Fundamentals of Focal Series Inline Electron Holography

A. Lubk, K. Vogel, D. Wolf, J. Krehl, and F. RöderTriebenberg Laboratory, Institute of Solid State Physics,

Technische Universität Dresden, 01062 Dresden, Germany

L. Clark, G. Guzzinati, and J. VerbeeckEMAT, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium

The loss of the quantum phase information in the measurement process constitutes a fundamentallimitation to transmission electron microscopy as the electron wave’s phase often contains valuableinformation about the studied specimen. Phase retrieval from focal series is a holographic techniquethat seeks to recover the lost information from a set of images recorded at different defoci. Inspite of its widespread use, in particular for wave reconstruction at atomic resolution, a number offundamental properties, e.g., regarding the conditions for a unique wave function reconstruction,the magnitude of the reconstruction error and the influence of inconsistencies or incomplete dataare not well-understood as of today. Here, we elaborate on the fundamentals of the technique,making extensive use of the tomographic representation of a focal series as tilt series in phase space.Using this perspective we discuss, among others, requirements on the focal series for a uniquereconstruction, such a the focus interval ranging from the far field at underfocus to the far field atoverfocus or the focus step size. We reveal that the prominent Gerchberg-Saxton iterative projectionalgorithm corresponds to a numerical integration of the quantum Hamilton-Jacobi equation in thesmall focus step limit. Moreover, we show that the topology of the starting guess divides the solutionspace of the Gerchberg-Saxton algorithm into equivalence classes, which mitigates the impact ofthe incompleteness of typical focal series data. To facilitate a focal series reconstruction meetingthe above theoretical requirements such as the long range focus interval, we develop a dedicatedcalibration procedure facilitating the determination of unknown electron optical parameters suchas the focal length of the principal imaging lens or the position of object and image planes. Thefindings are demonstrated with an example of a focal series reconstruction of an electron vortexbeam.

Keywords: Electron Holography, Quantum State Tomography, Focal Series Phase Retrieval

CONTENTS

I. Introduction 1

II. Thin Lens Imaging 3

III. Experimental Implementation 6A. Weak Magnetic Lens Approximation 6B. Calibration of Focal Series 7C. Numerical Propagation Algorithms 9

IV. Long Range Focal Series Reconstruction 9

V. Case Study 14

VI. Summary and Outlook 16

VII. Acknowledgment 17

References 17

A. Calibration of a Thick Lens 19

I. INTRODUCTION

Focal series wave reconstruction in the TransmissionElectron Microscope (TEM) is a well-established holo-

graphic technique for mapping electric, magnetic andstrain fields (Dietrich et al. 2014, Koch et al. 2010, Songet al. 2013) in solids with nanometer resolution and forstudying atomic configurations at crystal defects or grainboundaries (Allen et al. 2004, Koch 2014, Thust et al.1996). More recently, also the structure of phase vor-tices at caustics has been explored using the technique(Petersen et al. 2013). The schematic optical setup, de-picted in Figure 1, is an extension of Gabor’s original in-line holography (Gabor 1948), which used only one out-of-focus image for the reconstruction of the wave func-tion. Focal series reconstruction does not require anundisturbed reference wave and a biprism like off-axisholography, hence may be carried out at practically everyTEM. Furthermore, it may be conducted under relaxedpartial coherence provided that the latter is well-behavedand well-known in advance (Koch 2008). Moreover, fo-cal series holography possesses an intriguing connectionto quantum state tomography (Lubk and Röder 2015), atechnique that is successfully employed to study mixed(i.e., incoherent) quantum states of matter (e.g., atoms)and light (Breitenbach et al. 1997, Schleich 2001, Smitheyet al. 1993).

These advantages are opposed by ambiguities in the re-constructed wave function (Fienup and Wackerman 1986,Luke et al. 2002), e.g., due to inconsistent and incom-plete focal series data. Under experimental conditions

2

every focal series is inconsistent due to the presence ofpartial coherence (e.g., from the electron gun, inelasticinteraction and thermal diffuse scattering (Rother et al.2009)) or shot and detector noise (Niermann et al. 2012)as well as geometric and chromatic aberrations depend-ing on the defocus. Similarly, every focal series is in-complete because of a limited number and range of de-focus values, typically limited to the near field regime,and the restriction to isotropic foci, whereas astigmaticfoci are required for an unambiguous reconstruction ofa wave function (Lubk and Röder 2015). For instance,the problematic reconstruction of low spatial frequencies(Niermann and Lehmann 2016, Ophus and Ewalds 2012,Thust et al. 1996) can be traced back to missing focalseries data in the far field.

The design of reconstruction algorithms behaving wellin the presence of inconsistent and incomplete data in-cluding their characterization in terms of reconstructionerrors is highly non-trivial and subject to large and on-going efforts. Moreover, the accurate calibration of cru-cial experimental parameters such as the precise defo-cus values or aberrations and distortions in the seriesremain challenging. Thus, in spite of the relative ex-perimental convenience of this technique, the majorityof quantitative electric, magnetic and strain field studiesare conducted by means of off-axis electron holography(see (Dunin-Borkowski et al. 2004, Kasama et al. 2011,Lichte et al. 2013, McCartney et al. 2010, Pozzi et al.2014, Tonomura 1987, Völkl et al. 1999) and referencestherein), which has a less problematic linear and uniquereconstruction procedure including well-defined error es-timates (Lenz 1988, Röder et al. 2014).

In the following, we elaborate on analyzing focal se-ries reconstruction from the perspective of quantumstate tomography and use the obtained results to in-crease the scope of the technique in terms of convergenceand uniqueness in particular for low spatial frequencies.Moreover, we are able to explain a number of previousresults by exploiting the phase space analogy, and openpathways to further improvements.

The history of focal series reconstructions goes backto the first half of the 20th century, when Pauli raisedthe question, whether the amplitude of a complex (wave)function and of its Fourier transform completely definethe underlying function (Pauli 1933). This phase re-trieval problem can be considered as one particular in-stance of a reconstruction from a focal series consist-ing only of one in-focus and one far-field image. In thewake of finding whole classes of functions (up to constantphase factors) sharing the same amplitude both in posi-tion space and Fourier space (Jaming 2014, Luke et al.2002), the prospects of such reconstructions have beenconsidered rather pessimistic. For instance, if the Fouriertransforms are symmetric F {Ψ} (k) = F {Ψ} (−k), thecorresponding amplitudes are invariant under complex

objective lens

sample

condenser

source

acquisitionplanes

f

image plane

Figure 1. Principle optical setup of focal series inline hologra-phy. Accordingly, the defocus interval is ideally bounded bythe far field located below (i.e., in focal plane f) and abovethe image plane.

conjugation, i.e.,

|Ψ (x)| = |Ψ∗ (x)| (1)|F {Ψ} (k)| = |F {Ψ∗} (k)| .

In a seminal paper Gerchberg and Saxton, however,proposed a simple iterative projection algorithm, whichproduces surprisingly sensible wave reconstruction fromtwo-dimensional image and diffraction intensities (Ger-chberg and Saxton 1972). Subsequently, modificationsand extension of the original Gerchberg-Saxton algorithm(Boucher 1980, Cederquist et al. 1989, Combettes andTrussell 1990, Dong et al. 1997, Fienup 1982, Gonsalves1976, Levi and Stark 1984, Misell 1973, Ohneda et al.2001, Quatieri and Oppenheim 1981, Stark and Sezan1994, Takajo et al. 2002, Yang et al. 1994, Zou and Un-behauen 1997) have been successfully applied in a largenumber of different fields ranging from measuring theaberrations of telescopes (Baba and Mutoh 2001, Fienupet al. 1993, Lyon et al. 1997) to phase retrieval in elec-tron microscopy (Allen et al. 2004, Allen and Oxley 2001,Koch 2008, 2014). It is interesting to note that in par-ticular the unfortunate initial manufacturing error of theHubble space telescope’s optics (Burrows et al. 1991) andits subsequent characterization (Burrows 1991, Fienupet al. 1993, Lyon et al. 1997, 1991) lead to a boost in re-search on wave front reconstruction algorithms and theirproperties, which exhibits a remarkable parallel to Ga-bor’s invention of holography that had been stimulated

3

by the search for aberration corrected electron optics inthe TEM (Gabor 1948).

