Like a Bridge over ColoredWater: A Mathematical Reviewof The Rainbow Bridge:Rainbows in Art, Myth, andScienceReviewed by John A. Adam

1360 NOTICES OF THE AMS VOLUME 49, NUMBER 11

The Rainbow Bridge: Rainbows in Art, Myth, andScienceRaymond L. Lee Jr. and Alistair B. FraserPennsylvania State University Press , 2001393 pages, $65.00ISBN 0-271-01977-8

This is a magnificent and scholarly book, exquis-itely produced, and definitely not destined onlyfor the coffee table. It is multifaceted in character,

addressing rainbow-relevant as-pects of mythology, religion,the history of art, art criticism,the history of optics, the theoryof color, the philosophy of sci-ence, and advertising! The qual-ity of the reproductions andphotographs is superb. The au-thors are experts in meteoro-logical optics, but their bookdraws on many other subdisci-plines. It is a challenge, there-fore, to write a review about abook that contains no equa-tions or explicit mathematicalthemes for what is primarily amathematical audience.However, while the mathemat-

ical description of the rainbow may be hidden in thisbook, it is nonetheless present. Clearly, such a re-view runs the risk of giving a distorted picture ofwhat the book is about, both by “unfolding” the hid-den mathematics and suppressing, to some extent,

other important and explicit themes: the connec-tions with mythology, art, and science. To a de-gree this is inevitable, if such a weighted metric isto be used. Lee and Fraser are intent on exploringbridges from the rainbow to all the places listedabove, and in the opinion of this reviewer theyhave succeeded admirably. The serious reader willglean much of value, and mathematicians in par-ticular may benefit from the tantalizing hints ofmathematical structure hidden in the photographsand graphics. Following an account of severalthemes present in the book that I found particularlyappealing, some of the mathematical structure un-derlying and supporting the bridge between therainbow and its optical description will be empha-sized in this review.

I have included several direct quotations to illustrate both the writing style and the features ofthe book that I found most intriguing. Frankly, I found some of the early chapters harder to appreciate than later ones. This is a reflection, nodoubt, of my own educational deficiencies in theliberal arts; but upon completing the book, I wasmoved to suggest to the chair of the art departmentthat it might be interesting to offer a team-taughtgraduate seminar in art, using this book as a text.He is now avidly reading my copy of the book.

The ten chapters combined trace the rainbowbridge to “the gods” as a sign and symbol (“emblemand enigma”); to the “growing tension between scientific and artistic images of the rainbow”;through the inconsistencies of the Aristotelian description and beyond, to those of Descartes andNewton, and the latter’s theory of color; to claimsof a new unity between the scientific and artistic

John A. Adam is professor of mathematics at Old Do-minion University. His email address is jadam@odu.edu.

DECEMBER 2002 NOTICES OF THE AMS 1361

enterprises; the evolution of scientific models ofthe rainbow to relatively recent times; and the exploitation and commercialization of the rain-bow. All these bridges, the authors claim, are unitedby the human appreciation of the rainbow’s com-pelling natural beauty. And who can disagree? Anappendix is provided (“a field guide to the rainbow”)comprising nineteen basic questions about the rain-bow, with nontechnical (but scientifically accurate) answers for the interested observer. This is followed by a set of chapter notes and a bibliography, both of which are very comprehensive.The more technical scientific aspects of the distri-bution of light within a rainbow are scattered lib-erally throughout the latter half of the book; indeed,the reader more interested in the scientific aspectsof the rainbow might wish to read the last fivechapters in parallel with the first five (as did I). Thetechnical aspects referred to are explained withgreat clarity.

But what is a rainbow? Towards the end of this review, several mutually inexclusive but comple-mentary levels of explanation will be noted, reflecting the fact that there is a great deal ofphysics and mathematics behind one of nature’smost awesome spectacles. At a more basic obser-vational level, surely everyone can describe the colored arc of light we call a rainbow, we might suggest, certainly as far as the primary bow is con-cerned. In principle, whenever there is a primaryrainbow, there is a larger and fainter secondarybow. The primary (formed by light being refractedtwice and reflected once in raindrops) lies beneatha fainter secondary bow (formed by an additionalreflection, and therefore fainter because of light loss)which is not always easily seen. There are severalother things to look for: faint pastel fringes just be-low the top of the primary bow (supernumerarybows, of which more anon), the reversal of colorsin the secondary bow, the dark region between thetwo bows, and the bright region below the primarybow. This observational description, however, isprobably not one that is universally known; in some cultures it is considered unwise even to lookat a rainbow. Indeed, in many parts of the world,it appears, merely pointing at a rainbow is consid-ered to be a foolhardy act. Lee and Fraser statethat “getting jaundice, losing an eye, being struckby lightning, or simply disappearing are amongthe unsavory aftermaths of rainbow pointing.”

The historical descriptions are in places quitebreathtaking; we are invited to look over the shoul-ders, as it were, of Descartes and Newton as they work through their respective accounts of the rainbow’s position and colors. While the color theory of Descartes was flawed, his geometric theory was not. Commenting on the latter, the authors point out that “Descartes’[s] seventeenth-century analysis of the rainbow bears out Plato’s

great faith in observations simplified and clarified by the power of mathematics.” Newton, on the other hand, eschews Descartes’s cumbersome ray-tracing technique and “silently invokes his mathematical invention of the 1660s, differentialcalculus, to specify the minimum deviation rays ofthe primary and secondary rainbows.” Later in thebook Lee and Fraser remark that Aristotle and laterscientists in antiquity “constructed theories that pri-marily describe natural phenomena in mathemati-cal or geometric terms, with little or no concern forphysical mechanisms that might explain them.” Thiscontrast goes to the heart of the difference betweenAristotelian and mathematical modeling.

I was pleasantly surprised to learn that the Eng-lish painter John Constable was quite an avid am-ateur scientist: he was concerned that his paintingsof clouds and rainbows should accurately reflectthe science of the day, and he took great troubleto acquaint himself with Newton’s theory of therainbow (many other details can be found inpp. 80–7 of the book). In a similar vein, the writerJohn Ruskin was a detailed observer of nature, hisgoal being “that of transforming close observationinto faithful depiction of a purposeful, divinelyshaped nature.” The poet John Keats implies in hisLamia that Newton’s natural philosophy destroysthe beauty of the rainbow (“There was an awful rain-bow once in heaven: / We know her woof, her tex-ture; she is given / In the dull catalog of commonthings. / Philosophy will clip an Angel’s wings”).Keats’s words reflected a continuing debate: did scientific knowledge facilitate or constrain poeticdescriptions of nature? His contemporary WilliamWordsworth apparently held adifferent opinion, for he wrote,“The beauty in form of a plantor an animal is not made lessbut more apparent as a wholeby more accurate insight intoits constituent properties andpowers.”

