1. Introduction

Dual-phase x-ray grating interferometry (DP-XGI) [1] is an imaging technique that allows the study of microstructures at length-scales smaller than the spatial resolution of the imaging system [2,3]. In contrast to conventional grating interferometry, where one phase-grating is utilized to generate a Talbot carpet and one absorption-grating reads out the fringe formation at a fractional Talbot distance downstream [4,5], DP-XGI systems are composed of two identical phase-gratings with a high-frequency pattern, which produce a resolvable pattern downstream [6,7]. Working with pure phase gratings is an advantage as it increases the photon flux at the detector making the system more efficient without the analyzer grating. Furthermore, the low-aspect-ratio gratings are easier to fabricate [8].

The formation of the interference pattern at the detector is described by the Moiré effect, which is the superposition of two layers with periodically repeating patterns and slightly different periods [9]. This is analogous to the beating effect in acoustics, where two signals of slightly different frequencies interfere and result in modulation at a much lower frequency. For the case of DP-XGI, the superposition of the magnified periodic pattern of the first phase grating with the second phase grating generates a Moiré pattern large enough to be resolvable by the sensor. If an object is placed right before the first grating this Moiré pattern is modified in its amplitude, phase, and mean value. Analogous to other interferometry techniques, three different signals can be retrieved by a Fourier decomposition: the absorption, differential phase, and scattering or dark field signal [10–12].

The dark field signal (DFS) is generated by the scattered radiation produced by microstructures within the sample [13,14]. The reduction of visibility of the Moiré pattern results from the contribution of all the radiation scattered by structures of the object in different length-scales that are not resolvable by the imaging system. In grating interferometry, the sensitivity to a specific sized scattering feature is defined by the auto-correlation length of the interferometer [5,6,13,15]. The auto-correlation length is a parameter that links the scattering properties of the sample with the phase shift of the modulation function and is defined by geometrical parameters of the interferometer as $\xi = {\lambda \text {L}_{s}}/{\text {P}_\text {det}}$ [16] , where $\lambda$ is the wavelength, $\text {L}_\text {s}$ is the distance between the sample and the detector, and $\text {P}_\text {det}$ is the period of the interference pattern recorded by the detector. A change in any of these parameters makes the interferometer sensitive to scattering structures of different sizes. Specifically, in a DP-XGI setup, when the distance between the two phase gratings increases, the period of the Moiré pattern is reduced. Therefore, the auto-correlation length increases while the position of the sample is the same, and the magnification of the image remains constant [1,17]. A real space correlation function (RSCF) is constructed with the DFS retrieval of the sample at different auto-correlation lengths. This function provides quantitative information on the structure, i.e. size and shape, of the scattering particles [13,15,16,18].

Therefore, DP-XGI becomes a unique and robust approach for micro-structural characterization of materials maintaining large fields of view in contrast to conventional microscopy methods. Specifically, DFS retrieval has shown its potential among the medical and material science community. Mentioning the latest results, for example, DFS has been used in the improvement in the diagnosis of breast cancer [19], and in the detection of respiratory diseases in lungs [20], and for characterization of porosity in materials [21] or the orientation of fibers for industrial applications [22].

In the scope of designing a DP-XGI sensitive to scattering features on a nanometer length scale, a study of the parameter space of the DP-XGI, including the source, detector, and grating parameters is required. Different combinations of these parameters must be tested to optimize the contrast of the fringes in a range of auto-correlation lengths with the same length scale range as the scattering features of interest. The mean visibility of the fringes at the detector plane (${V}$), defined as the energy-weighted visibility due to source polychromaticity, is used as a figure of merit for the optimization of the interferometer, as a good DFS retrieval is related to the signal strength of the fringes [23,24]. Hence, visibility prediction is a key tool in the development of a grating-based imaging system.

A variety of simulation tools for GI have been developed to model image formation with Grating Interferometry (GI). For instance, Monte Carlo (MC) particle transport extended with the Huygens-Fresnel principle [25–27] , which includes interactions within the media, pure ray tracing algorithms [28,29] , and Hybrid models that combine MC with wave propagation [30–32]. Despite the substantial improvements in computation time reached by hybrid methods compared to pure MC approaches, or variance reduction techniques for MC [33] , a Fresnel-based wave propagation simulation tool is expected to outperform MC with respect to simulation time in most applications. Furthermore, radiation interaction with samples is out of the scope of this work with the main objective being to design a DP-XGI setup based on mean visibility optimization. Thus, a Fresnel wave propagation approximation is chosen to model the diffraction phenomena [34] , and evaluate the parameter space of the DP-XGI to optimize the mean visibility of the Moiré fringes at different auto-correlation lengths.

The simulation framework presented allows the modeling of a micro-focus source and describes well the wave propagation through two pure-phase gratings and polychromatic sources. Its versatility helps to find the optimal configuration of the setup that maximizes the mean visibility in a specific length scale in a range of hundreds of nanometers. Up to now, an analytical description of the dual-phase interferometer has been reported for both monochromatic [7] and polychromatic source [35] configurations. These contributions consider the Wigner distribution formalism to propose a theoretical description of the visibility reduction of the fringes and are validated by a numerical simulation. Also, theoretical frameworks have been presented for configurations with a source grating and have been validated with experiments [8,36]. Despite this, none of the present approaches have considered a rigorous study on the energy dependence of the visibility and the auto-correlation length. On top of this, neither an extended analysis nor a description of the system for an optimal design has been set out.

