Using Multiwavelength Spectroscopy. Alicia C. Garcia-Lopez

Transcription

1 Hybid Modl fo Chaactization of Submicon Paticls Using Multiwavlngth Spctoscopy by Alicia C. Gacia-Lopz A disstation submittd in patial fulfillmnt of th quimnts fo th dg of Docto of Philosophy Dpatmnt of Elctical Engining Collg of Engining Univsity of South Floida Majo Pofsso: Athu David Snid, Ph.D. Wilfido Mono, Ph.D. Knnth Buckl, Ph.D. Stanly Dans, Ph.D. Osca D. Cisall, Ph.D. Dat of Appoval: Mach 9, 5 Kywods: Paticl Analysis, Sphical Paticls, Non-sphical Paticls, In suspnsion Paticls, Rayligh-Dby-Gans Copyight 5, Alicia C. Gacia-Lopz

2 Ddication I would lik to ddicat this disstation to my young both Rodigo and my family.

3 Acknowldgmnts I would lik to acknowldg Pof. L. H. Gacia-Rubio fo th pmission to us his idas fo this sach pojct, th vast amount of consultation tim, and his xptis in th aa of paticl chaactization. I would also lik to acknowldg Pof. A. D. Snid fo giving m a wondful gaduat xpinc in pusuing my Ph.D. I hav gatly njoyd woking togth and you guidanc and mathmatics xptis hav gatly nhancd my lif. Finally, I would lik to acknowldg th PERC locatd at th Univsity of Floida fo thi financial suppot.

4 Tabl of Contnts List of Tabls List of Figus List of Symbols Abstact iii iv ix xii Chapt On: Intoduction and Mthods. Intoduction. Matials and Mthods.3 Ovviw of Chapts 3 Chapt Two: Scatting Thois and Modls 4. Backgound 6. Gnal Concpts and Equations 7.3 Mi Thoy and Modl.4 Rayligh-Dby-Gans Thoy and Modl 3.5 Hybid Thoy and Modl 5.6 Mthods Rviw 6 Chapt Th: Instumntation Coction Modl fo Tansmission 3. Instumntation Coction Modl 3. Implmntation of Instumnt Coctions 3 Chapt Fou: Nonsphical Paticls 6 4. Gomty and Notation fo Ellipsoids 6 4. Ellipsoid Simulations 3 Chapt Fiv: Validation Study of Rayligh-Dby-Gans Thoy Exploation of Thotical Limits Paticl Diamt << Wavlngth Paticl Diamt ~ Wavlngth, No Absoption Paticl Diamt ~ Wavlngth, Absoption κ > Conclusion 45 Chapt Six: Coctions to th Rfactiv Indx Rfactiv Indx 46 i

5 6. Hypochomic Effct Implmntation of Optical Coction fo Absoption Effctiv Rfactiv Indx Estimation Conclusion 54 Chapt Svn: Nw Hybid Thoy Gomty and Notation Intnal Fild Dipol Scatting Appoach Hybid Thoy Dtmining th Tansmission Scatting Intnsity Ratio and Tubidity 66 Chapt Eight: Validation and Snsitivity of Hybid Thoy Validation of Hybid Thoy Implmntation Cas : Rlativ Rfactiv indx n n ~ and Absoption κ = Cas : Rlativ Rfactiv indx n n and Absoption κ > Cas 3: Rlativ Rfactiv indx n n ~ and Absoption κ > Conclusions 8 Chapt Nin: Contibutions and Futu Wok Contibutions Rcommndations and Futu Wok 84 Rfncs 85 Appndics 87 Appndix A: Intnsity Ratio and Tubidity Modl 88 Appndix B: Optical Poptis 9 Appndix C: Validation fo Rayligh-Dby-Gans Thoy 98 C. Validation of Rayligh-Dby-Gans Thoy 98 Appndix D: Estimation of Absoption Cofficint and Hypochomism Modl D. Hypochomism Modl 4 About th Autho End Pag ii

6 List of Tabls Tabl. Simulation Paamts 3 Tabl 4. Tabl of Fom Factos 9 iii

7 List of Figus Figu. Diagam fo Complt Paticl Chaactization Modl 7 Figu. Diagam of Coodinat Systm usd in th Mi and RDG Modls Figu 3. Tansmission Systm (a) Opn Dtcto (b)pinhol Dtcto Figu 3. Calculatd Tansmission of Rayligh-Dby-Gans fo µm Hmoglobin Sphs with Pinhol Dtcto Stup 5 Figu 3.3 Calculatd Tansmission of Rayligh-Dby-Gans fo µm Hmoglobin Sphs with Opn Dtcto Stup 5 Figu 4. Light Scatting in Laboatoy Fam 7 Figu 4. Light Scatting in Paticl Fam 7 Figu 4.3 Fixd Ointations A, B, and C fo an Ellipsoid 3 Figu 4.4 Figu 4.5 Figu 5. Figu 5. Figu 5.3 Calculatd Tansmission of Soft Body Polat Ellipsoid ε=.3 with a Volum Equivalnt to a µm Sph 33 Calculatd Tansmission of Soft Body Polat Ellipsoid ε=.8 with a Volum Equivalnt to a µm Sph 33 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm AgB Sphs 36 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm AgCl Sphs 36 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm AgB Sphs 37 iv

8 Figu 5.4 Figu 5.5 Figu 5.6 Figu 5.7 Figu 5.8 Figu 5.9 Figu 5. Figu 5. Figu 5. Figu 5.3 Figu 6. Figu 6. Figu 6.3 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm AgCl Sphs 37 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm Soft Body Sphs 38 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of µm Soft Body Sphs 39 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5.5 µm Soft Body Sphs 39 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm Hmoglobin Sphs with κ= 4 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of µm Hmoglobin Sphs with κ= 4 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5.5 µm Hmoglobin Sphs with κ= 4 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of nm Hmoglobin Sphs 43 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm Hmoglobin Sphs 44 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of µm Hmoglobin Sphs 44 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo µm Hmoglobin Sph with Vshkin Coction 5 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo µm Hmoglobin Sph with % and % Hypochomocity 5 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo µm Hmoglobin Sph with Vshkin Coction to k c and an Effctiv n ff Calculatd though Kam-Konig Tansfom 5 v

9 Figu 6.4 Zoom in of Figu 6.4 of Calculatd Tansmission of Rayligh-Dby-Gans fo µm Hmoglobin Sph with Vshkin Coction to k c and an Effctiv n Calculatd though Kam-Konig Tansfom 5 Figu 7. Diagam of Scatt Point and Dtcto Location 55 Figu 7. Local Unit Vctos with Rspct to th Dtcto 56 Figu 7.3 Local Unit Vctos with Rspct to th Scatt 57 Figu 7.4 Diagam of th Volum at Hight z fo a Sph 64 Figu 8. Cofficints c,d, and d Vsus λ at th Limit whn k =k 68 Figu 8. Fom Facto f fo Hybid Thoy at k =k 69 Figu 8.3 Fom Facto f fo Rayligh-Dby-Gans Thoy 7 Figu 8.4 Figu 8.5 Figu 8.6 Figu 8.7 Figu 8.8 Figu 8.9 Figu 8. Figu 8. Ral Pat of Fom Facto f fo Hybid Thoy using Polystyn 7 Imaginay Pat of Fom Facto f fo Hybid Thoy using Polystyn 7 Ral Pat of Fom Facto f fo Hybid Thoy using Polystyn 7 Imaginay Pat of Fom Facto f fo Hybid Thoy using Polystyn 7 Fom Facto f fo Rayligh-Dby-Gans Thoy using Polystyn 7 Compaision of Calculatd Tansmission fo 5 nm Soft Body Sphs using RDG, Mi, and Hybid Thois 74 Compaision of Calculatd Tansmission fo nm Soft Body Sphs using RDG, Mi, and Hybid Thois 74 Compaision of Calculatd Tansmission fo 5 nm Soft Body Sphs using RDG, Mi, and Hybid Thois 75 vi

10 Figu 8. Figu 8.3 Figu 8.4 Figu 8.5 Figu 8.6 Figu 8.7 Figu 8.8 Figu 8.9 Figu 8. Compaision of Calculatd Tansmission fo 5 nm Soft Body Sphs using RDG, Mi, and Hybid Thois 75 Compaision of Calculatd Tansmission fo 5 nm Polystyn Sphs using RDG, Mi, and Hybid Thois 77 Compaision of Calculatd Tansmission fo nm Polystyn Sphs using RDG, Mi, and Hybid Thois 78 Compaision of Calculatd Tansmission fo 5 nm Polystyn Sphs using RDG, Mi, and Hybid Thois 78 Compaision of Calculatd Tansmission fo 5 nm Polystyn Sphs using RDG, Mi, and Hybid Thois 79 Compaision of Calculatd Tansmission fo nm Hmoglobin Sphs using RDG, Mi, and Hybid Thois 8 Compaision of Calculatd Tansmission fo 5 nm Hmoglobin Sphs using RDG, Mi, and Hybid Thois 8 Compaision of Calculatd Tansmission fo 5 nm Hmoglobin Sphs using RDG, Mi, and Hybid Thois 8 Compaison of Calculatd Tansmission fo 5 nm Hmoglobin Sphs using RDG, Mi, and Hybid Thois 8 Figu B. Optical Poptis fo Wat 9 Figu B. Optical Poptis fo Soft Body 9 Figu B.3 Optical Poptis fo Hmoglobin 93 Figu B.4 Optical Poptis fo Polystyn 93 Figu B.5 Optical Poptis of AgCl 94 Figu B.6 Optical Poptis of AgB 94 Figu B.7 Rlativ Rfactiv Indx of Soft Body in Wat 95 Figu B.8 Rlativ Rfactiv Indx of Hmoglobin in Wat 95 Figu B.9 Rlativ Rfactiv Indx of Polystyn in Wat 96 vii

11 Figu B. Rlativ Rfactiv Indx AgCl in Wat 96 Figu B. Rlativ Rfactiv Indx of AgB in Wat 97 Figu C. Figu C. Figu C.3 Figu C.4 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm Polystyn Sphs 99 Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm Polystyn Sphs Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of nm Polystyn Sphs Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm Polystyn Sphs Figu D. Stack Aangmnt of Chomophos along Chain Axis 4 viii

12 List of Symbols α β γ ε εˆ ε m κ c θ λ λ o π n ϕ τ τ n φ ψ ω χ Angl of dictional cosin Angl of dictional cosin Angl of dictional cosin Extinction cofficint fo a singl chomopho Avag xtinction cofficint fo a singl chomopho Mola xtinction cofficint fo a singl chomopho Coction imaginay pat of th factiv indx Angl btwn scatting and incidnt bam Wavlngth Wavlngth in vacuo Angl dpndnt function Solid angl Tansmission/Tubidity Angl dpndnt function Rotational Eul angl aound Z axis Rotational Eul angl aound Y axis Rotational Eul angl aound nw Z axis Siz paamt ix

