Abstract
The Fokker–Planck equation can be used in a partiallycoherent imaging context to model the evolution of the intensity of a paraxial xray wave field with propagation. This forms a natural generalisation of the transportofintensity equation. The xray Fokker–Planck equation can simultaneously account for both propagationbased phase contrast, and the diffusive effects of sampleinduced smallangle xray scattering, when forming an xray image of a thin sample. Two derivations are given for the Fokker–Planck equation associated with xray imaging, together with a Kramers–Moyal generalisation thereof. Both equations are underpinned by the concept of unresolved speckle due to unresolved sample microstructure. These equations may be applied to the forward problem of modelling image formation in the presence of both coherent and diffusive energy transport. They may also be used to formulate associated inverse problems of retrieving the phase shifts due to a sample placed in an xray beam, together with the diffusive properties of the sample. The domain of applicability for the Fokker–Planck and Kramers–Moyal equations for paraxial imaging is at least as broad as that of the transportofintensity equation which they generalise, hence the technique is also expected to be useful for paraxial imaging using visible light, electrons and neutrons.
Introduction
Suppose that coherent visiblelight plane waves normally illuminate a thin lens made of glass that is slightly cloudy. For such a lens, a large percentage of the light will be coherently focused or defocused upon propagation beyond the lens. Conversely, a small percentage of the light will be diffusely scattered by the random cloudy inclusions within the glass. In the hardxray analogue of this experiment, replace the glass lens with a thin semitransparent sample whose largescale projected structure locally resembles an aberrated converging or diverging lens (see Fig. 1(a)), and replace the cloudy inclusions with unresolved random microstructure within the volume of the sample (Fig. 1(b)). For hard x rays we will often have a situation where both coherent and diffuse scatter are narrowly peaked in the forward direction. With reference to Fig. 1(a), consider the “near field” regime where the objecttodetector distance Δ is significantly smaller than the focal length of the “local lenses” of which the sample is considered to be comprised: see e.g. the “local diverging lens” at A, and the “local converging lens” at C, the nominal focal length f for each lens being much larger than Δ. This is the key situation, of combined coherent and diffusive energy transport in a nearfield paraxial imaging setting, that we wish to consider in the present paper. While we focus on the case of hard x rays, the methods considered here will also be applicable to paraxial imaging using visible light, electrons, neutrons etc.
We make the simplifying assumptions that: (i) magnetic effects such as xray circular dichroism in the sample can be ignored; (ii) the sample is noncrystalline, allowing xray Braggdiffraction effects to be ignored; (iii) the sample is sufficiently thin that dynamical xray diffraction effects, within the volume of the sample, can also be ignored; (iv) all polarisationsensitive effects can be ignored. Many wellknown means exist, for the quantitative modelling of an xray beam interacting with such a sample and subsequently forming a paraxial image as it propagates through space^{1,2,3}. One of the most oftenemployed, and most elementary, means is to first use the projection approximation to quantify the passage of an incident coherent scalar xray beam through the sample, and then use the Fresnel diffraction formalism to propagate the resulting exitsurface wave field to the surface of a detector^{1,4}. The projection approximation assumes that: (i) the radiation is of sufficiently high energy that individual rays within the sample (more precisely, streamlines of the energyflow vector within the sample) are unchanged within the volume of the said sample in comparison to the streamlines that would have existed in the absence of the sample, and (ii) both phase and amplitude shifts are accumulated continuously and locally, as one traverses any such ray from the entrance surface of the sample to the exit surface–see e.g. Sec. 2.2 of Paganin^{4}. The Fresnel approximation, which is a consequence of the paraxial approximation, assumes that spherical Huygenstype wavelets can be approximated as having parabolic wave fronts. If the sampletodetector propagation distance is sufficiently small, and if diffuse scattering is negligible, Fresnel diffraction implies that the transportofintensity equation^{5} (TIE) may be used to model the evolution of image intensity with respect to propagation distance downstream of the sample, for a paraxial quasimonochromatic beam. The TIE, which is the continuity equation for paraxial wave fields, expresses local energy conservation^{6}. It forms a basis for several imaging techniques in coherent xray optics^{1,4,7}. Like the formalism of Fresnel diffraction upon which it is based, the TIE models what may be called “coherent energy transport” downstream of a thin sample. That is, at each point in the space downstream of the sample, there is one timeindependent energyflow vector (Poynting vector proportional to the intensity multiplied by the gradient of the wavefield phase^{8,9,10}). However, if there is unresolved spatially random microstructure within the sample^{11}, then, as hinted in our opening paragraph, smallangle xray scatter^{12} (SAXS) will also be present. This augments the previously mentioned coherent energyflow vector at each point in space, with an ensemble (“SAXS fan”) of energyflow vectors associated with diffusive energy transport. How can this additional effect be taken into account, in modelling paraxial hard xray imaging of thin objects?
One answer is provided by a more general equation than the TIE, namely the the Fokker–Planck equation^{13}. The latter equation, which can model flows in radiation and matter wave fields that have both a coherent and a diffusive component, may be obtained in a very general setting as a limit case of the still more general Boltzmann equation of classical statistical mechanics^{14}. The Fokker–Planck equation is used in many fields: examples include Brownian motion^{14}, hydrodynamics^{15}, electron and photon transport in biological tissues^{16,17} and homogeneous media^{18}, hightemperature atomiclevel disorder in premelting surfaces^{19}, gaseous microflows^{20}, droplet nucleation^{21}, dilute polymer dynamics^{22}, hot plasmas^{23}, biaxial fluids^{24}, cold turbulent gas jets^{25} and quark gluon plasmas^{26}. Notable also is the use of the Fokker–Planck equation in the nonimaging context of xray kinematical and dynamical diffraction from imperfect crystals with stochastically distributed defects^{27,28}. An extremely recent application, which is particularly close to that developed in the present paper, uses Fokker–Planck concepts to model the position probability density of a nanoparticle in a nearfield optical trap^{29}.
The Fokker–Planck equation is also useful for xray imaging, since this is a setting where both coherent and diffusive energy flows are present. This equation can simultaneously model four effects, resulting from the illumination of a thin object by normally incident coherent xray plane waves:
Local absorption by the object (attenuation contrast, known since the 1895 discovery of x rays by Röntgen^{30});
Local lensing by the object which concentrates or rarefies energy density due to local focusing or defocusing (propagationbased Laplaciantype phase contrast^{31});
Local prismlike effects of the object which transversely shift optical energy (propagationbased differential phase contrast^{31});
Local blurring with a positiondependent point spread function, associated with the SAXS fan emerging from each point at the exit of the object.
The first three effects are associated with the “TIE part” of the Fokker–Planck equation, with the fourth effect associated with the “diffusion equation” part. Of particular interest is the use of an xray form of the Fokker–Planck equation for modelling forward problems in paraxial xray imaging in a simple manner, e.g. by expressions valid in the near field (small propagation distance, Fresnel number^{32} \({N}_{{\rm{F}}}\gg 1\)) that are of firstorder accuracy with respect to the sampletodetector propagation distance. A Kramers–Moyal^{13} extension will also be of use when a more detailed treatment of the SAXS, emanating from each point over the exit surface of the object, is required. We anticipate that the resulting formalism will be of use in forward problems related to using x rays to image thin objects, as well as inverse problems such as phase/amplitude/diffusiontensor recovery in both two and three dimensions (the latter constitutes the field of tensor tomography^{33,34,35,36,37,38}).
