SURFACE WAVE SCATTERING FROM METALLIC NANO PARTICLES: THEORETICAL FRAMEWORK AND NUMERICAL ANALYSIS

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1 Uiversity of Ketucky UKowledge Uiversity of Ketucky Master's Theses Graduate School 006 SURFACE WAVE SCATTERING FROM METALLIC NANO PARTICLES: THEORETICAL FRAMEWORK AND NUMERICAL ANALYSIS Pradeep Kumar Garudadri Vekata Uiversity of Ketucky, Click here to let us kow how access to this documet beefits you. Recommeded Citatio Vekata, Pradeep Kumar Garudadri, "SURFACE WAVE SCATTERING FROM METALLIC NANO PARTICLES: THEORETICAL FRAMEWORK AND NUMERICAL ANALYSIS" (006). Uiversity of Ketucky Master's Theses This Thesis is brought to you for free ad ope access by the Graduate School at UKowledge. It has bee accepted for iclusio i Uiversity of Ketucky Master's Theses by a authorized admiistrator of UKowledge. For more iformatio, please cotact UKowledge@lsv.uky.edu.

2 ABSTRACT OF THESIS SURFACE WAVE SCATTERING FROM METALLIC NANO PARTICLES: THEORETICAL FRAMEWORK AND NUMERICAL ANALYSIS Recet advaces i ao techology have opeed doors to several ext geeratio devices ad sesors. Characterizig ao particles ad structures i a simple ad effective way is imperative for moitorig ad detectig processes at ao scale i a variety of eviromets. I recet years, the problem of studyig ao particle iteractios with surface plasmos or evaescet waves has gaied sigificat iterest. Here, a umerical model is preseted to characterize ao-size particles ad agglomerates ear a metal or a dielectric iterface. The methodology is based o a hybrid method, where the T-matrix approach is coupled with the image theory. The far field scatterig patters of sigle particles ad agglomerates subected to surface plasmos/evaescet waves are obtaied. The approach utilizes the vector spherical harmoics for the icidet ad scattered fields relatig them through a T-matrix. Effects of size, shape ad orietatio of the cluster o their scatterig patters are studied. A effort is made to distiguish particle characteristics from the scatterig iformatio obtaied at certai observatio agles. Uderstadig these scatterig patters is critical for the desig of sesors usig the surface plasmo scatterig techique to moitor ao self assembly processes. KEYWORDS: Particle Characterizatio, Surface Plasmos, Evaescet Waves, T- matrix, Nao Particle Agglomerates. Pradeep Kumar Garudadri Vekata /4/ Copyright Pradeep Kumar Garudadri Vekata 006.

3 SURFACE WAVE SCATTERING PATTERNS FROM METALLIC NANO PARTICLES: THEORETICAL FRAMEWORK AND NUMERICAL ANALYSIS By Pradeep Kumar Garudadri Vekata Dr. M. Piar Megüç Director of Thesis Dr. Mustafa M. Asla Co-Director of Thesis Dr. George P. Huag Director of Graduate Studies 01/4/006 Date

4 RULES FOR THE USE OF THESES Upublished thesis submitted for the Master s degree ad deposited i the Uiversity of Ketucky Library are as a rule ope for ispectio, but are to be used oly with dueregard to the rights of the authors. Bibliographical refereces may be oted, but quotatios or summaries of parts may be published oly with the permissio of the author, ad with the usual scholarly ackowledgemets. Extesive copyig or publicatio of the thesis i whole or i part also requires the coset of the Dea of the Graduate School of the Uiversity of Ketucky.

5 THESIS Pradeep Kumar Garudadri Vekata The Graduate School Uiversity of Ketucky 006.

6 SURFACE WAVE SCATTERING FROM METALLIC NANO PARTICLES: THEORETICAL FRAMEWORK AND NUMERICAL ANALYSIS THESIS A thesis submitted i partial fulfillmet of the requiremets for the degree of Master of Sciece i Mechaical Egieerig i the College of Egieerig at the Uiversity of Ketucky. By Pradeep Kumar Garudadri Vekata Director: Dr. M. Piar Megüç, Professor of Mechaical Egieerig Co-Director: Dr. Mustafa M. Asla, Assistat Professor of Mechaical Egieerig Lexigto, Ketucky 006.

7 MASTER S THESIS RELEASE I authorize the Uiversity of Ketucky Libraries to reproduce this thesis i whole or i part for the purpose of research. Siged: Pradeep Kumar Garudadri Vekata Date: 01/4/006

8 Ackowledgemets I would like to express my gratitude to my advisor Dr. M. Piar Megüç, whose costat support ad ecouragemet made this work possible. He was always supportive of my efforts ad was a source of ispiratio. His able guidace ad expertise helped immesely i shapig up this work. I leart a lot from him both academically ad persoally. I would like to express my gratefuless to my co-advisor Dr. Mustafa M. Asla without whose cotributios, this work would ot have bee possible. I would like to specially thak him for the cosiderable amout of time he spet with me despite his busy schedule. He was always available for discussios ad his valuable suggestios were very helpful to this work. I would also like to thak Dr. Gorde Videe ad Dr. Daiel Mackowski for makig available their computer programs to me to use i this work. Dr. Videe was istrumetal i several discussios about this work. I thak Dr. Todd Hastigs for takig time to serve o my committee. He was very helpful with my efforts with Femlab. I would like to thak my family for ecouragig me to take up higher educatio. Their support ad blessigs made everythig possible for me. Fially, I thak God for showerig his blessigs up o me. iii

9 Table of Cotets Ackowledgemets..iii List of Figures..vii Chapter 1. Itroductio Backgroud ad Motivatio Electromagetic Wave-matter Iteractios Plae-wave Propagatio Surface Waves Fresel Equatios Evaescet Waves Surface Plasmos Thesis Orgaizatio Chapter. Particle Characterizatio Literature Search Scatterig ad Absorptio By Small Particles Sigle, Multiple ad Depedet Scatterig Stokes Parameters ad the Amplitude Scatterig Matrix Other Methods Available for Modelig Scatterig By Small Particles Aalytical Approaches Differetial Approaches... a. Fiite Elemet Method b. Fiite Differece Time Domai Method... c. Poit Matchig Method Itegral Equatio Methods... 3 a. Method of Momets 4 b. Discrete Dipole Approximatio.4 iv

