Distributed Feedback Laser Diodes and Optical Tunable Filters

Distributed Feedback Laser Diodes and Optical Tunable Filters

Dr. H. Ghafouri–Shiraz The University of Birmingham, UK and Nanyang Technological University, Singapore

Distributed Feedback Laser Diodes and Optical Tunable Filters

Distributed Feedback Laser Diodes and Optical Tunable Filters

Dr. H. Ghafouri–Shiraz The University of Birmingham, UK and Nanyang Technological University, Singapore

Copyright # 2003

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777

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Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some of the content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Ghafouri–Shiraz, H. Distributed feedback laser diodes and optical tunable filters / H. Ghafouri–Shiraz. p. cm. Includes bibliographical references and index. ISBN 0-470-85618-1 (alk. paper) 1. Light emitting diodes. 2. Solid-state lasers. 3. Tunable lasers. 4. Light filters. I. Title. TK7871.89.L53G43 2003 621.360 6–dc21

2003050194

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-85618-1 Typeset in 10/12pt Times by Thomson Press (India) Limited, New Delhi Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

This book is dedicated to

My Father, the late Haji Mansour, for the uncompromising principles that guided his life and for his profound influence and inspiration. My Mother, Rahmat, for leading her children into intellectual pursuits. My Supervisor, the late Professor Takanori Okoshi, for his continuous guidance, encouragement, inspiring discussion and moral support. A distinguished scientist and a great teacher who made me aware of the immense potential of optical fibre communications. My Wife, Maryam, for her understanding support, affection and magnificent devotion to her family. My constant companion and best friend, she has demonstrated incredible patience and understanding during the rather painful process of writing this book while maintaining a most pleasant, cheerful and comforting home. My Children, Elham, Ahmad-Reza and Iman, for making everything worthwhile. To all of my research and undergraduate students since 1987, for their excellent and fruitful research work, and for many stimulating discussions, which encouraged and motivated me to write this book.

Contents Preface . . . . . . . . . . . . . Acknowledgements . . . . . Glossary of Abbreviations Glossary of Symbols . . . .

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Introduction to Optical Communication Systems . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Fibre Communication Systems . . . . . . . . . . . . . . . . 1.3.1 Intensity Modulation with a Direct Detection Scheme 1.3.2 Coherent Detection Schemes. . . . . . . . . . . . . . . . . . 1.4 System Requirements for High-Speed Optical Coherent Communication . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Spectral Purity Requirements . . . . . . . . . . . . . . . . . 1.4.2 Spectral Linewidth Requirements. . . . . . . . . . . . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2. Principles of Distributed Feedback Semiconductor Laser Diodes: Coupled Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Principle of Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Absorption and Emission of Radiation . . . . . . . . . . . . . . . 2.2.2 The Einstein Relations and the Concept of Population Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Dispersive Properties of Atomic Transitions . . . . . . . . . . . 2.3 Basic Principles of Semiconductor Lasers . . . . . . . . . . . . . . . . . . 2.3.1 Population Inversion in Semiconductor Junctions. . . . . . . . 2.3.2 Principle of the Fabry–Perot Etalon . . . . . . . . . . . . . . . . . 2.3.3 Structural Improvements in Semiconductor Lasers . . . . . . . 2.3.4 Material Gain in Semiconductor Lasers . . . . . . . . . . . . . . 2.3.5 Total Radiative Recombination Rate in Semiconductors . . . 2.4 Coupled Wave Equations in Distributed Feedback Semiconductor Laser Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 A Purely Index-coupled DFB Laser Diode . . . . . . . . . . . . 2.4.2 A Mixed-coupled DFB Laser Diode. . . . . . . . . . . . . . . . . 2.4.3 A Gain-coupled or Loss-coupled DFB Laser Diode . . . . . . 2.5 Coupling Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 A Structural Definition of the Coupling Coefficient for DFB Semiconductor Lasers. . . . . . . . . . . . . . . . . . . . . . .

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2.5.2 The Effect of Corrugation Shape on Coupling Coefficient . . 2.5.3 Transverse Field Distribution in an Unperturbed Waveguide . 2.5.4 Results Based upon the Trapezoidal Corrugation . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Structural Impacts on the Solutions of Coupled Wave Equations: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solutions of the Coupled Wave Equations. . . . . . . . . . . . . . . . . 3.3 Solutions of Complex Transcendental Equations using the Newton–Raphson Approximation . . . . . . . . . . . . . . . . . . . . . . . 3.4 Concepts of Mode Discrimination and Gain Margin. . . . . . . . . . 3.5 Threshold Analysis of a Conventional DFB Laser . . . . . . . . . . . 3.6 Impact of Corrugation Phase at Laser Facets. . . . . . . . . . . . . . . 3.7 The Effects of Phase Discontinuity along the DFB Laser Cavity . 3.7.1 Effects of Phase Shift on the Lasing Characteristics of a 1PS DFB Laser Diode . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Effects of Phase Shift Position (PSP) on the Lasing Characteristics of a 1PS DFB Laser Diode . . . . . . . . . . . 3.8 Advantages and Disadvantages of QWS DFB Laser Diodes . . . . 3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Transfer Matrix Modelling in DFB Semiconductor Lasers . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Brief Review of Matrix Methods . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Formulation of Transfer Matrices . . . . . . . . . . . . . . . . . 4.2.2 Introduction of Phase Shift (or Phase Discontinuity) . . . . 4.2.3 Effects of Finite Facet Reflectivities. . . . . . . . . . . . . . . . 4.3 Threshold Condition for the N-Sectioned Laser Cavity . . . . . . . . 4.4 Formulation of the Amplified Spontaneous Emission Spectrum using the TMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Green’s Function Method Based on the Transfer Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Determination of Below-Threshold Spontaneous Emission Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Numerical Results from Various DFB Laser Diodes. . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Threshold Analysis and Optimisation of Various DFB LDs Using the Transfer Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Threshold Analysis of the Three-phase-shift (3PS) DFB Laser . 5.2.1 Effects of Phase Shift on the Lasing Characteristics. . . .

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5.2.2 Effects of Phase Shift Position (PSP) on the Lasing Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . Optimum Design of a 3PS DFB Laser Structure . . . . . . . . . . . . 5.3.1 Structural Impacts on the Gain Margin. . . . . . . . . . . . . . 5.3.2 Structural Impacts on the Uniformity of the Internal Field Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Threshold Analysis of the Distributed Coupling Coefficient (DCC) DFB LD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Effects of the Coupling Ratio on the Threshold Characteristics . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Effects of the Position of Corrugation Change . . . . . . . . 5.4.3 Optimisation of the DCC DFB Laser Structure . . . . . . . . Threshold Analysis of the DCC þ 3PS DFB Laser Structure . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8. Circuit and Transmission-Line Laser Modelling (TLLM) Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Transmission-Line Matrix (TLM) Method . . . . . . . . . . . . . . . . .

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6. Above-Threshold Characteristics of DFB Laser Diodes: A TMM Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Determination of the Above-Threshold Lasing Mode using the TMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Features of Numerical Processing . . . . . . . . . . . . . . . . . . . . . 6.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Quarterly-Wavelength-Shifted (QWS) DFB LDs . . . . . . 6.4.2 Three-phase-shift (3PS) DFB LDs . . . . . . . . . . . . . . . . 6.4.3 Distributed Coupling Coefficient with Quarterly-Wavelength-Shifted (DCC þ QWS) DFB LDs 6.4.4 Distributed Coupling Coefficient with Three-Phase-Shift (DCC þ 3PS) DFB LDs. . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Above-Threshold Analysis of Various DFB Laser Structures Using the TMM . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Single-Mode Stability in DFB LDs . . . . . . . . . . . . 7.3 Numerical Results on the Gain Margin of DFB LDs 7.4 Above-Threshold Spontaneous Emission Spectrum . 7.5 Spectral Linewidth . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Numerical Results on Spectral Linewidth . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13

8.2.1 TLM Link Lines . . . . . . . . . . . . . . . . . . . . 8.2.2 TLM Stub Lines . . . . . . . . . . . . . . . . . . . Scattering and Connecting Matrices . . . . . . . . . . . . Transmission-Line Laser Modelling (TLLM). . . . . . Basic Construction of the Model . . . . . . . . . . . . . . Carrier Density Model . . . . . . . . . . . . . . . . . . . . . Laser Amplification . . . . . . . . . . . . . . . . . . . . . . . Carrier-induced Frequency Chirp . . . . . . . . . . . . . . Spontaneous Emission Model . . . . . . . . . . . . . . . . Computational Efficiency Baseband Transformation . Signal Analysis – Post-processing Methods. . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9. Analysis of DFB Laser Diode Characteristics Based on Transmission-Line Laser Modelling (TLLM) . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 DFB Laser Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 TLLM for DFB Laser Diodes . . . . . . . . . . . . . . . . . . . . . . 9.4 A DFB Laser Diode Model with Phase Shift. . . . . . . . . . . . 9.5 Analysis of TLLM for DFB Laser Diodes . . . . . . . . . . . . . 9.5.1 Scattering Matrix for a Uniform DFB LD . . . . . . . . 9.5.2 Scattering Matrix for the DFB Laser Diode with Phase Shift. . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Connection Matrix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Connection Matrix C for the Stubs Within a Section 9.7 Carrier Density Rate Equation. . . . . . . . . . . . . . . . . . . . . . 9.8 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.1 Dynamic Characteristics . . . . . . . . . . . . . . . . . . . . 9.8.2 Longitudinal Distribution . . . . . . . . . . . . . . . . . . . 9.8.3 Effects of the Number of Phase Shifts . . . . . . . . . . 9.8.4 Effects of Phase Position and Value on the 3PS DFB LD’s Characteristics. . . . . . . . . . . . . . . . 9.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10. Wavelength Tunable Optical Filters Based on DFB Laser Structures. 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Wavelength Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Solutions of the Coupled Wave Equations. . . . . . . . . . . . . . . . . . 10.3.1 The Dispersion Relationship and Stop bands . . . . . . . . . . 10.3.2 Formulation of the Transfer Matrix . . . . . . . . . . . . . . . . 10.3.3 Solutions of Complex Transcendental Equations using the Newton–Raphson Approximation . . . . . . . . . . . . . . . 10.4 Threshold Analysis of DFB Laser Diodes . . . . . . . . . . . . . . . . . . 10.4.1 Phase Discontinuities in DFB LDs . . . . . . . . . . . . . . . . . 10.4.2 Below-threshold Characteristics . . . . . . . . . . . . . . . . . . .

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10.5 10.6

Active Tunability DFB LD Amplifier Filters . . . . . . . . . . Structural Impacts on DFB LD Amplifier Filters . . . . . . . 10.6.1 Phase-shift-controlled DFB LD Amplifier Filters . 10.7 New Multi-section and Phase-shift-controlled DFB LD Optical Filter Structures . . . . . . . . . . . . . . . . . . . . . . . . 10.8 DFB LDs Versus DBR LDs . . . . . . . . . . . . . . . . . . . . . 10.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11. Other Wavelength Tunable Optical Filters Based on the DFB Laser Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Results and Discussions of Various Optical Tunable Filters. . . . . 11.3.1 A Quarter Wavelength Phase-shifted Double Phase-shift-controlled DFB LD-based Wavelength Tunable Filter . . . . . . . . . . . . . . . . . . . . . 11.3.2 A Single-phase-shift-controlled Double-phase-shift DFB Wavelength Tunable Optical Filter . . . . . . . . . . . . . . . . 11.3.3 A Multiple-phase-shift Controlled DFB LD-based Wavelength Tunable Optical Filter . . . . . . . . . . . . . . . . 11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Conclusion, Summary and Suggestions . . . . . . . . . . . . . . . . . . . 12.1 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The TMM and/or TLLM Analysis . . . . . . . . . . . . . . . . . . . 12.3 Future Research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Extension to the Analysis of Quantum Well Devices 12.3.2 Extension to Gain-Coupled Devices . . . . . . . . . . . . 12.3.3 Further Investigation of Optical Devices to be Used in WDM. . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 Switching Phenomena . . . . . . . . . . . . . . . . . . . . . 12.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preface It was in May 1976 that I published my second book entitled Distributed Feedback Laser Diodes: Principles and Physical Modelling. Since then we have witnessed rapid and dramatic advances in optical devices. To provide a comprehensive and up-to-date account of changes I decided to publish this new book, which is in fact an extension to the aforementioned title. The main objective of this new book is to serve both as a textbook and as a reference monograph. As a result, each chapter is designed to cover both physical understanding and engineering aspects of both DFB lasers and optical filters. The deployment of worldwide optical fibre networks during the 1980s has laid the cornerstone for a new information era. While millions of electronic messages are travelling along the present global network every day, the demand for broadband communications is expected to increase rapidly with the development of multimedia techniques. An integrated type of service that comprises interactive video-on-demand, real time video conferencing and information which may consist of voice, data, fax, images and motion pictures will be available soon to both corporate and domestic users. There is no doubt that this information technology is influencing our society in various ways. Behind all these technological changes in the information highway, the pace and advancement in the design and implementation of optical communication systems has played a crucial role. The continuing worldwide installation of single-mode fibres (SMFs) provides an extremely high bandwidth (about 1014 Hz) which is superior to the traditional radio and microwave systems. The glass-based SMF also offers immunity against electromagnetic interference, something which often affects other systems. In fact, the characteristic performance of optical fibres has been improved significantly in the past 30 years. In the early stages of research Kao and Hockham proposed the first dielectric circular waveguide used for optical communications. On the basis of this discovery, optical fibre technology has since improved dramatically, leading to the current use of SMF which has a typical attenuation coefficient of about 0.2 dB/km at a 1:55 mm operating wavelength. In addition to the use of SMF there is a second driving force behind the success of optical fibre communication technology. Owing to advances in fabrication and theoretical research, a wide variety of optical devices – transmitters, receivers, modulators, switches, amplifiers, mirrors, couplers and other optical components – have been developed and implemented in optical communication networks. In particular, the rapid growth of semiconductor technology has given rise to a family of optical transmitters better known as semiconductor laser diodes. By injecting an electric current directly across a well-designed semiconductor laser diode, coherent photons are generated as a result of stimulated recombination. Compared with gaseous and solid-state lasers, the semiconductor laser diode has a small physical size and a high modulation bandwidth which is compatible with those of SMFs.

