Analytical Approach to Polarization Mode Dispersion in Linearly Spun Fiber With Birefringence

Transcription

1 lyicl roch o Polrizio Mod Dirio i Lirly Su Fir Wih Birfrigc y Viod K. Mihr RL-TR-6869 Mrch rovd for ulic rl; diriuio i uliid.

2 NOTICES Diclir Th fidig i hi ror r o o corud officil Dr of h ry oiio ul o digd y ohr uhorizd docu. Ciio of ufcurr or rd do o coiu officil dor or rovl of h u hrof. Droy hi ror wh i i o logr dd. Do o rur i o h origior.

3 ry Rrch Lorory rd Provig Groud, MD RL-TR-6869 Mrch lyicl roch o Polrizio Mod Dirio i Lirly Su Fir Wih Birfrigc Viod K. Mihr Couiol d Iforio Scic Dircor, RL rovd for ulic rl; diriuio i uliid.

4 REPORT DOCUMENTTION PGE For rovd OMB No Pulic rorig urd for hi collcio of iforio i id o vrg hour r ro, icludig h i for rviwig irucio, rchig xiig d ourc, ghrig d iiig h d dd, d colig d rviwig h collcio iforio. Sd co rgrdig hi urd i or y ohr c of hi collcio of iforio, icludig uggio for rducig h urd, o Dr of Df, Whigo Hdurr Srvic, Dircor for Iforio Orio d Ror 7-88, 5 Jffro Dvi Highwy, Sui, rligo, V -. Rod hould wr h owihdig y ohr roviio of lw, o ro hll ujc o y ly for filig o coly wih collcio of iforio if i do o dily currly vlid OMB corol ur. PLESE DO NOT RETURN YOUR FORM TO THE BOVE DDRESS.. REPORT DTE DD-MM-YYYY Mrch. REPORT TYPE Fil. TITLE ND SUBTITLE lyicl roch o Polrizio Mod Dirio i Lirly Su Fir Wih Birfrigc. DTES COVERED Fro - To Jury Ocor 5. CONTRCT NUMBER 5. GRNT NUMBER 5c. PROGRM ELEMENT NUMBER 6. UTHORS Viod K. Mihr 5d. PROJECT NUMBER 5. TSK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGNIZTION NMES ND DDRESSES U.S. ry Rrch Lorory TTN: RDRL-CIH-N rd Provig Groud, MD PERFORMING ORGNIZTION REPORT NUMBER RL-TR SPONSORINGMONITORING GENCY NMES ND DDRESSES. SPONSORMONITOR S CRONYMS. SPONSORMONITOR'S REPORT NUMBERS. DISTRIBUTIONVILBILITY STTEMENT rovd for ulic rl; diriuio i uliid.. SUPPLEMENTRY NOTES. BSTRCT I lyicl Thory for Polrizio Mod Dirio of Su d Twid Fir, Wg l. rd o roxi lyicl rul for h olrizio od dirio for u fir. Thi ror dcri xio of h odl yod fir-ordr rurio d r o w lyicl d uricl rul. 5. SUBJECT TERMS olrizio od dirio, riodic i fucio, irfrigc, PMD corrcio fcor, PCF 6. SECURITY CLSSIFICTION OF:. REPORT Uclifid. BSTRCT Uclifid c. THIS PGE Uclifid 7. LIMITTION OF BSTRCT UU 8. NUMBER OF PGES 9. NME OF RESPONSIBLE PERSON Viod K. Mihr 9. TELEPHONE NUMBER Iclud r cod -78- Sdrd For 98 Rv. 898 Prcrid y NSI Sd. Z9.8 ii

5 Co Li of Figur Li of Tl ckowldg iv iv v. Iroducio. Thoricl lyi. Th Modl Wih Priodic Si Fucio.... Lir Sg roxiio o h Priodic Si Fucio: lyicl Soluio for h Jo liud Euio..... Th Modl..... Th Grl M-Sg Soluio..... Th Scific -Sg Soluio...5. Clculio of PMD Corrcio Fcor PCF...8. Nuricl Rul 9. Th Sll-Q Lii < L B...9. Th Lrg-Q Lii > L B.... Cocluio dix. Pruriv Clculio for Sg Diriuio Li 6 iii

6 Li of Figur Figur. Th hr-g roxiio o h riodic i fucio.... Figur. Th PCF curv for ruriv lii wih = d L B =. Th curv for xc d ruriv clculio r lo idicl.... Figur. Th PCF curv for ruriv lii wih = d L B = 5. Th curv for xc d ruriv clculio r lo idicl. No h fr = 5, h wo r divrgig lil.... Figur. Th PCF curv for oruriv lii wih = 5 d L B =. Th o d oo curv how xc d ruriv clculio, rcivly. No h ruriv roxiio udri h PCF i hi rgi.... Figur 5. Th PCF curv for oruriv lii wih = d L B =. Th o d oo curv how xc d ruriv clculio, rcivly. I i h ruriv roxiio udri h PCF i hi rgi.... Li of Tl Tl. For PCF v. z lo....9 Tl. For PCF v. z lo.... iv

