Module II: Part A. Optical Fibers

Transcription

1 Module II: Pat A Optical Fibes

2 Optical Fibes as Tasissio Mediu Mai Liitatio: Atteuatio Although fibes have bee kow sice the 8 s as ediu fo light tasissio, thei pactical use becae evidet whe losses whee educed though ipoved aufactuig e.g., iceasig puity of glass ad desig e.g., by itoducig a claddig laye. Low-loss tasissio widows of silica fibes i the wavelegth egios ea.8,.3 µ ad.55 µ. st geeatio OC-3 d geeatio, OC- 3d & 4 th geeatios, OC-48 &96

3 Optical Fibes as Tasissio Mediu Othe Liitatios: Dispesio---Leads to pulse boadeig ad liits the the axiu bit-ate B possible ove a tasissio distace L. We ofte talk about the badwidth-distace poduct BL. Dispesio causes BL to decease. We will discuss the details of dispesio late o. 3 Optical Nolieaities: These effects ca also cause futhe boadeig ad atteuatio. Howeve, this effect ca be exploited to aplify light iside the fibe, as doe i fibe aplifies! Nolieaities ca also be used fo oliea odulatio, such as phase odulatio i coheet optical couicatio.

4 Basic Fibe Types Covetioal Fibes: Step-Idex left ad Gaded- Idex ight Coss sectios of fibes. [Fo Agawal] I geeal, havig a claddig laye with a lowe efactive idex helps cofie the light light iside the fibe the coe laye.

5 Advaced Fibe Types Holey Fibes: ffective Idex-guidig Photoic Badgap Guidig Holey fibes ay be oe optio as a poject topic fo those who ae iteested i fibes!

6 Step-Idex Fibes Jacket Claddig Coe Woks o the basic piciple of siple Total Iteal Reflectio TIR TIR igoously valid oly if fo a>>λ Valid fo ultiode fibes Coss sectio ad efactive-idex pofile fo step-idex fibes

7 Step-Idex Fibes Geoetical-optics Desciptio Light cofieet though total iteal eflectio i step-idex fibes The idea is that optical ays that ete the fibe at a sufficietly sall agle ae totally eflected off the claddig-coe bouday. Reflectio at the ai-fibe iteface: TIR coditio: We wat < π/ o siθi siθ si < This is called Sell s law Reflectio at the coe-claddig: iteface: si si Citical case occus whe π/ Thus, c si

8 Step-Idex Fibes Geoetical-optics Desciptio NA Nueical apetue NA: / siθi cos c [we used the fact that π/-θ ] Fo /, NA ca be appoxiated by: NA, I coclusio, as log as a ay eteig the fibe coe satisfies the coditio siθ i < NA/, the ay will exhibit total iteal eflectio ad will be totally cofied withi the fibe as it popagates though it. O oe had, we would wat NA to be lage so as to allow oe light to be cofied; but o the othe had, the geate the ube of suvivig ays also called odes i electoagetics laguage, the highe is the so-called odal dispesio.

9 Modal dispesio: Step-Idex Fibes Geoetical-optics Desciptio The idea is that ays tavelig at diffeet agles have diffeet speeds alog the diectio of the fibe diectio of light popagatio, o -diectio. This is because the -diectio copoet of the speed depeds o the agle of the ay. Let us coside the axiu delay betwee the fastest ay eteig at θ i ad the slowest ay eteig at θ i NA/ :

10 Step-Idex Fibes odal-dispesio-liited tasissio ate Naely, let s coside the tie delay betwee the shotest ad logest ay paths: Coespodig the tie delay is give by: T c L si C L L c c is the speed of light i fee space: c3.x 8 /s Maxiu-delay citeio: To avoid ove-seaig of pulses, we equie T<T B, whee T B is the duatio of the pulse. By defiig B/ T B as the bit-tasissio ate, we obtai: Fudaetal liitatio of stepidex fibes: BL < c

11 Gaded-Idex Fibes Fo Reduced Modal Dispesios Idea: Foce the ays that ae away fo the fibe axis tavel at highe speeds i ode to ake up fo the educed speed caused by thei agled diectio. Solutio: Make the efactive idex of the coe decease as ays vee away fo the cete, as show i the gaded idex pofile below Jacket Claddig Coe Gaded efactive idex: Ray tajectoies i a gaded-idex fibe

12 Gaded-Idex Fibes Typical fo of a gaded efactive idex pofile: [ / ] α a ; ; < a a The tajectoy of a paaxial ay is deteied by: d d d d Geeal solutio fo < a ad α : ' cos p / psi p whee, p / a / ad ae the positio ad the diectio of the iput ay, espectively. Fo a deivatio of the tajectoy equatio see Appedix. We will ow look at soe cosequeces of gaded idexig. is the distace alog the fibe axis is the adial distace fo the -axis

13 Modal Dispesio i Gaded-idex Fibes Vaiatio of odal dispesio with the pofile paaete α fo a gaded-idex fibe [fo Agawal]. The iiu dispesio occus whe: α The iiu dispesio delay diffeetial i tie is: T / L / 8c The liitig bit ate-distace poduct: BL < 8c /

