MHz-XPCS: Ultrafast X-ray Photon Correlation

• MHz-XPCS is a coherent X-ray method that uses XFEL pulse trains and split-pulse techniques to capture dynamics from picoseconds to microseconds.
• It quantifies speckle correlations via autocorrelation functions to study molecular diffusion, beam-induced effects, and aggregation in dense samples.
• The approach integrates optimized detector designs and real-time data pipelines to handle high-volume, ultrafast measurements in both biological and soft-matter systems.

Megahertz X-ray Photon Correlation Spectroscopy (MHz-XPCS) is an XFEL-based extension of X-ray photon correlation spectroscopy in which coherent X-ray pulses and burst-mode timing are used to resolve speckle correlations on sub-microsecond to microsecond timescales, and, in split-pulse implementations, on picosecond timescales at atomic momentum transfers. In its European XFEL realization at the Materials Imaging and Dynamics instrument, the method exploits megahertz pulse trains so that successive scattering patterns within a train provide an internal time axis for correlation analysis, while train-wise sample translation refreshes the illuminated volume between bursts (Reiser et al., 2022). In the XFEL ultrafast context, the same term is also associated with split-pulse XPCS or X-ray speckle visibility spectroscopy, where two replicas of a femtosecond pulse separated by a controllable delay encode temporal decorrelation into speckle contrast rather than direct frame-to-frame readout (Shinohara et al., 2020).

1. Measurement concept and temporal regimes

XPCS uses fluctuations in coherent scattering speckle patterns to infer microscopic dynamics. When coherent X-rays scatter from a disordered sample, the speckle pattern changes over time as the molecules move. In the sequential MHz-XPCS formulation used for dense antibody protein solutions, the principal observable is the two-time correlation function

$$c_2(q,t_1,t_2)=\bigg\langle\frac{\langle I_p(q,t_1)I_p(q,t_2)\rangle_p}{\langle I_p(q,t_1)\rangle_p\langle I_p(q,t_2)\rangle_p}\bigg\rangle_j-1\,,$$

from which a time-resolved intensity autocorrelation function is constructed as

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$

The correlation functions are fit with a Kohlrausch-Williams-Watts form,

$$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$

with $\beta(q)$ the speckle contrast, $\Gamma(q)$ the relaxation rate, and $\alpha$ the stretching exponent. For simple Brownian diffusion,

$\Gamma(q)=D_0 q^2,\qquad \alpha=1.$

These relations make the conversion from speckle decorrelation to diffusive dynamics explicit (Reiser et al., 2022).

The defining practical feature of MHz-XPCS is that the XFEL timing itself supplies the correlation delay. At the European XFEL, pulse separations of $443$ ns or $886$ ns correspond to repetition rates of about $2.26$ MHz or $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$0 MHz, so decorrelation can be sampled from sub-microseconds to tens of microseconds. This timing matches the characteristic window for dense-protein diffusion coefficients around $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$1, which corresponds to characteristic times of roughly $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$2–$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$3 on nanometer length scales (Reiser et al., 2022).

In the split-pulse regime, direct detector readout is too slow for the relevant dynamics, so the observable is the speckle visibility of a summed two-pulse pattern as a function of delay $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$4. If the sample is essentially unchanged during $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$5, the two scattering patterns interfere coherently and the speckle contrast is high; if the sample has evolved, the patterns decorrelate and the contrast decreases. This is the basis of XSVS and is the route by which XFEL-based XPCS reaches sub-ps to ps dynamics at $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$6 (Shinohara et al., 2020).

2. Source structure, beamlines, and detector architecture

At the European XFEL MID instrument, MHz-XPCS has been implemented in SAXS geometry using a pink beam at $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$7 keV for the antibody-protein study and in a standard SAXS configuration with SASE-2 at a typical photon energy of about $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$8 keV for the soft-matter pipeline. The pulse-train structure is central: up to $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$9 pulses per train were used in the protein experiment, whereas the standardized MID pipeline reports that up to $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$0 pulses can be recorded in one train and that trains repeat at $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$1 Hz (Reiser et al., 2022, Leonau et al., 10 Jun 2025).