13 a a n b n C abs C xt C sca D E Ê E & f G h h n I o I s k k j n l M w m Radius Mi cofficint Mi cofficint Absoption coss sction Extinction coss sction Scatting coss sction Paticl diamt Extinction cofficint Avag xtinction cofficint Calculatd Extinction cofficint Fom facto Coss sctional aa Hypochomicity Hankl functions Incidnt intnsity Scatting intnsity Wav numb Quantity of chomophos Bssl functions Pathlngth of sampl Molcula wight Rlativ factiv indx x

14 N N N A N p n P P n Q abs Q xt Q sca Complx factiv indx of paticl Complx factiv indx of mdium Avogado s numb Numb of paticls Ral pat of th factiv indx of paticl Pobability of absoption by a photon Lgnd polynomial Absoption fficincy facto Extinction fficincy facto Scatting fficincy facto Q xt Appant xtinction fficincy facto Q sca Appant scatting fficincy facto R Rˆ S s V vf Coction facto Path avagd coction facto Distanc btwn Middl of Sampl and Dtcto Scatting Amplitud Function Effctiv Gomtic Aa of Chomopho Volum of Paticl Volum faction of xi

15 Hybid Modl fo Chaactization of Submicon Paticls Using Multiwavlngth Spctoscopy Alicia C. Gacia-Lopz ABSTRACT Th aa of paticl chaactization is xpansiv; it contains many tchnologis and mthods of analysis. Light spctoscopy tchniqus yild infomation on th joint popty distibution of paticls, compising th chmical composition, siz, shap, and ointation of th paticls. Th objctiv of this disstation is to dvlop a hybid scatting-absoption modl incopoating Mi and Rayligh-Dby-Gans thoy to chaactiz submicon paticls in suspnsion with multiwavlngth spctoscopy. Rayligh-Dby-Gans thoy (RDG) was chosn as a modl to lat th paticl s joint popty distibution to th light scatting and absoption phnomna fo submicon paticls. A coction modl to instumnt paamts of lvanc was implmntd to Rayligh-Dby-Gans thoy fo sphs. Bhavio of nonsphical paticls using RDG thoy was compad with Mi thoy (as a fnc). A multiwavlngth assssmnt of Rayligh-Dby-Gans thoy fo sphs was conductd wh stict adhnc to th limits could not b followd. Rpotd coctions to th factiv indics w implmntd to RDG to ty and achiv Mi s spctal pdiction fo sphs. xii

16 Th sults of studis conductd fo RDG concludd th following. Th angl of accptanc plays an impotant ol in bing abl to assss and intpt spctal diffncs. Multiwavlngth tansmission spcta contains qualitativ infomation on shap and ointation of non-sphical paticls, and it should b possibl to xtact this infomation fom cafully masud spcta. Th is disagmnt btwn Rayligh- Dby-Gans and Mi thoy fo tansmission simulations with sphical scatts of diffnt sizs and factiv indics. Finally, it is not possibl to adquatly o alistically compnsat fo th diffncs btwn Mi and RDG though th us of hypochomicity modls and/o ffctiv factiv indics. A hybid modl combining RDG and Mi thois was dvlopd and tstd fo sphs of diffnt sizs and factiv indics. Th sults of hybid modl is that it appoximats Mi thoy much btt than Rayligh-Dby-Gans fo paticl sizs small than th wavlngth and fo a boad ang of optical poptis in th contxt of multiwavlngth spctoscopy. Ovall, this nw modl is an impovmnt ov Rayligh-Dby-Gans thoy in appoximating Mi thoy fo submicon paticls and is computationally mo ffctiv ov oth mthods. Th dvlopmnt of th hybid sphical modl constituts a platfom fo dvloping nonsphical modls. xiii

17 Chapt On Intoduction and Mthods. Intoduction Chaactization of paticls ntails obtaining infomation about siz, shap, ointation and chmical composition. Paticl chaactization is a boad aa of undtaking which ncompasss many tchnologis, among thm light spctoscopy tchniqus. Light spctoscopy typically involvs scatting and absoption mthods. Scatting masumnts a pfomd at a singl wavlngth but masud as a function of th diction of obsvation. Fo absoption, th light is masud in th fowad diction as a function of wavlngth. Light scatting tchniqus typically us highly collimatd soucs (lass), whas absoption spctophotomtic tchniqus us boadband soucs to poduc multiwavlngth spcta. In ith cas th sulting spcta can b intptd with th thoy of lctomagntic adiation, which dscibs intaction of light with matt (Maxwll s Equations). Mi and Rayligh-Dby-Gans thois a solutions to Maxwll s Equations that lat th paticl s joint popty distibution to th light scatting and absoption phnomna. This connction is mad though th optical poptis that a chaactistic of th matials containd in th paticl. Th objctiv of this study is to dvlop a hybid scatting-absoption modl incopoating both thois to chaactiz submicon paticls with multiwavlngth spctoscopy. To accomplish this objctiv Mi thoy and Rayligh-Dby-Gans thoy

18 a visitd and xtndd to account fo th fild altation pdictd by Mi, and fo th dipol adiation mchanism mployd by Rayligh-Dby-Gans. Thoughout this disstation Mi thoy and Rayligh-Dby-Gans thois a mphasizd bcaus at this point thy nabl al tim paticl chaactization fo industial and biomdical applications. Th lagst aa of application that would pofit fom this study is in th biological and biomdical fild in th subjct of micobial and disas dtction in tissu and bodily fluids.. Matials and Mthods Th pogams fo Mi thoy, Rayligh-Dby-Gans thoy, instumnt modls and hypochomicity w dvlopd in Matlab v6.5.. Computations fo ths pogams w conductd using a Dll Inspion 4 with GHz Pntium III pocsso and 5 MB RAM. Th optical poptis (factiv indics) utilizd w povidd by D. Gacia-Rubio and th SAPD laboatoy though th Collg of Main Scinc at th Univsity of South Floida [7]. Th comput cods dvlopd fo th analysis of Rayligh-Dby-Gans and Mi paticls w tstd against publishd valus of th scatting functions [, 4]. In tsting and xploing th algoithms fo Rayligh-Dby-Gans th factiv indics slctd w thos of soft bodis and hmoglobin, wh soft bodis a dfind h as paticls whos lativ factiv indx is clos to on with no absoption componnt. Th valus of th indx of faction n+iκ fo biological paticls commonly usd a soft bodis (. 45 n.4 ) and hmoglobin (. 48 n. 6,. κ. 5 ) [7]. Polystyn (. 5 n.,. κ. 8 ), silv bomid (.6 n 3. 5,

19 . κ.6 ) and silv chloid ( n. 7,. < κ. 85 ) a matials found in industial applications whos poptis a usd as standads fo optical instumnts [7]. Wat (. 3 n.4 ) was usd as th suspnding mdium. Th factiv indics as function of wavlngth a potd in Appndix B. Th angs of paticl volums w chosn btwn 7 nm 3 and 87 µm 3. Th sphical diamt quivalnts to th volum ang a btwn 5 nm -5.5 µm. Th tabl blow givs th simulation paamts usd to dfin th suspnsions fo th analyss conductd in this disstation. Tabl.: Simulation Paamts Light Souc Wavlngth Paticl Concntation Paticl Dnsity -9 nm E-4 g/cc g/cc.3 Ovviw of Chapts This disstation is dividd into nin chapts. Chapt two psnts a viw of Mi and Rayligh-Dby-Gans thoy, citing th sulting fomulas fo th scattd fild and thi matix fomulations. Fo ach thoy th scatting intnsity atio that govns scatting masumnts/simulations and th tubidity fomula that govns tansmission masumnts/simulations a displayd. Th latt fomula contains a tm popotional to th scatting coss sction, which taks diffnt foms fo th two modls, and a tm popotional to th absoption coss sction, which is th sam fo both modls. Only simulations of tansmission a potd, at multiwavlngth. Chapt th dscibs aptu coction modls that account fo th fact that actual tansmission masumnts a invitably pollutd by th psnc of som nafowad scattd adiation. Th simulations of this ffct a fo th RDG modl only, fo µm hmoglobin sphs. 3

20 Chapt fou pots a study of nonsphical scatts, simulating soft body llipsoids. Tansmission cuvs a compad fo two ccnticitis, ach with th diffnt body ointations, using th RDG modl. Chapt fiv compas RDG thoy and Mi thoy fo tansmission with simulations fo sphical scatts of diffnt sizs and faction indics. Disagmnt btwn th two thois is dmonstatd. Th st of th disstation is concnd with attmpts to modify RDG to bing th tansmission simulations into clos agmnt with Mi (th xact solution fo sphs). In chapt six, two appoachs a dscibd to incas th computd RDG tubidity to that of Mi. Th fist appoach is to us hypochomicity as a coction to RDG to account fo absoption. Th scond appoach is basd on th obsvation that th RDG fomula fo th xtinction coss sction is (vy naly) a simpl quadatic function of n and κ; thfo on can invt this function and find "ffctiv" valus of κ o n that will sult in tubidity valus calculatd by RDG in agmnt with thos computd by Mi. Rasons fo jcting ths appoachs a citd. Chapt svn psnts th nw hybid modl, basd on th igoous Mi calculation of th intnal fild and th Rayligh-Dby-Gans appoach fo th scatting adiation. This thoy is dvlopd in full fo sphs. In chapt ight th hybid modl is tstd by simulatd compaisons with Mi and Rayligh-Dby-Gans thois, mploying th span of optical poptis of intst. Finally, conclusions, contibutions, and commndations a covd in chapt nin. 4

21 Chapt Two Scatting Thois and Modls This chapt dscibs light scatting thois and masumnts of submicon paticls in suspnsion. Th fist sction povids infomation on scatting and tubidity masumnts usd to chaactiz paticls. This sction also povids a dsciption of th modls usd to dscib th obsvd masumnt. Th subsqunt sctions povid an outlin of Mi thoy and RDG thoy. Ths thois dscib th scatting phnomna obsvd in tansmission and scatting masumnts. Spcifically, th scatting intnsity atio as a function of wavlngth (in ou cas a boad wavlngth ang) and angl of obsvation, and th tubidity as a function of wavlngth (again boad wavlngth ang) a quantifid. Oth mo computationally intnsiv, tchniqus fo solving light scatting and absoption poblms a discussd. Ths tchniqus includ th T-Matix and th Pucll-Pnnypack mthods.. Backgound Th a many typs of spctoscopy masumnt usd to chaactiz paticls in suspnsion. Most intst focuss on tansmission and scatting masumnts. In th fom th lctomagntic ngy of an incidnt wav is masud aft intaction with a paticl o suspnsion as it lavs th systm in th fowad diction. In contast, scatting masumnts captu th light aft intacting with a paticl as it lavs th 5

