Modeling, Design and Applications of Optical ...

Modeling, Design and Applications of Optical Amplifiers and Long Period Gratings

Thesis submitted to the University of Delhi for the award of degree of Doctor of Philosphy In Electronic Sciences

By Amita Kapoor Department of Electronic Science University of Delhi South Campus, New Delhi, India September 2010

Supervisors:

Prof. Enakshi Khular Sharma Department of Electronic Sciences University of Delhi South Campus Delhi, India.

Prof. Dr.-Ing Dr. h.c. Wolfgang Freude Institute of Photonics and Quantum Electronics, Karlsruhe Institute of Technology, Karlsruhe, Germany.

“For light I go directly to the source of light, not to any of the reflections” Peace Pilgrim (1908-1981)

Acknowledgement The six and half years of my doctoral program had been a great learning and unlearning experience. The results presented in this thesis have been realized with the support of a large number of people. Most of them are working or used to work at the Department of Electronic Sciences (DOES), University of Delhi South Campus, Delhi, India and Institute of Photonics and Quantum Electronics (IPQ), Karlsruhe Institute of Technology, Karlsruhe, Germany, where I had the pleasure to work for duration of one year under the DAAD “Sandwich Model” fellowship. In these pages I would like to take an opportunity to thank all those whose help and support made this work possible. I took care not to forget anybody, but in case I have forgotten to mention you, please forgive me. The first special thanks are to my supervisors Prof. Enakshi K. Sharma and Prof. Wolfgang Freude for giving me the opportunity to work under their guidance. Their insight, immense knowledge and enormous grasp of subject are unparalleled. Both being perfectionist, constantly pushed me to rise above my limitations. In the times of despair, it was the constant encouragement and motivation of Prof. Sharma that kept me going. She taught me not to give up and as a wonderful teacher always cleared all my doubts with patience. Prof. Freude taught me to question and to doubt, two great qualities for a scientist. The stimulating discussions with him every week helped me in understanding various aspects of semiconductor based optical devices. I am extremely grateful to Dr. S. Lakshmi Devi, Principal, Shaheed Rajguru College of Applied Sciences for Women (SRCASW), for her encouragement and support. As my mentor, she took special interest in my research progress. My sincere thanks are due to Prof. Avinashi Kapoor, Head, DOES for his continuous support, motivation and affection. His doors were always open, whenever I needed his guidance. My sincere thanks are also due to Prof. Juerg Leuthold, Head, IPQ for his support and invaluable discussions. Thanks are due to Prof. A. K. Verma, of Department of Electronic Sciences, UDSC for inspiring discussions and guidance. I am very grateful to Prof. Anurag Sharma, IIT Delhi, for allowing me to use IIT internet facility for accessing various online journals and his patience when I used to ring up Enakshi madam at wee hours. I am grateful to Prof. K. N. Tripathi, Prof. P. K. Bhatnagar, Prof. R. S. Gupta, Prof. R. M. Mehra and Dr. Mridula Gupta of DOES for their support. I would like to thanks all my colleagues at IPQ; they helped me survive in a foreign country and made me feel like at home. Very special thanks are due to Mr. René Bonk and Mr. Andrej Marculesque of IPQ for the stimulating discussions and providing me with experimental data, which helped me a lot in characterizing my simulations. I would especially like to thank Ms. B. Lehmann for helping me with all the administrative work and making my stay in Germany comfortable. I would also like to thank Sybille madam for her affection and for the wonderful dinner on my birthday.

I am thankful to all my colleagues at SRCASW for being my extending family, supporting me in both my personal and educational pursuit. I express my gratitude to the technical staff at DOES, with a special mention of Sh. D. V. Tyagi, for their support and cooperation. I would like to express my sincere thanks to all my colleagues at DOES with a special mention of my seniors Dr. Sangeeta Srivastava, Dr. Rashmi Singh and Dr. Geetika Jain for inspiring discussions, lot of support and their friendship. I would also like to thanks Dr. Jagneet Kaur for her help and support. I should not forget to thank my colleagues at DOES, Dr. Krishna Chandra Patra for letting me know the real India, and making me feel how lucky I am to be born in a metro, Nandan for his support and dear Jyoti Anand for her help and support during the end stage of my research work. I am thankful to DAAD (Deutscher Akademischer Austausch Dienst) for providing me the financial support during my stay in Germany, which made it possible for me to concentrate on my research activities. I am grateful to my maternal Uncles Er. K. Maini and Mr. H.C.D. Maini and my maternal Aunts Ms. Suraksha Maini and Ms. Rita Maini for their emotional support. I am very thankful to all my cousins for understanding why I could not be with them in their special life moments. I am very thankful to the first teacher of my life, my mother, Late Smt. Swarnlata Kapoor; I don’t have large memories of her, but still what all I remember, it was she who instilled in me the value of education, and desire to excel in whatever you do. My gratitude is to God and my guarding angels for helping me in fulfilling the dream of my mother despite many adversaries. My greatest debt of gratitude is to of my friend and mentor Dr. Narotam Singh for his patience, and unconditional support to me in this sometimes seemingly endless task. Thanks for understanding why I needed to spend my weekends in front of my computer instead of spending time with him. And finally I express my thanks to the unnamed forces, each one of them contributed in a unique way to make this work possible.

Amita Kapoor New Delhi, Sep 2010

Synopsis

Synopsis Communication using light rays is not new; as early as 490 BC, in the famous siege of Athens by Persia light rays were used to send messages. The modern optical communication systems today can boast of accessing data from any part of the earth, at a data rate of 30 Mbps. The foundation of optical communication as we know it today can be traced back to the year 1917, when Albert Einstein predicted the presence of stimulated emission in the paper entitled “Zur Quantentheorie der Strahlung”. Charles Townes, in USA, Nikholai Basov and Alexander Prochorov, in USSR, using the concept of stimulated emission and population inversion developed the world’s first MASER in the year 1960. This led way to the construction of oscillators and amplifiers based on the maser-laser principle. Soon, Theodore Maiman demonstrated the first functional laser, a Ruby laser in the year 1960, and Robert Hall developed the first semiconductor injection laser in the year 1962. Parallel to the development in optical sources and amplifiers, work was going on in choosing a right medium for optical transmission. In 1966, Charles K. Kao published his work in which he concluded that for optical fibers to be a viable communication medium the fundamental limit of attenuation would be 20 dB/km. Four years later, Corning Inc. (then known as Corning Glass works) announced that they have fabricated successfully single mode fibers with an attenuation of below 20 dB/km at 633 nm. Following these breakthroughs, world’s first commercial optic communication system was deployed in the year 1975. It had a bit rate of 45 Mbps with repeater spacing of up to 10 km. By 1987 second generation of optical communication system with bit rates of up to 1.7 Gbps and repeater spacing of 50 km was operating. A new generation of single-mode systems was just beyond the horizon operating at 1.55 m with fiber loss around 0.2 dB/km. This third generation of optical communication system was operating at 2.5 Gbps with repeater spacing in excess of 100 km. The fourth generation of optical communication systems employed optical amplifiers to

i

Synopsis

reduce the need for repeaters and wavelength division multiplexing (WDM) to increase data capacity. These technologies brought about a revolution, resulting in doubling of capacity every six months starting from 1992. With the spread of internet and World Wide Web, not to mention technologies like video conferencing and VOIP the demand for higher capacity is continuously growing. Today fiber-to-thehome/office is becoming a reality. In an essence, we can say that because of high speeds, better reliability and noise immunity; optical communication will continue to grow. It will change our society and make our lives more convenient, more enjoyable and more comfortable. Perhaps it is the recognition of the fact, how optical communication has changed the world for good, that in the year 2009 Charles K. Kao was awarded the Nobel Prize in Physics “for groundbreaking achievements concerning the transmission of light in fibers for optical communication”. Emergence of erbium doped fiber amplifier (EDFA) in 1987 was an important development that revolutionized optical telecommunication. EDFA mainly consists of a silica fiber (usually 4 m to 50 m long), in which core is doped with erbium ions. The erbium ions in a silica host when pumped by a 980 nm pump radiation amplify many wavelength channels within the C and L band, with a wavelength range of 1.531.6 m. Thus, one can use a single amplifier for all wavelengths in wavelength division multiplexed (WDM) or dense wavelength division multiplexed (DWDM) systems. However, EDFA suffers from a limitation, that the gain is not same for all signal wavelengths. This is a problem for WDM systems, as it can result in receiver imbalance. Various techniques have been proposed to flatten the gain spectrum. One of the popular technique for gain flattening is the use of an external optical gain flattening filter (GFF) having a loss profile that is reverse of the gain spectra of EDFA. However, in these GFF gain equalization is achieved by attenuating the wavelengths with higher gain, hence reducing the efficiency. Recently it was proposed that an appropriately chosen long period grating written through a length of the erbium doped fiber (EDF) itself can bring about gain flattening. As a result, in this configuration, there was gain flattening as well as increase in the average gain across the 1.54-1.56 m band.

ii

Synopsis

Another important issue in the EDFA is the presence of spontaneous emission which is also amplified as it propagates through the fiber. The amplified spontaneous emission (ASE) essentially contributes to noise and depletes the population inversion. ASE in the same direction as the signal is a major source of cumulative noise (reducing signal to noise ratio), while backward propagating ASEs can harm source lasers if not filtered out. We expect that LPG written in EDF itself would also affect the amplification of spontaneous emission, and hence, the noise characteristics of the amplifier. The spontaneous emission generated in each small section of fiber has a random phase. LPG is a phase sensitive device, and hence, it is necessary to take into account both the amplitude and phase of the propagating ASE. We have evolved a methodology to incorporate ASE taking into consideration the random phase of spontaneous emission. Our results show that LPG written in EDF itself, not only brings about gain flattening, but also suppresses the ASE. Amplified spontaneous emission is also used as a broadband source. In an EDF with the LPG written in it, the output power spectrum of this source over the1.53-1.56 m band is also flattened. In order to study the LPG written in the EDF, we had to develop an understanding of LPG and its applications. A specific feature of LPGs is the sensitivity of the transmission spectrum to the refractive index of ambient namb , i.e., the material surrounding the cladding of the fiber. The primary effect of change in the ambient refractive index is the consequent change in resonant wavelength. Hence, several authors have exploited this feature of a LPG to implement refractive index sensors based on the change in resonance wavelength, i.e., they measure the small shift in the resonance wavelength with change in ambient refractive index. The limitation of this technique is that the LPG has to be interrogated with a broadband source and, the measurement of such small wavelength shifts requires the use of relatively expensive high-resolution optical spectrum analyzers (OSA). We present an alternative approach for measurement of refractive index using a LPG. In our method the LPG is interrogated by a single wavelength source and, instead of measuring the shift in resonance, the change in the power retained in the core mode due to change in ambient index is measured. We present a criterion to design the grating based

iii

Synopsis

refractive index sensor, which takes into account the desired refractive index range and maximizes the sensitivity. Analogous to the EDFA is the erbium doped waveguide amplifier (EDWA) that uses a waveguide instead of a fiber to amplify the optical signal. One such waveguide amplifier is erbium doped titanium in-diffused lithium niobate waveguide amplifier. Titanium in-diffused optical waveguides in LiNbO3 (lithium niobate) are an attractive host material for the development of active and passive integrated optical components. Doping LiNbO3 waveguides with erbium ions, under appropriate pumping conditions forms an amplifying medium for wavelengths around 1.53 m. A rigorous modeling of the EDWA implies the computation of integrals containing modal intensity and erbium density distribution across the doped area. To obtain the gain and ASE characteristics of an EDWA one has to solve a large number of coupled differential equations each containing such integrals. Hence, obtaining the gain characteristics for one signal wavelength is time consuming, and when ASE, which is spread over the entire gain spectrum region, is considered the modeling is computationally expensive. We have proposed an approximation for the modal fields, which reduce these integrals to analytical forms. This results in a computationally efficient solution, especially when a large number of wavelengths are co-propagating. For the last few years, telecommunication institutes around the world have shown lot of interest in semiconductor optical amplifiers (SOA). There is a valid reason for this interest. Firstly, they are compatible with monolithic integration, hence offer a low cost option. Secondly, their gain bandwidth can be moved almost without limit over a wide range of wavelengths by choice of material composition, e.g., in InGaAsP the range is from 1.2 to 1.6 m. With photonics moving near to end user, SOAs are being explored as inline amplifiers or power boosters in metropolitan computer networks. Further, due to their strong nonlinear behavior, SOAs also find application in optical data processing. An SOA is essentially a semiconductor laser with no reflecting facets. It has a gain region (also called active layer) sandwiched between two cladding layers, one p-doped and the other n-doped. Under proper biasing conditions, there occurs a population inversion in the gain region, which is necessary for the amplification of light. Several

iv

Synopsis

models have been developed in the past to simulate SOAs and depending on the behavior we want to simulate, one model can be better than others. In general, they all solve rate equations, i.e., time dependent differential equations for the carrier and photons, and either, average the carrier density over the entire length of the device or divide the device into small segments. While the rate equations provide a fast means to predict the behavior of an SOA, for better understanding and designing one needs a tool which takes into account all the physics of the semiconductor. We need a tool, which calculates the basic material parameters and their dependence on the applied electric and optical field. Silvaco’s ATLAS is one such physics based tool. However, ATLAS supports only Lasers, LEDs, photodetectors and Solar cells among the optical devices. As mentioned above, ATLAS is a versatile simulation tool with the capability to include various physical effects important for semiconductor lasers. An SOA is essentially a semiconductor laser with low mirror reflectivities. Thus, theoretically speaking by reducing the mirror reflectivities in the ATLAS LASER module, we should be able to model an SOA. This can provide us with the information regarding gain and ASE spectrum of the SOA under no signal conditions. However, this information is insufficient. An SOA as an amplifier or as an optical signal processor operates on an input optical signal. For complete characterization of an SOA it is important to know how the gain and carrier densities of the amplifier change (gain saturation and gain recovery) in response to the optical signal. Unfortunately, ATLAS has no support for an optical input to an SOA. Hence, we had to improvise, from the existing models in ATLAS, a way that will have the similar effect on carriers and gain as an input optical signal has. We developed a technique of implementing a virtual optical source in ATLAS and using ATLAS characterized two experimentally well defined SOAs. Our results showed a good match with the experimental data, thus, proving that we can extend the capabilities of ATLAS in engineering SOA. Having established this extended capability, we further investigated the effect of various modifications in SOA design to improve device performance. Specifically, we considered the modifications in doping, active layer depth and ridge width. Our results show that these changes in the SOA design have a significant effect on the SOA

v

Synopsis

saturation and recovery behavior. A p-doped SOA on proper bias (high injection current) can be more suitable for amplifying applications, and an n-doped SOA more suitable for signal processing applications. Smaller ridge width reduces effective carrier lifetime, saturation powers, and at same time increase alpha factor thus is better they are better suited for optical signal processing applications. In case we want to design an SOA for in-line amplification application, the best design will be a gradually increasing tapered structure. Small width in the beginning of the SOA will enable a fast response, and growing width along the length of SOA will increase the saturation power. Decrease in the depth of active region, decreases the confinement factor, and reduces effective carrier lifetime. This results in increase of both input and output saturation powers, making an SOA with small active layer depths a better choice for in-line optical amplification. The scope of this thesis is modeling, design and application of optical amplifiers and long period gratings (LPGs). Typically, we concentrate on erbium doped fiber amplifiers, erbium doped lithium niobate waveguide optical amplifiers (EDWAs), and semiconductor optical amplifiers (SOAs). These three devices besides being employed as optical amplifiers also promise their feasibility to be used as optical signal processors. Modeling and simulation of these amplifiers is thus an important tool in the understanding and designing of these amplifiers. This can further aid in the development on structural modifications which can help in implementing new technologies and look for novel applications of the same structure. The salient features of the underlying work can be summarized as below: The thesis is structured as follows: After a brief introduction to the state of art in Chapter 1, Chapter 2 provides a review of the concepts used in subsequent chapters. Confinement and guidance of electromagnetic waves through waveguides and fibers is presented. Specifically, we discuss the variational method for channel waveguides and analysis for optical fibers taking into account the core, cladding and ambient refractive index regions. The basic characteristics of long period gratings are presented along with the coupled mode theory for determining the coupling between the fundament core mode and co-propagating cladding mode. Finally, the general principle of optical amplification is presented and some common optical amplifiers are discussed.

