1. Introduction

Propagation-based phase contrast imaging (PB-PCI) has become a promising method for X-ray microscopy and tomography of low absorption samples. The main benefit of the technique is an increased contrast given by edge enhancement, that is, by near-field interference fringes around sharp edges. This increased contrast can be used, for example, to avoid additional staining in biological samples and reduce measurement times, as has been demonstrated by a range of bio-medical [1–7] and material science [8,9] applications over the past two decades. Furthermore, the method is especially suitable for laboratory setups using X-ray micro-focus sources [10,11], since no additional optics are necessary [12], and the requirements on temporal and spatial coherence are moderate. The improved contrast, simplicity of the setup and its availability in laboratories as well as synchrotrons thus fuel a rising interest in the further development of the method. In this article we restrict ourselves to microfocus-source based laboratory setups, which have rather limited photon flux and coherence.

In PB-PCI the setup parameters, such as X-ray energy, source size, detector resolution and sample properties are usually pre-defined by the used equipment. However, geometry parameters, such as distances, can be optimized with respect to a chosen figure-of-merit, for example contrast, contrast-to-noise ratio (CNR) and fringe separation.

To predict the expected performance, wave propagation based on Fresnel diffraction theory can be modelled, as discussed by Nesterets et al. [13]. The theoretical models imply that trade-offs between contrast, CNR and resolution need to be considered, depending on the given imaging task, and that major gains in CNR or contrast can be achieved by correct optimization. However, experiments [14,15] to test the validity of the used assumptions were so far rather limited in scope and yielded only qualitative comparisons. Since the theory is based on several idealizations, such as a small energy bandwidth, paraxial geometry and projection approximation, systematical experimental studies are due. In particular, it is important but challenging to correctly describe the effect of the broadband spectrum of laboratory sources, accounting for the source spectrum, the energy-dependent refractive index of the sample as well as the scintillator efficiency.

Here, we present a comprehensive experimental and theoretical study of the geometry optimization for PB-PCI. We compare calculations based on the theoretical model [13] of a generic edge sample in 1D to experimental measurements of contrast, CNR and fringe separation for low-magnification geometries. We restrict this study to the case of microscopy of weakly scattering, low-absorption objects since this is the main use of PB-PCI. The necessary high resolution for this imaging task at laboratory setups is usually achieved by using a detector with sub-micron pixel size, because most commercially available X-ray tubes have significantly larger spot sizes, in the order of 2-50 micron. We only consider the propagated image, since evaluating and comparing phase retrieval approaches, e.g. transport of intensity equation (TIE) based filtering [12], would exceed the scope of this paper. While most of the results are based on a custom-built system, we also present results for a commercial system (Zeiss Xradia). We show that it is crucial to consider the X-ray spectrum and the scintillator efficiency, including the definition of effective values for absorption and phase shift [16], for a good agreement with the prediction.

2. Theory

2.1 Phase contrast formation

As illustrated in Fig. 1, the description of PB-PCI is based on the free space propagation of an object wave from the object plane ${z_1}$ to the detector plane ${z_{tot}} = {z_1} + {z_2}$. For simplicity, we consider a thin sample whose thickness is only varying in x. Assuming a monochromatic plane wave of wavelength $\lambda $ (parallel beam geometry), a weakly absorbing object and a small propagation distance ${z_2}$, the intensity distribution ${I_{det}}$ at the detector plane can be derived using the Fresnel propagator, yielding the well-known form [17]

 

Fig. 1. (a) PB-PCI lab setup geometry using a micro focus X-ray tube with a source spot of width ${\sigma _s}$ and a high-resolution pixel detector with PSF ${\sigma _d}$. The magnification $M\; = ({{z_1} + {z_2}} )/{z_1}$ should be low for a setup with ${\sigma _s} \gg {\sigma _d}$. (b) Example of a flat field corrected PB-PCI image of a tip of the sample (broken Si₃N₄ window). The red box indicates the used region of interest (ROI) for the data analysis (y width: 30 pixels). (c) Examples of measured data (black stars), fitted fringe (red) and absorption (black dashed line) for different propagation distances ${z_2}$ and ${z_{tot}} = 25.90\; $cm.

