Step-index silica fiber

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1 Modl lysis of step-idex fibes Itoductio C 46/566 Guided Wve Optics Step-idex silic fibe Mteil d fbictio Types d mig of modes Deivtio d solutio of the W Solutio of the W T/TM modes Hybid modes LP modes Robet R. McLeod, ivesity of Colodo 65

2 Modl lysis of step-idex fibes Mteils d fbictio C 46/566 Guided Wve Optics Opticl popeties SiO & SiO /GeO Loss [db/m] g D@ p sêh m m LD Absoptio: Lge t smll wvelegth due to Ryleigh sctteig off of ihomogeeities i the glss Lge t log wvelegth due to molecul vibtiol esoces (bsoptio) fthe out i the IR. Is embly low loss oud.5 mis Dispesio: Pue d doped silic hve ely ideticl dispesio Zeo dispesio oud.3 mis Robet R. McLeod, ivesity of Colodo 66

3 Modl lysis of step-idex fibes Mteils d fbictio C 46/566 Guided Wve Optics Fbictio of the pefom Robet R. McLeod, ivesity of Colodo 67

4 Modl lysis of step-idex fibes Mteils d fbictio C 46/566 Guided Wve Optics Dwig fibe - m/sec 4 o F Sevel m of fibe e typicl o sigle eel. Robet R. McLeod, ivesity of Colodo 68

5 Modl lysis of step-idex fibes Types of modes C 46/566 Guided Wve Optics Types of modes xct solutio Meidol ys (): T & TM Sew ys ( ): H & H Wely guidig ppoximtio LP modes C be expessed s sum of T, TM, H, H tht beme degeete fo smll δ Robet R. McLeod, ivesity of Colodo 69

6 C 46/566 Guided Wve Optics Robet R. McLeod, ivesity of Colodo 7 Wve equtio i cylidicl fo d H Moochomtic vecto W y x y x ˆ ˆ ˆ Ctesi vecto Lplci ε ε ε Scl simplifictio ˆ,, ˆ,, ˆ,,,, φ φ φ φ φ φ Wite i cylidicl odites: dil imuthl dil does ot uple to d φ fields: Modl lysis of step-idex fibes Deivtio of the wve equtio ε φ ε Scl W fo i cylidicl od. Note tht H obeys the sme W..

7 Modl lysis of step-idex fibes Solutio of the wve equtio C 46/566 Guided Wve Optics Solutio of the W Septio of vibles ( φ, ) R Φ( φ) Z( ), Septio of vibles Plug ito wve equtio. Note ow odiy diffeetil eq. d R ΦZ d dr Φ d d Z RZ RΦ ε RΦZ d dφ d Multiply by /RΦZ d R dr d Φ d Z ε R d d Φ dφ Z d Assume siusoidl depedece i : Z() exp[- ] ( ε ) d R dr Φ d R d d Φ dφ Depeds o Depeds o φ Depeds o R d Φ ( ε ) d R dr d d Φ dφ Robet R. McLeod, ivesity of Colodo 7

8 Modl lysis of step-idex fibes Solutio of the wve equtio C 46/566 Guided Wve Optics dφ Solutio of the W Rdil d imuthl fuctios Solutio of imuthl equtio. Sice must be peiodic, itege. d Φ Φ Φ ± ± Solutio of dil equtio. d R d dr d φ ( φ) A e ε R Note simility to d /d ( -(/) ). Solutios: [ f f ] ( ) f π ( ) ( ) ( ) π 4 π e lge lge s π eo Zeos of : Coe: > N Robet R. McLeod, ivesity of Colodo Coe: cl < N [ f f ] ( ) f

9 C 46/566 Guided Wve Optics Robet R. McLeod, ivesity of Colodo 73 Fom of fields > < > < e e D e e B H e e C e e A φ φ φ φ φ φ,,,, Foms of modes: cl cl N N whee, s befoe, Modl lysis of step-idex fibes Solutio of the wve equtio Though Mxwell s cul equtios, we c fid ll the tsvese fields i tems of these two logitudil fields: φ ε ωε φ φ φ ωµ φ φ ε ωε ωµ i i i i i H H i H H H H ie o clddig We thus hve 5 uows A,B,C,D d d fou tiuity ditios o, φ, H, d H φ levig oe fee vible which is the totl mode mplitude.

