53

Atmospheric Optics

Craig F. Bohren Pennsylvania State University, Department of Meteorology, University Park, Pennsylvania, USA Phone: (814) 466-6264; Fax: (814) 865-3663; e-mail: [email protected]

Abstract Colors of the sky and colored displays in the sky are mostly a consequence of selective scattering by molecules or particles, absorption usually being irrelevant. Molecular scattering selective by wavelength – incident sunlight of some wavelengths being scattered more than others – but the same in any direction at all wavelengths gives rise to the blue of the sky and the red of sunsets and sunrises. Scattering by particles selective by direction – different in different directions at a given wavelength – gives rise to rainbows, coronas, iridescent clouds, the glory, sun dogs, halos, and other ice-crystal displays. The size distribution of these particles and their shapes determine what is observed, water droplets and ice crystals, for example, resulting in distinct displays. To understand the variation and color and brightness of the sky as well as the brightness of clouds requires coming to grips with multiple scattering: scatterers in an ensemble are illuminated by incident sunlight and by the scattered light from each other. The optical properties of an ensemble are not necessarily those of its individual members. Mirages are a consequence of the spatial variation of coherent scattering (refraction) by air molecules, whereas the green flash owes its existence to both coherent scattering by molecules and incoherent scattering by molecules and particles.

Keywords sky colors; mirages; green flash; coronas; rainbows; the glory; sun dogs; halos; visibility.

1 Introduction 54 2 Color and Brightness of Molecular Atmosphere 55 2.1 A Brief History 55 54 Atmospheric Optics

2.2 Molecular Scattering and the Blue of the Sky 57 2.3 Spectrum and Color of Skylight 58 2.4 Variation of Sky Color and Brightness 59 2.5 Sunrise and Sunset 62 3 Polarization of Light in a Molecular Atmosphere 63 3.1 The Nature of Polarized Light 63 3.2 Polarization by Molecular Scattering 64 4 Scattering by Particles 66 4.1 The Salient Differences between Particles and Molecules: Magnitude of Scattering 66 4.2 The Salient Differences between Particles and Molecules: Wavelength Dependence of Scattering 67 4.3 The Salient Differences between Particles and Molecules: Angular Dependence of Scattering 68 4.4 The Salient Differences between Particles and Molecules: Degree of Polarization of Scattered Light 69 4.5 The Salient Differences between Particles and Molecules: Vertical Distributions 70 5 Atmospheric Visibility 71 6 Atmospheric Refraction 73 6.1 Physical Origins of Refraction 73 6.2 Terrestrial Mirages 73 6.3 Extraterrestrial Mirages 76 6.4 The Green Flash 77 7 Scattering by Single Water Droplets 78 7.1 Coronas and Iridescent Clouds 78 7.2 Rainbows 80 7.3 The Glory 82 8 Scattering by Single Ice Crystals 83 8.1 Sun Dogs and Halos 83 9 Clouds 86 9.1 Cloud Optical Thickness 86 9.2 Givers and Takers of Light 87 Glossary 89 References 90 Further Reading 90

1 atmosphere and the primary source of Introduction their illumination is the sun. Essentially all light we see is scattered light, even that Atmospheric optics is nearly synonymous directly from the sun. When we say that with light scattering, the only restrictions such light is unscattered we really mean being that the scatterers inhabit the that it is scattered in the forward direction; Atmospheric Optics 55 hence it is as if it were unscattered. glories) resulting from illumination of Scattered light is radiation from matter cloud droplets by sunlight. excited by an external source. When the This article begins with the color source vanishes, so does the scattered light, and brightness of a purely molecu- as distinguished from light emitted by lar atmosphere, including their variation matter, which persists in the absence of across the vault of the sky. This nat- external sources. urally leads to the state of polarization Atmospheric scatterers are either mole- of skylight. Because the atmosphere is cules or particles. A particle is an aggrega- rarely, if ever, entirely free of particles, tion of sufficiently many molecules that the general characteristics of scattering it can be ascribed macroscopic proper- by particles follow, setting the stage ties such as temperature and refractive for a discussion of atmospheric visibil- index. There is no canonical number of ity. molecules that must unite to form a Atmospheric refraction usually sits by bona fide particle. Two molecules clearly itself, unjustly isolated from all those at- do not a quorum make, but what about mospheric phenomena embraced by the 10, 100, 1000? The particle size corre- term scattering. Yet refraction is another sponding to the largest of these numbers manifestation of scattering, coherent scat- −3 µ is about 10 m. Particles this small tering in the sense that phase differences of water substance would evaporate so cannot be ignored. rapidly that they could not exist long under Scattering by single water droplets and conditions normally found in the atmo- ice crystals, each discussed in turn, yields sphere. As a practical matter, therefore, feasts for the eye as well as the mind. The we need not worry unduly about scatterers curtain closes on the optical properties in the shadow region between molecule of clouds. and particle. A property of great relevance to scat- tering problems is coherence,bothofthe 2 array of scatterers and of the incident light. Color and Brightness of Molecular At visible wavelengths, air is an array of Atmosphere incoherent scatterers: the radiant power scattered by N molecules is N times that 2.1 scattered by one (except in the forward ABriefHistory direction). But when water vapor in air condenses, an incoherent array is trans- Edward Nichols began his 1908 presiden- formed into a coherent array: uncorrelated tial address to the New York meeting of the water molecules become part of a single American Physical Society as follows: ‘‘In entity. Although a single droplet is a coher- asking your attention to-day, even briefly, ent array, a cloud of droplets taken together to the consideration of the present state of is incoherent. our knowledge concerning the color of the Sunlight is incoherent but not in an sky it may be truly said that I am inviting absolute sense. Its lateral coherence length you to leave the thronged thoroughfares is tens of micrometers, which is why of our science for some quiet side street we can observe what are essentially where little is going on and you may even interference patterns (e.g., coronas and suspect that I am coaxing you into some 56 Atmospheric Optics

blind alley, the inhabitants of which belong fell short of the mark. Yet his hypothesis to the dead past.’’ that ‘‘the blueness we see in the atmo- Despite this depreciatory statement, sphere is not intrinsic color, but is caused hoary with age, correct and complete by warm vapor evaporated in minute and explanations of the color of the sky still insensible atoms on which the solar rays are hard to find. Indeed, all the faulty fall, rendering them luminous against the explanations lead active lives: the blue infinite darkness of the fiery sphere which sky is the reflection of the blue sea; lies beyond and includes it’’ would, with it is caused by water, either vapor or minor changes, stand critical scrutiny to- droplets or both; it is caused by dust. day. If we set aside Leonardo as sui generis, Thetruecauseoftheblueskyisnot scientific attempts to unravel the origins of difficult to understand, requiring only a theblueskymaybesaidtohavebegunwith bit of critical thought stimulated by belief Newton, that towering pioneer of optics, in the inherent fascination of all natural who, in time-honored fashion, reduced it phenomena, even those made familiar by to what he already had considered: inter- everyday occurrence. ference colors in thin films. Almost two Our contemplative prehistoric ancestors centuries elapsed before more pieces in no doubt speculated on the origin of the the puzzle were contributed by the exper- blue sky, their musings having vanished imental investigations of von Brucke¨ and into it. Yet it is curious that Aristotle, the Tyndall on light scattering by suspensions most prolific speculator of early recorded of particles. Around the same time Clau- history, makes no mention of it in his sius added his bit in the form of a theory Meteorologica even though he delivered that scattering by minute bubbles causes pronouncements on rainbows, halos, and the blueness of the sky. A better theory mock suns and realized that ‘‘the sun was not long in coming. It is associated looks red when seen through mist or with a man known to the world as Lord smoke.’’ Historical discussions of the blue Rayleigh even though he was born John sky sometimes cite Leonardo as the first to William Strutt. comment intelligently on the blue of the Rayleigh’s paper of 1871 marks the sky, although this reflects a European bias. beginning of a satisfactory explanation If history were to be written by a supremely of the blue sky. His scattering law, the disinterested observer, Arab philosophers key to the blue sky, is perhaps the would likely be given more credit for most famous result ever obtained by having had profound insights into the dimensional analysis. Rayleigh argued that workings of nature many centuries before the field Es scattered by a particle small their European counterparts descended compared with the light illuminating it from the trees. Indeed, Moller¨ [1] begins is proportional to its volume V and his brief history of the blue sky with to the incident field Ei. Radiant energy Jakub Ibn Ishak Al Kindi (800–870), who conservation requires that the scattered explained it as ‘‘a mixture of the darkness field diminish inversely as the distance of the night with the light of the dust and r from the particle so that the scattered haze particles in the air illuminated by power diminishes as the square of r.To the sun.’’ make this proportionality dimensionally Leonardo was a keen observer of light in homogeneous requires the inverse square nature even if his explanations sometimes of a quantity with the dimensions of Atmospheric Optics 57 length. The only plausible physical variable 2.2 at hand is the wavelength of the incident Molecular Scattering and the Blue of the Sky light, which leads to Our illustrious predecessors all gave ex- EiV planations of the blue sky requiring the Es ∝ .(1) rλ2 presence of water in the atmosphere: Leonardo’s ‘‘evaporated warm vapor,’’ When the field is squared to ob- Newton’s ‘‘Globules of water,’’ Clausius’s tain the scattered power, the result is bubbles. Small wonder, then, that water Rayleigh’s inverse fourth-power law. This still is invoked as the cause of the blue law is really only an often – but not sky. Yet a cause of something is that with- always – very good approximation. Miss- out which it would not occur, and the sky ing from it are dimensionless proper- would be no less blue if the atmosphere ties of the particle such as its refractive were free of water. index, which itself depends on wave- A possible physical reason for attributing length. Because of this dispersion, there- the blue sky to water vapor is that, because fore, nothing scatters exactly as the inverse of selective absorption, liquid water (and fourth power. ice) is blue upon transmission of white Rayleigh’s 1871 paper did not give the light over distances of order meters. Yet complete explanation of the color and if all the water in the atmosphere at any polarization of skylight. What he did that instant were to be compressed into a liquid, was not done by his predecessors was to the result would be a layer about 1 cm thick, give a law of scattering, which could be which is not sufficient to transform white used to test quantitatively the hypothesis light into blue by selective absorption. that selective scattering by atmospheric Water vapor does not compensate for particles could transform white sunlight its hundredfold lower abundance than into blue skylight. But as far as giving nitrogen and oxygen by greater scattering the agent responsible for the blue sky is per molecule. Indeed, scattering of visible concerned, Rayleigh did not go essentially light by a water molecule is slightly less beyond Newton and Tyndall, who invoked than that by either nitrogen or oxygen. particles. Rayleigh was circumspect about Scattering by atmospheric molecules the nature of these particles, settling on does not obey Rayleigh’s inverse fourth- salt as the most likely candidate. It was not power law exactly. A least-squares fit over until 1899 that he published the capstone the visible spectrum from 400 to 700 nm to his work on skylight, arguing that air of the molecular scattering coefficient of sea- molecules themselves were the source of level air tabulated by Penndorf [2] yields an the blue sky. Tyndall cannot be given inverse 4.089th-power scattering law. β thecreditforthisbecauseheconsidered The molecular scattering coefficient , air to be optically empty:whenpurged which plays important roles in following of all particles it scatters no light. This sections, may be written erroneous conclusion was a result of the β = Nσs,(2) small scale of his laboratory experiments. On the scale of the atmosphere, sufficient where N is the number of molecules light is scattered by air molecules to be per unit volume and σs, the scattering readily observable. cross section (an average because air is 58 Atmospheric Optics