The detector platform is the Adaptive Gain Integrating Pixel Detector (AGIPD). In the antibody-protein experiment, the detector was placed $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$2 m downstream, with most of the path evacuated, and the beam was focused to about $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$3m FWHM using compound refractive lenses to enhance speckle contrast and signal-to-noise (Reiser et al., 2022). In the standardized soft-matter implementation, AGIPD comprises 16 modules, each $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$4 pixels, is operated in high-CDS mode, supports up to $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$5 MHz frame rate, and stores up to $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$6 frames per train (Leonau et al., 10 Jun 2025). A much earlier detector-simulation study already framed European XFEL XPCS around one coherent X-ray pulse every $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$7 ns in trains of up to $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$8 pulses, identifying the detector problem as one of matching large-pitch pixels to small speckles under sub-microsecond timing constraints (Becker et al., 2011).

The soft-matter geometry at MID reflects a deliberate compromise between speckle contrast and radiation sensitivity. The sample position is about $$g_2(q, \tau) = 1 +\beta(q)\,e^{-2(\Gamma(q)\tau)^\alpha}\,,$$9 m from the source, the detector is placed about $\beta(q)$0 m downstream of the sample, and the beam is only modestly focused, $\beta(q)$1m, because protein solutions typically tolerate only about $\beta(q)$2–$\beta(q)$3 kGy under relevant dose-rate conditions. Under typical conditions, the speckle size is $\beta(q)$4m while the AGIPD pixel size is $\beta(q)$5m, so the speckle is under-sampled, leading to only a few percent contrast (Leonau et al., 10 Jun 2025).

The split-pulse architecture differs fundamentally. At BL3 of SACLA, split-pulse XPCS with seeded X-rays used an 8-GeV electron beam with $\beta(q)$6 pC charge and $\beta(q)$7 fs duration, reflection self-seeding with a Si(220) microchannel-cut crystal monochromator, a photon energy of $\beta(q)$8 keV, and a wavefront-division split-and-delay optic based on Si(220) crystals. The reported delay range was $\beta(q)$9–$\Gamma(q)$0 ps, the split pulse pair was focused to about $\Gamma(q)$1, and scattering was recorded with three MPCCDs positioned $\Gamma(q)$2 m downstream of a continuous water jet (Shinohara et al., 2020).

3. Samples, length scales, and accessible dynamics

The dense biological system examined with sequential MHz-XPCS was a bovine immunoglobulin solution containing roughly $\Gamma(q)$3 IgG with polyethylene glycol added as a depletant to induce attractive interactions. The dense phase had a concentration of about $\Gamma(q)$4 mg/mL and was measured at $\Gamma(q)$5 K, above the binodal, in the single-phase regime. The sample is biologically relevant as a crowded antibody environment, has strong protein-protein interactions, and is sensitive both to equilibrium crowding dynamics and beam-induced effects (Reiser et al., 2022).

The measured length scales are set by the scattering vector. The protein study notes estimated mean-square displacements on the order of $\Gamma(q)$6 nm at $\Gamma(q)$7 and $\Gamma(q)$8 nm at $\Gamma(q)$9, which means the method is sensitive to motion within the first coordination shell of the protein molecules. This is the basis for the claim that MHz-XPCS provides direct access to collective molecular motion in dense biological solutions on nanometer and microsecond scales (Reiser et al., 2022).

The MID pipeline generalizes this regime to soft matter samples measured in capillaries on a scanner stage with standardized holders and the possibility of measuring in air using diamond windows. During a pulse train, the sample moves negligibly, $\alpha$0, so all pulses in a train probe essentially the same spot; between trains, the translation is large enough that the next train hits a fresh spot, separated by multiple beam diameters. Each train therefore yields an independent TTCF, and averaging over trains produces the final correlation estimate (Leonau et al., 10 Jun 2025).

In ultrafast split-pulse XPCS, the accessible regime shifts to atomic length scales and picosecond dynamics. The demonstration on water at $\alpha$1 showed that the measured contrast decreases with increasing delay exactly as expected if the water structure decorrelates on a picosecond scale, and that for $\alpha$2 ps the contrast is essentially at the baseline corresponding to uncorrelated beams. Water was chosen because it is both a canonical liquid for ultrafast dynamics and a weakly scattering benchmark, making it a stringent test of atomic-scale XSVS (Shinohara et al., 2020).