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The above description assumes a parallel beam geometry. For the case of a cone beam, as is common in laboratory setups, the magnification factor $M = {\; }{z_{tot}}/{z_1}$ needs to be considered. This is described by a scaling of the image (Fresnel scaling theorem [18]), and the free space propagation can then be related to an effective propagation distance ${z_{eff}} = {\; }{z_2}/M$ instead of ${z_2}$ [13].

Moreover, the limited spatial coherence of a lab source and the finite detector resolution need to be accounted for. To do so, the intensity needs to be convoluted with the point-spread-functions (PSF) of source S and detector D. In the cone beam geometry, the PSFs need to be scaled appropriately with the magnification before the convolution. It is convenient to scale all functions to the object plane [13], which in the following will be referred to as “back-projected” (BP). The convolution can be performed as a multiplication using the Fourier transforms $\hat{D}(u ),$ $\hat{S}(u )$ and ${\hat{I}_{det}}(u )$ (where u describes the spatial frequency in $x$):

2.2 Sample: blurred edge model

Real samples have a wide range of features, but we restrict ourselves to a straight and sharp edge to be able to accurately test the model. While PB-PCI is often used with tomography for 3D imaging, previous calculations show only minor deviations in 3D and 2D compared with the 1D case [19]. Thus, we reduce the problem to 1D. In the following the defining feature of the object will be an edge, which implies a sharp thickness change $T(x )$ from 0 to d. The sharpness of the edge is represented by an isolated error function of width ${\sigma _{obj}}$

The source S as well as the detector D are modelled as Gaussian shaped PSFs. The resolution of the whole system thus depends on the source size ${\sigma _s}$, the detector resolution ${\sigma _d}$ and the object blur ${\sigma _{obj}}$, as well as the magnification M (see Fig. 1):

To calculate the measured intensity, the convolutions in Eq. (2) can be performed using the chosen edge model for the object transmission and phase shift. Following the derivation in [13] the detected intensity, back-projected to the object plane, becomes

The positions of the first fringes relative to the edge, ${\pm} {x_0}$, are determined by the first zero crossing of the derivative of Eq. (5):

Three examples of measured intensity distributions are shown in Fig. 1, for three different magnifications, respectively. Depending on the magnification M, the fringe amplitudes and positions change. For small M, in the contact regime, only weak and barely separated fringes are visible. The fringes grow to a maximum value with increasing M, but beyond this point the fringes become wide and weak due to blurring from the finite source size. The experimental results and the fitting will be described in more detail in the results sections.

2.3 Contrast, CNR and fringe separation

When it comes to comparing images for different geometries, the contrast is an important figure-of-merit. For PB-PCI the relative fringe contrast ${c_f}$ can be used to describe the edge enhancement. It is given by the amplitude difference between the first positive and negative fringe, and will depend on the imaging geometry [13]:

In addition, the CNR should be considered, since it will affect the phase retrieval accuracy [20]. The absolute fringe contrast scales with the mean detected intensity $\overline {{I_0}} $. The noise N is modelled based on the detector response function for different exposure times. The CNR is thus given by

It must be noted that the high-resolution detectors often used in PB-PCI laboratory setups are usually not photon counting and can show a non-linear detector response. The noise model can thus not generally be considered to follow a simple Poisson distribution, and an experimental detector characterization is therefore advisable to get reliable results. This was done for the used Rigaku XSight, which confirmed the given model assumptions, yielding a factor $H = 7.16$, see Supplement 1 section S1.

As a third figure-of-merit it is worth looking at the separation of the phase fringes. The resolution of the imaging system is theoretically limited by the PSFs of source, detector, and sample edge, see Eq. (4). It is necessary that the separation of the first fringes $2{x_0}$ is larger than at least twice the resolution to adequately sample them, because otherwise the image blur will reduce the contrast.