a mixture) per molecule, approximately an ideal gas (to good approximation the obeys Rayleigh’s law. The form of this atmosphere is an ideal gas), scattering by expression betrays the incoherence of N molecules is N times scattering by one. scattering by atmospheric molecules. The This is the only sense in which the blue sky inverse of β is interpreted as the scattering can be attributed to scattering by fluctua- mean free path, the average distance a tions. Perfectly homogeneous matter does photon must travel before being scattered. not exist. As stated pithily by Planck, ‘‘a To say that the sky is blue because chemically pure substance may be spo- of Rayleigh scattering, as is sometimes ken of as a vacuum made turbid by the done, is to confuse an agent with a presence of molecules.’’ law. Moreover, as Young [3] pointed out, the term Rayleigh scattering has many 2.3 meanings. Particles small compared with Spectrum and Color of Skylight the wavelength scatter according to the same law as do molecules. Both can What is the spectrum of skylight? What is be said to be Rayleigh scatterers, but its color? These are two different questions. only molecules are necessary for the blue Answering the first answers the second sky. Particles, even small ones, generally but not the reverse. Knowing the color of diminishthevividnessofthebluesky. skylight we cannot uniquely determine its Fluctuations are sometimes trumpeted spectrum because of metamerism:Agiven asthe‘‘real’’causeofthebluesky.Pre- perceived color can in general be obtained sumably, this stems from the fluctuation in an indefinite number of ways. theory of light scattering by media in which Skylight is not blue (itself an impre- the scatterers are separated by distances cise term) in an absolute sense. When small compared with the wavelength. In the visible spectrum of sunlight outside this theory, which is associated with Ein- the earth’s atmosphere is modulated by stein and Smoluchowski, matter is taken Rayleigh’s scattering law, the result is a to be continuous but characterized by a spectrum of scattered light that is nei- refractive index that is a random function ther solely blue nor even peaked in the of position. Einstein [4] stated that ‘‘it is blue (Fig. 1). Although blue does not pre- remarkable that our theory does not make dominate spectrally, it does predominate direct use of the assumption of a discrete perceptually. We perceive the sky to be distribution of matter.’’ That is, he cir- blue even though skylight contains light of cumvented a difficulty but realized it could all wavelengths. have been met head on, as Zimm [5] did Any source of light may be looked upon years later. as a mixture of white light and light of The blue sky is really caused by scat- a single wavelength called the dominant tering by molecules – to be more precise, wavelength.Thepurity of the source is scattering by bound electrons: free elec- the relative amount of the monochromatic trons do not scatter selectively. Because air component in the mixture. The dominant molecules are separated by distances small wavelength of sunlight scattered according compared with the wavelengths of visible to Rayleigh’s law is about 475 nm, which light, it is not obvious that the power scat- lies solidly in the blue if we take this tered by such molecules can be added. Yet to mean light with wavelengths between if they are completely uncorrelated, as in 450 and 490 nm. The purity of this Atmospheric Optics 59

2.4 Variation of Sky Color and Brightness

Notonlyisskylightnotpureblue, but its color and brightness vary across the vault of the sky, with the best blues at zenith. Near the astronomical horizon the sky is brighter than overhead but of considerably lower purity. That this variation can be observed from an airplane flying at 10 km, well above most particles, suggests that the sky is inherently nonuniform in color and Fig. 1 Rayleigh’s scattering law (dots), the brightness (Fig. 2). To understand why spectrum of sunlight outside the Earth’s requires invoking multiple scattering. atmosphere (dashes), and the product of the two (solid curve). The solar spectrum is taken from Multiple scattering gives rise to observ- Thekaekara, M. P., Drummond, A. J. (1971), Nat. able phenomena that cannot be explained Phys. Sci. 229, 6–9 [6] solely by single-scattering arguments. This is easily demonstrated. Fill a blackened pan scattered light, about 42%, is the upper limit for skylight. Blues of real skies are less pure. Another way of conveying the color of a source of light is by its color temperature, the temperature of a blackbody having the same perceived color as the source. Since blackbodies do not span the entire gamut of colors, not all sources of light can be assigned color temperatures. But many natural sources of light can. The color temperature of light scattered according to Rayleigh’s law is infinite. This follows from Planck’s spectral emission function ebλ in the limit of high temperature, 2πckT hc e λ ≈ ,  kT,(3) b λ4 λ where h is Planck’s constant, k is Boltz- mann’s constant, c is the speed of light in vacuo,andT is absolute temperature. Thus, the emission spectrum of a black- Fig. 2 Even at an altitude of 10 km, well above body with an infinite temperature has the most particles, the sky brightness increases same functional form as Rayleigh’s scat- markedly from the zenith to the tering law. astronomical horizon 60 Atmospheric Optics

with clean water, then add a few drops of path, the optical thickness is about 35 times milk. The resulting dilute suspension il- greater (Fig. 4), which leads to several luminated by sunlight has a bluish cast. observable phenomena. But when more milk is added, the suspen- Even an intrinsically black object is sion turns white. Yet the properties of the luminous to an observer because of scatterers (fat globules) have not changed, airlight, light scattered by all the molecules only their optical thickness: the blue suspen- and particles along the line of sight sion being optically thin, the white being from observer to object. Provided that optically thick. this is uniformly illuminated by sunlight Optical thickness is physical thickness and that ground reflection is negligi- in units of scattering mean free path, and ble, the airlight radiance L is approxi- hence is dimensionless. The optical thick- mately −τ ness τ between any two points connected L = GL0(1 − e ), (5) by an arbitrary path in a medium populated by (incoherent) scatterers is an integral where L0 is the radiance of incident over the path: sunlight along the line of sight with optical
thickness τ.ThetermG accounts for 2 geometric reduction of radiance because τ = β ds.(4) 1 of scattering of nearly monodirectional sunlight in all directions. If the line of τ The normal optical thickness n of the sight is uniform in composition, τ = βd, atmosphere is that along a radial path where β is the scattering coefficient and d extending from the surface of the Earth is the physical distance to the black object. to infinity. Figure 3 shows τn over the If τ is small (1), L ≈ GL0τ.Ina visible spectrum for a purely molecular purely molecular atmosphere, τ varies τ atmosphere. Because n is generally small with wavelength according to Rayleigh’s compared with unity, a photon from law; hence the distant black object in the sun traversing a radial path in the such an atmosphere is perceived to be atmosphere is unlikely to be scattered more than once. But along a tangential

Fig. 4 Optical thickness (relative to the normal optical thickness) of a molecular atmosphere Fig. 3 Normal optical thickness of a pure along various paths with zenith angles between ◦ ◦ molecular atmosphere 0 (normal) and 90 (tangential) Atmospheric Optics 61 bluish. As τ increases so does L but not transmitted to the observer without be- proportionally. Its limit is GL0: The airlight ing scattered out of the line of sight. radiance spectrum is that of the source of For a long optical path, these two pro- illumination. Only in the limit d = 0is cesses compensate, resulting in a hori- L = 0 and the black object truly black. zon radiance spectrum which is that of Variation of the brightness and color of the source. dark objects with distance was called aerial Selective scattering by molecules is not perspective by Leonardo. By means of it we sufficient for a blue sky. The atmosphere estimate distances to objects of unknown also must be optically thin, at least for size such as mountains. most zenith angles (Fig. 4) (the black- Aerial perspective belongs to the same ness of space as a backdrop is taken family as the variation of color and for granted but also is necessary, as brightness of the sky with zenith angle. Leonardo recognized). A corollary of this Although the optical thickness along a path is that the blue sky is not inevitable: an tangent to the Earth is not infinite, it is atmosphere composed entirely of nonab- sorbing, selectively scattering molecules sufficiently large (Figs. 3 and 4) that GL0 is a good approximation for the radiance of overlying a nonselectively reflecting earth the horizon sky. For isotropic scattering (a need not be blue. Figure 5 shows calcu- condition almost satisfied by molecules), lated spectra of the zenith sky over black − ground for a molecular atmosphere with G is around 10 5, the ratio of the solid the present normal optical thickness as angle subtended by the sun to the solid well as for hypothetical atmospheres 10 angle of all directions (4π). Thus, the and 40 times thicker. What we take to horizon sky is not nearly so bright as be inevitable is accidental: If our atmo- direct sunlight. sphere were much thicker, but identical Unlike in the milk experiment, what in composition, the color of the sky is observed when looking at the hori- would be quite different from what it zon sky is not multiply scattered light. is now. Both have their origins in multiple scat- tering but manifested in different ways. Milk is white because it is weakly ab- sorbing and optically thick, and hence all components of incident white light are multiply scattered to the observer even though the blue component traverses a shorter average path in the suspension than the red component. White horizon light has escaped being multiply scat- tered, although multiple scattering is why this light is white (strictly, has the spec- trum of the source). More light at the short-wavelength end of the spectrum is Fig. 5 Spectrum of overhead skylight for the scattered toward the observer than at the present molecular atmosphere (solid curve), as long-wavelength end. But long-wavelength well as for hypothetical atmospheres 10 (dashes) light has the greater likelihood of being and 40 (dots) times thicker 62 Atmospheric Optics

2.5 scatterers not lying along the line of Sunrise and Sunset sight to the sun. A striking example of this is a horizon sky tinged with If short-wavelength light is preferentially oranges and pinks in the direction opposite scattered out of direct sunlight, long- the sun. wavelength light is preferentially trans- The color and brightness of the sun mitted in the direction of sunlight. changes as it arcs across the sky because Transmission is described by an expo- the optical thickness along the line of sight nential law (if light multiply scattered changes with solar zenith angle .Ifthe back into the direction of the sunlight Earth were flat (as some still aver), the is negligible): transmitted solar radiance would be −τ τ /  L = L0e ,(6) n cos L = L0e .(7) where L is the radiance at the observer This equation is a good approximation in the direction of the sun, L is the 0 except near the horizon. On a flat earth, radiance of sunlight outside the atmo- the optical thickness is infinite for horizon sphere, and τ is the optical thickness along paths. On a spherical earth, optical thick- this path. nesses are finite although much larger for If the wavelength dependence of τ is horizon than for radial paths. given by Rayleigh’s law, sunlight is red- The normal optical thickness of an dened upon transmission: The spectrum atmosphere in which the number density of the transmitted light is comparatively of scatterers decreases exponentially with richer than the incident spectrum in height z above the surface, exp(−z/H), is lightatthelong-wavelengthendofthe the same as that for a uniform atmosphere visible spectrum. But to say that trans- of finite thickness: mitted sunlight is reddened is not the
∞ same as saying it is red. The perceived τn = β dz = β0H,(8) color can be yellow, orange, or red, de- 0 pending on the magnitude of the optical β thickness. In a molecular atmosphere, where H is the scale height and 0 is the optical thickness along a path from the scattering coefficient at sea level. the sun, even on or below the horizon, This equivalence yields a good approx- is not sufficient to give red light upon imation even for the tangential opti- transmission. Although selective scatter- cal thickness. For any zenith angle, ing by molecules yields a blue sky, reds the optical thickness is given approxi- are not possible in a molecular atmo- mately by sphere, only yellows and oranges. This  τ R2 2R can be observed on clear days, when the = e 2  + e + 2 cos 1 horizon sky at sunset becomes succes- τn H H sively tinged with yellow, then orange, but Re not red. − cos , (9) H Equation (6) applies to the radiance only in the direction of the sun. Oranges and where Re is the radius of the Earth. reds can be seen in other directions AflatearthisoneforwhichRe is because reddened sunlight illuminates infinite, in which instance Eq. (9) yields Atmospheric Optics 63 the expected relation from direct sunlight by clouds (that do not, of course, occlude the distant cloud τ 1 lim = .(10) of interest). This may reduce the first →∞ τ  Re n cos term in Eq. (11) so that the second term For Earth’s atmosphere, the molecular dominates. Thus, under a partly over- scale height is about 8 km. According to cast sky, distant horizon clouds may be the approximate relation Eq. (9), therefore, reddish even when the sun is high in the horizon optical thickness is about the sky. 39 times greater than the normal optical The zenith sky at sunset and twilight thickness. Taking the exponential decrease is the exception to the general rule that of molecular number density into account molecular scattering is sufficient to ac- yields a value about 10% lower. count for the color of the sky. In the Variations on the theme of reds and absence of molecular absorption, the spec- oranges at sunrise and sunset can be trum of the zenith sky would be essentially seen even when the sun is overhead. The that of the zenith sun (although greatly radiance at an observer an optical distance reduced in radiance), hence would not τ from a (horizon) cloud is the sum bethebluethatisobserved.Thiswas of cloudlight transmitted to the observer pointed out by Hulburt [7], who showed and airlight: that absorption by ozone profoundly af- fects the color of the zenith sky when the −τ −τ L = L0G(1 − e ) + L0Gce ,(11) sun is near the horizon. The Chappuis band of ozone extends from about 450 to where Gc is a geometrical factor that 700 nm and peaks at around 600 nm. Pref- accounts for scattering of nearly monodi- erential absorption of sunlight by ozone rectional sunlight into a hemisphere of over long horizon paths gives the zenith directions by the cloud. If the cloud sky its blueness when the sun is near is approximated as an isotropic reflec- the horizon. With the sun more than tor with reflectance R and illuminated ◦ about 10 above the horizon, however, at an angle , the geometrical factor ozone has little effect on the color of G is  R cos/π,where is the c s s the sky. solid angle subtended by the sun at the Earth. If Gc > G, the observed ra- diance is redder (i.e., enriched in light of longer wavelengths) than the incident 3 radiance. If Gc < G, the observed radi- Polarization of Light in a Molecular ance is bluer than the incident radiance. Atmosphere Thus, distant horizon clouds can be red- dish if they are bright or bluish if they 3.1 are dark. The Nature of Polarized Light Underlying Eq. (11) is the implicit as- sumption that the line of sight is uniformly Unlike sound, light is a vector wave, an illuminated by sunlight. The first term electromagnetic field lying in a plane nor- in this equation is airlight; the second mal to the propagation direction. The is transmitted cloudlight. Suppose, how- polarization state of such a wave is deter- ever, that the line of sight is shadowed mined by the degree of correlation of any 64 Atmospheric Optics