xiv

PREFACE

Also the possible monolithic integration of laser diodes with other optical devices enhances the potential applications of semiconductor laser diodes. Following the development of optical communication systems, the design of semiconductor lasers has undergone several changes. From the earlier work based on the Fabry–Perot resonant cavity characterised by its broadband spectral output, single longitudinal mode oscillation has been achieved with a built-in wavelength selective grating fabricated inside the laser cavity to make it a distributed feedback (DFB) laser diode. The continuous research into materials has allowed better control over the quality and performance of semiconductor optical devices. The use of semiconductor laser diodes has also diversified a lot. Applications in laser printing, compact disc players, data storage and optical remote sensing techniques continue to expand. The continuous improvement in the performance of optical devices and, in particular, the wide application of laser diodes will further enhance the significance of optical device technology and hence the prospects for optical fibre communication systems. In viewing the growth of semiconductor laser diodes, it is our belief that scientists, students and field engineers should equip themselves with the necessary knowledge and tools to enrich this rapidly growing technology through innovative research. There are of course many good books on optical communication systems, but information related to the design and development of semiconductor laser diodes is often found scattered in journals and articles. Our vision is to consolidate the information behind this rapidly growing technology and to try to present it in a systematic way. In this book we concentrate on the basic working principles and applications of the DFB laser diode which is characterised by a narrow spectral output and single-mode oscillation. Also in this book, wavelength tunable optical filters which are useful components of wavelength division multiplexed systems (WDM) are discussed. This book is organised as follows. Chapter 1 outlines the historical progress of the fast-growing optical fibre communication systems. Performance requirements of the semiconductor laser diodes used in these optical systems are discussed. In Chapter 2 the general laser theory is reviewed. This is then followed by the operational principles of semiconductor laser diodes. On the basis of coupled wave theory, the working principles of the DFB laser diode are introduced. In Chapter 3, the lasing characteristics of conventional DFB lasers are presented. Using the eigenvalue approach, various physical conditions that affect the threshold characteristics of the DFB laser are discussed. The limitations of conventional DFB laser diodes are examined. To improve the immunity of DFB laser diodes against the single-mode deterioration induced by the spatial hole burning effect (SHB), the structural design of the conventional DFB laser diode has to be improved. This, too, is discussed in this chapter. In Chapter 4, a method known as the transfer matrix method (TMM) is introduced. Compared with the eigenvalue approach, the TMM technique is more general and flexible and hence can be applied to the analysis and structural design of DFB laser diodes. In Chapter 5 an attempt is made to optimise the structure of various DFB laser diodes. In addition to threshold analysis, the TMM can also be used to determine the above-threshold laser characteristics. In Chapter 6, a novel numerical technique is presented for the above-threshold analysis using TMM. Numerical results are presented for the above-threshold analysis using TMM. Numerical results obtained from various DFB laser structures have presented based on stable single-mode oscillation and narrow spectral linewidth, which are crucial to the development of optical fibre communication

ACKNOWLEDGEMENTS

xv

systems. In Chapter 7, the TMM is applied to evaluate both the spectral and noise properties of DFB laser diodes taking into account the effects of SHB and other nonlinear gain phenomena. Also, the single-mode stability and spectral linewidth of various DFB laser diode structures are studied. In Chapter 8, the transmission line laser model (TLLM) is introduced which provides additional insight into the operation of the laser diodes. Application of TLLM in the analysis of multi-phase-shift DFB laser diodes characteristic performances is discussed in Chapter 9. In Chapters 10 and 11, basic principles of wavelength tunable filters are introduced. These optical filters are based on DFB laser structures and play an important role in wavelength division multiplexed (WDM) systems. Conclusions, suggestions and possible future directions are presented in Chapter 12. The book is referenced throughout by extensive end-of-chapter sections which provide a guide for the enthusiast and indicate a source for those equations or expressions that have been quoted without derivation. A complete glossary of symbols and abbreviations used throughout the book is also listed. It is my hope that this book will be useful to final year undergraduate and postgraduate students who wish to explore advanced lightwave technology, and to researchers and scientists who wish to broaden their knowledge and continue the journey into optical communication technology. It is assumed that the reader is familiar with matrix equations, differential and integral calculus. In addition, foundations in solid-state physics and electromagnetism will be an asset. Dr H. Ghafouri–Shiraz The University of Birmingham, UK August 2003

ACKNOWLEDGEMENTS I owe particular debts of gratitude to my former research students, Drs B. S. Lo, C. Y. Chu, P. W. Tan and W. M. Wong, for their excellent research work on DFB laser diodes, optical filters and optical amplifiers. I am also very grateful indeed for the useful comments and suggestions provided by reviewers, which have resulted in significant improvements to this book. Thanks also must be given to the authors of numerous papers, articles and books which I have referenced while preparing this book, and especially to those authors and publishers who have kindly granted permission for the reproduction of some diagrams. I am also grateful to my very many undergraduate and research students who have helped me in my investigations.

Glossary of Abbreviations 1PS 3PS AlGaAs AR ASE ASK BC BER BH CCC CP CPM CSP CW DBR DCC DCPBH DD DFB DFT DPSK DSM DWDM EC EDFA EIM FDM FET FFT FP FSK FTS FWHM GaAs GP HBT HE/CP HE/IP IC IF

single phase shift three phase shift aluminium gallium arsenide anti-reflection amplified spontaneous emission amplitude-shift keying buried crescent bit error rate buried heterostructure cleaved-coupled cavity corrugation position continuous-pitch-modulated channelled substrate planar continuous wave distributed Bragg reflector distributed coupling coefficient dual-channel planar buried heterostructure direct detection distributed feedback discrete Fourier transform differential phase-shift keying dynamic single mode dense wavelength division multiplexing external cavity erbium-doped fibre amplifier effective index method frequency division multiplexing field-effect transistor fast Fourier transform Fabry–Perot frequency-shift keying filter transmission spectra full width at half maximum gallium arsenide geometric progression heterojunction bipolar transistor heterodyne receiver with coherent postdetection heterodyne receiver with incoherent postdetection integrated circuit intermediate frequency

xviii

IM InGaAsP InP ISDN ISI LD LPE MOCVD MPS MQW MZ NRZ OEIC OTDM PCM PMM PS PSK PSP QW QWS RF RG RW RZ SCH SHB SLA SLM SMF SMSR TDM TE TEM TG TLLM TLM TM TMM WDM WG

GLOSSARY OF ABBREVIATIONS

intensity modulation indium gallium arsenide phosphide indium phosphide integrated service digital network inter-symbolic interference laser diode liquid phase epitaxy metal–organic chemical vapour deposition multiple phase shift multiple quantum well Mach–Zehnder non-return to zero opto-electronic integrated circuit optical time-division multiplexing pulse code modulation power matrix model phase shift phase-shift keying phase-shift position quantum well quarterly wavelength-shifted radio frequency rectangular grating ridge waveguide return to zero separate confinement heterostructure spatial hole burning semiconductor laser amplifier single longitudinal mode single-mode fibre side mode suppression ratio time-domain model transverse electric field transverse electromagnetic field triangular grating transmission-line laser modelling transmission-line matrix transverse magnetic field transfer matrix method wavelength division multiplexing waveguide

Glossary of Symbols A0 A1 A2 A21 B B21 c c.c C CN CS d dn=dN D; DFF
e or exp E E; E1 E1 E2 Ea Eb Ec ; Ev Ef EFc ; EFv Eg Ei ER ðzÞ ES ðzÞ E! ðzÞ f f ðEÞ fij F F F(k) F! ðzÞ g gðE21 Þ

differential gain parameter governing the base width of the gain spectrum or the gain curvature wavelength shifting coefficient or differential peak wavelength spontaneous transition rate radiative bimolecular recombination Einstein coefficient of stimulated emission free space velocity complex conjugate Auger recombination coefficient weighted function weighted function active layer thickness differential index diffusion coefficient exponential electric field vector self-explanatory parameters occupied valence band at energy E1 occupied conduction band at energy E2 occupied conduction band at energy Ea occupied valence band at energy Eb conduction and valence band edges, respectively equilibrium Fermi level quasi-Fermi levels in the conduction band and valence band, respectively energy gap intrinsic Fermi level complex forward propagating wave at z complex backward propagating wave at z complex Fourier component of the electric field optical frequency Fermi–Dirac distribution function matrix elements (for i, j ¼ 1; 2) of the matrix F flatness forward transfer matrix (k ¼ 1 to N) forward transfer matrix ¼ T(m)P(m) Langevin noise term material gain gain parameter

xx

gij gpeak gth G G G(k) Gðz; z0 Þ h h f ¼ hc=
H! ðzÞ I I Iavg Ith IðzÞ j jth k k k0 K0 Kun Kð1Þ K (2) K (3) K tn L m m0 mn mp M21 Mb Menvpffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ ð"="0 Þ n0 n1 n2 nact nclad ng nini nsp N N0 N1 N2

GLOSSARY OF SYMBOLS

matrix elements (for i; j ¼ 1; 2) of the matrix G peak material gain threshold gain backward transfer matrix amplifying term (k ¼ 1 to N) backward transfer matrix Green’s function Planck’s constant (6:626  1034 J.s) photon energy complex Fourier component of the magnetic field identity matrix injection current average field intensity threshold current local p ffiffiffiffiffiffiffi field intensity 1 nominal threshold current density Boltzmann’s constant ð1:38  1023 J/K1 Þ propagation constant complex propagation constant free space propagation constant complex propagation constant for unperturbed structure self-explained parameter self-explained parameter self-explained parameter local frequency tuning efficiency overall laser cavity length order of Bragg diffraction mass of electron effective mass of electron effective mass of hole momentum matrix of the carrier transition average matrix element of the Bloch states slowly varying envelope function refractive index steady-state refractive index at threshold active layer refractive index cladding layer refractive index active layer refractive index cladding layer refractive index group refractive index refractive index at zero current injection population inversion factor carrier concentration (or electron density) carrier concentration at transparency carrier concentration at energy E1 inside the valence band carrier concentration at energy E2 inside the conduction band

GLOSSARY OF SYMBOLS

N th P P(m) Pmut Pnum PN q r r1 ; r2 ^r1 ; ^r2 r spon (E21 ) r stim (E21 ) R R0 R00 RðNÞ RðzÞ R1 ; R2 Ra!b Rb!a Ra!b (net) Rsp Rst S S0 S00 SðzÞ S1 ; S2 ^t1 ; ^t2 tij T T T(k) uij U U(k) vg V w W W WB WT x

xxi

carrier concentration or density at threshold optical power phase shift matrix mutual interaction between the coupled waves RðzÞ and SðzÞ total number of photons found inside the laser cavity amplified spontaneous emission power electronic charge coupling ratio 1 =2 complex reflection coefficients at the left and right facets, respectively amplitude reflection coefficients at the left and right facets, respectively spontaneous emission rate stimulated emission rate rate of non-coherent carrier recombination first-order derivative of RðzÞ second-order derivative of RðzÞ carrier recombination rate excluding stimulated emission complex amplitude term complex coefficients overall downward transition between two energy bands overall upward transition between two energy bands net downward transition between two energy bands rate of spontaneous emission rate of stimulated emission photon density first-order derivative of SðzÞ second-order derivative of SðzÞ complex amplitude term complex coefficients amplitude transmission coefficients at laser facets matrix elements (for i; j ¼ 1; 2) of the matrix T forward transfer matrix temperature in degrees Kelvin (k ¼ 1 to N) forward transfer matrix matrix elements (for i; j; ¼ 1; 2) of the matrix U backward transfer matrix (k ¼ 1 to N) backward transfer matrix group velocity volume of the active layer active layer width Wronskian term distance spans by the rising and the dropping edges of the corrugation distance spans by the bottom width of the corrugation distance spans by the top width of the corrugation transverse direction

xxii

y yij Y z zij Z Z1 ðxÞ; Z2 ðxÞ Z1 ðzÞ; Z2 ðzÞ  0 a c eff H loss sca ^  ^L  ^SM  b b0 g G  ^ij ^ ^L ^SM


" " "0 "0 "ini i  " k  avg i g RS

GLOSSARY OF SYMBOLS

lateral direction matrix elements (for i; j; ¼ 1; 2) of the matrix Y overall transfer matrix chain using the forward propagation matrix longitudinal direction matrix elements (for i; j; ¼ 1; 2) of the matrix Z overall transfer matrix chain using backward propagation matrix corrugation functions solutions of the homogeneous wave equation satisfying the boundary conditions at the left and right facets, respectively amplitude gain coefficient steady-state amplitude gain absorption loss in the active layer absorption loss in the cladding layer effective linewidth enhancement factor intrinsic linewidth enhancement factor internal cavity loss scattering loss at heterostructure interface average amplitude gain average amplitude gain for the lasing mode average amplitude gain for the most probable side mode propagation constant the Bragg propagation constant complex propagation constant optical confinement factor detuning coefficient Kronecker delta function average detuning coefficient average detuning coefficient for the lasing mode average detuning coefficient for the most probable side mode spin-orbit splitting gain margin perturbed relative permittivity induced by the corrugation non-linear gain coefficient complex permittivity permittivity of free space (8:854  1012 Fm1 ) average relative permittivity internal quantum efficiency relative phase difference between perturbation of the refractive index and the amplitude gain effective phase shift complex electric field phase shift between sections k and (k  1) coupling coefficient average coupling coefficient used in a DCC laser structure index coupling coefficient gain coupling coefficient forward coupling coefficient

GLOSSARY OF SYMBOLS

SR

0
B
c
i
th L
v
vsp
vNN
vNS  

N 1;

! 

