Slide 1OverviewSpectral Domain Immitance MethodSDI Method (cont.)Slide 5Slide 6Slide 7Slide 8Slide 9TMz PWTMz PW (cont.)Slide 12TEz PWWave ImpedancesTEN: TMz PWTEN: TMz PW (cont.)TEN: TEz PWSource ModelSource Model (cont.)Slide 20Slide 21TENExampleExample (cont.)Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Finite SourceFinite Source (cont.)Slide 33Slide 34Slide 35Spectral-Domain Green’s FunctionSlide 37Slide 38Slide 391Spring 2011Notes 19ECE 6345ECE 6345Prof. David R. JacksonECE Dept.2OverviewOverviewIn this set of notes we introduce the Spectral Domain Immitance (SDI) method, which is a powerful method for solving for the fields due to sources inside of layered media. The basic idea is developed here by decomposing a finite current sheet into a set of infinite phased current sheets. The fields are found from an infinite phased current sheet.The fields from the infinite current sheets are added together (spectral integration) to recover the fields of the finite current sheet.3Spectral Domain Immitance MethodSpectral Domain Immitance Method1e( , )sJ x y0e0ezWe initially consider a planar source inside of a layered structure. (The method can be extended to include vertical sources as well.)The figure shows a single layer, but the method works for any number of layers.4SDI Method (cont.)SDI Method (cont.)Considering the integrals as limits of sums, we can write( )( )( )( )( )( )2, ( , )1, ,2x yx yj k x k ys x y sj k x k ys s x y x yJ k k J x y e dxdyJ x y J k k e dk dkp+�+�+- �- �+�+�- +- �- �==����%%Introduce Fourier transform pair:( )( )( )( )21, ,2xm ynj k x k ys s xm yn x ym nJ x y J k k k k ep� �- +=- � =- �= D D��%1e( , )sJ x y0e0ez5SDI Method (cont.)SDI Method (cont.)( )( )( )( )21, ,2xm ynj k x k ys s xm yn x ym nJ x y J k k k k ep� �- +=- � =- �= D D��%xkyk( ),xm ynk kxkDykD6Denote( )( )21,2mns xm yn x yA J k k k kp= D D%( )( )( )( )21, ,2xm ynj k x k ys s xm yn x ym nJ x y J k k k k ep� �- +=- � =- �= D D��%( )( ),xm ynj k x k ys mnm nJ x y A e� �- +=- � =- �=��ThenThe finite-size current sheet is thus expressed as a superposition of infinite phased current sheets. SDI Method (cont.)SDI Method (cont.)7( ) ( ), ,mns sm nJ x y J x y� �=- � =- �=��( )( ),xm ynj k x k ymns mnJ x y A e- +=That is,where( , )mnsJ x yzSDI Method (cont.)SDI Method (cont.)8Consider a single sheet of current of the form:( )0x yj k x k yppssJ J e- +=( , )psJ x yz zTM TE+zWe wish to determine the amplitude of the plane waves that this current source launches, and the field at any point inside the structure.SDI Method (cont.)SDI Method (cont.)The superscript p stands for “phased current sheet.”Note: TMz and TEz waves reflect from the boundary and remain TMz and TEz, respectively.9pwfxyPWtktanypwxkkf =The current sheet launches a pair of plane waves that propagate in a transverse direction determined by the wavenumber.( , )psJ x yz zTM TE+ztop viewSDI Method (cont.)SDI Method (cont.)10TMTMzz PW PW3D viewtEtHtop viewThe upward-going TMz plane wave inside the layer is shown here.kpwfxPWyyPWHxzE11TMTMzz PW (cont.) PW (cont.)Denoteˆˆ ˆ ˆ ˆ ˆcos sinyxtpw pwt tkku k x y x yk kf� � � �= = + = +� � � �� � � �andNote: These unit vectors depend on the values of (m,n).