Chapr 6 Rflcion and Transmission of Wavs 6.1 Boundary Condiions A h boundary of wo diffrn mdium, lcromagnic fild hav o saisfy physical condiion, which is drmind by Maxwll s quaion. This is h boundary condiion o b applid o any lcromagnic fild. n ( H1 H) = JS (6.1.1) n ( 1 ) = (6.1.) n ( D1 D) = ρ S (6.1.3) n ( B B ) = (6.1.4) 1 From h Maxwll s quaion, w can find hs four boundary condiions. Hr n dnos h uni normal vcor o h boundary surfac. Physical manings of hs quaions can b dscribd as follows. Th angnial lcric fild is coninuous across h boundary surfac. Th disconinuiy in h angnial magnic fild H is qual o h surfac currn J S. Th normal componn B is coninuous across h boundary surfac. Th disconinuiy in h normal componn of D is qual o h surfac charg dnsiy ρ S. 6. Rflcion and Transmission a a Dilcric nrfac n scion 5., w dscrid a plan wav in wo-dimnsional spac. Now w considr a plan wav impings upon a plan dilcric inrfac, as shown in Figur. Whn h incidn wav impings on h boundary a an obliqu angl, h normal of h boundary and h incidn ray form a plan calld h plan of incidnc. Th fild of h incidn wav may b polarizd prpndicular or paralll o h plan of incidnc. incidn wav. Th incidn wav can b xprssd as: W now considr a prpndicularly polarizd = y x z i jk x (6..1) i jk x H xkz zkx x z = ( + ) ωµ 1 (6..) Th rflcd wav is givn by Chapr 6 Rflcion and Transmission 1
r = y R rx rz jk x+ (6..3) r R H xkrz zkrx rx + rz = ( + + ) ωµ 1 jk x (6..4) whr R is h rflcion cofficin for h wav. Th wav vcor of h rflcd wav is kr = xkrx zkrz (6..5) And h ransmid wav is giv by = y jkx x z T (6..6) T H xkz zkx x z = ( + ) ωµ jk x (6..7) whr T is h ransmission cofficin. Whn nihr of wo mdia is a prfc conducor, h surfac currn J S =. Thn, h boundary condiions (6.1.1) and (6.1.) rquir ha boh h angnial lcric fild and magnic fild componns b coninuous a z =. W hus hav jkxx jkxx jkxx + R = T (6..8) kz jk x krz jk x kz + R = T ωµ ωµ ωµ x rx jkxx 1 1 (6..9) hs quaions hav o b saisfid for all valus of x. Th consqunc is ha kx = krx = kx (6..1) mans ha h angnial componn of h hr wav vcors k, k r, and k ar qual. This condiion is known as h phas maching condiion. W can obain h magniud of h hr wav vcors by subsiuing h soluion for i, r and ino h wav quaion ( ωµε ) + R ST U VW = 1 1 r i (6..11) ( + ωµε ) = (6..1) W find k + k = ωµε = k (6..13) x z 1 1 1 Chapr 6 Rflcion and Transmission
k + k = ωµε = k (6..14) rx rz 1 1 1 and k + k = ωµε = k (6..15) x z From h phas maching condiion w find kx = krx = kx and kz = krz. Using hs rsul in (6..8) and (6..9) w obain 1+ R = T (6..16) µ 1kz 1 R = k T µ (6..17) z and solving hs quaions for R and T givs: R T µ kz µ 1k = µ k + µ k z z 1 z µ kz = µ k + µ k z 1 z Rfrring o h angl of incidnc θ and (5..), (6..18) (6..19) k = x k1 sinθ (6..a) k = k 1 sinθ (6..b) k rx x r = k sinθ (6..c) Subsiuing hs quaions ino h phas maching condiion, (6..1), w find k sinθ = k sinθ = k sinθ (6..1) 1 r 1 Th firs qual sign sas ha θ r = h dfiniion usd in opics, rfraciv indics θ, ha is h angl of rflcion is qual o h angl of incidnc. By using c n = c µε = k ω n 1 1 1 1 c = c µε = k ω h phas maching condiion k x = k givs ris o x (6..a) (6..b) n1sinθ = nsinθ (6..3) his is h Snll s law. Chapr 6 Rflcion and Transmission 3
