George Frederick Herbert Smith

REFLECTION, REFRACTION, AND DISPERSION

IT is obvious that, since a stone suitable for ornamental use must appeal to the eye, its most important characters are those which depend upon light; indeed, the whole art of the lapidary consists in shaping it in such a way as to show these qualities to the best advantage. To understand why certain forms are given to a cut stone, it is essential for us to ascertain what becomes of the light which falls upon the surface of the stone; further, we shall find that the action of a stone upon light is of very great help in distinguishing the different species of gem-stones. The phenomena displayed by light which impinges upon the surface separating two media[1] are very similar in character, whatever be the nature of the media.

Ordinary experience with a plane mirror tells us that, when light is returned, or reflected, as it is usually termed, from a plane or flat surface, there is no alteration in the size of objects viewed in this way, but that the right and the left hands are interchanged: our right hand becomes the left hand in our reflection in the mirror. We notice, further, that our reflection is apparently just as far distant from the mirror on the farther side as we are on this side. In Fig. 13 MM´ is a section of the mirror, and O´ is the image of the hand O as seen in the mirror. Light from O reaches the eye E by way of m, but it appears to come from O´. Since OO´ is perpendicular to the mirror, and O and O´ lie at equal distances from it, it follows from elementary geometry that the angle i´, which the reflected ray makes with mn, the normal to the mirror, is equal to i, the angle which the incident ray makes with the same direction.

Again, everyday experience tells us that the case is less simple when light actually crosses the bounding surface and passes into the other medium. Thus, if we look down into a bath filled with water, the bottom of the bath appears to have been raised up, and a stick plunged into the water seems to be bent just at the surface, except in the particular case when it is perfectly upright. Since the stick itself has not been bent, light evidently suffers some change in direction as it passes into the water or emerges therefrom. The passage of light from one medium to another was studied by Snell in the seventeenth century, and he enunciated the following laws:—

1. The refracted ray lies in the plane containing the incident ray and the normal to the plane surface separating the two media.

It will be noticed that the reflected ray obeys this law also.

2. The angle r, which the refracted ray makes with the normal, is related to the angle i, which the incident ray makes with the same direction, by the equation

n sin i = n´ sin r, (a)

where n and n´ are constants for the two media which are known as the indices of refraction, or the refractive indices.

This simple trigonometrical relation may be expressed in geometrical language. Suppose we cut a plane section through the two media at right angles to the bounding plane, which then appears as a straight line, SOS´ (Fig. 14), and suppose that IO represents the direction of the incident ray; then Snell’s first law tells us that the refracted ray OR will also lie in this plane. Draw the normal NON´, and with centre O and any radius describe a circle intersecting the incident and refracted rays in the points a and b respectively; let drop perpendiculars ac and bd on to the normal NON´. Then we have n.ac = n´.bd, whence we see that if n be greater than n´, ac is less than bd, and therefore when light passes from one medium into another which is less optically dense, in its passage across the boundary it is bent, or refracted, away from the normal.

We see, then, that when light falls on the boundary of two different media, some is reflected in the first and some is refracted into the second medium. The relative amounts of light reflected and refracted depend on the angle of incidence and the refractive indices of the media. We shall return to this point when we come to consider the lustre of stones.

We will proceed to consider the course of rays at different angles of incidence when light passes out from a medium into one less dense—for instance, from water into air. Corresponding to light with a small angle of incidence such as I₁O (Fig. 15), some of it is reflected in the direction OI´₁ and the remainder is refracted out in the direction OR₁. Similarly, for the ray I₂O some is reflected along OI´₂ and some refracted along OR₂. Since, in the case we have taken, the angle of refraction is greater than the angle of incidence, the refracted ray corresponding to some incident, ray I_cO will graze the bounding surface, and corresponding to a ray beyond it, such as I₃O, which has a still greater angle of incidence, there is no refracted ray, and all the light is wholly or totally reflected within the dense medium. The critical angle I_cON, which is called the angle of total-reflection, is very simply related to the refractive indices of the two media; for, since r is now a right angle, sin r = 1, and equation (a) becomes

n sin i = n´ (b)

Hence, if the angle of total-reflection is measured and one of the indices is known, the other can easily be calculated.

The phenomenon of total-reflection may be appreciated if we hold a glass of water above our head, and view the light of a lamp on a table reflected from the under surface of the water. This reflection is incomparably more brilliant than that given by the upper surface.

The refractive index of air is taken as unity; strictly, it is that of a vacuum, but the difference is too small to be appreciated even in very delicate work. Every substance has different indices for light of different colour, and it is customary to take as the standard the yellow light of a sodium flame. This happens to be the colour to which our eyes are most sensitive, and a flame of this kind is easily prepared by volatilizing a little bicarbonate of soda in the flame of a bunsen burner. A survey of Table III at the end of the book shows clearly how valuable a measurement of the refractive index is for determining the species to which a cut stone belongs. The values found for different specimens of the species do in cases vary considerably owing to the great latitude possible in the chemical constitution due to the isomorphous replacement of one element by another. Some variation in the index may even occur in different directions within the same stone; it results from the remarkable property of splitting up a beam of light into two beams, which is possessed by many crystallized substances. This forms the subject of a later chapter.

Upon the fact that the refractive index of a substance varies for light of different colours depends such familiar phenomena as the splendour of the rainbow and the ‘fire’ of the diamond. When white light is refracted into a stone it no longer remains white, but is split up into a spectrum. Except in certain anomalous substances the refractive index increases progressively as the wave-length of the light decreases, and consequently a normal spectrum is violet at one end and passes through green and yellow to red at the other end. The width of the spectrum, which may be measured by the difference between the refractive indices for the extreme red and violet rays, also varies, though on the whole it increases with the refractive index. It is the dispersion, as this difference is termed, that determines the ‘fire’—a character of the utmost importance in colourless transparent stones, which, but for it, would be lacking in interest. Diamond excels all colourless stones in this respect, although it is closely followed by zircon, the colour of which has been driven off by heating; it is, however, surpassed by two coloured species: sphene, which is seldom seen in jewellery, and demantoid, the green garnet from the Urals, which often passes under the misnomer ‘olivine.’ The dispersion of the more prominent species for the B and G lines of the solar spectrum is given in Table IV at the end of the book.

We will now proceed to discuss methods that may be used for the measurement of the refractive indices of cut stones.