n1 < n2 and n2 > n3: t =
n1 < n2 < n3: t =
Non-reflective: t =

#### Thin-film interference

Interference between light waves is the reason that thin films, such as soap bubbles, show colorful patterns. This is known as thin-film interference - interference between light waves reflecting off the top surface of a film with waves reflecting from the bottom surface. To obtain a nice colored pattern, the thickness of the film has to be comparable to the wavelength of light.

For completely constructive interference to occur, the two reflected waves must be shifted by an integer multiple of wavelengths relative to one another. This relative shift includes any phase shifts introduced by reflections off a higher-n medium, as well as the extra distance traveled by the wave that goes down and back through the film.

Note that one has to be very careful in dealing with the wavelength, because the wavelength depends on the medium the wave is in. For thin-film interference the key wavelength is ln, the wavelength in the film itself. For a film with index of refraction n, this wavelength is related to l, the wavelength in vacuum, by:

ln = l/n