NEGATIVE REFRACTIVE INDEX SEMICONDUCTOR CIRCULAR WAVEGUIDE FIBER

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2 Islic nivesity of G Deney of Gdute Science Fculty of Science Deptent of Physics NGATIV RFRACTIV INDX SMICONDCTOR CIRCLAR AVGID FIBR By Abdll Si Sdeh Supeviso Pof D Mohed M Shbt Subitted to the Fculty of Science s ptil Fulfillent of the Mste degee of Science pog M Sc in Physics Plestine G My- 3

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4 -To y de pents, bothes, nd sistes fo thei suppot, ntinuous enugeent, nd love - To y fily fo thei get ce nd help - To y fiends nd eltives - To y hoelnd Plestine Abdll Si Sdeh I

5 ACNOLGMNTS In the ne of Allh to who povide y endless suppot, enugeent nd help e to plete y thesis I would like to expess y deep thnks to ll those who helped e getly to ovee the obstes I exteely indebted to y supeviso Pofesso D Mohed M Shbt not only fo his ntinuous effots but lso fo his effective help e woked dy nd night to help e to nge nd wite y thesis Also, I would like to expess y deep thnks to D Sofyn Ty who suppoted e nd ws lwys by y side Also, I would like to thnk D Men Abdl too fo his ntinues guidnce duing y esech In fct, I would like to expess y deepest ppecition to the wondeful stff of Physics Deptent t the Islic nivesity of G Finlly, I deeply nd foeve indebted to y lovely pents nd y wife fo thei suppot nd enugeent though y entie life Plestine, G My, 3 Abdll Si Sdeh II

6 ABSTRACT In this wok, we will nside the electognetic wves popgtion long the syetic cylindi fibe with left hnded teil LM s the e lye bounded by seinducto d lye e will study the dispesion popeties in the nsideed opti fibe fo two cses In the fist cse, we nside the leding digonl eleents of the seinducto peittivity tenso e diffeent fo ech othe In the send cse, we nside s specil cse, the eleents e equl to ech othe In the two cses, the fst nd slow odes will be studied in T, TM, nd ode politions The dispesion eltions fo ll odes will be obtined fo Mxwells equtions The totl powe flow in the stuctue within e nd d lyes will be culted in the genel nd in specil cses fo T nd TM ode politions The dispesion cuves fo T, TM, nd odes will be studied fo diffeent petes of the stuctue III

7 List of Figue Cptions CAPTR ON Figue : ve pulse oving t nstnt speed 4 Figue : Light eflection t diffeent cses with c nd c n n, c TIR n n, c nd b n n, Figue 3: Fibe nsists of cylindi centl e d by teil 4 Figue 4: Light ys ipinging on the e -dding intefce t n ngle gete 5 thn the citi ngle e tpped inside the e of the fibe Figue 5: Peittivity-peebility ε μnd efctive index n dig fo ll 7 teils Figue 6: Oienttion of the electic nd gnetic vectos of the intensity of the 8 electognetic fields, nd, with espect to the wve vecto k nd Poynting vecto S fo LM nd RM CAPTR TO Figue : Stuctue of the opti fibe with LM e nd dielectic d Figue : Opti fibe in cylindi odinte syste,, Figue 3: Zeo- nd fist-ode Bessel functions of the send kind with, 7 Figue 4: Modified Bessel functions of fist I x nd the send x kind 8 with, Figue 5: Dispesion cuves descibed the fst ode AB cuvee, fst- slow ode 4 CF cuve nd slow ode PL cuve Fibe petes: dding 5, n, nd fibe dius c fte [5] Figue 6: Fst guided odes with cuve AB- type t, with 4 fibe petes 5,, nd fibe dius c by [5] Figue 7: The fst-slow nd slow low-ode guided odes, with fibe 4 petes : 5,, nd fibe dius c by [5] CAPTR TR Figue 3: Schetic fo opti fibe with LM e lye, seinducto 46 IV

8 d CAPTR FOR Figue 4: Dispesion eltion fo TM ode with diffeent fctionl e F 69 Fibe petes,, 3, p / G, 3G nd G c 7 Figue 4: Dispesion eltion fo TM ode with diffeent e fibe dius 69 Fibe petes,, F 56, p / G, 3G nd G c 7 Figue 43: Dispesion eltion fo T ode with diffeent fctionl e F 7 Fibe petes,,, p / G, 3 3G nd G c 7 Figue 44: Dispesion eltion fo T ode with diffeent e fibe dius 7 Fibe petes,, p / G, 3G, F 56 nd G c 7 Figue 45: Dispesion eltion fo ode with diffeent fctionl e F 7 Fibe petes,, 3, p / G 3G nd G c 7 Figue 46: Dispesion eltion fo ode with diffeent e fibe dius 7 Fibe petes,, F 56, p / G, 3G nd G c 7 Figue 47: Dispesion eltion fo ode with diffeent fctionl e F 73 Fibe petes,, 3, p / G, 3G nd G c 7 Figue 48: Dispesion eltion fo ode with diffeent e fibe dius 73 Fibe petes,, F 56, p / G, 3G nd G c 7 Figue 49: The totl el pt of the powe flowing though the stuctue in T 74 ode vesus the fequency with diffeent vlue of e dius Fibe petes F 56,, C=, p G, 3G nd G c 7 V

9 Figue 4: The totl iginy pt of the powe flowing though the stuctue 75 in T ode vesus the fequency with diffeent vlues of e dius Fibe petes F 56,, C=, p G, c 7G nd 3G Figue 4: The totl el pt of the powe flowing though the stuctue in T 76 ode vesus the fequency with diffeent vlues of fctionl e F, Fibe petes,, C=, p G, G, 3G nd 3 c 7 Figue 4: The totl iginy pt of the powe flowing though the stuctue in 76 T ode vesus the fequency with diffeent vlues of fctionl e F Fibe petes,, C=, p G, c 7G, 3G nd 3 Figue 43: The totl el pt of the powe flowing though the stuctue in T 77 ode vesus the popgtion nstnt with diffeent vlues of e dius Fibe petes F 56,, C=, p G, G nd 3G c 7 Figue 44: The totl iginy powe flowing though the stuctue in T odes 78 vesus the popgtion nstnt with diffeent vlues of e dius Fibe petes F 56,, C=, p / G, c 7G nd 3G Figue 45: The totl el pt of the powe flowing though the stuctue in TM 79 ode vesus the fequency with diffeent vlues of e dius Fibe petes F 56,, C=, p G, 3G nd c 7G Figue 46: The totl iginy pt of the powe flowing though the stuctue in 79 TM ode vesus the fequency with diffeent vlues of e dius Fibe petes F 56,,C=, p G, c 7G nd 3G Figue 47: The totl el pt of the powe flowing though the stuctue in TM 8 ode vesus the fequency with diffeent vlue of fctionl e F, Fibe petes,, C=, p G, G c 7 VI

