Considerations on measurement of bidirectional transmittance distribution function of thick samples over a wide range of viewing zenith angles

Abstract

We delve into theoretical and experimental considerations for determining the spectral bidirectional transmittance distribution function (BTDF) of thick samples across a broad viewing zenith angle range. Nominally, BTDF is defined as the ratio of transmitted radiance to incident irradiance measured from the same plane. However, when employing thick samples for BTDF measurements, the viewing plane of the transmitted beam may shift from the front to the rear surface of the sample, altering the measurement geometry compared to using the sample front surface as the reference plane. Consequently, the viewing zenith angle from the sample rear surface increases relative to the sample front surface, and the sample-to-detector-aperture distance decreases by an amount corresponding to the sample thickness. We introduce a method for determining the BTDF of thick samples, considering the transformation of practical measurement results to a scenario where the measurements are conducted at a very large distance from the sample. To validate the method, we utilize a BTDF facility equipped with two instruments that significantly differ in their sample-to-detector-aperture distances. We evaluate the impact of a 2 mm sample thickness on the BTDF by assessing the ratio of transmitted and incident radiant fluxes as a function of viewing zenith angle relative to the sample rear surface. The evaluation is conducted in the wavelength range from 550 nm to 1450 nm in 300 nm steps, and in the viewing zenith angle range from -70° to 70° in 5° steps. Measurements are performed in-plane at an incident zenith angle of 0°. It is concluded that consistent determination of BTDF of a thick sample is possible by converting the experimental parameters of the real measurements at relatively short distances from the sample to correspond to those that would be obtained from measurements at very large distances from the sample.