Interpretation of ptychographical reconstructions

43

Scanning Microscopy Vol. 11, 1997 (Pages 43-52) 0891-7035/97$5.00+.25Scanning Microscopy International, Chicago (AMF O’Hare), IL 60666 USA

THE INTERPRETATION OF PTYCHOGRAPHICAL RECONSTRUCTIONS FROMSPHALERITE STRUCTURES

Abstract

Ptychography is an electron diffraction techniquein which Bragg reflections are interfered coherently in orderto measure the phases of diffraction orders directly. Twosimple reconstruction methods exist, which require differentdegrees of coherence in the illuminating electron beam.Simulations on InP <110> for two electron acceleratingvoltages suggest that the asymmetric ‘dumbbells’ can bereconstructed for small thicknesses, but that thereconstruction methods break down at thicknesses around10 nm. At larger thicknesses one reconstruction methodyields images which show the phosphorus atom. The phaseobject approximation is shown to break down at thicknessesof about 3.3 nm.

Key Words: Ptychography, sphalerite, super-resolutionimaging, phase retrieval, phase object approximation,scanning transmission electron microscopy.

*Address for correspondence:T. PlamannCentre d’Etudes de Chimie Métallurgique CNRS,15 rue Georges Urbain, 94407 Vitry-sur-Seine, France

Telephone number: +33-1-46873593FAX number:+33-1-46750433E-mail: plamann@glvt-cnrs.fr

T. Plamann

Centre d’Etudes de Chimie Métallurgique CNRS, 94407 Vitry-sur-Seine, France

Introduction

Much research is currently being done to extract thephase information in electron microscope images and toreconstruct the electron wavefunction at the exit surface ofthe specimen in amplitude and phase. A project funded bythe European Community was set up, which aimed at therecovery of structural information at 0.1 nm resolution usingholography (Lichte, 1992) and focus variation (Coene et al.,1992; Van Dyck et al., 1993). A different class of imagereconstruction methods, comprising tilt series in theconventional high-resolution transmission electronmicroscope (HRTEM) (Kirkland et al., 1995) and super-resolution imaging in the scanning transmission electronmicroscope (STEM) (Rodenburg and Bates, 1992) arecurrently being developed in Cambridge, the latter methodbeing the subject of this paper. Although these advancesare being made in the quantitative analysis of imagecontrast, the interpretation of the new information in termsof specimen structure remains somewhat limited. This paperis concerned with the discussion of the limitations and theopportunities of super-resolution imaging in STEM as theseare determined by the dynamical interaction of the electronwave with the potential distribution in the local environ-ment of atoms. Throughout the paper, the assumption of aperfect crystal structure will be made, in which case super-resolution imaging is a straightforward extension to atechnique called “ptychography” which was introduced byHoppe (1969a, 1969b, 1982) as a means of measuring thephases of diffraction orders directly.

Nellist et al. (1995) showed that Si in the <110>orientation with its dumbbell features can be comfortablyreconstructed at a resolution of 0.136 nm using ptycho-graphy on data recorded in a STEM, which had a nominalresolution of only 0.42 nm. Here we examine the applicabilityof the same method to the imaging of zinc-blendesemiconductor compounds. We discuss the conditionsunder which it is possible to resolve and identify thesublattices of the different constituents in the <110>orientation.

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T. Plamann

Ptychography

Hoppe proposed to perform a diffraction experimentwith a coherent primary wave of suitable shape instead ofplane wave illumination, which is used in conventionaldiffraction experiments. If we assume that the structure is aperfect crystal, for which the scattered waves areconcentrated at discrete points in reciprocal space, thenthe philosophy behind the technique is to widen thereciprocal space points into regions which overlap. In theoverlap region a phase sensitive addition of the beams ofadjacent lattice points occurs. If the interaction of thespecimen with the illumination is multiplicative, as in thecase of a two-dimensional (infinitely flat) crystal, then thewidening of the diffraction spots can be easily described astheir convolution with the Fourier transform of the primarywavefunction which we call the aperture function. Theadvantage of this approach is that the interferences takeplace at the specimen level, from which the wave travels tothe detector without further manipulation. In contrast toholography where the phase information is supplied throughthe interference of the scattered waves with a singlereference wave which might have travelled a considerablydifferent path length through the optical system, the basisof the approach outlined here is the reinterference ofneighboring beams.

