How are glories formed?

MiePlot uses Mie theory to simulate glories, but it does not offer any explanation for their formation.  However, MiePlot can also use the Debye series reformulation of the Mie series to separate the contributions made by various scattering processes (e.g. by rays of order p that have suffered p - 1 internal reflections).

The Debye series goes far beyond the limitations of geometric optics by including the effects of diffraction and surface waves. It should be noted that the Debye series is NOT an approximation: the summation of the Debye series for all integer values of p from zero to infinity gives the same result as Mie theory. More information about the use of the Debye series is given here.


Fig. 1   Using the Debye series to investigate scattering of white light by r = 10 µm water droplets

The Debye series results shown in Fig. 1 demonstrate that p = 2 rays (i.e. those that have suffered one internal reflection in the sphere) are dominant in forming the glory, but substantial contributions in the vicinity of θ = 180° are also made by p = 11, p = 7 and p = 6 rays.  The coloured bars above Fig. 1 show only the Debye p = 2 contributions.  

Fig. 2   Simulation of glory caused by r = 10 µm water droplets
                Mie theory (left) and Debye series for p = 2 rays (right)

To facilitate a direct comparison of the calculation methods, Fig. 2 shows simulations for Mie scattering (which includes the contributions from ray paths for all values of p) and for the Debye series for p = 2 rays. The main difference is that a bright white central zone (θ → 180°) is visible with Mie scattering.  Fig.1 suggests that higher order rays (e.g. p = 6,  7 and 11) also contribute to the central feature.

Fig. 3   As Fig. 1, but using a logarithmic scale for intensity
The sequence of lines from top to bottom at 175° is p = 2 parallel polarisation, p = 2 perpendicular polarisation, p = 0 both polarisations, p = 11 perpendicular polarisation, p = 6 parallel polarisation, P = 11 parallel polarisation, p = 7 perpendicular polarisation, p = 7 parallel polarisation and p = 6 perpendicular polarisation

Fig. 3 is similar to Fig. 1, except that it uses a logarithmic scale for intensity - thus showing more clearly the contributions of higher order rays in the Debye series.  The coloured parts of the glory between θ = 175° and θ = 178° are obviously due to Debye p = 2 rays.  Note that the colours of the plotted lines represent the saturated colours, as described here. In particular, the p = 0 ray (caused by reflection from the exterior of the sphere and by diffraction) is essentially white.  However, the contributions from higher order rays show colours which are essentially identical to those for p = 2, 6 and 11 rays.  We will see later that this is not a coincidence .....

As explained elsewhere on this web site, ray tracing using geometric optics can be successfully used to provide simple explanations of primary and secondary rainbows - especially when interference between two or more rays of light is taken into account.  Can such techniques be used to explain the glory? 

Fig. 4 below shows the results of geometric optics calculations for p = 2 rays for a refractive index of 1.333.  The values marked along the curves correspond to the impact parameter b, where b = 0 indicates rays passing through the centre of the sphere (resulting in θ = 180°) and b = ±1 indicates rays incident at the edge of the sphere (resulting in θ = 165.6° for a refractive index of 1.333). The primary rainbow at θ = 137.8° is due to rays around b = 0.861.  Between θ = 138° and θ = 150°, two rays of similar intensity contribute to the scattered light: interference between these two rays produces the supernumerary arcs of the primary rainbow. For the glory, we need to concentrate on angles θ above 170° - where Fig. 4 shows that only central rays (i.e. |b| < 0.2) are relevant. However, the scattered light generated by such rays is not responsible for the glory.  Calculations for the scattering of sunlight by these rays results in relatively uniform white light for θ > 170°. 


Fig. 4    Graph of intensity versus scattering angle θ for geometric p = 2 rays with refractive index n = 1.333 (the values marked along the curves correspond to the impact parameter b)


In 1947, van de Hulst1 suggested that glories are caused by p = 2 rays with b = ±1 and postulated that the 14.4° gap between θ = 165.6° and θ = 180° could be bridged by surface waves - as indicated in Fig. 5 below. For simplicity, Fig. 5 shows this gap of 14.4° as the last part of the ray path before it emerges from the sphere, but this gap could be covered by three separate segments of surface waves covering a total of 14.4°. Because of symmetry of the sphere, ray paths of the form shown in Fig. 5 generate a toroidal wave front, with a diameter nearly equal to that of the sphere, propagating in the direction θ = 180°. van de Hulst explained the glory as the interference pattern corresponding to this toroidal wave front.