In a subsection of Chapter 4entitled “The Inescapable (andUnapproachable) Bow”, the au-thors address, amongst otherthings, some common misper-ceptions about rainbows. Justas occurs when we contemplateour visage in a mirror, what wesee is not an object, but animage; near the end of the chap-ter this rainbow image is beau-tifully portrayed as “a mosaic ofsunlit rain.” Optically, the rain-bow is located at infinity, even

violet

red

Alexander’sdark band

red

violetsuper-

numerarybows

brightregion

Figure 1. A sector illustrating the rainbowconfiguration, including both a primary bow (onthe bottom) and a secondary bow (at the top).The bright region is not shown to scale.

i

i

r

sin r= sin in

nred= 1 .3318

D = π (p− 1) + 2 i− 2pr

bi

(p= 2here)

D

Figure 2. A typical path througha raindrop for a ray of lightcontributing to the primaryrainbow, according togeometrical optics.

1362 NOTICES OF THE AMS VOLUME 49, NUMBER 11

though the raindrops or dropletssprayed from a garden hose are not.To place the rainbow in the sky, notethat the antisolar point is 180 from theSun on a line through the head of theobserver. As will be noted below, raysof sunlight are deviated from this line(in a clockwise sense in Figure 2); theray of minimum deviation (sometimescalled the rainbow or Descartes ray) isdeviated by about 138 for the pri-mary bow. Therefore there is a con-centration of deviated rays near thisangle, and so for the observer the pri-mary bow is an arc of a circle of radius180− 138 = 42 centered on the an-tisolar point. Thus the rainbow has afixed angular radius of about 42 de-spite illusions (and allusions) to thecontrary. The corresponding angle forthe secondary bow is about 51 ,though in each instance the anglevaries a little depending on the wave-length of the light. Without this phe-nomenon of dispersion there would beonly a “whitebow”!

It is perhaps an occupational hazardfor professional scientists and math-ematicians to be a little frustrated byincorrect depictions or explanationsof observable phenomena and mathe-matical concepts in literature, art, themedia, etc. From the point of view ofan artist, however (Constable notwith-standing), scientific accuracy is notnecessarily a prelude to artistic ex-pression; indeed, to some it may beconsidered a hindrance. Nevertheless,we can identify perhaps with the au-thors, who, in commenting on thepaintings of Frederic Church (in par-ticular his Rainy Season in the Tropics),write, “Like Constable, he in places

bends the unyielding rule that all shadows must beradii to the bow (that is, they must converge on therainbow’s center).” The same rule applies to sun-beams, should they be observable concurrentlywith a rainbow.

By the middle of the eighteenth century, thecontributions of Descartes and Newton notwith-standing, observations of supernumerary bowswere a persistent reminder of the inadequacy of cur-rent theories of the rainbow. As Lee and Fraser sopointedly remark, “One common reaction to beingconfronted with the unexplained is to label it in-explicable.” This led to these troubling featuresbeing labeled spurious; hence the unfortunate ad-dition of the adjective supernumerary to the rain-bow phenomenon. Now it is known that such bows“are an integral part of the rainbow, not a vexingcorruption of it”; an appropriate, though possiblyunintended, mathematical pun (see below). By fo-cusing attention on the light wavefronts incidenton a spherical drop rather than on the rays normalto them, it is easier to appreciate the self-interfer-ence of such a wave as it becomes “folded” ontoitself as a result of refraction and reflection withinthe drop. This is readily seen from Figures 8.7 and8.9 in the book (see Figures 5 and 6 here), fromwhich the true extent of the rainbow is revealed:the primary rainbow is in fact the first interferencemaximum, the second and third maxima being thefirst and second supernumerary bows respectively(and so on).

The angular spacing of these bands depends onthe size of the droplets producing them. The widthof individual bands and the spacing between themdecreases as the drops get larger. If drops of manydifferent sizes are present, these supernumeraryarcs tend to overlap somewhat and smear out whatwould have been obvious interference bands fordroplets of uniform size. This is why these pale blueor pink or green bands are then most noticeable nearthe top of the rainbow: it is the near-sphericity ofthe smaller drops that enable them to contributeto this part of the bow; larger drops are distortedfrom sphericity by the aerodynamic forces actingupon them. Nearer the horizon a wide range ofdrop size contributes to the bow, but at the sametime it tends to blur the interference bands. Inprinciple, similar interference effects also occurabove the secondary rainbow, though they are veryrare, for reasons discussed in the penultimate chap-ter. Lee and Fraser summarize the importance ofspurious bows with typical metaphorical creativity:“Thus the supernumerary rainbows proved to be the

Figure 3. The paths ofseveral rays of light

impinging upon araindrop with different

angles of incidence.Such rays contribute to

the primary rainbow.The ray of minimum

deviation (the rainbowray) is the darkest line.

Figure 4. The ray pathsfor the secondary

rainbow, similar toFigure 3 above. Again,

the ray of minimumdeviation is the darkest

line.

Graph of the Airy function Ai(x), which is a solution of the differential equation y′′ + xy = 0 . Below, according toAiry's theory, illumination is proportional to Ai(x)2, simulating a monochromatic bow along with several

supernumerary bows.

y = Ai(x)

DECEMBER 2002 NOTICES OF THE AMS 1363

midwife that delivered the wave theory of light toits place of dominance in the nineteenth century.”

It is important to recognize that, not only werethe Cartesian and Newtonian theories unable toaccount for the presence of supernumerary bows,but also they both predicted an abrupt transitionbetween regions of illumination and shadow (as atthe edges of Alexander’s dark band, when onlyrays giving rise to the primary and secondary bowsare considered). In the wave theory of light suchsharp boundaries are softened by diffraction, which occurs when the normal interference patternresponsible for rectilinear propagation of light isdistorted in some way. Diffraction effects are par-ticularly prevalent in the vicinity of caustics. In1835 Potter showed that the rainbow ray may beinterpreted as a caustic, i.e., the envelope of the sys-tem of rays constituting the rainbow. The wordcaustic means burning, and caustics are associatedwith regions of high-intensity illumination (withgeometrical optics predicting an infinite intensitythere). Thus the rainbow problem is essentiallythat of determining the intensity of (scattered) lightin the neighborhood of a caustic.

