A Thesis. Submitted to the Faculty. Drexel University. Yifei Li. in partial fulfillment of the. requirements for the degree. Doctor of Philosophy

Transcription

1 i Optically Geeratio of Rapidly Tuable Millimeter Wave Subcarrier Usig Microchip Lasers A Thesis Submitted to the Faculty of Drexel Uiversity by Yifei Li i partial fulfillmet of the requiremets for the degree of Doctor of Philosophy May 003

2 Copyright 003 Yifei Li. All Rights Reserved ii

3 ii Dedicatios To my family

4 iii Ackowledgemets First, I thak my thesis advisor, Dr. Peter R. Herczfeld, for his guidace, support, ad patiece throughout my graduate study. His help for the writig is essetial for the completio of this thesis work. Secodly, I thak Dr. Samuel Goldwasser for his persoal support, friedship, ad professioal help i the research, without which this thesis work would be less fruitful. Next, I like to thak Dr. Lorezo Narducci. His help ad guidace are vital i my uderstadig of the laser physics ad the completio of the theoretical part of the thesis work. I also feel very obliged to the friedship ad persoal support from Dr. Arye Rose ad Dr. Stewart D. Persoick. I also wish to thak Dr. Lida Mulle ad Dr. Maja Bystrom for their support ad helpful suggestios. I am extremely grateful to my frieds ad colleagues here i the Ceter for the Microwave/Lightwave egieerig. I would like specially to thak Dr. Afshi Daryoush, Mike Ermold, David Yoo, ad Mohammad Tofighi for the helpful discussios ad to Reee Cohe for her extreme good will. Fially, I would like to thak my family, for their support durig this ew step i my life.

5 iv Table of Cotets List of Tables...vii List of Figures...viii Abstract...xii Chapter 1: Itroductio The goal of this research effort Cotributios Thesis orgaizatio... Chapter : Backgroud ad review of literature Backgroud ad motivatios Hybrid lidar/radar system Mm-wave fiber radio...6. Review o mm-wave optical subcarrier geeratio A ew approach of geeratig low oise, tuable mm-wave optical subcarrier Microchip lasers Noise cotrol techiques....4 Laser itracavity FM dyamics Laser frequecy switchig Laser FM oscillatio Summary o laser FM dyamics review The objectives of this thesis work...8 Chapter 3: Dyamics of the tuable microchip laser Problem idetificatio Problem formulatio Modelig of compoets...34

6 v 3.. Overall cavity formulatio Model represetatio ad uiform field limit Liear dyamic model Liearized Maxwell-Bloch equatios Closed-form solutio i frequecy domai The trasiet characteristics of the laser respose with frequecy tuig Direct simulatio ad discussio Dyamic respose of the laser Laser dyamic behavior for differet cavity parameters Summary of laser respose for varyig voltage ramp ad cavity parameters...87 Chapter 4: Tuable trasmitter desig DTMOT cocept Trasmitter sigle logitudial mode operatio The microchip laser trasmitter sigle spatial mode operatio Laser tuig Electrical voltage tuig sesitivity Temperature tuig sesitivity Sesitivity to pump power tuig Laser trasmitter characterizatio Laser threshold ad efficiecy Microchip laser frequecy tuig Mm-wave operatio potetial Microchip laser chirp rate testig The free ruig phase oise of the tuable trasmitter Summary Chapter 5: Trasmitter phase oise cotrol subsystem... 11

7 vi 5.1 Digital frequecy sythesizer Sythesizer subsystem phase oise Frequecy modulatio withi digital sythesizer subsystem The delay lie optical frequecy-lockig loop OFLL operatio priciple OFLL implemetatio usig a o-ideally balaced mixer OFLL loop stability ad the loop filter desig Desig of the OFLL system usig a sigle loop Multi-loop OFLLs Noise cotrol subsystem characterizatio Digital sythesizer type oise cotrol Delay lie optical frequecy locked loop characterizatio Chapter 6: Digital FM fiber radio dowlik trasmissio Fiber radio dowlik performace Digital FM fiber radio dowlik experimet FM fiber radio up lik ad basestatio implemetatio issues Summary of fiber radio dowlik testig Chapter 7: Coclusios ad recommedatios for future work Coclusios Recommedatios for the future work List of Refereces Appedix A: The closed-form solutio of electro-optic sectio Appedix B: Simulatio method based o the direct itegratio of Maxwell-Bloch equatio Appedix C: The derivatio for fudametal mode cross sectio Appedix D: A ew discrimiator structure miimizig the AM to phase oise coversio Vita

8 vii List of Tables.1 Performace goal for the tuable trasmitter i various applicatios Review of literature table o mm-wave geeratio techologies Comparisos betwee differet mm-wave subcarrier geeratio techiques The compoets iside tuable microchip laser Field evelope equatios, boudary ad iitial coditios, i each sub-sectios of the laser cavity Pertiet laser parameters for the simulatio Lser parameters for electro-optic sectio legth simulatio Commo parameters for gai liewidth simulatio Parameters for total cavity legth simulatio The advatage ad disadvatages of the digital sythesizer approach The desig objectives of a sigle loop OFLL The parameters for the compoets i the sigle loop OFLL The digital FM fiber radio lik parameters

9 viii List of Figures.1 Hybrid CWFM lidar/radar system Fiber radio dowlik based o direct digital FM modulatio A ew approach for tuable mm-wave optical subcarrier geeratio The electro-optic tuable microchip laser A uidirectioal rig cavity with electro-optic tuability The electro-optic tuable laser formulatio process A ideal model of a lossless rig cavity uder electro-optic modulatio Trasformatio characteristics of the gai medium The trasfer fuctio of laser dyamic system The flow diagram for simulatio The laser respose uder a ramp sigal comparable to the experimetal value The lasig frequecy ad gai profile The laser respose uder moderate speed, low voltage ramp sigal The laser respose uder fast, low voltage ramp sigal Frequecy ripple observed uder fast, low voltage tuig The laser respose uder high voltage moderate speed ramp The laser respose uder very fast ramp sigal with high voltage The laser FM respose vs. differet electro-optic sectio legth The laser tuig sesitivity at steady state The maximum ripple magitude as a fuctio of the electro-optic sectio legth The ripple decay time as a fuctio of the electro-optic sectio legth The laser FM respose vs. differet gai liewidth... 81

10 ix 3.0 The ripple decay time vs. the gai medium liewidth The steady state frequecy tuig vs. the gai medium liewidth The laser FM respose vs. differet cavity legths or roud trip times The dyamically tuable millimeter wave optical trasmitter structure The heterodye trasmitter implemetatio Multimode operatio threshold vs. total cavity legths ad gai sectio legth Multimode operatio threshold vs. frequecy detuig ad gai sectio legth Multimode operatio threshold vs. frequecy detuig ad total cavity legth The cavity structure where gap exist betwee the gai sectio ad the mirror The multimode operatio threshold vs. gap legth ad gai sectio legth The multi-mode operatio threshold vs. gap legth ( 0. ~0.4 mm The multi-mode operatio threshold vs. gap legth ad frequecy detuig The coceptual laser cavity uder the pump iduced the thermal guidig ad thermal expasio Fudametal spatial mode radius vs. differet pump power The ormalized spatial mode radius vs. differet pump power level ad spatial mode The microchip laser threshold ad slope efficiecy characteristics Laser spectrum at m A trasverse mode beat toe captured by microwave spectrum aalyzer Voltage tuig sesitivity The frequecy of the two laser beat sigal as a fuctio of the temperature Two-microchip laser beat sigal vs. pump power chage The optical spectrum of 10GHz subcarrier 1340 m The microchip laser chirp rate measuremet system

11 x 4.1 The output characteristics of the frequecy discrimiator Applied ramp sigal (top ad the frequecy respose (bottom The laser phase oise measuremet setup The free ruig phase oise measuremet The digital frequecy sythesizer Delay lie optical frequecy locked loop (OFLL OFLL system block diagram The OFLL implemetatio scheme with reduced AM to PM oise coversio OFLL loop filter structure The performace of the loop filter with 1 st order compesatio The performace of the loop filter with d order compesatio OFLL system oise floor for differet fiber delay legths A multi-loop OFLL employ two loops The ope loop respose of the double loop OFLL at maximum stable gai Stability of the multi-loop OFLL The digital sythesizer phase oise cotrol subsystem The digital sythesizer phase oise spectrum of 8.447GHz The digital sythesizer phase oise vs. operatio frequecy The experimetal scheme for 40GHz operatio based o the digital sythesizer phase oise cotrol Optical spectrum of 40GHz subcarrier sigal Phase oise spectrum of the dowcoverted sigal (14GHz The respose of the digital sythesizer uder directio modulatio The OFLL experimetal setup

12 xi 5.0 Phase oise spectrum of sigle loop OFLL at lower frequecies The OFLL output sigle spectrum at 37GHz The OFLL phase oise measuremet results at mm-wave frequecies The digital FM fiber radio experimetal setup The fiber radio dowlik oise charactistics The demodulatio result captured by the HP VSA The parasitic AM respose The basebad sigal rigig Fiber radio basestatio A.1 A separate E/O tuig sectio A. Coordiate trasform from ( z ~,t plae to (u,v plae B.1 Normalized time ad space et to carry out the solutio process B. The flow diagram for two dimesio Euler method B.3 The flow chart for the d order two dimesioal R-K method C.1 The coceptual cavity structure uder thermal guidig effect... 19

13 xii Abstract Optically geeratio of rapidly tuable millimeter wave subcarriers usig microchip lasers Yifei Li Peter R. Herczfeld, PhD There is growig iterest i applyig photoic techiques to the geeratio of high fidelity millimeter wave sigals. This thesis cocers the optical domai geeratio of rapidly tuable, very low oise millimeter wave subcarrier sigals. Specifically, a optical trasmitter employig two heterodyed electro-optically tuable microchip lasers has bee desiged, fabricated ad characterized regardig its tuig rage, speed, ad oise performace. To explai the fast tuig speed observed i the experimet, a theoretical aalysis of the laser dyamics durig the itracavity frequecy tuig process is advaced usig the semi-classic Maxwell-Bloch formulatio. The theoretical aalysis cofirms that microchip laser has a virtually ulimited tuig speed. The output of the tuable trasmitter is cotamiated by the optical phase oise iitiatig i the microchip laser. A ovel phase oise cotrol approach, coied a optical frequecy locked loop (OFLL, is developed to achieve low oise operatio. Ulike covetioal techiques, this scheme utilizes a log fiber delay i place of a exteral referece oscillator to correct the phase error. The optical frequecy locked loop outperforms covetioal phased locks loops, particularly at higher millimeter wave frequecies. The tuable trasmitter was successfully evaluated i the cotext of a broadbad fiber radio dowlik experimet.

14 i

15 1 Chapter 1: Itroductio 1.1 The goal of this research effort This thesis is cocered with a ovel way of geeratig a fast dyamically tuable millimeter wave (mm-wave optical subcarrier for frequecy chirped hybrid lidar/radar ad frequecy modulated (FM mm-wave fiber radio lik applicatios. Traditioal mm-wave subcarrier techiques are limited by tuability ad speed, which affects the achievable resolutio i hybrid lidar/radar ad data rate of fiber radio liks. I respose to these shortcomigs, a dyamically tuable mm-wave optical trasmitter, which utilizes two heterodyed electro-optically tuable microchip lasers, is proposed, implemeted, ad evaluated. A theoretical aalysis of the laser FM dyamics is performed to explai the rapid tuig speed obtaied i the experimet. I order to obtai a low oise mm-wave subcarrier sigal throughout the trasmitter tuig rage, a phase oise cotrol system employig the fiber delay lie as a frequecy referece is proposed, implemeted ad tested. 1. Cotributios The key cotributios of this thesis work are as followig: A theoretical model for the electro-optic tuable laser based o the Maxwell-Bloch formulatio was developed. Usig o this model, we foud that the optical frequecy is tued istatly; however, frequecy ripples occur due to the o-uiformity of the laser cavity. Laser respose uder itracavity frequecy tuig was foud to be a strog fuctio of the relative size of the tuig elemet. We proved umerically ad

16 aalytically that the appearace of ripples is a trasiet effect ad the ripple decay time is calculated aalytically. A ovel high-speed tuable mm-wave optical trasmitter employig electro-optically tuable microchip lasers was desiged, implemeted ad tested. A ovel phase oise cotrol scheme capable of achievig low oise operatio over a wide rage of carrier frequecies from 0 to 40 GHz ad beyod, was proposed, implemeted ad tested. This achieved a performace better tha 10kHz offset, at least 10dB better tha ay other heterodyig system described i the literature. A fiber radio dowlik trasmissio usig a directly modulated tuable mm-wave optical trasmitter was performed for the first time with promisig results. 1.3 Thesis orgaizatio The orgaizatio of this thesis is the followig: Chapter addresses the backgroud ad reviews the literature o the mm-wave optic subcarrier geeratio techiques. A ew subcarrier geeratio approach usig the heterodyig of two or more electro-optic tuable microchip laser is proposed. Fially, the existig works o the laser itracavity FM dyamics are examied. Chapter 3 studies the dyamics durig the microchip laser itracavity frequecy modulatio. First, a formulatio based o the semi-classic Maxwell-Bloch equatios is derived specifically for the electro-optic tuable laser cavity. I the formulatio, the effect of the electro-optical tuig is modeled as a perturbatio o the boudary coditio

17 3 of the laser system. Based o the formulatio, the coditios for the uiform field limit or sigle mode operatio are addressed. For the case where the effective frequecy tuig is less tha the gai liewidth, a approximate closed-form solutio is derived to aalyze the trasiet behavior of the laser durig frequecy tuig ad its fial steady state. Fially, a umerical simulatio is performed to explore the speed limitatios of the laser. Chapter 4 ivestigates issues pertiet to the desig, ad performace characterizatio of the dyamically tuable mm-wave optical trasmitter icludig output power, tuig rage, tuig speed, ad oise. Chapter 5 examies two alterate schemes of the phase oise cotrol, amely digital sythesizer ad the optical frequecy locked loop (OFLL. The OFLL format is explaied usig a system model. The issues regardig the achievable oise performace of the OFLL, ad its practical implemetatio are addressed. Fially, empirical results are used to compare the performace of these two approaches. Chapter 6 focuses o a practical applicatio of the tuable mm-wave optical trasmitter i the cotext of a fiber-radio dowlik.

18 4 Chapter : Backgroud ad review of literature.1 Backgroud ad motivatios Microwave ad millimeter wave techiques are icreasigly beig adapted to optical systems, such as hybrid lidar/radar ad hybrid mm-wave wireless / fiber optic commuicatios. The motivatio of this thesis work is to develop a low oise trasmitter that ca deliver broadbad, rapidly tuable mm-wave optical subcarrier sigal for hybrid lidar/radar, ad for mm-wave fiber radio systems. I this subsectio, we ll first discuss briefly the hybrid lidar/radar system ad mm-wave fiber radio ad explai why the tuable mm-wave optic trasmitter is a eablig techology for those applicatios. Next we review the pertiet literature cocerig optical domai geeratio of mm-waves ad itroduce the otio of microchip lasers as potetial sources for the geeratio of high fidelity mm-wave sigals. This chapter culmiates i the precise defiitio of the objectives of this thesis..1.1 Hybrid lidar/radar system Sice the late 1970s, a umber of lidar systems have bee developed for the detectio of uderwater objects ad for tumors i huma tissues, as well as for aerial turbulece. Whe these lidar systems are used to probe targets submerged i turbulet media like ocea water ad body tissues, the itese scatterig (clutter from the media results i poor cotrast for target idetificatio [1]. Oe way of suppressig the clutter is to use hybrid lidar/radar [1], i which a Q-switched optical pulse is modulated by a microwave sigal, which is later processed by a coheret radar receiver. This hybrid approach is effective because the lidar clutter has a low pass trasfer characteristic. Thus,

19 5 the clutter effect is reduced whe sigal processig is performed at microwave frequecies, far above the clutter cutoff frequecy. However, this pulsed approach has limited resolutio. I a pulsed lidar/radar system, the resolutio is give by δ = u τ /, where u is the speed of light i the medium ad τ is the pulse width. For example, a typical medical imagig system requires a mm resolutio (equivalet to a ~0ps pulse width, which makes the radar sigal processig extremely difficult. I order to achieve such a high resolutio, we proposed employig cotiuous wave frequecy modulatio (CWFM [, 3, 4, 5] schemes i the hybrid lidar/radar system (see Figure.1. I this cofiguratio, the critical compoet is a rapidly tuable mm-wave optical trasmitter, which sweeps the mm-wave optical subcarrier frequecy. The target distace is determied by the frequecy differece betwee the trasmitted ad retured sigal. V Ramp Sigal t Freq Trasmitted sigal Reflected sigal Rapidly tuable mm-wave Optical trasmitter τ t Hspeed PD 1 Hspeed PD Processor Display Figure.1: Hybrid CWFM lidar/radar system

20 6 As show i Fig..1, the hybrid lidar receiver oly processes the low frequecy homodye output sigal from the mixer, which is easy compared with the pulsed system. The resolutio of this system is give by, R=u/ F, where F is frequecy excursio. I order to achieve mm resolutio, the excursio of the subcarrier frequecy eeds to be at least 37 GHz. This CWFM Lidar/Radar ca also be used for free-space laser ragig, where the target distace is much loger compared with biomedical imagig. The log target rage requires lower phase oise (arrower liewidth to icrease the coheret legth of the subcarrier sigal. Geeratio of a broadbad tuable low oise mm-wave optical subcarrier for the high-resolutio hybrid lidar/radar systems is oe of the motivatios for this thesis work..1. Mm-wave fiber radio Recetly, there is iterest i fiber radio systems [7, 8, 9, 10, 11, 1] for hybrid wireless ad fiber optic commuicatios, i which a cetral statio commuicates with oe or more base statios via a fiber optic lik (dowlik, while traffic betwee the base statio ad termials withi a local pico-cell uses mm-wave radio liks. This hybrid approach ejoys the advatages of both covetioal wireless (coveiece ad flexibility ad fiber optic (high badwidth commuicatios systems. Iside existig fiber radio system, the mm-wave subcarrier ad data modulatio are geerated separately ad the data modulatio format is limited to amplitude types, such as biary phase shift keyig (BPSK, quadrature amplitude modulatio (QAM, ad quadrature phase shift keyig (QPSK, etc. This results i complexity, but more importatly, icompatibility with existig wireless systems that use cotiuous phase modulatio (CPM such as

21 7 miimum shift keyig (MSK. Oe solutio to those problems is to use the digital FM fiber radio lik (figure. [13]. Ceter Statio Iformatio Base Statio Tuable mm-wave optical trasmitter High speed PD MSK receiver Figure.: Fiber radio dowlik based o direct digital FM modulatio I figure., the iformatio bit directly modulates the frequecy of mm-wave subcarrier, ad both a high quality mm-wave subcarrier ad CPM-type digital sigal are geerated simultaeously. The eablig compoet here is a tuable mm-wave optical trasmitter with high tuig sesitivity, high tuig speed, ad low phase oise to assure reliable, high data rate operatio. The desig ad implemetatio of such a trasmitter for mm-wave fiber radio is the secod cocer of the thesis. I coclusio, the key requiremet for both the CWFM hybrid lidar/radar ad directly FM mm-wave fiber radio is a rapidly tuable mm-wave optical trasmitter. The trasmitter should have a very wide tuig rage (>37 GHz for biomedical imagig, ad the low phase oise for the free space ragig. While i the fiber radio system, the trasmitter should be adopted to have high sesitivity, high tuig speed, ad low phase oise. Table.1 summarized the performace goals for the tuable trasmitter for four typical applicatios.

22 8 Table.1: Performace goal for the tuable trasmitter i various applicatios Applicatios Tuig rage Power Wavelegth Biomedical imagig >37GHz >10mW 700~ 900m Uder water >1GHz, depeds >1W ~500m detectio o resolutio Free space ragig Digital FM fiber radio requiremet >1GHz, depeds o resolutio requiremet >100MHz, depeds o data rate >1W 1.06/ 1.5m >1mW 1.06 /1.3/1.5 m Tuig speed >10GHz/µs Phase offset <-90dBc/Hz (deped o rage <-90dBc/Hz (deped o rage < -90dBc/Hz. Review o mm-wave optical subcarrier geeratio A extesive literature review has bee performed o the existig mm-wave optical subcarrier geeratio techiques. Approximately 10 papers o millimeter wave subcarrier geeratio were searched ad reviewed, from which the 40 most represetative papers are reviewed i detail with emphasis o the phase oise performace, tuig rage ad tuig speed. The review is summarized i Table.. Most of the existig millimeter-wave optical subcarrier geeratio techiques fall ito the followig six categories: A. Direct modulatio of a laser diode (LD B. Exteral modulatio of laser diode or other laser source C. Resoace modulatio D. Laser mode lockig E. Ijectio lockig

23 9 F. Lasers heterodyig The compariso of these approaches for mm-wave optical subcarrier geeratio techiques is summarized i Table.3. The examiatio of these methods suggests that the preferred way of geeratio of a low oise, rapidly tuable mm-wave optical subcarrier should be optical heterodyig of two laser sources with high spectral quality.

