Electromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology

Transcription

1 Electromagnetc catterng Graduate Coure Electrcal Engneerng (Communcaton) 1 t Semeter, Sharf Unverty of Technology

2 Content of lecture Lecture : Bac catterng parameter Formulaton of the problem Scatterng cro ecton Aborpton cro ecton Scatterng ampltude matrx Unt vector ytem Expreon for the cattered feld: Born approxmaton Relaton wth Fourer tranform Optcal theorem Bac catterng parameter

3 Content of lecture Complementary materal: Ihmaru Chapter 10, pp Tang, Kong, Dng Chapter 1, pp. 1-9 Bac catterng parameter 3

4 Introducton Often, one am to gan nformaton on an object far away by endng out a wave and analyzng the cattered wave At a uffcently large dtance uch far feld wave are een a plane TEM wave So, we have to tudy the catterng of plane TEM wave by object Scatterng: polarzaton and conducton charge are et nto moton by the ncomng feld Induced current radate an extra feld (cattered feld) Source Bac catterng parameter 4

5 Introducton We have to calculate thee nduced current and ue them to compute the reultng far feld, th becaue the cattered feld alo meaured at a large dtance for mot applcaton But, frt, we dcu a number of general parameter ued n catterng theory Source Meaurement Bac catterng parameter 5

6 Bac catterng parameter Conder an object wth a permttvty dfferent from the delectrc contant of the background. A plane wave wth a known polarzaton propagate along and ht the object. Far away from the object, the nduced cattered feld alo behave a a plane wave n each drecton kˆ kˆ E E k r exp,0 j ( ) p r E E k r exp,0 j Incdent feld E kˆ V E kˆ Scattered feld, locally behavng a a plane wave n the far-zone H H Bac catterng parameter 6

7 Bac catterng parameter The ncdent electrc feld E E exp jkkˆ r,0 E kˆ r E Far-zone cattered feld E E kˆ, kˆ,0 exp r jkr Far feld kˆ krˆ Ampltude vector of the cattered feld up to a factor 1/r, alo depend on the ncomng electrc feld vector Scatterng ampltude: f kˆ, kˆ ˆ, ˆ r E E k k E E,0,0 Bac catterng parameter 7

8 Bac catterng parameter Incdent Poyntng vector S E,0 kˆ Poyntng vector of the cattered wave ˆ ˆ k, k E f S k k r E ˆ,0 ˆ kˆ n co, n n, co Bac catterng parameter 8

9 Scatterng cro ecton Scattered power flowng through a mall urface normal to the oberved drecton ˆ ˆ dp S nˆ da S f k, k d dp k ˆ, k ˆ f d S Dfferental catterng cro ecton d n d d k ˆ, kˆ f k ˆ, kˆ d Bac catterng parameter 9

10 Scatterng cro ecton Dfferental catterng cro ecton ha the dmenon of area: dp S kˆ, kˆ d d Power cattered nto a mall normal urface wth a old angle d equal ncdent power pang through a urface normal to the ncdent wave wth the area kˆ, kˆ d d It a meaurable quantty f we know the ntenty of the ncdent wave and the dtance from the obervaton pont to the catterer Bac catterng parameter 10

11 Scatterng cro ecton Total catterng cro ecton: k ˆ, kˆ P S f d 4 P S 4 f k ˆ, kˆ d Scatterng cro ecton P k ˆ, kˆ d d S 4 d n d d Bac catterng parameter 11

12 Scatterng cro ecton Scatterng cro ecton: area of a urface normal to the ncdent wave whch capture an ncdent power equal to the total power cattered Not equal to the geometrcal cro ecton : area projected onto a plane perpendcular to the ncdent wave vector g kˆ g g Bac catterng parameter 1

13 Aborpton cro ecton Part of the power captured by the object may be dpated due to polarzaton lo or conductvty Total power aborbed by the object (magnetc lo neglected): P a ( ) nt dv E r V Feld nde the object Aborpton cro ecton: a P S a Bac catterng parameter 13

