Physics 74/84 Elecomagneic Theoy II G. A. Kaff Jeffeson Lab Jeffeson Lab Pofesso of Physics Old Dominion Univesiy Physics 84 Elecomagneic Theoy II -3-1
Pependicula Polaizaion = E + E E Tangenial E ( ) ε ( ) ε = E E cos i E cos Tangenial H µ µ E ncosi = E n cos i µ + n n sin µ i µ E ncosi n n sin i µ = E µ n cos i + n n sin i µ Physics 84 Elecomagneic Theoy II
Wave Guides: Suface Absobion Homogeneous Maxwell Equaions imply n B B c = n E E c = ( ) c ( ) Cul lhh Maxwell Equaion and lage bu finie i conduciviy i implies n H H c = ( ) 1 E = c H c σ i H c = E µω c c Physics 84 Elecomagneic Theoy II
ξ i ( + n H ) c = δ n H c = 1/ δ = µωσ c ξ / δ iξ / δ H = H e e H c pa pa ωµ E c i n H σ dp 1 Re * µωδ c = n E H = H da 4 c ( ξ = ) = ( 1 )( ) pa pa is he field a he suface Physics 84 Elecomagneic Theoy II
Alenaive Calculaion ˆ δ 1 * 1 powe/vol = J E = J σ K = Jd ξ = nˆ H 1 ( ) ( ) ( i) 1 / J = σ Ec = 1 i n H pa e ξ δ eff dp da pa 1 µωδ = Keff = σδ 4 c K eff suface cuen fo pefec conduco Physics 84 Elecomagneic Theoy II
Cylindical Sysems z z-axis along he cylinde diecion E = i ω B B = B = iµε E E = ( E + µεω ) = B ω B = B x, y e E = E x, y e ( ) ( ) ± ikz ± ikz ω Physics 84 Elecomagneic Theoy II
Tansvese Sepaaion E = E ˆ ( ˆ) zz + E E = E E z zˆ B = B ˆ ( ˆ ) zz + B B = B B z zˆ E + ( µεω ) k = B E + i ω zˆ B = ˆ ( ) Ez z E = i ω Bz z B iµεω zˆ E = B zˆ ( B ) = i µεω E z E z Bz E = B = z z z z Physics 84 Elecomagneic Theoy II
TEM Modes Soluions wih ansvese field only Eem = E, em Bem = B, em E = E =, em, em E em mus solve -D elecosaic poblem k = k = µεω B =± µε zˆ E em em Can hee be a soluion inside a single closed waveguide? No, need a leas wo conducos. No cuoff fffequency Physics 84 Elecomagneic Theoy II
Moe Geneal Case Tansvese field expessible in ems of z-field only! i E ˆ = k Ez ω z B z µεω k i B = k B + µεω zˆ E µεω k z z Tansvese Magneic (TM) B z = ; Bounday Condiion E = z S Tansvese Elecic (TE) E z B = ; Bounday Condiion z = n S Physics 84 Elecomagneic Theoy II
Waveguides ± 1 H = z ˆ E Z Wave Impedance Z k k µ = (TM) εω k ε = µω k µ = (TE) k k ε Physics 84 Elecomagneic Theoy II
Eigenvalue Poblem TM Waves TE Waves E H ik = ± ψ γ ik = ± ψ γ Tansvese Helmholz Equaion + = + + = x y γ ψ γ ψ ( ) Physics 84 Elecomagneic Theoy II
Bounday Condiions ψ S ψ = (TM) = (TE) nn S Specum of eigenvalues and eigenfuncions γ λ ψ λ ( x, y ) Wavelengh in mode λ k λ = µεω γ λ Physics 84 Elecomagneic Theoy II
Cuoff Fequency ω λ ω = γ λ µε No popagaion a fequencies below cuoff ( ) k λ = µε ω ω λ Fo single-mode popagaion choose fequency o be above cuoff fo lowes mode and below cuoff fo all ohe modes. Phase velociy infinie a cuoff! Physics 84 Elecomagneic Theoy II
Recangula Waveguide x y + + γ ψ = ψ mπ x nπ x = (TE) ψ mn ( x, y ) = Acos cos n a a S ω γ mn mn m = π + a n b π m = + µε a n b 1/ Physics 84 Elecomagneic Theoy II
Lowes Mode ω 1 = π µε m a π x H z = Acos e a ika π x H z = Asin e π a ω aµ π x Ey = i Asin e π a ikz iω ikz iω ikz iω Lowes TM mode has m and n one wih a sin soluion. Why? Is cuoff fo he lowes mode is a a fequency highe by ( 1/ 1 + a /b b ) Physics 84 Elecomagneic Theoy II