vi

Synopsis

In Chapter 3 power coupled equations for modeling of EDFA’s are discussed. Recently, researchers have proposed EDF's with LPG written in them for better gain flattening. These structures are analyzed using modified coupled mode analysis. We have evolved a methodology to incorporate ASE taking into consideration the random phase of spontaneous emission. Our results show that LPG written in EDF itself, not only brings about gain flattening, but also suppresses the ASE. Chapter 4 looks into the analysis of erbium doped titanium in-diffused lithium niobate waveguide optical amplifiers. The gain coupled differential equations involve integrals and depend explicitly on the modal fields, making it time consuming to solve. In this chapter we approximate the Hermite-Gaussian modal field obtained from the variational analysis by suitably chosen approximations. These approximations reduce the integrals to analytical forms. This results in a computationally efficient solution. In Chapters 5-7, we focus on semiconductor optical amplifiers. Chapter 5 discusses the basic semiconductor physics controlling the behavior of an SOA. The chapter also discusses the widely accepted Connelly model, to model an SOA in steady state. Chapter 6 investigates the possibility of using ATLAS, a Silvaco’s physics based simulator tool, to model the behavior of SOA. The results obtained by ATLAS are compared with the experimental data, validating the use of ATLAS to simulate SOAs. In Chapter 7 we further investigate using ATLAS, the effect of modification in SOA design on gain saturation and alpha factor. Specifically, we look into the effects of doping the active layer of the SOA, changing the depth of the active layer and changing the width of ridge. Our results show that it is possible to engineer the gain saturation and alpha factor of an SOA. During our study of the transmission spectra of LPG written in it, we observed that at a certain wavelength greater than the resonance wavelength, the transmitted core power varies significantly from 0 (no transmission) to 1 (full transmission) as the ambient index is varied. This motivated us to investigate the possibility of intensity based refractive index sensor using LPG. Hence, in Chapter 8, we look into the application of LPGs as refractive index sensors. We propose a design recipe to tailor a refractive index sensor with maximum sensitivity in the desired refractive index range.

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Synopsis

Finally we present an outlook on future research. .

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Table of contents

Table of contents SYNOPSIS .............................................................................................................................................. I TABLE OF CONTENTS .................................................................................................................... IX LIST OF SYMBOLS ........................................................................................................................ XIII 1 INTRODUCTION ............................................................................................................................ 1 1.1 ACHIEVEMENTS OF THE PRESENT WORK ...................................................................................... 12 2 WAVEGUIDANCE, MODE COUPLING IN FIBER GRATINGS AND OPTICAL AMPLIFIERS: THEORETICAL BACKGROUND ................................................................... 17 2.1 INTRODUCTION ............................................................................................................................ 17 2.2 MAXWELL’S EQUATIONS.............................................................................................................. 18 2.3 INTEGRATED OPTICAL WAVEGUIDES ............................................................................................ 21 2.3.1 Planar waveguides ......................................................................................................... 22 2.3.2 Diffused channel waveguides ......................................................................................... 23 2.4 OPTICAL FIBERS ........................................................................................................................... 27 2.4.1 Guided core mode........................................................................................................... 31 2.4.2 Cladding modes .............................................................................................................. 33 2.4.3 Comment on considering only the cladding-ambient interface for cladding modes....... 34 2.5 LONG PERIOD GRATINGS .............................................................................................................. 36 2.5.1 Coupled mode analysis ................................................................................................... 38 2.5.2 Applications .................................................................................................................... 41 2.6 OPTICAL AMPLIFICATION ............................................................................................................. 42 2.6.1 Einstein coefficients ........................................................................................................ 42 2.6.2 Optical gain .................................................................................................................... 45 2.6.3 Spectral broadening ....................................................................................................... 46 2.6.4 Types of optical amplifiers.............................................................................................. 48 3 ERBIUM DOPED FIBER AMPLIFIERS .................................................................................... 53 3.1 INTRODUCTION ............................................................................................................................ 53 3.2 ATOMIC STRUCTURE AND RELATED OPTICAL SPECTRUM ............................................................. 55 3.2.1 Cross sections ................................................................................................................. 57 3.2.2 Optical gain and rate equations ..................................................................................... 58 3.3 POWER COUPLED EQUATIONS....................................................................................................... 61 3.3.1 Amplified spontaneous emission ..................................................................................... 63 3.3.2 Simplified power equations............................................................................................. 65 3.4 ERBIUM DOPED FIBER AMPLIFIER WITH LONG PERIOD GRATING WRITTEN IN IT ............................ 70 3.4.1 Amplified spontaneous emission in phase sensitive structures ....................................... 70 3.4.2 Comparison of results ..................................................................................................... 73 3.5 ASE SUPPRESSION IN EDF WITH LPG WRITTEN IN IT................................................................... 76 3.6 SUMMARY .................................................................................................................................... 80 4 ERBIUM DOPED LITHIUM NIOBATE WAVEGUIDE AMPLIFIERS ................................ 83 4.1 INTRODUCTION ............................................................................................................................ 83

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Table of contents

4.2 ERBIUM IN LITHIUM NIOBATE ...................................................................................................... 84 4.2.1 Signal, pump and noise propagation .............................................................................. 87 4.2.2 Three variable variational fields .................................................................................... 89 4.3 SIMPLIFIED GAIN AND ASE CALCULATIONS ................................................................................ 92 4.3.1 Rectangular approximation............................................................................................ 93 4.3.2 Symmetric Gaussian approximation .............................................................................. 95 4.3.3 Asymmetric Gaussian approximation ............................................................................ 98 4.3.4 Comparison of results from three approximate fields .................................................. 100 4.3.5 ASE and multiple signal propagation .......................................................................... 102 4.4 SUMMARY ................................................................................................................................. 106 5 SEMICONDUCTOR OPTICAL AMPLIFIERS....................................................................... 107 5.1 INTRODUCTION .......................................................................................................................... 107 5.2 SEMICONDUCTOR PHYSICS ........................................................................................................ 108 5.2.1 Band structure of direct band gap semiconductors ...................................................... 109 5.2.2 Electron and hole concentration .................................................................................. 111 5.2.3 Generation and recombination processes .................................................................... 114 5.2.4 Intra band interactions ................................................................................................. 119 5.3 SEMICONDUCTOR OPTICAL AMPLIFIERS ..................................................................................... 121 5.3.1 Condition for amplification .......................................................................................... 121 5.3.2 Optical gain.................................................................................................................. 122 5.3.3 Rate equations .............................................................................................................. 125 5.3.4 SOA modeling: Connelly model ................................................................................... 127 5.3.5 Gain saturation ............................................................................................................ 133 5.3.6 Gain recovery ............................................................................................................... 136 5.3.7 Alpha factor.................................................................................................................. 137 6 SEMICONDUCTOR OPTICAL AMPLIFIERS: MODELING USING ATLAS .................. 141 6.1 ATLAS: AN INTRODUCTION ..................................................................................................... 141 6.2 SIMULATION MODEL AND MATERIAL PARAMETERS ................................................................... 142 6.3 ATLAS SIMULATION: ISSUES TO BE RESOLVED ......................................................................... 144 6.3.1 Effect of bimolecular coefficient................................................................................... 145 6.3.2 Virtual optical source ................................................................................................... 147 6.3.3 Simulating gain saturation ........................................................................................... 149 6.4 SIMULATION RESULTS ............................................................................................................... 150 6.4.1 Optical gain spectrum .................................................................................................. 156 6.4.2 Gain saturation ............................................................................................................ 157 6.4.3 Gain recovery ............................................................................................................... 160 6.4.4 Alpha factor.................................................................................................................. 161 6.5 SUMMARY ................................................................................................................................. 162 7 ENGINEERING BULK SEMICONDUCTOR OPTICAL AMPLIFIERS ............................. 165 7.1 GAIN SATURATION..................................................................................................................... 165 7.2 MODIFICATIONS IN DESIGN ........................................................................................................ 167 7.2.1 Doping the active layer ................................................................................................ 167 7.2.2 Modifying active layer width ........................................................................................ 174 7.2.3 Modifying active layer depth ........................................................................................ 183 7.3 SUMMARY ................................................................................................................................. 189

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Table of contents

8 LONG PERIOD GRATINGS: REFRACTIVE INDEX SENSOR........................................... 193 8.1 INTRODUCTION .......................................................................................................................... 193 8.2 CONVENTIONAL REFRACTOMETERS ........................................................................................... 195 8.2.1 Gratings based refractive index sensors ....................................................................... 196 8.2.2 Limitations .................................................................................................................... 197 8.3 MODIFIED SENSOR ..................................................................................................................... 198 8.3.1 Mathematical analysis .................................................................................................. 199 8.3.2 Design criteria for the refractive index sensor ............................................................. 202 8.4 SIMULATION RESULTS ................................................................................................................ 204 8.4.1 Sugar and salt solution ................................................................................................. 204 8.4.2 Xylene in heptane.......................................................................................................... 207 8.4.3 Effect of temperature and wavelength fluctuations ...................................................... 209 8.5 SUMMARY .................................................................................................................................. 212 SCOPE FOR FUTURE WORK ........................................................................................................ 213 APPENDIX A BIBLIOGRAPHY ..................................................................................................... 215 APPENDIX B MODIFIED COUPLED MODE ANALYSIS ......................................................... 227 APPENDIX C OPTICAL AND ELECTRICAL PARAMETERS FOR INDIUM GALLIUM ARSENIDE PHOSPHATE .......................................................................................................... 231 APPENDIX D EXPERIMENTAL SETUP ...................................................................................... 233 D.1 EXPERIMENTAL SETUP TO DETERMINE GAIN SATURATION IN SOAS........................................... 233 D.2 EXPERIMENTAL SETUP FOR MEASURING GAIN RECOVERY OF SOAS........................................... 233

xi

List of symbols

List of symbols E

Electric field vector (V/m)

H

Magnetic field vector (A/m)

,

0

,

0

r

Dielectric permittivity of material, free space, relative permittivity Magnetic permeability of free space Charge density (C/m3)

Ph

Energy density of photons (J/m3)

Er

Erbium ion density (/m3)

J

Current density (A/m2)

c

Speed of light (m/s)

n

Refractive index Propagation coefficient (m-1)

k0

Wave vector ( m-1)

neff

Effective index

,

Modal fields

X(x)

Modal field in x-direction

Y(y)

Modal field in y-direction

A(z )

Complex amplitude of the mode envelope

aco,acl

Core radius, cladding radius ( m) Relative refractive index difference

xiii

List of symbols

n

Induced change in refractive index

g

Optical gain (m-1) Gain coefficient (m-1)

G0

Small signal Amplifier gain

G

Amplifier gain Coupling coefficient (m-1)

ng

Group index

nT

Electron density (m-3)

p

Hole density (m-3)

V

Electrostatic potential (V)

I

Intensity (W/m2) Photon flux

f

L f

Line shape function

f

Frequency (Hz)

If

Intensity of light at frequency f (W/m2) Confinement factor; detuning factor Angular frequency (Hz) a

e

,

fc

,

f ,

mirr

a

f

Loss due to bulk absorption, free carrier absorption, mirror loss Emission cross-section, absorption cross-section (m2) Ratio of emission and absorption cross-section;

Wg

Bandgap energy (eV)

W1 , W2

Energy levels (eV)

N1, N2

Population density of energy levels W1, W2 (m-3)

WC, WV

Minima of conduction band, Maxima of valence band (eV)

xiv

List of symbols

Electric susceptibility L

Length of the amplifier (m)

Rsp,RASE,Rsig

Recombination rate due to spontaneous emission, amplified spontaneous emission, optical signal

RSRH,RAuger

Shockley Read and Hall recombination rate, Auger recombination rate

Q

Emission factor

me* , mh*

Effective mass of electron, hole

H

P

Alpha factor Power (W)

sat Pinsat , Pout

Input saturation power, output saturation power

P frac

Fractional power in the core

A, B

Einstein coefficients Delta function

S

Photon density (m-3)

xv

Chapter 1: Introduction

1 Introduction Communication using light rays is not new; as early as 490 BC, in the famous siege of Athens by Persia, light rays were used to send messages [68]. The modern optical communication systems today can boast of accessing data from any part of the earth, at a data rate as high as 30 Mbps. The foundation of optical communication as we know it today can be traced back to the year 1917, when Albert Einstein predicted the presence of stimulated emission in the paper entitled “Zur Quantentheorie der Strahlung” [43]. Charles Townes, in USA, Nikholai Basov and Alexander Prochorov, in USSR, using the concept of stimulated emission and population inversion developed the world’s first MASER in the year 19601. This led way to the construction of oscillators and amplifiers based on the maser-laser principle. Soon, Theodore Maiman demonstrated the first functional laser, a Ruby laser in the year 1960, and Robert Hall developed the first semiconductor injection laser in the year 1962. Parallel to the development in optical sources and amplifiers, work was going on in choosing a right medium for optical transmission. In 1966, Charles K. Kao [64] published his work in which he concluded that for optical fibers to be a viable

1

They shared the Nobel Prize for this contribution in the year 1964.

