X-ray Fokker--Planck equation for paraxial imaging

David M. Paganin; Kaye S. Morgan

REVIEW 3 major objections 4 minor 114 references

X-ray Fokker--Planck equation for paraxial imaging

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A single Fokker–Planck equation merges x-ray phase contrast with dark-field blur.

desk verdict A genuinely useful unification of phase contrast and SAXS diffusion, but the headline equation has an operator-ordering inconsistency that needs fixing before forward models should use it. read the letter →

arxiv 1908.01473 v1 pith:SV6SZYRT submitted 2019-08-05 physics.optics physics.app-ph

David M. Paganin , Kaye S. Morgan This is my paper · ORCID

classification physics.opticsphysics.app-ph

keywords x-rayimagingFokker-PlanckequationKramers-Moyaltransport-of-intensitysmall-anglescatteringphasecontrastdark-fieldparaxialoptics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Near-field x-ray images of thin samples are usually modelled either as coherent propagation-based phase contrast or as diffusive dark-field scattering, but real samples show both at once. The paper establishes a single equation, the x-ray Fokker–Planck equation, that describes the downstream intensity when coherent refraction and unresolved-sample small-angle scattering act together. It derives the equation twice: by fusing the transport-of-intensity equation with a diffusion equation through an energy-conserving current, and from an ensemble of unresolved random phases at the sample exit surface. The same machinery yields a Kramers–Moyal generalisation for anisotropic or structured scattering fans. If the claims hold, forward modelling becomes a single finite-difference update, and inverse problems can recover phase, thickness, and dark-field diffusion from one or two near-field images.

What carries the argument

The engine of the paper is a continuity equation for transverse energy flow in which the current has two parts: a coherent Poynting-vector current $I\nabla_\perp\phi/k$ and a diffusive Fick-type current $-\nabla_\perp[DI]$ entering with a weight $F$. Combining these gives the finite-difference Fokker–Planck update of Eq. (11). The diffusion coefficient is $\Delta$-dependent, $D=L^2/\Delta$, so that the small-angle-scattering blur expands as the propagation distance itself rather than as its square root. For the extended form, the Kramers–Moyal equation replaces the scalar $D$ with a hierarchy of diffusion tensors defined as moments of the position-dependent scattering kernel, allowing elliptical and structured scattering fans to be tracked. The hierarchy also supplies a principled truncation point, since Pawula's theorem forbids stopping at any order above second short of keeping everything.

What would settle it

Take a sample whose scattering fan is known to be broad or structured and image it at three or more propagation distances; if the projected thickness recovered from each pair of distances via Eq. (61) changes with the chosen pair, or if the inferred $D_{\rm eff}/\Delta$ is not constant, the second-order diffusion truncation behind the Fokker–Planck equation has broken down.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the intensity of a paraxial quasi-monochromatic beam evolves as $$ \frac{\partial I}{\partial z}=-\frac{1}{k}\nabla_\perp\cdot[I\nabla_\perp\phi]+F\nabla_\$perp^{2}$[DI]. $$ The first term is the transport-of-intensity contribution from coherent refraction; the second adds a diffusive blur whose strength is the local fraction of the beam converted to small-angle scattering times a diffusion coefficient. The paper derives this equation twice: first by adding the coherent and diffusive currents in an energy-conserving way, then from first principles starting from a decomposition of the exit-surface phase into a slowly varying part and an unresolved random part. The first-principles route makes the physics explicit: the scattering fraction is $F=1-\exp(-\sigma_{\varphi_f}^2)$, the diffusion coefficient is $D=L^2/\Delta$, and the blur width $L$ grows linearly with propagation distance. When the scattering fan is anisotropic or highly structured, the same construction produces the Kramers–Moyal equation, whose successive terms are moments of the local scattering kernel.

Load-bearing premise

The load-bearing premise is that each local small-angle scattering fan is narrow, smooth, and almost forward-peaked, so that the blur it produces can be captured by a second-order Taylor expansion in spatial derivatives over the propagation distances used.