10 3G nd 3 Figue 48: The totl iginy pt of the powe flowing though the stuctue in 8 TM ode vesus the fequency with diffeent vlues of fctionl e F Fibe petes c 7G nd 3G 3,, C=, p G, Figue 49: The totl el pt of the powe flowing though the stuctue in TM 8 ode vesus the popgtion nstnt with diffeent vlues of e dius Fibe petes F 56,, A=, p G, c 7G, 3G nd 3 Figue 4: The totl iginy pt of the powe flowing though the stuctue 8 in TM ode vesus the popgtion nstnt with diffeent vlues of e dius Fibe petes F 56,, A=, p G, G 3G nd 3 c 7 CAPTR FIV Figue 5: Dispesion cuves descibing the fst- slow ode cuve AC,fst ode 87 with cuve DF in the fibe Fibe petes, 5, fctionl e F 56, p G, c G nd / G tht be divided in two egion cuves Figue 5: Fst-slow guided odes desetion cuve of AC type in Fig t 88, Fibe petes, 5, fctionl e F 56, p G, c G nd / G Figue 53: Slow guided odes desetion cuve of DF type in Fig t 89 fibe petes,, 5, fctionl e F 56, p G, c G nd / G Figue 54: Dispesion eltion fo TM ode with diffeent fctionl e F Fibe 9 petes,, 5, p G, c G nd / G Figue 55: Dispesion eltion fo TM ode with diffeent e dius, 9 Fibe petes, fctionl e F 56, p G, c G nd / G VII

11 Figue 56: Dispesion eltion fo T ode with diffeent fctionl e F, 9 Fibe petes, 5, p G, c G nd / G Figue 57: Dispesion eltion fo T ode with diffeent e dius, 9 Fibe petes, fctionl e F 56, p G, c G nd / G Figue 58: Dispesion eltion fo ode with diffeent fctionl e F, 93 Fibe petes, 5, p G, c G nd / G Figue 59: Dispesion eltion fo ode with diffeent e dius, 93 Fibe petes, fctionl e unit cell F 56, p G, c G nd / G Figue 5: Dispesion eltion fo ode with diffeent fctionl e F, 94, Fibe pete c G nd / G, 5, p G, Figue 5: Dispesion eltion fo ode with diffeent e dius, 94 Fibe petes, fctionl e unit cell F 56, p G, c G nd / G Figue 5: The totl el pt of the powe flowing though the stuctue in 95 T ode vesus the fequency with diffeent vlues of e dius Fibe petes F 56,, C=, p G, c G nd / G Figue 53: The totl iginy pt of the powe flowing though the stuctue 96 in T ode vesus the fequency with diffeent vlues of e dius Fibe pete F 56,, C=, p G, c G nd / G Figue 54: The totl el pt of the powe flowing though the stuctue in 97 T ode vesus the popgtion nstnt with diffeent vlue of e dius Fibe petes F 56,, C=, VIII

12 p G, c G nd / G Figue 55: The totl iginy pt of the powe flowing though the stuctue 97 in T ode vesus the popgtion nstnt with diffeent vlues of e dius Fibe pete F 56,, C=,, p G, c G nd / G Figue 56: The totl el pt of the powe flowing though the stuctue 98 in TM ode vesus the fequency with diffeent vlues of e dius Fibe petes F 56,, A=,, p G, c G nd / G Figue 57: The totl iginy pt of the powe flowing though the stuctue 99 in TM ode vesus the fequency with diffeent vlues of e dius Fibe petes F 56,, A=, p G, c G nd / G Figue 58: The totl el pt of the powe flowing though the stuctue TM in ode vesus the popgtion nstnt with diffeent vlues of e dius Fibe petes F 56,, A=, p G, c G nd / G Figue 59: The totl iginy pt of the powe flowing though the stuctue in TM ode vesus the popgtion nstnt with diffeent vlue of e dius Fibe petes F 56,, A=, p G, c G nd / G Figue 5: The totl el pt of the powe flowing though the stuctue in T ode vesus the fequency with diffeent vlues of fctionl e F Fibe petes c G nd / G 5,, C=, p G, Figue 5: The totl iginy pt of the powe flowing though the stuctue in T ode vesus the fequency with diffeent vlues of fctionl e Fibe petes c G nd / G 5,, C=, p G, Figue 5: The totl el pt of the powe flowing though the stuctue in 3 TM ode vesus the fequency with diffeent vlues of fctionl IX

13 e Fibe pete 5,, A=, p G, c G nd / G Figue 53: The totl iginy pt of the powe flowing though the stuctue 3 in TM ode vesus the fequency with diffeent vlues of fctionl e F Fibe petes 5,, A=, p G, c G nd / G X

14 Contents CAPTR ON: INTRODCTION TO OPTICAL AVGIDS AND FIBRS Intoduction Light Popgtion Mxwells qutions nd Plne ves ve qution 4 3 Boundy Conditions t n Intefce 6 4 Poyntings Vectos 7 3 Polition 8 3 S- polied light T ode 8 3 P - polied light TM ode 9 4 Fibe Optics 9 4 istoy nd Definition of Opti Fibe 4Types of Opti Fibes 43Totl Intenl Reflection TIR 44 Stuctue of Opti Fibe 3 5 Left- hnded teils LM 5 5Intoduction to LM teils 5 5lectognetic Popeties of LM teils 7 CAPTR TO : GIDD MODS IN NGATIV RFRACTIV INDX FIBRS Stuctue Anlysis Solution of ve qutions in the Cylindi Coodinte 3 Solution Of Mxwells qutions in Cylindi Coodinte 4 xpessions fo,, Coponents 4, 5 xpessions fo nd Coponents 5 5 Solutions in the Coe Region 6 5 Solutions in the Cld Region 8 XI

15 6 xpessions fo the nd Coponents in Tes of Bessel 9 Function 6 xpessions fo the,,, in the Coe 3 Region 6 xpessions fo the,,, in the Cld Region 3 7 Dispesion qution of n Opti Fibe 3 8 Modes in Opti Fibes 36 8Meidionl Modes : 36 8 Skew Modes: 36 9 Nuei Results 38 CAPTR TR: MODS IN OPTCAL FIBRS IT SMICNDCTOR CLADDNG AND LM COR 3 Intoduction to Seinductos 4 3 Opti Popeties of Seinducto 43 3 lectognetic negy 44 3 Mxwells qution Stuctue Anlysis Solution of Mxwells qutions in Cld egion Seinducto xpessions fo,, in the Cld Seinducto 47, 36 Dispesion qution fo Modes T Mode TM Mode ybid Mode Opti Powe fo T nd TM Modes T ode 6 37 TM ode 6 CAPTR FOR: MODS IN OPTICAL FIBR T LM COR AND SMICNDCTOR CLAD; GNRAL CAS 4 Stuctue Anlysis 65 4 The Dispesion Reltions 66 XII

16 43 Powe Considetion T Mode Cse TM Mode Cse Nuei Results nd Discussions 68 CAPTR FIV: MODS IN OPTICAL FIBR IT LM COR AND SMICNDCTOR CLAD; SPCIAL CAS 5 Stuctue Anlysis 83 5 The Dispesion qutions 84 5 T nd TM Mode 84 5 ybid Mode nd Powe Considetions Fo T Mode Fo TM Mode Nuei Results 87 CAPTR SIX : CONCLSIONS Applictions 6 Advntges nd Disdvntges 7 Refeences 8 XIII

