1. Introduction

X-rays are a powerful tool in biomedical imaging, traditionally using the absorption properties of the structure in question to reveal features of morphological (e.g. anatomical, structural) interest. Such absorptive imaging has the capacity to easily display bones and highly attenuating materials [1]. In addition to this well established mode, propagation-based phase contrast x-ray imaging (PCXI) demonstrates that even soft tissue may be seen using sufficiently coherent x-rays, using the phase changes that occur when a wave passes through a structure [2,3]. In particular, PCXI makes use of the transverse phase differences that are seen in a wave when it exits a scattering volume containing regions of different materials. In propagation based phase contrast, these phase variations are observed as intensity variations upon free-space propagation from the object to the detector, producing marked light and dark intensity interference fringes along the boundary of the two differing regions [4]. These high contrast fringes make the edges of tissue regions, for example an airway lumen, easily seen. As x-ray detector technology develops, smaller pixels enable such structures to be observed at high resolution in excellent detail. The use of synchrotron x-rays, characterised by their brightness and coherence, has also played an important part in realising detailed and informative PCXI [2,3].

Having established methods for biomedical PCXI, the balance of work is now moving from qualitative observations to quantitative measures of biological function (see, e.g., [5]). While PCXI resolves the edges of soft tissue well [6], it is the phase contrast fringe which can reveal quantitative information about the phase changes effected by different materials, hence the spatial distribution and characteristics of those materials [5,7]. The projection approximation (PA) is a valuable tool in simulating the phase contrast process and in the development of phase retrieval algorithms (see, e.g., [7,8]). This approximation describes the passage of rays through an object, by defining a nominal exit surface, immediately “downstream” of the irradiated object, at which transverse phase and intensity changes are imprinted. The projection approximation assumes that all scattering within the object is fully described by this exit wave, with negligible diffraction within the scattering volume. This simplification is very useful in recovering quantitative information from phase contrast images, and as theory is pushed further, it is timely to re-examine the projection approximation and edge contrast detail.

Here we look at the case of propagation-based x-ray phase contrast imaging of a cylinder or cylindrical edge, a simple model for the edges of many biological samples. A cylinder can model airways, blood vessels and other anatomical passages. The boundary between tissue and air is particularly suited for PCXI; an airway may be easily seen which would appear near invisible if observing attenuation only [9]. More generally, a cylinder can also be a good approximation to a rounded edge in projection, such as features A, B, C and D of Fig. 1 . In this figure the imaging geometry is shown, with a monochromatic scalar electromagnetic plane wave propagating in the positive z direction, incident on a cylinder or cylindrically modelled edge. The incident wave incurs changes in phase during its passage through the various regions of the object. As mentioned earlier, propagation based PCXI makes use of these phase changes by converting them into intensity variations through free-space propagation after the scatterer [2,3,10]. For sufficiently short propagation distances, propagation-based phase contrast exhibits fringes that become more visible the greater the propagation distance and for a greater difference in projected refractive index between neighbouring regions in the scatterer.

 

Fig. 1 Imaging geometry showing the projected thickness of a sample object, as used by the projection approximation, along with the associated exit plane.

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As previously mentioned, the projection approximation assumes no diffraction occurs within the object volume, and hence phase and intensity variations at the scatterer's exit surface arise only from the projected complex refractive index. Further propagation from the exit plane to the imaging plane is required to then convert transverse phase gradients into phase contrast, visualised as intensity fringes. However, any diffraction within the object volume will mean that narrow intensity fringes will be observed at the “exit plane”, e.g. from each feature marked A-D in Fig. 1. In other words, the diffraction between the z position of edges A-D and the exit plane will be sufficient for the wave that has passed on one side of the feature to interfere with the wave passing on the other side of the feature. As shown by the red traces in Fig. 1, features such as A, which are further from the exit plane, will produce wider, more intense fringes than closer features such as B. Application of the projection approximation to simulate this process will predict only absorption contrast at the exit plane, requiring further propagation to produce phase contrast.