10 c. Fredholm Itegral Equatio Method.4 d. Superpositio Method T-matrix Method (TMM) Direct ad Iverse Problems... 6 Chapter 3. Light Scatterig From Particles o Substrates Previous Work Computatioal Models Double Iteractio Model Extictio Theorem Image Theory Ray Tracig Model Experimetal Methods Ellipsometry Surface Plasmo Reflectace Surface Plasmo Scatterig Chapter 4. T-matrix Formulatio for Particles o Substrates T-matrix Formulatio For Sigle Sphere o a Substrate T-matrix Method For Sphere Agglomerates: Evaluatio of the Code Chapter 5. Scatterig Patters of Nao-sized Particles o Substrates Gold Properties Idepedet Scatterig From Sigle ad Multiple Gold Particles: Depedet Scatterig From Gold Nao Particles Scatterig Patters For Silver Particles o Substrates v

11 Chapter 6. Coclusios ad Future Work Coclusios Future work Refereces 91 Vita 94 vi

12 List of Figures Figure 1.1 : Electromagetic spectrum 3 Figure 1. : Reflectio ad refractio at a iterface 7 Figure 1.3 : Depictio of a evaescet wave field 9 Figure 1.4 : Total iteral reflectio for a icidet TE mode wave 10 Figure 1.5 : Formatio of a surface wave 11 Figure.1 : Schematic of scatterig from a obstacle 16 Figure 3.1 : A schematic of double iteractio model 30 Figure 3. : A schematic of ellipsometry techique 33 Figure 3.3 : A schematic of surface plasmo reflectace method 34 Figure 3.4 : A schematic of surface plasmo scatterig method 35 Figure 4.1 : Geometry of the problem 38 Figure 4. : Hybrid method 41 Figure 4.3 : The schematic of the scatterig from a group of particles o ( h = 0 ) or ear ( h > 0 ) a thi film 4 Figure 4.4 : Compariso results for Lorez-Mie code ad the preset work 49 Figure 4.5 : Comparisos for a 0 m diameter particle o surface 51 Figure 4.6 : Comparisos for 0 m diameter particle placed at a height of 50 m from the surface 5 Figure 4.7 : Comparisos for 50 m diameter particle o surface 53 Figure 5.1 : Spectral variatio of gold refractive idex (+ik) 55 Figure 5. : Spectral variatio of critical agle at glass-gold iterface 56 Figure 5.3 : Schematic of idepedet scatterig from small particles o or above a surface 57 Figure 5.4 : Scatterig profiles from particles of differet diameters o surface 58 Figure 5.5 : Model I-a 59 Figure 5.6 : Model I-b 59 vii

13 Figure 5.7 : Scatterig profiles for cofiguratio show i Model I-a o a gold substrate (Top) ad o a quartz substrate (Bottom) 60 Figure 5.8 : Scatterig profiles for cofiguratio show i Model I-b o a gold substrate (Top) ad o a quartz substrate (Bottom) 61 Figure 5.9 : Model II 6 Figure 5.10 : Scatterig profiles for two particle cofiguratio (Model II) 6 Figure 5.11 : Model III-a-Three particles o surface 63 Figure 5.1 : Scatterig profiles for 10 m diameter particles o surface 64 Figure 5.13 : M 11 profile for 0 m diameter particles o surface 65 Figure 5.14 : M 11 profile for 50 m diameter particles o surface 65 Figure 5.15 : Three particles placed i z-directio (Model III-b) 66 Figure 5.16 : M 11 results for three 0 m particles placed o the surface i Model III-b (Top) ad III-a (bottom) arragemets 67 Figure 5.17 : M 1 results for 0 m particles o the surface placed i Model III-b (Top) ad III-a (Bottom) arragemets 68 Figure 5.18 : M 33 results for 0 m particles o the surface placed i Model III-b (Top) ad III-a (Bottom) arragemets 69 Figure 5.19 : M 34 results for 0 m particles o the surface placed i Model III-b (Top) ad III-a (Bottom) arragemets 70 Figure 5.0 : M 11 profiles for Model III-b, 0 m diameter moomer (Top), 50 m diameter moomer (Bottom) 71 Figure 5.1 : Spherical particles arraged i the form of squares 73 viii

14 Figure 5. : Scatterig profiles for particle cofiguratios IV-a, IV-b, IV-c ad IV-d 74 Figure 5.3 : Spherical particles arraged i triagular shapes (Models V-a, V-b) 75 Figure 5.4 : Scatterig profiles for cofiguratios i Models V-a ad V-b 76 Figure 5.5 : Spectral variatio of silver refractive idex 77 Figure 5.6 : Comparisos for gold ad silver particles o a surface 78 Figure 5.7 : Two 0 m diameter silver particles placed o the surface 79 Figure 5.8 : Scatterig profiles for silver particles i Models III-a ad III-b 80 Figure 5.9 : M 11 profiles for cofiguratio i Model III-a, gold particles (top) ad silver particles (bottom) 81 Figure 5.30 : M 1 profiles for gold (top) ad silver (bottom) particle cofiguratios preseted i Model III-a 8 Figure 5.31 : M 33 profiles for gold (top) ad silver (bottom) particle cofiguratios preseted i Model-III a 83 Figure 5.3 : M 34 profiles for gold (top) ad silver (bottom) particle cofiguratios preseted i Model-III a 84 Figure 5.33 : M 11 cotour plots for gold (top) ad silver (bottom) particle cofiguratios show i Model III-b 85 ix

