Tutorials on X-ray Phase Contrast Imaging: Some Fundamentals and SomeConjectures on Future Developments

David M. Paganin

School of Physics and Astronomy,Monash University, Victoria 3800,Australia∗

Daniele Pelliccia

Instruments & Data Tools Pty Ltd, Victoria 3178,Australia†

These tutorials introduce some basics of imaging with coherent X-rays, focusing onphase contrast. We consider the transport-of-intensity equation, as one particularmethod for X-ray phase contrast imaging among many, before passing on to theinverse problem of phase retrieval. These ideas are applied to two-dimensionaland three-dimensional propagation-based phase-contrast imaging using coherentX-rays. We then consider the role of partial coherence, and sketch a generic meansby which partially coherent X-ray imaging scenarios may be modelled, using thespace–frequency description of partial coherence. Besides covering fundamentalconcepts in both theory and practice, we also give opinions on future trends in X-rayphase contrast imaging including X-ray tomography, and comparison of differentphase contrast imaging methods. These tutorials will be accessible to those witha basic background in optics (e.g. wave equation, Maxwell equations, Fresnel andFraunhofer diffraction, and the basics of Fourier and vector analysis) and interac-tions of X-rays with matter (e.g. attenuation mechanisms and complex refractive index).

Fifteen video lectures, based directly on these notes, are at: https://bit.ly/2GdoVg8

We humbly dedicate these notes to the memory of Claudio Ferrero.

CONTENTS

I. Introduction to these tutorials 2

I X-ray imaging basics 2

II. Theory 2A. Vector vacuum wave equations 2B. Scalar vacuum wave equation & complex

wave-function 3C. Physical meaning of intensity and phase 3D. Fully coherent fields 3E. Coherent paraxial fields 4F. Projection approximation & absorption contrast 5G. Fresnel diffraction & propagation-based phase

contrast 6

III. Practice 8A. Validity of the projection approximation 8B. X-ray tomography beyond the projection

approximation 9C. Describing the propagation through thick samples:

multi-slice approach 10

II Elements of X-ray phase retrieval 11

∗ david.paganin@monash.edu† daniele@idtools.com.au

IV. Theory 11

A. Transport-of-intensity equation (TIE) 11

B. Arbitrary imaging systems 12

C. Arbitrary linear imaging systems 12

D. Arbitrary linear shift-invariant imaging systems 13

E. Transfer function formalism 14

F. Phase contrast 14

G. Forward and inverse problems 15

H. Two inverse problems 16

I. Transport-of-intensity phase retrieval 16

J. The inverse problem of tomography 17

V. Practice 17

A. A quick survey of modern X-ray phase-contrastimaging methods 17

B. Phase gradient methods 18

C. Analyser-based and grating-based imaging 19

III Partial coherence for arbitraryphase contrast imaging systems 20

VI. Theory 20

A. Partial coherence 20

B. Modelling a wide class of partially coherent X-rayphase contrast imaging systems 21

C. Speculations regarding future trends 24

Acknowledgements 26

References 26

arX

iv:1

902.

0036

4v2

[ee

ss.I

V]

27

Feb

2019

2

I. INTRODUCTION TO THESE TUTORIALS

We consider the field of X-ray phase contrast imagingfrom a tutorial perspective. We cover some basics of thefield, augmenting our discussions throughout with specu-lations regarding future lines of development of the field.The primary intended audience is those commencing re-search in the field, although it is our intention that theselargely self-contained notes be more broadly accessible.

Each of the three parts of our tutorial commences witha theory component focused upon the mathematical-physics underpinning of the topics treated therein. Thefirst two parts are rounded out with a complementarycomponent giving practical applications and examples.

The first part of these notes deals with X-ray imagingbasics, sketching a passage from the Maxwell equations ofclassical electrodynamics, through to the paraxial waveequation describing coherent scalar X-ray fields. We alsointroduce the projection approximation, Fresnel diffrac-tion, absorption contrast and phase contrast. We thenexamine, from a practical perspective, the validity con-ditions of the projection approximation, including theconditions under which this approximation is likely tobreak down. Some attention is also given to the ques-tion of tomography beyond the projection approxima-tion, including the roles of diffraction tomography andthe multi-slice approach.

The second part deals with elements of X-ray phasecontrast imaging and the associated inverse problemof phase retrieval. We begin with an outline ofthe transport-of-intensity equation, which is tied toone of the common phase-contrast methods, namelypropagation-based X-ray phase contrast. Rather thansubsequently considering in detail a multiplicity of otherpowerful methods for X-ray phase contrast imaging, weinstead generalise a wide class of such phase contrastimaging systems, by considering many of them to be par-ticular examples of shift-invariant coherent linear imag-ing systems. We give some time to considering the as-sociated transfer function concept, and the realisationof X-ray phase contrast imaging in such a general set-ting. We indicate some key concepts in the underpinningtheory of forward problems and inverse problems, be-fore considering the inverse problem of phase retrieval.Two particular examples of phase retrieval are brieflyconsidered, namely transport-of-intensity phase retrievalin both two and three spatial dimensions. The prac-tice component then gives broader consideration to thesuite of available X-ray phase contrast imaging methods,and discusses some similarities between certain of thesemethods. We emphasise that no one method of X-rayphase contrast imaging is superior to all others in all cir-cumstances, arguing rather that each have their relativestrengths and weaknesses.

The third and final part considers partial coherence,with particular reference to X-ray phase contrast imag-

ing using partially coherent radiation processed via arbi-trary linear imaging systems. We seek to give a generalmeans to theoretically and computationally model a verylarge class of such X-ray phase contrast imaging systems,both those that currently exist, and many of those thatmay be developed in the future. The key underpinningidea is the space–frequency description of partial coher-ence, whereby one has a statistical ensemble of strictlymonochromatic fields at each spatial frequency, indepen-dently propagating through a given generalised X-rayphase contrast imaging system. This allows one to deter-mine, in an efficient manner, the resulting spectral den-sity (i.e. ensemble averaged intensity) at any point in theimaging system, together with the transport through thesystem of coherence functions such as the cross-spectraldensity, the Wigner function and the ambiguity function.Lastly, we give a number of opinions and speculations re-garding possible future developments in the field.

Further detail, on many of the topics presented here,is available in the text by Paganin (2006). Note also thatthese tutorials do not claim in any way to be a represen-tative overview of the field. Rather, they are intendedas an introductory overview of certain key aspects of thefield, which contains enough entry points to the publishedliterature to empower a journey of further exploration.

Part I

X-ray imaging basicsII. THEORY

We cover some basics of coherent X-ray imaging, in-cluding the wave equations for X-ray waves and theirinteractions with matter, the projection approximation,Fresnel diffraction and phase contrast.

A. Vector vacuum wave equations

The Maxwell equations, which govern the evolution ofclassical electromagnetic fields in space and time, lead tothe following d’Alembert equations for the electric fieldE(x, y, z, t) and magnetic field B(x, y, z, t) in free space:(

1

c2∂2

∂t2−∇2

)E(x, y, z, t) = 0, (1)(

1

c2∂2

∂t2−∇2

)B(x, y, z, t) = 0. (2)

Here, (x, y, z) are Cartesian spatial coordinates, t is time,c is a speed given by

c =1

√µ0ε0

, (3)

3

the Laplacian in three spatial dimensions is

∇2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2, (4)

ε0 is the electrical permittivity of free space and µ0

is the magnetic permeability of free space. Systeme-Internationale (SI) units are used consistently. The no-tation used throughout is the same as Paganin (2006).

The above equations imply two facts which were quiterevolutionary when first discovered: (i) Electromagneticdisturbances propagate as waves in vacuum; (ii) thespeed c of these electromagnetic waves, given by theMaxwell relation (Eq. 3), coincides so closely with thespeed of light in vacuum, to suggest that light is an elec-tromagnetic wave. In the late nineteenth-century contextin which it was derived, this then-radical observation uni-fied what were previously thought to be three separatebodies of physics knowledge: electricity, magnetism and(visible light) optics.

This was indeed a colossal moment in the history ofphysics. Before the advent of the Maxwell equationsand the associated discovery that visible light is an elec-tromagnetic disturbance, there were five separate theo-ries describing aspects of the physical world: electricity,magnetism, (visible light) optics, thermodynamics andmechanics. With both the Maxwell equations and thediscovery that light is an electromagnetic wave, the firstthree of these theories united into one overarching theoryof electromagnetism and electromagnetic waves. Such aunification remains a guiding light in modern quests fora unification of quantum theory with Einstein’s gravita-tional theory of general relativity.

Returning to the main thread of our argument, we nowknow that the class of electromagnetic waves is not ex-hausted by those that are visible to the human eye. Ofparticular focus to us are X-ray electromagnetic waves.

B. Scalar vacuum wave equation & complex wave-function

Equations 1 and 2 are a pair of vector equations, or,equivalently, a set of six scalar equations: three for theCartesian components of the electric field, and three forthe Cartesian components of the magnetic field. Each ofthese six scalar vacuum field equations has the form:(

1

c2∂2

∂t2−∇2

)Ψ(x, y, z, t) = 0. (5)

It is convenient to treat Ψ(x, y, z, t) as a complex func-tion, termed the “wave function”, which describes the X-ray field. Only the real part of this wave-function is phys-ically meaningful, but we will not need to make use of thisfact at any point in these notes. By transitioning froma vector-wave description to a scalar-wave description ofthe X-ray field, polarisation is implicitly neglected (ora single linear polarisation is implicitly assumed). This

assumption is often reasonable in many paraxial imag-ing and diffraction contexts. However, note that thereare many cases (e.g. magnetic scattering of circularly-polarised X-rays, and dynamical diffraction from near-perfect crystals) where the effects of X-ray polarisationmust be taken into account.

C. Physical meaning of intensity and phase

At each point (x, y, z) in space, for each instant of timet, Ψ(x, y, z, t) = 0 is a complex number. Complex num-bers have magnitude and phase, so we may write:

Ψ(x, y, z, t) =√I(x, y, z, t) exp[iφ(x, y, z, t)]. (6)

The magnitude of Ψ(x, y, z, t) has been written as√I(x, y, z, t), so that

I(x, y, z, t) = |Ψ(x, y, z, t)|2, (7)

where I(x, y, z, t) is the intensity of the field. The phaseof Ψ(x, y, z, t) = 0 has been denoted φ(x, y, z, t). Forthe instant of time t, surfaces of constant phase maybe identified with wave-fronts of the X-ray field. Thesefronts move at (extremely close to) the speed of light invacuum1, in a direction that is typically away from thesource generating those waves – see Fig. 1(a).

D. Fully coherent fields

Assume the field to be strictly monochromatic, andtherefore perfectly coherent, so that its time developmentat any point in space oscillates with a fixed angular fre-quency ω:

Ψ(x, y, z, t) = ψω(x, y, z) exp(−iωt), (8)

where

ω = 2πf = ck, (9)

f denotes frequency, and k is the wave-number corre-sponding to vacuum wavelength λ:

k = 2π/λ. (10)

Substitution of Eq. 8 into Eq. 5 gives the Helmholtzequation in vacuum:(

∇2 + k2)ψω(x, y, z) = 0. (11)

1 We are here alluding to the subtlety that the speed of light invacuum is in general less than the speed of a plane wave in vac-uum. See Giovannini et al. (2015) for further information on thisfascinating point.

4

FIG. 1 X-ray wave-fronts. (a) At a given instant of timet = t0, the intensity of an X-ray field is I(x, y, z, t =t0). The wave-fronts, namely the surfaces of constant phaseφ(x, y, z, t = t0), are indicated by the series of curved surfaces.At any point A the wave-fronts move away from the sourceas time increases. The direction of energy flow at point A is∇φ(x, y, z, t = t0). The associated current density (Poyntingvector) is S(x, y, z, t = t0) ∝ I(x, y, z, t = t0)∇φ(x, y, z, t =t0), which is perpendicular to each wave-front. The currentdensity is everywhere tangent to the associated streamlines,such as the streamline BAC passing through the point A.(b) For a paraxial field, all wave-fronts may be obtained byslightly deforming planar surfaces perpendicular to the opticaxis (i.e. the z axis, which runs from left to right). All Poynt-ing vectors are close to being parallel to the optic axis (hencethe term paraxial). The associated streamlines such as DEare well approximated by straight lines parallel to the opticaxis. Image adapted from Paganin (2006).

This vacuum wave equation for coherent scalar elec-tromagnetic waves may be generalised to account forthe presence of material media. Such media will be as-sumed to be static, non-magnetic, and sufficiently slowlyspatially varying, so that they may be described by aposition-dependent refractive index nω(x, y, z). This re-fractive index alters the vacuum wavelength as follows:

λ −→ λ

nω(x, y, z), (12)

hence

k −→ k nω(x, y, z). (13)

The vacuum Helmholtz equation (Eq. 11) therefore be-comes the Helmholtz equation in the presence of non-magnetic scattering media:[

∇2 + k2n2ω(x, y, z)

]ψω(x, y, z) = 0. (14)

See e.g. Paganin (2006) for a full derivation of the aboveequation, which elaborates on the key assumptions that

the scattering medium be (i) linear, (ii) isotropic, (iii)static, (iv) non-magnetic, (v) have zero charge densityand (vi) zero current density, and (vii) be spatially slowlyvarying in its material properties.

As an interesting aside, note that the above equation ismathematically identical in form to the time-independentSchrodinger equation for non-relativistic electrons in thepresence of a scalar scattering potential (this latter equa-tion assumes that the effects of electron spin can be ig-nored, and that the material with which the electron in-teracts is non-magnetic). Hence, the research fields of co-herent X-ray optics and transmission electron microscopyhave much in common. Indeed, we are of the view thatfundamental progress in both fields would advance morerapidly if more workers from each field were to familiarisethemselves with work from both fields.

As a second aside, we recall the statement invokedin deriving Eq. 14, namely the three assumptions thatthe scattering “media will be assumed to be static, non-magnetic, and sufficiently slowly spatially varying, sothat they may be described by a position-dependent re-fractive index”. The breakdown of any or all of thesethree key assumptions leads to interesting generalisationsthat will not be considered here. For example,

(i) the breakdown of the first assumption enters us intothe very interesting realm of time-dependent samples, in-cluding those that experience radiation damage duringthe act of X-ray imaging;

(ii) the breakdown of the second assumption is keyto the study of magnetic materials using, for example,circularly polarised X-rays;

(iii) the breakdown of the third assumption will be-come progressively more important as X-ray imaging ispushed more and more often to regions of high resolution,e.g. on nm and smaller length scales.

E. Coherent paraxial fields

With reference to Fig. 1(b), assume our monochro-matic complex scalar X-ray wave-field to be paraxial,in the sense described in the caption to the said fig-ure. Under this approximation it is natural to expressthe complex disturbance ψω(x, y, z) as a product of a z-directed plane wave exp(ikz), and a perturbing envelopeψω(x, y, z). Without any loss of generality, we then have:

ψω(x, y, z) ≡ ψω(x, y, z) exp(ikz). (15)

Conveniently,

|ψω(x, y, z)|2 = |ψω(x, y, z)|2 = Iω(x, y, z), (16)

so that the intensity of the envelope ψω(x, y, z) is thesame as the intensity of ψω(x, y, z).

Now, if Eq. 15 is substituted into Eq. 14, and a termcontaining the second z derivative is discarded as being

5

FIG. 2 From entrance-surface to exit-surface wave-field, un-der the projection approximation. Adapted from Paganin(2006).

small compared to the other terms on account of theparaxial assumption, one obtains:(

2ik∂

∂z+∇2

⊥ + k2[n2ω(x, y, z)− 1]

)ψω(x, y, z) = 0,

(17)where

∇2⊥ ≡

∂2

∂x2+

∂2

∂y2(18)

is the transverse Laplacian (i.e., the Laplacian in the(x, y) plane perpendicular to the optic axis z, so that∇2 = ∇2

⊥ + ∂2/∂z2).We again draw a parallel with quantum mechanics,

noting that Eq. 17 is mathematically identical in formto the time-dependent Schrodinger equation in 2+1 di-mensions (i.e. two space dimensions and one time di-mension), in the presence of a time-dependent scalar po-tential V (x, y, t), if one replaces z with t, and considersn2ω(x, y, z → t)− 1 to be proportional to −V (x, y, t).

