for forming the beam in an appropriate way and hexapole elements for adjusting the chromatic aberration of the corrector such that it compensates for the chromatic aberration of the entire system. The combination of the dispersion with the hexapole fields produces a first-order chromatic aberration whose sign can be adjusted by means of the polarity of the hexapoles. Apart from the chromatic aberration, the dipole and hexapole fields also introduce second-order geometrical aberrations which limit the resolution. Hence a feasible corrector must be designed in such a way that it does not introduce any second-order aberration. This requirement poses an extremely challenging task on the design of the corrector. Symmetry conditions imposed on the arrangement of the magnetic elements and on the course of the fundamental paraxial rays offer
⁎
the most promising method for eliminating the second-order aberrations, as it is the case for the hexapole corrector employed in atomicresolution electron microscopes for eliminating the third-order spherical aberration of the objective lens [5,6]. To satisfy all symmetry requirements a large number of elements is necessary making it difficult to align the system with the required accuracy. Moreover, the dipole and the quadrupole fields introduce third-order axial aberrations which must also be eliminated in order to achieve an appreciable increase in resolution. Because the third-order axial aberration of non-round elements consists of three components, the spherical aberration, the star aberration and the fourfold axial astigmatism, we need at least three independent octupole elements for compensating these aberrations. However, in order to judge the final optical performance also higherrank chromatic aberrations have to be taken into account. 2. Design of a corrector In order that all second-order geometrical aberrations cancel out in the region behind the corrector, we impose symmetry conditions [7] with respect to the mid-plane zM the corrector and with respect to the central planes zC1 and zC2 of each half of the system as depicted in zM Fig. 1. In particular, we propose that the magnetic field is symmetric with respect to the mid-plane zM. The quadrupole component of the magnetic field is also symmetric with respect to the central plane of each half of the system, whereas the dipole and the hexapole components are anti-symmetric. Apart from the symmetry conditions for the
Corresponding author: Englerstr. 28, D-69126 Heidelberg. E-mail addresses: [email protected] (H. Rose), [email protected] (A. Nejati), [email protected] (H. Müller).
https://doi.org/10.1016/j.ultramic.2018.11.014 Received 3 September 2018; Received in revised form 17 November 2018; Accepted 23 November 2018 Available online 29 November 2018 0304-3991/ © 2018 Elsevier B.V. All rights reserved.
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
Fig. 1. Scheme of the doubly symmetric magnetic chromatic-aberration corrector consisting of four 90°-deflection magnets Mμν, μ, ν = 1,2, two central quadrupoles QPC1 = QPC2, four quadrupoles QP111 = QP121 = QP221 = QP211, three hexapoles HPM and HP111 = HP211, four superposed quadupoles and hexapoles QPμ12 + HPμ12= QPμ22 + HPμ12, μ = 1,2, and two optional hexapoles HPC1, HPC2 superposed with the central quadruples. The mid-plane of the corrector is indicated by M, the central plane of each half of the corrector by C1 and C2, respectively. Images of the diffraction plane are located at the nodal planes N1 and N2. At every HP there is an additional octupole field for third-order correction (see §6).
magnetic field, we impose that the fundamental axial rays xα and yβ are symmetric and the field rays xγ and yδ are anti-symmetric with respect to zM. In addition the rays xα and yδ are symmetric and the rays yβ and xγ are anti-symmetric with respect to the central planes zC1 and zC2 of each half of the system. Owing to the required course of the fundamental rays, a stigmatic image is formed at the mid-plane zM and two mutually orthogonal astigmatic images of the object and aperture plane are each formed at the central planes zC1 and zC2. For determining the arrangement of the multipole elements and the course of the fundamental rays, it suffices to consider the Gaussian optics of the first quarter of the doubly symmetric system shown in Fig. 1. The conditions imposed on the course of the fundamental rays are fulfilled for each quarter system consisting of a quadrupole followed by the deflection element and two quadrupoles with opposite excitation. The hexapoles do not affect the paraxial rays. They introduce a usable negative chromatic aberration and second-order geometrical aberrations which are compensated for the system as a whole owing to the symmetry conditions imposed on the arrangement of the hexapoles and on the course of the fundamental rays with respect to the mid-plane M and the central planes C1 and C2, respectively. The quadrupoles QP111, QP121, QP221, and QP211are located near the boundaries of the deflection magnets Mµ , µ , = 1, 2 . Since the strength of these quadrupoles is rather weak they can be substituted by edge quadrupoles obtained by tilting the entrance face of each homogeneous dipole magnet with respect to the optic axis [7]. The focal length of these quadrupoles is fixed and proportional to (a) the dipole strength of the deflection magnet and (b) the tangent of the tilt angle enclosed by the normal of the exit face with the optic axis.
The axial rays xα and yβ are straight lines running parallel to the optic axis in front of the quadruplet. These rays satisfy the initial conditions
x (z 0 ) = y (z 0) = 1, x (z 0) = y (z 0) = 0.
Owing to the conditions (1) and (2) imposed on the fundamental rays, the Helmholtz–Lagrange relations of these rays adopt the simple form
x x
x x = 1, y y
y y = 1.
(3)
The focal length of the first quadrupole placed at the entrance plane z = z1 of the deflection magnet is f1 = f y1 = fx1 > 0 , that of the second quadrupole located at the exit plane z2 is f2 = f y2 = fx 2 > 0 . The distance of this plane from the entrance plane along the curved optic axis is l2 = z2 z1 = 2 R where the radius of curvature R is inversely proportional to the strength Ψ1s of the homogeneous dipole magnet. Within the frame of validity of the Sharp Cut-Off Fringing Field (SCOFF) approximation the y-component of the paraxial rays is not affected by the dipole field. Within this homogeneous field the x-component of an arbitrary paraxial ray is given by
x (z ) = x1 cos[(z
x (z ) =
z1)/ R] + x1 R sin [(z
x1 sin[(z R
z1)/ R] + x1 cos [(z
z1)/ R], z1 < z < z2 ,
z1)/R],
(4) (5)
with
x1 = x (z1) = x 0 + x 0 l1, x1 = x (z1) = x 0 + x1/f1 .
