Electromagnetic. G. A. Krafft Jefferson Lab Jefferson Lab Professor of Physics Old Dominion University Physics 804 Electromagnetic Theory II

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1 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

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 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 v ω µε = µε ω = = 1 p g µε A da g A * ψψ da Physics 84 Elecomagneic Theoy II

18 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

19 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

20 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

21 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 =

22 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

23 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 R = µε R d ρ π z H z = AJ1 cosφ sin e R d iω Physics 84 Elecomagneic Theoy II

24 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

25 ( ) 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

26 Enegy Consevaion? Nomal Incidence E n = = E n n µε 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

27 Misakes fom Las Time Enegy Consevaion a Nomal Incidence ε E ε E ε E = + µ µ µ E n = = E n n µε µε µε 1 E µε n n = = E n n µε + µ ε + 1 Physics 84 Elecomagneic Theoy II

28 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 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

29 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

30 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

31 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

32 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

33 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

34 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

35 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

36 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

37 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

38 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

39 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

40 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

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