Optimization Tool for Erbium Doped Fibre Amplifiers · ix Abstract Erbium Doped Fibre Amplifiers...

Optimization Tool for Erbium Doped Fibre Amplifiers

Pedro Tavares Antunes

Thesis to obtain the Master Science Degree in

Electrical and Computer Engineering

Supervisors: Prof. Paulo Sérgio de Brito André

Eng. Carlo Marques

Dr. Paola Frascella

Examination Committee

Chairperson: Prof. José Eduardo Charters Ribeiro da Cunha Sanguino

Supervisor: Prof. Paulo Sérgio de Brito André

Members of the Committee: Prof. Mário José Neves de Lima

May, 2016

iii

To my family

v

Acknowledgments

Firstly, I would like to express my sincere gratitude to Prof. Paulo André, for selecting me for this

dissertation and grating me the opportunity to develop this work in a business environment.

I would like to thank Coriant for providing me with a working space in their facilities in Alfragide (Lisbon),

where most of the work took place. I would also like to leave a word of appreciation to my co-supervisors

within the company, Eng. Carlo Marques and Dr. Paola Frascella, for their constant support, patience

and knowledge which was instrumental to overcome some of the challenges I faced throughout this

dissertation. Thank you for steering me in the right direction throughout this process.

Special thanks to my friends and colleagues, and most importantly to my family, particularly my parents

and brother, for providing me with the possibility to be where I am today. Without your love,

encouragement, advice and endless support, none of this would have been possible.

vii

Resumo

Os amplificadores de Fibra Dopada com Érbio (EDFA) - caracterizados pelo elevado ganho e largura

de banda, baixas perdas intrínsecas e rápidos tempos de reposta - emergiram nas últimas décadas

como componentes essências de comunicações óticas com multiplexagem espectral (WDM). A

transição progressiva das redes atuais para redes puramente óticas de alta capacidade, aumentou a

complexidade dos elementos de rede, influenciando o custo global dos sistemas. Consequentemente,

a otimização de elementos ativos tornou-se um processo crucial em sistemas WDM modernos. O

objetivo desta dissertação é propor uma ferramenta de otimização eficaz para o desenho de EDFAs.

As principais contribuições desta dissertação incluem: i) teoria e criação de modelos representativos

da Fibra Dopada com Érbio e dos dispositivos passivos que compõem a arquitetura dos EDFAs, ii)

descrição das características do amplificador e a sua relação com diferentes componentes, topologias

e configurações de bombeamento, iii) avaliação dos principais parâmetros de desempenho, iv)

desenvolvimento de um simulador eficiente para EDFA, englobando várias topologias e opções de

bombeamento, v) elaboração de um algoritmo heurístico para testar e avaliar vários EDFAs, de acordo

com um conjunto de requisitos de rede, de modo a selecionar e otimizar a mais solução de menor custo.

Os resultados obtidos demonstram que a ferramenta proposta é capaz de determinar uma solução

otimizada de EDFA para diversas situações, indicando com precisão os parâmetros de desenho que

garantem os requisitos de rede predefinidos. A utilização desta ferramenta reduz a quantidade de testes

experimentais a efetuar para encontrar uma solução ótima, economizando tempo e dinheiro.

Palavras-chave — Comunicações óticas, WDM, amplificador ótico, EDFA, técnicas de otimização.

ix

Abstract

Erbium Doped Fibre Amplifiers (EDFA), featuring high gain over a large bandwidth, low intrinsic losses

and long fluorescence times, have emerged over the past decades as key enabling components in WDM

systems. The progressive transition towards high capacity, all-optical networks has increased the

complexity of each network element, thus impacting the overall cost of the systems. Therefore, optimum

design of network active elements became a crucial part of modern WDM systems. The goal of this

dissertation is to propose an effective optimization tool for steady-state EDFA design.

The main contributions of this dissertation include: i) theory and modelling of Erbium Doped Fibre and

passive devices that compose EDFA´s architecture, ii) description on EDFA´s characteristics and their

relation with amplifier’s components, topologies and pumping configurations, iii) assessment of key

design parameters, iv) development of an efficient simulator for EDFA, supporting several topologies

and pumping configurations, v) construction of a heuristic algorithm for testing and evaluating several

EDFA topologies according to a set of network requirements, in order to select and optimize the most

cost-effective solution.

The obtained results show that the proposed optimization tool is able to deliver an optimized EDFA

solution for all the three case studies, indicating an accurate determination of design parameters in order

to meet pre-establish network requirements. Additionally, employing the proposed tool reduces the

amount of experimental tests necessary to find an optimum solution, thereby saving both time and

money.

Keywords — Optical Communications, WDM, Optical Amplifier, EDFA, optimization techniques.

xi

Contents

Acknowledgments v

Resumo vii

Abstract ix

List of Figures xiii

List of Tables xv

Nomenclature xvii

Glossary xix

1 Introduction 1

1.1 Motivation and Background ..................................................................................................... 1

1.2 Thesis Outline and Objectives ................................................................................................. 3

1.3 Main Contributions ................................................................................................................... 4

2 Erbium Doped Fibre Theory 5

2.1 Fibre Properties ....................................................................................................................... 5

2.1.1 Fibre Losses ........................................................................................................................ 5

2.1.2 Properties of the Erbium Glass ............................................................................................ 8

2.1.3 Quantum processes .......................................................................................................... 12

2.2 Propagation and Rate Equations of the Erbium doped Fibre ................................................ 15

2.2.1 Amplification in three-level systems .................................................................................. 15

2.2.2 Reduction of the three-level system to the two-level system ............................................ 21

2.2.3 Amplified Spontaneous Emission (ASE) ........................................................................... 24

2.2.4 Excited State Absorption (ESA) ......................................................................................... 26

2.2.5 Final Propagation and Rate equations .............................................................................. 28

3 Erbium Doped Fibre Amplifiers 33

3.1 Importance of EDFAs in WDM systems ................................................................................ 33

3.2 Single-Stage EDFA ............................................................................................................... 34

3.2.1 Gain ................................................................................................................................... 35

3.2.2 Noise figure ........................................................................................................................ 40

3.2.3 Gain Flatness ..................................................................................................................... 45

3.2.4 Gain Tilt and Gain Ripple .................................................................................................. 46

3.2.5 Single Stage EDFA Limitations ......................................................................................... 48

4 Optimization Tool for Erbium Doped Fibre Amplifiers 51

4.1 Component Modelling ............................................................................................................ 51

4.2 Design Parameters ................................................................................................................ 52

xii

4.2.1 EDF Length ........................................................................................................................ 52

4.2.2 Gain Flattening Filter’s Attenuation ................................................................................... 54

4.2.3 Variable Optical Attenuator ................................................................................................ 56

4.2.4 EDF coil ratio and Power Splitter Ratio ............................................................................. 56

4.3 Optimization Tool for EDFA ................................................................................................... 57

4.3.1 Simulator structure............................................................................................................. 58

4.3.2 EDFA Operating Region: ................................................................................................... 59

4.3.3 Topology Chooser: ............................................................................................................ 60

4.3.4 Optimization ....................................................................................................................... 62

4.3.5 Performance Analysis ........................................................................................................ 62

5 Results 65

5.1 Case study: Loss compensation at the Reconfigurable Optical Add and Drop Multiplexer

(ROADM) ........................................................................................................................................... 65

5.1.1 Initial Conditions ................................................................................................................ 65

5.1.2 Topology Chooser ............................................................................................................. 66

5.2 Case study: Unregulated Tilt Cancelling ............................................................................... 68

5.2.1 Initial Requirements: .......................................................................................................... 68


5.2.3 Final Solution Optimization ................................................................................................ 70

5.3 Case study: Dispersion Compensation Module .................................................................... 72

5.3.1 Initial Requirements ........................................................................................................... 72


5.3.3 Final Solution Optimization……………………………………………………………………...74

6 Conclusions and future work 77

7 Bibliography 79

8 Appendix A 81

8.1 Multi-Stage EDFA .................................................................................................................. 81

8.1.1 Noise Figure ...................................................................................................................... 81

8.1.2 Pumping Techniques ......................................................................................................... 82

9 Appendix B 85

9.1 EDFA Topologies .................................................................................................................. 85

xiii

List of Figures

2.1 - Fibre loss spectrum [3]. ................................................................................................................... 6

2.2 - Stark splitting of energy levels (reproduced from [11]). .................................................................. 9

2.3 - Absorption and Emission Cross-section of an Erbium doped fibre [15]. ....................................... 11

2.4 - Absorption (left) and Spontaneous Emission (right) diagram (reproduced from [17]). ................. 13

2.5 - Stimulated Emission diagram (reproduced from [17]). .................................................................. 13

2.6 - Fluorescence diagram (reproduced from [17]). ............................................................................. 14

2.7 - Three-level System representation (reproduced from [11]). ......................................................... 15

2.8 - Fractional population inversion and pump threshold [11]. ............................................................ 17

2.9 - Saturation and Small Signal gain regions [11]. ............................................................................. 21

2.10 - Two-Level System representation (reproduced from [11]).......................................................... 21

2.11 - Total forward and backward ASE power as a function of position [11]. ...................................... 26

2.12 - Pump excited-state absorption representation (reproduced from [11]). ..................................... 27

3.1 - Block diagram of a Repeater. ........................................................................................................ 33

3.2 - Power, Line and Preamplifier position in a WDM link. .................................................................. 34

3.3 - Architecture of a typical Single-Stage EDFA. ............................................................................... 35

3.4 - Net cross section for different values of the fractional upper state population [11]. ..................... 36

3.5 - a) Fractional upper state population and b) Signal gain along the fibre for three distinct values of

pump power [11]. ................................................................................................................................... 36

3.6 - Gain as a function of pump power for a 14 meter EDF pumped at 980 nm and 1480 nm [11]. ... 37

3.7 - Fractional upper state population as a function of position along a 14 meter fibre pumped at a) 980

nm and at b) 1480 nm [11]. ................................................................................................................... 38

3.8 - Up: Signal gain and fractional upper state population as a function of pump power for an 8 meter

EDF. Down: Signal gain and fractional upper state population as a function of pump power for a 25

meter EDF [11]. ..................................................................................................................................... 39

3.9 - Noise Figure at 1550 nm as a function of Gain for a 980 nm and 1480 nm pump [11]. ............... 42

3.10 - Pump configurations for a Single-Stage EDFA [11]. ................................................................... 43

3.11 - Signal output power as a function of fibre length for a co propagating, counter propagating and

bidirectional pump [11]. ......................................................................................................................... 44

3.12 - Noise Figure as a function of pump power for a co and counter propagating configuration. An 8

and 12 meter fibre is tested [11]. ........................................................................................................... 44

3.13 - Gain spectrum analysis of single stage EDFA with GFF positioned after EDF [28]. .................. 45

3.14 - Gain Tilt and Gain Ripple of an EDFA output spectrum [20]. ..................................................... 46

3.15 - Gain tilt adjustment using a Variable Optical Attenuator (reproduced from [20])........................ 47

3.16 - Example of Unregulated tilt cancelling using EDFA in WDM systems (reproduced from [20]). . 48

4.1 - Tilted Output Gain profile, consequence of a poorly dimensioned GFF. ...................................... 55

xiv

4.2 - GFF’s Attenuation profile being adjusted to a Gain profile. .......................................................... 55

4.3 - Figure of Merit evaluated for different values of Relative Length of First Fibre Coil. .................... 56

4.4 - Figure of Merit evaluated in terms of EDF coil Ratio and Power Splitter Ratio. ........................... 57

4.5 - Optimization Tool general Diagram. .............................................................................................. 58

4.6 - Amplifier’s Operating Region delimited by power and gain conditions. ........................................ 59

4.7 - Double-Stage EDFA Simulation. ................................................................................................... 60

4.8 – Key performance data vs amplifier gain for different lengths of first EDF coil percentages. ....... 63

5.1 – Case Study 1: EDFA Operating Region. ...................................................................................... 65

5.2 - Cost Figure applied to potential solutions. .................................................................................... 67

5.3 – Case Study 2. EDFA Operating Region. ...................................................................................... 68

5.4 – Case Study 2: Cost Figure applied to potential solutions............................................................. 69

5.5 - Figure of Merit evaluated for different EDF Coil Ratio and Splitting Ratio. ................................... 71

5.6 – Case Study 2: Optimized Gain Flattening Filter Attenuation Profile. ........................................... 72

5.7 - Case Study 3. EDFA Operating Region. ....................................................................................... 72

5.8 - Figure of Merit for different values relative Length of EDF in the first Stage. ............................... 74

5.9 – Optimized Gain Flattening Filter Attenuation Profile. ................................................................... 75

8.1 - Example of Multistage Amplifier. ................................................................................................... 81

8.2 - Double Stage Amplifier design using either a Pump Power Splitter or a Pump Bypass [22]. ...... 83

9.1 - Single-Stage configuration 1a) ...................................................................................................... 85

9.2 - Single-Stage Configuration 1b) ..................................................................................................... 85

9.3 - Single-Stage Configuration 1c) ..................................................................................................... 86

9.4 - Double-Stage Configuration 2a) .................................................................................................... 86

9.5 - Double-Stage Configuration 2b) .................................................................................................... 86

9.6 - Double-Stage Configuration 2 c) ................................................................................................... 86

9.7 - Four-Stage Configuration 3 a) ....................................................................................................... 86

9.8 - Five-Stage Configuration 3 b) ....................................................................................................... 86

xv

List of Tables

4.1 - Components Insertion Losses. ...................................................................................................... 52

4.2 - EDF Length for different wavelength and population inversion values. ........................................ 53

4.3 - Optimum EDF Length as a function of fractional population inversion. ........................................ 54

5.1 – Case Study 1: Gain and Power Requirements. ........................................................................... 65

5.2 – Case Study 1: Noise Figure requirements. .................................................................................. 66

5.3 – Case Study 1: Additional specifications. ...................................................................................... 66

5.4 – Single-stage optimized solution. ................................................................................................... 68

5.5 – Case Study 2: Gain and Power requirements. ............................................................................. 68

5.6 – Case Study 2: Noise Figure requirements. .................................................................................. 68

5.7 – Case Study 2: Additional specifications. ...................................................................................... 69

5.8 - Double-Stage specifications after Topology Chooser. .................................................................. 70

5.9 - Double-Stage optimized solution. ................................................................................................. 71

5.10 – Case Study 3: Gain and Power Requirements. ......................................................................... 72

5.11 – Case Study 3: Noise Figure requirements. ................................................................................ 73

5.12 – Case Study 3: Additional specifications. .................................................................................... 73

5.13 – Double-Stage specifications after Topology Chooser. ............................................................... 74

5.14 – Double-Stage Optimized solution. .............................................................................................. 75

9.1 - Single-Stage configuration 1a) ...................................................................................................... 85

9.2 - Single-Stage Configuration 1b) ..................................................................................................... 85

9.3 - Single-Stage Configuration 1c) ..................................................................................................... 86

9.4 - Double-Stage Configuration 2a) .................................................................................................... 86

9.5 - Double-Stage Configuration 2b) .................................................................................................... 86

9.6 - Double-Stage Topology 2 c) ......................................................................................................... 86

9.7 - Four-Stage Configuration 3 a) ....................................................................................................... 86

9.8 - Five-Stage Configuration 3 b) ....................................................................................................... 86

xvii

Nomenclature

𝛼 Fibre attenuation coefficient

𝐿 Fibre section length

𝜆 Wavelength

𝛼𝑅 Rayleigh scattering attenuation coefficient

𝜏 Fluorescence lifetime

𝜏𝑟 Radiative lifetime

𝜏𝑛𝑟 Non-radiative lifetime

𝑛 Refractive index

𝜎21 Emission cross-section

𝜎12 Absorption cross-section

𝑐 Speed of light in vacuum

ℎ Planck constant

𝑘 Boltzmann constant

𝑇 Temperature

𝜐 Frequency

𝜇 Medium permeability

𝜑𝑝 Incident pump intensity flux

𝜑𝑠 Incident signal intensity flux

𝜙𝑡ℎ Required pump flux

Γ21 Transition probability from level 2 to level 1

Γ32 Transition probability from level 3 to level 2

𝑁 Total electron population

𝑁1 Electron population of level 1




I𝑠 Signal intensity

xviii

I𝑝 Pump intensity

𝐼𝑡ℎ Required light intensity

𝐼𝑆𝐴𝑇 Saturation Intensity

A Cross-sectional area

𝜌 Ion density

𝑃𝑠𝑎𝑡 Saturation Power

𝑃𝑧=𝐿 Output Power

𝑃𝑧=0 Input Power

𝑃𝐴𝑆𝐸0 Equivalent noise power

𝑃𝐴𝑆𝐸 Amplified spontaneous emission power

𝑃𝑝 Pump Power

𝑃𝑠 Signal Power

Γ Overlap factor

Δ𝜈 Bandwidth

𝜉 Fibre parameter

𝑁𝑠ℎ𝑜𝑡 Shot noise power

𝑁𝑠−𝑠𝑝 Signal-spontaneous noise power

𝑁𝑠𝑝−𝑠𝑝 Spontaneous-spontaneous noise power

𝐵𝑜 Optical Bandwidth

𝐵𝑒 Electrical bandwidth

𝑞 Elementary charge

𝑛𝑠𝑝 Population inversion parameter

xix

Glossary

ASE Amplifier Spontaneous Emission

CWDM Coarse Wavelength Division Multiplexing

DCF Dispersion Compensating Fibre

DCM Dispersion Compensating Module

DWDM Dense Wavelength Division Multiplexing

EDF Erbium Doped Fibre

EDFA Erbium Doped Fibre Amplifier

ESA Excited State Absorption

GFF Gain Flattening Filter

GTC Gain Tilt Control

IL Insertion Losses

LD Laser Diode

NF Noise Figure

OADM Optical Add-Drop Multiplexer

OEO Optical-Electrical-Optical

ODE Ordinary Differential equation

OSC Optical Supervisory Channel

OSNR Optical Signal to Noise Ratio

RFA Raman Fibre Amplifier

ROADM Reconfigurable Optical Add-Drop Multiplexer

SMF Single Mode Fibre

SNR Signal to Noise Ratio

SRS Stimulated Raman Scattering

VOA Variable Optical Attenuator

WDM Wavelength Division Multiplexing

WSS Wavelength Selective Switching

1

1 Introduction

In this chapter, the motivation, the objectives and the contributions of this dissertation are exposed.

1.1 Motivation and Background

Since the development of the first fully operational laser by Theodore Maiman in the early 1960’s and

the work of Charles K. Kao proposing optical fibres as a practical communication medium, optical fibre

transmission has been perceived as the future of telecommunications systems [1, 2]. However, it would

take about a decade until the technology had matured to the point where fibres with losses lower than

20 dB/km and laser diodes capable of working at room temperature were available commercially, thus

providing the level of performance required for practical applications in optical communications [4].

The launch of the internet in the 1980’s, and the consequently increase in the demand for capacity,

exposed the fragilities and limitations of the traditional copper wire transmission systems and triggered

the expansion of fibre optics communications, which were less susceptible to electromagnetic

interference and had a larger bandwidth than its predecessor [3].

By the end of the decade, fibre transmission losses had dropped to 0,2 dB/km and both speed and

range of fibre optics systems had increased [3]. Signal could now be transmitted at 2,5 Gb/s and

distances up to 100 km were achievable without requiring regeneration or amplification [4].

In 1994, the introduction of Wavelength Division Multiplexing (WDM) transmission technology

revolutionized the capacity of fibre optic systems. By assigning a specific wavelength to each signal and

then coupling them together into a single strand of fibre, the capacity of the link was multiplied and the

large cost of implementing new fibres could be avoided [4, 7]. Although the initial WDM solutions

(Broadband WDM) could only combine two channels – at 1310 𝑛𝑚 and 1550 𝑛𝑚 respectively – modern

solutions allow for several frequency patterns. Dense Wavelength Division Multiplexing (DWDM),

extensively used in long haul systems, can combine multiple signals, providing 40 to 80 channels with

a spacing of 100 GHz to 50 GHz, in the third optical transmission window [5]. Moreover, Coarse

Wavelength Division Multiplexing (CWDM) solutions are often employed whenever less information is

transmitted over shorter distances, providing up to 16 channels and taking advantage on a larger

spacing between them in order to reduce cost by using less sophisticated transceivers [6].

The development of optical amplifiers, particularly Erbium Doped Fibre Amplifiers (EDFA), played a

pivotal role in the success of WDM systems [4, 5]. Commercially available since the beginning of the

1990’s, EDFA were responsible for substantially increasing the distance of WDM solutions by

compensating the attenuation introduced by fibres and other components along the link. Prior to optical

amplifiers, compensating loss in optical fibre was done resorting to regenerators, placed periodically

along the link [6]. Their main advantage was guaranteeing that network impairments such as dispersion,

2

noise and nonlinearities were compensated at each node. However, the complex optical-electrical-

optical (OEO) conversion technology in these devices made them expensive and also unable to exploit

the large bandwidth of optical fibres [6]. Additionally, system upgrades, such as modulations formats

and bit rates were difficult and costly to implement, since it usually required replacing all regenerators

in the link.

Being a purely optical device, EDFA do not share most of these limitations. Their insensitivity to bit rates

and signal formats and their large bandwidth enabled them to accommodate and amplify numerous

WDM signal, along the C-Band, simultaneously.

For the most part of a decade, EDFA took centre stage in WDM systems, specifically long haul solutions,

due to their high deployed gain obtain with relatively low pump values [11]. During this period, EDFA

underwent progressive improvements and their initial bandwidth was extended from the C-band to L-

band, further enhancing the capacity, flexibility and cost of WDM networks [6].

By the end of the 1990’s, the technological advances on optical components and the deployment of high

power pump laser motivated a renewed interest for Raman Fibre Amplifiers [8]. Although research had

begun as far back as the 1970’s, the high pump powers required by these amplifiers (tens of milliwatts

per dB gain) put them at a disadvantage compared to EDFA (tenths of milliwatts per dB gain) [9]. Once

this limitation was lifted, Raman amplifiers were seen as a way to further extend the capacity of WDM

systems [8]. Unlike EDFA (lumped amplification scheme) where the maximum distance is limited by

span loss, as well as by nonlinear effects (whenever signal power is above the maximum admissible

power allowed in the fibre) and Optical Signal to Noise Ratio (OSNR) degradation (whenever power fell

below the minimum acceptable), Raman amplifiers have a distributed amplification scheme, where the

transmission fibre itself becomes the amplification medium [9]. Distributed amplification schemes allow

greater distances since the power level propagating inside the fibre can be contained and kept from

reaching both the limits imposed by nonlinearities and OSNR degradation.

