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The Mathematical Physics of Rainbows and Glories John A Physics Reports 356 (2002) 229–365 The mathematical physics of rainbows and glories John !" Adam Department of Mathematics and Statistics,Old Dominion University,Norfolk,VA 23529,USA Received May 2001& editors' : Eichler; T:): Gallagher Contents %" Introduction 231 5.7" Summary of the 8!$ theory for %"%" Structure and philosophy of the review 231 rainbows and glories 302 1.2" The rainbow' elementary physical 5.8" ! synopsis' di9ractive scattering, features 233 tunneling e9ects: shape resonances 1.3" The rainbow' elementary mathematical and Regge trajectories [89= 304 considerations 243 6" The electromagnetic problem 311 %"." Polarization of the rainbow 246 6.1" Polarization 311 1.5" The physical basis for the divergence 6.2" Further developments on polarization: problem 250 Airy theory revisited 314 2" Theoretical foundations 252 6.3" Comparison of theories 319 2.1" The supernumerary rainbows; 6.4" Non-spherical (non-pendant) drops 326 a heuristic account of Airy theory 252 6.5" Rainbows and glories in atomic, 2.2" Mie scattering theory 258 nuclear and particle physics 331 3" Glories 260 2" The rainbow as a di9raction catastrophe 335 3.1" The backward glory 260 0" Summary 343 3.2" Rainbow glories 267 0"%" The rainbow according to 8!$ theory 344 3.3" The forward glory 269 Acknowledgements 347 ." Semi-classical and uniform approximation Appendix !" Classical scattering& the scattering descriptions of scattering 270 cross section 347 5" The comple4 angular momentum theory: !"%" Semi-classical considerations' a preci7 s 351 scalar problem 276 Appendix ?" Airy functions and Foc1 functions 353 5.1" The quantum mechanical connection 276 Appendix 8" The Watson transform and 5.2" The poles of the scattering matri4 280 its modiÿcation for the 8!$ 5.3" The Debye expansion 283 method 354 5.4" Geometrical optics regime7 s 289 Appendix 6" The Chester–Friedman–Ursell 5.5" Saddle points 291 (CFU) method 359 5.6" The glory 297 References 360 E-mail address: [email protected], (J.A" Adam). 0370-1573/02/E3 see front matter c 2002 Published by Elsevier Science ?"F" PII' - 03203%523(0%)000263G 230 J.A$ Adam %Physics Reports 35( (2002+ 229–365 Abstract ! detailed qualitative summary of the optical rainbow is provided at several complementary levels of description: including geometrical optics (ray theory): the Airy approximation: Mie scattering theory: the comple4 angular momentum (CAM) method: and catastrophe theory" The phenomenon known commonly as the glory is also discussed from both physical and mathematical points of view' backward glories, rainbow-glories and forward glories" While both rainbows and glories result from scattering of the incident ◦ radiation: the primary rainbow arises from scattering at about 138 from the forward direction: whereas the (backward) glory is associated with scattering very close to the backward direction" +n fact: it is a more comple4 phenomenon physically than the rainbow: involving a variety of di9erent e9ects (including surface waves) associated with the scattering droplet" Both sets of optical phenomena—rainbows and glories—have their counterparts in atomic: molecular and nuclear scattering: and these are addressed also" The conceptual foundations for understanding rainbows: glories and their associated features range from classical geometrical optics: through quantum mechanics (in particular scattering from a square well potential& the associated Regge poles and scattering amplitude functions) to di9raction catastrophes" Both the scalar and the electromagnetic scattering problems are reviewed: the latter providing details about the polarization of the rainbow that the scalar problem cannot address" The basis for the comple4 angular momentum (CAM) theory (used in both types of scattering problem) is a modiÿcation of the Watson transform: developed by Watson in the early part of this century in the study of radio wave di9raction around the earth" This modiÿed Watson transform enables a valuable and accurate approximation to be made to the Mie partial-wave series: which while exact: converges very slowly at high frequencies" The theory and many applications of the 8!$ method were developed in a fundamental series of papers by Nussenzveig and co-workers (including an important interpretation based on the concept of tunneling): but many other contributions have been made to the understanding of these beautiful phenomena: including descriptions in terms of so-called di9raction catastrophes" The rainbow is a ÿne example of an observable event which may be described at many levels of mathematical sophistication using distinct mathematical approaches: and in so doing the connections between several seemingly unrelated areas within physics become evident. c 2002 Published by Elsevier Science ?"F" PACS: 42.15.i& 42.25.p& 42.25.FX& 42.68.