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X-ray grating interferometry has been highlighted in the last decade as a multi-modal X-ray phase-imaging technique for providing absorption, differential phase, and visibility-contrast images. It has been mainly reported that the visibility contrast in the visibility-contrast image originates from unresolvable random microstructures. In this paper, we show that the visibility contrast is even reduced by a uniform sample with flat surfaces due to the so-called “beam-hardening effect”, which has to be taken into account when X-rays with a continuous spectrum is used. We drive a criterion for determining whether the beam-hardening effect occurs or not, and propose a method for correcting the effect of beam hardening on a visibility-contrast image.
© 2015 Optical Society of America
1. Introduction
Grating-based X-ray phase-imaging techniques, which are represented by X-ray Talbot and Talbot-Lau interferometries, have attracted increasing attention in the last decade [1–7] because they allow for highly sensitive and quantitative X-ray phase imaging with a laboratory X-ray source. Another advantage of these interferometries over the conventional X-ray-imaging techniques based on X-ray absorption is that, as a multi-modal X-ray imaging technique, they provide three images, i.e., absorption, differential phase, and visibility-contrast images, from a series of experimentally obtained moiré images. The approach to yield the visibility-contrast image was first reported by F. Pfeiffer et al. [8]. In a previous paper, we theoretically described visibility contrast, and experimentally showed that it can be quantitatively related to angular distribution of ultra-small-angle X-ray scattering (USAXS) from unresolvable random microstructures (typically of the order of μm) [9]. That study paved the way for experimentally extracting quantitative structural information. Moreover, we recently reported that visibility contrast is even generated by an unresolvable sharp edge [10].
In the present study, we show that visibility contrast is, however, generated not only by such unresolvable random microstructures and an sharp edge, but also by a uniform sample with flat surfaces due to an effect called “beam hardening”. This effect can occur when X-rays with a continuous spectrum are used for imaging. It is experimentally demonstrated that this effect is indeed observed by using a laboratory X-ray source with a tungsten target. The results of this experimental demonstration give another interpretation of the visibility-contrast image and lead to the following suggestion concerning the usage of the terms that have been used so far: ‘dark-field contrast’ is included in a more general concept called ‘visibility contrast’ and the former term should only be used for contrast arising from USAXS.
In the next section, experimental results showing the effect of beam hardening on visibility contrast are presented. In Section 3, a criterion for judging whether the beam-hardening effect occurs or not is derived. After that, a method for correcting the beam-hardening effect and discriminating it from effects arising from other origins is proposed.
2. Experiment and Results
The setup of the experiment in which we observed the effect of beam hardening on visibility contrast is shown in Fig. 1. We constructed an X-ray Talbot-Lau interferometer with a rotating-anode tungsten multiline X-ray source [11], which provides an array of 10 μm line X-ray sources with a pitch of 30 μm in the plane perpendicular to the optical axis. The total size of the X-ray source in the plane was set to about 0.3 mm × 0.3 mm. The X-ray generator was operated with a tube voltage of 40 kV and a tube current of 45 mA.
The X-ray Talbot-Lau interferometer was designed for an X-ray wavelength of 0.46 Å, which corresponds to the peak of the continuous spectrum from the tungsten target. A 4.5-μm-pitch gold π/2 phase grating (G1) was located at a distance of 1464 mm from the X-ray source, and a 5.3-μm-pitch gold absorption grating (G2) with a height of 40 μm was superimposed on a self-image of G1 formed due to the Talbot effect [12,13] at a distance of 258 mm downstream of G1 (corresponding to a Talbot order of 0.5 with respect to G1), so that a moiré image was generated behind G2. Both the two gratings were fabricated by electroplating on 200-μm-thick silicon wafers. An X-ray image detector with an effective pixel size of 18 μm (Spectral Instruments), in which a charge coupled device (CCD) was connected to a 40-μm-thick Gd2O2S (GOS) scintillator screen with a 2:1 fiber coupling, was used for capturing the intensity distribution behind G2.