In this work, an exhaustive analysis of the parameter space of a DP-XGI is performed to obtain the optimal configuration of the system in order to maximize the mean visibility of the interference pattern for a certain range of auto-correlation lengths. Several parameters are evaluated, including source target materials, x-ray filters, grating design energy, period and materials, and detector scintillator and pixel pitch. Finally, experimental measurements of the mean visibility and quantification of auto-correlation lengths are performed in a lab setup to validate the simulation results.

2. Materials and methods

2.1 Dual-phase-grating interferometer

The configuration of a dual-phase x-ray grating interferometer setup is shown in Fig. 1. It consists of a lab-based x-ray tube (S), a detector with a pixel size $\Delta x$, and two phase-gratings with the same period ($P= P_{1}= P_{2}$). X-rays that are emitted by the source interact with the first grating $G_{1}$, which diffracts the incoming beam, generating a magnified intensity distribution with high frequency at the detector plane expressed as ${exp}(i2 \pi x/M_{1} P_{1})$ [7]. The magnification is given by the source-grating distance $L_{1}$ and the inter-grating distance D as

 

Fig. 1. Schematic representation of a dual-phase interferometer setup. The polychromatic source ($S$) with source size ($\sigma$) illuminates the first phase-grating ($\text {G}_{1}$). Afterward, a second phase-grating ($\text {G}_{2}$) with exactly the same properties as the first one, is placed to generate the Moiré pattern downstream where a sensitive detector ($Det$) is placed. $L_{1}$ is a fixed position between S and $G_{1}$, and $D$ is the distance between the two gratings that is changed in order to tune the auto-correlation length. The red lines indicate the planes where the real and reciprocal coordinates are defined for the Fresnel propagation description.

Download Full Size | PDF

Likewise, the second phase-grating $G_{2}$, which is placed at a distance D after $G_{1}$, generates a similar intensity distribution with a different magnified diffraction pattern with period $P_{2}$ at the detector plane expressed as ${exp}(i2 \pi x/M_{2} P_{2})$, where

In a first approximation, the final intensity modulation is then the superposition of the two patterns with different periods, $P'_{1} = M_{1} P_{1}$ and $P'_{2}=M_{2} P_{2}$, which can be expressed as

The modulation expressed in (3) represents the Moiré effect, in which the superposition of periodic patterns with two slightly different frequencies generates a beat pattern with low and high frequencies, similar to the mechanism utilized in [22]. The beat low-frequency patterns contribute to the formation of the fringes resolved by the detector [7] and therefore to the mean visibility, whereas the high-frequency component is too fine to be resolved and vanishes completely. The beat pattern corresponding to the first order resolved downstream, is expressed as

As it was demonstrated by Kagias et al. [1], tunability of the dark field signal [13] is performed smoothly by changing the inter-grating distance ($D$), as the frequency of the fringes $f_{det}$ in Eq. (4) increases with the inter-grating distance, allowing the system to be sensitive to different length scales. Therefore, with the retrieval of the DFS at different auto-correlation lengths, it is possible to generate the real space correlation function, which describes scattering properties of the microstructures that are below the spatial resolution of the imaging system in real space [16,17].

Aiming for the optimal design of a DP-XGI for DFS retrieval in a specific auto-correlation length scale range, the performance of the different parameters of the setup has to be evaluated. The properties of the source, such as the characteristic spectrum of the anode material, the source size, and spectrum filtering, and the detector properties such as the pixel size and the scintillator material, will define the space parameters which will be evaluated. When referring to the geometry of the DP-XGI, the physical distances between the source, the gratings, and the sensor are considered.

2.2 Numerical simulation

The numerical simulation framework is developed to estimate the sensitivity of the dual-phase interferometer to small-angle scattering while tuning the auto-correlation length. A wave propagation approach has been implemented to predict the behavior of the interference pattern downstream by describing the diffraction phenomena of the polychromatic imaging system. The mean visibility of the final pattern, calculated as the energy-weighted visibility of the fringes, is the figure of merit for optimization.

The simulation framework developed makes use of the Fresnel transfer function approach of wave propagation described by Goodman [37], and the Fresnel scaling theorem demonstrated in [38,39]. This theorem states that the diffraction pattern generated by a point source, with a magnification M related to the source-object-detector distances, is analogous to a diffraction pattern generated in a transformed coordinate system with a plane-wave illumination, scaled transversely by a magnification factor (M), a propagation distance scaled down (1/M), and a re-scaled intensity ($1/M^2$). Under this consideration, and with help of the sketch shown in Fig. 1, the proposed model is described as follows. A monochromatic plane-wave ($\psi (x,\lambda )$) interacting with the first grating ($G_{1}$) lying at the Cartesian plane with coordinate x and illuminated with a wave field with wavelength ($\lambda$) in the z direction, is defined as

After the interaction with the first grating, the wavefront $\psi _{G1+}$ in Eq. (8) is propagated through free space until it meets the second grating $G_{2}$ placed at a distance $D$ from $G_{1}$ reaching the Cartesian plane with coordinate $x'= x\cdot M_{g1}$ and propagation distance in the transformed coordinate system $D'= D/M_{g1}$ with $M_{g1}= L_{1}+D/L_{1}$.

Based on the Fresnel approximation, the resulting wave field is the convolution between the complex field from Eq. (8) and the diffraction impulse response

With $u$ the reciprocal of coordinate x, the kernel in the Fourier space is defined as

According the convolution theorem (with $\mathcal {F}$ the Fourier transform) the wavefront before passing through $G_{2}$ is expressed as,

From this expression, a new amplitude and phase ($\varphi$) define the wavefront $\psi _{G2-}$ before passing through the second grating.

