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:

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.