It has no vertical component tmz wave note also that

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

This is the end of the preview. Sign up to access the rest of the document.

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...
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