1. Introduction

Medical x-ray phase contrast imaging is based on refraction in matter and visualizes the traversed electron density. Phase contrast images can emphasize variations in soft tissue. Quite a few phase contrast techniques are available [1]; this article focusses on differential techniques, which allow regular medical detector resolutions and usually are compatible with x-ray tubes. Differential techniques measure the differential quotient $\Delta \Phi /\Delta x$ of the wavefront phase Φ along a lateral shear Δx, corresponding to a refraction angle of

This work aims at low-dose phase contrast imaging by improving the measurement range α_R of the refraction angle α. Basically, $1/{\alpha }_{\text{R}}$ is the angular sensitivity of an x-ray setup [2], hence α_R is strongly related to x-ray dose, and a reduction of α_R is an issue [3,4]. Formally, a small angular change Δα in the incident x-rays results in an intensity change at the detector which is proportional to $\Delta \alpha /{\alpha }_{\text{R}}.$ As an example, the mammography phantom of the American college of Radiology caused refraction angles of $\left|\alpha \right|\le 0.2\text{µrad}$ at 18 keV [5]. When imaging the phantom using a setup of α_R = 50 µrad, intensity changes will be tiny.

Another factor which influences the dose (mainly in grating-based contexts) is the contrast at the detector, which is called visibility: When α gets varied across the whole range $[-{\alpha }_{\text{R}}/2,{\alpha }_{\text{R}}/2],$ the intensity of a background detector pixel varies within a range [I_Lo, I_Hi]. The ratio of the variation-amplitude $(I_{\text{Hi}}-I_{\text{Lo}})/2$ and average $(I_{\text{Hi}}+I_{\text{Lo}})/2$ is called visibility V. Combined with α_R, the intensity signal ΔI(Δα) is proportional to $V(\Delta \alpha /{\alpha }_{\text{R}}).$

The required dose depends on the signal ΔI(Δα): Given a photon count D (“dose”) leaving the patient, the quantum noise is $\sqrt{D}$ (“Poisson distribution”). A transmission T of post-patient optics absorbs D (1−T) photons, hence just DT photons reach the detector, which also reduces the quantum noise to $\sqrt{DT}.$ The resulting signal to noise ratio (SNR) is proportional as follows:

1.1 Gap in available measurement ranges

A compact setup of a small measurement range would be both practical and dose efficient: A compact post-patient distance of less than 0.2 m is used in most commercial medical x-ray machines: It ensures sufficient patient-space in mammography [9], allows a low blur even for large focal spots, and it allows setups to fit into a computer tomography scanner. As for measurement range, a human finger in air caused refraction by less than ± 2 µrad at 30 keV [10]; phase unwrapping seems possible for a measurement range α_R of 2 µrad. Hence, an intermediate range of α_R = 2 µrad would be perfect for a small dose in case of the human finger. In Fig. 1, the area representing such favorable compact setups is marked by a circle.

 

Fig. 1 Measurement ranges α_R of several differential phase contrast imaging techniques. Smaller ranges save dose D since $D~{\alpha }_{\text{R}}{}^2.$ The aim of this paper is a combination of a low dose and a short post-patient distance d, see the encircled area. Gray numbers indicate the year of the corresponding literature, see the text below.

Download Full Size | PDF

Figure 1 also shows data of setups given in literature (see below). Among small measurement ranges, examples for low and for high post-patient distances d were selected. The given measurement ranges are optimistic ones, assuming a thin sample in the most sensitive position. The techniques are as follows:

The aim of this paper is to propose compact sensitive setups within the circle in Fig. 1.

1.2 Proposal: Combine large lateral shear with Talbot-Lau visualization

To reach a high sensitivity, this proposal increases the lateral shear using refraction in steep prisms. Ascending and descending prisms are arranged periodically like bars and slits in a regular line grating. The prism-period is similar to the grating-periods of Talbot-Lau setups. To be steep, the prisms have to be quite high, for example 50 to 100 µm. Such “prism-gratings” of ascending prisms alternating with descending prisms can be produced [22].

Fringe formation is a two-step setup [23]: Prisms first combine light along a large lateral shear, then a following Talbot-interferometer visualizes that shear. Interestingly, this scheme also works in simulation using polychromatic light. The reason is that a periodic arrangement of prisms causes a structured light output due to the Talbot effect. Hence, the prism steepness defines the refraction angle, but the Talbot effect defines the periodic structure in the output. As in Talbot-Lau setups, a source grating can allow x-ray tubes of large focal spots.

1.3 Structure of paper

1.4 Simulation method

The simulation code which performed the simulations in this work used the Fresnel-approximation of Kirchhoff-diffraction. The simulation iteratively propagated wavefronts u_G from one grating plane G to the next grating plane H. The wavefronts were one-dimensional, i.e., amplitude and phase were being simulated in the x/z-plane, where x was the phase sensitive dimension and z the optical axis. At each grating plane G, amplitude and phase were modified depending on the profile h_G(x) of the grating. The amplitude u_H(η) in front of grating H at the x-position η was

2. Monochromatic setups

This section introduces the working principle for monochromatic sources.

2.1 Talbot-Lau interferometer

Fringe formation is based on a $\pi /2$-shifting Talbot-Lau interferometer [16], see Fig. 2(a): Bars of the phase grating G1 shift the phase of light by $\pi /2.$ This creates a periodic intensity pattern (see item ①) in the G2-plane (Talbot effect). Refraction within the patient ② distorts the pattern. The distorted pattern is then sampled by the analyzer grating G2 of highly absorbing bars. Another absorption grating (the source grating G0) has periodic slits, such that light passing any slit in G0 (see item ③) creates the same pattern at G2.

 

Fig. 2 A Talbot-Lau interferometer (a) compared to the proposed setup “fringe-formation” (b). Fringe formation compares phases of wavefront sections which originally are separated by a lateral shear L. This is done be rearranging wavefront sections of height s_A ( = one strip): Refraction in GA gets compensated in GB, resulting in an rearranged wavefront. A following Talbot-Lau section translates phase differences into detectable intensity contrast. Gray numbers at the bottom indicate corresponding sections in this paper.