Fienup established the connection between theGerchberg-Saxton algorithm (Gerchberg and Saxton1972) and line-search methods. Indeed, the Gerchberg-Saxton algorithm is closely related to the discipline ofconvex optimization (Bauschke et al. 2002, Levi andStark 1984), where similar algorithms, referred to as pro-jection on convex sets or alternating projections, are em-ployed. Within the convex setting, these algorithms arewell-behaved and well-understood in their convergencebehaviour. However, the Gerchberg-Saxton algorithmoperates on non-convex sets, as will be discussed below.Thus, there remain a number of properties, which re-main not well understood till today, in spite of the largeprogress made in the field of focal series reconstructions.For instance, the theoretical results on the convergenceof convex projection algorithms do not apply to the prob-lem of phase retrieval from focal series (Combettes 1996,Luke et al. 2002, Seldin and Fienup 1990), resulting installing iterations or reconstruction artifacts. Therefore,a large number of studies has been concerned with theinvestigation of constraints on the reconstructed functionspace narrowing the number of potential solutions (e.g„pertaining to a different starting guess in the Gerchberg-Saxton algorithm), such as limited supports in positionor Fourier space (Hayes et al. 1980, Marchesini et al.2015) or the analyticity of the wave function (Huiser et al.1976, Huiser and Ferwerda 1976, Huiser et al. 1977). Im-portant aspects of these studies are whether the studiedconstraints are applicable to the experimental situation(e.g., taking into account inconsistent (noisy) data) andhow these constraints can be implemented in actual re-construction algorithms.

A particularly useful perspective is provided by lift-ing the phase retrieval problem into quantum mechani-cal phase space, where it corresponds to a matrix com-pletion problem in the discrete setting (Candès et al.2013), because the sets possessing the same image in-tensity are convex in phase space. Moreover, it is wellunderstood that a focal series corresponds to a tilt seriesin quantum mechanical phase space (Lubk and Röder2015, Raymer et al. 1994). Thus, the focal series re-construction translates to a tomographic reconstructionof the underlying phase space distribution. In the fol-lowing we explore this analogy and its manifold conse-quences for the Gerchberg-Saxton type reconstruction al-gorithms. Most importantly, we will show that the focalrange needs to extend over both near and far field tofacilitate a unique reconstruction. We implement sucha long-range focal series reconstruction using a variantof the Landweber iteration, well-known from (quantumstate) tomography (Natterer and Wübbeling 2001). Thesignificance of the large focal range for a unique recon-struction is also proven by showing that the Gerchberg-Saxton algorithm is closely related to the numerical inte-gration of the quantum Hamilton-Jacobi equation. More-over, we discuss several boundary conditions narrowing

the under-determined focal series reconstruction from alimited number of isotropic defocused images, such astopology or regularity constraints.

This article is organized as follows. We begin with abrief discussion of imaging with a thin lens, both waveoptically as well as in the phase space setting, therebyproviding the basis for the subsequent focal series recon-structions. We proceed with discussing some crucial ex-perimental aspects pertaining to the recording of a longrange focal series. Subsequently, we elaborate on theGerchberg-Saxton reconstruction algorithm. Finally, weperform a case study using a higher-order electron vorexbeam, not in the least because this topic has garneredmuch interest and development in the recent years. Suchbeams possess a non-trivial topology by design, and aretherefore nicely suited to discuss the crucial impact of(implicit) topology constraints. The phase retrieval ofsuch waves has proved elusive because of missing undis-turbed plane references for off-axis holography or non-trivial boundary conditions in direct reconstruction tech-niques such as the Transport of Intensity (TIE) phaseretrieval (Lubk et al. 2013b, Teague 1983).

II. THIN LENS IMAGING

We begin with introducing one particular representa-tion of quantum mechanical phase space - the Wignerfunction, which is computed from a wave function Ψ (astate vector) according to the following definition (e.g.(Schleich 2001))

W (r, k) :=1

2π

ˆ ∞

−∞Ψ∗(r − 1

2r′)

Ψ

(r +

1

2r′)e−ikr

′dr′

(2)

=1

2π

ˆ ∞

−∞Ψ∗(k − 1

2k′)

Ψ

(k +

1

2k′)eik′rdk′ ,

where the normalization factor 1/ (2π) ensuresˆ ∞

−∞dr

ˆ ∞

−∞dkW (r, k) = 1 . (3)

It is important to note that the above definition is just aspecial case of the more general

W (r, k) :=1

2π

ˆ ∞

−∞dr′ρ

(r +

1

2r′, r − 1

2r′)e−ikr

′, (4)

where any mixed quantum state (i.e., incoherent superpo-sition of pure state wave functions) represented by somedensity matrix

ρ

(r +

1

2r′, r − 1

2r′)

=∑n

Ψ∗n

(r − 1

2r′)

Ψn

(r +

1

2r′)(5)

is allowed. The parameter space (r, k) of the Wignerfunction defines a quantum mechanical phase space, with

4

W (r, k)

r

k

r

ρ(r) =´∞−∞W (r, k)dk

I(r)

k

ρ(k)=´

∞ −∞W

(r,k

)dr

I(k)

Figure 2. A Wigner function and its marginal densities.

coordinate k being the momentum coordinate. The twodimensional version required for electron optics followsin a straightforward manner by replacing the phase spacecoordinates r and k with 2D vectors r and k.

The definition (2) implies that the Wigner functionis strictly real. Projecting the Wigner function along k

yields a strictly positive quantity, which can be identifiedas the quantum mechanical density in position space

ρ (r) =

ˆ ∞

−∞W (r, k) dk . (6)

The same holds for Fourier space, i.e.,

ρ (k) =

ˆ ∞

−∞W (r, k) dr . (7)

The latter projection along r could be alternatively un-derstood as a rotation of phase space around 90° with asubsequent projection along the vertical axis (Figure 2).The generalization of this concept to arbitrary tilt anglesleads to the notion of fractional Fourier transforms (e.g.,(Almeida 1994, Ozaktas and Mendlovic 1995)), which isclosely related to the free space propagation in the parax-ial regime (Fresnel propagation) discussed subsequently.

In the following, we only require the behaviour of theWigner function under free space propagation. Withinthe paraxial regime it may be obtained by applying theFresnel propagator to the wave function (propagationlength z, wave number k0) and inserting the propagatedwave function in the above definition

Wδz(r,k) =1

4π2

ˆ ∞

−∞d2k′ei

δz2k0

(k− 12k′)

2

Ψ∗(k− 1

2k′)

Ψ

(k +

1

2k′)e−i

δz2k0

(k+ 12k′)

2

eik′r (8)

=1

4π2

ˆ ∞

−∞d2k′Ψ∗

(k− 1

2k′)

Ψ

(k +

1

2k′)eik′(r− δzk0

k)

= W0(r− δz

k0k,k) .

This result, also holding for mixed states, exhibits a re-markable simplicity. The paraxially propagated Wignerfunction is merely a sheared version of its initial stateW0, which is exactly the behaviour of a classical phasespace ensemble of particles with position r and momen-tum k. In other words, the quantum nature of the sys-tem is solely encoded in the underlying Wigner quasi-probability distribution.

It is now crucial that one may transform this shear intoa rotation by suitably shearing (along k) and rescalingthe propagated Wigner function. Note that these trans-formations do not affect the projections along k, i.e., therecorded intensities. Indeed, a focal series is nothing elsethen a scaled series of intensities of freely propagatedwaves (see below), and we may conclude that a focal se-ries reconstruction corresponds to a tomographic recon-struction of the Wigner function, referred to as quantumstate tomography (Raymer et al. 1994) (Figure 3). Be-cause the Wigner function describes both pure quantumstates (i.e., wave function) and mixed quantum states,

this is not just a mere reformulation of focal series phaseretrieval of wave functions in phase space. Indeed, quan-tum state tomography may be employed to reconstructquantum states of any coherence and the restriction towave functions (pure states) is given by constraining thephase space volume of the quantum state according to

4π2

¨ ∞

−∞d2rd2kW 2 (r,k) = 1 . (9)

Because the latter integral constraint only marginally re-duces the infinitely-dimensional space of quantum statesby one dimension, important properties of phase spacetomography, e.g., pertaining to the required focal range,may be directly transferred to focal series reconstruc-tion of wave functions, where they are much less obvi-ous. Most importantly, the 180° tilt range required for aunique phase space reconstruction translates into a focusinterval ranging from the far field at underfocus via thein-focus plane to the far field at overfocus. It is further-more readily verified in phase space that both far fields

5

underfocussh

ear

seri

es

r

kin focus

defocus

overfocusti

ltse

ries

rs

ks

far field ∼= −90◦

defocus

far field ∼= 90◦

Figure 3. Inline holography as quantum state reconstruction.The shear series of the defocused Wigner function is shownin the first row. The corresponding tilt series obtained bysuitably scaling the phase space is depicted in the second row.

(at over- and underfocus) may be identified by invertingthe spatial coordinates.