15 CHAPTER Itroductio 1.1. Backgroud ad Motivatio Scatterig from small particles has bee of iterest for a log time for may scietists ad egieers workig i diverse fields like astroomy, solid state physics, chemistry, bio-physics, egieerig ad others. The primary reaso for this iterest has bee to characterize micro/ao particles based o their light scatterig profiles. Characterizig particles ad structures at ao scale is imperative for uderstadig ad cotrollig self assembly processes i bottom up approach of buildig structures at ao scale. Availability of AFM, TEM ad the latest Scaig Near field Optical Microscopes (SNOM) ad Scaig Tuelig Microscopy (STM) has made it easier to a extet, to observe ad maipulate ao structures ad particles. However, these techiques are very expesive ad caot be readily used for o-lie moitorig applicatios. O the other had, sice ao particles are much smaller tha the wavelegth of light ad other commoly used radiatio i the electromagetic spectrum, traditioal electromagetic wave scatterig approaches caot be utilized easily. There is a defiite 1

16 eed to develop a better uderstadig of the iteractio betwee ao-size structures o or ear surfaces with light/em-waves ad to characterize such structures from their scatterig profiles. Scatterig of light by micro-sized particles has bee of sigificat iterest i several areas of egieerig ad applied scieces because of their impact o the effective ad apparet properties of media they are imbedded i, as well as for diagosis of their size, shape ad structures. Research o scatterig from small particles (size parameter x= π r/ λ 1) was iitiated by Lord Rayleigh [1], where he solved for scattered electromagetic field from a small sphere to explai the color of sky. Several studies o scatterig ad absorptio behaviors of small particles have bee ad are still beig coducted to uderstad may iterestig pheomea (e.g. atmospheric research o dust particles, astroomical studies o iterstellar dust ad radiatio, optical diagostics for idustrial aerosol processes, combustio research ad detectio of soot particles ad precursors, as well as selfassembly of ao-particles []). The radiative formulatio for sphere clusters usig superpositio method was formulated idepedetly by Bruig ad Lo [3] ad Borghese et al. [4]. A alterate, yet equivalet, approach usig the itegral formulatio of Maxwell s equatios to sphere clusters was developed by Iskader et al. [5]. These approaches are primarily used to study the radiative properties of soot ad aerosol particles. Scatterig from systems composed of a sphere or a dipole o or ear a iterface were explored by Bobbert ad Vlieger [6], Nahm ad Wolfe [7], ad Videe [8]. Recetly, surface plasmos/evaescet waves are beig explored to characterize aosize particles ear surface, which has gaied a great amout of attetio i the recet times [9, 10]. This recet research directio is based o the T-matrix methodology to characterize ao particles ad agglomerates subected to surface plasmos/evaescet waves i couctio with a custom built experimetal system. The overall goal of this work is to perform umerical simulatios for various cases of particle, agglomerate sizes, orietatio ad surface-particle separatio, ad compare some of them to the experimetal results. The preset thesis focuses o the theoretical challeges associated with this research program.

17 Reproductio of the experimetal results has bee made through umerical modelig of the problem for sigle particles earlier. The work has bee exteded for spherical agglomerates ad particles, ad the umerical results are preseted here. Durig the rest of this Chapter, we will first itroduce the fudametal cocepts, defiitios, ad equatios ecessary to follow the EM-wave-particle iteractios. This overview is ot extesive, as ay textbook i Optics or Physics may have more detailed discussio of the subect. 1.. Electromagetic Wave-matter Iteractios Maxwell s work o electromagetism proved that light is a electromagetic wave [11].The electromagetic spectrum is depicted i Figure 1.1. Electromagetic Spectrum Gamma rays X rays Ultraviolet rays Ifrared rays Radar FM TV Short wave Wavelegth i meters Visible Spectrum Wavelegth i m Figure 1.1: Electromagetic spectrum. Maxwell s equatios i S.I uits are expressed, from [11] as: 3

18 id = ρ F ib = 0 B E + = 0 t D H = JF + t The equatios are give for ay iterior poit i the media. E is the electric field ad B is the magetic iductio. ρ F is the charge desity, J F is the curret desity i the (1.1) medium. Eve whe J F is zero i a medium (as i air), there ca be iduced magetic field because of the term D. This observace is oe of the greatest cotributios of t Maxwell. The electric displacemet D ad magetic field H are defied by D = ε E+ P 0 B H = M µ 0 (1.) where P is the electric polarizatio (average electric dipole momet per uit volume) ad M the magetizatio (average magetic dipole momet per uit volume). The quadruple ad higher momets are egligible compared to the dipole momet. It is clear from Maxwell s equatios that the electric ad magetic fields are depedet o each other ad are mutually perpedicular. A time varyig E -field geerates a B -field i space ad a time varyig B -field geerates a E -field i the directio of chage of the B - field. A motioless charge has a electric field that stretches to ifiity. Whe the charge is accelerated, the field distributio i the viciity of the charge chages. This alteratio propagates out ito space at a fiite speed. This pulse is called a electromagetic wave that is geerated by a accelerated charge which oce geerated moves beyod its source ad idepedetly of it. The frequecy of a electromagetic wave depeds o the frequecy of the acceleratig charge ad the amplitude depeds o the acceleratio. Uless ecoutered by a free or a partially boud charge that ca iteract with a electromagetic wave, the wave propagates udisturbed 4

19 idefiitely. The speed of all electromagetic disturbaces is a costat ad equal to c= x10 8 m/s i vacuum or air [ Plae-wave Propagatio Ay wave ca be expressed as a combiatio of time-harmoic fields. A plae electromagetic wave ca also be expressed as the sum of sie ad cosie fuctios. iωt Sice e = cos( ωt) isi( ωt), the combiatios of sie ad cosie fuctios are expressed i terms of expoetial complex umbers for simplicity of calculatios. Yet, the fial results are iterpreted oly by cosiderig the real part of the calculatios. A time harmoicity of i t e ω is assumed so as to have a positive complex part of refractive idex for a metal. Oly electromagetic fields that satisfy Maxwell s equatio are physically realizable. Suppose we cosider plae waves of the form E = E exp( ikx iωt), (1.3) 0 where k is the wave vector give by k = π / λ. The wave is compatible with Maxwell s equatios oly whe k, E0 ad H 0 are mutually perpedicular to each other ad also whe the followig dispersio equatio is satisfied k k = ω εµ (1.4). Note that, k specifies the directio of propagatio ad E0 ad H0 are always perpedicular to the directio of propagatio. From equatio (1.4), oe ca coclude that ~ ~ ω N k = c N = c εµ = εµ ε µ 0 0 (1.5) where ~ N is the complex refractive idex of a medium, geerally expressed as ~ N=+ik. 5