We close this introduction by outlining the remainder of the paper. The xray Fokker–Planck model is derived in two complementary ways. The first derivation considers the Fokker–Planck equation to be a fusing of (i) the transportofintensity equation for coherent energy transport in paraxial monochromatic xray beams, with (ii) the diffusion equation for diffusive paraxial energy transport associated with smallangle xray scattering within a sample. The second derivation is a microscopic treatment based on first principles, which relies on the crucial assumption that the phase at the exit surface of the object can be written as a sum of (i) a component that varies slowly with respect to the detector pixel size, and (ii) a component that fluctuates many times over one detector pixel and is therefore unresolved. We then give a generalised form of the Fokker–Planck equation, known as the Kramers–Moyal equation, as a means for treating the insample SAXS in a more precise manner. Next we sketch some indicative means by which the Fokker–Planck and Kramers–Moyal equations may be used in the forward problem of modelling nearfield intensity contrast in coherent xray imaging, together with the associated inverse problem of phase retrieval and SAXS determination. We also consider some broader implications of the Fokker–Planck formalism.
Fokker–Planck model for xray imaging
Here we give two complementary derivations of the xray Fokker–Planck equation. The first is phenomenological, based on local conservation of energy in the presence of both coherent and diffusive paraxial xray energy transport. The second is a microscopic firstprinciples analysis. The former derivation is more general, while the latter is more physically illuminating.
Xray Fokker–Planck equation: Transportofintensity equation with a diffusive term
Xray beams “flow” in the sense that they stream optical energy as they propagate. As a descriptor of such energy flows, the Fokker–Planck equation may be viewed as a diffusive generalisation of coherent paraxial photon energy transport. Adopting this perspective, below we consider the transportofintensity equation^{5} for coherent paraxial xray energy transport and then separately consider the diffusion equation for diffusive paraxial xray energy transport. We then show that these two equations may be merged, in a manner that conserves energy both locally and globally, to give the Fokker–Planck equation for paraxial xray imaging.
Transportofintensity equation for coherent paraxial energy transport
The transportofintensity equation^{5} describes the intensity evolution of a paraxial complex coherent scalar wave field as it propagates. This continuity equation, arising from the parabolic equation of paraxial scalar wave optics, expresses local energy conservation. It states that the negative divergence – which may loosely be spoken of as the “convergence” – of the transverse Poynting vector (energy flow vector) is proportional to the longitudinal rate of change of intensity^{5}:
Here, I is the intensity of the monochromatic scalar wave field, ϕ is its phase, k is the wavenumber and ∇_{⊥} is the gradient in (x, y) planes perpendicular to the optical axis z. Equation (1) quantifies the fact that energy density (and hence intensity) increases with increasing propagation distance for locallyconverging waves, and decreases with increasing propagation distance for locallydiverging waves.
As previously mentioned, the TIE is a continuity equation expressing local energy conservation under coherent xray energy transport. The conserved (Noether^{39}) current is the transverse Poynting vector, which up to a multiplicative constant is^{4,8,10}
The corresponding constant Noether charge \({\mathfrak{N}}\) is the total integrated intensity over any plane perpendicular to the optical axis:
For propagation in vacuo through a distance Δ ≥ 0 that is sufficiently small for the Fresnel number to be much greater than unity, a longitudinal finitedifference approximation to the TIE can be used to model the resulting propagationbased phase contrast^{4}:
The underpinning physics is sketched in an imaging context in Fig. 1(a), whereby local specular refraction of xray radiation, by a thin object, leads to transverse redistribution of optical energy upon propagation through a distance Δ that is not too large. Here and henceforth, we use the term “specular refraction” in direct analogy to “specular reflection”, with the former term referring to the refraction associated with a mirrorlike refracting surface (i.e. smooth, lacking in the effects of roughness or spatial randomness; cf. Latin speculum). Rays passing through A have a positive divergence, hence the intensity at B is reduced compared to that which would have been registered if the rays passing through A had all been parallel to the optical axis z. Similarly, rays passing through C have a negative divergence (positive “convergence”), hence the intensity at D is increased compared to that which would have been obtained had the rays exiting C been parallel to the optical axis. This phase contrast effect is increased with increased propagation distance Δ, reduced xray energy (decreased wavenumber k) or faster spatial variations in wavefield phase (larger \({\nabla }_{\perp }\varphi (x,y,z=0)\)). We can also have effects such as a transverse deflection, e.g. with the local maximum of intensity at D being slightly shifted from the point x = x_{0}. Concentration or rarefaction of energy density (and hence intensity) may be thought of as due to the lensing term proportional to \(I(x,y,z){\nabla }_{\perp }^{2}\varphi (x,y,z)\) in Eq. (4), while the transverse shifts in intensity maxima or minima may be viewed as due to the prism term \({\nabla }_{\perp }I(x,y,z)\cdot {\nabla }_{\perp }\varphi (x,y,z)\) ^{31}.
Diffusion equation for diffusive paraxial energy transport
If there were only diffusive energy transport on account of unresolved microstructure in the sample (i.e. SAXS), but no specular refraction, we could instead introduce a diffusion coefficient D(x, y, z) to describe the effects at distance z = Δ downstream of an illuminated thin object in the plane z = 0. The fact that this coefficient depends on x and y reflects the fact that the SAXS fan will in general vary with position over the nominal exit surface z = 0 of the thin sample. Bearing all of this in mind, the intensity evolution of the beam may then be described by the diffusion equation:
with the very important proviso that the above expression is always to be understood in its finitedifference form
Note that a formal dependence on Δ has been added to the diffusion coefficient D, since SAXSinduced diffusive blur has a characteristic transverse width that scales with Δ rather than Δ^{1/2}.
The effect shown in one transverse dimension in Fig. 1(b) can be modelled using the above finitedifference form of the diffusion equation. Here, an xray beamlet of width Δx illuminates a slab containing SAXSinducing unresolved microstructure. This width Δx is assumed to be large compared to the characteristic length scale of the projected microstructure, but small compared to the characteristic transverse width associated with the macroscopic projected phase and attenuation coefficients of the object. The illuminated region E of the slab, located at transverse coordinate x = x_{0}, results in a smooth SAXS fan of opening angle θ(x_{0}), which smears the propagated intensity over the plane z = Δ. The characteristic width of this blur, which may be considered as a broadening of the local rocking curve due to the sample^{40,41}, is
Thus, local diffuse scattering by a thin object leads to a transverse smearing of intensity upon propagation. This blur increases with increased propagation distance Δ. At the level at which diffusion is here being considered, the opening angle θ(x_{0}) of the SAXS fan is taken to completely characterise its diffusive effects at each location x_{0}; a more sophisticated treatment which replaces the single number θ(x_{0}) (or its associated diffusion coefficient) with a hierarchy of tensors, will be given later.