F. Projection approximation & absorption contrast

Consider Fig. 2. Here, z-directed monochromatic com-plex scalar X-ray waves illuminate a static non-magneticobject, from the left. By assumption, the object is to-tally contained within the slab of space between z = 0and z = z0 ≥ 0. The object is described by its refrac-tive index distribution nω(x, y, z), which will only differfrom unity (i.e. the refractive index of vacuum) withinthe volume occupied by the object.

We wish to determine the complex disturbance (wave-function) over the plane z = z0, which is termed the“exit surface” of the object, as a function of both (i) thecomplex disturbance over the “entrance surface” z = 0and (ii) the refractive index distribution of the object.

We assume the object to be sufficiently slowly varyingin space, that all streamlines of the X-ray flow may bewell approximated by straight lines parallel to z. Underthis so-called projection approximation, the validity con-ditions for which are further discussed in Sec. III.A be-low, the transverse Laplacian may be neglected in Eq. 17.Thus:

∂

∂zψω(x, y, z) ≈ k

2i[1− n2

ω(x, y, z)]ψω(x, y, z). (19)

For each fixed point (x, y), this is a simple linear first-order ordinary differential equation, which can be imme-diately integrated with respect to z to give:

ψω(x, y, z = z0) ≈ (20)

exp

{k

2i

∫ z=z0

z=0

[1− n2ω(x, y, z)]dz

}ψω(x, y, z = 0).

At this point, it is convenient to introduce a complexform for the refractive index, the real part of which cor-responds to the refractive index; we shall see that theimaginary part of this complexified refractive index canbe related to the absorptive properties of a sample. Withthis in mind, write the complex refractive index as

nω = 1− δω + iβω, (21)

where

|δω|, |βω| � 1 (22)

since the complex refractive index for hard X-rays is typ-ically extremely close to unity. Hence:

1− n2ω(x, y, z) ≈ 2[δω(x, y, z)− iβω(x, y, z)], (23)

where we have discarded terms containing δ2ω, β2

ω andδωβω since these will be much smaller than the termsthat have been retained in the right side of Eq. 23.

If the above expression is substituted into Eq. 20, weobtain

ψω(x, y, z = z0) ≈ ψω(x, y, z = 0) (24)

× exp

{−ik

∫ z=z0

z=0

[δω(x, y, z)− iβω(x, y, z)]dz

}.

This shows that the exit wave-field ψω(x, y, z = z0) maybe obtained from the entrance wave-field ψω(x, y, z =0) via multiplication by a transmission function T (x, y).The transmission function is the second line of Eq. 24.

The position-dependent phase shift

arg T (x, y) ≡ ∆φ(x, y), (25)

due to the object, is:

∆φ(x, y) = −k∫δω(x, y, z)dz. (26)

6

The above expression quantifies the deformation of the X-ray wave-fronts due to passage through the object. Phys-ically, for each fixed transverse coordinate (x, y), phaseshifts (and the associated wave-front deformations) arecontinuously accumulated along energy-flow streamlines(loosely, “rays”) such as AB in Fig. 2. In making allof these statements, it is useful to look back to Fig. 1and recall the direct connection between the phase of acomplex wave-field, and its associated wave-fronts. Thephase shifts—associated with passage of an X-ray wavethrough an object—quantify the wave-front deformationsand associated refractive properties of the object. Also,since we are working with a wave picture rather thanthe less-general ray picture for X-ray light, refraction isassociated with wave-front deformation rather than raydeflection.

Refraction, due to the object, is a property that maybe augmented by the attenuation due to the object. Thislatter quantity may be obtained by taking the squaredmodulus of Eq. 24, to give the Beer–Lambert law:

Iω(x, y, z = z0) (27)

= exp

[−∫µω(x, y, z)dz

]Iω(x, y, z = 0).

Above, we have used the following expression relating theimaginary part βω of the refractive index, to the associ-ated linear attenuation coefficient µω:

µω = 2kβω. (28)

Note for later reference, that Eq. 27 may also be writ-ten in the logarithmic form:

loge[Iω(x, y, z = z0)/Iω(x, y, z = 0)] (29)

= −∫µω(x, y, z)dz.

Equation 27 forms the basis for absorption contrastimaging. In particular, if a two-dimensional position sen-sitive detector is placed in the plane z = z0 in Fig. 2, andthe illuminating radiation has an intensity Iω(x, y, z = 0)that is approximately constant with respect to x and y,then all contrast in the resulting “contact” image willbe due to local absorption of rays such as AB in Fig. 2.While the logarithm of this image is sensitive to the pro-jected linear attenuation coefficient

∫µω(x, y, z)dz, the

contact image contains no contrast whatsoever, due tothe phase shifts quantified by Eq. 26. This lack of phasecontrast, in conventional contact X-ray imaging, is unfor-tunate, since many structures of interest (such as soft bio-logical tissues) are close to being non-absorbing, meaningthat they are poorly visualised, if at all, in absorption-contrast X-ray imaging.

G. Fresnel diffraction & propagation-based phase contrast

Consider Fig. 3, which shows a source A radiating intospace. Optical elements and samples, which may lie be-

FIG. 3 Free-space propagation of X-ray waves. Imageadapted from Paganin (2006).

tween A and the plane z = 0, are not shown. The diffrac-tion problem seeks to determine the wave-field over theplane z > 0, given the disturbance over the plane z = 0.The space z ≥ 0 is assumed to be vacuum, and all wavesin this space are assumed to be both paraxial with re-spect to the optic axis z, and monochromatic.

In the space z ≥ 0 the waves will obey the“nω(x, y, z) = 1” special case of Eq. 17, namely:(

2ik∂

∂z+∇2

⊥

)ψω(x, y, z) = 0. (30)

The solution to the diffraction problem, based on theabove free-space paraxial equation, may be written as:

ψω(x, y, z = ∆) = D∆ψω(x, y, z = 0), ∆ ≥ 0. (31)

Here, D∆ is a (Fresnel) diffraction operator, which actson the unpropagated forward-travelling field ψω(x, y, z =0), propagating it a distance ∆, to give ψω(x, y, z = ∆).An expression for D∆, which may be readily derived fromthe free-space paraxial equation, will be given later.

From the squared magnitude of Eq. 31, it is clear thatthe intensity of the propagated field depends on both theintensity and phase of the unpropagated field. This pointis both trivial—because the right side, of the squaredmodulus of Eq. 31, obviously depends on the phase—andof profound importance, since it implies that the Fresneldiffraction pattern, namely the propagated intensity overthe plane z > 0 in Fig. 3, provides the phase contrastthat was missing from the contact image.

This mechanism, for obtaining intensity contrast (inthe plane z > 0) that is sensitive to phase variations(in the plane z = 0), is known as propagation basedphase contrast. This phenomenon has been known un-der different names for millennia in visible-light optics,

7

e.g. from the heat shimmer over a hot road, and knownfor many decades in both the visible-light microscopy(e.g. Zernike, 1942; Bremmer, 1952) and electron mi-croscopy (e.g. Cowley, 1959) communities. The X-raycommunity only became significantly aware of this phe-nomenon from the early 1990s (e.g. White and Cerrina,1992), including pioneering studies by a number of sci-entists from the European Synchrotron, such as thoseof Snigirev and colleagues (1995), and Cloetens and col-leagues (1996). Other noteworthy X-ray papers from thisperiod include those of Wilkins and colleagues (1996),and Nugent and colleagues (1996).

For the remainder of this subsection, we seek to furtherdevelop the intuition of the reader, regarding the quali-tative nature of propagation-based X-ray phase contrast.

With this end in mind, consider Fig. 4, in which asmall X-ray source S illuminates an object H shown ingrey. The source-to-object distance is denoted by R1

and the object-to-detector distance is denoted by R2.The distance R2 is assumed to be large enough thatpropagation-based phase contrast is manifest over thedetector plane B, but not so large that multiple Fres-nel diffraction fringes are present2. Assuming the objectto be sufficiently thin that the projection approximationholds, one may identify three different features within theobject, which are here labelled J , L and N .

• Features such as J correspond to either the thinobject in projection behaving locally like a con-vex lens, or a point within the volume of an ob-ject which has a local peak of density. Because thereal part of the complex refractive index is less thanunity for X-rays, convex X-ray lenses are defocusingoptical elements (cf. the case for visible light, whereconvex lenses are focusing optics since the real partof the refractive index is greater than unity). SinceJ may be considered as a defocusing feature in theobject, the local ray density at point K on the de-tector will be lessened via the refractive effects ofJ ; hence point K in the detector will have reducedbrightness, on account of the propagation distanceR2 that lies between J and K.

• Features such as L, which may be either a con-cave feature in the projected thickness of the sam-ple or a feature within the sample which has a local

2 More precisely, we are here assuming the Fresnel number NF =Ma2/(λR2) to be much greater than unity. Here, a correspondsto the smallest transverse characteristic feature size in the objectthat is not smeared out by the finite size of the source, R2 is theobject-to-detector distance and M = (R1 + R2)/R1 is the geo-metric magnification. Note also that the concept of the Fresnelnumber is used in a different but related context later in thesenotes, when it is used to consider the conditions under which theprojection approximation is valid for a given sample.

FIG. 4 Geometric-optics diagram to aid understanding ofpropagation-based X-ray phase contrast imaging. Here, S is asmall X-ray source, which illuminates an object shown in grey.The imaging plane is shown to the right of the image. Thesource-to-object and object-to-detector distances are denotedby R1 and R2 respectively. Reproduced with permission fromGureyev et al. (2009).

trough of density, will act as a converging lens forX-rays. Hence the intensity at point M will be in-creased by the effects of refraction by feature L,provided that R2 is large enough for the intensity-increasing effects of the focusing element L to bemanifest at point M on the detector. Note thecrucial role played by the object-to-detector dis-tance R2, through which the wave propagates be-fore reaching the detector.

• One also has propagation-based phase contrast dueto features such as N , which correspond to pointson the edge of the object. Here, “edge” refers to theedge of the object when projected along the opticaxis z. On account of Fresnel diffraction in the slabof vacuum between the object and the detector, thepropagation-based phase contrast signature of anedge such as N will be an increase of intensity atpoint P , with a corresponding decrease at point Q.Such “edge contrast” is a characteristic feature ofpropagation-based X-ray phase contrast.

Before proceeding, we need to briefly outline the ideaof image blurring due to non-zero source size. See Fig. 5.The fact, that the X-ray source is not a point, will leadto some blurring of images formed using such a source.For a so-called “extended incoherent source”, say a pla-nar source of diameter D, one can by definition considereach point on the source to be an independent radiatorof X-rays. Provided that the source is not too large,and that the assumption of independent radiators is rea-sonable, each of the radiators will form a separate im-age of the object, with images due to separate points onthe source being transversely displaced from one another.The net effect, of adding all of the slightly-displaced im-ages formed by each point on the extended incoherent

8

FIG. 5 An extended incoherent source S, with diameter D,leads to source-size blurring of spatial width Deff = DR2/R1

over the detector plane B.

source, is to blur the resulting image obtained over theplane B. The transverse length scale, over which thisblurring takes place, may be obtained via the similar-triangles construction in Fig. 5. Here, we see that thetransverse spatial extent Deff of the source-size inducedblurring, of the image of the object H that is obtainedover the detector plane B, is given by Deff = DR2/R1.This effect is known as “penumbral blurring”. Note thatthe source-size induced blurring becomes progressivelyworse as R2 increases, for fixed D and R1.

Following on from the above aside regarding source-size-induced blurring, we now return to the main threadof our discussion regarding propagation-based phase con-trast. To further develop the reader’s intuition regardingthe key features of such a mechanism for phase contrast—an image-sharpening effect which competes with penum-bral image blurring—consider the simulations shown inFig. 6. These simulations correspond to an X-ray wave-length of 0.5 A, and a fixed source-to-object distance R1

of 10 cm. The two variables are (i) the diameter of thesource D, which decreases from top to bottom in thefigure, and (ii) the object-to-detector propagation dis-tance R2, which increases from left to right. The sim-ulated sample is a solid carbon sphere with diameter0.5 mm. When the source has the relatively large di-ameter of D = 100µm, corresponding to the top rowof Fig. 6, increasing the object-to-detector distance R2

has the expected effect of progressively blurring the im-age of the sphere. A similar trend is seen in the secondrow, corresponding to halving the source diameter. Allof the images in the top two rows of Fig. 6 may be takenas demonstrating absorption contrast alone, with a dark“shadow” of the carbon sphere corresponding to the ab-sorption of X-rays that pass through the sphere. How-ever, in the bottom two rows of the figure, the sourcesize is sufficiently small to have reduced the source-size-induced blurring to such a degree that propagation-basedphase contrast is manifest. Edge contrast, in the sensedescribed earlier in this subsection, is clearly evident inboth of the bottom rows, for object-to-detector propaga-tion distances of 10 cm or greater (columns 2, 3 and 4 ofthe bottom two rows). If the object-to-detector propa-

FIG. 6 Simulated propagation-based X-ray phase contrastimages of a solid carbon sphere of diameter 0.5 mm, corre-sponding to X-rays with wavelength 0.5 A. The source-to-object distance R1 is fixed at 10 cm. The object-to-detectordistance R2 is increased as one moves from left to right. Thesource diameterD decreases as one moves from top to bottom.Reproduced with permission from Gureyev et al. (2009).

gation distance is zero, however, one has a contact imagethat displays no propagation-based X-ray phase contrast(column 1). In all of the above, one has an evident trade-off: R2 must be sufficiently large to obtain propagation-induced phase contrast, while being sufficiently small forthe penumbral blurring to be sufficiently mild that it doesnot wash out the sharpening effect of propagation-basedphase contrast.

Before proceeding, we strongly recommend to readerswho have not previously seen propagation-based X-rayphase contrast images, that they briefly study some ofthe images in one of more of the classic early papers(Snigirev et al., 1995; Cloetens et al., 1996; Wilkins etal., 1996; Nugent et al., 1996). This will further developthe reader’s intuition for the qualitative nature of suchcontrast, beyond what has been sketched in these notes.

III. PRACTICE

A. Validity of the projection approximation

The first practice tutorial discusses the limits ofvalidity of the projection approximation, and someof the consequences of the breakdown of the projec-tion approximation for high resolution X-ray microscopy.

Let us suppose we want to image a cell of thickness T ,

9

FIG. 7 Diagram and notations used in the discussion of theprojection approximation. T is the sample thickness and a isthe feature of interest. With some generalisation, one couldassume a to be the resolution of the imaging system. Imagedrawn by Kristina Pelliccia.

with the aim of resolving an organelle of size a within thecell. The projection approximation in this configurationis valid if we can neglect diffraction effects within thesample, i.e. we can assume that X-rays propagate alongstraight lines in the sample. Radiation of wavelength λscattered by the organelle will have a typical (maximum)diffraction angle of the order of

∆θ =λ

a. (32)

Therefore the maximum spread of the radiation at theexit face of the sample (assuming the organelle to beclose to the entrance face) will be ∆θ T . The projectionapproximation is valid if we can neglect the diffractionspread when compared to the resolution, i.e.