22 systm at any angl of obsvation. Thy diff bcaus tansmission masumnts captu both th (fowad) scattd light and th unscattd potion of th incidnt bam. Infomation concning th poptis of th scatting and absobing paticl is containd in th masud spcta, which a plots of th pow intnsity vsus fquncy o wavlngth (and, diction). Though th uss of th appopiat thois and modls, it is possibl to obtain stimats of th siz, shap, chmical composition, intnal stuctu, and sufac chag fom spctoscopic masumnts []. A complt schm fo paticl chaactization must tak into account vaious xpimntal conditions occuing in th lab systm whn spctoscopy masumnts a conductd. Ths includ th typ of masumnt, instumntation stup, paticllight intaction, and oth optical phnomna. Figu. illustats how ths componnts lat to on anoth. Th scatting intnsity atio quation and th tubidity quation a ngy balanc quations that a dvlopd fom th scatting thois studid. A dtaild dsciption of th tansmission masumnt and analysis is povidd in chapt th. Rfactiv indics and coctions a discussd in chapt six. Th dsi to chaactiz paticulat systms fo al-tim continuous monitoing has ld to th slction of Rayligh-Dby-Gans (RDG) thoy and Mi thoy. Computation tim bing th sticting facto, ths thois povid light scatting solutions in a suitabl tim. Th st of this chapt is ddicatd to th dsciption of Mi and RDG thois along with th dvlopmnt of th cosponding scatting intnsity atio and tubidity fomulas. 6

23 Figu.: Diagam fo Complt Paticl Chaactization Modl. Gnal Concpts and Equations Th mphasis of this sction is th utilization of th scatting matix fomalism to valuat th xtinction of light pdictd by th scatting intnsity and tubidity quations. Thoughout th cnt cous of light scatting histoy, th tms tubidity (o optical dpth) and optical dnsity hav causd much confusion [4]. Tubidity has bn taditionally dfind as an attnuation cofficint du to scatting (only) fo th tansmission of th incidnt bam. Hin, tubidity is dscibd as th total attnuation obsvd du to scatting and absoption. Th tm optical dnsity (O.D.) was oiginally usd synonymously with absoption; th units of O.D. a absoption unit p pathlngth (Au/cm). Tubidity will b dscibd in th units of optical dnsity. 7

24 Th amplitud scatting matix is usd to lat th incidnt and scattd filds. ) E s S3 i E s ik ( z = ik S S 4 E S E i. wh E i, E i, E s, E s a th asymptotic incidnt and scatting filds paalll and ppndicula to th scatting plan; is th distanc fom th scatting cnt to th dtcto, z is distanc along th axis of popagation of th incoming wav, and k is th popagation constant o wav numb in th mdium suounding th paticl; k ( π n o ) λ o =, wh λ o is th wavlngth in vacuo and n o is th factiv indx of th mdium. Van d Hulst [] dscibs in dtail vaious assumptions mad fo simplifying th scatting functions with gad to otation and symmty of paticls. Fo a sphical paticl S 3 and S 4 a qual to zo. S and S a complx amplitud scatting lmnts; thy dpnd on th indics of faction, paticl siz, and th scatting diction. S and S a givn in th Mi modl by fomidabl sis xpansions involving Bssl, Numann, and Lgnd functions. Th xpssions fo S and S pdictd by Mi and RDG thoy a povidd in subsqunt sctions. If th dtcto is not situatd in th fowad diction, it is illuminatd only by light that is scattd by th paticl; its constuction shilds it fom th incidnt bam. Th tm scatting masumnt fs to this configuation. This scattd intnsity I s is givn by 8

25 + = + = =,,,, R R R i i s s s s s E S E S k E E E I µ µ µ. wh is th pmittivity and µ is th pmability. If th incoming light is unpolaizd,,, i i E E = = o E so ( ) o o o s I k S S E k S S E S S k I R R + = + = + = µ µ.3 I s /I o is known as th scatting intnsity atio. On th oth hand, if th dtcto is alignd with th incoming bam, it masus th fowad scattd wav togth with th tansmittd incidnt wav. Th analysis of such a tansmission masumnt is most asily conductd by accounting fo th ngy loss suffd by th oiginal incidnt bam. Th a two loss mchanisms which attnuat th incidnt bam, scatting and absoption. Th pow scattd out of th bam by a paticl is valuatd by th intgating th scattd intnsity ov an nclosing sph at infinity (in sphical coodinats). ( ) θ φ θ φ θ λ π π d d I pow scattd s = sin,,,.4 9

26 It is convnint to dfin a scatting coss sction C sca as th aa ov which on would intgat th incoming intnsity I o to balanc th scattd pow; thus I o Csca = scattd pow.5 o C sca ( λ, θ, φ) π π I s, = sinθ dφdθ.6 I o Th scatting fficincy Q sca is th atio of th C sca to th actual coss sction G that th paticl psnts to th incoming bam: C Q = sca sca G.7 Th pow absobd by th paticl will b discussd in dtail in chapt fiv; it too can b xpssd using an quivalnt aa C abs : I o Cabs = absobd pow.8 Accoding to Van d Hulst [], th absoption coss sction fo a paticl of volum V and lativ factiv indx of faction m ( n + i ) no = κ is givn (in both thois) by m C abs = 3kV Im.9 m + Th absoption fficincy is xpssd analogously: Cabs Qabs =. G Thus on can mathmatically chaactizd th pow loss in th incidnt light as if an aa (C sca + C abs ) w blockd out of th incidnt bam by ach paticl. Th xtinction coss sction and fficincy, thn, a givn by

27 C = C + C, xt sca abs C xt Q xt =. G If th paticl dnsity in th mdium is N p, thn in an infinitsimal lngth dz th a N p dz paticls p unit aa; ach ffctivly dplts th bam pow by C xt I. Thfo th attnuation of th bam intnsity I(z) as a function of pathlngth z is govnd by o I ( z dz) I( z) = N ( dz) C I( z) +. p xt di z dz = N C I(z).3 p xt Th solution to th diffntial quation is I N C z ( z) = I( o) p xt.4 Th optical thom [] stats that th xtinction coss sction can b xpssd in tms of th lmnts of th scatting amplitud matix in th fowad diction 4π 4π C xt = R[ S( θ = ) ] = R[ S ( θ = )].5 k k Th tms tubidity, o optical dnsity τ a usd to chaactiz th total attnuation in a sampl of lngth l. τ = N C l N GQ l.6 p xt = In pactic, a dtcto placd in th fowad diction will hav a finit aptu, and thus captu som of th adiation scattd at small angls; coctions fo this ffct a discussd in chapt th. p xt

28 .3 Mi Thoy and Modl Th xact solution to th bounday valu poblm fo light scatting by a sph is gnally fd to as Mi Thoy []. Mi thoy assums that th sphical scatting objct is composd of a homognous, possibly absobing isotopic and optically lina matial iadiatd by an infinitly xtnding plan wav. Figu.: Diagam of Coodinat Systm usd in th Mi and RDG Modls In figu. (x,y,z) f to Catsian coodinats and (θ,φ, ) f to sphical coodinats. Th amplitud scatting matix lmnts S and S a xpssd xplicitly as S S ( θ ) ( θ ) = = n= n= n + n { an π n ( cosθ ) + bn τ n ( cosθ )} ( n + ) n + n { bn π n ( cosθ ) + an τ n ( cosθ )} ( n + ).7

29 wh π n (cosθ) and τ n (cosθ) a dfind in tms of th associatd Lgnd polynomial ; P n π τ n n sinθ d n dθ ( cosθ ) = P ( cosθ ) ( cosθ ) = P ( cosθ ) n.8 Not that th is no φ dpndnc in th Mi modl. To calculat th scatting intnsity atio using Mi thoy, th quations in.7 fo th amplitud scatting matix lmnts can b substitutd into quation. Th amplitud scatting matix lmnts fo Mi thoy at θ = a givn by: S o o ( ) = S ( ) = S ( ) = ( n + )( n= o a n + b n ).9 Th tubidity fomula fo Mi thoy is calculatd by substituting.l9, into quations.5 and.6. Though th dict calculation of th tubidity and scatting intnsity atio quations, Mi thoy has bn shown to b an ffctiv tool fo dtmining paticl siz distibutions of nonsphical shaps, intnal stuctus, and optical poptis [4]..4 Rayligh-Dby-Gans Thoy and Modl Th basis of th thoy fo Rayligh-Dby-Gans scatting is Rayligh scatting. Rayligh psntd an appoximat thoy fo paticls of any shap and siz having a lativ factiv indx na unity, Dby and Gans lat addd finmnts. Kk [5] stats that th fundamntal appoximation in th Rayligh-Dby-Gans appoach is that th phas shift, th chang of th phas of a light ay that passs though th sph is 3

30 ngligibl. A stiction is thfo put upon th paticl siz, th wavlngth, and th factiv indx. Fo a paticl of adius a th stiction is ka m <<. Th physical assumption of Rayligh-Dby-Gans scatting is that ach infinitsimal volum lmnt of th paticl givs is to Rayligh scatting and dos so indpndntly of th oth volum lmnts. Th wavs scattd in a givn diction by ths lmnts intf du to th diffnt positions of th volum lmnts in spac. Fo sphical and non-sphical paticls, a fom facto f iδ = dv V ( θ ). is intoducd that avags th phas diffnc δ thoughout th volums V of th (sphical and nonsphical) paticl; th scatting amplitud lmnts S and S thn tak th fom of 3 3ik m S V f ( θ ) π m =. 4 + S 3 3ik m V ( θ ) cosθ 4π m =. + f w S 3 = S 4 =. Kk lists fom factos fom vaious shaps, of which som a quotd in this disstation. In subsqunt sctions th spcific divation of th fom of f is givn, as wll as a discussion fo utilizing f to dduc ointation. A dtaild divation of th scatting intnsity atio quation and th tubidity quation fo RDG is dvlopd in Appndix A. Th scatting intnsity atio is asily obtaind by substituting quations. and. fo th amplitud scatting lmnts into quation.. Fo unpolaizd light,.3 bcoms 4