Despite Raman fibre amplifiers achieving greater distances, the majority of modern, high capacity WDM

systems still employ EDFA’s. The migration towards all-optical networks also led to the evolution of

network nodes. Switching and routing operations, previously performed in the electrical domain, are

done using optical components, namely Reconfigurable Optical Add/Drop Multiplexers (ROADM) and

Optical Cross Connects (OXC) [5]. The increasing number of purely optical components at network

nodes has increased the attenuation at these locations. This has extended the role played by these

amplifiers in modern optical networks. EDFA’s are now being deploy to compensate for loss in

transmission fibre and for component attenuation at network nodes.

This evolution of communications systems to purely optical networks revealed new challenges for

amplifier design, resulting in new topologies and configurations being studied. The most common way

to design new components is to simulate their behaviour and try to optimize their parameters for best

performance. Available software programs for EDFA design are divided into two groups: i) the first

group is composed of software programs such as Linksys, Comsys, Photos and Oasis and their

objective is to test and simulate EDFA operation in complete communication systems [10]. Since their

3

primary focus is on the overall behaviour of the network and not on the components itself, these

programs are limited on the quantity of EDFA configurations, as well as on the number of design

parameters that can be manipulate. Ii) the second group is made of software programs like

VPIComponentMaker and EDFA_Design that allow for more basic simulations, focusing of the on

amplifier and its components [10]. They possess a larger variety of components for assembling EDFA

configurations, and allow for more complex variations of the design parameters.

The Optimization Tool for EDFA Design proposed in this work will follow on the steps of software

programs mentioned in the second group. The ultimate goal, is for it to be capable of testing several

EDFA topologies and pump configurations, select and optimize the most cost-effective solution and

provide the user with data regarding the amplifier’s performance.

1.2 Thesis Outline and Objectives

The main objective of this work is to develop an Optimization Tool that employs optimization techniques

and algorithms in order to evaluate, test, select, study and deliver a fully optimized EDFA topology that

presents itself as the most cost-effective solution for a set of predefined network requirements. To fulfil

this objective, this dissertation is structured as follows.

In chapter 2, the main physical events and characteristics of the erbium glass are studied. The second

part of this chapter is dedicated to the formulation of the Erbium Doped Fibre model.

In chapter 3, the goal is to provide an introduction to EDFA’s architecture, specifically to describe the

influence of every component on amplifier’s characteristics (Gain, Noise Figure, Gain Flatness, Gain Tilt

and Ripple).

In chapter 4, the entire structure of the optimization tool is discussed, starting with the Matlab

representation of every component and EDFA topology, as well as the description of the design

parameters and how to optimize them.

In chapter 5, three case studies for three distinct set of requirements are presented and the methodology

of the Optimization Tool is analyzed as it converges towards a final solution.

In chapter 6, the final conclusions of this work and suggestions for future improvements on this topic are

presented.

4

1.3 Main Contributions

The main contributions of this work are:

Development of an efficient simulator for EDFA, supporting several topologies and pumping

configurations;

Description of methods and algorithms for the optimization of EDFA design parameters: i) Fibre

Length, ii) Pump Power, iii) Variable Optical Attenuator and Gain Flattening Filter Attenuation,

iv) Power Splitter Splitting Ratio and v) EDF Coil Ratio;

Construction of a heuristic algorithm for testing and evaluating several EDFA topologies

according to a set of network requirements, in order to select and optimize the most cost-

effective solution.

5

2 Erbium Doped Fibre Theory

In this chapter, a detailed model of the erbium doped fibre is derived. The first part of the chapter focuses

on the causes of attenuation in optical fibres and presents properties and characteristics of the erbium

glass (lifetime, transition cross-section, linewidth and broadening) essential for the amplification process

to take place. The second part of the chapter will be dedicated to the formulation of the erbium doped

fibre model. A three-level system representing the erbium ion structure will be introduced, possible

approximations and simplifications, as well as additional effects (Amplified Spontaneous Emission and

Excited State Absorption) will be considered. Finally, a system of differential equations composed by

rate and propagation equations describing the erbium doped fibre is solved.

2.1 Fibre Properties

2.1.1 Fibre Losses

As no optical fibre is perfectly transparent, optical signals sent through fibre will experience attenuation.

This fact is true for transmission fibre as well as for erbium-doped fibres. This section will focus on the

various loss mechanisms in optical fibres.

In general the loss resulting from “changes in the average optical power 𝑃 of a bit stream propagating

inside a typical optical fibre are governed by Beer’s Law” [3], which states:

𝑑𝑃

𝑑𝑧= −𝛼𝑃 (2.1)

Where 𝛼 represents the attenuation coefficient, or background loss coefficient.

The 𝛼 parameter includes material absorption, Rayleigh scattering and waveguide imperfections, which

will be discussed below. Although it is usually provided by fibre manufactures in units of 𝑑𝐵. 𝑘𝑚−1, 𝛼 can

also be expressed in linear units, as it is in equation (2.1). Furthermore, from the analytical solution of

the equation (2.1):

𝑃𝑜𝑢𝑡,𝑚𝑊 = 𝑃𝑖𝑛,𝑚𝑊 . 𝑒(−𝛼.𝐿) (2.2)

𝑃𝑜𝑢𝑡,𝑑𝐵𝑚 = 𝑃𝑖𝑛,𝑑𝐵𝑚 − 𝛼𝑑𝐵.𝑘𝑚−1 . 𝐿 (2.3)

An expression providing the relationship between the linear and logarithmic 𝛼 parameter can be derived.

𝛼𝑑𝐵.𝑘𝑚−1 =10

ln(10). 𝛼 = 4,343𝛼 (2.4)

6

Where 𝑃𝑖𝑛 and 𝑃𝑜𝑢𝑡 are the power launched into the fibre and the output power at the end of the fibre,

respectively. 𝐿 represents the fibre’s length.The 𝛼 parameter is also wavelength dependent 𝛼(𝜆) as it is

shown in Figure 2.1. Besides exhibiting a strong peak right around the 1,39 𝜇𝑚, and a loss of about

0,2 𝑑𝐵. 𝑘𝑚−1 in the third transmitting window (1,55𝜇𝑚) – near the physical limit of 0,16 𝑑𝐵. 𝑘𝑚−1 imposed

by Rayleigh Scattering – Figure 2.1 displays the two main factors responsible for background losses:

material absorption and Rayleigh scattering.

Material Absorption

Absorption occurs in optical fibres due to the presence of imperfections in the structure of the fibre

material (𝑆𝑖𝑂2 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠).Two types of material absorption are to be considered: intrinsic and extrinsic

absorption.

The first is a consequence of the vibration of the silicon-oxygen bonds on the infrared region (𝜆 > 7 𝜇𝑚)

and the electronic resonances in the ultraviolet region (𝜆 < 0,4 𝜇𝑚). “Because of the amorphous nature

of fused silica, these resonances are in the form of absorption bands whose tails extend into the visible

region” [3], as depicted in Figure 2.1.

The latter is a result of the presence of impurities in the host medium material. These impurities are

often transition-metal elements such as Fe, Cu, Co, Ni, Mn and Cr and produce strong absorption in the

0,6 to 1,6 𝜇𝑚 band. Other main source of extrinsic absorption is related to the presence of residual water

vapours in the silica structure. The OH ions cause a resonance vibrations near 2,73 𝜇𝑚, whose

harmonics have repercussions throughout the silica wavelength spectrum, showing peaks of absorption

at 0,95 𝜇𝑚, 1,24 𝜇𝑚 and 1,39 𝜇𝑚 (Figure 2.1). Nowadays, the adverse effects of OH absorption loss

have been mitigated in All Wave Fibre, a fibre featuring reduce attenuation at wavelengths where OH

Figure 2.1 - Fibre loss spectrum (Source: P. Agrawal) Figure 2.1 - Fibre loss spectrum [3].

7

absorption was previously dominant, consequence of a new manufacturing procedure where the OH

ions are removed from the silica structure.

Rayleigh Scattering

During fibre fabrication, silica molecules move randomly in a molten state. The material is then cooled

and the molecules freeze in place. This creates random density fluctuations, smaller than the

wavelength in scale, due to irregular microscopic structures, influencing the refractive index of the

medium. “Light scattering in such a medium is known as Rayleigh scattering” [3]. Intrinsic loss in silica

fibres from Rayleigh scattering can be written according to the following expression:

𝛼𝑅 =𝐶

𝜆4 (2.5)

Where C is a constant raging from 0,7 to 0,9 (𝑑𝐵. 𝑘𝑚−1. 𝜇𝑚−4), depending on fibre composition. In the

third transmission window (around 1,55 𝜇𝑚), 𝛼𝑅 ranges from 0,12 to 0,16 (𝑑𝐵. 𝑘𝑚−1) being the

predominant loss factor to take into account as seen in Figure 2.1.

Waveguide Imperfections

Since perfectly cylindrical guides are impossible to produce, there is always a percentage of energy that

escapes into the cladding layers. These losses due to core-cladding imperfections are described by Mie

Scattering which contrary to Rayleigh Scattering mentioned above, happens because of “refractive

index inhomogeneities on a scale longer that the optical wavelength” [3]. Nowadays fibre fabrication can

limit such index variations to about 1% making the Mie Scattering loss on the order of 0,03 𝑑𝐵. 𝑘𝑚−1.

Other important waveguide imperfections arise from the bending of optical fibres. Bends fall into two

categories: macro and microbends.

- Macrobends happen in situations where the fibre cable suffers a curvature which angle

surpasses the critical angle, and as a consequence the light propagating inside the fibre core

does not experience total internal reflection;

- Microbends are related to axial distortions that occur when fibre is pressed against a surface

that is not completely smooth. Such bends can drastically increase the attenuation in a system

(~100 𝑑𝐵. 𝑘𝑚−1) severely limiting it.

As a response to the macrobends attenuation issue, modern fibres insensitive to bends have

been created, in which an optical trench composed of medium with low refractive index is incorporated

in the fibre cable. The trench’s objective is to “trap” the light that would otherwise leave the core and

force it remain in the inside it, significantly reducing bend attenuation.

8

Microbends can be minimized by avoiding light propagating through fibre locations with axial

distortions. Since these distortions occur predominantly at the fibre’s surface, reducing microbends

attenuation can be accomplished by ensuring that the majority of the light propagating inside the fibre

stays concentrated at its core.

2.1.2 Properties of the Erbium Glass

Erbium is a chemical element with atomic number 68 (Z=68) discovered in 1843 by Swedish chemist

Carl Gustaf Mosander [25]. Rare earth elements, which Erbium is a part of, are divided in two groups:

the lanthanides (atomic number 57 through 71) and actinides (atomic number 89 to 103).

Most of today’s earth doped fibres and lasers use lanthanides as an active element. What makes them

so popular in modern communication systems is their optical behaviour resulting from their unique

atomic structure.

Bearing in mind the classic atomic model proposed by Bohr in 1913 - portraying an atom as a nucleus

surrounded by layers (shells) of electron, which are progressively occupied as we move along the

periodic table [16] – one would think, and correctly for the most part, that each added layer of electrons

would have a larger radius than the one before, meaning the radii would increase monotonically. This is

where the lanthanides’ structure differs from the rest. When “the 5s and 5p shells (5𝑠2 and 5𝑝6) are full

one adds next a 4f shell in which electrons are progressively inserted. The 4f shell, instead of having a

larger radius than 5s and 5p, actually contracts and becomes bounded by these shells” [11]. The

shielding of the 4f shells by the 5s and 5p ones is responsible for the lanthanides (and rare earth

elements in general) rich optical properties in a solid medium such as crystal or glass.

When inserted in a crystal or glass medium, the energy states (levels) that compose the 4f shell of an

erbium ion no longer possess the spherical symmetry characteristic of the vapour phase. As a result of

the molecular reorganization between the erbium ion and the silica host medium (𝑆𝑖𝑂2), local electric

fields modify the 4f shell energy states, splitting each one into a set of sub energy levels. Such effect is

referred to as Stark Splitting and is depicted in Figure 2.2.

The splitting of the energy states along with the broadening they experience, when erbium ions are

inserted in the silica medium host structure, make up the large energy gap between the ground level

𝐼415/2 and the upper level 𝐼4

13/2. The length of this transition is the key success of erbium doped

amplifiers. The long and mostly radiative lifetime of erbium ions means that population inversion in

erbium doped fibres can be achieve with a relatively weak, and thus practical, pump source.

9

Lifetimes:

Lifetimes is the designation given to the duration of time an ion stays in an excited energy level before

it decays. It is “inversely proportional to the probability per unit time of the exit of an ion from that excited

level” [11] and it is also the time constant that characterizes the exponential decay of the population in

a certain energy level.

The decay paths can be classified as radiative or non-radiative depending on their nature. Multiple decay

paths can exist for an energy level, so generically it is defined the total probability as a sum of the

radiative and non-radiative path, or:

1

𝜏=

1

𝜏𝑟

+1

𝜏𝑛𝑟

(2.6)

Where 𝜏 is the total lifetime and 𝜏𝑟 and 𝜏𝑛𝑟 are the radiative and non-radiative lifetime, respectively.

Radiative lifetimes occur when the decay between two energy levels originates the emission of photons

- the quantum of all forms of electromagnetic radiations, including light - carrying the difference of energy

between the involved levels. Radiative lifetimes tend to be long, on the order of microseconds to

milliseconds. In non-radiative lifetimes, however the excess energy is dissipated not by the emission of

light, but in the form of phonons, which are associated with lattice vibrations of the medium. Phonons

emission is a rather quick process when compared to photons emission, so when possible, the radiative

transition tends to be bypassed by a non-radiative one, and cannot be observed.

Figure 2.2 - Stark splitting of energy levels (reproduced from [11]).

Stark levels

Crystal field Atomic forces

4𝑓𝑁

2𝑆 + 1𝐿𝑗

10

Transition cross-sections

The cross-sections of emission and absorption are parameters that “quantify the ability of an ion to

absorb and emit light” [11] at a determined wavelength. These parameters are strongly dependent on

the medium host composition and play an important role on the final magnitude and shape of the gain

profile an amplifier can deliver, ultimately affecting its performance [13].

Traditionally, cross-sections are derived from experimental values of fluorescence and absorption

bandwidths using the Fuchbauer-Landerberg equation, which is based on Einstein A and B coefficients

[14]. According to Fuchbauer-Landerberg the cross-sections can be defined as:

𝜎21 =𝜆2

8𝜋𝑛2𝐴21𝑔(𝜐) (2.7)

𝜎12 =𝑔2

𝑔1

𝜆2

8𝜋𝑛2𝐴21𝑔′(𝜐) (2.8)

With 𝜆, the wavelength of peak emission-absorption, 𝑔(𝜐) and 𝑔′(𝜐) are the line shape for emission and

absorption respectively. 𝐴21 represents the spontaneous decay rate and is given by 1

𝜏𝑟 (assuming there

is no radiative decay), 𝑛 is the medium refractive index and 𝑔2, 𝑔1 the level degeneracies. Since the

effective linewidth is given by:

𝐼𝑝𝑘Δ𝜆𝑒𝑓𝑓 = ∫ 𝐼(𝜆) 𝑑𝜆∞

0

(2.9)

And

𝑔(𝜐) = 𝑔′(𝜐) =𝐼𝑝𝑘

∫ 𝐼 𝑑𝜐 (2.10)

Through some mathematical manipulation, equations (2.7) and (2.8) can be re-written as

𝜎21 =𝜆4

8𝜋𝑛2𝑐.

1

𝜏 Δ𝜆𝐸

(2.11)

𝜎12 =𝑔2

𝑔1

𝜆4

8𝜋𝑛2𝑐.

1

𝜏 Δ𝜆𝐴

(2.12)

Although expression (2.11) and (2.12) are meant to describe all fibres, in 1960’s it was observed that

this treatment did not match experimental results in the context of transition metal ions (of which erbium

is a part of). An alternative method for calculating cross-sections was then presented by McCumber and

become known as the McCumber relation [11].

The McCumber relation assumes that if “the probability of non-radiative transitions between the emission

and absorption manifold is small over the spontaneous emission lifetime of the excited state”, and if “the

11

thermal relaxation time within each manifold is negligibly small compared to relevant timescales” [12],

then by meeting these two conditions the absorption and emission cross-sections have the following

relation:

𝜎𝑎(𝜐) = 𝜎𝑒(𝜐). 𝑒ℎ(𝜐−𝜇)

𝑘𝑇⁄ (2.13)

Where ℎ𝜇 is the energy necessary to excite ions from the ground to the excited state at a constant

temperature 𝑇 and 𝑘 is the Boltzmann constant.

Another useful relation can be derived from equation (2.13), coupling the emission cross-section with

the radiative lifetime.

1

𝜏21

= 8𝜋𝑛2

𝑐2∫ 𝜐2 𝜎21(𝜐)𝑑𝜐 (2.14)

With this information it is possible to find the emission cross-sections of a medium host, by measuring

the lifetime and absorption cross-section.

The McCumber relation has led to excellent results when compared to experimental data. Results for

an erbium-doped fluorophosphate glass were also extremely accurate making this model very useful in

the modelling of the doped fibre amplifiers [13].

Linewidths and Broadening

The linewidth, or breadth, of a transition between two energy levels is influenced by two main factors:

homogeneous and inhomogeneous broadening.

Figure 2.3 - Absorption and Emission Cross-section of an Erbium doped fibre [15].

12

Homogeneous broadening arises from the interactions of photons with the host medium resulting on all

the ions exhibiting the same broadened spectrum. Inhomogeneous broadening however, is caused by

variations in glass sites where groups of ions are situated. At these locations ions experience different

local electric fields, and consequently will have different absorption and emission spectral shapes. The

final inhomogeneous fluorescence spectrum will, therefore, be an average of several homogeneous

spectral shapes.

When it comes to erbium-doped amplifiers the shape of the broadening line between the ground level

and level two plays an important role in the gain process, especially when it comes to gain saturation

and WDM amplification. “In the presence of homogeneous broadening, a strong enough signal can

extract all the energy stored in the amplifier, while for an inhomogeneously broadened amplifier only the

energy stored in the subset of ions interacting with the incident radiation can be extracted” [11]. This

makes homogeneously broadened amplifiers more efficient in providing energy to a signal when

compared to inhomogeneously broadened ones. This is no longer true when considering WDM systems

that suffer multiple adding and dropping of wavelengths channels. Since in inhomogeneously broadened

amplifiers, channels are well spaced between each other one can affirm they propagate quasi-

independently. This makes them more robust to channel variation.

2.1.3 Quantum processes

In this section the importance of quantum properties in atomic and molecular physics, as well as the

major photon processes that occur during the light amplification process are described.

According to Bohr atomic model depicted at the beginning of this section, radiation for a given frequency

can only be associated with one quantum energy value, respecting the Planck Relationship, as

expressed in the equation below.

𝐸 = ℎ. 𝜈 = ℎ.𝑐

𝜆 (2.15)

This means that energy levels of atoms and molecules have only certain quantized values. Transitions

between these quantized levels can occur through processes of absorption, emission - either

spontaneous or stimulated - and fluorescence.

Absorption and Spontaneous Emission

Absorption of a photon by the atoms or molecules of a medium, and the consequent transition of

electrons to a higher energy state, happens when the quantum energy of that photon matches the

energy gap between the atom’s initial and final state.

𝐸𝑝ℎ𝑜𝑡𝑜𝑛 = ℎ𝜐 = Δ𝐸 = 𝐸2 − 𝐸1 (2.16)

13

If by some reason there is no match between photon and the pair of energy levels of the atom in

question, then the medium is said to be transparent to that particular radiation and no absorption

phenomenon occurs.

Spontaneous emission is the opposite process to absorption. It describes the transition of a previously

excited electron, from its higher energy state to a lower one. During this relaxation, a photon is emitted

with an energy equivalent to the energy difference of the both levels.

Stimulated Emission

If an electron is already in an excited state, then by interacting with an incoming photon (with a quantum

energy that matches the energy difference between the current and lower levels), it can be “stimulated”

to transition to a lower level, while producing a second photon of the same energy as the first one. Light

amplification is the stimulated emission of multiple photons [17].

Fluorescence

Fluorescence transitions happen if an excited electron undergoes some interaction with the crystal

lattice of the medium or experiences some collisional process. As a result, energy is dissipated and the

Figure 2.4 - Absorption (left) and Spontaneous Emission (right) diagram (reproduced from [17]).

−

𝐸2

𝐸1

− 𝐸2

𝐸1

Figure 2.5 - Stimulated Emission diagram (reproduced from [17]).

− 𝐸2

𝐸1

− 𝐸2

𝐸1

Stimulated Emission

14

electron is transferred to a lower quantum state situated between the ground and upper level (𝐸2′ ). When

such electron transitions back to the ground level, it releases a photon of lower quantum energy than

the energy difference of levels 𝐸1 and 𝐸2 [17].

𝐸𝑝ℎ𝑜𝑡𝑜𝑛 = ℎ𝜐 = 𝐸2′ − 𝐸1 (2.17)

Figure 2.6 - Fluorescence diagram (reproduced from [17]).

−

𝐸2

𝐸1 −

𝐸2

𝐸1

𝐸2′

Fluorescence

15

2.2 Propagation and Rate Equations of the Erbium doped Fibre

2.2.1 Amplification in three-level systems

In order to derive the rate equations that model erbium doped fibre amplifiers, we will begin by studying

the fundamentals of the three level atomic system as represented in Figure 2.7. This system’s objective

is to highlight the part of the erbium ion energy structure (4f shell) that is predominant in the amplification

process.

