# Diffraction by a sphere

For forward scattering (i.e. for scattering angle θ close to 0°), the dominant mechanism is generally that of diffraction, which can be modelled by the following equation (see note 1): The characteristic rings of the corona are due mainly to the term J1[x sin θ] in the above equation.

This term has its first 4 maxima when x sin θ = 5.136, 8.417, 11.62 and 14.796, which correspond to sin θ = 0.8174 λ/r , 1.3398 λ/r , 1.8494 λ/r and 2.3549 λ/r, respectively.

These simple relationships based on the Fraunhofer equation allow us to predict the angular size of the rings of the corona for specific wavelengths λ as shown in Table 1 below.

 Corona ring Redλ = 0.65 μm Greenλ = 0.51 μm Blueλ = 0.44 μm θ1 30.4/r 23.9/r 20.6/r θ2 49.9/r 39.1/r 33.8/r θ3 68.9/r 54.0/r 46.6/r θ4 87.7/r 68.8/r 59.4/r

Table 1  Angular radius θ (measured in degrees) of the first 4 rings of the corona for the specified
monochromatic wavelength as a function of radius r (measured in μm)

The values in Table 1 show formulas giving the angular size of the 4 inner rings of the corona for illumination by monochromatic red, green and blue light. For example, for r = 10 μm and λ = 0.65 μm, maxima will occur at θ ≈ 3°, 5°, 6.9° and 8.8°. Strictly speaking, these values are valid only when sin θ = θ, but this approximation is reasonable for the small values of θ corresponding to the corona.

These values should NOT be used for illumination by white light because they can give very misleading results. To illustrate the nature of this problem, Fig. 1 below shows the results of calculations for the scattering of sunlight (i.e. 0.38 - 0.7 μm). Fig. 1     Scattering of sunlight (& lambda; = 0.38 - 0.7 μm) by a water droplet of radius r = 10 μm

The graph in Fig. 1 shows 6 individual curves at equally-spaced wavelengths across the spectrum of sunlight, whilst the upper curve shows the sum of the individual curves. The red, green and blue vertical lines show the positions of the maxima defined in Table 1. The coloured bars above the graph show simulations of the corona calculated using the Fraunhofer equation at 300 wavelengths between 0.38 μm and 0.7 μm. The brightness of the lower bar has been normalised to the peak intensity (at θ = 0°) but this means that the rings of the corona are barely visible. The brightness of the middle bar has been increased by a factor of 15 compared with the lower bar, thus allowing part of the corona to become visible. The upper bar shows the saturated colours of the corona (irrespective of the intensity of the corona).

The coloured vertical lines at the top of the graph show the predicted positions of the corona's rings for red, green and blue light (using the values from Table 1). The red lines show reasonable agreement with the red parts of the coloured bars at the top of Fig. 1 at θ1 ≈ 3°, θ2 ≈ 5°, θ3 ≈ 7° and θ4 ≈ 9°. However, it is important to note that the coloured bars show an additional red ring at θ ≈ 1.6° which does not coincide with a maximum in the Fraunhofer equation. As the angular size of the red rings in a natural corona around the sun or the moon is frequently used to estimate the size of the scattering particles, it is imperative to take account of this extra red ring at θ ≈ 16/r.

The green parts of the coloured bars at θ ≈ 2.6° and θ ≈ 4° are close to the values predicted by the Fraunhofer equations for green light - but the other green parts of the coloured bars do not match the values predicted in Table 1. Similarly, for blue light, only the blue-coloured part at θ ≈ 2° matches the values predicted in Table 1. The lack of agreement for the higher-order rings is because the colour of the scattered light is determined not by a single colour, but by a mixture of colours (e.g. the maximum for red light at θ ≈ 6.9° coincides with a maximum for green light).

Table 1 indicates that the angular size of the corona's rings is inversely proportional to the radius r of the scattering particle. This is illustrated in Fig. 2 below, which shows the scattered intensity for λ = 0.65 μm for r = 100, 10 and 1 μm. Fig. 2   Diffraction theory calculations for λ = 0.65 µm and r = 100, 10 and 1 μm Fig. 3   Calculations of scattering using Mie theory and diffraction for r = 100 µm, λ = 0.65 µm, n = 1.33257,  perpendicular polarisation Fig. 4   Calculations of scattering using Mie theory and diffraction for r = 10 µm, λ = 0.65 µm, n = 1.33257,  perpendicular polarisation > Fig. 5   Calculations of scattering using Mie theory and diffraction for r = 1 µm, λ = 0.65 µm, n = 1.33257,  perpendicular polarisation

Figs. 3 - 5 compare the results of Mie theory calculations with the diffraction pattern.  Note that the diffraction model is NOT dependent on the refractive index of the sphere, whereas Mie theory takes account of other scattering mechanisms (such as direct transmission through the sphere) which are dependent on refractive index.  In practice, the diffraction model is reasonably accurate for forward scattering for large drops (e.g. x = 1000), at least for the first few maxima and minima.  However, it becomes much less accurate for smaller drops. The next page contains much more information about the accuracy of the diffraction model.   In essence, the diffraction model should not be used for particles of radius r < 5 μm. It must also be emphasised that the diffraction model is appropriate only for calculations of forward scattering.

Note 1: Similar equations are frequently attributed to Fraunhofer, but Craig Bohren has pointed out that, without diminishing the importance of Fraunhofers pioneering work in experimental optics, there is no evidence to suggest that Fraunhofer developed any theoretical treatment of diffraction. Consequently, Fresnel-Fraunhofer-Airy-Schwerd might be a more appropriate designation. Schwerds contribution is described in Hoover, R.B. and Harris, F. S. "Die Beugungserscheinungen: a Tribute to F. M. Schwerds Monumental Work on Fraunhofer Diffraction" [Applied Optics, 8, 21612164 (1969)].

Page updated on 23 November 2012