The scattering of light may be thought of as the redirection of light that takes place when

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1 D.W.H. July 009 Itrducti Light Scatterig Thery David W. Hah Departmet f Mechaical ad Aerspace Egieerig Uiversity f Flrida (dwhah@ufl.edu) The terig f light may be thught f as the redirecti f light that takes place whe a electrmagetic (EM) wave (i.e. a icidet light ray) ecuters a bstacle r hmgeeity, i ur case the terig particle. As the EM wave iteracts with the discrete particle, the electr rbits withi the particle s cstituet mlecules are perturbed peridically with the same frequecy ( ) as the electric field f the icidet wave. The scillati r perturbati f the electr clud results i a peridic separati f charge withi the mlecule, which is called a iduced diple mmet. The scillatig iduced diple mmet is maifest as a surce f EM radiati, thereby resultig i tered light. The majrity f light tered by the particle is emitted at the idetical frequecy ( ) f the icidet light, a prcess referred t as elastic terig. I summary, the abve cmmets describe the prcess f light terig as a cmplex iteracti betwee the icidet EM wave ad the mlecular/atmic structure f the terig bject; hece light terig is t simply a matter f icidet phts r EM waves bucig ff the surface f a ecutered bject. Electric field Scattered wave ( ) Icidet EM wave ( ) Figure. Light terig by a iduced diple mmet due t a icidet EM wave.

2 D.W.H. July 009 Frmal light terig thery may be categrized i terms f tw theretical framewrks. Oe is the thery f Rayleigh terig (after Lrd Rayleigh) that is, strictly speakig as rigially frmulated, applicable t small, dielectric (-absrbig), spherical particles. The secd is the thery f Mie terig (after Gustav Mie) that ecmpasses the geeral spherical terig sluti (absrbig r -absrbig) withut a particular bud particle size. Accrdigly, Mie terig thery has size limitatis ad cverges t the limit f gemetric ptics fr large particles. Mie thery, therefre, may be used fr describig mst spherical particle terig systems, icludig Rayleigh terig. Hwever, Rayleigh terig thery is geerally preferred if applicable, due t the cmplexity f the Mie terig frmulati. The criteria fr Rayleigh terig is that << ad m<<, where is the dimesiless size parameter give by the expressi a, () where a is the spherical particle radius, ad is the relative terig wavelegth defied as, () m where is the icidet wavelegth with respect t vacuum, ad m represets the refractive idex f the surrudig medium. Fially, m is the refractive idex f the terig particle, ad is cmmly represeted by the cmplex tati defied as m = - ik. (3) I this tati, idicates the refracti f light (i.e. equals the speed f light i vacuum divided by the speed f light i the material), while the cmplex term is related t absrpti. The cmmly used absrpti cefficiet f the material (cm - ) is related t the cmplex part f the refractive idex via the relati

3 D.W.H. July absrpti cefficiet. (4) It is ted that the value f k is ever exactly zer fr ay material, but materials with a value apprachig zer are termed dielectrics. The magitude f the refractive idex, m, as eeded fr the Rayleigh criteria, is give by the expressi / m. (5) The Rayleigh criteria as related abve, amely << ad m<<, crrespd physically t the assumptis that the particle is sufficietly small such that the particle ecuters a uifrm electric field at ay mmet, accrdigly the time fr peetrati f the electric field is much less tha the perid f scillati f the EM wave. Figure shws the spherical crdiate terig gemetry used fr Mie ad Rayleigh light terig crrespdig t a sigle icidet light ray a sigle spherical particle. Usig this crdiate system, the terig parameters may be defied fr the Rayleigh ad Mie slutis. x S m E B S ic m z y Figure. Crdiate gemetry fr Rayleigh ad Mie terig. 3

4 D.W.H. July 009 Fr each terig agle (,), the Equatis (6) ad (7) represet the itesities (W/cm ) f tered radiati vertically ad hriztally plarized with respect t the terig plae, respectively, which is defied by the icidet ray (f itesity I ) ad the tered ray, tig the plarizati state f the icidet ray as shw i Figure, I I i si 4 r, (6) I I i cs 4 r. (7) Fr perfectly spherical particles, plarized icidet radiati prduces similarly plarized tered radiati; hece the terig prblem may be redefied i terms f the plarizati states with respect t the terig plae. Accrdigly, equatis (6) ad (7) may be recast i terms f the differetial terig crss sectis (cm /sr), amely I I HH I (8) r I HH. (9) r I these tw equatis, the subscripts refer t the state f plarizati f the icidet ad tered light, respectively, with rietati defied by the terig plae. Specifically, the subscripts refer t bth vertically plarized icidet light ad vertically plarized tered light with respect t the terig plae (i.e. = 90 ). Similarly, the subscripts HH refer t bth hriztally plarized icidet light ad hriztally plarized tered light with respect t the terig plae (i.e. = 0 ). Fr uplarized icidet light, the terig is give by the fllwig I I, (0) r 4