We start by showing all three levels, denoting the ground level ( 𝐼15/24 ) as 1, the upper level ( 𝐼13/2

4 ) as

2 and the intermediate level ( 𝐼11/24 ) as 3. Consequently 𝑁1, 𝑁2 and 𝑁3 is the electron population of each

level. In addition, other key parameters used in describing this model are list below:

- 𝜑𝑝, Incident pump intensity flux (number of photons per unit time per unit area);

- 𝜑𝑠, Incident signal intensity flux (number of photons per unit time per unit area);

- 𝜎𝑝, Absorption cross-section (unit area);

- 𝜎𝑠, Emission cross-section (unit area);

- Γ21, Transition probability from level 2 to level 1;

- Γ32, Transition probability from level 3 to level 2.

One of the goal of the entire amplification process is to obtain population inversion between levels 1 and

2. The population inversion is the relocation of the majority of level 1 electrons (at least half of them) to

level 2. In order to do so, lasers are used to pump power into the erbium doped fibre in order to promote

them into a higher energy state.

Let us now breakdown the amplification process, assuming as a starting point that the erbium ion

distribution is constant along the length of the fibre, and over its cross-sectional area. Observing Figure

2.7 six types of transition are possible. The first is related to population inversion by exciting the

population from level 1 to level 2 through absorption. This is represented as the product of the pump’s

Figure 2.7 - Three-level System representation (reproduced from [11]).

3

2

1

𝜑𝑝𝜎𝑝

𝜑𝑠𝜎𝑠

𝛤32

𝛤21

16

light density flux (𝜑𝑝) by the absorption cross section (𝜎𝑝) and describes the quantity of photons that

transfer from the ground to the upper level in a certain amount of time. Depending on the laser pump

wavelength used, two types of absorption are possible. By using a 1480 nm pump the absorbed energy

puts electrons directly in level 2, also called metastable level because of its long lifetime (10 𝑚𝑠).

However if a 980 nm pump is employed, according to quantum physics the energy of the incident pump

photon is higher (𝐸2 = 𝐸1 + ℎ𝑐

𝜆) and so, the ground level population tends to be excited to level 3, instead

of level 2. The intermediate level does not have the stability of the upper level due to its short lifetime

(100 𝜇𝑠) and so its population tends to rapidly decay to level 2 via a nonradiative transition, represented

in Figure 2.7 by the transition probability of level 3 to 2, Γ32 defined as Γ32 =1

𝜏3, 𝜏3 being the lifetime of

the intermediate level 3 [11].

After achieving population inversion two possibilities arise. electrons that make up 𝑁2 participate in the

process of stimulated emission, interacting with signal photons and relax back to level 1 emitting photons

that are identical to it which ultimately results in signal’s light amplification (represented by the term 𝜑𝑠𝜎𝑠,

where 𝜎𝑠 is defined as the emission cross-section). Otherwise they eventually decay to level 1 by

spontaneous emission denoted in Figure 2.7 by Γ21 (radiative transition) releasing photons uncorrelated

with the signal, thereby producing noise.

Rate equations

The rate equations for the erbium ion population changes describing the relationships between

all three levels are the following:

𝑑𝑁3

𝑑𝑡= −Γ32 𝑁3 + (𝑁1 − 𝑁3)𝜙𝑝𝜎𝑝 (2.18)

𝑑𝑁2

𝑑𝑡= −Γ21 𝑁2 + Γ32 𝑁3 − (𝑁2 − 𝑁1)𝜙𝑠𝜎𝑠 (2.19)

𝑑𝑁1

𝑑𝑡= Γ21 𝑁2 − (𝑁1 − 𝑁3)𝜙𝑝𝜎𝑝 + (𝑁2 − 𝑁1)𝜙𝑠𝜎𝑠 (2.20)

Considering a steady state situation

𝑑𝑁1

𝑑𝑡=

𝑑𝑁2

𝑑𝑡=

𝑑𝑁3

𝑑𝑡= 0 (2.21)

and knowing that the total population N is the sum of the population of all three energy states

𝑁 = 𝑁1 + 𝑁2 + 𝑁3 (2.22)

We are able to write the population of level 3, 𝑁3 as

17

N3 = N1

1

Γ32

ϕpσp+ 1

(2.23)

Since 𝜏3 is very small when compared to 𝜏2, the population on level 3 will be close to zero (N3 ≈ 0). Γ32

will be large when compared to ϕpσp (Γ32 ≫ 𝜙𝑝𝜎𝑝), so Γ32 + 𝜙𝑝𝜎𝑝 ≈ Γ32. By considering this, 𝑁2 and 𝑁1

are described by the following expressions:

𝑁2 = 𝑁1

(𝜙𝑝𝜎𝑝 + 𝜙𝑠𝜎𝑠 )

Γ21 + 𝜙𝑠𝜎𝑠

(2.24)

𝑁1 = 𝑁Γ21 + 𝜙𝑠𝜎𝑠

Γ21 + 2𝜙𝑠𝜎𝑠 + 𝜙𝑝𝜎𝑝

(2.25)

From the previous expressions and through the use of the population inversion relation 𝑁2 ≥ 𝑁1, the

threshold pump value for which amplification begins can be derived. First 𝑁2 − 𝑁1 is calculated

𝑁2 − 𝑁1 = 𝑁𝜙𝑝𝜎𝑝 − Γ21

Γ21 + 2𝜙𝑠𝜎𝑠 + 𝜙𝑝𝜎𝑝

(2.26)

Then the required (threshold) pump flux is defined as

𝜙𝑡ℎ =Γ21

𝜎𝑝

=1

𝜎𝑝𝜏2

(2.27)

And assuming the Γ21 is larger than 𝜙𝑠𝜎𝑠 when the signal intensity is very small, expression (2.26) can

be further simplified resulting in

𝑁2 − 𝑁1

𝑁=

𝜙𝑝

𝜙𝑡ℎ− 1

𝜙𝑝

𝜙𝑡ℎ+ 1

(2.28)

Figure 2.8 - Fractional population inversion and pump threshold [11].

18

Figure 2.8 is the plot of the normalized population inversion given by equation (2.28). Figure 2.8 shows

that below the marked pump threshold the inversion is negative and the signal experiences attenuation.

On the other hand, pump values higher than the threshold have a positive inversion and so signal would

show gain under these circumstances. The low pump threshold typical of EDFA is achieved because of

the erbium high absorption cross-section and lengthy lifetime as observed in equation (2.27). Such pump

values are “easily obtained with electrically pumped diode lasers” [11].

Propagation Equations

The propagation equations describe the interactions during propagation of the signal and pump light

fields with the ions present in the host medium. These equations can be derived, using the results of

equations (2.18) through (2.20) along with some considerations. The first assumption is based on the

fact that Single Mode Fibres (SMF) favour light propagation along one path nearly parallel to the fibre’s

axis, therefore both signal and pump light fields are assumed to propagate along the z axis, simplifying

the analysis by making this a one dimensional problem. Additionally, both signal and pump are assumed

to be co-propagating, meaning that they travel in the same direction inside the fibre. Finally, keeping in

mind that the variations in the fields occur mainly because of absorption (pump) and stimulated emission

(signal) effects – excluding at this stage the effects of noise - the propagations equations are given by:

𝑑𝜙𝑠

𝑑𝑧= (𝑁2 − 𝑁1)𝜙𝑠𝜎𝑠 (2.29)

𝑑𝜙𝑝

𝑑𝑧= (𝑁1 − 𝑁3)𝜙𝑝𝜎𝑝 (2.30)

Equations (2.29) and (2.30) can also be represented in terms of the pump and signal intensities:

𝜙𝑠 =I𝑠

ℎ𝜐𝑠

(2.31)

𝜙𝑝 =I𝑝

ℎ𝜐𝑝

(2.32)

Applying these relationships to equations (2.29) and (2.30) and through the use of some mathematical

manipulation the differential equations for the pump and signal intensities behaviour along an

infinitesimal length dz can be derived:

𝑑𝐼𝑠

𝑑𝑧=

𝜎𝑝𝐼𝑝

ℎ𝜐𝑝− Γ21

Γ21 + 2𝜎𝑠𝐼𝑠

ℎ𝜐𝑠+

𝜎𝑝𝐼𝑝

ℎ𝜐𝑝

. 𝑁𝜎𝑠𝐼𝑠 (2.33)

19

𝑑𝐼𝑝

𝑑𝑧= −

𝜎𝑠𝐼𝑠

ℎ𝜐𝑠+ Γ21

Γ21 + 2𝜎𝑠𝐼𝑠

ℎ𝜐𝑠+

𝜎𝑝𝐼𝑝

ℎ𝜐𝑝

. 𝑁𝜎𝑝𝐼𝑝 (2.34)

Small signal gain regime

Because the erbium doped fibre is nonlinear, a measurement of the saturated large-signal gain may not

provide a complete characterization of an amplifier. In fact, most of the erbium doped fibre interest lies

in its behaviour in small signal gain regime, meaning the signal’s power is “weak” when compared to the

pump’s (“strong”) [11, 15]. So as to better understand the signal power vs pump power relation with gain

equations (2.33) and (2.34) will be modified and written in a simpler fashion. First, the pump and signal

light intensities are “normalized” through the use of the expressions below:

𝐼𝑝′ =

𝐼𝑝

𝐼𝑡ℎ

(2.35)

𝐼𝑠′ =

𝐼𝑠

𝐼𝑡ℎ

(2.36)

Where 𝐼𝑡ℎ is derived from (2.27)

𝐼𝑡ℎ =ℎ𝜐𝑝

𝜎𝑝𝜏2

(2.37)

Then 𝜂 is defined as

𝜂 =ℎ𝜐𝑝𝜎𝑠

ℎ𝜐𝑠𝜎𝑝

(2.38)

And finally the saturation Intensity, 𝐼𝑆𝐴𝑇 as

𝐼𝑆𝐴𝑇 =1 + 𝐼𝑝

′

2𝜂 (2.39)

At this point, equation (2.33) and (2.34) are rewritten as the normalized signal and pump intensities

making use of the terms above.

𝑑𝐼𝑠′

𝑑𝑧=

1

1 +𝐼𝑠

′

𝐼𝑆𝐴𝑇

. (𝐼𝑝

′ − 1

𝐼𝑝′ + 1

) . 𝑁𝜎𝑠𝐼𝑠′

(2.40)

𝑑𝐼𝑝

′

𝑑𝑧= −

𝜂𝐼𝑠′ + 1

1 + 2𝜂𝐼𝑠′ + 𝐼𝑝

′ . 𝑁𝜎𝑝𝐼𝑝

′ (2.41)

20

Although equations (2.40) and (2.41) do not account some detrimental effects experienced by the

erbium doped fibre, which we will discuss later on, they still allows us to make some conclusions about

its behavior:

- The first conclusion is one we have stated before. In order for the signal to show gain 𝐼𝑝 ≥ 𝐼𝑡ℎ

otherwise signal would be attenuated;

- Assuming 𝐼𝑝 ≥ 𝐼𝑡ℎ , if 𝐼𝑠′ ≪ 𝐼𝑆𝐴𝑇 than we are within the limits of the small signal gain approach

and equation (2.40) takes the form

𝐼𝑠′(𝑧) = 𝐼𝑠

′(0). 𝑒(𝛼𝑝𝑧) (2.42)

and the signal power shows exponential growth;

- Since the pump power is considered to be “strong”, meaning that it is several times larger in

value than 𝐼𝑡ℎ , 𝛼𝑝 depends only on the emission cross-section and on the total electron

population (assumed inverted in this circumstance).

𝛼𝑝 ≅ 𝑁𝜎𝑠 (2.43)

Saturation regime

Contrary to the small signal situation we have discussed before, in saturation regime the signal power

is comparable to the pump’s power, and so equation (2.42) is no longer viable [11]. The bigger 𝐼𝑠′ gets,

the more signal intensity gets dissipated via the factor 1

1+𝐼𝑠′

𝐼𝑆𝐴𝑇

. In fact when 𝐼𝑠′ ≫ 𝐼𝑆𝐴𝑇 the growth of the

signal can be characterized by the following approximation:

𝑑𝐼𝑠

′

𝑑𝑧= 𝐼𝑆𝐴𝑇 . (

𝐼𝑝′ − 1

𝐼𝑝′ + 1

) . 𝑁𝜎𝑠𝐼𝑠′ (2.44)

It is clear that in saturation regime the signal growth is not exponential like in the small signal regime,

but rather linear. Figure 2.9 shows the signal growth in both saturated and small signal regime as a

function of the pump’s power. As expected, the small signal regime has a higher gain range than the

saturated regime. This can be easily explained by this next example. Consider a launched signal power

of about −40 𝑑𝐵𝑚 (100 𝑛𝑊). If we wanted to amplifying this signal by 3 𝑑𝐵 it would mean having

−37 𝑑𝐵𝑚 (roughly 200 𝑛𝑊) at the end of the amplification process, about twice the initial power. Keeping

in mind that the pump power is several milliwatts, raising the signal by 100 𝑛𝑊 is not a difficult task. Now

instead of −40 𝑑𝐵𝑚, consider that the signal launched into the fibre has −10 𝑑𝐵𝑚 (100 𝜇𝑊) and we still

want to have a 3 𝑑𝐵 gain, twice the power. If we compare both situations it is clear that amplifying 3 𝑑𝐵

of a high signal is much “harder” (by a factor of 1000 in this case) than raising 3 𝑑𝐵 of a small signal.

Besides needing a lot more pump power (which can also impose limitations), the amplifier has a

21

maximum output power that cannot be surpassed. In general, the higher the signal power launched into

the fibre, the less gain range the amplifier will have before it enters the saturated regime.

Another interesting fact is the relation between the saturation power and the pump power. Observing

Figure 2.9, one can notice that the saturation power (limit) increases with the increase of the pump

power. This happens because “the electrons driven down from the excited level 2 via stimulated

emission by the signal are immediately available for pump absorption and can be returned to the excited

level almost “instantaneously”, given a high enough pumping rate. Maintaining a high level of inversion

in the presence of a high signal power yields a high saturation value for the signal” [11].

2.2.2 Reduction of the three-level system to the two-level system

As described in previous sections, the energy levels of erbium (derived from its absorption and emission

cross-sections) are composed for the most part of well separated manifolds (multiplets). This fact comes

from the modifications the erbium structure experiences when inserted in the silica host, which causes

Figure 2.10 - Two-Level System representation (reproduced from [11]).

Pump

Signal

multiplet 2

multiplet 1

Figure 2.9 - Saturation and Small Signal gain regions [11].

22

broadening in between the levels and splits each of those levels into a certain number of broadened

sublevels (Stark splitting, see section 2.1.2) [11, 15].

We have also seen that when it comes to pumping, EDFA usually use pump lasers of 980 𝑛𝑚 and

1480 𝑛𝑚. Depending on the wavelength of the pump, electrons in erbium ions are excited to different

energy levels. 980 𝑛𝑚 pumps excite them from the ground level ( 𝐼15/24 ) to the level 3 multiplet ( 𝐼11/2

4 ),

while in 1480 𝑛𝑚 pumping the electrons travel directly to the upper level manifold ( 𝐼13/24 ). These facts

imply that the amplifier model is characterized by a pure three-level system when 980 𝑛𝑚 pump is used

and a quasi three-level system for 1480 𝑛𝑚 pump. In both cases an approximation to a two level system

can be taken and still effectively describe the amplifier. Such approximation is justified by “the

nonradiative decay rate being much larger than the stimulated emission rate from 𝐼11/24 to 𝐼15/2

4 ” [26].

Therefore, we can conclude that the population of level 3 can be neglected and so the rate equations

can be rewritten accounting for the populations of level 1 and 2 only.

Two level-system rate equations

Having reduced the amplifier model to a two-level system, the rate equations will reflect this changes,

particularly the equation of 𝑁2 that will now have an additional term corresponding to the fact that

pumping is now done directly to the upper level.

𝑑𝑁2

𝑑𝑡= − Γ21 𝑁2 + (𝑁1𝜎𝑠

𝑎 − 𝑁2𝜎𝑠𝑒)𝜙𝑠 − (𝑁2𝜎𝑝

𝑒 − 𝑁1𝜎𝑝𝑎)𝜙𝑝 (2.45)

𝑑𝑁1

𝑑𝑡= Γ21 𝑁2 + (𝑁2𝜎𝑠

𝑒 − 𝑁1𝜎𝑠𝑎)𝜙𝑠 − (𝑁1𝜎𝑝

𝑎 − 𝑁2𝜎𝑝𝑒)𝜙𝑝 (2.46)

Notice that 𝜎𝑠𝑒, 𝜎𝑠

𝑎, 𝜎𝑝𝑒 and 𝜎𝑝

𝑎 are the respective emission and absorption cross-sections of signal and

pump. The rest of the progress is relatively similar to the one presented in the previous section. Defining

𝑁 as the sum of the population density of both levels

𝑁 = 𝑁1 + 𝑁2 (2.47)

And again, considering a steady state situation

𝑑𝑁1

𝑑𝑡= −

𝑑𝑁2

𝑑𝑡 (2.48)

We can write the population density of level 2 as a function of z (the position along the fibre)

𝑁2(𝑧) =

𝜏2𝜎𝑠𝑎𝐼𝑠(𝑧)ℎ𝜐𝑠

+𝜏2𝜎𝑝

𝑎𝐼𝑝(𝑧)ℎ𝜐𝑝

𝜏2(𝜎𝑠𝑎 + 𝜎𝑠

𝑒)𝐼𝑠(𝑧)ℎ𝜐𝑠

+𝜏2(𝜎𝑝

𝑎 + 𝜎𝑝𝑒)𝐼𝑝(𝑧)

ℎ𝜐𝑝+ 1

𝑁 (2.49)

23

Where we have made use of expressions (2.31) and (2.32). The above equation has been calculated

assuming the simplest of situation: one pump and one signal wavelength amplification. In reality

however, and since EDFA are wildly used in WDM systems, a more precise approach would consider

the possibility of multiple pumps and signals. Equation (2.49) can be generalized as follows:

𝑁2(𝑧) =

∑𝜏2𝜎𝑠𝑖

𝑎𝐼𝑠𝑖(𝑧)

ℎ𝜐𝑠𝑖𝑠𝑖

+ ∑𝜏2𝜎𝑝𝑖

𝑎 𝐼𝑝𝑖(𝑧)

ℎ𝜐𝑝𝑖𝑝𝑖

∑𝜏2(𝜎𝑠𝑖

𝑎 + 𝜎𝑠𝑖𝑒 )𝐼𝑠𝑖

(𝑧)

ℎ𝜐𝑠𝑖𝑠𝑖

+ ∑𝜏2(𝜎𝑝𝑖

𝑎 + 𝜎𝑝𝑖𝑒 )𝐼𝑝𝑖

(𝑧)

ℎ𝜐𝑝𝑖𝑝𝑖

+ 1

𝑁 (2.50)

Propagations equations for the reduced two-level system

The propagation equations much like the rate equations suffer changes from the moment we reduce the

three-level system according to the considerations stated in the beginning of this section. Also the fact

that we are now more rigorous in distinguishing the signal and pump’s absorption and emission cross-

sections, which we had assume for simplicity, in the previous section to be equal, slightly modifies the

both the propagation equations as well as the pump’s threshold intensity condition.

𝑑𝐼𝑠(𝑧)

𝑑𝑧= (𝑁2𝜎𝑠

𝑒 − 𝑁1𝜎𝑝𝑎)𝐼𝑠(𝑧) (2.51)

𝑑𝐼𝑝(𝑧)

𝑑𝑧= (𝑁2𝜎𝑝

𝑒 − 𝑁1𝜎𝑠𝑎)𝐼𝑝(𝑧) (2.52)

Again, applying the condition for population inversion 𝑁2 ≥ 𝑁1, and assuming small signal regime, the

pump’s threshold value would be

𝐼𝑡ℎ =ℎ𝜐𝑝

(𝜎𝑝𝑎 − 𝜎𝑝

𝑒)𝜏2

(2.53)

Analytical Solutions to the Two-Level System

One of the advantages of the two-level system introduced before is the possibility to solve it analytically

[15, 18]. The interest lies in the fact, that this particular solution shows reasonably accurate results in

predicting gains up to 20 𝑑𝐵, before the saturation of the amplifier due to ASE becomes predominant.

In order to achieve the final transcendent equation for the pump and signal, the upper level differential

equation is “written in terms of the derivatives of the field with respect to the fibre length, so that the

propagation equations for the fields can be integrated along the entire fibre length” [11].

∂N2(z, t)

∂t= −

N2(z, t)

τ−

1

ρA (

∂Ps(z, t)

∂z+ u

∂Pp(z, t)

∂z) (2.54)

24

Where 𝑢 indicates whether the pump is co-propagating (𝑢 = 1) or counter-propagating (𝑢 = −1), 𝐴 is

defined as the cross-sectional area and 𝜌 the ion density. Notice that N2 is defined in the above

expression as the normalized population density of level 2 (N2 =𝑁2

𝑁). The equations representing the

pump and signal powers are

∂Pp(z, t)

∂z= uρΓp × [(σp

e + σpa )N2(z, t) − σp

a ] × Pp(z, t) (2.55)

∂Ps(z, t)

∂z= ρΓs × [(σs

e + σsa)N2(z, t) − σs

a] × Ps(z, t) (2.56)

As in sections 2.2.1 and 2.2.2, a steady state situation is considered, and so after substituting (2.54) in

(2.55) and (2.56) the propagation equations become

𝑢𝑑𝑃𝑝(𝑧, 𝑡)

𝑃𝑝(𝑧, 𝑡)= − [𝛼𝑝

𝑎 + 1

𝑃𝑝𝑠𝑎𝑡

(𝑑𝑃𝑠(𝑧, 𝑡)

𝑑𝑧+ 𝑢

𝑑𝑃𝑝(𝑧, 𝑡)

𝑑𝑧)] 𝑑𝑧 (2.57)

𝑑𝑃𝑠(𝑧, 𝑡)

𝑃𝑠(𝑧, 𝑡)= − [𝛼𝑠

𝑎 + 1

𝑃𝑠𝑠𝑎𝑡

(𝑑𝑃𝑠(𝑧, 𝑡)

𝑑𝑧+ 𝑢

𝑑𝑃𝑝(𝑧, 𝑡)

𝑑𝑧)] 𝑑𝑧 (2.58)

Where 𝛼𝑝,𝑠𝑎 = 𝜌Γ𝑝,𝑠𝜎𝑝,𝑠

𝑎 and 𝑃𝑝.𝑠𝑠𝑎𝑡 =

𝐴

(𝜎𝑝,𝑠𝑒 +𝜎𝑝,𝑠

𝑎 )𝜏Γ𝑝,𝑠. All that is left to do is to integrate both equations along

the length of the fibre L.