20 1.3. Surface Waves The speed of a plae electromagetic wave is a fuctio of idex of refractio of medium i which the wave propagates. If the refractive idex of the material is high, the speed of the wave is reduced. The relatio betwee the speed ad refractive idex is give by c = c/ (1.6) where c is the speed i vacuum ad c is the speed i a medium that has a refractive idex of. Accordig to this equatio, if a electromagetic wave ecouters a medium of ifiite refractive idex, its velocity approaches to zero.i.e. it comes to a rest. Such a pheomeo happes whe a electromagetic wave ecouters a metallic surface, peetrates to some extet, ad at the surface where the ecouter takes place, it yields a stadig wave called surface plasmo polaritos. This wave occurs primarily due to the oscillatio of free charges o the surface ad propagates o the surface. A similar effect happes whe total iteral reflectio occurs at a dielectric iterface. A wave setup o the surface of a dielectric iterface is called a evaescet wave. Such waves die dow rapidly at a short distace (withi m) above the surface. These stadig waves are ot observable uless they are tueled via some kid of a probe or a medium i the viciity of the waves Fresel Equatios Whe a plae wave ecouters a iterface, it udergoes both reflectio ad refractio as show i the Figure 1. below (for a trasverse magetic (TM) wave icidet at a agleθ i ). The trasverse magetic (TM) mode is othig but a mode of propagatio i which the magetic field oscillates perpedicular to the plae of observatio. Similarly, a trasverse electric (TE) mode is a propagatio mode i which the electric field oscillates perpedicular to the plae of observatio. Fresel equatios give the reflectio ad trasmissio coefficiets which are the ratios of the reflected ad trasmitted electric fields, respectively, with icidet electric field. The Fresel coefficiets for the icidet TM wave are give by [11] r TM Er t cosθi icosθ t = ( ) TM = E cosθ + cosθ i i t t i (1.7) 6

21 t TM Et icosθi = ( ) TM = E cosθ + cosθ i i t t i (1.8) Fresel coefficiets for the icidet TE wave are r t TE TE Er icosθi t cosθt = ( ) TE = E cosθ + cosθ i i i t t Et icosθi = ( ) TE = E cosθ + cosθ i i i t t (1.9) (1.10) y E i E r i B i k i θ i θ r B r k r x t Figure 1.: Reflectio ad refractio at a iterface Evaescet Waves Whe a wave passes from a higher refractive idex medium to aother with lower refractive idex, it udergoes total iteral reflectio. There is supposed to be o trasmissio of the icidet field beyod a certai critical agle. However it becomes impossible to satisfy the boudary coditios at the iterface without the cosideratio of a trasmitted wave. If a trasmitted wave is assumed at the iterface, E ik ( t. r ωt) t = E0te (1.11) where kr. = kx+ ky(see Figure 1..) t tx ty 7

22 k = k siθ (1.1) tx t t k = k cosθ. (1.13) ty t t However, si θ k = k = k (1.14) i 1/ t cos θt t(1 si θt) t(1 ) ( t / i) For the case of a total iteral reflectio, si θ > ( / ) (1.15) hece, i t i si θ =± = β (1.16) i 1/ kt cos θ ikt( 1) i ( t / i) k tx kt = siθi (1.17) ( / ) t i This yields a expressio for the trasmitted field as E = E e e t 0t kx t siθi i( ω t) β y ( t / i) (1.18) This trasmitted field decays expoetially at the iterface i the lesser dese medium. This is a evaescet wave or the boudary wave whose eergy oscillates at the iterface of the two media. A schematic of the evaescet waves is show i the Figure

23 Evaescet wave field a i a r Total Iteral Reflectio Figure 1.3: Depictio of a evaescet wave field. Simulatios usig Comsol Multiphysics 3. show the pheomeo of total iteral reflectio ad the formatio of surface waves at a quartz-air iterface (here, we oly preset sample results for the visualizatio of surface wave cocept). Fig. 1.4 depicts the total iteral reflectio occurrig i a triagular quartz slab. Fig. 1.5 shows the electric field from a TE mode wave over the surface. We observe that the wave decays dow over few hudred ao meters idicatig a surface wave. It should be oted that depedig o the idex of refractio of at a give wavelegth, this decay ca be differet (See eq. 1.18). 9

24 Figure 1.4: Total iteral reflectio for a icidet TE mode wave. 10

25 Figure 1.5: Formatio of a surface wave Surface Plasmos Whe a electromagetic wave ecouters a metallic iterface, a surface wave is geerated at the iterface provided there is a tagetial compoet of the wave vector at the iterface. This results i strog scatterig, ehaced electromagetic fields ad surface plasmo absorptio bads characteristic of the type of material at the iterface. [11] provides a mechaical oscillator aalogy to explai the electromagetic field iteractios with the metal. Accordigly, i metals, availability of free electros results i the displacemet of electro cloud from the uclei whe icidet up o by a electric field. The displaced electro cloud is restored by a restorig force give by F = mω x (1.19) res e 0 for small displacemets of x. The force exerted o a electro of charge qe by a icidet electric field E ic is F cos ext = qeeic = qee0 ωt (1.0) From Newto s secod law, 11