two orthogonal components into which observed brightness indicates some degree its electric (or magnetic) field is resolved. of partial polarization. Completely polarized light corresponds to In the analysis of any scattering prob- complete correlation; completely unpolar- lem, a plane of reference is required. This ized light corresponds to no correlation; is usually the scattering plane, determined partially polarized light corresponds to par- by the directions of the incident and scat- tial correlation. tered waves, the angle between them being If an electromagnetic wave is completely the scattering angle. Light polarized perpen- polarized, the tip of its oscillating electric dicular (parallel) to the scattering plane is field traces out a definite elliptical curve, sometimes said to be vertically (horizon- the vibration ellipse. Lines and circles are tally) polarized. Vertical and horizontal in special ellipses, the light being said to this context, however, are arbitrary terms be linearly or circularly polarized, respec- indicating orthogonality and bear no rela- tively. The general state of polarization is tion, except by accident, to the direction elliptical. of gravity. Any beam of light can be consid- ThedegreeofpolarizationP of light ered an incoherent superposition of two scattered by a tiny sphere illuminated by collinear beams, one unpolarized, the unpolarized light is (Fig. 6) other completely polarized. The radiance 1 − cos2 θ of the polarized component relative to P = ,(12) the total is defined as the degree of po- 1 + cos2 θ larization (often multiplied by 100 and where the scattering angle θ ranges from ◦ ◦ expressed as a percentage). This can be 0 (forward direction) to 180 (backward measured for a source of light (e.g., light direction); the scattered light is partially from different sky directions) by rotat- linearly polarized perpendicular to the ing a (linear) polarizing filter and noting scattering plane. Although this equation the minimum and maximum radiances transmitted by it. The degree of (linear) polarization is defined as the difference between these two radiances divided by their sum.

3.2 Polarization by Molecular Scattering

Unpolarized light can be transformed into partially polarized light upon interaction with matter because of different changes in amplitude of the two orthogonal field com- ponents. An example of this is the partial polarization of sunlight upon scattering Fig. 6 Degree of polarization of the light by atmospheric molecules, which can be scattered by a small (compared with the wavelength) sphere for incident unpolarized detected by looking at the sky through a po- light (solid curve). The dashed curve is for a larizing filter (e.g., polarizing sunglasses) small spheroid chosen such that the degree of ◦ while rotating it. Waxing and waning of the polarization at 90 is that for air Atmospheric Optics 65 is a first step toward understanding that in the atmosphere, as opposed to polarization of skylight, more often than the laboratory, multiple scattering is not not it also has been a false step, having led negligible. Also, atmospheric air is almost countless authors to assert that skylight is never free of particles and is illuminated ◦ completely polarized at 90 from the sun. by light reflected by the ground. We must ◦ Although P = 1atθ = 90 according to take the atmosphere as it is, whereas Eq. (12), skylight is never 100% polarized in the laboratory we often can eliminate at this or any other angle, and for everything we consider extraneous. several reasons. Because of both multiple scattering Although air molecules are very small and ground reflection, light from any compared with the wavelengths of visible directionintheskyisnot,ingeneral, light, a requirement underlying Eq. (12), made up solely of light scattered in a the dominant constituents of air are not single direction relative to the incident spherically symmetric. sunlight but is a superposition of beams The simplest model of an asymmetric with different scattering histories, hence molecule is a small spheroid. Although different degrees of polarization. As a it is indeed possible to find a direction consequence, even if air molecules were in which the light scattered by such a perfect spheres and the atmosphere were spheroid is 100% polarized, this direction completely free of particles, skylight would ◦ depends on the spheroid’s orientation. not be 100% polarized at 90 to the sun or In an ensemble of randomly oriented at any other angle. spheroids, each contributes its mite to the Reduction of the maximum degree of total radiance in a given direction, but polarization is not the only consequence each contribution is partially polarized to of multiple scattering. According to Fig. 6, varying degrees between 0 and 100%. It there should be two neutral points in the is impossible for beams of light to be sky, directions in which skylight is unpo- incoherently superposed in such a way that larized: directly toward and away from the the degree of polarization of the resultant sun. Because of multiple scattering, how- is greater than the degree of polarization ever, there are three such points. When ◦ of the most highly polarized beam. the sun is higher than about 20 above the ◦ Because air is an ensemble of randomly horizon there are neutral points within 20 oriented asymmetric molecules, sunlight of the sun, the Babinet point above it, the scattered by air never is 100% polarized. Brewster point below. They coincide when The intrinsic departure from perfection is the sun is directly overhead and move about 6%. Figure 6 also includes a curve apart as the sun descends. When the sun ◦ for light scattered by randomly oriented is lower than 20 ,theArago point is about ◦ spheroids chosen to yield 94% polarization 20 above the antisolar point, the direction ◦ at 90 . This angle is so often singled opposite the sun. outthatitmaydeflectattentionfrom One consequence of the partial polar- nearby scattering angles. Yet, the degree of ization of skylight is that the colors of polarization is greater than 50% for a range distant objects may change when viewed ◦ of scattering angles 70 wide centered through a rotated polarizing filter. If the ◦ about 90 . sun is high in the sky, horizontal airlight Equation (12) applies to air, not to will have a fairly high degree of polariza- the atmosphere, the distinction being tion. According to the previous section, 66 Atmospheric Optics

airlight is bluish. But if it also is partially scattered field, which in turn is the sum polarized, its radiance can be diminished of all the fields scattered by the individual with a polarizing filter. Transmitted cloud- molecules. For an incoherent array we may light, however, is unpolarized. Because the ignore the wave nature of light, whereas radiance of airlight can be reduced more for a coherent array we must take it than that of cloudlight, distant clouds may into account. change from white to yellow to orange Water vapor is a good example to ponder when viewed through a rotated polariz- because it is a constituent of air and ing filter. can condense to form cloud droplets. The difference between a sky containing water vapor and the same sky with the same 4 amount of water but in the form of a cloud Scattering by Particles of droplets is dramatic. According to Rayleigh’s law, scattering Up to this point we have considered only an by a particle small compared with the atmosphere free of particles, an idealized wavelength increases as the sixth power state rarely achieved in nature. Particles of its size (volume squared). A droplet of still would inhabit the atmosphere even diameter 0.03 µm, for example, scatters if the human race were to vanish from about 1012 times more light than does one the Earth. They are not simply by- of its constituent molecules. Such a droplet products of the ‘‘dark satanic mills’’ of contains about 107 molecules. Thus, civilization. scattering per molecule as a consequence All molecules of the same substance are of condensation of water vapor into essentially identical. This is not true of a coherent water droplet increases by particles: They vary in shape and size, about 105. and may be composed of one or more Cloud droplets are much larger than homogeneous regions. 0.03 µm, a typical diameter being about 10 µm. Scattering per molecule in such 4.1 a droplet is much greater than scatter- The Salient Differences between Particles ing by an isolated molecule, but not to and Molecules: Magnitude of Scattering the extent given by Rayleigh’s law. Scat- tering increases as the sixth power of The distinction between scattering by droplet diameter only when the molecules molecules when widely separated and scatter coherently in phase. If a droplet when packed together into a droplet is is sufficiently small compared with the that between scattering by incoherent and wavelength, each of its molecules is ex- coherent arrays. Isolated molecules are cited by essentially the same field and excited primarily by incident (external) all the waves scattered by them inter- light, whereas the same molecules forming fere constructively. But when a droplet a droplet are excited by incident light and is comparable to or larger than the wave- by each other’s scattered fields. The total length, interference can be constructive, power scattered by an incoherent array of destructive, and everything in between, molecules is the sum of their scattered and hence scattering does not increase powers. The total power scattered by a as rapidly with droplet size as predicted by coherent array is the square of the total Rayleigh’s law. Atmospheric Optics 67

The figure of merit for comparing different from the water vapor out of which scatterers of different size is their scatter- it was born that the offspring bears no ing cross section per unit volume, which, resemblance to its parents. We can see except for a multiplicative factor, is the scat- through tens of kilometers of air laden tering cross section per molecule. A scat- with water vapor, whereas a cloud a few tering cross section may be looked upon tens of meters thick is enough to occult as an effective area for removing radiant the sun. Yet a rainshaft born out of a energy from a beam: the scattering cross cloud is considerably more translucent section times the beam irradiance is the than its parent. radiant power scattered in all directions. The scattering cross section per unit 4.2 volume for water droplets illuminated The Salient Differences between Particles by visible light and varying in size and Molecules: Wavelength Dependence of from molecules (10−4 µm) to raindrops Scattering (103 µm) is shown in Fig. 7. Scattering by a molecule that belongs to a cloud droplet is Regardless of their size and composi- about 109 times greater than scattering by tion, particles scatter approximately as an isolated molecule, a striking example the inverse fourth power of wavelength of the virtue of cooperation. Yet in if they are small compared with the wave- molecular as in human societies there are length and absorption is negligible, two limits beyond which cooperation becomes important caveats. Failure to recognize dysfunctional: Scattering by a molecule them has led to errors, such as that yel- that belongs to a raindrop is about 100 low light penetrates fog better because times less than scattering by a molecule it is not scattered as much as light of that belongs to a cloud droplet. This shorter wavelengths. Although there may tremendous variation of scattering by be perfectly sound reasons for choosing water molecules depending on their state yellow instead of blue or green as the of aggregation has profound observational color of fog lights, greater transmission consequences. A cloud is optically so much through fog is not one of them: Scat- tering by fog droplets is essentially in- dependent of wavelength over the visible spectrum. Small particles are selective scatterers; large particles are not. Particles nei- ther small nor large give the reverse of what we have come to expect as nor- mal. Figure 8 shows scattering of visible light by oil droplets with diameters 0.1, 0.8, and 10 µm. The smaller droplets scatter according to Rayleigh’s law; the larger droplets (typical cloud droplet size) are nonselective. Between these two ex- µ Fig. 7 Scattering (per molecule) of visible light tremes are droplets (0.8 m) that scatter (arbitrary units) by water droplets varying in size long-wavelength light more than short- from a single molecule to a raindrop wavelength. Sunlight or moonlight seen 68 Atmospheric Optics

the forward and backward hemispheres, scattering becomes markedly asymmetric for particles comparable to or larger than the wavelength. For example, forward scat- tering by a water droplet as small as 0.5 µm is about 100 times greater than backward scattering, and the ratio of forward to back- ward scattering increases more or less monotonically with size (Fig. 9). The reason for this asymmetry is found in the singularity of the forward direc- tion. In this direction, waves scattered Fig. 8 Scattering of visible light by oil droplets of diameter 0.1 µm (solid curve), 0.8 µm by two or more scatterers excited solely (dashes), and 10 µm(dots) by incident light (ignoring mutual ex- citation) are always in phase regardless of the wavelength and the separation through a thin cloud of these intermediate of the scatterers. If we imagine a par- droplets would be bluish or greenish. This ticletobemadeupofN small sub- requires droplets of just the right size, and units, scattering in the forward direc- hence it is a rare event, so rare that it oc- tion increases as N2,theonlydirec- curs once in a blue moon. Astronomers, tion for which this is always true. For for unfathomable reasons, refer to the sec- other directions, the wavelets scattered by ond full moon in a month as a blue moon, the subunits will not necessarily all be but if such a moon were blue it would be in phase. As a consequence, scattering only by coincidence. The last reliably re- in the forward direction increases with ported outbreak of blue and green suns size (i.e., N) more rapidly than in any and moons occurred in 1950 and was other direction. attributed to an oily smoke produced in Canadian forest fires.

