Light scattering and absorption by atmospheric particulates. Part 2: Scattering and absorption by spherical particles.

Transcription

1 Lctu. Light cattig ad abpti by atmphic paticulat. at : cattig ad abpti by phical paticl. Objctiv:. Maxwll quati. Wav quati. Dilctical ctat f a mdium.. Mi-Dby thy. 3. Vlum ptical ppti f a mbl f paticl. Rquid Radig: L: 5., 3.3. Additial/Advacd Radig: Bh, G.F., ad D.R. Huffma, Abpti ad cattig f light by mall paticl. Jh Wily&, 983 (Mi thy ivati i giv pp.8-, a hadcpy will b pvidd i cla). Maxwll quati. Wav quati. Dilctical ctat f a mdium. Maxwll quati cct th fiv baic quatiti th lctic vct, E, magtic vct, H, magtic iducti, B, lctic diplacmt, D, ad lctic cut dity, j : (i cg ytm) D π H + j c t c B E c t D πρ B wh c i a ctat (wav vlcity); ad ρ i th lctic chag dity. [.] T allw a uiqu dtmiati f th lctmagtic fild vct, th Maxwll quati mut b upplmtd by lati which dcib th bhavi f ubtac u th ifluc f lctmagtic fild. Thy a j σe D εe B µ H [.] wh σ i calld th pcific cductivity; ε i calld th dilctical ctat ( th pmittivity), ad µ i calld th magtic pmability.

2 Dpdig th valu f σ, th ubtac a dividd it: cduct: σ dilctic ( iulat: σ (i.., σ i NOT gligibly mall), (f itac, mtal) (i.., σ i gligibly mall), (f itac, ai, al ad clud paticulat) Lt ci th ppagati f EM wav i a mdium which i (a) uifm, that ε ha th am valu at all pit; (b) itpic, that ε i idpdt f th dicti f ppagati; (c) -cductig (dilctic), that σ ad thf j ; (d) f fm chag, that ρ. With th aumpti th Maxwll quati duc t ε E H c t µ H E c t E H [.3] Elimiatig E ad H i th fit tw quati i [.3] ad uig th vct thm, w hav εµ E E [.] c t εµ H H c t Th abv quati a tadad quati f wav mti f a wav ppagatig with a vlcity wh c i th pd f light i vacuum. c v [.5] εµ NOTE: f vacuum: µ ad ε i cg uit, but i I ytm µ ο ad ε a ctat uch that c / ε µ.

3 F mt ubtac (icludig th ai µ i uity. Thu, th lctical ppti f a mdium a chaactizd by th dilctical ctat ε. Rfactiv idx ( ptical ctat) f a mdium i dfid a m ε [.6] aumig that µ. NOTE: tictly pakig, ε i Eq.[.6] i th lativ pmittivity f mdium (h it i lativ t vacuum). Rfactiv idx: Th factiv idx mm - im i i cmmly xpd a a cmplx umb. Th z imagiay pat m i f th factiv idx i pibl f abpti f th wav a it ppagat thugh th mdium; wha th al pat m f th factiv idx lat t th vlcity f ppagati f th EM wav. Th factiv idx i a tg fucti f th wavlgth. Each ubtac ha a pcific pctum f th factiv idx ( figu , Lctu 5) aticl f difft iz, hap ad idic f facti will hav difft cattig ad abbig ppti. Al paticl ft cit f val chmical pci (calld th ital mixtu. Th a val appach (calld mixig ul) t calculat th ffctiv factiv idx m f th itally mixd paticl uig th factiv idic f th idividual pci ( Lctu 5) 3

4 cattig dmai: Rayligh cattig: π/ << ad m i abitay (appli t cattig by mlcul ad mall al paticl); π Rayligh-Ga cattig: m << ad m << (t uful f atmphic applicati); Mi-Dby cattig: π/ ad m a bth abitay but f ph ly (appli t cattig by al ad clud paticl) Gmtic ptic: π/ >> ad m i al (appli t cattig by lag clud plt). Figu. Rlatihip btw paticl iz, adiati wavlgth ad cattig bhavi f atmphic paticl. Diagal dahd li pt ugh budai btw cattig gim.