2

xxiii

backward coupling coefficient lasing wavelength peak wavelength at zero gain transparency the Bragg wavelength complex wavelength
c ¼
þ
i imaginary wavelength lasing wavelength at threshold corrugation period spectral linewidth spectral linewidth due to spontaneous emission spectral linewidth due to local fluctuation of the carrier density spectral linewidth due to the correlation between the local carrier and photon changes mathematical constant self-explained parameter,  ¼ jk =ð j þ Þ linear recombination lifetime carrier recombination lifetime position factor complex corrugation phase terms associated with r1 and r2 angular frequency corrugation phase at z origin

1 An Introduction to Optical Communication Systems 1.1

INTRODUCTION

Communication is a process in which messages, ideas and information can be exchanged between two individuals. From the early days when languages were developed, the methods people use to communicate have experienced a dramatic evolution. Nowadays, rapid transmission of information over long distances and instant access to various information sources have become conspicuous and important features of our society. The rapidly growing information era has been augmented by a global network of optical fibre [1]. By offering an enormous transmission bandwidth of about 1014 Hz and a low signal attenuation, the low-cost, glass-based single-mode optical fibre (SMF) provides an ideal transmission medium. In order that information can be carried along the SMF, information at the transmitter side is first converted into a stream of coherent photons. Using a specially designed semiconductor junction diode with heavy doping concentration, semiconductor lasers have been used to provide the reliable optical source required in fibrebased lightwave communication. With its miniature size compatible to the SMF, the semiconductor laser diode has played a crucial role in the success of optical fibre communication systems. This chapter has been organised as follows: in section 1.2, the historical progress of optical communication is presented. Before exploring the characteristics of semiconductor lasers, various configurations of optical fibre-based communication systems are discussed in section 1.3. Depending on the type of detection method used, both direct and coherent detection schemes are discussed. Based upon the characteristics of coherent optical communication systems, the performance requirements of semiconductor lasers are presented at the end of the chapter. In particular, the significance of having an optical source that oscillates at a single frequency whilst having a narrow spectral linewidth is reviewed.

1.2

HISTORICAL PROGRESS

In the early days of human civilisation, simple optical communication in terms of signal fires and smoke was used. In those days, only limited information could be transferred within line Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

2

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

of sight distances. In addition the transmission quality was strongly restricted by atmospheric disturbances. This form of visual communication was extended and used in the form of flags and signal lamps until the early 1790s, when a French scientist, Claude Chappe [2] suggested a system of semaphore stations. Messages were first translated into a sequence of visual telegraphs. These were then transmitted between tall towers which could be as far as 32 km apart. These towers acted as regenerators or repeaters such that messages could be transmitted over a longer distance. However, this method was slow and costly since messages had to be verified between each tower. With the beginning of a modern understanding of electricity in the 19th century, scientists started to investigate how electricity might be used in long distance communication. The telegraph [3] and telephone [4] were two inventions best representing this early stage of the electrical communication era. During that period of time, optical communication in the atmosphere received less attention and the systems developed were slow and inefficient. The lack of suitable optical sources and transmission media were two factors that hindered the development of optical communication. It was not until the early 1960s when the invention of laser [5] once again stimulated interest in optical communication. A laser source provides a highly directional light source in which photons generated are in phase with one another. By modulating the laser, the coherent, low divergence laser beam enables the development of optical communication. Due to the atmospheric attenuation, however, laser use is restricted to short distance applications. Long distance communication employing laser sources became feasible after a breakthrough was reached in 1966 when Kao and Hockham [6] and Werts [7] discovered the use of glass-based optical waveguides. By trapping light along the central core of the cylindrical waveguide, light confined along the optical fibre could travel a longer distance as compared with atmospheric propagation. Despite the fact that the attenuation of the optical fibre used was so high, with virtually no practical application at that time, this new way of carrying optical signals received worldwide attention. With improvements in manufacturing techniques and intensive research, the attenuation of optical fibre continued to drop. Fibre loss of about 4.2 db km1 was reported [8] for wavelengths around 1 mm, whilst low-loss fibre jointing techniques also became available. In order to build an optical communication system based on optical fibres, researchers in the 1960s started focusing on the development of other optical components including optical sources and detectors [9–11]. A new family of optical devices based on semiconductor junction diodes was developed. By converting electrical current directly into a stream of coherent photons, semiconductor lasers are considered to be reliable optical laser sources. Based on similar working principles, efficient photodetectors based on the junction diode were developed. By responding to optical power, rather than optical electromagnetic fields, optical signals received are converted back into electrical signals. In this early phase of development, semiconductor lasers used were restricted to pulse operation at a very low temperature. It was not until the 1970s that practical devices operating in continuous wave at room temperature became feasible [12]. The availability of both low-loss optical fibre and reliable semiconductor-based optical devices laid the cornerstone for modern lightwave communication systems. In the late 1970s, lightwave systems were operated at 0.8 mm [13]. Semiconductor lasers and detectors employed in these systems were fabricated using alluminium gallium arsenide alloy AlGaAs [14]. Optical fibres used had a large core of diameter between 50 and 400 mm whilst typical attenuation was about 4 dB km1. At the receiver side of the system, direct detection was

HISTORICAL PROGRESS

Figure 1.1

3

Attenuation of silica-based optical fibre with wavelength (after [44]).

used in which optical signals were directly converted to baseband optical signals. The overall system performance was limited by the relatively larger attenuation and inter-modal dispersion of the optical fibre used. In order to reduce the cost associated with the installation and maintainence of electrical repeaters used in the lightwave communication systems, it was clear that the repeater spacing could be improved by extending the operating wavelength to a new region between 1.1 and 1.6 mm where the attenuation of the optical fibre was found to be smaller. Figure 1.1 shows the relation between the attenuation of a typical SMF and optical wavelength. For systems operating at a longer wavelength, semiconductor optical devices were fabricated using quantenary InGaAsP alloy. In order to avoid inter-modal competition associated with high-order oscillation modes inside the optical fibre, optical fibres having a smaller core diameter of about 8 mm were used. In this way, oscillation in an optical fibre was reduced to single mode. For systems operating in such a longer wavelength region, both wavelengths at 1.3 and 1.55 mm have received a lot of attention. For systems operating near 1.3 mm, it was found that the single-mode fibre used had minimum dispersion, and hence maximum bandwidth could be achieved. In the early 1980s, many systems were built using singlemode fibre at around 1.3 mm wavelength. An even lower fibre attenuation of about 0.2 dB km1 is found at around 1.55 mm. However, the deployment of lightwave systems in the 1.55 mm region was delayed due to the intrinsic fibre dispersion which limits the maximum bit rate the system can support. The problem was later alleviated by adopting dispersion-shifted or dispersion-flattened fibre [15,16]. Alternatively, semiconductor lasers oscillating in single longitudinal modes were developed [17,18]. By limiting the spread of the laser spectrum, this type of laser is widely used in upgrading the 1.3 mm lightwave

4

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

systems to 1.55 mm wavelengths in which conventional single-mode fibres were used. Since 1988, field trial tests for coherent lightwave communication systems have been carried out [19–21]. In order to improve the bit rate of the present lightwave system whilst utilising available fibre bandwidth in a better way, frequency division multiplexing (FDM) schemes [22] were implemented. Before information is converted into optical signals, electronic multiplexing is often applied in combining the signals. Such a system is normally referred to as coherent optical communication since heterodyne or homodyne detection is used at the receiver end. By mixing the incoming optical signal with an optical local oscillator, coherent detection employs a different technique as compared with the direct detection method. In the 1980s, the development of coherent optical communications was hindered due to poor spectral purity and frequency instability in semiconductor lasers. Due to advances in fabrication techniques, semiconductor lasers nowadays show improved performance. In long-haul optical fibre communication systems, fibre dispersion and intrinsic attenuation are two major obstacles that affect the system performance. In the 1990s, optical fibre communication systems continued to develop in order to tackle these obstacles. To circumvent the fibre dispersion, the non-linear optical soliton able to travel extremely long distances was proven both theoretically [23,24] and experimentally [25,26]. By using optical amplifiers [27,28] as pre-amplifiers, post-amplifiers and optical repeaters, one witnesses the deployment of these wideband amplifiers in optical communication networks. In the coming years, networks employing a densely spaced wavelength division multiplexing (WDM) scheme [29] are expected. As a result, more channels and hence more information will be transmitted over a single optical fibre link. There is no doubt that a new paradigm of communication comprising an optically transparent network is already on the way [30].

1.3

OPTICAL FIBRE COMMUNICATION SYSTEMS

By transferring information in the form of light along an optical fibre, a communication system based on optical fibres starts to grow rapidly. This system, like many other communication systems, consists of many different components. A simple block diagram as shown in Fig. 1.2 represents the various components required in an optical fibre communication system. At the transmitter side, information is encoded, modulated and is then converted into a stream of optical signals. At the receiver side, optical signals received are filtered and demodulated into the original information. For long distance applications, repeaters or regenerators have to be used to compensate the intrinsic attenuation of optical fibre. In order to maximise the amount of information that can be transferred over a single optical fibre link, various multiplexing schemes might also be applied. To ensure successful implementation of optical fibre communication links, careful planning and system consideration is necessary. Apart from the performance characteristics of every component used within the system, it is also necessary to consider interactions and compatibility between various components. Depending on the system requirements, the type of transmission (analogue or digital), required transmission bandwidth, cost and reliability, may vary from one system to another. According to the type of detection method used at the receiver end, it is common to categorise an optical fibre system into either a direct detection or a coherent detection scheme.

OPTICAL FIBRE COMMUNICATION SYSTEMS

5

Figure 1.2 Simple block diagram showing various components for optical fibre communication systems.

1.3.1 Intensity Modulation with a Direct Detection Scheme Simply by varying the biasing current injected into a semiconductor laser diode at the transmitter, the so-called intensity modulation with direct detection (IM/DD) scheme was widely adopted. The expression ‘intensity modulation’ derives from the fact that the intensity of the light emitted at the transmitter side is linearly modulated with respect to the input signal for either digital or analogue systems. The expression ‘direct detection’ is used because the optical detector at the receiver end responds to optical power, rather than electromagnetic fields as compared to radio or microwave links. In other words, all optical signals received at the optical detector are demodulated into baseband electrical signals. Due to its simplicity and low cost, the IM/DD transmission scheme has had great success, in particular in point-to-point transmission systems. In order to explore the potential of the optical spectrum, however, coherent detection has to be used.

1.3.2 Coherent Detection Schemes Compared to the IM/DD transmission scheme, coherent optical communication [31–33] is characterised by mixing the incoming optical signal with the local oscillator so that the baseband signal (for homodyne detection) or an intermediate frequency (IF) signal (for heterodyne detection) is generated at the receiver. Since spatial coherence of the carriers and local oscillators is exploited, the expression ‘coherent’ is used to describe such a system configuration. The advantages of coherent detection have long been investigated and were recognised in the 1960s [34], but it was not until the late 1970s that single-mode transmission from an AlGaAs semiconductor laser was demonstrated [35,36]. With a

6

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

narrower spectral output, fibre-based lightwave systems employing coherent detection became feasible. Various digital modulation methods have been used in coherent optical communication, including the amplitude-shift keying (ASK), the frequency-shift keying (FSK) and the phase-shift keying (PSK) methods [37,38]. They differ from one another in the way digital messages can be transmitted by variations in amplitude, frequency and phase, respectively. For any digital transmission scheme and receiver architecture, a bit error rate (BER) in the region between 109 and 1010 must be achieved at the receiver side for a satisfactory transmission. The coherent optical communication system using homodyne/heterodyne detection has several advantages over the IM/DD transmission scheme [39,40]. First of all, coherent detection can improve the receiver sensitivity by about 15 to 20 dB, depending on the modulation scheme adopted. As a result, spacing between repeaters is improved for long distance communication, whilst transmission rates can be increased in existing long distance links without reducing the repeater distance. Moreover, by using modulation like PSK or FSK, which are well known in communication theory, the receiver can push to reach the ideal quantum noise detection limit. In addition, by adopting densely spaced frequencydivision multiplexing (FDM) or wavelength division multiplexing (WDM), a wider fibre bandwidth can be utilised. In practice, however, the coherent optical system has a stringent requirement for device performance. In Fig. 1.3, a general block diagram for the coherent optical communication system is shown. As illustrated in Fig. 1.3, two injection lasers are involved in the system. One acts as a transmitter and the other as a local oscillator. The laser transmitter which acts as an optical frequency oscillator can be used directly in the FSK transmission. An external modulator is optional for the ASK and the PSK transmission before the optical signals are launched into the single-mode fibre (SMF). Optical amplifiers like semiconductor laser amplifiers (SLA) or erbium-doped fibre amplifiers (EDFA) are used in long distance transmission for boosting the signal. Under the heterodyne receiver category with non-zero IF frequency, two different types of postdetection process have been adopted. The name heterodyne receiver with coherent postdetection processing (HE/CP) is usually given to one that has IF carrier recovered at the receiver. Similarly, heterodyne receiver with incoherent postdetection processing (HE/IP) describes the system that has no IF carrier recovered. Comparatively, the HE/IP receiver

Figure 1.3

Schematic diagram for the coherent optical communication system.

