Dichography: Two-Frame Diffraction Imaging

• Dichography is a method for reconstructing two distinct images from a single diffraction pattern by separating incoherent scattering signals from ultrashort XFEL pulses.
• It employs advanced iterative phase retrieval techniques with independent real-space supports to resolve the overlapping diffraction contributions.
• The technique extends to chiroptical sensing and dichroic-mirror metrology, demonstrating versatility alongside challenges like high photon requirements and sensitivity to experimental conditions.

Dichography denotes several non-equivalent practices in current technical literature. In its most specific recent usage, it is a coherent diffraction imaging technique that reconstructs two distinct images from a single recorded diffraction pattern containing the incoherent sum of two scattering signals, typically produced by two ultrashort X-ray free-electron laser pulses (Hecht et al., 27 Aug 2025). In broader usage found in the supplied literature, the same word is also applied to structural inference from dichroic contrasts, to wavelength-selective metrology enabled by dichroic mirrors, and, in some cases, appears as a mistaken substitute for ptychography (Hao et al., 27 May 2026, Guo et al., 2024, He et al., 2 May 2025).

1. Dichography in the strict XFEL diffraction sense

In the strictest and most explicit sense, Dichography is a coherent diffraction imaging method designed for situations in which two ultrashort, time-delayed light pulses scatter from a sample, while the detector is too slow to resolve the pulses separately in time (Hecht et al., 27 Aug 2025). The detector therefore records only a single integrated diffraction image even though the sample has been probed in two distinct configurations. Dichography addresses this mismatch between femtosecond pulse structure and much slower detector response by computationally restoring two unique views of the sample from the superimposed scattering signals.

The central forward model is

$$I(\mathbf{q}) = I^A(\mathbf{q}) + I^B(\mathbf{q}) = |\psi^A(\mathbf{q})|^2 + |\psi^B(\mathbf{q})|^2,$$

with

$$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$

Here $\rho^A$ and $\rho^B$ are the two object functions or two time-separated electron densities, and the two fields are assumed to add incoherently at detection (Hecht et al., 27 Aug 2025). The method is therefore distinct from holography: there is no cross term carrying interference fringes between the two views.

The intended experimental setting is two-color XFEL operation, in which two collinear pulses with different photon energies and controllable femtosecond delay provide two snapshots of an evolving system. The same formalism also applies when two distinct particles are intercepted by the same pulse and the unresolved detector records an effective incoherent sum of their diffraction patterns (Hecht et al., 27 Aug 2025).

2. Forward model, constraints, and inversion

Dichography inherits the kinematic, far-field, coherent-beam assumptions of standard coherent diffraction imaging, but it enlarges the inverse problem from one object to two (Hecht et al., 27 Aug 2025). In place of a single unknown Fourier amplitude constrained by one measured intensity, each detector pixel now constrains a pair of amplitudes whose squared norms must sum to the observed intensity. Operationally, the unknowns are two complex Fourier fields $\psi^A$ and $\psi^B$, or equivalently two real-space densities $\rho^A$ and $\rho^B$.

The Fourier-space constraint is implemented by treating the per-pixel amplitudes $M^A_{ij}$ and $M^B_{ij}$ as a 2D vector and renormalizing that vector so that its norm equals $$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$0, while retaining the phases $$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$1 and $$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$2. This projector enforces

$$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$3

for every reciprocal-space pixel (Hecht et al., 27 Aug 2025). The two reconstructions then proceed in parallel through inverse Fourier transforms, separate real-space constraints, and independent support updates.

Real-space support is central. Each frame has its own support $$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$4 and $$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$5, refined by Shrink-wrap, and no structural linkage is imposed between them (Hecht et al., 27 Aug 2025). That independence is what permits Dichography to handle either two times of a single evolving object or two unrelated objects.

