7.2.   COLLIMATED TRANSMISSION METHOD

The extinction coefficient μ_t, which is defined as the sum of the absorption
coefficient μ_a and the scattering coefficient μ_s, can be measured by the collimated
transmission method. In this method, a collimated lightbeam is incident
perpendicularly on the surface of a sample. The sample can be a cuvette of
liquid (e.g., an Intralipid^® solution), a tissue-mimicking gel phantom (e.g., an
agar gel containing polystyrene spheres), a solid phantom (e.g., a solidified resin
containing TiO₂ particles), or a piece of biological tissue. The collimated (or
ballistic) portion of the transmitted light is selected by apertures and then measured
by a photodetector (Figure 7.1). First, a clear medium (e.g., water), the
refractive index of which closely matches that of the sample to be tested, is
measured to provide a reference ballistic-light signal I₀. Then, the sample is
measured, which yields a transmitted light signal I_s. According to Beer’s law,
we have

where d denotes the sample thickness. Here, light absorption by the clear medium
is neglected (see Problem 7.1). The ballistic transmittance T of the scattering

medium is defined as

Substituting Eq. (7.2) into Eq. (7.1), we obtain the extinction coefficient of the
sample:

A key assumption in Eq. (7.1) is that the detected scattered light is much
weaker than the detected ballistic light. Three factors affect this assumption:
(1) the scattering optical depth of the sample μ_sd, (2) the scattering phase function
p, and (3) the acceptance angle (half-angle) of detection θ_d . The first factor
determines the ratio of the number of scattered photons to the number of unscattered
transmitted photons. The next two factors determine the collection fraction
of the scattered photons to be detected χ.

7.2.1.   Distribution of Scattering Count

We first consider an ideal scattering slab with the following optical properties:
relative refractive index n_rel = 1 and scattering anisotropy g = 1. In this slab,
specular reflection does not exist, and scattering does not deflect the photon. If
the number of incident photons is N_in, the number of unscattered transmitted
photons N₀ can be computed from Beer’s law:

The number of transmitted photons that have experienced i scattering events N_i
is given by the following Poisson distribution (see Problem 7.2):