The Gain Performance of Erbium-Doped Fiber Amplifiers

Transcription

1 Republic of Iraq Ministry of Higher Education and Scientific Research Thi-Qar University College of Science Physics Department ج ك C H The Gain Performance of Erbium-Doped Fiber Amplifiers A Thesis Submitted to the Council of College of Science / Thi-Qar University in Partial Fulfillment of the Requirement of Master Degree in Physics By Muhammed Rahma Harb B.Sc. 013 Supervised by Prof. Dr. Hassan Abid Yasser Dr. Sadiq. J. Kadhim 015 A.D 1436 A.H

2 بسم ا الرحمن الرحيم ا ن ور ال سم او ات و الا ر ض م ث ل ن ور ه ك م ش ك اة ف يه ا م ص ب اح ال م ص ب اح في ز ج اج ة ال زج اج ة ك ا ن ه ا ك و ك ب د ر ي ي وق د م ن ش ج ر ة مب ار ك ة ز ي ت ون ة لا ش ر ق ي ة و لا غ ر ب ي ة ي ك اد ز ي ت ه ا ي ض يء و ل و لم تم س س ه ن ار ن ور ع ل ى ن ور ي ه د ي ا ل ن ور ه م ن ي ش اء و ي ض ر ب ا الا م ث ال ل لن اس ا و ب ك ل ش ي ء ع ل يم صدق ا العلي العظيم سورة النور (الا ية ٣٥) (٣٥) i

3 Supervisor Certification We certify that the preparation of this thesis entitled '' The gain performance of erbium-doped fiber amplifiers '' was prepared by (Muhammed Rahma Harb) under our supervision in the department of physics, College of Science University of Thi-Qar, as a partial fulfillment of the requirements of the degree of Master of Science in Physics. Signature : Name : Dr. Hassan A. Yasser Title : Professor Address : College of Science/ Thi- Qar University. Date : / /015 (Supervisor) Signature : Name : Dr. Sadiq. J. Kadhim Title : Address : College of Pharmacy / Al- Nahrain University. Date : / /015 (Supervisor) In view of the available recommendation, we forward this thesis for debate by the Examining Committee. Signature : Name : Dr. Emad A. Salman Title : Assistant Professor Address : Head of the Department of Physics Date : / /015 ii

4 Examining Committee Certificate We, the examining committee certify that this thesis entitled '' The Gain Performance of Erbium-Doped Fiber Amplifiers '' and examining the student (Muhammed Rahma Harb) in its content and that, in our opinion, it meet the standards of a thesis for degree of Master of Science in Physics with excellent degree Signature : Name : Dr. Ahmed H. Flayyih Title : Assistant Professor Address : College of Science/ Thi- Qar University. Date : / /015 (Member) Signature : Name : Dr. Hussein H. Warid Title : Assistant Professor Address : College of Science/ Thi- Qar University. Date : / /015 (Member) Signature : Name : Dr. Hassan A. Yasser Title : Professor Address : College of Science/ Thi- Qar University. Date : / /015 (Supervisor) Signature : Name : Dr. Sadiq. J. Kadhim Title :Lecturer Address : College of Pharmacy / Al- Nahrain University Date : / /015 (Supervisor) Signature : Name : Dr. Hashim Ali Yusr Title : Assistant Professor Address : College of Science / Wasit University. Date : / /015 (Chairman) Approved by the Council of the College of Science/ Thi-Qar University Signature : Name : Dr. Mohammed A. Auda Title : Professor Address : Dean of the College of Science / Thi-Qar University Date : / /015 iii

5 Dedication To : My family : Thank you for your unconditional support with my studies. Thank you for giving me a chance to prove and improve myself through all my walks of life. Iraqi Army and heroes of People's Mobilization (al- Hashd al-shaabi) iv Muhammed 015

6 Acknowledgments Praise be to Allah. Lord of the worlds, and best prayers and peace be unto his last messenger Mohammed, and his pure descendants, and his noble companions. I would like to express my deep sense of gratitude to my supervisors Prof. Dr. Hassan A. Yasser and Dr. Sadiq. J. Kadhim for their valuable advices, and for being helpful all the time, for the professional way in guiding me in this work, for suggesting the topic, and for their assistance throughout the course of this work. I would like to thank all the physics faculty members who stand behind the master program in this college, and who support the students in their study and research. Special thanks are to my family, for their support and patience. Thanks are also to the each person help me. Muhammed 015 v

7 List of Acronyms ASE EDFA FOPAS FP-SOA FWHM FWM HE,EH HNLF LED LP MCVD NF ZDSF OVD PCVD SBS SNR SOA SPM SRS TDM TM TE TW-SOA VAD WDM XPM Amplified pontaneous emission Erbium doped fiber amplifier Fiber-optic parametric amplifier Fabry-perot SOA Full width at half maximum Four-wave mixing Hybrid modes Highly nonlinear fiber Light-emitting diodes Linear polarization Modified chemical vapor deposition Noise figure Zero dispersion shifted fiber Outside vapor deposition Plasma chemical vapor deposition Stimulated Brillouin scattering Signal-to-noise ratio Semiconductor optical amplifier Self-phase modulation Stimulated Raman scattering time division multiplexing Transverse magnetic Transverse electric Traveling-wave SOA Vapor axial deposition Wavelength division multiplexing Cross-phase modulation vi

8 List of Symbols Symbol A 1 Spontaneous decay rate Definition Aeff A j a CR D DM DW E E(T ) E e f (R) G Go g() go g1 g H h I K L M Mw m Effective cross-sectional area Strength of jth resonance Fiber core radius Constant rayleigh scattering Mass density of the doped material Material dispersion Waveguide dispersion Electric field Excitation energy Optical intensity Charge of the electron Profile function Amplifier gain Peak value of the amplifier gain Gain coefficient Peak value of the unsaturated gain Degeneracy of the lower atomic state Degeneracy of upper atomic state Magnetic field Plank's constant Average current Boltzmann constant Fiber length Total number of modes Molecular weight radial mode number vii

9 N NA N N N t 1 ~ n(, E n g n sp n(r) no n 1 n P PASE PASE Pout sat Pout Pp Ps q R1 r S sp T T U V W W1 W1 ) Number of amplifier sections Numerical aperture Total erbium-ion density Population density of the ground state Population density of the metastable level Nonlinear refraction Group refractive index Spontaneous emission factor Refractive index distribution Air refractive index Refractive index in core Refractive index in cladding Total polarization Forward spontaneous emission power Backward spontaneous emission power Output powers of the signal being amplified Output saturation power of the signal Pump power Signal power Graded order Pump rate Radial distance Spectral density of spontaneous emission noise Absolute temperature Dipole relaxation time Normalized propagation constant in core Normalized frequency Normalized propagation constant in cladding Stimulated absorption rate of the ground state Stimulated emission rate of the metastable state viii

10 w w j z p R s p s A f g g o ղ a 1 ZD pa sa se shot sig sp Spot size Resonance frequency Distance Attenuation coefficient Absorption coefficient of pump Rayleigh scattering Absorption coefficient of signal Propagation constant Nonlinear effects Confinement factors of the pump Confinement factors of the signal Relative core-cladding index difference Amplifier frequency bandwidth Detector frequency bandwidth Gain angular frequency bandwidth Gain bandwidth Vacuum permittivity Quantum efficiency of detector Acceptance angle of fiber Light ray angle incident on the fiber core wavelength Zero-dispersion wavelength Pump absorption cross section Signal absorption cross sections Signal emission cross sections Mean-square shot noise current Signal-ASE beat noise ix

11 sp sp th spon o c NL () (3) ASE-ASE beat noise Mean-square thermal noise current Fluorescence lifetime Frequency Atomic transition frequency Angle at core-cladding interface Critical angle Nonlinear phase Second-order susceptibility Third-order susceptibility Transverse field component either E-field or H -field Angular frequency Laplacian operator Responsively of a photodetector Azimuthal mode number x

12 Table of Contents Section address Pages Dedication Acknowledgments v List of acronyms vi List of symbols vii Table of contents xi Abstract xiii CHAPTER ONE Introduction 1.1 Optical communication 1 1. Wavelength division multiplexing Optical amplifier Erbium doped fiber amplifier Semiconductor optical amplifier Raman amplifier Brillouin fiber amplifier Fiber-optic parametric amplifier Literature survey Aim of thesis Organization of thesis 14 CHAPTER TWO Basic Requirements for Fiber and Erbium Doped Fiber Amplifier.1 Introduction 15. Step-index fiber 16.3 Graded-index fiber 17.4 Fiber materials Glass fiber Plastic-clad silica core fibers Plastic fibers 0.5 Manufacturing of optical fiber 0.6 Ray transmission view.9 Fiber modes 3.8 Losses Attenuation Absorption losses Radiative losses 8.8. Bending losses Losses in fiber joins 9.9 Dispersion 30 xi iv

13 .10 Rare-earth-doped fiber Erbium concentration Fiber nonlinearities Nonlinear refraction Stimulated inelastic scattering 36.1 Fundamental of Erbium doped fiber amplifier Configurations of erbium doped fiber amplifier Operating principles for erbium doped fiber amplifier 38 CHAPTER THREE Theoretical Analysis of Erbium Doped Fiber Amplifier 3.1 Introduction Pumping and gain coefficient Gain spectrum and bandwidth Noise figure Gain saturation Rate equations Propagation equations Present analytical solution Confinement factor of graded-index fibers 58 CHAPTER FOUR Results and Discussion 4.1 Introduction The effect of confinement factor Graded order effects Gain of EDFA Optimization of EDFA The optimum length Amplified spontaneous emission 77 CHAPTER FIVE Conclusions and Future Works 5.1 Conclusions 9 5. Future works 93 Appendix A: Atomic term symbols 94 References 97 xii

14 Abstract The transmitted pulses experience an increase attenuation by increasing the transmission distance in optical communication system. So they need to be amplified to return to its initial power. The amplification process is performed using different types of optical amplifiers each with special advantages depends on physical medium and properties of the propagated pulses. In general, the purpose of all amplifiers is to achieve higher gain, maximum bandwidth, and better flatness. Erbium doped fiber amplifier (EDFA) is the best in terms of adapted to it is fully constituted of optical fiber. The core of any optical fiber from any type can be doped into erbium to get amplifier. As long as the work in optical communication requires a single mode fiber, we will restrict ourselves by using this type. Since the fiber can be graded or step-index, we analyzed the general situation to access to the best properties of the amplifier. In order to study EDFA amplification, the characteristics of the propagated mode (fundamental mode ) must be specified, where the Maxwell's equations in the core and cladding regions are solved to graded fiber and step-index fiber using power series method. The results proved that the pulse form in the core region follows the graded type. The well-known case, (Bessel's functions) may be presented if the fiber is step-index and the case (Laguerre functions) is presented for the graded order q, while all other cases do not give a specific closed form. An analytical solution to the equations of propagated pulses depends on the division of optical fiber to many concatenated sections was presented. The properties of the modes in the general situation of graded type included with this solution in order to study the properties of amplifier. In general, numerical results perfectly match the suggested model. Results proved that the step-index fiber shows more acceptable properties in comparison with graded index fibers for use in amplifier. In general, EDFA achieves higher gain and large gain bandwidth, but it suffers in a certain band xiii

15 from poor flatness. This property is inherent in the amplifier and cannot be canceled by controlling the operation circumstances. xiv

16 CHAPTER ONE Introduction

17 CHAPTER ONE Introduction 1.1 Optical Communication The physical phenomenon used to transfer the light in the optical fiber is the total internal reflection of the light inside the fiber [1]. In the past, the communication was accomplished primarily by using copper wires. In the present day, however, fibers are now much more often installed than copper wires []. Optical fiber is a medium for carrying information from one point to another in the form of light. Information revolution implies that the information can be presented, analyzed, transported in an efficient manner. The presentation and analysis of information is achieved by using the facilities of the computers, while the transportation of information is achieved using communication networks. The information moving over the networks can be video, voice information, data information or text[3]. Optical fiber provides low loss communication links as compared to radio or electrical cables [3]. In optical fiber communication there are many advantages such as high bandwidth ( 30THz ), the bandwidth increases exponentially at about 60% per year as in Fig.(1.1), low loss, and no electromagnetic interference, because the connection is not electrical, you can neither pick up nor create electrical interference (the major source of noise). As the transmission of information is through dielectric media, transmitting and receiving ends are electrically isolated. Optical links are more reliable and can support future applications due to inherently large available capacity [4]. Fiber technology is still in its beginning. Using just a single channel per fiber, researchers have trial systems in operation that communicate at speeds of 100Gbit per second. By sending many signals having different frequencies (channels) using wavelength division multiplexing (WDM) on a single fiber, this capacity can be increased to hundreds or perhaps thousands times. Dense WDM 1

18 and erbium doped fiber amplifier (EDFA) have been used successfully to further increase data rates to beyond a terabit per second excess of 100 km [5]. ( 100Gb/ sec) over distances in Fig.(1.1): Increase in bit rate-distance produce BL during the years [6]. A basic optical communication system consists of a transmitting, communication channel that carries the light, and receiver as in Fig(1.). Fig.(1.): Basic optical communication system. The application of optical fiber communications is generally possible in any area that requires transfer of information from one place to another. However, fiber-optic communication systems have been developed mostly for telecommunications application [6]. The telecommunication application can be broadly classified into two categories, short haul (i.e., local area network), and long-haul (i.e., long-distance telephone or cable television trunking), depending on whether the optical signal is transmitted over relatively long or short distances

19 compared with typical inter-city distances (~ 100km ). Optical Transmitters used to convert the electrical signal into optical form and launch the resulting optical signal into the optical fiber [6]. Fig.(1.3) shows the block diagram of an optical transmitter. It consists of an optical source, a modulator and a channel coupler. The optical signal is generated by modulating the optical wave. The semiconductor lasers or light-emitting diodes (LED) are used as optical sources because of their compatibility with the optical fiber communication channel. Fig.(1.3): Components of an optical transmitter[6]. 1. Division Multiplexing Systems The fiber material has a large bandwidth of 30 THz. If only one signal of 10MHz is used, than effectively it is the wastage of bandwidth. To use bandwidth effectively, different techniques are found like time division multiplexing (TDM) and WDM. But it is difficult to multiplex signal in time domain as it is very difficult to generate signals of femtoseconds. So WDM is the best technique that can be used to multiplex signals. Several techniques of achieving this exist, such as using a prism as a wavelength dispersion element or using a multilayer dielectric filter as a wavelength selective beam splitter. The devices that perform multiplexing and demultiplexing are commonly called couplers [7,8]. WDM is a modern practical method of increasing transmission capacity in fiber communication system [9,10]. In this technology, multiple wavelengths of light are modulated separately and sent into the fiber simultaneously. As long as the power within each signal is not too high, the fiber acts as a linear medium, the 3

20 interaction of different wavelengths with each other will be negligible, and each wavelength propagates in the fiber independent of the others [11]. Implementation of a typical WDM system employing N channels is shown in Fig.(1.4) [1], the desired number of lasers emitting a different wavelength, are multiplexed together by a wavelength multiplexer into the same high-bandwidth fiber. Each of N different-wavelength lasers is operating at the slower gigabits per second speeds. After being transmitted through a high-bandwidth optical fiber, the combined optical signals must be demultiplexed by wavelength demultiplexer at the receiver which distribute the total optical power to N output ports [13]. Fig.(1.4):Diagram of simple WDM system [1]. 1.3 Optical Amplifier Optical fiber have become an unavoidable part of any high speed communication system due to its high information carrying capacity, high bandwidth and extremely low loss. Losses are relatively small, especially in the wavelength region near 1.55 m ( 0.dB/ km). For this reason, losses can simply be ignored if fiber length is 1km or less. In the case of long-haul fiber optic communication systems, transmission distances may exceed thousands of kilometers. The transmission performance of the optical communication systems is limited by the various effects such as attenuation, dispersion, non-linearity, scattering...etc. To compensate for all these limitations, the signals have to be 4

21 regenerated within the transmission link. The losses can be compensated by using optoelectronic repeaters or optical amplifiers [14]. Using optical amplifiers in a fiber link has many advantages [10]: 1) the information-carrying signals are directly amplified in the optical domain without conversion to the electrical domain. ) optical amplifiers can be easily spliced to the telecommunication fiber link with minimal insertion losses. 3) since EDFAs have a large-gain bandwidth ( i.e., they can provide gain over a large spectral bandwidth ~ 40nm). 4) the noise added by the amplifier is close to the lowest possible level ( 3 4dB). 5) the gain provided by EDFA is polarization insensitive. Optical amplifiers can be used at many points in a communication link. Fig.(1.5) shows some typical examples. Booster amplifier is placed immediately after an optical transmitter to boost the transmitted power. This serves to increase the transmission distance by ( 10100km) depending on the amplifier gain and fiber loss [15]. They are designed to provide low optical output power levels for stability purposes [16]. In-line amplifier are used intermediate point in the link to overcome attenuation and other losses[16,17]. Preamplifier is placed just before the receiver that is used to increase the receiver sensitivity (the minimum power required by the receiver to function properly) [15,17]. Fig.(1.5) shows the block diagram of the fiber optic system with optical amplifiers booster, in-line, and preamplifier. There are several types of optical amplifiers that will be summarized in the next subsections. Fig.(1.5): Typical fiber optic system with optical amplifiers booster, in-line, and preamplifier [6]. 5