Download Full Size | PDF

Quantitatively, the smallest post-patient distance d₁ is the first fractional Talbot-distance $D_1=p_1{}^2/2\lambda ,$ scaled by the lens equation $1/D_1=1/d_0+1/d_1$ [2]. To get p₂, the period p₁ of G1 is scaled ① by magnification $M_1=(d_0+d_1)/d_0,$ hence $p_2=M_1{p}_1.$ The measurement range ③ is given by ${\alpha }_{\text{R}} = p_2/d_1 = p_0/d_0$ (since α = α_R shifts the pattern at G2 by one period p₂).

2.2 Proposed setup “fringe formation”

Figure 2(b) illustrates the proposed setup, which replaces the phase grating G1 by two prism gratings GA and GB. Light is refracted away from the center of each prism-pair (see the arrows ①), because the index of refraction $n=1-\delta$ is less than 1 for x-rays. In the following, a refraction towards the positive x-axis is called “upwards” and towards the negative x-axis is called “downwards”.

A period p consists of two prism-strips s, i.e. p_A = 2s_A and p_B = 2s_B.

The goal of the setup is to compare phases of wavefront-sections which originally are separated by a lateral shear L ②. The plan is to arrange such wavefront sections side-by-side by refraction in GA and propagation to GB. GB then is to make the phases comparable by compensating the refraction in GA, this way parallelizing adjacent wavefront-sections. Propagation within the Talbot-Lau-section then is to create an input-dependent contrast at G2.

To minimize the unintended sensitivity in the Talbot-part, the first fractional Talbot order is used for d_B (i.e., $m_{\text{B}}=1$ results in a short $D_{\text{B}}=m_{\text{B}}{p}_{\text{B}}{}^2/2\lambda ).$ Between GA and GB, the m_A-th fractional Talbot distance $D_{\text{A}} = m_{\text{A}}{p}_{\text{A}}{}^2/2\lambda$ can be used with $1/D_{\text{A}}=1/d_0+1/d_{\text{A}}$ to get d_A.

In this section, m_A has to be even (“monochromatic setup”); section 4 is about odd values of m_A. Details about the topics covered by each section are shown in gray at the bottom of Fig. 2(b). Basically, this section 2 is exact regarding the second part of the setup from GB to G2. Section 3 and appendix A are exact regarding the first part from G0 to GB.

2.3 Basic idea

Figure 3(a) illustrates the basic plan of creating sensitivity: Prisms in GA shift half of the input wavefront upward and the other half downward (see item ①). At GB, both halves merge, resulting in a wavefront with an added rectangular phase shift. This phase shift ΔΦ depends on the angle α of the incoming wavefront.

 

Fig. 3 (a) The basic idea is that x-shifts rearrange the wavefront ①, creating a rectangular phase of shift ΔΦ(α), which translates into contrast (②, Talbot effect). (b) Simplified setup to explain why wavefront sections reach matching prisms in GB. (c) Quantitative situation for an x-shift by S = 2 strips. Parallel-beam geometry is used for simplicity.

Download Full Size | PDF

The second section ② behind GB creates a maximum contrast for phase shifts of $\Delta \Phi =\pm \pi /2$ (i.e., like in a $\pi /2$-shifting Talbot-Lau interferometer just behind G1). The sign of ΔΦ decides whether the slits in G2 or the bars of G2 receive more intensity. No intensity contrast occurs for ΔΦ = 0 due to symmetry (here, α is also zero, hence input wavefront, GA and GB are axisymmetric with respect to a properly chosen optical axis).

By now, the plan is defined. The goal of the remainder of this section is to make plausible that the plan actually works. For simplicity, Fig. 3 uses plane-wave geometry (i.e., p_A = p_B).

Figure 3(b) illustrates why the Talbot effect causes the upward-shifted and downward-shifted parts to merge at GB. The shown situation is simplified as GA consists of downward-refracting prisms only; the other strips block all light (see item ①). Because of the periodicity of the wavefront behind GA, the Talbot effect occurs at the design energy for even fractional Talbot orders m_A: The wavefront behind GA exactly reoccurs in front of GB (see item ②), matching the GB-prisms. Hence, GB can compensate the refraction done in GA.

Due to the Talbot effect near the design energy, the periodic wavefront reoccurs at the same x-position in front of GB independently of the prism refraction angle [25]. However, the exact prism refraction angle is still relevant for intensity distributions (item ③).

Refraction in the sample ④ is not a p_A-periodic wavefront modification and thus shifts the pattern at GB along the (vertical) x-axis. This is the measurement principle of Talbot-interferometers, but here it causes a misalignment between the wavefront and the GB-prisms. Hence, refraction angles α have to stay small, so that the rectangular phase shift ΔΦ is the dominating effect.

2.4 Quantitative aspects

Figure 3(c) illustrates how a wavefront-angle α translates into a phase shift ΔΦ: Due to a x-shift of S strips, the total lateral shear is $L=S{p}_{\text{A}}$ ①. Across that shear L, a wavefront at angle α has a phase-difference of

The measurement range α_R is the offset to wavefront-angle α which causes an equivalent phase shift of $\Delta \Phi (\alpha +{\alpha }_{\text{R}})=2\pi +\Delta \Phi (\alpha ),$ resulting in identical intensities in front of G2. Combining this condition with Eq. (5), the measurement range is:

2.5 Phase stepping and measurement range

To measure the refraction within the sample α, phase stepping can be used in case of Talbot-Lau interferometers [15]: An x-ray source grating G0 moved by some Δx translates into a modified $\alpha (\Delta x) \approx \alpha + \Delta x/d_0,$ hence the intensity pattern in front of G2 is moved to position $(\alpha (\Delta x)/{\alpha }_{\text{R}}\text{})p_2.$ The resulting detected intensity $I_1(\Delta x)$ usually is quite sinusoidal.

To check if phase stepping is also possible in case of fringe formation, a simulation without sample (i.e., $\alpha (\Delta x) \approx \Delta x/d_0)$ was used. To focus on deviations from a sine, a point source was simulated along with a perfectly absorbing G2 for example setup G from section 6.1. The initial assumption was that extreme intensities and contrasts should occur for $\Delta \Phi =\pm \pi /2,$ which corresponds to $\alpha /{\alpha }_{\text{R}}=\pm 25\%$ due to a combination of Eqs. (5) and (6):

2.5.1 Simulation results

Figure 4(a) shows the simulated intensity I₁ (i.e., the intensity passing the slits of G2) and Fig. 4(b) shows the corresponding simulated contrast $C=(I_1-I_2)/(I_1+I_2)$ (where I₂ was the intensity in front of the G2-bars). All individual graphs look sinusoidal.