𝑃𝑝,𝑠𝑧=𝐿 = 𝑃𝑃,𝑠

𝑧=0 × 𝑒−𝛼𝑝,𝑠𝑎 𝐿𝑒

(𝑃𝑧=0−𝑃𝑧=𝐿)𝑃𝑝,𝑠

𝑠𝑎𝑡⁄ (2.59)

And then sum all equations of the signal and pumps so as to obtain a generalized expression

(transcendental in 𝑃𝑧=𝐿) for the total output power at the end of the erbium doped fibre.

𝑃𝑧=𝐿 = (𝑃𝑃𝑧=0 × 𝑒−𝛼𝑝

𝑎𝐿𝑒

𝑃𝑧=0𝑃𝑝

𝑠𝑎𝑡⁄) 𝑒

−𝑃𝑧=𝐿

𝑃𝑝𝑠𝑎𝑡

+ (𝑃𝑠𝑧=0 × 𝑒−𝛼𝑠

𝑎𝐿𝑒𝑃𝑧=0

𝑃𝑠𝑠𝑎𝑡⁄

) 𝑒−𝑃𝑧=𝐿

𝑃𝑠𝑠𝑎𝑡

(2.60)

An immediate conclusion is that by solving for 𝑃𝑧=𝐿 the equation above, the individual signal powers can

be calculated using equation (2.59). This makes the analytical approach very popular since it obtains

accurate results employing a simple analytical method that does not require an extensive iterative

process and thus is very practical [18].

2.2.3 Amplified Spontaneous Emission (ASE)

Amplified spontaneous emission is an important factor, and one that imposes the most limitations in

optical amplifiers. It is an unwanted by-product of the amplification process that happens when electrons

that had been excited to energy level 2 (upper state) spontaneously decay to the ground level. During

this relaxation to ground state a photon is emitted that has no correlation to the signal we aim to amplify.

Additionally a cascade effect takes place where “this spontaneously emitted photon can be amplified as

25

it travels down the fibre and stimulates the emission of more photons from excited ions, photons that

belong to the same mode of the electromagnetic field as the original spontaneous photon [11].

ASE is in fact, problematic not only for “wasting” the original photon through spontaneous emission

producing noise, but mainly for amplifying the process of spontaneous emission throughout the length

of the erbium doped fibre, limiting the amount of gain the amplifier can provide.

Model

In order to write the propagation equations regarding these ASE effects, it is useful to understand the

concept of spontaneous emission power or equivalent noise power.

𝑃𝐴𝑆𝐸0 = 2ℎ𝜐Δ𝜐 (2.61)

Let us then breakdown expression (2.61). First it his helpful to remember that according to quantum

mechanics, determining “the spontaneous emission rate into a given mode is the same as determining

the stimulated emission rate into that mode with one photon already present in that mode” [11]. This

means we need to calculate the power of the original photon that triggers the entire process.

Considering a photon occupying a volume of length L, with energy ℎ𝜐, and velocity 𝑐, it is easily seen

that the power of such photon in a given mode will be ℎ𝜐𝑐

𝐿 (1). The total of modes present in

bandwidth Δ𝜐, in frequency space, for a medium of length L is 2𝐿𝛥𝜐

𝑐. This expression is then multiplied

by a factor of 2 since each SMF supports two modes 4𝐿𝛥𝜐

𝑐 (2). Finally by multiplying (1) and (2) we get

4ℎ𝜐Δ𝜐 which represents the noise power traveling in both forward and backward directions. Expression

(2.61) is half of the total noise power since the equivalent noise power is defined as the noise power

propagating in only one direction of the fibre.

Keeping this in mind, the propagation equation for the ASE is described by the following expression:

𝑑𝑃𝐴𝑆𝐸(𝜐)

𝑑𝑧= (𝑁2𝜎𝑒(𝜐) − 𝑁1𝜎𝑎(𝜐))𝑃𝐴𝑆𝐸(𝜐) + 𝑃𝐴𝑆𝐸

0 (𝜐)𝑁2𝜎𝑒 (2.62)

That is composed by a first term, identical to the one in the propagation equations deduced before for

the signal power and pump power, and a second term representing the added local noise power 𝑃𝐴𝑆𝐸0

from the spontaneously emitted photons.

The propagation equation (2.62) can be re-written in a more accurate way considering that ASE travels

in two directions, one parallel to the signal and another opposite to it. The total ASE power is therefore,

the sum of both the backward ASE power and the forward ASE power.

𝑃𝐴𝑆𝐸(𝜐𝑗) = 𝑃𝐴𝑆𝐸+ (𝜐𝑗) + 𝑃𝐴𝑆𝐸

− (𝜐𝑗) (2.63)

26

𝑑𝑃𝑝


𝑒 − 𝑁1𝜎𝑝𝑎)Γ𝑝𝑃𝑝 − 𝛼𝑝

(𝑎0)𝑃𝑝 (2.64)

𝑑𝑃𝑠


𝑒 − 𝑁1𝜎𝑠𝑎)Γ𝑠𝑃𝑠 − 𝛼𝑠

(𝑎0)𝑃𝑠 (2.65)

𝑑𝑃𝐴𝑆𝐸

+ (𝜐𝑗)

𝑑𝑧= (𝑁2𝜎𝜐𝑗

𝑒 − 𝑁1𝜎𝜐𝑗𝑎 ) Γ𝑠𝑃𝐴𝑆𝐸

+ (𝜐𝑗) + 𝑁2𝜎𝜐𝑗𝑒 Γ𝑠ℎ𝜐𝑗Δ𝜐𝑗 − 𝛼𝜐𝑗

(𝑎0)𝑃𝐴𝑆𝐸

+ (𝜐𝑗) (2.66)


− (𝜐𝑗)

𝑑𝑧= − (𝑁2𝜎𝜐𝑗


− (𝜐𝑗) − 𝑁2𝜎𝜐𝑗𝑒 Γ𝑠ℎ𝜐𝑗Δ𝜐𝑗 + 𝛼𝜐𝑗


− (𝜐𝑗) (2.67)

Consequently, the equation describing the electron population of level 2 must also be modified to

accommodate the ASE effects. An extra term is added to expression (2.50) along with the terms for

pump and signal power.

𝑁2 =

∑𝜏2𝜎𝑠𝑖

𝑎

𝐴ℎ𝜐𝑠𝑖

Γ𝑠𝑖𝑃𝑠𝑖𝑠𝑖

+ ∑𝜏2𝜎𝜐𝑗

𝑎

𝐴ℎ𝜐𝑗Γ𝜐𝑗

𝑃𝐴𝑆𝐸(𝜐𝑗)𝜐𝑗+ ∑

𝜏2𝜎𝑝𝑖

𝑎

𝐴ℎ𝜐𝑝𝑖

Γ𝑝𝑖𝑃𝑝𝑖𝑝𝑖

∑𝜏2(𝜎𝑠𝑖

𝑎 + 𝜎𝑠𝑖

𝑒 )

𝐴ℎ𝜐𝑠𝑖

Γ𝑠𝑖𝑃𝑠𝑖𝑠𝑖

+ ∑𝜏2 (𝜎𝜐𝑗

𝑎 + 𝜎𝜐𝑗

𝑒 )

𝐴ℎ𝜐𝑗Γ𝜐𝑗

𝑃𝐴𝑆𝐸(𝜐𝑗)𝜐𝑗+ ∑

𝜏2(𝜎𝑝𝑖

𝑎 + 𝜎𝑝𝑖

𝑒 )

𝐴ℎ𝜐𝑝𝑖

Γ𝑝𝑖𝑃𝑝𝑖𝑝𝑖

+ 1

𝑁 (2.68)

Finally, it is interesting to show how the ASE power evolves along the length of an erbium doped fibre.

Figure 2.11 is the graphical representation of a solution to equations (2.66) and (2.67) showing the

behaviour of both backward and forward ASE power in a 14 metre fibre.

2.2.4 Excited State Absorption (ESA)

So far, we have represented the model of the erbium ion as a three-level system and even as a two-

level system under some considerations. Performance degrading effects such as the fibre background

losses, amplified spontaneous emission and the stimulated emission at the pump wavelength were

discussed.

Figure 2.11 - Total forward and backward ASE power as a function of position [11].

27

However, the erbium ion energy level structure is much more complex than this, meaning that many

other levels exist above the level 𝐼11/24 . Therefore, depending on the pump’s wavelength, excited-state

absorption effects are possible and must be taken into account in our model.

ESFA occurs when “either a pump or signal photon, respectively, is absorbed by an erbium ion in an

excited state, thereby promoting it to an even higher energy state” [11]. These effects, shown in Figure

2.12, are detrimental to the efficiency and performance operation of an EDFA since the “re-excited”

electrons usually move from an energy level where amplification is possible (metastable level) to another

which is outside this region of amplification.

ESA is determined by the wavelength of the laser used to pump the erbium doped fibre, and it has a

higher impact on amplifier performance on the 800 and 980 nm pump band, being negligible on lasers

of higher wavelength (1480 nm).

Model

A complete model, which include ESA, has an extra rate equation for level 4 ( 𝐼9/24 ), the designated

upper level for the excite-state transition.

𝑑𝑁4

𝑑𝑡= −Γ42𝑁4 + 𝑁2𝜑𝑝𝜎𝐸𝑆𝐴 (2.69)

As sketched in Figure 2.12, it has been assumed for simplicity purposes that the erbium ion electrons

relax back into level 2 (−Γ42𝑁4). The excited absorption effect is represented by the second term

𝑁2𝜑𝑝𝜎𝐸𝑆𝐴 , where 𝜎𝐸𝑆𝐴 is the excited-state absorption cross section. As a consequence, the equation for

level 2 is also altered to incorporate the new transitions with level 4.

Figure 2.12 - Pump excited-state absorption representation (reproduced from [11]).

3

2

1

4

ℎ𝜐𝑝

ℎ𝜐𝑝

28

𝑑𝑁2

𝑑𝑡= −Γ21𝑁2 + Γ42𝑁4 + (𝑁1𝜎𝑠

𝑎 − 𝑁2𝜎𝑠𝑒)𝜑𝑠 + (𝑁1𝜎𝑝

𝑎 − 𝑁2𝜎𝑝𝑒)𝜑𝑝 − 𝑁2𝜑𝑝𝜎𝐸𝑆𝐴 (2.70)

Lastly, the pump propagation equation (for the 980 nm pump) also needs to be amended to include the

effects of excited-state absorption.

𝑑𝜙𝑝

𝑑𝑧= (−𝑁1𝜎𝑝

𝑎 + 𝑁2𝜎𝑝𝑒)𝜙𝑝 − 𝑁2𝜑𝑝𝜎𝐸𝑆𝐴 (2.71)

This new system of equations is much more complex than the ones we have derived so far. Although

we could solve it numerically, a more interesting alternative is to try and simplify these equations in order

to make the entire system easier to handle. With this objective in mind, if we notice that the lifetime of

the electrons in level 4 is much shorter when compared to level 2, it would be reasonably accurate to

suggest that those ions presented in level 4 decay rapidly to levels below, meaning the effective

population in this levels could be considered as close to zero, 𝑁4 ≈ 0.

Applying this condition to the system stated above, we get rate equations similar to the ones derived in

the previous chapter and only a pump equation containing the term correspondent to the excited-state

absorption effect.

Consequences of ESA

In the presence of ESA, the gain of an amplifier will be smaller when compared to the case where ESA

is absent, ”because a pump photon that is absorbed by an electron in level 2 is no longer available to

excite an electrons form level 1 to level 2” [11]. Additionally, after reaching level 4 these electrons rapidly

decay in a fluorescent transition to the below energy levels emitting a green light characteristic of the

ESA effect.

Other parameters, such as Noise Figure are also affected due to the fact that the pump is contributing

both to population inversion and ESA.

We therefore conclude that in order to mitigate ESA effects, a correct choice in pump configuration is

necessary. Two common solutions (for small and moderate signals) are to use counter propagating

pumping or bidirectional pumping (co-propagating + counter propagating pumping) in order to create a

more uniform inversion, thus avoiding regions of high inversions, that result in strong ESA and “waste

of photons”.

2.2.5 Final Propagation and Rate equations

All the important concepts and effects were described in the previous sections and all together determine

a complete and precise model for the Erbium doped fibre’s behaviour. In this section it will be shown

how to derive the equations for 𝑁2, 𝑃𝑝, 𝑃𝑠, 𝑃𝐴𝑆𝐸+ and 𝑃𝐴𝑆𝐸

− .

29

Rate equations

The complete model considers the two-level approximation system depicted in section 2.2.2 with the

inclusion of the fourth level referent to Excited-State Absorption discussed in section 2.2.4.

𝑑𝑁1

𝑑𝑡= Γ21 𝑁2 + (𝑁2𝜎𝑠

𝑒 − 𝑁1𝜎𝑠𝑎)𝜙𝑠 − (𝑁1𝜎𝑝

𝑎 − 𝑁2𝜎𝑝𝑒)𝜙𝑝 (2.72)

𝑑𝑁2

𝑑𝑡= −Γ21𝑁2 + Γ42𝑁4 + (𝑁1𝜎𝑠

𝑎 − 𝑁2𝜎𝑠𝑒)𝜑𝑠 + (𝑁1𝜎𝑝

𝑎 − 𝑁2𝜎𝑝𝑒)𝜑𝑝 − 𝑁2𝜑𝑝𝜎𝐸𝑆𝐴 (2.73)

𝑑𝑁4

𝑑𝑡= −Γ42𝑁4 + 𝑁2𝜑𝑝𝜎𝐸𝑆𝐴 (2.74)

Keeping in mind the system above, as well as the considerations made in the referred sections, the

normalized population density of level two can be represented as

𝑁2

𝑁=

𝜎𝑠𝑎𝜑𝑠 + 𝜎𝑝

𝑎𝜑𝑝

Γ21 + 𝜑𝑠(𝜎𝑠𝑒 + 𝜎𝑠

𝑎) + 𝜑𝑝(𝜎𝑝𝑒 + 𝜎𝑝

𝑎) +𝜑𝑝

2𝜎𝑝𝑎𝜎𝐸𝑆𝐴

Γ42

(2.75)

Next, a similar procedure to the one employed in section 2.2.1 to re-write 𝑁2

𝑁 is used, making use of

equations (2.31) and (2.32), along with the quantities 𝑔 = 𝑁𝜎𝑒Γ, 𝛼 = 𝑁𝜎𝑎Γ (gain and loss spectrum also

known as Giles coefficients [15]) and 𝜉21 =𝐴𝑁

𝜏21 , 𝜉42 =

𝐴𝑁

𝜏42 (defined as fibre parameter of levels 2 and 4

respectively) commonly supplied my fibre manufactures.

𝑵𝟐

𝑵=

𝜶𝒔

𝝃𝟐𝟏𝒉𝝊𝒔𝑷𝒔 +

𝜶𝒑

𝝃𝟐𝟏𝒉𝝊𝒑𝑷𝒑 +

𝜶𝑨𝑺𝑬

𝝃𝟐𝟏𝒉𝝊𝑨𝑺𝑬𝑷𝑨𝑺𝑬

𝟏 +(𝒈𝒔 + 𝜶𝒔)

𝝃𝟐𝟏𝒉𝝊𝒔𝑷𝒔 +

(𝒈𝒑 + 𝜶𝒑)𝝃𝟐𝟏𝒉𝝊𝒑

𝑷𝒑 +(𝒈𝑨𝑺𝑬 + 𝜶𝑨𝑺𝑬)

𝝃𝟐𝟏𝒉𝝊𝑨𝑺𝑬𝑷𝑨𝑺𝑬 +

𝑷𝒑𝟐𝜶𝒑𝜶𝑬𝑺𝑨

𝝃𝟐𝟏𝝃𝟒𝟐𝒉𝟐𝝊𝒑𝟐

(2.76)

Propagation equations

The propagation differential equations for both pump and signal can be solved through basic integration

methods in order to get an exact formula.

𝑑𝑃𝑝


𝑒 − 𝑁1𝜎𝑝𝑎)Γ𝑝𝑃𝑝 − 𝛼𝑝

(𝑎0)𝑃𝑝 − 𝑁2𝜎𝐸𝑆𝐴Γ𝑝𝑃𝑝 (2.77)

𝑑𝑃𝑠


𝑒 − 𝑁1𝜎𝑠𝑎)Γ𝑠𝑃𝑠 − 𝛼𝑠

(𝑎0)𝑃𝑠 (2.78)

By noticing that both equation are of the type 𝑑𝑃

𝑑𝑥= 𝐴. 𝑃, the population density of level 1 can be

calculated using 𝑁1 = 𝑁 − 𝑁2 and that the average normalized population density of the upper level is

30

given by 𝑁2

𝑁

=

1

𝐿∫

𝑁2(𝑧)

𝑁 𝑑𝑧

𝐿

0. Integrating both equations over the length of the fibre, we can write them as

we see below

𝑷𝒑(𝑳) = 𝑷𝒑(𝟎). 𝒆𝒙𝒑 [(𝑵𝟐

𝑵

(𝒈𝒑 + 𝜶𝒑 − 𝜶𝑬𝑺𝑨) − 𝜶𝒑 − 𝜶𝒑

(𝒂𝟎)) . 𝑳] (2.79)

𝑷𝒔(𝑳) = 𝑷𝒔(𝟎). 𝒆𝒙𝒑 [(𝑵𝟐

𝑵

(𝒈𝒔 + 𝜶𝒔) − 𝜶𝒔 − 𝜶𝒔

(𝒂𝟎)) . 𝑳] (2.80)

The amplified spontaneous emission equation for the forward and backward direction requires a slightly

different treatment. Both equations are of the type 𝑑𝑃

𝑑𝑥= 𝑃. 𝐴 + 𝐵, which means an exact solution can be

found by separating each equation into the homogeneous and nonhomogeneous forms and then

applying boundary conditions (see Figure 2.11).


+ (𝜐𝑗)

𝑑𝑧= 𝑃𝐴𝑆𝐸

+ (𝜐𝑗) [Γ𝑠 (𝑁2𝜎𝜐𝑗𝑒 − 𝑁1𝜎𝜐𝑗

𝑎 ) − 𝛼𝜐𝑗

(𝑎0)] + 2𝑁2𝜎𝜐𝑗

𝑒 Γ𝑠ℎ𝜐𝑗Δ𝜐𝑗 (2.81)

- Homogeneous:


+ (𝜐𝑗)

𝑑𝑧= 𝑃𝐴𝑆𝐸

+ (𝜐𝑗) [𝑁2

𝑁(𝑔𝐴𝑆𝐸 + 𝛼𝐴𝑆𝐸) − 𝛼𝐴𝑆𝐸 − 𝛼𝐴𝑆𝐸

(𝑎0)] (2.82)

𝑃𝐴𝑆𝐸+ (𝜐𝑗) = 𝐾. 𝑒𝑥𝑝 [(

𝑁2

𝑁

(𝑔𝐴𝑆𝐸 + 𝛼𝐴𝑆𝐸) − 𝛼𝐴𝑆𝐸 − 𝛼𝐴𝑆𝐸

(𝑎0)) . 𝐿] (2.83)

- Nonhomogeneous:

𝑃𝐴𝑆𝐸+ (𝜐𝑗) [Γ𝑠 (𝑁2𝜎𝜐𝑗

𝑒 − 𝑁1𝜎𝜐𝑗𝑎 ) − 𝛼𝜐𝑗

(𝑎0)] + 2𝑁2𝜎𝜐𝑗

𝑒 Γ𝑠ℎ𝜐𝑗Δ𝜐𝑗 = 0 (2.84)

𝑃𝐴𝑆𝐸+ (𝜐𝑗) =

−𝑁2

𝑁𝑔𝐴𝑆𝐸2ℎ𝜐𝑗Δ𝜐𝑗

𝑁2

𝑁(𝑔𝐴𝑆𝐸 + 𝛼𝐴𝑆𝐸) − 𝛼𝐴𝑆𝐸 − 𝛼𝐴𝑆𝐸

(𝑎0) (2.85)

Boundary condition: 𝑃𝐴𝑆𝐸+ (𝜐𝑗) = 𝑃𝐴𝑆𝐸

𝑖𝑛 ⟶ 𝑧 = 0 (see Figure 2.11)

𝑷𝑨𝑺𝑬+ (𝝊𝒋) =

−𝒃

𝒂+ (𝑷𝑨𝑺𝑬

𝒊𝒏 +𝒃

𝒂) . 𝒆𝒙𝒑 [(

𝑵𝟐

𝑵

(𝒈𝑨𝑺𝑬 + 𝜶𝑨𝑺𝑬) − 𝜶𝑨𝑺𝑬 − 𝜶𝑨𝑺𝑬

(𝒂𝟎)) . 𝑳] (2.86)

A similar approach is used to derive the backward propagation equation for the amplified spontaneous

emission, with a boundary condition: 𝑃𝐴𝑆𝐸− (𝜐𝑗) = 0 ⟶ 𝑧 = 𝐿𝑚𝑎𝑥


− (𝜐𝑗)

𝑑𝑧= − (𝑁2𝜎𝜐𝑗


− (𝜐𝑗) − 2𝑁2𝜎𝜐𝑗𝑒 Γ𝑠ℎ𝜐𝑗Δ𝜐𝑗 + 𝛼𝜐𝑗


− (𝜐𝑗) (2.87)

31

𝑷𝑨𝑺𝑬− (𝝊𝒋) =

−𝒃

𝒂+

𝒃

𝒂 𝒆𝒙𝒑 [(

𝑵𝟐

𝑵

(𝒈𝑨𝑺𝑬 + 𝜶𝑨𝑺𝑬) − 𝜶𝑨𝑺𝑬

− 𝜶𝑨𝑺𝑬(𝒂𝟎)

) . 𝑳𝒎𝒂𝒙] . 𝒆𝒙𝒑 [(−𝑵𝟐

𝑵

(𝒈𝑨𝑺𝑬 + 𝜶𝑨𝑺𝑬) + 𝜶𝑨𝑺𝑬 + 𝜶𝑨𝑺𝑬

(𝒂𝟎)) . 𝑳]

(2.88)

The system of equations formed by (2.76), (2.79), (2.80), (2.86) and (2.88) are the base of the Matlab

function destined to reproduce the Erbium doped Fibre behaviour, and will be revisited in the chapter 4.

