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Modelling the Corona
(Diffraction or Mie theory)

The previous page indicated that the diffraction model is adequate for calculation of forward scattering from large spheres.  However, one of the most interesting natural phenomena associated with forward scattering is the corona, which appears as coloured rings surrounding the cloud-covered sun or moon.

Is the diffraction model appropriate for simulations of the corona?  The Lee diagrams shown below attempt to answer this question.  Figs. 1 - 3 have been calculated using the diffraction model, whereas Figs. 4 - 6 have been calculated using Mie theory.

Fig. 1    Diffraction Fig. 2    Diffraction (brightness x 10) Fig. 3    Diffraction (saturated colours)
Fig. 4    Mie theory Fig. 5    Mie theory (brightness x 10) Fig. 6    Mie theory (saturated colours)

Fig. 1 shows the light scattered by spheres of radius r between 1 µm and 10 µm when illuminated by sunlight.  Note the logarithmic horizontal scale.  As the brightest scattering occurs at 0°, Fig. 1 contains little information because the corona is very much darker than the forward scattered light - and computer displays cannot reproduce the necessary large dynamic range.  To overcome this problem, Fig. 2 shows what happens when the brightness of Fig. 1 is increased by a factor of 10.  The top part of Fig. 2 becomes "over-exposed", but the coloured rings of the corona become visible.

The data from Fig. 1 is replotted in Fig. 3 so as to remove the brightness information, whilst showing the saturated colour of each pixel: thus showing that, according to the diffraction model, the sequence of colours in the corona is independent of r.

In particular, the red rings of the corona appear at θ1 ≈ 16/r, θ2 ≈ 31/r, θ3 ≈ 50/r, θ4 ≈ 70/r and θ5 ≈ 88/r (where θ is measured in degrees and the radius r is measured in μm). For more information about the origin of these values, please see the previous page.

As the outer rings tend towards white, Fig. 3 suggests that no more than 2 rings of the corona will be visible even under optimum viewing conditions.  Although diagrams based on saturated colours (such as Fig. 3) are useful for comparing colours, it must be emphasised that they do not represent the appearance of scattered light.

Looking now at the equivalent diagrams (Figs. 4 - 6) produced using Mie theory, the key difference between Fig. 1 and Fig. 4 is that, for r between 1 µm and 2 µm, Mie theory predicts uniform bands of colour at scattering angles less than 8°.  Comparison of Fig. 2 and Fig. 5 indicates that Mie theory produces very complex patterns for r < 3 µm.  Comparison of Fig. 3 and Fig. 6 shows that Mie theory and diffraction calculations produce similar results for r > 5 µm, but the diffraction model is inadequate for r < 3 µm.  Fig. 6 also confirms the findings of Gedzelman & Lock 1 who reported that "The sequence of corona colors changes rapidly for small droplets but becomes fixed once droplet radius exceeds about 6 µm".

Fig. 7    Mie theory (saturated colours) for r between 1 µm and 2 µm

Fig. 7 provides a closer view of Fig. 6 for r between 1 µm and  2 µm.  Looking at Fig. 7, it seems that there are "islands" of fairly uniform colour.  White lines seem to form boundaries between some of these islands - as well as merging with other white lines at various points in the diagram.  What causes this complex pattern of colours?

Elsewhere on this site, the Debye series has been used to analyse the corona for monochromatic light.  A similar approach can be used to analyse the corona caused by scattering of white light for r = 1.4 µm - which corresponds to the dotted vertical line in Fig. 7.  For this value of r, Fig. 7 indicates that the scattered light is blue-green for scattering angles between 0° and 5°, yellow between 7° and 10°, white near 12° and various shades of pale blue between 13° and 20°.

Fig. 8 Calculations for scattering of sunlight from a sphere of radius r = 1.4 µm using Mie theory and the Debye series
Note that the stripes above the graph show "saturated colours"

The Mie theory curve in Fig. 8 is due to a combination of the Debye p = 0 and p = 1 contributions.  As indicated by the colours of the plotted lines and by the coloured horizontal stripes above the graph, the Debyep= 0 contribution produces very bright colours, whilst thep= 1 contribution is colourless.  The horizontal stripes also show the saturated colours predicted by other scattering models.  For example, there are only minor differences between Mie theory and "Debye 0 + 1", which corresponds to the vector sum of the Debyep= 0 and p = 1 contributions.  Similarly, diffraction calculations agree with the Debye p = 0 calculations for angles below 10°, but the results from two methods become progressively mis-aligned at higher angles.

Fig. 8 shows that the corona for r = 1.4 µm is due to mixture of modes - thus explaining why the colours in Fig. 7 and the left side of Fig. 6 do not follow a simple pattern.  On the other hand, the colours on the right side of Fig. 6 follow a regular pattern because coronas for r > 5 µm are essentially due to Debye p = 0 contributions.

Fig. 9 Calculations for scattering of unpolarised light of 7 wavelengths from a sphere of radius r = 1.4 µm using the diffraction model
Note that the stripes above the graph show "saturated colours"

Fig. 9 shows diffraction calculations for 7 wavelengths across the visible spectrum (actually at wavelengths of 402.9, 448.6, 494.3, 540, 585.7, 631.4 and 677.1 nm).  Note that the upper curve is the sum of the curves for individual wavelengths.  The variations in the colours of the upper curve correspond to the saturated colours.

Fig. 10     Calculations for scattering of unpolarised light of 7 wavelengths from a sphere of radius r = 1.4 µm using Mie theory
Note that the stripes above the graph show "saturated colours"

Fig. 10 shows the equivalent results of calculations using Mie theory.  Unlike Fig. 9, there is no obvious regular pattern of maxima and minima.  This complicated behaviour is caused by the mixture of Debye p = 0 and  p = 1 contributions.  For example, the sharp minimum for λ =448.6 nm shown in Fig. 10 at about 8.7° is caused by the Debye p = 0 and p = 1 contributions having similar amplitudes but a phase differenceof 180°.

In summary, Mie theory is required to simulate coronas when r < 5 µm, whilst diffraction calculations are adequate for larger values of r.

All of the above calculations have assumed that the scattering particle is a water droplet with refractive index n1 ≈ 1.33 in a medium of refractive index n0 = 1 (i.e in a vacuum).   However, the Fraunhofer equation (see the previous page) does not even mention refractive index! On the other hand, Mie theory takes full account of n1 and n0. In practice, Mie theory calculations show that diffraction is independent of the particle's refractive index n1. Many people are surprised by this result, but it makes sense because the process of diffraction does not involve light entering the particle (i.e. diffraction can be considered to occur around the the edges of an opaque particle). It is interesting to note that the glory which appears as a backscattering phenomenon when θ → 180° is also independent of n1.2

Nevertheless, Mie theory calculations show that the diffraction pattern is very dependent on the refractive index n0 of the surrounding medium. Although the Fraunhofer equation does not explicitly mention n0, it is based on the wavelength λ of the incident light in the surrounding medium: note that λ = n0 λ0 where λ0 is the wavelength in a vacuum. Consequently, calculations of diffraction using the Fraunhofer approximation show the correct results for the specified value of refractive index n0 of the medium.


References

1    S. D. Gedzelman & J. A. Lock, "Simulating coronas in color", Applied Optics, 42, 3, pp. 497 - 504, 20 January 2003

2    P. Laven, "Effects of refractive index on glories," Appl. Opt. 47, H133-H142 (2008) — available on this web site here

Page updated on 23 November 2012
 
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