## Abstract

Transmission electron microscope (TEM) images recorded under tilted illumination conditions transfer higher spatial frequencies than axial images. This super resolution information transfer is highly directional in a single image, but can be extended in all directions through the use of complementary beam tilts during exit wave function reconstruction. We have determined the optimal experimental tilt magnitude for aperture synthesis in an aberration-corrected TEM. It is shown that electron-optical aberration correction allows the use of larger tilt angles and reduces the constraints that are imposed on experimental data acquisition in an uncorrected microscope. We demonstrate that, in many cases, the resolution improvement achievable is now limited by the sample and not by instrumental parameters. An exit wave function is presented that has been successfully reconstructed from a dataset of aberration-corrected images, including images acquired at a beam tilt of 18 mrad, which clearly demonstrates a resolution improvement from 0.11 nm to better than 0.08 nm at 200 kV.

## 1. Introduction

The axial information limit, **k**_{d}, for phase contrast imaging in a transmission electron microscope (TEM) is ultimately determined by the effects of partial temporal coherence (for instruments fitted with field emission sources). Correction of the aberrations in the objective lens allows the interpretable resolution in an image to be extended to the axial information limit. Aberration correction/compensation is possible, either directly using suitable electron-optical elements (Haider *et al.*1995, 1998*a*,*b*), or indirectly by computational reconstruction of the specimen exit wave function. Combining these approaches allows compensation of the higher order aberrations (Tillmann *et al.* 2004; Houßen 2006; Houßen *et al.* 2006; Kirkland *et al.* 2006; Lovely *et al.* 2006), providing improved quantitative structural information. However, from a dataset consisting of a defocus series of images, the interpretable resolution is still limited to the axial information limit.

Using tilted imaging modes, the resolution of a TEM can be extended beyond the axial limit. Tilting the electron beam shifts the circular region of transfer in Fourier space such that it is no longer centred on the optic axis, but is offset in the opposite direction to that of the beam tilt. An illumination tilt angle, , results in information transfer that, although directional, extends to spatial frequencies up to twice the axial limit (2**k**_{d}) (Kirkland *et al.* 1995). However, this asymmetry in the tilted transfer function necessitates the inclusion of several tilted illumination images in an exit wave function reconstruction in order to cover the full azimuthal range of Fourier space. Since this approach synthesizes a large effective Fourier space aperture from several smaller ones, it has also been termed ‘aperture-synthesis’ for comparison with similar approaches in radio astronomy (Michelson 1890; Pease 1931) and radar (Ryle & Vonberg 1946).

## 2. Information transfer for tilted illumination

In determining the optimum conditions for exit wave reconstruction using tilted illumination images, it is first necessary to consider the effects of partial coherence that ultimately limit information transfer in any imaging mode. Functions *E*_{s}(**k**+**k**′,**k**′) and *E*_{f}(**k**+**k**′,**k**′) can be defined to model the effects of partial spatial and temporal coherence, respectively,
2.1
and
2.2
where *β* is the root-mean-square spread of the Gaussian beam profile (Hopkins 1953; Foschepoth & Kohl 1998), Δ is the e^{−1/2} value of the focal spread distribution and ∇*χ*, with *χ*=2*π**W*/*λ*, is the gradient of the phase change due to the wave aberration function, *W*, including all coherent aberrations (see equation (2.4)).

For a conventional TEM, it has been shown that the defocus *C*_{1} under which tilted illumination images for use in exit wave function reconstruction are recorded is critical (Kirkland *et al.* 1995), and is optimally chosen with reference to the beam tilt *τ* and spherical aberration *C*_{3} as
2.3

This coupling of beam tilt and defocus implies that the dataset required for exit wave function reconstruction using tilted images is extremely sensitive to focal drift (Kirkland *et al.*1995, 1997).

The above yields a typical tilt magnitude of approximately 8 mrad with a corresponding defocus of 39 nm for *C*_{3}=0.6 mm at 300 kV. Experimental exit wave function reconstructions using tilted illumination images have been performed successfully using a conventional TEM (Kirkland *et al.*1995, 1997; Kirkland & Meyer 2004), and have demonstrated an improvement in the limit of *continuous* information transfer from 0.2 nm to approximately 0.11 nm (Kirkland *et al.* 1997). However, this represents a relatively modest improvement over the axial information limit for the instrument used (0.14 nm). Moreover, due to the sensitive experimental conditioning described above, this approach has not found widespread experimental application. As discussed subsequently, electron-optical aberration correction relaxes the coupling of *C*_{3} and *C*_{1}, making larger tilt magnitudes accessible leading to potentially greater resolution gains and substantially more robust instrumental settings.

