The Limits of Resolution and Dose for Aberration-Corrected Electron Tomography
TThe Limits of Resolution and Dose for Aberration-Corrected Electron Tomography
Reed Yalisove, ∗ Suk Hyun Sung, ∗ Peter Ercius, and Robert Hovden
1, 3 Department of Materials Science and Engineering,University of Michigan, Ann Arbor, Michigan 48109, USA The Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Applied Physics Program, University of Michigan, Ann Arbor, Michigan 48109, USA † (Dated: June 12, 2020)Aberration-corrected electron microscopy can resolve the smallest atomic bond-lengths in nature.However, the high-convergence angles that enable spectacular resolution in 2D have unknown3D resolution limits for all but the smallest objects ( < ∼ >
20 nm) using available microscopes and modest specimen tilting ( < ° ). Furthermore,aberration-corrected tomography follows the rule of dose-fractionation where a specified total dosecan be divided among tilts and defoci.
125 character summary: Unprecedented 3D resolution of large specimens established by novelanalytic limits for aberration-corrected electron tomography.
I. INTRODUCTION
Among the greatest goals in experimental science is todirectly measure the complete 3D arrangement of atomsmaterials. However, fundamental sampling limits andan inextricable connection between lateral resolution anddepth-of-focus have strictly prohibited 3D atomic mea-surement of extended materials. The theoretical limitsof electron tomography have long been defined by (1)the Crowther-Klug criterion, which relates 3D resolutionto the number of projections acquired [1], and (2) thedose fractionation theorem [2], which states that the sig-nificance of a reconstructed object is independent of thedistribution of dose. These limits were developed on theassumption that each image in a tomographic tilt se-ries gives a perfect projection of the specimen. Over thelast half century, this assumption has sufficed for micro-scopes where the depth-of-focus is large relative to thespecimen[3–6].However, the defined limits of tomography are in-valid for the new era of aberration-corrected scanningtransmission electron microscopy (STEM) where highly-convergent electron beams confined to sub-˚Angstrom lat-eral dimensions [7] provide routine atomic imaging in2D [8–10]. These revolutionary microscopes no longerprovide simple projections of a specimen [11–13] and to-mography fails for objects larger than the depth-of-focus( < ∼ ∗ These two authors contributed equally † [email protected] coming the limitations of aberration-corrected STEMtomography requires collecting a through-focal imagestack at every specimen tilt [14]. Although experimen-tally demonstrated, the theoretical limits of aberration-corrected tomography remain undefined. Understandingthe tradeoff between resolution, object size, sampling,and dose using highly-convergent beams demands a newtheoretical definition.Here we present a theoretical foundation foraberration-corrected electron tomography that estab-lishes analytic descriptions for resolution, sampling, andobject size. We show that aberration-corrected tomog-raphy can far exceed the resolution limits of traditionaltomography and breaks the conventional Crowther-Klugcriteria.The 3D contrast transfer function (CTF) foraberration-corrected STEM tomography distinctly mea-sures a volume of information made from a superpositionof toroids with petal-shaped cross-sections (Fig. 1) at ev-ery specimen tilt. A remarkable feature of the 3D CTFis the overlapped regions that permit complete 3D in-formation collection up to a specified resolution (1 /k c )—unachievable with conventional tomography. This occurswhen the incremental tilt angle (∆ θ ) becomes smallerthan twice the beam convergance semi-angle (2 α ) whichis typical to instrument operation ( α > θ < ° ). This complete information trans-fer breaks the Crowther-Klug relationships and the max-imum reconstructable object size is unlimited up to acritical resolution (1 /k c ). Beyond this critical resolution,Crowther-like tradeoffs define the maximum object size( D ) allowed at a given 3D resolution ( d ). With more a r X i v : . [ c ond - m a t . m t r l - s c i ] J un FIG. 1.
3D contrast transfer function tomography using aberration-corrected scanning transmission electronmicroscopy. a) Through-focal CTF for typical aberration-corrected STEM ( α = 30mrad, 200keV). b) Internal structure of thetomographic CTF at k x = 0 is highlighted. Each petal-shaped lobe represents a single through-focal CTF. Complete transferof information is guaranteed within a spherical radius k c . c) 3D CTF of through-focal tomography (tilt axis, ˆ x ). Blue shelldenotes the information transfer limit in 3D. α and ∆ θ are exaggerated to 30 ° . The plane slices through the CTF at k x = 0. specimen tilts and higher convergence angles, 3D resolu-tion improves quickly and the maximum object size in-creases dramatically. Using a tilt increment that matchesthe convergence angle (∆ θ = α ) any object size can beresolved with 3D resolution at ∼
50% of the microscope’sdiffraction limited resolution.Despite the large amount of image data required byaberration-corrected electron tomography, the dose canbe chosen to mitigate total specimen exposure. Ex-tending the dose fractionation arguments presented byHoppe[2] and Saxton[15], we show aberration-correctedelectron tomography allows tunable dose allocationacross any number of tilts and focal planes when over-sampled. Thus, the signal-to-noise ratio of a 3D recon-struction is only dependent on the total dose imparted.The relationships defined herein for aberration-corrected tomography supplant the Crowther-Klugcriterion[1, 16] and the dose-fractionation theorem[2, 15]that have long defined traditional tomographic tech-niques.
