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Abstract
One of the key tasks in grating based x-ray phase contrast imaging is to accurately retrieve local phase gradients of a sample from measured intensity fringe shifts. To fulfill this task in dual phase grating interferometry, one needs to know the exact mathematical relationship between the two. In this work, using intuitive analysis of the sample-generated fringe shifts based on the beat pattern formation mechanism, the authors derived the formulas relating sample phase gradients to fringe phase shifts. These formulas provide also a design optimization tool for dual phase grating interferometry.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In x-ray interferometry based phase contrast imaging, one of the key tasks is to retrieve local phase gradients of a sample. More specifically, the local refraction angle of a sample, $\alpha (x, y)$, is proportional to its local phase gradient [1–5]:
where $\lambda$ is x-ray wavelength and $\Phi _{\mathrm {s}}(x, y)$ denotes local sample phase shift, which is equal to $\Phi _{\mathrm {s}}(x, y)=-\lambda r_{\mathrm {e}}\int \rho _{\mathrm {e}}(x, y, s)ds$. In this integral $\rho _{\mathrm {e}}$ denotes the sample electron density and $r_{\mathrm {e}}$ is the classical electron density which equals to $2.82\times 10^{-15}$m. In Eq. (1), $\partial \Phi _{\mathrm {s}}(x, y)/\partial x$ denotes the phase gradient of sample along the direction perpendicular to grating slits. In x-ray interferometry one measures the fringe shift $\Delta \phi$ generated by sample refraction. In order to retrieve local phase gradients of the sample, one must find out functional relationship between intensity fringe shifts and sample phase gradients. It turns out that the mathematical relation between them depends not only on the geometrical configuration of the interferometer and phase grating periods, but also on the position of the sample. So one important task in x-ray interferometry is to rigorously derive this functional relationship which will enable retrieval of sample phase gradients from measured interference fringe shifts. Combined with tomography, the retrieved sample gradients from angular projections can be used to reconstruct 3D maps of sample electron densities. In Talbot-Lau interferometry, where only a single phase grating is employed as a beam splitter, high-contrast interference fringes can be formed at certain distances downstream (Fig. 1(a)). Sample refraction distorts the intensity fringe pattern, and the mathematical relations between fringe shifts and sample phase gradient are well described in literature [1–5]. The fringe phase shift $\Delta \phi$ generated by sample refraction is given by T. Donath et al.[5]:Recently, dual phase grating x-ray interferometry emerges as an attractive alternative [6–9]. A typical setup of dual phase grating interferometers employs two phase gratings $G_1$ and $G_2$ as the beam splitters, as is shown in Fig. 1(b). The split waves transmitting through the phase gratings create different diffraction orders interfering with each other. The intensity fringe pattern includes a beat pattern [8]. The imaging detector $D$ has a pixel size much larger than the periods of both phase gratings (Fig. 1(b)). The detector just resolves the beat patterns of large periodicities, and renders other fine patterns to a constant background [8]. Hence, different from Talbot-Lau interferometry [1–5,10–20], the dual phase grating interferometers directly detect interference fringes without the need of absorbing analyzer grating. This advantage brings significant radiation dose reduction as compared to Talbot-Lau interferometry.
However, to retrieve local phase gradients of a sample, those formulas in Eq. (2) are not applicable to dual phase grating interferometers, in which refracted x-rays may pass through two phase-gratings rather than a single phase grating. To our knowledge, there is no any other report yet that analyzes the relationship between fringe phase shifts and sample phase gradients in dual-phase grating interferometry. In this work we set out to derive the corresponding formulas relating these two quantities. We hope this work will stimulate more study on this important topic for dual-phase grating interferometry. In section 2, using intuitive analysis of the sample-generated fringe shifts and the intensity beat pattern formation, we derived formulas that relate the sample phase gradients to fringe phase shifts. In section 3 we perform wave propagation simulations to validate the formulas derived in section 2. We also show how the angular sensitivity of a dual phase grating interferometer is determined in dual phase grating interferometry. We conclude the paper in section 4.
