[4] The sample is modeled as a series of layers, each of which is parallel to each other and perpendicular to the incident beam.
The representative layer theory gives a way of performing the calculations for new sample properties by changing the properties of a single layer of the particles, which doesn’t require reworking the mathematics for a sample as a whole.
The first attempt to account for transmission and reflection of a layered material was carried out by George G. Stokes in about 1860 [5] and led to some very useful relationships.
He was concerned with the cloudy atmospheres of stars, and developed a plane-parallel layer model in which the radiation field was divided into forward and backward components.
This same model was used much later by Paul Kubelka and Franz Munk, whose names are usually attached to it by spectroscopists.
Following WWII, the field of reflectance spectroscopy was heavily researched, both theoretically and experimentally.
as being equivalent to the linear absorption and scattering coefficients as they appear in the Bouguer-Lambert law, even though sources who derived the equations preferred the symbolism
[9] In 1966, in a book entitled Reflectance Spectroscopy, Harry Hecht had pointed out that the formulation
"against the wavelength or wave-number for a particular sample" giving a curve corresponding "to the real absorption determined by transmission measurements, except for a displacement by
However, in data presented, "the marked deviation in the remission function ... in the region of large extinction is obvious."
He listed various reasons given by other authors for this "failure ... to remain valid in strongly absorbing materials", including: "incomplete diffusion in the scattering process"; failure to use "diffuse illumination; "increased proportion of regular reflection"; but concluded that "notwithstanding the above mentioned difficulties, ... the remission function should be a linear function of the concentration at a given wavelength for a constant particle size" though stating that "this discussion has been restricted entirely to the reflectance of homogeneous powder layers" though "equation systems for combination of inhomogeneous layers cannot be solved for the scattering and absorbing properties even in the simple case of a dual combination of sublayers.
This means that the (Kubelka-Munk) theory fails to include, in an explicit manner, any dependence of reflection on particle size or shape or refractive index".
[11] The USDA discovered how to use NIR empirically, based on available sources, gratings, and detector materials.
Consequently, it was not seen as a rigorous science: it had not evolved in the usual way, from research institutions to general usage.
[12] Even though the Kubelka-Munk theory provided a remission function that could have been used as the absorption metric, Norris selected
[13] He believed that the problem of non-linearity between the metric and concentration was due to particle size (a theoretical concern) and stray light (an instrumental effect).
, and it appears that (this equation) can be used to explain the deviations in part", and "represents an improvement in the range of validity and shows the need to consider the particulate nature of scattering media in developing a more precise theory by which absolute absorptivities can be determined."
At this meeting was Harry Hecht, who may have at the time been the world's most knowledgeable person in the theory of diffuse reflectance.
In 1987, Birth and Hecht wrote a joint article in a new handbook,[16] which pointed a direction for future theoretical work.
In 1994, Donald and Kevin Dahm began using numerical techniques to calculate remission and transmission from samples of varying numbers of plane parallel layers from absorption and remission fractions for a single layer.
This cumulative mathematics was tested on data collected using directed radiation on plastic sheets, a system that precisely matches the physical model of a series of plane parallel layers, and found to conform.
While other assumptions could be made, those most often used are those of normal incidence of a directed beam of light, with internal and external reflection from the surface being the same.
For the special case where the incident radiation is normal (perpendicular) to a surface and the absorption is negligible, the intensity of the reflected and transmitted beams can be calculated from the refractive indices η1 and η2 of the two media, where r is the fraction of the incident light reflected, and t is the fraction of the transmitted light:
For the case of zero absorption in the interior, the total remission and transmission from the layer can be determined from the infinite series, where
Dahm has shown that for this special case, the total amount of light absorbed by the interior of the sheet (considering surface remission) is the same as that absorbed in a single trip (independent of surface remission).
Note that this time the calculation corresponds to three and a half layers, a thickness of sample that cannot exist physically.
But a particulate sample often looks a jumbled maze of particles of various sizes and shapes, showing no structured pattern of any kind, and certainly not literally divided into distinct, identical layers.
• The surface area fraction of each type of particle is the same in the representative layer as in the sample as a whole.
The values of the absorption and remission coefficients represent a challenge in this modeling approach.
used should ideally model the ability of the particle to absorb light, independent of other processes (scattering, remission) that also occur.