As we are dealing with a four-dimensional phase space

in general, the projection geometry is more complicatedthan the simple single-tilt-axis geometry (where the setof projections is referred to as Radon transform) typicallyconsidered in tomography (including two-dimensionalphase space tomography). In particular, each individ-ual defocused image is obtained by projecting along a2D plane in phase space, tilted by the defocus. Sucha projection transformation represents a generalizationof the 2D Radon transformation, referred to as K-planetransform K4

2 (e.g., (Keinert 1989)), if the projectionsare performed over all 2D planes in 4D phase space. Torealize arbitrary shears in 4D phase space electron opti-cally, line defoci (where one direction remains in focus)with varying orientation have to be realized.

We will now discuss the scaling of the focal series,i.e., the relation between free space propagation and theexperimental images in the series, for imaging with athin lens. The thick lens case, required further below,is treated in Appendix A. Following Abbe, the object,back focal plane and image plane of a single lens are con-nected in an ideal (aberration-free) system wave opticallyby means of a Fourier and an inverse Fourier transforma-tion, respectively. This result may be derived by propa-gating a wave given at some object distance zobj througha thin lens with focal length f (see Figure 4)

Ψ (r, z) =1

8π3

˚ ∞

−∞d2r′d2k′d2kΨ̃ (k′, zobj) e

−izobj2k0

k′2ei(k′−k)r′e−i

k02f r′2e−i

z2k0

k2

eikr (10)

=f

4π2ik0

¨ ∞

−∞d2kd2k′Ψ̃ (k′, zobj) e

−izobj2k0

k′2eif

2k0(k−k′)

2

e−iz

2k0k2

eikr

=f

2π (f − z)e−i

k02(f−z) r

2ˆ ∞

−∞d2k′e

i2k0

zobjz − f (zobj + z)

f − z︸ ︷︷ ︸δz(z)

k′2

Ψ̃ (k′, zobj) e

if

f − z︸ ︷︷ ︸M−1

rk′

=ik0Mδz

e−ik02M ( 1

f+1δz )r2

F−1{

Ψ (r, 0) e−ik02δz r

2}( k0

Mzr

).

In the second transformation we defined an effective de-focus δz, i.e., the corresponding propagation length infree space, and a magnification M , describing the rescal-ing with respect to a freely propagated wave. Insertingthe wave function propagating through a thin lens intothe definition of the Wigner function (2) and taking intoaccount (8), we obtain a succession of shears along theposition and the momentum coordinate in phase spaceas depicted in Figure 4. Again, this shear series may betransformed into a tilt series by suitably rescaling phasespace.

It follows that the diffraction pattern ρ =∣∣∣Ψ̃∣∣∣2 occurs

in the back focal plane (z = f) because (starting from

the second line in Eq. (10))

Ψ (r, f) =f

i4π2k0

ˆ ∞

−∞d2k′ei

f−zobj2k0

k′2Ψ̃ (k′, zobj) (11)

×(ˆ ∞

−∞d2ke

i(r− f

k0k′)k)

=f

ik0

ˆ ∞

−∞d2k′δ

(r− f

k0k′)eif−zobj

2k0k′2Ψ̃ (k′, zobj)

=1

ieik0(f−zobj)

2f2 r2

Ψ̃

(k0fr, zobj

).

Eq. (10) indicates three ways to change the effectivedefocus δz (facilitating a focal series) by varying zobj, for zimg, respectively. However, the object plane (i.e., thespecimen or some conjugated plane) and the image plane(i.e., the detector plane or some conjugated plane) remain

6

z

x

k

x

k

x

k

x

k

x

k

(a) object plane

f-fzobj

zimg

(e) image plane

(b) pre-lens

(d) back focal plane

(c) post-lens

field ray

axial ray

Figure 4. Imaging at a thin lens in phase space. Accordingly,the initial rectangular phase space distribution in the objectplane (a) passes through several (affine) shear transformations(b)-(d) yielding a 180° rotated and, along the momentum co-ordinate, sheared version in the image plane (e).

fixed in a typical TEM. Thus, using a single lens, focalseries are usually recorded by varying the lens excitation,hence f . It is important to note that by varying thefocal length f not only in the effective defocus δz butalso in the magnification M changes. This rescaling hasto be taken into account, when reconstructing the wavefunction from an experimental focal series. An accuratecalibration of the focal series in terms of defocus andmagnification are absolutely crucial therefore. This is aserious challenge and we present a dedicated calibrationprocedure adapted to the above focal series recorded withthe diffraction lens later on.

III. EXPERIMENTAL IMPLEMENTATION

As of today, the recording of a focal series rangingfrom the far field at underfocus via the focal plane tothe far field at overfocus that is free from additionalmodifications such as magnification changes, rotations,distortions or incoherent aberrations represents a seriouschallenge. Several authors noted the importance of tak-ing into account these imaging errors in the focal seriesreconstruction (Koch 2014, Meyer et al. 2004, 2002).

To mitigate some of the noted obstacles, we focus onmedium resolution focal series in the following. This al-lows us to use a low magnification mode, where the fo-cal series is facilitated by changing the excitation of thediffraction lens (Figure 5). In a first step we will treat

electron source

1st condenser lens

2nd condenser lenswith aperture

upper objective lens

lower objective lens

diffraction lens

projector lens(es)

objectplane

nearfield

back-focalplane

farfield

objective aperture

1st imageplane selected area aperture

detector plane

Figure 5. Conventional bright field TEM (CTEM) schemewith the electron source located in the far field (Fourier plane)of the object plane, which in turn is conjugated to the detectorplane. Conjugated object planes are indicated by arrows. Themedium resolution focal series considered here was recordedby varying the diffraction lens (highlighted) within the lowmagnification mode of the TEM.

the diffraction lens as a weak (thin) magnetic lens, whichgreatly simplifies the optics and the derivation of the per-tinent equations linking the change of the lens excitationat the microscope with an actual defocus. Based on that,we discuss the crucial calibration of the focal series even-tually also considering thick lenses. We illustrate thevarious steps at the example of a Philips CM200 TEMoperated at 200 kV.

A. Weak Magnetic Lens Approximation

In order to calibrate the focal length as well as theobject and image plane coordinates, we shall start withthe weak lens approximation before discussing a morerealistic optical model. For a thin lens the focal lengthsand focal planes coincide and the principal planes fall

7

into the symmetry plane of the lens. The focal lengthof a thin round magnetic lens is given by the integralof the squared magnetic induction along the optical axis(Hawkes 2013)

1

f=

e

8meU∗

ˆ ∞

−∞B2z (z) dz (12)

with the relativistically corrected acceleration voltagegiven by

U∗ = U

(1 +

eU

mec2

). (13)

Similarly, the image rotation ϕ is given by the integral ofthe induction

ϕ =e

2mev

ˆ ∞

−∞Bzdz (14)

with the electron velocity denoted by v.To evaluate the above integrals, we need to know the

magnetic induction inside of the diffraction lens. Inthe absence of accurate magnetic field configurationsfrom the manufacturer, we employ the GRAY model(α = 2.636) for the magnetic induction in the pole piece(diameter D, gap S)

Bz (z) =µ0I

2S

(tanh

(αS

2D

(1 + 2

z

S

))(15)

+ tanh

(αS

2D

(1− 2

z

S

))),

which provides a rather accurate approximation forweakly excited round magnetic lenses (Hawkes 2013). Inthe GRAY model the inverse focal length is proportionalto the squared lens excitation (lens current I in Ampere-turns) with the proportionality constant Cf

1

f=

eµ20

8meU∗2S

(1 + tanh2

(αS2D

)2 tanh

(αS2D

) − D

αS

)I2 = CfI

2 .

(16)Thus, we only have to determine Cf in our calibrationprocedure in order to know the focal length f as a func-tion of the lens excitation. The image rotation on theother hand is linearly proportional to I according to

ϕ =eµ0I

2mev. (17)

We use this dependency to calibrate the excitationstrength displayed by the TEM control interface in termsof Ampere-turns by measuring the image rotation at dif-ferent excitation strengths and solving Eq. (17) for I.Figure 6 exhibits the linear increase of the image rota-tion with the lens current in the diffraction lens at ourPhilips CM200 TEM operated at 200 kV. Accordingly,at 100% excitation strength the current amounts to 4904Ampere-turns.

1962 2942 3924 4904

diffraction lens current [A-turns]

0.3 0.4 0.5 0.6 0.7 0.8 0.9 140

60

80

100

120

140

160

diffraction lens excitation

imagerotation

[◦]

Figure 6. Image rotation as a function of the diffraction lensexcitation. Note the linear increase in agreement with Eq.(17).