5 . Mi-Dby thy. NOTE: Mi-Dby thy i ft calld Mi thy Ltz-Mi thy Mi thy utli: Aumpti: i) aticl i a ph f adiu ii) aticl i hmgu (thf it i chaactizd by a factiv idx mm - im i at a giv wavlgth); NOTE: Mi thy qui th lativ factiv idx factiv idx f a paticl dividd by th factiv idx f a mdium. But f ai m i abut, d t kw th factiv idx f th paticl (i.., factiv idx f th matial f which th paticl i cmpd). NOTE: If a paticl ha cmplx chmical cmpiti, th ffctiv factiv idx mut b calculatd at a giv wavlgth. tatgy: ) k a luti f a vct wav quati (Eq.[.]) f E ad H ε E E c t with th buday cditi that th tagtial cmpt f E ad H b ctiuu ac th phical ufac f a paticl. Aumpti th phical ufac f a paticl allw lvig th vct quati aalytically. ) R-wit th wav quati i phical cdiat ad xp lctic fild iid ad utid ph i vct phical hamic xpai. NOTE: Mi thy calculat th lctmagtic fild at all pit i th paticl (calld ital fild) ad at all pit f th hmgu mdium i which th paticl i mbddd. F all pactical applicati i th atmph, light cattig bvati a caid ut i th fa-fild z (i.., at th lag ditac fm a ph: 3) Apply buday cditi match tav fild at ph ufac t btai cattd phical wav Mi cfficit a ad b which d t dpd th agl 5

6 but dpd iz paamt x π/ ( i th adiu f th paticl) ad vaiabl y x m (m i factiv idx f th paticl). ) U i ivlvig a ad b t btai xticti ad cattig fficici ( ad ). 5) U i i Mi agula fucti π ad t btai cattig amplitud fucti (Θ) ad (Θ), fm which th cattig pha fucti i ivd. NOTE: Full ivati f Mi thy a giv i L, cti 5. (ad Bh&Huffma 983, pp.8-). Mi cattig amplitud (al calld cattig fucti) ivd fm Mi thy a ( Eq i L) + [ a π (cθ) + b (cθ ] ( Θ) ) ( + ) + [ b (c Θ) + a (c Θ ] ( Θ) ) ( + ) π [.7] wh Mi cfficit a ad b a ( Eq.5..7 i L) y ( y) y ( x) my ( y) y ( x) my ( y) y ( x) y ( y) y ( x) a ( x, y) b ( x, y) [.8] y ( y) x ( x) my ( y) x ( x) my ( y) x ( x) y ( y) x ( x) h th pim dt difftiati; x π/ ad y x m; πρ ( ρ) J / ( ρ) ad πρ () ξ ( ρ) H + / ( ρ) wh J ) ( +/ ρ i th half-itgal- ψ + () phical Bl fucti ad H i th half-itgal- Hakl fucti f th cd kid; +/ ad π ad a th Mi agula fucti wh π (cθ) (cθ) i( Θ) d (cθ) (cθ) [.9] dθ a th aciatd Lg plymial ( Appdix E)

7 I th fa-fild z (i.., at th lag ditac R fm a ph, Mi thy giv th luti f th vct wav quati a E E l xp( ikr + ikz) ikr 3 E E i l i [.] Eq.[.] i a fudamtal quati f cattd adiati icludig plaizati i th fa fild. ( Θ) ( Θ) 3( Θ) ( Θ) i th amplitud cattig matix (uitl) F ph: 3 (Θ) (Θ) Thu, f ph Eq.[5.] duc t E E l xp( ikr + ikz) ikr wh xp(ikz) i th icidt pla wav, ad wav. E E xp( ikr) ikr i l i [.] i th utgig cattd Fudamtal xticti fmula ( ptical thm) giv th xticti c cti f a paticl π σ R[ ( )], [.] k But f th fwad dicti (i.. Θ ) fm Eq.[.7], w hav [.3] ( ) ( ) ( + )( a + b ) Thu, xticti c cti i latd t cattig i fwad dicti. 7

8 Efficici ( fficicy fact f xticti, cattig ad abpti a dfid a σ π π a a σ [.] π wh π i th paticl aa pjctd t th pla ppdicula t th icidt bam. Mi fficicy fact a ivd fm th Mi cattig amplitud [.5] x ( + ) R[ a + ] b [.6] x ( + )[ a + ] b ad th abpti fficicy ca b calculatd a a [.7] Figu. Exampl f ad a calculatd with Mi thy f val factiv idx. 8

