Monte Carlo Modeling of Optical Properties of Integrating Spheres

The Monte Carlo method has been applied to numerical modeling of an integrating sphere designed for hemispherical-directional reflectance factor measurements. A newly developed algorithm [1,2] substantially improves the convergence of the computational process by analytical calculation of direct source-induced irradiation for every point of diffuse reflection of rays traced. The method developed was applied to an integrating sphere reflectometer for the visible and infrared spectral ranges (see Figure 1). The deviations of the measured sample reflectance from the actual reflectance as a result of various factors were computed. In addition, parametric studies of hemispherical radiance distributions for radiation incident onto the sample center were performed. Cylindrical projections of these distributions are shown in the color figures. Black color indicates that the radiance is less than the lower limit of the color scale at the right in each figure; white indicates a level above the upper limit.

The influence of critical factors such as the angular distribution of radiant intensity of the radiation source and the reflectance and specularity of the sphere’s internal surface was examined. Corrections for the systematic uncertainty that is due to specular components of reflection of the sample and the IS internal surface were computed. These radiance distributions were used to obtain uncertainty contributions to the eventual measurements of sample reflectance. No analytical methods for performing this type of analysis are available. The results were also used to improve the eventual sphere design. Some obvious routes to improvement include adjusting the baffle reflectances on both sides to better match the surrounding radiance levels and making a significant effort to produce as nearly as possible a Lambertian distribution for the input light source.

Local coordinate systems.

Figure 1. Integrating Sphere for Measurement of Hemispherical-Directional Reflectance Factor in Vis and NIR Spectral Range.

Cylindrical projection map for hemispherical relative distribution of incident radiance at the sample center.

In order to examine the possibilities of improvment of the integrating sphere performance, the integrating sphere without baffles but with the radiation source placed into a recess in the sphere wall has been modeled; the dependence of radiance map uniformity on the recess depth has been studied. One of these maps is shown in the figure at right.

For high-temperature measurements, the integrating sphere made of sintered PTFE has been equipped by a separate metallic insert coated with BaSO₄ and contains sample, reference, and source ports and baffles. The insert is water cooled to accommodate samples up to 1300 K (see picture below). The appropriate radiance map is depicted in the lower right-hand figure.

Integrating Sphere with High-Temperature Insert

Incident radiance map for integrating sphere with recessed source.

Incident radiance map for integrating sphere with high-temperature insert having lower reflectance

References

1. Evaluation of Performance of Integrating Spheres for Indirect Emittance Measurement.
   A. Prokhorov, S. Mekhontsev, and L. Hanssen,
   Proc. 8th International Symposium on Temperature and Thermal Measurements in Industry and Science, 1 (Berlin, Germany, 19-21 June 2001), pp. 277-282.


2. Monte Carlo modeling of an integrating sphere reflectometer, A.V. Prokhorov, S.N. Mekhontsev, and L.M. Hanssen, Appl. Opt. 42(19), 3832-3842 (2003).