Editorial extensions

If this is right

A near-field image of a thin sample with unresolved microstructure can be computed by one finite-difference Fokker–Planck update from the exit-surface intensity, phase, and diffusion coefficient, without resolving the sample's internal speckle.
Two near-field images at different propagation distances can be combined, through Eqs. (60)–(61), to recover the projected thickness of a single-material sample independently of its dark-field scattering; the dark-field coefficient then follows algebraically.
The same equations apply to paraxial imaging with visible light, electrons, and neutrons, because they depend only on the paraxial scalar wave equation and energy conservation.
The geometric-flow speckle-tracking equation gains a diffusive term, so a single augmented equation can in principle reconstruct both phase and effective diffusion coefficient from speckle images.
Anisotropic or structured scattering fans can be represented by the Kramers–Moyal tensors, which can be measured by raster-scanning a focused beam over the sample.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the two-distance inversion is stable, it suggests a practical protocol: collect two defocus distances and solve for thickness and dark-field simultaneously, rather than acquiring a full scattered-intensity map.
The diffusion-tensor moments in the Kramers–Moyal equation could be inverted directly from ordinary near-field images, connecting phase retrieval with tensor tomography without dedicated grating or raster-scan setups.
Because the Fokker–Planck update is linear in propagation distance for fixed $D$, time-lapse imaging of a slowly evolving sample could be modelled by the same equation with time as the evolution parameter, an extension the paper does not develop.
The same second-order expansion of the scattering kernel suggests that resolution-dependent classification of speckle as coherent versus diffuse could be used to tune detectors or binning to suppress dark-field blur in phase-contrast imaging.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something real: it assembles known ingredients from Nesterets, Yashiro, Goodman, and the authors' own prior work into an explicit Fokker–Planck equation for near-field x-ray imaging, and it gives two derivations plus a Kramers–Moyal extension. The writing is clear, the physical picture is sensible, and the microphysical identifications — F = 1 - exp(-σ_φf²) and D = L²/Δ — turn the phenomenological coefficients into something concrete. The Kramers–Moyal hierarchy of SAXS-fan moments gives a useful formal umbrella. I don't hold the lack of numerical or experimental demonstration against it; this is a theory paper and the scope limits are stated honestly.

The soft spot is the one the stress-test note flags, and it is real. Equation (9), the equation highlighted in the abstract, puts D inside the Laplacian: F∇²[DI]. The first-principles derivation in §2.2 and the Kramers–Moyal equation in Eq. (50) both give the second-order diffusion term with D outside the derivatives — in the isotropic case, F D ∇²I. The paper acknowledges in §2.2 that the microscopic derivation only reduces to the D-inside form when D commutes with the Laplacian, but Eq. (9) is stated unconditionally and is the form quoted as the central claim. For position-dependent D, the two forms differ by 2F∇D·∇I + FI∇²D. That is not a negligible higher-order correction; it is a different model. If the intended model is the D-outside form, Eq. (9) should be amended or qualified. If the D-inside Fick form is intended, the derivation does not justify it. This is fixable and the correct D-outside form is present in the paper, but it needs to be sorted out before the equation gets used in forward or inverse problems.

Citations: nothing offensive. The paper leans on the authors' prior work, but the cited results are standard in the field and the new synthesis is clearly distinguished. This is a paper for the x-ray phase-contrast and dark-field imaging community, and for anyone doing tensor tomography or speckle tracking. It deserves a serious referee and probably a major revision, not a desk rejection. I would bring it to a reading group because the discussion alone is worth the time.

Referee Report

3 major / 4 minor

Summary. The paper proposes an x-ray Fokker–Planck equation for paraxial near-field imaging of thin samples, which augments the transport-of-intensity equation (TIE) with a diffusive term that models unresolved small-angle x-ray scattering (SAXS). Two derivations are presented: a phenomenological merging of the TIE and a diffusion equation in Sec. 2.1, and a first-principles derivation in Sec. 2.2 based on Fresnel propagation, ensemble averaging over a Gaussian stationary fast phase, and a second-order narrow-fan expansion. Section 3 generalises the result to a Kramers–Moyal equation with a hierarchy of SAXS-fan moments, and Section 4 outlines forward- and inverse-problem applications, including a two-distance phase-retrieval formula. The central claim is that Eq. (9) (and its one-dimensional form, Eq. (10)) simultaneously captures propagation-based phase contrast and dark-field blurring, with the Kramers–Moyal form, Eq. (50), handling anisotropic scattering fans.

Significance. If the proposed equations are valid, they provide a compact and practical formalism for modelling combined coherent and diffuse transport in x-ray imaging, with potential applications to phase retrieval, dark-field tomography, and imaging with visible light, electrons, and neutrons. The paper's strengths include a transparent microscopic derivation that gives independent physical meaning to the scattering fraction F, the phase depth σ_φf, the blur width L, and the diffusion coefficient D (Eqs. (28), (39), and (40)), and a Kramers–Moyal hierarchy that connects the Fokker–Planck equation to a rigorous moment expansion of the SAXS fan. The discussion of detector-resolution dependence of what is classified as SAXS is also valuable. However, the headline equation Eq. (9) has an operator-ordering inconsistency with the microscopic derivations that needs to be resolved before the paper can be accepted as is.

major comments (3)

