## Abstract

We present a quantitative, non-interferometric, X-ray differential phase contrast imaging technique based on the edge illumination principle. We derive a novel phase retrieval algorithm which requires only two images to be acquired and verify the technique experimentally using synchrotron radiation. The technique is useful for planar imaging but is expected to be important for quantitative phase tomography also. The properties and limitations of the technique are studied in detail.

© 2012 Optical Society of America

## 1. Introduction

Conventional radiographs record the projection, along the direction of X-ray propagation, of an object’s absorption properties. Image contrast thus arises due to spatial variation of an object’s thickness and constituent material absorption coefficients. Since the pioneering work of Bonse and Hart [1], a variety of techniques have been developed to acquire X-ray images possessing contrast due to sample induced phase shifts [2–7]. Such images are referred to as phase contrast images. Since then, techniques have been developed which are capable of quantifying the phase shift. Such techniques, whilst being useful for planar imaging, are also essential for performing quantitative phase tomography.

Existing quantitative X-ray phase contrast imaging (XPCI) techniques may be divided into three categories. The first category may be denoted analyser based imaging (ABI) and develops contrast using the rocking curve of an analyser crystal [2]. Using this technique, contributions to image contrast due to the phase and absorption are able to separated. Furthermore, researchers have developed algorithms for quantifying the phase shift using various simplifying approximations [8–11]. This method requires a beam of high spectral purity and small angular divergence thus limiting it to synchrotron sources or monochromated laboratory sources [2, 12].

The second method for performing quantitative XPCI is based upon a technique known as in-line holography [4, 13]. This technique requires a source of high spatial coherence and so must be performed using synchrotron radiation or a microfocal source. Algorithms for phase extraction have largely employed transport of intensity formalism and generally require one or more assumptions to be made. For example, Nugent *et al.* [14] introduced a phase retrieval algorithm for the special case of uniform irradiance. Algorithms were introduced for performing quantitative phase imaging of phase objects [15] and weakly absorbing objects [16]. Paganin *et al.* [17] introduced a formalism for phase extraction based upon the assumption of a homogeneous sample. Gureyev later combined the transport of intensity formalism with the first Born formalism to obtain new phase retrieval algorithms [18, 19]. These techniques have all been extended and we refer the reader to a recent review [20] for a thorough account.

The final method, known as grating interferometry, employs two or three gratings and the principle of Talbot self-imaging. A sample’s phase and absorption information is thus encoded in the relative shift and amplitude, respectively, of fringes formed through self-imaging. Detectors with pixel sizes small enough to resolve the fringes are, however, unsatisfactory for imaging large fields of view with clinically compatible exposure times. Phase stepping [21] and the moiré [22] configuration, developed for use in optical interferometry, have both been employed to overcome this problem. The generation of fringes requires a spatially coherent source or an array or such sources in order to achieve Talbot self-imaging. This method has thus been employed using synchrotron radiation [23, 24] and apertured laboratory sources [25–27].

The principle employed in this paper is called edge illumination XPCI (EIXPCI) [28] and is depicted in Fig. 1(a). EIXPCI works by projecting an X-ray beam onto the edge of a sensitive region of a detector. When the arrangement in Fig. 1(a) is tiled as shown in Fig. 1(b), the coded aperture XPCI (CAXPCI) system results. Image formation is equivalent in both systems with the exception that the sample must be scanned through the beam in the EIXPCI case. The analysis presented in this paper thus applies equally to both cases. For the remainder of this paper we refer principally to EIXPCI assuming that the results also apply to CAXPCI.

Superficially, EIXPCI shares similarities with systems from two categories: ABI and grating interferometry. On first inspection, EIXPCI seems to be in the family of grating based interferometers due to the presence of two sets of apertures. The technique is, however, non-interferometric since it does not employ Talbot’s self-imaging phenomenon [29]. In this regard the edge illumination technique is related to that reported recently by Huang *et. al* [30] who, contrary to EIXPCI, employ phase stepping and do not make use of edge illumination. On the other hand, EIXPCI was first developed in analogy with ABI [28] due to the manner in which the pixel edge creates sensitivity to the angle of photon refraction. It is no surprise then, that the quantitative adaptation of EIXPCI shares more in common with ABI than grating interferometry since an image of the object is taken for two orientations of the detector apertures relative to the object. Furthermore, phase stepping is not employed.

