Lectue 15. Light catteing and aboption by atmopheic paticuate. Pat 1: Pincipe of catteing. Main concept: eementay wave, poaization, Stoke matix, and catteing phae function. Rayeigh catteing. Objective: 1. Pincipe of catteing. Main concept: eementay wave, poaization, Stoke matix, and catteing phae function.. Rayeigh (moecua) catteing. Requied Reading: L0: 1.1.4; 3.3.1 Additiona Reading: Buchotz, A. (1995), Rayeigh-catteing cacuation fo the teetia atmophee, App. Optic., 34(15), 765-773. 1. Pincipe of catteing and main concept. Figue 15.1 Simpified viuaization of catteing of an incident wave by a patice. Note: incident fied i epeented by a pane wave wheea catteed fied by a pheica wave. 1
How catteing wok: Conide a inge abitay patice compoed of many dipoe. The incident eectomagnetic fied induce dipoe ociation. Dipoe ociate at the fequency of the incident fied and theefoe catte adiation in a diection. n a given diection of obevation, the tota catteed fied i a upepoition of the catteed waveet of thee dipoe. Scatteing of the eectomagnetic adiation i decibed by the caica eectomagnetic theoy, conideing the popagation of a ight beam a a tanvee wave motion (coection of eectomagnetic individua wave). ectomagnetic fied i chaacteized by the eectic vecto and magnetic vecto H, which ae othogona to each othe and to the diection of the popagation. and H obey the Maxwe equation (wi be dicued in Lectue 16) Poynting vecto give the fux of adiant enegy and the diection of popagation a (in cg ytem) c S H [15.1] 4 S i in unit of enegy pe unit time pe unit aea (i.e. fux F); NOT: H mean a vecto poduct of two vecto. Since eectomagnetic fied ha the wave-ike natue, the caica theoy of wave motion can be ued to chaacteize the popagation of adiation. Conide a pane wave popagating in z-diection (i.e., ociate in the x-y pane). The eectic vecto can be decompoed into the paae and pependicua component a exp( i )exp( ikz it) [15.a] a exp( i )exp( ikz it) [15.b] whee a and a ae the ampitude of the paae and pependicua component, epectivey; and ae the phae of the paae and pependicua component,
epectivey; k i the popagation (o wave) contant, k = /and i the cicua fequency, = kc=c/ q.[15.] can be witten in coine epeentation a a a co( ) whee kzt and + i caed the phae. co( ) Then we have / a co( )co( ) in( )in( ) [15.3] and thu / a co( )co( ) in( )in( ) ( / a ) ( / a ) ( / a )( / a )co( ) in ( ) [15.4] whee = - i the phae diffeence (o phae hift). q.[15.4] epeent an eipe => eipticay poaized wave f = m (m =0, +/1; +/- ), then in() =0 and q.[15.4] become a a 0 o a [15.5] q.[15.5] epeent two pependicua ine => ineay poaized wave a f = m (m = +/-1; +/-3, ) and a = a = a, q.[15.4] become q.[15.6] epeent a cice => cicuay poaized wave a [15.6] n genea, ight i a upepoition of many wave of diffeent fequencie, phae, and ampitude. Poaization i detemined by the eative ize and coeation between two eectica fied component. Radiation may be unpoaized, patiay poaized, o competey poaized. 3
Figue 15. xampe of veticay poaized ight. Natua unight i unpoaized. f thee i a definite eation of phae between diffeent cattee => adiation i caed coheent. f thee i no eation in phae hift => ight i caed incoheent Natua unight i incoheent. The popety of incoheent adiation: The intenity due to a catteing cente i the um of individua intenitie. NOT: n ou coue, we tudy the incoheent catteing of the atmopheic adiation. NOT: The aumption of independent cattee i vioated if the patice ae too coey packed (pacing between patice houd be evea time thei diamete to pevent intemoecua foce fom cauing coeation between catteing cente). q.[15.4] how that, in the genea cae, thee independent paamete a, a and ae equied to chaacteize an eectomagnetic wave. Thee paamete ae not meaued => moe convenient to ue anothe et of paamete that ae popotiona to the intenity (caed Stoke paamete). Stoke paamete: o-caed intenity, the degee of poaization Q, the pane of poaization U, and the eipticity V of the eectomagnetic wave 4