14 Total cro ecton Total cro ecton P P a t a S S Albedo of the object 1 t a Bac catterng parameter 14

15 Scatterng ampltude matrx So far, we only dcued the relaton between the ncdent and cattered energy flow (power) But what about the feld themelve? What about ther ampltude and drecton? ( ) p r E kˆ V E kˆ H H Bac catterng parameter 15

16 Scatterng ampltude matrx ˆ The catterng ampltude f k, k wa ntroduced wthout any ˆ reference to the polarzaton of the ncdent and reflected wave E E exp jkkˆ r,0 E E E kˆ, kˆ,0 exp r jkr kˆ r Far feld kˆ E krˆ But, n realty, th functon depend on thee drecton Bac catterng parameter 16

17 Scatterng ampltude matrx General formulaton: chooe a ytem of coordnate for the electrc feld of the ncdent wave and for the electrc feld of the cattered wave for each catterng drecton The unt vector of the ncdent wave are perpendcular to and the unt vector of the cattered wave are perpendcular to k ˆ kˆ aˆ E aˆ E bˆ kˆ bˆ kˆ Bac catterng parameter 17

18 Scatterng ampltude matrx E aˆ bˆ kˆ a b E ˆ E ˆ exp jk ˆ E a b k r a b E E aˆ E bˆ f bˆ exp jkr k ˆ, kˆ f k ˆ, kˆ a a E aa ab E b b E f E ba bb k ˆ, kˆ f k ˆ, kˆ E r aˆ kˆ Scatterng ampltude matrx Bac catterng parameter 18

19 Scatterng ampltude matrx Note that the wave are not necearly lnearly polarzed. Dependng on ther coeffcent, they may have lnear, crcular or ellptc polarzaton n the plane perpendcular to the propagaton drecton Alo note that the ytem of unt vector for the cattered wave may depend on the catterng drecton condered Fnally, there are two man ytem of unt vector whch are uually ued a we ee next Bac catterng parameter 19

20 Unt vector ytem (1) Sytem baed on catterng plane: for every catterng k k drecton, and le on a plane wth the normal vector ˆ ˆ nˆ kˆ kˆ kˆ kˆ chooe aˆ aˆ nˆ bˆ kˆ aˆ aˆ nˆ bˆ kˆ aˆ bˆ kˆ aˆ Vector depend on catterng bˆ kˆ drecton condered Bac catterng parameter 0

21 Unt vector ytem () Vertcal and horzontal polarzaton: often there a preferred plane lke the earth urface (n geographcal ytem). Let normal to th plane be the ax z Then: nd unt vector choen to le n horzontal plane bˆ bˆ zˆ kˆ zˆ kˆ zˆ kˆ zˆ kˆ ẑ bˆ kˆ bˆ kˆ Bac catterng parameter 1

22 Unt vector ytem The other unt vector follow: a bˆ kˆ a bˆ kˆ ˆ ˆ Termnology: horzontal and vertcal polarzaton bˆ kˆ bˆ Later we may ue the noton TE and TM whch ẑ aˆ aˆ kˆ hould not be confued wth wavegude mode Bac catterng parameter

23 Unt vector ytem In term of cylndrcal coordnate ytem: 1 kˆ k co, k n, k k,,, z k, z z î bˆ zˆ kˆ n, co, 0 zˆ kˆ ˆ k ˆ kˆ aˆ bˆ kˆ 1 k k co, k n, k, z, z, x k, y Bac catterng parameter 3

24 Unt vector ytem In term of phercal coordnate ytem: kˆ n co, n n, co bˆ n, co, 0 θˆ aˆ co co, co n, n kˆ n co, n n, co bˆ n, co, 0 θˆ aˆ co co, co n, n ˆ ˆ Bac catterng parameter 4

25 Expreon for the cattered feld So far we jut dcued a number of parameter whch are ued n catterng problem But how can one expre the cattered feld n term of what happenng nde the object? Let u focu on an object whch may be a delectrc and/or conductve n free pace. Th object wll be characterzed by a complex permttvty 0 ( ) p r V j p ( r) p( r) p ( r) Bac catterng parameter 5

