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Characterization of erbium doped fibers Movassaghi, Mahan 1996

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C H A R A C T E R I Z A T I O N O F E R B I U M D O P E D FIBERS by M A H A N M O V A S S A G H I B.Sc. in Electrical Engineering (Electronics), Amir-Kabi r University of Technology, Tehran, Iran, 1992 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (The Department of Electrical Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A M a y 1996 © M a h a n Movassaghi, 1996 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for schol-arly 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 permission. Department of Electrical Engineering The university of British Columbia Vancouver, Canada D E - 6 (2/88) A B S T R A C T In this thesis the theory of operation of single-mode erbium-doped fiber amplifiers pumped at 980 nm is described. Details of the derivation of the general rate equation for the propagation of signal, pump, and amplified spontaneous emission are provided. Based on this equation, and McCumber 's theory of phonon-terminated optical masers, two closed form expressions are derived. In one of them, the fluorescence spectrum of an erbium-doped fiber is related to its spectral absorption coefficient. Based on this expression, a rigorous basis for the 4 4 assessment of the applicability of McCumber's theory to the study of I13/2 <=> ^15/2 transi-tions in erbium-doped fibers has been established. For the cases of five silica-based erbium-doped fibers, experiments were performed and the results were used to validate this expres-sion. Another important benefit of this expression is that it does not require a measurement of silica-based fiber's fluorescence spectrum, as it can be simply calculated from the spectral absorption coefficient, simplifying fiber characterization. The other closed form expression, provides a simple means for calculating absorption and emission cross-sections of erbium-doped fibers using the easily measured spectral absorption coefficient, the gain coefficient at one particular wavelength, and the fluorescence lifetime. Also , based on this expression, an analytical method for the simple determination of the erbium ion concentration inside the fiber core is proposed. Experiments were performed to evaluate the cross-sections of an erbium-doped fiber over the wavelength range 1400-1650 nm. To check the accuracy of the calculated cross-sections, ii saturation powers at the wavelengths 1530 nm and 1550 nm were measured and results com-pared with the ones calculated from the cross-sections, obtaining agreements within 6%. Problems and difficulties associated with the conventional techniques for the measurement of the spectral fluorescence and of the fluorescence lifetime are described. Also new experi-mental setups were designed to simplify the measurement of these parameters. Furthermore, experimental techniques for the measurement of other fiber parameters such as, spectral absorption coefficient, gain coefficient, and signal saturation power are described. The new theoretical and experimental techniques presented in this thesis can provide a much simpler and more accurate means for characterizing erbium-doped fibers, improving the accuracy of the numerical models commonly used for modeling erbium-doped fiber amplifi-ers. iii Table of Contents Abstract ii Table of Contents iv List of Tables v List of Figures vi Acknowledgments '. viii Chapter 1. Introduction 1 Chapter 2. Theory 5 2.1. Introduction 5 2.2. Fundamentals of erbium-doped fiber amplifiers , 7 2.3. Validation of the applicability of McCumber's theory to the study of silica-based erbium-doped fibers : spectral correlation between fluorescence and absorption coefficient 10 2.4. A new method for the simple determination of cross-sections 21 2.5. Evaluation of the erbium concentration inside the fiber core 32 Summary of Chapter 2 34 Chapter 3. Experimental Techniques and Measured Results 36 3.1. Introduction 36 3.2. Measurement of the spectral absorption coefficient 37 3.3. Measurement of the fluorescence spectrum 39 3.4. Measurement of the small signal gain coefficient 44 3.5. Measurement of the fluorescence lifetime 50 3.6. Measurement of the saturation power at signal wavelengths 54 Summary of Chapter 3 59 Chapter 4. Summary, Conclusions, and Recommendations for Future Work 60 4.1. Summary and Conclusions : 60 4.2. Recommendations for future research 62 References 63 Appendix A: Rate equation analysis of the erbium ion population 69 Appendix B: General rate equation for the propagation of signal, pump and amplified spontaneous emission in single-mode erbium-doped fibers 72 Appendix C: Previous attempts at the determination of cross-sections 80 iv List of Tables Table 2.1. Fiber parameters, estimated errors, and maximum discrepancies in the 1520-1560 nm range, for the four fibers, for which the absorption coefficient and normalized fluorescence spectra are given in F ig . 2.3(a)-(d) 16 Table 2.2. Measured and calculated values of Psat(X) at wavelengths X = 1530nm and A, = 1550rcm 32 v List of Figures Figure 2.1. Layout of a basic erbium-doped fiber amplifier 7 Figure 2.2. Energy levels of erbium ions in glass hosts : 9 Figure 2.3(a). Measured absorption coefficient (solid), and the measured (dashed) and calculated (dotted) fluorescence for Fiber #1 17 Figure 2.3(b). Measured absorption coefficient (solid), and the measured (dashed) and calculated (dotted) fluorescence for Fiber #2 18 Figure 2.3(c). Measured absorption coefficient (solid), and the measured (dashed) and calculated (dotted) fluorescence for Fiber #3 19 Figure 2.3(d). Measured absorption coefficient (solid), and the measured (dashed) and calculated (dotted) fluorescence for Fiber #4 20 Figure 2.4. Calculated absorption cross-section (solid) and emission cross-section (dotted) for Fiber #5 27 Figure 2.5. Measured spectral absorption coefficient of Fiber #5 28 Figure 2.6. Measured fluorescence of Fiber #5. The fluorescence spectrum is normalized with respect to its peak value 29 Figure 2.7. Comparison between the measured and calculated fluorescence. Both spectra are normalized with respect to their values at kpeak 30 Figure 3.1. Schematic of the experimental set up used for the measurement of the spectral absorption 38 Figure 3.2. Schematic of the experimental set up used for the measurement of the fluorescence spectrum 42 Figure 3.3. Spectral response of the W D M , normalized with respect to its peak value 43 Figure 3.4. Schematic of the experimental set up used for the measurement of the gain coefficient 44 Figure 3.5. Spectral input power and spectral output power of the EDF 48 vi Figure 3.6. Traditional set up for the measurement of the fluorescence lifetime (from reference [7]) 51 Figure 3.7. Schematic of the experimental set up used for the measurement of the fluorescence lifetime 52 Figure 3.8. Fluorescence decay curve for the 2cm long of Fiber #5 (circles). The solid line corresponds to the least-squares fit used to obtain x 53 Figure 3.9. Schematic of the experimental set up used for determination of the saturation power 55 Figure 3.10. Experimental transmission characteristics (Pou/Pin) of 3.781 meters of Fiber#5 at the wavelength 1530 nm (circles). The solid line corresponds to the analytical transmission characteristics where Pout is calculated from equation (3.5) using the values of , Pin, L, and Psat 57 Figure 3.11. Experimental transmission characteristics (Pou/Pin) of 3.781 meters of Fiber#5 at the wavelength 1550 nm (circles). The solid line corresponds to the analytical transmission characteristics where Pout is calculated from equation (3.5) using the values of oc^ , Pin, L, and Psat 58 Figure A . l . Energy level diagram corresponding to the first three levels of erbium ions in a glass host, and all the possible transitions between these levels 70 vii Acknowledgments M y deepest gratitude goes to the members of my family for their encouragement and sup-port throughout my education, including the course of this work. I would like to express my gratitude to Dr. P.. Vella, of the National Optics Institute (NOI), for suggesting this project and providing continuous support.throughout the course of this work. I also would like to thank Dr. N . A . F. Jaeger for his continual support and guidance during my research. M y gratitude extends to Yves Lacroix, research associate at the N O I , for his invaluable help with the experimental measurements, and for many productive discussions and sugges-tions. I am also grateful to Lisheng Huang, Dr. F. Chenard, Andre Croteau, Francois Gregoire, and Fernand Sylvain, all at the NOI , who have helped me in various ways throughout the course of this work. I also would like to thank Dr. M . K . Jackson for many fruitful discussions as well as Dr. D . L . Piilfrey and Dr. L . M . Wedepohl for their guidance and invaluable advice during my time at U B C . I would like to specially thank my friend, Shahram Tafazoli, for his help in computer pro-gramming and his invaluable support during my research. Finally, my appreciation goes to all those individuals at the National Optics Institute and the Department of Electrical Engineering who have helped me in numerous ways. viii Chapter 1 Introduction The field of optical fiber communications was introduced less than 20 years ago, when the first light-wave communications system operating near 0.8 | i m , at a bit rate up to 2.3 Mbit/s, and electronic repeaters spaced at about 32 km, was demonstrated in 1978 [1]. Since then, one of the major research activities within the area of optical communications has focused on reducing the attenuation in the signal path between the transmitter and the receiver, as well as reducing pulse dispersion. In the mid 1980s, the dispersion problem was overcome by using laser sources with ultra-narrow linewidths and dispersion-shifted fibers [2]. However, the real limitation has been the need to regenerate optical signals, typically every 70-100 km, as they undergo attenuation when propagating along a fiber link. The signal regeneration has been tra-ditionally performed using electronic repeaters in which the optical signal is converted into an electric current by a photodiode; the electric current is then amplified and converted back into an optical signal by a laser diode. The electronic speed of such repeaters is fixed once and for all ; this limits the information rate of such lightwave systems. Other disadvantages of elec-tronic repeaters are their high cost and complexity, especially for multi-channel lightwave sys-tems, as many parallel repeaters are needed to regenerate several optical channels in the same fiber. In 1985, just when many believed that the optical systems had approached their peak per-1 formance, a research group at Southampton University in the U K , showed that optical fibers doped with the rare earth erbium can exhibit signal gain at a wavelength near 1.55 (im [3], and could replace electronic repeaters. This realization spread to other laboratories soon after, and by early 1987 the first erbium-doped fiber amplifier (EDFA) , for telecommunication applica-tions, was demonstrated [4]. A few years later, several systems with erbium-doped fiber amplifiers, which were capable of communicating information at bit rates of 5 Gbit/s over 14,000 km long fiber finks, were demonstrated [5]; this performance was more than 140 times better than the performance of unrepeatered systems in 1985, which were operating at bit rates of up to 4 Gbit/s over distances of around 100 km [6]. Such radical improvements, together with the numerous desirable properties of erbium-doped fibers, e.g., high gain, low noise, high power conversion efficiency, high saturation power, polarization insensitivity, broad spectral bandwidth, very low coupling loss, low cost, and immunity to interchannel crosstalk [7], have created a general belief in the great impact that these amplifiers w i l l have in future optical fiber communications systems. Hence, many researchers around the world have placed con-siderable emphasis on improving the performance of E D F A s using numerical models based on rate and propagation equations [8]-[18]. For these models to be relevant, it is important to characterize the erbium-doped fibers accurately. In this regard, precise measurement of sev-eral fiber parameters such as the spectral absorption coefficients, the