Q [15.7] U V i( whee denote the compex conjugate vaue. They ae eated a Stoke paamete can be ao expeed a ) Q U V [15.8] a a Q [15.9] a a U a a co( ) V a a in( ) Actua ight conit of many individua wave each having it own ampitude and phae. The degee of poaization DP of a ight beam i defined a DP ( Q U V ) 1/ / [15.10] The degee of inea poaization LP of a ight beam i defined by negecting U and V a LP Unpoaized ight: Q U V 0 Q [15.11] Fuy poaized ight: Q U V Linea poaized ight: V 0 Cicua poaized ight: V 5
The catteing phae function P(co) i defined a a non-dimeniona paamete to decibe the angua ditibution of the catteed adiation 1 4 P(co ) d 1 whee i the catteing ange between the diection of incidence and obevation. NOT: Common notation fo the phae function P(co) = P(', ',, ), [15.1] whee (', ') and (, ) ae the pheica coodinate of incident beam and diection of obevation, and (ee L0: Appendix C): co( = co('co(in('in(co'-) [15.13] otopic catteing: P(co)=1 Fowad catteing efe to the obevation diection fo which < / Backwad catteing efe to the obevation diection fo which > /. Rayeigh catteing Conide a ma homogeneou pheica patice (e.g., moecue) with ize mae than the waveength of incident adiation 0. Then the induced dipoe moment p 0 i p 0 0 [15.14] whee i the poaizabiity of the patice. NOT: Do not confue the poaization of the medium with poaization aociated with the M wave! The catteed eectic fied at the age ditance (caed fa fied catteing) fom the dipoe i given (in cg unit) by 1 1 p in( ) [15.15] c t whee i the ange between the catteed dipoe moment p and the diection of obevation. 6
The dipoe moment i p p 0 exp( ik( ct)) [15.16] and thu the eectica fied i exp( ik( ct)) 0 k in( ) [15.17] 0 p Dipoe 1 0 Diection of incident adiation 1 =/; =/- p Diection of catteing (out of page) NOT: Pane of catteing (o catteing pane) i defined a a pane containing the incident beam and catteed beam in the diection of obevation. Decompoing the eectica vecto on two othogona component pependicua and paae to the pane of catteing, we have exp( ik( ct)) 0 k in( 1) [15.18] exp( ik( ct)) 0 k in( ) Uing 1 =/; =/-andthat 1 c 4, [15.19] pependicua and paae intenitie (o inea poaized intenitie) ae 4 0 k / [15.0] 4 0 k co ( ) / Uing that the natua ight (incident beam) in not poaized ( 0 = 0 = 0 /) and that k=, we have 7
4 1 co ( ) 0 [15.1] q.[15.1] give the intenity catteed by moecue fo unpoaized incident ight, caed Rayeigh catteing. Rayeigh catteing phae function fo the incident unpoaized adiation (foow fom q.[15.1]) i 3 P (co( )) (1 co ( )) [15.] 4 q.[15.] may be ewitten in the fom 5 0 18 P( ) (co( )) [15.3] 4 3 4 q.[15.1] may be ewitten in the tem of the catteing co ection 0 P( ) (co( )) [15.4] 4 Hee the catteing co ection (in unit o aea) by a inge moecue i 18 3 5 [15.5] 4 The poaizabiity i given by the Loentz-Loenz fomua (ee L0: Appendix D): 3 m 1 [15.6] 4N m whee N in the numbe of moecue pe unit voume and m = m i m i i the efactive index of ai. NOT: Fo ai moecue in oa pectum m i about 1 but depend on andm i =0. Thu the poaizabiity can be appoximated a 1 ( m 1) [15.7] 4N Theefoe the catteing co ection of ai moecue (q.[15.5]) become 8
3 8 ( m 1) f ( ) [15.8] 4 3 N whee f() i the coection facto fo the aniotopic popetie of ai moecue, defined a f() =(6+3)/(6-7) and =0.035 Uing thi catteing co ection of moecue, one can cacuate the optica depth of the entie atmophee due to moecua catteing a top ( ) ( ) N( z) dz [15.9] 0 Appoximation of moecua Rayeigh optica depth (i.e., optica depth due to moecua catteing) down to peue eve p in the ath atmophee: ( ) 0.0088 1013 p mb 4.150. [15.30] Rayeigh catteing eut in the ky poaization. The degee of inea poaization i ( Q co 1 in LP ) [15.31] co 1 co 1 Fowad and backwad catteing diection: unpoaized ight 90 o catteing ange: competey poaized 9