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

7

configuration is the simplest as IF carrier reconstruction is unnecessary. However, it shows the weakest receiver sensitivity among the three receiver designs. The incoherent postdetection process could be used in conjunction with several modulation schemes such as ASK, FSK and differential phase-shift keying (DPSK). In the HE/CP receiver design, IF signals are recovered at the receiver stage for further signal processing. The coherent postdetection process can improve the receiver performance and so it is applicable to any modulation method. However, it is substantially more complicated than the incoherent method and stringent device performance is required. For zero IF frequency, the homodyne receiver has the best receiver sensitivity as data is recovered directly from the optical mixing process at the receiver. A narrower receiver bandwidth and only baseband electronic processing are required. These offer significant advantages to the homodyne receivers. In practice, however, the technologies required in achieving these advantages in the homodyne receiver are demanding. An effective synchronous demodulation process is essential in phase locking the local oscillator and the received optical signal. Phase jitters caused by phase noise and shot noise could impair the system performance easily. It has been evaluated [41] that the phase variance must be limited to within  10 to ensure a lower power penalty for a BER  109. This sets an upper limit on the permissible laser spectral linewidth and other laser performance characteristics. In the coming sections, we are going to discuss some fundamental device characteristics and their impact on system performance.

1.4

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

1.4.1 Spectral Purity Requirements An ideal monochromatic laser source has been needed for some time. As a result, the spectral purity of the laser source has often been the first issue confronting users of semiconductor lasers in coherent optical communication. Due to the dispersive nature of the optical fibres used, digital pulses are broadened whilst propagating along the optical fibre. Such pulse spreading causes adjacent pulses to overlap so that errors occur as a result of inter-symbolic interference (ISI). Thus, apart from the power limitation due to the intrinsic fibre attenuation, the transmission distance is also limited by dispersion. The use of single-mode fibres has eliminated the severe inter-modal dispersion of multimode fibres. However, because of the finite spectral width of the optical sources, singlemode fibre is limited by chromatic dispersion (or intra-modal dispersion). Since the laser sources do not emit a single frequency but a band of frequency, each frequency component of the field propagates with a different time delay in the single-mode fibre, causing a broadening of the initial pulse width and hence intra-modal dispersion. The delay differences in single mode fibre may be caused by the dispersive properties of material through variation in the cladding refractive index (material dispersion) and also the guidance effects within the structure (waveguide dispersion). In order to minimise the effect of dispersion in singlemode fibre and hence improve the transmission distance, there are two different approaches. The first method involves the use of a dispersion-shifted or dispersion-flattened fibre. With a distinctive refractive index profile, these fibres can reduce the effect of dispersion at the

8

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

1.55 mm wavelength significantly. Another possible way involves the improvement of semiconductor laser sources. The ability to lase in single mode with a narrow linewidth can circumvent the effect of dispersion. In the rest of this section, the concept of single-mode operation, especially the possibility of a single longitudinal mode, will be discussed, whilst the impact and the control of spectral linewidth will be left for later sections.

(a) Single mode along the transverse plane It was shown in the previous section that the coherent optical communication system requires semiconductor lasers that can emit at a monochromatic frequency in order to achieve the required system BER. As a result, it is necessary to achieve a single-mode oscillation in each of the orthogonal directions inside the laser cavity. To understand the transverse waveguiding problem of semiconductor lasers, one must start with electromagnetic wave theory the basis for the study of electromagnetic wave propagation is provided by Maxwell’s equations [42]. For a medium with zero conductivity ~ the vector relationships may be written in terms of the electric field ~ E and magnetic field H as ~ r~ E ¼ j!H ~ ¼ j!"~ rH E

ð1:1Þ ð1:2Þ

where " and  are the permittivity and permeability of the medium. The above equations are expressed in the time harmonic form (with time variation term as e j!t ) and are true for source free and lossless media. By using the vector identity and taking the curl on both sides of eqn (1.1), one can arrive at the scalar wave equation for the electric field E such that r2 ~ E ¼ k2~ E ¼ !2 " ~ E 2 2 ~ ¼ k0 n ðx; yÞE

ð1:3Þ

where k is the propagation constant in the medium with the refractive index distribution of n (x,y) and k0 is the free space propagation constant. Similarly, by taking the curl on both sides of eqn (1.2), one ends up with the scalar wave equation for the magnetic field H. ~ þ k20 n2 ðx; yÞH ~¼0 r2 H

ð1:4Þ

Either eqn (1.3) or (1.4) can be used to determine the field components as they are related to one another by the Maxwell equations. Nevertheless, the scalar wave equation for the electric field is often used as the electric field is responsible for most physical processes and it is the principal field used by photodetectors. To determine the transverse modal field of the semiconductor laser, one must first find the thickness and the refractive indices of materials used in the fabrication process. Depending on the specific laser structure, it is quite possible to have three or four epitaxial layers lying on top of and below the active layer of the semiconductor laser. These laser structures may look complicated at first glance. In fact, their waveguiding properties can be explained with the use of a three-layer dielectric slab (or planar) waveguide. As shown in Fig. 1.4, the

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

9

Figure 1.4 Schematic cross-section of a slab dielectric waveguide. Refractive indices of different regions are shown.

asymmetric waveguide consists of three layers. The active layer, having refractive index n1 and thickness d, is sandwiched between the substrate and the cladding of the waveguide. Without loss of generality, it is assumed that the refractive indices of the slab waveguide obey the following inequality n1 > n2  n3

ð1:5Þ

where the equal sign implies a symmetrical waveguiding structure. With such a planar structure, the field variation along the y-axis can be ignored and so @=@y ¼ 0. By separating the Maxwell equations into different field components, the following equations are obtained [42] @Ey @z @Ex @Ez  @z @x @Ey @x @Hy  @z @Hx @Hz  @z @x @Hy @x

¼ j!Hx

ð1:6aÞ

¼ j!Hy

ð1:6bÞ

¼ j!Hz

ð1:6cÞ

¼ j!"Ex

ð1:6dÞ

¼ j!"Ey

ð1:6eÞ

¼ j!"Ez

ð1:6fÞ

The direction of wave propagation has always been assumed to be the longitudinal z direction. By inspecting the above equations carefully, one can separate the above equations into two groups. The first group includes Ey , Hx and Hz from eqns (1.6a), (1.6c) and (1.6e). The results generated from these equations are referred to as the TE mode since the electric field is found along the transverse y-axis (normal to the propagation direction). The other group includes Hy , Ex and Ez , which generates solutions for the TM mode. An inspection of the structure shows that either the TE or the TM mode is supported, but not both simultaneously. Since there is no physical boundary along the y direction, the continuity condition allows only Hz or Ez to exist.

10

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

For a travelling wave propagating along the z direction, the electric field takes the form Eð~ r; tÞ ¼ Eð~ x;~ yÞejbz~z

ð1:7Þ

where ~ r is the radial vector in space ðx; y; zÞ and the time harmonic term is omitted here for the sake of simplicity. bz is the propagation constant at a fixed angular frequency ! which can also be written as bz ¼ k0 neff

ð1:8Þ

with neff being the effective refractive index. The electric field component Ey for different layers in the slab dielectric can be obtained by substituting eqn (1.7) into eqn (1.3) and putting @=@y ¼ 0. This is @2 Ey ðxÞ ¼ ðk20 n21  b2z ÞEy ðxÞ ¼ h2 Ey ðxÞ; d  x  0 @x2 @2 Ey ðxÞ ¼ ðk20 n22  b2z ÞEy ðxÞ ¼ p2 Ey ðxÞ; 0x<1 @x2 2 @ Ey ðxÞ ¼ ðk20 n23  b2z ÞEy ðxÞ ¼ q2 Ey ðxÞ; 1 < x  d @x2

ð1:9aÞ ð1:9bÞ ð1:9cÞ

where h, p and q are constants defined as h2 ¼ k20 n21  b2z p2 ¼ b2z  k20 n22 q ¼ 2

b2z



ð1:10Þ

k20 n23

Depending on the relative values of n1 , n2 , n3 , k0 and bz , there are different regimes of propagation constant as shown in Fig. 1.5.

Figure 1.5

Different types of modal solutions determined by the constants p, q and h. (after [44]).

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

11

If p2 and q2 are negative whilst h2 is positive (i.e. 0 < bz < k0 n3 ), the propagation is said to be in the continuous radiation regime and the field solutions show sinusoidal behaviour in all three layers. For a guided mode to occur, the constants p2 , q2 and h2 must all be positive. In other words, the inequality k0 n2 < bz < k0 n1 holds so that sinusoidal oscillation is restricted to the central active layer while the electric field is decaying exponentially in other layers. For the TE mode, a general formulation of Ey ðxÞ takes the form Ey ðxÞ ¼ Aeqx ;

0x<1

¼ A cosðhxÞ þ B sinðhxÞ; ¼ ðA cosðhxÞ þ B sinðhxÞÞe

d  x  0 pðxþdÞ

;

ð1:11Þ

1 < x  d

where A and B are two arbitrary constants that are related to the power of the propagating mode. Since the tangential fields must be continuous at the dielectric interface, by rearranging eqn (1.6c) such that, Hz ¼

j @Ey ! @x

ð1:12Þ

the boundary conditions of the magnetic field Hz at x ¼ 0 and x ¼ d become Aq þ Bh ¼ 0 A½h sinðhdÞ  p cosðhdÞ þ B½p sinðhdÞ þ h cosðhdÞ ¼ 0

ð1:13aÞ ð1:13bÞ

On combining the above equations and eliminating the arbitrary constants A and B, we arrive at the eigenvalue equation for the guided mode [43] tanðhdÞ ¼

ðp þ qÞh h2  pq

ð1:14aÞ

From the above equation, it is clear that the active layer thickness d is decisive in determining the guided mode. By replacing n3 with n2 (hence q ¼ p), the slab waveguide becomes symmetrical and the above eigenequation becomes tanðhdÞ ¼

2ph 2p=h ¼ 2 h p 1  p2 =h2 2

ð1:14bÞ

By equating the last eigenvalue equation with the half angle trigonometric identity of tanðxÞ ¼

2 tanðx=2Þ 1  tan2 ðx=2Þ

ð1:15Þ

one can solve the quadratic equation in terms of tan ðhd=2Þ such that   hd p tan ¼ 2 h   hd h tan ¼ 2 p

ð1:16aÞ ð1:16bÞ

12

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

Equations (1.16a) and (1.16b) represent the even and odd TE modes of the slab dielectric waveguide. In order to solve the eigenequation for the TE mode, an extra equation is necessary. By equating the propagating constants in eqns (1.10), one could obtain [44] 

pd 2

2  2  2 hd 2 2 2 d þ ¼ ðn1  n2 Þk0 ¼ D2 2 2

ð1:17Þ

where D is defined as the normalised waveguide thickness. Therefore, for a guided mode, the constants p and h must satisfy both eqns (1.14b) and (1.17). Using a graphical method, these equations can be solved. Clearly, eqn (1.17) represents a circle with radius D on the ðhd=2Þ  ðpd=2Þ plane. By putting in the odd and the even TE mode eigenequations in the same ðhd=2Þ  ðpd=2Þ plane, we obtain the plot shown in Fig. 1.6. Each intersection (or solution) with p > 0 and h > 0 between the circles and the tangent function corresponds to a guided mode, from which the propagation constant could be determined according to eqn (1.15). Due to the periodicity of the trigonometric function, multiple modes may occur as the frequency, and hence the radius of the circle, keeps increasing. From Fig. 1.6, there is only one intersection point between the circle and the tangent functions when the normalised waveguide thickness D has value 0
 2

ð1:18Þ

Figure 1.6 Graphical method to solve the eigenequation for a symmetrical three-layer dielectric waveguide. (after [44]).

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

13

This mode, designated as the TE1, is the fundamental mode excited in the slab dielectric waveguide. By expanding D according to eqn (1.17), the inequality shown above becomes 0

2 ðn  n22 Þ1=2 2 1

ð1:19Þ

This inequality limits the maximum active layer thickness for single transverse mode excitation. Similarly, by equating the boundary conditions for the TM mode, we end up with the following eigenvalue equation for the asymmetric waveguide tanðhdÞ ¼

n21 hðn23 p þ n22 qÞ n22 n23 h2  n41 pq

ð1:20Þ

tanðhdÞ ¼

2n21 n22 hp n42 h2  n41 p2

ð1:21Þ

and

for the symmetrical slab waveguide. Following the analysis of the TE mode, the even and the odd TM eigenequations could be written as    2 hd n1 p tan ¼ 2 n22 h  2   hd n h tan ¼  22 2 n1 p

ð1:22aÞ ð1:22bÞ

Using the same graphical technique, the same limitation for the active layer thickness is obtained for single transverse mode excitation in the TM mode. The only difference between the TE and the TM mode is the refractive index correction term added in the TM mode eigenequations for the symmetrical slab waveguide. For most semiconductor lasers having lateral confinement, the width of the active layer W is finite. With the width dimension found comparable to the active layer thickness, a rectangular waveguide is formed. Nevertheless, one can still follow a similar procedure to that applied to the slab waveguide in solving the field solution of the rectangular waveguide. Figure 1.7 shows the schematic cross-section of a rectangular waveguide. The central core

Figure 1.7 Schematic cross-section of a rectangular waveguide. Refractive indices of the different regions are indicated.