1


communication medium the limit of attenuation would be 20 dB/km which is much higher than the lower limit of loss figure imposed by fundamental mechanisms in glassy materials. Four years later, Corning Inc. (then known as Corning Glass Works) announced the successful fabrication of single mode fibers with an attenuation below 20 dB/km at 633 nm. It is important to mention that around same time Manfred Börner [38] from Telefunken, Germany, was the first to propose a multi-stage transmission system for information presented in pulse code modulation. The proposed transmission system used optical fibers and had repeaters for long distance. Following these breakthroughs, the world’s first commercial optic communication system was deployed in the year 1975. It had a bit rate of 45 Mbps with repeater spacing of up to 10 km. By 1987 second generation of optical communication systems with bit rates of up to 1.7 Gbps and repeater spacing of 50 km were operating. A new generation of single-mode systems was just beyond the horizon operating at 1.55 m with fiber loss around 0.2 dB/km. This third generation of optical communication system, operating at 2.5 Gbps, had repeater spacing in excess of 100 km. The fourth generation of optical communication systems employed optical amplifiers to reduce the need for repeaters and wavelength division multiplexing (WDM) to increase data capacity. These technologies brought about a revolution, resulting in doubling of capacity every six months starting from 1992. With the spread of internet and World Wide Web, not to mention technologies like video conferencing and VOIP, the demand for higher capacity is continuously growing. Today fiber-to-the-home/office is becoming a reality. In an essence, we can say that because of high speeds, better reliability and noise immunity optical communication will continue to grow. It will change our society and make our lives more convenient, more enjoyable and more comfortable. Perhaps, it is the recognition of the fact that optical communication has changed the world for good, in the year 2009 Charles K. Kao was awarded the Nobel Prize in Physics “for groundbreaking achievements concerning the transmission of light in fibers for optical communication”. The most basic element of any guided wave optical communication system is the guiding channel for optical waves: a waveguide. In the simplest term a waveguide is

2


an optical interconnect, analogous to electrical wires in electronic circuits. The basic concept of waveguide is very simple. Light is confined in a dielectric medium of one refractive index, embedded in a dielectric medium of lower refractive index. The light travels through the medium by the principle of total internal reflection (TIR) (Figure 1.1). Thus, “an optical waveguide is a light conduit consisting of a slab, strip or cylinder of dielectric material surrounded by another dielectric material of lower refractive index” [110]. The dielectric medium where light is confined is referred as the guiding region or core and the surrounding lower refractive index medium is referred as substrate or cover or cladding.

Figure 1.1: Propagation of light by the principle of total internal reflection.

The simplest form of optical waveguide is the three layer slab waveguide (Figure 1.2 a) in which light is confined in one dimension only (y direction). Channel waveguides provide two dimensional (2D) confinement of light (both x and y direction). Depending on the fabrication technology employed there are different type of channel waveguides possible, viz., buried channel waveguide (Figure 1.2 b), diffused channel waveguide, (Figure 1.2 c), ridge waveguide (Figure 1.2 d) to mention a few. Planar and channel waveguides are an important component of integrated optical circuits (IOC). Light propagates in these waveguides in specific transverse field distributions, called modes, which do not change with propagation distance. These modes and there propagation characteristics are obtained as solutions of Maxwell’s equations with appropriate boundary conditions [5,26,47,48,110,122,130]. For slab

3


waveguides the equations reduce to an ordinary one dimensional differential equation for the transverse components of electric and magnetic fields, and can be solved as a boundary value problem. The solutions are part of almost all textbooks on Photonics [3,5,26,47,48,68,108,110,122,128,130]. However, for channel waveguides such direct solutions are not possible and a lot of early work was focused on development of approximate methods like Marcatili’s method [80] or effective index method [59] or variational

methods

[112,113,114].

With

increasing

computational

power,

increasingly numerically intensive methods [103,104] such as finite element (FE), finite difference (FD), beam propagation method (BPM) and finite difference time domain (FTDT) are being used to analyze waveguide structures. However, the approximate methods in general represent workhorses for design and modeling of optical waveguide structures [82]. In Chapter 2, we briefly discuss the variational method to obtain the modal fields of the diffused channel waveguides with a focus on channel waveguides formed by titanium in-diffusion into lithium niobate substrate. These channel waveguides form the host for waveguide amplifiers discussed in Chapter 4.

(a)

(b)

(c) (d) Figure 1.2: Different type of waveguides (a) Planar waveguide, (b) Buried channel waveguide (c) Diffused channel waveguide, (d) Ridge waveguide structure.

4


The most widely used optical waveguide is the low loss optical fiber, which forms the guiding channel in all long haul optical communication systems. An optical fiber basically is a cylindrical dielectric waveguide, made of a low loss transparent material, usually silica (SiO2) glass (Figure 1.3). It has a central core of slightly higher refractive index, n1, (usually GeO2 doped silica glass), in which the light is guided, surrounded by an outer cladding of slightly lower refractive index, n2, (pure silica) and a protective colored polymer jacket

Figure 1.3: Basic structure of an optical fiber.

.

Figure 1.4: Propagating core mode in the fiber.

Light propagates over long distances confined to the core of the fiber, essentially by TIR at the core-cladding interface (Figure 1.4). Like planar and channel waveguides, the optical fiber also supports discrete guided modes, obtained as solutions of Maxwell’s equations. Depending upon the number of guided modes supported the fiber is classified as single mode fiber (SMF) or multi-mode fiber (MMF). MMFs have large core dimensions (core radius a co ~ 50 μm ) and can also be well

5


understood by the simpler ray optics. SMFs on the other hand have core dimensions close to wavelength, a co ~ 5 μm and hence, it is necessary to understand propagation in them in terms of electromagnetic wave propagation based on Maxwell equations. Since most waveguides are made of glass, the so called weakly guiding condition, n1

n2

n1 is applicable and as a result a considerable simplification ensures in the

theoretical analysis. In particular, it is possible to find linearly polarized (LP) modes of the structure which form useful approximations to the true hybrid modes [3,122]. In an SMF, the cylindrical dielectric waveguide of radius, a co , formed by the corecladding interface supports only a single bound (LP01) mode confined to the core area also referred to as the core mode or core guided mode. Most textbooks solve for these LP modes using a two layer geometry [5,48,68,130], i.e., assuming the cladding to extend to infinity. However, a guiding structure is also formed by the claddingambient interface, provided that cladding index is greater than the ambient index. This highly multimoded waveguide supports bound modes extending over the whole cladding area, the so called cladding modes. In Chapter 2, we briefly review the modal analysis for LP modes of the optical fiber using the complete three layer geometry (core, cladding and ambient) [44,89,117] to obtain the propagation characteristics of the core and cladding modes. The characterization of cladding modes, in addition to core mode, is essential in the design of long period fiber gratings discussed in Chapter 2, 4 and 8.

Figure 1.5: Propagating cladding mode in the fiber.

In addition to being used as an optical interconnect; the optical fiber is also host for large number of active and passive fiber devices like fiber gratings and fiber amplifiers. If the core of the fiber has a periodic refractive index modulation, along the direction of propagation, a fiber grating is obtained. In 1978, Hill and his coworkers accidentally discovered photosensitivity of fibers while studying the non-linear

6


properties in germania (GeO2) doped silica fiber with visible argon ion laser radiation [57]. The major breakthrough came eight years later when Meltz and coworkers reported successful grating writing by placing a fiber in the interference pattern formed by a UV laser [87]. These fiber Bragg gratings (FBGs) with periods, 500 nm result in the reflection of the Bragg wavelength

B

,~

2n [48,66,121]. FBGs

are also called reflection gratings and find application as a wavelength selective element.

Figure 1.6: (a) Fiber Bragg grating (b) Long period grating.

In 1996, Vengsarkar et al. [134] in their paper “Long Period Gratings as Band Rejection Filters” (which has emerged as one of the most cited paper of the last decade) introduced a new type of grating to the optics community. Like FBGs these long period fiber gratings (LPGs) are also formed by the periodic modulation of the core refractive index, but with a periodicity ranging between 100 μm to 1 mm. LPGs couple light between the forward propagating core mode and several discrete forward propagating cladding modes [19,44,45,117,134]. In its conventional application, these cladding modes can be attenuated, leaving a series of loss bands in the transmission

7


spectrum (Figure 1.6). This property makes the LPG a band rejection filter and finds application in gain flattening filters for optical amplifiers [51,107,143]. Emergence of erbium doped fiber amplifier (EDFA) in 1987 [35,83,84] was one of the most important developments that revolutionized optical telecommunication. EDFA mainly consists of a silica glass fiber (usually 4 m to 50 m long), in which the core is doped with erbium ions. The erbium ions in a silica host when pumped by a 980 nm pump radiation can amplify many wavelength channels within the C and L band, i.e., a wavelength range of 1.53-1.6 m. Thus, one can use a single amplifier for all wavelengths in wavelength division multiplexed (WDM) or dense wavelength division multiplexed (DWDM) systems. However, EDFA suffers from two limitations: (i) that the gain is not same for all signal wavelengths, (ii) the presence of amplified spontaneous emission (ASE) noise. One of the techniques for gain flattening is the use of an external optical gain flattening filter (GFF) having a loss profile that is reverse of the gain spectra of EDFA [51,107,143]. However, in these gain equalization is achieved by attenuating the wavelengths with higher gain, hence reducing the efficiency. Recently, Singh et al. [119] proposed that an appropriately chosen long period grating written through a length of the erbium doped fiber (EDF) itself can bring about gain flattening. As a result, in this configuration, there was gain flattening as well as increase in the average gain across the 1.54-1.56 m band. We expect that LPG written in EDF itself would also affect the amplification of spontaneous emission, and hence, the noise characteristics of the amplifier. The amplified spontaneous emission essentially contributes to noise and depletes the population inversion. ASE in the same direction as the signal is a major source of cumulative noise (reducing signal to noise ratio), while backward propagating ASEs can harm source lasers if not isolated. The spontaneous emission generated in each small section of fiber has a random phase. LPG is a phase sensitive device, and hence, it is necessary to take into account both the amplitude and phase of the propagating spontaneous emission. We have evolved a methodology to incorporate ASE taking into consideration the random phase of spontaneous emission. Our results show that LPG written in EDF itself, not only brings about gain flattening, but also suppresses the ASE noise. Further, amplified spontaneous emission is also used as a broadband

8


source. If an EDF with the LPG written in it is used for such a source, our results show that, the output power spectrum of the source over the 1.53-1.56 m band is also flattened. In order to study the LPG written in an EDF for gain flattening and noise suppression, we had to develop an understanding of LPGs and its applications. In Chapter 2, we briefly review the coupled mode theory to analyze the coupling of power between the core mode and different cladding modes [121,122] of a LPG. A specific feature of LPGs is the sensitivity of the transmission spectrum to the refractive index of ambient namb , i.e., the material surrounding the cladding of fiber [19,134], because the cl , m effective indices of the cladding modes, neff , are strongly influenced by the ambient

refractive index. The primary effect of change in the ambient refractive index is the consequent change in resonant wavelength. Hence, several authors have exploited this feature of a LPG to implement refractive index sensors based on the change in resonance wavelength, [18,96,115,134,146] i.e., they measure the small shift in the resonance wavelength with change in ambient refractive index. The limitation of this technique is that the measurement of such small wavelength shifts requires the use of relatively expensive high-resolution optical spectrum analyzers (OSA). In Chapter 8, we present an alternative approach for measurement of refractive index using a LPG. In our method a LPG is interrogated by a single wavelength source and, instead of measuring the shift in resonance, the change in the power retained in the core mode due to change in ambient index is measured. We present a criterion to design the LPG based refractive index sensor, which takes into account the desired refractive index range and maximizes the sensitivity. Analogous to the EDFA is the erbium doped waveguide amplifier (EDWA) that uses a channel waveguide instead of a fiber to confine and amplify the optical signal. LiNbO3 (lithium niobate) has been an attractive host material for the development of active and passive integrated optical components and titanium in-diffused LiNbO3 waveguides form the backbone of integrated optical circuits (IOCs) in LiNbO3. Doping LiNbO3 waveguides with erbium ions, under appropriate pumping conditions forms an amplifying channel for wavelengths around 1.53 m. A rigorous modeling of the EDWA implies numerical solution of coupled differential equations, which

9


require the computation of integrals containing modal intensity and erbium density distribution across the doped area at each step. Hence, obtaining the gain characteristics for even one signal wavelength is time consuming, and when ASE, which is spread over the entire gain spectrum region, is considered the modeling is computationally expensive. In Chapter 4, we have proposed approximations for the modal fields, which reduce these integrals to analytical forms. This results in computationally efficient solution, especially when a large number of wavelengths are co-propagating and iterative solutions of the differential equations are needed for estimating the ASE.

Figure 1.7: Schematic of an SOA

For the last few years, telecommunication institutes around the world have shown lot of interest in semiconductor optical amplifiers (SOA). There is a valid reason for this interest. Firstly, they are compatible with monolithic integration and hence, offer a low cost option. Secondly, their gain bandwidth can be moved almost without limit over a wide range of wavelengths by choice of material composition, e.g., in InGaAsP the range is from 1.2 to 1.6 m. With photonics moving near to end user, SOAs are being explored as inline amplifiers or power boosters in metropolitan networks, with a tough competition between bulk-SOAs, quantum well (QW) SOAs and quantum dot (QD) SOAs [30,31,91]. On the other hand, due to strong nonlinear behavior SOAs appear to be a convenient tool for optical data processing [30,31,91].

