Dynamic Speckle Holography

We introduce dynamic speckle holography, a new technique that combines imaging and scattering to measure three-dimensional maps of displacements as small as ten nanometers over several centimeters, greatly extending the capabilities of traditional imaging systems. We attain this sensitivity by imaging speckle patterns of light collected at three scattering angles and measuring the decay in the temporal correlation due to local motion. We use dynamic speckle holography to measure the strain field of a colloidal gel undergoing fracture and establish the surprising role of internal tension in driving the fracture.

Figure 1

(a) DSH geometry (not to scale): scattered light is collected by a camera ($C$) through lens ($L$) and diaphragm ($D$), to image the speckles ($P^{\prime }$, $Q^{\prime }$). (b) Speckle image of flow channel. Channel edges are highlighted by dashed lines. Insets: expanded region of $200\times 200\mu {\mathrm{m}}^2$, recorded at times $t_1$ and $t_2$. (c) Correlation functions measured with $\overrightarrow{q}\parallel \overrightarrow{v}$ for flow rate $Q=5,10,20,50,100,200,500,1000{\mathrm{mm}}^3/\mathrm{h}$ (full symbols, blue to red), all in the laminar flow regime. Lines: analytical model combining Poiseuille flow and Brownian motion. (d) Correlation functions measured with $\overrightarrow{q}\perp \overrightarrow{v}$, for same $Q$ (open symbols). Line: correlation decay due to Brownian motion. (e) Symbols: data from panels (c),(d) plotted against $v_0\tau$. Solid line: correlation function for Poiseuille flow. Dashed line: correlation function for rigid speckle drift. (f) Poiseuille flow velocity $v_0$ as a function of expected value $v_e$. Line: $v_0=v_e$.

Figure 2

(a) Optical configuration for three-dimensional DSH. Two laser beams illuminate the sample, forming perpendicular scattering planes. Light scattered along the $\widehat{z}$ axis is collected from both sides by mirrored optical setups composed of a diaphragm, a lens, and a camera. (b) Orientation of the four scattering vectors formed by two incident wave vectors and 2 scattered wave vectors.

Figure 3

(a) Correlation map from a propagating crack, indicated by the dashed blue line, measured with $\overrightarrow{q}\parallel \widehat{x}$ for $\tau =1\mathrm{s}$. The scale bar corresponds to 1 mm. (b)–(d) Maps of the partial displacements $\mathrm{\Delta }\overrightarrow{r}$, accumulated over $\tau =1\mathrm{s}$. (b) Displacements along $\widehat{x}$, represented with a logarithmic color map; (c) Displacements along $\widehat{y}$; (d) Combined map of displacements in the $xy$ plane (arrows) and along $\widehat{z}$ (color map). (e) Map of total displacements $\overrightarrow{R}$ relative to the sample at rest. (f) Deformation profile as a function of distance $d$ from the crack. Full black symbols: in-plane displacements; red line: fit to the exponential tail; open black symbols: residuals of the exponential fit; blue symbols: out-of-plane displacements.