5 D.W.H. July 009 where is the average f ad HH, ad tig there is depedece. Fially, it is ted that the depedecy f the abve quatities the terig agle is thrugh the differetial crss sectis, as detailed belw. Equatis 8-0 prvide a expressi fr the itesity f light abut a sigle tered ray. These equatis may als be recsidered i terms f the rate f tered eergy it a defied slid agle, as shw i Figure 3. Figure 3. Agular terig itesity. Usig the differetial terig crss secti, the ttal tered eergy rate (W) strikig da is E I d, () where the slid agle d is related t the subteded area by d = da/r. The uits f equati () are readily apparet. Substituti f da/r fr the slid agle ad divisi f bth sides by da yields equati (0); hece the tered itesity abut the tered ray I. While the abve equatis accut fr the redistributi f icidet radiati due t light terig, icidet radiati may als be absrbed by the particle. The rate f the ttal amut f icidet eergy abstracted frm the icidet beam due t iteractis with a sigle particle is calculated directly frm the exticti crss secti (cm ), 5

6 D.W.H. July 009 E. () remved I ext The exticti crss secti represets lss f eergy frm the icidet beam due t bth terig ad absrpti; hece the exticti crss secti may be expressed as ext = abs +, (3) where abs ad are the absrpti ad ttal terig crss sectis (cm ), respectively. The latter quatity is calculated by itegratig the differetial crss secti ver 4 steradias. While the abve equatis allw calculati f the relevat terig ad exticti quatities based the icidet light itesity, it w falls t the Rayleigh ad Mie theries t prvide the apprpriate expressis fr calculati f the varius crss-sectis expressed abve. Rayleigh Thery I the Rayleigh regime, the differetial terig crss sectis are readily calculated frm the fllwig equatis: 6 m 4 m, (4) HH cs. (5) Examiati f equatis (4) ad (5) reveals several iterestig items. Fuctially, the differetial terig crss sectis are prprtial t the 6 th pwer f particle size, ad are iversely prprtial t the 4 th pwer f wavelegth. This latter depedecy gives rise t the blue clr f ur sky, as the air mlecules (e.g. N ad O ) are well withi the Rayleigh regime; hece the shrter blue light f the su is mre efficietly redirected ut f the direct path f sulight ad subsequetly redirected frm all directis as tered light. I additi, te that the vertical-vertical differetial terig crss secti is idepedet f the bservati 6

7 D.W.H. July 009 agle, while the hriztal-hriztal differetial terig crss secti has a miimum at 90. This implies that uplarized light will be strgly plarized at 90 bservati fr Rayleigh particles. The ttal terig crss secti (cm ) ad absrpti crss secti (cm ) are defied as 6 m sca, (6) 3 m 3 m abs Im. (7) m Fially, the ttal exticti crss secti (cm ) is defied as a sum f the terig ad absrpti crss sectis, amely, ext. (8) sca abs As represeted i equatis (6) ad (7), the terig crss secti scales with 6, while the absrpti crss secti is prprtial t 3. I the Rayleigh regime, the size parameter must be much less tha uity, therefre the ctributi f terig (i.e. sca ) t the ttal exticti crss secti is geerally eglected fr a absrbig particle ( k 0 ), ad it is therefre assumed that ext abs. Hwever, fr a dielectric particle ( k 0 ), the ext, as the ctributi f absrpti is idetically zer ( 0). abs Mie Thery Based the thery f Mie, the differetial terig crss sectis are defied i terms f the agular itesity fuctis i ad i, as give by the expressis i (9) 4 7

8 D.W.H. July 009 HH i. (0) 4 As befre, the abve tw equatis are averaged t defie the differetial terig crss secti fr uplarized icidet light, which gives the relati ( i i ) () 8 I this frmulati, the itesity fuctis are calculated frm the ifiite series give by i a b cs cs, () i a b cs cs. (3) I the equatis () ad (3), the agular depedet fuctis ad are expressed i terms f the Legedre plymials by () P cs cs, (4) si () dp cs cs, (5) d where the parameters a ad b are defied as a m m m m m m, (6) b m m m m m m. (7) The size parameter is defied usig Equatis () ad () as am. (8) 8

