Waveguide basics. Ray picture & normalized quantities. n cl. n co. Numerical aperture of guide measured in air

Transcription

1 Gude odes Bass C 66 ueal Mehods Phoos Wavegude bass Ray pue & oaled quaes θ a θ l o θ π λ o l Ray vew of gudg a slab wavegude. The os eee ay s apped va oal eal efleo a he oe/laddg bouday. ueal apeue of gude easued a π l λ π λ o A sθ o sθ o o o l o l Phase shf of lg wave aoss gude fae V θ a A a s o l a Popagao osa, effeve de j β ( y, ) ψ (, y) e β, Robe R. MLeod, Uvesy of Coloado 47

2 Gude odes Slab wavegudes C 66 ueal Mehods Phoos Sgle-laye slab wavegudes Radao odes, deve va wave eq Subs T ode o 3D wave eq: j ω β,, Ψ yˆ e yeldg ψ ( ) ( ) ( ) β ( ) (, ) C ep Aep C ep Robe R. MLeod, Uvesy of Coloado ( ) β ( ) + ( ) (, ) d + ( ) β ( ) Ψ d f s osa eah laye, he feld laye us be Ψ j β + j β ( ) C e + C e Le be he fequey he laddg, ad he oe l o Tae he -vaao of he feld fo ay o be * ( + j ) + D ep( j ) ( j) ( j ) + D ep( + j ) oe ha ay value of s allowed. o l a < a < a > a l o l he WG feld, s alled β. 48

3 Gude odes Slab wavegudes C 66 ueal Mehods Phoos Sgle-laye slab wavegudes Radao ode, M BCs & fo Tageal opoes of felds ( y ad H ) us be ouous. Se y H j, ωµ hs s equvale o y ad s slope ae ouous. Wh ou hoe of feld, oly eed o osde +a bouday: C C ep ep ( ja) + D ep( + ja) Aep( ja) ( j a) + D ep( + j a) A ep( j a) Ths s a a fo C ad D es of a sgle aplude A, so a be easly solved a ay. Soe eaples: These loo eaably le plae-waves hoogeeous aeal eep fo he aso hough he oe. Robe R. MLeod, Uvesy of Coloado 49

4 Gude odes Slab wavegudes C 66 ueal Mehods Phoos Sgle-laye slab wavegudes Radao odes, epaso bass We fd ha hese odes ae ohogoal. * * [ ( ) ] ψ (, ) d ( ) ψ (, ) d ψ (, ) * ( ) ψ (, ) ψ (, ) ( ) δ ( ) ( ) (, ) ψ (, ) d ψ (, ) ψ ( ) δ ( ) * ψ, d oe ha he la of a osa value o he gh eques oalao of he odes by pg a paula A. We a use hs ohogoaly as a bass fo a de feld: ( ) ( ) ψ (, ) d dd, * ( ) ( ) ψ (, ) d ψ ( ) ( ) Assued fo of epaso. Mulply by Ψ*, ʃ ove d Re-aage ʃ o RHS Use ohogoaly elaoshp Whh gves us he epaso oeffes desed We ow have a fowad ad evese asfo pa. Robe R. MLeod, Uvesy of Coloado 5

5 Gude odes Slab wavegudes C 66 ueal Mehods Phoos Sgle-laye slab wavegudes Radao odes, wha s ssg How do hese odes evolve? Jus le hoogeeous spae, eah egefuo Ψ(, ) s assoaed wh oe : ( ) β ( ) l A β vs loos vey fala β Thus we ow how o popagae he felds., d * jβ ( ) (, ) (,) ψ (, ) d e ψ ( ) Oe poble: The odes we have foud ae ohogoal, bu o que oplee all ases. We e ssg soehg. Robe R. MLeod, Uvesy of Coloado 5

6 Gude odes Slab wavegudes C 66 ueal Mehods Phoos Sgle-laye slab wavegudes Guded odes, oduo As a fuo of we have foud he values show bold: o l The (possbly) ssg values ae hee. Cosde he oe/laddg efae fo whh s oseved. Thee ae a values of whh epese a wave popagag he oe bu evaese he laddg. l a o l o l > > l o Robe R. MLeod, Uvesy of Coloado 5

7 Gude odes Slab wavegudes C 66 ueal Mehods Phoos Sgle-laye slab wavegudes Guded odes, deve va wave eq Subsue a T -depede ode o 3D wave eq: ( ) j β (, ) (, ) Ψ( ) yˆ e yeldg d + ( ) β ( ) Ψ d a f s osa eah laye l o Ψ j β + j β ( ) C e + C e l A popagag ode us have β >, β < l o Fo he sye slab wavegude, we ay he we Ψ ( ) D ep As C ep ( + α ) + B os ( α ) < a < a > a whee β α j o, l β Robe R. MLeod, Uvesy of Coloado P. Yeh, Opal Waves Layeed Meda, Chape 53