Similarly, ${\psi } _{G2-}(x^{\prime },\lambda )$ interacts with $G_{2}$ and propagates a distance $L_{2}=z_{G_{2}-Det}$ downstream before it reaches the detector plane. It again corresponds to a multiplication with the transmission function of $T_{G2}(x')$ followed by a convolution with a similar propagator as in Eq. (9). Hence, the wavefront at the detector plane with spatial Cartesian coordinate $(x'')$, after the back Fourier transformation results in

Then, the intensity profile downstream for one single wavelength ($\lambda$) is calculated from

The influence of the source size and the polychromatic nature of the source are included in the final expression of the intensity pattern. As the DP-XGI is a spatially invariant system, any displacement of the point source in the transverse direction yields the same displacement of the fringes at the detector plane [7]. Furthermore, as a monochromatic wavefront is a solution of the wave equation, the addition of different wavefronts is also a solution of this equation. Therefore, the source profile at the detector plane ($S_{\sigma _{sen}}$) can be modeled as a Gaussian distribution with a $\sigma _{sen}$ equal to the projected source size [24,40,41]. It is convolved with the intensity pattern obtained with Eq. (13) and integrated over all the energies (or its corresponding wavelength) of the target spectrum, weighted by its intensity ($S(\lambda )$) to account for polychromaticity as follows

Finally, this simulation framework evaluates the influence of the detector efficiency sensitivity ($\epsilon (\lambda )$), which includes both the absorption efficiency and the deposited dose for photon [42], and the influence of spectrum filters, air, and grating substrates ($F(\lambda )$). For that purpose, an extra term $W(\lambda )$ that weights the interference pattern for each $\lambda$ is defined as

Then, the intensity pattern at the detector results in

The final goal of the DP-XGI optimization is to generate a good dark field signal retrieval, which is defined as the visibility reduction of the interference pattern at the detector. The figure of merit for the DP-XGI optimization is the mean visibility (V) of these fringes [6,7] because a good contrast of the interference pattern downstream increases the reliability of the dark field signal retrieval. The mean visibility (V) of an interference pattern is defined as the ratio of the first ($a_{1}$) and zero-order ($a_{0}$) of the Fourier coefficients of the interference pattern, and it is equal to the ratio

 

Fig. 2. Representation of the intensity variation in one pixel while stepping $G_{2}$ in the x direction without (black) and with (red) a sample in the beam. Here, $a_{0}$ and $a_{1}$ represent the first and second Fourier coefficients, from which the visibility is calculated.

Download Full Size | PDF

The effective energy of the DP-XGI is weighted by the properties of the spectrum expressed previously in terms of spectrum intensity $(S)$, properties of the filters and gratings $(F)$, and the sensor response $(\epsilon )$ for each energy. Additionally, a term that accounts for the visibility per energy ($V_{E}$) is included for the effective energy calculation as it accounts for the interferometer response to each wavefront with characteristic energy at different inter-grating distances. It is calculated as

As noted in Eq. (14), due to the polychromatic illumination configuration of the DP-XGI, the fringe pattern is a spectrum-weighted intensity. In the analytical model proposed by Yan et. al., [35], the visibility harmonics of each energy-weighted interference pattern contribute to the mean visibility of the final fringes detected downstream. Nevertheless, the contribution of harmonics is limited to the second order as higher frequencies cannot be resolved by the detector due to pixel limitations and a decrease of coherence at higher orders. If $V_{n}$ is defined as the contribution of the n-harmonic, the intensity pattern resolved by the detector up to the second order is expressed as

2.3 Simulation constraints

The simulation framework was evaluated for a combination of different parameters keeping a symmetric design, i.e. $L_{1}=L/2$, which should maximize the auto-correlation length range [6,43] and the source-to-detector distance was limited to 1 m for compactness. A tungsten (W) target with $k_{\alpha _1}$ line at 59.32 keV, and $L_{\alpha _1}$ and $L_{\beta _1}$ lines at 8.397 and 9.275 keV respectively, and molybdenum (Mo) target with $K_{\alpha _1}$-line of 17.479 keV, both targets most commonly found in microfocus x-ray sources were evaluated. The corresponding spectra is shown in Fig. 9 in the Appendix. In addition, a beryllium (Be) filter of 0.2 $mm$ thickness was added to account for the influence of the exit window in typical directional tubes in combination with source sizes of 8 $\mu m$ and 20 $\mu m$.

Two detector responses were tested. The first one is composed of a gadolinium oxysulfide ($Gd_{2}O_{2}S$) scintillator of 6.6 $g/ cm^{2}$ mass thickness and a 9 $\mu m$ pixel size. The second one is a cesium iodide (CsI) scintillator of 45 $g/ cm^{2}$ mass thickness with a pixel size of 16.4 $\mu m$ and 32.8 $\mu m$ in binning mode. Figure 10 in the Appendix shows the plots of the scintillator responses implemented in the simulation and calculated with Monte Carlo simulations reported by Dhaene et al. [42]. The gratings were modeled with three different periods of 1.0 $\mu$m, 1.2 $\mu$m, and 1.5 $\mu$m, all with a duty cycle of 0.5. Different design energies were evaluated, and the height of the grating was calculated to generate a $\pi$-phase shift of the wavefront at specific design energy. The thickness of the substrate and the influence of air is included in the final calculation of the intensity of the fringes (See Table 1). The mean visibility (V) is calculated as defined in Eq. (17) over the 1D Moiré pattern generated downstream. The sampling rate of the aperture is 273 $1/ \mu m$, and the simulated field of view (FOV) is 0.3 mm large.

Table 1. Summary of the parameters used for the simulation and the experimental validation.

