The motivation for the development of full-resolution wave reconstruction (GPFRWR) assisted by gradient inversions and phase prediction is that, in most TEM investigations of specimens that do not generate magnetic and electrostatic fringe fields, the phase in the vacuum regions within the field of view has a constant value. The slab geometry of many TEM samples can also result in roughly flat phase regions over large parts of the sample, at least at medium spatial resolution. In other words, the phase gradient is often quite sparse, especially when high spatial frequencies are excluded.
In its original application, the charge reversal algorithm for solving crystal structures from X-ray diffraction data30 is very effective in finding a sparse solution in the charge density domain by changing the signs of values small while retaining values above a certain threshold and imposing consistency with measured diffraction intensities. After demonstrating its feasibility for removing low spatial frequency noise in TIE reconstructions23,24, we adapted the principle to nonlinear online electron holography by inserting a phase modification procedure every few iterations (e.g. every third iteration) in an iterative reconstruction algorithm (the FRWR25,26 algorithm), inverting the signs of small values of each of the two components of the phase gradient and reducing their amplitudes, obtaining a modified gradient \({\overrightarrow {G}}^{^{\prime} }\left(\overrightarrow{r}\right)\). This procedure was implemented by simply multiplying these values by a scaling factor β that was slightly greater than − 1 (eg, β = − 0.97). This operation was only performed within the field of view defined by the experimental data. The size of the matrix defining the reconstructed phase was larger than the field of view, in order to accommodate non-periodic boundary conditions. While the larger matrix has periodic boundary conditions, the matrix corresponding to the field of view of the experimental data can have any boundary because it lies within the typically 1.5 to 2 times wider reconstruction matrix that has conditions of periodic limits16. After the small gradients have been inverted, the modified phase is obtained by integrating the modified gradient \({\overrightarrow{G}}^{^{\prime}}\left(\overrightarrow{r}\right)\) The Fourier transform of the modified phase \({\phi }^{^{\prime}}\left(\overrightarrow{q}\right)\) is obtained from \({\overrightarrow{G}}^{ ^{\prime }}\left(\overrightarrow{r}\right)\) by the following operation:
$${\phi }^{^{\prime}}\left(\overrightarrow{q}\right)=\phi \left(\overrightarrow{q}\right)\left[1-\mathit{exp}\left(-{r}_{c}^{2}{q}^{2}\right)\right]+FT\left[\overrightarrow{\nabla }\cdot {\overrightarrow{G}}^{^{\prime}}\left(\overrightarrow{r}\right)\right]\frac{\mathit{exp}\left(-{r}_{c}^{2}{q}^{2}\right)}{-{q}^{2}},$$
(3)
where rc is the length scale below which gradient inversion has very little effect, i.e. this value can be chosen to apply gradient inversion only to spatial frequencies for which the iterative reconstruction algorithm would converge only very slowly. Divide by – q2 effectively implements an inverse Laplace operator. At what = 0, we multiply by 0 instead of dividing by q2. This approach can be justified by the argument that absolute phase is not a well-defined physical quantity. Multiplication by 0 a what = 0 will cause the average of the reconstructed phase to be set to 0. After reconstruction, an offset can be subtracted that corresponds to the average of the input phase gap. This subtraction was applied to the phase maps shown in this paper.
Using the above expression, the gradient change mostly affects the spatial frequencies of the phase that are greater than rcusing a Gaussian taper to transition between intervals r> rc ir< rc . Since the iterative part of the focal series reconstruction algorithm mainly reconstructs high spatial frequencies in phase and requires many iterations to have an effect at lower spatial frequencies16, Eq. (3) ensures that gradient flipping does not significantly affect the convergence of the iterative reconstruction algorithm. It can even help speed up convergence, especially in regions where large areas of the phase are flat (for examplefor nanoparticles on homogeneous supports or if the field of view contains large areas of empty space).
In order to reduce the detrimental effect of incoherent scattering (e.g., electron-phonon and electron-plasmon interactions or core electron excitation) on the interpretability of experimental TEM images using elastic scattering theory, the our algorithm is based on the following flow preservation. image model, which is modified to include an incoherent background:
$${I}_{\Delta f}\left(\overrightarrow{r}\right)=\left[{\left|{\Psi }_{\Delta f}\left(\overrightarrow{r}\right)\right|}^{2}+{I}_{incoherent}\left(\overrightarrow{r}\right)\right]\otimes {E}_{s,\Delta f}\left(\overrightarrow{r}\right),$$
(4)
wherePSf( r ) is the coherent electronic wave function defocused by an amount Δfin the plane of the detector, Es, Δf (r) is the inverse Fourier transform of the spatial coherence envelope15 i meincoherent(r ) is the intensity distribution of the incoherent background which is determined in addition to the electron wave function PS0( r ) on the exit surface of the sample. Since only a single incoherent image is assumed, this residual image is considered to be the same for all images in the focal series. Incoherent intensity is strictly positive. At each iteration, after the mean of the remaining difference between the forward simulated image and the experimental image has been added to the incoherent intensity matrix, this matrix is multiplied by a damping factor, which is set to 0.98 by default, but that can be changed when calling the rebuild program. Thus, only a fraction of the minimal observed difference between simulated and experimental images at each pixel is assigned to this incoherent intensity matrix, ensuring that only image counts that cannot be attributed to the coherent imaging process are considered for allocation to the inconsistent fund.
