We demonstrate high spatial resolution phase retrieval of a non-periodic gold nano-structure using the method of Fresnel coherent diffractive imaging. The result is quantitative to better than 10% and does not rely on any a priori knowledge of the sample.
© 2008 Optical Society of America
Coherent diffractive imaging (CDI) offers a way to reconstruct the exit surface wave of a sample with diffraction-limited resolution. Recent demonstrations of the technical feasibility of the method offer the possibility of imaging small clusters of molecules, one of the main science drivers for the x-ray free electron laser projects around the world. In particular, it has now been shown that sufficient flux from precursors to these fourth-generation sources can be imparted to certain samples quickly enough in order to allow the collection of a high resolution diffraction pattern before a sample is damaged . Research into CDI is addressing issues such as improving the iterative algorithm [2,3] used to invert the diffraction pattern, the quality of the solution in the presence of noise [4,5] and the effect of partial coherence [6,7]. Other issues involve the question of the uniqueness of the solution  and the limitation of a finite sample support . The method of Fresnel CDI (FCDI) has been explored recently as a way forward with many of these issues [10–14].
FCDI is differentiated from CDI by the fact that the phase of the illuminating wave field is non-planar. For instance, in the case of a sample a short distance from a focal point, the phase of the illuminating beam can be approximated as spherical and the resulting far-field diffraction pattern contains a mixture of both spatial and spatial-frequency information, reducing the number of degrees of freedom in the solution space so that the iterative procedure converges to a unique solution, and does so much faster than for the corresponding plane wave CDI case .
FCDI requires that the illuminating beam be well characterized and, in order that the spatial registration of the illumination with the sample be maintained over the course of an experiment, that there be a high degree of stability between the beam and the sample. Interestingly, the method can actually be used to perform the required beam characterization  and the successful recovery of the shape of a test object  shows that the required stability can be achieved.
There has been relatively little work in either FCDI or CDI in demonstrating that the complex transmission function of a sample can be quantitatively recovered. This is a crucial requirement before CDI methods can be employed to directly quantify variations in sample phase, corresponding, for example, to thickness for a sample of uniform, known composition and density. Accordingly, in this paper we demonstrate that the complex exit surface wave of a sample retrieved by FCDI can be used to quantitatively obtain the sample thickness of an object.
The Fresnel coherent diffractive imaging procedure involves both the characterization of a wavefield, such as that focused by an optic like a Fresnel zone-plate, and the use of that wavefield as the illumination in a coherent diffractive imaging experiment. A procedure for determining the illuminating field has been described elsewhere , while the recovery of the exit surface wave of a sample illuminated with such a field by phase retrieval of its diffraction pattern has also been demonstrated . In both cases, the retrieval of phase from intensity data involves the judicious use of a priori information and the numerical propagation of wavefields between fixed planes.
The imaging procedure employed here defines the exit surface wave in the plane of the sample, ψ exit(r)=ψ inc(r)[T(r)+1], to be composed of two components: the illuminating field, ψ inc(r), and the modification of that field by the sample, described by T(r), where r defines position in the plane of the sample and perpendicular to the direction of propagation of the field, which is assumed to be predominantly in the z-direction. The field component modified by the sample undergoes absorption and phase advance according to the index of refraction of the material. If the projection approximation holds for the experiment, which requires that the maximum thickness of the sample, τmax, obeys the condition that τmax<1/kNA 2 where k is the wavenumber of the incident light and NA is the numerical aperture of the illumination, then the optical path through the sample has a negligible deviation from the sample thickness and we may directly relate T(r)to the sample thickness distribution via the index of refraction by using the paraxial form of the inhomogeneous Helmholtz equation. That is, we may write:
where n(r) is the complex index of refraction. For x-rays the index of refraction can be written as n=1-δ+iβ, where δ≪1 and β≪1 so that we may reduce Eq. (1) to:
If we may further assume that the sample is homogeneous, then the exponents in Eq. (2) become exp[-iδkτ(r)]exp[-βkτ(r)]. The reconstruction algorithm employed here  involves the subtraction of the complex illuminating field from the field leaving the sample to yield T(r)ψinc(r) as the reconstructed quantity, it is apparent that the phase and amplitude of the reconstruction of a thin homogeneous object is related to the index of refraction in a straight-forward way, yielding a direct experimental determination of τ(r).
In the case of plane wave CDI the illuminating wave is implicitly assumed to be uniform in amplitude and phase and so plays no role in the reconstruction process; indeed, it is often eliminated from the data through the introduction of a beam stop. It follows that the phase of the reconstructed object is only known relative to the rest of the sample - a priori information concerning the sample is required if an absolute thickness measurement is to be obtained. In contrast, Fresnel CDI information cannot be understood without explicit reference to the illuminating wave and so the self-consistency imposed by the iterative procedure necessarily yields a reconstruction with a phase relative to the incident field. If the composition is known, for example, the phase of the quantity on the left hand side of Eq. (2) provides measurement of the thickness distribution, τ(r), of the object.
The experiment was carried out at beam line 2-ID-B of the Advanced Photon Source. The experimental geometry is shown in Fig. 1 and is identical to that described in . The x-ray beam was monochromatized by a spherical grating monochromator before illuminating a 160 µm diameter Fresnel zone plate (FZP) with an outer zone width of 50 nm giving a focal length of 11.8 mm at 1.83 keV, the energy used in the experiment. To remove the higher diffraction orders produced by the FZP, a 15µm order-sorting aperture was placed at the focus.