View Table

2.4 Experimental validation

In order to validate the simulation framework performance, the fringe visibility was measured utilizing a dual-phase x-ray grating interferometer (DP-XGI) with the following characteristics: The source is a transmission-type x-ray tube FXT-160.51 (FEINFOCUS GmbH), with a tungsten (W) target (thickness 3 $\mu$m) operated at 40 kVp and 100 $\mu$A. The detector is a sCMOS camera with 9 $\mu$m pixel size and a $Gd_{2}O_{2}S:Tb$ (Gadox) scintillator of 6.6 $g/cm^2$ mass thickness (XR Gsense, Photonics Science). Two identical Silicon phase-gratings were fabricated in-house (Paul Scherrer Institute) using the Displacement Talbot Lithography method and Deep Reactive Ion Etching [44,45], with a period of 1.0 $\mu$m and 0.5 duty cycle. At the design energy $\text {E}_{D}$=22 keV, the height of the gratings is calculated to be 28.17 $\mu m$ for a phase-shift of $\pi$ of the wavefront [46,47]. A scanning electron microscopy (SEM) image of the profile of one of the gratings is shown in Fig. 11 in the Appendix. The source-detector distance is kept one meter long and the $G_{1}$ grating is placed at 0.5 m from the source.

The mean visibility was obtained by the Fourier decomposition of the phase-stepping curve. Based on the theoretical description of fringe formation in a polychromatic dual-phase interferometer presented by Yan et al. [7,35], the first and second Fourier coefficients are evaluated as they contribute to the mean visibility of the fringes. The PSC was built with scans over 8 steps covering one period of the grating with a linear stage (CLS-5282 SmarAct GmbH), each scan has an exposure time of 30 s. Furthermore, the inter-grating distance (D) was tuned from 3 mm to 15 mm in steps of 1 mm with a linear stage (CLS-5252 SmarAct GmbH), covering auto-correlation lengths from 50 nm to 280 nm. The initial inter-grating distance was calculated with an algorithm developed in-house which performs simultaneously the alignment of the gratings and the integrating distance calculation. Through the alignment algorithm, the two gratings can be aligned accurately with errors not larger than hundredths of a degree, meanwhile, the distances can be accurately determined up to the micrometer range. This allows calculating the inter-grating distance and the source to G1 distance with high accuracy.

3. Results and discussion

3.1 Experimental validation

In Fig. 3, the mean visibility of the interference pattern obtained with the simulation (red line) is compared with the mean visibility of the experimental data (dots in black) of the interferograms, as a function of the inter-grating distances on the top axes and the corresponding calculated auto-correlation length in the bottom one.

 

Fig. 3. Comparison of the fringe visibility between the numerical simulation (red line) and the experimental results(black). For both the simulation and the experiment a W target was considered, phase-gratings with period of 1.0 $\mu m$, same source-grating and grating-detector distance, and a pixel size of 9 $\mu m$ is set.

Download Full Size | PDF

For comparison purposes with the simulated mean visibility, the calculation has been limited to the center of the detector where the influence of the duty cycle of the gratings on the cone beam illumination is neglected and the wavefront is comparable to the one of a plane wave. The simulated mean visibility is obtained with Eq. (17), the variation of the interference pattern at each inter-grating distance did not preset deviation to report statistics. The experimental mean visibility is calculated by evaluating the second harmonic ($V_{2}$) of the Fourier coefficients of the phase stepping curve for each pixel of the detector and at different inter-grating distances. The values reported for mean visibility in the graph and the corresponding standard deviation are obtained with the average of several pixels within the center of the FOV.

The plots show a high agreement between both the simulated and the measured pattern when comparing them in a FOV of 0.3 mm, validating the implementation of the simulation for the DP-XGI optimization. Furthermore, these results present an experimental validation in our setup of the analytical prediction proposed by Yan et al. in which the contribution of the second harmonic dominates over the first coefficient for $\pi$ shift phase gratings [35]. Table 1 summarizes the parameters implemented in both, the experiment and the simulation.

The discrepancy between the experimentally measured mean visibility and the simulation performance has several causes. First, the precision in measuring the source-first grating and the source-detector distance cannot be calculated by the algorithm, leading to uncertainties in the calculation of the auto-correlation length. Second, the difference between the values of the modeled grating thickness of the substrate in which the grating is etched, and the value of the height of the grating. Third, the limitation in modeling the exact shape of the grating, since the difference between the effective shape and the simulated rectangular shape leads to differences between the simulated and experimentally measured mean visibility [48]. In addition, the spectrum and sensitivity of the scintillator are simulated curves implemented in the model, which are another source of error compared to the real spectra and scintillator response. Finally, the average mean visibility is calculated by averaging the pixels in a 2D array at the center of the detector, the corresponding standard deviation is plotted in Figure 3. The experimental error also takes into account the mechanical instability and the accuracy of the motors, which affect the step size of the grating and thus the formation of the phase-stepping curve. Although these errors produce a difference between the experiment and the model, the results agree well, making the simulation framework reliable for evaluating the parameter spectra of other DP-XGI configurations.

3.2 System optimization

The performance of the dual-phase interferometer is evaluated with the developed simulation framework considering various parameters of the setup. The objective is to find a suitable combination of parameters to obtain good mean visibility according to the Eq. (17) in a nanoscale range of auto-correlation lengths for a given design energy of the optical system. Therefore, different combinations of source, detector system, and grating parameters were considered. The selection criteria for the optimal configuration is a visibility range greater than $10\%$ over the full range of required auto-correlation lengths. The following results show the optimization process for the design of a DP-XGI, which operates in a low energy range. However, the simulation framework is versatile for optimizing any DP-XGI under different design requirements such as geometry, design energy, and range of auto-correlation length.

