1. Introduction

Several X-ray phase-contrast imaging (XPCi) techniques have been developed in the last decades [1–10]. These all share the same general principle: they are based on the conversion of the X-ray phase shifts introduced by the sample into beam intensity variations measurable by a detector. Since phase related effects can be much larger than the attenuation ones exploited by conventional X-ray methods (especially for materials made of low-atomic number elements), XPCi techniques have the potential to provide considerably improved contrast-to-noise ratio and, therefore, to strongly enhance the visualization of sample structures.

The edge illumination (EI) XPCi technique was first developed and applied in the late nineties as a synchrotron method [6], but was later demonstrated to be efficiently applicable also to non-microfocal and polychromatic laboratory sources [7,11–15]. The independence from polychromaticity and the robustness to relatively large source sizes are due to the incoherent nature of the method. EI does not rely on coherent wave effects and, in fact, its main principles can be described by a purely geometrical point of view, as we will see in the following. Thanks to the applicability to conventional X-ray sources, EI bears the potential to be widely used for a large number of applications in several fields of X-ray imaging, potentially including clinical diagnostics. In fact, following early observations on the method’s resilience against vibrations and potential for reduced exposure times [7], more recently the method has been applied to cartilage [12] and breast [13] imaging, showing that high phase sensitivity can be achieved at doses compatible with clinical requirements. It should also be noted that the low aspect ratio of the masks makes them scalable, cost effective and easy to fabricate, and makes the setup resilient to mechanical instabilities [16].

The signal in EI XPCi has been modelled in previous publications using both wave [17,18] and geometrical optics [19,20]. In particular, it was shown in [17] that the signal can be well described by using the approximated geometrical optics when the method is implemented with extended laboratory sources and in the presence of sufficiently slowly varying sample features. Models based on geometrical optics provide certain advantages compared to wave optics ones. Not only are they faster and simpler to apply, but they lend themselves to an easier interpretation and can thus be used to model and analyse the dependencies of the signal upon the various experimental parameters in a more straightforward way.

In this article, we propose a new geometrical optics approach to the modelling of the signal in EI, based on the so-called transport of intensity (TIE) equation [21–27]. We show that this treatment complements the results obtained by previous models, and we use it to clarify the different sources of contrast and analyse their contributions. In particular, the present model allows taking into account the signals provided by both large and small sample features, and studying the dependencies of the signal upon several experimental parameters.

2. The EI principle

The setup employed by the EI technique when implemented with laboratory sources is schematically presented in Fig. 1. Two absorption masks are used, one just upstream the sample (the ‘sample’ mask) and the other in contact with the detector (the ‘detector’ mask). Although EI implementations which can be sensitive to phase effects in two dimensions exist [28], we focus here on the typical EI setup featuring one-dimensional sensitivity, where the mask apertures are extended in one direction (the direction x orthogonal to the plane of drawing in Fig. 1). The first mask produces a series of laminar beamlets (with a size of few to tens of µm), which lead to a structured illumination of the sample. The aim of the second mask is to create insensitive regions between adjacent detector pixels. The two masks are slightly misaligned, so that each beamlet straddles the edge of an aperture in the detector mask (this is the realization of the so-called edge illumination condition, see Fig. 1).

 

Fig. 1 Diagram of the EI experimental setup in its laboratory implementation (not to scale).

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In addition to attenuating the intensity of each beamlet, the sample can also produce beam refraction. This typically happens at the sample interfaces, where the X-ray phase shift is varying more rapidly. The component of the refraction angle along y (direction orthogonal to the mask lines) is equal to $\Delta {\theta }_y\left(x,y;\lambda \right)=\left(\lambda /2\pi \right)\partial \varphi \left(x,y;\lambda \right)/\partial y$, where $\varphi \left(x,y;\lambda \right)=-\left(2\pi /\lambda \right){\displaystyle \int dz\delta \left(x,y,z;\lambda \right)}$ is the phase shift, λ is the wavelength and δ is the refractive index decrement of the sample. Due to the presence of the second mask, the beamlet refraction is converted into a change of intensity on the corresponding detector pixel. Depending on its direction, in fact, refraction can either deviate onto the detector a certain number of photons previously incident on the absorbing part of the mask (therefore increasing the counts on the detector), or the opposite may happen (Fig. 1).

If the object is kept in a fixed position during the acquisition, the object sampling rate is equal to the period of the first mask (typically, tens of µm). However, it is possible to further increase the sampling rate and therefore the spatial resolution by “dithering” the object, i.e. by acquiring multiple images at different sub-period displacements of the object and by combining them into a single, oversampled image [29,30]. In the following theoretical modelling, we will consider for simplicity that only a single slit is present in both masks and that the sample is scanned through it along its entire length. Note that, from the point of view of the signal, this is equivalent to the case of multiple apertures (if the dithering step is the same), so long as no cross-talk between them exists.

We first model the EI signal by using a rigorous wave optics approach. The complex amplitude E_out of the wave exiting the object can be expressed as a function of the wave E_in entering the first mask by:

$q\left(x,y;\lambda \right)=\sqrt{T\left(x,y;\lambda \right)}\exp \left[i\varphi \left(x,y;\lambda \right)\right]$ is the complex transmission function of the object, where $T\left(x,y;\lambda \right)=\exp \left[-{\displaystyle \int dz\mu \left(x,y,z;\lambda \right)}\right]$ and µ is the linear attenuation coefficient. The complex transmission function of a perfect, fully absorbing mask is given by $m\left(y\right)=rec{t}_a\left(y-a/2\right)$, where $rec{t}_a\left(y-a/2\right)$ indicates a rectangular function of width a and centre in a/2, defined as 1 in the range $\left(0,a\right)$ and 0 elsewhere. Useful properties of this rectangular function, which will be extensively used in the following, include $rec{t}_{a/M}\left(y\right)=rec{t}_a\left(My\right)$ and $\partial rec{t}_a\left(y-a/2\right)/\partial y=\delta \left(y\right)-\delta \left(y-a\right)$, where $\delta \left(y-a\right)$ is a Dirac delta function centred in a. The coordinates of the first mask are denoted by $\left(x,y\right)$ and the ones of the sample by $\left(x,y^{\prime }\right)$, where $y^{\prime }=y-p$ and $p=n_y\cdot \Delta y$, with $\Delta y$ the dithering step size and $n_y$ an integer defining the number of the step. The definition of these two different coordinate systems describes the fact that the object is scanned through the aperture, and therefore the region of the object that is sampled in the measurement changes at each dithering step. Note that, in case of multiple apertures and assuming no dithering is performed, the step size $\Delta y$ is equal to the period of the first mask.

The complex amplitude of the wave incident on the detector mask can be described by Fresnel diffraction. In particular, in the case of a spherical X-ray beam emitted by a point source set at a distance z₁ upstream the object, the intensity (i.e. spatial density of photons) incident on the detector aperture is equal to [31]:

In real experimental conditions, however, the X-ray source will have a given finite size, leading to a blurring of the intensity pattern. This can be taken into account by convolving the ideal point-source intensity by the projected source intensity distribution $G\left(\tilde{x},\tilde{y}\right)$. The signal registered by a given detector pixel, at a certain dithering position p, is then given by the total beam intensity incident within the sensitive area defined by the detector aperture, i.e.:

Note that the function K effectively takes into account both the blurring of the intensity incident onto the detector mask due to the projected source size, and the integration within the limits of the detector aperture. Finally, the detector signal obtained with a polychromatic beam can be obtained by summing the monochromatic components given by Eq. (5), weighted by the energy spectrum [32]. In the following, however, we will consider only the case of a monochromatic beam, as generalization of the developed model to the polychromatic case is straightforward. We will also drop, for simplicity of notation, all dependencies from the wavelength, as well as the dependence of S and K upon the edge position $y_e$.

