Heterodyne Scattering Model¶

This section presents the core theoretical model implemented by the heterodyne package: a two-component interference scattering model derived from PNAS 2025 SI Section F (Eqs. S-77 through S-98). The heterodyne geometry — where scattered light from a moving sample interferes with that from a static reference — produces characteristic oscillations in the cross-correlation whose frequency directly encodes the sample velocity.

Multi-Component Scattered Field¶

For a system of \(N\) scattering components, the total scattered field at wavevector \(q\) and time \(t\) is (Eq. S-77):

\[E(q, t) = \sum_{n=1}^{N} x_n(t)\, E_n(q, t)\]

where \(x_n(t)\) is the field amplitude fraction of the \(n\)-th component and \(E_n(q, t)\) is its scattered field. Each component has its own transport coefficient \(J_n(t)\) and mean velocity \(\langle v_n(t)\rangle\).

Two Key Assumptions¶

The derivation proceeds under two physical assumptions (Eq. S-84):

Uniform scattering contrast: all components scatter with the same contrast factor, so that intensity fractions are determined solely by composition \(x_n(t)\).
No cross-composition spatial correlation: the positions of particles in different components are statistically independent, so cross-component field correlations vanish.

Under these assumptions, the multi-component intensity correlation takes a compact form.

General N-Component Correlation¶

The second-order two-time correlation for an \(N\)-component system is (Eq. S-94):

\[c_2(q, t_1, t_2) \;=\; 1 + \frac{\beta}{f^2} \sum_{n=1}^{N} \sum_{m=1}^{N} x_n(t_1)\, x_n(t_2)\, x_m(t_1)\, x_m(t_2) \;\cdot\; A_{nm}(t_1, t_2)\]

where the cross-correlation amplitude \(A_{nm}\) is:

\[A_{nm}(t_1, t_2) \;=\; \exp\!\left(-\frac{q^2}{2}\int_{t_1}^{t_2} \left[J_n(t') + J_m(t')\right] dt'\right) \cos\!\left(q\cos(\varphi)\int_{t_1}^{t_2} \left[\langle v_n(t')\rangle - \langle v_m(t')\rangle\right] dt'\right)\]

where \(\beta\) is the optical contrast, \(f\) is a normalization factor, and \(\varphi_n\) is the angle between the velocity of component \(n\) and the scattering vector \(q\).

Note

The cosine term arises from the real part of the product \(c_1^{\mathrm{(ex)},n} \times c_1^{\mathrm{(ex)},m*}\), which reduces the complex exponentials to a single cosine of the velocity difference. Same-component terms (\(n = m\)) have zero velocity difference, so their cosine factor is unity.

Two-Component Simplification¶

The heterodyne package implements the \(N = 2\) specialization with a reference component (\(r\)) and a sample component (\(s\)):

Reference (\(r\)): static scatterer with transport \(J_r(t)\) and zero mean velocity
Sample (\(s\)): moving scatterer with transport \(J_s(t)\), mean velocity \(\langle v(t)\rangle\), and flow angle \(\varphi\)

The sample fraction is \(x_s(t) \in [0, 1]\) and the reference fraction is \(x_r(t) = 1 - x_s(t)\).

Two-Time Form (Eq. S-95)¶

The full two-time correlation is:

\[c_2(q, t_1, t_2) \;=\; 1 + \frac{\beta}{f^2} \left[ \bigl[x_r(t_1)\, x_r(t_2)\bigr]^2\, A_{rr} \;+\; \bigl[x_s(t_1)\, x_s(t_2)\bigr]^2\, A_{ss} \;+\; 2\, x_r(t_1)\, x_r(t_2)\, x_s(t_1)\, x_s(t_2)\, A_{rs} \right]\]

with:

\[\begin{split}A_{rr} &= \exp\!\left(-q^2 \int_{t_1}^{t_2} J_r(t')\, dt'\right), \\ A_{ss} &= \exp\!\left(-q^2 \int_{t_1}^{t_2} J_s(t')\, dt'\right), \\ A_{rs} &= \exp\!\left(-\frac{q^2}{2} \int_{t_1}^{t_2}\left[J_r(t') + J_s(t')\right] dt'\right) \cos\!\left(q\cos(\varphi) \int_{t_1}^{t_2}\langle v(t')\rangle\, dt'\right)\end{split}\]

This expression contains three distinct physical contributions, described below.