This suggests that the label “MHz-XPCS” encompasses two experimentally distinct but conceptually related temporal regimes. One regime uses sequential megahertz pulse trains to sample microsecond and sub-microsecond dynamics directly; the other uses split-delay optics to encode picosecond decorrelation into contrast changes when direct frame-resolved timing is not practical (Reiser et al., 2022, Shinohara et al., 2020).

4. Dose-rate physics, beam-induced motion, and “correlation before aggregation”

A central experimental issue in biological MHz-XPCS is that the observed dynamics are a combination of intrinsic diffusive motion and XFEL-induced effects. The antibody-protein study separates these by tracking static scattering and dynamic correlations independently while varying both total absorbed dose $\alpha$3 and average dose rate $\alpha$4. The dose is written as

$\alpha$5

and the average dose rate as

$\alpha$6

The tested average dose rates lay between $\alpha$7 and $\alpha$8 (Reiser et al., 2022).

Two distinct dynamic signatures were identified. With increasing dose rate at low total dose, the correlation decays faster, the extracted $\alpha$9 increases, and the overall shape changes only modestly; this is interpreted as XFEL-driven motion, possibly associated with local heating, radiolysis gradients, changes in ionic strength, protein charging, and damage to PEG. With increasing total dose, the correlation function becomes more stretched, $\Gamma(q)=D_0 q^2,\qquad \alpha=1.$0, the effective diffusion coefficient decreases, and low-$\Gamma(q)=D_0 q^2,\qquad \alpha=1.$1 scattering increases; this is interpreted as aggregation and beam-induced structural change (Reiser et al., 2022).

The static damage threshold was quantified through the Porod invariant $\Gamma(q)=D_0 q^2,\qquad \alpha=1.$2, which is flat up to about $\Gamma(q)=D_0 q^2,\qquad \alpha=1.$3 kGy and then decreases by more than $\Gamma(q)=D_0 q^2,\qquad \alpha=1.$4, signaling structural change. The threshold is reported as essentially dose-rate independent within the tested range. At the lowest dose rate, $\Gamma(q)=D_0 q^2,\qquad \alpha=1.$5, the $\Gamma(q)=D_0 q^2,\qquad \alpha=1.$6 kGy threshold corresponds to a time window of about $\Gamma(q)=D_0 q^2,\qquad \alpha=1.$7, which is why the work emphasizes a $\Gamma(q)=D_0 q^2,\qquad \alpha=1.$8 window before beam-induced aggregation becomes noticeable (Reiser et al., 2022).

Aggregation was further monitored through the hydrodynamic-radius relation

$\Gamma(q)=D_0 q^2,\qquad \alpha=1.$9

When plotted against RMSD, all dose-rate curves collapse, showing that aggregation onset depends largely on how far proteins have moved; the reported onset is after an RMSD of about $443$0 nm. The authors therefore describe the usable regime as “correlation before aggregation”: the XPCS signal still reflects meaningful protein dynamics, but beam-induced aggregation has not yet significantly altered the sample (Reiser et al., 2022).

A common simplification is to treat faster decorrelation at higher repetition rate as a direct indicator of faster intrinsic diffusion. The dose-rate study shows that this is not generally valid for radiation-sensitive biological solutions. Faster apparent diffusion coefficients can instead reflect a transient non-equilibrium state created by the pulse train, whereas slower and more stretched relaxation at larger accumulated dose tracks aggregation rather than equilibrium crowding dynamics (Reiser et al., 2022).

5. Split-pulse implementations and the problem of beam mismatch

Split-pulse XPCS was proposed because XFEL pulses can be divided into two replicas with tunable delay $443$1, so that the correlation between two rapid successive scattering events reveals how much the structure changed during the delay. However, practical beam splitting in hard X-rays introduces beam dissimilarities that complicate the usual visibility-based interpretation. The detailed realization paper shows that both wavefront splitting and wavelength splitting are feasible, but both generally make the same detector pixel correspond to slightly different momentum transfers for the two pulses. As a result, reduced speckle visibility can be caused by geometry rather than by sample dynamics, and conventional visibility spectroscopy can decouple from the underlying material correlation function (Sun et al., 2020).

