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The influence of different physical parameters, such as the source size and the energy spectrum, on the functional capability of a grating interferometer applied for phase-contrast imaging is discussed using numerical simulations based on Fresnel diffraction theory. The presented simulation results explain why the interferometer could be well combined with polychromatic laboratory x-ray sources in recent experiments. Furthermore, it is shown that the distance between the two gratings of the interferometer is not in general limited by the width of the photon energy spectrum. This implies that interferometers that give a further improved image quality for phase measurements can be designed, because the primary measurement signal for phase measurements can be increased by enlargement of this distance. Finally, the mathematical background and practical instructions for the quantitative evaluation of measurement data acquired with a polychromatic x-ray source are given.

X-ray imaging is an important technique for medical and technical applications. Normally, the absorption contrast is used for imaging. However, for imaging of weakly absorbing or thin structures, the applicability of this method is limited. Phase-sensitive x-ray imaging can overcome these shortcomings because it has the potential for a significantly increased contrast (Fitzgerald, 2000; Momose, 2003, 2005). For phase-contrast imaging, propagation-based methods are often applied at synchrotron x-ray sources (Snigirev et al., 1995; Nugent et al., 1996; Cloetens et al., 1999; Peele et al., 2005), but can also be well combined with commercial or custom-made laboratory x-ray sources (Wilkins et al., 1996; Mayo et al., 2003) in order to determine the phase shift of x rays transmitted through matter. Using recently developed grating-based methods (David et al., 2002; Momose, 2003; Weitkamp et al., 2005a, b, 2006; Pfeiffer et al., 2006, 2007b), the differential phase contrast (first derivative of the phase shift) is measured. This is an advantage in comparison to propagation-based methods, where the data contain essentially the second derivative of the phase shift.

Differential x-ray phase-contrast imaging using a grating interferometer was combined with a magnifying cone beam geometry using a conventional high-brilliance microfocus x-ray source. By contrast to propagation-based methods and previous grating-based methods, this offers the possibility to apply an efficient low-resolution detector for phase-contrast imaging with high-spatial resolution. It was shown experimentally how the actual measurement values depend on the magnification factor (Engelhardt et al., 2007a). The potential of the method was demonstrated by a phase tomography of an insect (Engelhardt et al., 2007a; Pfeiffer et al., 2007a). Furthermore, quality validation measurements of refractive x-ray lenses were performed (Engelhardt et al., 2007b).

Grating-based interferometer

With grating-based methods, the beam deflection angle, induced by the differential phase shift, is determined. In Fig. 1, the principle of the grating interferometer used for differential phase imaging is outlined. The phase grating g_{1} consists of lines that cause a phase shift of π and a negligible absorption. It serves as a beam splitter and essentially divides the incident beam into the +1st and –1st diffraction orders. As described by the fractional Talbot effect (Guigay et al., 2004; Pfeiffer et al., 2005), these diffracted beams interfere and form a periodic interference pattern in planes parallel to g_{1}. For a parallel beam set-up, the interference pattern has a maximum modulation at distances d_{m} of (Suleski, 1997)

(1)

where λ is the wavelength, p_{1} is the period of g_{1} and m is an odd integer, corresponding to the order of the fractional Talbot distance. For a cone beam set-up, these distances re-scale to d_{m}* (Cowley, 1995a):

(2)

where l is the distance between source and g_{1}. For a parallel beam set-up, the interference pattern has a lateral period of p_{2} (Suleski, 1997):

(3)

and for a cone beam set-up of p_{2}*:

(4)

where d is the inter-grating distance (in general, for all beam geometries). If d equals a fractional Talbot distance d_{m}*, for a cone beam geometry, Eq. (4) can be re-written using Eq. (2) to (Weitkamp et al., 2005a)

(5)

As shown in Fig. 1(b), perturbations of the incident beam due to an object in the beam path yield a change Δα_{PG} of the first-order diffraction angles at the plane of the phase grating g_{1} of (Pfeiffer et al., 2006):

(6)

where ∂φ_{PG}/∂x_{PG} is the differential phase shift at g_{1} due to the object in the beam path. This causes a shift s of the interference pattern of

(7)

The shift of the interference pattern can be expressed in the radian measure by

(8)

Using Eqs (6), (7) and (8), the differential phase shift ∂φ_{PG}/∂x_{PG} follows from the shift ϕ of the interference pattern by

(9)

If the inter-grating distance d equals a fractional Talbot distance d_{m}*, for a cone beam geometry, Eq. (9) can be re-written using Eqs (2) and (5) to

(10)