spectral gain coefficients, the spectral fluorescence, the fluorescence lifetime, the erbium density distribution inside the fiber core, the spectral absorption cross-sections, the spectral emission cross-sections, and sat-uration powers are particularly important. Among these parameters the absorption and emis-sion cross-sections at signal wavelengths and the erbium density distribution inside the fiber 2 core are the ones which are very difficult to measure accurately. Regarding the values of cross-sections, discrepancies of up to 50% are reported for the different methods used to determine these parameters (see Appendix C for details) [8], [19]. Recently a new technique was pro-posed which allows determination of the quotient of cross-sections [20]. Nevertheless, lack of access to an accurate method for determination of the absolute values of cross-sections still exists. Likewise, previous attempts to determine the erbium density distribution inside the fiber core have not been successful (see Appendix C for details). This lack of success is prima-rily due to the small size of the fiber core. Recently D . Uttamchandani et. al. [21] have pro-posed a technique which allows direct determination of the erbium profile in the fiber core. However, as far as we know, there is no means for the accurate determination of erbium ion concentration inside the fiber core. In this thesis, a new technique for simple and accurate determination of absorption and emission cross-sections is described. Also , an analytical method for the simple determination of the erbium ion concentration inside the fiber is proposed. The work presented in this thesis was performed at the National Optics Institute (NOI). The work on erbium-doped fiber ampli-fiers at N O I started in early 1990. Since then, the activity has focused on the fabrication of s i l -ica-based erbium-doped fibers. The Institute, with its well-equipped fiber fabrication laboratories, which are known to be among the best in their category worldwide [22], now has a mature technology in the fabrication of a large variety of erbium-doped fibers. In order to predict the behavior of amplifiers made using these erbium-doped fibers, and furthermore to improve the fiber design, establishing accurate models for the E D F A s is of particular impor-tance. The work presented in this thesis concerns itself with the full characterization of 3 erbium-doped fibers, which is the initial and most important step towards establishing a com-prehensive model for E D F A s . In Chapter 2 of this thesis the theory of operation of erbium-doped fiber amplifiers, pumped at 980 nm, is described. A new technique for the simple assessment of the applicabil-ity of McCumber 's theory (see Appendix C for details) to the study of silica-based erbium-doped fibers, a new method for the simple determination of absorption and emission cross-sections, and a new approach to the evaluation of the erbium ion concentration inside the fiber core are presented. Rate equation analysis of the erbium ion population based on a three level system model is provided in Appendix A . In Appendix B , details of the derivation of the gen-eral rate equation for the propagation of signal, pump, and amplified spontaneous emission in single-mode erbium-doped fiber amplifiers are given. In Appendix C , previous attempts at the determination of the absorption and emission cross-sections and the erbium density distribu-tion inside the fiber core are described. In Chapter 3 details of the experimental procedure for the determination of several erbium-doped fiber parameters such as, spectral absorption coefficient, gain coefficient, and signal saturation power are described. Problems and difficulties associated with the conven-tional techniques for the measurement of the spectral fluorescence and the fluorescence life-time are explained. A l so details of the new experimental setups which were designed to simplify the measurement of these parameters are described. Finally, a summary of the work, conclusions, and suggestions for improving the work are given in Chapter 4. 4 Chapter 2 Theory 2.1. Introduction A n erbium-doped fiber amplifier (EDFA) is a system in which combined characteristics of a single-mode fiber and a laser glass are utilized. Buil t upon this fact, many researchers, have over the past few years, developed several theoretical models for the study of these fiber amplifiers [9]-[18]. These models are important for predicting the behavior and for optimizing the performance of E D F A s . In these models, knowledge of several fiber parameters is required; among these parameters the absorption and emission cross-sections at signal wave-lengths and the erbium density distribution inside the fiber core are the ones which are very difficult to measure accurately (see Appendix C for details). Despite extensive efforts placed on the evaluation of cross-sections by several researchers, relatively large uncertainty still exists throughout the literature regarding their actual values, and discrepancies of up to 50% are reported for the different techniques used to determine these parameters [8], [19]. In fact, accurate determination of cross-sections is recognized to be the most challenging task in char-acterizing erbium-doped fibers [8]. The traditional approach for the determination of cross-sections is based on the Futchbacher-Ladenberg (FL) relationship [23] (see Appendix C for details). However, in 1991, several researchers reported its failure for the case of erbium-5 doped glasses [19],[24], [25] (see Appendix C for details). Alternately, in 1991 Miniscalco and Quimby [19], [26], suggested that McCumber's theory of phonon-terminated optical masers [27] could be used to determine these parameters. However, their attempt to validate this application of McCumber's theory to the case of silica-based erbium-doped fibers was not quite successful (see Appendix C for details). Furthermore, their approach for determining cross-sections is complicated and requires a knowledge of many fiber parameters, some of which are very difficult to obtain accurately, e.g., erbium ion density distribution, and Stark-level energies. Recently H . Zech [20] has proposed a new technique which allows determina-tion of the quotient of cross-sections. Nevertheless, as far as we know, still there is no accurate method for determination of the absolute values of cross-sections. Likewise, measurement of the erbium density distribution inside the fiber core is a challenging task. This is because of the small size of the fiber core. Previous attempts to measure this parameter have not been quite successful and the accuracy of the measurements is estimated to be about 18% [28] (see Section 2.5 and Appendix C for more details). Recently D . Uttamchandani et. al. [21] showed that confocal optical microscopy can be used for direct measurement of the erbium profile inside the fiber core. Nevertheless, as far as we know, still there is no means for the accurate determination of erbium ion concentration inside the fiber core. In this chapter, the theory of operation of erbium-doped fiber amplifiers pumped at 980 nm is described. Based on the general rate equation for the propagation of the signal, pump and amplified spontaneous emission, for which details of its derivation are described in Appendix B , and on McCumber 's theory of phonon-terminated optical masers, a simple means for deter-mining the spectral absorption and emission cross-sections is presented. Finally, a new 6 approach to the evaluation of the erbium ion concentration inside the fiber core is described. 2.2. Fundamentals of erbium-doped fiber amplifiers The basic architecture of an erbium-doped fiber amplifier is illustrated in Figure 2.1. The system includes a pump laser diode (LD) , a wavelength selective coupler or combiner (WSC), which is also called a wavelength division multiplexer ( W D M ) , and a length of a single-mode erbium-doped fiber (EDF). EDF signal @1530nm Figure 2.1. Layout of a basic erbium-doped fiber amplifier. The erbium-doped fiber is made by incorporating erbium dopants together with germania ( G e 0 2 ) , and/or alumina (AI2O3) as co-dopants into the fiber core using various techniques. Among these techniques are, modified chemical vapor deposition ( M C V D ) [29], [30], vapor axial deposition (VAD) [31], outside vapor deposition (OVD) [32], and solution doping [33]. 7 Alumina and germania are used as index-raising codopants. Furthermore, alumina improves the solubility of the erbium into the fiber core, which results in high concentration doped fibers [34]. The erbium-doped fibers most commonly used are silica-based. The benefits of using s i l -ica as the host glass are that it is directly compatible with telecom fibers and hence fusion splicing is possible, yielding ultra low loss and low reflection joints, and that it is an excep-tionally durable glass. The energy levels of erbium ions in silica glasses are shown in figure 2.2. The cross-hatched rectangles in the figure indicate that the main energy levels are split into multiple sub-levels due to the Stark effect. The Stark effect is induced by the permanent electric field, called a crystal or ligand field, which is generated by the charge distribution in the glass host [35]. The transition corresponding to the 1520-1570 nm signal band, known as the third tele-4 4 4 communications window, is the one from / 1 3 / 2 to / 1 5 / 2 . T h e 7 1 3 / 2 level is the metastable 4 state, and the I15/2 level, is the ground state. The population inversion between these two levels is achieved by optically pumping the erbium ions from the ground state to some higher state from which the ions relax to the metastable state. Once the population inversion is achieved, signal amplification in the 1520-1570 nm band is accomplished by stimulated emis-sion [36]. 8 F 9 / 2 X///////////////////////////A 650 nm V///////////////////////777A 800 nm \vX////////////////////////7m 980 nm \,y/////////////////////////7A 1530 nm 4 I 1 5 / 2 V//////////////////////////A Ground Level Figure 2.2. Energy levels of erbium ions in a silica glass host. 4 4 Among the several transitions that can be used to pump E D F A s , the I15/2 —> In/2 t r a n " sition, which corresponds to the 980 nm pump band, offers the highest gain efficiencies [37], [38], and the lowest noise figures [39], [40]. Recent developments in the field of semiconduc-tor lasers has made it possible to make 980 nm laser diodes with very high output powers (near 450 mW) [41]. Hence, 980 nm is currently the preferred pump band and throughout the work presented in this thesis 980 nm pumping was used. To analyze the interactions between the erbium ions and the signal and pump photons in 9 erbium-doped fiber amplifiers pumped at 980 nm, rate equations corresponding to a basic three level laser system have been used throughout the literature. These rate equations are described in Appendix A. Subsequently, in Appendix B, details of the derivation of the gen-eral rate equation for the propagation of signal, pump, and amplified spontaneous emission in single-mode EDFAs is provided. In subsequent sections of this chapter, mainly based on this general rate equation, equation (B.21), several closed-form expressions are derived by which a number of difficult-to-measure parameters, e.g., absorption and emission cross-sections at sig-nal wavelengths and the erbium ion concentration in the fiber core, are related to easily mea-surable parameters. 