1. Introduction

Since their discovery in 1895 by Röntgen, x-rays have been utilized for visualizing internal structures non-destructively in a wide range of fields including those of physics, chemistry, biology, and medicine. The absorption of x-rays has been conventionally used for imaging applications, but in the early 1990s several imaging techniques using x-ray phase shift were proposed. These so-called x-ray phase imaging techniques [1, 2] have been highlighted because they can have a sensitivity to light elements three orders of magnitude higher than that of absorption-contrast imaging. However, most of these techniques have essentially required a highly brilliant synchrotron x-ray source. A great deal of attention has recently been paid to x-ray phase imaging techniques using transmission gratings [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], represented by Talbot and Talbot-Lau interferometry, because they work even with polychromatic and cone beam from a compact laboratory source.

We can retrieve two kinds of quantitative images with x-ray Talbot (-Lau) interferometry, i.e. absorption and differential-phase, from a series of experimentally obtained moiré images. Pfeiffer et al. has recently proposed another approach to forming image contrast, where a relative reduction in the visibility of the moiré image is quantified by defining normalized visibility [6]. They reported that this visibility contrast was formed through the mechanism of small-angle x-ray scattering from microstructures with a scale much smaller than the spatial resolution of the imaging system. Their approach is fascinating because it can provide structural information that is inaccessible from the absorption and differential-phase images, and it shows promise of offering a broad range of applications [14, 15, 16, 17, 18, 19, 20]. However no general formulation of the phenomenon, which is essential for quantitative structure analysis, has thus far been provided.

This paper explains how the reduction in the visibility can generally be formulated by an autocorrelation function describing the spatial fluctuations of a wavefront due to the microstructures. Visibility reduction obtained experimentally with microspheres and a melamine sponge was successfully explained by the formula. The experimental results were also consistent with results from ultra-small angle x-ray scattering measurements in the far field. Three parameters are required to characterize the wavefront fluctuations: variance, correlation length, and the Hurst exponent. We demonstrate from analytical and numerical calculations that the correlation length represents the average size of the microstructures, while the Hurst exponent is relevant to their shape. The results of least-squares fitting to the experimental data were well explained by this interpretation. Thus, the visibility contrast is quantitatively related to the structural parameters of the microstructures. Our approach can directly provide two- and three-dimensional structural information in real space, a scale that is only accessible by ultra-small-angle x-ray scattering (the near-field speckle region [21]), and it is expected to be broadly applied to medical, biological, and material sciences.

2. Theoretical description of the reduced visibility

First, we derive a general formula for the reduced visibility on the basis of the spatial fluctuations of a wavefront. Let us consider the x-ray Talbot interferometer system schematically illustrated in Fig. 1. Here, we assume that a plane wave with wavelength λ is illuminating a sample located just in front of the first grating; the following can easily be extended to the case of polychromatic- and spherical-wave illumination [22].

The complex refractive index of the sample is expressed by 1 − δ + iβ, and the electric field just behind the sample can be written in the projection approximation by E ₀ exp[−α(x,y)/2]exp[−iΦ(x,y)], where E ₀ is the amplitude of the incident x-rays, and α and Φ are given by (4π/λ) ∫ β (x,y,z)dz and (−2π/λ) ∫ δ (x,y,z)dz. To discuss the effect of phase fluctuations, we present α and Φ as superpositions of smooth (resolvable) and fine (unresolvable) features [23], α = α _s + α _f and Φ = Φ_s + Φ_f. In the following, we assume that α _f is negligible. We also assume, for the sake of simplicity, that the size of the first grating is infinite.

The electric field just in front of the second grating, located at distance z _T from the first grating, can be given in the paraxial approximation by [10]

Here, a′_n is the nth Fourier coefficient of the electric field generated in front of the second grating (n = 0,±1, …), given by a_n exp(−iπpn ²), where a_n is the nth Fourier coefficient of the amplitude transmission function of the first grating and p is the Talbot order [24], d is the pitch of the gratings, and Φ_n(x,y) is defined by Φ(x − npd,y).

The intensity just behind the second grating detected by a pixel located at point (x,y) is approximated by

 

Fig. 1. Experimental setup for x-ray Talbot interferometry.

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where

Here, I ₀ ≡ ∣E ₀∣², c_N is the Nth Fourier coefficient of the intensity transmission function of the second grating (N = 0,±1, …), µ_m is the complex coherence factor of x-rays [25] at two points separated by distance mpd on the first grating (m = 0,±1, …), χ is the relative displacement of the second grating to the first grating in the x direction, and PSF(x,y) is the normalized point spread function (PSF) of the detector. Note that, in Eq. (2), we have assumed that the two gratings are parallel to the y axis without loss of generality and that the scale resolved by the image system is sufficiently larger than the pitch of the gratings.

