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In a number of techniques that measure weak fluxes of optical radiation, it is frequently necessary to keep a detector in a medium different from that of the radiating source by separating it from the source with a planar transparent window. However, sources such as systems of light-emitting diodes, large-fiber illuminators, and microscopic living objects that emit biological luminescence may sometimes be regarded as multiple-point sources. To estimate the fluxes of optical radiation illuminating a surface from a nonuniformly distributed multiple-point source, a method for calculating fluxes from a single off-axis point source is needed. A formula is derived to estimate a flux of temporally incoherent optical radiation incident on a circular disk from a single off-axis point source separated by a plane-parallel plate (PPP). This formula is expressed by a series of single integrals of some superposed elementary functions. These functions depend on the variables that characterize the point-source-plane-parallel-plate-circular-disk geometry and on the optical properties of the media that separate the source from the PPP and the PPP from the disk. The solution was obtained for isotropic media. For illustrative purposes some examples of the use of the formula are presented. The selected results are illustrated by three-dimensional surface plots and compared with the values of the fluxes calculated for radiation incident on the disk from a point source not separated by a PPP.

A general multidomain integral formula is presented for calculating fluxes of radiation striking a circular disk from various off-axis point source types embedded in an attenuating or nonattenuating medium. This formula is expressed by double line integrals of radiant intensity and sine functions with respect to the polar and horizontal angles determining the angular distribution of the emitted radiation. The formula reduces to single line integral expressions when radiation does not depend on the horizontal angle and is directly applicable for calculating fluxes of revolutional symmetry around the optical axis of the source perpendicular to the disk. The applicability of this reduced formula is tested by computing radiant fluxes from Lambertian and Gaussian point sources using a simple numerical procedure for single integrals. The computed data are illustrated graphically, tabulated, and validated using OSLO. Finally, the accuracy, similarity, and applicability of the results provided by the integral formula and the OSLO program are analyzed. Numerical results have shown the effectiveness of the presented formulas for calculating radiant fluxes from various on- and off-axis point sources passing through a nonattenuating or attenuating homogeneous isotropic media and incident on a circular disk perpendicular to optical axes of these sources. Practical applications of these formulas include optical sensing and metrology, optical coupling, fiber optic for biomedical measurements, and creative lighting design.

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