James L. Keef, John F. Clare, and Kurtis J. Thome, "Analytical solution for integrating sphere spectral efficiency inclusive of
atmospheric attenuation," Appl. Opt. 47, 253-262 (2008)

An analytical solution to the attenuation of flux within an integrating sphere due to spherical integrating source coating, exit port escape, and atmospheric absorption is derived employing a geometric probability distribution of completed sphere transits. This is used to determine the mean number of completed sphere transits and its variance. Equations that provide the attenuation ratios for the three extinction mechanisms are derived using the energy balance and summation of reflection methods. The mean length of a transit of the sphere and its variance are presented and used to derive expressions for the mean and variance of photon path lengths in the sphere.

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Tabulated values of $\u3008n\u3009$, partial mean reflections
$\u3008{n}_{i}\u3009$, i = a, b, or c;
$\hat{l}$; fractional efficiency ${F}_{S}$ of a gas-purged sphere; uncompensated (for transits n ≥ 2) partial absorption ratios a, b, c, and compensated (for transits n ≥ 1) partial absorption ratios a′, b′, c′ for SIS parameters diameter ${D}_{S}$, exit port radius ${R}_{\mathit{XP}}$, spectral coating reflectance ${\rho}_{\lambda}$, and atmospheric attenuation ${\alpha}_{\lambda}$. The results calculated are notional values for one arbitrary wavelength only. The results cited are independent of the choice of self-consistent length units for the SIS parameterizations ${R}_{\mathit{XP}}$ and ${D}_{S}$.

Tabulated values of $\u3008n\u3009$, partial mean reflections
$\u3008{n}_{i}\u3009$, i = a, b, or c;
$\hat{l}$; fractional efficiency ${F}_{S}$ of a gas-purged sphere; uncompensated (for transits n ≥ 2) partial absorption ratios a, b, c, and compensated (for transits n ≥ 1) partial absorption ratios a′, b′, c′ for SIS parameters diameter ${D}_{S}$, exit port radius ${R}_{\mathit{XP}}$, spectral coating reflectance ${\rho}_{\lambda}$, and atmospheric attenuation ${\alpha}_{\lambda}$. The results calculated are notional values for one arbitrary wavelength only. The results cited are independent of the choice of self-consistent length units for the SIS parameterizations ${R}_{\mathit{XP}}$ and ${D}_{S}$.