Volume Scattering. Our challenge is to determine how a monochromatic plane wave might scatter and propagate through such a medium.

Transcription

1 11/3/4 Volum Scattrng 1/4 Volum Scattrng In many radar rmot nng applcaton, th llumnatd targt a random collcton of rrgular partcl or lmnt, dprd throughout om 3- dmnonal volum. Our challng to dtrmn how a monochromatc plan wav mght cattr and propagat through uch a mdum. I. lctromagntc Scattrng Frt, lt rvw om background matral, nvolvng th lctromagntc Scattrng from a ngl objct or partcl. * W fnd that th cattrd far-fld from any objct can b xprd n trm of t Scattrng Matrx. * W can lkw dcrb th cattrd powr dnty (or ntnty) from an objct n trm of t Scattrng Cro Scton.

2 11/3/4 Volum Scattrng /4 II. Scattrng from a Random Mda Now, lt condr th ca whr multpl cattrr ar prnt. * Wth knowldg of th cattrng matrx and locaton of ach partcl, w can, provdd w condr all Multpl Scattrng Mchanm, dtrmn th farfld cattrng from th collcton. * For many (or mot!) applcaton, our cattrng volum cont of a random collcton of cattrr, dcrbd wth tattcal maur. Thu, w mut lkw charactrz th Scattrng from Random Mda wth tattcal maur. * Although a cattrng volum can cont many dffrnt typ, hap, and z of random cattrr, w fnd that th tattc of th cattrd fld can uually b wll dcrbd n trm of Raylgh Fadng Stattc.

3 11/3/4 Volum Scattrng 3/4 III. Propagaton through a Random Mda Whn a plan wav ntract wth a partcl, t rduc th nrgy n th wav. * Th rat at whch a ngl partcl rmov nrgy from a plan wav pcfd by t xtncton Cro- Scton. * Th wav attnuaton xhbtd by a collcton of partcl dcrbd by an xtncton Coffcnt. * Ung th Optcal Thorm, w can lkw dtrmn th ffctv propagaton contant for a collcton of random partcl. * W mut account for xtncton f w wh to accuratly dtrmn th avrag cattrng from a larg volum of cattrng partcl. W can accomplh th by mplmntng th Dtortd Born Approxmaton.

4 11/3/4 Volum Scattrng 4/4 IV. Volum Scattrng from Collcton of Raylgh Scattrr * Scattrng from partcl that ar mall wth rpct to a wavlngth dcrbd by a thory known a Raylgh Scattrng. * A Raylgh Scattrr compltly charactrzd by a Polarzablty Tnor. * Th propagaton through a random volum of Raylgh cattrr can b dcrbd n trm of an quvalnt dlctrc contant drvd from Mxng Modl. * W can apply all of our acqurd knowldg to dtrmn th Volum Scattrng from a Layr of Raylgh Scattrr.

5 1/5/4 lctromagntc Scattrng 1/5 lctromagntc Scattrng Condr om known lctromagntc fld, xtng throughout mpty pac: ( r,t ), ( r,t) H ε,µ If w nrt om objct nto th pac, th lctromagntc fld ar modfd: ( r,t ), ( r,t) H σ 1,ε,µ ε,µ

6 1/5/4 lctromagntc Scattrng /5 Th dffrnc btwn th ntal fld ( r,t ), ( r,t) and th modfd fld ( r,t ), ( r,t) cattrd fld ( r,t ), H ( r,t) : H, H, ar dfnd a th ( r,t) ( r,t) ( r,t) ( r,t) ( r,t) ( r,t) H H H Rarrangng, w can altrnatvly tat that: ( r,t) ( r,t) + ( r,t) ( r,t) ( r,t) + ( r,t) H H H Thu, th modfd fld ar formd whn th cattrd r,t, H r,t ar addd to th orgnal (ncdnt) fld ( ) ( ) fld ( r,t ), ( r,t) mot oftn rfrrd to a total fld ( r,t ), ( r,t) H. A a rult, th modfd fld ar H : ( r,t) ( r,t) + ( r,t) ( r,t) ( r,t) + ( r,t) H H H σ 1,ε 1,µ 1 ε,µ

7 1/5/4 lctromagntc Scattrng 3/5 Q: Why ar th ncdnt fld modfd? Whr do th cattrd fld com from? A: Th ncdnt fld nduc currnt (conducton, polarzaton, and/or magntzaton) wthn th objct th currnt n turn crat nw (.., cattrd) fld! W fnd that th total fld can b rlatd to th cattrd fld a: r,t G r,t ; r,t r,t dv dt ( ) ( ) ( ) H r,t G r,t ; r,t H r,t dv dt ( ) ( ) ( ) whr G ( r,t;r,t ) and ( r,t;r,t ) G m Grn functon (not mportant!). m ar calld dyadc Not n th xpron, th cattrd fld ar dpndnt on th total fld (th charg and dpol wthn th objct can t tll th dffrnc btwn a cattrd and ncdnt fld!). But rcall w lkw dtrmnd that total fld ar dpndnt on th cattrd fld: ( r,t) ( r,t) + ( r,t) ( r,t) ( r,t) + ( r,t) H H H

8 1/5/4 lctromagntc Scattrng 4/5 Th crcular logc xprd whn w combn th rult: ( r,t) ( ) + G ( ) ( r,t ) ( t) H ( r,t) + G ( r,t ) H( r,t ) Hr, r,t r,t ; r,t dv dt m ; r,t dv dt Th quaton dmontrat th dffculty n fndng mcrowav cattrng oluton not th unknown total fld ar lkw part of th ntgraton! Th xpron ar thrfor known a ntgral quaton, and fndng thr oluton ar xcdngly dffcult. In fact, human hav found oluton only for th mplt of objct (.g., phr, cylndr)! Q: So thn, how do w dtrmn lctromagntc cattrng? A: Thr ar bacally thr mthod: numrcal approxmaton, aymptotc approxmaton, and drct maurmnt. Numrcal Approxmaton - Gvn th computatonal rourc avalabl today, th mthod ha bcom th mot popular mthod. Tchnqu uch a th momnt mthod allow