22 1.3.1 Erbium Doped Fiber Amplifier A device that boosts the signal in an optical fiber introduced in the late 198's. The EDFA was the first successful optical amplifier. It was a major factor in the rapid development of fiber-optic networks in the 1990's, because it extended the distance between costly regenerators. An EDFA is a fiber segment, a few meters long, that is doped with the rare earth element. The erbium ions may be excited to a higher atomic state by pumping these atoms by suitable optical frequencies. The two more convenient excitation wavelengths are 1450 nm[9,15]. Some of the drawbacks of EDFAs are [10]: 1) they are limited currently to 1550nm systems only. ) they require high pump powers ( mW). 3) very short lengths are not possible. Fig.(1.6) shows the internal structure of amplifier EDFA. 980nm and Fig.(1.6): Shows the internal structure of EDFA Semiconductor Optical Amplifier Semiconductor optical amplifier (SOA) is very similar to a laser except it has no reflecting facets [18]. The signal amplification is achieved by electric carrier injection into a semiconductor to provide a population inversion [15]. In order to get only the amplification function, it is necessary to prevent selfoscillation (lasing) of the device. This is accomplished by eliminating cavity reflections through the use of an antireflection coating [16]. 6

23 The main advantage of a SOA is the possibility of very large gains ( 0dB) in a short (less than ~ 400 m ) semiconductor. The main disadvantage of the SOA compared to the EDFA is the short lifetime of the carrier recombination. The short recombination lifetime in the SOA can cause crosstalk between communication channels [16,19]. SOAs can be divided into two categories, The Fabry-perot SOA (FP-SOA) and traveling-wave SOA (TW-SOA). The facet reflectivities in FP-SOA amplifiers are lower than that of a laser so that self-oscillation is not possible while the facets reflectivities in TW-SOA amplifier are negligible [18] Raman Amplifier Stimulated Raman scattering (SRS) amplifiers are non-doped fiber amplifiers that employ high-power pumps to take advantage of the non-linear properties of the fiber [0]. In the case of stimulated emission an incident photon stimulates emission of another identical photon without losing its energy, in the case of SRS the incident pump photon gives up its energy to create another photon of reduced energy at a lower frequency (inelastic scattering); the remaining energy is absorbed by the medium in the form of molecular vibrations by optical phonons. Thus, Raman amplifiers must be pumped optically to provide gain [6]. The pump and signal are injected into the fiber through a fiber coupler. The energy is transferred from the pump beam to the signal beam through SRS as two beams co-propagating inside the fiber. The pump and signal beams counterpropagate in the backward pumping configuration commonly used in practice [17,0] Brillouin Fiber Amplifier The operating principle of this amplifier is same as Raman amplifiers except that optical gain is obtained by stimulated Brillouin scattering (SBS). SBS arises when a strong optical signal generates an acoustic wave that produces variations 7

24 in the refractive index. These index variations cause light waves to scatter in the backward direction towards the transmitter. This backward scattering light experiences gain from the forward propagating signals, which leads to the depletion of the signal power. Brillouin gain spectrum is extremely narrow with bandwidth 100MHz [1,] Fiber-Optic Parametric Amplifier Fiber-optic parametric amplifier (FOPA) show great potential for applications in high-speed optical communication systems. FOPA exploits nonlinear optical properties of optical fiber. Parametric amplification is a well- () known phenomenon in materials providing (second-order susceptibility) nonlinearity. However, parametric amplification can be also be obtained in optical (3) fibers exploiting the ( third-order susceptibility) nonlinearity [3,4]. FOPAs are based on the four-wave mixing (FWM) process. FOPAs today has been significantly improved by the advances in both highly nonlinear fiber (HNLF) and high power. However, the gain produced by FOPAs cannot be described by the FWM process itself. Important FOPA properties such as available gain bandwidth and gain uniformity depend on the phase-matching condition [3]. It offers a wide gain bandwidth and may in similarity with the Raman amplifier [4]. Fig.(1.7) explains a comparison among the gain profiles for various types of amplifier. Fig.(1.7): Gain-bandwidth characteristics of different optical amplifiers[5]. 8

25 1.4 Literature Survey In 1971, D. Gloge [6] studied the optical signals transmitted through cladded glass fibers that are subject to delay distortion. He isolated these effects and evaluated their significance for cases of practical interest. These concern fibers in which the refractive index of the cladding is slightly lower than that of the core. In 1976, K. Eilner and A. Heuvel [7] explained that the glass fibers can be made with attenuation coefficients as low as in 0.4 db / km for electromagnetic energy 1m wavelength region. It is shown that this property, together with the potentially very large bandwidth attainable. In 1978, D. Messerschmitt [8] showed that the optimum minimum Meansquare error (MSE) equalization, both linear and decision feedback, of a digital fiber optic transmission system with Poisson signal statistics and additive widesense stationary noise is considered. In 198, M. Tur and et al [9] showed that a new fiber-optic signal processor is proposed to implement systolic matrix-vector multipliers and lattice filters multiplications/sec can be achieved with currently available components for matrix-vector multiplications that involve Toeplitz matrices. In 1986, B. Kim and et al [30] demonstrated all-fiber-optic frequency shifter that used mode coupling between the LP 01, LP11 modes by a traveling acoustic flexural wave guided along the optical fiber. The input and output leads of this frequency shifter are single-mode fibers. In 1989, M. Farries and et al [31] described an optical amplifier consisting of an erbium-doped germane-silicate fiber optically pumped at 53 nm. Negligible excited-state absorption at 53nm allows efficient pumping, enabling a gain of 34dB at 1536nmto be obtained for only 5mW pump power. In 1990, G. Agrawal [3] showed that the pulse amplification in EDFA is studied by considering a general model that includes both gain saturation and gain dispersion. The effects of gain dispersion are studied numerically for the case in 9

26 which a fundamental soliton is launched at the amplifier input. The results show that fiber amplifiers may be useful for simultaneous amplification and compression of weak optical pulses. In 1991, B. Pedersen, et al [33] pumped the EDFA at nm and 1480 nm, the optimum cutoff wavelength for step profiles with arbitrary numerical aperture is shown to 800nm and 900 nm, respectively. The use of a confined erbium profile can improve the gain coefficient up to 45%. In 199, K. Rottwitt, et al [34] showed that the comprehensive theoretical analysis on the design of a distributed EDFA for long-distance transmission has been carried out, using a highly accurate model. The dispersion of the optical fiber as a function of included. the numerical aperture and the cutoff wavelength is In 1993, K. Rottwitt, et al [35] illustrated that the theoretical limits in noise figure for a long-haul transmission line based on lumped amplification are contrasted with distributed amplification. The letter results in a reduction of approximately 60% of the required number of pump power stations. In 1994,Y.Lai, et al [36] experimentally demonstrated WDM optical-timedomain-reflectometry supervisory technique in a 00km transmission system. The technique can simultaneously achieve in-line EDFA performance monitoring and diversity protection, and fault location in optical fiber. In 1995, M. Zervas and R. Laming [37] studied the effect of the internal Rayleigh scattering on the gain efficiency and noise of the EDFA. It is shown that excess background loss due to Rayleigh scattering at the signal and pump wavelength dramatically reduces the optimum gain efficiency of either high NA, low concentration or confined EDFA. In 1996, S. Feming and T.Whitley [38] described an accurate and sensitive experimental determination of the pump dependence of the refractive index in an EDFA. In a doped fiber at wavelength close to an absorption, and the refractive index, hence the dispersion, is expected to be a strong function of wavelength, as described by the Kramers-Kronig relationship.

27 In 1997, Y. Sun, et al [39] presented a detailed study of a set of models for characterizing the gain, the input and output powers of single EDFAs and networks of EDFAs. The time dependent gain is described by a single ordinary differential equation for the average inversion level of an EDFA propagation directions. In 1998, H. Kim, et al [40] demonstrated an actively gain-flattened EDFA using an all-fiber gain-flattening filter with electronically controllable spectral profiles. A good gain flatness ( 0.7dB) over a broad wavelength span ( 35nm) is achieved for a wide range of operational gain levels as well as input signal and pump power. In 1999, Y. sun, et al [41] explained that the WDM systems with optical amplifiers, can cause power changes in the existing channels of WDM systems with optical amplifiers. One present an analytical formula for the transient power of the existing channels as function of time when one or more input channel is dropped or added on an EDFA. In 000, K. song, et al [4] provided the significance of self-phase modulation (SPM) and cross-phase modulation (XPM) in L-band EDFAs that are evaluated through numerical simulations of 0Gb/ s nonzero dispersion-shifted fiber (NZDSF) transmission link. The results show the effects of SPM/XPM in NZDSF. In 001, B. Choi, et al [43] measured the gain coefficient and noise figure (NF) of the L-band EDFA for pump wavelengths between The gain coefficient with a 1530nm and 1560 nm. 1545nm pump is more than twice as large as with a 1480 nm. The noise figure of 1530nm pump is 6.36dB at worst, which is 0.75dB higher than that of 1480nm pumped EDFA. In 00, R. Olivares and J. Souza [44] presented a time-domain modeling of EDFAs, which is adequate for the analysis of the power transients resulting from the dynamic variation of the number of channels in wavelength routed optical networks. 11

28 In 003, S. Harun, et al [45] demonstrated that the gain in the long wavelength L-band EDFA using dual forward pumping scheme in double-pass system, the small signal gain for However, a noise figure 1580nm signal can be improved by 13.5dB..9dB was obtained due to the backward C-band amplified spontaneous emission (ASE) from second stage. In 004, F. Vasil and P. Schiopu [46] presented a method for determination of the saturation power in the EDFA. This method uses the analytical expressions in different approximations to obtain the saturation power of the EDFA. In 005, D. Thomas and J. Wied [47] showed that the sustained relaxation oscillations in a gain-clamped EDFA can be generated depending on the traffic loading in packet-switched WDM optical network. The effects are dependent on the channel wavelength, on the packet power and width, as well as on the time delay between traveling packets. In 005, A. Tran, et al [48] proposed and demonstrated a simple technique to control rapid EDFA transients in optical burst switched networks. They have demonstrated a 6dB transient reduction in a 10Gb/ s over 80km long WDM system with three EDFAs. They have shown through simulation that the technique can be used at the first EDFA in links of as many as ten EDFAs to control transients to within db. In 006, Y. Ben, et al [49] studied all-optical method for automatic gain controlling of transients in EDFAs using a SOA-based negative feedback loop for adjusting the EDFA pump power. A dynamic model for the EDFA-SOA system is developed and used for numerical simulation of the transient behavior of this system. In 006, H. Chang, et al [50] presented an EDFA-free all-optical R( reamplification and reshaping) regeneration scheme based on a compact self-seeded fabry-perot laser diode. The proposed R regenerator achieves a straight line transmission at 10 Gb/s over 76km without either the EDFA or the external probe laser, both of which are traditionally required. 1

29 In 007, L. Qiao, et al [51] described theoretical model to calculate the ASE at the output of an optical amplifier. The technique can be used to determine the usable optical signal power at the output of an EDFA operating in the constantoutput-power control mode. The model was experimentally verified by automating an EDFA to operate in the constant-signal-power control mode using an ASE correction based on this model. In 008, L. Vincetti, et al [5] developed a numerical model for the analysis and design of S-band EDFAs. The model can to accurately predict the amplifier performances by taking into account the ASE suppression due to the bending, as well as leakage losses of the fiber used as active medium. In 009, C. Berkdemir and S. Özsoy [53] provided an efficient temperaturedependent analysis to study the effect of cooperative upconversion on the temperature-dependent gain performance of the C-band EDFA at highconcentration in forward pumping configuration at a signal wavelength of and in the temperature range of o o 40C to 80C nm In 010, M. Paul, et al [54] demonstrated the wideband EDFA using an erbium-doped zirconia fiber as the gain medium. With a combination of both Zr and Al, a high erbium doped concentration of achieved without any phase separations of rare-earth. 430ppm in the glass host has been In 011, A. Naji, et al [n55] presented a theoretical analysis model of dual stage quadruple pass EDFA. This analysis is carried out by the broad input signal power starting from power. 50dB to 0dB input signal with 10mW to 0mW pumping In 01, H. Sariri, et al [56] studied the effect of ASE on the gain modulation in EDFAs. The derivation of an analytical model for EDFA over modulation response has been presented. In 013, P. Sharma and et al [57] discussed various gain flattening techniques. EDFA is widely used amplifier due to its transient suppression, wideband variable gain operation. Electronic feedback control is designed for achieving flat gain at the top of required gain range.

30 In 014, G. Ivanovs, et al [58] shown that the characteristics of the EDFA are investigated. The amplification and noise figure dependences on different EDFA parameters in a simulated and measured..5gb/ s one-channel WDM transmission system are 1.5 Aim of the Thesis The aim of this work is to present a theoretical analysis to EDFA operating under co-directional configuration. In turn, the parameters that maximize the characteristics of EDFA will be explained. However, the main issues to be addressed are: 1) to analyze the gain spectrum and bandwidth. ) to solve the rate equation with related propagation equations. 3) to highlight the ASE effects. 4) to investigate the effect of the confinement factor using Maxwell's equation, which may be affected by the graded order. 1.6 Organization of the Thesis The work of this study falls into five chapters. After the introduction in chapter one, chapter two presents the basic requirements for the fiber and EDFA that will be used in the next chapters. Chapter three presents a theoretical analyze the EDFA. An analysis has been presented for the rate equations and the related propagation equations to enhance the characteristics of EDFA. Chapter four contains the results and discussion, while chapter five summarizes the main conclusions for the results obtained from this study and gives suggestions for future work. 14

31 CHAPTER TWO Basic Requirements for Fiber and Erbium Doped Fiber Amplifier

32 CHAPTER TWO Basic Requirements for Fiber and Erbium Doped Fiber Amplifier.1 Introduction In its simplest form, an optical fiber consists of a central glass core surrounded by a cladding layer whose refractive index ( n ) is slightly lower than the core index ( n 1 ). The refractive index of the core decreases gradually from center to core boundary [1,59,60,61]. Although, in principle, a cladding is not necessary for light to propagate along the core of the fiber, it serves several purposes. The cladding reduces scattering loss resulting from dielectric discontinuities at the core surface, it adds mechanical strength to the fiber, and it protects the core from absorbing surface contaminants with which it could come in contact [60,61]. Two parameters that characterize an optical fiber are the relative core-cladding index difference [59] n n n n 1 1 n1 n1 (.1) Considerable insight into the guiding properties of optical fiber can be gained by using a ray picture based on geometrical optics. The geometrical-optics description, although approximate, is valid when the core radius ( a ) is much larger than the light wavelength ( ), this situation corresponds to the multi-mode fiber. When the magnitudes of the two quantities are similar as happens with the single-mode fiber, it is necessary to use the electromagnetic theory to describe adequately the propagation of light along the waveguide[5]. In the manufacturing process, the glass used in the core has impurities such as germanium or phosphorous added to raise the refractive index under controlled conditions, while cladding layer be lower refractive index to make the optical fiber work. Optical fiber core is manufactured in different diameters for different 15

33 applications. The cladding and core are manufactured together from the same silicon dioxide-based material in permanently fused state [6]. The manufacturing process adds different amounts of impurities to the core and cladding to maintain a difference in refractive indices between them of about 1 percent [6]. There are two types of optical fiber (fiber types according to refractive index), step-index fibers and graded-index fibers. Both the step- and graded-index fibers can be further divided into single-mode and multimode [60].. Step-Index Fiber Step-index multimode fibers contain hundreds of modes of propagation. It is used in applications that require high bandwidth ( 1GHz ) over relatively short distances ( 3km ) such as a local area network [60]. The major benefits of multimode fiber are: it is relatively easy to work with, it can be used with both lasers and LEDs as sources, the larger core radius of multimode fibers make it easier to launch optical power into fiber, and coupling losses are less than those of the single-mode fiber. Many modes are allowed to propagate, so it suffers from modal dispersion. The result of modal dispersion is bandwidth limitation, which translates into lower data rates [60]. Single-mode step-index fiber allows for only one path, or mode, for light to travel within the fiber [6]. The core diameter is reduced to 16 10m, so that the normalized frequency V of the fiber is smaller than the cutoff. 405 of the mode that is the next higher mode from the dominant mode []. The major benefits of single-mode fiber are [63]: 1) the most common choice for long distance communication. ) it does not have modal dispersion, which distorts the signal pulse at long distance. around Fig.(.1a) shows a multimode step index fiber with a core diameter of 00m or greater, which is large enough to allow the propagation of many modes within the fiber core. Fig.(.1b) shows a single-mode step index fiber which allows the propagation of only one transverse electromagnetic mode

34 (typically HE 11 ), and hence the core diameter must be of the order of to 10m [5]. Optical fibers designed to satisfy this condition are called single-mode and multimode fiber. The core radius is typically However, single-mode fiber with require a 5m [,59]. 00m for multimode fibers. Fig.(.1): The refractive index profile and ray transmission in step index fibers: a) multimode step index fiber, b) single-mode step index fiber [5]..3 Graded-Index Fiber Graded-index fiber was developed in order to minimize the effects of intermodal dispersion, the dispersion of the graded index fiber is much smaller than that of the step index fiber []. In the gradient-index fiber design the core refractive index decreases continuously with increasing radial distance r from the center of the fiber but is generally constant in the cladding [60,64]. The refractive index of a graded-index fiber can be described by the expression [10] n( r) n n 1 r 1 ( ) a q r a core (.) r a cladding where q is profile index or the graded order, r is the radial distance from the fiber axis, n 1 and n are refractive indices at the core and cladding, respectively. 17

35 The case q refractive index. For, as in Fig.(.), corresponds to parabolic profile of the q, Eq.(.) reduces to the step-index profile [10]. Fig.(.): the relationship between the radius and refractive index of various q [60]. A multimode graded index fiber with a parabolic index profile core is illustrated in Fig.(.3). It may be observed that the rays shown appear to follow curved paths through the fiber core [5]. Fig.(.3): The refractive index profile and ray transmission in a multimode graded index fiber [5]..4 Fiber Materials The majority of fibers are made of glass consisting either of silica (SiO ) or a silicate. The variety of available glass fibers ranges from high-loss fibers used for short-transmission distances to very low-loss fibers employed in long-haul 18

