﻿ MasteringPhysics 2.0: Problem Print View
Why Butterfly Wings Shimmer
Description: Basic conceptual and quantitative questions on thin film interference, followed by an application to the iridescence of butterfly wings.
Learning Goal: To understand the concept of thin-film interference and how to apply it.

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.

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 due to reflection from a medium with higher index of refraction. (Because of the structure of the butterflies' 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 . How much farther does the light reflected from the back surface travel than the light reflected from the front surface?
 2*d
Part B
For constructive interference to occur, the difference between the two paths must be an integer multiple of the wavelength of the light (as is true in any interference problem), i.e. the general criterion for constructive interference is , where is a positive integer. This is usually stated in the slightly more explicit form

.

Given the thickness of the film , what is the longest wavelength that can exhibit constructive interference?
 Â  Â = 2*d
Part C

If you have a thin film of thickness , what is the third-longest wavelength of light that exhibits constructive interference with the reflected light?

Note that this corresponds to .

 Â  Â = 200 Â
Part D

The criterion for destructive interference is very similar to the criterion for constructive interference. For destructive interference to occur, the difference between the two paths must be some integer number of wavelengths plus half a wavelength:

,

or

,

where is a nonnegative integer. What is the second-longest wavelength that will not be visible (i.e., will have strong destructive interference for the reflected waves) when reflected from a film of thickness ?

Note that the longest wavelength corresponds to for destructive interference. This is why the notation used for the second-longest wavelength is instead of .

 Â  Â = 400

The blue morpho butterfly lives in tropical rainforests and can have a wingspan greater than 15 centimeters. The brilliant blue color of its wings is a result of thin-film interference. A pigment would not produce such vibrant, pure colors. What cannot be conveyed by a picture is that the colors vary with the viewing angle, which causes the shimmering iridescence of the actual butterfly.

The scales of the butterfly's wings consist of two thin layers of keratin (a transparent substance with index of refraction greater than one), separated by a 200-nanometer gap filled with air.

Part E
What wavelength of light would be strongly reflected at normal incidence? The keratin layers are thin enough that you can think of them simply as marking the surfaces of a 200-nanometer "film" of air.

 Â  Â = 400 Â

This wavelength is near the cutoff between visible (violet) and ultraviolet light, so the shorter wavelengths that are strongly reflected will not be relevant to what humans see when they look at the butterfly.

To understand why the color changes with viewing angle, try drawing a diagram of light incident on a thin film at a large angle. The distance within the film will be increased, but the light reflected from the front surface will have to travel further to the observer outside of the film than the light reflected from the back surface. The increased distance outside of the material for the front surface reflection actually makes the net path-length difference smaller than it would be for normal incidence. As viewing angle increases, the largest wavelength that experiences constructive interference gets shorter. Thus, while the butterfly is blue at normal incidence, at large angle of incidence no visible light is short enough wavelength to be strongly reflected.

Part F

The wavelength that we have been discussing is technically the wavelength of light within the medium of the film. It is important to remember this, since most thin-film problems will involve films with index of refraction different from that of air. Suppose that the butterfly gets wet, thus filling the gaps between the keratin sheets with water (). What wavelength in air will be strongly reflected now?

Note that the wavelength within the film was already determined in the previous problem.

Hint F.1 Hint not displayed