Modeling fluorescence reemission in the one-dimensional radiative
transfer problem using the P3 approximation

Luke J. Sandilands

doi:10.1364/JOSAA.434354

Journal of the Optical Society of America A
Vol. 38,
Issue 11,
pp. 1631-1639
(2021)
•https://doi.org/10.1364/JOSAA.434354

Modeling fluorescence reemission in the one-dimensional radiative transfer problem using the P3 approximation

Luke J. Sandilands

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Luke J. Sandilands, "Modeling fluorescence reemission in the one-dimensional radiative transfer problem using the P3 approximation," J. Opt. Soc. Am. A 38, 1631-1639 (2021)

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Table of Contents Category
- Scattering and Propagation
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About this Article
History
- Original Manuscript: June 23, 2021
- Revised Manuscript: September 22, 2021
- Manuscript Accepted: September 24, 2021
- Published: October 18, 2021

Abstract

A model of radiative transport in fluorescent, scattering media that accounts for fluorescence reabsorption and reemission effects is discussed. The model is studied in a simplified one-dimensional geometry using the P3 approximation. An example calculation of a model system, sintered polytetrafluoroethylene doped with rhodamine 6G, is used to illustrate the features of the model.

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Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Equations (18)

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Quantity	Symbol	Units
Wavelength	$λ$	nm
Position inside slab	$\vec{r}$	cm
Direction inside slab	$\hat{s}$	Dimensionless
Distance from slab front surface	$z$	cm
Angular coordinates	$θ$ , $ϕ$	Radian
$c o s θ$	$μ$	Dimensionless
Spectral radiance	$L$	$W m^{- 2} n m^{- 1} s r^{- 1}$
Total attenuation coefficient	$σ_{t}$	$c m^{- 1}$
Absorption coefficient	$σ_{a}$	$c m^{- 1}$
Scattering coefficient	$σ_{s}$	$c m^{- 1}$
Scattering phase function	$f$	$s r^{- 1}$
Fluorophore absorption coefficient	$σ_{f}$	$c m^{- 1}$
Fluorescence spectral profile	$q / 4 π$	$n m^{- 1} s r^{- 1}$
Wavelength interval	$Δ λ$	nm
Excitation irradiance	$E_{o}$	$W m^{- 2}$
Legendre coefficients of $L$	$L_{n}$	$W m^{- 2} n m^{- 1}$
Legendre coefficients of $f$	$f_{n}$	Dimensionless
Spectral radiance vector	$\vec{l}$	$W m^{- 2} n m^{- 1}$
Source vector	$\vec{s}$	$W m^{- 2} n m^{- 1} c m^{- 1}$
Matrix of differential equation coefficients	$D$	$c m^{- 1}$
Matrix of the eigenvectors of $D$	$Φ$	$W m^{- 2} n m^{- 1}$
Eigenvalues of $D$	$ϵ_{k}$	$c m^{- 1}$
Diagonal matrix with elements exp [ $ϵ_{k} z$ ]	$E$	Dimensionless
Coefficient vector	$\vec{c}$	Dimensionless
Particular solution vector	${\vec{l}}_{p}$	$W m^{- 2} n m^{- 1}$

Quantity	Symbol	Units
Scattering anisotropy parameter of sintered PTFE	$g$	Dimensionless
Intrinsic absorption coefficient of sintered PTFE	$σ_{a o}$	$c m^{- 1}$
Molar absorption coefficient of rhodamine 6G	$ϵ$	${c m}^{2} m o l^{- 1}$
Concentration of fluorophore rhodamine 6G	$n$	$c m^{- 3} m o l$
Hemispherical reflectance	$R$	Dimensionless
Excitation–emission matrix	$β$	$n m^{- 1}$
Apparent quantum yield	AQY	Dimensionless
Cosine of the spectral angle	cos $θ_{s}$	Dimensionless

Quantity	Symbol	Units
Wavelength	$λ$	nm
Position inside slab	$\vec{r}$	cm
Direction inside slab	$\hat{s}$	Dimensionless
Distance from slab front surface	$z$	cm
Angular coordinates	$θ$ , $ϕ$	Radian
$c o s θ$	$μ$	Dimensionless
Spectral radiance	$L$	$W m^{- 2} n m^{- 1} s r^{- 1}$
Total attenuation coefficient	$σ_{t}$	$c m^{- 1}$
Absorption coefficient	$σ_{a}$	$c m^{- 1}$
Scattering coefficient	$σ_{s}$	$c m^{- 1}$
Scattering phase function	$f$	$s r^{- 1}$
Fluorophore absorption coefficient	$σ_{f}$	$c m^{- 1}$
Fluorescence spectral profile	$q / 4 π$	$n m^{- 1} s r^{- 1}$
Wavelength interval	$Δ λ$	nm
Excitation irradiance	$E_{o}$	$W m^{- 2}$
Legendre coefficients of $L$	$L_{n}$	$W m^{- 2} n m^{- 1}$
Legendre coefficients of $f$	$f_{n}$	Dimensionless
Spectral radiance vector	$\vec{l}$	$W m^{- 2} n m^{- 1}$
Source vector	$\vec{s}$	$W m^{- 2} n m^{- 1} c m^{- 1}$
Matrix of differential equation coefficients	$D$	$c m^{- 1}$
Matrix of the eigenvectors of $D$	$Φ$	$W m^{- 2} n m^{- 1}$
Eigenvalues of $D$	$ϵ_{k}$	$c m^{- 1}$
Diagonal matrix with elements exp [ $ϵ_{k} z$ ]	$E$	Dimensionless
Coefficient vector	$\vec{c}$	Dimensionless
Particular solution vector	${\vec{l}}_{p}$	$W m^{- 2} n m^{- 1}$

Quantity	Symbol	Units
Scattering anisotropy parameter of sintered PTFE	$g$	Dimensionless
Intrinsic absorption coefficient of sintered PTFE	$σ_{a o}$	$c m^{- 1}$
Molar absorption coefficient of rhodamine 6G	$ϵ$	${c m}^{2} m o l^{- 1}$
Concentration of fluorophore rhodamine 6G	$n$	$c m^{- 3} m o l$
Hemispherical reflectance	$R$	Dimensionless
Excitation–emission matrix	$β$	$n m^{- 1}$
Apparent quantum yield	AQY	Dimensionless
Cosine of the spectral angle	cos $θ_{s}$	Dimensionless

Abstract

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Tables (2)

Equations (18)

Journal of the Optical Society of America A