2. Methods
We start our derivation from explaining the fringe formation mechanism in dual phase grating interferometry. Figure 1(b) shows the geometrical configuration of a dual phase grating interferometer. It consists of a source $S$, two phase gratings $G_1$ and $G_2$, and an imaging detector $D$. The periods of the first and second phase gratings are $p_{1}$ and $p_{2}$ respectively, $R_s$ is the source-to-$G_1$ distance. Note that $R_g$ is the spacing between the two phase gratings, and $R_d$ denotes the $G_2$-to-detector distance (Fig. 1(b)). For sake of convenience in discussion, we define several magnification factors as follows:
where $M_{g_1}$ represents the geometric magnification factor from $G_1$ to detector plane, and $M_{g_2}$ is the geometric magnification factor from $G_2$ to detector plane. Obviously, in absence of $G_2$-grating, a diffraction order of the intensity pattern generated by $G_1$ phase grating alone would be represented by $\exp \left [i2\pi (l \cdot x)/(M_{g_1}\cdot {p_1})\right ]$, where $l$ is an integer and indexes the diffraction order. Similarly, in absence of $G_1$-grating, a diffraction order of the intensity pattern generated by $G_2$ grating alone would be represented by $\exp \left [i2\pi (m \cdot x)/(M_{g_2} p_{2})\right ]$, where $m$ indexes the diffraction order of the fringe in absence of $G_1$-grating. As is shown from our theory of dual phase grating interferometry [8], x-ray irradiance at detector entrance is a result of cross-modulation between the fringe patterns generated by phase gratings $G_1$ and $G_2$ respectively. Hence the irradiance pattern at the detector entrance is a weighted sum of different diffracted orders. Each of the orders in dual phase grating interferometry is indexed by two integers $(l, m)$ and represented by a product as:Once the fringe formation mechanism is understood, we are now ready to derive the mathematical relationship between fringe phase shift and sample phase gradient. First, we consider a relatively simple case of sample being downstream of the second phase grating $G_2$. Let the local refraction angle of the sample be $\alpha (x, y)$, and the sample-detector distance be $L_{\mathrm {D}}$. Referring to Fig. 2(a), the fringe lateral shift caused by the refraction is equal to $\alpha L_{\mathrm {D}}$. Since the period of the $m$-th diffracted order in the beat pattern is ${p_{\mathrm {fr}}}/m$, using Eqs. (1) and (5) we found the corresponding fringe phase shift $\Delta \phi _{\mathrm {down}}(x, y, m)$ as
Fig. 2. Geometric configurations for fringe shift analysis. (a) Sample positioned downstream of $G_2$. (b) and (c) Sample positioned upstream of $G_1$. In (b) $G_2$ is assumed absent. In (c) $G_1$ is assumed absent.
On the other hand, there are different ways to position the sample. Assume now that the sample is placed upstream of $G_1$ and the source-sample distance is $L_{\mathrm {S}}$. Let us firstly consider how the sample phase gradient would affect $G_1$-associated fringe in absence of $G_2$ grating. Different from the dual grating setups discussed earlier, this is a hypothetical single phase grating setup, so Eq. (2) can be used for the fringe phase shift generated by the sample. Rephrasing $R_1$ and $R_2$ in Eq. (2) with $R_s$ and $R_g+R_d$ in Fig. 2(b) respectively, we find the $G_1$-associated fringe phase shift generated by sample fraction is given by
A similar derivation can be applied to $G_2$-fringe phase shift, assuming absence of $G_1$ grating. Referring to Fig. 2(c) and rephrasing $R_1$ and $R_2$ in Eq. (2) with $R_s+R_g$, and $R_d$ respectively, we find the $G_2$-associated fringe phase shift generated by sample refraction is given by
In dual phase grating interferometry the interference fringe pattern is indexed by two integers $(l, m)$ and represented by a product as $\exp \left [i2\pi (l \cdot x)/(M_{g_1}\cdot p_{1})\right ]\times \exp \left [i2\pi (m \cdot x)/(M_{g_2}\cdot p _{2})\right ]$, thus the refraction-generated fringe phase shift of the order $(l, m)$ is
In dual phase grating x-ray interferometry, one usually adopts a common imaging detector of pixels much larger than grating periods. As is discussed above on fringe formation mechanism, due to the pixel averaging effects, the detector resolves only the beat patterns of diffracted orders $(l=-m, m)$. This being so, Eq. (9) gives the sample-refraction generated fringe phase shift for the $m$-th order beat patterns as follows:Obviously, this proportional constant in Eq. (12) is different to that in Eq. (6), which corresponds to the setups with sample positioned downstream of the $2^{\mathrm {nd}}$ phase grating.