Like the TIE, the diffusion equation may be viewed as a continuity equation expressing local energy conservation, albeit under diffusive xray energy transport. Here, the conserved current is given by Fick’s first law as^{13,42}
Fokker–Planck equation for combined coherent and diffusive paraxial energy transport
If both specular refraction and localSAXS effects are present simultaneously, we can add the TIE and diffusion equations together, in an energypreserving manner. This gives the continuity equation:
Here, F(x, y) is the fraction of the optical energy converted to SAXS^{43} when illuminating a thin sample in the plane z = 0 at the point (x, y). If \(F(x,y)\ll 1\) (as is often the case for both soft^{11} and hard^{44} xrays) one can replace 1 − F(x, y) in the above expression by 1. The resulting convection–diffusion equation (forward Kolmogorov equation^{13}) is the Fokker–Planck equation in two transverse dimensions. It models both effects in Fig. 1, i.e. it models the effects of both local sampleinduced specular refraction and local sampleinduced SAXS.
If the expressions for the coherent current \({{\bf{J}}}_{\perp }^{(1)}\) and diffusive current \({{\bf{J}}}_{\perp }^{(2)}\) are written explicitly, and the approximations made that (i) \(F(x,y)\ll 1\) and (ii) F(x, y) is slowly varying with x and y, we obtain the 2 + 1dimensional Fokker–Planck equation:
In one transverse dimension, this becomes the onedimensional Fokker–Planck equation:
The corresponding finitedifference forms, which are to be preferred for reasons that have already been outlined, are respectively given by the following pair of equations:
We close this section by noting that, if D may be considered to be sufficiently slowly varying with respect to transverse coordinates that D commutes with the transverse Laplacian, we may instead work with the simpler finitedifference expressions:
In the above two finitedifference Fokker–Planck equations, FD plays the role of an effective diffusion coefficient
that also accounts for the fraction of the incident radiation converted to SAXS.
Xray Fokker–Planck equation: Derived from first principles
Here we derive the xray Fokker–Planck equation from first principles. We consider two length scales of sampleinduced wavefield variation. The larger scale corresponds to the length scales in the sample that are sufficiently large for their absorptive and refractive effects to lead to detectorresolvable intensity variations. The smaller length scale corresponds to diffuse scattering from unresolved microstructure resulting in SAXS. This derivation clarifies the physical mechanisms underpinning both coherent and diffusive xray energy transport, in a paraxial xray imaging context.
We work in one transverse dimension for simplicity, ignore polarisation and assume monochromatic x rays. Let ψ(x, z = 0) be the spatial part of the complex scalar amplitude at the exit surface of a sample. Subsequent Fresnel diffraction gives^{4}
where P_{Δ}(x) is the Fresnel propagator corresponding to freespace diffraction through a distance Δ ≥ 0, and paraxial radiation is assumed throughout. The squared modulus of the above convolution integral gives the propagated intensity as the following linear combination of contributions from every pair of points x′, x″ in the unpropagated wave field^{45}:
The sample is assumed to be thin, normal to the incident planewave illumination, and with nominal exit surface z = 0. Many models exist for incorporating unresolved microstructure into this sample^{46,47}. Follow Voronovich^{47}, Nesterets^{45} and Yashiro et al.^{48,49} in decomposing the phase ϕ(x, z = 0) of ψ(x, z = 0) as
where (i) ϕ_{s}(x) is a phase component that varies slowly with respect to the detector pixel size, and (ii) ϕ_{f}(x) is a fastvarying spatially random phase due to the sample microstructure, which fluctuates many times over one detector pixel and is thus unresolved (see also Gureyev et al.^{31}). Further assume a slowlyvarying intensity I_{s}(x), together with xrays that are of sufficiently high energy for the projection approximation to hold for a given sample. Hence
and so Eq. (18) becomes^{45}:
Average over an ensemble of realisations of the unresolved fast phasemaps ϕ_{f}(x)^{45,50,51,52}. Denoting this average by an overline,
Assuming the fast phase maps ϕ_{f} to be spatially statistically stationary over transverse displacements that are small compared to the characteristic length scale over which ϕ_{s} fluctuates appreciably, the overlined expression in Eq. (22) depends only on coordinate differences x′ − x″. Assuming ϕ_{f}(x − x′) and ϕ_{f}(x − x″) to be Gaussian variables, with zero mean, this correlation function is^{45,48,49,51,52,53}
where \({\sigma }_{{\varphi }_{{\rm{f}}}}^{2}(x)\) is the xdependent variance of the ensemble of fast phase maps {ϕ_{f}(x)}, and
is the normalised fastphase autocorrelation function^{54} as a function of coordinate separation Δx and transverse coordinate x.
The quantity \({\sigma }_{{\varphi }_{{\rm{f}}}}(x)\) is called the “phase depth”. If this is not significantly greater than unity, the right side of Eq. (23) does not decay to zero as \(x^{\prime} x^{\prime\prime} \to \infty \). Rather, this correlation function asymptotes to \(\exp \,[\,\,{\sigma }_{{\varphi }_{{\rm{f}}}}^{2}(x)]\): see Fig. 2(a). The physical reason for this decay to a nonzero constant is the phase depth not being sufficiently high for \(\exp \,(i{\varphi }_{{\rm{f}}})\) at two different locations to crosscorrelate to zero, irrespective of how widely separated these locations may be (cf. Prade et al.^{55}).
To proceed further, follow Goodman^{51} in separating out the asymptote term from \(\overline{\exp \,\{i[{\varphi }_{{\rm{f}}}(xx^{\prime} ){\varphi }_{{\rm{f}}}(xx^{\prime\prime} )]\}}\) in Eq. (23), and renormalising the part of the correlation that does decay to zero. Thus:
where C(x′ − x″; x) is a renormalised correlation function
that does decay to zero for coordinate separations \(\Delta x\equiv x^{\prime} x^{\prime\prime} \gg \tau \), and τ is the transverse correlation length (see Fig. 2(b)).
Substituting Eq. (25) into Eq. (22) gives^{45}
where \({{\mathscr D}}_{\Delta }^{({\rm{F}})}\) is the Fresnel diffraction operator^{4} acting on the smooth component of the complex amplitude, and \( {\mathcal B} (x,z=\Delta )\) represents smallangle xray scattering. Thus a physical consequence of the lack of decay to zero in Eq. (23), is that the diffraction pattern in Eq. (27) contains two components^{45,51,52,56,57}:
A “specular diffraction” component that undergoes Fresnel diffraction as if there were no microstructure present, but with intensity damped by the decoherence factor^{45} shown in Fig. 2(c);
A diffuse component \( {\mathcal B} (x,z=\Delta )\) associated with SAXS.
This splitting of xray energy flows – whereby an incident coherent xray energy flow upstream of the object bifurcates into superposed coherent and diffusive energy flows downstream of the object – corresponds directly to the factors of F and 1 − F introduced “by hand” in the previous subsection (see Eq. (9)), if we take
Note that a similar bifurcation of energy flows occurs, for similar reasons, in statistical dynamical xray diffraction theory^{58,59} and the frozenphonon model of electron diffraction theory. Note also that a nonzero speculardiffraction component (which will typically be the dominant component in the present context) indicates the SAXSinduced speckle to be “partially developed”^{60}; this may be contrasted with the case of “fully developed” SAXS speckle, which is not relevant to our context, in which the specular component would be fully extinguished.