Ptychography is a general technique which inprinciple could be performed with all types of radiation, butthe requirement of a high degree of coherence favorselectrons over all other types of radiation because field-emission guns can produce electron beams of suitablebrightness and coherence. Furthermore, electrons, beingcharged particles, can be easily deflected by magnetic andelectric field configurations. In fact, modern high-resolutionelectron microscopes equipped with field-emission gunsare ideally suited for the ptychographical technique, as theyroutinely produce electron probes with diameters below onenanometer. We will follow the nomenclature of the STEMwhere we can record coherent microdiffraction patternsconsisting of coherently overlapping diffraction discs inthe so-called microdiffraction plane. Hoppe suggested touse only one or two such patterns. Indeed, one patternalone may contain enough information for a successfulphase assignment if the examined structure consists of atwo-dimensional lattice of a suitable geometry. Generally,one encounters a phase ambiguity, which is most easilyresolved if either the beam is defocused - leading to straightfringes perpendicular to the disc separation vector asexperimentally observed by Vine et al. (1992), Steeds et al.(1992), Vincent et al. (1993), Tanaka et al. (1994) - or if aseries of patterns are recorded. Figure 1 shows calculatedmicrodiffraction patterns for different probe positions bothclose to focus and with a large amount of defocus. The

variation of the intensity distribution in the overlap regionsdepends on the direction of the probe movement with respectto the disc separation vectors. In Figure 1 the probe ismoved in a direction indicated by the arrow and it isnoticeable that there are interference regions in which theintensity distribution does not change. In the following,we will examine a four-dimensional data set ofmicrodiffraction patterns |M(µ′, ρ)|2 recorded as a functionof the probe position ρ. In this notation reciprocal spacecoordinates are denoted by a dash, µ′ being a two-dimensional vector describing a position in themicrodiffraction plane (i.e., a certain scattering angle) and ρbeing a two-dimensional vector describing the position ofthe probe in the specimen plane. In the case of a perfectcrystal, the probe movement gives rise to a sinusoidalintensity variation in the interference regions, and it is thephase of this intensity variation which we interpret as thephase difference between the diffraction orders. After theFourier transformation of the data set with respect to theprobe position interference regions as those depicted inFigure 2 show up for different spatial frequenciescorresponding to different disc separation vectors. Theyyield the phase information which allows us to reconstructthe diffracted beams in amplitude and phase. For non-crystalline materials, applying a super-resolution algorithm(Rodenburg and Bates, 1992) to the four-dimensionalmicrodiffraction data-set |M(µ′ ,ρ)|2 allows for thereconstruction of the wavefunction in amplitude and phasebeyond the spatial frequency cut-off as defined by theaperture of the probe-forming lens.

Let us express the intensity distribution in themicrodiffraction plane mathematically. If we assume thatthe illuminating aperture is fully coherently filled and thatthe interaction of the electron wave with the specimen ismultiplicative, then the wave function in the microdiffractionplane is a convolution of the specimen function Ψ(a′) with

the aperture function Α(a′) in reciprocal space such that theintensity distribution is given by

|M(µ′,ρ)|2 =

∫∫ A(µ′-a′) A*(µ′-b′) Ψ(a′) Ψ*(b′)exp[2πiρ(b′-a′)] da′db′ (1)

When the size of the objective aperture is such thatthe diffraction discs have just single overlap, the intensityin the overlap region between the disc G and G+H can beeasily shown to be

|M(µ′,ρ)|2 = |ΨG|2 + |Ψ

G+H|2

+ 2|ΨG| |Ψ

G+H|cos [(α

G+H- α

G)

+ (χ(µ′-(G+H)) - χ(µ′-G)) + 2πρH] (2)

where αG and α

G+H are the phases of the beams G and G+H,

respectively. The lens aberrations are expressed by the

Interpretation of ptychographical reconstructions

45

function χ(a′), which is related to the aperture function A(a′)by A(a′) = |A(a′)| exp (iχ(a′)). The intensity in the overlapregion is seen to vary sinusoidally as the probe is scannedparallel to the disc separation vector H. After taking theFourier transform with respect to ρ, G(µ′, ρ′) has considerable

magnitude only for values of µ′ within the overlap region

between the two discs when ρ′ is equal to H or -H. For ρ′ =H we have

G(µ′, ρ′) = |ΨG| |Ψ

G+H|exp [i(α

G+H - α

G)

+ {χ(µ′-(G+H)) - χ(µ′-G)}]

Figure 1. Calculated microdiffractionpatterns for different probe positions: (a),(b) close to focus; (c), (d) with a largeamount of defocus. The variation of theintensity distribution in the overlap regionsdepends on the direction of the probemovement with respect to the discseparation vectors. Here the probe ismoved in a direction indicated by the arrow.There are interference regions in which theintensity distribution does not change.

