FWM in One-dimensional Nonlinear Photonic Crystal and Theoretical Investigation of Parametric Down Conversion Efficiency (Steady State Analysis)

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1 Procdings of h Inrnaional MuliConfrnc of nginrs and Compur Sciniss 9 Vol II IMCS 9 March 8-9 Hong Kong FWM in On-dimnsional Nonlinar Phoonic Crysal and Thorical Invsigaion of Paramric Down Convrsion fficincy (Sady Sa Analysis) M. Booarjmhr Absrac W sudy h hory of ligh propagaion in on-dimnsional phoonic crysal via nonlinar Four Wav Mixing (FWM) procss. Th linar and nonlinar rfraciv indics ar chosn o b priodic. A sysm of h nonlinar coupld mod quaions (NLCMs) including pump filds dplion is drivd for FWM procss and sady sa analysis is prsnd numrically. Also som of imporan sysm paramrs' ffcs on FWM procss ar invsigad. I has bn shown ha alhough w hav considrd pump dplion () () χ χ and usd a mdium which is small compard o -no for cnrosymmric marials- convrsion fficincy is nhancd a las ims o h prvious works which had () χ usd undpld pump approximaion and a mdium. Indx Trms Fibr opics and opical communicaion Four- wav mixing Nonlinar Opics Paramric Procsss. I. INTRODUCTION Paramric inracions in nonlinar priodic srucurs play an imporan rol in all-opical nworks. FWM procss is on of hs inracions which hav long bn sudid []-[]. Rcnly FWM procss is usd in inrsing aras of quanum informaion chnology o gnra singl and nangld phoons via nonlinar phoonic crysals [[-[9]. Ths crysals includ bulk marials such as BBO and GaAs bu sinc h nonlinar suscpibiliy of GaAs is much grar han BBO crysals which affc h convrsion fficincy of FWM procss GaAs crysals ar widly usd in quanum opics aras of rsarch bu bcaus of lack of birfringnc phas maching condiion is no saisfid asily so o obain phas maching condiion asily nonlinar phoonic crysals ar usd insad of bulk GaAs. FWM is a phoon scaring procss during which wo phoons from a rlaivly high-innsiy bam calld pump bams scar hrough hird ordr nonlinariy of a marial o gnra wo corrlad phoons calld signal and idlr phoons rspcivly [] [][]. In homognous nonlinar mdia (such as bulk marial) fficin xchang of nrgy bwn inracing mods of h lcromagnic Manuscrip rcivd Ocobr 8 8. M.Booarjmhr is wih h Phoonics Group Physics Faculy Univrsiy of Tabri and wih h Phoonics and Nanocrysals Rsarch Lab. (PNRL) Faculy of lcrical and Compur nginring Univrsiy of Tabri Tabri 566 Iran (corrsponding auhor; -mail: maryambooarjmhr@ gmail.com). fild is drmind by h linar and nonlinar suscpibiliis of h mdium. So succssful achivmn of h proposd applicaions srongly dpnds on h nonlinariy srngh and h mdium srucur. Bu hs marials suffr from problms which som of hm ar mniond blow: () (). χ nonlinariy is small compard o χ xcp h fac ha in cnrosymmric marials du o h symmry of () ponial funcion χ vanishs and bcoms ro [].. In h cas of small convrsion fficincy vn small amouns of pump bam scaring gnras larg background coun ras ha mask h dcion of corrlaions bwn signal and idlr phoons. (on h ohr hand scaring of h pump filds nds o mask h dsird quanum ffcs). () Th nonlinariy is an off-rsonanc χ Krr ffc wih an ulra fas frquncy rspons xnding from dc o wll abov TH. Alhough wak i can giv ris o vry larg nonlinar ffc in long fibrs. I is obvious ha h marial s prmiiviy drmins how phas machd is a givn paramric procss whras h acual coupling of nrgy bwn h mods is a funcion of h marial s nonlinar polariabiliy. In an amp o circumvn marial consrains (scond alrnaiv) much works hav bn focusd on h possibiliy of using priodic mdia o mdia nonlinar procsss. Th inroducion of h priodic nonlinar modulaion lads o boh flxibiliy in phas maching and also maks accssibl a marial s largs nonlinar cofficin. I has bn shown ha priodic modulaion of a nonlinar marial