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43lue because it was computationally demanding.C. van de Hulst published a bibliography5 of Mie

sults available in 1957. Such results were mainlythe form of tables of computed values for small

ops with size parameter x 30 x 2r wheres the drop radius and is the wavelength of thecident light. H. C. van de Hulst also published6me graphs of intensity versus scattering angle for 1x 5. For scattering of light from water drops ine atmosphere we are interested in much larger

puter program for Mie scattering of light has beendeveloped. Based on the algorithm of Bohren andHuffman,11 the MiePlot program can provide full-color simulations of optical effects caused by spheri-cal water drops, such as rainbows, coronas, andglories. It can also produce a wide range of graphs ofscattered intensity for perpendicular andor parallelpolarizations as functions of r, , scattering angle ,and refractive index n. This program, whichruns under Microsoft Windows, can be downloadedat no charge from http:www.philiplaven.commieplot.htm.

An example of the graphical output of the MiePlotprogram is given in Fig. 1, which demonstrates thatMie scattering of monochromatic light is very depen-dent on the size of the water drops. For light ofwavelength 0.65 m, scattering from a spherical dropwith r 1000 m or r 100 m produces a primaryrainbow at approximately 138 and a secondary

. Laven [email protected] can be reached c/o the Euro-n Broadcasting Union, Ancienne Route 17A, CH-1218 Grand

connex, Geneva, Switzerland.eceived 16 January 2002; revised manuscript received 25rch 2002.003-693503030436-09$15.0002003 Optical Society of America

6 APPLIED OPTICS Vol. 42, No. 3 20 January 2003imulation of rainbows, corony use of Mie theory

ilip Laven

Mie theory offers an exact solutionUntil recently, most applications oflength. Mie theory can now be useatmospheric optical effects, such aswith observations of natural glories

OCIS codes: 010.1290, 290.4020

Introduction

attering of sunlight by spherical drops of watery seem to be a relatively simple process, but it

uses a surprising variety of complicated optical ef-ts, such as primary and secondary rainbows, coro-s, and gloriessome of which, even today, are notlly understood. In the seventeenth century Des-rtes and Newton used geometrical optics to explaine formation of rainbows.1,2 In 1838 Airy3 ex-ded the earlier theories by including the effects offraction and interference, thus explaining the for-tion of the rainbows supernumerary arcs that are, and glories

e problem of scattering of sunlight by spherical drops of water.theory to scattering of light were restricted to a single wave-

modern personal computers to produce full-color simulations ofbows, coronas, and glories. Comparison of such simulationscloudbows is encouraging. 2003 Optical Society of America

lues of x, perhaps from x 50 for fog droplets to x,000 for rain during thunderstorms. Because Mieeory becomes even more forbidding for largeheres, significant effort has been spent on search-g for approximations to Mie theory.79 Eventu-y, the increasing power of computers and thevelopment of efficient computer algorithms10,11rmitted wider use of Mie theory,1216 but such uses generally restricted to scattering of light of agle wavelength. However, it was not until thee 1990s that Mie theory was, at last, used to sim-

ate atmospheric optical effects in color.1719The research described here was initiated as a di-t predicted by geometrical optics.

In 1908 Gustav Mie4 provided a rigorous solution toe problem of scattering of electromagnetic planeves by a homogeneous sphere. However, forny years this theory seemed to have little practicalct result of Lees seminal paper17 in which he com-red simulations of rainbows using Mie theory andry theory. With the potential of modern personalmputers with good-quality color displays, a com-rainand.

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rainbow at approximately 129, together withAlexanders dark band between them. Other fea-tures, such as the corona for 10 and the glory for 170, are visible on the curve for r 10 m. Thecalculations of intensity as a function of were madeat intervals of 0.1 for Fig. 1, but all other calculationsreported in this paper used an interval of 0.01.

