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Thseprsente pour obtenir le grade de

Docteur de lUniversit Louis Pasteur

Strasbourg IDiscipline : Sciences Pour lIngnieur - Photonique

par Sylvain Lecler

Etude de la diffusion de la lumirepar des particules sub-microniques

Soutenue publiquement le 18 novembre 2005

Membres du juryDirecteur de thse : M.Patrick Meyrueis, Professeur, ULP (Strasbourg)Rapporteur interne : Charles Hirlimann, Directeur de Recherche, ULP (Strasbourg)Rapporteur externe : Frdrique De Fornel, Directrice de recherche, UB (Dijon)Rapporteur externe : Pinar Mengc, Professor, Univ. of Kentucky (Lexington USA)Examinateur : Yoshitate Takakura, HDR, ULP (Strasbourg)

Laboratoire des Systmes photoniquesBoulevard Sbastien BRANT http://lsp.u-strasbg.frBP 10413 Phone : +33(0)3 90 24 46 14F- 67 412 Illkirch Cedex - France Fax : +33(0)3 90 24 46 19

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Thesisto obtain the

Natural Philosophy Degree

of Louis Pasteur University - Strasbourg I

by Sylvain Lecler

Light scattering by sub-micrometric particles

Public presentation the 18th November 2005

Jury membersThesis supervisor : M.Patrick Meyrueis, Professor, ULP (Strasbourg)Internal examinator: Charles Hirlimann, Research Director, ULP (Strasbourg)External examinator: Frdrique De Fornel, Research Director, UB (Dijon)External examinator: Pinar Mengc, Professor, Univ. of Kentucky (Lexington, USA)Scientific adviser : Yoshitate Takakura, HDR, ULP (Strasbourg)

Photonics Systems LaboratoryBoulevard Sbastien BRANT http://lsp.u-strasbg.frBP 10413 Phone : +33(0)3 90 24 46 14F-67 412 Illkirch Cedex - France Fax : +33(0)3 90 24 46 19

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Rsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Figures and tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Rsum long en franais . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1 Introduction 281.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2 Light scattering: short history and background . . . . . . . . . . . . . . 301.3 Necessity of understanding light-matter interactions . . . . . . . . . . . 331.4 Objective and outline of the dissertation . . . . . . . . . . . . . . . . . 34

2 Light scattering 352.1 Single and independent scattering . . . . . . . . . . . . . . . . . . . . . 372.2 Rayleigh scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3 The Lorenz-Mie theory: a summary . . . . . . . . . . . . . . . . . . . . 432.4 Multiple scattering and aggregates . . . . . . . . . . . . . . . . . . . . 452.5 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.6 Electromagnetic solvers: libraries and comments . . . . . . . . . . . . . 492.7 Comments on the choice of the T-matrix approach . . . . . . . . . . . . 54

3 Light scattering by spheres via T-matrix approach 57

3.1 Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Scattering by a single sphere : the Lorenz-Mie theory . . . . . . . . . . 583.3 The integral formulation of the electromagnetic solution . . . . . . . . . 653.4 A demonstration of T-matrix algorithm . . . . . . . . . . . . . . . . . . 683.5 Convergence and limitations of the algorithm . . . . . . . . . . . . . . . 713.6 Evaluation of computing time . . . . . . . . . . . . . . . . . . . . . . . 753.7 Scattering phase function and cross sections . . . . . . . . . . . . . . . 783.8 Analysis of the polarization response of an aggregate . . . . . . . . . . 82

4 Application: The Photonic Jet 864.1 A photonic jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Focusing with a micro-spherical lens . . . . . . . . . . . . . . . . . . . . 894.3 High intensity concentration . . . . . . . . . . . . . . . . . . . . . . . . 91

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E Matlab Programs 153E.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

E.2 Algorithm structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154E.3 Input and output of the program . . . . . . . . . . . . . . . . . . . . . 156E.4 Graphic user interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Micrometric particles light scattering 5/163 PhD, Sylvain Lecler 2002-2005

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Acknowledgments

I would like to thank the following people:

Yoshitate Takakura for having accepted to spend time following my scientific workthrough and for the scientific method he has tried to teach me.

Patrick Meyrueisfor having made my PhD work possible and for the material thathis laboratory has trustfully placed at my disposal.

Pinar Mengc, Frdrique de Fornel, Charles Hirlimann for having acceptedto be in my jury of thesis and to consider my work.

Patrice Twardowskifor the long and interesting discussions we had.

Jean-Claude Wormsfor having made me aware of the ICAPS project.

The PhD students of the laboratory for having helped me with the process thatwe have tried to initiate in the laboratory and for their liking.

Alexis Bonyfor always being in a good mood and for putting up with being with mein the same office.

Vinvent Francois and Sarah T. for having helped me to proofread this thesis.

Thierry Engel and Joel Fontainefor the teaching that they permitted me to do inthe INSA of Strasbourg.

My parents for their support and for having made my university education possi-ble.

Keywords

Light scattering, T-matrix, multipole method, spherical vectorial functions, near-field,sub-wavelength focusing, electromagnetic couplings, Mie, spheres, aggregate of spheres,photonic jet.


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Abstract

The objective of this work has been to contribute to the physical understanding of in-teractions of visible light with sub-micrometric spherical particles, in particular in thenear field. Two different phenomena have been studied, the possibility to concentratelight in a photonic jet and the electromagnetic couplings between two close particles.The Lorenz-Mie theory and the T-matrix algorithm, which is an extension to simulaterigorously light interaction with several spheres, have been used for simulations. In thetwo cases, the Maxwell equations have been analytically solved.

First, we have observed the possibility with spherical dielectric particles to highlyconcentrate energy under the diffraction limit and to reach the diffraction limit in the

near field. We have shown that the laws of these focusing are different from the geo-metrical optics. This near field focusing occurs when the radius size can be comparedto the wavelength but also for larger spheres. When the focus point is just on thesurface of the sphere or a few wavelengths behind, the width (FWHM) of the beamcan be smaller than the wavelength, the beam has a low divergence and the energy ishighly concentrated. Such a focused beam is called a photonic jet. We have shown itsexistence for spherical particles and described its main physical properties according tothe optical properties of the sphere. Several possible applications have been presented.

This energy concentration in the near field has raised the question of possible elec-tromagnetic couplings between particles inside an aggregate of dielectric spheres. Tostudy these electromagnetic couplings, we have simulated couples of micrometric par-ticles. The study has been performed for two dielectric and two perfectly conductivespheres for several orientations. Our objective has been to propose physical interpre-tations of the possible electromagnetic couplings between two close particles.

For a couple of particles, which would be orthogonal to the incident plane wave vec-tor, a comparison with circular Young slits has been made. In single scattering, thescattered intensity in the far field can be described as interferences and diffraction. Wehave extended this comparison to multiple scattering regimes and we have shown thatinteractions due to multiple scattering mainly change the ratio of the incident wave

that interacts with the particles.

We have also considered a couple of particles parallel to the incident wave vector.In multiple scattering and in backward direction, we have pointed out a shadow effectfor perfectly conductive spheres that makes the interference intensity decreases. Inopposition, a Perot-Fabry effect (cavity resonances) has been observed for a couple ofdielectric spheres. This effect makes the interference intensity increases and can beused in future applications.


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Rsum

Lobjectif de ce travail a t de contribuer la comprhension physique de linteractionde la lumire visible avec des particules sub-microniques, en particulier en champproche. Deux phnomnes particuliers ont t tudis, la possibilit de concentrerla lumire en un jet photonique et les couplages lectromagntiques entre particulesproches. La thorie de Lorenz-Mie et lalgorithme de la T-matrice, qui est une exten-sion pour simuler rigoureusement linteraction de la lumire avec plusieurs sphres, ontt utiliss pour les simulations. Dans les deux cas, les quations de Maxwell ont trsolues analytiquement.

En premier lieu, nous avons observ la possibilit quavaient des sphres dilectriques

de focaliser jusqu la limite de diffraction et de fortement concentrer lnergie enchamp proche. Cette focalisation en champ proche a lieu quand le rayon de la sphrea une taille comparable avec la longueur donde, mais aussi pour des sphres de tailleplus grande. Quand le point focal est juste sur la surface de la sphre ou quelqueslongueurs donde devant, la largeur (FWHM) du faisceau peut tre plus petite que lalongueur donde, le faisceau est faiblement divergent et lnergie peut tre fortementconcentre. Un tel faisceau focalis a t appel un jet photonique. Nous avons montrson existence pour des particules sphriques et avons dcrit ses principales propritsen fonction des proprits optiques de la sphre. Plusieurs applications possibles ontt prsentes.

Cette concentration dnergie en champ proche a pos la question des couplages lectro-magntiques qui peuvent intervenir entre particules au coeur dun agrgat de sphresdilectriques. Pour tudier ces couplages, nous avons simul des couples de particules.Ltude a t ralise avec des couples de sphres dilectriques et des couples de sphresconductrices parfaites et ceci pour plusieurs orientations. Notre but a t de proposerune interprtation physique des phnomnes de couplages qui peuvent avoir lieu entredeux particules proches.