The first parameter evaluated is the period of the grating. Three different period sizes were considered: 1.0 $\mu$m, 1.2 $\mu$m, and 1.5 $\mu$m. The design energy of the grating is kept at 22 keV and the source voltage at 40 kVp. The source size is 20 $\mu$m, the detector has a pixel size of 16 $\mu$m, and the scintillator is made of CsI. Figure 4 plots the mean visibility as a function of auto-correlation length for the two target materials molybdenum (Mo) and tungsten (W). For comparison, the dotted black line shows the threshold for good visibility ($> 10\%$). The curves show that the visibility increases when the period of the gratings is smaller, peaking at a smaller auto-correlation length as well. Thus, the optimal period would be 1.0 $\mu$m for both types of targets. A comparison of the two targets shows that Mo is also readily visible at higher auto-correlation lengths and has a higher peak at an auto-correlation length of approximately 100 nm. The difference in auto-correlation length range is expected because the effective energy of the imaging system calculated with each target is different.

 

Fig. 4. Mean visibility at different auto-correlation lengths (inter-grating distance) as a function of the period of the gratings for a molybdenum (Mo) target (continuous lines) and a tungsten target (marked lines). Design energy is set to 22 keV, the voltage at 40 kVp, pixel size of 16 $\mu$m, and a CsI scintillator.

Download Full Size | PDF

The second parameter evaluated is the source voltage. Figure 5 shows the dependence of the mean visibility on different source voltage values (kVp) at different auto-correlation lengths for the tungsten target. Although, good visibility is presented in all four cases evaluated, at the smaller voltage the visibility increases and the peak gets wider, covering a larger range of auto-correlation lengths.

 

Fig. 5. Mean visibility as a function of the auto-correlation length for different source voltage for the tungsten target with a 20 $\mu m$ source size. Period of the grating is 1.0 $\mu m$, $\text {E}_\text {D}= 22 \text { keV}$ , scintillator is CsI with a pixel size of 16 $\mu$m

Download Full Size | PDF

Likewise, various setup design energies ($E_\text {D}$) were evaluated and the corresponding height of the grating was calculated in order to cause a phase shift of $\pi$ to the wavefront. Figure 6 shows how the design energy of the setup affects the variation of the mean visibility profile for (a) tungsten and (b) molybdenum targets at 40 kVp. The mean visibility curves for the W target have a very similar profile along different auto-correlation lengths with good visibility ranging from 40 nm to 150 nm auto-correlation length and peaking at 90 nm with $30 \%$ for the design energies of 20 keV, 22 keV, and 24 keV. The lowest mean visibility corresponds to a design energy of 15 keV, and the highest is found for a design energy of 22 keV. Figure 6(b) shows that for the Mo target, the highest visibility is about 40 $\%$ for a design energy corresponding to the $K_{\alpha }$ energy of the spectrum. For lower design energies, e.g., 15 keV, or higher design energies such as 20 keV or 22 keV that have been evaluated, the visibility at the peak is reduced from 40 $\%$ to 30 $\%$, meaning a reduction of 25 $\%$ when using a design energy of 22 keV instead of one of 17.4 keV.

 

Fig. 6. Evaluation of the mean visibility at different auto-correlation lengths for (a) W target and (b) Mo at different design energies for the gratings, to generate a $\pi$- phase shift to the incoming beam. Period of the grating is 1.0 $\mu m$, $\text {E}_\text {D}= 22 \text { keV}$ , scintillator is CsI with a pixel size of 16 $\mu$m.

Download Full Size | PDF

The influence of the source size is evaluated simultaneously with the spectrum filtering. The voltage is set at 40 kVp, design energy of 22 keV, and a 1.0 $\mu$m gratings period. Figure 7(a) compares an 8 $\mu m$ and a 20 $\mu m$ source size in combination with a beryllium filter of 0.2 mm thickness. Figure 7(a) shows how the visibility decreases when the source size increases as the spatial coherence are lost. Also, a local maximum presented around 350 nm decreases for higher source sizes and almost vanishes at 8 $\mu$m source size. Therefore, the smallest the source size, the higher the mean visibility.

 

Fig. 7. Mean visibility dependency on the source size for (a) a Mo target and a W target with the spectrum filtered with 0.2 mm of beryllium and (b) the mean visibility evaluated without filtering for different source sizes. Both figures were calculated using a CsI scintillator with a pixel size of 16$\mu m$.

Download Full Size | PDF

Two different sensor responses in combination with pixel size have been compared. In Fig. 8(a) the visibility profiles are plotted for both targets with a CsI scintillator and a pixel size of 16 $\mu m$. Figure 8(b) shows the comparison with a Gadox scintillator and 9 $\mu m$ pixel size. The combination of parameters was chosen according to the fabrication characteristics of two different detectors available in the market. In both cases, the mean visibility and the interval of auto-correlation length with good mean visibility are lower when the source size increases. This evidences that, as expected, the visibility is reduced with the spatial coherence of the wavefront. The better performance of the Mo target in higher auto-correlation length values is also perceived when comparing the two detectors.

 

Fig. 8. Comparison of two detectors: Gadox scintillator with 9 $\mu m$ pixel size and CsI scintillator with 16 $\mu m$ pixel size for (a) tungsten target and (b) molybdenum target. Targets were filtered with 0.2 mm of beryllium.

