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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.

A straightforward method for estimating the position of the optical receiving plane of a spherical, dome-shaped diffuser from its spatial responsivity data is presented. The method is tested with two diffusers, types J1002 and J1015 from CMS Schreder, commonly used in solar UV spectroradiometers. The shift of the receiving plane from its nominal position determines a potential measurement error that occurs when measurements and calibrations are carried out with sources at different distances from the diffuser. Such information is particularly valuable for voluminous solar UV monitoring spectroradiometers that cannot easily be transported to laboratory calibrations. The results suggest that systematic measurement errors are at least of the order of 1%, if the position of the receiving plane is not properly taken into account, thus indicating a need to study the effect more carefully. This method can also be used to minimize measurement errors when designing diffusers.