Fiber-Optic Parametric Amplifiers: Their Advantages and Limitations by Fatih Yaman Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Professor Govind P. Agrawal The Institute of Optics The College School of Engineering and Applied Sciences University of Rochester Rochester, New York 2006 ii Curriculum Vitae The author was born in Diyarbakir, in 1978 and grew up in Aydin, Turkey. He grad- uated first in his class with B.S. degrees in both Physics and Mathematics from Koc University, Istanbul in 2000. He then began graduate studies at the Institute of Optics same year. He has carried out his doctoral research in nonlinear fiber optics under the directionofProfessorGovindP.Agrawal. iii Publications Journal Articles F.Yaman,Q.Lin,andG.P.Agrawal,“Anoveldesignforpolarization-independent single-pumpfiber-opticparametricamplifiers,”submittedtoIEEEPhoton. Tech- nol. Lett. Q. Lin, F. Yaman, and G. P. Agrawal, “Photon-pair generation in optical fibers through four-wave mixing: role of Raman scattering and pump polarization,” submittedtoPhys. Rev. A. F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, “Impact of polarization-mode dispersion and polarization-dependent loss on fiber-optic parametric amplifiers,” acceptedforpublicationinJ.LightwaveTechnol. Q. Lin, F. Yaman, and G. P. Agrawal, “Raman-induced polarization-dependent gain in parametric amplifiers pumped with orthogonally polarized lasers,” IEEE Photon. Technol. Lett. 18,397(2006). Q.Lin,F.Yaman,andG.P.Agrawal,“Photon-pairgenerationbyfour-wavemix- inginsideopticalfibers,”Opt. Lett. 31,1286(2006). A.Sennaroglu,U.Demirbas,S.Ozharar,andF.Yaman,“Accuratedetermination of saturation parameters for Cr4+-doped solid-state saturable absorbers,” J. Opt. Soc. Am. B23,241(2006). iv F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, “Impact of pump-phase modula- tion on dual-pump fiber-optic parametric amplifiers and wavelength convertors,” IEEEPhoton. Technol. Lett. 17,2053(2005). F.Yaman,Q.Lin,S.Radic,andG.P.Agrawal“Pumpnoisetransferindual-pump fiber-opticparametricamplifiers: walk-offeffects,”Opt. Lett. 30,1048(2005). F.Yaman,Q.Lin,S.Radic,andG.P.Agrawal,“Impactofdispersionfluctuations on dual-pump fiber-optic parametric amplifiers,” IEEE Photonic Technol. Lett. 16,1292(2004). F.Yaman,Q.Lin,andG.P.Agrawal,“Effectsofpolarizationmodedispersionin dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 16, 431(2004). A. Levent, S. G. Rajeev, F. Yaman, and G. P. Agrawal, “Nonlinear theory of polarization-mode dispersion for fiber solitons,” Phys. Rev. Lett. 90, 013902 (2003). Book Chapters F.Yaman,Q.Lin,andG.P.Agrawal,“Fiber-opticparametricamplifiersforlight- wave systems,” Guided Wave Optical Components and Devices, B. P. Pal Ed. (AcademicPress,SanDiego,CA,2005),Chap.7,pp. 101–117. v Presentations F.Yaman,Q.Lin,andG.P.Agrawal,“Anoveldesignforpolarization-independent single-pump fiber-optic parametric amplifiers,” in Proc. Conf. on Lasers and Electro-Optics(CLEO)2006,LongBeach,CA,PaperJWB59. F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, “Impact of pump-phase modu- lation on fiber-optic parametric amplifiers and wavelength converters,” in Proc. Conf. onLasersandElectro-Optics(CLEO)2005,Baltimore,MD,PaperCTuJ6. Q. Lin, F. Yaman, S. Radic, and G. P. Agrawal, “Fundamental noise limits in dual-pumpfiber-opticparametricamplifiersandwavelengthconverters,”inProc. Conf. onLasersandElectro-Optics(CLEO)2005,Baltimore,MD,PaperCTuT4. F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, “Impact of pump-phase modula- tionontheperformanceofdual-pumpfiber-opticparametricamplifiers,”inProc. Conf. OpticalFiberCommunications(OFC)2005,Anaheim,CA,PaperOWN3. F.Yaman,Q.Lin,andG.P.Agrawal,“Impactofpolarizationmodedispersionon dual-pump fiber optic