Enegy Flow 1 * S = E H γ * ε zˆ ψ + i ψ ψ ω k k S = 4 γ γ * µ zˆ ψ i ψ ψ k ω k ε * P = S zda ˆ = 4 ψ ψ da γ µ A A ωk ε ψ P = ψ dl ψ ψ da 4 + γ µ * * n C A Physics 84 Elecomagneic Theoy II
Enegy and Goup Velociy P U P U v v 1/ λ 1 1 ω ω λ ε = µε ωλ ω µ 1 ω ε = ω λ µ * ψ ψ ω λ = k 1 1 1 v ω µε = µε ω = = 1 p g µε A da g A * ψψ da Physics 84 Elecomagneic Theoy II
Aenuaion Field aenuaion consan given by β λ ( ) = λ P z P e β λ β z 1 dp = P dz dp 1 = nˆ H dl dz σδ C 1 ψ 1 µ ω λ n = dl σδ C 1 ω λ ωλ 1 nˆ ψ ψ + µ ωλ ω ω Physics 84 Elecomagneic Theoy II
ψ n nˆ ψ µεω ψ λ C 1 ω λ ψ n C dl = ς λ µε A A ψ da ( / ) 1/ ε 1 C ω ω λ ω λ βλ = ς 1/ λ + ηλ µ σδ λ A 1 ω / ω ω λ ( 1 ω / ω ) λ Physics 84 Elecomagneic Theoy II
Resonan Caviies Pu end conducos on a cylindical waveguide. Example: Cylindical caviy of lengh d and adius R. In geneal, z dependence d is Asin kz + B cos kz ( ) ( ) π BCs k = p, p =,1,, K d TM pπ Ez = ψ ( x, y) cos, p =,1,, K d TE pπ H z = ψ ( x, y ) sin, p = 1,, K d Physics 84 Elecomagneic Theoy II
Field Paens TM sin p p E π π ψ = sin ˆ cos E d d i p H z ψ γ εω π ψ = TE d i ψ γ ˆ sin i p E z d ωµ π ψ γ = ( ) cos / p p H d d d π π ψ γ = Physics 84 Elecomagneic Theoy II ( ) / p d γ µεω π =
1 ω = λ p γ pπ d µε + TM Eigenvalue Equaion ( ( / ) ) λ (, ) = AJ ( ) ψ ρ φ γ ρ m mn e ± imφ xmn γ mn = J m ( xmn ) = R ω mnp 1 x p π.45 mn = + ω = µε µε 1 R d R.45ρ ε.45ρ E z = AJ e H = i AJ e R µ R iω iω φ 1 Physics 84 Elecomagneic Theoy II
TE (, ) = AJ ( ) ψ ρ φ γ ρ γ ω ω m mn mn mn m mn mnp e ± imφ x = J ( x ) = R 1 x mn = + µε R p π d 1.841 R = 1 +.91 µε R d 111 1.841ρ π z H z = AJ1 cosφ sin e R d iω Physics 84 Elecomagneic Theoy II
Caviy Losses Q = du d Soed enegy ω boh popoional o ψ Powe loss ω = U Q U () = U e ω () Q / ( ω ω) ω / Q i + E = Ee e Physics 84 Elecomagneic Theoy II
( ) Paallel Polaizaion = cos i E E cos E Tangenial E ε ( ) ε = E + E E Tangenial H µ µ E nn cosi = E µ n cos i + n n n sin µ i µ E n cos i n n n sin i µ = E µ n cos i + n n n sin i µ Physics 84 Elecomagneic Theoy II
Enegy Consevaion? Nomal Incidence E n = = E n n µε + + 1 µε µε 1 E µε n n = = E n n µε + 1 µε + ε E ε E ε E = + µ µ µ Physics 84 Elecomagneic Theoy II
Misakes fom Las Time Enegy Consevaion a Nomal Incidence ε E ε E ε E = + µ µ µ E n = = E n n µε + + 1 µε µε 1 E µε n n = = E n n µε + µ ε + 1 Physics 84 Elecomagneic Theoy II
Bewse s Angle Calculaion n cos i = n n n sin i B ( sin ) B n cos i = n n n i n 4 B B 4 4 + n ± n 4 n cos i sin i + n ± n ( 1 cos i ) B B B = = cos ib cos i B i Bewse = an 1 n n Zeo eflecion in pependicula polaizaion implies n' = n Physics 84 Elecomagneic Theoy II
µ = µ Bewse s Angle If efleced paallel polaizaion ampliude vanishes when inciden a he Bewse angle i Bewse = an 1 n n Refleced wave compleely plane-polaized (polaizaion pependicula o plane of incidence) if mixed-polaizaion beam inciden a Bewse angle. i Bewse o n = 56 fo = 1.5 n Physics 84 Elecomagneic Theoy II
