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

Transcription

1 Lctu Light catting and abptin by atmphic paticulat. at : catting and abptin by phical paticl. Objctiv:. Maxwll quatin. Wav quatin. Dilctical cntant f a mdium.. Mi-Dby thy. 3. Optical ppti f an nmbl f phical paticl. Rquid Rading: L: 5. Additinal/Advancd Rading: Bhn, G.F., and D.R. Huffmn, Abptin and catting f light by mall paticl. Jhn Wily&n, 983. Excllnt wb it n Th ptic f a wat dp: Mi catting Maxwll quatin. Wav quatin. Dilctical cntant f a mdium. Maxwll quatin cnnct th fiv baic quantiti th lctic vct, E, magntic vct, H, magntic inductin, B, lctic diplacmnt, D, and lctic cunt dnity, j : (in cg ytm) D H + c t B E c t D πρ B wh c i a cntant (wav vlcity); and ρ i th lctic chag dnity. π c j [.]

2 T allw a uniqu dtminatin f th lctmagntic fild vct, th Maxwll quatin mut b upplmntd by latin which dcib th bhavi f ubtanc und th influnc f lctmagntic fild. Thy a j E D E [.] B µ E wh i calld th pcific cnductivity; i calld th dilctical cntant ( th pmittivity), and µ i calld th magntic pmability. Dpnding n th valu f, th ubtanc a dividd int: cnduct: dilctic ( inulat): (i.., i NOT ngligibly mall), (f intanc, mtal) (i.., i ngligibly mall), (f intanc, ai al and clud paticulat) Lt cnid th ppagatin f EM wav in a mdium which i (a) unifm, that ha th am valu at all pint; (b) itpic, that i indpndnt f th dictin f ppagatin; (c) nn-cnducting (dilctic), that and thf j ; (d) f fm chag, that ρ. With th aumptin th Maxwll quatin duc t E H c t µ H E c t E H [.3] Eliminating E and H in th fit tw quatin in [.3] and uing th vct thm, w hav

3 µ E E c t µ H H c t [.] Th abv quatin a tandad quatin f wav mtin f a wav ppagating with a vlcity v c µ [.5] wh c i th pd f light in vacuum. NOTE: f vacuum: µ and in cg unit, but in I ytm µ ο and a cntant uch that c / µ. F mt ubtanc (including th ai) µ i unity. Thu, th lctical ppti f a mdium i chaactizd by th dilctical cntant. Rfactiv indx ( ptical cntant) f a mdium i dfind a m [.6] auming that µ. NOTE: tictly paking, in Eq.[.6] i th lativ pmittivity f mdium (h it i lativ t vacuum). Th factiv indx mm +im i in a cmplx numb. Th nnz imaginay pat m i f th factiv indx i pnibl f abptin f th wav a it ppagat thugh th mdium; wha th al pat m f th factiv indx lat t th vlcity f ppagatin f th EM wav. Th factiv indx i a tng functin f th wavlngth. Each ubtanc ha a pcific pctum f th factiv indx. 3

4 aticl f diffnt iz, hap and indic f factin will hav diffnt catting and abbing ppti. Rfactiv indic f wat, ic and m al pci Figu. Ral and imaginay pat f th factiv indx f wat and ic in th IR.

5 Figu. Ral and imaginay pat f th factiv indx f wat and ic in th viibl and na-ir. NOTE: wat ha lw imaginay pat in th viibl > ngligibl abptin by wat dp in th viibl 5

6 Figu.3 Imaginay pat f th factiv indx f m al matial (Bhn and Huffman, Fig.5.6). NOTE: Main abbing pci in th W a black cabn (t) and hmatit (dut), but in th LW vaiu pci hav high imaginay pat f th factiv indx. But vall abptin (i.., abptin cfficint) i al cntlld by paticl iz. 6