24 10 Table.: Review of literature table o mm-wave geeratio techologies Id Articles (Authors, title, source, page umber 1 Larkis, E C, et al., Improved performace from pseudomorphic IyGa1-y As-GaAs MQW lasers with low growth temperature AlxGa 1-xAs short-period superlattice claddig, IEEE Photoics Techology Letters, Vol 7, No. 1, 1995, pp Bez, W.,"Dampig-limited modulatio badwidths up to 40 GHz i udoped shortcavity I0.35Ga0.65As-GaAs multiplequatum-well lasers", IEEE Photoics Techology Letters, Vol 8 Issue: 5, May 1996, pp Zhag X., 0.98 um Multiple-Quatum- Well Tuelig Ijectio Laser with 98 GHz Itrisic Modulatio Badwidth, IEEE Joural of selected topics i quatum electroics, Vol 3., No., April, 1997, pp Baba, S. et al., Millimeter-wave fiber optics systems for persoal radio commuicatio, IEEE Trasactios o Microwave Theory ad Techiques, Vol 40, Issue: 1, Dec 199, pp Cat A A A B Descriptio Ultra-high-speed direct modulated MQW laser diodes. Decreasig the o radioactive recombiatio ceter cocetratio by the low temperature AlGaAs growth temperature, which result i decrease i the laser threshold ad icrease i laser efficiet 00 um log ridge waveguide laser structure is employed. Ultra-high-speed directly modulated MQW laser diodes Improved MBE growth parameters ad dopig sequece Short ridge waveguide laser diode structure: 130 um Theoretical ad experimetal ivestigatio of MQW tuelig ijectio laser diode, which ijects electros ito the active regio via tuelig ad avoid the carrier heatig effect The tuelig barrier also helps to cofie the electros from goig to claddig layer 00 um ridge waveguide desig Discuss several lik cofiguratios for mm-wave subcarrier trasmissio over fiber Outlie fiber radio for persoal commuicatio cocept Evaluate both the RF (Exteral E/O modulatio 6GHz ad IF lik direct LD modulatio 70MHz / QPSK ad 300 MHz / aalog FM Critique Positive attributes Negative attributes 5GHz 3dB direct Noise modulatio badwidth performace: ot High efficiecy clear. 40GHz 3dB direct modulatio badwidth High efficiecy High speed: 48GHz 3dB badwidth 98GHz itrisic modulatio badwidth Low threshold: <3 ma Outlie the fiber radio system Successfully evaluate both digital (QPSK ad aalog FM fiber radio lik Same as above Same as above mm-wave subcarrier ad data modulatio is geerated separated. High isertio loss: > 7dB 10

25 11 Table. (cotiued 5 Ackerma, E, et al., Maximum dyamic rage operatio of a microwave exteral modulatio fiber-optic lik, IEEE Trasactio o Microwave Theory ad Techiques, Vol 41 Issue 8, Aug 1993, pp Kojucharow, Kostati Simultaeous electrooptical upcoversio, remote oscillator geeratio, ad air trasmissio of multiple optical WDM chaels for a 60- GHz high-capacity idoor system IEEE Trasactios o Microwave Theory ad Techiques, v47, 1, Dec, 1999, pp Kuri, Toshiaki et al., Fiber-optic millimeter-wave dowlik system usig 60 GHz-bad exteral modulatio, Joural of Lightwave Techology, v 17, 5, 1999, p B B B Ivestigatig theoretically ad experimetally the exteral modulator based microwave fiber lik performace: lik gai (s1, ad dyamic rage 60GHz mm-wave subcarrier is geerated by exteral E/O modulatio. Digital IF sigal directly modulatio diode mm-wave fiber radio system based o WDM. 60GHz mm-wave subcarrier ad modulatio is simultaeously geerated by exteral E/A modulator. Experimetally achieved 0dB lik gai 99.1dB 1dB compressio dyamic rage 77.0dB spurious free dyamic rage WDM Phase oise: 10kHz Simultaeously geeratio of mm-wave subcarrier ad iformatio modulatio Higher oliear respose, which limits its spurious free dyamic rage High isertio loss Noliear respose Low modulatio idex High cost High isertio loss: >9dB Low modulatio idex Noliear respose 8 Noguchi, K, et al., Desig of ultra-broadbad LiNbO3 optical modulators with ridge structure, IEEE Trasactios o Microwave Theory ad Techiques, Vol: 43 Issue: 9, Sep 1995, pp: Udupa, A. et al., Polymer modulators with badwidth exceedig 100 GHz, 4th Europea Coferece o Optical Commuicatio, Vol: 1, Sep -4, 1998, pp B Optimizig the ridge structure LiNbO3 MZM 100GHz bad width is achieved B Exteral MZM optical modulator based o DR 19 polymer 113GHz badwdith Higher isertio loss Noliear respose High isertio loss Noliearity 11

26 1 Table. (cotiued Georges, Joh B, et al., Multichael millimeter wave subcarrier trasmissio by 10 resoat modulatio of moolithic semicoductor lasers IEEE PLT, v7, 4, Apr, 1995, p Georges, Joh B, et al., Optical trasmissio of arrowbad millimeterwave sigals IEEE Trasactios o Microwave Theory ad Techiques, v43, 9/, Sept, 1995, p Oho, Tetsuichiro, et al. Applicatio of DBR mode-locked lasers i millimeterwave fiber-radio system, IEEE Joural of Lightwave Techology, v18, 1, Ja, 000, p Ahmed, Z, et al., Lockig characteristics of a passively mode-locked moolithic DBR laser stabilized by optical ijectio, IEEE Photoics Techology Letters, Vol 8 Issue 1, Ja 1996, pp Kuri, T, et al., Log-term stabilized millimeter-wave geeratio usig a highpower mode-locked laser diode module, IEEE Trasactios o Microwave Theory ad Techiques, Vol 47 Issue: 5, May 1999, pp: C C, E mm-wave subcarrier ad iformatio modulatio is geerated simultaeously by resoace modulatio Resoace modulatio badwidth, ad system dyamic rage is ivestigated experimetally Compare three schemes of digital sigal modulated mm-wave subcarrier geeratio: LD resoace modulatio LD heterodyig with feed-forward modulatio Passive mode lockig with PLL stabilizatio D 40GHz mm-wave subcarrier geeratio by 4- modes DBR LD AM Mode lockig Reduce dispersio pealty by fiber gradig filter D, E Usig optical ijectio lockig to stabilized passive mode locked laser diode 37GHz mm-wave subcarrier is geerated D, E Passive mode locked is stabilized by optical ijectio lockig 60GHz mm-wave subcarrier is geerated Simultaeous geeratio of 40GHz subcarrier ad sigal modulatio (~5MHz Simplicity LD heterodyig: higher tuig rage: 100 GHz. Feedforward modulatio: easy to implemet for 1MHz liewidth LD Passive mode lockig with PLL: phase oise: offset Good phase oise: -109 offset Low isertio loss With optical ijectio lockig the phase oise decreased by 0dB (from 1MHz to 1MHz Thefrequecy stability of the 60GHz carrier is withi 50Hz over 1500 hour period Narrow badwidth: <5MHz No-flat respose Resoace modulatio: arrow badwidth LD heterodye: complexity. Passive mode lockig: o tuability, arrorw badwidth (<10MHz No tuability Itesity oise Less ideal phase oise performace No phase oise measuremet 1

27 13 Table. (cotiued 15 Park, Joh, et al. Broad-bad millimeterwave optical modulator usig a passively modelocked semicoductor laser with phase oise compesatio, IEEE Photoics Techology Letters, v 9, 5, May, 1997, p Amarildo J. C. Vieira, "Optical trasmitter with Millimeter-wave subcarrier", Ph.D. Thesis, Drexel Uiversity, Philadelphia, PA, Tog, D.T.K et al. Cotiuously tuable optoelectroic millimetre-wave trasmitter usig moolithic mode-locked semicoductor laser Electroics Letters, v3, 1, Oct 10, 1996, p Daryoush, Afshi S, Large-sigal modulatio of laser diodes ad its applicatios i idirect of optical ijectio lockig of millimeter wave oscillators, Ph.D. dissertatio, 1986, Drexel Uiversity, Philadelphia 19 Ogusu, M et al., 60 GHz millimeter-wave source usig two-mode ijectio-lockig of a Fabry-Perot slave laser, IEEE Microwave ad Wireless Compoets Letters, v 11, 3, March, 001, p We, Y.J, et al., Optical sigal geeratio at millimeter-wave repetitio rates usig semicoductor lasers with pulsed subharmoic optical ijectio, IEEE Joural of Quatum Electroics, v 37, 9, September, 001, p D mm-wave subcarrier geeratio by passive mode lockig LD OPLL is employed to suppress the phase oise Simplicity Good oise potetial: CNR 119 dbc/hz after reducig the PLL loop delay D Microchip laser mode lockig Low phase oise ad RIN oise Decet power Moolithic desig D D E E Cotiuously tuable mm-wave subcarrier geeratio by the combiatio of LD mode lockig, optical domai mode selectio ad exteral E/O modulatio Mode Lockig laser diode to geerate microwave subcarrier Lockig sigal is sub harmoics of cavity resoace frequecy Geeratig 60GHz mm-wave subcarrier by ijectio lockig mode F-P LD usig referece optical source Geeratig mm-wave subcarrier by ijectio optical pulse sequece with repetitio rate at the cavity resoace subharmoics Large tuability: >100GHz Simplicity Good tuig potetial Good phase oise: - for 56GHz for 35 GHz subcarrier Limited tuability: 400MHz Limited tuability Dyamic frequecy tuig is limited Less idea phase oise: CNR ~75dBc/Hz Complexity & lossy No tuability Less ideal phase oise due to the poor LD liewidth Phase oise: -9dBc/Hz at 100KHz offset Complex: eed a DFB master laser ad exteral phase modulator: Limited tuig rage: ~400MHz Complex 13

28 14 Table. (cotiued 1 Fukushima, S, et al., Optoelectroic sythesis of milliwatt-level multi-octave millimeter-wave sigals usig a optical frequecy comb geerator ad a uitravelig-carrier photodiode, IEEE Photoics Techology Letters, v 13, 7, July, 001, p 70-7 * Hog, Ji, et al., Tuable millimeter-wave geeratio with sub harmoic ijectio lockig i two-sectio strogly gaicoupled DFB lasers, IEEE PLT, v1, 5, 000, p Ralf Peter Brau, et al., Optical microwave geeratio ad trasmissio experimets i the 1- ad 60-GHz regio for wireless commuicatios, IEEE Trasactios o Microwave Theory ad Techiques, Vol 46 Issue: 4, Apr 1998, pp Ka-Sue Lee, et al., Geeratio of optical millimeter-wave with a widely tuable carrier usig Fabry-Perot gratig-les exteral cavity laser, IEEE Microwave ad Guided Wave Letters, Vol: 9, Issue: 5, May 1999, pp: E Geeratig ijectio source by optical comb Two frequecy compoets are selected from the optical comb ad ijected to two tuable laser diodes E Ijectio lockig (/mode lockig two modes of strogly gai coupled dual mode LD by sub harmoics of the mode spacig The mode spacig is cotrolled by the bias curret E, F 60GHz FM fiber radio lik based o two LDs heterodyig is evaluated. Discussio o two multi-carrier fiber radio schemes: Three LDs heterodyig Heterodyig a mode locked laser with referece diode laser 6GHz MM-wave subcarrier geeratio by ijectio-lockig the side bads of a mode locked laser diode source. F Heterodyig two logitudial modes of a FP gratig les exteral cavity laser to geerate 119GHz mm-wave subcarrier Usig gratig to adjust wavelegth Very broad tuig rage: 10 ~ 60GHz Tuable: mode spacig ca be varied from: 18 to 40GHz, idicatig GHz tuig rage Liewidth: <30Hz Evaluated ovel FM fiber radio lik, i which sigal modulatio ad mm-wave carrier is simultaeously geerated. Laser diode FM has higher FM idex. Tuable: > 110GHz Demostratig a PIC for two laser heterodyig Phase oise: 100Hz > 60 m wavelegth tuig Less ideal phase oise: 100Khz for 60GHz carrier Complex Lack dyamic tuability Need tuable referece source Laser diode has higher phase oise Phase lockig scheme is complicated No phase oise cotrol 14

29 15 Table. (cotiued 5 Wake, D, et al., Optical geeratio of millimeter-wave sigals for fiber-radio systems usig a dual-mode DFB semicoductor laser Davies, IEEE Trasactios o Microwave Theory ad Techiques, Vol: 43 Issue: 9, Sep 1995, pp Wis e, F.W. et al., Low phase oise GHz sigal geeratio usig multilaser phase-locked loops, IEEE Photoics Techology Letters, Vol: 10 Issue: 9, Sep 1998, pp: Matsuura, Shuji, et al., Tuable cavitylocked diode laser source for terahertz photomixig, IEEE Trasactios o Microwave Theory ad Techiques, v48, 3, 000, p Simois G. J. et al., Optical Geeratio, Distributio, ad cotrol of Microwaves Usig Laser Heterodye, IEEE Tras. MTT, vol. 38, No. 5, pp , May, Ramos R. T., A. J. Seeds, " Delay, Liewidth ad badwidth limitatios i optical phase-locked loop desig", Electroic Letters, vol. 6, pp , March Doi, Y. et al.; Frequecy stabilizatio of millimeter-wave subcarrier usig laser heterodye source ad optical delay lie IEEE Photoics Techology Letters, v 13, 9, September, 001, p E Ijectio lockig (mode lockig dual mode laser diode Geeratig 60GHz & 40GHz mm-wave subcarrier Phase oise: 10kHz offset F Three exteral cavity LDs heterodyig Geerates two subcarriers simultaeously Itegrated phase oise (0 to 5MHz offset rage: 0.01 rad rms F F F F THz sigal geeratio by laser diode heterodyig Usig high fiesse F-P cavity as the frequecy referece mm-wave subcarrier geeratio by direct heterodyig usig low oise solid state laser No lockig mechaism is employed Aalysis o liewidth limitatio o optical PLL for LD heterodyig LD heterodyig stabilized by frequecy discrimiator structure No electroic referece Terahertz sigal rage sigal Dc to 5GHz tuig rage Noise: 300KHz offset Ivestigatig the laser liewidth vs. the allowed PLL loop delay No eed for exteral referece sigal Potetially 100GHz tuig Narrowbad Not tuable No tuig mechaism is metioed. Noisy Liewidth <=1MHz Slow tuig speed: temperature tuig Cotiuous tuig rage limited by laser FSR Poor liwidth of LD makes the PLL implemetatio difficult Phase oise: very poor 15

30 16 Table. (cotiued 31 Fa, Zheca Frak, et al., Highly coheret RF sigal geeratio by heterodye optical phase lockig of exteral cavity semicoductor lasers, IEEE Photoics Techology Letters, v 10, 5, May, 1998, p Loh, W.H. et al., 40 GHz opticalmillimetre wave geeratio with a dual polarizatio distributed feedback fibre laser, Electroics Letters, v 33, 7, Mar 7, 1997, p Pajarola, S., Optical geeratio of millimeter-waves usig a dual-polarizatio emissio exteral cavity diode laser, IEEE Photoics Techology Letters, v8, 1, Ja, 1996, p Olbright, G. R, et al., Liewidth, tuability, ad VHF-millimeter wave frequecy sythesis of vertical-cavity GaAs quatum-well surface-emittig laser diode arrays, IEEE PTL v3, 9, Sep, 1991, p Yao, X. Steve, Optoelectroic oscillator for photoic systems, IEEE Joural of Quatum Electroics, v3, 7, Jul, 1996, p F F F-P exteral cavity LD heterodyig PLL is employed PLL is facilitated by additioal frequecy discrimiator feedback circuit Heterodyig dual polarizatio mode fiber laser Fiber laser is fabricated by DFB birefrigece Er doped fiber Large tuig rage potetial Large tuig sesitivity potetial Good phase oise: - 00kHz offset 1GHz carrier 40GHz beat betwee two polarizatio mode is observed Temperature sesitivity is decet low: 40MHz/C F Heterodyig dual polarizatio mode LD Measured beat frequecy tuable from 1 to GHz, with potetial 1330GHz rage F A, B, C Heterodyig the of outputs from a VCSEL array to sythesis frequecy statig from VHF to mm-wave Explore the potetial for THz operatio Optoelectroics oscillator aalysis ad experimetal results Differet lockig schemes is ivestigated Sythesis differet frequecy by ijectio curret modulatio ad VCSEL array locatio Large tuig rage (- 0GHz No eed for referece Phase oise potetial: - 140dBc/Hz at 10KHz up to 75GHz. Frequecy sythesis potetial Complex Lack cotiuous tuig rage due to the frequecy discrimiator feedback No phase stabilizatio is employed. Poor mm-wave liewidth: ~300Hz plus kilohertz rage driftig No tuability is metioed Poor phase oise Poor phase oise Complicated No dyamic tuig 16

31 17 Table. (cotiued 36 Yao, et al., Multiloop optoelectroic oscillator, IEEE Joural of Quatum Electroics, Vol 36 Issue: 1, Ja 000, pp: Davidso, T, et al., High spectral purity CW oscillatio ad pulse geeratio i optoelectroic microwave oscillator, Electroics Letters, v 35, 15, 1999, p Wag, Xiaolu, et al., Microwave/millimeter-wave frequecy subcarrier lightwave modulatios based o self-sustaied pulsatio of laser diode, Joural of Lightwave Techology, v 11,, Feb, 1993, p Judith Dawes et al., Dual-polarizatio frequecy-modulated laser source, IEEE PLT, Volume: 8 Issue: 8, Aug 1996 p Lightwave electroics corp.., catalog, Lightwave Electroics Corporatio, Moutai View CA 94043, 003 B B Multi loop OEO: the first loop for the resoator, ad the secod o for frequecy selectio Presetig a carrier suppressio schemes based o the fiber delay lie frequecy discrimiator for further phase oise suppressio Optoelectroics oscillator 1GHz --- Studies o LD self-sustaied pulsatio dyamics Experimetal demostratio of microwave subcarrier geeratio by LD self-sustaied pulsatio (SSP Direct frequecy modulatio o LD SSP F F Beatig two lasig modes of orthogoal polarizatio of a dual polarizatio solid state laser The frequecy of oe mode is electrooptically tued Beatig two o-plae rig lasers Beat frequecy is stabilized by a laser offset lockig accessory Employig multi-loops Measured phase oise: 10kHz for 10GHz carrier No eed for referece Good phase oise: - offset Frequecy sythesis potetial Measured tuability: 7GHz by varyig the ijectio curret SSP frequecy tuig sesitivity: 100MHz/mA Stable beat toe Easy couplig Wide tuig rage Low RIN oise: <165dBc/Hz above 10MHz Same as above Same as above Poor oise performace of LD SSP, free ruig liewidth: 50MHz. After optoelectroic feedback the liewidth is reduced to: 5KHz still ot good at all Noliear FM respose Sigificet parasitic AM Affected by the modal competitio oise Phase oise is ot clear Small tuig sesitivity, <1MHz/V Phase oise is high: 10kHz Cotiues tuig rage is small: <5GHz limited by FSR Slow tuig speed 17

32 18 Table.3: Comparisos betwee differet mm-wave subcarrier geeratio techiques Geeratio Approach Direct laser diode modulatio Exteral modulatio Laser mode lockig Resoace modulatio Ijectio lockig Descriptio Advatage Disadvatage The state-of-the-art Usig a mm-wave carrier source to directly modulate a high-speed laser diode Exterally modulatio of a clea CW laser source usig a Mach-Zeders modulator or electro-absorptio modulator at mm-wave frequecies Usig period perturbatio (FM or AM to sychroize logitudial modes of lasers (LD, microchip laser, etc ad geerate mm-wave pulses Amplitude modulatio of a laser diode usig a arrow bad source i the viciity of the laser diode frequecy spectrum rage. Two modulatio tues of a master laser are seeded ito oe or two slave laser (s, which produces mm-wave subcarrier output after photo mixig. Simple Efficiet Tuable with a mmwave source Low oise (depeds o mmwave drivig source Tuable with a microwave source Simple High modulatio idex Low oise High operatio frequecy High AM modulatio idex Capable of geeratig very high frequecy sigal Tuable with a microwave source Speed limited by LD badwidth High RIN oise High isertio loss Noliear respose Complex High cost 98GHz direct modulatio badwidth 110GHz modulatio badwidth 10KHz offset for 40GHz subcarrier Narrow bad process >100GHz carrier frequecy 10KHz offset for 40GHz subcarrier Narrow badwidth (less tuability No-flat respose Complex Noise depeds o seedig source >40GHz operatio frequecy ~10Mhz modulatio badwidth THz badwidth 10kHz offset Laser heterodyig Optical mixig of two sigle mode lasers to produce mm-wave sigal. Extremely broadbad tuability Potetially high tuig speed Potetially simple low cost implemetatio Complexity depeds o the laser oise spectral quality Noise depeds o laser spectral quality ad oise cotrol sub system THz badwidth < 10kHz 18

33 19.3 A ew approach of geeratig low oise, tuable mm-wave optical subcarrier As show i table.3, the curretly employed methods have some weakesses. Therefore, a ew approach that is better for geeratig a low oise mm-wave optical subcarrier with a wide tuig rage is cosidered i this thesis work. Specifically, we explore a optical trasmitter (Fig..3 comprised of two sigle mode electro-optically tuable microchip lasers, the outputs of which are heterodyed to yield a mm-wave optical subcarrier. The microchip laser cavity is comprised of a short gai sectio ad a log electrooptic modulator sectio. Whe a voltage is applied across the electro-optic modulator sectio, its wavelegth is shifted. Cosequetly, the frequecy of the mm-wave optical subcarrier is chaged. Sice the chage occurs i the optical domai (>10 14 Hz, sigificat frequecy tuig ca be achieved at mm-wave frequecies with oly a miute chage i optical wavelegth. I order to assure low oise, the trasmitter icludes a oise cotrol apparatus to suppress the phase oise from the optical sigal. It could be a optical phase locked loop or some other ovel techique, such as optical frequecy locked loop (Sectio 5.. As will be show i the Chapters 4 ad 5, this approach eables us to geerate a low oise 10kHz offset idepedet of frequecy, rapidly tuable (>8.8GHz/µs experimetal, >8.8GHz/s theoretical mm-wave subcarrier with wide tuig rage (cotiues tuig rage >40GHz experimetal. The performace meets the requiremet for various applicatios (Table.1.

34 0 Tuig voltage potetial Electro-optic tuable microchip laser Electro-optic tuable microchip laser Tuable mm-wave optical subcarrier Phase oise cotrol Figure.3: A ew approach for tuable mm-wave optical subcarrier geeratio I the followig paragraphs, we review the pertiet literature regardig microchip lasers ad oise suppressio techiques for laser heterodyig..3.1 Microchip lasers Microchip lasers are comprised of a short moolithic plao-plao cavity formed by a gai material ad possibly other elemets such as a electro-optic or oliear crystal, or Q-switch. A microchip laser combies some of the best features of solid-state lasers (high spectral quality ad semicoductor laser diodes (such as compactess ad efficiecy. Zayhowski, et al., performed much of the pioeerig research o microchip lasers i the early 1990s [14, 15]. Microchip lasers were origially used oly as sigle mode (trasverse ad logitudial laser sources. Later o, with the iclusio of oliear ad electro-optic elemets withi the optical cavity, more sophisticated applicatios emerged. The itracavity secod harmoic geeratio (SHG [16] ad third harmoic geeratio (THG [17] have bee demostrated. Microchip lasers have bee Q-switched actively

35 1 [18] or passively [19] to geerate optical pulse trais with high repetitio rates. They have already bee used i microwave photoics as well. Vieira A. J., et al., used a actively mode locked microchip laser to geerate a low oise mm-wave subcarrier sigal [0]. I the followig, we will review the importat operatio priciples ad the state-ofthe-art for microchip laser systems. i. Sigle mode operatio Microchip lasers geerally operate i a sigle logitudial mode because their cavity legths ad pump absorptio depths are short ad thus the spatial hole burig effect iside the F-P (Fabry-Perot stadig wave cavity is reduced [1,]. The microchip lasers trasverse mode cofiemet is give by the thermal guidig / or thermal expasio effect from the pump[3]. A microchip laser will operate i sigle trasverse mode if the pump beam cross-sectio is smaller tha the fudametal spatial mode cross sectio. ii. Frequecy tuig of a microchip laser Microchip lasers ca be tued by thermal expasio [4], the piezo-electric effect [5], ad the electro-optic effect [6]. State-of-the-art microchip laser system use electrooptical tuig. A frequecy modulatio of GHz has bee demostrated usig composite Nd:YAG / LiTaO 3 electro-optic tuable microchip lasers. The cotiuous tuig rage of microchip lasers is limited by the cavity free spectral rage (FSR, which is the iverse of the legth cavity legth. A smaller cavity legth is required i order to achieve a larger tuig rage.

