Interference Involving Multiple Reflections

Some of the most beautiful effects of interference result from the multiple reflection of light between the two surfaces of a thin film of transparent material. These effects require no special apparatus for their production or observation and are familiar to anyone who has noticed the colors shown by thin films of oil on water, by soap bubbles, or by cracks in a piece of glass. We begin our investigation of this class of interference by considering the somewhat idealized case of reflection and refraction from the boundary separating different optical media. In Fig. 14A(a) a ray of light in air or vacuum incident on a plane surface of a transparent medium like water is indicated by a. The reflected and refracted rays are indicated by ar and at, respectively. A question of particular interest from the standpoint of physical optics is that of a possible abrupt change of phase of waves when they are reflected from a boundary. For a given boundary the result will differ, as we shall now show, according to whether the waves approach from the side of higher velocity or from that of lower velocity. Thus, let the symbol a in the left-hand part of Fig. 14A represent the amplitude (not the intensity) of a set of waves striking the surface, let r be the fraction of the amplitude reflected, and let t be the fraction transmitted. The amplitudes of the two sets of waves will then be ar and at, as shown, Now, following a treatment given by