2. Simulation of Rainbows

Figure 2a shows Mie scattering of monochromaticlight from a drop with r 100 m for between 137and 145. Anyone familiar with Airy theory willprobably consider the high-frequency ripples to beunnatural, but such ripples might be observable un-der laboratory conditions of monochromatic lightscattered from a single drop. These ripples aremainly the result of interference between rays thathave undergone one internal reflection p 2 andrays that have undergone one external reflection p0.20,21 Airy theory does not predict such ripples onthe primary rainbow, because it ignores anythingother than p 2 rays.

Atmospheric optical effects are caused not by asingle water drop but by millions of water drops ofnonuniform size. Figure 2b shows the result of av-eraging the Mie intensities from 50 drops with a log-normal size distribution with a median radius of 100m and a standard deviation of 0.1 m 0.1% of themedian radius. Although this size variation is ex-tremely small, it has a dramatic effect: Comparisonof Figs. 2a and 2b shows that scattering from dropsof nonuniform size removes the high-frequency rip-ples.

Natural atmospheric optical effects are caused bythe Sun or the Moon, which are circular light sourceswith an apparent diameter of 0.5. Because Mietheory assumes a plane wave, it is necessary to inte-grate the intensities resulting from the Mie calcula-tions over the diameter of the light source. Applying

this additional step to the curve shown in Fig. 2aproduces the monodisperse intensity curve shown inFig. 2c, where the high-frequency ripples have al-most been removed.

Figure 2 is based on illumination by monochro-matic light, whereas light from the Sun has a contin-uous broad spectrum. To simulate the spectrum ofsunlight, the MiePlot program performs calculationsat N equally spaced wavelengths between 0.38 and0.7 m where the value of N can be selected by theuser. The relative intensity of each wavelength isbased on the measured solar spectral radiance shownin Fig. 3 of Lees paper.17 Although it can be arguedthat this spectral illuminant is not ideal because theSuns elevation was 45, it was adopted because itfacilitated comparisons with Lees results.

Figures 3a and 3b show the relative luminancecurves resulting from addition of Mie calculations forN 10 and N 1000, respectively. These curvesare similar in shape, but the curve for N 10 has acomplicated ripple structure, which is caused by thefact that the high-frequency ripples for monochro-matic light vary rapidly with changing wavelength.When intensity curves for different wavelengths areadded together, the ripple structure will be reinforcedif the maxima and minima for the various wave-lengths coincide; for example, the ripples shown inFig. 2a for 650 nm coincide approximately withthose for 650.9 nm, 651.8 nm, and so on. Figure3b indicates that the ripple structure almost disap-pears for N 1000, but this value of N is very largeby normal colorimetric standards.

Figure 4 shows intensity curves plotted on a loga-rithmic scale for primary and secondary rainbows forr 100 m. This figure shows that the intensity forperpendicular polarization is generally much greaterthan for parallel polarization. An important excep-tion occurs near the minima of the primary rainbows

Fig. 1. Graphs of intensity versus scattering angle . Radius of spheres r 1, 10, 100, and 1000 m; wavelength of light 0.65 m;refractive index n 1.332; unpolarized light.

20 January 2003 Vol. 42, No. 3 APPLIED OPTICS 437

supernumeraries, which coincide with maxima of thesupernumeraries for parallel polarization.22

Figure 4 was calculated with N 2000 to avoid anyproblems with the ripple structure. For each valueof , the intensity curve is the sum of the intensitiesfor each of the 2000 wavelengths. Color simulationsrequire the scattered intensities for each wavelengthto be converted into RGB red, green, blue valuesused by computer displays. For each value of , thegamma-corrected sum of the resulting RGB values isused to generate the three horizontal stripes abovethe graph in Fig. 4, which represent the brightnessand the color of light at specific values of : The top

stripe is for perpendicular polarization, the middlestripe for parallel polarization, and the bottom stripefor unpolarized light. Since rainbows are stronglypolarized, most of the middle stripe is very dark.