10


An SOA is essentially a semiconductor laser with no reflecting facets [147]. It has a gain region (also called active layer) sandwiched between two cladding layers, one pdoped and the other n-doped (Figure 1.7). Under proper biasing conditions, there occurs a population inversion in the gain region, which is necessary for the amplification of light. Several models have been developed in the past to simulate SOAs [23,30,31,62,81], and depending on the behavior we want to simulate, one model can be better than others. In general, they all solve rate equations, i.e., time dependent differential equations for the carrier and photon densities, and either, average the carrier density over the entire length of the device or segment the SOA into small segments. While the rate equations provide a fast means to predict the behavior of an SOA, for better understanding and designing one needs a tool which takes into account all the physics of the semiconductor. (Some of the basic semiconductor physics relevant to the understanding of an SOA and its modeling is given in Chapter 5). We need a tool, which calculates the basic material parameters and their dependence on applied electric and optical fields. Silvaco’s ATLAS is one such physics based tool. However, ATLAS supports only Lasers, LEDs, photodetectors and Solar cells among the optical devices. We have developed a methodology to extend the capability of ATLAs for modeling of SOAs as described in Chapter 6. As mentioned above, ATLAS is a versatile simulation tool with the capability to include various physical effects important for semiconductor lasers. An SOA is essentially a semiconductor laser with low mirror reflectivities [30,31,42,91]. Thus, theoretically speaking by reducing the mirror reflectivities in the ATLAS LASER module, we should be able to model an SOA. This can provide us with the information regarding gain and ASE spectrum of the SOA under no signal conditions. However, this information is insufficient. An SOA as an amplifier or optical signal processor operates on an input optical signal. For complete characterization of an SOA it is important to know how the gain and carrier densities of the amplifier change (gain saturation and gain recovery) in response to the optical signal. Unfortunately, ATLAS has no support for an optical input to an SOA. Hence, we had to improvise, from the existing models in ATLAS, a way that will have the similar effect on carriers

11


and gain as an input optical signal has. We developed a technique of implementing a virtual optical source in ATLAS and characterized two experimentally well defined SOAs. Our results show a good match with the experimental data, thus, proving that we can extend the capabilities of ATLAS to engineer SOAs. Having established this extended capability, we further investigated the effect of various modifications in SOA design to improve device performance. Specifically, we considered the modifications in doping, active layer depth and ridge width. Our results show that these changes in the SOA design have a significant effect on the SOA saturation and recovery behavior. A p-doped SOA on proper bias (high injection current) can be more suitable for amplifying applications, and an n-doped SOA more suitable for signal processing applications. Further, smaller ridge width reduces effective carrier lifetime, saturation powers, and at same time increases the alpha factor and is hence better suited for optical signal processing applications. Decrease in the depth of active region, decreases the confinement factor, and reduces effective carrier lifetime. This results in increase of both input and output saturation powers, making an SOA with small active layer depths would be a better choice for in-line optical amplification. The results of SOA engineering are presented in Chapter 7.

1.1 Achievements of the present work The scope of this thesis is modeling, design and application of optical amplifiers and long period gratings (LPGs). Typically, we concentrate on erbium doped fiber amplifiers, erbium doped lithium niobate waveguide optical amplifiers (EDWAs), and semiconductor optical amplifiers (SOAs). These three devices, besides being employed as optical amplifiers, also promise their feasibility to be used as optical signal processors. Modeling and simulation of these amplifiers is thus an important tool in the understanding and designing of these amplifiers. This can further aid in the development on structural modifications which can help in implementing new technologies and look for novel applications of the same structure. In summary in the present thesis we have achieved the following: Evolved a methodology to work with complex amplitude coupled differential equations instead of conventional power coupled equations for use in the study

12


of ASE in erbium doped fiber amplifiers. This was made possible by developing a recipe to incorporate both, the amplitude and the random phase of the spontaneous emission generated along the propagation length of the EDF. The method was needed to analyze the possible ASE noise suppression in EDF with the LPG written in it, a novel structure proposed by Singh et al. [119] for gain flattened EDFs. We established the validity of our method by comparing the results with well established power propagation equations for EDF in the absence of LPG. Our results show that the novel structure not only brings about gain flattening, but also suppresses the ASE noise. Proposed field approximations for simplified gain and ASE calculations in EDWA. Field approximations obtained by the variational analysis, for titanium indiffused lithium niobate channel waveguides, by various authors [3,60,61,112,113,114] were studied. These were further approximated to reduce the integrals involved in gain and ASE noise characterization to analytical forms. The reduction to analytical forms reduces the computation time significantly for gain calculations when a large number of signals are multiplexed and for ASE noise evaluation. Developed a technique for implementing a virtual optical source in ATLAS and characterized two experimentally well defined SOAs. ATLAS was chosen because of its capability to accurately predict the electrical and optical characteristics associated with specified bias conditions. Though the ATLAS LASER module supports the simulation of semiconductor lasers, it cannot provide optical input necessary for simulating SOAs. We modeled a virtual optical source by manually changing the radiative recombination parameter to simulate the decrease of carrier concentration that would physically result from an external optical signal. Our results showed a good match with available experimental data, thus, proving that we can extend the capabilities of ATLAS in modeling SOA. Used the extended capability of ATLAS in SOA engineering. We investigated the effect of various modifications in SOA design to improve device performance. We considered the modifications in doping, active layer depth and ridge width. Our results show that these changes in the SOA design have a

13


significant effect on the SOA saturation and recovery behavior. A p-doped SOA on proper bias (high injection current) can be more suitable for amplifying applications, and an n-doped SOA more suitable for signal processing applications. Smaller ridge width reduces effective carrier lifetime, saturation powers, and at same time increase alpha factor. Decrease in the depth of active region, decreases the confinement factor, and reduces effective carrier lifetime. Thus depending upon the SOA application we can engineer the SOA design. Developed a criterion to design a long period grating for refractive index sensing around desired ambient index when only one interrogating wavelength is to be used. The design is based on power measurement and maximizes the sensitivity around the desired refractive index range.

The above work has resulted in the following publications: 1. G. Jain, Amita Kapoor and E. K. Sharma, Er-LiNbO3 Waveguide: field

approximation for simplified gain calculations in DWDM applications, J. Opt. Soc. America B, 26(4), 633-639, (2009). 2. Amita Kapoor and E. K. Sharma, Long Period Grating Refractive Index

Sensor: Optimal Design For Single Wavelength Interrogation, Appl. Opt. 48, G88-G94, (2009). 3. R. Singh, Amita Kapoor and E. K. Sharma, Long Period Gratings in erbium

Doped Fibers: Gain Flattening and ASE Reduction, Photonics 2004 (Seventh International conference on Optoelectronics, Fiber optics and Photonics), Cochin, India, 9-11th December (2004). 4. G. Jain, Amita Kapoor, and E. K. Sharma, Simplified Modeling of titanium

indiffused LiNbO3 Waveguide Amplifiers, Photonics 2006 (Eighth International Conference on Optoelectronics, Fiber-optics and Photonics), University of Hyderabad, Hyderabad, India, 12-16th December (2006). 5. Amita Kapoor, G. Jain and E. K. Sharma, Simplified Gain Calculations in

erbium doped LiNbO3 waveguides. Proc. SPIE 6468, 646808, (2007).

14


6. Amita Kapoor, R. Singh and E. K. Sharma, Suppression of Amplified

Spontaneous Emission in erbium Doped Fiber with Long Period Grating written in it, Microwave Conference, 2007. APMC 2007. Asia-Pacific, 1-4, (2007). 7. Amita Kapoor, R. Singh and E. K. Sharma, Estimation of Amplified

Spontaneous Emission in erbium Doped Fiber with Long Period Grating Written in it, The Second Research Forum of Japan-Indo Collaboration Project on Infrastructural Communication Technologies Supporting Fully Ubiquitous Information Society, Kyushu University, Fukuoka, Japan, Forum Digest, 5761, (2007). 8. Amita Kapoor, G. Jain and E. K. Sharma, Er-LiNbO3 Waveguide: Simplified

Gain Calculations for DWDM Application, 2007 Japan-Indo Workshop on Microwaves, Photonics and Communication Systems, Kyushu University, Fukuoka, Japan. Workshop Digest, 113-118, (2007). 9. Amita Kapoor and E. K. Sharma, Long Period Grating Refractive Index

Sensor: Optimal Design For Single Wavelength Interrogation, 2008 JapanIndo Workshop on Microwaves, Photonics and Communication Systems, Kyushu University, Fukuoka, Japan. Workshop Digest, (2008). 10. G. Jain, Amita Kapoor and E. K. Sharma, Er-LiNbO3 Waveguide: Field

Approximations for Simplified Gain Calculations in DWDM Application, Photonics 2008 (Ninth International Conference on Optoelectronics, Fiberoptics and Photonics), Delhi, India, 13-17th December (2008). 11. Amita Kapoor and E. K. Sharma; Long Period Grating Refractive Index

Sensor: Optimal Design For Single Wavelength Interrogation, Photonics 2008 (Ninth International Conference on Optoelectronics, Fiber-optics and Photonics), Delhi, India, 13-17th December (2008). 12. Amita Kapoor, E. K. Sharma, W. Freude and J. Leuthold, Saturation

Characteristics of InGaAsP-InP bulk SOA, Proc. SPIE 7597, 75971I (2010). 13. W. Freude, R. Bonk, T. Vallaitis, A. Marculescu, Amita Kapoor, C. Meuer,

D. Bimberg, R. Brenot, F. Lelarge, G. H. Duan and J. Leuthold Semiconductor

15


optical amplifiers (SOA) for linear and nonlinear applications, Deutsche Physikalische Gesellschaft e.V., Regensberg , 21-26th March 2010. 14. W. Freude, R. Bonk, T. Vallaitis, A. Marculescu, Amita Kapoor, E. K.

Sharma, C. Meuer, D. Bimberg, R. Brenot, F. Lelarge, G. H. Duan, C. Koos and J. Leuthold, Linear and nonlinear semiconductor optical amplifiers, 2010 12th International Conference on Transparent Optical Networks (ICTON), 1-4, (2010).

16

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background

2 Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background 2.1 Introduction Waveguides are an essential component of all optical communication systems. In addition to being the physical channel for guided optical communication systems in the form of optical fibers (optical interconnect) they also form the host for active and passive optical devices. Wave propagation through these waveguides in well defined electromagnetic modes is studied by Maxwell’s equations using the appropriate boundary conditions. In this chapter, for completeness, we first review the basic Maxwell’s equations governing the propagation of the electromagnetic modes in optical waveguides. We then outline the procedure for obtaining the propagation characteristics of the modes propagating in the single mode diffused channel waveguides using the variational analysis. Specifically we look into the channel waveguides formed by the in-diffusion of titanium into lithium niobate to obtain an analytical description of the modal fields of the diffused channel waveguide and these results are later used in Chapter 4 to

17

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background obtain simplified analytical expressions for the gain and noise characteristics of the erbium doped waveguide amplifier (EDWA). After channel waveguides, we move onto the most widely used optical waveguide: the optical fiber. We briefly review the modal analysis of the optical fiber in the weakly guiding approximation using all the three layers, i.e., core cladding and ambient to obtain the propagation characteristics of the linearly polarized (LP) core guided and cladding guided modes of the optical fiber. These results are used to understand the mode coupling in long period fiber gratings. Application of long period gratings written in erbium doped fibers for gain flattening and noise suppression is discussed in Chapter 3. Finally, we review the underlying physics behind the process of optical amplification. The main focus of this thesis is on optical amplifiers, hence this discussion forms the foundation.

2.2 Maxwell’s equations In a charge free, non-magnetic dielectric medium the electric field vector E ( x, y, z, t ) , the magnetic field vector H ( x, y, z , t ) , the electric flux density D( x, y, z , t ) and the magnetic flux density B( x, y, z, t ) are related by Maxwell’s equations: D

0

(2.1)

B

0

(2.2)

E H with D

E and B

material.

0

and

0

0

H , where

H t

0

(2.3)

D t 0

(2.4)

r

is the permittivity of free space,

is the dielectric permittivity of the r

x, y, z is the relative permittivity,

is the magnetic permeability of the non magnetic medium. For an isotropic

medium the dielectric permittivity

x, y, z is a scalar and the relative permittivity is

18


related to the refractive index, n(x,y,z), by relation,

n 2 . However, in any general

r

anisotropic medium, in the principal coordinate system of the crystal,

x, y, z is a

diagonal tensor given by: 1r

0 0

0

0

0

2r

0

0

3r

(2.5)

and the electric flux density D( x, y, z , t ) can be written as: Dx Dy Dz

1r 0

0 0

0

0

Ex

2r

0

0

3r

Ey Ez

In a uniaxial medium, like lithium niobate,

1r

(2.6)

no2 , and

2r

3r

nex2 , where no is

the ordinary refractive index, nex is the extra ordinary refractive index and the crystal z-axis is the optic axis. Waveguidance in dielectric waveguides is made possible due to the difference in refractive index of core (guiding medium) and the cladding (substrate or cover). Most practical waveguides are weakly guiding, i.e., the refractive index of cladding (substrate) is not very different from that of core. In general for an inhomogeneous medium, using the Maxwell’s equation one can easily derive the wave equation describing the propagation of an electromagnetic wave: 2

1 n2

E

2

n2 E

0

D

(2.7)

t2

For an infinitely homogenous medium, the second term on the LHS is zero everywhere and each Cartesian component, (Ex, Ey or Ez.), obeys the scalar wave equation: 2

where

n2 c2

2

(2.8)

t2

x, y, z , t may represent Ex, Ey or Ez and c 1 /

0

0

is the speed of light in

free space. The solutions of above equation are in the form of uniform plane waves with only transverse electric and magnetic components. It should be noted that in an anisotropic medium different polarizations see a different refractive indices, e.g. in

19

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background lithium niobate, the x or y polarized waves propagating along the z-direction see the ordinary refractive index no , while a z-polarized wave propagating along the x or y direction will see the extra ordinary index, nex . For medium with gradually varying dielectric properties, i.e., when the refractive index varies sufficiently slowly so that it can be assumed constant within distances of the order of wavelength, the second term on the LHS of Eq. (2.7) is negligible in comparison. For such a medium with weak homogeneity the electromagnetic waves are nearly transverse in nature with transverse component of the electric field satisfying the scalar wave equation (2.8). In a typical waveguide with propagation along z-direction, the refractive index is independent of z and varies only with the transverse coordinates, i.e.,: n 2 x, y, z

n 2 x, y , and assuming time harmonic fields

the solution for the transverse component can be written as: x, y , z , t where k0

x, y e i

is the propagation constant, neff

2

0

,

0

t

(2.9)

z

k 0 is called the effective index, where

being the free space wavelength,

x, y determines the transverse

field distribution (also called modal field) of the guided mode and satisfies the following differential equation, also known as Helmholtz equation, with appropriate boundary conditions: 2 t

with

2 t

x2

y2

( x, y ) k 02 n 2

2 neff

( x, y )

0

(2.10)

. The solution of this Helmholtz equation defines the modal

fields in a guided structure. The Eq. (2.10) is in fact an eigenvalue equation with 2 x, y being the eigenfunction and neff or

2 t

2

being the eigenvalue of the operator

k 02 . Only certain discrete solutions, known as the guided modes and a continuum

of solutions referred as radiation modes are allowed. A mode represents a field configuration which propagates along the waveguide without any change in polarization or in the field distribution except for change in phase. Guided modes are confined within the guiding medium, while radiation modes are not bound to the

20

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background guiding region. Depending upon the refractive index profile, dimensions of the waveguide and wavelength, a waveguide may support a number of guided modes, each with a different propagation constant

and spatial mode

x, y .

