Scattering cross section (scattering width)

Transcription

1 Scatterng cro ecton (catterng wdth) We aw n the begnnng how a catterng cro ecton defned for a fnte catterer n ter of the cattered power An nfnte cylnder, however, not a fnte object The feld radated by ource nde the cylnder (cattered feld) doe not drop a 1/ r, but a 1/ We have to reforulate our defnton of catterng cro ecton for nfnte cylndrcal object Scatterng fro cylndrcal object 77

2 Scatterng cro ecton (catterng wdth) Agan conder an ncdent wave E e E j r 0 exp Now, the cattered far feld alway of the type E f ˆ, ˆ E j j z e 0 exp,, z Note: catterng apltude ay depend on the polarzaton of the ncdent wave! Scatterng fro cylndrcal object 78

3 Scatterng cro ecton (catterng wdth) Correpondng far feld Poyntng vector are S 0 E ˆ ˆ, ˆ 0 f 0 E E S ˆ ˆ, z, It portant to bear n nd that ρ z ˆ,, z ˆ ˆ Scatterng fro cylndrcal object 79

4 Scatterng cro ecton (catterng wdth) The Poyntng vector ndependent of z, t the ae for all vertcal coordnate We, therefore, cannot defne the total cattered power nce t nfnte But, we can defne the cattered power per unt length by ntegraton over the angle P d S f d, ˆ L S ρ, 0 0 Scatterng fro cylndrcal object 80

5 Scatterng cro ecton (catterng wdth) Reeber that, co, Total catterng wdth, z now defned a P, L W f, d S 0 Scatterng fro cylndrcal object 81

6 Scatterng cro ecton (catterng wdth) A an exaple conder the TE cae E hˆ E exp j r hˆ ˆ 0 ˆ ˆ The cattered far feld: TE ˆ 0 ˆ E E h jc exp( j j z) () 0, J R,, H R, exp j z Scatterng fro cylndrcal object 8

7 Scatterng cro ecton (catterng wdth) The catterng apltude f TE (),, J, R, exp j H R Scatterng wdth for TE wave: J R H R TE, TE 4, W f, d () 0, Scatterng fro cylndrcal object 83

8 Scatterng cro ecton (catterng wdth) TM cae: vˆ E vˆ E exp j r 0 ˆ ˆ v ρ z, z, ˆ ˆ ˆ Scattered feld:,, 0 exp( 0,, ) TM z jc j j zz E ˆ ˆ z E v ˆ J R, H R (), exp j, Scatterng fro cylndrcal object 84

9 Scatterng cro ecton (catterng wdth) Scatterng apltude: f TM J R, exp, j (), H, R Scatterng wdth: J R TM W H R, TM 4, f, d () 0, Scatterng fro cylndrcal object 85

10 Scatterng cro ecton (catterng wdth) Exaple: thn cylnder 4 J TE, R W (), R O, R H 4, R TM W, ln 4 J R 1 O () H, R, R, R 4 Scatterng fro cylndrcal object 86

11 Scatterng cro ecton (catterng wdth) Nuercal reult for the TE cae TE W, R Scatterng fro cylndrcal object 87

12 Scatterng cro ecton (catterng wdth) Plotted n a dfferent way TE W R, R Scatterng fro cylndrcal object 88

13 Scatterng cro ecton (catterng wdth) For the TM cae TM W, R Scatterng fro cylndrcal object 89

14 Scatterng by an nfntely long delectrc cylnder So far we dcued a (perfectly) conductng cylnder What about a delectrc cylnder? We can actually ue the ae achnery to olve the proble Incdent plane wave: J um, E ( r) (,, z), z v N J,, z (,, z) Scatterng fro cylndrcal object 90

15 Scatterng by an nfntely long delectrc cylnder For the cattered wave (outde cylnder) we ue the ae expanon a before H am, E ( r) (,, z) b N, z H,, z (,, z) Inde the cylnder we ue oluton baed on Beel functon of 1 t nd (why?) J J c,, E ( r ) c M (,, z) d N (,, z), z, z Scatterng fro cylndrcal object 91

16 Scatterng by an nfntely long delectrc cylnder Next tep to atch the feld at the boundary: ˆ E E ˆ E c R zˆ E E zˆ E c E z E Th agan lead to algebrac equaton for coeffcent whch can be olved Note that nde the cylnder, c, d, z d 0 d Scatterng fro cylndrcal object 9

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