Figure 5 shows the results of an exercise to deter-mine the minimum value of N for Mie simulations ofthe primary and secondary rainbows for r 100 m.Each horizontal stripe shows the results of calcula-tions by use of a specific value of N. Figure 5ademonstrates that simulations of the primary rain-bow i.e., between 137 145 are fairly consis-tent for 10 N 2000. Nevertheless, becausethere are some subtle variations when N 30, it issuggested that N 30 would be a safe choice for theprimary rainbow when r 100 m. The secondaryrainbow is shown in greater detail in Fig. 5b, whichindicates the need for even higher values of N e.g.,600 to avoid visible ripples on such simulations.This discrepancy in the required value of N is due todifferences in the ripple structures: Successivemaxima of the ripples are separated by0.15 on theprimary rainbow, compared with 0.6 on the sec-ondary rainbow. The smoothing effect caused by the0.5 diameter of the Sun effectively removes the 0.15ripples but has no effect on the 0.6 ripples. Conse-quently, more wavelengths are needed to suppressthe visibility of the ripple structure on simulations ofthe secondary rainbow.

Lee17 introduced a powerful technique to illustrate

Fig. 2. a Graph of intensity versus scattering angle . r 100m, 0.65 m, n 1.332, perpendicular polarization. bAs ina but intensity is the average intensity of scattering by 50 sphereswith different values of radius with a log-normal distribution witha median radius of 100 m and a standard deviation of 0.1 m0.1% of the median radius. cAs in a but intensity is averagedto take account of the 0.5 diameter of the Sun.

Fig. 3. a Graph of intensity versus scattering angle . r 100m, ten values of equally spaced between 0.38 and 0.7 m, nvaries as function of , perpendicular polarization. b As in abut with 1000 values of equally spaced between 0.38 and 0.7 m.

438 APPLIED OPTICS Vol. 42, No. 3 20 January 2003

how the appearance of rainbows varies with dropsize. The MiePlot program can also generate theseLee diagrams, as shown in Fig. 6 where each coloredpoint represents the color of light scattered in a spe-cific direction by a drop of radius r. The brightnessof each vertical line of colors representing the rain-bow caused by a drop of radius r has been normalizedby the maximum luminance for that value of r. Fig-ure 6 is based on the assumption that the Sun is thelight source, which implies that the Mie calculationshave been smoothed to take account of the 0.5 ap-parent diameter of the Sun.

Figure 6 can be compared with Fig. 15 of Lees

paper17: As Lee noted, the latters only literally un-natural feature seems to be the Mie maps ripplestructure, which appears as a subtle color marblingon the cloudbow primaries and their supernumerar-ies Ref. 17, p. 1514. Severe marbling is visible inFig. 6, which was calculated with N 7 for all valuesof r between 10 and 1000 m, but Fig. 7 does notexhibit such marbling. The value of N used in pro-duction of Fig. 7 was a function of drop size: N 600 for r 10 m drops, reducing to N 30 for r 200 m drops or larger. Comparison of Figs. 6 and7 demonstrates that marbling on such simulationscan be avoided by use of higher values of N. It seems

Fig. 5. a Simulation of primary and secondary rainbows caused by scattering of unpolarized sunlight from r 100 m water drops withthe specified number N of wavelengths. b As in a but limited to the secondary rainbow.

Fig. 4. Graph of intensity versus scattering angle for r 100 m, together with colored stripes simulating primary and secondaryrainbows for perpendicular polarization, parallel polarization, and for unpolarized light N 2000.


that marbling is another manifestation of the ripplesdiscussed above. Because the ripples occur at dif-ferent values of for different values of r, they appearon Lee diagrams as lighter or darker diagonalstripes.

From the foregoing it is obvious that Mie theorycan be demanding in terms of the number of wave-lengths required when it is used to simulate the scat-tering of a continuous spectrum. It is important tounderstand that this is not due to the spectral char-acteristics of the incident light, such as sunlight.Similar problems occur even with a flat spectrum.The fundamental cause is that the high-frequencyripples correctly predicted by Mie theory for mono-chromatic light disappear when monochromatic lightis replaced with light that has a continuous spectrum.Simulating this process of averaging requires Miecalculations at many wavelengths.