8 Gude odes Slab wavegudes C 66 ueal Mehods Phoos Sgle-laye slab wavegudes Guded ode, M BCs Tageal opoes of felds ( y ad H ) us be ouous. Se y H j, ωµ hs s equvale o feld ad s slope ae ouous. Yelds fou odos fo fve uows (A,B,C,D,β), leavg pea aplude as a fee paaee. The odos ae: As Aos As a Aos a + Reaagg slghly a + B os B s a + B os B s a C ep a α C ep a D ep a α D ep ( α a) ( α a) ( α a) ( α a) As Aos B os Bs a a a a ( C D) ep( α a) α ( C D) ep( α a) ( C + D) ep( α a) ( C + D) ep( α a) α Robe R. MLeod, Uvesy of Coloado 54

9 Gude odes Slab wavegudes C 66 ueal Mehods Phoos Sgle-laye slab wavegudes Guded ode, T asedeal eq. Two lasses (sye ad a-sye) A, C D, a a α B, C D, o a α To solve hese asedeal equaos gaphally, le U a, W α a a a α U au W o a α U ou W Tasedeal egevalue equaos. Oly dsee soluos, ule adao odes whh allowed ay. U + W ( ) a V o l 6 Obsevaos 5 ew ode a V > π/,,.. # T odes (V/π/) + o uoff fo lowes ode Modes of eah ype aleae W v u U Robe R. MLeod, Uvesy of Coloado P. Yeh, Opal Waves Layeed Meda, Chape 55

10 Ray aalyss of wavegudes Fo of waves sep de 5 Guded odes C 66 ueal Mehods Phoos α o α 5 o α 5 o α l Robe MLeod, Uvesy of Coloado 56

11 Ray aalyss of wavegudes Fo of waves sep de C 66 ueal Mehods Phoos Radao odes Saples of ode ouu 5 5 l l l Robe R. MLeod, Uvesy of Coloado 57

12 Gude odes Slab wavegudes C 66 ueal Mehods Phoos Sgle-laye slab wavegudes Guded ode, TM asedeal eq. We follow he sae appoah bu ow fo H y Ψ H j β (, ) Ψ( ) yˆ e wh egeodes ( ) j D ep As C ep ωε H ( + α ) + B os ( α ) y < a Robe R. MLeod, Uvesy of Coloado 58 < a > a subje o he ouy of H y ad oe ha slope of H y s dsouous due o ε gevalue equaos ow slghly odfed A, B, C D, C D, ~ o α β l a a o a l o l Defe a ew fo of α ad use pevous aalyss: o l α α

13 C 66 ueal Mehods Phoos Robe R. MLeod, Uvesy of Coloado 59 Vaaoal ehque Devao ( ) ψ ε Subsue adally-sye ode o sala 3D wave eq: ( ) j e β ψ ψ ( ) [ ] ( ) ( ) ( ) d d d d ψ β ψ β yeldg ow ulply by ψ() ad e-aage he fs e ( ) ( ) ( ) ( ) ( ) ( ) + d d d d d d d d d d ψ ψ ψ ψ ψ ψ egae fo eo o fy ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) [ ] ( ) d d d d d d d d d d d d ψ β ψ ψ ψ ψ β ψ ψ ψ ( ) ( ) ( ) ( )d d d d ψ ψ ψ β Yeldg epesso fo β Gude odes Vaaoal ehod A.W. Syde, J.D. Love, Opal Wavegude Theoy, Seo 5 Whh loos a lo le weghed by he esy y +

14 Gude odes Vaaoal ehod C 66 ueal Mehods Phoos Gaussa odes of Gaussa gudes Gude pofle ε ( ) ( ) ( ) ( ) ( ) + e + A e lad oe lad lad Assued guded-ode shape ψ β ( ) ( ) ψ e Yeldg a aaly soluo A lad + + gude /e adus powe /e adus Depeds o ode adus whh s uow, so o eal useful ye. Typal plo of effeve de β/ vs he ode adus b Mode oo sall, dffao doag Soluo vald Mau β s fudaeal ode Mode oo lage, ovelappg wh laddg 4 µ δ..5 lad λ. µ V 4. Robe R. MLeod, Uvesy of Coloado 6

