Problem te polarization exp jki z z ki z h r

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Unformatted text preview: lution of the problem � We now consider the scattering of a plane wave by an infinitely long, perfectly conducting cylinder � When the incident wave hits the cylinder, surface currents (and charges) are induced. � ki These currents create the ‘scattered’ field. At any point, the total electric field is E ( r ) � Ei ( r ) � E s ( r ) Ei ( r ) � Ei0 exp � � jki � r � Scattering from cylindrical objects 35 Exact solution of the problem � We saw how the incident plane wave can be represented in terms of cylindrical vector solutions � The scattered field (outside the cylinder) can also be expanded in terms of those solutions � But: for the scattered field we should ki use vectors with the right condition at the infinity � �� Scattering from cylindrical objects 36 Exact solution of the problem � We should use the Hankel function of the 2nd kind for these waves which satisfy the radiation condition (behave as outgoing waves at infinity) � Also, since the system is uniform in z-direction, kz is the same as that of the incident wave: k z � ki , z . Of course the wave will be scattered along different angles (different �’s) Es (r ) � M H m , ki , z � � m ��� H am M m,ki ,z ( � , � , z ) � 1 H ˆ � �� �� m ,ki ,z z � � � k Scattering from cylindrical objects � � m ��� N H bm N m,ki ,z ( � , � , z ) H m , ki , z 1 H � �� M m,ki ,z k 37 Exact solution of the problem � The vector functions are explicitly given by H M m , ki , z � exp � � jm� � jki , z z � � jm (2) ˆ ki , � H (2)� � k � � � m i,� � ˆ k � H m � ki , � � � � � k � � H N m, ki ,z (r ) � exp � � jm� � jki , z z � � � jki , z ki , � (2) ki2,� (2) mki , z (2) � � ki , � � � � ˆ ˆ Hm H m � ki , � � � � z 2 H m � ki , � � �� �� ˆ 2 2 k k� k � � � Note that we have used Scattering from cylindrical objects k � � k 2 � k z2 � k 2 � ki2z � ki ,� , 38 Exact solution of the problem � The next step is to choose the coefficients of the expansion such that the condition at the surface of the cylinder is satisfied � Zero tangential component of total electric field E z ( a, � , z ) � E� (a , � , z ) � 0 � a 1st Result: bm � �vm J m � ki , � a � H (2) m �k a� Ez E� i ,� k J m � ki , � a � ˆ � � E � vi � ( � j )m exp � jm�i � ( ki ,� H m2) � ki ,� a � 0 i Scattering from cylindrical objects 39 Exact solution of the problem � 2nd result: am � �um � J m � ki , � a � (2) � H m �� � k i , � a � � � � k J m � ki ,� a � 0ˆ � j Ei � hi ( � j )m exp � jm�i � ( ki ,� H m2)� � ki ,� a � � � � Note that a TE incident wave excites TE polarized waves (horizontal polarization), and a TM wave excites TM polarized waves (vertical polarization) Scattering from cylindrical objects 40 Exact solution of the problem (TE polarization) � TE scattering (horizontal electric field): EsTE ( r ) � � � m ��� H am M m , ki , z ( � , � , z ) � um J m � ki ,� a � � ��� m ��� � � ˆ jk � E � hi ki , � 0 i (2) H m � � ki , � a � � � m ��� Scattering from cylindrical objects H M m , ki , z ( � , � , z ) � J m � ki , � a � (2) H m � � ki , � a � H ( � j ) m exp �...
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