Integrating sphere theory is developed for restricted field of view (FOV) detectors using a simple series solution technique. The sphere throughput, sample reflectance, and sphere wall reflectance are calculated. The effects of the sample’s scattering characteristics on sphere measurements are determined. It is shown that although the generalized equations incorporating detector FOV dependence reduce to the hemispherical FOV equations in some cases, in general integrating sphere behavior is altered through restriction of the detector FOV.
Leonard M. Hanssen, "Effects of restricting the detector field of view when using integrating spheres," Appl. Opt. 28, 2097-2103 (1989)