A quantitative theory of integrating spherethroughput: Comparison with experiments

We present a rigorous approach for measuring the throughput of an integrating sphere, from which the so-called sphere multiplier M can be derived. The critical ingredients of this approach are: (i) the transmitted power is measured at the base of an integrating port to avoid non-ideal port effects associated with reflections on the port wall; (ii) to implement this last point, optical fibers are used for light collection, providing a well-defined collection area and numerical aperture; (iii) the angular-dependent fiber throughput and detector sensitivity are determined experimentally and accounted for. We demonstrate in particular that a more realistic theory, accounting for the propagation of skew rays through the fiber, is needed to quantitatively model the fiber effect on the measured sphere throughput. We show experimentally that failure to fulfill these three points produces erroneous results, by as much as 50%. With an accurate experimentally derived sphere multiplier, agreement with theory is then obtained only if realistic ports (with non-zero reflectivity) are assumed. This provides experimental evidence for recent theoretical predictions of the importance of realistic ports [Tang et al., Appl. Opt.57, 1581 (2018)APOPAI0003-693510.1364/AO.57.001581]. Using the same experimental techniques, we also present clear experimental proof of two other predictions from that study: that the angular distribution exiting the port is strongly altered and that the overall port transmittivity is drastically reduced for high aspect ratio ports. This work will provide a solid basis for future quantitative measurements of absolute throughput and for further developments of the theory of integrating spheres.