26 d x ext res e 0cosω eω0 e F + F = q E t m x= m (1.1) dt The solutio for this equatio is give by ( q / m ) x() t = E ( ω ) e e 0 ω ic (1.) ad the dispersio relatio ca be calculated makig use of the electric polarizatio relatios. For large values of frequecy the dispersio relatio is ~ Nqe N ( ω) = 1 ε m ω (1.3) 0 e where N is the umber of electros i the system, is the real part of the complex refractive idex. The free electros ad the positive ios i metals are equivaletly thought to be plasma Nqe with a plasma frequecy of ω = p ε m ω. So we have ~ ω N ( ω) 1 ( ) ω 0 e p = (1.4) For icidet frequecies less tha the plasma frequecy, a complex refractive idex results ad the peetratig wave drops off expoetially i magitude. For icidet frequecy greater tha the plasma frequecy, the metal has a real refractive idex ad is trasparet to the icidet field. For frequecies equal to the plasma frequecy, resoace occurs resultig i strog absorptio bads ad selective reflectios. The dispersio relatio for SPPs is give by K sp ω ( ω) = c ε m( ωε ) s ε ( ω) + ε m s (1.5) where K sp is the wave-vector of SPP, ω is the agular frequecy of the SPP, ε m is the dielectric costat of the metal ad ε s is the dielectric costat of the surroudig medium. A SPP is associated with a evaescet wave at a dielectric iterface, travelig at a speed lower tha that of the evaescet wave [1] ad ca be geerated by meas of a evaescet wave, which i tur ca be geerated by total iteral reflectio or by ay other subwavelegth structure. 1

27 1.4. Thesis Orgaizatio This thesis is orgaized ito six Chapters. The first Chapter preseted the motivatio for the work ad a itroductio to the fudametals ad defiitios of electromagetic waves, Fresel equatios ad surface waves. The secod chapter summarizes previous work o particle characterizatio. A brief itroductio to various theoretical ad computatioal methods available for spherical ad o-spherical particle characterizatio is also provided. The third Chapter presets a literature search to characterizatio of particles o substrates, which requires iteractio of surface waves with particles ad structures. Brief details of the available experimetal methodologies are also preseted i Chapter three. The fourth Chapter presets the formulatio ad the solutio methodology adapted usig a T-matrix method. Details of the umerical modelig of the problem are also preseted alog with a evaluatio of the code that represets the problem formulatio. Chapter five presets umerical results for ao particles ad D agglomerates o the surface. Numerical results for gold ad silver ao particles o gold substrate, ad some comparisos for gold ad silver particles are also preseted i this chapter. Fially, Chapter six summarizes the work completed ad presets a list of future tasks to be accomplished. 13

28 CHAPTER Particle Characterizatio.1. Literature Search The problem of absorptio ad scatterig of a plae wave by a homogeous sphere was first studied by Lorez (1891) ad the idepedetly by Mie (1908). This exact solutio to this problem of scatterig by small particles is kow as the Lorez-Mie theory. The solutio is a elegat mathematical formulatio of the problem i which the icidet, iteral ad scattered fields are expaded i terms of vector spherical harmoics. The scattered fields are obtaied from the icidet ad iteral fields by makig use of the Bessel fuctios ad the Legedre polyomials. Though the solutio is preseted for spherical particles, it acts as a guide to describe first order optical effects i o spherical small particles also. The computer algorithms for the Lorez-Mie theory are well documeted [13]. Several extesios ad geeralizatios of the method are available to treat ihomogeeous spherically symmetric ad axially symmetric ospherical particles. There are also several methods available for o-spherical ad 14

29 clusters of o-iteractig particles, all of which are based o solvig Maxwell s equatios either aalytically or umerically [14]. With the advet of high speed computers, several umerical techiques were developed ad solved for agglomerates of particles of a variety of shapes. The availability of solutios to the scatterig problem led researchers to characterize particles based o light, for soot particles [15], fie particles [16], cotto fibers [17], idustrial aerosol particles ad soot agglomerates [18]. Characterizatio of ao particles ad structures has become crucial to uderstad ad study ao self assembly processes ad bottom up buildig processes. The problem has iterestig applicatios i characterizig micro structures o flat substrates ad developmet of surface scaers [19]. Most of the previous work o particle characterizatio is cocetrated o particles embedded i a dielectric medium rather tha o particles iteractig with a iterface. This chapter summarizes the previous importat work o light scatterig by particles ad particle clusters, without the presece of ay iterferece... Scatterig ad Absorptio by Small Particles Scatterig ad absorptio of light by small particles is a very importat tool for visualizatio ad diagostic applicatios. Scatterig of electromagetic wave occurs because of the presece of ay heterogeeity i a give medium. Every system is composed of discrete charges amely electros ad protos. Whe these charges come i cotact with the electromagetic waves, they udergo oscillatios i tue with the icidet field. Fractioal eergy of these excited charges is reradiated as electromagetic eergy, which is scattered i all directios ad some of it is lost as thermal eergy, which cotributes to the iteral eergy of the system, called absorptio. Each atom or molecule i a particle acts like a dipole to the icomig radiatio with a slight polarizatio of positive ad egative charges. The dipoles oscillate with the frequecy of the icidet radiatio ad are capable of scatterig secodary wavelets. The scattered field by each dipole, whe superimposed i a particular directio gives the scatterig i that directio. I geeral, scatterig by these dipoles i a give directio is i phase, i.e. coheret, which meas that there is o phase differece i the scattered wavelets of the each dipole i that 15

30 give directio. Such a scatterig is oly true for a smaller particle. As the particle size icreases, the possibilities for other iteractios icrease givig rise to strog variatios i the scattered patter. A cartoo of the scatterig pheomeo is depicted i Fig..1. Icidet Field Scattered Field Obstacle Figure.1: Schematic of scatterig from a obstacle...1. Sigle, Multiple ad Depedet Scatterig Sigle scatterig by a particle occurs whe the icidet field o the particle acts i isolatio with other particles. I this case, oe particle is separated by a large distace from aother particle i such a way that the scattered field from oe particle is small ad does ot ifluece the field icidet o aother particle. Multiple scatterig is a case whe the scattered field from oe particle adds to the total electric field i such a way that the icidet field o ay other particle is ot i isolatio from oe aother. Particles are sufficietly away from each other i such a way that the scattered field from oe particle is icidet upo aother particle as a plae wave. Scatterig by clouds is a case of multiple scatterig where water droplets are large i umber ad scatter sufficietly [13]. 16