To determine the position of the interference pattern, it is scanned with the analyzer grating g_{2}. This grating consists of highly absorbing lines with a period that matches p_{2}*. While the grating lines move over the interference pattern, an oscillation of the offset corrected grey values Γ can be observed for each pixel. A series of images is acquired at different relative lateral positions of the gratings, when the object is in the beam path. This procedure is repeated without the object. Thus, for each pixel, two series of grey values Γ_{data} and Γ_{flat} are obtained (cf. Fig. 2). The intensity of a pixel averaged over a period of the pattern is proportional to the beam intensity incident on the interferometer. Thus, the transmission τ_{meas} through the object is calculated by

(11)

where χ is the position of the analyzer grating g_{2}. The average value over an arbitrary number of complete periods p_{2}* of the pattern is denoted by . From the shift of the oscillations of the measured grey values, which corresponds to the shift ϕ of the interference pattern, the differential phase shift ∂φ_{PG}/∂x_{PG} can be determined using Eqs (9) and (10). The total phase shift can be derived from the differential phase shift by integration.

The evaluation of the measurement data can be accomplished using a discrete Fourier transform. In this case, the shift of the interference pattern ϕ is determined by

(12)

where is the Fourier component at the spatial frequency ν_{p}= 1/p_{2}* of the interference pattern and arg is the phase angle of a complex number.

For an extended source, the source size ξ in the direction perpendicular to the grating lines must be chosen in a way that the transversal coherence length Λ_{c}=λ·l/ξ at the plane of g_{1} is much larger than the separation of the interfering beams of h_{m}=m·p_{1}/4. Only in this case, the interference pattern exhibits a reasonable modulation.

So far, the discussion has been conducted for monochromatic radiation. However, the method described earlier can be applied to a polychromatic energy spectrum as well, because the interference pattern has a high modulation for a relatively wide energy interval (Weitkamp et al., 2005a). A simple method to quantitatively evaluate a measurement with a polychromatic energy spectrum is to assign an effective energy E_{eff} (Pfeiffer et al., 2006). In this paper, this approach is justified numerically. Previously, E_{eff} was often estimated from transmission data (Pfeiffer et al., 2006; Engelhardt et al., 2007a).

Simulation principle

The complex wave function u(x, y) and thereby the intensity distribution of the interference pattern downstream of g_{1} were determined in different planes parallel to g_{1}. The wave function u(x, y) can be propagated along the z axis from the plane L_{1} to the next plane L_{2} with the Rayleigh Sommerfeld integral (Goodman, 1968a):

(13)

where k= 2π/λ is the wave number, u_{1} (x_{1}, y_{1}) is the wave function at the position (x_{1}, y_{1}) in L_{1}, u_{2} (x_{2}, y_{2}) is the wave function at the position (x_{2}, y_{2}) in L_{2}, is the normal vector of L_{1}, is a vector that connects (x_{1}, y_{1}) and (x_{2}, y_{2}) and . Thereby, it is assumed that the wave function u_{1} is undisturbed within a certain area (e.g. an aperture) and that outside this area (e.g. an aperture plate) u_{1}= 0.

Based on the assumptions that the distance D between the planes L_{1} and L_{2} is much larger than the non-zero part (aperture) of u_{1} and much larger than the part of u_{2}, which is of interest, the following approximations are made in order to further simplify Eq. (13) (for details, cf. Goodman, 1968a):

1: The accuracy of this approximation is within 5%, if the angle is smaller than 18°.

2Fresnel diffraction: The distance r is approximated to second order by , where D is the distance between the planes L_{1} and L_{2}. A sufficient condition for the validity of this equation is . This condition may be violated for small propagation distances of the interference patterns presented in this context. However, the Fresnel approximation often holds valid, if this condition is infringed, because in this case the major contribution to the integral will arise from a zone near the point (x_{1}=x_{2}, y_{1}=y_{2}) according to the stationary phase principle (Goodman, 1968a; Born & Wolf, 2002).

Under these assumptions, Eq. (13) can be reduced to a convolution operation (Goodman, 1968a):

(14)

where ⊗ denotes the convolution operation.

Thin objects can be reduced to layers and characterized by their transmission function T(x, y):

(15)

(16)

where T_{1} (x, y) is the transmission function of the object in L_{1}, u_{incident,1} is the incident wave function in L_{1}, u_{exit,1} is the wave function exiting L_{1} after interaction with the object, ΔD is the thickness of the object and n= 1 −δ+iβ is the complex refractive index of the object. Objects characterized by Eqs (15) and (16) are reduced to layers. This implies that interference effects within the object are neglected. For the simulations presented, Eqs (15) and (16) are used for simulation of the phase grating g_{1} and the absorption of the analyzer grating g_{2}. The sample itself is described in section, ‘Quantitative evaluation of measurement data’, as a uniform phase gradient ∂φ_{PG}/∂x_{PG}, which occurs in the same plane as the phase grating g_{1} and is sufficiently small in order that the fringe shift remains lower than π for the major part of the dominant energies of the photon energy spectrum. Effects related to more complex structures of the sample, especially crystalline media and strong space-variations or strong anisotropies of the refractive index and associated propagation effects, are not considered in the present work. As outlined in Engelhardt et al. (2007a), if the sample is placed at a certain distance from the phase grating g_{1}, ∂φ_{PG}/∂x_{PG} and ϕ are reduced.