15 Gude odes Vaaoal ehod C 66 ueal Mehods Phoos Gaussa odes Aaly esul Mae β wh espe o A Shows ode se a V, should be V. Seod ode ous fo V > ~.4 V! D w A V V lad lad + A + ( ) oe V lad V Mode adus vs gude segh ffeve de ages fo laddg o oe. oe fo β ( ) ψ ( ) d ψ d ( ) oe LD lad lad H - la d D w V ffeve de vs gude segh Wea gudg So effeve de s ~ powe weghed aveage of de V D w Robe R. MLeod, Uvesy of Coloado 6 V Sog gudg oe lad

16 Foue bea popagao agay dsae ehod C 66 ueal Mehods Phoos agay dsae ehod fo fdg odes Le us epess he felds a he pu o ou wavegude as a su of he wavegudes odes (, y,) aψ (, y) + (, y ) Ψ(, y,, y ) dd y We ow how hs feld wll evolve as popagaes he gude: jβ jβ (, ) y (, y, ) aψ (, y) e + (, y ) Ψ(, y,, y ) e dd y Reebe ha he popagao osas ae sly odeed β > > β > Kβ odes β (, ) age ow ha we popagaed a dsae j ζ : β ( ) ζ β, y (, y, ζ ) aψ (, y) e + (, y ) Ψ(, y,, y ) e ζ dd y ah ode s ow aplfed as popagaes bu he lowes ode ode s aplfed he fases. Thus: ( y, ζ ) Ψ ( y),, The gowh ae of he ode should gve us he popagao osa: (, y, ζ + ζ ) ddy l (, y ζ )ddy β ζ l, Guded Howeve dd fd hs o be vey auae. Robe R. MLeod, Uvesy of Coloado 6 y Radao Yaauh, Chape

17 Foue bea popagao agay dsae ehod C 66 ueal Mehods Phoos Sye slab eaple.8 ζ ζ 8 (,ζ) R The ehod does a pey good job of ahg he ea ode shape: R e@d a 5[ µ ] λ [ µ ] o.5 δ. V odes 4 ag. Ds. a s also faly good. The vaaoal ehod s a b hgh (oe he hghe wgs of he ode). The BPM ehod s odued e. opaso: agay dsae.5945 Vaaoal.599 BPM phase ehod.5943 a.594 Robe R. MLeod, Uvesy of Coloado 63

18 Foue bea popagao agay dsae ehod C 66 ueal Mehods Phoos Valdao ad va BPM Lauh ode o egula BPM sulao. Aplude uhagg wh popagao:»» Z Z Re Plo eal pa of wh plae wave laddg dvded ou. Reag phase s dffeee bewee l ad. Use hs o alulae : j j l [ e e ] os ( ) π λ ( ) l L π [ ] λ + l L Re [ ( ) e ] j l, L 5 Robe R. MLeod, Uvesy of Coloado 64

19 Foue bea popagao agay dsae ehod C 66 ueal Mehods Phoos Sye slab eaple Asye () has o oe..5 ζ ζ 8 (, ζ) 6 R The ehod ow fds he lowes ode asye ode: R e@d.5 ag. Ds. a agay dsae ad vaaoal ae ow pey bad, bu he BPM phase s o oo fa off: opaso: agay dsae.5945 Vaaoal.5967 BPM phase ehod.5787 a.5767 Robe R. MLeod, Uvesy of Coloado 65

20 Foue bea popagao agay dsae ehod C 66 ueal Mehods Phoos Sye slab eaple Hghe ode odes Sa wh odal epaso of abay feld: Use ohogoaly o fd oe of hs feld (D fo sply): (, y,) aψ (, y) + (, y ) Ψ(, y,, y ) * * (,) Ψ ( ) d a Ψ ( ) + ( ) Ψ(, ) d Ψ ( ) dd y a * (,) Ψ ( ) d a Ψ (, y) * (,) Ψ ( ) Ψ (, y) d d Ψ Ψ d ( ) (,) ( ) Ψ ( ) ad eove hs so ha ew de feld has o oe: (, ) (, ) a Ψ ( ) d.5.5 ag. Ds. a R e@d ζ ζ R e@d Robe R. MLeod, Uvesy of Coloado opaso: BPM phase ehod.55 a.54867

21 Gude odes Modal paaees fo BPM () C 66 ueal Mehods Phoos BPM -spae ode fde FFT BPM sulao paaees. -ell ABC used. d a d λ / d lad 3 / 4 µ λ. µ lad.5 δ. A.56 V Modes (,) Off-as Gaussa eao, ω 5 d 8 seps 496 seps Robe R. MLeod, Uvesy of Coloado 67 Kelv Wage.

22 Gude odes Modal paaees fo BPM () C 66 ueal Mehods Phoos FFT of (,) (, ) Radao odes dffag o a le of adus l Boud odes wh o dffao ad l < < o Robe R. MLeod, Uvesy of Coloado 68

23 Gude odes Modal paaees fo BPM () C 66 ueal Mehods Phoos Poje fo he boud odes 6 oe ha we see fou odes wh effeve de l < < o. The paula off-as sulao has sogly eed he lowes wo odes ad wealy eed hd ad foh. Tae seod dffeee of pojeed o loae peas ( ), d Robe R. MLeod, Uvesy of Coloado 69

24 Gude odes Modal paaees fo BPM () C 66 ueal Mehods Phoos Reove he ode shapes a eah defed sle of (, ) ad vese FT Ψ ( ) F { (, β )} [AU] [AU] [µ] [µ] [AU] a egeodes FFT Bea pop ehod [µ] Robe R. MLeod, Uvesy of Coloado [µ] 7 [AU] 3.54

25 Mullaye slab wavegudes Devao of odes C 66 ueal Mehods Phoos Mullaye slab wavegudes Seup Le hee be + oal layes of de l, l.. sepaaed by boudaes l, l..- 3 The felds eah laye ae a fowad ad evese avelg plae wave wh aplude A l ad B l. These plae waves have a wave veo wh opoe l l β whee β s he opoe of he wave veo. The egeode feld aplude s hus Ψ ( ) Ae Al e A + e j j l ( ) + j ( ) + B e ( ) + j ( ) j l + B e l l ( ) ( ) + j( + ) ( ) + B l + s o y fo T ad H o y fo TM < l + e < < < l, l We ow eed o apply he wo bouday odos of ouy of asvese ad H a eah of he + boudaes, yeldg + equaos fo + A s, + B s ad β o a oal of + 5 uows. Two oe equaos ae he owledge of he feld a ±, yeldg as befoe a se of equaos fo β ad all bu oe aplude, leavg he aplude of he wave as a fee paaee. Robe R. MLeod, Uvesy of Coloado 7

26 Mullaye slab wavegudes Devao of odes C 66 ueal Mehods Phoos Devao of he asfe a () Cosde a paula laye bewee p ad q whee qp+ (o splfy oao). A p we a we he wo M bouday odos as epessos fo he waves avelg away fo he bouday es of he waves avelg owads he bouday ad he Fesel efleo ad assso oeffes: A B q p pq pq A A p p + + qp qp B q B whee he Fesel oeffes ae pq p p + q p p q q q + q q q p p fo s o T waves fo p o TM waves q p Aq A q B q Ap Bq Bp q p fo efleos of waves p off of q ad ba o edu p pq p fo s o T waves p + q pq p fo p o TM waves pq + q p fo asssos of waves p o edu q Robe R. MLeod, Uvesy of Coloado 7

27 Mullaye slab wavegudes Devao of odes C 66 ueal Mehods Phoos Devao of he asfe a () lae qp ad qp by og ha he Fesel oeffes have he followg elaoshps: q pq qp, pq A B q p pq pq A A p p + pq p q B p q pq B q qp yeldg a a equao ha oly depeds o (,) pq he fs equao a be e-aaged o epess A p es of ped vaables. To solve fo B p es of oly ped vaables, suba he op equao ulpled by pq fo he boo equao ulpled by pq : pq B p pq A q + pq + q p pq B q pq A + whh we a ow we as a a equao q ( R + T ) B A + B pq pq pq q A B p p A q B q A q B q pq hs asfes felds aoss Tp pq pq bouday p The fal pee we eed s o elae he felds o oppose sdes of slab q A q B q e j q d q e j q d q A B q q P q A B q q hs popagaes felds aoss laye q bewee p ad q whee d q q p s he laye q hess. Robe R. MLeod, Uvesy of Coloado 73 P. Yeh, Opal Waves Layeed Meda, Chape

28 C 66 ueal Mehods Phoos Robe R. MLeod, Uvesy of Coloado 74 Syse a M ad popees of popagao Cobg he pevous esuls, we a elae he felds he wo sefe eeal layes ad elae all eedae vaables: B A B A B A M T P P T T K Popees of M: * *, M M M M f + Reals,, + Reals ( ) + M Feld efleae ad asae: M B B M M B A M A A M M A B A A B B M Powe efleae ad asae: ( ) ( ) ( ) ( ) T M M T R M M M M R M Mullaye slab wavegudes Devao of odes

29 Mullaye slab wavegudes Devao of odes C 66 ueal Mehods Phoos B M M Guded odes of ullaye slabs Fo a guded ode, β/ > ad + so ha he felds hese eeo egos ae evaese ad deay o eo a fy. l l β wll beoe agay. Thee ae wo oos (+ ad -). We us sply hoose he sae oo o boh sdes. Fo he egave hoe, he felds ae ow A + Ψ ( ) + Be Al e A + e j M M β l ( ) ( ) + j ( ) l β Applyg ou syse a + B e l ( ) + < l l < l < < l, l whh ples he dspeso elao (asedeal eq fo β) s M To eosu he felds, assue A + ad wo bawads Al Bl T P T l l KP T l+ By solvg he egevalue equao fo he dsee se of boud odes ad usg he a equao o fd he apludes eah laye, he odes of a paula wavegude ae opleely spefed. Robe R. MLeod, Uvesy of Coloado 75