17 CAPTR ON INTRODCTION TO OPTICAL AVGIDS AND FIBRS In this chpte, the bsic ncepts nd equtions of electognetic wve theoy equied in the tetent of opti wveguides e pesented Mxwells equtions, wve equtions, Boundy nditions nd Poyntings Vecto e descibed in section T nd TM Politions e discussed in section 3 Opti fibe popeties nd its stuctue, dvntges nd disdvntges, Typs of opti fibe, nd totl intenl eflection TIR e pesented in section 4 Finlly, left hnded etteil LM nd thei electognetic popeties e explined in section 5 Intoduction Couniction iplies tnsfe of infotion fo one point to nothe hen it is necessy to tnsit infotion, such s speech, iges, o dt, ove distnce, one genelly uses the ncept of cie wve uniction In such syste, the infotion sent odultes n electognetic wve such s dio wve, iwve, o light wve, which cts s cie This odulted wve is then tnsitted to the eceive though chnnel nd the eceive deodultes it to etieve the ipinted signl This is due to the fct tht, in ny uniction syste eploying electognetic wves s the cie, the ount of infotion tht cn be sent inceses s the fequency of the cie is incesed The ide of using light wves fo uniction cn be tced s f bck s 88 when Alexnde Gh Bell [] invented the photophone shotly fte he invented the telephone in 876 ockh [] in 966 suggested tht opti fibes bsed on silic glss uld povide the necessy tnsission ediu if etllic nd othe ipuities uld be eoved In 97 pon et l, [] hd successful by poduced silic fibes with loss of bout 7 db/k t wvelength of 633 n Since then, the technology hs dvnced with teendous pidity By 985 [] glss fibes wee outinely poduced with exteely low losses < db/k Along the pth of the opti fibe e splices, which e penent oints between sections of fibes, nd epetes tht boost the signl nd ect ny distotion tht y hve occued long the pth of the fibe

18 Light Popgtion Mxwell s qutions nd Plne ves All the nlysis of wve popgtion is bsed on Mxwell s equtions which goven the tie dependence of the intensity of the electic nd gnetic fields nd, espectively These two field vectos e often used to descibe the electognetic fields Mxwell s equtions cn be witten s follows -Fdy s lw [3] B t -Apee s lw with Mxwell s ection, [3] D t 3-Guss s lw [3] D 3 4- Non existnse of gnetic onopole [3] B 4 The quntities D nd B e led the electic displceentnd the gnetic induction,espectively The quntities nd e the electic chge density nd the cuent density, espectively In ediu with no fee chges = nd no cuents = q nd q ewitten s: D t 5 6 e cn lso intoduce the electic peittivity nd the gnetic peebility These two petes chcteie the esponse of the teil to n extenl electic nd gnetic field s [4]:

19 D P 7 B M 8 whee P nd M e the electic nd gnetic politions, espectively hen n electognetic field is pesent in teil, the electic field cn petub the otion of electons nd poduce dipole polition P The ppliction of gnetic field to teil cn induce gnetition M Fo plne wve, the electic nd gnetic fields cn be expessed s [5]: i t, t e k 9 i t, t e k whee is the fequency of the field nd k is the wve vecto, is the position vecto, t is the tie, nd nd define the plitude nd the diection of the vectos nd espectively Now, we substitute q 7 nd q 8 into Mxwell s equtions with the use of q 9 nd q, Mxwell s equtions bee [6]: k t k t k 3 k 4 q 3 nd q 4 show tht both the electic field vecto nd the gnetic field vecto e pependicul to the wve vecto k whee k nd k Also, fo q nd q we see tht the electic nd gnetic field vectos e pependicul to ech othe, nd k fo tiplet of utully pependicul vectos By ultiplying both sides of q by Thus, we obtin tht the thee vectos,, k we obtin k 5 3

20 Siilly, ultiplying both sides of q by k to obtin k εω 6 Dividing q 5 by q 6, we obtin eltion between the bsolute vlues of the intensities of the electic nd gnetic fields 7 ve qution Conside now wve pulse popgting in the x-diection s shown in figue It is given by the eltion y=a sin kx 8 whee A is the wve plitude y A y Asin kx x Figue ve pulse oving t nstnt speed The wve vecto cn be defined s : k 9 And soe eltion to the pulse is given by 4

21 T v f T f T whee T is the peiodic tie, f is the fequency, is the ngul fequency, is the wve length nd v is the speed nstnt Fo qs, 5, 7, nd 8, we cn deive the wve eqution fo the electic nd gnetic fields Applying the opeto to both sides of q we get: 3 t The left-hnd side of q 3 cn be siplified with use of the identity 4 The ight-hnd side of q 3 cn be ewitten using qs 5 nd 7, finlly the eqution cn be witten s: 5 t sing q 9 into q 5 the wve eqution is witten s [5,8]: nk 6 which is led the wve eqution of the electic filed, whee n is the efctive index n Siil eqution cn be obtined fo the gnetic field by pplying the opeto to both sides of q 5 7 t The left-hnd side of q 7 cn be siplified with use of the identity 5

22 8 The ight-hnd side of q 7 cn be ewitten by using qs nd 8, finlly we obtin the eqution 9 t sing q into q 9 the wve eqution is witten s [5,8]: nk 3 which is led the wve eqution to the gnetic filed The fields nd popgte though epty spce with with speed equl to the velocity of light, c oweve, in ediu with nd μ diffeent fo, the electognetic fields popgte with velocity v s: v c c n 3 3 Boundy Conditions t n Intefce At the intefce between two edi, the following boundy nditions e stisfied s [3,5]: n B n B D D n n ee, n is the unit vecto pependicul to the plne sufce nd indices, efe to the fist nd send ediu The qs 3 35 follow diectly fo Mxwell s equtions They ply key ole in ou nlysis of popgtion of electognetic wves though vious stuctues Fo qs 34 nd 35, it follows tht the tngentil ponents of both nd e ntinuous t the intefce s [5]: 6

23 t t nd t t 36 Siilly, by using equtions 3 nd 33 with q7 nd q 8, the nol ponents of the electic displceent nd the gnetic inductnce e ntinuous t the intefce s in [5]: D n Dn nd Bn Bn 37 The ntinuity nditions, given by q 36 nd 37, e vey ipotnt in the nlysis of the popgtion of electognetic wves 4 Poyntings Vecto Poynting s theoe sttes tht the tie te of flow of the electognetic enegy pe unit e is given by the vecto S, led the Poynting vecto which is defined s the ss poduct of the electic nd gnetic fields s [4]: S 38 Since S hs the physi ening of enegy flow, it ust be el Theefoe, in q 38, only the el pts of ll thee vectos,, nd S should be nsideed: S Re 39 ee, we e inteested only in the popgtion of electognetic wves whee both the electic nd the gnetic fields oscillte s e i t Theefoe, it is oe nvenient to vege the Poynting vecto S ove one peiod in q of the oscilltion of electic nd gnetic fields Fo the plne wve in q 9 nd q, we get: i k t i k t i t k t Re e e * e 4 i k t i k t i t k t Re e e * e 4 Afte substitution of these two expessions into the RS of q 39, we get: 7

