Rayleigh scattering

Rayleigh scattering is applicable when the radius (r) of the scattering sphere is much smaller than the wavelength (λ) of the incident light. For example, for red light (λ = 0.65 µm), calculations usingRayleigh scattering when r < 0.01 µm are essentially identical to the rigorous results obtained using Mie theory.

Rayleigh scattering is defined by the following equations:

**Fig. 1** Rayleigh scattering calculations for r = 0.01 µm, λ = 0.65 µm, n = 1.33257 (logarithmic scale for intensity)

Fig. 1 shows that the intensity for perpendicular polarisation is independent of scattering angle - whereas the intensity for parallel polarisation varies with scattering angle with a minimum at 90°.

**Fig. 2** Rayleigh scattering calculations for r = 0.01 µm, λ = 0.65 µm, n = 1.33257 (linear scale for intensity)

These results in Fig. 1 are re-plotted in Fig. 2 using a linear scale for intensity. Fig. 2 illustrates clearly the variation of the intensity for parallel polarisation.

**Fig. 3** Rayleigh scattering calculations for r = 0.01 µm, λ = 0.65 µm, n = 1.33257
Polar plot with linear scale for intensity.

Alternatively, these results can be shown as a polar plot - as in Fig. 3. If the incident light is unpolarised, Fig. 3 indicates that the light scattered at angles near 90° will be strongly polarised.

**Fig. 4** Rayleigh scattering calculations for r = 0.01 µm and r = 0.001 µm, λ = 0.65 µm, n = 1.33257 (logarithmic scale for intensity)

Fig. 4 illustrates that when the radius of the sphere is increased by a factor of 10 (e.g. from r = 0.01 µm to r = 0.001 µm) the intensity of Rayleigh scattering increases by a factor of 10⁶: thus implying that the scattered intensity is proportional to r⁶.

**Fig. 5** Rayleigh scattering at 0° or 180° for r = 0.01 µm as a function of wavelength for scattering from a drop of water and from a sphere of fixed refractive index (logarithmic scales for intensity and for wavelength)

The variation of scattering as a function of wavelength is complicated by the fact that the refractive index of water varies slightly across the visible spectrum, varying from 1.34658 at 380 nm to 1.33141 at 700 nm. The intensity of Rayleigh scattering from a drop of water (taking into account this variation in refractive index) is shown by the red line in Fig. 5, whereas the result that would be obtained for a sphere with a constant refractive index n of 1.33257 is shown by the blue line. The blue line shows that increasing the wavelength λ by a factor of 2 (e.g. from 350 nm to 700 nm) changes the intensity by a factor of 1/16. In other words, since 2^-4 = 1/16, Rayleigh scattering is proportional to λ^-4.

**Fig. 6** Rayleigh scattering calculations for r = 0.01 µm, Sunlight (linear scale for intensity)

For sunlight incident on spheres of radius r = 0.01 µm, the scattered light is blue because Rayleigh scattering is more intense at shorter wavelengths. Note that the coloured bars show very clearly that the scattered light is strongly polarised at angles near 90°. Although the blue colour of the sky is caused by Rayleigh scattering from aerosols or from molecules in the atmosphere, rather than from tiny droplets of water, the polarisation effects shown in Fig. 6 can be readily observed on cloudless days by anybody wearing polarising sun glasses!

Page updated on 14 May 2003

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