Dielectric Wave Guide

Transcript

1 Electrmagnetc Fels Delectrc Wave Gue A electrc wavegue s a structure whch eplts ttal reflectn at electrc nterfaces t gue electrmagnetc raatn. The smplest case s the symmetrc electrc slab wave gue clang y cre 1 clang Fr guance ne must have > 1 Amangawa, 006 Dgtal Maestr Seres 10

2 Electrmagnetc Fels Smlarly t the parallel plate wavegue, we assume prpagatn alng the rectn an unfrm cntns alng the yrectn. Gue waves are launche wth angle f ncence larger than the crtcal angle, s that ttal (nternal) reflectn takes place. Waves launche at smaller angles suffer partal refractn nt the clang an eventually the pwer n the cre regn wll sappear fr suffcently lng wave gues. θ > θ c θ < θ c Amangawa, 006 Dgtal Maestr Seres 11

3 Electrmagnetc Fels We cnser agan TE an TM mes. Only certan angles f ncence are allwe, but here the reflectn ceffcent fr ttal reflectn s a cmple quantty, ntrucng a phase shft n the reflecte fel, whch epens n the angle f ncence. In the case f metal plates, nstea, there s always a phase shft f 180 fr the tangental electrc fel. In rer fr the angle t be accepte, the wave nees t establsh a selfcnsstent cnstructve nterference pattern fr any pnt nse the cre, as ncate n the fgure belw A θ θ Amangawa, 006 Dgtal Maestr Seres 1

4 Electrmagnetc Fels Cnser a pnt A n the cre f the wave gue an a wave frnt mvng frm t reachng pnt B. The phase shft fr the phase planes mvng frm A t B s π ϕ1 = β1 ABcsθ = AB csθ λ / B r1 A θ θ AB csθ θ C λ s the wavelength n vacuum at the gven frequency f peratn. Amangawa, 006 Dgtal Maestr Seres 13

5 Electrmagnetc Fels The wave frnt reflecte at pnt B eperences a phase jump equal t the phase f the cmple reflectn ceffcent. Assumng a TE wave, r perpencular plaratn, ϕ = Γ ( E) = B = csθ j sn θ csθ j sn θ csθ j sn θ ( ) θ + j θ 1/ 1 csθ j sn θ + + = cs sn / / / / 1 = tan 1 sn θ 1/ csθ Amangawa, 006 Dgtal Maestr Seres 14

6 Electrmagnetc Fels Then, the reflecte wave eperences a phase shft when mvng frm B t C π ϕ3 = β1 BCcsθ = BCcsθ λ / r1 B BC csθ A θ θ θ C Amangawa, 006 Dgtal Maestr Seres 15

7 Electrmagnetc Fels The wave frnt reflecte at pnt C eperences agan a phase jump equal t the phase f the cmple reflectn ceffcent. Fr a symmetrc wavegue ϕ = ϕ 1 1csθ + j sn θ 1 ϕ 4 = Γ ( E) C = 1 1csθ j sn θ 1 = 4 tan sn θ 1 1 csθ Amangawa, 006 Dgtal Maestr Seres 16

8 Electrmagnetc Fels The reflecte wave eperences a phase shft mvng frm C back t A π ϕ5 = β1 CAcsθ = CAcsθ λ / r1 B CA csθ A θ θ C Amangawa, 006 Dgtal Maestr Seres 17

9 Electrmagnetc Fels Fr cnstructve nterference (selfcnsstency), the sum f all the phase shft cmpnents must be equal t a multple f π π ( AB + BC + CA) csθ + ϕ + ϕ4 = mπ, λ/ r1 m = 0, 1, wth ( AB BC CA) + + = λ π / r1 csθ mπ = tan sn θ 1 1 csθ m = 0, 1, Amangawa, 006 Dgtal Maestr Seres 18

10 Electrmagnetc Fels Takng the tangent f all terms we btan the characterstc equatn fr the TE mes. π tan cs =, m = 0, 1, λ sn θ r1 mπ θ 1 csθ In terms f even an slutns, we can rewrte tan π = r1 csθ = λ = f ( csθ ) sn θ Even mes 1 g(csθ ) m = 0,, csθ O mes csθ 1 m = 1, 3, g(csθ ) sn θ 1 Amangawa, 006 Dgtal Maestr Seres 19