Pour un couple de particules qui serait orthogonal au vecteur donde de londe planeincidente, une comparaison avec des trous dYoung a t faite. En diffusion simple,

lintensit diffuse en champ lointain peut tre dcrite en terme dinterfrences et dediffraction. Nous avons tendu cette comparaison aux cas o il y a diffusion multipleet nous avons montr que les interactions dues la diffusion multiple influaient essen-tiellement sur la fraction de londe incidente qui interagissait avec les particules.

Nous avons galement considr le cas o le couple de particules serait parallle auvecteur donde de londe plane incidente. En diffusion multiple et dans la direction dertro diffusion, nous avons mis en vidence un phnomne dombrage dans le cas desphres conductrices parfaites, qui fait dcrotre lintensit des interfrences observes.A linverse, un effet de type Fabry-Perot (rsonances dune cavit) a t observ pour un

couple de particules dilectriques, effet qui fait augmenter lintensit des interfrencesobserves et qui pourra tre utiliss pour des applications futures.


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Figure 1: Jacques Harthong.

I would like to dedicate this thesis to Jacques Harthong,who honored me by accepting to be in the jury of thesis,but whose life was cut short.

You are sorely missed.

See his web site : http://moire4.u-strasbg.fr

"Peut-tre vous demandez-vous pourquoi cette" These"... est en anglais. ...La vraieraison est simple : ...langlais est la seule langue qui peut tre comprise aussi bien parun Allemand, un Italien, un Nerlandais, un Espagnol, un Anglais, un Sudois, ou unFranais. La langue utilise ici nest dailleurs pas vraiment langlais, ni lamricain;cest tout simplement le Basic English, la langue internationale."

Extrait de la page daccueil du site web de Jacques Harthong.

What could have been translated in English:Perhaps you ask yourself why this thesis... is in English. The reason is easy to under-stand: ... English is the only language that can be understood by a German, an Italian,a Dutchman, a Spaniard, a British, a Swedish or a French. However, the used languageis not really the English, nor American, it is only the Basic English, the internationallanguage.(Inspired from the first page of the website of Jacques Harthong).

Unfortunately some spelling mistakes are still in this thesis. Pardon me...


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Foreword : PhD work and scientific work

I did not do a PhD just to have a new degree, that was not an opportunity, butsomething substantial I had wanted to do for a long time. A PhD has been one ofthe possibility to have time to observe and try to understand the reality with rationalarguments. Therefore among all the scientific domains, mine was necessarily physics.Physics not in opposition to biology or chemistry, but as Physis, the word used byAristote to describe the nature. Because I am very surprised by the accuracy of sciencebut I am more surprised by nature itself.

In scientific work as a thesis, there is a technical aspect: making programs, formu-lae demonstrations, plotting curves, but all these indispensable steps are not scientific

results, there are tools to find meaning. A real result is what we call the physicalunderstanding of a phenomenon, probably not the true explanation of a phenomenonbut something that makes sense.

Mathematics are one of the tools that physicists use. They often consider that physicshas its reasons that mathematics ignore. One of the things that I have learned duringmy PhD, is that a good (experimental or theoretical) physicist does not have to bea specialist in mathematics but has to apply the same strictness as in mathematic.What we call an approximation, can be inspired by physical intuitions but must bemathematically controlled. What we call an experimental error is necessary but mustbe rigorously determined.

Another thing that I have learned during my PhD is that independent of our results orknowledge, independent of the apparently power of science, one of the main qualitiesof a scientist must be his humility. This quality is necessary to learn, to ask oneselfquestions, to observe and to be able to progress. Paradoxically teaching can be a goodschool for humility, with this condition, to be open to all questions.

At the beginning of my PhD work, I hoped to progress more rapidly. I did not knowthat understanding needs more time than doing. I would have liked to plan my work,but I was not aware that in science, it is the unexpected results that have more interest

than the other ones. Actually, during my PhD work, I probably asked myself more ques-tions than I found answers, but I have tried to explain what I have come to understand.

Sylvain Lecler

September 2005


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List of Figures

1 Jacques Harthong. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Section efficace normalise dune sphre de rayonR et dindice de rfrac-

tion n2 = 1.52. Onde incidente plane linairement polarise. . . . . . . 20

3 Interface permettant de dcrire lagrgat et londe incidente pour ensuitecalculer le champ diffus. . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 (a) Section efficace de rtrodiffusion (b) Comparaison entre la diffractionpar des trous dYoung ( ) et la lumire diffuse (). Systme diffusant: 2 sphres, rayon a: ka = 2, conductivit 2 =, distance entre lecentre des 2 sphres d: kd = 45. Le vecteur donde k et le champlectrique, linairement polaris, sont orthogonaux laxe des 2 particules. 23

5 Section efficace de rtrodiffusion en fonction de la distance d entre lessphres. Rayona: ka = 2. Ici les deux sphres sont dans laxe du vecteurdonde incidentk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Intensit du champ lectrique autour dune sphre de rayon R = 5et dindice n2 = 1.3. La largeur totale mi-hauteur du faisceau estinfrieure . Le calcul a t fait avec un ordre Lmax= 45 et pour uneonde incidente plane linairement polarise Hi= Hy and ki=kz. . . . 25

1.1 Lord Rayleigh: J.W. Strutt 1842-1919 Cambridge [1] . . . . . . . . . . 301.2 Microscope view of the scattering structure on the wing of a butterfly

calledpieris brasicae[2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1 Refraction, reflection and diffraction in scattering phenomena. . . . . . 362.2 Several different scattering regimes as functions of R and n [3]. . . . . . 38

2.3 (a) I(, ) for one sphere a= 0.1, n = 1.52, Hi = Hy, k = kz (anglesin degree). (b) Spherical coordinates. . . . . . . . . . . . . . . . . . . . 39

2.4 I(, = 0)/(k2Csca) for a sphere with n= 1.52, Hi=Hy, k = kz and 3different radii a. = 0 corresponds to the forward scattering. Cscais anormalization constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5 Normalized scattering cross section of a sphere of radiusR and n = 1.52,Hi= Hy,k= kz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 (a) Geometrical description of Rayleigh scattering. (b) Scattered electricfield vector. (c) Stokes parameters. a= 0.1,n2 = 1.52,Hi= Hy,k =kz. 42

2.7 ParameterS3of Stokes for a sphere of radiusa = 1, indexn = 1.52and

for an linear polarized incident wave Hi = Hy, k = kz. Black: circularpolarization, white: linear polarization, gray: elliptic polarization. . . . 44

11

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LIST OF FIGURES

2.8 The Euler angles allow to describe the orientation of a particle. is inthe (O,x,y) plane and describe the line . z is orthogonal to and

makes an angle with z. is the angle between and y in the planeorthogonal toz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.9 Typical curves of intensity and linear polarization degree illustratingnegative polarization and opposition effect. . . . . . . . . . . . . . . . . 47

2.10 Energy balance in a small volume of a scattering medium used in radia-tive transfer method. L is the luminance, the optical thickness, theextinction coefficient,the angular direction. . . . . . . . . . . . . . . 52

2.11 Non-exhaustive classification of rigorous algorithms used to study lightscattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1 (a) Description of the sphere of radiusa, refractive indexn2, conductivity2. (b) Electromagnetic fields. . . . . . . . . . . . . . . . . . . . . . . . 583.2 Perfectly conductive sphere of radius a = 3, Hi = Hy and ki = kz.(a)

scattered intensity, (b) total intensity, (c) tangential component of thescattered magnetic field Hs . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3 k, the propagation vector of the incident field is in the (O,x,z) plane.e1is an unit vector along the y axis. e2is an unit vector orthogonal tok and e1. Eand Hare in the (O, e1, e2) plane. . . . . . . . . . . . . . 65

3.4 2D representation of a 3D contour of integration. Case of a half-infinitemedium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Two equivalent contours of integration for spheres. . . . . . . . . . . . . 68

3.6 Translation of the coordinate frame from O to O. d + r= r. . . . . . . 713.7 Example of smallest virtual sphere of radiusaagg containing all scatterers. 723.8 First 6 diagonal components of the T-matrix of a perfectly conductive

sphere as function of its radiusR. T11: diagonal component correspond-ing to l = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.9 First 6 diagonal components of the T-matrix of a dielectric sphere withn= 1.52 as function of its radius. . . . . . . . . . . . . . . . . . . . . . 74

3.10 (a) Expansion coefficients of an incident plane wave on SVF in free space.i= 0,Hi=Hy andki=kz. (b) and (c) Incident intensity in backwarddirection forR= 5 (b) and R= 5(c). . . . . . . . . . . . . . . . . . 76

3.11 Geometrical description of sphere positions and possible incident wavevector Each case corresponds to particular possible couplings. . . . . . 77