4.3 The Salient Differences between Particles and Molecules: Angular Dependence of Scattering

The angular distribution of scattered light changes dramatically with the size of the scatterer. Molecules and particles that are small compared with the wavelength are nearly isotropic scatterers of unpo- larized light, the ratio of maximum (at ◦ ◦ ◦ 0 and 180 ) to minimum (at 90 )scat- Fig. 9 Angular dependence of scattering of visible light (0.55 µm) by water droplets small tered radiance being only 2 for spheres, compared with the wavelength (dashes), and slightly less for other spheroids. Al- diameter 0.5 µm (solid curve), and diameter though small particles scatter the same in 10 µm(dots) Atmospheric Optics 69

Many common observable phenom- ena depend on this forward-backward asymmetry. Viewed toward the illumi- nating sun, glistening fog droplets on a spider’s web warn us of its presence. But when we view the web with our backs to the sun, the web mysteriously disap- pears. A pattern of dew illuminated by the rising sun on a cold morning seems etched on a windowpane. But if we go outside to look at the window, the pattern vanishes. Thin clouds sometimes hover Fig. 10 Degree of polarization of light scattered over warm, moist heaps of dung, but may by water droplets illuminated by unpolarized go unnoticed unless they lie between us visible light (0.55 µm). The dashed curve is for a and the source of illumination. These are droplet small compared with the wavelength; the solid curve is for a droplet of diameter 0.5 µm; but a few examples of the consequences the dotted curve is for a droplet of diameter of strongly asymmetric scattering by sin- 1.0 µm. Negative degrees of polarization gle particles comparable to or larger than indicate that the scattered light is partially the wavelength. polarized parallel to the scattering plane

4.4 The Salient Differences between Particles and Molecules: Degree of Polarization of Scattered Light

All the simple rules about polarization upon scattering are broken when we turn from molecules and small particles to particles comparable to the wavelength. For example, the degree of polarization of light scattered by small particles is a simple function of scattering angle. But simplicity gives way to complexity as particles grow Fig. 11 Degree of polarization at a scattering ◦ (Fig. 10), the scattered light being partially angle of 90 of light scattered by a water droplet polarized parallel to the scattering plane of diameter 0.5 µm illuminated by unpolarized light for some scattering angles, perpendicular for others. ◦ The degree of polarization of light say, 90 may vary considerably over the scattered by molecules or by small particles visible spectrum (Fig. 11). is essentially independent of wavelength. In general, particles can act as polarizers But this is not true for particles comparable or retarders or both. A polarizer transforms to or larger than the wavelength. Scattering unpolarized light into partially polarized by such particles exhibits dispersion of light. A retarder transforms polarized light polarization: The degree of polarization at, of one form into that of another (e.g., 70 Atmospheric Optics

linear into elliptical). Molecules and small 4.5 particles, however, are restricted to roles The Salient Differences between Particles as polarizers. If the atmosphere were and Molecules: Vertical Distributions inhabited solely by such scatterers, skylight could never be other than partially linearly Not only are the scattering properties of polarized. Yet particles comparable to particles quite different, in general, from or larger than the wavelength often those of molecules; the different vertical are present; hence skylight can acquire distributions of particles and molecules by a degree of ellipticity upon multiple themselves affect what is observed. The scattering: Incident unpolarized light is number density of molecules decreases partially linearly polarized in the first more or less exponentially with height scattering event, then transformed into z above the surface: exp(−z/Hm), where partially elliptically polarized light in the molecular scale height Hm is around subsequent events. 8 km. Although the decrease in number Bees can navigate by polarized sky- density of particles with height is also ap- light. This statement, intended to evoke proximately exponential, the scale height great awe for the photopolimetric pow- for particles Hp is about 1–2 km. As a ers of bees, is rarely accompanied by consequence, particles contribute dispro- an important caveat: The sky must be portionately to optical thicknesses along clear. Figures 10 and 11 show two rea- near-horizon paths. Subject to the approxi- sons – there are others – why bees, re- mations underlying Eq. (9), the ratio of the markable though they may be, cannot do tangential (horizon) optical thickness for the impossible. The simple wavelength- particles τtp to that for molecules τtm is independent relation between the posi-  tion of the sun and the direction in τtp τnp Hm which skylight is most highly polarized, = ,(13) τ τ H an underlying necessity for navigating tm nm p by means of polarized skylight, is oblit- erated when clouds cover the sky. This where the subscript t indicates a tangential was recognized by the decoder of bee path and n indicates a normal (radial) path. dances himself von Frisch, [8]: ‘‘Some- Because of the incoherence of scattering times a cloud would pass across the area by atmospheric molecules and particles, of sky visible through the tube; when this scattering coefficients are additive, and happened the dances became disoriented, hence so are optical thicknesses. For equal and the bees were unable to indicate the normal optical thicknesses, the tangential direction to the feeding place. Whatever optical thickness for particles is at least phenomenon in the blue sky served to ori- twice that for molecules. Molecules by ent the dances, this experiment showed themselves cannot give red sunrises and that it was seriously disturbed if the sunsets; molecules need the help of blue sky was covered by a cloud.’’ But particles. For a fixed τnp, the tangential von Frisch’s words often have been for- optical thickness for particles is greater gotten by disciples eager to spread the themoretheyareconcentratednear story about bee magic to those just as the ground. eager to believe what is charming even At the horizon the relative rate of change though untrue. of transmission T of sunlight with zenith Atmospheric Optics 71 angle is 1 dT Re = τn ,(14) T d H where the scale height and normal opti- cal thickness may be those for molecules or particles. Not only do particles, be- ing more concentrated near the surface, give disproportionate attenuation of sun- light on the horizon, but they magnify the angular gradient of attenuation there. A perceptible change in color across the ◦ sun’s disk (which subtends about 0.5 ) on the horizon also requires the help of particles. Fig. 12 Because of scattering by molecules and particles along the line of sight, each successive ridge is brighter than the ones in front of it even 5 though all of them are covered with the same Atmospheric Visibility dark vegetation

On a clear day can we really see for- ever? If not, how far can we see? To account the portion of it that stimu- answer this question requires qualifying lates the human eye or by what relative it by restricting viewing to more or less amount it does so at each wavelength. horizontal paths during daylight. Stars Luminance (also sometimes called bright- at staggering distances can be seen at ness) is the corresponding photometric night, partly because there is no sky- quantity. Luminance and radiance are light to reduce contrast, partly because related by an integral over the visi- stars overhead are seen in directions ble spectrum: for which attenuation by the atmosphere
is least. B = K(λ)L(λ) dλ, (15) The radiance in the direction of a black object is not zero, because of light scattered where the luminous efficiency of the hu- along the line of sight (see Sec. 2.4). At man eye K peaks at about 550 nm and sufficiently large distances, this airlight is vanishes outside the range 385–760 nm. indistinguishable from the horizon sky. The contrast C between any object and An example is a phalanx of parallel dark the horizon sky is ridges, each ridge less distinct than those in front of it (Fig. 12). The farthest ridges − = B B∞ ,() blend into the horizon sky. Beyond some C 16 B∞ distance we cannot see ridges because of insufficient contrast. where B∞ is the luminance for an infinite Equation (5) gives the airlight radi- horizon optical thickness. For a uniformly ance, a radiometric quantity that de- illuminated line of sight of length d, scribes radiant power without taking into uniform in its scattering properties, and 72 Atmospheric Optics

with a black backdrop, the contrast is sky assuming a contrast threshold of 0.02
and ignoring the curvature of the earth.

KGL0 exp(−βd) dλ We also observe contrast between ele- C =−
.(17) ments of the same scene, a hillside mottled with stands of trees and forest clearings, KGL0 dλ for example. The extent to which we can resolve details in such a scene depends on The ratio of integrals in this equation sun angle as well as distance. defines an average optical thickness: The airlight radiance for a nonreflecting =   C =−exp(−τ). (18) object is Eq. (5) with G p( ) s,where p() is the probability (per unit solid angle) This expression for contrast reduction that light is scattered in a direction making with (optical) distance is mathematically, an angle  with the incident sunlight and s is the solid angle subtended by the sun. but not physically, identical to Eq. (6), ◦ When the sun is overhead,  = 90 ;with which perhaps has engendered the mis- ◦ conception that atmospheric visibility is the sun at the observer’s back,  = 180 ; for an observer looking directly into the reduced because of attenuation. Yet as ◦ there is no light from a black object to be sun,  = 0 . attenuated, its finite visual range cannot The radiance of an object with a finite re- be a consequence of attenuation. flectance R and illuminated at an angle  The distance beyond which a dark is given by Eq. (11). Equations (5) and (11) object cannot be distinguished from the can be combined to obtain the contrast be- horizon sky is determined by the contrast tween reflecting and nonreflecting objects: threshold: the smallest contrast detectable Fe−τ by the human observer. Although this = , C + ( − ) −τ depends on the particular observer, the 1 F 1 e angular size of the object observed, R cos  F = .(20) the presence of nearby objects, and the πp() absolute luminance, a contrast threshold of 0.02 is often taken as an average. This All else being equal, therefore, contrast  value in Eq. (18) gives decreases as p( ) increases. As shown in Fig. 9, p() is more sharply peaked in the − ln |C|=3.9 =τ=βd.(19) forward direction the larger the scatterer. Thus, we expect the details of a distant To convert an optical distance into a scene to be less distinct when looking physical distance requires the scattering toward the sun than away from it if the coefficient. Because K is peaked at around optical thickness of the line of sight has 550 nm, we can obtain an approximate an appreciable component contributed by value of d from the scattering coefficient particles comparable to or larger than the at this wavelength in Eq. (19). At sea wavelength. level, the molecular scattering coefficient On humid, hazy days, visibility is in the middle of the visible spectrum often depressingly poor. Haze, however, corresponds to about 330 km for ‘‘forever’’: is not water vapor but rather water the greatest distance at which a black that has ceased to be vapor. At high object can be seen against the horizon relative humidities, but still well below Atmospheric Optics 73

