1. INTRODUCTION

In recent years various x-ray phase contrast imaging techniques have been introduced for laboratory setups [1–5]. Experimentally, the most convenient of these techniques is propagation-based phase contrast imaging (PB-PCI), since it does not require any additional optical elements, such as gratings or masks, or multiple exposures [6]. By introducing a propagation distance between the sample and detector the phase information is encoded in an additional intensity modulation in the image: fringes appear around interfaces, an effect that is commonly termed “edge enhancement.” This near-field interference phenomenon can be described by the transport-of-intensity equation (TIE) [7]. Since it is quite insensitive to longitudinal coherence and the spatial coherence requirements are moderate [1], PB-PCI is well suited to laboratory x-ray imaging. Therefore, the technique has become widely popular in systems based on micro- and nano-focus lab sources, although it is also popular at synchrotrons [2,8–14]. To date, PB-PCI has found applications in many fields, such as biomedicine [15–19], material science [20,21], and archeology [22].

Recently a new generation of x-ray lab sources has become available, based on nano-focus transmission targets [23]. These sources reach a much smaller electron spot of a couple of hundred nanometers at a similar target power density compared to traditional micro-focus sources. Please note that nano-focus here refers to the electron spot size of the source which emits a divergent beam, not the focusing of the x-rays, as is done at nano-focus beamlines at synchrotron radiation facilities. Accordingly, sufficient coherence and negligible source-blurring can already be realized at much shorter source–sample distances, thus allowing for geometries with very high geometrical magnifications. Unlike micro-focus source setups, the edge enhancement regime cannot be avoided with nano-focus sources for practical geometries. As we have shown in our earlier study, the relative height of the edge enhancement fringes, and thus the contrast-to-noise ratio (CNR), depends on the magnification and can be optimized accordingly [24]. However, this was performed at low magnifications, between 1.01 and 1.6, with a micro-focus source and it is not clear whether these conclusions hold for the much larger magnifications typical for a nano-focus source setup. Here, we present an experimental investigation of the influence of the magnification on edge enhancement fringes performed at a high-magnification setup using a nano-focus source [25], in a range of magnifications between 17 and 175. Despite the significant change of geometry, we find similar results for this regime, with a constant magnification maximizing the relative fringe contrast for any total source–detector distance. However, a key difference is that the influence of the object’s feature sharpness on the optimal magnification becomes stronger in high-magnification geometries.

2. THEORY

Propagation-based phase contrast can be described in terms of Fresnel free-space propagation. If a plane wave of wavelength $\lambda$ illuminates a sample of refractive index $n = 1 - \delta + i\beta$ and thickness $T({x,y})$ located at a source–sample distance ${z_1}$ it will be damped according to the Beer–Lambert law ${I_{{\rm exit}}}({x,y}) = \def\LDeqbreak{}{I_0}{\rm exp}({-\mu T({x,y})})$, with $\mu = 2k\beta$, as well as phase-shifted by ${\phi _{{\rm exit}}}({x,y}) = - k\delta T({x,y})$. For homogeneous, weakly absorbing samples under the paraxial and projection approximation and propagation distances ${z_2}$ in the near field, the intensity distribution at the detector plane can be described by the TIE [7]:

Accordingly, besides the pure absorption contrast, the image shows an additional intensity modulation that is proportional to the propagation distance, wavelength, and second derivative of the phase shift $\nabla _ \bot ^2{\phi _{{\rm exit}}}$. This means the phase contrast will be strongest around sharp edges, thus the name “edge enhancement.”