14

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

Figure 1.8 Procedures in analysing rectangular waveguides using EIM. (a) The first slab approximation along the x-axis; (b) the second slab approximation along the y-axis with neff used.

region, which is surrounded by four cladding regions, has the highest refractive index ðn1 Þ. With the propagation mode mainly confined to the central core, the field penetration in the corners (shaded regions in the figure) of the structure can be neglected [45]. An exact analytical solution for the propagation characteristics of the strip waveguide is not possible and a certain degree of approximation is necessary. A very convenient technique known as the effective index method (EIM) can be used in providing an accurate analytical solution for the rectangular waveguide. In this method, the equivalent slab waveguide in one direction is solved first (see Fig. 1.8) so that the effective refractive index of the central slab can be generated. This effective refractive index is then used to solve for the other slab waveguide which is found perpendicular to the original one. In this way, the solutions for the two slab waveguides are coupled. In a similar way to the slab waveguide analysis, the number of excited transverse modes is determined by both the active thickness (d) and width (w) of the central core region. Therefore, it can be shown that proper controls over the active thickness and width are necessary for single transverse mode operation. To show how the EIM works, a symmetrical rectangular waveguide is used instead. With n3 ¼ n2 and n5 ¼ n4 , we can solve for the central slab first by using the graphical approach shown earlier in the TE mode of the slab waveguide. For a single transverse mode along the x-axis, one ends up with the following inequality [44] d  D ¼ k0 ðn21  n22 Þ1=2 < 2 2

ð1:23aÞ

or d<


2 ðn  n22 Þ1=2 2 1

ð1:23bÞ

with D being the normalised waveguide thickness as defined earlier. Then, the second slab waveguide analysis is carried out along the perpendicular y-axis. Based on the TM mode analysis of the slab waveguide, one ends up with [46] W ¼ k0

w 2  ðn  n24 Þ1=2 < 2 eff 2

ð1:24aÞ

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

15

or w<


2 ðn  n24 Þ1=2 2 eff

ð1:24bÞ

for single lateral mode excitation (along the y-axis) with W being the normalised waveguide width. In eqn (1.24), the effective refractive index neff which was derived earlier from the first slab approximation, is used instead. So far, a purely index-guided structure has been used, where all possible gain or loss in the structure has been ignored. Nevertheless, the single-mode operating conditions are still feasible since the imaginary part of the refractive index due to gain or losses has a much smaller magnitude when compared with the corresponding real refractive indices. As a result, there is a slight difference between the field solutions of the purely index-guided structure and the one that processes gain/loss [28]. The description of the waveguiding properties in the index guided waveguide is now complete. If the active region dimensions are chosen to satisfy both eqn (1.23b) and eqn (1.24b), only a single mode along the transverse plane is supported.

(b) Single longitudinal mode (SLM) In semiconductor lasers, electron movements occur between two energy bands that consist of a finite number of discrete energy levels. Rather than a discrete energy transfer like the gaseous laser, semiconductor lasers are characterised by a wider gain spectrum. In an inhomogeneously broadened laser, the gain spectrum may be found several times wider than the longitudinal mode spacing. In the simplest type of optical resonator, better described as the Fabry–Perot (FP) cavity, photons escape from a cavity which has two partially reflected mirrors facing one another. Without any frequency discrimination, photons at any frequency could escape from the cavity. As a result, it is common for several oscillation modes, as depicted in Fig. 1.9, to be observed; this is particularly true when it is under rapidly pulsed operation. As will be shown in detail in chapter 2, the longitudinal mode spacing between each frequency component in the FP semiconductor laser cavity is given as f ¼

c 2ng L

ð1:25Þ

where c is the free space velocity, ng is the group refractive index and L is the laser cavity length. In order to have a better control over the longitudinal modes in the FP cavity, several methods have been proposed. By using a shorter cavity one can increase the mode spacing. However, a higher threshold current is expected and several longitudinal modes may occur at severe modulation. Besides, as we will discuss later, such a shorter cavity will lead to a wider spectral width [17]. A different method involves an external light injection from another laser [47]. In this way, single longitudinal mode (SLM) oscillation can be achieved. However, careful tuning is essential in order that the lasing modes of the two lasers be matched [48]. By adding a reflecting surface outside the FP cavity, an external coupled cavity has also been used [49]. Due to the interference with the external cavity, the overall modal loss becomes frequency dependent and hence limits the oscillation mode from the FP

16

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

Figure 1.9 (a) Possible gain profile of an inhomogeneously broadened laser; (b) the resulting intensity spectrum (after [44]).

laser cavity. On the other hand, both the external light injection method and the external coupled cavity laser lack long-term stability. These devices are vulnerable to mechanical vibration, temperature and pressure changes and hence the lasing mode tends to shift from time to time. While they are useful in coherent system experiments, it has become evident that they are difficult to realise as commercial products. To include a wavelength or frequency selective mechanism, one popular technique is to include an etched diffraction grating within the laser waveguide structure. The grating-based single-frequency lasers are classified into two categories. If the active layer and the grating extend along the whole length of the laser cavity, the device is known as a distributed feedback (DFB) laser. If the grating or feedback sections are passive such that the gain region is located in a separate planar gain section, a distributed Bragg reflector (DBR) laser structure is formed. In Fig. 1.10, the structural difference between the DFB and DBR lasers is illustrated. One of the most convenient methods of generating gratings is to use a holographic exposure technique [12]. With a two arm interferometer set-up, the interference pattern of two coherent lightwaves cross each other at a predetermined angle, and a periodic

Figure 1.10

Schematic diagrams of (a) DFB and (b) DBR laser diodes.

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

17

grating is formed. Other methods like electron beam lithography are also possible. Then, followed by the conventional wet chemical etching or the dry ion beam etching process, the grating pattern is delineated on the substrate. In general, the grating can be placed under the active layer or in the upper cladding layer. In DBR lasers, the Bragg grating sections are separated from the active section where major carrier recombination occurs. In other words, the frequency at which the grating section reflects does not depend on the bias current and so non-linear influence on the guided refractive index (due to injected carriers) is rare. However, the use of end passive gratings in DBR lasers implies extra etching processes during fabrication. Besides this, the effective length of the planar section, which is crucial in deciding the oscillation frequency of DBR lasers, is difficult to control precisely. Due to the tolerance inherited in fabrication, the relative grating phases of the two Bragg reflectors become unpredictable. In general, DBR lasers are more complex, and effective measures are needed in tackling the problem of yield and reliability. In order to suppress any possible FP mode in these grating structures, it is quite usual to have anti-reflective coatings on the laser facets of DFB and DBR lasers. The working principle of a uniform-grating DFB laser with perfectly anti-reflective facet coatings (sometimes called a conventional DFB laser) was first explained by Kogelnik and Shank in 1972 [50] using coupled wave theory, but it was not until 1975 [51] when the first DFB laser that operates at room temperature (300 K) under continuous wave (CW) operation was built. Details of coupled wave theory will be covered in the next chapter. Normally in DFB lasers, the grating period L must be carefully chosen to satisfy the Bragg condition L¼

m
B 2ng

ð1:26Þ

where ng is the usual group refractive index and
B is sometimes called the Bragg wavelength. For a 1.55 mm InGaAsP laser, with a first-order grating ðm ¼ 1Þ, a typical value of ng is 3.4 and L is 0.23 mm. Figure 1.11 shows the difference between the operation of FP and DFB lasers. For DFB lasers, the longitudinal mode which is found closest to the Bragg wavelength will lase, while other side modes having larger losses are severely suppressed as a wavelength/frequency-dependent loss is introduced. The net gain difference (or gain margin) between the dominant lasing mode and the most probable side mode is found to be much higher than that for FP lasers.

1.4.2 Spectral Linewidth Requirements (a) Origin of phase noise and the formulation of spectral linewidth Due to the dispersive nature of the optical fibre, laser sources that operate at a single frequency will be necessary for long-haul transmission at speeds higher than a few GHz. However, even the output spectrum of single-frequency semiconductor lasers (such as DFB semiconductor lasers) is never ‘pure’ but always contaminated by a finite spectral width. The finite spectral width is mainly due to the random phase of the spontaneous emission that couples into the lasing mode. Consequently, the fluctuation in gain broadens the spectral linewidth as a result of lasing frequency shift [52]. To measure the spread of the laser spectrum, the spectral linewidth ð
Þ which is defined as the full width half maximum

18

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

Figure 1.11 The typical gain, loss and spectral profiles of (a) A Fabry–Perot laser diode; (b) a distributed feedback (DFB) laser diode.

(FWHM) of the optical power spectrum is used. Based on the delayed self-heterodyne technique [53], the spectral linewidth can be measured easily. To understand the spectral behaviour of laser light, one can use the classical description [54]. Consider a laser with a spatial mode FðxÞ, the transverse optical field of the laser (which is a real quantity) could be expressed as Eðx; tÞ ¼ B½ ðtÞFðxÞ þ ðtÞ
FðxÞ


ð1:27Þ

where ðtÞ is the time dependence complex wave amplitude and B is a constant. The asterisk * indicates the complex conjugate. Usually, the complex amplitude ðtÞ can be expressed in terms of two real quantities: intensity IðtÞ and phase ðtÞ such that [54,55]

ðtÞ ¼ IðtÞ1=2 e j’

ð1:28Þ

where the time harmonic term expðj!0 tÞ is omitted. The real factor B is chosen such that the energy term which is found proportional to the product


, is determined by the expression given by  h!0 IðtÞ. As a result, IðtÞ as shown in the previous equation corresponds to the total number of photons found inside the laser cavity. To understand the origin of phase noise, one must understand the nature of spontaneous emission. There are mainly two mechanisms which lead to the change of phase in semiconductor lasers. 1.

The first contribution to phase noise ð
0 Þ comes when a photon is injected spontaneously into the lasing mode. Both the intensity and phase of the optical field in

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

19

the mode will undergo changes. These fluctuations of field due to the spontaneous emission will contribute directly to phase noise. 2.

In semiconductor lasers, there is a second contribution to phase noise ð
00 Þ associated with the intensity change as described in mechanism 1 above. According to the Kramer– Kroenig relation [56], any change of gain inside a semiconductor will alter the refractive index of the semiconductor. Consequently, the optical field experiences an additional phase change, which induces extra linewidth broadening. Note that this is not readily observed in a stable CW gas laser, since the time taken for the intensity to return to its initial steady state value is shorter than that of the semiconductor laser (of the order of 109s). Within such a short period of time, the above processes rarely happen in gas lasers and so the spectral linewidth of the gas laser is narrower than that of the semiconductor laser.

In Fig. 1.12, a phasor diagram that follows Henry’s work [54,55], shows how an arbitrary i-th spontaneous emission can produce a change
I in the intensity (or photon number) and a phase change of
0i as described by mechanism 1 above. Using simple geometrical considerations, it can be shown that the changes of intensity and phase can be written as
i ¼ I 1=2 sin i

ð1:29Þ


Ii ¼ 1 þ 2I

ð1:30Þ

1=2

cos i

Figure 1.12 The spontaneous change of phase and intensity of the optical field induced by the i-th spontaneous event (after [54–55]).

20

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

where the i-th spontaneous photon changes the complex wave amplitude ðtÞ by
ðtÞ, where
ðtÞ is represented by a unit magnitude and a random phase of i . In this simple analysis, it is assumed that a single photon is emitted for each spontaneous emission. Besides this, over a large number of spontaneous events, the average phase change is assumed to be zero. In order to understand the contribution to phase noise of the second contributing element
00 , it is necessary to determine the mutual coupling between the intensity and the phase of the optical mode. From the time-dependent wave equation, one ends up with the following rate equation for the complex wave amplitude [54,55] d vg ðGg  0 Þ ð1 þ jH Þ ¼ 2 dt

ð1:31Þ

where vg is the group velocity, G is the optical confinement factor, g is the amplitude gain coefficient due to material gain and 0 is the cavity absorption. In the equation shown above, H is the intrinsic linewidth enhancement factor [57–59] defined as H ¼ 

d00 ðNÞ=dN d0 ðNÞ=dN

ð1:32Þ

where 0 and 00 are the real and imaginary parts of the susceptibility, while N is the injected carrier concentration. In all practical cases, both the carrier-induced refractive index change and that induced by the imaginary part of permittivity are much smaller than the intrinsic refractive index of the semiconductor. As a result, the intrinsic linewidth enhancement factor can also be expressed as H ¼ 2k0

dn=dN dg=dN

ð1:33Þ

where the relative change of refractive index (n) and gain (g) due to variations in injected carrier concentration (N) are considered. The negative sign included in the above equations is important in keeping H positive as the refractive index is found to have a negative change with increasing carrier concentration [58] (i.e. dn=dN is negative). The intrinsic linewidth enhancement factor was first introduced by Henry [54] and used to explain the experimental results of Fleming and Mooradian [60] on AlGaAs semiconductor lasers. In the case of InGaAsP/InP devices, the value of H ranges from 4.5 to 7 [58,61]. By examining eqn (1.32), it is obvious that the intrinsic linewidth enhancement factor depends on several parameters of the medium. The H dependence on carrier concentration and photon density was first suggested by Vahala et al. [57]. At a fixed carrier concentration, they found that H decreases with increasing photon energy, while H increases with carrier concentration at a fixed photon energy. On the other hand, H remains constant at the peak gain for different values of carrier concentration and photon energies. The results for the quaternary InGaAsP with bandgap of  0:8 eV (or
 1:55 mm) are shown in Fig. 1.13. It is interesting to see how H changes when the lasing wavelength of the single-frequency laser (e.g. the DFB laser) is allowed to shift with respect to the peak gain. It can be seen from Fig. 1.13 that H becomes larger at wavelengths longer than the peak gain, and becomes