36 applications. Plastic fibers are less widely used because of their substantially higher attenuation than glass fibers [60]..4.1 Glass Fiber It is the material of choice for low-loss optical fiber. Synthesized by fusing SiO molecules. The refractive index difference between the core and cladding is realized by selective use of dopants during the fabrication process [59]. The largest category of optically transparent glasses from which optical fibers are made consists of the oxide glasses. Of these the most common is silica (SiO ), which has a refractive index of at 850 nm. To produce two similar materials having slightly different indices of refraction for the core and cladding, either F or various oxides ( referred to as dopants) such as B O 3, GeO, or P O 5 are added to the silica. The addition of GeO or P O 5 increases the refractive index whereas doping the silica with F or B O 3 decreases it as in Fig.(.4). Since the cladding must have a lower index than the core [60]. Fig.(.4): Refractive index as a function of doping Concentration in silica glass [60]..4. Plastic-Clad Silica Core Fibers These fiber have a glass core and plastic cladding. The core is larger than all-glass fiber, but the cladding is much thinner in relation to the core. Plastic-clad silica fiber is designed to provide a compromise between the high performance 19

37 signal-carrying ability of glass fiber and the durability, low cost, and relatively easy handling of plastic fiber. These fiber are used in area where a moderate degree of ruggedness, and high performance over short distances are important[6]..4.3 Plastic Fibers Optical fiber transmission principles also work successfully with optical fibers made from plastic. They are, however, limited to a step index multimode construction. Because of the nature of plastics, the fabrication process produces a core and cladding with significantly different refractive indices [5]. There are three main advantages associated with using plastic fibers: 1) they are a lot more robust than glass fibers. They can sustain significantly more shock, pressure and stress without damage than glass fibers, ) they are far more flexible, easier to handle and therefore, easier than glass fibers, and 3) plastic is a low-cost material. On the other hand, there are a number of significant disadvantages. They have a significantly higher attenuation than glass fibers and are generally used over very short distances ( wavelength of 100m maximum) only. Plastic has an optimum operating 650 nm (red LEDs are used), which has an attenuation of bandwidths with maximum operating data speeds up to approximately over a maximum of 50 m [1,65]. 10Mbps.5 Manufacturing of Optical Fiber Optical fiber is manufactured to meet very high standards, because the core diameter and refractive indices of the core and cladding must remain consistent over stretches of up to 80km [6]. There are four different methods commonly used to make optical fiber: 1) modified chemical vapor deposition (MCVD), ) outside vapor deposition (OVD), 3) vapor axial deposition (VAD), and 4) plasma chemical vapor deposition (PCVD) [,66]. Optical fiber are made by two stages. In the first stage a vapor-deposition method is used to make a cylindrical perform with the desired refractive index 0

38 profile. The perform is typically 1m long and cm in diameter and contains core and cladding layers with correct relative dimensions. In the second stage, the perform is drawn into a fiber by using a precision-feed mechanism that feeds the perform into a furnace at the proper speed [,59]. The dimensional and mechanical properties of the optical fiber are all determined by the drawing process. The typical core diameters are 5m for single mode fiber and 50m for multimode fiber. The typical cladding outside diameter is about 15m for both single mode and multimode fibers. In order to get good quality, during manufacturing the fiber diameter is feedback controlled by varying the drawing speed using fiber diameter monitoring signals. The drawing speed is typically m / sec [66]. Fig.(.5) shows the drawing apparatus schematically, the perform is fed into a furnace in a controlled manner where it is heated to a temperature of about precision-feed mechanism [6,66]. o 000 C. The melted perform is drawn into a fiber by using a Fig.(.5): Drawing the perform into fiber [6]. 1

39 .6 Ray Transmission View To consider the propagation of light within an optical fiber utilizing the ray theory model, it is necessary to take account of refractive index of dielectric medium. The refractive index of a medium is defined as the ratio of the velocity of light in a vacuum to the velocity of light in the medium [5]. A ray of light incident upon the interface between two transparent optical materials having different indices of refraction, will be totally internally reflected if: 1) the ray is incident upon the interface from the direction of the more dense material, and ) the angle made by the ray with the normal to the interface is greater than some critical angle, the latter being dependent only on the indices of refraction of the media. This is the mechanism by which light at a sufficiently shallow angle (less than fiber with low loss[5]. o 90 c ) may be considered to propagate down an optical One of the primary parameters of a fiber is its numerical aperture ( NA). This parameter is used in equations describing the coupling losses of the source light entering the fiber, the mode excitation, connector and splice losses, and other system performance equations [67]. NA is a basic descriptive characteristic of specific fibers. NA can represent the size or ''degree of openness'' of the input acceptance cone. It is possible to continue the theory analysis to obtain a relationship between the acceptance angle and the refractive indices of the three media involved, namely the core, cladding, and air. This leads to the definition of a more generally used term: The numerical aperture of the fiber [5]. Fig.(.6) shows a light ray incident on the fiber core at an angle 1 to the fiber axis which is less than the acceptance angle for fiber a. The ray enters the fiber from a medium (air) of refractive index n o, and the fiber core has a refractive index n 1, which is slightly greater than the cladding refractive index n. The NA is defined as [1,66] NA sin a n 1 n (.3)

40 The NA can be made as a function of using Eq.(.1) to yield NA n 1 (.4) This relationship is very useful measure of the light-collecting ability of a fiber[1,5]. Fig.(.6): The ray path for a meridional ray launched into an optical fiber in air at input angle less than the acceptance angle for fiber [5]..7 Fiber Modes Modes are mathematical and physical ways of describing the propagation of electromagnetic waves in an arbitrary medium. In its mathematical form, the theory of electromagnetic modes derives from Maxwell's equations. With his equations, Maxwell showed that electric and magnetic energy are two forms of the same electromagnetic energy. A mode is an acceptable solution to Maxwell's equations. For the sake of simplicity, a mode can be described as a possible direction (route) that a light wave will follow down, for example, an optical fiber. The number of possible modes or energy directions that can occur in a fiber ranges from one to over a hundred thousand. Exactly how many modes can be carried by a fiber is determined by the fiber's geometric properties (dimensions) together with the fiber's optical parameters [68]. In general, electromagnetic wave in a step optical cylindrical fiber can be divided into three distinct categories[5,1,69]: 1.Transverse electric ( TE ): TE modes exist when the electric field is perpendicular to the direction of propagation. they are characterized by E 0 and H 0. z 3 z

41 . Transverse magnetic (TM ): in a TM mode the magnetic field is perpendicular to the direction of propagation, they are characterized by E 0 and H Hybrid modes ( HE, EH ): here both the electric and magnetic fields are perpendicular to the z-direction; characterized by E 0 H 0. In more detail, the modes are commonly referred to as 4 z z z z TE m, TM m [1]. Where is the azimuthal mode number and takes integer values and m value is designated by its radial mode number [70]. For m 0, the modes are analogous to the TE and TM modes of a planar waveguide because the axial component of the electric field, or the magnetic field, vanishes. However, for 0, fiber modes become hybrid, i.e., all six components of the electromagnetic field are nonzero. Linear polarization ( LP ) Modes: the hybrid modes have a complicated polarization, although the HE11 mode is almost plane-polarized. However, for weak guides, it can be shown that all the other modes can be grouped together in to combinations, which are degenerate. A linear combination of these modes can then be used to construct a new set of nearly plane-polarized fields. These modes are called "linearly polarized". The relationship between the traditional HE, EH, TE, and TE mode designations and the LP m mode designations is shown in Fig.(.7) and Table(.1) shows the constituents of some low-order LP modes. The mode subscripts and m are related to the electric field intensity profile for a particular LP mode [68,71]. The total number of modes that can propagate in a multimode fiber M is given by [67] q M V q 1 where V is known as the normalized frequency, or the V -number, which relates the fiber size, the refractive index, and the wavelength via the relation. V a n 1 n where k0 is the propagation constant, a is the core radius and λ is the wavelength of light. It is shown that a step-index fiber supports a single mode if V. 405 (.5) (.6)

42 [66]. For values above.405, the fiber is multimode [63]. For a step index fiber with q, the number of modes M is approximated by M V /. For a graded index fiber with q number of modes M is approximated by V / 4 (normal parabolic graded index fiber), the M. Fig.(.7): The electric field configuration for the three lowest LP modes: (a) LP mode, (b) exact mode designations, (c) electric field distribution, (d) intensity distribution[7]. Table(.1): Constituents of some low-order LP modes [7]..8 Losses There are many mechanisms that may be caused losses such as: attenuation, coupling and bending. In the next subsections, these mechanisms will be summarized..8.1 Attenuation All fibers attenuate the power of wave passing through them [3]. The rate at which light is absorbed is dependent on the wavelength of the light and the 5

43 characteristics of the particular glass [7]. Attenuation is commonly characterized in decibel per kilometer ( db/ km ). Attenuation is a very important factor in designing effective long distance fiber optical networks [11]. If Po is the power launched at the input of a fiber of length L, the transmitted power P(L) is given by [59] P( L) Po exp( L) (.7) where the attenuation constant α is a measure of total fiber losses from all sources. The attenuation coefficient ( db / km) is related to ( 1/ km) via the relation [69] db 1 ( ) ( ) (.8) km km The principle sources of attenuation in on optical fiber can be broadly classified into two groups: absorptive and radiative [10] Absorption Losses Absorption caused by impurities in the glass refers to the conversion of optical energy into either optoelectronic activity or molecular vibration [7]. This component of absorption can be attributed to the interaction of light with the atoms and molecules that make up the bulk of material in a glass fiber [11]. Although glass fiber are extremely pure, some impurities will still remain as residues after purification. These impurities are in the form of ionized molecules. Metal ions such iron, copper and nickel are the main offenders. They absorb the light particles (photons) and in the energy exchange process the fiber will heat up. The absorption losses caused by metal ion impurities are substantial in poor quality glass [65]. The presence of 1ppm of Fe + would lead to a loss of 0.68dB/km at 1.1μm [10]. Glass will also contain significant amounts of water ion ( OH ) impurities that resonate at certain frequencies. Most signal attenuation due to water ions occurs in the 0.85 m wavelength region [65]. The presence of OH ions leads to 6

44 absorption peaks at (0.7, 0.88, 0.95,1.13,1.4,and 1.38) m. The broad peaks at 1.4 and 1.38 m in Fig.(.8) are due to the presence of OH ions. Fortunately, the absorption bands are narrow enough that ultrapure fiber exhibit losses 0.dB/ km at o 1. 55m [10]. An absolutely pure silicate glass has little intrinsic absorption due to its basic material structure in the near-infrared region. It may be observed that there is a fundamental absorption edge, the peaks of which are centered in the ultraviolet wavelength region. This is due to the stimulation of electron transitions within the glass by higher energy excitation. The tail of this peak may extend into the window region at the shorter wavelength, as in Fig.(.8) [5,67]. Fig.(.8): Spectrum of losses in fiber [1]. Extrinsic losses are not associated with the fundamental properties of material. There are many sources of extrinsic losses, for example presence of additional impurities such iron, nickel, and chromium, introduced into the fiber material during manufacture or caused by dissolved water in the glass, as ( OH ). Another sources of extrinsic loss is surface irregularities at the core-cladding 7

45 boundary. Some other related losses are due to bending and source-fiber coupling[1,73] Radiative Losses In an amorphous material such as glass, the density of the material is not exactly uniform throughout, there are bound to be local variations. An absolutely pure material cannot be manufactured. Glass will never be completely transparent. A light ray travelling through such a material will be scattered in other directions[68]. Rayleigh Scattering is a fundamental loss mechanism arising from density fluctuations frozen into the fused silica during manufacture. Random local variations of index of refraction resulting from random defects, material composition fluctuations, and various inhomogeneities. Rayleigh scattering scales as λ -4, and therefore drops rapidly at longer wavelengths [11,59]. Rayleigh scattering shown by a dashed line in Fig.(.8) [59]. R R C R 4 4 where the constant C is in the range ( ) db/( km m ) depending on the constituents of the fiber core. As R = silica fiber are dominated by Rayleigh scattering [10,59]. (.9) db / km near 1. 55m, losses in An ideal single-mode fiber with a perfect cylindrical geometry guides the optical mode without energy leakage into the cladding layer. In practice, imperfections at the core-cladding interface can lead to additional losses which contribute to the net fiber loss. The physical process behind such losses in Mie scattering [6]. Mie scattering is caused by imperfections in the fiber of index inhomogeneites on a scale longer than the optical wavelength [5]. The resulting scattering loss is typically below 0.03 db/km[6]..8. Bending Losses Micro bends and Macro bends are common problems in installed cable systems because they can induce signal power loss [64]. Losses due to the 8

46 curvature and losses caused by an abrupt change in radius of curvature are referred to as "bending losses" care must be taken when cabling and jointing fiber otherwise the attenuation of the fiber can be significantly affected [7]. Micro bending is due to small fiber imperfection caused in the manufacturing process [8]. Micro bends cause an increase in the fiber loss for both multimode and single-mode fiber and can result in an excessively large loss (~100dB/km) if precautions are not taken to minimize them [6]. Macro bending occurs when the fiber is bent past its minimum bend radius, which results in light escaping its waveguide confines [8]. Macro bending losses are depends on significantly wavelength and are a losses that has been caused by large-scale bending of the fiber [7]..8.3 Losses in Fiber Joins Losses in fiber joins are commonly classified into two types: extrinsic losses and intrinsic losses. Extrinsic losses are those losses caused by factors concerned in joining the fiber but are unrelated to the properties of the fiber itself [5], such as; numerical aperture differences, angular fiber misalignment, and longitudinal separation [65,10,68]. Intrinsic losses are those losses caused by some property inherent in the construction of the fiber [5], such as; differences in core diameters, different cladding diameters (loss due to transverse misalignment), concentricity error, and core shape (ellipticity) [5,10,65]. Fig.(.9) explains different cases for fiber joins. Fig.(.9): Losses in joins fiber [5]. 9

47 .9 Dispersion The most important fiber characteristic after transmission loss is dispersion. This refers to the broadening of optical pulses as they propagate along the fiber. As pulses broaden, they tend to interfere with adjacent pulses [63]. Modal dispersion occurs only in multimode fiber. It arises because rays follow different paths through the fiber and consequently arrive at the other end of the fiber at different time. Thus different rays take a shorter or longer time to travel the length of the fiber. The ray that goes straight down the center of the core without reflecting, arrives at the other end first, other rays arrive later. Thus light entering the fiber at the same time exit the other end at different times. The spreading of light is called modal dispersion. Modal dispersion is that type of dispersion that results from the varying modal path lengths in the fiber [64,74]. In the fact that the refractive index of the fiber glass, just like that of any other material, depends on wavelength. The wavelength dependence on the refractive index is behind two different contributions to delay distortion. They are collectively called chromatic dispersion. The first is the material dispersion (D M ) that arises directly from the wavelength dependence on the index. It is independent of geometry and depends only on wavelength. Variation of refractive index with respect to wavelength are described by the Sellmeier equation which is expressed as follows [74] n ( w) 1 3 j1 w A w j j j w (.10) where w j is the resonance frequency and A j is the strength of jth resonance [59,74]. Fig.(.10) shows the wavelength dependence of n and ng in the range m for fused silica. D M is related to the slope of g n by the relation [6] D M 1 dng c d (.11) 30

48 Fig.(.10): Variation of n and ng with wavelength [6]. It turns out dn g / d 0at 1. 76m. This wavelength is referred to as the zero-dispersion wavelength ZD. The dispersion parameter D M is negative below ZD and positive above that [69]. ZD of optical fiber also depends on the core radius and the relative core-cladding index difference Δ [6,69]. The second is called waveguide dispersion ( group velocity D W ) that exists because the v g depends on the wavelength [1]. Waveguide dispersion is cause by power distribution in the fiber's core and cladding, which is wavelength dependent. Light travels faster in the core and slower in the cladding [8]. Different power distribution causes different wavelength travels at different speed. Waveguide dispersion is important in single-mode fibers, where modal dispersion is not exhibited [6,10]. Fig.(.11) shows D M, D W,and their sum D D M D W, for single mode fiber. The total dispersion is zero near 1.31μm [6,69]. In the manufacturing process allow fiber to created with various dispersion flattened or dispersion shifted fiber[8]. DM only affected ratios impurities while DW depends on fiber parameters such as the core radius a and the index difference, it is possible to design the 31

49 fiber such that ZD dispersion-shifted fibers. is shifted into the vicinity of 1.55μm. such fiber are called Fig.(.11): Relative contributions of D M typical single mode fiber [69]. and D W, well as total dispersion D for Within a single mode fiber, the linear polarization light really has two components, x-component and y-component each of them perpendicular to each other. The two components exhibit a different refractive index to the different birefringence, this situation is called polarization mode dispersion (PMD). The PMD leads not only to random changes of the polarization state of light, but also to pulse broadening [73]. As in Fig.(.1). Fig.(.1): Effect of PMD when a signal is introduced in a short fiber with equal components along the two principal axes [5]. 3

