Holographically aided iterative phase retrieval

1. Introduction

Fourier transform holography (FTH) is a lensless nanometer spatial resolution imaging technique which is ideally suited for single-shot imaging at free-electron laser (FEL) sources due to its spatial resolution, noise-resistance, and the simplicity of the reconstruction process allowing real-time analysis of diffraction data. The otherwise absent phase information in the measured diffraction pattern is provided via interference from a reference object [1, 2] placed at a distance at least three sample radii from the center of the sample. FTH is particularly suited to the imaging of magnetic domains at soft x-ray wavelengths using resonant magnetic circular dichroism. FTH was first demonstrated for magnetic samples by Eisebitt et al [3], and has for example been applied on a femtosecond timescale at an FEL [4, 5], and for tomographic imaging of biological samples [6]. One obtains a reconstruction by computing the Fourier transform of a measured diffraction pattern, which produces a combined autocorrelation of the object and reference in the center and two cross correlations between the object and the reference – these cross correlations being the reconstruction. The spatial separation between the reference and the object is necessary so that the cross correlations appear distinct from the autocorrelation, and the resolution of the reconstruction is limited by the size of the reference object.

Coherent diffractive imaging (CDI) [7] is a similar technique which uses an iterative phase retrieval algorithm to recover the phase information instead of the holographic phasing with a reference beam. This allows one to illuminate only the sample thereby increasing the photon density on the sample compared to FTH, as the available photon flux can be focused to a correspondingly smaller area. In this case (CDI), the reconstruction resolution is determined by the maximum momentum transfer caused by scattering at the sample, while in FTH the size of the reference object is an additional limiting factor should an iterative refinement not be applied [8, 9]. The disadvantage of CDI in contrast to FTH is however, that any iterative phase retrieval algorithm is extremely noise sensitive [10], and obtaining a unique reconstruction is not always possible. The uniqueness problem has inspired a number of modifications to the traditional plane-wave CDI regime. Where one wishes to image an extended object, ptychography may be used where one scans an apertured beam across different overlapping regions of the sample, using the overlap between different exposures as a further constraint [10, 11]. Fresnel CDI [12, 13] is another method where the sample is illuminated with a diverging beam producing a Fresnel diffraction pattern instead of a Fraunhofer diffraction pattern. Fresnel CDI is much less sensitive to noise in comparison with regular CDI, but because a spatial shift between the beam and the sample produces a qualitative change in the diffraction pattern, nanometer sample stability is required. With Fresnel CDI, one also requires a measurement of the naked beam without the sample, which is problematic for its application to single-shot imaging at free-electron laser (FEL) sources. One may also improve the reliability of regular CDI by incorporating any additional a priori information into the reconstruction algorithm, with an example shown in reference [14].

Combining CDI with FTH [9, 15] provides one with the resolution benefits of CDI and the ability to use all the coherently scattered photons, along with the noise-resistance of FTH. Whereas the authors of references [9, 15] used a standard FTH geometry, we move our reference object closer to the object in order to increase photon efficiency by allowing one to focus the illumination on the sample alone. A further benefit is that this allows us to profit from an experimental setup where the position of the reference scatterer in relation to the sample is not determined prior to an experiment, for example when both are injected simultaneously from a liquid-jet at an FEL [16]. When the reference is in close proximity to the object, there will now be an overlap between the object reconstruction and its autocorrelation, necessitating the use of iterative phase retrieval to separate the two. Despite this, as our results will demonstrate, it is possible to visually pick out the FTH reconstruction well enough to define the constraint for the iterative refinement. In comparison to CDI without a reference, the presence of the reference object provides a significant defense against the effects of noise, which reduces the otherwise potentially onerous task of data-preprocessing [17] characteristic of plane-wave CDI experiments. Full coherence is also not necessary, provided one has sufficient knowledge of the coherence of the illuminating beam to incorporate this information into the iterative phase retrieval algorithm [18].

