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Characterization and modeling of erbium-doped fiber amplifiers and impact of fiber dispersion on semiconductor… Movassaghi, Mahan 1999

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Characterization and Modeling of Erbium-Doped Fiber Amplifiers and Impact of Fiber Dispersion on Semiconductor Laser Noise By MAHAN MOVASSAGHI B.Sc. Amir-Kabir University of Technology, Tehran, Iran, 1992 M.A.Sc. University of British Columbia, Vancouver, Canada, 1996 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Electrical and Computer Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1999 © Mahan Movassaghi, 1999 ln presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, 1 agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of tfalfiCcJ! <iW Centflukr' ^ yinarJr}^ The University of British Columbia Vancouver, Canada D a t e fl(*3: /?7f DE-6 (2/88) ABSTRACT This thesis describes theoretical and experimental studies on two subjects: first is characteriza-tion, design and modeling of erbium-doped fiber amplifiers (EDFAs); second is the effect of fiber dispersion on the noise of distributed feedback (DFB) lasers and the impact of this effect on the performance of 1550nm video lightwave transmission systems. A simple electrical measurement technique for EDFA noise figure characterization is devel-oped which has significantly better accuracy than other methods reported. This is achieved by noise measurements at identical detected optical power levels, with and without EDFA present. This approach ensures that the system noise level is identical in both measurements, thereby even small EDFA noise levels can be separated accurately from the large noise of the measure-ment system by subtracting the two noise measurements. Using this technique an excellent agreement is obtained between optically- and electrically-measured noise figures of saturated EDFAs. This result is in contrast to earlier reports by Willems and van der Platts from Bell Lab-oratories, showing significant discrepancies between optically- and electrically-measured noise figures of a saturated EDFA which sparked* a serious controversy over the appropriate approach to model and measure the noise figure of EDFAs. Using the general, radially dependent rate-equation EDFA model, it is shown that highest-efficiency operation of saturated EDFAs is achieved with erbium distributed throughout the entire fiber core, in contrast to generally-accepted design principles. A simplified one-dimen-sional steady-state model for gain and noise in such EDFAs is derived which is accurate for any arbitrary distribution of erbium doping inside the fiber core. It is shown that the saturation parameters normally included in conventional models can be eliminated without loss of accu-racy, with the resulting model requiring only small-signal gain and loss coefficients as parame-ters. This great simplification eases fiber characterization, and enhances accuracy in predicting amplifier performance. DFB laser relative intensity noise (RIN) variation induced by fiber chromatic dispersion is measured in the range of frequencies relevant to cable television systems. For two analog lasers tested, RFN degradation as large as 15dB is observed after 48km of standard fiber at a baseband frequency of 800MHz. The degradation increases with frequency, affecting higher channels the most. The experimental results are in excellent agreement with a simple theory by Yamamoto, which only requires knowledge of the laser linewidth to determine the RIN degradation. It is shown that this RIN degradation can significantly impair system carrier-to-noise ratio. i i i Table of Contents Abstract ii List of Tables vii List of Figures viii Publications x Acknowledgements xi 1 Introduction 1 1.1 Introduction to Thesis 1 1.1.1 Overview 1 1.1.2 Outline of Chapter 2 1.2 Motivation . 3 1.3 Operating Principles of CATV Systems 4 1.4 Erbium-Doped Fiber Amplifiers 5 1.4.1 Operating Principles 6 1.4.2 Noise Sources 10 1.5 Conventional EDFA Noise Figure Measurement Techniques . . . . 11 1.5.1 Optical Measurement Techniques 11 1.5.2 Electrical Measurement Technique 14 1.6 Modeling Gain and Noise Figure in EDFAs 15 1.7 Effect of Fiber Dispersion on DFB Laser RfN 21 iv 1.8 Laser RIN Measurement Techniques 24 1.9 Outline of the Thesis 25 2 Noise Figure of Erbium-Doped Fiber Amplifiers in Saturated Operation 27 2.1 Introduction 27 2.2 Motivation 28 2.3 A New Electrical Noise Figure Measurement Technique 31 2.4 Optical Noise Figure Measurements 39 2.5 NF Measurement Results and Discussions 43 2.6 Summary and Conclusions 48 3 Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 49 3.1 Introduction 49 3.2 Motivation 50 3.3 Effects of Erbium Confinement on the Performance of Saturated EDFAs . 52 3.4 Assessment of Saleh's Model Accuracy for Non-Confined EDFAs. 56 3.5 A Simple and Accurate Model for Saturated EDFAs 60 3.6 EDF Characterization Methods and Measurements 68 3.7 Generalization of the Proposed Model 76 3.8 Conclusions 77 4 Dispersion-Induced RIN Degradation and its Impact on 78 1550nm AM Video Lightwave Transmission Systems 4.1 Introduction 78 4.2 Motivation 79 V 4.3 Laser RIN Measurements 79 4.4 Modeling and Discussion 84 4.5 Summary and Conclusions 94 5 Summary and Conclusions 95 Bibliography 99 Appendix 105 A Rate and Propagation Equations for 980nm-pumped EDFAs 105 List of Tables 2.1 Comparison between the mean values of the NF 48 3.1 Theoretical comparison of the gain and noise figure 56 4.1 Parameters used in the CNR calculations 92 vii List of Figures 1.1 Basic system configuration for CATV systems . 5 1.2 A basic architecture of an EDFA 7 1.3 The first three energy levels of erbium in silica 8 1.4 Experimental setup for optical NF measurements 13 1.5 Measured RIN at the 1550nm DFB laser 23 2.1 Noise figure of a 980nm co-propagating EDFA 30 2.2 Experimental setup for broadband EDFA noise figure 34 2.3 Calibration measurements for EDFA noise figure. . . 38 2.4 Experimental setup for optical NF measurements . 41 2.5 Experimental results of polarization nulling technique 42 2.6 Measured noise figure as a function of input signal power 44 2.7 Frequency-resolved noise figure 46 3.1 Theoretical gain and noise figure comparison 54 3.2 Theoretical gain and noise figure comparison 55 3.3 Theoretical results of monochromatic loss data 58 3.4 Theoretical gain and NF comparison 59 3.5 Normalized population of the excited level 64 3.6 Theoretical gain and NF comparison 69 3.7 Normalized pump and signal photon numbers 70 v i i i 3.8 980nm loss measurement results 73 3.9 comparison between theoretical and experimental gain and NF 75 4.1 Experimental setup for RIN measurement 80 4.2 Measured electrical noise power versus average photodiode c u r r e n t . . . . 82 4.3 Frequency resolved RIN for DFB1 83 4.4 RIN versus fiber length 85 4.5 Experimental setup for linewidth measurement 88 4.6 Measured laser spectrum for linewidth characterization 89 4.7 Calculated CNR versus frequency 93 IX Publications I hereby declare that I am the sole author of this thesis. In accordance with University of British Columbia thesis guidelines, I hereby declare that parts of this thesis has been or will be published under the following titles: M. Movassaghi, M. K. Jackson, V. M. Smith, J. F. Young, and W. J. Hallam, "Noise figure of sat-urated erbium-doped fiber amplifiers: electrical versus optical measurement," in Conference on Optical Fiber Communication OFC'97, Vol. 6, 1997, OS A Technical Digest Series (Optical Society of America, Washington, D.C, 1997), paper WA2. M. Movassaghi, M. K. Jackson, V. M. Smith and W. J. Hallam, "Noise figure of erbium-doped fiber amplifiers in saturated operation," IEEE Journal of Lightwave Technology, vol. 16, no. 5, May 1998. M. Movassaghi, M. K. Jackson, V. M. Smith, J. F. Young, and W. J. Hallam, "Accurate fre-quency-resolved measurements of EDFA noise figure," in Optical Amplifiers and Their Applica-tions, 1997 Technical Digest (Optical Society of America, Washington, D.C, 1997), paper TuDl. M. Movassaghi and M. K. Jackson, "Design and modeling of saturated erbium-doped fiber amplifiers," to appear in Tech. Dig. IEEE/LEOS Summer Topical Meet., San Diego, CA, USA, July 1999. M. Movassaghi and M. K. Jackson, "Simple and accurate modeling of high-efficiency saturated erbium-doped fiber amplifiers," submitted for publication to J. Lightwave Technol. on July 27, 1999. M. Movassaghi, M. K. Jackson and V. M. Smith, "DFB laser RIN degradation in CATV light-wave transmission," Tech. Dig. IEEE/LEOS'98, vol. 2, paper FB2, Dec 1998. M. Movassaghi, M. K. Jackson and V. M. Smith, "Dispersion-Induced RIN degradation and its impact on 1550nm AM video lightwave transmission systems," revised manuscript under prepa-ration for submission to /. Lightwave Technol. Acknowledgement My deepest gratitude goes to the members of my family for their encouragement and support throughout my education. I would also like to express my deep gratitude to my Ph.D. advisor Dr. Mike Jackson, for his knowledgeable guidance, patience and support throughout the course of this work and during my entire graduate studies. I found Mike to be a unique individual possess-ing a combination of many great qualities, including professionalism, dedication and brilliance. It was truly a great pleasure to work with Mike as I was continually learning new things from him. With no doubt, he played a major role in enhancing my knowledge, skills, and character; I owe him a great deal of gratitude. I also wish to thank Dr. Jeff Young, from Department of Phys-ics and Astronomy, for his fruitful comments and suggestions during the course of my Ph.D. program. I would like to express my gratitude to Vince Smith, Senior Development Engineer at Tho-mas&Betts Photon Systems, who helped me in many practical issues concerning my Ph.D. work. Vince was one the most skillful engineers I had ever known and working with him was a significant learning experience. My gratitude extends to John Maycock, former Vice President of Thomas&Betts Photon systems, who kindly provided me with the opportunity to conduct my Ph.D. research at Photon Systems. I would also like to acknowledge Wes Hallam, Development Engineer at Thomas&Betts Photon Systems, for his assistance in my experimental setups. I would like to sincerely thank Dr. David Pulfrey for his help and support during difficult times associated with my M.A.Sc. program which dragged into the beginning of my Ph.D. Pro-gram. I am also grateful to Dr. Bob Donaldson and Dr. Mike Davies, former and present heads of the Department of Electrical and Computer Engineering, for trusting me and providing me with the opportunity to teach EE474 for six consecutive terms during my Ph.D. program. I would like to thank all of my friends especially, Dr. Shahram Tafazoli, Dr. Mohammad Sameti, Dr. Siavash Jahromi, Roberto Rosales, Dr. Mehrdad Sharifzadeh, and Sohrab Sigarian for their encouragement, help and support during the course of this work. I would also like to thank many wonderful staff members of the Department of Electrical and Computer engineering who helped me in various ways during the course of my graduate studies at UBC; in particular I am very grateful to Doris Metcalf, Cathleen Holtvogt, Leslie Leroux, Tony Leungner, Kathy Brindamour and Gail Schmidt. x i i Chapter 1 Introduction 1.1 Introduction to Thesis 1.1.1 Overview This thesis describes theoretical and experimental studies on two subjects: first is characteriza-tion, design and modeling of erbium-doped fiber amplifiers operating in saturation; second is the impact of fiber dispersion on the relative intensity noise of distributed feedback (DFB) lasers. The above subjects are studied in the context of 1550nm optically amplified AM (amplitude modulated) video lightwave transmission systems, which are of particular interest to the indus-trial collaborator involved in this work, Thomas&Betts Photon Systems in Burnaby, B.C. How-ever, the results shown in this thesis are relevant and applicable in general to many modern analog and digital fiber optic communication systems including the rapidly growing field of wavelength division multiplexing, in which saturated erbium-doped fiber amplifiers play a key role. The work described in this thesis is composed of three major parts. The first part, concerning the erbium-doped fiber amplifier (EDFA) noise figure, describes a new technique for electrical noise figure measurements of EDFAs and comparison of results versus the conventional optical measurement technique. The second part, concerning design, modeling and characterization of 1 Chapter 1. Introduction. 2 EDFAs, describes the design of highly efficient saturated EDFAs, describes a simplified yet accurate model for theoretical prediction of gain and noise figure in saturated EDFAs, and finally describes characterization techniques for obtaining the erbium-doped fiber parameters required for the model. The third part, concerning DFB laser noise, presents measurements of DFB laser relative intensity noise (RIN) in the frequencies relevant to the video lightwave trans-mission systems and shows that significant degradation of laser RIN can occur with dispersive fiber propagation. Finally, it describes modeling of laser RIN variation with fiber dispersion and discusses the consequence of this effect on the overall performance of the 1550nm optically amplified AM video lightwave transmission systems. 1.1.2 Outline of Chapter The purpose of this chapter is to describe the motivation of the work and provide relevant back-ground information on topics presented in this thesis. Section 1.2 describes motivation and incentives of the work presented in this thesis. In Section 1.3 operating principles of 1550nm optically-amplified AM video lightwave transmission systems are described. In Section 1.4 fun-damentals of EDFA operation and EDFA noise sources are described. Section 1.5 describes con-ventional EDFA noise figure measurement techniques. In Section 1.6 conventional approaches for modeling EDFAs are presented. Section 1.7 describes DFB laser noise and its variation with fiber dispersion. In Section 1.8 laser RIN measurement techniques are described and finally Sec-tion 1.9 describes the organization of the remainder of the thesis. Chapter 1. Introduction 3 1.2 Motivation 1550nm optically-amplified AM video lightwave transmission systems, which will be referred to as CATV (cable television) systems, are being extensively deployed around the world to replace coaxial networks. The underlying reason is that fiber-based systems can achieve higher quality transmission with much lower installation and maintenance costs. The coaxial networks contain many cascaded amplifiers to compensate the cable attenuation in the magnitude of sev-eral hundreds of decibel (dB) and accumulation of the noise of these amplifiers limits transmis-sion span of coaxial networks. In comparison, the low loss of optical fibers at 1550nm, and the advent of erbium-doped fiber amplifiers (EDFAs), which optically amplify the signals in the 1550nm window, have enabled CATV systems to transmit up to 80 video channels over trans-mission span lengths of 120km or more [1]. In addition, at the end of the transmission line, high power and low noise erbium-doped fiber amplifiers allow the signal to be split among many users maintaining its high quality. A stringent requirement for CATV systems is that a high signal-to-noise ratio is required at the receiver, and therefore in designing such systems the noise contribution from various compo-nents of the system must be carefully analyzed and modeled. Beside the receiver shot noise, two of the essential noise contributors in these systems are the erbium doped fiber amplifier (EDFA), used as power and/or inline amplifier, and the DFB laser in the transmitter. In addition, as EDFAs play the role of amplifying and boosting the optical signal, modeling of the gain of these amplifiers is also of particular importance. The work presented in this thesis concerns theoretical and experimental analyses on several important issues regarding the EDFA gain and noise and Chapter I. Introduction. 4 also the noise of DFB lasers. 1.3 Operating Principles of CATV Systems A layout of a basic 1550nm optically-amplified AM video lightwave transmission system (CATV systems) is shown in Fig. 1.1. It consists of a DFB laser operating in CW (continuous wave) mode employed as an optical carrier. A number of amplitude-modulated vestigial side-band (AM-VSB) TV channels, typically between 40 to 120 channels, which have frequencies between 50 to 800 MHz, with 6MHz channel spacing and each having 4MHz bandwidth are combined using an RP power combiner. The resulting composite signal is then applied to a lin-ear electro-optic modulator to intensity modulate the optical carrier. The output of the modulator is optically amplified by an erbium doped fiber amplifier and the EDFA output is launched to the transmission fiber at the other end of which the optical signal is converted back to an electrical signal using an optoelectronic receiver. As the TV channels are formed by AM modulation, the performance of CATV systems is significantly affected by the noise of the different components in these systems. For optimum design of CATV systems the noise characteristics of different components of these systems should be analyzed. In this thesis, several issues concerning the noise of the EDFA and DFB laser, which are important contributors of noise in CATV systems, are analyzed. Chapter 1. Introduction 5 video (80-120 channels) Figure 1.1: Basic system configuration for 1550nm optically-amplified AM video lightwave transmission systems (CATV systems). Components are described in the text. 1.4 Erbium-Doped Fiber Amplifiers (EDFAs) The advent of erbium-doped fiber amplifiers in 1987 revolutionized the field of optical fiber communication. Prior to the advent of EDFAs, the standard way to compensate for the attenua-tion of the optical fiber transmission line was to use electronic regenerators. In these regenera-tors the weak optical signal is converted into an electrical signal using an optoelectronic receiver, the electrical signal is then amplified and converted back into an optical signal by an electro-optic transmitter. Besides the complexity and high cost, the bandwidth of such regenerators is limited by the speed of their electronic components, which is much lower than the bandwidth of the optical fiber, thus limiting the transmission capacity of the entire fiber optic communication system. In contrast, EDFAs allow direct optical amplification of the weak signals along the fiber transmission link and therefore are not limited by any electronic bandwidth, thus allowing a dra-matic increase in transmission capacity compared to electronic regenerators. In addition, EDFAs have several other attractive features such as high gain, high output power, high efficiency, polar-Chapter 1. Introduction. 