The mathematical treatments of absorption spectroscopy for scattering materials were originally largely borrowed from other fields.
Treatments which involve different assumptions and which yield incompatible results are the Giovanelli[9] exact solutions, and the particle theories of Melamed[10] and Simmons.
[11] George Gabriel Stokes (not to neglect the later work of Gustav Kirchhoff) is often given credit for having first enunciated the fundamental principles of spectroscopy.
In 1862, Stokes published formulas for determining the quantities of light remitted and transmitted from "a pile of plates".
Stokes used the term "reflexion", not "remission", specifically referring to what is often called regular or specular reflection.
For a sample that consists of n layers, each having its absorption, remission, and transmission (ART) fractions symbolized by {a, r, t } , with a + r + t = 1, one may symbolize the ART fractions for the sample as {Αn, Rn, Tn} and calculate their values by where and In 1905, in an article entitled "Radiation through a foggy atmosphere", Arthur Schuster published a solution to the equation of radiative transfer, which describes the propagation of radiation through a medium, affected by absorption, emission, and scattering processes.
He used the symbols k and s for absorption and isotropic scattering coefficients, and repeatedly refers to radiation entering a "layer", which ranges in size from infinitesimal to infinitely thick.
In 1931, Paul Kubelka (with Franz Munk) published "An article on the optics of paint", the contents of which has come to be known as the Kubelka-Munk theory.
While symbols and terminology are changed here, it seems clear from their language that the terms in their differential equations stand for absorption and backscatter (remission) fractions.
This turns out to be incorrect for layers of finite thickness, and the equation was modified for spectroscopic purposes (below), but Kubelka-Munk theory has found extensive use in coatings.
Deane Judd was very interested the effect of light polarization and degree of diffusion on the appearance of objects.
In the 1920s and 30s, Albert H. Taylor, Arthur C. Hardy, and others of the General Electric company developed a series of instruments that were capable of easily recording spectral data "in reflection".
In 1946, Frank Benford[2] published a series of parametric equations that gave results equivalent to the Stokes formulas.
This book came to dominate thinking of the day for 20 years in the emerging fields of both DRIFTS and NIR Spectroscopy.
He developed an approach based on Schuster's work by ignoring the emissivity of the clouds in the "foggy atmosphere".
In practice these are found to be at least qualitatively confirmed, and suitable conditions fulfilling the assumptions made, quantitatively as well."
Melamed of Westinghouse Research Labs, "abandoned the idea of plane parallel layers and substituted them with a statistical summation over individual particles.
One of the features of the remission function emphasized by Hecht was the fact that should yield the absorption spectrum displaced by -log s. While the scattering coefficient might change with particle size, the absorption coefficient, which should be proportional to concentration of an absorber, would be obtainable by a background correction for a spectrum.
Proposed culprits included: incomplete diffusion, anisotropic scatter ("the invalid assumption that radiation is returned equally in all directions from a given particle"), and presence of regular reflection.
[3][22] In his book, Hecht reported the mathematics of Stokes and Melamed formulas (which he called “statistical methods”).
E.L. Simmons used a simplified modification of the particle model to relate diffuse reflectance to fundamental optical constants without the use of the cumbersome equations.
In 1976, Hecht wrote a lengthy paper comprehensively describing the myriad of mathematical treatments that had been proposed to deal with diffuse reflectance.
To correct this deficiency of the Kubelka–Munk approach, for the case of an infinitely thick sample, Hecht blended the particle and layer methods by replacing the differential equations in the Kubelka–Munk treatment by finite difference equations, and obtained the Hecht finite difference formula: Hecht apparently did not know that this result could be generalized, but he realized that the above formula "represents an improvement … and shows the need to consider the particulate nature of scattering media in developing a more precise theory".
He teamed up with Harry Hecht (who was active in the early meetings of IDRC) to write the Physics theory chapter, with many photographic illustrations, in an influential Handbook edited by Phil Williams and Karl Norris:[25] Nearinfrared Technology in the Agriculture and Food Industries.
The Dahms argued that the conventional absorption and scattering coefficients, as well as the differential equations which employ them, implicitly assume that a sample is homogenous at the molecular level.
All light leaving the sample on the same side as the incident beam is termed remission, whether it arises from reflection or back scatter.
(In a three-flux or higher treatment, such as Giovanelli's, the forward scatter is not indistinguishable from the directly transmitted light.
They developed a scheme, subject to the limitations of a two-flux model, to calculate the "scatter corrected absorbance" for a sample.
In spectroscopy, the term "plane parallel layers" may be employed as a mathematical construct in discussing theory.
Implicit in the representative layer theory is that absorption occurs at the molecular level, but that scatter is from a whole particle.