Diffusing-wave spectroscopy

One-speckle diffusing-wave spectroscopy

Laser light is sent to the sample and the outcoming transmitted or backscattered light is detected by an optoelectric sensor. The light intensity detected is the result of the interference of all the optical waves coming from the different light paths.

The signal is analysed by calculating the intensity autocorrelation function called g₂. ${\displaystyle g_2(\tau )=\frac{⟨I(t)I(t+\tau ){⟩}_t}{⟨I(t){⟩}_t^2}}$

For the case of non-interacting particles suspended in a (complex) fluid a direct relation between g₂-1 and the mean squared displacement of the particles <Δr²> can be established. Let us note P(s) the probability density function (PDF) of the photon path length s. The relation can be written as follows:

${\displaystyle g_2(\tau )-1=[\int dsP(s)\exp (-(s/l\ast )k_0^2⟨\mathrm{\Delta }{r}^2(\tau )⟩){]}^2}$

with ${\displaystyle k_0=\frac{2\pi n}{\lambda }}$ and ${\displaystyle l\ast }$ is the transport mean free path of scattered light.

For simple cell geometries, it is thus possible to calculate the mean squared displacement of the particles <Δr²> from the measured g₂-1 values analytically. For example, for the backscattering geometry, an infinitely thick cell, large laser spot illumination and detection of photons coming from the center of the spot, the relationship between g₂-1 and <Δr²> is:

${\displaystyle g_2(\tau )-1=\exp \left(-2\gamma \sqrt{⟨\mathrm{\Delta }{r}^2(\tau )⟩k_0^2}\right)}$, γ value is around 2.

For less thick cells and in transmission, the relationship depends also on l* (the transport length).

For quasi-transparent cells, an angle-independent variant method called cavity amplified scattering spectroscopy makes use of an integrating sphere to isotropically probe samples from all directions, elongating photon paths through the sample in the process, allowing for the study of low turbidity samples under the DWS formalism.