This was exactly what Airy attempted to do several years later in 1838. The principle behindAiry’s approach was established by Huygens in theseventeenth century: Huygens’s principle regardsevery point of a wavefront as a secondary sourceof waves, which in turn defines a new wavefront andhence determines the subsequent propagation ofthe wave. As pointed out by Nussenzveig [9], Airyreasoned that if one knew the amplitude distribu-tion of the waves along any complete wavefront ina raindrop, the distribution at any other point could

be determined by Huygens’s principle. Airy choseas his starting point a wavefront surface inside theraindrop, the surface being orthogonal to all the raysthat constitute the primary bow; this surface hasa point of inflection wherever it intersects the rayof minimum deviation, the “rainbow ray”. Usingthe standard assumptions of diffraction theory,he formulated the local intensity of scattered lightin terms of a “rainbow integral”, subsequently re-named the Airy integral in his honor; it is relatedto the now familiar Airy function. It is analogousto the Fresnel integrals which also arise in diffrac-tion theory. There are several equivalent repre-sentations of the Airy integral in the literature; following the form used by Nussenzveig [10], wewrite it as

(1) Ai(C) = (31/3π−1)∫∞

0cos(t3 + 31/3Ct)dt.

While the argument of the Airy function is ar-bitrary at this point, C refers to the set of controlspace parameters in the discussion below on dif-fraction catastrophes. In this case it represents the deviation or scattering angle coordinate. Onesevere limitation of the Airy theory for the opticalrainbow is that the amplitude distribution along the initial wavefront is unknown: based on certainassumptions it has to be guessed. There is a nat-ural and fundamental parameter, the size para-meter, β , which is useful in determining the domainof validity of the Airy approximation; it is definedas the ratio of the droplet circumference to thewavelength η of light. In terms of the wavenum-ber k this is

Figure 5. A wavefront version of Figure 3illustrating the constructive and destructiveinterference patterns of the light as it interactswith a raindrop. The width of each band of color isrelatively narrow, resulting in fairly pure rainbowcolors. The primary rainbow and the first twosupernumerary bows are identified here.

Rep

rod

uce

d w

ith

per

mis

sion

of

R. L

ee a

nd

A. F

rase

r.

Figure 6. Similar to Figure 5, but for a much smallercloud droplet. The resulting bands of color are so broadthat additive color mixing produces a whitish“cloudbow”.

1364 NOTICES OF THE AMS VOLUME 49, NUMBER 11

β = 2πaη

≡ ka,

a being the droplet radius. Typically, for sizesranging from fog droplets to large raindrops, βranges from about one hundred to several thou-sand. Airy’s approximation is a good one only forβ 5000 and for angles sufficiently close to thatof the rainbow ray. In light of these remarks it isperhaps surprising that an exact solution doesexist for the rainbow problem, as indicated below.

In the epilogue to the final chapter of the book, onesubsection is entitled “Airy’s Rainbow Theory: TheIncomplete ‘Complete’ Answer”. This theory did gobeyond the models of the day in that it quantified thedependence upon the raindrop size of (i) the rain-bow’s angular width, (ii) its angular radius, and (iii) the spacing of the supernumeraries. Also, un-like the models of Descartes and Newton, Airy’s predicted a nonzero distribution of light intensity in Alexander’s dark band and a finite intensity at theangle of minimum deviation (as noted above, theearlier theories predicted an infinite intensity there).However, spurred on by Maxwell’s recognition thatlight is part of the electromagnetic spectrum andthe subsequent publication of his mathematical treatise on electromagnetic waves, several mathe-matical physicists sought a more complete theoryof scattering, because it had been demonstrated bythen that the Airy theory failed to predict preciselythe angular position of many laboratory-generatedrainbows. Among them were the German physicistGustav Mie,1 who published a paper in 1908 on thescattering of light by homogeneous spheres in a homogeneous medium, and Peter Debye, who independently developed a similar theory for thescattering of electromagnetic waves by spheres. Mie’stheory was intended to explain the colors exhibitedby colloidally dispersed metal particles, whereas Debye’s work, based on his 1908 thesis, dealt withthe problem of light pressure on a spherical particle.The resulting body of knowledge is usually referredto as Mie theory, and typical computations based onit are formidable compared with those based on Airytheory, unless the drop size is sufficiently small. Asimilar (but nonelectromagnetic) formulation arises

in the scattering of sound waves by an impenetrablesphere, studied by Lord Rayleigh and others in thenineteenth century [6].

Mie theory is based on the solution of Maxwell’sequations of electromagnetic theory for a mono-chromatic plane wave from infinity incident upon ahomogeneous isotropic sphere of radius a. The sur-rounding medium is transparent (as the sphere maybe), homogeneous, and isotropic. The incident waveinduces forced oscillations of both free and boundcharges in synchrony with the applied field, and thisinduces a secondary electric and magnetic field, each of which has components inside and outside the sphere. Of crucial importance in the theory arethe scattering amplitudes (Sj (k,θ), j = 1,2) for thetwo independent polarizations, θ being the angularvariable; these amplitudes can be expressed as aninfinite sum called a partial-wave expansion. Eachterm (or partial wave) in the expansion is defined interms of combinations of Legendre functions of the first kind, Riccati-Bessel functions, and Riccati-Hankel functions (the latter two being rather simply related to spherical Bessel and Hankel functions respectively).

It is obviously of interest to determine under whatconditions such an infinite set of terms can betruncated and what the resulting error may be byso doing. However, it turns out that the number ofterms that must be retained is of the same orderof magnitude as the size parameter β , i.e., up to sev-eral thousand for the rainbow problem. On theother hand, the “why is the sky blue?” scatteringproblem—Rayleigh scattering—requires only oneterm, because the scatterers are molecules muchsmaller than a wavelength of light, so the simplesttruncation—retaining only the first term—is per-fectly adequate. Although in principle the rainbowproblem can be “solved” with enough computertime and resources, numerical solutions by them-selves (as Nussenzveig points out [9]) offer little orno insight into the physics of the phenomenon.