26 Expreon for the cattered feld When the ncdent feld radate the object, current are nduced. The um of nduced polarzaton and conducton current J ( r p ) j ( r p ) 0 E ( r ) J c( r ) p ( r ) E ( r ) J ( r ) J ( r p ) J c ( r ) j p ( r ) 0 E ( r ) What the far feld generated by th current? (Th the catterng feld) ( ) p r V Bac catterng parameter 6

27 Expreon for the cattered feld Far feld expreon for the cattered feld: f exp jkr E ( ) ˆ ˆ ˆ r jk r 4 r r F r f jk exp jkr H ( ) ˆ r r F r 4 r ˆ jk ˆ ˆ F r exp r r J ( r) dv V Note that n all drecton the cattered wave along the poton vector k ˆ r ˆ Bac catterng parameter 7

28 Expreon for the cattered feld (Far-zone) cattered electrc feld: for mplcty let u defne ˆ, ˆ F k ˆ Q k k jk 4 f exp jkr E ( r ) kˆ kˆ Q k ˆ, kˆ r Scatterng from equvalent current n a delectrc: k Q k, k exp k r ( r ) E ( r ) dv ˆ ˆ j p 4 0 V ( r) ( r) p p 0 Th the total feld nde the object: ncdent plu the feld generated by the object telf Bac catterng parameter 8

29 Expreon for the cattered feld Equvalently: f exp jkr E ( r ) Q ˆ, ˆ k k r,,, Q k ˆ k ˆ Q k ˆ k ˆ ˆ ˆ ˆ ˆ k Q k k k Compare wth the defnton of catterng ampltude: E 1 f r f E E ˆ ˆ k, ˆ, ˆ ˆ, ˆ k k k Q k k,0 Bac catterng parameter 9

30 Born approxmaton In prncple, we hould know the total feld nde the catterng object n order to be able to compute the cattered feld But, f ( ) an approxmaton can be made p r 0 In that cae the feld generated by the object mall and the total feld cloe to the ncdent feld E ( r) E ( r) E exp jk r,0 Bac catterng parameter 30

31 Born approxmaton Reult: ˆ ˆ k Q k, k exp j k k r ( r) dv E p,0 4 0 V f exp jkr E ( r ) kˆ kˆ Q k ˆ, kˆ r eˆ kˆ r eˆ kˆ Bac catterng parameter 31

32 Born approxmaton Equvalently f exp jkr E ( r ) Q ˆ, ˆ k k r ˆ ˆ k Q k, k exp j p( ) dv 4 k k r r 0 V E E kˆ kˆ,0,0 Bac catterng parameter 3

33 Born approxmaton So that E exp ( r jkr ), r E E k k k k ˆ ˆ S ˆ ˆ f,0,0 ˆ ˆ k S k, k exp j k k r ( r ) dv p 4 0 V The polarzaton of the cattered feld n the drecton kˆ gven by the drecton of the vector E E kˆ kˆ,0,0 Bac catterng parameter 33

34 Born approxmaton From the defnton of the catterng ampltude t follow that f 1 E k ˆ ˆ ˆ ˆ ˆ ˆ, k E,0 E,0 k k S k, k,0 Note that the polarzaton of the cattered wave n th approxmaton doe not depend on materal properte Alo the dependence of f on polarzaton the ame for all object becaue S ndependent of polarzaton. Fnally note that S the Fourer tranform of for p k k k k kˆ kˆ d Bac catterng parameter 34

35 Born approxmaton Example: phere of radu R wth contant Ung phercal coordnate, and choong a lnearly polarzed ncdent wave propagatng along z: k ˆ kˆ k 1 S S k R k R k R k d r, n co 3 d d d k d k d k n x k Becaue of the ymmetry of the tructure we do not need to k z conder other ncdent drecton y Bac catterng parameter 35