The visibility is approximately proportional to the ratio of the modulus of the 1st order Fourier coefficient term, q ₁, in Eq. (2) (corresponding to N = 1) to that of the 0th order, q ₀, (corresponding to N = 0), both of which are experimentally obtained by using the fringe-scanning technique [26]. The 0th order Fourier coefficient, q ₀, is given by

The factor, Ã _n,m,0, is rewritten by

where Φ_f,n(x,y) ≡ Φ_f(x − npd,y). Here, we have considered that the resolvable feature, Φ_s, is a slowly varying function of x and that the derivative of Φ_s higher than the first can be neglected. If we assume that the unresolvable features, Φ_f, are distributed randomly and that the width of PSF is sufficiently larger than the characteristic scale of the unresolvable features, Ã _n,m,0(x,y) can be further approximated by [7]

Here, the bar means the averaging around (x,y):

where D is the width of the PSF. Since the phase Φ_f,n(x,y) − Φ_f,n−m(x,y) in Eq. (7) is random, the terms of à _n,m,0(x,y) for m ≠ 0 are much smaller than à _n,0,0(x,y). Hence,

Thus, ∣q ₀∣ is finally given by

where C ₀ ≡ ∑_nμ ₀∣a′_n∣² c ₀.

We can similarly obtain an approximate form of ∣q ₁∣. From Eq. (2),

Similar to Eqs. (5), (6), and (7), the factor, Ã _n,m,1, is rewritten by

Since the phase, Φ_f,n(x,y) − Φ_f,n−m(x,y), in Eq. (15) is random, the terms, à _n,m,1(x,y), for m ≠ −1 are much smaller than à _n,−1,1(x,y). Hence,

If a grating with a Ronchi ruling is used as the first grating, the even order Fourier coefficients of the amplitude transmission function of the grating except for the 0th order vanish. Then

Expressing a′₀ a′^* ₁ by |a′₀ a′^* ₁|exp[iΩ], and using the fact a′₋₁ a′^* ₀ = (a′₀ a′^* ₁)^*, we can finally obtain an approximate form of ∣q ₁∣:

where C ₁ ≡ 2∣μ ₋₁∣∣c ₁∣∣a′₀ a′^* ₁∣ cosΩ and ∆Φ_f(x,y; pd) ≡ Φ_f,0(x,y) − Φ_f,1(x,y). Here, we have assumed that the average phase, $\overline{{\Phi }_{f,0}(x,y)-{\Phi }_{f,1}(x,y)}$ , is zero, and that

because D was assumed to be much larger than pd.

Hence, the normalized visibility, which is the ratio of visibility with the sample (V) to that without the sample (V ₀), is given by

In the derivation of Eq. (22), we assumed that Φ_f could be modeled as a random Gaussian process [23, 27], the width of which is given by σ. The term, γ, is the normalized autocorrelation function given by

As wavefront fluctuations can be treated as surface roughness, we can use the simplest general model for γ that was proposed by Sinha et al. to deal with height fluctuations on surfaces [28]:

where H is the Hurst exponent (0<H <1) and ξ is the correlation length of phase fluctuations. Thus, we can generally formulate the reduced visibility in terms of the autocorrelation function of spatial fluctuations of a wavefront. Note that, from Eq. (22), V/V ₀ approaches zero when σ increases. In addition, for a given sample (with a given ‘scattering power’), V/V ₀ is a function of pd.

3. Verification of formulation through experiments

3.1. Dependence of V/V0 on pd

We measured the dependence of V/V ₀ on pd to corroborate Eq. (22) by using monochromatic x-rays. The experiment was carried out with synchrotron x-rays at the beamline 14C, Photon Factory (PF), Japan. Cross-linked PMMA microspheres (EPOSTAR MA, Nippon Shokubai Co., Ltd.) with various radii and a melamine sponge were used as the samples. Microspheres with a fixed radius were dispersed in glycerin liquid, which was dispensed into a 10 mm-thick plastic cell. The volume fraction of the microspheres was fixed at 0.057. The melamine sponge was 1.6 mm thick. Gold gratings with a Ronchi ruling (a π/2-phase grating for the first grating and a 30 µm-thick absorption grating) were used. They were aligned parallel to each other. The images were recorded using a charge-coupled device (CCD)-based x-ray image detector (Spectral Instruments), where the CCD (4096 × 4096 pixels) was connected to a 40 µm GOS screen with a 2:1 fiber coupling. The effective pixel size was 18 µm, and the width of the line spread function was 70 µm.