λ

aT � a. (33)

The previous inequality can be redefined in terms of theFresnel number

NF =a2

λT� 1. (34)

Figure 8 shows a contour map of the Fresnel numberas a function of both a and T , calculated setting thewavelength to λ = 1.5 A. Somewhat arbitrarily, the con-tour NF = 10, marked with a thicker line, is chosen asa boundary for the validity of the projection approxima-tion. The region on the right of this line (warm colourssuch as yellow, orange and red) is where the projection

FIG. 8 Contour map of the Fresnel number NF , as a functionof resolution a and sample thickness T , calculated at the X-ray wavelength λ = 1.5 A. The black line marks the contourNF = 10. As a rule of thumb, a Fresnel number below thisvalue is a situation where the projection approximation mightnot hold.

approximation is generally valid. This region correspondsto the range of values attained, for instance, by modernmicro-CT (computed tomography) systems, where reso-lution of a few micrometers and sample thickness of a fewmillimetres are the state of the art.

The region on the left of the NF = 10 contour (coldcolours) is where the projection approximation is at risk.In this region, namely the domain of ultra high reso-lution X-ray microscopy systems typical of synchrotronbeam-lines, the sample thickness becomes very large com-pared to the resolution. Admittedly, in this region laysone of the major strength of X-rays when compared withother probes for microscopy: X-rays can visualise minutedetails within larger samples—for instance single cellswithin a larger tissue—in a less invasive fashion.

Quantitative analysis at such a high resolution levelhowever, requires one to take into account that the pro-jection approximation may no longer hold. Let us brieflydiscuss two consequences of this fact. The first dealswith solving the inverse problem of tomography; the sec-ond implication is relevant when modelling X-ray opticalelements.

B. X-ray tomography beyond the projection approximation

One of the main consequences of the projection ap-proximation is that the attenuation and phase shift ex-perienced by an X-ray monochromatic beam can be ex-pressed as line integrals, as in Eqs 26 and 27. That is,X-ray tomography is based on a geometrical model of the

10

propagation of X-rays through samples.

Therefore, one of the most important consequences—as far as X-ray imaging is concerned—of the failure of theprojection approximation, is that conventional tomogra-phy algorithms must be revisited. Specifically, the wellknown Fourier slice theorem is no longer valid. In thiscase, one must turn to what has been termed “diffrac-tion tomography”, which has found wide applications inoptical 3D imaging of semi-transparent samples.

The concept of diffraction tomography was first intro-duced by Wolf (1969). A recent review of the literatureof diffraction tomography can be found in Muller et al.,2016. Diffraction tomography makes use of the so-calledFourier diffraction theorem, which reduces to the Fourierslice theorem in the geometrical-optics limit (Gbur andWolf, 2001).

C. Describing the propagation through thick samples:multi-slice approach

Simulating and modelling high resolution transmissionX-ray optics, or reflective optics, is a second example ofsituations where the projection approximation generallydoes not hold. Transmission optics such as refractiveX-ray lenses or Bragg–Fresnel lenses can be, to someextent, considered thin in the medium resolution range.High resolution applications however, demand extremelyfine X-ray optical structures (for instance outermostzone of Fresnel or Bragg–Fresnel lenses are in the nmrange). This fact can be appreciated in Fig. 9, which isa close-up view of the contour map in Fig. 8, applicablein the region relevant to high resolution X-ray optics.

Modelling X-ray propagation through such elementsalways requires dropping the projection approximationin favour of a more accurate approach. Furthermore,conventional reflective optics must be considered “thick”in all cases, as obviously the beam angular deviation inreflection is always significant. In all those cases, themulti-slice approximation is a very useful approach.

Originally introduced by Cowley and Moodie (1957and 1959) in the context of Transmission Electron Mi-croscopy, the multi-slice approximation is being increas-ingly used to simulate high resolution X-ray optics andimaging. See for instance Paganin (2006), Martz etal. (2007), Doring et al. (2013) or Li et al. (2017). In-cidentally, and building upon a deliberately-provocativeremark made earlier in these notes, contemporary workin X-ray multi-slice gives an excellent example of howprogress in X-ray optics would be accelerated by moreworkers in this field being familiar with electron op-tics, since the multi-slice method was brought to a veryhigh state of development by the electron-optics commu-nity, decades before the method began to be employed inearnest by the X-ray optics community.

FIG. 9 Contour map of the Fresnel number NF , as a functionof resolution a and sample thickness T , calculated at the X-raywavelength λ = 1.5 A, for the high resolution case applicableto modelling X-ray diffractive optics. As before, the black linemarks the contour NF = 10.

In the multi-slice approach, the thick sample is de-composed into a number of slices along the optic axisdirection. The thickness of each slice should be chosento guarantee that such a slice can be considered opticallythin. This corresponds to NF � 1 for each individualslice. Therefore, for each slice one can assume the pro-jection approximation to be valid.

Following Eq. 24, and dropping the subscript ω forclarity, the transmission function of the slice j can bewritten as:

Tj(x, y) = exp {−ik nj(x, y) ∆z} . (35)

In Eq. 35,

nj(x, y) =

∫ zj

zj−1

n(x, y, z)dz/∆z

≈ n(x, y, z = zj) (36)

is the complex refractive index of slice j, located at thelongitudinal position z = zj and, with similar notation,

Tj(x, y) ≡ T (x, y, z = zj). (37)

The slice thickness is

∆z = zj − zj−1. (38)

Note that, in deriving Eq. 35, by passing from the firstto the second line of Eq. 36, we assumed the refractiveindex of each slice to be independent of z, within thevolume occupied by the said slice. This will be a goodapproximation if the slices are thin enough (compared tothe length scale over which n varies).

11

Under these assumptions, the propagation of the wavefield to the next slice can be performed using Fresnelpropagation in vacuum, using Eq. 31:

ψj+1(x, y) = D∆z [ψj(x, y)Tj(x, y)] . (39)

The multi-slice algorithm applies this procedure itera-tively, to propagate through all slices of the sample. Thepreviously-cited papers of Martz et al. (2007), Doringet al. (2013) and Li et al. (2017) give excellent exam-ples of the application of this very powerful and gen-eral method for considering X-ray interactions with sam-ples, in situations where the projection approximationhas broken down. For those seeking to apply the multi-slice method in an X-ray setting, much can be learnedfrom the electron-optics text by Kirkland (2010).

Part II

Elements of X-rayphase retrievalThe second part of our notes introduces the transport-of-intensity equation, as a means for quantifying the con-trast present in propagation-based X-ray phase contrastimages. We then consider generalised phase contrast X-ray imaging systems, these being an infinite variety ofimaging systems that yield phase contrast in the sensethat they are sensitive to the refractive (phase) effects ofX-ray-transparent samples. Finally, the inverse problemof phase retrieval (namely the decoding of X-ray phasecontrast images to obtain information regarding the ob-ject that resulted in such images) is considered, andapplied to both two-dimensional and three-dimensionalphase-contrast X-ray imaging.

IV. THEORY

A. Transport-of-intensity equation (TIE)

Substitute Eq. 6 into Eq. 30, expand, cancel a commonfactor, and then take the imaginary part. This gives acontinuity equation expressing local conservation of op-tical energy (Teague, 1983; cf. Madelung, 1927), calledthe transport of intensity equation (TIE):

−∇⊥ · [I(x, y, z)∇⊥φ(x, y, z)] = k∂I(x, y, z)

∂z. (40)

Physically, this equation asserts that the divergenceof the transverse Poynting vector (transverse energy-flowvector) S ∝ I∇⊥φ governs the longitudinal rate of changeof intensity. If the divergence of the Poynting vector ispositive, because the wave-field is locally behaving as anexpanding wave, optical energy will be moving away from

the local optic axis and so the longitudinal derivative ofintensity will be negative (local defocusing; see points Jand K in Fig. 4 for an example). Conversely, if the di-vergence of the Poynting vector is negative, because thewave-field is locally contracting, optical energy will bemoving towards the local optic axis and so the longitudi-nal derivative of intensity will be positive (local focusing;see points L and M in Fig. 4 for an example). Indeed,if we speak of the negative divergence “−∇⊥·” as the“convergence”, then the TIE merely makes the intuitivestatement that “the convergence of the transverse Poynt-ing vector is proportional to the z rate of change of in-tensity”: thus (i) a converging wave (positive convergenceor negative divergence) has a positive rate of change ofintensity with respect to z because optical energy is be-ing concentrated (focused) as z increases (see again thepoints L and M in Fig. 4); (ii) conversely, a divergingwave (negative convergence or positive divergence) hasa negative rate of change of intensity with respect to zbecause optical energy is being rarefied (defocused) as zincreases (points J and K in Fig. 4).

The above comments also pertain to the form of theTIE obtained if the finite-difference approximation

∂I(x, y, z)

∂z≈ I(x, y, z + δz)− I(x, y, z)

δz(41)

is substituted into Eq. 40, before being solved for thepropagated intensity, to give the following approximatedescription for propagation-based phase contrast, in theregime of sufficiently small propagation distance δz:

I(x, y, z + δz) ≈ I(x, y, z) (42)

−δzk∇⊥ · [I(x, y, z)∇⊥φ(x, y, z)] .

Propagation-based methods are not the only means bywhich phase contrast can be achieved. Many other ex-tremely important methods exist, including methods util-ising crystals (e.g. Forster et al. (1980)), diffractive imag-ing from far-field patterns (Miao et al. 1999), perfectgratings (Momose et al. 2003; Weitkamp et al. 2005;Pfeiffer et al. 2008), random gratings (Berujon et al.,2012; Morgan et al., 2012; see Zdora (2018) for a com-prehensive review), edge illumination (e.g. Diemoz etal., 2017, and references therein), ptychography (Pfeif-fer, 2018) and of course interferometry (Bonse and Hart,1965). Due to time limitations, these will not be reviewedhere, but we note that (i) some of these methods will bebriefly covered in the practice sessions, in Sec. V; (ii)many of these methods can be considered to be specialcases of the set of all possible linear shift invariant phasecontrast imaging systems, which will be treated later inthe present text. Taken together, the previously listedsuite of methods forms a powerful toolbox for the X-rayimaging of samples, with each method having its partic-ular strengths and limitations. No method is superior toall others in all scenarios and circumstances.

12

FIG. 10 Generalised phase-contrast imaging system. X-raysfrom a source to the far left (not shown) pass through a sample(not shown), the exit surface of which is denoted A. The wave-field over this plane A, which is assumed to be a paraxial beamtravelling in the z direction, is then input into an arbitraryimaging system denoted by the black box. The correspondingoutput complex wave-field exists over the plane B. The stateof the black box is schematically denoted by the coloureddials, representing the control parameters τ1, τ2 etc.

B. Arbitrary imaging systems

We have already seen that the act of free-space prop-agation, from plane to plane, can achieve phase con-trast in the sense that the propagated image (overthe downstream plane, such as that given by the de-tector B in Fig. 4) has a transverse intensity distri-bution that depends on the transverse X-ray phaseshifts is an upstream plane (such as the plane at theexit-surface of the object in Fig. 4). What happensif we generalise this propagation-based X-ray phase-contrast-imaging scenario, to a more general X-rayphase-contrast-imaging setup, by interposing an opticalimaging system in between the object and the detector?

Consider an arbitrary coherent X-ray imaging systemthat takes a two-dimensional monochromatic paraxialcomplex X-ray wave-field ψIN(x, y) as input: this cor-responds to the (x, y) plane labelled “A” in Fig. 10,which is perpendicular to the optic axis z. Assumealso that the state of the imaging system can be char-acterised by a set of real control parameters τ1, τ2, · · · ,with ψOUT(x, y, τ1, τ2, · · · ) being the corresponding com-plex output wave-field.

We may consider the action of this imaging system, inoperator terms3. That is, we may consider the imagingsystem to be described by an operator D(τ1, τ2, · · · ) thatacts on the input field to give the output field. This may

3 For our purposes, an operator “acts” on a given function to givea new function. Thus, if the operator A acts on the functionf to give a different function g, this would be written as Af=g.We follow the usual convention that each operator acts on theelement to the right of it, with the rightmost operator actingfirst: for example, if A,B are two operators, then BAf is thesame as B(Af), so that f is first acted upon by A to give Af ,with the result being subsequently acted upon by B to give BAf .

be written in the following way:

ψOUT(x, y, τ1, τ2, · · · ) = D(τ1, τ2, · · · )ψIN(x, y). (43)

At this stage our imaging system has a very high degreeof generality: its arbitrariness is limited only by the im-plicit assumptions associated with a forward-propagatingmonochromatic scalar input being mapped to a forward-propagating monochromatic scalar output that has thesame energy4.

C. Arbitrary linear imaging systems

Make the further assumption that the imaging systemis linear, i.e. that the output field is a linear functionof the input field. Stated differently, we are here as-suming the superposition principle to hold: if the in-put field is given by the sum of two particular input

fields αψ(1)IN (x, y) + βψ

(2)IN (x, y), where α and β are ar-

bitrary complex weighting coefficients, then the outputfield will (by assumption) always be equal to the sum of

corresponding outputs, i.e. D[αψ(1)IN (x, y)+βψ

(2)IN (x, y)] =

αDψ(1)IN (x, y) + βDψ(2)

IN (x, y) + γ. The complex constantγ, while consistent with the assumption of linearity, willbe set to zero since it is natural to assume that a zero in-put field corresponds to a zero output field. Assume fur-ther that any magnification, rotation and shear is takeninto account by appropriate choice of coordinates for theplane occupied by the output wave-field. The action ofthe imaging system can then be described by the follow-ing linear integral transform5, which may be viewed as acontinuous form of matrix multiplication:

ψOUT(x, y, τ1, τ2, · · · ) (44)

=

∫∫dx′dy′G(x, y, x′, y′, τ1, τ2, · · · )ψIN(x′, y′).

The kernel of the above linear integral transform hasbeen denoted by G, since it is a Green function. It mayalso be interpreted as a generalised Huygens wavelet.

4 These implicit assumptions include the imaging system beingtime-independent, elastically scattering and non back-scattering.

5 An integral transform is an integral that transforms one functioninto another. A linear integral transform is an integral that (i)transforms one function into another, and (ii) has the property oflinearity. The linearity property, by definition, requires the linearintegral transform of a sum of two functions, to be equal to thesum of the corresponding transforms. Example: For a functionf(x) of one variable x, an arbitrary linear integral transformcould be written as g(x) =

∫f(x′)K(x, x′)dx′ + L(x), where

K(x, x′) and L(x) are arbitrary functions. In the main text, weuse linear integral transforms to represent the action of linearimaging systems. Here, the linear integral transform serves tochange (transform!) the field input into the imaging system,into the field that is output by the imaging system. Finally, wenote that: (i) The function K(x, x′) is often called the kernel ofthe linear integral transform; (ii) if one can assume that a zeroinput gives a zero output, then L(x) = 0.

13

To see this latter point, choose the special case

ψIN(x′, y′) = δ(x′ − x0, y′ − y0) (45)

in the above expression, where δ(x, y) is a two-dimensional Dirac delta, corresponding to a single point

(x, y) = (x0, y0) (46)

being illuminated in the input plane of the imag-ing system. Via the sifting property of theDirac delta6, Eq. 44 gives the associated out-put field as G(x, y, x0, y0, τ1, τ2, · · · ). ThereforeG(x, y, x0, y0, τ1, τ2, · · · ) is the output field as a func-tion of x and y coordinates in the output plane, whichwould be obtained if a unit-strength point source wereto be located at position (x0, y0) in the input plane,and the imaging system interposed between input andoutput plane were to have the state characterised bythe particular control parameters τ1, τ2, · · · . HenceG(x, y, x0, y0, τ1, τ2, · · · ) is indeed a generalised Huygens-type wavelet, with the form of the wavelet depending onboth the state of the imaging system and on the position(x0, y0) of the input “pinpoint of X-ray light”.