24 i k t i k t i k t e * e e * e 4 S S * * 4 S Re * i k t whee * is the plex nugte of electic nd gnetic wve S cn be expessed in tes of the plex fields nd in the fo of q 44 By using q nd q into q 44, we cn find the new two eltions fo poynting vecto S s k kˆ 45 S Also tht k kˆ 46 S 3 Polition Polition of light hs nueous pplictions, ny of these pplictions cn esily be seen though poliing fil, which selectively psses light with pticul diection of the vecto, nd fo ny given diection of popgtion thee e two independent polition vectos, which cn be in ny two utully othogonl diections nol to k The fist cse is tht in which the vecto of the electic field, of the incident wve is pllel to the boundy plne This cse is led tnsvese electic o T polition we will use lso the ne s-wve The send cse is tht in which the vecto of the gnetic field, of the incident wve is pllel to the boundy plne This is led tnsvese gnetic o TM polition This type lso l the TM polied wve the p-wve - S- polied light By definition, in the cse of the T polition the electic intensity is pllel to the intefce, thee is no electic field in the diection of popgtion In the nottion tht vecto hs only one ponent, 8

25 , y, 47 nd vecto hs two ponents only, x,, 48 - P- Polied light Fo the TM polition, the gnetic field vecto pllel to the intefce nd, thee is no gnetic field in the diection of popgtion theefoe, it hs only one ponent, y, 49 The electic field vecto hs two ponents s x,, 5 4 Opti Fibe 4 istoy nd Definition of Opti Fibe One of the ost ipotnt types of wveguides e opti fibes An opti fibe is bsily cylindi dielectic wveguide with cicul ss section whee highindex wve guiding e is suounded by low-index dding Opti fibes e usully de of silic SiO glss The index step nd pofile e ntolled by the ncenttion nd distibution of dopnts Light popgtes in n opti fibe by epeted totl intenl eflection t the e dding glss boundy Opti fibes hve phenoenlly lge cpcity to cy infotion nd e ble to delive this infotion to exteely distnt loctions They e theefoe suitble fo opti unictions nd ost lse pplictions in this nge of the spectu Opti fibes de of othe teils e lso developed fo specil pplictions In 87 ohn Tyndll [] deonstted tht light cn tvel within cuved et ste of wte fo hole de on the side of wte pil Popgtion ws bsed on epeted totl intenl eflection t the i wte boundy oweve In 936 Cson et l [7] hve shown tht cicul dielectic wveguide cn suppot hybid doinnt ode with no cutoff fequency, it ws lost fo 3 yes until 966, when o nd ockh [] pooted the use of glss fibe s the tnsission ediu, tht the doos wee opened fo using the pinciple of totl intenl eflection s vible ens of uniction In 97 the whole field of opti uniction links though 9

26 opti fibes ws wkened by the successful developent of low-loss fibes with losses less thn db k [7] Fibe-optic uniction systes possess such dvntges s: - Low tnsission loss - Lge cpcity of infotion tnsission 3- No electognetic intefeence 4 -Lighte weight thn ppe 5- No spks even when shot-cicuited 6- ighe elting point thn ppe 7- Pctily inexhustible w teil supply On the othe hnd, the disdvntges e: - Connections nd tps e oe difficult to ke thn fo ppe wie - Fibe is not s flexible s ppe wie 4 Types of Opti Fibes Multiode Step-Index Fibe In this type the e diete is 5, nd the efctive index diffeence is equl to 5 % The nolied dius V is 3, nd the nube of odes is on the ode of hundeds The dvntge of this fibe is the ese of upling to souce o nnecting to nothe fibe becuse of the lge e diete nd lge nuei petue NA vlue But this fibe hs liited cpcity fo infotion tnsission due to ode dispesion nd is used piily fo shot-distnce uniction The cpcity bndwidth length poduct is bout 65 Mb/sk with NA = -Single-Mode Fibe The e diete is educed to 8, nd efctive index diffeence 3 5, so tht the nolied dius V of the fibe is slle thn the cutoff V = 4 of the ode tht is the next highe ode fo the doinnt ode Coupling is difficult in the single ode fibe, but the infotion tnsission cpcity is significntly lge

27 3- Dispesion-Shifted Fibe The dispesion-shifted fibe is not only noinlly fee fo dispesion, but lso the wvelength of opetion is t 55, The dispesion of the fibe is eoved by choosing efctive index distibution such tht wveguide dispesion cncels teil dispesion t wvelength 55 Tnsission loss is bout db/k nd the pcti liit on the cpcity of infotion tnsission is bout Tb/sk The efctive index distibution is eithe one lge pek of odultion in the dding lye 6 9, o ncentic step 4-Silic Coe Fluoine-Added Cldding Fibe The dding glss is nolly pue SiO Geniu dioxide GeO is usully used to ise the efctive index of the e glss with espect to tht of the dding glss The inusion of GeO is n dditionl inhoogeneity in the e nd incese the tnsission loss of the fibe 5-Plstic Fibe This fibe is de out of low-tnsission-loss plstic teil Being de of plstic, the diete of the e cn be s lge s nd lge NA vlue of 5 It is piily used fo shot-distnce uniction Plstic fibe whose e is de of poly ethyl eth cylte PMMA nd whose dding is de of fluointed lkyl eth cylte polye is useful in the egion of wve length 6 8 iniu tnsission loss is db/k t 68 nd the 6- oley Opti Fibe F oley opti fibe F is single teil fibe with peiodic y of cicul o ellipti i holes unning in the xil diection of the opti fibe The effective efctive index diffeence cn be chieved fo single teil This type of fibe is single ode ove n exceptionlly wide wve length nge fo 458 to 55 n, s deteined by esuing the nuei petue NA It ws found tht the vlue of NA lost linely inceses with wvelength NA = 3 t =458 n, nd NA = 36 t =, 55 n

28 43 Totl Intenl Reflection TIR At the het of n opti uniction syste is the opti fibe which cts s the tnsission chnnel cying the light be loded with infotion As entioned elie, the guidnce of the light be though the opti fibe tkes plce becuse of the phenoenon of totl intenl eflection TIR e fist define the efctive index n of ediu in q 3 As you know, when y of light is incident t the intefce of two edi like i nd glss, the y undegoes ptil eflection nd ptil efction s shown in Figue,b,c stisfying the ndition: nsin nsin 5 n n n n n n, c b n n, c n n c n n, c TIR Figue : Light eflection t diffeent cses with c nd c n n, c TIR n n, c nd b n n,