11 Electrmagnetc Fels The characterstc equatn fr TM mes s btane by usng the reflectn ceffcent fr parallel plaratn n the ervatn π tan cs =, m = 0, 1, λ θ r, n terms f even an slutns tan sn θ r1 mπ θ 1 ( / 1) cs π sn θ Even mes 1 = g(csθ ) m = 0,, r1 ( / 1) csθ O mes ( / ) cs 1 m = 1, 3, c g(csθ ) csθ = λ 1 θ = f ( sθ ) sn θ 1 Amangawa, 006 Dgtal Maestr Seres 0

12 Electrmagnetc Fels tan ( θ ) ( f ) cs Eample f graphcal slutn f the characterstc equatn fr the mes n a symmetrc slab electrc wave gue m = 0 θ = g π ( θ ) cs m = m = 1 m = 3 m = 4 θ m = 5 m = = θ g c 6 1 cs csθ ( θ ) Amangawa, 006 Dgtal Maestr Seres 1

13 Electrmagnetc Fels The cut-ff frequences fr the mes are btane by bservng that at cut-ff the angle f ncence s mnmum (crtcal angle). At the crtcal angle, the characterstc equatn s TE ) TM ) sn θ c π tan r1 mπ cs 1 θc = = 0 λc csθc sn θ c π tan r1 mπ csθ 1 c = = 0 λc ( / 1 ) csθ c fr bth TE an TM mes snce 1 1 π r1 mπ csθ c = λ c θ = sn c Amangawa, 006 Dgtal Maestr Seres

14 Electrmagnetc Fels The cut-ff wavelengths (reference t free space as usual n ptcal wave gues) an the crrespnng cut-ff frequences fr the gue mes are λ θ θ m m r1 1 r = = r1 r m m r1 1 cs r c = c = 1 sn c r1 f c mc = = λ c r1 r, m = 0, 1, The funamental mes are the TE 0 an the TM 0 wth er cut-ff frequency. TE an TM mes wth the same ne frm egenerate pars wth entcal cut-ff frequences. Amangawa, 006 Dgtal Maestr Seres 3

15 Electrmagnetc Fels Fel epressns Even TE mes α ( /) cs 1 / E e e cs( 1 ) Ey = E β e α ( ) ( /) cs 1 / + E β e e 0 1 O TE mes α ( /) sn 1 / E e e sn ( 1 ) Ey = E β e α ( ) ( /) sn 1 / + E β e e 0 1 Amangawa, 006 Dgtal Maestr Seres 4

16 Electrmagnetc Fels Even TM mes α ( /) cs 1 / H e e cs ( 1 ) Hy = H β e α ( ) ( /) cs 1 / + H β e e 0 1 O TM mes α ( /) sn 1 / H e e sn ( 1 ) Hy = H β e α ( ) ( /) sn 1 / + H β e e jβ jβ 0 1 Amangawa, 006 Dgtal Maestr Seres 5

17 Electrmagnetc Fels In meum 1 ( β ) 1 = β1 + β = ω µ r1 In meum ( β ) = α + β = ω µ r We have β = β snθ 1 β = β csθ 1 1 Fr each me the angle f ncence s btane frm the slutn f the characterstc equatn. Amangawa, 006 Dgtal Maestr Seres 6

18 Electrmagnetc Fels Eamples f prfles fr the transverse electrc fel f TE mes. TM mes have smlar prfles fr the magnetc fel. f just abve cut-ff f >> f c attenuatn TE 0 attenuatn attenuatn TE 1 attenuatn Amangawa, 006 Dgtal Maestr Seres 7

19 Electrmagnetc Fels f just abve cut-ff f >> f c attenuatn TE attenuatn attenuatn TE 3 attenuatn Amangawa, 006 Dgtal Maestr Seres 8

20 Electrmagnetc Fels Electrc Fel f TE me n electrc wave gue splls ver nt the clang where t ecays epnentally Electrc Fel f TE mes n parallel plate wave gue ges t er at the metal bunares Amangawa, 006 Dgtal Maestr Seres 9