II. BACKGROUND
In 1970, Crowther et al. established the fundamentaltradeoff between 3D resolution, specimen size, and thenumber of projections measured[16]. Bracewell and Rid-dle showed the same relationship for radio astronomy in1967[17]. With evenly spaced specimen tilts about a sin- gle axis of rotation, the expression is compactly stated: d = πD/N , where d is the smallest resolved feature sizein three-dimensions, D is object size, and N is the num-ber of projections acquired with equal angular spacing(∆ θ = π/N ). This sampling criterion is the most strin-gent requirement that can be adopted and ensures speci-men features are measured and the entire reconstructionis equally sharp and free of aliasing.Conceptually, 3D resolution is limited by tomography’sinability to collect complete information about the spec-imen. Projection images at each tilt map to a plane ofinformation in frequency space (k-space)—as defined bythe projection slice theorem[18]. The missing informationbetween planes limits the 3D resolution and object sizeof a tomographic reconstruction. For specimens tiltedabout a single axis of rotation, the planes of informationintersect along one axis ( k x ) on a cylindrical coordinatesystem as illustrated in Supplemental Figure SI3. Sam-pling is maximal radially and along the axis of rotation,however undersampling occurs azimuthally along k θ andworsens at higher frequencies ( k r ). The azimuthal gapbetween adjacent measurement planes (∆ k θ ≈ k r · ∆ θ )limits the largest resolvable object: ∆ k θ = 1 /D . Thus,collecting more specimen tilts reduces the distance be-tween measurement planes and allows higher resolution(larger k r ) or larger object sizes in 3D. This theorem iswell suited for traditional S/TEM tomography where thedepth-of-focus is larger than the object size (Supplemen-tal Figure SI2).Six years after Crowther et al. defined the resolutionand sampling limits of tomography, Hegerl and Hoppe es-tablished a dose fractionation property for tomography,which Saxberg and Saxton further refined after debate[2, 15]. Their work showed when an object is sufficientlysampled (i.e. better than Crowther-Klug requirements)the signal-to-noise ratio (SNR) of a reconstruction de-pends only on the total dose imparted. Maintaining anequivalent total dose, one may divide that dose acrossmore or fewer projections. Both derivations are based onweak-contrast imaging—however we will show a weak-contrast approximation is not required. III. CONTRAST TRANSFER FUNCTION OFABERRATION-CORRECTED TOMOGRAPHY
Due to the highly convergent nature of aberration-corrected electron beams, Crowther’s derivation of sam-pling requirements do not hold for aberration-correctedtomography. The CTF is no longer a 2D plane; its formbecomes a 3D toroid (Fig. 1a) derived analytically by In-taraprasonk, Xin, Muller [19] and reproduced in Supple-mental Information SI1. Thus for aberration-correctedtomography, the projection planes in k-space are replacedby toroidal CTFs. Figure 1b,c show the total tomo-graphic CTF in 3D. Summing a rotated set of toroidalCTFs describes the region of k-space from which infor-mation is collected in aberration-corrected tomography: H ( k r , k z ) = (cid:88) θ h ( k θr , k θz ) (1)Here h ( k θr , k θz ) is the radially symmetric toroidal CTFfrom a single specimen tilt, θ , measured by through-focalimaging. For an aberration-free beam this through-focalCTF is: h ( k θr , k θz ) = 12 π αk θr (cid:115) − (cid:18) k θr λ α + | k θz | αk θr (cid:19) (2) | k θz | ≤ λ k θr (cid:18) αλ − k θr (cid:19) (3)where α is the convergence semi-angle of the electronbeam, k r is the radial frequency, and λ is the wave-length of the electron. By parameterizing the equationfor the through-focal CTF (Eq. 2), we express a CTFthat has been tilted about an axis ( k x ) perpendicularto the optical axis ( k z ) in k-space. This rotation gives k θr = k r cos( θ ) − k z sin( θ ) and k θz = k r sin( θ ) + k z cos( θ )for θ ∈ ( − π/ , π/ ° tilt intervalabout a single axis of rotation. IV. 3D RESOLUTION AND OBJECT SIZELIMITS
For aberration-corrected tomography, the sampling re-quirements that limit object size and resolution are deter-mined by the missing information between the toroidalbounds of the through-focal CTF (Eq. 3). For infinitelylarge objects, measuring specimen structure at a singlefrequency ( k (cid:48) ) requires the information to lie within thetomography CTF. However, for objects of finite size, D ,the condition loosens and CTF bounds only need to mea-sure the neighborhood, k (cid:48) + δ where | δ | = 1 /D (Ap-pendix Fig. A1). Thus, the size of missing informationin k-space limits the detectable object size. Similar to theanalysis by Crowther and Klug, we aim to calculate themaximum object size for a tomographic reconstructionby calculating the k-space distance, ∆ k θ , between theinformation collected at sequential specimen tilts. Thenon-planar geometry of aberration-corrected tomographyreduces distances of missing information in k-space thatpermit measurement of larger real space objects at higherresolutions than the Crowther-Klug limit.The strictest resolution requirement ensures measure-ment of objects of any shape or symmetry without anyprior information. Although the tomographic CTF isnon-isotropic and resolution is higher along the axis of ro-tation, we define 3D resolution by the worst measurableresolution. For single tilt-axis tomography, undersam-pling occurs along k θ and defines the largest k-space dis-tance between adjacent through-focal CTFs. The miss-ing information (∆ k θ ) increases at higher frequencies, k r ,and thus limits resolution ( d = 1 /k r ). We show (Ap-pendix A) the distance between measured information ismaximal on the k y k z -plane (Fig. 1) and is used to calcu-late the limits for aberration-corrected tomography.The most striking feature of the aberration-correctedtomography CTF is the continuum of information it canpermit. From Figure 1b we see that with small tilt incre-ments and large convergence angles, adjacent through-focal CTFs will overlap, allowing complete informationtransfer up to a critical frequency, k c : k c = 2 α − ∆ θλ (4)where ∆ θ is the angular spacing between specimentilts. Equation 4 defines this critical frequency undera first-order small-angle approximation as derived in Ap-pendix A and is valid when k c is positive (i.e. ∆ θ < α ).This critical frequency splits the problem into tworegimes, so the resolution limit on object size is definedpiece-wise. For k r ≤ k c , the structure of the specimenis completely measured—this corresponds to unboundedmaximum object sizes. For k r > k c , there is a finite dis-tance between adjacent regions of information (AppendixEq. A2), which relates the maximum frequency, k y , in areconstruction to the maximum object size, D (shown inFigure 2c). The piece-wise expression is: FIG. 2.
Relationships between tilt angle, resolution, and maximum object size
The CTF of aberration-correctedtomography hosts overlap regions which permit complete information transfer—and therefore unlimited object size with reso-lution d = 1 /k c . a) Full tomographic CTF in cross-section for 200 keV, 30 mrad, 2 ° tilts. b) Subregion of full tomographic CTFhighlights the maximum frequency of complete information transfer, k c , and ∆ k θ denotes separation between through-focalCTFs at each tilt. c) Spatial frequency vs. maximum object size for several tilt step-sizes with teal curve matching conditionsin a). For k y < k c , the maximum object size is unbounded. D = (cid:40) d λ (1 − k c d ) , λ < d < k c ∞ , k c ≤ d < ∞ (5)Equation 5 defines a new limit relating resolution, ob-ject size, and sampling for aberration-corrected electrontomography, analogous to the Crowther-Klug limit forconventional tomography. It shows higher beam energies(i.e. smaller wavelengths) and higher convergence anglesallow higher resolution and larger object sizes.Remarkably, when ∆ θ < α , there is always a resolu-tion at which an infinite object size can be reconstructed(neglecting multiple scattering). This behavior is not pre-dicted by the Crowther criterion and exceeds the previ-ously expected limits. As illustrated in Figure 3, witheven smaller tilt increments the reconstructable objectsize diverges at high resolution. Notably, when the tiltincrement matches the convergence angle (∆ θ = α ) anyobject size can be resolved in 3D at ∼
50% of the micro-scope’s diffraction limit. For ∆ θ = α/ k c ,aberration-corrected tomography still outperforms tradi-tional tomography due to the reduced missing informa-tion between lobes in the tomography CTF. Figure 3b,shows the trade-off between object size and resolution isfavorably non-linear. It provides the object size that can be reconstructed at a given resolution for different spec-imen tilt increments. For example, a 75 nm specimen,imaged with a 30 mrad convergence angle and 3 ° (50mrad) increments between tilts allows 2 ˚A resolution in3D at 200 keV.Moreover, atomic resolution imaging, with 1.5 ˚A reso-lution in 3D, is possible over a 15 nm object if sampledat 3 ° using a 200 keV beam and 30 mrad convergencesemi-angle. 3D atomic resolution imaging of extendedobjects has been computationally verified (Fig. 3a) us-ing quantum mechanical multiple scattering simulationsof aberration-corrected tomography performed on crys-talline nanoparticles within a 20 nm volume (See Sup-plemental Information SI2). This simulation computedover 500 million elastically scattered electron wavefunc-tions to generate images at 13 defocus positions at eachof 105 tilts, using over 15,000 GPU core hours.Despite some missing information at higher frequen-cies ( k r > k c ), a significant portion of k-space is mea-sured and aberration-corrected tomography provides asuperior reconstruction compared to traditional tomog-raphy. However, due to the finite periodic samplingof specimen tilts, aberration-corrected tomography maystill permit weak aliasing. Aliasing can occur azimuthallyat high radial frequencies, k r > k c , when reconstructedobject sizes exceed the Crowther-Klug relation, even ifthe requirements for resolution (Eq. 5) are met. For-tunately, if present, aliasing is substantially attenuatedby the amount of information collected with aberration- FIG. 3.
Aberration-corrected electron tomography en-ables unprecedented high-resolution of extended ob-jects. a) 3D atomic resolution tomography of three nanopar-ticles in a 20nm volume—reconstructed here from quantummechanical scattering simulations. b) Tradeoff between 3Dresolution and object size is plotted for different specimentilt increments, ∆ θ . The Crowther limit is surpassed when∆ θ ≤ α (grey) such that objects of any size may be recon-structed. Star denotes the size and resolution of reconstruc-tion in (a). c) The percent of information collected at eachresolution. corrected tomography. The intensity of azimuthal alias-ing is proportional to the percentage of information col-lected at each radial frequency k r (plotted as a functionof resolution in Figure 3c). With more measured infor-mation, the reconstruction quality improves and aliasingbecomes negligible. Thus, no aliasing occurs at low fre-quencies ( k r < k c ) and at the microscope’s transfer limit( k r = k max ) the aliasing is significant and matches tra-ditional tomography. V. DEFOCUS SAMPLING REQUIREMENT
At each specimen tilt, aberration-corrected electron to-mography acquires a through-focal stack of images. Thisovercomes the limited depth-of-focus ( < ∼ z , becomes an an additional sam-pling requirement.The defocus step must be smaller than the micro-scope’s depth-of-focus. This sampling requirement is de-scribed by the widest portion of a through-focal CTF( k max z ) along the beam direction ( k z ). The largest defo-cus step size is: ∆ z max = 12 k max z = λα (6)This equation is the well-known depth-of-focus relation-ship, and is analytically derived via wave optics in Ap-pendix B.Ideally, the focal range should not exceed the objectbeing measured as images captured beyond the objectbounds increases dose to the sample without adding in-formation. The most dose-efficient measurement has afield-of-view and defocus range that matches the objectsize. VI. DOSE FRACTIONATION
Surprisingly, aberration-corrected tomography is notnecessarily dose intensive. Expanding the dose fraction-ation theorem the total dose can be chosen and dis-tributed among both specimen tilts and defoci. Hegerland Hoppe’s original construction for conventional elec-tron tomography states that the SNR of reconstructedvoxels depends only on the total dose imparted, not thedistribution of dose. It assumes weak contrast imagingwith additive noise [2, 15]. We use a more complete noisemodel with Poisson statistics [20]. For a Poisson limitedsignal, each noisy image (cid:101) p ( x, y ) of projected object p ( x, y )has a signal-to-noise ratio of SNR[ (cid:101) p ( x, y )] = 1+ χt p ( x, y )for acquisition time t and dose-rate χ (See Appendix C).For tomography, the signal and noise variance from pro-jections at each tilt add linearly to the final reconstruc-tion because the noise from each image is uncorrelated. Itshows the SNR for projection images—and tomographicreconstructions thereof—depends on both the dose andthe specimen.3D reconstruction quality is independent of the num-ber of specimen tilts only when k-space is sufficientlysampled (i.e. oversampled) and the total dose is evenlydistributed across equally spaced tilts. The same is truefor aberration corrected tomography where the signal andnoise from volumetric through-focal CTFs add linearly ink-space. The SNR of the final reconstruction depends onthe total dose imparted onto the specimen not the num-ber of specimen tilts—so long as k-space is sufficientlysampled. Notably, oversampling is guaranteed below k c (Eq. 4). FIG. 4.