9 1/5/4 lctromagntc Scattrng 5/5 u to numrcally approxmaton th ntgraton (.g., a a ummaton). Th rult ar oftn vry accurat, although nc thy ar numrc, thy do not provd much paramtrc nght (.g., th cattrng varaton wth rpct to dlctrc contant), unl w run multpl ca. Aymptotc Approxmaton - Oftn, w fnd that w can olv th ntgral quaton f on or mor of th phycal charactrtc of th objct tak om xtrm valu. For xampl, f th volum of th cattrng objct zro, thn ( r ) and ( r ) ( r ). Although th xact oluton (lk th xampl) ar gnrally not partcularly uful, w can oftn dtrmn from thm approxmat oluton, vald whn th cattrng problm approach th xtrm ca. Accordngly, from th abov xampl, w mght dtrmn an approxmaton that accurat whn th cattrng objct vry mall. Drct Maurmnt - Som cattrng objct ar o complx (.g. arcraft) that cattrng oluton can only b dtrmnd by drct maurmnt n an anchoc chambr. Addtonally, w oftn mak drct maurmnt of mplr objct, o a to valdat a numrc or aymptotc approxmaton.

10 1/5/4 Th Scattrng Matrx 1/8 Th Scattrng Matrx Th ncdnt fld for mot cattrng problm aumd to b a monochromatc plan wav of th form: whr { jω t } ( ) ( ) r,t R r + ( r ) j k r ( ) v v ˆ ˆ h h + j k r and: π k k ˆ k k ˆ λ v ˆ h ˆ k ˆ v ˆ h ˆ Th complx valu v and h thu dfn wav polarzaton, and th unt vctor ˆk dcrb th propagaton drcton. Say now w plac a fnt objct n th ncdnt fld and locat t at th orgn. A cattrd fld mut b gnratd!

11 1/5/4 Th Scattrng Matrx /8 ˆv ĥ ˆk z y If th cattrng objct mad of mpl (.., lnar) matral, and tm nvarant (.., t not movng!), thn th cattrd fld wll hav th form: whr: x { jω t } ( ) ( ) r,t R r + j r ( r ) ( k ) k d k Th mply tat that th cattrd fld a uprpoton of plan wav, propagatng n all pobl drcton k ˆ. Th cattrr multanouly cattr n all drcton! Howvr, th cattrr wll not cattr qually n all drcton, nor wth th am polarzaton. Th dtrbuton of cattrd nrgy acro drcton and polarzaton

12 1/5/4 Th Scattrng Matrx 3/8 dcrbd by th cattrng pctrum ( k ). Th cattrng, a t (ung th ba pctrum complt dcrb th cattrd fld ( r ) ntally th Fourr tranform of ( r ) functon j k r ). Not that th ncdnt fld can lkw b dcrbd n trm of a cattrng pctrum. Snc: ( ) ( ) ( ) k ˆ + ˆ h r k d k v h j r j k r v t apparnt that: ( ) ( ) ( ) ˆ + ˆ ˆ ˆ h δ k v h k k v ( ˆ ˆ ) δ k k Th of cour mpl tat what w alrady know: th ncdnt fld pctrum cont of prcly on plan wav! Q: So w can xpr th ncdnt fld a ( k ) cattrd fld a ( k ) two functon?, and th. I thr om way to rlat th S : A: Y! Thy ar rlatd by th cattrng tnor ( k ; k ) k k ; k k d k ( ) S ( ) ( )

13 1/5/4 Th Scattrng Matrx 4/8 Ung th pctrum of our ngl ncdnt plan wav, th bcom: k k k k k ( ) S ( ; ) ( ) d S ( ) ( ) S ( k ; k ) Th cattrng tnor ( k ; k ) k ; k δ k k d k S ntrly dpndnt on th cattrng objct (but th nclud z, hap, orntaton and matral!). Thu, f w know th cattrng tnor S ( k ; k ) mply fnd th cattrd fld a: ( ) ( ) j k r r k d k S k k k ( ˆ ) j k r ; d Q: Yk! Th don t look mpl at all!, w can A: Actually, valuatng th ntgral typcally mpobl at lat wthout th ad of thr numrc or aymptotc approxmaton!

14 1/5/4 Th Scattrng Matrx 5/8 So hr w wll apply our frt aymptotc approxmaton. Condr th cattrd fld at a pont dnotd a r r k ˆ (.., at a dtanc r from th orgn, n th drcton ˆk ). W wll dtrmn th cattrd fld a th pont approach an nfnt dtanc from th cattrng objct (.., a r r approach ): lm r r k lm k ; k d k ˆ r whr k k ˆ k. ( ˆ ) S ( ) j k ˆ r k r j k r S ( k; k ) r W u th rult to approxmatly dtrmn th cattrd fld at pont a gnfcant dtanc from th cattrng objct. Not th approxmaton ay that th cattrd fld at th dtant pont appar to b a plan wav of th form: j k r k k r j k r r ( r ) S ( ; ) Th approxmaton mply th far-fld approxmaton!

15 1/5/4 Th Scattrng Matrx 6/8 W can furthr xpr th cattrd fld a: ( r ) j k r r j k r + r ( ) v v ˆ ˆ h h whr v ˆ ˆ ˆ h k and v ˆ ˆ h. Thrfor: S ( k; k ) ( ) ( ) ( ) v ˆ + v h h ˆ S k ; k v ˆ + ˆ v h h It vdnt that th far-fld cattrng tnor can b xprd a: Q: Huh? S ( ; ) S vh ( ; ) ˆ ˆ + S hh ( ; ) + S ( ; ) ˆˆ + S hv ( ; ) k k k k v h k k h ˆ h ˆ k k v v k k h ˆ v ˆ vv A: At th pont, t mplr to jut to u matrx notaton. W frt dfn th cattrng matrx: ( k ˆ k ˆ ) ( ) vh k ˆ k ˆ ( k ˆ ˆ ) ( ˆ ˆ ) k hh k k S ; S ; ( ˆ ˆ ) S k ; k S vh ; S ; vv