(3)

The magnitude of G(µ′, ρ′) is given by the amplitude productof the reinterfering beams and its phase by their phasedifference (including the aberrations introduced by the lens).For the measurement of the phase we shall extract theinformation from the point midway between the discs, i.e.,from the centre of the interference regions, for which µ′ =G+H/2. In this point the two terms representing the lensaberrations in Equation (3) cancel if the aperture function iscentro-symmetric. All the relative phases determined inthis way are insensitive to lens characteristics (McCallumand Rodenburg, 1993). Using reconstruction methods asthose discussed in the next section the Fourier space wavefunction can be retrieved, which via a Fourier transformyields a reconstructed real-space wave function in amplitudeand phase.

Simple Reconstruction Methods

In the simple case of a perfect crystal of unit cell

dimensions such that the discs in the microdiffraction planejust partly overlap as in Figures 1 and 2, all the diffractiondiscs can be successively phased (Spence and Cowley, 1978;Spence 1977, 1978). Such a reconstruction method - whichwe will refer to as reconstruction method I (RM I) - is allowedif the discs appear uniform, in which case the wavefield inthe microdiffraction plane can be described as a convolutionof the diffraction peaks of the specimen function with anaperture function - a characteristic which lent its name tothe technique (πτυξ means ‘fold’). In other words, theinteraction of the electron probe with the specimen shouldbe multiplicative. This is only valid in the case of a verythin but not necessarily only kinematically scattering crystal(e.g., when the phase object approximation holds) or in thelimit as the wavelength of the scattering radiation tends tozero. For a finite wavelength amplitudes and phases of thescattered wave will vary across the disc diameter, when thecrystal thickness exceeds a certain value. Broeckx et al.(1995) have related the assumption of a multiplicativeinteraction to the channelling of the STEM probe. In aforthcoming publication we will discuss the problems withthe description of ptychography using dynamical diffractiontheory in more detail.

Another simple approach to reconstruct the objectwave function becomes apparent if we increase the radiusof the aperture such that there is overlap between severaldiffraction orders and we consider the intensity variationmidway in the overlap between the central disc and eachdiffracted disc. After Fourier transformation with respectto the probe position we obtain interference regions of theform depicted in Figure 3 for the case of the <110> zone-axisof a zinc-blende structure. In the reconstruction methoddiscussed in the following, amplitudes and phases of the

46

T. Plamann

reconstructed beams are taken from the centre of the overlapbetween the central disc and the pertinent diffracted disc(these points are indicated by arrows in Figure 3). We shallrefer to this approach as reconstruction method II (RM II).In the case of a strongly-scattering crystal we have to ensurethat the centre of overlap between the central disc and eachdiffracted disc does not coincide with any part of the overlapregion of two other discs, as this would lead to mixing of theinterference regions. However, as Figure 3 indicates, wecan find an aperture size for which the centers of allinterference regions are not obscured. In particular, the twopictures on the right of Figure 3 correspond to the (002) and(004) reflections, respectively. We can conclude that for asuitable geometry of the diffraction pattern amplitude andphase of the reflections of the types G and 2G can be ob-tained, even if the overlap of the reflection 2G inevitablyoverlaps with the disc of the reflection G. If the aperturesize was increased even further in Figure 3, then reflectionshigher than (004) could be reconstructed, but the methodwould break down for the low-order reflections (in this casethe (111) reflection). It should be noted that points midwaythe central disc and the different diffraction discs, whichare indicated by arrows, correspond to the Bragg conditionfor each beam. In the kinematical theory, it can be shown

that the reconstructed wave function corresponds to aperfect projection of the specimen structure (Plamann andRodenburg, 1994). It should be noted that the requirementson the degree of coherence in the illuminating electron beamis different for the two reconstruction methods. While it isonly necessary that neighboring discs interfere coherentlyin RM I, the coherence in RM II must be sufficient to ensuredetectable interference between the central disc and thehighest-order disc which we want to phase.