3.12 Computing time needed to calculate the T-matrix for one and two spheresaccording to the angle i of the incident wave vector. The peaks in (a)are numerical artifacts. The difference between curves in (b) are onlydue to an additional constant. . . . . . . . . . . . . . . . . . . . . . . . 78

3.13 Computing time needed to calculate the field in a rectangular spatialzone (20 20 or 40 40) in several cases for one sphere and regressionlaws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.14 Energy conservation: Cext = Csca+ Cabs. . . . . . . . . . . . . . . . . . 81

3.15 Normalized scattering cross section as function of its radius of (a) aperfectly conductive sphere (b) a dielectric sphere with refractive indexn2 = 1.2. For a large sphere Csca tends to 2. . . . . . . . . . . . . . . . 82


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LIST OF FIGURES

3.16 Schematic explanation of why the normalized cross section approachesto 2 when the sphere is large. . . . . . . . . . . . . . . . . . . . . . . . 82

3.17 Example for 2 spheres of planes where the scattered wave polarizationis linear for two incident linear polarized wave. . . . . . . . . . . . . . . 84

3.18 Linear polarization degree,|S1/S0|, in the far field as a function of thedirection of observation for two different geometries (a) and (b) describedin figure 3.17. The scatterer is made of 2 spheres a = ,n = 1.5,d = 4.We observe that the scattered field stays linear (white regions) in the

(k, Ei) and (k, Hi) planes, because these planes are also the symmetricplanes of the couple of particles. . . . . . . . . . . . . . . . . . . . . . . 85

4.1 Notations to describe the micro-spherical lens. n1 = 1 and 2 = 0. ki

is the incident wave vector, a and n2 are respectively the radius and therefractive index of the sphere. . . . . . . . . . . . . . . . . . . . . . . . 864.2 Definition of the FWHM of a beam. R is the radial position in a plane

transverse to the propagation axis. . . . . . . . . . . . . . . . . . . . . 874.3 Focusing of an incident plane wave (Hi=Hy andki=kz) by a dielectric

sphere of index n2 and radius a= 5(Lmax= 40). . . . . . . . . . . . . 884.4 Electric field scattered by a dielectric sphere of indexn2 = 2.5and radius

a= 5 of an unitary plane wave (Hi =Hy and ki =kz). Only the fieldoutside the particle is represented. . . . . . . . . . . . . . . . . . . . . . 89

4.5 Electric field intensityHi = Hy and ki = kz (top) and FWHM (down)of a dielectric sphere of index n2 = 1.3 and radius a= 5. The focusedbeam is smaller than the wavelength. . . . . . . . . . . . . . . . . . . . 90

4.6 Focus position as a function of the refractive index for several sphereradii a. At the bottom a = 2, then a= 4, until a= 18 at the top.The points show the case when the focal point is just on the spheresurface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.7 Difference of focus position between electromagnetic computing (fig. 4.6)and geometrical law (4.1) for several sphere radii a, according to therefractive index n2. At the bottom a= 2, then a= 4, until a= 18at the top. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.8 Difference between the calculated focus and the theoretical geometrical

focus for a large refractive index (n2>2.5, asymptotic value). . . . . . . 924.9 Intensity of the total electric field for a sphere witha= 5andn2 = 1.63.

Calculations have been made with Lmax = 45 and the incident planewave isHi=Hy and ki=kz. . . . . . . . . . . . . . . . . . . . . . . . 93

4.10 Electric field intensity for a sphere with a = 5 and n2 = 1.3, a sub-wavelength FWHM is observed. Calculations have been made withLmax= 45 orders and the incident plane wave is Hi= Hy and ki=kz. . 94

4.11 Abbes law for an imaging system of focusfand apertureD. Image bya lens of a point placed at infinity. . . . . . . . . . . . . . . . . . . . . . 95

4.12 Total electric field around a dielectric sphere of radius a = 0.1 and

optical index n2 = 1.5, for an incident field: Hi = Hy and ki = kz: Inthe Rayleigh case there are two maxima at two sides of the particle. . . 96


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LIST OF FIGURES

4.13 Distance along thez-axis (see geometry in figure 4.1) where the FWHMstays smaller than the wavelength and the intensity over the half global

maximum for 4 radii. The curves are regular only for small indexesbecause for larger indexes a near field effect occurs: the global intensitymaximum, which is close to the sphere, jumps from a local maximum ofa stationary case to another when the refractive index changes. . . . . . 97

4.14 Intensity profile of the total electric field for a sphere with a= 5 andn2 = 1.63 in two orthogonal planes. Calculations have been made withLmax= 45 and the incident plane wave is taken as Hi = Hy and ki=kz. 98

4.15 Total electric field intensity forward a sphere of radiusa= 15and indexn2 = 1.3. Hi = Hy and ki=kz. The sphere is centered in z= 0. . . . . 99

4.16 FWHM of the beam forward a sphere of radius a = 15 and index

n2 = 1.3. The discontinuities are only due to the space sampling. Thesphere is centered in z= 0. . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.17 Total electric field intensity forward a sphere of radiusa= 15and indexn2 = 1.9. Hi = Hy and ki=kz. The sphere is centered in z= 0. . . . . 99

4.18 Electromagnetic field components around a dielectric sphere of radiusa= 5 and indexn2 = 1.63 (Lmax= 115). Hi=Hy and ki = kz. . . . . 101

4.19 Total electric field intensity on the optical axis forward a sphere of radiusa= 15and index n2= 1.3. Hi=Hy and ki=kz. . . . . . . . . . . . . 102

4.20 If a nano-particle is beside a micro-sphere, according to the incidentwave vector direction, the nano-particle will be detectable (ki1) or not

(ki2) (in backscattering). . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.1 Geometrical description of the couple of particles considered. . . . . . . 1055.2 Normalized backscattering cross section of two spheres with ka = 2,

2= , i= 90o (ki=kx) and two incident linear polarizations. . . . . 1065.3 Normalized backscattering cross section of two spheres with ka = 2,

n2 = 1.5,i = 90o and two incident linear polarizations. . . . . . . . . . 107

5.4 (a) Geometrical description of the couple of particles. (b) Equivalentcircular Young slits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.5 Comparison between circular Young ( ) slits and single scattering(). ka = 2, 2 =

, i = 90

o (ki = kx), kd = 45, for the two linear

incident polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.6 Comparison between circular Young slits and forward scattering.ka = 2,

kd = 10,2 = ,i= 90o (ki = kx), for the two incident polarizations. 1105.7 Comparison between circular Young slits and forward scattering.ka = 2,

kd = 10 or kd = 45,n2 = 1.5,i = 90o (ki=kx) and Ei = Ey. . . . . . 111

5.8 Comparison between circular Young slits and backscattering. ka = 2,kd = 45 or kd = 10,n2 = 1.5,i = 90

o (ki=kx) and Ei = Ey. . . . . . 1125.9 Comparison between circular Young slits and backscattering. ka = 2,

kd = 45 or kd = 10,n2 = 1.5,i = 90o (ki=kx) and Ei = Ey. . . . . . 112

5.10 Geometrical description of the couple of spheres. The incident wave

vector is parallel to the axis of the two spheres. . . . . . . . . . . . . . 1135.11 Backscattering cross section as a function of kd. ka = 2 or ka = 6,2= , i= 0o (ki = kz). . . . . . . . . . . . . . . . . . . . . . . . . . 114


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LIST OF FIGURES

5.12 Backscattering cross section as a function of kd. ka = 2, n2 = 1.5,i= 0

o (ki=kz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A.1 Distortion balance of Coulomb. The forces between charged spherescreate a proportional distortion in the fiber. . . . . . . . . . . . . . . . 135

B.1 Change of coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

C.1 Translation of the coordinate frame from O to O. d + r= r. . . . . . . 144C.2 Change of spherical coordinates after a translation along the z-axis. . . 146

E.1 General structure of the algorithm. . . . . . . . . . . . . . . . . . . . . 156E.2 Interface to describe the incident field and the aggregate. . . . . . . . . 160

E.3 Interface to choose what kind of post treatment we want. . . . . . . . . 161


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2.1 Comparison of rigorous algorithms for electromagnetic problems. x= kais the size parameter. The computing time is given as a proportion of

the parameter size, in some case different algorithm can be used. . . . . 54

3.1 In figure 3.9, for a given refractive index of the sphere, all the diagonalcomponents of the T-matrixTll(a)have the same period. This table givesthe period of these components as a function of the refractive indexn ofthe sphere. A proportionality with n 1 seems to appear. . . . . . . . 74

3.2 Order of convergence for an aggregate of two identical spheres whichcenters are separated by a distance of 4 times their radius R and withi= 0 (independent of the refractive index) . . . . . . . . . . . . . . . . 75

3.3 Number of needed SVF in different cases. For a linear incident wave,this number is two times ((e1, o2) or (o1, e2)) the number of couples (l, m). 75

3.4 Possible couplings between SVF.o1 e2 means that the (aio1) compo-nent of the incident wave has an influence on the component (ase2)of thescattered wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1 Properties of the focus point as a function of the refractive indexn2 andof the radius aof the sphere. (n1= 1). . . . . . . . . . . . . . . . . . . 97