The oversampling requirement is stricter than in ordinary CDI. With detector size $$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$6,

$$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$7

and the supplied discussion states that a condition roughly $$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$8 is needed because each intensity value must carry information about three phases rather than one (Hecht et al., 27 Aug 2025). If the two frames have similar size, each may occupy at most about $$\psi^A(\mathbf{q}) = \mathcal{F}\{\rho^A(\mathbf{x})\}, \qquad \psi^B(\mathbf{q}) = \mathcal{F}\{\rho^B(\mathbf{x})\}.$$9 of the total field of view, compared with up to $\rho^A$0 in standard CDI.

The practical reconstruction engine is an extension of Memetic Phase Retrieval, implemented as “Equinox,” combining iterative phase retrieval, parallel random starts, evolutionary selection, crossover, mutation, and separate Shrink-wrap supports for the two frames (Hecht et al., 27 Aug 2025). In the especially low-count helium-droplet experiments, additional prior structure is introduced through Droplet CDI, which constrains the helium envelope to a spherical profile obtained from two-color Mie-theory fitting.

3. Demonstrations, performance, and current limits

The first experimental class demonstrated with Dichography involved “double-hit” diffraction from silver nanoparticles at SwissFEL (Hecht et al., 27 Aug 2025). Single-color pulses at 1000 eV, about $\rho^A$1 fs, about $\rho^A$2 mJ, and focused to about $\rho^A$3 produced diffraction patterns from silver nanoparticles of varied morphology, including cubes, triangles, hexagons, and agglomerates. When two particles were simultaneously intercepted and their interference fringes were too fine to be resolved, the recorded pattern was effectively the sum of two independent diffraction patterns. Dichography then recovered paired frames such as two cubes of slightly different size and orientation, a cube plus a triangular nanoplate, or a hexagon plus an agglomerate of two cubes connected by low-density buffer medium (Hecht et al., 27 Aug 2025).

The second demonstration used two-color EuXFEL data from xenon-doped superfluid helium nanodroplets (Hecht et al., 27 Aug 2025). The two undulator sections were tuned to $\rho^A$4 eV and $\rho^A$5 eV, and a magnetic chicane introduced a delay of $\rho^A$6 fs between pulses. In representative reconstructions, the earlier and later frames both contained continuous xenon structures extending through the droplet, consistent with vortex-aligned dopant aggregates. The reconstructed images were presented as evidence that xenon structures survive up to $\rho^A$7 fs after interaction with the first shot (Hecht et al., 27 Aug 2025).

Performance is strongly brightness-dependent. The supplied discussion reports that the method performs best when overall diffraction signal is high and the photon yields of the two frames are not too different, with several examples showing frame-to-frame photon imbalance factors of about $\rho^A$8 and $\rho^A$9 (Hecht et al., 27 Aug 2025). Ghosting can occur, but in bright silver-particle examples the residual traces were described as two orders of magnitude weaker than the genuine signal. In the helium experiments, the spatial resolution was limited to about $\rho^B$0 nm by the momentum transfer range with usable signal (Hecht et al., 27 Aug 2025).

The current limitations are explicit. Dichography is more photon-hungry than standard CDI, more sensitive to missing central data and dynamic-range limitations, and presently formulated for two frames rather than more. The supplied material also states that mathematical uniqueness of the general solution has not yet been proven, although the second frame becomes standard CDI if the first frame is already known exactly (Hecht et al., 27 Aug 2025). Proposed extensions include attosecond two-color operation, three-beam experiments with an optical pump plus two XFEL probes, improved detector-side energy separation, brighter two-color modes, lower photon energies, polarization-resolved Dichography, and same-color multi-bunch schemes (Hecht et al., 27 Aug 2025).

4. Chiroptical dichography as combined dichroic structural sensing

A broader, non-titular usage in the supplied material treats dichography as structural characterization from dichroic contrasts, especially the combined use of circular dichroism and circular differential scattering (Hao et al., 27 May 2026). In that setting, circular dichroism is the differential attenuation of left- and right-circularly polarized light,

$\rho^B$1

while circular differential scattering is the analogous contrast in scattered light intensity. The stated motivation is that CD probes chiral absorption or extinction, whereas CDS probes chiral scattering, which is more sensitive to longer-range structure, orientational order, and multipolar electromagnetic modes.

