From The Gemology Project
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As explained in nature of light, light travels in transversal waves.

Fig.1: Lightwave

The peaks of the waves are called crests, the valleys are named troughs.

The amplitude defines the intensity of the color, the wavelenght is the distance between two crests (expressed in nanometers (nm) ).

When white light hits the surface of a transparent material, like a gemstone, part of the light is reflected while another part is refracted inside the gemstone. The refracted ray will eventually reach another boundary and the same process repeats itself.

The part of the light that was refracted in the first run will be partically reflected and leave the gemstone.
These two parts (both the origional reflected and refracted/reflected waves) will combine again after leaving the stone, trying to join together.

This can best be explained by a thin film (such as oil) laying on the street (Fig.2).

Fig.2: Thin film

The incident light (I) reaches the oil and part gets reflected (R1) while another part gets refracted inside the oil. The refracted part then reaches the boundary of oil/street and gets reflected out of the oil (R2).
Both R1 and R2 travel parallel to each other and try to combine.

When R1 and R2 travel with the crests and troughs next to one and other, the two rays will constructively interfere with one and other, creating a more intense color. Creating a play of color as seen in oil slick or bath bubbles.

When one ray lags behind the other at a wavelength that is a half, they will destructively interfere and cancel each other out. So no color will be seen.

Fig.3: Waves in phase

Here the two waves travel with their crests and troughs with synchronized wavelengths and they will enforce one and other (Fig.3).
As seen in the picture, the result will be a doubling in amplitude, creating a more intense wave. This is what causes the play of colors due to interference.

A doubling of amplitude means a quadrupling of the intensity of the wavelength. Meaning that the color will be more intense. Wavelengths with low amplitude will make them appear duller or even black.

Note that the wavelength doesn't change.

Fig.4: Waves out of phase

In Fig.4 you see that one ray is lagging behind the other at half a wavelength. The crests of the one are at the same position as the troughs of the other. This is what we call retardation.
Retardation is the lagging of one ray behind the other.

In this case the retardation is half a wavelength, so they cancel each other out. The result is a wave that has no amplitude.

A wave with no amplitude is black.

When two rays lag behind each other at a full wavelength (or 2 or 3 etc), we say that they are in phase and will constructively interfere, so play of color will be observed.
If the retardation is a half a wavelength (or 1.5 etc), the waves will cancel each other out by destructive interference and we say the two rays are out of phase.