High-resolution three-dimensional crystalline microscopy
HHigh-resolution three-dimensional crystalline microscopy
Marc ALLAIN , Virginie CHAMARD and Stephan O. HRUSZKEWYCZ . Aix-Marseille Univ, CNRS, Centrale Marseille, Institute Fresnel, 13013 Marseille, France. Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA.
Abstract—
In this communication, we discuss how 3D infor-mation about the structure of a crystalline sample is encoded inBragg 3DXCDI measurements. Our analysis brings to light therole of the experimental parameters in the quality of the final re-construction. One of our salient conclusions is that these param-eters can be set prior to the ptychographic 3DXCDI experimentand that the spatial resolution limit of the 3D reconstruction canbe evaluated accordingly.
Since its introduction in the early 2000s [1, 2], three-dimensional X-ray Coherent Diffraction Imaging (3DXCDI)has widely demonstrated its ability to provide non-destructivethree-dimensional (3D) images of complex nanostructures.Two key features of 3DXCDI are noteworthy: 1) 3DXCDI of-fers the possibility to measure data either in a Bragg [2] or ina forward geometry, the former case providing 3D images ofstrains in crystalline materials [3, 4]; and 2) 3DXCDI can be ex-ecuted with a ptychographic (spatial) scan, hence providing im-ages of extended samples [5, 6, 7]. For these reasons, 3DXCDIopens a wide playground for x-ray microscopy. In this commu-nication, we discuss how 3D information about the structure ofa sample is encoded in Bragg 3DXCDI measurements. In par-ticular, our analysis brings to light the role of the experimentalparameters in the quality of the final reconstruction.
Any 3DXCDI approach is inherently a lens-less tomographic modality: it is lens-less because the dataset (a series of coherentintensity patterns) is numerically inverted by a phasing algo-rithm, and tomographic since each collected diffraction inten-sity is drawn from the sample via a tomographic measurement.Some details about the Bragg ptychographical experimentare now provided. Let us introduce first the exit-field ψ m = p m × ρ (1)where m ∈ { , · · · M − } is the position index during the spa-tial (ptychographical) scan. In the relation above, ρ : R → C denotes the scattering density [11, Sec. 7.1.2] of the diffract-ing crystal and p m : R → C is the m -th coherent probe illuminating the sample. When a Bragg condition is met ,both quantities are conveniently expressed with coordinates r := ( r ⊥ , r z ) ∈ R within a 3D frame in (direct-)spacematching the detection geometry, see Fig. 1. In addition, forthe sake of simplicity, we consider that the probing field is According to the usual convention in coherent Bragg diffraction, the originof reciprocal space ( q x = 0 , q y = 0 , q z = 0) corresponds to a reciprocalspace Bragg peak denoted by the reciprocal space vector G HKL for a given ( H, K, L ) Bragg reflection, see for instance [8]. generated from a probe function p shifted along the scatter-ing direction e z only i.e., we have p m ( r ) := p ( r − r m ) with r m := 0 × e x + 0 × e y + m ∆ e z where ∆ ∈ R is the step sizeof the spatial scan. In Bragg ptychography, as in any coher-ent diffraction method, the measurement is the intensity of thescattered-field collected by an array detector. Under the first-order born approximation [9, Sec. 8.10.1], the scattered wave-field collected in the “far-field” reads at the detector plane Ψ( q ⊥ ; r z = m ∆) = (cid:101) ψ m ( q ⊥ , q z = 0) (2)where ˜ ψ m is the 3D Fourier transform of ψ m and q := ( q ⊥ , q z ) is the 3D frequency (or reciprocal-space ) coordinates. Finally,the expected ( i.e., noise-free) measurements at the detectorplane are given by the intensity of the scattered field (2) I ( q ⊥ ; r z = m ∆) = | Ψ( q ⊥ ; r z = m ∆) | . (3)Although the relations (2) and (3) are useful in deriving ac-tual Bragg ptychography reconstruction algorithms (see for in-stance [6, 7]), it does not tell anything about the spatial infor-mation extracted from the sample via the ptychographical mea-surements. A substancial leverage to address this question isprovided by the following result, easily deduced from the SliceProjection Theorem [10, Sec 6.3.3] F − ⊥ Ψ = ρ ⊗ // p. (4)In the relation above, F ⊥ is the bi-dimensional (2D) Fouriertransform with respect to r ⊥ , and ⊗ // is the one-dimensionalconvolution operator acting along the scattering direction e z .In other words, the 3D quantity g ( r ) := ( ρ ⊗ // p )( r ) (5)is an approximation of the scattering-density ρ built on a fil-tering