33

3 Erbium Doped Fibre Amplifiers

In this Chapter, the architecture of Erbium Doped Fibre Amplifiers is introduced. The function of every

component in both Single-Stage and Multi-Stage EDFA’s is discussed, along with their influence in the

amplifiers characteristics (Gain, Noise Figure, Gain Flatness, Gain Tilt and Gain Ripple).

Simultaneously, the analysis of each component will lead to design considerations that are the basis of

this dissertation.

3.1 Importance of EDFAs in WDM systems

Wavelength division multiplexing (WDM) technology was developed in order to increase the capacity of

single channel optic communications employed at the time. By allocating a different wavelength to each

channel and then multiplexing them into a single fibre, WDM is able to exploit the large bandwidth offered

by optical fibres, dramatically increasing the capacity of a link.

WDM solutions, either DWDM or CWDM, became extensively used in long haul transmission systems.

Originally the architecture of WDM systems used electronic devices called repeaters, periodically placed

along the link. Repeaters reconstruct and retransmit optical signals through optical-electrical-optical

(OEO) conversion, a process that converts optical signals into the electrical domain, regenerates them

using a 3R scheme (Retiming, Reshaping and Rescaling) before converting them back into the optical

domain for transmission (see Figure 3.1) [19]. One of the main advantages of repeaters is that they

ensure that network impairments such as noise, attenuation, dispersion and nonlinearities are

compensated at each network node.

However, using repeaters presents two main issues:

- First, the OEO conversion is a complex process and increases the overall cost of the system;

- Second, the system is not transparent because of the OEO conversion, and so it cannot be

used for parallel transmission of different data format on different wavelength (WDM).

A way of overcoming these disadvantages would be to avoid the OEO conversion and develop a purely

optical device. EDFAs are transparent devices insensitive to bit rates or signal formats. Their low intrinsic

losses, long fluorescence times and high gain over a large bandwidth means they can accommodate

and amplify numerous WDM signals simultaneously. Additionally, EDFAs are not only cheaper to

produce, but also easier to upgrade once implemented [6]. Their inclusion in WDM links increased the

Figure 3.1 - Block diagram of a Repeater.

o/e e/o Electrical receiver

Electrical transmitter

34

distance between repeaters, allowing for optical signals to the transmitted over distances of more than

a 1000 kilometres [4].

Consequently, optical amplifiers are widely deployed in today optical communication systems,

particularly EDFAs that are ideally suited to operate in the third optical window. Figure 3.2 illustrates a

DWDM network using EDFAs.

Depending on their function on the network, EDFA can be classified as [4]:

- Power amplifiers (Booster): Used at the starting point of a link, meant to amplify the signals

coming out of the multiplexer to an appropriate level suitable for transmission over the fibre;

- Line amplifiers: Used for long distance transmission in the middle of a link, as a way to

compensate for the loss caused by long fibre spans;

- Preamplifiers: Positioned at the end of the link, their function is to amplify the signals coming

from the last fibre span so that they can be detected by the receiver.

The following section discusses the main parameters of the EDFA, while explaining the function of all

the EDFA components as well as their relationship with the amplifiers characteristics.

3.2 Single-Stage EDFA

In its most basic form EDFA consist of an EDF spool (typically ranging from 10 to 30 meters), a

semiconductor laser diode (either a 980 𝑛𝑚 or a 1480 𝑛𝑚 pump) and a WDM coupler, a device that

separates or combines optical signals at a certain operating wavelength [11, 24]. WDM couplers show

high isolation between two determined wavelengths with low excess loss making them extensively used

in EDFA architecture, as a way of efficiently combining the pump input with signals in the third

transmission window. A manageable amplifier however, has additional devices in its structure as shown

in Figure 3.3:

- Tap coupler: device which function is to divert a small percentage of the signals power, usually

about 1%, to a photodetector connecter to the EDFA’s control unit. The exact number of tap

couplers depends on the size and complexity of the amplifier in question, but at least two are

inserted so as to monitor the signals’ power at the input and output of the amplifier [27].

Figure 3.2 - Power, Line and Preamplifier position in a WDM link.

EDFA EDFA EDFA Transmitter Receiver

Power amplifier Line amplifier Preamplifier

35

- Isolator: optical component capable of allowing light to propagate in a favoured direction,

severely attenuating light travelling in directions opposite to it. For this reason they are classified

as unidirectional devices and its insertion in EDFA structure serves two main purposes: To

ensure that lasing cannot take place within the EDF and simultaneously, to act as a filter and

prevent forward propagation of the laser pump’s light outside of the doped fibre [27].

The EDFA structure presented in Figure 3.3 can be subject to modifications, either by the choice of

different pumping configurations, types of laser pumps or even the inclusion of other optical devices

(VOA, GFF) that alter the amplifier main characteristics: Gain, Noise Figure, Gain Flatness, Tilt and

Ripple. These interactions between EDFA’s characteristics/components will be studied in the following

subsections.

3.2.1 Gain

The gain of any amplifier is generally expressed in 𝑑𝐵 by a ratio between the signal’s output and input

power level.

𝐺[𝑑𝐵] = 10 log10 (𝑃𝑜𝑢𝑡

𝑃𝑖𝑛

) (3.1)

As stated in the beginning of this chapter, when it comes to WDM systems, EDFA have the ability to

amplify several optical signals simultaneously. However not all the wavelengths experience the same

degree of gain. If one were to recall Figure 2.3 in the previous chapter, it would be easy to conclude that

the gain profile of an EDF is critically affected by the absorption/emission cross sections at each

wavelength. This creates (for a highly inverted fibre in small gain situation) a gain peak at 1530 𝑛𝑚 and

a flatter area around 1550 𝑛𝑚 as shown in Figure 3.4.

Figure 3.3 - Architecture of a typical Single-Stage EDFA.

36

Besides the EDF proprieties privileging some wavelengths more than others throughout the

amplification process, expressions (2.79) and (2.80) suggest there is a very strict relation between pump

power/upper state inversion/gain. The next two figures illustrate this fact. Figure 3.5 a) shows the

variation of the fractional upper state population along the length of the EDF for three distinct values of

pump power. The second Figure shows how the percentage of population inversion achieved conditions

the overall gain profile.

From the design standpoint the goal is to determine the optimum amount of EDF and pump power

needed in order to fulfil a gain requirement, since the EDF characteristics are provided by the fibre

manufacturer. Consequently, the next two subsections will be dedicated to analyse the effects these

parameters - fibre length and pump power - have on gain profile.

Figure 3.5 - a) Fractional upper state population and b) Signal gain along the fibre for three distinct values of pump power [11].

Figure 3.4 - Net cross section for different values of the fractional upper state population [11].

37

Pump characteristics

Throughout section 2.2 a model for the EDF was constructed assuming, among other considerations,

that either a 980 𝑛𝑚 or a 1480 𝑛𝑚 pump could be used to transition electrons from the ground to the

upper level, thus achieving population inversion. We will now focus on the “pro and cons” of each type

of pump with the objective of establishing a set of guidelines to help choose the best pump option in any

given situation.

Let us begin by considering an EDF with 14 m of length and assume both pump and signal to be co-

propagating in this particular fibre. On the next Figures, we will be comparing the two wavelength

mentioned above (1530 𝑛𝑚 𝑎𝑛𝑑 1550 𝑛𝑚) so as to get a feel on how this two regions of the spectrum

behave with each type of pumping. Both signals will be launched into the fibre with −40 𝑑𝐵𝑚 (small

signal gain situation). The first figure plots the evolution of signal gain corresponding to both wavelength

when either a 980 𝑛𝑚 (filled line) or a 1480 𝑛𝑚 (dashed lined) pump is employed.

A detailed analysis of Figure 3.6 reveals several important results:

- The first quick conclusion is that at low pump values (up to 5 𝑚𝑊 in this case) a higher signal

gain is achieved by the 1480 𝑛𝑚 pump. At high pump values this is no longer true and the

980 𝑛𝑚 becomes dominant.

- The pump’s emission cross sections are largely responsible for this results [15]. At 980 𝑛𝑚 the

emission cross section of the EDF is zero and so this type of pump is able to acquire a high

level of inversion. However, at 1480 𝑛𝑚 this value is no longer zero, which means that this type

of pump is at a disadvantage because some the upper state population is being drained back

Figure 3.6 - Gain as a function of pump power for a 14 meter EDF pumped at 980 nm and 1480 nm [11].

38

to ground state via a nonzero emission cross section, ultimately limiting the overall inverted

population [11].

- When it comes to signal wavelength gain, in both cases the 1530 𝑛𝑚 wavelength experienced

higher gain than 1550 𝑛𝑚 wavelength for high pump values. This again, demonstrates the

significance of the emission cross section on signal gain at high population inversion levels,

showing that in these situations “the gain factor is simply proportional to the emission cross

section” [11].

- At low pump values however, since the population in both ground and upper level are roughly

the same, “the signal gain coefficient becomes proportional to the difference between the

absorption and emission cross sections” [11].

It is also useful to observe the contrast in the population density of the upper level created by the two

pumps and how they are distributed along the length of the fibre (see Figure 3.7):

- While the 1480 𝑛𝑚 pump tends to be evenly distributed throughout the entire length of the fibre

(higher pump conversion efficiency), the same cannot be said for the 980 𝑛𝑚 pump which

displays fluctuations of upper level population along the fibre.

- More specifically, at low pump values most of the 980 𝑛𝑚 pump power is absorbed in the first

meters of EDF, locally creating a better inversion than the 1480 𝑛𝑚 counterpart. However, this

is not a long-lasting effect as the pump’s power is depleted after the first few meters and the

remaining fibre is under pumped, causing a significant decrease in upper level population

towards the last section of the fibre.

- With the increase of pump power, ASE is no longer dismissible and consequently it drains the

upper state population, particularly in the 980 𝑛𝑚 pump case [11]. Curiously enough, the ASE

effects are more evident in the first meters of fibre, supposedly at a location where the pump

has its best performance. The reason behind this, is that the backward ASE, due to the

increasing pump power, has reached its highest value at the beginning of the fibre as well,

resulting in a depletion of upper state population.

Figure 3.7 - Fractional upper state population as a function of position along a 14 meter fibre pumped at a) 980 nm and at b) 1480 nm [11].

39

- Regarding the 1480 𝑛𝑚 pump, the population density distribution is uniform with a tendency of

getting flatter as the pump power increases. Also the ASE forward and backward powers in this

type of pump tend to be flatter along the length of the fibre, therefore not significantly impacting

the upper state population distribution [11].

Gain as a function of Fibre Length

So far, the behaviour of each pump (980 𝑛𝑚 and 1480 𝑛𝑚) has been studied assuming an EDF with a

constant length of 14 meters. The effects of both low and high pump power values were discussed, as

well as the impact of ASE in the population density distribution of the upper level throughout the fibre.

Since we have already established that the population density distribution along the fibre varies with the

type of pump used, it becomes essential to analyse the effects that different lengths of EDF cause on

signal gain.

To do so, the next figures aim to describe not only the evolution of signal gain as a function of pump

powers for the 980 𝑛𝑚 and 1480 𝑛𝑚 pumps but also to highlight the contrast of using two distinct lengths

of fibre. For the purpose of this discussion an 8 and 25 meter fibre were considered.

Although most of the previous considerations still hold true, different fibre lengths introduce some news

aspects worth mentioning:

Figure 3.8 - Up: Signal gain and fractional upper state population as a function of pump power for an 8 meter EDF. Down: Signal gain and fractional upper state population as a function of pump power for a

25 meter EDF [11].

40

- First, in the 8 meter fibre case, the fibre is not considerate long enough (“short fibre”), for ASE

to build up into a degree where it can drain upper level population. As a result the 980 𝑛𝑚 pump

inverts almost all the fibre and attains signal gains on the 20 to 35 𝑑𝐵 range with relatively low

pump values .

- The situation changes with the 25 meter fibre. This particular fibre is already consider a “long

fibre”, at least from the 980 𝑛𝑚 pump viewpoint. As high pump values are necessary in order to

achieve population inversion, a large amount of ASE power is generated, decreasing the

efficiency of this pump and as a result the pump’s power needs to be higher to get signal gains

comparable with the ones in the previous case.

- Finally, we see that a longer fibre is actually a better fit for the 1480 𝑛𝑚 pump. Although this

pump has a more uniform upper level population distribution along the fibre than the 980 𝑛𝑚

pump, it cannot obtain high population inversion due to its finite emission cross section. Thus,

a longer fibre enables it to attain comparable signal gains as the 980 𝑛𝑚 pump in the 8 meter

fibre case [11].

3.2.2 Noise figure

In optical communication systems, all optical signals must be converted back into the electric domain at

the end of the transmission process. It is the receiver’s responsibility to recreate the original signal by

turning all the incident photons into the correspondent electrical signal. However, from the moment the

signal is generated at the transmitter and travels along the fibre until it reaches the receiver’s detector,

the so called “useful” signal photons are impaired by noise photons. This becomes increasingly relevant

when optical amplifiers are employed. The spontaneous emission photons, either the ones created

during the amplification process or the ones that arrive at the amplifier already mixed with the signal

photons, will give rise to a portion of the final electrical signal that is designated as noise [11]. As this

noise power, composed by photons of random frequency that contain no viable information, becomes

comparable to the signal’s power, it interferes with the ability of the receiver to reconstruct the intended

signal. The phenomenon limits the receiver’s sensitivity, introducing errors in the final bit stream and

ultimately affecting the overall quality of service.

Noise introduces an additional restriction to amplifier design. Not only does the amplifier need to meet

the designated gain requirements, it also has to control the noise generated along a link, under penalty

that the signal cannot be correctly recovered at the receiver because of high ASE power levels. As a

result, noise metrics such as Signal to Noise Ratio (SNR) and Noise Figure (NF) of an amplifier become

of immense importance, allowing us to measure and quantify the level of degradation the signal

experiences at a specific point along the link.

This section will focus on the noise properties of EDFA, as well as the derivation of the metrics Noise

Figure and Signal to Noise Ratio. Finally, pump configurations options will be discussed as a way to

minimizing the noise effects on the signal’s transmission.

41

Signal to Noise Ratio and Noise Figure Derivation

Noise Figure is the designation given to the representation of the Noise factor in logarithmic units, and

is a performance metric from which any amplifier can be characterized. The Noise factor is defined as

a quotient between the SNR at the input and the SNR at the output of the device.

𝑁𝐹[𝑑𝐵] = 10 log10 (𝑆𝑁𝑅𝑖𝑛

𝑆𝑁𝑅𝑜𝑢𝑡

) (3.2)

The Signal to Noise Ratio compares the level of a desired signal with the level of background noise.

𝑆𝑁𝑅 =𝑝

𝑛 (3.3)

Let us now consider an EDFA working as a preamplifier. In this scenario the noise at the receiver is

composed of three dominant terms: shot noise, signal-spontaneous beat noise, spontaneous-

spontaneous beat noise and thermal noise [11]. Expressions describing all three noise terms mentioned

above can be found by integrating the noise power densities over the receiver’s electrical bandwidth

(𝐵𝑒)

𝑁𝑠ℎ𝑜𝑡 = 2𝐵𝑜(𝐺𝐼𝑠 + 𝐼𝑠𝑝)𝑞 (3.4)

𝑁𝑠−𝑠𝑝 = 2𝐺𝐼𝑠𝐼𝑠𝑝

𝐵𝑒

𝐵𝑜

(3.5)

𝑁𝑠𝑝−𝑠𝑝 =1

2𝐼𝑠𝑝

2𝐵𝑒(2𝐵𝑜 − 𝐵𝑒)

𝐵02 (3.6)

𝑁𝑡ℎ =4kT

R (3.7)

Where 𝐺 is the gain of the amplifier, 𝑞 represents the elementary charge, 𝐵𝑜 the optical bandwidth and

𝐼𝑠 and 𝐼𝑠𝑝 are the photocurrent generated at the detector by signal and spontaneous emission photons

respectively. Additionally, 𝐼𝑠 is related to the signal’s optical power by 𝐼𝑠 =𝑃

ℎ𝜐𝑞 and 𝑅 is the resistance

of the detector load resistor.

Under the assumption of a shot-noise-limited source, the Signal to Noise Ratio at the input and output

of the amplifier can be written as

𝑆𝑁𝑅𝑖𝑛 =𝐼𝑠

2

2𝑞𝐼𝑠𝐵𝑒

=𝐼𝑠

2𝑞𝐵𝑒

(3.8)

𝑆𝑁𝑅𝑜𝑢𝑡 =(𝐺𝐼𝑠)2

𝑁𝑠−𝑠𝑝 + 𝑁𝑠𝑝−𝑠𝑝 + 𝑁𝑠ℎ𝑜𝑡 + 𝑁𝑡ℎ

(3.9)

Where each noise power is defined by expressions (3.4), (3.5), (3.6) and (3.7). Consequently, the Noise

Figure will have the following expression.

42

𝑁𝐹 =

𝐺𝐼𝑠𝑝𝐼𝑠

2

𝑞𝐵0+

𝐼𝑠

4𝑞𝐼𝑠𝑝

2 (2𝐵0 − 𝐵𝑒)

𝐵02 + 𝐼𝑠(𝐺𝐼𝑠 + 𝐼𝑠𝑝) +

𝐼𝑠

𝑞𝐵𝑒

2kTR

(𝐺𝐼𝑠)2

(3.10)

It is possible to further simplify the equation above by making use of the expression for the spontaneous

emission power current, 𝐼𝑠𝑝 = 2𝑛𝑠𝑝(𝐺 − 1)𝑞𝐵0, where 𝑛𝑠𝑝 is the inversion parameter, defined as 𝑛𝑠𝑝 =

𝑁2𝜎𝑒

𝑁2𝜎𝑒−𝑁1𝜎𝑎. Expression (3.10) then becomes

𝑁𝐹 = 2𝑛𝑠𝑝

(𝐺 − 1)

𝐺+

1

𝐺+

𝑛𝑠𝑝(𝐺 − 1)2𝑞(2𝐵0 − 𝐵𝑒)

𝐺2𝐼𝑠

+2𝑛𝑠𝑝(𝐺 − 1)𝑞𝐵0

𝐺2𝐼𝑠

+

2kTq𝐵𝑒R

𝐺2𝐼𝑠

(3.11)

Finally, a closer look at equation (3.11) reveals that when 𝐺 ≫ 1 only the first two terms on the right

hand side are relevant. With this in mind, the expression for Noise Figure can be simply expressed as

𝑁𝐹 =𝑃𝐴𝑆𝐸

ℎ𝜐Δ𝜐𝐺+

1

𝐺 (3.12)

Where the total ASE power is written as a function of 𝐺, 𝑃𝐴𝑆𝐸 = 2𝑛𝑠𝑝ℎ𝜐Δ𝜐(𝐺 − 1). Expression (3.12)

provides several important results on how we can mitigate the effects of noise in EDFA through a

criterious choice of components:

- Consider an ideal amplifier. If 𝐺 ≫ 1 and we assume total population inversion is achieved

(𝑁2

𝑁= 1) , then

(𝐺−1)

𝐺≈ 1, 𝑛𝑠𝑝 ≈ 1 and the noise factor will equal 2. Consequently the Noise

Figure will hit its lowest possible value of 3 𝑑𝐵, called Noise Figure Quantum Limit. Although

this value has been attained experimentally, normal Noise Figure in EDFA tends to range

Figure 3.9 - Noise Figure at 1550 nm as a function of Gain for a 980 nm and 1480 nm pump [11].

43

between 4 and 6 𝑑𝐵 in a situation of small signal gain and even higher when the amplifier is

working in the saturation region (Boost Amplifiers).

- Another vital information is the fact that the wavelength of the laser used to pump the EDF

influences directly the overall value of NF, via the inversion parameter 𝑛𝑠𝑝. This means that a

pump that promotes a higher level of population inversion will achieve a lower NF value, which

favours a 980 𝑛𝑚 pumping instead of a 1480 𝑛𝑚 one.

- Finally, as equation (3.12) is also a function of 𝐺, observing Figure 3.9, it is easy to notice that

for lower gains we will have higher NF values and as the gain increases the NF drops

significantly.

Effects of co-propagating, counter propagating and bidirectional pumping in EDFA’s gain and

noise figure.

When it comes to pump configurations, EDFA have three possibilities. The EDF can either be pumped

in the same direction of propagation as the signal (co-propagating pump), or be pumped in the opposite

direction in relation to the signal (counter propagating pump). The third option is a hybrid solution where

both co and counter propagation schemes are used simultaneously to pump the EDF and is designated

as bidirectional pumping. In a situation of small signal gain, commonly found in EDFA serving as

preamplifiers – input signal on the −40 𝑑𝐵𝑚 range – there is no significant benefit in using a counter

propagating configuration compared to a co propagating one. The reason behind it, is that the ASE

patterns along the length of the fibre “generated by the two pumps are mirror images of each other and

so the average upper state population is the same in both cases” [11]. However, if a bidirectional

Figure 3.10 - Pump configurations for a Single-Stage EDFA [11].

44

pumping configuration was used, the resulting ASE pattern would be different, consequence of a more

uniform population distribution of the upper level. This fact also makes bidirectional pumping a preferable

choice when ESA is present, since the lack of highly inverted regions diminishes the probability of this

phenomenon to occur. Therefore, the bidirectional configuration gets the highest output power (for a

long enough fibre) of all the three configurations for a signal power in between −40 and −20 𝑑𝐵𝑚 as

shown in Figure 3.11 a). It is also interesting to notice that for a short fibre the choice of pump

configuration dos not matter, since they all yield the same results.