Examples for the function K are reported in Fig. 2, where both cases of point and extended sources are considered. In the first situation, K is equal to the rectangular function describing the detector aperture, while in the latter this is blurred by the projected source distribution. The relevance of this function for the characterization of an EI setup will be clarified in section 3, and its relationship with the angular sensitivity analyzed.

 

Fig. 2 Example profiles for function K, in the case of point and extended sources. Considered parameters of the experimental setup: z₁ = 1.6 m, z₂ = 0.4 m, y_e = 7.5 µm (which corresponds to a 50% illumination for an aperture a = 12 µm), d = 20 µm, extended source of 70 µm full width at half maximum (parameters matching one of the experimental setups installed at University College London (UCL)).

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A useful simplification of the previous general model was proposed in [33], under the assumption that the object attenuation and refraction are approximately constant within the size of the sample aperture. In this case, the spatial distribution of the intensity incident on the detector slit is simply reduced in amplitude due to attenuation and rigidly shifted due to refraction, i.e. $I_{point}\left(\overline{y};p\right)=T\left(p\right)I_{ref,point}\left(\overline{y}-z_2\Delta {\theta }_y\left(p\right)\right)$, where $I_{point}\left(\overline{y}\right)$ and $I_{ref,point}\left(\overline{y}\right)$ are the intensities obtained, with a point source, with and without the object. By inserting this expression into Eq. (4), we obtain:

 

Fig. 3 Example profiles for the illumination curve C, in the case of both point and extended sources. Considered parameters of the experimental setup: z₁ = 1.6 m, z₂ = 0.4 m, a = 12 µm, d = 20 µm, X-ray energy = 20 keV, detector mask period = 85 µm, extended source of 70 µm full width at half maximum (parameters matching one of the experimental setups installed at UCL). The peak of the curves is reached at y_e = −2.5 µm, corresponding to perfectly aligned slits.

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3. The transport of intensity equation and its application to EI

3.1 The transport of intensity equation in free-space propagation

The Fresnel diffraction integral can be simplified, in the near-field regime, by using the well-known transport of intensity equation (TIE) [21]. This type of approximation has been extensively used in the literature, particularly in the framework of the free-space propagation (FSP) XPCi technique. FSP XPCi does not make use of any optical elements along the beam path, but simply produces image contrast by illuminating the sample and recording the intensity pattern generated at a certain distance downstream of it [3,4,22–27,31]. Under the TIE approximation, this intensity can be written as [21,23]:

The validity conditions of the TIE have been studied in detail by Gureyev and associates [27]. The source distribution and detector point spread function were modelled as Gaussian functions of standard deviations ${\sigma }_{src}$ and ${\sigma }_{det}$, and an object phase edge expressed by $\varphi \left(y\right)={\varphi }_{max}H\left(y\right)\ast f_{obj}\left(y\right)$ was considered, where $H\left(y\right)$ is the step function and $f_{obj}\left(y\right)$ is a Gaussian function of standard deviation ${\sigma }_{obj}$ (which defines the smoothness of the edge). It was found that a necessary and sufficient condition for the validity of the TIE is represented by:

3.2 Application of the transport of intensity equation to EI

We now consider the case of an EI setup, and make the assumption that the intensity incident on the detector aperture can be expressed by the TIE. The validity conditions expressed in the previous section show that this can provide a good approximation when a laboratory setup making use of a non-microfocal source is considered (i.e. for low spatial coherence at the object plane), provided that the sample-detector distance is sufficiently small and the object transmission function is sufficiently slowly varying. Note that the use of the TIE seems in contradiction with the assumption of perfectly sharp edges for the sample slit. However, as we will see in the simulations at the end of the present section, in practical cases the TIE still provides a good approximation if the other conditions are satisfied. Moreover, the following theoretical treatment will be generalized in section 5 to the case of smooth mask edges, providing a more rigorous theoretical foundation to the validity of this approach.

By combining Eqs. (5) and (9), and realizing that in the case of EI the complex transmission function that needs to be considered in Eq. (9) is that of the combined slit – object system, expressed by $q\left(y^{\prime }\right)\cdot m\left(y\right)$ (so that $I_T\left(y\right)=I_{0}T\left(y^{\prime }\right){\left|m\left(y\right)\right|}^2$), we obtain:

By calculating the derivative of the quantity in the square parenthesis in Eq. (12) and inserting the result into Eq. (11), we obtain:

 

Fig. 4 Plots of (a) f_T and (b) f_R for different source sizes and illumination fractions. Considered experimental parameters: z₁ = 1.6 m, z₂ = 0.4 m, a = 12 µm, d = 20 µm, detector mask period = 85 µm. y_e/M = 6 µm corresponds to + 50% slits misalignment, y_e/M = 9 µm to + 75% slits misalignment and y_e/M = −2 µm to perfectly aligned slits.

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The second term of Eq. (13) incorporates the dependence upon refraction (or, equivalently, upon the first derivative of the phase). In particular, at every point, the value of the refraction angle is weighted by the intensity transmitted through the object (i.e. by the density of photons at that point), and by the value of the function $f_R\left(y+p\right)$, which can then be interpreted as a position-dependent sensitivity function for the refraction signal. Examples of $f_R$ profiles are shown in Fig. 4(b), in the cases of a 30 µm source with 50% slits misalignment (violet curve), a 70 µm source with 50% and 75% misalignments between the two slits (red and green curves, respectively), and a 70 µm source with the two slits aligned (i.e. centered with each other, blue curve). In the first three cases, the refraction sensitivity function is peaked at a position close to the detector edge, and its width increases with the projected source size. The latter quantity therefore determines the size of the sample region that contributes to the signal. Note that this quantity is strictly linked to the achievable spatial resolution, as described in [30]. In the limiting case of a point source, the refraction sensitivity function is equal to $f_R\text{}\left(y\right)=z_{def}\left[\delta \left(y_e/M-y\right)-\delta \left(y_e/M+d/M-y\right)\right]rec{t}_a\text{}\left(y-a/2\right)$ and, if the two slits are not completely misaligned, it reduces to $f_R\left(y\right)\equiv z_{def}\delta \left(y-y_e/M\right)$, which would imply infinite spatial resolution. This non-physical result is due to the fact that the geometrical optics approximation breaks down for a point source. As shown in [30], the width of diffraction provides a limit to the resolution in this case.

In the ‘aligned’ configuration (blue curve in Fig. 4(b)), positions on either side of the sample aperture have opposite refraction sensitivities. This expresses the fact that a certain refraction angle will produce opposite effects at the two edges: when photons are gained (or lost) at one edge, photons will be lost (or gained) at the other. Likewise, the refraction sensitivity function will be negative in the case of negative misalignment between the slits, as the refraction contrast is reversed in this case.

Let us now rewrite Eq. (11) by developing the derivative in its third term, to obtain:

Equation (14) allows us to model the signal in the two opposite cases of small and large object features. In the first case (small object boundary, fully contained within the sample aperture), the refraction is equal to zero at the edges of the sample aperture, while the first and second derivatives of the phase are non-negligible in between. The second term then vanishes and Eq. (14) can be written as:

The signal is therefore expressed as the cross-correlation of the function f_T with the object intensity profile that would be provided at the detector plane by pure FSP. Obviously, a complete collection of this type of signal requires the use of a small dithering step (the corresponding condition in FSP would be the need to use a very small detector pixel), as the signal width is by definition smaller than the sample aperture. It is important to notice that this type of refraction signal does not depend on the use of the sample mask, as it would be measured even without it. As pointed out in [19,20], however, the presence of the sample mask facilitates the detection of this signal, as it reduces the amplitude of the background signal (i.e. the signal measured without the object) thus increasing the contrast.