Used as a sample, a cuboid block of single-crystalline silicon was located in front of G1. As shown in Fig. 1, the block was aligned so that two adjacent faces (including an edge) were inclined at an angle of 45 degrees with respect to the optical axis. The three images on the left in Fig. 2 are images of transmittance (Fig. 2(a)), moiré-phase (proportional to differential phase) (Fig. 2(b)), and visibility-contrast (normalized visibility [8]) (Fig. 2(c)) taken near the edge of the block. They were obtained from a series of moiré images captured in an equal-sampling 10-step fringe scanning [14–16] with an exposure time of 7 sec for each moiré image.
Fig. 2 Left figures: (a) transmittance (log(I/I0)), (b) moiré-phase, and (c) normalized-visibility (log(V/V0)) images near an edge of a single-crystalline silicon block with two orthogonal faces inclined at an angle of 45 degrees with respect to the optical axis, as shown in Fig. 1. Right graphs: thickness dependences experimentally obtained from the line profiles along the red lines in the left images (filled circles) and results of numerical calculations (green lines).
Filled circles in the right graphs in Fig. 2 are dependences of transmittance (log(I/I0)), moiré phase, and normalized visibility (log(V/V0)), which were experimentally obtained from the line profiles along the red lines in the left figures, on the thickness of the silicon sample. For monochromatic X-rays, log(I/I0) linearly decreases with increasing sample thickness, as is well known according to the Beer-Lambert law. However, for X-rays with a continuous spectrum, the log(I/I0) plot is “concave up” because the center of the X-ray spectrum shifts toward higher energies. This effect is known as ‘beam-hardening’ [17, 18]. The plot of experimentally obtained log(I/I0) (Fig. 2(a)) has a concave-up shape with increasing thickness. This result can be qualitatively explained by the beam-hardening effect.
It was reported that a beam-hardening effect also occurs in a differential-phase (moiré-phase) image [19]. In the case of a moiré-phase image, it is expected that the moiré phase for a sample with linearly increasing thickness is constant for monochromatic X-rays because the phase gradient of the sample is constant. The experimentally obtained moiré phase for the silicon sample (Fig. 2(b)) slightly decreased with increasing thickness. We expected that this result can also be explained by the shift in the center of the continuous X-ray spectrum.
As shown in the right graph in Fig. 2(c), the experimentally obtained normalized visibility, which is generally defined at each pixel by the ratio of the visibility of the intensity oscillation during the fringe scanning with a sample (V) to that without the sample (V0), also depends on the thickness of the sample. When the thickness was less than 1 mm, the normalized visibility was almost unity, but it started to decrease at thickness around 1 mm. This behavior is different from those previously reported for unresolvable random microstructures, for which log(V/V0) linearly decreases from zero thickness [9, 20]. In addition, it cannot be explained by the effect of an unresolvable sharp edge [10], which can only be seen very near to the edge (see Fig. 2(c)). This behavior of the normalized visibility is also expected to be a beam-hardening effect because the visibility V0 with no sample in X-ray grating interferometry generally depends on X-ray wavelength and, when the center of the continuous X-ray spectrum changes, averaged visibility over the spectrum can also change.
To confirm that the thickness dependences shown in Fig. 2 are due to beam hardening, transmittance, moiré phase, and normalized visibility were numerically calculated for the silicon sample. In the calculations, a continuous spectrum experimentally obtained by a CdTe diode detector (AMPTEX, XR-100T-CdTe X-Ray & Gamma Ray Detector) after an escape-peak correction [21] was used. The effects of X-ray absorptions by the air, the sample, the substrates of the two gratings, and the windows of the image detector as well as the energy dependence of detection efficiency of the X-ray image detector were taken into account.
