The amplitude in all directions is the same

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)...
View Full Document

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.

Ask a homework question - tutors are online