Fig. 5  van de Hulst’s surface wave associated with a p = 2 ray for a sphere with refractive index n = 1.333


Following the pioneering work of van de Hulst, the consensus is that glories are caused by surface waves - but the precise mechanism remains rather obscure.  Glories are usually explained by lots of scientific "arm waving": for example, in their book “Color and light in nature”, Lynch and Livingston2 remark:

“Although the glory pattern is correctly predicted by Mie theory, a good physical explanation is, in our opinion, lacking. In some way light is backscattered after traversing the periphery of droplet. Examined in detail, each drop is found to shine uniformly around its edge with an annulus of light that is coherent (the waves are in phase).”

 Robert Greenler3 observes on page 145 of “Rainbows, halos and glories”:

“In one sense, the glory is now well understood. A mathematical theory (Mie scattering theory) enables us to calculate the intensity variation in the glory pattern. Unfortunately, it gives us little physical insight into the process that produces the rings.   . . . . . . .  I wonder if there is no simple model containing the physical essence of the glory.”

Similarly, on page 389 of “Absorption and scattering of light by small particles”, Bohren and Huffmann4 state:

“Unlike the rainbow, the glory is not easy to explain, other than to say that it is a consequence of all of the thousands of terms in the scattering series, a correct but unsatisfying statement.”

Nussenzveig5 has drawn attention to the importance of resonances in forming the glory, for example, in 2003, he wrote

“Tunneling is the dominant effect in backscattering. It produces the meteorological glory  . . . .  The glory provides direct and visually stunning experimental evidence of the importance of resonances and light tunneling in clouds”.

In another paper6 entitled “Does the glory have a simple explanation?”, Nussenzveig tried to respond to the above requests, but he concluded that:

Mie theory describes the glory by the sum of a large number of complicated terms within which the physical mechanisms cannot be discerned. CAM [Complex Angular Momentum] theory brings out the dominant physical effects and provides an accurate representation for each of them. That it does so by analytic continuation seems inevitable. I know of no other way of quantitatively representing tunneling.”

Despite this discouraging conclusion, it is perplexing that the glory cannot be explained (even by eminent scientists) except by resorting to mathematical formulations that offer little insight into the mechanisms that actually cause the glory.  In an attempt to overcome this problem, I wrote a paper7 entitled “How are glories formed?”.  The following text and graphics show a simplified version of the paper published in Applied Optics.

Surface waves in optics may seem slightly mysterious. Nevertheless, the existence of surface waves in other areas of electromagnetic wave propagation is not in doubt: for example, vertically-polarized radio transmissions at frequencies around 1 MHz propagate via “ground waves”.  Unfortunately, no rigorous method seems to be available for calculating the intensity of scattering caused by surface waves. Various authors have proposed approximate methods applicable to scattering of light by small spheres – but all warn that their approximations are not valid at θ = 180° or, indeed, near 180°. These limitations are especially problematic for investigations of the glory! Although these approximate methods are fairly similar, the surface wave calculations shown here are based on Khare’s method.8

    

Fig. 6a  Comparison of calculation methods for p = 1 scattering of light of wavelength λ = 650 nm from a sphere of radius r = 100 µm and refractive index n = 1.33

Fig. 6b  As Fig. 6a but with radius r = 10 µm


As the scattering contributions made by surface waves are generally much weaker than those due to other scattering mechanisms, experimental verification is obviously difficult. On the other hand, Debye series calculations can be used to isolate specific scattering processes even when they generate only very weak scattering. For example, the Debye series p = 1 term accurately defines transmission through a sphere: for small values of θ, the Debye series term closely matches calculations based on geometric optics, as shown in Figs. 6a and 6b. As geometric optics cannot make any contribution to the p = 1 scattered intensity when θ > 180° – 2 sin-1[1/n] ≈ 82.8° for n = 1.333, another mechanism must be responsible for scattering when θ > 82.8°. Figs. 6a and 6b show that Khare’s calculation method for surface waves gives a good approximation to the Debye series term for p = 1 when θ > 82.8°. There is a slight error in intensity in Fig. 6b (for r = 10 µm), but the differing slopes of the curves for the two polarizations are correctly reproduced. This close agreement confirms that surface waves are the dominant p = 1 scattering mechanism for θ > 82.8°.

As surface waves shed light continuously as they travel along the surface of the droplet, the intensity of the scattering due to surface waves reduces exponentially with the length of the path taken by the surface wave – thus explaining why the intensity of the surface waves reduces much more rapidly for r = 100 µm (Fig. 6a) than for r = 10 µm (Fig. 6b). This exponential decay for surface waves can also be recognized by the fact that the “curves” of intensity versus θ in Figs. 6a and 6b are almost straight lines (because the intensity scale is logarithmic whilst the angular scale is linear). In general, the surface wave intensity due to parallel polarization is significantly higher than that due to perpendicular polarization. As most natural glories seem to be caused by scattering from water droplets with r between 4 µm and 25 µm, the following examples are based on the scattering of red light (λ = 650 nm) from r = 10 µm water droplets.