2. Theory

In a typical scattering experiment, an object with an exit surface wave $O$ scatters x-rays to the detector which are recorded with intensity $I={\left|FT\left(O\right)\right|}^2$. In practice however, one measures the intensity along with an error term $I+\Delta Ia$. With FTH, the reconstruction is obtained simply by calculating the Fourier transform of the measured intensity, and with the linearity of the Fourier transform operator one obtains $\Delta O=FT\left(\Delta I\right)$.

With iterative CDI, the reconstruction process demands that

3. Experiment

We tested our theory by imaging out of plane magnetic domains of approximately 80 nm width in a Cobalt/Platinum multilayer thin film. When imaged at a photon energy tuned to the Co L₃-edge (780 eV), this class of sample displays strong magnetic circular dichroism contrast when imaged with circularly polarized light. We collected diffraction patterns using circularly polarized illumination of both positive and negative helicities – with the difference of the two being used to display only the magnetic part of the signal [19]. The experiment was performed in the ALICE diffraction chamber [20] at BESSY II in Berlin at the soft x-ray undulator beamline UE56-2-PGM1. The scattered x-rays were collected on a charge coupled device (CCD) detector (2048 × 2048 pixels, 13.5 µm pixel size) 41 cm downstream of the sample, and unscattered light was stopped by a beam-stop in front of the detector. Our experimental setup was the same as used in a recent experiment [21], and is single-shot compatible except for the collection of diffraction data using two helicities chosen in this particular case. Implementation of the light polarizing setup proposed in [22] would allow us to perform this experiment at an FEL source today. We defined the field of view by a 2 µm circular aperture produced by focused ion beam milling in a gold film, with the reference aperture placed approximately 500nm away from the edge of the object aperture as shown in the scanning electron microscope (SEM) images in Fig. 1(a) . Two samples with these 2 µm object apertures were used – one with an 80 nm diameter reference, (approximately the magnetic domain size) and another with 200 nm reference. Diffraction patterns for both reference aperture sizes (with positive helicity circular polarization) are shown in Fig. 1(c) and 1(d). In order to control the quality of the iterative reconstructions, FTH references were placed at a distance of approximately 4µm from the object aperture, and subsequently closed by platinum deposition prior to collecting the data sets to be analyzed using iterative phase retrieval.

 

Fig. 1 (a) SEM image of the object aperture and reference aperture. The bright region on the right hand side of the object aperture is a closed reference aperture from a previous measurement. (b) the aperture array used for measuring the coherence. (c) The diffraction pattern from the sample with the small (80nm) reference, and (d) is the diffraction pattern from the sample with the larger (200nm) reference aperture.

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Prior to collecting the sample diffraction patterns, we characterized the coherence of the beam using the aperture structure shown in Fig. 1(b). The coherence lengths $\left({\sigma }_x,{\sigma }_y\right)$ can then be measured by comparing our experimental results with those predicted if the illumination was uniform and fully coherent, according to the theory in reference [23]. The application of this theory to our diffraction data from the aperture array shown in Fig. 1(b) put a lower bound on the horizontal and vertical coherence lengths of 3.3 and 4.2 µm respectively. From these measurements and using the assumption that the degree of coherence varies according to a Gaussian model in the horizontal and vertical directions, we calculated the coherent modes of the incident beam according to reference [24]. Over the 2.5 µm square region of the object aperture and reference aperture, the occupancy of the primary mode was 92%.