The green curves in the right graphs in Fig. 2 are the results of the calculations. Here, the line width of G1 was set to 2.4 μm, which was evaluated by a laser microscope, and the full-width at half maximum (FWHM) of the line X-ray sources and the line width of G2 were set to 12.9 μm and 3.4 μm, respectively, so that these values consistently give the experimentally obtained V0 without the sample (0.38) and the transmittance, moiré phase, and normalized visibility shown in Fig. 2. The size of the line X-ray sources are slightly larger than the designed width of the lines (10 μm), but this deviation can be explained by the oscillation of the rotating anode. These curves are in good agreement with the experimental results, supporting that the reduction in the normalized visibility for the sample with thickness of more than 1 mm is attributed to beam hardening.
3. Discussion
Here, we device a criterion for judging whether or not the beam hardening affects visibility contrast, and propose a method for correcting the beam-hardening effect and discriminating it from effects arising from other origins.
3.1. Universal curve and universal parameter
Before the criterion is devised, a universal parameter that describes the beam-hardening effect on normalized visibility is demonstrated. Dependences of normalized visibility (V/V0) on transmittance (I/I0) is shown in Fig. 3. Black line and blue line respectively show the experimental measurements and calculation results for silicon presented in the proceeding section (respectively corresponding to filled circles and green lines in Fig. 2). In the figure, results of numerical calculations for gold and copper (obtained using the same spectrum as for silicon) are indicated by red broken line and yellow broken line, respectively; the results for silicon, copper, and gold appear to be on a universal curve.
Fig. 3 Dependences of normalized visibility (V/V0) on transmittance (I/I0) for several materials. Black line and blue line: experiment and calculation results for silicon shown in Section 2; red, yellow, green, and gray broken lines: results of calculations for gold, copper, PMMA, and silver.
V/V0-I/I0 curves for various materials were calculated and found to lie on the same universal curve when photoelectric absorption is dominant and the effects of absorption edges are negligible. Gray broken line in Fig. 3 is the curve for silver, which has an absorption edge at an energy almost at the center of the spectrum (25.5 keV). In this case, the normalized visibility can even be more than unity because the band width of the spectrum is reduced and the center of the spectrum is shifted towards lower energies with increasing thickness of silver. The green broken line in the figure is the result for PMMA (chemical formula: (C5O2H8)n; density: 1.18 g/cm3), for which the contribution of Compton scattering to the reduction in transmittance is not negligible. This line indicates that the effect of beam hardening on normalized visibility becomes smaller when the contribution of Compton scattering becomes larger. In the case of PMMA, the beam-hardening effect is not remarkable when the transmittance is more than 0.2, corresponding to a thickness of 40 mm.
To explain the universal behavior of normalized visibility as described above, the simple model shown in Fig. 4(a) is considered. In this figure, intensity S(λ) averaged over a fringe scanning and visibility V0(λ) of the intensity oscillation during the fringe scanning with no sample present (λ: X-ray wavelength; both S(λ) and V0(λ) can be given for a pixel) are schematically shown. For simplicity, it is assumed that S(λ) and V0(λ) are even functions with respect to λ and that both of them have a center at λ = λ0. If transmittance T(λ) of the sample can be regarded as a slowly varying function of λ around the center of the spectrum, T(λ) can be well approximated by a Taylor series up to the third order. Since the first- and the third-order derivatives of T(λ) are canceled when averaged over even functions S(λ) and S(λ)V0(λ), the normalized visibility V/V0 can be approximated as
where and T″ is the second derivative of T with respect to λ. Equation (2) is a good approximation of V/V0 even when the contribution of Compton scattering is not negligible.Fig. 4 (a) Simple models of intensity S(λ) averaged over a fringe scanning and the visibility V0(λ) of the intensity oscillation during the fringe scanning with no sample present (λ : X-ray wavelength; both S(λ) and V0(λ) can be given for a pixel), and (b) T″(λ0)/T(λ0) as a function of ρtC.