s rfraciv indx can lad o nhancd convrsion fficincis in paramric procsss []. In his papr w propos a compl s of coupld wav quaions dscribing FWM in on-dimnsional nonlinar phoonic crysal. Th drivd rlaions includ all sysm paramrs and inpu saus. Our considraion concnra on.55μm which is inrsing for opical communicaion. Also h proposd priodic srucur can b imagind as nonlinar fibr Bragg Graing. Afr drivaion of h coupld wav quaions for all fild componns (forward and backward componns) numrical mhods hav bn usd o simula h procss. W ry o nhanc convrsion fficincis using FWM procss. This papr is organid as follows: In scion II mahmaical modl and drivaion of FWM procss in on dimnsional nonlinar phoonic crysal is discussd. Scion III includs numrical valuaion of quaions and som of imporan sysm paramrs' ffcs ISBN: IMCS 9

2 Procdings of h Inrnaional MuliConfrnc of nginrs and Compur Sciniss 9 Vol II IMCS 9 March 8-9 Hong Kong on FWM procss fficincy ar invsigad. Finally his papr nds wih summary of h work and conclusion in scion IV. II. MATHMATICAL MODLING W now prsn a mahmaical modl for FWM procss in D nonlinar phoonic crysal. Schmaic skch of his srucur for modling FWM procss is illusrad in Fig.. Pump Λ B Fig.. On-dimnsional nonlinar phoonic crysal and h indics of rfracion disribuion N l N nl Fig.. Profils of linar and nonlinar rfraciv indics For h proposd srucur h rfraciv indx profil is givn as follows: n n a cos( k δ ) a cos( k δ) () whr n a k δ a and ar h avrag rfraciv indxs of crysal h firs harmonic cofficin of Fourir xpansion for h linar indx of rfracion avrag incidn wav vcor duning bwn incidn wav vcor and priodic srucur's wav vcor h firs harmonic cofficin of Fourir xpansion for h nonlinar indx of rfracion and h applid lcric fild rspcivly. quaion. () shows ha boh linar and nonlinar componns of h rfraciv indxs ar chosn o b priodic as shown in Fig.. W hav wrin h following fild disribuion for FWM procss in h priodic srucur ik ik iω ( ) ( ( ( ik Pump y L ik ik ) ) ) c. c. whr ± i k i and ωi ar ampliuds of h forward and backward pump signal and idlr filds hir wav vcors x Signal Idlr () and frquncis for all componns rspcivly. In wriing h lcric fild disribuion phas mismach condiion bwn four wav vcors should b saisfid Δ k k k k k () which k k k and k ar wo pump signal and idlr wav vcors rspcivly. Nonlinar polariaion is [] PNL ε [ A( )( i. j ) k B( )( i. j) k ] c. c. () whr A () and B() ar nonlinariis rlad o nonlinar mdium disribuion profils or krr ffc dscripion in our modl and givn as follows i( k ) ( ) ( ) [ δ A B A i(k δ ) ] (5) whr k k k k k. (6) and δ is h duning of h wav vcor of priodic srucur ( Λ B ) wih avrag applid filds' wav vcors( k ). Indd for a wav vcor of k whn δ is ro h phoonic crysal works in h cnr of h forbiddn gap so ha ligh dos no pnra ino h srucur which is drminabl for nrgy ransfr in h WDM (Wavlngh Division Muliplxing)h mos inrsing cas occurs whn h frquncy ω (rlad o k ) lis a h band dg which nhancs h ffciv opical nonlinariy and consqunly h FWM fficincy. Now for obaining h coupld wav quaions h lcric fild and h nonlinar polariaion should saisfy h Maxwll's wav quaion n PNL μ (7) c whr c is spd of ligh in vacuum and n is h rfraciv indx of mdium. Bcaus of small prurbaion in h rfraciv indx h following approximaion is usd in Maxwll's wav quaion ha is h cofficins such as a a ar nglcd. i(k δ ) i(k δ ) n n na ( ) i(k ) ( ) (8) δ i k δ n a ( ) Finally by subsiuing quaions () () and (8) ino q. (7) and using h slowly varying funcion approximaion and doing som mahmaical simplificaions for xampl all h snncs having h rm xp( i( ω ω ω)) xp( iδ) ar nglcd bcaus hy can no oscilla and coupl insid h crysal so h following coupld wav quaions ar obaind ISBN: IMCS 9