36 iii. Microchip laser spectral purity Superior spectral purity is the major advatage of (solid state microchip lasers over (semicoductor laser diodes. The microchip laser liewidth (or phase/frequecy istability is cotributed by two idepedet processes: 1 quatum oise due to spotaeous emissio [8]; ad oise due to crystal thermal vibratios [14]. For a microchip laser system, the laser liewidth due to the spotaeous emissio is i the rage of 1 Hz, ad has a Lorezia shape, which is give by (Schawlow ad Towes [8]. The thermal vibratio of the crystal lattice results i radom variatio of effective cavity legth, which is the pricipal cotributio of liewidth for all microchip laser systems. The liewidth broadeig by thermal vibratios is i the rage of 10 khz. iv. Microchip laser review summary Microchip lasers have superior spectral quality over laser diodes. They are more compact ad efficiet whe compared with traditioal solid-state laser systems. Whe a electro-optic elemet is placed iside the laser cavity, the microchip laser ca be tued at high speed. The electro-optically tuable microchip laser is a ideal cadidate for the purpose of geeratig rapidly tuable mm-wave optical subcarriers..3. Noise cotrol techiques The optical heterodye processes dowcoverts optical phase oise ito the mmwave subcarrier. I order to assure a clea sigal, a meas of cotrollig phase oise is required. Most of the existig heterodye schemes employ a phase locked loop (PLL system to lock the heterodye beat toe to a low oise referece. The suppressio of laser phase oise by PLL is a fuctio of its gai ad loop badwidth. Due to the good spectral

37 3 quality of microchip lasers, a PLL with moderate loop gai ad badwidth is sufficiet for the phase oise cotrol of the microchip lasers. However, the performace of all the existig phase locked loops is ultimately limited by the stability of the referece source. Whe the tuable trasmitter is operatig at frequecy above 30GHz, such a referece source becomes icreasigly difficult to attai. This is a commo problem for all PLLs at higher frequecies. I the literature, the oly scheme that produces frequecy idepedet phase oise performace is the Optoelectroic oscillator (OEO, i which a log fiber delay lie is employed to serve as a resoator to provide frequecy referece. The bestrecorded phase oise performace [55] of a OEO is 10kHz. I sectio 5., a ew scheme ispired by the OEO is explored to achieve the frequecy-idepedet phase oise performace..4 Laser itracavity FM dyamics I this thesis work, oe fudametal questio eeds to be aswered: how fast ca the mm-wave subcarrier be tued? The electro-optic effect has practically ulimited speed [30]. However, it remais a questio as to how the dyamics of the laser system will respod to the itracavity modulatio. I this sectio, we review the pertiet work o the theoretical backgroud of laser itracavity frequecy modulatio (FM. The research effort i laser itracavity FM dyamics is divided ito two related areas, amely the laser frequecy switchig from oe steady state to aother by a tuig sigal, ad the laser FM oscillatio, where the laser is uder periodic perturbatio i the viciity of its cavity free spectral rage (FSR.

38 4.4.1 Laser frequecy switchig Laser frequecy shiftig by meas of perturbig the refractive idex of a itracavity electro-optic elemet was suggested first by Yariv [31]. Usig a steady state argumet, he treated the laser frequecy shift as the cosequece of the tuig i cavity resoace frequecy, ad o iformatio o the trasiet behavior betwee the steady states was preseted. Later, Geack et al., proposed a improved theory [3]: Durig electro-optic tuig, two related processes occur: first, whe the optical wave propagates through the electro-optic sectio, the optical phase is chaged by the phase modulatio ad the time-varyig phase chage results i a chage i istataeous frequecy; secodly, the electro-optic phase modulatio perturbs the cavity legth ad iduces a chage i the cavity resoace frequecy. These two processes are sychroized, ad lead to the followig dyamic behavior: The optical frequecy icreases stepwise every cavity roud trip time; ad, A periodic frequecy oscillatio occurs eve after the rampig. By takig ito cosideratio the phase modulatio o the istataeous optical frequecy chage, Geach s theory is a great improvemet. However, it does ot cosider the fiite legth of the electro-optic sectio ad it also does ot icorporate the dyamical cotributios due to the laser gai medium. Ufortuately, from the literature review, o other research work appears to have addressed these importat effects..4. Laser FM oscillatio Aside from the laser frequecy switchig, there is a sigificat amout of research work o laser FM oscillatio. Laser FM oscillatio is studied both i the frequecy domai [33, 34, 35] ad i the time domai [36, 37].

39 5 i. Frequecy domai approach The frequecy domai theory of laser FM oscillatio uses the coupled mode method [35]. I this approach, the optical field is decomposed ito a superpositio of the Fourier modes, ad the electro-optic perturbatio (FM modulatio itroduces a couplig betwee them. Three regios of laser FM oscillatios are distiguished based o the detuig from the laser free spectral rage (FSR. They are: 1 FM oscillatios, distorted FM oscillatios, ad 3 FM mode lockig. Whe the perturbatio frequecy is far away from the cavity FSR, the laser operates i a pure FM oscillatio at steady state, i which the laser output behaves as a ideal frequecy modulated optical sigal. The FM modulatio idex is give by Γ = c 1 δ L v where c is speed of light i a vacuum, L is the cavity legth, v is the detuig betwee the modulatio sigal ad the cavity FSR, ad δ is the couplig betwee the Fourier modes due to the electro-optic perturbatio. The FM idex icreases rapidly as the detuig is reduced. Whe the detuig is reduced below a certai value, the FM operatio eters the secod regime: distorted FM, where amplitude modulatio at twice the modulatio frequecy occurs. Whe the detuig is further reduced, the laser geerates a pulse trai with a repetitio rate equal to the modulatio frequecy. This mode of operatio is laser FM mode lockig. The disadvatage of the coupled mode approach is that the coupled mode equatios become icreasig difficult to solve umerically whe the perturbatio is very close to the cavity resoace. [35]

40 6 ii. Time domai approach Compared with the coupled mode approach, the time domai approach is much simpler for aalyzig the laser behavior i the distorted FM ad mode-lockig regios [36]. I essece, the time domai approach uses the self-cosistece coditio (per roud trip of the laser complex field evelope to aalyze the steady-state behavior uder periodic perturbatio. For a homogeously broadeed fast gai laser, the self-cosistet equatio is [37]: u( τ + TR = [1 + g l + Dg τ + i cos( ω τ ] u( τ m where u is the complex evelope of the optical sigal, τ is the time variable, T R is the cavity roud trip time, g is the gai per roud trip, l is the cavity loss per roud trip, D g accouts for the frequecy limitig effect, ad is the phase shift per pass itroduced by the electro-optic perturbatio. Usig the self-cosistet equatio, the laser steady state behavior ca be calculated. I the exact FM regio, the time domai approach yields the same result as the coupled mode approach. I the distorted FM regio, the steady state amplitude distortio ca be obtaied aalytically i the time domai approach [37] as: where Γ is the FM idex, ad 3 D g Γ ω m A ( τ = exp si( ω mτ 4 ω m is the modulatio frequecy. Whe the laser is operatig i the FM mode lockig regio, the self-cosistecy coditio implies a Gaussia pulse trai propagatig iside the laser cavity. The idividual pulse shape is [38], u( t = exp( γ t

41 7 where γ = 1± i ω ( /16D 1/. ( m g However, it should be emphasized that the self-cosistet equatio does ot fully iclude the dyamical cotributio of the gai medium. Istead, it assumes a costat gai ad a Gaussia form frequecy limitig of the active medium. This approximatio is ot valid for microchip laser systems where the gai liewidth is comparable to the cavity mode spacig ad the gai profile has a Lorezia shape. iii. Compariso of the two approaches. The coupled mode approach provides a meas to aalyze the laser FM dyamics both at trasiet ad i steady state. However, whe the detuig is small, simulatio employig the coupled mode approach becomes difficult. The time domai method is simpler i explaiig the steady state behavior. However, it is ot able to aalyze the trasiet behavior. More importatly, the uderlyig assumptio for the time domai approach is ot proper for microchip laser systems Summary o laser FM dyamics review A satisfactory theory of the dyamical behavior of a microchip laser tuig does ot exist. The laser frequecy switchig theory does ot icorporate the dyamics of the gai medium ad it also eglects the fiite legth of the electro-optic tuig sectio, which is ot appropriate i the case of a microchip laser. O the other had, the laser FM oscillatio theory oly studies siusoidal modulatio, which is ot the pricipal mode of operatio for the tuable microchip laser (e.g., high-speed radom bit stream for MSK fiber radio ad high excursio ramp sigal for hybrid lidar/radar. The existig time domai method is oly useful for steady state situatios ad its treatmet of the dyamics of the gai medium is oversimplified for microchip lasers, where the cavity FSR is

42 8 comparable to the gai badwidth of the laser medium. The coupled mode approach is more accurate i modelig the gai medium. However, it becomes icreasigly difficult to study umerically ear the resoace frequecies. Furthermore, oe of the existig coupled mode approaches is based o the state-of-the-art theories, amely the Maxwell- Bloch [40] equatios. I summary, a ew FM dyamics theory suitable for a microchip laser system is eeded..5 The objectives of this thesis work The thesis work has to two objectives: Desig, fabricate, ad characterize a low oise, rapidly tuable mmwave optical trasmitter usig electro-optically tuable microchip lasers. Develop a ew model for microchip laser FM dyamics based o the Maxwell-Bloch formulatio that is i agreemet with the empirical characterizatio of the electro-optically tuable microchip laser

43 9 Chapter 3: Dyamics of the tuable microchip laser The pricipal goal of the research preseted i this thesis is the optical geeratio of low oise mm-wave sigals by heterodyig two microchip lasers. I this ivestigatio, oe of the two lasers is operated i steady state ad the other laser is tued via a embedded electro-optic crystal. This chapter cocers the dyamic model of the tuable microchip laser. The itracavity electro-optic tuig of the microchip laser represets a oliear dyamic pheomeo, i which the laser gai medium, electrooptic tuig sectio ad cavity resoace all cotribute importat fuctios. I this chapter, we develop a model that takes ito cosideratio the complicated iteractio betwee the laser gai medium, electro-optic tuig, ad cavity resoace. The developmet of the model cosists of the followig steps: Problem idetificatio, i.e., the experimetal microchip laser system, its iput ad output parameters are defied as the basis for the theoretical aalysis Formulatio usig the Maxwell-Bloch equatios for a uidirectioal rig laser with a embedded electro-optic tuig elemet. Derivatio of a approximate aalytic solutio for the case where the amout of frequecy tuig is small compared with the active medium gai badwidth. Simulatio based o the direct umeric itegratio of the Maxwell-Bloch equatios.

44 Problem idetificatio The tuable microchip laser we use i the experimets is depicted i Fig V(t Tuig Voltage Dielectric mirror Dielectric mirror Optic Pump Gai: Nd: YVO4 E/O Modulator LN Laser Output Figure 3.1: The electro-optic tuable microchip laser The laser employs a composite cavity structure comprised of a short Nd:YVO 4 gai sectio that is followed by a log LiNbO 3 electro-optic tuig sectio. The Nd:YVO 4 is a very efficiet gai material [4] with very short pump absorptio depth so that most of the pump power is absorbed i the regio very close to the laser mirror, where all the logitudial modes have a commo ull poit. Therefore, the spatial hole burig effect that leads to multi-logitudial mode operatio is reduced ad the sigle logitudial mode operatio is achieved [1]. Whe a voltage is applied to the LiNbO 3, it perturbs the refractive idex iside this sectio, which results i a optical frequecy shift i the laser output. The idividual compoets i the laser system are summarized i Table 3.1.

45 31 Table 3.1: The compoets iside tuable microchip laser Compoets Material Fuctios Gai sectio Nd:YVO4 Provide gai iside the cavity Electro-optic tuig sectio LiNbO3 Itroduce iteractio betwee the laser field ad applied tuig electric field (voltage Modify the cavity resoace coditio Ed mirrors Two dielectric mirrors Provide for a F-P resoace cavity Diode pump 808m ope heat sik diodes Excite the gai medium The goal of this chapter is to aalyze the microchip laser system ad predict its output whe subjected to a applied tuig voltage. From the system s poit of view we defie the followig iput/output parameters, which reflect the experimetal variables. Iput parameters: Semicoductor diode pump which is held costat (808m wavelegth ad approximately 00mW itesity Laser temperature, which is held costat Applied voltage, which is cosidered arbitrary for the theoretical developmet. However, from the perspective of applicatios three types of applied potetial fuctios are of iterest: o Voltage ramp of the form V = V 0 t, which produces a chirp output o Siusoidal sigal of the form V=V 0 cos(ωt. o Biary digital basebad sigal of the form: V = I g ( t T, where I is digital data, I (t is the basebad waveform, ad T s is the symbol rate. g I s

46 3 Laser system output: Amplitude, frequecy (wavelegth of the optical field. 3. Problem formulatio The microchip laser used i the experimets employs a Fabry-Perot (F-P stadig wave cavity, as show i Fig However, this cavity coceptually resembles a uidirectioal rig because of the reduced spatial hole burig effect. Therefore, we choose a stadard uidirectioal rig cavity structure for the laser modelig to reduce the complexity of the derivatios [40]. Pump Isolator Sectio 4 Sectio 1 Gai Sectio Z=0 Z=z 1 =L E/O Modulator Partial reflector: R Sectio Output Z=z 3 Z=z Sectio 3 Figure 3.: A uidirectioal rig cavity with electro-optic tuability The uidirectioal rig structure for the laser modelig is depicted i Fig. 3.. A ideal optical isolator is placed iside the cavity to assure that light oly propagates i the clockwise directio. The spatial distat z is refereced alog the light propagatio directio iside the cavity, ad the origi (z=0 marks the start of the gai sectio. The

47 33 gai (sectio 1 ad electro-optic tuig (sectio 3 regios are separated by two segmets of pure delay paths (sectio ad 4. For the curret derivatio these two sectios have o sigificat fuctio. However, i the log term we aticipate the isertio of two fuctioal compoets ito the laser cavity, a optical filter ad a piezoelectric tuig elemet. The filter could be used to tailor the gai profile if eeded, particularly if alterate gai media are used (e.g. Er:glass. The piezo-electric tuer could provide a efficiet meas to adjust the cavity resoace to the desired iitial value. Except for the electro-optic tuig sectio, Fig. 3. represets a stadard cavity structure used i laser studies [40]. The approach followed here has 4 steps: i. Modelig of each elemet idividually iside the laser cavity ii. Solvig the delay ad electro-optic sectios aalytically, ad determiig the closed-form mappigs betwee optical field at the iput ad output boudaries of the electro-optic sectors. iii. Obtaiig the boudary coditio for the gai sectio, i.e., the mappig betwee the time varyig field at z=l to the field at z=0, based o the closed-form solutio of the electro-optic segmet. iv. Itroducig the variable ad coordiate trasforms to obtai the boudary coditio i periodic form. The gai sectio ad electro-optic tuig sectio are discussed first i sectio The gai sectio is modeled usig the stadard Maxwell-Bloch formulatio [40] ad the optical wave propagatio equatio iside a electro-optic medium is employed to describe the electro-optic tuig sectio.

48 34 The modelig of the etire tuable laser system is performed i sectio 3.., i which we first derive a aalytic solutio of the electro-optic tuig sectio ad the use this result to determie the boudary coditios for the tuable laser system ad obtai the equatios of motio for the etire tuable laser system Modelig of compoets I this sectio, we model the idividual compoets of the laser. As show i Fig. 3., the overall laser system is comprised of: the gai, electro-optic, ad delay sectios. The delay sectios itroduce a costat time delay, which ca be directly specified (see Table 3.. I the modelig process, we do ot cosider the trasverse profile of the lasig mode. The space coordiate, z, i the subsequet derivatios is the physical distace multiplied by the iitial idex of refractio, or the effective optical legth. The optical field iside the laser cavity ca be expressed i terms of a plae wave carrier ad a slowly varyig field evelope 1 E( t, z = [ ξ ( z, t e i ( kc z ωct + C. C.] where ξ ( z, t is the evelope of the optical field, ω c π c = m is the iitial cold cavity Λ (i.e., zero applied voltage across the electro-optic sectio resoace frequecy that is closest to the atomic resoace, m is a iteger, Λ is the iitial optical legth of the cavity, c is the speed of light i a vacuum, ad k c is the wave umber, k c ωc =. It should c be oted that eve though the selectio of the plae wave carrier frequecy ca be ay value ear the atomic resoace, we choose the frequecy for the cavity resoace to reduce complexities i the boudary coditio.

49 35 Next, we are goig to model each elemet of the cavity. i. Gai sectio Our aalysis of the dyamic behavior iside the gai medium is based o the Maxwell-Bloch formulatio [40] for a plae wave optical field ad a homogeeously broadeed two-level active medium. I order to obtai a simpler form of the equatios of motio, we employ the dimesioless field evelope, i.e., the Rabi frequecy, ~ µξ Ω = h /( γ γ 1/ [40], where γ ad γ are the decay rates of drivig polarizatio ad populatio iversio, respectively. The Rabi frequecy will be used i the subsequet sectios as well. Uder the slowly varyig field evelope approximatio (SVEA, the equatios of motio for the dimesioless field evelope ad atomic variables have the followig form [40]: ~ ~ Ω( t, z Ω( t, z + c = c α P( t, z (3.1.a t z P( t, z t ~ ~ = γ (1 + iδ P( t, z + Ω( t, z d( t, z (3.1.b AC γ d ( t, z 1 ~ * eq = γ ( Ω( t, z P ( t, z + c. c. γ ( d ( t, z d ( t, z (3.1.c t where P(t,z is the dimesioless evelope of the macroscopic drivig polarizatio, d is the populatio differece per atom betwee the upper ad lower lasig levels, µ is the microscopic dipole momet of the laser medium, d eq is the equilibrium iversio differece per atom, α is the usaturated gai per uit legth, adδ ~ AC is the ormalized detuig betwee the atomic resoace ad the selected cavity resoace i uits of γ. Next, we ll discuss the modelig of the electro-optic tuig sectio.

50 36 ii. The electro-optic modulatio sectio Usually, the electro-optic sectio is dealt with as a lumped elemet phase modulator [3-38], where the time varyig voltage itroduces a optical phase chage proportioal to the voltage. However, i the case where the legth of the electro-optic tuig sectio is comparable to the etire laser cavity legth, or if the chage i the modulatig waveform caot be eglected durig the trasit time withi the electro-optic sectio, this assumptio is o loger valid because the travelig wave effects may be sigificat. Therefore, we employ a rigorous wave propagatio model, i which the optical field iside is described by the followig wave propagatio equatio [43]: z E( z, t c t E( z, t = 1 c ε 0 t P e / o ( z, t (3. where is the refractive idex of the electro-optic material whe o voltage is applied, E(z,t is the optical field iside the electro-optic sectio, P e/o is the drivig polarizatio resposible for electro-optic effect. For a z cut LiNbO 3 electro-optic crystal, P e / o = ε d E( z, t E ( t ( tuig where d 33is the oliear optical coefficiet ad E tuig is the tuig field. The oliear polarizatio term o the right had side of Eq. 3. is resposible for the electro-optic modulatio. Applyig the SVEA to Eq. 3. ad assumig the applied tuig voltage is slow-varyig compared with the oscillatio of the optical field (~10 15 Hz, we obtai the propagatio equatio for the complex evelope of the optical field ξ ( z, t t + c ξ ( z, t z ω c = i d33 E ( t ( z, t tuig ξ (3.4

51 37 where ξ is the complex field evelope. Sice the electro-optic elemet is placed iside the laser cavity, it is more coveiet to employ the dimesioless field evelope Ω ~ (i.e., Rabi frequecy. The result is Eq ~ ~ Ω( z, t Ω( z, t ω + c = i t z 0 0 d 33 E tuig ( t Ω ~ ( z, t (3.5 Recall that for the laser modelig, we use the effective optical legth for the space coordiate. Therefore, there is o explicit depedece o the refractive idex o the left had side of Eq Table 3. summarizes the modelig equatios for the optical field evelope ad iitial coditios i each sub-sectio of the laser cavity. I the ext sectio, we will derive the aalytical solutio of the electro-optic sectio, ad use it to obtai the boudary coditio for the gai sectio ad the formulatio for the dyamics of the etire tuable laser system.

52 38 Table 3.: Field evelope equatios, boudary ad iitial coditios, i each sub-sectios of the laser cavity Sectio id Modelig equatio(s Field cotiuity coditios Iitial Coditios Gai Sectio Maxwell-Bloch Equatio for plae waves: Maxwell field: Iitial steady state ~ ~ Ω Ω E( z = 0, t = E( z = Λ, t Field field: + c = c α P evelope: Delay sectio (sectio t z P ~ ~ = γ ( 1 + iδ AC P + γ Ω d t d 1 ~ * ~ * = γ ( Ω P + Ω P γ ( d d t Delay: ~ Ω ~ ( z, t = Ω( t z / c eq ~ Ω ~ ( z = 0, t = Ω( z = Λ, t Maxwell field: + E( z = z1, t = R E( z = z1, t Field evelop: ~ + ~ Ω( z = z1, t = R Ω( z = z1, t ~ Ω stead ( z, t = 0 The iitial steady state field: ~ Ω stead ( z, t = 0 Commets Ω: Rabi frequecy P: the drivig polarizatio d: the iversio probability d eq : the equilibrium iversio probability c: speed of light α : usaturated field absorptio coefficiet δ ~ AC : Detuig betwee the cavity resoace frequecy ad atomic resoace γ, γ C : decay costat R: The output mirror reflectivity. Electro-optic sectio Delay Sectio (sectio 4 Wave propagatio equatios uder slowly varyig evelop approximatio: ~ ~ Ω Ω ω0 ~ + c = i d 33 E ( t Ω tuig t z Delay: ~ Ω ~ ( z, t = Ω( t z / c 0 Maxwell field: + E( z = z, t = E( z = z, t Field ~ evelope: + ~ Ω ( z = z, t = Ω( z = z, t Maxwell field: + E( z = z3, t = E( z = z3, t Field ~ evelope: + ~ Ω ( z = z3, t = Ω( z = z3, t The iitial steady state field: ~ Ω stead ( z, t = 0 The iitial steady state field: ~ Ω stead ( z, t = 0 ω 0 : optical frequecy; d 33 : oliear efficiet of LN; E tuig : tuig electric field. 38

53 Overall cavity formulatio The overall cavity formulatio of the composite rig laser cavity requires cocateatio of the idividual system blocks. From Table 3., we fid that the equatios represetig the gai sectio are complicated ad have o closed-form solutio, while the electro-optic tuig sectio has a relatively simple field equatio. Therefore, the reasoable approach may be comprised of the steps outlied i the followig flow chart. Obtai the steady state solutio for the electro-optic ad gai sectios Steady state t=<0 E tuig (t=0 Solve E/O sectio aalytically ad derive the closed form relatioship betwee the iput ad output boudary of E/O sectio subject to the applied field ad the iitial coditio (t=0 provided by the steady state solutio E/O sectio Dyamic t>0 E tuig (t!=0 Derive the formal boudary coditio for the tuable laser cavity based o the solutio for the E/O sectio. The complete E/O tuable laser dyamics becomes the dyamics problem iside the gai sectio Gai sectio Dyamic t>0 E tuig (t!=0 Itroduce variable ad coordiate trasform to obtai the formulatio equatios with the ormal boudary coditio Figure 3.3: The electro-optic tuable laser formulatio process

54 40 For the time period t=0, o exteral electric field is applied ad the system is i steady state, with a kow solutio. This steady state serves as the iitial coditio for the system. At t=0, a electric field is applied, which perturbs the system. We aticipate that the pricipal effect of the applied field is to iduce a time depedet phase delay, or frequecy shift as the optical field propagates across the electro-optical sectio. I subsectio 3.., we solve the electro-optic sectio aalytically ad obtai the closedform fuctioal mappig from its iput to its output field evelope. Later, the mappig is employed to acquire a boudary coditio for the gai medium. Up to this poit, the complicated dyamics ivolvig the iteractio betwee the electro-optic sectio ad the oliear gai medium becomes a more maageable problem for the gai medium. Next, we explai this process i detail. i. Closed-form solutio of the electro-optic tuig sectio The objective of this sectio is to solve the propagatio equatio iside the electro-optic sectio (Eq. 3.5 with the iitial coditio, ad the kowledge of the tuig potetial. The propagatio equatio is a first-order partial differetial equatio for which a closed-form solutio exists (Appedix A. The complex field evelope at the output boudary of the electro-optic sectio is give by ~ ~ i Φ( ( z = z3, t = Ω( z = z, t 3 e t Ω τ (3.6 where t ω0 Φ( t = β Vtuig( u du, β = d 33, h is the thickess of the electro-optic t τ 3 h 0 modulator, ad V tuig (t is the applied voltage. As show i Eq. 3.6, the time depedet exteral field produces a time-varyig phase shift,? (t, ad a costat time delay, τ 3. The derivative of the time depedet

55 41 phase leads to the frequecy shift, dφ, which is resposible for frequecy modulatio dt iside the laser cavity. Next, we will use the closed-form solutio for the electro-optic sectio to obtai the boudary coditio for the gai sectio. ii. Boudary coditio for the electro-optic tuable laser I subsectio a, we solved the mappig betwee the iput ad output complex fields of the electro-optic tuig sectio. Now, we use this closed-form mappig to cocateate the delay paths, electro-optic tuig sectio, ad gai sectio to acquire a self-cosistet boudary coditio for the gai sectio. First, we write dow the optical field at ed of the gai sectio, z=l, E( z = L, t = 1 Ω ~ ( z = L, t [ µ /( γ γ h 1/ e i ( kc L ωct + c. c.] (3.7a The, from the result for the delay sectio 1 i table 3., we fid the optical field at z=z, is E( z = z, t = 1 Ω ~ ( z = L, t τ R[ µ /( γ γ h 1/ e i ( k z ω ct c. c.] c + (3.7b Employig the closed-form solutio (Eq. 3.6 of the electro-optical sectio, we get the expressio for the optical field at z=z 3, E( z = z 3 ~ 1 Ω( z = L, t τ τ, t = R [ µ 1/ /( γ γ h 3 e iφ( t e i ( k z 3 ω ct c. c.] c + (3.7c Fially, the optical field at z= Λ, ca be calculated similarly usig the result for delay sectio 4 (Table 3..