7 Al paticl ftn cnit f val chmical pci (calld th intnal mixtu). Th a val appach (calld mixing ul) t calculat th ffctiv factiv indx m f th intnally mixd paticl uing th factiv indic f th individual pci: A) Vlum ( ma) wightd mixing: m m f [.7] wh m j i th factiv indx f j-pci and f j i it vlum factin. B) Buggman appximatin f tw andmly mixd pci: f j j j + f [.8] + + wh i a th dilctic cntant f tw matial and f i a thi vlum factin. Rcall that th factiv indx i m C) Maxwll-Gantt appximatin f tw pciu whn n i a matix (ht matial) with th dilctic cntant and anth i an incluin with : [.9] f + + f + + NOTE: B) and C) appach can b xtndd f th n-cmpnnt mixtu. catting dmain: Rayligh catting: π/λ << and m i abitay (appli t catting by mlcul and mall al paticl); π Rayligh-Gan catting: m << and m << (nt uful f atmphic λ applicatin); Mi-Dby catting: π/λ and m a bth abitay but f ph nly (appli t catting by al and clud paticl) Gmtic ptic: π/λ >> and m i al (appli t catting by lag clud dplt). 7

8 . Mi-Dby thy. NOTE: Mi-Dby thy i ftn calld Mi thy Lntz-Mi thy. Mi thy utlin: Aumptin: i) aticl i a ph f adiu a; ii) aticl i hmgnu (thf it i chaactizd by a ingl factiv indx mm + im i at a givn wavlngth); NOTE: Mi thy qui th lativ factiv indx factiv indx f a paticl/factiv indx f a mdium. But f ai m i abut, n nd t knw th factiv indx f th paticl (i.., factiv indx f th matial f which th paticl i cmpd). NOTE: If a paticl ha cmplx chmical cmpitin, th ffctiv factiv indx mut b calculatd at a givn wavlngth. tatgy: ) k a lutin f a vct wav quatin (Eq.[.]) f E and H E c E t with th bunday cnditin that th tangntial cmpnnt f E and H b cntinuu ac th phical ufac f a paticl. Aumptin n th phical ufac f a paticl allw lving th vct quatin analytically. ) R-wit th wav quatin in phical cdinat and xp lctic fild inid and utid ph in a vct phical hamnic xpanin. NOTE: Mi thy calculat th lctmagntic fild at all pint in th paticl (calld intnal fild) and at all pint f th hmgnu mdium in which th paticl i mbddd. F all pactical applicatin in th atmph, light catting bvatin a caid ut in th fa-fild zn (i.., at th lag ditanc fm a ph): 3) Apply bunday cnditin match tanv fild at ph ufac t btain cattd phical wav Mi cfficint a n and b n which dn t dpnd n th angl 8

9 but dpnd n iz paamt x πa/λ (a i th adiu f th paticl) and vaiabl y x m (m i factiv indx f th paticl). Th xpin f a n and b n a givn by Eq.[5..7] in L. ) U i invlving a n and b n t btain xtinctin and catting fficinci ( and ). 5) U i in Mi angula functin π n and τ n t btain catting amplitud functin (Θ) and (Θ), fm which th catting pha functin i divd. Mi catting amplitud (al calld catting functin) a intducd a n + [ a n π (c Θ ) + b τ (c Θ ] ( Θ ) n n n ) n n ( n + ) n + [ b n (c Θ ) + a τ (c Θ ] ( Θ ) n n n ) n n ( n + ) wh π n and τ n a Mi angula functin π [.] π (c n Θ ) in( Θ ) n (c Θ ) wh d τ n (c Θ ) n (c Θ ) [.] dθ n a th aciatd Lgnd plynmial. In th fa-fild zn (i.., at th lag ditanc fm a ph), th lutin f th vct wav quatin can b btaind a E E l xp( ik + ikz ) ik 3 E E i l i [.] Eq.[.] i a fundamntal quatin f cattd adiatin including plaizatin in th fa fild. ( Θ) ( Θ) 3 ( Θ) ( Θ) i th amplitud catting matix (unitl) F ph: 3 (Θ) (Θ) 9

10 Thu, f ph Eq.[.] duc t wh xp( ikz ) wav. E E l xp( ik + ikz ) ik i th incidnt plan wav, and E E xp( ik ) ik i l i [.3] i th utging cattd Fundamntal xtinctin fmula ( ptical thm) giv th xtinctin c ctin f a paticl π R[ ( )], [.] k But f th fwad dictin (i.. Θ ) fm Eq.[.], w hav [.5] ( ) ( ) (n + )( a n + b n ) n Thu, xtinctin c ctin i latd t catting in fwad dictin. Efficinci ( fficincy fact) f xtinctin, catting and abptin a dfind a a a π a π a [.6] π a wh πa i th paticl aa pjctd nt th plan ppndicula t th incidnt bam. Mi fficincy fact a divd fm th Mi catting amplitud [.7] x ( n + ) R[ a + ] n b n n [.8] x ( n + )[ a + ] n b n n and th abptin fficincy can b calculatd a a [.9]