Dose fractionation for through-focal acquisi-tion a) Oversampling of focal planes contains the same infor-mation as fewer focal planes, so long as the defocus samplingrequirement is met. b) Defocus sampling requirement is set bythe depth-of-focus (∆ z max , 28˚A for 200keV, 30mrad). Multi-slice simulation with Poisson noise shows the equivalence ofSNR in c,d) the sum of adjacent low-dose defocused imagesand e) a single high dose image. However, for aberration-corrected tomography, doseis not only divided among tilts, but also among de-foci. Here, dose fractionation also holds for through-focal image acquisitions. SNR of a through-focal stackdescribes the quality of data at a given dose and dose dis-tribution (See Appendix D). In an oversampled through-focal image stack (∆ z (cid:28) ∆ z max ), adjacent defocusedimages can be summed without loss of information(Fig. 4a,b). The SNR after summing M adjacent images(SNR[ (cid:80) ∆ z (cid:101) p ( x, y, z f +∆ z )] = 1+ M χt ¯ p ( x, y )) across de-foci z f + ∆ z with dose-per-image χt matches that of asingle image, ¯ p ( x, y, z f ), taken with the same total dose( M χt ). It is the total dose across all images, not thenumber of images, that determines the SNR of useful in-formation so long as the defocus sampling requirement ismet (Eq. 6).Fully quantum-mechanical multislice simulations withPoisson noise demonstrate the dose fractionation theo-rem for through-focal imaging in Figure 4. A simulatedhigh-dose image (Fig. 4e) and a binned through-focalstack with same total dose (Fig. 4c,d) have compara-ble SNR and carry the same information. We expand this over the full range of defoci. When the through-focal stack is evenly oversampled along defocus, the re-construction quality is dependent only on the total dose,not the distribution.Thus, aberration-corrected tomography does not in-herently require high doses. The desired total dose for agiven specimen determines the SNR of a 3D reconstruc-tion. The total dose may be chosen and divided acrosstilts and defoci, so long as all sampling requirements aresufficiently met. The traditional dose fractionation the-orem is upheld in aberration-corrected tomography buthas the added dimension of defocus sampling. Unfor-tunately we anticipate aberration corrected tomographywill still adhere to traditional dose requirements where3D resolution scales inversely with dose / [15, 21] andatomic resolution requires substantial beam exposure. VII. CONCLUSION
Aberration-corrected tomography’s volumetric CTFbreaks the traditional sampling requirements for objectsize and resolution as famously set by Crowther andKlug [1]. Accounting for the highly-convergent imagingprobes, a novel limit on resolution, object size, and sam-pling is presented in Equation 4 and 5. Up to a criticalspatial frequency, aberration-corrected tomography canreconstruct an object of any size, and above that fre-quency the limits on object size still exceed conventionaltomography. This is critically significant for the nextgeneration of electron microscopes with ever increasingconvergence angles ( >
60 mrad) and diminishing depth-of-focus. Lastly, the signal-to-noise of a tomographic re-construction is determined by the total dose of the mea-surement and that dose may be distributed among defociand specimen tilt.Moreover, this work extends beyond scanning trans-mission electron tomography and is applicable to any in-coherent linear imaging technique that uses highly con-vergent beams where the depth-of-focus is small com-pared to the 3D object size.With the theoretical limits defined herein, we can pro-ceed to higher resolution across larger fields-of-view toknow the atomic structure of extended specimens in allthree dimensions.
