A typical plane wave front in 1D with amplitude 1 and initial reference phase  is represented by the following complex expression

Where,  is the plane wave function,  is the wave vector, and x is the spatial dimension over which the amplitude of the stationary wave is spread with cosinusoidal oscillation. The wave nature of light is observed e.g., during the passage through the slits under appropriate experimental conditions such as coherent illumination, monochromaticity, relative dimension of illumination wavelength, slit and periodicity. In the case of an electron, which is a matter wave, the wave nature manifests in the form of the diffraction pattern of crystal, Fresnel edge pattern and off-axis electron holography fringes.

The Fraunhofer pattern in the diffraction plane can be calculated within the celebrated Fresnel-Huygens construction. The origin of diffraction phenomena is due to the superposition of various spherical wavelets emanating from object forming envelope waves with modulation in amplitude and phase along different momentum directions. However, the geometry of interference and associated phase correlation between various wave vectors is different for the near-field and the far-field regimes (sections S1.1 and 1.2 (available online at stacks.iop.org/JPCO/6/045007/mmedia)). There exist analytical expressions for various types of aperture geometries through the evaluation of the Fraunhofer integral [29]. The Fraunhofer pattern can also be calculated by absolute Fourier transformation (abs-FT) of the real aperture function [figure 1], but the intensity pattern is given as a function of frequency k of the plane wave basis and needs to be calibrated either with the scattering angle  derived from analytical formula or through a calibrated diffraction pattern.

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The Fourier based method is analogous to the physical picture of Abbe’s hypothesis of image formation, where the point of interaction between probe and specimen generates outgoing waves with a continuous range of frequencies, waves with higher frequency propagate along larger scattering angles (section S1.3). Whereas, in the Fourier method, the object function is expanded into various frequencies of plane waves with no information on the scattering angle. Both the scattering processes involve the presence of various frequency components of the outgoing waves that are reminiscent of the picture of Bloch wave propagation of probe electrons in a periodic crystal but with a discrete range of frequencies having symmetry with the crystal periodicity [39]. The inverse Fourier transformation returns the aperture function, if the  and  components of the Fourier waves are known. The phase of the wave function that carries the object information, can be calculated by arctan function, i.e.,  corresponding to diffraction plane and image plane with variables k and x, respectively.

For atomic systems, the electrostatic potential replaces the slit object function. The FT of isolated and periodic atomic potential in the diffraction plane is the atom scattering factor  and structure factor  respectively. In this case, the transmitted wave function can be derived from the Schrödinger integral equation involving interference between incident plane wave and outgoing spherical wave with amplitude factor given by the atom scattering factor (supp. section 1.4) [31, 40]. The image intensity of the atom can then be calculated following Huygens’s principle in Fraunhofer approximation as given in Equation S16 with the consideration of resolution limiting aberrations acting as coherent envelopes. The phase contrast image calculated varies weakly with atomic number and the peak phase shift  follows ∼  where Z is the atomic number [32]. This is possible as the formulation of the scattering factor has the information on the atomic potential which corresponds to both crystallographic and object phase information. The strength of the scattering potential or object phase will only modify the magnitude of specific intensity pattern due to crystallographic phase. Generally, this so-called phase information is lost in the experimental diffraction pattern and only the absolute of R e and Im components or abs-FT are known. However, it is shown in section 4.B that it is possible to retrieve the image intensity function of certain types directly from the diffraction pattern. Once the image function is known, the object phase can be determined according to the standard procedures as described in sec. C or by the alternative method presented in section 4.A.