(6)
The slope of the ray changes abruptly by its transition through the entrance plane z1due to the refraction of the infinitely short quadrupole located at this plane. This quadrupole changes the x- and y-component of the slope as
3. Gaussian optics of the multipole quadruplet In order to survey the Gaussian optics of the corrector, it suffices to consider the first quarter of the system. The course of the paraxial rays within this subsystem is determined by the multipole quadruplet consisting of a single deflection element and three quadrupoles. For reasons of simplicity we assume short quadrupoles, the first of which is directly placed at the non-tilted entrance plane, the second at the exit plane of a homogeneous 90°-deflection magnet. Accordingly, the distances of the quadrupoles from the boundary planes are zero (d1 = 0, d2 = 0) . The field rays xγ and yδ start from the center of the plane z0 = zN1 which is the front nodal plane N1 of the corrector. This plane is located at distance d 0 = l1 = z1 z 0 in front of the entrance plane z1 of the deflection magnet M11. The field rays satisfy the initial conditions
x (z 0 ) = y (z 0) = 0, x (z 0) = y (z 0) = 1.
(2)
x1 = x1/f1 ,
y1 =
(7)
y1 / f1 ,
but not their lateral distance from the optic axis. At the exit plane z2 of the bending magnet the second quadrupole changes the slope components of the ray by
x 2 = x2 / f2 ,
y2 =
(8)
y2 /f2 .
Hence behind the second quadrupole the slope components of the ray are
x2 =
(1)
where 140
x1 x + 2 , y2 = y1 R f2
y2 , f2
(9)
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
y1 = y (z1) = y0 + y0 l1, y1 = y (z1) = y0
y1 /f1 .
must be placed in the plane z2 behind the bending magnet. Owing to the imposed symmetries on the paraxial rays, the field of the second quadrupole of the first subunit must be superimposed onto the field of the first quadrupole of the second subunit. We achieve this by substituting the two quadrupole doublets of the two neighboring subunits of each half of the corrector by a symmetric quadrupole triplet such that the excitation strength of the central qudrupole is twice as large and opposite to that of the outer quadrupoles. Because the sextupole fields of each half of the corrector also overlap at its mid-plane zM, we replace the two sextupoles at this plane by a single hexapole whose excitation is twice as large and opposite to that of the sextupoles centered about the plane z0 and its mirror plane, respectively. By taking into account these measures we find that the chromatic aberration corrector exhibits double symmetry with respect to the geometry and excitation strength of its constituent elements. This behavior does not hold true for the course of the fundamental paraxial rays which are either symmetric or anti-symmetric with respect to the mid-plane and/or the central plane of each half of the corrector. By taking into account these measures, the entire chromatic corrector consists of four dipole magnets six magnetic quadrupoles and seven sextupole elements. Each specific multipole element has the same structure. The result of the short-quadrupole approximation only serves as a guide for the principle design of the corrector. However, for the actual realistic design we must take into account the finite extension of the quadrupole and hexapole elements and the precise shape of the dipole and quadrupole fields in the paraxial domain. This can be accomplished by using the extended fringe-field (EFF) approximation [8]. Moreover, due to alignment difficulties resulting from hysteresis effects it may be necessary to generate the quadrupole and hexapole fields located at the same place behind the dipole magnet by two spatially separated elements. Although these measures will slightly change the geometry of the system, they will not significantly change the results of our thin-lens approximation. In particular, the symmetry required for the fields and for the fundamental rays in order to eliminate the second-order aberrations will be preserved. In order to verify the validity of our conjecture, we have investigated a realistic symmetric system depicted in Fig. 2. Each half of the doubly symmetric system consists of two deflection magnets with dipole strength Ψ1s(z) and of six extended quadrupoles with strength Ψ2s(z) including fringing fields. The course of the multipole strengths, of the dispersion ray xκ and that of the axial rays xα and yβ along the optic axis within this system satisfy all symmetry requirements and convincingly verifies our supposition. The dispersion ray xκ describes the displacement of the electron trajectory due to its relative energy deviation κ in paraxial approximation [9]. The symmetry conditions are also fulfilled for the field rays xγ(z) and yδ(z), as shown in Fig. 3. The corrector shown in Fig. 1 also enables the correction of the third-order axial aberration. For this purpose we introduce six dodecapole elements which allow the excitation of superposed quadrupole, hexapole, and octupole fields within each element. The six dodecapole elements are arranged as two triplets, one of which is centered at the central plane C1, the other at C2. The two octupole fields, each of which is excited within the central element of the triplets, compensate for the x-component of the axial third-order aberration whereas the octopole fields exited within the outer elements of the triplets compensate for the y-component. The remaining mixed x,y-component of the third-order axial aberration is eliminated by two octupoles, one is located close to the front plane z0, the other at its conjugate mirror plane behind the corrector. Owing to the symmetry of the fundamental rays, the proposed incorporation of the symmetrically arranged octupole fields does neither introduce third-order off-axial coma nor three-fold field astigmatism. However, the interaction of the dispersion with the octupole fields produces an axial chromatic aberration of first degree and second order which limits the resolution in the case large energy spreads. Fortunately, we can reduce this aberration by placing a slit aperture at the mid-plane of the magnetic Cc/Cs-corrector where spatially separated monochromatic images of the effective
(10)
Within the frame of the short-lens approximation the lateral distance of the paraxial ray is not changed at the plane z2. At this plane the components of the paraxial ray are
x2 = x (z2) = Rx1 =
2
l2 (x 0 + x1/f1 ),
y2 = y (z2) = y1 + y1 l2 = y1 + l2 (y0
y1 / f1 ).