2.4 Optimization of geometry

To optimize the geometry, we want to find the maximum of the fringe contrast with respect to M for a fixed sample and wavelength, but this is not trivial from Eq. (8). As shown in the Supplement 1 section S2, it is possible to rewrite the fringe contrast Eq. (8) as being only a function of the Fresnel number F (similar to the parameter $A/B{\; }$in [13]) defined as:

Furthermore, we demonstrate that the fringe contrast ${c_f}(F )$ is a monotonously decreasing function of F, for reasonable parameters in lab setups (see Supplement 1 section S2). To maximize the contrast, we can thus either increase ${z_{tot}}$ or, if ${z_{tot}}$ is fixed, find the minimum of F with respect to M. Following this reasoning (see Supplement 1 section S2), a rather simple expression for the magnification ${M_{opt}}$ to reach the highest possible contrast for a given ${z_{tot}}$ can be found:

In Fig. 2, we have calculated the maximum relative fringe contrast ${\; }{c_f}({{z_{tot}},{M_{opt}}} )$ and the corresponding CNR as functions of ${z_{tot}}$, at ${M_{opt}} = 1.1$, for $\phi = {\; } - 0.35$, $\mu = {\; }0.0067$, $\lambda = {\; }1.24{\; {\AA}}$ and ${I_0} = 2760$. The relative fringe contrast increases monotonically when ${z_{tot}} \to \infty $, while the CNR decreases.

 

Fig. 2. Calculated maximum relative phase contrast ${c_f}({{z_{tot}},\; {M_{opt}}} )$ (dashed red) and maximum CNR (solid blue) depending on the overall distance ${z_{tot}}$, for constant exposure time ${t_{exp}}$. We used the parameters for our sample and Cu source setup as described in the main text. While the contrast increases with the total distance, the CNR decreases due the reduced intensity.

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The optimal magnification ${M_{opt}}\; $for the highest contrast differs from the magnification ${M_{res}}$ that yields the best system resolution (see Eq. (4)). For setups with a larger source spot than detector resolution, ${\sigma _{d\; }}/{\sigma _s} < 1$, this means that ${M_{opt}} > {M_{res}}$. Only if detector and source have the same PSF (for an infinitely sharp edge), the same magnification ${M_{opt}} = 2$ will optimize both resolution and contrast, yielding ${\sigma _{min}} = {\sigma _{d\; }}/\sqrt 2 \; .$

Note that even though increasing the overall distance ${z_{tot}}$ increases the relative fringe contrast, the above derivations for PB-PCI are only valid for the near-field Fresnel regime $F \ge 1$, meaning small propagation distances ${z_2}$. For larger ${z_{tot}}$ the same magnification ${M_{opt}}$ will eventually lead to propagation distances that exceed the validity of the TIE and higher order fringes will appear. In case of sufficient coherence this can be treated in terms of far-field Fraunhofer diffraction $F \ll 1$, which is not part of this discussion. To stay in the Fresnel regime, we can find a condition for the maximum overall distance ${z_{tot,\; max}}$ that still conforms with the derivation above

2.5 Polychromatic corrections

So far, the derivations assume monochromatic X-rays, which is generally far from true for X-ray tubes. For an X-ray tube spectrum with dominant characteristic lines and a broad bremsstrahlung background, the refractive index values will usually be dominated by the former. The first approximation therefore is to use the characteristic line energy of the source, but in most cases this does not suffice to get a correct prediction of the contrast [16]. To fully cover the polychromatic case, an additional integration over the wavelength $\lambda $ in Eq. (2) must be performed. It has been shown that this has the effect of an additional damping of higher spatial frequencies, which is moderate if $\mathrm{\Delta }\lambda \sim {\lambda _0}$, but which will eventually limit the spatial resolution in the image [21].