7 ckowldg I hk Nick Frigo forrly T&T L d ow h U.S. Nvl cdy for gig ird i hi oic. v

8 INTENTIONLLY LEFT BLNK. vi

9 . Iroducio I, Wg l. drivd xrio for h Diffril Grou Dly DGD of rdoly irfrig fir i h Fixd Modulu Modl FMM. Th DGD h oh odulu d h. Th FMM u h h odulu of h irfrigc vcor i rdo vril. Wg l. rd lyicl rul wih h followig uio: Th i fucio i riodic. Th riodiciy lgh of h fir i uch llr h h fir corrlio lgh L F or << L F. Lr hy lo grlizd h FMM d rd h Rdo Modulu Modl RMM, which lo iclud h rdo i h dircio of h irfrigc vcor. Now h RMM uio could oly olvd uriclly. I h r work, h full ilicio of h FMM hv xlord udr h followig codiio: Th << L F roxiio h rlxd. ozro wi h icludd. co i r h ddd. W giv h lyicl oluio of h xc FMM uio d r o uricl rul howig h ffc of diffr hyicl codiio.. Thoricl lyi. Th Modl Wih Priodic Si Fucio Th rig oi i h wll-kow vcor uio dcriig h chg i h Jo locl lcric fild vcor, z wih h gulr frucy d dic z log wid fir. d z dz i d z dz i z i z z z Wg, M.; Li, T.; Ji, S. lyicl Thory for Polrizio Mod Dirio of Su d Twid Fir. Oic Exr, 9,.

10 Hr i h irfrigc d z i z i h riodic i rofil fucio wih i giud d gulr frucy of il odulio. Th oudry codiio r, d dz, d dz i L z diiol vril. W u d dz d d o rwri uio. i ici ici Th ucri do diffriio c r diiol co. d d, d d. lo, β η d W xr ll rr i r of h lgh giv lgh L B, i riod, d coulig lgh l. Th w c wri LB, c LB l. Th w oudry codiio r, 5, i 5 Th uio uio or, uivlly, uio do o hv lyicl oluio. I h ruriv roxiio dix, lyicl rul h drivd rlir. I h r work, w driv lyic oluio y roxiig h i fucio y lir g d cor h o h ruriv oluio for h g.. Lir Sg roxiio o h Priodic Si Fucio: lyicl Soluio for h Jo liud Euio.. Th Modl Th riod of h righ li g how i figur roxi h riodic i fucio. igl riod wih hr-g roxiio i how.

11 Θ π I π II π III π Figur. Th hr-g roxiio o h riodic i fucio. Th fild liud for giv g ify h followig uio. i i i 6 Th urcri d ucri oh idic h g for which h could uio hold. Th lii of g follow. W ruir h h doi of hould h h of h i fucio i rofil ci c~ c o h h d-oi codiio for for ll g. Dfi i g hold. For =, Sg I:, c ~ i, i c For =, Sg II:, c ~ c i c, i c

12 For =, Sg III:, 5 5, ~ c c c c i i.. Th Grl M-Sg Soluio Th oluio for h h g hv h followig grl for: i i i i i i i i co 7 wih d d, Th xc oluio for h could uio i o g r rld o ho i h rviou djc g y h followig chi-rlio og h coffici. Dfi,, v u d h h chi-rlio r giv y v u 8 Hr, h rix l r. i i, co i, i co, co co Th rix chi-rlio c wri cocly y xrig h ric our roduc dod y h yol of wo ric

13 5 v u 9.. Th Scific -Sg Soluio Th dil ou oluio for -g follow. Sg I: Th uio r i i i Th oudry codiio r, i, L ~ L B l c h h lyicl oluio r iilr o ho giv i cio. i i ic ic i co ~ ~ Corio wih grl xrio giv h followig coffici:,,, For clculig Polrizio Mod Dirio PMD Corrcio Fcor PCF, h liud hv o diffrid wih rc o ω, which will dod y ucri ω. So uful rlio dd for hi r d d d d d d,,,, 5

14 Th w c wri c ~ ic ~ 5 i i 6 7 i i 8 co i 6 whr,,, 7 5, 6, 7, So irig rlio r foud 8 7 Sg II: Th uio r, z, 8 i i i 9 Th oudry codiio r Siilr xrio xi for. Uig h chi-rlio wih =, h lyicl oluio r oid. i c ~ c i c ~ c Th ω -diffrid liud r foud co i i i i co co co i i i whr i c ~ c i c ~ c i i co i i i co i i i co co i co co c co i i d 6

