EE243 Advanced Electromagnetic Theory Lec # 23 Scattering and Diffraction. Reading: Jackson Chapter , lite

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1 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 43 Advancd lctromagntic Thory Lc # 3 cattring and Diffraction calar Diffraction Thory Vctor Diffraction Thory Babint and Othr Principls Optical Thorm ading: Jackson Chaptr.5-.9,.-. lit Copyright 6 gnts of Univrsity of California

2 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 Ovrviw Obcts larg compard to a wavlngth ar gnrally tratd by approximat intgrals ovr th assumd filds on thir surfacs. In many cass (whr th polarization is not important scalar diffraction can b usd. Whr polarization ffcts ar important a vctor formulation is ndd. Th two ky factors in th approximation Th assumd filds on th surfacs or aprturs Th sourc fr Grn s function usd in th intgral Copyright 6 gnts of Univrsity of California

3 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 calar Intgral prsntation for Far Fild ψ G ψ ( x = [ ψ ( x n G( x, x G( x, x n ψ ( x ] ( x, x = ( x ψ ik = 4π x x = f 4π ik n ψ + ik + 4π 4πr ψ ψ r ( θ, φ ik Gnral prsntation for solution to scalar wav quation Choos scalar Grn s function ( to simplifis notation Intgral that closs surfac at infinity gos to zro radiation condition f(θ,φ is th radiation pattrn Copyright 6 gnts of Univrsity of California r i k ψ da da Jackson.5 3

4 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 ψ ψ Kirchhoff Approximation prsntation GN D ( x = 4π πi ik ik n ik ψ i n k ( x = ψ ( x da ψ da n Apply to crn with aprtur Assumptions ψ and its normal drivativ vanish xcpt on opning ψ and its drivativ ar qual to th thos incidnt on aprtur with no scrn Inhrnt inconsistncis inc scattrd fild is zro vrywhr on scrn it is zro vrywhr Intgral dos not yild th assumd valus on th opnings nforcing ithr Dirichlt or Numan Boundary Conditions rsults in a consistnt formulation i k Copyright 6 gnts of Univrsity of California Jackson.5 4

5 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 ψ ψ Kirchhoff Approximation: Grn s Function GN ( x ( x = ϑ( θ, θ (, θ (, θ ϑ θ ϑ θ (, θ = ( cosθ + cosθ ϑ θ = 4π k πi = cosθ = cosθ r ik r n ψ + ik + da ψ da xampl for a point sourc on on sid of crn Approximating ψ, δψ/δn or kping both (Kirchhoff givs th sam intgral xcpt for th obliquity factor ϑ(θ,θ that wights th rays by th cos of th arrival or takoff angl. i k Copyright 6 gnts of Univrsity of California Jackson.5 aprtur n 5

6 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 Vctor Intgral prsntation for Far Fild G ˆ ( x = [ ( n G G( n ] da ( x = [ iω( n B G + ( n G + ( n ] * s ( x F ( k, k F 4πr r iknˆ x i ik x * * ( k, k = [ ωˆ ( n Bs + ˆ ( k ( n s ] da 4π tart with x in volum and intraction intgral Trat x as singular point plus rst of volum Apply divrgnc thorm Us fr spac Grn Function Intgral on surfac at infinity gos to zro writ in transvr only componnts of and B on surfac Copyright 6 gnts of Univrsity of California G da Jackson.7 6

7 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 Diffraction by crn with Aprtur A B ( x = ( nˆ B ( x = ( nˆ B diff π π scrn π scrn da da ( x = ( nˆ da aprturs ik B is givn by intgrating B valus on th scrn gomtry. is givn by intgrating filds on th aprturs This suggst that dual problms ar rlatd (Babint s principl ik ik scrn aprtur n Copyright 6 gnts of Univrsity of California 7

8 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 Vctor Thorms and Concpts quivalnc thorm: Contributions from sourcs outsid of a volum can b found from tangntial and H on th surfac of th volum. action intgral: Intgral of tangntial filds on th surfac is sam as calculating dot J and H dot M throughout th volum. Grn s Function choic: Th rgion outsid th volum could b filld with p..c. matrial to cancl and doubl ffct of H or magntic matrial to cancl H and doubl th ffct of Babint s Principl: For prfctly conducting thin scrn and its complmnt th lctric and magntic filds for complmntary problms ar givn by th sam intgral. For xampl in th cas of a slot th magntic fild in a an aprtur is usd and th complmntary cas is a mtallic bar (scrn and th lctric fild ovr th bar is usd. Copyright 6 gnts of Univrsity of California Jackson.8 8