Instead of performing the full wavelength integration in Eq. (4), which can be challenging, using effective values for the projected absorption ${\mu _{poly}}(x ){\; }$and phase shift ${\phi _{poly}}(x )$ already considerably improves the theoretical model and subsequent phase retrieval [16]. These effective values are calculated by integrating the refractive index weighted by the source spectrum. Since $\beta (\lambda )$ and $\delta (\lambda )$ show different wavelength dependencies, the effective wavelength ${\lambda _{eff}}$ corresponding to the calculated ${\mu _{poly}}$ and ${\phi _{poly}}$ will be different. Here the given ${\lambda _{eff}}\; $therefore only refers to the weighted average of the used spectrum (Supplement 1 section S3). Using this approach, the formulas for contrast, CNR and fringe separation can be easily adjusted to include the polychromatic spectrum. Moreover simple phase retrieval based on filtering (e.g. TIE filtering) will yield more accurate results [16].

However, the recorded image will not only depend on the wavelength dependence of the refractive index of the sample, but also on the wavelength dependent sensitivity of the detector. The scintillators used in high-resolution X-ray detectors are usually very thin [22] and their efficiency is strongly reduced for higher energies [23]. Therefore, the low energy part of the source spectrum will contribute more to the image. To account for this effect, we adjusted the calculations for the effective values with an additional spectral filtering based on the detection efficiency. We find that this improves the results considerably for the broad W spectrum, while it was not necessary for the Cu source (see results section).

3. Experimental methods

Most of the experiments were performed at a custom build laboratory setup that uses the low-magnification geometry, see Fig. 1. It uses a microfocus Cu X-ray tube (Rigaku) with a nominal spot size of 25 µm (FWHM @ 45 kV). The images were recorded with a Rigaku XSight high resolution detector, a scintillator-based lens-coupled CCD camera with an effective pixel size of 0.55 µm (lens unit LC 0540). The magnification was measured using an edge scan with the encoded transversal motors and the known effective pixel size of the detector. The exposure time for all measurements was kept constant at 600 s, except for the shortest overall distance which was measured at 300 s. The detector was cooled to -20°C.

Similar measurements were performed with a commercial system, a Zeiss Xradia Versa 520. The W transmission target tube was set to 80 kV (7 W), for which the manufacturer specifies a source spot size of 2.5 µm. No spectral filter was used. To acquire the images the high-resolution detector with 40x objective, binning 2, was chosen, to reach an effective pixel size of 0.66 µm. The exposure time was 120 s per image.

The test sample was an intentionally broken Si₃N₄ window (Silson) of $d = $1 µm membrane thickness, showing sharp isolated edges, see Fig. S3 in Supplement 1.

4. Results and discussion

4.1 Measurements with the Cu source system

The theoretical models described above are based on several assumptions, for instance monochromatic X-ray energy, and it is not clear how well they describe experiments. To investigate this, we performed five measurement series at different overall distances ${z_{tot}}$. For each series, we took images at about 25 different $M$, and in total we acquired images for 130 geometries. In each image the intensity was averaged over a ROI of 30 pixels along the $y$-direction for the subsequent one-dimensional analysis in the $x$-direction (Fig. 1(b)).

Since Eq. (5) is too complex for automatic fitting, we developed a simplified fit function based on the second derivative of the error function model Eq. (3) (see Supplement 1 section S5). For each measurement, a least-mean-square fit was performed on the average along the edge. Three examples are shown in Fig. 1(c), demonstrating that the developed fit function is well suited to quantitatively model the fringes. The predicted general trend for varying propagation distances can be observed already in this small dataset: narrow and weak fringes in the close contact regime, well pronounced fringes at the optimal magnification, and broad and weak fringes at large propagation distances.