2.3. Validation of the applicability of McCumber's theory to the study of silica-based erbium-doped fibers - Spectral correlation between fluo-rescence and absorption coefficient. Until now, McCumber's theory of phonon-terminated optical masers has been shown to be 4 4 ideally appropriate for the study of / 1 5 / 2 and / 1 3 / 2 levels of erbium in glass hosts. How-ever, previous attempts to experimentally verify this application of McCumber's theory have not been successful. This lack of success is primarily due to the fact that the approach taken in those experimental assessments requires knowledge of many fiber parameters which cannot be measured accurately. Essential among these are the erbium ion concentration inside the fiber core and Stark level energies (see Appendix C for details). In this section, based on McCumber's theory and the general rate equation, equation 10 (B.21), we derive a closed form expression which relates the fluorescence to the absorption 4 4 4 coefficient for the / 1 3 / 2 <=> ^ 1 5 / 2 transitions, the transition of erbium ions from / 1 3 / 2 level 4 to / 1 5 / 2 and vice-versa, in erbium-doped glasses. Since in erbium-doped fibers the fluores-cence and absorption coefficients can be measured with good accuracy (less than 2% error) [8], comparison of the calculated fluorescence, calculated from the absorption coefficient using our expression, with the measured one provides a reliable means for assessing the appli-cability of McCumber 's theory to the study of these fibers. Finally, using this approach, we 4 4 validate the applicability of McCumber 's theory to the study of I13/2 <=> A 5 / 2 transitions in silica-based erbium-doped fibers. We start with the general rate equation, equation (B.21). Let us assume that in an amplifier the following conditions are achieved: 1- The pump power is very high, i.e., ~B—TT^T » 1 ; sats'"pJ 2- The signal power at any fiber coordinate z is much lower than the pump power, i.e., PS(X, z) « Pp(hp> z)> also that p — « 1 but is still strong enough such that amplified spontaneous emission can be neglected, i.e., PS(X, z) ^ P A S E & > Z) • Under these operating conditions, equation (B.21) can be simplified to: dPJX,z) 2 f \p(r)ce(X)Psa,zmkr)rdr . (2.1) d z o>2s(X) dPs(X,z) = d[\nPs(X,z)] Ps(X,z)dz " dz gs(X) . Therefore, in terms of gs(X) equation (2.1) can be written as: where the term — ^ — — = ^ is defined as the signal gain coefficient A ,z) z z 11 gs(X) = ae(X) f p(r)x|/,(A, r)rdr . (2.2) On the other hand, let us assume an amplifier in which the pump is turned off, i.e., z) y y P (A , z) = 0 , and the signal power is so small that p—7T\ " 1 . Under this condition, s j sat ASE can be neglected, and equation (B.21) can be simplified to: dPXk,z) ^ . - MPJKz)Vs(X,r)rdr , (2.3) dz fJ 0 CD:(X) a P(r)aa( * dP.(k,z) d[]nP(X,z)] where the term - — — r — — = is defined as the signal absorption coeffi-Ps{k,z)dz dz cient, a (A) . In terms of as(k) , equation (2.3) can be written as: as(X) = -2— oa(\) | p ( r ) V , ( \ , r)rrfr . (2.4) co'(A) Now, let us consider an amplifier in which the signal is turned off, i.e., PS(X, z) = 0 , and the Pp(\,z) pump power is so high that ~£—7y~T " 1 • Under this operating condition, equation (B.21) sat^'p can be simplified to: dPASE^z) 4P0(A)a e(A) f J7 = 1— p(r)ys(X,r)rdr . (2.5) a z CO^(A) J 0 Accordingly, the A S E power of a length L of an erbium-doped fiber at wavelength A and wavelength interval AA can be obtained by integration of equation (2.5): 12 4LPJX)aJX) f PASE(X,L) = Q-2 e— p(r)V,(A,,.r)rrfr . (2.6) The A S E power at low gains, i.e., very short fiber lengths, is the fluorescence F(X) [44]. Thus, from the combination of equations (2.2) and (2.6), we obtain: im 2LP0a) = , (2.7) 8 A") X which gives a relationship between fluorescence and gain coefficient. From equations (2.2) and (2.4) the relation ^ ) _ £ ^ ™ is easily obtained. Now, from equation (2.8) and the McCumber relation, equation (C.3), we obtain the expression: _ _ he , _ ixM Equation (2.9) provides an alternative means for the simple determination of e, the parameter which is defined in equation (C.4). To evaluate e, one can simply measure the absorption and gain coefficients at a particular wavelength X. In order to minimize measurement errors, we choose X to be at the peak of the absorption spectrum Xpeak. This is because, in deriving equation (B.21) the effect of fiber background loss has not been taken into account. Therefore, as(X) , in equation (2.4), represents only the absorption coefficient due to erbium dopants, and gs(X) , in equation (2.5), represents the gain coefficient, assuming the fiber background 13 loss is zero. On the other hand, as OLs(X) and gs(X) cannot be measured directly, instead, the fiber absorption coefficient, OL's(k) = <xs(X) + ab(X) , and fiber gain coefficient, g's(X) = gs(X) - ab(X) , are measured using the conventional cutback method [43], where ab(X) is the background loss coefficient of the fiber. Thus, to obtain as(X) and gs(X) , ab(X) has to be determined first. Since the fiber background loss in the spectral range X = 1400 - 1650 nm is masked by the loss due to erbium dopants, the value of ab(X) is determined by extrapolating the fiber background loss coefficients measured at wavelengths away from the erbium absorption band [44]. Hence, ab(X) cannot be determined accurately. However, since values of ab(X) are very small (typically around 0.06 dB/m for silica-based fibers [50]) compared with the values of as(X) and gs(X) , the errors introduced by uncer-tain values of cxb(X) into the calculated values of as(X) and gs(X) are extremely small. Obviously, the errors are minimum at the peak values of <xs(X) and gs(X) , and since the peaks of these two are very close together, around 1 nm apart in silica-based erbium-doped fibers, in equation (2.9) by choosing X to be at the peak of the absorption spectrum, we can minimize the error in the calculation of e. Accordingly, equation (2.9) takes the form: e = _ ^ + ^ r i n F ( ^ ) 1 . (2.10) Kpeak \-asK^peak>^ Now, from the combination of equations (2.7), (2.8), (2.10), and (C.3), we can write: F(X)X3AXpeak = as(X) fhc{\/Xp,ak-\/Xy ^ which provides the correlation between the absorption coefficient and the fluorescence spec-trum. Since, in the measurement of the fluorescence spectra, using an optical spectrum ana-1 4 lyzer, the resolution of the instrument can be chosen to be fixed (e.g., AX = 0.5nm) over the entire sweep of the fluorescence spectrum, AX = AXpeak and equation (2.11) takes the form: F(A)A 3 cc,(X) exp M l / A 'peak - 1 / X ) S (2.12) peak To verify equation (2.12) one would first measure the fluorescence and the absorption coeffi-spectrum would be calculated using equation (2.12). Finally, the calculated and measured flu-orescence spectra would be compared. We measured the spectral fluorescence and absorption coefficient of four germano-alumino silicate erbium-doped fibers in the wavelength range 1450-1650 nm (details of the experimental procedures for measuring the absorption coeffi-cient and fluorescence are explained in sections 3.2 and 3.3, respectively). These fibers each had a different erbium concentration, see Table 2.1. Figures 2.3 (a)-(d) show the measured absorption and the measured and calculated fluorescence spectra; both fluorescence spectra have been normalized with respect to their values at Xpeak. Table 2.1 also gives estimates of the measurement errors (for the case of absorption these were obtained by performing several cutbacks and for the case of fluorescence by performing several measurements on different pieces of fiber). We have also calculated the maximum discrepancy between the calculated and measured fluorescence over the wavelength range of interest, 1520-1560 nm, for each fiber. A s can be seen in Table 2.1,the peak discrepancies range from a low of 3% to a maxi-mum of 6%. cient spectra. Then, using the measured absorption coefficient spectrum, the fluorescence 15 Table 2.1. Fiber Parameters, estimated errors, and maximum discrepancies in the 1520-1560 nm range, for the four fibers, for which the absorption coefficient and normalized fluorescence spectra are given in Fig. 2.3(a)-(d). Fiber Peak Absorption (dB/m) Estimated Fluorescence Meas. Error Estimated Absorption Meas. Error Erbium Concentration (ppm-wt) Max. Discrepancy Between Measured and Calculated Fluorescence in the 1520-1560 nm Range 1 1.45 1.1% 1.7% 450 5.7% 2 5.26 2.0% 0.8% 950 3.1% 3 16.01 2.3% 1.1% 2000 4.8% 4 35.49 2.6% 1.4% 4400 3.2% 16 17 1450 1500 1550 1600 Wavelength (nm) Figure 2.3 (b): Measured absorption coefficient (solid), and the measured (dashed) and calculated (dotted) fluorescence for Fiber #2. 18 Figure 2.3 (c): Measured absorption coefficient (solid), arid the measured (dashed) and calculated (dotted) fluorescence for Fiber #3. 19 Figure 2.3 (d): Measured absorption coefficient (solid), and the measured (dashed) and calculated (dotted) fluorescence for Fiber #4. 20 The close agreement between the calculated and measured fluorescence spectra of these fibers validates our closed form expression, equation (2.12), and, thereby, the apphcability of 4 4 McCumber's theory to the study of the I13/2 <=> ^15/2 transitions in silica-based erbium-doped fibers. A direct, and important, benefit of our validation of equation (2.12) is that it is no longer necessary to measure a silica-based fiber's fluorescence spectrum when the absorp-tion spectrum is available, as it can be simply calculated. Since both the absorption and fluo-rescence spectra are among the essential parameters that have to be measured, when characterizing a fiber, our expression provides a simpler means for fiber characterization as the fluorescence measurement can be eliminated. 2.4. A new method for the simple determination of cross-sections As was discussed in section 2.2, despite extensive efforts directed towards evaluating the absorption and emission cross-sections, relatively large uncertainty is still being expressed in the literature with regard to their actual values. This is primarily due to the fact that the evalu-ation of these parameters depends on the exact determination of the erbium density distribu-tion within the fiber, which is very difficult to measure accurately. In this section, a new technique will be presented by which the absorption and emission 4 4 cross-sections of / 1 3 / 2 <=> ^15/2 transitions can be conveniently and accurately determined using the easily measured spectral absorption coefficient, the gain coefficient at one particular 21 wavelength, and the fluorescence lifetime. In our technique, based upon McCumber's theory and the general rate equation, equation (B.21), we derive a closed form expression in which we have removed the need to determine the overlap integral, containing the difficult to mea-sure erbium density distribution, by substituting an equivalent expression that contains more easily measured parameters. This vastly simplifies and improves the accuracy of the calcula-tion of the absorption and emission cross-sections. We start with equation (C.6) from McCumber's theory of phonon-terminated optical masers, which when combined with equation (2.2) gives: X2 •»=W | Z'WGA) A ; (2.13) X ^ X4 jp(r)\\fs(X,r)rdr 4 4 where X1 is at one end of the / 1 3 / 2 —> A 5 / 2 emission spectrum and X2 is at the other end of the spectrum. Now, from the mean value theorem [45], equation (2.13) can be written as: 00 X<2 \p(r)ys(X*,r)rdr = 4nxcn 5 *' dX., (2.14) 0 Xl where Xx < X* < X2. In equation (2.14) the overlap integral at wavelength X*, the mean over-lap integral, is expressed in terms of x, (Os(X) , and gs(X) . Therefore, in order to evaluate the mean overlap integral, these three parameters have to be determined. The fluorescence lifetime, x, can be simply measured by monitoring the decaying fluorescence from a length of erbium-doped fiber pumped by a chopped laser beam operating at X = 980 nm. The details 22 of the experimental procedure are explained in section 3.3. To evaluate the mode power radius, co^ , we consider the case of a weakly guiding step-index fiber (applicable to the case of commonly used single-mode erbium-doped fibers [46]), for which the fundamental mode is well approximated by the L P 0 i solution [47], [48]. The L P 0 i solution for the fundamental mode envelope is given by [46], [48]: where a is the core radius, J0 and K0 are the Bessel and modified Bessel (or Macdonald's) functions, respectively, Umd W are the transverse propagation constants of L P 0 1 mode, which are given by [47]: (2.15) and 2 u = 1 + (4 +V 4 ) 0.25 and (2.16) 2 2 2 W = (V - U ) , where V is the normalized frequency, and is defined as [47]: V = 2%aNA (2.17) and NA is the fiber numerical aperture given by [48]: 23 NA = ( n c o r e - n c l a d ) 2 , (2.18) where n c o r e and n c l a d are the refractive indices of core and cladding, respectively. Now, from the definition of co^ . according to equation (B.9), and by using equation (2.15), co5 can be written as [8]: VKX{W) UKQ(W) ° > ^ ) = « TTlFTTin W ) • (2.19) Therefore, by knowing the fiber core radius and its numerical aperture, tos(X) can be simply calculated from equation (2.19). The last parameter in equation (2.14), gs(X) , cannot be measured easily, as it should be 4 4 determined over the entire / 1 3 / 2 —> 7 1 5 / 2 emission spectrum, which is a wide spectral range (typically from 