Optical Fibers: History, Structure and the Weakly Guided Solution. May 30, 2008 Physics 464 Dr. La Rosa Portland State University.

Transcription

1 Optica Fibes: Histoy, Stuctue and the Weaky Guided Soution May 30, 008 Physics 464 D. La Rosa Potand State Univesity by Tayo Biyeu

2 Abstact In this pape, I wi tak about the histoy and deveopment of fibe optics, chaacteistics and uses fo singe mode fibes, as we as SI and GI mutimode fibes. The buk of the pape wi be the deivation of the vectoia wave equation and finding the ineay poaized modes of a SI fibe. To finish, I used the weaky guided appoximation to compute the cutoff waveength fo a coning SMF-8 optica fibe, which I found to be ~1500nm, wheeas the actua opeating ange is fom about , with the east attenuation in the 1550nm ange. Intoduction In today s wod, the tansmission of infomation ove both shot and ong distances is vita. Optica fibes pay a key oe in these tansmissions, and wi continue to do so as we move fowad. It is a gowing fied with job oppotunities spanning a vaiety of scientific fieds, and I beieve knowedge of the topic wi seve me we in the futue. In this pape, I wi begin by taking biefy about the discovey and oigins of optica fibes. I wi then move on to discuss the stuctue and uses fo some diffeent types of fibe, such as singe-mode, muti-mode, step index, and gaded index fibes, as we as sevea modes of infomation oss in these fibes. Using Maxwe s equations as a stating point, I wi use a seies of substitutions to deive a vectoia wave equation/hemhotz equation fo an optica fibe. Then, assuming <1% change in efactive index between the coe and cadding, I use the weaky guided appoximation to find ineay poaized soutions of ou wave equation. Finay, as an exampe I wi use ou soution aong with the physica specifications of a SMF-8 optica fibe to compae some cacuated vaues with the manufactues infomation. Content Histoy The eaiest attempts at optica communication date back to the 1790 s, when Caude Chappe of Fance invented what is now temed as the optica teegaph. Vaious ights mounted atop age towes with human opeatos

3 woud fash signas to neaby towes. In the 1840 s, two physicists, Danie Coodon and Jacques Babinet demonstated the tansmission of ight aong tubes of wate in fountain dispays. This ed to the now popuaized 1854 expeiments of John Tynda, which is seen by many as the tue beginning of fibe optics. In this demonstation, a jet of wate was aowed to fow out of a pipe in the side of a containe; ight was anged such that it enteed and then exited the containe though the jet of wate fowing fom the pipe, and bent with the wate as the jet fe to a containe beow. was essentiay an a gass fibe that was In the 1950 s, Bian O Bien, used to tansmit images. The ack of a Nainde Kapany and coeagues cadding esuted in excess. deveoped the fibescope, which sive oss, and pompted the deveopment of moe sophisticated and moden fibes. Successive impovements on the genea design wee made ove the next 0 yeas, and in 1970 scientists at Coning Gass Woks doped extemey pue siica gass mateia meant that fo the fist time with titanium, esuting in a gass optica fibe was a viabe means fo age with ess than 0 db/km scae tansmission of infomation, and the attenuation. Such a ow oss moden ea of fibe optics had begun. Types of Fibes In genea an optica fibe consists of an inne coe of doped siica gass, with an oute coe of simia mateia but with a sighty owe index of efaction. This causes a ight signa to be tapped within and tansmitted aong the coe. In the simpest case, the efactive indices ae unifom

4 within the coe and cadding espectivey. This is caed a step-index(si) fibe. In a gaded-index(gi) fibe, the index of the coe can have a vaiety of diffeent pattens, aowing fo speciaized fibes fo diffeent appications. Ony modes of ight enteing the coe though the cone of acceptance wi popagate in the fibe. The cone of acceptance is simpy the cone, concentic with the atea axis of the fibe and bounded by an ange α eative to the axis, above which the ight beam wi not achieve tota intena efection. One of the most common types of fibes is the singe mode SI optica fibe. In this case, the coe of the fibe is vey sma, geneay 8 micons o ess, with the esut that the cone of acceptance ony aows one mode to popagate. This fibe is most often used fo tansmission ove age distances since thee is no intefeence between modes and the shaowe ange esuts in fewe contacts with the cadding and thus ess dispesion.