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

21

Figure 1.13 Dependence of linewidth enhancement factor  on photon energy for undoped 1.55 mm InGaAsP at various levels of electron concentration, n (after [49]).

smaller on the shorter wavelength side of the spectrum. The detuning effect of the DFB laser was confirmed by Ogita [62] and a narrower spectral linewidth was found at the shorter wavelength side of the gain peak. Besides carrier concentration and photon energies, the intrinsic linewidth enhancement factor also depends on temperature. With decreasing temperature, the change in refractive index with carrier concentration is large, and so is the gain variation. However, the gain variation is faster than that of the refractive index, and consequently H decreases as temperature drops. Since there is no possible way to measure H directly, it is still not clear to what extent the linewidth enhancement factor is related to structural change and other experimental uncertainty. Various methods like the AM/FM noise spectrum measurement [61], and the spectral linewidth measurement [63] have been used to find the value of H for DFB semiconductor lasers. However, the relative merits of these methods are still uncertain and so it is difficult to make any direct comparison between these results. On the other hand, it is believed that the intrinsic linewidth enhancement factor H is less structurally dependent in a purely index-guided structure than in a gain-guided structure. Due to the lateral guiding characteristics of the gain-guided structure, the variation of the real and imaginary part of the permittivity will not be the same in the lateral direction. In that case, an effective linewidth enhancement factor is used instead [64]. Detailed analysis of the effective linewidth enhancement factor will be discussed further in later chapters. In a similar way to eqn (1.31), the conjugate of the complex wave amplitude also satisfies the following equation d
vg ðGg  0 Þ ð1  jH Þ
¼ 2 dt

ð1:34Þ

22

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

From eqn (1.28) and its complex conjugate, one can express the intensity and phase in terms of the complex wave amplitude such that I ¼
1  ¼ lnð =
Þ 2j

ð1:35Þ

Therefore, by taking the derivative of eqn (1.35) and using eqns (1.31) and (1.34), one obtains dI ¼ vg ðGg  0 ÞI dt d H vg ðGg  0 Þ ¼ 2 dt

ð1:36Þ

Combining the above two equations gives d H dI ¼ 2I dt dt

ð1:37Þ

From eqn (1.37), it is clear that a change in intensity due to spontaneous emission will induce a second phase alternation. If the change in intensity and phase occurs within a short period of time, the second phase change
00 could be determined by taking an integral so that eqn (1.37) becomes H
Ii 2I H ð1 þ 2I 1=2 cos i Þ ¼ 2I


00i ¼ 

ð1:38Þ

For a laser operating at above threshold, the intensity changes instantly from I to I þ
Ii and then drops back to a final value of I after the relaxation oscillation has died out. Thus, a minus sign is included in the above equation as the intensity change becomes 
Ii . As a result, the total change in phase due to the i-th spontaneous emission becomes
 ¼
0 þ
00 H þ I 1=2 ðsin i  H cos i Þ ¼ 2I

ð1:39Þ

So far, the total phase change due to a single spontaneous photon has been considered. Suppose that a total number of M spontaneous emission events happen within a period of time t, then the total phase fluctuation is obtained by summing eqn (1.39) over M events such that
 ¼ 

M X H M þ I 1=2 ðsin i  H cos i Þ 2I i¼1

ð1:40Þ

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

23

Sometimes, instead of using the total number of spontaneous events, it is more convenient to use the spontaneous emission rate Rsp . By replacing M with Rsp t, the first term of the above equation will result in an average phase change of h
i ¼ 

H Rsp t 2I

ð1:41Þ

or a frequency shift of
! ¼

H Rsp d h
i ¼  dt 2I

ð1:42Þ

This frequency shift is solely caused by the spontaneous emission. From the summation term shown in eqn (1.40), the mean square phase variation term can be determined as h
2 i ¼

Rsp ð1 þ 2H Þt 2I

ð1:43Þ

as all phase crossing terms vanish. With Gaussian probability distribution assumed for the phase change, the power spectrum of the laser becomes Lorentzian in shape [55]. As a result, the spectral linewidth
becomes
¼

h
2 i Rsp ð1 þ 2H Þ ¼ 2 t 4 I

ð1:44Þ

To facilitate experimental measurements on the spectral linewidth of semiconductor lasers, it is better to express the intensity in terms of optical power such that I¼

ðP1 þ P2 Þ
vg m hc

ð1:45Þ

where P1 and P2 are the optical power output from the laser facets, whilst m is the mirror loss of the laser cavity. For a closed cavity, such as a Fabry–Perot laser with high reflecting ends, the total spontaneous emission rate can be expressed as Rsp ¼ vg G g nsp ¼ vg gth nsp

ð1:46Þ

where gth is the threshold gain and nsp is the population inversion factor defined as h!0 eV kT

1=nsp ¼ 1  e

ð1:47Þ

where eV is the distance between the quasi-Fermi levels of the conduction band and the valence band of the semiconductor lasers. In the above equation, nsp characterises the incomplete inversion of the laser transition. If there is complete population inversion, the spontaneous emission factor will become unity.

24

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

On substituting eqns (1.44) and (1.45) into eqn (1.43), the spectral linewidth becomes
¼

v2g nsp gth m

hc ð1 þ 2H Þ 4ðP1 þ P2 Þ


ð1:48Þ

where the intrinsic linewidth enhancement and the inversion factor nsp are assumed constants and homogeneous at least over the active region of the laser. Although the spectral linewidth formula above is derived for the FP type laser, it also holds for index-guided DFB semiconductor lasers [63,65], providing that the loss term m is interpreted as the output loss. In gain-guided lasers, maximum gain is found along the centre of the active layer. Rather than having a plane phase front, the phase fronts of gain-guided lasers are curved and so a wider far field spread is expected. Therefore, it is intuitively reasonable to assume that the lasing mode of gain-guided lasers may capture a larger fraction of spontaneous emission than the lasing mode of index-guided lasers. Such an enhancement in spontaneous emission Ktr in a transversely single-mode gain-guided laser was first proposed by Petermann [56] and is given here as  Ð Ð 2  j t j dx dy2   Ktr ¼  Ð Ð 2

dx dy 

ð1:49Þ

t

where t is the transverse electric field distribution of the lasing mode. Ktr is commonly defined as Petermann’s gain guiding factor. For index-guided semiconductor lasers having plane phase fronts, the transverse electric field is purely real and so Ktr becomes unity. Meanwhile, it has also been shown that the spontaneous emission coupled to the lasing mode is enhanced by the output coupling, due to the low facet reflectivities in semiconductor lasers. This oscillator loading effect was first noticed by Ujihara [66] and later Henry [67] but in a different format. According to Wang et al. [68], such a longitudinal correction factor Kz is defined as  Ð L  j 2 j dz 0 z  ð1:50Þ Kz ¼  Ð L   2 dz  0

z

where z is the longitudinal electric field distribution of the lasing mode and L is the laser cavity length. In order to define eqns (1.49) and (1.50), the laser itself must show a constant transverse mode pattern along the longitudinal direction so that the electric field can be separated into the transverse and longitudinal factors. Finally, the corrected spectral linewidth of eqn (1.48) becomes
 ¼

v2g nsp gth m

hc ð1 þ 2H Þ Ktr Kz 4ðP1 þ P2 Þ


ð1:51Þ

(b) System requirements In general, the requirement of spectral linewidth broadly depends on the modulation scheme, receiver design and transmission bit rate ðRb Þ of the system. Detailed analysis was carried

SYSTEM REQUIREMENTS FOR HIGH-SPEED OPTICAL COHERENT COMMUNICATION

25

Table 1.1 Laser spectral linewidth requirements for a coherent optical communication system. Rb is the transmission bit rate. One dB power penalty is assumed at the receiver (after ref. [69]) Receiver Type

IF processing

Modulation

D =Rb (%)

Heterodyne Heterodyne Heterodyne Heterodyne Homodyne

incoherent incoherent incoherent coherent coherent

ASK FSK DPSK PSK PSK

9% 2–9 % 0.165 % 0.226 % 0.031 %

out by Kazovsky [69] in which the maximum permissible laser linewidth for various coherent communication systems was reported. Both heterodyne (HE) and homodyne (HO) systems were considered. Table 1.1 summarises the results and compares different receiver designs. From Table 1.1, it is clear that the homodyne system, which provides the best receiver sensitivity, requires a laser linewidth of 0.031% Rb . With such a demanding spectral linewidth, it is more practical to switch to other schemes like the FSK system in which the linewidth requirement is more flexible. Inspection of Table 1.1 also reveals that the HE incoherent post-detection (ASK, FSK) type receivers are most robust (which permits the largest laser linewidth), followed by the HE coherent post-detection (PSK) type receivers and then by the homodyne (PSK) receivers. Figure 1.14 shows variations of the maximum permitted laser linewidth
with data bit rate for different optical communication systems (after [69]).

Figure 1.14

Comparison of maximum laser linewidth with the system transmission rate [after 69].

26

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

(c) Limitations of the linewidth formula According to eqn (1.51), the spectral linewidth formula for a single-mode semiconductor laser is inversely proportional to the total output power. However, experimental results reveal that even in a single-frequency laser such as a DFB laser, the spectral linewidth tends to saturate or rebroaden at high output power when the laser is severely biased [65,70]. These results were difficult to understand based on Henry’s linewidth formula. It was suggested [71,72] that the presence of other non-lasing side modes might contribute to the spectral linewidth for the single-mode DFB semiconductor laser. At above-threshold bias, a gain depression of the lasing main mode and hopping in lasing wavelength have also been observed [73,74]. The longitudinal correction term as shown in eqn (1.51) may not be a constant but may be sensitive to biasing current and the structural design of laser diodes. For DFB laser diodes operating at high biasing currents, such linewidth saturation or rebroadening phenomena have to be considered since the performance and reliability of a coherent optical fibre communication system depends greatly on the spectral linewidth of the laser source. A more comprehensive approach, which includes the variation due to structural changes and longitudinal variations of both carrier and photon densities will be discussed in a later chapter.

1.5

SUMMARY

From a historical prospect, the development of optical communication, in particular fibrebased communication systems, has been reviewed. By trapping light inside the core of a cylindrical fibre as a result of the total internal reflection, optical fibres with a low attenuation loss serve as an ideal medium for optical communication. At the transmitter side, information is first converted into a stream of photons, which are then launched into the optical fibre. The compact size semiconductor laser diodes serve this purpose by generating coherent photons. The availability of both single-mode optical fibre and reliable semiconductor optical devices paved the way for the global spread of optical fibre networks. According to the type of detection method used at the receiver end, one can categorise an optical fibre system into either a direct or a coherent detection scheme. For limited peer to peer applications, direct detection schemes can be very simple. However, with the increased demand in wide-bandwidth, long-haul applications, the coherent detection scheme is a better alternative. Using the spatial coherence of the optical carriers, coherent detection schemes show improved receiver sensitivity, whilst the available optical spectrum of optical fibre can be utilised in a better way. Using appropriate multiplexing schemes such as FDM or WDM, more information can be loaded into a single fibre link. In practice, however, optical fibre communication systems using the coherent detection scheme require a more careful design and demand stringent device performance. From the system requirement of coherent optical communication systems, the device characteristics of optical devices, in particular the semiconductor laser diodes, were briefly reviewed. In order to minimise the effect caused by the dispersive property of optical fibre, semiconductor laser diodes used should have a stable single-mode oscillation. By controlling the width and thickness of the active layer, transverse single-mode oscillation can be controlled easily. However, single longitudinal mode oscillation requires special attention.

REFERENCES

27

With the gain spectrum in semiconductors usually found to be broader than the longitudinal mode spacing, multiple mode oscillation in the conventional Fabry–Perot cavity is common. By incorporating a specially designed grating inside the laser cavity, a built-in wavelength selective mechanism is created. This wavelength filtering effect forms the core of the distributed feedback laser diodes. Due to the characteristics of semiconductors, a laser diode oscillating at single mode does not generate a pure monochromatic spectrum. The finite spectral spread, formally known as the spectral linewidth, has to be considered in high-speed optical links. At the end of this chapter, the origin and the system requirements on spectral linewidth inherent in semiconductor lasers were reviewed.