30 Mullaye slab wavegudes Devao of odes C 66 ueal Mehods Phoos Two gudes sepaaed by 6 µ dvdual gudes pe BPM -spae odefde eaple Ψ [AU] Ψ [AU] [µ] [µ] Ψ [AU] Ψ [AU] [µ] [µ] Ψ [AU] Ψ [AU] [µ] [µ] Ψ [AU] Ψ [AU] [µ] [µ] Robe R. MLeod, Uvesy of Coloado 76

31 Popagao peod eda Devao of Bloh waves C 66 ueal Mehods Phoos Popagao peod eda A sog hoogeey Assue a se-fe edu ade of wo hoogeeous soop layes of de ad ad hess a ad b. Gve a feld dsbuo (,), wha s he feld a soe deph o he aeal? ( ) a A - B - Λ a + b b, (, )? C D A B We ow he felds eah slab us obey (, ) whee Ae Ce ( Λ) + j ( Λ) + B e j jβ e j + l l β ( Λ+ a) + j ( Λ a) + D e ( ) Λ a < < Λ Λ < < Λ a Usg he asfe ad popagao aes jus deved, we ay we A B oe ha M A PT P B T A M B Robe R. MLeod, Uvesy of Coloado 77

32 Popagao peod eda Devao of Bloh waves C 66 ueal Mehods Phoos Floque heoy aa Bloh s heoy ( ulple ds) The Floque heoe saes ha he egefuos of a fe peod edu us have a evelope of he sae peody as he edu. Le all of ou egefuos, hey also have a phase fao of popagao, K. β s ow he ouous gve vaable. A B j β K (, ) Ψ( ) e e j e j K Λ A B whee ( + Λ) Ψ( ) Due o hs assued fo of he feld, he plae-wave apludes us be elaed by by obg he Floque assupo wh he a popagao we ge a egevalue poble A M B e j K Λ A B The egevalue of a a a be we Λ e j K ( M + M ) ± j [ ( M + M )] os( K Λ) + j s( KΛ) o K os [ ( M + M )] Λ The oespodg egeveo yelds he feld apludes ad hus Ψ( ) A B e jkλ M M Robe R. MLeod, Uvesy of Coloado 78 Ψ Dspeso elao fo Bloh waves C D A T P B

33 Popagao peod eda aples C 66 ueal Mehods Phoos voluo of he K sufae as eases a. µ,.5 b. µ,.5 L Λ K K soop wh ( + )/ T TM λ µ a a a + b b. µ,.8 a. µ,.5 + b b. µ,. a. µ,.5 + b.45λ.5λ.57λ Λ L K Λ L K / λ µ - - / - - / λ µ Robe R. MLeod, Uvesy of Coloado 79

34 Popagao peod eda aples L Λ K K Λ L K C 66 ueal Mehods Phoos Relaoshp of K sufae o soop wh ( + )/ T TM bad dagas K sufae - - / λ µ λ µ Bad daga Λ K π / λ µ Λ L K.5.5 / - - Cuoff, o / Robe R. MLeod, Uvesy of Coloado 8

35 Popagao peod eda aples C 66 ueal Mehods Phoos Cople bad daga Tasvese spaal fequey oaled o Re[K Λ] [K Λ] b. µ,. a. µ,.5 S a + b.57 µ Λ a + b.3µ S / Λ.9 Re[K Λ] [K Λ] 4 h 3 d TR d Sop bads TR s Robe R. MLeod, Uvesy of Coloado 8

36 Popagao peod eda aples C 66 ueal Mehods Phoos Tadoal bad daga Tasvese spaal fequey o oaled o b. µ,. a. µ,.5 Re[K Λ] [K Λ] S a + b.57 µ Λ a + b.3µ S / Λ.9 Re[K Λ] [K Λ] 4 h 3 d Lgh le d Sop bads TR s TR Robe R. MLeod, Uvesy of Coloado 8

37 Popagao peod eda aples C 66 ueal Mehods Phoos Why ae bads be? Re[K Λ] [K Λ] 4 h 3 d TR d Sop bads TR s φ a [ ] a + b + b Opal phase aoss sgle peod as fuo of 4 h 3 d d s So bads ou whe oud-p phase oal () deo s π Robe R. MLeod, Uvesy of Coloado 83

