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# Airy theory and rainbows

George Biddell Airy's paper (1) entitled "On the intensity of light in the neighbourhood of a caustic"was published in 1838.  In this paper, he showed that the intensity of light in a rainbow could be modelled using a cubic wave-front. He provided a table of values of what is now universally known as the "Airy Integral" and wrote:

"The extent of this table for the positive values of m is not so great as I would wish; but it goes far enough to enable us to point out the most remarkable circumstances of the distribution of illumination."
The rather wistful first part of this remark suggested that Airy had devoted substantial personal effort to evaluating his integral.  With the aid of modern computers, calculations using Airy theory have now become almost trivial.  Despite the development of much more complex mathematical models, Airy theory remains a useful tool.  Although it was originally developed as a model for the primary rainbow (p = 2), Airy theory has been extended to apply to arbitrary values of p (2,3) and to deal with both perpendicular and parallel polarisations.(4) Fig. 1   Comparison of p = 2 rainbow calculated by Mie theory and Airy theory             r = 103.4057 µm, λ = 0.65 µm,  x = 2 π r / λ = 1000             n = 1.333,  perpendicular polarisation

Fig. 1 compares Mie theory and Airy theory for the primary rainbow for x = 1000 and n 1.333.

Fig. 1 demonstrates that Airy theory provides a good approximation of the general features of the primary rainbow, such as the broad maximum around 139° and the maxima of the supernumerary arcs around 141.1°, 142.6° and so on.  The ripples on the Mie theory calculations are due to interference between the p = 2 rainbow rays (which have suffered one internal reflections within the sphere) and p = 0 rays (which are reflected from the exterior of the sphere) - as explained here.  As the calculations using Airy theory only concern the effects of p = 2 rays, it is perhaps rather unfair to complain that Airy theory does not reproduce the ripples predicted by Mie theory.  It is probably fairer to compare Airy theory with calculations using Debye series for a specific value of p (such as p = 2 for the primary rainbow) - as shown below in Fig. 2. Fig. 2   Comparison of p = 2 rainbow calculated by Airy theory and Debye series             r = 103.4057 µm, λ = 0.65 µm,  x = 2 π r / λ = 1000             n = 1.333,  perpendicular polarisation

Fig. 2 shows that Airy theory agrees very closely with the results provided by the rigorous Debye series - despite the relative simplicity of the Airy calculations.  However, Airy theory miscalculates the scattering angles for the maxima and minima of the supernumerary arcs above 142°.

Note that Figs. 2 - 6 on this page are equivalent to Figs. 3 - 7 in the paper by Hovenac and Lock. (3) Fig. 3   Comparison of p = 3 rainbow calculated by Airy theory and Debye series             r = 103.4057 µm, λ = 0.65 µm,  x = 2 π r / λ = 1000             n = 1.333,  perpendicular polarisation

Fig. 3 extends the comparison to the secondary rainbow (p = 3).  In this case, Airy theory is quite accurate in predicting the principal maximum around 127.5°, but much less accurate for the supernumerary arcs - in terms of scattering angles and intensity. Fig. 4   Comparison of p = 4 rainbow calculated by Airy theory and Debye series             r = 103.4057 µm, λ = 0.65 µm,  x = 2 π r / λ = 1000             n = 1.333,  perpendicular polarisation

Similar comparisons of Airy theory and the Debye series for higher order rainbows (p = 4, 5 and 6) are shown in Figs. 4 - 6 respectively.  In each case, Airy theory seems adequate for calculation of the principal maximum, but much less accurate for the supernumeraries. Fig. 5   Comparison of p = 5 rainbow calculated by Airy theory and Debye series             r = 103.4057 µm, λ = 0.65 µm,  x = 2 π r / λ = 1000,             n = 1.333  perpendicular polarisation Fig. 6   Comparison of p = 6 rainbow calculated by Airy theory and Debye series             r = 103.4057 µm, λ = 0.65 µm,  x = 2 π r / λ = 1000,             n = 1.333  perpendicular polarisation

Of course, "monochromatic" rainbows caused by illumination of a sphere by light of a single wavelength do not correspond to the popular notion of coloured rainbows.  It is therefore appropriate to examine the performance of Airy theory in predicting natural rainbows (i.e. those due to the scattering of sunlight). Figs. 7 and 8 show the results of calculations using Mie theory and Airy theory.  Fig. 7   Simulation of primary and secondary rainbows using Mie theory             r = 100 µm, sunlight  Fig. 8   Simulation of primary and secondary rainbows using Airy theory             r = 100 µm, sunlight

Comparison of Figs. 7 and 8 confirms that Airy theory can be successfully used to simulate rainbows caused by sunlight, but also reveals several key differences:

• Mie theory predicts that Alexander's dark band between the primary and secondary rainbows will be much brighter than that predicted by Airy theory. This discrepancy can be understood by examining Fig. 9 below, which uses the Debye series to demonstrate that contributions from p = 0 rays are dominant in this region.
• For parallel polarisation, there are some minor errors on the primary rainbow  - but much more significant errors on the secondary rainbow.  In practice, these errors can probably be ignored because the rainbow caused by perpendicularly polarised light is much stronger.
• The coloured bars above Figs. 7 and 8 representing the visual appearance of the rainbows are effectively identical, except that Airy theory slightly misplaces the supernumerary arcs above 145°.  Fig. 9   Simulation of primary and secondary rainbows using Debye series             r = 100 µm, sunlight

As outlined by Lee (5), an excellent method of comparing Airy theory with Mie theory is to plot "Lee diagrams" (which are described elsewhere on this web site).  Figs. 10 and 11 are Lee diagrams showing primary rainbows caused by sunlight falling on water drops for r between r = 10 µm and r = 1000 µm.  Fig. 10   Lee diagram calculated using Mie theory Fig. 11   Lee diagram calculated using Airy theory

Comparison of Figs. 10 and 11 shows very close agreement between the results obtained with Mie theory and Airy theory.  Nevertheless, there are some subtle differences, such as the dark gaps between the supernumerary arcs (e.g. near 142° for r = 50 µm) being darker and more clearly defined when using Airy theory.  Fig. 11 is not as accurate than Fig. 10 - but Fig. 11 was computed in less than 2 hours, whilst Fig. 10 took almost 1 week!

Airy theory can be successfully used to model scattering in the vicinity of rainbow angles - at least for the primary (p = 2) and secondary (p = 3) rainbows.  However, Mie theory is essential for more complicated scattering mechanisms, such as those causing the glory.

MiePlot offers the option of calculations based on:

• Mie theory
• Debye series
• Ray tracing (based on geometrical optics)
• Ray tracing including the effects of interference between rays
• Airy theory
• Rayleigh scattering
• Diffraction

References:
1    G. B. Airy, "On the intensity of light in the neighbourhood of a caustic", Transactions of the Cambridge Philosophical Society, 6, 3, pp. 397-402 (1838)
2    R. T. Wang & H. C. van de Hulst, "Rainbows: Mie computations and the Airy approximation", Applied Optics, 30, 1, pp. 106-117 (1991)
3    E. Hovenac and J. A. Lock, "Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series", Journal of Optical Society of America A, 9, 5, pp. 781-795 (1992) Free download
4    G. P. Können and J. H. de Boer, "Polarized rainbow", Applied Optics, 18, 12, pp. 1961-1965 (1979) Free download
5    "Mie theory, Airy theory, and the natural rainbow",  Raymond L. Lee Jr.   Applied Optics, 37, 9, pp.1506 - 1520 (1998) Free download

Page updated on 23 April 2010

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