The most interesting and fundamental research was carried out in our laboratory on the modification and improvement of the two-flux Kubelka-Munk (KM) approach. It allowed us to prove, contrary to the well-known opinion in publications on its limited applicability, that with the use of the improved KM method it is possible to obtain absolutely rigorous and strict analytical solutions for fluxes on boundaries of light-scattering media in the one-dimensional (1D) problem. It became possible by means of small reformulation of initial differential KM equations and by means of more correct initial definition of local optical properties of the medium. It was shown that the main problem in the KM approach as well as in the general Radiative Transport Theory (RTT) consisted in a wrong phenomenological assumption of the existence of two independent optical properties of turbid media - absorption and scattering coefficients. In the general case of a turbid medium with continuous absorption and discrete scattering, when both the absorption and scattering of light exist, the first coefficient in the right side of KM equations cannot be separated in two independent coefficients - absorption and scattering ("K" and "S" in the KM notations; “mu_a” and “mu_s” in the RTT notations) - and must be considered as one united attenuation coefficient “AC”. The absorption coefficient "K" (mu_a) is included into “AC” as well as into “S” (mu_s), but not additively. Without absorption K=0 and AC=S; without scattering S=0 and AC=K, like it must be in the classical RTT, but if both absorption and scattering phenomenon are presented in the medium, then the classical phenomenological assumption AC=K+S is wrong (see our papers in Proc. SPIE here and in Biomedical Engineering here). It is the main source of errors of the classical KM approach if the boundary fluxes are calculated. Only if the absorption in a medium is small, much less than the scattering, or the single scattering approximation is used, then the classical assumption can take place. Moreover, such modification of KM equations showed us an equivalence between absorption coefficients in both theories (RTT and KM), i.e. allowed us to estimate exactly that K=mu_a, what is contrary to the well-known literature data, as well. But in our opinion, it is more reasonable than K=2*mu_a, which is widely used today in a majority of publications. Thus, in our opinion, this small mistake in the classical KM approach, which existed in the literature around 70 years (!) and which has led to errors in calculations, was found out and was corrected by us. Moreover, this new exact solution of a 1D light scattering problem has forced to reflect a correctness of some definitions of transport optical properties of turbid media in the classical RTT. On its basis it has been shown that the more absorption presents in the medium, the more erroneous values for optical coefficients turn out at the usage of classical approach and definitions. First of all, a revision has to be made for a definition of scattering coefficient and for albedo. Scattering coefficient turn out to be not only the function of optical properties of the medium, but depends on the selected mathematical approach, as well. Moreover, unlike the classical assumption by A.Ishimaru, the scattering coefficient of the sum of scatterers is not equal to the sum of the scattering coefficients of each scatterer. For more information on this topic see our article in three parts here, here and here. The Monte Carlo simulation, by the way, without taking these features into account, also leads to errors in calculations, contrary to the established opinion that this is the exact numerical method (see our articles here and here). Errors are also possible when using classical RTE to simulate the emission of fluorescence in the medium. To calculate the fluorescence-radiation field, the knowledge of the absorption coefficient in the medium for the excitation radiation absorption is required. However, for non-separable coefficients of equations this task has some peculiarity. See our publication in Journal of Fluorescence (2015) here.

     In the general case, the approach proposed by us turned out to be promising for solving a number of 2D and 3D scattering problems? as well. Today, it has been used to obtain the exact analytical solution of the scattering problem in the approximation of a narrow (pencil-like) illuminating beam and under a single scattering approximation (see our last article here). In addition, based on all these solutions, an improved and more theoretically substantiated approach to the description of spatial light scattering by an elementary light-scattering volume of a biological tissue was proposed. The change in the shape of the scattering phase function from sets of scatterers in an elementary light-scattering volume was shown in comparison with the phase function of a single scatterer (see details here and here). A new form of the transport equation for the problem of 2D orthogonal scattering is derived and justified (see details here ).
     At one of the stage of our research it was additionally shown that the mathematical formalism of the Markovian processes theory in application to the "photon-migration" problem in the general RTT for a perfectly scattering media (without absorption) allows one to obtain one simple, important, closed-form and analytical solution, as well (see
here). The solution has, in its turn, a number of important consequences for the next development of the light scattering theory and its experimental study. For example, one of them consists in follows: if the power of incident light is very small and a photodetector has a short time resolution, then the detected amount of photons will be different from sampling to sampling because of a stochastic nature of the photon migration and scattering. It will cause a random amplitude modulation of the measured photocurrent that can probably be an additional technique to measure the scattering optical properties of turbid media.
     Development of the classical theory of diffraction of electromagnetic waves on a randomly-rough surfaces with reference to the problems of light scattering by rough surface of biological tissues has allowed us to develop the theoretical foundations of classic photometry regarding a description of photometric indicatrix of reflection and scattering, including for a beam of radiation passing through rough dielectric boundary into a medium. On the basis of this approach it has appeared possible to show theoretically that Lambertian character of light scattering can be simulated by a randomly-rough and perfect conducting metal surface with the Gaussian correlation function of heights of a roughness. Previously, the Lambert law in a photometric literature was considered as not having a reliable theoretical substantiation. In application to theoretical problems of biomedical optics and laser noninvasive diagnostics the analytical expression for indicatrix of transmitted radiation for a randomly-rough dielectric surface  in a case of a normal incidence of radiation has been obtained (see our paper
here). This solution allows anyone to set competently boundary conditions in multidimensional problems of RTT.
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