16 1/5/4 Th Scattrng Matrx 7/8 whr th valu S vv,s vh,s hh,s hv ar complx cattrng coffcnt. Th coffcnt compltly dcrb th farfld cattrng n drcton ˆk, gvn om ncdnt fld of drcton ˆk. Now, xprng and a vctor: v v and h h w can ay: ( k ˆ k ˆ ) ( ) vh k ˆ k ˆ ( k ˆ ˆ ) ( ˆ ˆ ) k hh k k v S vv ; S ; v S h vh ; S ; h S Thrfor, th far-fld cattrd fld xprd a: j k r ( r ) r j k r r S

17 1/5/4 Th Scattrng Matrx 8/8 Q: What f th cattrr not locatd at th orgn? Say t locatd at th pont dnotd by r? A: In that ca, th cattrd far-fld : ( r ) ( ) j k r r r j k r r j k r ( ) ( ( ) ) k k S j r

18 1/5/4 Radar Cro Scton 1/3 Radar Cro Scton From th Poyntng vctor, w can how that th powr dnty of th ncdnt wav n fr-pac : ( ˆ Wk) k ˆ η W m whl th powr dnty of a cattrd fld : 1 ( ˆ ) k ˆ r η W k 1 S ˆ W k r η m W can lkw dfn th cattrd powr dnty of on polarzaton componnt ˆp (.g., p ˆ h ˆ or p ˆ v ˆ ) a: W p 1 r p ˆ S T η k ˆ Th cattrng cro cton σ of an objct can b dfnd a: W ( ) k ˆ ( ˆ, ˆ S σ k k) lm r m r ( ˆ Wk)

19 1/5/4 Radar Cro Scton /3 Not th valu dpndnt on th ncdnt wav polarzaton, a wll a th cattrng objct. Accordngly, w typcally dfn cattrng cro cton n trm of an xplct ncdnt wav polarzaton, a wll a on polarzaton componnt of cattrd fld. For xampl, th four tandard cro-cton valu ar: W ( k ˆ ) v σ lm r v T ˆ S vv ( k ˆ, k ˆ ) whr v v ˆ r Wk ( ˆ ) ( ) W ( k ˆ ) v σ ( k ˆ, k ˆ ) lm r whr h ˆ vh v T ˆ S r Wk ( ˆ ) ( ) h v T W ( k ˆ ) h ˆ S hv ( k ˆ, k ˆ ) whr ˆ v v r Wk ( ˆ ) ( ) σ lm r σ v T W ( k ˆ ) h ˆ S hh ( k ˆ, k ˆ ) lm r whr h ˆ r Wk ( ˆ ) ( ) h Of partcular rlvanc to radar problm th backcattrng cro-cton. Th back cattrng crocton mply th cattrng cro-cton valuatd for th ca whn: k ˆ k ˆ (backcattrng condton)

20 1/5/4 Radar Cro Scton 3/3 In othr word, th ca whn th cattrd wav travlng back toward th ourc of th ncdnt wav. Bcau of t rlvanc to th radar problm, th backcattrng cro-cton oftn rfrrd to a th radar cro cton.

21 1/5/4 Multpl Scattrng and th Born Approxmaton 1/7 Multpl Scattrng and th Born Approxmaton Now, lt condr th ca whr w hav two cattrng objct (at locaton 1 r and r ), ach llumnatd by th am ncdnt wav. Q: So n t th rultng cattrd fld jut th um of th cattrd fld from ach: j k r ( ) ( ) ( ) ( ) j 1 r j r 1( ˆ ˆ k k r ; ) ( ˆ ˆ k k S k k + S k; k)??? r 1 1 t -ordr cattrng mchanm

22 1/5/4 Multpl Scattrng and th Born Approxmaton /7 A: NO! If t wr only that ay! Th problm that th cattrd fld from objct 1 crat a cond ncdnt fld at objct, and th cattrd fld from objct crat a cond ncdnt fld at objct 1. 1 ˆk 1 ˆk 1 nd -ordr cattrng mchanm ( ) ( ) ( ) j k r ( ) 1 1( ˆ ˆ k r ; ) ( ˆ ˆ k k k S k k + S k; k) r j r j r j k r + ; ; r r r jkˆ k1 ( r 1 r) j 1 r + k j r 1( ˆ ˆ ; 1) ( ˆ ˆ k + S k k S k 1; k) 1 r r jkk ˆ 1 ( r 1 r) + j k r j 1 r ( ˆ ˆ 1) 1( ˆ ˆ k S k k S k1 k) 1 + But wat that not all! Th nw cattrd fld from objct 1 crat a thrd ncdnt fld on objct, and vc vra.

23 1/5/4 Multpl Scattrng and th Born Approxmaton 3/7 1 ˆk 1 ˆk 1 3 rd -ordr cattrng mchanm ( r ) j k r ( ( ) ( ) ) k 1 1( ˆ ˆ k k k ; k) + ( k ˆ ˆ k S S ; k) r + j k r j r j r ; ; r r r jkk ˆ 1 ( r 1 r) + j k r j 1 r ( ˆ ˆ 1) 1( ˆ ˆ k S k k S k1 k) 1 jkk ˆ 1 ( 1 r r ) j 1 r + k j r 1( ˆ ˆ ; 1) ( ˆ ˆ k + S k k S k 1; k) 1 r r j k ˆ r jk 1( 1 r r) k + j k 1 r + S1( k ˆ ˆ ; k1) S( k ˆ ˆ 1; k1) r 1 r r j k k ˆ r r j k ˆ r r ( ) k ( ) ˆ ˆ ˆ ˆ ; + ; r r r r j k 1 r + j k r S ( k k ) S ( k k ) jkk ˆ 1 ( 1 r r ) 1( ˆ j r 1; ˆ 1) ( ˆ ˆ k S k k S k1; k ) ) r r 1 +

24 1/5/4 Multpl Scattrng and th Born Approxmaton 4/7 Hopfully, you can that th analy can contnu forvr. Thr ar an nfnt numbr of cattrng mchanm, and all of thm ar rqurd to provd th prc cattrng oluton. Of cour, th xampl ncludd only two cattrr magn th m w would crat tryng to dtrmn all th cattrng mchanm aocatd wth multpl cattrr! ( ) r?? Fortunatly, w gnrally fnd that ach uccv cattrng ordr (trm) l gnfcant than th prvou on. Thrfor, w vntually fnd that w can truncat th nfnt ummaton (calld th Born Sr), wll lttl mpact on th oluton accuracy w don t hav to condr an nfnt numbr of cattrng trm!