For larger signal input power, like the ones found in inline EDFAs employed in long haul communication

systems, the results are somewhat different. Inline amplifiers operate in conditions propitious to the

increase of noise. Consequently, they often have short lengths of fibre as a way to control the building

of ASE power and ultimately Noise Figure. Additionally, they operate with significantly higher signal

powers, in the 0 𝑑𝐵𝑚 range [11]. Observing Figure 3.11 we can easily see for these situations that all

pumping schemes have identical results for the so called “short fibre” length. Interestingly enough, once

the fibre length increases past the 20 meters, the counter propagating pump’s performance decreases

due to an increase of forward ASE that ends up consuming a portion of its gain.

Now let us analyse how a particular pump configuration impacts Noise Figure. Figure 3.12 demonstrates

the evolution of NF as a function of pump power for an 8 and 12 meter fibre with a co and counter

propagating pump configuration. The plot confirms a previous result deducted from expression (3.12),

Figure 3.11 - Signal output power as a function of fibre length for a co propagating, counter propagating and bidirectional pump [11].

Figure 3.12 - Noise Figure as a function of pump power for a co and counter propagating configuration. An 8 and 12 meter fibre is tested [11].

45

stating that a 980 𝑛𝑚 pump produced a lower NF than a 1480 𝑛𝑚 pump. It also shows that a co

propagating scheme always has better noise figure than a counter propagating one. In simple words, in

a co propagating configuration the first portion of EDF is more inverted than the last portion of fibre

where signal exits. “Thus the signal undergoes more gain per unit length at the beginning of the fibre

than at the exit” [11]. The opposite situation is verified in the counter propagating case, where the signal

first travels through a section of EDF that is poorly inverted, or not inverted at all. As we already know,

this causes attenuation in the signal’s power which in turn will degrade the noise figure.

3.2.3 Gain Flatness

Throughout the previous sections we have scrutinized the aspects that make EDFAs one of the most

successful amplifiers for WDM applications. For instance, Long Haul Communications Systems employ

this optical amplifier due to its large bandwidth, relatively low noise and high gain range attained with

fairly low pump powers. Multichannel amplification, however, is also where EDFA display their main

limitation: spectral nonuniformity of the gain profile.

“Ideally an optical amplifier should provide the same gain for all the channels under all possible operating

conditions” [3]. Nevertheless, this is not the case for the majority of optical amplifiers, EDFA being no

exception. In Figure 3.4 it is shown that the gain is far from flat, resulting in different wavelengths being

amplified by different amounts. These gain discrepancies between channels get worse when EDFA are

inserted in a chain of cascaded amplifiers, like in the Long Haul Communications Systems example. For

large enough links, this difference of power among channels can grow into a factor that puts them

beyond the dynamic power range the receiver can cover, which is unacceptable in practice.

One way to counteract these effects, is to try and group the wavelength channels into a “flatter” region

of the gain spectrum [11]. This may work for some CWDM solutions, where the system operates with a

small number of channels, but it hardly constitutes a solution in situations where 80 plus channels

(DWDM) are employed and practically all of the EDFA bandwidth is necessary. A more effective solution

would be to equalize the entire EDFA bandwidth (40 𝑛𝑚) with the help of an external gain flattening

element.

Figure 3.13 - Gain spectrum analysis of single stage EDFA with GFF positioned after EDF [28].

46

The concept behind gain flatness is rather simple. The Gain Flattening Filter (GFF) acts as an attenuator

which is wavelength selective. Its function is to match the gain profile from a predefined reference level

and to attenuate each individual channel accordingly (high attenuation in high gain regions and low

attenuation in low gain regions). Figure 3.13 illustrates this process where the device is position after

the EDF.

Many types of flattening techniques have been investigated and developed along the years: Mach-

Zehnder s, acoustic-optic and thin film interference filters are some examples [21]. The study of these

devices falls outside of the scope of this dissertation because the interest, from the design viewpoint,

lies only in determining the amount of attenuation each channel requires.

GFF in EDFA architecture

The insertion of GFFs in EDFA architecture influences the gain and noise figure characteristics of the

amplifier. If the filter is located after the EDF as represented in Figure 3.13, it will reduce the maximum

output power the amplifier can deliver, whereas placing it before the EDF causes a relative increase of

the forward ASE power compared to the signal’s power, which as we already know, worsens Noise

Figure [11]. Clearly a compromise must be found, and in this case, the solution is to incorporate the GFF

strategically in a multistage EDFA, an approach studied in the next section.

3.2.4 Gain Tilt and Gain Ripple

The main limitation of passive gain flattening elements, such as GFF, is that they “are usually designed

for a specific operating point of the EDFA. Thus they are not necessarily that robust when system

parameters vary and the spectral gain they are compensating for changes” [11].

Figure 3.14 - Gain Tilt and Gain Ripple of an EDFA output spectrum [20].

47

This means that in situations where the amplifier is required to operate outside of the optimum point

under different power and gain conditions, gain flatness is not attainable simply because the gain profile

generated under these conditions does not “match” the previously dimensioned GFF attenuation. As a

result, the gain spectrum at the end of the process is tilted as shown in Figure 3.14, by a factor

denominated tilt coefficient (in 𝑑𝐵. 𝑇𝐻𝑧−1).

Gain Tilt

The contribution of gain tilt to gain spectrum nonuniformity is an effect that can be compensated by the

EDFA itself through the use of a Variable Optical Attenuator (VOA). Unlike GFFs which are wavelength

selective, a VOA attenuates all wavelengths equally. When placed before the EDF it is able to indirectly

manipulate the shape of the gain profile by acting on the signal’s input power. Since the signal’s output

power is a fixed condition, a change in its input power represents a change in the gain requirement. A

different gain requirement, usually means different pump power values being pumped into the EDF,

ultimately generating different gain profiles. By correctly choosing the attenuation in the VOA, a gain

profile can be found that matches the GFF’s attenuation, thus cancelling the initially tilted spectrum (see

Figure 3.15). Gain flatness across the EDFA’s operating region can be attained by changing the VOA’s

attenuation accordingly [23].

Nonetheless, there are situations where gain tilt may prove to be advantageous. Some WDM systems

suffer from problems of unregulated tilt. In these systems the “unwanted tilt” is generated along the

transmission fibre by two effects intrinsic to it: fibre Background Loss and Stimulated Raman Scattering

(SRS) between signal wavelengths [20]. While the first effect has already been introduced in the

previous chapter and is based on the irregularities on the attenuation spectrum of the typical silica fibre,

𝑉𝑂𝐴𝑎𝑡𝑡 = 𝐺𝑑𝑒𝑠𝑖𝑔𝑛 − 𝐺

𝐺 < 𝐺𝑑𝑒𝑠𝑖𝑔𝑛

𝑉𝑂𝐴𝑎𝑡𝑡 𝑎𝑑𝑗𝑢𝑠𝑡𝑚𝑒𝑛𝑡

Figure 3.15 - Gain tilt adjustment using a Variable Optical Attenuator (reproduced from [20]).

48

the second is a scattering type effect, based on photon interactions, that results in “higher wavelengths

suppressing signal of lower wavelengths” [29]. Figure 3.16 illustrates this scenario.

In these situations, the EDFA is designed with a tilt requirement. In practice, this means the GFF is not

dimensioned to produce gain flatness directly, but to compensate the unregulated tilt with an equal yet

opposite tilt. This way EDFA’s can correct problems associated with unregulated tilt and stop them from

propagating throughout the rest of the system.

Gain Ripple

Gain ripple can be best described as a peak to peak gain error function, from which one can analyse

the maximum gain flatness of the transmitted signals. Gain ripple is directly associated with the precision

of the GFF, and is usually used to measure its quality. From the design view point a gain ripple limit is

established, meaning that in order for a certain gain profile to be classified as flat, its spectrum

fluctuations must remain confined within that limit.

3.2.5 Single Stage EDFA Limitations

It has already been shown the interplay of EDF sizes, pump types and configurations on design

requirements such as Gain and Noise Figure. Let us now discuss the main issue that an amplifier with

a single stage topology faces.

As it was mentioned in the previous sections, Noise figure and Gain requirements influence the length

of EDF and the amount of pump power needed to be launched into the fibre. Preamplifiers and Inline

amplifiers often demand medium to high gain ranges with strict values for Noise figure. In these

situations, single-stage EDFA struggle to simultaneously attain both conditions [11]. At first glance, one

Figure 3.16 - Example of Unregulated tilt cancelling using EDFA in WDM systems (reproduced from [20]).

EDFA EDFA

DCU Span 1 = 25 dB Span 2 = 15 dB

Unregulated Tilt Tilt Reference ≤ 0

Provisioned Tilt

49

would think that a higher gain requirement would simply imply additional meters of EDF. While that

would still be true, the overall situation is slightly more complex. A longer piece of EDF would consume

all the pump’s light and inevitably lead to a decrease in the average population inversion, therefore

increasing NF. A shorter fibre would evenly be inverted, offering a good NF value but limiting the gain

range the amplifier would deliver. The ideal solution would be to successively increase the pump’s power

as a response to the increase in EDF length. However, this is not practical since there is a limit to the

laser pump powers.

Another issue with single-stage EDFA is the fact that Gain Tilt cannot be controlled over an entire gain

range. As discussed in section 3.2.4, VOAs can be inserted in EDFAs architecture, providing Gain Tilt

Control by manipulating the internal gain of the amplifier to match a predefined GFF’s attenuation profile.

However, for this device to be able to correct a tilted gain profile, it has to be placed in between

amplification stages. This way the first stage is tasked with providing a rough estimation of the final gain

profile, allowing the VOA to “fine tune” the signal’s power at the input of the second amplification stage,

thus achieving the required tilt.

This conflict between Noise Figure and high gain ranges, together with the fact that Single-Stage

topologies are incapable of providing Gain Tilt Control over an entire gain range is the starting point for

the discussion on Multi-stage EDFA’s (see Appendix A).

51

4 Optimization Tool for Erbium Doped Fibre Amplifiers

4.1 Component Modelling

The first challenge when building an Optimization Tool for EDFA is deciding on how to portray EDFA’s

components. Since one of the criteria of this dissertation, was for the tool to be implemented using

Matlab, a function was assigned to each element. This way, representing an amplifier becomes the

simple process of organizing these individual functions, similarly to building a block diagram.

Erbium Doped Fibre

At the core of the EDFA Optimization Tool is EDF modelling. An accurate representation of the doped

fibre’s behaviour and its effects on optical signals is essential, otherwise any form of optimization has

no practical use whatsoever. In light of this, chapter 0 was entirely dedicated to the study of EDF

properties and to find the rate and propagation equations that best describe the effects optical signals

experience in that medium. The result was a system of Ordinary Differential Equations (ODE) of which

the general solution is shown by equations (2.76), (2.79), (2.80), (2.86) and (2.88) here repeated.

𝑃𝑠(𝐿) = 𝑃𝑠(0). 𝑒𝑥𝑝 [(𝑁2

𝑁

(𝑔𝑠 + 𝛼𝑠) − 𝛼𝑠 − 𝛼𝑠

(𝑎0)) . 𝐿] (4.1)

𝑃𝑝(𝐿) = 𝑃𝑝(0). 𝑒𝑥𝑝 [(𝑁2

𝑁

(𝑔𝑝 + 𝛼𝑝 − 𝛼𝐸𝑆𝐴) − 𝛼𝑝 − 𝛼𝑝

(𝑎0)) . 𝐿] (4.2)

𝑃𝐴𝑆𝐸+ (𝜐𝑗) =

−𝑏

𝑎+ (𝑃𝐴𝑆𝐸

𝑖𝑛 +𝑏

𝑎) . 𝑒𝑥𝑝 [(

𝑁2

𝑁

(𝑔𝐴𝑆𝐸 + 𝛼𝐴𝑆𝐸) − 𝛼𝐴𝑆𝐸 − 𝛼𝐴𝑆𝐸

(𝑎0)) . 𝐿] (4.3)

𝑃𝐴𝑆𝐸− (𝜐𝑗) =

−𝑏

𝑎+

𝑏

𝑎 𝑒𝑥𝑝 [(

𝑁2

𝑁

(𝑔𝐴𝑆𝐸 + 𝛼𝐴𝑆𝐸) − 𝛼𝐴𝑆𝐸

− 𝛼𝐴𝑆𝐸(𝑎0)

) . 𝐿𝑚𝑎𝑥] . 𝑒𝑥𝑝 [(−𝑁2

𝑁

(𝑔𝐴𝑆𝐸 + 𝛼𝐴𝑆𝐸) + 𝛼𝐴𝑆𝐸 + 𝛼𝐴𝑆𝐸

(𝑎0)) . 𝐿]

(4.4)

𝑁2

𝑁=

𝛼𝑠

𝜉21ℎ𝜐𝑠𝑃𝑠 +

𝛼𝑝

𝜉21ℎ𝜐𝑝𝑃𝑝 +

𝛼𝐴𝑆𝐸

𝜉21ℎ𝜐𝐴𝑆𝐸𝑃𝐴𝑆𝐸

1 +(𝑔𝑠 + 𝛼𝑠)

𝜉21ℎ𝜐𝑠𝑃𝑠 +

(𝑔𝑝 + 𝛼𝑝)𝜉21ℎ𝜐𝑝

𝑃𝑝 +(𝑔𝐴𝑆𝐸 + 𝛼𝐴𝑆𝐸)

𝜉21ℎ𝜐𝐴𝑆𝐸𝑃𝐴𝑆𝐸 +

𝑃𝑝2𝛼𝑝𝛼𝐸𝑆𝐴

𝜉21𝜉42ℎ2𝜐𝑝2

(4.5)

Both the rate equation’s dependence on signals’ power (𝑁2

𝑁(𝑃𝑠, 𝑃𝑝, 𝑃𝐴𝑆𝐸)) as well as each propagation

equation dependence on the average fractional population on level energy level 2 𝑃 (𝑁2

𝑁), mean that the

system cannot be solved directly, which implies that a iterative approach must be used. Taking

advantage of the fact that 𝑁2

𝑁 is a multivariable function, a solution can be approximated by employing

an unconstrained nonlinear optimization algorithm like fminsearch. Given a set of initial values for

52

𝑃𝑠, 𝑃𝑝 , 𝑃𝐴𝑆𝐸 and provided the length of fibre 𝐿𝑚𝑎𝑥, fminsearch is able to minimize the error between

consecutive iterations of 𝑁2

𝑁, until a the required precision has been met.

𝑒𝑟𝑟𝑜𝑟 (𝑁2

𝑁)

𝑛=

𝑁2

𝑁 𝑛−1−

𝑁2

𝑁 𝑛

𝑁2

𝑁 𝑛

(4.6)

After 𝑁2

𝑁 has been calculated along the entire length of the doped fibre, the average fractional population

of energy level 2 and each propagation equation can finally be computed. Through this method, it is

possible to faithfully describe the evolution of each optical signals’ power as it propagates inside the

EDF.

Passive devices:

The remaining component beside EDF – and excluding at this point the GFF and VOA, which will

deserve a more detailed analyses in the next section - are modelled in terms of how they affect the

incoming signals’ power. This means that from the signals’ point of view, every device listed below is

only perceived by its insertion losses, according to expression (4.7).

𝑪𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕 𝑰𝒏𝒔𝒆𝒓𝒕𝒊𝒐𝒏 𝑳𝒐𝒔𝒔

[𝒅𝑩]

Tap Monitor 0,65

Isolator 0,2

WDM Coupler 0,07

Optical Switch 0,8

Power Splitter 0,07

OSC Add/Drop 0,6

𝑃𝑠𝑜𝑢𝑡𝑝𝑢𝑡 [𝑑𝐵𝑚] = 𝑃𝑠𝑖𝑛𝑝𝑢𝑡 [𝑑𝐵𝑚] − 𝐼𝐿𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 [𝑑𝐵] (4.7)

4.2 Design Parameters

4.2.1 EDF Length

Correct EDF length determination is an essential step towards fulfilling a predefined gain requirement

in EDFA design. An optimized EDF length must guarantee that, at least the referenced gain is

achievable for all wavelengths channels, while trying simultaneously to minimize the stress put on the

Table 4.1 - Components Insertion Losses.

53

amplifier’s pump system. Various methods exist to determine the correct amount of EDF needed for a

specific gain. Most of them are based on performing gain simulation for increasing values of EDF length

while the pump power is treated as a constant value [30]. However, these methods are both time

consuming, due to the amount of simulations they require, and also discard possible solutions by

choosing the pump value beforehand. The method proposed and utilized in the EDFA Optimization Tool

does not support an a priori knowledge of the pump’s power, allowing it to accommodate a broader

range of possible solutions. Furthermore, it does not require an exclusive number of simulations, which

ultimately makes it more expedite than the previous one.

In order to optimize the EDF length, this approach takes advantage of equation (2.80) derived in chapter

2, rewriting it as:

𝐿 =𝑙𝑛(𝑔)

𝑁2

𝑁


(𝑎0)

(4.8)

Two considerations are worth mentioning, at this point:

- 𝑔 in the expression above does not stand for the required gain the EDFA must achieve, but

rather the amplifier’s internal gain. Since every component in EDFA’s architecture introduces

additional attenuation to the incoming signal power, to meet a certain gain requirement (external

gain), the actual level of amplification inside the device, provided by the EDF, must be higher

(𝑔𝑖𝑛𝑡 > 𝑔𝑒𝑥𝑡) so as to overcome component’s attenuation.

- Also 𝑁2

𝑁 is unknown at this stage, and so 𝐿 in expression (4.8) is a matrix with columns

representing the length of EDF for each wavelength channel and lines showing the EDF length

variation for different values of 𝑁2

𝑁

.

𝑳 [𝒎] 𝑾𝒂𝒗𝒆𝒍𝒆𝒏𝒈𝒕𝒉 𝑪𝒉𝒂𝒏𝒏𝒆𝒍𝒔 [𝒏𝒎]

1528,8 1535,8 1544,5 1552,1 1560,2 1566,7

𝑵𝟐

𝑵

0,57 37,96 28,75 23,49 20,30 19,35 21,73

0,59 25,25 22,28 19,59 17,87 17,15 19,65

0,61 18,91 18,06 16,64 15,82 15,48 17,94

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0,99 3,97 4,76 5,35 5,79 6,28 7,77

Analysing Table 4.2, it becomes clear that the absorption and emission cross-sections characteristic of

EDF that privilege amplification of some wavelengths more than others has consequences on the fibre

length each channel requires. The channels that experience less amplification end up needing more

Table 4.2 - EDF Length for different wavelength and population inversion values.

54

fibre to meet a specified gain condition and vice-versa. Therefore, from the design viewpoint – for a

given value of population inversion - only the maximum length can guarantee that all wavelength

channels meet the required gain.

𝑵𝟐

𝑵

𝑳𝒎𝒂𝒙 [𝒎]

0,57 37,96

0,59 25,25

0,61 18,91

.

.

.

.

.

.

0,99 7,77

These pair of 𝐿𝐸𝐷𝐹/𝑁2

𝑁

values are than inputted in equation (4.9), generating a matrix (𝑔𝑜𝑝𝑡) representing

possible gain profile shapes associated with the gain requirement.

𝑔𝑜𝑝𝑡 = 𝑒𝑥𝑝 [(𝑁2

𝑁


(𝑎0)) . 𝐿𝑚𝑎𝑥] (4.9)

The final step consists in using the information from Table 4.3 and matrix 𝑔𝑜𝑝𝑡, to determine which

𝐿𝐸𝐷𝐹/𝑁2

𝑁

combination generates a gain profile with the lowest deviation (lowest variation of power across

wavelengths) possible throughout the entire EDF bandwidth. This way, a fibre length 𝐿 is chosen, that

is long enough to allow that all channel have at least the required gain and short enough in order to

minimize gain spectrum deviation.

4.2.2 Gain Flattening Filter’s Attenuation

Gain Flattening Filters are used to flatten or smooth the top layer of gain spectrum, where characteristic

EDF gain fluctuations are more pronounced, restoring all wavelength channels to approximately the

same intensity. Consequently, GFF design is critical in order to achieve strict values of gain flatness,

gain tilt and ripple conditions. When it comes to optimization, one of the options is to take a similar

approach to the one describe above for in the EDF length case, and find the GFF that “matches” the

gain profile with the lowest deviation. However, calculating GFF´s attenuation in this matter has proven

to be unreliable because by choosing the attenuation profile that minimizes gain deviation, the method

indirectly assumes to know the value for the average fractional population inversion. In practice however,

depending on the pump scheme and on the pump algorithm used to determine the power of every pump,

Table 4.3 - Optimum EDF Length as a function of fractional

population inversion.

55

the 𝑁2

𝑁

value might end up being higher or lower than the one predicted by the method, thus causing a

poor match between gain and GFF profiles (see Figure 4.1).

To overcome this limitation, the Optimization Tool employs an iterative process that determines the

GFF’s attenuation profile while pump algorithm converges towards a final solution. For this to work a

“database” or “library” of possible GFF profile is created, taking advantage of matrix 𝑔𝑜𝑝𝑡 defined

previously.

𝐺𝐹𝐹𝑎𝑡𝑡 [𝑑𝐵] = 𝐺𝑟𝑒𝑓 − 𝐺𝑜𝑝𝑡 (4.10)

Having a matrix 𝐺𝐹𝐹𝑎𝑡𝑡, containing different shapes of attenuation profiles, the final step consists in

adjusting the GFF’s attenuation profile in each pump algorithm iteration. This adjustment is a comparison

of the gain profile with the listed attenuation profiles of 𝐺𝐹𝐹𝑎𝑡𝑡 and leads to finding the attenuation profile

that better compensates gain non-uniformities within a predefined error margin.

Figure 4.1 - Tilted Output Gain profile, consequence of a poorly dimensioned GFF.

Figure 4.2 - GFF’s Attenuation profile being adjusted to a Gain profile.

Gain Profile

GFF’s Attenuation Profile

56

4.2.3 Variable Optical Attenuator

Once GFF and EDF length are properly dimensioned for operating at the Optimum Point, it is important

that the amplifier is able to reproduce these gain tilt and ripple features outside of the Optimum Point

and at various gain conditions. In these situations, gain adjustments, also designated as Tilt control, are

achieved by varying the VOA’s attenuation [23].