The study of edge contrast from such shapes is approached through a simulation which takes the exit wave as predicted by the projection approximation, and propagates to the detector surface, as described in section 2. Section 3 then looks at the signature of the resulting complex wavefield in the Argand plane. The simulation results are then validated by comparison with x-ray phase contrast images of a perspex cylinder taken at the SPring-8 synchrotron in Japan, in section 4.

2. X-ray image simulation using the projection approximation

The projection approximation is a consequence of the paraxial equation in an inhomogeneous medium (called the inhomogeneous paraxial equation hereafter), under the assumption that the scattering introduced by the sample is not strong enough to significantly disturb the ray paths compared to the ray paths which would have existed within the same volume in the absence of the scatterer. This “semi-classical” approximation, which is somewhat reminiscent of the geometrical theory of diffraction described by Keller [11] in so far as it ascribes a phase to each ray path, approximates the phase and amplitude variation of a wave on travelling through a scatterer (for a textbook account see e.g. reference 7). The inhomogeneous paraxial equation, for a single wavelength, is given in Eq. (1):

The projection approximation states that waves passing through space with this refractive index n will undergo a phase shift, as denoted by the second term on the right-hand side of Eq. (2) and experience attenuation, as denoted by the third term of Eq. (2), where T_j is the projected thickness of each material j along the z direction [7]:

The projection approximation may be used to simulate an x-ray phase contrast image, similar to the use of the projected charge-density approximation in electron microscopy [12]. Using this approach, the projection approximation was applied to the x-ray plane wave incident upon the object in question to give the exit wavefield. We look at the case of a cylinder, such as those approximating edges in Fig. 1 (e.g. B), so n is defined throughout the object and the total projected thickness (T) of each material may be determined by the position x on the plane where the image is to be evaluated. As the projection approximation effectively projects all material to the exit surface, the diffraction and interference of waves is not described until after propagating the wavefunction from the exit plane to the image plane. This propagation was done by applying the wavefield propagator, then the modulus squared of the projected wavefield was calculated to obtain the intensity phase contrast image. The propagation was implemented using the angular spectrum representation of the Rayleigh-Sommerfeld diffraction integral of the first kind [13,14]. This method for simulation has been previously used in the field of synchrotron imaging (see, e.g [15]) and biomedical x-ray imaging (see, e.g [16]). The technique is particularly useful in that phase contrast images may be quickly and easily simulated. This allows optimisation of experimental configurations when the sample composition is well-known.

Simulations produced images showing the edge of a cylinder (or rounded edge), as seen in Fig. 2 , where fringes grow wider and more intense as a result of increased propagation (up to several metres for hard x-rays). Images experimentally obtained using a monochromatic 25keV synchrotron source of low divergence, as seen in Fig. 3 , agree with simulations when simulated images are smoothed with the point spread function (PSF) appropriate to the detector characteristics, after adding noise and sampling at the same pixel size - see details below. The simulated and observed profiles show the same width and positioning of the multiple fringes from the cylinder edge, as well as the same maximum intensity from the fringe set. The smoother fringe envelope in the observed image is likely due to a PSF with broader tails than was measured experimentally (by knife-edge image) and used to smooth the simulated image. Here the projected edge of a 3 mm diameter cylinder is imaged with 50 cm and 100 cm propagation before the detector, distances typical of PCXI.

 

Fig. 2 Phase contrast fringes from the simulation for light of wavelength λ = 0.5 Angstrom incident on the left edge of a 3 mm diameter perspex cylinder (using δ_air = 4.13 × 10⁻¹⁰, β_air = 0, δ_perspex = 4.00 × 10⁻⁷, β_perspex = 3.1998 × 10⁻¹⁰), propagated 50 cm.

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Fig. 3 Phase contrast images show the projection approximation correctly predicts the fringes seen at sufficiently long propagation distances. Images taken at the upstream hutch of BL20XU, SPring-8, of a 3 mm diameter perspex cylinder at 25keV for a propagation of a) 50 cm and b) 100 cm. The observed profile is shown in black and the simulated in blue.

