The Bidirectional Reflectance Distribution Function (BRDF) is a fundamental quantity describing the reflectance properties of a target as a function of illumination and viewing geometries. However, there are only a few publications that describe properly the nature, fundamentals and applicability of the BRDF in optical sciences and engineering. The authors of this Applied Optics article performed a remarkable study on establishing a 0/45 reflectance factor scale in the shortwave infrared (SWIR) spectral range. This newly established reflectance factor scale can be used in numerous government and industrial calibration facilities and, most importantly, in secondary calibration facilities working not only on optical science and engineering but also on computer modeling and model validation.

The National Institute of Standards and Technology (NIST) provides a reflectance scale for both directional-hemispherical and bidirectional measurement geometries through calibrated standards using the Spectral Tri-function Automated Reference Reflectometer (STARR). The authors describe the establishment and application of a 0/45 reflectance factor scale in the SWIR range from 1100 nm to 2500 nm through design, characterization, validation and uncertainty budget calculation.

BRDF depends on wavelength and is determined by the structural and optical properties of the surface, such as specular or diffuse, volume or surface scattering, etc. BRDF describes the variation of reflectance within the illumination and scattered light directions. BRDF can be approximated by:

BRDF = ^P_s⁄_Ω⁄_PicosΘs
where P_i and P_s are the incident and scatter powers, respectively, Ω= A/R² is the solid angle determined by the detector aperture area, A, and the radius from the sample to the detector, R, and Θ_s is the scatter angle. The geometry of the BRDF measurements is designated by its incident angle θi and detection angle Θ_s as Θ_i/Θ_s. For example, a bidirectional geometry in which the sample is illuminated at 0° from its normal and the reflected radiant flux is detected at an angle 45° from the normal is referred to as 0/45.

A single grating monochromator – lamp assembly was used as a light source. A xenon arc lamp was used for wavelengths of 400 nm and shorter and a quartz-tungsten-halogen incandescent lamp for longer wavelengths. Silicon end extended InGaAs photodiodes based receivers were used at their respective spectral ranges. For detection with a silicon photodiode and picoammeter, dark signals are obtained by closing the shutter on the monochromator, and the net signals for the incident and reflected positions are obtained by subtracting the dark signals. For detection with an ex- InGaAs photodiode, the signals are collected in AC mode using a mechanical chopper and phase-sensitive detection. The values for both polarizations are averaged to yield the reflectance factor for unpolarized incident radiant flux. The ex- InGaAs photodiode based radiometer consists of a 4-stage thermo-electrically cooled photodiode, a low noise preamplifier, an aperture, and a 25 mm diameter, bi-convex calcium fluoride singlet lens with a 50 mm focal length. The radiant flux is focused onto the 3 mm diameter active area of the detector. The photodiode is cooled to approximately −55 °C. The temperature of the radiometer body is stabilized using an external chiller connected to the heat-sink of the thermoelectric cooler with a set point of 19 °C. An aperture is placed directly in front of the lens. The radius of the aperture, which is used to evaluate the area for BRDF calculation was measured by the NIST Aperture Area Facility and was determined to be 10.17763 mm with a relative standard uncertainty u(r) of 0.007 %. The distance, d, between the sample and the aperture of the receiver is 560.4 mm with an uncertainty u(D) of 0.3 mm. The uncertainty budget is very well organized and of great use to others who try to follow and/or replicate the same or similar measurement setups. The sources of uncertainty, assumed to have normal probability distributions, such as aperture distance, aperture area, viewing angle, solid angle calculation, radiometer linearity, radiometer gain ratio, and repeatability, are all discussed. The uncertainty budget for 0/45 reflectance factor values obtained using STARR with an ex-InGaAs radiometer is presented.

The new capability of measuring 0/45 reflectance factor in the SWIR is demonstrated by measuring three different diffuser materials: sintered PTFE, pressed PTFE, and ceramic. Both forms of PTFE are commonly used as materials for reflectance standards. The 0/45 reflectance factors of the three materials were determined in the spectral range from 250 nm to 2500 nm and compared to their respective 6/di directional/hemispherical reflectance factors. Prediction bands were calculated for a confidence level of 95 %. They are 0.011 for pressed PTFE, 0.006 for sintered PTFE, and 0.008 for ceramic.
Although this paper focuses on the establishment of the reflectance factor scale over the shortwave infrared, it also covers to some extend the UV-VIS-NIR spectral range BRDF measurements for more detailed instrumental setup description. The exceptionally well-presented and detailed work constitutes a substantial contribution to BRDF measurement efforts in support of remote sensing and other industry applications.

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