50 .10 Rare-Earth-Doped Fiber Rare earth ions not rare in the earth crust. They are typically dispersed and not found in concentrated form. That is the reason why they are named rare earth. Although fiber were made as early as 1964, their use became practical only after 1986 when the techniques for fabrication and characterization of low-loss, rareearth-doped fibers were perfected [14,59]. Rare earth atoms are divided in two groups: the lanthanides with atomic number 57 through 71, and the actinides with atomic number 89 through 103 [9]. Lanthanides form a group of 14 similar elements with atomic numbers. When these elements are doped in silica or other glass fiber [14], they become triply ionized by removal of the two outer6s electrons and an inner 4 f electron [75]. Rare earth are characterized by their 4 f electrons in seven 4 f -orbital's. The optical properties of the dopants are then determined by the partially filled 4f orbital and since the outer from the field of the host material [75]. 5p and 5s electrons effectively shield the 4f electron Many different rare earth ions, such as erbium (work around 1.55m in EDFA) [63], holmium, neodymium, samarium, thulium, and ytterbium, can be used to make fiber amplifier that operate at wavelengths covering a wide range from visible to infrared [14]. EDFAs have attracted the most attention as they operate near 1.55μm and are useful for modern fiber-optic communication systems [14] Erbium Concentration The erbium doping concentration in doped fiber is specified usually in terms 3 of mole ppm (parts per million) or in terms of number of Ee ions per cm 3. Let M represent the mole ppm of Er O in a doped pure silica fiber. This implies that in 1 mole of the composite material there is ( M) mole of SiO and Er O 3. Since the molecular weight of SiO and Er O 3 are [10]: 10 6 M mole of Mw Mw SiO ErO g / mol g / mol (.1) 33

51 where Mw is the molecular weight of any substance. The weight (m) of 1mole of doped SiO is 6 6 (1 10 M ) M g / mol 1mol 60.1 (.13) m SiO g Since 1mol of any substance contains molecules, 60.1g of the composite material will contain molecules. If D(g/cm 3 ) is mass density 3 of the doped material, then this implies that 60.1 D cm of the material contains molecules. The weight of Er O 3 in this volume is M g. Now, 38.6g of Er O 3 contains molecules of Er O 3. Hence, M g of Er O 3 will contain M molecules of Er O 3. Since every molecule of Er O 3 contains two ions of Er +3, the number of Er +3 ions in this will be M. Hence, the Er +3 ion density is [9,10] ( ion) D( g / cm ) M 3 N ( ions / cm ) m ( g) SiO (.14) where one have used the density D of SiO, which is about N.86g / cm 3 ions/ m M ( mol ppm) (.15) So, M=100mol ppm corresponds to about ion/m Fiber Nonlinearities The response of any dielectric to light becomes nonlinear for intense electromagnetic fields, and optical fiber are no exception [59]. The origin of nonlinear response is related to a harmonic motion of bound electrons under the influence of an applied field. Even though silica is intrinsically not a highly nonlinear material. The waveguide geometry that confines light to a small cross section over long fiber lengths makes nonlinear effects important in the design of modern light wave systems [6]. As a result, the total polarization P induced by electric dipoles is not linear in the electric field E, but satisfies the more general relation P o (1) () (3) E : EE EEE (.16) 34

52 where ( j) is the vacuum permittivity and ( j 1,, ) o is the jth order susceptibility. In general, ( j) is a tensor of rank j 1. The linear susceptibility (1) represents the dominant contribution to P. Its effects are included through the refractive index n and the attenuation coefficient susceptibility (). The second-order is responsible for such nonlinear effects as second-harmonic generation and sum-frequency generation. However, it is nonzero only for media that lack an inversion symmetry at the molecular level. As SiO is a symmetric molecule, () vanishes for silica glasses [59]. Nonlinear effects in fibers are mainly due to two causes. First cause is the nonlinear refraction, this dependence gives rise to effects such as four wave mixing (FWM), self-phase modulation (SPM) and cross-phase modulation (XPM). The second cause is the non-elastic scattering of photon in fiber, which results in stimulated Raman scattering (SRS) and stimulated Brillion scattering (SBS) phenomena [11] Nonlinear Refraction The lowest-order nonlinear effects in optical fiber originate from the thirdorder susceptibility (3), which is responsible for phenomena such as thirdharmonic generation, and FWM. These nonlinearities are essentially based on the light-induced nonlinear electronic polarization of the medium. The refractive index can be expressed as [13,76] ~ ) n( ) n E n(, E (.17) where n() is the linear part given by Eq.(.10), E is the optical intensity inside the fiber, and n is the nonlinear refractive index. In silica, [64], that is related to (3) by the relation [76,59] n 3 (3 ) xxxx Re( 8 35 ) 0 n 3.10 m / W (.18) where Re stands for the real part and the optical field is assumed to be linearly (3) polarized so that only one component of the fourth-rank tensor contributes to xxxx

53 the refractive index. Practically, the coefficient that determines the magnitude of the nonlinear effects is [76] n A eff (.19) where is the nonlinear coefficient, is the wavelength and Aeff is the effective 1 1 core area. For a typical single mode fiber, (10 100) W km [14,76]. SPM refers to the self-induced phase shift experienced by an optical field during its propagation in optical fiber. The intensity-dependent nonlinear phase shift [59] NL n o k L E (.0) With high signal intensities, the light itself induces local variable changes in the refractive index of the fiber known as the Kerr effect. This phenomenon produces a time-varying phase in the same channel. The time-varying refractive index modulates the phase of the transmitted wavelength, broadening the wavelength spectrum of the transmitted optical pulse [64,68]. XPM refers to the nonlinear phase shift of an optical field induced by another field having a different wavelength, direction, or state of polarization [59]. It occurs when two or more optical channels are transmitted simultaneously inside an optical fiber using the WDM technique [6]. XPM effect only arises when transmitting multiple channels on the same fiber [64]. NL n E1 (.1) k o L E FWM happens when three pulses at three different frequencies travel side by side each other for a long distance in the fiber. If the frequencies of these three pulses are f, and f 3, they tend to mix and generate waves at a new frequency 1, f f FWM given by [11,68] f FWM f 1 f f 3 (.).11. Stimulated Inelastic Scattering A second class of nonlinear effects results from stimulated inelastic scattering in which the optical field transfers part of its energy to the nonlinear 36

54 medium. Two important nonlinear effect in optical fibers fall in this category; both of them are related to vibration excitation modes of silica. These phenomena, known as SRS and SBS [59]. The main difference between the two is that optical phonons participate in SRS while acoustic phonons participate in SBS [68]. A fundamental difference is that SBS in optical fibers occurs only in the backward direction whereas SRS can occur in both directions [64,68]..1 Fundamental of Erbium Doped Fiber Amplifier An optical amplifier that used a length of fiber doped with erbium and energized with pump laser to inject energy into a signal [6]. The EDFA had a significant impact on optical fiber communication systems. EDFA has a broad gain bandwidth near the 1.55μm wavelength region that coincides with the lowloss transmission wavelength of the silica glass fibers [7]. The EDFA has made tremendous progress since its invention in It replaced the rather involved process of the fiber optic repeater station. It created a revolution in long in distance optical communication systems. The main condition that should be fulfill to ensure the optical power transfer to the signal is that the erbium atoms are to be in the excited state [58]. The proportion of the erbium atoms that are excited into the higher energy states is called the inversion level of the EDFA. An atom that is not excited remain in ground state. Incident signal photons will interact with the excited Er +3 ions to release energy as photons in the 1550nm band. The resultant released photon propagates with the same phase, polarization and direction as the incident photon, hence, signal amplification is realized [77]. The source of amplification energy for the EDFA comes from a pump, which is a laser light at the input operating at 980nm or 1480nm wavelengths. ASE is a specific EDFA effect. The excited erbium atoms can release their energy spontaneously without the catalyst of an incident photon. This results in photons with a random phase, polarization and direction. These random photons also 37

55 become amplified by the same process that amplifies the input signal. Since does not carry signal information, it contributes to noise [78]..1.1 Configurations of Erbium Doped Fiber Amplifier In the Fig.(.13) the amplifier is pumped by a semiconductor laser, which is complemented with a wavelength selective coupler (also known as the WDM coupler) which combines the pump laser light with the signal light. Optical isolators used unwanted reflection. The center part of an EDFA is called "gain block", which consists mainly of EDFA. Three types of pumping schemes exist, namely forward (co-directional), backward (counter-directional) and dual (bidirectional) pumping[58]. Fig.(.13): simplified view of EDFA[58]. A forward-pumped scheme will ensure high population inversion at the input of the EDFA will ensure less spontaneous. On other hand, a backwardpumped schemes will have high ASE noise but also high output power. A dualpumped scheme combines the two schemes, thus providing an optimized version as in Fig.(.14). Often a dual-pumped scheme will be used [79]..1. Operating Principles for Erbium Doped Fiber Amplifier The EDFA operating principles can be described by considering the first three energy levels for the erbium ions, which are shown in Fig.(.15). The figure indicates that each main energy level is split into a manifold of multiple energy 38

56 sublevels due to the Stark effect. There are three stats that are ground state ( 4 I 15 / ), the metastable state ( 4 I 13/ ), and pump state ( 4 I 11/ ) [7]. Fig.(.14): EDFA pumping schemes: (a) co-directional pumping, (b) counter-directional pumping (c) dual pumping [9]. Fig.(.15):Schematic diagram of erbium ion energy levels and the spontaneous lifetimes of excited levels [58]. 39

57 4 4 Among the several transition that can be used to pump EDFAs, I15/ I11/ 4 4 and I15/ I13/ transitions, corresponding to 980nm and 1480nm pumping, respectively, are the most efficient pump bands [19]. For 980nm pumping, the EDFA behaves like a three level system; the energy from the pump laser boosts the erbium ions from the ground state ( 4 I15 / manifold) to the pump state ( 4 I11/ manifold), from which the ions relax to the metastable state ( 4 I13/ manifold). Since the relaxation rate of ions from the level 3 to level is much higher than the pumping rate, the population of erbium ions in the pump state is negligible and a high degree of population inversion can be achieved. Metastable state often has a long lifetime is around (10ms) in the case of a good amplifier [9]. For 1480nm pumping, the energy from the pump laser excites the erbium ions in the low levels of the 4 I15 / to the high level in the 4 I 13/, leading to a population inversion between the two main energy levels, the 1480nm pumped EDFAs can also be analyzed by considering only two levels [13,7]. Amplification at wavelengths approximately between 150 to 1560nm takes place between the low levels of the 4 I15 / and the high in the 4 I 13/. Once in the metastable state, the ions are stimulated by the signal propagation along the EDF, causing them to decay back to the ground state; this results in the emission of photon in phase with the signal, thus amplifying the signal by simulated emission. Besides the stimulated emission, there exists a competing process of spontaneous [9,80]. The photons arising from this process have no coherence characteristics with respect to the signal; some of the spontaneously emitted photons are captured by the fiber and then amplified, leading to ASE at the EDFA output. Therefore, the output spectrum of an EDFA consists of the amplified input spectrum, and the broadband ASE, which is known to be the major EDFA noise contributor [7]. 40

58 CHAPTER THREE Theoretical Analysis of Erbium Doped Fiber Amplifier

59 CHAPTER THREE Theoretical Analysis of Erbium Doped Fiber Amplifier 3.1 Introduction As signals travel in optical communication systems, they are attenuated by optical fiber. Eventually, after some distance they can become too weak to be detected. One possible way to avoid this situation is to use optical amplifiers to increase the amplitude of the signal [1]. Doping a part of the optical fiber core by (Er 3+ ) ions in presence of external pumping power will lead to form the EDFA. The performance of this optical amplifier depends on (the power and the wavelength of the pumping laser, the power and wavelength of the input signal, amplifier length, ion concentration). These parameters will affect the characteristics of EDFA such as amplifier gain, gain saturation, noise figure and output power [1,13,17,80]. The gain is provided by pumping a medium, which undergoes population inversion when stimulated by an optical signal. Thus, an optical amplifier functions in much the same way as a laser. These devices are characterized by their gain efficiency, or the gain as a function of input power ( db / mw ). Only a certain range of optical frequencies, called the gain bandwidth, can be amplified by a particular device. All amplifiers have an upper limit on their output power, increasing the input power further will not produce any change in the output beyond this point. This effect is called gain saturation. There are various types of noise sources that affect optical amplifiers; in particular, these devices are sensitive to the polarization of the input light. Polarization sensitivity refers to the variations in amplifier gain with changes in the signal polarization [63]. In this chapter, the EDFA properties are examined using the rate equations and the propagation equations. The propagation equations are solved analytically using a novel method which are coincided the numerically methods. The graded order is very important factor that may be influenced the EDFA characteristics. 41

60 The proposed analytical solution may be found for any graded order, while the well known solutions presented only for the step index fiber. 3. Pumping and Gain Coefficient Amplification in optical amplifiers is through stimulated emission, the same mechanism that is used by lasers [19]. The process of stimulated emission can be happen if the photon energy h of the incident light is the same as the energy difference E E E1, the photon is absorbed by the atom, raise to the excited state. The excited atoms eventually return to their normal "ground" state and emit light in the process. Light emission can occur through two fundamental processes known as spontaneous emission and stimulated emission. The remarkable feature of stimulated emission is that the emitted photon matches the original photon not only in energy, but also in its other characteristics. Indeed, an optical amplifier is just a laser device without feedback. Its main ingredient is the optical gain, occurring when the amplifier is pump optically to realize population inversion. Depending on the energy levels of the dopant, pumping schemes can be classified as a three level or four level scheme[,6]. Fig.(3.1) shows the two kinds of pumping schemes. In both cases, dopants absorb pump photons to reach a higher energy state and then relax rapidly to a lower-energy excited state. The main difference between the three-and four-level pumping schemes is related to the energy state occupied by the dopant after each stimulated emission event. In the case of a three-level scheme, the ion ends up in the ground state, whereas it remains in an excited state in the case of a four level pumping scheme [14]. EDFA mark use of a three level pumping scheme [7]. The term symbols of energy levels is presented in appendix A. The McCumber relationship then yields the emission cross section in terms of the absorption cross section [9]. The emission and absorption between the two energy levels are characterized by the emission and absorption cross-section, ( ) and ( ), which satisfy the relation [78] 1 1 4

61 Fig.(3.1): Energy level diagram of erbium ions [19]. where E(T ) K ( ) ( ) 1 ( ) exp 1 E T K T (3.1) is the Boltzmann constant, T is the absolute temperature in Kelvin, and is the excitation energy required to excite one ion to its excited state at temperature T, and is the angular frequency. As an example, Fig.(3.) shows the spectra of ( ) and ( ) at T 95K (typical room temperature) for an 1 1 EDFA. Note that at higher photon energy (shorter wavelength), the absorption cross section is greater than the emission cross section compared to lower photon energy the reverse is true [77]. The emission and absorption cross-sections can be used to calculate an important amplifier parameter called the gain coefficient and is defined as [19,77] where N1 and N g( ) 1( ) N 1( ) N1 (3.) denote the atomic densities of the lower and upper states, respectively. This expression can be simplified by making some simple assumption, often satisfied in practice. Assume that the emission and absorption 43

62 cross-sections have similar shapes governed by () but different peak values, then it is possible to write the gain coefficient as [19] Fig.(3.): Absorption and emission-cross section of EDFA at T 95K [14]. where g1 and g g ( ) ( ) 1 g N N (3.3) g1 are the degeneracy of the lower and upper atomic state, respectively. For erbium ions, it is known that g g 7/ 8 [19]. It is possible to approximate the gain coefficient by [19] g( ) ( 1 / 1 )N N (3.4) where () is the transition cross section. The gain coefficient of g() can be calculated for both the three- and four-level pumping schemes by using the appropriate rate equations. The gain coefficient of a homogeneously broadened gain medium can be written as [19,81] g o g(, P) ( o ) 1 o P P sat (3.5) where go is the peak value of the unsaturated gain, o is the atomic transition frequency, o is the gain bandwidth, P and are the optical power and 44

63 frequency of the amplifier signal, Psat is the saturation power. Local gain can also be written as [14, 19, 81] g o g(, P) 1 ( ) T o P P sat (3.6) where is the angular frequency and T is known as dipole relaxation time and is typically quite small (~0.1ps) for fiber amplifier. Eq.(3.6) can be used to discuss the important characteristics of optical amplifiers such as gain bandwidth, amplification factor, and output saturation power. 3.3 Gain Spectrum and Bandwidth The role of the term P / Psat in Eq.(3.6) is to reduce the gain as signal power increases in the fiber. This phenomenon is common to all amplifiers and is referred to as gain saturation. If the amplifier operates at power levels such that P / P sat 1, the amplifier is said to operate in the unsaturated regime. The unsaturated gain coefficient has a Lorentzain shape that is characteristic of homogenously broadened systems [1,19] g o g ) 1 ( ) ( o T (3.7) The following conclusions can be derived from the above equation: 1) maximum gain corresponds to transition with angular frequency o, ) for o, gain spectrum is described by Lorentzian profile, see Fig.(3.3) that is characteristic of homogenously broadened systems, 3) gain bandwidth, which is defined as the full width at half maximum (FWHM), is g / T. Gain bandwidth g is defined by points in frequency where gain takes half the value at the maximum. In terms of frequency it can be written as [19] 1 g T g (3.8) As an example, THz when T 0. 1ps. Amplifiers with a relatively large g 3 bandwidth are preferred for optical communication systems. Because the gain is 45

64 then nearly constant over the entire bandwidth of even signal[6,14]. a multichannel The concept of amplifier bandwidth is commonly used in place of the gain bandwidth. The difference becomes clear when one considers the amplifier gain G, known as the amplification factor and defined as where Pin and Pout G P P out in are the input and output powers of the signal being amplified. (3.9) Fig.(3.3): Lorentzian gain profile g( ) and the corresponding amplifier gain spectrum G() for a two-level gain medium [6]. Let P(z) described as [1] be the optical power at a distance z from the input end. Its change is dp( z) g(, P) P( z) dz Assume a linear device (here power levels P Psat the signal power. Integration of the above equation gives (3.10) ) where gain independent of 46