 

Fig. 4 Simulated consequences at the detector of varying the wavefront angle α within measurement range α_R in monochromatic example setup G (point source; design energy 30 keV): (a) shows the average intensity I₁ within the slits of G2, (b) the contrast $C=(I_1-I_2)/(I_1+I_2)$ (where I₂ is the average intensity in front of the bars of G2). All graphs look sine-like, the contrast graphs are exactly point-symmetric to the origin. Based on the average $\overline{I}$ and on the amplitude V, a synthetic sine was defined for each graph and the resulting small differences “sine − graph” are shown as dotted lines using the scale on the right hand side of each diagram. For $\left|\alpha /{\alpha }_{\text{R}}\right|<50\%$ the graphs closely match the sines.

Download Full Size | PDF

Sine-functions were fitted to each graph based on the values at $\alpha /{\alpha }_{\text{R}}=\pm 25\%;$ the dashed graphs show the (magnified) differences between the fitted sines and the simulation result. All graphs and all difference-graphs look smooth. For the contrasts, the largest deviations from a sine are about 2% of the visibility for $\left|\alpha /{\alpha }_{\text{R}}\right|<25\%$ (the largest differences occurred for the design energy). For larger angles, the differences grew due to a misalignment between phase pattern and GB (and due to slight sine-frequency differences). The fitted sine-graphs indicate these relationships, where V is the visibility and $\overline{I}$ is the mean intensity:

2.6 Simulated height profiles

To demonstrate the basic idea, two height profiles were simulated: A phase jump by $\pi /2$ at x = 0 [see Fig. 5(a)] and a steep wedge in the right half of Fig. 5(b), both intended to result in $\Delta \Phi =\pi /2$ for a maximum contrast C. The setup was a plane-wave variant (i.e., $p/2=s_A=s_B=s_2)$ of the monochromatic example G from section 6.1 (i.e., prisms shifted by 8π at 30 keV). The scales at the top are a model for ΔΦ: The setup basically compares the phase across the lateral shear $L=S{p}_{\text{A}}$ for S = 16. Hence, in both cases, the changes at x = 0 influence a region of width L (see the pair of pink vertical lines).

 

Fig. 5 Monochromatic simulation results for (a) a phase step and (b) a wedge for an x-shift in GA by S = 16 strips. The scales at the top illustrate that the phases separated by L = 16 periods were being compared. ① shows that the phase behind GB had the added rectangular phase shift ΔΦ, ② shows the strip-averaged intensity I and ③ the contrast C in front of G2.

Download Full Size | PDF

2.6.1 Simulation results

The detail graphs first show ① the phase right behind GB (vertical lines separate strips of width s₂). This graph in particular shows the added rectangular phase shift ΔΦ.

The second detail graph ② is the “single-strip-intensity” in front of G2, i.e., the raw intensity averaged across a strip-width s₂. The last graph ③ is the contrast C in graph ②, measured exactly between adjacent strip-centers. The contrast graph ③ of Fig. 5(b) looks sinusoidal, as expected from the relation $C=V\sin \Delta \Phi$ of Eq. (8).

3. Setup geometry

3.1 Section overview

An important aspect in this section is the hierarchical source grating G0: Compared to a Talbot-Lau setup, the source grating has fewer and smaller slits, i.e., a smaller duty cycle. Effectively, this reduces the available x-ray intensity by some integer factor. Exposure times will change by a smaller factor, depending on the saved dose.

However, the section first calculates the height of the prisms in GA and GB depending on the x-shift of S strips:

3.2 Prism heights

Substituting the magnification $M_{\text{A}}=(d_0+d_{\text{A}}\text{})/d_0$ in the lens equation $1/D_{\text{A}}=1/d_0+1/d_{\text{A}}$ results in $M_{\text{A}}{D}_{\text{A}}=d_{\text{A}}.$ A combination with the scaling $M_{\text{A}}{p}_{\text{A}}=p_{\text{B}}$ yields $p_{\text{A}}/D_{\text{A}}=p_{\text{B}}/d_{\text{A}}.$ Regrouping the expression $D_{\text{A}}=m_{\text{A}}{p}_{\text{A}}{}^2/2\lambda$ and then inserting the previous expression yields:

GB has to compensate the phase-shift in GA, hence it also has to shift the wavefront by Nπ (this was used and confirmed by all simulation setups in this paper).

3.3 Measurement range in terms of periods and distances

Most techniques described below Fig. 1 had measurement ranges of the form ${\alpha }_{\text{R}}=p/d$ for some grating period p and the post-patient distance d. Substituting Eq. (9) in Eq. (6) yields

3.4 Source grating

The source grating G0 depends on three periods called P₀, Q₀ and p₀. Geometrically, each slit in G0 must project GA-periods onto GB-periods and GB-periods in turn onto G2-periods.

To project GB onto G2, P₀-periodic slits in G0 must satisfy an angular condition equivalent to ③ in Fig. 2(a):

The above conditions are about geometry; a final condition is due to sensitivity: The smallest period p₀ depends on the measurement range in the same way as for Talbot-Lau setups, i.e.,

3.5 Example geometry

Figure 6 shows a specific example including the source grating periods. A short d₀-distance was used with a shift of just S = 3 strips and the first fractional Talbot order (m_A = 1). Using D_A = D_B = D as horizontal scale, d₀ = 6D results in $d_{\text{A}}+d_{\text{B}}=3D.$ In each P₀-group, two $p_0/2$-slits are open; the total width of $3p_0/2=P_0/4$ is exactly the proposed limit width $Q_0/4.$

 

Fig. 6 Hierarchical source grating G0 for an x-shift of $S=N{m}_{\text{A}}=3\cdot 1$ prisms; the geometry is the same as in Fig. 2(a). Source grating period P₀ and width Q₀/4 depend on distances and periods, the finest period p₀ additionally depends on the prism height. Groups of p₀-periodic slits (up to a width of $Q_0/4)$ can be opened within each P₀-period in monochromatic setups.

