## Abstract

We demonstrate use of a complex constraint based on the interaction of x-rays with matter for reconstructing images from coherent X-ray diffraction. We show the complementary information provided by the phase and magnitude of the reconstructed wavefield greatly improves the quality of the resulting estimate of the transmission function of an object without the need for *a priori* information about the object composition.

©2010 Optical Society of America

## 1. Introduction

Coherent diffractive imaging (CDI) is a promising form of high-resolution lensless microscopy. It has been shown [1] that the far-field diffraction pattern from plane-wave illumination of an isolated object almost always gives a unique solution (apart from trivial ambiguities) [2]. The CDI method typically relies on iterative means to find an object distribution that is consistent with the measured diffraction pattern and any physical constraints that are known about the object. The constraints that are available to be applied to the data are critical to the ability to recover a high-quality image. The first, and obvious, constraint is that the diffraction pattern of the object must match the data – this is known as the “modulus constraint”. The diffraction pattern is measured in the far-field and so can only be properly measured if this object is finite in extent. If the extent of the object is known or can be estimated, then this is an additional constraint and is known as the “support constraint”. Additionally the support can be updated dynamically via the “shrinkwrap” method [3]. Other possible constraints include a knowledge that the object might be purely absorptive (i.e., it has a real transmission function) or that the object has a known composition.

Significant work has also been undertaken seeking to optimize the iterative algorithms. Accordingly, a number of phase retrieval algorithms have been proposed. These include error reduction (ER) [4], hybrid input output (HIO) [4], difference map (DM) [5], saddle point optimization [6, 7], hybrid reflection projection (HPR) [8], relaxed averaged alternating reflections (RAAR) [9], averaged successive reflections (ASR) [10] and solvent flipping [11]. Several successful demonstrations of CDI have been published [12–20], including applications to biological imaging [13], materials science [14, 17, 18, 20] and ultrafast imaging using free-electron lasers [15].

It is also possible to use illumination with significant phase curvature [21]; in this case, the diffraction pattern must be dealt with in the Fresnel approximation. A diffraction pattern produced in this way has been shown to provide a unique solution [22]. This approach (Fresnel Coherent Diffractive Imaging or FCDI) has been shown to be fast converging [23] as well as robust in the presence of partial coherence [24, 25], and has been used to image a malaria-infected red blood cell [26]. The known illumination can be used to supply a reference phase that can provide a quantitative estimate of the transmission function of the sample [27, 28]. The finite illumination can be used to define the support, so that it is possible to image an extended object [29], removing the need for a physically finite object.

Imaging by FCDI begins with a determination of the illuminating wavefield [21]. With the illuminating wave subtracted from an estimate of the diffracted wave, the iterative method proceeds as for the plane-wave case with repeated applications of the Fourier modulus constraint and the support constraint. The optimisation of the iterative process with these two constraints has been explored [4, 6].

In this paper we exploit the complex response of matter to x-rays as an additional constraint. The phase and magnitude of the wave exiting a sample typically have a close relationship in the x-ray regime. A logical limit of this observation is the case of a sample of known and uniform composition in which the phase change in the transmitted wave is proportional to the logarithm of the transmitted wave magnitude. The idea of homogeneity has been used with considerable success in conventional x-ray phase imaging [30–32]. We generalize this concept and test the method using simulations and experimental data.

## 2. Method

Consider a plane, highly-coherent, quasi-monochromatic wave incident on a sample with complex transmission function $T\left(\text{r}\right)=A(\text{r})\mathrm{exp}[i\varphi (\text{r})]$ where *A* is the scalar amplitude (which here is synonymous with magnitude) and *ϕ* the phase modification respectively due to the sample and ** r** is a two dimensional vector in the plane of the sample. In the projection approximation [33], the phase modification and magnitude of the wave leaving the sample can be written in terms of the decrement to the real and the imaginary parts of the refractive index, $n=1-\delta +i\beta $, respectively as follows:

*z*is distance in the direction of propagation along the optical axis. Note that in the case of a homogenous sample then $\varphi (\text{r})\to -k\delta \tau (\text{r})$ and $A(\text{r})\to \mathrm{exp}\left[-k\beta \tau (\text{r})\right]$, where $\tau (r)$is the projected thickness of the sample. The wavefield leaving the object has the form:

In the following, we will first consider the projection of a single-material object. The method is readily extended to multiple materials and to three-dimensional imaging. We note that during the iterative procedure the thickness estimates obtained from the phase and magnitude of the transmission function can converge at different rates. Accordingly, we exploit the relationship between the phase and magnitude for a given material as in Eq. (4) to introduce an additional constraint into the phase retrieval algorithm. We further expect that the combination of phase and magnitude components of the wavefield will enforce a greater degree of self consistency in the iterated wavefield, providing an additional constraint and enhancing convergence.