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An experimental image will usually not show many more than a few Fresnel fringes, due to the limited coherence of the source as well as the PSF of the imaging set-up, which is dependent on both the source size and the detector system (here most significantly due to scattering in the phosphor screen used to convert x-rays to visible/UV light for CCD capture as a digital image). A biomedical phase contrast image (for example, of an airway in a mouse or rabbit) often shows only one or two fringes, due to the overlying textured tissue and scattering within the volume [17]. The detector system used consisted of a phosphor screen, lens and a CCD, resulting in a pixel size of 0.45 microns and observed point spread function (PSF) of full width half maximum 3.8 microns. The cylinder was at an angle θ to the columns of the pixel array. Hence, to improve the signal to noise ratio, each row was shifted by aligning the fringe peaks to allow an average of the edge fringe over many rows. Since a true profile of the fringes should be taken perpendicular to the edge, correction by a factor of cos(θ) was made to the horizontal axis (i.e. via the pixel size).

3. Argand representation of the complex wavefield at the imaging plane

The wavefield at the exit plane or at the image plane (see Fig. 1) may be mapped to the Argand plane. For the case of a cylinder edge, a characteristic curve is formed from a Cornu spiral and a kind of hypocycloid (a cycloid inscribed on a circle). The Argand trace is parameterized by position in the imaging plane, with intensity described as square of the distance from the Argand origin and phase as the angle from the positive real axis.

Depending on the wavelength, object size and propagation, the wavefield across a plane downstream of the cylindrical edge will trace out an Argand-plane trace similar to that seen in Fig. 4 . The centre of the blue spiral corresponds to far outside the geometric shadow of the cylinder (i.e. large x), so has the uniform intensity and phase of the unscattered plane wave over a plane of constant z. Cycling outwards around the blue spiral increases and decreases the intensity in increasingly large bands, producing the increasingly wide, intense Fresnel fringes seen when approaching (from the outside) the phase contrast fringe arising from the edge of the cylinder.

 

Fig. 4 Argand plot of 1 Angstrom waves incident on a 3 mm cylinder, propagated 5 mm, corresponding to 8 micron either side of the cylinder's geometric shadow, producing a connected Cornu spiral (blue) and hypocyloid (black).

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The black hypocycloid in Fig. 4 describes the wavefield when moving inside the geometric shadow of the cylinder. In this example the absorption contrast is negligible compared to the phase contrast, so the trace moves along a circle of almost uniform radius/intensity, slowly changing in phase, according to the projection approximation. Oscillations in and out of this circle become light/dark bands of decreasing width and intensity, as seen in the phase contrast image (Fig. 2).

The diffraction pattern and hence the Argand plot can be explained by conceptually separating the x-ray wavefield scattered by the cylinder into two components, as seen in Fig. 5 .

 

Fig. 5 Propagation based phase contrast imaging of a cylinder separated into edge diffraction and a distorted transmitted wave.

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The first component, that of a plane wave diffracting around an opaque black screen A (bounded by the edges of the projected cylinder), is seen as the blue Cornu spiral in Fig. 4 [18]. The second component, the cylindrically distorted wave diffracted between two black screens, B₁ and B₂, is seen as the hypocycloid; this hypocycloid results from a combination of the projection approximation and diffraction from the edge of the cylinder/opaque barrier. Note that screen A is complementary to the screen formed by B₁ and B₂. We examine each of the diffracted components in detail below.

A Cornu spiral is typically plotted with C(u) and S(u) as the Fresnel integrals on the x and y axes respectively [18], where u is a reduced variable proportional to z, the distance across the imaging plane. In order to calculate the diffracted wavefield along a plane which is downstream of an opaque edge, the Cornu spiral is shifted by + 1/2 in both x and y directions, rotated about the origin by –π/4 and divided by √2 as in Eq. (3) [18]:

This will place the centre of the spiral a distance from the origin equal to the square root of the wave intensity. The square of the distance from the origin will then give the intensity of the wave, giving local maxima and minima while moving around the spiral, closer to and further from the origin. This links to the elegant edge diffraction model described by Margaritondo and Tromba [19], although the cylindrical nature of the edge considered here will slightly distort the diffraction pattern from that which would be seen from a rectangular edge, as they describe. Their model also uses Fresnel integrals to look at diffraction of a wave, this time from an absorbing, but not completely opaque edge. The wavefield is then altered to Eq. (4) [19]:

The addition of the projection approximation, multiplied by a Cornu spiral around the origin, will produce phase contrast fringes within the geometric shadow of the non-opaque edge. This signature, as a trace in the Argand plane, is seen in our simulations of a cylindrical edge using the projection approximation (cf. Fig. 4 and Fig. 7 ).

