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As noted on the previous page, non-absorbing materials have real values of refractive index m = n + i k. Absorbing materials have complex values of refractive index.: for example, in the visible spectrum, the refractive index of water has a very small imaginary component, such as 1.33257 + i 0.0000000164 at a wavelength λ = 0.65 µm. In practice, it is usual to ignore the imaginary
component in calculations of scattering from water - thus assuming that water is non-absorbing.

Scattering
of red light
(wavelength λ = 0.65 µm) from a sphere of radius 100 µm for
refractive indices of 1.33257 and 1.33257 + i 1000Fig. 1 |

Fig. 1 shows the effect of absorption on scattering of red light
from a sphere with *r* = 100 µm: the red/brown curves show
results for a refractive index of 1.33257 (i.e. non-absorbing), whilst the blue
curves show results for a refractive index of 1.33257 + i1000 (i.e.
very absorbing). For the non-absorbing sphere, the primary rainbow and
its supernumerary arcs are clearly visible at scattering angles around
140°.
For the absorbing sphere, the scattered intensity is constant between
30°
and 180° - and, perhaps surprisingly, almost as strong as the
primary rainbow for the non-absorbing sphere. For the
absorbing sphere, the scattered light is the vector sum of light
diffracted by the sphere and light reflected from the exterior of the
sphere. Diffraction is the dominant mechanism between 0° and
30° but Fig. 2 below shows that the intensity of the light
reflected from the exterior of an absorbing sphere is independent of
scattering angle - coinciding with the constant intensity shown in Fig.
1 between
30°
and 180°.

As
Fig. 1, but showing only external reflection from a sphere of radius
100 µm for
refractive indices of 1.33257 and 1.33257 + i 1000Fig. 2 |

Debye series (Fig. 3 p = 2) scattering of red light (wavelength λ = 0.65 µm) from a sphere with radius r = 100
µm for real refractive index of 1.33257 and different values of imaginary refractive index. |

Fig. 3 demonstrates that the *p* = 2 rays (those suffering one internal reflection and generating the primary rainbow) reduce in
intensity as the imaginary value of refractive index is increased, in line with the following formula:

where

λ is the wavelength

Debye series Fig. 4 p = 0 and p = 2 scattering of red light (wavelength λ = 0.65 µm) from an absorbing sphere with radius r = 100 µm (refractive index = 1.33257 + i 0.001) |

Fig. 4 shows that, for refractive index = 1.33257 + i 0.001 (mildly absorbing), the intensity of the Debye *p* = 0 contribution (due to
diffraction and external reflection) is greater than that of the *p* = 2 rays.

Mie scattering of red light (wavelength λ = 0.65 µm) from an absorbing sphere with radius Fig 5 r = 100 µm (refractive index = 1.33257 + i 0.001) |

Comparison of Mie scattering with the vector sum of Debye Fig. 6p = 0 and p = 2 contributions for scattering of red light (wavelength λ = 0.65
µm) from an absorbing sphere with radius r = 100 µm (refractive index = 1.33257 + i 0.001) |

Comparison of Mie scattering with the vector sum of Debye Fig. 7p = 0, p = 2 and p = 3 contributions for scattering of red light (wavelength λ = 0.65 µm) from an absorbing sphere with radius r = 100 µm
(refractive
index = 1.33257 + i 0.001) |

Figs. 5, 6 and 7 compare Mie theory and Debye series calculations for scattering angles between 120° and 150° (for the same conditions as Fig. 4). In all of graphs, the results of Mie theory calculations are shown by red (and brown) curves. In Fig. 6, the blue curves showing the vector sum of the Debye series

*Page updated on 30 November 2003*

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