6 ization insensitivity, and low noise figure. These attributes have made the EDFA an exception-ally important component in most analog and digital lightwave communication systems. 1.4.1 Operating Principles A basic architecture of an EDFA is illustrated in Fig. 1.2. The light from the signal laser (between 1520nm-1560nm) and the light from the pump laser (typically either at 980nm or 1480nm) are coupled to the erbium-doped fiber (EDF) with a wavelength division multiplexer (WDM). The EDF is made by incorporating erbium dopants together with germania (Ge02), and/or alumina (A1203) into the silica fiber core using various techniques [2]-[3]. Germania is used as an index-raising codopant and alumina improves the solubility of erbium into silica fiber, resulting in higher concentration doped fiber. An isolator at the end of the EDF blocks any back-reflections to the EDFA from the downstream line. The architecture shown in Fig. 1.2 is called a co-directionally pumped configuration as the pump is travelling in the same direction of the signal inside the EDF. In another EDFA architecture, signal and pump travel in opposite directions, which is called counter-directionally pumped configuration. There are several other architectures for EDFA which are obtained by using one or more co- and counter- propagating pumps, isolators, optical filters and cascading stages. However, the principles of their operation are generally identical. The focus of the work presented in this thesis is on the single stage co-directionally pumped EDFAs, which are normally used in CATV systems. Chapter 1. Introduction 7 EDF Signal Laser W D M ># Isolator Pump Laser fusion splice fusion splice Figure 1.2: A basic architecture of an EDFA The EDFA operation principles can be described by considering the first three energy levels for the erbium in a silica glass, which are shown in Fig. 1.3. The shaded rectangles in the figure indicate that each main energy level is split into a manifold of multiple energy sublevels due to the Stark effect. The Stark effect is induced by the permanent electric field, which is generated by charge distribution in the glass host [4]. Nevertheless, due to the effect of intramanifold ther-malization, which maintains a constant population distribution within the manifolds (Boltz-mann's distribution), each manifold can be considered as a single energy level [2]-[4]. 4 4 Among the several transitions that can be used to pump EDFAs, / 1 5 / 2 —> I\\/2 an<^ 4 4 7 1 5 / 2 —> /13/2 transitions, corresponding to 980 and 1480nm pumping, respectively, are the most efficient pump bands. For 980nm pumping, the EDFA behaves like a three level system; 4 the energy from the pump laser boosts the erbium ions from the ground state ( / 1 5 / 2 manifold) 4 4 to the pump state ( In/2 manifold), from which the ions relax to the metastable state ( 7 1 3 / 2 manifold). Since the relaxation rate of ions from the pump state to the metastable state is much higher than the pumping rate, the population of the erbium ions in the pump state is negligible Chapter 1. Introduction h\/2 t-13/2 98()nm 15/2 fast decay 148()nm 1550nm Figure 1.3: The first three energy levels of erbium in a silica glass host Chapter I. Introduction. 9 and a high degree of population inversion can be achieved. Therefore, the 980nm pumped EDFAs can be analyzed by considering only two levels. For 1480nm pumping, the energy from 4 the pump laser excites the erbium ions in the low lying levels of the / 1 5 / 2 manifold to the high lying ones in the 4 / 1 3 / 2 manifold, leading to a population inversion between the two main energy levels; the 1480nm pumped EDFAs can also be analyzed by considering only two levels. Once in the metastable state, the ions are stimulated by the signal propagating along the EDF, causing them to decay back to the ground state; this results in the emission of photons in phase with the signal, thus amplifying the signal by stimulated emission. Amplification at wavelengths 4 approximately between 1520 to 1560nm takes place between the low lying levels of the 7 1 3 / 2 4 manifold and the high lying ones in the / 1 5 / 2 . Besides the stimulated emission, there exists a competing process of spontaneous emission. The photons arising from this process have no coherence characteristics with respect to the signal; some of the spontaneously emitted photons are captured by the fiber and then amplified, leading to amplified spontaneous emission (ASE) at the EDFA output. Therefore, the output spectrum of an EDFA consists of the amplified input spectrum, and the broadband ASE, which is known to be the major EDFA noise contributor [2]-[4]. Besides the transitions shown in Fig. 1.3 there exist additional transitions associated with an undesirable phenomena called pump excited state absorption (ESA). In this process, the pump 4 photons at frequency are not absorbed from the ground state Il5/2, but are absorbed from 4 the metastable level / j 3 / 2 . This is due to existence of another level whose energy gap with the metastable level happens to closely match the pump photon energy hvp, where h is Planck's Chapter 1. Introduction. 10 constant. For 980nm pumping, ESA occurs between the metastable level (level 2) and level 4 4 and for 1480nm pumping this occurs between level 2 and Iu/2 manifold (level 3). The pump ESA process would therefore result in an excess loss for the pump, which reduces the amplifier gain. However, it has been shown that for the 980nm and 1480nm pumping, the pump ESA is very small and almost negligible [4]-[5]. Operating conditions of EDFAs are generally grouped into two regimes, unsaturated and sat-urated regimes. Unsaturated regime corresponds to the conditions where the amplifier gain is within 3dB of its small signal gain and saturated regime refers to the conditions where the amplifier gain is reduced more than 3dB from its small signal gain. Small signal gain is the high-est gain of an EDFA. 1.4.2 Noise sources The noise that will be seen on a photodetector placed after an EDFA is comprised of signal shot noise, ASE shot noise, signal-ASE beat noise, ASE-ASE beat noise, and interferometric noise. The first two of these are the shot noise generated in the receiver's photodetector due to the EDFA output signal power level and the ASE power level. The signal-ASE term is caused by heterodyne mixing between the amplified signal and the ASE at the receiver's photodetector. This is generally the dominant noise source of EDFAs. The ASE-ASE term is generated due to heterodyne mixing of the ASE with itself. Interferometric noise is caused by heterodyne mixing of the main optical output signal with the multiply backscatterd optical fields which are gener-ated by reflections from EDFA components and/or reflections due to fiber Rayleigh scattering. Chapter I. Introduction. 11 The relative intensity noise (RIN), defined in Section 1.5.2, corresponding to the interferometric noise, is given by [3],[6]-[7]: *™tiPi(f)~-r±—i ( L 1 ) f +Av where MPI denotes multi-pass interference, Av is the source laser linewidth, and / is the noise frequency. Therefore, interferometric noise is frequency dependent and its influence on the total noise of an EDFA reduces at higher frequencies; the rate of this reduction is greater for laser sources with smaller linewidths. 1.5 Conventional EDFA Noise Figure Measurement Techniques The noise figure (NF) of an amplifier is a measure of the degradation of the signal-to-noise ratio for a signal passing through the amplifier, with the signal and noise measured in the electrical domain. However, due to difficulties in making such measurements in the electrical domain, an alternative approach called the optical method has prevailed and is generally used for determina-tion of the EDFA noise figure [3], [7]-[8]. In this section both the conventional electrical and optical methods are described. 1.5.1 Optical Measurement Techniques The optical NF measurement techniques are based on measuring the amplifier signal gain and ASE power, using an optical spectrum analyzer, and then calculating the noise figure from a the-oretical relationship given by [3]-[4], [7]-[9]: Chapter 1. Introduction 12 N p = PASE j _ (1.2) hvAvG G where PASE is the total EDFA amplified spontaneous emission power within the optical band-width of Av , v is the optical signal frequency and G is the EDFA gain at the signal wavelength. However, the above expression only includes the shot noise and signal-ASE noise contributions. In fact, it is assumed that the other EDFA noise terms are negligible. Fig. 1.4 shows the experimental setup for measuring PASE and G. The main experimental apparatus is an optical spectrum analyzer (OSA) which measures the power spectrum. To deter-mine the noise figure, two measurements are performed. The first is a measurement of the laser source spectrum that is used to determine the input signal level to the EDFA (Pin) and the source spontaneous emission level at the signal wavelength (SSE). The second is a measurement of the output spectrum of the EDFA to determine the signal power level and the output spontane-ous emission (OSE). The difference in the signal levels from the two measurements is the EDFA gain (G). The OSE contains the SSE multiplied by the amplifier gain plus the PASE, i.e. OSE = SSExG + PASE . (1.3) Chapter I. Introduction. 13 DFB OSA Figure 1.4: Experimental setup for optical NF measurements. To precisely determine the noise figure, PASE must be determined at the signal wavelength. However, this cannot be measured directly as the signal power level masks the ASE level at the signal wavelength; instead it is determined using the linear interpolation technique [10]. In this technique, the SSE and OSE levels are measured at a wavelength just above and just below the signal wavelength. Knowing the amplifier gain, PASE at the signal wavelength is then deter-mined from equation (1.3) and using a linear interpolation. This technique is based on the assumption that the ASE produced by the EDFA is linear (on a linear scale) over a small wave-length range (typically +/- lnm) about the signal. This assumption is well justified as experi-mental evidence for typical EDFAs indicates that the maximum ASE deviation from a straight line fit over a +/- 1 nm range is less than 1% at all wavelengths between 1542 to 1565 nm; this is the wavelength region in which EDFAs are commonly used [10]. However, although the linear interpolation technique is relatively simple and expected to be highly accurate for EDFAs oper-ating in the unsaturated regime, it has been argued that it is not accurate for amplifiers operating in saturation [8]-[12]. The argument is based on the fact that in saturation, amplified sideband components of the signal become comparable to ASE, due to the reduction of the ASE power level in saturation, and distort the shape of the ASE spectrum around the signal, therefore mak-Chapter 1. Introduction. 14 ing it difficult to measure the ASE level accurately. A polarization nulling technique was pro-posed to improve the accuracy of noise figure measurements in saturation [9]. This technique, which has become widely accepted, is built upon the fact that the ASE is randomly polarized, whereas the input signal is polarized in one direction. Therefore by adding a polarization state controller and a linear polarizer between the EDFA and the optical spectrum analyzer (OSA), ideally it is possible to suppress the amplified signal and its sidemodes and, thereby, measure the ASE level at the signal wavelength. However, experimental measurements have shown large residual components of the signal even at 40 dB signal suppression [11]-[ 12]. Therefore in prac-tice the interpolation technique is used in conjunction with polarization nulling to accurately determine PASE and thereby the EDFA noise figure [10]. 1.5.2 Electrical Measurement Technique (RIN Subtraction Method) The conventional electrical noise figure measurement technique is based on the method pro-posed by Willems et. al. which is known as the RIN (relative intensity noise) subtraction method [7], [13]-[15]. In this method, the NF is related to the RIN at the input and output of the EDFA through the expression, AfF - output ~ 'sjgnai) Pjn 1 2hv G ' U j where Pin is the optical input power, and G is the amplifier gain. RIN0Utput and RINsignal are the RIN measured at the output and input of the EDFA, respectively. The RIN of a signal is 2 2 2 defined as AP /P expressed in units of dB/Hz, where AP is the mean square optical inten-Chapter 1. Introduction 15 sity fluctuations (in a 1 Hz bandwidth) at a particular frequency, and P is the square of the optical power [3]. The RIN is measured using a commercial Hewlett-Packard 71400 lightwave signal analyzer, which is an optoelectronic receiver connected to an RF spectrum analyzer. This equipment determines the RIN automatically by subtracting the signal shot noise and the receiver thermal noise from the total noise measured, and from measurements of optical signal power and several frequency dependent system parameters; these parameters are photodetector, amplifier, and spectrum analyzer frequency responses as well as mismatch losses between them [10], [16]. The accuracy of the RIN measurement using the HP 71400 is estimated to be ±2dB [14]. This rather large uncertainty in RIN measurement directly limits the accuracy of the elec-trical measurement technique, specially for the saturated EDFAs where RINoutput approaches RINsignal. In Chapter 2 of this thesis a new electrical NF measurement technique is described which has an absolute accuracy of ±0.3 dB. 1.6 Modeling Gain and Noise Figure in EDFAs To predict the gain and noise properties of EDFAs, models based on propagation and rate equa-tions of a homogeneous, two level system have been used [2]-[4]. In these models, the broad-band ASE spectrum, having a total spectral width of Av, is discretized into k optical beams having a frequency bandwidth of Av^ centered at the optical wavelength Xk = c /vk. Avk are the frequency steps used in the simulation to resolve the ASE spectrum, and therefore k = Av/Av^. To model the evolution of signal, pump, and ASE, a set of differential-integral equations, each corresponding to one optical beam in the spectral slot of Av^ and center wave-Chapter 1. Introduction 16 length Xk, have to be solved. By considering that ASE propagates in two directions (forward and backward), the total number of equations would then become 2k + n, where n is the total num-ber of signal and pump sources [2]-[4]. These equations are coupled and have to be solved numerically, requiring complex programming and substantial computational time. For future ref-erence, the above approach will be referred to as the "comprehensive model". The number of equations and therefore the complexity of the numerical calculations in the comprehensive model can be significantly reduced by considering ASE as only two optical beams at the signal wavelength with an effective bandwidth, propagating in the forward and backward directions. This model is called the effective ASE model and for 980 pumping the equations describing propagation of the signal, Ps, pump Pp and total ASE propagating in the forward and backward directions, P* and Pa , can be written as [4], [17] (see Appendix A for details of derivations): dP dz Csen2(r) - a M/»,(/•)] \|/5(r) rdr - a'sPs (1.5) P( r ) \.<5ESAn2{r) + opan^r)] \|#p(r) rdr - a'pPp (1.6) ± + ~ dz dP a <*se MO [Pa + 2hyfs A v e / / ] - °sa n\(r) Pa\vs(r)rdr (1.7) ± a Chapter I. Introduction 17 where subscripts s and p denote the signal and pump, respectively, and \\fp are the radial mode envelope distributions with unity peak, a>s and (£>p are the power mode sizes, and z is the fiber longitudinal coordinate. p(r) is the erbium density distribution, errand Gsa are the emis-sion and absorption cross sections at the signal wavelength, respectively; apa and GESA are the absorption cross section and excited state absorption cross section at the pump wavelength, respectively. a's and a'p are the fiber back-ground loss coefficients at the signal and pump wavelengths, respectively. nx{r) and n2(r) are the ground and upper level population densities normalized by p(r), which satisfy «j(r) + n2(r) = 1. n2(r) is given by [4] (see Appendix A for details of derivation): sat TP V„(0 + Ps(z)+P.(z) + Pa(z) n2(r,z) = _ p °sa + °se .sat 1 + f ^ - V p ( r ) + — p s (1.8) where psat = hvsii co5 (1.9) and Chapter 1. Introduction. 18 are the intrinsic saturation powers at the signal and pump wavelengths, respectively, and vp are the optical frequencies, h is Planck's constant, and x is the spontaneous emission lifetime. In the above model, the ASE is modeled as forward and backward propagating optical beams cen-tered at the signal wavelength with an effective ASE bandwidth given by [10], [20], Aveff = J(ae(v)/ae(v,)) dv . (1.11) o The noise figure, NF, is calculated using NF = , P a } L ) + 1 (1.12) hvsAveffG G where Pa (L) is the total ASE power at the output end of the amplifier. The effective ASE model provides an accurate estimate of the noise figure, and allows approximate evaluation of the impact of ASE on the population n2(r). The comprehensive model and the effective ASE model described above, although are most suitable for accurate theoretical prediction of gain and noise figure of EDFAs, but they are not very practical as they include many parameters some of which are difficult to measure accu-rately. Among these parameters are erbium ion concentration and distribution inside fiber core, optical mode distributions, emission and absorption cross sections and fluorescence lifetime [3]-[4], [17]-[18]. Alternately, several models based on various assumptions have been developed Chapter 1. Introduction. 