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# Lecture 4 - Electromagnetic scattering Graduate Course...

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Electromagnetic scattering Graduate Course Electrical Engineering (Communications) 1 st Semester, 1390-1391 Sharif University of Technology

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Scattering from cylindrical objects 2 Contents of lecture 4 Contents of lecture 4: Scattering from cylindrical objects ± Introduction ± Scalar waves in cylindrical coordinates ± Vector wave equation ± Vector solutions in cylindrical coordinates ± Expansion of plane waves ± Scattering by perfectly conducting infinitely long cylinder ± Scattering width ± Scattering by a dielectric cylinder
Scattering from cylindrical objects 3 Introduction Scattering from flat, layered media is an example of an exactly solvable problem Another class of exactly solvable systems is the scattering by cylinders of infinite length In case of dielectric cylinders they are solvable for a constant dielectric constant i k

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Scattering from cylindrical objects 4 Introduction The system has rotational symmetry but, of course, the incident plane wave is not rotationally symmetric Nonetheless, we can still analyze the problem using the ‘natural’ solutions of the wave equation in cylindrical coordinates First, consider scalar waves in a homogeneous medium x y z ± ² ³ 2 2 0 k ´ µ ¶ · 2 2 2 2 2 2 1 1 0 k z ¸ ¹ º º º º · » ¼ º º º º ½ ¾ In cylindrical coordinates 2 2 k ¿ À ·
Scattering from cylindrical objects 5 Scalar waves in cylindrical coordinates Try solutions of the type ± ( , , ) ( )exp z z f jm jk z ² ³ ´ ³ ´ µ General solution: ± 2 2 2 2 2 2 ( ) ( ) ( ) 0 d f df k m f d d · · µ 2 2 z k k k µ ± ± , ( ) z m k m m f AJ k BY k µ · ¸ ± ± ± , ( , , ) exp z m k m m z z AJ k BY k jm jk z ² ³ ´ ¹ º µ · » ¼

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Scattering from cylindrical objects 6 Scalar waves in cylindrical coordinates Alternatively, one can represent the solution in terms of Hankel functions of the 1 st and 2 nd kind ± ± (1,2) ( ) m m m H k J k jY k ² ³ ´ ± ± ± (1) (2) , ( , , ) exp z m k m r m r z z AH k BH k jm jk z µ ² ¶ · ¸ ³ ¹ º º » ¼ Hankel functions preserve the ‘wave’ picture. For large radial distances they represent inward and outward moving waves ± ± (1,2) 2 : ( ) exp / 4 m m H k j jk j k ½ · ¸ ¾ ¿ ´ º » ¼ ±
Scattering from cylindrical objects 7 Vector wave equation These were the solutions in a homogeneous medium At the first sight the scattering problem for a cylinder can now be easily solved: just solve the Bessel and Hankel waves outside and inside the cylinder (if dielectric) and match them at the interface But there are two problems: We have to deal with a vector problem and a vector wave equation: EM fields are vector fields and have a polarization. Incident wave is not a ‘cylindrical’ wave, but a normal plane wave These facts make the analysis ‘a bit’ more complicated!

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Scattering from cylindrical objects 8 Vector wave equation Remember vector wave equation in a homogeneous medium ± ± 2 2 2 0 k k ²³ ²³ ´ µ ´² ¶ ² ²· ´ µ E E E E E Instead of trying to move to cylindrical coordinates and directly
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Lecture 4 - Electromagnetic scattering Graduate Course...

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