100%, small soluble particles in the gas and the scattering cross section σs of a atmosphere accrete liquid water to become gas molecule: solution droplets (haze). Although these = + 1 α ,() droplets are much smaller than cloud n 1 2 N 21 droplets, they markedly diminish visual 4 k 2 rangebecauseofthesharpincreasein σs = |α| ,(22) 6π scattering with particle size (Fig. 7). The same number of water molecules when where N is the number density (not mass aggregated in haze scatter vastly more than density) of gas molecules, k = 2π/λ is the when apart. wave number of the incident light, and α is the polarizability of a molecule (induced dipole moment per unit inducing electric 6 field). The appearance of the polarizabil- Atmospheric Refraction ity in Eq. (21) but its square in Eq. (22) is the clue that refraction is associated with 6.1 electric fields whereas lateral scattering Physical Origins of Refraction is associated with electric fields squared (powers). Scattering, without qualification, Atmospheric refraction is a consequence often means incoherent scattering in all of molecular scattering, which is rarely directions. Refraction, in a nutshell, is co- stated given the historical accident that herent scattering in a particular direction. before light and matter were well un- Readers whose appetites have been derstood refraction and scattering were whetted by the preceding brief discussion locked in separate compartments and sub- of the physical origins of refraction are sequently have been sequestered more directed to a beautiful paper by Doyle [9] rigidly than monks and nuns in neigh- in which he shows how the Fresnel boring cloisters. equations can be dissected to reveal the Consider a beam of light propagating in scattering origins of (specular) reflection an optically homogeneous medium. Light and refraction. is scattered (weakly but observably) later- ally to this beam as well as in the direction 6.2 of the beam (the forward direction). The Terrestrial Mirages observed beam is a coherent superposi- tion of incident light and forward-scattered Mirages are not illusions, any more so light, which was excited by the incident than are reflections in a pond. Reflections light. Although refractive indices are of- of plants growing at its edge are not ten defined by ratios of phase velocities, interpreted as plants growing into the we may also look upon a refractive index water. If the water is ruffled by wind, as a parameter that specifies the phase the reflected images may be so distorted shift between an incident beam and the that they are no longer recognizable forward-scattered beam that the incident as those of plants. Yet we still would beam excites. The connection between not call such distorted images illusions. (incoherent) scattering and refraction (co- And so is it with mirages. They are herent scattering) can be divined from the images noticeably different from what they expressions for the refractive index n of a would be in the absence of atmospheric 74 Atmospheric Optics

refraction, creations of the atmosphere, shallow surface layers, in which the pres- not of the mind. sure is nearly constant, the temperature Mirages are vastly more common than gradient determines the refractive index is realized. Look and you shall see them. gradient. It is in such shallow layers that Contrary to popular opinion, they are mirages, which are caused by refractive- not unique to deserts. Mirages can be index gradients, are seen. seen frequently even over ice-covered Cartoonists by their fertile imaginations landscapes and highways flanked by deep unfettered by science, and textbook writers snowbanks. Temperature per se is not by their carelessness, have engendered what gives mirages but rather temperature the notion that atmospheric refraction can gradients. work wonders, lifting images of ships, for Because air is a mixture of gases, the example, from the sea high into the sky. polarizability for air in Eq. (21) is an A back-of-the-envelope calculation dispels average over all its molecular constituents, such notions. The refractive index of air at although their individual polarizabilities sea level is about 1.0003 (Fig. 13). Light are about the same (at visible wavelengths). from empty space incident at glancing The vertical refractive index gradient can incidence onto a uniform slab with this be written so as to show its dependence on refractive index is displaced in angular pressure p and (absolute) temperature T: position from where it would have been intheabsenceofrefractionby d 1 dp 1 dT  ln(n − 1) = − .(23) δ = ( − ). ( ) dz p dz T dz 2 n 1 24 This yields an angular displacement of Pressure decreases approximately ex- ◦ about 1.4 , which as we shall see is a rough ponentially with height, where the scale upper limit. height is around 8 km. Thus, the first term Trajectories of light rays in nonuniform on the right-hand side of Eq. (23) is around media can be expressed in different ways. 0.1 km−1. Temperature usually decreases According to Fermat’s principle of least with height in the atmosphere. An average lapse rate of temperature (i.e., its decrease ◦ with height) is around 6 C/km. The aver- age temperature in the troposphere (within about 15 km of the surface) is around 280 K. Thus, the magnitude of the second term in Eq. (23) is around 0.02 km−1.On average, therefore, the refractive-index gra- dient is dominated by the vertical pressure gradient. But within a few meters of the surface, conditions are far from average. On a sun-baked highway your feet may ◦ be touching asphalt at 50 Cwhileyour ◦ nose is breathing air at 35 C, which cor- Fig. 13 Sea-level refractive index versus ◦ ◦ responds to a lapse rate a thousand times wavelength at −15 C(dashes)and15 C(solid the average. Moreover, near the surface, curve). Data from Penndorf, R. (1957), J. Opt. temperature can increase with height. In Soc. Am. 47, 176–182[2] Atmospheric Optics 75 time (which ought to be extreme time), the actual path taken by a ray between two points is such that the path integral
2 n ds (25) 1 is an extremum over all possible paths. This principle has inspired piffle about the alleged efficiency of nature, which directs light over routes that minimize travel time, presumably freeing it to tend to important business at its destination. Fig. 14 Parabolic ray paths in an atmosphere The scale of mirages is such that in with a constant refractive-index gradient (inferior analyzing them we may pretend that the mirage). Note the vastly different horizontal and vertical scales Earthisflat.Onsuchanearth,with an atmosphere in which the refractive index varies only in the vertical, Fermat’s familiar highway mirage, seen over high- principle yields a generalization ways warmer than the air above them. The downward angular displacement is n sin θ = constant (26) 1 dn δ = s ,(28) of Snel’s law, where θ is the angle between 2 dz the ray and the vertical direction. We could, of course, have bypassed Fermat’s where s is the horizontal distance between principle to obtain this result. object and observer (image). Even for If we restrict ourselves to nearly hori- a temperature gradient 1000 times the zontal rays, Eq. (26) yields the following tropospheric average, displacements of differential equation satisfied by a ray: mirages are less than a degree at distances of a few kilometers. d2z dn If temperature increases with height, = ,(27) dy2 dz as it does, for example, in air over a cold sea, the resulting mirage is called where y and z are its horizontal and vertical a superior mirage. Inferior and superior are coordinates, respectively. For a constant not designations of lower and higher caste refractive-index gradient, which to good but rather of displacements downward approximation occurs for a constant tem- and upward. perature gradient, Eq. (27) yields parabolas For a constant temperature gradient, for ray trajectories. One such parabola for one and only one parabolic ray tra- a constant temperature gradient about 100 jectory connects an object point to an times the average is shown in Fig. 14. image point. Multiple images therefore Note the vastly different horizontal and are not possible. But temperature gra- vertical scales. The image is displaced dients close to the ground are rarely downward from what it would be in the linear. The upward transport of energy absence of atmospheric refraction; hence from a hot surface occurs by molecular the designation inferior mirage. This is the conduction through a stagnant boundary 76 Atmospheric Optics

layer of air. Somewhat above the surface, originating from outside it had to have however, energy is transported by air in been incident on it from an angle δ below motion. As a consequence, the tempera- the horizon: ture gradient steepens toward the ground   2H 2H if the energy flux is constant. This vari- δ = − − 2(n − 1), (29) able gradient can lead to two observable R R consequences: magnification and multi- where R is the radius of the Earth. Thus, ple images. when the sun (or moon) is seen to be on According to Eq. (28), all image points the horizon it is actually more than halfway ◦ at a given horizontal distance are dis- below it, δ being about 0.36 , whereas the placed downward by an amount propor- angular width of the sun (or moon) is ◦ tional to the (constant) refractive index about 0.5 . gradient. A corollary is that the closer Extraterrestrial bodies seen near the an object point is to a surface, where horizon also are vertically compressed. The the temperature gradient is greatest, the simplest way to estimate the amount of greater the downward displacement of the compression is from the rate of change of corresponding image point. Thus, non- angle of refraction θr with angle of inci- linear vertical temperature profiles may dence θi for a uniform slab magnify images. dθ cos θ Multiple images are seen frequently r =  i ,(30) θ 2 2 on highways. What often appears to d i n − sin θi be water on the highway ahead but evaporates before it is reached is the wheretheangleofincidenceisthatfor inverted secondary image of either the a curved but uniform atmosphere such horizon sky or horizon objects lighter than that the refracted ray is horizontal. The dark asphalt. result is  dθ R r = 1 − (n − 1), (31) 6.3 dθ H Extraterrestrial Mirages i according to which the sun near the When we turn from mirages of terrestrial horizon is distorted into an ellipse with objects to those of extraterrestrial bodies, aspect ratio about 0.87. We are unlikely most notably the sun and moon, we to notice this distortion, however, be- can no longer pretend that the Earth causeweexpectthesunandmoonto is flat. But we can pretend that the be circular, and hence we see them atmosphere is uniform and bounded. that way. The total phase shift of a vertical ray The previous conclusions about the from the surface to infinity is the same downward displacement and distortion of in an atmosphere with an exponentially the sun were based on a refractive-index decreasing molecular number density as in profile determined mostly by the verti- a hypothetical atmosphere with a uniform cal pressure gradient. Near the ground, number density equal to the surface value however, the temperature gradient is the up to height H. prime determinant of the refractive-index A ray refracted along a horizon path gradient, as a consequence of which the by this hypothetical atmosphere and sun on the horizon can take on shapes Atmospheric Optics 77

6.4 The Green Flash

Compared to the rainbow, the green flash is not a rare phenomenon. Before you dismiss this assertion as the ravings of a lunatic, consider that rainbows require raindrops as well as sunlight to illuminate them, whereas rainclouds often completely obscure the sun. Moreover, ◦ the sun must be below about 42 .Asa consequence of these conditions, rainbows are not seen often, but often enough that they are taken as the paragon of color variation. Yet tinges of green on the upper Fig. 15 A nearly triangular sun on the horizon. rim of the sun can be seen every day The serrations are a consequence of horizontal at sunrise and sunset given a sufficiently variations in refractive index low horizon and a cloudless sky. Thus, the conditions for seeing a green flash more striking than a mere ellipse. For are more easily met than those for seeing example, Fig. 15 shows a nearly triangu- a rainbow. Why then is the green flash lar sun with serrated edges. Assigning considered to be so rare? The distinction a cause to these serrations provides a here is that between a rarely observed lesson in the perils of jumping to con- phenomenon (the green flash) and a rarely clusions. Obviously, the serrations are the observable one (the rainbow). result of sharp changes in the temper- Thesunmaybeconsideredtobea ature gradient – or so one might think. collection of disks, one for each visible Setting aside how such changes could be wavelength. When the sun is overhead, produced and maintained in a real at- all the disks coincide and we see the mosphere, a theorem of Fraser [10] gives sun as white. But as it descends in the pause for thought. According to this the- sky, atmospheric refraction displaces the orem, ‘‘In a horizontally (spherically) ho- disksbyslightlydifferentamounts,thered mogeneous atmosphere it is impossible less than the violet (see Fig. 13). Most of for more than one image of an extrater- each disk overlaps all the others except restrial object (sun) to be seen above the for the disks at the extremes of the visible astronomical horizon.’’ The serrations on spectrum. As a consequence, the upper the sun in Fig. 15 are multiple images. rim of the sun is violet or blue, its lower But if the refractive index varies only rim red, whereas its interior, the region in vertically (i.e., along a radius), no mat- which all disks overlap, is still white. ter how sharply, multiple images are not This is what would happen in the ab- possible. Thus, the serrations must owe sence of lateral scattering of sunlight. But their existence to horizontal variations of refraction and lateral scattering go hand in the refractive index, a consequence of hand,eveninanatmospherefreeofpar- gravity waves propagating along a tem- ticles. Selective scattering by atmospheric perature inversion. molecules and particles causes the color 78 Atmospheric Optics