The instrumental realization is a dual-channel spectrometer that acquires CD and CDS concurrently from the same solution (Hao et al., 27 May 2026). Its optical train uses a $\rho^B$2 W xenon lamp, a $\rho^B$3 cm grating monochromator covering $\rho^B$4–$\rho^B$5 nm, a Glan–Thompson polarizer set at $\rho^B$6 to the PEM axis, a photoelastic modulator operated at $\rho^B$7 kHz, and a $\rho^B$8 mm pathlength cuvette. The CD channel uses a forward photomultiplier behind a $\rho^B$9 mm pinhole giving acceptance half-angle $\psi^A$0, while the CDS channel uses a second photomultiplier at $\psi^A$1 with scattering collection half-angle $\psi^A$2 (Hao et al., 27 May 2026). The AC/DC lock-in ratios are converted to ellipticity in both channels using a shared calibration factor $\psi^A$3.

A major methodological issue is CDS baseline correction. The supplied discussion identifies CDS baseline drift with PMT high-voltage adjustment and stray-light sensitivity, then introduces a “scattering spectral matching method” in which an achiral reference sample is chosen so that its wavelength-dependent scattering closely matches that of the chiral sample (Hao et al., 27 May 2026). Baseline subtraction is then performed using the matched achiral spectrum rather than a simple buffer.

Two model systems illustrate the structural interpretation. In mixtures of ammonium d-10 camphor sulfonate and polystyrene nanoparticles, the CD bands remain dominated by molecular ACS absorption, whereas the CDS bands have the opposite sign because the PSNP scattering is modulated by the chiral absorption upstream (Hao et al., 27 May 2026). In plasmonic gold helicoid nanoparticles, both CD and CDS exhibit matched resonance wavelengths and the same handed response, indicating that chiral absorption and chiral scattering arise from the same plasmonic resonance modes. The paper states that this is the first experimental demonstration of concurrent acquisition of ensemble-averaged CD and CDS spectra from the same sample under identical conditions (Hao et al., 27 May 2026).

5. Polarimetric and nonlinear extensions

Another supplied usage closely aligned with dichographic aims is the Stokes-vector method for spherical chiral particles (Trigo et al., 22 Sep 2025). Instead of measuring forward extinction as in conventional CD, this approach measures the full Stokes vector of the scattered field at any non-forward angle for incident helicities $\psi^A$4. From the Stokes data, one reconstructs quadratic combinations of electric, magnetic, and chiral polarizabilities, then forms the Stokes Chirality Measure,

$\psi^A$5

In the supplied discussion, SCM is described as eliminating achiral background noise, being independent of both concentration and optical path length, revealing which enantiomer predominates in a mixed solution, and being verifiable in situ by repeating the reconstruction at two different non-forward angles (Trigo et al., 22 Sep 2025).

A related nonlinear version appears in second-harmonic generation spectroscopy, where circular dichroism of the SHG response is analyzed through multipole symmetry (Lovesey et al., 2019). Two dichroic signals are identified: natural circular dichroism from a parity-odd, time-even tertiary process, and magnetic circular dichroism from a parity-even, time-odd tertiary process. In the notation of the supplied discussion,

$\psi^A$6

and

$\psi^A$7

where $\psi^A$8 is the helicity Stokes parameter, $\psi^A$9 is a polar quadrupole, and $\psi^B$0, $\psi^B$1 are magnetic multipoles (Lovesey et al., 2019). The supplied material interprets these formulas as a basis for using SHG circular dichroism to map chiral and magnetic textures.

Taken together, these polarimetric and nonlinear approaches expand dichography beyond scalar transmission differences. They shift the central observable toward full Stokes-vector structure, angle-resolved scattering, or higher-order optical response, while preserving the underlying goal of recovering structural chirality or magnetism from dichroic asymmetries (Trigo et al., 22 Sep 2025, Lovesey et al., 2019).

