Diffuse reflectance spectroscopy


Diffuse reflectance spectroscopy, or diffuse reflection spectroscopy, is a subset of absorption spectroscopy. It is sometimes called remission spectroscopy. Remission is the reflection or back-scattering of light by a material, while transmission is the passage of light through a material. The word remission implies a direction of scatter, independent of the scattering process. Remission includes both specular and diffusely back-scattered light. The word reflection often implies a particular physical process, such as specular reflection.
The use of the term remission spectroscopy is relatively recent, and found first use in applications related to medicine and biochemistry. While the term is becoming more common in certain areas of absorption spectroscopy, the term diffuse reflectance is firmly entrenched, as in diffuse reflectance infrared Fourier transform spectroscopy and diffuse-reflectance ultraviolet–visible spectroscopy.

Mathematical treatments related to diffuse reflectance and transmittance

The mathematical treatments of absorption spectroscopy for scattering materials were originally largely borrowed from other fields. The most successful treatments use the concept of dividing a sample into layers, called plane parallel layers. The treatments are generally those consistent with a two-flux or two-stream approximation. Some of the treatments require all the scattered light, both remitted and transmitted light, to be measured. Others apply only to remitted light, with the assumption that the sample is "infinitely thick" and transmits no light. These are special cases of the more general treatments.
There are several general treatments, all of which are compatible with each other, related to the mathematics of plane parallel layers. They are the Stokes formulas, equations of Benford, Hecht finite difference formula, and the Dahm equation. For the special case of infinitesimal layers, the Kubelka–Munk and Schuster–Kortüm treatments also give compatible results. Treatments which involve different assumptions and which yield incompatible results are the Giovanelli exact solutions, and the particle theories of Melamed and Simmons.

George Gabriel Stokes

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". He described his work as addressing a "mathematical problem of some interest". He solved the problem using summations of geometric series, but the results are expressed as continuous functions. This means that the results can be applied to fractional numbers of plates, though they have the intended meaning only for an integral number. The results below are presented in a form compatible with discontinuous functions.
Stokes used the term "reflexion", not "remission", specifically referring to what is often called regular or specular reflection. In regular reflection, the Fresnel equations describe the physics, which includes both reflection and refraction, at the optical boundary of a plate. A "pile of plates" is still a term of art used to describe a polarizer in which a polarized beam is obtained by tilting a pile of plates at an angle to an unpolarized incident beam. The area of polarization was specifically what interested Stokes in this mathematical problem.

Stokes formulas for remission from and transmission through a "pile of plates"

For a sample that consists of layers, each having its absorption, remission, and transmission fractions symbolized by, with, one may symbolize the ART fractions for the sample as and calculate their values by
where
and

Franz Arthur Friedrich Schuster

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. His mathematics used a two flux approximation; i.e., all light is assumed to travel with a component either in the same direction as the incident beam, or in the opposite direction. He used the word scattering rather than reflection, and considered scatter to be in all directions. 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 his treatment, the radiation enters the layers at all possible angles, referred to as "diffuse illumination".

Kubelka and Munk

In 1931, Paul Kubelka published "An article on the optics of paint", the contents of which has come to be known as the Kubelka-Munk theory. They used absorption and remission constants, noting that "an infinitesimal layer of the coating absorbs and scatters a certain constant portion of all the light passing through it". 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 fractions. They also noted that the reflectance from an infinite number of these infinitesimal layers is "solely a function of the ratio of the absorption and back-scatter constants, but not in any way on the absolute numerical values of these constants". This turns out to be incorrect for layers of finite thickness, and the equation was modified for spectroscopic purposes, but Kubelka-Munk theory has found extensive use in coatings.
However, in revised presentations of their mathematical treatment, including that of Kubelka, Kortüm and Hecht, the following symbolism became popular, using coefficients rather than fractions:
  • is the Absorption Coefficient ≡ the limiting fraction of absorption of light energy per unit thickness, as thickness becomes very small.
  • is the Back-Scattering Coefficient ≡ the limiting fraction of light energy scattered backwards per unit thickness as thickness tends to zero.