31 I I s o 4 9k m ( λ, θ ) = V f ( θ )( + cos θ ) 3 π m +.3 Th scatting coss sction C sca is xpssd fom.6 as π 4 9k V m C sca = f + sinθ dθ.4 6 m + π ( θ )( cos θ ) Th gnal xpssion fo th absoption coss sction is, fom quation.9, m C abs = 3kV Im.5 m + Th xtinction coss sction is th sum of ths two (quation.4 and.5): C xt 4 π = 9 + k V m m f 6π m ( θ )( + cos θ ) sinθ dθ + 3kV Im m +.6 Th xpssion fo th tubidity fo a monodispsd systm can thn b xplicitly xpssd by th following quation. 4 π 9k V m m τ = N p l 6 + f.7 π m ( θ )( + cos θ ) sinθ dθ + 3kV Im m +.5 Hybid Thoy and Modl Th nw scatting modl poposd in this disstation is a hybid combination of Mi thoy and Rayligh-Dby-Gans thoy. Chapt six is ddicatd to th dsciption and dvlopmnt of th hybid thoy and modl. 5

32 .6 Mthods Rviw In sctions.3 and.4 th modls psntd fo chaactizing paticls hav bn Mi thoy fo sphical paticls and Rayligh-Dby-Gans thoy fo abitay shapd paticls. To summaiz, Mi thoy is an xact mathmatical solution to Maxwll s Equations fo light scattd by sphs. It has bn usd xtnsivly to chaactiz nonsphical paticls appoximatly fo a boad ang of sizs and optical poptis. Although Mi thoy povids good stimats of th siz and chmical composition it dos not posss th ability to stimat th actual shap and ointation of non-sphical paticls. Rayligh-Dby-Gans thoy povids infomation on shap and ointation fo sphical and nonsphical paticls; howv, its applicability has limitations with gads to siz and optical poptis of systms. Fo th cas of nonsphical paticls, many xact and appoximat mthods hav bn dvlopd. Singham and Bohn discuss advantags and disadvantags of sval [6, 7]. Two mthods considd significant to this invstigation a th T-Matix and Pucll-Pnnypack mthod. Th T-Matix is th lina tansfomation conncting th cofficints of th ignfunctions in th scattd fild and thos fo th incidnt fild. It is though th linaity of Maxwll s quations and th bounday condition satisfid by th lctomagntic fild that ths cofficints a linaly latd. Bohn stats that in pincipl th cofficints of th T-matix a obtainabl by intgation; howv, computational difficultis ais if th paticls a highly absobing o thi shaps xtm. Rcnt pogss has bn mad fo computing th T-matix as discussd by Mishchnko t. al [8]. Mishchnko povids a viw and dsciption of sval 6

33 numical tchniqus dvlopd fo singl and agggatd paticls. Th scop of this disstation is on singl paticls and as such a bif ovviw follows. Th standad appoach fo computing th T-matix fo singl scatts is basd on th xtndd bounday condition mthod dvlopd by Watman fo homognous paticls. Naly all numical sults computd by th T-matix lats to bodis of volution. Th fist computational advanc of th T-matix is using nonsphical paticls of fixd ointation using th xtndd bounday condition mthod. This mthod povd to b fast than th convntional spaation of vaiabls mthod fo sphoids and disct dipol appoximation intgal quation fomulation. Th disadvantag to using th xtndd bounday condition mthod is its poo numical stability fo paticls with vy lag al and/o imaginay pats of th factiv indx, lag siz compad to th wavlngth, and/o lag xtm gomtis. Mishchnko discusss th itativ xtndd bounday condition as anoth appoach fo ovcoming th poblm of numical instability in computing th T-matix fo highly longatd sphoids. Th disadvantag of this tchniqu is that th numical stability coms at th xpns of considabl incas in computation cod, complxity, and CPU tim. Th last computational appoach discussd by Mishchnko is that of using xtndd pcision instad of doubl pcision floating point vaiabls. This tchniqu povids an incas in siz paamt fo sphoids and btt accuacy. Th us of th xtndd pcision vaiabl quis only a ngligibly small additional mmoy and its appoach is simpl with littl additional pogamming. Th disadvantag to th xtndd pcision is th CPU tim. 7

34 In th Pucll-Pnnypack mthod a paticl is appoximatd by a lattic of dipols, ach small compad with th wavlngth but still lag nough to contain many atoms. Each dipol is xcitd by th incidnt fild and by th filds of all th oth dipols. Two mthods a usd to solv fo ths quations; ith itation o matix invsion. Th itativ mthod is slow fo lag paamts, dipols and valus of th factiv indx. Th matix invsion mthod, although usful fo calculating scatting by a paticl in mo than on ointation and fo ointational avaging ov an nsmbl of paticls, is limitd to small numb of dipols. An mbdding mthod using a scatting-ods appoach was dvlopd by Singham and Bohn [6]. Th scatting-ods ptubation mthod is an xtnsion of th sis fo th ptubation mthod which looks upon a nonsphical paticl as a sph, th bounday of which is distotd o ptubd by diffnt amounts at diffnt points []. In th cas of Singham and Bohn, th scatting-od ptubation mthod is usd to fomulat th coupld dipol mthod. This mthod uss th dipola intactions as infinit sis in scatting-ods. Two intactions xist, thos btwn dipols within th sph and all oths. Th scatting filds sulting in th dipol intaction within th sph a dscibd by Mi thoy o any oth quivalnt thoy. It was dmonstatd that this mthod wokd and shotnd computation tim. Howv, it was limitd by th scatting-od sis, which can divg. Th gat th factiv indx, th small th paticl siz fo which it divgs. 8

35 Th ndavo to chaactiz biopaticulats in th contxt of ngining application quis idntifying th poblm, finding solutions, analyzing, dsigning and tsting of th solution all taking plac in al-tim. Mi, Rayligh-Dby-Gans, and th nw hybid thoy povid a mo palatabl mans to chaactiz biological systms as compad to thos afomntiond. Th st of this disstation dscibs how ths thois a utilizd and xplod. 9

36 Chapt Th Instumntation Coction Modl fo Tansmission Th sach psntd in this documnt utilizs many modls to psnt thos conditions psnt whn taking xpimntal masumnts. As statd in th pvious chapt and illustatd in figu., light scatting, instumntation and optical fomulas a intgatd to mak up a complt modl fo xpimntal conditions obsvd whn taking tubidity masumnts. Th pvious chapt dscibd th light scatting quations fo th intnsity atio and tansmission spctoscopy masumnts. This chapt is ddicatd to discussing th instumnt fomulas availabl fo simulating tansmission masumnts and implmntation of th fomulas to RDG thoy. 3. Instumntation Coction Modl Dpak t al. [9] dfins th xpssion fowad scatting fo a singl scatting phnomnon as scattd adiation aching th dtcto aft bing scattd only onc by a scatt situatd within th path of th dict adiation. Idally, th conditions fo tubidity masumnts ntail a non-divgnt bam illuminating a homognous mdium and a dtcto dictly, opposit th light souc, possssing an accptanc angl clos to zo (to captu light solly in th fowad diction). Th tubidity quation psntd in th pvious chapt dos not account fo th instumnt stup, thfo an instumnt modl o coction to th tubidity is intoducd in this chapt. On th basis

37 of th scatting thois psntd and th instumnt stup usd fo tansmission masumnts, th coction facto povidd by Dpak t al. was chosn. Th coction facto is dpndnt upon th gomty of th tansmission masumnt though th angl θ, subtndd at th scatt by th dtcto window (figu 3.). Thfo th instumnt dsign has to b takn into account. Fo compltnss th two dsigns povidd by Dpak t al. w implmntd [9]: th opn dtcto and pinhol dtcto. Th gomtis a illustatd in figu 3. (a) and (b). Figu 3.: Tansmission Systm (a) Opn Dtcto (b)pinhol Dtcto Dpak t al. stats that th tu optical dpth τ fom tansmission masumnts cannot b obtaind dictly. Dpak t al. nots that th finit aptu of th dtcto, in a tansmission masumnt, picks up som additional light that has bn scattd into th aptu. As sn in chapt two th tubidity fomula (quation.6)

38 p ( Csca + Cabs ) = N plg( Qsca + Qabs ) N plqxt τ = N l = 3. accounts fo light that has bn scattd out of, o absobd fom, th incidnt bam. Rcall that th pow scattd by a paticl quals (quation.4) π π s ( λ, θ, φ) sinθ dφd CscaI o scattd pow = I, θ = 3. Howv, if th finit aptu captus scattd light within a con of half-angl δ in th fowad diction, thn th bam dpltion du to scatting only π π I s δ giving an ffctiv scatting coss sction of ( λ, θ, φ ) sinθ dφdθ,, 3.3 C sca = C sca π π δ I s ( λ,, θ, φ) I o sinθ dφdθ 3.4 Th appant xtinction coss sction of th paticl can b wittn C xt = C sca + C abs = C sca + C abs π π δ I s ( λ,, θ, φ) I o sinθ dφdθ 3.5 o C C xt ( λ,, θ, φ) π π I s = = sinθ dφdθ P( δ ) 3.6 C I xt xt δ In th opn dtcto systm (figu 3.a) a adiation souc is placd at th focal lngth of th tansmitt lns L, which tansmits a paalll bam though th mdium of thicknss l, which is in tun masud by an opn dtcto of adius R. Th path avagd coction facto Pˆ fo paticls on axis is dfind as o

39 π ( χ, δ ) P ˆ = tanδ R csc φ dφ 3.7 φ wh th angl δ fo th scatts satisfis tanδ = R L. Th pinhol dtcto systm (Figu 3.b) consists of a adiation souc placd at th focal point of a lns L. A scond lns L, with focal lngth f, focuss th light though an aptu of adius in th focal plan and onto th dtcto. Th path avagd coction facto dtcto is simply Pˆ fo th pinhol ( ) P ˆ = P δ 3.8 Th sult of quation 3. stms fom th fact that δ is dtmind by tan δ = f, which stays constant. Th coction facto cosponding to th dsign st up can b intoducd into th tubidity quation, quation., by multiplying though by Pˆ ( δ ): ( δ ) τ ( λ) = N l π a Q ( χ, m) Pˆ 3.9 p xt 3. Implmntation of Instumnt Coctions Th coctions fo scatting dvlopd by Dpak [9] hav bn valuatd fo RDG thoy fo hmoglobin sphs. Hmoglobin sphs a hypothtical sphs whos factiv indics a thos of hmoglobin. Th factiv indics of hmoglobin w usd to tst ths ffcts fo stong scatting and absoption lativ factiv indics ( n n o.,. κ n. ). Th accptanc angl fo spctomt dtctos a o typically lss than o qual to dgs. Th accptanc angls studid fo both dtcto 3