The simulation starts at a point source. The wave field in the next plane behind the point source can be calculated efficiently using Eq. (13), because the integration over the source plane may be reduced to a single term. The wave field in the subsequent planes results from Eq. (14), which can be implemented efficiently with a fast Fourier transform (FFT). To minimize rounding errors, the number of planes is kept as small as possible. Therefore, all planes of the spatially resolved intensity distribution behind g_{1} are calculated by one propagation step from g_{1}.

The x-ray intensity distribution I(x, y, z) is linked to the wave function u(x, y, z) by

(17)

The spatially resolved intensity distribution of the interference patterns behind the phase grating g_{1} is – apart from a certain stretch due to the divergent beam as described by Eqs (2), (4) and (5)– periodic in directions perpendicular to the grating lines. Thus, the validity of the Fresnel approximation for small propagation distances can be verified by comparing the interference pattern for small propagation distances to that for large propagation distances.

An extended or polychromatic source can be simulated with many monochromatic point sources, which reproduce the spectral and spatial intensity distribution of the source. The intensity distributions I(x, y) of the interference patterns of these monochromatic point sources are summed up incoherently, giving the intensity distribution for an extended polychromatic source (Cowley, 1995b). In the following, the grey value measured with the detector is determined quantitatively for an extended or polychromatic x-ray source.

The photon energy spectrum w(E) of an x-ray tube can be defined as the probability density function of the energy of an emitted photon. Spectra w(E) have been tabulated for various x-ray tubes (cf. e.g. Ankerhold, 2000). We define an effective spectrum w_{eff}(E), which also comprises the probability that the photon is detected by the scintillator. Furthermore, the x-ray attenuation of objects in the beam path, which give the same spectral shift (beam hardening) for all detector pixels under consideration and no significant diffraction or refraction effects, can also be included in w_{eff}(E), which is calculated by

(18)

where τ_{bh}(E) ∈[ 0;1] is the x-ray transmission of objects in the beam path, which are to be considered in the spectrum, and γ_{sci}(E) ∈[ 0;1] describes the scintillator detection efficiency. In this paper, γ_{sci}(E) is approximated using the photo absorption of the scintillator material. For an energy-integrating detector, it is assumed that the grey value Γ follows the relationship:

(19)

where I_{norm}(E, x, y) is the intensity distribution of the interference pattern at the detector plane for a point source of energy E with normalized intensity.

For an extended polychromatic source, the grey value Γ_{ext} measured with the detector can be calculated from the grey value Γ_{point} obtained for a point source with the same photon energy spectrum by (in analogy to Weitkamp et al., 2006)

(20)

where S_{spatial}(x, y) is the spatial source intensity distribution, l is the distance between the source and the phase grating g_{1,} and d is the inter-grating distance.

Simulations

In the following, the intensity distribution downstream of a phase grating g_{1} with a duty cycle of 0.5 (line width to period ratio of the grating) is discussed. This is done in several steps: First, the intensity distributions for monochromatic point sources with different energies are investigated. Then, the interference patterns for a polychromatic energy spectrum and for an extended source are studied.

The set-up assumed for the simulations resembles the set-up applied for the measurements presented in Engelhardt et al. (2007b).

The interferometer has the same design energy of E_{0}= 17.5 keV. At this energy, the silicon phase grating g_{1} gives a phase shift of π. Furthermore, the fractional Talbot distances are calculated for E_{0}. The photon energy spectrum used for the simulations as well as the source size (section Interference pattern for a polychromatic point source and an extended polychromatic source) resemble the parameters of the measurements. It can be assumed that the detector applied is energy integrating.

In the measurements, the distance l+d between the source and the analyzer grating g_{2} and the period p_{2}*= 2 μm of the analyzer grating g_{2} were kept constant. Experimentally, two different inter-grating distances were investigated by variation of the distance l between the source and the phase grating g_{1} and the period p_{1} (cf. Eqs (4) and (5)) of g_{1}. According to Eqs (1–5), the dimensions of the interference pattern change, if l or p_{1} is varied.