The Watson transform, originally introduced byWatson in connection with the diffraction of radiowaves around the Earth (and subsequently modi-fied by Nussenzveig in his studies of the rainbowproblem), is a method for transforming the slowlyconverging partial-wave series into a rapidly con-vergent expression involving an integral in thecomplex angular-momentum plane. The Watsontransform is intimately related to the Poisson sum-mation formula

(2)∞∑l=0

g(l + 12, x) =

∞∑m=−∞

e−imπ∫∞

0g(λ, x)e2πimλ dλ

for an “interpolating function” g(λ, x), where xdenotes a set of parameters and λ = l + 1

2 is nowconsidered to be the complex angular momentumvariable.

1I am grateful to J. D. Jackson, who informed me that “Lud-vig Lorenz, a Danish theorist, preceded Mie by about fif-teen years in the treatment of the scattering of electro-magnetic waves by spheres.” His contributions toelectromagnetic scattering theory and optics are ratheroverlooked, probably because his work was published inDanish (in 1890). Further details of his research, includ-ing his contributions to applied mathematics, may befound in reference [5] (the article immediately followingthat one on p. 4696 is about Gustav Mie). There is also avaluable historical account by Logan [6] of other contri-butions to this branch of (classical) mathematical physics.

DECEMBER 2002 NOTICES OF THE AMS 1365

But why angular momentum? Although theypossess zero rest mass, photons have energyE = hc/η and momentum E/c = h/η, where h isPlanck’s constant and c is the speed of light invacuo. Thus for a nonzero impact parameter bi, aphoton will carry an angular momentum bih/η (bibeing the perpendicular distance of the incident rayfrom the axis of symmetry of the sun-raindropsystem). Each of these discrete values can be iden-tified with a term in the partial-wave series ex-pansion. Furthermore, as the photon undergoesrepeated internal reflections, it can be thought ofas orbiting the center of the raindrop. Why com-plex angular momentum? This allows the abovetransformation to effectively “redistribute” thecontributions to the partial wave series into a fewpoints in the complex plane—specifically poles(called Regge poles in elementary particle physics)and saddle points. Such a decomposition meansthat angular momentum, instead of being identi-fied with certain discrete real numbers, is now permitted to move continuously through complexvalues. However, despite this modification, thepoles and saddle points have profound physical interpretations in the rainbow problem.

In the simplest Cartesian terms, on the illumi-nated side of the rainbow (in a limiting sense) thereare two rays of light emerging in parallel directions:at the rainbow angle they coalesce into the ray ofminimum deflection, and on the shadow side, ac-cording to geometrical optics, they vanish (this isactually a good definition of a caustic curve or sur-face). From a study of real and complex rays, it hap-pens that, mathematically, in the context of thecomplex angular momentum plane, a rainbow is thecollision of two real saddle points. But this is not all:this collision does not result in the mutual anni-hilation of these saddle points; instead, two com-plex saddle points are born, one corresponding toa complex ray on the shadow side of the causticcurve. This is directly associated with the diffractedlight in Alexander’s dark band.

Lee and Fraser point out that as far as most aspectsof the optical rainbow are concerned, Mie theory isesoteric overkill; Airy theory is quite sufficient for de-scribing the outdoor rainbow. I found particularlyvaluable the following comment by the authors; ithas implications for mathematical modeling in general, not just for the optics of the rainbow. Intheir comparison of the less accurate Airy theory ofthe rainbow with the more general and powerful Mie theory, they write, “Our point here is not thatthe exact Mie theory describes the natural rainbowinadequately, but rather that the approximate Airytheory can describe it quite well. Thus the suppos-edly outmoded Airy theory generates a more nat-ural-looking map of real rainbow colors than Mietheory does, even though Airy theory makes sub-stantial errors in describing the scattering of mono-

chromatic light by isolated small drops. As in manyhierarchies of scientific models, the virtues of a simpler theory can, under the right circumstances,outweigh its vices” [emphasis added].

Earlier in this review the question, What is arainbow? was answered at a basic descriptive level. But at an explanatory level a rainbow is much, much more than such an answer mightimply. Amongst other things a rainbow is: (1) a concentration of light rays corresponding to a minimum (for the primary bow) of the deviation or scattering angle as a function of the angle of incidence; this minimum is identified as theDescartes or rainbow ray; (2) a caustic, separatinga 2-ray region from a 0-ray (or shadow) region; (3) an integral superposition of waves over a (lo-cally) cubic wavefront (the Airy approximation);(4) an interference problem (the origin of the supernumerary bows); (5) a coalescence of two real saddle points; (6) a result of scattering by aneffective potential consisting of a square well anda centrifugal barrier (a surprising macroscopicquantum connection); (7) associated with tunnel-ing in the so-called edge domain [4]; (8) a tangen-tially polarized circular arc; and (9) a fold diffraction catastrophe. These nine topics are very much interrelated, and the interested readerwill find many references to them in [1], [4], [9], and [10]. The books by Grandy and Nussenzveig are excellent: they are primarily written by theo-retical physicists for theoretical physicists and so do not possess the kind of rigor analysts seek (such as in the book by Pearson [11]); perhapssuch a transition will be made in the future. But whynot read The Rainbow Bridge first?

A Mathematical AppendixSome descriptive material has been presented aboveconcerning complementary explanations of therainbow. It is of interest to put some mathemati-cal flesh on those bones, so to that end a brief sum-mary is provided of three aspects of the rainbowproblem at the level of geometrical optics, as ascalar scattering problem, and as a diffraction catastrophe. These accounts touch on items (1)–(3),(5), (6) and (9) in particular, but as noted above itis difficult to separate them in a precise way.