(11)
The third quadrupole is centered at the plane z 3 = z C1 which coincides with the central plane C1of the first half of the corrector. This plane is located midway between the exit face of the first deflection magnet and the entrance plane of the second deflection magnet, as shown in Fig. 1. The polarity and the focusing (f3 = fy3 = fx3 < 0) of the third quadruple are opposite to that of the first and second quadrupole. Hence, the central quadrupole focuses the ray in the xz-section and defocuses it in the yz-section. As a result, the slope components are abruptly changed at the central plane by
x 3 = x3 /f3 ,
y3 =
(12)
y3 / f3 ,
x3 = x2 + x 2 l3, y3 = y2 + y2 l3, where l3 = z3
(13)
z2 .
4. Properties of the fundamental paraxial rays The focal lengths fν of the quadrupoles and the axial distances l , = 1, 2, 3 are related with each other by the conditions imposed on the fundamental rays at the central plane z C1 = z 3:
y 3 = y (z 3) = 0, x x
3
= x (z3) = 0,
(14)
= x (z 3) = 0, y 3 = y (z 3) = 0.
(15)
3
The fundamental paraxial rays also satisfy the initial conditions (1) and (2). By employing these conditions, the course of the rays in the region between the starting plane z0 and the central plane z3 is defined by the relations (4) to (15). We can eliminate the first quadrupole by taking f1 → ∞, so that the quadruplet collapses to a triplet. Therefore, it can be shown that
x
3
=
1 l 1+ 3 , R f3
(16)
l3 = 0, f2
(17)
y3 = 1
x
3
=R 1+
y3 = 1
l3 f2
(l1 + l2)
l1 l3 = 0, R 1 1 + f3 f2
(18)
l3 f2 f3
l3 = 0, f3
(19)
from which it is readily seen that
f2 = x
3
f3 = l3, l1 1.36R, l3 1.47R, l = 3 x1 1.47x1 = 1.47, y 3 = l3. R
(20)
The relations above show that each of the four subunits of the chromatic aberration corrector is a triplet composed of a homogeneous 90° bending magnet and an anti-symmetric magnetic quadrupole doublet. The entire system is composed of four of these triplets and has a total extension of 4(l1 + R) 9.4 R in axial direction. For arbitrarily adjusting both the x-component and the y-component of the axial chromatic aberration without introducing any second-order aberrations and an off-axial chromatic aberration, we must incorporate two hexapole elements into each subunit. One must be centered about the optic axis in the plane z0 where the field rays are zero, the other 141
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
Fig. 2. Course of the dipole strength Ψ1s(z), the quadrupole strengthΨ2s(z), the dispersion ray xκ(z), and of the axial fundamental rays xα(z) and yβ(z) along the straightened optic axis within the doubly symmetric corrector in the absence of the quadrupoles QP111, QP121, QP221 and QP211 shown in Fig. 1.
source are formed. In this case the system acts as a combined monochromator and Cc/Cs corrector.
As a result, a large energy spread will limit the optical resolution. The second-degree dispersion is a direct consequence of the correction principle based on hexapole fields. This unwanted effect is caused by a combination of the rather large first-degree dispersion with the multipole field. In order to improve the aberration corrector for applications with large energy spread, we can compensate for the second-degree dispersion by deviating from the strictly double-symmetric course of the hexapole fields. To eliminate the second-degree dispersion we place additional hexpole fields HPC1 and HPC2 with symmetric excitation at
5. Elimination of the second-degree dispersion The system which we have considered so far is free of second-order aberrations and off-axial chromatic aberration and provides an adjustable axial chromatic aberration. Unfortunately, the correction of the chromatic aberration introduces a second-degree dispersion which is of the same order of magnitude as the third-order geometrical aberration.
Fig. 3. Course of the field rays xγ(z)and yδ(z) which represent the nodal rays of the telescopic corrector. 142
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
Table. 1 The nine hexapole elements are excited according to five independent patterns Hi with i=1, …, 5. The symbols ‘+’ and ‘-’ indicate a positive or negative unit excitation, respectively. Each pattern of excitations is symmetric with respect to the mid-plane of the entire corrector zM. The positions of the hexapole elements is depicted in Fig. 1. HP111 H1
HP112
HPC1
HP122
HPM
HP222
HPC2
HP212
+
HP211
Effects
+
A2
H2
+
–
–
+
C1c, A1c, A0cc
H3
+
+
+
+
B2, A2, C1c, A1c, A0cc
H4
+
H5
C1c, A1c, A0cc
+
+
the planes zC1 and zC2, respectively. This measure breaks the antisymmetry of the hexapole fields with respect to the planes zC1 and zC2. Unfortunately, sacrificing the symmetry introduces axial second-order coma. To eliminate this unwanted effect we must excite all five hexpoles within the first half of the system differently to eliminate a set of five aberrations, namely chromatic aberration C1c, chromatic astigmatism A1c, three-fold astigmatism A2, axial coma B2, and second-degree dispersion A0cc. The symmetry with respect to the mid-plane zM of the entire system is maintained. Table. 1 shows the five excitation patterns for the hexapole fields which are used as independent free parameters to compensate for the set of five aberrations. This strategy enables the elimination of all second-rank axial aberrations together with the firstand second-degree dispersion.
