Index of refraction by total reflection

One of the simplest devices for determining an index of refraction was invented by A. H. Pfund. A description appears in Physical Optics by R. W. Wood (p. 70, third revised edition) from which this experiment is drawn.

A glass plate of thickness $t$ is illuminated from the top. In our experiment we will use a He-Ne laser and a mirror as shown in the sketch in Figure 3.1.

Figure 3.1: Without optical contact at the paper, the steepest angle for light inside the plate is just the critical angle. The paper is illuminated by a circular disk of light. With glycerin between the glass and the paper, light reenters the plate at all angles after it scatters from the paper. Total reflection at the upper surface illuminates the paper with a pattern complementary to the one without a liquid.

The laser light passes through the upper surface and strikes the lower surface of the plate. In Pfund's original design, this surface was painted white, but in our experiment we will use a white piece of paper and later add some glycerin to illustrate just how this works. If you have just a piece of paper, the light passes through the lower surface and is scattered by the paper. This spot now is a new source, sending rays back through the plate. The rays continue to the upper surface where some are reflected back down. If the returning rays encounter the lower surface at less than the critical angle they leave the plate and strike the paper again. The paper is therefore illuminated with a disk of light, centered on the laser spot.

For light incident on a surface from inside the glass, there is a critical angle of incidence $\theta_c$ above which all light is totally reflected and none is transmitted. The critical angle determines the size of the circles of light seen in the Pfund refractometer. The diameter of the disk is given by

$$D = 4 t / {\sqrt{n^2 - 1}}$$ (31)

Here $n$ is the index of refraction of the glass relative to the surrounding air and $t$ is the thickness of the plate.