65 P( L) G( ) exp L P(0) g( ) (3.11) where P ( 0) Pin is the signal input power. P L) Pout ( is the output power. Using Eqs.(3.5) and (3.11), yields [1] g ( ) exp o L G 1 ( o ) / o (3.1) This represents a small signal gain that depends exponentially on the amplifier length [77]. Both the amplifier gain G() and the gain coefficient g( ) maximum when o and decrease with the signal detuning o. However, are G() decreases much faster than g( ) seen in Eq.(3.1) [6,14]. because of the exponential dependence Amplifier bandwidth A is evaluated using the above solution. It is defined by two frequency points where power drops by 50%, i.e. P db 0 P 3. 5 max, which translates into is[1] 1 G G 3 db max is the maximum value of gain evaluated at o. It A o ln() ln( G ) ln() o (3.13) where Go is the peak value of the amplifier gain gol Go e [1]. Fig.(3.3) shows the gain profile g( ) and the amplification factor G() by plotting g / go and G / Go as function of 0 ) T. ( The amplifier bandwidth is smaller than the gain bandwidth, and the difference depends on the amplifier gain itself [6,14]. 3.4 Noise Figure The basic manifestation of noise in optical amplifiers is in the form of spontaneous emission. This incoherent radiation also propagates with the signal and interferes with the signal when it is detected by a photodetector. Spontaneous emission is present in a spectral interval corresponding to the gain spectrum of the amplifier, and the spectral density of the noise is proportional to gain [9,10]. The 47

66 noise that will be seen on a photodetector placed after an EDFA is comprised of signal shot noise, signal-ase beat noise, ASE-ASE beat noise, and interferometric noise. The first two of these are the shot noise generated in the receiver's photodetector due to the EDFA output signal power level and the ASE power level. The signal-ase term is caused by heterodyne mixing between the amplified signal and the ASE at the receiver's photodetector. The ASE-ASE term is generated due to heterodyne mixing of the ASE with itself [8]. Shot noise current arises due to random fluctuations, that are generated randomly when light falls on a photodetector. The electrical shot noise power will be related to the shot noise, a mean-square shot noise current given by [10] shot eif (3.14) 19 where e is the charge of the electron ( C ), I is the average current produced due to the incident light beam, and f which the noise is being considered. is the detector bandwidth over Thermal noise is generated as result of the random motion of electrons brought about by the finite temperature of the system. The mean-square thermal noise current is given by [63] where th 4k Tf (3.15) 3 k is the Boltzmann constant ( J / K ), T is the absolute temperature, and is the responsively of a photodetector. The dominant contribution to the noise current comes from the beating of spontaneous emission with the signal. This beating phenomenon is similar to heterodyne detection. The variance in the photocurrent to the signal-ase beat noise is given by [14] sig sp 4( GP in )( S sp ) f (3.16) The variance in the noise current due to the ASE-ASE beat noise is given by [9] sp sp 4 S sp f ( f / ) (3.17) 48

67 Signal-to-noise ratio ( SNR ) of optical amplifiers is degraded because spontaneous emission adds to the signal during its amplification. Amplifier noise figure NF is defined [8] ( SNR) NF ( SNR) in out (3.18) SNR refers to the electrical power generated when the signal is converted to electrical current by using a photodetector. The optical detector is assumed to be ideal and ղ=1. Thermal and shot noises are in shot th (3.19) The thermal noise contribution can be easily subtracted from the measurement because thermal noise is constant for a given load resistance at fixed temperature. Therefore, under the assumption of a shot noise limited source, the NF[9] 1 n NF G sp ( G 1) S G sp ( f hvg P in / ) (3.0) where N N ). The first two terms of Eq.(3.0) are dominant. Then, it n sp ( N1 is easy to show that the noise factor is given by [1] 1 n NF G sp ( G 1) G (3.1) The noise figure will always be greater than one, due to the fact that the amplifier adds noise during the amplification process. One can show that for an ideal amplifier for which n 1 and G 1 sp i.e., the best value achievable is NF n (3dB). sp 3.5 Gain Saturation The origin of gain saturation lies in power dependence of the g(, P) in Eq.(3.6). Since g(, P) is reduced when P becomes comparable to P sat, the amplification factor G decreases with an increase in the signal power. This phenomenon is called gain saturation. Consider the case in which incident signal frequency is exactly tuned to the gain peak ). Substituting g(, P) from Eq.(3.6) into (3.10), yields [1] ( 0 49

68 dp dz g o P 1 P / Psat (3.) Eq.(3.) can be easily integrated over the amplifier length. Using the initial conditions P( 0) Pin Pout / G and P( L) Pout, one may be found the following implicit relation for the large signal amplifier gain [6] G G o G 1 P exp G P out sat (3.3) Eq.(3.3) shows that the amplification factor G decreases from its unsaturated value Go when Pout becomes comparable to P sat. Fig.(3.4) shows the saturation characteristics by plotting G as a function of P out / Psat for several values of o G. A quantity of practical interest is the output saturation power sat P out, that defined as the output power for which the amplifier gain G is reduced by a factor of (or by 3dB) from its unsaturated value G. Using the property G / in Eq.(3.3) gives[1] o G 0 Here sat Pout P sat out Go ln P G o sat is smaller than Ps by about 30%. Indeed, by noting that Go (3.4) in sat practice ( G o 100 for 30dBamplifier gain), Pout ln( ) Psat 0. 69Psat. Note that, becomes nearly independent of Go for G o 0dB as in Fig.(3.4) [81]. sat P out Fig.(3.4): Saturated amplifier gain G as a function of the output power for several values of the unsaturated amplifier gaing o [1]. 50

69 3.6 Rate Equations The gain of EDFAs depends on many parameters such as erbium-ion concentration, amplifiers length, core radius, and pump power [6,14]. The amplification process in EDFA based on t he three - level model is shown in Fig.(3.1) for the case of a 980nm pump. A pump photon at a 980nm wavelength is absorbed by an erbium-ion in the ground state and jumps into the highest energy level. Thus through a non-radiative decay the ion loses its energy and arrives into a metastable state (level). Once there, a photon having wavelength of 1530nm can force a stimulated transition of the ion into its ground state, creating one additional photon and an amplification [1]. Here the three-level system has been reduced to an effective two-level system [9]. Population inversion is achieved by injecting power into the system through an external energy source, which is known as optical pumping. Pumping will excite Er atoms into the upper energy level [83]. Under the assumption that the pump wavelength is in the 1480nm region. The EDFA can be modeled using the propagation and rate equations for a homogeneous two-level laser medium. The transition rate of an erbium doped fiber can be defined as [83] Transition rate I h (3.5) where is the cross-section of the fiber, h is the Plank's constant, is the frequency, and I is the light intensity that can be defined as I P A eff (3.6) where P is the light power and erbium ions. Aeff is the effective area of the distribution of The stimulated absorption and emission rates between the ground state and the metastable state are denoted by W 1 and W 1, respectively and transition rates such as pump rate are denoted by R 1. From Eqs.(3.6) and (3.5), R 1, W 1, and W1 are defined as [79] 51

70 W W R s s p as h A s s p ap eff es h A h A eff eff P P s P s p ASE P P ASE P P ASE ASE (3.7) (3.8) (3.9) where se, sa are the emission and absorption cross sections for signal frequency s c s, and pa is the absorption cross section for pump frequency p c p. h is the plank's constant and c is the speed of light in vacuum. Aeff is the effective area of the distribution of erbium ions. Ps and Pp are the signal and pump powers, P and P ASE ASE are the forward and backward spontaneous emission power. s and p are the confinement factors of the signal and pump, which represents the overlapping of the optical mode and the erbium-ion distribution. Only the portion of the optical mode which overlaps with the erbium ion 3 distribution will stimulate absorption or emission from the Er transitions. Since the erbium is confined to the core of the optical fiber the mode intensity can more readily invert the erbium ions. However, the confinement factor is related to frequency and graded order. In section (3.9), we will present a theoretical management to deduce the accurate value of depending on the frequency and the graded order. Spontaneous decay rate A1 of the excited energy level, hence, it is defined as [77] A 1 1 spon where spon has been measured as 10m sec. depends on the fluorescence lifetime (3.30) The population densities of the two states, N1 and N, satisfy the following two rate equations [10,16]: dn dt dn dt 1 W W 1 1 N N 1 1 W W 1 1 N N R R 1 1 N N 1 1 A A 1 1 N N (3.31) (3.3) 5

71 W N 1 1 number of absorption from level 1 to level due to the signal at s. W N 1 R N 1 1 number of stimulated emission from level to level 1 due to the signal. number of absorption from level 1 to level due to the pump at p. A number of spontaneous emission from the level to level 1. 1 N The total erbium-ion density per unit volume is defined as N t N 1 N. Steady-state analysis is applicable when the rate of change of variable is much slower than the inherent time constants of the system. Mathematically, the steadystate solutions are obtained by setting the time derivatives in the rate equation to zero[11] dn dt 1 dn dt 0 Substituting Eq.(3.33) into (3.31) and (3.3) yields, 0 W1 N1 W1N R1 N1 A1N (3.33) (3.34) 0 W1N1 W1N R1N1 A1N (3.35) Using these equation, the population densities of the 4 I15 / and 4 I13 / will be [84] N 1 N t 1 W 1 1 W spon 1 W 1 spon spon R 1 spon (3.36) N N R W 1 spon 1 spon t 1W1 spon W1 spon R1 spon (3.37) 3.7 Propagation Equations The equations that describe the propagation of P, P, P, and P s p ASE ASE are written as bases on the Giles and Desurvire model [84] dp dz dp dz dp dz s p ASE P P p s P p ASE s N N P (3.38) se sa 1 s s N N P (3.39) s pe pa 1 p p N N N h v P (3.40) se sa 1 se s s s ASE 53

72 where z is the coordinate along the EDFA, The signal positive (+) means a forward propagation beam and signal negative (-) for backward propagation, P s is the forward and backward signal, respectively. Pp is the pump power of the EDFA in forward direction, PASE are the forward and backward ASE of the EDFA, respectively. The second term on the right hand side of Eq.(3.40) is the spontaneous noise power ( P o ASE h v ) of the EDFA within the EDFA s homogeneous bandwidth ( v ), this bandwidth noise can be estimated from v ( ( v) / dv. Thus the two-level amplifier system can be fully 0 e e. peak ) characterized using Eqs.(3.38) to (3.40) that describe the propagation of the signal, pump and ASE along the EDFA. Note that, s and p are the absorption coefficients per unit length of signal and pump, respectively, which are defined as[6] N s s sa t (3.41) p p pa N t (3.4) These equations govern the evolution of the attenuation to the signal and pump powers inside an EDFA. In short fibers, these losses are negligible. However, they should be taken into account for long fibers especially for the distributed erbium doped fiber. 3.8 Present Analytical Solution There are many published studies in the scientific literatures [19,84,85] that analyzed the performance of EDFA and solved Eqs.(3.38) to (3.40), numerically. In this section, we propose an analytical solution. We see that the behavior of the erbium ion concentration of level 1 and is constant along most of the length of the amplifier ( and that is depends on the pump power) after that the concentration decreases (for N ) or increases (for N 1 ) smoothly. So, for short length, we can assume that the values of N and N 1 are constants. Our model assumes that the amplifier consists of many equal short concatenated segments. In 54

73 this model, the concentrations may be assumed as constants at each segment of EDFA. Our assumption is necessary to consider the differential equations with constant coefficient. So these equations may be solved analytically. To clarify the derivations, let A p B C pe N N s se sen s N h v s sa pa N 1 1 s p Substituting A, B, and C into Eqs.(3.38)to (3.40), yields dp dz dp dz p s AP BP p s (3.43) (3.44) dp dz ASE BP ASE C Note that, according to our assumption, the parameters A, B, (3.45) and C will be invariant through each segment. Their variations are limited in the end points only. The solution of the pump equation for the concatenated sections will be P P P P p p p p 1 ( z) P (0) e p (z) P p A z ( z) e (( N 1) z) P p p Az AN 1z ( N ) ze AN z ( N 1 ze ( N z) P ) These equations may be generalized to yield P ( L) P p p AN z ( N 1) z e (3.46) where L N. z represents the length, N is the number of sections, P (0) p is initial pump power that inters in the forward direction. It is important to note that the coefficients A's may be positive or negative depending on the operating conditions. The entire description of variation depends on the recursive solution in the concatenation segments, which is P ( iz) P p p Ai z ( j 1) z e (3.47) 55

74 where 1 j N is an integer number. For backward pump, the recursive relation will be P ( L p jz) P p AN j1z L ( j 1) z e (3.48) Using a similar procedure, the recursion formula of the signal will be P S ( jz) P S B jz ( j 1) z e (3.49) The propagation equations of P and P ASE ASE are non-homogeneous differential equations which can be solved as following: First, for forward ASE, we have the solution P ASE ( z) ke Bz C B (3.50) where the constant k may be determined using the boundary conditions at z 0 to yield C k P ASE (0) B Substituting Eq.(3.51) into (3.50), yields P ASE C ( z) P (0) Bz ASE e B (3.51) e Bz 1 (3.5) The last equation may be reformed for the concatenated segment to yield P P P ASE ASE ASE B1z C1 B1 ( z) PASE (0) e e B ( Nz) P 1 z Bz C B ( z) PASE ( z) e e B ASE 1 z B z C N N BN z ( N 1) ze e 1 1 From these equations one can deduce the recursion formula P ASE ( jz) P ASE C B N Bjz j Bjz ( j 1) ze e 1 (3.53) B j where P (0) ASE refers to the initial spontaneous emission power for the forward direction. Second, for backward ASE, using a similar steps for forward ASE, the reconstructed PASE will be 56

75 P P ASE ASE B Z C N N BN z ( L z) PASE ( L) e ( e 1) B B 1Z C N N 1 BN 1 ( L z) PASE ( L z) e ( e B N N 1 z 1) P ASE B1Z C1 B1 ( L Nz) PASE ( L ( N 1) z) e ( e B 1 z 1) P ASE B1Z C1 B1 ( 0) PASE ( z) e ( e B In this case, the resulted recursion relation will be P ASE 1 z 1) B 1Z C N j N j 1 BN j1 ( L jz) PASE ( L ( j 1) z) e ( e B N j1 z 1) (3.54) The initial value of P (0) ASE in forward direction is zero and the initial value P ASE (L) in the backward direction is zero. There is no problem in calculating PASE because the initial condition is known in z 0, so we can find the ( jz) iteratively using P (0) P ASE ASE. The calculated PASE of the first segment becomes the initial value for the second segment and so on. This is correct in the numerical methods or our proposed method. But when we start to calculate PASE in the first segment, we found that there is no known initial value in z 0, so, the solution of Eq.(3.54) for PASE must begin from the end of the fiber ( z L ) because the initial value ( L) 0. And by solving Eq.(3.54) beginning from the last segment ( N ) in backward direction with taking the value of P ASE ( L) 0 as initial value we get the value of ( L z) P ASE P ASE for ( N 1) segment. We take ( L z) P ASE as the initial value to calculate the value of ( L z) P ASE for the ( N ) segment, and so on. We are obliged to calculate Eqs.(3.43) to (3.45) and the related equations three times. In the first one, we calculate P only and ASE P, p PS for all segments in forward direction. We put PASE 0 for all segment. In the second time, we begin the calculation from the last segment of EDFA using the result of the forward calculation to find the values of P using ( L) 0 as initial value. In the third time, we repeat the first ASE P ASE 57

76 calculation using the stored values of PASE beginning from the first segment in forward direction. 3.9 Confinement Factor of Graded-Index Fibers Similarly to all other electromagnetic phenomena propagation of light in optical fibers is described by the wave equation. The wave equation can be derived from Maxwell equations. In the following assumptions: 1) nonlinear polarization and dispersion are negligible. ) the imaginary component of dielectric constant ( ) ( n( ) ic / ) neglected, because the loss in optical fiber is low in spectrum range of interest for fiber optics techniques. The wave equation takes form of equation known as Helmholtz equation [59] ~ ~ ~ E n ( ) k o ~ E 0 where E E( r, ) is Fourier transform of the electric field 1 ~ E( r, t) (, ) E r e i t d is (3.55) (3.56) is the Laplacian operator that takes the following form in the cylindrical coordinates r 1 r r 1 r z (3.57) Similar relations exist for the magnetic field H ( r, t). E (or H ) has six components. Two of these components are independent. It is customary to choose as the independent components and express 58 ~ E ~ ~, E, H and E ~ z and H ~ z H ~ in terms of E ~ z H ~ z. The wave equation in cylindrical coordinates for the electric field E ~ z component along the fiber axis is given by [79] ~ ~ ~ ~ Ez 1 Ez 1 Ez Ez n ( r) k r r r r z The wave equation for variables. Solution for E ~ z E ~ z o ~ E z 0 and (3.58) is easily solved by using the method of separation of in cylindrical coordinates takes the from

77 ~ E z ( r) ( ) Z( z) (3.59) Substituting Eq.(3.59) into (3.58), yields 1 d Z 1 d 1 d 1 d n k o Z dz dr r dr r d (3.60) Right side of the above equation does not depend on z, this means that the left side of the equation must be a certain constant. Let us denote this constant as where i, (3.61) where is the propagation constant, and is the attenuation. As the is low, it can be neglected in the above equation. So, Eq.(3.60) may be separated to the equations 1 d Z Z dz (3.6) d dr 1 d r d r d n k r r o dr (3.63) The left side depends on only while the right side depends on the radial distance only. So, each side may be equal to a constant, say dependence will be where C1 and C ( ) C1 cos( ) C sin( ). The solution of (3.64) are the constants determined from boundary condition. The function ( ) must meet the condition of rotational symmetry ( ) ( ), therefore must be an integer. The radial dependence part will be r d d r dr dr k n ( r) r 0 (3.65) o Eq.(3.65) may be normalized using the assumption R=r/a to yield d dr 1 R d a dr n ( R) k 0 (3.66) o R 59