For the simulations, the distance l= 1689 mm between the source and g_{1} and the period p_{1}= 4 μm of g_{1} were kept constant, whereas the position and the period p_{2}* of the analyzer grating g_{2} (cf. Eqs (4) and (5)) were varied for the discussion of different inter-grating distances d. This was done in order to investigate different inter-grating distances d based on a single image of the interference pattern. These deviations of the dimensions of the simulated set-up from the real set-up only have a small effect on the dimensions of the interference pattern according to Eqs (1–5), but no influence on the effects observed.

Interference pattern for a monochromatic point source

For a monochromatic point source with an energy E that matches the grating design energy E_{0}, the interference pattern behind a Si-phase grating is shown in Fig. 3. Apparent from this figure, at fractional Talbot distances of odd order m, rectangular intensity oscillations with a ratio of the widths of the zero part and the non-zero part of 1:1 can be observed. The lateral period p_{2}* of the pattern is given by Eqs (4) and (5). At fractional Talbot distances of even order m, the intensity distribution is flat.

The interference pattern for x rays of E= 23.3 keV photon energy behind a phase grating designed for E_{0}= 17.5 keV is shown in Fig. 4. For a monochromatic point source with an energy E that differs from the design energy E_{0} of the interferometer, three effects can be observed:

1The pattern is stretched (cf. Fig. 4 (a)) in the propagation direction z because the fractional Talbot distances depend on the photon energy according to Eqs (1) and (2).

2The pattern is distorted (cf. Figs 4(a) & (b)). This is due to the fact that the phase shift caused by the lines of the phase grating designed for E_{0}= 17.5 keV x rays is only approximately 0.75 π for 23.3 keV x rays.

3Apart from a longitudinal and transversal stretch (cf. Eqs (2), (4) and (5)), due to the cone beam geometry and a lateral shift of p_{2}*, the pattern is periodic in distances d_{m}* (calculated for the energy E of the incident photons) with , where is a natural number. This is the (non-fractional) Talbot effect (Guigay et al., 2004).

The pattern exhibits a strong contrast, even at fractional Talbot distances calculated for E_{0}= 17.5 keV (cf. Fig. 4(c)). To characterize the contrast of the interference pattern quantitatively, the scanning process with the analyzer grating is simulated and the visibility υ is determined, which is defined in analogy to Cowley (1995c) by

(21)

where Γ_{max} is the maximum value and Γ_{min} the minimum value of the detector grey-value oscillations during the scanning process. For an interferometer laid out for E_{0}= 17.5 keV, the visibility, as a function of the incident photon energy of a monochromatic point source, is displayed in Fig. 5(a–c) by the green graph for the 1st, 3rd and 15th fractional Talbot distance, respectively. An ideal analyzer grating with a duty cycle of 0.5 and lines that attenuate the incident x rays by 100% was assumed. The incident photon energies E range from 4.375 to 50 keV. At E=E_{0}= 17.5 keV, the visibility is almost unity, as expected. The graphs show that the interference fringes exhibit a high contrast for a wide range of incident photon energies E around E_{0}, except for certain zero points and the vicinity thereof. The curves are determined by the effects, which were discussed in the current section for the interference pattern in Fig. 4:

1If the incident photon energy E is varied, the interference pattern is stretched in propagation direction z. As the analyzer grating is placed at a fixed position, the interference pattern will normally not be scanned at the odd fractional Talbot distances for the incident photon energy. This yields a decrease of the fringe visibility, which depends on the incident photon energy E. For example, for an inter-grating distance of 89 mm, corresponding to the third fractional Talbot distance for E_{0}= 17.5 keV, a visibility close to zero is observed for a photon energy of E= 26.25 keV (cf. Fig. 5(b), green graph), since for this energy, the second fractional Talbot distance is at 89 mm. With increasing fractional Talbot distance order m, the intervals between the zero points of the graphs get smaller, because a certain stretch of the pattern in the propagation direction yields a larger shift of the pattern at higher distances from the phase grating.

2If the energy of the incident photons E differs from the interferometer design energy E_{0}, the interference pattern is distorted, and thus the visibility is decreased. For example, at all fractional Talbot distances, a visibility close to zero is observed, if the phase shift due to the phase grating lines is , where is a natural number or zero. This is the case for photon energies of approximately 4.375 keV, 8.75 keV and for E→∞(cf. Fig. 5(a–c), green graphs). If the phase shift due to the phase grating lines equals , and the pattern is scanned at a fractional Talbot distance of odd order for the photon energy E, the visibility is almost unity, for example for photon energies E of approximately 5.83 keV and 17.5 keV (cf. Fig. 5(a–c), green graphs).