Combining Eqs. (6) and (12) together, we find the measured fringe phase shift $\Delta \phi$ is proportional to sample phase gradient with a proportional constant $\xi$ given by
In addition, for theoretical completeness, we present the corresponding formula for the case of sample placed between the two phase gratings in appendix. It is inconvenient to place sample in this way, since the spacing between the two phase gratings is typically small.
To validate the fringe shift formula of Eq. (13), we conduct numerical simulations. The scheme is as follows. Assuming a point x-ray source, the projected sample attenuation $A^{2}(x,y)$ and phase map $\Phi _{\mathrm {s}}(x,y)$ are simulated with ray-tracing technique. Once the phase map is simulated, the phase gradient $\partial \Phi _{\mathrm {s}}(x,y)/\partial x$ can be computed through Fourier transform. We simulate the intensity fringe image as results of Fresnel wave diffraction propagating from the source point, through the two phase gratings, and the sample, finally to the imaging detector [21]. We then employ the Fourier fringe analysis method pioneered in [22] for retrieval of the fringe phase shift $\Delta \phi (x, y)$ and attenuation map $A^{2}(x, y)$. The value of the retrieved fringe phase shift is then compared to the theoretical value computed from Eq. (13).
3. Results
To validate the fringe shift formula of Eq. (13), we compare the calculated fringe phase shift of formula Eq. (13) to that determined through numerical simulations. We employed four different interferometer setups in the simulation study. In the first simulation, a point x-ray source of $20$keV design energy and dual-$\pi$ phase gratings of period $p_{1}=p_{2} = 1\mu$m are employed. The geometric setup is $R_s=R_d=450$mm and $R_g=5$mm. With this geometric setup, the fringe can attain a high visibility pattern and large fringe period ${p_{\mathrm {fr}}}=p_0=181\mu$m as well [8]. So with a detector of ${p_{\!_{\mathrm {D}}}}=36.2\mu$m pitch, one period of the resolved fringe will take up five pixels. To validate the fringe shift formula of Eq. (13), a sample is placed half way between the source and $G_1$-plane, i.e. $L_{\mathrm {S}}=R_s/2=225$mm. The sample is assumed a sphere of diameter 4 mm filled with $100\%$ adipose tissue. We employed ray-tracing to get the projected sample attenuation $A^{2}(x, y)$, and the phase map of the sample. From sample phase map we computed sample phase gradients. Using Eq. (13) and the sample phase gradients we theoretically predict a map of the fringe shift $\Delta \phi (x, y)$. The theoretical map of sample attenuation $A^{2}(x, y)$ and map of fringe phase shift $\Delta \phi (x, y)$ are shown in Figs. 3(a) and 3(d) respectively. We then employed Fresnel propagation down stream from sample plane through the gratings $G_1$, $G_2$ and finally to the detector plane to get the projected image. We employed the Fourier fringe analysis method derived in [22] for retrieval of the fringe phase shift $\Delta \phi (x, y)$ and attenuation map $A^{2}(x, y)$. The maps of retrieved attenuation and phase shift are shown in Figs. 3(b) and 3(e). As comparison, profiles of the attenuation $A^{2}(x)$ and phase shift $\Delta \phi (x)$ along the central line across grating are shown in Figs. 3(c) and 3(f) respectively. The dashed blue curves represent the theoretical value, while the solid red curves are the retrieved values from the intensity fringes. The good match between the theoretical and retrieved values of $\Delta \phi (x)$ validates the fringe phase shift formula of Eq. (13) for sample upstream $G_1$.