3 Procdings of h Inrnaional MuliConfrnc of nginrs and Compur Sciniss 9 Vol II IMCS 9 March 8-9 Hong Kong iδ iω ( ) na iω n a A kc kc iδ iδ [ α α α whr α α iδ iδk α. quaion. (9) illusras h coupld wav quaion for h firs pump fild propagaing from lf o righ insid h crysal. [ Γ whr Γ iω na k c Γ Γ Γ iω ( na A) k c Γ iδ ] ] Δ i k. (9) () () () quaion. () illusras h coupld wav quaion for h backward componn of h firs pump fild propagaing from righ o lf insid h crysal. W hav h similar quaion forms for h co-propagaion filds- which can b obaind sraigh forwardly. W hav solvd hs quaions numrically and h ffc of sysm paramrs on convrsion fficincy (For ach on of pump phoons which ar annihilad signal and idlr phoons ar crad corrlaly) is considrd and illusrad in h nx scion. Th convrsion fficincy for forward ravling signal componns can b dfind as follows () P η () P () W will call his fficincy co-propagaion fficincy in h nx scions. P() and P( ) sands for h forward pump and signal powr rspcivly which signal and pump filds ar applid from righ hand sid and lf hand sid o our sysm rspcivly. Also w dfin convrsion fficincy paramr for h backward signal powr as follows ( ) P L η () P () whr P ( L) is h backward signal powr a h righ hand sid of crysal. III. SIMULATION RSULTS AND DISCUSSION In his scion w prsn som of numrical rsuls for boh co-propagaing and counr-propagaing fficincis in FWM procss. Our simulaions show ha h co-propagaing fficincy is largr han h counr-propagaing fficincy; his is du o h basic principls of nrgy ransfr bwn forward and backward propagaing mods in Brag graings. Figur. () shows h ffc of numbr of priods on co-propagaion convrsion fficincy. Convrsion fficincy is dcrasd by dcrasing of h nonlinar rfraciv indx cofficin. As i is shown in Fig. () w hav incrasd h layrs f crysal up o 6 layrs in simulaion procss w obsrvd ha by choosing h numbr of layrs o mor han 6 layrs co-propagaing fficincy has oscillaory bhavior so i is ncssary o kp in mind ha h convrsion fficincy is nonlinarly dpndan on h numbr of priods and his subjc should b considrd in dsign procss of a phoonic crysal o avoid dcrasing of h convrsion fficincy. fficincy x Numbr of priods Fig.. fficincy vs. numbr of priods for diffrn valus of ) a ) a ) a n.5 Δk ( m ) a. δ ( m ) Figur. () illusras h ffc of phas mismach bwn wav vcors of four opical filds (wo pumps signal and idlr filds) on convrsion fficincy. I is shown ha by incrasing of phas mismach h convrsion fficincy is dcrasd du o h wak nrgy ransfr bwn propagaing mods insid h crysal. fficincy.5 x Numbr of Priods Fig.. fficincy vs. numbr of priods for diffrn valus of a Δ k ISBN: IMCS 9

4 Procdings of h Inrnaional MuliConfrnc of nginrs and Compur Sciniss 9 Vol II IMCS 9 March 8-9 Hong Kong ) Δk.k ( m ) ) Δk.k ( m ) 5 ) a n.5 δ ( m ) a. Figurs. (5) and (6) dmonsra h ffc of phas mismach bwn h avrag wav vcor of opical filds and Graing wav vcor on h convrsion fficincy as δ incrass h convrsion fficincy dcrass. So in ordr o obain high fficincis in Bragg graings δ should b considrd as small as possibl. Also carful choic of numbr of layrs is imporan du o h nonlinar dpndnc bwn numbr of layrs and h convrsion fficincy as discussd bfor. Th sam simulaions hav bn illusrad for counr-propagaion cas. Figurs.(7) and (8) show h ffc of phas mismach bwn h avrag wav vcors of h opical filds and graing wav vcor on h convrsion fficincy and as i is xpcd by incrasing δ h counr-propagaing fficincy dcrass..6 x -8.. Counr-Propagaion x fficincy fficincy Numbr of Priods Numbr of Priods Fig. 5. fficincy vs. numbr of priods for diffrn valus of δ ) δ.k ( m )) δ.k ( m )) δ.k ( m ) 5 n.5 Δk ( m ) a. a Fig. 7. fficincy vs. numbr of priods for diffrn valus of δ ) δ ( m )) δ.k ( m )) δ.k ( m ) 5 n.5 Δk ( m ) a. a W obsrvd h sam rsuls from our simulaions for counr-propagaing filds. As an xampl w show h nonlinar rfraciv indx's cofficin in diffrn numbr of layrs on h counr-propagaion convrsion fficincy. Th ffc of phas mismach bwn Graing wav vcor and h avrag wav vcor of four opical filds on convrsion fficincy in broad rangs is shown in Fig. 6.I is apparn ha h highr δ is h smallr h fficincy..6 x -8.. Counr-Propagaion fficincy.5 x δ x Fig. 6. fficincy vs. phas mismach bwn graing and avrag applid filds' wav vcors. 