Scatterng by a perfectly conductng nfnte cylnder Reeber that ths s the full soluton everywhere. We are actually nterested n the scatterng n the far feld lt. We agan use the asyptotc relatonshp exp exp (2) 2 j j jk j jk H ( k 1) exp jc0 4 k k We fnd that n the far feld lt the felds are parallel to: E s ( k 1) ˆ E TM s k z k ( k ˆ ˆ 1) z k k Sae as ˆθ Scatterng fro cylndrcal objects 49
Scatterng by a perfectly conductng nfnte cylnder More specfcally: ˆ 0 ˆ s E E h jc exp( jk jk z) 0,, 0 exp( 0,, ) TM k z k jc jk jk z z E ˆ ˆ s z k k E v k ˆ Scatterng fro cylndrcal objects 50 k (2) J k R H k R (2) J k R z H k R exp j exp j
Scatterng by a perfectly conductng nfnte cylnder The nterestng pont here s the behavor of far feld as functon of the angle, whch s governed by the functons (2) J k R H k R exp j J k R H k R (2) exp j TM Reeber that H ( 1) H J ( 1) J (2) (2) Scatterng fro cylndrcal objects 51
Scatterng by a thn conductng cylnder To get soe nsght let us consder the lt of a thn cylnder k R kr 1 For sall arguents (suffcent to consder postve or zero s) J 0 z z 2 1 0 ( ) z J z J 1 z J z z 2 ( 1)! H (2) 0 z j2 z j 2! z z H (2) z H (2) ( z) H (2) z 0 1 1 Scatterng fro cylndrcal objects 52
Scatterng by a thn conductng cylnder To the lowest order we have for the case (2) J k R H k R exp j 2 k R j 1 2cos 4 E s (,, z) ˆC 4 0 0 E ˆ h k R k exp( jk jk zz) 1 2cos 2 Scatterng fro cylndrcal objects 53
Scatterng by a thn conductng cylnder Consder the scattered electrc feld. It s phase does not depend on angle. It s apltude s gven below E (,, z) ˆE ( ) exp( jk jk z) s s, z E 0 E ˆ h k R C 4 k 0 s, ( ) 1 2cos 2 Es, y k x Scatterng fro cylndrcal objects 54
Scatterng by a thn conductng cylnder Note that scatterng s largest when cos 1 Ths s the backward scatterng case It s zero when 2 Es, y x Scatterng fro cylndrcal objects 55
Nuercal experents Let us return to the exact result for case and agan focus on the azuthal coponent of the electrc feld 0 ˆ E ( r) j E h exp jk z s, z (2) ( j) J k R H k j H k R exp (2) We would lke to copare ths wth the asyptotc result Ths a dffcult functon to copute because of bad behavor of Bessel functons of large order Scatterng fro cylndrcal objects 56
Scatterng by a perfectly conductng nfnte cylnder So let us consder the exact nuercal result for the azuthal coponent of the electrc feld We plot ths functon when y Es, 0 k k,0, k, k k x z x hˆ ˆ yˆ E hˆ E 0 0 y E k, x x ˆ E ( r) E yˆ exp jk r E yexp jk x jk z 0 0 y y x z Scatterng fro cylndrcal objects 57
Nuercal experents For a thn cylnder we have the apltude profle E / 3 k R 0.25 k, x Scatterng fro cylndrcal objects 58
Nuercal experents It s also nstructve to look at constant phase fronts, at whch phase( Q) 0 k R 0.25 Scatterng fro cylndrcal objects 59
Nuercal experents So n the far feld, the phase fronts are alost crcular n the x- y plane: there s no angle-dependence of the phase of the scattered feld, as found fro the thn-fl approxaton Also the apltude s largest n case of backscattered waves, as found fro the sae approxaton The apltude becoes nearly zero when 3 Agan ths s n lne wth what we found before So ths approxaton s qute accurate Scatterng fro cylndrcal objects 60
TM Scatterng by a thn conductng cylnder For the TM case fro a thn cylnder we have k R kr 1 For sall arguents (consder postve or zero s) J z 1 z! 2 (2) 2 j H0 z ln z H (2) 0 z j 2 ( 1)! z Lowest order n k R : J k R j exp (2) j 2ln H k R k R Scatterng fro cylndrcal objects 61
TM Scatterng by a thn conductng cylnder Resultng far scattered feld for a thn conductng cylnder E E k k (,, z) E ˆ zˆ exp( jk jk z) k k C ˆ E v k 2ln k R TM TM z s s, v z TM 0 0 s, v In the lowest order approxaton there s no angle dependence for the TM scattered wave! The apltude n all drectons s the sae Scatterng fro cylndrcal objects 62
TM Scatterng by a thn conductng cylnder Let us copare the scatterng strength n the two cases E ˆ 0 2 k C E 0 h k R s, ( ) 1 2cos k 4 k E E vˆ TM 0 s, v C 0 k 2 ln k R For equal horzontal and vertcal coponents of the ncdent feld we have E s, E ( ) 3 2 TM s, v k k 2 k R ln k R Scatterng fro cylndrcal objects 63
