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Multiple Rayleigh Scattering of Electromagnetic Waves E Multiple Rayleigh Scattering of Electromagnetic Waves E. Amic, J. Luck, Th. Nieuwenhuizen To cite this version: E. Amic, J. Luck, Th. Nieuwenhuizen. Multiple Rayleigh Scattering of Electromagnetic Waves. Journal de Physique I, EDP Sciences, 1997, 7 (3), pp.445-483. 10.1051/jp1:1997170. jpa-00247338 HAL Id: jpa-00247338 https://hal.archives-ouvertes.fr/jpa-00247338 Submitted on 1 Jan 1997 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. (1997) Phys. FFance J. I 7 445-483 1997, 445 MARCH PAGE Multiple Scattering Rayleigh Electromagnetic of Waves Antic J.M. Luck E. (~), Nieuwenhuizen and Th.M. (~i*) (~) Saclay, CEA Physique Thdorique, Service de Cedex, (~) Gif-sur~Yvette 91191 France Laboratorium, Waals-Zeeman Van der Valckenierstraat 65, (~) Amsterdam, XE The Netherlands 1018 (Recei>'ed August 1996) final form 1996, 22 accepted 1996, recei>'ed November in November 19 7 PACS.42.68.Ay Propagation, transmission, attenuation, and radiative transfer propagation, PACS.42.25.Bs ivave absorption transmission and PACS.42.68.Mj Scattering, polarization Multiple scattering polarized electromagnetic diffusive of Abstract. media is in in- waves by vestigated approach theory. of radiative transfer summing ladder This the to amounts means diagrams diffuse reflected for intensity, cyclical transmitted for the the the of or ones or cone backscattering. becomes enhanced The interest, such method in of several situations exact as (slab experiment thick-slab wavelength path &ee thickness L £ The A). » present » a mean study Rayleigh scattering. incorporates dependence restricted is It natural the in to way a on polarizations, and detected the incident reflections and takes full of the internal the account at sample, optical possible boundaries the of index of due the the mismatch between to n mean surroundings. rely of and that the This work the diffusion the medium does not approx- ni on therefore correctly layers, describes imation. It radiation the skin where takes in crossover a Quantities place interest, free diffusive of propagation, and such between and the as nice-versa. angle-resolved polarization-dependent, transmission, diffuse intensity reflection in and in mean backscattering, shape predicted and of the of the of enhanced solutions in terms to cone are analytically, Schwarzschild-MiIne of equations. The latter both the absence obtained in are (n/ni in ni), large 1). reflections regime internal and in the of index mismatch I « » or a = Introduction 1. undergoes through inhomogeneous Light multiple media dis- propagating scattering when over of path variety situations, larger free I. This much than wide in tances may one mean occur a multiple biological theory of atmospheres planets from the of and The ranging tissues. stars to [I-4], which has been electromagnetic physics classical of scattering of is old waves area an developed by subject experienc- mostly astrophysicists. This has been for almost century, one decade, motivated for ing experimental activity important of theoretical and revival one an localization) (weak by the analogy random disorder between effects of the the strong on or quantum-mechanical (electromagnetic, seismic) of classical and of propagation acoustic, waves (electrons solids). first weak-localization discovered has been the The effect be in to waves angular backscattering phenomenon, place celebrated enhanced which takes in cone narrow a backscattering around the of direction [5j. exact (e-mail: luck©split.saclay.cea.fr) (*) correspondence Author for (ditions © Physique Les de 1997 PHYSIQUE JOURNAL N°3 DE I 446 light multiple Typical polysty- of scattering laboratory experiments suspensions involve on paint) (white path Ti02 fluids. free of these t In the grains spheres situations in or mean rene light, wavelength larger optically samples often thick of usually much than the ~ and the is are by slabs, L, I. I I characterized diffusive regime interest, of thickness The of L I. is « » « e., a through multiple diffusive admits scattering. of radiation This three different regime transport approach description. diffusion theoretical where approximation, of The crudest the levels is (I) by multiply effective diffusion of intensity equation. described the scattered The is means an I, length supplemented by larger boundary only than valid scales latter that is has be is to on so keeps approach phenomenological As this character. conditions. somehow (ii consequence, a a (RTT), approach, theory long by mesoscopic used for The