small. The same holds true for the residual off-axial aberrations. The complete system is most efficiently realized with four 90°-sector magnets and nine dodecapole elements using standard technology, as known from conventional multipole aberration correctors [8]. 7. Conclusion The correction of chromatic aberration in entirely magnetic systems is possible but necessitates the incorporation of magnetic deflection elements. Because these elements curve the optic axis, magnetic systems corrected for chromatic aberration are always curved. Nevertheless, the asymptotic optic axis of the whole system can be straight if the asymptote of the optic axis in front of the system matches the exit asymptote. This is the case for the omega-type doubly-symmetric aberration corrector shown in Fig. 1 which can be considered as the magnetic equivalent of the electric and magnetic Cc/Cs correctors with straight optic axis incorporated in the TEAM [2], PICO [10], and SALVE [1] microscopes. While the application of these correctors is limited to voltages lower than about 300 kV the proposed magnetic corrector can handle voltages up to several MV. Due to its fully magnetic nature and its high degree of achromaticity the proposed correction system can be used for ultra-high voltage imaging and probe-forming instruments as recently described by Wan [3]. Our proposals for a feasible magnetic aberration corrector with a total length smaller than one meter is suited to compensate for the chromatic and the spherical aberration at 3 MeV provided that the associated coefficients are in the range of a few centimeters. In this case a resolution of about 0.1 nm can be achieved at energy spreads up to 100 eV. The corrector can be used for both probe-forming and imaging systems. In the latter case about 103 equally-well-resolved image points per diameter can be transferred simultaneously. Our calculations show that the excitations of the multipole elements are feasible with conventional technology as used in other multipole aberration correctors [8]. Also the stability requirements can be met. The most critical are the 90° deflectors. If driven by individual and independent current drivers a stability in the range of 10−8 with an upper frequency limit of about 100 Hz would be required. Fortunately, it is easily possible to drive all deflectors by the same current driver with all four coils connected in a row. In this case the stability requirement is relaxed by a factor of about 103. Summarizing we have shown that by employing symmetry requirements realistic aberration correctors can be achieved even in the case of a curved optic axis.
6. Elimination of the chromatic three-fold astigmatism A similar methods also works to reduce the third-rank aberrations. After we have eliminated the axial third-order aberrations C3, S3 and A3 the chromatic three-fold astigmatism A2c remains as the dominating aberration owing to the double symmetry of the octupole fields. We nullify the remaining A2c by using four independent octupole excitations in the first half of the system as free parameters. This measure breaks the symmetry with respect to zC1 and zC2, respectively, but maintains the symmetry with respect to the mid-plane zM, as in the case of the hexapoles. The octupole fields are superimposed to all hexapole fields. We employ the same octupole excitation OP112 = OP122 at the position of HP112 and HP122, respectively. Table. 2 shows the four excitation patterns for the octupole fields which are used as independent free parameters to compensate for the set of four aberrations. With the optimized choice of the hexapole and octupole excitations we end up with a very capable fully magnetic correction system for the chromatic and third-order axial spherical aberrations of the objective. The residual aberrations chromatic axial coma B2c and second-degree chromatic aberration C1cc can be tolerated since they are sufficiently Table. 2 The nine octupole elements are excited according to four independent patterns Oi with i = 1, …, 4. The symbols ‘ + ’ and ‘-’ indicate a positive or negative unit excitation, respectively. Each pattern of excitations is symmetric with respect to the mid-plane of the entire corrector zM. The positions of the octupole elements are the same as the positions of the hexapole elements with the same subscript depicted in Fig. 1. OP111 O1 O2 O3 O4
+
OP112
OPC1
+
OP122
OPM
+ +
OP222
OPC2
+ +
OP212 +
+
OP211
Effects
+
A3 C3, S3, A3 A3, A2c C3, S3, A3, A2c
B2, A2, C1c, A1c, A0cc
Acknowledgement The authors would like to thank S. Uhlemann and M. Haider for comments and stimulating discussions. 143
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
References
transmission electron microscope, Optik 85 (1990) 19–24. [6] M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius, K. Urban, Electron microscopy image enhanced, Nature 392 (6678) (1998) 768. [7] H. Rose, Aberration correction of homogeneous magnetic deflection systems, Optik 51 (1978) 15–38. [8] M. Haider, H. Müller, S. Uhlemann, Present and future hexapole aberration correctors for high-resolution electron microscopy, Adv. Imaging Electron Phys. 153 (2008) 43–120. [9] H. Rose, D. Krahl, Electron optics of imaging energy filters, Energy-filtering Transmission Electron Microscopy, Springer, Berlin, Heidelberg, 1995, pp. 43–149. [10] L. Jin, J. Barthel, C.L. Jia, K.W. Urban, Atomic resolution imaging of YAlO3: Ce in the chromatic and spherical aberration corrected PICO electron microscope, Ultramicroscopy 176 (2017) 99–104.
[1] M. Linck, P. Hartel, S. Uhlemann, F. Kahl, H. Müller, J. Zach, M. Haider, M. Niestadt, M. Bischoff, J. Biskupek, Z. Lee, Chromatic aberration correction for atomic resolution TEM imaging from 20 to 80kV, Phys. Rev. Lett. 117 (7) (2016) 076101. [2] M. Haider, P. Hartel, H. Müller, S. Uhlemann, J. Zach, Information transfer in a TEM corrected for spherical and chromatic aberration, Microscopy Microanal. 16 (4) (2010) 393–408. [3] W. Wan, F.R. Chen, Y. Zhu, Design of Compact Ultrafast Microscopes for Single-and Multi-Shot Imaging with MeV Electrons, Ultramicroscopy 194 (2018) 143–153. [4] H. Rose, Über die Korrigierbarkeit von Linsen für schnelle Elektronen, Optik 26 (1967) 289–298. [5] H. Rose, Outline of a spherically corrected semiaplanatic medium-voltage
144
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Magnetic Cc/Cs-corrector compensating for the chromatic aberration and the spherical aberration of electron lenses
T
H. Rosea, A. Nejatib, H. Müllerb,
⁎
a b
Ulm University, Meyerhofstr. 27, D-89081 Ulm, Germany CEOS GmbH, Englerstr. 28, D-69126 Heidelberg, Germany
ABSTRACT
Aberration correction in transmission electron microscopy has proven feasible and useful over a large range of acceleration voltages. The spherical aberration has been corrected for beam energies from 15 kV [1] up to 1.2 MeV [2] while the correction of the chromatic aberration has been achieved for beam energies ranging from 20 kV[3] up to 300 kV[4]. Above this threshold the conventional correction principle based on mixed electric and magnetic focusing elements becomes infeasible with present technology [5]. For conventional electron sources at high voltages the relative energy width of the beam gets so small that chromatic correction becomes less important. Nevertheless, for new applications with pulsed electron sources with energy spreads in the order of 100 eV chromatic aberration will become a limiting factor even at high energies [6]. To enable chromatic aberration correction for such systems a novel type of a feasible, purely magnetic multipole aberration corrector with curved optic axis is proposed which is capable of compensating for the chromatic and spherical aberration up to several MeV.
1. Introduction The chromatic aberration of magnetic systems with a straight optic axis is unavoidable [4]. In order to correct for this aberration in systems with a straight optic axis one must introduce mixed electric and magnetic quadrupole elements. Unfortunately, the electric field strength is limited to values smaller than about 7 kV/mm. Owing to this limitation the application of electric-magnetic correctors is limited to electron energies smaller than about 300 kV provided that the length of the corrector cannot surpass 1 m. Correction of chromatic aberration is possible in magnetic systems with a curved optic axis which is produced by the dipole component of the magnetic field. In addition, we must incorporate quadrupole elements for focusing and for forming the beam in an appropriate way and hexapole elements for adjusting the chromatic aberration of the corrector such that it compensates for the chromatic aberration of the entire system. The combination of the dispersion with the hexapole fields produces a first-order chromatic aberration whose sign can be adjusted by means of the polarity of the hexapoles. Apart from the chromatic aberration, the dipole and hexapole fields also introduce second-order geometrical aberrations which limit the resolution. Hence a feasible corrector must be designed in such a way that it does not introduce any second-order aberration. This requirement poses an extremely challenging task on the design of the corrector. Symmetry conditions imposed on the arrangement of the magnetic elements and on the course of the fundamental paraxial rays offer
⁎
the most promising method for eliminating the second-order aberrations, as it is the case for the hexapole corrector employed in atomicresolution electron microscopes for eliminating the third-order spherical aberration of the objective lens [5,6]. To satisfy all symmetry requirements a large number of elements is necessary making it difficult to align the system with the required accuracy. Moreover, the dipole and the quadrupole fields introduce third-order axial aberrations which must also be eliminated in order to achieve an appreciable increase in resolution. Because the third-order axial aberration of non-round elements consists of three components, the spherical aberration, the star aberration and the fourfold axial astigmatism, we need at least three independent octupole elements for compensating these aberrations. However, in order to judge the final optical performance also higherrank chromatic aberrations have to be taken into account. 2. Design of a corrector In order that all second-order geometrical aberrations cancel out in the region behind the corrector, we impose symmetry conditions [7] with respect to the mid-plane zM the corrector and with respect to the central planes zC1 and zC2 of each half of the system as depicted in zM Fig. 1. In particular, we propose that the magnetic field is symmetric with respect to the mid-plane zM. The quadrupole component of the magnetic field is also symmetric with respect to the central plane of each half of the system, whereas the dipole and the hexapole components are anti-symmetric. Apart from the symmetry conditions for the
Corresponding author: Englerstr. 28, D-69126 Heidelberg. E-mail addresses: [email protected] (H. Rose), [email protected] (A. Nejati), [email protected] (H. Müller).
https://doi.org/10.1016/j.ultramic.2018.11.014 Received 3 September 2018; Received in revised form 17 November 2018; Accepted 23 November 2018 Available online 29 November 2018 0304-3991/ © 2018 Elsevier B.V. All rights reserved.
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
Fig. 1. Scheme of the doubly symmetric magnetic chromatic-aberration corrector consisting of four 90°-deflection magnets Mμν, μ, ν = 1,2, two central quadrupoles QPC1 = QPC2, four quadrupoles QP111 = QP121 = QP221 = QP211, three hexapoles HPM and HP111 = HP211, four superposed quadupoles and hexapoles QPμ12 + HPμ12= QPμ22 + HPμ12, μ = 1,2, and two optional hexapoles HPC1, HPC2 superposed with the central quadruples. The mid-plane of the corrector is indicated by M, the central plane of each half of the corrector by C1 and C2, respectively. Images of the diffraction plane are located at the nodal planes N1 and N2. At every HP there is an additional octupole field for third-order correction (see §6).
magnetic field, we impose that the fundamental axial rays xα and yβ are symmetric and the field rays xγ and yδ are anti-symmetric with respect to zM. In addition the rays xα and yδ are symmetric and the rays yβ and xγ are anti-symmetric with respect to the central planes zC1 and zC2 of each half of the system. Owing to the required course of the fundamental rays, a stigmatic image is formed at the mid-plane zM and two mutually orthogonal astigmatic images of the object and aperture plane are each formed at the central planes zC1 and zC2. For determining the arrangement of the multipole elements and the course of the fundamental rays, it suffices to consider the Gaussian optics of the first quarter of the doubly symmetric system shown in Fig. 1. The conditions imposed on the course of the fundamental rays are fulfilled for each quarter system consisting of a quadrupole followed by the deflection element and two quadrupoles with opposite excitation. The hexapoles do not affect the paraxial rays. They introduce a usable negative chromatic aberration and second-order geometrical aberrations which are compensated for the system as a whole owing to the symmetry conditions imposed on the arrangement of the hexapoles and on the course of the fundamental rays with respect to the mid-plane M and the central planes C1 and C2, respectively. The quadrupoles QP111, QP121, QP221, and QP211are located near the boundaries of the deflection magnets Mµ , µ , = 1, 2 . Since the strength of these quadrupoles is rather weak they can be substituted by edge quadrupoles obtained by tilting the entrance face of each homogeneous dipole magnet with respect to the optic axis [7]. The focal length of these quadrupoles is fixed and proportional to (a) the dipole strength of the deflection magnet and (b) the tangent of the tilt angle enclosed by the normal of the exit face with the optic axis.
The axial rays xα and yβ are straight lines running parallel to the optic axis in front of the quadruplet. These rays satisfy the initial conditions
x (z 0 ) = y (z 0) = 1, x (z 0) = y (z 0) = 0.
Owing to the conditions (1) and (2) imposed on the fundamental rays, the Helmholtz–Lagrange relations of these rays adopt the simple form
x x
x x = 1, y y
y y = 1.