For a point source emitting a spherical wave this expression needs to be scaled by the magnification $M = ({z_1} + {z_2})/{z_1}$, according to the Fresnel scaling theorem:

For an ideal point source and detector, the effective propagation distance ${z_{{\rm eff}}} = {z_2}/M$ reaches its maximum at $M = 2$, thus maximizing the phase term in Eq. (1). However, this is not sufficient to describe a real experiment, since the source size and detector resolution will cause a magnification-dependent blurring of the image, which influences the contrast of the fringes [26]. The measured intensity can be described by a convolution with the respective magnified point spread functions (PSFs) of source $S$ and detector $D$:

Accordingly, the resolution of the system is given by the convolution of the scaled PSFs. For Gaussian-shaped PSFs with standard deviations (STDs) ${\sigma _s}$, ${\sigma _{{\rm obj}}}$, and ${\sigma _d}$, for source, object, and detector, respectively, this amounts to an overall PSF width a STD of

The convolution in Eq. (3) can be solved analytically only for relatively simple models of source, object, and detector [27]. For the case of a Gaussian-shaped source and detector and a step edge with a small Gauss blur (error function), an integral solution has been proposed by Nesterets et al. (back-projected (BP) to the sample plane):

Two characteristics of the fringes can be directly extracted from Eq. (5): the fringe position relative to the edge, ${\pm}{x_0}$, as determined by the first zero crossing of the derivative of Eq. (5),

As we have shown in our previous study [24], a theoretical optimum for the magnification that maximizes the relative fringe contrast can be derived. It is independent of the total distance between the source and detector and only depends on the STDs of the source, object, and detector:

If ${\sigma _{{\rm obj}}} \ll {\sigma _d},\;{\sigma _s}$, this can be further simplified. Note that for high-magnification setups such as the one investigated here, ${\sigma _s} \ll {\sigma _d}$ and the inherent STD of the object feature ${\sigma _{{\rm obj}}}$ will have a stronger influence on the value of ${M_{{\rm opt}}}$ than in a low-magnification setup, since the second term in the sum will be dominant.

 

Fig. 1. Propagation-based phase contrast imaging with a divergent source in a high-magnification geometry. (a) Experimental geometry using a small source spot and high magnification $M = ({{z_1} + {z_2}})/{z_1}$. (b) X-ray image of a splinter of a broken ${{\rm Si}_3}{{\rm N}_4}$ window (thickness 1 µm). The white ROI was used to extract the edge enhancement fringes shown in (c) at different magnifications. Note how the fringes initially grow with larger magnification (sample moved toward source at constant ${z_{{\rm tot}}}$), reach a maximum height and then wash out. The gray band shows the ${3 {\text -}}\sigma$ uncertainty of the fit. The error bars of the data points represent the standard deviation of the pixel average in the ROI along the edge.

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Please note that the magnification that maximizes the fringes differs from the magnification that provides the highest resolution (minimizes $\sigma$), which is

For high-magnification setups ${M_{{\rm res}}}$ is very high (several thousand), meaning that the sample should be as close to the source as possible.

Besides the contrast itself the CNR is an important figure of merit since it will determine the information content of the image. The CNR is defined as the quotient of absolute fringe contrast and the image noise $N({{{\bar I}_0}}):$

In the simplest case the noise can be modeled as a Poisson distribution, with $N({{{\bar I}_0}})\sim\sqrt {{{\bar I}_0}}$, as for the example shown in [24]. However, in many cases, this model does not describe the detector well. Accordingly, a characterization of the noise in dependence of the recorded intensity is necessary (see Section 4 and SI).

A. Polychromatic Radiation

Both the TIE [Eq. (1)] and the integral solution for the edge model [Eq. (5)] were originally derived for monochromatic radiation. Nevertheless, it has been shown that PB-PCI does not rely on longitudinal coherence and the fringe positions are relatively insensitive to the wavelength [1], which is why PB-PCI has become so popular for laboratory setups with a polychromatic source. In theory the image generated by a polychromatic incoherent source will be a superposition of the images of each wavelength component. A common approach to model this without performing the propagation for each wavelength component is by using a spectrally weighted average for wavelength and refractive index [28]. In our previous study [24] we demonstrated that such effective values were sufficient to describe the experiment if both source spectrum and detector sensitivity were considered (see SI, Fig. S4). Here we additionally include the calculated energy-dependent air absorption based on the total distance, which will lead to an increase of effective energy (beam hardening) with increasing air path. Air scattering was not included in the simulations.