The plots in Fig. 2 (a) represent the experimental results for the microspheres. The wavelength of the x-rays was fixed at 0.71 Å. Here, 5.3 µm-pitch gratings were used and the Talbot order, p, was changed by changing the distance between the first and second gratings. The curves in the figure plot the results of least-squares fitting to the experimental data by using Eqs. (22) and (24). The experimental data are in good agreement with the fitting curves. This means that the description of the reduced visibility obtained by using Eqs. (22) and (24) is valid.

Figure 2(b) plots the experimental results for the melamine sponge, where the results for three wavelengths and two pitches of the gratings are shown. The experiments were performed at the beamline 14C, PF, and at the beamline 20XU, SPring-8, Japan. At SPring-8, an x-ray camera consisting of a phosphor screen (10 µm-P43, Gd₂O₂S:Tb+ fine powders), a relay lens, and a CCD camera (Hamamatsu Photonics C4742-98-24A, 1344 × 1024 pixels) was used. The effective pixel size of the detector was 3.14 µm. The open and filled circles in Fig. 2 (b) plot the results for λ = 1.0 and 0.5 Å (SPring-8), while the triangles and squares plot those for another sample with the same thickness for d = 5.3 and 8.0 µm at 0.7 Å (PF). At SPring-8, 20 × 20 pixel binning was implemented to make D almost the same as that of the detector used at PF. The good agreement between the experimental results for the 5.3 µm and 8.0 µm pitches corroborates that Eq. (22) is correct. In addition, the experimental data were well fitted by using Eqs. (22) and (24) for all the wavelengths. Note that the results did not depend on the direction of the sample. The fitting results in Figs. 2(a) and 2(b) are summarized in Table 1.

Table 1. Results of least-squares fitting to −ln (V/V₀) data in Fig. 2

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3.2. Ultra-small-angle scattering measurement

The angular distribution of small-angle scattering in the far field should be related to the Fourier transform of Eq. (21). Hence, we can also check the validity of Eq. (22) through small-angle x-ray scattering measurements. The experiment was performed using Mo Kα characteristic x-rays from a laboratory x-ray source. The non-dispersive double-crystal arrangement seen in Fig. 3(a) was employed. The intensity, I_x, integrated in the y-direction was measured with this setup, which should agree with the Fourier transform of Eq. (22) in the x-direction. The plots in Fig. 3(b) represent the measured intensity of the (ultra) small-angle scattering for the microspheres samples. The curves in Fig. 3(b) are the Fourier transforms of the V/V ₀ of the fitting curves in Fig. 2(a) convoluted by the measured rocking curve, the width of which was 7 µrad. The good agreement between the plots and curves substantiates the validity of Eq. (21).

 

Fig. 2. Dependence of −ln (V/V ₀) on pd. (a) Results for microspheres with various radii a and (b) those for melamine sponge with its optical microscope image. Plots are experimental data and curves are results of least-squares fitting to experimental data.

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Fig. 3. (a) Experimental setup for (ultra) small-angle scattering measurement and (b) results of measured angular distribution for microspheres with various radii.

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4. Relation between fitting parameters and structural parameters of sample

Finally, we will explain how the parameters that characterize the spatial fluctuations of a wavefront are related to the structural parameters of a sample. First, let us discuss σ and γ. For simplicity, we approximate the sample by a matrix containing a large number of unresolved microscopic particles. If the positions and directions of the particles are uncorrelated in Eq. (23), there is also no correlation between the projections of the particles. Then, γ(x,y;∆x) in Eq. (22) can simply be given by the autocorrelation function of the projection of a particle. In addition, since γ is normalized as it satisfies γ(x,y;0) = 1, γ is determined only by the shape of a particle. Other parameters such as the number density, N, of particles and the thickness, T, of the sample are contained in σ ². For example, in microspheres with a radius of a, γ and σ ² can be analytically given as: $\gamma =(1+\frac{\Delta {x\prime }^2}{2})\sqrt{1-\Delta {x\prime }^2}$

 

Fig. 4. Hurst exponent is plotted against ratio of diameter to height for cylinder particles. Correlation length ξ normalized by average radius ā is also plotted.

