\,. Si2:nal Detection and. Optical AmpUfler or Signal Regenentor ""' Fiber La~er Laser Coupler Driver Diode,-~ [> I I ~ : Modulator. Splice.

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2 Sieal Geeratio!! Ietroi Fiber Laer Laser Coupler Driver Diode, [> I I : Modulator Iterfae I I: t A Trasissio Mediu Splie ptial ApUfler or Sigal Regeetor ""' I N C y Coditioig ' Eletrois I Eletroil Detetor \, Si2:al Detetio ad D /1 I Iterfae I 'rj I I F / I b er Coditioi2 Aplifier Couplr

3 FIBER PTIC SENSRS Extrisi: Sigal I Sigal ut Eviroetal Chages Itrisi: Eviroetal Fators: Tep, Press,Che

4 ( 1iQ (D I N > '0 'E 0" "' Q t") :r t") '(;, (;; "' 0, :!) 0 ::;, '0 VI ", V1 V1 X " Z V1 < "' 4 x "'0 z \I\ < > VI 1 >< "'0 Z VI < '" J8 I z J 4 z ẕ 4 o o Ut z CJJ Ut < Ut 0 4 z U\ x 3: g "T1 4 r "'0 :1: z 4 z r 4 "' )80 0 "' a z G') o " " " 4 0' 4 > 0' "' < < ' o o :3: :1: H 0 W G) :1: H I 3 v C) C') :I: H I 3 A o o :1: N 3 > z 4 x < r < < r < > Q r < < > G) > Q \ft > "' VI C "' > "' > V\ "'U 0 " :' < "'U D :' " o D :' C 0 D 0 Q ) z o >< ", tt u 0! az a >< "' z Q _ I "!

5 ANATMY fptical FIBERS' Gradedidex fiber r r + Core radius (a); laddig thikess (ha) Step idex profile Graded idex profile There are two approahes to aalyzig the propagatio of light through fibers a) Field propagatio usig exat solutios to Maxwell's equatios b) Ray traig Whe the ore radius is uh larger tha the wavelegth A (ieee a la), a use geoetrial optis desriptio with high auraye Geoetrial optis is very isightful for ay situatios 2

6 Geoetria' Desr Bea Pro a atio i tial,fibers: The geoetrial desriptio of light propagatio i fibers is based o the pheoea Whe a bea is iidet to a iterfae fro a ediu with higher refrative idex a ritial agle ours at whih light is refrated at 90 to the surfae oral betwee to dieletris Beyod this agle light is o loger trasitted ito the seod ediu (appreiably) 3

7 Sell's Law: (}1 = si Si(}2 l Sie l > 2 there is a liit to 2 (ie 2 = 900) ad this liit is the ritial agle It ours whe 2 = () l Surroudig a ediu with refrative idex l with a ediu of lower idex 2 will for a trap for light that is iidet at agles > 8o This is the basis for a optial waveguide ptial Waveguide 2 1 TE Polarized Fields: 4

8 TE Itesity Coeffiiets: Ti = TM Polarized Fields: TM Itesity Coeffiiets: I = 1' = 12 It II I Ca also osider that for agles of iidee that are equal ad greater tha the ritial agle e : l 2, )2 2 SIll (71 JI2 write: 5

9 B = [ ( ;; ) si2 el 1 ]112 oeffiiets as os ' j = exp(2j' ) si ' exp(j' ) ta' =! = 2 [ ( l I 2 )2 si2 01 _l JI/2 A l os 01 = [ si201(2/i)2j/2 os 01 Siilarly for TM refletio: Therefore for eah ase there is a of the iidet field 6

10 ad 2 = 10 is show below: $ 1 2 II L 401ioLL Iidet agle FIGURE 313 The phase hage upo total refletio for eah of the two polarizatios For this alulatio we assued that the idex of the dese ediu was 15 ad the idex of the less dese ediu was ""! q I'C ", "'r,; "? f /I1f#J"' :, (, :\,1 2 '11' 2 ( r

11 Fiber Nuerial Aperture: The aeptae agle for a fiber defies its uerial aperture (NA) It is a iportat paraeter for deteriig the bea propagatio ad ouplig harateristis of optial waveguides The NA is defied as: NA sa 1 1 where ai is the largest aeptae agle that is oupled ito the waveguide This a be deteried fro the TIR oditio at the iterfae betwee l ad 2 ill( ) si1 1 J 90 5

12 This leads to the siple result: Exaple: III = 150; ll2 = 49; NA = ad ai = 996, A iportat paraeter based o the uerial aperture is the frequey or v# This paraeter is used to deterie ode harateristis of the fiber struture The V# is defied as: a = 2J'[ NA A Notie that a fiber with a partiular radius ad NA a hage siply by usig a differet wavelegth 6