The iterative procedure used in CDI and FCDI is described by the portion of the flow chart in Fig. 1 that is in blue and our complex constraint adds in the green sections. The following numbered paragraphs describe the boxes and actions in Fig. 1. Following terminology we have used earlier [34] we refer to quantities before the operation of the support constraint, as estimates and the corresponding quantities after operation of the support constraint as iterates. Quantities or operations that take place in the detector space are denoted by the caret, ^, symbol.

- 1. The square root of the measured intensity, ${I}_{\text{measured}}$, is combined with a guess for the phase, ${\varphi}_{\text{guess}}$. Other starting points are possible; one that is commonly used is to begin at box 5 with a phase guess and uniform intensity filling the support.
- 2. This gives the starting estimate for the complex wavefield at the detector, ${\widehat{\Psi}}_{j}$, where we have dropped the explicit position dependence on the quantities in the boxes for simplicity.
- 3. In Fresnel CDI the first step, ${\widehat{\pi}}_{A\Psi}^{-1}$, is to subtract the illuminating wavefield from the estimate of the wavefield in the detector plane. Assuming single scattering, this produces an estimate of the scattered wavefield, ${\widehat{\Psi}}_{j}^{s}$, which, in the projection approximation, is related to the current estimate for the transmission function, ${T}_{j}$, by:$${\widehat{\Psi}}_{j}-{\widehat{\Psi}}_{0}=\stackrel{\wedge}{\left({\Psi}_{0}{T}_{j}\right)}-{\widehat{\Psi}}_{0}={\widehat{\Psi}}_{j}^{s}\text{.}$$
In CDI, subtraction of the wavefield is not performed explicitly, but the presence of a beamstop that blocks the central diffracted spot in the measured data amounts to the same thing in practice. The missing data can be recovered during the reconstruction process or inserted from

*a priori*information. Not having to deal with this problem is one of the advantages of using the FCDI approach. - 4. The inverse propagation operator, ${\widehat{\pi}}_{P}^{-1}$, is applied to the estimate of the scattered wavefield at the detector to give the corresponding estimate for the scattered wavefield, ${\Psi}_{j}^{s}$, in the sample plane. In CDI the propagation operator is the Fourier Transform; in FCDI Fresnel diffraction is used.
- 5. Following the dashed blue line, a support constraint, ${\pi}_{S}$, is applied to produce the iterate of the scattered wavefield in the sample plane, ${\Psi}_{j}^{{s}^{\prime}}$. The form of the support constraint can vary; a simple method is to set the value of the scattered wavefield to zero outside a defined area and to leave the scattered wavefield inside unchanged. The complex constraint proposed here (green boxes) can be viewed as a modification to the support constraint.
- 6. The illuminating wavefield is then added to the iterate of the scattered wavefield, ${\widehat{\pi}}_{A\Psi}$, to produce the iterate of the wavefield in the detector plane. Again in CDI this step is usually not performed as only the scattered intensity is measured at the detector.
- 7. The modulus constraint, ${\widehat{\pi}}_{m}$, which replaces the magnitude of the wavefield iterate with the measured magnitude, is then applied to complete one cycle of the loop and to update to iteration
*j*+ 1 in box 2.- 4a. In the approach proposed here we diverge from the FCDI path by replacing the dotted blue arrow with the green arrows. The estimate of the scattered wavefield in the sample plane (box 4) is divided by the illuminating wavefield and unity added, ${\pi}_{M\Psi -1}^{-1}$, to give the corresponding estimate of the transmission function as per Eq. (6). This can be written as:
where the magnitude and phase terms are the current estimates thereof.

- 4b. An iterate of the transmission function is then created via the combined operation of the support constraint and our new complex constraint, ${\pi}_{C}$, which is described below. Note that as we are operating on the transmission function the support, ${\pi}_{{S}^{\prime}}$, is defined as unity outside the support, rather than zero as for the scattered wavefield case. The iterate of the transmission function is multiplied by the illuminating wavefield and unity added to give a new iterate of the scattered wavefield in box 5 and the loop is rejoined as before.

