# Interference Colors

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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## Interference Criteria

 As the two beams of polarized light pass through a crystal, they travel at different speeds and get out of phase. The slow ray is said to be retarded and the phase difference is called retardationIf the retardation is a whole number of wavelengths (left), the beams recombine with the same orientation as when they entered the crystal. These wavelengths will be blocked by the upper polarizer. If the retardation is a whole number of wavelengths plus one-half(right), the beams recombine with an orientation perpendicular to the original direction of polarization. These wavelengths will be fully transmitted by the upper polarizer.

### Retardation and Interference Color

What determines the retardation of a crystal? Consider a crystal of thickness h. The index of refraction of its slow ray is ns and that of its fast ray is nf. The value of ns is greater than nf.

We can calculate:

• Velocities of the Beams
• Velocity of fast ray: vf = c/nf
• Velocity of slow ray: vs = c/ns
• Travel time through the crystal:
• For the fast ray: tf = h/vf = hnf/c
• For the slow ray: ts = h/vs = hns/c
• Lag time of the slow ray: ts - tf = h(ns - nf)/c
• Amount fast ray leads slow ray once both exit
• Lead = lag time x c = h(ns - nf) = Retardation

Thus Retardation = h(ns - nf). Retardation depends only on the thickness of the material and the difference in refractive index for the two beams, called the birefringence. Thickness is controlled by grinding thin sections to a standard thickness, usually .035 mm. This figure is not arbitrary; it is designed to give quartz a consistent appearance in thin sections.

For a given retardation, some wavelengths will satisfy the transmission condition (retardation = integer plus one-half wavelength), while others will be blocked (retardation = whole number of wavelengths). Consider the case below.

 Note that we may have to consider wavelengths outside the visible range to get a complete picture, as for the red end of the spectrum here. Also, the rules of color combination are those for light, not paint. Red and green make yellow, not a mess; blue and yellow make white, not green. This particular case will result in a fairly bright blue.

A given retardation always results in the same combination of wavelengths and always results in the same color. The sequence of colors that results from increasing retardation is one of the basic facts of optical mineralogy.

Note that zero retardation satisfies the blockage criteria for all wavelengths. If there is zero retardation, the light recombines in the same orientation as it had originally and is blocked by the upper polarizer. Some materials have the same refractive index in all directions and always produce zero retardation. Such materials are called isotropic. Noncrystalline materials like glass are isotropic, so are isometric minerals like garnet or fluorite. All other materials are termed anisotropic.

 For low retardations, no visible wavelengths are blocked or fully transmitted. We see part of the whole spectrum and a neutral gray or white. As retardation reaches 4000 A, the first blockage reaches the blue end of the spectrum and the transmitted color shifts to yellow, then red.At 5500 A the blockage is in the green, and we see magenta, the result of mixing red and violet. Each multiple of 5500 A will result in a magenta hue. The sequence of colors from one magenta to the next is termed an order. For very large retardations, several wavelengths are blocked while others are transmitted. As more and more windows in the spectrum appear, the colors become progressively more pale. Finally they approach white, but a warmer off-white rather than the cold neutral white of low retardation.
Retardation A Order Colors Notes
0 Zero Black
0-5500 First Gray, White, Yellow, Red Neutral colors are cold, yellows dull.
5500-11000 Second Violet Through Spectrum to Red Purest colors, though not totally pure
11000-16500 Third Violet Through Spectrum to Red Have a "fluorescent" appearance
16500 and up Fourth and higher Mostly greens and pinks Colors become more washed out with increasing retardation

### Anomalous interference colors

Some minerals show colors that are not part of the standard interference color sequence. Such colors are termed anomalous. Anomalous colors can result when:

• The mineral is strongly colored and its natural color combines with the interference color.
• The refractive index of the material changes markedly with wavelength, so that different wavelengths satisfy different interference criteria. In some cases the mineral may be isotropic for some wavelengths but not others.
• The privileged directions of the material vary with wavelength, so the mineral never goes completely to extinction at any one time.

Anomalous interference colors tend to be dark. Blue, violet, green and brown are all possible. These are actually useful diagnostic features. Chlorite tends to show anomalous greens and browns. Epidote commonly shows a dark "denim" blue color near extinction. When combined with its normal bright yellow and red interference colors, the effect is quite colorful.

Created 15 Sep 1997, Last Update 14 Dec 2009

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