Correlation Functions¶

The heterodyne scattering framework defines a hierarchy of correlation functions that connect microscopic dynamics to the experimentally measured intensity correlations [BerneXPCS] [SuttonXPCS]. These follow from PNAS 2024 Eqs. 1–3 and SI Eqs. S-1 through S-20.

First-Order Two-Time Correlation¶

The first-order (field) correlation function between times \(t_1\) and \(t_2\) is defined as (PNAS 2024 Eq. 1):

\[c_1(q, t_1, t_2) = \frac{\langle E^*(q, t_1)\, E(q, t_2) \rangle} {\bigl[\langle |E(q, t_1)|^2 \rangle\, \langle |E(q, t_2)|^2 \rangle\bigr]^{1/2}}\]

where \(E(q, t)\) is the scattered electric field at wavevector \(q\) and time \(t\), and angle brackets denote ensemble averaging.

Second-Order Correlation and the Siegert Relation¶

The second-order (intensity) correlation function is (PNAS 2024 Eq. 2):

\[c_2(q, t_1, t_2) = \frac{\langle I(q, t_1)\, I(q, t_2) \rangle} {\langle I(q, t_1) \rangle\, \langle I(q, t_2) \rangle}\]

For Gaussian scattered fields, the Siegert relation [SiegertRelation] connects the two orders:

\[c_2(q, t_1, t_2) = 1 + \beta\, |c_1(q, t_1, t_2)|^2\]

where \(\beta\) is the optical contrast (speckle contrast), determined by the coherence properties of the beamline optics. In practice, \(\beta \in (0, 1]\).

One-Time Correlation (Equilibrium)¶

At thermal equilibrium, the system is time-translation invariant and the correlation functions depend only on the lag time \(\tau = t_2 - t_1\):

\[g_2(q, \tau) = 1 + \beta\, |g_1(q, \tau)|^2\]

where \(g_1(q, \tau) = c_1(q, t, t+\tau)\) and \(g_2(q, \tau) = c_2(q, t, t+\tau)\) are the standard XPCS autocorrelation functions. The two-time formulation reduces to the one-time formulation in this limit.

Factorization of the Field Correlation¶

The first-order correlation factorizes into internal (diffusive) and external (advective) contributions (PNAS 2024 Eq. 7):

\[c_1(q, t_1, t_2) = c_{1,\mathrm{in}}(q, t_1, t_2) \cdot c_{1,\mathrm{ex}}(q, t_1, t_2)\]

This factorization holds when the internal (thermal) fluctuations are statistically independent of the external (flow-driven) displacement.

Internal Correlation¶

The internal contribution arises from thermal diffusion and is expressed through the transport coefficient integral (PNAS 2024 Eq. 8):

\[c_{1,\mathrm{in}}(q, t_1, t_2) \;=\; \exp\!\left(-\frac{q^2}{2} \int_{t_1}^{t_2} J(t')\, dt'\right)\]

where \(J(t)\) is the transport coefficient. The factor of \(\frac{1}{2}\) arises because the integral gives the full position variance growth, while \(c_1\) involves the one-dimensional projection along \(q\).

Important

The implementation always computes the integral numerically via trapezoid_cumsum, even when \(J(t)\) is constant. It never substitutes analytical antiderivatives. This ensures correctness for the general power-law parameterization \(J(t) = D_0 t^\alpha + D_\mathrm{offset}\), where no useful closed-form antiderivative exists.

External Correlation¶

The external contribution encodes the deterministic (mean) flow velocity (PNAS 2024 Eq. 9):

\[c_{1,\mathrm{ex}}(q, t_1, t_2) \;=\; \exp\!\left(i\, q \int_{t_1}^{t_2} \langle v(t')\rangle\, dt'\right)\]

where \(\langle v(t)\rangle\) is the expected velocity at time \(t\). This term produces a phase shift proportional to the mean displacement, leading to the oscillatory interference pattern that is the hallmark of heterodyne detection.

When the field correlation enters the Siegert relation, only the modulus \(|c_1|^2\) matters for homodyne detection, eliminating the phase information. Heterodyne detection preserves the phase through interference with a reference beam.