The method described here yields an image intensity pattern at or near the Gaussian image plane. The definition of weak phase object is introduced here and its implication on the recorded intensity pattern. For pure phase or weakly scattering object, the one-dimensional object function is mathematically represented by a complex function of the form  or  for small  where  is a real function corresponding to the discrete or periodic transparent object and is known asWPOA. The effect of the Zernike phase plate modifies the intensity of the object wave that depends linearly on the phase change due to object according to  This is the foundation of the Zernike’s phase contrast theory [29]. In fact, the object function is real, it is the replica of the object carried by the probe plane wave that is a complex wave function (similar to the Fourier waves propagation with the information of the object function) and absolute of this function after the interaction is imaged on the recording device. Therefore, in the recorded image pattern the information on the object function or crystallographic phase is preserved. The phase corresponding to OEW may be determined from this pattern using the arctan function if the corresponding object wave is known.

However, a similar approach for the atomic system incorporated the change in phase in the plane wave illumination in terms of change in magnitude of the wave vector  [30, 31]. This brings about transmission function and under axial imaging condition is further convoluted with the point spread function (psf) in the image plane or multiplication with the phase contrast transfer function (PCTF) function which acts as a coherent envelope in the diffraction plane incorporating the effect of aberration phase shift to form the final image [27, 30]. The intensity depends linearly on the phase change due to object potential after ignoring the higher order interaction terms (equation S21) that is similar to Zernike like transfer if the complete envelope function is set to 1. Considering that the spatial resolution criteria of the instrument is being met, again, the information on the crystallographic phase is not lost in the image intensity pattern. However, if the WPOA approximation is not considered, then the information related to object function  and complete phase information is lost [section S1.4.2]. Kindly note that, the phase shift due to aberration cannot be added in the trigonometric function in the diffraction plane as that shifts the wave, rather it is used as coherent envelope or a frequency filter function. In real space, the psf due to PCTF gives weight to the intensity (e.g. Scherzer transfer will have maximum weight for the optimum value of spherical aberration and defocus) that depends on the integration value of PCTF over the limits of reciprocal vector acting as a frequency filter and other coherent aberration e.g., aperture function determines the broadening or full width at half maximum (FWHM) of the potential function [27, 41]. As the convolution procedure changes the magnitude of the resultant function significantly, a flux balance approach is helpful to observe the decrease in intensity and increase in FWHM due to aberration in comparison to the ideal image free from any aberration (figure S7) [41].

However, the above description of WPOA does not draw any analogy in terms of interference geometry between Gabor’s in-line holography, Fresnel diffraction geometry and defocus HRTEM image except transmission function between in-line holography and transmission function derived based on the change in magnitude of electron wavelength (compare equation (S24) and (S25)). However, instead of considering the change in momentum vector direction due to interaction with the object potential provides insights in all three pictures in terms of the geometry of interference (sections S1.1 and S1.2). An account of comparative image simulation based on various methods described above can be found in [41]. In brief, due to various ways of considering the flow of phase information by different methods, the simulated image patterns do not correspond to each other for the same atom type.

Reconstructions methods associated with the HRTEM through focal image series aim to retrieve the unknown phase  of the OEW function of the form  The OEW function can be used to interpret the object structure  based on the model methods as described in the previous sections. Generally, the illumination is considered to be monochromatic and for semi-monochromatic waves, partial coherence theory appears into the stage. While the theoretical model behind the off-axis electron holography technique to retrieve the OEW function is straightforward (section S2.2), the methods corresponding to inline holography using through focal HRTEM image series are much more elaborate involving intensive data refinement and fitting procedures [17]. This is due to the nature of the fitting equation considered to retrieve the  that attempts to reduce the effect of complex conjugate  and non-linear image components  from the recorded image set [equations (5) and (7)].