### (a) Resolution limits due to incoherent aberrations

At large beam tilt magnitudes, partial temporal coherence causes a loss of transfer at the centre of the tilted transfer function. For a beam tilt, *τ*_{0}=*λ***k**_{0}, beams at the same angle *τ*_{0} with respect to the tilt axis are perfectly transferred, as and the temporal coherence envelope does not damp information transfer (). However, for large beam tilts, the transfer falls away significantly inside this ring of perfect transfer (the achromatic circle). Reducing the tilt angle prevents this central transfer loss, but reduces the resolution improvement compared to the axial case.

Figure 1 compares the modulus of the wave transfer function for axial imaging with that for tilted illumination imaging. Allowing a reduction at the centre of the tilted transfer function of 50 per cent of the axial value gives a maximum tilt magnitude of 16 mrad for aberration-corrected images. This 50 per cent limit is an arbitrary choice, however, less conservative values of 30 per cent or 10 per cent of the axial value still provide acceptable transfer, whilst allowing maximum tilt magnitudes of 19 or 26 mrad, respectively.

This reduction in transfer for tilted illumination can, in principle, be recovered in the final reconstruction, either by increasing the number of tilt azimuthal angles or by including more than one tilt magnitude. However, either of these approaches increases the number of images required with consequent implications for experimental stability and overall radiation dose. The tilt magnitude used to record the images forming the dataset used for reconstruction ultimately determines the highest spatial frequency that is transferred to the reconstruction. For a dataset consisting of six equally spaced beam tilts of equal magnitude, the highest transfer occurs along the beam tilt direction and the weakest transfer direction occurs midway between two tilt directions (figure 2).

### (b) Resolution limits due to coherent aberrations

Using the notation provided by Typke & Dierksen (1995), the wave aberration function, *W*, under the isoplanatic approximation expanded to sixth order in the complex scattering angle *ω*=*k**λ* is given as
2.4

Tilting the illumination by an angle, *τ*, changes the origin of the Taylor expansion in equation (2.4) and thus the magnitudes of the aberration coefficients (Thust *et al.* 1996).

The new tilted aberration coefficients (indicated with a prime) are related to the axial coefficients through 2.5

A given set of experimentally measured axial aberration coefficients can be used to predict how the values of selected coefficients vary with illumination tilt (figure 3) by expansion of equation (2.5) with respect to the complex aberration coefficients and illumination tilt angle, and by taking the real part of this expansion.

The aberration coefficients for tilted illumination conditions have also been measured experimentally within the CEOS correction software (Uhlemann & Haider 1998), as shown in figure 4. Using a thin amorphous foil as a specimen, at an initial magnification of 400 000 times and an axial defocus of approximately −400 nm (under-focus defined as negative), the software acquires a tableaux of diffractograms for defined illumination tilt angles, and measures *C*_{1}′ and *A*_{1}′ values using a pattern recognition algorithm applied to the first 12 zeros in each diffractogram (Uhlemann & Haider 1998). Subsequently, a least-squares fitting routine operating on these values is used to determine all aberration coefficients defined by equation (2.5). For the data reported in this paper, the tilt coil calibration was determined independently. The spherical aberration coefficient was set to −4 μm and the other coefficients were corrected to zero levels within the measurement accuracy.

The errors shown in figure 4 are twice the standard deviation and thus refer to a 95 per cent confidence assuming a normal distribution (bib33). At tilt magnitudes greater than 50 mrad, the errors rapidly become very large as the experimental diffractograms recorded become too distorted to provide reliable *C*_{1}′ and *A*_{1}′ measurements. In addition to the errors in the experimental measurements, there is an error associated with the theoretical predictions since these rely on the accuracy of the measured axial coefficients.

Given the large beam tilts used and the associated experimental errors, figure 4 shows a good match between theory and experiment, particularly for tilt magnitudes less than 30 mrad. This indicates that, for tilt angles less than 30 mrad, both the experimental (CEOS) measurements and theoretical predictions of the aberration coefficients are reliable, at least to third order in the wave aberration function.