As mentioned above, natural atmospheric effectsinvolve scattering from millions of drops of varyingsizes. Having calculated the data for many values ofr to produce a Lee diagram for monodisperse drops,one can easily generate a Lee diagram for dispersedrops. Figure 8 shows the results for drops with alog-normal size distribution with a standard devia-tion of 20%. Comparison of Fig. 7 with Fig. 8 showsthat this variability in drop sizes suppresses the rain-bows supernumeraries, especially for r 100 m.

Figures 9a and 9b show simulations of primaryand secondary rainbows caused by scattering of sun-light from water drops of r 200 m and r 500 m,respectively as might be observed with a 35-mm filmcamera with a lens of focal length 70 mm. In na-ture, Alexanders dark band would be much lighterthan shown in Fig. 9, because these simulations ig-nore other sources of light such as diffuse or multiplescattering, skylight, and reflected light. Such mech-anisms have been widely studied, but recent researchby Gedzelman and Lock is particularly relevant be-cause it combines Mie theory with multiple scatter-ing.19,23,24 Furthermore, Fig. 9 is based on theunrealistic assumption of monodisperse raindrops:Variations in drop size will inevitably reduce the vis-ibility of the supernumerary arcs.

For scattering of sunlight from natural waterdrops, it seems that the ripples calculated so exactlyby Mie theory must be smoothed away, thus givingresults similar to Airy theory. Consequently, it iseasy to understand why the Airy method is preferredfor simulation of rainbows. Lees paper concludes asfollows:

Thus, far from being an outdated irrelevancy, Airytheory shows itself to be a simple, quantitativelyreliable model of the natural rainbows colors andluminances. Provided that we restrict ourselves

Fig. 7. Lee diagram of the primary rainbow for values of r be-tween 10 and 1000 m for unpolarized sunlight N 6000r for 10m r 200 m and N 30 for r 200 m.

Fig. 6. Lee diagram of the primary rainbow for values of r be-tween 10 and 1000 m for unpolarized sunlight N 7.


to spectrally integrated luminances or radiances,it can be used to predict accurately the visual ap-pearance of most naturally occurring water-dropbows and supernumeraries. Equally important,we see that Mie theory for monodispersions shouldnot always be the model of first resort, as it pro-

duces details not seen in the natural rainbow anddoes so with much more computational effort thanAiry theory. Ref. 17. p. 1518

3. Simulation of Coronas and Glories

Figure 10 shows MiePlot simulations of the coronaand the glory for r 10 m. In real life it is difficultto observe the corona around the Sun, because theglare of the Sun obscures the colored rings. Hencethe brightness of the simulated corona in Fig. 10 hasbeen increased by a factor of 20 so that the Sun andits immediate surroundings are overexposed. Thecorona and the glory have different sizes, but thesequence of colors is almost identical. The key dif-ferences are that, for r 10 m, the glory has a darkring around 179.3 and that, in general, the outerrings of glories are relatively bright compared withthose of coronas.

Although similar in appearance, these two phe-nomena are caused by completely different mecha-nisms. The corona is the result of simple diffractionand, thus, can be satisfactorily modeled by use ofBessel functions. On the other hand, there are nosimple theories to explain the glory. Following thepioneering research in 1947 of van de Hulst,25 themodern consensus is that the glory is caused by acombination of surface waves and rays that have un-dergone many internal reflections.2628

Figure 11 shows simulations of the glory caused byscattering of unpolarized sunlight from a water dropof r 10 m with the specified number of wave-lengths. At least 600 wavelengths are needed toachieve consistent results. Figure 11 indicates thatthe inner red ring of the glory occurs at approxi-mately 177.6, with a second red ring at 176.3and much fainter red rings at 174.4 and 172.5.