We want to investigate the applicability of the tworeconstruction methods, and we discuss the case of InP inthe <110> orientation as an example below. The choice ofthis structure is motivated by recent, yet to be publishedwork which demonstrated that the atomic columns of galliumphosphide can be clearly reconstructed using RM I. Thereconstruction of the phase showed an excellent match withthe projected structure. Direct resolution of the atomiccolumns along the <111> and <100> direction is morestraightforward given their greater projected separationdistance and the presence of a centre of inversion in theseorientations (Ourmazd et al., 1986; see Kawasaki andTonomura (1992) for reconstructions of InP <100> usingholography). It should be noted that for a satisfyingstructural analysis of these materials, we require the abilitynot only to resolve the atomic columns but also to identifythe atomic species occupying a given column. This isimportant for defect analysis, as defects in differentsublattices show different mechanical and electronicproperties.

Simulation Method

If we wanted to simulate the probe propagationthrough the crystal for each probe position, then we wouldhave to make great demands on the size and the sampling ofthe used supercell. The computer time required would beproportional to Nµ′2Nρ′2 log Nµ′, where Nµ′

is the number ofsampling points in the microdiffraction plane (that is, numberof beams included in the calculation) and Nρ′ the number ofprobe positions. This would be identical to the simulationof annular dark field (ADF) images (Kirkland et al., 1987),which is notoriously slow. However, the propagation of theprobe does not have to be calculated, at least not in thecase of a perfect crystal, because the phase determinationrelies on the interference of only two beams from oppositepoints in the illuminating convergent beam. These beamspropagate through the crystal independently of all otherscattered beams and can thus be modelled by (tilted) planewave illumination conditions in ordinary multi-sliceprograms. In the case of aperiodic specimens mixing of allthe beams in the scattering process occurs and the fullpropagation of the probe has to be simulated.

Multi-slice simulations were carried out using the

Figure 2. Interference regions which show up after Fouriertransformation of the super-resolution data set with respectto the probe position for different spatial frequencies ρ′: (a)

ρ′ = 0; this is the sum of all patterns and corresponds to theincoherent convergent beam electron diffraction (CBED)pattern; (b, c, d) finite ρ′, corresponding to different discseparation vectors and containing the phase information.

Interpretation of ptychographical reconstructions

47

software package CERIUS (developed by MolecularSimulations, Cambridge, UK). The electron scatteringfactors of Doyle and Turner (1968) were used, and DebyeWaller factors were taken from Reid (1983). Illuminationtilts K

t of the incoming plane wave were modelled by the

alteration of the propagation function P(K) in Fourier spaceaccording to:

P(K)=exp πi λ ∆z(K+Kt)2

Figure 3. Interference regionsfor the case of a bigger aperture;in reconstruction method IIamplitudes and phases of thereconstructed beams are takenfrom the centre of the overlapbetween the central disc and thepertinent diffracted disc, indi-cated by arrows.

Figure 4 (on next two pages).Comparison of differentreconstruction methods on InP<110>.

(4)

which corresponds to a slight shear of the specimen suchthat every successive slice is shifted by a small amount(Self et al., 1983).

In contrast to conventional lattice imaging we donot have to calculate the effect of the lens aberrations onthe lattice interference fringes, as we always choose diffrac-tion information from the centre of overlap between thediscs, which is not affected by the lens transfer function,provided that the objective lens is perfectly aligned (see

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T. Plamann

Interpretation of ptychographical reconstructions

49

50

T. Plamann

above). Hence, the defocus is not a free parameter in thecalculations.

Simulations on InP <110>

We introduced two different reconstructionsmethods above. In Plamann and Rodenburg (1997) it isshown that the reconstructed object wave functions do notcorrespond to the exit surface wave functions (ESWFs)obtained under axial illumination conditions, but to‘backpropagated’ wave functions pertaining to the middlesection of the crystal. These functions are somewhat closerto pure phase objects than the ESWFs. In the following, wewill make a comprehensive comparison between the differentreconstruction methods for two accelerating voltages withthe strict phase object approximation (POA), using InP <110>as an example. The examination of the thickness limitationsof the different reconstruction methods is important for thedetermination of instrumental parameters such asaccelerating voltage or coherence width which are mostsuited for the performance of super-resolution experiments.