5.1 NBSCS and NFSCS for one and two spheres withka = 2 and 2 = .i= 90

o (ki = kx) and Hi = Hy. . . . . . . . . . . . . . . . . . . . . . . 1065.2 NBSCS and NFSCS for one and two spheres with ka = 2 and n2= 1.5.

i= 90o (ki = kx) and Ei= Ey. . . . . . . . . . . . . . . . . . . . . . . 107

E.1 Comparison of notations between several references. . . . . . . . . . . . 153E.2 Comparison of notations between several references. . . . . . . . . . . . 154

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Rsum long en franais

Etude de la diffusion de la lumire par des particulessub-microniques

Thse soutenue par Sylvain Lecler en novembre 2005 - LSP - Strasbourg I - FranceFrench abstract of 10 pages

Introduction

La lumire et plus gnralement les ondes lectromagntiques sont un moyen privilgipour sonder et observer le monde qui nous entoure. Cependant pour que le scientifiquepuisse extraire partir de ses mesures optiques une information objective sur le milieuquil tudie, il est ncessaire quil connaisse les lois dinteraction de la lumire avec

la matire. Au cours de lhistoire, afin den dterminer les lois, ces interactions ontt divises en sous-familles dont les principales sont la rfraction, la diffraction et ladiffusion [3]. Cette dernire, la diffusion, est sans aucun doute la plus gnrale car elleenglobe les deux autres. Cest elle qui en faisant des objets des sources secondaires, lesrend perceptibles par notre oeil. Quand un faisceau lumineux interagit avec une sur-face rugeuse ou un milieu htrogne, sa puissance va tre redistribue dans toutes lesdirections de lespace. Cest cette redistribution de lintensit dans toutes les directionsqui semble le mieux caractriser la diffusion lumineuse. Dans la lumire diffuse, ondistinguera la diffusion spculaire, qui correspond la fraction de lumire rflchie ausens de la formule de Snell-Descartes et la partie lie aux formes des objets rencontrset qui correspond la diffraction.

Les premires tudes de la diffusion lumineuse ont t faites dans le cadre de lobservationastronomique [3]. En effet le flux lumineux qui vient des toiles jusque dans nos tle-scopes a subi une attnuation due la lumire diffuse dans les nuages de poussiresquil a rencontr sur sont parcours. La diffusion apparat donc l comme un dfaut queles astronomes ont voulu estimer. Mais les physiciens se sont trs vites rendu compteque cette diffusion navait pas seulement attnu le faisceau de lumire mais avaitchang ses proprits, entre autre de polarisation. A partir de l, la diffusion ntaitplus vue comme un dfaut. Ltude des proprits de la lumire collecte allait devenirun moyen de dduire les proprits des nuages interstellaires traverss : proprits de

densit du gaz, de taille et de forme des particules [4, 5, 6], mais la condition davoirde bons modles.

17

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LIST OF TABLES

Les premiers modles dvelopps (fin XIX et dbut XX), et qui sont encore trs util-

iss, sont la diffusion de Rayleigh [7] et la thorie de Lorenz-Mie [8, 9]. La diffusion deRayleigh concerne la diffusion de la lumire par des particules trs petites compares la longueur donde. La thorie de Lorenz-Mie dcrit, elle, la diffusion de la lumirepar une particule sphrique de taille quelconque. Ces deux modles sont trs perfor-mants mais possdent deux limites importantes. Dabord ils ne rendent pas compte desproprits de diffusion de particules nayant pas une symtrie sphrique, ensuite ils nepeuvent tre utiliss que dans le cas o les particules sont suffisamment espaces entreelles pour que leurs interactions puissent tre ngliges. Cest dire que ce sont desmodles adapts ltude de la diffusion de milieux peu denses. On parle de diffusionsimple.

Cependant, de nouveaux besoins ont vu le jour concernant la diffusion de la lumirepar des particules non forcment sphriques et dans des milieux plus denses. Commeexemples peuvent tre cites ltude du rayonnement thermique dans des milieux densesen combustion [10, 11] et linteraction de la lumire avec des matires biologiques [12].Dans ces cas l, on parle de diffusion multiple. Une particule ne diffuse pas seule-ment londe incidente, mais galement le champ diffus par les autres particules. Denouveaux modles et algorithmes ont t dvelopps dabord pour tenir compte de laforme des particules diffusantes, ensuite pour prendre en considration leurs interac-tions. Les deux grandes familles dalgorithme [13] sont dune part les algorithmes bass

sur lexpression des quations diffrentielles dans un espace discrtis et dautre part lesalgorithmes modales qui dcrivent avec le thorme intgral le lien entre les fonctionsdune base sur laquelle les champs lectromagntiques sont dcomposs.

L encore, ces algorithmes ont une limite. Sils permettent de dcrire la diffusionde la lumire par des agrgats de particules complexes, ils ne permettent pas toujoursdidentifier les phnomnes physiques lmentaires qui ont lieu. Cest dans ce cadreque cest plac mon travail de thse. Notre but a t damener des lments de com-prhension concernant les couplages lectromagntiques entre particules proches et enparticulier de sonder ce quil se passait en champ proche. Ceci afin de mieux compren-dre la diffusion par des milieux complexes et de permettre une meilleure exploitation

de la lumire pour sonder le rel. Nous nous sommes restreints au cas de particulessphriques, dilectriques ou parfaitement conductrices et ayant une taille de lordre degrandeur de la longueur donde. Les algorithmes rigoureux que nous avons utiliss sontla thorie de Mie et lalgorithme de la T-matrice.

Les premiers rsultats que nous revendiquons concernent linterprtation physique ducouplage entre particules en fonction de leurs natures, de leurs orientations et de ladistance qui les spare [14]. Le deuxime rsultat important concerne la mise en vi-dence et ltude dune concentration dintensit en champ proche que lon nomme jetphotonique [15]. Quand les conditions sont runies, ce faisceau focalis se manifeste

la surface de particule dilectrique et permet datteindre la limite de diffraction. Cefaisceau est intense, peu large et peu divergence, cest ce qui fait son intrt.


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Proprits et tude de la diffusion lumineuse

Quand la lumire interagit avec un agrgat de particules, la lumire diffuse dpendde nombreux paramtres [3, 13]: la taille des particules compares la longueurdonde, leurs proprits optiques (absorption, indice de rfraction, conductivit), leurforme, lorientation de lagrgat par rapport londe incidente, la densit en partic-ules de lagrgat, la nature plus ou moins homogne et isotrope de la rpartition et delorientation des particules dans lagrgat, la forme de lagrgat, etc.

Si lagrgat est beaucoup plus petit que la longueur donde, on pourra utiliser la dif-fusion de Rayleigh [7, 3]. Dans ce cas lintensit de lumire diffuse est inversementproportionnelle la 4 ( est la longueur donde de la lumire incidente). La lumire

diffuse a la mme polarisation que londe incidente et si londe incidente est linaire-ment polarise le diagramme de diffusion sera similaire au diagramme dmission dundiple oscillant. Quand lagrgat est plus volumineux, on peut dans de nombreux caslassimiler une sphre et utiliser la thorie de Mie.

Dans la thorie de Mie [9, 8], on dcompose les ondes incidentes, diffuses et le champ lintrieur de la sphre sur la base des fonctions vectorielles sphriques. Ces fonctionssont les solutions de lquation de propagation vectorielle exprime en coordonnessphriques. En appliquant la continuit des composantes tangentielles du champ lasurface des particules on trouve les lois qui dcrivent le lien entre les coefficients dedcomposition de londe incidente et ceux de londe diffuse et interne la sphre. Lenombre dordre prendre en compte dans cette dcomposition des champs, dpend dela taille en unit de longueur donde de lagrgat de particules. Si la particule est trspetite compare la longueur donde, on retombe sur la diffusion de Rayleigh. Seul lepremier ordre est excit. Plus la particule est grande compare la longueur dondeplus le nombre de modes excits sera grand. Pour certaines longueurs donde et pourune taille de particule donne, des rsonances peuvent apparatre. En diffusion de Mieplusieurs lobes de diffusion peuvent tre observs dont le nombre, la taille et lintensitdpendent les proprits de la particule. Si londe incidente est linairement polarise,londe diffuse pourra, elle, avoir une polarisation linaire, circulaire ou elliptique selonla direction dobservation.

