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How many wavelengths are needed to produce a good simulation?

The continuous spectrum of sunlight can be approximated by using N discrete wavelengths spaced evenly between 0.38 µm and 0.7 µm. Normal colorimetric practice uses a relatively small number of wavelengths (e.g. 7 - 30) but Mie calculations can give misleading results when only a few wavelengths are used.  It must be emphasised that this problem is not caused by the spectral characteristics of the incident light: indeed, such problems occur even if a flat spectrum is used. 

The fundamental cause is that the fine ripple structure visible in graphs of Mie scattering of monochromatic light is very dependent on wavelength. As calculations for multiple wavelengths involve the addition of intensity curves for different wavelengths, the ripple structure will be reinforced if the maxima and minima for different wavelengths coincide: for example, the maxima and minima of the ripple structure shown here for 650 nm coincide approximately with those for 650.9 nm, 651.8 nm and so on. 

The following diagrams explore the effect of variations of N on simulations of rainbows, glories and coronas.
 

Fig. 1   Simulation of the primary rainbow (for unpolarised light) for a water drop of radius 200 µm 
              for the specified numbers of wavelengths spaced evenly between 0.38 µm and 0.7 µm

Fig. 1 indicates that simulations of the primary rainbow for r = 200 µm are almost identical when calculated using more than 15 wavelengths, apart from minor changes in brightness.  Hence, there is little value in using more than 30 wavelengths for this condition. 
Fig. 2   Simulation of the primary and secondary rainbows (for unpolarised light) 
              for a water drop of radius 100 µm for the specified number of wavelengths
Fig. 3   Simulation of the secondary rainbow (for unpolarised light) for 
              a water drop of radius 100 µm for the specified number of wavelengths

For r = 100 µm, Fig. 2 shows minor variations in the appearance of the primary rainbow (137° - 150°) for 10 or more wavelengths, but the secondary rainbow (120° - 129°)  shows significant ripples when less than 100 wavelengths are used.  The secondary rainbow is shown in much greater detail in Fig. 3, indicating that perhaps 300  wavelengths are needed for consistent simulations of the secondary rainbow.
Fig. 4   Simulation of the glory (for unpolarised light) for a water drop 
               of radius 10 µm for the specified  number of wavelengths
Fig. 4 indicates that 10 µm drops result in a glory consisting of a bright red circle centred on the anti-solar point with a radius of ~2.1° (scattering angle of 177.9°) and a weaker red circle with a radius of ~3.7° (176.3°).

The variations between the horizontal stripes of Fig. 4 show that MiePlot simulations of the glory are very sensitive to the number of wavelengths used in the calculations.  Whereas the simulations of the primary and secondary rainbows in Figs. 1 - 3 are fairly consistent with 30 or more wavelengths, it is obvious from Fig. 4 that many more wavelengths (more than 500) are required to achieve an accurate simulation of the glory.
 

Fig. 5   Simulation of the corona (for unpolarised light) for a water drop 
                of radius 3 µm for the specified  number of wavelengths
Fig. 5 indicates that, unlike the glory and the rainbow, consistent simulations of the corona can be produced using only a small number of wavelengths.

Updated on 22 July 2002
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