A rainbow occurs when the the scattering angleD, as a function of the angle of incidence i, passesthrough an extremum (a minimum for the primarybow; see Figure 3). The “folding back” of the cor-responding scattered or deviated ray takes placeat this extremal scattering angle (the rainbow angleDmin ≡ θR; note that sometimes in the literature therainbow angle is defined as the complement of thedeviation, π −Dmin) . Two rays scattered in thesame direction with different angles of incidenceon the illuminated side of the rainbow (θ > θR)fuse together at the rainbow angle and disappear

1366 NOTICES OF THE AMS VOLUME 49, NUMBER 11

as the dark side (θ < θR) is approached. This is oneof the simplest physical examples of a fold cata-strophe in the sense of Thom. As is noted below,rainbows of different orders are associated with so-called Debye terms of different orders; the primaryand secondary bows correspond to p = 2 and p = 3respectively. For p − 1 internal reflections, p beingan integer greater than 1, the total deviation is, fromelementary geometry,

Dp−1(i) = 2(i − pr )+ (p − 1)π,

where by Snell’s law the angle of refraction r = r (i)is also a function of the angle of incidence. Thusfor the primary rainbow (p = 2),

D1(i) = π − 4r + 2i,

and for the secondary rainbow (p = 3),

D2(i) = 2i − 6r

(modulo 2π). Removing the dependence on r , theangle through which a ray is deviated for p = 2 is

D1(i) = π + 2i − 4 arcsin(

sin in

),

where n > 1 is the refractive index of the raindrop.In general

Dp−1(i) = (p − 1)π + 2i − 2p arcsin(

sin in

).

The minimum deviation occurs for p = 2 when

i = ic = arccos

√n2 − 1

3

and in general when

i = ic = arccos

√n2 − 1p2 − 1

.

For the primary rainbow p = 2, so ic ≈ 59 , usingan approximate value for water of n = 4/3; fromthis it follows that D(ic ) = Dmin = θR ≈ 138. Forthe secondary bow p = 3 and ic ≈ 72 ; nowD(ic ) = θR ≈ 231 = −129. The rainbows each lieon the surface of a cone, centered at the observer’seye, with axis the line from that point to the anti-solar point (delineated by the shadow of the ob-server’s head). The cone semiangles are about 42

and 51 respectively for the primary and secondarybows. For p = 2, in terms of n alone, D(ic ) = θR (therainbow angle) is defined by

D(ic ) = θR = 2 arccos

1n2

(4− n2

3

)3/2 .

This approach can be thought of as the elemen-tary classical description. Surprising as it may seem,there is also a wave mechanical counterpart to the

optical rainbow; indeed, such effects are well knownin atomic, molecular, and nuclear physics [1], [10].On the way to this, so to speak, there is a “semi-classical” description. In a primitive sense, the semi-classical approach is the “geometric mean” betweenclassical and quantum mechanical descriptions ofphenomena in which interference and diffractioneffects enter the picture. The latter do so via the tran-sition from geometrical optics to wave optics. A mea-sure of the scattering (by raindrop or atom) that oc-curs is the differential scattering cross-sectiondσ/dΩ = |f (k,θ)|2 (essentially the relative particleor wave “density” scattered per unit solid angle ata given angle θ in cylindrically symmetric geome-try), where f (k,θ) is the scattering amplitude definedbelow; this in turn is expressible as a partial-waveexpansion. The formal relationship between thelatter and the classical differential cross-section isestablished using the WKB approximation, and theprinciple ofstationary phase is used to evaluateasymptotically a certain phase integral. A point ofstationary phase can be identified with a classicaltrajectory, but if more than one such point is pre-sent (provided they are well separated and of thefirst order), the corresponding expression for|f (k,θ)|2 will contain interference terms. This is adistinguishing feature of the “primitive” semiclas-sical formulation. Indeed, the infinite intensitiespredicted by geometrical optics at focal points,lines, and caustics in general are breeding groundsfor diffraction effects, as are light/shadow bound-aries for which geometrical optics predicts finitediscontinuities in intensity. Such effects are mostsignificant when the wavelength is comparablewith (or larger than) the typical length scale for vari-ation of the physical property of interest (e.g., sizeof the scattering object). Thus a scattering objectwith a “sharp” boundary (relative to one wave-length) can give rise to diffractive scattering phe-nomena [10].

There are critical angular regions where thissemiclassical approximation breaks down and dif-fraction effects cannot be ignored, although theangular ranges in which such critical effects be-come significant get narrower as the wavelength de-creases. Early work in this field contained transi-tional asymptotic approximations to the scatteringamplitude in these critical angular domains, but theyhave very narrow domains of validity and do notmatch smoothly with neighboring noncritical an-gular domains. It is therefore of considerable im-portance to seek uniform asymptotic approximationsthat by definition do not suffer from these failings.Fortunately, the problem of plane-wave scatteringby a homogeneous sphere exhibits all of the criti-cal scattering effects (and it can be solved exactly,in principle) and is therefore an ideal laboratory inwhich to test both the efficacy and accuracy of thevarious approximations. Furthermore, it has

DECEMBER 2002 NOTICES OF THE AMS 1367

relevance to both quantum mechanics (as a squarewell or barrier problem) and optics (Mie scattering);indeed, it also serves as a model for the scatteringof acoustic and elastic waves and, as noted earlier,was studied in the early twentieth century as amodel for the diffraction of radio waves around thesurface of the earth. Light waves obviously areelectromagnetic in character, and so the full weightof that theory is necessary to explain features suchas the polarization of the rainbow, but the essenceof the approach is well captured by the scalar the-ory.

The essential mathematical problem for scalarwaves can be thought of either in terms of classi-cal mathematical physics, e.g., the scattering ofsound waves, or in wave-mechanical terms, e.g., thenonrelativistic scattering of particles by a squarepotential well (or barrier) of radius a and depth (orheight) V0. In either case we can consider a scalarplane wave impinging in the direction θ = 0 on apenetrable (“transparent”) sphere of radius a. Thewave function ψ(r , θ) satisfies the scalar Helmholtzequation

(3)∇2ψ+ n2k2ψ = 0 r ≤ a∇2ψ+ k2ψ = 0 r ≥ a,

where again k is the wavenumber and n > 1 is therefractive index of the sphere (a similar problemcan be posed for gas bubbles in a liquid, for whichn < 1). The boundary conditions are that ψ(r , θ)and ψ′(r , θ) are continuous at the surface. Fur-thermore, at large distances from the sphere(r a) the wave field can be decomposed into anincident wave + scattered field, i.e.,

ψ ∼ eikr cosθ + af (k,θ)eikr

r,

the dimensionless scattering amplitude being de-fined as

(4) f (k,θ) = 12ika

∞∑l=0

(2l + 1)(Sl(k)− 1)Pl(cosθ),

where Sl is the scattering function for a given l andPl is a Legendre polynomial of degree l. For a spher-ical square well or barrier,