Download Full Size | PDF

4. Polychromatic setups

4.1 Simulation conditions

This section contains simulation results of example setups. Example setups will be identified by a letter (F, G, H or J) and the corresponding design energy, for example F (poly 20 keV) for the polychromatic setup F. Details of the example setups are given in section 6.1.

All following simulations use the same spectrum called $"\Delta E/E=50\%"$ [see Fig. 7(c)]: For a design-energy E_D, this spectrum has a sine-shaped intensity distribution with a full width half maximum (FWHM) of 50% E_D, i.e., $I(E)~1+\cos (2\pi (E/E_{\text{D}}-1))$ for $50\%\le E/E_{\text{D}}\le 150\%.$ There are two reasons for this choice: First, it allows to compare three polychromatic example setups although they have three different design energies. Second, this kind of spectrum is also used for the other example setups in section 6.1.

 

Fig. 7 Simplified intensity transfer (a) at design energy and (b) at a lower photon energy for monochromatic setups (“even m_A”). Parallelized wavefronts contribute to visibility. (c) Point-source visibilities (simulated) for the setups G (monochromatic) and H (polychromatic).

Download Full Size | PDF

All simulations which show the contrast C depending on the photon energy E used the source position $\Delta x=p_0/4$ of maximum contrast at the design energy. Because of Eq. (7) and ${\alpha }_{\text{R}}=p_0/d_0,$ that position can also be specified by $\Delta x/p_0=\alpha /{\alpha }_{\text{R}}=\Delta \Phi /2\pi =25\%,$ see Fig. 4.

4.2 Polychromatic problem: Gaps at GB

Figure 3(b) illustrated that the wavefront-sections formed by GA will find their matching GB-prism due to the Talbot effect for an even fractional Talbot order m_A. This is illustrated again in Fig. 7(a) for the design energy. Figure 7(b) illustrates the problems occurring for other photon energies: A first problem is, that (on average) only half of the intensity reaches the correct prisms. A second problem is, that the energy reaches the matching prisms off-center: Gaps remain between the “correct intensity beamlets”, which further reduce the visibility.

To compare the problems and the proposed solution, setups G (mono 30 keV) and H (poly 30 keV) were simulated using a point source and a perfectly absorbing G2 at the position of maximum contrast. A variant of setup G called G' had a GB which was x-shifted by one strip.

4.2.1 Simulation results

In Fig. 7(c), the two simulated contrast graphs G and G' show wide valleys between the contrast peaks. Both visibilities (“spectrum-weighted contrast averages”) were about 21% (see the arrows). On the other hand, polychromatic setup H had a good contrast between 25 keV and 33 keV; the polychromatic visibility was 30%, i.e., larger than for graphs G and G'.

4.3 Polychromatic solution: m_A = 1st fractional Talbot order

The key difference between setups G and H was that setup H had an odd fractional Talbot order of m_A = 1. The odd fractional Talbot order means that the wavefront sections refracted in GA did not merge well at GB; this was different from the case of even m_A in Fig. 3(b). However, in an odd fractional Talbot order behind an amplitude grating as in Fig. 3(b), the phase (instead of the amplitude) is structured due to the Talbot effect [25].

To check the polychromatic influence on the phase, a variation of Fig. 3(b) was simulated based on setup H (poly 30 keV) using a point source: GA consisted of blocked upward-prisms and GB was a single wide non-absorbing prism, which parallelized the whole wavefront.

4.3.1 Simulation results

Figure 8(a) shows that exactly at the design energy of 30 keV, the simulated phase behind the single large GB-prism was a $\pi /2$-shifting rectangle wave, as predicted by the Talbot effect. Figure 8(a) also shows that for a narrow spectrum, the phase behind each second GB-strip was quite flat (i.e., within a $\pi /2$-band, see the bright strips), despite energy-dependent refraction in GA and GB. Figure 8(b) shows the same situation for a wider spectrum: The pattern of Fig. 8(a) occurred several times, but the flat regions remained in place.

 

Fig. 8 Setup H (poly 30 keV; m_A = 1; point source) had a flat phase behind GB for (a) a narrow and (b) a wide spectrum. GB consisted of a single large prism and in GA, all non-matching prisms blocked all light.

Download Full Size | PDF

Hence, a proper alignment of GA relative to GB means that contributions from GA to a matching GB-prism have a quite flat phase behind GB. That is, the phase of the input to the Talbot-Lau section is quite good, which explains the improved visibility of setup H. Contributions of GA to non-matching GB-prisms are refracted further by GB, resulting in an unstructured reduction of visibility, i.e., the first problem stays unsolved.

4.4 Simulated energy-dependent visibility

To check the visibility and its dependence on the photon energy, a point-source was simulated using a perfectly absorbing G2. The source position of maximum contrast $\Delta x=p_0/4$ was used. The polychromatic setups F (20 keV), G (30 keV) and H (40keV) were simulated.

4.4.1 Simulation results

Figure 9(a) shows the result; photon energies are given relative to the design energy (“100%”). For all three setups, the contrast was good in the central energy-interval labeled “design”. In the regions labeled “low” and “high”, the contrast varied, but was still usable. In the outside regions “bot” and “top”, the contrast usually was small and of varying sign. The contrasts in the lower energy ranges varied with a high frequency.

 

Fig. 9 (a) Contrasts of different polychromatic setups are quite similar near the design energy; the overall visibility indicated by the arrows is about 30% in each case. (b) The visibility depended on the slit-count in G0: Increasing the number of p₀-periodic slits within a P₀-group increased the x-ray flux but reduced the resulting visibilities shown next to the arrows.

Download Full Size | PDF

All spectrum-weighted visibilities were about 30% in all three cases, as indicated by the arrows on the vertical axis. That is, contrast differences did not affect the overall visibility.

4.5 Simulated source size

To check the influence of the source size on the contrast, the source size was varied in the previous simulation of setup H (poly 30 keV). This still was a synthetic simulation, because it used a perfectly absorbing G2 and a perfectly absorbing G0. In G0, different numbers of open slits per P₀-group (40% p₀ wide slits) were simulated.

