Analysis Modes¶

The heterodyne package implements a 14-parameter two-component correlation model with 2 additional per-angle scaling parameters. This section defines each parameter, its physical role, and contrasts the heterodyne analysis mode with the simpler homodyne alternatives.

Heterodyne Model Parameters¶

The 14 physics parameters are organized into five groups:

Reference Transport¶

Transport rate for the static reference component: \(J_r(t) = D_{0,r}\, t^{\alpha_r} + D_{\mathrm{offset},r}\)

Parameter	Unit	Default	Description
`D0_ref`	\(\text{\AA}^2/\text{s}^\alpha\)	\(10^4\)	Reference diffusion prefactor
`alpha_ref`	—	0.0	Reference transport exponent
`D_offset_ref`	\(\text{\AA}^2/\text{s}\)	0.0	Reference transport rate offset

Sample Transport¶

Transport rate for the moving sample component: \(J_s(t) = D_{0,s}\, t^{\alpha_s} + D_{\mathrm{offset},s}\)

Parameter	Unit	Default	Description
`D0_sample`	\(\text{\AA}^2/\text{s}^\alpha\)	\(10^4\)	Sample diffusion prefactor
`alpha_sample`	—	0.0	Sample transport exponent
`D_offset_sample`	\(\text{\AA}^2/\text{s}\)	0.0	Sample transport rate offset

Velocity¶

Velocity rate for the sample component: \(v(t) = v_0\, t^\beta + v_\mathrm{offset}\)

Parameter	Unit	Default	Description
`v0`	\(\text{\AA}/\text{s}^\beta\)	\(10^3\)	Velocity prefactor
`beta`	—	0.0	Velocity exponent (0 = constant velocity)
`v_offset`	\(\text{\AA/s}\)	0.0	Velocity offset (negative allowed for reversal)

Sample Fraction¶

Time-dependent sample fraction:

\[f_s(t) = \operatorname{clip}\!\bigl( f_0 \, \exp\!\bigl(f_1 (t - f_2)\bigr) + f_3,\; 0,\; 1\bigr)\]

Parameter	Unit	Default	Description
`f0`	—	0.5	Fraction amplitude
`f1`	\(\text{s}^{-1}\)	0.0	Exponential rate (0 = constant fraction)
`f2`	s	0.0	Time shift
`f3`	—	0.0	Baseline offset

When \(f_1 = 0\), the fraction is constant: \(f_s = \operatorname{clip}(f_0 + f_3, 0, 1)\).

Flow Angle¶

Parameter	Unit	Default	Description
`phi0`	degrees	0.0	Flow angle offset relative to \(q\)-vector

The total flow angle is \(\varphi = \varphi_\mathrm{detector} + \varphi_0\), where \(\varphi_\mathrm{detector}\) is the known detector geometry and \(\varphi_0\) is a fitted correction.

Per-Angle Scaling Parameters¶

Two scaling parameters are fit independently for each detector angle:

Parameter	Unit	Default	Description
`contrast`	—	0.5	Optical contrast \(\beta\) (speckle contrast)
`offset`	—	1.0	Baseline offset

The full model is:

\[c_2^\mathrm{model} \;=\; \mathrm{offset} + \mathrm{contrast} \times \frac{C_\mathrm{ref} + C_\mathrm{sample} + C_\mathrm{cross}}{f^2}\]

Rate Functions¶

The two rate functions entering the correlation model are:

Transport rate (per component):

\[J(t) \;=\; D_0\, t^\alpha + D_\mathrm{offset}\]

This is evaluated by compute_transport_rate and floored at zero to ensure non-negative transport.

Velocity rate:

\[v(t) \;=\; v_0\, t^\beta + v_\mathrm{offset}\]

This is evaluated by compute_velocity_rate and is not floored, because the velocity integral enters a cosine and is naturally bounded.

The velocity enters the phase factor as:

\[\mathrm{phase}(t_1, t_2) \;=\; q \cos(\varphi) \int_{t_1}^{t_2} v(t')\, dt'\]

Both integrals are always evaluated numerically via trapezoid_cumsum. No analytical antiderivatives are used.

Comparison with Homodyne Modes¶

Mode	Parameters	Description
Homodyne static	3	Single component: \(g_2 = 1 + \beta \exp(-2q^2 \int J\, dt)\). Parameters: \(D_0\), \(\alpha\), \(D_\mathrm{offset}\).
Homodyne laminar flow	7	Single component with shear: adds a \(\mathrm{sinc}^2\) shear broadening term and 4 flow parameters.
Heterodyne	14 + 2/angle	Two components (reference + sample) with relative velocity. The cross-correlation oscillation encodes velocity information that is invisible in homodyne detection.

The fundamental advantage of heterodyne analysis is access to the phase of the field correlation through interference with a reference. Homodyne detection measures only \(|c_1|^2\), losing all velocity phase information. The cost is a larger parameter space (14 vs. 3–7) and the requirement that a coherent reference scattering component be present in the measurement geometry.