Assume I_{s}(x) to be sufficiently slowly varying that the squareroot term in Eq. (28) may be replaced by I_{s}(x). For the complex exponent in Eq. (28), make the more sensitive approximation:
where \({\varphi ^{\prime} }_{{\rm{s}}}(x)\) denotes the derivative of ϕ_{s}(x) with respect to x; the above equation makes implicit use of the fact that ϕ_{f} varies much more quickly than ϕ_{s}. Then write P_{Δ}(x′) in Eq. (28) as a Fourier integral using the notation and convention specified by
where k_{x} is the Fourier coordinate corresponding to x. Also write \({P}_{\Delta }^{\ast }\) and C in Eq. (28) as Fourier integrals, with C being Fourier transformed with respect to its first argument. With all of these steps, together with the fact that \({\breve{P}}_{\Delta }=1\) for the Fresnel propagator^{4}, one obtains (cf. similar equations in related contexts^{45,52,61,62}, especially those due to Nesterets^{45}):
Physically, the above equation states that the SAXS contribution, at a given transverse location x, integrates over a fan of rays making angles \([{\varphi ^{\prime} }_{{\rm{s}}}(x){k}_{x}]/k\) with respect to the optical axis. This fan of rays is due to scattering from unresolved microstructure associated with the ensemble of fast phase maps {ϕ_{f}(x)}. The whole fan of SAXS rays is rotated by the angle \({\varphi ^{\prime} }_{{\rm{s}}}(x)/k\), due to the specular refraction. All SAXSscattered x rays, due to illumination of the point at x = x_{0}, are integrated over a single pixel. See Fig. 2(d). The angular spread of the SAXS fan will typically be on the order of a few degrees (50 milliradians) for hard x rays^{63}, while the fan rotation angle \({\varphi ^{\prime} }_{{\rm{s}}}(x)/k\) will typically be on the order of microradians^{64}. The function \(\breve{C}\), being the Fourier transform of a correlation function^{43}, is proportional to the power spectrum of the ensemble of fastphase fields \(\exp \,\{i{\varphi }_{{\rm{f}}}(x)\}\) (Wiener–Khinchin theorem^{65}); this power spectrum (local differential scattering cross section^{54,55,66}; cf. Eq. (44) below) gives the weights for the diffracted “rays” in the SAXS fan. The fraction \(F(x)=1\exp \,[\,\,{\sigma }_{{\varphi }_{{\rm{f}}}}^{2}(x)]\) of the beam converted to SAXS is complementary to the fraction \(1F(x)=\exp \,[\,\,{\sigma }_{{\varphi }_{{\rm{f}}}}^{2}(x)]\) channelled into the specular Fresnel diffraction (cf. Eq. (29)).
If the objecttodetector distance Δ is large enough that the SAXSinduced fan of rays in Fig. 2(d) spreads over more than one pixel, we have the scenario shown in Fig. 2(e). Equation (32) generalises to the convolutiontype integral:
where d(x; x_{0}) is a diffusive blur kernel given by
The function d is simply a transverse rescaling of \(\breve{C}\), from a function of Fourierspace coordinates k_{x} (i.e. angular coordinates k_{x}/k, in the paraxial approximation) to a function of the transverse coordinates x over the detector plane. Stated differently, d(x; x_{0}) is the intensity distribution of SAXS, as a function of position x on the detector, due to x rays incident at position x_{0}. If the objecttodetector distance Δ is sufficiently small, two additional approximations can be made.
 (i)
Small Δ implies d(x; x_{0}) to be sufficiently narrow for its Fourier transform \(\breve{d}({k}_{x};{x}_{0})\) with respect to x to be sufficiently broad that it can be represented by a secondorder Taylor expansion in k_{x}. The term in the Taylor series for \(\breve{d}({k}_{x};{x}_{0})\) that is linear in k_{x} will vanish since C(Δx; x_{0}) is approximately even in Δx (cf. Eq. (47)). Given all of the above, the convolution operator \(\sqrt{2\pi }\,d(x;{x}_{0})\otimes \) in Eq. (33) is well approximated by
$$\sqrt{2\pi }\,d(x;{x}_{0})\otimes \approx Q+{L}^{2}({x}_{0};\Delta )({d}^{2}/d{x}^{2}),$$(35)where \(L({x}_{0};\Delta )\) is an x_{0}dependent blurring width (this blurring being due to SAXS), and Q is a positive constant. Invoke conservation of energy, which implies that the total scattered intensity, due to an incident xray pencil beam of intensity I_{s}(x) and with small thickness Δx, will have the same integrated intensity after scattering from the object; see Fig. 2(e). This shows that Q = 1, so that \(\sqrt{2\pi }\,d(x;{x}_{0})\otimes \approx 1+{L}^{2}({x}_{0};\Delta )({d}^{2}/d{x}^{2})\) (cf. Gureyev et al.^{67}). Hence we may approximate Eq. (33) as:
$$ {\mathcal B} (x={x}_{0},z=\Delta )=\{1\exp \,[\,\,{\sigma }_{{\varphi }_{{\rm{f}}}}^{2}({x}_{0})]\}\{[1+{L}^{2}(x;\Delta )\frac{{d}^{2}}{d{x}^{2}}]{I}_{{\rm{s}}}(x){\}}_{x={x}_{0}}.$$(36)Before proceeding, let us backtrack a little, and point out that in Eq. (33) we consider \({I}_{{\rm{s}}}(x)\otimes d(x;{x}_{0})\) to be equivalent to \(\int \,{I}_{{\rm{s}}}({x}_{0})d(x{x}_{0};{x}_{0})\,d{x}_{0}\). Strictly speaking, this linear integral transform is not a convolution integral since the blurring kernel d depends on x_{0}. We use the convolution notation for clarity, however, to denote such smearing with a position dependent point spread function that results from the positiondependent SAXS. See also the next section, for an analysis that considers the same linear integral transform using a more conventional and explicit notation.
 (ii)
A second consequence of small Δ is that the TIE approximation^{5} may be applied to the first term on the right of Eq. (27):
If Eqs (36) and (37) are substituted into Eq. (27), we obtain
Assume only a small fraction of the incident x rays to be converted to SAXS, so that \({\sigma }_{{\varphi }_{{\rm{f}}}}^{2}(x)\ll 1\). Thus:
Upon comparing the final term of Eq. (39) with the final term in Eq. (15), and then noting from the far right side of Eq. (29) that \({\sigma }_{{\varphi }_{{\rm{f}}}}^{2}(x)\approx F(x)\), we obtain:
The corresponding positiondependent effective diffusion coefficient D_{eff}(x; Δ) obeys Eq. (16), so that
Equation (39) then becomes the 1D form of the finitedifference xray Fokker–Planck equation in Eq. (13), provided the diffusion coefficient varies sufficiently slowly with x that it commutes with the transverse Laplacian. Note also that a Kramers–Moyal generalisation of the Fokker–Planck equation would have resulted if all terms had been included in the expansion for \(\sqrt{2\pi }\,d(x;{x}_{0})\otimes \) (see next section). We again emphasise that Eq. (39) is to be considered with Δ being small but finite and fixed. Thus, the simplicity of using diffusion concepts to describe SAXS, must be balanced against the fact that freespace propagation blurwidth L scales as Δ rather than \(\sqrt{\Delta }\); formally, this is accounted for by having a Δdependent diffusion coefficient for the xray Fokker–Planck equation, which is only ever used in its finitedifference form for small nonzero Δ.