19 which provide a more practical way to model EDFAs [3]-[4], [17]-[20]; among these a model developed by Saleh et. al. [19]-[20], has been widely accepted and used for modeling of EDFAs operating in saturation [21]-[27]. This model is derived from the comprehensive model based on neglecting the effect of amplified spontaneous emission on erbium ions population inversion, excited state absorption, and fiber background losses at the signal and pump wavelengths. In addition it has been assumed that the area of the erbium-doped active region is so small, i.e. erbium is well confined near the center of the core, such that the radial mode envelope distribu-tions at the pump and signal wavelengths are identical. Accordingly, the amplifier gain is obtained from the following equations: nout nin ~0.kL (.Pj„-Pou,)/Pk" Pk = pk e e O- 1 3 ) where P\n and P°kut are the input and output powers of the kth optical beam and where Pin - I Pf a n d Pou<= l PJ"' d'14) are the total EDFA input and output powers, respectively, ak is the small signal absorption coef-ficient of the kth beam and L is the length of the amplifier. By summing both sides of equation (1.13) over k, Pout can be obtained by solving a transcendental equation based on knowing the parameters oc^ ., Pskat, and the input powers. Once Poul is obtained, it is inserted into equation (1.13) to obtain the output P°ku', and hence the gain, at each wavelength, ak and P™' at each wavelength can be obtained by monochromatic loss measurements at several input powers to the Chapter 1. Introduction 20 EDF and fitting equation (1.13) to the results [19]. For noise figure calculations, the power of amplified spontaneous emission PASE at the EDFA output is required (see equation (1.2)). This is obtained by multiplying the spontaneous emission from each infinitesimal section of the EDFA by the gain it experiences to the end of the amplifier, and integrating that over the length of the amplifier. Accordingly, PASE at the EDFA output is given by: z= L ASE f Pk (I) = Av gk I N2(z) Gk(z, L) dz (1.15) z= 0 where tajpjW N2(z) = — (1.16) c j^u+Pjizyp5;'] is the population of the second level, Gk(z, L) is the gain of the fcth optical beam from position z to the end of the amplifier. £ = pA/x, where A is the area of the doped region; C, is obtained from £ = oc^  P ™' at A, = 980nm. gk is the small signal gain coefficient which is obtained from gk = (t, /P™') - ak. In Chapter 3 of this thesis it is shown that the efficient design of a saturated EDFAs is achieved by not confining the erbium and distributing it across the entire fiber core. Given that Saleh's model is derived based on the assumption that the erbium is well confined, it cannot Chapter 1. Introduction. 21 accurately predict the performance of non-confined EDFAs. In Section 3.5 a simple model is derived which can closely predict the gain and noise figure of non-confined EDFAs. 1.7 Effect of Fiber Dispersion on DFB Laser RIN A fundamental source of noise in lasers is due to the spontaneous emission which continually adds new power to the laser oscillation field generated by the stimulated emission. The electro-magnetic field associated with the spontaneous emission is not coherent with the field generated by the stimulated emission, and therefore the laser output field fluctuates both in amplitude and in phase. Therefore, the actual electric field of a single mode laser can be represented by E(t) = ( £ 0 + A £ ( 0 ) exp[z'(co0f + (p(0)] (1.17) where EQ is the amplitude, AE(t) is intrinsic amplitude fluctuation (intrinsic amplitude noise), co0 is the central angular frequency, and cp(f) is the phase fluctuation (phase noise) of the laser electric field. It should be noted that since both amplitude and phase fluctuations originate from the same source, spontaneous emission, they are correlated. At the laser output the measured intensity noise is only due to the intrinsic amplitude noise and the phase noise does not cause any intensity noise. However, through dispersive fiber propagation, different frequency compo-nents of the laser electric field, caused by the phase noise, travel at different velocities, thus reaching to the end of the fiber at different times. In a photodiode placed at the end of fiber, these delayed components are mixed interferometically causing amplitude noise. Since this amplitude noise originates from the laser phase noise, this phenomena is called phase to intensity or PM Chapter I. Introduction. 22 (phase modulation) to AM (amplitude modulation) noise conversion. Therefore, after dispersive propagation, the laser intensity noise is due to a combination of the laser intrinsic amplitude noise and an additional amplitude noise created by the PM to AM noise conversion. The effect of fiber dispersion on the semiconductor laser intensity noise has been analyzed with two different approaches. In one approach, by Yamamoto et. al. [28], the laser intrinsic amplitude fluctuation AE(t) in equation (1.17) is neglected and only phase fluctuations are con-sidered. The result of this analysis leads to a simple expression showing a proportional increase in the laser intensity noise by the fiber dispersion and laser linewidth. In the second approach, by Marshall et. al. [29], both of the laser intrinsic amplitude and phase noise together with the pro-cess correlating these two noise sources are considered. The resulting model is thus comprehen-sive, but it is complicated and contains several laser parameters which are difficult to measure accurately. Their results show that over moderate distances (several km for standard single mode fiber at 1550nm) the laser intensity noise is reduced over a wide range of frequencies, but for longer fiber distances the intensity noise is proportionally increased with fiber length. Fig. 1.5 shows their measurement results for relative intensity noise (RIN) of a 1550nm DFB laser and its variation after 4.1 km and 20 km of standard single mode fiber. The reduction and increase of laser RIN for the two lengths of fiber shown in Fig. 1.5 is clearly visible for frequencies above 1GHz; however, the measurement results become uncertain below 1 GHz as it is difficult to mea-sure RIN accurately because of its low level. Chapter 4 of this thesis, presents a detailed analysis of the effect of fiber dispersion on the DFB laser RIN for frequencies relevant to CATV systems (below 800 MHz). Chapter 1. Introduction . . . >"| 30 r • i i • • 11.1 i i i i i 1 1 i l 1 — i i i i 11il 0.1 1 10 100 Frequency, Q/2n (GHz) Fig. 1.5: Measured RIN at the 1550nm DFB laser (circles), and after propagation in the standard single mode fiber of length 4.1 km (triangles) and 20 km (diamonds). From Ref. [29]. Chapter 1. Introduction. 24 1.8 Laser RIN Measurement Techniques The conventional laser RIN measurement technique is based on the approach described in Sec-tion 1.5.2. The limitation of this technique is that it requires knowledge of several frequency dependent system parameters, which are difficult to measure. In another RIN measurement approach, developed by Nazarathy, et. al. [30], laser RIN is obtained independently of these parameters. In this method the laser voltage noise spectral density is measured, using an opto-electronic receiver attached to an RF spectrum analyzer, at several optical received powers; laser RIN is then extracted form least squares fitting of the following equation to the measured results [30]: NT-Ndark = RIN R2 H P2 + 2e HRP (1.18) where NT is the voltage noise spectral density, and Ndark is the voltage noise spectral density in the absence of light to the receiver (laser is disconnected from the receiver). R is responsivity of the photodiode in the optoelectronic receiver, H is the transimpedance gain from the photo-diode's current to the input voltage of the spectrum analyzer, and P is the optical power input to the receiver1. The measurement approach by Nazarathy et. al., provides a simpler means for laser RIN measurements than the conventional method as it requires knowledge of only three parameters. 1. The above definitions given for the parameters contained in equation (1.18) are from Ref. [30]; how-ever it is believed that the proper definitions for NT and Ndark are that they denote the noise power spec-tral densities and H is equal to Z 2 / Z , „ , where Z is the transimpedance of the network connecting the photodiode to the spectrum analyzer and Zin is the spectrum analyzer input impedance (See Chapters 2 and 4 for more details). Chapter I. Introduction. 25 In Chapter 4 of this thesis, a similar method is used to measure laser RIN. 1.9 Outline of the Thesis Chapter 2 describes a new electrical noise figure measurement technique which can provide accurate determination of the EDFA noise figure. A comparison between electrically- and opti-cally- measured noise figures of two EDFAs is presented showing excellent agreement between the two measurement approaches; this confirms the applicability of conventional homoge-neously broadened two level system model for EDFA noise prediction. The agreement obtained between the two measurement approaches is in contrast to earlier reports showing significant discrepancies between electrically- and optically- measured noise figure of a similar saturated EDFA. In Chapter 3, it is shown that highest-efficiency operation of saturated EDFAs is achieved with erbium distributed throughout the entire fiber core, in contrast to generally-accepted design principles. It is shown that the commonly used Saleh model cannot accurately predict the perfor-mance of non-confined EDFAs and a simplified EDFA model is derived which can accurately predict gain and noise figure of confined and non-confined EDFAs. A comparison between the predictions of the proposed model and the experimental gain and NF of a 980nm-pumped EDFA is presented showing an excellent agreement between them. In Chapter 4, an enhanced measurement method for simple and accurate characterization of laser RIN is described. Measurements of laser RIN are presented in the range of frequencies rel-evant to CATV systems, and for the first time, lowest RIN values as low as -172 dB/Hz are Chapter 1. Introduction. 26 shown for an analog 1550nm DFB laser. Laser RIN measurements versus several lengths of standard fiber, up to 48 km, are presented and it is shown that for the case of CATV systems, RIN degradation by fiber dispersion can be significant. Finally, excellent agreement between the experimental results with predictions of a simple theory which relates laser RIN degradations to the laser linewidth is presented. Chapter 2 Noise Figure of Erbium-Doped Fiber Amplifiers in Saturated Operation 2.1 Introduction As described in Chapter 1, one of the main objectives of the Ph.D. research being presented in this thesis is to develop a model for accurate prediction of the gain and noise figure in saturated erbium-doped fiber amplifiers (EDFAs). One of the important factors involved in achieving this goal was having access to reliable techniques for accurate measurements of EDFA gain and noise figure (NF), allowing reasonable comparison to be made between model predictions and experimental results. While conventional techniques were sufficiently accurate for gain mea-surements, there was a serious controversy over the appropriate technique for measuring NF of saturated EDFAs. Obviously, prior to any attempt to develop the model, the possibilities of being able to make accurate NF measurements should have been assessed. In this chapter a new tech-nique for accurate measurement of EDFA noise figure is presented which has an absolute accu-racy within ±0.3 dB for noise figure measurements. This chapter is organized as follows. In Section 2.2, the motivation of this work, which fur-nishes an outline of the previous attempts in measuring EDFA noise figure and controversy sur-rounding the appropriate approach for its measurement, is described. In Section 2.3, a new electrical noise figure measurement technique is described. This technique is based upon mea-27 Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 28 surement of the electrically-detected noise levels with and without the amplifier present. By adjusting the optical power levels at the detector to be identical for these two measurements, it is ensured that contributions from shot, thermal, and input RIN noise are identical; therefore the EDFA noise contribution can be recovered by subtracting the results of the two measurements. In Section 2.4, the method used for optical NF measurements is described. This method is based on the conventional polarization nulling technique, together with a calibration technique which allows accurate determination of the loss incurred by the polarizer and the polarization control-ler. In Section 2.5, the results of electrical and optical NF measurements are presented, com-pared and discussed. Finally, summary and conclusions of the results presented in this chapter are in Section 2.6. 2.2 Motivation As described in Chapter 1, conventional techniques for NF measurement can be grouped into two categories: electrical and optical methods. The definition of noise figure for an optical amplifier refers explicitly to measurements made with electrical detection [31], making the link between the results of electrical characterization and noise figure straightforward. However, accurate measurement of the very low levels of noise is challenging, and most reports of N F measurement have been based on optical measurements [3], [7]-[9]. Hentschel et. al. have com-pared optical and electrical NF characterization for amplifiers operating in the unsaturated regime, and found good agreement between the two results [7]. However, later in 1995, Willems and van der Platts questioned the validity of optical NF measurements of saturated EDFAs [13]-Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 29 [14]. For a 980nm-pumped copropagating EDFA operating in the signal-saturated regime, these authors found that the results of optical NF measurements were significantly higher than those derived electrically. Fig. 2.1 illustrates the results of their measurements. The upper curve shows the measured optical NF and the lower curve shows the electrical NF versus optical input signal power to the EDFA. As can be seen from this figure, at high input signal powers where the EDFA is saturated, the electrical NF decreases while the optical NF increases for higher input signal powers. They attributed the decrease of the electrical NF to a reduction in statistical fluc-tuations due to gain saturation predicted by a qualitative consideration of nonlinear photon sta-tistics in the EDFA [4]. Consequently, they concluded that optical measurement methods can lead to overestimation of the actual NF of saturated EDFAs. This was also an indication that the commonly-used semi-classical EDFA models cannot provide valid noise figure predictions for saturated EDFAs; this is because these models commonly predict that the EDFA noise figure rises as the amplifier is driven deeper into saturation. These conclusions, created a significant controversy over the appropriate approach to measure and model the noise figure of saturated EDFAs. In this Chapter, a new electrical NF measurement technique is presented which has signifi-cantly better accuracy compared to the conventional method. Using this method, a similar com-parison between optically- and electrically-measured NF of two amplifiers, similar to the one used in Ref. [13]-[14], is performed. The results show excellent agreement between the two measurement approaches, which is in contrast to the results of Willems and van der Platts [32]-[34]. Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 30 Figure 2.1: Noise figure of a 980-nm co-propagating EDFA as a function of signal input power measured with the conventional Optical Method (upper curve) and the conventional Electrical Method, RIN subtraction method (lower curve). (From Ref. [14]) Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 31 2.3 A New Electrical Noise Figure Measurement Technique In this section an expression is derived which relates the EDFA noise figure to parameters which can be measured simply and accurately. An electrical measurement technique for accurate deter-mination of these parameters is then described. The NF of an optical amplifier is defined as the ratio of the input signal-to-noise ratio (SNR), to output SNR, with the SNRs measured at the output of an ideal photodiode (IPD). With the SNRs expressed in terms of the mean squares of noise photocurrent at the output of the IPD, the NF equation is given by [7], J.JJ-. _ ^^^input _ (^signal) in ^'noise1* out _ 1 ^'noise*out (n -\\ ~ SNR " 7^2 r~ 72 r~ " ~i ~i r~ • { Z A ) ou'Put \ l noise) in \ l signal* out G \ l noise) in 2 2 where (inojse)in and (inoise)out correspond to the amplifier input and output mean square noise currents in a unit bandwidth, respectively, and G is the amplifier gain. By definition, the ideal photodiode has a quantum efficiency of one. In addition, the only source of noise detected when measuring the input signal is assumed to be shot noise. Therefore, (inoise)in is given by, <Lse)in = 2 ^ * P , „ (2.2) where e is the electron charge, P I N is the amplifier optical input signal power, R* = e/hv is 2 the responsivity of the IPD, h is Planck's constant, and v is the optical frequency. ( i n o i s e ) o u t is the output shot noise plus the additional noise from the EDFA in a unit bandwidth; this can be written as: Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 32 EDFA, out (2.3) where N EDFA, out is the EDFA mean square noise photocurrent in a unit bandwidth at the IPD, and can be written as, EDFA, out = (R*f ((^optical)2) (2.4) where ( (k-Popticad ) * s m e m e a n square power fluctuation in a unit bandwidth of the EDFA output signal. By considering that real photodetectors (PD) have a quantum efficiency r|, less than one, and the coupling efficiency between the EDFA output and PD is Kin (e.g. due to the connector loss), the EDFA mean square noise photocurrent in a unit bandwidth measured by a real photodiode, NEDFA, can be written as, where R = r\e/hv is the responsivity of the real photodiode. From equations (2.1)-(2.5), and considering that KinR can be obtained by dividing the photodiode average current I by the EDFA output power P • x G, the EDFA noise figure equation can be written as: N EDFA = R2K-n < i^Poptical)2 > (2.5) N EDFA P; in NF = 2 hvl2 (2.6) Equation (2.6) is of particular significance as it shows that the EDFA noise figure can be Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 33 expressed in terms of three parameters Pin, G, and / , which can be measured easily and accu-rately. Therefore, the difficult problem of measuring NF has been reduced to an accurate deter-mination of NEDFA, which is considered next. The factor NEDFA was determined from measurements of the noise power at the input and output of EDFA using the experimental setup shown in Fig. 2.2. Using this setup, the noise power is measured using an HP 437B broadband radio-frequency average power meter (RFPM) over the 50-450 MHz passband of an electrical bandpass filter (EBPF). A distributed feedback laser (DFB) operating at the wavelength 1547.4nm is used as the EDFA input signal source, and optical attenuator A l is used to vary the signal level at the EDFA input. Optical