B. Calibration of Focal Series

The principal part of the calibration procedure consistsof three measurements. First, we record two in-focus im-ages of a Selected Area (SA) aperture (radius 50 µm) anda copper grid in the object plane (objective lens switchedoff) of known size and determine their respective magni-ficationsM1,2. These magnifications are composed of thepost-magnification of the projector lenses and the actualmagnification of the Dif lens according to

M1,2 = MprojM(1,2)dif (18)

= Mprojf1,2 − zimg

f1,2

= Mproj

(1− CfI21,2zimg

),

which can be solved for

Cfzimg =M1 −M2

M1I22 −M2I21(19)

and the post-magnification of the projector lenses

Mproj =M1,2

1− CfI21,2zimg. (20)

To calibrate the remaining unknown optical parame-ters, namely Cf , zimg and zobj, we record a referencefocal series of a small grating (lattice constant d = 100nm, objective lens switched off) because the correspond-ing large far field permits the following approximation fora large range of lens excitations

Ψ (r, z) =1

2πMe−i

k02(f−z) r

2

(21)

׈

d2k′ei

2k0δzk′2Ψ̃ (k′, zobj) e

iM−1rk′

SPA∼ Ψ̃

(k0Mδz

r, zobj

).

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-12

-1

-8

-6

-4

-2

0

2

diffraction lens excitation

refle

ctiondistan

ce[m

m]

experimentthin lens fitthick lens fit

Figure 7. Distance between systematic reflections as a func-tion of the lens excitation

In the last line we used the stationary phase approxima-tion (SPA) assuming an effective defocus large enough tobe in the far field of the grating. Consequently, trackingthe difference s = rn− rn−1 between adjacent diffractionorders (q = 2πn/d)

s = Mprojλ

d

(zobjzimg − f (zimg + zobj)

f

)(22)

= Mprojλ

d

(zobjzimgCfI

2 − zimg − zobj)

= Mprojλ

d

(zobj

M1 −M2

M1I22 −M2I21I2 − zimg − zobj

)permits the fitting of the unknown parameters zobj andzimg. Figure 7 shows the dependency of the distance sto the lens excitation I. Accordingly, we obtain a goodagreement with the predicted quadratic behaviour forweak lens excitation only. Combining (22), (19) and (20)we can compute Cf and finally the effective defocus

δz (I) =zobjzimg − f (I) (zobj + zimg)

f (I)− zimg(23)

and magnification

M (I) =f (I)− zimg

f (I)(24)

within a long-range focal series recorded by varying theexcitation of the diffraction lens.

Figure 7 exhibits that the propagation of the wavewithin the magnetic field of the lens, i.e., thick lens ef-fects, cannot be neglected anymore for higher excitations.In Appendix A we thus consider the propagation througha thick lens modeled as a sum of two refracting surfacesat a distance

D (I) = CDI (25)

depending linearly on the lens current, and derive the cor-responding quantities measured in the above calibration

parameter value

zO 18.3 cm

zSA 3.6 cm

zimg 47.3 cm

CD 7.9µm/A

Cf 2.8 (mA)−2 /m

Mproj 88

Table I. Table of optical parameters of the diffraction lens ina Philips CM200 TEM operated in a low magnification mode.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

diffraction lens excitation

focallen

gth[m

]

Figure 8. Focal length calibrated from the calibration experi-ments using the thick lens model for the fitting of the opticalparameters.

experiments (see Tab. I). Such a thick lens model yieldsvery good agreement for the distance between the diffrac-tion orders as a function of the lens excitation (see Fig-ure 7 ). The calibrated optical parameters of our PhilipsCM200 TEM are in the expected range although estima-tions of the distance between diffraction lens and objectplane suggest an error in the range of 20%. The lattertranslates to a deviation between the predicted and theexperimentally observed position of the in-focus plane inthe actual focal series, which has to be taken into accountthe when discussing the reconstruction results.

We conclude that the characterization of the opticalsystem and hence the parameters of the focal series ob-tained from the above procedure necessarily representan approximation because of several assumptions, whichmight be violated to some extent in reality. For instance,the quadratic dependency of the reciprocal focus and thelinear dependency of the thickness of the lens from thelens current (16,25) are expected to break down at higherexcitations. Similarly, very weak excitations may be af-fected by hysteresis effects, disturbing the proportional-ity between I2 and f−1 as the lens is operated outside ofits typical regime. More precise models, however, wouldrequire ray tracing simulations within realistic magneticfields based on the exact geometry of the lenses.

9

0.2 0.4 0.6 0.8 1−2

−1

0

1

diffraction lens excitation

effective

defocus

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−20

−10

0

diffraction lens excitation

effective

magnification

Figure 9. Effective defocus and magnification calibrated fromthe calibration experiments using the thick lens model for thefitting of the optical parameters.

To account for the inaccurate calibration, we there-fore implement a self-consistent adaption of the effectivemagnification in the focal series reconstruction, similarlyto (Koch 2014). Note, however, that such a proceduremight also modify the reconstructed wave function andshould be only employed to correct for calibration er-rors. Alternatively, it is possible to fit the effective defo-cus and magnification by comparing a reference series ofsome well-defined beam shape (e.g., plane wave througha round aperture) to a simulation.

C. Numerical Propagation Algorithms

When performing a focal series reconstruction fromdata covering both the near and the far field the tworegimes must be treated separately numerically to avoid aspread of the image beyond the simulated domain, whilemaintaining both a sufficient accuracy and speed in thenumerical Fresnel propagation. In the near field regime,the numerical Fresnel propagation is performed in Fourierspace exploiting both the small slope of the quadraticFresnel phase (i.e., slow spreading) and the speed of theFast Fourier transform:

Ψ (r, z) = F−1{ei

z2k0

k2

F {Ψ (r, 0)}}. (26)

When entering the far field regime (characterized by theFresnel number F = a2/ (λz) � 1 with a the extent ofthe wave function), the reconstruction algorithm directlyevaluates the convolution with the position space Fresnelpropagator according to (García et al. 1996)

Ψ (r, z) =1

iλzeik02z r

2

ˆ ∞

−∞Ψ (r′, 0) ei

k02z r′2e−i

k0z rr′d2r′(27)

=2π

iλzeik02z r

2

F{

Ψ (r, 0) eik02z r

2}(k0

zr

).

By separating the different r and r′ dependencies we canuse Fast Fourier transformations for efficiently comput-ing the above convolution. Note, however, that the latter

implicitly rescales the defocused wave function by k0/z,counteracting the spread upon free propagation. As therescaling is generally different from the spread of the wavefunction, the sampling has to be chosen carefully to en-sure an adequate scaling. In order to facilitate a matchingscale between experimentally recorded images and sim-ulated ones, both the near field and the far field wavefunctions computed by means of (26) and (27) need tobe rescaled with M to fit to the experimental intensityrecorded in the corresponding plane.

IV. LONG RANGE FOCAL SERIESRECONSTRUCTION

In the following, we consider the focal series wave re-construction as a quantum state tomography restrictedto pure states (i.e., wave functions). Note, however, thata certain violation of the pure state assumption, e.g.,due to inelastic scattering at the sample, the partial co-herence of the electron emitter, or the point spread ofthe detector (Niermann et al. 2012), can never be com-pletely avoided, and must therefore be taken into accounttherefore (Martin et al. 2006).

In a hardware-corrected TEM or at medium (nanome-ter) resolution the defocus remains as the main sourceof aberrations. Thereby, we assume that the effects oftemporal incoherence may be sufficiently suppressed dueto chromatic aberration correction and energy filtering.In order to compute the impact of a finite source sizeon the Wigner function in the presence of defocus only,it suffices to incoherently sum the underlying coherentWigner function over the transverse momentum distribu-tion of the source ρS (k)(∼=effective source size) to obtainthe corresponding mixed quantum state

W (r,k) =

ˆ ∞

−∞W (coh)(r,k− k′)ρS (k′) d2k′ , (28)

which is defocused in a second step

W (r,k, z) = W (r− z

k0k,k, 0) (29)

=

ˆ ∞

−∞W (coh)(r− z

k0k,k− k′, 0)ρS (k′) d2k′ .

This convolution commutes with the projection of theWigner function along the momentum coordinate (i.e.,the measurement) according to

ρ (r, z) =

ˆ ∞

−∞W (r,k, z)d2k (30)

=

¨ ∞

−∞W (coh)(r− z

k0k,k− k′, 0)ρS (k′) d2kd2k′

=

ˆ ∞

−∞ρ(coh)(r− z

k0k′, 0)ρS (k′) d2k′ ,

hence can be inverted (deconvoluted) to a certain extentdirectly in the recorded image within the limits deter-mined by noise (Koch 2008, Lubk and Röder 2015). From

10

r

k

ρ(r)

W (coh)

Figure 10. Impact of partial spatial coherence in combinationwith defocus on the average mixed quantum state. Spatialincoherence in combination with a defocus leads to a convo-lution in phase space, which may be deconvolved from therecorded image therefore.

a phase space perspective, such a deconvolution corre-sponds to a purification of the quantum state, as thepartial trace over a subspace of the total configurationspace (i.e., the incoherent emitter surface) is reverted.