From the fits, we determined three figures-of-merit: the relative fringe contrast ${c_f}$, the CNR, and the fringe separation $2{x_0}$ based on the definitions in section 2.3. We have plotted the results as functions of the magnification in Figs. 3, 4 and 5, respectively, showing excellent quantitative agreement with the theoretical predictions in Eqs. (6), (8) and (11), respectively. The error bars represent the propagated uncertainties of the fit parameters (for the fit function see Supplement 1 section S5). The calculated curves for contrast, CNR and fringe separation based on the theoretical model are plotted as dashed lines alongside the datapoints from the measurements.

 

Fig. 3. Relative fringe contrast vs. magnification $M$ for 5 different total distances ${z_{tot}}$, showing simulations as (dashed lines) and measured data as symbols. The vertical dashed line indicates the optimal magnification ${M_{opt}} = 1.1$ that is constant for all five series. The error bars represent the propagated uncertainties of the fit parameters (see Supplement 1 section S5). The results are shown separately for each ${z_{tot}}$ in Fig. S5 in Supplement 1.

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Fig. 4. CNR vs. magnification M, simulations (dashed lines) and fits from measured data for 5 different total distances ${z_{tot}}.$ Note that the order of the five datasets is reversed compared with Fig. 3. The error derived from the fit uncertainties was about 0.35 for all datapoints (error bars omitted for readability) (see Supplement 1 section S5).

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Fig. 5. Fringe separation $2{x_0}$ over magnification M, simulations (lines) and measured data for 5 different total distances ${z_{tot}}$. The black dashed line indicates the calculated system resolution $\sigma $ based on Eq. (4), with a minimal value of ${\sigma _{min}} = 1.25$ µm.

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Note that the calculated curves are not fits. The same setup parameters, ${\sigma _s} = 12$ µm, ${\sigma _{obj}} = 1.1$ µm and ${\sigma _d} = 0.6$ µm and effective refractive index ${\mu _{poly}} = \; 0.0067\; \mu {m^{ - 1}},\; {\phi _{poly}} = \; - 0.35\; $ and ${\lambda _{eff}} = 1.24\; {\AA}$, were assumed for all curves. For the calculation of the effective values see Supplement 1 section S3 and [16]. The choice of the source and detector parameters was solely based on values specified by the manufacturer. The relatively large edge width ${\sigma _{obj}}$ is justified by the averaging along the edge, which cannot be assumed to be fully parallel to the pixel rows and thus introduces an additional blur, see SEM image (Fig. S3 in Supplement 1). A detailed discussion of the polychromatic corrections will follow at the end of this section.

After determining that the measurements show excellent agreement with the theoretical models, we can study the results in more detail. The relative fringe contrast as function of magnification is shown in Fig. 3, confirming the same trend for all five overall distances ${z_{tot}}$. Each curve shows a single maximum, whose position is always at the same magnification ${M_{opt}} = 1.1$ as predicted by Eq. (13). In the close contact regime absorption dominates the image contrast. The calculated absorption contrast for ${\lambda _{eff}} = 1.24\; {\AA}$ and a 1 µm thick Si₃N₄ membrane is $7.1 \times {10^{ - 3}},$ which matches with the measured relative fringe contrast of about $({7 \pm 0.1} )\times {10^{ - 3}}$. With increasing propagation distance, interference fringes start to form around the edges. The relative fringe contrast reaches its maximum before slowly decreasing towards larger magnifications. The fringe contrast ranges from $7 \times {10^{ - 3}}$ in the close contact regime to $50 \times {10^{ - 3}}$ at the optimum. This large variation demonstrates the importance of optimizing the geometry.

Although the position of the maximum is always the same, the value at the maximum increases with larger overall distances ${z_{tot}}$. This can be understood in terms of the system resolution and spatial coherence at the sample. The closer the sample is to the source, the smaller the coherence length on the sample, which in turn lowers the visibility of interference fringes. Using larger overall distances ${z_{tot}}$ therefore allows for larger source-sample distances ${z_1}$ and reduces the source size dependence of the contrast. This generally raises the relative contrast.

