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with the wave vector k � ˆ � k z z ˆ E ˆ for every angle �. They all
make an angle sin � � k z / k with the zaxis
� � They behave as TEM waves:
fields are perpendicular to ˆ
k� ˆ � kz z Wave front each other and to the wave
vector in all directions Scattering from cylindrical objects �
21 Vector waves in cylindrical coordinates
� Magnetic field has no �
component, electric field has
only a � component � Again the electric and magnetic ˆ
k� ˆ � k z z
H ˆ
z ˆ E ˆ fields are orthogonal
� At a large distance these wave
behave as TEM waves � Remember: the above analysis of the asymptotic behavior is
only valid for solutions based on Hankel function of the 2nd kind Scattering from cylindrical objects 22 Vector waves in cylindrical coordinates
� The behavior of the TM solution at ˆ
k� ˆ � kz z
ˆ
z large distance is similar to TE but
now the electric field has no � ˆ H ˆ
E component and magnetic field only
has a � component E � N 0, kz k� � � j m k � � exp(� jk � � � jm� � jk z z )
� kz
ˆ
� � � j � �� ˆ � z � �
C0 �
�k
�
k ��
k� �
�k
� j
H � M m, kz
� � j m k� � exp(� jk � � � jm� � jk z z )
jˆ
C0 �
�
�
�k
�
��
k� �
� Scattering from cylindrical objects 23 Expansion of a plane wave
� So far we have analyzed the ‘natural’ solutions to the vector
wave equation in cylindrical coordinates. We expect them to be
useful when treating scattering by cylindrically symmetric objects � We have seen that asymptotically, these solutions behave as
TEM waves at large distance � But the actual problem we are interested is not the scattering of
these waves, but the scattering of a simple plane wave such as Ei ( r ) � Ei0 exp � � jki � r � ki � k
Scattering from cylindrical objects 24 Expansion of a plane wave
� To be able to use the cylindrical
vector solutions, we first have to
expand the plane wave into these ki functions
� Let us write z ki � � ki , x , ki , y , ki , z � � � ki , � cos �i , ki , � sin �i , ki , z � k 2
i,� �k �k
2
i, z ki , z y 2 �i x Scattering from cylindrical objects ki ki , � 25 Expansion of a plane wave
� z Remember that r r � � x, y, z � � � � cos � , � sin � , z � y ki � r � ki , � � cos �� � �i � � ki , z z
� We next use the following relationship � x � from the theory of Bessel functions exp � � jz cos � � � � � m ��� exp � � jki � r � � exp � � jki , z z �
Scattering from cylindrical objects (� j )m J m ( z ) exp � � jm� � � � m ��� (� j ) m J m � ki , � � � exp � � jm �� � �i � �
�
�
26 Expansion of a plane wave
z
� Remember that r J
� m,ki ,z ( � , � , z ) � J m � ki , � � � exp � � jm� � jki , z z �
� y � is one solution of the scalar wave
equation with the Bessel function of the x � first kind exp � � jki � r � �
� � � m ��� J
(� j )m� m, ki ,z ( � , � , z ) exp � jm�i � This is the expansion of a scalar plane wave for an incident
wave with ki , z , �i . What about the vector plane wave? Scattering from cylindrical objects 27 Expansion of a plane wave
� 1st result:
�
1
J
ˆ
� � � z exp � � jki � r �� � � (� j ) m M m, ki ,z ( � , � , z ) exp � j...
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 Spring '13
 akbabi
 Electromagnet

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