# Z we next use the following relationship x from the

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Unformatted text preview: ting ˆ 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 z-axis � � 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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