Download Full Size | PDF

The spectral response of the detector is evaluated with two different sensors. The first one is an x-ray detector with a CsI scintillator of 100 $\mu m$ thickness, 16.4 $\mu m$ pixel size. The second one is an x-ray detector with a GdOS scintillator of 6.6 $gr/cm^{2}$ mass thickness and 9 $\mu m$ pixel size. In Fig. 8 the mean visibility is plotted for (a) tungsten and (b) molybdenum for each combination of scintillator and pixel size. According to these curves, the performance of the two sensors is similar, peaking slightly higher for the gadox scintillator at a lower auto-correlation length. The influence of the pixel size in the behavior of the mean visibility is presented for the molybdenum target with the CsI scintillator using the 16.3 $\mu m$ pixel size and the binning mode with 32.8 $\mu m$ (Fig. 8(b)). In general, better performance is found with the Gadox scintillator and 9 $\mu m$ pixel size. For both types of scintillators, the smaller the pixel size and the source size, the higher the mean visibility.

In summary, the good performance of a DP-XGI relies on the appropriate combination of the setup components as all of them influence the mean visibility behavior simultaneously. A system designed for good dark field signal retrieval in a specific range of correlation lengths can benefit from this analysis of parameters. The evaluation of the parameters can be extended to optimize a DP-XGI working in higher energy ranges for the analysis of denser materials. Also, different grating properties can be adjusted according to fixed source-detector parameters, or alternative combinations of parameters can be tested according to the components available. The optimization procedure presented aimed to optimize a DP-XGI to retrieve good visibility in a nanoscale range with a source that operates at 40 kV with a tungsten target. For that configuration, the optimal parameters found for the gratings are a period of 1.0 $\mu$m with $\text {E}_\text {D}=$ 22 keV. Although not optimal, these parameters also work well for the Mo target. Either target has a good visibility response for a CsI sensor, small pixel size, and small source size.

4. Conclusion

The performance of a dual-phase x-ray grating interferometer (DP- XGI) is evaluated to optimize its sensitivity to the dark field signal generated by scattering samples of a sample in the sub-pixel length scale. A Fresnel wave propagation simulation framework is presented as a tool to optimize the imaging system operated in an auto-correlation length range that allows the investigation of scattering features with the size of a few hundreds of nanometers. As the quality of the interference pattern is related to the dark field signal retrieval, the mean visibility of the Moiré fringes is used as a figure of merit for the design of the system. An expression to calculate the effective energy and the auto-correlation length based on the influence of visibility at different inter-grating distances is proposed and the auto-correlation length is set as the proxy for the feature size being investigated. Different components of the DP-XGI were evaluated to reach good visibility (i.e. $> 10 \ \%$) in a nanoscale range of the auto-correlation length. The study evaluates different source spectra and detector setups, taking into account the practical constraints of an experimental setup. Sensitivities in the auto-correlation range from 50 to 480 nm are reported for the simulated DP-XGI, and experimentally validated up to 260 nm. The latter results validate the simulation framework, indicating the accuracy of the model. Although the validation is only performed in the design energy range between 17 keV and 24 keV, there are no obstacles that prevent this simulation framework to be used for higher energies or considering other geometries. In further work, we pursue an experimental implementation of a DP-XGI to work in both low and high-energy ranges, based on the results of the optimization presented here. Also, aiming to reduce the experimental errors an automatic alignment of the gratings and calculation of the distance between sample gratings and the inter-grating distance is pursued. In order to enable parameter optimization, the evaluation of the mean visibility was limited to the center of the gratings. However, the influence of the cone beam illumination as well as some other experimental considerations were recently studied [48]. Future work is focused on modeling the influence of a scattering object right before the first grating in the visibility behavior of the fringes and evaluating the dark field signal at different correlation lengths.

5. Appendix

5.1. Spectrum of the target materials

Two different x-ray spectra were implemented in the simulation for two different target materials (W and Mo). Figure 9 shows the spectra for 40 kVp simulated by Dhaene $et \ al$. [42]

 

Fig. 9. Simulated spectrum for (a) W and Mo at 40 kVp.

Download Full Size | PDF

5.2. Scintillator response

The curves in Fig. 10 show the statistical energy deposition for Caesium iodide (CsI) scintillator and Gadox scintillator implemented in the simulation.

 

Fig. 10. Statistical energy deposition of CsI and Gadox scintillators of different thicknesses used to calculate the detector responses in the simulation.

Download Full Size | PDF

5.3. SEM images of the phase-gratings

The experimental evaluation of visibility was performed with two phase-gratings with a period of $1.0 \ \mu$m and theoretical height of 28.17 $\mu m$ to generate a $\pi$-shift at a design energy of $22$ keV. A scanning electron microscopy (SEM) profile of the grating is shown in Fig. 11.

 

Fig. 11. SEM profile of one of the Silicon phase-gratings with period of 1.0 $\mu m$ and 28 $\mu m \pm 10 \%$ height for a $\pi$-shift at a design energy of $22\; \text {keV}$.

Download Full Size | PDF

Funding

Fonds Wetenschappelijk Onderzoek (3179I12018); Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (159263, R’Equip 189662, Sinergia Nr. CRSII5_183568); SwissLOS Lottery Fund of the Kanton of Aargau, Switzerland; Interreg Vlaanderen-Nederland (Smart*Light); Regional Development Funds (Smart*Light); Provincie Oost-Vlaanderen (Smart*Light) (0386).

Acknowledgments

We acknowledge our technicians Gordan Mikuljan and Philipp Zuppiger from the TOMCAT group at the Paul Scherrer Institute for their technical support. Matias Kagias acknowledges the Swiss National Foundation for financial support (grant Nr. P400P2$\_$194371). The authors would like to thank the anonymous reviewers for their useful comments.

