## Wednesday, 2 June 2010

### Thin-film interference

Constructive and destructive interference of light waves is also the reason why thin films, such as soap bubbles, show colorful patterns. This is known as thin-film interference, because it is the interference of light waves reflecting off the top surface of a film with the waves reflecting from the bottom surface. To obtain a nice colored pattern, the thickness of the film has to be on the order of the wavelength of light.
Consider the case of a thin film of oil floating on water. Thin-film interference can take place if these two light waves interfere constructively:
1. the light from the air reflecting off the top surface
2. the light traveling from the air, through the oil, reflecting off the bottom surface, traveling back through the oil and out into the air again.
An important consideration in determining whether these waves interfere constructively or destructively is the fact that whenever light reflects off a surface of higher index of refraction, a 180� phase shift in the wave is introduced.

Light in air, reflecting off just about anything (glass, water, oil, etc.) will undergo a 180� shift. On the other hand, light in oil, which has a higher n than water does, will have no phase shift if it reflects off an oil-water interface. Note that a shift by 180� is equivalent to the wave traveling a distance of half a wavelength.
To get constructive interference, the two reflected waves have to be shifted by an integer multiple of wavelengths. This must account for any phase shift introduced by a reflection off a higher-n material, as well as for the extra distance traveled by the wave traveling down and back through the film. With the oil film example, constructive interference will occur if the film thickness is 1/4 wavelength, 3/4 wavelength, 5/4, etc. Destructive interference occurs when the thickness of the oil film is 1/2 wavelength, 1 wavelength, 3/2 wavelength, etc.

In the case of 1/4 wavelength, the wave reflected off the top surface is shifted by 1/2 a wavelength by the reflection. The wave traveling through the film has no phase shift, but travels a total down-and-back distance of 1/2 wavelength, meaning that it will be in phase with the wave reflected from the top. On the other hand, if the film thickness is 1/2 wavelength, the first wave gets a 1/2 wavelength shift and the other gets a wavelength shift; these waves would cancel each other out.
Note that one has to be very careful in dealing with the wavelength, because the wavelength depends on the index of refraction. The film thickness, for constructive interference in the example above, has to be 1/4 (or 3/4 or 5/4 or ...) of the wavelength of the light in the oil. This wavelength is related to the wavelength in vacuum (which differs negligibly from the wavelength in air) by:

The cancellation (destructive interference) of reflected light waves is utilized to make non-reflective coatings. Such coatings are commonly found on some camera lenses or binocular lenses, and often have a bluish tint. The coating is put over glass, and the coating material generally has an index of refraction less than that of glass. In that case, then, both reflected waves have a 180� phase shift, and a film thickness of 1/4 wavelength (in the film) would produce a net shift of 1/2 wavelength, resulting in cancellation.

For non-reflective coatings, then, the minimum film thickness required is:

where n is the index of refraction of the coating material.
Note that you have to be very careful to account for whether a phase shift occurs at an interface where reflection is taking place. In some cases, the minimum film thickness required for constructive interference is a quarter of the wavelength; in other cases, the minimum film thickness must be half a wavelength for constructive interference to take place. It all depends on whether or not a phase shift occurs for reflections at both interfaces, one interface, or neither interface.