The parameter
in Eq. (2), which is normally positive, is determined by the configuration of the interferometer and independent of the presence of the sample. When the width of S(λ) is sufficiently narrower than V0(λ), V0(λ) in the integral in Eq. (5) can be regarded as a constant. As a result, Cvar = 0 and the normalized visibility is not reduced when the thickness of the sample increases. On the other hand, when the width of S(λ) is comparable to that of V0(λ), the beam-hardening effect on the normalized visibility can occur.It is well-known that the contribution of photoelectric absorption to the mass absorption coefficient is roughly approximated by Cλ3 (C > 0) in a range of X-ray wavelength in which the contributions of both absorption edges and Compton scattering are negligible. Transmittance T and factor T″/T in Eq. (2) can therefore be approximated by
where ρ and t are respectively the density and the thickness of the sample. From Eqs. (6) and (7), it is deduced that T and T″/T are determined by factor ρtC.Similarly to Eq. (2), average transmittance I/I0 of the sample over the whole range of the spectrum is approximated by
Hence, both I/I0 and V/V0 are functions of a universal parameter ρtC; accordingly, the I/I0–V/V0 curves for various materials are on the same universal curve.
3.2. Critical transmittance and critical thickness
A criterion that determines whether or not the beam hardening affects normalized visibility is presented as follows. First, it is assumed that effects of absorption edges and Compton scattering are negligible. (Note that this assumption is later relaxed for the case where the contribution of Compton scattering is not negligible.)
T″ (λ0)/T(λ0) given by Eq. (7) is plotted as a function of ρtC in Fig. 4(b). Since T″/T is quadric with respect to ρtC, it takes a value of zero at two points (ρtC = 0 and 2/(3λ03)), corresponding to inflection points of T(λ), and a minimum value of −1/λ02 at ρtC = 1/(3λ03). In fact, the effects of beam hardening on transmittance and the normalized visibility are not remarkable when 0 ≤ ρtC ≤ 2/(3λ03). This fact can be confirmed by considering the maximum absolute value of in Eqs. (2) and (8), which is less than 1/8 in the range of ρtC even when is λ0/2. On the other hand, when ρtC > 2/(3λ03), T″/T increases significantly with increasing ρtC, thereby reducing V/V0 in Eq. (2). It thus follows that a solution of
in other words, ρtC = 2/(3λ03) provides a good criterion, above which the effects of beam hardening become remarkable.From Eqs. (6) and (8), when ρtC = 2/(3λ03),
That is, the beam-hardening effects become remarkable when the transmittance of the sample is less than this critical transmittance. This fact is supported in reference to Fig. 3, where V/V0 for silicon, gold, and copper start to decrease at I/I0 ≈ 0.5. It should be noted that the critical transmittance is around 0.5 even when the mass absorption coefficient is more generally given by Cλn (3 ≤ n ≤ 4).
It might therefore be natural to define a critical thickness tc, above which the beam-hardening effects become remarkable, as a thickness giving a transmittance of 0.5 at λ0:
Critical thicknesses for several materials for λ0 = 0.5Å are listed in Table 1.
Table 1. Critical thicknesses for several materials for λ0 = 0.5Å.
Equation (9) is a good criterion for predicting the beam-hardening effects even when Compton scattering reduces T. In the table, the critical thickness tc for PMMA for λ0 = 0.5Å is also given. Here, the dependence of T on λ for PMMA was fitted by a polynomial, from which the second derivative T″ with respect to λ (and tc) was obtained. The critical transmittance for PMMA was obtained from tc as 0.27, which is much smaller than 0.5 and consistent with the result for PMMA shown in Fig. 3.
