# Resonant scattering from a spherical particle Fig. 1   Mie calculations of intensity of scattering of red light (λ = 650 nm) by a spherical droplet as a function of refractive index n of the droplet.
Droplet radius = 10 μm (corresponding to size parameter x = 96.664). Refractive index of the medium = 1. Scattering angle θ = 150°.

Fig.1 shows various sharp maxima as a function of refractive index n of the droplet - for example, the four maxima between n = 1.33 and n = 1.34 are marked by the letters A - D (A and B for perpendicular polarisation, C and D for parallel polarisation). A and C are relatively broad maxima, whereas B and D are extremely narrow. It is interesting to note that the maxima shown in Fig. 1 (and also in the other graphs on this page) tend to occur in pairs - in which a broad maximum (e.g. A) occurs close to a narrow maximum (e.g. B) leading to the observation that A and B (or C and D) seem to be linked in some way. (a) Refractive index of droplet = 1.33 (b) Refractive index of droplet = 1.331 (c) Refractive index of droplet = 1.332 (d) Refractive index of droplet = 1.333 (e) Refractive index of droplet = 1.334

Fig. 2 (a) - (e)    Intensity of scattering of red light (λ = 650 nm) by a spherical droplet as a function of size parameter x = 2 π r / λ of the droplet,
where r is the radius of the droplet and λ is the wavelength of the light.
Refractive index of the medium = 1. Scattering angle θ = 150°.

Fig. 1 shows the maxima occurring as a function of the refractive index n of the droplet. However, Fig. 2 shows that similar patterns occur when intensity is plotted as a function of size parameter x at a fixed value of refractive index n. Fig. 2 consists of five separate graphs (a) - (e) showing results for values of n between 1.33 and 1.334.: Fig. 3   As Fig. 2(a) except that the scattering angle θ = 170°.

Note that the locations of the maxima seem to be almost independent of the scattering angle θ - as can be seen by comparing Fig. 2(a) for θ = 150° with Fig. 3 for θ = 170°. Although the graphs show different intensities, the maxima A, B and D occur at essentially the same values of size parameter x - but the maximum C seems to disappear as θ increases from 150° to 170°.

Visual inspection of Fig. 2 (a) - (e) suggests that the locations of the maxima A - D move downwards in terms of x as the refractive index n is increased. This behaviour is quantified in Table 1 below:

 A B C D Refractive index n x x * n x x * n x x * n x x * n 1.33 96.744 128.6695 96.8315 128.7859 97.0205 129.0373 97.204 129.2813 1.331 96.6745 128.6738 96.761 128.7889 96.9535 129.0451 97.134 129.2854 1.332 96.6055 128.6785 96.6905 128.7917 96.8865 129.0578 97.0645 129.2899 1.333 96.536 128.6823 96.62 128.7945 96.82 129.0611 96.9955 129.295 1.334 96.4665 128.6863 96.55 128.7977 96.7535 129.0692 96.926 129.2993

Table 1   Locations of the maxima A, B, C and D as identified in Figs. 2 (a) - (e)

Table 1 shows that the products of x * n for each of the maxima A - D is roughly constant (e.g.maximum A seems to occur when the product x * n ≈ 128.68, whereas B seems to occur when x * n ≈ 128.79). Using these values of the products x * n, we can now identify the maxima A - D in Fig. 4 below which shows a false-colour map of the intensity scattered at θ = 150° as a function of refractive index n of the droplet and its size parameter x. Fig. 4   False-colour map showing the intensity of scattering of red light (λ = 650 nm)
by a spherical droplet as a function of refractive index n of the droplet and as a function of the size parameter x.
Refractive index of the medium = 1. Scattering angle θ = 150°. Fig. 5   As Fig. 4, except that scattering angle θ = 170°.

Although these sharp maxima appear as a function of refractive index n of the droplet and as a function of its size parameter x, it is noteworthy that such maxima do not appear on graphs of intensity as a function of scattering angle θ.

Page updated on 4 June 2010