(3)
The focal length of the first quadrupole placed at the entrance plane z = z1 of the deflection magnet is f1 = f y1 = fx1 > 0 , that of the second quadrupole located at the exit plane z2 is f2 = f y2 = fx 2 > 0 . The distance of this plane from the entrance plane along the curved optic axis is l2 = z2 z1 = 2 R where the radius of curvature R is inversely proportional to the strength Ψ1s of the homogeneous dipole magnet. Within the frame of validity of the Sharp Cut-Off Fringing Field (SCOFF) approximation the y-component of the paraxial rays is not affected by the dipole field. Within this homogeneous field the x-component of an arbitrary paraxial ray is given by
x (z ) = x1 cos[(z
x (z ) =
z1)/ R] + x1 R sin [(z
x1 sin[(z R
z1)/ R] + x1 cos [(z
z1)/ R], z1 < z < z2 ,
z1)/R],
(4) (5)
with
x1 = x (z1) = x 0 + x 0 l1, x1 = x (z1) = x 0 + x1/f1 .
(6)
The slope of the ray changes abruptly by its transition through the entrance plane z1due to the refraction of the infinitely short quadrupole located at this plane. This quadrupole changes the x- and y-component of the slope as
3. Gaussian optics of the multipole quadruplet In order to survey the Gaussian optics of the corrector, it suffices to consider the first quarter of the system. The course of the paraxial rays within this subsystem is determined by the multipole quadruplet consisting of a single deflection element and three quadrupoles. For reasons of simplicity we assume short quadrupoles, the first of which is directly placed at the non-tilted entrance plane, the second at the exit plane of a homogeneous 90°-deflection magnet. Accordingly, the distances of the quadrupoles from the boundary planes are zero (d1 = 0, d2 = 0) . The field rays xγ and yδ start from the center of the plane z0 = zN1 which is the front nodal plane N1 of the corrector. This plane is located at distance d 0 = l1 = z1 z 0 in front of the entrance plane z1 of the deflection magnet M11. The field rays satisfy the initial conditions
x (z 0 ) = y (z 0) = 0, x (z 0) = y (z 0) = 1.
(2)
x1 = x1/f1 ,
y1 =
(7)
y1 / f1 ,
but not their lateral distance from the optic axis. At the exit plane z2 of the bending magnet the second quadrupole changes the slope components of the ray by
x 2 = x2 / f2 ,
y2 =
(8)
y2 /f2 .
Hence behind the second quadrupole the slope components of the ray are
x2 =
(1)
where 140
x1 x + 2 , y2 = y1 R f2
y2 , f2
(9)
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
y1 = y (z1) = y0 + y0 l1, y1 = y (z1) = y0
y1 /f1 .
must be placed in the plane z2 behind the bending magnet. Owing to the imposed symmetries on the paraxial rays, the field of the second quadrupole of the first subunit must be superimposed onto the field of the first quadrupole of the second subunit. We achieve this by substituting the two quadrupole doublets of the two neighboring subunits of each half of the corrector by a symmetric quadrupole triplet such that the excitation strength of the central qudrupole is twice as large and opposite to that of the outer quadrupoles. Because the sextupole fields of each half of the corrector also overlap at its mid-plane zM, we replace the two sextupoles at this plane by a single hexapole whose excitation is twice as large and opposite to that of the sextupoles centered about the plane z0 and its mirror plane, respectively. By taking into account these measures we find that the chromatic aberration corrector exhibits double symmetry with respect to the geometry and excitation strength of its constituent elements. This behavior does not hold true for the course of the fundamental paraxial rays which are either symmetric or anti-symmetric with respect to the mid-plane and/or the central plane of each half of the corrector. By taking into account these measures, the entire chromatic corrector consists of four dipole magnets six magnetic quadrupoles and seven sextupole elements. Each specific multipole element has the same structure. The result of the short-quadrupole approximation only serves as a guide for the principle design of the corrector. However, for the actual realistic design we must take into account the finite extension of the quadrupole and hexapole elements and the precise shape of the dipole and quadrupole fields in the paraxial domain. This can be accomplished by using the extended fringe-field (EFF) approximation [8]. Moreover, due to alignment difficulties resulting from hysteresis effects it may be necessary to generate the quadrupole and hexapole fields located at the same place behind the dipole magnet by two spatially separated elements. Although these measures will slightly change the geometry of the system, they will not significantly change the results of our thin-lens approximation. In particular, the symmetry required for the fields and for the fundamental rays in order to eliminate the second-order aberrations will be preserved. In order to verify the validity of our conjecture, we have investigated a realistic symmetric system depicted in Fig. 2. Each half of the doubly symmetric system consists of two deflection magnets with dipole strength Ψ1s(z) and of six extended quadrupoles with strength Ψ2s(z) including fringing fields. The course of the multipole strengths, of the dispersion ray xκ and that of the axial rays xα and yβ along the optic axis within this system satisfy all symmetry requirements and convincingly verifies our supposition. The dispersion ray xκ describes the displacement of the electron trajectory due to its relative energy deviation κ in paraxial approximation [9]. The symmetry conditions are also fulfilled for the field rays xγ(z) and yδ(z), as shown in Fig. 3. The corrector shown in Fig. 1 also enables the correction of the third-order axial aberration. For this purpose we introduce six dodecapole elements which allow the excitation of superposed quadrupole, hexapole, and octupole fields within each element. The six dodecapole elements are arranged as two triplets, one of which is centered at the central plane C1, the other at C2. The two octupole fields, each of which is excited within the central element of the triplets, compensate for the x-component of the axial third-order aberration whereas the octopole fields exited within the outer elements of the triplets compensate for the y-component. The remaining mixed x,y-component of the third-order axial aberration is eliminated by two octupoles, one is located close to the front plane z0, the other at its conjugate mirror plane behind the corrector. Owing to the symmetry of the fundamental rays, the proposed incorporation of the symmetrically arranged octupole fields does neither introduce third-order off-axial coma nor three-fold field astigmatism. However, the interaction of the dispersion with the octupole fields produces an axial chromatic aberration of first degree and second order which limits the resolution in the case large energy spreads. Fortunately, we can reduce this aberration by placing a slit aperture at the mid-plane of the magnetic Cc/Cs-corrector where spatially separated monochromatic images of the effective
(10)
Within the frame of the short-lens approximation the lateral distance of the paraxial ray is not changed at the plane z2. At this plane the components of the paraxial ray are
x2 = x (z2) = Rx1 =
2
l2 (x 0 + x1/f1 ),
y2 = y (z2) = y1 + y1 l2 = y1 + l2 (y0
y1 / f1 ).