We close this sub-section by reversing the chain of logicthat is given above, so as to physically motivate the writ-ing down of Eq. 44 for an arbitrary linear imaging system.We characterise such an imaging system by the fact that,if the input is a “pinpoint of X-ray light” δ(x−x′, y−y′) atsome point (x′, y′) in the entrance plane A of Fig. 10, thenthe corresponding output field—considered as a functionof coordinates (x, y) over the output plane B—will begiven by G(x, y, x′, y′, τ1, τ2, · · · ). In this expression forthe output field G, the coordinates (x′, y′) of the input“pinpoint of X-ray light” are considered to be fixed, withthe parameters τ1, τ2, · · · describing the state of the imag-ing system also being fixed. To proceed further, we canused the sifting property of the Dirac delta to decomposean arbitrary input field ψIN(x, y) as a superposition (de-scribed by the continuous sum, namely the integral signbelow) of X-ray pinpoints of light, each such pinpointhaving the form δ(x− x′, y − y′), so that:

ψIN(x, y) =

∫∫dx′dy′ψIN(x′, y′)δ(x− x′, y − y′). (47)

In order to map inputs to outputs, namely to convertψIN(x, y) in the above integral (superposition of pinpointinputs, each of which have the form δ(x − x′, y − y′)

6 Here and elsewhere, the reader is assumed to be familiar withthe basics of Fourier analysis in an optics context. Such basicsinclude the sifting property of the Dirac delta, the concept ofconvolution, the convolution theorem, and the Fourier derivativetheorem. See e.g. Appendix A in Paganin (2006), for an overviewof these basics that employs a notation consistent with thesenotes.

multiplied by a weighting coefficient ψIN(x′, y′)) intoψOUT(x, y), we need only replace each of the pinpointinputs δ(x − x′, y − y′) under the integral sign, with itscorresponding output G(x, y, x′, y′, τ1, τ2, · · · ). This di-rect employment of the superposition principle—whichis justified on account of our key assumption that theimaging system is linear—leads directly to Eq. 44.

D. Arbitrary linear shift-invariant imaging systems

We specialise still further, by assuming the linear imag-ing system to be shift invariant. This augments theprevious assumptions, with the additional assumptionthat, if there is a transverse shift of the input wave-field,this merely serves to transversely shift the output wave-field. Such an assumption cannot hold for arbitrarilylarge transverse shifts, but is often approximately truefor a sufficiently small range of transverse shifts in thevicinity of the centre of the field of view of a coherentlinear imaging system. The assumption of (transverse)shift invariance implies that Eq. 44 may be simplified to:

ψOUT(x, y, τ1, τ2, · · · ) (48)

=

∫∫dx′dy′G(x− x′, y − y′, τ1, τ2, · · · )ψIN(x′, y′).

This will be recognised as a two-dimensional convolu-tion (folding, Faltung) integral, and hence may be morecompactly written as:

ψOUT(x, y, τ1, τ2, · · · ) (49)

= ψIN(x, y) ? G(x, y, τ1, τ2, · · · ),

where ? denotes two-dimensional convolution.

A very rich variety of imaging systems in coher-ent X-ray optics may be described using the formal-ism based on Eq. 48, including propagation-based X-ray phase contrast, analyser-crystal-based phase con-trast, imaging/microscopy using compound refractivelenses, imaging/microscopy using Fresnel zone plates, in-line holography, off-axis holography, Zernike phase con-trast imaging, imaging/microscopy using Kirkpatrick–Baez mirrors, interferometry, grating-based X-ray imag-ing, speckle-tracking X-ray imaging and various forms ofimaging system that perform optical encryption. How-ever, systems such as reflective optics and X-ray wave-guides, where multi-slice is required to describe passageof X-rays through optical elements, need the more gen-eral form given by Eq. 44.

14

E. Transfer function formalism

Fourier transform7 both sides of Eq. 49 with respectto x and y, indicated by the operator F . Invoke theconvolution theorem of Fourier analysis, to convert con-volution to multiplication. The inverse Fourier transformof the resulting expression is the following operator-typedescription of the action of the imaging system:

ψOUT(x, y, τ1, τ2, · · · ) = D(τ1, τ2, · · · )ψIN(x, y) (50)

where the Fourier transform of our Huygens-type wavelethas been termed the transfer function:

T (kx, ky, τ1, τ2, · · · ) ≡ 2πF [G(x, y, τ1, τ2, · · · )], (51)

(kx, ky) denotes Fourier-space (spatial frequency) coordi-nates corresponding to real-space coordinates (x, y), andthe generalised diffraction operator quantifying our imag-ing system is the following Fourier-space filtration:

D(τ1, τ2, · · · ) = F−1T (kx, ky, τ1, τ2, · · · )F . (52)

In the above, it is important to recall that all operatorsact from right to left: i.e. if the operator D(τ1, τ2, · · · ) isapplied to an input field, that input field is first actedupon by the Fourier transform F , then multiplied bythe transfer function T , and then inverse Fourier trans-formed. We previously stated this as, “We follow theusual convention that each operator acts on the elementto the right of it, with the rightmost operator actingfirst.”

In words, Eqs 50 and 52 state the following: In or-der to map the input field ψIN(x, y) to the correspondingfield ψOUT(x, y) output by a linear shift-invariant imag-ing system, a sequence of three steps may be used:

1. Apply the Fourier transform operator F to the in-put field;

2. Multiply the resulting object, which will be a func-tion of the Fourier coordinates (kx, ky), by thetransfer function T (kx, ky, τ1, τ2, · · · ) correspond-ing to the linear shift-invariant imaging system be-ing in a state described by the control parameters(τ1, τ2, · · · );

7 We use the Fourier-transform convention from Appendix Aof Paganin (2006). In one spatial dimension, the Fouriertransform F of a function g(x) is denoted by F [g(x)] ≡g(kx), where kx is the Fourier coordinate corresponding tox, and g(kx) = (1/

√2π)

∫∞−∞ g(x) exp[−ikxx]dx, with g(x) =

(1/√

2π)∫∞−∞ g(kx) exp[ikxx]dkx denoting the corresponding in-

verse Fourier transform. In two dimensions, and in an ob-vious extension of the notation, the forward Fourier trans-form becomes g(kx, ky) = (1/(2π))

∫∫∞−∞ g(x, y) exp[−i(kxx +

kyy)]dxdy, and the inverse transform becomes g(x, y) =(1/(2π))

∫∫∞−∞ g(kx, ky) exp[i(kxx+ kyy)]dkxdky .

3. Apply the inverse Fourier transform operator.

This verbal description may be considered as pseudo codefor a computational simulation of a linear shift-invariantimaging system; the resulting computer codes are typi-cally rendered extremely efficient by the use of the fastFourier transform (FFT) to implement both the forwardand inverse Fourier transform operators. From a morephysical perspective: Step 1 is a decomposition of theinput field into its constituent plane-wave components(Fourier components), Step 2 is a filtration of these plane-wave components in which each such plane-wave compo-nent is weighted by a different multiplicative factor thatis given by the transfer function T , and Step 3 is a syn-thesis in which all of the resulting weighted plane wavesare added up to give the output field. For more on thesynthesis–decomposition concept in optics, we refer thereader to Gureyev et al. (2018).

An important special case of a linear shift-invariantimaging system, is the previously considered case of free-space propagation through vacuum by a distance ∆, inwhich case we write D(τ1, τ2, · · · ) −→ D∆, with

D∆ = exp(ik∆)F−1 exp

[−i∆(k2

x + k2y)

2k

]F . (53)

This is a two-Fourier-transform version of the Fresneldiffraction integral. For more detail on this connection,we refer the reader to Paganin (2006).

A second important special case corresponds toanalyser-based X-ray phase contrast, where the X-rayfield transmitted through a sample is reflected from thesurface of a near-perfect crystal before having its inten-sity registered by a position-sensitive detector. In thiscase, upon suitable rotation of the (x, y) coordinates,

D(τ1, τ2, · · · ) −→ A = F−1A(kx)F , (54)

where the analyser-crystal transfer function A(kx) is apolarisation-dependent function of kx whose exact formis not needed here (cf. Paganin et al., 2004a,b).

F. Phase contrast

The squared magnitude of the input–output equation

ψOUT(x, y, τ1, τ2, · · · ) = D(τ1, τ2, · · · )ψIN(x, y) (55)

gives the intensity, output by our shift-invariant linearimaging system, as:

IOUT(x, y, τ1, τ2, · · · ) = |D(τ1, τ2, · · · )ψIN(x, y)|2. (56)

Evidently—and with the important exception of the“perfect imaging system” case where D is equal tounity—the output intensity typically depends upon boththe intensity and phase of the input, since the right side

15

of Eq. 56 will typically couple the phase of the input fieldto the intensity of the output field. Any state (τ1, τ2, · · · )of the imaging system, which generates an output in-tensity that is influenced by the input phase, is said toexhibit phase contrast. Again, most states of an imper-fect imaging system described by the operator D 6= 1 willyield both intensity contrast and phase contrast.

Let us re-iterate a most important point. If we definea “perfect” imaging system as one which perfectly repro-duces the input field, up to magnification, then such asystem will have

D(τ1, τ2, · · · )→ 1. (57)

Equation 56 reduces to

IOUT(x, y, τ1, τ2, · · · ) = IIN(x, y). (58)

Therefore, an imaging system which is perfect at thefield level, in the sense that the diffraction operator thatmaps input field to output field is given by D = 1, yieldsno phase contrast. This trivial statement may be com-pared with the rather important statement that imperfect(aberrated) imaging systems typically do exhibit phasecontrast. See e.g. Paganin and Gureyev (2008) and Pa-ganin et al. (2018) for further information, regarding thenature of the phase contrast that may be associated witharbitrary linear shift-invariant imaging systems.

G. Forward and inverse problems

So-called forward problems, in physics, seek to deter-mine effects from causes. Examples of such forward prob-lems include:

1. Solving the Schrodinger equation of non-relativisticquantum mechanics, to determine the allowed en-ergy levels of a hydrogen atom;

2. Determining the spectrum of different soundpitches that would be created if a guitar string ofa given length and tension etc. were to be pluckedat a particular position;

3. Using the transfer-function formalism to calculatethe intensity distribution of a propagation-basedphase contrast image, for a specified sample withknown three-dimensional complex refractive index,under the projection approximation, for known ex-perimental parameters such as X-ray wavelength,source-to-detector distance etc.

Inverse problems, on the other hand, seek to determinecauses from effects. Examples include:

1. Schrodinger’s inferring of his famous equation,based on data available at the time, such as themeasured energy levels of the hydrogen atom;

2. Determining the position at which a guitar stringof known length is plucked, given a measurementof the spectrum of different sound pitches createdby the plucked string;

3. Determining both the magnitude and the phaseof the projected complex refractive index createdby a sample, under the projection approximation,for known experimental parameters such as X-raywavelength, source-to-detector distance etc., and aknown propagation-based phase contrast intensityimage.

If the underlying fundamental physics equations areknown, and enough reasonable initial data is specified,the forward problems of classical physics are typicallysoluble. This broad statement is based on the fact that,in performing an experiment to model a given classical-physics scenario, nature always chooses a “solution”—namely the actual physical state for a classical system ata given specified time in its future—for a specified startingstate of the system8.

Inverse problems are harder, in general, than their as-sociated forward problems. Solutions to specified inverseproblems do not necessarily exist; even if they do exist,they may not be unique; even if a unique solution exists,it may not be stable with respect to perturbations in thedata due to realistic amounts of experimental noise, andother imperfections present in any real experiment. If aninverse problem is indeed such that there exists a uniquesolution that is stable with respect to perturbations inthe input data, it is said to be well posed in the sense ofHadamard (Hadamard, 1923; Kress, 1984). While such aproperty is desirable from both an analytic and aestheticperspective, the class of inverse problems that scientists

8 This solution may not necessarily be uniquely obtained from thestarting point (e.g. in dissipative systems with a point-like at-tractor, whereby a family of state-space trajectories may con-verge upon a single point in state space; see Ruelle 1989), and itmay exhibit sensitive dependence upon initial conditions (e.g. innon-dissipative chaotic systems with strange attractors; againsee Ruelle 1989), but such subtleties do not change the fact that,classically speaking, “nature always chooses a solution”. More-over, if one’s systems of equations, which model a given scenarioin the physical world, should evolve to states that are singular(e.g. the infinite energy densities associated with ray caustics ingeometric optics), then this lets one know that a more generaltheory is needed (e.g. wave optics, which smooths out the in-finities predicted by crossing rays in geometric optics; see Berry& Upstill 1980, Berry 1998, and Paganin 2006); again, the exis-tence of singularities in one’s physical model does not contradictthe earlier statement regarding nature always finding a solution.Also, there may be the more subtle problem that, for a givensystem of equations, it may not be rigorously known whether so-lutions to the equations as posed even exist for certain specifiedclasses of initial condition (e.g. such questions remain outstand-ing for the Navier–Stokes equations of classical fluid mechanics;see Kreiss & Lorenz 1989); again, such interesting subtleties donot contradict our earlier statement.

16

and engineers may wish to solve, is rather broader thanthe class of inverse problems that are well posed in thesense of Hadamard. In this latter context, various formsof optimisation method are very powerful, although atreatment of such methods is beyond the scope of thesenotes.

H. Two inverse problems

We open this sub-section by revising what we havelearned so far regarding the forward problem of imagingusing generalised shift-invariant linear (phase contrast)imaging systems. We separately consider the forwardand inverse problems at the levels of (i) fields, (ii) inten-sities. Note that the former problem is somewhat ide-alised, since complex X-ray wave-fields are not measureddirectly. Rather, it is time-averaged intensities that aredirectly measured by X-ray detectors, with the time aver-age being taken over the acquisition time of the detector.

(i) At the field level for an arbitrary linear shift in-variant imaging system, we learned that the input fieldmay be related to the output via Eq. 50, with the input-to-output operator D(τ1, τ2, · · · ) given by Eq. 52. Theassociated inverse problem, namely the determination ofthe input field given the output field, is solved by:

ψIN(x, y) = F−1 1

T (kx, ky, τ1, τ2, · · · )F (59)

×ψOUT(x, y, τ1, τ2, · · · ).

Often there are division-by-zero issues associated withspatial frequencies (kx, ky) at which the transfer functionT (kx, ky, τ1, τ2, · · · ) vanishes. This amounts to informa-tion loss in the forward problem, leading to instabilityin the associated inverse problem. Sometimes one can“regularise” the above expression by replacing 1/T withT ∗/(|T |2 + ℵ) where ℵ is a small positive real number.A more sophisticated solution is to consider several out-puts associated withN > 1 different states of the imagingsystem, leading to the following solution to the field-levelinverse problem (Schiske 1968; Paganin et al. 2004c):

ψIN(x, y) = F−1

j=N∑j=1

T ∗j (kx, ky)∑p=Np=1 |Tp(kx, ky)|2

F (60)

×ψ(j)OUT(x, y).

Here, Tj(kx, ky) denotes the transfer function associated

with the jth state of the imaging system, and ψ(j)OUT(x, y)

denotes the corresponding output. The above expressionwill have no division-by-zero issues if

∑p=Np=1 |Tp(kx, ky)|2

is non-zero at every spatial frequency (kx, ky). If division-by-zero issues remain, one can always regularise theabove expression, or increase the number of differentstates of the imaging system that is utilised.

(ii) The inverse problem of phase retrieval, or moreproperly of phase–amplitude retrieval, seeks to recon-struct both the intensity and phase of the input field,given only the intensity of the output field correspond-ing to one or more states (τ1, τ2, · · · ) of the imagingsystem. This problem is vastly more difficult than thepreviously-considered field-level inverse problem. Indeed,no closed form solution exists in general, to the phase–amplitude retrieval problem. Note the evident parallelswith the concept of inline holography as conceived byGabor (1948), in which imaging is viewed as a two-stepprocess: data recording, followed by reconstruction. The“holographic” spirit of this latter point implicitly runsthrough many of our subsequent discussions regardingphase retrieval. Before proceeding, however, we makethe following general remark, which again has parallelswith holography: Since imperfect shift-invariant aber-rated imaging systems typically yield measurable outputimages IOUT(x, y) that are affected by the phase of theinput complex field ψIN(x, y), the output intensity maybe viewed as containing encrypted or encoded informa-tion regarding the phase of the input field. Under thisview, the phase-retrieval problem corresponds to seek-ing a means to decyrpt or decode one or more measuredoutput-intensity maps, so as to infer the phase distribu-tion (or, more generally, both the phase and the ampli-tude/intensity) of the input field.

