MATHEMATICAL MODEL OF AN INTEGRATING SPHERE BASED DIFFUSE LIGHT SOURCES WITH COSINE EMITTERS

Abstract

This paper presents a mathematical model of a diffuse radiation source based on an integrating sphere employing light-emitting diode (LED) sources with a cosine-approximated radiation pattern. The proposed approach is aimed at improving the accuracy of analyzing the formation of spatial luminance distribution and spectral characteristics of radiation, which is essential for the calibration of modern optoelectronic systems, digital cameras, photometric, and radiometric measurement equipment. Based on the multiple-reflection method, analytical expressions are derived that account for the LED radiation pattern, the geometric parameters of the integrating sphere, the diffuse reflectance of the inner coating, the output aperture size, and the contribution of successive reflection orders to the total illuminance. The influence of the LED radiation directivity, the reflectance coefficient of the coating, and the structural parameters of the integrating sphere on the output luminance and spatial uniformity of the generated light field is investigated. The results demonstrate that narrowing the radiation pattern increases the luminance of the output aperture while simultaneously imposing stricter requirements on the positioning accuracy of the radiation sources and the optical properties of the diffuse coating. Practical recommendations are proposed regarding the selection of LEDs, the inner coating material, the number of radiation sources, and the regulation of their operating conditions for the design of high-stability diffuse radiation sources. To validate the proposed model, an experimental prototype of an integrating-sphere-based diffuse radiation source equipped with CREE XML-L2 LEDs was developed. The experimental results show good agreement with the theoretical predictions and confirm the capability of the proposed design to generate a stable and spatially uniform luminance field within a compact structure. The developed mathematical model and engineering recommendations can be applied to the design of calibration radiation sources for precision photometry, radiometry, and the metrological support of advanced optoelectronic systems.