by the probe profile along e z . The scattering-densityapproximation g given in (5) appears in a previous publicationfrom the authors [12]. In this communication, this relation isa pivotal tool in deriving resolution limits and sampling condi-tions for Bragg ptychography experiments.Let us assumed that the probe profile along e z is a band-limited function with its support strictly contained in the do-main Ω z := [ − ¯ q z , ¯ q z ] with ¯ q z ≥ . We deduce from (2) and(5) that the scanning step-size ∆ should be set at least suchthat the spatial information is preserved in g ( r ⊥ , r z = m ∆) , m ∈ N . The Shannon-Nyquist sampling rate is then driven bythe maximal frequency ¯ q z in the (assumed) band-limited probeprofile. In addition, because the spatial sampling is performedover wave-field intensities (3), it is not difficult to show that In general, the 3D shifting of the probe can be casted within a non-orthogonal frame ( e (cid:48) x , e y , e z ) with e (cid:48) x pointing along the direction of the in-coming probe, see Fig. 1. In the x-ray regim, the probe p ( r ) is invariant along e (cid:48) x and any shift along this direction does not provide any spatial informationabout the sample. a r X i v : . [ phy s i c s . c o m p - ph ] S e p igure 1: A simplified Bragg ptychography experiment. (Upper) When theincoming beam and the scattering direction define a specific Bragg angle θ B ,the Bragg condition for the chosen ( H , K , L ) Bragg peak is met and a diffrac-tion (intensity) signal shows up at the detection plane. As the sample is shiftedin a focused probe, the diffraction signal is recorded in each spatial position.(Lower) A series of spatial scan can be performed with various tilts of the sam-ple. This result in more 3D spatial information extracted from the sample. the sampling rate should be at least twice the Shannon-Nyquistlimit for the wave-field, i.e., the following condition ∆ > / (4¯ q z ) (6)ensures that the approximation g given by (5) can be retrievedfrom the series of (noise-free) intensity measurements (3). Theresolution bounds one may achieve in practice are also pro-vided by g , as this latter function is the best 3D approximationof the scattering density one can expect from the spatial scan.Because the convolution in (5) acts as a pointwise multiplica-tion with respect to the coordinates r ⊥ , the spatial resolutionbounds along the directions e x and e y are not restricted by theprobe. In the third direction e z , the spatial resolution is re-stricted by the convolution kernel in (5), leading to the bound R z = 1 / ¯ q z . (7)The so-called “rocking-curve” can nevertheless extend furtherthis resolution limit via angular diversity; this topic is devel-oped in the next section. When Bragg 3DXCDI was introduced in the early 2000’s , themethod relied on the sample rotation to explore the 3D Fouriercomponents of the sample to retrieve. In this context, a single3D Bragg peak is probed by the camera plane while the sampleis tilted. It is nevertheless a very peculiar tomographic modal-ity: as the Bragg peak sits at a given point in the 3D reciprocal Bragg 3DXCDI was introduced as a natural extension of standard CDItechniques. In this context, the method aimed at imaging isolated nano-crystals,with restricted supports small enough so that the unfocused coherent beam il-luminating the sample can be considered as a single plane wave. The methodwas then mostly understood as a 3D Fourier synthesis strategy [2, 3]. lattice of the probed crystal, the whole 3D Bragg peak is probedwith unusually small angular ranges. In addition, the samplerotation results in a cartesian , rather than polar sampling of theBragg peak, see Fig. 1-Lower. When Bragg 3DXCDI is per-formed with a scanning (focused) probe, the relation (5) clearlystates that a 3D reconstruction can still be obtained withoutsample rotation. If the sample rotation is also performed, wecan expect more spatial information to be extracted. This is thequestion we aim at addressing in this section.In Bragg geometry, a small rotation of the sample, by anangle δ θ , results in a frequency shift by w of the 3D Fouriertransform of the scattering-density ρ , see Fig. 1-Lower. Thescattered field at the camera plane reads then Ψ w ( q ⊥ ; r z = m ∆) = (cid:101) ψ m ; w ( q ⊥ , q z = 0) (8)where ψ m ; w ( r ) := p m ( r ) × ρ ( r ) e j π w t r is the modulated exit-field. The relation (4) is modified accordingly F − ⊥ Ψ w = ( ρ × e j π w t • ) ⊗ // p. (9)The relation above shows that the accessible frequency