#### (i) The effect of axial coma

The experimental measurements shown in figure 4*d* demonstrate that, for aberration-corrected conditions, the tilt-induced changes in axial coma (*B*_{2}) are significantly reduced compared to those in a conventional microscope. This suggests that there exists a range of axes that are effectively coma free in a corrected instrument, which is consistent with the aplanatic property of the hexapole corrector (Rose 2002). This property relaxes the coupling between *C*_{1} and *τ* and allows the use of large beam tilts of 20–25 mrad for exit wave function reconstruction from aberration-corrected tilted illumination images without inducing significant image shift, defocus, astigmatism or axial coma. It also provides an additional experimental advantage where the beam tilt coils are used to correct for small errors in specimen orientation (Lentzen *et al.* 2002; Lentzen 2004).

### (c) Resolution limits due to geometric parallax

In contrast to wave function reconstruction using focal series data, the individual images within a tilt series dataset are not simply different measurements of the same projected specimen potential. Under tilted illumination, the specimen potential is projected along a slightly different direction compared to the axial case, and this introduces an additional phase shift in the image that can be estimated using a simple geometric parallax model. A phase shift of ±*π*/2 results in a contrast inversion, and choosing this value as a maximum phase variation, for a resolvable distance, *d*, the maximum parallax allowed between the top and bottom specimen surfaces relative to the middle of the specimen is *d*/4 (figure 5). For a beam tilt, *τ*, this leads to a maximum specimen thickness of approximately *d*/2*τ*. The dependence of maximum thickness on both tilt magnitude and minimum resolvable distance is shown in figure 6.

In the data reported here, the restored wave function corresponds to the central plane of the crystal so it is reasonable to average this phase shift over both the top and bottom surfaces of the crystal. However, these thickness limitations are only an approximation, and alternative limits produce different thickness constraints by considering the exit surface of the specimen as the reconstruction plane or by choosing a different maximum phase shift (Smith *et al.* 1983; Haider *et al.* 2000).

The simple geometric parallax argument described above is independent of the imaging conditions, but when higher resolution information is present in images, the limits imposed are more significant. Table 1 shows that, in order to restore a projected distance of better than 0.08 nm in all directions from tilted illumination data, a tilt magnitude of at least 14 mrad is required (figure 2*b*), and consequently the specimen must be less than 2.8 nm thick (figure 6). Alternatively, for a specimen for which the minimum achievable specimen thickness is 2 nm, a resolution of 0.0728 nm is achievable in all azimuthal directions (0.0645 nm along the tilt direction) using a tilt magnitude of 18 mrad.

### (d) Overall limiting conditions

The various limits for wave function reconstruction using tilted illumination images discussed in the previous sections are summarized in table 2. The parallax limit is the result of a simple geometric argument, and this has been found to be more stringent than the limit determined from a full dynamical calculation (Meyer 2002). However, these calculations clearly demonstrate that the aperture synthesis approach to image reconstruction can use larger tilt magnitudes under aberration-corrected imaging conditions. This gives the possibility of greater resolution improvement, significantly beyond the axial information limit of the microscope. At these very high resolutions, parallax considerations become increasingly significant, leading to a new regime where the sample limits the resolution attainable in tilt series exit wave function reconstruction.

## 3. Experimental

All data reported in this paper were acquired using a 200 kV instrument (JEOL JEM-2200MCO) (Hutchison *et al.* 2005) fitted with probe and image hexapole correctors. Prior to the collection of suitable image datasets for exit wave function reconstruction, an initial microscope alignment was performed using images of a thin amorphous germanium film supported on a thicker holey carbon film. Initially, the two-fold astigmatism was adjusted manually and the objective lens defocus was set to a suitable under-focus condition (typically −500 nm for magnifications of 600 000 times).

The imaging corrector was aligned by iterative acquisition, and correction using Zemlin tableaus of diffractograms recorded at several tilt angles between 18 and 40 mrad (Zemlin *et al.* 1978; Zemlin & Zemlin 2002). After final correction of the aberration coefficients to third order, the residual phase plate had a *π*/4 phase shift limit at a radius of 18–25 mrad. The specimen area of interest was then brought to focus using a piezoelectric stage in order to avoid altering the objective lens excitation.