1400-1650 nm for the case of silica-based erbium-doped fibers). In order to avoid this, we determined gs(X) , from the fluorescence spectrum and by measuring the gain coefficient at only one particular wavelength, say gs(Xx), through the relation that is derived earlier in equation (2.7). In other words, equation (2.7) can be written as: 8sfr) A 3 F ( A ) (2.20) Giles et. al. [44], give a similar, but different, relation between the gain coefficient and the flu-orescence, gs(X)/gs(Xx) = F(X)/F(XX) , which we believe is not exact. In equation (2.20), F(X)/F{XX) can either be determined experimentally, or it can be simply obtained from the measured absorption coefficient spectrum and, using the relation that is derived in equation (2.12), and gs{Xx) can be easily measured using the cutback 24 method [8]; details of the measurement procedure are given in section 3.4. As a result of the above discussion, it can be seen that the mean overlap integral can be calculated from its equivalent expression shown in equation (2.14), which contains the param-eters x, (£>(X), and gs(X), that can be determined both simply and accurately. In commonly used erbium-doped fibers, erbium is confined to 20% - 50% of the fiber core area [46]. For such fibers, it can be shown that the mode power distribution, \\fs(X, r) , over the entire emission spectrum (typically 1400 < A, < 1650 for silica-based erbium-doped fibers) varies by only a few percent. As an example, let us take a typical erbium-doped fiber with a core diameter of 4 | im, a numerical aperture of 0.2, and a step like erbium distribution with 50% confinement, then the absolute value of the maximum difference among the overlap integrals in the 1400-1650 nm wavelength range is only 2.36%. These discrepancies for cases in which the erbium-doped fiber has the same core radius and numerical aperture, but confine-ments of 40%, 30%, and 20%, were calculated to be 1.51%, 0.85%, and 0.37%, respectively. Consequently, using the mean overlap integral as the value of the overlap integral for the entire emission spectrum, introduces only a few percent error. In addition, since in the mean overlap integral, X* is a wavelength between the limits and X2, intuitively near the middle of the emission spectrum, the errors caused by replacing it as the value of the overlap integral for the entire emission spectrum would be even less than the maximum discrepancy evaluated before. For the fiber in the above example with 50% erbium confinement, by assuming X* = 1525 nm, the absolute value of the maximum difference between the mean overlap inte-gral at X*, and the ones in the wavelength range 1400-1650 nm is only 1.22%. This shows that the mean overlap integral, which can be simply calculated using equation (2.14), can be 25 actually substituted for the overlap integrals over the entire emission spectrum of the 4 4 ^13/2 ~~* hs/2 transition in erbium-doped fibers. Once the mean overlap integral is calcu-lated, one can simply calculate the emission cross-sections using equation (2.2). The absorp-tion cross-sections can then be simply calculated either using equation (C.3), or equation (2.4). Figure 2.4 shows calculated emission and absorption cross-sections for a germano-alu-mino silicate erbium-doped fiber (fiber #5) having a cutoff wavelength of 850 nm, a core radius of 1.8 \x.m, a numerical aperture of 0.18, and a fluorescence lifetime of x = 10.15 ms. The measured spectral absorption coefficient and the normalized fluorescence of this fiber are shown in Figures 2.5 and 2.6, respectively. Also, Figure 2.7 shows a comparison between the measured fluorescence and the one calculated from the absorption coefficient data using equa-tion (2.12). 26 27 1400 1450 1500 1550 1600 1650 Wavelength (nm) Figure 2.5. Measured absorption coefficient of fiber #5. 28 Fiber #5 Q l ^ 1 L _ 1 1 1400 1450 1500 1550 1600 1650 1700 Wavelength (nm) Figure 2.6. Measured fluorescence of fiber #5. The fluorescence spectrum is normalized with respect to its peak value. 29 1.2 F i b e r #5 A t ' Q I I III ^ I I I I I 1400 1450 1500 1550 1600 1650 Wavelength (nm) Figure 2.7. Comparison between the measured fluorescence (dashed) and calculated fluorescence (dotted). Both spectra are normalized with respect to their values at X k . 30 The saturation power at signal wavelengths, p (X) » defined in equation (B.17), is a function of the absorption and emission cross-sections, mode power radius, and fluorescence lifetime. But, due to the uncertainty which exists regarding the actual values of cross-sections, this parameter is commonly calculated by a technique first proposed by Saleh et. al. [49]. According to this technique, Psat(X) can be determined based on measured monochromatic absorption data using: Pout(V = PinW e e , (2.21) where Pin(X) is the power of a monochromatic beam at wavelength A into a piece of erbium-doped fiber, P0Ut(X) is the power of the beam at the output of the fiber, and L is the length of the fiber. To check the accuracy of the cross-sections calculated using our method, we calculated Psat(X) at A = 1530 nm and A = 1550 nm using equation (B.17), and com-pared those with the measured Psat(X) using the above technique proposed by Saleh et. al. (Details of the measurement procedure for obtaining Psat(X) are given in section 3.6). Table 2.2 gives the calculated and measured values of Psat(X) at the two wavelengths, together with the absolute value of the maximum discrepancy between them. 31 Table 2.2. Measured and calculated values of Psat(X) at wavelengths X = 1530 nm and X = 1550 nm. Wavelength (nm) Measured P s a t (mW) Calculated P s a t (mW) Discrepancy Between Measured and Calculated P s a t 1530 0.1767 0.1826 3.3% 1550 0.2921 0.3099 6.1% The close agreement between the measured and the calculated Psat(X) confirms the accuracy of the calculated cross-sections using our method. 2.5. Evaluation of the erbium concentration inside the fiber core The erbium density distribution, p(r) , in the core of the erbium-doped fibers is one of the parameters that greatly affects the performance of an optical amplifier. Theoretical studies, [16], [46], [50], have shown that a change in erbium density distribution would result in a change in gain, gain coefficient, and optimal length of an optical amplifier. Therefore, knowl-edge of this parameter is of great importance for accurate modeling of fiber amplifiers and is also important for improving the techniques used to design and manufacture erbium-doped fibers. 32 A direct measurement of the erbium concentration profile in the fiber core is a challenging task. This is due to the small size of the fiber core (typically having a diameter of about 4 \im). Previous attempts to determine this parameter, such as secondary ion mass spectrometry (SIMS) and electron probe microanalysis ( E P M A ) in conjunction with X-ray fluorescence ( X R F ) , have been based on measurements performed on slices of partially drawn, thick pre-forms [28], [51]. The accuracy of these measurements is estimated to be about 18% [28] and, as was explained in section 2.3, the final draw of the preform into the actual fiber w i l l cause redistribution of the erbium ions, resulting in additional errors. Recently D . Uttamchandani et. al. [21] have proposed a technique which allows direct determination of the erbium profile in the fiber core. In this technique short lengths of erbium-doped fibers (approximately 1 mm long) are pumped by a laser beam, and the generated fluorescence at the output of the fiber, across the fiber core, is imaged using a confocal optical microscope. Since the intensity of the generated fluorescence is proportional to the erbium concentration, the erbium distribution inside the fiber core can be determined. Although this technique allows the shape of the erbium distribution to be determined, it cannot provide the actual concentration. Here, we show that, once the erbium distribution is known, the erbium concentration inside the fiber core can be calculated using our closed-form expression, equation (2.13). We define, p 0 to be the peak value of p(r ) . Accordingly, equation (2.13) can be written as: 33 p 0 = Alien x dl , (2.22) r)rdr where p(r)/p0 , is the normalized erbium distribution, i.e., the shape of the erbium profile inside the fiber core. Since all of the parameters on the right hand side of equation (2.22) can be either measured or evaluated, by using this equation p0 can be determined. Summary of Chapter 2 In this chapter the theory of operation of single-mode erbium-doped fiber amplifiers pumped at 980 nm has been described. Based on the general rate equation, and McCumber's theory of phonon-terminated optical masers, two closed form expressions have been derived. In one of them, the fluorescence spectrum of an erbium-doped fiber is related to its spectral absorption coefficient. Based on this expression, a rigorous basis for the assessment of the 4 4 applicability of McCumber's theory to the study of / 1 3 / 2 ^15/2 transitions in erbium-doped fibers has been established. For the cases of five silica-based erbium-doped fibers, experiments were performed and the results were used to validate this expression. The other closed form expression, provides a simple means for calculating absorption and emission 34 cross-sections of erbium-doped fibers using the easily measured spectral absorption coeffi-cient, the gain coefficient at one particular wavelength, and the fluorescence lifetime. Also, based on this expression, an analytical method for the simple determination of the erbium ion concentration inside the fiber core has been proposed. Experiments were performed to evaluate the spectral cross-sections of an erbium-doped fiber over the wavelength range 1400-1650 nm, and the accuracy of the calculated cross-sections was ensured by comparing the measured values of saturation powers at 1530 nm and 1550 nm with the ones calculated using the values of the cross-sections. 3 5 Chapter 3 Experimental Techniques and Measured Results 3.1. Introduction In order to evaluate the absorption and emission cross-sections of an erbium-doped fiber at signal wavelengths, using the technique described in chapter 2, several fiber parameters such as the spectral absorption coefficient, as(X) , spectral fluorescence, F(k) , gain coefficient, gs(X) , and fluorescence lifetime, x, have to be measured. In the evaluation of these parameters, while as(k) and gs(X) were determined using the conventional cutback technique, for the measurement of spectral fluorescence, a new setup was designed to measure the fluorescence in the backward direction, as opposed to the for-ward direction which is the direction of pump light propagation. This technique provides a simpler, and more accurate means for the measurement of spectral fluorescence as compared with the conventional techniques by which the fluorescence is measured either in the forward direction or from the side of the fiber [7], [43], [44], [52]. Also, in determining the fluores-cence lifetime, x, unlike conventional techniques in which x is determined from the fluores-cence intensity measured in the forward direction [15], [44], [53] or from the side of the fiber [7], [54], we determined x using the fluorescence intensity measured in the backward direc-tion. In this way we have managed to simplify the experimental procedure for determining this parameter. 36 In order to examine the accuracy of the calculated cross-sections using our technique, as was described in section 2.4, the saturation power at signal wavelengths was determined by performing monochromatic absorption measurements. 