21 Electrmagnetc Fels Magnetc fel cmpnents fr TE mes are btane frm Faraay s law E =jω µ H ˆ ˆ ˆ Ey = jωµ H y et E E = jωµ H y = 0 y E = 0 Ey E = 0 Ey =jωµ H Amangawa, 006 Dgtal Maestr Seres 30

22 Electrmagnetc Fels Fr eample, the transverse magnetc fel cmpnent s prprtnal t the (transverse) electrc fel. In the gue cre: E = jωµ H y ( ) Even) Ecs β1 e =jω µ H ( ) O) Esn β1 e βe ( ) cs β1 e (Even) β ωµ H = Ey = ωµ βe ( ) sn β1 e (O) ω µ Amangawa, 006 Dgtal Maestr Seres 31

23 Electrmagnetc Fels Electrc fel cmpnents fr TM mes are btane frm Ampere s law H = jω E ˆ ˆ ˆ H E y y = jω et H H = 0 = jω E y H =0 H y H =0 H y= jωe y Net, s a summary f all the fel cmpnents parallel t the plane f ncence. Amangawa, 006 Dgtal Maestr Seres 3

24 Electrmagnetc Fels Even TE mes β ωµ 1 ( β ω µ ) cs ( β ) 1 1 ( /) E cs / e e H = E e β ωµ ( + /) E cs / e e α α 0 1 jα ωµ 1 ( β ω µ ) sn ( β ) ( /) E cs / e e H = j E e jα ωµ ( + /) E cs / e e α α 0 1 Amangawa, 006 Dgtal Maestr Seres 33

25 Electrmagnetc Fels O TE mes β ωµ 1 ( β ω µ ) sn ( β ) 1 1 ( /) E sn / e e H = E e β ωµ ( + /) E sn / e e α α 0 1 jα ωµ 1 ( β ω µ ) cs ( β ) ( /) E sn / e e H = j E e jα ωµ ( + /) E sn / e e α α 0 1 Amangawa, 006 Dgtal Maestr Seres 34

26 Electrmagnetc Fels Even TM mes E β ω 1 ( β ω ) H cs ( β ) 1 1 β ω 1 ( /) H cs / e e ( + /) H cs / e e α = α e 0 1 E jα ω 1 ( β ω ) sn ( β ) ( /) H cs / e e = j H e jα ω ( + /) H cs / e e α α 0 1 Amangawa, 006 Dgtal Maestr Seres 35

27 Electrmagnetc Fels O TM mes β ω 1 ( β ω ) sn ( β ) ( /) H sn / e e E = H e β ω ( + /) H sn / e e α α 0 1 jα ω 1 ( β ω ) cs ( β ) ( /) H sn / e e E = j H e jα ω ( + /) H sn / e e α α 0 1 Amangawa, 006 Dgtal Maestr Seres 36

28 Cnser a wave enterng the en f the wave gue frm ar. r Electrmagnetc Fels θ r1 θ t θ r A classcal prblem f ptcal wavegues s t etermne the mamum angle f entrance θ that satsfes the cntn f ttal nternal reflectn (guance). Amangawa, 006 Dgtal Maestr Seres 37

29 Electrmagnetc Fels r θ = θ c θ m The crtcal angle at the cre/clang r1 r nterface s reache when the angle n ar s the mamum angle permttng guance. θ > θ m r r1 r θ < θ c When the angle n ar ecees the mamum value necessary fr guance, transmssn (leakage) nt the clang takes place. Amangawa, 006 Dgtal Maestr Seres 38

30 Electrmagnetc Fels At the arcre nterface snθ t = 1 sn θ rar snθ = r1 r1 θ + θ = 90 csθ = snθ t t θ t θ At the crtcal angle snθ = snθ = c r r1 sn = 1 cs = 1 sn = 1 = θ r tm θtm θc r1 snθ = = m r1 r 1 m = sn r1 r θ numercal aperture sn θ r1 m Amangawa, 006 Dgtal Maestr Seres 39