36 Born approxmaton k S j dv Detal of calculaton: ˆ ˆ p k, k exp k r d 4 0 phere Change coordnate ytem to one where z- along k d R ˆ ˆ k p S k, k exp jkd r co r n drd d 4 k R p exp d co n k k n k r R 1 R p p d exp jk d ru r drdu k 0 d k k v n vdv n k d R k d R co k d R kd R p p 3 3 0k d 0 0k d jk r r drd Bac catterng parameter 36 rdr

37 Born approxmaton Forward catterng lmt: 0 kd 1 V S S R k k 3 4 ˆ ˆ 3 k, k f r 1 r 1 f k ˆ, ˆ ˆ, ˆ k S k k 0 k R V k 4 R 3 3 Backward catterng lmt: kd k ˆ ˆ r 1 S k, k S b n kr kr co kr 8k f k ˆ, ˆ ˆ, ˆ k S k k Bac catterng parameter 37 k k

38 Born approxmaton A functon of catterng angle k k kr 0.5 kr 1 kr 5 S S f / / / Bac catterng parameter 38

39 Born approxmaton Sphere mall compared to k wavelength mall change wth angle Large phere how ocllatory k behavor wth decreang envelop In all cae the maxmum S n kr 15 forward drecton 0 dfferent rad) S V k 4 f r 1 (dfferent for S S f / Bac catterng parameter 39

40 Born approxmaton If the phere large compared to wavelength (kr>>1) then S larget n the forward drecton wthn the range gven by k d R kr R kr 15 k R S S f / Bac catterng parameter 40

41 Born approxmaton Note: n all thee dcuon, the catterng ampltude f follow from S by ncludng 1 E E kˆ kˆ n : angle between kˆ and E E,0,0,0 Forward, backward catterng: we look n a drecton along ncdent wave ( kˆ normal to kˆ kˆ ) o that the ncdent polarzaton and f and S are the ame In other drecton th not the cae, and we have to nclude the relatve angle wth repect to ncdent polarzaton to fnd the true catterng trength Bac catterng parameter 41

42 Born approxmaton Let u conder ome catterng parameter of a delectrc phere n Born approxmaton Dfferental catterng cro ecton kˆ, kˆ f kˆ, kˆ n S k ˆ, kˆ d Total catterng cro ecton x d kˆ, kˆ kˆ, kˆ n dd d d k y k z Bac catterng parameter 4

43 Born approxmaton n S n dd 0 0 Snce ncdent wave along z-drecton, take t polarzaton to be on the x-y plane makng an angle Then n 1 eˆ kˆ 1 n co 0 0 wth x-ax eˆ y 0 x k z Bac catterng parameter 43

44 Born approxmaton 0 S 1 co n d k p S n krn krn co krn 3 0 k n For a mall phere kr 1 k R 1 S ~ d k 3 p 0 R Bac catterng parameter 44

45 Born approxmaton 3 k pr 1 co n d k pr p k R V Bac catterng parameter 45

46 Born approxmaton Why the ky blue and the unet red? Becaue the catterng cro ecton proportonal to the 4 th order of frequency (k). Blue lght cattered more than red by mall partcle n the atmophere (dlute ga) Lght receved from drecton other than that of the ncdent beam reache u by catterng. Thu hfted to blue. Drect lght tend to be red a t the lght whch dd not undergo catterng procee n the atmophere. Bac catterng parameter 46

47 Born approxmaton* Scatterng ampltude matrx (n the catterng plane) E exp ( r jkr ), r E E k k k k ˆ ˆ S ˆ ˆ f,0,0 a b E ˆ E ˆ exp jk ˆ E a b k r aˆ nˆ exp f f, a f, b ˆ jkr E ˆ E a E b r faa fab 1 0 S fba f bb 0 co bˆ kˆ bˆ aˆ kˆ Bac catterng parameter 47

48 Born approxmaton* Note that a a E 1 0 E S b b E 0 co E Component normal to catterng plane (depend on obervaton drecton) get multpled by S. The other component frt projected on the unt ax of the ncdent wave. a E nˆ b E kˆ a E b E kˆ Bac catterng parameter 48