29 The veti dotted line epesents the nol to the sufce The ngles is the incident ngle, is the efction ngle nd, epesents the eflection ngle Futhe, the incident y, eflected y, nd efcted y lie in the se plne In Figue, since n n we ust hve fo Snell s lw, ie, the y will bend towd the nol On the othe hnd, if y is incident t the intefce of e ediu n n, the y will bend wy fo the nol see Figue b The ngle of incidence, fo which the ngle of efction is 9 o, is known s the citi ngle c nd is denoted by [6]: n c sin 5 n o 9 hen the ngle of incidence exceeds the citi ngle ie, when c, thee is no efcted y nd we hve totl intenl eflection s in Figue c 44 Stuctue of Opti Fibe Light popgtes in n opti fibe by epeted totl intenl eflection t the e dding cylindi glss boundy An opti fibe nsists of e nd dding cylindi nd is xilly syetic Since the efctive index of the e is slightly highe thn tht of the dding, the opti field is lgely nfined to the e The esponding efctive index distibution in the tnsvese diection is given by: n n fo n n fo 53 whee n epesent the efctive index of e nd n efctive index of dding nd epesents the dius of the e The eltive index diffeence o eltive pofile height, though the following equtions s in [7]: n n 54 n 3

30 hen s is indeed tue fo silic fibes whee n is vey nely equl to n we y wite s in [7]: n n n n n n n n 55 n n n Now, fo y enteing the fibe e t its end, if the ngle of incidence t the intenl e-dding intefce is gete thn the citi ngle c, the y will undego TIR t tht intefce Futhe, becuse of the cylindi syety in the fibe stuctue, this y will suffe TIR t the intefce lso nd theefoe be guided though the e by epeted totl intenl eflections ven fo bent fibe, light guidnce cn occu though ultiple totl intenl eflections Fo tnsission of light fo one plce to nothe, the fibe ust be suppoted Suppoting stuctues, howeve, y nsidebly distot the fibe, thee by ffecting the guidnce of the light wve This is voided by choosing sufficiently thick dding Futhe, in fibe bundle, in the bsence of the dding, light cn lek though fo one fibe to nothe Cld Coe Cld Figue 3: Fibe nsists of cylindi centl e d by teil 4

31 Ai A Clding n B Coe n Clding n C Ai Figue 4: Light ys ipinging on the e-dding intefce t n ngle gete thn the citi ngle e tpped inside the e of the fibe 5 Left nded Mteils LM 5 Intoduction to LM Mteils Veselgo [8] ws the fist who theoetily pedicted such edi e eveled tht teil with siultneously negtive dielectic peittivity nd gnetic peebility is equivlent to negtive efctive- index ediu, nd investigted the electognetic wve popgtion though such edi Mteil cn be divided cding to the sign of gnetic peebility nd electic peittivity into ny goups s in figue 5, the fist type hve two positive sign vlues of gnetic peebility μ nd electic peittivity be ned ight hnded teil RM The send type hs positive sign of gnetic peebility nd negtive sign of peittivity led etl The thid type hs negtive sign vlues of gnetic peebility nd positive sign peittivity be led feognetic teil, nd the lst type teil hs two negtive sign vlues of gnetic peebility nd peittivity be led left hnded teil LM o etteil Also we know tht the efctive index depends of the gnetic peebility nd peittivity whee n s [9] Metteils MTMs e bodly defined s tificil effectively hoogeneous electognetic stuctues with unusul popeties not edily vilble in ntue An effectively hoogeneous stuctue is stuctue whose stuctul vege cell sie p is uch slle thn the guided wvelength 5

32 g Theefoe, this vege cell sie should be t lest slle thn qute of wve length, p g / 4 e will efe to the ndition p g / 4s the effective hoogeneity liit o effective-hoogeneity ndition, to ensue tht efctive phenoen will dointe ove sctteing/diffction phenoen when wve popgtes inside the MTM ediu If the ndition of effective-hoogeneity is stisfied, the stuctue behves s el teil in the sense tht electognetic wves e essentilly yopic to the lttice nd only pobe the vege, o effective, spic nd well-defined nstitutive petes, which depend on the ntue of the unit cell, the stuctue is thus electognetily unifo long the diection of popgtion The nstitutive petes e the peittivity nd the peebility, which e elted to the efctive index n by n, whee nd e the eltive peittivity nd peebility elted to the fee spce peittivity nd 7 peebility by / 8854 nd / 4, espectively In the sign ± fo the double vlued sque oot function hs been pioity ditted fo genelityl teils, s nsequence of thei double negtive petes, e chcteied by ntipllel phse nd goup velocities, o negtive efctive index NRI L stuctues e ely MTMs, cding to the definition given bove, since they e tificil, effectively hoogeneous p g / 4, nd exhibit highly unusul popeties, It should be noted tht, lthough the te MTM hs been used ost often in efeence to L stuctues in the litetue, oweve, L stuctues hve been by f the ost popul of the MTMs, due to thei exceptionl popety of negtive efctive index 6

33 μ, II n I Plss pe Metls t opti fequencies evnescent wve, n R, n Isotopic dielectics Right-hnded RM fowd-wve popgtion I, III n R, n Veselgo s teils Left-hnded LM bckwd-wve popgtion, IV n I feites p feignetic teils evnescent wve Figue 5: Peittivity-peebility ε μ nd efctive index n dig fo ll teils 5 lectognetic Popeties of Left-nded Mteils A-ve qution The wve eqution fo n electognetic wve in ediu with ε nd μ is given by []: k 56 c fo which it follows tht the plne wve in q 9 cn popgte in the ediu if the poduct εμ is positive This is possible if eithe both ε nd μ e positive, o both petes e negtive Thus we nude tht popgtion of electognetic wves is possible in left-hnded ediu 7

34 B- Left-nded Rule The electic field vecto, gnetic field vecto nd wve vecto k follow the ight hnded tiplet of vectos, nd the Poynting vecto S pllel to k oweve, if both ε nd μ e negtive, we obtin fo Mxwell s equtions tht [9]: k 57 k 58 It is e tht the vectos,, nd k follow now the left-hnded tiplet of vectos, nd vectos k nd S hve opposite oienttion s shown in figue 6 x S k k S y RM LM Figue 6: Oienttion of the electic nd gnetic vectos of the intensity of electognetic fields nd with espect to the wve vecto k nd Poynting vecto S fo LM nd RM Note tht the diection of the popgtion of the electognetic wve is deteined by the oienttion of the Poynting vecto S, nd not by the oienttion of the wve vecto k The opposite sign of k only ens tht the phse velocity of the wve inside the left hnded ediu is negtive 8

35 C- Dispesion The enegy of the electognetic field y be witten [] s: 8 59 e ieditely see tht both peittivity ε nd peebility ust depend on the fequency Othewise, the enegy educes to the eltion / 8 which would be negtive fo both ε nd e negtive On the othe hnd, the enegy is lwys positive this en tht eqution stisfy s [9]: nd 6 The fequency dependence of ε nd ens, due to the es-onig eltions, tht both the peittivity nd peebility ust be plex The noneo iginy pts of the peittivity nd peebility led unvoidbly to bsoption losses of the electognetic wve popgting though the left-hnded teil The dielectic peittivity nd gnetic peebility in soe teil of LM bee siultneously negtive in the fequency nge fo 4 to 6 G 9