9 cattig pha matix Rcall dfiiti f tk paamt ( Lctu 5), which uiquly chaactiz th lctmagtic wav. Lt I,, U ad V b th tk paamt f icidt fild ad I,, U ad V b th tk paamt f cattd adiati V U I R V U I π [.8] wh i th cattig pha matix [.9] wh ach lmt dpd th cattig agl (/R i fm lid agl) F ph: ad 33 NOTE: I gal, f a paticl f ay hap, th cattig pha matix cit f 6 idpdt lmt, but f a ph thi umb duc t fu. Thu f ph, Eq.[.8] duc t V U I R V U I π [.] wh ach lmt f th cattig pha matix i xpd via th cattig amplitud (Θ) ad (Θ) [ ] k + π [ ] k π [.] 9

10 [ ] 33 k + π [ ] 3 k π (Θ) (Θ) i th cattig pha fucti dfid i Lctu. Figu.3 Exampl f cattig pha fucti calculatd with Mi thy f val iz paamt f abbig ph. Nt icaig cillatig bhavi with icaig iz paamt.

11 m highlight f Mi cattig ult: Exticti fficicy v. iz paamt x (aumig NO ABORTION): ) mall i Rayligh limit: x ) lagt wh paticl ad wavlgth hav imila iz 3) -> i th gmtic limit ( x ) ) Ocillati ( Fig.5.3) fm itfc f tamittd ad diffactd wav id i x f itfc cillati dpd th factiv idx. Abpti duc itfc cillati ad kill ippl tuctu. cattig ad abpti fficici v. iz paamt with ABORTION: A x : ad, tig ay a abbd iid paticl. mall imagiay pat f th factiv idx qui lag paticl t fully abb ital ay. cattig pha fucti: fwad pak hight ica amatically with x. F igl paticl umb f cillati i (Θ) ica with x. 3. Vlum ptical ppti f a mbl f paticl. Mi thy giv th xticti, cattig ad abpti c-cti (ad fficici) ad th cattig pha matix f a igl phical paticl. NOTE: Rcall Lctu wh th paticl iz ditibuti w itducd f atmphic al ad clud. If th paticl chaactizd by a iz ditibuti N(, th vlum xticti, cattig ad abpti cfficit (i uit LENGTH - ) a calculatd a β max mi σ ( N( max β ( N( [.] mi β a max mi σ ( N( a

12 wh σ i th cpdig c cti f a paticl f adiu ad N( i th paticl iz ditibuti (.g., i uit m -3 µm - ). igl cattig albd (uitl) i dfid a ω β [.3] β Th igl cattig albd giv th pctag f light which will b cattd i a igl cattd vt. cattig pha fucti i [ + ] N( max ( Θ) π k β [.] mi ( Θ) max mi ( Θ) N( β [.5] Aymmty paamt i dfid a th fit mmt f th cattig pha fucti g (cθ)c( Θ) d(cθ) [.6] g f qual fwad ad backwad cattig; g f ttally fwad cattig Th Hyy-Gti cattig pha fucti i a mdl pha fucti, which i ft ud i adiativ taf calculati t appximat al cattig: g HG ( Θ) 3/ ( + g g cθ) [.7] wh g i th aymmty paamt.

13 Optical ppti f clud p: F may pactical applicati, th ptical ppti f wat clud a paamtizd a a fucti f th ffctiv adiu ad liquid wat ctt (LWC). Th ffctiv adiu i dfid a 3 N( N( wh N( i th paticl iz ditibuti (.g., i uit m -3 µm - ). Th liquid wat ctt (LWC) wa dfid i Lctu 5 ( Eq.[5.7]): 3 [.8] LWC w V w N( 3 π [.9] Uig that th xticti cfficit f clud plt i σ ( N( β π N( ) ad that f wat plt at la wavlgth, w hav LWC β 3 [.3] w Effctiv ptical ppti f a atmphic lay citig f ga, al ad/ clud paticl: Effctiv (al calld ttal) ptical dpth: wh M a, ad M M A A a, +, + a, +, [.3] M, a ptical dpth du t abpti by ga ad mlcula (Rayligh) cattig, pctivly; A A a, ad, a ptical dpth du t abpti ad cattig by al (ad/ clud) paticl, pctivly. 3

14 Effctiv igl cattig albd: ω A M,,, + [.3] Effctiv cattig pha fucti: A M A A M M,,,, ) ( ) ( ) ( + Θ + Θ Θ [.33] Effctiv aymmty paamt: A M A A g g,,, + [.3]