4.5.1 Simulation results

Figure 9(b) shows several contrast graphs; the graph for the point source is identical to Fig. 9(a). For one slit per P₀-group, the simulated visibility was 23%; simulated Talbot-Lau setups (of m₁ = 7^th and 9^th fractional Talbot order, identical conditions such as 40% duty cycle) had a visibility of 36% = 1.56 × 23%. Increasing the slit-count caused the visibility to decrease, however the contrasts of photon energies near the design energy were quite unaffected.

The interpretation is that period p₀ depends on the photon energy, hence different slits have different phase-stepping curves (depending on the photon energy).

4.6 Simulated phase-stepping G0

To show the measurement process using phase-stepping, G0 was phase-stepped in simulation without sample. That is, polychromatic dispersion by the measurement process was included, but dispersion due to the refraction in the sample was not included.

This simulation is intended to predict what can be expected in practice for setup H (poly, 30 keV). Hence, the absorption gratings consisted of 150 µm high gold absorbers. Duty cycles were 50% for G2 and 40% for p₀-periodic slits in G0. For G0, only one period p₀ was a regular period, the other 15 p₀-periods within a P₀-Group consisted of 400 µm of gold to reduce the leakage (see the coarse “blocker G0” in Fig. 6). The contrasts $C=(I_1-I_2)/(I_1+I_2)$ were determined by measuring the intensity twice: Too measure I₂, G2 was moved by $p_2/2.$

4.6.1 Simulation results

Figure 10(a) shows contrast graphs corresponding to the energy sub-ranges in Fig. 9(b); the graphs represent simple contrast-averages of these energy intervals. All contrasts (even single monochromatic contrasts) were smooth and looked sine-like. However, all contrasts had different periods and amplitudes. The spectrum-weighted average contrast $"\Delta E/E=50\%"$ reached a visibility of V = 21% for $x/p_0=\pm 25\%.$ When approximating that average contrast using a sine of matching period and amplitude, the difference between average contrast and sine grew with $\left|\Delta x\right|.$ The thin dotted graph shows 30 times of the difference; the difference of the spectrum-weighted contrast to the sine stayed below 0.0042 = 2% V for $|x/p_0|<40\%.$

 

Fig. 10 (a) Stepping G0 without sample and (b) a changing sample refraction with a fixed G0. Setup H (polychromatic; 30 keV) was simulated using imperfect absorption gratings, however a “blocker” G0 of larger opening width and larger absorber height was assumed. All graphs are sine-like near the origin x = 0 p₀. The difference to the sine and the energy dependency are larger for the varied sample gradient in (b) due to energy-dependent refraction in the sample.

Download Full Size | PDF

4.7 Simulated sample gradients

The previous simulation (phase stepping without sample) was repeated, this time including a sample of varying steepness ( = gradient $\Delta z/\Delta x),$ but without phase stepping (i.e., $x=0p_0).$

4.7.1 Simulation results

Figure 10(b) shows contrasts of similar amplitude as in the previous simulation. However, the “sine”-frequencies varied much stronger than in Fig. 10(a) and only one sine period in each case was actually sine-like.

An optimistic interpretation is that for actual medical images, the refraction in a given sample can be quite low for most pixels. Hence, only a small percentage of pixels might be strongly affected by sample-dispersion.

5. Grating production and alignment

X-ray gratings are often built using lithography/electroplating or reactive ion etching. These processes create a 3D geometry using a binary 2D mask. Since this proposal demands a high aspect ratio, such proven processes should be used to build the required prism gratings.

One option is to use a line grating mask [see Fig. 11(a)] and shear within the x-direction during lithography, so that the layout is displaced by half an x-period at the bottom [22].

 

Fig. 11 Sheared production: (a) x-shear, (b) y-shear layout mask and (c) resulting 3D-prism.

Download Full Size | PDF

However, to reach a good visibility, a good alignment between GA, GB and G2 is necessary (since the contrast is sine-like only near x = 0 in Fig. 10). Hence, for real setups, the scaling $p_{\text{A}}/d_0=p_{\text{B}}/(d_0+d_{\text{A}})=p_2/(d_0+d_{\text{A}}+d_{\text{B}})$ and any rotation around the optical axis are critical. Here, a possible first mitigation is a line scanner, i.e., gratings which are short in one dimension and long in the other direction, combined with a moving sample [9,26,27].

However, the ability to align pairs of gratings (GA⬄GB or GB⬄G2) also seemed important. As a solution, this work proposes to manufacture the prism gratings by shearing in the y-direction during lithography [28], so that the layout is displaced by a whole y-period at the bottom of the grating. The advantage is that y-shearing allows to mix line grating layouts with prism grating layouts, see the (synthetic) example in Fig. 11(b): The outermost parts are not affected by the y-shear and result in regular line gratings. They are intended to act as calibration areas. These areas can represent (imperfect) phase and absorption gratings to help align GA relative to GB, or to align GB relative to G2 (“imperfect Talbot-Lau setups”).

The inner main region looks like meandering line gratings, but actually consists of lines of opposing triangles (see the dark individual triangles of height p_y). Each vertical strip of triangles will act as a single prism strip. The reason is shown in Fig. 11(c): A single (red) triangle T (“top”) of the mask is projected onto a triangle B (“bottom”) shifted by one y-period. This pattern is also projected onto the cross sectional plane C between B and T. The reason is that when moving T towards B (see area I “intermediate”), the entire pattern T slides across the cross sectional plane. When choosing the green prism as y-period T, the cross section C contains the same pattern with a different spatial phase (“the red triangle in two parts”). The changed phase does not matter, since gratings are thin compared to propagation distances. The area of all cross-sections (red, green,…) along the y-axis is the same, and so is the total material height profile. Hence, the total material height only varies along the x-axis and in the same way as in Fig. 11(a).

6. Dose savings of concrete example setups

This section first defines setups of various design energies and grating materials. These setups are then related to the setups from literature in Fig. 1 and a dose advantage is assessed.