25 1/5/4 Multpl Scattrng and th Born Approxmaton 5/7 In fact, w oftn nd to condr only a fw cattrng trm to gt accptabl accuracy. Th numbr of rqurd trm dpnd on vral thng, but motly on th cattrng ntnty and dnty of th partcl. If th partcl ar lghtly cattrng and parly populatd, thn w can aum th total fld at ach partcl approxmatly that of th orgnal ncdnt fld only. Th approxmaton known a th Born approxmaton, and t rult n th cattrd fld bng approxmatd by th frt-ordr cattrng trm only. Thu, for a collcton of N cattrr: N n ( r ) ( r ) n 1 j k r N ( k k ) n n( ˆ ˆ S k; k) r n 1 j r Q: So, do th man that th cattrng cro-cton of th collcton of cattrr lkw th ummaton of th cro-cton of ach cattrr: N σ σ??? n 1 n A: NO! Th dfntly not tru. Th cattrd powr dnty from th collcton of objct :

26 1/5/4 Multpl Scattrng and th Born Approxmaton 6/7 N 1 1 ( ) ( ˆ j k k rn W k) ( ˆ ˆ Sn k; k) r η n 1 N 1 1 ( ˆ ˆ ) Sn k ; k r η n 1 N n 1 1 j ( k k ) ( rn rm) H H n ( ˆ ˆ ) m( ˆ ˆ + R S k ; k S k; k) r η n 1 m 1 N 1 1 n r η n 1 N n 1 1 j ( k k ) ( rn rm) H + R n m r η n 1 m 1 Not that th frt trm rprnt th um of powr dnty from ach cattrr. Howvr, thr cond trm n th xpron! Th trm known a th cohrnt trm. It a ral valu, but t can b potv or ngatv. A a rult, th total cattrd powr dnty can b much gratr, or much l, than mply th um of th cattrd powr from ach objct. In fact, th total powr dnty can vn b zro! Th total cattrng cro-cton for th collcton of cattrr thrfor: n m N N n 1 j ( k k ) ( r r ) H σ σn + R n m n 1 n 1 m 1

27 1/5/4 Multpl Scattrng and th Born Approxmaton 7/7 Agan, w that th cro-cton can b much mallr, or much largr, that mply th um of ndvdual cattrng cro-cton.

28 11/3/4 Scattrng from Random Mda 1/5 Scattrng from Random Mda * Typcally, f w ar ntrtd n dtrmnng th cattrng from a larg collcton of dcrt objct, w dcrb th charactrtc of th collcton wth tattcal maur. * That, w trat om, mot, or vn all of th phycal dcrptor a random varabl. Th random varabl can nclud partcl locaton, orntaton, matral, z, and hap. * W can xplctly dcrb th random varabl wth a jont probablty dnty functon (pdf), or mor mply n trm of tattcal momnt uch a man, varanc, and covaranc. Q: How can w dtrmn cattrng from a collcton of cattrr f w don t know prcly what th collcton?? A: W hav to dcrb th cattrd fld n th am way w dcrb th collcton of cattr ung tattcal maur! In othr word, w mut lkw trat th cattrd fld tlf a a random proc (ovr 3-dmnon of pac).

29 11/3/4 Scattrng from Random Mda /5 Typcally, w mply dcrb th random fld n trm of thr frt two tattcal momnt: ( r ) th valu of ( r ) man ( r ) th of ( r ) varanc Not that th varanc ha a mor phycal ntrprtaton, a t proportonal to th avrag powr dnty of th wav. For xampl, condr a collcton of cattrr whr th Born approxmaton applcabl. W know that th cattrd fld approxmatly: N n ( r ) ( r ) n 1 ˆ ( k k ) jk k r N jk ˆ ˆ rn ˆ ˆ r n 1 S ( k ; k ) n Aumng th random varabl of dmlar lmnt ar ndpndnt, th man cattrd fld : N n ( r ) ( r ) n 1 ˆ ( k k ) jk k r N jk ˆ ˆ rn ˆ ˆ Sn( k; k) r n 1 Th valu S ( k ˆ ; k ˆ ) th cattrng matrx avragd n acro th dtrbuton of partcl z, hap, matral, tc.

30 11/3/4 Scattrng from Random Mda 3/5 Th valu ( kˆ ˆ k ) jk rn dpndnt on th dtrbuton of partcl poton r n (hr t ha bn aumd that partcl poton lkw ndpndnt of othr paramtr uch a z and hap). It turn out, f th partcl ar dtrbutd throughout om volum that, n ach dmnon, gratr than a wavlngth (.., k V ), thn: 3 1 jk ( kˆ ˆ k ) n r and thrfor: ( r ) In othr word, th cattrd fld from a random cattrng mdum typcally on avrag zro. Not th do not man that th cattrd fld tlf typcally zro t almot nvr! Not thn that th avrag total fld n/from a random cattrng mda : ( r ) ( r ) + ( r ) jk k r + ˆ jk kˆ r

31 11/3/4 Scattrng from Random Mda 4/5 In othr word, th avrag total fld mply qual to th ncdnt fld. Not that th ncdnt fld not a random fld! Now lt condr th varanc of th cattrd fld: N nh m ( r ) ( r ) ( r ) n 1 m 1 1 r 1 r N n 1 m 1 n 1 1 r n ( k k ) ( n m) N N jk ˆ ˆ r r N ( k k ) n ( k k ) N n 1 jk ˆ ˆ r + jk ˆ ˆ r m H + R n m n 1 m n Hr w hav aumd that: H n ( kˆ ˆ ) ( ) ˆ ˆ ˆ ˆ k ( k k ) ( k k ) jk r r jk r + jk r f n m n m n m m In othr word, w hav aumd that th locaton of dmlar partcl ar ndpndnt. Th actually cannot b tru! Th raon for th that two partcl cannot occupy th am locaton. Howvr, for parly dtrbutd partcl, w fnd that th ndpndnt aumpton approxmatly tru.