The sample mask has instead a central role in generating the signal in the opposite situation, i.e. that of an object edge larger than the aperture. In this case, the refraction angle and attenuation can be considered to be almost constant in the range $\left(-p,-p+a\right)$, so that the third and fourth terms of Eq. (14) vanish and the signal becomes:

Note that this type of refraction signal, unlike the previous one, is strictly related to the presence of the sample mask and would vanish without it, as $K\left(Ma\right)-K\left(0\right)$ would be equal to zero ($K\left(-\infty \right)=K\left(+\infty \right)=0$). In analogy to the “position-dependent” sensitivity functions f_T and f_R, the quantities $F_T\left(y_e\right)\equiv {\displaystyle {\int }_{-\infty }^{+\infty }dy{f}_T\left(y;y_e\right)}={\displaystyle {\int }_0^{a}dy}K\left(My\right)$ and $F_R\left(y_e\right)\equiv {\displaystyle {\int }_{-\infty }^{+\infty }dy{f}_R\left(y;y_e\right)}=z_{def}\cdot \left[K\left(Ma\right)-K\left(0\right)\right]$ can be interpreted as integral sensitivity functions for the attenuation and refraction signals. In fact, while f_T and f_R represent the contributions of a given position in the object plane to the attenuation and refraction signals, F_T and F_R represent the sum of contributions from all points.

The expressions for F_T and F_R can be used to study analytically the dependence of the sensitivity upon the different acquisitions parameters. This can be convenient both for characterizing an EI setup, and for designing and/or optimizing a new one, as it can guide the choice of the optimal parameters for a given application. As an example, we report in Fig. 5(a) the refraction integral sensitivity as a function of the full width at half maximum (FWHM) of the source distribution, for two different positions of the detector edge. The values of the considered experimental parameters are indicated in the figure caption. In Fig. 5(b), we study instead the variation of the refraction integral sensitivity with the position of the detector edge, for two values of the source FWHM. The plot clearly indicates that different positions provide very different sensitivities, confirming the importance of such an optimization procedure.

 

Fig. 5 Plots of the refraction integral sensitivity function F_R (a) as a function of the source FWHM for two different positions of the detector edge, and (b) as a function of the detector edge position for two source FWHMs. The values for the other experimental parameters are: z₁ = 1.6 m, z₂ = 0.4 m, a = 12 µm, d = 20 µm, detector mask period = 85 µm.

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By using the definition of the function K in Eq. (6) and the definition of the illumination curve C, and by exchanging the order of the integrations in the first term of Eq. (16), it can be verified that this is equal to $a{I}_T\left(p\right)C\left(y_e\right)$. The second term can also be expressed as a function of the illumination curve C, by noting that:

Therefore, Eq. (16) is equivalent to:

In order to validate the TIE-based model developed in this article, we compare in Fig. 6 the signals calculated with it (through the use of Eq. (13)) against those obtained from a rigorous wave optics calculation [18]. We consider two objects characterized by different size and shape. The first (Fig. 6(a)) is a 200 µm diameter filament made of polyethylene terephthalate (PET) plastics. The second sample (Fig. 6(b)) is a 30 µm thick slab of PET, whose edge is modelled as a step function convolved with a Gaussian of 1 µm standard deviation (the latter parameter defines the smoothness of the edge). The dithering step size considered in the two cases is 1 µm and 0.5 µm, respectively. Despite the very different shape and size of the two objects, and the relatively large contrast obtained (about ± 20% and almost – 10%, respectively), a reasonable agreement with wave optics is found for both cases. In the first plot, the profiles match almost entirely except at the edges of the wire. This is due to the fact that, at these positions, the refraction angle is large, thus breaking the validity condition for the TIE-based model. This leads to a slight shift in the position of the peaks, as well as an overestimation of their values. It is worth mentioning that a similar situation was shown to arise also for the FSP case, where the use of the TIE outside of its range of validity overestimates the signal on the detector, because the signal increase as a function of the propagation distance is not linear anymore (see, for example [27],). In the second plot, the correct value of the peak is preserved, but its position is slightly shifted in the TIE model. Additional simulations show that the agreement between the two models becomes better/worse as the contrast is made smaller/larger, as expected from the theory.

 

Fig. 6 Normalized signal profiles obtained using wave optics and TIE-based models for (a) a 200 µm diameter cylinder made of PET, and (b) a 30 µm thick PET slab, with an edge shape expressed as a step function convolved with a Gaussian function of 1 µm standard deviation.

