HOW TO CALCULATE COLOR FROM SPECTRA OF UNIAXIAL GEMSTONES

Color is the most important quality for colored stones, one that is widely discussed and studied by gemologists. In general, most light sources are unpolarized. Therefore, in order to quantitatively describe the color of gem materials, it is most appropriate to consider them under unpolarized light conditions. For uniaxial gemstones, when unpolarized light is transmitted parallel to the optic axis, all of the light passes through as o-rays, with their electric field oscillation direction perpendicular to the optic axis. When unpolarized light passes through the material perpendicular to the optic axis, it splits into two different rays: an o-ray and an e-ray, vibrating perpendicular and parallel to the optic axis, respectively. When unpolarized light passes through a uniaxial gemstone in any general direction, it will also split into two rays, with one ray constrained to vibrate along the o-ray and the other vibrating perpendicular to both the o-ray oscillation direction and the path of the light ray. Based on the authors' knowledge, however, the exact means of determining the unpolarized absorption spectrum of a uniaxial mineral in any general direction has never been clearly elucidated. In this research, we demonstrate that the unpolarized spectra can be reproduced by adding weighted polarized transmittance spectra. Adding weighted polarized absorption spectra does not correctly predict unpolarized spectra and is not mathematically equivalent to the correct method of combining transmittance spectra. This has been verified by comparison with a synthetic V-bearing sapphire as well as a synthetic Cr-bearing sapphire that was cut and polished into wafers at various angles relative to the optic axis. Our results showed that estimated spectra (based on the transmittance method) agree well with the measured spectrum. Therefore, the mathematically and physically correct way to reconstruct unpolarized spectra of uniaxial materials in any direction is based on the square of the cosine and sine functions of o-ray and e-ray, provided the incident light angle relative to the optic axis is known. This was derived theoretically and verified experimentally. The extension of this theoretical framework from polished wafers to faceted gemstones represents a significant challenge and will be the focus of future work.