Note that the Debye series term in Fig. 6b shows a series of maxima and minima as θ → 180°. What causes these ripples? The top part of Fig. 7 shows a ray with an impact parameter b = 1 taking a shortcut through 82.8° and then propagating 92.2° clockwise along the surface of the sphere, resulting in a scattering angle θ = 82.8° + 92.2° = 175°. The lower part of Fig. 7 shows a ray with b = – 1 taking a shortcut through 82.8° and then propagating anti-clockwise 102.2° along the surface of the sphere, resulting in deflection of 82.8° + 102.2° =185°, which is equivalent to scattering angle θ = 360° – 185° = 175°. Of course, the value of θ = 175° in Fig. 7 has been chosen solely as an illustrative example. More generally, scattering in a specific direction θ = 180° – δ can be caused by two surface wave components: the “short” path generated by incident rays with impact parameter b = 1 involving a deflection of 180° – δ and the “long” path generated by incident rays with impact parameter b = – 1 involving a deflection of 180° + δ.

Fig. 7   Surface wave paths (p = 1) resulting in scattering angle θ = 175°: the upper part of this diagram shows the “short” path, whilst the lower part shows the “long” path.


Fig. 8a below shows in greater detail the ripples on the Debye p = 1 term, together with calculations of the intensity due to the short and long path surface waves. The long path contributions are weaker than the short path contributions simply because the longer path gives greater attenuation. The difference in path length between the long path and the short path also causes a phase difference between the two contributions. Constructive interference will occur when this phase difference is a multiple of 360°, whereas destructive interference will occur when it is an odd multiple of 180°. Consequently, a series of maxima and minima will occur in the scattering pattern as a function of θ. Note that there is an additional phase shift of 90° between the short and long paths due to the fact that the long path crosses one focal line more than the short path.9

Fig. 8b compares the Debye p = 1 term with the vector sum of the contributions from the short and long path surface waves. The general shape of the ripples in Fig. 8b is reassuringly similar to the Debye series calculation. Note that Fig. 8a provides an explanation for the increasing amplitude of the ripples as θ → 180° in Fig. 8b. When θ ≈ 150°, Fig. 8a indicates that the short path surface wave is dominant and, hence, there are no ripples in this part of Fig. 8b. However, as θ is increased towards 180°, Fig. 8a shows that the intensity of the long path surface wave increases relative to the short path surface wave and, consequently, the ripples in Fig. 8b become larger. Note that these ripples correspond to the circular rings of a glory caused by p = 1 scattering. This “mathematical” result may be surprising, but p = 1 glories cannot be observed in practice because the intensity of the p = 2 glory is greater by more than 5 orders of magnitude. Nevertheless, as surface waves are the only p = 1 scattering mechanism applicable to θ > 82.8°, the Debye p = 1 term provides a crucial test of the accuracy of surface wave calculations.

    

Fig. 8a  As Fig. 6b, but showing the separate contributions from short path and long path surface waves (see Fig. 7)


Fig. 8b  As Fig. 8a, but showing the vector sum of the short path and long path surface wave contributions


At first sight, this explanation may seem similar to that of van de Hulst – but it is instructive to examine the differences between the two explanations. Using Huygens’ principle, van de Hulst showed that the interference pattern corresponding to the toroidal wave front propagating in the direction θ = 180° is defined by:

I1 = [C1 {J1(u) – J2(u)} + C2 {J1(u) + J2(u)}]2
     I2 = [C2 {J1(u) – J2(u)} + C1 {J1(u) + J2(u)}]2

where I1 and I2 are the intensities in the direction θ = 180° – δ for perpendicular and parallel polarization respectively, C1 and C2 are proportional to the amplitudes of the components with perpendicular and parallel polarization, J1 and J2 are Bessel functions and u = 2 π r sin (δ) / λ.   In assessing the accuracy of van de Hulst’s method, an immediate problem is that we do not know the values of C1 and C2. Although van de Hulst suggested some notional values, such as 0 > C1 / C2 > – 0.25, the above equations do not yield any quantitative predictions of the intensity of the glory.

Fig. 9a shows a comparison of the Debye series for the p = 1 glory with van de Hulst’s method: in this case, the value of C1 / C2 = – 0.12 has been chosen because it reproduces the minimum for perpendicular polarization at θ ≈ 177.4°, whilst the value of C1 has been set to give the correct intensity at θ = 180°. Fig. 9a shows that van de Hulst’s diffraction pattern agrees closely with the Debye series calculations when θ is near 180°, but becomes increasingly inaccurate as θ is reduced.

