# The amplitude in all directions is the same

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Unformatted text preview: H H s ( r ) � � � vm (2) M m,ki , z ( � , � , z ) � m ��� H m � ki ,� a � Scattering from cylindrical objects 47 Exact solution of the problem (TM polarization) H TM s ,� ˆ ( r ) � � E � vi � 0 i � � m m ��� H TM s ,� exp � � jki , z z � � ki , � � ( � j ) m J m � ki , � a � (2) H m � ki , � a � (2) H m � ki ,� � � exp � � jm �� � �i � � � � j ˆi � exp � � jki , z z � (r ) � � � E � v � 0 i � � m ��� ( � j ) m J m � ki , � a � (2) H m � ki , � a � Scattering from cylindrical objects (2) H m � � ki ,� � � exp � � jm �� � �i � � � � 48 Scattering by a perfectly conducting infinite cylinder � Remember that this is the full solution everywhere. We are actually interested in the scattering in the far field limit. � We again use the asymptotic relationship m j m exp � � jk � � � 2 j� � j exp � � jk � � � � (2) H m ( k � � � 1) � exp � � � jC0 � � k� � k� � �4� � We find that in the far field limit the fields are parallel to: E (ki , � � � 1) � ˆ TE s Scattering from cylindrical objects E TM s ki , � � ki , z ˆ� (ki , � � � 1) � � � k �k ˆ Same as θ � ˆ z� � 49 Scattering by a perfectly conducting infinite cylinder � More specifically: E TE s � ˆ � ˆ Ei0 � hi � jC0 exp( � jk � � � jk z z ) k� � � � m ��� E TM s ki , � � ki , z ˆ� � �� k �k (2) H m � � ki , � a � exp � jm ��i � � � � � � � � jC0 � exp( � jki ,� � � jki ,z z ) �0 ˆ ˆ z � � Ei � vi � ki ,� � � � � m ��� Scattering from cylindrical objects � J m � ki , � a � J m � ki ,� a � H (2) m �k a � i ,� exp � jm ��i � � � � � � 50 Scattering by a perfectly conducting infinite cylinder � The interesting point here is the behavior of far field as function of the angle � , which is governed by the functions � J m � ki ,� a � � � m ��� H � � ki , � a � � J m � ki ,� a � � m ��� � (2) m H (2) m �k a � i ,� exp � jm ��i � � � � � � TE exp � jm ��i � � � � � � TM Remember that (2) (2) H � m � (�1) m H m J � m � (�1) m J m Scattering from cylindrical objects 51 TE Scattering by a thin conducting cylinder � To get some insight let us consider the limit of a thin cylinder k i , � a � ka � 1 � For small arguments (sufficient to consider positive or zero m’s) z � J0 � z � � � 2 (2) H0 � � z � � � j2 �z Scattering from cylindrical objects m z m�1 � J m�0 � z � � � J m�1 ( z ) � J m � z � � m z 2 (m � 1)! m (2) j 2m m ! (2) (2) H m � 0� � z � � � H m �1 ( z ) � H m � z � � � z � z m �1 52 TE Scattering by a thin conducting cylinder � To the lowest order we have for the TE case � � m ��� � J m � ki , � a � H � � ki , � a � (2)...
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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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