3 Theoretical Models of Light Scattering and Absorption
51
The equations… have been deduced under the assumption that the radiation throughout
the absorbing mass is uniformly distributed in such a way that it does not depend on the
angle between any direction considered and the normal drawn toward the same side. This
supposition is obviously incorrect, for it appears that, even if it were to hold at any surface,
e.g., the first surface of the layer dx, absorption in that layer would destroy the uniformity
owing to the greater absorption which the oblique rays suffer. To some extent, the effect of
scattering would act in the sense of partly restoring the equality of distribution…
Kubelka published a later paper with a mathematical treatment that was intended to
be more general [10]. Rather than using a two-flux model, he assumed that scatter was
isotropic and accounted for the various angles of travel within a layer. He concluded
that Eq. 3.13 was an “exact” mathematical solution only in two cases: When the
incident light striking the sample was perfectly diffuse, and when the incident light
was striking the front surface of the sample at an angle of exactly 60°.
Other authors have gone beyond the “two-flux” approach. Burger et al. used a
three-flux approximation in which the three fluxes were each modeled at 120° from
each other [11]. Giovanelli published exact solutions for the cases of directed and
diffuse illumination encountering a semi-infinite slab [12]. However, even these more
general treatments used continuous mathematics, which is itself a limitation when
applied to particulate samples.
In the Shuster and Kubelka–Munk approaches, as in the derivations of Beer’s Law
and Eq. 3.11 for non-scattering samples, the individual plane parallel layers were
assumed to be infinitesimal in thickness. Stated in physical rather than mathematical
terms, it is assumed that the fractions of light absorbed and scattered by a single interaction with a single layer are extremely small. In effect, the mathematics used treat
the sample as a homogeneous continuum, in which either absorption or scattering
can occur at any location. However, as outlined in Sect. 3.4, it is the discontinuities
(surfaces) within a sample that are responsible for the phenomenon of scatter, so
the assumption of a homogeneous continuum is a limitation in such a case. Kubelka
and Munk in their original paper were investigating a system in which the individual
particles that made up the sample were small, and therefore it was justifiable to model
them mathematically as infinitesimal. Shuster was investigating a system in which
the density of particles was extremely low, and it was therefore quite reasonable to
say that only a tiny fraction of light would be absorbed or scattered within any single
layer. However, what if no such justification exists? Sect. 3.8 addresses this question
by outlining an approach that uses discontinuous mathematics.
3.8 Quantifying Absorption, Transmission, and Remission
in Plane Parallel Layers
Section 3.6 introduced the concepts of the plane parallel layer and the two-flux model.
Here, we outline a two-flux modeling approach that quantifies each of the individual
“forward” and “backward” fluxes traveling between individual layers. Throughout
this section, we will define A as the fraction of incident light absorbed by a layer, T as
51
The equations… have been deduced under the assumption that the radiation throughout
the absorbing mass is uniformly distributed in such a way that it does not depend on the
angle between any direction considered and the normal drawn toward the same side. This
supposition is obviously incorrect, for it appears that, even if it were to hold at any surface,
e.g., the first surface of the layer dx, absorption in that layer would destroy the uniformity
owing to the greater absorption which the oblique rays suffer. To some extent, the effect of
scattering would act in the sense of partly restoring the equality of distribution…
Kubelka published a later paper with a mathematical treatment that was intended to
be more general [10]. Rather than using a two-flux model, he assumed that scatter was
isotropic and accounted for the various angles of travel within a layer. He concluded
that Eq. 3.13 was an “exact” mathematical solution only in two cases: When the
incident light striking the sample was perfectly diffuse, and when the incident light
was striking the front surface of the sample at an angle of exactly 60°.
Other authors have gone beyond the “two-flux” approach. Burger et al. used a
three-flux approximation in which the three fluxes were each modeled at 120° from
each other [11]. Giovanelli published exact solutions for the cases of directed and
diffuse illumination encountering a semi-infinite slab [12]. However, even these more
general treatments used continuous mathematics, which is itself a limitation when
applied to particulate samples.
In the Shuster and Kubelka–Munk approaches, as in the derivations of Beer’s Law
and Eq. 3.11 for non-scattering samples, the individual plane parallel layers were
assumed to be infinitesimal in thickness. Stated in physical rather than mathematical
terms, it is assumed that the fractions of light absorbed and scattered by a single interaction with a single layer are extremely small. In effect, the mathematics used treat
the sample as a homogeneous continuum, in which either absorption or scattering
can occur at any location. However, as outlined in Sect. 3.4, it is the discontinuities
(surfaces) within a sample that are responsible for the phenomenon of scatter, so
the assumption of a homogeneous continuum is a limitation in such a case. Kubelka
and Munk in their original paper were investigating a system in which the individual
particles that made up the sample were small, and therefore it was justifiable to model
them mathematically as infinitesimal. Shuster was investigating a system in which
the density of particles was extremely low, and it was therefore quite reasonable to
say that only a tiny fraction of light would be absorbed or scattered within any single
layer. However, what if no such justification exists? Sect. 3.8 addresses this question
by outlining an approach that uses discontinuous mathematics.
3.8 Quantifying Absorption, Transmission, and Remission
in Plane Parallel Layers
Section 3.6 introduced the concepts of the plane parallel layer and the two-flux model.
Here, we outline a two-flux modeling approach that quantifies each of the individual
“forward” and “backward” fluxes traveling between individual layers. Throughout
this section, we will define A as the fraction of incident light absorbed by a layer, T as