10 Modl lysis of step-idex fibes Chcteistic equtio C 46/566 Guided Wve Optics Chcteistic equtio d eltios betwee mplitudes Steps:. Fid the fields φ, d H φ fom the foms of d H usig the expessios o the lst pge.. Set them equl ss the boudy to geete fou equtios tht e lie i the fou uows A,B,C,D d tscedetl i. 3. Requie tht the fou equtios be stisfied by settig the detemit of the 4 4 mtix equtio equl to eo. 4. The esultig equtio ivolves oly d is the chcteistic equtio fo the modes. (See sectio 4.5 fo the detils) ( ) ( ) ( ) ( ) ( ) ( ) cl ( ) Fo ech,, thee will be solutios with m,, dil eos. This is logous to the slb WG but with ew mode umbe. sig the sme B.C.s oe c elte the field mplitudes: Cotiuity of : Cotiuity of H : Cotiuity of φ : Cotiuity of H φ : A B Robet R. McLeod, ivesity of Colodo 74 ( ) C ( ) ( ) D ( ) A ωµ ( ) ( ) ω cl A ( ) ( ) B B The lst two e edudt, which is to be expected sice oe vible should emied udetemied. We eed both foms, s we ll see.

11 C 46/566 Guided Wve Optics Robet R. McLeod, ivesity of Colodo 75 T d TM modes Modl lysis of step-idex fibes T d TM modes cl Fo meidiol ys tht pss though the xis,. The chcteistic equtio bemes: If this tem, the A by the tiuity of φ : B A ωµ A C d by the tiuity of : so fo ll spce d thus these e T modes. The chcteistic equtio fo the T modes is thus: Yieldig the eigevlues T m A logous gumet fo the sed tem yields BD so H fo ll spce d thus TM modes. The chcteistic equtio fo is: TM m cl Lie the slb, this loos lie T but with tem lie the tio of the idices squed.

12 Modl lysis of step-idex fibes T d TM modes C 46/566 Guided Wve Optics T mode φ, H d H If A d, (see q. 4. i boo) d φ vies lie. lectic field. Colo plot, ows x d y Note tht (T). Mgetic field. Colo plot H, ows H x d H y Note tht H (ot TM). Note tht powe is flowig out of the scee. Robet R. McLeod, ivesity of Colodo 76

13 Modl lysis of step-idex fibes T d TM modes C 46/566 Guided Wve Optics TM mode, d H φ If B d, H (see q. 4. i boo) d H φ vies lie. lectic field. Colo plot, ows x d y Note tht (ot T). Mgetic field. Colo plot H, ows H x d H y Note tht H (TM) Note tht powe is flowig out of the scee. I the we guidig limit cl these two modes will be ~ degeete. Robet R. McLeod, ivesity of Colodo 77

14 Modl lysis of step-idex fibes T d TM modes C 46/566 Guided Wve Optics Gphicl solutio of T/TM chcteistic equtio γ P ( ) ( ) W ( W ) ( W ) T/TM chcteistic equtio with omlitios d politio fcto defied fo slb γ p fo TM, fo T W V cl N N λ [ µ m] V mode 5[ µ m].5 δ.5.8 λ [ µ m] mode 5[ µ m] V.5 δ. 7.7 Light lie TM diffeece exggeted fo clity LHS, T LHS, TM RHS Robet R. McLeod, ivesity of Colodo 78 V Cutoff

15 Modl lysis of step-idex fibes T d TM modes C 46/566 Guided Wve Optics T/TM modes Obsevtios lie the slb, thee is defiite cutoff V below which T d TM modes do ot popgte. The fist two (oe T d oe TM) modes e llowed whe V the fist eo of.45 The sed two (oe T d oe TM) modes e llowed whe V the sed eo of 5.5 TM modes should hve gete th T, θ thus N TM < N T. x si N T dispesio digm fo fibe with sme popeties of slb wveguide exmple, pevious. N cl Not llowed Boud Rditio ω / c Robet R. McLeod, ivesity of Colodo 79

16 C 46/566 Guided Wve Optics Robet R. McLeod, ivesity of Colodo 8 Hybid modes Ntue of fields W W W W W W W W W N cl cl Modl lysis of step-idex fibes Hybid modes If, we must use the mplete chcteistic equtio. Nomliig: If, the eithe of the tems i bcets is eo. Fom the boudy ditios, B A ωµ cl B A ω If eithe A o B is eo, the both must be eo, d ll fields e eo eveywhee. Thus, the modes hve both d H d e eithe T o TM. If B < A (H < ), the modes e lbeled H. If A < B ( < H ), the modes e lbeled H.