Th phas maching condiion can b rprsnd graphically. Whn n1 < n, w can find vcors k r, and k for h givn k. Howvr, whn n > n, for h angl grar han θ c, k x is largr han h magniud of k. n ha cas, z x 1 k = k k < (6..4) or wih α = k x k k z =± jα (6..5) bing a posiiv ral numbr. n his cas, h wav anuas xponnially in h + z dircion. Th ransmid lcric fild can b givn by = y T αz jkx x (6.6.6) which rprsns a no uniform plan wav or a surfac wav. For xampl, h prmiiviy of h war a opical frquncy is 177. ε and h criical angl is givn by θ c = sin ( ) = 49. 177. 1 1 Chapr 6 Rflcion and Transmission 4
An incidn wav of arbirary polarizaion can b dcomposd ino wo wavs having prpndicular and paralll polarizaions. Th lcric fild of h prpndicularly polarizd wav is prpndicular o h plan of incidnc, and h paralll polarizd wav s lcric fild is paralll o ha plan. W now considr a paralll polarizd incidn wav. Th siuaion of lcric and magnic fild is shown in h figur and hy can b dscribd by i jkxx z H = yh (6..7) i H jk x xkz zkx x z = ( ) (6..8) ωε 1 r jkrx x+ jkrzz H = yr H (6..9) r R H jk x xkrz zkrx rx + rz = ( ) (6..3) ωε 1 jkx x z H = yth (6..31) TH xkz zkx x z = ( ) ωε jk x (6..3) Whr R and T ar, rspcivly, h rflcion and ransmission cofficins for h magnic fild vcor for h paralllly polarizd wav. Thy can b givn by applying h boundary condiion o (6..7) o (6..3) as: T R ε kz = ε k + ε k z 1 z ε k = ε k ε k + ε k z 1 z z 1 z (6..33) (6..34) From (6..33), whn µ 1 = µ, R = givs ω µ ε cosθ ω µ ε cosθ = (6..35) 1 b 1 1 Chapr 6 Rflcion and Transmission 5
and h phas maching condiion givs ω µ ε sinθ ω µ ε sinθ = (6..36) 1 1 b 1 Solving h abov wo quaions, w find θ + θ = and b π θ ε ε b = an 1 1 (6..37) whr h incidn angl θ b is calld Brwsr angl. Rflcion powr as a funcion of incidn angl. Th marial is glass wih ε = 5. ε. Th Brwsr angl is 56. A gas lasr wih Brwsr window Chapr 6 Rflcion and Transmission 6
6.3 Sanding Wavs Th complx prmiiviy is dfind as (4.6.4) and is givn by ε = ε σ j (6.3.1) ω Th prfc conducor is a mdium wih infini conduciviy. And from (6.3.1) w find ha h prfc conducor can b rgardd as a mdium wih infini prmiiviy. Subsiuing ε and k z ω µε w obain R = 1 (6.3.) R = 1 (6.3.3) Considr a prfcly conducing half-spac as shown in figur. A uniform plan wav impings normally on h boundary is givn by i = x (6.3.4) H i y F = H G K J η (6.3.5) And h rflcd wav is givn by: H r = x (6.3.6) r = F H G K J y η (6.3.7) Th oal lcromagnic fild in mdium 1 is h sum of h incidn and h rlcd wavs = x ( ) = x j sin kz (6.3.8) H y ( ) y coskz η η = + = F H G K J (6.3.9) Th boundary condiion (6.1.1) sas a z= givs h surfac currn on h prfc conducor. JS = ( z) H = x η (6.3.1) Chapr 6 Rflcion and Transmission 7
Th insananous valus of lcric ad magnic fild ar givn by: = x sin kzsinω (6.3.11) F H = y H G η KJ coskz cos ω (6.3.1) and hy ar plod in h figur. Ths pars ar calld sanding-wav pars bcaus h wavform dos no shif in spac as im procsss. 6.4 Sanding Wav in fron of a Dilcric Mdium Considr a uniform plan wav impings on a plan dilcric inrfac. Th oal fild in mdium 1 is givn as: 1 1 = y + R ( ) (6.4.1) 1 and h fild in mdium is = y T (6.4.) Th rflcion and h ransmission cofficins ar givn by (6..18) and (6..19). Assum ha mdium 1 is air and mdium is soil characrizd by ε = 1ε, σ = 1. / w hav: S m, and µ µ =. Whn h frquncy is 5MHz, k z k k 1 148 (6.4.3) k1z = k kx = k = 3365. j587. (6.4.4) R T = 537. =. 471 j173. 4 j587. z (6.4.5) (6.4.6) Subsiuing h R valu in (6.4.) w hav j 1 = 1+ 537. 1y + ( + 1734. ) k z (6.4.7) and similarly, =. 471 y 587. z (6.4.8) Chapr 6 Rflcion and Transmission 8