15 co i i 5 6 f i co co g 7 8 h Sg III: Th uio r i i i Th oudry codiio r 5 5 Siilr xrio xi for oluio r oid.. Uig h chi-rlio wih =, h lyicl i c i ~ c c ~ c co i i i i co co i i i i i co i co co co i i i i co co co 6 Th ω -diffrid liud r foud whr i c ~ c i c ~ c i i co 7 i i i co i i 8 i i i co co co co co co 8 7

16 8 co co i 8c i i i i i i co co co 8d i i i i co co 5 8 co co co co co co i i i 6 8f co co co co i i 7 8g i i i i i i co co co 8 8h. Clculio of PCF Th u of ur of h ω-diffrid liud i iilr o owr d c clculd y h followig xrio: i co Hr, =,, rfr o g i uil r. For clculig h orlizd PCF, w d iilr xrio for uu fir: fir uu

17 Th h xrio for h PCF co PCF uu fir Th lf-hd id of uio i fucio of rr d d rgu. I grl, h xrio r ui colicd, u for h fir g h PCF i ily clculd d i giv y i PCF. Nuricl Rul Th hyicl co,,, or uivlly L B, l,, d h rr, rig i h PCF xrio r rld y l l L B, l L B W how rul for of rr i wo lii o hiz h diffrc w h xc d ruriv clculio.. Th Sll-Q Lii < L B how i l d figur d, i hi lii, wo of rr wr cho o g ll -vlu l h. Thi corrod o lgh ig lrgr h i riod. Tl. PCF v. z lo wih ll--lii rr. Prr:, L B, l Vlu,,,.9978,.679 << L B,5,.9879,.6 < L B Co 9

18 PMD Chg Fcor PMD Chg Fcor PCF v Diiol dic PCF, r PCF, x Figur. Th PCF curv for ruriv lii wih = d d ruriv clculio r lo idicl. PCF v L B =. Th curv for xc Diiol dic PCF, r PCF, x Figur. Th PCF curv for ruriv lii wih = d L B = 5. Th curv for xc d ruriv clculio r lo idicl. No h fr = 5, h wo r divrgig lil.

19 PMD Chg Fcor. Th Lrg-Q Lii > L B how i l d figur d 5, i hi lii, wo of rr wr cho o g lrg -vlu uch lrgr h. Thi corrod o lgh ig llr h i riod. Tl. PCF v. z lo wih lrg--lii rr. Prr:, L B, l Vlu, Co 5,,.786,.75 > L B hyicl oruriv lii,,.786, 9.79 >> L B hyicl vry oruriv lii PCF v Diiol dic PCF,x PCF,r Figur. Th PCF curv for oruriv lii wih = 5 d L B =. Th o d oo curv how xc d ruriv clculio, rcivly. No h ruriv roxiio udri h PCF i hi rgi.

20 PMD Chg Fcor PCF v Diiol dic PCF,xc PCF,r Figur 5. Th PCF curv for oruriv lii wih = d L B =. Th o d oo curv how xc d ruriv clculio, rcivly. I i h ruriv roxiio udri h PCF i hi rgi.. Cocluio I w how hrough h clculio h h ruriv roxiio d i Glro l. h liid vlidiy cord wih xc clculio. Th hr-g roxiio giv hr c xdd o y ur of g. h x lvl, h g xrio c drivd for y giv rofil of h i fucio. Th xc lyic xrio llow hyicl udrdig of h lii of h roxiio loyd rlir. Glro,.; Pliri, L.; Pizzi,. Oiizd Siig Dig for Low PMD Fir: lyicl roch. J. Lighwv Tchol., 9, 5 5.

21 dix. Pruriv Clculio for Sg

22 Th ruriv roch i d o h followig uio: Th coulig w h olrizio i o ll h h uio co dcould. Th o coo i co,,,, d oly h cod coo chg. Th oudry codiio ri uchgd. Udr h uio, h diiol co co c ~, which i rld o h hyicl lgh c~ l Th w uio d hir oluio k h followig for: Sg I: Pruriv uio: i ic ~ ic ~ Soluio: ic i i c c Th u of ur of h ω-diffrid liud: c~ r co c ~ i c~ So PCF i c ~ r c r ~ uu fir 5

23 Sg II: Pruriv uio: i i c ~ c i c ~ c 6 Soluio: i c c ~ co cco c ~ i c ii c ~ c~ 7 Th u of ur of h ω-diffrid liud: c c c c ~ co co co co c c c c r ~ i i i ~ 8 Exrio for PCF i oid for. Sg III: Pruriv uio: i i c ~ c i c ~ c 9 Soluio: co c co c co6c co c ~ i i c i c i 6c i c ~ c c~ i c i c i 6c i c ~ i co c co c co6c Th u of ur of h ω-diffrid liud: 5 co c c c c c ~ r c~ co co8 co 6 co i c i8 c i 6 c i c ~ Th PCF c clculd for. 5

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