6.1 Example setups

Table 1 shows proposals for setups sorted by design energy. The setups were optimized for compactness and sensitivity within the design limits given below the table. The underlined setups D, F, H and J are intended for x-ray tubes; setups E and G are proposed for the compact light source (CLS) [29]. Visibilities V are simulation results for a point source, a perfectly absorbing G2 (i.e., T₂ = 50%), and a sine-shaped spectrum of the listed bandwidth $\Delta E/E$ full width half maximum (FWHM). As an exception, the spectrum for setup D assumes a molybdenum anode at 36 kVp, filtered by 40 µm of niobium. The sine-shaped tube-spectra for $\Delta E/E=50\%$ are compatible to tungsten-anodes filtered by 1 to 2 mm aluminum. Simulated polychromatic visibilities were about 66% of simulated visibilities of Talbot-Lau setups (for large fractional Talbot orders m ≥ 7).

Table 1. Example configurations corresponding to Fig. 12 (which visualizes columns α_R and d_A + d_B)

View Table

6.2 Dose savings

The aim defined in the introduction was to propose compact sensitive setups to save dose.

Figure 12 shows that setups A, G, H and J of Table 1 are within the circle of favorable setups of Fig. 1. The 20 keV-setups D, E and F currently are outside the circle because the lightly absorbing grating materials required at 20 keV need the highest prisms of all example setups. These high prisms are currently producible only for larger grating periods, i.e., larger post-patient distances.

 

Fig. 12 Sensitivity depending on compactness of the setups proposed in Table 1.

Download Full Size | PDF

Next, the dose savings of the proposed setups within the circle are compared to existing setups; the comparison is to be done using Eq. (3), i.e., $D~{\alpha }_{\text{R}}^2/(V^{2}T).$ For setup H (poly 30 keV), the visibility V = 21% (see section 4.6.1) and phase grating transmission T_AB = 49% are about half of what can be expected of Talbot-Lau setups (in particular for high fractional Talbot orders), hence the term $V^{2}T$ is a factor of about 8 worse for fringe formation compared to Talbot-Lau setups. The measurement ranges α_R of setups A, G, H and J are a factor of 6.6 (for setup J) to 10.5 (for setups A and G) below the graph for Talbot-Lau setups. Using the proposed factor 8 to compare $V^{2}T,$ this translates into a dose-advantage over Talbot-Lau setups of at least 5.4 ($={6.6}^2/8$ for setup J).

7. Conclusion

Existing medical x-ray phase contrast imaging techniques do not cover measurement ranges (“angular sensitivities”) of about 2 µrad refraction within the sample, in particular when considering only post-patient distances below 0.2 m. To close the gap, this work proposes a setup of prism gratings. The chance is a dose reduction of a factor of 5.4 compared to Talbot-Lau setups of same post-patient distance (using regular x-ray tubes). Phase stepping can be done similar to Talbot-Lau setups; the resulting phase stepping curve however is based on a changing contrast (“fringe formation”) and not on a moving intensity pattern.

The use of prisms may feel wrong at first because of the energy-dependency of refraction. However, the prisms are arranged periodically, allowing the use of the Talbot effect. Compared to Talbot-Lau setups of fractional Talbot orders of 7 or more, about half of the visibility can be expected for polychromatic setups (based on simulation).

In practice, the main challenge is expected not to be the production of prism gratings, but the exact alignment of the four-grating setup. As a mitigation, a sample-scanning setup would allow a reduction in grating-width. Furthermore, prism gratings containing calibration areas allow to sequentially align individual pairs of gratings (GA vs. GB, GB vs. G2).

Prism gratings made from nickel absorb up to half of the x-ray intensity. Here, improved production capabilities might later allow to use silicon or even photoresist instead.

Appendix A Contributing effects and energy-dependency

Section 2.4 described the sensitivity of fringe formation due to “wavefront beamlets” which get shifted vertically against each other. This contribution to sensitivity is called ① SHIFT in Fig. 13 and is one of the three contributions ① SHIFT, ② PATH and ③ HEIGHT. In the following, all contributions will be shown to be of same magnitude and energy-dependency. Finally, ② PATH is shown to have a different sign, this way compensating ③ HEIGHT.

 

Fig. 13 (a) The three contributions to sensitivity along with all variables used within the text.

Download Full Size | PDF

A plane-wave geometry is used here to discuss the contributions (i.e., $s=p_{\text{A}}/2=p_{\text{B}}/2,$ prism height h = h_A = h_B, distance d = d_A). Furthermore, half of the lateral shear L is called $\gamma \cdot d$ here, i.e., $L/2=Ss=\gamma d$ for a prism of refraction-angle $\gamma =\delta h/s$ (where δ is the refractive index decrement). A plane wave incident at angle α is assumed.

Contribution ① SHIFT is given by Eq. (5), which is

(18)$$\Delta \Phi =L\alpha (2\pi /\lambda \text{})=2\gamma d\alpha (2\pi /\lambda \text{})=2\Delta {\mathcal{l}}_1(2\pi /\lambda \text{})$$

for the total optical path length difference 2Δℓ₁:

(19)$$2\Delta {\mathcal{l}}_1=2\alpha \cdot \gamma d.$$

Contribution ② PATH is due to path length differences for a “beamlet” between GA and GB: The sample refraction angle α lengthens one path and shortens the other one at the same time. Using the Taylor-series approximation $\sqrt{d^2+\Delta x^2}\approx d+\Delta x^2\text{}/2d$ for the x-component $\Delta x=(\alpha \pm \gamma )d,$ the term $2\Delta {\mathcal{l}}_2=\sqrt{d^2+{(\alpha +\gamma )}^2{d}^2}-\sqrt{d^2+{(\alpha -\gamma )}^2{d}^2}$ simplifies to

(20)$$2\Delta {\mathcal{l}}_2\approx [{(\alpha +\gamma )}^2-{(\alpha -\gamma )}^2]d^2/2d=2\alpha \cdot \gamma d.$$

Contribution ③ HEIGHT is due to path length differences within the grating material. In Fig. 13, this difference is shown at GB, but (as for ② PATH), all parallel paths are equivalent. Using the prism steepness $h/s,$ the material height changes by $\Delta h=\alpha d\cdot (h/s).$ This corresponds to a path length difference of $\Delta {\mathcal{l}}_3=\Delta h\delta$ for the refractive index decrement δ. Using the prism refraction angle $\gamma =\delta (h/s)$ for both light paths, this again results in

(21)$$2\Delta {\mathcal{l}}_3=2(\alpha d)\cdot (h/s)\delta =2\alpha \cdot \gamma d.$$