ˆ ˆˆv z u= �pwfxyˆuˆvtk12For the TMz plane wave, we then haveˆˆt ut vE u EH v H==The “t” subscript means “transverse (perpendicular) to the z direction.TMTMzz PW (cont.) PW (cont.)pwfxytEtHtk13TETEzz PW PWThe upward-going TEz plane wave is polarized as shown here.ˆˆt ut vH u HE v E==kyPWHxzEyPWHxzEpwfxytEtHtk14Wave ImpedancesWave ImpedancesTMz :TEz :The wave impedances are calculated for waves traveling upward. pwfxytEtHtkpwfxytEtHtkTMuzvEkZH we= =TEvu zEZH kwm-= =15TEN: TMTEN: TMzz PW PWConsider the TMz plane wave that gets launched by the current sheet:zTMWe wish to use a TEN model to find the plane-wave field inside the layered structure.zzTM( , )psJ x y16TEN: TMTEN: TMzz PW (cont.) PW (cont.)We introduce the following modeling equations: ( ) ( ) ( )( ) ( ) ( )000 00 0TMu uTMv vV z E , ,z E zI z H , ,z H z= == =TEN modelsourceNote that the voltage (tangential electric field) must be continuous at the source location, so the source model is a parallel element.The zero subscript indicates that the field has the exponential phase term suppressed.( )TMI z+-( )TMV z( )( )1/22 2 21/22 2zi i x yi tk k k kk k= - -= -0TMZ0TMZ1TMZ1TMZTMsI17TEN: TETEN: TEzz PW PW( ) ( ) ( )( ) ( ) ( )000 00 0TEv vTEu uV z E , ,z E zI z H , ,z H z=- =-= =We introduce similar modeling equations for the TEz case: TEN modelsource( )TEI z( )TEV z+-( )( )1/22 2 21/22 2zi i x yi tk k k kk k= - -= -0TEZ0TEZ1TEZ1TEZTEsI18Source ModelSource Model( ) ( )( ) ( ), , , ,0, , , ,0zzjk zt tjk zt tH x y z H x y eH x y z H x y e+ +-- -+==By symmetry,   0,,0,, yxHyxHttConsider the current sheet in an infinite homogeneous medium:( , )psJ x y1e1ez+ wave (TMz+TEz)- Wave (TMz+TEz)19Source Model (cont.)Source Model (cont.)Denote( ) ( ) ( ), , ,TM TEp p ps s sJ x y J x y J x y= +( ) ( ) ( )( )( ) ( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )ˆ, , ,0 , ,0ˆ ˆˆ, ,0 , ,0ˆ, ,0 , ,0ˆ0,0,0 0, 0, 0ˆˆTM TM TMx yx yx ypt tsv vv vj k x k yv vj k x k yTM TMj k x k yTMsJ x y z H x y H x yz v H x y v H x yu H x y H x yu e H Hu e I z I zu e I+ -+ -+ -- ++ -- ++ -- += � -= � -=- -=- -=- -=-Conclusion: The current that launches the TMz plane wave is polarized in the u direction.The TM current is that part of the total current that launches only a TMz plane wave, while the TE current launches only a TEz plane wave.20Source Model (cont.)Source Model (cont.)Hence( ) ( )( )ˆ ˆ, ,TMp ps sJ x y u u J x y= �Similarly,( ) ( )( )ˆ ˆ, ,TEp ps sJ x y v v J x y= �fxytkTMpsJTEpsJ( ) ( )()( )()ˆ ˆ, , ,TM TEp p ps s sJ x y u J x y v J x y= +21Source Model (cont.)Source Model (cont.)We can now determine the source amplitude in the TEN model:so( )()ˆ ˆ ˆ ˆTMx yj k x k yp pTMs ssu J u J u u e I- +� = � = �-or( )( )ˆ,x yj k x k ypTMssI u J x y e+=- �0ˆpTMssI u J=- �Similarly,0ˆpTEssI v J= �The zero subscript indicates that the current has the exponential phase term suppressed.From …