2.3 Integrated optical waveguides

(a)

(b)

(c) (d) Figure 2.1: Different type of waveguides (a) Step index planar waveguide, (b) Buried channel waveguide (c) Diffused channel waveguide, (d) Ridge waveguide.

As mentioned in Chapter 1, there are different types of integrated optical waveguides viz: slab or planar, ridge and diffused channel waveguide. The planar waveguides in which refractive index has only y dependence, n 2

n 2 y confines light in only one

dimension the y direction (Figure 2.1 a). The channel and ridge waveguides on the other hand have a refractive index varying with both transverse coordinates, i.e., n2

n 2 x, y and thus confines light in two dimensions (Figure 2.1 b, c and d). In this

section, for completeness we first present a brief introduction to planar waveguides and then move on to the diffused channel waveguides.

21


2.3.1

Planar waveguides

Planar waveguides confine light in only one direction. The refractive index profile for planar waveguide is given by: n2

(2.11)

n 2 ( y)

To simplify, calculations, we assume that the y-axis is normal to the plane of waveguide, and the waveguide is semi-infinite along the x-direction. For a planar waveguide, the Maxwell equations (2.3) and (2.4) lead to two independent set of field solutions: the TE modes and TM modes.

Figure 2.2: A simple step index planar waveguide structure.

With field components Ex, Hy and Hz related by the following set of equation respectively: i Ex

i

Ex / y i Hy

Hz / y

0

i

Hy 0

i

Hz 0

(2.12)

n2 y Ex

And, Ey, Ez and Hx related by:

22


i Hx

i

Hx / y i Ey

0

i

n2 y Ey 0

Ez / y

(2.13)

n2 y Ez i

0

Hz

The transverse component of the field vectors satisfy the Helmholtz equation (2.10) which reduces to an ordinary differential equation. For TE waves the transverse component Ex satisfies: .

d 2 Ex dy 2

k 02 n 2 y

2 neff Ex

(2.14)

0

This equation can always be solved as a boundary value problem to obtain the TE modes,

such

solutions

are

part

of

almost

all

textbooks

on

photonics

[3,5,26,47,48,68,108,122,128]. 2.3.2 Diffused channel waveguides

In channel waveguides, where there is confinement in both transverse dimensions hybrid modes are supported, in which all the three components of electric field vector and magnetic field vector are non-zero. In weakly guiding waveguides [122], the hybrid modes are essentially almost TEM modes polarized along x and y directions. The guided modes supported by the channel waveguides, therefore are also classified depending upon whether the main component of the electric field lies in the x or y direction. The mode with main electric field component in the x- direction, Ex, is x y called the E pq mode, which resembles TE modes in the slab waveguide. E pq mode is

polarized predominantly in the y direction and resembles a TM mode in the slab waveguide.

Figure 2.3:Typical refractive index profile of a titanium diffused LiNbO3 waveguide.

23

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background In Chapter 4 we present a simplified analysis of gain and ASE evolution, for an Erdoped waveguide amplifier in which the erbium ions are hosted in a diffused channel waveguide fabricated by titanium indiffusion into z-cut LiNbO3 [39] The typical refractive index profile n( x, y ) of the channel waveguide obtained by titanium diffusion into LiNbO3 (Figure 2.3) can be represented as: n 2 x, y

n s2

2n s n f x w g y h

nc2

y

0

y

0

(2.15)

where n s is the substrate refractive index, nc is the cover refractive index (air in this n is the maximum index change from the substrate index. The profile function

case),

exp

f x w

x 2 / w 2 and g y h

exp

y 2 / h 2 model the index variation along

the width and depth respectively, and w and h are the (1/e) profile depth and width respectively. The transverse component of electric field satisfies the Helmholtz equation (2.10): H ( x, y )

with H given by H

2

2

x2

y2

2

(2.16)

( x, y )

k 02 n 2 x, y and the modal field are normalized as

2

x, y dx dy 1 , n 2 ( x, y ) will correspond to the ordinary index for the TE mode and extraordinary index for TM modes. In our analysis we consider TE modes and

h

use

the

following

profile

parameters: n s

2.297 ,

w

7.5 μm

and

6.5 μm [39,60]. The corresponding refractive index profile is shown in Figure 2.4.

Eq. (2.16) cannot be solved analytically for a non-separable refractive index profile of the type given by Eq. (2.15). While numerically intensive methods like beam propagation method, finite difference/element method [24,104] etc are more accurate, they do not provide us with analytical expressions for the modal fields. On the other hand, a semi-analytical procedure, developed over the last decade, based on the variational principle leads to analytical form for the modal fields. This facilitates the analysis of waveguide devices like amplifiers. We briefly discuss the variational analysis in the next section.

24


(a)

(b)

Figure 2.4: Refractive index profile along (a) x-direction at y=0 and (b) y-direction at x=0 for the channel waveguide with ns=2.297, n=0.0048, w=7.5 m, and h=6.5 m.

2.3.2.1 Variational principle From Eq. (2.16), multiplying by

*

( x, y ) on both sides and integrating over the entire

cross section of waveguide, a stationary expression for the effective index is obtained: 2 neff

1 k 02

*

x, y

2 t

x, y dx dy

2

n 2 x, y

An approximation is chosen for the modal field

t

x, y dx dy

(2.17)

( x, y; a1 , a 2 ,....a n ) , generally

known as the trial field, with a1 , a 2 ,....a n as n adjustable variational parameters. This trial field is substituted in the RHS of Eq. (2.17), which is then maximized with respect to the variational parameters. The maximum value gives an estimate of the effective index of the lowest mode and the corresponding

t

( x, y; a1 , a 2 ,....a n ) with

the corresponding parameters an approximation for the modal field. Accuracy of the variational method depends on the closeness of the assumed trial field to the exact modal field of the guiding structure. It has been shown by earlier authors that one can assume a separable variational trial field [60,61,129,112,113,114] and the following three parameter variational field, with a Gaussian variation along the width and an evanescent Hermite-Gaussian variation along the depth gives satisfactory results for channel waveguides: t

x, y

X xY y

(2.18)

with

25


1

X x

wd x

exp

a1

x2 w2

2

(2.19)

and 1

Y y

hd y

1 a2

y exp h

a 32

y2 h2

y

(2.20)

1

y exp a 2 h hd y

where d x

a1 1

2 , and d y

y

2a 2

1

2a 3

0

1

0

a 2 2a32

2

1

a 22 8a 33

1

2,

and a1 , a 2 and a3 are the three variational parameter. Substituting the trial field in the stationary expression of Eq. (2.17), closed form expression for the normalized effective index is obtained: b

2 neff

n s2

r s d xd y

2n s n

(2.21)

with r and s as: r with s x

a1

2 , and s y

Vy

k 0 h 2n s n ,

ry

2

1 2a32

rx ry and s

p

a 22

n s2

pd x 2a 2

1 Vy

2 3a 2

2

2

4a 3

nc2 2n s n ,

4a 2 1 2a32

12

1

sydx

Vx 2

2

(2.22)

sx d y

4a 2 a3 8a3 , V x

1 2a12

rx 4 1 2a32

3

2

k 0 w 2n s n ,

and

,

.

The three variational parameters are obtained by maximizing the expression for b. For the channel waveguide in discussion at =1.532 m the three Variational parameters obtained by maximizing b are a1 = 1.188, a2 = 47.46, and a3 = 1.099. And the corresponding effective index value is 2.29807.

26


(a)

(b)

Figure 2.5: Modal field profile for a channel waveguide, with ns=2.297, n=0.0048, w=7.5 m, h=6.5 m for a signal at =1.532 m using three Variational parameters. (a) Along xdirection at y = 0 (b) Along y-direction at x=0

Figure 2.5 shows the modal field variation along x and y direction respectively. A close look at the Y(y) modal field shows that it can be approximated by two Gaussians with different width around the field maxima y0. In Chapter 4, we make use of this to further approximate the modal fields in the analysis of erbium doped waveguide amplifier.

2.4 Optical fibers The optical fiber is a cylindrical dielectric waveguide, made of a low loss material. The most widely used fiber has a central guiding core, of GeO2 doped silica glass, surrounded by an outer cladding of slightly lower refractive index silica glass and a polymer jacket for protection. As per the EIA 598 standards the jacket is color coded to easily identify the fiber type. The fibers are classified on the basis of either the refractive index profile or the number of propagating modes supported. The index variation in the core can be either uniform with an abrupt change at core-cladding interface, the so called step index fiber. The index can also vary as a function of the radial distance from the core center and these fibers are called graded index fibers.

27


Figure 2.6: Different types of optical fiber [3]

Like planar and channel waveguides an optical fiber also supports a set of discrete guided modes. Depending upon the number of propagating modes supported, the fibers are classified as single mode fiber (SMF) and multimode fiber (MMF). SMF supports only a single electromagnetic mode of propagation, while MMF can sustain many hundreds of modes. MMFs have large dimensions can be well understood by the simpler ray optics. SMFs on the other hand have dimensions close to wavelength a co

5 μm and hence to understand propagation through them it is necessary to

consider the wave optics. Most in-line fiber components like erbium doped fibers, fiber gratings, couplers, multiplexers etc. are based on single mode step index fibers. The refractive index profile n(r ) of a step index fiber can be written as: nr

n1

0

n2

a co

n amb

r

r

a co r

a cl

(2.23)

a cl

where n1 is core refractive index, n 2 is cladding refractive index, namb is refractive index of the medium surrounding the fiber, a co is core radius and a cl is the cladding radius. Figure 2.7 shows the cross-sectional view of the fiber, showing the three regions.

28


Figure 2.7: Cross-sectional view of a single mode optical fiber

Conventionally the refractive index profile is defined in terms of the relative refractive index difference

r

n(r ) n 2 n1 expressed in percent. Figure 2.8 shows the

relative refractive index profile for the SMF fiber, whose parameters are listed in Table 2-1.

Figure 2.8: Relative refractive index difference for the SMF fiber (Table 2-1)

Most of the fibers in practice are weakly guiding, i.e. n1

n 2 or

1 , and the

modes are assumed to be nearly transverse and can have an arbitrary state of polarization. The two independent sets of modes can be assumed to be x-polarized and

29

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background y-polarized and they have the same propagation constants. These linearly polarized

modes are usually denoted as the LPlm modes. Table 2-1: Fiber Parameters Parameter Core index Clad index Core radius

Symbol n1 n2 aco acl

Value 1.458 1.450 2.5 m

Clad radius

62.5 m

In the weakly guiding approximation, the transverse component of the electric field (Ex or Ey) of these LPlm modes satisfies the Helmholtz equation (2.10). Since the fiber has a cylindrical symmetry it is advantageous to work in cylindrical co-ordinates, and the transverse component of the electric field can be written as: r, e i

r , , z, t

with

r,

t

(2.24)

z

satisfying the equation:

1 r r r

r, r

1 r2

2

r, 2

k 02 n 2 (r )

2

r,

(2.25)

0

Figure 2.9: Three layer refractive index profile for calculation of cladding modes.

The

transverse

2

r,

2

field

rdrd

profile

r,

is

assumed

to

be

normalized

as

1 . Since the refractive index depends only on the radial

0

30

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background R r e il , where l is an

r,

coordinate, r, the field solution can be written as

integer defining the azimuthal variations; l=0 for the azimuthally symmetric modes. R(r) satisfies the equation: d 2R r dr 2

1 dR r r dr

k 02 n 2 (r )

l2 Rr r2

2

0

(2.26)

with R(r) and its derivative continuous at the core-cladding and cladding-ambient interfaces. Most text books [48,48,66,121,128,130] solve the above equation for the core guided modes considering only the core-cladding interface and assuming an infinite cladding. Following this analysis, early authors used a similar analysis for cladding modes by assuming these to be the guided modes of a fiber with a large core of index n2 and infinite cladding of index namb, i.e. they neglected the presence of the small core. It was later observed that considering only cladding-ambient interface for cladding modes results in erroneous values for effective indices and modal fields [117]. This can have serious effects on the designing of devices employing cladding modes like LPGs. Thus in the analysis present below we consider the complete three layer geometry [89,117] (Figure 2.9). In SMF only the fundamental mode LP01 is the core guided mode, i.e., l=0 and can couple only to l=0 cladding modes due to the orthogonality constraint. Hence, we confine our modal analysis to l=0 modes for which Eq. (2.26) reduces to: d 2R r dr 2

1 dR r r dr

k 02 n 2 (r )

2

Rr

0

(2.27)

For our illustrative calculations in this section we have used the fiber parameters listed in Table 2-1. It is the fiber we investigate later in Chapter 3 as the erbium doped fiber amplifier. 2.4.1 Guided core mode

For n12

2 neff

n 22 , one obtains solutions that are oscillatory in the core and decay in

the cladding and ambient, [1,89,101,111,117]:

31


Aco J 0 Rr

where U 2

k 02 n12

Bco I 0

2

Ur a co

r

0

Wr a cl

C co K 0

Dco K 0

W1 r a cl

a co2 , W 2

Wr a cl

a co r

2

a co r

(2.28)

a cl

a cl

k 02 n22 a cl2 , and W12

2

2 k 02 namb a cl2 . J 0

and I 0 are the Bessel and modified Bessel function of the first kind respectively, while K 0 is the modified Bessel function of the second kind. Using the continuity of R r and its derivative at interfaces ( r

a co and r

a cl ) we obtain the eigenvalue

equations for determining the effective indices: ~ J U ~ J U

I 1 Wac K 1 W K 1 Wac I 1 W

where a c

~ ~ K Wac K W1 ~ ~ I Wa c K W1

~ I W ~ KW

(2.29)

Z0 x , Z represents the different Bessel functions. xZ 1 x

~ aco / a cl , and Z x

Figure 2.10: Fundamental Core Modal field LP01 for the fiber with parameters listed in Table 2-1 .

The constants Bco, Cco and Dco can also be expressed in terms of Aco, which is given by 2

the normalization condition 2

2

R r rdrd

1 . Figure 2.10 shows the field profile

0

32

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background of the core mode for the fiber with parameters given in Table 2-1 at wavelength =1.53 m. The corresponding effective index is 1.45205. 2.4.2 Cladding modes

For n 22

2 neff

2 n amb , one obtains solutions which are oscillating in the core and

cladding but decay in the ambient: Ur a co Ur Ur Bcl J 0 1 C cl Y0 1 a cl a cl Wr Dcl K 0 1 a cl Acl J 0

Rr

where U 12

k 02 n22

2

0

r

a co r

aco r

(2.30)

a cl

a cl

a cl2 and. J 0 is the Bessel function of the first kind, Y0 and K 0

are the Bessel and modified Bessel function of the second kind respectively.