3.2. Measurement of the spectral absorption coefficient The spectral absorption coefficient of erbium-doped fibers is an essential parameter used in numerical models for EDFAs [15]. It is also used to determine the location and the strength of the possible pump bands [43]. This parameter was measured by the standard cutback method [55]. A schematic of the experimental setup is illustrated in figure 3.1. The light source is a MRV Communications MREDSP5000 broad-band L E D , 1400-1700 nm, with an output power of 5 \iW, and having a single-mode fiber pigtail. In order to couple the light of the L E D into the erbium-doped fiber a fusion splice was used. To prepare the fibers for fusion splicing, the end of the LED's fiber and one end of an erbium-doped fiber, were soaked in trichloroethylene for about one minute to loosen their jackets. Afterwards about 6 centimeters of the jackets were removed from each of the fiber ends, the ends were cleaved using York FK11 fiber cleaver and, finally, the cleaved ends of the fibers were fusion spliced together using a Northern Telecom NT7L30AB video fusion splicer. To strip out any light that may have been launched into the cladding of the erbium-37 doped fiber, the regions of the doped fiber adjacent to the splice were covered with index-matching fluid index-matching fluid. To monitor the spectral output power, the other end of the erbium-doped fiber was con-nected to an Anritso MV02 optical spectrum analyzer (OSA), using an Anritso MA915A fiber adaptor and, to record the data, the OSA was connected via its GP-IB to an IBM PC compatible computer. OSA Computer Figure 3.1. Schematic of the experimental setup used for the measurement of the spectral absorption. To determine the fiber's absorption coefficient, the transmitted optical power PA(X), as a function of the optical wavelength, was measured for a given fiber length. The fiber was then cutback, by cutting away a length, L, of the fiber at its output end, and the corresponding transmitted optical power spectra PB(k) was measured again. The absorption coefficient, OLs(k) (in decibels per meter), was then calculated from the relation [7], [52]: 38 log cc,(X) = 10 " P f a ( X L . (3.1) To ensure the reproducibility and consistency of the results for each fiber, several cutbacks were performed. Typical absorption coefficient spectra are shown in figures 2.3 (a)-(d) and 2.5. Finally, in order to verify me validity of the assumption, that PS(X, z) ^ Psat(k) ,made in the derivation of equation (2.4), the remaining length of the doped fiber was cut and the spectral power, which in this case was the signal power at the input of the erbium-doped fiber (EDF), i.e., Ps(k, z =0) , was measured. For the resolution of the OSA set to 1 nm, the maxi-mum value of the spectral input power was -39 dBm = 0.126|iW, at the wavelength X = 1556 nm. Since the measured Psat(X = 1550 nm) is 0.2921 mW, see Table 2.2, the above assumption is valid. 3.3. Measurement of the fluorescence spectrum The fluorescence spectra provide information on the possibility of obtaining gain over a particular wavelength range of interest [7]. Furthermore, from the fluorescence spectrum and by measuring the gain coefficient at only one particular wavelength, the spectral gain coeffi-cient, an essential parameter in numerical models for EDFAs [15], can be easily evaluated 39 using our closed form expression derived in section 2.3, equation (2.20). Traditionally the fluorescence spectrum has been measured either in the forward direction, the direction of the pump light propagation, or from the side of the fiber [7], [43], [44], [52], [54]. In the forward detection technique [7], [43], [44], the fluorescence is measured at the output of a very short length of erbium-doped fiber (typically a length sufficiently short so that the peak of the small signal absorption is less than 0.2 dB), pumped with a laser operating at 980 nm having a typical output power of 100 mW. The length of the fiber is chosen to be short, in order to keep the total fiber gain small, thus avoiding distortion of the fluorescence profile due to the stimulated emission. Furthermore, high pumping powers allow full medium inver-sion to be achieved along the entire fiber length, avoiding distortion of the fluorescence profile 4 4 due to the absorption corresponding to the / 1 5 / 2 —> ^13/2 transitions of the erbium ions. However, since the fluorescence and the pump light are propagating in the same direction, when detecting the fluorescence the laser noise in the spectral range of the fluorescence profile is also being detected; as the power of the fluorescence is very weak, the laser noise may dis-tort its profile. In the side detection technique [7], [52], [54], the fluorescence is, instead, being detected from the side of an erbium-doped fiber, using a very sensitive detector. In this tech-nique, since the optical spectrum analyzer cannot be used to record the fluorescence spectrum, a somewhat complicated setup which includes a transimpedance amplifier, a lock-in amplifier, a monochromator, and data processing equipment are used [7]. In addition, since the mea-sured spectrum is normally distorted by the wavelength dependence of the detector and by the efficiency of the monochromator, subsequent corrections have to be made [7], making the pro-cedure even more complicated. 40 To avoid the problems associated with the above-mentioned techniques, a new setup was designed to measure the fluorescence in the backward direction. Figure 3.2 shows a schematic of the experimental setup used. In this experiment, the pump source was a Seastar Optics LD-2000 InGaAs-GaAs laser diode (LD) operating at 980 nm with 90 mW of output power, and had a connectorized single-mode fiber pigtail. The L D was connected to the 980 nm arm of a 2 x 1 , single-mode, 980/1550 nm Gould Electronics wavelength division multiplexer (WDM). The connection between the ends of the connectorized fibers of the LD and the 980 nm arm of the W D M was made by using an OZ Optics PMPC-03 fiber adaptor. A short piece of doped fiber, having a peak small signal absorption less than 0.2 dB, was fusion spliced to the 980/1550 nm arm of the W D M , its exit end was broken and immersed in an index-match-ing fluid to avoid reflections. The backward fluorescence, F*(k) , was then collected at the WDM's 1550 nm arm using the optical spectrum analyzer and was recorded using an IBM PC compatible computer attached to the OSA. By using this technique, we managed to eliminate the laser noise in the measurement of the fluorescence spectrum, as the laser light was propa-gating in the forward direction. Since the W D M normally has a non-uniform spectral response, accordingly the measured fluorescence had to be corrected to give the actual fluorescence spectrum, F ( A ) . Therefore, the spectral response of the W D M was measured over the wavelength range 1400-1700 nm. To measure the W D M spectral response, the output end of the broadband L E D , used in the absorption measurement, was fusion spliced to the 980/1550 nm arm of the W D M , and the transmitted fight, 7 \ (A) , was collected at the WDM's 1550 nm arm using the OSA and recorded using the attached computer. Separately, the spectral response of the LED, T2(X), 41 Fiber Connectors L D „ | | f you nm arm W D M 980/1550 nm arm \ , ,. 1550 nm arm '•I i \-Computer fusion splice EDF index matching fluid Figure 3.2. Schematic of the experimental setup used for the measurement of the fluorescence spectrum. was measured by directly connecting the LED's fiber end to the OSA. To eliminate the effect of coupling loss due to fusion splicing in the measurement of TX{X), both small-signal gain the measured spectra, i.e., TX(X) and T 2 ( A ) , were normalized with respect to their peak val-ues. Finally, the normalized spectral response of the W D M , TWDM, was evaluated by dividing Tj(A) by T 2 (A) and normalizing the result with respect to its peak value. Figure 3.3 shows the normalized spectral response of the W D M in the wavelength range 1400-1700 nm. By knowing TWDM, the actual shape of the fiber's fluorescence spectra, F(k) , was calculated from the measured fluorescence spectra, F*(A) , using the relationship F(A) = F*(A) I r w z ) M ( A ) . The normalized fluorescence spectrum was then obtained, by normalizing F(A) with respect to its peak value. Figure 2.6 shows a typical normalized fluorescence spectrum 42-measured in the wavelength range 1400-1700 nm. 1400 1450 1500 1550 1600 Wavelength (nm) 1650 1700 Figure 3.3. Spectral response of the W D M , normalized with respect to its peak value. 43 3.4. Measurement of the small-signal gain coefficient The small-signal gain coefficient, gs(k) , is one the essential parameters in the numerical models for E D F A s [15]. A s is the case for the absorption coefficient, this parameter is mea-sured using the standard cutback technique [8]. Figure 3.4 shows a schematic of the experi-mental setup used for the measurement of the gain coefficient. single-mode Fiber With Angled Tip Figure 3.4. Schematic of the experimental setup used for the measurement of the gain coefficient. To be specific, the experimental procedure for the measurement of the gain coefficient is 44 described using a particular erbium-doped fiber (EDF), fiber #5, as an example. This fiber had a cutoff wavelength of 850 nm, a core radius of 1.8 | im, and a numerical aperture of 0.18. The absorption coefficient spectrum and the fluorescence spectrum of this fiber are shown in Fig-ures 2.5, and 2.6, respectively. In this experiment, the signal source was an NOI tunable erbium-doped fiber laser (EDF laser) operating in the wavelength range 1520-1564 nm. To be able to adjust the level of the signal power, the EDF laser was connected to a JDS FTTEL adjustable attenuator. The other end of the attenuator was connected to the 1550 nm arm of a 2 x 1, single-mode, 980/1550 nm Gould Electronics, W D M . The pump source was the same laser diode used in the fluorescence measurement, and was connected to the 980 nm arm of the W D M . A length of the EDF, 3.081 m , was fusion spliced to the end of the 980/1550 nm arm of the W D M and, to strip out the light that may have been launched into the cladding of the EDF, the regions of the doped fiber adjacent to the splice were covered with index-matching fluid. Also, in order to avoid reflections, the exit end of the EDF was fusion spliced to one end of a standard single-mode o fiber having a connectorized, angled tip at its exit end (having an angle of 10 with respect to the fiber longitudinal axis). To monitor the output spectral power of the signal, the other end of the standard single-mode fiber was connected to the OSA. The experiment was begun by tuning the signal source, the EDF laser, to operate at the peak of the fiber fluorescence spectrum, X = 1530 nm (see Figure 2.6). As was explained in section 2.3, this minimizes the error introduced by uncertain values of fiber background loss ctb(k) or when ignoring ab(k) in the calculation of gain coefficient. To satisfy the assump-tions, Ps(k,z) « Psat(^ ^ d Ps(k,z) « Pp(k,z) , made in the derivation of equation 45 (2.2), the signal power had to be adjusted appropriately. In order to do so, the pump laser was turned on and, while the spectral power of the signal was being monitored using the OSA, the attenuator was adjusted such that the measured output signal power at A = 1530 nm was -23.5 dBm. By neglecting the background loss of the standard single-mode fiber (~ 0.2 dB/ km at A = 1550 nm), and the splice loss (typically 0.05 dB), we can write Ps(k, z=L=3.081m) = -23.5 dBm. This is 188 times larger than the saturation power of fiber #5, which is 0.1767mW=-0.75 dBm (see Table 2.2). Since Ps(X,z=L) is the maximum value of the Ps(k, z), it is clear that the assumption PS(X, z) ^ Psat(X) is valid. To mea-sure the pump power, the end of the standard single-mode fiber was disconnected from the OSA, and connected to a Newport 840 optical power meter. The pump power was measured to be 71 mW=18.75 dBm. Since this is the minimum value of Pp(Xp, z), and Ps(k, z=L) is the maximum value of Ps(A, z), the assumption Ps(k,z) ^ Pp(k,z) is clearly valid. The end of the standard single-mode fiber was reconnected to the OSA, and the spectral power of the output signal was monitored using the OSA. In Figure 3.5 this spectrum is shown by a plot designated as "output". From this spectrum, the signal power level was measured PA*(X) = -23.5 dBm. The spectrum also shows the level of the noise power around the sig-nal wavelength. The level of the noise power at the signal wavelength cannot be measured directly because the signal power level masks the noise level at this wavelength; instead it can be determined using the interpolation technique [56]. In this technique, the noise level at the signal wavelength is determined by measuring the noise level at wavelengths just above and just below the signal wavelength and then, by interpolation, the noise level at the signal wave-length is determined. Using this technique, the noise power level of the spectral output power, 46 designated as "output" in Figure 3.5, was measured to be Nout = -44.25 dBm. This amount of noise, which is the total output noise power within the optical spectrum analyzer's resolu-tion bandwidth at the signal wavelength, is related to the A S E power of the EDFA, P A S E , through [56], Nout = GainxNin + PASE , (3.2) where Nin is the noise power level at the input of the EDF. To obtain the actual signal power level, P A & ) ' Nout has to be subtracted from PA{X) , which gives PA{X) =4.43 [lW = -23.54 dBm. To calculate the gain coefficient, the EDF was cutback, by cutting away 1.084 m of the fiber at its output end, and the corresponding spectral output power was monitored using the OSA. The signal power level at X = 1530 nm was measured Pg(^ = -28.2 dBm, and the noise power level at this wavelength was measured -48.95 dBm; this gives the actual signal power of PB(X)=1.5 [iW = -28.24 dBm. The gain coefficient, gs(X) (in decibels per meter), was then calculated from the relation [7], [52]: gs(X) = 10 V L M , (3.3) giving g (k= 1530 nm) = 4.35 dB/m for fiber #5. 