78 where r is radial coordinate, a is core radius, n(r) is the refractive index distribution in the fiber, is the propagation constant, is a parameter coming from the azimuthal part after separation of variables, and is any transverse field component either E -field or H -field. In fiber with graded index core, n(r) falls from n 1 the refractive index value on the axis of the fiber to n the refractive index value at the core-cladding interface. The definition in Eq.(.1) may be reformed to obtain f q n n f ( R) n 1 R (3.67) n( R) n1 1 1 q ( R) R is called the profile function. The exponent q in f (R) gives the core index profile. The distribution for q 1 is called triangular profile, for q called parabolic profile, and for q it is, represents the profile of step index fiber [86]. Fig.(3.5) illustrates the index profile for many values of the graded order q. The exponent q is very sensitive in the analysis of modes in optical fibers, where the reconstructed modes are affected by q in terms of number of modes and the corresponding distribution. It controls numerical aperture, intermodal dispersion, zero dispersion wavelength etc. 1.5 q= q=10 refractive index q=1 q= q=5 q= q= r/a Fig.(3.5): The refraction index profile for different value of q. 60

79 61 Eq.(3.66) may be rewritten in the form (3.68) 0 ) ( 1 R R f V U dr d R dr d where (3.69) 1 n k a U o (3.70) n 1 n ak V o V is the normalized frequency and U is the normalized propagation constant at core region. Eq.(3.66) or (3.68) has two linearly independent solutions. For particular, the solutions are denoted as m LP. The mode with highest is the fundamental mode. Within the core, one has to find the solution of Eq.(3.68) with q R R f ) (. The solution in the core region can be found using power series technique as in following form (3.71) ) ( 0 n n n core R a R As in the solution of Bessel's equation, we take 0 0 a and 0 1 a. For any q,!) ( 1 0 a. Here we used the same normalization factor used in the solutions of Bessel's equation, because for q, the series must converge to Bessel function. Using the first and second derivatives of Eq.(3.71) into Eq.(3.68), yields (3.7) n n n n n q n n n n R a R R V U R n a R n n a Eq.(3.7) may be reformed to explain (3.73) 0 0 q n n q n n n n n n n R a V R a U R n n a The recursion relation that may be deduced using Eq.(3.73) consists of two parts (3.74) ) ( 1 ) ( q n if n n a V a U q n if n n a U a q n n n n

80 6 Note that; for even q all the odd coefficients are zero. For odd q, the odd coefficients are zero up to q n ; all the coefficients are non-zero for q n. Note that, our series solution can be found only for integer values of q, while the other values may be extracted through the simulations. For step index fiber q, Eq.(3.74) will be (3.75) n n a U a n n The series solution for this case will be the well known relation [86] (3.76) ) ( ) ( 0 UR J R a R n n n core For parabolic profile (i.e. q ), the solution will be Laguerre polynomials 70 (3.77) ) ( ) ( 1 m core L e R where V R, m is the radial mode number and (3.78)!!! 1)!( ) ( 0 n s s s n s s n s n L is the generalized Laguerre polynomial when for convenience the substitution m 1 n has been mode. For other index profiles, the radial function takes the form (3.79) ) ( ) ( ) ( 1 0 q n q n n n q n n core R n a V a U R n n a U R a R The determination of radial function for each index profile is very important to explain the spot size w that will control the confinement factor. In the cladding region, the refraction index is constant, i.e. 1 ) ( R f. So Eq.(3.68) will be (3.80) 0 1 R V U dr d R dr d The normalized propagation constant in cladding is defined as (3.81) n k a W o So, W U V and Eq.(3.80) will be

81 d 1 d W dr R dr The general solution of last equation clad (RW) R 0 where (RW) is the standard modified Bessel function. For q (3.8) (3.83), in order to calculate the mode fields it is necessary to determine the values of. This may be achieved by matching the tangential field components at the core/cladding ( R 1) interface, resulting in the following characteristic equation [86] U J U ) W J ( U ) ( 1 1 ( W ) ( W ) (3.84) The required values of correspond to the crossing points of Eq.(3.84) and the curves of U V W, see Fig.(3.6). For a 50 m, NA 0.1, 850nm, then are 11 modes for this particular azimuthal mode number. Each discrete mode with the same value is designated by its radial mode number m, when m 1 to the fundamental mode with highest value of and m 11 mode nearest cut-off. is[87] corresponds correspond to the For q, the index profile will be parabolic and the characteristic equation V L 1( V ) W ( V ) 1 m 1 V Lm ( W ) ( W ) (3.85) Fig.(3.6):Matching tangential fields at core/cladding interface to determine propagation constants in step-index fiber [70]. 63

82 In our general case, when q 1,,3,, the characteristic equation takes the form R) ( R) ( W ) W ( W ) core ( 1 core where the boundary conditions ( R) ( WR) R 1 with the identities ( W ) and ( R) ( WR) R 1 1 z) (3.86) were used ( z) ( ) (3.87) ( 1 1 z 1( z) ( z) 1( z) z The confinement factor of graded index fiber may be defined as (3.88) q a 0 q R1 b 0 q R rdr a rdr R rdr (3.89) where q R is the radial distribution in the core, which may be described by Bessel functions, Laguerre functions, or the general distribution. R modified Bessel function. is described by the Note that, the confinement factor is determined analytically in the step index fiber, which may be approximated by 1 e b w where b is radius of doping. But our analysis take the general formula, such that the confinement factor will be computed using the numerical management. The determination of NA for graded index fiber is more complex than for step index fiber. In graded index fiber, NA is a function of position across the core end face. This is in contrast to the step index fiber, where NA is constant across the core. Geometrical optics consideration shows that the light incident on the fiber core at position r will propagate as a guided mode only if it is within the local numerical aperture NA(R) defined as [60] at that point. The local numerical aperture is NA(0) NA( R) 0 1 q R R 1 R 1 (3.90) where the axial local numerical aperture NA(0) 64 is defined as

83 NA( 0) n1 n n1 (3.91) It is clear that NAof a graded index fiber decreases from NA(0) from the fiber axis to the core cladding boundary. As the use of mode distribution (R) to zero as moves given by the previous management is cumbersome in practice, the fundamental mode is often approximated by Gaussian distribution of the form F( R) e r w (3.9) where the width parameter is determined by curve fitting or by following a variational procedure. The quality of fit is generally quite good for values of V in the neighborhood of. The spot size (w) can be determined from an analytic approximation accurate to within 1% for 1. V. 405 and given by [6] w a V.879V 3 6 (3.93) The spot size w is different for each graded order q, where the smaller q is the larger spot size w. So, it is predicated that the confinement factor will be larger for the smallest graded order q. The factor controls the EDFA equations, so the graded order will be very important to enhance the EDFA properties. 65

84 CHAPTER FOUR Results and Discussion

85 CHAPTER FOUR Results and Discussion 4.1 Introduction There are different parameters that may restrict the EDFA performance such as; fiber length, input powers, wavelengths of pump and signals, fiber type, and another. This work focuses on the solution of the propagation equations of pump, signal, and ASE signals and the related rate equations. The suggested model is simulated in this chapter in order to maximize the EDFA performance. There are many scientific papers that studied these topics using different assumptions and restrictions [33,39,84]. In this work, the simulations based on the determination of the EDFA characteristics that related to the fiber graded order. This issue is very sensitive since the EDFA parameters are altered by changing the fiber type. However, the procedure in this work depends on the examination of the results that relate to the different graded orders and then the best results are used to enhance the EDFA performance. The procedure of extraction as follows: for a certain value of q, Eq.(3.79) used to determine the radial distribution. Eq.(3.86) and 66 U V W will be plotted to determine the intersection points that will result the corresponding numbers and m. These numbers will be substituted again into Eqs.(3.79) and (3.83) and the resulted radial distribution is used into Eq.(3.89) to yield the field distribution that corresponds to the parameters, m, and q. For each pair, m : the relation between V and the normalized propagation constant b W /V be plotted. The above procedure will be continued to explain the related parameters such as: confinement factor and attenuation coefficients which in turn, are used in the solutions of rate and propagation equations. Fig.(4.1) summarizes the steps that use to construct the fiber characteristic for any graded order. can

86 start Input a,n 1,n q=1 λ=λ 1 Fig.(4.1): The general flow chart to construct the modes characteristics. Compute V l=0 Compute the intersections between the characteristic equation and V =U +W to find U,V,W for each intersection point m=1 the radial number Find; ψ core (R) & ψ clad (RW) core ( R) f ( R) ( RW ) R 1 R 1 Construct the different parameters Yes m=m+1 is m 10 No l=l+1 is 10 Yes No λ= λ 1 +Δλ Yes is 1650nm Yes is q 00 q=q+1 No No stop 67

87 4. The Effect of Confinement Factor Fig.(4.) shows the radial distribution of the light field for the fundamental mode in many fibers with different graded order. It is clear that the different cases have a semi-gaussian distribution with different widths. Generally, the different fundamental modes that corresponds to the light fields in fibers with different graded order have a maximum peak at the mode center. The width of distribution is different depending on the graded order. The intensity at the fiber center has the same value, whereas for q 1, the field penetrates more into the cladding as compared to the case q. Also, It can be concluded that when the value of q is high, then the part indeed core of the field will be more. The curves are found to spread outward in the cladding region as q value decreases. Fig.(4.3) shows the variation of the confinement factor () as a function of the graded order of single mode fiber for 5 light fields of wavelengths ( )nm. It is clear that the value of confinement factor increases by increasing the value of the graded order. The confinement factor will tend to be constant for the higher q. Moreover, high wavelengths have low curves. After the value q 10, the confinement factor becomes almost constant. We find that the confinement factor is 67% for q 1, while 85% for q 10 at the wavelength 1550nm, this wavelength has a minimum attenuation and EDFA amplifier works efficiently at this wavelength. Spot size is one of the important characteristics of graded index fiber as it can be used to determine some quantities like substantial splice loss, and micro bending loss that is sensitive to fiber bend. For single mode fibers, the spot size may be estimated from the V number for various values of graded order as in Fig.(4.4). This shows that the normalized spot size (to fiber radius) decreases by increasing V. As V value approaches.405 the spot size (w) is equal to or slightly greater than the core radius (a) for various q. For V. 405 the spot size is significantly larger than the core radius for each values of q, therefore that part of the beam penetrates into cladding. It can be deduced that For V. 4 the value of 68

88 spot size increases and the value of confinement factor decreases with decreasing V. Generally, the variation of graded order change the characteristics of mode. That is; for each graded order there is different: mode distribution, confinement factor, and spot size. Fig.(4.5) shows the effective area ( A eff ) as a function of wavelength for different values of q, where effective area depends also on fiber parameters such as the core radius and the core cladding index difference. Note that the effective area must be large in case q 1 and less whenever q is increased, for example if we focused on the wavelength 1550nm, you'll notice that A eff 56m when q 1 whereas may its value A eff 38m when q which is almost Aeff the same value which is equal to 37m when q, we conclude from this that in the case of Aeff the graded order when graded parabola q, is similar in the case of Aeff stepindex fiber when using light with a wavelength of 1550nm. Fig.(4.6) shows the normalized propagation constant (b) as a function of V for different values of graded order (q) for the fundamental modes. The cut off value of V for single mode operation in a fiber with a graded refractive index profile that follows from Eq.(3.67) is different from the.405 value used for step index fiber. For a parabolic index fiber ( q ), the cutoff value is 3.53; for a triangular index profile ( q 1), it is The estimated cut off value of V for the fundamental mode propagation may be approximated by V ( / q). If other parameters are the same, the core diameter of the graded index single mode fiber can be a factor of 1 (/ q) larger than the equivalent step index fiber. A parabolic index fiber ( q ) provides an improvement by a factor of, and the triangular profile fiber ( q 1) provides a 3 improvement. Note that, the characteristics curves of the modes are shifted to right by decreasing the graded order. That is; the cut-off frequency will be increased for the smaller values of graded order. The higher values of q leads gradually to the step index fiber that has the minimum cut-off frequency. Note that for q is higher 69

89 than 100, the profile will be step index. To get the results for step index fiber with power series, the series is convert to Bessel function as in Eq.(3.74) by making a nq 0, where the parameter b which varies between 0 and 1 is particularly useful in the theory of single mode fiber because the relative index difference is very small. Fig.(4.7) shows the confinement factor at different values of q and V as a function of wavelength. It is clear that all curves in this figure are linear. That is; the confinement factor is inversely proportional with light wavelength for all values of V and graded order. In general, the largest confinement factor can be satisfied for the higher graded orders and higher V. This behavior is due to the increasing of the core radius by increasing V. That is; the fiber core may contain a more power. In turn, the confinement factor may be raised. There are many scientific papers [6,19,88] that supposed a fixed value of the confinement factor, but this correction proved that the confinement factor is wavelength bandwidth for all q, but this figure proves that the confinement factor is not constant with wavelength bandwidth and for all q. Consequently, our results introduce a correction of the well known EDFA properties such as gain bandwidth and power variation. Note that the confinement factor decreases by increasing wavelength while it increases with increasing q, and V. Although nearly (83-93)% of the mode power resides in the core for V. 6339, percentage of confinement factor drops for V , at all values of the q. 4.3 Graded Order Effects Fig(4.8a,b) shows the power of maximum ASE forward and backward as a function of wavelength for different values of q for single mode fiber when the pump power, input signal power, erbium doping density and length of EDFA are constants. For each value of q, it is clear that the forward and backward ASE powers increase up to maximum value by increasing the wavelength to 1560nm. After that, these power will be dropped rapidly. That is; the maximum range 70

90 happens around 1500nm where the larger power has the larger q. Note that, in case q 1, the maximum ASE power forward of about 0.15W and ASE power backward of about is 0.18W while in case q W and ASE power backward is.38w 1560 m. Where power of ASE forward at 0.04W while in case backward at a rate of.w 71 the maximum ASE power forward 0 are obtained at wavelength q higher than the q 1 at a rate of 0. ASE arise spontaneously during the amplification process but their value remains a small and ineffective. Fig.(4.8c) shows the behavior the output signal power with respect to the wavelength for different values q. The maximum output signal power is at 0.115mW q and 0.09mW at q 1 in the wavelength 1560 nm. The effect of q is obvious, by increasing the value of q, the spectrum curve peaks will be high, especially as the peak in case of wavelengths that are less than q. Note that the differences between the curves 1560nm for any value of q are minor differences and can be neglected, but the differences increase for the wavelengths more than 1560nm. Fig.(4.8d) shows gain spectrum of EDFA as a function of wavelength for different values of q for signal mode fiber. A gain peak of gain peak of 0dB for q and 19dB for q 1 are obtained at wavelength 1565nm and other are 16.5dB for q and 11.5dB for q 1 are obtained at wavelength 1590 nm. The presence of double peaks at 1560nm and 1600nmas depends on a large number of parameters such as N t, L, a, P p0 and P s0. Note that, for higher values of q, the gain curves will be better. Hence the value of q increases, gain becomes almost similar. Therefore, the gain in step index is the best. Note that, each graphics in Fig(4.8) significantly affect by both emission and absorption cross-section values of the interaction of light at different wavelengths with erbium ions. Fig.(4.9a) shows the gain of EDFA as a function of pump power for different q when the input signal power, erbium doping density, wavelength and length of EDFA are constants. Note that the gain of EDFA sharply increases with

91 the increasing pump power. The gain increases from increased from achieved with 1dB to 1dB at pump power 10mW to 1000mW for all values of q. A higher gain can be q. The gain will be constant for sat Pp P out, where the upper state population reaches almost to a constant level, the increase in gain becomes smaller with pump power. Fig.(4.9b) shows the optimal length as function of pump power for different values of q. The optimal length increases in increasing the pump power. After certain pump power the optimal length become approximately fixed in the point at which the pump power becomes equal to the threshold pump. The graph between L and P p becomes lower in increasing q. Fig.(4.10a) shows the gain of EDFA as function of input signal power for different q when the pump power, erbium doping density, wavelength and length of EDFA are constants. The gain values are gradually decreased with the increasing input signal power but it increases with q. At higher input signal power, the gain significantly depletes in the population inversion and the occurrence of saturation of the EDFA and the pump will be decreased as a result of the gain decreases rapidly with input signal power. Fig.(4.10b) shows the optimal length as function of input signal power for different values of the q. The optimal length decreased with increasing input signal power and q. Note that the optimal length at q is much less than q 1, where optimal length of the EDFA amplifier is best when it is a smaller length. 4.4 Gain of EDFA Fig.(4.11) shows the amplifier gain as a function of length for different values of the input signal power and wavelength in the step index fiber, i.e. q, for certain value of pump power and erbium doping density. It can be seen that the gain is maximum at a certain length for each wavelength and input signal power. The length that exhibits the maximum gain is called the optimum length. This may be attributed to the insufficient population inversion that happen due to 7