In this paper we begin by deriving an algorithm for extracting the gradient of the phase shift due to a sample imaged using the EIXPCI system. We then derive the limitations and important properties of the technique before showing experimental results obtained using synchrotron radiation. Although the theoretical and experimental results of this paper imply the use of highly coherent synchrotron sources, the technique is in fact amenable to use with laboratory sources using the coded aperture technique [31–33].

## 2. Derivation of quantitative method

#### 2.1. Outline of the problem

In the X-ray regime, the refractive index is normally expressed as *n* = 1
− *δ* +*iβ* where
*δ* and *β* are the refractive index decrements.
*δ* can be as large as 10^{−6} whilst
*β* can be as much as three orders of magnitude less than
*δ*. A wavefront which propagates through a thin object will be perturbed by
the object’s complex transmission function *T*(*x,y*)
= exp(−*iϕ*(*x,y*) −
*μ*(*x,y*)) where *ϕ* and
*μ*, both real valued, are defined as:

*k*is the wave number.

*μ*results in attenuation of the wavefront amplitude, which is directly detectable. Please note that our definition of

*μ*differs from the attenuation coefficient as traditionally used in X-ray imaging which is represented by the same symbol. For the sake of reference, the attenuation coefficient is given by 2

*kβ*(

*x,y,z*). Generally only the gradient or Laplacian of

*ϕ*can be measured using one of the previously mentioned quantitative XPCI techniques.

As mentioned previously, the proposed quantitative imaging technique is based upon the EIXPCI method [28]. The system is designed to perform XPCI with laboratory sources, however in the first instance we develop the theory of quantitative imaging assuming an idealised monochromatic point source. In practice, a synchrotron X-ray source can be assumed to closely approximate this condition. EIXPCI is generally performed using a system illustrated schematically in Fig. 1(b), which employs two sets of apertures. The set of apertures labeled *A*_{1} in Fig. 1(b) is called the pre-sample aperture as it is placed immediately upstream of the sample. The set of apertures labeled *A*_{2} is called the detector aperture as it is positioned as close to the flat panel detector as possible. The projected pitches of *A*_{1} and *A*_{2} match that of the flat panel pixel width.

The key to EIXPCI is the principle of pixel edge illumination [28, 34]. This condition is achieved by offsetting apertures *A*_{1} and *A*_{2} from each other as demonstrated in Fig. 1(b). The partial illumination condition is usually quantified by the illuminated pixel fraction (IPF) [29] which is defined as the total integrated X-ray intensity normalised by its maximum. The maximum pixel intensity is usually observed when the two apertures are in alignment. Although an EIXPCI system sensitive to phase gradients in two dimensions has been developed [35, 36], the system in Fig. 1(b) is sensitive to phase gradients in one direction only. As a result it is henceforth assumed, for the mathematical derivations only, that the imaging system and objects extend uniformly out of the page in Fig. 1(b). In practice, however, the stationary phase approximation [37, Pgs. 29–34] shows that to within a good approximation, a region of a sample generates contrast only in the row of pixels that it projects onto. This allows the assumption of a sample uniform in one direction to be made whilst at the same time forming an image of the sample.

#### 2.2. Derivation for object with linear phase and absorption profile

For simplicity and without loss of generality, while deriving the equations for quantitative EIX-PCI we consider a single pair of apertures as demonstrated in Fig. 2. Quantitative EIXPCI is performed by imaging the same object using two complementary positions of *A*_{2}. Both configurations are set to an IPF of 0.5, yet will result in inverted contrasts. For a non-absorbing, prism-like, object depicted in Fig. 2, the configuration denoted by *I*_{−} will result in a detected signal which is lower than the flat field whilst the *I*_{+} configuration will result in a detected signal which is greater than the flat field. If an absorbing object were employed, *I*_{−} and *I*_{+} would also be affected by absorption. We will derive the equations of the quantitative method by assuming a monochromatic point source. Within the paraxial (i.e., small angle) approximation, a point source results in an X-ray intensity which is uniform in the *y* direction of Fig. 2. Then, by assuming that the imaging system and object are also uniform in the *y* direction we need only consider variations of intensity in the *x* direction. If the complex amplitude of X-rays incident upon *A*_{2} is given by *U*(*x*) then, assuming a pixel of height *P*, *I*_{−} and *I*_{+} are given by:

*M*is the system magnification given by (

*z*+

_{so}*z*)/

_{od}*z*and

_{so}*z*and

_{so}*z*are defined in Fig. 2.

