The optics of a water drop
Mie scattering

by Philip Laven

Fig. 1 Primary and secondary rainbows at sunset in New York

Fig. 2 A glory surrounding the shadow of an aircraft on a cloud

The scattering of sunlight by spherical drops of water causes a surprising variety of complicated optical effects, such as those shown in Figs. 1 and 2. 

In the 17th century, Descartes and Newton used geometrical optics to explain the formation of rainbows.  In 1838, Airy extended our understanding of rainbows by including the effects of diffraction and interference - thus explaining the formation of supernumerary arcs which were not predicted by geometrical optics.  Although such techniques can be used to provide excellent simulations of rainbows, they are inadequate for simulation of glories.

In 1908, Gustav Mie developed a rigorous method to calculate the intensity of light scattered by uniform spheres.  Although Mie's solution was precise, it involved a huge number of calculations and was rarely used until about 20 years ago when supercomputers became available for scientific research. 

Rapid advances in computer power mean that now you can explore the optics of spherical raindrops using the Mie algorithm on a personal computer - for example, by using the freely available MiePlot computer program.  Some examples of the output of the MiePlot program are shown below:

Fig. 3    MiePlot graph of intensity of primary and secondary rainbows 
for r =100µm water drops

The curves in Fig. 3 show the intensity of the scattered light as a function of scattering angle, whilst the coloured horizontal stripes above the graph show the resulting brightness and colour  of the scattered light: the top stripe is for perpendicular polarisation, the middle stripe is for parallel polarisation and the bottom stripe is for unpolarised light.  Note that the stripe for parallel polarisation is almost black, indicating that rainbows are strongly polarised. 

In addition to graphs of intensity, the MiePlot program can also provide simulations matching photographs of optical effects caused by scattering of light from spherical drops of water - as shown in Fig. 4 below:

Fig. 4  MiePlot simulation for r = 500 µm water drops superimposed on a photograph of a primary and secondary rainbow

Although Mie theory is not necessary for simulations of rainbows, Mie theory comes into its own for simulations of glories as shown in Fig. 5 below:

Fig. 5   MiePlot simulation of scattering of sunlight from r = 4.8 µm water drops superimposed on a digital image of a glory taken from a commercial aircraft.

The other pages on this web site show the results of different types of Mie calculations performed by the MiePlot computer program. 

Page updated on 13 December 2002

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