Infinite space Green’s function of the time-dependent radiative transfer equation

André Liemert,Alwin Kienle

doi:10.1364/boe.3.000543

Abstract

This study contains the derivation of an infinite space Green’s function of the time-dependent radiative transfer equation in an anisotropically scattering medium based on analytical approaches. The final solutions are analytical regarding the time variable and given by a superposition of real and complex exponential functions. The obtained expressions were successfully validated with Monte Carlo simulations.

Highlights

The radiative transfer equation (RTE), an integro-partial differential equation, is widely used in many areas of physics to model the propagation of waves in random scattering media
Due to this fact the RTE is usually solved by numerical approaches such as the Monte Carlo method [5], the discrete ordinate method [6], the finite-element method [7] or is approximated by the diffusion equation [8]
Monte Carlo simulations are in the limit of an infinitely large number of simulated photons an exact solution of the RTE

Summary

Introduction

The radiative transfer equation (RTE), an integro-partial differential equation, is widely used in many areas of physics to model the propagation of waves in random scattering media. Examples are the light propagation in biological tissue [1], the neutron transport theory [2, 3], the electromagnetic waves propagation in plasmas or in the atmosphere This equation is used since many decades complete analytical solutions for the steady-state and time domains are only available for the simplest cases [2,3,4]. By performing numerically an inverse Fourier transform of the solution in the frequency domain the solution of the time-dependent RTE is obtained This procedure involves in many cases difficulties due to the fact that the Green’s function of the steady-state RTE is already not a square integrable function [10]. The derived Green’s function cannot only be used for applications in infinitely extended scattering media but for all applications where the well-known approximated solution obtained from the diffusion equation is used [13], as was demonstrated for the case of isotropic scattering [14]

Theory

Numerical implementation

Results

Conclusion