ACKNOWLEDGMENTS
We thank Yi Jiang for helpful discussions. R.Y., S.H.S.acknowledge support from the DOE BES(j) (SubawardNo. K002192-00-S01). R.H. acknowledges support fromthe Keck Foundation. Simulations made use of the Ad-vanced Research Computing Technology Services’ sharedhigh-performance computing at the University of Michi-gan and the Molecular Foundry (supported by the Officeof Science, U.S. Department of Energy under ContractNo. DE-AC02-05CH11231). [1] A. Klug and R. Crowther, Nature , 435 (1972).[2] R. Hegerl and W. Hoppe, Z. Naturforsch. A , 1717(1976).[3] D. J. De Rosier and A. Klug, Nature , 130 (1968).[4] P. A. Midgley, M. Weyland, J. M. Thomas, and B. F. G.Johnson, ChemComm , 907 (2001).[5] M. C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu,P. Ercius, U. Dahmen, B. C. Regan, and J. Miao, Nature , 444 (2012).[6] Y. Yang, C. C-C, M. C. Scott, C. Ophus, R. Xu, A. Pryor,L. Wu, F. Sun, W. Theis, J. Zhou, M. Eisenbach, P. R. C.Kent, R. F. Sabirianov, P. Zeng, H. Ercius, and J. Miao,Nature , 72 (2017).[7] O. L. Krivanek, N. Delby, and A. R. Lupini, Ultrami-croscopy , 1 (1999).[8] P. E. Batson, N. Delby, and O. L. Krivanek, Nature ,617 (2002).[9] P. D. Nellist, M. F. Chisholm, N. Dellby, O. L. Krivanek,M. F. Murfitt, Z. S. Szilagyi, A. R. Lupini, A. Borisevich,W. H. Sides, and S. J. Pennycook, Science , 1741(2004).[10] D. A. Muller, L. F. Kourkoutis, M. Murfit, J. Song, H. Y.Hwang, J. Silcox, N. Dellby, and O. L. Krivanek, Science , 1073 (2008).[11] G. Behan, E. Cosgriff, A. Kirkland, and P. Nellist, PhilosTrans R Soc A-Math Phys Eng Sci , 3825 (2009).[12] R. Hovden, H. L. Xin, and D. A. Muller, Microsc. Mi-croanal. , 75 (2011).[13] H. Yang, J. G. Lozano, T. J. Pennycook, L. Jones, P. B.Hirsch, and P. D. Nellist, Nature Communications , 1(2015).[14] R. Hovden, P. Ercius, Y. Jiang, D. Wang, Y. Yu, H. D.Abru˜na, V. Elser, and D. A. Muller, Ultramicroscopy , 26 (2014).[15] B. E. H. Saxberg and W. O. Saxton, Ultramicroscopy ,85 (1981).[16] R. Crowther, D. J. DeRosier, and A. Klug, Proc. R. Soc.London, Ser. A , 319 (1970).[17] R. N. Bracewell and A. C. Riddle, The AstrophysicalJournal , 427 (1967).[18] R. N. Bracewell, Science , 697 (1990).[19] V. Intaraprasonk, H. L. Xin, and D. A. Muller, Ultrami-croscopy , 1454 (2008).[20] A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian,IEEE Transactions on Image Processing , 1737 (2008).[21] B. F. McEwen, M. Marko, C.-E. Hsieh, and C. Mannella,Journal of Structural Biology , 47 (2002).[22] E. J. Kirkland, Advanced Computing in Electron Mi-croscopy , 2nd ed. (Springer US, 2010).[23] A. Jeffrey and D. Zwillinger,
Table of Integrals, Series,and Products , 8th ed. (Elsevier, 2007).[24] A. Pryor Jr, C. Ophus, and J. Miao, Adv. Struct. Chem.Imaging , 15 (2017).[25] Y. Yang, C.-C. Chen., M. C. Scott, C. Ophus, R. Xu,A. Pryor, L. Wu, F. Sun, W. Theis, J. Zhou, M. Eisen-bach, P. R. C. Kent, R. F. Sabirianov, P. Zeng, H. Ercius,and J. Miao, Materials Data Bank FePt00001 . Appendix A: Resolution, Sampling, Object-SizeRelationship for Aberration-Corrected Tomography
The information measured in reciprocal space byaberration-corrected STEM tomography is described bya superposition of contrast transfer functions (CTFs)from each specimen tilt about a single axis of rotation(Eq. 1- 3). The maximum object size that can be re-constructed with a given resolution is determined by thearc length between adjacent tilted CTFs, ∆ k θ . The up-per bound of a CTF in cylindrical coordinates about the k z -axis ( k r , φ, k z ) is k z = λ k r (cid:18) αλ − k r (cid:19) (A1)This upper bound is radially symmetric about the k z -axis and the lower bound is a reflection across the k z = 0plane. Equivalent points (same k x and k y ) on the boundsof a CTF in Cartesian coordinates are located at k =( k x , k y , k z ) and k = ( k x , k y , − k z ).Fixing k and rotating k by an angle ∆ θ about the k x -axis we find the vector between adjacent CTFs, ∆ k θ .∆ k θ = k y (cos ∆ θ −
1) + k z sin ∆ θk y sin ∆ θ − k z (cos ∆ θ + 1) The magnitude of this vector accurately approximatesthe arc length between equivalent points on two adjacentCTFs.∆ k θ = (cid:0) [ k y (cos ∆ θ −
