Dispersion and Scattering

When a wavelength changes index of refraction, there is some scattering of the wavelengths and this phenomena is called dispersion.

Dispersion

Different frequencies of light contain different phase velocities because of the material they pass through's properties. When the velocity differs, dispersion occurs. The most recognizable act of dispersion is seen in most transparent materials, when there is a decrease in index of refraction, which leads to an increase in wavelength.

Normal Dispersion

Dispersion that occurs in wavelength ranges where the material does not absorb light is normal dispersion. When a light travels through a transparent material, there is a decrease in index of refraction, which leads to an increase in the wavelength of the light. This is an example of normal dispersion, because the material doesn't absorb the light. Another example is the separation of colors that occurs when light passes through a glass prism. When the light comes into the surface of the prism, the light incident to the normal, at a certain angle, θ, will be refracted at an angle that is equal to arcsin(sin(θ)/n), according to Snell's law. This demonstrates why there is a rainbow patter, because a blue light, for example, has a high refractive index so its wavelength would be bent more strongly than red light, because red light has a smaller refractive index.

Material Dispersion

When light has a wavelength range that results in the medium having significant absorbing, the index of refraction can increase with wavelength, and result in material dispersion. This type of dispersion is characterized by the Abbe's number: [math]\displaystyle{ {\frac{{n}_{D} - 1}{{n}_{F} - {n}_{C}}} }[/math], which gives a simple measure of dispersion based on the index of refraction at three specific wavelengths. The Abbe number is also known as the v value, a low v value implies high dispersion ("Abbe).

Dispersion Delay Parameter

The different kinds of dispersion cause changes in the group characteristics of the wave. These group characteristics are the features of the wave packet that change with the same frequency as the amplitude of the wave. "Group velocity dispersion" occurs as a spreading-out of the signal "envelope" of the radiation. This dispersion can be calculated with a group dispersion delay parameter: [math]\displaystyle{ D = {\frac{1}{{v}_{g}^2}} {\frac{d{v}_{g}}{dλ}} }[/math], and we know that group velocity, [math]\displaystyle{ {v}_{g} }[/math], is [math]\displaystyle{ {\frac{c}{n-λ{\frac{dn}{dλ}}}} }[/math], where n is the index of refraction and c is the speed of light through a vacuum. Combining these two equations gives us a simpler form of the dispersion delay parameter:[math]\displaystyle{ D = {\frac{-λ}{c}} {\frac{d^2n}{dλ^2}} }[/math]

If D is less than zero, the medium is said to have positive dispersion or normal dispersion. If D is greater than zero, the medium has negative dispersion. If a light pulse is propagated through a normally dispersive medium, the result is the higher frequency components slow down more than the lower frequency components. The pulse therefore becomes positively chirped, or up-chirped, increasing in frequency with time. This causes the spectrum coming out of a prism to appear with red light the least refracted and blue/violet light the most refracted. Conversely, if a pulse travels through an anomalously (negatively) dispersive medium, high frequency components travel faster than the lower ones, and the pulse becomes negatively chirped, or down-chirped, decreasing in frequency with time.

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References

1. "Abbe Number." Wikipedia. Wikimedia Foundation. Web. 4 Dec. 2015. <https://en.wikipedia.org/wiki/Abbe_number>.

2. "Dispersion." HyperPhysics. HyperPhysics. Web. 1 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/dispersion.html>.

3. "Optics." Wikipedia. Wikimedia Foundation, 2015. Web. 2 Dec. 2015. <https://en.wikipedia.org/wiki/Optics#Dispersion_and_scattering>.