We introduce theoretically and demonstrate experimentally a contrast transfer function based phase retrieval algorithm that reconstructs the projected thickness of an homogeneous sample using a polychromatic x-ray source. We show excellent quantitative recovery of test samples in 2D using a synchrotron source with significant harmonic contamination, and in 3D using a laboratory source.
© 2011 OSA
Since the discovery of X-rays in 1895, X-ray imaging has been a powerful tool in the non-destructive investigation of materials. In the intervening century the attenuation of an x-ray beam by the sample through photo-electric absorption has been the dominant method used to produce image contrast. More recently coherent scattering, which at a microscopic level is characterized by the real part of the refractive index, has been used to produce contrast from sources with a high degree of temporal and/or spatial coherence .
Since then a variety of methods [2–5] have been demonstrated which retrieve a quantitative map of the sample scattering potential – so called phase retrieval methods. Typically phase retrieval methods rely on both spatial and temporal coherence in the illuminating field though recent approaches have also been demonstrated in the presence of partial spatial  and temporal [7, 8] coherence. In the so-called Fresnel imaging regime, a technique based on an extension to the Transport of Intensity equation (TIE)  has been used to demonstrate phase retrieval for a polychromatic source in cases where Here λ is the wavelength, z is the propagation distance and u is the spatial frequency of phase features in the sample. The validity condition is satisfied for sufficiently small propagation distances. Consequently, the approach is referred to as applying in the “near Fresnel” regime.
Here we propose an extension of the conventional contrast transfer function (CTF) based phase retrieval , which extends the validity condition to the intermediate regime for homogeneous objects, i.e. objects composed of a single material [4, 10]. The technique requires weak absorbing, homogeneous objects with slowly varying phase . We show that the new technique can achieve quantitative image reconstruction for homogeneous objects using polychromatic sources of known spectra, demonstrated using a polychromatic lab based source. We then apply the technique for a quantitative reconstruction of a two dimension object using a synchrotron beam with a significantly higher harmonic content and a three dimension object using a polychromatic x-ray source whose spectrum has been well characterized.
Just as the previous polychromatic approach  generalized the TIE, which operates in the near Fresnel regime, our approach generalizes the contrast transfer function (CTF)  method which operates for slowly varying phase into the intermediate regime. The requirement for homogeneity is imposed where we wish to acquire a single image and that will provide a quantitative result .
2. Contrast transfer function based phase retrieval for polychromatic sources
It is well known that for a homogeneous sample, given the average density for each voxel of a homogeneous sample is described by the projected thickness, along a projection through the sample is described by the following equation
Consider a monochromatic plane wave, of wavelength λ arriving at a thin, homogeneous sample. The exit wave field after the sample is
The diffracted intensity at the detector plane of a distance z from the sample, can be determined using the following relation Eq. (3) can be linearized by applying the assumptions 
The conditions shown in (4) refer to the requirement of a weak absorbing sample of slowly varying phase.
After the linearization, Eq. (3) can be rewritten as
In an experiment with a polychromatic source, the measured intensity after the sample isEq. (9) is valid here, in which scales the source strength. Therefore, Eq. (5) is re-written equivalently as
We define as the weighted average of due to spectral and detection effects
Then Eq. (11) can be written
The thickness, is obtained as:
In order to evaluate Eq. (14) to reconstruct the thickness of a homogeneous sample by using the Fast Fourier Transform algorithm we must adequately sample the argument of the Fourier transform. Under the assumption that the intensity is slowly varying compared to the CTF, a condition whereby the CTF is sampled adequately for the whole spectrum can be derived using the Shannon-Nyquist sampling criterion is two samples per period. Thus the maximum phase gradient of the CTF for a particular wavelength, λ, across the entire array must be less than π per sampling interval, which here is given by the pixel size,and the number of pixel, After some derivations we have In order to apply for the whole spectrum, the requirement in Eq. (14) must be correct for every wavelength. This means that
Note that, in addition to the requirement described in Eq. (4), Eq. (15) sets the maximum propagation distance z for which Eq. (14) can be applied numerically. Furthermore, when we want to reconstruct a feature with the size larger than a pixel size, the value of z can be extended. In this case, can be considered as the smallest feature size that can be reconstructed.
3.1. Polychromatic CTF based phase retrieval for a synchrotron source with harmonic contamination
It is often assumed by many users that synchrotron radiation after a monochromator is monochromatic. However, it is not uncommon that a significant fraction of higher order harmonics is present in the beam, particularly when the sample and/or the optics are highly absorbing. The degree of harmonic contamination can be quantified using a series of filters of known thicknesses following the relationship .
The experiment was conducted at the SAXS/WAXS beamline of the Australian Synchrotron . The fundamental energy was set to 9030 eV and the beam was heavily attenuated (by rougly 3 orders of magnitude) by a filter located upstream of the experiment hutch. At this level of attenuation we assume that there is a measureable fraction of 3rd order harmonic photons in the beam after a Si 111 monochromator. We used 9 Aluminum filters with thicknesses in the range from 11.2 µm to 350 µm to characterise the harmonic content in the beam. The fraction of the 3rd order harmonic in the beam was found to be 11.3%±0.1%.
A sample of Cu with a nominal thickness of 100 nm, satisfying the weak absorption condition and slowing varying phase conditions, was used. The sample had a width of 100 μm. A diffraction pattern was recorded with a scintillator-coupled CCD detector (2048 x 2048 pixels, 13.5 μm) located at a distance of 3.37 m downstream of the sample. A 10x lens was used to image the scintillator giving an effective pixel size of 1.35 μm. A profile of the diffraction pattern is shown in Fig. 1(b) .
The reconstructed thickness from the diffraction pattern following Eq. (14) and Eq. (5) are shown in Fig. 1(c). An error of approximate 10% occurred in the thickness reconstruction (only about 90 nm ± 1 nm) using the monochromatic CTF (Eq. (5)) at energy of 9030 eV compared with the nominal thickness. The polychromatic formalism described in the paper produced a quantitative reconstruction which is in very good agreement with the expected value, which was 100 nm ± 1 nm.
3.2. Polychromatic CTF based phase retrieval for a polychromatic laboratory source
We applied the new algorithm to 3D tomographic imaging using a laboratory polychromatic x-ray unfiltered source (Xradia Inc. micro-XCT) in the Department of Physics, La Trobe University. The source consisted of a closed x-ray tube with a tungsten target. The source size is about 6 μm at a tube voltage of 40 keV. The spectrum was measured using a wavelength sensitive detector (XR-100T-CdTe, AMPTEK Inc.) shown in Fig. 2 , the measurement was described in .
The sample was made from a polyimide tube (Kapton, C22H10N2O4) with average density of 1.42 ±0.02 g/cm3. The inner diameter was 122 µm and the outer diameter was 168 µm. Accordingly, the projected thickness through the middle of the tube is expected to be approximately 46 µm.
The sample was located at a distance of 40 mm from the source and a CCD detector was located at a distance of 360 mm downstream from the sample. The CCD detector was described in section 4.1. With the combination of the geometric magnification, the objective magnification 10x, and the 2x2 binning of the CCD, the effective pixel size was 0.27 µm for images taken by the detector with this configuration.
The diffraction pattern of a typical projection recorded at the detector is shown in Fig. 3a . The measured intensity was corrected for the measured dark current and normalized by the corresponding white field. From the diffraction pattern, the observed noise level, estimated as the standard deviation in the background area of the diffraction pattern, was about 1.2%. The maximum net absorption of the tube is less than 2%, which satisfies the weak absorption requirement for the sample.
As the experiment is in a point projection geometry we use the Fresnel scaling theorem  to recast the distance and pixel scales into corresponding effective quantities for plane wave geometry. Here, the effective propagation distance is 3.6 cm. This satisfies the condition for the distance required for adequate scaling as shown in Eq. (15).
Figure. 3b shows the 2D thickness reconstruction from Fig. 3a. In Fig. 3c, the black dashed line shows the nominal projected thickness for the ideal tube. The red line shows the quantitative result using polychromatic CTF (Eq. (14)). The dotted line shows the monochromatic CTF reconstruction using Eq. (5) for an energy corresponding to the peak energy of 8.3 keV in the measured spectrum, which is the combined spectral and detection response distribution shown in Fig. 2a, the measurement of this spectrum has been discussed in . The reconstructed projected thickness in the central region obtained using Eq. (14) was 45.9 μm ± 0.2 μm, which agrees very well with the expected value.
From Eq. (1), the 3D density of the sample,can be reconstructed with an additional tomography technique, the standard technique is excellently discussed in [15, 16]. Here, we reconstruct the 3D distribution of the object density, rather than reconstruct the 3D refractive index of that, because the density is independent on energy. This means that for each projection, a projected thickness is reconstructed. Then, the corresponding column density is calculated (here, a known average density of the sample is of 1.42 ± 0.02 g/cm3). The process is repeated for every other projection to calculate the corresponding column densities. Finally, these column densities are combined by applying standard filtered back projection tomographic reconstruction [15, 16] to reconstruct assuming that an equivalent plane wave geometry to the point projection geometry was used - valid here as the divergence of the source subtended by the sample was rad.
Indeed, for this particular sample, we used 361 projections from 0° to 180°. The 3D distribution of the density of the sample was reconstructed and is shown in Fig. 4a . Figure 4b shows the distribution of the density in a slice, which is a 2D across section of the sample. Figure 4c shows a 1d plot along the dashed line in (b).
In the Fig. 4c, the maximum of the second peak for in a slice of the sample shows the value of 1.38 ± 0.02 g/cm3. The differences between two peaks and between the measured and the average density show that the density of the sample as well as the wall thickness is not uniform. We used the reconstructed 3D density of the sample to calculate the column density along the projection described in the 2D reconstruction above (Fig. 3b), and then calculate the 2D projected thickness with the average density of 1.42 ± 0.02 g/cm3. From that we obtained again the 2D result shown in Fig. 3b, and then the plot shown and c, which has the value along the dashed line (in Fig. 3b) through the middle, was approximately 46.0 μm ± 0.2 μm. This is an excellent agreement between the nominal thickness and the reconstructed thickness using polychromatic CTF.
We have demonstrated an applicability of a polychromatic version of the CTF approach for phase retrieval that relies only on the measured (polychromatic) diffraction pattern and knowledge of the source spectrum, the detector response and, where significant, additional detection related effects. We applied the new technique to obtain quantitative reconstructions of homogeneous samples using real polychromatic sources. Once again, the polychromatic CTF approach can be used to extend the validity condition from “near Fresnel” regime to the intermediate regime to overcome the distance limitations of TIE, whose validity condition only valid in the near field.
The authors acknowledge the support of the Australian Research Council through the Centre of Excellent for Coherent X-ray Science. The authors also acknowledge the support the Australian Synchrotron, particularly all the SAXS/WAXS beamline scientists.
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