of the sun to change. In particular, the theories. For example, coronas are said to violet-bluish upper rim of the low sun can be caused by diffraction and rainbows by be transformed to green. refraction. Yet both the corona and the According to Eq. (29) and Fig. 13, the rainbow can be described quantitatively to angular width of the green upper rim of high accuracy with a theory (the Mie the- ◦ the low sun is about 0.01 , too narrow to ory for scattering by a sphere) in which be resolved with the naked eye or even to diffraction and refraction do not explicitly be seen against its bright backdrop. But appear. No fundamentally impenetrable depending on the temperature profile, the barrier separates scattering from (specu- atmosphere itself can magnify the upper lar) reflection, refraction, and diffraction. rim and yield a second image of it, thereby Becausethesetermscameintogeneral enabling it to be seen without the aid of a use and were entombed in textbooks be- telescope or binoculars. Green rims, which fore the nature of light and matter was well require artificial magnification, can be understood, we are stuck with them. But seen more frequently than green flashes, if we insist that diffraction, for example, is which require natural magnification. Yet somehow different from scattering, we do both can be seen often by those who know so at the expense of shattering the unity what to look for and are willing to look. of the seemingly disparate observable phe- nomena that result when light interacts with matter. What is observed depends 7 on the composition and disposition of the Scattering by Single Water Droplets matter, not on which approximate theory in a hierarchy is used for quantitative de- All the colored atmospheric displays that scription. result when water droplets (or ice crystals) Atmospheric optical phenomena are are illuminated by sunlight have the same best classified by the direction in which underlying cause: light is scattered in they are seen and by the agents respon- different amounts in different directions sible for them. Accordingly, the following by particles larger than the wavelength, sections are arranged in order of scattering and the directions in which scattering is direction, from forward to backward. greatest depends on wavelength. Thus, When a single water droplet is illumi- when particles are illuminated by white nated by white light and the scattered light, the result can be angular separation light projected onto a screen, the result of colors even if scattering integrated over is a set of colored rings. But in the atmo- all directions is independent of wavelength sphere we see a mosaic to which individual (as it essentially is for cloud droplets and droplets contribute. The scattering pattern ice crystals). This description, although of a single droplet is the same as the correct, is too general to be completely mosaic provided that multiple scattering satisfying. We need something more is negligible. specific, more quantitative, which requires theories of scattering. 7.1 Because superficially different theories Coronas and Iridescent Clouds have been used to describe different op- tical phenomena, the notion has become A cloud of droplets narrowly distributed in widespread that they are caused by these size and thinly veiling the sun (or moon) Atmospheric Optics 79 can yield a spectacular series of colored concentric rings around it. This corona is most easily described quantitatively by the Fraunhofer diffraction theory, a sim- ple approximation valid for particles large compared with the wavelength and for scattering angles near the forward direc- tion. According to this approximation, the differential scattering cross section (cross section for scattering into a unit solid angle) of a spherical droplet of radius a illuminated by light of wave number k is Fig. 16 Scattering of light near the forward |S|2 direction (according to Fraunhofer theory) by a ,() µ 2 32 sphere of diameter 10 m illuminated by red and k green light where the scattering amplitude is 1 + cos θ J (x sin θ) S = x2 1 .(33) droplet size, gives sufficient dispersion to 2 x sin θ yield colored coronas. Suppose that the first angular maxi- The term J1 is the Bessel function of µ first order and the size parameter x = mum for blue light (0.47 m) occurs for a µ ka. The quantity (1 + cos θ)/2 is usually droplet of radius a. For red light (0.66 m) approximated by 1 since only near-forward a maximum is obtained at the same an- + scattering angles θ are of interest. gle for a droplet of radius a a.That The differential scattering cross section, is, the two maxima, one for each wave- which determines the angular distribution length, coincide. From this we conclude of the scattered light, has maxima for that coronas require narrow size distri- x sin θ = 5.137, 8.417, 11.62, ... Thus, butions: if cloud droplets are distributed the dispersion in the position of the first in radius with a relative variance a/a maximum is greater than about 0.4, color separation is not possible. dθ 0.817 ≈ ( ) Because of the stringent requirements λ 34 d a for the occurrence of coronas, they are and is greater for higher-order maxima. not observed often. Of greater occur- This dispersion determines the upper limit rence are the corona’s cousins, iridescent on drop size such that a corona can be clouds, which display colors but usually observed. For the total angular dispersion not arranged in any obviously regular ge- over the visible spectrum to be greater ometrical pattern. Iridescent patches in ◦ than the angular width of the sun (0.5 ), clouds can be seen even at the edges of the droplets cannot be larger than about thick clouds that occult the sun. 60 µmindiameter.Dropsinrain,even Coronas are not the unique signatures of in drizzle, are appreciably larger than spherical scatterers. Randomly oriented ice this, which is why coronas are not seen columns and plates give similar patterns through rainshafts. Scattering by a droplet according to Fraunhofer theory [11]. As a of diameter 10 µm (Fig. 16), a typical cloud practical matter, however, most coronas 80 Atmospheric Optics

probably are caused by droplets. Many of the differential scattering cross section clouds at temperatures well below freezing at which the conditions contain subcooled water droplets. Only dθ b if a corona were seen in a cloud at a = 0, = 0 (36) ◦ db sin θ temperature lower than −40 C could one assert with confidence that it must be an are satisfied. Missing from Eq. (35) are ice-crystal corona. various reflection and transmission coeffi- cients (Fresnel coefficients), which display 7.2 no singularities and hence do not deter- Rainbows mine rainbow angles. A rainbow is not associated with rays In contrast with coronas, which are seen externally reflected or transmitted without looking toward the sun, rainbows are internal reflection. The succession of seen looking away from it, and are rainbow angles associated with one, two, caused by water drops much larger than three ... internal reflections are called those that give coronas. To treat the primary, secondary, tertiary ... rainbows. rainbow quantitatively we may pretend Aristotle recognized that ‘‘Three or more that light incident on a transparent sphere rainbows are never seen, because even is composed of individual rays, each of the second is dimmer than the first, and which suffers a different fate determined so the third reflection is altogether too only by the laws of specular reflection and feeble to reach the sun (Aristotle’s view refraction. Theoretical justification for this was that light streams outward from the is provided by van de Hulst’s ([12], p. 208) eye)’’. Although he intuitively grasped that localization principle, according to which each successive ray is associated with terms in the exact solution for scattering by ever-diminishing energy, his statement a transparent sphere correspond to more about the nonexistence of tertiary rainbows or less localized rays. in nature is not quite true. Although Each incident ray splinters into an infi- reliable reports of such rainbows are rare nite number of scattered rays: externally (unreliable reports are as common as dirt), reflected, transmitted without internal re- at least one observer who can be believed flection, transmitted after one, two, and so has seen one [13]. on internal reflections. At any scattering An incident ray undergoes a total angle θ, each splinter contributes to the angular deviation as a consequence of scattered light. Accordingly, the differen- transmission into the drop, one or more tial scattering cross section is an infinite internal reflections, and transmission out series with terms of the form of the drop. Rainbow angles are angles of (θ) minimum deviation. b db .() θ θ 35 Forarainbowofanyordertoexist, sin d  2 The impact parameter b is a sin i,where n − 1  cos i = (37) i is the angle between an incident p(p + 1) ray and the normal to the sphere. Each term in the series corresponds to one must lie between 0 and 1, where i is of the splinters of an incident ray. A the angle of incidence of a ray that gives rainbow angle is a singularity (or caustic) a rainbow after p internal reflections and Atmospheric Optics 81 n is the refractive index of the drop. A to the pencil-and-paper variety, are nec- primary bow therefore requires drops with essarily observed in an atmosphere, the refractive index less than 2; a secondary molecules and particles of which scat- bow requires drops with refractive index ter sunlight that adds to the light from less than 3. If raindrops were composed the rainbow but subtracts from its purity of titanium dioxide (n ≈ 3), a commonly of color. used opacifier for paints, primary rainbows Although geometrical optics yields the would be absent from the sky and we positions, widths, and color separation wouldhavetobecontentwithonly of rainbows, it yields little else. For secondary bows. example, geometrical optics is blind to If we take the refractive index of water to supernumerary bows, a series of narrow be 1.33, the scattering angle for the primary bands sometimes seen below the primary ◦ rainbow is about 138 .Thisismeasured bow. These bows are a consequence from the forward direction (solar point). of interference, and hence fall outside Measured from the antisolar point (the the province of geometrical optics. Since direction toward which one must look supernumerary bows are an interference in order to see rainbows in nature), this phenomenon, they, unlike primary and ◦ scattering angle corresponds to 42 ,the secondary bows (according to geometrical basis for a previous assertion that rainbows optics), depend on drop size. This poses (strictly, primary rainbows) cannot be the question of how supernumerary bows ◦ seen when the sun is above 42 .The can be seen in rain showers, the drops ◦ secondary rainbow is seen at about 51 in which are widely distributed in size. In from the antisolar point. Between these a nice piece of detective work, Fraser [16] two rainbows is Alexander’s dark band,a answered this question. region into which no light is scattered Raindrops falling in a vacuum are spher- according to geometrical optics. ical. Those falling in air are distorted by The colors of rainbows are a conse- aerodynamic forces, not, despite the de- quence of sufficient dispersion of the pictions of countless artists, into teardrops refractive index over the visible spectrum but rather into nearly oblate spheroids with to give a spread of rainbow angles that their axes more or less vertical. Fraser ar- appreciably exceeds the width of the sun. gued that supernumerary bows are caused The width of the primary bow from violet by drops with a diameter of about 0.5 mm, ◦ to red is about 1.7 ; that of the secondary at which diameter the angular position ◦ bow is about 3.1 . of the first (and second) supernumerary Because of its band of colors arcing bow has a minimum: interference causes across the sky, the rainbow has become the position of the supernumerary bow the paragon of color, the standard against to increase with decreasing size whereas which all other colors are compared. Lee drop distortion causes it to increase with and Fraser [14, 15], however, challenged increasing size. Supernumerary patterns this status of the rainbow, pointing out contributed by drops on either side of the that even the most vivid rainbows are minimum cancel, leaving only the contri- colorimetrically far from pure. bution from drops at the minimum. This Rainbows are almost invariably dis- cancellation occurs only near the tops of cussed as if they occurred literally in a rainbows, where supernumerary bows are vacuum. But real rainbows, as opposed seen. In the vertical parts of a rainbow, a 82 Atmospheric Optics

horizontal slice through a distorted drop at the end of our journey: the glory.Because is more or less circular, and hence these it is most easily seen from airplanes it drops do not exhibit a minimum supernu- sometimes is called the pilot’s bow.Another merary angle. name is anticorona, which signals that it According to geometrical optics, all is a corona around the antisolar point. Al- spherical drops, regardless of size, yield though glories and coronas share some the same rainbow. But it is not necessary common characteristics, there are differ- for a drop to be spherical for it to yield ences between them other than direction rainbows independent of its size. This of observation. Unlike coronas, which may merely requires that the plane defined by be caused by nonspherical ice crystals, glo- the incident and scattered rays intersect ries require spherical cloud droplets. And the drop in a circle. Even distorted a greater number of colored rings may be drops satisfy this condition in the vertical seen in glories than in coronas because part of a bow. As a consequence, the the decrease in luminance away from the absence of supernumerary bows there is backward direction is not as steep as that compensated for by more vivid colors away from the forward direction. To see of the primary and secondary bows [17]. a glory from an airplane, look for colored Smaller drops are more likely to be rings around its shadow cast on clouds be- spherical, but the smaller a drop, the low. This shadow is not an essential part less light it scatters. Thus, the dominant of the glory, it merely directs you to the contribution to the luminance of rainbows antisolar point. is from the larger drops. At the top of a Like the rainbow, the glory may be bow, the plane defined by the incident and looked upon as a singularity in the dif- scattered rays intersects the large, distorted ferential scattering cross section Eq. (35). drops in an ellipse, yielding a range of Equation (36) gives one set of conditions rainbow angles varying with the amount of for a singularity; the second set is distortion, and hence a pastel rainbow. To θ = , (θ) = .() the knowledgeable observer, rainbows are sin 0 b 0 38 no more uniform in color and brightness That is, the differential scattering cross than is the sky. section is infinite for nonzero impact Although geometrical optics predicts parameters (corresponding to incident that all rainbows are equal (neglecting rays that do not intersect the center of the ◦ background light), real rainbows do not sphere) that give forward (0 ) or backward ◦ slavishly follow the dictates of this approx- (180 ) scattering. The forward direction imate theory. Rainbows in nature range is excluded because this is the direction from nearly colorless fog bows (or cloud of intense scattering accounted for by the bows) to the vividly colorful vertical por- Fraunhofer theory. tions of rainbows likely to have inspired For one internal reflection, Eq. (38) leads myths about pots of gold. to the condition  n 2 7.3 sin i = 4 − n ,(39) The Glory 2 which is satisfied only for refractive indices Continuing our sweep of scattering direc- between 1.414 and 2, the lower refractive tions, from forward to backward, we arrive index corresponding to a grazing-incidence Atmospheric Optics 83 ray. The refractive index of water lies 8 outside this range. Although a condition Scattering by Single Ice Crystals similar to Eq. (39) is satisfied for rays un- dergoing four or more internal reflections, Scattering by spherical water drops in the insufficient energy is associated with such atmosphere gives rise to three distinct dis- rays. Thus, it seems that we have reached plays in the sky: coronas, rainbows, and an impasse: the theoretical condition for glories. Ice particles (crystals) also can in- a glory cannot be met by water droplets. habit the atmosphere, and they introduce Not so, says van de Hulst [18] in a sem- two new variables in addition to size: shape inal paper. He argues that 1.414 is close and orientation, the second a consequence enough to 1.33 given that geometrical op- of the first. Given this increase in the tics is, after all, an approximation. Cloud number of degrees of freedom, it is hardly droplets are large compared with the wave- cause for wonder that ice crystals are the length, but not so large that geometrical source of a greater variety of displays than optics is an infallible guide to their optical are water drops. As with rainbows, the behavior. Support for the van de Hulstian gross features of ice-crystal phenomena interpretation of glories was provided by can be described simply with geometrical Bryant and Cox [19], who showed that the optics, various phenomena arising from dominant contribution to the glory is from the various fates of rays incident on crys- the last terms in the exact series for scat- tals. Colorless displays (e.g., sun pillars) tering by a sphere. Each successive term are generally associated with reflected rays, in this series is associated with ever larger colored displays (e.g., sun dogs and halos) impact parameters. Thus, the terms that with refracted rays. Because of the wealth give the glory are indeed those correspond- of ice-crystal displays, it is not possible to ing to grazing rays. Further unraveling of treat all of them here, but one example the glory and vindication of van de Hulst’s should point the way toward understand- conjectures about the glory were provided ing many of them. by Nussenzveig [20]. It sometimes is asserted that geometrical 8.1 optics is incapable of treating the glory. Sun Dogs and Halos Yet the same can be said for the rainbow. Geometrical optics explains rainbows only Because of the hexagonal crystalline struc- in the sense that it predicts singularities for ture of ice it can form as hexagonal plates scattering in certain directions (rainbow in the atmosphere. The stable position of angles). But it can predict only the angles a plate falling in air is with the normal of intense scattering, not the amount. toitsfacemoreorlessvertical,whichis Indeed, the error is infinite. Geometrical easy to demonstrate with an ordinary busi- optics also predicts a singularity in the ness card. When the card is dropped with backward direction. Again, this simple its edge facing downward (the supposedly theory is powerless to predict more. aerodynamic position that many people Results from geometrical optics for both instinctively choose), the card somersaults rainbows and glories are not the end in a helter-skelter path to the ground. But but rather the beginning, an invitation when the card is dropped with its face par- to take a closer look with more powerful allel to the ground, it rocks back and forth magnifying glasses. gently in descent. 84 Atmospheric Optics