Figure 2.11: Modal fields of the LP0m cladding modes (m>1)for the single mode fiber with parameters listed in Table 2-1.

The continuity of R r and its derivative at interfaces ( r

a co and r

acl ) leads to

the eigenvalue equations for determining the effective indices:

33

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background ~ Y U 1ac ~ J U 1ac

J 1 U 1 a c Y1 U 1 Y1 U 1 a c J 1 U 1

~ J U ~ J U

~ K W1 ~ K W1

~ J U1 ~ Y U1

(2.31)

Again Bcl, Ccl and Dcl can be expressed in terms of Acl, which is obtained by the normalization condition. Figure 2.11 shows the modal field profile of different cladding modes for the fiber with the parameters given in Table 2-1, at =1.53 m . The corresponding effective indices are tabulated in the last column of Table 2-2.

2.4.3

Comment on considering only the cladding-ambient interface for cladding modes

Figure 2.12: (a) Two layer geometry for calculation of cladding modes. (b) Core index as perturbation on the two layer geometry.

As mentioned earlier, some early authors used the two layer geometry in refractive index given by: n( r )

n2

0 r

n amb

r

a cl

(2.32)

a cl

for obtaining the propagation characteristics of cladding modes. The solution of this geometry is well known and is given by [48,48,66,130]:

R (r )

with U 12

(k 02 n22

2

Ur A J0 1 J 0 (U 1 ) acl

r

Wr A K0 1 K 0 (W1 ) acl

r

a cl

(2.33) a cl

)acl2 and the eigenvalue equation is given by:

34


J 1 (U 1 ) J 0 (U 1 )

U1 Since

W1

K1 (W1 ) K 0 (W1 )

(2.34)

1 and aco1) cladding modes of the actual fiber.

35

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background Table 2-2: Effective indices for fiber with refractive index profile of Eq. (2.32) and first order perturbation correction compared with the fiber with refractive index profile Eq. (2.23) at = 1.53 m

LPlm mode

Effective Index (2 Layer) neff

Fractional power in core

LP01 LP02 LP03 LP04 LP05 LP06 LP07 LP08 LP09

1.449970 1.449842 1.449611 1.449277 1.448841 1.448302 1.447660 1.446915 1.446067

0.0058793 0.0135523 0.0209219 0.0278120 0.0340747 0.0395870 0.0442547 0.0480150 0.0508386

P frac

First order correction 2 neff

0.000137 0.000315 0.000487 0.000647 0.000793 0.000920 0.001029 0.001117 0.001183

Corrected Effective Index

Effective Index (3 Layer) neff

1.45002 1.44995 1.44978 1.44950 1.44911 1.44862 1.44802 1.44730 1.44648

1.449948 1.449775 1.449486 1.449083 1.448568 1.447943 1.447210 1.446368

It is important to reiterate that, for cladding modes the propagation constants and hence the modal fields obtained using the two layer geometry, significantly differ from that obtained using the complete three layer geometry.

2.5 Long period gratings A long period grating has periodic modulation of refractive index in the core of the fiber along the direction of propagation, usually represented by a sinusoidal zdependent variation

n 2 ( z)

n 2 sin Kz [48,117,130,134], where K

2 /

,

being the grating period. As mentioned in Chapter 1, a LPG couples power between co-propagating core and cladding modes. Coupling between modes occurs when the phase matching condition is satisfied [18,19,45,48,66,115,117,134], i.e.,: co

where

co

and

(m) cl

(m) cl

2

(2.38)

are the propagation constants of the fundamental core mode

(LP01) and the phase matched cladding mode (LP0m). This leads to the resonant coupling wavelength given by:

36

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background ( m) res

co neff

cl , m neff

(2.39)

cl , m co where neff and neff are the effective indices of the fundamental core mode (LP01) and

the phase matched cladding mode (LP0m). The grating periods corresponding to coupling to different cladding modes at different wavelengths can be determined from “phase matching curves”. Figure 2.13 shows the phase matching curves for the Corning SMF28 with n1

1.46 , a relative refractive index difference of 0.36 %, and a

4.1μm , cladding radius of a cl

core radius of a co

62.5 μm [32].

Figure 2.13: Phase matching curve for coupling to different LP0m cladding modes of the Corning SMF28 fiber (m is marked on the curve).

A grating of period

~ 320 μm will result in four resonance bands corresponding to

cladding modes LP05, LP06, LP07 and LP08 with resonance wavelengths of ( 5) res

1.09 μm ,

( 6) res

1.15 μm ,

(7) res

1.257μm , and

(8) res

1.525μm and a typical

transmission spectrum for the grating period 320 m is shown in Figure 2.14.

37


Figure 2.14: Transmission spectrum of SMF28 for a grating of 320 m with namb=1.0

2.5.1

Coupled mode analysis

The most popular method for studying power exchange between core and cladding modes of a LPG is the coupled mode theory [28,29,48,66,117,130]. In most cases, the individual resonances are sufficiently narrow and spectrally well-separated, thus at a time, coupling between the core mode and a single cladding mode well describes the transmission spectrum in a specified band of wavelengths. For such cases, a simple two-mode coupled mode theory is employed. For cases, where the core modecladding mode resonances overlap one another or a large number of resonances fall in the specified spectral band, all the cladding modes which are resonant in the band must be included simultaneously in the coupled mode theory [95]. We summarize the results of the simple two-mode coupled mode theory in this Section. If

co

mode,

r and m cl

co

r and

are the normalized modal field and propagation constant of the core m cl

are the normalized modal field and propagation constant of the

phase matched cladding mode of the fiber, then the total field at any value of z can be described as:

38


A( z )

re

co

i

co

z

m cl

B( z)

i

re

m cl

z

(2.40)

Following the analysis in various text books [48, ,121,122], in the slowly varying approximation coupled differential equations which describe the change of amplitudes with propagation distance are obtained as: dA z dz

12

dB z dz

where

12

21

B z ei

K z

i

K z

Aze

12

21

a

k 02 2 co

co

n

2

m cl

B z ei

r dr ,

21

0

k 02 2 clm

(2.41)

z

Aze

(2.42)

i z

a m cl

n2

co

r dr and Γ known as

0

detuning or phase mismatch factor is defined as 2

co neff

2

cl , ( m ) neff

(2.43)

Differentiating equation (2.41) with respect to z and using (2.42), we obtain a second order differential equation for A: d2A z dz 2 where

12 21

i

dA z dz

2

Az

(2.44)

0

is the coupling coefficient.

Employing the boundary conditions that A(z=0)=A0 and B(z=0)=B0, we obtain the following analytical solutions for A(z) and B(z) : Az

e

Bz

e 2

where

4

i

z 2

i z 2

2

A0 cos( z ) i A0 i

2

sin( z )

sin( z )

sin( z )

B0 i

(2.45) B0 cos( z ) i

2

sin( z )

. If we assume that power is initially launched only in the core

mode, i.e., A (z=0)=1 and B(z=0)=0 and the above equations reduce to a following simplified form:

39


Az

e Bz

i

z 2

cos( z ) i i

e

z 2

i

sin( z )

2

(2.46)

sin( z )

The power in the core mode and the cladding mode at any z can now be expressed as: 2

Pco z

Az A z

1

2

sin 2

z (2.47)

2

Pcl z

BzB z

2

sin 2

z

At phase matching wavelength the detuning factor

0 and

. Hence, the

power in the core mode and cladding mode can be expressed as: 1 sin 2

Pco z Pcl z

sin

2

z

(2.48)

z

Power is continuously exchanged between the core mode and the phase matched cladding mode and complete power transfer occurs at coupling length l c is given by: lc

(2.49)

2

In Section 2.4.3 we briefly commented on the use of two layer geometry for characterizing the cladding modes and tabulated the error introduced by this simplified picture in calculations for the fiber with parameter given in Table 2-1. To further emphasize the necessity of using complete all the three layers for the characterizing the cladding modes we also investigate the transmission spectrum obtained by the two calculations. We consider a LPG written in the fiber with parameters given in Table 2-1. With index modulation 297 μm . A two layer calculation predicts coupling while the correct

res

res

n2

2 10

4

and

1.53 μm for the LP01-LP09

1.438 μm . The transmission spectrum for a grating of

length 4.6 cm, calculated by the two layer geometry shows a complete resonance at 1.53 m, while the correct spectrum shows an incomplete resonance at 1.438 m.

40


Figure 2.15: Treansmission spectrum of 4.6 cm long LPG with grating period of 298 m. Dashed curves are obtained using two layer geometry, the solid curves are obtained considering the complete three layer geometry for the calculation of propagation constant and modal fields for the cladding modes.

2.5.2 Applications

The band reject property of LPGs make them suitable for in line filters. They have been extensively studied for use as in-line gain flattening filters (GFF) for optical amplifiers [119,134]. Recently, Singh et al. [119] proposed that a GFF made by writing an appropriate LPG in the EDF itself not only provides gain flattening, but also increases the gain in 1.54-1.56 m band. In Chapter 3, we briefly review this and further analyze the effect of the LPG on the ASE noise characteristics of the EDF. Our results show that EDF with LPG written in it also suppresses the ASE noise along with gain flattening. In the recent years LPGs have also been used extensively in strain, temperature and refractive index sensing applications. Since the cladding mode effective index depends on the ambient index, any change in the ambient index will cause the resonance wavelength to change. This property of LPGs has been harnessed in various biological and chemical sensing applications [8,27,70,137]. In this thesis we present an alternative approach (Chapter 8), in which the LPG is interrogated by a single

41

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background wavelength source and instead of measuring the shift in resonance wavelengths, the change in the power retained in the core is measured.

2.6 Optical amplification Before proceeding to analyze various optical amplifiers in the following Chapters it is important to understand the basic process of optical amplification. An atom or molecule may emit (create) or absorb (annihilate) a photon by undergoing downward or upward transitions between its energy levels, conserving energy in the process. In this section we briefly discuss the various processes by which photons interact with an atomic system and consequent optical amplification [135]. 2.6.1

Einstein coefficients

Let N1 and N2 represent the number of atoms per unit volume in energy states 1 and 2, respectively; the states corresponds to energies W1 and W2 respectively. In 1917, Einstein identified three radiative processes that affect the concentration of atoms in the two energy levels, viz: absorption, stimulated emission and spontaneous emission (Figure 2.16).

Figure 2.16: Three radiative processes affecting the concenteration of atoms in the two energy states.

2.6.1.1 Spontaneous emission An atom in the upper energy state W2 may decay spontaneously to the lower energy state W1 and release its energy in the form of a photon of frequency f 0

W2 W1 h

42

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background (Figure 2.16 c). The process is called spontaneous emission. The emitted photon has a random phase and direction. The spontaneous emission rate (number of spontaneous transitions per unit time per unit volume) from state 2 to state 1 is given by: R sp

(2.50)

A21 N 2

where A21 is the Einstein’s A coefficient for the transition from state 2 to state 1.

dN 2 dt

Hence, only due to spontaneous emission

N2 t

N2 0 e

t/

sp

where

A21 N 2

which simplifies to

1 A21 is the spontaneous lifetime of state 2.

sp

2.6.1.2 Absorption When a photon of energy W

hf

W2 W1 impinges on the atomic system, an atom

in the lower energy level W1 can absorb the energy and make an upward transition to upper energy level W2 (Figure 2.16 a). The rate at which this process takes place must depend on the density of absorbing atoms (N1) as well as the energy density of the incident photons,

Ph

f , at the center frequency f, and is given by: a Rstim

B12 N 1

Ph

f

(2.51)

where B12 is the Einstein B coefficient for transition from energy state 1 to state 2. If

NPh denotes the number of photons per unit volume at frequency f, then Ph

f

N Ph hf

2.6.1.3 Stimulated emission An atom in the upper energy level W2, can be induced by the impinging photons, to make a downward transition to lower energy level W1 (Figure 2.16 b). This process is called stimulated or induced emission, since it is induced by the presence of photons. The emitted photon is at the same frequency, same phase, and same polarization and propagates in the same direction as the photon that induced the atom to undergo this type of transition. The rate of stimulated emission will depend on the density of atoms in the upper energy level and the energy density of the incident photons.

43

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background e Rstim

B21 N 2

Ph

(2.52)

f

where B21 is the Einstein B coefficient for transition from energy state 2 to state 1.

2.6.1.4 Relationship between Einstein coefficients Let us consider a cavity of volume V

Adz . Under thermal equilibrium, the energy

density of photons at frequency f, in the cavity is given by Planck’s blackbody radiation expression:

Ph

8 n 2 ng f

f

c

2

hf

3

hf

e

k BT

(2.53) 1

where n is the refractive index of the medium, ng is the group index defined as:

ng

dn d

n

(2.54)

Under thermal equilibrium, the rate of upward transitions, defined by Eq (2.51), should be equal to the downward transitions, defined by Eqs. (2.50) and (2.52). Thus we may write:

A21 N 2 or

B21 N 2 Ph

f

Ph

f

B12 N 1

Ph

f

A21 N B12 1 B21 N2

(2.55)

The atoms in the cavity at thermal equilibrium follow Boltzmann distribution i.e.

N2 N1

g2 e g1

hf k BT

, where g 2 (1) is the degeneracy of the energy2 state W2 (1) . Comparing

the above relation with the Eq. (2.53) for the energy density of incident radiation, gives:

2

For a simple atom the quantity g2(1) is related to the total angular momentum number J2(1) by g2(1)=2J2(1)+1

44

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background g 2 B21

and where

A21 B21

g1 B12

8 n 2 n g hf 3 c3

(2.56)

ng 1 8 n 2 hf c 2

c / f is the wavelength of the electromagnetic wave. The importance of

these relations lies in the fact that it if one of the coefficients is known, the others can be evaluated. It is very important to realize that Einstein coefficients are characteristics of an atom. The atom has no way to know whether it is in a thermodynamic equilibrium environment of a cavity or in the presence of an intense optical field (a laser) generated by other atoms. It responds to an electromagnetic radiation as determined by Eqs. (2.50-2.52). Hence these relations are always valid. 2.6.2 Optical gain

We now describe the process of amplification of radiation by its interaction with the atom. Photon flux

f is defined as the number of photons crossing per unit area per

f

N Ph n g / c . Eq. (2.51) and (2.52) describe the number of upward

unit time, i.e.,

and downward transitions respectively, in response to the photon flux,

f . The net

increase in the photon flux per unit length along the z-direction is determined by the difference between stimulated emission and absorption rates: d

f dz

e Rstim

a Rstim

N2

g2 N 1 B21 g1

f

From this relation we can see that there is net optical gain only if N 2

(2.57) g2 N 1 , i.e., g1

population inversion is necessary for optical amplification. Population inversion does not occur naturally, some external means is needed to create this state of population inversion. We can achieve population inversion either by pumping by another optical source (as in EDFA and EDWA), or by electrical injection (as in SOAs). The optical gain of the medium is the fractional increase in photon flux per unit length is given by [147]:

45

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background 1 d f f dz

g f

(2.58)

Thus, the optical gain at signal frequency f is obtained as: ng

g f

2.6.3

c

hf B21 N 2

g2 N1 g1

(2.59)

Spectral broadening

The above analysis assumed sharp energy levels involved in the transition. However, from uncertainty principle we know that there is no such thing as precise energy state, which implies it is not possible to have all the transitions at exactly f such that hf

W2 W1 . Instead the real systems emit a narrow band of frequencies whose

width

f is much less than the center frequency f.