92 the pump depletion. In turn, the losses will be higher than the gain, which leads to decreasing the resulted gain. The maximum gain is achieved within a narrow range of the fiber length for each wavelength and signal power. The optimum length is shifted to right by increasing the signal wavelength. Generally, the optimum gain is controlled by the parameters such as wavelength, length and input signal power. Fig.(4.1) illustrates the gain as functions of wavelength for different values of input pump power and different lengths in step index fiber when the erbium doping density and input signal power are constant. In this figure, the performance is represented by the gain bandwidth and the flatness. Note that these issues are affected by the parameters: input pump power, fiber length and wavelengths. Note that, the gain spectrum changes and become more flat at L=60m when P p =100mW and this flatness will be less for P p =(00,300,400)mW. The best case may be determined according to the specified application where flatness and gain bandwidth are the keys for optimization. However, the best amplifier is affected by the erbium doped density and the signal power where the maximum gain satisfies at the maximum output signal power that affects by the pump depletion. All these parameters may be balanced to enhance the EDFA characteristics. Fig.(4.13) shows the gain as function of erbium ion density for different input signal power and fiber length in step index fiber when the pump power and wavelength are constants. It is clear that the maximum gain may be controlled by the optimum N t, L, s0 P. That is; the pump depletion may be affected by these parameters. However, to enhance the EDFA properties the parameters P s0, N t, L, P p, may be calibrated. For all cases, the EDFA gain does not has the best flatness. This property is well known in the literatures when then are many configurations that may be used to satisfy the required flatness. 73

93 4.5 Optimization of EDFA Fig.(4.14a,b) shows the ASE power forward and backward as a function of wavelength for different values of the pump power in step index fiber when the input signal power, erbium doping density and length of EDFA are constants. The power determination significantly depends on emissions and absorption crosssection for interaction the incident light with Er+3 ions. Therefore, the gain spectrums will be approximately similar to the cross-sections spectrum. There are peaks in these spectrums, its values increases with increasing pump power. Note that, the P ase- peaks values are greater than P ase+ peaks values, where the all peaks fall in wavelength 1560 nm, after this value it begins decreasing. Fig.(4.14c) shows the output signal power as a function of the wavelength for different values of the pump power at step index fiber. Note that the output signal power increases with increasing pump power, and spectrum peak increases with increasing pump power where all peaks fall in the wavelength nm, after this value it begins decreasing. Fig.(4.14d) shows the gain of EDFA as a function of wavelength for different values of the pump power at step index fiber. Note that the EDFA gain increases with increasing pump power. Note that, the gain spectrum peak is 4.5dB at 1560nm when pump power 1400 mw. Fig.(4.15a,b) shows the ASE forward and backward power as a function of wavelength for different values of the erbium doping density in step index when the input signal power, pump power and length of EDFA are constants. ASE power increases by increasing erbium doping density. Note that, increasing the Er +3 ions concentration will make the great spectrum. Note that, the P ase forward spectrum has less bandwidth and the peak will be less than P ase backward spectrum. Fig.(4.15c) shows the output signal power as a function of the wavelength for different values of the erbium doping density in step index fiber. Note that the increasing of erbium doping density leads to increase spectrum peak for output signal power. Fig.(4.15d) shows the gain of EDFA as a function of wavelength for

94 different values of the erbium doping density in step index fiber. Note that the EDFA gain increases with increasing erbium doping density and ions 4 3 concentration become fixed after exceeds 110 m. Note that, the gain spectrum peak is 1dB at 1560nm when Er ion concentration is 110 m. Fig.(4.16a,b) shows the ASE forward and backward power as a function of wavelength for different input signal power in step index fiber when the pump power, Er +3 ion concentration and length of EDFA are constants. As it is expected, higher stimulated emission and less spontaneous emission occur with increasing input signal power and therefore ASE power decreases, especially around wavelength of 1560 nm (because both emission and absorption crosssections in this wavelength are maximum). Fig.(4.16c) shows the output signal power as a function of the wavelength for different input signal power in step index fiber. Note that the input signal power increases this lead to increases the stimulated emission, in turn, means increases in output signal power. Fig.(4.16d) shows the gain of EDFA as a function of wavelength for different values of the input signal power in step index fiber. It can be seen that the EDFA gain decreases with increasing input signal power. The reason of this attributes to a higher input signal power significantly depletes the population inversion and leads to saturation of the EDFA. EDFA can provide amplifications of greater than at 1560nm when input signal power 1 W. 0dB 4.6 The Optimum Length Fig.(4.17a,b) shows the ASE power forward and backward as a function of length of EDFA for different values of the pump power in step index fiber when the input signal power, wavelength, erbium-ion concentration are constants. The ASE power is created and grown when traveling over the length of EDFA. Note that, the ASE power increases with EDFA length increase. The reason for this behavior attributes to the high value of ASE power in length when N 1 =N, then the carriers number in the level will be decreased, so the stimulated and spontaneous emission will be decreased. It is worth to be noted that lower pump 75

95 powers is not enough to invert the entire population and therefore the ASE might grow to a value where the upper population should be significantly depleted. Backward ASE travels over a longer distance and become much higher at the beginning of the EDFA which depletes the inversion and reduces the gain at the expense of the signal. Fig.(4.17c) shows the output signal power as a function of the length of EDFA for different values of the pump power in step index. The output signal power increases by increasing pump power and length of EDFA due to the stimulated emission. Output signal power peak is mW at pump power 500 mw. Fig.(4.17d) shows the upper state population ( N ) and ground state population ( N 1) as a function of length of EDFA for different values of pump power in step index. The two curves N 1, N intersect at distances (47, 55, 60)m at pump powers (100, 300, 500)mW respectively. Amplifier lengths for lesser lengths and higher pump powers, the N 1, N curves do not intersect. This means that the pump power is higher than the threshold power. For example, using a 30m long EDFA, N1 and N at m results shows that, at the end of than the N1 and hence N1 and N hand, if the length of EDFA is is m and 110 m, respectively. These 30m long EDFA the N is substantially higher curves do not intersect each other. On the other 60m at 500 mw, pump power at the end of EDFA is less than the threshold pump power which is not able to invert the population. As a result, N1 and N curves intersect each other at the end of the 60m long EDFA. Fig.(4.18a,b) shows the forward and backward ASE power as a function of position along a 80m long EDFA at different values of the wavelength in step index fiber when the input signal power, pump power, erbium-ion concentration are constants. The backward ASE in Fig.(4.18b) travels over an inverted section of fiber even at low pump power. ASE power forward is ASE power backward is 0.W at 1500 nm W at 1500nm while Fig.(4.18c) shows the output signal power as a function of the length of EDFA for different values of the wavelength in step index fiber. The output signal

96 power increases by increasing length of EDFA up to the peak and then it begins to drop because of an insufficient population inversion. The optimum length is at point at which the achieve higher output signal power for curve signal power peck is 0.06mW at wavelength 1574 nm. 55m 1574 nm. Output Fig.(4.18d) shows the population in the upper state ( N ) and ground stat( N 1) as a function of position along 80m long EDFA at different values of the wavelength in step index fiber. Note that the N remains fixed up to the length 40m and then decreases up to N 1. So that, the population inversion does not get. This behavior attributes to that the pump power is absorbed inside the EDFA due to the population inversion. Note that if the length of EDFA is shorter than 55 m, and 60m at 1574 nm, 1500 nm, and 1600nm respectively, N1 and N 5 m, curves do not intersect each other. On the other hand, N1 and N intersect together in 5 m, 55 m, and 60m then the optimum length is obtained for wavelength listed above. As a result, the optimum length for EDFA is 55m at 1500 nm. 4.7 Amplified Spontaneous Emission Fig.(4.19) shows the ASE forward and backward power as function of length of EDFA for different values of the input signal power and erbium doping density for step index fiber when the pump power and wavelength are constants. The Er +3 ions concentration, P ase+, P ase- and the difference P ase+, P ase- will be increased, The increasing in P s leads to increase the value of P ase+ and P ase- in all cases. The ASE power increases with increasing fiber length and erbium doping density due to the gain provided inside EDFA. Note that the backward ASE output is more effective than the forward ASE output. Power of ASE forward up to 0.05m and power of ASE backward up to 0.05W at W 4 3 and m Er +3 ion concentration. 1 input signal power 77

97 Fig.(4.): Field distributions for different graded orders. Fig.(4.3): Confinement factor as a function of the graded order for different wavelengths. Fig.(4.4): Normalized spot size as function of the normalized frequency for different graded orders. 78

98 Fig.(4.5): The effective area as a function of wavelength for different graded order. Fig.(4.6): The normalized propagation constant as function of the normalized frequency for different graded order. 79

99 Fig.(4.7): The confinement factor as a function of the wavelength for different values of q and V. 80

100 Fig.(4.8): The power of ASE, P s, and gain as function of wavelength for different values of the graded order, when N t = m -3, L=60m, P p0 =00mW, P s0 =1µW. 81

101 Fig.(4.9): The gain as a function of pump power for different values of the graded order, when N t = m -3, L=100m, P s0 =1µW and λ=1574nm. Fig.(4.10): The gain as function of input signal power for different values of the graded order, when N t = m -3, L=100m, P p0 =00mWand λ=1574nm. 8

102 Fig.(4.11): The gain as function of length for different values of the input signal power and wavelength, when N t = m -3, P p =00mW, q=. 83

103 Fig.(4.1): The gain as function of wavelength for different values of the length and pump power, when N t = m -3, P s =1µW, q=. 84

104 Fig.(4.13): The gain as a function of erbium doping density for different values of pump power and length of EDFA, when λ=1574.5nm, P p =00mW, q=. 85

105 Fig.(4.14): The power of ASE, P s, and gain as function of wavelength for different values of the pump power, when N t = m -3, L=60m, P s0 =1µW,and q=. 86

106 Fig.(4.15): The power of ASE, P s, and gain as function of λ for different values of the erbium doping density, when, L=60m, P s0 =1µW, P p0 =00mW. 87

107 Fig.(4.16): The power of ASE, P s, and gain as function of λ for different values of the P s0, for step index, when N t = m -3, L=60m, and P p0 =00mW. 88

108 Fig.(4.17): The power of ASE, P s, and N 1,N as function of L for different values of the pump power, for step index, when N t = m -3, P s0 =1µW, and λ=1550nm. 89

109 Fig.(4.18):The power of ASE, P s, and N 1,N as function of L for different values of the wavelength, for step index, when N t = m -3, P s0 =1µW, andp p0 =00mW. 90

110 Fig.(4.19): The power of ASE as function of L for different values of the P s0 and erbium doping density, for step index, when λ=1550nm and P p0 =00mW. 91

111 CHAPTER FIVE Conclusions and Future Work

112 5.1 Conclusions CHAPTER FIVE Conclusions and Future Work The main conclusions draw from this work are: 1. The best gain occurs by increasing graded order where the optimum gain will be at the step index fiber.. The achieved gain does not enhance if the input signal power was exceeded the value P s mW. 3. The larger graded order has the smaller optimum length. That is; the step index fiber has the minimum fiber length. 4. The power at the core is maximum for the step index fiber, while the decreasing of graded order will raise the power at the cladding. 5. The confinement factor increases with graded order for any wavelength. However, the confinement factor may be constant for q For a certain value of V, the spot size at the step index fiber will be maximum. So, the effective area is the best for step index fiber. 7. The gain may be increased by increasing Pp 0 or decreasing P s0. The gain sat will be constant beyond P out. So, the optimum length is minimum for step index fiber. 8. The amplifier characteristics: gain, gain bandwidth and the flatness may be controlled by using many stages. 9. The ASE powers are affected by the parameters: N t, P s0, P p0, L and wavelength. However, the P decreases with length but the P ase ase increases, up to maximum value depending on the same parameters. However, the ASE powers do not affect the amplifier characteristics for smaller values of input pump power. 9

113 5. Future Works There are still a lot of remaining issues that are related to present theoretical treatment, such as: 1. The effects of dispersion and nonlinearity are very important phenomena which may be included to enhance the present analysis.. The others rare-earth elements may be studied and comparing the different amplifiers to determine the characteristics of each element. 3. The flatness of EDFA may be enhanced using a cascade EDFA's. 4. The flatness may be achieved using a hybrid amplifiers, such as Raman- EDFA and SOA-EDFA. 5. The temperature can be affect the population inversion. So it can be change the resulted power. 6. The state of polarization of each input signal is very important factor. It may be included to enhance the modeling of EDFA. 7. The effect of FWM in WDM systems may be introduced to analyze the multi-channel amplification. 8. The new fibers, such as photonic crystal fibers, have good benefits, which can be introduced to enhance EDFA. 93

114 Appendix A Atomic Term Symbols

115 A single term symbol has the form number, and S 1 Appendix A Atomic Term Symbols S 1 LJ, where S is the total spin quantum is generally referred to as spin multiplicity due to the fact that it is also the number of possible J states for a given ( L, S ). J is the total angular momentum quantum number, and L is the total orbital quantum number, using the notation S 0, 1 P, D,. A term symbol represents a set of microstates which have the corresponding values of L, S, J,can be distinguished as such[89]. Erbium has the electronic configuration [Xe] 4 f 1 6s. It exists in 3+ charge state in most of the solid materials. Er basically emits in the green region ( 53 nm) of the visible spectrum and in the infrared region ( 1.5m ). The infrared emission is due to the electronic transition from the excited energy level 4 I15 and for the visible range emission due to the electronic transition of 4 I15 to H 11. The infrared ( 1.5m ) emission is very suitable for optical fiber communication due to minimum loss for this wavelength. For this purpose it is essential to know about its emission properties [9]. Given a particular electron configuration, the following procedure may be used to determine the term symbols: 1.Write down every possible, Pauli-allowed microstate corresponding to given electron configuration. As a check, you can use the formula (l 1)! N e!((l 1) e)! (A.1) where N is the number of microstate you should end up with for e electrons in a subshell with angular momentum l..catalog ml and m S for each microstate, and make a table which describes the number of states that were found which each value of ml and m S. 94

116 3.Remove symmetric, rectangular blocks of maximum possible area, of 1's from the table, and make an block of ml for L and the maximum value of 4.Once the table has been exhausted for each possible S 1 L term for each, using the maximum value of your S 1 L terms using L S,..., L 1 J J. 95 m S for S. Some points in this procedure really deserve clarification S 1 L term you created, create all 1.Removing symmetric, rectangular block of 1's. the logic here is that if an angular momentum L to symmetric about the center of the table. L, this correspond to a block of state which is.taking blocks of maximum possible area. Taking a particular column in a row when you could have expanded to larger rectangular block corresponds to selecting a particular ml or m S or S that generated it. You need the entire range of make a well-defined determination of L or S. Some trick to note for doing these evaluations are: 1.Filled subshells never contribute, their total ml and value at random, and trying to guess the value L m S or ml in every case to m S are universally 0..There is a symmetry between electrons and ''holes'' or the absence of electrons. That is, the term symbols corresponding to terms from s p are the same as 4 from s p. 3.These tables are basically a lot of show for incorporating the Pauli Principle. If your electrons reside in different orbitals, you can generate the possible values of L and S from L L1 L,..., L1 L and S S1 S,..., S1 S. Now that the term symbols have been determined, it is useful to determine the set of states with the lowest energy, or the ''Ground State Term Symbol''. This is done systematically with Hund's Rules, which are stated in order of importance as: 1.For a given electron configuration, the term with maximum multiplicity has the lowest energy. Since multiplicity is equal to S 1, this is also the term with maximum S..For a given multiplicity, the term with the largest value of L has lowest energy.

