It has no vertical component tmz wave note also that

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Unformatted text preview: m ,kz � z � � k k �� k ��� �� m,kz jm ˆ �ˆ � m , kz � k� k �� N m ,k z 1 �2 k �� m,kz � � 2 ˆ mk z � ˆ � � � ˆ jk z m , k z � z k �� m , kz � �� � � � Scattering from cylindrical objects 14 Vector waves in cylindrical coordinates � Let us see what kind of fields they represent. 1st solution: E � M m ,k z � df ( � ) � � jm ˆ m,kz f m, k z ( � ) � � exp(� jm� � jkz z ) � ˆ � k� kd � � � This field has no vertical component (TEz wave). It has an elliptic polarization in the x-y plane. � Its corresponding magnetic field is 1 j j H �� �� E � N m,kz � exp(� jm� � jkz z ) j�� � � 2 � � k� jkz df m,kz ( � ) ˆ k z m ˆ � 2 f m , k z ( � ) � z 2 f m, k z ( � ) � �� ˆ 2 k d� k� k � � Scattering from cylindrical objects 15 Vector waves in cylindrical coordinates � The 2nd solution for the electric field is: E � N m,k z � exp(� jm� � jk z z ) 2 � � df m,kz ( � ) k� jk z ˆ kz m f ( � ) � z ˆ 2 f m, kz ( � ) � �2 �� ˆ 2 m ,kz k d� k� k � � � Here, the electric field does have a vertical component. � It again has elliptic polarization, but in a plane rotated around the radial unit vector. Scattering from cylindrical objects 16 Vector waves in cylindrical coordinates � The corresponding magnetic field is � � 1 1 j H �� �� E � � � � � � M m, k z � M m , kz j�� j�� k � df ( � ) � � j jm ˆ m , kz f m,kz ( � ) � � exp(� jm� � jkz z ) � ˆ � � k� kd � � � � Now the magnetic field lies in the horizontal plane. It has no vertical component (TMz wave) � Note also that in both cases M m, kz � N m, kz � 0 � E � H � 0 Scattering from cylindrical objects 17 Vector waves in cylindrical coordinates � Example: m = 0 corresponds to the cylindrically symmetric fields M 0,kz N 0,kz � � � ˆ exp(� jk z z ) df m,k z ( � ) kd � 2 � � k� jk z df m,kz ( � ) ˆ � exp(� jk z z ) � � ˆ 2 � z 2 f m,k z ( � ) � k d� k � � Example: k z � 0 corresponds to solutions uniform along z M m,0 � jm ˆ df m,0 ( � ) � f m,0 ( � ) � � exp(� jm� ) � ˆ k� kd � � � � ˆ N m,0 � z exp(� jm� ) f m,0 ( � ) Scattering from cylindrical objects k� � k 18 Vector waves in cylindrical coordinates � For later use, let us consider the vector functions when (2) f m , kz ( r ) � H m � k � � � � This choice is useful for representing scattered wave since it satisfies the radiation condition � Consider the functions at a large distance from the center. Asymptotic relation for the Hankel function k� � � 1 � H (2) m (k � � ) � j Scattering from cylindrical objects m 2 j� � � exp � � jk � � � � � k� � 4� � 19 Vector waves in cylindrical coordinates � The TE solution has the asymptotic behavior: k � � � 1 � E � M m, kz j H � N 0,kz � j m k� exp(� jk� � � jm� � jk z z ) ˆ � C0 k k� � k� � � j m k� � exp(� jk� � � jm� � jk z z ) � kz ˆ � �� ˆ � z � � C � �k 0 � � k �� k� � �k � C0 Scattering from cylindrical objects 2 � j� � exp � � j � �4� 20 Vector waves in cylindrical coordinates � At large distance these are H ˆ k� ˆ � kz z ˆ z conical waves propaga...
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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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