These two effects are superimposed. For an ideal analyzer grating with lines of 100% x-ray attenuation that is shifted in a way that it remains at a fractional Talbot distance of odd order for the energy E of the incident photons the visibility as a function of E is displayed in Fig. 5(a–c) by the black curve. It forms an envelope for the visibility determined for the analyzer grating placed at a fixed position (Fig. 5(a–c), green graphs), showing that the visibility is maximal at the fractional Talbot distances of odd order.

In particular for low x-ray energies, the calculated graphs of the visibility do not fall to zero and do not reach unity. This is due to the fact that the visibility is calculated for discrete energies only and – in particular for low energies, for example for E≈ 4.375 keV – due to the fact that the lines of the simulated Si phase grating show a certain x-ray attenuation yielding a distortion of the pattern.

So far, the discussion has been conducted for an ideal analyzer grating with lines that attenuate the incident x rays by 100%. As shown in Fig. 5(a–c) by the brown graphs, for an analyzer grating with lines of 24 μm gold, the fringe visibility is lowered, if the x-ray absorption 1 −τ_{line} (Fig. 5(d)) of the analyzer grating lines is less than 100%.

An analytical description of this effect can be obtained by considering the two borderline cases in the scanning process of the interference pattern with the analyzer grating g_{2}: (i) The analyzer grating positions χ_{min} for which the absorbing lines are positioned at the maxima of the interference pattern and the minima Γ_{min} of the grey-value oscillations of a detector pixel are observed and (ii) the analyzer grating positions χ_{max} for which the absorbing lines are positioned at the minima of the interference pattern and the maxima Γ_{max} of the grey-value oscillations of a detector pixel are observed. It is assumed that the analyzer grating has a duty cycle of 0.5 and that the analyzer grating positions χ_{min} are at the centre in between the analyzer grating positions χ_{max}. In this case, the lines of an analyzer grating positioned at χ_{min} are exactly at the same position as the gaps of an analyzer grating positioned at χ_{max}. Furthermore, they have the same width. Therefore, x rays absorbed by the lines of the analyzer grating positioned at χ_{min} are transmitted through the gaps of the analyzer grating positioned at χ_{max}. Based on these considerations, the maximum Γ_{max}(τ_{line}) and the minimum value Γ_{min}(τ_{line}) of the detector grey-value oscillations during the scanning process for an analyzer grating with lines showing an x-ray transmission of τ_{line} are calculated by

(22)

(23)

where Γ_{max}(τ_{line}= 0) and Γ_{min}(τ_{line}= 0) are the corresponding detector grey values during the scanning process for an ideal analyzer grating. Using Eqs (22) and (23), the following relation can be derived from Eq. (21):

(24)

where υ (τ_{line}) is the visibility obtained for an analyzer grating with lines showing an x-ray transmission of τ_{line} and υ (τ_{line}= 0) for an ideal analyzer grating. The behaviour displayed in Fig. 5 is predicted well by Eq. (24): The maximum deviation between the visibility υ (τ_{line}) calculated by Eq. (24) and the simulated data is 0.008.

Interference pattern for a polychromatic point source and an extended polychromatic source

By incoherent superposition of the interference patterns of several monoenergetic x-ray sources, according to Eq. (19), the interference pattern for a polychromatic point source was simulated. From the data, the interference pattern for an extended polychromatic source was derived using Eq. (20).

A tabulated photon energy spectrum of a W target irradiated by 50 keV electrons was used (Ankerhold, 2000). To calculate the effective energy spectrum w_{eff}(E) according to Eq. (18), furthermore, the scintillator detection efficiency (Hubbell & Seltzer, 1995) and beam hardening (Hubbell & Seltzer, 1995) were taken into account (cf. Fig. 6). The effective photon energy spectrum w_{eff}(E) attains its maximum value at approximately 24 keV and its half-maximum values at 16 keV and 40 keV, respectively. The intensity distribution of the source assumed is a Gaussian distribution with a full width half maximum (FWHM) size of ξ_{FWHM}= 7.5 μm, which corresponds to a source size of 8 μm, if measured according to DIN EN 12543–5 under idealized conditions. This value matches the manufacturer's data for the x-ray source applied in Engelhardt et al. (2007b).

The resulting interference pattern for a point source is displayed in Fig. 7(a) and for the extended source in Fig. 7(b). As the images were calculated according to Eq. (19) with an effective spectrum that comprises the detector efficiency and beam hardening, they show the interference pattern in the form of the grey values that the x radiation would cause for the detector applied.