The proposed remedy is to move from visibility-only analysis to the spatial autocorrelation of the summed speckle image. If the recorded intensity is

$443$2

then the relevant spatial autocorrelation at the offset corresponding to the branch mismatch is

$443$3

with

$443$4

For equal pulse intensities, $443$5, giving

$443$6

In this formulation the time correlation is encoded in the side band of the spatial autocorrelation of the summed pattern, and the side-band amplitude decreases as the two-time correlation decays (Sun et al., 2020).

The same work derives a compensation condition for combined angle and wavelength mismatch,

$443$7

and for the example parameters reports $443$8, within the bandwidth of a typical SASE XFEL pulse. This geometry cancels the out-of-detector-plane mismatch while leaving an in-plane shift that is then used by the side-band autocorrelation method (Sun et al., 2020).

The seeded-X-ray demonstration at SACLA addressed a separate bottleneck: insufficient pulse energy after splitting, especially at high $443$9. Without self-seeding, the average pulse energy at the sample would be only $886$0; with self-seeding, the average total pulse energy after the split-and-delay optic is $886$1, with a single-shot bandwidth of about $886$2 eV FWHM and seed selection at $886$3 keV with $886$4 eV FWHM in the seeding stage. At $886$5, the reported dual-pulse contrast at $886$6 is $886$7, the baseline single-pulse contrast is $886$8, and the uncorrelated-beam baseline is $886$9. The contrast becomes baseline-like by $2.26$0 ps (Shinohara et al., 2020).

The split-pulse literature also makes clear that heating from the first pulse is not a secondary issue. In water, scattering profiles are essentially independent of fixed-branch energy at $2.26$1 or very small delays, but for longer delays and higher fixed-branch energy the scattering peak shifts to higher $2.26$2, indicating heating. The heating effect becomes noticeable after about $2.26$3 ps, and the data regarded as reliable are those with $2.26$4 and delays up to about $2.26$5 ps (Shinohara et al., 2020).

6. Data volumes, automated pipelines, and real-time analysis

MHz-XPCS generates data at scales that make analysis and data movement part of the experimental method. At MID, for a typical run with about $2.26$6 trains, the AGIPD data volume is estimated as

$2.26$7

for only about $2.26$8 minutes of collection, and over a full multi-day experiment this scales to several petabytes. The pipeline paper therefore describes a highly automated framework in which raw AGIPD output is calibrated into photonized HDF5 “proc” files, yielding compression ratios of about $2.26$9, or about $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$00 when the calibration mask is included (Leonau et al., 10 Jun 2025).

The first XPCS-specific stage is mean-intensity calculation,

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$01

which supports outlier detection, geometry refinement, and later filtering. On the DESY Maxwell cluster, the default for this step is $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$02 cores, with a simultaneous memory load of about $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$03 GB. The train/pulse-resolved TTCF is written as

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$04

and because each train hits a fresh sample position, the main physically relevant correlations are within the same train, followed by averaging over trains (Leonau et al., 10 Jun 2025).

The pipeline includes beam-center refinement by “pizza slicing,” outlier-pixel detection based on normalized RMS in about $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$05 annular $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$06-regions with a default empirical window $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$07 to $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$08, and intensity filtering in which trains deviating by more than $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$09 from the median XGM-normalized intensity are excluded. Off-correlations between neighboring trains are also computed and subtracted from the TTCF to suppress residual artifacts from jumping pixels and memory-cell instabilities; this shifts the TTCF baseline from $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$10 to $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$11. A practical limit is explicitly stated: if the average photon count per pixel per pulse falls to roughly $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$12, residual detector artifacts dominate and the data are considered unusable for XPCS with AGIPD (Leonau et al., 10 Jun 2025).

A related computational development is homomorphic data compression for XPCS. In this framework the two-time correlation matrix is expressed as

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$13

where $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$14 is the full time series of detector frames. With the SVD

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$15

lossless compression projects onto the full right-singular-vector basis,

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$16

so that

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$17

Lossy compression retains only the top $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$18 singular vectors,

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$19

and the homomorphic identity becomes

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$20

The central point is that the correlation can be computed directly on compressed data without decompression (Strempfer et al., 2024).

