3.4 Interference in thin films  (Page 3/7)

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Strategy

Solution

Thus, the smallest constructive thickness $t_{\text{c}}$ is

The next thickness that gives constructive interference is $t_{\text{c}}^{\prime }=3{\lambda }_n\text{/}4$ , so that

Finally, the third thickness producing constructive interference is $t_{\text{c}}^{\prime }=5{\lambda }_n\text{/}4$ , so that

b. For destructive interference, the path length difference here is an integral multiple of the wavelength. The first occurs for zero thickness, since there is a phase change at the top surface, that is,

the very thin (or negligibly thin) case discussed above. The first non-zero thickness producing destructive interference is

Substituting known values gives

Finally, the third destructive thickness is $2t_{\text{d}}^{\text{″}}=2{\lambda }_n$ , so that

Significance

Another example of thin-film interference can be seen when microscope slides are separated (see [link] ). The slides are very flat, so that the wedge of air between them increases in thickness very uniformly. A phase change occurs at the second surface but not the first, so a dark band forms where the slides touch. The rainbow colors of constructive interference repeat, going from violet to red again and again as the distance between the slides increases. As the layer of air increases, the bands become more difficult to see, because slight changes in incident angle have greater effects on path length differences. If monochromatic light instead of white light is used, then bright and dark bands are obtained rather than repeating rainbow colors.