$j Keywords: Rainbow& Glory& Mie theory& Scattering& Comple4 angular momentum& Di9raction catastrophe J.A$ Adam % Physics Reports 35( (2002+ 229–365 231 Like the appearance of a rainbow in the clouds on a rainy day: so was the radiance around him" This was the appearance of the likeness of the glory of the Lord... The Boo1 of Ezekiel: chapter %: verse 20 (New International Version of the Bible). 1.Introduction 2$2$ Structure and philosophy of the review The rainbow is at one and the same time one of the most beautiful visual displays in nature and: in a sense: an intangible phenomenon" +t is illusory in that it is not of course a solid arch, but like mirages: it is nonetheless real" +t can be seen and photographed: and described as a phenomenon of mathematical physics: but it cannot be located at a speciÿc place: only in a particular direction" What then is a rainbowJSince many levels of description will be encoun- tered along the way: the answers to this question will take ,s on a rather long but fascinating journey in the footsteps of those who have made signiÿcant contributions to the subject of “light scattering by small particles”" Let ,s ÿrst ‘listenNto what others have written about rainbows and the mathematical tools with which to understand them. “Rainbows have long been a source of inspiration both for those who would prefer to treat them impressionistically or mathematically" The attraction to this phenomenon of 6escartes,7 Newton: and Young: among others: has resulted in the formulation and testing of some of the most fundamental principles of mathematical physics.L P" Sassen <%=" “The rainbow is a bridge between two cultures' poets and scientists alike have long been challenged to describe it: : : Some of the most powerful tools of mathematical physics were devised explicitly to deal with the problem of the rainbow and with closely related problems" Indeed: the rainbow has served as a touchstone for testing theories of optics. With the more successful of those theories it is now possible to describe the rainbow mathematically: that is: to predict the distribution of light in the sky" The same methods can also be applied to related phenomena: such as the bright ring of color called the glory: and even to other kinds of rainbows: such as atomic and nuclear ones.L Q"$" Nussenzveig [2]. “Probably no mathematical structure is richer: in terms of the variety of physical situations to which it can be applied: than the equations and techniques that constitute wave theory. Eigenvalues and eigenfunctions: Hilbert spaces and abstract quantum mechanics: numerical Fourier analysis: the wave equations of Helmholt/ (optics: sound: radio): SchrodingeR r (elec- trons in matter): Dirac (fast electrons) and Klein–Gordon (mesons): variational methods, scattering theory: asymptotic evaluation of integrals (ship waves: tidal waves: radio waves around the earth: di9raction of light)—examples such as these jostle together to prove the proposition.L $"F" Berry [3]. The three quotations above provide a succinct yet comprehensive survey of the topic addressed in this review' the mathematical physics of rainbows and glories" !n attempt has been made to provide complementary levels of description of the rainbow and related phenomena& this mirrors 232 J.A$ Adam % Physics Reports 35( (2002+ 229–365 to some extent the historical development of the subject: but at a deeper level it addresses the fact that: in order to understand a given phenomenon as fully as possible: it is necessary to study it at as many complementary levels of description as possible" +n the present context: this means both descriptive and mathematical accounts of the rainbow and related phenomena: the latter account forming the basis of the paper' it is subdivided into the various approaches and levels of mathematical sophistication that have characterized the subject from the investigations of 6escarte7 s down to the present era. There are several classic books and important papers that have been drawn on frequently throughout this paper' the boo1 by Greenler on rainbows: halos and glories <.= has proved invaluable for the descriptive physics in this introduction& the article by Nussenzveig [2= from which the second quotation is taken is an excellent introduction to both the physics and the qualitative description of the various mathematical theories that exist for the rainbow" The two papers [5,6= by the same author constitute a major thread running throughout this article: but particularly so in Section 5 (comple4 angular momentum theory)" Van de Hulst’s boo1 on light scattering by small particles <2= is a classic in the ÿeld: and for that reason is often cited: both in this article and in many of the references" Indeed: to
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