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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 xy 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.
 Spring '13
 akbabi
 Electromagnet

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