40 configuations w,, and 5 dgs fo a sphical paticl of µm. Th paamts usd to conduct th simulations a povidd in tabl.. Figus 3. and 3.3 show th simulatd tansmissions using Rayligh-Dby-Gans thoy with and without accptanc angl coctions fo ach dtcto configuation. Th (uncoctd) Mi thoy simulation is includd as a fnc. Th pinhol dtcto stup (figu 3.) shows small subtl changs in th coctd spcta fo all angls (RDG, RDG, RDG 5 ) compad to uncoctd (RDG) spcta. Th opn dtcto stup (figu 3.3) shows damatic changs in th spcta as th aptu angl incass. It is vidnt that th angl of accptanc plays an impotant ol in bing abl to assss and intpt spctal diffncs. Bohn has mad simila statmnts concning th angl of accptanc though h calculations hav bn povidd. Having dmonstatd th mthodology fo cocting fo finit dtcto aptus in tansmission masumnts, w nxt tun to a study of th ffcts of paticl shap and ointation. Aptu coctions a omittd in th simulations potd in th maind of th disstation, xcpt wh notd. 4

41 RDG RDG RDG RDG 5 Optical Dnsity Wavlngth (nm) Figu 3.: Calculatd Tansmission of Rayligh-Dby-Gans fo µm Hmoglobin Sphs with Pinhol Dtcto Stup RDG RDG RDG RDG 5 Optical Dnsity Wavlngth (nm) Figu 3.3: Calculatd Tansmission of Rayligh-Dby-Gans fo µm Hmoglobin Sphs with Opn Dtcto Stup 5

42 Chapt Fou Nonsphical Paticls Fom factos hav bn dvlopd fo many shaps such as cylinds, disks, ods, sphs, and llipsoids. W a intstd in biological systms, whos paticls a bst psntd by llipsoids. 4. Gomty and Notation fo Ellipsoids It is convnint to cay out th psnt analysis in two non-standad, lft handd coodinat systms, th laboatoy and paticl fam. Th laboatoy fam stablishs th position and ointation of th paticl lativ to th souc and dtcto. Th paticl fam xploits th convninc of paticl coodinats in dscibing th scatting. Thfo scatting calculations a in th paticl fam and latd back to th laboatoy fam. Unit vctos along th x L, y L, and z L axs in th lab fam a dnotd by i L, jl, and k L ; and similaly fo th paticl fam. 6

43 Figu 4.: Light Scatting in Laboatoy Fam Figu 4.: Light Scatting in Paticl Fam Th bhavio of light scattd fom th paticl is dpndnt upon its siz, shap, ointation and chmical composition. Figu 4. illustats th incidnt bam appoaching th paticl in th z L -diction in th laboatoy fam. Th light scattd fom th paticl can adiat in all dictions. Th lin in th scatting plan that biscts th angl π θ btwn th incidnt and scattd bam is calld th bisctix [], dnotd 7

44 by BIS in figus 4. and 4.. Th plan though th bisctix ( BIS ) and ppndicula to th plan of th scatting ( SCA ) will b calld th bisctix plan. Equation 4. dscibs th dictions of incidnt light, scattd light and th bisctix fom th laboatoy fam. IN = k L SCA = sin θil cosθk L BIS = SCA IN = sin θi L ( cosθ+ ) k L 4. Th dsciption can also b xpssd in th paticl fam. Figu 4. shows th angls α, β, γ usd to lat th bisctix to th paticl axs. Calculation of th scattd light intnsity at diffnt angls in th laboatoy fam can b constuctd though its lation to th paticl fam. i j k L L L i k p = E j 4. p p wh i p =, j p =, k p = in th paticl fam, th Eul angls ψ, φ, ω a usd to lat th fams using th zyz Eul otation squnc, and E is givn by. E = cosψ cosω cosφ sinψ sinφ cosψ cosω sinφ sinψ cosφ cosψ sinω sinψ cosω cosφ + cosψ sinφ sinψ cosω sinφ + cosψ cosφ sinψ sinω sinω cosφ sinω sinφ cosω 4.3 Using this lationship, th dictional cosins of angls α, β, γ can b found in th laboatoy fam to mathmatically dscib th bisctix (quation 4.). BIS BIS BIS cos α = il, cos β = jl, cosγ = k L 4.4 BIS BIS BIS 8

45 In th cas of an llipsoid with a fixd ointation, w can s th us of th dictional cosins in th fom facto computd by Kk: 9πJ 3 f ( θ ) = u u = ha A = a 3 ( u) cos α + b cos β + c cos γ 4.5 wh A is a vcto dscibd by th dictional cosins and a, b, c a th smiaxs fo llipsoid. Th vaiabl h quals 4π θ h = sin. λ Th following is a shot tabl of vaious fom factos, ptinnt to this wok, takn fom Kk [4]. Th quations psntd in this sction w pogammd; simulations conductd a potd in th nxt sction. Tabl 4.: Tabl of Fom Factos Shaps Dfinition of vaiabls Fom facto f (θ) Sph Radius=a 9πJ 3 ( u) u=ha 3 u Concntic Inn adius=a 3 9π J ( v) Sph with Out adius=b m m a 3 sphical shll u=ha 3 v m b v=hb Inn factiv indx=m Inn factiv indx=m Ellipsoid of Smi axs a a,b, β is angl Rvolution 9πJ 3 ( u) btwn figu axis & 3 bisctix u u=h(a cos β + b sin β) / Ellipsoid Smi axs a a,b,c 9πJ 3 ( u) u=ha A =a cos α + b cos 3 β+ u c cos γ α,β,γ a th dictional cosins of th bisctix J 3 u u 3 ( ) + 3 9

46 4. Ellipsoid Simulations Fo th simulation of th ffcts of shap and ointation on th multiwavlngth tansmission spcta, polat llipsoids with factiv indics of soft paticls hav bn slctd. Ellipsoids hav bn usd to modl th scatting bhavio of a lag vaity of biological systms such as micooganisms and d blood clls, and off th possibility of xploing gomtical xtms btwn sphs and ndl-lik paticls. Th smimajo and smimino axs fo polat llipsoid w dtmind fo a volum quivalnt to µm diamt sph. Fo absoption it is appant fom quation.9 that th pathlngth fo th paticl is not lvant sinc th absobd pow is popotional to th volum. Nvthlss, scatting is popotional to th coss sctional aa and thfo ointation. Fo a polat llipsoid with lngth a along its smimajo axis and lngths b=c (<a) along its smimino axs, th ccnticity ( ε = c / a ) valus tstd w.3 and.8. Figu 4.3 illustats th ointations studid fo th llipsoid. Consid th xtinction coss sctional aa of th paticl as th shadow pojctd on th fowad plan (Gomtical Optics). Ointation A illuminats th llipsoid along a smimino axis, h th lagst coss sctional aa (football shap) is pojctd with th light passing though th shotst pathlngth of th llipsoid. Ointation B illuminats th llipsoid along th smimajo axis; h th smallst coss sctional aa (cicl) is pojctd with th light passing though at th longst pathlngth of th paticl. Finally, ointation C illuminats acoss th llipsoid along a nonaxial diction, h th pojction of th paticl and th light passing though a intmdiat to thos of ointation A and B. Th ointations statd psnt xtm cass fo a paticl fixd in spac. 3

47 Ointation A Ointation B Ointation C Figu 4.3: Fixd Ointations A, B, and C fo an Ellipsoid Fo ach of ths ointations th pojctd coss sctional aa in th laboatoy xy plan and avag pathlngth along th laboatoy z axis w calculatd. Ths pojctd valus w usd to comput th tansmission (using th RDG fom factos as psntd ali). Th algoithms fo Rayligh-Dby-Gans calculations fo nonsphical paticls with volums quivalnt to that of a µm diamt sph, w tstd against Mi calculations fo th sph, using th lativ factiv indics in th ang wh th RDG assumptions a mt; fo ths cas studis th factiv indics of soft 3

48 bodis w usd. Th multiwavlngth tubidity spcta w calculatd fo soft body llipsoids fo two ccnticity valus and th fixd ointations. Th multiwavlngth tubidity spcta fo th soft body llipsoids a plottd along sid th calculatd tubidity using Mi thoy. Figu 4.4 shows th ffct of th paticl ointation (A, B, C) fo a polat llipsoid with an ccnticity of.3. Figu 4.5 shows th tubidity fo an llipsoid with an ccnticity of.8. (An ccnticity of fo an llipsoid sults in a sph; thus th pdictd spcta fo RDG should b th sam as fo Mi thoy.) Compaison of llipsoids with vaying ccnticitis (figus 4.4 and 4.5) shows that paticl ointation and shap hav a significant ffct on th fatus and amplitud of th multiwavlngth tubidity, in th RDG modl. Multiwavlngth tansmission spcta contain quantitativ and qualitativ infomation on shap and ointation of non-sphical paticls, and it should b possibl to xtact this infomation fom cafully masud spcta. This conclusion is in agmnt with th sults potd in Buhl [8]. 3

49 RDG A RDG B RDG C Mi 3 Optical Dnsity Wavlngth (nm) Figu 4.4: Calculatd Tansmission of Soft Body Polat Ellipsoid ε=.3 with a Volum Equivalnt to a µm Sph 3.5 RDG A 3.5 RDG B RDG C Mi Optical Dnsity Wavlngth (nm) Figu 4.5: Calculatd Tansmission of Soft Body Polat Ellipsoid ε=.8 with a Volum Equivalnt to a µm Sph 33

50 Chapt Fiv Validation Study of Rayligh-Dby-Gans Thoy This chapt scutinizs Rayligh-Dby-Gans thoy by simulating a boad ang of factiv indics and paticl sizs, pobing th limitations imposd by th assumptions and appoximations implicit in th thoy. 5. Exploation of Thotical Limits Th limits of Rayligh-Dby-Gans modl w discussd in chapt two. To summaiz: th lativ factiv indx m must b clos to on and th siz of th paticl must b much small than λ m. Th xists a tad off fo th limits of RDG thoy; fist, if m is clos to on and no absoption is psnt thn th siz of th paticl can b th sam od of magnitud as th wavlngth. Convsly, if absoption is psnt and m is gat than on, th paticl siz must b small than th wavlngth. Although ths assumptions a invokd in th divation of th thoy, scop of th psnt sach and th complxity of its modls calls fo a valuation of ths stictions fo multiwavlngth masumnts wh stict adhnc to th limits cannot b followd. Th appoachs w takn to xplo, though simulation, th constaints of this thoy fo sphs. Fist, th sizs of th sphical paticls w kpt small compad to th wavlngths, but th wavlngth-dpndnt lativ factiv indx was allowd to significantly xcd on, typical of actual matials. 34