Fig. 3. Simulation results for validation of Eq. (13). In the simulation, the setup consists of two $\pi$-gratings and a 20keV point source. A sphere shaped sample of adipose tissue is placed upstream of the $1^{\mathrm {st}}$ grating and downstream of the $2^{\mathrm {nd}}$ grating respectively. The theoretical maps of attenuation $A^{2}(x, y)$ and fringe phase shift $\Delta \phi (x, y)$, when sample is placed half way between the source and the $1^{\mathrm {st}}$ grating, are shown in Figs. 3(a) and 3(d); while the retrieved $A^{2}(x, y)$ and $\Delta \phi (x, y)$ are shown in Figs. 3(b) and 3(e) respectively. For the purpose of comparison, the profiles of the attenuation and fringe phase shift along the central line across grating are shown in Figs. 3(c) and 3(f). The blue curves correspond to the theoretical values, while the red curves correspond to the values retrieved from intensity fringes. Figure 3(g) is the profiles of the fringe phase shift $\Delta \phi (x)$ along the central line across grating, when sample is placed to the plane half way between $G_2$ grating and detector. The good agreement between the theoretical values and the fringe-pattern retrieved values provide a validation of Eq. (13). For details, see text.
In the second simulation, we keep the same setup as in the previous simulation but placed sample to the half way between the $G_2$-grating and detector plane, i.e, $L_{\mathrm {D}}=R_d/2=225$mm. The diameter of the sample sphere is also increased to $12$mm to increase the resolution. The profiles of the phase fringe shifts $\Delta \phi (x)$ along the central line across grating are shown in Fig. 3(g), in which the dashed blue line represents the theoretical value and the solid red line is the retrieved one. The closeness of the two profiles validates the fringe phase shift formula of Eq. (13) for sample downstream $G_2$. Comparing Figs. 3(f) and 3(g) in previous page one finds the profiles of $\Delta \phi (x)$ are reflected, in other words, the values of $\Delta \phi (x)$ change signs when the sample moves from upstream to downstream, as is predicted by the sign difference in Eq. (13) for the two scenarios. Noting this interesting feature is important in practice for accurately retrieving sample phase gradients from measured fringe phase shifts.
In the third and fourth simulations, we replace the second phase grating $G_2$ with a $p_{2}=1.1\mu$m period $\pi$-phase grating and keep the first grating $G_1$ the same. Corresponding to this change, the geometric setup is changed accordingly for attaining good fringe visibility [8]. So in the third and fourth simulations we set the interferometer with $R_s=450$mm, $R_d=380$mm, and $R_g=40$mm. In this configuration, the fringe period, as is determined by Eq. (5), is ${p_{\mathrm {fr}}}=-191.4\mu$m and $p_0=11.67\mu$m [8]. In addition, the detector pixel size is changed to ${p_{\!_{\mathrm {D}}}}=31.9\mu$m, thereby one fringe period can take up 6 pixels. We then simulate the fringe formation processes by placing the sample upstream to plane $L_{\mathrm {S}}=R_s/2=225$mm in the third simulation, and downstream to plane $L_{\mathrm {D}}=R_d/2=190$mm in the fourth simulation. The results are shown in Fig. 4. The profiles of $\Delta \phi (x)$ along the central line for sample upstream of $G_1$ are shown in Fig. 4(a) and those for sample downstream of $G_2$ are shown in Fig. 4(b), in which the dashed blue lines are the theoretical values and the solid red lines correspond to the retrieved ones. The close match between the red and blue lines again validated Eq. (13).
