1. Introduction
2. Diffuse Transmission Spectroscopy (DTS)
3. Diffusing Wave Spectroscopy (DWS)

One of the hallmark features of soft-condensed matter is the presence of structure at length scales that are intermediate between the molecular and the macroscopic. Diverse examples include the arrangement of colloidal particles into liquid, glassy, or crystalline structures in paint, ink, and dairy products; the packing of gas or liquid bubbles inside foams and emulsions; the arrangement of grains in a pile of sand; or the configuration of cells in a tissue. These mesoscopic structures are interesting both in terms of the microscopic physics underlying their formation, and in terms of the macroscopic properties they impart to the material. Since the individual components typically have different refractive indices, they can strongly scatter visible light and hence cause bulk samples to have an opaque, white appearance. Physically, photons travel ballistically between successive scattering events whose cumulative effect is to produce a random walk. Similar multiple scattering processes occur for photons in astrophysical structures, electrons and phonons in solids, neutrons in reactors, and molecules in a gas. In soft condensed matter, this can preclude characterization by traditional optical means such as video microscopy or angle-resolved static and quasi-elastic light scattering. Fortunately, experimental techniques that exploit multiple light scattering have been developed for probing the structure and dynamics of the mesoscopic structures. In diffuse-transmission spectroscopy (DTS), the transmission probability, T_d, for incident light to be transmitted through an opaque slab is measured as a function of wavelength and analyzed in terms of the transport mean free path, L*, or stepsize in the random walk, of the photons. Structural details are then deduced from the value and wavelength dependence of L*. In diffusing-wave spectroscopy (DWS), fluctuations in the intensity of a portion of the multiply-scattered light are measured and expressed in terms of a normalized electric field autocorrelation function,

For both DTS and DWS, our favorite experimental geometry is to shine a light beam at normal incidence onto a slab of material with thickness L much greater than L* and with lateral dimensions so large that essentially no photons escape out the side. We then place a photodetector on the opposite face of the sample and arrange the optics such that the detection probability is independent of precisely where a diffusing photon happens to emerge. This gives an effectively one-dimensional symmetry (plane-wave in, plane-wave out); the diameter of the light beam is thus not an important parameter, but is always chosen to be much greater than L* to ensure good statistics.

As shown above, many length scales are important in analyzing the physics of the photon propagation via multiple scattering. Besides the thickness and transport mean free path, there is also the penetration depth (which depends on the scattering length l_s) and the extrapolation length (which depends on boundary reflectivity). The value of the transport mean free path is given by

with

In this technique we measure the probability T_d that an incident photon will be diffusely transmitted through an opaque sample. This is often accomplished by comparing light emerging within a narrow solid angle with that for a calibration sample with known T_d. However, this makes a mistake in assuming that both the angle dependence of the emerging light and the values of z_e are the same in the known and unknown samples. Therefore we have devised a new scheme that does not rely on comparison with a calibration sample. The basic idea is to record the detected intensity at finely spaced angles around the entire sample, and then deduce T_d from integrating these distributions for both transmitted and backscattered light. Our apparatus looks like this, though we often use light from a monochrometer rather than from a laser:

To analyze the results, we use the diffusion theory prediction

In this technique we measure fluctuations in the intensity, rather than just the average as in DTS. The idea is first to form a speckle pattern with the transmitted light. If the scattering sites are not moving, then the speckle will be static just as in the familiar result of shining a laser beam at a rough surface or through a piece of scotch tape. If the scattering sites move around, then the speckle will also move around in response. To characterize this, simple measure fluctuations in the intensity received over an area comparable to a speckle diameter. In practice, we do this digitally via photon counting. Thus the PMT signal is amplified and discriminated so that each detected photon produces a square TTL pulse that is then fed directly into a digital correlation board for real-time computation of the intensity autocorrelation function, ·I(0)I(t)Ò. Our apparatus looks like:

For this to work quantitatively, it is important that the incident light be very coherent as from a laser with an etalon.

To analyze the results, we first convert to the normalized electric field autocorrelation, g₁(t), using the Siegert relation