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 (DRIFTS) and diffuse reflectance ultraviolet–visible spectroscopy.

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. They 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. 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 (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". 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.

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".

The Kubelka–Munk equation describes the remission from a sample composed of an infinite number of infinitesimal layers, each having a0 as an absorption fraction, and r0 as a remission fraction.

Deane 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 Sd, where d 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 a0/r0 in terms of R∞. This led to a very early (perhaps the first) use of the term "remission" in place of "reflectance" when Judd defined a "remission function" as , where k and s are absorption and scattering coefficients, which replace a0 and r0 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.

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 in 1954.

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.

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 (long in preparation and translation) 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 (or specular) 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 α as the fraction of incident light absorbed and σ as the fraction scattered isotropically by a single particle (referred to by Kortüm as the "true coefficients of single scatter"), 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 k and s as the absorption and scattering coefficient per centimeter of the material and substituting: K ≡ 2k and S ≡ 2s, while pointing out in a footnote that S 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."

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. Their plan was to "start with a simple model, treat the problem numerically rather than analytically, then look for analytical functions that describe the numerical results. Assuming success with that, the model would be made more complex, allowing more complex analytical expressions to be derived, eventually, leading to an understanding of diffuse reflection at a level that appropriately approximated particulate samples." Successful application of theory to a real world sample using the mathematics of plane parallel layers requires assigning properties to the layers that are representative of the sample as a whole (which does not require extensively reworking the mathematics). Such a layer was termed a representative layer, and the theory was termed the representative layer theory. The decadic absorbance of a scattering sample is defined as −log10(R+T) or −log10(1−A). For a non scattering sample, 1=R = 0, and the expression becomes −log10T or , which is more familiar. In a non scattering sample, the absorbance has the property that the numerical value is proportional to sample thickness. Consequently, a scatter corrected absorbance might reasonably be defined as one that has that property. If one has measured the remission and transmission fractions for a sample, Rs and Ts, then the scatter corrected absorbance should have half the value for half the sample thickness. By calculating the values for R and T for successively thinner samples using the Benford's equations for half thickness, a place will be reached where, for successive values of n (0,1,2,3, ), the expression 2^(n) (−log(R+T)) becomes constant to within a some specified limit, typically 0.01 absorbance units. This value is the scatter corrected absorbance.

In spectroscopy, remission refers to the reflection or back scattering of light by a material. While seeming similar to the word "re emission", it is the light which is scattered back from a material, as opposed to that which is "transmitted" through the material. The word "re emission" connotes no such directional character. Based on the origin of the word "emit", which means "to send out or away", "re emit" means "to send out again", "transmit" means "to send across or through", and "remit" means "to send back".

In spectroscopy, the term "plane parallel layers" may be employed as a mathematical construct in discussing theory. The layers are considered to be semi infinite. (In mathematics, semi infinite objects are objects which are infinite or unbounded in some, but not all, possible ways.) Generally, a semi infinite layer is envisioned as a being bounded by two flat parallel planes, each extending indefinitely, and normal (perpendicular) to the direction of a collimated (or directed) incident beam. The planes are not necessarily physical surfaces which refract and reflect light, but may just describe a mathematical plane, suspended in space. When the plane parallel layers have surfaces, they have been variously called plates, sheets, or slabs.

The term "representative layer" refers to a hypothetical plane parallel layer that has properties relevant to absorption spectroscopy that are representative of a sample as a whole. For particulate samples, a layer is representative if each type of particle in the sample makes up the same fraction of volume and surface area in the layer as in the sample. The void fraction in the layer is also the same as in the sample. Implicit in the representative layer theory is that absorption occurs at the molecular level, but that scatter is from a whole particle.