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Light dispersion

This applet shows the phenomenon of the chromatic dispersion of light. Firstly it shows light dispersion through a prism. It also shows rainbows as an example of light dispersion.

'Dispersion in a prism' window

This first window displays the dispersion of light when it goes through a isosceles triangular prism. In order to simplify the study of the problem of dispersion, we only consider transmissions in the lateral surfaces, and do not take into account the reflections or the transmission in the base of the prism.

Thus, a light beam arrives at the first surface of the prism, it is refracted by the material of the prism, and finally the light exits refracted by the second surface of the prism, giving a spectral decomposition of the light, since the refractive index of the prism depends on the wavelength. This phenomenon is known as light dispersion.

The parameters that can be varied are:

• "Angle of the prism". It corresponds to the angle of the vertex of the prism between the exit and entrance surfaces.
  
• "Angle of incidence". It is the angle between the incident ray and the normal direction to the surface of incidence of the prism.
  

The other parameter that can be altered by the applet is the nature of the material. This can be done by varying the refractive index of the material. For this the applet has some sliders to modify the values of n0, A and B. These values correspond to the terms of the Cauchy equation to represent the refractive index of a material:

n = n0 + A / λ² + B / λ⁴

Another option is to select a known material from the list that is offered upon these sliders, so that the parameters n0, A and B are fixed for the type of glass that has been selected from the possible to be chosen: fused silica, 2 BK 7, BAK 1, F or LaK 10.

For the incident light in the prism, the program allows a spectral decomposition of the light of up to seven different wavelengths, that they can be selected with the sliders at the right part of the window. There is also the option to select a predetermined spectral distribution for the 7 wavelengths instead of using the free selection option. In the list that can be selected upon the sliders appears the options of white light, which presents seven wavelengths equally distributed in the visible interval; Hg, that presents the main wavelengths of the spectral emission of a Mercury source; Cd, for the main lines of a cadmium source; and Zn, for the main lines of a zinc source.

The results obtained using these parameters are displayed on first place in the window in top left hand corner of this tab, where a representation of the dispersion of the light through the prism is displayed. Under this representation the angular distribution for the different refracted wavelengths is displayed in respect to the angle of deviation. This angle corresponds to the deviation between the emergent ray and the incident one. Clicking the zoom button this representation is increased between the extreme values of the angles of deviation. These angles of deviation are also calculated for each wavelength.

The applet also presents the refractive indexes for two main lines of emission of Hydrogen and one of the Helium, which are usually used to calibrate the refractive indexes of different materials. nd corresponds to the line of 587,6 nm of Helium, and nF to the line of of 486.1 nm and nC to the one of 656.3 nm of the Hydrogen.

These values can be used to calculate the number of Abbe:

ν_d = (nd-1) / (nF-nC)

which gives an idea of the chromatic dispersion of the glass. If, ν_d>50 the glass is little dispersive and is defined as a crown type, otherwise when ν_d<50 the glass is very dispersive and is defined as a flint type glass.

At the right of the prism display there are two other graphs. The first, "Refraction index", presents the refraction index of the glass as a function of the wavelength, noting the positions of the lines of emission of the Hydrogen and the Helium. The second graph, "Angle of deviation", presents the angle of deviation as a function of the angle of incidence for the different wavelengths. In this graph can be observed that there is a angle of incidence for which the angle of deviation is minimum. This minimum angle of deviation is different for each wavelength, in this case what is marked is the angle of incidence for the minimum deviation for the orange line of the emission spectrum of the Helium, λ_d=587.6 nm. Also it is possible to see in the same graph that there is not dispersion until certain angle of incidence, because there is total reflection in the second surface when the limit angle of refraction is surpassed.

'Rainbow' window

This second window presents the study of the formation of a rainbow, consequence of the dispersion of the light through the water drops present in the atmosphere.

When a light ray arrives at a water drop takes place the chromatic dispersion of the light. This applet allows the user to vary the wavelength, λ, of the incident ray and the parameter of impact, b, (the height where the ray hits in relation to the radius) on two drops of water. in top left hand corner of this tab we can see the representation of two water drops, the first one represents a drop in which the ray hits over the horizontal diameter and have only one inner reflection before leaving, in this case the primary rainbow is obtained. The second drop represents the case in which the impact takes place below the horizontal diameter and, after two internal reflections, the light is transmitted exiting the drop and giving the secondary rainbow as a result. Because the refractive index of the water depends on the wavelength, the angle in which the light leaves the drop is different for each colour although the light has the same input angle. These angles can be calculated from the impact parameter, sin α=b and sin β= b/n.

From these angles also the angles of deviation can be calculated, δ, for the primary and secondary rainbows:

δ_p = π + 2α - 4β

δ_s = 6β - 2α

In the bottom right corner there is a graph of the angles of deviation as a function of the impact parameter. The curve in the top corresponds to the primary rainbow and the lower curve to the secondary one. It is possible to appreciate that these two curves do not cross giving a dark zone between the primary and secondary rainbows denominated Alexander dark band.

The rainbow is only visible when the angle of view corresponds to the situation in which the light is reflected in the conditions of minimum deviation, i.e., a minimum or a maximum in the curves of the previous graph. These will be different for the primary and the secondary rainbows. In the right superior part we can see a scheme of the formation of the two rainbows, the primary and the secondary ones, where the angle of view is shown for each rainbow respect the horizontal for each wavelength. This angle of view corresponds to the complementary of the angle of minimum deviation. The angular width of the primary rainbow is 1.84º, the width of the secondary rainbow is 3.21º and the separation between both rainbows is 8º.

So for the light to leave the drop with an angle of minimum deviation it only depends on the parameter of impact, which in this position can be expressed as a function of the refractive index, therefore slightly different for each wavelength, which causes the chromatic dispersion. For the primary rainbow we have:

And for the secondary rainbow: