Integrated XRM-SAXS Method

• Integrated XRM-SAXS Method is an imaging technique that simultaneously measures refractive phase shifts and SAXS contrast using X-ray speckle-tracking.
• It leverages a paraxial Fokker–Planck formalism coupled with optical-flow-based MIST to efficiently compute phase and dark-field signals from minimal images.
• The approach enables rapid, dose-efficient mapping with high spatial resolution for both synchrotron and laboratory X-ray microscopy applications.

Integrated X-ray phase–Small-Angle X-ray Scattering (XRM-SAXS) employing Multi-modal Intrinsic-Speckle-Tracking (MIST) constitutes a methodology for simultaneous measurement of both the refractive phase shift and position-dependent small-angle X-ray scattering (SAXS) properties of a phase object using X-ray speckle-tracking. By leveraging a paraxial Fokker–Planck formalism along with optical-flow-based speckle analysis, the integrated XRM-SAXS method enables rapid, dose-efficient acquisition of co-registered maps of refractive and SAXS contrast from a minimal set of images, with direct application to synchrotron and laboratory X-ray microscopy (Pavlov et al., 2019).

1. Theoretical Foundations

The integrated XRM-SAXS method is grounded in the paraxial transport-of-intensity equation (TIE) for describing propagation of coherent X-rays: $$\frac{\partial I}{\partial z} + \nabla_\perp\cdot(I \mathbf{v})=0,\qquad \mathbf{v}=\frac{1}{k}\nabla_\perp\phi$$ where $I$ is the intensity, $\phi$ is the phase, and $\nabla_\perp$ denotes the transverse gradient. To accommodate unresolved micro-structures (such as SAXS, source, and detector blurring), a diffusive term is added, yielding the paraxial Fokker–Planck equation: $$\frac{\partial I(x,y,z)}{\partial z} + \nabla_\perp\cdot\bigl(I\,\mathbf{v}\bigr) = D(x,y)\,\nabla_\perp^2 I$$ Here, $D(x,y)$ quantifies local diffusivity due to SAXS-induced broadening and blur. The MIST formalism employs optical-flow speckle-tracking: thin, random masks (speckle generators) are illuminated, producing reference and sample speckle images at two distinct transverse positions. For thin objects and under the paraxial regime, the intensity difference satisfies: $$I_R(x,y)-I_S(x,y) = \frac{\Delta}{k}\nabla_\perp\cdot\bigl(I_R\nabla_\perp\phi\bigr) - \Delta\nabla_\perp^2\bigl[D I_R\bigr]$$ Under the "random-speckle × smooth-phase" and slowly varying $D(x, y)$ approximations, cross terms are negligible and $\nabla_\perp^2[D I_R] \approx D \nabla_\perp^2 I_R$, simplifying inversion.

2. Experimental Configuration

The XRM-SAXS setup consists of a speckle generator (such as P800 sandpaper or 10–20 µm powder, mounted on an $x,y$ piezoelectric stage) placed upstream of the sample, with a detector positioned downstream. In a canonical synchrotron implementation (e.g., ESRF BM05 beamline):

• The X-ray beam is monochromatic ($E=17$ keV, $\Delta E/E\sim10^{-4}$).
• The speckle mask is located 0.5 m before the sample at $z=0$.
• The sample is imaged with a detector 1.0 m downstream (yielding total propagation distance $\Delta = 1.0$ m).
• Effective pixel size is typically 5.8 µm, and signal-to-noise ratio (SNR) exceeds 500 for $10^8$ photons/pixel.

Data acquisition protocol consists of recording two reference speckle images and two sample images at corresponding transverse mask positions, separated laterally by 20–50 µm. High reproducibility of the mask position ($\ll 2-5$ µm, the speckle grain size) is critical for stability.