In each case (RM I, RM II and POA) the object wavefunction was calculated in Fourier space, yielding thecomplex values of the 19 innermost beams. Then a Fouriertransform was taken to obtain a periodic function in realspace. It is obvious that if we multiply the wave functionsby a constant phase, we obtain another possible solution.We assigned a phase of -2.5 rad to the central beam in orderto avoid a phase wrap-around on the indium column forsmall thicknesses. The object wave functions in Fourierspace were calculated as follows:

RM I: First, we calculated amplitudes of all beamsincluding the {004} reflections for axial illumination, that isK

t = 0. This produced the beam amplitudes for the ptycho-

graphical reconstruction which are extracted from the centersof the diffraction discs. Second, we calculated phasedifferences between pertinent beams by applying tiltedplane wave illumination conditions. All beams were phasedsuccessively, using the shortest routes available.

RM II: For the reconstruction of each Fouriercomponent of the reconstructed wave function, each beamwas set into the Bragg condition, successively. Theamplitude of the reconstructed Fourier component was setto the calculated amplitude product of the central beam andthe pertinent beam, and its phase was set to the calculatedrelative phase. The amplitude of the central beam in thereconstruction was set to unity.

POA: In the POA the potential distribution isprojected onto a two-dimensional surface. The thicknessvalues in Figure 4 do not correspond to an actual distance,but to the scaling factors of the projected potential. In thecalculation, the propagation between successive slices wasneglected. Phase transmission functions were calculated

at an accelerating voltage of 100 keV. The propagator inEquation (4) was set to unity. It is apparent that noillumination tilts had to be modelled, since in the POA thebeam amplitudes are the same for each illumination tilt.

Figure 4 shows the magnitude (m) and phase (p)components of our reconstructions at thickness steps of1.7 nm. For t = 1.7 nm the image features in the five rowslook somewhat similar. This is expected because at smallthicknesses the POA can be made. However, even at athickness as small as 3.3 nm the images start to lookconsiderably different: While for 300 keV the reconstructedphase for both reconstruction methods matches the phasein the POA, the indium column shows up much stronger inthe phase of the reconstructions for 100 keV, since the POAbreaks down more rapidly for smaller accelerating voltages.At slightly larger thicknesses (5.0 - 6.6 nm) the phase of thereconstructions consistently shows a close resemblance tothe projected structure, with a greater phase shift on theupper indium column and a clear expression of the crystalpolarity. In contrast to the phase, the magnitudecomponents of the reconstructions do not match thestructure reliably. We note that there is an apparentinversion of the dumbbell asymmetry in the magnitudecomponent between 5 and 6.6 nm for RM I at 100 keV.Surprisingly, the “dumbbells” look almost symmetrical inthe phase of the POA. This is a consequence of thetruncation of the wave function in Fourier space. For aperfect phase object the indium column scatters into higherdiffraction orders which are not included in the wave functionshown here. If the POA wave function comprised morebeams, the asymmetry would become more apparent.

At thicknesses exceeding 8.3 nm the image recon-structions break down successively and do not show theatomic positions reliably. The first reconstruction methodto break down is RM I at 100 keV (for RM II at bothaccelerating voltages and t = 8.3 nm the indium columnappears dark as a consequence of a phase wrap-around),but all the others follow quickly. The images at these largerthicknesses cannot be called structure images. However,let us consider the phases of the RM II wave functions forthicknesses greater than 10 nm. They all show strong brightpatches on top of the phosphorus column, while the indiumcolumn is not visible except for the case of 300 keV, in whichit shortly re-appears at 13.3 nm. The striking appearance ofthe phosphorus column in the reconstructions is not yetunderstood. The dynamical scattering for RM II is rathercomplicated, since all beams are brought into the Braggcondition, successively. However, the reason for thedominance of the phosphorus column has probably to dowith the strong excitation of its s-states. The result alsoseems to be in agreement with simulations of annular dark-field images of InP (Hillyard et al., 1993), in which thechannelling peak on top of the indium columns was shown

Interpretation of ptychographical reconstructions

51

to disappear at thicknesses around 10 nm.

Conclusions

Simulations on InP <110> for two electronaccelerating voltages have shown that the asymmetric“dumbbells” can be reconstructed for small thicknessesbelow 8 nm. In this thickness regime the choice of thereconstruction methods and of the accelerating voltage doesnot make any significant difference. This is an essentialresult as the experiments can be performed at relatively lowvoltages and with only modest coherence. It was shownthat at larger thicknesses the reconstruction method whichrequires a high degree of coherence yields images whichshow the phosphorus columns very clearly and partly alsothe indium columns, though very weakly.