51 Scond, th lativ factiv indx was kpt clos to on whil th absoption was hld at zo, and paticl sizs compaabl to th wavlngths w considd. Thid, th contibution of absoption in th lativ factiv indx, kpt clos to, was invstigatd fo paticl sizs compaabl to th wavlngths. Th following subsctions dscib in mo dtail th paamts usd and th conclusions and obsvations dawn. 5. Paticl Diamt << Wavlngth Th fist of th snsitivity studis conductd tstd th limits of Rayligh-Dby-Gans fo lativ factiv indics gat than on and th absoption gat than zo, whil kping small sizd sphical paticls, compad to th wavlngths ( nm-9 nm). Th multiwavlngth tansmission spcta w calculatd fo Mi and Rayligh-Dby- Gans using sphs of silv bomid (. n n. 4,. κ. 85 ) and sphs of silv chloid (. n n. 4,. κ. 6 ). Th sphical diamt sizs o chosn w 5 nm and 5 nm. Paticl concntation, paticl dnsity, and wavlngth ang w kpt constant Figus 5., 5. show that Rayligh-Dby-Gans givs an adquat appoximation to Mi fo paticl sizs much small than th wavlngth. Figus 5.3, 5.4 vals that fo slightly lag paticls Rayligh-Dby-Gans no long closly follows Mi Thoy. Notic that in th spctal gion wh absoption is small (3-9 nm) both thois coincid vn though n/n o >. Howv, wh stong absoption is psnt, th thois apidly divg (s th optical poptis of AgB and AgCl potd in Appndix B), claly suggsting that absoption plays an impotant ol in th dispaity btwn th o thois. 35

52 45 4 RDG Mi 35 3 Optical Dnsity Wavlngth (nm) Figu 5.: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm AgB Sphs 5 RDG Mi Optical Dnsity Wavlngth (nm) Figu 5.: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm AgCl Sphs 36

53 6 5 RDG Mi 4 Optical Dnsity Wavlngth (nm) Figu 5.3: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm AgB Sphs 5 RDG Mi Optical Dnsity Wavlngth (nm) Figu 5.4: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm AgCl Sphs 37

54 5.3 Paticl Diamt ~ Wavlngth, No Absoption Th stiction of Rayligh-Dby-Gans thoy with spct to siz was tstd though th calculation of tansmission spcta fo nonabsobing sphical paticls with lativ factiv indx clos to on. Th factiv indics chosn w soft bodis ( n =. 4 ) and hmoglobin (. n n. ). Only th al pat of th factiv indx was usd o fo hmoglobin. Paticl diamts usd w 5 nm, µm, and 5.5 µm. Figus 5.5 and 5.6 show that Rayligh-Dby-Gans thoy appoximats Mi thoy fo 5nm and µm. Figu 5.7 shows that fo a paticl siz of 5.5 µm Rayligh- Dby-Gans and Mi no long coincid. Th combination of zo absoption and factiv indx atio clos to incass considably th paticl siz angs fo which RDG is applicabl; this is in agmnt with th sults potd in Kk [5]..7.6 RDG Mi.5 Optical Dnsity Wavlngth (nm) Figu 5.5: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm Soft Body Sphs 38

55 .9 RDG Mi.8.7 Optical Dnsity Wavlngth (nm) Figu 5.6: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of µm Soft Body Sphs 3.5 RDG Mi Optical Dnsity Wavlngth (nm) Figu 5.7: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5.5 µm Soft Body Sphs 39

56 Th multiwavlngth tansmission calculations conductd with only th al pat of th factiv indx of hmoglobin show that fo 5 nm diamt paticls (figu 5.8), th thois follow on anoth closly in spctal shap but th a quantifiabl diffncs in amplitud. If th tubidity is usd fo analysis th spctal diffncs btwn th two thois would sult in considabl vaiation in th stimat of paticl siz and concntation. With incasing of th paticl diamt to µm (figu 5.9), th spctal shap fo Mi thoy lativ to Rayligh-Dby-Gans flattns considably at th shot wavlngths. Figu 5. shows a smi-logaithmic tubidity plot of 5.5 µm paticls to show th diffncs in shap and amplitud fo th two thois. Th ffct of a lativ factiv indx gat than on with no absoption sults in a limitd paticl siz ang fo RDG thoy, in contast to paticls with a factiv indx clos to on with no absoption. 4

57 .5 RDG Mi Optical Dnsity Wavlngth (nm) Figu 5.8: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm Hmoglobin Sphs with κ= 3.5 RDG Mi Optical Dnsity Wavlngth (nm) Figu 5.9: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of µm Hmoglobin Sphs with κ= 4

58 RDG Mi Optical Dnsity Wavlngth (nm) Figu 5.: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5.5 µm Hmoglobin Sphs with κ= 5.4 Paticl Diamt ~ Wavlngth, Absoption κ > Th limits of validity Rayligh-Dby-Gans thoy with a lativ factiv indx clos to on and an absoption valu gat than zo w tstd though th calculation of th tansmission spcta fo sphical paticls whos sizs a compaabl to thos of th multiwavlngth ang. Th factiv indics of whol hmoglobin (. n n.,. κ.), maning th al and imaginay pat of th complx factiv indx w considd. Th paticl diamt sizs usd w: nm, 5 nm, and µm. Figu 5. shows that Rayligh-Dby-Gans and Mi closly follow on anoth fo a nm sph. As th paticl siz was incasd to 5 nm and µm th calculatd tubidity fom Rayligh-Dby-Gans slowly dviats fom Mi (s figus 5. and o 4

59 5.3). As th siz incass, th fatus of th spcta calculatd with Mi thoy flattn. This obsvd diffnc appas to b causd by absoption coss sction C abs which is popotional to th volum in cas fo RDG thoy, but not in th cas of Mi thoy..6.4 RDG Mi. Optical Dnsity Wavlngth (nm) Figu 5.: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of nm Hmoglobin Sphs 43

60 .5 RDG Mi Optical Dnsity Wavlngth (nm) Figu 5.: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of 5 nm Hmoglobin Sphs RDG Mi 3 Optical Dnsity Wavlngth (nm) Figu 5.3: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo a Suspnsion of µm Hmoglobin Sphs 44

61 5.5 Conclusion Th is disagmnt btwn Rayligh-Dby-Gans and Mi thoy fo tansmission simulations with sphical scatts of diffnt sizs and factiv indics. Th disagmnt is most sv whn absoption is psnt. Th st of this disstation will b concnd with attmpts to modify RDG thoy to bing th tansmission simulations into clos agmnt with Mi. 45

62 Chapt Six Coctions to th Rfactiv Indx This chapt is ddicatd to xamining th ffct of th complx indx of faction on th tansmission chaactistics of a paticl. Schms fo adjusting th complx factiv indx to bing RDG pdictions into agmnt with Mi thoy a psntd. 6. Rfactiv Indx Th complx factiv indx is givn by N = n + iκ 6. wh n and κ a non ngativ valus,n is th factiv indx (al), κ is th absoption cofficint (imaginay). Th scatting of light is du to diffncs in factiv indics btwn th mdium and th paticl. Th factiv indx of th paticl (N ) lativ to th suspnding mdium (N ) is, m N N = = 6. n n + iκ + iκ fd to as th lativ factiv indx, as psntd in chapt two Th al and imaginay pats of th complx factiv indx xpssd as function of fquncy, a latd though th intgal Kams-Konig lations. 46

63 ( Ω) Ωk n( ω ) = P dω 6.3 π Ω ω ( Ω, ) ω n κ ( ω) = P dω 6.4 π Ω ω h ω is th angula fquncy masud and P is th pincipal valu of th intgal []. In pincipl, if ith n(ω) o κ (ω) is known o can b masud, th oth can b calculatd dictly though quations 6.3 and 6.4. Masumnts ov th complt ang of fquncy ( to ) a quid whn applying this tansfom. 6. Hypochomic Effct Th obsvabl light scatting phnomna dpnds on th instumntation configuation and optical poptis of th matial. Th optical poptis (al and imaginay pats of th factiv indx) a intinsic poptis of matt. It is known that th optical poptis dpnd on th stat of agggation []. Howv, und ctain conditions (i.. infinit dilution) th optical poptis a additiv and indpndnt of concntation. Th psnc of absobing goups (chomophos) in high concntation within paticls givs is to a concntation dpndnc of th obsvd optical phnomna. This phnomnon gnally sults in a dcas of th imaginay componnt of th factiv indx κ lativ to its valu in solution (hypochomicity). Hypochomicity is a phnomnon in which an individual molcul, containing sval chomophos, has a ctain absoptivity at a givn wavlngth that is lss than th sum of th absoptivitis of th individual chomophos at that sam wavlngth. Whn analyzing paticulat systms, th stat of agggation of chomophos within th 47

64 paticls may sult in hypochomic ffcts that bias th stimation of thi concntation. Fo this ason, hypochomism was usd to dtmin a coction facto fo th imaginay pat κ of th factiv indx. Th most cnt modls fo hypochomicity a thos dvlopd by Vshkin [,,and 3] and tak into considation th molcula stuctu and th numb of chomophoic goups p unit volum of paticl. Th pocdu of Vshkin was xtndd to th multiwavlngth scnaio and implmntd; dtails a givn in Appndix D. Th diffncs in bhavio obsvd btwn th spcta calculatd with Mi and with RDG thoy suggsts, in agmnt with th wok of Latim [5], that it may b possibl to compnsat RDG thoy though th us of ffctiv optical poptis stimatd fom paticls of known shap and composition. 6.3 Implmntation of Optical Coction fo Absoption Rayligh-Dby-Gans is limitd to small changs in factiv indx n(λ) clos to on and small valus of absoption κ (λ). Rayligh-Dby-Gans thoy assums ach dipol absobs and scatts indpndntly and only consids th intfnc of th scatting wav. As a sult, th angula scatting intnsity is shap and ointation dpndnt, whas th absoption coss-sction is indpndnt of th paticl shap (quation.9); in oth wods, th total absoption is only dpndnt on th paticl volum (th total numb of chomophoic goups in th paticl). Whn compad with Mi th latt causs a lag discpancy wh th absoption fficincis calculatd with Mi thoy always small (hypochomic) than th valus calculatd with RDG fo lag absoption cofficints (i.., Hmoglobin, DNA). This appant hypochomicity suggsts that th 48