There exist a few different focus variation methods e.g., Wiener formulation by Schiske, 3D paraboloid method (PM), maximum likelihood method (ML) and various numerical schemes associated with them [2, 3, 42, 43]. Saxton showed the equivalence between the different reconstruction techniques in terms of equivalence in the form of restoring filter applied to the intensity expression in Fourier space to retrieve the wave function. The simplified form of the restoring filter is given as

and the associated aberration function

finally, the restored wave function is written as

Where,  is the third order spherical aberration,  is the wavelength,  is the defocus corresponding to the nth image, and  is the spatial frequency. The application of restoring filter is similar to deconvolving the effect of aberration. The inverse-FT of equation (4) will return the wave function in the image plane. This is only possible if one starts with the intensity pattern recorded in the image plane with sufficient spatial resolution and during inverse-FT, spatial information on the crystallographic phase is preserved through  and  of Fourier waves. The pattern of the wave function in terms of phase  and amplitude  thus obtained required to be fitted with the model calculation for further interpretation [9].

However, Saxton derived the final form of the wave function without considering any restoration filter from a different start in the form of the intensity expression as given in equation (5). The approach provides insight into the effect of restoration on dispersing the effect of the conjugate wave function and non-linear image component. Under perfect coherence, the image intensity at some focus z near the Gaussian image plane is written as

Where,  is the wanted part or the OEW,  and  are the unwanted non-linear term and complex conjugate, respectively.  is the twin image of  Kindly note that the reference wave  is set to 1 by considering axial illumination () in the above expression.

The 2D Fourier transformation of equation (5) and then explicitly adding the dependence on defocus results in the following expression.

Where,  is the non-linear component that describes the autocorrelation between the two linear terms in the reciprocal space. The paraboloid method as proposed by van Dyck can be derived from the above expression by taking 3D-FT with respect to defocus where the wave function and its complex conjugate follow the reflected parabola from the reference diffraction plane [2].

Now, within the coherent detection i.e., multiplying by corresponding phase conjugate and summing over N images after assuming  is non-zero which allows to omit the delta function and for a constant  the image intensity becomes

From the above expression, it was concluded that the wave functions  and  accumulate to N and  times to its original value, respectively [3]. The non-linear term  behave randomly with the defocus. Thus, by dividing the above equation by N, the wanted OEW function may be recovered. Equation (7) is widely considered in almost all the through focus image series OEW reconstruction. And all the efforts on developing image reconstruction codes primarily deal with eliminating the non-linear and complex conjugate terms and finding the best fit with the model calculation. However, the presence of complex conjugate and non-linear terms will always be present depending on the extent their weights are subdued.

Nonetheless, there exists a contention, after complete evaluation of equation (5), yields the following final form

Where,  is replaced with the wave function of the form  describing the image intensity pattern based on self-interference between reference incident and scattered waves within the picture of single electron wave interference phenomena [Equation S49 & S50]. The expression has the similarity with the wave interference between the reference and object waves in off-axis electron holography (section S2.2). However, in off-axis geometry, there is an additional phase term  due to wave interference at an angle that gives rise to spatial modulation in the interference field [5]. Moreover, equation (5) is in an intermediate state, which is used to eliminate the effect of twin image and non-linear terms by working in the diffraction plane [section S2.1]. However, the existence of the final form of the expression implies that fitting the intensity equation alone and evaluating the phase term should, in principle, allow extracting the relative phase change from the image plane. For more details on associated twin image wave functions and applicability of equation (8), see section 4. The results based on the existing schemes can be found in [6, 17] and do not show any systematic trends with the sample thickness.

Finally, the approach based on partial coherence theory considers the effect of finite source size, chromatic defocus spread, current voltage fluctuation of the instrument, objective aperture size and wave aberration function [1, 44, 45]. The reconstruction method based on partial coherence theory is known as an iterative linear restoration that addresses the residual non-linear term. Repeatedly applying a linear restoring filter from the subtraction of the calculated non-linear term improves the initially guessed wave function. Various derivations available based on partial coherence theory are given in section S2.4. One can notice that the coherence and interference phase shift in the formalism has the origin in convolution procedure in real space and cross correlation in Fourier space.

The method described here works on atomic resolution HRTEM image recorded under suitable imaging conditions i.e., with a particular combination of spherical aberration coefficient  and defocus  that sets the optimum contrast and resolution.