38 Popagao peod eda aples C 66 ueal Mehods Phoos Polaao T TM [K Λ] Re[K Λ] Bewse s agle [ aa( )] s[ aa( )] +. sθ s Robe R. MLeod, Uvesy of Coloado 84

39 Popagao peod eda aples C 66 ueal Mehods Phoos Bloh waves 3.5 λ µ 3.5 soop wh ( + )/ T TM L Λ K / [V/] a.3 µ,.5 b.4 µ,. 3 [µ] Robe R. MLeod, Uvesy of Coloado 85

40 Popagao peod eda aples C 66 ueal Mehods Phoos Bloh waves λ µ 3.5 soop wh ( + )/ T TM L Λ K / / Ψ [AU] a.3 µ,.5 b.4 µ,. 3 [µ] Robe R. MLeod, Uvesy of Coloado 86

41 Popagao peod eda aples C 66 ueal Mehods Phoos Bloh waves λ µ 3.5 soop wh ( + )/ T TM L Λ K / / Ψ [AU] a.3 µ,.5 b.4 µ,. 3 [µ] Robe R. MLeod, Uvesy of Coloado 87

42 Popagao peod eda aples C 66 ueal Mehods Phoos Bloh waves. 3.5 λ µ 3.5 soop wh ( + )/ T TM L Λ K / / Ψ [AU] a.3 µ,.5 b.4 µ,. 3 [µ] Robe R. MLeod, Uvesy of Coloado 88

43 Popagao peod eda The bg pue C 66 ueal Mehods Phoos Popagao peod eda Coeo wh Foue popagao We a bea ay ele feld dsbuo o he bouday up o a se of se waves va he Foue asfo. ah of hese ahes oo ehe a T o a TM Bloh wave depedg o s polaao ad asvese spaal fequey, β. Se we ow he Bloh waves, we a popagae eah of hese asvese susods fowad o he edu, he add he all ba up wh a vese Foue asfo. { } ( ) [ ( )] ( ) j K ( β ), F F, Ψ, K e Real spae Bloh spae β ( ), K Robe R. MLeod, Uvesy of Coloado 89

44 geode epaso oduo C 66 ueal Mehods Phoos geode epaso ehod oduo Good Bass se s oe physally eagful ha plae waves heely opaaal whe usg opaaal odes Moe aual bdeoal veso ha BPM, geeal Whe oe seo hages, oly ha a hages. Maes opao fase. Syse feaue se dvoed fo sulao feaue se. Bad Lage asvese oss-seo (vey ulode) suues effe Algoh oe ople ha BPM eeds oe ae ad feedg D. F.G. Gallaghe, T. P. Fel Phoos Wes, Sa Jose, 3. Pape Robe R. MLeod, Uvesy of Coloado 9

45 geode epaso oduo C 66 ueal Mehods Phoos Types of odes a ope suue ( eff ) Re( eff ) Popagag adao odes wh ouous eal eff < M( lad ). Susodal a ±. vaese adao odes wh agay eff. Susodal a ±. Boud odes wh eal eff. Ma( oe ) > eff > Ma( lad ). poeally deayg a ±. Leay odes wh ople eff. Re( eff ) < M( lad ). poeally gowg a ±. Robe R. MLeod, Uvesy of Coloado P. Besa, Ph.D. Dsseao, U. Ge, 9

46 geode epaso oduo C 66 ueal Mehods Phoos Types of odes a losed suue ( eff ) Re( eff ) The egals ove he ouous adao odes ae ueally ovee, so we wll eae he asvese spae soe bouday (PC, PMC, PML ). Ths eoves he leay odes ad us all ohe odes o boud odes of he supesuue. f he bouday s popely appled (e.g. dsa) he odes wll be vey lose o he sae shape ad popagao osa as he ope suue. The dsee se s sll ohogoal ad oplee fo fuos ha ee he bouday odos of he supesuue. Robe R. MLeod, Uvesy of Coloado 9

47 geode epaso Devao C 66 ueal Mehods Phoos Loe epoy heoe () Sa wh he Mawell ul equaos fo wo egeodes of he sae suue Fo H () (4) + () H (3) : H H H + J J S ( H H) ( jωµ H) ( J + jωε ) J + jωε ) H ( jωµ H ) + jωµ H H J + jωε jωµ H H J + jωε H ( ) ( ) ( 3) ( 4) Robe R. MLeod, Uvesy of Coloado 93 H egae ove boh sdes ad use Gauss heoe o LHS ( H H ) ds ( J J )dv V Le S S + S + S 3. The LHS equals eo ove S 3 ehe beause odes ae guded (felds deay o eo) o he bouday odos. S S 3 Veo dey S LH sdes RH sdes Cael es