Figure 7 shows that, for a realistic polychromatic x-ray spectrum, the intensity fluctuations in propagation direction become smooth. As the fractional Talbot distances depend on the photon energy E (cf. Eqs (1) and (2)), for a polychromatic energy spectrum, several interference patterns with different fractional Talbot distances are superimposed and the intensity fluctuations in propagation direction are smeared out. Hence, for a wide polychromatic x-ray spectrum, inter-grating distances of ideal contrast are not limited to the fractional Talbot distances for the interferometer design energy E_{0}.

Furthermore, Fig. 7 demonstrates that, for a realistic polychromatic x-ray spectrum, the interference pattern maintains a high lateral fringe contrast. This means that current interferometers can be well combined with a realistic polychromatic x-ray spectrum, at least if the phase gradient induced by the sample is moderate (giving a low fringe shift) and varies smoothly with position (complex structures of the object and associated propagation effects were not considered).

Finally, Fig. 7(b) shows that for an extended source, the interference fringes become more and more blurred as the distance from the phase grating increases.

For a quantitative characterization of the fringe contrast of the interference patterns displayed in Fig. 7, the scanning process with the analyzer grating was simulated and the visibility υ_{poly} of the obtained grey-value oscillations was determined using Eq. (21). It is displayed in Fig. 8 as a function of the distance d from the phase grating.

To explain the results, in the following, the visibility υ_{poly} for the polychromatic energy spectrum w_{eff}(E) is calculated from the visibility υ (E) for monochromatic x-ray sources. It is assumed that the maximum Γ_{max} and the minimum Γ_{min} of the grey-value oscillations of a detector pixel for a polychromatic x-ray source can be calculated in analogy to Eq. (19) from the corresponding maxima I_{max}(E) and minima I_{min}(E) determined for monochromatic x-ray sources. This implies that the analyzer grating positions for which I_{max}(E) and I_{min}(E) are observed do not depend on the photon energy E. This requires that the contrast of the interference pattern does not invert, like it does for the Talbot effect. For the Talbot effect, the pattern is shifted by half of its lateral period when it repeats itself. However, for the application of the fractional Talbot effect outlined in this paper, contrast inversion was not observed (cf. Figs 3 & 4).

Based on these assumptions, using Eqs (19), (21), (22) and (23) and the corresponding assumptions, the following basic relationship can be derived for the visibility:

(25)

where υ_{poly} is the visibility for the polychromatic energy spectrum w_{eff}(E) and υ (E) is the visibility for the photon energy E. The results displayed in Fig. 8 are predicted well by Eq. (25): The visibility υ_{poly} for polychromatic radiation, calculated with Eq. (25) from the data υ (E) displayed in Fig. 5, deviates from the values displayed in Fig. 8 by not more than 0.009. The high accuracy of the results obtained using Eq. (25) indicates that the assumptions made for the derivation of Eq. (25) are valid for dominant energies of the photon energy spectrum.

By Eq. (25), the high lateral fringe contrast for a polychromatic x-ray source (cf. Figs 7 & 8) is shown to be a direct consequence of the high lateral fringe contrast obtained in the monochromatic case.

Apparent from the green and brown graphs in Fig. 8, for the polychromatic point source, the fringe visibility υ does not drop for large inter-grating distances. This is due to the fact that the visibility υ_{poly} for a polychromatic radiation can be calculated based on an integration of the visibility υ (E) for monochromatic x rays over the photon energy spectrum, since no contrast inversion was observed for the application of the fractional Talbot effect outlined in this paper. Even though υ (E) oscillates faster within its black envelope when the inter-grating distance d is increased (cf. Fig. 5(a–c)), the integral of υ (E) over a wide energy interval is not significantly lowered with increasing inter-grating distance.

Hence, no general limitation of the acceptable bandwidth ΔE/E by the inter-grating distance was observed. This finding differs from the predictions of Weitkamp et al. (2005a) and implies that current interferometers have a considerable potential for improvement: For current interferometers, the fringe shift ϕ, which is the primary measurement signal, is low, with typical values of much less than 0.1 π (Engelhardt et al., 2007a), giving a moderate signal-to-noise ratio. As shown in Engelhardt et al. (2007a), the fringe shift can be increased by selecting a larger inter-grating distance d. The results in this paper clearly demonstrate that this approach is limited by the source size, but not in general by the width of the energy spectrum, making interferometers with further improved image quality for phase measurements possible.

However, the maximum obtainable fringe shift can be limited, because, for a high fringe shift, the interference pattern can become blurred as the x-ray refraction caused by the object and thus the fringe shift depend on the energy. Thus, for a very high fringe shift, the contrast of the interference pattern is lowered, if a polychromatic x-ray source with a wide photon energy spectrum is used. Furthermore, a fringe shift of 2 ·π+ϕ cannot be distinguished from a fringe shift of ϕ.