The technique used to compensate for gain tilt deviations, and thus maintain the gain profile constant is

displayed below.

Δ𝑉𝑂𝐴 [𝑑𝐵] = 𝑇𝑖𝑙𝑡 − 𝑝𝑟𝑒_𝑇𝑖𝑙𝑡

𝑇𝑖𝑙𝑡𝑐𝑜𝑟𝑟

(4.11)

𝑉𝑂𝐴[𝑑𝐵] = 𝑉𝑂𝐴 + Δ𝑉𝑂𝐴 (4.12)

Where 𝑝𝑟𝑒_𝑇𝑖𝑙𝑡 is the EDFA tilt at Optimum Operating conditions, 𝑇𝑖𝑙𝑡 stands for the actual gain tilt we

aim to cancel and 𝑇𝑖𝑙𝑡𝑐𝑜𝑟𝑟 is a proportionality constant between the gain tilt and VOA deviation, attained

experimentally. This iterative method produces good results, effectively correcting unregulated Tilt, after

3 iterations.

4.2.4 EDF coil ratio and Power Splitter Ratio

EDF coil and Power splitters ratios can be determined by resorting to a parameter called Figure of Merit.

Figure of merit characterizes the amplifiers performance and is defined by:

ℱ𝑀𝑒𝑟𝑖𝑡 = 𝑚𝑖𝑛{𝑃 − 𝑁𝐹} (4.13)

𝑃 stands for channel power in 𝑑𝐵 units and 𝑁𝐹 is the noise figure associated with it. Depending on the

pump configuration used, two situations are possible:

Figure 4.3 - Figure of Merit evaluated for different values of Relative Length of First Fibre Coil.

57

The first corresponds to a multistage amplifier fed by independent pumps. In this case, no power splitter

is employed and so the determination of the optimum EDF coil ratio becomes a two dimensional

problem. A sweep along the possible values for EDF coil percentage is done - typically from 20% to

80% - and the Figure of Merit of each point is calculated. The optimum percentage value for the EDF

coil will be associated with the maximum value of Figure of Merit (see Figure 4.3).

The other possibility occurs in multistage amplifiers with a pump shared between both stages. The

introduction of a new variable (splitting ratio of the Power Splitter) increases the complexity of the

problem, and imposes a different approach in order to determine both variables simultaneously. A set

of contour plots showing the behaviour the Figure of Merit as function of gain and splitting ratio are

created for different possible values of EDF coil ratio. This information is then averaged over the

amplifier’s gain range, resulting in one final contour plot illustrating the Figure of Merit as function of EDF

coil percentage and splitting ratio, where again, the maximum value of the Figure of Merit corresponds

to the optimum values for both parameters (see Figure 4.4).

4.3 Optimization Tool for EDFA

The aim of the next sections is to provide an insight on the structure of the optimization tool. To this

effect, flowcharts will be used to help understand the organization behind the main blocks that compose

the Tool, as well as to describe some of the algorithm within those blocks.

Figure 4.4 - Figure of Merit evaluated in terms of EDF coil Ratio and Power Splitter Ratio.

58

4.3.1 Simulator structure

The flowchart in Figure 4.5 illustrates a general representation of the Optimization Tool, specifically the

key blocks upon where the main processes take place. The logic behind the Tool itself is rather simple.

First, a set of initial conditions, characterizing the amplifier the user aims to obtain, are inputted into the

program. Based on this data, the Tool defines the Operating Region of the wanted amplifier and selects

key Operating Points to be tested. Different EDFA topologies and Pump configurations are then

simulated and a list compiling the possible solutions to the problem at hand is created. The most cost-

effective solution is chosen, according to some predefined Cost criteria, and this solution is then

optimized for best performance. The final results are assembled in a report and stored in a database. If

by any chance, the initial conditions coincide with an EDFA report already present in the database, then

the Tool simply loads the information and displays it to the user.

Figure 4.5 - Optimization Tool general Diagram.

Optimization Tool Structure

New Challenge?

Initial Conditions

Search Database for Solution

Define EDFA’s Operating Region

Topology Chooser

Optimization

Performance Analysis

Save Data

Display Results

END

NO

YES

59

The next sections provide a more detailed analysis of each block, describing the relationship between

the optimization algorithms employed and the design parameters introduced in section 4.2.

4.3.2 EDFA Operating Region:

The first step towards selecting and optimizing any amplifier is to determine the possible Operating

Points it might be subject to. An EDFA’s Operating Region is defined based on signal power conditions,

specifically:

- Minimum and Maximum total input power that reaches the amplifier;

- Minimum and Maximum total output power an EDFA must deliver;

- Minimum and Maximum Gain provided by the device.

From these set of conditions, a polygon type region is constructed (see Figure 4.6), and the Optimum

Operating Point is selected – typically corresponding to a situation where an amplifier is working at

maximum capacity, with maximum output power and highest gain – from which design parameters

discussed in the previous section, namely GFF’s attenuation profile and EDF length, are dimensioned.

Modern WDM systems used in commercial applications, are usually put into operation with a small

number channels, although links are dimensioned for maximum capacity (40 to 96 channels). As a

consequence, the total input power is quite small at the beginning and can reach quite large values later

on. Therefore, it becomes of vital importance that the amplifier can guarantee network requirements in

all those situations. To accomplish this, the EDFA’s Operating Region can be used to help determine

other points of interest at which the amplifier must be tested. These set of points are often comprised of

the Operating Region’s vertices and correspond to situations where it is often demanding for the

amplifier to fulfil the network requirements.

Figure 4.6 - Amplifier’s Operating Region delimited by power and gain conditions.

60

4.3.3 Topology Chooser:

Topology chooser is the designated name of a portion of the code, whose task is testing and selecting

possible EDFA architectures that provide a solution to the initial requirements.

The algorithm, illustrated by the flowchart in Figure 4.7 is based on a heuristic method of “trial and error”,

where a list containing all available EDFA topologies, is analysed and tested until all viable solutions are

identified. The flowchart below synthesizes this approach for the general case of a double-stage

amplifier, although the process is identical for other types of EDFA. The selection process incorporates

two distinct steps.

Figure 4.7 - Double-Stage EDFA Simulation.

Double-Stage EDFA Simulation

New Pump Requirements

New Operating Region Point

Check Requirements

Tilt Control

END

Pump Algorithm

Double-Stage EDFA

Unregulated Tilt?

NO

YES

All Operating Points Tested?

NO

YES

Save EDFA Report

All Pump Configurations

Tested? NO YES

61

In the first step, the optimum EDF Length is obtained for Optimum Operating Point conditions and a

Pump Algorithm – employing a modified bisection convergence method - is applied to the amplifier in

question. Each iteration, increases or decreases the pump’s power fed to the EDFA, gradually

approximating its gain to the one requested by the Optimum Operating Point conditions. Furthermore,

every pump iteration is followed by several GFF’s attenuation profile adjustments, guarantying a power

spectrum compliant with Gain Flatness and Gain Tilt requirements at the end of the convergence

process. If the pump configuration used involves multiple pumps, then a power ratio between pumps is

added to the convergence method rules, favouring an increase of pump power at the amplifier’s first

stage, thus minimizing Noise Figure. Other possible modification that directly affects the convergence

method is related to the inclusion of DCF at the mid-stage access point. In this case, the total amount

of signal’s power admissible at the mid-stage access is limited, under penalty that the DCF will not

perform as specified. In these situations, the Pump Algorithm must redirect the remaining power to the

second half of the amplifier once the limit at the mid-stage access is reached.

Once EDF length, GFF’s attenuation profile and maximum pump power have been dimensioned, the

second step consists in testing the remaining key Operating Region (see section 4.3.2) to make sure

that all the requirements are respected across the EDFA’s Operating Region. Different Operating points

often produce different gain profiles that do not match the GFF dimensioned for optimum conditions,

leading to inevitable gain Tilt deviations outside of the established boundaries. In these situations, a Tilt

Control feedback loop is activated and the VOA’s attenuation, detailed in section 4.2.3, is calculated to

compensate these effects.

During the routines discussed above, not all amplifier/pump configurations will meet the designated

requirements, and therefore end up being discarded. The main factors for discarding an amplifier/pump

configurations include:

- Pump power exceeding the predefined maximum;

- Pump power lower than minimum power at which the laser pump is stable;

- VOA value outside of the pre-established interval;

- GFF’s attenuation higher than what is commercially available.

After all Operating points have been tested, the EDFA’s characteristics - Gain, Noise Figure, Gain

Flatness, Tilt and Ripple – are evaluated against the initial requirements and the amplifier is categorized

as viable or not viable. All data regarding the solution is then stored in a .mat file and the process is

repeated until all pump configurations have been tested.

By the time the Topology Chooser has gone through all possible EDFA’s and Pump Configurations, a

list compiling all viable solution will have been created. The priority then becomes choosing from the

viable EDFA listed, the one that is best suited for the problem at hand. This is done by applying a Cost

Figure to each viable EDFA and choosing the most cost-effective solution. This approach is based on

the fact that point-to-point links often employ several optical amplifiers, and in order to lower the overall

cost of a link, the network designers tend to choose a solution that reduces the cost without sacrificing

functionality.

62

𝐶𝑜𝑠𝑡 𝐹𝑖𝑔𝑢𝑟𝑒 = 𝐶𝑜𝑠𝑡𝑇𝑜𝑝𝑜𝑙𝑜𝑔𝑦 + ∑ 𝐶𝑜𝑠𝑡𝑃𝑢𝑚𝑝 (4.14)

Equation (4.14) is applied to every viable EDFA, and a final solution is found that minimizes the cost of

the amplifier.

4.3.4 Optimization

In case the Topology Chooser’s algorithm yields a solution involving a single-stage EDFA, the

optimization process is considerate complete as all relevant design parameters, namely EDF length,

GFF’s attenuation and pump powers, have been specified. On the contrary, if the solution is based on

a multi-stage architecture, depending on the complexity of the amplifier itself and on the pump

configuration scheme deployed, an algorithm is used applied to determine the optimum percentage of

EDF in each coil and, if necessary, the splitting ratio in the power splitter that ensures the best

performance.

The amplifier in question is analysed by a series of simulations across its entire gain range, at a power

level predefined in the initial conditions. Throughout this process, a sweep comprised of various values

of EDF coil and splitting ratios is executed and the results of each simulation are stored in .mat files.

Once all possible combinations have been tested, the data of every .mat file is gathered and evaluated

using the Figure of Merit metric introduced in section 4.2.4, producing plots like the ones in Figure 4.3,

and Figure 4.4. Finally, the GFF’s attenuation and the values for each pump are adjusted according to

the newly optimized parameters, using a process similar to the one illustrated by the flowchart in Figure

4.7.

4.3.5 Performance Analysis

With a completely characterized EDFA, this section’s objective is to provide the user with an idea of the

resulting amplifier’s performance, through some additional graphical illustrations relating Gain, Noise

Figure, Optical Signal to Noise Ratio and Maximum Output Power, specifically:

- Maximum Output Power vs Gain;

- Noise Figure vs Gain;

- Figure of Merit vs Gain;

- OSNR vs Gain;

- Pump Power first Pump vs Gain;

- Pump Power second Pump vs Gain;

- VOA vs Gain.

63

Figure 4.8 – Key performance data vs amplifier gain for different lengths of first EDF coil percentages.

65

5 Results

The objective of this section is to analyse and evaluate the performance of the proposed Optimization

Tool for EDFA Design. Three case studies will be executed, each one with distinct network

requirements, to illustrate three possible design challenges the Optimization Tool might be subjected to.

5.1 Case study: Loss compensation at the Reconfigurable Optical Add

and Drop Multiplexer (ROADM)

In modern WDM systems, ROADM are tasked with remotely switching traffic at the wavelength level.

This functionality is nowadays achieved with Wavelength Selective Switches (WSS) modules,

introducing additional attenuation to the channel wavelength being dropped or added. To compensate

this effect, EDFA can be used for ensure a balanced optic power at the output of the network element.

The initial conditions for this scenario are displayed below.

5.1.1 Initial Conditions

From this information, the Tool can construct the amplifier’s Operating Region and the Operating points

can be derived (see Figure 5.1). Additionally, each point is associated with a Noise Figure condition that

every possible solution must respect:

Power and Gain Requirements

𝑃𝑖𝑛𝑚𝑖𝑛𝑡𝑜𝑡𝑎𝑙 = −20 𝑑𝐵𝑚

𝑃𝑖𝑛𝑚𝑎𝑥𝑡𝑜𝑡𝑎𝑙 = −4 𝑑𝐵𝑚

𝑃𝑜𝑢𝑡𝑚𝑖𝑛𝑡𝑜𝑡𝑎𝑙 = 3 𝑑𝐵𝑚

𝑃𝑜𝑢𝑡𝑚𝑎𝑥𝑡𝑜𝑡𝑎𝑙 = 17 𝑑𝐵𝑚

𝐺𝑚𝑎𝑥 = 23 𝑑𝐵


𝑁𝑐ℎ,𝑚𝑎𝑥 = 24

Figure 5.1 – Case Study 1: EDFA Operating Region. Table 5.1 – Case Study 1: Gain and Power Requirements.

66

Operating Region Point [𝒅𝑩𝒎] Number of channels Noise Figure Limit [𝒅𝑩]

1 (−6 | 17) 24 5

2 (−4| 17) 24 5

3 (−4 |1 3) 24 5

4 (−14 | 3) 24 6

5 (−20 | 3) 24 6,5

The maximum number of channels each Operating point can support is determined by limits of minimum

input and maximum output channel power. While the first is imposed by receiver’s sensitivity, the latter

is associated with the maximum power per channel that can be injected in the transmission fibre without

triggering nonlinear effects. Comparing these two limits with every key Operating Region Point, the

maximum number of channels allowed can be found.

5.1.2 Topology Chooser

Once the Operating Region Points have been determined, the algorithm starts testing a list of EDFA

and multiple Pump Configurations (see Appendix B) and compiles a list of viable EDFA that meet the

initial requirements. In this particular case, the candidate solutions are:

Single-Stage EDFA with:

1) GFF located before the EDF, 1 independent 980 𝑛𝑚 co-propagating pump;

2) No GFF, 1 independent 980 𝑛𝑚 co-propagating pump;

3) GFF located after the EDF, 1 independent 980 𝑛𝑚 co-propagating pump;

Double-Stage EDFA with:

4) VOA, GFF located after the EDF, 2 shared 980 𝑛𝑚 co-propagating pumps;

5) VOA, GFF located after the EDF, 2 independent 980 𝑛𝑚 co-propagating pumps + 1480 𝑛𝑚

counter-propagating pump;

Additional Specifications

𝑀𝑎𝑥 𝑃𝑢𝑚𝑝 𝑃𝑜𝑤𝑒𝑟 = 700 𝑚𝑊

𝑀𝑖𝑛 𝑃𝑢𝑚𝑝 𝑃𝑜𝑤𝑒𝑟 = 10 𝑚𝑊

𝑃𝑖𝑛,𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑚𝑖𝑛 = −35 𝑑𝐵𝑚

𝑃𝑜𝑢𝑡,𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑚𝑎𝑥 = −6,3 𝑑𝐵𝑚

Table 5.2 – Case Study 1: Noise Figure requirements.

Table 5.3 – Case Study 1: Additional specifications.

67

6) VOA, GFF located after the EDF, 2 shared 980 𝑛𝑚 co-propagating pumps + 1480 𝑛𝑚


7) VOA, GFF located before the EDF, 1 shared 980 𝑛𝑚 co-propagating pump;

8) VOA, GFF located after the EDF, 1 shared 980 𝑛𝑚 co-propagating pump;

9) VOA, GFF located after the EDF, 1 shared 980 𝑛𝑚 co-propagating pump + 1480 𝑛𝑚


10) VOA, GFF located before the EDF, 1 shared 980 𝑛𝑚 co-propagating pump + 1480 𝑛𝑚


11) VOA, GFF located after the EDF, 2 independent 980 𝑛𝑚 co-propagating pumps;

12) VOA, GFF located after the EDF, 2 shared 980 𝑛𝑚 co-propagating pumps;

The compiled list includes a large set of Double-Stage EDFA and a few more basic Single-Stage

topologies, which is to be expected, since the initial conditions presented no Gain Tilt, Gain Ripple or

Gain Flatness requirements.

A final solution is selected based on the Cost Figure criteria introduced in section 4.3.3. In this case,

Figure 5.2 shows option 2 to be the most cost-effective solution. For this specific topology, the design

parameters are solely the Length of the EDF and the Pump Power, which have already been determined

by the testing portion of the algorithm. Therefore, no further optimization is required and the results are

displayed in Table 5.4.

The selected solution guarantees all gain requirements, ensuring a minimum Noise Figure of 4,75 𝑑𝐵

for a 23 𝑑𝐵 gain in Operating point 1 conditions and worst Noise Figure of 5,46 𝑑𝐵 in the Operating point

5 case. The simple design of this particular topology allows it to achieve gain and Noise Figure

requirements with small pump values, resulting in the employment of cheaper laser pumps.

Figure 5.2 - Cost Figure applied to potential solutions.

68

EDF Length: 19,2237 m

Operating Points 1 2 3 4 5

Gain [𝒅𝑩] 22,9961 21,0070 17,0005 17,0027 23,0090

Noise Figure [𝒅𝑩] 4,7548 4,7764 4,9922 5,2713 5,4618

Pump Power [𝒎𝑾] 158,2422 162,1719 71,6699 16,8115 22,3779

5.2 Case study: Unregulated Tilt Cancelling

In a point-to-point link, whenever a span of transmission fibre shows signs of unregulated tilt caused by

the effects mentioned in section 3.2.4, one or more EDFA can be used to act on the propagating signals’

tilt. The current case study illustrates the scenario, where an EDFA is required with a specific Tilt

requirement and the ability to act on the Tilt.

5.2.1 Initial Requirements:

Gain and Power Requirements


𝑃𝑖𝑛𝑚𝑎𝑥𝑡𝑜𝑡𝑎𝑙 = 5 𝑑𝐵𝑚






.


1 (−6 | 24) 96 5

2 (4| 24) 96 8

3 (−17 |3) 48 11

4 (−18 | 3) 48 10

5 (−18 | 12) 48 6

Table 5.4 – Single-stage optimized solution.

Figure 5.3 – Case Study 2. EDFA Operating Region. Table 5.5 – Case Study 2: Gain and Power requirements.


69

A Tilt requirement introduces some additional specifications that have to be taken into account, during

the optimization process. The necessity of Gain Tilt Control means a VOA and a GFF will be part of the

EDFA’s topology. Limits for the GFF and VOA are established based on typical values commercially

available for these devices, further restricting the number of possible solutions.


Most of the optimization process is identical to the one described in the previous case study. From the

requirements of Table 5.5, Table 5.6 and Table 5.7, the topology chooser delivers two possible solutions:

1) Double-Stage with VOA, GFF located before the second EDF and 1 shared 980 𝑛𝑚 co-

propagating pump + 1480 𝑛𝑚 counter-propagating pump;

2) Four-Stage topology with 2x (1 shared) 980 𝑛𝑚 co-propagating pump + 1480 𝑛𝑚 counter-

propagating pump;






𝑉𝑂𝐴 𝑚𝑎𝑥 𝑣𝑎𝑙𝑢𝑒 = 15 𝑑𝐵

𝐺𝐹𝐹𝑎𝑡𝑡𝑒𝑛𝑢𝑎𝑡𝑖𝑜𝑛𝑚𝑎𝑥 = 10 𝑑𝐵

𝐺𝑎𝑖𝑛 𝑇𝑖𝑙𝑡𝐶𝑜𝑒𝑓𝑓 ≤ 0,4 𝑑𝐵 𝑇𝐻𝑧−1

𝑒𝑟𝑟𝑜𝑟 = ±0,02 𝑑𝐵 𝑇𝐻𝑧−1

𝐺𝑎𝑖𝑛𝑅𝑖𝑝𝑝𝑙𝑒 ≤ 0,5 𝑑𝐵


Figure 5.4 – Case Study 2: Cost Figure applied to potential solutions.

70

Once again, a Cost Figure is applied to both potential solutions and the relative cost of each EDFA is

calculated. According to Figure 5.4, the cost is minimum for option 1.

Option 1 achieves maximum gain of 30 𝑑𝐵 with 4,7 𝑑𝐵 of Noise Figure in Operating point 1 conditions,

and has a worst Noise Figure of 9,7 𝑑𝐵 in the Operating Point 3 case. Both Gain Tilt and Gain Ripple

requirements are met across the EDFA’s Operating Region with VOA’s attenuation values below the

established limit.

5.2.3 Final Solution Optimization

The Double-Stage selected uses a shared pump between stages indicating the presence of a Power

Splitter in the amplifier architecture, which means that in addition to the optimum EDF Coil Ratio the

Power Splitter’s Splitting Ratio must also be defined. The amplifier is than tested for a series of EDF

Coil Ratio/Splitting Ratio combinations along its gain range (usually at the maximum Output Power).

The result, displayed in Figure 5.5, shows that the maximum Figure of Merit (discussed in section 4.2.4)

is obtained for about 55% EDF Coil in the first stage and a Splitting Ratio in the [85 ; 90]% interval.

Since the optimized Splitting Ratio is very high, it is clear that the gain and Noise Figure requirements

could have been achieved with a Single-Stage EDFA. However the Gain Tilt Control requirement across

the EDFA’s gain range can never be achieved with a Single-Stage topology.

EDF Length: 25.5238 m

EDF Coil Ratio: 50%/50%

Power Splitter Ratio: 60% /40%


Gain [𝒅𝑩] 29,9973 19,9906 19,9975 20,9916 30,0050

Noise Figure [𝒅𝑩] 4,7083 7,9437 9,7449 9,0416 5,2235

Gain Tilt coefficient [𝒅𝑩 𝑻𝑯𝒛−𝟏] 0,4087 0,4224 0,4202 0,4187 0,4213

Gain Ripple [𝒅𝑩] 0,3892 0,3861 0,4641 0,4644 0,4569

Pump 1 Power [𝒎𝑾] 700 700 15,3833 14,9805 51,6016

Pump 2 Power [𝒎𝑾] 254,1406 380,3125 10 10 20,8398

VOA [𝒅𝑩] 2 12,0595 12,7694 11,7702 2,7856

Table 5.8 - Double-Stage specifications after Topology Chooser.