do-main along u z is now Ω z ( w ) := Ω z ⊕ w t u z (where ⊕ isthe Minkowski sum ). In practice, a series of N tilts is usuallyperformed, inducing an equivalent series of frequency shiftsdenoted W := { w n } Nn =1 . The best approximation one canachieve is then of the form (5) and reads g ( r ; W ) = [ ρ ⊗ // p ( · ; W )]( r ) (10)where the equivalent probe p ( r ; W ) := (cid:80) n p ( r ) × e j π w tn r defines the spatial information extracted from the joint spa-tial/angular scan. The set of frequency that are extracted bythis equivalent probe are Ω z ( W ) := Ω z ⊕ (cid:80) n w tn u z , and theresolution limit along e z is obviously better than (7) and reads R (cid:48) z = 1 / ¯ q (cid:48) z with ¯ q (cid:48) z := ¯ q z + (cid:80) n | w tn u z | . (11)We underline that the sampling condition (6) is actually un-changed when angular diversity is considered. A remaining,potential issue is that Ω z ( W ) may not be a compact set, hencecreating un-probed “holes” in the frequency space of the ap-proximation (10). The condition || w n || cos θ B ≤ ¯ q z , ∀ n, with θ B the Bragg angle nevertheless ensures a continuous probingof the frequency domain.This last section clearly connects Bragg ptychography toother super-resolved imaging techniques, e.g., structured illu-mination microscopy [13], synthetic aperture [14] strategies.We also stress that these resolution limits are reached only inthe asymptotic limit, with noise-free intensity measurements.In practice, both the photon shot-noise and the physical exten-sion of the camera will reduce the effective resolution in allthree directions. Acknowledgement
This work was supported by the European Research Coun-cil (European Union’s Horizon H2020 research and innovationprogram grant agreement No 724881). Work at Argonne Na-tional Laboratory (development of the Bragg ptychography for-ward model) was supported by the U.S. Department of Energy(DOE), Office of Basic Energy Sciences (BES), Materials Sci-ence and Engineering Division. If the chosen (probed) Bragg peak is not the one that sits at the origin of3D reciprocal lattice, the angular range required for a full 3D scan is ∼ ◦ . eferences [1] J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai,and K. O. Hodgson, “High Resolution 3D X-Ray Diffrac-tion Microscopy”, Phys. Rev. Lett. , 088303, 2002[2] G. J. Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K.Robinson, “Three-Dimensional Imaging of Microstruc-ture in Au Nanocrystals”, Phys. Rev. Lett. , 175501,2003[3] M. A. Pfeifer, G. J. Williams, Ivan A. Vartanyants, R.Harder and I. K. Robinson, “Three-dimensional mappingof a deformation field inside a nanocrystal”, Nature ,63-66, 2006[4] J. N. Clark, J. Ihli, A. S. Schenk, Y.-Y. Kim, A. N. Ku-lak, J. M. Campbell, G. Nisbet, F. C. Meldrum and I.K. Robinson, “Three-dimensional imaging of dislocationpropagation during crystal growth and dissolution”, Na-ture Materials , 780-784, 2015[5] M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M.Kewish, R. Wepf, O. Bunk and F. Pfeiffer, “PtychographicX-ray computed tomography at the nanoscale”, Nature , 436-439, 2010[6] P. Godard, G. Carbone, M. Allain, F. Mastropietro, G.Chen, L. Capello, A. Diaz, T.H. Metzger, J. Stangl andV. Chamard “Three-dimensional high-resolution quanti-tative microscopy of extended crystals”, Nature Comm. , 568-570, 2011[7] F. Mastropietro, P. Godard, M. Burghammer, C. Cheval-lard, J. Daillant, J. Duboisset, M. Allain, P. Guenoun, J.Nouet and V. Chamard, “Revealing crystalline domainsin a mollusc shell single-crystalline prism”, Nature Mate-rials , 946-952, 2017[8] I. A. Vartanyants and I. K. Robinson, “Partial coherenceeffects on the imaging of small crystals using coherent x-ray diffraction”, Journal of Physics – Condensed Matter , 10593, 2001[9] W. C Chew, “Waves and Fields in Inhomogeneous Me-dia”, IEEE Press (1995).[10] A. C. Kak and M. Slaney, “Principles of ComputerizedTomographic Imaging”, IEEE Press (1988).[11] J. Stangl, C. Mocuta, V. Chamard and G. Carbone,“Nanobeam X-Ray Scattering”, Wiley-VCH (2014).[12] S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray,J. R. Holt, P. H. Fuoss and V. Chamard, “High-resolutionthree-dimensional structural microscopy by single-angleBragg ptychography”, Nature Materials , 244-251,2017[13] M. G. L. Gustafsson, “Surpassing the lateral resolutionlimit by a factor of two using structured illumination mi-croscopy ”, Journal of Microscopy , 82-87, 2000[14] M. E. Testorf and M. A. Fiddy, “SuperresolutionImaging–Revisited ”, Advances in Imaging and ElectronPhysics163