I. Transport-of-intensity phase retrieval

For propagation based X-ray phase contrast imaging ofa single-material object with projected thickness T (x, y)that is normally illuminated by plane waves of uniformintensity I0, under the projection approximation, it is ev-ident from Eqs 26 and 27 that both the phase and theamplitude at the exit surface of the object may be ob-tained from T (x, y)9. This opens the logical possibilitythat the projected thickness of the single-material samplemay be obtained from a single propagation-based phasecontrast image I(x, y, z = ∆), obtained at a distance ∆downstream of the object that is sufficiently small for theFresnel number10 NF to be much greater than unity (thiscorresponds to the “single edge fringe” regime exempli-fied by the propagation-based phase contrast images in

9 For a single-material object illuminated by normally inci-dent plane waves of uniform intensity I0, the phase shift inEq. 26 becomes ∆φ(x, y) = −k δωT (x, y), and the absorption-contrast intensity map in Eq. 27 becomes I(x, y, z = z0) =I0 exp[−µωT (x, y)]. Here, T (x, y) is the projected thickness ofthe single-material sample, and the wavenumber k is equal to 2πdivided by the X-ray wavelength λ.

10 Here, the Fresnel number is as defined in Eq. 34, but with theimportant difference that the T in the denominator is replacedby the object-to-detector propagation distance ∆.

17

the bottom right of Fig. 6). With the previously men-tioned approximations, but no further approximationsof any kind, the transport-of-intensity equation (Eq. 40)may be solved exactly, to give the projected thickness ofthe sample from a single propagation-based phase con-trast image (Paganin et al., 2002):

T (x, y) = − 1

µloge

(F−1

{F [I(x, y, z = ∆)] /I01 + (δ∆/µ)(k2

x + k2y)

}).

(61)

The above algorithm has been widely utilised, andis now known as “Paganin’s algorithm” or “Paganin’smethod”. Its advantages, bought at the price of thepreviously stated strong assumptions, include simplicity,speed, very significant noise robustness and the ability toprocess time-dependent objects frame-by-frame. Whilethe method provides quantitative results when its keyassumptions are sufficiently well met, qualitative recon-structions obtained under a broader set of conditions areoften of utility where non-quantitative morphological in-formation is sufficient.

A variant of the Paganin algorithm has been devel-oped for analyser based phase contrast imaging and otherphase contrast imaging systems that yield first-derivativephase contrast (Paganin et al., 2004b). Another varianthas has been developed for phase contrast imaging sys-tems that simultaneously yield both first-derivative andsecond-derivative phase contrast (Pavlov et al., 2004 and2005).

J. The inverse problem of tomography

Suppose that a static non-magnetic three-dimensionalobject is placed upon a spindle about which the objectcan be rotated through a set of azimuthal angles ϕ whichare (say) equally spaced throughout the interval from 0to π radians. Suppose further that the sample is nor-mally illuminated with uniform intensity monochromaticscalar X-ray plane waves, and that all of the assumptionsneeded for the projection approximation are valid.

As we have previously learned in our discussions relat-ing to the projection approximation, both the phase andthe logarithm of the intensity, of the exit surface wavefieldfor each orientation, may be obtained via a simple linearprojection of the complex refractive index (see Eqs 26and 29 respectively). This process may be inverted inthe process of tomography, with the imaginary part ofthe three-dimensional complex refractive index being ob-tainable from measurements of the logarithm of the exit-surface intensity over the set of azimuthal angles of thespindle. Similarly, if a suitable phase retrieval can beperformed for each orientation of the object, the recov-ered set of two-dimensional phase maps may be invertedto give the real part of the complex refractive index ofthe sample.

Note that when the Paganin method is utilised in atomographic context, its domain of utility broadens sincemany objects may be viewed as locally composed of a sin-gle material of interest, in three spatial dimensions, thatcannot be described as composed of a single material inprojection (Beltran et al., 2010 and 2011). In such atomographic setting, the algorithm is sufficiently robustwith respect to noise that Beltran et al. (2010, 2011)noted it could exhibit signal-to-noise ratio (SNR) boostsof up to 85 (resp. 200). More recent studies have shownthat this SNR boost has, as an approximate upper limit,0.3 δω/βω if Poisson statistics are assumed (Nesterets andGureyev, 2014; Gureyev et al., 2014). Interestingly, thisboost in SNR can be even more marked for very small ex-posure times (Kitchen et al., 2017). Since signal-to-noisevaries with the square root of dose, this SNR-boost im-plies that reduction in dose of a factor of 3002 = 90, 000 ispossible, at least in principle, when the Paganin methodit applied to tomography (Kitchen et al., 2017). Thisfact is of importance in dose-sensitive applications of themethod (e.g. to biomedical imaging), as well as time-sensitive applications where imaging speed is an issue.Largely on account of its SNR-boosting properties, newapplications of the Paganin method are regularly pub-lished in a variety of fields; we refer the reader to

https://bit.ly/2FtA3Fw

for the latest articles applying the method. Note alsothe important caveat that the method trades simplic-ity against resolution (and often numerical precision), inthe sense that the often rather gross approximation ofa single-material object will often break down when oneseeks to image at sufficiently high resolution, or in many(but not all) imaging scenarios where quantitative infor-mation is required. In such circumstances, more sophisti-cated approaches—such as the holotomography methodreported in Cloetens et al. (1999)—are required. Notealso that an approximate solution derived e.g. using thetransport of intensity equation may always be used asstarting point for a more sophisticated iterative recon-struction (Gureyev et al., 2004).

V. PRACTICE

A. A quick survey of modern X-ray phase-contrast imagingmethods

Here we describe a few phase contrast methods thathave received attention in recent years, and are currentlyused in the X-ray imaging community. This survey is byno mean exhaustive, and does not cover all the experi-mental details of the various techniques. A recent keyreview on the subject, providing insights on the relativestrengths and weaknesses of each method, is in Wilkins etal. (2014). Bravin et al. (2013) not so long ago published

18

a review discussing preclinical and clinical applications ofphase contrast imaging.

Our approach is to look at the different techniques fol-lowing the Transport of Intensity Equation (TIE), andspecifically its version valid for small object-to-detectorpropagation distances δz, as written in Eq. 42. To makeour approach clearer, we expand the divergence operatoron the right hand side of Eq. 42:

I(x, y, z + δz) ≈ I(x, y, z) (62)

−δzk

[∇⊥I(x, y, z) · ∇⊥φ(x, y, z) + I(x, y, z)∇2

⊥φ(x, y, z)].

Under the approximations used to derive this finite-difference version of the TIE, the terms in the squarebrackets describe the phase contrast contribution to theimage. (i) The phase gradient corresponds to the direc-tion of a local stream line (Fig. 1), whereas (ii) the Lapla-cian measures the curvature of the wave front (Fig. 4).Stated differently: (i) The first term in square bracketscontains the (transverse) phase gradient, and representsa prism-like effect that transversely displaces optical en-ergy in a manner proportional to the local deflection an-gle ∇⊥φ/k. (ii) The second term in the square bracketsis a lensing term that contains the transverse Laplacian,which describes the local concentration or rarefaction ofoptical energy density (and hence intensity) due to thesample locally focusing or defocusing the X-ray radia-tion streaming through it (cf. features J and L in Fig.4). With the exception of direct methods to measurethe phase—such as interferometry—many (but certainlynot all!) commonly used phase contrast methods mea-sure phase derivatives, and many such methods can bedescribed using Eq. 62.

Methods such as X-ray grating interferometry (Mo-mose et al. 2003, Weitkamp et al. 2005, Pfeiffer etal. 2008) or analyser-based X-ray imaging (Chapmanet al. 1997, Wernick et al. 2003, Rigon et al. 2007) pro-vide image contrast dependent upon the first derivativeof the phase in the transverse plane (first term in thesquare bracket). Propagation-based methods (Snigirevet al. 1995, Cloetens et al. 1996, Wilkins et al. 1996)measure the second derivative of the phase, described bythe second term in the square brackets.

Before going into some more detailed analysis it isworth pointing out two general facts about phase con-trast X-ray imaging, which descend straight from Eq. 62.

Fact 1: All phase contrast imaging techniques requirepropagation.

Fact 2: Both gradient and Laplacian of the phase can bepresent, at the same time, in phase contrast images.

Fact 1 is the obvious consequence of the observationthat the second line of Eq. 62 vanishes if δz = 0. Morephysically, phase contrast signal—for cases when it is notgenerated by interferometry—is generated by refraction.

X-rays passing through a sample are refracted as wellas absorbed. Normally refraction effects goes unnoticedas the refraction angle is extremely small. To becomeappreciable, measuring refraction requires the detector tobe placed some distance away from the sample to analysethe wave front.

Fact 2 is strictly speaking correct only for methodsthat are sensitive to the phase gradient. Since all of thesemethods still requires propagation, they will always mea-sure a combination of gradient and Laplacian of the phase(Pavlov et al. 2004, 2005; Diemoz al., 2017).

B. Phase gradient methods

In this category we find methods such as analyser-based imaging (ABI), grating interferometry (GI) and itsvariants, edge illumination (EI) and its variants (Olivo etal. 2011, Munro et al. 2013), as well as speckle track-ing (Berujon et al. 2012, Morgan et al. 2012, Zdora2018). At the same time, scanning methods using a fo-cused beam as probe (Sayre and Chapman 1995, Schnei-der 1998) also yield phase gradients and can be includedin this description.

Looking once again at Eq. 62, we immediately un-derstand what all of these methods have in common.To measure the phase transverse gradient ∇⊥φ(x, y, z)one must introduce a transverse intensity gradient∇⊥I(x, y, z) and allow for some propagation distance δz.Interestingly, such an intensity gradient can be intro-duced in a single frame or throughout multiple frames.Techniques such as speckle tracking or single-image phaseretrieval using a grating before the object work by intro-ducing spatial intensity variations in the field of view.Methods such as ABI or grating interferometry (for in-stance when using diffraction-enhanced imaging or fringescanning respectively) rely on intensity gradients gener-ated across several images. In this case the phase re-trieval method will require more that one image to work.Typically single-image methods are quicker and enablelower X-ray dose, while multi-image methods can attainbetter spatial resolution.

Another case of multi-image phase retrieval is repre-sented by scanning methods. In this case the intensitygradient is given by the beam itself, that can be shapedeither before (i.e. in STXM, Scanning Transmission X-ray Microscopy) or after the sample (EI) to yield thedesired phase gradient.

Given the common basis we are here describing, it isnot surprising that different methods may share simi-lar approaches. In the next subsection we will focus onthe complementarity between ABI and GI, showing howsimilar approaches have been discovered and applied in-dependently by the two communities.

19

C. Analyser-based and grating-based imaging

In the spirit of the unified view of gradient-based phasecontrast imaging methods, we discuss here in more de-tail the strong analogy between two popular methods forX-ray phase-contrast imaging, namely ABI and GI. Bothmethods, in their general form, work by realising an an-gular scan of either a crystal or one of the gratings. Notethat a relative transverse shift x of a grating with re-spect to the other in GI can be seen as a relative angularchange x/p where p is the grating period.

The analogy therefore begins by considering the an-gular transmission function of both systems: a rockingcurve for ABI and a periodic transmission function forGI (see Fig. 11). Historically the development of ABIand GI followed two different approaches, where ABI wasbrought to fame by a two-image phase retrieval methoddeveloped by Chapman et al. (1997), while GI phaseretrieval was initially based on the fringe scanning (alsoknown as phase stepping) technique (Momose et al. 2003,Weitkamp et al. 2005). The fringe scanning method isbased on the assumption of periodicity of the GI trans-mission function. A rocking curve used in ABI on theother hand is not periodic, and therefore fringe scanningis unique to GI.

Thus, with the exception of fringe scanning, we candraw a strong analogy between other phase retrievalmethods independently developed for ABI and GI. Wecan divide these methods into two categories, which wewill here call the geometric approach and convolution ap-proach respectively.

The geometric approach is based on Taylor expansionof the transmission function T . We here consider thetransmission function to be a function of the angle whichan X-ray makes with respect to the optic axis; while thereare two such angles, corresponding to each of the two in-dependent orthogonal transverse directions, for simplic-ity we here consider T to be a function of only one an-gle. Taylor expansion truncated at the first order meansthat the transmission function, considered as a functionof angle, is locally approximated by a straight line. Theintensity transmitted at each point by a sample in thiscase can be approximated by:

I(θ) ≈ I0(θ)T (θ0 −∆θR). (63)

Here I0(x, y) is the intensity before the sample, ∆θR isthe angular shift due to local refraction and |T | < 1accounts for the angle-dependent attenuation. This ap-proximation holds for instance for the flanks of the rock-ing curve (hence the DEI11 method by Chapman et al.1997) or the linear part of the sinusoidal transmission

11 “Diffraction enhanced imaging”.

FIG. 11 (a) Sketch of the transmission function of an ABIsystem (rocking curve). The black curve represents the rock-ing curve without the sample, while the red curve representsthe rocking curve with the sample. The sample action is toattenuate, refract and scatter the incident beam. (b) Corre-sponding (simplified) transmission function for a GI systembased on two gratings (optionally with a source grating). Inthis case the transmission function is in general a periodicMoire pattern.

function of the GI (hence the two-image methods devel-oped in a tomographic setup for GI by Zhu et al. 2010).

A better approximation is represented by truncatingthe Taylor series at the quadratic term, which meansapproximating the transmission function locally as aparabola. In this case the sample action is modelledthrough a combination of attenuation, refraction andscattering, and the intensity after the sample:

I(θ) ≈ I0(θ)

∫T (θ0 −∆θR −∆θS) f(∆θS)d∆θS. (64)

In this case the sample is assumed not only to attenuateand refract, but also to scatter with a typical scatteringwidth ∆θS . A typical imaging detector is insensitiveto such an angular spread generated by scattering,which therefore will be integrated in detection. It ishowever possible to separate attenuation, refraction andscattering width by acquiring at least three images.A three-image algorithm was developed by Rigon etal. (2007) in ABI and by Pelliccia et al. (2013) in GI.The algorithm is in fact formally the same in both cases.

The methods described above rely on a Taylor expan-sion of the transmission function. A more accurate ap-proach, which on the other hand requires more images, isrepresented by the convolution approach. In this case the

20

intensity after the sample is modelled by the convolutionproduct

I(θ) ≈ I0(θ) ? S(θ), (65)

where the overall effect of the sample is modelled by thefunction S(θ) which has the effect of attenuating, shiftingand broadening the transmitted intensity. A phase re-trieval algorithm based on the convolution approach wasproposed by Wernick et al. (2003) for ABI and by Mod-regger et al. (2012) in GI. In both cases the algorithmworks by determining the shape of S(θ) by a deconvolu-tion procedure. The technique obviously requires morethan three images to produce a reliable estimate of thesystem transmission function, but has the advantage ofincreased accuracy (no Taylor expansion involved) in thephase retrieval process. It is worth noting that the con-volution method reduces to the geometric method in thelimit specified by Eq. 63 (Pelliccia et al., 2013).