3. METHODS

The setup uses a high-magnification geometry with a tungsten transmission target source (NanoTube N2 60 kV, Excillum) and a sCMOS detector with a GdOS scintillator (Photonic Science). The source was operated at 60 kV acceleration voltage, a target current of 13.7 µA, and the measured electron spot had a FWHM of 0.52 µm (${\sigma _s} = 0.22\;{\unicode{x00B5}{\rm m}}$), yielding an electron power density of 1580 W/m². The small electron spot size comes with the advantage of being able to use a high geometrical magnification without introducing source blurring in the images. Accordingly, the detector pixel size can be more relaxed while still achieving high resolution. The detector has a physical pixel size of 9 µm and ${2} \times {2}$ hardware binning was used in all images, resulting in ${\sigma _d} = 22.98\;{\unicode{x00B5}{\rm m}}$ as an estimated STD of the detector PSF, as specified by the manufacturer. A “mixed gain” mode of the detector was used, that balances between high signal (high gain) and low noise (low gain). The source spectrum, distance-dependent air absorption, and efficiency of the scintillator were modeled and included in the calculation of effective values for wavelength and refractive index.

The sample was a 1 µm thick ${{\rm Si}_3}{{\rm N}_4}$ window, intentionally broken to form sharp, isolated edges (density of thin film ${{\rm Si}_3}{{\rm N}_4}$ is $\varrho = 2.71\;{\rm g}/{{\rm cm}^3}$ [29]). It was mounted on a sample tower comprised of a dual-axis high-precision piezo stage on top of an air bearing stage on top of a hexapod. Fifteen different source–sample distances ${z_1}$ were measured for each of the four total distances ${z_{{\rm tot}}}$ (14.3 cm, 19.3 cm, 24.3 cm, and 34.3 cm), each resulting in a different magnification between 17 and 175. The integration time was 5 s for the shortest total distance (14.3 cm) and 8 s for all other total distances. It was verified that the detector reacted linearly to the integration time (see SI) and all images were normalized to the same exposure time. For every sample and detector position 20–30 images were recorded, which were analyzed individually.

The edge enhancement fringes were fitted into lineouts extracted from a ROI for every image [see Fig. 1(b)1(c)]. Afterwards, the relative fringe contrast ${c\!_f}$ and separation ${x_0}$ shown in Fig. 2 were calculated as weighted averages from the set of fits. The error bars represent the propagated uncertainties of the averaged fit parameters.

 

Fig. 2. Relative fringe contrast and fringe separation vs. magnification. (a) Relative fringe contrast over magnification for four different total distances. The data points were extracted from fits of fringes around an edge of a ${{\rm Si}_3}{{\rm N}_4}$ window [see Fig. 1(b)/1(c)]. The curves are simulations (not fits) based on the blurred edge model given in Eq. (7) [27] for effective values of wavelength and refractive index that include the source spectrum, air absorption, and detector sensitivity (see Table 1). The dashed vertical line indicates the calculated peak position at $M = 42$ [see Eq. (8)]. (b) Fringe separation over magnification, measured (data points) and simulated (curves). The legend in (b) applies for both panels.

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4. RESULTS AND DISCUSSION

As can be seen in Fig. 2(a), the relative fringe contrast for all four total distances follows the same trend. When going from low magnifications to higher magnifications (decreasing ${z_1}$) the contrast increases quickly, reaches a maximum at around ${M_{{\rm opt}}} = 42$, and then decreases slowly toward very high magnification; also see Fig. 1(c). The initial increase can be explained by the increase in propagation distance ${z_2}$ with higher magnification (at constant ${z_{{\rm tot}}}$). The decrease toward high magnification is due to the blurring from the source.

The maximum peak position of the contrast is independent of ${z_{{\rm tot}}}$, which agrees with our previous findings for low-magnification setups [24] and matches the theoretical value [see Eq. (8)] using the measured source size and detector PSF and assuming an inherent edge blurring of ${\sigma _{{\rm obj}}} = 0.5\;{\unicode{x00B5}{\rm m}}$. This value is reasonable considering that the ROI along the edge was large (${\sim}200\;{\rm px}$) and any small curvature of the edge within the ROI would lead to an apparent widening of the integrated edge in the lineout.

