P₃ solution for the total steady-state and time-resolved reflectance and transmittance from a turbid slab

Abstract

In this paper, we derive some explicit analytical solutions to the P 3 equations for the slab geometry that is illuminated by a collimated plane source. The resulting expressions for the total reflectance and transmittance are compared with the corresponding transport theory solution predicted by the Monte Carlo method. Further, for the special case of a non-absorbing anisotropically scattering slab, simple and accurate expressions in the P 1 approximation are obtained, yielding for optically thick slabs, the typical behavior of Ohm's law. In view of the time domain, we present an alternative method to the classical frequency-domain approach avoiding the use of complex numbers.

Experimental imaging and Monte Carlo modeling of ultrafast pulse propagation in thin scattering slabs

“… 47 Additionally, ongoing progress is being made with a number of analytical and semi-analytical methods to solve the radiative transfer equation (RTE) in the time domain, some of which can also be applied to model the transmittance through bounded geometries. Efficient solutions exist for the time-domain transmittance through a slab geometry, 48 while other solutions are being developed following the rotated reference frame method, which however are used to study either the spatial domain or semi-infinite geometries 49 , 50 and may still exhibit small deviations for samples with . Computationally efficient solutions have been also reported using a colocated radial basis function method, 51 the pseudospectral method, 52 or within the framework of the broad class discrete ordinate methods, 53 e.g., by combining it with spectral methods and Laplace transforms, 54 convection schemes, 55 finite-volume, 56 or Fourier continuation methods.…”

A Photosynthetic Light Acclimation Model Accounting for the Effects of Leaf Age, Chlorophyll Content, and Intra-Leaf Radiation Transfer

“…This approach assumes perfectly diffused radiation streams, with equal probability of backward and forward scattering of photons, setting the anisotropy parameter for scattering g to zero. However, an accurate approximation to the radiation transfer equation for a scattering and absorbing slab was recently derived ( Liemert et al, 2019 ). This solution could be a useful asset in improving parameter calibration of Equation 2 or similar functions, which eventually can lead to the derivation of better approximations; even different incidence angles and refraction index changes at the leaf surface can be accounted for Liemert et al (2019) .…”

“…However, an accurate approximation to the radiation transfer equation for a scattering and absorbing slab was recently derived ( Liemert et al, 2019 ). This solution could be a useful asset in improving parameter calibration of Equation 2 or similar functions, which eventually can lead to the derivation of better approximations; even different incidence angles and refraction index changes at the leaf surface can be accounted for Liemert et al (2019) . For that, an independent spectral parameterization of the anisotropy parameter g(λ) (or the scattering phase function) would be required.…”

Broadband Time Domain Diffuse Optical Reflectance Spectroscopy: A Review of Systems, Methods, and Applications

“…This approximation, in turns, brings to a coupled equation between Φ(r, t) (the fluence) and J(r, t) (the flux), whose solution allows one to derive the radiance, which in turn help the derivation of the signal transmitted or reflected. Solutions to the P3 approximation have been recently derived in a semi-infinite medium [79]. Moreover, by introducing the approximations of (…”