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Exact and approximate analytical solutions of the two- und three-dimensional radiative transfer equation

Subject Area Optics, Quantum Optics and Physics of Atoms, Molecules and Plasmas
Term from 2015 to 2024
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 284841045
 
Final Report Year 2024

Final Report Abstract

Within the project, we have obtained numerous new analytical solutions concerning the twoand three-dimensional radiative transfer equation for different geometries and in all relevant measurement domains. In this context, we could partly use the preliminary work we have performed in the past, but also new analytical approaches such as the method of characteristic has been successfully applied. Although this method itself is well known in the field of partial differential equations (especially for solving first order equations), its application in the light of the radiative transport equation (subject to reflection boundary conditions) seems not to have been considered until now. This method is particularly interesting for the determination of the angular radiance as function of the scattering order. In this context, we could derive the first analytical solutions for the case of a three-dimensional reflecting half-space. The corresponding publication has been selected as Editors’ Pick in the Journal of the Optical Society of America (JOSA A). ’Editor’s Picks serve to highlight articles with excellent scientific quality and are representative of the work taking place in a specific field.’ In general, the presented solutions are general in view of the incident direction (obliquely incident sources) and the scattering phase functions (e.g. obtained from Mie theory). Furthermore, is also possible to describe the light propagation of fluorescence, phosphorescence or Raman-scattered light, which is important for a lot of applications, such as in microscopy or drug investigations in small animal studies. Besides this, the methodologies can also be applied in the context of the correlation RTE, with which the light propagation in turbid media containing moving particles can be desribed. We have also extended our analysis to scattering media that exhibits arbitrary coefficient functions. The first step was the derivation of an appropriate diffusion model that can be evaluated quite fast. Furthermore, in view of the radiative transport equation, we have derived the first analytical solutions for the single-scattered radiance (in the spatial frequency domain as well as in space domain) in scattering media with depth-dependent optical properties. We furthermore can report about new results concerning the mean-square displacement (MSD). Studies of the mean-square displacement are prevalent in many scientific areas, such as in heat conduction, in ballistic and diffusive Brownian motion, in light transport in clouds or in glaciers, in transport in disordered media, in animal movement and in further stochastic problems in physics, chemistry, and electrical engineering. The MSD of particles in an infinitely extended random medium has been studied for more than 100 years, where Einstein derived his famous (approximated) formula in the time domain investigating Brownian motion. In this project, we could derive the exact transport theory MSD in the steady state and time domain as a function of the number of scattering events. In addition to the program of the proposal, we were also able to derive analytical solutions of the vector radiative transport equation, which are applicable for describing the propagation of polarized light. It should also be mentioned that we could perform several important steps forward concerning the classical three-dimensional transport equation. Based on our earlier work, it was possible to evaluate several quantities of practical importance such as the fluence and the reflectance. The evaluation of the angle-resolved radiance, the main quantity of the RTE, near boundaries and/or sources was difficult due to numerical instabilities. Within the project, we took the first steps in resolving this problem by making use of a hybrid ansatz as well as a completely new solution approach. To find out the improvements obtained with the new methods, we have implemented these approaches for the case of isotropic scattering media. We have observed very promising results that are in perfect agreement with the Monte Carlo method. A further important aspect is that parts of these (analytical) methods can again be used within other approaches such as the Monte Carlo method. It is planned to extend this solution approach to the important case of anisotropic scattering within a future project.

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