We close this section with Fig. 3. This indicates the relation between various key quantities: the divergence angle θ of the SAXS fan, the characteristic transverse length scale \(l\) of the spatiallyrapid wavefront fluctuations induced by unresolved microstructure at the sample exit surface, the phase depth \({\sigma }_{{\varphi }_{{\rm{f}}}}\) of the associated spatiallyrapid phase fluctuations, the diffusion coefficient D and the effective diffusion coefficient D_{eff} in the Fokker–Planck equation, and the smear width L associated with the SAXS fan. We make the following remarks on each equation in Fig. 3.
 (a)
The link between L and θ follows from elementary geometry;
 (b)
The link between l and θ follows from the optical uncertainty principle^{68};
 (c)
The link between l and L follows upon eliminating θ from relations (a) and (b);
 (d)
The link between D and L follows from Eq. (40);
 (e)
The link between D and l follows upon eliminating L from relations (c) and (d);
 (f)
The link between D and θ follows upon eliminating L from relations (a) and (d);
 (g)
The link between D and D_{eff} follows from Eqs (16) and (41); see also Eq. (29).
All physical dependencies, for the quantities in Fig. 3, are intuitively reasonable: (a) the blur width L is directly proportional to the angular spread θ of the SAXS fan; the blur width L is directly proportional to the sampletodetector distance Δ; (b) the angular width θ of the SAXS fan is inversely proportional to the transverse length scale l of the projected unresolved microstructure, etc.
Kramers–Moyal extension of xray Fokker–Planck equation
The preceding analysis gives only a crude treatment of the angular distribution of the local SAXS fan, which is described merely via its angular spread θ(x, y) at each point over the exit surface z = 0 of a thin sample. Here, we show that a lesscrude treatment leads to the Kramers–Moyal generalisation of the Fokker–Planck equation, in a form suitable for xray nearfield imaging of thin objects.
Recall Eq. (29), which links the local phase depth associated with the projected microstructure of a thin object, to the associated fraction F(x, y) of the xray beam which is channelled into SAXS. The twotransversedimensional forms of Eqs (27), (33) and (37), upon writing the linear integral transform in Eq. (33) explicitly and setting \(\sqrt{2\pi }d\equiv K\), then become:
Again, the first term on the right side of Eq. (42) is associated with the coherentlyscattered component of the xray field downstream of the thin object in the plane z = 0, while the second term \( {\mathscr B} \) is associated with the diffusely scattered component. In the zeroSAXS limit where F(x, y) → 0, Eqs (42) and (43) reduce to Eq. (4).
Recall the usual definition of a differential scattering cross section \([d\sigma /d\Omega ]({\theta }_{x},{\theta }_{y})\), where dΩ is an infinitesimal element of solid angle and θ_{x}, θ_{y} describe angular deflections (scattering angles) in the x, y transverse directions, respectively^{69}. Hence conclude that \(K=\sqrt{2\pi }d\) in Eq. (43) is proportional to the local differential scattering cross section^{66} associated with unresolved microstructure in the sample, namely a differential scattering cross section \([d\sigma /d\Omega ](x^{\prime} ,y^{\prime} ;{\theta }_{x},{\theta }_{y})\) that depends upon an illuminating beamlet’s centre (x′, y′), as well as upon the scattering angles θ_{x}, θ_{y}. Thus we have the proportionality
where the paraxial approximation has again been assumed in taking \(\tan \,{\theta }_{x,y}\approx {\theta }_{x,y}\). Equations (42) and (43) then have the simple interpretation that the intensity downstream of a sample, which is contained in the plane z = 0, superposes a microstructureindependent nearfield Fresnel diffraction pattern that is damped with a decoherence factor^{45} 1 − F, with a diffuselyscattered signal \( {\mathscr B} \) arising from the superposition of each SAXS fan emerging from each point over the exit surface of the sample. Such a family of SAXS fans is measured e.g. in Schaff et al.^{36}; this amounts to a measurement of \(K(x,y,x^{\prime} ,y^{\prime} ,z=\Delta )\).
In the form given by Eqs (42) and (43), the objecttodetector distance Δ is assumed to be sufficiently small that the detector is in the nearfield of the slowlyvarying part \(\sqrt{{I}_{{\rm{s}}}(x,y,z=0)}\,\exp \,[i{\varphi }_{{\rm{s}}}(x,y,z=0)]\) of the complex wave field at the sample exit surface z = 0. In these same equations, Δ may be sufficiently large that the SAXS fan \(K(x,y,x^{\prime} ,y^{\prime} ,z=\varDelta )\)—emerging from the point (x′, y′, 0) over the exit surface z = 0 of the thin object, and considered as a function of coordinates (x, y) over the detector in the plane z = Δ—can have a width that is several pixels or more (cf. Fig. 2(e)). If these SAXS fans are highly structured, e.g. if they exhibit SAXS satellite peaks from highly directional unresolved structures such as collagen fibres^{70}, then Eqs (42) and (43) can be used without any need for further approximation, to model image contrast in a regime where the detector plane is in the nearfield of the resolved structure in the sample, but in the farfield of the unresolved microstructure.
However if Δ is indeed small enough that the SAXS fan spreads over no more than a few pixels, and if the SAXS fans are not too highly structured, an intermediate simplification is possible, with a domain of validity between that of the lessgeneral Fokker–Planck equation and the more general expression in Eqs (42) and (43). This intermediate model arises from a more precise consideration of the angular scattering distribution in the local SAXS differential cross sections, compared to that given in the preceding section. With this end in mind, assume that the angular widths θ(x′, y′) of all differential scattering cross sections contained in \(K(x,y,x^{\prime} ,y^{\prime} ,z=\Delta )\) obey \(\theta (x^{\prime} ,y^{\prime} )\ll 1\), with all such narrow SAXS fans being strongly peaked in the forward direction. If this is the case, we are justified in Taylor expanding \({I}_{{\rm{s}}}(x{\rm{^{\prime} }},y{\rm{^{\prime} }},z\,=\,0)\) about (x, y), so that Eq. (43) becomes:
In obtaining the above expression, we have made use of the following three points:
We have assumed that each SAXS fan \(K(x,y,x^{\prime} ,y^{\prime} ,z=\Delta )\)—emanating from the point (x′, y′) over the exit surface of the sample to produce an intensity distribution that is a function of detector coordinates (x, y) in the plane z = Δ—obeys:
$$\iint \,K(x,y,x{\rm{^{\prime} }},y{\rm{^{\prime} }},z=\Delta )\,dx{\rm{^{\prime} }}dy{\rm{^{\prime} }}=1.$$(46)This normalisation condition is appropriate since the multiplicative factor of F is a scattering fraction which accounts for the fact that only a certain percentage of the incident beam is converted to SAXS upon passage through the sample.