The object phase can be recovered by applying equation (9) as given below, which is a modified form of equation (8) describing the image intensity pattern in HRTEM within few nanometers [section S2.2.1] from the Gaussian image plane.

Where,  is the mean vacuum intensity. The factors  and  represent the fractions of direct and scattered part of the intensity and can be determined by analyzing the image pattern, where  Typical values of  and  are found to be ∼ 0.88/0.95 and 0.12/0.05, respectively from the experimental images of MoS₂/BN [section S5].

And the intensity expression corresponding to the off-axis electron hologram is given by

The appearance of sinusoidal function in equation (9) in contrast to equation (8) is due to the OEW phase term having a relative phase term with respect to vacuum phase  in addition to  due to object potential in the case of HRTEM. This phase difference is absent in off-axis electron holography [compare equations (9) and (10)] and is generally considered to be  i.e.,  that sets the vacuum phase value to zero. In case of off-axis geometry, the two halves of the wave on either side of the bi-prism carry the same vacuum reference phase term and gets eliminated in the final intensity except for part of the wave that carries the object phase information [section S2.2]. For an intuitive physical picture on the origin of such a  phase shift between diffracted and primary incident waves based on interference geometry is provided in section S 2.2.1 [figure S11]. The modifications incorporated in equation (9) are essential and yield excellent results from various experimental images with different types of atoms present for a given crystal. For more discussion on the factors  and  and the particular use of amplitude terms and various forms of equation (10) and their relationship with experimental image see section S5.

Now, both the equations are founded on the physical picture of self-interference of single electron wave function involving interaction in the form of phase change with the object potential [28]. The interaction is mostly elastic due to the fast probe electron, and the small probability of inelastic interaction is useful for the analytical techniques. The primary difference between equations (8) and (10) is that the additional phase term of  that appears in the trigonometric function in off-axis electron holography due to wave interference at an angle (section S2.2). This particular phase term oscillates with the spatial coordinate x over the field of the electron hologram. The phase term  corresponding to OEW phase is acquired by another half of the wave while passing through the sample. This OEW phase manifests as holographic fringe bending in the image plane, e.g., relative fringe bending between vacuum and MgO crystal and differently striped thickness of object structure, that allows to determine the mean inner potential (MIP) at medium resolution [4, 46].

Another difference between the two holography techniques is the twin image. It is well known that for in-line holography, separation of twin images is an issue, whereas, for off-axis geometry, they get separated at  in the frequency space. Ptychography has also been used to eliminate the twin image issue. However, in the image plane, they superimpose on top of each other in both the techniques. In the standard practice of OEW reconstruction, equations (9) and (10) are generally written in an intermediate state (see equation (5) for in-line and Equation S54 for off-axis) having both OEW function  and its conjugate  in consistent with the holographic principle of image formation [29]. The motivation behind this was based on the concept that the twin image components can be separated from the direct component (DC) if allowed to propagate along the scattering direction i.e., from the image plane to the diffraction plane and emphasizing the direct retrieval of the wave function. However, it is argued here that this does not pose an issue if one wishes to work in the image plane of the intensity pattern to retrieve directly both the phase and amplitude of OEW directly. The choice and limitations between the diffraction plane and image plane can be understood by Heisenberg’s uncertainty principle, i.e., only momentum and position variables of the OEW are known in the diffraction plane and image plane at a time, respectively.