1.6

REFERENCES

1. Cochrane, P. and Heatley, D. J. T., Future directions in long-haul optical-fibre systems, Br. Telecom. Technol. J., 9, 268–280, 1991. 2. Microsoft Corporation, Microsoft Encarta 94, CD-ROM 1994. 3. Ryder, J. D. and Fink, D. G., Engineer and Electronics. New York: IEEE Press, 1984. 4. Bell, A. G., U.S. Patent 174456, 1876. 5. Maiman, T. H., Stimulated optical radiation in ruby, Nature, 187, 493– 494, 1960. 6. Kao, C. K. and Hockham, G. A., Dielectric fiber surface waveguides for optical frequency, Proc. IEE, 113(7), 1151–1158, 1966. 7. Werts, A. Propagation de la lumiere coherente dans les fibres optiques, L’Onde Electrique, 46, 967–980, 1966. 8. Takahashi, S. and Kawashima, T., Preparation of low loss multi-component glass fiber, Tech. Dig. Int. Conf. Integra. Opt. and Opt. Fiber Commun., p. 621, 1977. 9. Kressel, H. and Butler, J. K., Semiconductor Lasers and Heterojunction LEDs. New York: Academic Press, 1977. 10. Melchior, H., Fisher, M. B. and Arams, F. R., Photodetectors for optical communication systems, Proc. IEEE, 58, 1446–1486, 1970. 11. Newman, D. H. and Ritchie, S., Sources and detectors for optical fibre communications application: the first 20 years, IEE Pro., Pt. J, 133, 1213–229, 1986. 12. Adams, M. J., Steventon, A. G., Delvin, W. J. and Henning, I. D., Semiconductor Lasers for Long-Wavelength Optical-Fibre Communications Systems, IEE material & devices series, ed. Dr. N. Parkman and Prof. D. V. Morgan, no. 4, London: Peter Peregrinus, 1987. 13. Shumate, P. W., Chen Jr., F. S. and Dorman, P. W., GaAlAs laser transmitter for lightwave technology, Bell Syst. Tech. J., 57, 1823–1836, 1978. 14. Alferov, Z. I, Andreev, V. M., Korol’kov, V. I., Portnoi, E. L. and Tret’yakov, D. N., Injection properties of n-AlxGa1xAs-p-GaAs heterojunctions, Soviet Phy. Semicon., 2(10), 3733–3740, 1969. 15. Neumann, E. G., Single-mode Fibers: Fundamentals, Springer-Verlag, Berlin, 1988. 16. Noda, J., Okamoto, K. and Sasaki, Y., Polarization-maintaining fibers and their applications, J. Lightwave Technol., LT-4, 1071–1089, 1986. 17. Bell, T. E., Single-frequency semicondcutor lasers, IEEE Spectrum, 20, 38, 1983. 18. Kobayashi, K. and Mito, I., Single frequency and tunable laser diodes, J. Lightwave Technol., LT-6(11), 1623–1633, 1988. 19. Creaner, M. J., Steele, R. C., Marshall, I., Walker, G. R., Walker, N. G., Mellis, J., Al-Chalaba, S., Sturgess, I., Rutherford, M., Davidson, J. and Brain, M., Field demonstration of 565 Mbit/s DPSK coherent transmission system over 176 km of installed fiber, Electron. Lett., 24, 1254–1356, 1988.

28

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

20. Brian, M. C., Creaner, M. J., Steele, R. C., Marshall, I., Walker, G. R., Walker, N. G., Mellis, J., Al-Chalaba, S., Davidson, J., Rutherford, M. and Sturgess, I., Progress towards the field deployment of coherent optical fiber systems, J. Lightwave Technol., LT-8, 423–437, 1990. 21. Bryan, E. G., Carter, S. F., Wright, J. V., Boggis, J. M. and Cox, J. D., A 1.2 Gbit/s optical FSK field trial demonstration, Br. Telecom. Technol. J., 8, 18–26, 1990. 22. DeLange, O. E. Wide band optical communication systems: Part II–frequency-division multiplexing, Proc. IEEE., 58, 1683–1690, 1970. 23. Hasegawa, A. and Tappert, F., Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion, Appl. Phys. Lett., 23(3), 143–144, 1973. 24. Ghafouri-Shiraz, H., Shum, P. and Nagata, M., A novel method for analysis of soliton propagation in optical fibers, IEEE J. Quantum Electron., QE-31(1), 190–200, 1995. 25. Nakazawa, M., Yamada, E., Kubota, H. and Susuki, K., 10 Gbits/s soliton data transmission over one million kilometers, Electron. Lett., 27(14), 1270–1272, 1991. 26. Mollenauer, L. F., Neubelt, M. J., Haner, M., Lichtman, E., Evangelides, S. G. and Nyman, B. M., Demonstration of error-free soliton transmission at 2.5 Gbit/s over more than 14000 km, Electron Lett., 27(22), 2055–2056, 1991. 27. Eisenstein, G., Semiconductor optical amplifiers, IEEE Circuits Devices Mag., pp. 25–30, July 1989. 28. Ghafouri-Shiraz, H. and Chu, C. Y. J., Analysis of waveguiding properties of travelling-wave semiconductor laser amplifiers using perturbation techniques, Fiber and Int. Opt., 11, 51–70, 1992. 29. Brackett, C. A., Dense wavelength division multiplexing networks: principles and applications, IEEE J. Select. Areas Commun., SAC-8(8), 948–965, 1990. 30. Cochrane, P., Heckingbottom, R. and Heatley, D., Optical transparency: A new paradigm for telecommunications, Opt. and Phot. News, 5, 15–19, and 54, 1994. 31. Linke, R. A., Optical heterodyne communications systems, IEEE Communications Magazine, pp. 36–41, 1989. 32. Wagner, R. E. and Linke, R. A., Heterodyne lightwave systems: moving towards commercial use, IEEE LCS Magazine., 1(4), 28–35, 1990. 33. Yamamoto, Y. and Kimura, T., Coherent optical fiber transmission system, IEEE J. Quantum Electron., QE-17(6), 919–934, 1981. 34. DeLange, O. E., Optical heterodyne detection, IEEE. Spectrum, pp. 77–85, 1968. 35. Namizaki, H., Single mode operation of GaAs-GaAlAs TJS-laser diodes, Trans. IECE Japan, E59, 8–15, 1978. 36. Kajimura, M., Aiki, K., Chinone, N., Ito, R. and Umeda J., Longitudinal-mode behaviours of mode stabilized AlGaAs injection lasers, J. Appl. Phys., 49, 4644–4648, 1978. 37. Stein, S. and Jones, J. J., Modern Communication Theory. New York: McGraw Hill, 1964. 38. Couch II, L. W., Digital and Analog Communication Systems, 4th edition. New York: Macmillan, 1993. 39. Yamamoto, Y., Receiver performance evaluation of various digital optical modulation–demodulation systems in the 0.5–10 mm wavelength region, IEEE J. Quantum Electron., QE-16(11), 1251–1259, 1980. 40. Hodgkinson, T. G., Smith, D. W., Wyatt, R., and Maylon, D. J., Coherent optical fibre transmission systems, Br. Telecom. Technol. J., 3, 5–18, 1985. 41. Kazovsky, L. G., Coherent optical receivers: performance analysis and laser linewidth requirements, Opt. Eng., 24(4), 575–579, 1986. 42. Balanis, C. A., Advances in Engineering Electromagnetics. New York: John Wiley & Sons, 1989. 43. Adams, M. J. and Wyatt, R., An Introduction to Optical Waveguides. London:Wiley, 1981. 44. Agrawal, G. P. and Dutta, N. K., Long-Wavelength Semiconductor Lasers. Princeton, NJ: Van Nostrand, 1986. 45. Marcatilli, E. A. J., Dielectric rectangular waveguides and directional couplers for integrated optics, Bell Syst. Tech. J., 48(7), 2071–2102, 1969.

REFERENCES

29

46. Ghafoori-Shiraz, H., Single transverse mode condition in long wavelength SCH semiconductor laser diodes, Trans. IEICE, E 70(2), 130–134, 1987. 47. Tsang, W. T., Olsson, A., Linke, R. A., and Logan, R. A., 1.5 mm wavelength GaInAsP C3 lasers: single-frequency operation and wide-band frequency tuning, Electron. Lett., 19, 415–416, 1983. 48. Kobayashi, K. and Mito, I., Single frequency and tunable laser diodes, J. Lightwave Technol., LT6(11), 1623–1633, 1988. 49. Wyatt, R. and Devlin, W. J., 10-kHz linewidth 1.5 mm InGaAsP external cavity laser with 55-nm tuning range, Electron. Lett., 19, 110–112, 1983. 50. Kogelnik, H. and Shank, C. V., Coupled-wave theory of distributed feedback lasers, J. Appl. Phys., 43(5), 2327–2335, 1972. 51. Nakamura, M., Aiki, K., Umeda, J. and Yariv, A., CW operatiion of distributed feedback GaAs/ GaAlAs diode lasers at temperatures up to 300 K, Appl. Phys. Lett., 27, 403–405, 1975. 52. Yariv, A., Optical Electronics, 4th edition. Orlando, FL: Saunders College Publishing, 1991. 53. Okoshi, T, Kikuchi, K. and Nakayama, N., A novel method for high revolution measurement of laser output spectrum, Electron. Lett., 16, 630–631, 1980. 54. Henry, C. H., Theory of the linewidth of semiconductor lasers, IEEE J. Quantum Electron., QE18(2), 259–264, 1982. 55. Henry, C. H., Phase Noise in semiconductor lasers, J. Lightwave Technol., LT-4(3), 298–311, 1986. 56. Petermann, K., Laser Diode Modulation and Noise. Tokyo, Japan: KTK Scientific and Kluwer Academic Publishers, 1988. 57. Vahala, K., Chiu, L. C., Margalit, S. and Yariv, A., On the linewidth enhancement factor  in semiconductor injection lasers, Appl. Phys. Lett., 42, 631–633, 1983. 58. Osinski, M. and Buus J., Linewidth broadening factor in semiconductor lasers – an overview, IEEE J. Quantum Electron., QE-23(1), 9–28, 1987. 59. Sugimura, A., Patzak, E. and Meissner, P., Homogeneous linewidth and linewidth enhancement factor for a GaAs semiconductor laser, J. Phys. D (Appl. Phys.), 19, 7–16, 1986. 60. Fleming, M. W. and Mooradian A., Fundamental line broadening of single-mode A1GaAs/GaAs diode lasers, Appl. Phys. Lett., 38(7), 511–513, 1981. 61. Kikuchi, K. and Okoshi, T., Measurement of FM noise, AM noise and field spectra of 1.3 mm InGaAsP/InP DFB lasers and determination of their linewidth enhancement factor, IEEE J. Quantum Electron., QE-21(6), 1814–1818, 1985. 62. Ogita, S., Yano, M., Ishikawa, H. and Imai, H., Linewidth reduction in DFB lasers by detuning effect, Electron. Lett., 23, 393–394, 1987. 63. Fujise, M., Spectral linewidth estimation of a 1.5 mm range InGaAsP/InP distributed feedback laser, IEEE J. Quantum Electron., QE-22(3), 458–462, 1986. 64. Furuya, K., Dependence of linewidth enhancement factor  on waveguide structure in semiconductor lasers, Electron. Lett., 21, 200–201, 1985. 65. Kojima, K., Kyuma, K. and Nakayama T., Analysis of spectral linewidth of distributed feedback laser diodes, J. Lightwave Technol., LT-3(5), 1048–1055, 1985. 66. Ujihara, K., Phase noise in a laser with output coupling, IEEE J. Quantum Electron., QE-20(7), 814–818, 1984. 67. Henry, C. H., Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers, J. Lightwave Technol., LT-4(3), 288–297, 1986. 68. Wang, J., Schunk, N. and Petermann, K., Linewidth enhancement factor for DFB lasers due to longitudinal field dependence in the laser cavity, Electron. Lett., 23, 715–717, 1987. 69. Kazovsky, L. G., Coherent optical receivers: performance analysis and laser linewidth requirements, Opt. Eng. 24(4), 575–579, 1986. 70. Kikuchi, K., Effects of 1/f-type FM noise on semiconductor laser linewidth residual in high-power limit, IEEE J. Quantum Electron., QE-25, 312–315, 1989. 71. Kru¨ger, U. and Petermann, K., The semiconductor laser linewidth due to the presence of side modes, IEEE J. Quantum Electron., QE-24(12), 2355–2358, 1988.

30

AN INTRODUCTION TO OPTICAL COMMUNICATION SYSTEMS

72. Miller, S. E., The influence of power level on injection laser linewidth and intensity fluctuations including side-mode contributions, IEEE J. Quantum Electron., QE-24(9), 1837–1876, 1988. 73. Takano, S., Murata, S., Mito, I. and Kobayashi K., Threshold gain difference dependence of the spectral linewidth in DFB DC-PBH LD, Tech. Dig. 1st Optoelectronics Conf., pp. 204–205, 1986. 74. Ketelsen, L. J. P., Hoshino, I. and Ackerman, D. A., Experimental and theoretical evaluation of the CW suppression of TE side modes in conventional 1.55 mm InP-InGaAsP distributed feedback lasers, IEEE J. Quantum Electron., QE-27(4), 965–975, 1991.

2 Principles of Distributed Feedback Semiconductor Laser Diodes: Coupled Wave Theory 2.1

INTRODUCTION

The rapid development of both terrestrial and undersea optical fibre networks has paved the way for a global communication network. Highly efficient semiconductor injection lasers have played a leading role in facing the challenges of the information era. In this chapter, before discussing the operating principle of the semiconductor distributed feedback (DFB) laser diode (LD), general concepts with regard to the principles of lasers will first be presented. In section 2.2.1, general absorption and emission of radiation will be discussed with the help of a simple two-level system. In order for any travelling wave to be amplified along a two-level system, the condition of population inversion has to be satisfied and the detail of this will be presented in section 2.2.2. Due to the dispersive nature of the material, any amplification will be accompanied by a finite change of phase. In section 2.2.3, such dispersive properties of atomic transitions will be discussed. In semiconductor lasers, rather than two discrete energy levels, electrons jump between two energy bands which consist of a finite number of energy levels closely packed together. Following the Fermi–Dirac distribution function, population inversion in semiconductor lasers will be explained in section 2.3.1. Even though the population inversion condition is satisfied, it is still necessary to form an optical resonator within the laser structure. In section 2.3.2, the simplest Fabry–Perot (FP) etalon, which consists of two partially reflecting mirrors facing one another, will be investigated. A brief historical development of semiconductor lasers will be reviewed in section 2.3.3. The improvements in both the lateral and transverse carrier confinements will be highlighted. In semiconductor lasers, energy comes in the form of external current injection and it is important to understand how the injection current can affect the gain spectrum. In section 2.3.4, various aspects that will affect the material gain of the semiconductor will be discussed. In particular, the dependence of the carrier concentration on both the material gain and refractive index will be emphasised. Based on the Einstein relation for absorption, spontaneous emission and stimulated emission rates, the carrier recombination rate in semiconductors will be presented in section 2.3.5.