32 11/3/4 Scattrng from Random Mda 5/5 At any rat, w hav alrady dtrmnd that f th cattrng volum uffcntly larg, thn: ( kˆ ˆ ˆ ˆ k ) n ( k k ) jk r + jk r m and thu: ( r ) 1 r N n 1 n ( k k ) n ( k k ) 1 R r n 1 m n N 1 n r N n 1 jk ˆ ˆ r + jk ˆ ˆ r m H + n m n 1 Th ay that th total avrag cattrd powr dnty mply th um of th avrag cattrd powr from ach partcl. Th man that th avrag cattrng cro-cton from th collcton of partcl lkw mply th um of th avrag cattrng cro-cton of ach partcl: σ N n 1 σ n But b carful! Th rult ar only tru for a larg volum of par, ndpndnt cattrr.

33 11/3/4 Raylgh Fadng Stattc 1/5 Raylgh Fadng Stattc Q: So w now know th avrag (.., man) valu of th cattrng cro-cton of a collcton of cattrng partcl. But what about th varanc of σ? Can w dcrb σ wth mor tattcal pcfcty? A: A a mattr of fact, w can dtrmn (approxmatly) th ntr probablty dnty functon (pdf) of σ! Condr jut on calar componnt of th cattrd fld v r r ): ( )( ay ( ) v n Now, rcall that ( ) ( ) v ˆ ( ) r r v ˆ N n 1 N n 1 N n 1 vn n v ˆ n ( r ) ( r ) ( r ) r a complx functon, that can b xprd n trm of t ral and magnary part: ( ) ( ) + ( ) r r j r v r v v N N nr v n 1 n 1 n ( ) ( ) r + j r v

34 11/3/4 Raylgh Fadng Stattc /5 In othr word th ral part of th cattrd fld th um of th ral part of th cattrd fld from ach ndvdual cattrr! Oh, by th way, th am tru for th magnary part. Now watch out, hr com th tattc! From th cntral lmt thorm, f N (th numbr of r cattrng partcl) larg, thn th pdf of v ( r ) and v ( r ) ar dntcal, ndpndnt Gauan probablty dnty functon! W hav alrady dtrmnd th man valu ar zro (rcall ), o th pdf ar: ( r ) vr ( ) p 1 πσ vr ( ) σ whr: v ( ) p 1 πσ vr vr ( ) ( ) v ( ) σ σ Snc th two dtrbuton ar ndpndnt, w can wrt th jont dtrbuton a:

35 11/3/4 Raylgh Fadng Stattc 3/5 vr v vr v ( ) ( ) ( ) p, p p 1 πσ vr v ( ) ( ) + σ But rcall: ( ) ( ) v vr v + and thu t can b hown that: v ( ) p 1 σ v σ From th pdf, w can dtrmn that: o thrfor: and thu: v σ σ v v ( ) p 1 v v v Snc v σ, w can conclud that:

36 11/3/4 Raylgh Fadng Stattc 4/5 p ( σ ) 1 σ σ σ Th dtrbuton calld th xponntal dtrbuton. p ( σ ) σ σ Th dtrbuton how th wd varanc n pobl valu of cattrng cro-cton σ. Thu, although w mght know th avrag valu of th cattrng cro-cton of a random collcton of cattrr, th actual valu of σ oftn vry dffrnt than th man valu!

37 11/3/4 Raylgh Fadng Stattc 5/5 Th tattcal dcrpton know a Raylgh Fadng Stattc, a nam drvd from th pdf of th valu σ : ( ) σ σ σ p σ σ A probablty dnty functon known a th Raylgh dtrbuton.

38 11/3/4 Th xtncton Cro Scton 1/ Th xtncton Cro-Scton Say a loy partcl llumnatd by a plan wav wth powr dnty W ( r ): W( r ) η k ˆ ˆk Th partcl wll: 1. Aborb nrgy at a rat P a Watt.. Scattr nrgy (n all drcton) at a rat P Watt. Bcau of conrvaton of nrgy, th powr dnty of th ncdnt wav mut b dmnhd aftr ntractng wth th partcl: ˆk ˆk P a W( r ) η k ˆ P W( r + r ) < η k ˆ

39 11/3/4 Th xtncton Cro Scton / W can thrfor dfn th aborpton cro-cton of th partcl a: σ a Pa W ( r ) m and lkw th total cattrng cro-cton a: σ P W ( r ) m Not that ung th dfnton w fnd that: ( ) and ( ) P σ W r P σ W r a a W can lkw dfn a partcl xtncton cro-cton a: P + P P P σ σ σ m a a + a + W ( r ) W ( r ) W ( r ) Th xtncton cro cton (a wll a a σ and σ ), dpnd on th phycal proprt (z, hap, matral) of th partcl.

40 11/3/4 Th xtncton Cofcnt 1/6 Th xtncton Coffcnt Condr a plan wav that propagat a dtanc x through a thn volum wth cro-ctonal ara A. Ara A W ( x + x) ˆx ˆx W( x ) η x ˆ Volum of rgon x V A x

41 11/3/4 Th xtncton Cofcnt /6 Say that th volum flld wth N partcl. Th partcl dnty n o thrfor dfnd a: N N partcl no 3 V A x m Now, w know that nrgy flowng nto th front of th volum at a rat of: n ( ) [ ] P W x A W whl th powr of th plan wav xtng th back of th volum : P W x + x A W n ( ) [ ] Of cour, th partcl wthn th volum wll xtract powr from th plan wav (du to aborpton and cattrng). Thu, th powr lot du to xtncton can b wrttn a: P P P out n ( ) W( ) W x + x A x A W A Whr W W( x + x) W( x ). Not nc P out < P n, both P and W wll b ngatv valu (.., th powr dcra a t pa through th volum).

