Module II, Part C. More Insight into Fiber Dispersion

Transcription

1 Moule II Par C More Insigh ino Fiber Dispersion

2 . Polariaion Moe Dispersion Fiber Birefringence: Imperfec cylinrical symmery leas o wha is known as birefringence. Recall he HE moe an is E x componen which is given in erms of Bessel funcions see Eq...4 in he ex. From Maxwell s equaions we can obain he moe s H y componen as H y n e /µ / E x. These wo componens consiue one polariaion sae of he moe. There is anoher polariaion sae for his moe which is perpenicular o he firs polariaion for example is E componen is along he y-irecion an is H componen is along he x irecion. However by cylinrical symmery of he fiber hese o polariaion saes are ienical. lso noe ha since E an H are assume negligible compare o he ransverse componens E φ is also negligible since E Z H Eφ µ Z ρ ρ φ ρ

3 Fiber Birefringence The fac ha E φ is negligible implies ha he E fiel also he H fiel is linearly polarie an i will remain linearly polarie wih he same polariaion vecor as long as he refracive inex on he fiber has cylinrical symmery. In acualiy such symmery is isurbe by fiber bening sress asymmeric oping ec. This causes he polariaion sae o change from linear o ellipical hen back o linear in a perioic manner see skech in he nex page. The perio is eermine by he ifference in he moe inices in he x an y irecions. This is given by he birefringence parameer B m : B m nx ny

4 Fiber Birefringence The bea lengh L B is λ/b m Typically for a polariaion-mainaining fiber L B is abou m for λ µm.

5 Ranom Fiber Birefringence In realiy B m varies ranomly along he propagaion irecion which means ha we nee o o a saisical analysis of he bea lengh. lso he birefringence varies for each wavelengh an his leas o ifferen group velociies for each polariaions which in urn leas o polariaion-moe ispersion PMD. PMD is an imporan limiing facor in long-haul communicaions

6 PMD in Consan-birefringence Fibers Le us neglec for now he change in he polariaion along he fiber as in polariaion-mainaining fibers. ssume furher ha we have a single-moe fiber. If he launche ligh has wo polariaion componens wih group velociies v gx an v gy hen he ime elay beween he wo polariaions T afer propagaing a isance L is T L / v gx /v gy L x / y L Typically for polariaion mainaining fibers: ~ ns/km

7 D p is calle he PMD parameer. Typically D p is below. ps/km in moern fibers. Luckily he effec of PMD grows as he square roo of isance in conras o GVD effecs which grow linearly wih he isance. PMD General If he birefringence is ranom as a resul of he ranom change in he refracive inex as a funcion of hen we nee o rea T as a ranom variable. We can alk abou is variaion abou is mean σ T or simply he variance of T as a funcion of propagaion isance : σ T [ / ] / l l e l where l c is he correlaion lengh for he wo polariaion componens. Typically l c ~ m For >> l c we have he approximaion c c σ T l c c D p

8 B. Pulse Broaening Due o Dispersion an Chirp We now consier he propagaion of a Gaussian-shape pulse hrough he fiber an esimae he broaening as a funcion of isance. Consier he ime-ransforme elecric fiel ~ E r ~ i xf ˆ x y B e ~ xf ˆ x y B where x-ha is a uni vecor poining in he irecion of polariaion an ~ ~ B B FourierTrans.{ B } ~ B FourierTrans.{ B } where B is he inpu ampliue ime-profile a.

9 For narrow-ban pulses i.e. when - << we can approximae as Bu 6 / g v

10 If we now subsiue his approximaion ino he inverse-fourierransform expression for B we obain: using change of variable - Now pull ou he facors ha on epen on inegrae ou he res call he resul of he inegraion an finally obain: { } exp ~ 6 π i B B i e B

11 Now recall he efining equaion of i follows from i ha ~ ~ B Hence we have π ~ exp { i } 6 To calculae we nee o firs calculae which is he Fourier ransform of he Gaussian iniial pulse shape. We will consier specific examples laer. ~

12 Propagaion Equaion is calle he slowly-varying envelope ampliue of he pulse. I is sraighforwar o check ha if we iffereniae he efining equaion for wih respec o we obain he so-calle pulsepropagaion equaion [vi. iffereniae uner he inegral an recognie ha muliplying he Fourier ransform of by correspons o i/ of ]: Pulse Propagaion Equaion Now o he change of variable - an obain he alernaive form 6 i ' 6 ' i

13 Pulse Propagaion Special case: If GVD ispersion is absen i.e. hen Hence as expece he pulse unergoes NO BRODENING in his special case since oes no change as a funcion of he moion compensae ime. To see he effec of broaening in he general case le s assume ha he launche pulse has he chirpe Gaussian profile e ic i / T Here C is a chirp frequency variaion parameer an T is he wih of he pulse. sie: For a Gaussian pulse T is relae o he emporal FWHM of he pulse T FWHM by he relaion T FWHM.665T.

14 Gaussian Pulse Propagaion The iea now is o fin. We will o his nex. Recall ha { } exp ~ 6 π i

15 To calculae we nee o firs calculae which is he FT of he Gaussian iniial pulse shape. Calculaions yiel We now subsiue ino he expression for an finally obain: We nex calculae he pulse broaening / / where T i C Q e Q Q T ic ~ / / ~ ic T e ic T π

16 Gaussian Pulse Broaening Le T be he wih of he Gaussian pulse afer propagaing a isance. Then by using he previous equaion i can be shown ha he broaening facor is given by / T T C T T

17 Furher Generaliaion If he pulse is no Gaussian hen is wih is eermine by is rms wih a isance fer eious mah one obains he following maser equaion for pulse broaening which is vali for a source wih infiniesimally narrow specrum i.e. almos monochromaic ligh maser equaion for pulse broaening for monochromaic sources σ 4 σ σ σ σ σ C C

18 Banwih Limiaion The broaening is srongly epenen on he specral wih of he source. If σ is he specral wih of he source hen σ σ C V C V σ σ 4 σ maser equaion for pulse broaening for chromaic sources where V σ σ. To avoi inersymbol inerference we wan σ < T B /4 where T B is he bi uraion use in communicaion eros an ones. This is simply a rule of humb an he 4 provies a safey facor. The above boun can be use o euce a boun for BL.

19 Banwih Limiaion There are wo imporan special cases: V << an V >> The V << case: source wih small specral wih like a isribue feeback laser The V >> case: source wih broa specral wih like an LED chirp is ofen negligible in his case. One can calculae he upper bun prouc BL for each of hese cases. Furher we can unersan he behavior of he BL limi in cases where we operae near he ero-ispersion wavelengh we se Laer you will be aske o erive expressions o upper boun BL is such special cases in he nex homework.

20 Reaing Rea pages This inclues maerial on fiber loss an main scaering mechanisms.