We have assumed that
$$\iint \,(x{\rm{^{\prime} }}x)K(x,y,x{\rm{^{\prime} }},y{\rm{^{\prime} }},z=\Delta )\,dx{\rm{^{\prime} }}dy{\rm{^{\prime} }}=\iint \,(y{\rm{^{\prime} }}y)K(x,y,x{\rm{^{\prime} }},y{\rm{^{\prime} }},z=\Delta )\,dx{\rm{^{\prime} }}dy{\rm{^{\prime} }}=0,$$(47)consistent with the fact that any transverse shift in the centroid of the SAXS fan may be interpreted as a local gradient in \({\varphi }_{{\rm{s}}}(x,y)\) that has already been accounted for in Eq. (42). This eliminates the terms \({D}_{01}^{(1)}(x,y;\Delta )\) and \({D}_{10}^{(1)}(x,y;\Delta )\) that would otherwise have appeared in Eq. (45). There is an intrinsic ambiguity here: it is a question of semantics whether we consider \({\varphi }_{{\rm{s}}}(x,y)\) to possess a linear phase gradient and assume the SAXS fan to have a centreofmass that is undeflected, or whether we instead consider there to be no local gradient in \({\varphi }_{{\rm{s}}}(x,y)\) but view the centreofmass of the SAXS fan to be transversely shifted on account of correlations that are present in the sample’s unresolved microstructure. In writing Eq. (47) we take the former of these two equivalent points of view.
We have defined the following hierarchy of diffusion tensors (SAXSfan moments, cf. Gureyev et al.^{67}), which arise from terms of progressively higher order in the Taylor expansion of I_{s}(x′, y′, z = 0) about (x, y):
Here, \((\begin{array}{c}M\\ m\end{array})\) denotes a binomial coefficient \(M!/[m!(Mm)!]\). The moments in Eq. (48) quantify the ellipticity of the SAXS fan, via the three independent diffusiontensor components \({D}_{02}^{(2)}(x,y;\Delta ),{D}_{11}^{(2)}(x,y;\Delta ),\) \({D}_{20}^{(2)}(x,y;\Delta )\), which may be mapped to the semimajor axis of the SAXS ellipse, the semiminor axis and a rotation angle^{42}. The higherorder diffusion tensors in Eqs (49) and (50) have an analogous interpretation.
Finally, if we insert Eq. (45) into Eq. (42), we obtain the following finitedifference form of a Kramers–Moyal equation^{13}:
The F(x, y) = 0 case of Eq. (51) is the commonlyemployed firstorder finitedifference form of the transportofintensity equation^{5} (see Eq. (4)). The final lines of this equation incorporate SAXSinduced blurring, in a manner which takes the anisotropy of the local SAXS fans into account. Pawula^{71} has shown that such Kramers–Moyal expansions may be either truncated at most to second or lower orders in spatial derivatives, or not truncated at all. In our context, this amounts to working with one of five possibilities, with increasing order of generality:
Keeping only the first two terms on the righthand side of Eq. (51) (TIE approximation), with F = 0, which ignores SAXS altogether;
Setting D^{(3)} and all higherorder diffusion tensors to zero, and assuming rotationallyinvariant SAXS fans via
leaving us with the Fokker–Planck form given in Eq. (14) if we can also assume that \(F\ll 1\);
Setting D^{(3)} and all higherorder diffusion tensors to zero, thereby reducing Eq. (51) to an anisotropic Fokker–Planck equation incorporating the diffusion tensor \({D}_{mn}^{(2)}(x,y;\Delta )\) (assumption of an elliptic SAXS fan);
Keeping all orders in Eq. (51), which contains all SAXSfan moments. Note that this hierarchy of SAXSfan moments may be measured, e.g. by raster scanning a focused beam through a thin sample to obtain a family of SAXS patterns^{36,72} from which each component of each diffusion tensor at each location (x, y) may be obtained using Eqs (48)–(50);
Eschewing the Kramers–Moyal and Fokker–Planck approximations and instead working with Eqs (42) and (43) directly.
Discussion
The primary goal of this paper is to give several means by which the Fokker–Planck equation (and its extension to the Kramers–Moyal equation) may be used to model forward and inverse problems of nearfield xray imaging, in the presence of both phase–amplitude shifts and nonnegligible smallangle xray scattering from unresolved microstructure in a thin sample.
A detailed study of any particular application or applications is beyond the scope of this paper. One such application, gratingbased xray phase contrast imaging^{73,74,75,76}, is studied in detail in a companion paper^{77} that gives a number of analytical expressions based on the finitedifference expression in Eq. (11). More generally, Eqs (11) and (10) may be used to calculate, in one and two transverse dimensions respectively, the nearfield paraxial image corresponding to a thin sample with known positiondependent exitsurface intensity, phase and diffusion coefficient. As we have already seen, a sufficiently slowly varying fraction F(x, y), this being the fraction of the incident radiation that is converted to SAXS, may be brought into the second square brackets on the right hand side of Eqs (10) and (11), and thereby incorporated into an effective diffusion coefficient. See Eqs (16), (29) and (41), in addition to Eq. (g) in Fig. 3. These expressions can be used for calculating propagationbased phase contrast xray images^{78,79,80} and speckletracking phase contrast images^{81,82,83}, as well as the previously mentioned gratingbased phasecontrast images. Note, also, that if one is in the intermediatefield rather than the nearfield regime for \(\sqrt{{I}_{{\rm{s}}}}\,\exp \,(i{\varphi }_{{\rm{s}}})\), expressions such as (i) Eqs (27) and (36); or (ii) Eqs (27) and (43); or (iii) Eqs (27) and (45) may instead be used. All of these expressions are amenable to analytical implementation if the functional forms of intensity, phase and diffusion coefficients are known, or to computational implementation if the intensity, phase and diffusion are numerically modelled e.g. using Geant4^{84}.
The above indications, of the forward problem of nearfield and intermediatefield xray imaging for a thin sample in the presence of both coherent and diffusive scattering, set up Fokker–Planck and Kramers–Moyal equations that can be used to consider the associated inverse problem of reconstructing intensity, phase and diffusion from measured intensity data. It is again beyond the scope of our paper to explore this point further, beyond the indicative suggestions given in the three examples below.
Example #1
As a first example of inverse problems that may be tackled based on the formalism of this paper, observe that the diffusiontensor moments may be measured e.g. using the previouslymentioned technique of raster scanning a focused quasimonochromatic xray probe over a lattice of points at the entrance surface of a thin object^{36,72}, and the scattering fractions F(x, y) obtained using the same data. Since both \(\overline{I}(x,y,z=\Delta )\) and \({I}_{{\rm{s}}}(x,y,z=0)\) can be measured using quasimonochromatic x rays, and since both the objecttodetector distance Δ and the radiation wave number k are known, the only remaining unknown in Eq. (51) would be the phase ϕ_{s}. This phase could then be obtained using existing techniques for numerically solving the transportofintensity equation (e.g. a full multigrid method^{85} or a method using four Fourier transforms^{86}).