17 Modl lysis of step-idex fibes Hybid modes C 46/566 Guided Wve Optics Gphicl solutio Sigle d multimode (left d ight) λ [ µ m] 5[ µ m].5 δ.5 V [ µ m] λ [ µ m].5 δ. V 7.7 m m m m m,3 m4,5 4 6 m m,3 4 6 m m,3 4 6 m m Robet R. McLeod, ivesity of Colodo 8

18 Modl lysis of step-idex fibes Hybid modes C 46/566 Guided Wve Optics Numbe of modes H mode does t cut off. Fist T mode cuts i t V fist ull of Fo we guidig, ~ cl d RHS d ew mode ppes with evey eo of. Tig the lge gumet ppoximtio fo π ( ) ( ) π s lge π 4 Note simility to the slb wveguide. Fo ech, the umbe of dil modes will thus be ppoximtely # of modes m fo ech SINGL MOD CONDITION: V <.45 π π π 4 At cut off ~ V, so the cut off ditio fo ech mode, m is V Totl umbe of eigevlues Times two oiettios d politios π m V π V π Which is the sque of the slb cse. Robet R. McLeod, ivesity of Colodo 8 V π V π m

19 Modl lysis of step-idex fibes Hybid modes C 46/566 Guided Wve Optics H mode Politio lectic field. Colo plot, ows x d y Note tht. Note tht ely lie polied. Mgetic field. Colo plot H, ows H x d H y Note tht H. Note tht powe is flowig out of the scee. Robet R. McLeod, ivesity of Colodo 83

20 Modl lysis of step-idex fibes Hybid modes C 46/566 Guided Wve Optics H mode Politio lectic field. Colo plot, ows x d y Note tht. Note tht ely lie polied. Mgetic field. Colo plot H, ows H x d H y Note tht H. Note tht powe is flowig out of the scee. Robet R. McLeod, ivesity of Colodo 84

21 Modl lysis of step-idex fibes Hybid modes C 46/566 Guided Wve Optics Hybid mode shpes lectic fields, oly oe politio d oiettio m m 3 4 Robet R. McLeod, ivesity of Colodo 85

22 Modl lysis of step-idex fibes LP modes C 46/566 Guided Wve Optics LP modes Deivtio of the chcteistic equtio Stt with the mplete chcteistic equtio ( ) ( ) ( W ) ( W ) ( ) ( ) ( W ) ( W ) N cl W W cl W Ad let be ely cl. N must thus lso be ely cl d the equtio simplifies to W sig Bessel fuctio idetities lie ( ) ( ) ( ) ( ) W W ± ± ± ( W ) ( W ) ( W ) ( W ) ( ) ( ) ( ) The LHS c be ccelled, leve eltio much lie tht of the T/TM: Robet R. McLeod, ivesity of Colodo 86 H - H The boo simplifies this oe moe step, but I c t get thei fomul, so I ll stop hee. We c see fom this expessio tht yields ou T modes. Sice we ve ppoximted cl, TM m must be degeete with T m. These modes e lso degeete with H m. Fo, the uppe d lowe sigs yield diffeet equtios, ssocited with the modes s show. H, m is degeete with H -, m. We my thus liely mbie modes withi y degeete set to cete moe veiet set of modes, still with the sme effective idex.

23 Modl lysis of step-idex fibes LP modes C 46/566 Guided Wve Optics LP modes Field distibutios T H ch degeete set c be liely mbied to cete ew set with pedomitely lie politois. The xil politios ( d H ) e much smlle th the tsvese. This is sistet with the wely guidig ppoximtio the gle of popgtio must be vey e the xis (NA vey smll). Fo modes with o gul vitio, thee e two othogol politios. Fo modes with gul vitio (e.g. si( φ)), thee is lso sed gul phse (e.g. s( φ)), which is fuctiolly othogol. Robet R. McLeod, ivesity of Colodo 87

possible gul m phses d ll hve two possible politios m

24 Modl lysis of step-idex fibes LP modes C 46/566 Guided Wve Optics LP modes of pbolic fibe ch mode > hs two possible gul m phses d ll hve two possible politios m 9 Robet R. McLeod, ivesity of Colodo Michelso, Chpte 5 88

25 Modl lysis of step-idex fibes Summy C 46/566 Guided Wve Optics Summy The solutio of the wve-equtio i cylidicl odites yields i the e (lie s) d i the clddig (lie e -x ). Mtchig these fuctios ss the boudy with the M boudy ditios yields mplex chcteistic equtio fo which hs diffeet solutios m fo ech gul mode umbe. If thee is o gul vitio (), the solutios e T m d TM m. Othewise the solutios ivolve ll 6 field mpoets d e lbeled H m d HM m. The H mode is ~liely polied d eve cuts off. The umbe of modes i tems of V is oughly equl to the umbe i slb wveguide of the sme pmetes, squed. T/TM modes occu fo V >.45. Iceses i the decese the mximum m by ~/. I the limit of we guidig, sets of these modes hve degeete effective idices d c thus be liely mbied. The esultig LP modes e liely polied, hve smll xil fields d hve two degeete politios d (fo modes with gul vitio) two gul phses. Robet R. McLeod, ivesity of Colodo 89

PROGRESSION AND SERIES

INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of