From Fig. 2(a) it is apparent that the simulations based on Eq. (7) show a qualitatively good agreement to the measured data. However, it can be noted that the predicted fringe height does not match as well as in our previous study [24]. Similarly, the fringe separation is slightly underestimated compared to the measurement; see Fig. 2(b). This could be due to the used effective values for wavelength and refractive index since the fringe height is very sensitive to them [24]. We used calculated values based on a simulated source spectrum (provided by the manufacturer), tabulated values from the Chantler database [30] for the refractive index of ${{\rm Si}_3}{{\rm N}_4}$ ($\varrho = 2.71\;{\rm g}/{{\rm cm}^3}$) and air, and the spectral sensitivity of the GdOS detector (provided by the manufacturer). Due to the distance-dependent air absorption the values differ slightly for different ${z_{{\rm tot}}}$ (listed in Table 1). Since all of these are calculated values, not measurements, it is possible that the actual effective wavelength is slightly different, causing the observed mismatch. The overestimation of fringe height in the simulations would then indicate that the calculated effective wavelength is too large (underestimating the energy; see SI). An example of the effect of effective energy can be found in the SI for comparison. Please note that in the simulations the effective values were calculated separately for wavelength and refractive index. Since the refractive index itself has a wavelength dependence, the effective refractive index is not identical with the refractive index at the effective wavelength. For example, the effective energy in the simulation for ${z_{{\rm tot}}} = 14.3\;{\rm cm}$ was 13.21 keV (see Table 1) but the effective values for $\phi$ and $\mu$ would correspond to 10.3 keV and 9.2 keV, respectively.

Table 1. Effective Values of Energy, Wavelength, Absorption Coefficient, and Phase Shift for the Simulations Shown in Figs. 2–4^a

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Another reason could be that the real source and/or detector PSF do not have a purely Gaussian shape. In that case the used model would not fully describe the experimental situation. For example, an additional wide, low-intensity “foot” in the source distribution would reduce the spatial coherence and contrast [31]. However, this effect should be relatively weak, considering the overall good match of the peak position, which indicates that ${\sigma _s}$ was modeled correctly.

Experimentally the magnification can be changed by either changing the source–sample distance ${z_1}$ or the sample–detector distance ${z_2}$. If the total distance is kept constant and ${z_1}$ is changed, ${z_2}$ changes automatically. For experimental recommendations it is therefore instructive to also look at the contrast over ${z_1}$ (see Fig. 3). Larger total distances ${z_{{\rm tot}}}$ move the peak position toward larger ${z_1}$, which can be an experimental advantage, for example if there are restrictions due to the source housing or the sample mounting. However, lager total distances come at the cost of photons (also see CNR discussion below).

 

Fig. 3. Relative fringe contrast versus source–sample distance ${z_1}$. The peak position moves to larger ${z_1}$ for larger ${z_{{\rm tot}}}$ [but stays at the same magnification, see Fig. 2(a)]. Moreover, the peak is asymmetric, with a flat tail toward large ${z_1}$.

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Fig. 4. Contrast-to-noise ratio for different magnifications and total distances. (a) CNR over magnification using the relative fringe contrast from Fig. 2(a) and a polynomial model for the detector noise (see SI). Note that the CNR is highest for the shortest total distance, but generally relatively insensitive to changes in total distance. (b) Theoretical dependence of relative fringe contrast and CNR on the total distance, for constant refractive index and wavelength $\lambda = 1.2\;{\mathop{\rm A}\limits^\circ}$, ${\Delta}\mu = 0.0034\;{{\unicode{x00B5}{\rm m}}^{- 1}}$, and ${\Delta}\phi = - 0.278\;{{\unicode{x00B5}{\rm m}}^{- 1}}$. While the fringe contrast improves with larger total distance, the absolute CNR is relatively unaffected. The feature at ${z_{{\rm tot}}} = 18\;{\rm cm}$ is due to a change in the noise response of the detector for the used mixed gain mode (see SI). Moreover, please note that the changes in effective refractive index and wavelength due to the changing air absorption for different distances are not included in panel (b). All distances assume the same effective wavelength and refractive index. The beam hardening due to air absorption will cause the effective wavelength to decrease with larger ${z_{{\rm tot}}}$ (see Table 1). Accordingly, the fringe contrast will decrease (see SI), and the CNR shown here is overestimated for large ${z_{{\rm tot}}}$ compared to reality [see Fig. 4(a)].