The energy-dependency of each contribution is $\Delta \mathcal{l}~{\lambda }^2,$ because $\Delta \mathcal{l}~\gamma ~\delta$ and for x-rays in general, $\delta ~{\lambda }^2$ holds. Combining Eq. (18) with Eq. (6) $1/{\alpha }_{\text{R}}=L/\lambda$ and $\Delta \mathcal{l}~{\lambda }^2$ means that there is a reduced sensitivity for higher photon energies E:

(22)$$\Delta \Phi ~1/{\alpha }_{\text{R}}~\Delta {\mathcal{l}}_1/\lambda ~\lambda ~1/E$$

In particular, Eq. (15) ${\alpha }_{\text{R}}=p_0/d_0$ yields an energy-dependent source grating period $p_0~E.$

As for the sign of the contributions, the red downward-path in Fig. 13 is a longer ② PATH and delays the phase. On the other hand, both ① SHIFT and ③ HEIGHT advance the phase (due to an increased grating height and $n=1-\delta <1),$ hence $\Delta {\mathcal{l}}_1=-\Delta {\mathcal{l}}_2=\Delta {\mathcal{l}}_3.$ In summary, ① SHIFT describes the sensitivity as the other two contributions cancel out.

Appendix B Justifications for periods p₀ and Q₀

Starting with the requirement $Q_0/d_0=p_{\text{B}}/d_{\text{A}},$ Eq. (14) follows from scalings (SC) and lens equations (LE), finally expanding the geometric magnification M_A on the right-hand-side:

(23)$$\begin{array}{l}\frac{Q_0}{d_{\text{0}}}:=\frac{p_{\text{B}}}{d_{\text{A}}}=\frac{p_{\text{A}}}{D_{\text{A}}}=\frac{p_{\text{A}}{M}_{\text{A}}^2}{D_{\text{B}}{m}_{\text{A}}}=\frac{p_{\text{B}}}{D_{\text{B}}}\cdot \frac{M_{\text{A}}}{m_{\text{A}}}=\frac{p_{\text{2}}}{d_{\text{B}}}\cdot \frac{M_{\text{A}}}{m_{\text{A}}}\\ (\text{def)}\begin{array}{l}(SC\text{)}{p}_{\text{B}}=M_{\text{A}}{p}_{\text{A}}\hfill \\ \text{(LE)}{\text{d}}_{\text{A}}=M_{\text{A}}{D}_{\text{A}}\hfill \\ M_{\text{A}}=(d_0+d_{\text{A}}\text{})/d_0\hfill \end{array}\begin{array}{l}{D}_{\text{B}}/D_{\text{A}}\hfill \\ =p_{\text{B}}^2/(m_{\text{A}}{p}_{\text{A}}^2)\hfill \\ =M_{\text{A}}^2/m_{\text{A}}\hfill \end{array}\begin{array}{l}\text{(SC)}{M}_{\text{B}}{p}_{\text{B}}=p_2\hfill \\ \text{(LE)}{M}_{\text{B}}{D}_{\text{B}}=d_{\text{B}}\hfill \\ M_{\text{B}}=1+d_{\text{B}}/(d_0\text{}+d_{\text{A}}\text{})\hfill \end{array}\end{array}$$

To justify Eq. (16), Eq. (6) is used to start as $(p_0/d_0)/M_{\text{A}}={\alpha }_{\text{R}}/M_{\text{A}}=\lambda /(S{p}_{\text{B}}):$

(24)$$\begin{array}{l}\frac{p_0}{d_0+d_A}=\frac{\lambda }{pB}/S=\frac{p_B}{2D_B}/S=\frac{p_2}{d_B}/2S\\ [\text{Eq}\text{.}\text{(6})]\begin{array}{c}{D}_B=p_B^2/2\lambda \iff \\ \lambda /p_B=p_B/2D_B\end{array}\begin{array}{c}\text{(SC)}{M}_{\text{B}}{p}_{\text{B}}=p_2\\ \text{(LE)}{M}_{\text{B}}{D}_{\text{B}}=d_{\text{B}}\end{array}\end{array}$$

Appendix C Visible light prototype

A simple prototype illuminated by regular laser pointers was built to get some practical validation without the need for an expensive x-ray setup. Prisms of periods p_A = 1.05 mm and p_B = 1.8 mm were milled 0.522 mm deep into polymethyl methacrylate (PMMA), see Fig. 14(a). The gratings were immersed in line oil (air-bleached at 120°C). The resulting refractive index difference (line oil vs. PMMA) was Δn ≈ 0.01, and prisms shifted the phase by $N\pi \approx 16\pi$ at a wavelength of λ = 651 nm. A small Δn was used for similarity to the x-ray case and because it allows prism surface deviations of about 10 µm. A problem of line-oil immersion was that the four glass plates containing the oil were not perfectly planar, causing image distortions.

 

Fig. 14 (a) Milled PMMA grating. Images taken in G2-plane (the phase sensitive x-axis is the horizontal axis): (b) Monochromatic setup M (651 nm), (c) Polychromatic setup P (651 nm), (d) P (532 nm), (e) P (420 nm).

Download Full Size | PDF

Two setups were built: The monochromatic setup M for λ = 651 nm had a total length of 10.8 m, m_A = 2 and α_R = 19 µrad. The polychromatic setup P had a total length of 10.6 m.

Figure 14 shows images of the G2-plane without sample for both configurations. GB was rotated slightly around the optical z-axis. This made the inter-grating x-offset vary along the vertical y-axis. The effect is as if the source-position and thus ΔΦ changed along the y-axis.

The images look like the sinusoidal intensity pattern which can be expected from Eq. (8) for a ΔΦ increasing along the y-axis. The pattern looked sinusoidal for quite a few periods, due to the large phase shift of 16π and $Q_0=2N{p}_0$ of Eq. (17). A phase object (i.e., hot air blown into the ray in front of GA) caused the patterns to move vertically.

Funding

Karlsruhe Nano Micro Facility (KNMF).

Acknowledgments

I would like to thank Gisela Anton, Thomas Weber, Christoph Pflaum, Christian Riess and Ralf Hofmann for discussions and feedback.

Disclosures

OP declares no commercial relationships regarding x-ray hardware.