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4. Relationship with other theoretical models for EI

We have already analyzed in section 3.2 the relationship of the TIE-based model with that based on the illumination curve, which was developed in [14,33,34]. In that case, the attenuation and refraction functions were assumed to be approximately constant within the sample aperture (i.e. to be slowly varying functions). The present model does not require this assumption and therefore enables the modelling of much smaller object features. However, it assumes that the beam at the detector can be expressed through the TIE, which neglects all diffraction effects and thus is not expected to correctly model the signal in the case of high X-ray coherence (for instance, in synchrotron setups). The two models therefore appear to be complementary, and the choice of one or the other depends on the type of structures considered and on the setup characteristics. Moreover, we have seen that, under appropriate conditions, the two models converge into Eq. (18).

The model developed in [20] appears to be a special case of the present theoretical framework, corresponding to the result in Eq. (15). In fact, it considers that the object boundary is smaller than the sample aperture (sample with rapidly varying attenuation and refraction functions), so that the beam impinging on the detector aperture is equivalent to that produced by FSP. The signal on the detector is thus the cross-correlation of this intensity with the function f_T (Eq. (15)).

The model employed in [11,35] to retrieve the object attenuation and refraction maps assumes the refraction angle to be constant within the sample aperture. However, compared to the illumination curve model, it does not impose that the attenuation is constant, but just assumes that it varies linearly within the aperture. The beam incident on the detector aperture is modelled by using geometrical optics and, although it is not introduced explicitly, effectively a linear approximation of the illumination curve is considered. Therefore, this model is expected to have a more limited range of validity compared to the illumination curve one [14,33,34].

The effect of scattering, i.e. of a distribution of refraction angles within the sample aperture size, was considered in [15]. There, it was shown that, by combining three input images acquired at different positions on the illumination curve, the sample attenuation, the mean of the angular distribution (i.e. the refraction angle) and its width (the so-called scattering parameter) can be extracted. This method represents a generalization of the 2-images retrieval based on the illumination curve. However, the latter is assumed to have the shape of a Gaussian, in order to analytically retrieve the three object components. The present TIE model also includes the possibility of having a distribution of refraction angles (with a certain width describing the scattering parameter) within the aperture size; however only small refraction and scattering angles can be considered, due to the above-mentioned validity conditions.

5. Effect of the smoothness of sample boundary and mask edges

We now utilize the theoretical framework developed in section 3 to study the effect on the signal of smooth mask apertures, which may be encountered in practical experimental situations. Let us first suppose that only the detector mask has smooth edges. We can express this unsharpness by replacing the rectangular function describing the detector aperture with the following smoothed rectangular function, $rec{t}_{smooth,}{}_d\left(y_e+d/2-y\right)=\left[f_d\ast rec{t}_d\right]\left(y_e+d/2-y\right)$, where $f_d$ is the smoothing function (for instance, but not necessarily, a Gaussian). This impacts on the definition of K (Eq. (6)), which needs to be rewritten as:

Equation (19) can be alternatively expressed as:

On the contrary, we will see in the following that smooth sample aperture edges do not have, in general, an effect equivalent to that of the source blurring. The position-dependent sensitivity functions (cfr. Eq. (13)) become, in fact, $f_T\left(y\right)\equiv K_d\left(My\right)rec{t}_{smooth,a}\left(y-a/2\right)$ and $f_R\left(y\right)\equiv z_{def}\frac{\partial K_d\left(My\right)}{\partial y}rec{t}_{smooth,a}\left(y-a/2\right)$, where $rec{t}_{smooth,}{}_a\left(y-a/2\right)$ $=\left[f_a\ast rec{t}_a\right]\left(y-a/2\right)$ and f_a is the smoothing function. The two functions can be written equivalently as:

It is apparent from the previous expressions for $f_T$ and $f_R$ that the smoothing of detector and sample masks are not equivalent, as the two quantities $f_a\left(y\right)$ and $f_{d,r}\left(y\right)$are not interchangeable. Their effects on the spatial resolution, for instance, are different. In fact, in the case of sample mask smoothing, even points of the object outside the range $\left(o,a\right)$ can contribute to the signal, which does not happen in the latter situation, where the function $f_T$ is non-zero only in the range $\left(o,a\right)$. The influence of the two terms on the signal will also be different in general, and will depend on both the experimental setup and the sample. Let us consider, for example, the limiting case, described by Eq. (15), of a sample aperture much larger than the detector aperture. If, for simplicity, the object transmission is assumed to be constant, the signal is given by (see Eq. (15)):

A case exists, however, where the two contributions have the same effect on the signal. Let us assume that the attenuation and refraction are slowly varying on the length scale of the sample aperture (approximation of large object boundary, described by Eq. (16)). With reference to Eq. (16), the integral sensitivity functions are now $F_T\left(y_e\right)={\displaystyle {\int }_{-\infty }^{+\infty }dy{K}_d\left(My\right)\left[rec{t}_a\ast f_a\right]\left(y-a/2\right)}$ and $F_R\left(y_e\right)=z_{def}{\displaystyle {\int }_{-\infty }^{+\infty }dy\left[\partial K_d\left(My\right)/\partial y\right]\cdot }\left[rec{t}_a\ast f_a\right]\left(y-a/2\right)$. By using the definition of $K_d\left(My\right)$ in Eq. (20) and assuming for simplicity that $f_a\left(y\right)$ is symmetric, the sensitivity functions can be expressed more directly as a function of the various quantities, as:

In the special case of large object boundary, therefore, the demagnified projected source size, the smoothing function of the sample aperture and the demagnified smoothing function of the detector aperture are convolved with each other, and have thus the same effect on the signal. This important result can also be further compacted by defining an effective K equal to:

We now turn our attention to the dependence of the signal upon the smoothness of the object boundary. In particular, we consider for simplicity a phase object, and model its phase function as $\varphi \left(y\right)={\varphi }_{step}\cdot \left[H\ast g_{obj}\right]\left(y\right)$, where H is the Heaviside step function and $g_{obj}$ is a Gaussian function of width ${\sigma }_{obj}$. Then, if we additionally consider smooth sample and detector apertures, Eq. (13) becomes:

6. Conclusions

We have developed a new theoretical model for EI XPCi, based on the use of the transport of intensity equation (TIE). The presented theoretical framework clarifies the origin of the various sources of contrast in EI. Moreover, it enables the derivation of simple expressions relating the signal to the various parameters pertaining to the EI setup and to the object itself. For this reason, the developed model is expected to be useful in the design and optimization of future EI experimental setups, as it also avoids the need for time-consuming wave optics calculations. The present TIE model is shown to provide fairly accurate results, in particular, in the case of X-ray beams of low spatial coherence and for limited image contrast provided by the object, which makes it especially interesting for simulating biological samples with laboratory setups based on conventional X-ray tubes. In addition to allowing simple forward simulations, moreover, this model could also be used as the basis for the development of new phase retrieval algorithms, in particular aimed at extracting the object profile at length scales smaller than the sample aperture (since it does not assume attenuation and refraction to be constant within the sample aperture).

Furthermore, the derivation of the principles at the basis of the signal in EI from the TIE formally strengthens the relationship of this technique with FSP XPCi. Both techniques share, in fact, important features, among which the fact that, in the near-field regime, they are intrinsically incoherent methods. Important consequences are, in particular, the possibility to use fully polychromatic beams (as no degree of temporal coherence is required) and the possibility to model the signal with pure geometrical optics (as coherent wave effects can be neglected).

Compared to previous models, the present theoretical framework based on the TIE has also the advantage to enable the modelling of both large and small features, i.e. of both low and high spatial frequencies, therefore allowing the description of a much wider class of objects. In the present article, the developed model was also used to quantify the impact on the signal of smooth mask apertures, which may be encountered in practical setups and reduce the achievable sensitivity, as well as the effect of sharp/smooth sample boundaries.