56 4 E(z =?, t = 1 R O ~ (z = L, t t t [ µ /(?? h 1/ t e 4 i (k c?? t e + c.c.] 3 4 iφ(t t c (3.7d Whe expressed i term of a field evelope ad a carrier, the optical field at z=0 is, E( z = 0, t = 1 R [ µ h Ω ~ ( z = 0, t /( γ γ 1/ e iφ( t τ 4 i ( kc 0 ω ct e + c. c.] (3.7e Sice E(z =?, t ad E ( z = 0, t describes the field at the same physical locatio, the followig boudary coditio for the field evelope has to be satisfied, Φ Λ Ω ~ ~ i ( t 4 c ( z = 0, t = Ω( z = L, t ( τ + + R e e i k 3 τ τ 4 τ (3.8a Noted that the carrier frequecy is selected to be the cavity resoace, i.e., k c π = m, m is a iteger, the boudary coditio is reduced to the followig form, Λ ~ ~ i Φ ( t 4 Ω( = 0, t = Ω( z = L, t ( τ 3 + τ + τ 4 R e τ z (3.8b The ew boudary coditio, Eq. 3.8b, together with the Maxwell-Bloch equatios, Eq. 3.1, defies the complete dyamic behavior of the etire tuable laser system. Except for the additioal time depedet phase lag iduced by the electro-optic modulatio, Eq. 3.8b is very similar to the boudary coditios of the lasers without electro-optic tuig [40]. The curret microchip laser system has a egligible air gap (< % total cavity legth. For simplicity of the discussio, we let the delay time i sectios ad 4 vaish, i.e., τ =τ 0. The the boudary coditio is reduced to, 4 = ~ ~ i Φ( t Ω( z = 0, t = Ω( z = L, t τ 3 R e (3.9

57 43 With a time delay ad a time depedet phase i (t e Φ, the boudary coditio give by Eq. 3.9 is complicated. Next, we will use coordiate ad variable trasforms to obtai a boudary coditio i ormal (periodic form. It should be emphasized that the techique employed to simplify the boudary coditio is valid eve whe the air gap exists. To elimiate the explicit time delay, the coordiate trasformatio itroduced by Beza ad Lugiato [44], is employed z = z t = t + τ 3 z / L (3.10 I (t,z coordiates, the boudary coditio is simplified: ~ ~ i Φ( t Ω ( z = 0, t = Ω( z = L, t R e (3.11 At this stage, it is coveiet to itroduce a field variable trasform to elimiate the time varyig phase ad loss i the boudary coditio: ~ lr z L i G( z, t Ω ( z, t = F ( z, t e e P( z lr z L i G( z, t, t = p( z, t e e (3.1 where F(z,t ad p(z,t are the ew field evelope ad polarizatio evelope variable, ad G(z,t is a ukow fuctio that satisfies the followig costrait: G ( z = 0, t = G( z = L, t + Φ( t (3.13 The, the boudary coditio for the ew field variable F(z,t is i the ormal form, F ( z' = 0, t' = F ( z' = L, t' (3.14

58 44 I order to complete the defiitio of the ew field ad polarizatio variables, the exact form of the fuctio G(z,t eeds to be determied. However, the costrait (Eq is rather loose ad a ifiite umber of cadidates exist for G(z, t. To select the most appropriate oe, we substitute the trasforms (Eq.3.1 ito the Maxwell-Bloch equatios (3.1 ad obtai the ew equatio of motio for the ew field ad polarizatio variables as the followig, c l R G G [(1 + b + c ] F( z', t' = c α p + F i F (1 + b + c (3.15a t' z' L t' z' p( z', t' t = γ ~ (1 + iδ G( z', t' p i p + t' AC γ F d (3.15b d ( z', t' 1 * eq = γ C ( F p + C. C.exp( z' (l R / L γ C ( d d (3.15c t where b = τ /τ, ad L 3 1 τ 1 = is the propagatio delay i the gai sectio. c Ispectig Eqs. 3.15, we fid that the equatio of motio for the tuable laser system will have a simple form whe imposig the followig additioal costrait for G(z, t, i G ( z e (1 + b t', t i G ( z e + c z', t = 0 (3.16 At first sight, the costrait give by Eq is just a mathematical coveiece. However, we will show later its physical iterpretatio. Equatio 3.16 yields, G ( z', t' = Y ( t' z'(1 + b / c (3.17 where Y(t is a fuctio to be determied by the iitial coditio ad the boudary coditio (3.13. We assume iitially Y(t is zero so that the ew field evelope variable

59 45 F(z,t is cosistet with the origial field evelope Ω ~ ( z', t', whe there is o electrooptic perturbatio. Thus, the iitial coditio for Y(t is the followig, Y ( t' = 0, t '< 0 (3.18 Substitutig Eq ito the boudary coditio (3.13 ad employig the iitial coditio (Eq. 3.18, we obtai the closed-form solutio for G(z,t : m= u / τ G ( z', t' = Φ( u m τ, m= 0 (1 + b z' u = t' (3.19 c whereτ = τ 1 + τ 3 is the roud trip cavity delay. Thus, we uiquely determied G(z,t. At this stage, we ca write dow the fial equatio of motio for the ew field ad atomic variables uder the ew coordiate system (t, z : + c' F ( z', t' = k(c p + F t' z' (3.0a p( z', t' t ~ = γ (1 + iδ p i f ( z', t' p + AC γ F d (3.0b d ( z', t' 1 * eq = γ C ( F p + C. C.exp( z' (l R / L γ C ( d d (3.0c t with the periodic boudary coditio (Eq. 3.14, F ( z' = 0, t' = F ( z' = L, t' where c c =, 1 + b k c l R =, ad L C αl = are the effective speed of light, cavity l R decay rate costat, ad ormalized small sigle pass gai of the laser system, ad m= u / τ G( z', t' f ( z', t' = ( = β [ V( u m τ V( u m τ τ3 ], t' m= 0 (1 + b z' u = t' (3.1 c

60 46 where V(t is the applied tuig voltage. f ( z', t' ca be treated as a modulatio of the cavity resoace frequecy. The equatios of motio (Eq. 3.0 together with the boudary coditio (Eq represet the formulatio of the etire composite cavity tuable laser system. Before proceedig further, we discuss the iterpretatio of e ig( z', t'. I the (t,z coordiate system, Eq represets a complex evelope propagatig i a passive, dispersioless ad lossless medium with speed of c/(1+b. While i the real (t, z coordiates, Eq describes a evelope that is propagatig i such a medium with a speed of c. Therefore, ig( z ', t' e represets the optical field evelope iside a passive, lossless, uidirectioal rig cavity with electro-optic modulatio as show i Fig perfect reflector Sectio 1 perfect reflector z=0 z=l V(t Sectio 4 Sectio perfect reflector E/O Modulator Z=z 3 Z=z Sectio 3 perfect reflector Figure 3.4: A ideal model of a lossless rig cavity uder electro-optic modulatio Based o this iterpretatio, the ew field variable F(z,t ca be treated as a modificatio imposed by the gai medium o the optical field propagatig i the ideal lossless, passive cavity i Fig The fial equatios of motio (Eqs. 3.0 are similar to the laser modelig equatios obtaied by previous work [40], except for the iclusio

61 47 of modulatio term if ( z', t'. The modulatio term iduced a perturbatio o the detuig betwee the cavity resoace ad atomic resoace. I sectio 3.4, we will simulate the behavior of the dyamically tuable laser system usig the equatios derived above. But ext, the model represetatio of the laser would be derived based o the periodic boudary coditio (Eq. 14. Meawhile, the sigle mode operatio coditio for the laser uder itracavity frequecy modulatio will be addressed Model represetatio ad uiform field limit Based o the periodic boudary coditio (3.14, F(z,t, p(z,t ad d(z,t ca be expeded i Fourier series [40], F ( z', t' = F ( t' exp( i k z' p ( z', t' = P ( t' exp( i k z' d ( z', t' = d ( t' exp( i k z' where k = π L, is a iteger. It should be oted that the terms F (t, P (t, ad d (t are Fourier coefficiets represetig the modes of the laser i (z,t coordiates. The associate orthoormal modal fuctios are u ( z = 1 e L ik z with 1 ( δ L L * u, um = dz u u = m, 0

62 48 Substitutig the above equatios ito the equatios of the motio (Eq. 3.0, we fid that the ifiite set of time-depedet variables F (t, P (t, ad d (t obey the followig equatios: m m m m m F k c i P C k kf t F + = ' (3.a = m m m AC m d F P f i P f i i t P γ γ γ δ γ 0 0 ~ (1 ' (3.b (.. ( 1 ' * m eq m m d d C C P F L t d Γ + = C C γ γ (3.c where is the covolutio operator, = L eq z ik eq m d e dz L d 0 1 = L z ik z t f e dz L t f 0, ( 1 ' ( + = = Γ L R L z z ik L k i R R R d e dz L 0 l / ( l 1 1 1, is a iteger Thus, we obtaied the modal represetatio of the electro-optic tuable laser system, which ivolves couplig of a ifiite umber of modes ad a exact solutio is very difficult to obtai. I reality, the tuable laser operates as a sigle mode device, which leads to the expectatio that the 0 th order mode (=0 domiates. If this is true, the sigle mode approximatio, or the uiform field limit i the (t,z coordiate system, ca be employed to simplify the simulatio. By ispectig the model represetatio (Eq. 3., we fid that the sigle mode approximatio is valid if o modal couplig exists. That is the followig approximate coditio should be satisfied,

63 49 Γ Γδ,0 (3.3a f ( t ' 0 γ << 1 (3.3b The first approximatio (Eq. 3.3a repeats the previous result obtaied by Narducci et al. [40]. I order to satisfy the Eq. 3.3a, we must have: R 1 α L 0 (3.4 l R / αl O(1 The sigle mode requiremet specifically for the laser uder electro-optic tuig is give by Eq. 3.3b, which idicates that the electro-optic perturbatio oly itroduces a egligible couplig betwee the differet Fourier modes of the medium polarizatio. By Eq. 3.1, the coefficiet f (t' is calculated, β t' f 0 ( t' = V ( u du (3.5a τ t'-τ 3 f β [ V ( u V ( u τ ] ( t' t' = 3 e iω u du - τ (3.5b where ω π = τ. As show i Eq. 3.5, the requiremet for sigle mode operatio relies o the history, or the etire waveform of the applied tuig voltage. Next, we would discuss briefly the sigle mode operatio coditio for differet perturbatio sigals, amely, the liear ramp, ad periodic sigal. i. The liear ramp If the tuig potetial is a liear ramp, i.e., V ( t = α t, for t>τ 3, Eq. 3.5 yields,

64 50 β f 0 ( t' = τ 3 α ( t' + τ 3 / τ f ( t' = β τ α τ 3 iω e iω t' + (1 e ω iω τ 3 / τ 3 for 0 where α is the voltage rampig rate. I order to satisfy the requiremet for the uiform field limit uder electro-optic perturbatio (Eq. 3.4b, we must have, γ βατ 3 1 (1 e >> max, ω τ ω j ω τ 3 1 τ / τ 3 for 0 (3. 6 I practice, the medium gai liewidth ( γ of solid-state lasers are much bigger tha the frequecy modulatio durig a roud trip period ( βατ. Therefore, Eq. 3.6 is 3 always satisfied for the perturbatio uder liear voltage ramp. ii. The periodic perturbatio sigal Whe the perturbatio is periodic, the situatio becomes more complicated. Sice ay periodic sigal ca be treated as a Fourier series, we oly study the siusoidal sigal, i.e., V ( t = 1 Ae i( ωc +δω t + C. C. where A is the amplitude, ω c π = τ is the th cavity resoace frequecy, ad δω is the detuig betwee the frequecy of the perturbatio sigal ad cavity resoace. Usig Eq. 3.5, for t>τ 3, we get i δω t ' i ( δω ω t ' β e 1 i( ω + δω τ β e 1 3 i ( ω + δω τ3 f ( t = A (1 e + A (1 e for 0 τ i δω τ i ( δω ω I order to satisfy the requiremet for sigle mode operatio (Eq. 3.3b, we should have,

65 51 β 1 iω τ δω >> A (1 e 3 (3.7 τ γ If the etire cavity is uiformly filled with electro-optic medium, i.e., τ 3 = τ, the sigle mode coditio (Eq. 3.7 always holds. Otherwise, as the detuig is reduced to zero, the costrait give by Eq. 3.7 caot be satisfied. The uiform field limit, or the sigle mode approximatio is o loger valid. The laser the operates i distorted FM oscillatio or eve FM mode lockig regio [33]. Uder the sigle mode approximatio, the equatios of motio i modal form (Eq. 3. are reduced to the form of simple O.D.E., F ( t ' 0 = kf 0 + k t ' C P 0 (3.8a P ( t' ~ 0 = γ (1 i P0 F0 d 0 i f 0 ( t P0 ( t' t + δ AC + γ + (3.8b d 0 ( t' 1 * eq = γ C ( F0 P0 + C. C. γ C ( d0 d (3.8c t' Comparig Eq. 3.8 with the equatios of motio i the uiform field limit for lasers without electro-optic tuig, we fid that the electro-optic tuig simply itroduces a time-depedet perturbatio to the cavity resoace frequecy, amely, f (. However, the cavity resoace frequecy tuig is ot a simple fuctio of the tuig potetial. Istead, it assumes a itegral form (averaged over a time period τ 3. This is the result of the distributed modelig of the electro-optic tuig sectio. Whe the tuig voltage is kept costat, the cavity resoace frequecy tuig is reduced to the 0 t covetioal form, that is. f 0( V 3 t = β τ / τ.

66 5 Next, we will derive a approximate, closed-form solutio for the tuable laser with the assumptio that the frequecy tuig is small compared with the gai liewidth.

67 Liear dyamics model From sectio 3., we show that the solutio iside a coceptual lossless cavity ca be treated as the 0 th -order approximatio of the laser field by eglectig the frequecy pullig effect of the gai medium. The ew field variable F(z,t provides the correctio to this estimatio. As show i Eq. 3.1, because the electro-optic sectio does ot occupy the etire laser cavity, a periodic ad persistet oscillatio i the 0 th order estimatio of the optical frequecy, f(z,t will occur eve after the tuig voltage reaches a steady value. This raised the cocer whether or ot this oscillatio is a trasiet effect for the real optical field ad its decay time. I this sectio, we provide the aswer to this problem aalytically. The Maxwell-Bloch equatios are complicated ad the exact closed-form solutio is impossible to obtai. However, i may cases of iterest for microchip lasers system, the modulatio to the cavity resoace frequecy f ( z', t' is much smaller tha the gai liewidth ( γ. Thus, we ca liearize the dyamic system i the proximity of the iitial steady state (see sectio 3.3.1; the we will seek a approximate aalytical solutio i the frequecy domai (sectio 3.3. to prove aalytically that the frequecy ripple effect is trasiet; fially we calculate the time costats that gover the trasiet behavior as the laser frequecy reaches the steady state (sectio The objective is to idetify the ew steady state value of optical frequecy after the electro-optic perturbatio ad fid out how log it takes to reach this value.

68 Liearized Maxwell-Bloch equatios Whe the optical field is experiecig a frequecy offset from the iitial state, a sigificat amout of deviatio of its field evelope from the iitial value will occur. Therefore, it is improper to use perturbatio method directly for the field evelope. Istead, we employ the perturbatio techique for the modulus of the field evelope ad the relative phase. First, we express the evelopes of the field F(z,t, ad polarizatio, p(z,t, i terms of a modulus ad a phase F( z', t' = A F ( z', t' e i φf ( z', t' p( z', t' i φp ( z', t' = AP ( z', t' e (3.9 Substitutig Eq. 3.9 ito Eq. 3.0, we get: ( + c' AF ( z', t' = k(c Ap cosθ + AF (3.30a t z' Ap ( + c' φf ( z', t' = k C si θ t z' A F (3.30b Ap = γ AP + γ AF d cosθ (3.30c t ' ( φ t ' F + θ = γ ~ ( δ AC + ~ f ( z', t' γ AF d A P si θ (3.30d z lr/ L eq d = γ II AF AP cosθ e γ II ( d d (3.30e t ' where θ = φ p φ is the relative phase betwee the medium polarizatio P(z,t ad the F ~ 1 field variable F(z,t, ad f = f ( z', t' is the ormalized effective perturbatio. γ The dyamic system is i steady state before the voltage is applied. The, after the electro-optic tuig, the field ad atomic variables become:

69 55 A F = A + δa (3.31a st F F A P = A + δa (3.31b st P P d st = d + δd (3.31c θ = θ st + δθ (3.31d φ = φ + δφ (3.31e F st F F where the subscript st idicates steady state value. Based o the assumptio that the electro-optic perturbatio is much less tha the medium gai badwidth, ~ γ f ( z', t' << γ, we assume F δ A, δ AP, δ d, ad δθ << 1. It should be emphasized that o assumptio is made regardig the phase chage, δφ, which is ot ecessarily small. Substitutig Eq ito Eq. 3.30, we get the liearized equatios of motio for the termsδ AF, δ AP, δ d, δθ ad δφ F : st st st ( + c' δaf = kδaf k C cosθ δap + k C AP si θ δθ (3.3a t z' F + c' δφ t z' st P ( F = k C st ( AF A A k C A st P st F si θ cosθ st st δθ δa F 1 k C A st F si θ st δa P (3.3b δa t ' p st st st st st = γ δap + γ d cosθ δaf γ AF d si θ δθ (3.3c ( δφ t ' F + δθ = γ γ ~ f ( z', t' γ A A st F st P si θ st d A st st P si θ δd γ st A δa st F d A st P F st + γ A cosθ st F ( A st d st st P δθ si θ st δa P (3.3d

70 56 st st zl R/ L st st zlr / L δd = γ II AP cosθ e δaf γ II AF cosθ e δap t ' (3.3e st st st zl R/ L + γ A A si θ e δθ γ δd II F P Substitutig Eq..9 ito Eq. 3.14, we obtai the boudary coditio for A F ad φ F, II A ( z' = 0, t = A ( z' L, t (3.3f F F = φ F ( z' = 0, t = φf ( z' = L, t + π (3.3g For simplicity, we assume that iitially, the cavity resoace ad atomic resoace overlap, ad the laser is i the uiform field limit, which is valid for a high Q laser cavity. Uder this situatio, the steady state field, polarizatio, ad iversio desity ca be evaluated as: A A d θ st F st P st st = 1 = 1/ C = 1/ C = 0 (3.33 where C = c' α / k. The, Eq. 3.3 becomes, ( + c' δaf = kδaf k C δap (3.34a t z' δa t ' p = γ δa P + γ d st δa F (3.34b δd = γ t ' II st A δa P F γ II A st F δa P γ II δd (3.34c ( + c' δφf kδθ t z' = (3.34d

71 57 ~ ( δφf + δθ = γ f ( z', t' γ δθ (3.34e t ' Eqs. 3.34a~c ad Eqs. 3.34d~e are two idepedet sets of equatios. The first set is self-cosistet ad homogeeous. Thus, if stable, they will rest i the steady state. Therefore, the first order effect of electro-optic modulatio would ot chage the modulus of the field, polarizatio, ad iversio desity. The frequecy/phase iformatio is determied by Eqs. 3.34d~e with the boudary coditio. Equatios 3.34d~e ca be rewritte as: ( + c' δω t z' F = k δθ t' (3.35a δω + δθ = γ ~ F f ( z', t' γ δθ (3.35b t' where δω F is the istat frequecy chage of the ew field variable, δω F = F t' δφ. It satisfies the boudary coditio: δω ( z' = 0, t = δω ( z' L, t (3.35c F F = Thus, we have obtaied the liearized equatios describig the evolutio of optical frequecy as a fuctio of the applied voltage. Next, we derive its closed-form solutio i the frequecy domai Closed-form solutio i frequecy domai It is coveiet to study the equatios for the optical frequecy (Eq i the frequecy domai where Eq has a simpler form as show below, ( iω + c' δωf = k iω δθ (3.36a z' ~ δωf + i ω δθ = γ F ( z ', ω γ δθ (3.36b

72 58 where ', ( ~ ω z F is the Fourier trasform of ', ( ~ t z f. The correspodig boudary coditio i the frequecy domai is, ' ( 0, ' ( ω δω ω δω L z z F F = = = (3.36c Combiig Eqs. 3.36a ad 3.36b we get, ω γ ω γ ω ω δω ω δω ω γ ω i z F i k z z c z i k i F F + = ', ( ~ ', ( ' ' ', ( 1 ( (3.37 With the boudary coditio, Eq yields the followig closed-form solutio: 0, ' ( ~ 1 1 0, ' ( (1 ω γ ω δω ω γ ω τ ω γ ω τ = = = z F e e z i k i i k i F (3.38 where τ is the cavity roud trip delay. Note that the real evelope of the optical field is give by (see Eq. 3.1:, ( l, (, ( t z G i z L R e e t z F t z = Ω. Therefore, the tuig i the istataeous optical frequecy is: 0, ( ~ , ' ( 0, ( (1 ω γ ω δω ω δω ω γ ω τ ω γ ω τ = = = = = z F e e z z i k i i k i (3.39 where the coordiate trasformatio 3.10 is employed to chage Eq.39 back to the (z, t space. Thus, we obtaied the fial result of this derivatio, which gives the istataeous optical frequecy as a fuctio of 0, ( ~ ω = z F. The dyamics of the etire tuable laser reduces to a liear system, as is depicted i Fig I this simplified approach, the gai medium act as a liear filter H(?, which filters the error produced by the 0 th -order

73 59 estimatio of the optical frequecy. A sample of the trasfer fuctio, H(ω, is plotted agaist the ormalized frequecy i Fig. 3.6, where the cavity roud trip time is equal to 1/ γ ad the frequecy ad time parameters are ormalized with respect to γ. Figure 3.5: Trasformatio characteristics of the gai medium Figure 3.6: The trasfer fuctio of laser dyamic system.