References and links

1. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013). [CrossRef]   [PubMed]  

2. 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]  

3. D. Stutman and M. Finkenthal, “K-edge and mirror filtered x-ray grating interferometers,” AIP Conf. Proc. 1466, 229–236 (2012). [CrossRef]  

4. L. B. Gromann, D. Bequé, K. Scherer, K. Willer, L. Birnbacher, M. Willner, J. Herzen, S. Grandl, K. Hellerhoff, J. I. Sperl, F. Pfeiffer, and C. Cozzini, “Low-dose, phase-contrast mammography with high signal-to-noise ratio,” Biomed. Opt. Express 7(2), 381–391 (2016). [CrossRef]   [PubMed]  

5. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997). [CrossRef]   [PubMed]  

6. V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81(7), 073709 (2010). [CrossRef]   [PubMed]  

7. R. Raupach and T. G. Flohr, “Analytical evaluation of the signal and noise propagation in x-ray differential phase-contrast computed tomography,” Phys. Med. Biol. 56(7), 2219–2244 (2011). [CrossRef]   [PubMed]  

8. K. J. Engel, D. Geller, T. Köhler, G. Martens, S. Schusser, G. Vogtmeier, and E. Rössl, “Contrast-to-noise in x-ray differential phase contrast imaging,” Nucl. Instrum. Methods Phys. Res. A 648, S202–S207 (2011). [CrossRef]  

9. E. Roessl, H. Daerr, T. Koehler, G. Martens, and U. van Stevendaal, “Clinical boundary conditions for grating-based differential phase-contrast mammography,” Philos Trans A Math Phys Eng Sci 372(2010), 20130033 (2014). [CrossRef]   [PubMed]  

10. D. Stutman, T. J. Beck, J. A. Carrino, and C. O. Bingham, “Talbot phase-contrast x-ray imaging for the small joints of the hand,” Phys. Med. Biol. 56(17), 5697–5720 (2011). [CrossRef]   [PubMed]  

11. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91(7), 074106 (2007). [CrossRef]  

12. P. C. Diemoz, M. Endrizzi, C. E. Zapata, Z. D. Pešić, C. Rau, A. Bravin, I. K. Robinson, and A. Olivo, “X-ray phase-contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110(13), 138105 (2013). [CrossRef]   [PubMed]  

13. A. Zamir, M. Endrizzi, C. K. Hagen, F. A. Vittoria, L. Urbani, P. De Coppi, and A. Olivo, “Robust phase retrieval for high resolution edge illumination x-ray phase-contrast computed tomography in non-ideal environments,” Sci. Rep. 6(1), 31197 (2016). [CrossRef]   [PubMed]  

14. M. Endrizzi, F. A. Vittoria, L. Rigon, D. Dreossi, F. Iacoviello, P. R. Shearing, and A. Olivo, “X-ray phase-contrast radiography and tomography with a multiaperture analyzer,” Phys. Rev. Lett. 118(24), 243902 (2017). [CrossRef]   [PubMed]  

15. 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]   [PubMed]  

16. 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]  

17. F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52(23), 6923–6930 (2007). [CrossRef]   [PubMed]  

18. H. Wen, A. A. Gomella, A. Patel, S. K. Lynch, N. Y. Morgan, S. A. Anderson, E. E. Bennett, X. Xiao, C. Liu, and D. E. Wolfe, “Subnanoradian X-ray phase-contrast imaging using a far-field interferometer of nanometric phase gratings,” Nat. Commun. 4, 2659 (2013). [CrossRef]   [PubMed]  

19. H. Miao, A. A. Gomella, K. J. Harmon, E. E. Bennett, N. Chedid, S. Znati, A. Panna, B. A. Foster, P. Bhandarkar, and H. Wen, “Enhancing tabletop x-ray phase contrast imaging with nano-fabrication,” Sci. Rep. 5(1), 13581 (2015). [CrossRef]   [PubMed]  

20. 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]   [PubMed]  

21. H. Suhonen, M. Fernández, A. Bravin, J. Keyriläinen, and P. Suortti, “Refraction and scattering of X-rays in analyzer-based imaging,” J. Synchrotron Radiat. 14(6), 512–521 (2007). [CrossRef]   [PubMed]  

22. A. Yaroshenko, M. Bech, G. Potdevin, A. Malecki, T. Biernath, J. Wolf, A. Tapfer, M. Schüttler, J. Meiser, D. Kunka, M. Amberger, J. Mohr, and F. Pfeiffer, “Non-binary phase gratings for x-ray imaging with a compact Talbot interferometer,” Opt. Express 22(1), 547–556 (2014). [CrossRef]   [PubMed]  

23. O. Preusche, patent US 9,763,634 B2 (2017).

24. C. T. Chantler, K. Olsen, R. A. Dragoset, J. Chang, A. R. Kishore, S. A. Kotochigova, and D. S. Zucker, “X-ray form factor, attenuation and scattering tables,” (version 2.1, 2005) http://physics.nist.gov/ffast.

25. J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta (Lond.) 18(9), 677–682 (1971). [CrossRef]  

26. C. Kottler, F. Pfeiffer, O. Bunk, C. Grünzweig, and C. David, “Grating interferometer based scanning setup for hard X-ray phase contrast imaging,” Rev. Sci. Instrum. 78(4), 043710 (2007). [CrossRef]   [PubMed]  

27. S. Bachche, M. Nonoguchi, K. Kato, M. Kageyama, T. Koike, M. Kuribayashi, and A. Momose, “Laboratory-based X-ray phase-imaging scanner using Talbot-Lau interferometer for non-destructive testing,” Sci. Rep. 7(1), 6711 (2017). [CrossRef]   [PubMed]  

28. O. Preusche, “Lens gratings for dose optimization of medical X-ray phase contrast imaging,” Opt. Express 24(23), 26161–26174 (2016). [CrossRef]   [PubMed]  

29. M. Bech, O. Bunk, C. David, R. Ruth, J. Rifkin, R. Loewen, R. Feidenhans’l, and F. Pfeiffer, “Hard x-ray phase-contrast imaging with the compact light source based on inverse Compton x-rays,” J. Synchrotron Radiat. 16(1), 43–47 (2009). [CrossRef]   [PubMed]