The angle of refraction is about 35°, clearly less than 45°, just as was predicted in part (b).
Note: The function sin-1 is of course the arcsin. We will use the sin-1 notation since that is what is found on scientific calculators.3. Critical angle and total internal reflection.
When light travels from a medium of higher index to one of lower index, we encounter some interesting results. Refer to Figure 3-10, where we see four rays of light originating from point O in the higher-index medium, each incident on the interface at a different angle of incidence. Ray 1 is incident on the interface at 90° (normal incidence) so there is no bending.
The light in this direction simply speeds up in the second medium (why?) but continues along the same direction. Ray 2 is incident at angle i and refracts (bends away from the normal) at angle r. Ray 3 is incident at the critical angle ic, large enough to cause the refracted ray bending away from the normal (N) to bend by 90°, thereby traveling along the interface between the two media. (This ray is trapped in the interface.) Ray 4 is incident on the interface at an angle greater than the critical angle, and is totally reflected into the same medium from which it came. Ray 4 obeys the law of reflection so that its angle of reflection is exactly equal to its angle of incidence. We exploit the phenomenon of total internal reflection when designing light propagation in fibers by trapping the light in the fiber through successive internal reflections along the fiber. We do this also when designing “retroreflecting” prisms. Compared with ordinary reflection from mirrors, the sharpness and brightness of totally internally reflected light beams is enhanced considerably.
The calculation of the critical angle of incidence for any two optical media—whenever light is incident from the medium of higher index—is accomplished with Snell’s law. Referring to Ray 3 in Figure 3-10 and using Snell’s law in Equation 3-2 appropriately, we have
where ni is the index for the incident medium, ic is the critical angle of incidence, nr is the index for the medium of lower index, and r = 90° is the angle of refraction at the critical angle. Then, since sin 90° = 1, we obtain for the critical angle,