Fig. 4. Yttrium-aluminum garnet refraction index values for various wavelengths. Click for enlarge.

In the blue - violet range, the index of refraction value grows, with decreasing wavelength, much faster than in other spectrum ranges. This relationship in the visible part of the spectrum is sometimes described with the hyperbola set as follows: ,
where .

Such behavior of n(l) curve is determined by proximity of the edge of ultraviolet characteristic electronic absorption. We shall neglect this peculiarity for the sake of simplicity in further estimations.

Capacity of a prism to decompose a white ray is called angular dispersion of the prism, expressed in degree/mm; that is the change of ray angle per micron of wavelength change.
Let us consider the observer's eye (see Fig.3) focused at the facet of the prism nearest to it, that is, looking at the prism. The sharp image of the facet is thereat formed on the eye's retina, each point on the facet surface being mapped with a retina point. The Figure shows passage of rays through one such point on the prism's facet. The beam leaving the prism forms a fan-shaped bundle of rays of various colors.

Actually, the ray begins to disperse upon entering the prism, but this is much less than at exit. Because the prism size is negligible compared to its distance from the eye, we will neglect the input dispersion. Should all this 'fan' enter the pupil of the eye, the eye's lens will focus all of it into a white-colored dot on the retina.

However, if the fan area exceeds pupil size at the eye location, then only a part of the 'fan' cut-off with the pupil would be focused on the retina.

The distance from the gem to the observer can vary, so instead of linear pupil size it is more convenient to employ its angular size, i.e. the angle at which the pupil diameter is seen from the gem. If the angular 'fan' size exceeds the angular pupil diameter, then only a part of complete visible spectrum radiation of certain wavelengths will hit the focused dot on the retina.

The higher the prism angular dispersion and the narrower the pupil, the narrower would be the wavelength band Dl cut off from the complete spectrum, resulting in higher saturation of the colors observed.

As gems are usually viewed in good illumination conditions and from convenient distances, the pupil diameter can be assumed 3 mm and the distance to the gem being that of normal viewing, that is, 25 cm. Then the pupil angular size will be 0.7 degrees. To isolate for this size the wavelength interval Dl = 50 nm providing for obtaining "pure colors" will require the gem angular dispersion of at least 14 degree/mm. Angular dispersion values inferior to 3.5 degree/mm leads to complete loss of green color, the other colors being observed rather rarely. It should be emphasized that the fire improves with increasing distances between the gem and the observer and diminishes at lower illumination conditions, which make the pupil expand.

The above is true only if the prism is illuminated with a plane light wave. Such illumination is generated by a point source, that is, the one located at the distance much greter than its own size; in other words, it is a source of a small angular size. For an expanded source of illumination, the rays emitted from its different points and entering the prism from different directions would mix their colors on the retina, degrading the purity of the observed color. Hence, good fire requires illumination of the gem from a light source of small angular size. The source angular size affects the color purity similarly to the observer's eye pupil size.

Since the angular size of the sun, a lamp or a candle does not exceed 0.5 degree, the fire depends primarily on the pupil angular size. The fire of colorless gems is perfect in the sunlight and even better in illumination coming from multiple point sources, such as a chandelier with many lamps without matted shades or a candelabrum with candles. Then the number of color "fans" from a gem will increase by, as many times as there will be the sources of light. The fire from a faceted gem will be completely lost in the light of a dull day under cloudy sky.

All this should be always kept in mind by those engaged in sales or advertising of precious stones or those choosing jewelry for tonight's party.