 

Fig. 7 Argand plot of 0.5 Angstrom waves incident on a 1 mm diameter cylinder, propagated 1 mm, corresponding 8 micron either side of the cylinder's geometric shadow, with significant attenuation by the cylinder (β _{Fig. 7} = 500 × β _{Fig. 4}). The dotted circle indicates the uniform intensity of the unscattered wave.

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The spiral shape of the Cornu spiral (the blue trace in Fig. 4) is due to the phasor addition of the unscattered plane wave passing outside the cylinder in Fig. 6 with the cylindrical wave scattered from the edge of the cylinder, such as ray AB. These can also be described by the interference of the incident plane wave and a cylindrical wave scattered from the edge of the cylinder, using the rays seen in Fig. 6. The projection approximation will incur a phase change, as given in Eq. (2), on the incident plane wave travelling through the cylinder and landing at C. This distorted plane wave, as estimated via the projection approximation, will then interfere with the cylindrical edge wave AC. The resulting wave, as described by Eq. (5), will then be the sum of a plane wave (e^ikz) multiplied by the projection-approximation phase shift e^-ikδT, and this spherical wave scattered from the edge of the cylinder;

 

Fig. 6 Geometry of rays incident on a cylinder, showing the rays which interfere to form a Cornu spiral and a hypocycloid.

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R will be increased as the point C moves deeper into the geometric shadow of the cylinder, and in the projection approximation, $T=\sqrt{a^2-x^2}\mathrm{,}$ will also be increasing. This means that as the projection approximation traces out a uniform-intensity circle in the Argand plane, interference between the distorted plane wave landing at C and the spherical wave AC will cycle with decreasing amplitude in the same way as a Cornu spiral. The effect of a Cornu spiral moving anti-clockwise around the Argand plane, following the projection approximation, is therefore seen as a hypocycloid in which the “epicycle” has reducing radius. If there is significant attenuation from the cylinder or rounded edge, the projection approximation will also show a decrease in the amplitude of the wavefield, as seen in Fig. 7. Interestingly, this Argand-plane trace exhibits a transition from classic hypocycloid-type cusps (e.g. feature α in Fig. 7) to a looped structure (e.g. feature β in Fig. 7), indicative of “retrograde” Argand-plane motion. The amplitude of this scattered edge wave, S, as related to where the phase begins “retrograde” Argand-plane behaviour, is explored mathematically in the appendix.

As the wavefield is further propagated from the cylinder to an image plane, the size of the Cornu spiral increases, as does the amplitude of the hypocycloid oscillations. This will produce more intense, wider Fresnel fringes. The pixel size and point spread function, relative to the size of the fringes, will be significant when phase contrast is observed with many fringes. Given small enough pixels, a holographic region fringe set will be observed, as would be seen by taking the absolute value squared of the Argand trace in Fig. 4. Larger pixels would give an image in the “edge detection” region, showing a single light/dark fringe at the boundary. As the pixel size increases further, the image will be less able to detect fringes, eventually leading to an absorption-only image. This smoothing of fringes is also observed with a decrease in the transverse coherence of the beam, the effect of which is well approximated by smoothing of the simulated image with a demagnified image of the source. As well as decreasing fringe visibility, this could smooth out the phase “loops” seen in Fig. 7.

Regardless of pixel size, the projection approximation's prediction of no intensity fringes due to phase contrast at the “exit” surface can be seen in an Argand plot for zero object-to-detector propagation distance.

4. Underestimation of fringes by the PA at short object-to-detector propagation distance

Here we examine how the PA underestimates phase contrast fringe visibility and width close to the object, both through the Argand trace and in practice. The origin of this underestimation is that the projection approximation omits the Young-type boundary wave [18] given by the final term of Eq. (5), over the exit-plane of the object. The associated underestimation of phase-contrast fringe visibility carries over to sufficiently-small object-to-detector propagation distances, as examined below.