42 11/3/4 Th xtncton Cofcnt 3/6 Q: What happnd to th mng nrgy? A: Th partcl wthn th volum xtract th nrgy by aborbng or cattrng th ncdnt plan wav. If P n th rat at whch nrgy xtractd by th n-th partcl, thn th total rat of nrgy xtncton by th ntr collcton : P N n 1 N n 1 P σ n n W ( x ) By conrvaton of nrgy, w can conclud that: P P P P n out And thrfor: P P N σ n 1 n W ( x ) Q: Yk! How ar w uppod to know σ for ach and vry on of th N partcl? A: Look clor at th quaton! W don t nd to all th valu σn --w mply nd to know th um of all th valu N (.., σn )! n 1

43 11/3/4 Th xtncton Cofcnt 4/6 To dtrmn th um, w jut nd to know th avrag valu of th xtncton cro-cton ( σ ), dfnd a: Thrfor: σ N 1 N n 1 N σ n P σ W ( x ) n 1 W W n ( x ) N n 1 σ ( x) ( N σ ) N W ( x) σ n Rcall, howvr, that: N n V n A x o o Thrfor: P N W ( x) σ naw( x) σ x o and thu: P W o W A n ( x) σ x Fnally (whw!) w can ay: W o ( ) x n W x σ

44 11/3/4 Th xtncton Cofcnt 5/6 And takng th lmt a x, w hav dtrmnd th followng dffrntal quaton: d W dx ( x) ( n σ ) W ( x ) o Th dffrntal quaton aly olvd: ( ) ( ) ( n σ ) o W x W x x And thu: ( ) ( ) ( n σ ) o W x W x x x ˆ Th valu n o σ obvouly vry mportant and calld th xtncton coffcnt κ of a random mdum κ noσ Thu, th powr dnty of a plan wav pang through a random collcton of partcl (wth partcl dnty n o and avrag xtncton cro-cton σ ) :

45 11/3/4 Th xtncton Cofcnt 6/6 ( r ) ( r ) κ ( η k ˆ r ) k ˆ r ) κ ( W W k ˆ k ˆ Thu, th powr dnty of th ncdnt fld wthn a collcton of cattrr wll dmnh xponntally wth propagaton dtanc! Th xponntal bhavor wll dpnd ntrly on th xtncton coffcnt κ, whch n turn dpnd on th dnty and avrag xtncton cro-cton of th partcl.

46 11/3/4 Th Optcal Thorm 1/6 Th Optcal Thorm Q: Now I m confud! W arlr concludd that th avrag fld n a collcton of random cattrr wa mply th ncdnt fld: j r r r k ( ) ( ) But th would rult n a contant powr dnty throughout th cattrng volum n othr word no xtncton! W ( r ) j k r ( r ) ˆ κ( k r ) η η η η??? η η or, η κ ( k ˆ r )? A: Rcall w concludd that th cohrnt (.., avrag) fld wa qual to th ncdnt fld ung th rult of th Born Approxmaton. It turn out that th Born Approxmaton gnrally a bad approxmaton whn appld to th propagaton problm!

47 11/3/4 Th Optcal Thorm /6 Rcall th Born Approxmaton condr only frt ordr cattrng mchanm. Th total oluton mply a uprpoton of all th ndvdual frt-ordr oluton and th frt-ordr oluton ffctvly aum that no othr partcl ar prnt. η η But f no othr partcl ar prnt, thn xtncton do not occur! In othr word, xtncton a dcddly multpl-ordr cattrng ffct th ncdnt wav ncountr many partcl a t propagat through th mdum. Q: So jut what th cohrnt wav ( r )? A: Ung an analy mlar to that whch arrvd at th xtncton coffcnt, w can dtrmn that th cohrnt wav mut atfy th dffrntal quaton:

48 11/3/4 Th Optcal Thorm 3/6 v ( x) dx h ( x) dx v j k ( x) ff v h j k ( x) ff h whr k ff th complx valu: v π no kff k + S, k ( k ˆ k ˆ ) vv h π no kff k + S hh, k ( k ˆ ˆ ) k Not that th complx valu S ( ) vv k ˆ ˆ, k and S ( ) hh k ˆ ˆ, k dcrb th forward cattrng coffcnt (.., k ˆ ˆ k ) of a partcl. Th xpron abov ar du to a rult known a th optcal thorm. Although th rult lkw an approxmaton vald only for a par collcton of ndpndnt cattrr t much mor accurat than th cohrnt wav oluton ung th Born approxmaton.

49 11/3/4 Th Optcal Thorm 4/6 Th optcal thorm thu provd th oluton: v ff ( x) ˆ + h v jk x h jkff x ˆ v h or mor gnrally: v ff ( ) ˆ + h v jk r h jkff r ˆ r v h whr: k k k ˆ and k k v v h h ff ff ff ff k ˆ Not that th powr dnty aocatd wth ach componnt of th cohrnt wav : v v ( ) v v v k + k Im{ kff } v r ˆ r ( ) ˆ j r j r ˆ k Wv r k ˆ k k η η η h h ( r ) h h h k + k Im{ kff } ˆ h j r j r k r Wh ( r ) k ˆ ˆ ˆ k k η η η whr dnot complx conjugat.

50 11/3/4 Th Optcal Thorm 5/6 Compar th wth th rult n trm of th xtncton coffcnt: W ( r ) κ ( k ˆ r ) η k ˆ It vdnt thn that w can u th optcal thorm to dtrmn th xtncton coffcnt of a collcton of cattrr: { ff } Im { kff } κ Im k and κ v v h h Addtonally, nc κ n o σ, and: w can conclud that: { } ( ) v π no ˆ ˆ Im kff Im S vv k, k k 4π n Im S, { ( ˆ ˆ ) } k k o vv k { ( ˆ ˆ ) } k k 4π σ Im S, v vv k { ( ˆ ˆ ) } hh k k 4π σ Im S, h k

51 11/3/4 Th Optcal Thorm 6/6 Or, mor pcfcally: { ( ˆ ˆ )} k k 4π σ Im S, v vv k { ( ˆ ˆ )} hh k k 4π σ Im S, h k Thu, w can u th optcal thorm to dtrmn th xtncton coffcnt of a collcton of cattrr, a wll a th xtncton cro-cton of an ndvdual partcl.