49 Born approxmaton So, f Born approxmaton apple, t eem that f we can meaure the catterng ampltude, we can recontruct the permttvty profle by an nvere Fourer tranform: 4 ( r ) S kˆ, kˆ exp k r k j dk ( ) 0 d p d 3 But the range of obervaton n k-pace lmted: k k kˆ kˆ kkˆ k d d k n 0 k k d kkˆ k d Bac catterng parameter 49

50 Born approxmaton Let u be more pecfc by conderng an example. We had k S k, k exp jk r ( r ) dv ˆ ˆ d p 4 0 V Let u conder a delectrc lab wth an nternal delectrc contant only dependng on x, kkˆ L y L ( r ) ( ) p p x kkˆ z w x Bac catterng parameter 50

51 Born approxmaton Then w / L / L / k S, exp j ( x ) dx dydz k ˆ kˆ k r k d p 4 0 w / L / L / k d, y L k d, z L n / n / k k 0 d, y d, z w / w / exp jk x ( x ) dx d, x p Hence, we can extract the followng functon from meaurement w /,, I k exp jk x ( x) dx d x d x p w / k k kˆ kˆ xˆ k k k d, x d, x Bac catterng parameter 51

52 Born approxmaton We apply the nvere Fourer tranform to th functon, neglectng the value outde the range of obervaton: / / k 1 e ( x ) I k exp jk x dk w k x x ( x ) dx w k k d, x d, x d, x 1 x x exp jk x x dk n k x x x x Bac catterng parameter 5 p d, x d, x

53 Born approxmaton The recontructed functon e(x) not the ame a. It a moothened veron averaged over the effectve wdth of the nc functon w / w / n k x x e( x) p ( x) dx x x Wdth of the nc functon x k 4 p x n k x x x x Bac catterng parameter 53

54 Relaton wth Fourer tranform Remember that A( r ) G r r J r dv 0 0 V Aume that thee current are nde the object. Conder two nfnte plane on the two de of the object x z y, 0 0 Bac catterng parameter 54

55 Relaton wth Fourer tranform Let u apply a Fourer tranform on a plane A( r )exp jk x x 0 0 V y jk y dxdy x, y exp x y J r G z z k k jk x jk y dv G z k k G jk x jk y dxdy, exp x y r x y 0 0 exp jk z z, k k k k jk z z x y Not necearly real! But hould atfy k k Re 0, Im 0 Bac catterng parameter 55 z z

56 Relaton wth Fourer tranform For the plane on the rght: z z exp 0 jkzz (, ) exp R A z k x ky jk r J rdv jk z V k R ( k, k, k ) x y z On the left: z z exp 0 jkzz (, ) exp L A z k x ky jk r J rdv jk z V k L ( k, k, k ) x y z Bac catterng parameter 56

57 Relaton wth Fourer tranform Invere Fourer tranform R, L R, L x A ( r) A ( z k, k )exp jk x jk y x y x y dk dk y Magnetc feld R, L 1 R, L 0 x H ( r) A ( z kx, ky )exp jkxx jky y dk dk y Fourer tranform of the magnetc feld jk H ( z k, k ) ( z k, k ) R, L R, L R, L x y A x y 0 Bac catterng parameter 57

58 Relaton wth Fourer tranform Smlarly, Fourer tranform of the electrc feld 1 E z k k k k A z k k R, L R. L R, L R, L R, L ( x, y) ( x, y) j 0 0 R, L exp jkzz R. L R, L R, L k k exp jk r J rdv kz 0 V Bac catterng parameter 58

59 Relaton wth Fourer tranform Conder the cae where k k k k R, L x y : real vector kˆ R. L kˆ Let and ue ˆ R, L R exp, L j F k k r J r dv V k exp jk z E ( z, ) k k F k R, L R, L z ˆR. L ˆR, L ˆR, L kx k y k z f exp jkr E ( r) jk kˆ kˆ F k ˆ 4 r Bac catterng parameter 59

