Radiative flux from a multiple-point bioluminescent or chemiluminescent source within a cylindrical reactor incident on a planar-circular coaxial detector. II. Rotationally symmetric radiation

Stanislaw Tryka

Stanislaw Tryka^{1}

^{1}University of Life Sciences in Lublin, Department of Agricultural Sciences, Laboratory of Physics, Szczebrzeska 102, PL-22-400 Zamosc, Poland (stanislaw.tryka@up.lublin.pl)

Stanislaw Tryka, "Radiative flux from a multiple-point bioluminescent or chemiluminescent source within a cylindrical reactor incident on a planar-circular coaxial detector. II. Rotationally symmetric radiation," J. Opt. Soc. Am. A 28, 147-156 (2011)

In the previous paper [J. Opt. Soc. Am. A 28, 126 (2011)], an analytical formula was presented for calculating radiative fluxes from arbitrarily distributed and arbitrarily radiating multiple-point emitters of bioluminescent or chemiluminescent sources within cylindrical reactors, when the radiation from these point emitters propagates through two homogeneous isotropic media and reaches a planar-circular coaxial detector. This formula was based on two assumptions. The first is that radiation passes across a planar boundary interface between the two media. The second is that the surface reflections on the lateral surface and on the reactor base opposite the detector may be neglected. In this paper, the formula obtained previously was simplified for the case of uniformly distributed point emitters of bioluminescent or chemiluminescent sources emitting an identical rotationally symmetric radiation. The simplified formula is suitable for optimizing and calibrating the analyzed reactor–detector sys tem, which is most commonly used to study the bioluminescence emitted by small biological objects and the chemiluminescence from chemical reactions. Representative data were calculated, illustrated graphically, and tabulated.

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Average Fluxes $\u3008{\mathrm{\Phi}}_{\lambda ,{V}_{c}\to {s}_{2}}\u3009$ Computed for ${I}_{\lambda ,0}=1\text{}\mathrm{nW}\xb7{\mathrm{sr}}^{-\mathsf{1}}\xb7{\mathrm{nm}}^{-\mathsf{1}}$ at $h=1.0\text{}\mathrm{cm}$, ${H}_{2}=1.0\text{}\mathrm{cm}$, $R=1.0\text{}\mathrm{cm}$, ${n}_{1}=1.33$, ${n}_{2}=1.00$, and ${\alpha}_{\lambda ,a{t}_{2}}=0$^{
a
}

The data were calculated within the spectral bandwidth $\Delta \mathit{\lambda}=1\text{}\mathrm{nm}$ for a and ${H}_{1}$ given in centimeters and the attenuation coefficient ${\alpha}_{\lambda ,a{t}_{1}}$, given in ${\mathrm{cm}}^{-1}$. The identical values of $\u3008{\mathrm{\Phi}}_{\lambda ,{V}_{c}\to {s}_{2}}\u3009$ will be obtained for all geometrical variables expressed in millimeters and the attenuation coefficients expressed in ${\mathrm{mm}}^{-\mathrm{1}}$ or for all geometrical variables given in meters and the attenuation coefficients expressed in ${\mathrm{m}}^{-1}$.

Table 2

Average Fluxes $\u3008{\mathrm{\Phi}}_{\lambda ,{V}_{c}\to {S}_{2}}\u3009$$\mathrm{\Delta}\lambda =1\text{}\mathrm{nm}$ Computed for ${I}_{\lambda ,0}=1\text{}\mathrm{nW}\xb7{\mathrm{sr}}^{-1}\xb7{\mathrm{nm}}^{-1}$ at $h=1.0\text{}\mathrm{cm}$, ${H}_{1}=0.75\text{}\mathrm{cm}$, $R=1.0\text{}\mathrm{cm}$, ${n}_{1}=1.33$, ${n}_{2}=1.00$, and ${\alpha}_{\lambda ,a{t}_{2}}=0$^{
a
}

The data were calculated within spectral bandwidth Δλ=1 nm for a and ${H}_{1}$, given in centimeters and the attenuation coefficient ${\alpha}_{\lambda ,a{t}_{1}}$ given in $c{\mathrm{m}}^{-1}$. The identical data will be obtained for a and ${H}_{1}$ given in millimeters and the attenuation coefficient ${\alpha}_{\lambda ,a{t}_{1}}$ expressed in ${\mathrm{mm}}^{-\mathrm{1}}$ or for a and ${H}_{1}$ given in meters and the attenuation coefficient ${\alpha}_{\lambda ,a{t}_{1}}$ expressed in ${\mathrm{m}}^{-1}$.

Tables (2)

Table 1

Average Fluxes $\u3008{\mathrm{\Phi}}_{\lambda ,{V}_{c}\to {s}_{2}}\u3009$ Computed for ${I}_{\lambda ,0}=1\text{}\mathrm{nW}\xb7{\mathrm{sr}}^{-\mathsf{1}}\xb7{\mathrm{nm}}^{-\mathsf{1}}$ at $h=1.0\text{}\mathrm{cm}$, ${H}_{2}=1.0\text{}\mathrm{cm}$, $R=1.0\text{}\mathrm{cm}$, ${n}_{1}=1.33$, ${n}_{2}=1.00$, and ${\alpha}_{\lambda ,a{t}_{2}}=0$^{
a
}

The data were calculated within the spectral bandwidth $\Delta \mathit{\lambda}=1\text{}\mathrm{nm}$ for a and ${H}_{1}$ given in centimeters and the attenuation coefficient ${\alpha}_{\lambda ,a{t}_{1}}$, given in ${\mathrm{cm}}^{-1}$. The identical values of $\u3008{\mathrm{\Phi}}_{\lambda ,{V}_{c}\to {s}_{2}}\u3009$ will be obtained for all geometrical variables expressed in millimeters and the attenuation coefficients expressed in ${\mathrm{mm}}^{-\mathrm{1}}$ or for all geometrical variables given in meters and the attenuation coefficients expressed in ${\mathrm{m}}^{-1}$.

Table 2

Average Fluxes $\u3008{\mathrm{\Phi}}_{\lambda ,{V}_{c}\to {S}_{2}}\u3009$$\mathrm{\Delta}\lambda =1\text{}\mathrm{nm}$ Computed for ${I}_{\lambda ,0}=1\text{}\mathrm{nW}\xb7{\mathrm{sr}}^{-1}\xb7{\mathrm{nm}}^{-1}$ at $h=1.0\text{}\mathrm{cm}$, ${H}_{1}=0.75\text{}\mathrm{cm}$, $R=1.0\text{}\mathrm{cm}$, ${n}_{1}=1.33$, ${n}_{2}=1.00$, and ${\alpha}_{\lambda ,a{t}_{2}}=0$^{
a
}

The data were calculated within spectral bandwidth Δλ=1 nm for a and ${H}_{1}$, given in centimeters and the attenuation coefficient ${\alpha}_{\lambda ,a{t}_{1}}$ given in $c{\mathrm{m}}^{-1}$. The identical data will be obtained for a and ${H}_{1}$ given in millimeters and the attenuation coefficient ${\alpha}_{\lambda ,a{t}_{1}}$ expressed in ${\mathrm{mm}}^{-\mathrm{1}}$ or for a and ${H}_{1}$ given in meters and the attenuation coefficient ${\alpha}_{\lambda ,a{t}_{1}}$ expressed in ${\mathrm{m}}^{-1}$.