48 geode epaso Devao C 66 ueal Mehods Phoos Loe epoy heoe () Fo oveee defe F S F ds S S S F ds F ds + H H F ds ds [ F ( + ) F ( ) ] S S F ds. The LHS s hus S S The RHS s depede of fo a sall dsae V ( J ) dv ( J J ) J S Dvde by ad le appoah eo o ge ou esul S ( H H ) ds ( J J ) ˆ S ds ds Robe R. MLeod, Uvesy of Coloado 94

49 geode epaso Devao C 66 ueal Mehods Phoos Ohogoaly of odes Le hee be wo egeodes of a suue, ad. These ould be dsee (boud) o ouous (adao) odes. H H jβ (, y, ) (, y) e jβ (, y, ) H (, y) e jβ (, y, ) (, y) e jβ (, y, ) H (, y) e a soue-fee (lossless) ego, J ad hus j ds ( β + β ) ( H H ) ˆ S Oly he ageal (,y) opoes of he felds a obue: ( β + β ) ( H H ) ˆ ds S Robe R. MLeod, Uvesy of Coloado 95, y, y, y, y Fo ode, hee ess a evese avelg wave wh egave β ad, ( β β ) ( H + H ) ˆ ds S, y, y, y, y Assug β β we a add he wo equaos o ge S ( H ) ˆ ds, y, y Fo T ad TM odes of + desoal syses (o es. lossless) * ds δ T H Whh s ou ohogoaly odo. H * ε ds δ TM

50 geode epaso Devao C 66 ueal Mehods Phoos Saeg vs. Tasfe a Pevously, we used he saeg a, S, whh desbes he feld apludes avelg away fo a deve (o efae) es of he aplude of he waves avelg owads he deve. A B A B A S B A M B,,,, A B We used hs o deve he asfe a (syboled hee by M) whh elaes feld apludes o oe sde of he deve o feld apludes o he ohe sde. geeal, hey ae elaed by M S S M S M S M S S M S M S M us ou ha he S a s uh oe ueally sable (as well as sple fo). Ufouaely, does asade easly. So we wll alulae S fo eah laye, aslae o M s so we a asade, he aslae BACK o S fo he whole syse fo ueal sably. Robe R. MLeod, Uvesy of Coloado 96 M A B Rego whee ad ae geealed efleo ad assso oeffes A B A B Rego

51 geode epaso Bdeoal soluo C 66 ueal Mehods Phoos Bdeoal soluo Poble seup Ths devao s a eeso of he pevous ul-laye suue devao. The a dffeees ae ) hee ae ow possble fowad ad evese avelg odes ad ) he odes of oe ego ae o auoaally ohogoal o he odes of he e laye, equg soe pojeos o alulae he assso ad efleo oeffes. oao:, y(, y) s he asvese ( ad y) veo opoe of he ele feld of h egeode of ego.,y de feld ode of ego,, Robe R. MLeod, Uvesy of Coloado 97, (, y), T ( ),, y, y ( y) y R,, y Refleo oeffes of de ode o efleed ode fo he odes of suue. H, y, y Tassso oeffes of de ode suue o ased ode fo he odes of suue. Couy of ageal ad H o he bouday he gves:,, (, y) + R (, y) T (, y),, (, y) R H (, y) T H (, y),,, y, y,,, y, y Sg hage se hs s bawads avelg wave. Mode pofles ego Mus be ue a all asvese oods (,y) Fo odes, hs gves 4 equaos vaables. Bad.

52 geode epaso Bdeoal soluo C 66 ueal Mehods Phoos Bdeoal soluo Devao of R ad T Le s dop he y subsp fo hee ad jus eebe ha all ode pofles ae jus he ageal opoes. ow, oss ulply he fs equao by H ad he seod by fo a abay ode ego ad egae ove all asvese spae o podue a se of lea equaos fo he R s ad T s: o, H + R, H T, H H o, H R,,,,, H Robe R. MLeod, Uvesy of Coloado 98 T,,,, whee <,H > s he e podu elao, H, y, y S, H ( H ) ds ˆ f whh a be used o ollapse he su ove s. The equaos ae ow fo oe de ode ad oe efleed ode,, o, H + R,, H T,, H de. efleed,, H o, H R, H T, H ased,, Fo a paula de ode, hee ae uows (R, ad T, fo, ) ad equaos. Good. Ths poedue a he be epeaed fo evey de ode o yeld R, ad T,. The h olu of hese aes epesses how he aplude of he h ode ego efles o he.. odes of ego ad ass o he.. odes of ego. The poedue a be epeaed fo he ohe sde o also fd R, ad T,. ow jus lea equaos (o y depedee).