65 thotical modls dvlopd to account fo hypochomic o scning ffcts may b abl to bing RDG into a btt agmnt with Mi. To xplo th potntial application of Vshkin s coction th volum faction of chomophoic goups (v f in quations ) is tatd as an adjustabl paamt. Two cass a considd: v f = cosponds to % hypochomicity which tanslats to th coctd κ c (λ) bing qual to zo; and v f = cosponds to using th valu of κ (λ) dictly. Sphical hmoglobin paticls with a diamt of µm w wh Vshkin s coction was applid only to κ (λ). Th volum faction valus usd in this study w.5,.,.33, and.5. Th molcula diamt of hmoglobin is 68 Å with th coss sctional aa of Å and a molcula wight of 6 [6]. Th ointation valu q was st to on, maning th molculs a andomly ointd []. Th sults of th hypochomicity coctions implmntd in RDG thoy a shown in figus 6. and 6.. Figu 6. shows th spcta calculatd with RDG and Mi without any coctions fo hypochomicity, togth with th spcta calculatd with RDG and sval lvls of hypochomicity (i.., volum factions). Notic that, although intmdiat lvls of hypochomicity sult in impovd RDG-calculatd spcta, Vshkin s modl is not vy ffctiv in ducing th diffncs btwn Mi and RDG thois. This point is dmonstatd mo damatically whn % hypochomicity is considd. Figu 6. shows th xtm cass of % and % hypochomicity applid to both thois. 49

66 Optical Dnsity Mi-No coction RDG-No coction κ c5 κ c κ c33 κ c Wavlngth (nm) Figu 6.: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo µm Hmoglobin Sph with Vshkin Coction 4 Mi κ=% 3.5 RDG κ=% Mi κ=% 3 RDG κ=% Tansmission Wavlngth (nm) Figu 6.: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo µm Hmoglobin Sph with % and % Hypochomicity 5

67 Th us of Vshkin s modl fo th coction of th absoption cofficint bings about th poblm of th inconsistncy in tms of th Kams-Konig tansfoms sinc, aft th coction, quations 6.3 and 6.4 will no long hold. To dmonstat this inconsistncy, an ffctiv valu of n ff (λ) was calculatd though th Kam-Konig tansfom aft κ (λ) was coctd, using Vshkin s modl. All th conditions w kpt th sam fo calculating th tansmission as pviously in this sction. Figu 6.3 shows th sults of calculating th tansmission fo Rayligh-Dby-Gans with an ffctiv n ff and a coctd κ c using Vshkin s modl compad, to uncoctd valus of κ and n fo RDG and Mi thoy. Optical Dnsity RDG-No coction Mi-No coction n ff &κ 5 n ff &κ n ff &κ 33 n ff &κ Wavlngth (nm) Figu 6.3: Calculatd Tansmission of Mi and Rayligh-Dby-Gans fo µm Hmoglobin Sph with Vshkin Coction to k c and an Effctiv n ff Calculatd though Kams-Konig Tansfom 5

68 Using th ffctiv n calculatd fom coctd κ c, on would xpct th tansmission by RDG mo closly th tansmission calculatd by Mi; howv, th contay is obsvd. A clos look at th tansmission valus of figu 6.3 can b sn in figu 6.4. At diffnt volum factions of th chomopho lating to k c and n ff valus, th a distinct diffncs in th shap of th spcta. Th diffncs in th spcta a ootd in dtmining th valus n fom a visd κ using th Kams-Konig tansfom..7 n ff &κ c5.6.5 n ff &κ c n ff &κ c33 n ff &κ c5 Optical Dnsity Wavlngth (nm) Figu 6.4: Zoom in of Figu 6.4 of Calculatd Tansmission of Rayligh-Dby-Gans fo µm Hmoglobin Sph with Vshkin Coction to k c and an Effctiv n Calculatd though Kams-Konig Tansfom An altnat appoach fo binging togth Mi and RDG thois is th mathmatical adjustmnt of th factiv indics at ach wavlngth. This is xplod in th nxt sction. 5

69 6.4 Effctiv Rfactiv Indx Estimation Th concpt bhind calculating ffctiv factiv indics is that givn th absoption fficincy Q abs dtmind by Mi thoy th is a st of n(λ) and κ(λ) valus that would allow Rayligh-Dby-Gans to pdict som xtinction fficincy Q xt to coincid with th xtinction fficincy calculatd by Mi. Th absoption fficincy fo ith Rayligh-Dby-Gans o Mi thoy was xpssd in chapt two as: m 8 Q abs = 4ka Im 4ka Im ( m ) = kaκ m Th scatting fficincy Q sca in RDG is xpssd (quation.4): Q sca π 4 m = ( ka) f ( θ )( cos θ ) sinθ dθ m Fo simplification intoduc Λ = π f ( )( ) θ cos θ sinθ dθ, which isindpndnt of n and κ fo m, ( Q sca ( ka) Λ ( n ) + i κ = ( ka) Λ ( n ) + κ ) Thfo, th xtinction fficincy can thn b xpssd as th sum of quations 6.5 and 6.7 Q xt 8 = 3 4 = ( + κ ) 4 ( ka) κ + ( ka) Λ ( n ) ( ka) Λ( n ) + ( ka) Λκ + ( ka)κ 6.8 If w assum Q xt and n a known, thn w can solv fo κ xplicitly using th quadatic fomula. 53

70 Th sam typ of algba manipulation can b don to solv fo th factiv indx n if Q xt and κ a known. Q xt ko n, = + no 6.9 q q q q Q ± + 4 xt, no κ = no q 6. Equations 6.9 and 6. a wittn in tms of lativ valus. Not that fo quations 6.9 and 6. to giv al ffctiv valus fo n and κ ; Q xt, no, Q xt,ko, q, and q must always b positiv. Th atificiality of this mathmatical juggling is cla; changing only κ o n, and not th oth, would lad to a violation of th Kam-Konig lations. Thfo th implmntation of this adjustmnt is jctd. 6.5 Conclusion Th ffct of changs in th factiv indics has bn xplod as a mans to xtnd th ang of application of RDG and to bing it into btt agmnt with Mi thoy fo lag paticls and fo paticls containing stong chomophoic goups. It was concludd that, it is not possibl to adquatly o alistically compnsat fo th diffncs btwn Mi and RDG though th us of hypochomicity modls and/o ffctiv factiv indics. Thfo, a diffnt typ of appoach is quid. Th following chapt discusss xploiting th intnal fild calculation of Mi thoy as a vhicl to impov th Rayligh-Dby-Gans and Mi conflict in th psnc of absoption. 54

71 Chapt Svn Nw Hybid Thoy Th hybid thoy dvlopd in this chapt uss th Mi solution to comput th intnal fild of a sph, and th Rayligh-Dby-Gans appoach to solv fo th scatting filds. Th inducd Rayligh-Dby-Gans dipol momnt is computd fom th intnal Mi fild, ath than th incoming fild. Fom this w solv fo th scatting filds in tms of th paalll and ppndicula componnts of th incoming light. Fom factos w gnatd though th scatting amplitud functions. Th following is th mathmatical dvlopmnt of th hybidizd thoy fo a sphical paticl. 7. Gomty and Notation Figu 7.: Diagam of Scatt Point and Dtcto Location 55

72 Fo dscibing th lctic fild scattd by a paticl in th laboatoy systm th a two objcts of intst, th dtcto and scatt. Figu 7. illustats th dtcto locatd at with sphical coodinats (,θ,φ) o Catsian coodinats (x,y,z). Points within th scatt a idntifid by R with coodinats (R,Θ,Φ) o (X,Y,Z). Th cuvilina unit vctos attachd to th dtcto in figu 7. can b xpssd in ctangula coodinats though th following quations: v v v θ φ x = sinθ cosφ y = sinθ sinφ z = cosθ v v v = sinθ cosφ x + sinθ sinφ y + cosθ z v v v = cosθ cosφ x + cosθ sinφ y sinθ v v = sinφ + cosθ x y z Figu 7.: Local Unit Vctos with Rspct to th Dtcto 56

73 Figu 7.3 similaly dpicts th unit vctos attachd to a point insid th scatt. Th tansfomation quations fo ths vctos a idntical to quations 7. and 7., with cosponding subscipts and angls. Figu 7.3: Local Unit Vctos with Rspct to th Scatt As indicatd, th incidnt wav movs in th z-diction and is psumd to b planpolaizd in th x-diction. It impings upon th paticl and is scattd. Th scattd wav is dtctd at som angl θ and φ masud fom th diction of popagation of th incidnt wav; s figu 7.. Th following sction povids a mathmatical dsciption of th filds inducd by th paticl. As will b sn, th scatting dynamics a bst dscibd using th vctos R, Θ, ; th scattd adiation is bst dscibd by Φ, θ, φ. Thfo th tansfomation quations play an impotant ol in unifying th dsciption. 57

74 7. Intnal Fild Th incoming fild fo light illuminating a sphical paticl, popagating in th z- diction, and polaizd in th x-diction, is dscibd in gnic Catsian coodinats as v E i, 7.3 ikς iωt ( ξ η, ς, t) = Eo x wh k is th wav numb in th mdium. Th tim facto following. i t ω will b omittd in th Th sulting fild insid th sph is givn by Mi thoy as wh M and N v E ( R) = E o n= n n + ( ) ( ) cnm O + n id n N E n n a th solutions to th vcto wav quation in tms of Bssl 7.4 functions and sphical hamonics []. W tuncat th sis abov in th following mann E E ( R) o 3 cm O 3 idn E 5 i d N 6 E a + O λ 7.5 wh a is th adius of th sphical paticl and λ is th wavlngth. Bohn and Huffman povid th gnal xpssions fo th tms M and N as sis thmslvs, which w also tuncat as: k v k v ( R) R cosφ Rsin Φ Θ O( [ k R] ) M O Θ cos Φ v v v N E ( R) cosφ sin Θ R + cosφ cosθ Θ sin Φ Φ + O( [ kr] )

75 N E 6 v 3 ( R) k R cosφ sin Θ cosθ + k R cosθ ( cos Θ ) 5 3 krsin Φ cosθ 5 v Φ R + O 5 ([ k R] ) v Θ 7.8 wh k is th wav numb insid th sph. Th cofficints fo c n and d n a calculatd though d c n n = = µ j µ m µ j n n [ n ] µ hn ( ka) [ kajn ( ka) ] [ kah ka ] µ h ( ka) mkaj ( mka) ( ka) kah ( ka) ( mka) ( ) µ mj k j n n n ( a) kah ( ka) n [ ] [ n ] µ mhn ( ka) [ kajn ( ka) ] [ kah ka ] µ h ( ka) mkaj ( mka) ( mka) ( ) n n n [ ] n wh µ is th pmability of th sph and is psumd to qual µ, th pmability of th mdium, and k is th wav numb in th mdium. Th pims dnot diffntiation with spct to ka. Th xpssions of 7.6, 7.7 and 7.8 a tanslatd to ctangula coodinats as M k v k v X z O Z x + v N E x 7. 3 N E 3 v 3 v kzx + kxz sulting in v E E ( R) ( d + c ) v ( d c ) o = d + ikz x + v ik X z + O ([ k R] ) 7.4 This xpssion can b wittn in xponntial fom to th sam od of accuacy; sinc ( ) x = + x + O x