A 28 µm central stop was placed upstream of the FZP which served to remove the direct beam so there was no need for a beam stop on the CCD detector. The direct detection CCD detector had an array of 1024×1024 pixels, each square, with a side-length of 24 µm. The sample was placed 1.1 mm from the focus of the FZP. The x-ray beam remained in vacuo until the central stop and reentered vacuum immediately after passing through the sample whereupon it entered a 0.48 m long evacuated flight tube.
The transverse coherence of the beam has been previously studied  and the entrance and exit slits of the monochromator were set so as to provide coherence lengths of at least 135 µm in both transverse directions at the FZP thus producing a coherence length larger than the sample at the sample plane .
The sample consisted of a collection of approximately oval shapes with sizes ranging from 300 to 800 nm across distributed over an area of approximately 5×5 µm2. The sample was fabricated in gold by electroplating from a pattern produced by electron beam lithography and was supported on a 1µm thick Si3N4 substrate. The density of the gold was calculated by measuring the x-ray transmission through an area that had been uniformly plated under identical conditions to the sample used in the experiment. Using tabulated values  for the attenuation coefficient and after measuring the thickness of the plated region we obtained a value of 18.41±0.09 g cm-3 for the density. Using the real part of the atomic scattering factor from the same tabulation then gives a value of δ=6.13±0.03 ×10-4 for the decrement to the real part of the refractive index. The errors were estimated based on the variation in the measured transmission. Possible systematic errors include the uniformity of the electroplated area and fundamental limitations in available form factor tabulations. Profilometry indicated that the first effect is relatively small (less than 5%) and the range across available form factor tabulations is also small (of the order of 5%).
The dynamic range of the CCD camera was limited by the well-depth in the CCD chip and so the required dynamic range was obtained by summing 50 or more sets of data. We found that the sample moved over the time that the exposures were acquired and so the datasets were correlated with each other and only those sets that displayed a high degree of correlation were included in the sum. Figure 2(a) shows an example of the resulting data set.
where n represents pixels on the detector, I is the measured intensity [Fig. 2(a)], andFig. 3(c). Note the apparent lower thickness visible as a lower grey-level around the edges of the gold structures.
Figure 3(b) shows an image of the same sample obtained using the same experimental setup configured as a scanning transmission x-ray microscope (STXM). That is, the sample was scanned transversely in the focal plane of the FZP and the transmitted intensity recorded at each point. This gave an image of the absorption contrast in the sample at 75 nm resolution, corresponding to the step size used in the scan, which is slightly worse than the nominal zone plate resolution of 63 nm. The reconstructed FCDI image agrees well with the STXM image as to the overall shape of the sample but it can be seen that the resolution in the reconstruction is significantly better than the STXM image.
Resolution in coherent diffractive imaging has been estimated by determining the scattering angle at which the reproducibility of the phase solution drops below a certain level . However, we find that in FCDI different random start guesses converge to almost identical phase solutions. Accordingly, we define the resolution based on the extent of the
measured scatter. This is measured with respect to the centre of the diffraction pattern corresponding to the gold features, which can be seen in Fig. 2(a). In our case this gives an estimated resolution of 52 nm – significantly better than the 63 nm possible with this STXM and the 75 nm actually obtained. Figure 3(d) shows a lineout comparison of the reconstructed thickness with the height measured using an atomic force microscope (AFM). The AFM scan is broadened significantly, as shown in Fig. 3(a), due to tip convolution with the steeply rising sides of the sample. Consequently, only the central portions of each shape are expected to give a reliable AFM value – and it can be seen that in this region the AFM heights and reconstructed thicknesses agree well. The lower grey-level region around each shape in Fig. 3(c), mentioned above, is explained by the “mushroom” profile of the samples seen in the tilted scanning electron microscope images shown in Fig. 3(e). As a final test of the quantitative nature of our results we averaged the height obtained by AFM in the central regions of the four shapes shown in Fig. 3(c) and compared this to the average reconstructed thickness obtained in the same regions. The results are summarized in Table 1. The errors are estimated from the variance in the heights recovered within each region; as this includes intrinsic height variation the errors are therefore conservative. It can be seen that the values are in good agreement, with less than a 10% maximum variation between all measurements. The STXM transmission can also be used to estimate the thickness and the results are on average about 20% lower than for the AFM results. We attribute this difference to that fact that there are only a few STXM pixels across the measurement area which includes a significant overlap with the thinner portions of the sample away from the central region.
This ability of the method to recover the thickness of the samples emphasizes the important place that FCDI has in the pantheon of imaging techniques. Scanning electron microscopy is sensitive only to the surface shape of a sample; STXM provides less resolution and will provide a poor thickness estimate when absorption through a sample is negligible. Transmission electron microscopy can provide high resolution images, but multiple scattering can prevent meaningful thickness measurements for samples more than about 100 nm in thickness. FCDI images can provide information that cannot be obtained by these other methods and which can be used as projections for high resolution tomography.
The authors acknowledge the support of the Australian Research Council Centre of Excellence for Coherent X-ray Science and the Australian Synchrotron Research Program. Use of the APS was supported by the U.S. Department of Energy, under Contract No. DE-AC02-06CH11357. Thanks to Andrew Stevenson for assisting with the gold density measurements.
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