MATHEMATICAL MODELS
OF LINEAR INTERACTION
OF LASER RADIATION WITH DIFFERENT MEDIA
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On one simple and analytical “photon-migration”
solution in 1-D light scattering theory
The mathematical formalism of the Markov processes theory in application to the "photon-migration" task in the general light scattering and transport theory for a perfectly turbid media allows anyone to obtain one simple, important, closed-form and analytical solution. The solution has two important consequences for the next development of the light scattering theory and its experimental study. The first consequence for the general theory of light propagation in turbid media relates to the question of the right understanding of the term of transport scattering coefficient. The second one consists in follows: if the power of light source 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 process. It will cause of amplitude modulation of the measured photocurrent that can be an additional technique to measure a scattering optical properties of the turbid media.
For more information, please, see our paper:
1. Rogatkin
D.A., Tchernyi V.V., On one simple and analytical photon-migration solution in
1-D light scattering theory and its consequences for the laser medical
diagnostic problems / Proc. SPIE, v.5319, 2004. – pp. 385-390.
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THEORY OF DIFFRACTION OF ELECTROMAGNETIC WAVES BY RANDOMLY ROUGH SURFACES IN APPLICATION TO THE OPTICS, PHOTOMETRY AND TRANSPORT THEORY
A progress of the classic theory of the electromagnetic waves diffraction by randomly rough surfaces in application to the problem of light scattering by randomly rough boundary surfaces of different materials and media allows to extend and supplement the theoretical basis of classic photometry, namely, in a part of formulation and description of the photometric scattering and reflecting indicatrix and, in that number, for the fluxes penetrating the medium through the impedance rough boundary interface. It was shown, for example, that the Lambertian scattering can be modeled exactly by a randomly rough metal surface with random Gaussian fluctuations of roughness and a perfect conductivity. The exact solution for the transmitting indicatrix in the case of randomly rough dielectric surfaces and a normal incidence of the optical beam was obtained as well.
The more recent and completed information you can find in the publication by D.A.Rogatkin "Scattering of electromagnetic waves by a randomly rough surface as a boundary problem of laser radiation interaction with light-scattering materials and media", published in Russian translated journal "Optics and spectroscopy", v.97, No.3, 2004. - pp.455-463.
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For more information, please, see our publication:
1. Rogatkin D.A., Bulavskii Yu.V., Konyakhin V.V. The theoretical analysis of opto-physical reflectance parameters for perfect conducting and rough surfaces // Prep. of Sc. Acad. of the USSR, Siberian Dep. – Novosibirsk, 1989 (in Russian).
2.
Rogatkin D.A., Tchernyi V.V. "Scattering of electromagnetic waves by
rough surfaces as a part of interaction of laser light with
biotissues", SPIE Proc. “Laser-Tissue Interaction XIII”,
vol. 4617, 2002. – p. 257-266.
3. Rogatkin D.A. Metal sphere as a standard reflector for goniophotometer // Optical Journal, No.9, 1992. - pp.72-74.
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DEVELOPMENT AND MODIFICATION OF THE FLUXES KUBELKA-MUNK THEORY
It is proposed a general modification of the Kubelka-Munk two-fluxes approach in the general
theory of light scattering and
propagation in turbid media. The Kubelka-Munk (KM) two-flux light transport one-dimensional (1D)
model is, evidently, the most widely used Radiative Transport Theory (RTT)
approach in a modern optics of scattering and turbid media because of its
simplicity and the existence of its clear analytical solution. Moreover, the KM
approach is the best approximation of the general radiative transport equation (TE)
in the case of 1D theoretical tasks. But it is well known from the literature
that the KM model doesn’t allow anyone to obtain an exact solution, especially
in the case of highly-absorbing and weakly-scattering media. In the most of
publications it is assumed that the light must be diffuse on a surface as well
as within the medium for a correct application of the KM equations. However, in
our opinion, there are no any reasons to separate light radiation on diffuse and
collimated components for a simple 1D theoretical model. In a 1D theoretical
task there are no any differences between diffuse and collimated beams because
both of them penetrates a medium along 1D “x”-axis only. So, a root of the
problem may be located in a far another field. To find out any reasons of the 1D
KM approach's low accuracy we have examined the general KM equations on the
theoretical model of 1D scattering and absorbing medium, for which an exact
photometric analytical solution of the light scattering problem can be obtained.