Because of its particular link to fundamental laws ofquantum mechanics and its preeminent practical signif-icance, we focus in the following on the archetypical re-construction algorithm in the field, the Gerchberg-Saxtonalgorithm. The latter consists of propagating a trial wavefunction into the various experimental focal planes, whereits modulus is iteratively updated to the experimentalvalue. In the limit of an infinitesimally small focal stepsize, we may derive the following analytical descriptionof this iterative procedure. We start be writing down the

paraxial Klein-Gordon equation in field-free space

i∂zΨ(r, z) = − 1

2k0∆Ψ(r, z) , (31)

which adequately describes the propagation of an elec-tron beam of wave vector k0 along the optical axis zin the TEM. The structure of this equation correspondsto a two-dimensional time-dependent Schrödinger equa-tion with z taking the place of the time coordinate. Infull analogy to the time-dependent Schrödinger equa-tion one can rewrite (31) as a coupled system of twodifferential equations for the wave’s probability densityρ = ΨΨ∗ = A2 and the phase ϕ

∂ρ(r, z)

∂z= − 1

k0∇(ρ(r, z)∇ϕ(r, z))︸ ︷︷ ︸

j

(32)

∂ϕ(r, z)

∂z= − 1

2k0(∇ϕ(r, z))

2+

1

2k0

4A(r, z)

A(r, z)︸ ︷︷ ︸quantumpotential

. (33)

The first line equates the change of the probability den-sity (or intensity) along z with the lateral divergence ofthe probability current j. Consequently, it represents thecontinuity equation for the paraxial regime. In the con-text of the corresponding holographic technique (Lubket al. 2013b, Teague 1983), Eq. (32) is also referred to asTransport of Intensity Equation (TIE). The second equa-tion is the quantum version of the paraxial Hamilton-Jacobi equation (QHJE) differing from the classical oneby the so-called quantum potential.

If we now start the Gerchberg-Saxton iteration witha deliberately chosen guess for the phase Ψ0 (z = 0) =√ρ (r, 0) exp (iϕ0(r, 0)) at the initial plane z = 0, the

propagation over an infinitesimal small defocus step δzat some point z = nδz along the optical axis reads

Ψn(r, z + δz) =

(1 +

iδz

2k0∆

)Ψn(r, z) (34)

=

(1− δz

2k0

(∇A(r, z)∇ϕn(r, z)

A(r, z)− i∆A(r, z)

A(r, z)+ i (∇ϕn(r, z))

2

))A(r, z)eiϕn(r,z)

= e−δz

2k0(∇A(r,z)∇ϕn(r,z)

A(r,z) −i∆A(r,z)A(r,z) +i(∇ϕn(r,z))2)A(r, z)eiϕn(r,z) .

In the last line, we used the smallness of the focal stepδz to write the expression appearing in the bracket ofthe second line as an exponential. Note that this trans-formation is only possible if the amplitude A 6= 0. Thisreflects the fact that a large relative change of the wavefunction may occur upon small propagation steps atpoints, where the amplitude is zero. The Gerchberg-Saxton algorithm proceeds by replacing the amplitudein the plane z + δz with the experimental amplitude

A(r, z + dz) =√ρ(r, z + δz) in that plane

Ψn+1(r, z + δz) = e−iδz

2k0( ∆A(r,z)A(r,z) −(∇ϕn(r,z))

2) (35)

× A(r, z + δz)eiϕn(r,z) .

11

Consequently, the iterated phase reads

ϕn+1(r, z + δz) = ϕn(r, z) (36)

− δz

2k0

(∆A(r, z)

A(r, z)− (∇ϕn(r, z))

2

)= ϕn(r, z) +

∂ϕn(r, z)

∂zδz ,

where we inserted (33) in the last line.The last equation reveals that the ordered Gerchberg-

Saxton iteration with small step size corresponds to the(numerical) integration of the quantum Hamilton-Jacobiequation starting from some deliberately chosen initialwave function. Thus, the Gerchberg-Saxton algorithmmay be considered as conjugate to the phase retrievalbased on solving the Transport of Intensity Equation asthe coupled system of TIE and QHJE is equivalent tothe paraxial Klein-Gordon equation (31). Noting thatany starting guess ϕ0 yields a solution to the above inte-gration, reconstructing the correct wave function under-lying the observed intensities in the various focal planescorresponds to single out the correct starting guess. Thelatter may be accomplished by starting the integrationat the far field at underfocus and integrating until reach-ing the far field at overfocus, because only the correctsolution(s) may be identified in the far field at overfocuswith the corresponding starting guess at underfocus af-ter inversion of spatial coordinates. In other words, thefar field data is necessary to identify the self-consistentsolution(s) (and the pertaining periodic Bohm trajecto-ries forming the characteristics of the quantum Hamilton-

Jacobi equation).Relating the Gerchberg-Saxton algorithm with the nu-

merical integration of the (quantum) Hamilton-Jacobiequation allows us to use the comprehensive Hamilton-Jacobi theory for characterizing the algorithm. Withinthe scope of this work, however, we are content with find-ing an alternative justification for the long focal rangeand leave questions, such as how many self-consistent so-lutions, i.e., how many wave function possess the sameintensities in a long-range isotropic focal series to futurework. Nevertheless, we will encounter an example be-low, where at least two different wave functions (i.e., notdiffering by a mere phase offset only) may not be distin-guished from their intensities in isotropic focal planes.

In practice, the focal step employed in a focal series istoo large to allow for a linear approximation of the propa-gation as in (34). Therefore, the full (Fresnel) propagatorhas to be employed to propagate the wave function fromfocal plane to focal plane. Under these circumstances,the direct interpretation of the algorithm as numericalintegrator of the quantum Hamilton Jacobi equation isviolated to a certain extent and one may not prove in arigorous way anymore that the true solution can be ob-tained by exploring the space of starting conditions. Infact, the larger defocus steps cause the Gerchberg-Saxtoniteration to deviate from the QHJ solution pertaining toone particular starting wave by meandering in a com-plicated way through QHJE solutions corresponding todifferent starting waves while iterating. To get more in-sight into this convergence behaviour we first note thefollowing bound for the distance between iterated wavefunctions

‖Ψn(r, z)−Ψn−1(r, z)‖2 =∥∥∥An(r, z)eiϕn(r,z) −A (r, z) eiϕn(r,z)

∥∥∥2 (37)

=

ˆ

A2n(r, z) +A2(r, z)− 2A2 (r, z)A (r, z) d2r

≤ˆ

A2n(r, z) +A2(r, z)− 2A2 (r, z)A (r, z) cos (ϕn(r, z)− ϕ(r, z)) d2r

=∥∥∥An(r, z)eiϕn(r,z) −A (r, z) eiϕ(r,z)

∥∥∥2 ,

which shows that one iteration step corresponds to an or-thogonal projection on the closest representative from aset of wave functions (with arbitrary ϕ) sharing the samemodulus A with the experimental value in that plane(Fienup 1982). Consequently, the Gerchberg-Saxton al-gorithm can be considered as an iterative projection al-gorithm as used in convex optimization, with the crucialdifference between both settings being the non-convexityof the set of wave functions sharing the same modulusin some optical plane (see Figure 11). This feature pre-vents the transfer of existence and convergence theoremsfrom the convex setting to the typical long-focal step

Gerchberg-Saxton iteration (cf. Figure 12 for a visualrepresentation of the problem). Accordingly, the algo-rithm may fail to converge even if a consistent solution tothe focal series exists (which is typically not the case dueto noise, partial coherence, etc.). Note furthermore thatthe solution depends on the initial guess in the under-determined case (corresponding to incomplete focal seriestypically recorded in the TEM) even in the convex set-ting. These two characteristic properties again show thesusceptibility of the Gerchberg-Saxton algorithm with re-spect to the starting guess, which has to be taken intoaccount in the focal series reconstruction.