47 -10 -20 -30 co. I -50 -60 -70 -80 • • OUt[ \ )Ut / 1 r ~ i r i i n p u t j L _ 1527.5 1528 1528.5 1529 1529.5 1530 1530.5 1531 1531.5 1532 1532.5 Wavelength (nm) Figure 3.5. Spectral input signal power to the EDF and spectral output signal power of the EDF. To determine the signal power and the noise power at the input of the EDF, the remaining length of the EDF was cut, and the spectral power of the input signal was monitored using the OSA. In Figure 3.5 this spectrum is shown by a plot designated as "input". From this spec-trum, the signal power level at 1530 nm was measured, PQ (A.) - -36.63 dBm, and the level 4 8 of the noise, using the extrapolation technique, was measured Nin = -76.37 dBm. Therefore, the actual input power is equal to Pc(k = 1530 nm) = 0.22(1, W = -36.63 dBm. From the above results, the A S E power of the E D F A can be calculated, using equation (3.2), as P A S E = Nout ~ G a i n x W,M=3.76 x 10" 8 -18.97 x 2.31 x 10" 1 1 = 3.72 x 10~ 8 W = -44.29 dBm; this gives a ratio of 119 between the output signal power PA = Ps(k,z=L), and the total A S E power of the E D F A , PASE(X, z=L). From the above, and also from the fact that the signal to noise ratio improves for z < L, it is clear that the other assumption, Ps( A, z) ^ PASE(^> Z) > m a d e i n m e derivation of equation (2.2) is valid. Finally, the last assumption, Pp(Xp, z) ^ P s a A p ) » made in the derivation of equation (2.2) also had to be validated. The value of Psat(kp = 980 nm) can be estimated as follows. From the combination of equations (B.17), (B.18), (2.2), and (2.4) we can write: oo p(r)yCkp, r)rdr Psat(\) = <*A)+8A) A o Psat(X) a(Xp) Xp - • ^ ; \p(r)\\fs(X,r)rdr o Let us consider A = 1530 nm, then from the measurement of the absorption coefficient we have oCy(A=1530 nm) =4.43 dB/m =1.02 m 1 and from the measurement of the gain coeffi-cient we have g^(A=1530 nm) =4.35 dB/m=1.00 m _ 1 . Besides, a(Xp) was measured to be 3.6 dB/m = 0.83 m - 1 (measured by Mr. A . Croteau), using the same procedure as that described in section 3.2 except using an Anritso M G 9 2 2 A white light source as the signal source around 980 nm. The saturation power at signal wavelength,P5flI(A=1530 nm), is 49 0.1767 mW (see Table 2.2). From the fact that the mode power distribution shrinks as the wavelength of the light becomes shorter, it is obvious that the ratio of the overlap integrals on the right side of equation (3.4) is less than 1. Therefore, from the above, Psat(Xp =980 nm) < 0.67mW . Comparing this value with the lowest level of the pump power, the pump power at the output end of the E D F , which was measured to be 71 mW=18.75 dBm, clearly validates the assumption Pp(kp,z) » PSat^p^ • 3.5. Measurement of the fluorescence lifetime The fluorescence lifetime, T , of erbium ions in the metastable level, is another important parameter in numerical models for E D F A s [15]. This parameter can be measured by pumping a short length of erbium-doped fiber, and monitoring the decaying fluorescence intensity ver-sus time after the pump light is chopped by a mechanical chopper. Since the fluorescence intensity decays exponentially according to e x p ( - / / x ) , where t denotes time, x is simply the time interval over which the fluorescence intensity decays to 1 /e of its peak value. Traditionally, x is measured using a setup illustrated in Figure 3.6 [7]. 50 Chopped . pump light o Launching fiber Ellipsoidal mirror ^ E D F ^ Splice Hollow needle Data-processing equipment Detector 0 Tra impec pream ins-lance Lplifier Postamplifier Digital oscilloscope Figure 3.6. Traditional setup for the measurement of the fluorescence lifetime (form reference [7]). In this setup, the pump light is fed through one end of a launching fiber, and a short piece of E D F is spliced to the other end. The E D F is placed in an ellipsoidal mirror in such a way that a part of the E D F coincides with one of the focal points of the mirror and, to prevent the pump light from reaching the detector, which is positioned at the other focal point of the mir-ror, the end of the doped fiber is placed in a hollow needle. Using this configuration, the fluo-rescence from sides of the doped fiber is collimated on the detector. In order to simplify the experimental procedure, based on our new design for measuring fluorescence in the backward direction, the setup illustrated in Figure 3.7 was used to measure x. 51 Objective Lenses Figure 3.7. Schematic of the experimental setup used for the measurement of the fluorescence lifetime. In this experiment, the pump light was coupled into the 980 nm arm of the W D M , using two 20 x objective lenses positioned before and after an Alpha Omega Instruments mechani-cal chopper. A short piece of an erbium-doped fiber (the length was chosen such that the peak small signal absorption was less than 0.2 dB) was fusion spliced to the end of the 980/1550 nm arm of the W D M and, in order to avoid reflections, the other end of the E D F was broken and immersed in index-matching fluid. The backward fluorescence was collected from the 1550 nm arm of the W D M and was measured using a Germanium detector. To monitor the shape of the decaying fluorescence, the detector was connected to an H P 400Msa/s Digit izing Oscillo-52 scope, and the data was recorded using an I B M personal computer connected to the oscillo-scope. The value of x was obtained, by least-squares fitting of an exponentially decaying function to the measured fluorescence decay profile; x is simply the time constant of the fitted curve. Figure 3.8 shows the measured fluorescence decay curve and the fit used to obtain x ; for this fiber x was calculated to be 10.15 msec. 141 1 1 1 1 1 1 1 1 Time (ms) Figure 3.8. Fluorescence decay curve for the 2 cm long of the fiber #5 (circles). The solid line corresponds to the least-squares fit used to obtain x . 53 3.6. Measurement of the saturation power at signal wavelengths The saturation power at signal wavelengths, Psat(X) , is another important parameter in numerical models of E D F A s [49], [57]. A s was discussed in section (2.4), to check the accu-racy of the absorption and emission cross-sections calculated using our proposed technique, Psat(k) was determined experimentally at A = 1530 nm and A = 1550 n m , and the results were compared with the values of Psat(X) calculated from equation (B.17), see Table 2.2. The experimental values of Psat(k) were determined based on a technique proposed by Saleh et. al. [49]. In this technique, P.sat(k) is determined by performing a simple monochro-matic absorption measurement; the absorption data is then used for the evaluation of Psat(X) using: Pout&) = Pin(V e e (3-5) where Pin(X) is the power of a monochromatic beam at wavelength A at the input of a piece of erbium-doped fiber, Pout(X) is the power of the beam at the output of the fiber, and L is the length of the fiber. The experimental setup for the measurement of Psat(X) is illustrated in Figure 3.9. In this experiment the monochromatic beam at the wavelength A (in this case, A =1530 nm and 1550 nm) was provided by using the same tunable E D F laser used in the measurement of the gain coefficient. To be able to adjust the power level of the signal, the E D F laser was con-nected to one end of a JDS F I T E L adjustable attenuator, and the other end of the attenuator was connected to the input arm of a 1 x 2 single-mode 3 dB fused coupler (from Gould Elec-54 tronics). To monitor the signal power level, one of the output arms of the coupler was con-nected to a Newport 840 optical power meter; the other output arm of the coupler was fusion spliced to a length of erbium-doped fiber. To strip out any light that may have been launched into the cladding of the erbium-doped fiber, the regions of the doped fiber adjacent to the splice were covered with index-matching fluid. The signal power at the output end of the E D F , P0Ut(X) , was then measured using a Newport 818-IR S /N 746 detector attached to a Newport 835 optical power meter. Figure 3.9. Schematic of the experimental setup used for determining the saturation power. P0Ut(X) was measured over a range of input signal power levels, by a sequential adjust-ment of the attenuator. To keep track of these adjustments, the reading on the monitoring opti-55 cal power meter was recorded each time. Finally, the E D F was cut and PIN(X) was measured over the same range of input signal power levels as it was in the measurement of P0UT(X) (this was done by a sequential adjustment of the attenuator such that the readings in the moni-toring optical power meter remained the same as the ones recorded for the measurement of Pout(k)). Figures 3.10 and 3.11 show the experimental transmission characteristics, Pout(k)/Pin(k) , of 3.781 m of the fiber #5 at wavelengths of 1530 nm and 1550 nm as func-tions of input power, PIN(X) , respectively. Knowing the value of CLs(k) from the absorption coefficient measurement, PSAT{X) was obtained by solving the equation (3.5) in a least squares sense [73]. For fiber #5, the values of PSAT(X) , determined according to the above approach, at wavelengths 1530 nm and 1550 nm are listed in Table 2.2 under the measured Psat(k) . After obtaining Psat(k) , P0UT(X) was calculated from equation (3.5) and using the values of OLS{X) , PIN(X) , and L, at both wavelengths. The solid lines in Figures 3.10 and 3.11 correspond to the analytical solutions of equation (3.5), which, in fact, represent the least square fits to the experimental transmission characteristics. 56 Oi 1 1 1 1 1 1 1 r -| 3' 1 1 1 1 1 1 1 1 I -40 -35 -30 -25 -20 -15 -10 -5 0 5 Input Power, P p , (dBm) Figure 3.10. Experimental transmission characteristics (Pou/Pin) of 3.781 meters of fiber#5 at the wavelength 1530 nm (circles). The solid line corresponds to the analytical transmission characteristics where Pout is calculated from equation (3.5) using the values of oc^, Pin, L, and P s a t . 57 Figure 3.11. Experimental transmission characteristics (Pou/Pin) of 3.781 meters of fiber#5 at the wavelength 1550 nm (circles). The solid line corresponds to the analytical transmission characteristics where Pout is calculated from equation (3.5) using the values of as, Pin, L, and Psat. 58 Summary of Chapter 3 In this chapter problems and difficulties associated with the conventional techniques for the measurement of the spectral fluorescence and the fluorescence lifetime have been described, as well as the new setups which were designed to simplify the measurement of these parameters. Also, experimental procedures for the measurement of other fiber parame-ters, such as spectral absorption coefficient, gain coefficient, and signal saturation power, have been described. 