71

Finally, having established EDF Coil Ratio and Pump Splitting Ratio, the EDFA is retested in order to

adjust Pump Power, VOA and GFF values. Final results are displayed in Table 5.9.

EDF Length: 25.5238 m


Power Splitter Ratio: 85% /15%


Gain [𝒅𝑩] 30,0055 20,0080 19,9952 21,0005 30,0071

Noise Figure [𝒅𝑩] 4,6975 7,8961 7,8208 7,1599 5,2179

Gain Tilt coefficient [𝒅𝑩 𝑻𝑯𝒛−𝟏 ] 0,4052 0,4044 0,4104 0,4132 0,4021

Gain Ripple [𝒅𝑩] 0,4774 0,4867 0,4855 0,4952 0,4805

Pump 1 Power [𝒎𝑾] 700 700 18,7891 18,0566 70,9375

Pump 2 Power [𝒎𝑾] 368,5938 629,6875 10 10 27,1875

VOA [𝒅𝑩] 2 11,9893 12,6949 11,7054 2,5723

With this choice of EDF coil Ratio and Power Splitter splitting ratio, the presented solution is able to

achieve all gain, tilt and ripple requirements, while reducing the Noise Figure values, particularly in when

the amplifier is working under Operating Point 3 and 4 conditions.

Figure 5.5 - Figure of Merit evaluated for different EDF Coil Ratio and Splitting Ratio.

Table 5.9 - Double-Stage optimized solution.

72

5.3 Case study: Dispersion Compensation Module

This case study illustrates the design of an EDFA in a situation where a dispersion compensation module

(DCM) is required in the amplifier architecture. The challenge lies in achieving the network requirements

while guaranteeing that a power at the input of the DCM does not surpass the maximum predefined

limit.

5.3.1 Initial Requirements

Gain and Power

Requirements


𝑃𝑖𝑛𝑚𝑎𝑥𝑡𝑜𝑡𝑎𝑙 = 8 𝑑𝐵𝑚






Figure 5.6 – Case Study 2: Optimized Gain Flattening Filter Attenuation Profile.

Figure 5.7 - Case Study 3. EDFA Operating Region. Table 5.10 – Case Study 3: Gain and Power Requirements.

73

The total power at the input of the DCM cannot surpass 16 𝑑𝐵𝑚 for a scenario of 80 channels, due to

fibre nonlinearities that may be triggered above this threshold power.


Once again all EDFA configurations are tested, and the viable solutions are compiled. For this particular

network requirements, the potential solutions are:

1) Double-Stage with VOA, GFF located before the second EDF coil and 1 shared 980 𝑛𝑚 co-

propagating pump + 1480 𝑛𝑚 counter-propagating pump;

2) Double-Stage with VOA, GFF located before the second EDF coil and 2 independent co-

propagating pumps + 1480 𝑛𝑚 counter-propagating pump;

A Cost figure is applied to both topologies and option 2 is found to be the most cost-effective EDFA for

the specified network requirements.


1 (−9| 23) 80 5,2

2 (2 | 23) 80 10

3 (−16 | 5) 72 11

4 (−25 | 5) 6 6.5

5 (−25 | 7) 6 6.5






𝐷𝐶𝐹 𝑚𝑎𝑥 𝑖𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 = 16 𝑑𝐵𝑚

𝑉𝑂𝐴 𝑚𝑎𝑥 𝑣𝑎𝑙𝑢𝑒 = 15 𝑑𝐵

𝐺𝐹𝐹𝑎𝑡𝑡𝑒𝑛𝑢𝑎𝑡𝑖𝑜𝑛𝑚𝑎𝑥 = 10 𝑑𝐵

𝐺𝑎𝑖𝑛𝐹𝑙𝑎𝑡𝑛𝑒𝑠𝑠 ≤ 0,75 𝑑𝐵

𝐺𝑎𝑖𝑛 𝑇𝑖𝑙𝑡𝐶𝑜𝑒𝑓𝑓 ≤ 0,0 𝑑𝐵 𝑇𝐻𝑧−1

𝑒𝑟𝑟𝑜𝑟 = ±0,02 𝑑𝐵 𝑇𝐻𝑧−1

𝐺𝑎𝑖𝑛𝑅𝑖𝑝𝑝𝑙𝑒 ≤ 0,5 𝑑𝐵



74

Option 2 achieves a maximum gain of 32 𝑑𝐵 with 4,7 𝑑𝐵 of Noise Figure for Operating Point 1 conditions,

exhibiting a worst Noise Figure of 10,5 𝑑𝐵 for the Operating Point 3 case. In all Operating points tested

the values of Gain Tilt, Gain Ripple and Gain Flatness are within the pre-established limits, ensuring a

“flat gain profile” at the output of the device.

5.3.3 Final Solution Optimization

For this particular case, the final solution presents a Pump configurations composed entirely by

independent pumps. Consequently, the only design parameter left to optimize is the percentage of EDF

distributed between the first and second stages. As discussed in section 4.2.4, the optimization algorithm

performs a sweep along the possible values of EDF Coil Ratio and the Figure of Merit is calculated for

each point. Figure 5.8 displays the result of this procedure.




Gain [𝒅𝑩] 32,003 20,9991 21,0036 30,0029 32,0098

Noise Figure [𝒅𝑩] 4,7653 8,9300 10,5508 5,9181 5,6272

Gain Flatness [𝒅𝑩] 0,2974 0,5093 0,5651 0,3070 0,5343

Gain Tilt coefficient [𝒅𝑩/𝑻𝑯𝒛] 0,0784 0,1468 0,1583 -0,0188 0,1604

Gain Ripple [𝒅𝑩] 0,2947 0,3052 0,3856 0,2624 0,2848

Pump 1 Power [𝒎𝑾] 300 300 11,7212 12,7832 85

Pump 2 Power [𝒎𝑾] 300 300 10 10 10,1563

Pump 3 Power [𝒎𝑾] 155,3125 211,5625 10 10 10

VOA [𝒅𝑩] 2 12,6395 13,2328 5,0283 2

Table 5.13 – Double-Stage specifications after Topology Chooser.

Figure 5.8 - Figure of Merit for different values relative Length of EDF in the first Stage.

75

It is clear, that for this particular topology, variations of the EDF Coil Ratio do not pose a major issue

since the Figure of Merit is practically stable throughout the [35; 65] % interval. Consequently, any final

optimize solution will not show significant improvements from the depicted in Table 5.14.




Gain [𝒅𝑩] 32,0061 21,0095 20,9983 30,0057 31,9951

Noise Figure [𝒅𝑩] 4,7688 8,9787 10,2313 5,9004 5,6683

Gain Flatness [𝒅𝑩] 0,2515 0,4618 0,4345 0,3351 0,5020

Gain Tilt coefficient [𝒅𝑩/𝑻𝑯𝒛] 0,0641 0,1320 0,1166 -0,0344 0,1501

Gain Ripple [𝒅𝑩] 0,2840 0,2946 0,3706 0,2535 0,2768

Pump 1 Power [𝒎𝑾] 300 300 12,7100 13,8086 85

Pump 2 Power [𝒎𝑾] 300 300 10 10 10,1563

Pump 3 Power [𝒎𝑾] 158,4375 231,8750 10 10 10

VOA [𝒅𝑩] 2 12,6375 13,3886 5,0377 2

Table 5.14 – Double-Stage Optimized solution.

Figure 5.9 – Optimized Gain Flattening Filter Attenuation Profile.

77

6 Conclusions and future work

In this work, an Optimization Tool for EDFA design was proposed, studied and developed. With the

evolution of fibre optic communication systems, EDFA designs are becoming more complex. Along with

this complexity comes the need for optimization methods that allow the user to verify all the features of

an amplifier and design it for best performance. The focus on EDFA optimization, as a complex system

composed of several components, is a unique harmonization of existing scattered practical techniques

that EDFA designers use nowadays. This thesis work sets the bases for precise optimization on a

component level as well as on an EDFA-system level, which are of crucial importance for the design of

modern WDM systems.

The development of such a tool meant a complete characterization of EDFA components, particularly

an EDF model. It was demonstrated that once an accurate model of the EDF was established, that a

simulator for EDFA could be constructed, serving as a reliable support for the optimization methods

employed.

Another objective of this dissertation was to construct a heuristic algorithm that, in association with the

EDFA simulator mentioned above, would test and evaluate several amplifier topologies and

configurations in order to select and optimize the most cost-effective solution. The results of the case

studies show that this algorithm (see Figure 4.5) has proven to be accurate enough to support design

of current optical amplifiers (gain, noise, tilt and ripple requirements). Additionally, an accurate

optimization tool means only special cases require experimental testing, saving both time and money.

Although there may be other optimization tools available commercially, to the best of our understanding

they either do EDFA optimization looking at the overall behaviour of the network or they specialized in

optimizing a set of design parameters in a particular EDFA topology. Excluding the possibility that there

may exist private and proprietary tools, property of the various telecom equipment suppliers and/or

telecom services suppliers, none other optimization tool publicly available performs the combine tasks

discussed throughout this dissertation.

Consequent to the developed work, several suggestions for future work are presented:

- Implementation of a method for simultaneously controlling and EDFA and an amplifier

arrangement;

- Development of a new pump algorithm based on iterative methods with a faster convergence

rate than the implemented bisection method;

- Increase the accuracy of the results, and adjust them to experimental measures;

- Integrate this optimization Tool with another that optimizes EDFA according to different

parameters (optimization focused on Tilt Control and OSNR).

79

7 Bibliography

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[2] Hecht, Jeff. City of light: the story of fiber optics. Oxford University Press on Demand,

2004.

[3] Agrawal, G.P., Fiber-optic Communication Systems. 4th ed. 2010: Wiley.

[4] Lee, S. C. J. Dynamical Performance of Optical Amplifiers for Long-Haul/Ultra LongHaul

Transmission Systems. Eindhoven: Eindhoven University of Technology, 2005.

[5] "Introduction to DWDM Technology." Http://www.cisco.com/. Cisco, 4 June 2001. Web. 13

Apr. 2016.

[6] Neto, B. "Técnicas alternativas para amplificação de Raman em

telecomunicações." Doutoramento Departamento de Física, Universidade de Aveiro,

Aveiro (2010).

[7] Gringeri, Steven, et al. "Flexible architectures for optical transport nodes and

networks." Communications Magazine, IEEE 48.7 (2010): 40-50.

[8] Islam, Mohammed N. "Raman amplifiers for telecommunications." Selected Topics in

Quantum Electronics, IEEE Journal of 8.3 (2002): 548-559.

[9] Deben Deben Lamon. "Raman Amplification." Web.fe.up.pt. 2008. Web. 13 Apr. 2016.

<https://web.fe.up.pt/~ext07025/documents/Deben_Lamon__Raman_amplification.pdf>.

[10] Kozak, M. M., R. Caspary, and U. B. Unrau. "Computer aided edfa design, simulation and

optimization." Proceedings of the 3rd International Conference on Transparent Optical

Networks. 2001.

[11] Becker, Philippe M., Anders A. Olsson, and Jay R. Simpson. Erbium-doped fiber

amplifiers: fundamentals and technology. Academic press, 1999.

[12] Foster, S., and A. Tikhomirov. "Absorption and emission cross sections for erbuim doped

silica glass." OECC/ACOFT 2008-Joint Conference of the Opto-Electronics and

Communications Conference and the Australian Conference on Optical Fibre Technology.

2008.

[13] Miniscalco, William J. "Erbium-doped glasses for fiber amplifiers at 1500 nm."Lightwave

Technology, Journal of 9.2 (1991): 234-250.

[14] Barnes, William L., et al. "Absorption and emission cross section of Er 3+ doped silica

fibers." Quantum Electronics, IEEE Journal of 27.4 (1991): 1004-1010.

[15] Giles, C. Randy, and Emmanuel Desurvire. "Modeling erbium-doped fiber

amplifiers." Lightwave Technology, Journal of 9.2 (1991): 271-283.

[16] Bohr, Niels. "I. On the constitution of atoms and molecules." The London, Edinburgh, and

Dublin Philosophical Magazine and Journal of Science 26.151 (1913): 1-25.

[17] "Quantum Processes." Hyperphysics.phy. R Nave. Web. 14 Apr. 2016.

<http://hyperphysics.phy-astr.gsu.edu/hbase/mod5.html#c1>.

80

[18] Saleh, A. A. M., et al. "Modeling of gain in erbium-doped fiber amplifiers."Photonics

Technology Letters, IEEE 2.10 (1990): 714-717.

[19] Luís, Miguel Antunes Da Silva Alves. "Geração E Amplificação Em Sistemas De Fibra

Óptica." Instituto Superior Técnico, Lisboa (2012).

[20] "Cisco ONS 15454 DWDM Engineering and Planning Guide." Http://www.cisco.com/. 3

Sept. 2007. Web. 13 Apr. 2016.

[21] Rapp, Lutz. "Reconfigurable gain-flattened erbium-doped fiber amplifiers with variable

gain at improved noise characteristics." Journal of optical communications 23.4 (2002):

127-131.

[22] Rapp, Lutz. "Comparison of EDFA stages using pump power splitting or pump bypass

technique with respect to steady-state performance." Journal of optical

communications 27.4 (2006): 194-200.

[23] Rapp, Lutz, and Dario Setti. "Comparison of Tilt Control Techniques for Erbium–doped

Fiber Amplifiers." Journal of Optical Communications 28.3 (2007): 162-167.

[24] Zimmerman, Donald R., and Leo H. Spiekman. "Amplifiers for the masses: EDFA, EDWA,

and SOA amplets for metro and access applications." Journal of lightwave

technology 22.1 (2004): 63.

[25] The discovery of the elements. XVI. The rare earth elements. Journal of Chemical

Education, 1932, 9.10: 1751.

[26] Alegria, Carlos Feio Gama. All-fibre devices for WDM optical communications. Diss.

University of Southampton, Faculty of Engineering and Science, 2001.

[27] "Instruction to EDFA (Erbium-Doped Fiber Amplifiers) Technology."Http://www.fs.com/. 9

Sept. 2014. Web. 13 Apr. 2016

[28] "Optical Amplification Source: Master 7_5. Optical Amplifiers Optical Amplifier An Optical

Amplifier Is a Device Which Amplifies the Optical Signal Directly."Http://slideplayer.com/.

2008. Web. 13 Apr. 2016. <http://slideplayer.com/slide/4834038/>

[29] Alwayn, Vivek. "Fiber-optic technologies." Optical Network Design and Implementation

(Cisco Press, Indianapolis, IN, 2004) (2004).

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EDFA." Journal of Applied Sciences 12.19 (2012): 2071.

81

8 Appendix A

8.1 Multi-Stage EDFA

Multistage EDFA architecture consists of successive lengths of EDF separated by passive components

with the objective of allowing signal’s gain to grow at the expenses of a reduction in ASE power. This

way, multistage amplifiers (see Figure 8.1) overcome one of the main drawbacks pointed out in single-

stage EDFA, achieving higher gains while ensuring a low Noise Figure.

Multi-stage design also offers functionalities that would otherwise be impossible to implement in a single

stage architecture. One of them is the possibility to incorporate Dispersion Compensating Fibre (DCF)

at the amplifier’s mid stage access, whenever the signal experience strong dispersive broadening over

the transmission fibre, causing the modulated signal to overlap with its neighbour signals (Inter-Symbol

Interference). Other possible modification is the inclusion of optical switches, devices that allow all

signals in an optical fibre to be selectively redirected between two optical circuits. It is usually employed

to include or exclude an extra amplification stage, turning a N into a N+1 stage amplifier with all the

benefits associated with it.

In general, Multi-stage amplifiers have revealed themselves very useful in system applications, showing

tilt control and gain flattening for extended gain rages, as well as higher output powers in exchange for

increasing complexity and cost.

8.1.1 Noise Figure

Noise Figure determination in Multi-stage EDFA is similar to the general approach taken in cascades

amplifiers [11].

𝐹𝑠𝑦𝑠 = 𝑆𝑁𝑅0

𝑆𝑁𝑅1

.𝑆𝑁𝑅1

𝑆𝑁𝑅2

. . . 𝑆𝑁𝑅𝑁−1

𝑆𝑁𝑅𝑁

(8.1)

Figure 8.1 - Example of Multistage Amplifier.

82

Where 𝑆𝑁𝑅 represents the Signal to Noise Ratio of each amplifier multiplied by 1

𝐿𝑖 , the spacing between

them. In the case of Multi-stage amplifiers 𝐿𝑖 = 1, so equation (8.1) degenerates into the following

expression:

𝐹𝑠𝑦𝑠 = 𝐹1 +𝐹2

𝐺1

+ ⋯ +𝐹𝑁

𝐺1𝐺2 … 𝐺𝑁−1

(8.2)

Leading us to two conclusions:

- Noise Figure is dominated by the portion of NF in the first stage. Consequently, to minimize the

amplifiers Noise Figure the pump feeding the first stage must guarantee low values of ASE

power through a well inverted fibre.

- The second conclusion is that, however low the Noise Figure in multi-stage amplifiers may be,

it will always be worse than that of a single-stage, which achieves the optimum Noise Figure

due to its simple design.

8.1.2 Pumping Techniques

On pair with the various forms of multi-stage amplifiers reported in the literature, there are multiple

pumping configurations possible of being employed. Assuming, for simplification purposes, a double

stage EDFA described in Figure 8.1, in the most convection pump setup, the power for both coils of EDF

are provided by independent pumps. A golden rule when designing EDFA with several stages is that it

is advantageous to have the first stage pumped with a 980 𝑛𝑚 co-propagating pump to minimize Noise,

as stated in expression (8.2). The second stage can also be pumped in similar fashion or with a

bidirectional configuration adding a 1480 𝑛𝑚 counter-propagating pump for higher output powers.

Other pumping technique were developed as pump technology matured to the point where pumps could

provide several hundreds of milliwatts (≥ 700 𝑚𝑊). It became interesting and cost effective to explore

solutions where one pump would be used to feed more than one amplifying stage (see Figure 8.2):

- One of the methods studied was to reutilize the leftover pump power not absorbed throughout

the first stage and have it feed the coil of EDF in the second stage. This would be done by

implementing a bypass between both stages so that the pump’s power would not be attenuated

by the isolator located at the end of the first coil [22].

- The other method used a power splitter immediately after the pump light source, guiding a

predefined percentage of the total (𝛼) to the first coil and feeding the remaining power (1 − 𝛼)

to the second coil [22]. New pumping configurations also present new challenges as EDFA

optimization is concerned. Using a bypass method, optimal performance is achieved by

adjusting the relative length of the first EDF coil, whereas using a power splitter increases the

complexity of the optimization process with the addition of a new variable 𝛼.

83

Figure 8.2 - Double Stage Amplifier design using either a Pump Power Splitter or a Pump Bypass [22].

85

9 Appendix B

9.1 EDFA Topologies

In this sections, the EDFA topologies implemented in the Optimization Tool for EDFA Design are

displayed and the main advantages and disadvantages of each topology is presented.

Single-stage EDFA Topology 1 a)

Best Noise Figure No Flat Gain

Suitable for single channel amplifier No Tilt Control

Low Gain - Has worse Noise figure

Power limited by single laser

Single-stage EDFA Topology 1 b)

Best Noise Figure No Flat Gain control

Flat Gain – for a specific gain No Tilt Control

Low Gain - Has worse Noise figure

Table 9.1 - Single-Stage configuration 1a)

Figure 9.1 - Single-Stage configuration 1a)

Table 9.2 - Single-Stage Configuration 1b)

Figure 9.2 - Single-Stage Configuration 1b)

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Single-stage EDFA Topology 1 c)

Best Noise Figure Limited Tilt Control range

Flat Gain – for a range of gain Limited Flat Gain

Limited Tilt Control

Double-stage EDFA Topology 2 a)

Improved Noise Figure Limited Tilt Control range

Flat Gain – for a limited range of gain Limited Flat Gain

Improved Tilt Control

Double-stage EDFA Topology 2 b)

Improved Noise Figure Laser power loss – due to extra IL

Improved Tilt Control Limited output power

Improved Flat Gain – comparing with 2a

Single Laser design

Table 9.3 - Single-Stage Configuration 1c)

Figure 9.3 - Single-Stage Configuration 1c)

Table 9.4 - Double-Stage Configuration 2a)

Figure 9.4 - Double-Stage Configuration 2a)

Table 9.5 - Double-Stage Configuration 2b)

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Double-stage EDFA Topology 2 c)

Improved Noise Figure Laser power loss – due to extra IL

Improved Tilt Control

Improved Flat Gain – comparing with 2a

Laser redundancy

Laser balancing

Pump Power Control with priorities

Figure 9.5 - Double-Stage Configuration 2b)

Table 9.6 - Double-Stage Topology 2 c)

Figure 9.6 - Double-Stage Configuration 2 c)

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Four-stage EDFA Topology 3 a)

Improved Noise Figure comparing with 2 – extended

Many components

Improved Tilt Control comparing with 2 – extended range

Many Lasers

Improved Flat Gain – comparing with 2 – extended range

Pump Power with priorities (Achieve high inversion at beginning of erbium doped fibre by maintaining the average inversion level)

Possibility to accept a DCF – dispersion compensation fibre

Extended Output Power

Table 9.7 - Four-Stage Configuration 3 a)

Figure 9.7 - Four-Stage Configuration 3 a)

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Five-stage EDFA configuration 3 b)

Extended Gain Range Many components

Best Noise Figure through Gain Range Many Lasers

Improved Tilt Control comparing with 2 – extended range

Complex Control

Improved Flat Gain – comparing with 2 – extended range

Expensive

Pump Power with priorities (Achieve high inversion at beginning of erbium-doped fibre by maintaining the average inversion level)

Possibility to accept a DCF – dispersion compensation fibre

Extended Output Power

Table 9.8 - Five-Stage Configuration 3 b)

Figure 9.8 - Five-Stage Configuration 3 b)

Optimization Tool for Erbium Doped Fibre Amplifiers · ix Abstract Erbium Doped Fibre Amplifiers...

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