Sl = −h(2)l (β)

h(1)l (β)

ln′ h(2)

l (β)− n ln′ jl(α)

ln′ h(1)l (β)− n ln′ jl(α)

,

where ln′ represents the logarithmic derivative op-erator, jl and hl are spherical Bessel and Hankelfunctions respectively, the size parameter β = kaplays the role of a dimensionless external wavenum-ber, and α = nβ is the corresponding internalwavenumber. Sl may be equivalently expressed interms of cylindrical Bessel and Hankel functions.The lth partial wave in the solution is associated

with an impact parameter bl = (l + 12 )/k; i.e., only

rays hitting the sphere (bl a) are significantlyscattered, and the number of terms that must beretained in the series to get an accurate result isof order β . As implied earlier, for visible light scat-tered by water droplets in the atmosphere, β ∼several thousand. This is why, to quote ArnoldSommerfeld, “The series converge so slowly thatthey become practically useless.” This problem canbe remedied by using the Poisson summation for-mula (2) above to rewrite f (k,θ) as

(5)f (β,θ) = i

β

∞∑m=−∞

(−1)m

×∫∞

0[1− S(λ,β)]Pλ− 1

2(cosθ)e2imπλλdλ.

For fixed β , S(λ,β) is a meromorphic function ofthe complex variable λ = l + 1/2, and in particularin what follows it is the poles of this function thatare of interest. In terms of cylindrical Bessel andHankel functions, the poles are defined by the con-dition

ln′H (1)λ (β) = n ln′ Jλ(α)

and are called Regge poles in the scattering theoryliterature [7], [8]. Typically, they are associated withsurface waves for the impenetrable sphere problem,but for the transparent sphere two types of Reggepoles arise—one type (Regge-Debye poles) leadingto rapidly convergent residue series, representingthe surface wave (or diffracted or creeping ray) con-tributions to the scattering amplitude; and the other

virtual front

true front

virtu

al ca

ustic

rainbow ra

y

Figure 7. Wavefronts in a raindrop. The caustic isbest seen in the virtual rays (projected back from thefinal exterior rays), as is the basis of Airy’s analysis.After the virtual (or Airy) wavefront emerges, thepart which is convex forward continues to expand,while the concave forward part collapses to a focusand then expands. This is the reason for the cuspedwavefront at the bottom left of the figure.

1368 NOTICES OF THE AMS VOLUME 49, NUMBER 11

type associated with resonances via the internalstructure of the potential, which is now of courseaccessible. They are characterized by an effectiveradial wavenumber within the potential well. Manyin this latter group are clustered close to the realaxis, spoiling the rapid convergence of the residueseries. They correspond to different types of res-onances depending on the value of Re(λj ) . Thusmathematically these resonances are complexeigenfrequencies associated with the poles λj of thescattering function S(λ, k) in the first quadrant ofthe complex λ-plane for real values of β . The imag-inary parts of the poles are directly related to res-onance widths (and therefore lifetimes). As β in-creases, the poles λj trace out Regge trajectoriesand Imλj tends exponentially to zero. When Reλjpasses close to a “physical” value, λ = l + 1/2, it isassociated with a resonance in the lth partial wave;the larger the value of β , the sharper the reso-nance becomes for a given node number j . Theother type of significant points to be consideredin the complex λ-plane are saddle points (which canbe real or complex). In the complex angular mo-mentum method, contour integral paths are de-formed to concentrate the dominant contributionsto the poles and saddle points. A real saddle pointis also a point of stationary phase, and stationar-ity implies that Fermat’s principle is satisfied. Sincethe phase in many cases at such points can be ap-proximated by the WKB phase, the latter can beidentified with the rays of geometrical optics (some-times called classical paths). The asymptotic con-tributions from these points are the dominant oneswithin geometrically illuminated regions [10].

In [8] it is shown that

(6)

S(λ,β) = H (2)λ (β)

H (1)λ (β)

R22(λ,β)

+T21(λ,β)T12(λ,β)H (1)λ (α)

H (2)λ (α)

∞∑p=1

[ρ(λ,β)]p−1,

where

ρ(λ,β) = R11(λ,β)H (1)λ (α)

H (2)λ (α)

.

This is the Debye expansion, arrived at by expand-ing the expression [1− ρ(λ,β)]−1 as an infinite geo-metric series. R22, R11, T21, and T12 are respectivelythe external/internal reflection and internal/ex-ternal transmission coefficients for the problem.This procedure transforms the interaction of “wave+ sphere” into a series of surface interactions. Inso doing it unfolds the stationary points of the in-tegrand so that a given integral in the Poisson sum-mation contains at most one stationary point. Thispermits a ready identification of the many termsin accordance with ray theory. The first term rep-resents direct reflection from the surface. The pth

term in the summation represents transmissioninto the sphere (via the term T21) subsequentlybouncing back and forth between r = a and r = 0a total of p times with p − 1 internal reflections atthe surface (this time via the R11 term in ρ). The fi-nal factor in the second term, T12, corresponds totransmission to the outside medium. In general,therefore, the pth term of the Debye expansionrepresents the effect of p + 1 surface interactions.Now f (β,θ) can be expressed as

(7) f (β,θ) = f0(β,θ)+∞∑p=1

fp(β,θ),

where

(8)f0(β,θ) = i

β

∞∑m=−∞

(−1)m∫∞

0

(1− H

(2)λ (β)

H (1)λ (β)

R22

)

× Pλ− 12(cosθ) exp(2imπλ)λdλ.

The expression for fp(β,θ) involves a similartype of integral for p ≥ 1. The application of themodified Watson transform to the third term(p = 2) in the Debye expansion of the scatteringamplitude shows that it is this term which is as-sociated with the phenomena of the primary rain-bow. Residue contributions also arise from theRegge-Debye poles. More generally, for a Debyeterm of given order p, a rainbow is characterizedin the λ-plane by the occurrence of two real sad-dle points, λ and λ′ , between 0 and β in some do-main of scattering angles θ, corresponding to thetwo scattered rays on the lighted side. As θ → θ+Rthe two saddle points move toward each otheralong the real axis, merging together at θ = θR. Asθ moves into the dark side, the two saddle pointsbecome complex, moving away from the real axisin complex conjugate directions. Therefore, froma mathematical point of view, a rainbow can be de-fined as a coalescence of two saddle points in thecomplex angular momentum plane.