It was estimated, that the classical two-flux KM model could give absolutely
exact values for the boundary fluxes (transmitted and backscattered radiation)
for any cases of scattering and absorbing media by means of small reformulation of initial differential KM equations and
by means of more correct initial definition of the transport optical properties
of media. It was shown that the main problem of the KM approach as well as of
the general RTT consisted in a wrong phenomenological assumption of the
existence of two independent optical transport properties of media - absorption
and scattering. In the general case of a turbid medium, where both the
absorption and the scattering of light is presented, a first coefficient in the
right side of the TE and/or KM equations cannot be separated into the two
independent transport coefficients - absorption and scattering ("K"
and "S" in the KM notations; “mu_a” and “mu_s”
in the TE notations) - and must be considered as one, united attenuation
coefficient “AC”. The absorption transport property "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 an absorption and scattering are presented,
then the classical phenomenological assumption AC=K+S is wrong. It is the main
source of errors of the KM approach, for example, if the boundary fluxes are
calculated. Only if the absorption into a medium is small, more less than
scattering, then the classical assumption can take place. Moreover, such
modification of the KM equations showed us an equivalence between absorption
coefficients in both theories, 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,
is more reasonable than K=2*mu_a, what is widely used today in a lot of
publications. Thus, in our opinion, it was found out and
was corrected a small mistake it the Kubelka-Munk approach, which has existed in
the literature around 70 years (!) and has led to errors in calculations.
The more recent and completed information you can find out in our papers:
L.G. Lapaeva, D.A. Rogatkin "Improved Kublka-Munk approach for determination of tissues' optical properties in biomedical noninvasive reflectance spectroscopy", Proc. SPIE, vol. 6536, 65360Z, 2007.
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D.A.Rogatkin "A specific feature of the procedure for determination of optical properties of turbid biological tissues and media in calculation for noninvasive medical spectrophotometry", Biomedical engineering, v.41, No. 2, 2007. – p.59-65.
For more information, please, see our previous publications:
1. Rogatkin D.A., Svirin V.N., Tchernyi V.V. "Problems of theoretical calculation of the laser light distributions in biological tissues", SPIE Proc., v.4162, 2000. - pp.189-193.
2. Dmitriev M.A., Feducova M.V., Rogatkin D.A. "On one simple backscattering task in the general light scattering theory", SPIE Proc., v.5475, 2004. - pp.115-122.
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3. Rogatkin D.A., "Development of the two-fluxes Kubelka-Munk approach to solve 1D task of the light transport into biological scattering tissues and media", Optics and spectroscopy, v.87, No. 1, 1999. - pp.109-114.
The extending of 1D two-fluxes Kubelka-Munk approach up to the two- or three-dimensional scattering problems (2D, 3D, etc.) allows anyone to develop a new approach to formulate a multidimensional (spatial) task of light scattering in the 2D or 3D turbid media. In this case new initial differential equations (system of equations) of the problem can be yielded. General way and the general idea of that were published in our paper:
D. A. Rogatkin, “An approach to the solution of multidimensional problems of the theory of light scattering in turbid media”, Quantum Electronics, vol. 31, No.3, pp. 279-281, 2001.
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Thus, the generally improved two-flux Kubelka-Munk approach shows that there are in the transport theory a number of
incorrect-understandable definitions of transport optical properties of turbid
media. For example, - the definition of the scattering properties of the
medium as well as the definition of albedo
of that. So, we have studied
more detailed a difference between classic and improved values for transport
coefficients and terms mentioned above. Some results of the study was presented
in 2009 at PIERS'2009 Moscow Symposium. In
our opinion, we have obtained quite interesting results...
Rogatkin D.A., Tchernyi V.V., "Revised optical properties of turbid media on a base of general improved two-flux Kubelka-Munk approach" // Abstr. book of the Progress in Electromagnetics Research Symposium “PIERS’2009”, August 18-21, Moscow, Russia, 2009. - p.385.
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DEVELOPMENT OF METHOD OF MOMENTS TO SOLVE THE GENERAL INTEGRAL-DIFFERENTIAL TRANSPORT EQUATION
Development of Method of Moments to solve the general classic integral-differential Transport equation for the problem of biological layered scattering media with fluorescence is proposed. The analytical exact solution of the problem has been obtained. Method of Moments is considered today as the most exact method in the Radiative Transport theory because it doesn't need any limits on the optical properties of medium and can give the solution with any level of accuracy by means of taking into account more higher order of members of decomposition of the initial Transport equation. For more information, please, see our paper:
Hachaturian G.V.,
Rogatkin D.A. "Methods of Moments in Calculation of the
Autofluorescence of Biological Tissues". // Optics and Spectroscopy,
v.87, No. 2, 1999. – p.240 -246.
Have a look at this paper (Russian version) in PDF (220Kb)
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