14 Fibe Modes We ow tu to udestadig which odes ae cadidates to be tasitted. To do so, we eed to udetake a oe sophisticated teatet tha geoetic optics. We eed to use electoagetic, o wave, optics. We stat by statig Maxwell s quatios: / / B D t D H t B whee ' ', ',, dt t t t t P χ ε H B P D µ ε

15 Take Fouie tasfos i t, e.g., Fibe Modes Now the fequecy-depedet dielectic costat ε is give by: ε iαc/ω whee α is the absoptio coefficiet of the fibe. Also, ε is elated to the Fouie tasfo of Χ, Χ ~ by ~, ω, t exp iω t dt ~ ε Χ Thus, afte soe substitutios we obtai the wave popagatio equatio: with ~ ω k Re ~ χ / ad k ~ ω / c π / λ

16 Fibe Modes Cotiued Wave ca ow cast the popagatio equatio i cylidical coodiates: k ad obtai thee sepaated diffeetial equatios: Fo step-idex fibes: > a a We ca solve the popagatio equatio usig the ethod of sepaatio of vaiables. That is, put Z F Φ / / Φ Φ F k d df d F d d d Z d Z d

17 Fibe Modes Cotiued Solutios to the diffeetial equatios ae: > Φ a ; ' a ; ' exp exp q I C q CK p A Y p AJ F i i Z k q k p Usig bouday coditios: > a ; exp exp a ; exp exp i i q CK i i p AJ > a ; exp exp a ; exp exp i i q DK i i p BJ H The othe copoets i the coe egio ae: whee, ω µ ω µ H p i H p i ω ε ω ε H p i H H p i H J, K, ad I, ae diffeet types of Bessel fuctios [see Ref. 3 i the text fo details]

18 Fibe Modes Applyig the cotiuity of H ad at the coe-claddig bouday, we get the eigevalue equatio: ' ' ' ' q p q p a pa qk pa K pa pj pa J pa qk pa K pa pj pa J Fo give set of paaetes a, k,, ad, the eigevalue equatio ca be solved yieldig the popagatio costat. We ay obtai ultiple solutios fo each. We deote these solutios, fo each, as,.,. ach coespods to a ode. ach ode is uiquely deteied ad idetified by its popagatio costat.

19 Fibe Modes Fo each, we wite the coespodig solutio fo H o as H o H, depedig o whethe H o doiates the othe. I the special case whe, the odes ae called T ad TM odes, coespodig to tasvese electic i.e., ad tasvese agetic H odes, espectively.

20 Fibe Modes Noalied popagatio costat b as a fuctio of oalied fequecy V fo low-ode fibe odes Noalied fequecy: Defie V k / a π / λ a Noalied popagatio costat: / k b V deteies how ay odes a fibe ca suppot: see figue o the left. Fo sigle ode opeatio, oly H called the fudaetal ode should be suppoted.

21 Sigle Mode Fibes Noalied popagatio costat b as a fuctio of oalied fequecy V fo low-ode fibe odes V Fo sigle ode opeatio, oly H called the fudaetal ode ca be suppoted. Thus, sigle ode iplies that both T ad TM ust ot be suppoted. Thus, T ad TM ust both be below thei cutoff V s. See the hoiotal itecept fo T ad TM ad coclude that <.45 Must hold fo sigle ode opeatio

22 Appedix o Gaded-Idex Fibes Deivatio of the tajectoy equatio We stat suppessig tie i the field expessios: set a exp-ik S; ad H h exp-ik S, whee S is a eal fuctio of the positio vecto. By substitutig these ito Maxwell s equatio ad assuig k ifiity, we obtai the so-called ikoal equatio i ectagula coodiates: S x, y, o, S / Now, geoetical light ays ae defied as the othogoal tajectoies to the geoetical wave fot, defied by S costat. Let s deote the positio vecto of a poit P, say, o a light ay, whee s is the ac-legth alog the ay, the d s Sˆ, whee Sˆ ds is a uit vecto i the diectio of S.

23 Appedix o Gaded-Idex Fibes Deivatio of the tajectoy equatio But, S ˆ S / S S / Hece, we obtai the equatio of the ay d s S. ay equatio ds Ray tacig: Take d/ds of both sides of the ay equatio ad use the fact that d / ds d / ds to obtai d d d S. ds ds ds Sice d ds S /, we obtai afte soe algeba: d ds d. ds Let s focus o the -copoet of the above equatio, whee ad the gadiet becoes a deivative with espect to. Thus we have d ds d ds.

24 Appedix o Gaded-Idex Fibes Deivatio of the tajectoy equatio Hece, d ds d ds costat b ad b d d Let s ow focus o the x- copoet x of equatio : d ds dx ds. x 3 b Usig i 3 yields o d d b b dx d d x d x x x x Fo adially syetic fibes, x ca be thought as, ad the ay-tacig equatio follows.

25 Next topics Now we kow what coditio guaatees sigle ode opeatio, ad this esues us that thee will be o odal dispesio. Howeve, thee ae othe types of dispesio such as goupvelocity dispesio ad ateial dispesio that we will study i the ext copoet of Module II. Please ead the text, pages 3-37.