Fig. 4. In this simulation, the second grating is replaced with a $p_{2}=1.1\mu$m period $\pi$ grating but the first grating is kept the same as that in Fig. 3. The geometric setup is also changed to $R_s=450$mm, $R_g=40$mm, and $R_d=380$mm and detector pixel pitch ${p_{\!_{\mathrm {D}}}}=39.1\mu$m. By placing the sample to plane $L_{\mathrm {S}}=R_s/2=225$mm and to plane $L_{\mathrm {D}}=R_d/2=190$mm, one gets the retrieved plots of $\Delta \phi (x)$ for sample upstream of $G_1$ (Fig. 4(a)) and for sample downstream of $G_2$ (Fig. 4(b)).
Carefully examining Figs. 4(a) and 4(b), one may notice that the values of $\Delta \phi (x)$ do not change signs when the sample moves from upstream to downstream, as their features differ from that exhibited in Figs. 3(e)–3(f). This difference is exactly predicted by Eq. (13). In fact, because of the different interferometer configuration, ${p_{\mathrm {fr}}}$ and $p_0$ in these simulations are changed to ${p_{\mathrm {fr}}}=-191.4\mu \mathrm {m}<0$ and $p_0=11.67\mu \mathrm {m}>0$. Equation (13) dictates that $\Delta \phi (x)$ will not change sign when sample is moved from upstream of $G_1$ to downstream of $G_2$. This example demonstrates again that Eq. (13) gives accurate mathematical relation between sample phase gradient and fringe phase shift, accurate not only for predicting their magnitudes but also for their signs. Hence Eq. (13) provides a useful tool in quantitative phase contrast imaging based on dual phase grating interferometry.
In literature the ratio $\left \vert {\xi }\right \vert /\lambda =\left \vert {\Delta \phi /(2\pi \alpha )}\right \vert$ is called the angular sensitivity. It is a measure of fringe phase shift generated per unit sample-refraction angle. Apparently, the larger the $\left \vert {\xi }\right \vert$ is, the larger the magnitude of the fringe phase shift is. Equation (13) can be used to compute the angular sensitivity of a given setup for design optimization of an interferometer. For example, Tab. 1 lists the angular sensitivity values computed by using Eq. (13) for the setups investigated in the four simulations. The notations used for describing the setups are explained in the text.
Table 1. Angular sensitivities of the setups in simulations 3 and 4.
For the first two setups, the angular sensitivities are equal since the two setups are symmetric: $L_s=L_d$, $R_s=R_d$, and the two phase grating periods are identical. While for simulations 3 and 4, the angular sensitivity of the setup with the sample placed upstream of the $1^{\mathrm {st}}$ phase grating is about nineteen times that of the setup with sample positioned downstream of the $2^{\mathrm {nd}}$ phase grating. With a given geometric setup, the closer the sample is placed to the first grating for sample upstream or the closer the sample is to the second grating for sample downstream, the larger the angular sensitivity is. For given sample, the setup with a higher angular sensitivity will generate larger magnitude of fringe phase shifts for measurement.
4. Discussion and conclusions
One important task in x-ray interferometry is to rigorously derive the functional relationship between measured interference fringe shifts and sample phase gradients. Combined with tomography, the retrieved sample gradients from different angular projections can be used to reconstruct quantitative volumetric images of sample electron densities. The mathematical relationship between fringe shifts and sample phase gradients depends not only on the grating periods and geometric configuration of the interferometer, but also on the sample position within the interferometer. The formulas for retrieving sample phase gradient in Talbot-Lau interferometry are well known as summarized in Eq. (2). However, there is a need to provide corresponding formulas applicable to dual phase grating interferometry with various sample position. In this work we fulfill this need using a novel intuitive analysis based on the fringe pattern formation mechanism. The derived formulas of Eq. (13) provides a useful tool for retrieving sample phase gradients in dual phase grating interferometry with any sample position. These formulas have been validated by extensive simulations, not only in the magnitudes but also in the signs of the retrieved sample phase gradients, as is shown in section 3. In addition Tab. 1 shows that sample position has a significant effect on angular sensitivity of a setup.