The reported performance is specifically relevant to future MHz-XPCS workflows: the lossy compression reduces computational time by a factor of $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$21, enabling real-time calculation at kHz framerate; for the online example, encoding a new image with $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$22 takes $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$23 ms on an NVIDIA GeForce RTX 3090 GPU, and TTC calculation for a compressed time series of $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$24 takes $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$25 ms. The maximum raw data rate per beamline is stated to exceed $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$26 GB/s, or about $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$27 PB/week, which is why compression designed to preserve the downstream correlation algebra is treated as a practical route to real-time feedback rather than only offline storage reduction (Strempfer et al., 2024).

7. Detector optimization, theoretical foundations, and interpretive limits

Detector design strongly shapes what MHz-XPCS can measure. A simulation study for European XFEL XPCS derived the speckle-size estimate

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$28

and used the geometric contrast expression

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$29

to compare standard AGIPD pixels $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$30, an apertured AGIPD with effective $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$31 pixels, and a hypothetical $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$32 detector. In the simulated case the speckle FWHM was about $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$33m. At high intensity, the observed contrast was around $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$34–$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$35 for $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$36 pixels and around $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$37–$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$38 for $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$39 pixels. Aperturing was found not to be beneficial at low intensity because data are thrown away, but to be beneficial above about $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$40 photon per $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$41 because charge sharing effects are excluded. The hypothetical $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$42 detector produced significantly better results than the $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$43 system once the average intensity exceeded about $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$44 photons per $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$45 (Becker et al., 2011).

That study also emphasized that detector optimization is not reducible to a single monotonic rule. Smaller pixels improve contrast, but the relative error of the extracted correlation constants depends on photon statistics, fit stability, sensitive area, and frame-storage depth. The paper therefore concluded that the detector trade-off is not simply “smaller pixels are always better,” a point that remains consistent with the later MID pipeline’s emphasis on under-sampling, sparse occupancy, and artifact correction [(Becker et al., 2011); (Leonau et al., 10 Jun 2025)].

At the level of theory, XPCS is often interpreted through the classical Siegert relation, but a microscopic quantum treatment shows that this is not always sufficient. Starting from

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$46

and reducing the nonresonant scattering operator to the density operator $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$47, the quantum theory derives a four-point electron-density correlator for the two-photon signal and decomposes it into Siegert-relation, opposite-momentum, and exchange-correlation contributions. The generalized Siegert relation is

$$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$48

with the extra $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$49 term representing a channel absent from the classical textbook form. The work states that the original Siegert relation can break down even for a non-interacting Fermi gas because of exchange correlations (Siriviboon et al., 2024).

This places an important limit on a common assumption about MHz-XPCS. Faster acquisition does not change the underlying observable into a purely classical one; it means that higher-order density correlations are sampled at shorter delays and with tighter timing control. In quantum materials, especially fermionic systems, a measured coincidence signal is not automatically governed only by $$g_2(q,\tau, \mathcal{D}(t_0))=\langle c_2(q,t_1=t_0+\tau,t_2=t_0\pm \Delta t)\rangle_{\Delta t}+1\,.$$50, because exchange and momentum-transfer structure can add extra oscillatory contributions (Siriviboon et al., 2024).

Taken together, the available literature defines MHz-XPCS as a family of coherent X-ray correlation methods enabled by XFEL timing, detector architecture, and correlation-specific data reduction. At MID it is an operational workflow for burst-mode soft-matter and biological measurements on sub-microsecond scales, constrained by dose, low speckle contrast, and PB-scale data handling (Leonau et al., 10 Jun 2025). In split-pulse realizations it is a route to picosecond and atomic-scale dynamics, constrained by coherent flux, beam mismatch, and pulse-induced perturbation (Shinohara et al., 2020). The broader implication is that future progress depends simultaneously on optics, detector design, online computation, and a theory of intensity correlations that remains valid when quantum exchange and nonclassical coincidence channels matter [(Becker et al., 2011); (Siriviboon et al., 2024)].