Immediately prior to data acquisition, a final measurement of the aberration coefficients was performed using a local region of amorphous material. Although the accuracy of this measurement is generally poor for the third-order coefficients, it is useful to have an accurate, local measure of the first- and second-order coefficients, which change over several hours of operation (bib33; Haider *et al.* 2000), whereas the third-order coefficients are stable over several days.

All images were zero loss filtered using an in column post-specimen energy filter (Tsuno *et al.* 1997) with an energy selecting slit width of 10 eV, and which was aligned to ensure isochromaticity across the full field of view of the 4096×4096 pixel charge-coupled device (CCD) camera.

Accurate calibration of the illumination tilt coils is an essential step in data acquisition for tilt series reconstruction. In this work, tilt coil calibration was performed by measurement of the tilt-induced shift in a selected area diffraction pattern of a known sample. This method has been found to agree with an alternative calibration approach using diffractograms calculated from images of amorphous specimens (Meyer 2002; Meyer *et al.* 2002). The magnitude and direction of both *x* and *y* tilt coils were measured independently and were found to have a linear dependence on deflector current for tilt magnitudes up to 26 mrad. For acquisition of the tilt-defocus dataset used for exit wave function restoration (Meyer *et al.* 2002), customized acquisition scripts were written using Digital Micrograph.

Information transfer from the CCD camera was optimized using 2×2 pixel binning of the full 4096×4096 pixel field, and images were recorded at magnifications of 1 500 000 times corresponding to a pixel size in the image plane of approximately 0.015 nm. For the 4096×4096 pixel Gatan digital camera fitted to the instrument, these conditions represent an optimization of a representative field of view (approx. 4000 nm^{2}), usable exposure, camera read out and processing times (Meyer & Kirkland 2000).

The exit wave function reconstructions presented here were calculated using a linear Wiener filter which, if the aberration coefficients are known, optimally suppress transfer of the conjugate wave in the presence of noise (Wiener 1949; Saxton 1988). The aberrations were measured locally from a dataset of images of known tilts, to a high accuracy using a novel approach (Meyer *et al.*2001, 2002, 2004; Meyer 2002) for final input into the exit wave function restoration. In this approach, images were initially aligned relative to each other using a phase correlation function (PCF), and then absolute two-fold astigmatism and defocus values for each image were determined (as a function of tilt magnitude and angle) using a phase contrast index (PCI) function. By comparison with diffractogram fitting, this approach is less reliant on the presence of amorphous material, and hence, can be applied to largely crystalline samples. The tilt magnitude was determined during prior calibration, but the azimuthal tilt angle was determined as part of the fitting routine as described. For the restorations presented subsequently, consistency with the forward imaging process was established by simply calculating, from the restoration, the images expected for the imaging conditions determined for each of the experimentally observed images (Kirkland *et al.* 1995).

### (a) Comparison of exit wave function reconstruction using tilt and focal series data

As already discussed, geometric parallax limitations imply that samples must be thinner for reconstructions where tilted images are included, compared to wave function reconstructions using purely focal series data. However, the improvement in information transfer obtained using tilt-defocus datasets compared to focal series is evident from comparisons of the calculated effective wave transfer functions for these two geometries.

These can be generated from experimental datasets acquired with reference to the optimal conditions described earlier, and usefully predict the spatial frequencies that are transferred to the reconstructed exit wave function for a particular dataset. Comparison of the effective wave transfer functions for tilt and focal series geometries demonstrates that information transfer falls below 10 per cent at a resolution of 8.5 nm^{−1} (0.117 nm) for the focal series reconstruction, but persists to 14.1 nm^{−1} (0.071 nm) for the tilt series reconstruction (14.7 nm^{−1} (0.068 nm) along the tilt direction).

Information transfer at low spatial frequencies is, in general, worse for aberration-corrected data than for conventional images. Using uncorrected focal series data, transfer falls below 90 per cent of the maximum transfer for spatial frequencies less than 0.36 nm^{−1} (2.8 nm) at 200 kV. However, using aberration-corrected tilt or focal series data, information transfer falls below 90 per cent of the maximum for spatial frequencies less than 1.2 nm^{−1} (0.84 nm) and 1.4 nm^{−1} (0.7 nm), respectively. Transfer of the unwanted conjugate wave is also suppressed in the exit wave function reconstruction for all but very low spatial frequencies (to less than 10% transfer at spatial frequencies less than 1.2 and 1.4 nm^{−1} for aberration-corrected focal and tilt series exit wave function reconstructions, respectively).

