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# Class Note for ECE 6341 with Professor Jackson at UH 3

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Date Created: 02/06/15

ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 36 Radiation Physics in Layered Media O x Note TMZ and also TEy since O 82 F O or y gt W 2 AZ w 010 wi1 FTE kx ajkyoye jkxxdkx 47 kyo Reflection Coefficient 21 kx ZTE kx 1 TEkX zTE Z0TE in x 0 x where ZizE kx jZlTE tanky1h my 2 2 12 25E k 0 kyo k0 kx yO a 12 ZITE 10 kyl k12 43 yl This is the same equation as the TRE for finding the wavenumber of a surface wave W kxp roots of TRE kaW Poles cont Complex kx plane kx 2 km 1km km S W C I X I I I I I I x I er k1 k0 k0 SW k1 If a slight loss is added the SW poles are shifted off the real axis as shown Poles cont For the lossless case two possible paths are shown here Review of Branch Cuts and Branch Points In the next few slides we review the basic concepts of branch points and branch cuts Branch Cuts and Points cont Consider 212 Z I 616 Choose E 9 0 212 1 There are two possible values Branch Cuts and Points cont The concept is illustrated for 212 Z r 6 9 12 192 Z e Consider what happens if we y encircle the origin r 1 1 B C Branch Cuts and Points cont 2 3962 ZlZ ej y Branch Cuts and Points cont Now consider encircling the origin twice point 6 z 2 A 1 B 72 j C 27239 1 D 37239 j E 47239 1 11 12 3992 Z ej y K7r1 D C B A E x We now get back the same result Hence the squareroot function is a doublevalued function Branch Cuts and Points cont The origin is called a branch point we are not allowed to encircle it if we wish to make the squareroot function singlevalued In order to make the squareroot function singlevalued we must put a barrier or branch cut y branch cut W x Here the branch cut was chosen to lie on the negative real axis an arbitrary choice 12 Branch Cuts and Points cont We must now choose what branch of the function we want 396 12 3962 zzre Z xe Z1 121 Branch Cuts and Points cont Here is the other choice of branch 396 12 3962 zzre Z xe Branch Cuts and Points cont Note that the function is discontinuous across the branch cut 396 12 3962 zzre Z xe branch cut 657 W 1 x Z 1 9 7r z Zlz 2 ZlZ 1 15 Branch Cuts and Points cent The shape of the branch cut is arbitrary Z r 619 212 ejQZ y o x Z 1 branch cut 212 1 Branch Cuts and Points cont The branch cut does not even have to be a straight line Z r ejg In this case the branch is determined by requiring that the 2 62 squareroot function and hence the angle 6 change Z e continuously as we start from a specified value eg z 1 y Z 1 Z Z J 12 j 41 J Z12 Z 9 1 0 o x Z 1 branch cut 12 Z 1 0 Z J39 17 18 Branch Cuts and Points cont Consider this function similar to our wavenumber function What do the branch points and branch cuts look like for this function Branch Cuts and Points cont fZ 22 12 Z112Z112 Z112 Z112 y There are two branch cuts we are not allowed to encircle either branch point Branch Cuts and Points cont Geometric interpretation Z 112z 112 2 WI 2 win w1z 1rlequot9i W2 2 z 1 r2 6 62 the functionfz is unique once we specify the value at any point 20 Riemann Surface The concept of the Riemann surface is illustrated for fzzl2 zrej6 The Riemann surface is really two complex planes connected together The function 212 is continuous everywhere on this surface there are no branch cuts It also assumes all possible values on the surface Consider this choice Top sheet 7239 lt 9 lt 7239 1 Bottom sheet 7239 lt 9 lt 37239 1 21 emann Surface comm X side View top V39eW Riemann Surface cont top sheet branch point x 0 bottom sheet 24 Riemann Surface cont connection between sheets 27239 47239 12 Z Branch Cuts in Radiation Problem Now we return to the original problem wAZ Branch Cuts 1 k k02 432 k0 kxk0 kx yO kx k0 NIH jkx k0 Note it is arbitrary that we have factored out a j instead of a j since we have not yet determined the meaning of the square roots Branch points appear at kx iko No branch cuts appear at kx ikl Theintegrand iS an even function of kyl 26 Branch Cuts cont Branch cuts are lines we are not allowed to cross Branch Cuts cont For kxzrealgtk Z 0 Choose argkx k0 0 kyO J lky0 argkx k0 0 at this point kyO k02 x2 12 km O XI This choice then uniquely defines kyo everywhere in the complex plane 28 Branch Cuts cont gt k ky0j lkxk0ej 2H lkxk l C 7r 0 For kx 2 real we have afg kx k 0ltkxltkO mg x C xi Riemann Surface top sheet bottom sheet There are two sheets joined at the blue lines 30 Proper Improper Regions Let kx er kxz39 r k n The goal is to figure out which k0 k0 J 0 regions of the complex plane are quotproperquot and quotimproperquot proper region 1m kyo lt O improper region 1m k gt O boundary 1m kyo 0 gt kjoszZ kf realgt0 31 Proper Improper Regions cont Hence ko39 jlcoquot2 er ew2 2 real gt 0 ko392 k0quot2 kfr kxf j 2k039k0quot 2erkxi 2 real gt 0 Therefore hyperbolas One point on curve er k0 I 0 k k0 XI kakO 32 Proper Improper Regions cont Also k 2 ICO 2 k kxiz gt 0 0 xi The solid curves satisfy this condition Proper Improper Regions cont k xi Complex plane top sheet k XI proper improper region On the complex plane corresponding to the bottom sheet the proper and improper regions are reversed from what is shown here 34 Sommerfeld Branch Cuts k xi XI k0 hyperbola Complex plane corresponding to top sheet proper everywhere Complex plane corresponding to bottom sheet improper everywhere 35 Sommerfeld Branch Cuts k k x1 xi Com lex lane p 390 Riemann surface Note we can think of a single complex plane with branch cuts or a Riemann surface with hyperbolicshaped ramps connecting the two sheets The Riemann surface allows us to show all possible poles both proper surfacewave and improper leakywave 36 Sommerfeld Branch Cut k Let k0quot gt 0 xi XI The branch cuts now He along the imaginary axis and part of the real axis 37 38 Path of Integration k xi The path is on the complex plane corresponding to the top Riemann sheet Numerical Path of Integration LeakyMode Poles 1m kyo gt 0 ReVIew of frequency behaVIor improper k f fc ko SW er ISW v k1 Note TM0 never 39 I becomes Improper I LW f 2 fs I l 40 Riemann Surface We can now show the k leakywave poles x 41 42 SW and CS Fields k XI SW field CS field k0 k1 kXIquot Cp LW SW Total field surfacewave SW field continuousspectrum CS field Note the CS field indirectly accounts for the LW pole Leaky Waves LW poles may be important if LW k0 s Rekxp s k0 Im kxp ltlt k0 Physical Interpretation LW k0 511160 The LW pole is then Close to the path on the Riemann surface radiation 9 m Reef W 43 leaky wave Improper Nature of LWs Region of strong I leakage fields kpr z ja leakage rays The rays are stronger near the beginning of the wave this gives us exponential growth vertically 44 Improper Nature cont Mathematical explanation of exponential growth improper behavior 1 Icng k02 4pr 25 gt 1ng szoz 41 le 2 y jay2 kOZ a2 Equate imaginary parts 06 gt 0 gt 05y lt 0 improper 45

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