La figure 2 reprsente la section efficace normalise dune particule sphrique dindice1.52 en fonction de son rayon en unit de longueur donde. Par dfinition le produit dela section efficace normalise par la section gomtrique de la particule (R2) et par leflux incident, donne le flux total diffus. Quand la particule est trs petite (R

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0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

R in unit

N

Normalized scattering cross section

Rayleigh scattering

Figure 2: Section efficace normalise dune sphre de rayon Ret dindice de rfractionn2 = 1.52. Onde incidente plane linairement polarise.

les lments finis ou les diffrences finies comme la FDTD [16] rsolvent les quationsde Maxwell dans un espace discrtis. Ce sont des mthodes gnriques mais qui n-cessitent de grande capacit de calcul et beaucoup de place mmoire. La DDA [17] est

une autre mthode qui dcrit la particule comme une matrice de diples en interactionet qui considre le flux diffus comme la somme des contributions de tous ces diples.Enfin les mthodes modales [13] sont bass sur une dcomposition des champs sur unebase de fonctions orthogonales. Lapplication des thormes intgral [18] permet deretrouver les liens entre les coefficients de dcomposition des champs. Lalgorithme dela T-matrice, que jai cod pendant la thse, fait partie de cette catgorie de mthode.Il constitue un outil idal car rapide et rigoureux pour permettre ltude des couplagesentre particules.

Lalgorithme de la T-matrice

Lalgorithme de la T-matrice [19, 13, 20] est considr comme une extension de lathorie de Mie au cas dun agrgat de particules sphriques. Cest une mthode modale.La base des fonctions sur lesquelles les champs sont dcomposs est la mme que cellede la thorie de Mie. Il sagit de la base des fonctions vectorielles sphriques. La T-matrice est la matrice qui lie les coefficients de dcomposition de londe diffuse auxcoefficients de dcomposition de londe incidente.

Pour calculer cette T-matrice, le champ en un point quelconque, hors de lagrgat

est exprim comme une somme de londe incidente et (lintgrale) des composantestangentielles du champ la surface des particules multiplies par la fonction de Greenqui contient un terme de dphasage. Une formulation lgrement diffrente permet


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LIST OF TABLES

davoir lexpression du champ aux points situs lintrieur de lagrgat. Ces deuxquations permettent dtablir le lien entre londe incidente et londe diffuse con-

dition dtre capable de calculer les intgrales dcrites. Le champ tant dcompossur une base de fonctions adapte aux coordonnes sphriques, les termes intgralessont faciles calculer quand le repre est plac au centre de la sphre sur laquelle estralise lintgration. Comme il y a plusieurs sphres et un seul repre, le thormede translation addition est utilis. Il permet dexprimer les fonctions donnes dans unrepre sphrique comme une combinaison linaire des mmes fonctions donnes dansun autre repre sphrique.

Toutes ces formules tant connues sous leurs formes littrales les coefficients de laT-matrice peuvent tre calculs analytiquement, ce qui fait de cette mthode, une des

mthodes les plus rapides et prcises. Ltude des temps de calcul a t faite. Deuxapproximations sont faites dans le calcul. Les champs sont dcomposs sur une basetronque, cest dire sur un nombre fini de fonctions. Une inversion de matrice, nces-saire dans le processus de calcul, est ralise numriquement. Ces deux approximationsont une consquence sest de limiter la taille maximale de lagrgat simul. Les agr-gats simuls pourront atteindre une centaine de longueurs donde de diamtre, ce quifait de cette mthode une des mthodes capable de simuler les agrgats les plus gros [13].

Nous avons cod notre propre algorithme sous Matlab afin den avoir une bonne matrise(interface figure 3). Notre travail sest bas sur la publication de Peterson et Strom

[20]. Une fois calcul le champ diffus, plusieurs autres programmes ont t cods pourexploiter les rsultats obtenus. Parmi les grandeurs physiques que nous avons calcules,on peut trouver, les composantes du champ lectromagntique, le vecteur de Poynting,le diagramme de diffusion 2 et 3D, les lments de Stokes dcrivant les tats de polari-sation, les axes de polarisation, ainsi que les sections efficaces de diffusion, dextinctionet de rtrodiffusion. Lensemble constituant loutil de base pour notre tude.

Figure 3: Interface permettant de dcrire lagrgat et londe incidente pour ensuitecalculer le champ diffus.


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Couplages lectromagntiques entre particules

Notre but tait de pouvoir identifier et interprter les phnomnes physiques lmen-taires qui surviennent quand il y a des couplages lectromagntiques entre des partic-ules. Cest pour cette raison que nous nous sommes restreints ltude dun systme deux particules identiques. Ltude a t ralise avec des particules similaires, pourplusieurs tailles, avec deux orientations des particules et deux polarisations incidentesdiffrentes et cela pour des particules dilectriques et parfaitement conductrices [21, 14].Ltude a t ralise avec lalgorithme rigoureux de la T-matrice.

Pour savoir si nous tions en diffusion simple ou multiple, cest dire pour savoirsil y avait interaction o pas entre les particules, nous avons tudi la rtrodiffusion

du couple de particules en fonction de la distance qui les sparait (figure 4a). Nousavons compar le rsultat avec ce que lon aurait eu en sommant les champs (modlecohrent) rtrodiffuss par chaque particule seule. Nous avons videment tenu comptedu dphasage qui pouvait venir de leurs positions diffrentes. Quand la distance qui s-pare les particules est grande compare au rayon des particules, lintensit rtrodiffuseest quivalente la somme (cohrente) des champs rtrodiffuss par les deux particules.Par contre quand les particules sont proches lune de lautre, par exemple quand leurscentres sont spars de quelques rayons, il apparat une diffrence. Cest ce cas, o il ya interaction, que nous avons voulu tudier afin de donner une interprtation physiquede linteraction. Comme nous lavons constat linterprtation physique est diffrenteselon lorientation des particules et la nature des particules.

Nous avons dabord considr le cas o londe incidente se propage dans le plan normal laxe portant les deux sphres. Nous avons compar le diagramme de diffusion verslavant et vers larrire du couple de sphres avec la figure de diffraction cre par deuxouvertures circulaires ayant le mme rayon que les sphres et tant spares par lamme distance qui spare les deux sphres (Trous dYoung [8]). Lavantage de cettecomparaison est quelle permet une interprtation physique du diagramme de diffusion.Quand les sphres sont suffisamment loignes lune de lautre pour quil ny ait pasdinteraction, les deux courbes (intensits en fonction de langle) correspondent bien(figure 4b). Le maximum dintensit correspond la fraction du flux incident qui a

interagit avec les particules ( la section efficace autrement dit). La distance entre deuxmaxima conscutifs est li au phnomne dinterfrence entre les ondes diffuses parchaque particule, cest dire dans notre cas la distance inter-billes. Enfin lenveloppeglobale reprsente le terme de diffraction, cest dire la gomtrie et la taille des struc-tures vues par londe. Dans le cas sans interaction, la diffraction est lie la sectiongomtrique dune particule.

Quest ce que le couplage entre particules va changer ? Pour rpondre cette ques-tion, nous avons refait la comparaison entre le diagramme de diffusion et la diffractionpar des trous dYoung, mais dans le cas o nous avions identifier quil y avait inter-

action. Notre conclusion est la mme pour les particules dilectriques et parfaitementconductrices. Le phnomne dinterfrence reste peu prs identique au cas sans in-


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teraction. La taille et la gomtrie de la structure vue par londe reste similaire lasection gomtrique dune particule. Le paramtre qui change le plus est celui de la

section efficace du couple de particules, cest dire la fraction de londe incidente quiinteragit avec les particules. Selon la distance qui spare les particules, elle peut treplus grande ou plus faible.

Nous nous sommes galement pos la question de linterprtation du couplage dans

0 10 20 30 40 502

2.5

3

3.5

4

4.5

5

5.5

kd

bn

Normalized backscattering cross section

kR=2, =Ei=Ey

1 0.5 0 0.5 10

5

10

15

20

25

sin()

Intensity

(1) Forward Scattering and (2) Young slits

(1)(2)

(a) (b)

Figure 4: (a) Section efficace de rtrodiffusion (b) Comparaison entre la diffraction pardes trous dYoung ( ) et la lumire diffuse (). Systme diffusant : 2 sphres, rayona: ka = 2, conductivit 2 =, distance entre le centre des 2 sphres d: kd = 45.Le vecteur donde k et le champ lectrique, linairement polaris, sont orthogonaux laxe des 2 particules.

le cas o les deux sphres sont dans laxe du vecteur donde incident. Dans ce caslinterprtation physique nest pas la mme selon que la particule est dilectrique ou par-

faitement conductrice. Pour les sphres parfaitement conductrices, un effet dombragese manifeste qui fait dcrotre la section efficace des deux sphres quand elles se rap-prochent (figure 5(a)). A linverse quand les deux sphres sont dilectriques, la sectionefficace des deux sphres aura tendance augmenter quand elles se rapprochent (figure5(a)). Dans ce cas, il y a un effet similaire un Fabry-Perot [8], lespace entre les deuxparticules se conduit comme une cavit avec des maxima de transmission.

Cette tude montre bien les couplages lectromagntiques qui apparaissent entre desparticules illumines quand celles-ci se rapprochent. Nous avons interprt physique-ment comment se manifeste ce couplage, mais pour mieux comprendre sa cause, nous

avons tudi la rpartition dintensit en champ proche, cest dire autour dune par-ticule.


