Determining the irradiance signal from an asymmetric source with directional detectors: application to calibrations of radiometers with diffusers

The energy transfer integral between radiating rectangular and detecting circular parallel plates having nonideal angular characteristics is solved for modeling the distance dependence of the irradiance signal. The equation derived for the irradiance signal, which is called the modified inverse-square law, depends on the position, shape, size, and angular characteristics of the light source and the detector. We apply the new model equation to the calibration of a spectroradiometer to determine accurately the distance offsets, which fix the positions of the effective receiving apertures of diffusers used in the entrance optics of spectroradiometers. Earlier measurement results, e.g., for solar UV irradiance, may include uncorrected effects and can be corrected reliably as diffuser offsets and other correction factors are determined with the modified inverse-square law. Simplifications of the modified inverse-square law for analyzing the distance offsets and the correction factors are studied. Simplified equations for the diffuser offset analysis may be used without losing the accuracy when the cosine response of the diffuser is reasonably good. However, for diffusers whose angular responsivities deviate much from the cosinusoidal angular responsivity, large approximation errors in the diffuser offset values may appear if the angular effects are not properly taken into account.