36 CAPTR TO GIDD MODS IN NGATIV RFRACTIV INDX FIBRS In this chpte we need to nlye the pevious studies epesented by AV Novitsky nd LM Bkovsy in [5], which inuding electognetic wve popgte long the fibe nsist of LM e nd dielectic teil in the d, solution of wve eqution nd Mxwells equtions to the opti fibe in cylindi odinte e obtined in sections nd 3, the ponent of electic nd gnetic fields with, e poved in section 4, solution of electic nd gnetic fields t - diection in the e nd d egions e descibed in section 5, the expessions vlue to the electic nd gnetic fields ponents in tes of Bessel function fo the e nd d egions e suied in section 6, dispesion equtions fo the fst nd slow ode e defined in section 7, dispesion eltion fo T, TM,, nd odes in the fibe e inuded in section 8 Finlly the electognetic popeties in the fibe e studied whee the dispesion cuves fo ll odes e plotted in section 9 Stuctue Anlysis AV Novitsky nd LM Bkovsy [] hve studied the popeties of electognetic wve popgting in opti fibe with dius t the - diection The nsideed opti fibe nsist of LM in the e in the egion which chcteied by n electic eltive peittivity nd gnetic eltive peebility s []: p F whee p is the pls fequency in the LM, is the esonnt fequency nd F is the fctionl e of the unit cell The d egion is filled with dielectic teil which hve n electic eltive peittivity nd gnetic eltive peebility given by []:

37 dielectic Cld Coe LM Figue : Stuctue of the opti fibe with LM e nd dielectic d Solution of ve qutions in the Cylindi Coodinte Conside n opti fibe cylindi odinte with e dius nd efctive index e is n bounded by dding lye with dius b, the ssuption of b is de The d efctive index is n s shown in Figue Coe Cld P,, R Z y x Figue : Opti fibe in cylindi odinte syste,, The electic filed nd gnetic filed in cylindi odintes e witten s[]: ˆ θˆ ˆ 3 ˆ θˆ ˆ 4

38 To solve the wve eqution in q 6, in the cylindi odintes ust use the identity s in [3]: θ A ˆ ˆ ˆ A A A A A A A 5 sing q 5 nd substituting into q 6 to find the ponents of the electic filed in the cylindi odintes to get: ˆ ˆ ˆ θ nk nk nk nk 6 Siilly to solve the wve eqution in the gnte field q 3, using the se wy to find the ponents of the gnetic filed in the cylindi odintes, whee tht the vlue of in the cylindi odinte given by []: 7 Note tht in q 6, the ˆ ponent ntins both nd likewise with the θˆ ponent, but the ẑ ponent ntins only Becuse of this fct, the ponent is fist solved nd then the othe ponents nd will be obtined diectly fo Mxwell s eqution s we wnt to explin lte in this chpte 3 Solution of Mxwells qutions in Cylindi Coodinte Conside tht n electognetic wve popgtinglong the -diection s in []: e t, nd e t, 8 whee is the popgtion nstnt sing q 8 nd q 8 then substituting into Mxwell eqution, q bee

39 9 q 9 cn be ewitten s in {3]: ˆ θˆ ˆ The expnded fo q gives tht 3 Fo q 8 we find tht: 4 sing q 4 nd substituting into qs nd 3 we get: 5 6 qs 5, q 6 nd q 3 e the solutions of Mxwell eqution give q in cylindi odintes Siilly use the se wy to solve q with no cuents =, using q 8 nd q 7 then substituting into Mxwell eqution, q s in {3]: 7 3

40 4 θ ˆ ˆ ˆ 8 The expnded fo q 8 gives 9 sing q 4 nd substituting into qs 9 nd in ode to get: 3 qs, q 3 nd q e the solution of Mxwell eqution give q 5 in cylindi odintes 4 xpessions Fo,,, Coponents Afte solving Mxwell equtions it is esy to find the ponents of electic nd gnetic fields,, nd sing q 6 nd substituting into q, getting: 4 q 4 epesents the vlue of sing q 5 nd substituting into q 3 we get:

41 5 q 5 epesents the vlue of sing q nd substituting into q 6 we get: 6 q 6 epesents the vlue of se q 3 nd substituting into q 5 getting tht: 7 q 7 epesents the vlue of 5 xpessions of nd Coponents qution 6 being vecto eqution, ech ponent hs to vnish individully in ode fo the vecto su of the ponents to be eo sing q 7, the ponent of q 6 is expessed s in [,3]: nk 8 nk 9 The diffeentil eqution, q 9, is solved by the ethod of seption of vibles s poduct of thee functions, ech,, letting tht s []: F Z 3 5

42 Inseting q 3 into q 9 then use the seption of vibles ethod we get:: Z Z 3 3 n k F F F F 33 5 Solutions In The Coe Region Fist, the nsideed solutions of q 3 inside the e egion is given in []: Z e be 34 lso, the solution of the q 3 is given by []: c s + d sin 35 lso, the solution of the q 33 in tes of Bessel function given by []: F e f N 36 whee, b, c, d, e nd f is nstnt, is the ode of Bessel function, is Bessel function of the fist kind, N is Bessel function of the send kind with n n Since the vition the thn exponentil in is inside the e, positive vlue s [,]: n k 37 whee is the wve nube in the e qution 3 nsists of fowd nd bckwd wves Fo the ost pt, fibes e ecipo Thee do exist soe nonecipo effects tht e eployed in cetin fibe sensos, but these e unusul Recipocity ens tht, upling nditions being 6

43 equl, it does not tte which end of the fibe is nnected to the souce It is sufficient to choose one te fo the ight-hnd side of q 65 in this cse Z e 38 qution 33 epesents two skew ys tht e otting in opposite senses Mode pttens e stnding wves geneted by oppositely winding skew ys nd both tes in q 66 e needed The loction of nd e the se Fo q 33 with th ode Bessel function The cuves of N x e shown in Figue 3, bees negtive infinity t nd physily cnnot be ccepted, nd f in q 33 hs to be eo nd the solution of q 33 bee N F e 39 Figue 3: Zeo- nd fist-ode Bessel functions of the send kind with, In the e egion, nd e given s in []: s e A 4 sin e C 4 7

44 5 Solutions In The Cld Region The type of solution in the dding egion is one whee the vlue of the function pidly deceses with n incese in As entioned elie, the solution of q 33 bees decying function if the vlue of the sque bcket is negtive with, whee is the wve nube in the d s in [,]: n n n k 4 The solution fo F is then F e I f 43 whee the functions I e led the odified Bessel functions of the fist kind nd e odified Bessel functions of the send kind of the ode The vlue of I x inceses with n incese in x but tht of x deceses with n incese in x, s shown in Figue 4 Since I x bees indefinitely lge with n incese in, it is eected on physi gounds, nd in q 43 should be eo Figue 4: Modified Bessel functions of fist I x nd the send x, kinds with 8

