Diffuse reflection theory

Guthrie, A.J., Narayanaswamy, R. and Russell, D.A. (1988). Application of Kubelka-Munk diffuse reflectance theory to optical fibre sensors. Analyst 113,457-461. Narayanan, S. (1989). Principles and Applications of Laboratory Instrumentation. ASCP Press, Chicago, pp. 192-195. 

Provides new chapters on diffuse reflection theories, the calibration and validation of process sensors, and the NIR spectra of gases... 

It can also happen, and is perhaps more common, in the case of diffuse reflectance. In that measurement technique, absent a rigorous theory to describe this physical phenomenon, the concept of a variable pathlength is used as a first approximation to the nature of the change in the measurements. 

There are other sources of noise, whose behavior cannot be described analytically. They are often principally due to the sample. A premier example is the variability of the measured reflectance of powdered solids. Since we do not have a rigorous ab initio theory of diffuse reflectance, we cannot create analytic expressions that describe the variation of the reflectance. Situations where the sample is unavoidably inhomogeneous will also fall into this category. In all such cases the nature of the noise will be unique to each situation and would have to be dealt with on a case-by-case basis. 

III. THEORY ASSOCIATED WITH DIFFUSE REFLECTANCE MEASUREMENTS... 

The theory associated with diffuse reflectance has been developed in great detail, and its full exposition is beyond the scope of this chapter. Interested readers are referred to the texts by Kortum 1], and by Wendlandt and Hecbt [2], where the general theory is presented in sufficient detail. The most generally accepted theory concerning diffuse reflectance was developed by Kubelka and Munk [3,4],... 

The Kubelka-Munk theory treats the diffuse reflectance of infinitely thick opaque layers [4], a situation achieved in practice for UV/VIS spectroscopy through the use of powder path lengths of at least several millimeters. In this instance, the Kubelka-Munk equation has the form... 

Spectra of solid samples are usually recorded in the units of reflectance (R) or percent reflectance (%/ ), which is analogous to percent transmittance in that reflectance equals the ratio of the reflected radiation to the incident radiation. With diffuse reflectance, the reflected signal is attenuated by two phenomena absorption (coefficient k) and scattering (coefficient s). Lollowing the Kubelka-Munk theory, these two coefficients are related to the reflectance of an infinitely thick sample, by... 

While this is not the place for a complete description of the theory, uses, and ramihcations of all possible processing treatments that can be applied to chemical imaging data, the majority of NIR chemical images recorded to date have been obtained from solid or powder samples, measured predominantly by diffuse reflectance. As such, the following discussions may serve as a general outline of the more common procedures and their applications to these types of imaging data. 

PR. Griffiths and l.M. Olinger, Continnnm theories in diffuse reflection, in Handbook of Vibrational Spectroscopy, l.M. Chalmers and P.R. Griffiths (Eds), Sampling Techniques Vol. 2, John Wiley Sons, Ltd, Chichester, 2002. 

J.M. Olinger, PR. Griffiths and T. Burger, Theory of diffuse reflectance in the NIR region. In Handbook of Near-Infrared Analysis, 2nd edition, D. Burns and E.W. Ciurczak (eds), Marcel Dekker, New York, 19-52, 2001. 

Opaque minerals like iron oxides are frequently examined in the reflectance mode - and usually give diffuse reflectance spectra. Reflectance spectra provide information about the scattering and absorption coefficients of the samples and hence their optical properties. The parameters of reflectance spectra may be described in four different ways (1) by the tristimulus values of the CIE system (see 7.3.3) (2) by the Kubelka-Munk theory and (3) by using the derivative of the reflectance or remission function (Kosmas et al., 1984 Malengreau et ak, 1994 1996 Scheinost et al. 1998) and, (4) more precisely, by band fitting (Scheinost et al. 1999). 

The dielectric function of a metal can be decomposed into a free-electron term and an interband, or bound-electron term, as was done for silver in Fig. 9.12. This separation of terms is important in the mean free path limitation because only the free-electron term is modified. For metals such as gold and copper there is a large interband contribution near the Frohlich mode frequency, but for metals such as silver and aluminum the free-electron term dominates. A good discussion of the mean free path limitation has been given by Kreibig (1974), who applied his results to interpreting absorption by small silver particles. The basic idea is simple the damping constant in the Drude theory, which is the inverse of the collision time for conduction electrons, is increased because of additional collisions with the boundary of the particle. Under the assumption that the electrons are diffusely reflected at the boundary, y can be written... 

The generally accepted theory of diffuse reflectance was developed originally by Kubelka and Munk [43,44] for application to infinitely thick, opaque layers. 

Griffiths, P.R. 8t Olinger, J.M., Continuum Theories in Diffuse Reflection. In Chalmers, J.M. 8c Griffiths, P.R. (eds) Handbook of Vibrational Spectroscopy, Sampling Techniques, Volume 2 John Wiley 8t Sons Chichester, 2002 pp. 1125-1139. 

Farrell TJ, Patterson MS, Wilson B. A diffusion-theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo. Medical Physics 1992, 19, 879-888. 

In fluorescence spectroscopy, however, diffuse reflectance correction of spectral distortions in biological media has been studied extensively. Analytical models based on photon migration theory,44 diffusion theory46,60,61 as well as empirical models,62 have been reported to obtain intrinsic fluorescence. In the following, we will review a particular correction method based on photon migration theory for fluorescence spectroscopy and introduce its Raman counterpart. 

The Kubelka-Munk theory of diffuse reflectance is a good description of the optical properties of paper. The two parameters of the theory, absorption and scattering coefficient, are purely phenomenological, but are closely related to basic properties of paper. The absorption coefficient is approximately a linear function of the chrcmgphore concentration in the paper. The scattering coefficient is related to the nonbonded fiber surface area in the paper, or the area "not in optical contact," and the Fresnel reflectivity of that surface. 

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