9 D.W.H. July 009 The Ricatti-Bessel fuctis ad are defied i terms f the half-iteger-rder Bessel fucti f the first kid (J +/ (z)), where z z J z. (9) Equati (30) describes the parameter z z H z zi z, (30) where H +/ (z) is the half-iteger-rder Hakel fucti f the secd kid, where the parameter X is defied i terms f the half-iteger-rder Bessel fucti f the secd kid, Y +/ (z), amely z z Y z. (3) Fially, the ttal exticti ad terig crss sectis are expressed as (3) ext ( ) Rea b 0 ), (33) ( )( a b 0 tig that the absrpti crss secti is readily calculated frm the abve tw. 9

10 D.W.H. July 009 Refereces This paper primarily fllws the treatmet ad tati f Kerker (969) frm his classic textbk light terig. Fr thse iterested i the develpmet f light terig histry, Kerker prvides a excellet summary i Secti 3.4, Histrical Perspective. I csider the fllwig three refereces t be the classic texts light terig: M. Kerker. The terig f light ad ther electrmagetic radiati. Academic, New Yrk, 969. H.C. va de Hulst. Light terig by small particles. Jh Wiley & Ss, New Yrk, 957. (Paperback by Dver Publicatis, New Yrk, 98) C.F. Bhre ad D.R. Huffma. Absrpti ad terig f light by small particles. Jh Wiley & Ss, New Yrk, 983. (Paperback by Wily Sciece Paperback Series, 998) Light Scatterig Behavir The fllwig plts briefly depict the behavir f Mie ad Rayleigh terig fr spherical particles fr a rage f cditis, icludig agular depedecy ad size depedecy. 0

11 D.W.H. July 009 Diff. Scatterig Crss Secti (cm sr - ) (x0-3 ).7 m 70 m m=.4-0i =53 m (x0 6 ) 7 m Scatterig Agle (deg) Figure 4. This plt presets the differetial terig crss-secti ( ) as a fucti f terig agle (see Figure ), hece 0 is frward terig ad 80 is backterig. The smallest particle size f 7 m ( = 0.) is well withi the Rayleigh regime, ad therefre the agular depedece is egligible. The 70 m particle ( =.0) is departig frm the Rayleigh regime, ad therefre displays sme agular variati, with frward terig begiig t emerge. Nte, hwever, that the fucti is still mtic. The.7 m particle ( = 0) is clearly i the Mie regime, where frward terig dmiates. Hwever, te the strg ripple structure as a fucti f terig agle. This is characteristic f Mie terig, ad is due t the cmplex iteractis f tered ad refracted rays that result i cstructive ad destructive iterferece alg differet paths (i.e. differet terig agles). Nte the scale factrs i paretheses fr the l7 m ad.7 m particle plts, hece these crss-sectis vary by mre tha 0 rders f magitude fr the tw-rder f magitude variati i particle size.

12 D.W.H. July m=.4-0i =53 m 4 3 Q ext Size Parameter () Figure 5. This plt presets the exticti efficiecy as a fucti f size parameter. The exticti efficiecy is the exticti crss secti ( ext ) rmalized by the gemetric crss secti (a ). Fr Rayleigh sized particles, the exticti efficiecy is much less tha. Fr particles i the Mie regime, tw characteristics are ted. The ripple r resace structure bserved i the agular terig plt is als preset with exticti, althugh t as pruced. I additi, the asympttic limit f the abve plt is, meaig that i the large particle limit, twice as much eergy is remved as expected based the gemetric crss secti. This is ctrary t bservati, ad is referred t as the exticti paradx. I actuality, e half f the eergy is remved by terig/exticti, while the secd half is remved via ear-frward agle diffracti. Sice a bserver geerally sees the diffracted eergy, it appears that a large particle ly remves eergy equal t its actual gemetric crss-secti.

13 D.W.H. July 009 Diff. Scat. Crss Secti at 5 (cm sr - ) = 53 m m =.4-0.0i m=.4-0.i m =.4 -.0i m =.4 -.4i Particle Diameter (m) Figure 6. This plt presets the differetial terig crss secti at a fixed agle f 5 fr vertical-vertical terig ( ) as a fucti f terig particle diameter fr varius particle refractive idices. Data are preseted fr a dielectric, as well as icreasigly absrptive particles, ad fially fr a particle with ehaced refracti ad absrpti. The mai pit illustrated frm these plts is the cmplex ature f the depedecy f Mie terig refractive idex. It is very difficult t make a priri assumptis abut the relative magitude f terig fr a give particle size ad terig agle with regard t the value f refractive idex m. 3