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Mathematical models of scattering

Although Mie theory and the Debye series offer exact solutions for scattering of electromagnetic waves from a homogeneous sphere, these calculations can be very time consuming.  Table 1 shows the approximate computation time using MiePlot on a PC with a 600 MHz Pentium processor.

Calculation method
r = 10 µm
r = 100 µm
r = 1000 µm
Mie theory
Debye series
Airy theory

Table 1  Duration (seconds) of MiePlot calculations using different mathematical models
Wavelength λ = 0.65 µm,  Refractive index n = 1.33257
Scattering angles: 120° - 150°,  Angular resolution: 0.01°

For Mie theory and the Debye series, the computation time is roughly proportional to the radius r of the sphere: thus increasing the radius by a factor of 10 increases the computation time by a factor of about 10.  On the other hand, calculations using Airy theory are extremely quick and essentially independent of r.

Under certain circumstances, much simpler mathematical models can yield accurate results in a fraction of the time.  For example, Airy theory can provide good simulations of rainbows.  Other calculation methods, such as ray tracing, are important from the perspective of the history of science - as well as offering insights into the process of scattering.

MiePlot offers the following additional mathematical models of scattering of light by a sphere:

When is it safe to use these other mathematical models?

The key parameter is the radius r of the scattering sphere compared with the wavelength (λ) of the incident light.

Fig. 1   Mie theory calculations of scattering at 0° and 139° as a function of radius r for λ = 0.65 µm, n = 1.33257 and perpendicular polarisation

Fig. 1 shows Mie theory calculations for scattering of light with wavelength λ = 0.65 µm: the red line represents scattering at 0°, whilst the blue line represents scattering at 139° (i.e. close to the geometric primary rainbow angle of 137.8° for refractive index n = 1.33257).  Careful examination of Fig. 1 shows that much of the red line could be approximated by two straight lines.  As the blue line coincides with the red line when r < 0.05 µm, Fig. 1 suggests that (for perpendicular polarisation) the scattered intensity is independent of scattering angle when r << λ.

Fig. 2   Calculations of scattering at 0° and 139° as a function of radius r
for λ = 0.65 µm, n = 1.33257 and perpendicular polarisation

Fig. 2 shows the results obtained using different mathematical models. The red and grey lines show calculations of forward scattering (i.e. at 0°).  The red line corresponds to Rayleigh scattering, which is valid when r << λ.  The grey line is based on diffraction, which is the dominant mechanism for forward scattering when r >> λ.  Note that, in both cases, the scattered intensity increases with r: the intensity is proportional to r6 for Rayleigh scattering and proportional to r4 for diffraction.

The blue line in Fig. 2 shows the results obtained with Airy theory for a scattering angle of 139° (for p = 2 rays which have been subject to one internal reflection in the sphere) .  The maximum near r = 200 µm is due to the primary rainbow, whereas the maxima at higher values of r are actually supernumeraries.  For r = 480 µm, the primary rainbow has a maximum at 138.2°, with the first supernumerary arc at 139°.  Similarly, for r = 875 µm, maxima occur at 138.1°, 138.65° and 139°.  Comparison of the blue lines in Figs. 1 and 2 suggests that Airy theory offers a reasonable approximation for calculation of the primary rainbow and its supernumeraries.  Further comparisons of Airy theory and Mie theory are shown on the next page.

Fig. 3   Calculations of scattering at 0° as a function of radius r
for λ = 0.65 µm and n = 1.33257

Fig. 3 concentrates on the range between r = 0.1 µm and r = 10 µm where neither Rayleigh scattering nor diffraction calculations are appropriate models for forward scattering (for λ = 0.65 µm).  When r and λ are similar, accurate calculations demand use of Mie theory.

Page updated on 4 August 2003

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