Toal Inenal Reflecion Examine Snell s Law in case n > n' i = sin 1 n Fo angles of incidence geae, hee is no ansmied wave soluion o aach o, only an exponenially damped soluion. This implies oal eflecion, also called oal inenal eflecion. Opical communicaion sysems ae based on his phenomenon! n Physics 84 Elecomagneic Theoy II
Goup Velociy Unil now, we have assumed ha he elaive pemiiviy and pemeabiliy ae independen of fequency. This may be fa fom he case. Relaxing he equiemen of consan phase velociy as a funcion of fequency leads o moe geneal wave phenomena. Allow he fequency o depend on wavelengh in 1 dimension: 1 ikx ω k (, ) ( ) ( ) u x = A k e dk π The funcion ω(k) is known as he dispesion funcion. A sicly linea dispesion funcion, as we ve had up o now, does no lead o pulse speading, o dispesion. Physics 84 Elecomagneic Theoy II
ikx A k u x e dx ( ) (,) = dω ω ω ω ω ( k) = + ( k k ) + L = ( k ) (, ) dk e ( ω/ ) i k d dk ω i x ( dω / dk) k u x A k e dk π ( ω/ ) i k d dk ω = ( ) ( ( ω / ), ) e u x d dk The pulse shape avels a he goup velociy v g = dω dk Physics 84 Elecomagneic Theoy II
Dispesion 1 ikx ω ( k ) u( x, ) = A( k) e dk π Have exac calculaion fo modulaed Gaussian funcion ( ) ( ), = exp / u x x L e ikx A( k) = u( x, ) e dx ik x ( )( ) = π L exp L / k k a k ω = ν + ( k ) ν 1 Physics 84 Elecomagneic Theoy II
Pulse Speading, o Dispesion v g dω = = ν dk a k L ( L /)( k k ) ikx iv 1 + ( a k /) u( x, ) = e e dk π ( x ν a k ) exp ia ν L 1+ L ( = exp ik ) 1/ x iν 1 a k / + ia ν 1 + L Physics 84 Elecomagneic Theoy II
dω L L a L dk () ( = = + ν / ) d ω ν a v g = k = dk L ( ) ( ) ( ) x = x + v g Physics 84 Elecomagneic Theoy II
Causaliy D x E x (, ω ) = ε ( ω) (, ω) Convoluion Theoem (Fahlung Theoem) implies non-localiy in ime. 1 iω iω D( x, ) = D( x, ) e d = ( ) E( x, ) e d π 1 π ω ω ε ω ω ω ε ω e d e E x dω = iω + iω ( ) (, ) = ε E( x, ) + G( τ ) E( x, τ ) dτ Physics 84 Elecomagneic Theoy II
Geen funcion fo connecion 1 ( ) ( ) iωτ G τ = ε ω / ε 1 e dω π Damped oscillao connecion ( ) ( / 1= i ) 1 ε ω ε ω ω ω γω p / sin ν τ G τ = ω pe γ Θ τ ν ( ) ( ) ν ω γ = /4 Vanishes fo negaive τ,, cause canno pecede effec. Causal Geen s funcions mus be analyic in uppe ½ of complex plane. Physics 84 Elecomagneic Theoy II
Kames-Konig Relaions iωτ ε ( ω) / ε 1= G( τ ) e dτ Is auomaically causal fo a wide vaiey of choices fo G. Analyiciy in UH-ωP implies a elaionship beween eal and imaginay pa of he pemiiviy. Cauchy s heoem fo z inside a closed cuve C 1 ε ( ω ) / ε 1 ε ( z ) / ε = 1+ dω π i ω z 1 = 1 + πi C ( ) ε ω / ε 1 d ω ω z whee he inegal is now along he eal axis Physics 84 Elecomagneic Theoy II
1 1 = P + πδ i ( ω ω) ω ω iδ ω ω 1 Im ε ( ω ) / ε Re ε ( ω) / ε = 1+ P dω π ω z ( ) ε 1 Re ε ω / 1 Im ε ( ω) / ε = P dω π ω z ( ) ω Im ε ω / ε Re ε ( ω) / ε = 1+ P dω π ω ω ( ) ε ω Re ε ω / 1 Im ε ( ω) / ε = P dω * ( ) ( * = ) ε ω ε ω π ω ω Physics 84 Elecomagneic Theoy II
Sum Rules Sum Rules fo oscillao senghs Second Sum Rule ω p = 1+ P ω Im ε ( ω) / ε dω π 1 N N ( ) p d Re ε ω / ε ω = 1+ ω N Physics 84 Elecomagneic Theoy II