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 using Rayleigh 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 replotted 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 10^{6}
: thus implying that the scattered intensity is proportional to r^{6}.
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
Previous page: Airy theory and rainbows 