However, larger overall distances also lead to a lower mean intensity, and thus degrade the signal compared to the noise. This can be clearly seen in the plot of CNR vs. M in Fig. 4, where the order from top to bottom is reversed and the lowest ${z_{tot}}$ shows the highest CNR. Due to the cone beam geometry the intensity decreases quadratically with distance (see Eq. (10)). The lower mean intensity thus reduces the CNR at larger overall distances. The highest CNR is reached for ${z_{tot}} \to 0\; $ and ${M_{opt}}$. Due to experimental constraints the smallest overall distance was limited to ${z_{tot}} = \; 12.8$ cm. For this overall distance, the maximum at ${M_{opt}} = 1.1\; $is CNR = 2.95, which is about 1.6 times the highest CNR reached for ${z_{tot}}\; = \; 48.7\;$cm. Note that the CNR scales with the square root of the measurement time, which means that optimizing the geometry corresponds to significant gains in time.

The third extracted figure-of-merit is the fringe separation, shown in Fig. 5. The fringe separation reaches its minimum at a magnification very close to 1, smaller than the magnification at the fringe contrast maximum ${M_{opt}}$. For small magnifications, the fringe separation curves for different overall distances are very similar. Significant differences can only be reached at high magnifications, in which case the larger overall distances yield better results. The fringe separation is only relevant as a figure-of-merit if it is in the same range as the system resolution $\sigma $. If it is smaller than the system resolution, the fringes will not be adequately sampled, and the fitting procedure described above is unreliable.

The polychromatic source spectrum mainly influences the effective refractive index, which affects the height of the relative fringe contrast curve. The simulations in Figs. 3–5 were performed using effective values for ${\mu _{poly}} = \; 0.0067\; \mu {m^{ - 1}},\; {\phi _{poly}} = \; - 0.35\; $ and ${\lambda _{eff}} = 1.24\; {\AA}$ [16] (see Supplement 1 section S3). These values were calculated from the well-defined membrane composition and thickness $d = 1$ µm and the measured Cu spectrum for a tube voltage of 45 kV [16], while the scintillator efficiency was considered constant in this spectral range.

As can be seen in Fig. 3, the simulations based on the given model fit slightly worse to the measurement series with larger overall distances. This can be explained by additional air absorption and beam hardening, which changes the spectrum and thus the effective refractive index and wavelength. Since the corrections are small at the given experimental configuration, a detailed account of air absorption as in [14] was not included here. Instead, the effective values were kept constant for all simulations.

4.2 Xradia measurements

The photon yield of the scintillator used in the detector will show an energy dependence. For thin scintillators, as commonly used in high-resolution detectors, higher X-ray energies will yield less visible-light photons [19,23]. If the source spectrum spans a large energy interval, this will considerably change the spectrum contributing to the image formation. For the Cu source a constant scintillator efficiency was sufficient for a good match between simulations and measurement, since the spectral range is rather small and dominated by the Cu K-lines (for comparison see Fig. S5 in Supplement 1). The commercial Zeiss Xradia system, like many X-ray microscopy systems, uses a tungsten (W) target, which creates a much broader spectrum than the Cu source. The W spectrum shows high-intensity emission for the K-lines around 59 keV and the L-lines around 8.4 keV, and a broad bremsstrahlung contribution in between. It is not obvious that the theoretical predictions will hold for such a broadband spectrum.

Two datasets where acquired with the Xradia, at 5 cm and 10 cm source-detector distance ${z_{tot}}$, and analyzed in the same fashion as the measurements above. The same sample STD (${\sigma _{obj}} = 1.1$ µm) and a detector STD including the binning were assumed (${\sigma _d} = 0.66$ µm) but a shorter measurement time (120 s) was sufficient because of the shorter distances.