Figure 8 shows the Argand-plane trace for 25keV x-rays falling on the edge of a 3 mm diameter cylinder with increased propagation before the image plane, as used in our experiment. The intensity and phase of the trace has been blurred with a Gaussian of standard deviation 0.18μm to describe the effect on the observable fringes of using 0.18μm effective pixel size. As mentioned earlier, the angular-spectrum formalism has been used to numerically propagate from the nominal planar exit surface of the object, to the surface of the detector.

 

Fig. 8 Argand plot of 0.5 Angstrom waves incident on a 3 mm cylinder, smoothed by a Gaussian of pixel size 0.18 micron, showing the introduction of multiple intense, wider fringes with propagation, viewing 1 micron either side of the interface.

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In Fig. 8, it can be seen that the projection approximation shows no fringes at 0 mm propagation, simply uniform intensity with slowly varying phase behind the cylinder. With 1.5 mm propagation from the exit plane, using the projection approximation, a single faint light/dark fringe is predicted, closer to that theoretically expected at the exit plane, which will itself sits 1.5 mm from the edge of the 1 mm cylinder. This illustrates the assertion that the propagation within a scattering volume upstream of the nominal exit plane will not be taken into account by the projection approximation. Propagations of 4 mm and greater are predicted to show multiple observable fringes both behind and outside the cylinder shadow.

A comparison between Fig. 8 and Fig. 4 shows that, as predicted by the projection approximation, a greater phase change is incurred with a short wavelength or a stronger phase object. This is seen when across the imaging plane a short wavelength (e.g. the 0.5 Angstrom hard x-rays of Fig. 8) passes through a full 2π phase change within a micron of the cylinder edge shadow, while a longer wavelength (e.g. the 1 Angstrom softer x-rays of Fig. 4) wraps more slowly (requiring 4 microns across the image plane for a 2π phase change in Fig. 4). It is for this reason that small propagation lengths on the order of millimeters (hence narrow fringes) and small pixels have been used in Fig. 8, to avoid wrapping the trace around upon itself.

The inability of the projection approximation to predict fringes at very short propagations was confirmed by imaging the same 3 mm perspex cylinder at propagation lengths comparable to the cylinder radius. Figure 9(a) shows that at 25keV there was no fringe predicted by the projection approximation at contact, but a fringe was observed in the image, as predicted by Eq. (5). When the propagation distance is increased to the radius, and then to the diameter of the cylinder, the fringes predicted by the PA become much more similar to those observed. The simulated fringes are not only more similar in amplitude, but also in fringe width. At 1m propagation, a typical distance as would be used for medical phase contrast imaging to produce good edge contrast (see, e.g., [2–6]), the images are correctly simulated using the projection approximation, as was seen in Fig. 3(b). This demonstrates how the projection approximation will only underestimate phase contrast intensity fringes when a significant amount of the total diffraction occurs within the scattering volume.

 

Fig. 9 Experimental phase contrast images show the projection approximation does not predict the fringes seen at very short propagation distances, but the simulated fringes become more accurate as the propagation distance increases. Images taken of the same 3 mm diameter perspex cylinder as Fig. 3, at 25keV for a propagation of a) contact, b) 1.5 mm, c) 3 mm. At these short propagation distances, the pixel size and point spread function prevent multiple fringes being resolved at the edge interface. The shorter propagation distance means that the signal to noise ratio is reduced. The observed profile is shown in black and that simulated using the projection approximation (then smoothed with detector characteristics as in Fig. 3) in blue.

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Therefore, for typically-large propagation distance as used in small animal PCI (as seen in Fig. 3), where the relative error in the propagation distance is small, the projection approximation is a fast and accurate tool in the simulation of x-ray phase contrast imaging.

5. Conclusion

It has been seen that the projection approximation is indeed accurate in predicting propagation-based phase contrast x-ray images of a cylinder given that the propagation distance is significant compared to the radius of the cylinder. This observation can be extended to other volumes in a general sense, in that the approximation is valid provided that there is not significant space for diffraction within the scattering volume compared to the propagation distance. The projection approximation is therefore very useful in efficiently simulating phase contrast as part of the journey to recovering quantitative information from images. We have also seen the “signature” of the projection approximation in the Argand plane, winding the phase around at uniform intensity turning a Cornu spiral into a hypocycloid in the geometric shadow of the rounded edge. The magnitude of these Cornu/hypocycloid oscillations was seen to increase with propagation.