13 Approxiatg/!f ethod _for M odes: Cosider the optial waveguide show i the figure below with l > 2 02 A 2a 01 e,! i i r" k = 27tDl /A B A ray propagates fro A to B to C refletig loatios A ad B at iterfaes betwee 1 ad 2 at I order jror a wavefrot to be stable withi the waveguide the wavefrot ust be otiuous at all loatios withi the guide The total phase delay betwee the wave frot at poits A ad : <PI is thelryhase hage upo refletio (there are two refletios), ad <P2 is the phase hage due path ABC 7

14 The geoetrial path ARC: L ABC = Covertig to a (j)2 = Therefore a exist whe: 2 8JZ'"1 rptot = rpl + a as () = 2JZ'" with = 0, 1, 2, t;?! J o{a tn\ A J, (::! f=,;,;t {A) p, \I1A /,; 6' M A, I AJ fio'1, 8

15 Exaple: l = = 140 a= d/2 = 25 Jl; A = 10 Jl Assue that <p 1 00 I this ase e = 6896 ; ai = ad :il1 w > lajtx rouded to the ext lowest iteger There will also be oe ode for M = akig the total estiated odes equal to 55 It shoulrll be oted that this is oly a approxiate expressio to provide a ituit jive oept of odes The atual uber of odes is deteried fro solutio to the field equatios ad boudary oditios for the speifi waveguide If the r radius ad NA is kow it a quikly be deteffilied if the fiber is operatig i a sigle ode oditio sie all odes exept the fudaetal will be ut off whe v

16 EXAMPLE: For a SMF the NA 01 Therefore if the ore idex l = 150, 2 = With NA = 010 a fiber with A = 085 JL a= 3254 JL A = 130 JL a= 4976 JL A = 155 JL a= 5933 JL # Modes V2 2 The zopoet of the propagatio ostat k = 27tl/A is referred to as the eigevalue of the waveguide ad is usually assiged the 10

17 The field propagatig i a waveguide (axis alog the z) a be expressed as where ko = 27t/A Notie that the arguet of the radial a beoe < 0 K IS ad sigifies a osillatig field It is defied as: This applies whe < ko I sillatory (propagatig) for for the eletri field Siilarly whe J3 > ko2: y speifies a atteuatio oeffiiet The field deays does ot propagate The values for 13 are deteffilied fro solutios to the field equatios ad boudary oditios A harateristi equatio a be fored that speifies the allowed odes for the waveguide 14

18 F or a syetri waveguide the odes are: for TE eve ad odd ta KQ = y K K y ad for the eve ad odd ta KQ = y K l 2 K 2 y The harateristi equatios are solved either uerially or graphially by settig the lefthad side of the equatio equal to the right had side The itersetio of the two urves gives the e K is foud, the oeffiiets /3 ad yare deteried to speify the allowed fiber odes 12

19 ALTERNATIVE REPRESENTATIN: Syetri waveguide propagatio ostats: Let a equal the of the waveguidig ( ore) regio Multiply both sides by a2 ad obie expressios: 2 Let X = Ka ad y = ya This allows the relatio to be writte i the for: R2 = X2 + y2 = 2lZ"a A 2 The harateristi equatios beoe: ay == ak ta ay == ay ta or Y=XtaX y=xotx 13

20 Yad t: I TEa TE 1 / R 0 > II >< TE2 hi;i1, rzt,r /t,jiy'!1f 1ttf" tl L /JI'C L, r 4?e >!1 >< TEJ :1\J Qf r A 1'1 4"+ :1 J gj )( o j 11 )( I j ) o TE X b f :/ v*, i ; d('l"'(127' Iry,, v'" If!4 v J I e F : r1 La;)tJ ) 21T x = 1=( d Iiure 2 CharlL"l<'risliL" <'4U;ltiIJ Ji;lgr;l T[: \\J j l)i Ji l:l'trll" 1;1h \I iv<,l1ijt: ftd (}! I L,LY L = /?""l f( v v',1' $11I/51l IV{' J r ( 1\C vv r y:::x t x ("<vx) P L)(C,yt,i 1 j (YY"t" \/ ; lr //1rf) " k te1(,,,r fr ;>(" 1i" /L<C J;ll C7'"'+'I''V t: 1L Wort()

21 This gives a ore ituitive represetatio for the harateristi equatio A Modes Ya Ka SM, Cutoff The allowed values for K ad yare give by the itersetio of the two urves All fos of harateristi equatios basially work i this aer Notie that a sigle ode operatig oditio ours whe the V# is less tha a value orrespodig to the seod to lowest allowed ta or otaget futio 14