Velocity gradients in spatially resolved laser Doppler flowmetry and dynamic light scattering with confocal and coherence gating

Dynamic light scattering (DLS) is widely used to characterize diffusive motion to obtain precise information on colloidal suspensions by calculating the autocorrelation function of the signal from a heterodyne optical system. DLS can also be used to determine the flow velocity field in systems that exhibit mass transport by incorporating the effects of the deterministic motion of scatterers on the autocorrelation function, a technique commonly known as laser Doppler flowmetry. DLS measurements can be localized with confocal and coherence gating techniques such as confocal microscopy and optical coherence tomography, thereby enabling the determination of the spatially resolved velocity field in three dimensions. It has been thought that spatially resolved DLS can determine the axial velocity as well as the lateral speed in a single measurement. We demonstrate, however, that gradients in the axial velocity of scatterers exert a fundamental influence on the autocorrelation function even in well-behaved, nonturbulent flow. By obtaining the explicit functional relation between axial-velocity gradients and the autocorrelation function, we show that the velocity field and its derivatives are intimately related and their contributions cannot be separated. Therefore, a single DLS measurement cannot univocally determine the velocity field. Our extended theoretical model was found to be in good agreement with experimental measurements.

Figure 1

Geometry of the experimental setup, with fully developed parabolic flow of an intralipid solution inside silicone tubing. (Inset) The estimated signal-to-noise ratio of the tomograms for each ${\theta }_n$ angle.

Figure 2

(a) Experimental $g^{\left(1\right)}\left(\tau \right)$ for ${\theta }_n=-{10}^{\circ }$ and 5 mL/min (green, solid), and ${\theta }_n={15}^{\circ }$ and 5 mL/min (violet-blue, dashed). (b) The same autocorrelation functions after normalization. Note the nonlinear color mapping to ease visualization of the beginning of each curve.

Figure 3

(a) Unwrapped phase of the experimental $g^{\left(1\right)}\left(\tau \right)$ for nominal angles ${\theta }_n={[-10,0,15,30]}^{\circ }$ and $[10,15,20,25,30]$ mL/min. (b) Unwrapped phase of $g^{\left(1\right)}\left(\tau \right)$ used to determine the actual experimental angles $\theta ={[-8.4,0.1,13.2,30.4]}^{\circ }$. Only the first 32 values of $\tau$ are shown for each flow rate.

Figure 4

(a) Scatter plot of $g_{\mathrm{mod}}^{\left(1\right)}$ vs $g_{\exp }^{\left(1\right)}$ for all flow rates at normal incidence used to determine $w_x$. (b) Scatter plot of $g_{\mathrm{mod}}^{\left(1\right)}$ vs $g_{\exp }^{\left(1\right)}$ for the gradient-free region at all angles and flow rates used to determine $w_z$.

Figure 5

(a) Absolute value of the experimental $g^{\left(1\right)}\left(\tau \right)$ for nominal angles ${\theta }_n={[-10,0,15,35]}^{\circ }$ and 15 mL/min. (b) Absolute value of $g^{\left(1\right)}\left(\tau \right)$ according to our model that includes gradient effects.

Figure 6

Scatter plots of $g_{\mathrm{mod}}^{\left(1\right)}$ vs $g_{\exp }^{\left(1\right)}$ with (a) no gradient effects, (b) only axial gradient contribution, (c) lateral and axial gradient contributions, and (d) lateral and axial gradient contributions considering optical aberrations according to our model.

Figure 7

Inverse $g^{\left(1\right)}$ decorrelation time for angles $-{10}^{\circ }$ (a), $0^{\circ }$ (b), ${15}^{\circ }$ (c), and ${35}^{\circ }$ (d) for experimental data (circles) and our model (lines) that includes gradient effects.