One can also perform Taylor expansions of the complextransfer function, to first order (Paganin et al., 2004b)and second order (Pavlov et al., 2004 and 2005) in spa-tial frequency, for arbitrary shift invariant linear imag-ing systems, in the contexts of both the forward prob-lem of generalised phase contrast and the associated in-verse problem of phase retrieval. Thus, for example,a first-order Taylor expansion of the complex transferfunction T (kx, ky, τ1, τ2, · · · ) in Eq. 51, about a Fourier-space point (k0

x, k0y) located near the centre of the Fourier

transform of the input wave-field, would approximate thetransfer function as T ≈ γ1 + γ2(kx − k0

x) + γ3(ky − k0y),

where the constants γ1,2,3 are functions of the aberra-tion coefficients τ1, τ2, · · · . This approach yields equa-tions for the output intensity, obtained from a fairlygeneral perspective, that are very closely related to the“phase gradient methods” considered earlier (Paganinet al., 2004b). Similarly, if the transfer function isexpanded to second order in spatial frequency, usingT ≈ γ1 + γ2(kx − k0

x) + γ3(ky − k0y) + γ4(kx − k0

x)2 +γ5(ky − k0

y)2 + γ6(kx − k0x)(ky − k0

y), one obtains equa-tions for the output intensity that simultaneously exhibitboth phase-gradient contrast, and Laplacian-type phasecontrast (Pavlov et al., 2004 and 2005).

Part III

Partial coherence forarbitrary phase contrastimaging systems

The final part considers partial coherence in the contextof arbitrary linear (phase contrast) imaging systems.

VI. THEORY

A. Partial coherence

The reader is assumed to be familiar with elemen-tary concepts of partial coherence, including the con-cepts of spatial and temporal coherence. Hence we as-sume a more advanced (albeit very intuitive, if one thinksabout it for long enough!) perspective based on thecoherence-optics equivalent of the density-matrix formal-ism of quantum mechanics. This formalism is knownas the space–frequency description of partial coherence(Wolf, 1982; see also Mandel and Wolf, 1995, and refer-ences therein).

Our intent here is rather modest. We will briefly in-troduce the space–frequency description of partial co-herence, in which a given partially coherent field maybe described via a two-point correlation function knownas the cross spectral density. The cross-spectral densitymay be obtained at any given angular frequency via asuitable averaging procedure over an ensemble of strictlymonochromatic fields, all of which have the same angularfrequency12. We then use this space–frequency descrip-tion of partially coherent X-ray fields to describe the ac-tion of any phase contrast imaging system, provided thatthe system is linear and the effects of polarisation maybe neglected.

An advantage of the particular formalism adoptedhere, apart from its broad applicability to a rich varietyof both existing and future phase-contrast X-ray imagingand other coherent X-ray imaging scenarios, is the readi-ness with which the resulting mathematical expressionsmay be implemented in computer code. Indeed, much ofthe mathematics that will follow is, in essence, computerpseudo-code rather than explicit calculations per se.

Since the context is phase-contrast imaging, or imag-ing more broadly, we describe a given paraxial complexscalar partially coherent X-ray field via a stochastic pro-cess that can be characterised via an ensemble of strictlymonochromatic fields

{ψ(j)ω (x, y), cj} (66)

at each angular frequency ω, with all fields in each en-semble having the same angular frequency ω (Wolf, 1982;Mandel and Wolf, 1995). Here, the j index labels eachmember of the ensemble, with the associated real statis-tical weight cj . Each of these weights lies between zeroand unity inclusive, with∑

j

cj = 1. (67)

12 Cf. the Gibbs-type statistical ensembles that are often used inthe formalism of thermodynamics.

21

In general, these weights will depend on angular fre-quency, although for compactness our notation does notexplicitly indicate this dependence.

The cross spectral density Wω(x1, y1, x2, y2) is a two-point correlation function that quantifies the degree ofcorrelation between the optical disturbance at the pairof points (x1, y1) and (x2, y2), at the specified angularfrequency ω. It is given by the ensemble average over allof the N members of the statistical ensemble:

Wω(x1, y1, x2, y2) =

j=N∑j=1

cjψ(j)∗ω (x1, y1)ψ(j)

ω (x2, y2)

≡ 〈ψ(j)∗ω (x1, y1)ψ(j)

ω (x2, y2)〉ω. (68)

The associated spectral density, which may be viewedas the “diagonal of the cross spectral density”, is:

Sω(x, y) ≡Wω(x, y, x, y) = 〈|ψ(j)ω (x, y)|2〉ω. (69)

Note that the spectral density is simply the averageintensity, being a weighted average over the intensitiesof the individual fields that make up the statistical en-

semble {ψ(j)ω (x, y), cj}. We note also that the measured

intensity I(x, y), registered by a detector which integratesover angular frequencies ω, will be given by the followingweighted average of the spectral density:

I(x, y) =

∫Sω(x, y)ℵ(ω)dω, (70)

where ℵ(ω) is a measure of the variable efficiency of thedetector, as a function of angular frequency ω. Note thatthe shape of the energy spectrum is implicitly taken intoaccount in the cross-spectral density itself, consideredas function of energy E = ~ω where ~ is Planck’s con-stant h divided by 2π. Note also that the set of weights{cj}, used to construct each of the ω-dependent quanti-ties Sω(x, y) in the above spectral sum, will in general bedifferent for each ω. Thus, for the purposes of this para-graph only, cj would be better notated as cj,ω. Notwith-standing this, for the sake of notational simplicity we willdenote the statistical weights at a given fixed angular fre-quency ω, as cj , henceforth. Lastly, we point out that theenergy spectrum in the above equation is implicit in the

normalisation of the ψ(j)ω (x, y) functions; thus, if ω takes

a value ω which is such that there is very little X-ray

power at energy ~ω, all of the ψ(j)ω (x, y) functions will be

multiplied by a suitably small normalisation factor.We will take the cross spectral density as the descrip-

tor of partially coherent X-ray fields whose statistics areindependent of time (a property known as statistical sta-tionarity). We have also implicitly made the assump-tion of ergodicity, namely the assumed equality of ensem-ble averages with time averages. Moreover, if Gaussianstatistics may be assumed, then all higher-order corre-lation functions may be determined from the two-pointfield correlation function W with which we are working.

Proper treatments of these important points will not begiven here; they are available in standard texts such asthat of Mandel and Wolf (1995).

B. Modelling a wide class of partially coherent X-ray phasecontrast imaging systems

We now turn to the question of how the space–frequency description for partial coherence may be usedto study how cross-spectral densities are transformedupon passage through linear imaging systems utilisingpartially coherent X-ray radiation. Given the previously-mentioned trivial relation between cross-spectral densityand spectral density, and the fact that spectral densitymay be averaged over angular frequency to give the totaldetected intensity distribution, the formalism outlinedbelow will allow one to determine the intensity distri-bution output by any linear phase contrast imaging sys-tem. This covers a rather broad class of existing X-rayphase contrast imaging systems as well as many othercoherent-X-ray-optics imaging systems, and we dare saythat it also covers a rather large class of coherent imag-ing systems that will be developed in the future. Whilewe ignore the effects of polarisation, these can be readilytaken into account in a generalisation of the ideas pre-sented here, if required.

Rather than writing down mathematical expressions,we instead list a sequence of steps that is able to calcu-late the intensity distribution produced by an arbitrarylinear X-ray phase contrast imaging system, illuminatingan object, utilising partially coherent radiation under thespace–frequency description of partial coherence:

• Assume a particular realistic model of the X-raysource after it has passed through whatever condi-tioning optics, such as monochromators, that maybe present in a given X-ray phase contrast imagingsystem. This will imply a given statistical ensembleof strictly monochromatic fields

{ψ(j)ω (x, y), cj}, (71)

at the entrance surface of an object that is to beimaged.

(a) One simple example, of such a statistical ensem-ble, is an ensemble of monochromatic X-ray planewaves whose directions of propagation are uni-formly distributed within some narrow cone whoseaxis coincides with the optic axis (Pelliccia and Pa-ganin, 2012).

(b) A second example, of a suitable ensemble ofstrictly monochromatic fields, could be an ensem-ble of z-directed plane waves that each have a spa-tially random continuous phase perturbation im-printed upon them (Irvine et al., 2010; Morgan et

22

al., 2010). This spatially random phase pertur-bation could be chosen to have a specified trans-verse correlation length, and a specified root-mean-square phase excursion. This provides an excellentmodel, for example, for partially coherent X-rayimaging systems that utilise a random phase dif-fuser to “clean up” an illuminating beam that con-tains unwanted structure on account of imperfectoptics.

• Assume that the object is static13, non-magnetic14,elastically scattering15 and sufficiently gently spa-tially varying16 for the projection approximation to

be valid. The ensemble of fields {ψ(j)ω (x, y), cj} at

the entrance surface of the object then leads to theensemble of fields

{ψ′(j)ω (x, y), cj} = {ψ(j)

ω (x, y)T (j)ω (x, y), cj} (72)

at the nominal exit surface of the object, where

T (j)ω (x, y) is the complex transmission function cor-

responding to illumination of the object by the

strictly monochromatic field ψ(j)ω (x, y). For suffi-

ciently paraxial fields, T (j)ω (x, y) may be taken to be

independent of j, and equal to the complex trans-mission function that is predicted by the projectionapproximation, for a z-directed plane wave of an-gular frequency ω.

• Each member in the ensemble of fields{ψ′(j)

ω (x, y), cj} at the exit surface of the ob-ject may then be individually propagated througha specified linear imaging system, to give theensemble of strictly monochromatic fields

{ψ′′(j)ω (x, y), cj} (73)

= {D(j)ω (τ1(ω), τ2(ω), · · · )ψ′(j)

ω (x, y), cj}

at the exit surface of the imaging system. Here,

D(j)ω (τ1(ω), τ2(ω), · · · ) (74)

denotes the operator which maps the jth input fieldat angular frequency ω, to the jth output field at

13 This ignores potentially important effects such as a moving sam-ple, or the time-dependent accumulation of radiation damage bya radiation-sensitive sample.

14 This allows polarisation effects to be ignored, for example.15 This enables the assumption that the energy of the X-rays is not

changed upon passing through the sample. We thereby ignoreoften-important phenomena such as fluorescence.

16 If the projection approximation breaks down, one could employa better approximation, such as the multi-slice method that wasmentioned earlier in these notes.

FIG. 12 Sample experimental setup, to illustrate the meansfor modelling X-ray imaging systems that is outlined in themain text.

the same angular frequency, when the imaging sys-tem is in a state characterised by the control param-eters τ1(ω), τ2(ω), · · · . Note that the control pa-rameters of the imaging system will in general varywith angular frequency, even though the said imag-ing system would typically have the same physi-cal configuration for all angular frequencies. How-ever, for paraxial fields, one could often assume

D(j)ω (τ1(ω), τ2(ω), · · · ) to be independent of j.

• The exit-surface, of the previously mentioned gen-eralised imaging system, is assumed to coincidewith the surface of the detector. The cross-spectraldensity W , the spectral density S, and the av-erage intensity I, over the surface of the detec-tor, can then be calculated using the ensemble

{ψ′′(j)ω (x, y), cj}, together with the previously spec-

ified formulae for W , S and I.

As an example of the logic outlined in general termsabove, consider the X-ray phase-contrast imaging setupthat is sketched in Fig. 12. Here, a monochromated X-raysource (energy E = ~ω) illuminates a thin object, beforepassing through a linear shift-invariant imaging system(LSI), and having the resulting intensity distribution be-ing measured by a position sensitive two-dimensional de-tector.

Example for strictly monochromatic case: If onecan assume a strictly monochromatic paraxial fieldand a thin object, the projection approximation canbe used to map the complex field ψω(x, y) over theentrance-surface α of the object, to the field

Tω(x, y)ψω(x, y) : plane β (75)

over the exit surface β (see Eqs 26 and 27 forhow to calculate the complex transmission func-tion Tω(x, y) under the projection approximation).To propagate through vacuum by the distance Z1

between the exit surface β of the object and the en-trance surface γ of the linear shift-invariant imagingsystem (LSI system), one can apply the free-spacediffraction operator DZ1

to the field over the plane

23

β: see Eqs 31 and 53. This gives:

DZ1Tω(x, y)ψω(x, y) : plane γ (76)

over the entrance surface γ of the LSI system. [Asan aside, we re-iterate the fact that operators actfrom right to left, thus e.g. the above equationshould be read as meaning “take the complex fieldψω(x, y) over then plane α, then multiply by thecomplex transmission function of the object thatis given by Tω(x, y) under the projection approxi-mation, then apply the diffraction operator DZ1 tothe resulting field so as to propagate it through adistance of Z1.”] Assuming the LSI system to bein a state characterised by the control parameters(τ1, τ2, · · · ), one can then use Eq. 52 to propagateto the plane δ, over which the complex disturbancewill be:

D(τ1, τ2, · · · )DZ1Tω(x, y)ψω(x, y) : plane δ. (77)

Free-space propagation through a distance Z2 thengives the complex disturbance over the surface ofthe detector:

DZ2D(τ1, τ2, · · · )DZ1

Tω(x, y)ψω(x, y) : plane ε.(78)

The squared modulus of the above expression givesthe intensity measured by the detector. Alterna-tively, one could calculate the phase φω(x, y, z = ε)of the above expression, to determine the phaseof the coherent X-ray wave-fronts impinging uponthe detector. Other quantities can also be derivedfrom the above expression for the complex fieldψω(x, y, z = ε) over the plane ε coinciding withthe surface of the detector, such as the transverseenergy-flow vector (Poynting vector)

S⊥(x, y) ∝ |ψω(x, y, z = ε)|2∇⊥φω(x, y, z = ε)(79)

or other related quantities such as the angular-momentum density and the vorticity of the trans-verse energy-flow vector (see e.g. Berry 2009 forhow these last-mentioned quantities may be calcu-lated).

Example for partially-coherent case: The abovecase showed how to propagate the strictlymonochromatic field ψω(x, y) over the entrancesurface α of the object in Fig. 12, through boththe object and an LSI system, to the surfaceε of a two-dimensional detector. If the field ispartially coherent, then for each angular frequencyω corresponding to each energy E via E = ~ω,one can instead have an ensemble of strictlymonochromatic fields of the form specified by

Eqs 66 and 67. Each field ψ(j)ω (x, y) is propagated

through the optical system in exactly the same way

as for the coherent case, with the associated statis-tical weights cj being unchanged via propagationthrough the optical system. [Note, however, thatwhile the weights cj do not change, the normalisa-tion of the individual wave functions may change,as they may be transmitted through the sampleor LSI system with different efficiency17.] Thisimmediately yields an expression for convertingthe ensemble of strictly monochromatic inputfields (over plane α) in Eq. 66, to the followingensemble of output fields over the plane ε:

{DZ2D(τ1, τ2, · · · )DZ1

Tω(x, y)ψ(j)ω (x, y), cj}. (80)

The resulting statistical ensemble can be used tocalculate a number of derived quantities of interest:

• The cross-spectral density Wω(x1, y1, x2, y2)(see Eq. 68);

• The spectral density Sω(x, y) (see Eq. 69),which may be integrated over ω to give thetotal spectral density (see Eq. 70);

• The position-dependent ensemble-averagedangular momentum density and vorticity, viaensemble averaging the expressions for thesequantities that appear in a coherent-opticscontext in Berry (2009);

• The Wigner function, the ambiguity func-tion, the generalised radiance function etc. aretwo-point correlation functions that can allbe derived from the ensemble in Eq. 8018.This amounts to a computationally simplemeans for modelling the important “coherencetransport” problem, especially when the just-mentioned correlation functions are calculatedfor different points along the optic axis, as onetraverses the various optical elements in se-quence from the source to the detector.