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LIST OF TABLES

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

kd

bn


0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

kd

bn


(a) Sphres parfaitement conductrices (b) Sphres dilectriquesn2 = 1.5

Figure 5: Section efficace de rtrodiffusion en fonction de la distancedentre les sphres.Rayon a: ka = 2. Ici les deux sphres sont dans laxe du vecteur donde incident k.

Le jet photonique

En gnral, lorsquon tudie linteraction de la lumire avec des particules de taillecomparable la longueur donde, on ne considre que le champ diffus et lobservationest faite en champ lointain [3]. Pour mieux comprendre la cause des couplages entreparticules, nous avons tudi la rpartition dintensit autour dune particule dilec-trique en champ proche [15]. Dans ce cas, dans la thorie de Mie, le champ hors dela particule est la somme du champ diffus et du champ incident. Ce que nous avonscalcul et qui corrobore les simulations de Taflove effectues par FDTD sur des cylin-dres dilectriques [22], est quil pouvait y avoir une trs forte concentration dintensiten champ proche (figure 6). Cette forte concentration dintensit est appele un jetphotonique. Nous avons voulu mieux comprendre ses proprits et son origine.

Nos simulations montrent quune sphre dilectrique de quelques longueurs donde,selon son indice de rfraction peut concentrer (focaliser) lintensit dans la sphre (fortindice de rfraction) ou hors de la sphre (faible indice). Si on appelle point focal lelieu sur laxe optique o lintensit est maximale, alors ce point focal nobit pas auxlois habituelles de loptique gomtrique. Pour une sphre de taille donne, nous avonsmontr que le maximum dintensit tait atteint quand le point focal tombait juste la surface de la sphre. Si la sphre ne fait que quelque longueur donde, nous avonsmontr quil fallait utiliser un matriau ayant un indice de rfraction de lordre de 1.6.Si la particule est plus large, lindice ncessaire tendra vers 2.

La deuxime observation importante tait que ce jet photonique pouvait atteindrela limite de diffraction, cest dire atteindre une largeur totale mi hauteur dune


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Figure 6: Intensit du champ lectrique autour dune sphre de rayon R = 5 etdindice n2 = 1.3. La largeur totale mi-hauteur du faisceau est infrieure . Lecalcul a t fait avec un ordreLmax = 45et pour une onde incidente plane linairementpolarise Hi= Hy and ki = kz.

demi-longueur donde. Nous avons alors recherch dans quel cas la largeur du faisceaufocalis allait garder une largeur infrieure la longueur donde sur la plus longue dis-tance de propagation possible. Pour avoir un tel phnomne, nous avons montr quil

fallait focaliser, non plus juste la surface de la particule, mais quelques longueursdonde derrire (figure 6). Indpendamment de la taille de la particule, la longueurmaximale de propagation en dessous de la longueur donde est atteinte pour un indicede rfraction de lordre de 1.3.

Nous nous sommes galement intress la nature lectromagntique dun jet pho-tonique. Nous avons montr que sa polarisation (en champ proche) tait la mme quecelle de londe incidente et cela malgr que le champ dans une direction quelconqueproche de la particule nest plus rigoureusement transverse. En effet, en champ proche,la composante radiale du champ lectrique nest pas nulle dans toutes les directions.

Enfin nous avons voulu savoir pour quelle taille de particules le phnomne du jetphotonique pouvait avoir lieu. Il na pas lieu dans le rgime de Rayleigh (kR < ).Dans ce cas, quand la particule est trop petite compare la longueur donde, les max-ima dintensit ne sont plus sur laxe optique mais de part et dautre de la particule[23]. Si nous avons trouv une taille minimale, en revanche nous navons pas trouvde taille maximale. Un jet photonique avec les mmes proprits que celles dcritesci-dessus semble pouvoir tre ralis avec des particules de grandes tailles (R >>20).Ce qui semble montrer que lapparition dun jet photonique ne serait pas due la taillede la particule mais bien la proximit du point focal avec la surface de la particule.

Pour rsumer, ds quune particule dilectrique a un rayon suprieur ou de lordrede la longueur donde, il est possible en choisissant un indice de rfraction n2 qui per-


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mette de focaliser juste sa surface (1.6 < n2 < 2) ou bien juste derrire la sphre,davoir un jet photonique, cest dire un faisceau focalis, avec une largeur infrieure

la longueur donde, avec une trs forte intensit et une faible divergence. Cela permetmieux de comprendre pourquoi deux particules proches peuvent se comporter commeune petite cavit avec un effet Fabry-Perot. Mais le jet photonique seul pourrait aussitre utilis dans de nombreuses applications comme nous allons le voir.

Conclusion

Afin dtudier les couplages lectromagntiques en diffusion multiple et pour en obtenirune interprtation physique nous avons cod deux algorithmes rigoureux de rsolutiondes quations de Maxwell: la thorie de Mie et lalgorithme de la T-matrice.

Grce lalgorithme de la T-matrice nous avons pu mettre en vidence et inter-prter physiquement le couplage entre deux particules qui survient quand elles sonttrs proches [14]. Nous avons montr que quand les particules sont orthogonales ladirection de propagation, le couplage se traduisait majoritairement par un changementde la fraction de londe incidente qui tait diffuse. Les phnomnes de diffraction etdinterfrence taient trs faiblement affects par linteraction. Nous avons galementtudi le cas o les particules sont alignes dans la direction de propagation de londeincidente. Un phnomne dombrage, se traduisant par une baisse de la section efficacede diffusion est observ si les particules sont parfaitement conductrices. A linverse

un phnomne similaire aux pics de transmission dune cavit Fabry-Perot est observsous la forme dune augmentation de la section efficace de diffusion lors de linteractionde deux particules dilectriques.

Une tude complmentaire de lintensit en champ proche autour dune particule dilec-trique a permis de mettre en vidence la possibilit quelle avait de fortement concentrerlintensit. Ceci en plus de permettre une meilleure comprhension des couplages entreparticules proches, constitue un phnomne intressant en soi. Ce faisceau focalis,appel jet photonique, survient quand le point de focalisation est juste sur ou justederrire la surface de la sphre. Il atteint une trs grande intensit, peut rester avec

une largeur infrieure la longueur donde sur plusieurs longueurs donde de propaga-tion et a une faible divergence [15].

Ce jet photonique pourrait avoir plusieurs applications intressantes :

amlioration de lusinage laser [23], amplification de phnomnes non linaires optiques, augmentation des capacits des mmoires optiques (CD),

dtection de nano-particules [24],

amlioration de la rsolution en microcopie [25], SNOM par exemple.


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LIST OF TABLES

Lexploration de ces pistes semble prometteuse. La mise en vidence exprimentaledun jet photonique reste galement un dfi intressant. Indpendamment de cela

lamlioration des algorithmes dtude de la diffusion multiple sera une tape nces-saire pour simuler des agrgats plus gros, plus ralistes et pour permettre de mieuxcomprendre les couplages lectromagntiques dans les milieux htrognes.


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Chapter 1

Introduction

1.1 Introduction

In the Bible, if God created light the first day, that was probably to observe his cre-ation. Actually, light may be one of the best means of observing reality. Light is ableto go through a medium, to have various interactions according to the shape and theproperties of the matter. Often these interactions do not affect the observed object.The information propagates fast and ends up in our eyes or sensors. However our smallbrain must know the laws of interaction of light with matter to deduct informationabout this light. In order to know that we see an object before us, we must (uncon-

sciously) know that the surface of the object scatters light and behaves as secondarysources. We must know the law of refraction and reflection of the propagating light,etc. The physicist must understand the same kind of knowledge about light to beable to extract objective data from the reality. That is the reason for explaining theinterest for studying interaction between light and matter. In this work, we focus ourattention on the description of one of these interactions: light scattering. I will explainthis choice more in depth below.

Since the first laws of Lord Rayleigh in 1871 which deal with light scattering of parti-cles with a size small compared with the wavelength, significant improvement has beenmade for a better description of this phenomenon [3, 26]. We can cite, for example, theMie theory in 1908 [9, 27, 28], and before the work of Lorenz [29, 30], which describesrigorously light scattering by spherical particles. However, most of these developmentsonly deal with single scattering. That is, when each particle is supposed to scatter onlythe incident light independently of the other particles. This hypothesis is true when thedistance between particles is large compared to the wavelength. When light is scatteredby a dense cloud of particles, multiple scattering must be taken into account. Eachparticle scatters the incident light but also the light scattered by the other particles.The study of these electromagnetic interactions between the particles is more difficultto describe and often needs the use of a numerical resolution of Maxwell equation.

Several specific algorithms have been developed these two last decades (see section2.6) to describe such multiple scattering interactions. The need to understand multiple

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1.1. INTRODUCTION

scattering that occurs in many new applications can probably explain the increase ofinterest for this subject.