74 60 Referrig to Fig. 3.6, we mote that close to zero frequecy the trasfer fuctio H(? is slightly below uity, which represets the precise shift of the cavity resoat frequecy. This meas that the chage i the istataeous optical frequecy is slightly smaller tha the chage i the cavity resoace. The behavior is attributed to the wellkow mode pullig effect[45]. More importatly, the trasfer fuctio is zero at the cavity FSR or its multiples. This suggests that the oscillatio existig i the 0 th order estimatio (i.e., f(t z ca oly be a trasiet effect. The gai medium will evetually smooth out these oscillatios. Next, we derive a closed-form estimatio for the decay time costat of the frequecy ripples, amely the time for the istataeous frequecy to reach the steady state The trasiet characteristics of the laser respose with frequecy tuig I this sectio, we study the trasiet characteristics of the laser frequecy as it reaches the ew steady state. The discussio here is i real (t, z coordiates. Examiig Eq. 3.1, we fid that after the electro-optic tuig, the effective electro-optical perturbatio f ( t, z is a periodic fuctio with a period equal to the cavity roud trip time. I the trasiet period, therefore, f ( t, z = 0 ca be expaded usig the Fourier series, f iω t ( t > 0, z = 0 = f e (3.40 where ω = π FSR, ad it is assumed here that the tuig voltage remais costat after t=0.

75 61 The Fourier compoet f iωt e ca be cosidered as a idepedet excitig source for the liear system, H(?. The, we expad the laser frequecy turig, δω ( z = 0, t, i terms of a Fourier series with a slow time-varyig Fourier coefficiet. δω i ω t ( z = 0, t = δω ( t e (3.41 If the chage of δω (t is egligible durig the cavity roud trip time, each spectral compoet δω jω t ( t e does ot iterfere. Thus, (t δω ca be determied from the local characteristics of the trasfer fuctio (Eq i the viciity of ay, we ca solve δω (t usig, { H ( ω } ω. Thus, for 1 δω ( t f u( t I (3.4 where u(t is a uit step fuctio, H ω ω is a approximatio of the origial trasfer fuctio H (ω ear ω, i.e., ( H ( ω ω H ( ω ω ω Next, we derive H (ω ad usig Eq. 3.4 to obtai the slowly varyig Fourier coefficiet, δω (t durig the trasiet period. To simplify the discussio, we assume the cavity decay rate, k, is much less tha the gai liewidth, which is valid for the lasers i cosideratio ( k~1ghz, γ ~ 300GHz. i. Time evolutio of the DC Fourier compoet Close to DC, the trasfer fuctio H(? ca be approximated usig, iω τ k γ + iω e 1 k H ( ω = γ (1 1 0 k iω τ + γ + i + k + i ω (3.43 (1 γ ω e 1 ω 0

76 6 The impulse respose of the approximatig system is, t k e k t t h + = ( 0 ( ( γ δ (3.44 The the Fourier compoet, ( 0 t δω should satisfy, + = = + (1 1 ~ ( ( ~ ( ( t k e k k f t h t u f t γ γ γ γ δω (3.45 As show i Eq. 3.45, the DC compoet decays expoetially to a ew steady state value (1 ~ 0 + γ γ k k f, with a time costat + γ γ 1 1 k. The steady state value is exactly the same as the result from the mode pullig formula obtaied i the steady state. ii. The time evolutio of the o-dc Fourier compoets Close to the o-zero multiples of the cavity FSR, the trasfer fuctio H(? ca be approximated as, /( 1 1 (1 ( (1 i k i i k i i k i i i e e H ω γ ω δω δω γ γ δω ω ω ω γ ω τ ω γ ω τ + + = (3.46 whereδω is the offset from ω, ω ω δω = Therefore, the th slowly-varyig Fourier coefficiet should satisfy, ( ( ~ ( t h t u f t = γ δω (3.47 where (t h is the impulse respose of H (d?. Usig Fourier aalysis, we get, t k t k i e e f i k i i f t + + = + + = I ~ /( ~ ( 1 ω γ ω ω γ ω γ γ ω γ ω δω γ δω (3.48

77 63 It is show i Eq that the th Fourier coefficiet experieces rigig, ad evetually decays to zero. Both the decay rate ad oscillatio frequecy are modedepedet (if we treat the Fourier compoets as modes, ad are give by γ ω + ω k ad γ γ ω + ω k, respectively. Higher order modes have a faster decay rate. Sice the optical frequecy ripples vaish as the o-dc Fourier compoets δω(t decay to zero, the decay rate of the frequecy ripples should be the slowest decay rate of the etire Fourier coefficiets, which is the decay rate of the first order Fourier mode δω (, i.e., 1 t γ ripple = γ ( π FSR + (π FSR k (3.49 If the laser cavity is very small, i.e., ( π FSR >> γ, the frequecy ripple decay rate is close to the laser cavity decay rate, k. iii. Discussio Usig the approximate closed-form solutio (Eq of the laser dyamics, we obtaied the decay rate for the 0 th ad higher order Fourier compoet iduced by the electro-optical perturbatio. The 0 th order compoet will coverge to a o trivial value give by the mode pullig formula. Therefore, it should be regarded as the sigal iduced by the electro-optical perturbatio. The speed of the respose is give by Eq. 3.44, i.e., γ, which is very fast for microchip lasers (>00GHz. The higher-order Fourier compoets vaish as the laser attais the steady state. However, it is a slow process compared with the output sigal. Therefore, they should be best treated as a effective oise source. Usig the result for sectio 3., we ca

78 64 approximate the oise level, which is a fuctio of the cavity roud trip time, relative legth of electro-optic sectio, ad the rate of ramp potetial. Next, we discuss the simulatio results. 3.4 Direct simulatio ad discussio I this sectio, the laser dyamic system (Eq. 3.8 is solved umerically by direct itegratio. The objective is to aalyze the trasiet behavior of the laser whe subjected to exteral force (applied potetial. We use a d -order two-dimesioal Ruge-Kutta method for the umerically solutio. A brief flow chart of the simulatio program is illustrated i Fig The detailed explaatio of this algorithm is attached i Appedix B. Start Iitial coditio: Fm, =0 Pm, =0 dm, =0 Calculate the field, polarizatio, ad iversio desity values of the ext time step (F,P,d m>0,+1 from the previous value (F,P,dm, usig d order R-K estimatio Calculate the field, polarizatio, ad iversio desity at the boudary: (F,P,dm=0,+1, from the boudary coditio No =+1 Eough time duratio Note: m represets the discrete space coordiate represets the discrete time coordiate Ed Figure 3.7: The flow diagram for simulatio

79 65 I sectio 3.1 we idicated that three types of applied sigals are o iterest: liear ramp (chirp, siusoidal, ad digital. We will focus o the laser respose uder liear ramp perturbatio because that describes best the laser trasiet dyamics. The simulatios are carried out for differet laser parameters ad ramp rate, ad the results are summarized i this sectio Dyamic respose of the laser I this sectio we examie the dyamic respose of the laser system for differet applied ramp voltages. The pertiet laser parameters are summarized i Table 3.3 ad, i geeral, these values correspod to the attributes of the lasers used i the experimets. However, the upper level lifetime is chose to be 4 orders of magitude smaller i order to make the umeric solutio less stiff [46]. Oe disadvatage of this chage is that it icreases the speed of the amplitude respose compared with the real laser. Table 3.3: Pertiet laser parameters for the simulatio Parameters Value Log(R: cavity loss 0.0 C: ormalized sigle pass gai 1.1 γ : gai liewidth 00GHz γ : decay rate of the iversio desity 1e+8 β: electro-optic modulator sesitivity 0MHz/Volt Effective total cavity legth electro-optic 4. mm 80% cavity legth

80 66 For these simulatios, the speed (rate of chage ad magitude of the applied ramp sigal are varied. The speed rages from slow (comparable to the curret experimetal value, slew rate < 1V/s, to moderate (faster, but achievable usig stadard electroics, slew rate < 0V/s, to fast (requires special high-speed electroics, slew rate > 100V/s, where the frequecy tuig per roud trip is ot egligible compared with the cavity FSR. i. Laser respose for slowly chagig ramp voltage The time depedece of the applied potetial, the drivig force, is show i Fig. 3.8 a. There is o electric field for t<0. From t=0 to t=.05 µsec the liear ramp is applied to the electro-optic sectio of the laser. For t>.05 µsec the potetial is held costat at 40V. (a The applied voltage sigal Figure 3.8: The laser respose uder a ramp sigal comparable to the experimetal value

81 67 (b Laser frequecy ad amplitude vs. time. No frequecy ripples is clearly idetified durig the trasiet time. Figure 3.8 (cotiued The respose of the laser to this slow ramp perturbatio is depicted i Fig.3.8 b. For t<0 the laser is i steady state equilibrium. From t=0 to t=.05 µsec the optical frequecy varies liearly with the applied potetial yieldig a total frequecy shift of.64 GHz. The slope of the frequecy variatio, 16 MHz/V, is the sesitivity of the electrooptic regio. The frequecy shift is remarkably liear ad istataeous, just like the experimetal results preseted i the ext chapter. For t>.05 µsec the laser frequecy remais costat idicatig that a ew steady state has bee reached. The amplitude variatio of the laser, also show i Fig. 3.8 b, is slightly more complex. After the potetial is applied, a short period of rigig behavior (relaxatio is

82 68 observed. Approximately 1 µsec later, steady state is reached, with a slight (<0.01% decrease i the laser amplitude. This decrease is due to the fact that the gai profile is ot completely flat. Iitially the lasig occurs at the ceter of the gai profile with maximum gai. As the frequecy shifts away from the ceter the gai slightly decreases. This is illustrated graphically i Fig The amplitude relaxatio oscillatio is a welluderstood behavior for solid-state lasers [45]. Gai profile Optical frequecy after tuig Iitial Optical frequecy Optical frequecy Figure 3.9: The lasig frequecy ad gai profile ii. Laser respose uder moderate ramp speed It is istructive to ivestigate the laser dyamics as the ramp speed, amely the speed of the exteral perturbatio is icreased. The laser respose uder a moderate speed ramp sigal, e.g. 40volt 5s ramp sigal is show i Fig The frequecy variatio of the laser follows the modulatio sigal istatly, just as before. The optical field amplitude slowly reaches a ew steady state after a period of rigig. The total optical field amplitude variatio is egligible (less tha 0.1%.

83 69 (a The applied voltage sigal (b Laser frequecy ad amplitude vs. time Figure 3.10: The laser respose uder moderate speed, low voltage ramp sigal

84 70 iii. Respose uder fast ramp perturbatio The laser respose uder fast ramp perturbatio is also ivestigated. The applied ramp (0.1s duratio, 40 volt is depicted i Fig. 3.11a. The simulated laser respose is illustrated i Fig. 3.11b~c. (a The applied voltage sigal (b The amplitude ad frequecy respose Figure 3.11: The laser respose uder fast, low voltage ramp sigal

85 71 (a Frequecy ripples observed as steady state is reached (b A zoom-i illustratio of the frequecy ripple Figure 3.1: Frequecy ripple observed uder fast, low voltage tuig Whe the laser is perturbed usig a very fast ramp (0.1s duratio, 40 volt, the optical frequecy of the laser output still follows the ramp istatly. However, small ripples with a period of the cavity roud trip time (~0.014s appear, as show i Fig.

86 The maximum value for the ripple is less tha % of the frequecy shift betwee the two steady states. The frequecy ripple is attributed the fact that the electro-optic material does ot fill the etire cavity. The frequecy ripples rapidly decay ad vaish i about a time period that correspods to 180 cavity roud trips. I additio, the optical field amplitude experieced a slight variatio of less tha 0.1%. iv. Respose with high voltage, moderate speed ramp perturbatio It is also iformative to ivestigate the laser respose for various magitudes of the exteral voltage. A 400volt, 50s ramp potetial (Fig. 3.13a is selected for this purpose. The simulatio results are illustrated i Fig. 3.13b~c. (a The applied high voltage, moderate speed tuig sigal Figure 3.13: The laser respose uder high voltage moderate speed ramp

87 73 (b The laser amplitude ad frequecy respose Figure 3.13 (cotiued Uder high voltage moderate speed ramp perturbatio, the laser frequecy follows the voltage istatly. No sigificat frequecy ripple is detected. Further simulatios show the frequecy ripples do ot deped o the magitude of the voltage, but are strog fuctio of the ramp speed. I additio, as show i Fig. 3.13c, uder high voltage perturbatio, the amout of amplitude variatio is icreased to 0.3%, which is cosistet with larger frequecy detuig from the gai ceter. v. Respose uder the high voltage, high-speed ramp sigal perturbatio Next, we use a high voltage, high-speed ramp (1s, 400volt as a perturbatio potetial i the simulatio. The result is illustrated i Fig b~d.

88 74 (a The applied voltage sigal (b Laser amplitude ad frequecy respose Figure 3.14: The laser respose uder very fast ramp sigal with high voltage

89 75 Figure 3.14 (cotiued (c The observed frequecy ripple durig the trasiet period Whe the laser is uder the fast ad high voltage ramp (1s duratio, 400volt max, the laser frequecy still follows the voltage sigal almost istataeously (Fig. 3.14b, except that optical frequecy ripple occurs as before (Fig. 3.13c. The ripple disappears after s. The ripple behavior is cosistet with pervious theoretical results (sectio 3.. Also show i Fig. 3.14b, there is a slow amplitude variatio of about 0.3% Laser dyamic behavior for differet cavity parameters I this subsectio, we study the laser dyamics (mode pullig, frequecy ripples, etc vs. differet cavity parameters. For these studies the speed of the drivig force, the slope of the applied ramp potetial, is held costat (300Volt/s, ~1s duratio.

90 76 i. The impact of the electro-optic sectio legth The laser s dyamic behavior depeds o the relative legth of electro-optic sectio. Ideally, if the etire cavity is homogeously electro-optic, the FM respose is expected to be very smooth with o observable frequecy ripples. The laser parameters used i the simulatio are selected to maximize the frequecy pullig effect. They are show i table below. The results are depicted i Figs ~3.18. The variatio of the optical frequecy for three differet electro-optic legths is illustrated i Fig The steady state tuig sesitivity as a fuctio of the ratio betwee the legths of the electrooptic sectio ad the etire laser cavity is plotted i Fig The ripple level ad ripple decay time (1/e are depicted i Fig ad 3.18 respectively. Table 3.4: Laser parameters for electro-optic sectio legth simulatio Parameters Log(R: cavity loss 0. C: Normalized sigle pass gai 1.5 Value γ : Gai liewidth 10GHz γ : 1 / Upper level life time 1e+8 β: electro-optic modulator sesitivity 10MHz/Volt L: total cavity legth 67 mm

91 77 (a The laser FM respose whe electro-optic sectio is 19% total cavity legth (b The laser FM respose whe electro-optic sectio is 81% total cavity legth Figure 3.15: The laser FM respose vs. differet electro-optic sectio legth

92 78 (c The laser FM respose whe electro-optic sectio is 94% total cavity legth Figure 3.15 (cotiued The istataeous frequecy of the laser experieces very large oscillatio with short electro-optic sectio (e.g. 0% the legth of the cavity. I additio, for this case the frequecy shift is small (.5 GHz. I cotrast as log as the electro-optic sectio of the laser is comparable to the cavity legth (80% or higher the laser approaches steady state rather quickly, ad with very little oscillatios (see Figs 3.15 b ad c.. Ad for these cases the frequecy shift is larger. I additio, the mode pullig effect [45] is observed as the laser reaches the steady state, but it is idepedet of the relative legth of the electro-optic sectio.

93 79 Figure 3.16: The laser tuig sesitivity at steady state As show i Fig. 3.16, the steady state frequecy tuig sesitivity, the frequecy shift for a give applied voltage, is a liear fuctio of the relative legth of the electrooptic sectio. Figure 3.17: The maximum ripple magitude as a fuctio of the electro-optic sectio legth

94 80 Figure 3.18: The ripple decay time as a fuctio of the electro-optic sectio legth The simulated characteristics of the frequecy ripple are displayed i Figs. 3.17, ad The ripple amplitude is a triagular fuctio of the relative electro-optic sectio legth, with maximum ripple occurrig whe the electro-optic sectio is half of the cavity legth. The simulatios suggest that the ripple decay time is urelated to the relative legth of the electro-sectio, which agrees with the theoretical predicatio (see Eq The predicated decay time is less tha the simulated value by 30% ad this is caused by the error i measurig the decay time from the simulated time respose. Next, we will study the effect of the gai liewidth o the laser FM respose. ii. Effect of gai liewidth γ o the laser dyamics The laser respose is simulated for differet gai medium liewidths. The laser parameters for this simulatio are selected to maximize the ripple ad frequecy pullig effect. They are listed i table 3.5.

95 81 Table 3.5: Commo parameters for gai liewidth simulatio Parameters Log(R: cavity loss 0. C: Normalized sigle pass gai 1.5 Value γ 1e+8 : 1 / Upper level life time β: electro-optic modulator sesitivity 10MHz/Volt L: total cavity legth 67 mm electro-optic sectio legth 54 mm (a Gai Liewidth: γ =5GHz Figure 3.19: The laser FM respose vs. differet gai liewidth

96 8 (b Gai liewidth: γ =50GHz (c Gai liewidth: γ =100GHz Figure 3.19 (cotiued

97 83 Figure 3.0: The ripple decay time vs. the gai medium liewidth Figure 3.1: The steady state frequecy tuig vs. the gai medium liewidth

98 84 Simulatios of the laser dyamics correspodig to three typical optical gai liewidths are illustrated i Fig The magitude of the ripples appears to be idepedet of the gai medium liewidth. However, it takes loger for the ripple to vaish for lasers with larger gai medium badwidth. The ripple decay time is depicted as a fuctio of the gai liewidth i Fig The simulatio result (based o the direct itegratio of the equatios of motio agrees with the theoretical result (Eq As idicated i Fig. 3.1, the steady state frequecy tuig depeds o the gai liewidth, ad smaller liewidth results i a larger frequecy pullig effect. Next, we ll study the effect of cavity legth. iii. Dyamic respose as a fuctio of cavity legth (or roud trip time The laser FM respose is simulated for differet total cavity legths. The laser parameters for this simulatio are selected to maximize the frequecy pullig ad ripple effects ad they are listed i table 3.6. The simulatio results of the optical frequecy progressio for three differet cavity legths are show i Fig. 3.. The relative legth of the electro-optic sectio throughout these simulatios is held costat at 81%.

99 85 Table 3.6: Parameters for total cavity legth simulatio Parameters Log(R: cavity loss 0. C: Normalized sigle pass gai 1.5 Value γ : Gai liewidth 50GHz γ : 1 / Upper level life time 1e+8 β: electro-optic modulator sesitivity electro-optic sectio legth 10MHz/Volt 81% cavity legth (a Cavity roud trip: τ = 10 ps Figure 3.: The laser FM respose vs. differet cavity legths or roud trip times

100 86 (b Cavity roud trip: τ = 0 ps (c Cavity roud trip: τ = 40 ps Figure 3. (cotiued

101 87 The laser respose is a fuctio of the cavity legth. Both the theoretical aalysis (sectio 3., ad extesive simulatios idicate that shorter cavities will produce smaller frequecy ripples. I additio, from the simulatio we ca also coclude that the ripple decay time is also a fuctio of the cavity roud trip time. With a 40ps cavity roudtrip, the decay time is aroud 0.s. Whe the cavity roud trip is reduced to 10 ps, the ripple lasts less tha 0.05 s. Thus, to obtai smooth FM respose, the cavity roud trip time should be kept as small as possible Summary of laser respose for varyig voltage ramp ad cavity parameters The laser dyamic respose uder voltage ramp excitatio has the followig characteristics: The laser frequecy respose (shift to the exteral ramp voltage sigal is practically istat The dyamic frequecy respose of the laser icludes small ripples, which have the period of cavity roud trip time. These ripples itesify with icreasig slope of the applied ramp perturbatio. The laser cavity parameters also have sigificat ifluece o the laser dyamic respose. The relative legth of the electro-optic segmet should be kept as log as possible to assure a smooth trasiet respose (i.e., small ripple amplitude ad higher tuig sesitivity. Ideally, the etire cavity should be uiformly electro-optic. The ripple decay time decreases for arrower gai liewidths, but at the expese of higher frequecy pullig ad lower tuig sesitivity.

102 88 Shorter cavity legths assure smaller ripples ad more rapid covergece to steady state. Therefore, a shorter cavity laser is preferred for laser frequecy tuig applicatios. I the microchip laser system built i this thesis work, the ripples oly itroduce very small chages to the phase of the optical field evelope. Thus, we ca cosider the laser tuig respose istataeous, accompaied by a effective frequecy oise (i.e., the ripples i the simulatio. The oise level depeds o the tuig speed, electro-optic sectio legth, ad other factors eumerated above, ad it is ca be approximated usig the 0 th -order theory for the electro-optic passive cavities (sectio 3., = β V& mi{ LE / O, ( Λ LE / O / c where V & is the ramp rate, Λ is the total cavity legth, L E/O is the electro-optic sectio legth. Next, we will discuss the microchip laser trasmitter desig issues ad experimetal results.