    The Kubelka–Munk equation

The Kubelka–Munk equation describes the remission from a sample composed of an infinite number of infinitesimal layers, each having as an absorption fraction, and as a remission fraction.

Deane B. Judd

was very interested the effect of light polarization and degree of diffusion on the appearance of objects. He made important contributions to the fields of colorimetry, color discrimination, color order, and color vision. Judd defined the scattering power for a sample as, where is the particle diameter. This is consistent with the belief that the scattering from a single particle is conceptually more important than the derived coefficients.
The above Kubelka–Munk equation can be resolved for the ratio in terms of. This led to a very early use of the term "remission" in place of "reflectance" when Judd defined a "remission function" as, where and are absorption and scattering coefficients, which replace and in the Kubelka–Munk equation above. Judd tabulated the remission function as a function of percent reflectance from an infinitely thick sample. This function, when used as a measure of absorption, was sometimes referred to as "pseudo-absorbance", a term which has been used later with other definitions as well.

General Electric

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". Their display preference for the data was "% Reflectance". In 1946, Frank Benford published a series of parametric equations that gave results equivalent to the Stokes formulas. The formulas used fractions to express reflectance and transmittance.

Equations of Benford

If,, and are known for the representative layer of a sample, and, and are known for a layer composed of representative layers, the ART fractions for a layer with thickness of are
If, and are known for a layer with thickness, the ART fractions for a layer with thickness of are
and the fractions for a layer with thickness of are
If, and are known for layer and and are known for layer, the ART fractions for a sample composed of layer and layer are

Giovanelli and Chandrasekhar

In 1955, Ron Giovanelli published explicit expressions for several cases of interest which are touted as exact solutions to the radiative transfer equation for a semi-infinite ideal diffuser. His solutions have become the standard against which results from approximate theoretical treatments are measured. Many of the solutions appear deceptively simple due to the work of Subrahmanyan Chandrasekhar. For example, the total reflectance for light incident in the direction μ0 is
Here is known as the albedo of single scatter, representing the fraction of the radiation lost by scattering in a medium where both absorption and scattering take place. The function is called the H-integral, the values of which were tabulated by Chandrasekhar.

Gustav Kortüm

was a physical chemist who had a broad range of interests, and published prolifically. His research covered many aspects of light scattering. He began to pull together what was known in various fields into an understanding of how “reflectance spectroscopy” worked. In 1969, the English translation of his book entitled Reflectance Spectroscopy was published. This book came to dominate thinking of the day for 20 years in the emerging fields of both DRIFTS and NIR Spectroscopy.
Kortüm's position was that since regular reflection is governed by different laws than diffuse reflection, they should therefore be accorded different mathematical treatments. He developed an approach based on Schuster's work by ignoring the emissivity of the clouds in the "foggy atmosphere". If we take as the fraction of incident light absorbed and as the fraction scattered isotropically by a single particle, and define the absorption and isotropic scattering for a layer as and then:
This is the same "remission function" as used by Judd, but Kortüm's translator referred to it as "the so-called reflectance function". If we substitute back for the particle properties, we obtain and then we obtain the Schuster equation for isotropic scattering:
Additionally, Kortüm derived "the Kubelka-Munk exponential solution" by defining and as the absorption and scattering coefficient per centimeter of the material and substituting: and, while pointing out in a footnote that is a back-scattering coefficient. He wound up with what he called the "Kubelka–Munk function", commonly called the Kubelka–Munk equation:
Kortüm concluded that "the two constant theory of Kubelka and Munk leads to conclusions accessible to experimental test. In practice these are found to be at least qualitatively confirmed, and suitable conditions fulfilling the assumptions made, quantitatively as well."
Kortüm tended to eschew the "particle theories", though he did record that one author, N.T. Melamed of Westinghouse Research Labs, "abandoned the idea of plane parallel layers and substituted them with a statistical summation over individual particles."