1) + k z sin ∆ θ ] +[ k y sin ∆ θ − k z (cos ∆ θ + 1)] (cid:1) = √ (cid:0) k y (1 − cos ∆ θ ) − k y k z sin ∆ θ + k z (1 + cos ∆ θ ) (cid:1) = √ (cid:16) k y √ − cos ∆ θ − k z √ θ (cid:17) Transforming to cylindrical coordinates ( k y = k r sin φ ),substituting Equation A1, and using sine and cosine half-angle identities, the distance between two adjacent CTFsis ∆ k θ = 2 (cid:20) k r sin φ sin ∆ θ λ k r (cid:18) k r − αλ (cid:19) cos ∆ θ (cid:21) FIG. A1. Illustration of two adjacent tilted CTFs in the planeof maximum separation. On the right, ∆ k θ separates CTFs,and on the left, the neighborhood about k (cid:48) with diameter | δ | = 1 /D limits the maximum object size Information is maximally sampled along k r , k x and un-dersampled along k θ due to a finite number of specimentilts. The strictest limit for resolution and object size isset by the path in k-space along which ∆ k θ is largest.We seek to find the plane which ∆ k θ is maximal and alsopasses through the origin of k-space, so we maximize ∆ k θ with respect to angle φ about the k z -axis. We need notconsider the equivalent angle about the k y -axis, as thetomogaphic CTF is symmetric. ∂ ∆ k θ ∂φ = 2 k r cos φ sin ∆ θ ∂ ∆ k θ /∂k x = 0, we find an extremum when φ = π/ k r = 0 and ∆ θ = 0). Thesecond derivative test shows that φ = π/ k θ is largest when k r is in the k x = 0plane, making the problem 2D. The tomography CTF isformed by rotating individual CTFs about the k x -axis,so k ρ (a polar coordinate representing distance from thetilt axis) can be substituted for k r . The spacing betweenadjacent CTFs now simplifies to∆ k θ = 2 k ρ (cid:18) sin ∆ θ λ (cid:18) k ρ − αλ (cid:19) cos ∆ θ (cid:19) (A2)The maximum measurable object size is inversely relatedto the distance between CTFs as D = 1 / ∆ k θ . We canuse equation A2 to define the maximum size, D , of a re-construction at a given convergence semi-angle, electronwavelength, tilt step, and maximum spatial frequency. D = 1 λk ρ (cid:16) cos ∆ θ + λk ρ (2 sin ∆ θ − α cos ∆ θ ) (cid:17) (A3)Introducing d = 1 /k ρ to relate maximum spatial fre-quency to resolution, we change A3 to D = d λ (cid:0) cos ∆ θ + dλ (2 sin ∆ θ − α cos ∆ θ ) (cid:1) (A4)Complete information transfer occurs when adjacentCTFs overlap. The distance between adjacent CTFs iszero at this point (labelled k c in Fig. 2), so we set Equa-tion A2 to zero to find k c . k c = 2 λ (cid:18) α − tan ∆ θ (cid:19) Under the small angle approximation tan( ∆ θ ) ≈ ∆ θ , k c = 2 α − ∆ θλ (A5)Equations A3 and A4 are only valid for k c ≤ k ρ ≤ k max or 1 /k max ≤ d ≤ /k c . For 0 ≤ k ρ ≤ k c and 1 /k c ≤ d ,complete information is collected and the maximum ob-ject size is unbounded, giving piecewise equations. Us-ing first-order small angle approximations (sin(∆ θ/ ≈ ∆ θ/ θ/ ≈ D = (cid:40) ∞ , ≤ k ρ ≤ k c λk ρ (1 − kckρ ) , k c < k ρ ≤ αλ (A6) D = (cid:40) ∞ , ∞ > d ≥ k c d λ (1 − k c d ) , k c > d ≥ λα (A7)Equations A A Appendix B: Sampling From of Single Focal Plane
To understand how information is sampled using defo-cus, consider a single projection taken at focal plane, ∆ z .Under an incoherent linear imaging model an image, I ,at defocus ∆ z is a slice of the convolution of the object, O ( r ), with the electron probe, h ( r ) : I ( r ) = [ O ( r ) ⊗ h ( r )] · δ ( z − ∆ z ) (B1)= (cid:20) ⊗ (cid:21) ·I ( k ) = [ O ( k ) · h ( k )] ⊗ e − ik z ∆ z δ ( k x ) δ ( k y ) (B2)= (cid:20) · (cid:21) ⊗ In k-space, this is a multiplication followed by convolu-tion with a rod whose phase oscillates according to defo-cus ( Eq. B2. Evaluating the convolution in Equation B2and rearranging, we find I ( k x , k y , ∆ z ) = (cid:90) ∞−∞ dk (cid:48) z O ( k x , k y , k (cid:48) z ) h ( k x , k y , k (cid:48) z ) e − i ∆ zk (cid:48) z (B3)This is a Fourier transform in only one dimension from k z -space to ∆ z -space. Figure B2 shows this function foran aberration free beam and a point object—it illustrateshow information in the midband is only measured withina limited focal range (2∆ z max ). Nyquist sampling setsthe depth resolution at ∆ z max = 1 /k maxz = λ/α whichdefines the depth-of-focus—derived herein using wave op-tics. Appendix C: Poissonian–Gaussian Noise Modelingof S/TEM Images
An experimentally measured noisy S/TEM image, (cid:101) p ( x, y ), of a projected specimen, p ( x, y ), is adequatelymodeled with both Poisson and Gaussian noise. Withdose rate χ , acquisition time t , a noisy image becomes (cid:101) p ( x, y ) = χt p ( x, y ) + χt n p (0 , χtp ( x, y )) + tn g (0 , σ t )(C1) FIG. B2. a) CTF of a through-focal image stack and theb) CTF Fourier transform along k z for a 200keV, 30mradaberration-free electron beam. This illustrates that the mid-band frequencies set the maximum defocus step required forsampling with defocus. When the beam is out of focus froma specimen feature only low frequencies are transferred. Sur-prisingly, the highest frequencies are also transferred but theinformation intensity is too low to be useful. The first two terms describe Poisson statistics and thelast term follows Gaussian statistics [20]. The Poissonnoise is a function of both the specimen and dose ( χt )and the noise term n p is mean centered. The Gaus-sian noise n g is specimen independent, dose independent,mean-centered, and has variance σ t described by the cen-tral limit theorem. The measured image is the expectedvalue of our measurement, E[ (cid:101) p ( x, y )] = χt p ( x, y ), andnoise adds signal variance, Var[ (cid:101) p ( x, y )] = χt p ( x, y )+ tσ .The variance has two terms from the gaussian and Pois-son noise.The signal to noise ratio (SNR) of a S/TEM image is:SNR[ (cid:101) p ( x, y )] = E[ (cid:101) p ( x, y )]Var[ (cid:101) p ( x, y )] = 1 + (E[ (cid:101) p ( x, y )]) Var[ (cid:101) p ( x, y )] (C2)= 1 + χ t p ( x, y ) χt p ( x, y ) + tσ A S/TEM image is Poisson limited ( χt p ( x, y ) (cid:29) tσ )for large signals or high detector efficiency and the SNRdepends only on the object and total dose.SNR[ (cid:101) p ( x, y )] = 1 + χt p ( x, y ) (C3) Appendix D: Dose Fractionation for Through-FocalImaging