A hexagonal ice plate falling through air and illuminated by a low sun is ◦ like a 60 prism illuminated normally to its sides (Fig. 17). Because there is no mechanism for orienting a plate within the horizontal plane, all plate orientations in this plane are equally probable. Stated another way, all angles of incidence for a fixed plate are equally probable. Yet all scattering angles (deviation angles) of rays refracted into and out of the plate are not equally probable. Figure 18 shows the range of scattering Fig. 18 Scattering by a hexagonal ice plate (see angles corresponding to a range of rays Fig. 17) in various orientations (angles of ◦ incident on a 60 ice prism that is part of a incidence). The solid curve is for red light, the dashed for blue light hexagonal plate. For angles of incidence ◦ less than about 13 , the transmitted (θ) θ ray is totally internally reflected in the P of scattering angles by prism. For angles of incidence greater p(θ ) ◦ (θ) = i .() than about 70 , the transmittance plunges. P 40 dθ/dθi Thus, the only rays of consequence are ◦ those incident between about 13 and At the incidence angle for which ◦ θ/ θ = (θ) 70 . d d i 0, P is infinite and scat- All scattering angles are not equally tered rays are intensely concentrated probable. The (uniform) probability near the corresponding angle of mini- mum deviation. distribution p(θi) of incidence angles θi is related to the probability distribution The physical manifestation of this singu- larity (or caustic) at the angle of minimum ◦ deviation for a 60 hexagonal ice plate is ◦ abrightspotabout22 from either or both sides of a sun low in the sky. These bright spots are called sun dogs (because they accompany the sun) or parhelia or mock suns. The angle of minimum deviation θm, hence the angular position of sun dogs, ◦ depends on the prism angle (60 for the plates considered) and refrac- tive index:  

−1 Fig. 17 Scattering by a hexagonal ice plate θm = 2sin n sin − . (41) illuminated by light parallel to its basal plane. 2 The particular scattering angle θ shown is an angle of minimum deviation. The scattered light Because ice is dispersive, the separation between the angles of minimum deviation is that associated with two refractions by ◦ the plate for red and blue light is about 0.7 (Fig. 18), Atmospheric Optics 85 somewhat greater than the angular width to be randomly oriented by Brownian of the sun. As a consequence, sun dogs motion, they are too small to yield sharp may be tinged with color, most noticeably scattering patterns. toward the sun. Because the refractive But completely randomly oriented plates index of ice is least at the red end of the are not necessary to give halos, especially spectrum, the red component of a sun dog ones of nonuniform brightness. Each part is closest to the sun. Moreover, light of any of a halo is contributed to by plates with two wavelengths has the same scattering a different tip angle (angle between the angle for different angles of incidence normal to the plate and the vertical). if one of the wavelengths does not The transition from oriented plates (zero correspond to red. Thus, red is the purest tip angle) to randomly oriented plates color seen in a sun dog. Away from its red occurs over a narrow range of sizes. In inner edge a sun dog fades into whiteness. the transition region, plates can be small With increasing solar elevation, sun enough to be partially oriented yet large dogs move away from the sun. A falling ice enough to give a distinct contribution to plate is roughly equivalent to a prism, the the halo. Moreover, the mapping between prism angle of which increases with solar tip angles and azimuthal angles on the elevation. From Eq. (41) it follows that the halo depends on solar elevation. When angle of minimum deviation, hence the the sun is near the horizon, plates can sun dog position, also increases. give a distinct halo over much of its At this point you may be wondering why azimuth. ◦ only the 60 prism portion of a hexagonal plate was singled out for attention. As evident from Fig. 17, a hexagonal plate ◦ couldbeconsideredtobemadeupof120 prisms. For a ray to be refracted twice, its angle of incidence at the second interface must be less than the critical angle. This imposes limitations on the prism angle. For a refractive index 1.31, all incident rays are totally internally reflected by prisms ◦ with angles greater than about 99.5 . A close relative of the sun dog is the ◦ ◦ 22 halo, a ring of light approximately 22 from the sun (Fig. 19). Lunar halos are also possible and are observed frequently (although less frequently than solar halos); even moon dogs are possible. Until Fraser [21] analyzed halos in detail, the conventional wisdom had been that they obviously were the result of randomly oriented crystals, yet another example of jumping to conclusions. By combining ◦ optics and aerodynamics, Fraser showed Fig. 19 A22 solar halo. The hand is not for that if ice crystals are small enough artistic effect but rather to occlude the bright sun 86 Atmospheric Optics

When the sun is high in the sky, days a year over a 16-year period, with hexagonal plates cannot give a sharp halo extremes of 29 and 152 halos a year. Al- ◦ but hexagonal columns – another possible though the 22 halo was by far the most form of atmospheric ice particles – can. frequently seen display, ice-crystal displays The stable position of a falling column is of all kinds were seen, on average, more with its long axis horizontal. When the often than once every four days at a loca- sun is directly overhead, such columns tion not especially blessed with clear skies. can give a uniform halo even if they all lie Although thin clouds are necessary for ice- in the horizontal plane. When the sun is crystal displays, clouds thick enough to not overhead but well above the horizon, obscure the sun are their bane. columns also can give halos. A corollary of Fraser’s analysis is that halos are caused by crystals with a range of 9 sizes between about 12 and 40 µm. Larger Clouds crystals are oriented; smaller particles are too small to yield distinct scatter- Although scattering by isolated particles ing patterns. can be studied in the laboratory, parti- More or less uniformly bright halos with cles in the atmosphere occur in crowds the sun neither high nor low in the sky (sometimes called clouds). Implicit in the could be caused by mixtures of hexagonal previous two sections is the assumption plates and columns or by clusters of bullets that each particle is illuminated solely (rosettes). Fraser opines that the latter is by incident sunlight; the particles do not more likely. illuminate each other to an appreciable de- One of the by-products of his analysis is gree. That is, clouds of water droplets or an understanding of the relative rarity of ice grains were assumed to be optically ◦ the 46 halo.Aswehaveseen,theangleof thin, and hence multiple scattering was minimum deviation depends on the prism negligible. Yet the term cloud evokes fluffy angle. Light can be incident on a hexagonal white objects in the sky, or perhaps an ◦ column such that the prism angle is 60 overcast sky on a gloomy day. For such ◦ for rays incident on its side or 90 for clouds, multiple scattering is not negligi- rays incident on its end. For n = 1.31, ble, it is the major determinant of their Eq. (41) yields a minimum deviation angle appearance. And the quantity that deter- ◦ ◦ ◦ of about 46 for = 90 .Yet,although46 mines the degree of multiple scattering is halos are possible, they are seen much less optical thickness (see Sec. 2.4). ◦ frequently than 22 halos. Plates cannot ◦ give distinct 46 halos although columns 9.1 can.Yettheymustbesolidandmost Cloud Optical Thickness columns have hollow ends. Moreover, the range of sun elevations is restricted. Despite their sometimes solid appearance, Like the green flash, ice-crystal phenom- clouds are so flimsy as to be almost ena are not intrinsically rare. Halos and nonexistent – except optically. The fraction sun dogs can be seen frequently – once of the total cloud volume occupied by you know what to look for. Neuberger [22] water substance (liquid or solid) is about reports that halos were observed in State 10−6 or less. Yet although the mass College, Pennsylvania, an average of 74 density of clouds is that of air to within Atmospheric Optics 87 a small fraction of a percent, their optical usually translucent, not transparent, yet thickness (per unit physical thickness) is not completely opaque. much greater. The number density of air The scattering coefficient of cloud molecules is vastly greater than that of droplets, in contrast with that of air water droplets in clouds, but scattering per molecules, is more or less independent molecule of a cloud droplet is also much of wavelength. This is often invoked as greater than scattering per air molecule the cause of the colorlessness of clouds. (see Fig. 7). Yet wavelength independence of scatter- Because a typical cloud droplet is much ing by a single particle is only sufficient, larger than the wavelengths of visible light, not necessary, for wavelength indepen- its scattering cross section is to good dence of scattering by a cloud of particles approximation proportional to the square (see Sec. 2.4). Any cloud that is optically of its diameter. As a consequence, the thick and composed of particles for which scattering coefficient [see Eq. (2)] of a cloud absorption is negligible is white upon having a volume fraction f of droplets is illumination by white light. Although ab- approximately sorption by water (liquid and solid) is not identically zero at visible wavelengths, and d2 selective absorption by water can lead to β = 3f ,(42) d3 observable consequences (e.g., colors of the sea and glaciers), the appearance of all where the brackets indicate an average but the thickest clouds is not determined over the distribution of droplet diameters by this selective absorption. d. Unlike molecules, cloud droplets are Equation (42) is the key to the vastly distributed in size. Although cloud parti- different optical characteristics of clouds cles can be ice particles as well as water and of the rain for which they are the droplets, none of the results in this and the progenitors. For a fixed amount of water following section hinge on the assumption (as specified by the quantity fh), optical of spherical particles. thickness is inversely proportional to mean The optical thickness along a cloud diameter. Rain drops are about 100 times path of physical thickness h is βh for larger on average than cloud droplets, and a cloud with uniform properties. The hence optical thicknesses of rain shafts are ratio d3/d2 defines a mean droplet correspondingly smaller. We often can see diameter, a typical value for which is through many kilometers of intense rain − 10 µm. For this diameter and f = 10 6,the whereas a small patch of fog on a well- optical thickness per unit meter of physical traveled highway can result in carnage. thicknessisaboutthesameasthenormal optical thickness of the atmosphere in the 9.2 middle of the visible spectrum (see Fig. 3). Givers and Takers of Light Thus, a cloud only 1 m thick is equivalent optically to the entire gaseous atmosphere. Scattering of visible light by a single A cloud with (normal) optical thickness water droplet is vastly greater in the ◦ about 10 (i.e., a physical thickness of about forward (θ<90 ) hemisphere than in the ◦ 100 m) is sufficient to obscure the disk of backward (θ>90 ) hemisphere (Fig. 9). the sun. But even the thickest cloud does But water droplets in a thick cloud not transform day into night. Clouds are illuminated by sunlight collectively scatter 88 Atmospheric Optics