59 Chapter 4 Summary, Conclusions, and Recommendations for Future Work 4.1. Summary and Conclusions The theory of operation of single-mode, erbium-doped fiber amplifiers pumped at 980 nm has been described. Details of the derivation of the general rate equation for the propagation of signal, pump, and spontaneous emission has been provided. Based on this rate equation and McCumber 's theory of phonon-terminated optical masers, a closed form expression relating the fluorescence spectra of a single-mode erbium-doped fiber to its spectral absorption coeffi-cient has been derived. Accordingly, a rigorous basis for the assessment of the applicability of 4 4 McCumber 's theory to the study of / 1 3 / 2 <=> ^15/2 transitions in erbium-doped fibers has been established. For the cases of five silica-based, erbium-doped fibers, experimental results were used to validate this closed form expression and, thereby, the applicability of M c C u m -4 4 ber's theory to the study of / 1 3 / 2 <=> ^15/2 transitions in silica-based, erbium-doped fibers. Previous attempts, made by several researchers, to determine the absorption and emission cross-sections of erbium-doped fibers have been described, and the reasons for their lack of success with regard to the accurate evaluation of these parameters have been explained. Bui ld-ing upon the application of McCumber 's theory to the general rate equation for the propaga-tion of signal, pump, and amplified spontaneous emission, an analytical expression has been 60 derived by which the absorption and emission cross-sections can be conveniently and accu-rately determined using the easily measured spectral absorption coefficient, the gain coeffi-cient at one particular wavelength, and the fluorescence lifetime. Experiments were performed to evaluate the spectral cross-sections of an erbium-doped fiber over the wavelength range 1400-1650 nm, and the accuracy of the calculated cross-sections was ensured by comparing the measured values of saturation powers at 1530 nm and 1550 nm with the ones calculated using the values of the cross-sections. Existing difficulties with regard to the accurate estimation of the erbium ion concentration inside fiber cores have been described, and an analytical method for its simple determination has been proposed. Problems and difficulties associated with the conventional techniques for the measurement of the spectral fluorescence and the fluorescence lifetime have been described, and new setups were designed to simplify the measurement of these parameters. Also , experimental proce-dures for the measurement of other fiber parameters, such as spectral absorption coefficient, gain coefficient, and signal saturation power, have been described. In conclusion our new techniques for the determination of absorption and emission cross-sections, erbium ion concentration inside the fiber core, evaluation of the normalized fluores-cence spectra from the spectral absorption coefficient, and measurement of the spectral fluo-rescence and fluorescence lifetime, can provide a much simpler and potentially more accurate means for characterizing erbium-doped fibers. This directly upgrades the accuracy of the numerical models commonly used for predicting the behavior and optimizing the performance of erbium-doped fiber amplifiers. 61 4.2. Recommendation for future research A s a first step towards continuing the work presented in this thesis, determination of the shape of the erbium distribution inside the fiber core using a recently proposed technique by D . Uttamchandani et. al. [21] is recommended. 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Krupke, "Radiative transition probabilities within the 4 / ground configuration of N d : Y A G , " IEEEJ. Quantum Electron, vol . 7, no. 4, p. 153, 1971. 4 4 [70] E . Desurvire and J. Simpson, "Evaluation of I\^/2 a n d ^ 1 3 / 2 Stark-level energies in erbium-doped alumino silicate glass fiber," Optics Lett, vol . 15, p. 547, 1990. [71] W. Barnes, R. Laming, P. Morkel , and E . Tarbox, "Absorption-emission cross-section ratio for Er^-doped fibers at 1.55 | i m , " in Proc. Conference on Lasers and Electro-Optics, C L E O ' 90, paper J T U A 3 , p. 50, Optical Society of America, Washington D C , 1990. [72] K . Dyhal , N . Bjerre, J. Pederson, and C . Larsen, "Spectroscopic properties of erbium-doped silica fibers and preforms," in proc. SPIE Conference on Fiber Laser Sources and Amplifiers, vol , 1171, p. 209, 1989. [73] M A T L A B User's Guide for Un ix Workstations, The Math Works, Inc., Natick, Mass, 1992. 68 Appendix A: Rate equation analysis of the erbium ion population A s was discussed in section 2.2, by choosing the 980 nm pump band, only the first three energy levels of erbium need to be considered. Here, in accordance with the literature, a three level rate equation model, which has been traditionally used for some lasers [58] (e.g., a Ruby laser), is adapted for modeling light amplification in erbium doped, single mode fibers. Figure A . 1 shows these three levels as well as all of the possible transitions between them. Here R is the absorption rate from level 1 to 3, corresponding to the 980 nm pump absorption. Wj2 and W2j are, respectively, the absorption rate and the stimulated emission rate between levels 1 and 2 in the 1520-1570 nm signal band. A21 represents the spontaneous decay rate from level 2 to level 1. For the case of erbium-silica glass, experimental measurements have shown that 4 4 the radiative quantum efficiency for the I13/2 —> 115/2 transition is 99.93% [42], [59]. Therefore, A21 can be assumed to be only the radiative spontaneous decay rate, with A 2 1 = 1 /x , where x , by definition, is the fluorescence lifetime [8]. A 3 2 and A31 are the two possible decay rates, for ions, from the pump level. Experimental studies for the case of 4 4 erbium-silica glass have shown that the / n / 2 — * ^13/2 transition is essentially nonradiative, 4 - l with a rate of 5.53 x 10 s . Furthermore, the total decay rates of A 3 1 + radiative decay rate of A 3 2 = 125 s'1 [60]. A s a result we can ignore the decay rate A 3 1 , and we can also assume that the transitions corresponding to the decay rate A 3 2 are totally nonradiative. 69 11/2 13/2 1 15/2 i 2 R i w 1 2 i w 2 1 A 2 1 1 A 3 1 flevel(3) 1 level(2) level (1) Figure A . l . Energy level diagram corresponding to the first three levels of erbium ions in the glass host, and all of the possible transitions between these levels. Now, we can write the rate equations corresponding to the population concentrations of erbium ions in the ground level Nl, metastable level N2, and pump level N3 as [8]: dNl dt = -RNl-Wl2Nl + W21N2 + A2lN2 , (A.1) dN2 dt = Wl2Nx-W2XN2-A2lN2 + A32N3 , (A.2) dN, -jl =RNl-A32N3 (A3) dN; If we consider the steady state regime of operation, i.e., —^• = 0 (i = 1, 2, 3), as well as dt the fact that A32 » R [8], then the population density of ions in the three levels can be written as: 70 i + w?1x : (A.4) 1 " P 1 + (R + Wu + W21)x ' (R + W12)x ^ = P X + ( R + wl2 + W2l)z ' i ( A , 5 ) and ^3 = *i - 0 , ' (A.6) A 3 2 • | where p = N1 + N2 + N3 is the erbium ion density. Equation (A:6) shows that the pump level population, N3, is approximately zero. This is due to the fast nonradiative decay rate of ions from the pump level to the metastable level ( A 3 2 ) . ! While , in the above analysis, each energy level is assumed to be nondegenerate, in reality such is not the case, and, as was mentioned in section 2.2, due to the Stark effect, each single energy level of the erbium splits into a manifold of g energy sublevels, where g is the total level degeneracy. Nevertheless, through a more complex rate equation model [8], it is shown that the nondegenerate assumption, of the main energy levels of erbium in glass hosts, still remains accurate. This is because of the fast thermalization effect which prevails within the Stark manifolds. The thermalization effect maintains a constant population distribution within the manifolds (Boltzman's distribution), which eventually makes it possible to consider each manifold as a single energy level. 71 i i Appendix B: General rate equation for the propagation of signal, pump, and amplified spontaneous emission in single mode erbium doped fibers. In Appendix A the steady state populations of erbium ions in the first three energy levels were shown. Now, here, by considering the effect of light confinement in an erbium doped, single mode fiber, and relating the fiber waveguide parameters to the transition rates of erbium ions between the three levels, a general rate equation for the propagation of signal, pump, and amplified spontaneous emission w i l l be derived. When a light signal at wavelength X with intensity Is (power per area) passes through an active medium of length dz, and atomic population densities of A 7 ^ , for the ground level, and N2, for the metastable level, the intensity change dls is given by [8], [61]: dls = ( o e ( W 2 - <*a(*W 7 , dz , (B . l ) where oe(X) is the emission cross section and Ga(X) is the absorption cross section, both at the signal wavelength A, . Now we consider the fact that the signal light is actually guided in a single mode fiber. If the signal power coupled into the mode is PS(X), and the mode power distribution at X is WS(X, r, 9 ) , then the light intensity distribution, IS(X, r, 9 ) , i n the fiber transverse plane is given as [8], VA> r, 9) IS(X, r, 9) = PS(X) —-^ , (B.2) \\\fs(X,r,Q)rdrdQ s where r is the radial coordinate, 9 is the azimutal coordinate, and S denotes that the integral should be taken over the entire transverse plane. 72 From equations (B.l) and (B.2), we can write: dPM dz = PS{X) j{ae(X)N2(X,r,Q)-aa(X)Nl(X,r,Q)}^s(l,r,Q)rdrdQ , (B.3) where \\fs is the normalized mode power at wavelength X defined as: V|/„(A, r, 0) ij? (A, r, 0) = - — ^ — . (B.4) Jv,(k» r, Q)rdrdQ Equation (B.3) represents the rate of signal power change, as light propagates through the fiber in the guided mode. In deriving the above rate equation, equation (B.3), it is assumed that all of the erbium ions present in the active medium at any point (z, r, 0) have identical cross sections. This is equivalent to assuming homogeneous broadening, i.e., every ion in the collec-tion of ions has the same center frequency and the same lineshape and frequency response, so that a signal applied to the transition has exactly the same effect on all the ions in the collec-tion [58]. In reality, such is not the case for laser systems in which the levels are split by the Stark effect. This is because the lower and upper energy levels are split into manifolds of gl and g2 sublevels. Therefore, the laser line corresponding to the two main levels, are made of the superposition of glxg2 possible laser transitions, where each laser transition between the two Stark manifolds has a different characteristic. Furthermore, variation of the crystalline electric field from site to site in the glass host causes a nonuniform Stark splitting effect. These effects induce inhomogeneous broadening of the laser line. However, in silica based erbium doped fibers the Stark sublevels have small energy gaps and therefore are strongly 73 coupled by the effect of thermalization [8], [62]. For the case of erbium-silica, the thermaliza-tion rates A^R and A^R, which correspond to upward excitation (absorption) and downward nonradiative relaxation (creation of a phonon) of ions between the Stark sublevels respec-tively, are on the order of 1 ps or less [8]. Since these thermalization rates generally overtake the pumping and stimulated emission rates, the condition of thermal equilibrium within the Stark manifolds is always maintained [8]. Therefore, since the relative populations within the Stark manifolds are unchanged by pumping or saturation, the spectral characteristics of 4 4 / 1 3 / 2 « ^15/2 transition cross sections also remain unchanged. A s a result the homoge-neous broadening approximation, made in the derivation of equation (B.3) for the case of commonly used silica based erbium doped fibers, is justifiable. In order to derive a general rate equation for the signal, the effect of amplified spontaneous emission (ASE) also has to be taken into account [8]. The generation of A S E is an effect of radiative spontaneous deexcitation of the erbium ions from the metastable level. The photons arising from this process have no coherence characteristics with respect to the incoming signal light. The generated photons are then multiplied as they travel along the amplifier, creating background noise which is added to the signal light. This background noise is called the amplified spontaneous emission. The rate of creation of spontaneous emission power coupled into the fiber mode, and in the direction of the incoming signal (here the positive z direction) in the frequency interval between v and v + Av ( A v represents a frequency slot of arbi-trary width) is given by [8], [63], [64]: 74 dP™W = 2 P 0 ( v ) a e ( v)JiV 2 ( v , r, 6)vj7J(v, r, Q)rdrdQ , (B.5) dz s where P0(v) = hvAv is defined as the equivalent input noise power corresponding to one photon per mode in a bandwidth of Av . Using the relation |Av| = \c AX / X2\, we can write P0(X) = hc2AX/X3 . Now, using equations (B.3) and (B.5) we can express the evolution of the total