Figure 12 is a Lee diagram showing how the ap-pearance of the glory varies with the radius r of thewater drop. The sequence of ring colors is essen-tially independent of r. To a first approximation, the

Fig. 8. As in Fig. 7, except that drops of nominal radius r have alog-normal size distribution with a standard deviation of 20% ofthe nominal value.

Fig. 9. a Simulation of primary and secondary rainbows caused by scattering of unpolarized sunlight from r 200 m water drops. bSimulation of primary and secondary rainbows caused by scattering of unpolarized sunlight from r 500 m water drops.


radius of a given ring from the antisolar point 180 is inversely proportional to r; for example, thefour inner red rings have radii measured in degreesof approximately 24r, 37r, 56r, and 75r where ris measured in m.

The MiePlot program allows comparisons of simu-lations with photographs of atmospheric optical ef-fects. Figure 13 shows a digital image of a gloryobserved from a commercial aircraft, together with asuperimposed MiePlot simulation of scattering ofsunlight by water drops with r 4.8 m. Apartfrom adding a background color roughly matchingthat of the original image, no other adjustments havebeen made to the simulation. It should be notedthat the close agreement between the original imageand the simulation does not necessarily prove thatthe simulation is accurate, because we have no inde-pendent confirmation of the size of the water dropscausing the glory. Nevertheless, another simulationfor r 4.7 m produces rings that are larger than theobserved rings, whereas a simulation for r 4.9 mproduces rings that are smaller. Subject to the ac-curacy of the cameras calibration, we can be moder-ately confident that water drops of 4.8 m wereresponsible for this particular glory.

Figure 14 shows a digital image of a faint glorysurrounded by a portion of a cloudbow, in which thereis a dark ring around 144 and an almost-whitering around 141. Figure 15 is a MiePlot simu-

lation of the same image based on scattering of sun-light by water drops with r 20 m, plus a uniformbackground color. The details provided by the sim-ulation are much clearer than in the original image,but it is reasonable to assume that water drops withr 20 m were dominant in the observed clouds.

4. Sources of Error

Although Mie theory is rigorous, various assump-tions made within the MiePlot program can affect thevalidity of simulations of atmospheric optical effects.One assumption mentioned above concerns the spec-tral radiance of sunlight: Different models of spec-

Fig. 10. Simulation of corona top and glory bottom caused byscattering of unpolarized sunlight from r 10 m water drops.

Fig. 11. Simulation of the glory caused by scattering of unpolarized sunlight from r 10 m water drops with the specified number ofwavelengths equally spaced between 0.38 and 0.7 m.

Fig. 12. Lee diagram of the glory caused by scattering of unpo-larized sunlight from water drops of nominal radius r with a log-normal size distribution and a standard deviation of 5% of thenominal value.


tral radiance would obviously affect the color balanceof simulations.

It is also necessary to define the refractive indexn of water as a function of wavelength . There aresurprisingly wide variations in the value of nquoted by various sources.2933 For example, thequoted values of n for 400 nm vary between1.3427 and 1.3498, corresponding to primary rain-bow angles of 139.3 and 140.3. Because mostatmospheric effects are dependent on the absoluteand relative values of n as a function of , suchdifferences are, to say the least, disconcerting.The MiePlot program now allows the user to selectone of several equations for n as a function of , butthe default equation is that defined by the Interna-tional Association for the Properties of Water andSteam.33

Even if the Mie simulations could be made math-ematically precise, there is a substantial problemwith colorimetry. First, it is necessary to definethe colors corresponding to specific wavelengths oflight in terms of RGB values used within the com-

puter.34 The MiePlot program uses the simple al-gorithm devised by Bruton.35 Second, full-colorsimulations depend very much on the characteristicsof the display devices, such as computer screens,projectors, or printers. In practice, differences inthe inherent characteristics of computer displayse.g., primary colors and gamma are compoundedby user settings, such as brightness, contrast, andcolor balance. Consequently, simulations in colorcan look dramatically different on different equip-ment. Further research is needed to address boththese issues.

