Refraction and Double Refraction

We may now consider how the reflection varies with the angle of incidence. Take the sine formula first. For perpendicular incidence the formula becomes ambiguous, but may be replaced by (n'—n)/(n/H-n). For water the refractive index is about and so, when light falls perpendicularly on water the ampli tude of the reflected wave is 4. of that of the incident, and therefore the intensity is only about 2 per cent. With increasing angle of incidence, the reflection increases somewhat (for inci dence on water at 45° it becomes about 5 per cent), and, as the incidence approaches grazing, it increases rapidly up to unity, which means perfect reflection. The phase of the reflected light is always the same as that of the incident. The tangent formula behaves quite differently. For perpendicular incidence it has the same value, it starts diminishing, and finally vanishes at the polarizing angle, when tan 0=n'/n, at which angle the reflected and refracted waves are perpendicular to one another. For water the angle of incidence is about 54°. After this it increases again and reaches unity at grazing incidence. The phase is the same as that of the incident light up to the polarizing angle, and from there onwards differs from it by These phenomena have been much studied experimentally. The behaviour of a transparent medium depends only on its re fractive index, which is most accurately "determined by the re fractometer (essentially, a prism of the substance) ; so that ex periments on reflection do not provide new information, but serve as a very valuable check on the theory. Suppose, for example, that we illuminate a glass surface obliquely with light polarized in some direction neither in nor perpendicular to the plane of in cidence. To find the reflection this light must be resolved into two components, one of which obeys the sine and the other the tangent formula. They will be unequally reflected, but in neither case is the phase changed at the reflection (or only through 180°, which does not matter) ; and so the reflected light will again be plane polarized, but in a new direction. If the light is incident at the polarizing angle ',he reflected light will contain only the com ponent polarized in the direction at right-angles to the plane of incidence. If ordinary light is incident the same is true, and this process is often used for obtaining polarized light. The most refined experiments have revealed the fact that the polarization is never perfect, but the unwanted component can usually be at tributed to the presence of grease on the surface. If elaborate precautions are taken to remove this grease the effect becomes very small, but it never quite vanishes. This is probably to be attributed to the fact that the surface atoms are necessarily in a different state from those inside, so that it is not possible for a medium to remain truly homogeneous up to the boundary.

We have described what happens when the incident light is in the medium of lower refractive index. In the contrary case the intensity of reflection follows similar rules for the two com ponents, but with one very important difference. In this case the angle of refraction is larger than the angle of incidence, and the reflection becomes complete for both polarized components when the refracted ray is at 9o°, at which point the inclination of the incident ray is given by sin0=il/n. For greater angles of inci dence the phenomenon of total internal reflection supervenes, and this we must now consider.

When sinO>n7n there is no angle 0' for which the equation n sin0= n'sin0' can be satisfied, and so there can be no progressive wave in the second medium. The appropriate solution involves instead a real exponential factor. If the boundary is the z plane, and if the incident wave is at angle 0, its phase will be given by evidently fits the boundary condition and may be verified to satisfy the wave equations. The real exponential implies that the disturbance only penetrates a very short distance into the second medium, roughly not much more than a wave-length.

When the amplitudes are worked out, the reflected wave is found to have amplitude equal to the incident, but with a changed phase, and the change is unequal for the two polarizations. Thus, if in cident plane polarized light is totally reflected, the emergent light is polarized elliptically. Working on this principle Fresnel de vised an instrument which turns plane into circularly polarized light.

For water and air the angle of total reflection is about Thus, when the surface of a glass of water is viewed obliquely from below, it looks like mercury. Total reflection has a curious effect on the field of view of a fish, for however close it is to the surface, everything outside the water must be crowded into a cone of angle 49°, the edge of which will represent the horizon; while, on account of the total reflection, the fish will be able to see the bottom, except for parts nearly underneath, quite as well reflected in the surface as directly. Total internal reflection is much used in optical instruments, as it provides more perfect reflection than any silvering. In many types of binocular the rays are internally reflected no less than four times between the two lenses of each telescope (see BINOCULAR INSTRUMENT).

Refraction in Absorbing Media.

To discuss the passage of light through metals, we take both a dielectric constant and a conduction current and, by Ohm's law, the latter will be pro portional to the electric force. The first electromagnetic equation is now while the remainder are unchanged. The presence of the con duction term has an effect something like what we found in total internal reflection, for it compels us to introduce a real ex ponential. For a wave of frequency v going along z a solution can be found in which n is the refractive index, and K is called the absorption co efficient. Considered at a given instant of time, the wave is a damped sine-curve of wave-length c/nv and the amplitude de creases to a fraction of itself for each successive crest. For actual metals K/n is quite large, so that the light can only pene trate a very short distance. The value of n could be determined experimentally from the deflection of light by a prism, if one could be made so thin as to transmit light, and K could be deter mined by finding how much the light is attenuated in passing through a plate; but in view of the extreme opacity of metals such methods are very troublesome, and it is more convenient to deduce n and K from experiments on reflection.

The principle of reflection is just the same as for transparent media, but the details are very different on account of the real exponential in the internal wave. There is a change of phase in the reflected wave, and it is different for the two polarizations. Consequently, if plane polarized light is reflected, it becomes elliptic, and the study of this ellipticity is the most powerful method of evaluating n and K. In the case of perpendicular incidence it can be shown that the intensity reflected is I +2n). For all metals K is consider ably larger than n, and so the reflection is not far from complete. We see how it comes about that strong absorption, or large K, means strong reflection. The refractive indices of metals vary over a much wider range than those of transparent substances. Thus, while the latter range roughly speaking between 1 and 2.4, silver has refractive index 0.18, associated with absorption coefficient 3.67. More remarkable still is sodium, which, if it can be used untarnished, is an even better reflector than silver. Here n=o•oo5 and K =a•61, and 99.7 per cent of the light is reflected at per pendicular incidence. In so far as wave-velocity has a meaning in such a substance, the wave-velocity is two hundred times the velocity of light.