The above formalism may also be trivially generalisedto the case of four-point correlation functions, six-pointcorrelations functions and so on. Four-point correlationfunctions are primarily of interest in the X-ray context ofthe Hanbury Brown–Twiss effect, which plays a role inX-ray ghost imaging (Yu et al. 2016; Pelliccia et al. 2016;Schori and Shwartz 2017; Zhang et al. 2018; Pelliccia etal. 2018; Schori et al. 2018; Ceddia and Paganin 2018;Gureyev et al. 2018; Kingston et al. 2018; Kim et al. 2018;Kingston et al. 2019). While interesting, X-ray ghostimaging will not be discussed in these tutorials. Also,

17 We thank C. Detlefs for bringing this point to our attention.18 See e.g. Alonso (2011) for a description of how the Wigner, am-

biguity, generalised-radiance and related functions may be cal-culated from the statistical ensemble of fields in Eq. 80.

24

as previously mentioned, for the common special case ofGaussian statistics, all higher-order correlation functionsare either determined by the two-point correlation func-tions that have been our main focus, or they vanish (seee.g. Mandel and Wolf, 1995). We also note that the cor-relation functions of various orders would be of pivotalimportance in the context of the nascent field of quan-tum X-ray optics (Adams et al., 2013; Kuznetsova andKocharovskaya, 2017), which has yet to mature to thesame stage of development as quantum optics in the con-text of visible light. This maturation of quantum X-rayoptics in coming years is an obvious emerging researchopportunity for coming generations, as one can readilyascertain by studying the most recent literature.

C. Speculations regarding future trends

Below we give a sequence of speculations regardingpossible future trends in X-ray phase contrast imagingin particular, and coherent X-ray imaging more broadly.These speculations, which range from the self-evident tothe tentatively hypothetical, are intended as both stim-ulus and guide for future research in the field.

(a) As one moves to progressively higher resolution,e.g. in tomography, the projection approximation willbecome increasingly ill behaved. One will eventually needto embrace fully dynamical models such as the multi-slice approximation, in the context of the inverse problemof tomography, or diffraction-tomography methods, to alarger degree than is the case at present. This transitionis already in progress, as even a cursory survey of recentliterature will show. On a related note, and again as onemoves to ever-higher resolution, the scalar approxima-tion for the X-ray wave-field may begin to break downe.g. when large scattering angles or magnetic phenomenaare considered (Detlefs, 2019). In all of this, much guid-ance is to be gleaned from the existing literature on elec-tron tomography, which has—very broadly speaking—been forced to grapple with such problems at an earlierstage than the X-ray tomography community. Much canalso be learned from the very well established field ofX-ray scattering and absorption by magnetic materials—see e.g. the text by Lovesey and Collins (1996)—togetherwith any aspects of X-ray physics in which the effects ofmagnetism and/or polarisation are important.

(b) Some emphasis has been given in these notesto X-ray phase contrast imaging using shift-invariantlinear imaging systems that possess arbitrary aberra-tions. This was done, in part, because a large frac-tion of the existing suite of phase contrast imaging sys-tems (e.g. propagation-based phase contrast, Zernikephase contrast, analyser-crystal phase contrast, edge-illumination phase contrast etc.) may be viewed as aber-rated imaging systems, in the sense of possessing a com-plex transfer function that is not unity. This emphasis

also had the following future speculation in mind: As co-herent X-ray imaging systems become more mature, theability to tune particular aberrations will become increas-ingly refined, in light of the fact that non-zero aberrationsare a necessary condition for the existence of phase con-trast. Rather than considering oneself to be limited tothe particular aberrations associated with a particularphase-contrast imaging system, it may be fruitful to in-stead consider a near future where a suite of differentaberration parameters may be accessed or tuned at will.Again, this speculated future state of affairs—for coher-ent X-ray imaging—is a current reality for electron imag-ing, now that transmission electron microscopes that arecorrected for spherical aberration and have tunable aber-rations have become available (Pennycook, 2017).

(c) Here we remark on possible future developmentsregarding the complementarity between iterative and de-terministic methods for solving inverse imaging problemsin coherent X-ray optics.

• Iterative methods have the virtue of applicabilityto a wider class of inverse imaging problems, andin particular to imaging problems for which de-terministic closed-form solutions are not known.Iterative methods typically rely on minimisationof an error metric, which is often rather easy towrite down but typically rather computationallyexpensive to minimise. Such computational ex-pense, which typically seeks error-metric minimi-sation in a function space of high dimension, willbecome less of a limitation in light of the avail-ability of cheap ever-more-powerful computing ma-chines. The ideas of compressive sensing and ma-chine learning are both readily incorporated intoiterative approaches, which again speaks in theirfavour regarding future applications. One negativeof iterative approaches is that they often leave ob-scure the question of the uniqueness, and there-fore the correctness, of the resulting reconstruc-tion. They also often leave obscure the fundamen-tal questions of the stability of reconstructions withrespect to imperfections in the input data. Thesenegatives must be balanced against the very richvariety of successful iterative image reconstructionsin the context of coherent X-ray imaging that areextant, a state of affairs that will surely continuein future for a wide variety of imaging scenarios.

• Deterministic approaches seek closed-form solu-tions to the inverse problems of coherent X-ray op-tics. While applicable to a smaller class of (sim-pler) problems compared to the larger set of prob-lems that may be treated iteratively, deterministicapproaches often but not always have the advan-tage of greater conceptual clarity and speed of re-construction. The conceptual clarity is providedby knowledge that the provided solution is both

25

unique and stable with respect to perturbations inthe input data that will be present for any realisticexperiment, while speed is provided by applicationof an often-simple closed-form formula. It is pos-sible that, in the future, some problems that werehitherto only thought to be soluble via iterativemeans, may turn out to be addressable by deter-ministic means, after all.

• Lastly, we remark that iterative and determinis-tic methods are not mutually exclusive, as thereare many situations in which the approximate so-lutions provided by the latter may be iterativelyrefined by the former. Both iterative and determin-istic methods are important, and the just-describedcomplementarity is likely to prevail in future. Seee.g. Pavlov et al. (2018) for more information.

(d) The key ideas of compressed sensing, machinelearning and artificial intelligence are likely to play anincreasing role in both the quantitative and qualitativeanalysis of X-ray imaging data, on a variety of levels. Forexample:

• Compressed sensing comes with the promise ofsignificant improvements in the efficiency of X-ray data collection, permitting reconstructions ofa sample of interest to be performed, under certaincircumstances, with fewer data (and therefore lessdose) than was previously thought possible.

• Machine learning (an approach that is surging mas-sively across many fields) comes with the promise ofvery holistic and flexible analyses which make fulleruse of the totality of available X-ray data, than hasbeen the case hitherto. In this context we note theilluminating comment of Carsten Detlefs (2019):“Machine learning depends critically on the avail-ability of training data. In many cases a forwardsimulation of a known (virtual) sample is much eas-ier to compute than the reconstruction of an un-known sample. Thus, virtual experiments mightbe used to train machine learning algorithms with-out the need for (costly) synchrotron experiments.On the other hand, the quality of the machine anal-ysis depends critically on the quality of the train-ing data. Therefore any bias, systematic error, orfailure to represent a systematic error in the realexperiment may lead to flaws in the analysis.”

• Artificial intelligence, besides the role it can play inmachine learning, brings the promise of automatedor semi-automated assessment of large volumes ofX-ray imaging data, e.g. in a medical-diagnosis oran industrial-testing context.

(e) We have nowhere mentioned speckles, in our dis-cussions relating to partial coherence. This is remiss,

given that the concept of unresolved speckle lies at theheart of most phenomena relating to partial coherence(see e.g. Nugent et al. 2003; Vartanyants & Robinson2003; Nesterets 2008). The following comment fromVartanyants & Robinson (2003), regarding unresolvedspeckle, is particularly illuminating: “It is important tonote that the ‘decoherence’ effect of optics is not a degra-dation of the inherent source coherence, but instead thecreation of an entirely new component to the coherencefunction with a dramatically reduced coherence length.”Typical scalar partially coherent X-ray fields are litteredwith a profusion of speckles that are too rapidly varyingin both space and time to be resolved by one’s detec-tors. The idea that one averages over such speckles, witha spatial average over the detector element and a tem-poral average over the detection time, gives a means forvisualising and indeed modelling partially coherent X-ray fields. With current computing power, and increasedcomputing power projected for the future, such directspatio-temporal modelling of partially coherent fields interms of underpinning fluctuating speckle fields, forms avery promising and conceptually illuminating avenue forfuture research. Note that, at the scalar-field level, suchspeckle fields are threaded by typically complicated—indeed fractal (O’Holleren et al., 2008)—random net-works of nodal-line-threaded phase singularities associ-ated with X-ray vortices. For more on X-ray phase vor-tices and their associated nodal lines (“threads of dark-ness”), see e.g. Chapter 5 of Paganin (2006), togetherwith references therein. If one ascends to a vectorial de-scription of the classical X-ray electromagnetic field forpartially-coherent X-ray imaging scenarios, the specklesin the instantaneous energy density will remain, but thephase vortices will be threaded by a random network ofvectorial singularities (see e.g. Nye (1999), and referencestherein, for a discussion of the singularities of vector fieldssuch as the electric and magnetic fields).

(f) Virtual optics, in which the computer forms an in-trinsic component of an X-ray optical imaging system,has of course long been a reality, e.g. in X-ray com-puted tomography (Diemoz et al. 2017), interferometry(Bonse & Hart, 1965), holography (Gabor, 1948), crystal-lography (Hammond, 2001), coherent diffractive imaging(Miao et al. 1999), ptychography (Pfeiffer, 1999) etc. Invirtual-optics systems, the computer forms an intrinsiccomponent of the imaging system, with (i) hardware op-tics streaming and manipulating X-ray optical informa-tion at the field level, supplemented by (ii) digital com-puters that stream and manipulate digital optical infor-mation at the level of bits. Stated differently, an analoguecomputer (optical hardware streaming and manipulatingphysical X-ray wave-fields) is coupled with a digital com-puter (digital information processing unit). As comput-ers become both more powerful and cheaper, it is likelythat further impetus will be given to enhancing the roleof computers as an intrinsic component of X-ray imaging

26

systems, with a corresponding simplification of the asso-ciated optical hardware. For an example of virtual opticsrelated to X-ray phase contrast and phase retrieval, seee.g. Paganin et al. (2004a).

(g) Much thought has been given to the implicationsof future advances in the brightest-available X-ray sourcetechnology, such as X-ray free-electron lasers. We willnot add to this discussion here. Rather, we explore adifferent class of implications of advances in source tech-nology, regarding off-the-shelf X-ray sources available tosmall research facilities, hospitals and industry. In thiscontext, we state an obvious fact: while third-generationX-ray synchrotron sources will become increasingly diver-gent from fourth-generation and higher-generation X-raysources, they are likely to become increasingly conver-gent towards off-the-shelf compact sources available insmaller scale research laboratories and hospitals. Thisis important in contexts such as medical and industrialimaging, enhancing the ability of proof-of-concept third-generation-synchrotron experiments to be translated tomedical and industrial settings.

(h) What will be the implications of advances in detec-tor technology, for example due to future pixel detectorsbeing able to measure a full X-ray spectrum at each pixelof a two-dimensional image? Surely there are immensefuture opportunities here, to mine the very information-rich data-sets that such detectors will provide.

ACKNOWLEDGEMENTS

This tutorial was presented as three two-hour sem-inars, delivered to the European Synchrotron (ESRF)community, on May 31 – June 2, 2017. We thank theEuropean Synchrotron for facilitating and videotapingthese lectures, as well as making them available to all onYouTube (see link in abstract).

Both authors offer sincere thanks to Alexander Rackand the ESRF Directors for supporting their visits toESRF, and for the privilege of working with AlexanderRack, Margie Olbinado and Yin Cheng on X-ray ghostimaging.

Thanks to all of you who stopped by, for inspiring con-versations and meetings during our ESRF visits in 2017and 2018.

Figure 7 was artistically drawn and coloured byKristina Pelliccia.

Thankyou to Carsten Detlefs (ESRF), for his closereading of the first arXiv version of this text, and for hisdetailed comments. Incorporating these corrections andclarifications has led to many significant improvements.

REFERENCES

B.W. Adams, C. Buth, S.M. Cavaletto, J. Evers,Z. Harman, C.H. Keitel, A. Palffy, A. Picon, R.Rohlsberger, Y. Rostovtsev and K. Tamasaku, X-rayquantum optics, J. Mod. Opt. 60, 2–21 (2013).

M.A. Alonso, Wigner functions in optics: describingbeams as ray bundles and pulses as particle ensembles,Adv. Opt. Photonics 3, 272–365 (2011).

M.A. Beltran, D.M. Paganin, K. Uesugi and M.J.Kitchen, 2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance, Opt. Ex-press 18, 6423–6436 (2010).

M.A. Beltran, D.M. Paganin, K.K.W. Siu, A. Fouras,S.B. Hooper, D.H. Reser and M.J. Kitchen, Interface-specific X-ray phase retrieval tomography of complex bi-ological organs, Phys. Med. Biol. 56, 7353–7369 (2011).

M.V. Berry and C. Upstill, Catastrophe Optics: Mor-phologies of caustics and their diffraction patterns,Prog. Opt. 18, 257–346 (1980).

M.V. Berry, Much ado about nothing: Optical dis-location lines (phase singularities, zeros, vortices ...),Proc. SPIE 3487, 1–5 (1998).

M.V. Berry, Optical currents, J. Opt. A: Pure Appl.Opt. 11, 094001 (2009).

S. Berujon, E. Ziegler, R. Cerbino and L. Peverini,Two-dimensional X-ray beam phase sensing, Phys. Rev.Lett. 108, 158102 (2012).

A. Bravin, P. Coan and P. Suortti, X-ray phase-contrast imaging: from pre-clinical applications towardsclinics, Phys. Med. Biol. 58, R1–R35 (2013).

H. Bremmer, On the asymptotic evaluation of diffrac-tion integrals with a special view to the theory of defocus-ing and optical contrast, Physica 18, 469–485 (1952).

U. Bonse and M. Hart, A X-ray interferometer, Appl.Phys. Lett. 6, 155–156 (1965).

D. Ceddia and D.M. Paganin, Random-matrix bases,ghost imaging and x-ray phase contrast computationalghost imaging, Phys. Rev. A 97, 062119 (2018).

D. Chapman, W. Thomlinson, R. E. Johnston, D.Washburn, E. Pisano, N. Gmur, Z. Zhong, R. H. Menk,F. Arfelli and D. Sayers, Diffraction enhanced X-rayimaging, Phys. Med. Biol. 42, 2015–2025 (1997).

P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay andM. Schlenker, Phase objects in synchrotron radiation hardX-ray imaging, J. Phys. D: Appl. Phys. 29, 133–146(1996).

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J.Van Landuyt, J. P. Guigay and M. Schlenker, Holotomog-raphy: Quantitative phase tomography with micrometerresolution using hard synchrotron radiation x rays, Appl.Phys. Lett. 75, 2912–2914 (1999).

J.M. Cowley and A.F. Moodie, The scattering of elec-trons by atoms and crystals. I. A new theoretical ap-proach, Acta Cryst. 10, 609–619 (1957).

27

J.M. Cowley and A.F. Moodie, The scattering of elec-trons by atoms and crystals. III. Single-crystal diffrac-tion patterns, Acta Cryst. 12, 360–367 (1959).

C. Detlefs, Private communication to D.M. Paganin,2019.

F. Doring, A.L. Robisch, C. Eberl, M. Osterhoff, A.Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M.Bartels, T. Salditt and H.U. Krebs, Sub-5 nm hard X-ray point focusing by a combined Kirkpatrick-Baez mirrorand multilayer zone plate, Opt. Express 21, 19311–19323(2013).

P. C. Diemoz, C. K. Hagen, M. Endrizzi, M. Minuti,R. Bellazzini, L. Urbani, P. De Coppi and A. Olivo,Single-shot X-ray phase-contrast computed tomographywith nonmicrofocal laboratory sources, Phys. Rev. Appl.7, 044029 (2017).