3. Reconstruction Workflow

The computational workflow for phase and SAXS dark-field retrieval proceeds as follows:

1. Pre-processing: Perform flat-field and dark-field correction on all four images.
2. Laplacian Estimation: Calculate $\nabla_\perp^2 I_{R_1}$ and $\nabla_\perp^2 I_{R_2}$ using finite differences or a $5\times5$ Savitzky–Golay filter.
3. Pixel-wise Inversion: At each $(x, y)$, form the linear system:

$$\begin{pmatrix} \frac{\Delta}{k}I_{R_1} & -\Delta\nabla^2I_{R_1}\ \frac{\Delta}{k}I_{R_2} & -\Delta\nabla^2I_{R_2} \end{pmatrix} \begin{pmatrix} \nabla^2\phi\ D \end{pmatrix} = \begin{pmatrix} I_{R_1}-I_{S_1}\ I_{R_2}-I_{S_2} \end{pmatrix}$$

Solve via direct inversion; for $N>2$ positions, a least-squares approach is used.

1. Phase Recovery: Integrate $\nabla^2\phi$ in Fourier space:

$$\widehat{\phi}(k_x,k_y) = -\widehat{\nabla^2\phi}(k_x,k_y)/(k_x^2+k_y^2)$$

with Tikhonov regularization to suppress low-frequency noise.

2. Post-processing: Negative values of $D$ are set to zero; optional Gaussian smoothing over 1–2 pixels is applied.

An optional step is to initialize correlation-based UMPA or XSVT workflows with the phase and $D$ maps for improved flexibility in handling mixed-phase/attenuation properties, at the expense of computational speed.

4. Spatial Resolution and Sensitivity

Performance characteristics are dictated by speckle structure and data acquisition parameters. In the phase channel, spatial resolution is approximately equal to the pixel size (5.8 µm) if speckle grains span at least 5 pixels, with achievable phase sensitivity in the range of 0.1–0.2 mrad for $10^8$ photons/pixel. In the SAXS (dark-field) channel, pixel-wise inversion limits spatial resolution to 10–20 µm. The contrast-to-noise ratio (CNR) for $N=2$ mask positions is approximately 10, improving to 20–25 for $N=4$. The method remains stable with a single-pass inversion, provided linear independence of speckle shifts.

5. Practical Constraints and Limitations

The integrated XRM-SAXS method, as realized using the MIST approach, is dose-efficient—requiring only two mask positions and achieving analysis in a few seconds per $1 \mathrm{k} \times 1 \mathrm{k}$ image. Essential applicability assumptions include:

• Predominant pure-phase or weak attenuation in the sample.
• Slowly varying diffusivity $D(x, y)$.
• Negligibility of the cross term $(\nabla I_R\cdot\nabla\phi)$.

Table: Key Practical Attributes

Parameter	Typical Value (Synchrotron)	Note
Mask reproducibility	≪2–5 µm	≪speckle grain size
Detector pixel size	5.8 µm	Matches phase resolution
SNR	>500	With >10⁸ photons/pixel
Exposure (lab sources)	1–10 s	10⁶–10⁷ photons/pixel
Phase channel CNR (N=2)	≈10	≈20–25 for N=4

A plausible implication is that while synchrotron sources enable full performance (maximum resolution, SNR, and SAXS sensitivity), laboratory sources may necessitate longer exposures and typically exhibit diminished high-$q$ SAXS contrast due to increased blur.

6. Comparative Context and Extensions

Compared to correlation-based methods such as UMPA and XSVT—which are applicable to mixed-phase/attenuating samples and strong scatterers but require $N\geq5$ images and have compute times exceeding 10× that of MIST—the integrated XRM-SAXS approach using MIST facilitates expedited, minimally invasive imaging with fewer measurements. Limitations include the need for assumptions regarding sample properties and the restriction to slowly varying diffusion. The integration of rapid MIST-based phase/SAXS mapping with further UMPA/XSVT refinement workflows provides extensibility for more challenging imaging regimes (Pavlov et al., 2019).

The overall workflow enables dose-minimizing, rapid, and objectively quantifiable retrieval of co-registered phase and SAXS signals for weakly attenuating specimens, with practical constraints that define its optimal operational domain in X-ray microscopy.