Appendix

The scattered wavefield from the cylinder edge will produce the Argand plane loops as seen in Fig. 7. Whether the cyloid shape seen in the complex wavefield behind the geometric shadow of the cylinder will be seen to “loop the loop” (i.e., exhibit a retrograde-like trace) will be determined by the propagation distance, refractive index and edge gradient of the cylinder. This looping will mean that the Argand-plane trace due to phase along the image plane is moving slightly backwards (due to the scattered field from the edge of the cylinder) before continuing to increase while moving further behind the increasingly thick volume of the cylinder.

Given Eq. (5), we look at the relationship between the slow moving phase due to the projection approximation for the increasingly thick object and the small fast cycles due to edge cylindrical waves. In Fig. 7 a transition from where the hypocycloid goes from showing cusps (e.g. feature α in Fig. 7) to loops (e.g. feature β in Fig. 7) is seen. This is where the gradient of the cylinder is sufficiently small that the projection approximation is no longer moving the phase around the Argand plane fast enough to keep up with the fast cycles of the fringes from edge wave interference. Figure 10 shows a perfect hypocycloid, where a small circle moves along the inside of a bigger circle without any slipping. In order to avoid slipping, the distance along which it travels between peaks must be 2πd, the circumference of the cylinder. If the distance is smaller (or the circle turns faster, slipping as it moves across the surface), loops will be seen.

 

Fig. 10 A hypocycloid is traced by a circle turning, a) without slipping and b) with slipping, around the inside of a bigger circle.

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In the case considered, looking at the loops in the Argand plane of the wavefield observed along an image plane, the wave is given by Eq. (A1) [c.f. Eq. (5)]:

(A1)$${\psi }_{\text{image plane}}=e^{ikz}\times e^{i{\phi }_{\text{object}}}+\genfrac{}{}{0.1ex}{}{S}{\sqrt{R}}{e}^{i{\phi }_{\text{edge}}}.$$

Note that the object phase, ${\phi }_{object}=-k\delta T$, is decreasing when moving further behind the cylinder, while the edge phase, ${\phi }_{edge}=kR$, is increasing. In the Argand plane considered, the large circle (see e.g. dotted line in Fig. 4 or Fig. 7) is created by the first term of Eq. (A1) varying slowly in phase and the smaller circle comes from the second term varying quickly in phase. The critical case of a perfect hypocycloid with no slipping [Fig. 10(a)] will require two conditions to be fulfilled for a section of the trace (hence a section of the image plane);

• • The slowly moving phase from the first term in Eq. (A1), φ_object, changes by the circumference of the smaller circle (2πd) so that a full cycloid cycle can be traced by the edge of a “non-slipping” circle.
• • The fast moving phase from the second term in Eq. (A1), φ_edge, changes by a full cycle (2π) within the same distance.

This distance or section of the image plane is defined as Δx, in addressing the first requirement. The smaller circle will have radius d = S/√R (from the second term in Eq. (A1) and will require the arc length 2πd in the Argand plane on which to complete a full rotation without slipping. This arc length will come from the slow moving phase due to the slowly increasing projected thickness of the object. The change in phase across Δx is given in Eq. (A2):

(A2)$${\phi }_{object}\left(x+\Delta x\right)-{\phi }_{object}\left(x\right)={\phi }_{object}\text{'}\left(x\right)\times \Delta x,$$

with a dash denoting differentiation with respect to x. This angle swept out by the phase change given in Eq. (A2) will also be the arc length traversed in the Argand plane for a circle of radius 1 (i.e. intensity = 1). This means the critical case of a perfect hypocycloid is now described by Eq. (A3):

(A3)$$2\pi d=2\pi \genfrac{}{}{0.1ex}{}{S}{\sqrt{R\left(x\right)}}=\phi {\text{'}}_{object}\left(x\right)\times \Delta x.$$