The rainbow light/shadow transition region isthus associated physically with the confluence ofa pair of geometrical rays and their transformationinto complex rays; mathematically this correspondsto a pair of real saddle points merging into a com-plex saddle point. The next problem is to find theasymptotic expansion of an integral having two sad-dle points that move toward or away from eachother. The generalization of the standard saddle-point technique to include such problems wasmade by Chester et al. [3]. Using their method,Nussenzveig was able to find a uniform asymptoticexpansion of the scattering amplitude which wasvalid throughout the rainbow region and whichmatched smoothly onto results for neighboringregions [10], [8]. Unsurprisingly perhaps, the low-est-order approximation in this expansion turns out

DECEMBER 2002 NOTICES OF THE AMS 1369

to be the Airy approximation, which, as alreadynoted, was the best prior approximate treatment.However, Airy’s theory has a limited range of ap-plicability as a result of its underlying assump-tions, e.g., β 5000, |θ − θR| 0.5 . By contrast,the uniform expansion (and more generally thecomplex angular momentum theory) is valid overmuch larger ranges.

Finally, we examine the rainbow as a diffractioncatastrophe following the account by Berry andUpstill [2]. Optics is concerned to a great degreewith families of rays filling regions of space; thesingularities of such ray families are caustics. Foroptical purposes this level of description is im-portant for classifying caustics using the conceptof structural stability: this enables one to classifythose caustics whose topology survives perturba-tion. Structural stability means that if a singular-ity S1 is produced by a generating function φ1,and φ1 is perturbed to φ2, the correspondinglychanged S2 is related to S1 by a diffeomorphism ofthe control set C (that is, by a smooth reversibleset of control parameters, a smooth deformation).In the present context this means physically thatdistortions of the raindrop shape to incoming wave-fronts from their “ideal” spherical or planar formsdo not prevent the formation of rainbows, thoughthere may be some changes in the features. Anotherway of expressing this concept is to describe thesystem as well posed in the limited sense thatsmall changes in the input generate correspond-ingly small changes in the output. For the ele-mentary catastrophes, structural stability is ageneric property of caustics. Each structurally sta-ble caustic has a characteristic diffraction pattern,the wave function of which has an integral repre-sentation in terms of the standard polynomial de-scribing that catastrophe. From a mathematicalpoint of view, these diffraction catastrophes are es-pecially interesting, because they constitute a newhierarchy of functions distinct from the specialfunctions of analysis [2].

As noted above, at the level of geometrical opticsthe scattering deviation angle D has an extremumcorresponding to the rainbow angle (or Descartesray) when considered as a function of the angle ofincidence i. This extremum is a minimum for theprimary rainbow. Clearly the point (i,D(i)) corre-sponding to this minimum is a singular point (ap-proximately (59,138)) insofar as it separates atwo-ray region (D > Dmin = θR) from a zero-ray re-gion (D < θR ) at this level of description. This is asingularity or caustic point. The rays form a direc-tional caustic at this point; this is a fold catastro-phe, the simplest example of a catastrophe. It is theonly stable singularity with codimension one (thedimensionality of the control space (one) minus thedimensionality of the singularity itself (zero)). In

physical space the caustic surface is asymptotic toa cone with semiangle 42 .

Diffraction is discussed in terms of the scalarHelmholtz equation (3) at the point R for the com-plex scalar wavefunction ψ(R). The concern in cat-astrophe optics is to study the asymptotic behav-ior of wave fields near caustics in the short-wavelimit k→∞ (semiclassical theory). In a standardmanner, ψ(R) is expressed as

ψ(R) = A(R)eiκχ(R),

where the modulus A and the phase κχ are bothreal quantities. The integral representation for ψis

(9)ψ(R) = e−iNπ/4

( κ2π

)N/2

×∫· · ·

∫b(s;R) exp[ikφ(s;R)]dNs,

where N is the number of state (or behavior) vari-ables s and b is a weight function. According to theprinciple of stationary phase, the main contribu-tions to the above integral for given R come fromthe stationary points, i.e., those points si for whichthe gradient map ∂φ/∂si vanishes; caustics are sin-gularities of this map, where two or more station-ary points coalesce. Because k→∞ , the integrandis a rapidly oscillating function of s , so other thannear the points si , destructive interference occursand the corresponding contributions are negligible.The stationary points are well separated providedR is not near a caustic; the simplest form of sta-tionary phase can then be applied and yields a se-ries of terms of the form

ψ(R) ≈∑µAµ exp[igµ(k,R)],

where the details of the gµ need not concern ushere. Near a caustic, however, two or more of thestationary points are close (in some appropriatesense), and their contributions cannot be sepa-rated without a reformulation of the stationaryphase principle to accommodate this or by usingdiffraction catastrophes. The problem is that theray contributions can no longer be considered sep-arately; when the stationary points approach closerthan a distance O(k−1/2) , the contributions are notseparated by a region in which destructive inter-ference occurs. When such points coalesce, φ(s;R)is stationary to higher than first order and quadraticterms as well as linear terms in s − sµ vanish. Thisimplies the existence of a set of displacements dsi ,away from the extrema sµ, for which the gradientmap ∂φ/∂si still vanishes, i.e., for which∑

i

∂2φ∂si∂sj

dsi = 0.

1370 NOTICES OF THE AMS VOLUME 49, NUMBER 11

The condition for this homogeneous system of equa-tions to have a solution (i.e., for the set of controlparameters C to lie on a caustic) is that the Hess-ian

H(φ) ≡ det(∂2φ∂si∂sj

) = 0

at points sµ(C) where ∂φ/∂si = 0. The caustic de-fined by H = 0 determines the bifurcation set forwhich at least two stationary points coalesce (in thepresent circumstance this is just the rainbow angle).In view of this discussion there are two other waysof expressing this: (i) rays coalesce on caustics,and (ii) caustics correspond to singularities of gra-dient maps.

To remedy this problem, the function φ is re-placed by a simpler “normal form” Φ with the samestationary-point structure, and the resulting dif-fraction integral is evaluated exactly. This is wherethe property of structural stability is so impor-tant, because if the caustic is structurally stable, itmust be equivalent to one of the catastrophes (inthe diffeomorphic sense described above). The re-sult is a generic diffraction integral which willoccur in many different contexts. The basic dif-fraction catastrophe integrals (one for each cata-strophe) may be reduced to the form

(10) Ψ (C) = 1(2π )N/2

∫· · ·

∫exp[iΦ(s;C)]dNs,

where s represents the state variables and C thecontrol parameters (for the case of the rainbow

there is only one of each, so N = 1). These integralsstably represent the wave patterns near caustics.For the fold,

(11) Φ(s;C) = 13s3 + Cs.