Yet another great change can explain the increase of light scattering interest. Fora long time, light scattering has been considered a cause of loss of information andpower during propagation. For example, the light scattered by cosmic particles ex-plains a part of the decrease of light collected by astrophysicists telescopes, what iscalled the extinction phenomenon [3]. Scattering can be considered too, as a reason fordecrease of polarization degree of a beam propagating in a medium. However it is nowunderstood that the optical properties of the scattered field can be used to learn in-formation on the media. The intensity scattered according to the direction (scatteringphase function) and the degree of linear polarization of the scattered field can be used

to find the density, the shape, the optical properties of particles or aggregates of par-ticles. Scattering phenomena are no longer a drawback but become a non-destructivemeans to investigate matter.

The need to understand multiple scattering effects and to solve inverse problems toobtain information about media, can probably explain the increased interest for lightscattering. As a result many new numerical methods have been developed to improvethe understanding of light scattering. Because of the size of particles, which can be com-pared to the wavelength, these methods are based on the rigorous Maxwells equations.Differential methods as Finite Element Method (FEM) [31][32] or Finite Difference

Time Domain Method (FDTD) [33, 16, 34] have been used in 2D. New methods basedon a physical description of matter have been developed such as the Discrete DipoleApproximation (DDA) [35, 36]. These methods allow the study of 3D problems, al-though only for small particles or aggregates. Actually only the integral methods, suchas the T-matrix algorithm [37], have shown to be capable of modelling light scatteredby large irregular particles or aggregates.

In this context, the main objective of this PhD work is to contribute to bring com-plementary physical explanations of some basic phenomena that occur in visible lightscattering for sub-micrometric particles, in particular in the near field region. The workis theoretical but has been carried out by considering experimental conditions. In single

scattering, we have analyzed the possibility with dielectric spheres to focus light untilthe diffraction limit and to obtain a high spatial concentration of energy in the nearfield, what is called a photonic jet. This possibility can have applications in microscopyor in laser processing, but it also illustrates the high electromagnetic coupling that canappear inside an aggregate of particles. That is the reason why we have also studiedin the far field the electromagnetic coupling between a couple of close particles. Ourobjectives were not only to study the influence of several optical parameters but togive physical interpretations of these effects. In particular the transition between sin-gle and multiple scattering has been described as interferences, diffraction and energycoupling. Such a knowledge may have applications for aggregate characterization by

light scattering. To carry out these theoretical studies, a T-matrix algorithm has beencoded and will be described in this report.


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1.2. LIGHT SCATTERING: SHORT HISTORY AND BACKGROUND

Before introducing our work, we would like to summarize the main historical steps

in the understanding of volume light scattering and corresponding applications.

1.2 Light scattering: short history and background

This section deals with the main steps in the understanding and applications of lightscattering in order to situate our work in a more general context.

-XIXth century : the birth of electromagnetismSeveral observations and famous experiments in the XVIIIth century led to the appari-tion of two different domains, the electrostatic and the magnetostatic. They respec-tively describe strength created by charges and magnets on other charges and magnets.Because of new experiments such as the experiment of Oersted, these two domains havebeen brought together by Maxwell [38] at the end of the XIX th century to create theelectromagnetism. The main historical contributions in electromagnetism are summa-rized in appendix A.1. The understanding of light scattering was mainly to be carriedout in the XXth century.

- 1871 The Rayleigh scatteringLord Rayleigh (figure 1.1) is one of the first to rigorously describe light scattering byparticles smaller than wavelength [7, 1]. By using symmetries and a dimensional study,

he deduced that the scattered intensity of such a particle is inversely proportional to4, where is the wavelength of light (see section 2.2). Thus he explained that thesmall water drops in the sky scatter more blue light than red, that is the reason whythe sky is blue (scattered light) in the day and red (direct light) immediately beforeand after the sunset.

Figure 1.1: Lord Rayleigh: J.W. Strutt 1842-1919 Cambridge [1]

-Physical colorsColors are often a consequence of absorption properties of material. However colors can


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also be a consequence of physical phenomena, for example the "prism effect". Scatter-ing is another example of physical colors [39]. We have yet pointed that the scattered

intensity in the Rayleigh case depends on the wavelength and that it explains why thesky is blue. The white color of ice and paper can also be explained by multiple scat-tering. Their components (crystal and chalk) are smaller than the visible wavelengths,thus all the colors are scattered and create together a white appearance. The colorof some butterflies (figure 1.2) can also be explained by scattering [2] and the color ofsome glasses are produced by including nano-size metallic objects in the glass.

Figure 1.2: Microscope view of the scattering structure on the wing of a butterfly calledpieris brasicae[2].

-1891 Lorenz and 1908 the Mie scatteringGustave Mie (1868-1957) used the analogy with sound to solve the propagation equa-tion of electromagnetic waves in a dielectric sphere [8, 9, 27, 28](see section 2.3). Asimilar work had been made before by Lorenz [29, 30]. The Mie theory makes thecalculation of light scattering possible for spheres larger than the wavelength. Thistheory is also called the Lorenz-Mie theory.

-Atmospheric applicationsDuring half a century, all scattering problems were described as the sum of light scat-tered by spheres. The effective shape of a particle and the electromagnetic interactionbetween particles were considered as negligible [3]. The main family of applicationswere the atmospheric ones. The effects (extinction, polarization change) of aerosols,

cloud or haze scattering about atmospheric optics [40] as radar and satellite commu-nications were studied. Light scattering was also used to observe atmosphere for themeteorology and pollutant quantification.

-1966 Waterman: arbitrary shape particlesThe conceptual contribution of Waterman [19] is very important because it lead theway to the future methods. By using an integral formulation and the vectorial Greenfunction, he described light scattering by a particle with an arbitrary shape (will be dis-cussed in chapter 3). The Green function describes an electromagnetic impulse source.The integral formulation is the vectorial analogue of the Huygens-Fresnel integral in

scalar theory. This new theory is considered as the birth of the T-matrix algorithm andis also called the Extended Boundary Condition Method (EBCM). At the beginning,this formulation was applied for spheroidal particles in order to be able to carry out


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analytical calculations. Recently because of advance of computers this method hasbeen applied for more arbitrary shapes (discus in section 2.6).

-1973 Peterson and Strm: scattering by an aggregate of particlesThe contribution of Peterson and Strm is the second great contribution in light scatter-ing description related to this work. The integral formulation of Waterman is extendedfor the scattering of several particles [20]. Thus dependent (that is with phase effects)and multiple scattering can be rigorously taken into account. Other algorithms havebeen developed to simplify computing by using assumption (see section 2.6). MorerecentlyMishchenko has contributed to the development of the T-matrix algorithmand of its use in several domains [37, 13].

-Astrophysical observations and particle characterizationLight scattering is of a great interest in astrophysics. Initially what has been under-stood is that light scattering was a reason for the decrease of the incident flux in ourtelescope. This extinction of light is due to absorption and light scattering by theinterstellar particles. Auguste Comte in the XIXth century considered the studies ofastrophysical object impossible because of their distance and cited them as examplesof the limits of science [41]. Electromagnetism has contradicted his point of view.

The scattering phenomena change the properties of the transmitted light. More re-cently physicists have understood that the study of these properties (scattering phase

function, polarization) was a means to study these interstellar particles (size, shape,density, etc.). Among the particles of interest are the stellar particles, the comets (sur-face and tail), the regolith (dust on the planet surfaces).

In order to have a better understanding the interaction of light with matter in micro-gravity, to describe planetary formations and to understand the light scattering prop-erties in function of the scattering aggregates, several projects have been carried out.PROGRA2 [4, 5][Web1] and the ICAPS (Interaction in Cosmic and Atmospheric Parti-cle Systems) project [6], where some experiments will be carried out in the InternationalSpace Station (ISS), are two of these projects.

- Dense media and biological applicationsBecause of the possibility to describe more realistic scattering phenomena, light scatter-ing is now used in new domains such as oceanography, colloidal chemistry, biophysics,among others.

One of the more recent aims is to describe light scattering by dense media of par-ticles or propagation of light in dense media of particles (powder, dust, heterogenicmedia). In this case, electromagnetic couplings between particles occur, there is mul-tiple scattering. Such descriptions of scattering by dense media often use radiativetransfer (see section 2.6) and need the calculation of the scattering phase function in

the material. To be computed, this last function may need assumption. In particularthe propagation of near infrared light in biological tissues [12] (they are slightly ab-


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1.3. NECESSITY OF UNDERSTANDING LIGHT-MATTER INTERACTIONS

sorbed) is a new cheap studied means, that could be used to make 3D reconstructionsof organs (optical tomography)[42].

1.3 Necessity of understanding light-matter interac-

tions

As we have seen in the previous part, light scattering can be used for particles char-acterizations. For example, polarization response are linked to the shape of scatterers.A comparison between polarization response of spherical and spheroidal particles ispresented in reference [43] and illustrates this possibility.

However, we must underline the limitation of such an inverse problem. The propertiesof the light scattered by an aggregate depend on a large number of parameters:

shape and dispersion of shape of particles, size and dispersion of size of particles, density, that is the number of particles per volume unit, structure of aggregates (fluffy, fractal, dense, etc.),

optical properties of particles (permittivity, absorption, isotropy),

quality of particle surfaces (roughness, buffing, etc.).