To satisfy the second condition, we require the condition given in Eq. (A4), that the phase changes from the cylindrical wave [φ_edge in Eq. (A1)] change by 2π (or less, to avoid loops) over the length Δx:

(A4)$$\phi {\text{'}}_{edge}\left(x\right)\Delta x\le 2\pi \mathrm{.}$$

Substituting Eq. (A3) into Eq. (A4) to eliminate Δx gives Eq. (A5):

(A5)$$\phi {\text{'}}_{edge}\left(x\right)\genfrac{}{}{0.1ex}{}{2\pi S}{\sqrt{R\left(x\right)}\phi {\text{'}}_{object}\left(x\right)}\le 2\pi \mathrm{.}$$

Simplifying produces Eq. (A6)

(A6)$$S\phi {\text{'}}_{edge}\left(x\right)\le \sqrt{R\left(x\right)}\phi {\text{'}}_{object}\left(x\right)\mathrm{.}$$

Putting in the expressions for each of φ_object and φ_edge as determined by the object and by the edge wave {Eq. (5) gives Eq. (A7)]:

(A7)$$Sk{R}^{\prime }\left(x\right)\le \sqrt{R\left(x\right)}k\delta T^{\prime }\left(x\right)\mathrm{.}$$

where k is the wavenumber, R(x) the distance from the edge and T(x) the projected thickness of the object, as previously defined. In all edge cases, $R\left(x\right)=\sqrt{z^2+x^2}$ for the origin in x (i.e. x = 0) set at the edge of the object. This will produce a requirement for a cycloid without loops, Eq. (A8):

(A8)$$T^{\prime }\left(x\right)\ge \genfrac{}{}{0.1ex}{}{Sx}{\delta {\left(z^2+x^2\right)}^{3/4}}\mathrm{.}$$

Assuming that x is small compared to z, i.e. fringes are seen close to the edge for long propagations, this simplifies to Eq. (A9) for a non-retrograde Argand-plane trace:

(A9)$$T^{\prime }\left(x\right)\ge \genfrac{}{}{0.1ex}{}{Sx}{\delta z^{3/2}}\mathrm{.}$$

This describes how the sample thickness gradient must be sufficiently large that the slowly varying phase from the projection approximation moves around the Argand plane quickly enough to keep up with the fast loops from the edge wave. More specifically, tighter loops will be seen for any of shorter propagation z (where the spherical edge wave has greater variations across Δx), smaller δ (a weak phase object will not wrap the slow moving phase from the object around as quickly) or greater distance (x) from the edge, where the reduced radius of the hypocycloid means it can complete a circuit more quickly. The increase in retrograde behaviour with position x in the imaging plane when moving behind the cylinder has already been observed in Fig. 7.

For the specific case of the edge of a cylinder, the projected thickness T(x) can be described as $T\left(x\right)=2\sqrt{a^2-{\left(x-a\right)}^2}$, where a is the radius of the cylinder and |x|≤ a. Substituting into Eq. (A9) and again assuming that x is close to the edge, and small compared to the radius a, gives Eq. (A10);

(A10)$$x\le z\sqrt[3]{\genfrac{}{}{0.1ex}{}{2a{\delta }^2}{S^2}}.$$

This means that the scattered edge wave coefficient, S [from Eq. (5)], could be determined from a plot such as Fig. 7, simply from the x coordinate at which the hypocycloid begins to loop around. With significant attenuation, this x coordinate will change slightly, with the condition given in Eq. (A9) now divided by $e^{-k\beta T}$ on the right hand side.

Acknowledgments

The authors thank the Japan Synchrotron Radiation Research Institute and beamline 20XU scientists Kentaro Uesugi, Yoshio Sukuzi and Akihisa Takeuchi for the privilege of using the SPring-8 facility in obtaining the experimental images. We acknowledge funding for the travel to SPring-8 from the Access to Major Research Facilities Program, which is supported by the Commonwealth of Australia under the International Science Linkages program. Kaye Morgan acknowledges the support of an Australian Postgraduate Award and a J.L. William Scholarship from Monash University. David Paganin acknowledges the Australian Research Council, together with useful advice from Christian Dwyer, Timur Gureyev and Freda Werdiger. Karen Siu acknowledges the National Health and Medical Research Council.

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