5 n.5 Δk ( m ) a. a N fficincy Numbr of Priods Fig. 9. fficincy vs. numbr of priods for diffrn valus of ) a ) a ) a n.5 Δk ( m ) a. δ ( m ) IV. SUMMARY AND CONCLUSION Th coupld mod quaions for ligh propagaion hrough on-dimnsional nonlinar phoonic crysals using FWM a ISBN: IMCS 9

5 Procdings of h Inrnaional MuliConfrnc of nginrs and Compur Sciniss 9 Vol II IMCS 9 March 8-9 Hong Kong procss hav bn drivd and numrical rsuls for h sady sa sudy hav bn discussd. W hav considrd h Bragg as a losslss and inhomognous mdium. Th linar and nonlinar rfraciv indics ar considrd o b priodic. I has bn shown ha alhough w hav considrd ha all h filds dpl via propagaing in D phoonic crysal-which spcially bcoms imporan whn h numbr () of layrs incras - and w hav usd a mdium wih χ nonlinariy an nhancmn of a las ims in boh co-propagaing and counr-propagaing fficincis hav bn obsrvd compard o h prvious works ha us () undpld pump filds and χ mdium []. In summary NCMs ar drivd via D nonlinar phoonic crysal using h following cass: [9] D. Prosyan and G. Kuriki "Phoon-phoon Corrlaions and nanglmn in Dopd Phoonic Crysals" Phy. Rv. A Vol [] A. Yariv Quanum lcronics rd d. John Wily Nw York 989. [] R. W. Boyd Nonlinar Opics Boson Acadmic Prss 99. [] A. N. Vamivakas B.. A. Salh A. V. Srginko and M. C. Tich "Thory of Sponanous Paramric Down-convrsion from Phoonic Crysals" Phys. Rv. A Vol [] S John "Srong Localiaion of Phoons in Crain Disordrd Dilcric Suprlaics" Phys. Rv. L Vol (987). []. Yablonovich "Inhibid Sponanous mission in solid-sa Physics and lcronics" Phys. Rv. L. Vol. 58pp W hav usd coninuous wav frquncy rahr han shor puls cas.. All filds (pump signal and idlr filds) ar affcd by Bragg graing so boh SPM (Slf-phas modulaion) and XPM (Cross-phas modulaion) phnomna ar obsrvd in qs.( 9)-() and similar quaions for.. Pump dplion hav bn considrd alhough his is no ncssarily srious bcaus of low convrsion fficincis bu i affcs h fficincy spcially in larg numbr of layrs.. Boh linar and nonlinar rfraciv of indics ar priodic. 5. In simulaion procsss boh backward and forward componns of h filds ar akn ino accoun and non of hm ar nglcd. ACKNOWLDGMNT Th auhor would lik o xprss hr snior hanks o Prof.Mir Kamal Mirnia from h Mahmaics dparmn a h Univrsiy of Tabri for h usful and fruiful discussions on numrical solving mhods of nonlinar coupld quaions. RFRNCS [] M. J. Sl and C. Marijn d Srk "Paramric amplificaion of shor pulss in opical fibr Bragg graings" Phys. Rv. Vol [] M. J. Sl and C. Marijn d Srk "Coninuous-wav paramric amplificaion in Bragg graings" JOSA B Vol [] G. P. Agrawal Nonlinar Fibr Opics rd d. Acadmic Prss Nw York. [] H. Taksu K. Inou "Gnraion of polariaion nangld phoon pairs and violaion of Bll's inqualiy using sponanous four-wav mixing in fibr loop". Arxiv:quan-ph/8 V. [5] B. Sandrs J. Vuckoric and P. Grangir "Singl phoons on dmand" urophysics nws March 5. [6] M. C. Booh M. Aaur G. Di Giuspp B.. A. Salh A. V. Srginko and M. C. Tich "Counr-propagaing nangld Phoons from a Wavguid wih Priodic Nonlinariy" Phys. Rv. A Vol [7] L. J. Wang C. K. Hong and S. R. Fribrg "Gnraion of Corrlad Phoons via Four-wav Mixing in Opical Fibrs" J. Op. B: Quanum Smiclass. Op [8] W. T. M. Irvin M. J. A. d Dood and D. Bouwmsr "Bloch Thory of nangld Phoon Gnraion in Nonlinar Phoonic Crysals" Phys. Rv. A Vol ISBN: IMCS 9