(11)
The third quadrupole is centered at the plane z 3 = z C1 which coincides with the central plane C1of the first half of the corrector. This plane is located midway between the exit face of the first deflection magnet and the entrance plane of the second deflection magnet, as shown in Fig. 1. The polarity and the focusing (f3 = fy3 = fx3 < 0) of the third quadruple are opposite to that of the first and second quadrupole. Hence, the central quadrupole focuses the ray in the xz-section and defocuses it in the yz-section. As a result, the slope components are abruptly changed at the central plane by
x 3 = x3 /f3 ,
y3 =
(12)
y3 / f3 ,
x3 = x2 + x 2 l3, y3 = y2 + y2 l3, where l3 = z3
(13)
z2 .
4. Properties of the fundamental paraxial rays The focal lengths fν of the quadrupoles and the axial distances l , = 1, 2, 3 are related with each other by the conditions imposed on the fundamental rays at the central plane z C1 = z 3:
y 3 = y (z 3) = 0, x x
3
= x (z3) = 0,
(14)
= x (z 3) = 0, y 3 = y (z 3) = 0.
(15)
3
The fundamental paraxial rays also satisfy the initial conditions (1) and (2). By employing these conditions, the course of the rays in the region between the starting plane z0 and the central plane z3 is defined by the relations (4) to (15). We can eliminate the first quadrupole by taking f1 → ∞, so that the quadruplet collapses to a triplet. Therefore, it can be shown that
x
3
=
1 l 1+ 3 , R f3
(16)
l3 = 0, f2
(17)
y3 = 1
x
3
=R 1+
y3 = 1
l3 f2
(l1 + l2)
l1 l3 = 0, R 1 1 + f3 f2
(18)
l3 f2 f3
l3 = 0, f3
(19)
from which it is readily seen that
f2 = x
3
f3 = l3, l1 1.36R, l3 1.47R, l = 3 x1 1.47x1 = 1.47, y 3 = l3. R
(20)
The relations above show that each of the four subunits of the chromatic aberration corrector is a triplet composed of a homogeneous 90° bending magnet and an anti-symmetric magnetic quadrupole doublet. The entire system is composed of four of these triplets and has a total extension of 4(l1 + R) 9.4 R in axial direction. For arbitrarily adjusting both the x-component and the y-component of the axial chromatic aberration without introducing any second-order aberrations and an off-axial chromatic aberration, we must incorporate two hexapole elements into each subunit. One must be centered about the optic axis in the plane z0 where the field rays are zero, the other 141
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
Fig. 2. Course of the dipole strength Ψ1s(z), the quadrupole strengthΨ2s(z), the dispersion ray xκ(z), and of the axial fundamental rays xα(z) and yβ(z) along the straightened optic axis within the doubly symmetric corrector in the absence of the quadrupoles QP111, QP121, QP221 and QP211 shown in Fig. 1.
source are formed. In this case the system acts as a combined monochromator and Cc/Cs corrector.
As a result, a large energy spread will limit the optical resolution. The second-degree dispersion is a direct consequence of the correction principle based on hexapole fields. This unwanted effect is caused by a combination of the rather large first-degree dispersion with the multipole field. In order to improve the aberration corrector for applications with large energy spread, we can compensate for the second-degree dispersion by deviating from the strictly double-symmetric course of the hexapole fields. To eliminate the second-degree dispersion we place additional hexpole fields HPC1 and HPC2 with symmetric excitation at
5. Elimination of the second-degree dispersion The system which we have considered so far is free of second-order aberrations and off-axial chromatic aberration and provides an adjustable axial chromatic aberration. Unfortunately, the correction of the chromatic aberration introduces a second-degree dispersion which is of the same order of magnitude as the third-order geometrical aberration.
Fig. 3. Course of the field rays xγ(z)and yδ(z) which represent the nodal rays of the telescopic corrector. 142
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
Table. 1 The nine hexapole elements are excited according to five independent patterns Hi with i=1, …, 5. The symbols ‘+’ and ‘-’ indicate a positive or negative unit excitation, respectively. Each pattern of excitations is symmetric with respect to the mid-plane of the entire corrector zM. The positions of the hexapole elements is depicted in Fig. 1. HP111 H1
HP112
HPC1
HP122
HPM
HP222
HPC2
HP212
+
HP211
Effects
+
A2
H2
+
–
–
+
C1c, A1c, A0cc
H3
+
+
+
+
B2, A2, C1c, A1c, A0cc
H4
+
H5
C1c, A1c, A0cc
+
+
the planes zC1 and zC2, respectively. This measure breaks the antisymmetry of the hexapole fields with respect to the planes zC1 and zC2. Unfortunately, sacrificing the symmetry introduces axial second-order coma. To eliminate this unwanted effect we must excite all five hexpoles within the first half of the system differently to eliminate a set of five aberrations, namely chromatic aberration C1c, chromatic astigmatism A1c, three-fold astigmatism A2, axial coma B2, and second-degree dispersion A0cc. The symmetry with respect to the mid-plane zM of the entire system is maintained. Table. 1 shows the five excitation patterns for the hexapole fields which are used as independent free parameters to compensate for the set of five aberrations. This strategy enables the elimination of all second-rank axial aberrations together with the firstand second-degree dispersion.
