Fig. 1 Results of Mie theory calculations for scattering of light of λ = 650 nm from a bubble of radius 10 μm and refractive index n1 = 1
immersed in water of refractive index n0 = 1.33257
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Much of this web site is concerned with scattering of light by spherical droplets of water (e.g. by rain or fog). However, this section examines the scattering of red light (λ = 0.65 μm = 650 nm) by a spherical bubble immersed in water.
Note that the refractive index n1 of the bubble is assumed to be 1 (corresponding to a vacuum) and that the refractive index n0 of the surrounding water is 1.33257. Of course, this does not correspond to a practical situation since the bubble would collapse! A more realistic situation would be an air bubble (i.e. refractive index n1 ≈ 1.0003) in water - but the results of scattering calculations are essentially the same whether the bubble is filled with air or "contains" a vacuum.
These pages will consider the scattering mechanisms relevant to air bubbles in water - as well as answering the question "Do air bubbles in water cause rainbows, coronas and/or glories?"
Fig.2 shows that the dominant scattering contributions come from the p = 0, p = 1 and p = 2 terms of the Debye series but, as θ → 180°, higher order scattering cannot be neglected. As Fig. 2 is very crowded, it is difficult to identify the various scattering contributions. Therefore, the following set of graphs show each term of the Debye series results individually, as well as comparing the results with the geometrical optics approximation.
The forward-scattering peak of intensity at θ = 0° in Fig. 3 corresponds to diffraction - which is included in the Debye p = 0 term along with reflection from the exterior of the sphere. However, the geometrical optics approximation for p = 0 considers only reflection from the exterior of the sphere.
Note that horizontal blue line in Fig. 3 between θ = 0° and θ ≈ 82.8° corresponds to isotropic scattering due to the fact that incident rays with impact parameter b > [n1 / n0] ≈ 0.75 do not penetrate into the sphere because all of the incident light is reflected from the exterior of the sphere. This is similar to the concept of total internal reflection, which may be familiar from studies of scattering at plane boundaries such as between water and air. If light is travelling into a medium of lower refractive index, light rays with angles of incidence greater than the critical angle are subject to total internal reflection but, in this case, it might be more appropriate to call it "total external reflection"!
Fig. 4 shows that the geometrical optics approximations gives unsatisfactory results for p = 1 scattering when θ > 60°. In this region, the Debye p = 1 curves show the effects of surface waves – as indicated by parallel polarisation being dominant, and by the fact that the curve of scattered intensity is approximately straight on this log-linear graph (thus implying a roughly exponential attenuation of the surface waves as a function of scattering angle). The ripples seen on the the Debye p = 1 curves when θ > 160° could be considered to be a glory, but this glory will not be visible in practice because the intensity of such scattering is very much weaker than the p = 0, p = 2 and p = 3 terms.
Whereas p = 2 scattering from water droplets results in the primary rainbow and the glory, Fig. 5 does not show any such effects for p = 2 scattering from an air bubble in water. However, Fig. 5 does show maxima at intervals of approximately 2.2° centred on the forward scattering direction θ = 0°. Given the very close agreement between Debye and geometric optics calculations (at least when θ < 30°), these maxima must be explainable by interference between geometric p = 2 rays. .
Taking an arbitrary example of scattering angle θ = 10°, Fig. 6 shows that there are two geometric p = 2 rays that result in scattering at θ = 10° from an air bubble in water. As θ changes, the path difference between these two types of rays also varies, causing constructive and destructive interference which results in successive maxima and minima as a function of θ. Similar explanations can be applied to the regular patterns of maxima and minima seen on Figs. 7 and 8 below.
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The graphs in Fig. 9 summarise the behaviour of geometric rays for 0 ≤ p ≤ 4. The graph on the left shows the angle through which incident rays are deviated. Due to symmetry of the spherical bubble, these deviation angles can be reduced into the range 0° ≤ θ ≤ 180°, as shown in the graph on the right.
Similar graphs for scattering from a water droplet in air show maxima or minima for values of p > 1, but no such maxima or minima appear in Fig. 9. As these maxima and minima correspond to rainbows, it can be concluded that scattering of light from air bubbles in water does not produce rainbows.
As noted above in Fig. 7, several incident rays can produce scattering at a given angle θ - for example, scattering at θ = 30° can be due to two geometrical p = 2 rays, three p = 3 rays and four p = 4 rays.
On the other hand, as shown in the following sequence of diagrams below in Fig. 10, as the impact parameter b approaches 0.75, high-order scattering (such as p = 8) results in the light rays following progressively shorter chords until their paths approximate the circumference of the bubble. Furthermore, when b = 0.75, θ ≈ 82.8° for ALL values of p.
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