Here we show the SNR of useful information in athrough-focal stack of images is only dependent on thetotal dose, so long as the through-focal stack is over-sampled along defoci. Each image is taken at a beam0defocus value ( z f ) and, for a through-focal acquisitionoversampled by a factor of M ( M ∆ z < λ/α ), M adja-cent defoci can be summed without any loss of informa-tion. δz is the defocus step size. The SNR after sum-ming M adjacent defoci, each with dose-per-image χt atdefoci z f + m δz , describes the quality of useful informa-tion. Because each acquired image is independently mea-sured the expected value and variance of the sum addslinearly: E[ (cid:80) m (cid:101) p ( x, y, z f + m ∆ z )] = M χt ¯ p ( x, y, z f ),Var[ (cid:80) m (cid:101) p ( x, y, z f + m ∆ z )] = M χt ¯ p ( x, y, z f ) + M tσ .¯ p ( x, y, z f ) denotes the averaged image. Therefore, SNRafter summing adjacent defocused images isSNR[ (cid:88) m (cid:101) p ( x, y, z f + m ∆ z )] = 1 + ( M χt ¯ p ( x, y, z f )) M χt ¯ p ( x, y, z f ) + M tσ (D1)For Poisson noise limited images the SNR becomesSNR[ (cid:88) m (cid:101) p ( x, y, z f + m ∆ z )] = 1 + M χt ¯ p ( x, y, z f )(D2)Therefore, the SNR of oversampled through-focal stackonly depends on the total dose ( M χt ). SUPPLEMENTARY MATERIALS
Appendix SI1-SI3Figure SI1-SI4Equation SI1-SI51
Supplemental Information forThe Limits of Resolution and Dose for Aberration-Corrected Electron Tomography
Reed Yalisove, Suk Hyun Sung, Peter Ercius, and Robert Hovden
Appendix SI1: 3D Contrast Transfer Function of Highly Convergent Electron Beams
Following the work from Intaraprasonk, Xin, and Muller [19], we start with the point spread function (PSF). Thisis found by taking the inverse Fourier transform of a disk of radius k max in k-space, then taking the magnitude andnormalizing the result.The aberration and defocus of the beam can be accounted for by including an aberration function, χ ( k r , z ) in thePSF. The PSF is found with an inverse Fourier transform of an aperture: f ( k r ) = (cid:115) πk (cid:40) , ≤ k r ≤ k max , otherwiseThe aperture is normalized such that (cid:82) ∞−∞ | f ( k r ) | = 1. Adding the aberration function and taking a Fourier transform, ψ ( r, z ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π ) (cid:90) k max k r dk r (cid:90) π dθ (cid:115) πk e − iχ ( k r ,z ) e − irk r cos( θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 πk π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) k max k r dk r J ( rk r ) e − iχ ( k r ,z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) The 3D contrast transfer function (CTF) of the electron beam is simply the Fourier transform of the PSF.Ψ( k r , k z ) = 2 π (cid:90) ∞ rdr (cid:90) ∞−∞ dzψ ( r, z ) e izk z J ( rk r )To complete this integral, first expand the magnitude of the PSF using dummy variables k and k .Ψ( k r , k z ) = 12 π k (cid:90) ∞ rdr (cid:90) ∞−∞ dze izk z J ( rk r ) (cid:32)(cid:90) k max k dk J ( rk ) e − iχ ( k ,z ) (cid:33) ∗ (cid:32)(cid:90) k max k dk J ( rk ) e − iχ ( k ,z ) (cid:33) = 12 π k (cid:90) ∞ rdr (cid:90) ∞−∞ dze izk z J ( rk r ) (cid:90) k max k dk (cid:90) k max k dk J ( rk ) J ( rk ) e i ( χ ( k ,z ) − χ ( k ,z )) As noted by [22], we can split the aberration function into a z -dependent and non- z -dependent component. The z -dependent term is defocus. Defining z = 0 as the in-focus plane, we get χ ( k, z ) = − λzk / C ( k ), where C ( k )represents the higher order aberrations. Substituting this into the previous equation, we find thatΨ( k r , k z ) = 12 π k (cid:90) ∞ rdr (cid:90) ∞−∞ dze izk z J ( rk r ) (cid:90) k max k dk (cid:90) k max k dk J ( rk ) J ( rk ) e iλz ( k − k ) / e i ( C ( k ) − C ( k )) To complete this integral, we assume that all higher order aberrations are 0, i.e. C ( k ) = 0. First we integrate withrespect to z . (cid:90) ∞−∞ dze izk z e iλz ( k − k ) / = δ (cid:18) k z + λ k − k ) (cid:19) = 1 λk (cid:34) δ (cid:32) k − (cid:114) k z λ + k (cid:33) + δ (cid:32) k + (cid:114) k z λ + k (cid:33)(cid:35) This uses two identities: δ ( ax ) = | a | δ ( x ) and δ ( x − a ) = a ( δ ( x − a ) + δ ( x + a )). We can then integrate with respectto k . (cid:90) k max k dk J ( rk ) 1 λk (cid:34) δ (cid:32) k − (cid:114) k z λ + k (cid:33) + δ (cid:32) k + (cid:114) k z λ + k (cid:33)(cid:35) k max ≤ (cid:113) k z λ + k , which implies k ≤ (cid:113) k − k z λ ( ∗ ). The value of the integral is1 λ (cid:90) k max dk J ( rk ) δ (cid:32) k − (cid:114) k z λ + k (cid:33) = 1 λ J (cid:32) r (cid:114) k z λ + k (cid:33) We will next integrate with respect to r .