Fig. 9b compares the Debye series for the p = 1 glory with the calculation method based on simple interference between short and long path surface waves. The latter method gives inaccurate results when θ → 180°, but it correctly reproduces the maxima and minima for other values of θ – especially for the dominant parallel polarization.

In essence, Fig. 9a shows that van de Hulst’s diffraction pattern is not a satisfactory model for the rings of the glory, whereas Fig. 9b and Fig. 8b act as reminders that surface waves shed light continuously as a function of θ – unlike van de Hulst’s diffraction pattern which is based on θ = 180° being a preferential direction.

    
Fig. 9a  Comparison of Debye series calculations for p = 1 scattering with calculations based on van de Hulst’s diffraction pattern
Fig. 9b  Comparison of Debye series calculations for p = 1 scattering with calculations based on two-ray interference between short and long path surface waves

The fact that the Debye series calculations predict a glory for the p = 1 term, as well as for the p = 2 term, may be unanticipated. Glories caused by interference between "short path" and "long path" surface waves are not restricted to p = 1 and p = 2 terms, thus providing an explanation why Fig. 3 shows almost identical colour sequences for the scattering of sunlight for various other values of p – although they are much weaker than the p = 2 term. 

The discussion about surface waves has so far focused on p = 1 scattering because this facilitates calibration of the approximate calculation methods for surface waves against the rigorous Debye series calculations. As indicated earlier, p = 2 scattering is responsible for the coloured rings of the natural glory. In practice, calculations for p = 2 scattering must take account of the fact that there are three separate p = 2 ray paths that, for refractive index n = 1.333, result in θ = 175° - as shown in Fig. 10 below.

Fig. 10 Diagram showing p = 2 rays that result in a scattering angle θ = 175° for a sphere with refractive index n = 1.333

Having identified a mechanism that could cause the glory, it is necessary to check whether the calculations for p = 2 glory correspond to the results given by the rigorous Debye series calculations as shown in Fig. 11a below. As noted earlier, Khare's method for calculating the intensity of surface waves fails as θ → 180° but  Fig. 11a shows that it reproduces the general features of the glory when the amplitude given by Khare's method is adjusted by an arbitrary factor of 0.5 (to correct the intensity errors seen in Fig. 6b).  As Khare's method does not reproduce the correct phase of the surface wave contributions, an arbitrary phase correction of 40°  has been applied, as well as including the effects of the geometric optics contribution shown in Fig. 11a have been included in Fig. 11b.


    

Fig. 11a Comparison of Debye series, surface wave and geometric optics calculations for parallel polarization for p = 2 scattering for refractive index n = 1.333
(N.B. Amplitude from Khare’s formula multiplied by factor of 0.5)


Fig. 11b    Comparison of Debye series calculations with the vector sum of surface wave and geometric optics calculations for parallel polarization for p = 2 scattering for refractive index n = 1.333 (N.B. Amplitude from Khare’s formula multiplied by factor of 0.5 with a phase correction of +40°)

SUMMARY  

The coloured rings of the glory are caused by two-ray interference between “short” and “long” path surface waves – which are generated by p = 2 rays entering the droplets at diametrically opposite points, as illustrated in Fig. 10.  Such interference causes glories for other values of p - but these are much weaker and, in practice, can be ignored for scattering from water droplets



References


1    H. C. van de Hulst, “A theory of the anti-coronae,” J. Opt. Soc. Am. 37, 16 (1947).

2    D. K. Lynch and W. Livingston, Color and Light in Nature (Cambridge U. Press, UK, 2001).

3    R. Greenler, Rainbows, halos and glories (Cambridge U. Press, Cambridge, UK, 1980).

4    C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

5    H.M. Nussenzveig, “Light Tunneling in Clouds”, Appl. Opt. 42, 1588-1593 (2003).

6    H. M. Nussenzveig, “Does the glory have a simple explanation?”, Opt. Lett. 27, 1379-1381 (2002).

7    P. Laven, “How are glories formed?” Appl. Opt. 44, 5675–5683 (2005). Free download

8    V. Khare, "Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere"; Ph.D. thesis (University of Rochester, Rochester, N.Y, 1976).

N.B.  Reference 8  may not be readily available, but the calculation method is summarized in E. A. 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,” J. Opt. Soc. Am. A, 9, 781-795 (1992)

9    H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981; reprint of 1957 Wiley edition). See section  12.22


Page updated on 15 July 2008
 

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