### (b) Experimental exit wave reconstruction of a 〈1 2 3〉 gold foil

Initial experimental verification of the resolution improvement using tilted illumination images in an exit wave function reconstruction, as discussed in the previous section used a specimen with a real space lattice separation beyond the axial information limit. A gold foil oriented along a 〈1 2 3〉 direction satisfies this requirement (figure 7) and in addition, a 〈1 2 3〉 orientation has the advantage that, although gold has a high atomic number such that significant nonlinear image information transfer can be expected (Spence 2002), the {3 3 1} reflections cannot be produced from any lower order components of the image intensity spectrum.

For the experimental exit wave function reconstructions reported here, a thin gold foil specimen (provided by Christian Kisielowski of National Center for Electron Microscopy, Lawrence Berkeley National Laboratory, Berkeley, CA) was used, and experimental data were recorded using the conditions described in a previous section. Figure 8 compares the moduli of the Fourier transforms of exit wave functions restored using both focal series and tilt-defocus datasets. The transform shown in figure 8*a* only contains the {1 1 1} reflections since all other reflections have spacings smaller than the 0.1 nm axial information limit. Thus, the only lattice detail present in the phase and modulus of the exit wave function (figure 9*a*,*b*) consists of the vertical planes corresponding to the 0.235 nm lattice spacing.

In contrast, the transform in figure 8*b* shows intensity at positions corresponding to the {3 3 1}, {4 2 0} and {2 4 2} reflections, with lattice spacings of 0.093, 0.091 and 0.083 nm, respectively. These spacings are beyond the 0.117 nm axial information limit and are transferred in both the phase and modulus of the resulting specimen exit wave function, as demonstrated by the presence of the weak horizontal fringes in figure 9*c*,*d*. Of particular note are the {3 3 1} reflections that cannot be produced from any lower order components of the image intensity spectrum, although the {4 2 0} and {2 4 2} may be generated from interferences between the {3 3 1} and {1 1 1} diffracted beams. The {3 3 3} spots are not present in the diffractogram due to an imperfect crystal orientation.

Images recorded of a boundary adjacent to the area of the sample used for exit wave function reconstruction with the crystal tilted to an 〈0 0 1〉 orientation, reveal an unusual thickness profile in the foil, with a relatively constant thickness for the majority of the specimen, terminating in a steep wedge at the sample edge (figure 10). Both exit wave function reconstructions show a rapid contrast oscillation at the specimen edge arising from this wedge profile. Contrast arising from the small contamination layer at the edge of the specimen can be removed by shifting the focal plane of the restored exit wave function by 16 nm. This degrades the quality of the reconstruction in the crystal, but indicates that the contamination layer is on the top surface of the specimen, as illustrated schematically in figure 10.

## 4. Conclusions

In this paper, we have detailed the critical experimental factors for exit wave reconstruction from a tilt-defocus dataset of aberration-corrected images using six complementary tilts of equal magnitude. Consideration of geometric parallax limits and the tilt-induced changes in both the coherent and incoherent aberrations suggests an optimal tilt magnitude of 15–18 mrad at 200 kV. Exit wave function reconstructions have also been shown for an experimental tilt-defocus dataset including images with a large tilt magnitude of 18 mrad. A 〈1 2 3〉 oriented gold foil sample demonstrates that, by including tilted images in the reconstruction, information transfer in the exit wave function can be improved beyond the 0.11 nm axial information limit of the instrument used to acquire the data. Transfer of the 0.093 nm lattice fringes is observed in this sample in both real and reciprocal space, clearly demonstrating the resolution improvement provided by this new technique.

## Acknowledgements

The authors thank Dr C.J.D. Hetherington and JEOL Ltd for technical assistance and Dr C. Kieselowski for the provision of the gold sample used. Financial support from EPSRC (Grant EP/F048009/1) is gratefully acknowledged.

## Footnotes

One contribution of 14 to a Discussion Meeting Issue ‘New possibilities with aberration-corrected electron microscopy’.

- © 2009 The Royal Society