12

Ψ ∼ eiϕ Ψ ∼ e−iϕ

Figure 11. Polya plots of two vortex wave functions with thesame amplitude A = |Ψ |, where each vector corresponds toa complex number in the complex plane. The complete setof wave functions with the same amplitude is obtained byarbitrarily changing the orientation of the vectors. Accord-ingly, linear combinations of wave functions from this set,Ψ =

∑i ciΨi with

∑ci = 1, generally do not belong to this

set. Consequently, the set of wave functions with the samemodulus is not convex.

overlapping convex sets

0

1

23

iterationpath

non-overlapping non-convex sets

0

Figure 12. Projection on overlapping convex versus non-overlapping (i.e., inconsistent) non-convex sets. As long asthe convex sets are overlapping the iterative projection con-verges to the intersection of the sets closest to the initial guessof the iteration in the here depicted under-determined case.In the non-convex setting the iteration may get trapped atpoints of close distance between the sets, not reaching other(and possibly better) solutions.

In spite of the above noted complicated convergencebehaviour, the large number of sensible solutions, whichhave been obtained previously, shows that increased focalstep sizes do not necessarily invalidate the reconstructionalgorithm as such. Indeed, various strategies and con-straints, mainly derived from the phase space perspec-tive, are suited to stabilize the algorithm to a certainextend as will be discussed below.

(A) First of all, we note that the phase space per-spective is intimately connected to matrix completionstrategies (Candès et al. 2013), which have been success-fully employed to transfer projections on non-convex sets

to underdetermined projections on convex sets, recently.The crucial point here is that the set of phase space distri-butions possessing the same projection in some directionis convex, which allows us to lift the original Gerchberg-Saxton projection on non-convex sets of complex wavefunctions to convex sets of phase space distributions per-taining to the same modulus in some plane. However,the mixed state formulation of through focus reconstruc-tions (i.e., quantum state tomography) discussed aboverevealed that a set of line foci ranging from the far fieldat underfocus via the in-focus plane to the far field atoverfocus and comprising all possible orientations of theline focus is necessary to obtain a unique quantum statereconstruction. Presently used stigmators do not permitthe acquisition of a line focal series from the near to thefar field, however. Thus, focal series typically compriseonly isotropically defocused images, although it has beennoted that the use of line foci in the near field reduces theambiguity of the reconstruction (Henderson et al. 2009,Petersen and Keast 2007). Furthermore, in spite of sig-nificant progress being achieved in the fields of imageregistration (Meyer et al. 2002, Saxton 1994), aberra-tion correction and aberration assessment (Meyer et al.2002), it remains a formidable challenge to record andregister a long range isotropic defocus series free fromspurious aberrations, distortions, rotations and magnifi-cation changes. To mitigate these issues focal series aretypically recorded in or close to the near field of the ob-ject taking into account that the limited focal range limitsthe focal series reconstruction of low spatial frequencies(Haigh et al. 2013, Niermann and Lehmann 2016). Morerecently, also non-linear defocus variations extending fur-ther into the far field have been employed successfully invarious inline holography studies (Haigh et al. 2013, Koch2014, Song et al. 2013) to increase the reconstructed spa-tial frequency band. Because equal tilt intervals in phasespace correspond to non-equally spaced focal steps a non-linear sampling of the defocus

δz (α) =k0k2σ

(tan (α+ δα)− tanα) (38)

δα�1≈ 2k0

k2σ

1

cos (2α) + 1δα

decreasing towards the far field is optimal in these studiesif seeking a minimal number of images in the focal series.Here kσ corresponds to the characteristic width of thewave function in Fourier space.

(B) To fulfill the support theorem of tomography (Hel-gason 2011), the complete electron wave to be recon-structed should be contained within the field of viewthroughout the entire tilt series. In practice, however,focal series reconstructions, in particular at the atomicresolution regime, are frequently carried out for electronbeams being larger than the recorded field of view, henceviolating the tomographic support theorem. It has beennoted that the corresponding artifacts in the reconstruc-tion may be mitigated to some extend by numericallypadding the intensity images with zeros, i.e., artificially

13

introducing a boundary to the beam (Lin et al. 2006,Ophus and Ewalds 2012).

(C) The alternating projection on convex sets inthe phase space setting corresponds to the Kaczmarz(Kaczmarz 1937) (or Algebraic Reconstruction Tech-nique (Gordon et al. 1970)) algorithm in the parlanceof tomography (Wei 2015). This analogy also suggeststhat a non-convergence of the Gerchberg-Saxton algo-rithm due to ubiquitous inconsistencies in the recordeddata, may be mitigated by combining all Kaczmarz it-erations within one cycle, i.e, by projecting on all focalplanes simultaneously. This strategy is an adaption ofthe Landweber algorithm or Sequential Iterative Recon-struction Technique (SIRT) well-known from tomogra-phy (Natterer and Wübbeling 2001) and forms the ba-sis behind Allen’s improved Gerchberg-Saxton algorithm(Allen et al. 2004). Figure 13 shows a modified version ofthe latter, including the previously mentioned iterationover the starting guess and some additional processingstep, which is used in the course of this work. VariousTEM studies employed similar variants of the Gerchberg-Saxton algorithm and a commercial version of Allen’sIFWR algorithm is available (Allen et al. 2004, Allen andOxley 2001, Koch 2008, 2014).

The above lines show that typically recorded focal se-ries are not sufficient to guarantee a unique and error-freereconstruction, which requires additional care in the in-terpretation of the results, in particular because errorestimates are less straight-forward compared to off-axisholography. Therefore, the application of additional as-sumptions restricting the set of possible quantum statescan be very useful to further improve the quality of thefocal series reconstructions. Which assumptions can beused and how they may be incorporated into efficient fo-cal series reconstruction algorithms represents an activearea of research, bearing close analogies to the regulariza-tion strategies discussed within the context of tomogra-phy and compressive sensing in general. In the followinga short overview of possible strategies is given.

We first note that each step of the Gerchberg-Saxtonalgorithm does not modify the topology of a wave func-tion, which is given by the total winding number

w =1

2π

˛

∂D

ds · ∇ϕ (39)

computed on the outer boundary ∂D of the electronbeam because only the amplitude is updated in the differ-ently defocused planes. Thus, the topology of the start-ing guess for the wave function is preserved throughoutthe iteration (Martin and Allen 2007) and it is importantto endow the starting guess with the correct topology toensure convergence to the true solution. Unless workingwith electron vortex beams (Lubk et al. 2013b, McMor-ran et al. 2011, Verbeeck et al. 2010) the wave functionsmaking up a conventional TEM beam posses a trivialtopology (winding number equals zero), because elasticscattering at the object does not change the topology ofthe initial wave function. Consequently, a wave of trivial

initial wave functionΨ(n=0)(r, z = 0)

propagateΨ

(n)m (r, zm)

compute R-factorR(n) =

∑m

´

| Im(r)−‖Ψ(n)m (r)‖2 | d2r

update (project)Ψm(r) =

√Im(r)e(i arg(Ψ

(n)m (r)))

back-propagateΨ

(n)m (r, z = 0)

assemble averageΨ(n+1)(r) =

∑m

Ψ(n)m (r)

Ψ(n+1)(r) = Ψ(n+1)(r)/‖Ψ(n+1)(r)‖

solutionΦ = Ψ(n+1), RΦ = R(n+1)

R(n

+1)−R

(n)>ε

affine

alignm

ent

regu

larization

iterateover

initialw

aves

tofin

dΦ

withminR

Φ

Figure 13. Modified Gerchberg-Saxton algorithm, where thewave functions Ψm in the various focal planes (index m) arecomputed simultaneously by propagating the actual iteratedwave function (iteration index n) from some predefined (fo-cal) plane (e.g., z = 0). The affine alignment of the recordedintensities with respect to the simulated wave functions mayoptionally be inserted to correct for spurious image rotations,magnification changes, distortions, etc. The regularizationbefore computing the updated wave function may be appliedto impose additional constraints, e.g., on the smoothness (reg-ularity) of the wave. The iteration stops if some convergencecriteria (e.g., R-factor limit ε or maximal iteration number)is reached.

topology has to be used as starting guess under these typ-ical circumstances. This implicit restriction of the solu-tion space present in almost all focal series reconstructionfrom TEM images reported to date of this publication isone of the main reasons for reasonable reconstructionsfrom restricted focal ranges. In case of an unknown wavetopology, it may also be possible to single out the correctone by comparing the R-factors of the reconstructionspertaining to different topologies. Note, however, thatsuch a test requires well-converged and unique solutionsfor the different topologies, in order to ensure that theinconsistencies due to an erroneous topology are not over-

14

shadowed by partial coherence, alignment issues, noise,etc.

Additional assumptions may pertain to a possible spar-sity of the wave function in some basis, possible smooth-ness (i.e., regularity) restrictions (Allen et al. 2004,Parvizi et al. 2016), support constraints of the wave func-tion (Latychevskaia et al. 2010) or positivity of the phaseshift. If inconsistent with the underlying wave function,these constraints introduce additional regularization er-rors (e.g., adding large gradients in the wave), which needto be balanced against the dampened reconstruction er-ror.