In the real system, we have a distribution of energies around levels W1 and W2. There can be transitions from the manifold W2 to manifold W1 at various frequencies centered around f. The broadening of each energy level can be different and may or may not be symmetrical. Thus, there exists a distribution of photon frequencies that can be emitted spontaneously. This relative distribution is called the line shape function L f . L f df gives the probability that a given transition between the two energy levels will result in the emission (or absorption) of a photon whose frequency lies between f and f +df. Thus

f L f df determines the energy density of photons

causing the transitions between f and f+df. Here we should also mention that lineshape function is normalized, i.e.

L f df

1.

In this thesis, we are mainly dealing with monochromatic sources/signals, i.e., only one frequency. For such signals the spectral width of L f , thus

f can be approximated by a

f is very small compared to

function:

46


f'

(2.60)

f' f

f

Thus the net increase in photon flux per unit length around center frequency f in the interval df will be given by integrating over the all the range of transitions covered in the frequency interval df [147]: e Rstim

ng

a Rstim

c

B21 N 2

g2 N1 g1

(2.61)

f L f df

to represent the optical gain including the spectral

To avoid confusion, we use broadening: f

1 d f f dz

L f

ng

g f df

c

hfB21 L f

N2

g2 N1 g1

(2.62)

The quantity in curly braces has the dimension of area (m2) and is referred to as the stimulated emission cross section

e

e

f . ng

f

In similar fashion we also define absorption cross section

a

f

(2.63)

hfB21 L f

c

g2 g1

e

a

f as: (2.64)

f

In terms of absorption and emission cross-section the gain coefficient can be written as:

f

e

f N2

a

f N1

(2.65)

Lasers and optical amplifiers are the two important optical devices based on the amplification of light by stimulated emission. While lasers generate optical signals, optical amplifiers amplify an optical signal in the optical domain. It is more common to express change in intensity or power as light propagates through the gain medium in terms of gain coefficient. Let If be the intensity of irradiated light of frequency f (considering monochromatic source), it depends on both the photon flux

f

and photon energy hf, I f

f hf . We can determine the increase

(decrease) in intensity as it passes through the gain (loss) medium using Eq. (2.62) as:

47


dI f dz

2.6.4

f If

(2.66)

Types of optical amplifiers

Optical amplifiers amplify an optical signal without any conversion of light into electrical signal. Though, the first generation of optical amplifiers was developed in 1960 using a Neodymium doped fiber [74,120], a major breakthrough in optical amplification came about twenty years later. In 1985, at Southampton University David Payne and his colleagues reported a high gain fiber amplifier using erbium doped silica fiber [83,84]. It provided new life to the optical fiber low loss transmission window centered at 1.55 m and motivated research into technologies like wavelength division multiplexing (WDM), that allow high bit rate transmission over long distances. The success of EDFA led to the development of many alternative amplifier technologies. While, EDFA remains the choice for long haul communication system, due to small size and ease of integration, deployment of semiconductor optical amplifiers in optical networks is increasing. In this section, we briefly describe some important optical amplifiers.

2.6.4.1 Erbium doped fiber amplifiers EDFA typically consists of an erbium doped fiber gain medium, a pump laser (980 nm or 1480 nm) to excite the gain medium and couplers to couple the pump and signal powers to the erbium doped fiber. The pump can be coupled at either end of the fiber resulting in either co-propagating or counter-propagating pump. The erbium ions present in the fiber core are excited to a high energy state by the pump. When photons from the input signal in the range of about 1.53-1.60 m strike the excited erbium ions, some of erbium ions return to a lower energy state transferring the energy to the optical signal. This results in amplification of the optical signal via stimulated emission. The EDFA has many desirable features the most important being that it amplifies in the 1.55 m wavelength window where fiber loss is minimum. It provides a gain bandwidth of 40 nm, which implies that it can amplify many channels simultaneously in the C-band, making it a key device in WDM/DWDM technology.

48

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background EDFA has two major disadvantages, the first being its non-uniform gain spectrum. The non-uniform amplification of WDM signals can cause signal distortion and poor signal-to-noise ratio performance. This led to the development of various types of gain flattening methods (GFF) [15,34,51,71,79,107,128,143]. The second disadvantage of EDFA is the amplification of spontaneous emission (ASE). ASE emerges from both ends of the doped fiber and must be dealt with. ASE in the same direction as the signal is a major source of cumulative noise (reducing signal to noise ratio), while backward propagating ASEs can harm source lasers if not filtered out. The presence of ASE also reduces the available optical gain for the signal field. Though undesirable, spontaneous emission is an inherent process of optical amplification and hence, ASE cannot be eliminated completely. Many methods have been explored to mitigate ASE, like using filters or isolators [15,34]. Besides erbium, other rare earth elements have also been used to provide gain in different wavelength regions [69]: Praseodymium (around 1.3 m), Neodymium (around 1 m) [78] and Thulium in the S-band (1.45-1.50 m) [109]. The design and operation of such amplifiers is, in principle, quite similar to that of the EDFA, although they require different pump wavelengths and typically much higher pump powers.

2.6.4.2 Erbium doped waveguide amplifiers The need for small size and great functionality optical components motivated interest in the erbium doped waveguide amplifiers (EDWA). EDWAs offer several advantages over EDFAs: They are of low cost and small size and allow integration of passive components like couplers etc on the same chip. In order to achieve same gain values in short length (few cm), the EDWAs require high erbium ion concentration. The high erbium ions concentration has some limitations, as the erbium ion concentration increases the ions come closer this increases the probability of the formation of clusters. When excited, the erbium ions in the clusters start exchanging energy this reduces their efficiency. A lot of attention is focused on lithium niobate (LiNbO3) waveguides doped with erbium [52,63]. These waveguides can be obtained from a variety of procedures, the

49

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background most common being titanium diffusion. These waveguides are pumped by 1484 nm radiation and provide amplification around 1.53 m.

2.6.4.3 Semiconductor optical amplifiers Semiconductor optical amplifier (SOA) is essentially a semiconductor laser with no reflecting facets [31,147]. In EDFA, EDWG and Raman amplifiers the pump is provided optically, but in a SOA an electrical current is used to excite the gain medium (also called active region or active layer). A typical SOA has a p-doped semiconductor layer, an n-doped semiconductor layer and a gain region (also called active layer) sandwiched between the two. The p and n-doped semiconductor layers have higher band gap and lower index of refraction than the active region. These layers behave as a cladding and tend to confine the optical mode within active region. The ends of the waveguide are usually treated to avoid optical feedback. An antireflective coating is often applied to the facets to decrease the reflection. When an optical signal is injected from the input facet, the light is amplified by the gain of the active layer. The gain spectrum of an SOA strongly depends on the bandgap of the semiconductor constituting the active region. Due to their compact size, reduced power consumption and reduced cost of fabrication, semiconductor optical amplifiers have begun to replace EDFAs in Metro access and local access networks. The disadvantages of SOAs include much narrower wavelength bands, reduced amplification, and higher noise figure than erbium-doped optical amplifiers The SOAs can also be used to perform different processing tasks [30,31,91] in all optical networks. Most of these functional applications are based on SOA nonlinearities. The main reason for nonlinearity of an SOA is that the gain of the SOA depends on the input signal. The four main types of nonlinearities are: Cross gain modulation (XGM): When the strong signal at one wavelength affects the

gain of a weak signal at another wavelength. Cross phase modulation (XPM): When the refractive index changes induced by a

strong optical signal at one wavelength affect the output phase of a weak signal at another wavelength.

50

Chapter 2: Waveguidance, mode coupling in fiber gratings and optical amplifiers: Theoretical background Self phase modulation (SPM): When the refractive index change induced by a signal

affects the output phase of the same signal. Four wave mixing (FWM): When the mixing of two or more signals propagating

along the SOA generate optical signal at new frequencies. These nonlinearities of an SOA can be exploited to implement functions like wavelength conversion, optical switches, optical logic gates, and multiplexers.

2.6.4.4 Raman amplifiers The discussion on types of optical amplifiers will be incomplete without the discussion of Raman amplifiers. The principle of Raman amplifiers is not same as other amplifiers (EDFA, EDWG or SOAs), instead the amplification is achieved by a nonlinear interaction between the signal and a pump laser within an optical fiber, the stimulated Raman scattering (SRS). The process of SRS generates scattered light at a wavelength longer than that of the incident light (pump wavelength). If another signal is present at this longer wavelength, the SRS light will amplify it Each Raman amplifier may contain one or more pumps. Raman amplifiers can easily adjust the amplification band by properly choosing the wavelength of the pumping light.

Raman

amplifiers

are

becoming

increasingly

important

in

optical

communication systems, particularly, in high-bit rate WDM/DWDM systems. An important advantage of Raman amplification is that the signal-to-noise ratio is much lower than that of an EDFA having the same gain. Raman amplifiers can cover a much wider spectral range than rare-earth based amplifiers. Furthermore, Raman amplifiers have lower noise levels than rare-earth amplifiers. These advantages make Raman amplifiers desirable for long haul WDM systems where the transmission bandwidth may be broad. However, the Raman amplifier not only has very low optical amplification efficiency but also needs a high power pump source, thereby increasing the entire size of the optical amplifier module and the price of the optical amplifier module

51


2.7 Summary In this chapter we reviewed the basic electromagnetic theory needed to understand the propagation of light in integrated optical waveguides and optical fibers. The chapter started with the review of Maxwell equations. Variational method for the analysis of diffused channel waveguides was briefly discussed. The solution for the core and cladding modes using the complete three layers was reviewed. In the end we discussed the basic physics involved in the process of optical amplification, and finally described some popular optical amplifiers.

52

Chapter 3: Erbium doped fiber amplifiers

3 Erbium doped fiber amplifiers 3.1 Introduction Emergence of erbium doped fiber (EDF) amplifier in 1987 [35,83,84] was one of the most important development that revolutionized optical telecommunication. The basic EDF configuration consists of a short length of silica (4 m to 50 m) fiber doped with low concentration of erbium ions in the core and an optical pump to excite these ions. EDF provides amplification in the C-band (1.53 m to 1.565 m).

Figure 3.1: Basic EDF configuration with co-propagating pump.

53


Figure 3.1 shows the typical configuration of an erbium doped fiber amplifier. A wavelength selective coupler at the input combines the input signal and the pump signal (typically at a wavelength of 980 nm or 1480 nm). The coupler at the output separates the amplified signal from the residual pump power. EDF can amplify many wavelength channels within its gain band. Thus, one can use a single amplifier for all wavelengths in wavelength division multiplexed (WDM) systems. However, EDF suffers from some limitations; the first is that the gain is not same for all signal wavelengths. This is a problem for WDM systems, as it can result in signal imbalance. The second is the presence of amplified spontaneous emission noise. It emerges from both ends of the doped fiber and must be dealt with. ASE in the same direction as the signal is a major source of cumulative noise (reducing signal to noise ratio), while backward propagating ASEs can harm source lasers if not filtered out. The presence of ASE also reduces the available optical gain for the signal field. Various techniques have been proposed to flatten the gain spectrum. One of the popular technique for gain flattening is the use of an external optical gain flattening filter (GFF) having a loss profile that is reverse of the gain spectra of EDF [51,107,143]. Singh et al. [119] proposed an alternative technique of gain flattening by an appropriately chosen long period grating written through a length of the EDF itself. The LPG was chosen so as to couple the 1.53 m wavelength experiencing peak gain into the cladding; thus inversion in core is made available to other wavelengths. As a result, in this configuration, there was gain flattening as well as increase in the average gain across the 1.54-1.56 m band. We expect that this GFF proposed should also affect the ASE noise characteristics. The spontaneous emission generated into the propagating mode in each small section of fiber has a random phase. LPG is a phase sensitive device and hence, it is necessary to consider both the amplitude and phase of the propagating modes. In this Chapter we present the methodology evolved by us to incorporate the spontaneous emission generated in each small section of fiber along with its random phase and study its amplification in the EDF with the LPG written in it. For completeness, we first review several aspects of erbium doped fiber amplifiers; specifically we delve into some basic physics, the principle of operation and device performance in terms of gain flattening and noise response. Then we propose the

54


method to evaluate the ASE in the EDF with LPG written in it. To validate our approach we compare our results with the conventional power coupled equations used to model EDF. Our analysis shows a good match between the two. Further, on investigating the effect of LPG written in the EDF itself, we find that LPG written in EDF itself, not only brings about gain flattening, but also suppresses the ASE noise.

3.2 Atomic structure and related optical spectrum The first step in understanding the behavior of erbium doped fiber is to understand the energy structure of Er+3 ions and how the host material modifies the energy diagram. Erbium, a rare earth element belongs to the group lanthanides. Its atomic number is 68. Its electronic shell configuration is [Xe]4f126s2, where [Xe] represents the ground state configuration of Xenon, i.e. [Xe] is 1s22s2p63s2p6d104s2p6d10f125s2p66s2. When present in a solid, erbium forms a trivalent ion Er+3 by removal of valence electrons: the two 6s electrons and one 4f electron. The resulting configuration is [Xe]4f11. Figure 3.2 shows the energy level diagram of the Er+3 ions in silica host glass and associated transitions [37]. The levels are labeled with the spectroscopic notation 2S+1

LJ, where S corresponds to the spin quantum number, L represents the orbital

quantum number and J=L+S is the total angular momentum3. Each energy level splits into a multiplicity of levels due to the electric field of the adjacent ions in the glass matrix and due to the amorphous nature of the silica glass matrix. The energy difference between the ground state and successive excited states corresponds to the wavelength around 1530 nm, 980 nm, 800 nm, 670 nm, 532 nm and 514 nm.