attenuator A2 is used to adjust the optical power level at the receiver input (PRX)- The optical bandpass filter (OBPF) has a passband of 2nm and is used to reduce the amplified spontaneous emission (ASE) power at the EDFA output, minimizing the effect of ASE-ASE beat noise. The receiver is an Ortel 2620A-E01 and consists of a photodiode (PD), and a transimpedance amplifier (TA); Ld is the total coupling loss from receiver input to PD. RFA is a wideband RF amplifier having a gain of 30dB and is used to minimize the effect of thermal noise in the RF power meter on the total measured noise power, enhancing the measurement dynamic range. IM is a 75Q to 50Q. con-verter which matches the 75Q. output impedance of the receiver to the 50Q input impedance of the RF amplifier. To minimize the influence of interferometric noise in the NF measurements, the EDFA in the setup has an input and output isolator to prevent multiple amplification of back scattered or reflected light, caused mainly by fiber Rayleigh scattering in the gain medium; also all the components in the setup have angled connectors with a return loss of more than 60 dB Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 34 and attenuators have reflectivities less than -55 dB. Furthermore, the DFB laser has a linewidth of approximately 1.5MHz which ensures that the interferometric noise has dropped significantly for frequencies above approximately 50MHz (see Section 1.4.2 for details or Ref. [35]). The DFB laser has a relative intensity noise (RIN) of -170+2 dB/Hz in the 50-450 MHz frequency range and such a small laser RIN makes the beat noise due to the interaction of the laser RIN with the ASE negligible [31]. DFB Receiver OBPF A2 PD IM EBPF RFPM RFA A 9 Ammeter Figure 2.2: Experimental setup for broadband EDFA noise figure measurements; the compo-nents are described in the text. The noise power measured by the RFPM, PN, is obtained by integrating the noise over the entire signal bandwidth. By considering that the interferometric noise is significantly suppressed in the 50-450 MHz frequency range, it can be assumed that the noise current is constant over the measurement bandwidth (this assumption will be also justified experimentally in Section 2.5), and hence PM can be written as: Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 35 "total Z2 B~ PN = -J£!2L + Pth , (2.7) where Ntotal is the total mean square noise photocurrent per unit bandwidth at the output of the photodiode receiver, Z is the transimpedance of the network which connects the photodiode to the power meter; Bn is the effective bandwidth of this network, which would normally be deter-mined by the electrical bandpass filter shown in Fig. 2.2. Zin is the input impedance of the power meter, and Pth is the noise power measured due to thermal noise generated in the power meter and the network connecting it to the photodiode. The total noise Ntotal is given by Ntotal = " EDFA + NRIN + Nshot • (2-8) In this equation NRIN describes source RIN noise per unit bandwidth and is given by "UN = (PRxLdR)2RIN. (2.9) The shot noise per unit bandwidth Nshot is given by "shot = 2ePRXLdR. (2-10) To obtain NEDFA, the contributions of the RIN, shot, and thermal noise have to be excluded from the measured noise power PN. However, the noise power due to these three extraneous noise terms is constant for a constant received power PRX. Therefore NEDFA can be determined from the measurement of noise power with and without the EDFA, maintaining identical Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 36 received optical power using attenuator A2. Thus N EDFA Z 2 B„ (2.11) where PN2 and PNl are the noise powers measured with and without the EDFA present, respec-tively. Finally, from equations (2.6) and (2.11) the following expression for the noise figure is obtained TLjT-i (PN2~ Pin ^in 1 NF = —— + — 2 h vi Z B„ . G (2.12) In the final noise figure equation, equation (2.12), all quantities, except the factor 2 (Z Bn)/Zin, can be obtained simply and accurately by direct measurement. Direct measure-2 ment of the factor (Z Bn)/Zin is not an easy task, as it requires access to internal circuitries of 2 the receiver and RF power meter. Alternately, determination of the product (Z Bn)/Zin can be accomplished accurately using the following calibration technique. This calibration technique relies upon the fact that, with the EDFA removed from the setup, the measured noise power PN is proportional to the sum of source RIN, photodiode shot, and thermal noise terms; from equa-tions (2.7), (2.9)-(2.10) and considering that LdR =I/PRX this proportionality can be written as: PN = f 2 \ RINZ B f2eZ2Bn^ I + P th (2.13) Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 37 2 The second coefficient of the above equation contains the product (Z Bn)/Zin; this product can be obtained from a least squares fit of a quadratic function to a series of measurements of PN at varying received power levels. The results of such a series of measurements are shown in Fig. 2.3 by open circles, and the solid line shows the quadratic fit. Accounting for the absolute uncer-tainties in the measurements of PN and / , which are ±0.052 dB and ±0.022 dB, respectively, the accuracy of the (Z Bn)/Zin determination is estimated to be ±0.076 dB [36]. Equation (2.13) is similar to equation (1.18), except that it replaces the two parameters of optical power and photodiode responsivity contained in equation (1.18) by a single parameter of photodiode average current. This enhances the accuracy for the following reasons: first, it elimi-nates the requirement of knowing the coupling loss between receiver input and photodiode, which is difficult to measure; second, it eliminates photodiode responsivity from the relation; and third, it replaces the optical power with average current, which can be measured with much higher accuracy. The broadband NF measurement technique described above is appropriate if the NF is con-stant over the passband of the electrical bandpass filter. However, in many practical situations where the signal laser has a broader linewidth and/or optical isolators are not employed at the input and output of EDFA, multiple pass interference caused by various effects, such as multiple Rayleigh back scattering and reflections from downstream and upstream components of the transmission line, creates interferometric noise [37] and, since this noise is frequency dependent, frequency-resolved measurements of noise figure is desirable [35]. The electrical measurement technique described above can be extended to this case by replacing the electrical bandpass filter Figure 2.3: Calibration measurements: noise power PN as a function of photodetector current / with the EDFA removed. The open circles show the measured data, and the solid line shows the least squares fit described in the text. Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 39 and the RF power meter with an electrical spectrum analyzer (ESA) in the measurement setup shown in Fig. 2.2. In this case, the spectrum analyzer measures the average noise power over a selected frequency span and within a selected resolution bandwidth and hence frequency-2 resolved measurements of NF can be achieved. Since the product (Z Bn)/Zin could be fre-quency dependent, in frequency-resolved measurements of NF the calibration technique described for the broadband measurements should be performed for each baseband frequency. 2.4 Optical Noise Figure Measurements To be able to compare the results of the electrical NF measurement technique with the optical NF, measurements of the optical NF were also performed. The measurement method used is based on the conventional polarization nulling technique, which is described in Section 1.5.1. In addition, a calibration technique was developed to enhance measurement accuracy. As described in Section 1.5.1, in the optical NF measurement technique, the ASE optical power PASE is mea-sured at the signal wavelength using an optical spectrum analyzer (OSA), and then the optical NF is calculated from equation (1.2). The experimental setup for the measurement of PASE is shown in Fig. 2.4. The DFB laser is the same laser used in the electrical measurements. The optical bandpass filter (OBPF) has a passband of 2nm and is used for calibration which will be discussed later. Attenuator A l is used to adjust the EDFA input power. The EDFA output is passed through a polarization controller and a linear polarizer (MPB Technologies model B-001) with an extinction ratio greater than 40dB. The spectrum at the output of the polarizer is measured using an ANDO 6315B optical Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 40 spectrum analyzer. Using the polarization controller, the polarization state of the amplified sig-nal is set orthogonal to the polarizer, and thereby the signal is suppressed by approximately 40dB, allowing measurements of the PASE • However, in highly saturated EDFAs where the ratio between the output signal power and PASE increases rapidly for higher input powers, this is not sufficient to eliminate the signal completely. As an example, the measured spectrum at the out-put of the polarizer for an EDFA input signal power of 0 dBm is shown in Fig. 2.5 by the lower curve; residual signal is evident at 1547.4nm. Due to the existence of this signal residue, a linear fit to values measured ±0.5 nm away from the signal wavelength is also used to estimate the ASE level at the signal wavelength [8]. However, due to the optical loss of the polarizer and the polarization controller, the measured ASE level is lower than the actual PAsE- Since this loss factor varies at different settings of the polarization controller, its direct measurement is difficult. The appropriate method to determine this loss factor has not been discussed by previous authors; here, this loss is determined by performing the following simple calibration. First, the polariza-tion controller and the polarizer are removed from the setup, and the EDFA output spectrum is measured directly; for the input signal power of 0 dBm this spectrum is shown as the upper curve in Fig. 2.5. At a wavelength separated from the signal wavelength by approximately 5 nm, the input signal is almost completely eliminated by the optical bandpass filter and the output spectrum at this wavelength represents the actual ASE level. Therefore at this wavelength, the difference between the two measured spectra shown in Fig. 2.5 is equal to the insertion loss. Finally, PASE is obtained by adding this insertion loss to the measured ASE level. Accounting for the accuracies of the OSA level measurements (±0.3 dB), linearity (±0.05 dB), polarization Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 41 dependency (±0.05 dB), and resolution bandwidth (±0.1 dB), the absolute uncertainty in the optical NF measurements is estimated to be ±0.5 dB. DFB 2 nm Optical Filter Polarization Controller Polarizer O Q Q OSA Figure 2.4: Experimental setup for optical NF measurements. Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 42 -60 1542 1544 1546 1548 Wavelength (nm) 1550 1552 Figure 2.5Experimental results of polarization nulling technique. The upper curve shows the spectrum at the EDFA output; the lower curve is the spectrum at the polarizer output, showing residual signal at 1547.4nm due to finite extinction ratio of the polarizer. The resolution of the optical spectrum analyzer is 0.1 nm. Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 43 2.5 NF Measurement Results and Discussions In this section results of optical and electrical NF measurements for two copropagating 980nm pumped EDFAs are presented and discussed. Both amplifiers use Er/Al/Ge/P doped fiber. EDFA1 has an input pump power of 20 dBm, and is built from 12 meters of doped fiber with an erbium concentration of 960 ppm-wt, core radius of 1.9|im and numerical aperture (NA) of 0.2. EDFA2 has an input pump power of 19 dBm and is built from 5 meters of the doped fiber having an erbium concentration of 2000 ppm-wt, core radius of 1.6|im and an NA of 0.24. Fig. 2.6(a) and 2.6(b) show results of NF and optical gain measurements for EDFA1 and EDFA2 respectively, as a function of the amplifier input signal power. Results of optical and broadband electrical NF measurements are shown as asterisks and open circles, respectively. The results of optical gain measurements are shown as squares. For broadband electrical measurements the uncertainty in determination of NF is estimated to be within ±0.3 dB. This estimate is based on the uncertainties in the measurements of different parameters in equation (2.12). These uncer-tainties are ±0.052 dB for electrical noise power measurements, ±0.13dB for optical power measurements, ±0.022 dB for DC current measurements, and ±0.076 dB for determination of 2 the calibration parameter (Z Bn)/Zin. For both amplifiers, the electrical and optical NF shown in Fig. 2.6 increase in saturation and there is an excellent agreement between the two measurements in this regime. Similar behavior has also been observed for several other EDFAs tested. This result is in contradiction to the results of Ref. [13]-[14] and throws into question the relevance of nonlinear photon statistics to saturated EDFAs. The agreement between the electrical and optical measurements worsens for Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation AA 6.0 m 5.5 = 5.0 h ffl 4.5 V CO S 4.0 3.5 (a) . ° o o o $ * * * * * * * • $ # • 9 -20 -15 -10 -5 0 10 -10 -5 0 Input Signal(dBm) Figure 2.6: Measured gains and noise figures as a function of input signal power: (a) and (b) show results for EDFA1 and EDFA2, respectively. The open circles and asterisks show results of broadband electrical and optical NF measurements, respectively. The squares show results of optical gain measurements. Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 45 the lower input signal powers, where the electrical NF becomes larger than the optical NF. By considering that optical NF represents only signal-ASE beat noise, while electrical NF includes two additional terms of ASE-ASE beat noise and interferometric noise, one might expect that the discrepancy between the two measurements might be due to these two additional terms. To investigate the influence of ASE-ASE beat noise on the NF of EDFA1 and EDFA2, the follow-ing expression was used [31] NF, = + { P a s e ) 1 , +1 . (2.14) ' ^ A v G 2hvBP;G2 G where NFt represents the optical noise figure due to signal-ASE beat noise and ASE-ASE beat noise (the second term in the above expression). B is the passband of the optical filter, which is 2nm for this case and Pase is the amplifier ASE power within the passband of the optical filter. Pase was calculated from equation (1.2) by replacing Av with B and using the measured opti-cal NF results. Calculations of NFt for both EDFA1 and EDFA2 have shown that the maximum contribution of ASE-ASE term is only 0.005 dB, which occurs at lowest input signal power of -15 dBm. Therefore, ASE-ASE beat noise should have negligible effect on the NF results pre-sented in Fig. 2.6. To investigate the influence of the interferometric noise on the NF of EDFA1 and EDFA2, frequency-resolved noise figure measurements were also performed. These mea-surements were obtained with a Rohde&Schwarz FSEA 20 spectrum analyzer. In these mea-surements, the resolution bandwidth of the spectrum analyzer was 10 MHz, and the video filter bandwidth was 5Hz. Fig. 2.7 shows results of the frequency-resolved noise figure measurements Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 46 6.5 S 6 S5.5 <D 3 5 D) L L 4.5 CD SD. 4 3.5 V P i n = 5.2 dBm P. = - 0.9 dBm i n P j n = - 8 . 9 dBm 0 100 200 300 400 500 600 Frequency (MHz) Figure 2.7: Frequency-resolved noise figure for EDFA1 at input powers of -8.9, -0.9, and 5.2 dBm, respectively, as labeled. Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 47 for EDFA1 at three input signal powers. For frequencies below 30MHz the NF increases consid-erably with frequency, but for frequencies above 30MHz the noise figure is almost constant hav-ing an inconsiderable negative slope. This increase of the NF at low frequencies is due to the presence of the interferometric noise; however, as it was explained earlier, since the laser line-width is 1.5MHz, the effect of this noise term becomes almost negligible for higher frequencies. The fact that the NF is almost flat in the frequency span of the electrical bandpass filter (50-450MHz) on one had confirms the validity of the broadband NF measurements and on the other hand suggests that the contribution of the interferometric noise to the total NF of EDFA1 and EDFA2 should be negligible. As a result, the discrepancy between the electrical and optical NF measurements in Fig. 2.6 can be neither due to ASE-ASE beat noise nor interferometric noise. Therefore this discrepancy is expected to be due to some unknown measurement artifacts; one possibility being the OSA and RFPM nonlinearity. However, considering the measurement uncertainties discussed earlier, it is obvious that there is an excellent agreement between the optical and electrical NF results presented in Fig. 2.6. In Table 2.1, the average value of the frequency-resolved measurements of NF in the fre-quency range of 50-450 MHz are compared with the ones obtained from the broadband mea-surements. This comparison shows an excellent agreement between the two approaches. Neglecting the nonlinearity of the ESA level measurements, the accuracy of the frequency-resolved NF measurements is also within ±0.3 dB. Chapter 2. Noise Figure of Saturated Erbium-Doped Fiber Amplifiers in Saturation 48 TABLE 2.1: Comparison between the mean values of the NF measured using an electrical spectrum analyzer with those measured with a broadband RF power meter. EDFA Input Power Mean of Frequency-Resolved NF Broadband NF (dBm) (dB) (dB) -8.9 3.92 3.83 -0.9 4.21 4.13 5.2 5.23 5.16 2.6 Summary and Conclusions A simple electrical NF measurement method was described, which has an absolute accuracy of ±0.3 dB for the measurements presented; this accuracy is significantly better than conven-tional RIN subtraction technique results reported to date. The accuracy of the conventional polarization nulling technique for optical NF measurements was enhanced by introducing a sim-ple calibration method. For the case of several EDFAs tested, an excellent agreement was found between the electrical and optical NF measurements in saturation. These results are in contrast to previous reports where significant differences between the two measurements have been observed, and throws into question the applicability of nonlinear photon statistics in erbium-doped fiber amplifiers. Chapter 3 Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 3.1 Introduction As described in Chapter 1 erbium-doped fiber amplifiers operating in signal-saturated regime have become the key components of the rapidly growing wavelength division multiplexed (WDM) and cable television (CATV) optical communication systems. Therefore, design for high efficiency, and simple and accurate theoretical modeling of gain and noise figure in satu-rated EDFAs is of prime importance. This chapter describes new approaches for efficient design of saturated EDFAs and for the modeling of gain and noise figure in EDFAs. The organization of this chapter is as follows. In Section 3.2, motivation of the work pre-sented in this chapter is described. In Section 3.3, using a general model, it is shown that in sig-nal-saturated EDFAs the effect of ASE on the gain and NF performance of amplifiers is negligible and that highest-efficiency operation of saturated EDFAs is achieved with erbium dis-tributed throughout the entire fiber core. In Section 3.4, gain and noise figure predictions of Saleh's model, for a non-confined 980nm pumped EDFA, are compared with predictions of the general model, and it is shown that Saleh's model cannot accurately predict the performance of non-confined EDFAs. In Section 3.5, a simple model for co-propagating 980-nm-pumped EDFAs is derived. This model incorporates commonly neglected effects of pump excited state 49 Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 50 absorption and fiber background losses at the signal and pump wavelengths. It is shown that the saturation parameters normally contained in conventional models can be eliminated without compromising the accuracy, with the resulting model requiring only small signal gain and loss coefficients as parameters. Gain and noise figure predictions of the resulting model are shown to be in excellent agreement with the predictions of the general model. In Section 3.6 a new method for determining the pump excited state absorption coefficient is presented and experi-mental methods used for measuring the parameters contained in the proposed model are described. Excellent agreement between predictions of the proposed model with the measured gain and noise figure of a 980-nm pumped EDFA is demonstrated. In Section 3.7 it is shown that the proposed model can be extended to model a wide variety of practical EDFAs, including commonly used pump wavelengths and configurations. Finally, conclusions are presented in Section 3.8. 