45 Then the expessions fo nd in the dding egion, e s e B 44 sin e D 45 6 xpessions fo nd Coponents in Tes of Bessel Function Fo the boundy ndition tht the tngentil filed ponents nd should be ntinues t the e - dding intefce, we obtin the two flowing expition nditions to find the nd ponents in te of Bessel function Then we hve defined two pete nd, whee is the nolied tnsvese wve nube in the e is given s in []: n k 46 Also, is the wve nube in the d is given by nk 47 whee is the nolied tnsvese wve nube in the d Fo q 46, we find / nd fo q 47 we find / then using the fist boundy ndition t to get: B A 48 whee the function descibes the longitudinl ponent of the electic nd gnetic filed in the e, nd the function descibes the longitudinl ponent of the electic nd gnetic filed in the d sing the send boundy ndition t tht s in [,] we get: 9

46 3 C D 49 sing q 48 nd q 49 whee e ws suppessed, then qs 4, 4, 44 nd 45 cn be ewitten to find nd in the e nd de Fist the electic filed in the e nd the d bee s in [4]: A A s s 5 Also, the gnetic filed in the e nd the d beess in [4]: C C sin sin 5 6 xpessions fo,,, In The Coe Region By using the qs 46, 5 nd q 5 fo nd substituting into qs 4, 5, 6 nd q 7, we hve: s C A 5 sin C A 53 sin C A 54

47 3 s C A 55 6 xpessions fo the,,, in the d egion Siilly using qs 47, 5 nd q 5 fo nd Substituting into qs 4, 5, 6 nd q 7 to get: s C A 56 sin θ Cω μ Aβ θ 57 sin C A 58 s C A 59 7 Dispesion qution of n Opti Fibe Boundy nditions e pplied t the intefse The tngentil ponents of nd hve to be ntinuous t The tngentil ponents of e nd nd those of e nd At the fist stting with the initil boundy ndition s[,]: t 6 sing q 53 nd q 57 nd substituting into q 6 we get: C A 6

48 3 sing the send boundy ndition t s in [,], we get: t 6 sing q 55 nd q 59 then substituting into q 6 we get: A C 63 sing the eltions s in [] sing q 64 nd q 65, nd ultiplying both q 6 nd q 63 to ech othe,we obtin tht: 66 e get the eltions s in [6]: c k nd c 67 sing q 67 then substituting into q 66 we get: k 68 The LS of q 68 cn be ewitten s: k k k 69

49 33 Fo q 46 nd q 47 we get: n k 7 n k 7 sing q 7 nd q 6 then substituting into q 69, getting tht: n n k 7 sing q 63 then q 68 ewitten s n n 73 Dividing both side of q 73 by nd using the eltion n we get: n n 74 Tking the LS of q 74 to be siplify S L 75 Renging q 75 we get: S L 76

50 Now tking the RS of q 74 to be siplify we get: R S 4 4 n n n n 4 4 n 77 Now tking the nueto fo the q 77 RS to siplify, using the q 46 nd q 47 the fist te bees: n k n k n n n k n n Fo q 77 the send te of the RS witen s: n k n k n n n k n n By dding the two q 78 nd q 79 to get: n n k n n k n n n n k n n n n Fo q 77 the thid te of the RS bees: n k n 4 4 n n k nn 8 Fo q 77, the lst te of the RS bees: n k nn n k nn 8 By dding the two q 8 nd q 8, we get: n n n n n n k n n n n k n n Then dding q 8 nd q 83 to get: n n n k 84 n sing q 84 nd substituting into q 77, now RS ed s: 34

51 35 k S R sing q 76 nd q 85 nd substituting into q 74 to get: k q 5 led the dispesion eltion which stisfy the ndition this en tht the poduct of in fst ode But when the poduct then it hs n iginy vlue, so leding in this cse i to be Substituting i into q 86 nd using the eltions s [,5]: ix I i x 87 I i i I i 88 I i i I i 89 sing qs 87 q 89 then substituting into q 86 we get: i k I i I i i I i I i i Afte soe cultion, q 9 bees: k I I I I q9 descibed the dispesion eltion to the sufce ode fo the fibein slow ode with

52 8 Modes in Opti Fibes q 86 is bintion of Bessel functions The solutions of the chcteistic eqution depend of the ode, whethe o ence, they will be teted in two septe subsections 8 Meidionl Modes The chcteistic eqution is significntly siplified with the ode of the Bessel function educes, nd the ight-hnd side of q 86 bees eo so tht q 86 educes to the two odes T nd TM ode se the eltions fo the Bessel functions s [3,5]: x 9 x x 93 x Fo q 9 nd q 93 then the LS of q 9 bees s [,3]: T ode 94 TM ode 95 8 Skew Modes: hen, the ight-hnd side of q 86 is not eo nd the pocedue fo obtining the solution of the chcteistic eqution is uch oe plicted This ens tht the solution of the q 86 one of the hve + sign nd nothe hve - sign to give two hybid ode nd odes so thee is two solution of 86 nd letting tht: q 36

53 37 96 L 97 By using qs96 nd 97, q 86 bee: k L L Multiply both side with 4 to find L k L Adding to the both side of q 99 L find L L k L L Afte soe thte cultion q ewitten s:

54 k L 4 4 L 4 4 Tking the sque oot to the booth side in the eqution, we get: L 4 k 4 L 4 4 q hve two solutions ned hybid ode, one + sign ode nd the sign ode s in [,4]: L 4 k 4 L 4 4 ode 3 L 4 k 4 L 4 4 ode 4 9 Nuei Results In the nlysis below, Novitsky nd Bkovsy [] nsideed the popgtion of electognetic wve long cicul fibe with e LM hve dius bounded by dielectic teil in the d s shown in figue The electic peittivity nd gnetic peebility in the e e given by q nd q The d hs dielectic teil with electic peittivity nd gnetic peebility Novitsky nd Bkovsy in [] studied the fst nd slow guided odes in the fibe 38

55 They lso studied the ode popeties dependence on the fibe petes The fibe pete in [], pls fequency p / G, esonnt fequency / 4 G, fctionl e unit cell F 56, e dius c, dielectic peittivity 5 nd gnetic peebility In Figue 5, thee e the thee egions of ode existence in dispesive fibe In egion I the nditions, hold tue nd fst guided odes ise, the fst odes espond to two types of dispesion cuves One of theis in egion I pletely cuve AB, nd the othe lies in both egions I nd II cuve CF Point D divides the fst ode fo the slow one nd stisfies the ndition Theefoe, the dispesion cuve segent CD descibes fst ode nd DF slow one Fo figue 6, The dispesion cuve of AB-type cnnot be fundentl guided ode nd t the se tie the ode politions e nged in the following wy: t the T ode ltentes with the TM one, stting with T; t the hybid ode ltentes with the one, stting with Fo Figue 7 The high-ode fst odes exist t lowe fequencies ose to the esonnt one oweve, fst fundentl guided odes in cicul left-hnded fibe e possible They ppe t nd espond to the dispesion cuves of CD-type Thee e two fst fundentl odes, the fist is T nd the send is TM polied Region II in figue 5 is chcteied by, nd inudes only slow odes Dispesion cuve DF t esponds to the polied wve, the T ode deteines the syptotic of lowfequency cuves of DF-type t the dispesion cuve intesects the T-ode cuve, fo dispesion cuves ppe t highe fequencies thn the T-ode cuve, TM ode deteines the syptotic of high-ode cuves of PLtype ll these dispesion cuves e highe thn the TM ode The dispesion cuve PL lwys esponds to slow ode It cn lie in egions II nd III in figue 5, The cuve G in egion II is chcteied by polition The dispesion cuve -PL in egion III is polied 39