small. The same holds true for the residual off-axial aberrations. The complete system is most efficiently realized with four 90°-sector magnets and nine dodecapole elements using standard technology, as known from conventional multipole aberration correctors [8]. 7. Conclusion The correction of chromatic aberration in entirely magnetic systems is possible but necessitates the incorporation of magnetic deflection elements. Because these elements curve the optic axis, magnetic systems corrected for chromatic aberration are always curved. Nevertheless, the asymptotic optic axis of the whole system can be straight if the asymptote of the optic axis in front of the system matches the exit asymptote. This is the case for the omega-type doubly-symmetric aberration corrector shown in Fig. 1 which can be considered as the magnetic equivalent of the electric and magnetic Cc/Cs correctors with straight optic axis incorporated in the TEAM [2], PICO [10], and SALVE [1] microscopes. While the application of these correctors is limited to voltages lower than about 300 kV the proposed magnetic corrector can handle voltages up to several MV. Due to its fully magnetic nature and its high degree of achromaticity the proposed correction system can be used for ultra-high voltage imaging and probe-forming instruments as recently described by Wan [3]. Our proposals for a feasible magnetic aberration corrector with a total length smaller than one meter is suited to compensate for the chromatic and the spherical aberration at 3 MeV provided that the associated coefficients are in the range of a few centimeters. In this case a resolution of about 0.1 nm can be achieved at energy spreads up to 100 eV. The corrector can be used for both probe-forming and imaging systems. In the latter case about 103 equally-well-resolved image points per diameter can be transferred simultaneously. Our calculations show that the excitations of the multipole elements are feasible with conventional technology as used in other multipole aberration correctors [8]. Also the stability requirements can be met. The most critical are the 90° deflectors. If driven by individual and independent current drivers a stability in the range of 10−8 with an upper frequency limit of about 100 Hz would be required. Fortunately, it is easily possible to drive all deflectors by the same current driver with all four coils connected in a row. In this case the stability requirement is relaxed by a factor of about 103. Summarizing we have shown that by employing symmetry requirements realistic aberration correctors can be achieved even in the case of a curved optic axis.
6. Elimination of the chromatic three-fold astigmatism A similar methods also works to reduce the third-rank aberrations. After we have eliminated the axial third-order aberrations C3, S3 and A3 the chromatic three-fold astigmatism A2c remains as the dominating aberration owing to the double symmetry of the octupole fields. We nullify the remaining A2c by using four independent octupole excitations in the first half of the system as free parameters. This measure breaks the symmetry with respect to zC1 and zC2, respectively, but maintains the symmetry with respect to the mid-plane zM, as in the case of the hexapoles. The octupole fields are superimposed to all hexapole fields. We employ the same octupole excitation OP112 = OP122 at the position of HP112 and HP122, respectively. Table. 2 shows the four excitation patterns for the octupole fields which are used as independent free parameters to compensate for the set of four aberrations. With the optimized choice of the hexapole and octupole excitations we end up with a very capable fully magnetic correction system for the chromatic and third-order axial spherical aberrations of the objective. The residual aberrations chromatic axial coma B2c and second-degree chromatic aberration C1cc can be tolerated since they are sufficiently Table. 2 The nine octupole elements are excited according to four independent patterns Oi with i = 1, …, 4. The symbols ‘ + ’ and ‘-’ indicate a positive or negative unit excitation, respectively. Each pattern of excitations is symmetric with respect to the mid-plane of the entire corrector zM. The positions of the octupole elements are the same as the positions of the hexapole elements with the same subscript depicted in Fig. 1. OP111 O1 O2 O3 O4
+
OP112
OPC1
+
OP122
OPM
+ +
OP222
OPC2
+ +
OP212 +
+
OP211
Effects
+
A3 C3, S3, A3 A3, A2c C3, S3, A3, A2c
B2, A2, C1c, A1c, A0cc
Acknowledgement The authors would like to thank S. Uhlemann and M. Haider for comments and stimulating discussions. 143
Ultramicroscopy 203 (2019) 139–144
H. Rose et al.
References
transmission electron microscope, Optik 85 (1990) 19–24. [6] M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius, K. Urban, Electron microscopy image enhanced, Nature 392 (6678) (1998) 768. [7] H. Rose, Aberration correction of homogeneous magnetic deflection systems, Optik 51 (1978) 15–38. [8] M. Haider, H. Müller, S. Uhlemann, Present and future hexapole aberration correctors for high-resolution electron microscopy, Adv. Imaging Electron Phys. 153 (2008) 43–120. [9] H. Rose, D. Krahl, Electron optics of imaging energy filters, Energy-filtering Transmission Electron Microscopy, Springer, Berlin, Heidelberg, 1995, pp. 43–149. [10] L. Jin, J. Barthel, C.L. Jia, K.W. Urban, Atomic resolution imaging of YAlO3: Ce in the chromatic and spherical aberration corrected PICO electron microscope, Ultramicroscopy 176 (2017) 99–104.
[1] M. Linck, P. Hartel, S. Uhlemann, F. Kahl, H. Müller, J. Zach, M. Haider, M. Niestadt, M. Bischoff, J. Biskupek, Z. Lee, Chromatic aberration correction for atomic resolution TEM imaging from 20 to 80kV, Phys. Rev. Lett. 117 (7) (2016) 076101. [2] M. Haider, P. Hartel, H. Müller, S. Uhlemann, J. Zach, Information transfer in a TEM corrected for spherical and chromatic aberration, Microscopy Microanal. 16 (4) (2010) 393–408. [3] W. Wan, F.R. Chen, Y. Zhu, Design of Compact Ultrafast Microscopes for Single-and Multi-Shot Imaging with MeV Electrons, Ultramicroscopy 194 (2018) 143–153. [4] H. Rose, Über die Korrigierbarkeit von Linsen für schnelle Elektronen, Optik 26 (1967) 289–298. [5] H. Rose, Outline of a spherically corrected semiaplanatic medium-voltage
144