TM Scatterng by a thn conductng cylnder Ths rato s qute sall for thn cylnders But ths result s to be expected: a vertcally polarzed ncdent electrc feld has a coponent along the wre and easly nduces electrc currents along the wre. These currents, n turn, generate the scattered feld. A horzontally polarzed ncdent wave has no longtudnal coponents and cannot excte such currents TM E E Scatterng fro cylndrcal objects 64
Scatterng by a thck conductng cylnder Now, let us nvestgate the other lt, that of a thck conductng cylnder whch satsfes k R R k k 2 2 z 1 Reeber: scattered electrc feld Far feld lt (2) jc exp( jk jk z) J k R E E h ˆ 0 ˆ 0 z s exp (2) j k H k R Scatterng fro cylndrcal objects 65 uj k R H Es ( r) M, k (,, z), z H k R
Scatterng by a thck conductng cylnder Let us approxate the denonator (2) (2) 1 2 j z 1 H z jh ( z) j exp jz z 4 J k R k R j exp exp (2) j j jk R 2 4 H k R, ( j) J k R exp j Next, use the relaton j cos exp jz cos ( j) J ( z) exp j Scatterng fro cylndrcal objects 66
Scatterng by a thck conductng cylnder The far scattered feld becoes (,, ) ˆ R Es z E h cos 0 ˆ exp jk ( R) jk R cos jk zz Reeber that ths result s also based on the far-feld behavor of the vector solutons (used here), whch n turn, was based on the asyptotc behavor of the Hankel functon when k 1 Scatterng fro cylndrcal objects 67
Nuercal results for a thck cylnder So let us agan look at soe nuercal results for the azuthal coponent of the electrc feld We agan plot ths functon when y Es, 0 k k,0, k, k k x z x E k, x x hˆ ˆ yˆ E hˆ E 0 0 y ˆ E ( r) E yˆ exp jk r E y exp jk x jk z 0 0 y y x z Scatterng fro cylndrcal objects 68
Nuercal results for a thck cylnder The apltude profle s shown below E k xr 10 k, x Scatterng fro cylndrcal objects 69
Nuercal results for a thck cylnder Phase fronts: E k xr 10 k, x Scatterng fro cylndrcal objects 70
Nuercal results for a thck cylnder Note that n front of the cylnder the scattered wave looks noral, t has a crcular phase front But behnd the cylnder, at a not too far dstance, the phase fronts are flat! Besdes, ts apltude s uch larger than the waves scattered back (to the left). But ths s just a sconcepton. To understand ths pont, one should realze that the total feld s E ( r) E ( r) s Scatterng fro cylndrcal objects 71
Nuercal results for a thck cylnder Let plot the coponent of the total feld The apltude of the scattered wave behnd the cylnder s large because t has to partally cancel the ncdent wave, to for a shadow regon Scatterng fro cylndrcal objects 72
Nuercal results for a thck cylnder Nevertheless, the nuercal results are very dfferent fro the asyptotc result we found earler f we only deand k 1 To understand why, consder the exact electrc feld 0 E hˆ exp E j jk z s, z (2) ( j) J k R H k j H k R exp (2) Asyptotc behavor of Hankel functon depends on whether arguent s larger, saller, or coparable to order Scatterng fro cylndrcal objects 73
Scatterng by a thck conductng cylnder The correct asyptotc result s 2 k H ( k ) j exp jk j / 4 2 (2) k For a thn cylnder k R only sall s contrbute ( =0,1) 1 Then, far feld asyptotc s vald snce k 1 (2) J k R H k R k R 10 k R 1 k R 0.25 Scatterng fro cylndrcal objects 74
Scatterng by a thck conductng cylnder But, for a thck cylnder, all orders wth k R have a contrbuton so that large orders are also portant The used asyptotc s then only accurate when k k R 2 2 R 2 Ths s slar to the defnton of the far feld regon (why?) So our earler result (and the earler dscusson of the far-feld behavor of vector wave solutons) s, strctly speakng, only applcable n ths lt Scatterng fro cylndrcal objects 75
Scatterng by a thck conductng cylnder Slar dscusson apples to TM scatterng: whereas n the far feld lt the scattered wave behaves as M waves wth a concal wave front (crcular n x-y plane), the behavor outsde ths lt can be totally dfferent One can have flat wave fronts, a shadow regon, etc In partcular, note that when the wavelength s very short copared to the densons of the object, the far feld result s only vald so far away that t ay be useless for practcal applcatons Scatterng fro cylndrcal objects 76
Scatterng cross secton (scatterng wdth) We saw n the begnnng how a scatterng cross secton s defned for a fnte scatterer n ters of the scattered power An nfnte cylnder, however, s not a fnte object The feld radated by sources nsde the cylnder (scattered feld) does not drop as 1/ r, but as 1/ We have to reforulate our defnton of scatterng cross secton for nfnte cylndrcal objects Scatterng fro cylndrcal objects 77