_{od}We now turn to the task of calculating the complex amplitude
*U*(*x*). We begin by dropping the *y* dependence of
the complex transmission function thus re-writing it as *T*(*x*).
Furthermore, in the object space we use the symbol *ξ* instead of
*x* to distinguish between the object and detector spaces. We also assume that the
phase and absorption functions may be described locally by a linear function giving the complex
transmission function as:

*ξ*is used to represent a shift of the sample, the Taylor series expansion is taken about

_{s}*ξ*= 0 and the parameter

*ξ*is used to shift the Taylor series representation of the transmission function. The series expansion is centered on

_{i}*ξ*= 0 as this coincides with the centre of the pre-sample aperture. Note that Eq. (3) means that we assume that the sample has constant phase and absorption gradients within the transmitting region of the pre-sample aperture. Then, by applying the paraxial approximation to the Fresnel-Kirchhoff diffraction integral the complex amplitude incident upon

*A*

_{2}may be found according to [38]:

*U*(

*x*)|, complex phase terms dependent upon

*y*have been omitted and

*U*

_{0}represents the amplitude of the incident wave emitted by the source. As in Ref. [38] we employ the stationary phase approximation (SPA) to yield the first order asymptotic solution to Eq. (4) as

*x*∈ [−

*MW*/2 − 1/

*k*(

*∂ϕ*/

*∂ξ*)|

^{ξs}

*z*,

_{od}*MW*/2 − 1/

*k*(

*∂ϕ*/

*∂ξ*)|

^{ξs}

*z*]. Outside of this range the first order asymptotic solution is zero. Returning now to the ultimate objective of evaluating

_{od}*I*

_{−}and

*I*

_{+}, substituting Eq. (6) into Eqs. (2) yields:

*U*

_{0}|

^{2}

*P*exp(2(

*∂μ*/

*∂ξ*)|

^{ξs}

*ξ*− 2

_{i}*μ*(

*ξ*))/((

_{s}*z*+

_{so}*z*)2

_{od}*z*(

_{so}*∂*

*μ*/

*∂ξ*)|

^{ξs}) and ℬ = exp(−2(

*∂μ*/

*∂ξ*)|

^{ξs}

*z*(1/

_{so}z_{od}*k*)(

*∂ϕ/∂ξ*)|

^{ξs}/(

*z*+

_{so}*z*)). Any X-ray transmitted by

_{od}*A*

_{1}and not absorbed by the sample should be detected in either the

*I*

_{−}or

*I*

_{+}configurations, so long as it is not scattered beyond the sensitive region of the pixel of both configurations. Further, any redistribution of X-rays between the

*I*

_{−}and

*I*

_{+}configurations not due to absorption may be assumed to be due to a sample phase gradient. These assumptions, and the mathematical form of Eqs. (7), lead us to form the sum and difference of these two quantities as:

*I*

_{0}as it is equivalent to the flat field in a traditional absorption image. In order to perform quantitative XPCI we must extract the quantity 1/

*k*(

*∂ϕ/∂ξ*)|

^{ξs}. We proceed by noting that an image, as opposed to a profile, may be obtained using one of two methods, either independently or in combination. One method makes use of periodic apertures and a flat panel detector as demonstrated in Fig. 1(b). The other method employs a technique known as dithering whereby the object is scanned relative to the imaging system. These approaches are equivalent and, in keeping with the current derivation, we choose to scan the object through a single aperture pair. We thus assume that we have access to

*I*

_{−}and

*I*

_{+}for object positions

*ξ*= −Δ

_{i}*ξ*, 0, Δ

*ξ*. This allows us to determine (

*∂μ*/

*∂ξ*)|

^{ξs}as:

*ξ*= −Δ

_{i}*ξ*, 0 in which case (

*∂μ*/

*∂ξ*)|

^{ξs}may be found slightly less accurately as:

*ξ*= 0 as:

*∂μ*/

*∂ξ*)|

^{ξs}and since

*W*is known, (1/

*k*)(

*∂ϕ/∂ξ*)|

^{ξs}may be found according to:

*x*) ≈

*x*, cosh(

*x*) ≈ 1 +

*x*

^{2}/2 and log(1 +

*x*) ≈

*x*for

*x*< 0.5, Eq. (15) can be found in a simpler form, valid so long as ${\frac{\partial \mu}{\partial \xi}|}^{{\xi}_{s}}W<1/2$, as:

*β*is non-zero. When

*β*is zero, the entire derivation is considerably simpler resulting in:

## 3. Examples and analysis

#### 3.1. Limitations of the technique

The first restriction of the technique is that the object’s phase and absorption functions (i.e., *ϕ* and *μ*) should be well approximated locally by a linear function. Furthermore, this linear approximation should be reasonable over the range of the sample aperture opening (*W*). This requirement is, however, common to all grating techniques since a projected fringe samples the average phase and absorption properties over a region of a sample. Secondly, Eqs. (11) and (15) were derived under the assumption that −*MW*/2 − 1/*k*(*∂ϕ*/*∂ξ*)|^{ξs} *z _{od}* ≥ −

*MW*and

*MW*/2 − 1/

*k*(

*∂ϕ*/

*∂*

*ξ*)|

^{ξs}

*z*≤

_{od}*MW*. This assumption is employed in the evaluation of Eqs. (7) but it also follows from energy conservation. In particular, if photons deviate outside of the union of the two exposed pixel regions in Fig. 2 then the quantity

*I*

_{+}+

*I*

_{−}will not be equivalent to the conventional absorption image of the object. These restrictions require that:

^{−5}/

*δ*

*m*

^{−1}. Given that

*δ*is seldom greater than 10

^{−6}this restriction will be satisfied in most practical applications. Furthermore, if a typical laboratory based system were used with

*z*= 1.6m and

_{so}*z*= 0.4m [36], the object slope should not exceed the less restrictive value of 4 × 10

_{od}^{−5}/

*δ*, assuming the same value of

*W*.

The next restriction concerns the gradient of the absorption function *μ*(*ξ*). For an object with properties described precisely by Eq. (3), the measured values of *I*_{+} and *I*_{−} will differ slightly from those predicted by Eqs. (7) due to the SPA employed in the evaluation of Eq. (6). *I*_{+} and *I*_{−} are, however, very well approximated by the expressions in Eqs. (7) and we demonstrate this by an example, employing a wedge-like object with thickness described by *h*(*ξ*) = 5*ξ* + 1.5 × 10^{−4} where all quantities are in m. *δ* was fixed at 10^{−6} whilst *β* was varied. The system parameters were *z _{so}* = 1.6m,

*z*= 0.4m,

_{od}*W*= 40

*μ*m, Δ

*ξ*= 10

*μ*m and the source had a photon energy of 20keV. In this analysis, the phase extraction algorithm was applied to data found by direct numerical evaluation of Eq. (4), rather than applying the SPA. We then seek to quantify the error in (

*∂ϕ*/

*∂ξ*)|

^{ξs}which results from employing the SPA in deriving the phase and absorption extraction formulae.

For simplicity, we introduce the quantity *ρ* to denote the ratio (*I*_{+} − *I*_{−})/(*I*_{+} + *I*_{−}), evaluated using the SPA. We then introduce the quantity *ε* to represent the perturbation to *ρ*, which becomes *ρ*(1+*ε*), when *I*_{+} and *I*_{−} are calculated by evaluating Eq. (4) directly. Figure 3 contains a plot of *ε* for a range of values of *β* and shows that the SPA indeed leads to a low value of *ε* across the range of values of *β* considered. *ε* decreases as *β* increases to 10^{−7}. This is because the oscillations present in the actual X-ray intensity incident upon the detector aperture, but not represented by the SPA, become smaller due to the increased absorption. *ε* begins to increase again when the absorption becomes so high that the oscillation in the X-ray intensity achieves a magnitude on the same order as the average intensity predicted using the SPA.

In the next example, we show that the value of (*∂ϕ*/*∂ξ*)|^{ξs}, extracted from field data calculated by direct evaluation of Eq. (4), departs steadily from the true value as *β* increases, as shown in Fig. 4, even though *ε* decreases. In the absence of noise, the technique calculates (*∂**μ*/*∂ξ*)|^{ξs} very accurately. Thus, the error in the predicted value of (*∂ϕ*/*∂ξ*)|^{ξs} is due to the absolute perturbation to *ρ* which occurs when field data calculated using direct evaluation of Eq. (4) is used. The absolute error in (*∂ϕ*/*∂ξ*)|^{ξs} is thus the difference between the values found when *ρ* and *ρ*(1 +*ε*) are used as input to Eq. (15). By assuming that, for large *x*, cosh(*x*) ≈ sinh(*x*) and for small *x*, log(1+*x*) ≈ *x*, it can be shown that, to a good approximation, the absolute error in (*∂ϕ*/*∂ξ*)|^{ξs} is given by:

*I*

_{−}and

*I*

_{+}found by evaluating Eq. (4) directly. Examining this equation it is evident that the error is introduced due to the large disparity between

*I*

_{+}and

*I*

_{−}which arises for large values of (

*∂μ*/

*∂ξ*)|

^{ξs}. This is demonstrated for this example by the plot of Fig. 5 which shows how (1 −

*I*

_{−}/

*I*

_{+})/((1/

*k*)(

*∂μ*/

*∂ξ*)|

^{ξs}) varies with

*β*for the wedge-like example considered. The acceptable value for (1 −

*I*

_{−}/

*I*

_{+})/((1/

*k*)(

*∂μ*/

*∂ξ*)|

^{ξs}) will vary with the accuracy required for the extracted phase. For design purposes, if an upper bound is placed upon (

*∂μ*/

*∂ξ*)|

^{ξs}, the maximum allowable value of

*W*can be found easily by assuming

*δ*= 0 and substituting Eqs. (7) into the expression (1 −

*I*

_{−}/

*I*

_{+})/((1/

*k*)(

*∂μ*/

*∂ξ*)|

^{ξs}).

As a final restriction we note that even in the limit of zero absorption, the estimated value for (*∂ϕ*/*∂ξ*)|^{ξs} in Fig. 4 is slightly below the real value of 5×10^{−6}. This comparatively small error is again due to the SPA which does not model diffraction by the pre-sample aperture. As a result, the estimated value of (*∂ϕ*/*∂ξ*)|^{ξs} is seen to have a small dependence on wavelength. Figure 6 contains a plot of (*∂ϕ*/*∂ξ*)|^{ξs} against photon energy for the wedge-like object considered previously with *β* = 10^{−8} and all other parameters as specified previously. In practice, this error will be less significant than in the example as the effects of diffraction are less significant when sources with finite spectral width and spatial extent are employed.

#### 3.2. Experimental setup

Experimental verification of the phase extraction technique was performed on the SYRMEP bending magnet of the Elettra synchrotron radiation facility in operation in Trieste, Italy. The beamline is described in more detail elsewhere [39, 40], we give here only the details relevant to the experiment. The sample stage was located, in the experimental room, approximately 22m from the apparent X-ray source which has full width at half maximum dimensions of 0.280(horizontal)×0.080(vertical) mm^{2}. This results in a usable beam of dimension approximately 120×4 mm^{2}. A channel-cut Si (1,1,1) crystal monochromates the beam to nominal photon energy of 20 keV with a fractional bandwidth of 0.2%. A photon counting, linear array silicon microstrip detector known as PICASSO was employed [41]. The detector works in the so-called “edge-on” configuration and provides an array of 2368 pixels 50*μ*m wide and 300*μ*m high. An important property of the PICASSO detector is that global and channel specific thresholds can be adjusted in order to have negligible pixel cross talk [42, 43].

The experimental system is shown schematically in Fig. 7. The diagram shows the narrow dimension of the beam and thus the beam extends 120mm into the page. The system in Fig. 7 is sensitive to the phase gradients in the vertical direction and this is also the direction in which the sample was scanned. A 20*μ*m slit was used as the pre-sample aperture, thus setting the value of *W* to 20*μ*m. A single edge was sufficient to act as the detector aperture and this was simply translated vertically to switch between the *I*_{+} and *I*_{−} configurations. A single edge is able to be used since a single X-ray beam is used, in contrast to the plurality of beams in the system shown in Fig. 1(b). If the system in Fig. 1(b) were employed, a larger value of *W* would need to be employed to ensure that nearly all photons which propagate through the object are counted in either *I*_{+} or *I*_{−}. With the edge in place against the detector, each pixel recorded approximately 4.5×10^{4} counts per second.