*N*pixels containing the sample:

*Ψ*. It should be noted that, while ${\overline{C}}_{j}$can be regarded as an estimate of the effective $\beta /\delta $ratio for the projected object, knowledge of the value of that ratio is not required in this algorithm. Like many phase retrieval methods the approach here will require some form of phase unwrapping where there are large phase excursions. However, for many x-ray imaging experiments with samples that obey the projection approximation phase excursions are small enough that unwrapping is usually unnecessary. The new transmission function is then used in Eq. (3) to generate the input for calculating the next iterate of the wavefield at the detector. We refer to this modification of the usual iterative procedure as the application of a “complex” constraint.

The extension to multi-material imaging is straightforward. The calculation of $\overline{{C}_{j}}$is performed for each region of the sample that has a fixed value of the ratio calculated in Eq. (4). The transmission function is then updated by applying the appropriate $\overline{{C}_{j}}$values to the relevant regions. Using Eq. (11), which averages over a region, avoids the problem where due to noise or non-converged iterates there are zeros in the phase of an individual pixel. The procedure for identifying different regions could be performed manually, but is also readily automated using image segmentation methods common in image processing. If the support region is denoted by*S*, then this can be broken up into *W* sub-regions, ${S}^{w}$ such that each sub-region element is an element of *S* (${S}^{w}\in S,\forall w$), no element of one sub-region is in another (${S}^{w}\notin {S}^{v}$for $w\ne v$) and S is formed by the union of all the sub-regions ($S={S}^{0}\cup {S}^{1}\mathrm{...}\cup {S}^{W}$). At each iteration, the constraint is applied over some or all of the sub-regions with the appropriate $\overline{{C}_{j}}$value, ${\overline{{C}_{j}}}^{w}$, calculated via Eq. (11) for the elements in the corresponding sub-region. The constraint is applied via Eq. (8) and Eq. (9) with all other regions left unchanged. In order to assist the segmentation of non-homogeneous regions, one approach is to perform a standard FCDI iteration (boxes 2, 3, 4, 5, 6, 7) for a fixed number of iterations to the point where regions are identifiable. The complex constraint can then be applied to segmented regions to refine the solution. In the following sections we simulate and then experimentally test this approach.

## 3. Simulations

We simulated the diffraction pattern of an object illuminated by a point source of spherical wavefronts with a Fresnel number of 24, defined as the number of *π*phase variations from the center to the corner of the 270 x 270 pixel support shown in Fig. 2(a)
. The illumination was simulated to be compatible with the geometry shown in Fig. 2(b). The array used was 896 x 896 pixels. A homogenous sample with surface roughness characterised by a root mean square value of 4.8% of the sample thickness as shown in Fig. 2(a) was used in order to approximate the actual sample used later. A number of different sample compositions were simulated so that the material had a *β/δ* ratio ranging from 0.2 to 5.0, typical for many materials in the x-ray band – with values greater than 1 usually only found near absorption edges. This represents a range from a pure phase object to a purely absorptive object.

Reconstructions were performed using the FCDI version [34] of error reduction (ER) and error reduction with our complex constraint (ER + C). Each reconstruction was run for 3000 iterations with a random phase as the initial guess. In order to demonstrate the method without *a priori* knowledge, all reconstructions using ER + C assumed no knowledge of the value given in Eq. (5). The value from Eq. (11) was calculated every iteration.

Reconstruction quality was measured using a conventional measure that assesses the level of agreement of the *j ^{th}* estimate of the intensity, ${\left|{\widehat{\Psi}}_{j}\right|}^{2}$, with the measured intensity,

*I:*

*N*represents all pixels on the detector.

Initial reconstructions using the complex constraint were carried out using four sets of values for (α_{1}, α_{2}): (0, 0), (1, 0), (1, 1) and (0, 1). These correspond to ER, using the phase estimate to update the magnitude estimate, mixing both equally and using the magnitude estimate to update the phase estimate respectively. These simulations allow a test of the effectiveness of the different constraint schemes assessed against objects ranging from phase dominated to absorption dominated. The final error metric values from Eq. (12) are plotted in Fig. 3
for each ratio of *β/δ*.

It is clear that the effectiveness of the different weighting schemes depends on the ratio of the linear attenuation and phase terms (*β/δ*). For low values of *β/δ*, the scheme updating the magnitude with the phase succeeds very well whereas for high values of *β/δ* the scheme updating the phase with the amplitude outperforms the others. When the ratio is close to one, using the phase estimate to update the magnitude and vice versa simultaneously works toprovide a good reconstruction. In Table 1
, the number of iterations required, for a range of values of α_{1:}α_{2} to reach an error metric of 2×10^{−30} are shown for the ER+C method. In each case, the ER method reaches an error metric of no better than 10^{−6}. In all cases, it can be seen that the appropriate weighting scheme far outperforms the conventional ER approach. To emphasize the quality of the reconstructions, Fig. 4
shows a profile through the magnitude and phase of the recovered object for ER, ER+C and the original object. The lines for ER+C and the original object are indistinguishable, while the errors from ER for the same number of iterations are obvious.