52 11/3/4 Th Dtortd Born Approxmaton 1/3 Th Dtortd Born Approxmaton Q: So, th Born Approxmaton do not account for xtncton wthn a collcton of partcl. Mut w compltly abandon th Born Approxmaton? A: Not ncarly! W can abandon th Born Approxmaton wth rpct to propagaton, but w can tll u t wth rpct to cattrng. In othr word, w can u th Optcal Thorm to account r, but for xtncton whn dtrmnng th man-fld ( ) thn only account for frt-ordr cattrng trm whn dtrmnng th avrag cattrd powr ( r ). Th approach known a th Dtortd Born Approxmaton, and thu th cattrd fld from th n-th partcl n th mdt of a collcton of N partcl pcfd a: 1 ff ( ) ( r ) S ( k ˆ ; k ˆ ) r ff j k r ff ff j( k k ) rn n r n j k r rn jk rn n whr n S n( k ˆ ˆ ; k ) and w hav aumd that v h k k. ff ff ff Jm Stl Th Unv. of Kana Dpt. of CS

53 11/3/4 Th Dtortd Born Approxmaton /3 Thu, th man cattrd fld : N n ( r ) ( r ) n 1 k r nc agan w fnd that: ff ff ff n j r N n 1 ( k k ) j r n ff ff j( ) rn k k. A a rult, th avrag total fld (.., cohrnt fld) : ( r ) ( r ) + ( r ) j k r + j k ff ff r Jm Stl Th Unv. of Kana Dpt. of CS

54 11/3/4 Th Dtortd Born Approxmaton 3/3 Lkw, th avrag cattrd powr thu: N nh m ( r ) ( r ) ( r ) n 1 m 1 + N ff ff ( k k ) j r r N n 1 m 1 ff ff ff ff ( k k ) ( k k ) ( k-k ) κ kˆ r N ˆ ˆ κ r n n r n 1 N κ kˆ r N n j rn j rm H 1 ( k k ) ( k k ) R n 1 m n ff ff ff ff n + m r + j r j r H n m n m But nc: ff ff j( ) rn k k W fnd that: ( r ) ( k-k ) κ k ˆ r N κ ˆ ˆ r n n r n 1 Not that th valu: ( k-k ) ˆ ˆ r κ n wll b ral and potv! Jm Stl Th Unv. of Kana Dpt. of CS

55 11/11/4 Raylgh Scattrng 1/4 Raylgh Scattrng Rcall that xact, prc, oluton to cattrng ntgral quaton: ( r,t) ( ) + G ( ) ( r,t ) ( t) H ( r,t) + G ( r,t ) H( r,t ) Hr, r,t r,t ; r,t dv dt m ; r,t dv dt ar vry dffcult to fnd ntally mpobl to dtrmn! In fact, about th only prfct oluton w hav for a fnt cattrr for that of a phr. Th oluton (.., for a phr), known a th M cattrng oluton. Although a phr a vry bac and mpl hap, t cattrng oluton nthr bac nor mpl! In fact, th oluton can only b wrttn a a wghtd uprpoton of an nfnt numbr of vctor ba functon ψ n ( r ): phr ( r ) an ψ n ( r ) n Th oluton altrnatvly rfrrd to a th M r.

56 11/11/4 Raylgh Scattrng /4 * Th vctor ba functon ψ ( r ) n ar vry complx functon, formd from phrcal Bl functon and Lgndr Polynomal (not mportant!). * Th r coffcnt a n ar known a M coffcnt, and thy dpnd on: 1) th ncdnt wav drcton and polarzaton (.., k ˆ, ), ) th matral proprt of th phr (..,σ,ε,µ), and 3) th lctrcal radu of th phr ka. Q: lctrcal radu? ka? What that? A: Th valu a mply th radu of th phr (.g., a.5 mtr). Th valu ka thu: π a ka a π λ λ Th valu ka ffctvly xpr th z (radu) of th phr wth rpct to on wavlngth. Th of cour frquntly th mportant dcrptor n lctromagntc t not how bg omthng, t how bg t compard to on wavlngth! Mor pcfcally, ka dcrb th radu n trm of lctrc pha. For xampl, f ka π radan, th radu of th phr λ.

57 11/11/4 Raylgh Scattrng 3/4 Q: So n ordr to gt th xact cattrng oluton, would w hav to add up an nfnt numbr of M cattrng trm? A: Yup! To gt an xact oluton, w mut condr an nfnt numbr of trm. Howvr, lk th Born r, w fnd that at om pont (.., om valu n) th M coffcnt wll bcom ngnfcant (.., narly zro). In othr word, th M r wll convrg, o that w can truncat th r and gt a vry good (although not prfct) cattrng oluton. Q: So, how many M cattrng trm mut w condr? A: It dpnd on two thng: 1) th matral proprt of th phr, and ) th z ka of th phr. A phr that hghly cattrng (.., a phr whr any of th matral proprt σ,ε, or µ ) rqur mor M cattrng trm than on that l cattrng. Lkw, a larg phr (.., ka 1) rqur many mor cattrng trm than a mall phr (.., ka 1). In fact, w can apply anothr aymptotc approxmaton: phr n n ka ka n ( ) ( ) ( ) lm r lm a ψ r a ψ r

58 11/11/4 Raylgh Scattrng 4/4 In othr word, for lctrcally mall phr (.., ka 1), th cattrd fld approxmatly qual to th frt M cattrng trm. Th approxmaton known a th Raylgh approxmaton, and gratly mplf th cattrng oluton not only for lctrcally mall phr, but for any lctrcally mall partcl! In fact, any lctrcally mall partcl (.., all partcl dmnon ar mall wth rpct to a wavlngth) condrd to b a Raylgh Scattrr, wth t cattrng oluton pcfd n trm of a polarzablty tnor.