103 89 Chapter 4: Tuable trasmitter desig I this chapter, we will discuss the desig ad performace of the low oise dyamically tuable mm-wave optical trasmitter (DTMOT. First, we ll provide the DTMOT cocept (sectio 4.1; the we ll address the detailed desig issues related to sigle logitudial mode (sectio 4. ad sigle spatial mode operatio (sectio 4.3 of the microchip laser. The tuig of the microchip laser is covered i sectio 4.4 ad fially the experimetal results are show i sectio DTMOT cocept The origial DTMOT cocept is show i Fig. 4.1, where P 1 ad P are two idepedet pumps, G is the Nd:YVO 4 gai sectio, M is the MgO:LiNbO 3 phase modulatio sectio, E 1 ad E are electrodes. Figure 4.1: The dyamically tuable millimeter wave optical trasmitter structure

104 90 As idicated i Fig. 4.1, the mm-wave optical trasmitter is based o optical heterodyig of two electro-optically tuable sigle mode microchip lasers. The critical compoet iside the DTMOT is the electro-optically tuable sigle-mode microchip laser composite crystal assembly (Nd:YVO 4 /Mgo:LiNbO 3 from VLOC, Ic. The gai sectio of the composite crystal is a 3% doped 0. mm thick Nd:YVO 4 crystal. A 0.8 mm log MgO:LiNbO 3 crystal acts as a itracavity phase modulator. Dielectric mirrors are directly deposited o each side of the laser system to form a plao-plao resoator. Two side-by-side laser sectios are implemeted iside a sigle microchip laser crystal by two idepedetly drive 808 m pump laser diodes. The Nd:YVO 4 provides a homogeous gai medium that may lead to multimode lasig i the presece of spatial hole burig. However, the short pump absorptio legth [1] reduces the chace for spatial hole burig thus assurig sigle mode lasig. I additio, the pump beam produces a weak thermal waveguidig effect [3] i the Nd:YVO 4 resultig i a sigle trasverse mode i the cavity. Two electrodes are deposited o the top ad bottom of the MgO:LiNbO 3 sectio for the tuig sigals. The outputs of the two lasers are combied, coupled ito a sigle mode fiber, ad trasmitted to a high-speed photodiode, where the optical sigals heterodye resultig i a microwave or millimeter-wave sigal. The moolithic cofiguratio gives the device simplicity, ad compactess. Origially we expected that these two lasers i oe crystal cofiguratio assured a better stability, ad reduced sesitivity to exteral temperature fluctuatios. However, later we foud that the beat frequecy fluctuatios caused by the exteral temperature fluctuatios ca be reasoably cotrolled by simple electroics. Havig two lasers separated, we ejoy the beefit of easily settig the operatig frequecy from 0 to over a hudred GHz by temperature

105 91 tuig (sectio 4.4. Therefore, i the later implemetatios, two tuable lasers are i separate crystals, which are idepedetly temperature cotrolled. The physical implemetatio of the trasmitter is show i Fig. 4.. This cofiguratio cosists of a 0. mm log Nd:YVO 4 crystal (the gai medium ad a 0.8 mm log MgO:LiNbO 3 crystal (the tuig sectio. The laser has a 80mW threshold ad a slope efficiecy of better tha 0%. Figure 4.: The heterodye trasmitter implemetatio Importat desig cosideratios regardig this optical trasmitter iclude laser sigle (logitudial ad trasverse mode operatio, frequecy-tuig rage, speed ad sesitivity, ad oise. I this chapter, we are goig to discuss these issues idividually. The laser efficiecy issues are ot goig to be addressed here sice they are well represeted i the literature [47].

106 9 4. Trasmitter sigle logitudial mode operatio Sigle logitudial mode operatio is a prerequisite for stable millimeter heterodye output. The sigle logitudial mode operatio is determied by the cavity geometry (i.e., gai sectio legth, ad total cavity legth. Iside the microchip laser cavity, several cavity modes (~5 modes are withi the gai badwidth (00GHz. Although Nd:YVO 4 is a homogeeously broadeed gai medium, oe would expect that the ihomogeeous gai caused by spatial hole burig [45] iside the F-P resoator will drive the laser ito multimode operatio. However, for the electro-optically tuable microchip laser, the pump absorptio oly occurs i the regio very close to the reflectio mirror where all the cavity modes have a commo ull poit. As a result, the localized ihomogeeous gai available to differet cavity modes is greatly reduced. To a certai extet, the cavity modes see mostly the same gai medium ad by modal competitio oly the mode closest to the gai ceter would oscillate. The sigle mode operatio of microchip laser ca be studied quatitatively by ormalized multimode operatio threshold ζ (1, [1], which is the ratio of the maximum sigle-mode iversio desity of the laser medium to the lasig threshold iversio desity. The microchip laser operates i sigle logitudial mode whe pump power is less tha ζ times laser threshold. Here ζ (1, (1, is give by [1]: β (1, < ψ (1, > [ β(1, < ψ (1, > ] ζ ( 1, = { 1} (4.1 < ψ > 1 < ψ > 1 (1, (1,

107 93 where β (1, ad ψ (1. are the discrimiatio ad the correlatio factors betwee the first ad secod oscillatig modes, respectively. For homogeeous broadeed microchip laser, β (1, is: 1+ [( f f / f ] 0 3dB β (1, = (4. 1+ [( f1 f 0 / f3db] where f 1 ad f are the optical frequecies of the first ad secod oscillatig modes, respectively, f 0 is the ceter of the gai spectrum, ad f 3dB is the gai spectrum 3dB badwidth. by If the gai sectio is located close to mirror, the correlatio factor ψ (1. is give 1 L g < ψ ( 1, >= [ N0 ( z cos ( k1 k z] dz (4.3 L < N > 0 g where L g is the effective optical legth of gai sectio; k 1 ad k are the wave umbers of the first ad secod oscillatig modes; N 0 (z is the excited state distributio alog the laser axis ad < N > is the average iversio desity, 1 < N >= L g L g 0 N 0 ( z dz For logitudially pumped laser system, the iversio desity has a expoetial distributio: N 0 ( 0 p z z = N (0 exp( α (4.4 where α pis the pump adsorptio coefficiet, ad N 0 (z is the iversio desity at the begiig of the gai sectio. I this case, we calculate the correlatio factor as:

108 94 αl k g 1 e (cos klg si klg < ψ (1, >= α (4.5 α + ( k αlg (1 e α where k is the wave umber differece betwee the first ad secod cavity mode. As show i Eqs. 4.1, 4., ad 4.5, the sigle mode operatio is a fuctio of the legths of the etire cavity ad the gai sectio, the rate of pump absorptio, ad the gai medium badwidth. The gai medium badwidth is approximately 0.5 m for Nd:YVO 4. The pump absorptio rate is a fuctio of Nd ios dopig cocetratio. For the 3% doped crystal used i the experimets the pump absorptio rate is 100/cm [4]. The microchip laser sigle mode operatio performace is studied for differet cavity structures usig Eqs. 4.1, 4., ad 4.5. The results are summarized i Figs 4.3, 4.4, ad 4.5. Fig. 4.3 depicts the multimode operatio threshold of the microchip laser whe the lasig mode is aliged at the ceter of the gai spectrum. The sigle mode operatio uder laser frequecy tuig is show i Fig. 4.4 ad 4.5. The multimode operatio threshold vs. frequecy detuig whe total cavity legth is 1 mm is depicted i Fig. 4.4, while Fig. 4.5 depicts the multimode operatio threshold vs. total cavity legth whe the gai sectio legth is fixed (0. mm

109 95 Figure 4.3: Multimode operatio threshold vs. total cavity legths ad gai sectio legth Figure 4.4: Multimode operatio threshold vs. frequecy detuig ad gai sectio legth

110 96 Figure 4.5: Multimode operatio threshold vs. frequecy detuig ad total cavity legth Discussio i. The cavity geometry for sigle mode operatio Short cavity geometry, especially short gai sectio, provides for better sigle mode operatio. As idicated i Fig. 4.3, the sigle mode operatio depeds very strogly o the gai sectio legth. Specifically, for a 0. mm 3% doped Nd:YVO 4 gai sectio, the laser will remai sigle mode eve whe the pump power is 10 times larger tha laser threshold. Therefore, we selected 0. mm gai sectio i the tuable microchip laser system. ii. The tuig rage cosideratios The multimode operatio threshold decreases whe the lasig frequecy is detued from the gai ceter. Without exceptio, whe the frequecy detuig is oe half

111 97 of the cavity FSR, the multimode operatio threshold becomes uity, which idicates the oset of the secod lasig mode. Therefore the cotiuous frequecy tuig rage is always smaller tha cavity FSR. A very iterestig poit from Fig. 4.5 is that the sigle mode performace uder ormalized frequecy detuig (with respect to cavity FSR is ot a strog fuctio of the overall cavity legth. Extesive simulatios show that this statemet holds whe the effective pump absorptio legth is short compared to the overall cavity legth. Sice shorter cavity laser would have larger FSR, it will have a better frequecy tuig rage. iii. Effect of the dielectric mirror thickess The high reflectio (HR dielectric mirror deposited o the gai crystal has fiite thickess, which itroduces a small gap betwee the mirror reflectio poit ad the begiig of gai sectio (Fig To iclude this effect, Eq. 4.5 should be modified as, k α l k cos k d si k d e [cos k( Lg + d si k( Lg + d] < ψ α α (1, >= (4.6 α + ( k αl (1 e g α where d is the distace betwee the mirror reflectio poit ad the start of the gai sectio. Usig Eq. 4.6, the effect of small gap o the laser sigle mode operatio performace is studied ad the results are depicted i Figs Fig. 4.7 shows the multimode operatio threshold whe the air gap is varied from 0 to 0.4 mm. Fig. 4.8 is a close-i illustratio of Fig 4.8. The max frequecy tuig vs. differet gap legth is show i Fig The sigle mode operatio depeds strogly o the gap legth. The multimode operatio threshold decreases by approximately 50% with 0 µm gap, which is the typical mirror thickess. With a loger gap (>00 um, oe would observe the oset

112 98 of multi-mode operatio slightly above laser threshold. As a result, for microchip laser desig, the gap betwee the reflectio poit ad gai sectio must be kept miimum. L M1 Gai L g M d Figure 4.6: The cavity structure where gap exist betwee the gai sectio ad the mirror Figure 4.7: The multimode operatio threshold vs. gap legth ad gai sectio legth

113 99 Figure 4.8: The multi-mode operatio threshold vs. gap legth ( 0. ~0.4 mm Figure 4.9: The multi-mode operatio threshold vs. gap legth ad frequecy detuig

114 The microchip laser trasmitter sigle spatial mode operatio Sigle spatial mode operatio is aother importat cosideratio for microchip laser desig. The microchip laser uses a plao-plao resoator structure ad its spatial mode is defied by the thermal effects of the pump beam (thermal lesig or thermal expasio at mirror surface [3]. I order to guaratee the sigle spatial mode operatio, the pump beam should have a smaller cross-sectio tha the fudametal cavity lasig mode. Otherwise, the usaturated iversio desity will provide gai for higher order trasverse modes, resultig i a possible multi-trasverse mode operatio. The sigle spatial mode lasig limits the choice of pump system desig. Zayhowski et al. [3], postulated a expressio for the microchip laser fudametal mode cross sectio. However, their expressio is ot valid for the electro-optically tuable microchip laser cavity, where two separate crystals exist ad the pump absorptio oly occurs iside the extremely short gai sectio (as see i Fig Thermal expasio L eff Pump power L g Thermal waveguide Figure 4.10: The coceptual laser cavity uder the pump iduced the thermal guidig ad thermal expasio Therefore, we derive a approximate expressio (see appedix C for the fudametal mode cross-sectio ω for the tuable microchip laser:

115 101 ω eff 1/ 4 ( α eff α P(0 1/ 4 π L (4.7 g ( k c 1 λ L where α is the heat-geeratig efficiecy of the pump, P(0 is the absorbed pump power desity at the begiig of the gai sectio, k c is the thermal coductivity of the gai crystal, is the refractive idex of the gai crystal, α eff is the effective temperature coefficiet, ad α eff + α = α e, where e α ad α are the thermal expasio coefficiet ad refractive idex temperature coefficiet of the gai crystal, respectively, l ad L eff are the effective optic legths of the gai crystal ad the etire cavity, respectively. Usig Eq. 4.7, the fudametal spatial mode radius uder differet pump power for a typical cavity cofiguratio is simulated. The result for a microchip laser system with 50 um pump radius, 0. mm log 3% doped Nd:YVO 4, ad 1 mm total cavity legth, is depicted i Fig Figure 4.11: Fudametal spatial mode radius vs. differet pump power

116 10 Pump spot size should be less tha the fudametal spatial mode size for guarateed sigle mode operatio. Hece, it is more advatageous to study the mode cross sectio ormalized with respect to the pump beam size. Fig. 4.1 depicts the ormalized spatial mode radius vs. differet pump power levels ad spatial modes for microchip laser used i the experimet (0. mm log 3% doped Nd:YVO 4 gai medium, ad 1 mm etire cavity legth. Whe the laser operates i the area above the dotted lie iside Fig. 4.1, where the pump beam is smaller tha the lasig mode cross sectio, sigle spatial mode operatio is guarateed. Figure 4.1: The ormalized spatial mode radius vs. differet pump power level ad spatial mode

117 103 As idicated i Fig. 4.1, small pump size is desired for microchip laser sigle spatial mode operatio. However, there is a trade off. Smaller pump size geerally meas more complicated pump optics. From Fig. 4.1 we ote that a 50 um diameter pump beam would lead to sigle spatial mode operatio up to 700 mw pump power, which is adequate for most realistic systems. Thus, a pumpig system usig 50 um diameter wide pump diode butt-coupled to the gai sectio represets a suitable solutio for the E/O tuable microchip laser system. However, we are uable to obtai such diodes i a ope heat sik package suitable for the laser system. This leads to uwated multi-spatial operatio at a higher pump level i the experimets. 4.4 Laser tuig I this sectio we discuss the voltage, pump power ad temperature tuig of the microchip laser. The discussio here is based o the steady state behavior of microchip lasers Electrical voltage tuig sesitivity Voltage is the desired tuig mechaism of the microchip laser because it has rapid frequecy respose ad it is highly liear. Whe voltage is applied across the LiNbO 3 sectio of oe microchip laser, the effective cavity legth, ad thus the resoat optical frequecy chages by: 1l1 V δ f = η 1 r33 foptical (4.8 l + l d 1 1 where 1 is the refractive idex of the extraordiary wave i the LiNbO 3 sectio, is the refractive idex of a π polarized wave i the Nd:YVO 4 gai sectio, l 1 is the electro-optic sectio (LiNbO 3 legth, l is the gai sectio (Nd:YVO 4 legth, d is the thickess of the

118 104 LiNbO 3 sectio, r 33 is the LiNbO 3 electro-optic coefficiet alog the z-axis, f optical is the optical frequecy, η is the overlap efficiecy betwee the applied electric field ad laser cavity mode, ad V is the applied voltage. Eq. (4.8 idicates a ideal liear relatioship betwee the frequecy tuig ad applied voltage. The FM sesitivity is liearly proportioal to the optical frequecy (iversely proportioal to the optical wavelegth. The wavelegth depedece implies a 30% higher sesitivity at 1.06 µm compared to 1.3 µm wavelegths. It eeds to be poited out that the FM sesitivity does ot deped o the absolute legth of tuig elemet. However, as Eq. 4.8 idicates, it is a fuctio of the ratio betwee the effective optical legths of E/O tuig sectio ( 1l1 ad etire laser cavity ( l + 1l1. Specifically, if we have 0. mm log gai sectio ad 0.8 mm log 0.5 mm thick tuig sectio, ad we cosider the 1.06 µm lasig wavelegth, the theoretical voltage tuig sesitivity is 36 MHz/volt assumig 100% overlap betwee the electric ad optical fields Temperature tuig sesitivity The absolute laser frequecy is extremely sesitive to exteral temperature variatio. Accurate modelig of exteral temperature iduced laser frequecy variatio ivolves the umerical solutio of temperature field equatio iside the laser cavity with proper boudary coditio, which is beyod the goal of the thesis work. However, if we assume that the microchip laser crystal has a costat temperature T alog its boudary, the temperature iduced laser frequecy variatio satisfies: δ1 ( T l1 + δ( T l + 1δ l1 ( T + δl( T optidal = f optical 1l 1 + l δ f (4.9

119 105 The thermal expasio coefficiets of MgO:LiNbO 3 ad Nd:YVO 4 are K -1 [48] ad K -1, respectively[49]. The temperature iduced refractive idex variatio coefficiet is K -1 for MgO:LiNbO 3 [48] ad K -1 for Nd:YVO 4 [49]. The calculated thermal tuig sesitivity is 13 GHz/ o K. This model is overly simplistic sice the microchip laser crystal has oly oe surface boded to the heat sik. Therefore, we ca cotrol the temperature of oly oe surface ad the actually thermal sesitivity could be sigificatly smaller. However, the predictio from the simple model still provides a good estimate. Accurate temperature tuig sesitivity is to be determied by experimet (sectio Sesitivity to pump power tuig Pump power variatio will itroduce localized tempareture variatio iside the laser crystal, which will chage the refractive idex ad therefore modify the cavity resoace coditio ad hece the optical frequecy [50]. For tuable microchip lasers, sigficat FM could be achieved with very small amplitude variatios. I additio, pump power tuig is useful i settig the trasmitter iitial frequecy. A comprehesive theorectical modelig o pump power tuig is give by Zayhowski et al. [50]. We will determie the pump power tuig experimetally as show i the subsequet sectios. 4.5 Laser trasmitter characterizatio Laser threshold ad efficiecy A microchip laser is fabricated usig a 0. mm 3 atom% doped Nd:YVO 4 gai sectio ad a 0.8 mm MgO:LiNbO 3 phase modulator sectio. By depositig two differet dielectric coatigs (1064 m HR / 1064m 98%R, ad 1034 m HR/ 1034m 99% R, both 1064m ad 1340 m microchip lasers are fabricated. The microchip lasers are

120 106 directly pumped by 808m 1W ope heatsik laser diodes (Spectra-physics SCI z1-01. The microchip laser optical output power was measured for differet pump curret levels, ad the result is plotted i Fig The threshold of the 1064m laser is approximately 80mW pump power ad its slope efficiecy is 0%. The threshold ad slope efficiet of the 1340 m laser is slightly iferior (threshold: 95mW, Slope: 16% due to the smaller emissio cross sectio [4] of Nd ios at 1340 m. Figure 4.13: The microchip laser threshold ad slope efficiecy characteristics

121 Optical spectrum quality ad spatial quality The optical spectrum of the microchip laser was ivestigated usig a Ado optical spectrum aalyzer ad Fig illustrates a typical optical spectrum. Figure 4.14: Laser spectrum at m The resolutio badwidth of the optical spectrum aalyzer is 0.1 m, ad the free spectral rage of the microchip laser is 60GHz (0. m i terms of optical wavelegth. Therefore the optical spectrum aalyzer is capable of resolvig idividual logitudial modes. I the optical spectrum measuremet (Fig. 4.14a, oly oe peak exists, which idicates sigle logitudial mode operatio. However, it should be metioed here that multi-mode behavior has bee observed whe the lasig mode is tued (by temperature to the regio close to the mode hoppig frequecy. This behavior is caused by the small mode discrimiatio ad it agrees with the theoretical predictios.

122 108 The microchip laser spatial profile was ispected first visually, ad the a microwave spectrum aalyzer was employed to idetify more accurately the occurrece of multiple trasverse modes by observig the possible beat sigal betwee differet trasverse modes. Figure 4.15: A trasverse mode beat toe captured by microwave spectrum aalyzer. I the visual measuremet, the laser has a close to ideal circularly shaped beam cross sectio at up to 6 times the threshold pump power. However, whe measured by the microwave spectrum aalyzer, the trasverse mode beatig (~ 60MHz is observed after oly 3 times the threshold pump power (Fig This agrees with our aalysis, (sectio 4.3., sice the simple direct laser diode pump produces a large elliptical pump beam cross sectio (~100 µm diameter, which results i a smaller fudametal lasig mode

123 109 cross sectio ad udeleted gai for higher order modes. Further improvemet i pump optics (small pump beam would result i better trasverse mode performace Microchip laser frequecy tuig I this sectio, we discuss the microchip laser frequecy tuig (voltage tuig, temperature tuig ad pump power tuig characteristics. i. DC voltage tuig sesitivity Microchip laser voltage tuig has very fast respose speed. Thus it is the primary operatio mechaism to geerate a rapidly tuable mm-wave optical subcarrier usig the DTMOT. The voltage tuig sesitivity of the trasmitter was characterized by measurig the heterodye sigal frequecy i the microwave domai versus the applied voltage. Varyig the voltage from 0 to 350 volt we measured a 7.7 GHz variatio i beat frequecy, correspodig to a tuig sesitivity of MHz/Volt, as show i Fig The differece betwee the measured ad theoretical sesitivities (3 MHz/Volt for this specific crystal is caused by the iadequate field overlap ad parasitic piezo effects of LiNbO 3. Further improvemet i tuig sesitivity ca be made by reducig the thickess of the modulator. The sesitivity is ultimately limited by the breakdow voltage of LiNbO3 (10kV/mm, which is 10 times larger tha the curret maximum tuig field.

124 110 Figure 4.16: Voltage tuig sesitivity ii. Microchip laser temperature tuig sesitivity The microchip laser frequecy is very sesitive to the temperature variatio, although the tuig by temperature has a very slow respose speed. Thus, we use the temperature tuig to set the DTMOT operatio offset frequecy. The microchip laser temperature tuig sesitivity is determied experimetally i the mm-wave / microwave domai by moitorig the two beat toe frequecy variatio vs. the temperature of oe laser crystal. The result is show i Fig With 6 degrees of temperature variatio, a 37GHz beatig toe tuig is observed ad it reveals a sesitivity of 6.GHz/C. The measured temperature tuig sesitivity is about two times less tha the calculated value (13GHz/ o C, as show i Eq ad it is a result of the overly simplistic theoretical model, which assumes that all the surfaces of the crystal have the same temperature.

125 111 Figure 4.17: The frequecy of the two laser beat sigal as a fuctio of the temperature iii. Microchip laser pump power tuig sesitivity The microchip laser frequecy is also sesitive to the pump power variatio. Sice the pump tues the laser frequecy via the localized temperature field variatio iduced by pump power[50], it has a faster respose tha overall temperature tuig. I the DTMOT, pump power tuig ca be used to set ad stablized the offset frequecy. The disadvatage of pump power tuig is that it itroduces parasitic amplitude modulatio. The sesitivity to the pump power variatio was experimetally determied by raisig the pump power of oe microchip laser from 90 mw to 170 mw ad measurig the two laser beat frequecy variatio. The resultig 5.6 GHz frequecy shift is show i Fig. 4.18, correspodig to a sesitivty of 70 MHz / mw.

126 11 Figure 4.18: Two-microchip laser beat sigal vs. pump power chage mm-wave operatio potetial Nd:YVO 4 has a gai liewidth over 00GHz. I priciple, the microchip laser ca operate at ay frequecy withi this 00GHz, which implies a 00GHz subcarrier tuig rage. I a practical tuable microchip laser system, by tuig temperature ad pump power settig, we ca set the differece frequecy betwee the two lasers from 0 to 150GHz limited by the fiite laser free spectrum rage. Fig depicts a example of the optical spectrum of the beat sigal i the 10GHz rage.

