Ki these currents create the scattered field at any

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online