The fact that for an extended source, the fringe visibility decreases as the inter-grating distance is increased is characterized quantitatively by the grey and purple graphs in Fig. 8. Assuming a sinusoidal interference pattern, the decrease of the visibility due to the extended source of Gaussian profile follows the relation (Weitkamp et al., 2006):

(26)

where υ_{point} is the visibility for a point source, υ_{ext} is the visibility for the extended source of Gaussian profile and ξ_{FWHM} is the full width half maximum size (FWHM) thereof. The results displayed in Fig. 8 are predicted well by Eq. (26), with a maximum deviation of the visibility for the extended source calculated using Eq. (26) from the values determined by simulation smaller than 0.008.

Eq. (26) shows that the acceptable source size is lowered as the inter-grating distance d is increased. However, the limited acceptable source size can be allowed for. For example, for the set-up assumed in the simulation study, a source size of ξ_{FWHM}= 12.6 μm and ξ_{FWHM}= 2.5 μm is required in order to achieve υ_{ext}= 0.7 ·υ_{point} for the 3rd and 15th fractional Talbot distance, respectively. These are realistic source sizes for commercial microfocus x-ray sources.

By simulation, for an analyzer grating with lines of 24 μm gold, a fringe visibility of υ= 0.24 and of υ= 0.14 was determined for the 1st and the 3rd fractional Talbot distance (calculated for E_{0}= 17.5 keV photon energy), respectively. This is in good agreement with the measured values of υ≈ 0.15 and υ≈ 0.25 for the set-up used in Engelhardt et al. (2007b).

Quantitative evaluation of measurement data

The simulations give clear evidence that a wide energy interval contributes to the measurement of the differential phase shift. This is normally accounted for by assigning an effective energy E_{eff} (Pfeiffer et al., 2006) for the evaluation of measurement data by Eqs (9), (10) and (12). In the following, this approach is justified mathematically by derivation of a correlation between the fringe shift ϕ_{poly} for a polychromatic x-ray spectrum w_{eff}(E) and the fringe shift ϕ (E_{0}) for the grating design energy E_{0}. In analogy to Eq. (19), the grey values Γ_{data} in Eq. (12) can be calculated by

(27)

where I_{norm}(E, χ) is the x-ray intensity incident on the detector pixel under consideration as a function of the grating position χ and the photon energy E of a monochromatic x-ray source with normalized total intensity. Eq. (27) is used to transform Eq. (12) into

(28)

Thereby, the phase offset term in Eq. (12) was neglected, because it will be constant for the following calculations. The linear behaviour of the Fourier transform (Goodman, 1968b)

In the following, the absolute value and the phase of are determined separately. The absolute value

(31)

is calculated based on a simulation of the interference pattern and the scanning process thereof with the analyzer grating g_{2}. The function ϒ (E) describes to which extent the interference fringes are detected, depending on the energy E of the incident photons.

In analogy to evaluation algorithms for propagation-based methods (Wilkins et al., 1998), the beam deflection angle induced by the sample is assumed to be proportional to 1/E^{2}, which is a good assumption in the absence of absorption edges. Therefore, the fringe shift (at fixed analyzer grating z position) is also proportional to 1/E^{2} and the phase of is calculated to be

Equation (33) is the correlation between the fringe shift ϕ (E_{0}) for the grating design energy E_{0} and the fringe shift ϕ_{poly} for the polychromatic energy spectrum, which is sought-after for quantitative data evaluation. Apparent from Eq. (33), the product E·w_{eff}(E) ·ϒ (E) describes to which degree a specific energy interval contributes to the measurement result.

In analogy to the visibility, ϒ (E) and E·w_{eff}(E) ·ϒ (E) depend on the inter-grating distance d.

The size of an extended source will not have any effect on the shape of ϒ (E), as long as the spatial source intensity distribution is independent of the energy. This can be explained by the fact that the blurring of the interference fringes by the extended source corresponds to a convolution thereof with the re-scaled spatial source intensity distribution (cf. Weitkamp et al., 2006 and Eq. (20)). As this convolution corresponds to a point by point multiplication in Fourier space, it yields a multiplicative factor for ϒ (E), which is independent of the energy, as long as the spatial source intensity distribution is independent of the energy.

A real object will not only give a phase shift, but also a certain attenuation depending on the energy. This can either be considered in ϒ (E) or in w_{eff}(E). Therefore, the factor E·w_{eff}(E) ·ϒ (E) will depend on the x-ray absorption 1 −τ (E) of the sample and thus on the position.