5. Conclusions

Mie theory provides a rigorous solution to the prob-lem of scattering of sunlight from spherical rain-drops, but it involves heavy computation. Simplertheories, such as Airy theory for rainbows or Besselfunctions for coronas, can give almost identical re-sults with much less computation. In some respects,using Mie theory can be equivalent to using a sledge-hammer to crack a nut. Nevertheless, Mie theoryremains the benchmark against which other theoriesare judged, and it is an invaluable tool for simulationof glories.

Advances in computer power mean that Mie the-ory can be used to produce full-color simulationsof atmospheric optical effects. This paper hasdescribed the MiePlot computer program, whichis freely available from the author. On a modernpersonal computer this program can produce simu-lations of glories and fogbows within a few minutes,whereas detailed simulations of rainbows mighttake 1 h or so.

Calculations with Mie theory can be erroneous ifthe scattering of a continuous spectrum is modeled bya small number of discrete wavelengths. For exam-ple, Mie calculations require 600 wavelengths forsimulation of the glory for r 10 m or 30 wave-lengths for simulation of the primary rainbow for r 100 m.

Fig. 13. Digital image of a glory with a superimposed simulationof scattering of unpolarized sunlight from r 4.8 m water drops.

Fig. 14. Digital image of a faint glory and a cloudbow.Fig. 15. Simulation of scattering of unpolarized sunlight from r20 m water drops.


Other sources of errors include

uncertainty about the values of refractive indexn as a function of wavelength ;

colorimetry issues associated with display de-vices;

relative intensity of the solar radiance as afunction of ;

the effects of indirect illumination, such as mul-tiple scattering, have been ignored.

Despite these potential sources of error, comparisonof MiePlot simulations with digital images of naturalglories is encouraging.

The author expresses his gratitude to RaymondLee Jr. for his stimulating paper17 and for his per-sonal encouragement. Les Cowley provided invalu-able suggestions and feedback during the writing andtesting of the MiePlot program.

References1. C. B. Boyer, The Rainbow: From Myth to MathematicsPrinceton University, Princeton, N.J., 1987; reprint of 1959Thomas Yoseloff edition.

2. R. L. Lee, Jr., and A. B. Fraser, The Rainbow Bridge: Rain-bows in Art, Myth, and Science Penn State, University Park,Pa., 2001.

3. G. B. Airy, On the intensity of light in the neighbourhood of acaustic, Trans. Cambridge Philos. Soc. 6, Part 3, 3974031838.

4. G. Mie, Beitrage zur Optik truber Medien, speziell kolloidalerMetallosungen, Ann. Phys. Leipzig 25, 377445 1908.

5. H. C. van de Hulst, Light Scattering by Small Particles Dover,New York, 1981; reprint of 1957 Wiley edition, pp. 167171.

6. Ref. 5, pp. 152153.7. H. M. Nussenzveig, The theory of the rainbow, in Atmo-

spheric Phenomena Freeman, San Francisco, Calif., 1980, pp.6071.

8. H. M. Nussenzveig, Complex angular momentum theory ofthe rainbow and the glory, J. Opt. Soc. Am. 69, 106810791979.

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10. J. V. Dave, Scattering of visible light by large water spheres,Appl. Opt. 8, 155164 1969.

11. C. F. Bohren and D. R. Huffman, Absorption and Scattering ofLight by Small Particles Wiley, New York, 1983.

12. K. Liou and J. E. Hansen, Intensity and polarization for singlescattering by polydisperse spheres: a comparison of ray op-tics and Mie theory, J. Atmos. Sci. 28, 9951004 1971.

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14. A. Ungut, G. Grehan, and G. Gouesbet, Comparisons betweengeometrical optics and LorenzMie theory, Appl. Opt. 20,29112918 1981.

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18. R. K. Brandt and R. G. Greenler, Color simulation of size-dependent features of rainbows, presented at the SeventhTopical Meeting on Meteorological Optics, Boulder, Colorado,58 June 2001; available at http:www.asp.ucar.eduMetOp-ticsPreprints.pdf.

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