3.2 Motivation It is generally believed that efficient EDFAs should be designed with confined erbium doped fibers (EDFs), that is, fibers where the erbium doping is confined near the center of the EDF core [2]-[4], [17], [38]-[41]. The logic behind this argument is that the pump mode envelope is maxi-mum at the fiber center and therefore a higher population inversion can be achieved by concen-trating the erbium ions near the core center. This is claimed to result in a larger ratio of signal to amplified spontaneous emission (ASE), which will result in higher amplifier gain and lower noise figure. This claim is correct for small-signal amplifiers. However, here it is shown that in Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 51 signal-saturated EDFAs, the effect of ASE on the population inversion is negligible, and that in the presence of fiber background loss, non-confined EDFAs can have improved efficiency, gain and noise figure in saturated operation. Accurate modeling of gain and noise figure in non-confined EDFAs has not been considered in previous work. The comprehensive model and the effective ASE model (described in Section 1.6) which include the radial optical fields and erbium distribution are most suitable for accurate modeling of such amplifiers. However, these models are not very practical as they include many parameters, some of which are difficult to measure accurately. On the other hand, the applicabil-ity of the widely used Saleh's model (described in Section 1.6) to non-confined EDFAs was never discussed in previous work. Nevertheless, it is not expected that this model provide accu-rate prediction of the performance of non-confined EDFAs. This is because the fundamental assumption in this model is that the erbium is so confined near the center of the fiber, such that the radial mode envelope distributions at the signal and pump wavelengths are identical in the doping region. When the EDF is non-confined, and especially when it is pumped at 980nm, the discrepancy between the pump and signal mode envelopes becomes large, and causes errors in predictions of the model [42]. In addition, the particular structure of this model does not allow inclusion of pump excited state absorption and fiber background loss at the signal and pump wavelengths. The parameters required in Saleh's model are the small signal loss coefficients and intrinsic saturation powers at the signal and pump wavelengths. While the loss coefficients can be precisely obtained using only relative measurements, the saturation powers, which are gener-ally determined from monochromatic loss measurements using a transcendental equation (equa-Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 52 tion (1.13) derived in Ref. [19], are less accurately known. There are two reasons for this inaccuracy: first, measurement of absolute optical powers is required, second the transcendental equation used in obtaining the saturation powers from the monochromatic loss measurements ignores the effect of ASE which can be important. Therefore, the accuracy of the Saleh's model is also limited by the accuracy of the intrinsic saturation powers. In Section 3.5 a simplified one-dimensional steady-state model for gain and noise in saturated EDFAs is derived which is accu-rate for both confined and non-confined EDFAs [43]. 3.3 Effect of Erbium Confinement on the Performance of Saturated EDFAs In this Section, it is shown that in contrast to generally-accepted EDFA design principles, high-est-efficiency operation of EDFAs is achieved by not confining the erbium but distributing it throughout the fiber core. First, using the effective ASE model, which will be referred to as the general model, the impact of the ASE on the gain and noise figure performance of a saturated 1550nm amplifier incorporating non-confined EDF is analyzed. This amplifier, which will be referred to as EDFA1, has the following parameters: fiber core radius 1.8p:m, numerical aper-24 -3 ture 0.2, step-like erbium doping with a concentration of 5 x 10 m . The erbium confine-ment factor is 1, i.e. erbium is distributed across the entire fiber core. The spontaneous emission lifetime is taken to be 10ms, the effective ASE bandwidth as 2.6THz [17], [44]. The signal -25 -25 2 absorption and emission cross sections are taken as 2 x 10 and 3x10 m , respectively, -25 2 and pump absorption cross section is 2 x 10 m . The EDF length is 7m and the input pump power is 125 mW. For the present comparison, background losses and ESA are ignored. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 53 Fig. 3.1 shows the gain and NF of EDFA1 versus input signal power for two different effec-tive ASE bandwidths: the asterisks show results for the nominal value, while the circles show the results where Av e ^ is set to a very small value, effectively eliminating the ASE influence on the population inversion. The near perfect agreement of the two results in Fig. 3.1 shows that even in the present case where the erbium doping extends throughout the core, no error is incurred by ignoring the ASE effect on population inversion. To confirm that this result is not highly sensitive to the exact value of the effective ASE bandwidth, similar calculations were performed with AveJy artificially increased by a factor of three; results are shown in Fig. 3.2 and are nearly indistinguishable from those in Fig. 3.1. Second, the general model is used to compare the saturated performance of two amplifiers with differing confinement, which will be referred to as EDFA2 and EDFA3. In making a realis-tic comparison, it is essential to include the effect of fiber background loss coefficients at the signal and pump wavelengths. For the present case these are taken as 0.03 and 0.08 dB/m, respectively, which are typical values for aluminum co-doped EDFs [45]-[46]. EDFA2 has a length of 10m, and other parameters are the same as in EDFA1 described above; EDFA3 is iden-tical, except for a confinement factor of 0.5 and a length of 25m. The comparison is performed at an input signal power of OdBm. The two EDF lengths have been chosen such that the residual pump powers are approximately lOdBm; this value is somewhat arbitrary but is typical for the pump and signal powers chosen here; however, the exact value chosen does not affect the con-clusion drawn below. Table 3.1 shows the result of the comparison, and EDFA2, which is non-confined, has approximately ldB higher gain compared to EDFA3, which is confined. The supe-Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 54 22 20 CQ 18 _ 16 -O 14 -12 -10 - i 1 1 1 r n r j i i _ 6 - 4 - 2 0 2 -6 - 4 - 2 0 2 Input Signal (dBm) _ i i _ 4 6 8 8 Fig. 3.1: Theoretical results, using the general model, for gain and noise figure versus input sig-nal power for EDFA1 which has an erbium confinement factor of 1. Open circles and asterisks show the results for cases that ASE effects on population inversion are neglected and included, respectively. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 55 •12 -10 3.7 r-3.6 -3.5 -CO 3.4 -LL 3.3 -z 3.2 -3.1 -3 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8 Input Signal (dBm) Fig. 3.2: Theoretical results, using the general model, for gain and noise figure versus input sig-nal power for EDFA1 which has an erbium confinement factor of 1. Open circles show the result for the case that ASE effect is neglected and asterisks show the results for the case that ASE is included and its effective bandwidths is artificially increased by a factor of 3. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 56 rior performance of the non-confined amplifier is due the fact that its optimized length is 2.5 times shorter compared to the confined case, and therefore fiber background losses at the signal and pump wavelengths have less deleterious effects on its performance. The shorter optimized length of the unconfined EDFA is due to its higher gain per unit length which occurs because of its larger erbium content per core area. Table 3.1: Theoretical comparison of gain and NF of two EDFAs; EDFA2 and EDFA3 which have erbium confinement factor of 1 and 0.5, respectively. Other parameters are given in the text. Amplifier Length(m) Gain(dB) NF(dB) Residual Pump(dBm) EDFA2 10 17.72 3.28 9.76 EDFA3 25 16.73 3.31 9.80 3.4 Assessment of Saleh's Model Accuracy for Non-Confined EDFAs In the previous section it was shown that non-confined EDFAs operating in signal saturated regime have a superior performance compared to confined ones. In this section, the accuracy of the widely used Saleh's model for predicting gain and noise figure of non-confined EDFAs is examined. To assess the accuracy of Saleh's model for non-confined EDFAs, gain and noise figure pre-Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 57 dictions of this model for the case of EDFA1 are compared versus predictions of the general model. Since Saleh's model does not include the effect of fiber background losses and ESA, they are ignored in this comparison. The required parameters for Saleh's model are c ,^ ap, P*"', and Pp"' which are small signal loss coefficients and intrinsic saturation powers at the signal and pump wavelengths, respectively. These parameters are generally obtained from monochro-matic loss measurements, at the signal and pump wavelengths, over a length of EDF, and fitting of the transcendental equation (1.13) to the measured results [19]-[20]. Here, these parameters are calculated based on the same approach described above, but using the general model to sim-ulate the loss measurements. A 3 m length of EDF was considered, and the monochromatic loss at the signal and pump wavelengths were calculated for input powers ranging from -40 to OdBm. sat scit Subsequently, the parameters as, ap, Ps , and Pp were obtained from the least-squares fit of equation (1.13) to the monochromatic loss calculation and were calculated as 2.25dB/m, 3.77dB/m, 0.489 mW and 1.036mW, respectively. These parameters were then used in equations (1.13)-(1.16) of Saleh's model to calculate gain and noise figure of EDFA1. In Fig 3.3 results of monochromatic loss calculation and the least-squares fits used to obtain the small signal loss coefficients and saturation parameters are illustrated. The open circles show the monochromatic loss and the solid curves are the least-squares fits; the upper and lowers panels show results for the signal and pump wavelengths, respectively. Fig. 3.4 shows theoretical prediction of the two models for the gain and noise figure of EDFA1; asterisks show predictions of the general model and squares show predictions of Saleh's model. From the results shown in Fig. 3.4 it is evident that Saleh's model cannot accu-Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 58 -12 ' i i i i i i i i i I -40 -35 -30 -25 -20 -15 -10 -5 0 Input Power (dBm) Fig. 3.3: Theoretical results of monochromatic loss (open circles) versus input power for 3m of the EDF used in EDFA1, and the least square fits (solid curves) used to obtain small signal loss coefficients and saturation parameters. The upper panel shows the results for the signal wave-length at 1550nm and the lower panel shows the results for the pump wavelength at 980nm. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 59 24 22 20 m 18 O 16 14 12 10 3.7 3.6 3.5 S T 11 3.4 3.3 3.2 3.1 3 - * * i 1 * i — * 1 -• • • • * • * • * -- • * • 12 -10 -8 -6 -2 0 8 - i 1 1 1 1 1 1 1 r • ft . 8 . 8 . 6 . 8 • • • 12 -10 -8 -6 - 4 - 2 0 2 Input Signal (dBm) 8 Fig. 3.4: Theoretical gain and NF comparison of the general model (asterisks) with the Saleh's model (squares) for the case of EDFA1 with an erbium confinement of 1. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 60 rately predict the gain and NF performance of non-confined EDFAs. This is due to the fact that in Saleh's model it is assumed that the erbium is well confined inside the fiber core such that the overlap integrals between the erbium and the optical modes are equal at the signal and pump wavelengths. When EDF is non-confined and especially when it is pumped at 980nm, the dis-crepancy between the overlap integrals at the signal and pump wavelengths become large, caus-ing errors in predictions of the model. 3.5 A Simple and Accurate Model for Saturated EDFAs In this section, from the general model, a one-dimensional model is derived which can closely predict the gain and noise figure of confined and non-confined saturated EDFAs and as parame-ters contains only small signal gain and loss coefficients. Starting from the general model, the first simplification is made by assuming that ASE has negligible effect on the population inversion, which is well justified for the case of saturated EDFAs as shown in Section 3.3. Considering the general, radially-dependant propagation equa-tions (1.5) and (1.6), four propagation constants that correspond to various specialized situations can be identified. The first, for no pump and small signal, i.e. n2(r) ~ 0, is (3.1) where Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 61 = P(r)ys(r)rdr (3.2) r is the overlap integral for the signal wavelength. The propagation constant is the signal absorption coefficient in the absence of the fiber background loss. The second, for small signal and strong 980nm pumping, where complete population inversion is achieved, i.e. « 2 ( r ) ~ 1, is where gs is the small signal gain coefficient in the absence of fiber background loss. The third, for no signal and small pump power, i.e. n2(r) ~ 0, is (3.4) where (3.5) r is the pump overlap integral and ap is the pump absorption coefficient in the absence of the fiber background loss, and finally, Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 62 « £ « = C3.6) is the excited state absorption coefficient, which obtains in the case of full inversion and negligi-ble signal and ASE power. Note that all the absorption coefficients are positive quantities. The above equations (3.1)-(3.6) can be used to eliminate microscopic cross sections in the propagation equations (1.5) and (1.6) yielding dPs Ps •I = f (as + gs)\p(r) n2(r) ys(r)rdr - (as + a's)Ps (3.7) and ^ = fr(<*p-<*ESA)jp(r) n2(r) VP(r)rdr -(ap + a'p)Pp. (3.8) The next step is to eliminate the radially dependent functions in equations (3.7)-(3.8). This can be easily achieved if one assumes that n2(r) is constant all across the doped region. Obviously this is not generally the case; however a close examination of equation (1.8) shows that for the case of saturated EDFAs, where the signal and pump are relatively large, n2(r) remains almost constant across the entire core of the erbium-doped fiber. This is due to fact that n2(r) is deter-mined by the ratios of two functions which are nearly similar. For a quantitative assessment of this claim, for the case of EDFA1, n2{r) was calculated versus EDF core radius at three loca-Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 63 tions along the length of the EDF. The results are shown in Fig. 3.5 depicting n2{r) versus EDF core radius at the input, middle and at the end of the EDF. Fig. 3.5 shows that n2(r) remains nearly constant across the entire fiber core. Taking advantage of this fact, an effective population density for the second level N2 is defined which, with good approximation, can simultaneously satisfy the following relations: N2 = jj- Jp(r) n2(r) rdr = ^  Jp(r) n2(r) ys(r) r dr = (3.9) p Jp(r) n2(r) yp(r)rdr where P, = \9(r)rdr (3.10) From equations (3.7)-(3.9), the propagation equations for the signal and pump can be written as: d-^ = ( a s + gs)N2Ps-(as + a's) Ps (3.11) and iP = («„ + a'p)(N2-\)Pp-(a'p +aESA)N2Pp. (3.12) Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 64 1.1 0.9 CCM 0.7 0.5 0.3 z = 3.5 m z = 7.0 m 0 0.2 0.4 0.6 0.8 1 r ( u r n ) z = 0 1.2 1.4 1.6 1.8 Fig. 3.5: Normalized population of the excited level n2(r) versus fiber radial coordinate r at three different locations (entrance, middle and exit) of EDFA1 as indicated in the figure. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 65 The equation for N2 can be obtained from equation (1.8) by using equations (3.1)-(3.6) to replace the cross sections, multiplying by rp(r), integrating, and finally using equation (3.9) to yield Pp(z) a, PS(Z) psat + U s + gs psat N2(z) = — — (3.13) PD(z) Pfz) 1 + -sat -sat r p r s where psat = 2KhVspt (3.14) and -psat = IjZhV^ ( 1 1 5 ) are effective saturation powers at the signal and pump wavelengths, respectively. The resulting set of equations (3.11)-(3.13) is suitable for modeling saturated, 980nm-pumped, copropagating EDFAs including those with unconfined erbium doping [42]. The above model depends on the determination of saturation parameters which, as men-tioned in Section 1.6, are less accurately known compared to loss and gain coefficients. There-fore, the accuracy of the model can be improved if saturation parameters can be eliminated from Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 66 the model. In addition, it is obvious that eliminating these parameters is desirable if accuracy can be maintained. In the following, it is shown that how this can be achieved. First consider-ation is that for typical EDFs, the saturation parameters at the signal and pump wavelengths, Ps and Pp , are on the order of 0.5 and lmW, respectively, and second is that in almost all practical saturated EDFAs, the input signal power and input pump power are so large that the sum of Pp/Pp and P"VPS which are the normalized number of pump and signal photons entering the amplifier, respectively, is much bigger than 1. Since in saturated EDFAs the ASE power is negligible, and the background loss and ESA coefficients are small, as the signal and pump propagate inside the amplifier only a small fraction of the total input photons will be lost. In other words, it is almost true that for each pump photon that disappears, one signal photon will be created. On the other hand, combination of (3.14) and (3.15) gives the following propor-tionality -£— + -£—..«* aDN+(a+gs)Ns (3.16) — sat — sat P P v s ° s / s v ' Pp Ps where, Np and Ns are the number of pump and signal photons, respectively. By considering that as + gs > ap, from (3.16) it is apparent that the factor (Pp(z)/Pp + Ps(z)/Ps ) grows proportionally with conversion of pump photons to signal photons, and therefore it always remains bigger than (P'p/Ppat + P'sn/P*a') over the entire amplifier length. Consequently, 1 in the denominator of equation (3.13) can be ignored and using equations (3.14) and (3.15), N2 can be written as: Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 67 % Pp(z) (as + 8s) Ps(z) (3.17) Hence, the saturation parameters can be eliminated from the model, not only making the EDF characterization easier but also making the model more accurate. This simplification is equiva-lent to letting the factor pt/x in equations (3.13)-(3.15) tend to zero, which is only possible if T tends to infinity; this physically means that the spontaneous emission has negligible effect in determining the inversion level, which is the case for saturated EDFAs as shown in Section 3.2. The proposed model is based on equations (3.11), (3.12), and (3.17): by solving these equa-tions amplifier gain and population of the excited state are obtained. Then the noise figure can be calculated from equation (1.2) and using equation (1.15) to calculate the amplified spontane-ous emission at the amplifier output: where PS(L)/Ps(z) corresponds to the gain the spontaneous emission, generated in an infinites-imal length of dz, experiences from position z to the amplifier output. To compare predictions of the proposed model with the results of the general model, gain and NF of a copropagating 980nm-pumped amplifier, EDFA4, were calculated versus input sig-L (3.18) Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 68 nal power. This amplifier is assumed to have the same parameters as EDFA1 plus an of 0.03 dB/m and a'p of 0.08 dB/m. The general model was used to determine the small signal gain and loss coefficients required for the proposed model and the results were as follows: 0^=2.25 dB/m, ,^ =3.37 dB/m, ap=3.11 dB/m. Fig. 3.6 shows theoretical predictions of the two models for the gain and NF of EDFA4, respectively; asterisks show predictions of the gen-eral model and circles show predictions of the present model. Despite the fact that EDFA4 is non-confined, the agreement between the two models is excellent, showing the applicability of the proposed model to saturated EDFAs, even in the case of an erbium confinement factor as great as 1. To show the validity of the assumption made in simplifying the expression for N2, in — sat — scit Fig. 3.7 the normalized pump and signal photon numbers, Pp(z)/Pp and PS{z)/Ps , are plotted versus EDF4's longitudinal coordinate z, at an input signal power of OdBm; the solid line shows the normalized pump photon numbers and the dashed line shows the normalized sig-sdl sat nal photon numbers. The parameters Ps and Pp were calculated from equations (3.14) and (3.15) and were determined to be 0.50 and 1.19mW, respectively. As can be seen from Fig. 3.7, sat sat while Pp(z)/Pp decreases, Ps(z)/Ps increases, and the sum of the two always remain well above 1 justifying the assumption made in deriving equation (3.17). 3.6 EDF Characterization Methods and Measurements In this section methods of characterization for the EDF parameters contained in the proposed model of Section 3.5 are described for the case of 980nm pumping. In the experiments con-ducted, an Er/Al/Ge/P doped fiber, which will be referred to as EDF5, was used in amplifier Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 69 C D O 22 20 18 16 14 12 10 1 1 1 1 1 1 1 - 9 Q 1 i 9 9 9 i . . . I I i i i i - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 3.7 3.6 I-3.5 | 3.4 Z 3.3 3.2 3.1 3 9 9 9 9 9 - 1 2 - 1 0 - 8 - 6 - 4 - 2 0 2 Input Signal (dBm) 8 1 r j i_ 4 6 8 Fig. 3.6: Theoretical comparison between gain and noise figure predictions of the general model (asterisks) with the ones predicted by the proposed model (circles) for the case of EDFA4 with an erbium confinement factor of 1. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 70 Fig. 3.7: Normalized pump (solid line) and signal (dashed line) photon numbers versus amplifier length for the case of EDFA4 at an input signal power of OdBm. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 71 EDFA5. It had an estimated erbium confinement factor of 0.7; the signal wavelength was 1551 nm. Fiber background loss coefficients o^  and a'p can be determined by measuring the EDF's absorption coefficient at a wavelength far away from the erbium absorption band. Using the empirical relation for the Rayleigh scattering loss aR = C/X4 dB/km, where C is a constant [10], [46], the loss at the signal and pump wavelengths can be estimated. The loss coefficient was measured at a wavelength of 1310 nm using the standard cutback method [4] and for EDF5, a loss coefficient of 0.02 dB/m was obtained; consequently the corresponding background loss coefficients are 0.01 and 0.06 dB/m at wavelengths of 1551 and 980nm, respectively. In the standard cutback method, one end of the fiber under test is connected to the signal source and the other end is connected to an optical spectrum analyzer. First, the spectral output power PA(A) is measured; then a segment of fiber is removed, by cutting away a length, L, of the fiber at its end, and the corresponding optical power spectra PB(A.) is measured again. The fiber loss coefficient, oc(A.) (in dB per meter), is then calculated from: a(X) = 10 & A B (3.19) Li The excited state absorption coefficient is traditionally measured using the pump-probe tech-nique [4]-[5]. In this method, a strong pump laser is used to create full population inversion along the EDF under test, and a white light source at the signal wavelength, propagating in the opposite direction of the pump, is used as a probe to measure the ESA coefficient. This measure-ment technique is difficult as it requires free space optics and careful calibration of the experi-Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 72 mental components. Here a'p + <*ESA was measured using the following method. Consider a piece of EDF which is pumped by a 980nm source while the signal is turned off, and the pump source is so strong that near-complete population inversion can be achieved, i.e. N2 is 1 or close to unity. Under such a situation, equation (3.12) simplifies to: d-^ = -(a'p +aESA)Pp (3.20) and, therefore, the factor ( a'p + aESA) is equal to the loss of any incremental pump power beyond the pump power at which the complete population inversion has occurred. Accordingly, to obtain the factor ( a'p + aESA ) for the EDF5, the transmission of 980nm pump was mea-sured over a 3 m length of fiber at several input pump powers. The upper panel of Fig. 3.8 shows the experimental results of pump loss (Pin/Pout) versus input pump power. In addition, the dif-ferential pump loss (APin/APout) was calculated, which is shown in the lower panel of Fig. 3.8. The factor ( a'p + o.ESA ) is equal to the differential loss at high input pump powers, where the near-complete inversion occurs, and was determined to be 0.07 dB/m. This, together with the fact that a'p is 0.06 dB, shows that excited state absorption at 980nm is very small, in agree-ment with the results presented in Refs. [4]-[5]. The small signal gain and loss coefficients at the signal and pump wavelengths were deter-mined using the standard cutback method [4]. Measurement of the small signal gain coefficient using the cutback method is similar to loss measurement described above; the only difference is that the input end of the fiber under test is spliced to the output of a 980/1550nm WDM which combines the signal and pump. For this measurement, the pump power should be high enough to Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 73 Fig. 3.8: 980nm loss measurement results over 3 m of EDF5. The upper panel shows results of pump loss (pin/pout) versus input pump power and the lower panel shows the differential pump loss (APin/APout) versus input pump power. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 74 create full population inversion along the length of the EDF under test. Due to the presence of the fiber background loss, the cutback method does not provide direct measurement of the gain and loss coefficients, and instead it gives ( gs - a's), (as + ) and ( ap + a'p). From cut-back measurements and the background loss coefficients given above, gs, as, and ap were cal-culated as 2.52 dB/m, 1.89 dB/m, and 2.90 dB/m, respectively. Next, to compare the predictions of the proposed model versus experimental results, the gain and noise figure of a co-propagating 980-nm-pumped amplifier, EDFA5, were measured versus input signal power ranging from -11 to +7 dBm. EDFA5 was built around 8.94 m of the EDF5, with an input pump power of 144 mW. The EDFA gain was measured, using the optical method described in Section 1.5.1 [10], with an ANDO 6315B optical spectrum analyzer. The EDFA noise figure was measured using the frequency-resolved method described in Chapter 1 at the baseband frequency of 800 MHz using a Rohde&Schwarz FSEA 20 spectrum analyzer. Mea-surement at 800 MHz together with the fact that the DFB laser signal source had a linewidth of 850kHz ensures that the interferometric noise has negligible effect on the measured noise figure [35]. Also, the presence of an optical bandpass filter at the end of the EDFA makes the contribu-tion of the ASE-ASE beat noise negligible [31]. Therefore, the measured noise figure is prima-rily due to the signal-ASE beat noise, the same term which gives the theoretical NF, making the comparison between theory and experiment appropriate. Fig. 3.9 shows the results of the com-parison; the asterisks show the experimental results and open circles show model predictions. From, Fig. 3.9 it is evident that for both gain and NF, the proposed model is in excellent agree-ment with the experimental results confirming the theoretical comparison presented in Fig. 3.6. Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 7 5 22 r-20 -18 -00 2 , 16 -CD 14 -12 -10 --10 - 8 - 6 - 4 - 2 0 2 4 Input Signal (dBm) 6 8 Fig. 3.9: Comparison between measured gain and NF of EDFA5 (asterisks) with the predictions of the proposed model (circles). Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 76 3.7 Generalization of the Proposed Model The focus of the analyses presented so far was on copropagating single-wavelength signal 980-nm-pumped EDFAs, and the applicability of the proposed model was demonstrated for saturated operation. However, from the effective ASE model for multiple signal and pump wavelengths [17], it is easy to show that the proposed model can be extended to include ASE, both 980 and 1480nm pumping, and as many as co- and counter-propagating signals and pumps. Accordingly, the model propagation equations in this general case and for an arbitrary number of k forward propagating (+) and backward propagating (-) signals, pumps and ASE can be written as: 4TT = ( ak + 8k) ^ 2 Pk + 2 8k N2 h vk Avk - aESA(vk) N 2 p\ -( a, + a'k )P\ (3.21) N 2 = _k X<a* +Sk)p±k/Vk k (3.22) where Av^ is zero for signal and pump and for ASE is the frequency interval used in the simula-tion to resolve the ASE spectrum [17]. This model is also applicable to the case of small signal amplifiers, if the denominator of equation (3.22) is greater than 1, which is satisfied for most practical small-signal amplifiers. This can be explained by considering the fact that the noise performance of small signal amplifiers improves with higher pump powers, and that commer-cially available pump laser diodes have a typical output power more than 90mW; this amount of pump power is sufficient to justify the assumption made in deriving equation (3.22), making the Chapter 3. Design and Modeling of Saturated Erbium-Doped Fiber Amplifiers 11 proposed model applicable to almost all practical EDFAs. 3.8 Conclusions In conclusion, using the general rate equation model including background loss at the signal and pump wavelengths, it was shown that the design of efficient saturated EDFAs is achieved by dis-tributing the erbium doping over the entire fiber core, in contrast to generally-accepted design principles. From the radially-dependent rate equation model a simple model was derived which can be used for accurate prediction of gain and NF in saturated EDFAs even with erbium con-finement factors as great as 1. Excellent agreement was shown between the gain and noise figure predictions of this model and those of the general rate equation model and also experimental results. A new measurement method was developed which can be used for simple and accurate characterization of excited-state absorption in erbium-doped fibers. Finally, it was shown that the proposed model can be extended to include modeling of a wide variety of 980 and 1480nm pumped EDFAs. Chapter 4 Dispersion-Induced RIN Degradation and its Impact on 1550nm AM Video Lightwave Transmission Systems 4.1 Introduction In Chapter 1, the operating principles of 1550nm optically-amplified AM video lightwave trans-mission systems (CATV systems) were described and it was shown that the performance of such systems is strongly affected by the noise of the different components in these systems. One source of noise in such systems is due to the DFB laser transmitter. In this chapter the effect of fiber dispersion on the relative intensity noise of analog DFB lasers is analyzed theoretically and experimentally and its impact on the overall performance of CATV systems is discussed. This chapter is organized as follows. In Section 4.2, the motivation of the work presented in this chapter is described. In Section 4.3, the laser RIN measurement technique is described and results of laser RIN measurements versus several lengths of dispersive fibers in frequencies rel-evant to CATV systems are shown. In Section 4.4, modeling of laser RIN variation with disper-sive propagation is described and predictions of the theory are compared with the experimental results. Finally, summary and conclusions of the results presented in this chapter are in Section 4.5. 7 8 Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm 79 4.2 Motivation A stringent requirement for CATV systems is that a high carrier-to-noise ratio (CNR) be main-tained along the transmission fiber link. One of the parameters affecting the system CNR is the relative intensity noise (RfN) of the distributed feedback (DFB) laser transmitter. In conven-tional CNR analysis of analog lightwave transmission systems, it is assumed that the laser RIN remains constant over the transmission fiber link [1], [47]-[48]. However, as shown in Section 1.7, it has been well known that fiber dispersion can affect laser RIN. Nevertheless, there has been no quantitative measure of the degree of laser RIN degradation with fiber dispersion in the frequency range relevant to CATV, nor has there been any discussion of the impact of this effect on overall performance of CATV systems. In this chapter, laser RIN variation with fiber disper-sion is carefully analyzed in the frequency range relevant to CATV systems [49]-[50]. 4.3 Laser RIN Measurements In this section, laser RIN measurement technique is described and results of RIN measurements for two analog 1550 DFB lasers versus fiber lengths up to 48 km are presented. The laser RIN measurement technique used is based on the method developed by Nazarathy, which was described in Section 1.8. However, as discussed in Section 2.3, for achieving higher accuracy, optical power and photodiode responsivity parameters contained in equation (1.18) can be replaced by the photodiode average current, and the resulting equation can be written as: PN(f) = H{f) RIN(f) I2 + 2eH(f)I + Pth(f) (4.1) Chapter 4. Dispersion Induced RIN degradation and its Impact on I550nm . 