Moreover, the formulas of Eq. (13) can also be used to compute the angular sensitivity $\left \vert {\xi }\right \vert /\lambda =\left \vert {\Delta \phi /(2\pi \alpha )}\right \vert$, which represents the fringe phase shift generated per unit sample-refraction angle in a given setup. As is discussed in section 3, although it is generally desirable to have a large $\left \vert {\xi }\right \vert /\lambda$ for a setup, but it is only a nominal sensitivity measure. The true sensitivity of a interferometer setup is the minimum detectable refraction angle, which depends on the fringe visibility and photon quantum noise with the setup. Detail discussion of this topic is out the scope of this paper.
In Eq. (13), the proportion constant $\left \vert {\xi }\right \vert$ between fringe phase shift and sample phase gradient is also called the auto-correlation length of an interferometer setup. This quantity $\left \vert {\xi }\right \vert$ characterizes the sensitivity of the setup for detecting small angle x-ray scattering from the fine structures of the sample. Eq. (13) provides a quantitative tool for adjustment $\xi$, namely length-scale specific dark-field sensitivity. It should be noted that in the process of adjusting, a change of inter-grating spacing $R_g$ can significantly affect fringe period and fringe visibility. For details about dark field imaging readers are referred to the literature [23–25]. In addition, we found that the auto-correlation length $\left \vert {\xi }\right \vert$ is inversely proportional to fringe period ${p_{\mathrm {fr}}}$ of Eq. (5), if the sample is positioned downstream of $G_2$ grating. On the other hand, when the sample is positioned upstream of $G_1$, the auto-correlation length $\left \vert {\xi }\right \vert$ is inversely proportional to a length-quantity $p_0$ of Eq. (11). The size of $p_0$ has a physical meaning in a different context. Namely, if a source grating were incorporated, then the periodicity of the source grating should be set to $p_0$ for achieving good fringe visibility [21].
A downside of the dual phase grating setup is that the large fringe period sharply limits sensitivity if the sample is placed downstream of the gratings. However, Eq. (13) dictates that upstream sample placement can give much higher angular sensitivity than downstream sample placement, since one can have large fringe period ${p_{\mathrm {fr}}}$ but small $p_0$ in dual phase grating interferometry setups. On the other hand, in common Talbot-Lau interferometry, one has small fringe period ${p_{\mathrm {fr}}}$ but large $p_0$. Hence, Eq. (2) dictates that upstream sample placement does not give advantage as compared to downstream sample placement. In addition to optimal sample placement in Talbot-Lau interferometry, one strategy for enhancing sensitivity is to use very fine phase gratings of sub-micrometer periods [6]. Implementing this strategy in Talbot-Lau interferometry is difficult, since it means that one must use an absorbing grating of sub-micrometer period, which not only block x-ray and increases radiation dose, but is also hard to fabricate. The dual phase grating interferometry can implement such a strategy. As a disadvantage of dual phase grating interferometry, note that its fringe visibility is generally lower than that of Talbot-Lau interferometry, as is shown in [8]. Detailed comparison of dual phase grating and Talbot-Lau interferometers is out of the scope of this work.
In conclusion, in this work, using intuitive analysis of the sample-generated fringe shifts based on the beat pattern formation mechanism, the authors derived the formulas relating sample phase gradients to fringe phase shifts. These formulas provide also a design optimization tool for dual phase grating interferometry.
Appendix
Usually one does not place the sample between $G_1$ and $G_2$ grating, since the inter-grating distance is typically very small. However, for theoretical completeness we briefly discuss this case in this appendix. Assume that the sample is placed between $G_1$ and $G_2$ and the $G_1$-sample distance is $L_{g_1}$. Using the same derivation method as that for the derivations of Eqs. (7)–(8), the $G_1$-associated fringe phase shift generated by sample refraction is now changed to:
Funding
National Institutes of Health (1R01CA193378).
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