31 Depedet scatterig is a case whe the scatterig by each particle iflueces the particles surroudig it i a direct way. It is a case i which particles are sufficietly ear to each other or touch each other i which case, the scattered field by a particle is icidet o aother particle as a spherical wave ad ot as a plae wave. So each particle is i tur exposed directly to the origial icidet field ad the scattered field from particles surroudig it..3. Stokes Parameters ad the Amplitude Scatterig Matrix The electromagetic field iside the particle ad at all poits outside the particle ca be obtaied by makig use of Maxwell s equatio ad applyig suitable boudary coditios.. The geeral Maxwell s equatios for a plae wave i vector form are give by: E = 0 H = 0 E = iωµ H (.1) H = iωε E Where E ad H are the electric ad magetic fields respectively. The boudary coditios imposed at the particle-medium iterface are [ E ( s) E ( s)] = 0 1 [ H ( s) H ( s)] = 0 1 ^ ^ (.) From equatios (1.3.1) we ca derive the vector wave equatios for electric ad magetic fields which are ad E+ ωεµ E = 0 (.3) H + ωεµ H = 0 (.4) Except i the rectagular coordiate system, E ad H do ot satisfy scalar wave equatio. The poit s lies o the surface of the particle. Usig Maxwell s equatios ad the boudary coditios (.1), the relatio betwee the icidet ad the scattered fields ca be obtaied i spherical co-ordiate system as 17

32 ETMs ad ETEs E ik ( r z) TMs e 3 TMi = ETEs ikr S4 S4 ETEi S S E are the parallel (TM) ad trasverse (TE) compoets of the scattered (.5) electric field respectively. The subscript i deotes the icidet compoet of the field ad s for the scattered electric field. k is the wave vector for the wave ad r is the radius vector. We see that the scattered ad icidet fields are related by a simple matrix called the amplitude scatterig matrix owig to the liearity of Maxwell s equatios. The elemets S i (i=1,,3,4) are the elemets of the scatterig amplitude matrix which are geerally depedet o the scatterig agle ad the azimuthal agle. The electric field or for that matter ay other property of a wave is ot measurable by the available experimetal systems due to the rapid oscillatios of these properties, as their frequecy(ω ) is i the order of for visible wavelegth spectrum. So, we eed a set of properties characteristic of a wave that ca be measured by a commo detector. A detector ca measure itesity, i.e. Poytig vector or the square of the amplitude of the electric field. Stokes defied a set of parameters that are related to itesity ad polarizatio of a wave which serve as a equivalet descriptio of a polarized wave. Such a set of parameters are measurable by a experimetal device. Below, we itroduce a series of defiitios for clarificatios of aalysis to be preseted later. 1. The irradiace recorded by a detector whe icidet upo by a upolarized beam of light is defied as I = E E + E E *( k/ ωµ ) (.6) * * s TMs TMs TEs TEs. Whe a beam of light passes through a horizotal polarizer (a polarizer that filters the perpedicular or the TE compoet of the wave ad oly allows the T.M compoet to pass through), the irradiace detected is: I = E E *( k/ ωµ ) * TM TM TM Whe a beam passes through a vertical polarizer (a polarizer that filters out the T.M compoet ad oly allows the T.E compoet of the wave to pass through), the irradiace detected is

33 I = E E *( k/ ωµ ). * TE TE TE 0 The differece betwee these two irradiace measured by the detector is I = E E E E *( k/ ωµ ) (.7) * * TMs TMs TEs TEs 3. If a beam passes through a polarizer (a polarizer which filters out the compoet of all electric field oscillatios i the agle of the plae), ad a polarizer (a polarizer which filters out the compoet of all electric field oscillatios i the agle of the plae), the differece of irradiaces measured by a detector from the two cases is I = E E + E E *( k/ ωµ ) (.8) * * TMs TEs TEs TMs 4. If a beam passes through a right haded polarizer (a polarizer that oly trasmits electric field oscillatios rotatig i aticlockwise directio whe viewed towards the source), ad a left haded polarizer (a polarizer that oly trasmits electric field oscillatios rotatig i the clockwise directio whe viewed towards the source), the differece of irradiaces measured by a detector from the two cases would be I = i E E E E *( k/ ωµ ) (.9) * * TMs TEs TEs TMs Equatios (.6)-(.9) represet the four ellipsometric parameters that represet a plae wave. By droppig the ( k / ωµ 0) factor that is commo to all the four equatios, we have the Stokes parameters give by I = E E + E E * * s TMs TMs TEs TEs Q = E E E E * * s TMs TMs TEs TEs U = E E + E E * * s TMs TEs TEs TMs V = i E E E E * * s TMs TEs TEs TMs (.10) The superscript * deotes the complex cougate of a parameter. The Stokes parameters defied by equatio (.10) are represeted i a colum vector called the Stokes vector. The scattered ad icidet Stokes vectors are related by a matrix that follows from the amplitude scatterig matrix as: 19

34 I S S S S I s i Q s 1 S1 S S3 S 4 Qi = U s kr S31 S3 S33 S 34 U i V S s 41 S4 S43 S 44 V i (.7) The 4x4 matrix cosistig of S i (i=1,,3,4 ad =1,,3,4) elemets is called the scatterig matrix whose elemets are obtaied from the elemets of amplitude scatterig matrix (i eq. (.5)). This matrix is also called the Mueller matrix whe the scatterig is by a sigle particle. For a sigle homogeous sphere, the scatterig matrix reduces to S S S S S33 S 34 S34 S33 with oly four sigificat terms S 11, S 1, S 33, S 34, which ca be sufficiet eough to eve characterize agglomerates ad particles of differet shapes [0]. Stokes vector ad the scatterig matrix are the most importat parameters required for obtaiig the complete scatterig profile of a particle. The scatterig matrix elemets are defied i terms of the scatterig amplitude matrix (eq..5) as 1 11 = ( ) S S S S S 1 1 = ( ) S S S S S S = Re( S S + S S ) * * S = Im( S S S S ) * * = ( ) S S S S S 1 = ( ) S S S S S S = Re( S S S S ) * * S = Im( S S + S S ) * * S = Re( S S + S S ) * *