signal power, including the effect of added ASE, at wavelength X and in the wavelength interval Ak , by: dPAX) f ' =1 {a e(A)iV 2(A ,r,0)[P J(A) + 2P o(A)] dz ,' L e v -aa(X)Nl(X,r,Q)PsiX)}ws(Kr,B)rdrdQ , (B.6) where Pt(X) = PS(X) + PSE(X) . By using the radial symmetry property of the fundamental mode in the fiber, and also assuming radial symmetry for the erbium density distribution inside the fiber core, equation (B.6) can be simplified as: d P ' ( X ) =2K j {Ge(X)N2(X,r)[Ps(X) + 2P0(X)] (B.7) dz 0 -c^XW^X, r)Ps(X)}ys(X, r)rdr . Equation (B.7) is expressed in terms of steady state atomic populations Nx and N2, which cannot be determined experimentally. In the following procedure these population densities 75 w i l l be related to the experimentally measurable signal and pump powers coupled into the fiber mode. For the case of single mode fibers, a mode power radius (£>s, corresponding to the mode power distribution \\fs, can be defined through [65], - y (k, r, Q)rdrdQ 1 (B.8) B y using the radial symmetry of the fundamental mode, we can write: r)rdr (B.9) where it is assumed that \\fs(X, r = 0) = 1 . Therefore, equation (B.4) can be written as: (B.10) Now, from equations (B.2) and (B. 10), the signal intensity at fiber coordinate z and distance r from the core axis, ISCK, r, z), can be written as: Ps(X,z)Vs(kr) T(k,r,z) = ( B . l l ) where Ps(k, z) is the signal power at wavelength X coupled into the fiber mode. The proportionality between IS(X, r, z) and the stimulated emission rate W21(X, r, z) is given by [8], [61]: 76 I3 W2l(X,r,z) = \ Is{X,r,z)^{X) , (B.12) 8 Tin h cz where h is Plank's constant, n is the refractive index of the fiber core, and %(A) is the line-shape function defined as: 8 %n x aJX) = r - * ^ . (B,13) A From equations (B.11)-(B.13) we can write: A \ | / t (A, r) (T (A) W 2 1 ( A , r , z ) = — ^ P , ( A , z ) . (B.14) 7t n c C O , (A) Using the relationship W2l = — Wl2 [8], and equation (B.14) we can also write: A \ | / r (A, r) aJX) Wl2(X,r,z) = — j ^ — Ps{X,z) . (B.15) Tt h c co/A) Likewise, for the pumping rate R(Xp, r,z) a definition similar to equation (B.15) can be introduced [8]: L \\f (X„, r) cJX„) R(Xp, r, z) = P V P 2 } a K p ) Pp(Xp, z) , (B.16) Khc(ap(Xp) where Ga(Xp) is the pump absorption cross section at wavelength Xp, Pp(Xp,z) is the pump power at wavelength Xp coupled into the fundamental fiber mode, MJp(r) the mode power distribution at wavelength Xp, and 0)p is the mode power radius at frequency Xp defined similarly to equation (B.9). The saturation power at signal wavelength X is defined by [8]: 77 2 hcTt (iiAX) p^--i^rt(xm- (B17) Likewise, the saturation power at pump frequency, considering the fact that ce(Xp) = 0 for 980 nm pumping, can be written as: 2 rtC7tQ)n(A_) W - • CB..8) Using equations (B.14)-(B.18), we can write the steady state atomic populations Nx and N2 expressed in equations (A.4) and (A.5) as functions of optical signal and pump powers: Ge(X)ys(X, r) Ps(X,z) . v a ^ nr ^ + ae(X) + Ga(X)' Psat(X) N I ( X ' R ' Z ) = P ( 0 , P,frrz) a -P,(Kz) a ^ ' ( B - 1 9 ) Ppt\,z) Ga(X)ys{X, r) Ps{X,z) Psat(Xp) V P { V r ) + ce(X) + ca(X) ' Psat{X) n ^ = ^ ; , ( x„<) • Pl(L) :\ • (B-20) Hence, the general rate equation for the propagation of signal, pump, and A S E (equation (B.7)) combined with equations (B.10),(B.19), and (B.20) takes the form: 78 J dPt(X,z) 2 I p ( r ) L ( X ) ^i + 2^ [ f (x,z) + 2P 0(X)] (B.21) Z <»;(X) J I +<=l + <=3 0 c2 = 2 o f l(\) + o,(X) P,fl,(X) C4 = a f l(X) + ce(l) P,atCk) 79 Appendix C: Previous attempts at the determination of cross sections A s can be inferred from the general rate equation for the propagation of signal, pump, and A S E , equation (B.21), the gain and noise performance of the three level erbium doped fiber amplifiers are quite sensitive to the values of absorption and emission cross sections at signal wavelengths. Therefore, knowledge of these parameters is essential to modeling erbium doped fiber amplifiers for predicting signal gain and amplified spontaneous emission power. How-ever, despite extensive efforts placed on the evaluation of cross sections by several research-ers, relatively large uncertainty still exists throughout the literature regarding their actual values [19], [24], [28], [66], [67]. One of the principal approaches for the determination of cross sections is based on the Futchbacher-Ladenberg (FL) relationships [23]. However, several researchers have reported 4 4 its failure for the / 1 3 / 2 <=> / 1 5 / 2 transitions in erbium-glasses [19], [24], [25]. Generalized to the case in which the lower level and upper level are split into multiple sub-levels, the F L relations take the form [8],[44]: o.(X)= . (CD » 2 K n c x F(k)dk band C a ( 3 l ) = £ 2 — , ( C 2 ) L(X)ak K nc x J" a( band where F(X) is the spectral fluorescence over the 4 / 1 3 / 2 —> 4 ^ i s / 2 emission band, a(k) is 80 the absorption spectrum over the / 1 3 / 2 —> ^15/2 absorption band, gx is the degeneracy of the ground level, and g2 is the degeneracy of the metastable level. The integrals in equations (C l ) and (C.2) are taken over the entire emission and absorption bands, respectively. The F L relations which are derived from Einstein's A and B coefficients [68] can be applied only to cases where the populations of all the Stark sublevels are equal [69]. This condition is met only when the total Stark splitting energy, AE, is less than kBT, where kB is the Boltzman constant and T is the absolute temperature [8]. However, experimental evidence shows that 4 4 the Stark splitting for the 1 1 5 / 2 and / 1 3 / 2 states of erbium doped glasses are typically 300-400 c m - 1 [25], [70], nearly twice kBT ~ 200 c m - 1 at room temperature; this suggests that the F L relations are inaccurate to some extent [8]. The inaccuracy of the F L relations was actually confirmed experimentally, [11], [19], [24], [71], as it was found that the measured peak cross sections r\peak = Ge(kpeak)/Ga(kpeak) = 0.84 - 0.94 , while the F L relations predicts val-ues of r \ p e a k = 1.17-1.28. For the case of commonly used Alumino-Phosphorous-silica fibers (Al/P -silica fiber) this discrepancy is estimated to be even greater than 50% [19]. Finally, in view of the above results, the Fuchtbauer-Ladenberg approach to the determination 4 4 of the cross sections of / 1 3 / 2 <=> ^15/2 transitions in erbium-glass looks flawed. Alternatively, in 1991 Miniscalco and Quimby [19], [26], suggested that McCumber's the-ory of phonon terminated optical masers [27] could be used to determine either the absorption or emission cross sections if the other is known. In the first step they argued that McCumber's theory, in which a simple model is used to describe the operating properties of the insulating 4 4 crystals doped with impurities, is appropriate for the study of the / 1 3 / 2 / 1 5 / 2 transitions of erbium doped glasses. In the model used in McCumber's theory, the energy levels of the 81 impurity ions are grouped into two manifolds, where each manifold is split into multiple sub-4 4 levels; this is identical to the / 1 5 / 2 and / 1 3 / 2 levels of erbium in a glass host. Furthermore, it is assumed that the nonradiative transition rate between the two manifolds is negligible compared with the radiative transition rate, and the time required to establish a thermal equi-librium within each manifold is very short compared to the spontaneous emission lifetime (spontaneous emission lifetime = 1 / radiative transition rate). These assumptions are satisfied 4 4 for the case of erbium-glass, as the / 1 3 / 2 —> ^15/2 transition is essentially radiative and the fluorescence lifetime of the metastable level is very long (around 10 ms). Accordingly, they concluded that McCumber 's two level model is, in fact, ideally appropriate for the study of 4 4 7 1 5 / 2 and / 1 3 / 2 levels of erbium in glass hosts. In the next step, based on McCumber 's the-ory, they attempted to determine the values of emission cross sections for several types of erbium doped fibers. In their approach, the absorption cross section oa(X) was determined from measurements of the absorption coefficient, the erbium density distribution, and the opti-cal mode field power distribution [28]. The emission cross section was then calculated from the measured absorption cross section using the McCumber relation given by [19],[27]: ae(X)= aa(X)exp / hc\ (C.3) [kBT j where e is the net free energy required to excite one erbium ion from the I 4 7 1 3 / 2 at temperature T. This parameter is defined by [19],[27]: 15/2 level to e = kBT l n 7 T > (C.4) 82 where Nl/N2 is the equilibrium population ratio between the two levels at temperature T in the absence of optical pumping. If the Stark sublevel energies in the ground level are labeled E l m (m = 1 , . . . , g^ and in the metastable level are labeled E2n (n = 1 , . . . , g 2) , for the case of erbium ions where gx = 8 and g2 = 1, e can be calculated according to [8],[19]: where AE2l is the separation between the lowest component of each manifold, and A £ l m , AE2n are the energy differences between Stark sublevels with respect to the lowest energy level in the corresponding manifold. The evaluation of Stark sublevel energies is very difficult and often inaccurate. In addition, generally the positions of all the Stark sublevels are not known [19],[70]. This induces inaccu-racy in the calculation of e, and consequently the emission cross section. For the case of an Al/P-silica fiber the discrepancy between the measured peak value of the emission cross sec-tion and that calculated from equation (C.3) from the measured absorption cross section, and using an estimated value for the e, was 28% [19]. This discrepancy for the other types of glasses varies from 1.4% for a fluorophosphate glass to 16% for a silicate glass [19]. In cases where e was not known, they measured the absorption cross sections over the whole spectral absorption band. The measured absorption cross sections were then transformed into relative emission cross section spectra using equation (C.3). To obtain the absolute emission cross sec-tion, this relative spectra were then scaled by using the following expression from McCum-( e = kr,T In (C5) 83 ber's theory [19], [27]: . 1 2 f ^ ( X ) - = Sncn dk , (C.6) x J. X 4 band where x is the fluorescence lifetime and n is the refractive index of the core. In the above approach, although the requirement of evaluating e is eliminated, there is still no guarantee that the calculated emission cross section spectrum is accurate. This could be attributed to errors introduced into the calculation from the uncertain values of the absorption cross sections. As was mentioned earlier, absorption cross sections have been determined based on the measured values of the erbium dopant concentration and its distribution within the fiber core [28]. Since the diameter of the actual fiber core is very small, this measurement is performed on slices of the fiber preform using secondary ion mass spectrometry (SIMS), where the accuracy of the measurement is estimated to be about 18% [28]. In addition to this error the final draw of the preform, into fiber, may create considerable changes in the erbium density distribution; this is attributable to changes in glass network structure that occur during preform collapse and fiberization, which are produced by various effects of phase transitions, codopant migration or diffusion, and erbium site rearrangements [72]. In order to assess the accuracy of the calculated cross sections using McCumber's theory, for several glasses, comparisons were made between the measured emission cross sections with that calculated from the absorption cross sections and the fluorescence lifetime using equation (C.6) [8], [19]. While excellent agreement was obtained over the wavelength range 1450-1650 nm, for the case of a fluorophosphate glass [19], large discrepancies (e.g., as much 8 4 as 13% to 25% in the wavelength range of 1520-1560 nm) were observed for the cases of sev-eral silica based erbium doped fibers [8]. Due to these rather large discrepancies, uncertainty still remains regarding the applicability of McCumber's theory for the study of 4 4 112/2 hs/2 transitions in erbium doped fibers. In Section 2.3 of this thesis, a new approach is presented, by which the apphcability of McCumber's theory to the study of silica based erbium doped fibers is validated. 85 

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