11 Figu. Exampl f calculatd with Mi thy f val factiv indx. (m in Lab 7). catting pha matix Rcall dfinitin f tk paamt ( Lctu 3), which uniquly chaactiz th lctmagntic wav. Lt I,, U and V b th tk paamt f incidnt fild and I,, U and V b th tk paamt f cattd adiatin V U I V U I π [.] wh i th catting pha matix.

12 [.] wh ach lmnt dpnd n th catting angl (/ i fm lid angl) F ph: and 33 NOTE: In gnal, f a paticl f any hap, th catting pha matix cnit f 6 indpndnt lmnt, but f a ph thi numb duc t fu. Thu f ph, Eq.[.] duc t V U I V U I π [.] wh ach lmnt f th catting pha matix i xpd via th catting amplitud (Θ) and (Θ) [ ] k + π [ ] k π [.3] [ ] 33 k + π [ ] 3 k π (Θ) (Θ) i th catting pha functin dfind in Lctu 3.

13 Figu.5 Exampl f catting pha functin calculatd with Mi thy f val iz paamt f nnabbing ph (m in Lab 7). m highlight f Mi catting ult Extinctin fficincy v. iz paamt x (auming NO ABORTION): ) mall in Rayligh limit: x ) lagt whn paticl and wavlngth hav imila iz 3) --> in gmtic limit ( x ) ) Ocillatin ( Fig..) fm intfnc f tanmittd and diffactd wav id in x f intfnc cillatin dpnd n th factiv indx. Abptin duc intfnc cillatin and kill ippl tuctu. catting and abptin fficinci v. iz paamt with ABORTION: A x : and, nting ay a abbd inid paticl. mall imaginay pat f th factiv indx qui lag paticl t fully abb intnal ay. catting pha functin: fwad pak hight inca damatically with x. F ingl paticl numb f cillatin in (Θ) inca with x. 3

14 3. Optical ppti f an nmbl f phical paticl. Mi thy giv th xtinctin, catting and abptin c-ctin and th catting pha matix f a ingl phical paticl. Rcall Lctu wh th al paticl iz ditibutin w intducd. If th paticl chaactizd by a iz ditibutin N(), th vlum xtinctin, catting and abptin cfficint (in unit LENGTH - ) a calculatd a max min ( ) N ( ) d max ( ) N ( ) d [.] a min max min ( ) N ( ) d ingl catting albd (unitl) if dfind a ω a [.5] Th ingl catting albd giv th pcntag f light which will b cattd in a ingl cattd vnt. catting pha functin i ( Θ ) k ( Θ ) π max [ ] N ( ) d + [.6] min max min ( Θ ) N ( ) d [.6a] Aymmty paamt i fit mmnt f th catting pha functin and i dfind a g (c Θ) c( Θ) d (c Θ) [.6b] g f qual fwad and backwad; g f ttally fwad F many pactical applicatin, th ptical ppti f wat clud a paamtizd a a functin f th ffctiv adiu and liquid wat cntnt (LWC).

15 Th ffctiv adiu i dfind a 3 N ( ) d N ( ) d wh N() i th dplt iz ditibutin (.g., in unit m -3 µm - ). Th liquid wat cntnt (LWC) wa dfind in Lctu : [.7] 3 LWC ρ w V ρ w N ( ) d 3 π [.8] Uing that th xtinctin cfficint f clud dplt i N d N d ( ) ( ) π ( ) and that f wat dplt at la wavlngth, w hav 3 LWC ρ [.9] w Figu.6 Exampl f ptical ppti f typical cumulu and tatu clud (f a clud dplt iz ditibutin ff 6 µm). H th nmalizd xtinctin cfficint i ( λ ) / (.5 m ) and (.5 ).8km. µm µ 5