3.3. Beam hardening correction
Equations (2) and (8) indicate that both V/V0 and (I/I0) · (1/T(λ0)) are determined only by T″ (λ0)/T(λ0) if the configuration of the interferometer is given. As a result, it is expected that V/V0 and (I/I0) · (1/T(λ0)) also exhibit a universality. Here, we call (I/I0) · (1/T(λ0)) “normalized I/I0” for convenience. An example of a V/V0-normalized I/I0 graph obtained from the calculation results shown in Fig. 3 is shown in Fig. 5. As expected, the curves for silicon, gold, copper, and PMMA lie on a universal curve. Here, λ0 is set to 0.52 Å so that it gives I/I0 = e−2/3 for silicon, but, in fact, this universality is kept when λ0 is set to a value around the center of the continuous spectrum: the shape of this universal curve depends on λ0, but the calculated curves for the materials are on a curve once λ0 is given. The universal curve shown here is more general than that shown in Fig. 3 because it applies even to the curve for PMMA, for which the contribution of Compton scattering is not negligible. It should be noted that this universality cannot be applied to materials for which effects of absorption edges are not negligible. In the figure, V/V0-normalized I/I0 curve for silver is also given as such a non-applicable example.
Fig. 5 V/V0-normalized I/I0 graph for the results of the calculations shown in Fig. 3 for silicon, gold, copper, PMMA, and silver.
The universal relation between V/V0 and normalized I/I0 makes it possible to correct for the effect of beam hardening on V/V0 independently of the kind of materials. This means that reduction in normalized visibility can be discriminated from those reductions due to unresolvable random microstructures and an unresolvable sharp edge. In Fig. 6, the experimentally obtained thickness dependence of normalized visibility for silicon (black filled circles) above the critical thickness is corrected by using the universal curve shown in Fig. 5: here, the experimentally obtained normalized visibility for silicon was further normalized by the normalized visibility obtained from the experimentally obtained normalized I/I0 using the universal curve. The corrected normalized visibility has values of almost unity even for a thickness of 5 mm (orange filled circles). Thus, we can retrieve V/V0 without the beam-hardening effect from normalized I/I0 once the universal relation is obtained.
Fig. 6 Experimentally obtained thickness dependence of normalized visibility for silicon (black filled circles) and that after beam-hardening correction by using the universal curve in Fig. 5 (orange filled circles).
4. Conclusion
Reduction in normalized visibility in X-ray grating interferometry was observed even in the case of a uniform sample with flat surfaces that has no unresolvable random microstructures (Fig. 3(c)). This reduction in normalized visibility was reproduced by a numerical calculation using an experimentally obtained continuous spectrum for the X-ray source used. It is concluded from this result that normalized visibility is reduced by beam hardening.
It was found that V/V0-I/I0 (Fig. 3) and V/V0-normalized I/I0 (Fig. 5) graphs exhibit universal curves. The latter (V/V0-normalized I/I0) is more general, namely, it applies even in the case that the contribution of Compton scattering is not negligible; however, neither of the two universal curves applies to a sample for which the effects of absorption edges are not negligible. From a simple modeling, a criterion to determine whether the beam-hardening effect occurs or not was derived, and critical transmittance and critical thickness (Table 1), respectively, below and above which beam hardening becomes remarkable, were defined. When the effects of both Compton scattering and absorption edges are negligible, critical transmittance is given as 0.5. When the contribution of Compton scattering is larger, the critical transmittance is smaller.
The universal relation between V/V0 and normalized I/I0 makes it possible to correct for the effect of beam hardening on V/V0 independently of the kind of materials; in other words, it makes it possible to retrieve normalized visibility without the beam-hardening effect from experimentally obtained I/I0 (Fig. 6). It is thus possible to discriminate reduction in normalized visibility from those reductions due to unresolvable random microstructures and an unresolvable sharp edge.
Acknowledgments
This research was in part supported by the Nano-Macro Materials, Devices and System Research Alliance Project from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), and Grants-in-Aid for Scientific Research (B) ( 15H03590) and Challenging Exploratory Research ( 26600137) from Japan Society for the Promotion of Science (JSPS).
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