much more in the backward hemisphere their surroundings. As so often happens, (reflected light) than in the forward more is not always better. Beyond a hemisphere (transmitted light). In each certain cloud optical thickness, the diffuse scattering event, incident photons are irradiance decreases. For a sufficiently deviated, on average, only slightly, but thick cloud, the sky overhead can be darker in many scattering events most photons than the clear sky. are deviated enough to escape from the Why are clouds bright? Why are they upper boundary of the cloud. Here is an dark? No inclusive one-line answers can be example in which the properties of an given to these questions. Better to ask, Why ensemble are different from those of its is that particular cloud bright? Why is that individual members. particular cloud dark? Each observation Clouds seen by passengers in an airplane must be treated individually; generaliza- can be dazzling, but if the airplane were tions are risky. Moreover, we must keep to descend through the cloud these same in mind the difference between bright- passengers might describe the cloudy sky ness and radiance when addressing the overhead as gloomy. Clouds are both givers queries of human observers. Brightness is and takers of light. This dual role is a sensation that is a property not only of exemplified in Fig. 20, which shows the the object observed but of its surround- calculated diffuse downward irradiance ings as well. If the luminance of an object below clouds of varying optical thickness. is appreciably greater than that of its sur- On an airless planet the sky would be black roundings, we call the object bright. If the in all directions (except directly toward luminance is appreciably less, we call the thesun).Butiftheskyweretobefilled object dark. But these are relative rather from horizon to horizon with a thin cloud, than absolute terms. the brightness overhead would markedly Two clouds, identical in all respects, increase. This can be observed in a partly including illumination, may still appear overcast sky, where gaps between clouds different because they are seen against (blue sky) often are noticeably darker than different backgrounds, a cloud against the horizon sky appearing darker than when seen against the zenith sky. Of two clouds under identical illumi- nation, the smaller (optically) will be less bright. If an even larger cloud were to ap- pear, the cloud that formerly had been de- scribed as white might be demoted to gray. With the sun below the horizon, two identical clouds at markedly different elevations might appear quite different in brightness, the lower cloud being shadowed from direct illumination by sunlight. A striking example of dark clouds can Fig. 20 Computed diffuse downward irradiance below a cloud relative to the incident solar sometimes be seen well after the sun irradiance as a function of cloud has set. Low-lying clouds that are not optical thickness illuminated by direct sunlight but are Atmospheric Optics 89 seen against the faint twilight sky may spectral response of the human ob- be relatively so dark as to seem like server. Also sometimes called photometric ink blotches. brightness. Because dark objects of our everyday .5.5 lives usually owe their darkness to absorp- tion, nonsense about dark clouds is rife: Mirage: An image appreciably different they are caused by pollution or soot. Yet of from what it would be in the absence of all the reasons that clouds are sometimes atmospheric refraction. seen to be dark or even black, absorption is not among them. Neutral Point: A direction in the sky for which the light is unpolarized.

Normal Optical Thickness: Optical thick- Glossary ness along a radial path from the surface of the earth to infinity. Airlight: Light resulting from scattering by all atmospheric molecules and particles Optical Thickness: The thickness of a scat- along a line of sight. tering medium measured in units of photon mean free paths. Optical thick- Antisolar Point: Direction opposite the nesses are dimensionless. sun.

Astronomical Horizon: Horizontal direc- Radiance: Radiant power crossing a unit tion determined by a bubble level. area and confined to a unit solid angle about a particular direction. Brightness: The attribute of sensation by which an observer is aware of differences Scale Height: The vertical distance over of luminance (definition recommended which a physical property of the at- by the 1922 Optical Society of America mosphere is reduced to 1/e of its Committee on Colorimetry). value.

Contrast Threshold: The minimum rela- Scattering Angle: Angle between incident tive luminance difference that can be and scattered waves. perceived by the human observer. Scattering Coefficient: The product of scat- Inferior Mirage: A mirage in which images tering cross section and number density of are displaced downward. scatterers.

Irradiance: Radiant power crossing unit Scattering Cross Section: Effective area of a area in a hemisphere of directions. scatterer for removal of light from a beam by scattering. Lapse Rate: Therateatwhichaphysical property of the atmosphere (usually tem- Scattering Plane: Plane determined by perature) decreases with height. incident and scattered waves.

Luminance: Radiance integrated over the Solar Point: The direction toward the visible spectrum and weighted by the sun. 90 Atmospheric Optics

Superior Mirage: A mirage in which im- [15] Lee, R. (1991), Appl. Opt. 30, 3401–3407. ∗ ages are displaced upward. [16] Fraser, A. B. (1983), J. Opt. Soc. Am. 73, 1626–1628. 1972 29 Tangential Optical Thickness: Optical [17] Fraser, A. B. ( ), J. Atmos. Sci. , 211, 212. thickness through the atmosphere along [18]∗ van de Hulst, H. C. (1947), J. Opt. Soc. a horizon path. Am. 37, 16–22. [19]∗ Bryant,H.C.,Cox,A.J.(1966), J. Opt. Soc. Am. 56, 1529–1532. References [20]∗ Nussenzveig, H. M. (1979), J. Opt. Soc. Am. 69, 1068–1079. [21]∗ Fraser, A. B. (1979), J. Opt. Soc. Am. 69, Many of the seminal papers in atmospheric 1112–1118. optics, including those by Lord Rayleigh, are [22] Neuberger, H. (1951), Introduction to Phys- bound together in Bohren, C. F. (Ed.) (1989), ical Meteorology. University Park, PA: Col- Selected Papers on Scattering in the Atmosphere, lege of Mineral Industries, Pennsylvania Bellingham, WA: SPIE Optical Engineering State University. Press. Papers marked with an asterisk are in this collection.

[1] Moller,¨ F. (1972), Radiation in the at- Further Reading mosphere, in D. P. McIntyre (Ed.), Me- teorological Challenges: A History. Ottawa: Minnaert, M. (1954), TheNatureofLightand Information Canada, pp. 43–71. Colour in the Open Air. New York: Dover [2]∗ Penndorf, R. (1957), J. Opt. Soc. Am. 47, Publications, is the bible for those interested 176–182. ∗ in atmospheric optics. Like accounts of natural [3] Young, A. T. (1982), Phys. Today 35(1), phenomena in the Bible, those in Minnaert’s 2–8. book are not always correct, despite which, [4] Einstein, A. (1910), Ann. Phys. (Leipzig) 33, 175; English translation in Alexan- again like the Bible, it has been and will der, J. (Ed.) (1926), Colloid Chemistry, continue to be a source of inspiration. Vol. I. New York: The Chemical Catalog A book in the spirit of Minnaert’s but with a Company, pp. 323–339. wealth of color plates is by Lynch, D. K., Liv- [5] Zimm, B. H. (1945), J. Chem. Phys. 13, ingston, W. (1995), Color and Light in Nature. 141–145. Cambridge, UK: Cambridge University Press. [6] Thekaekara, M. P., Drummond, A. J. A history of light scattering, From Leonardo to (1971), Nat. Phys. Sci. 229, 6–9. the Graser: Light Scattering in Historical Per- [7]∗ Hulburt, E. O. (1953), J. Opt. Soc. Am. 43, spective, was published serially by Hey, J. D. 1983 79 113–118. ( ), S. Afr. J. Sci. , 11–27, 310–324; 1985 81 [8] von Frisch, K. (1971), Bees: Their Vision, Hey, J. D. ( ), S. Afr. J. Sci. , 77–91, 1986 82 Chemical Senses, and Language, revised 601–613; Hey, J. D., ( ), S. Afr. J. Sci. , edition, Ithaca, NY: Cornell University 356–360. The history of the rainbow is re- Press, p. 116. counted by Boyer, C. B. (1987), The Rainbow. [9] Doyle, W. T. (1985), Am. J. Phys. 53, Princeton, NJ: Princeton University Press. 463–468. A unique, beautifully written and illustrated [10]∗ Fraser,A.B.(1975), Atmosphere 13, 1–10. treatise on rainbows in science and art, [11] Takano, Y., Asano, S. (1983), J. Meteor. both sacred and profane, is by Lee, R. L., Soc. Jpn. 61, 289–300. Fraser, A. B. (2001), The Rainbow Bridge, [12] van de Hulst, H. C. (1957), Light Scattering University Park, PA: Penn State University by Small Particles. New York: Wiley- Press. Interscience. Special issues of Journal of the Optical Society [13] Pledgley, E. (1986), Weather 41, 401. of America (August 1979 and December 1983) [14] Lee, R., Fraser, A. (1990), New Scientist and Applied Optics (20 August 1991 and 20 July 127(September), 40–42. 1994) are devoted to atmospheric optics. Atmospheric Optics 91

Several monographs on light scattering by a few relevant chapters. Two popular science particles are relevant to and contain exam- books on simple experiments in atmospheric ples drawn from atmospheric optics: van physics are heavily weighted toward atmo- de Hulst, H. C. (1957), Light Scattering by spheric optics: Bohren, C. F. (1987), Clouds in Small Particles. New York: Wiley-Interscience; a Glass of Beer. New York: Wiley; Bohren, C. F. reprint (1981), New York: Dover Publica- (1991), What Light Through Yonder Window tions; Deirmendjian, D. (1969), Electromag- Breaks? New York: Wiley. netic Scattering on Polydispersions.NewYork: For an expository article on colors of the sky Elsevier; Kerker, M. (1969), The Scattering see Bohren, C. F., Fraser, A. B. (1985), Phys. of Light and Other Electromagnetic Radiation. Teacher 23, 267–272. New York: Academic Press; Bohren, C. F., An elementary treatment of the coherence Huffman, D. R. (1983), Light Scattering by properties of light waves was given by For- Small Particles. New York: Wiley-Interscience; rester, A. T. (1956), Am.J.Phys.24, 192–196. Nussenzveig, H. M. (1992), Diffraction Effects This journal also published an expository arti- in Semiclassical Scattering.Cambridge,UK: cle on the observable consequences of multiple Cambridge University Press. scattering of light: Bohren, C. F. (1987), Am. J. The following books are devoted to a wide Phys. 55, 524–533. range of topics in atmospheric optics: Although a book devoted exclusively to atmo- Tricker, R. A. R. (1970), Introduction to Mete- spheric refraction has yet to be published, orological Optics. New York: Elsevier; McCart- an elementary yet thorough treatment of mi- ney, E. J. (1976), Optics of the Atmosphere.New rages was given by Fraser, A. B., Mach, W. H. York: Wiley; Greenler, R. (1980), Rainbows, (1976), Sci. Am. 234(1), 102–111. Halos, and Glories.Cambridge,UK:Cambridge Colorimetry, the often (and unjustly) neglected University Press. Monographs of more limited component of atmospheric optics, is treated scope are those by Middleton, W. E. K. (1952), in, for example, Optical Society of Amer- Vision Through the Atmosphere. Toronto: Uni- ica Committee on Colorimetry (1963), The versity of Toronto Press; O’Connell, D. J. K. Science of Color. Washington, DC: Optical (1958), The Green Flash and Other Low Society of America. Billmeyer, F. W., Saltz- Sun Phenomena. Amsterdam: North Holland; man, M. (1981), Principles of Color Technol- Rozenberg, G. V. (1966), Twilight: A Study in ogy, (2nd ed.), New York: Wiley-Interscience. Atmospheric Optics. New York: Plenum; Hen- MacAdam, D. L. (1985), Color Measurement, derson, S. T. (1977), Daylight and its Spectrum, (2nd ed.), Berlin: Springer. (2nd ed.), New York: Wiley; Tricker, R. A. R. Understanding atmospheric optical phenom- (1979), Ice Crystal Haloes. Washington, DC: ena is not possible without acquiring at Optical Society of America; Konnen,¨ G. P. least some knowledge of the properties (1985), Polarized Light in Nature.Cambridge, of the particles responsible for them. To UK: Cambridge University Press; Tape, W. this end, the following are recommended: (1994), Atmosphere Halos. Washington, DC: Pruppacher, H. R., Klett, J. D. (1980), Micro- American Geophysical Union. physics of Clouds and Precipitation.Dor- Although not devoted exclusively to atmospheric drecht, Holland: D. Reidel. Twomey, S. A. optics, Humphreys, W. J. (1964), Physics of the (1977), Atmospheric Aerosols.NewYork: Air. New York: Dover Publications, contains Elsevier.