# Ki these currents create the scattered field at any

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Unformatted text preview: m�i � � � k m ��� M J m , ki , z � � J �� m , ki , z 1 jm J J ˆ � � � � m , ki , z z � ˆ � m , ki , z � ˆ k k� k �� � jm ˆ ki , � J � � k � � � exp � � jm� � jk z � m i,� i,z � ˆ k � J m � ki , � � � � � k � � dJ m � z � � Jm � z� � dz Scattering from cylindrical objects 28 Expansion of a plane wave � The first vector relation now follows: � � � ˆˆ z � ki exp � � jki � r � � � j � � m ��� J (� j ) m M m , ki ,z ( � , � , z ) exp � jm�i � For the 2nd vector relation take the curl of the above equation � � � 1 J ˆˆ � � � z � ki exp � � jki � r � � � � j � (� j ) m N m ,ki ,z ( � , � , z ) exp � jm�i � � � k m ��� N Scattering from cylindrical objects J m, ki , z 1 J � � � M m,ki ,z k 29 Expansion of a plane wave � Explicit form of N: J N m,ki ,z � exp � � jm� � jki , z z � � � jki , z ki , � ki2, � mki , z ˆ � ˆ J m � ki , � � � � J m � ki , � � � � z 2 J m � ki , � � �� �� ˆ 2 2 k k� k � � � The 2nd vector relation now becomes � � � J ˆ � z � ki � ki � exp � � jki � r � � � � (� j )m N m ,k ( � , � , z ) exp � jm�i � ˆˆ i ,z � � m ��� Scattering from cylindrical objects 30 Expansion of a plane wave � Now, we define the unit vector system ˆˆ z � ki ˆ hi � ˆˆ z�k ˆˆ z � ki ˆ ˆˆˆ vi � hi � ki � � ki ˆˆ z�k i i ˆ z ˆ ki ˆˆ z � ki ˆ z ˆ ki � zˆ � kˆ � � kˆ i Scattering from cylindrical objects i 31 Expansion of a plane wave � Let us see what these vectors are: z ki � � ki , x , ki , y , ki , z � � � ki , � cos �i , ki , � sin �i , ki , z � ˆ1 ki � � ki , � cos �i , ki , � sin �i , ki , z � k ˆˆ z � ki ˆ hi � � � � sin �i , cos �i , 0 � � ˆi ˆˆ z � ki ˆ � k � ˆ � k � ki , z ρ � ki , � z ˆ ˆ ˆi ˆ vi � hi ˆi i i k k ˆ ki , z i ki �i x ˆ ρi y ki , � These cylindrical unit vectors are defined with respect to ki. They have a subscript i. Do not confuse them with the cylindrical unit vectors for position! ˆ ρi � � cos �i ,sin �i , 0 � Scattering from cylindrical objects 32 Expansion of a plane wave � We thus have the relations ˆ hi exp � � jki � r � � ˆi exp � jki � r � � k �j ki , � � � m ��� J (� j ) m M m, ki ,z ( � , � , z ) exp � jm�i � ki , � � � ki , z ˆ ˆ ˆ vi exp � � jki � r � � � ρi � z � exp � � jki � r � � k� �k k� J � (� j ) m N m , ki ,z ( � ,� , z ) exp � jm�i � � ki , � m ��� Scattering from cylindrical objects 33 Expansion of a plane wave � For any polarization of the incident plane wave: � � ˆˆ ˆˆ Ei (r ) � Ei0 exp � � jki � r � � � Ei0 � hi hi � � Ei0 � vi � vi � exp � � jki � r � � � Ei ( r ) � � � � m ��� um M ˆ um � � j E � hi 0 i � J m , ki ,z (� ,� , z ) � � � m ��� k (� j ) m exp � jm�i � ki , � k ˆ vm � � � E � vi � (� j ) m exp � jm�i � ki , � 0 i Scattering from cylindrical objects J vm N m,ki ,z ( � , � , z ) Related to TE: horizontal electric field Related to TM: horizontal magnetic field 34 Exact so...
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