80 where, H(f) is the transfer function of the network connecting the photodiode to the spectrum 2 analyzer, which is equivalent to the factor (Z Bn)/Zin described in Section 2.3. The RIN measurement setup is shown in Fig. 4.1. It consists of the DFB laser under test, a length of single-mode fiber (SMF), an optical attenuator, a receiver which contains a photodiode and a transimpedance amplifier, an ammeter to measure the average photodiode current, and an RF spectrum analyzer (ESA) which measures the noise power spectral density PN{f) • A series of measurements of PN{f) and / at various settings of the optical attenuator is made. The RIN spectrum is then obtained from least-squares fit of equation (4.1) to the resulting data at each baseband frequency. This simple "self-calibrating" approach eliminates the need to determine several parameters normally required in conventional RIN measurement technique [10], [16] some of which are difficult to measure accurately, and thereby significantly improves measure-ment accuracy. SMF DFB Variable Attenuator Receiver > Ammeter RF Spectrum Analyzer Figure 4.1: Experimental setup for RIN measurement; see text for description. Chapter 4. Dispersion Induced RIN degradation and its Impact on I550nm 81 Fig. 4.2 shows typical raw measurement results at one particular frequency, which is 800MHz in this case. The symbols show measured noise acquired at various settings of the opti-cal attenuator. The solid line shows the quadratic fit, from which the RIN was determined -170.5 dB/Hz. For illustration, the noise power due to the sum of shot and thermal noise is shown by the dashed line; laser RIN is being determined by the difference between the solid and dashed lines. The absolute error in the resulting RIN values depends upon RIN level, which affects the deviation between the quadratic fit and the dashed line, by the accuracy of noise power measure-ments, and by the accuracy of the average current measurement; accounting for the uncertainties of ±0.052 dB and ±0.022 dB for the last two parameters, respectively, the worst-case RIN mea-surement accuracy is ±1.5 dB at -172dB/Hz. Using the above technique the RIN of two analog DFB lasers, DFB1 and DFB2, were mea-sured for several lengths of standard single mode fiber. In all cases, the launched power into the fiber was kept below 4.5 dBm to avoid nonlinear effects. Fig. 4.3 shows frequency-resolved measurements of the RIN of DFB 1 at the laser, indicated as 0km, and after 32 and 48km of stan-dard single-mode fiber. At 0 km the laser RIN is quite low; the slight increase with frequency reflects the beginning of the large increase in RIN seen at the relaxation oscillation frequency, which is much higher than the range of frequencies shown in Fig. 4.3. After fiber propagation, two effects are apparent: the first is seen at low frequencies, where the RIN has increased because of the interferometric noise caused by double Rayleigh scattering in the transmission fiber. This effect has been extensively studied [51]-[52], and will not be considered here. The second effect seen in Fig. 4.3 at higher frequencies, is an increase in RIN with fiber length; for a Chapter 4. Dispersion Induced RIN degradation and its Impact on I550nm 82 ^ 2.5 I 1 1 1 1 1 1 1 r LU 0 ' 1 1 1 1 1 1 ' ' 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Average Photodetector Current (mA) Figure 4.2:Typical measured electrical noise power versus average photodiode current, for a baseband frequency of 800MHz. The open symbols show the measured results, the solid line is the quadratic fit discussed in the text, and the dashed line shows the noise power due to the sum of shot and thermal noise. The RIN value is determined to be -170.5 dB/Hz from these data. Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm 83 -140 -180 I 1 ' 1- 1 1 1 1 1 0 100 200 300 400 500 600 700 800 Frequency (MHz) Figure 4.3: Frequency-resolved RIN for DFB 1 at fiber lengths of 0, 20, and 48km. Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm 84 particular fiber length this degradation increases with increasing frequency. Therefore, the impact of this effect will be greatest on the higher frequency channels. From Fig. 4.3 it is appar-ent that, after 48km of fiber RIN has been degraded by 15dB at 800MHz. To more clearly show the distance dependence, in Fig. 4.4 the measured RIN as a function of fiber length is shown. The open circles in Figs. 4.4(a) and (b) show results for DFB1 and DFB2, respectively, at the baseband frequency of 800MHz. For both lasers RIN increases substantially and has a similar form; the degree of degradation is less for DFB2. As will be shown later, the lasers have linewidths less than 1MHz, which ensures that the noise due to fiber double Ray-leigh scattering has a negligible effect on the measured RIN at 800MHz [51] and therefore the RIN degradations shown in Fig. 4.4 are solely due to the phase to intensity noise conversion. 4.4 Modeling and Discussion As described in Section 1.7, modeling of laser RIN variation due to fiber dispersion has been already accomplished. The comprehensive modeling approach is based on the model developed by Marshall et. al. [29] which considers the coherent addition of the intrinsic RIN with the PM-to-AM converted RIN. However, this model is involved and requires knowledge of several laser parameters, some of which can be challenging to measure accurately. Given that analog DFB lasers demonstrate low intrinsic RIN at frequencies relevant to CATV (as shown in Fig. 4.3), and that in CATV systems the fiber link is generally longer than 40km, the coherent effects described by Marshall et. al. can be ignored and laser RIN after dispersive propagation can be calculated by an incoherent addition of laser intrinsic AM noise and PM-to-AM converted noise; Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm . 85 T 1 1 1 1 1 1 r J i i i i i i i L 0 10 20 30 40 50 Fiber Length (km) Figure 4.4:RIN versus fiber length (a) for DFB1 and (b) for DFB2. Open circles show measured results, and the solid curves show the RIN calculated using equation 4.3 and the measured intrinsic RIN. Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm . 86 as will be shown this greatly simplifies the analysis. The RIN degradation ARIN due to PM-to-AM noise conversion mediated by fiber dispersion can then be obtained from the simplified model by Yamamoto et. al. [28], described in Section 1.7, and is given by, JUN(f) = ^ ^ . . (I / 2 Av A M . ( 1 12 Av Af } . J (2n+n ,2. 2 \ n = 0 v y y (4.2) where 7 N is the Bessel function of the first kind, Av is the laser linewidth, Af is the frequency interval over which RIN is calculated and is equal to 1Hz, A, is the laser wavelength, D is the fiber dispersion, L is the fiber length, and c is the velocity of light in vacuum. For cases that 1 / 2 Av Af ^ Q Q | which is always satisfied for the range of frequencies relevant to CATV , equation 4 .2 can be well approximated by ARIN(f) = 1 ^ sin 2 (7t /Vl>L/c) . (4.3) The significance of equation (4.3) is that it is only required to know the laser linewidth and wavelength to characterize the laser and to calculate laser RIN degradation with fiber dispersion. Now comparison of the predictions of equation (4.3) with the experimental results is consid-ered. To determine the linewidths of the two lasers, the modulation sideband technique was used [10],[53]. With this method, the linewidth measurement can be performed at an intermediate fre-quency, overcoming the low frequency limitation of the detection electronics. In this technique, the laser diode current is directly modulated by small sinusoidal signals, creating sidebands Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm 87 around the carrier; a delayed self-homodyne technique is used to mix the carrier with the side-bands; the laser linewidth is then obtained from the detected signal which contains the laser line shape at the modulating frequency. The experimental setup for the linewidth measurement is shown in Fig. 4.5. The measurement setup consists of the laser under test, a signal generator for direct modulation of the laser DC current, a variable attenuator for adjusting the optical signal power, an interferometer, a receiver and an RF spectrum analyzer. The interferometer consists of two couplers, 7km of SMF fiber as a delay line, and a polarization controller for adjusting the level of the detected electrical signal. The receiver and ESA are the same as ones used in the RIN measurement setup. The modulation frequency was 200MHz and the ESA resolution band-width was 10kHz. Fig. 4.6 shows the experimental results for the two lasers tested, the dotted curves show the measured spectrum of the two lasers around the modulating frequency; the solid lines are the Lorentzian fits used to obtain the laser linewidth and yielded linewidths of 850 and 310 kHz for DFB1 and DFB2, respectively. The accuracy of these linewidth measurements are estimated to be approximately ±10 %. Inserting the measured linewidth value in equation (4.3), the RIN was calculated as a func-tion of distance by combining the measured laser intrinsic RIN and the calculated RIN degrada-tion, ARIN. Results are shown by the solid lines in Fig. 4.4; the fiber dispersion D was taken to be 17 ps/nm-km and the measured laser wavelengths of 1552 and 1547 nm were used for DFB1 and DFB2, respectively. Given that the ARIN is incoherently combined with the laser intrinsic RIN, the predicted RIN represents the upper limit. Nevertheless, as can be seen from Fig. 4.4 there is an excellent agreement between the estimated RIN and the experimental results, with the Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm 88 Signal Generator Variable Attenuator SMF 0 Polarization Controller RF Receiver Spectrum Analyzer Figure 4.5: Experimental setup for linewidth measurement; see text for description. Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm 89 -56 i 1 1 1 1 1 1 1 1 r 195 197 199 201 203 205 195 197 199 201 203 205 Frequency (MHz) Figure 4.6:Measured laser spectrum and the Lorentzian fit for laser linewidth measurement (a) for DFB1 and (b) for DFB2. The dotted curves are the experimental results and the solid curves are the Lorentzian fits which yield linewidths of 850 and 310kHz for DFB1 and DFB2, respec-tively. Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm . 90 predicted values lying slightly higher than the measured ones, as expected. This suggests that for DFB lasers demonstrating low intrinsic RIN on the order of -170dB/Hz, which is typical for ana-log 1550nm DFB lasers, the simplified theory adopted is appropriate and can give an accurate estimate of the laser RIN degradation by dispersive propagation. All analyses of overall CATV system CNR presented to date have assumed the laser RIN unaffected by fiber propagation, which is substantially in error, as shown above. To assess the impact of the RIN degradation on typical CATV systems, CNR calculations are performed for two representative 1550nm optically-amplified AM video transmission systems: the first, which is called System A, incorporates one erbium-doped fiber amplifier (EDFA) as a power amplifier and has a link length of 60km; the second, System B, has a 120km fiber link and includes a power amplifier and an in-line amplifier. For AM-VSB video lightwave systems with an EDFA, the AM CNR can be written as [1], [48] p CNR = s i g n a l . (4.4) "RIN + "shot + "thermal + " EDFA As described in Refs. [1], [48], the signal power is given by P signal = (mRPRX)2/2, the thermal noise by "thermal ~ n B e ' the shot noise by Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm . 91 N shot = 2 e R PRX Be and the EDFA noise by N EDFA = (2hv(NF-(l/G)))/PL in However, in contrast with previous analyses, the RIN noise must be described by NR1N = (RINint + ARIN) Be R P 2 RX ' where the additional parameter ARIN is introduced, to describe the effect of laser RIN degrada-tion caused by fiber dispersion. In the above equations, m is the peak AM modulation depth per channel, R is the photodiode responsivity, PRX is the received optical power, RINint is the laser intrinsic relative intensity noise measured at the laser output, Be is the bandwidth of each video channel, and n is the receiver equivalent thermal noise current. NF, G, and Pin are the EDFA noise figure, gain and optical input power respectively. System parameters used in the calcula-tions are given in Table 4.1, which are typical values for conventional CATV systems. The cal-culation results are shown in Fig. 4.7 (a) and (b) for systems A and B, respectively; the dashed lines show the conventional results where ARIN is neglected, and the solid lines show the results including RIN degradation for three different laser linewidths of 0.5, 1, and 1.5 MHz, as indicated. The results clearly show that laser RIN degradation caused by fiber dispersion can significantly impair CNR performance of the CATV systems; this impairment is most severe at higher baseband frequencies and can be several dB for the system parameters shown in Table 4.1. Chapter 4. Dispersion Induced RIN degradation and its Impact on 1550nm . TABLE 4.1: Parameters used in the CNR calculations of System A and System B Parameters Value Peak A M Modulation Index, m 3% Photodiode Responsivity, R (AAV) 0.9 Receiver Input Power, PRX (dBm) 0 Laser Intrinsic RIN, RINint (dB/Hz) -170 Bandwidth of Each Channel, Be (MHz) 4 Receiver Equivalent Noise Current, n ((pA/JWz)) 1 EDFA Noise Figure, NF (dB) 5 EDFA Optical Gain, G (dB) 13 EDFA Optical Input Power, Pin (dBm) 4 Chapter 4. Dispersion Induced RIN degradation and its Impact on I550nm . 93 Figure 4.7:CNR versus frequency calculated for (a) System A, and (b) System B. Dashed lines show the conventional analysis. Solid lines include RIN degradation for linewidths of 0.5, 1, and 1.5 MHz, as indicated. Chapter 4. Dispersion Induced RIN degradation and its Impact on I550nm . 94 In the analyses presented above only linear fiber propagation was considered. This is because the objective of the work presented here was to study only the effect of fiber dispersion. However, in practical CATV systems, the optical power launched into the transmission fiber can be as large as 18dBm, and therefore fiber nonlinearity becomes important. However, a recent theoretical study by Cartaxo et. al. [54] has shown that for frequencies less than 3GHz, fiber nonlinearity has no effect on the laser phase to intensity noise conversion. However, it would be interesting to investigate this issue experimentally in future research work. 4.5 Summary and Conclusions A sensitive self-calibrating technique for laser RIN measurements was described. Using this technique, the RIN of two analog 1550nm DFB lasers were measured and for the first time, RIN values as low as -172 dB/Hz were reported for analog DFB lasers. The RIN of the two lasers were also measured, in the frequency range relevant to CATV systems, for several lengths of standard single-mode fiber. The measurements show that laser RIN increases with increasing frequency and fiber length, and that this degradation is a function of laser linewidth. The mea-surement results are in close agreement with a simple theory which only requires knowledge of laser linewidth to calculate the RIN degradation. The effect of laser RIN degradation by disper-sive propagation on the CNR performance of two analog CATV systems was analyzed theoreti-cally, and shown that it can severely impair the CNR of these systems. It is especially important in the case of long distance CATV systems to account for the effects of RIN degradation, and for optimal CNR, analog lasers should be chosen on the basis of both RIN and linewidth. Chapter 5 Summary and Conclusions In Chapter 2, a simple electrical measurement technique for characterization of EDFA noise fig-ure was presented. This technique is built upon equation (2.12), which relates the EDFA noise figure to five parameters which can be measured easily and with high accuracy. An absolute accuracy of ±0.3 dB was achieved for the electrical NF measurements presented, which is sig-nificantly better than conventional RIN subtraction results reported to date. For optical noise fig-ure measurements, the accuracy of the conventional polarization nulling technique was enhanced by introducing a simple calibration technique; this technique allows accurate determi-nation of the loss incurred by the polarizer and the polarization controller used in the measure-ment setup. For the case of two copropagating 980nm pumped EDFAs, an excellent agreement was shown between the optical and electrical noise figure measurements in saturation. This results are in contrast to earlier reports [13]-[14], by Willems and van der Platts, where signifi-cant differences between the two measurements have been observed for the case of an EDFA similar to ones used in the work presented in Chapter 2. Reports by Willems and van der Platts caused a great deal of controversy and excitement, as they were claiming: first, that it is possible to achieve significant reduction of EDFA noise figure in saturation; second, that the conven-tional semi-classical models are not applicable to EDFAs operating in saturation; and third, that 95 Chapter 5. Summary and Conclusions 96 optical noise figure measurements were inappropriate and unreliable. The careful establishment of high accuracy in both electrical and optical noise figure measurements described in Chapter 2, and excellent agreement obtained between the two measurement approaches, strongly sug-gests that the first two of the conclusions of Ref. [13]-[14], outlined above, are incorrect. The conclusion concerning an appropriate measurement approach for EDFA noise figure character-ization is that the electrical method proposed in Chapter 2 is much simpler and can provide more accurate results compared to commonly used optical measurement techniques; it seems reason-able to expect widespread adoption of electrically-based noise measurement techniques for EDFA noise figure characterization. In Chapter 3, using a general, radially dependent rate-equation model it was shown that amplified spontaneous emission has negligible effect on the gain and noise performance of a sig-nal-saturated EDFA having an erbium confinement factor of one. This, coupled with the fact that erbium-doped fibers have rather large background losses at the signal and pump wavelengths, and that for a given amplifier the required EDF length increases with erbium confinement, leads one to the conclusion that, in contrast to the generally-accepted design principles, saturated EDFAs with non-confined EDFs would have better performance compared to ones with con-fined EDFs. Using the general model, it was shown that the gain of a typical 980nm pumped EDFA can be upgraded from 16.73 to 17.72dB if the EDF confinement factor is changed from 0.5 to 1. It was shown that the widely accepted Saleh model cannot accurately predict the gain and noise figure performance of EDFAs containing non-confined EDFs. This was attributed to the fact that the derivation of this model has been based on the principle assumption that the Chapter 5. Summary and Conclusions 97 erbium is confined near the center of fiber core. From the general model, a simplified one-dimensional steady-state model was derived which can accurately predict gain and noise figure of both confined and non-confined saturated EDFAs. In this model, the relatively hard to mea-sure intrinsic saturation powers, normally contained in the conventional models, were elimi-nated without compromising accuracy; this simplification is equivalent to assuming that the spontaneous emission lifetime is infinity, which physically means that the spontaneous emission has no impact on the