transfer been radiative has known as astrophysicists [1-4]. keeping equation, balance the of based local community It track of is a on (iii) of approach microscopic direction the of propagation intensity. systematic the The consists expanding diagrammatic of the solution equations the the random medium Max~v.ell in in as a (multiple-scattering) Born series. light multiple scattering diffuse quantities of In the experiments the interest most are mean shape backscattering. peak reflected of the of enhanced intensity, and the and transmitted For diagrammatic approach simplifies observables the greatly regime these t As the ~ L. in « < quantities, far averaged positions concerned, of random the the scatterers, mean over as are which, by diagrams, the diffuse radiation of the ladder is described the in turn, amounts sum backscattering RTT; by cyclical phenomenon the enhanced described the so-called is to or diagrams, maximally-crossed by adaptation On which summed of RTT be also [6, 7]. up can an validity hand, the other the of established has been rigorous starting basis RTT vector [8], on a perturbative from extending developments of Maxwell's equations, earlier treatment [9] a on electromagnetic multiple of scattering plasmas. the regime In where the random in waves short-range fluctuations of dielectric approach the rigorously correlations, have this constant justifies of Schwarzschild-MiIne the the Rayleigh phase with of equation RTT the vector use function, lv~hich the be work. of ~v-ill the present purpose principles of taking The electromagnetic RTT polar- for their into vector account waves, iii, exposed izations, by Chandrasekhar book the in formal which also contains are a ana- lytical of diffuse the derivation Rayleigh for scattering, intensity the absence of internal in reflections. approach This predictions level, needed obtain order is quantitative in at to a concerning observables reflection like the intensity diffuse transmission, and and in the in en- backscattering hanced particular diffusion In yield approximation the alone such cannot cone. chiefly predictions, boundary because fully dealt conditions be with accurate in cannot sat- a Surprisingly isfactory multiple enough, only modern of scattering the few in way. era very authors RTT. Thus far have used light, investigations major 1v-eak the of of the localization including effects, polarization [10-12] approximation have used either the diffusion rather or numerical simulations multiple Several bulk electromag- properties of of scattering other [13]. along including netic especially- investigated lines, Faraday have been these effects of the ,vaves [12,14,15] absorption effects rotation and of polarization the results backscat- Exact [16]. on on ii?] only tering general recently. appeared have properties has derived Mishchenko cone very of polarizations the of obtaining behaviour for the first reversal, under thus time time con- a of sistent the backscattering. enhancement direction derivation factors the of in The exact backscattering shape full of the investigated then been by has Ozrin who did [18], not, cone however, with analytical problem. of latter full In works, previous solution the up a come we undergoing considered of the multiple arbitrary have and isotropic scalar [19, 20] case waves scattering anisotropic layers, We of have takes the skin shown how RTT account [21]. proper light converted paths from diffusive field and few free beam where free is to over a a mean a and allows how it the effects of internal reflections due the index deal vice-versa, with to to RAYLEIGH SCATTERING ELECTROMAGNETIC MULTIPLE OF N°3 WAVES 447 Definitions wef~l for Table1. kinematic and other and q~antities. notations outside medium medium inside optical index ni mm n = mu/c 2~/~i nuJ/c 2~/~ hi wavenumber k = = = = angle incidence 9 91 fi 9 91 /1 cos cos = = @@ sine sin 91 iv v " = = parallel cos9 hi cos91 k P wave,~ector p = = I/m and sine total reflection and sin I I 91 > > < > m m m imaginary.) imaginary) (I.e. (i.e. condition P p sin91 ksin9 hi kv transverse wavevector (q( q " = = = angle azimuthal ~2 subject of sample. effect been much mismatch the boundaries of latter the The has the at [22-26]. recently activity scattering multiple of investigations the goal The of the extend is present to to paper our of obtaining complete analytic description for first electromagnetic thus the t.ime waves, a including L, both backscattering regime of ~ f intensity diffuse and the the in the « « cone Rayleigh analysis reflections. shall the polarization
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