60 Relaton wth Fourer tranform Hence, wth rˆ kˆ kˆ R, L R, L jkr jk exp z jkzz f, R, L exp R, L E ( r) E ( z k, x, k, y ) r The far electrc feld along any drecton n pace proportonal to the D Fourer tranform of the electrc feld on an arbtrary plane (between the object and the oberver) wth the Fourer component: k k, k k x, x y, y Bac catterng parameter 60

61 Relaton wth Fourer tranform From the defnton of the catterng ampltude t follow that ˆ ˆ k z f ( k, k ) E ( z k, x, k, y ) Alo note that k exp jk z E ( z k, k ) F k R, L R, L z ˆ R, L, x, y k z exp jkr E ( r) jk F k ˆ 4 r f, R, L R, L Bac catterng parameter 61

62 Optcal theorem Now, conder the catterng problem when the ncdent wave along z, normal to the fcttou plane Total feld: E E E H H H Not necearly far-zone, but the feld generated by nduced current at any pont k kzˆ x z, 0 0 Bac catterng parameter 6

63 Optcal theorem x Poyntng theorem appled to the urface contng of two nfnte plane normal to the z-ax, and urroundng the object k kzˆ ˆn ˆn z 1 Re S L S R * E H n ˆdS P L l =0 why? S S R, * 1 * Re ˆ Re ˆ E H nds E H nds S L S R S L S R 1 * 1 * Re ˆ Re ˆ ds ds E H n E H n S L S R S L S R Bac catterng parameter 63

64 Optcal theorem Reult: Scattered power Dpated power 1 Re S L S R * E ˆ H nds Pl 1 * 1 * Re ˆ Re ˆ E H nds ds E H n SL SR SL SR Bac catterng parameter 64

65 Optcal theorem Ung the defnton of a Fourer tranform: * R * E ˆ ˆ H n z exp E H,0 S S S L R R ds jkz ds L * jkz,0 zˆ exp E H R R L L exp jkz ˆ z ;0,0 exp jkz ˆ z ;0,0 * R R H ˆ,0 z exp jkz E L L z ;0,0 exp jkz E z ;0,0 z E H z E H * *,0,0 1 * R R L L E,0 exp jkz z ;0,0 exp jkz z ;0,0 E E Bac catterng parameter 65 S 1 * k E,0 F z E kz L L ˆ exp jkz z ;0, 0 k 0 nce k k 0 z x y L ds

66 Optcal theorem Ung the defnton of a Fourer tranform: * R * E ˆ ˆ H n z exp E,0 H S S S L R R ds jkz ds L * jkz,0 zˆ exp E H S L R * R L * L jkz zˆ E,0 ˆ H z jkz z E,0 H z R * R exp jkz L * L ;0, 0 ˆ z exp jkz H ;0,0 ˆ z z exp ;0,0 exp ;0,0 E,0 H z 1 R * R L * L E,0 exp jkz z ;0, 0 exp jkz z ;0,0 E E ds 1 k E,0 F z E kz L L ˆ exp jkz z ;0,0 * * Plu gn becaue t the returnng wave Bac catterng parameter 66

67 Optcal theorem Collectng the reult: 1 * 1 * P ˆ ˆ Pl Re Re E H nds E H nds SL SR SLSR 1 * L L Re,0 ˆ E exp jkz z ;0,0 F z E * L * L,0 ˆ E F z exp jkz E z ;0,0 1 * Re,0 ˆ E F z Bac catterng parameter 67

68 Optcal theorem Concluon: the total cattered plu aborbed power related to the forward catterng ampltude becaue f we look at the cattered feld along the ncdent feld f, R exp jkr E ( r) jk F 4 r In term of catterng cro ecton: zˆ Total cro ecton P 1 Re E F z l * t,0 S S ˆ Scatterng cro ecton Aborpton cro ecton a Bac catterng parameter 68

69 Optcal theorem Rewrtng th equaton: S E E F z E,0 * t Re,0,0 ˆ Same reult could have been ealy obtaned ung the conervaton of energy nde the object! Bac catterng parameter 69

70