population inversion of the system. The proposed model contains only small signal gain and loss coefficients which can be measured easily and with high accuracy. Finally it was shown that the proposed model, can be extended to include modeling of almost all practical EDFAs. In Chapter 4, an enhanced measurement method for simple and accurate characterization of laser RIN was described, and RIN values as low as -172 dB/Hz were shown for an analog 1550nm DFB laser. This is the lowest RIN value ever reported, and it is significantly lower than the normally reported values for analog 1550nm DFB lasers, which are in the order of -155 dB/ Hz. Measurement results for RIN of two analog DFB lasers were shown for several lengths of standard single mode fibers up to 48 km, and in the frequency range relevant to CATV systems for the case of linear propagation. The results show that laser RIN increases with increasing fre-quency and fiber length. These results are in contrast to conventional CNR analysis of CATV systems where it is assumed that laser RIN remains constant over the entire fiber transmission link. This degradation was attributed to the well known laser phase to intensity noise conversion by dispersive propagation. For the particular case of CATV systems, it was shown that, instead Chapter 5. Summary and Conclusions 98 of using the complex model of Marshall et. al., the simplified model of Yamamoto et. al. is suffi-ciently accurate for predicting RIN degradation of DFB lasers due to fiber dispersion. The par-ticular advantage of this model is that it contains only laser linewidth as the laser parameter. Finally, theoretical results of the effect of laser RIN degradation by dispersive propagation on the CNR performance of two analog CATV systems were presented. These results show that, even in optically-amplified CATV systems, RIN degradation by fiber dispersion can severely impair CNR performance of CATV systems, and for achieving optimal CNR performance of CATV systems, analog lasers should be chosen on the basis of both RIN and linewidth. Bibliography [1] H. Dai, S. Ovadia, and C. Lin, "Hybrid AM-VSB/M-QAM multichannel video transmis-sion over 120-km of standard single-mode fiber with Er-doped fiber amplifiers," IEEE Photon. Technol. Lett, vol. 8, no. 12, pp. 1713-1715, 1996. [2] A. Bjarklev, "Optical Fiber Amplifiers: Design and System Applications," Artech House, Boston, MA, 1993. [3] P. C. Becker, N. A. Olsson, and J. R. Simpson, "Erbium-doped fiber amplifiers: fundamen-tals and technology," Academic Press, San Diego, CA, 1999. [4] E. Desurvire, "Erbium-Doped Fiber Amplifiers," John Wiley & Sons, New York, 1994. [5] R. I. Laming, S. B. Poole, and E. J. Tarbox, "Pump excited-state absorption in erbium-doped fibers," Opt. Lett, vol. 13, no. 12, pp. 1084-1086, 1988. [6] M. Tachikura, "Numerical calculation of multiple reflected optical power in optical fiber transmission lines," IEEE Photon. Technol. Lett., vol. 6, No. 1, pp. 109-111, 1994. [7] C. Hentschel, E. M. Muler, and E. Leckel, "EDFA noise figure measurements - Compari-son between optical and electrical technique," Hewlett Packard, Boblingen Instruments Division, in 1994 Lightwave Symp. [8] S. Poole, "Noise Figure Measurement in Optical Fibre Amplifiers," in Symposium on Opti-cal Fiber Measurements, 1994, NIST Special Publication 864 (National Institute of Stan-dards and Technology, Boulder, CO, 1994), pp. 1-6. [9] J. Aspell, J. Federici, B. Nyman, D. Wilson, and D. Shenk, "Accurate noise figure mea-surements of erbium-doped fiber amplifiers in saturation conditions," in Optical Fiber Communication Conference, vol. 5, 1992, OSA Technical Digest Series (Optical Society of America, Washington, D.C, 1992), pp. 189-190. 99 Bibliography 100 [10] D. Derickson, "Fiber optic test and measurement," Hewlett-Packard Company, Prentice-Hall, NJ, 1998. [11] K. Bertilsson, P. Andrekson, and B. Olsson, "Noise Figure of Erbium Doped Fiber Amplifi-ers in the Saturated Regime," IEEE Photon. Technol. Lett., vol. 6, No. 2, pp. 199-201, 1994. [12] T. Kashiwada, M. Shigematsu and M. Nishimura, "Accuracy of noise figure measurement for erbium-doped fiber amplifiers by the optical method," Technical Digest Symposium on Optical Fiber Measurements, 1992, National Institute of Standards and Technology Spe-cial Publication 839, pp. 209-212. [13] F. Willems and J. van der Plaats, "EDFA noise-figure reduction in the saturated operation regime," in Optical Fiber Communication Conference, vol. 8, 1995, OS A Technical Digest Series (Optical Society of America, Washington, D.C, 1995), pp. 44-45. [14] F. Willems and J. van der Plaats, "Experimental Demonstration of Noise Figure Reduction Caused by Nonlinear Photon Statistics of Saturated EDFA's," IEEE Photon. Technol. Lett., vol. 7, No. 5, pp. 488-490, 1995. [15] F. Willems, J. van der Plaats, C. Hentschel, and E. Leckel, "Optical amplifier noise figure determination by signal RIN subtraction," in Symposium on Optical Fiber Measurements, 1994, NIST Special Publication 864 (National Institute of Standards and Technology, Boulder, CO, 1994), pp. 7-9. [16] Product Note 71400-1, "Lightwave Signal Analyzers Measure Relative Intensity Noise," Hewlett Packard Publication, 1996. [17] C. Giles and E. Desurvire, "Modeling erbium-doped fiber amplifiers," J. Lightwave Tech-nol, vol. 9, pp. 271-283, 1991. [18] C. Barnard, P. Myslinski, J. Chrostowski, and M. Kavehrad, "Analytical models for rare-earth-doped fiber amplifiers and lasers," J. Lightwave Technol, vol. 30, no. 8, pp. 1817-1830, 1994. Bibliography 101 [19] A. Saleh, R. Jopson, J. Evankow, and J. Aspell, "Modeling of gain in erbium-doped fiber amplifiers," IEEE Photon. Tech. Lett., vol. 2, no. 10, pp, 714-717, 1991. [20] R. Jopson and A. Saleh, "Modeling of gain and noise in erbium-doped fiber amplifiers," SPIEVol. 1581 Fiber Laser Sources and Amplifiers III, pp. 114-119, 1991. [21] I. Habbab, A. Saleh, N. Frigo, G. Bodeep, "Noise Reduction in Long-Haul Lightwave All-Amplifier Systems," J. Lightwave Technol., vol. 10, No. 9, pp. 1281-1289, 1992. [22] Y. Sun, A. K. Srivastava, J. L. Zyskind, J. W. Sulhoff, C. Wolf and R. W. Tkach, "Fast power transients in WDM optical networks with cascaded EDFAs," Electron. Lett., vol. 33, no. 4, pp. 313-314, 1997. [23] E. Desurvire, "An explicit analytical solution for the transcendental equation describing saturated erbium-doped fiber amplifiers," Optical Fiber Technology, no. 2, pp. 367-377, 1996. [24] Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, "Model for gain dynamics in erbium-doped fiber amplifiers," Electron. Lett., vol. 32, no. 16, pp. 1490-1491, 1996. [25] Y. Sun, J. L. Zyskind, and A. K. Srivastava, "Average inversion level, modeling, and phys-ics of erbium-doped fiber amplifiers," IEEE J. Sel. Top. Quantum. Electron., vol. 3, no. 4, pp. 991-1007, 1997 [26] A. Bononi and L. A. Rusch, "Doped fiber amplifier dynamics: a system perspective," J. Lightwave Technol., vol. 16, no. 5, pp. 945-957, 1998. [27] F. Lai, C. Liu and J. Jou, "Analyses of distortions and cross modulations in erbium-doped fiber amplifiers," IEEE Photon. Tech. Lett., vol. 11, no. 5, pp. 545-547, 1999. [28] S. Yamamoto, N. Edagawa, H. Taga, Y. Yoshida, and H. Wakabayashi, "Analysis of laser phase noise to intensity noise conversion by chromatic dispersion in intensity modulation Bibliography 102 and direct detection optical-fiber transmission," /. Lightwave Technol., vol. 8, no. 11, pp. 1716-1722, 1991. [29] W. K. Marshall, J. Paslaski, and A. Yariv, "Reduction of relative intensity noise of the out-put field of semiconductor lasers due to propagation in dispersive optical fiber," Appl. Phys. Lett., vol. 68, pp. 2496-2498, 1996. [30] M. Nazarathy, J. Berger, A. J. Ley, I. M. Levi, and Y. Kagan, "Progress in externally mod-ulated AM CATV transmission systems," J. Lightwave Technol., vol. 11, no. 1, pp. 82-105, 1993. [31] I. Jacobs, "Dependence of optical amplifier noise figure on relative-intensity-noise," J. Lightwave Technol., vol. 13, No. 7, pp. 1461-1465, 1995. [32] M. Movassaghi, M. K. Jackson, V. M. Smith, J. F. Young, and W. J. Hallam, "Noise figure of saturated erbium-doped fiber amplifiers: electrical versus optical measurement," in Con-ference on Optical Fiber Communication OFC'97, Vol. 6, 1997, OSA Technical Digest Series (Optical Society of America, Washington, D.C, 1997), paper WA2. [33] M. Movassaghi, M. K. Jackson, V. M. Smith and W. J. Hallam, "Noise figure of erbium-doped fiber amplifiers in saturated operation," IEEE Journal of Lightwave Technology, vol. 16, no. 5, May 1998. [34] M. Movassaghi, M. K. Jackson, V. M. Smith, J. F. Young, and W. J. Hallam, "Accurate fre-quency-resolved measurements of EDFA noise figure," in Optical Amplifiers and Their Applications, 1997 Technical Digest (Optical Society of America, Washington, D.C, 1997), paper TuDl. [35] F. W. Willems, J. C. van der Plaats, and D. J. DiGiovanni, "EDFA noise figure degradation by amplified signal double Rayleigh scattering in erbium doped fibers," IEE Electron. Lett., vol. 30, no. 8, pp 645-646, 1994. [36] P. Bevington and D. Robinson, "Data Reduction and Error Analysis for the Physical Sci-ences" McGraw-Hill, New York, 2nd ed, 1992. Bibliography 103 [37] M. N. Zervas and R. I. Laming, "Rayleigh scattering effect on the gain efficiency and noise of erbium-doped fiber amplifiers," IEEE J. Quantum Electron., vol. 31, no. 3, pp. 468-471, 1995. [38] E. Desurvire, "An explicit analytical solution for the transcendental equation describing saturated erbium-doped fiber amplifiers," Optical Fiber Technology, no. 2, pp. 367-377, 1996. [39] B. Pedersen, M. L. Dakss, B. A. Thompson, W. J. Miniscalco, T. Wei, and L. J. Andrews, "Experimental and theoretical analysis of efficient erbium-doped fiber power amplifiers," IEEE Photon. Tech. Lett., vol. 3, no. 12, pp. 1085-1087, 1991. [40] B. Pedersen, A. Bjarklev, O. Lumholt, and J. H. Povlsen, "Detailed design analysis of erbium-doped fiber amplifiers," IEEE Photon. Tech. Lett., vol. 3, no. 6, pp. 548-550, 1991. [41] M. Ohashi, and M. Tsubokawa, "Optimum parameter design of Er-doped fiber for optical amplifiers," IEEE Photon. Tech. Lett., vol. 3, no. 2, pp. 121-123, 1991. [42] M. Movassaghi and M. K. Jackson, "Design and modeling of saturated erbium-doped fiber amplifiers," to appear in Tech. Dig. IEEE/LEOS Summer Topical Meet., San Diego, CA, USA, July 1999. [43] M. Movassaghi and M. K. Jackson, "Simple and accurate modeling of high-efficiency sat-urated erbium-doped fiber amplifiers," submitted for publication to J. Lightwave Technol on July 27 1999. [44] K. Bertilsson and P. Anderkson, "Modeling of Noise in Erbium-Doped Fiber Amplifiers in the Saturated Regime," J. Lightwave Technol., vol. 12, No. 7, pp. 1198-1206, 1994. [45] C. Giles, C. Burrus, D. DiGiovanni, N. Dutta, and G. Rayban, "Characterization of erbium-doped fibers and application to modeling 980-nm and 1480-nm pumped amplifi-ers," IEEE Photon. Tech. Lett., vol. 3, no. 4, pp. 363-365, 1991. Bibliography 104 [46] S. P. Craig-Ryan, B. J. Ainslie and C. A. Miliar, "Fabrication of long lengths of low excess loss erbium-doped optical fiber," Electron. Lett., vol. 26, no. 3, pp. 185-186, 1990. [47] S. Ovadia, "CNR limitations of Er-doped optical fiber amplifiers in AM-VSB video light-wave truhking systems," IEEE Photon. Technol. Lett., vol. 9, no. 8, pp. 1152-1154, 1997. [48] I. M. I. Habbab and L. J. Cimini, "Optimized performance of erbium-doped fiber amplifi-ers in subcarrier multiplexed lightwave AM-VSB CATV systems," J. Lightwave Technol, vol. 9, no. 10, pp. 1321-1329, 1991. [49] M. Movassaghi, M. K. Jackson and V. M. Smith, "DFB laser RIN degradation in CATV lightwave transmission," Tech. Dig. IEEE/LEOS'98, vol. 2, paper FB2, Dec 1998. [50] M. Movassaghi, M. K. Jackson and V. M. Smith, "Dispersion-Induced RIN degradation and its impact on 1550nm AM video lightwave transmission systems," revised manuscript under preparation for submission to J. Lightwave Technol. [51] D. A. Atlas, R. Pidgeon, and F. Little, "Rayleigh backscatter effects on 1550-nm CATV distribution systems employing optical amplifiers," J. Lightwave Technol, vol. 13, no. 5, pp.933-946, 1995. [52] A. Yariv, H. Blauvelt, D. Huff, and H. Zarem, "An experimental and theoretical study of the suppression of interferometric noise and distortion in AM optical links by phase dither," J. Lightwave Technol, vol. 15, no. 3, pp. 437-443, 1997. [53] R. D. Esman and L. Goldberg, "Simple measurement of laser diode spectral linewidth using modulation sidebands," Electron. Lett., vol. 24, no. 22, pp. 1393-1395, 1988. [54] A. Cartaxo, B. Wedding, and W. Idler, "Influence of fiber nonlinearity on the phase noise to intensity noise conversion in fiber transmission: theoretical and experimental analysis," J. Lightwave Technol, vol. 16, no. 7, pp. 1187-1194, 1998. Appendix A Rate and Propagation Equations for 980nm-pumped EDFAs In this appendix a more detailed analysis of the rate and propagation equations of a three level system is presented; this analysis leads to derivation of equations which correspond to equation (1.5)-(1.8) used for modeling 980nm-pumped EDFAs. All the analysis and derivations presented here closely follows Ref. [4]. A.l Population Density of the Metastable level As described in Section 1.4, for 980nm pumping, the EDFA behaves like a three level sys-tem. Figure A.l shows these three levels as well as all the important transitions between them. In this figure R denotes the absorption rate (pumping rate) from level 1 to level 3, corresponding to the 980nm pumping. WJ2 and W21 are the absorption rate and the stimulated emission rate between levels 1 and 2, respectively. A2] represents the radiative spontaneous decay rate from level 2 to level 1, and A32 signifies the non radiative decay rate from level 3 to level 2. By defini-tion level 1 is the ground level, level 2 is the metastable level characterized by a long lifetime x, where T = 1 /A 2 1 , and level 3 is the pump level. The atomic rate equations corresponding to the populations of these three levels can be written as: 105 Appendix A. Rate and Propagation Equations for 980nm-pumped EDFAs 106 11/2 13/2 15/2 ^ 3 2 R . i i w 1 w21 r !A 2 1 level 3 level 2 level 1 Fig. A . l . Energy level diagram corresponding to the first three energy levels of erbium in the glass host, and all the important transitions between these levels. Appendix A. Rate and Propagation Equations for 980nm-pumped EDFAs 107 ^ = -RNl-WnN] + W2lN2 + A2lN2 (AA) iN7 — — — — = Wl2 N{ - W2l N2 - A 2 1 N2 + A 3 2 N: (A.2) dN3 -dt = RNl-A32N3 (A.3) where N^, N2, and N3 are the densities of the electrons in the energy levels 1, 2, and 3, respec-tively. In the steady state regime of operation where the populations are time invariant, i.e., dN: -j- = 0 (i = 1, 2, 3), and by considering the fact that A32 » R, the populations of the three levels can be written as: — 1 + W 2 1 T 1 K 1 +(R + Wl2 + W2l)x v ' (R + Wl2)x N > = p i + (jg + w 1 2 + w 2 1 ) t ( A - 5 ) N3 = -f - «! » 0 (A.6) A32 where p = N] + N2 + A 3^ is the erbium ion density. Equation (A.6) shows that the pump level population, N3, is approximately zero. As described in Section 1.4, this is due to the fast non radiative decay rate of electrons from the pump level to the metastable level, and therefore Appendix A. Rate and Propagation Equations for 980nm-pumped EDFAs 108 980nm-pumped erbium doped fibers can be analyzed by considering only levels 1 and 2. For a single mode fiber, R, W]2 and W2j , as a function of fiber coordinate z and radial axis r, are given as: PD(z) . * = - T ^ V P ( 0 (A.7) x P p Wl2 = —( r • \|/,(r) (A.8) W 21 = — , ~ ; -^—7 ¥ , ( 0 (A.9) 2 1 T (<*«,+ P™' By substituting equation (A.7)-(A.9) into equation (A.5), the population density of the second level can be obtained as: N2{r, z) = P p(r) S-Pp(z) , Gsa PS(Z) , — — w (r) + — vi/ (r) sat yPK ' rj +0 n^at Vs^'J (A. 10) sat Equation (A. 10) corresponds to equation (1.8), with the difference that n2(r, z) is a normalized population and that it includes the ASE. Appendix A. Rate and Propagation Equations for 980nm-pumped EDFAs 109 A.2 Propagation Equations for the Signal and Pump When a light signal at wavelength X with intensity Is (power per area) passes through an active medium of length dz, and population densities of , for the ground level, and N2, for the metastable level, the intensity change dls is given by: df = ( c ^ N l - a ^ W ^ f d z (AM) where oe(X) is the emission cross section and ca(X) is the absorption cross section, both at the signal wavelength X . For single mode propagation and for signal power of PS(X), the light intensity distribution, IS(X, r, 0) , in the fiber transverse plane is given as MfAX, r, 0) IS(X, r, 0) = PS(X) — , (A.12) \ys(X, r, Q)rdrdQ s where 0 is the azimuthal coordinate, and S denotes that the integral should be taken over the entire transverse plane. From equations (A.l 1) and (A.12), the propagation equation for the sig-nal can be written as: dpm f — — = PS(X) I {ae(X)N2(X, r, Q)-Ga(X)N](X, r, Q)}ys(X, r, d)rdrdd (A.13) s where \\fs is the normalized mode power at wavelength X defined as: Appendix A. Rate and Propagation Equations for 980nm-pumped EDFAs 110 ? A r, 0) = ¥ A r, 6) TC G£(X) (A. 14) \|f5(X, r, 6)r<irJe 5 By considering the radial symmetry of the optical mode, equation (A. 13) simply leads to equa-tion (1.5). The above analysis also directly applies to the pump power. By considering that for 980nm pumping, GE(X)=0, and that the excited state absorption for 980nm pumping occurs between levels 2 and 4, equation (1.6) is then easily obtained from equation (A. 13) at X = 980 nm. A.3 Propagation Equation for the ASE The rate of creation of spontaneous emission power (P$E) within an infinitesimal volume of active medium and within the frequency interval of Av is given by: where PQ = h v Av is the power of one spontaneous noise photon in bandwidth Av. The total ASE power at any position z of the fiber is the sum of the ASE power from the previous sections of the fiber and the added spontaneous emission power at the position z. Therefore, from equa-tion (A. 15), which gives the rate of creation of the spontaneous emission and equation (A.B) which is the propagation equation for an optical beam propagating in the direction of positive z, (A. 15) s Appendix A. Rate and Propagation Equations for 980nm-pumped EDFAs the propagation equation for the ASE propagating in positive z direction can be written as: 111 dP -jf = ^{ce(l)N2(X,r,Q)(PASEa) + 2P0) 5 - ca(X) N^d, r, 0 ) } v j / 5 ( ? i , r, 6 ) PASE rdrdQ (A.16) By considering that ASE is also generated in the direction of negative z, equation (1.7) is easily obtained from equation (A.16). 

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