5 In a mutimode SI fibe, the coe is significanty bigge when compaed to a singe mode fibe, about 100 micons. The age diamete esuts in a age cone of acceptance and theefoe moe modes ae aowed to popagate inside the fibe. One pobem with this type of fibe is moda dispesion. Fo instance, diffeent pats of the same image can be tansmitted down a fibe by a numbe of diffeent modes. Some modes wi tave faste than othes, esuting in intefeence and diffeent pats of the image being eceived at diffeent times. Fo this eason, step-index mutimode fibes ae geneay used fo iumination o data tansmission ove eativey shot anges. One way to pevent moda dispesion when tansmitting mutipe modes is to use a GI coe mutimode fibe. In this, the efactive index gaduay owes fom the cente of the coe to the cadding. What happens then is that a mode wi speed up the futhe it gets fom the cente of the coe, and it gaduay bends athe than efecting in a cusp at the cadding inteface. The inceased speed nea the cadding aows modes that come in at a steepe ange to popagate down the fibe at oughy the same speed, and intefeence, as we as tansmission deay is minimized. Dispesion sti exists nevetheess, and GI mutimode fibes ae aso used fo tansmission ove shot anges, but moe eiaby. A summay of the types of fibes taked about hee is given in the tabe beow

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7 Soutions Now that I have taked about a few types of fibes, I use Maxwe s equations to find a soution fo the modes popagating in a SI fibe. To begin, I wi use Maxwe s equations fo eectomagnetic waves in matte, aong with a few othe we known equations of eectomagnetism, to deive a vectoia wave equation. E & M Equations : D = ε, ε = ε0ε E B = μ, μ = μ0μ H ρ J = t Maxwe's equations in matte : D = ρ B E = (1.1) t B =0 D H = + J t Fo this deivation, we must aso use the phaso epesentations of the eectic and magnetic fieds and fux densities, as shown beow. iωt E(, t) = Re E() e iωt H(,)=Re t H() e iωt D(, t) = Re D() e iωt B(,)=Re t B() e By substituting these expessions into Maxwe s equations (1), we get the foowing set of equations: E = iωe = iωμoh (assumed μ = 1 and ρ = 0) (1.) H = iωd= iωεe (1.3) H =0 (1.4) =0 (1.5) ( ε E) Next, we can appy a vectoia otation opeato to equation (1.), and foow that by appying a vecto poduct ue to the eft-hand side.

8 ( E) = iωμ H (1.6) 0 ( E) = ( E) E (vecto poduct ue) (1.7) Next, we appy anothe vecto poduct ue to equation (1.5) and eaange it, giving us equation (1.9) beow. ( ε E) = 0 = ε E + ε E (vecto poduct ue) (1.8) ε E= E (1.9) ε Pugging equation (1.9) into the ight hand side of equation (1.7) gives us the foowing: ε ( E) = E E ε Howeve, we can aso substitute equation (1.3) into the ight hand side of equation (1.6), giving us ω iωμo H = ω μoεe= ko εe k0 =, k = kon= ko ε c0 Theefoe, we have the foowing eation: ε E+ E + k E= 0 ε This is the vectoia wave equation that we set out to find. Using a simia method, we woud get the foowing wave equation fo the magnetic fied. ε H+ ( H) + k H= 0 ε We can futhe simpify these equations by making the assumption that the fibe we wi be evauating is axiay unifom, aong the diection of popagation. If this is tue, which it must be, and we et the diection of