V. CASE STUDY

In the following we perform an exemplary focal se-ries reconstruction of a higher order vortex beam, tostudy the impact of the above principles, such as includ-ing the far field images or predefining the topology ofthe starting guess. The electron vortex beam has beencreated by inserting a forked hologram grating (Grilloet al. 2014, McMorran et al. 2011, Verbeeck et al. 2010)(diameter= 10µm) into the condenser aperture of a FEITitan3 TEM operated at 300 kV. Such masks produce aset of vortex beams of increasing order corresponding tothe various diffraction orders of the grating (Hecht 1987).Here, we cut out one of the |w| = 3 sidebands with asquare aperture (side length= 2µm) in the selected areaplane. The focal series was recorded by varying the ex-citation of the diffraction lens (see (Clark et al. 2016)for details). A total number of 20 defocused images wasrecorded by changing the excitation of the diffraction lensfrom 32% to 60%. The calibration of the effective propa-gation length and magnification was conducted with theabove methods. To further improve the accuracy of thecalibration, we employed a comparison with a simulatedreference series (see (Clark et al. 2016) for details). Inspite of using a large range of diffraction lens excitations,the defocus values ranged from 0 (in-focus) to 0.14 m inthe far field at overfocus only. Thus, 90° of the associatedphase space tilt series are missing, which translates intoconsiderable missing information and needs to be takeninto account when evaluating the results.

To register the images, we aligned their center of massand image rotation with a non-linear fitting procedureprior to the focal series reconstruction. Figure 14 showsa subset of registered images from the series used for theactual reconstruction. Subsequently, we carried out aninitial reconstruction with the help of the algorithm de-picted in Figure 13 using a starting wave with w = 3topology, where we performed an image registration stepafter each 100 steps to remove spurious magnificationand rotation errors in the experimental data. Figure 14exhibits the convergence behaviour of that run. The re-sultant series was subsequently used in the following re-constructions, where 500 iterations proved sufficient toensure a stable solution, when incorporating a small de-

focus into the starting guess.In the middle row of Figure 14, we compare recon-

structions pertaining to different starting waves withthe help of their R-factor (see Figure 13). Accordingly,the solutions fall into different equivalence classes distin-guished by the topology of the starting wave. This be-haviour is due to the preservation of topology within eachGerchberg-Saxton iteration, as noted previously. In fullagreement with the imprinted topology due to the forkaperture, the w = 3 reconstruction yields the best matchbetween experiment and reconstruction. The phase vor-tices appear separated in the reconstruction, which is aconsequence of small manufacturing faults of the forkaperture and a small misalignment of the square aper-ture, which slightly breaks the 4-fold symmetry (Lubket al. 2013a). As a consequence of the only slightly bro-ken four-fold symmetry, however, the w = −3 solutionmay not be ruled out definitely. To clearly distinguishbetween both solutions an astigmatic focal series wouldbe required.

In the following we use the correct w = 3 topologyto elaborate on various characteristics of the focal seriesreconstructions. The above reconstructions have beenobtained without constraining the regularity (i.e., bandlimit). Figure 15 shows two reconstructions with increas-ing band limits imposed by multiplying a Gaussian lowpass in each iteration cycle, where the left column in Fig-ure 15 corresponds to the right column in Figure 14. Inparticular, if no regularity constraint is applied (maximalspatial resolution), one observes several vortex-antivortexpairs at the aperture corners. Their appearance is a hall-mark of the above noted instability of the Gerchberg-Saxton iteration at amplitude zeros, which get more pro-nounced at higher resolutions progressively resolving theinterference phenomena at the aperture edges. Imple-menting a regularity constraint allows us to smooth outthese interference effects, hence minimizing the numberof vortices. This effect of the low-pass filter resemblesthat of partial coherence smearing out the zeros in therecorded intensity (Allen et al. 2004). Reducing the bandlimit beyond a certain limit eventually severely degradesthe reconstruction quality at the aperture edges, henceincreasing the R-factor. The observed variation in theactual vortex positions indicates susceptibility of the vor-tices to the iteration path and small changes in the re-constructed amplitude.

Figure 16 displays reconstructions obtained from avariation of different starting conditions. Again, the leftcolumn displays the best fitting solution from Figure 14for comparison. The other two solutions exhibit the sus-ceptibility with respect to the initial defocus of the start-ing wave, eventually leading to an expulsion of vortexlines out from the ring-shaped intensity maximum, whichmay not be reversed, even if extending the number of iter-ations. On top of that, additional vortex-antivortex pairsappear. These observations highlights the importance ofsingling out a suitable starting wave in the focal series re-construction. In the absence of previous knowledge one

15

defocus

focus

F � 1

F = 1

F � 1

Experiment

10 µm

Topological Test (reconstruction with varying w)

w = 0 w = 1 w = 2 w = 3 w = 4

−4 −2 0 2 40.2

0.22

0.24

0.26

w

R

residual vs. topology

0 200 400 600 8000

0.5

1

1.5

affine registration

number of iterations

R

convergence vs. iteration at w = 3

Result (w = 3)

amplitude

phase

Figure 14. Focal series reconstruction of the vortex beam. Left column: Three recorded densities from the defocus series.Central column: Solutions to w = 0, 1, 2, 3, 4 vortices with w = 3 fitting best to the experimental data according to R-factorplotted for the different topologies (highlighted by small red circle). The convergence of the w = 3 solution exhibits largeimprovements after the first few affine image registration steps correcting for spurious misalignments of the magnification androtation. Right column: Amplitude and phase in the aperture plane (selected area plane) pertaining to the w = 3 solution.

full bandwidth

R = 0.212

ampl

itud

eph

ase

slightly smoothed

R = 0.225

strongly smoothed

R = 0.341

Figure 15. Focal series reconstruction of the vortex beamusing different regularity constraints (band limits).

may employ computationally demanding brute force test-ing or more advanced strategies, such as employing TIEreconstructions from a set of slightly defocused imagesclose to the starting plane of the iteration.

Figure 17 displays reconstructions obtained from trun-cated focal series, where an increasing number of imagesfrom the far field has been removed. As usual, the left col-umn contains the original solution for comparison. Uponremoving far field images from the focal series data, oneobserves almost no degradation of the quality of the re-construction, in spite of the previously mentioned prob-lematic reconstruction of small spatial frequencies in caseof restricted defocus intervals. This result demonstratesthat the conservation of topology implicitly restricts thesolution space of the Gerchberg-Saxton algorithm evenin the presence of an incomplete isotropic focal series.However, the missing far field information eventually be-comes obvious when comparing the R-factors pertainingto different topology classes (Figure 17). Accordingly, thedifferent topologies may not be distinguished by their R-

16

underfocusam

plit

ude

phas

e

R = 0.212

plane

R = 0.282

overfocus

R = 0.228

Figure 16. Reconstructions of the vortex beam using differentstarting conditions, namely a small defocus, no defocus anda small overfocus of the initial wave function.

N = 20

ampl

itud

eph

ase

−4−20 2 40.2

0.220.240.26

w

R

N = 19

−4−20 2 40.180.2

0.220.24

w

N = 18

−4−20 2 40.160.170.170.180.18

w

Figure 17. Reconstructions of the vortex beam removing thenumber of images in the focal series starting at the far field.

factors anymore, if the last two mostly defocused imageshave been removed in our case.

VI. SUMMARY AND OUTLOOK

In summary, we elaborated on the theoretical founda-tions as well as the algorithmic and experimental imple-mentation of focal series inline holography. Employingthe correspondence between focal series phase retrievaland tomographic reconstruction of the correspondingWigner function, we showed that a complete reconstruc-tion may only be obtained from a long-range focal series

ranging from the far field at underfocus via the in-focusimage to the far-field at overfocus. Moreover, the elec-tron beam has to be completely contained within thefield of view throughout the focal series. Typically, thefocal ranges employed in focal series reconstructions inthe TEM do not comply with these principles. In com-bination with the ubiquitous experimental noise and theviolation of the pure state assumption due to partial co-herence, reconstruction errors due to incompleteness (inparticular at low spatial frequencies) and inconsistencies(in particular at large spatial frequencies) must, there-fore, be taken into account.

