To leading order in a small parameter range and for a single dominant control pp (e.g. \kappa), we may linearize:
(p)0+(pp0)+O((pp0)2),(C3)\Phi(p) \simeq \Phi_0 + \beta\, (p-p_0) + \mathcal{O}((p-p_0)^2), \tag{C3}
with =Ap(p0)\beta=\alpha \, \mathcal{A}_p(p_0). This is the canonical linear synthetic-phase law used in preceding sections.
2. Observable effect on fringes
Consider an interference observable (single counts or coincidence fringes) of the form
I(x;p)=I0[1+V(p)cos(kx+(p))],(C4)I(x;\mathbf{p}) = I_0\Big[1 + V(\mathbf{p})\cos\big(kx + \Phi(\mathbf{p})\big)\Big], \tag{C4}
where xx parameterizes the detector coordinate or scanning phase, kk the spatial/phase frequency, V(p)V(\mathbf{p}) the visibility, and (p)\Phi(\mathbf{p}) the total (optical + synthetic) phase. The predicted synthetic phase shift produced by a change p\Delta p in the control parameter is
=(p0+p)(p0)p.(C5)\Delta\Phi \;=\; \Phi(p_0+\Delta p)-\Phi(p_0) \;\simeq\; \beta\,\Delta p. \tag{C5}
Importantly, \beta is in principle measurable independently of the visibility envelope: one can fit fringe phases ^(p)\hat\Phi(p) as a function of pp to extract \beta even when V(p)V(p) varies.
3. Joint visibility--phase model and estimation
Fitting fringes across parameter values benefits from a joint model that captures both amplitude and phase dependence. A practical parametrization is