35 S = Re( S S S S ) * * S = Re( S S + S S ) * * S = Im( S S + S S ) * * S = Im( S S + S S ) * * S = Im( S S S S ) * * S = Im( S S S S ) * * S = Re( S S S S ) * * (.8).4. Other Methods Available for Modelig Scatterig By Small Particles I additio to the Lorez-Mie theory, several other theories ad umerical techiques have bee developed over the years to study the scatterig pheomea by particles. The aalytical solutio cosists of usig the separatio of variables techique to solve vector Helmholtz equatio for time-harmoic electric fields. Maxwell s equatios are solved by a combiatio of aalytical ad umerical methods. The book by Mishcheko, Hoveier ad Travis [14] provides a excellet guide to the previous work o light scatterig by o spherical particles Aalytical Approaches a. Separatio of Variable Method This method cosists of expadig the icidet, iteral ad scattered fields i vector spherical harmoics but i spherical coordiate system. The icidet field expasio coefficiets are calculated aalytically, where as the scattered field, iteral field expasio coefficiets are calculated from icidet field coefficiets ad boudary coditios. Because of the mathematical complexities ivolved with vector spherical harmoics, the method ca oly be applied to particles with semi-maor-axis size π d parameter ( ) less tha 40. The method was pioeered by Oguchi [1] ad Asao ad λ 1

36 Yamamoto [] for sigle homogeous ad isotropic particles. Later, it was further modified by Voshchiikov ad Farafoov [3]. The method was further exteded to a esemble of spheroids by Schulz et al. [4] ad was used primarily for spheroids. The advatage of the approach is its accuracy whereas its disadvatages iclude mathematical complexity, o applicability to large particles ad particles with large refractive idices..4.. Differetial Approaches a. Fiite Elemet Methods (FEM) This method cosists of solvig the vector Helmholtz differetial equatio umerically, subect to boudary coditios at a particle s surface. A fiite computatioal domai is selected, which is discretized ito small volume elemets with about 10 to 0 elemets per wavelegth i which, the scatterer is embedded. Usig the boudary coditios, the differetial equatio is coverted to a matrix form, which the ca be solved usig Gaussia elimiatio or cougate gradiet method. A modificatio of the method called the uimomet method by Morga et al. [5] uses a spherical computatioal domai with the scattered field expaded usig the spherical wave fuctios outside the computatioal domai. The most importat advatage of the method is the simplicity i modelig ihomogeeous ad arbitrary shaped particles where as its disadvatages iclude icreased computatioal time ad limitatios i size parameters. Aother disadvatage is that sice the domai here is limited i size, it does ot accout for the far field zoe ad so absorbig boudary coditios have to be eforced at the outer boudary to take care of wave reflectios back ito the domai. b. Fiite Differece Time Domai (FDTD) Method This method solves the time-depedet Maxwell s equatios to calculate electromagetic scatterig i both time ad space domais. Maxwell s equatios cotai space ad time derivatives of electric ad magetic fields. These derivatives are approximated by a fiite differece scheme ad discretized i both space ad time domais. The equatios are solved for i the discretized domai umerically with appropriate boudary coditios ad particle properties. A fully explicit scheme is

37 geerally used. This method also computes the solutio i a fiite domai, like the FEM, ad so a far zoe trasformatio has to be ivoked to calculate fields i far field. The method is popular because of its coceptual simplicity ad ease of implemetatio but has similar disadvatages as that of the FEM icludig limitatios i accuracy, mathematical complexity ad the eed to repeat computatios for differet agles of icidece. c. Poit Matchig Method (PMM) The poit matchig method is a differetial equatio techique cosistig of expadig icidet ad iteral fields i vector spherical wave fuctios regular at origi. The icidet field expasio coefficiets are calculated aalytically ad the iteral ad scattered fields are calculated by a boudary matchig techique at a selected umber of poits by makig use of the appropriate boudary coditios. The method makes use of Rayleigh hypothesis, which dictates that for particles much smaller tha the wavelegth of icidet light (whe the size parameter << 1), the icidet field ear the particle ad iside it behaves as electrostatic field ad the field iside is homogeous. A modificatio of the method called the geeralized poit matchig method makes use of boudary matchig at may more poits tha required ad forms a over determied system of equatios i order to be able to accurately determie the outgoig spherical waves for the scattered field outside the particle boudary. The method is umerically stable ad accurate for rotatioally symmetric particles ad ca also be applied to very large size particles Itegral Equatio Methods The method cosists of expressig the exteral field to a obect i terms of a itegral equatio. The equatio cotais a iteral field term that is ukow. The equatio represets the iteral field at each poit as a sum of the icidet field ad the field iduced by all other iterior poits. The same equatio ca be used to obtai iteral field by ay umerical method that uses discretizatio, usig a explicit scheme. Oce the iteral field is obtaied, the exteral field is calculated with the help of the itegral equatio. There are several modificatios available for the method called with differet ames such as discrete dipole approximatio, digitized Gree s fuctio by Goedecke 3

38 ad O Brie [6], the volume itegral equatio formulatio by Iskader et al. [7]. The advatage of the method is that it has few ukows ad ca be applied to aisotropic ad ihomogeeous scatterers. The disadvatages iclude low computatioal accuracy ad risig computatioal time with icreasig size parameters. a. Method of Momets (MOM) This is oe of the versios of the itegral equatio methods [8]. The iteral field of the itegral equatio is calculated by discretizig the iterior regio ito elemets with about 0 elemets per wave legth. A assumptio of homogeous field iside the volume of the scatterer is made. With the availability of the iteral field, the exteral field is foud out from the itegral equatio. The scattered field ca be obtaied by subtractig the icidet field from the exteral field. b. Discrete Dipole Approximatio (DDA) The method is based o dividig the idividual particle ito a large umber of polarizable uits called dipoles. The respose of a dipole to a icidet electromagetic field is kow. The icidet field o each dipole is the sum of origial icidet field ad the scattered field by all other dipoles. The problem reduces to obtaiig liear equatios ad fidig a umerical solutio. The method was origially derived by Purcell ad Peypacker i 1973 [9]. From the o the method uderwet may improvemets icludig a additio of magetic dipole terms. The importat advatage of this method is its applicability to particles of ay shape or property, ihomogeeous ad aisotropic. The disadvatages are that it is time cosumig, has slow covergece, has limited accuracy ad eeds computatios to be repeated for each agle of icidece. c. Fredholm Itegral Equatio Method The method is similar to itegral equatio methods i the sese that a modified itegral equatio i which the volume itegral is coverted to the wave umber coordiate space ad a Fourier trasform is applied to the iteral field. The itegral is discretized ito a matrix equatio which is solved umerically usig a explicit scheme to obtai the scattered field. This method is more stable tha those usig the itegral equatio method ad has good covergece results for particles with large aspect ratios 4

