Quantitative theory of integrating sphere throughput: 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.

Realistic ports in integrating spheres: reflectance, transmittance, and angular redirection.

We use Monte Carlo ray-tracing modeling to follow the stochastic trajectories of rays entering a cylindrical port from inside an integrating sphere. This allows us to study and quantify properties of realistic ports of non-negligible length, as opposed to the common thin-port assumption used in most theoretical treatments, where the port is simply considered as a hole in the spherical wall. We show that most practical ports encountered in integrating sphere applications cannot be modeled as thin ports. Indeed, a substantial proportion of rays entering the port can be reflected back into the sphere, with port reflectances as high as 80% demonstrated on realistic examples. This can have significant consequences on estimates of the sphere multiplier and therefore pathlength inside the sphere, a critical parameter in many applications. Moreover, a nonzero port reflectance is inevitably associated with reduced transmittance through the port, with implications in terms of overall throughput. We also discuss angular redistribution effects in a realistic port and the consequences in terms of detected throughput within a fixed numerical aperture. Those results highlight the importance of real port effects for any quantitative predictions of optical systems using integrating spheres. We believe that those effects can be exploited to engineer ports for specific applications and improve the overall sphere performance in terms of pathlength or throughput. This work carries important implications in our theoretical understanding of integrating spheres and on the practical design of optical systems using them.