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Unformatted text preview: m ï¿½ï¿½j exp ï¿½ jm ï¿½ï¿½i ï¿½ ï¿½ ï¿½ ï¿½
ï¿½
ï¿½ ï¿½ ï¿½ ki , ï¿½ a ï¿½ Ë† ï¿½ C0
E ( ï¿½ ,ï¿½, z) ï¿½
4
TE
s 2 4 ï¿½ Ë†
E ï¿½ hi
0
i ï¿½1 ï¿½ 2cos ï¿½ï¿½i ï¿½ ï¿½ ï¿½ ï¿½
ï¿½
ï¿½ ï¿½ ï¿½k a ï¿½
i ,ï¿½ ki ,ï¿½ ï¿½ 2 ï¿½ exp( ï¿½ jki ,ï¿½ ï¿½ ï¿½ jki , z z ) ï¿½1 ï¿½ 2 cos ï¿½ï¿½i ï¿½ ï¿½ ï¿½ ï¿½
ï¿½
ï¿½ Scattering from cylindrical objects 53 TE Scattering by a thin conducting cylinder
ï¿½ Consider the scattered electric field. Itâ€™s phase does not
depend on angle. Itâ€™s amplitude is given below ï¿½
E sTE ( ï¿½ , ï¿½ , z ) ï¿½ Ë† E sTE (ï¿½ ) exp( ï¿½ jk i , ï¿½ ï¿½ ï¿½ jk i , z z )
,ï¿½
ï¿½ TE (ï¿½ ) ï¿½ ï¿½ C0
Es ,ï¿½
4 ï¿½ ï¿½ Ë† ï¿½ k a ï¿½2
E ï¿½ hi i ,ï¿½
0
i ki ,ï¿½ ï¿½ ï¿½1 ï¿½ 2 cos ï¿½ï¿½ ï¿½ ï¿½i ï¿½ ï¿½
ï¿½
ï¿½ EsTE
,ï¿½ y ki
ï¿½ï¿½
Scattering from cylindrical objects ï¿½i ï¿½ ï¿½ ï¿½i x
54 TE Scattering by a thin conducting cylinder
ï¿½ Note that scattering is largest when cos ï¿½ï¿½ ï¿½ ï¿½i ï¿½ ï¿½ ï¿½1 ï¿½ ï¿½ ï¿½ ï¿½i ï¿½ ï¿½
ï¿½ This is the backward scattering case ï¿½ ï¿½
It is zero when ï¿½ ï¿½ ï¿½i ï¿½
2 EsTE
,ï¿½ y
ï¿½ ï¿½ ï¿½i ï¿½ ï¿½
ï¿½ï¿½ Scattering from cylindrical objects ï¿½i x
55 Numerical experiments
ï¿½ Let us return to the exact result for TE case and again
focus on the azimuthal component of the electric field ï¿½ ï¿½ Ë†
EsTE ( r ) ï¿½ ï¿½ j Ei0 ï¿½ hi exp ï¿½ ï¿½ jki , z z ï¿½
,ï¿½
ï¿½ ï¿½ m ï¿½ï¿½ï¿½ ï¿½
( ï¿½ j )m J m ï¿½ ki ,ï¿½ a ï¿½
(2)
H m ï¿½ ï¿½ ki ,ï¿½ a ï¿½ (2)
H m ï¿½ ï¿½ ki ,ï¿½ ï¿½ ï¿½ exp ï¿½ ï¿½ jm ï¿½ï¿½ ï¿½ ï¿½i ï¿½ ï¿½
ï¿½
ï¿½ ï¿½ We would like to compare this with the asymptotic result ï¿½ This a difficult function to compute because of â€˜badâ€™ behavior of
Bessel functions of large order Scattering from cylindrical objects 56 Scattering by a perfectly conducting infinite cylinder
ï¿½ So let us consider the exact numerical result for the azimuthal
component of the electric field ï¿½ y We plot this function when ï¿½i ï¿½ 0 ï¿½ ki ï¿½ ï¿½ ki , x ,0, ki , z ï¿½ , ki , ï¿½ ï¿½ ki , x Ei ï¿½
ki , x Ë†
Ë†
hi ï¿½ Ë† ï¿½ y
i EsTE
,ï¿½ x Ë†
Ei0 ï¿½ hi ï¿½ Ei0, y Ë†
Ë†
Ei (r ) ï¿½ Ei0, y y exp ï¿½ ï¿½ jki ï¿½ r ï¿½ ï¿½ Ei0, y y exp ï¿½ ï¿½ jki , x x ï¿½ jki , z z ï¿½
Scattering from cylindrical objects 57 Numerical experiments
ï¿½ For a thin cylinder we have the amplitude profile ï¿½ /3 Ei ki ,ï¿½ a ï¿½ 0.25 ki , x Scattering from cylindrical objects 58 Numerical experiments
ï¿½ It is also instructive to look at constant phase fronts, at which phase(Q) ï¿½ 0 ki ,ï¿½ a ï¿½ 0.25 Scattering from cylindrical objects 59 Numerical experiments
ï¿½ So in the far field, the phase fronts are almost circular in the xy plane: there is no angledependence of the phase of the
scattered field, as found from the thinfilm approximation ï¿½ Also the amplitude is largest in case of backscattered waves,
as found from the same approximation ï¿½ ï¿½
The amplitude becomes nearly zero when ï¿½ ï¿½ ï¿½i ï¿½
3 ï¿½ Again this is in line with what we found before ï¿½ So this approximation is quite accurate Scattering from cylindrical objects 60 TM Scattering by a thin conducting cylinder
ï¿½ For the TM case from a thin cylinder we have k i , ï¿½ a ï¿½ ka ï¿½ 1
ï¿½ For small arguments (consider positive or zero mâ€™s) 1 ï¿½zï¿½
Jm ï¿½ z ï¿½ ï¿½ ï¿½ ï¿½
m! ï¿½ 2 ï¿½
ï¿½ m Lowest order in 2j
(2)
H 0 ï¿½ z ï...
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This note was uploaded on 02/26/2013 for the course EE 25227 taught by Professor Akbabi during the Spring '13 term at Sharif University of Technology.
 Spring '13
 akbabi
 Electromagnet

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