Figure 8 shows a photograph of the sample which was used in the experiment. It was composed of six filaments of varying diameter and material properties as outlined in Table 1 in which each filament is labeled with a letter from A to F which is used throughout the remainder of this paper. The diameters were estimated from the experimental results shown in Fig. 9. The sample holder was positioned in the experimental setup of Fig. 7 such that the wires were approximately normal to the page and were thus roughly parallel with the beam and aperture edges. 770 vertical scan positions were taken for both the *I*_{+} and *I*_{−} configurations. All 770 scan positions were acquired for the *I*_{+} configuration before changing into the *I*_{−} configuration and returning the sample to its initial position. A scan step of 10*μ*m was employed however several scan steps were passed, without acquiring a detector signal, when the wires were not within the beam. An exposure time of 1s was employed and each of *I*_{+} and *I*_{−} took approximately 35 minutes in total to acquire. A motorised sample stage (Newport IMS300V) was employed which has a unidirectional reputability of 0.5*μ*m and a guaranteed on-axis accuracy of 10±5*μ*m. The reputability of the sample position is important to ensure that the *I*_{+} and *I*_{−} images are correctly registered.

#### 3.3. Experimental results

Figure 9 shows images representing the raw data acquired during the experiment with the filaments arranged in the same order as in Fig. 8. Each row of pixels represents an acquisition from the detector for a particular vertical position of the sample. The *I*_{+} and *I*_{−} images were normalised independently by a row of pixel values found by averaging over a sample-free region, denoted by boxes in each image. The images exhibit a transition from predominantly phase contrast, in the case of boron (lower filament), through to strong absorption contrast in the case of titanium (upper filament). Note that the 5*μ*m diameter tungsten core is clearly visible in the image of the boron filament. Figure 10 shows line plots of the intensity along a single column of both images in Fig. 9 and a plot of their sum. Note that the left side of the line plots correspond to the upper part of the images in Fig. 9.

Figure 11 contains images of both *μ* and *∂ϕ*/*∂ξ* which have been found by using the data from Fig. 9 as input to Eqs. (14) and (15) respectively. Note that all filaments are easily resolved in the phase image, whilst only the three most absorbing filaments are resolvable in the absorption image. The absorption profiles for filaments D, E and F (i.e., the right most three) in Fig. 10 exhibit anomalies in the form of erroneous peaks. It is likely that these are due to small misalignments arising due to the *I*_{+} and *I*_{−} images being taken during separate scan sequences. It should, however, be noted that this anomaly impacts minimally upon extracted phase in Fig. 12. This demonstrates the robustness of the phase retrieval algorithm.

Fig. 12 compares the extracted phase gradient with the analytic value for (*∂ϕ*/*∂ξ*)|^{ξs} found by differentiating *ϕ* in Eq. (1) directly, assuming a cylindrical sample. The analytic value for the boron filament was calculated assuming a 5*μ*m diameter tungsten core. Each of the analytically calculated phase gradients reach infinity at the filament edges. The extracted phase gradient does not, however, reach infinity. There are a few reasons for this, firstly, a degree of smoothing is inevitable since the X-ray beam used in the imaging technique has a finite width and thus senses a segment of the filament. Secondly, the extracted value of (*∂ϕ*/*∂ξ*)|^{ξs} relies upon the extracted value of (*∂μ*/*∂ξ*)|^{ξs} which is calculated by a central difference calculation. The central difference calculation is accurate only for smooth functions and breaks down when a sharp transition occurs, such as at the the filament edge. Despite these limitations, the extracted phase gradient is very close to the analytic phase gradient. Plot F of Fig. 12 contains two sharp peaks around *ξ* = 0 due to the 5*μ*m tungsten core on which the boron is constructed. The core is below the spatial resolution of the imaging system and thus why it appears as a spike.

## 4. Conclusions

We have presented the derivation, analysis and experimental verification of a quantitative X-ray phase contrast imaging technique. The experiment was performed using synchrotron radiation, yet the technique has been adapted to be used with laboratory sources using the coded aperture technique [33]. The synchrotron experiments were important in establishing the absolute accuracy of the method by eliminating spectral averaging. The experimentally determined phase gradients matched the theoretically expected results very closely. The technique also works well in close proximity to the infinite phase gradients near the filament edges. This work provides the theoretical foundation for a number of experiments and theoretical developments, in both planar and three dimensional imaging, currently in progress within our laboratory.

## Acknowledgments

This work was funded by the UK Engineering and Physical Sciences Research Council ( EP/G004250/1 and EP/I021884/1). K.I. was supported by the Wellcome Trust ( 085856/Z/08/Z) and P.M. is supported by a Discovery Early Career Research Award from the Australian Research Council ( DE120101331). F.L. is supported by the Prof. Giulio Brautti PhD Memorial Fellowship. We would like to thank the personnel from the Elettra synchrotron and the University of Trieste working on the SYRMEP beamline for assistance with obtaining the experimental results.

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