Another simulation was carried out using a sample that consisted of three regions, each having a different constant ratio of *β/δ*. The reconstruction was performed using ER and ER+C. Each reconstruction was run for 1000 iterations. The support consisted of the three regions shown in Fig. 5(e)
. The constraint using ER+C was applied to the three regions individually. For each of the three regions the ratio from Eq. (11) was calculated at each iteration. For each region, we used α_{1}=1 and the expected optimal α_{2} for the homogenous *β/δ* values obtained by interpolating from Table 1 using:

*χ*value was 8×10

^{2}^{−7}and 4×10

^{−16}for ER and ER+C respectively. The reconstructed magnitude and phase for the transmission function for ER and ER+C can be seen in Fig. 5.

## 4. Experiment

The experiments were carried out at beamline 2-ID-B at the Advanced Photon Source [35]. A coherent 2.535 keV x-ray beam illuminated a 160 μm diameter Fresnel zone plate with a 50 nm outer-zone width and focal length of 16.3 mm at this energy. A 43 μm diameter central stop in conjunction with a 30 μm diameter order sorting aperture placed at the focus blocked the zeroth and high-order diffraction, allowing the first diffraction order to illuminate the sample. The sample was placed ~1 mm from the focus (Fig. 2(b)) corresponding to a Fresnel number of 44 taken from the center to the edge of the illumination. This FCDI geometry is similar to that used in previous work [34]. The diffraction patterns were recorded approximately 0.67 m from the sample with a Peltier-cooled CCD camera with 2048 x 2048 13.5 μm pixels. This corresponds to a pixel scale in the reconstruction of 13.3 nm. The resolution will depend on the extent and signal to noise of the scattered signal and will in general be worse than the pixel scale. The zone plate, sample, and CCD were maintained *in vacuo*. The beamline exit slit, located 11.5 m upstream was set to illuminate the zone plate nearly coherently [36]. The sample consisted of a 150 ± 15 nm thick gold “X” pattern, shown as a schematic at Fig. 4(a), supported on a 100 nm thick Si_{3}N_{4} membrane (X40-30-2 calibration target, Xradia, Inc.).

In order to increase the dynamic range of the diffraction pattern 26 frames of data were summed together. The data for the gold X can be seen in Fig. 6(a) . The diffraction patterns were chosen by cross-correlating the diffraction patterns with each other from a larger data set and choosing the data frames with the highest correlation coefficient (> 0.9958) [34].

The illuminating wavefield was reconstructed [21] before the diffraction patterns were phased. To test the improvement the complex constraint brings, a reconstruction was done using ER and another using ER+C. In principal no knowledge of the composition of the material is required for the complex constraint to operate as Eq. (11) calculates the material dependant parameter at each iteration. In practice, when the material has a known uniform composition using a known β/δ ratio is expected to provide a stronger constraint by enforcing the actual value for ${\overline{C}}_{j}$at each iteration. The values of the complex refractive index for gold at 2.535 keV are *δ*= 3.21 × 104 and *β*= 1.88 × 104 [37], giving a *β/δ* ratio of 0.59 [37]. The complex constraint was enforced using α1 = 1.0 and α2 = 0.46. Reconstructions using ER+C and ER were performed. In both cases, shrinkwrap [3] was also used to update the support with the iterate convolved with a 1.25-pixel-standard deviation Gaussian and thresholded at 10% of the maximum value of the image. The support was updated for the first 25 iterations, after which it remained fixed. Figures 6(b) and (c) show the final support for ER and ER+C respectively. The support generated for ER using the same parameters as ER+C fails to find a good outline of the sample. Increasing the threshold resulted in the support excluding regions inside the X. The final array used for reconstruction was 1820 x 1820 13.5 μm pixels. The reconstructions using ER+C and ER were run to convergence and both stopped at 100 iterations. The convergence with the complex constraint is smooth and if it is switched off the solution remains stable.