ϒ (E) and E·w_{eff}(E) ·ϒ (E) were calculated for incident photon energies ranging from 4.4 to 50 keV for the set-up assumed for the first and third fractional Talbot distance, respectively (cf. Fig. 9), under the assumption of a pure phase object. Based thereon, the fringe shift ϕ_{poly} for the polychromatic energy spectrum w_{eff}(E) was calculated as a function of ϕ (E_{0}). In Fig. 10, the results are illustrated in the form of a correction factor η

(34)

This correction factor can be used for the quantitative evaluation of measurement data acquired with polychromatic x rays.

For many experiments, the fringe shift is low. For example, for the measurements presented in Engelhardt et al. (2007b), the fringe shift has typical values of a few degrees. In this case, η is almost constant and can be determined directly using the following expression:

(35)

which was derived from Eqs (33) and (34) by a power series development of the complex exponent in Eq. (33) under the assumption that ϕ (E_{0}) ·E_{0}^{2}/E^{2} is small.

If η is almost constant, an effective energy E_{eff} can be assigned to a measurement accomplished with polychromatic x rays. E_{eff} is calculated by

(36)

(37)

Thereby, it is again assumed that the beam deflection angle induced by the sample is proportional to 1/E^{2}.

These calculations clearly show that ϕ (E_{0}) can be derived from ϕ_{poly}. For a specific set-up, the correction factor η is ideally determined by a measurement of the differential phase shift of a calibration sample of known shape. The calibration sample must give a similar spectral shift as the sample under investigation. The correction factor η can also be determined based on numerical simulations. However, in this case, all differences between the real set-up and the idealized set-up, assumed for the simulations, may yield an error of the correction factor η determined. The correction factor η comprises the interference fringe shift ϕ (E) weighted with a parameter ϒ (E), which characterizes the detected modulation of the interference pattern. Therefore, an estimation of this factor based on transmission data – which comprise the attenuation coefficient, but not ϕ (E) or ϒ (E)– can only yield a very rough estimate of the correction factor η and the effective energy E_{eff}, respectively.

For the set-up used in Engelhardt et al. (2007b), an effective energy of E_{eff}= 21 keV was determined from experimental data for the third fractional Talbot distance and a small fringe shift ϕ_{poly} based on a measurement of the differential phase shift of a Be paraboloid of known shape, which had a maximum thickness of 1 mm. By simulation, for the similar set-up, E_{eff}= 22.6 keV was derived for a small fringe shift ϕ_{poly} under the assumption of a pure phase object. If the additional beam hardening due to 1 mm Be is considered in the calculations, the effective energy calculated by simulation is increased by 0.054 keV, which is normally negligible. The deviations of the numbers derived by simulation from the numbers determined for the real set-up are addressed to differences between the real set-up and the idealized set-up discussed in the simulation study.

Conclusion

The influence of different physical parameters, such as the source size and the energy spectrum, on the functional capability of a grating interferometer applied for phase-contrast imaging were characterized based on numerical simulations.

First, the interference pattern for a monochromatic point source was investigated for the case that the energy of the incident photons E matches the interferometer design energy E_{0} and for the case that it does not. The results demonstrate that the interference fringes have a high contrast for a wide range of incident photon energies E around E_{0}, except for certain zero points and the vicinity thereof.

Second, the interference patterns for a polychromatic point source and an extended x-ray source were discussed. The results demonstrate that, for a realistic polychromatic x-ray spectrum, the interference pattern maintains a high lateral fringe contrast, whereas the intensity fluctuations in propagation direction become smooth. The high lateral contrast explains why the interferometer could be well combined with polychromatic laboratory x-ray sources in recent experiments (Pfeiffer et al., 2006; Engelhardt et al., 2007a, b; Pfeiffer et al., 2007a). Because of the smoothing of the intensity fluctuations in propagation direction, inter-grating distances of ideal lateral contrast are not restricted to the exact fractional Talbot distances for the interferometer design energy E_{0}. Furthermore, for a realistic interferometer, it was shown that the inter-grating distance d is limited by the source size, but not in general by the width of the energy spectrum and can, therefore, be significantly enlarged. Since the primary measurement signal can be increased by the selection of a larger inter-grating distance d, this makes interferometers with further improved image quality for phase measurements possible. The results for polychromatic x-ray sources were explained by the results for monochromatic x-ray sources discussed in the first part of the paper.

Third, it was shown analytically that grating-based phase measurements accomplished with a polychromatic energy spectrum can be evaluated quantitatively by the introduction of a correction factor η or by assigning an effective energy E_{eff}. This gives the theoretical background and practical instructions for the quantitative evaluation of measurements accomplished with polychromatic x-ray sources.

Acknowledgements

The authors would like to thank Peter Böni and Burkhard Schillinger of Technische Universität München as well as Matthias Goldammer and Lothar Bätz of Siemens AG for fruitful discussions.