How to Measure the Optical Thickness of Scattering Particles from the Phase Delay of Scattered Waves: Application to Turbid Samples

We present a method based on the optical theorem that yields absolute, calibration free estimates of the optical thickness of scattering particles. The thickness is determined from the phase delay of the zero angle scattered wave. It uses a heterodyne scattering scheme operating in the Raman-Nath approximation. The phase is determined by the position of Talbot-like modulations in the two dimensional power spectrum $S(q_x,q_y)$ of the transmitted beam intensity distribution. The method is quite insensitive to multiple scattering. It is successfully tested to provide quantitative verification of the optical theorem. Exploratory tests on soft matter samples are reported to suggest its wide applicability to turbid samples.

Figure 1

The optical theorem scheme for an assembly of small particles (Rayleigh regime). Top: We show the incoming plane wave $A_0$ and the reconstructed plane wave $S(0)$ due to the sum of scattered waves in the forward direction. The projection on $A_0$ generates partial destructive interference that accounts for beam losses. The important term is the $2/3k^6{\alpha }^2$, due to radiation reaction, while the leading term in the scattered amplitude $i$ $k^3\alpha$ has no effect being in quadrature. The angle $\varphi$ is shown in green, and goes from zero to $\pi /2$ for larger particles. Bottom: Optical scheme. Main beam of diameter $D$ impinges on a thin sample and spherical scattered waves are generated. Scattered light and main beam fall onto a CCD at a distance $z$. Typical fringe patterns for particles at three positions are shown. Cases (a) and (b) generate phase shifted Talbot oscillations in the power spectra, while a particle in (c) does not contribute.

Figure 2

Speckle intensity distribution and associated power spectra from colloidal latex particles. (a) and (b) represent the intensity distribution from 0.6 and $2.0\mu \mathrm{m}$ particles. By visual inspection they appear indistinguishable, however their power spectra (c) and (d) show the typical fringelike structure. Fractional order on center is clearly different, and the two fringe systems are staggered almost completely out of phase.

Figure 3

Azimuthally averaged power spectra as a function of $q$ and for various delays for $2\mu \mathrm{m}$ colloidal particles. Talbot-like oscillations are clearly visible in both plots. (a) and (b) for $c_1={10}^{-5}$ and $c_2=7\times {10}^{-4}$, respectively, same vertical scale. Significantly smaller signals for (b) are observed because of strong beam attenuation. Below 1 s, little dynamics is observed in (a), while significant level changes are observed for (b), although spectra appear free from Talbot features. Talbot-free contributions are due to multiple scattering, while Talbot oscillations are due to single scattering alone, and their amplitude can be accurately determined even with rampant multiple scattering like in (b) (see text).

Figure 4

Plots of experimental data (no adjustable parameters) for the phase delay (left, open circles) and of the scattering cross section (right, open squares) as a function of the optical thickness $s$ (above) and the particles diameter $d$ (below). The continuous lines are Mie theory curves.