Shown in Fig. 7 is the thickness calculated from the reconstructed magnitude (left) and phase (right) for ER. Figure 8 shows the corresponding values obtained using ER+C. The reconstructed thickness shown is the average from ten independent reconstructions each using a random start guess for the phase. In each case the thickness, $\tau (r)$, is calculated under the projection approximation for a homogeneous object using:

The reconstructed phase and magnitude using the complex constraint clearly show an improvement in the overall shape of the test object. The outline of the object is more clearly defined and the bar towards the bottom is considerably sharper in both the magnitude and phase. A profile (Fig. 9 ) through the bar feature below the X compares the reconstructed thickness for both the ER and ER+C schemes. Not only does ER+C reconstruct the shape of the bar much better than for ER alone but the values of the reconstructed thickness from the magnitude and phase estimates agree quite well.

The horizontal and vertical lines present in the reconstruction are artifacts due to fringes caused by clipping of the beam by the edges of a Si_{3}N_{4} window in the beamline, approximately 3.5 m upstream of the zone plate. In principle, structure in the beam can be characterised and removed in the analysis. However, in this case the fringe position varied during measurement of the diffraction patterns due to drift in the beam trajectory or the window position. This instability may also be responsible for the radial structures seen in the reconstructions. The presence of the line artifacts also provides support for our contention that using extra *a priori* knowledge, in the form of the *β/δ* ratio, provides a stronger constraint. When instead of fixing the ratio we allowed it to float the final *β/δ* ratio was incorrect (0.83) and the thickness estimated from Eq. (14) disagreed between the phase and magnitude estimates.

## 5. Extension to non-homogenous samples

We performed a second diffraction measurement using the same parameters as described in section 4, but where the sample was replaced with a pattern fabricated in gold and chromium as shown in Fig. 10(a) and (b)
. The zone plate was also placed in a different region of the illumination where the fringes from the upstream window were less noticeable, allowing us to let ${\overline{C}}_{j}$float in the operation of the complex constraint. 100 frames of data with a correlation coefficient >0.995 were summed to produce the measured data set for analysis. The reconstruction was performed using ER (200 iterations) and ER+C. For ER+C, ER was initially run for 200 iterations, the object was then segmented manually into three regions and ER+C was then run for a further 200 iterations. During both ER and ER+C shrinkwrap was used with the support generated from a thresholded image of $\left[-\mathrm{ln}\left(A\left(\text{r}\right)\right)-\varphi \left(\text{r}\right)\right]>0$. The constraint using ER+C was applied to the three regions individually. For each of the three regions the ratio from Eq. (11) was calculated at each iteration. For each region we used α_{1} = 1 and used the expected optimal α_{2} for the homogenous *β/δ* values as per Eq. (13). The refractive index at 2.535 keV for chromium with a density of 7.19 g.cm^{−3} is *δ*= 2.12 × 10^{−4} and *β*= 1.85 × 10^{−5} [37], giving a *β/δ* ratio of 0.0873. The numbers for the gold are as before. The final ${\beta}_{Au}/{\delta}_{Au}$calculated from the iterative scheme was 0.57 which compares well to the actual value of 0.59 and for ${\beta}_{Cr}/{\delta}_{Cr}$ we obtained 0.098, which compares well to the actual value of 0.087.

## 6. Discussion and conclusion

Incorporation of the complex constraint in FCDI enhances reconstructions of objects without problems of stagnation. The demonstration here can be readily extended to plane wave illumination (CDI). This is important since visually good reconstructions are readily achievable whereas high resolution *quantitative* reconstructions with a loose support have been much harder to achieve from single diffraction patterns for complex objects. The complex constraint can also be used for objects that have regions with diffuse boundaries or smooth gradients in *β* and *δ*. In such cases constraining the ratio in Eq. (4) to be within a known range – which in practice can be very broad – will improve convergence. In addition many objects that are currently investigated using x-ray tomography have regions of difference that researchers wish to segment. Examples include agglomerated objects and objects with voids or porous structure. In these cases the approach described in Section 2 is applied to voxels instead of pixels. In practice this could done by choosing regions based on a histogram of ${\overline{C}}_{j}$or even of *β* or *δ* directly as in three-dimensions thickness variations are dealt with by the known voxel length. In three-dimensions, the segmentation of the sample into discrete regions should also more readily correspond to actual regions composed of different materials than in the projection cases demonstrated in this paper as tomographic imaging avoids the projection overlap of different regions. Consequently, it is expected that using the complex constraint in three dimensions should improve the reconstruction quality.

## Acknowledgements

We 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 Advanced Photon Source is supported by the U.S. Department of Energy, Office of Science, and Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

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