53 geode epaso Bdeoal soluo C 66 ueal Mehods Phoos We a fo Solvg fo R ad T Fo a gve de ode, osu he lea syse: δ δ, H, H, H, H R T,,,,, H, H o H o Fo eah de ode, hs fds oe olu of R, ad T,. The epea wh evesed wh : δ δ, H, H, H, H R T,,,, f odes have bee oaled o uy e podu, hese splfy o: δ δ δ δ, H, H, H, H R T R T,,,,,,,, δ δ δ δ o H o o H o, H, H o H o oe ha blo aes seod equao ae jus asposed vesos of hose he fs hs splfes he alulao. Robe R. MLeod, Uvesy of Coloado 99

54 geode epaso Bdeoal soluo C 66 ueal Mehods Phoos Bdeoal soluo Pug all ogehe We a ow we a blo S a ha elaes felds egos ad, epeseed as a se of ode apludes:,, A, A T R A S,, B B R T B whee he feld eah ego ae we jβ + jβ, y, A e + B e A B [ ] (, y) ( ) ( ) ( ) jβ ( ) ( ) + jβ (, y, A e + B e ) S, [ ] (, y), S S, S S, S S aes ae dfful o asade o a full syse. Oe way o do hs s o aslae o he T aes, whh ae easy o asade:,, A B Robe R. MLeod, Uvesy of Coloado 3 S, T, A B Ths ges he apludes A,B aoss a bouday. We a geeale he popagao a P o ove hough a ouous suue A B dag [ ] jβ e dag A [ ] jβ e B Copae hs o he ullaye equao o page 6. Copae o page 6. We a ow asade hese wo aes o desbe a ee syse. f we wa o fd he boud odes of hs syse, we a aslae ba o a S a epeseao, whh s oe ueally sable. A B A B A B A B Ths s he vese asfo Copae o page 6.

55 geode epaso Udeoal soluo C 66 ueal Mehods Phoos Splfao o udeoal Wea logudal vaao U aplude de feld ode of ego,y ( y), y, Assued ~ eo Mode pofles ego T,,,, y ( y) Tassso oeffes of ode suue o ode fo he odes of suue. Fowad popagao s desbed by a fs-ode D so oly eed oe bouday odo. Cosevao of asvese :,, y, ( y) T ( y),,, y Le us fuhe assue we ae wealy gudg a + (,y,) desoal syse o puely T o TM a + (,) desoal syse. hs ase he ohogoaly elao a be we as a podu ove sala ele felds. By ag he e podu wh he ego odes (o he efleed odes of ego, as befoe), we fd he T a dely:, τ, T,, τ T,, τ, τ δ τ T, τ,, τ τ, τ Modal apludes ae foud by pojeo of old bass oo ew bass, τ f odes ae oaled. Robe R. MLeod, Uvesy of Coloado 3

56 geode epaso Udeoal soluo C 66 ueal Mehods Phoos Splfao o udeoal Pug all ogehe Fs we fd he boud ad adao odes of he syse. Teae hs sees odes ad we he felds eah seo as L L L L jβ ( ) ( ) ( ), y, A, y e L,, The de feld apludes ego ae foud va he odal epaso (, ), A, Advae hese apludes o he, bouday wh he popagao osas β A A e jβ Poje he odal apludes ego o ego A Repea. p T,, p A p Robe R. MLeod, Uvesy of Coloado 3 D. F.G. Gallaghe, T. P. Fel Phoos Wes, Sa Jose, 3. Pape 4987-

57 geode epaso aple udeoal poble C 66 ueal Mehods Phoos aple poble (/3) Couplg of λ. µ lgh hough a sye slab wavegude wh a.667 µ, o.6, l.5, o a seod sye gude wh a 5 µ, o.55, l.5 Modes of Modes of Robe R. MLeod, Uvesy of Coloado Jaque Lou, MP hoe wo, 4 33

58 geode epaso aple udeoal poble C 66 ueal Mehods Phoos aple poble (/3) Lauh o ego Off as Gaussa de feld pojeed oo he odes of. oe he vese asfo (dos) s o a pefe epeseao of he Gaussa (le) se we dd o lude he adao odes whh aes he speu oplee. Popagao dow oe of gude usg he β s of he boud odes. Robe R. MLeod, Uvesy of Coloado 34

59 geode epaso aple udeoal poble C 66 ueal Mehods Phoos aple poble (3/3) Couplg o ego The feld a he ed of ego ad he epeseao as a su of he odes of ego. Aga, he epeseao s pefe beause we have o used he oplee odal se ludg adao odes. Coued popagao dow oe of ego Robe R. MLeod, Uvesy of Coloado 35