4. Results and Discussion

Figure 2(a) shows the FTH reconstruction of the sample with the 80nm close reference. This is the control image which Figs. 2(b) and 2(c) should be compared against. Figure 2(b) is the Fourier transform of the measured intensity after the distant FTH reference had been closed, and the domain pattern can be clearly seen overlaid upon its autocorrelation. In Fig. 2(c), the iteratively extracted magnetic domain pattern can be seen to compare favourably with the FTH control image, achieving a superior contrast to what was achieved with FTH. Further work will need to be done however to quantitatively asses under what parameters this scheme truly outperforms FTH. Figure 2(d) shows the best attempt at reconstructing a similar magnetic domain sample where the reference was approximately 200nm in diameter (significantly larger than the domain size). The poor quality of the reconstruction except for the aperture edge is consistent with the argument presented in the theory section that the enhanced reconstruction stability is provided only up to a resolution comparable to the reference object, and at higher resolutions one is working in the less stable regime of standard plane-wave CDI. Although CDI reconstructions of magnetic domains have been published [21, 25], one requires a higher quality scattering pattern in the absence of a reference to produce a publication quality image. The systematic inclusion of a reference object in future CDI experiments where possible is therefore expected to increase the quantity of publishable data obtained during synchrotron beamtimes. The iterative reconstructions shown (Figs. 2(c) and 2(d)) were obtained from the average of 20 reconstructions from random starting guesses using the Fienup hybrid input-output algorithm [26], modified to account for the partial coherence [18]. It should be noted here that the visible difference between each of the 20 reconstructions is minimal for the small reference data set, and it would have sufficed to have only run the algorithm once. For the large reference data set, the non-uniqueness of the solution at resolutions below 200nm is due to us being in the regime of standard phase retrieval, where a higher signal to noise ratio is required in order to obtain a reliable reconstruction.

 

Fig. 2 (a) FTH reconstruction of the object, (b) Fourier transform of the measured diffraction after the distant reference objects have been closed, (c) the iterative reconstruction obtained using the smaller 80nm reference object, (d) the iterative reconstruction of a similar magnetic domain sample but with a larger 200nm reference object.. The object aperture has a diameter of 2µm, and the pixel sizes in panels (a), (c) and (d) is 23 nm.

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The main difference between this method and standard phase retrieval is that the real space constraint is split into two parts. The sample constraint is drawn by marking the boundaries of the FTH reconstruction as shown in Fig. 2(b), and treated in the same way as with standard phase retrieval, but the reference set as a parameterized function (in this case a Gaussian or a pair of Gaussians for the large reference case) centered on the origin. The key here is that the reference contribution is kept fixed during the iterative reconstruction, with adjustments to the parameters made to optimize the fit to the experimental data at the end of each set of iterations. For this work, the parameters were optimized manually, but for future work we plan to automize the optimization process to further increase the speed and reliability of the technique.

In Figs. 2(a), 2(b), and 2(c), the difference of the positive and negative helicity images is shown in order to provide better visual contrast by excluding the non-magnetic parts of the object. It is our aim in a future experiment to test the ease of reconstruction as the reference is moved even closer to the object, to the point of sitting on the object perimeter.

It is also worth commenting on the ease at which the reconstruction of Fig. 2(c) was obtained following the experiment. This was the first experimental data analysed using this technique, and was obtained about 4 hours after uploading the experimental data into MATLAB, including the time needed to manually optimise the reference object parameters. By comparison, approximately 2 months work was required to process the data presented in our previous paper without a reference [21]. In future experiments once the reconstruction code has been refined to automatically optimize the reference object parameters we are confident to be able to obtain reconstructions taking of the order of 10 mins.

5. Conclusion

We have taken the photon efficiency and resolution benefits of CDI, and combined them with the reconstruction reliability of FTH to produce a robust and versatile imaging technique. The gain in efficiency over regular FTH is of special relevance to single-shot imaging, where sample dose limitations can be circumvented in a “diffract before you destroy” approach [27] and thus the high pulse intensity destroys the sample after a single shot. We see large potential in the application of our method on biological samples for example by placing a gold nanoparticle reference in close proximity to or potentially even on the surface of the sample. In a FEL setup, the samples (and reference objects) could then be injected using a droplet setup similar to the one developed in reference [16] Use of other types of reference objects such as sharp edges similar to reference [28] or a phase shifting reference [9] for use with hard x-rays, are obvious future extensions of this work.