Disclosures

The authors declare no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110(1), 014105 (2017). [CrossRef]  

2. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef]  

3. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]  

4. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]  

5. T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Dev. X-Ray Tomogr. V 6318, 63180S (2006). [CrossRef]  

6. H. Miao, A. Panna, A. A. Gomella, E. E. Bennett, S. Znati, L. Chen, and H. Wen, “A universal moiré effect and application in X-ray phase-contrast imaging,” Nat. Phys. 12(9), 830–834 (2016). [CrossRef]  

7. A. Yan, X. Wu, and H. Liu, “Quantitative theory of x-ray interferometers based on dual phase grating: fringe period and visibility,” Opt. Express 26(18), 23142–23155 (2018). [CrossRef]  

8. J. Bopp, V. Ludwig, M. Seifert, G. Pelzer, A. Maier, G. Anton, and C. Riess, “Simulation study on x-ray phase contrast imaging with dual-phase gratings,” Int. J. CARS 14(1), 3–10 (2019). Int. journal computer assisted radiology surgery [CrossRef]  

9. E. Gabrielyan, “The basics of line moiré patterns and optical speedup,” arXiv, arXiv preprint physics/0703098 (2007).

10. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef]  

11. M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55(18), 5529–5539 (2010). Phys. Med. Biol. [CrossRef]  

12. Z. T. Wang, K. J. Kang, Z. F. Huang, and Z. Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95(9), 094105 (2009). [CrossRef]  

13. W. Yashiro, Y. Terui, K. Kawabata, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18(16), 16890 (2010). [CrossRef]  

14. M. Kagias, Z. Wang, G. Lovric, K. Jefimovs, and M. Stampanoni, “Simultaneous reciprocal and real space x-ray imaging of time-evolving systems,” Phys. Rev. Appl. 15(4), 044038 (2021). [CrossRef]  

15. S. K. Lynch, V. Pai, J. Auxier, A. F. Stein, E. E. Bennett, C. K. Kemble, X. Xiao, W.-K. Lee, N. Y. Morgan, and H. H. Wen, “Interpretation of dark-field contrast and particle-size selectivity in grating interferometers,” Appl. Opt. 50(22), 4310–4319 (2011). [CrossRef]  

16. M. Strobl, “General solution for quantitative dark-field contrast imaging with grating interferometers,” Sci. Rep. 4(1), 7243 (2015). [CrossRef]  

17. A. Pandeshwar, M. Kagias, Z. Wang, and M. Stampanoni, “Modeling of beam hardening effects in a dual-phase x-ray grating interferometer for quantitative dark-field imaging,” Opt. Express 28(13), 19187–19204 (2020). [CrossRef]  

18. R. P. Harti, M. Strobl, B. Betz, K. Jefimovs, M. Kagias, and C. Grünzweig, “Sub-pixel correlation length neutron imaging: Spatially resolved scattering information of microstructures on a macroscopic scale,” Sci. Rep. 7(1), 44588 (2017). [CrossRef]  

19. J. Emons, P. A. Fasching, M. Wunderle, et al., “Assessment of the additional clinical potential of x-ray dark-field imaging for breast cancer in a preclinical setup,” Ther. Adv. Med. Oncol. 12, 175883592095793 (2020). [CrossRef]  

20. K. Willer, A. Fingerle, W. Noichl, F. De Marco, M. Frank, T. Urban, R. Schick, A. Gustschin, B. Gleich, J. Herzen, T. Koehler, A. Yaroshenko, T. Pralow, G. Zimmermann, B. Renger, A. Sauter, D. Pfeiffer, M. Makowski, E. Rummeny, P. Grenier, and F. Pfeiffer, “X-ray dark-field chest imaging can detect and quantify emphy-sema in copd patients,” medRxiv, PMC8565798 (2021). [CrossRef]  

21. B. K. Blykers, C. Organista, M. N. Boone, M. Kagias, F. Marone, M. Stampanoni, T. Bultreys, V. Cnudde, and J. Aelterman, “Tunable X-ray dark-field imaging for sub-resolution feature size quantification in porous media,” Sci. Rep. 11(1), 18446 (2021). [CrossRef]  

22. M. Kagias, Z. Wang, M. E. Birkbak, E. Lauridsen, M. Abis, G. Lovric, K. Jefimovs, and M. Stampanoni, “Diffractive small angle x-ray scattering imaging for anisotropic structures,” Nat. Commun. 10(1), 5130 (2019). [CrossRef]  

23. M. Chabior, T. Donath, C. David, M. Schuster, C. Schroer, and F. Pfeiffer, “Signal-to-noise ratio in x ray dark-field imaging using a grating interferometer,” J. Appl. Phys. 110(5), 053105 (2011). [CrossRef]  

24. T. Thuering and M. Stampanoni, “Performance and optimization of X-ray grating interferometry,” Phil. Trans. R. Soc. A. 372(2010), 20130027 (2014). [CrossRef]  

25. A. Prodi, E. Knudsen, P. Willendrup, S. Schmitt, C. Ferrero, R. Feidenhans, and K. Lefmann, “A monte carlo approach for simulating the propagation of partially coherent x-ray beams,” in Advances in Computational Methods for X-Ray Optics II, vol. 8141 (SPIE, 2011), pp. 54–61.

