ECE 634 Sping 06 Pof. David R. Jacson ECE Dept. Notes 5
TM x Sface-Wave Soltion Poblem nde consideation: x h ε, µ z A TM x sface wave is popagating in the z diection (no y vaiation).
TM x Sface-Wave Soltion TEN: TM Z0 x h R TM Z00 The efeence plane is pt at the top of the sbstate. Z = Z = TM x TM x0 0 00 ωε ωε 0 = ( ) = ( ) = / / x z x0 0 z z 0 TM TM in = 0 tan( x ) in = 00 Z jz h Z Z 3
TM x Sface-Wave Soltion (cont.) TRE: Z in = Z in so jz tan( h) = Z TM TM 0 x 00 Hence j tan( h) = ωε x x0 x ωε0 o ε x = j x0 tan( x h) x x Note: tan( h) is always eal ( egadless of whethe is eal o imaginay) x Note: Assming a eal z, a soltion will only exist if x0 is imaginay. > z 0 4
TM x Sface-Wave Soltion (cont.) Let = jα, α = x0 x0 x0 z 0 Then we have ε x = α x0 tan( x h) o ε / ( z ) = tan ( ) z 0 ( / ) z h Note: This mst be solved nmeically. 5
Popeties of Sface-Wave Soltion Assmptions: A lossless stcte A pope sface-wave soltion (the fields decay at x = ) 6
Popeties of SW Soltion (cont.) Popety ) z is eal Othewise, consevative of enegy is violated: x P ot = P (No adiation can escape since the mode decays at infinity.) in P in P ot z L z Assme Giding stcte = β jα P ot = P e α in L α = 0 7
Popeties of SW Soltion (cont.) Popety ) z 0 Othewise ε = imaginay (no soltion possible) ε x = α x0 tan( x h) α = x0 z 0 Recall: x tan( x h) is always eal 8
Popeties of SW Soltion (cont.) Popety 3) z < Othewise, = ( ) = jα = j / x z x z In this case, x jα x ε = tan( xh) = tan ( jαx) h αx0 α x0 α x = j ( j) tanh( α xh) α x0 α x = α x0 tanh( α xh) =negative eal nmbe 9
Popeties of SW Soltion (cont.) Hence, we have fo a lossless laye: z = eal < < 0 z Genealization to an abitay nmbe of layes: max 0
TE x Soltion fo Slab TM Z0 x h R TM Z00 Z Z TE 00 TE 0 ωµ 0 = x0 ωµ = x TRE: Z in = Z in o ωµ j tan( h) µ ωµ = 0 x x x0 x0 = j x tan( x h)
TE x Soltion fo Slab (cont.) Using = jα α = x0 x0 x0 z 0 we have o o µ µ jα x0 = j x α x0 = x tan( x h) tan( x h) µ = tan ( ) ( ) z 0 / z ( / ) z h
Gaphical Soltion fo SW Modes Conside TM x : α ε = tan( h) x0 x x o α h= ( h )tan( h ) x0 x x ε Let v α h x x0 h Then v= tan ε 3
Gaphical Soltion (cont.) We can develop anothe eqation by elating and v: = h z v = h z 0 Hence = h ( ) z v = h ( ) z 0 add 4
Gaphical Soltion (cont.) + v = h ( ) 0 = ( h) ( n ) 0 Define: R ( h) n 0 Note: R is popotional to feqency. Then + v = R 5
Gaphical Soltion (cont.) v R TM 0 π / π 3π / v= tan ε + v = R R ( h) ( n ) 0 6
Gaphical Soltion (cont.) v Gaph fo a Highe Feqency TM 0 TM R π / π 3π / Impope SW (v < 0) = x h v= α h x0 7
Pope vs. Impope Recall: v = α If v > 0 : pope SW (fields decease in x diection) If v < 0 : impope SW (fields incease in x diection) x0 h Ctoff feqency fo an open stcte: the tansition feqency between a pope and impope mode z = at ctoff 0 Note: This definition is diffeent fom that fo a closed wavegide stcte (whee z = 0 at the ctoff feqency). Ctoff feqency: TM mode: v= 0, = π 8
TM x Ctoff Feqency v TM : R = π h n 0 = π R π h λ = / n 0 Fo othe TM n modes: TM : n n = h λ = n/ 0 0,,,... n 9
Fthe Popeties of SW Soltions (obtained fom the gaphical soltion) Popety ) = at f z 0 c Poof: v= α h= h x0 z 0 At f = f : v= 0 = c z 0 0
Popeties of SW Soltions (cont.) Popety ) at f z Poof: π + nπ a constant ( ) Hence = h= h constant x z so h z constant Theefoe, z
TM 0 Mode The TM 0 mode has two special popeties: TM 0 #) No ct-off ( f 0) c Poof: see gaphical soltion
TM 0 Mode (cont.) TM 0 #) Poof: Hence z f 0 as 0 v= tan ε ε h h ( ) z 0 z ε h h n z z 0 ( 0 ) 0 ε 0 z z ( h 0 ) n 0 ε 0 0 3
Dispesion Plot z / 0 n.0 TM 0 TM TM f c f 4
v TE x Modes TE TE µ α x0 = x tan( x h) o R / µ π / π 3π / α h = h h ( ) cot ( ) x0 x x µ v= cot µ 5
TE x Modes (cont.) No TE 0 mode ( f c = 0) TE ct-off feqency at (R = π / ): In geneal, ( h) n 0 h λ = 0 π = /4 n TE n : h ( n ) λ = /4 n 0 n =,,3,... 6
v TE x Modes (cont.) v= cot µ R = h z / µ π / π 3π / v = h z 0 At this feqency, = 0. Fo lowe feqencies, becomes imaginay ( z > ). If we wish to tac the TE ISW fo lowe feqencies, we need to efomlate the gaphical soltion. 7
R / µ v TE x Modes (cont.) Low-feqency TE soltion z > Thee is always a soltion (intesection point). Let = jh = j z z v= cot µ Note: The ed cve stats ot above the ble cve, bt ends p below the ble cve. v= µ v= coth µ + v = R v= R + ( v < 0) 8
R / µ v TE x Modes (cont.) Low-feqency TE soltion z > v= coth µ Feqency loweed v= R + v= µ As the feqency is loweed, the point of intesection moves fthe ot, maing the mode incease moe apidly in the ai egion. 9