The FRWR25,26 algorithm accelerates the retrieval of low spatial frequency phase information using a “phase prediction” mechanism. In addition to directly minimizing the difference between simulated and experimental images in a manner somewhat similar to conventional Gerchberg-Saxton-type focal series reconstruction algorithms25, the FRWR algorithm also explicitly updates the phase. This is done by first computing a phase update that is motivated by the intensity transport equation (TIE) as follows:
$$\begin{array}{c}\Delta \varphi \left(\overrightarrow{r}\right)={\varphi }^{exp}\left(\overrightarrow{r}\right)-{\varphi } ^{sim}\left(\overrightarrow{r}\right)\\ =\frac{2\pi }{\lambda }\frac{{\nabla }^{-1}{I}_{0}^{ -1}\left(\overrightarrow{r}\right){\nabla }^{-1}\left({I}_{+\Delta f}^{\mathrm{exp}}\left(\overrightarrow{ r}\right)-{I}_{-\Delta f}^{\mathrm{exp}}\left(\overrightarrow{r}\right)\right)}{2\Delta f}-\frac{2 \pi }{\lambda }\frac{{\nabla }^{-1}{I}_{0}^{-1}\left(\overrightarrow{r}\right){\nabla }^{-1 }\left({I}_{+\Delta f}^{sim}\left(\overrightarrow{r}\right)-{I}_{-\Delta f}^{sim}\left(\overrightarrow{ r}\right)\right)}{2\Delta f}\\ =\frac{2\pi }{\lambda }\frac{{\nabla }^{-1}{I}_{0}^{ -1}\left(\overrightarrow{r}\right){\nabla }^{-1}\left({I}_{+\Delta f}^{\mathrm{exp}}\left(\overrightarrow{ r}\right)-{I}_{+\Delta f}^{sim}\left(\overrightarrow{r}\right)\right)}{2\Delta f}-\frac{2\pi }{ \lambda }\frac{{\nabla }^{-1}{I}_{0}^{-1}\left(\overrightarrow{r}\right){\nabla }^{-1}\left( { I}_{-\Delt af}^{\mathrm{exp}}\left(\overrightarrow{r}\right)-{I}_{-\Delta f}^{sim}\left(\overrightarrow{r }\right)\right) }{2\Delta f}\end{array}$$
(5)
in other words that isa TIE-like update of the phase that takes everything into accountNthe focal planes can be calculated using the expression
$$\Delta \varphi (\overrightarrow{r})=\frac{2\pi }{N\lambda }{\sum }_{n=1}^{N}\frac{{\nabla }^{- 1}{I}_{0}^{-1}(\overrightarrow{r}){\nabla }^{-1}\left({I}_{\Delta {f}_{n}}^{ exp}(\overrightarrow{r})-{I}_{\Delta {f}_{n}}^{sim}(\overrightarrow{r})\right)}{\Delta {f}_{n} }.$$
(6)
Before adding this phase update to the phase of the current estimate of the recovered wavefunction, its magnitude is limited to a maximum valueϕmax, the so-called phase prediction threshold (PPT), by using exponential compression for high values of Δϕ(r) to avoid a sharp threshold:
$$\varphi \left(\overrightarrow{r}\right)= \left\{\begin{array}{ll}sign\left(\varphi \left(\overrightarrow{r}\right)\right)\frac {{\varphi }_{max}}{2}\left[2-exp\left(\frac{\frac{{\varphi }_{max}}{2}-\left|\varphi \left(\overrightarrow{r}\right)\right|}{\frac{{\varphi }_{max}}{2}}\right)\right]& if \left|\varphi \left(\overrightarrow{r}\right)\right|\ge \frac{{\varphi }_{max}}{2}\\ \varphi \left(\overrightarrow{r} \right)& if \left|\varphi \left(\overrightarrow{r}\right)\right|< \frac{{\varphi }_{max}}{2}\end{array}\right.$$
The use of this limit is important because incoherent contributions to signal and noise prevent a perfect match between experimental and simulated intensities from being obtained, causing the expression in parentheses in Eq. (6) to practically never reach zero. Therefore, continuously adding a non-zero update to the phase would result in divergent behavior of the reconstruction algorithm.