[§2.1.3, Eq. (9)] The diffusion term F(x,y)∇²⊥[D(x,y;Δ)I] places the diffusion coefficient inside the Laplacian, but the first-principles derivation in §2.2 (Eq. (38)) and the Kramers–Moyal expansion in §3 (Eq. (50)) both place D outside the derivatives, giving FD∇²⊥I (in the isotropic case). For spatially varying D the two forms differ by 2F∇D·∇I + FI∇²D. These are not higher-order corrections in Δ; they survive at fixed small Δ. Concretely, for I_s(x)=1 with constant F and varying D(x), Eq. (9) predicts I(z=Δ)−I_s ≈ ΔFD''(x), whereas the kernel model Eq. (42) leaves a uniform intensity unchanged and Eq. (13) gives zero. Eq. (9) is stated unconditionally and is highlighted in the abstract, so this is a load-bearing inconsistency that should be corrected by replacing ∇²⊥(DI) with D∇²⊥I (or with ∇⊥·(D∇⊥I)) whenever D varies, or by explicitly restricting Eq. (9) to ∇D≈0.
[§2.1.2, Eq. (7)] The phenomenological diffusive current J(2)_⊥ = −∇⊥(DI) in Eq. (7) is nonstandard: combined with continuity it produces ∇²(DI), which is not the usual Fick-law form ∇·(D∇I). The uncritical use of this flux in the merging argument is the origin of the ordering discrepancy with the microscopic derivation. The authors should either justify why the current should contain I∇D or align the phenomenological derivation with the microscopic result by using the standard diffusive current −D∇I, which would yield F∇⊥·(D∇⊥I).
[Abstract] The claim that the domain of applicability of the Fokker–Planck and Kramers–Moyal equations is 'at least as broad as that of the transport-of-intensity equation' is overstated. The diffusive term relies on the second-order narrow-fan truncation (Eqs. (34)–(35) and Eq. (44)), which requires each local SAXS fan to be narrow, smooth, and forward-peaked and the propagation distance Δ to be small enough for the Taylor expansion to be accurate. The TIE itself does not require these additional conditions. The abstract and conclusion should be qualified, for example by saying 'as broad as the TIE in the near-field limit of narrow SAXS fans.'

minor comments (4)

[Introduction, p. 2] There is a typographical repetition: 'the the Fokker–Planck equation' should read 'the Fokker–Planck equation'.
[Eq. (28)] The arrow notation 'weak SAXS−−−−−−→' is unusual and could be replaced with a clearer statement such as 'in the weak-SAXS limit σ²_φf ≪ 1, F(x) ≈ σ²_φf(x)'.
[Eq. (33)] The notation kx=φ′s(x)−kx/Δ in the definition of d(x;x0) is confusing because kx appears on both sides with different meanings; using a separate symbol for the Fourier variable would improve readability.
[§2.2, Eq. (34)] The argument that Q=1 follows solely from energy conservation is terse; a sentence clarifying that the kernel integrates to unity (cf. Eq. (45)) would make the step easier to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Fokker–Planck equation is derived from Fresnel propagation plus a controlled small-angle expansion, with self-citations only to standard non-load-bearing results.

full rationale

The central derivation is not circular. Section 2.2 begins from Fresnel diffraction (Eq. 16), ensemble-averaged Gaussian fast-phase correlations (Eqs. 21–25), and a second-order Taylor expansion of the narrow SAXS blur kernel (Eq. 34). This yields the propagated intensity in Eq. (38), whose diffusive increment contains σ²_φf L² d²I/dx²; the subsequent identifications D = L²/∆ and F ≈ σ²_φf give D and F microphysical content rather than fitting them to the target equation. The phenomenological derivation in Sec. 2.1 is an explicit model-building step (adding the TIE and diffusion currents), not a disguised prediction. Self-citations to Paganin's textbook, to Paganin et al. (2002) for phase retrieval, and to the authors' speckle-tracking work are to standard, externally supported results and are not load-bearing for the new Fokker–Planck/Kramers–Moyal claims; the statistical-optics steps cite Nesterets, Yashiro, Goodman, and Voronovich. The operator-ordering discrepancy between Eq. (9)'s F∇²[DI] and the D-outside form of Eq. (38) is a genuine correctness or qualification issue for position-dependent D, but it is not circularity: Eq. (38) is an independent approximation, and the paper only asserts equivalence to the simpler D-commuting form under a stated slow-D assumption. No step reduces to its own input by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The derivation introduces no new physical entities. It does assume a hierarchy of length scales: resolved phase and intensity vary slowly, unresolved microstructure varies quickly, and the detector pixel sets the boundary. The model's free functions are the sample's phase, intensity, and scattering statistics. The central derivation rests on six stated assumptions, most of which are conventional in statistical optics and are acknowledged in the text.

free parameters (5)

