Analytical solution for integrating sphere spectral efficiency inclusive of atmospheric attenuation.

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.