Fringes of Equal Thickness

one of the effects of the optics of thin films. In constrast to fringes of equal inclination, they are observed directly on the surface of a transparent film of varying thickness (Figure 1).

Figure 1. The path difference of interfering rays reflected from the upper and lower boundaries of a thin film depends on the angles of incidence of the illuminating rays. Even in the case of extended light sources, however, the spread of these angles is so small that the path difference acquired at point M of the film by rays 1–1′ and 2–2′, which are emitted by different parts (S₁ and S₂) of the source, are practically the same. The fringes of equal thickness are therefore localized directly on the surface of the film and can be observed without auxiliary optical instruments—the lens in the figure could be the crystalline lens of the eye. M’ is the point on the retina (or, when a supplementary lens is used, on the screen) where the image of the point M of the surface of the film—that is, one of the points on a line of equal thickness—is focused.

The occurrence of fringes of equal thickness is due to interference of light reflected from the front and rear boundaries of the film (fringes of equal thickness in reflected light) or of light transmitted directly through the film and light twice reflected at its boundaries (fringes of equal thickness in transmitted light). Only fringes of equal monochromatic or nearly monochromatic light—light whose wavelengths λ lie in a comparatively small interval—are usually fringes in the strict sense of the word—that is, alternate dark and light distinct bands. In addition, the maxima and minima of the fringe intensity coincide with those lines on the surface of the film along which the path difference of the interfering rays is the same and is equal to an integral number of half-wavelengths. The geometric thickness of the layer is the same along these lines—hence the term “fringes of equal thickness.” When white light is used for illumination, superposition of the fringes corresponding to rays with different λ creates a complex, rainbow-colored pattern in which the fringes of rays with certain λ are often indistinguishable.

Fringes of equal thickness are responsible for the rainbow coloring of thin films, such as soap bubbles, patches of oil and gasoline on water, and films of oxides on metals; they are the cause of temper color. Such fringes are often used for determining the microrelief of thin plates and films and for making precise measurements with different kinds of interferometers and other instruments. Newton’s rings are an example of fringes of equal thickness.