Figure 1

The observed probability distribution of the wrapped phase evolution at five different time intervals $\tau$ (symbols) and the corresponding theoretical predictions (solid curves). The dashed curves in (a) and (b) are the small $\tau$, small $\Delta \Phi$ predictions (see text). The only fitting parameter [via the dependence of $P(\Delta \Phi )$ on $g_1$] is the path phase variance $⟨\Delta {\varphi }_{\mathrm{path}}^2⟩$ at time $\tau$, which gives ${\tau }_{\mathrm{DAWS}}=89\mathrm{ms}$. For these data, the number of scattering events $n=34\pm 2$. Note the wide variation in vertical scales from (a) to (d).Reuse & Permissions

Figure 2

(a) The universal relationship (solid curve), calculated from Eq. (2), between the variance of the phase shift along one path ($y$ axis) and the measured phase of the transmitted field ($x$ axis). Dashed lines and dotted lines apply when the phase is unwrapped, making the relation explicitly dependent on the motion of the particles. (b) The relative mean square displacement of the particles, $⟨\Delta r_{\mathrm{rel}}^2(\tau )⟩$ (left axis), determined from the wrapped phase via the corresponding $⟨\Delta {\varphi }_{\mathrm{path}}^2⟩$ (right axis). We compare the results from the wrapped phase shift distribution (⊙) and variance (□) with traditional DAWS measurements (★). The inset shows the effect of amplitude noise (see text).Reuse & Permissions

Figure 3

Comparison of theory and experiment for the phase derivative distribution functions $P({\Phi }^{(n)})$, where $n=1$, 2, or 3 denotes the $n$th derivative of $\Phi$ with respect to evolution time $T$. From $Q=40{\mathrm{rad}}^2/{\mathrm{s}}^2$, we find ${\tau }_{\mathrm{DAWS}}=91\mathrm{ms}$.Reuse & Permissions

Figure 4

Comparison of theory and experiment for (a) the phase derivative correlation function $C_{{\Phi }^{\prime }}$ and (b) the cumulative phase shift variance $⟨\Delta {\Phi }_c^2(\tau )⟩$.Reuse & Permissions