parametric amplifiers,” in Proc. Conf. Nonlinear Guided Waves(NLGW)2004,Toronto,PaperMC31. Q. Lin, F. Yaman, and G. P. Agrawal, “Impact of randomly varying fiber disper- sion on dual-pump fiber optic parametric amplifiers,” in Proc. Conf. Nonlinear GuidedWaves(NLGW)2004,Toronto,MC36. vi Acknowledgements I would like to express my gratitude to all those who gave me the opportunity to com- pletethisthesis. Firstandforemost,Iwouldliketothankmyadvisor,ProfessorGovind P.Agrawal. Henotonlytaughtmehowtodoscientificresearch,healsotaughtmehow to think coherently and write coherently. I am also grateful to him for leading me into theexcitingfieldofnonlinearfiberoptics. IamgratefultoProfessorStojanRadicforhisvaluablediscussionsandsuggestions, as well as providing the point of view of an experimentalist. I would like to thank Professor Sarada G. Rajeev for helping me see the concept of soliton propagation in fibers from a whole new perspective. I am grateful to Professor John C. Howell for lettingmeusehislaboratoryequipment. Amongmycolleagues,IamparticularlygratefultoQiangLinforhisconstantcon- tributions to my work, through passionate and fruitful discussions. He not only helped me develop a sound physical intuition about nonlinear processes in fibers but he also helpedmefindmywaythroughthecomplicatedmathematicsofpolarization-modedis- persionandstochasticprocesses. IwouldliketothankNickUsechakforthestimulating discussions about mode-locked lasers and pulse characterization. I am grateful to him for answering my questions about laboratory equipment. I thank Dr. Ekaterina Pout- rina and Dr. Jayanthi Santhanam for helping me have a better understanding of fiber communication systems. I am grateful to Dr. Levent Akant for introducing to me the exciting mathematical tools that were so helpful in understanding soliton propagation infibers. IthankIrfanAliKhanforhelpingmeinProfessorHowell’slaboratory. vii Abstract Fiber-opticparametricamplifiers(FOPAs)canbeusedinlightwavesystemsforseveral signal-processing applications including optical amplification, phase conjugation, and wavelengthconversion. Inprinciple,FOPAscanprovidehighgainuniformoverawide wavelength range (>100 nm). What is more, FOPAs add little noise to the amplified signal. FOPAscanhavenoisefigureaslowas0dBwhenoperatedinthephase-sensitive mode and 3 dB in the phase insensitive mode. However, in practice, these advantages of FOPAs are compromised. In this work, I investigate several factors that limit the performanceofFOPAs,andproposepracticalschemestominimizethoselimitations. FOPAs can provide a relatively large gain bandwidth because the gain spectrum of FOPAs is not determined by material resonances but by the phase-matching condition. For the same reason, FOPAs are very sensitive to perturbations stemming from fiber irregularities. One such irregularity is that fiber dispersion varies randomly along the fiber length. My numerical modeling showed that, because of such variations, FOPA gainspectrumcannotmaintainitsflatnessandalsothatFOPAgainprofilechangesfrom one fiber to the other. Using stochastic methods, an analytic theory is developed that can predict an “average gain spectrum.” This analytic theory can be used to show that flatnessofFOPAgainisrecoveredattheexpenseofreducingthegainbandwidth. Another fiber irregularity that affects FOPA gain spectrum is the residual birefrin- gence. Duringthefiber-drawingprocess,thecrosssectionoffibercoreinevitablydevi- atesfromperfectcircularsymmetry. Asaresult,allnon-polarizationmaintainingfibers exhibitresidualbirefringence. Boththemagnitudeofbirefringenceandthedirectionof viii itsprincipalaxisvaryalongthefiberlengthaswellasintime. Becauseofresidualbire- fringence, state of polarizations of the propagating