9 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 Diffraction by a Circular Aprtur Far Fild k ψ CALA VCTO k r = k knˆ x + ( x = 4πr i πr [ r ( nˆ x ] ( x = k nˆ ( x [ nˆ ψ ( x + ik nψ ( x ] ik x ( da ik x +... planar scrn da Jackson.9 x aprtur n Approximat k by Taylor sris Us calar or Us Vctor for A plus B curl A, curl B x Copyright 6 gnts of Univrsity of California 9

10 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 Diffraction by a Circular Aprtur Far Fild ξ = π a π ikρ sinα cosβ sinθ cos( φ β ( x = ρdρ dβ πr ( sin θ + sin α sinθ sinα cosφ π dβ ( kρξ i J ( ( ( kaξ x = a cosα k dp dω i = P cosα Pi = Z i r πa cosα ikρξ cosβ [ ] ( ka J ( ( kaξ 4π = cosα J cos θ + cos kaξ φ sin θ kaξ Plan wav in x-z plan incidnt from blow TAN rducd by cos α; linar phas in x dirction Find fild in dirction k linar phas in x and y dirctions Combin all phass; rcogniz azimuthal intgral as J ; intgrat in ρ => J sult is J (v/v with wighting for tangntial componnts of arrival and scattring Copyright 6 gnts of Univrsity of California a x k α inc z θ k y φ B inc in y dir

11 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 Diffraction by a Circular Aprtur Far Fild: i J ( ( ( kaξ x = a cosα k dp dω dp dω α = dp dω = P cosα Pi = Z = ψ i P P r ( x i i πa Vctor vrsus calar ( ka J ( ( kaξ ( kaξ ( ka cosα cosθ J ( kaξ 4π 4π = ik r cosα ( ka J ( kaξ π a cos + cosα cosα kaξ θ + cos kaξ φ sin cosα + cosθ J kaξ θ kaξ Compar calor Kirchhoff Vctor Diffrnc: Obliquity typ factors Agr whn α = Copyright 6 gnts of Univrsity of California kaξ a x k α inc z θ k y φ B inc in y dir

12 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 cattring in th hort Wavlngth Limit z shadow illuminatd y hadowd gion Contribution x Boundary Condition s = - inc ; B s = -B inc mall Av xcpt forward => dpnd only on proctd ara (diffraction pattrn from th shadow Illuminatd gion Contribution Boundary Conditions s = - inc ; B s = -B inc AM as Ill.!!! Normal diffrnc givs sign diffrnc and diffrnt rsult tationary phas brings our spcular surfac contributions hadow diffraction can dominat in forward dirction Figur.6 Copyright 6 gnts of Univrsity of California

13 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 Optical Thorm duction in forward dirction proportional to th total powr radiatd Know Far Fild xprssion Look at Nar Fild Poynting Vctor to dtrmin th total powr takn from th wav With substitution, som manipulation th intgral pf th total powr bcoms proportional to th intgral for th scattring amplitud in th forward dirction. Physical intrprtation: Th total powr takn from th wav must appar in a commnsurat rduction in th total fild in th forward dirction. How can an lctric fild b rlatd to Powr? (as a prturbation tot = inc + y and y << inc th powr * ~ inc inc *(- s / inc Powr rduction is proportional to s Copyright 6 gnts of Univrsity of California 3

14 Applid M Fall 6, Nuruthr Lctur #3 Vr /5/6 Dilctric Proprtis and Crystal Opalsnc ε ε ε ε ( ω = + ε m ( ω iωγ wω ( ω 4πN * = + f ( k = k p = m * f N k f ( ω iωγ wω k f ( k = k = m ( 4πε ω iωγ wω f duction in forward dirction proportional to th total powr radiatd Mdia Modl Chaptr 7.3.A (Harmonic Oscillator lat to Optical Thorm. (valuat absorption Includ mchanical intraction from forc on atoms.3.d Crystal Opalsnc occurs whn th forcs btwn atoms caus mchanical linkag of nrgy btwn atoms that produc dnsity variations that affct M absorption and scattring. Copyright 6 gnts of Univrsity of California 4