The diffraction catastrophes Ψ (C) provide transi-tional approximations, valid close to the caustic andfor short waves but increasingly inaccurate farfrom the caustic. By substituting the cubic term (11)into the integral (10), we obtain

(12) Ψ (C) = 1√2π

∫∞−∞

exp[i(s3/3+ Cs)]ds,

which is closely related to the Airy integral Ai(C)(equation (1), with s = 31/3t).

Let us now end where we began: at the rainbowbridge, but this time accompanied by a quotationfrom H. M. Nussenzveig [9] which aptly summarizesmuch of the subject matter addressed in this arti-cle.

The rainbow is a bridge between twocultures: poets and scientists alike havelong been challenged to describe it. Thescientific description is often supposedto be a simple problem in geometricaloptics.… This is not so; a satisfactoryquantitative theory of the rainbow hasbeen developed only in the past fewyears. Moreover, that theory involvesmuch more than geometrical optics; itdraws on all we know of the nature oflight.…

Some of the most powerful tools ofmathematical physics were devised ex-plicitly to deal with the problem of therainbow and with closely related prob-lems. Indeed, the rainbow has served asa touchstone for testing theories of op-tics. With the more successful of thosetheories it is now possible to describethe rainbow mathematically, that is, topredict the distribution of light in thesky. The same methods can also be ap-plied to related phenomena, such asthe bright ring of color called the glory,and even to other kinds of rainbows,such as atomic and nuclear ones.

AcknowledgementsI am grateful to Harold Boas, Bill Casselman, andAllyn Jackson for their valuable assistance in thepreparation of this article, and to the authors ofthe book, Raymond Lee and Alistair Fraser, fortheir permission to use Figures 8.7 and 8.9 from

René Descartes correctly explained the reasonsfor the primary and secondary rainbows withinthe confines of geometrical optics. This image

is from a copy of the first edition of “Lesmétéores” at the Thomas Fisher Rare Book

Library of the University of Toronto.

DECEMBER 2002 NOTICES OF THE AMS 1371

their book. I thank Raimondo Anni for his helpfulcomments at the proof stage of this article.

Note: For information about recent develop-ments in atomic, molecular, and nuclear scattering,visit http://cl.fisica.unile.it/~anni/Rainbow/Rainbow.htm.

References[1] J. A. ADAM, The mathematical physics of rain-

bows and glories, Phys. Rep. 356 (2002), 229–365.[2] M. V. BERRY and C. UPSTILL, Catastrophe optics:

morphologies of caustics and their diffractionpatterns, Progress in Optics, vol. 18 (E. Wolf, ed.),North-Holland, Amsterdam, 1980, pp. 257–346.

[3] C. CHESTER, B. FRIEDMAN, and F. URSELL, An exten-sion of the method of steepest descents, Proc.Cambridge Philos. Soc. 53 (1957), 599–611.

[4] W. T. GRANDY, Scattering of Waves from LargeSpheres, Cambridge University Press, Cambridge,2000.

[5] H. KRAGH, Ludvig Lorenz and nineteenth centuryoptical theory: The work of a great Danish sci-entist, Appl. Optics 30 (1991), 4688–95.

[6] N. A. LOGAN, Survey of some early studies of thescattering of plane waves by a sphere, Proc. IEEE53 (1965), 773–85.

[7] R. G. NEWTON, Scattering Theory of Waves andParticles, Springer-Verlag, New York, 1982.

[8] H. M. NUSSENZVEIG, High-frequency scattering bya transparent sphere. I. Direct reflection andtransmission, J. Math. Phys. 10 (1969), 82–124;High-frequency scattering by a transparentsphere. II. Theory of the rainbow and the glory,J. Math. Phys. 10 (1969), 125–76.

[9] ——— , The theory of the rainbow, Sci. Amer. 236(1977), 116–127.

[10] ——— , Diffraction Effects in Semiclassical Scat-tering, Cambridge University Press, Cambridge,1992.

[11] D. B. PEARSON, Quantum Scattering and Spec-tral Theory, Academic Press, London, 1988.

size of the drops in the particular rainbow atthe top indicated by the slice. Smaller drop radiiare at the left, and the slice corresponds to aradius of about a quarter of a millimetre.

The significance, perhaps even the reality, ofthese ‘extra’ bows was not appreciated for a longtime, quite likely since they were not some-thing Newton’s theory could account for. Eventoday, few people seem to be aware of them.They are faint, but not difficult to observe if onelooks carefully.

Alistair Fraser tells us that the historical roleplayed by such bows in motivating the devel-opment of the wave theory of light is greaterthan usually suggested in physics books. Healso points out that in nature rain drops arerarely homogeneous in size or shape, and thatthis has the effect that supernumerary bows areseen only in the top of the bow, where inter-ference arising from the variety of drops doesnot cancel. He says further, “As supernumerarybows are an interference pattern strongly de-pendent upon drop size, and as rain showershave a wide range of drop sizes, one would notexpect these bows to be seen in showers. Theyare, nonetheless, seen. Curiously, not all dropscontribute equally—through a mixture of thephysical optics of small drops and the non-spherical shape of large drops, a narrow band-pass filter is produced which eliminates the su-pernumerary bows produced by all but a smallrange of sizes.”

The color map is that produced by the the-ory of George Biddell Airy, dating from roughly1825. The book includes also a similar map(figure 10-29) deduced from the more exact Mietheory, but emphasizes that the fine detail thispredicts, although confirmed under laboratoryconditions, is imperceptible in nature, whereperturbations blur it.

—Bill Casselman(covers@ams.org)

About the Cover

Supernumerary Bows along with Airy’sExplanationThis month’s cover accompanies John Adam’sreview of The Rainbow Bridge. Both images aretaken from the book (figures 8-3 and 8-11). Thetop photograph was taken in 1979 near Koote-nay Lake, British Columbia, by Alistair Fraser,one of the book’s authors. It is one of hundredsof rainbow photographs he has taken in hislife, and shows well a set of supernumerarybows. At the bottom is a plot of rainbow colorconfigurations versus rain drop size, with the