Already explained how light is affected when it strikes the surface of a gem; i.e., part is reflected from the surface at an angle equal to the angle of incidence and the remainder is refracted into the gem, the angle of refraction depending on the angle of incidence and the reduction in velocity as it enters the gem.
We are now concerned with the behavior of light that enters a stone. It is this light that accounts for a gem's dispersion, or "fire" (the display of prismatic colors), its brilliancy (the proportion of white light the stone is capable of returning to the eye), and most of its body color; in short, its beauty. If the extent to which these three properties are displayed in fashioned stones were constant for each gem species, it would be unnecessary to study grading. Unfortunately, however, these properties depend on style of cutting, proportions, perfection of polish, and orientation of the stone in relation to its crystal graphic axes, factors that are controlled by the cutter. A trained gemologist is able to look at a stone and, with the aid of a few simple tests, determine the extent to which cutting has exploited its inherent beauty. To accomplish this step in appraising requires an understanding of light and its behavior in a stone and of optical properties, which are characteristic for each gem species. The following discussion of critical angle and total reflection is important not only for an understanding of the cause of beauty in a gem, but it concerns the theory underlying the construction and use of the refractometer, one of the most important gem testing instruments.
Total Reflection And Critical Angle
Already that light entering a gemstone from air perpendicular to the surface is un-deflected as it enters a gem; it is merely slowed down. Since light striking at any angle other than perpendicular to the surface is bent on entering the stone, it is obvious that the greater the angle to the perpendicular, the greater the amount of bending that takes place within the stone. The extreme angle from the perpendicular at which light can still enter the stone is reached when the beam just grazes the surface. At this angle it is subject to maximum bending when refracted into the stone. From this it can be seen that light approaching a point at the surface from all angles, ranging from perpendicular to almost horizontal, will, when entering a stone, be confined within the stone to a restricted angle that is determined by the degree to which the gem is capable of bending light. The greater the gem's ability to bend light, the smaller will be the angle within the stone that the light entering at that point on the surface will be confined.
To determine the behavior of light leaving a gem rather than entering it, it is only necessary to reverse its direction. In other words, if light could be turned back on itself just after entering the gem, it would follow the same path out of the stone as it took on entering. Stated another way, if a beam of light is directed perpendicular to the inner surface and the angle of incidence is then slowly changed while still directing the beam at the same point on the surface, for each increasing angle away from the perpendicular the amount of bending as the light leaves the stone will increase to such an extent that the light is finally bent, or refracted out, parallel to the surface. For light incident at a still greater angle, there is nowhere for it to go except to be TOTALLY REFLECTED from the inner face, thus rebounding back into the stone. The angle to the perpendicular within a stone for which light is refracted out parallel to the surface is called the CRITICAL ANGLE, for it is the angle beyond which all light striking an inner surface will be totally reflected within the stone. The size of the critical angle of a given stone is the important factor in determining the proper angles for the stone's facets, since maximum brilliancy can only be achieved if light is totally reflected from the pavilion facets and refracted back out through the crown facets.
Figures I, illustrate this discussion. In Figure 10 the wave fronts of the beam leaving the stone are parallel to the surface; thus they leave the stone without any change in direction. In Figure II, however, the beam strikes the surface at an angle. Since one side of each wave front enters the air before the other side, the greater speed in air causes the front, by the time it is completely out of the stone, to have changed directions; therefore, the beam is bent. Figure 12 shows even greater inclination, so that the refracted beam just grazes the surface. This is a limiting condition. Beyond this point (that is, at a greater angle to the perpendicular) the light must be totally reflected; there is nowhere else for it to go. This is demonstrated more fully in Figure 13, which shows beams 10, II, 12, 13, with beam 4 totally reflected. The critical angle for the material illustrated is indicated by the shaded area. The approximate critical angles for some of the common transparent gem species are as follows: diamond 24½°; zircon 30½°; corundum 34½°; spinel 35½°; topaz 38°; quartz 40½°.
Visual Observation Of Total Reflection
In order to understand total reflection more thoroughly, fill a transparent glass tumbler with water and observe it toward a strong light. First, look at the surface of the water with the glass held just below eye level; a reflection of less intensity than the light source will be seen (Figure 14). However, when the glass is held above eye level, it will be noted that the light is totally reflected (i.e., the full intensity of the light source) from the undersurface of the water (which will appear more or less mirror like), just as it is totally reflected from the inside surface of a polished gem (Figure 15). Water has a higher R.I. (1.33) than air, but much lower than any gemstone. Therefore, the undersurface of the water has a smaller angle in which total reflection may occur than does the inner surface of any gem. Similarly, it has a larger angle in which leakage may occur.
Critical-Angle Cone
For simplification, the illustrations used thus far have been confined
to cross section drawings of stones. In practice, the behavior of light
in stones should be considered in three dimensions. In other words,
it is obvious that light can impinge on a surface from all directions
about a given point, therefore, we can visualize and illustrate the
critical angle as a cone rather than an angle in a single plane (Figure
16). In brief, light striking an inner surface within the critical angle
cone will be refracted out of the stone (Figure 17) whereas light striking
outside of the cone will be totally reflected back into the stone (Figure
18).