9 popagation be the z-diection, we can teat the deivative as an opeato = iβ. β hee is the z-component of the wave numbe k. Futhemoe, z we can use the notation + =, giving us: z ε β E+ E +( k β )E= 0 n = eff ε k0 ε E+ E + k0 ( ε neff )E= 0 (1.10) ε ε H+ ( H) + k ( ε n )H = 0 (1.11) 0 eff ε Now that we have a wave equation, we wi appy it to the simpest case. Next I find what is known as the weak soution o ineay poaized modes of an optica fibe, but we must make a vey paticua assumption. It must be assumed that the change in the index of efaction between the coe and cadding is vey sma, <1% as a genea ue of thumb. This gives us ony an appoximation and it is ony vaid fo SI fibes, but the esuts ae vey teing, and we end up with a simpe fomua that aows one to design a SI fibe so that it wi be eithe singe mode o muti mode. We wi assume that n 1 is the index within the coe of adius (a) and n is the index of the cadding. To begin with, we assume that the eative pemittivity ε is constant in the medium, which educes equations (1.10) and (1.11) to the foowing equations, othewise known as Hemhotz equations: E + k ( ε n )E = 0 (.1) 0 eff H + k ( ε n )H = 0 (.) 0 eff Next, I wi ewite the Lapacian in cyindica coodinates, a eativey easy appication of the chain ue which can be found in the coves of many books, and so it is not impotant to cove in detai hee.

10 1 1 = θ z We wi sove equation.1, using the assumption that the tangentia eectic fied components in the x o y diections ae given by the foowing: E =R( ) Θ( θ ) x,y Substituting this expession aong with the cyindica epesentation of the Lapacian into equation (.1), and mutipying the esuting equation by, we get: R( ) Θ ( θ ) R( ) 1 R( ) 1 Θ( θ ) + + k0 ( ε neff )= R( ) Θ( θ) θ Since thee is now an expession in tems of on the eft side, and one in tems of θ on the ight, it must be tue that both sides ae equa to some constant, which can be caed. R( ) 1 R( ) 1 Θ( θ ) + + k0 ( ε neff )= = (.3) R( ) Θ( θ) θ This gives us two sepaate odinay diffeentia equations: d Θ dθ ( θ ) + Θ ( θ )=0 d R( ) 1 dr( ) + +R( ) k0 ( ε neff ) 0 = d d The fist equation is quite simpe and easiy soved, so hee I just pesent the soution, in this case is an intege and φ is an abitay phase constant.

11 Θ( θ ) = sin( θ + φ) (.4) The second equation is moe difficut, and to stat with I use a few substitutions and then appy the chain ue to get a simpe equation with egads to new vaiabes. % u k0 ( ε neff ) ξ u % d d dξ % d = = u d dξ d dξ d % d dξ % d = u = u d dξ d dξ Substituting these expessions into the second diffeentia equation above, we get % % % % d R( ) 1 dr( ) u + u + R( ) 0 u % = dξ ξ u d ( ξ u) d R( ) 1 dr( ) R( ) 0 = dξ ξ d ξ Soutions fo this equation, which I have again eft out fo the sake of bevity, ae the th-ode Besse functions beow. A u u J BN a + a fo a R ()= (.5) CK w + DI w a a fo a

12 J and N ae th-ode Besse functions of the fist and second kinds, whie K and I ae th-ode modified Besse functions of the fist and second kinds. The vaiabes u and w in equation (.5) ae defined beow, aong with a new vaiabe v. u = k a ( ε n ) 1 eff 0 w = k a ( n ε ) eff u + w = v 0 o 1 v= k a ε ε (.6) The paamete v is the nomaized fequency, whie u and w ae the nomaized atea popagation constant in the coe and the nomaized atea decay constant in the cadding, espectivey. Since the functions N and I ae divegent ove cetain anges, thei coefficients must be zeo, giving us the foowing expession: AJ R ()= CK u a w a fo a fo a To sove fo the emaining coefficients A and C, we use the bounday condition that the function R must have the same vaue when appoaching the coe/cadding inteface fom both the eft and ight sides, and it aso must be continuous. These conditions ead to the foowing eations: + Ra ( )= Ra ( ) A J( u) CK( w)=0 + dr( a ) dr( a ) = A uj ( u) CwK ( w) d d