76 w implmnt th following, ( d c ) ( d c ) d + ( d + c ) ik X = ik X ik Z = d + O ( d + c ) d ik Z ([ k R] ) + O ([ k R] ), sulting in th following appoximation fo th Mi fild insid th sph. E E ( R) o = d ik ( d + c ) ( d c ) d Z v x + ik X v z Not that in th limit as k k, if w hav d, c, d thn E ikz ( R) Eo x, th incoming fild valu. In chapt ight w dmonstat by comput studis that, indd, c, d, and d all qual whn k =k. This is consistnt; if th dilctic poptis of th scatt match thos of th mdium th incoming fild is unaltd. 7.3 Dipol Scatting Appoach p R Elctomagntic thoy stats that a dipol locatd at R iωt of intnsity ( ) adiats in th fa fild accoding to th following []: E s ik R 3 ik ik R 4πε iωt = R R [ p( R) ] 7.8 wh E s is th scattd lctic fild adiatd by th dipol and ε is th pmittivity o dilctic constant of th mdium. It also stats that a small dilctic sph of adius ρ placd in a unifom static lctic fild E gnats a dipol momnt. Th inducd dipol momnt is popotional to th fild and is givn by 6

77 EdV m m E m m p = + = ε π ρ ε 7.9 wh ε is th pmittivity, m is th lativ factiv indx, and dv is th volum of th scatt. Rayligh scatting assums that an oscillating, nonunifom fild ( ) t i R E ω gnats a dipol momnt in a sphical volum givn by th sam xpssion in quation 7.9 and that th dipol -adiats accoding to quation 7.8. Following th RDG appoach, w assum that ach infinitsimal volum within th scatt (not th nti scatt itslf) bhavs in this fashion. By substituting quation 7.9 into 7.8 w obtain th following xpssion fo th incmntal lctic fild adiatd by th infinitsimal dipol locatd at R : ( ) + = R dv E m m ik R ik E d R R R ik s v ε πε 7. Fo R <<, Rayligh appoximats R, R, and ( ) { } ikr i R ik R ik v = k. Whn substituting ths appoximations into quation 7. th following xpssion is obtaind. ( ) [ ] R E dv m m k E d ikr ik s + = 4 3 π 7. 6

78 7.4 Hybid Thoy Th diffnc btwn Rayligh-Dby-Gans thoy and th hybid thoy psntd hin is as follows: RDG assums that th local fild ( ) R E gnating th infinitsimal dipol in quation 7. is givn by th incoming fild, whas th hybid thoy taks th intnal fild that Mi thoy givs fo th sph as th fild inducing th dipol momnt. By using th intnal Mi fild, w a taking som account of th ffct of th suounding dipol fild altations to th incoming fild (such as attnuation, which is highlightd in chapt fiv as a majo shotcoming in RDG thoy). Th validity of ith appoach psums that th incoming lctic fild is oughly unifom ov th sph, so that th adius a of th sph must b a small faction of th wavlngth (a<< λ). If w substitut th xpssion fo th intnal lctic fild, quation 7.7, into quation 7. an xplicit fomula fo th scattd lctic fild is obtaind. + + = + o z Z d c d ik x Z d c d ik ikr ik s E d dv m m k E d v v v v 4 3 π 7. In od to valuat [ ] x v v and [ ] z v v on has to us th idntitis in quations 7.,. Th following xpssion is a sult of th convsion and mathmatical manipulation, with th idntifications z R Z = and x R X =. ( ) + + = + θ θ φ θ θ φ φ π d dv m m k E de x R d c d ik R d c d ik ikr ik o s v v v v v sin cos cos sin

79 As in Rayligh-Dby-Gans thoy, w sum (intgat) this ov th total scatt volum. W intoduc f and f as fom factos fo th sph: E s wh ik 3k m = des = Eo V θ θ φ 4π m + [ f cosθ cosφ + f sinθ f sinφ ] 7.4 f (,φ) ( d + c ) ( d + c ) ik R z ir k z k d ikr d = d dv = d dv θ 7.5 V V f ( θ, φ) = V = V ik ( d c ) ir k R x ( d c ) x k ikr dv V dv ir ( k ) dv 7.6 Not that if k = k, sinc (as notd abov) c = d d = =, f ducs to ikr ( Z ) R dv V, th fom facto f in th RDG thoy. Futhmo obsv that th facto f, which dos not appa in th RDG thoy, gos to zo whn k = k. Th poblm now bcoms how to calculat th intgals in quations 7.5 and 7.6. Thy all hav th fom ir S dv with constant S. Consid a local coodinat systm in th sph with its z axis alignd with S. If w look at figu 7.4, th lmnt of volum at hight z is ( x' y' ) dz ' dv = bas hight = π

80 Figu 7.4: Diagam of th Volum at Hight z fo a Sph Howv x ' + y' + z' = a and z uns fom -a to a; thfo, v ir S dv = v v i S R z ' dv = a z' = a i S z' ( π a z' ) dz' 7.8 W can us Mapl to pfom ths intgals. Th sults a iaa iaa iaa iaa iaa + iaa + f = dπ V i A f = V iba iba iba iba ica ica ica ica iba + iba + ica + ica + π i B i C A = B = C = k k k k + k + = k ( d + c ) kk ( d + c ) 4d ( d c ) 4 kk d ( d c ) cosθ cosφsinθ

81 Obsv fom figu 7. that θ and φ a th dtcto angls and that θ is in th scatting plan whil φ is ppndicula. Thfo paalll and ppndicula componnts of th scattd fild, quation 7.4, a xpssd in tms of th fom factos as Th scatting intnsity is givn by E ik 3k m = E o V [ f cos θ cos φ sin φ ] 7.3 4π m +, s + f ik 3k m E, s = E o V [ f sin φ ] π m + I s = R µ [ E, s + E s ] Scatting Amplitud Matix Fomulation fo th Hybid Modl Not that in this nw modl th scattd fild can still b xpssd using a scatting matix in th mann of Van d Hulst, Bohn and Huffman, and Kk. To do so, th incoming fild must b xpssd in tms of its componnts paalll and ppndicula to th scatting plan. In sphical coodinats th incoming fild is givn by E i = E o ikz ( sinθ cosφˆ + cosθ cosφˆ sinφˆ ) θ φ 7.35 H φ is ppndicula to th scatting plan whil th unit vcto lis in th plan. As a sult th incoming fild can b wittn as: cos θ θ + sinθ E E, i, i = E o = E o ikz ikz cosφ sinφ

82 Aft som manipulation th scattd fild (quations 7.3, 7.33) can b latd to th incidnt fild in a scatting matix fomat: E E, s, s 3k m = 4π m V + f cosθ + f sinθ cosφ f E E, i, i Scatting Intnsity Ratio and Tubidity Th scatting intnsity atio is xpssd using quation 7.4. I I s o [ f cosθ cosφ+ f sinθ f sinφ ] 4 Es 9k m = = V 7.38 E 3π m + o It can b wittn in tms of th scatting amplitud matix quation 7.37; howv this is not commndd du to th singulaity ( ) cosθ. Th fomula fo tubidity in th hybid modl is divd by calculating C sca fom instion of quation 7.38 into quation.6 fom chapt two; th absoption coss sction C abs mains as in quation.9: m C abs = 3kV Im 7.39 m + th tubidity is finally dtmind by (quation.6) p ( Csca + Cabs ) N plgqxt τ = N l =

83 Chapt Eight Validation and Snsitivity of Hybid Thoy Th pvious chapt gav a dtaild mathmatical dsciption of th hybid modl. This chapt is ddicatd to pfomanc valuation of th modl in compaison to thos of Rayligh-Dby-Gans and of Mi though a sis of tansmission simulation studis. Fist th hybid modl was tstd with th popagation constant of th mdium qual to that of th paticl to vify coct pogamming implmntation. Scond th validity of th hybid thoy using vaious paticl sizs was tstd fo lativ factiv indics clos to on. Th thid study tsts th hybid thoy s ffctivnss by intoducing absoption though th imaginay pat of th factiv indx. Th last study invstigats th bhavio of th hybid thoy fo factiv indics xcding th conditions quid fo Rayligh-Dby-Gans thoy, that is, having stong scatting and absoption componnts, fo vaious diamt sizs. Th oth paamts (concntation, pathlngth, tc.) quid to calculat th tansmission w kpt constant,as listd in tabl Validation of Hybid Thoy Implmntation As w indicatd in chapt six, whn k k th intnal Mi fild appoachs th incoming fild; c, d, and d all appoach. Th calculations displayd by figu 8. confim that c = d = d whn k = k =. 67

84 .8.6 c d d.4 Cofficints Wavlngth (nm) Figu 8.: Cofficints c, d, and d Vsus λ at th Limit whn k =k Anoth ffct of taking k qual to k is that th fom facto f of th hybid modl fo sphs ducs to th fom facto f of Rayligh-Dby-Gans fo sphs (and f gos to zo). Figus 8. and 8.3 a 3-D imags of ths fom factos gaphd by wavlngth λ and azimuthal angl of obsvation θ (with φ = ). Figu 8. shows f of th hybid modl. Th fom facto f of Rayligh-Dby-Gans is shown in figu 8.3. As an xampl of th cas wh k (paticl) is diffnt fom k (mdium), th factiv indics of polystyn (. κ. 6 ) in wat ( κ = ) w usd to illustat th bhavio of th fom factos fo th hybid modl compad to that of Rayligh-Dby-Gans. Figus 8.4, 8.5, 8.6, and 8.7 a th al and imaginay pats of th fom factos f and f fo th hybid modl. Figu 8.8 is th fom facto fo Rayligh- 68

85 Dby-Gans. Not that th hybid modl s factos contain al and imaginay pats vn if κ = (nonabsobing). Thfo ou woking pogam fo th hybid thoy has bn validatd. Th nxt sctions will val th supioity of th hybid thoy in appoximating th xact (Mi) solution, fo diffnt paticl diamts and lativ factiv indics. Figu 8.: Fom Facto f fo Hybid Thoy at k =k 69

86 Figu 8.3: Fom Facto f fo Rayligh-Dby-Gans Thoy Figu 8.4: Ral Pat of Fom Facto f fo Hybid Thoy using Polystyn 7

87 Figu 8.5: Imaginay Pat of Fom Facto f fo Hybid Thoy using Polystyn Figu 8.6: Ral Pat of Fom Facto f fo Hybid Thoy using Polystyn 7