26. D. G. Fischer, S. A. Prahl, and D. D. Duncan, “Monte carlo modeling of spatial coherence: free-space diffraction,” J. Opt. Soc. Am. A 25(10), 2571–2581 (2008). [CrossRef]  

27. S. Cipiccia, F. A. Vittoria, M. Weikum, A. Olivo, and D. A. Jaroszynski, “Inclusion of coherence in monte carlo models for simulation of x-ray phase contrast imaging,” Opt. Express 22(19), 23480–23488 (2014). [CrossRef]  

28. E. Bergbäck Knudsen, A. Prodi, J. Baltser, M. Thomsen, P. Kjaer Willendrup, M. Sanchez del Rio, C. Ferrero, E. Farhi, K. Haldrup, A. Vickery, R. Feidenhans’l, K. Mortensen, M. M. Nielsen, H. Friis Poulsen, S. Schmidt, and L. Kim, “Mcxtrace: a monte carlo software package for simulating x-ray optics, beamlines and experiments,” J. Appl. Crystallogr. 46(3), 679–696 (2013). [CrossRef]  

29. J. P. Wilde and L. Hesselink, “Modeling of an x-ray grating-based imaging interferometer using ray tracing,” Opt. Express 28(17), 24657–24681 (2020). [CrossRef]  

30. S. Peter, P. Modregger, M. K. Fix, W. Volken, D. Frei, P. Manser, and M. Stampanoni, “Combining monte carlo methods with coherent wave optics for the simulation of phase-sensitive x-ray imaging,” J. Synchrotron Radiat. 21(3), 613–622 (2014). [CrossRef]  

31. P. Bartl, J. Durst, W. Haas, E. Hempel, T. Michel, A. Ritter, T. Weber, and G. Anton, “Simulation of x-ray phase-contrast computed tomography of a medical phantom comprising particle and wave contributions,” in Medical Imaging 2010: Physics of Medical Imaging, vol. 7622 (SPIE, 2010), pp. 252–260.

32. J. Sanctorum, J. De Beenhouwer, and J. Sijbers, “X-ray phase contrast simulation for grating-based interferometry using gate,” Opt. Express 28(22), 33390–33412 (2020). [CrossRef]  

33. S. Tessarini, M. K. Fix, P. Manser, W. Volken, D. Frei, L. Mercolli, and M. Stampanoni, “Semi-classical monte carlo algorithm for the simulation of x-ray grating interferometry,” Sci. Rep. 12(1), 2485 (2022). [CrossRef]  

34. R. P. Harti, C. Kottler, J. Valsecchi, K. Jefimovs, M. Kagias, M. Strobl, and C. Grünzweig, “Visibility simulation of realistic grating interferometers including grating geometries and energy spectra,” Opt. Express 25(2), 1019–1029 (2017). [CrossRef]  

35. A. Yan, X. Wu, and H. Liu, “Predicting fringe visibility in dual-phase grating interferometry with polychromatic x-ray sources,” J. X-Ray Sci. Technol. 28(6), 1055–1067 (2020). [CrossRef]  

36. Y. Ge, J. Chen, P. Zhu, J. Yang, S. Deng, W. Shi, K. Zhang, J. Guo, H. Zhang, H. Zheng, and D. Liang, “Dual phase grating based x-ray differential phase contrast imaging with source grating: theory and validation,” Opt. Express 28(7), 9786–9801 (2020). [CrossRef]  

37. J. W. Goodman, Introduction to Fourier optics, 5 (W.H. Freeman and Company, 2017).

38. D. Paganin, Coherent X-ray optics, 6 (Oxford University Press on Demand, 2006).

39. P. R. T. Munro, “Rigorous multi-slice wave optical simulation of x-ray propagation in inhomogeneous space,” J. Opt. Soc. Am. A 36(7), 1197–1208 (2019). [CrossRef]  

40. A. Yan, X. Wu, and H. Liu, “Clarification on generalized lau condition for x-ray interferometers based on dual phase gratings,” Opt. Express 27(16), 22727–22736 (2019). [CrossRef]  

41. J. Vignero, N. Marshall, K. Bliznakova, and H. Bosmans, “A hybrid simulation framework for computer simulation and modelling studies of grating-based x-ray phase-contrast images,” Phys. Med. Biol. 63(14), 14NT03 (2018). [CrossRef]  

42. J. Dhaene, E. Pauwels, T. D. Schryver, A. D. Muynck, M. Dierick, and L. V. Hoorebeke, “A realistic projection simulator for laboratory based x-ray micro-ct,” Nucl. Instrum. Methods Phys. Res., Sect. B 342, 170–178 (2015). [CrossRef]  

43. T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys. 106(5), 054703 (2009). [CrossRef]  

44. Z. Shi, K. Jefimovs, L. Romano, and M. Stampanoni, “Optimization of displacement talbot lithography for fabrication of uniform high aspect ratio gratings,” Jpn. J. Appl. Phys. 60(SC), SCCA01 (2021). [CrossRef]  

45. K. Jefimovs, L. Romano, J. Vila-Comamala, M. Kagias, Z. Wang, L. Wang, C. Dais, H. Solak, and M. Stampanoni, “High-aspect ratio silicon structures by displacement Talbot lithography and Bosch etching,” in Advances in Patterning Materials and Processes XXXIV, vol. 10146C. K. Hohle, ed., International Society for Optics and Photonics (SPIE, 2017), pp. 140–146.

46. Z. Shi, K. Jefimovs, L. Romano, and M. Stampanoni, “Towards the fabrication of high-aspect-ratio silicon gratings by deep reactive ion etching,” Micromachines 11(9), 864 (2020). [CrossRef]  

47. M. Kagias, Z. Wang, V. A. Guzenko, C. David, M. Stampanoni, and K. Jefimovs, “Fabrication of au gratings by seedless electroplating for x-ray grating interferometry,” Mater. Sci. Semicond. Process. 92, 73–79 (2019). Material processing of optical devices and their applications. [CrossRef]  

48. R. Tang, C. Organista, G. Wannes, W. Stolp, M. Stampanoni, J. Aelterman, and M. N. Boone, “Detailed analysis of the interference patterns measured in lab-based x-ray dual-phase grating interferometry through wave propagation simulation,” Optics Express (Manuscript accepted November 2022).