Example #2
A second application of the formalism of the present paper, is to the field of xray speckle tracking^{81,82}. The technique of xray speckle tracking illuminates an object with a known resolved reference speckle field, and then measures one or more distorted forms of that speckle field which arise when a sample is placed in the illuminating speckle beam. The diffusescatter signal, considered in the present paper, arises naturally in xray speckle tracking – via unresolved sampleinduced speckles that should be carefully distinguished from the titular illuminating speckles – and gives useful information that may be extracted using a variety of techniques. These techniques are typically based on the reduction in illuminatingspeckle visibility due to SAXSinduced diffusion: see e.g. the review by Zdora^{83}, and references therein. The geometricflow approach to speckle tracking, which has recently been developed for xrays^{87} but has also been very recently applied to visiblelight imaging^{88}, takes as a startingpoint the following equation:
This relates the intensity I_{R}(x, y) of a reference speckle image obtained in the absence of a sample, to the distorted/coded/encrypted form of that speckle image, denoted I_{S}(x, y), obtained in the presence of a purephaseobject sample; D_{⊥}(x, y) is a transverse displacementvector field, which distorts the reference speckle image I_{R}(x, y) into the speckle image I_{S}(x, y) obtained in the presence of the sample. The above equation, modelled on the continuity equation that has played such a dominant role in the present paper, conserves photon energy both globally and locally. It may be used to reconstruct the speckle displacement D_{⊥}(x, y) induced by the phase object, and hence a map of the phase shift ϕ(x, y) due to the object, in a simple and efficient manner that implicitly rather than explicitly tracks speckles^{87}. Note that the displacement field is related to the phase shift of the object, via
where Δ is the sampletodetector distance and k is the wavenumber of the xrays. A Fokker–Planck form of Eq. (53), suitable for a phase object described by both its phase shift ϕ(x, y) and effective diffusion coefficient D_{eff}(x, y; Δ), is:
This incorporates both coherent flow and diffuse flow, into the geometricflow method for xray speckle tracking. It would be interesting to investigate an augmented form of the geometricflow xray speckletracking method, which takes Eq. (55) rather than Eq. (53) as a starting point, and seeks to reconstruct both ϕ(x, y) and D_{eff}(x, y; Δ) rather than just ϕ(x, y). Of possible relevance in such a context – namely, of separating out the signal due to coherent versus diffusive energy flow – is the fact that D_{eff}(x, y; Δ) will transform as a scalar upon rotation of the sample through 180 degrees (about a rotation axis that is perpendicular to the optical axis, e.g. in a computedtomography setting), whereas D_{⊥}(x, y) will transform as a vector under the same rotation.
Example #3
As a third and somewhat more detailed example of inverse problems that may be tackled using the formalism outlined in the present paper, consider a static nonmagnetic noncrystalline sample that is made of a single material. This sample may and in general will have varying spatial density, with complex refractive index^{4}
which is such that the ratio \(\delta (x,y,z)/\beta (x,y,z)\) is independent of position at all points within the sample^{7,89}. The associated unresolved microstructure is left arbitrary, and the exit surface of the sample is assumed to correspond to the plane z = 0. Further assume normallyincident unitintensity illumination by zdirected quasimonochromatic xray plane waves. This singlematerial approximation, together with the projection approximation^{4}, implies that the x rays at the exit surface of the sample have an intensity \(I(x,y,z=0)\) and phase \(\varphi (x,y,z=0)\) given by
Here μ = 2kβ is the linear attenuation coefficient of the material from which the sample is composed, and T(x, y) is the projected thickness of the sample along the z direction. The fact that^{7}
enables the 2 + 1dimensional finitedifference Fokker–Planck equation in Eq. (14) to be transformed to:
Here, we have used Eq. (16) to incorporate the effective diffusion coefficient D_{eff}(x, y; Δ). Equation (59) is a linear elliptic partial differential equation for the unknown exponentiated projected thickness exp[−μT(x, y)] of the singlematerial sample with projected thickness T(x, y), and is an algebraic equation for the unknown effective diffusion coefficient (SAXS term, or socalled “darkfield” signal^{11,44,76}) D_{eff}(x, y; Δ). Note that the D_{eff}(x, y; Δ) → 0 limit of Eq. (59) is the basis of a commonlyused singleimage method for xray phase retrieval^{7}. When we cannot assume D_{eff}(x, y; Δ) to vanish, we may solve Eq. (59) for both D_{eff}(x, y; Δ) and T(x, y), as follows. Suppose propagationbased phase contrast images are obtained at two different propagation distances, Δ_{1} and Δ_{2}. From Eq. (f) in Fig. 3, we can write down the scaling relation:
This scaling results from the alreadymentioned fact that the slab of vacuum, in between the exit surface z = 0 of the object and the entrance surface z = Δ of the detector, is not a diffusive medium. Thus the width of the SAXS fans scales linearly with Δ, rather than being proportional to Δ^{1/2} (as would be the case for true diffusion). Setting D ∝ Δ ensures that the final term on the right side of Eq. (6) is proportional to Δ^{2}, ensuring the width of the SAXS fan scales as (Δ^{1/2})^{2} = Δ, as required. As is the case throughout the paper, the artifice of a Δdependent diffusion coefficient is a price to be paid for the simplicity of being able to work simultaneously with both diffusion concepts and freespace coherent propagation concepts, in the context of the Fokker–Planck and Kramers–Moyal equations for xray imaging. To proceed we write the z = Δ_{1} and the z = Δ_{2} cases of Eq. (59), and then use Eq. (60) to eliminate D_{eff}(x, y; Δ_{1,2}). Hence:
Equation (61) is mathematically identical in form to Eq. (7) of Paganin et al.^{7}, and may therefore be solved using the same Fouriertransform method. This gives:
Here, \({\mathscr{F}}\) denotes Fourier transformation with respect to x and y, in any convention that transforms ∂/∂x into ik_{x} and ∂/∂y into ik_{y}; (k_{x}, k_{y}) are Fourier coordinates corresponding to (x, y), and \({ {\mathcal F} }^{1}\) is the inverse of \( {\mathcal F} \). Equation (62) shows how a linear combination of two images, taken at two different distances Δ_{1} and Δ_{2}, may be combined to yield the projected thickness T(x, y) of a thin singlematerial sample, in a manner that is independent of SAXS scattering by the sample (cf. Pagot et al.^{40}). Once T(x, y) has been so obtained, Eq. (59) can then be solved algebraically for D_{eff}(x, y;Δ). Such a phase/darkfield retrieval approach that incorporates darkfield effects might benefit samples from a range of fields including materials development (e.g. microcrack detection in highstrength materials^{90}) and biomedical imaging (e.g. diagnostic imaging of the lungs^{91}).
In the third example above, and indeed throughout the paper, the effective diffusion coefficient (and, more generally, the diffusion tensors as defined in Eqs (48)–(50)) will have at least three independent contributions:
The first contributor to the hierarchy of diffusion tensors is the sampleinduced scatter due to unresolved microstructure, that has been a main theme of the present paper.
The second contributor to the diffusion tensors is an edgescattering signal^{11,44} that is now known to be due to unresolved sharp edges^{92}, and which may be viewed as a Young–Maggi–Rubinowicz type boundary wave^{65,93,94,95,96,97} or a Kellertype^{98} diffracted ray. This is examined in further detail in the previously mentioned companion paper^{77}.
The third contribution to the diffusion tensors is associated with incoherent aberrations (including the modulation transfer function of the detector, and penumbral blurring effects due to finite source size) of the imaging system that is used to measure images of the sample. As an example: take Eq. (59), set D_{eff}(x, y; Δ) to a constant D_{eff}(Δ) that is independent of x, y so as to model penumbral blur due to finite source size in a linear shiftinvariant imaging system, and then note that the resulting equation has a solution identical to that recently published by Beltran et al.^{99}.
All three effects will in general be present simultaneously, so that e.g. the SAXS fans associated with sampleinduced blurring will need to be convolved with the intensity pointspread function of a given optical imaging system, and similarly with the signal associated with unresolved sharp edges. These interesting complications, which may be summarised in the statement that “partial coherence can come from the source, or the object itself”, are beyond the scope of the present paper.
