Thin-film interference is a commonly observed phenomenon. it causes the bright colors in soap bubbles and oil slicks. it also leads to the iridescent colors on many insects and bird feathers. in this problem, you will learn how to work with thin-film interference and see how it creates the dazzling display of a tropical butterfly's wings.

(figure 1) when light is incident on a thin film, some of the light will be reflected at the front surface of the film, and the rest will be transmitted into the film. some of the transmitted light will be reflected from the back surface of the film. the light reflected from the front surface and the light reflected from the back surface will interfere. depending upon the thickness of the film, this interference may be constructive or destructive. we will be studying the interference of light normal to the surface of the film. the figure shows the light entering at a small angle to normal only for the purpose of showing the incident and reflected rays.

for this problem, you will only be concerned with the geometric aspects of thin-film interference, so ignore phase shifts caused by reflection from a medium with higher index of refraction. (because of the structure of a butterfly's wings, such phase shifts do not contribute much to what you actually see when you look at the butterfly.)

part a

assume that light is incident normal to the surface of a film of thickness d. how much farther does the light reflected from the back surface travel than the light reflected from the front surface?

express your answer in terms of d.