Ψ( k r , k z ) = 12 π λk (cid:90) ∞ rdr (cid:90) k max k dk × J ( rk r ) J ( rk ) J (cid:32) r (cid:114) k z λ + k (cid:33) From [23], we find the formula (cid:90) ∞ xdxJ ( ax ) J ( bx ) J ( cx ) = 12∆ π Here, ∆ is the area of a triangle with sidelengths a , b , and c , given by ∆ = (cid:112) [ c − ( a − b ) ][( a + b ) − c ]. Here, a = k r , b = k , and c = (cid:112) k z /λ + k . This allows us to showΨ( k r , k z ) = 1 π λk k r (cid:90) k max k dk × (cid:32) k − (cid:18) k z λk r − k r (cid:19) (cid:33) − / (SI1)This integral is nonzero when the triangle inequality holds for a , b , and c . In particular, we are interested in two ofthe three possible inequalities: k r + k ≥ (cid:114) k z λ + k k + (cid:114) k z λ + k ≥ k r These can be rearranged to show that k ≥ (cid:12)(cid:12)(cid:12)(cid:12) k z k r λ − k r (cid:12)(cid:12)(cid:12)(cid:12) (SI2)From ( ∗ ) and Eq. SI2 we see that the bounds on the integral in Eq. SI1 reduce toΨ( k r , k z ) = 1 π λk k r (cid:90) √ k − kzλkzkrλ − kr k dk × (cid:32) k − (cid:18) k z λk r − k r (cid:19) (cid:33) − / = 14 π λk k r (cid:115) k − (cid:18) k z λk r − k r (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ k − kzλkzkrλ − kr = 12 π λk max k r (cid:115) − (cid:18) k r k max + k z λk max k r (cid:19) (SI3)Using the identity α = λk max from [22], where α is the aperture semi-angle, Eq. SI3 becomesΨ( k r , k z ) = 12 π αk r (cid:115) − (cid:18) λk r α + k z αk r (cid:19) (SI4)3The leading coefficient of this result varies slightly from [19] due to differences in the Fourier transform convention.We can find the bounds on k r and k z for which this equation is nonzero by consider Eq. SI2 and ( ∗ ). Combining theseequations, we find that 2 k z λ ≤ k r k + k r ≤ k r (cid:114) k − k z λ + k r (cid:18) k z λ − k r (cid:19) ≤ (cid:32) k r (cid:114) k − k z λ (cid:33) (cid:18) k z λ + k r (cid:19) ≤ k r k | k z | ≤ − λ k r ( k r − k max ) | k z | ≤ − λ k r (cid:18) k r − αλ (cid:19) (SI5) Appendix SI2: Methods: Tomographic Simulation using Quantum Mechanical Multislice Scattering
SFIG. SI1. 3D atomic resolution aberration corrected tomographic reconstruction of from quantum mechanical multislicesimulation. a,b,c) Orthographic view of the 3D reconstruction highlights atomic resolution in multiple viewing angles.
Fully quantum mechanical multislice simulation of three synthetic FePt nanoparticles spanning (15nm) was per-formed at incident electron energy of 200keV and convergence semi-angle of 30mrad, pixel size (0.25˚A) using PRIS-MATIC software [24]. Images were calculated on GPU accelerated computing clusters at University of MichiganAdvanced Research Computing and Technology Center and Lawrence Berkeley National Labs. The atomic coordi-nates for the FePt nanoparticles used were experimentally acquired by Yang et al. [25]. Each through-focal stackcontains 13 defoci images with a 1.25nm defocus step; 105 through-focal stacks were simulated at each tilts with a30mrad (1 . ° ) tilt step. In creating over 1300 images, the simulation computed 500 million wavefunctions over15,000 GPU core hours.To weight the through-focal stack in Fourier space, it is divided by the aberration-free CTF at 300keV, 30mrad, thisis multiplied with distance from the tilt axis (analogous to weighted back projection). The weighted through-focalstack from each tilt angle is mapped onto a universal Fourier space by bilinear extrapolation, which distributes thecomplex value of an input point to its four nearest neighbors on the output Cartesian grid. The final reconstructionis obtained directly from the 3D inverse Fourier transform. Appendix SI3: Additional figures SFIG. SI2. 2D slices through the PSF (top) and CTF (bottom) of electron wave functions with 10, 30, and 60mrad aperturesemiangles. Confinement of the PSF in the z-direction corresponds to extent in the k z -direction of the CTF.SFIG. SI3. 3D Contrast Transfer Function of the CTF for conventional tomography. The left image is a slice through the k y k z -plane of the 3D CTF. SFIG. SI4. Tomography CTFs for a) traditional tomography with 2 ◦ tilt, where information is collected along planes and b)aberration-corrected tomography with 200keV, α = 30mrad, and 2 ◦ tilts. In aberration corrected tomography, information iscompletely sampled for frequencies below k c . In each figure, the distance between adjacent regions of information (∆ k θθ