56 fst ode Slow ode Figue 5 : Dispesion cuves descibing the fst odes AB cuve, fst- slow ode CF cuve nd slow ode PL cuve Fibe petes: dding 5, with n, nd fibe dius c s [] = = Figue 6: Fst guided odes with cuve AB- type t, Fibe petes : 5,, nd fibe dius c s [] 4

57 = = Figue 7: The fst-slow nd slow low-ode guided odes, Fibe petes : 5,, nd fibe dius c s [] 4

58 CAPTR TR MODS IN OPTCAL FIBRS IT SMICNDCTOR CLADDNG AND LM COR In this chpte, the wve popgtion chcteises in opti fibe nsisting of LM e bonded by seinducto e investigted theetily The bsic ncepts nd equtions, opti nd electognetic popeties fo the seinducto e pesented in section 3, Solution of Mxwells equtions in the seinducto d will be deived in section 34 The electognetic fields ponents in the seinducto d e obtined in section 35 The dispesion equtions fo T, TM, nd odes e poved in section 36 Finlly, the totl powe in cses of T nd TM ode is explined in section 36 3 Intoduction to Seinductos Optoelectonics bings togethe optics nd electonics within single device Metls e excellent electi nductos, but do not llow light to tvel inside Glss nd elted dielectic teils cn codte nd guided light wves, like in opti fibes, but they e electi insultos Seinductos e in between these two teil types, s they cn cy electi cuent s well s light wves Seinductos cn be designed to llow fo the tnsfotion of light into cuent nd vice ves The nduction of electi cuent is bsed on the flow of electons Most electons e ttched to single tos nd e not ble to ove feely Only soe loosely bound electons e elesed nd bee nduction electons The se nube of positively chged tos ions is left behind; the net chge is eo The positive chges cn lso ove, s vlence electons up fo to to to Thus, both vlence electons holes nd nduction electons e ble to cy electi cuent Both the cies e septed by n enegy gp; ie, vlence electons need to eceive t lest the gp enegy g to bee nduction electons In seinductos, the gp enegy is on the ode of ev The enegy cn be povided, eg, by light hving wvelength of less thn the gp wvelength In the wve pictue, light is epesented by peiodic electognetic fields with the wvelength In the ptie 4

59 pictue, light is epesented by te of enegy pckets photons with the enegy is ph in [6] hc ph h 3 whee, h/ with h is Plnck nstnt The photon enegy ust be t lest s lge s the bnd gp g to genete electonhole pi Vice ves, nduction electons cn lso elese enegy in the fo of light nd bee vlence electons This enegy exchnge between electons nd photons is the key physi echnis in optoelectonic devices Fo n toic point of view, vlence electons belong to the outeost electon shell of the to, which is fully occupied in the cse of seinductos; ie, no oe electons with the se enegy e llowed As these tos e oined togethe in seinducto cystl, the electons stt to intect nd the vlence enegy levels septe slightly, foing vlence enegy bnd lectons within this bnd cn exchnge plces but no chge flow is possible unless thee is hole To genete holes, soe electons ust be excited into the next highe enegy bnd, the nduction bnd, which is initilly epty The ncenttion N of electons in the nduction bnd nd the ncenttion P of holes in the vlence bnd ntol the electi nductivity of seinductos s [6]: qn qp 3 n P with q is the eleenty chge nd the obility n nd p e the obility of the hole nd electons especctively 3 Opti Popeties of Seinducto 3 lectognetic negy The flow of electognetic enegy is given by the Pointing vecto s in q 4 whose tie vege gives the intensity c of the opti wve I opt s [6]: I opt 33 4 opt 43

60 The photon flux density ph is given s [7]: ph I opt / 34 The tie-veged lo opti enegy density opt is given s [7]: opt opt 35 3 Mxwells qutions Mxwells equtions e the fundentl bsis fo the ssi tetent of electi nd gnetic fields Fo high-fequency opti wves nd seinducto wveguides, sevel siplifictions e possible s discussed in the following The chge density cts s souce of the electosttic field vecto F nd it cn be neglected in the cse of high-fequency electic fields vecto Mgnetic field vitions in tie genete culs tie / t of the electi field Vice ves, electi field vitions in / t s well s genete the cuent culs of the gnetic field The stedy-stte o low-fequency cuent density F cn be exuded hee nductivity t low fequencies The eining "opti" cuent density opt depends on the seinducto nductivity opt t high fequencies This ens tht t high fequencies we get [7]: F opt And fo non-gtic seinducto sing q 7 nd q 8 nd subsisting into q to q 4 fo = nd = we get s [8]: i 38 i 39 opt opt opt

61 33 Stuctue Anlysis e nside n electognetic wve popgting in the opti fibe long - diection The stuctue nsisting of LM in the e with dius which is chcteied by n electic eltive peittivity s in q nd gnetic eltive peebility s in q bounded by seinducto teil in the d tht is chcteied by n electic eltive peittivity nd gnetic eltive peebility The seinducto peittivity tenso cn be witten s [8,9]: xx x x yy 3 whee xx,, yy, x nd x cn be witten s{9,]: p i xx 33 i c x x pc i 34 i c yy p i 35 whee p the pls fequency, c is the cyoton fequency, is the llection fequency nd the gnetic eltive peebility tenso ii This stuctue is shown in figue 3 45

62 46 Cld seinducto Coe LM Figue 3: Schetic fo n opti fibe with LM e lye nd seinducto d 34 Solutions of Mxwells qutions in Cld Region Seinducto Mxwell q 5 cn be solved with tenso electic eltive peittivity by using q 3 nd substituting into q 7, we get: yy x x xx ˆ ˆ ˆ θ 36 Tking the expntion of q 36, we get: x xx 37 yy x seing q 4 nd substituting into qs 37 38, we get: x xx 3 yy x 3

63 47 qs 39 3 e the solution of Mxwell eqution q 5 in cylindi odintes fo seinducto 35 xpessions fo,,, in the Seinducto Cld Afte solving the Mxwells eqution fo seinducto in the d egion s show in the pevious section we cn find xpessions fo,,, stting with sing q 3 fo nd substituting into q 3, the electic filed cn be witten s: x xx 3 q 3 epesents the vlue of sing q 5 fo nd substituting into q 3, we get: yy yy 33 q 33 epesents the vlue of sing q 6 fo nd substituting into q 3, we get: x xx xx 34 q 34 epesents the vlue of sing q 5 fo nd substituting into q 3 we get: x yy yy 35