Now, from equations (9) and (10), it is evident that the phase information is preserved in the final form of the intensity equation that is responsible for the intensity modulation as described by the trigonometric function. At the same time, no information on the wave functions or twin images and DC component are available. The application of the final form of intensity expression instead of wave function based intermediate state of the equation can be guided by the Born rule in Quantum Mechanics. Born rule states that it is the probability density given by the square of the probability amplitude of particle’s wave function  that is real and observable quantity during the measurement and not the associated complex wave function which acts as a state vector [47, 48]. There exists controversies in the literature on the measurement of such state vector which is a complex quantity but can be constructed based on the information of probability density [48–50]. Born rule is an important link between the abstract mathematical formalism of quantum theory based on the complex wave function and the experimental measurement and constitutes an essential part of the Copenhagen interpretation of quantum mechanics [51]. Therefore, the final form of the intensity equation is equivalent to measuring the total probability density due to contribution from all the three components of wave functions, i.e., DC part, wanted wave function  and its complex conjugate  And in the process of recording, the information on the wave functions is lost, recovering of which is emphasized in all the reconstruction approaches through intermediate state of the equations. Retaining the phase term is inherent to self-interference and resulting modulation in intensity pattern and similar to the off-axis electron holography fringe bending due to object potential in the image plane. The predicament of evaluating the local phase information from real space image in terms of fringe bending primarily comes from the restriction on spatial resolution and will be addressed in the forthcoming discussion along with additional aspects already mentioned earlier. In standard practice, the image is Fourier transformed and then a digital aperture function is used in that plane that takes care of the resolution. The use of intermediate state of the equation to disperse the twin images and DC part in the diffraction plane made the earlier methods elaborate and complex for HRTEM reconstruction where twin images overlap on the same reciprocal space and required to enhance the weight of  over the  and  terms through summation over many images. Though the information on the wave functions is lost, however, the phase term is preserved in the final state of the equation which are the same for both  and  but having opposite sign.

Now, the twin images are the exact copy of each other and superimposes at the Gaussian image plane and do not cause any loss in information in terms of phase and amplitude. This can be understood with the help of off-axis electron holography, where the inverse FT corresponding to each side band preserves the same intensity pattern. In inline holography, the twin images propagate along the opposite direction with defocus similar to the parabola picture of OEW reconstruction where complex conjugate pairs lie on a reflected parabola with respect to the diffraction plane and focus variation. This picture is reminiscent of the transactional interpretation of quantum mechanics where the wave function and its complex conjugate experience phase change in the opposite directions equivalent to the forward and backward propagation in time [52–54].

Now proceeding for the experimental reconstruction of the OEW function, at first, the phase can be determined directly by applying equation (9). If the quantity inside the arcsine function is more than 1, then this should be divided by 1 and the computer program should read the quotient as phase jump by the amount (quotient × ) plus the phase corresponding to the remainder and then calculate the total phase change and corresponding counts of the atoms accordingly. The  phase jump does not appear for one-layer thick MoS₂ and BN but is encountered in 3D crystal like ZnO where thickness includes several atoms in the column. Next, the amplitude corresponding to  needs to be evaluated from the image intensity. The total amplitude of the recorded image is simply the square root of it but not equal to the amplitude of the single OEW function. As already mentioned that the total amplitude in the image intensity pattern has three components, DC, and two twin image components for off-axis and DC and only diffracted components in case of in-line holography. In the case of in-line holography, the twin image components overlap with the DC part. The DC component has approximately one order of magnitude higher total intensity compared to individual twin images from SBs and diffracted FT spots as measured directly from the FT of the images by placing a region of interest around various spots and SBs and evaluating the sum. Thus, the intensity of the OEW function is approximately given by the factor  for HRTEM, and this will be further divided into halves between two twin image components in case of off-axis holography. The amplitude of  is given by the square root of  and only amplitude component  associated with trigonometric function is shown in reconstructed wavefunctions, where  and  are the magnitudes of individual diffracted and reference complex wave functions, respectively. Details on the evaluation on the amplitude from the intensity pattern, discussion on the interpretation of amplitude of isolated wavefunction, and on the appropriate form of conventional intensity expression of atomic resolution off-axis hologram and HRTEM are given in section S2.2. Kindly note that the subtraction procedure to adjust the intensity value is automatically taken care of by equation (9) while calculating the phase. Therefore, it is possible to reconstruct the OEW function from the information obtained from the measurement as described above i.e., phase and amplitude.