An inversion problem is possible if only some of the parameters are unknown and onlyif the dispersion of the characteristic of particles is low. The particle description ob-tained by inversion may have significant errors. Different kinds of scatterers can alsogive the same scattering phase function or polarization properties. That is the reasonwhy for a given optical response there is not necessarily only one solution to an inver-sion problem. These inversion algorithms use numerical methods, whose limitations donot permit sufficient modelling compared with the complexity of reality. Therefore the

better understanding of scattering that we would like to reach in this work, can alsohave interest to develop better inversion algorithms.

To understand the interaction between matter and light, we need:

new more accurate scattering measurements, a better characterization of the scattered particles (microscopy), development of more powerful algorithms (able to describe interactions between

larger aggregates or media),

a better physical comprehension of the phenomena.


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1.4. OBJECTIVE AND OUTLINE OF THE DISSERTATION

The objective of this work is to contribute to this last point, in particular by under-standing what happens in the near field region, just around the particle. What is

the intensity map around a single dielectric particle? What are the electromagneticcouplings between particles close compared to the wavelength?

1.4 Objective and outline of the dissertation

In this dissertation, first the fundamentals of light scattering phenomena will be out-lined. After that, different methods used to model light scattering by particles will bediscussed. Then, in chapter 3, we will describe the electromagnetic approach that wehave used to model the interaction of visible light with close sub-micrometric particles.In the context of this work only a small number of particles, and particles with a sizesmaller or comparable with a few wavelengths will be considered. The wavelengths areassumed to be in the visible light. Most of the considered particles will be dielectricand without absorption. The case of perfectly conductive sphere, or of dielectric spherewith absorption, will only be studied in few cases. The physical measurable values willalso be described.

Our studies have been carried out with a T-matrix algorithm and the Lorenz-Mietheory that have been programmed during the thesis with Matlab (programs will beaccessible in Internet). The principles and limitations of this algorithm will also bepresented in chapter 3. Then two main sets of results will be presented. The first

result described in chapter 4 (and published in [15]) is the possibility with a simplesphere, that scatters in the far field, to focus in the near field in a point smaller thanthe wavelength and to highly concentrate energy the near field. This focused beamin the near field has been called a photonic jet. This new observation can open theway to applications but also points out the particular phenomena that can occur in-side an aggregate of particles. The second main result, described in chapter 5 (andpresented in [14]): is about a proposed physical interpretation of the electromagneticcoupling between two spheres. If a single dielectric sphere is able to concentrate energyin the near field, importance of electromagnetic couplings between close particles canbe understood. The chapter 6 will summarize our main results and outline the future

problems and potential research directions.


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Chapter 2

Light scattering

After giving a general definition of what light scattering is, here we will discuss the fun-damentals equations and properties induced with light scattering phenomena. Scatter-ing regimes will be described depending on size and refractive index, for single particleand aggregates. Then a-state-of-the-art in measurement techniques and rigorous elec-tromagnetic solvers for light scattering, will be presented. The transition from singleto multiple scattering will also be discussed. More detailed technical aspect will be leftto the next chapter.

Light scattering is the general concept to point out light interaction with matter. Firstwe must distinguish several particular cases that are also included in light scattering:

We discuss refraction and reflection when the boundaries between materialsare plane or with smooth variation on a large scale compared to the wavelength.The material must be homogeneous and able to be described by the macroscopicconcept of refractive index. In this case the Snell-Descartes laws (geometric op-tics) and reflective coefficient of Fresnel can be used to describe light path.

We discussdiffraction, when we consider the optical effect of a spatial change inthe optical property of a material, if this change has a size of the same order of thewavelength and if this change has a regular shape or is periodically repeated. Inthis case the electric field must be considered as a wave: Fourier optics, gratingsetc. [44, 45, 46].

We discuss scattering when we consider small, irregular spatial variations ofthe optical properties of the matter (permittivity, shape). When these variationscannot be easily described (random media, roughness, inhomogeneities, etc.). Asrepresented in figure 2.1 the three previous regimes are included in light scatter-ing phenomena.

We discussquantum opticwhen we must take into account the quantification ofthe energy exchanges that occur during light material interactions. That is when

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the incident flux contains only few photons. (Laser is not necessary consideredquantum optics).

Figure 2.1: Refraction, reflection and diffraction in scattering phenomena.

Diffraction is included in light scattering phenomena, however the difference betweenscattering and diffraction is not always clear. For example the interaction betweenlight and one spherical particle or aligned spherical particle may simply be consideredas diffraction, whereas the interaction of light with randomly positioned spheres is lightscattering. In the case of random spheres, if there is independent and single scattering,the laws will be very similar to the case of light scattering by one particle. The onlydifference will be the use of a function describing the density of particles. In thesetwo cases (random and regular structures) the Lorenz-Mie theory can be used to de-scribe the phenomenon, what illustrates similarities between diffraction and scatteringin some cases.

In this work, the considered medium are random with irregular change of optical prop-erties. This change is assumed to be small compared to the wavelength but large enough

compared to molecular size. The flux are assumed to be enough to neglect quantumeffects. It is why we consider the word "scattering" as being more appropriate for theapproach discuss here.

Usually surface and volume scattering are separately studied. Surface scattering dealswith surface quality and roughness [47, 48], whereas volume scattering describe thelight scattering by particles or inhomogeneities [13]. A lot of works have already beenreport on surface scattering [47, 48], but in this thesis we will deal only with volumescattering.


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2.1. SINGLE AND INDEPENDENT SCATTERING

2.1 Single and independent scattering

We must also distinguish dependent and independent scattering, that is when phaseeffects can occur or not:

Independent scattering: if the scattering particles are far from each other andif the incident wave is slightly coherent, no phase effect will appear (no interferencebetween the wave scattered by each particle). The global scattered intensity is the sumof the intensities scattered by each particle.As approximative criterium, it can be considered that independent scattering occurswhen the distance between particles is larger than 3 times de radius of one particle (forfew wavelengths radius particles) [3, 21].

Light scattering by the clouds (Drops of1mm, density : 1drop/cm3

) is an example.Another case can justify independent scattering when the incident field is coherent.There is independent scattering with a coherent incident wave, if the speed of motionv of particles are larger compared to the integration time t of the quadratic sensor(/v > t with the wavelength); for example, a gas of particles [7].

Dependent scattering: if the scattering particles are close together, scattered wavesinterfere [3]. The global scattered intensity must be computed as the mean square ofthe electric field. Dependent scattering can also partially occur for an incoherent in-cident wave. This is the case in the opposition effect (in backscattering, see section 2.4).

The scattering laws, as the electromagnetic ones, do not depend on the size of the parti-cle but of the ratioa/wherea is the characteristic size of the particle and the wave-length. That is the reason why one talks about the size parameter x= ka= 2a/(k is the wave vector). We will use these notations in the following. We first considerparticles that scatter independently of the other particles. Depending on the refrac-tive index and of the size parameter of these particles, different approximations can bemade. Each of these approximations correspond to a particular family of scattering.These families are represented in figure 2.2.

For single independent scattering, we distinguish [3]:Rayleigh scattering (1871): the particle is very small compared to the wavelength(a

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2.2. RAYLEIGH SCATTERING

Figure 2.2: Several different scattering regimes as functions of R and n [3].

2.2 Rayleigh scattering

The Rayleigh scattering is historically the first quantitative description of light scatter-ing phenomena. Its success came from its ability to justify why the sky appears blue.This model assumes single scattering and incoherent sum of intensities in the far field.

We will discuss Rayleigh scattering below for the cases of polarized and unpolarizedincident field.

Basic characteristics[3]Consider particles with radiusa, which is small compared with the wavelength (a

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2.2. RAYLEIGH SCATTERING

Now if we put a small particle in an unitary incident plane wave, we would like toknow what is the ratio of the incident flux that is scattered? To find the answer, we

study the normalized scattered cross section. The scattered cross section multipliedby the incident flux yields the total scattered power by the particle (definition). Thiscross section can be normalized by dividing by a2 (see section also 3.7). Figure 2.5shows that the part of flux that will be scattered does not depend only on the effectivesection a2 of the particle (if it was the case the normalized cross section would beequal to 1) but falls when the radius of the particle decreases.

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Intensity along x

Phase function

Intensity along y

Intensity

along

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Figure 2.3: (a)I(, ) for one sphere a= 0.1, n= 1.52, Hi = Hy, k = kz (angles indegree). (b) Spherical coordinates.

Qualitative demonstrationTo understand the phenomenon of light scattering by small particles, we will first usethe same qualitative reasoning given by Lord Rayleigh in 1871 [7].One assumed that because of the particles motions, no phase effect (interferences) aretaken into account (Note: it is not true in the forward and backward direction, seetheopposition effectin section 2.4). Therefore the total scattered intensity is the sumof the intensity scattered by each particle. Because of the very small size of particlescompared to the

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