The fringe contrast, shown in Fig. 6, displays a maximum at the calculated magnification, ${M_{opt,Xradia}} = 1.47$, see Eq. (13). Plots of the CNR and fringe separation can be found in Supplement 1 section S7. The optimum position is shifted towards the source, since the transmission tube used in the Xradia has a smaller spot size of ${\sigma _s} = 2.5$ µm. In this respect the experiments again confirm the theoretical predictions. The peak is broader, meaning that the optimization is less sensitive to the magnification since the source and detector PSF are more similar in size.

 

Fig. 6. Measurements and fits of the relative fringe contrast for the commercial Xradia system. The symbols show the relative fringe contrast acquired from the fits of the measurements, vs. magnification, for two different total distances: ${z_{tot}}$ = 5 cm (orange circles) and 10 cm (blue crosses). The lines show simulations with different assumptions: Dotted line for monochromatic X-rays at the W L-line (8.4 keV), and dashed line for a simulated spectrum from a W X-ray source. The solid line shows a simulation using the W spectrum including the energy dependent detection efficiency using a YAG:Ce scintillator. The vertical dashed line shows the calculated optimal magnification ${M_{opt,{\; }Xradia}} = 1.47$, which is the same for both series. The results for the CNR and fringe separation can be found in Supplement 1 section S7.

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However, the simple monochromatic approach using the tungsten L-line energy of 8.4 keV obviously leads to an overestimation of the magnitude of the contrast, see Fig. 6 (dotted line). Assuming a calculated generic W spectrum for an 80 kV tube voltage [24], however, similar to how the calculations were done for the Cu source, leads to a strong underestimation (dashed line). The reason is that the K-lines at around 59 keV start to dominate the effective values. To correctly model the contribution of the higher energy part of the spectrum, it is therefore critical to include the scintillator efficiency. The scintillators used for high resolution detectors are usually very thin and show a steep decline in efficiency for high energies [22]. The exact composition and characteristics of the scintillators used in the Xradia and Xsight detectors are not publicly available. However, assuming the same source spectrum as before, weighted with a typical YAG:Ce scintillator detection efficiency [23], already achieved excellent quantitative agreement with the measured data. Incidentally, the effective values were calculated to be ${\mu _{poly}} = 0.006\; \mu {m^{ - 1}},\; {\phi _{poly}} = \; - 0.33$ and ${\lambda _{eff}} = 1.13\; {\AA},$ which is similar to the values for the Cu source setup above. Thus, we find that while the scintillator efficiency can be ignored for the Cu source system (see Fig. S5 in Supplement 1), it must be taken into consideration for the W source system.

5. Discussion

The experiments show excellent agreement with the theoretical models, but only when the energy dependencies of the refractive index and the scintillator efficiency are considered. The presented measurements confirm the prediction that there exists a magnification ${M_{opt}}$ which yields the highest relative fringe contrast, and that it is independent of the overall source-to-detector distance ${z_{tot}}$. Furthermore, the general trends of relative fringe contrast and CNR with respect to ${z_{tot}}$ are confirmed.

Using sources with a broad spectrum reaching to short wavelengths, such as W, require care in the theoretical modelling. Although using an effective wavelength and an effective refractive index can partially account for the polychromacy of the source, our results show that the theoretical model will fit less well to the experimental data and the contrast will generally be lower for shorter wavelengths. Note that the short wavelength part of the spectrum will contribute to the dose on the sample, even though it may not add to the image contrast. We also note that it is reasonable to assume that the modelling of polychromatic effects is important for the phase retrieval, although this is out of the scope of this work.

The decision of a suitable overall distance will heavily depend on the system parameters and the imaging task. For tomography scans of low-contrast samples with laboratory sources, the CNR will often be the more important figure-of-merit, to achieve a reasonable exposure time per image. On the other hand, for high resolution 2D imaging the exposure time might be of lesser concern and larger distances can be used to maximize the contrast within the range of PB-PCI. For determining the optimal source-detector distance for a sample in a specific setup we suggest starting at rather small distances maximizing the CNR and increasing the distance, if more contrast is required. If high phase contrast is wanted the presented results can be used for making an informed choice, especially concerning the used magnification.