3

In Er+3 the ground state is 4f, i.e. the principal quantum number n=4, the azimuthal quantum number l=3, resulting in 7 possible magnetic quantum number m=[-3 to 3], with each having two possible spins. With this information we get that for the ground state of Er+3 L= m=3+2+1+0=6, the Spin quantum number S=(1/2+1/2+1/2)=3/2, (three missing electrons), therefore, J=6+3/2=15/2. The ground state being thus 4I15/2.

55


Figure 3.2: Energy level diagram of Er+3 ions in silica host glass.

As shown in Figure 3.2, only the transitions between E2(4I13/2) and E1(4I15/2) manifold are radiative; all other relaxations are non-radiative. The lifetime of radiative transition between E2(4I13/2) and E1(4I15/2) is of the order of 10 ms, while the nonradiative lifetime for the E3 to E2 transitions is 6.6 s. Thus Er+3 ions pumped by 980 nm wavelength from the ground state E1 (4I15/2) to an excited state E3 (4I11/2), and relax rapidly to a metastable state E2(4I13/2) which has a long lifetime. The ions at E2 decay radiatively (either spontaneous or stimulated) to the ground state E1. Population

56


inversion is hence possible between levels E2 and E1 and hence, erbium ions in silica host pumped by 980 nm radiation form a three level laser system. 3.2.1 Cross sections

The emission and absorption cross sections

e,a

defined in Eqs. (2.63) and (2.64)

characterize the E2-E1 manifold transitions. They are the fundamental parameters for the EDF normally specified by the manufacturer [36]. For illustrative computations we will use all the parameters corresponding to the erbium doped fiber codoped with alumino silicate designated as Type III by Desurvire [34]. The variation of absorption and emission cross sections of this Type III EDF with wavelengths are shown in Figure 3.3. They are obtained from the following empirical formula [34] in the 1.4 to 1.6 m range: `a i exp

peak i

where ai ,

i

and

i

(3.1)

i

4 ln 2 i

are the fitting parameters.. The typical values of these

parameters for type III EDF are tabulated in Table 3-1. The peak values (at

1.53 μm ) are given by

7.0 10

e peak

pump wavelength of 980 nm,

a

p

25

2.0 10

m 2 and 25

a peak

0.92

e peak

. For the

m2

Table 3-1: Fitting parameters for absorption and emission cross section Absorption Emission i

ai

1 2 3 4 5 6 7 8

0.03 0.31 0.17 0.37 0.74 0.28 0.30 0.07

i

1.440 1.482 1.492 1.515 1.530 1.544 1.555 1.570

i

0.0400 0.0500 0.0290 0.0290 0.0165 0.0170 0.0250 0.0350

ai 0.06 0.16 0.30 0.73 0.38 0.49 0.20 0.06

i

1.470 1.500 1.520 1.530 1.542 1.556 1.575 1.600

i

0.0500 0.0400 0.0250 0.0125 0.0130 0.0220 0.0250 0.0600

57


Figure 3.3: Absorption and emission cross section for Type III EDF from [34].

3.2.2

Optical gain and rate equations

As mentioned earlier, the erbium ions in silica host can be modeled as a three level homogenously broadened laser system (Figure 3.4) with N1, N2 and N3 representing the population densities of erbium ions in energy states E1, E2 and E3 respectively. The transitions from energy levels E3 to E2 are very rapid and non radiative; thus population in level E3 is assumed to be effectively zero.

Figure 3.4: Three levels of an EDF.

58


When the erbium doped glass is irradiated with light of wavelengths around the s

~ 1.53 μm (the C-band), the light can undergo amplification resulting in net increase

in photon flux, provided the pump intensity I p (of

p=980

nm) has brought about

population inversion. From our discussion in Section 2.6.2 and 2.6.3 we have: dI s dz

s

Is

e

s

N2

a

s

N1 I s

(3.2)

dI p p

dz

Ip

a

N1 I p

p

where ( ) is the gain/loss coefficient at wavelength . Though the pump can be either co-propagating or counter-propagating, in this thesis we consider only co-propagating pump. The total erbium ion density is

Er

N 2 . The gain coefficient can also be

N1

written in terms of relative inversion parameter D

s

s

1 2

e

s

1 D

a

N2

s

N1 /

1 D

Er

as: (3.3)

Figure 3.5: Gain coefficient for different values of the relative inversion parameter D, as it increases from -1 to +1 in steps of 0.2. The lowest curve corresponds to D=-1, and top most curve to D=1. The fiber in consideration is the Type III [34].

59


It is clear from Eq. (3.3) that gain is possible only when the relative inversion parameter satisfies D

a

s

e

s

a

s

e

s

, Figure 3.5 shows the

variation of gain coefficient as the inversion parameter increases from its minimum value -1 to +1 in step of 0.2. To determine the gain coefficient we need to know the population density N1 and N2 of energy states E1 and E2 respectively in the presence of light intensity corresponding to pump and signal wavelengths. Since erbium doped glass can amplify signals of wavelengths lying within its gain bandwidth, we consider the presence of a large number of signals (as would be in a WDM system). The rate equation describing the change of population density N1 can be written as: dN1 dt

where

sp

a

p

p

Ip

hc

a

N1

I

k

k k

hc

k

e

N1

I

k

k k

hc

k

N2

N2

(3.4)

sp

(=10 ms) is the spontaneous emission lifetime of transition from energy

levels E2 to E1. The first term in Eqs. (3.4) corresponds to the stimulated absorption of the pump, the second term is a sum of absorption at all signal wavelengths, the third term represents the stimulated emission at all signal wavelength and the last term corresponds to the spontaneous emission from E2 to E1. The index k denotes all the simultaneously propagating signals. Under steady state condition, dN 1 dt

0 , and

one can obtain: ~ I

1

N1

~ Ip

k k k

~ 1 Ip

~ Ik 1

Er

, N2

I k I sat

k

k

~ 1 Ip

k

k

~ where I k

~ Ik ~ Ik 1

(3.5)

Er k

k

~ and I p

I p I sat

p

are normalized intensities at signal and

pump wavelengths respectively, with saturation intensities I sat

k

and I sat

p

defined as:

60


I sat I sat and

k

e

wavelength

k k

a

k

hc

sp

a

k

k

p

hc

sp

a

p

p

(3.6)

is the ratio of emission and absorption cross-section at

k

. Substituting the expressions of N1 and N2 in Eqs. (3.2) the change in

intensity Ij corresponding to jth wavelength, along propagation length are obtained as:

s

dI j

~ Ip

~ Ik k

Er

dz

a

~ 1 Ip

s

~ I

1

k k k

~ Ik 1

Ij

(3.7)

k

k

~ I

1

dI p Er

dz

a

p

k k k

~ 1 Ip

Ip

~ Ik 1

(3.8)

k

k

3.3 Power coupled equations In an erbium doped optical fiber with n1=1.458, n2=1.45, aco=2.5 m, and acl=62.5 m the pump and signal powers travel in well defined LP01 modes. The radial dependence of the modal fields is given by [47,48,110]:

(r )

aco (k 02 n12

where U

U r A J0 J 0 (U ) aco

r

Wr A K0 K 0 (W ) aco

aco

2 1/ 2

)

and W

aco (3.9)

aco (

2

r

k 02 n22 )1 / 2 .and

is the propagation

constant. The constant A can be determined using the normalization of the modal field 2

as 2

r rdr 1 as:

0

A

2 a

2 co

J 12 (U 1 ) J 02 (U 1 )

K12 (U 2 ) K 02 (U 2 )

12

(3.10)

61


Figure 3.6:Normalized modal field profiles corresponding at signal wavelength (S) and at pump wavelength

=1.53 m

=980 nm (P), with n1=1.458, n2=1.45, aco=2.5 m and

acl=62.5 m.

=1.53 m and pump

Figure 3.6 shows the modal fields for signal wavelength,

wavelength, =980 nm. The intensity distribution at any wavelength Ij r

2 j

Pj

r where Pk is the total power in the fiber at

k

j

is given by

and

k

r is the

corresponding modal field distribution, thus equations describing the variation of propagation power can be written as:

dPj z

k j

z

2

Er 0

a

j

~ Pp

2 p

~ Pk

r

~ 1 Pp

2 k

r

k 2 k

(3.11)

z Pj z

j

dz

1 ~ Pk

r

k k

2 k

r 1

~ Pk

2 k

r 2 j

r rdr

(3.12)

k

k

Similarly we have for pump power:

dPp z dz

p

z Pp z

(3.13)

62


1 p

z

2

Er

a

0

p

~ 1 Pp

k k

2 p

~ Pk ~ Pk

r

2 k 2 k

r r 1

Pp z

2 p

r rdr

(3.14)

k

k

~

and Pj , p

Pj , p I sat

j, p

. The gain coefficient

j, p

depends on propagation distance z.

Eqs. (3.11) and (3.13) form a set of coupled differential equations; the number of equations is determined by the number of signal wavelengths propagating simultaneously. These equations can be solved using the Fourth order Runge-Kutta method. If we consider 125 signals in wavelength range 1.5
0

(1)

y0

(7b)

y>0 +

γy 2α y2

+

γ y2

π

8α 3y

2

. There are three variational parrameters,

gives the width of the Gaussian field in x direction. It is observed

that in our range of interest 1.484µm≤λ≤1.6µm the three variable modal fields do not vary significantly (figure 3). Hence, for use in equation (4) and (5) the variational fields , Ψk ( x, y ) for all k, can be replaced by the Ψ ( x, y ) the variational field for λ=1.532µm. 0.5 0.45

0.3

0.4

0.25

0.35 0.3

0.2

Y (y )

X (x )

0.35

0.15

0.25 0.2 0.15

0.1

0.1 0.05

0.05 0

0 -20

-10

0

10

20

0

5

10

15

20

y (µm)

x (µm)

(a)

(b)

Fig. 3 (a) Modal field profile X(x) at y=0. (b) ) Modal field profile Y(y) at x=0. The dotted curves show the modal fields as obtained from variational analysis for λ=1.484µm(above) and λ=1.6µm(below). The smooth curves represents the fields calculated using Gaussian approximation.

The RHS of the equations (4) and (5) describing the signal and ASE power propagation contain integrals of these modal fields and it is not possible to solve the integrals analytically for the modal fields of the form in equation (6).A look at Y(y) shows that the field can be approximated by a Gaussian function centered at its maxima. Hence, we approximate the variational field by an appropriately chosen Gaussian approximation given by: ⎛α ΨG ( x, y ) = ⎜ x ⎜ w ⎝

2 ⎞⎟ π ⎟⎠

1/ 2

⎛ x 2 ⎞⎛ α ′y exp⎜⎜ − α x 2 2 ⎟⎟⎜ w ⎠⎜⎝ h ⎝

⎛ ( y − y0 ) 2 2 ⎞⎟ exp⎜ − α ′y 2 ⎜ π ⎟⎠ h2 ⎝

⎞ ⎟ ⎟ ⎠

Proc. of SPIE Vol. 6468 646808-4 Downloaded from SPIE Digital Library on 12 Sep 2010 to 122.162.62.21. Terms of Use: http://spiedl.org/terms

(8)

where y 0 =

h ⎛ 2 ⎜ α y + 2γ 2α y ⎝

y

2

− α y ⎞⎟ is the y value for which Y(y) variational field has maximum value, the ⎠

variational parameters α x , α y and γ y correspond to λ=1.532µm. To obtain the value of α ′y we maximize the overlap between the Y(y) field and the Gaussian field, i.e, we maximize the integral ∞⎛

I = ∫ ⎜1 + γ y 0⎝

2 ⎞⎛ α ′ ⎛ y ⎞⎛⎜ 1 ⎞⎟ ⎜ − α y 2 y ⎟⎜ y exp ⎟⎜ ⎜ h ⎠⎜ d y h ⎟⎟ h 2 ⎟⎠⎜⎝ h ⎝ ⎠ ⎝

2 ⎞⎟ π ⎟⎠

1/ 2

⎛ ( y − y0 ) 2 exp⎜ − α ′y 2 ⎜ h2 ⎝

⎞ ⎟dy ⎟ ⎠

Figure 3 shows the field as obtained from variational analysis and the Gaussian approximated fields in both x and y directions respectively. It can be observed that by appropriately choosing the Gaussian approximation we are able to retain the shape of modal field and at the same time the RHS integrals of equations (4) and (5) can now be solved analytically, as shown in section 3.

3. SIMPLIFIED GAIN CALCULATIONS On using the approximated Gaussian modal fields the gain coefficient γk for signal of wavelength λk and the ASE power calculations as defined in equations 4 and 4 of section 1, can be simplified as N p Ψ 2 ( x, y ) − 1 ∑ (η − η ) ~ k

±

j

G

j =0

−∞ 0

1+ ∑ η j +1 ~ p j ΨG 2 (x, y )

N

j =0

dP ASE ± ( λ k ) = P ASE dz

j

∞ ∞

γ k ( z ) = σ a (λ k ) ρ 0 ∫ ∫ ΨG 2 ( x, y )

(

)

N p j Ψ G 2 ( x, y ) ∑ ~

∞ ∞

( λ k )γ k ( z ) ± 2 P0 k σ e (λ k )ρ 0 ∫ ∫ Ψ G 2 ( x , y ) −∞ 0

(9)

dxdy

j=0 N

1+ ∑

j=0

Transforming coordinated from [x,y] plane to [ζ , η ] plane such that ζ = 2

(η j αxx w

(10)

dxdy +1 ~ p j Ψ G 2 (x , y )

)

and η = 2

modal field can be re-written as ΨG (ζ , η ) = A exp[−(ζ 2 + η 2 )] with constant A =

2α x α ′y whπ

α ′y ( y − y 0 ) h

the Gaussian

. The above expressions then

become:

γ k ( z) =

N A exp[−(ζ 2 + η 2 )] ∑ (η k − η j ) ~ p j −1

σ a (λ k ) ρ 0 ∞ ∞ j =0 2 2 dζ dη ∫ ∫ A exp[−(ζ + η )] N π −∞ −∞ 1 + A exp[−(ζ 2 + η 2 )] ∑ (η j + 1)~ pj j =0

2 P0 k σ e (λ k )ρ 0 dP ASE ( λ k ) = PASE ± ( λ k )γ k ( z ) ± ∫ ∫ A exp[ − (ζ dz π −∞ −∞ ±

∞ ∞

A exp[ − (ζ 2

+ η 2 )] 1 + A exp[ − (ζ

2

2

N + η 2 )] ∑ ~ pj j =0

N

(

)

+ η )] ∑ η j + 1 ~ pj 2

j =0

dζ dη

The limits of η field have been taken as [-∞,∞] since the modal fields are negligible after y

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