6. Dichroic-mirror metrology and simultaneous channel separation

In a further instrumental sense, the supplied literature uses “dichography” for multi-wavelength interferometric metrology built around dichroic optics (Guo et al., 2024). The cited example is a temporal phase-shift digital shearography system for simultaneous measurement of first-order out-of-plane displacement derivatives in orthogonal shear directions.

The optical architecture uses two single-mode lasers at $\psi^B$2 nm and $\psi^B$3 nm, a three-splitter prism structure, two dichroic mirrors with complementary spectral responses, a $\psi^B$4 relay with $\psi^B$5 mm, and a single color CMOS camera (Guo et al., 2024). Mirror $\psi^B$6 transmits $\psi^B$7 nm and reflects $\psi^B$8 nm, while $\psi^B$9 transmits $\rho^A$0 nm and reflects $\rho^A$1 nm. By adjusting the two dichroic mirrors so that the reference image is misregistered against the wavelength-selected return in orthogonal directions, the system creates simultaneous shear in $\rho^A$2 and $\rho^A$3.

Phase is extracted by temporal stepping of the reference mirror and Carré processing of four frames before loading and four frames after loading (Guo et al., 2024). After channel separation into red and green bands, denoising, and phase unwrapping, the displacement derivatives are obtained as

$\rho^A$4

The reported shear distances are $\rho^A$5 mm and $\rho^A$6 mm, with a field of view of $\rho^A$7 mm/pixel (Guo et al., 2024).

The experimental object is a round aluminum plate of diameter $\rho^A$8 mm and thickness $\rho^A$9 mm under central loading. Reported displacement-derivative ranges are $\rho^B$0 in one direction and $\rho^B$1 in the other, while the overall displacement integral peak-to-valley error between the two directions is $\rho^B$2–$\rho^B$3 (Guo et al., 2024). In this usage, dichography denotes wavelength-selective multiplexing and demultiplexing of simultaneous interferometric measurements rather than diffraction-based image separation.

7. Terminological boundaries and neighboring concepts

The supplied literature also documents a recurring terminological ambiguity with ptychography. In the discussion accompanying “Diffuse Optical Ptychography,” the term “dichography” is stated to be “almost certainly a reference to ptychography” (He et al., 2 May 2025). Diffuse Optical Ptychography itself is an imaging method for objects embedded in highly scattering media that uses a conventional camera, a scanning laser, overlapping but minimally correlated illumination patterns, and an inverse model based on two diffusion point spread functions. Its forward model is

$\rho^B$4

In the supplied account, DOP reconstructs binary and grayscale objects through media thicker than $\rho^B$5 transport mean free paths, reaches $\rho^B$6 mm resolution when diffusion properties are calibrated, and still resolves $\rho^B$7 mm features without calibration (He et al., 2 May 2025). Conceptually, it “transplants” the ptychographic principle of overlapping probes into a diffusive tomography regime, but the paper does not use the word dichography.

A separate, similarly sounding but conceptually unrelated use is the graph-theoretic “star dichromatic number” of a digraph (Hochstättler et al., 2018). That quantity is a fractional refinement of the dichromatic number defined through acyclic $\rho^B$8-colourings and satisfies

$\rho^B$9

with

$M^A_{ij}$0

It belongs to the theory of circular colourings and acyclic decompositions of digraphs rather than to optics or imaging (Hochstättler et al., 2018).

Accordingly, “dichography” does not yet denote a single stabilized cross-disciplinary concept in the supplied corpus. Its most explicit technical meaning is the two-frame XFEL diffraction method introduced in 2025, but the same label is also used more loosely for dichroic-contrast structural spectroscopy, dichroic-mirror-based measurement multiplexing, and, occasionally, as a mistaken rendering of ptychography (Hecht et al., 27 Aug 2025, Hao et al., 27 May 2026, Guo et al., 2024, He et al., 2 May 2025).