127 113 Figure 4.19: The optical spectrum of 10GHz subcarrier 1340 m Microchip laser chirp rate testig The microchip laser chirp rate is measured usig the experimetal setup show i Fig Two microchip lasers are used i the slew rate measuremet. A high-speed high voltage ramp sigal is applied o oe laser sectio to geerate frequecy chirpig, while the frequecy of the other sectio is kept at costat level. The two laser sectios were iitially set to geerate a heterodye frequecy of 1.9GHz. Heterodyig trasfers the optical domai chirpig ito the microwave domai chirpig that is more coveietly detected by a microwave homodye frequecy discrimiator [51]. A very low frequecy compoet of the discrimiator output (<100Hz is selected ad feedback to the laser driver to stabilized the heterodye sigal frequecy. The microwave homodye discrimiator respose is calibrated ad the result is show i Fig The discrimiator respose is oliear (siusoidal. Close to the ideal operatio poit, the discrimiator respose is 0.9mV/MHz. Fig. 4. reveals the

128 114 discrimiator output sigal whe, a 0 MHz, 10 V peak-to-peak triagular sigal is applied to the laser. The recovered sigal, lower trace i Fig. 4., matches with the applied voltage ramp, i.e., the upper trace. The scale of the lower trace is 100mV/ divisio, which idicates a MHz frequecy excursio based o the discrimiator characteristics. Therefore, a frequecy excursio of MHz over a 5 s time period was measured, ad a chirp rate of 8.8 GHz/µsec is achieved. It should be emphasized that the available voltage ramp geerator curretly limits the chirp rate, but the trasmitter is capable of producig higher speed chirps, if required.

129 115 LD pump 1 LD pump Fiber Optical Coupler Microwave power splitter Microwave spectrum aalyzer High-speed PD LD cotroller Ramp Geerator Digital Oscilloscope Variable delay lie Chirp detectio Low pass filter Mixer Microwave Hybrid Figure 4.0: The microchip laser chirp rate measuremet system 115

130 116 Figure 4.1: The output characteristics of the frequecy discrimiator Figure 4.: Applied ramp sigal (top ad the frequecy respose (bottom

131 The free ruig phase oise of the tuable trasmitter I the laser heterodyig process, the microchip laser optical phase oise will be dowcoverted ito the mm-wave subcarrier domai. As show i the ext chapter, a phase oise cotrol system is required to obtai a clea beat toe. I this subsectio, we will determie the free ruig phase oise of the tuable trasmitter, which is vital iformatio i desigig the phase oise cotrol system as show i the ext chapter The experimetal setup for the laser phase oise measuremet is depicted i Fig. Laser 1 Laser 1.04GHz PD PLL Badwith<50kHz Microwave spectrum aalyzer Figure 4.3: The laser phase oise measuremet setup As show i Fig. 4.3, a PLL with very small closed loop badwidth is employed to stabilize the frequecy driftig of the beat toe. With the PLL i place, the free ruig phase oise is give by, φ free( ω = H closedloop( ω φ beat ( ω

132 118 where (ω H closed loop is the PLL closed loop respose, ad φ beat ( ω is the phase oise of the beat toe with the PLL locked. The phase oise spectrum of the beat toe is show i Fig. 4.4a. The PLL closed-loop respose is plotted i Fig. 4.4b. Thus, we obtai the free ruig phase oise as show i Fig. 4.4c. (a The phase oise of the beat toe at 1.04GHz from 100kHz to 1MHz. Figure 4.4: The free ruig phase oise measuremet

133 119 (b The PLL ope loop respose from 100kHz to 1MHz. (c The calculated free ruig phase oise spectrum from 100kHz to 1MHz. Figure 4.4 (cotiued Limited by the miimum PLL badwidth, we oly measured the phase oise spectrum from 100kHz to 1MHz. As show i Fig. 4.4, the phase 100kHz offset is 70dBc/Hz ad the phase oise spectrum has a slope of -40dB/dec. The

134 10 experimetal results (the shape of the phase oise spectrum i Chapter 5 suggest that 40dB/dec is still valid below a 100kHz offset. Therefore, we estimate that the phase oise at 10kHz offset should be 30dBc/Hz. However, the measured slope rate of the phase oise spectrum is 40dB/dec, which does ot comply with results (i.e., 0dB/dec slope for the quatum oise limited laser system. Oe possible cause is the thermal vibratio of the crystal lattice. However, at this stage, we are uable to cofirm this explaatio. 4.6 Summary The dyamically tuable mm-wave optical trasmitter employig electro-optical tuable microchip laser heterodyig is desiged ad tested. The issues regardig sigle logitudial mode operatio, sigle spatial mode operatio, ad frequecy tuig are addressed i detail. By applyig differet dielectric mirror coatigs, both 1.06 um ad 1.34 um wavelegth lasig have bee achieved. The trasmitter has very wide tuig rage (>100GHz ad good voltage tuig sesitivity (0MHz/volt. The trasmitter (laser tuig speed is characterized usig a 10MHz 0Volt ramp sigal. I cotrast with the covetioal laser itesity modulatio, the tuig respose is ot limited by laser relaxatio oscillatio (~1MHz. No speed limitatio imposed by laser dyamics is observed, which agrees with the theoretical predictio of Chapter 3. I laser heterodyig, the laser phase oise ad frequecy jitter are directly dowcoverted to the subcarrier. For low oise operatio, a oise cotrol subsystem is required. I the ext chapter, we ll discussio the oise cotrol techiques for this heterodye trasmitter.

135 11 Chapter 5: Trasmitter phase oise cotrol subsystem The dyamically tuable mm-wave optical trasmitter (DTMOT is coceptually a ideal voltage cotrolled oscillator with high-speed ad very large tuig rage. However, its free ruig phase oise is high because the heterodyig process directly coverts the laser phase oise ito the mm-wave subcarrier domai. I order to geerate a low oise subcarrier sigal, a phase oise cotrol subsystem must be itroduced. I this chapter, we ll discuss two distictive yet coceptually related subsystems desiged for the dyamic tuable trasmitter phase oise cotrol, amely (1 digital frequecy sythesizer (sectio 5.1, ad ( delay lie optical frequecy lockig loop (OFLL (sectio 5.. The emphasis is placed o the OFLL subsystem due to its superior performace at higher frequecies. Fially, we will preset the experimetal results regardig oise suppressio (sectio Digital frequecy sythesizer We first discuss the digital frequecy sythesizer phase oise cotrol subsystem. A simplified sythesizer diagram is depicted i Fig. 5.1.

136 1 DTMOT Loop Filter PD Prescaler N Couter Phase Detector Modulatio Sigal Output Optioal d referece: f ref Referece f ref1 R Couter Figure 5.1: The digital frequecy sythesizer Except for the fact that the voltage-cotrolled oscillator is replaced by the DTMOT, the digital frequecy sythesizer phase oise cotrol subsystem is idetical to a covetioal digital sythesizer [5], the output of which is: P N f out = f ref 1 + f ref (5.1 R where P is the prescaler dividig umber, ad N ad R are the coutig umbers of the N couter ad R couters, respectively. I the followig, we ll briefly discuss the issues that are pertiet to the digital sythesizer phase oise cotrol system Sythesizer subsystem phase oise The oise sources of the output sigal phase oise of sythesizer subsystem are: the dowcoverted microchip laser phase oise, upcoverted referece phase oise, prescalar ad couter jitter/phase oise, ad AM oise from other electroic compoets coverted ito PM oise [5]. Mathematically, the combied oise is give by, φ ( ω = φ ( ω + φ ( ω + φ ( ω + φ ( ω + φ ( ω + φ ( ω φ ( ω (5. out optical ref 1 ref N R P + AM

137 13 The seve terms o the RHS of Eq. 5. are the phase oise cotributio from the laser optical phase oise, referece 1 phase oise, referece phase oise, N couter phase oise/jitter, R couter phase oise/jitter, prescaler phase oise / jitter, ad total amplitude oise of the electroic compoets respectively. They are listed as [5]: φ φ φ φ φ φ optical 1 ( ω = [ φml1( ω + φ ML ( ω] (5.3a 1+ H ( ω ref 1 ω H ( ω P N = φ1( (5.3b 1+ H ( ω R ref ω N R P H ( ω = φ( (5.3c 1+ H ( ω H ( ω P N ( ω = φ Neff ( ω (5.3d 1 + H ( ω R H ( ω P N ( ω = φre ff ( ω (5.3e 1 + H ( ω R H ( ω ( ω = P φpeff ( ω (5.3f 1 + H ( ω φ AM k / jω ( ω = V ( ω (5.3g 1+ H ( ω where H(ω is the ope loop gai of the feedback system; φ ML 1ad φ ML phase oise of the first ad secod microchip lasers; φ 1 ad φ are optical are phase oise cotributios of the first ad secod referece sources; φ Neff, Re ff φ ad φ Peff is the effective phase oise of N couter, R couter, ad prescaler; ad (ω is the effective amplitude oise of the electroic system at loop filter output; ad, k V is the trasmitter voltage tuig sesitivity.

138 14 As idicated i Eq. 5.3, higher ope loop gai would reduce the phase oise cotributio from the laser. However, the other phase oise iputs are early idepedet of the ope loop gai. For a well-desiged system, the oise from the referece source represets the ultimate performace limit, which icreases 0dB per decade with icreasig frequecy. This is the pricipal disadvatage of the digital sythesizer (PLL phase oise cotrol system Frequecy modulatio withi digital sythesizer subsystem Based o Fig. 5.1, the modulatio respose of the PLL ca by calculated as, f out V ( ω kv ( ω = ( H ( ω Sice the ope loop gai H(ω is lowpass, i order to accommodate direct modulatio, the sigal badwidth should be larger tha the PLL ope loop gai badwidth. Table 5.1 summarizes the advatages ad disadvatages of the digital sythesizer approach. Table 5.1: The advatage ad disadvatages of the digital sythesizer approach Advatages * Simple low cost circuitry. * Direct cotrol of output frequecy. * Relatively fast frequecy selectio. * Automatic recovery from loss of lock. * Frequecy accuracy determied by referece Disadvatages * Phase oise icreases at 0dB/decade with icreasig output frequecy. * Jitter ad oise of PLL dividers traslates to phase oise * Digital electroics is difficult to obtai at high frequecy rage

139 15 The digital frequecy sythesizer works well at lower frequecies (<0GHz. However, as the frequecy icreases, this approach becomes less suitable. I order to geerate high frequecy, high quality sigal, we propose a delay lie optical frequecylockig loop (OFLL. I the ext sectio, the OFLL cocept ad desig issues are explaied. 5. The delay lie optical frequecy-lockig loop The coceptual diagram of the delay lie optical frequecy-lockig loop (OFLL is depicted i Fig. 5., where the DTMOT is also treated as a voltage cotrolled oscillator (VCO. The output of the VCO is split ito two paths. The sigal i the first path is detected by a high-speed photo diode (PD, while the sigal i the secod path is detected after a log delay though a sigle mode fiber. The VCO phase error is recovered by mixig the outputs from the two paths, ad the fed back ito the VCO (DTMOT via a special loop filter. Ulike the covetioal sythesizer, the OFLL does ot eed a referece source to recover ad correct the phase / frequecy error from the VCO. Istead, it is accomplished by optical processig (the homodye delay. I the subsequet sectios, we ll discuss the OFLL operatio priciple, practical implemetatio cocers, ad OFLL system level desig. Direct Modulatio DTMOT 3dB Coupler PD1 Fiber stretcher PD Loop Filter Delay lie frequecy / phase discrimiator (DFPD Figure 5.: Delay lie optical frequecy locked loop (OFLL

140 OFLL operatio priciple I this sectio, we discuss the OFLL priciples: i. Mathematic modelig ii. Frequecy tuig ad modulatio scheme iii. Noise performace Each of these will be discussed i subsequet sectios. i. A aalytic model for the OFLL As a first step, we preset the mathematical model for the delay lie homodye frequecy discrimiator (DFD. The output of the delay homodye discrimiator is give by, Vo ( t = km A1 A[1 + ( t + ( t τ ]cos[ φ( t φ ( t τ + ω0τ ] (5.5 where k mis the mixer oliear coefficiet related to the mixer coversio gai, A 1 ad A are the subcarrier amplitudes of the first ad secod paths, respectively, (t is amplitude oise of the subcarrier, φ (t is the subcarrier phase oise, ad τ is the time delay of the secod (delayed path. I the steady state, the delay lie discrimiator has a vaishig DC output, which implies that the DC phase offset (ω 0 τ of the discrimiator output should satisfy the relatio ω o τ = ( kπ + π / (5.6 where k=0,+/-1, +/-, I additio, i order to maitai feedback stability, the k values must be either all eve or all odd depedig o the polarity of the loop filter output. For simplicity, we assume a egative polarity of the loop filter output (i.e., egative

141 17 feedback, k has to be eve, i.e., of the OFLL. k = i, where i is a iteger called the modal umber I the steady state (i.e., ω = ( iπ + π / / τ, the delay lie homodye o frequecy discrimiator output (Eq. 5.5 ca be simplified as, Vo ( t = km A1 A [ φ ( t φ( t τ ] (5.7 where the approximatio ( si θ θ, if θ << 1 is employed ad the higher order terms (the cross product betwee the phase ad AM oise are eglected. It is more coveiet to preset Eq. 5.7 i the frequecy domai, jω τ jω τ V ( jω = k [ φ ( jω φ ( jω e ] = k (1 e φ ( jω o p p (5.8 where k = k A A, ad φ ( jω p m 1 is the spectrum of the phase oise term φ (t. Based o the frequecy domai presetatio of the delay lie frequecy discrimiator, the etire optical frequecy lockig loop system ca be treated precisely as a liear feedback cotrol system, as show i Fig φ out(jω TMOT (VCO: Kv/jω ODFD H(ω=k p[1-exp(jωτ] φ opt(jω Loop Filter F(jω o(jω Figure 5.3: OFLL system block diagram

142 18 The tuable trasmitter is cosidered as a voltage-cotrolled oscillator with free ruig phase oise, the cotrol of which is the pricipal task of the OFLL system. The amplitude oise from the electroic compoets is modeled joitly as the equivalet output oise ( j of the delay lie homodye frequecy discrimiator. Next we 0 ω aalyze tuig (i.e., slow variatio of operatig frequecy ad direct modulatio (i.e., rapidly frequecy modulatio by applied aalog or digital sigal of the OFLL. ii. OFLL frequecy tuig ad direct modulatio give by, The OFLL operatio frequecy ca be tued by, Mode hoppig Delay time tuig by the fiber stretcher The first method chages the frequecy i stepwise fashio with a resolutio FSR = π /τ The secod method cotiuously chages the operatig frequecy, however, its rage is limited by the max fiber stretchig, δω = ω 0 where ω0 is the curret operatig frequecy, l is the max stretch of the stretcher, ad L is the total delay legth. I order for the OFLL to cover the etire frequecy rage, the stretcher tuig rage has to be larger tha delay lie free spectrum rage, which implies the requiremet for the miimum stretchig legth, l L π l >= L = λ τ ω 0 0 (5.9

143 19 where λ0is the mm-wave subcarrier wavelegth iside the fiber. I the experimets discussed later, both mode hoppig ad fiber stretchig were employed to obtai a cotiuum of operatig frequecies from 0 to over 40GHz. Like the digital sythesizer system, the OFLL ca be directly modulated. The OFLL direct modulatio is give by, f out V ( ω ( ω = kv ( G( ω where G(jω is the system ope loop gai, iωτ G( ω = k k (1 e F( ω iω (5.11 v p / I order to accommodate the direct modulatio of the OFLL output sigal, the modulatio frequecy must be larger tha the OFLL ope loop cutoff frequecy. iii. OFLL oise performace Usig the OFLL liear system diagram (Fig. 5.3, its output phase oise φ out ( jω ca be calculated as, φ out ( jω φ ( ω 0 ( ω k F ( ω/ iω v = + ( G( ω 1+ G( ω where F(ω is the loop filter gai, k v is the VCO voltage tuig sesitivity Uder the desirable OFLL operatio, G(ω >> 1, Eq. 5.1 ca be simplified as: φ out ( ω φ ( ω 0 ( ω = + (5.13 iωτ G( ω k p (1 e The first term o the RHS of Eq. 5.1 is defied as the residual phase oise, which represets the phase oise cotributio from the lasers. The secod term describes

144 130 the phase oise cotributio from the OFLL electroics, which is the oise floor of the OFLL system. Two issues cocerig the OFLL oise performace will be addressed ext. They are the suppressio of the free ruig VCO phase oise, ad the oise floor of the OFLL. (a Suppressio of free ruig VCO phase oise Suppressig the free ruig VCO phase oise is the pricipal task of the OFLL. A larger ope loop gai provides better optical domai phase oise suppressio. However, the ope loop gai has to be carefully desiged to satisfy the stability requiremet. The system stability issue is to be addressed i detail i the loop filter desig sectio (Sectio (b The OFLL oise floor The OFLL oise floor is give by the oise from the electroic compoets. I this sectio, the OFLL oise floor is calculated aalytically. From Eq. 5.11, the OFLL oise floor is give by, φ oise_ floor NEP ( jω = (5.14 iωτ 1 e where NEP is defied as the oise equivalet power per Hz at the discrimiator output, NEP = ( j /. 0 ω k p If we eglect the flicker oise (<1kHz, the followig electroic system oise sources cotribute to the overall oise at the output of the delay lie homodye frequecy discrimiator: The thermal oise [53] from the m th path: m T = k T NF B m G m The shot oise [54] from the m th path: m s = e I pd R m G m

145 131 The equivalet iput oise of loop filter: filter where k B is the Boltzma costat, T is the temperature i Kelvi, I pd is the photocurret of the photodiode, R is the termiatio impedace (50 ohm of the PD, NF m is the oise figure of the m th path, ad G m is the gai of the m th path. The total oise power is give by combiatio of all these oise cotributio, m ( ω 0 = filter + ( k BT NFm Gm + e I pd R Gm (5.15 where the total sigal power is give by: m k v = ( I R G, m = 1, (5.16 m pd i Neglectig the loop filter oise, which is always the case for a well-desiged OFLL loop filter, we obtai the formula for the OFLL system oise floor as show, φ m m k BT NF + e I pdr oise_ fllor( ω /1 exp( iωτ m m ( I pd R = (5.17 Three distictive features of the OFLL system are revealed i the aalytic expressio for the oise floor. First, the OFLL oise floor depeds o optical power (i.e., photocurret I pd. At a lower optical power, the oise floor is thermal oise limited (0dB decrease per 10 times icrease i optical power. At a higher optical power, the oise floor is limited by shot oise (10dB decrease per 10 time icrease i optical power. Secodly, the oise floor strogly depeds o the OFLL delay time, τ. A loger time delay meas better close-i carrier phase oise performace. However, a loger delay also results i smaller mode spacig, which leads to deterioratio of the phase oise far away from the carrier frequecy. More importatly, as we ll see i the loop filter desig sectio, a loger delay time reduces the maximum loop stable gai ad poses serious challege for the desig of the high gai loop filter.

146 13 Fially, the OFLL uses the fiber delay time as a frequecy referece. If we eglect the stochastic variatio of the fiber delay time, the OFLL output oise is a strog fuctio of the sigal-to-oise ratio at the delay lie discrimiator output. The OFLL oise floor does ot deped o the laser RIN assumig that the balaced mixer is ideal. However, i practice, the balaced mixer is ot ad therefore it coverts the laser RIN ito subcarrier phase oise, which cotamiates the sigal spectrum. I the followig sub-sectio, a special techique is itroduced to suppress this uwated effect. 5.. OFLL implemetatio usig a o-ideally balaced mixer The OFLL coceptual diagram (Fig. 5. is overly simplistic i terms of practical implemetatio. A practical balaced mixer at higher frequecies is ot ideally balaced, thus it itroduces a subcarrier AM oise ito the frequecy discrimiator output, ad sigificatly raises the OFLL phase oise floor. For a o-ideally balaced mixer, the homodye discrimiator output becomes, Vo ( t = km A1 A si[ φ ( t φ ( t τ + ω0τ ] + k1a1 (1 + ( t + k A (1 + ( t τ where k 1 ad k are costats related to the o ideally balaced mixer. (5.18 The DC term i Eq ca be compesated for by a proper iput offset i the OFLL loop filter. The AM oise term, however, will cotamiate the phase oise spectrum. I this sectio, we ll discuss a OFLL implemetatio scheme (show i Fig. 5.4 that suppresses subcarrier AM-to-PM oise coversio due to a ubalaced mixer.

147 133 Direct Modulatio PD 3dB Coupler PD1 Loop Filter M3 M M1 Low frequecy modulatio sigal Figure 5.4: The OFLL implemetatio scheme with reduced AM to PM oise coversio I Fig. 5.4, the mm-wave carrier sigal of the secod (delayed path is by a fixed low frequecy sigal (f m usig mixer (double sidebad up-coverter M1, whose output is later mixed with the mm-wave carrier sigal from the first path by the secod mixer M, ad the M output sigal is mixed with the low frequecy modulatio sigal to recover the subcarrier phase error iformatio. We ca prove that usig this approach (see Appedix D, the amout of the reductio i the subcarrier AM oise to phase oise coversio is give by, where Reductio i AM PM coversio = 0log(Ide x AM + 0log(A 1/A db (5.19 Idex AM is the modulatio idex of the first mixer (modulator. For a system with equal sigal amplitude i both paths, the AM-to-PM oise coversio is suppressed by 0 db with 0 db AM modulatio idex. This is sufficiet uder most situatios of iterests.

148 OFLL loop stability ad the loop filter desig The pricipal task of the OFLL is to suppress the phase oise cotributio by the laser. I order to reduce the residual laser phase oise at the OFLL output, the OFLL loop filter must provide a sufficiet ope loop gai without compromisig the loop stability. I this subsectio, we address the issues o the OFLL loop filter desig. I the loop filter desig, the pricipal difficulty emerges from the repetitious phase respose of the delay lie homodye discrimiator, which periodically varies its phase from 90 to +90. To deal with this periodic phase respose, a uique loop filter structure was devised as show i Fig OFLL loop filter C(s F1(s Adjustable gai Figure 5.5: OFLL loop filter structure The OFLL loop filter has three stages. The first stage is a compesatio block, which flattes the repetitious phase respose of the homodye discrimiator; the secod block is stadard prototype PLL loop filter; ad the fial stage is a adjustable gai. The three filter desig steps are idetified as: Step 1. Desig a prototype PLL loop filter Step. Desig compesatio block Step 3. Check the stability, ad loop gai

149 135 Next, we explai the loop filter desig process for a OFLL with 150-meter delay. Step 1. Prototype PLL loop filter We choose a 3 rd order PLL loop filter as our prototype filter (Eq Cosiderig a 150-meter fiber delay, which gives a free spectrum rage of 1.33MHz, we 6 put the ull of the prototype filter at rad /sec (approximately at the ceter of the frequecy spectrum rage related to a 150 meter delay. I additio, we set the pole 7 frequecy to rad / sec. F1( s s + ω s 1 1+ s / ω z = (5.0 p where ω z ad ω p are the ull ad pole frequecies of the prototype filter respectively. Step : Compesatio block desig The basic idea regardig the compesatio block desig is to approximate the siusoidal term (i.e., [ 1 exp( iωτ ] i Eq. 5.8 of the delay homodye discrimiator by a ratioal system, ad the usig the iverse of the ratioal system as the compesatio block. We use lower order (< Pade approximatio method [56] to approximate the siusoidal respose. As a compariso, the compesatio block is desiged usig both the 1 st ad d order Pade method. For a system with a 150-meter fiber delay, the delay time is 0.75 s. Usig the 1 st order Pade method, the siusoidal term [ 1 exp( i ω 0.75 s], ca be approximated by, s G 1( s = (5.1 s Thus, the compesatio block based o the first order approximatio is the iverse of G (, 1 s

150 136 6 s C1 ( s = (5. s Similarly, the compesatio block based o the d order approximatio ca be calculated as show i Eq s s C ( s = + (5.3 7 s Compared with 1 st order compesatio, d order compesatio itroduces a additioal differetiator (the last term of Eq. 5.3, which provides a 90-degree phase lead at high frequecies. As will be show later, the OFLL loop filter with d order compesatio have approximately 13 db more gai while maitaiig the loop stability due to the 90-degree phase lead. Step 3: Check the loop stability ad gai The objective of the loop filter desig is to achieve maximum ope loop gai without compromisig the loop stability. The last step i the loop filter desig is to optimize the adjustable gai to attai the maximum suppressio of the dowcoverted optical phase oise. Fig. 5.6 depicts the Bode plot ad Nyquist plot of the optimized ope loop respose of OFLL system with 1 st order compesatio. Nyquist plot idicates a stable system. However, the Nyquist tracer is very close to the ustable poit (-1,0 idicatig the stable margi. The Bode plot reveals a ope loop gai of 68 10kHz offset.

