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 ("Dispersion).

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 of the three specific wavelengths of the Fraunhofer lines. 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 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 value of D is less than zero, then the medium has positive, also known as normal, dispersion. If a light wave is propagated through a normally dispersive medium, then the initial waves’ higher frequency components are slowed down more than its lower frequency components. Because of this, the pulse becomes positively chirped, which means that the frequency increases with time. This is the reason for the spectrum that comes out of a prism to appear with red light as the least refracted and blue/violet light as the most refracted.

One the other hand, if D is greater than zero, the medium has negative dispersion. This means that when a light travels through an negatively (anomalously) medium, the initial waves’ high frequency components travel faster than its lower frequency componenets, and its frequency decreases with time, meaning that the pulse becomes negatively chirped ("Optics).

Example

Given the indices of refraction, calculate the v values for crown glass and flint glass. The refraction indices corresponding to the wavelengths of the Fraunhofer lines, F, D, and C, are 1.5293, 1.5230, and 1.5204, respectively, for crown glass. For flint glass, the refraction indices corresponding to the wavelengths of the Fraunhofer lines, F, D, and C, are 1.7378, 1.7200, and 1.7130, respectively. Once the v values have been calculated, explain which of the two glasses has a higher dispersion.

[math]\displaystyle{ {v}_{crown} = {\frac{{n}_{D} - 1}{{n}_{F} - {n}_{C}}} = {\frac{1.5230 - 1}{1.5293 - 1.5204}} = 58.76 }[/math]

[math]\displaystyle{ {v}_{flint} = {\frac{{n}_{D} - 1}{{n}_{F} - {n}_{C}}} = {\frac{1.7200 - 1}{1.7378 - 1.7139}} = 29.03 }[/math]

Since the v value of crown class is higher, it has a lower dispersion. So the glass with the highest dispersion is flint glass.

External links

Dispersion on HyperPhysics

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>.