We have analyzed selected experimental images of MoS₂, BN and ZnO using equation (9) and the reconstructed phase and amplitude images are presented in figure 2 and 3 for MoS₂ and BN, respectively. One can see from the phase images that the vacuum phase is zero and phase values are high and low depending on the brightness in the dots which represents the periodic arrangement of atoms in the lattice. Theoretical peak phase shift values extracted from [32] for the resolution range 1–0.5 Å are given in table 1 for the atoms of interest in the present investigation. According to this table, the peak phase shift  follows ∼  where Z is the atomic number [32]. The exponent of Z increases with increasing spatial resolution resulting in a higher peak phase shift value for a given atomic number. Though the trend can be complicated depending on the valence electron filling and for specific atoms with higher Z can have smaller contrast compared to atoms with lower Z next to each other in the periodic table. From the present reconstruction method, a peak phase shift value of ∼ 0.56 and 0.45 rad are obtained from the Mo and S atom positions, respectively throughout the image, which are little higher than the theoretical values. Kindly note that for MoS₂ there are two S atoms along the [0001] projection. Therefore, the phase shift value is almost double compared to the single S theoretical phase shift value. The values are 0.06 and 0.21 rad, for B and N atoms, respectively which are close match with the theoretical estimation. However, the value for N atom appear to be almost double than the theoretical reference value, this might because of the strong dependence of  and  values on the local intensity maximum, sample contamination and resolution dependence of theoretical reference values. Results on ZnO and bi-layer MoS₂ are given in figure S13 and S14. For ZnO thin film, as the sample thickness increases from the vacuum edge of the specimen, a systematic increase in phase and atoms numbers can be observed. As an example the phase value corresponding to Zn column changes from 1.23 to 2.3 rad from the edge to the interior of the specimen. This corresponds to a change in atom number/thickness from 4 (1.3 nm) 8 (2.6 nm), respectively. For the MoS₂ bilayer sample, a AB type of stacking is observed. The weak spot in between the strong hexagonal white dots are from S atoms. The intensity of dots corresponding to joint Mo and S atom columns are stronger and intensity of dots corresponding to S columns are weaker. The phase value corresponding to weaker dots ∼ 0 and from bright dots phase values of 0.67 and 0.5 rad are obtained, does not help to identify the atoms. This is because of image intensity depends on defocus ([41] for more details) and the intensity has a strong role to play to obtain the correct phase number according to equation (9). This will be part of a future communication on the reconstruction based on thicker crystal. The uncertainty in phase and amplitude values from the experimental measurement is ∼0.006.

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The reconstructed phase numbers calculated by equation (9) for Zn and O atoms in ZnO are in good agreement with the earlier report using atomic resolution off-axis electron holography [6]. However, the reconstruction from the HRTEM through focus series using the Gerchberg-Saxton scheme as implemented in MacTempas package did not yield expected variation in phase with sample thickness unlike the application of equation (9) yield systematic variation in phase with sample thickness [6, 17]. For thicker sample, channeling theory and diffraction imaging based reconstructions are being utilized at present [20, 21, 27]. However, more robust and straight forward reconstruction scheme considering relatively thicker specimen and large defocus will be part of a separate communication as this requires a different fundamental start than traditional approaches [41].

The method described above using equation (9), in fact translates the intensity information into a phase value and an excellent trend is obtained for images with low and high Z compounds giving lower and higher phase shift values, respectively without any additional data refinement procedures. Reference vacuum is not required as one can approximate the mean of atomic resolution image intensity as mean vacuum intensity. Intensity pattern if at all present other than OEW, e.g., Fresnel fringe, electron interference fringe in off-axis electron holography should be removed from the image data before applying equation (9). These unwanted patterns are generally removed from the image before the final analysis in case of off-axis electron holography.