151 137 (a. Bode diagram of the optimized OFLL ope loop respose with 1 st order compesatio block. The ope loop 10kHz is 68dB. (b. Nyquist diagram of the optimized ope loop gai. Figure 5.6: The performace of the loop filter with 1 st order compesatio I compariso, the Bode ad Nyquist plot of the optimized d order compesated system ope loop respose is show i Fig The Bode plot reveals a ope loop gai of kHz offset, which is 13dB higher tha the first order system, idicatig that the d order compesatio provide 13dB more optical domai phase oise suppressio. It occurs because the additioal differetiator itroduces a additioal 90- degree phase lead to compesate the phase lag from the VCO (DTMOT.

152 138 (a. Bode diagram of the optimized OFLL ope loop respose with d order compesatio block. The ope loop 10kHz is 81.3dB. (b. Nyquist diagram of the optimized (max gai OFLL ope loop gai. The Nyquist trace does ot ecircle the ustable poit (-1,0, idicatig a stable respose. Figure 5.7: The performace of the loop filter with d order compesatio It should be metioed that i priciple higher order compesatio (> should allow higher loop gai, ad result better optical domai phase oise suppressio. However, a higher order compesatio block would itroduce ustable behavior (poles i the right had s plae ad complicates the filter desig process. I most situatios, the simple d order compesatio is sufficiet. I additio, extesive simulatios were performed o the loop filters for differet fiber legths. It was foud that for both the first order ad the secod order compesated loop filter, the maximum stable gai decreases 40 db per 10 times icrease i the fiber delay legth. Thus, eve though icreasig fiber delay legth decreases the OFLL system oise floor, it reduces the stable ope loop gai ad results i a sigificat icrease i the residual laser phase oise. Therefore, uless usig a OFLL with multiple loops (see

153 139 Sectio 5.5, the legth of the fiber delay eeds to be carefully selected to trade off the residual oise ad the oise floor Desig of the OFLL system usig a sigle loop I this sectio, we are goig to discuss the OFLL system desig usig sigle loop. The results from the previous sectios will ow be applied to determie the OFLL system parameters such as system cofiguratio, total delay legth, etc. First, the desig objective, amely, the phase oise, was idetified. The, based o the trade off betwee the residual oise ad the oise floor, we selected a suitable legth of the fiber delay, from which the parameters for the loop filter is determied. i. OFLL desig objectives Table 5. summarizes the desig objectives of the OFLL system ad Table 5.3 summarizes the parameters of the DTMOT. Table 5.: The desig objectives of a sigle loop OFLL Output phase oise Requiremets Operatig frequecy: Objective 10kHz kHz DC to 40 GHz

154 140 Table 5.3: The parameters for the compoets i the sigle loop OFLL Compoets VCO (Microchip laser assembly VCO optical power VCO free ruig phase oise (Estimated Photodiode output delayed path Photodiode output direct path Lik oise figure Parameter k V = 1.56 *10 4 rad / s / volt both wavelegth -40 db/ dec slope kHz (by estimatio kHz MHz -6 dbm -1 dbm 6dB ii. Optimum delay legth The OFLL implemetatio scheme show i Fig. 5.4 is employed. I order to select the optimum legth of the fiber delay, the phase oise performace correspodig to differet fiber delay legths is depicted i Fig Figure 5.8: OFLL system oise floor for differet fiber delay legths

155 141 As show i Fig. 5.8, i the close-i carrier regio the phase oise is limited by the OFLL oise floor, ad it has a decay rate of 0dB/dec. However, whe the frequecy is further away from the ceter, the phase oise spectrum becomes flat, which idicates that i this regio the phase oise of the OFLL is domiated by the residual laser phase oise (i.e., ope loop gai limited. I order to optimize the phase oise from 10kHz to 100kHz regio, a fiber legth of 90 meters is selected, which yields the followig oise performace: 88 1KHz, kHz, 100kHz. I additio, usig the procedure i sectio 5..3, the loop filter for 90-meter fiber delay employig a secod order compesatio block is give by, s F ( s = e007 s e013 s + 6e e007 s s 1+ s / 3e + 7 I sectio 5.3, we will examie the experimetal results for this sigle loop OFLL. But ext, we will discuss a more sophisticated OFLL architecture that allows idepedet optimizatio of the ope loop gai ad the oise floor Multi-loop OFLLs I a sigle loop OFLL system, although a loger fiber delay lowers the oise floor, it reduced the stable feedback gai ad leads to a sigificat icrease of the residual phase oise. I this sectio, we itroduce a multi-loop OFLL to elimiate this problem. A geeral diagram of the multi-loop OFLL system is depicted i Fig Although the loop filter for the loger loop is geerally ot ecessary ad ca be elimiated to reduce the system complexity, it is still cosidered because we wat to obtai a geeral model for the multi-loop system ad it is foud i the simulatios that this filter does improve the system performace.

156 14 DTMOT Loger delay: ~ 1km PD3 1 x 3 Coupler Loop Filter for the short loop PD Short delay: ~ 50m PD1 Loop Filter for the loger loop Figure 5.9: A multi-loop OFLL employ two loops i. System ope loop gai ad stability Usig the aalytic model similar to the sigle loop OFLL, the ope loop gai of the double loop OFLL is calculated as, G( ω = k v = G 1 iωτ 1 iωτ [ k (1 e + k (1 e F ( ω ] 1 p k ( ω 1 + p p iωτ (1 e F ( ω 1 iωτ 1 k p(1 e F ( ω/ iω 1 (5.4 where k v is the trasmitter sesitivity, 1 k p ad k p are the discrimiator sesitivity of the short ad log loops respectively, τ 1 ad τ are the time delays of the short ad log loops respectively, F ( ad F ( are the loop filters of the short ad log loops 1 ω ω respectively, ad G 1 is the ope loop respose of the short loop. The loop stability requires that o poles of the followig fuctio exist i the right half of the s plae,

157 143 ( 1 (1 ( (1 ( G 1 1 ( 1 1 ( 1 1 ( ω ω ω ω ω ω ωτ ωτ G e k F e k G G H i p i p = + = (5.5 Thus, the stability of the double loop OFLL system is determied by the stability of the short loop OFLL (i.e., ( 1 1 G 1 ω + ad the stability of a seemigly more complex system ( ω H ~, ( ( 1 (1 ( (1 ( G 1 1 ~ ω ω ω ω ωτ ωτ G e k F e k j H i p i p + + = (5.6 We called ( ω H ~ stability fuctio of the Multi-loop OFLL. Although the fuctio ( ω H ~ looks complicated, it is well behaved. Extesive simulatio idicates that it is geerally stable if the ratio betwee the legths of the short loop to the loger loop is below 0. ii. The oise performace of the multi-loop OFLL Just like its sigle loop couterpart, the multi-loop OFLL has a output phase oise give by: 1 ( 1 / ( ( ( 1 ( ( ω ω ω ω ω ω φ ω φ G i F k G v o out = (5.6 where φ is the free ruig phase oise of the trasmitter, ad o is the equivalet iput oise at the discrimiator for the short loop. The expressio for phase oise of the multiloop OFLL has the same explicit form as that of the sigle loop OFLL. However i the

158 144 multi-loop OFLL, the equivalet iput oise is the combiatio of the oise for both loops. The first term o the right had side of the Eq. 5.6 is the OFLL residual phase oise, ad the secod term is the oise floor. For the regio close to the carrier frequecy, the ope loop gai is much bigger tha 1. The, the oise loop for the multi-loop OFLL ca be simplified as, φ out ( ω = k 1 p (1 e ( ω o iωτ 1 iωτ + k (1 e F ( ω p (5.7 Assumig equal carrier power i both loops ad a uity filter for the loger loop, it is show i Eq. 5.7 that the oise floor of the double loop OFLL is determied by the loger loop. iii. A practical example of double loop OFLL To coclude the discussio o the multi-loop OFLL, we discuss the desig of a double loop OFLL system employig 50 meter ad 1km fiber delay. The first step is to determie the filter parameters for the short loop. Applyig the result for the sigle loop OFLL, the loop filter with d order compesatio is give by: F1(s = k (s/.4e (s + 0.8e + 7/s ((s + 5e + 6/s (1 + s/5e + 7 where k is the adjustable gai. Ideally, the loop filter for the loger loop should be obtaied i the same maer as the short loop. But for simplicity of the circuitry, it is elimiated. As show below, good performace is still obtaied this way. Assumig both loops have equal power, we obtai the maximum stable ope loop gai i Fig The Nyquist diagrams for the ~ ope loop gai of both the short loop ad the stability fuctio H ( jω are depicted i

159 145 Fig Both diagrams idicate a stable respose. As show i Fig. 5.10, the ope loop gai for the double loop OFLL at 10kHz is 19 db. Figure 5.10: The ope loop respose of the double loop OFLL at maximum stable gai. (a The Nyquist plot for the short loop Figure 5.11: Stability of the multi-loop OFLL

160 146 (b The Nyquist plot for the multi-loop stability fuctio H ~ Figure 5.11 (cotiued The oise floor of this double loop OFLL is calculated by assumig similar lik parameters as show i table 5.3. The oise floor is foud to be 130dBc/Hz at 10kHz offset with a slope of 0dB/dec. It should be poited out that the multi-loop OFLL ca iclude more tha loops to further improve its performace. Their stability ad oise ca be similarly aalyzed usig the techiques explored i this subsectio. Next, we will discuss the experimetal result of the two oise cotrol systems. 5.3 Noise cotrol subsystem characterizatio I this sectio, we discuss first the experimetal results o the digital sythesizer phase oise cotrol subsystem ad the the results o the OFLL Digital sythesizer type oise cotrol The digital sythesizer type phase oise cotrol subsystem is depicted i Fig Usig the digital sythesizer, we lock the DTMOT output to a 133MHz ultra stable referece source (Miteq XTO Limited by the microwave prescaler, the digital

161 147 sythesizer has a operatig rage from 3GHz to 18GHz. The oise spectrum whe the DTMOT is operatig at GHz is depicted i Fig. 5.13, while Fig shows the phase oise at 10kHz for differet operatig frequecies. The phase oise, as expected, degrades with icreasig carrier frequecy.

162 148 LD curret source PD 16GHz prescaler Curret drivig D/A C Active filter Digital FPD N-couter R-couter A/D C 133MHz referece Figure 5.1: The digital sythesizer phase oise cotrol subsystem 148

163 149 Figure 5.13: The digital sythesizer phase oise spectrum of 8.447GHz Figure 5.14: The digital sythesizer phase oise vs. operatio frequecy

164 150 Usig a secodary referece source, the DTMOT ca be set to operate i the mmwave frequecy rage. The experimetal setup is depicted i Fig The DTMOT output is mixed dow to 14GHz by a 6GHz low oise sigal, ad the 14GHz dowcoverted sigal is locked by the digital sythesizer. The phase oise measuremet was carried out at the lower frequecy (14GHz to elimiate the oise cotributio from the secodary referece source. The optical spectrum at 40GHz is revealed i Fig The measured phase oise at a 10kHz offset is 90dBc/Hz (Fig Mm-wave optical Trasmitter Digital sythesizer sub-system 14 GHz 40 GHz PD 6GHz Ref 14 GHz Figure 5.15: The experimetal scheme for 40GHz operatio based o the digital sythesizer phase oise cotrol

165 151 Figure 5.16: Optical spectrum of 40GHz subcarrier sigal Figure 5.17: Phase oise spectrum of the dowcoverted sigal (14GHz The direct modulatio of the digital sythesizer was also examied ad the result is illustrated i Fig I this experimet, the DTMOT is set to operate at GHz. A 3volt, 1.5MHz triagular wave sigal is applied to the DTMOT which geerates a

166 15 liearly chirped subcarrier sigal, which is the demodulated by a Agilet vector sigal aalyzer. The low speed is chose sice we are iterested i idetifyig the lower boud of permitted modulatio sigal ad the digital sythesizer oly limits the lower frequecy modulatio sigal [5] (i.e., < its loop badwidth. (a Sigal spectrum captured by a HP VSA. (b Istataeous frequecy Figure 5.18: The respose of the digital sythesizer uder directio modulatio

167 Delay lie optical frequecy locked loop characterizatio I this sectio we characterize the performace of the 90-meter sigle loop OFLL desiged i sectio It should be metioed here that the iitial experimetal ivestigatio o multi-loop OFLL has bee performed with promisig results. However, at this stage the less ideal performace of some critical compoets, specifically directivity of the 3x3 fiber optic beam splitter, impaired the system performace. We expect this problem to be solved i the ear future. The experimetal setup of the sigle loop OFLL is show i Fig The expected phase oise of the OFLL is so low 10kHz regardless of operatio frequecy that we have o meas i our laboratory to verify precisely its phase oise performace at higher frequecies. Therefore, we first characterize the OFLL performace at lower frequecy (<1.5GHz. The, we demostrate the OFLL operatio at higher frequecy (0GHz ~ 40GHz. i. OFLL operatio at lower frequecy By tuig the temperature, the DTMOT is set to operate from 100MHz to 1.5GHz. The phase oise spectrum is measured usig HP 8564E spectrum aalyzer. Two samples of the phase oise spectrum for 300MHz operatio are depicted i Fig. 5.0.

168

169 155 Driver Driver TE Cotrol TE Cotrol ML ML d order compesated OFLL loop filter 3dB 90 m fiber delay PD DC offset adjustmet Mixer 1 Phase shifter Output PD Amp Amp1 RF Power divider 130MHz referece P out Microwave spectrum aalyzer Mixer Amp3 Mixer 3 Figure 5.19: The OFLL experimetal setup 155

170 156 (a The phase oise spectrum for 345MHz carrier (b The phase oise spectrum for GHz carrier sigal. Figure 5.0: Phase oise spectrum of sigle loop OFLL at lower frequecies

171 157 The measured phase oise spectrum agrees with the theoretical predictio. No sigificat variatio of the phase oise performace is observed through out the OFLL operatio frequecy (100MHz to 1.5GHz. ii. OFLL higher frequecy operatio Next, the DTMOT is tued to the mm-wave frequecy. A sample of the DTMOT sigal at 37GHz is show i Fig The phase oise is measured usig the HP 8564E spectrum aalyzer ad the results are show i Fig. 5.. Figure 5.1: The OFLL output sigle spectrum at 37GHz

172 158 (a. The measured phase oise spectrum for 9.1GHz carrier sigal (b The measured phase oise spectrum for 38Ghz carrier sigal Figure 5.: The OFLL phase oise measuremet results at mm-wave frequecies

173 159 At higher frequecies, the measurig equipmet (the oise floor of HP 8564E limited the precisio of the phase oise measuremet. The measured phase oise is extremely close to the oise floor of the measuremet equipmet. Therefore, we expect the actual phase oise of the OFLL output to be eve better.

174 160 Chapter 6: Digital FM fiber radio dowlik trasmissio This chapter cosiders a specific applicatio of the tuable microchip laser. The mm-wave fiber radio system [7] uses a fiber optical lik (dowlik to distribute high data rate traffic ad a mm-wave carrier sigal from the cetral statio to the basestatios. The traffic betwee the mobile uits ad basestatios is carried out via the mm-wave radio liks (uplik. The dowlik implemetatio is oe of the pricipal cocers of the mm-wave fiber radio system. A commo approach is to first geerate the mm-wave subcarrier ad the employ a exteral modulator to superimpose the digital/aalog iformatio. I additio to its high cost ad complexity, this two-step approach has a limited umber of data modulatio schemes (AM type because exteral optical modulators ca oly impose AM modulatio o the mm-wave subcarrier. It makes iterfacig betwee the fiber radio ad the wireless liks less trasparet, sice more sophisticate modulatio formats [57] (such as cotiuous phase PM (CPM, miimum shift keyig (MSK, etc, are extesively used i the curret wireless systems due to their badwidth efficiecy ad good immuity to cross talk, fadig, etc of wireless chaels. A ew approach i the fiber radio trasmissio is proposed employig the dyamic tuable mm-wave optical trasmitter [13] (Fig..1, i which the high quality mm-wave subcarrier ad high data rate CPM-type digital sigal are simultaeously geerated by directly frequecy modulatig the DTMOT. I this sectio, we will address the FM fiber radio dowlik performace, amely the maximum achievable data rate ad preset iitial experimetal results.

175 Fiber radio dowlik performace As the backboe of the hybrid fiber optic ad wireless commuicatio system, the dowlik must have a high data rate. Theoretically (see Chapter 3, the laser trasmitter itself has a fast speed. I practice, however, the data rate is limited by the trasmitter voltage tuig sesitivity, as explaied below. I digital FM schemes, the frequecy excursio betwee idividual symbols icreases with the data rate [57]. For example, i the case of miimum shift keyig (MSK, the frequecy excursio betwee the symbols has to be exactly oe half of the symbol rate. If the fiber radio lik is ruig at a data rate of 00Mbps, the frequecy excursio eeds to be 100MHz, which requires a +/-.5 volt 00Mbps basebad drivig voltage whe the trasmitter has a sesitivity of 0MHz/volt. However, if the data rate is icreased to 1Gbps, the compulsory frequecy excursio becomes 500MHz, which ecessitates a voltage driver to deliver a +/-1.5 Volt 1Gbps basebad sigal. This is difficult to implemet i practice. Higher data rates require larger frequecy excursios. However, geerally, the basebad sigal has a limited output voltage at high-speed. Thus, the best approach to icrease the data rate is to obtai a higher tuig sesitivity by reducig the electro-optic sectio thickess. Curretly, the voltage tuig sesitivity of the DTMOT is MHz/Volt. However, we expect that the thickess of the electro-optic sectio ca be reduced to 0. mm, which should provide sesitivity close to 100MHz / volt. The, just with a +/-5 volt basebad sigal, the data rate of Gbps for MSK modulatio scheme ca be achieved.

176 16 Aother way of improvig the data rate is to use a multi-level CPM modulatio format (m>, i which oe symbol trasmits more tha oe bit of iformatio. However, the multi-level CPM scheme does come with a price, it requires higher SNR to achieve the same bit error rate (BER compared to the two level CPM format (such as MSK. However, sice the mm-wave optic trasmitter ca deliver a very low oise carrier sigal, the multi-level CPM scheme is still a viable optio. 6. Digital FM fiber radio dowlik experimet The digital FM fiber radio dowlik trasmissio is demostrated at a 8.4GHz carrier frequecy usig the tuable mm-wave optic trasmitter. For simplicity, miimum shift keyig (MSK is used as the digital FM format. The experimetal setup is depicted i Fig As show i Fig. 6.1, the tuable mm-wave optical trasmitter is directly modulated by a basebad sigal. A digital sythesizer type phase oise cotrol subsystem is employed to lock the carrier frequecy at 8.4GHz. Later, a 6GHz referece sigal is used at the receiver ed to mix dow the 8.4GHz sigal. The resultig MSK modulated RF sigal at.4ghz is the coveietly aalyzed via a HP 89640A vector sigal aalyzer (VSA, which operates below.7ghz with a badwidth of 36MHz. Limited by the VSA badwidth (36MHz, a MSK symbol rate of 0Mbps is chose so that the mai lobe of the MSK modulated fiber radio sigal is withi the aalyzer badwidth. The basebad modulatio source is tued to attai a frequecy excursio of 10MHz betwee symbol 0 ad 1, a prerequisite for 0Mpbs MSK modulatio. The experimetal parameters are summarized i Table 6.1. The oise characteristics of the dowlik subcarrier sigal are revealed i Fig. 6.. The MSK demodulatio result captured by the

177 163 vector sigal aalyzer is show i Fig. 6.3, where the upper left trace is the sigal costellatio, the lower left trace is the symbol phase error, ad the trace o the upper right is the group delay (istat frequecy, ad the lower right trace is the sigal trellis. Table 6.1: The digital FM fiber radio lik parameters Parameters Value Subcarrier optical power 3dBm Subcarrier modulatio idex 0.97 Carrier frequecy 0.84GHz Carrier phase oise 10kHz offset Carrier AM oise -30dBam from 10kHz to 100MHz (Affected by laser relaxatio DTMOT sesitivity 0MHz / Volt Digital FM format MSK Data rate 0Mpbs Frequecy excursio 10MHz Data sequece repeat Digital patter geerator Atteuator Tuable trasmitter GHz PD 6GHz referece.4ghz Agilet 89640A VSA Figure 6.1: The digital FM fiber radio experimetal setup

178 164 Figure 6.: The fiber radio dowlik oise charactistics 164

179 165 Figure 6.3: The demodulatio result captured by the HP VSA 165

180 166 The MSK demodulatio result looks very promisig. The sigal costellatio ad trellis are decetly clustered together. As show i the sigal costellatio diagram, however, the costellatio poits spread slightly i both tagetial ad radial directios. They idicate the existece of the amplitude ad phase errors i the fiber radio sigal. The amplitude error is caused by the laser relaxatio oscillatio ad laser parasitic amplitude modulatio accompaied with the frequecy modulatio. The relaxatio oscillatio is 30dB below sigal level ad should have oly small effect. However, laser parasitic AM ca be sigificat because it will be resoatly ehaced close to the laser relaxatio frequecy (1~5 MHz, which is withi the MSK sigal bad. A sample of the measured parasitic AM respose is depicted i Fig. 6.4, where the Y axis uit is Am/Volt. The domiat source of laser parasitic AM is the piezo effect of the LiNbO3, which distorts the shape of the crystal ad modulates the cavity loss whe applyig a tuable voltage. Figure 6.4: The parasitic AM respose

181 167 The phase error show i the sigal costellatio is due to rigig of the basebad source. A sample of the basebad waveform is show i Fig Figure 6.5: The basebad sigal rigig I summary, a digital FM fiber radio trasmissio employig miimum shift keyig modulatio was successfully demodulated. Except for the slight spread i the sigal costellatio, the results are promisig. 6.3 FM fiber radio up lik ad basestatio implemetatio issues The digital FM fiber radio system requires a differet scheme for the basestatio implemetatio compared with the traditioal fiber radio systems [7]. The basestatio scheme for the FM fiber radio system is show i Fig. 6.6.