To reduce reconstruction errors introduced by inaccu-rate focal values and magnifications, we developed a cal-ibration scheme for the diffraction lens (used here to fa-cilitate the focal series), which may be adapted to otherlenses in a straight-forward manner. To improve the cali-bration and hence the quality of the reconstruction, moreaccurate wave propagation schemes through the extendedmagnetic fields of the lenses are required, however. Incombination with more open information policies of themicroscope manufacturers that could eventually facilitatethe precise modeling and calibration of a whole TEM’soptics including lenses, deflectors and apertures. Sucha virtual TEM would provide the means for a rapid de-sign of novel optical setups (e.g., split illumination) orthe automated control of the TEM (fast focal series ac-quisition). In the particular case of the focal series recon-struction, it would be possible to identify optimal com-binations of multiple lens excitations for a complete farfield at underfocus - in focus - far field at overfocus focalseries, presently not available within the TEM.

Subsequently, the archetypal Gerchberg-Saxton itera-tive projection algorithm has been identified as a numer-ical integration of the quantum Hamilton-Jacobi equa-tion in the limit of small focus increments. It is read-ily derived that the dependency of the integration onthe starting wave transfers to the Gerchberg-Saxton al-gorithm. Again, the far-field data are crucial to singleout the correct starting wave in the iteration because in-correct starting waves in the far-field at underfocus arenot matched by the corresponding quantum Hamilton-Jacobi solution at the far field at overfocus. More ad-vanced implications following from the analogy betweenGerchberg-Saxton and quantum Hamilton-Jacobi remainto be explored in the future.

We further showed that the topology of the startingguess is conserved throughout the Gerchberg-Saxton iter-ation. Hence, the solutions pertaining to different start-ing conditions may be separated into topology classes.This implicit constraint mitigates incompleteness prob-lem, if knowing the topology of the solution prior to thereconstruction. This is usually the case as the typicaltopology of a TEM beam is the trivial one. Moreover,we showed that the different topology classes may be dis-tinguished with the help of their reconstruction quality, ifthe focal series comprises sufficient images in the far field.The latter facilitates the topological characterization of

17

beams in singular electron optics, which can serve as afingerprint for the orbital angular momentum of vortexbeams, even if the overall quality of the reconstructionremains low (e.g., due to low signal-to-noise).

We conclude by noting that one of the biggest issuesof the above focal series reconstructions was the pres-ence of partial coherence, violating the pure wave func-tion assumption used throughout. The implementationof a quantum state reconstruction from the focal series,i.e., quantum state tomography can resolve that problemfacilitating the reconstruction of intrinsically incoherentbeams such as resulting from inelastic scattering (e.g., at

plasmons).

VII. ACKNOWLEDGMENT

L.C., G.G. and J.V. acknowledge funding from the Eu-ropean Research Council under the 7th Framework Pro-gram (FP7), ERC Starting Grant no. 278510 VORTEX.A.L., K.V., J. K., D.W., and F.R. acknowledge fundingfrom the DIP of the Deutsche Forschungsgesellschaft.

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Appendix A: Calibration of a Thick Lens

In the following a symmetric thick lens is modeled astwo identical refracting (i.e. phase shifting) surfaces atdistance D. The focal length of each surface is f = 2fthinto ensure that the resulting thin lens in the limit D →0 has focal length fthin. To establish the wave opticaltransfer through such a thick lens, we first propagatethe wave from the object plane (located at distance zobjto the plane of the first refracting plane) to that of thesecond refracting plane, employing the previously derivedthin lens formula (10)

Ψ (r, D) =f

2π (f −D)e−i

k02(f−D)

r2ˆ ∞

−∞d2k′e

i2k0

zobjD−f(z1+D)

f−D k′2Ψ̃ (k′, z1) eif

f−D rk′ . (A1)

The corresponding Fourier transform reads

Ψ̃ (k, D) =f

4π2 (f −D)

ˆ ∞

−∞d2re−i

k02(f−D)

r2ˆ

d2k′ei

2k0

zobjD−f(zobj+D)f−D k′2Ψ̃ (k′, z1) ei(

ff−Dk′−k)r (A2)

=f

2πik0

ˆ ∞

−∞d2k′ei

f−D2k0

( ff−Dk′−k)

2

ei

2k0

zobjD−f(zobj+D)f−D k′2Ψ̃ (k′, z1)

=f

2πik0eif−D2k0

k2ˆ ∞

−∞d2k′e−i

fk0

k′kei

2k0

zobjD−f(zobj+D−f)f−D k′2Ψ̃ (k′, z1) ,

20

and we use that result to perform the second propagation via the second refracting surface to the (conjugate) recordingplane

Ψ (r, z) =f

2π (f − z)e−i

k02(f−z) r

2ˆ ∞

−∞d2ke

i2k0

−fzf−zk

2

Ψ̃ (k, d) eiff−z rk (A3)

=f2

4π2ik0 (f − z)e−i

k02(f−z) r

2ˆ ∞

−∞d2k′Ψ̃ (k′, z1) e

i2k0

z1d−f(z1+D−f)f−D k′2

ˆ ∞

−∞d2ke−

i2k0

( fzf−z−f+D)k2

e−ifk0

k′keiff−z rk

=f2

2π(fzf−z − f +D

)(f − z)

e−ik0

2(f−z) r2ˆ ∞

−∞d2k′Ψ̃ (k′, z1) e

i2k0

z1D−f(z1+D−f)f−D k′2e

ik0

2( fzf−z−f+D)

(fk0

k′− ff−z r

)2

=f2

2π(fzf−z − f +D

)(f − z)

e−i k0

2(f−z)

(1+ f

( fzf−z−f+D)

)r2

׈

d2k′Ψ̃ (k′, z1) e

i2k0

(z1D − f (z1 +D − f)

f −D+

f2

fzf−z − f +D

)︸ ︷︷ ︸

δz

k′2

e

if2

fz − (f −D) (f − z)︸ ︷︷ ︸M−1

k′r

.

The obtained expression has a similar structure than the thin lens result in that an effective free-space propagationtogether with an effective rescaling may be identified.

The product of these two terms

Mδz =

(z1D − f (z1 +D − f)

f −D+

f2

fzf−z − f +D

)fz − (f −D) (f − z)

f2(A4)

=D(f − z1)(f − z) + f (f(z1 + z)− 2z1z)

f2

occurs in the stationary phase approximation of the wave function used for evaluating the focal series of the smallgrating for calibration. Taking into account that the associated focal length of one refracting plane reads f =2/(CfI

2), for the spacing between two diffraction orders we obtain

s = Mprojλ

d

(z + z1 +D −

(z1z +

D

2(z1 + z)

)CfI

2 +D

4z1zC

2fI

4

). (A5)

It can be readily verified that this result converges to the thin lens expression (22) in the limit D → 0. Expressing thisresult in the image and object distance from the symmetry plane of the Dif lens zimg = z+D/2 and zobj = z1 +D/2,we obtain

s = Mprojλ

d

(zimg + zobj −

((zobj −

D

2

)(zimg −

D

2

)+D

2(zobj + zimg −D)

)CfI

2 (A6)

+D

4

(zobj −

D

2

)(zimg −

D

2

)C2fI

4

)If assuming a linear proportionality between the distance of the two refracting planes and the lens current

D (I) = CDI , (A7)

such as valid for the Glaser model of moderately excited lenses (Reimer 1989), we obtain a fourth-order polynomialdependency of the distance between two adjacent diffraction orders and the lens current

s = Mprojλ

d

(zimg + zobj −

((zobj −

CDI

2

)(zimg −

CDI

2

)+CDI

2(zobj + zimg − CDI)

)CfI

2 (A8)

+CDI

4

(zobj −

CDI

2

)(zimg −

CDI

2

)C2fI

4

).

We used this model to fit the unknown optical parametersof the thick lens, yielding a very good agreement with the

experimental data (see Figure 7).

21

Last but not least, we note the thick lens expressionsfor the other two calibration experiments, i.e., the in-focus images of the object and SA plane objects of knownsize

M1,2 = MprojMdif1,2 (A9)

= Mprojf1,2 − zimg +D/2

f1,2

= Mproj

(1− CfI21,2

(zimg −

CDI1,22

)).

Eliminating Mproj then yields

M2CfI21

(zimg − CDI1

2

)−M2

= M1CfI22

(zimg − CDI2

2

)−M1 ,

(A10)

which can be solved for

Cf =M2 −M1

M2I21(zimg − CDI1

2

)−M1I22

(zimg − CDI2

2

)(A11)

and the post-magnification of the projector lenses

Mproj =M1,2

1− CfI21,2(zimg − CDI1,2

2

) . (A12)

Accordingly, the three calibration experiments employedfor the “thin” Diffraction lens are also sufficient for model-ing the Diffraction lens as a thick lens. All optical param-eters shown in the main text are obtained from this thicklens model. Possible sources for the remaining errors inthe calibrated parameters are the experimental errors ofthe calibration experiments and the limited validity ofthe proportionality between the lens thickness and thecurrent as well as the reciprocal focal length (refractivepower) and the squared lens current.