There is some relation between the differential form of the diffusion operators employed in this paper, and the idea of Laplacianbased unsharpmask image sharpening^{100,101}. From an unsharpmask image processing perspective, the linear differential operator^{102}
locally sharpens an image I(x, y) to which it is applied, with a characteristic positiondependent partialdeconvolution transverse length scale equal to w(x, y); cf. Eq. (59). The approximate inverse of this operator, which locally blurs an image over the same transverse length scale w(x, y), is (cf. Eqs (6) and (35))^{102}:
This blurring is rotationally symmetric. If this rotational symmetry is not present, we can use a Kramers–Moyal type linear differential operator to effect local blurring with a positiondependent pointspread function, namely
The notation above is analogous to the diffusive terms in Eq. (51), and we have set ∂/∂x ≡ ∂_{x}, ∂/∂y ≡∂_{y}. This gives another perspective on the use of derivative operators to effect blurring associated with a positiondependent point spread function. Similar ideas are used in image sharpening via coherent defocus^{67} and image blurring operations of the convolution type^{103}.
The two derivations of the Fokker–Planck equation for paraxial imaging, given in the present paper, may be augmented by other approaches. An approach based on Wigner functions^{104} would be an interesting avenue for future research, since Wigner functions naturally capture the idea that there can be a distribution of energy flow vectors at each point in space^{105}. See e.g. Eq. (19) of Nugent and Paganin^{106}, which was obtained via a Wignerfunction formulation of paraxial optics, and which may be readily manipulated into a Fokker–Planck form given by Eq. (14). From a different perspective again, the intensity transport equations associated with arbitrary aberrated linear shiftinvariant optical imaging systems^{107} may be rendered into Fokker–Planck and Kramers–Moyal forms via the slight generalisation of replacing shiftinvariant local differential operators with their corresponding shiftvariant forms, along very similar lines to those indicated in the preceding paragraph.
Unresolved speckle lies at the heart of many phenomena observed in imaging using partially coherent optical fields^{4,108}. Here, “speckle” is used in the general sense of optical fields whose spatial and/or temporal intensity fluctuations have a random component; this differs from the more usual usage which equates “speckle” with “fully developed speckle”. Unresolved speckles have two distinct origins: (i) the partial coherence of the illuminating beam and its associated spatiotemporal speckles, and (ii) unresolved speckle associated with sample microstructure. In the present paper, the spatial width w of the pointspread function (PSF) of the positionsensitive detector—a lower limit on which is given by the pixel size—defines a natural transverse length scale. Speckles much smaller in size than w will be spatially averaged when registering an image. If ergodicity of the associated stochastic process can be assumed, over the area of each pixel, then by definition the spatiallyaveraged intensity at each pixel location can be replaced with a corresponding ensemble average over many realisations of the ensemble of fields illuminating each pixel. Physically, and as has been used in the present paper, this corresponds to a statistical ensemble of objects, each of which has the same coarsegrained projected complex refractive index distribution, but differing spatiallyrandom microstructure. This construct permits us to approximate the resulting measured intensity map, with a probability distribution obeying the 2 + 1dimensional Kramers–Moyal equation, including the special case of the 2 + 1dimensional Fokker–Planck equation. Refraction and prism terms are associated with the resolved coarsegrained structure of the ensemble of objects and the ensemble of illuminations, with the diffusion term being associated with the unresolved speckles. What is deemed SAXS is a detectorresolutiondependent construct associated with unresolved sampleinduced speckles. Thus, the component of the scattered radiation attributed to SAXS is dependent on w, and thereby detector dependent. This last statement ties in with the broader idea that there is a close link between partial coherence and the spatiotemporal resolution of one’s detector—see e.g. Sec. 3.6.3 of Paganin^{4}. It is also closely related to what component of the xray energy flow is classed as “coherent energy transport”, and what is classed as “diffusive energy transport”, in the Fokker–Planck and Kramers–Moyal constructions: the finer the spatiotemporal resolution of the intensity detection, the smaller the fraction of the xray energy flow that will be classed as diffusive. Both the TIEtype and SAXStype scattering (resolved and unresolved structures) arise from the same fundamental physics, as is clear in the firstprinciples derivation of the xray Fokker–Planck equation. Again, what length scales ultimately fall into either category is determined by the resolution of the detector.
We close this discussion by emphasising the broader applicability, beyond the margins delineated in the present paper, of the Fokker–Planck and Kramers–Moyal equations to (i) partially coherent paraxial xray imaging in particular, and (ii) partiallycoherent paraxial imaging more generally. Context may be given to this claim, by the huge domain of applicability of Fokker–Planck and Kramers–Moyal concepts. Of equal relevance is the likelihood that the domain of validity of the paraxialimaging Fokker–Planck and Kramers–Moyal equations is at least as broad at that of the transportofintensity equation which they generalise; and, since the transportofintensity equation has been applied to visible light^{109}, electrons^{110}, neutrons^{111} and xrays^{7}, there is every expectation that the formalism of the present paper may be utilised for paraxialimaging scenarios using all of the justlisted radiation and matter wave fields. Extensions of the present paper, to imaging applications beyond those listed previously, include a Fokker–Planck approach to neutron phase contrast imaging^{111,112}, xray magnetic circular dichroism imaging^{113}, xray edgeillumination imaging^{114}, xray Zernike phase contrast imaging^{115}, neutron grating imaging^{116}, electron outoffocus contrast imaging (Fresnel contrast imaging)^{117} and visiblelight outoffocus contrast imaging^{109}. As with the present study, a core motivation for such an approach is its potential simultaneously to model, under the aegis of a single powerful formalism, both deterministic and stochastic aspects of paraxial thinobject imaging using partially coherent radiation and matter wave fields.
Conclusion
Fokker–Planck and Kramers–Moyal equations were obtained, to model nearfield and intermediatefield paraxial quasimonochromatic xray imaging of thin samples exhibiting both phase–amplitude and smallanglescattering contrast. Two complementary derivations were given. Possible applications, to both forward and inverse imaging problems, were discussed.
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Acknowledgements
We acknowledge useful discussions with Mario Beltran, Jeremy Brown, Tim Davis, Carsten Detlefs, Scott Findlay, Timur Gureyev, Alexander Kozlov, Kieran Larkin, Thomas Leatham, Heyang (Thomas) Li, Andrew Martin, Konstantin Pavlov, Tim Petersen and Florian Schaff. We acknowledge the following funding sources: ARC Future Fellowship FT180100374; Veski Victorian Postdoctoral Research Fellowship (VPRF); German Excellence Initiative and European Union Seventh Framework Program (291763).
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D.M.P. and K.S.M. worked on this paper in close collaboration. D.M.P. prepared all figures and performed all mathematical calculations, arising from discussions between both authors. The paper was mainly written by D.M.P., with input from K.S.M.
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Paganin, D.M., Morgan, K.S. Xray Fokker–Planck equation for paraxial imaging. Sci Rep 9, 17537 (2019). https://doi.org/10.1038/s41598019522845
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Applying the Fokker–Planck equation to gratingbased xray phase and darkfield imaging
Scientific Reports (2019)
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