Three-Term Structure¶

Reference self-correlation (first term):

\[C_\mathrm{ref} \;=\; \bigl[x_r(t_1)\, x_r(t_2)\bigr]^2\; \exp\!\left(-q^2 \int_{t_1}^{t_2} J_r(t')\, dt'\right)\]

Describes the decorrelation of the static reference scattering due to its own internal dynamics (thermal diffusion). This term decays monotonically with lag time.

Sample self-correlation (second term):

\[C_\mathrm{sample} \;=\; \bigl[x_s(t_1)\, x_s(t_2)\bigr]^2\; \exp\!\left(-q^2 \int_{t_1}^{t_2} J_s(t')\, dt'\right)\]

Describes the decorrelation of the sample scattering. Like the reference term, this decays monotonically but typically faster due to flow-enhanced transport.

Cross-correlation (third term):

\[C_\mathrm{cross} \;=\; 2\, x_r(t_1)\, x_r(t_2)\, x_s(t_1)\, x_s(t_2)\; \exp\!\left(-\frac{q^2}{2} \int_{t_1}^{t_2} \left[J_s(t') + J_r(t')\right] dt'\right)\; \cos\!\left(q \cos(\varphi) \int_{t_1}^{t_2} \langle v(t')\rangle\, dt'\right)\]

This is the signature heterodyne term. The cosine factor produces oscillations whose frequency is proportional to \(q \cos(\varphi) \cdot \langle v \rangle\) — the projection of the sample velocity onto the scattering vector. The oscillation amplitude is modulated by the geometric mean of the transport decays from both components, and is maximized when the reference and sample fractions are balanced (\(x_s \approx 0.5\)).

Equilibrium One-Time Form (Eq. S-98)¶

At equilibrium, where the composition fractions, transport coefficients, and velocity are all time-independent, the two-time correlation reduces to a function of lag time \(\tau = t_2 - t_1\) only. Denoting the equilibrium sample fraction \(x \equiv I_s / (I_s + I_r)\):

\[g_2(q, \tau) \;=\; 1 + \beta \left[ (1 - x)^2\, \exp\!\left(-q^2 \int_0^{\tau} J_r(t')\, dt'\right) + x^2\, \exp\!\left(-q^2 \int_0^{\tau} J_s(t')\, dt'\right) + 2\, x(1 - x)\, \exp\!\left(-\frac{q^2}{2} \int_0^{\tau} \left[J_r(t') + J_s(t')\right] dt'\right) \cos\!\left(q \cos(\varphi) \int_0^{\tau} \langle v(t')\rangle\, dt'\right) \right]\]

This is the one-time specialization of Eq. S-95, obtained by setting \(t_1 = 0\) and \(t_2 = \tau\) with time-independent fractions.

Important

The implementation always evaluates the integrals numerically via trapezoid_cumsum, even in equilibrium. It never substitutes analytical antiderivatives (e.g., \(\int J\, dt = 2D\tau\) for constant \(J\)). This avoids silent approximation errors for the general power-law parameterization \(J(t) = D_0 t^\alpha + D_\mathrm{offset}\), which has no useful closed-form antiderivative.

Normalization¶

The normalization factor \(f^2\) in the correlation expression ensures that \(c_2(q, t, t) = 1 + \beta\) on the diagonal. For the two-component system:

\[f^2(t_1, t_2) \;=\; \left[x_s^2(t_1) + x_r^2(t_1)\right] \cdot \left[x_s^2(t_2) + x_r^2(t_2)\right]\]

This normalization accounts for the fact that the total scattered intensity is not simply the sum of individual intensities when the component fractions are time-dependent.

Angle Dependence¶

The flow angle \(\varphi\) controls the projection of velocity onto the scattering direction:

\(\varphi = 0\): maximum velocity sensitivity (\(\cos(\varphi) = 1\))
\(\varphi = 90^\circ\): zero velocity sensitivity (\(\cos(\varphi) = 0\)), reducing to a purely diffusive model

By measuring at multiple detector angles \(\varphi\), the full velocity vector can be reconstructed. The implementation supports simultaneous multi-angle fitting where the 14 physics parameters are shared across angles while per-angle contrast and offset are independently varied.