Scattering from cylindrical objects 78 scattering

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Unformatted text preview: � � ln z � j 2m ( m � 1)! (2) H m�0 � z � � � � zm ki , � a : J m � ki , � a � j� � H (2) � k a � exp � jm ��i � � �� � 2 ln � k a � � � m ��� m i ,� i ,� � Scattering from cylindrical objects 61 TM Scattering by a thin conducting cylinder � Resulting far scattered field for a thin conducting cylinder ki , � � ki , z ˆ� E ( � ,� , z) � E � � k �k C0 � TM 0 ˆ Es ,v � � Ei � vi � ki ,� � 2 ln � ki ,� a � TM s � TM s ,v � ˆ � exp( � jki ,� � � jki , z z ) z � In the lowest order approximation there is no angle dependence for the TM scattered wave! � The amplitude in all directions is the same Scattering from cylindrical objects 62 TM Scattering by a thin conducting cylinder � Let us compare the scattering strength in the two cases ˆ � k a �2 k � C0 E � hi i ,� TE Es ,� (� ) � 1 � 2cos �� � �i � ki , � 4 ki , � � 0 i E � TM s ,v C0 ˆ � E � vi 0 i � ki ,� � 2 ln � ki ,� a � For equal horizontal and vertical components of the incident field we have EsTE (� ) ,� EsTM ,v Scattering from cylindrical objects 2 3k � � ki,� a � ln � ki ,� a � 2 ki , � 63 TM Scattering by a thin conducting cylinder � This ratio is quite small for thin cylinders � But this result is to be expected: a vertically polarized incident electric field has a component along the ‘wire’ and easily induces electric currents along the EiTE wire. These currents, in turn, generate the scattered field. � EiTM A horizontally polarized incident wave has no longitudinal components and cannot excite such currents Scattering from cylindrical objects 64 TE Scattering by a thick conducting cylinder � Now, let us investigate the other limit, that of a thick conducting cylinder which satisfies ki ,� a � a k 2 � ki2,z � 1 � Remember: scattered electric field � E (r ) � � � TE s � E TE s m ��� � um J m � ki ,� a � (2) H m � � ki , � a � H M m ,ki , z ( � , � , z ) Far field limit � ˆ � ˆ E � hi 0 i � jC0 exp( � jki ,� � � jk z z ) Scattering from cylindrical objects ki , � � � � m ��� � J m � ki , � a � H � � ki , � a � (2) m exp � jm ��i � � � � � � 65 TE Scattering by a thick conducting cylinder � Let us approximate the denominator (2) (2) z � 1 � H m � � z � � � jH m ( z ) � � j m �1 2 j� � � exp � � jz � � �z 4� � � J m � ki , � a � � ki , � a j� � � � H (2)� k a exp � jm ��i � � � � � j 2 exp � jki ,� a � 4 � � � � � m ��� m � i ,� � � � � m ��� � � ( � j )m J m � ki ,� a � exp � jm ��i � � � � � � Next, use the relation � j cos � exp � � jz cos � � � � � m ��� Scattering from cylindrical objects � (� j ) m J m ( z ) exp � � jm� � 66 TE Scattering by a thick conducting cylinder � The TE far scattered field becomes � ˆ E ( � , � , z ) � � ˆ Ei0 � hi TE s � a cos �� � �i � � � exp � � jki ,� ( � � a ) � jki ,� a cos �� � �i � � jki , z z � � � � Remember that this result is also based on the far-field behavior of the vector solutions (used here), which in turn, was based on the asymptotic behavior of the Hankel function when k� � � 1 Scattering from cylindrical objects 67 Numerical results for a thick cylinder � So let us again look at some numerical results for the azimuthal component of the electric field y � We again plot this function when Ei �i � 0 � ki � � ki , x ,0, ki , z � , ki , � � ki , x ˆ ˆ hi � ˆ � y i EsTE ,� � ki , x x ˆ Ei0 � hi � Ei0, y ˆ ˆ Ei (r ) � Ei0, y y exp � � jki � r � � Ei0, y y exp � � jki , x x � jki , z z � Scattering from cylindrical objects 68 Numerical results for a thick cylinder � The amplitude profile is shown below Ei ki , x a...
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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.

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