59 11/11/4 Th Polarzablty Tnor 1/6 Th Polarzablty Tnor ntally, vry mall partcl (.., Raylgh Scattrr) cattrr lk a mpl lctrc dpol. p In othr word, th ncdnt fld wll polarz th partcl, cratng dpol wth momnt p. Th rlatonhp btwn p and pcfd wth th partcl polarzablty tnor P : p ε P Th polarzablty tnor of a partcl compltly dpndnt on partcl proprt uch a z, hap, and matral. Convrly, t compltly ndpndnt of ncdnt or cattrng drcton, or polarzaton, or vn frquncy ω! Th cattrng matrx for an lctrc dpol wth momnt p :

60 11/11/4 Th Polarzablty Tnor /6 k ˆ ˆ k ˆ ˆ ( k ) k p S 4πε ( ) k k P 4π Th lmnt of th cattrng matrx can b vn mor mply tatd a: S S k k ˆ 4π 4π ( v ˆ P v ˆ ) S ( ˆ ) hv h P v vv vh k 4π 4π ( ) k v ˆ ( ) P h ˆ ˆ ˆ Shh h P h Not that th cattrng matrx lmnt ar dpndnt on ncdnt and cattrng drcton ˆk and ˆk, but only bcau thy affct th drcton of polarzaton vctor h ˆ, v ˆ, tc. W now can conclud many thng about Raylgh cattr, bad on our prvou analy and dcuon.

61 11/11/4 Th Polarzablty Tnor 3/6 For xampl, th cattrng cro-cton of a Raylgh partcl : vv ( k ˆ, k ˆ ) 4 vv ( k ˆ, k ˆ ) ˆ ˆ v P v σ π S k σ ( k ˆ, k ˆ ) 4 π S ( k ˆ, k ˆ ) k v h ˆ vh vh ˆ P σ ( k ˆ, k ˆ ) 4 π S ( k ˆ, k ˆ ) k h ˆ v ˆ hv hv P σ ( k ˆ, k ˆ ) 4 π S ( k ˆ, k ˆ ) k h ˆ h ˆ hh hh P Whl th xtncton cro-cton approxmatly: 4π σ Im S ˆ, ˆ k Im ˆ ˆ { ( )} k k { v P v } v vv k 4π σ Im S ˆ, ˆ k Im ˆ ˆ { ( )} { } hh k k h P h h k And thu th ffctv propagaton contant n a random collcton of Raylgh Scattrr : π n ˆ ˆ nk k k + S k, k k + v ˆ P v ˆ ( ) v o o ff vv k π n ˆ ˆ nk k k + S k, k k + h ˆ P h ˆ ( ) h o o ff hh k

62 11/11/4 Th Polarzablty Tnor 4/6 Q: OK, I that th polarzaton tnor all w nd to dcrb Raylgh cattrng, but what th polarzaton tnor? A: For th mplt Raylgh Scattrr a dlctrc phr w fnd that th polarzablty tnor : P εr 1 vo 3 I ε + r 3 whr v o th volum of th phr ( vo 4π a 3), ε r th rlatv dlctrc of th phr, and I th dntty tnor. Th valu P, dfnd a: εr 1 P 3 ε + r th normalzd polarzablty of a phr (.., th polarzablty pr unt volum). Thrfor, w can conclud that: σ ˆ ˆ k ˆ ˆ k v P ˆ vv ( k, k ) v P v o v v σ ( k ˆ, k ˆ ) k v h ˆ k v P v h ˆ vh ˆ P ˆ o σ ( k ˆ, k ˆ ) k h ˆ v k v P h ˆ v hv P ˆ o σ ( k ˆ, k ˆ ) k h ˆ h ˆ k v P h ˆ h ˆ hh P o

63 11/11/4 Th Polarzablty Tnor 5/6 and that: v σ kim kvo Im P ka Im P 3π { } { } ( ) 3 λ v ˆ P v ˆ { } h λ σ kim kvo Im P ka Im P 3π Not that σ v { } h ˆ P h ˆ { } ( ) 3 { } σ, and that th xtncton coffcnt v ncra proportonally wth ( ka ) 3. W fnd that th propagaton contant of a collcton of phrcal cattrr thrfor: v nk o fv kff k + vop k 1 P + h nk o fv kff k + vop k 1 P + whr f v th fractonal volum, dfnd a: fv no vo Not that th valu approxmatly th fracton of th total volum that flld wth phr. Typcally, a cattrng mdum condrd to b par f f 1. (.., l than 1 % flld). v

64 11/11/4 Th Polarzablty Tnor 6/6 Q: OK, o w now know th cattrng oluton for phrcal Raylgh cattr, but what about non-phrcal partcl? A: W fnd that th polarzablty tnor for partcl that ar roughly phrcal (.g., an c crytal) typcally vry clo to that of a phr. A a rult, w oftn u th phrcal Raylgh cattrng oluton to approxmat th cattrng and xtncton of non-phrcal Raylgh cattrr!

65 11/11/4 Mxng Formula 1/ Mxng Formula Th propagaton contant for a wav propagatng n a homognou dlctrc mdum ε : ω k k ˆ ˆ ω εµ k v p Thu, for a random mdum wth ffctv propagaton contant k ff, w can dfn an ffctv prmttvty ε ff : k ff ω ε µ k ˆ ff For a random collcton of phrcal Raylgh cattrr, wth volum fracton f v, t can b hown that th ffctv prmttvty approxmatly: ε ff ( ) ( ) ( ) ( ) 1+ f v ε ε ε + ε ε 1 fv ε ε ε + ε whr ε th dlctrc contant of th phrcal cattrr, and ε th dlctrc of th background matral (.., th matral that th phrcal partcl) ar mbddd n).

66 11/11/4 Mxng Formula / Th prvou quaton an xampl of a dlctrc mxng formula. Th partcular formula vald whn th Raylgh cattrr ar par (.., f v < 1. ) and th Raylgh partcl ar vry mall ka 1. Thr ar othr mxng formula oluton, whch ar vald for mor dnly packd and/or largr cattrr.

67 11/11/4 Volum Scattrng from a Layr 1/ Volum Scattrng from a Layr of Scattrr Condr now th cattrng from a layr of Raylgh Scattrr abov a dlctrc half pac. d Γ Thr ar actually four frt-ordr cattrng mchanm aocatd wth th total cattrng from th layr.

68 11/11/4 Volum Scattrng from a Layr / d Γ W can u th optcal thorm or mxng formula to dtrmn th ffctv propagaton contant of th man fld n th cattrng layr. W can thn u th Dtortd Born approxmaton to dtrmn th avrag cattrd powr from th random cattrr, carfully condrng ach of th four frt-ordr cattrng mchanm. Typcally, w k to fnd th backcattrng coffcnt σ of th random mda, whch th backcattrng crocton of on quar mtr of th layr.