fields change randomly also. Since the underlying four-wave mixing (FWM) process depends on the state of polarizations of the interacting fields, FOPA gain profile cannot maintain its flatness, and changes in time. The impact of residual birefringence on dual-pump FOPAs are investigated by means of numerical simulations. It is shown that residual birefringence also causes a trade off between flatness of FOPA gain spectrum and FOPA gain bandwidth. The FOPAgainbandwidthrequiredtomaintainaflatgainprofileisdetermined. FOPAs canbe used for all-opticalsignal processing applications sincethe underly- ing FWM process responds almost instantaneously. However, the ultrashort response time of FWM makes FOPAs susceptible to pump noise. This becomes a limitation on thenoisepropertiesofFOPAs. Iinvestigatedtheimpactofpumpnoiseontheamplified signal in the context of intensity noise transfer. It is shown that the group-velocity dif- ference between the signal and pumps play an important role and reduces the transfer ofintensitynoisefromthepumptothesignal. Another limitation of FOPAs is that, since the efficiency of FWM process depends onpolarizationstatesoftheinteractingfields,FOPAgainbecomessensitivetoinputpo- larizationofthesignal. Thereareseveralschemesthathavebeenproposedtoeliminate polarization dependence of FOPA gain. However, in the case of single-pump FOPAs theexistingmethodsarecumbersometoimplement. Inthiswork,Iproposearelatively simplewaytoachievepolarization-insensitivegainbyusingahighlybirefringentfiber. ix Table of Contents CurriculumVitae ii Publications iii Acknowledgements vi Abstract vii ListofTables xii ListofFigures xiii 1 Introduction 1 1.1 HistoricalReviewofPreviousWork . . . . . . . . . . . . . . . . . . . 1 1.2 ThesisObjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 ThesisOrganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 TheoreticalFramework 5 2.1 BasicPropertiesofFWM . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Maxwell’sandHelmholtz’sEquations . . . . . . . . . . . . . . . . . . 7 2.3 ModesofOpticalFibers . . . . . . . . . . . . . . . . . . . . . . . . . 10 x 2.4 Third-orderNonlinearResponse . . . . . . . . . . . . . . . . . . . . . 13 2.5 VectorialNLSEquation . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 VectorialFWMEquations . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 ScalarFWMEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 ParametricAmplifiers 27 3.1 Single-PumpParametricAmplifiers . . . . . . . . . . . . . . . . . . . 28 3.2 Dual-PumpParametricAmplifiers . . . . . . . . . . . . . . . . . . . . 36 3.3 PolarizationDependenceofFOPAGain . . . . . . . . . . . . . . . . . 41 3.3.1 TheoreticalModel . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.2 FWMinShortFibers . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.3 FWMinLongFibers . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 ApplicationsandLimitations . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.1 WavelengthConversionandOtherApplications . . . . . . . . . 52 3.4.2 PracticalLimitations . . . . . . . . . . . . . . . . . . . . . . . 55 4 ImpactofFiberImperfections 57 4.1 FluctuationsofZero-DispersionWavelength . . . . . . . . . . . . . . . 57 4.1.1 NumericalSimulations . . . . . . . . . . . . . . . . . . . . . . 58 4.1.2 NumericalResults . . . . . . . . . . . . . . . . . . . . . . . . 60 4.1.3 MitigationofZDWLFluctuations . . . . . . . . . . . . . . . . 62 4.1.4 AnalyticalModel . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 EffectofResidualFiberBirefringence . . . . . . . . . . . . . . . . . . 70 4.2.1 TheoreticalModel . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.2 NumericalModel . . . . . . . . . . . . . . . . . . . . . . . . . 72