39 as well. The implemetatio of the umerical scheme becomes simple for homogeeous ad symmetric particles. The mai disadvatage is that the matrix elemets have to be evaluated aalytically, which requires a differet program. Hece the implemetatio is limited to oly a few models of particles. d. Superpositio Method The separatio of variables techique for spheres ca be exteded to a cluster of spheres ad spheroids. By usig the traslatioal additio theorem for vector spherical wave fuctios (VSWF), the field illumiatig the cluster ad the fields scattered by each sphere i the cluster are expaded i VSWFs at idividual sphere ceters ad at oe sphere at secod origi [30]. The procedure gives rise to a matrix equatio for scattered field expasio coefficiets for each sphere. The umerical solutio to the equatio gives the idividual scattered fields for each sphere for a give icidet field. The superpositio of all the idividual scattered fields gives the total scattered field. The field icidet o each sphere is the sum of exteral field ad the idividual scattered fields which get reduced to a system of umerical equatios. This method s computatioal complexity depeds o the idividual sphere size parameter ad the umber of spheres i the cluster. The method is very efficiet for liear chai of spheres. For a cluster of spheres, the method produces a matrix at a fixed origi i the sphere cluster that ca trasform the icidet field ito scattered field which is a equivalet of a T-matrix. Other advatage is that it ca produce very accurate results T-matrix Method (TMM) The method was first itroduced by Waterma i 1971 [31]. Of all the methods, the T-matrix method (TMM) has the advatage of beig highly accurate, fast ad applicable to all kids of particles. The method cosists of expadig the icidet field i vector spherical wave fuctios (VWSF) regular at origi ad the scattered field i VWSFs, regular at ifiity i the medium outside the particle. The icidet field expasio coefficiets ca be calculated aalytically. A particle respose matrix called the T-matrix trasforms the icidet field expasio coefficiets to the scattered field coefficiets. The T-matrix depeds oly o the particle shape, size, refractive idex ad the orietatio of the particle with respect to a referece frame ad is idepedet of the 5

40 icidet ad scattered fields. This makes the method highly advatageous because oce the T-matrix is kow for a particle or a cofiguratio of particles, the scattered field ca be calculated for ay icidet field, for ay directio. The T-matrix has spherical waves as basis fuctios ad is usually computed usig the exteded boudary coditio method. The method was geeralized to multilayer scatterers ad arbitrary clusters of o spherical particles by Peterso ad Strom i 1973 [3]. The method ca be applied to particles of ay shape ad size although the computatios become quite complex for particles that have o symmetry. The computatio of T-matrix for a cluster makes use of the traslatioal additio theorems for VWSFs ad uses the T-matrices for all idividual spheres. Mackowski & Mishcheko [33], Wriedt ad Doicu [34] exteded the method to a cluster of spheres ad o spherical particles. The aalytical orietatio averagig method was developed by Mishcheko [35] for radomly orieted, rotatioally symmetric particles. The procedure was exteded to a cluster of spheres by Mackowski ad Mishcheko i 1996 [36]. Because of its high umerical accuracy, the T-matrix method has gaied more iterest i the past few years. The oly drawback of the method is that it produces low efficiecy results for particles with large aspect ratios or with particle shapes lackig axial symmetry..5. Direct ad Iverse Problems The problem of particle characterizatio usig electromagetic waves ca be carried out as solutio of the direct ad the iverse scatterig problems i sequetial way. The direct problem cosists of obtaiig the scatterig profiles for a particle or a system iteractig with electromagetic waves. Here, the particle compositio, shape, size ad other characteristics are kow before had alog with the properties of the icidet light like the wavelegth, polarizatio ad the agle of icidece. Much work has bee ad is beig doe i this area to obtai scatterig profiles from several complicated systems, fairly accurately usig umerical modelig schemes, as discussed above. The iverse problem cosists of obtaiig particle characteristics like the shape, size ad compositio by examiig the scatterig profiles of the particle. This problem though ot impossible is ot early as easy as the direct problem because the amout of 6

41 data required, like the phase, vector amplitude of the wave i all directios ad the field iside the particle, is ot always available. However, by careful costructio of the experimetal setup ad by cotrollig the icidet wavelegth, it is possible to get sufficiet data to iterpolate the rest of the data. This iformatio about scatterig amplitudes, by compariso with the already available scatterig profiles from the previous umerical modelig work, ca be used to determie the particle characteristics. Ellipsometry is oe such techique that ca be applied to obtai sesitive iformatio about the polarizatio ad itesity of scattered light from a system of particles. Elliptically polarized radiatio was successfully used i a Ellipsometry device [37] to obtai scatterig matrix elemets to characterize soot particles by Megüç ad Maickavasagam [18]. 7

42 CHAPTER Light Scatterig From Particles o Substrates I Chapter, we have outlied the cocepts ad models related to scatterig of electromagetic waves by particles suspeded i a medium. The premise of the approaches discussed is that a plaar wave is icidet o a particle which, is placed far away from other particles. This cocept of sigle scatterig ca be expaded to multiple scatterig media, such as clouds or aerosols, if the particles are sufficietly far from each other, such that the scattered wave frot oe particle appears plaar by the time it arrives o the ext oe. The, the problem is simply a additive oe. O the other had, if the particles are close to each other or touchig as i the case of agglomerates, the the scattered wave caot be cosidered plaar upo arrival to ext oe. This case, which is called as depedet scatterig, requires a more careful treatmet, as the superpositio of scattered waves ad phase propagatio make the problem more complicated. The applicatio of those approaches to ao-scale particles is straightforward, as the scalig relatioship ca be carried o with the size parameter, x = pd/l. Yet, if such ao-sized particles are o a substrate, the problem becomes differet 8