13 These equations can be witten in matix notation, J( u) K( w) A = 0 uj ( u) wk ( w) C If A and C ae nontivia, then the deteminant of the coefficient matix must be zeo, so. J ( u) K ( w) uj ( u) wk ( w) =0. Evauating the deteminant and eaanging, we get the foowing chaacteistic equation: uj ( u) wk ( w) = (.7) J ( u) K ( w) Equation (.6) can be ewitten fo modes, which as I stated eaie ae caed ineay poaized modes. The manipuation of (.6) is dependent on a vaiety of popeties of the Besse functions, so I wi not incude the mathematics invoved. LP ( = 0 and m 1) : 0m J0( u) K0( w) = uj ( u) wk ( w) 1 1 LP ( 1 and m 1) : m J( u) K( w) = uj ( u) wk ( w) 1 1 Now, we get something moe meaningfu fom these chaacteistic equations, we see what happens ove a ange of the paametes. This again

14 equies moe in-depth knowedge of the Besse function, namey the asymptotic appoximations of said functions. I am again foced to skip the intemediate mathematics, and go staight to the consequence of taking the imits of ou paametes w and u. Fist, fo the LP modes: 0m As w 0, u v: K0( w) J0( v) wk ( w) vj ( v) 1 1 The second expession impies that eithe v o J () 1 v appoach 0. If we et v=0, the expession hods, and so the cutoff nomaized fequency v c = 0. This means that the LP01mode has no cutoff condition. Fo highe ode LP0m whee m>1, we define j, m 1 as the ( m 1) th zeo of the Besse function of the fist kind. Evauating at the imits of ou paametes + as we did above, we see that as v j1, m 1, the condition above is met and vc = j1, m 1. Simiay, fo LP m modes, we get vc = j 1, m. The most impotant featue of the peceding obsevations is that thee is no cutoff fequency fo the LP 01 mode, and the fist zeo of the Besse functions is j 0,1 (.40486). This means that the LP 01 is the fundamenta mode, and aso that v c = gives the cutoff condition sepaating singe mode and muti mode opeation in a SI optica fibe. One shoud keep in mind though that thee ae technicay fundamenta modes, coesponding to two possibe poaizations. Now, ook back at equation (.6) aong with the foowing expessions:

15 π ko = λ ε = n ε 1 1 = n NA = n n 1 Pugging these into equation (.6), aong with ou obsevations of cutoff fequencies above, we can make the foowing concusions: π a NA < λ ( SINGLE MODE) π a NA > λ ( MULTIPLE MODES) The waveength coesponding to the vaue of v is the cutoff waveength λ c and is given by the foowing equation: π a λ c = NA To use a SI fibe in singe mode opeation, the waveength of the ight being tansmitted must exceed that of λ c. As an exampe, we wi ook at the Coning SMF-8 optica fibe. Accoding to manufactue data, the adius of the coe is 4.1 micometes, with a numeica apetue (NA) of The cutoff fequency is then: λ = c 6 π ( ) (0.14) = = nm 6 metes The fibe in question is actuay opeated at a ange of diffeent waveengths, incuding waveengths sighty beow the cutoff. This is not too supising consideing that the inea poaized modes ae ony an appoximation. It is

16 sti a good appoximation though, and in fact the fibe gets the east attenuation at about 1550nm. Concusions Thee is a vast amount of diffeent fibes that ae idea fo diffeent appications. Existing fibes ae being constanty pefected, and new designs atogethe ae aso being woked on. The design of these fibes equies a woking knowedge of what modes wi popagate unde what conditions. In the case of singe mode SI fibes, the weaky guided soution whose modes ae ineay poaized is a decent appoximation of the hybid modes which can be attained though a moe exact anaytica examination of Maxwe s equations. Any mode with a nomaized fequency beow about.405 wi esut in singe mode opeation. The Coning SIF-8 fibe woks best at about the 1550nm ange.

17 Refeences micons%9.pdf Chai Yeh, Handbook of Fibe Optics: Theoy and Appications. Academic Pess, Inc 1990 Katsunai Okamoto, Fundamentas of Optica Waveguides. Academic Pess, Inc, 000 Kenji Kawano and Tsutomu Kitoh, Intoduction to Optica Waveguide Anaysis. Wiey- Intescience Pubishing, 001 Step-Index Mutimode Fibe