117 3.For a given term, in an atom with outermost subshell half-filled or loss, the level with the lowest value J of lies lowest in energy. If the outermost shell is more than half-filled, the level with highest value of J is lowest in energy. Erbium is the trivalent Er +3 state, which has an electronic configuration [Xe] f. The Er +3 ion has an incompletely filled 4f-shell, allowing for different electronic configurations with different energies due to spin-spin and spin-orbit interactions. Radiative transitions between most of these energy levels are parity forbidden for Er +3 ions. When Er is incorporated in a solid however, the surrounding material perturbs the 4f wave functions. This has two important consequences. Firstly, the host material can introduce odd-parity character in the Er 4f wave functions, radiatve transitions weakly allowed. Secondly, the host material causes Stark-splitting of the different energy levels, which results in a broadening of the optical transitions. Fig.(A.1) shows a schematic level diagram of the Stark-split Er +3 energy levels. Since radiative transition in Er +3 are only weakly allowed, the cross sections for optical excitation and stimulated emission are quite small, and the radiative lifetimes of the excited states are long, up to several milliseconds. When Er is excited in one of its higher lying levels it rapidly relaxes to lower energy levels via multi-phonon emission. This results in typical excited state lifetimes ranging from 1ns to 100ms. The transition from the first excited state ( 4 I 13 ) to the ground state ( 4 I 15 ) is an exception to this rule [19,89]. Fig.(A.1): Schematic representation of the splitting of the 4f N ground configuration under the effect of progressively weaker perturbations. 96

118 References

119 References [1] B. Elliott and M. Gilmore, "Fiber Optic Cabling", Second Edition, Newnes, 00. [] K. Iizuka, "Elements of Photonics", Volume II For Fiber and Integrated Optics, John Wiley and Sone, Inc., 00. [3] Y. Singh, " Studies on Placement of Semiconductor Optical Amplifiers in Wavelength Division Multiplexed Star and Tree Topology Networks", Ph.D. Thesis, Electrical Engineering Department, Indian Institute of Technology, [4] S. Shahi, "Analysis and Mitigation of the Nonlinear Impairments in Fiber- Optic Communication Systems", Ph.D. Thesis, The School of Graduate Studies of McMaster University, 013. [5] H. Dutton, "Understanding Optical Communications", First Edition, International Technical Support Organization, IBM Corporation, USA, [6] G. Agrawal, " Fiber-Optic Communication Systems", Third Edition, John Wiley and Sons, Inc., 00. [7] R. Tricker, "Optoelectronics and Fiber Optic Technology", First Edition, Newnes, 00. [8] B. Chomycz, "Planning Fiber Optic Networks", The McGraw-Hill, Inc., 009. [9] P. Becker, N. Olsson, J. Simpson, "Erbium-Doped Fiber Amplifiers Fundamentals and Technology", Academic Press, [10] A. Ghatak and K. Thyagarajah, "Introduction to Fiber Optics", Cambridge University Press, [11] M. Azadeh, "Fiber Optics Engineering", Springer, 009. [1] M. Watak, "Computational Photonics an Introduction with Matlab", Cambridge University Press, 013. [13] M. Bass, C. MacDonald, G. Li, C. DeCusatis, V. Mahajan, "Handbook of Optics", Volume V, Third Edition, The McGraw-Hill Companies, 010. [14] G. Agrawal, "Applications of Nonlinear Fiber Optics", Academic Press,

120 [15] M. Johnson, "Optical Fiber, Cables and Systems", ITU, 010. [16] A. Yariv and P. Yeh, "Photonics Optical Electronics in Modern Communications", Sixth Edition, Oxford University Press, 007. [17] K. Thyagarajan and A. Ghatak, "Fiber Optic Essentials", John Wiley& Sons, Inc, 007. [18] N. K. Dutta and Q. Wang, '' Semiconductor Optical Amplifiers'', World Scientific, 006. [19] M. Premaratne, and G. Agrawal, "Light Propagation in Gain Media Optical Amplifiers", Cambridge University Press, 011. [0] C. Headley and G. Agrawal, "Raman Amplification in Fiber Optical Communication Systems", Elsevier Academic Press, 005. [1] A. Ruffin, M.Li, X. Chen, A. Kobyakov, and F. Annunziata, ''Brillouin Gain Analysis for Fibers With Different Refrctive Indices'', Optics Letters, Vol.30, No.3, 005. [] M. Nikles, L. Thevenaz, and P. Robert, '' Brillouin Gain Spectrum Characterization in Single-Mode Optical Fibers'', Journal of Lightwave Thchnology, Vol.15, No.10, [3] M. Marhic, "Fiber Optical Parametric Amplifiers, Oscillators and Related Devices", Cambridge University Press, 008. [4] J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P. Hedekvist, " Fiber- Based Optical Parametric Amplifiers and Their Applications", IEEE Journal of Selected Topics in Quantum Electronics, Vol.8, No.3, 00. [5] J. Senior and M. Jamro, "Optical Fiber Communications Principles and Practice", Third Edition, Pearson Edition Limited, Edinburgh Gate, Harlow, Essex CM0 JE, England, 009. [6] D. Gloge, " Dispersion in Weakly Guiding Fibers", Applied Optics, Vol.10, No.11, [7] K. Wilner and A. Heuvel, Fiber-Optic DeLay Lines for Microwave Signal Processing, Proceedings of the IEEE, Vol.64, No.5,

121 [8] D. Messerschmitt, " Minimum MSE Equalization of Digital Fiber Optic Systems", IEEE Transaction on communications, Vol.6, No.7, [9] M. Tur, J. Goodman, B. Moslehi, J. Bowers, and H. Shaw, "Fiber-Optic Signal Processor with Applications to Matrix-Vector Multiplication and Lattice Filtering", Optics Letters, Vol.7, No.9, 198. [30] B. Kim, J. Blake, H. Engan, and H. Shaw, "All-Fiber Acousto-Optical Frequency Shifter", Optics Letters, Vol.11, No.6, [31] M. Farries, P. Morkel, R. Laming, T. Birks, D. Payne, and E. Tarbox, " Operation of Erbium-Doped Fiber Amplifiers and Lasers Pumped with Frequency-doubled Nd:YAG Lasers", Journal of Lightwave Technology, Vol.7, No.10, [3] G. Agrawal, '' Amplification of Ultrashort Solitons in Erbium-Doped Fiber Amplifiers'', IEEE Photonics Technology Letters, Vol., No.1, [33] B. Pedersen, A. Bjarklev, O. Lumholt, and J. Povlsen, " Detailed Design Analysis of Erbium-Doped Fiber Amplifiers", IEEE photonics technology letters, Vol.3, No.6, [34] K. Rottwitt, A. Bjarklev, J. Povlsen, O. Lumholt, and T. Rasmussen, " Fundamental Design of a Distributed Erbium-Doped Fiber Amplifier for Long- Distance Transmission", Journal of Lightwave Technology, Vol.10, No.11, 199. [35] K. Rottwitt, J. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and T. Rasmussen, " Noise in Distributed Erbium-Doped Fibers", IEEE Photonics Technology Letters, Vol.5, No., [36] Y. Lai, Y. Chen, and W. Way, " Novel Supervisory Technique Using Wavelength-Division-Multiplexed OTDR in EDFA Repeatered Transmission Systems" IEEE Photonics Technology Letters, Vol.6, No.3, [37] M. Zervas and R. Laming, "Rayleigh Scattering Effect on the Gain Efficiency and Noise of Erbium-Doped Fiber Amplifiers", IEEE Journal of Quantum Electronics, Vol.31, No.3,

122 [38] S. Fleming, and T. Whitley, " Measurement and Analysis of Pump- Dependent Refractive Index and Dispersion Effects in Erbium-Doped Fiber Amplifiers", IEEE Journal of Quantum Electronics, Vol.3, No.7, [39] Y. Sun, J. Zyskind, and A. Srivastava, " Average Inversion Level, Modeling, and Physics of Erbium-Doped Fiber Amplifiers" IEEE Journal of Selected Topics in Quantum Electronics, Vol.3, No.4, [40] H. Kim, S. Yun, H. Kim, N. Park, and B. Kim, "Actively Gain-Flattened Erbium-Doped Fiber Amplifier Over 35nm by Using All-Fiber Acousto-optic Tunable Filters", IEEE Photonics Technology Letters, Vol.10, No.6, [41] Y. Sun, J. Zyskind, A. Srivastava, and L. Zhang, " Analytical Formula for the Transient Response of Erbium-Doped Fiber Amplifiers", Applied Optics, Vol.38, No.9, [4] K. Song, and M. Premaratne, " Effects of SPM, XPM, and four-wave-mixing in L-Band EDFAs on Fiber-Optic Signal Transmission", IEEE Photonics Technology Letters, Vol.1, No.1, 000. [43] B. Choi, M. Chu, H. Park, and J. Lee, " Performances of Erbium-Doped Fiber Amplifier Using 1530nm-Band Pump for Long Wavelangth Multichannel Amplification", ETRI Journal, Vol.3, No.1, 001. [44] R. Olivares and J. Souza, " Analysis of Power Transients in Erbium-Doped Fiber Amplifiers with Application to Wavelength Routed Optical Networks", Journal of Microwaves and Optoelectronics, Vol., No.6, 00. [45] S. Harun, N. Tamchek, P. PooPalan, and H. Ahmad," High Gain L-Band Erbium-Doped Fiber Amplifier with Two-Stage Double-Pass Configuration", Pramana-Journal of Physics, Vol.61, No.1, 003. [46] F. Vasile, and P. Schiopu, " The Determination of the Saturation Power for Erbium Doped Fiber Amplifier", Journal of Optoelectronics and Advanced Materials, Vol.6, No.4, 004. [47] D. Thomas and J.von, " Impairments of EDFA Dynamic Gain-Fluctuations in Packet-Switched WDM Optical Transmissions" IEEE Photonics Technology Letters, Vol.17, No.5,

123 [48] A. Tran, C. Chae, R. Tucker, and Y. Wen, "EDFA Transient Control Based on Envelope Detection for Optical Burst Switched Networks", IEEE Photonics Technology Letters, Vol.17, No.1, 005. [49] Y. Ezra, M. Haridim, and B. Lembrikov, " All-Optical AGC of EDFA Based on SOA", IEEE Journal of Quantum Electronics, Vol.4, No.1,006. [50] H. Chien, C. Lee, and S. Chi, " EDFA-Free All-Optical R Regeneration Using a Compact Self-Seeded Fabry-Perot Laser Diode", IEEE Photonics Technology Letters, Vol.18, No.9, 006. [51] L. Qiao, and Paul. Vella, " ASE Analysis and Correction for EDFA Automatic Control", Journal of Lightwave Technology, Vol.5, No.3, 007. [5] L. Vincetti, M. Foroni, F. Poli, M. Maini, A. Cucinotta, S. Selleri, and M. Zoboli, "Numerical Modeling of S-Band EDFA Based on Distributed Fiber Loss", Journal of Lightwave Technology, Vol.6, No.14, 008. [53] C. Berkdemir and S. Özsoy, " On the Temperature-Dependent Gain and Noise Figure Analysis of C-Band High-Concentration EDFAs with the Effect of Cooperative Upconversion", Journal of Lightwave Technology, Vol.7, No.9, 009. [54] M. Paul, S. Harun, N. Huri, A. Hamzah, S. Das, M. Pal, S. Bhadra, H. Ahmad, S. Yoo, M. Kalita, A. Boyland, and J. Sahu, " Wideband EDFA Based on Erbium Doped Crystalline Zirconia Yttria Alumino Silicate Fiber", Journal of Lightwave Technology, Vol.8, No.0, 010. [55] A. Naji, M. Mohammed, B. Zaidan, W. Al-Khateeb, K. Al-Khateeb, and M. Mahdi, " Anovel theoretical analysis of quadruple pass Erbium-Doped Fiber Amplifier", International Journal of the Physical Sciences, Vol.6, No.10, 011. [56] H. Sariri, M. Karkhanehchi, A. mohammadi, and F. Parandin, ''A Dynamic Simulink Model for Erbium-Doped Fiber Amplifiers Overmodulation in Presence of Amplified Spontaneous Emission Effect", Journal of Basic and Applied Scientific Research, Vol., No.3,

124 [57] P. Sharma, B. Kaushal, and S. Jain, "Study of Single and Multi Wavelength (WDM) EDFA Gain Control Methods", International Journal of Engineering Trends and Technology, Vol.4, No.5, 013. [58] G. Ivanovs, V. Bobrovs, S. Olonking, A. Alsevska, L. Gegere, R. Parts, P. Gavars, and G. Lauks, "Application of the Erbium-Doped Fiber Amplifier (EDFA) in Wavelength Division Multiplexing (WDM) Transmission Systems", Vol.9, No.5, 014. [59] G. Agrawal, "Nonlinear Fiber Optics", Fourth Edition, Academic Press, 007. [60] G. Keiser, "Optical Fiber Communications", Second Edition, McGraw-Hill, Inc., [61] M. Calvo, '' Optical Waveguides from Theory to Applied Technologies'', CRC press, Taylor & Francis Group, 007. [6] B. Woodward and E. Husson, "Fiber Optics Installer and Technician Guide", Sybex, 005. [63] C. DeCusatis and C.J.S. DeCusatis, "Fiber Optic Essentials", Academic Press is an Imprint of Elsevier, 006. [64] J. Laferrière, G. Lietaert, R. Taws, S. Wolszczak, "Reference Gulde to Fiber Optic Testing", Second Edition, Volum 1, JDS Uniphase Corporation, 011. [65] D.Bailey and E. Wright, "Practical Fiber Optics", Elsevier, 003. [66] Francis T.S.Yu, S. Jutamulia, and S. Yin, " Introduction to Information Optics", Academic Press, 001. [67] J. Powers, "An Introduction to Fiber Optic Systems", Second Edition,Irwin, Homewood, [68] S. Gistvik, "Optical Fiber Theory for Communication Networks", Third Edition, Hudiksvall Sweden, 006. [69] M. Ferreira, '' Nonlinear Effects in Optical Fibers '', John Wiley and Sone, Inc., 011. [70] A. Hallam, "Mode Control in Multimode Optical Fiber and its Applications", Ph.D. Thesis, Aston University,

125 [71] R. Syms and J. Cozens, "Optical Guided Waves and Devices", Mc Graul Hill International (UK) Ltd, 199. [7] K. Okamoto, " Fundamentals of Optical Waveguides", Second Edition, Academic Press, 006. [73] R. Paschotta, ''Field Guide to Optical Fiber Technology'', SPIE Press, 010. [74] F. Mitschke, "Fiber Optics Physics and Technology", Springer-Verlag Berlin Hedelberg, 009. [75] T. Andersen, "Applications of nonlinear Optics and Optical Fibers", Ph.D. Thesis, University of Aarhus, Department of Physics and Astronomy, 006. [76] J. Toulouse, " Optical Nonlinearities in Fibers: Review, Recent, Examples, and Systems Applications", Journal of Lightwave Technology, Vol.3, No.11, 005. [77] R. Quimby, " Photonics and Lasers An Introduction", John Wiley and Sone, Inc., 006. [78] P. Lecoy, "Fiber-Optic Communications", John Wiley and Sons, Inc., 008. [79] R. Hui, and M. O'Sullivan, "Fiber Optic Measurement Techniques", Elsevier Academic Press, 009. [80] M. Bass and E. Stryland, "Fiber Optics Handbook, Fiber, Devices, and Systems for Optical Communications", McGraw-Hill Companies, 00. [81] W. Shieh, and I. Djordjevic, "Orthogonal Frequency Division Multiplexing for Optical Communications", Academic Press, 010. [8] G. Agrawal, '' Lightwave Technology Telecommunication Systems'', John Wiley and Sons, Inc., 005. [83] J. Palais, "Fiber Optic Communications", Prentice-Hall, Fifth Edition, 005. [84] F. Akhter, M.Ibrahimy, A.Naji, and H. Siddiquei, '' Modeling and Characterization of All Possible Triple Pass EDFA Configurations'', International Journal of the Physical Sciences, Vol.7, No.18, 01. [85] P. Wahab, R. Asif, M. Imtiaz, F. Alam, '' Theoretical Modeling of Erbium Doped Fiber Amplifier Using Multi Quantum Well Laser'', International Journal of Science and Advanced Technology, Vol1, No6,

126 [86] C. Chen, "Foundations for Guided-Wave Optics", John Wiley and Sons, Inc, 007. [87] S. Bianchi, "Digital Holography Through Multimode Fibers", Ph.D. Thesis, Department Di Fisica, University Sapienza, 013. [88] C. Cheng, M. Xiao, " Optimization of an Erbium-Doped Fiber Amplifier with Radial Effects", Optics Communications, 54, 005. [89] W. C. Martin, R. Zalubas, and L. Hagan, '' Atomic Energy Levels- The Rare- Earth Elements'', NSRDS-NBS 60,

127 الخلاصة مع زیادة مسافة الا رسال أو سعة الا رسال في نظام الاتصالات البصریة فا ن النبضات المرسلة تعاني من زیادة التوھین وعلیھ فھي بحاجة إلى التضخیم لا عادتھا إلى صیغتھا الا ولیة. أن عملیة التضخیم تتم باستخدام أنواع مختلفة من المضخمات البصریات لكل منھا مزایا خاصة تعتمد على الوسط الفیزیاي ي وخواص النبضات المرسلة. عموما فا ن جمیع المضخمات غایتھا تحقیق أعظم ربح أعظم عرض ربح وأفضل تسطح. أن مضخمات الا لیاف البصریة المشوبة بالا ربیوم تعد الا فضل من حیث المواي مة كونھا تشكل بالكامل من الا لیاف البصریة. یمكن استخدام لیف بصري من أي نوع ویشوب القلب بالا ربیوم للحصول على مضخم. وطالما أن العمل في الاتصالات البصریة یتطلب ألیاف منفردة النمط فا ننا سنحدد أنفسنا باستخدام ھذا النوع. وحیث أن اللیف یمكن أن یكون متدرج graded أو غیر متدرج step-index فا ننا حللنا الحالة العامة للوصول إلى أفضل خواص للمضخم. من اجل دراسة التضخیم یجب تحدید سلوك النمط المنتشر (النمط الا ساسي) حیث تم حل معادلات ماكسویل في القلب core والطبقة المغلفة cladding لحالتي اللیف المتدرج وغیر المتدرج باستخدام طریقة الحل المتسلسلة. حیث أثبتت النتاي ج أن خواص النبضة في القلب تتبع نوع التدرج. حیث أن الحالة المعروفة وھي دوال بزل Bessel's تنتج عند اللیف غیر المتدرج وحالة دوال لیكوري Laguerre عندما تكون رتبة التدرج =q في حین أن جمیع الحالات الا خرى لا تعطي صیغة مغلقة معینة. تم اقتراح حل تحلیلي لمعادلات الانتشار للنبضات یعتمد تقسیم طول المضخم إلى مقاطع صغیرة. خواص الا نماط في الحالة العامة للتدرج اعتمدت مع ھذا الحل من اجل دراسة خواص المضخم. عموما النتاي ج العددیة تطابق تماما النموذج المقترح. أثبتت النتاي ج أن اللیف غیر المتدرج یظھر خواص أكثر مقبولیة لاستخدامھ في المضخم على عكس الا لیاف المتدرجة. عموما فا ن مضخم الا لیاف البصریة المشوب بالاربیوم یحقق ربح عالي وعرض حزمة ربح كبیر ولكنة یعاني في قسم منھ من عدم التسطح وھذه الخاصیة متا صلة في المضخم ولا یمكن إلغاؤھا بتغییر ظروف التشغیل.

128 C ك ج H جمھوریة العراق وزارة التعلیم العالي والبحث العلمي جامعة ذي قار - كلیة العلوم قسم الفیزیاء أداء الربح لمضخمات الا لیاف البصریة المشوبة بالا ربیوم رسالة مقدمة إلى مجلس كلیة العلوم جامعة ذي قار وھي جزء من متطلبات نیل درجة الماجستیر في علوم الفیزیاء من قبل محمد رحمه حرب بكالوریوس علوم / علوم الفیزیاء ٢٠١٣ أشراف أ.د. حسن عبد یاسر د. صادق جعفر كاظم ١٤٣٦ ھ ٢٠١٥ م