D(x,y;Δ) diffusion coefficient
Introduced in Eq. (5) as the coefficient of the diffusion term; physically it encodes the local width of the SAXS fan and is tied to L²/Δ in Eq. (39). Not fitted in this paper; it is a user-supplied or experimentally retrieved function.
F(x,y) scattering fraction
Introduced in Eq. (8) as the fraction of optical energy converted to SAXS; related to phase depth by F=1-exp(-σ_φf²) in Eq. (28). It is a sample property to be measured.
σ_φf(x) phase depth
Variance of the unresolved fast phase fluctuations φ_f, defined after Eq. (22). It controls the decoherence factor and the scattering fraction; assumed small in the weak-SAXS limit.
L(x;Δ) blur width
Local transverse width of the SAXS fan, introduced in Eq. (6) and used in the second-order expansion Eq. (35). It scales as Δ, not √Δ, and is related to D by Eq. (39).
Higher-order SAXS-fan moments D^(M)_m,M-m(x,y;Δ)
The hierarchy of diffusion tensors in Eqs. (47)-(49) for the Kramers-Moyal extension; they are moments of the local scattering kernel and must be measured or modeled for anisotropic scattering.

assumptions (6)

domain assumption Paraxial scalar Fresnel propagation
All results assume the exit-surface field propagates via the Fresnel convolution, Eq. (16), so non-paraxial and polarization effects are excluded.
domain assumption Projection approximation for a thin sample
The exit-surface wave field is written as √I_s exp(i(φ_s+φ_f)) in Eq. (19), assuming rays are not deviated inside the sample and phase/amplitude are accumulated along straight lines.
domain assumption Gaussian, stationary statistics for the fast phase φ_f
Eq. (22) assumes φ_f is Gaussian and spatially statistically stationary over small displacements, giving the decoherence factor exp(-σ²[1-γ]). Non-Gaussian microstructure would change the correlation splitting in Eq. (24).
domain assumption Slow variation of I_s and φ_s
Eq. (29) linearizes the smooth phase difference as φ_s(x-x')-φ_s(x-x'') ≈ (x''-x')φ'_s(x); this requires the resolved scale to be much larger than the correlation length τ.
domain assumption Narrow-fan second-order expansion for the blur kernel
Eqs. (34)-(35) approximate the SAXS convolution by 1+L² d²/dx², which is the step that produces a diffusion term; it requires small Δ and weakly structured scattering.
standard math Pawula's theorem on Kramers-Moyal truncation
Used in Sec. 3 to state that the expansion can be truncated at second order or kept to all orders, but not at intermediate orders.

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Pith. "Pith review of X-ray Fokker--Planck equation for paraxial imaging." pith.science (2026). https://pith.science/paper/SV6SZYRT

@misc{pith2026190801473,
  author       = {Pith},
  title        = {Pith review of: X-ray Fokker--Planck equation for paraxial imaging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SV6SZYRT}},
  note         = {Machine review of arXiv:1908.01473}
}

read the original abstract

The Fokker--Planck Equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation. This forms a natural generalisation of the transport-of-intensity equation. The x-ray Fokker--Planck equation can simultaneously account for both propagation-based phase contrast, and the diffusive effects of sample-induced small-angle x-ray scattering, when forming an x-ray image of a thin sample. Two derivations are given for the Fokker--Planck equation associated with x-ray imaging, together with a Kramers--Moyal generalisation thereof. Both equations are underpinned by the concept of unresolved speckle due to unresolved sample micro-structure. 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 x-ray 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 transport-of-intensity equation which they generalise, hence the technique is also expected to be useful for paraxial imaging using visible light, electrons and neutrons.

Figures

Figures reproduced from arXiv: 1908.01473 by the authors.

**Figure 1.** (a) Specular refraction by a thin object and associated propagation-based phase contrast (local concentration and rarefaction of photon energy density) associated with coherent energy transport downstream of the object. (b) Diffuse scattering by a thin object and associated propagation-based blurring associated with diffusive energy transport downstream of the object. Paraxial coherent energy transport may be modell… view at source ↗

**Figure 2.** (a) Auto-correlation of exp[iφf(x)] in Eq. (24), with x considered fixed; (b) Re-normalised fast-phase correlation function in Eq. (25), for a given fixed x; (c) Decoherence factor exp[−σ 2 φf (x)] appearing in Eq. (26) and onwards, for a given fixed x; (d) Physical interpretation of Eq. (31); (e) Specular and diffuse scattering for a single x-ray beamlet of width ∆x. Average over an ensemble of realisations of the … view at source ↗

**Figure 3.** Relations between several key physical quantities used in this paper. All physical dependencies, for the quantities in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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Pith tools

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