E. Forster, K. Goetz and P. Zaumseil, Double crys-tal diffractometry for the characterization of targets forlaser fusion experiments, Kristall und Technik, 15 937–945 (1980).

D. Gabor, A new microscopic principle, Nature 161,777–778 (1948).

G. Gbur and E. Wolf, Relation between computed to-mography and diffraction tomography, J. Opt. Soc. Am.A 18, 2132–2137 (2001).

D. Giovannini, J. Romero, V. Potocek, G. Ferenczi, F.Speirits, S.M. Barnett, D. Faccio and M.J. Padgett, Spa-tially structured photons that travel in free space slowerthan the speed of light, Science 347, 857–860 (2015).

J.W. Goodman, Introduction to Fourier Optics, 3rdedn, Roberts & Company, Englewood Colorado (2005).

T.E. Gureyev, A. Pogany, D.M. Paganin and S.W.Wilkins, Linear algorithms for phase retrieval in the Fres-nel region, Opt. Commun. 231, 53–70 (2004).

T.E. Gureyev, S.C. Mayo, D.E. Myers, Ya.Nesterets,D.M. Paganin, A. Pogany, A.W. Stevenson and S.W.Wilkins, Refracting Rontgen’s rays: Propagation-based x-ray phase contrast for biomedical imaging, J. Appl. Phys.105, 102005 (2009).

T.E. Gureyev, S.C. Mayo, Ya.I. Nesterets, S. Moham-madi, D. Lockie, R.H. Menk, F. Arfelli, K.M. Pavlov,M.J. Kitchen, F. Zanconati, C. Dullin and G. Tromba,Investigation of the imaging quality of synchrotron-basedphase-contrast mammographic tomography, J. Phys. D:Appl. Phys. 47, 365401 (2014).

T.E. Gureyev, D.M. Paganin, A. Kozlov, Ya.I.Nesterets and H.M. Quiney, Complementary aspects ofspatial resolution and signal-to-noise ratio in computa-tional imaging, Phys. Rev. A 97, 053819 (2018).

J. Hadamard, Lectures on Cauchy’s Problem in Lin-ear Partial Differential Equations, Yale University Press,New Haven (1923).

C. Hammond, The Basics of Crystallography andDiffraction, 2nd ed., Oxford University Press, Oxford(2001).

S.C. Irvine, K.S. Morgan, Y. Suzuki, K. Uesugi, A.Takeuchi, D.M. Paganin and K.K.W. Siu, Assessment ofthe use of a diffuser in propagation-based phase contrastimaging, Opt. Express 18, 13478–13491 (2010).

Y.Y. Kim, L. Gelisio, G. Mercurio, S. Dziarzhyt-ski, M. Beye, L. Bocklage, A. Classen, C. David,O.Yu. Gorobtsov, R. Khubbutdinov, S. Lazarev, N.Mukharamova, Yu.N. Obukhov, B. Roesner, K. Schlage,I.A. Zaluzhnyy, G. Brenner, R. Roehlsberger, J. von Zan-thier, W. Wurth, and I.A. Vartanyants, Ghost Imagingat an XUV Free-Electron Laser, pre-print available athttps://arxiv.org/abs/1811.06855 (2018).

A.M. Kingston, D. Pelliccia, A. Rack, M.P. Olbinado,Y. Cheng, G.R. Myers and D.M. Paganin, Ghost tomog-raphy, Optica 5, 1516–1520 (2018).

A.M. Kingston, G.R. Myers, D. Pelliccia, I.D. Svalbeand D.M. Paganin, X-ray ghost-tomography: Artefacts,dose distribution, and mask considerations, IEEE Trans.Comput. Imaging 5, 136–149 (2019).

E.J. Kirkland, Advanced Computing in Electron Mi-croscopy, 2nd ed., Springer, Berlin/Heidelberg (2010).

M.J. Kitchen, G.A. Buckley, T.E. Gureyev, M.J. Wal-lace, N. Andres-Thio, K. Uesugi, N. Yagi and S.B.Hooper, CT dose reduction factors in the thousands usingX-ray phase contrast, Sci. Rep. 7, 15953 (2017).

H.-O. Kreiss and J. Lorenz, Initial-Boundary Problemsand the Navier–Stokes Equations, Academic Press, SanDiego (1989).

R. Kress, Linear Integral Equations, Springer–Verlag,Berlin, p. 221 (1989).

E. Kuznetsova and O. Kocharovskaya, Quantum opticswith X-rays, Nat. Phot. 11, 685–686 (2017).

K. Li, M. Wojcik, and C. Jacobsen, Multislice does itall – calculating the performance of nanofocusing X-rayoptics, Opt. Express 25, 1831–1846 (2017).

S.W. Lovesey and S.P. Collins, X-ray Scattering andAbsorption by Magnetic Materials, Oxford UniversityPress, Oxford (1996).

E. Madelung, Quantentheorie in hydrodynamischerForm, Z. Phys. 40, 322–326 (1927).

L. Mandel and E. Wolf, Optical Coherence and Quan-tum Optics, Cambridge University Press, Cambridge(1995).

H.E. Martz Jr., B.J. Kozioziemski, S.K. Lehman, S.Hau-Riege, D.J. Schneberk and A. Barty, Validation ofradiographic simulation codes including X-ray phase ef-fects for millimeter-size objects with micrometer struc-tures, J. Opt. Soc. Am. A 24, 169–178 (2007).

J. Miao, P. Charalambous, J. Kirz and D. Sayre, Ex-tending the methodology of X-ray crystallography to allowimaging of micrometre-sized non-crystalline specimens,Nature 400, 342–344 (1999).

P. Modregger, F. Scattarella, B. R. Pinzer, C. David,R. Bellotti and M. Stampanoni, Imaging the ultrasmall-angle X-ray scattering distribution with grating interfer-ometry, Phys. Rev. Lett. 108, 048101 (2012).

28

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi,K. Takai and Y. Suzuki, Demonstration of X-ray Talbotinterferometry, Jpn. J. Appl. Phys. 42, L866–L868(2003).

K.S. Morgan, S.C. Irvine, Y. Suzuki, K. Uesugi, A.Takeuchi, D.M. Paganin and K.K.W. Siu, Measurementof hard x-ray coherence in the presence of a rotatingrandom-phase-screen diffuser, Opt. Commun. 283, 216–225 (2010).

K.S. Morgan, D.M. Paganin and K.K.W. Siu, X-rayphase imaging with a paper analyzer, Appl. Phys. Lett.100, 124102 (2012).

P.R.T. Munro, L. Rigon, K. Ignatyev, F.C.M. Lopez,D. Dreossi, R.D. Speller and A. Olivo, A quantitative,non-interferometric X-ray phase contrast imaging tech-nique, Opt. Express 21, 647–661 (2013).

P. Muller, M. Schurmann and J. Guck, The theory ofdiffraction tomography, https://arxiv.org/abs/1507.00466 (2016).

Ya.I. Nesterets, On the origins of decoherence and ex-tinction contrast in phase-contrast imaging, Opt. Com-mun. 281, 533–542 (2008).

Ya.I. Nesterets and T.E. Gureyev, Noise propagationin x-ray phase-contrast imaging and computed tomogra-phy, J. Phys. D: Appl. Phys. 47 105402 (2014).

K.A. Nugent, T.E. Gureyev, D. Cookson, D. Paganinand Z. Barnea, Quantitative phase imaging using hardX-rays, Phys. Rev. Lett. 77, 2961–2964 (1996).

K. A. Nugent, C. Q. Tran and A. Roberts, Coherencetransport through imperfect x-ray optical systems, Opt.Express 11, 2323-2328 (2003).

J.F. Nye, Natural Focusing and Fine Structure ofLight, Institute of Physics Publishing, Bristol (1999).

K. O’Holleran, M.R. Dennis, F. Flossmann and M.J.Padgett, Fractality of Light’s Darkness, Phys. Rev. Lett.100, 053902 (2008).

A. Olivo, K. Ignatyev, P.R.T. Munro and R.D. Speller,Noninterferometric phase-contrast images obtained withincoherent X-ray sources, Appl. Opt. 50, 1765–1769(2011).

D.M. Paganin, S.C. Mayo, T.E. Gureyev, P.R. Millerand S.W. Wilkins, Simultaneous phase and amplitude ex-traction from a single defocused image of a homogeneousobject, J. Microsc. 206, 33–40 (2002).

D. Paganin, T.E. Gureyev, S.C. Mayo, A.W. Steven-son, Ya.I. Nesterets and S.W. Wilkins, X-ray omni mi-croscopy, J. Microsc. 214, 315–327 (2004a).

D. Paganin, T.E. Gureyev, K.M. Pavlov, R.A. Lewisand M. Kitchen, Phase retrieval using coherent imagingsystems with linear transfer functions, Opt. Commun.234, 87–105 (2004b).

D. Paganin, A. Barty, P.J. McMahon and K.A. Nu-gent, Quantitative phase-amplitude microscopy. III. Theeffects of noise, J. Microsc. 214, 51–61 (2004c).

D.M. Paganin, Coherent X-ray Optics, Oxford Univer-sity Press, Oxford (2006).

D.M. Paganin and T.E. Gureyev, Phase contrast,phase retrieval and aberration balancing in shift-invariantlinear imaging systems, Opt. Commun. 281, 965–981(2008).

D.M. Paganin, T.C. Petersen and M.A. Beltran, Prop-agation of fully coherent and partially coherent complexscalar fields in aberration space, Phys. Rev. A 97,023835 (2018).

K.M. Pavlov, T.E. Gureyev, D. Paganin, Ya.I.Nesterets, M.J. Morgan and R.A. Lewis, Linear systemswith slowly varying transfer functions and their applica-tion to x-ray phase-contrast imaging, J. Phys. D: Appl.Phys. 37, 2746–2750 (2004).

K.M. Pavlov, T.E. Gureyev, D. Paganin, Ya.I.Nesterets, M.J. Kitchen, K.K.W. Siu, J.E. Gillam, K.Uesugi, N. Yagi, M.J. Morgan and R.A. Lewis, Uni-fication of analyser-based and propagation-based X-rayphase-contrast imaging, Nucl. Inst. Meths Phys. Res.A 548, 163–168 (2005).

K.M. Pavlov, K.S. Morgan, V.I. Punegov and D.M.Paganin, Deterministic X-ray Bragg coherent diffractionimaging as a seed for subsequent iterative reconstruction,J. Phys. Commun. 2, 085027 (2018).

D. Pelliccia and D.M. Paganin, Coherence vortices invortex-free partially coherent x-ray fields, Phys. Rev. A86, 015802 (2012).

D. Pelliccia, L. Rigon, F. Arfelli, R.H. Menk, I.Bukreeva and A. Cedola, A three-image algorithm forhard X-ray grating interferometry, Opt. Express, 21,19401–19411 (2013).

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli and D.M.Paganin, Experimental X-ray ghost imaging, Phys. Rev.Lett. 117, 113902 (2016).

D. Pelliccia, M. P. Olbinado, A. Rack, A. M. Kingston,G. R. Myers and D. M. Paganin, Towards a practicalimplementation of X-ray ghost imaging with synchrotronlight, IuCrJ 5, 428–438 (2018).

S.J. Pennycook, The impact of STEM aberration cor-rection on materials science, Ultramicroscopy 180, 22–33(2017).

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eiken-berry, Ch. Bronnimann, C. Grunzweig and C. David,Hard-X-ray dark-field imaging using a grating interfer-ometer, Nat. Mater. 7, 134–137 (2008)

F. Pfeiffer, X-ray ptychography, Nat. Photonics 12,9–17 (2018).

L. Rigon, F. Arfelli and R.-H. Menk, Three-imagediffraction enhanced imaging algorithm to extract absorp-tion, refraction and ultrasmall-angle scattering, Appl.Phys. Lett. 90, 114102 (2007).

D. Ruelle, Chaotic Evolution and Strange Attractors,Cambridge University Press, Cambridge (1989).

B.E.A. Saleh and M.C. Teich, Fundamentals of Pho-tonics, 2nd edn, Wiley-Interscience, Hoboken New Jersey(2007).

29

D. Sayre and H. N. Chapman, X-ray microscopy, ActaCryst. A51 237–252 (1995).

P. Schiske, Image reconstruction by means of focus se-ries, J. Microsc. 207, 154 (2002). Note that this is atranslation from the German, of a paper that first ap-peared in 1968, in the Proceedings of the Fourth RegionalCongress on Electron Microscopy, Rome 1968, vol. 1, pp.145–146.

G. Schneider, Cryo X-ray microscopy with high spa-tial resolution in amplitude and phase contrast, Ultrami-croscopy 75 85–104 (1998).

A. Schori and S. Shwartz, X-ray ghost imaging with alaboratory source, Opt. Express 25, 14822–14828 (2017).

A. Schori, D. Borodin, K. Tamasaku and S. Shwartz,Ghost imaging with paired x-ray photons, Phys. Rev. A97, 063804 (2018).

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov andI. Schelokov, On the possibilities of X-ray phase contrastmicroimaging by coherent high-energy synchrotron radi-ation, Rev. Sci. Instrum. 66, 5486–5492 (1995).

M.R. Teague, Deterministic phase retrieval: a Green’sfunction solution, J. Opt. Soc. Am. 73, 1434–1441(1983).

I.A. Vartanyants and I.K. Robinson, Origins of deco-herence in coherent X-ray diffraction experiments, Opt.Commun. 222, 29–50 (2003).

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stam-panoni, P. Cloetens and E. Ziegler, X-ray phase imagingwith a grating interferometer, Opt. Express 13, 6296–6304 (2005).

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong,N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M.A. Anastasio and C. Muehleman, Multiple-image radiog-raphy, Phys. Med. Biol. 48, 3875–3895 (2003).

V. White and D. Cerrina, Metal-less X-ray phase-shiftmasks for nanolithography, J. Vac. Sci. Technol. B 10,

3141–3144 (1992).S.W. Wilkins, T.E. Gureyev, D. Gao, A. Pogany and

A.W. Stevenson, Phase-contrast imaging using polychro-matic hard X-rays, Nature 384, 335–338 (1996).

S.W. Wilkins, Ya.I. Nesterets, T.E. Gureyev, S.C.Mayo, A. Pogany and A.W. Stevenson, On the evolutionand relative merits of hard X-ray phase-contrast imagingmethods, Phil. Trans. R. Soc. A 372, 20130021 (2014).

E. Wolf, Three dimensional structure determination ofsemi-transparent objects by holographic data, Opt. Com-mun. 1, 153–156 (1969).

E. Wolf, New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra ofsteady-state sources, J. Opt. Soc. Am 72, 343–351(1982).

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao andD. Zhu, Fourier-Transform Ghost Imaging with Hard XRays, Phys. Rev. Lett. 117, 113901 (2016).

B. Yu, L. Weber, A. Pacureanu, M. Langer, C. Olivier,P. Cloetens and F. Peyrin, Phase retrieval in 3D X-raymagnified phase nano CT: Imaging bone tissue at thenanoscale, 2017 IEEE 14th International Symposium onBiomedical Imaging (ISBI 2017), 56–59 (2017).

M.-C. Zdora, State of the art of X-ray speckle-basedphase-contrast and dark-field imaging, J. Imaging 4, 60(2018).

F. Zernike, Phase contrast, a new method for the mi-croscopic observation of transparent objects, Physica 9,686–698 (1942).

A.-X. Zhang, Y.-H. He, L.-A. Wu, L.-M. Chen andB.-B. Wang, Tabletop x-ray ghost imaging with ultra-lowradiation, Optica 5, 374–377 (2018).

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu,S. A. McDonald, F. Marone and M. Stampanoni, Low-dose, simple, and fast grating-based X-ray phase-contrastimaging, Proc. Natl. Acad. Sci. U.S.A. 107, 13576–13581 (2010).