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Generally, it is beneficial to be on the slowly decreasing tail of the peak toward larger ${z_1}$. This might seem counterintuitive considering that the resolution improves for shorter ${z_1}$. In our setup the magnification that provides the best resolution is about ${M_{{\rm res}}} = 10911$ [see Eq. (9)], meaning a ${z_1}$ of only 31 µm (for ${z_{{\rm tot}}} = 34.3\;{\rm cm}$). Besides this being experimentally impossible due to the target geometry (the W layer is deposited on a 100 µm tick diamond window), we see here that it is also not beneficial in terms of phase contrast.

Besides the relative fringe contrast, which is normalized to the mean image intensity, the CNR is an important figure of merit since the absolute contrast and noise level will change with the geometry. Figure 3 shows the CNR in dependence of magnification and total distance. Since the CNR simulations are based on ${c\!_f}$ [see Eq. (5)], a similar small quantitative mismatch between measurement and simulation can be observed. The noise was modeled based on the detector response for different exposure times (see SI). Interestingly, in the used setup the CNR is relatively insensitive to changes in total distance. This can be understood by looking at the dependence of the detector noise on the mean intensity. In contrast to the simple assumption of a Poisson noise, this detector shows a more complicated dependency (modeled section-wise as second-degree polynomials; see SI). This dependency is most likely due to the used mixed gain mode, which changes the gain depending on the signal strength to keep the signal-to-noise level stable over a large range of intensities. This change of gain is visible in the peak around ${z_{{\rm tot}}} = 18\;{\rm cm}$ in Fig. 3(b) where the increasing CNR is suddenly decreasing again. Please note that the absolute changes in CNR in the whole considered range of ${z_{{\rm tot}}}$ are very small. Reducing the total distance and thereby increasing the intensity therefore causes only a small increase in CNR. Nevertheless, the higher intensity at short distances allows using shorter integration times, which is experimentally favorable, especially in tomographic imaging.

Some additional observations can be made for high-magnification setups. It is noteworthy that the object ${\sigma _{{\rm obj}}}$ has a strong influence on the value of optimal magnification, since ${M_{{\rm opt}}}$ is now dominated by the second term in Eq. (8). This means that the fringe contrast for different features will be maximal at different magnifications. For high-magnification geometries where ${\sigma _s} \ll {\sigma _d}$, a small ${\sigma _{{\rm obj}}}$ will move the peak position toward higher magnifications, widen the peak, and increase the contrast (see Fig. S2 in the SI). In the limit of a perfectly sharp edge ${\sigma _{{\rm obj}}} = 0$, the optimal magnification becomes $M = 1 + \frac{{{\sigma _d}}}{{{\sigma _s}}} = 105$. Accordingly, a large ${\sigma _{{\rm obj}}}$ moves the peak toward a lower magnification, narrows it, and decreases the contrast. This means that any geometry will just be optimal for a specific spatial frequency, while fringes around other features are suppressed. Optimizing all fringes of complex features at the same time, even in a one-material sample, is thus not possible.

5. CONCLUSION

We have studied the influence of the experimental geometry on the PB-PCI fringe contrast of an isolated straight edge imaged with a high-magnification laboratory x-ray microscope. Like for low-magnification setups, the highest relative contrast is observed at an optimal magnification that only depends on the source size, detector resolution, and sample PSF. The peak position is in good agreement with the simulations based on the experimental parameters, while the peak contrast and fringe separation vary slightly from the simulations. This is most likely due to an overestimation of the effective wavelength in the simulations. A stronger dependence of the optimal magnification on the spatial frequencies of the object is observed in a high-magnification setup. However, in the used setup the CNR shows only a weak dependence on the total distance since the detector changes the gain according to the mean intensity. We conclude that the theory presented earlier for low-magnification setups can also be applied to high-magnification setups, providing some simple rules for choosing a suitable magnification to maximize edge enhancement fringes.