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where ∆x′ ≡ ∆x/(2a), ∆ρ is the number density of electrons, and r _e is the classical electron radius. Here we assumed that TNπa ² ≫ 1. These results were also confirmed by numerical calculations for a system where microspheres disperse. It should be noted that the first factor on the right hand side of Eq. (26) corresponds to the number of trials in the random walk problem, while the second factor originates from the phase shift by a microsphere. Since the number of trials is proportional to N and T and the phase shift by a particle is proportional to λ far from an absorption edge, σ ² and ln(V/V ₀) are proportional to TNλ ² for any homogeneous sample. It can be seen that σ ² is proportional to λ ² for the melamine sponge (see Table 1). We also confirmed that ln(V/V ₀) is proportional to the thickness of the melamine sponge.

The correlation length, ξ, and the Hurst exponent, H, are related to the average-structure parameters of the microscopic particles. In fact, Eq. (25) can also be fitted by using Eq. (24), resulting in ξ/a = 0.95 and H = 0.97. The former suggests that correlation length ξ represents the average size of the particles. On the other hand H should be relevant to the average shape of the particles. The results of numerical calculations of H for cylinder particles with random directions are indicated in Fig. 4 by the crosses, where ξ and H are plotted against the ratio of the diameter (2a) to the height (h). The plot indicates that H represents the asymmetry of the shape of the particle, i.e., if we define asymmetry factor β by abs (log (2a/h)), H should be approximately given by exp[−0.5β]. Note that, if we define the average radius, ā, of a cylinder by the radius of a sphere with the same volume as the cylinder, the deviation of ξ/ā from unity increases when β increases (open circles in the figure).

Thus, we can relate normalized visibility to the structural parameters of microstructures. Our experimental results showed that H ~ 1 for microspheres, which was consistent with our calculations. On the other hand H ~ 0.7 for the melamine sponge, which corresponds to β ~ 0.7. This was also consistent with its real shape; it has a thin fiber structure (see Fig. 2 (b)). Our results of ξ for microspheres indicate that the possibility of microspheres aggregating increases when their radius is smaller. This is also supported by the results from ultra-smallangle scattering in Fig. 3 because otherwise the experimentally obtained angular broadening, representing the inverse size of microstructures, cannot be explained. The results of ξ for the melamine sponge were also consistent with observations through an optical microscope.

Finally, we discuss the case where the size and shape of the microscopic particles are not uniform in a sample. Let us consider a sample consisting of uniform domains, each of which contains particles with a different size and shape from the others. For simplicity, we assume that each domain size is larger than the size that can be resolved by the detector, and each domain has no structural correlation with the others. From Eqs. (21) and (22), V/V ₀ at a point (x,y) on the detector is given by

where j represents the contribution from the jth domain on the path to the point (x,y) along the z-axis. Because σ ² _j should be proportional to the thickness of the jth domain, Eq. (29) has a similar form to the Beer-Lambert law. Hence, −ln [V/V ₀] can be given by

where σ and γ are expressed as a function of z. Note that ∂ (σ ²)/∂z is proportional to λ ² and (∆ρ)². Thus, we can also carry out tomography and determine the three-dimensional distribution of (1 − γ)∂ (σ ²)/∂z. Obtaining tomograms for ∂(σ ²)/∂_z, ξ, and H is also possible by performing scans for at least three pds.

5. Conclusion

In conclusion, we formulated the reduction in visibility in x-ray Talbot interferometry [Eqs. (22) and (24)]. The formulae quantitatively related the visibility contrast to the spatial fluctuations of a wavefront due to unresolved microstructures. The experimental results for the microspheres and the melamine sponge were consistent with our formulation. Three parameters were required to characterize the wavefront fluctuations: variance, correlation length, and the Hurst exponent. We showed from analytical and numerical calculations that the correlation length represents the average size of microstructures, while the Hurst exponent is relevant to their shape. The results of least-squares fitting to the experimental data (Table 1) were well explained by this interpretation. Thus, the visibility contrast is quantitatively related to the structural parameters of the microstructures.

Defining (1 − γ)∂ (σ ²)/∂z, we can also carry out tomography if γ is not regarded as being dependent on z in each domain. Obtaining tomograms for ∂ (σ ²)/∂z, ξ, and H is also possible by performing scans for at least three pds. Furthermore, we can use two-dimensional gratings to obtain structural parameters in both x and y directions. Our approach can directly provide twoand three-dimensional structural information on unresolved microstructures, and we expect it to be broadly applied to medical, biological, and material sciences.