References

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Coene W, Janssen G, Op de Beeck M, Van Dyck D(1992) Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron micros-copy. Phys Rev Lett 69: 3743-3746.

Doyle PA, Turner PS (1968) Relativistic Hartree-FockX-ray and electron scattering factors Acta Cryst A24: 390-397.

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Hoppe W (1969b) Beugung im InhomogenenPrimärstrahlwellenfeld. III. Amplituden- und Phasen-bestimmung bei unperiodischen Objekten. (Diffraction inan inhomogenous primary beam wavefield. III. Determinationof amplitudes and phases in non-periodic objects). ActaCryst A25: 508-514.

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Kawasaki T, Tonomura A (1992) Direct observationof InP projected potential using high-resolution electronholography. Phys Rev Lett 69: 293-296.

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Lichte H (1992) Electron Holography. I. Can electronholography reach 0.1 nm resolution? Ultramicroscopy 47:223-230.

McCallum BC, Rodenburg JM (1993) Error analysisof crystalline ptychography in the STEM mode. Ultrami-croscopy 52: 85-99.

Nellist PD, McCallum BC, Rodenburg JM (1995)Resolution beyond the “information limit” in transmissionelectron microscopy. Nature 374: 630-632.

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Plamann T, Rodenburg J M (1994) Double resolu-tion imaging with infinite depth of focus in single lensscanning microscopy. Optik 96: 31-36.

Plamann T, Rodenburg J M (1998) Electron ptycho-graphy II: Theory of three-dimensional propagation effects.Acta Cryst A54: 61-73.

Reid J (1983) Debye-Waller Factors of zinc-blende-structure materials - a lattice dynamical comparison. ActaCryst A39: 1-13.

Rodenburg JM, Bates RHT (1992) The theory ofsuper-resolution electron microscopy via Wigner-distri-bution deconvolution. Phil Trans Roy Soc Lond A339: 521-553.

Self PG, O’Keefe MA, Buseck PR, Spargo AEC (1983)Practical computation of amplitudes and phases in electrondiffraction. Ultramicroscopy 11: 35-47.

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Discussion with Reviewers

P.W. Hawkes: In the comments on coherence, I think thatspatial coherence (corresponding to source size) is meant.How will the conclusions be affected by temporalcoherence?Author: The effects of temporal coherence can be thoughtof as a rapid variation of defocus. In the midpoint of theoverlap region between two diffraction discs, in which thephase is measured, the defocus term cancels as well as allother centro-symmetric terms of the aperture function [seeEquation (4) for µ′ = G+H/2]. As a consequence, temporalcoherence does not affect the measured phase, as long asthe pixel size of the detector is chosen fine enough.

P.W. Hawkes: What specimens will be studied in the nextstage of the project?Author: Previous experiments on silicon and galliumphosphide have been very successful. It would now bevery interesting to apply the technique to materials withlarge unit cells. It is clear that for a given reflection, severalroutes can now be taken to determine its phase. Theredundancy in phase information should give a means ofreducing the error of the phase estimate.

R. Hegerl: Following the original ideas of W. Hoppe,ptychography should be applied to two-dimensionalcrystals of biological macromolecules. Apart from the muchlarger lattice periods, problems may arise from the need of alow electron dose and the resulting low signal-to-noise ratio.Assuming a tolerable electron dose of say 1000 el/nm2

(summed over all exposures of the specimen area), is there achance to obtain reliable phase information?Author: The applicability of ptychography to biologicalspecimens has yet to await experimental evidence. However,it is worth considering the effect of the sampling of theprobe position. When we record our four-dimensional dataset of microdiffraction patterns |M(µ′, ρ)|2 as a function ofthe probe position ρ, we have to ensure that all the probepositions are equi-distant, but the distance between themcan be chosen quite freely. The aim would be to choose thedistance as great as possible. That reduces the effect ofspecimen damage. However, one has to make sure that thethickness and orientation of the crystal does not changetoo much across the pertaining area. Furthermore, thenumber of probe positions required may be quite small. Onthe other hand, one would probably use neither of thereconstruction methods described above, in which onlycertain points of the microdiffraction plane are considered.In order to account for every electron contained in the data-set, a deconvolution as described in Rodenburg and Bates(1992) could be favorable.