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

32

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

The FP etalon, characterised by its wide gain spectrum and multi-mode oscillation, has limited use in the application of coherent optical communication. On the other hand, a single longitudinal mode (SLM) oscillation becomes feasible by introducing a periodic corrugation along the path of propagation. The periodic corrugation which backscatters all waves propagating along one direction is in fact the working principle of the DFB semiconductor laser. The periodic Bragg waveguide acts as an optical bandpass filter so that only frequency components close to the Bragg frequency will be coherently reinforced. Other frequency terms are effectively cut off as a result of destructive interference. In section 2.4, this physical phenomenon will be explained in terms of a pair of coupled wave equations. Based on the nature of the coupling coefficient, DFB semiconductor lasers are classified into purely index-coupled, mixed-coupled and purely gain- or loss-coupled structures. The periodic corrugations fabricated along the laser cavity play a crucial role since they strongly affect the coupling coefficient and the strength of optical feedback. In section 2.5, the impact due to the shape of various corrugations will be discussed. Results based on a five-layer separate confinement structure and a general trapezoidal corrugation function will be presented. A summary is to be found at the end of this chapter.

2.2

BASIC PRINCIPLE OF LASERS

2.2.1 Absorption and Emission of Radiation From the quantum theory, electrons can only exist in discrete energy states when the absorption or emission of light is caused by the transition of electrons from one energy state to another. The frequency of the absorbed or emitted radiation f is related to the energy difference between the higher energy state E2 and the lower energy state E1 by Planck’s equation such that E ¼ E2  E1 ¼ h f

ð2:1Þ

where h ¼ 6:626  1034 Js is Planck’s constant. In an atom, the energy state corresponds to the energy level of an electron with respect to the nucleus, which is usually marked as the ground state. Generally, energy states may represent the energy of excited atoms, molecules (in gas lasers) or carriers like electrons or holes in semiconductors. In order to explain the transitions between energy states, modern quantum mechanics should be used. It gives a probabilistic description of which atoms, molecules or carriers are most likely to be found at specific energy levels. Nevertheless, the concept of stable energy states and electron transitions between two energy states are sufficient in most situations. The term photons has always been used to describe the discrete packets of energy released or absorbed by a system when there is an interaction between light and matter. Suppose a photon of energy (E2  E1 ) is incident upon an atomic system as shown in Fig. 2.1 with two energy levels along the longitudinal z direction. An electron found at the lower energy state E1 may be excited to a higher energy state E2 through the absorption of the incident photon. This process is called an induced absorption. If the two-level system is considered a closed system, the induced absorption process results in a net energy loss. Alternatively, an electron found initially at the higher energy level E2 may be induced by the incident photon to jump back to the lower energy state. Such a change of energy will cause the release of a single photon at a frequency f according to Planck’s equation. This process is called stimulated

BASIC PRINCIPLE OF LASERS

Figure 2.1

33

Different recombination mechanisms found in a two-energy level system.

emission. The emitted photon created by stimulated emission has the same frequency as the incident initiator. In addition, output light associated with the incident and stimulated photons shares the same phase and polarisation state. In this way, coherent radiation is achieved. Contrary to the absorption process, there is an energy gain for stimulated emissions. Apart from induced absorption and stimulated emissions, there is another type of transition within the two-level system. An electron may jump from the higher energy state E2 to the lower energy state E1 without the presence of any incident photon. This type of transition is called a spontaneous emission. Just like stimulated emissions, there will be a net energy gain at the system output. However, spontaneous emission is a random process and the output photons show variations in phase and polarisation state. This non-coherent radiation created by spontaneous emission is important to the noise characteristics in semiconductor lasers.

2.2.2 The Einstein Relations and the Concept of Population Inversion In order to create a coherent optical light source, it is necessary to increase the rate of stimulated emission while minimising the rate of absorption and spontaneous emission. By examining the change of field intensity along the longitudinal direction, a necessary condition will be established. Let N1 and N2 be the electron populations found in the lower and higher energy states of the two-level system, respectively. For uniform incident radiation with energy spectral density f , the total induced upward transition rate R12 (subscript 12 indicates the transition from the lower energy level 1 to the higher energy level 2) can be written as R12 ¼ N1 B12  f ¼ W12 N1

ð2:2Þ

where B12 is the constant of proportionality known as the Einstein coefficient of absorption. The product B12 f is commonly known as the induced upward transition rate W12 .

34

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

An excited electron on the higher energy state can undergo downward transition through either spontaneous or stimulated emission. Since the rate of spontaneous emissions is directly proportional to the population N2 , the overall downward transition rate R21 becomes R21 ¼ A21 N2 þ N2 B21 f ¼ A21 N2 þ W21 N2

ð2:3Þ

where the stimulated emission rate is expressed in a similar manner as the rate of absorption. A21 is the spontaneous transition rate and B21 is the Einstein coefficient of stimulated emission. Subscript 21 indicates a downward transition from the higher energy state 2 to the lower energy state 1. Correspondingly, W21 ¼ B21 f is known as the induced downward transition rate. For a system at thermal equilibrium, the total upward transition rate must equal the total downward transition rate and therefore R12 ¼ R21 , or in other words N1 B12 f ¼ A21 N2 þ N2 B21 f

ð2:4Þ

By rearranging the previous equation, it follows that f ¼ 

A21 =B21  B12 N1 1 B21 N2

ð2:5Þ

At thermal equilibrium, the population distribution in the two-level system is described by Boltzmann statistics such that N2 ¼ eE=kT N1

ð2:6Þ

where k ¼ 1:381  1023 JK1 is the Boltzmann constant. Substituting eqn (2.6) into (2.5) gives f ¼ 

A21 =B21  B12 E=kT e 1 B21

ð2:7Þ

Since the two-level system is in thermal equilibrium, it is usual to compare the above equations with a blackbody radiation field at temperature T which is given as [1] f ¼

8pn3 hf 3 1 3 E=kT c 1 e

ð2:8Þ

where n is the refractive index and c is the free space velocity. By equating eqn (2.7) with (2.8), one can derive the following relations B12 ¼ B21 ¼¼> W12 ¼ W21 ¼ W

ð2:9Þ

BASIC PRINCIPLE OF LASERS

35

and A21 8pn3 hf 3 ¼ B21 c3

ð2:10Þ

From eqn (2.7), it is clear that the upward and downward induced transition rates are identical at thermal equilibrium. Therefore, using eqn (2.9), the final induced transition rate, W, becomes W¼

A21 c3 A21 c2  ¼ I f 8pn3 hf 3 8pn2 hf 3

ð2:11Þ

where I ¼ cf =n is the intensity (Wm2) of the optical wave. Since energy gain is associated with the downward transitions of electrons from a higher energy state to a lower energy state, the net induced downward transition rate of the twolevel system becomes ðN2  N1 ÞW. Therefore, the net power generated per unit volume V can be written as dP0 ¼ ðN2  N1 ÞW hf dV

ð2:12Þ

In the absence of any dissipation mechanism, the power change per unit volume is equivalent to the intensity change per unit longitudinal length. Substituting eqn (2.12) into (2.11) will generate dI dP0 A21 c2 ¼ ¼ ðN2  N1 Þ IðzÞ 8pn2 f 2 dz dV

ð2:13Þ

The general solution of the above first-order differential equation is given as IðzÞ ¼ I0 eI ð f Þz

ð2:14Þ

where I ð f Þ ¼ ðN2  N1 Þ

A21 c2 8pn2 f 2

ð2:15Þ

In the above equation, I ð f Þ is the frequency-dependent intensity gain coefficient. Hence, if I ðf Þ is greater than zero, the incident wave will grow exponentially and there will be an amplification. However, recalling the Boltzmann statistics from eqn (2.6), the electron population N2 in the higher energy state is always less than that of N1 found in the lower energy state at positive physical temperature. As a result, energy is absorbed at thermal equilibrium for the two-level system. In addition, according to eqns (2.8) and (2.10), the rate of spontaneous emission ðA21 Þ is always dominant over the rate of stimulated emission ðB21 f Þ at thermal equilibrium.

36

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

Mathematically, there are two possible ways one can create a stable stream of coherent photons. One method involves negative temperature which is physically impossible. The other method is to create a non-equilibrium distribution of electrons so that N2 > N1 . This condition is known as population inversion. In order to fulfil the requirement of population inversion, it is necessary to excite some electrons to the higher energy state in a process commonly known as ‘pumping’. An external energy source is required, which in a semiconductor injection laser, takes the form of an electric current.

2.2.3 Dispersive Properties of Atomic Transitions Physically, an atom in a dielectric acts as a small oscillating dipole when it is under the influence of an incident oscillating electric field. When the frequency of the incident wave is close to that of the atomic transition, the dipole will oscillate at the same frequency as the incident field. Therefore, the total transmitted field will be the sum of the incident field and the radiated fields from the dipole. However, due to spontaneous emissions, the radiated field may not be in phase with the incident field. As we shall discuss, such a phase difference will alter the propagation constant as well as the amplitude of the incident field. Hence, apart from induced transitions and photonic emissions, dispersive effects should also be considered. Classically, for the simple two-level system with two discrete energy levels, the dipole moment problem can be represented by an electron oscillator model [2]. This model is a well-established method used long before the advent of quantum mechanics. Based upon the electron oscillator model, an oscillating dipole in a dielectric is replaced by an electron oscillating in a harmonic potential well. The effect of dispersion is measured by the change of relative permittivity with respect to frequency. In the electron oscillator model, any electric radiation at angular frequency near to the resonant angular frequency !0 is characterised by a frequency-dependent complex electronic susceptibility ð!Þ which relates to the polarisation vector Pð!Þ such that ~ E Pð!Þ ¼ "0 ð!Þ~

ð2:16Þ

ð!Þ ¼ 0 ð!Þ  j00 ð!Þ

ð2:17Þ

where

0 and 00 being the real and imaginary components of the electronic susceptibility . To start with, a plane electric wave propagating in a medium with complex permittivity of "0 ð!Þ will be considered. The wave, which is travelling along the longitudinal z direction, can be expressed in phasor form such that 0

EðzÞ ¼ E0 e j!t ejk ð!Þz

ð2:18Þ

where E0 is a complex amplitude coefficient and k0 ð!Þ, the propagation constant, can be expressed as k0 ð!Þ ¼ !

pffiffiffiffiffiffiffi "0

ð2:19Þ

BASIC PRINCIPLE OF LASERS

37

From Maxwell’s equations, the complex permittivity of an isotropic medium, "0 , is given as   "0 ð2:20Þ "0 ð!Þ ¼ " 1 þ ð!Þ " where " is the relative permittivity of the medium when there is no incident field.  is the same complex electronic susceptibility as mentioned previously. Using eqn (2.20) and assuming ð"0 ="Þ jj 1, one can obtain   "0 ð2:21Þ k0 ð!Þ  k 1 þ ð!Þ 2" where pffiffiffiffiffiffi k ¼ ! " By expanding ð!Þ with eqn (2.21), the propagation constant k0 becomes    00  0 ð!Þ k ð!Þ 0 k ð!Þ  k 1 þ j 2 2n 2n2

ð2:22Þ

ð2:23Þ

1=2

where n ¼ ð"="0 Þ is the refractive index of the medium at a frequency far away from the resonant angular frequency !0. Substituting eqn (2.21) back into eqn (2.18), the electric plane wave becomes EðzÞ ¼ E0 ej!t ejðkþ
kÞz eðgint Þz=2

ð2:24Þ

where int is introduced to include any internal cavity loss and 0 ð!Þ 2n2 00 ð!Þ gð!Þ ¼ k 2 n


kð!Þ ¼ k

ð2:25Þ ð2:26Þ

In semiconductor lasers, it is likely that free carrier absorption and scattering at the heterostructure interface may contribute to internal losses. In the above equation,
k corresponds to a shift of propagation constant which is frequency dependent. Unless the electric field oscillates at the resonant angular frequency !0, there will be a finite phase delay and the new phase velocity of the incident wave becomes !=ðk þ
kÞ. Apart from the phase velocity change, the last exponential term in eqn (2.24) indicates an amplitude variation with g as the power gain coefficient. When ðg  int Þ is greater than zero, the electric plane wave will be amplified. Rather than the population inversion condition relating the population density at the two energy levels as in eqn (2.14), the imaginary part of the electronic susceptibility 00 ð!Þ is used to establish the amplifying condition. Sometimes, the net amplitude gain coefficient net is used to represent the necessary amplifying condition such that net ¼

g  int >0 2

ð2:27Þ

38

2.3

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

BASIC PRINCIPLES OF SEMICONDUCTOR LASERS

Before the operation of the semiconductor laser is introduced, some basic concepts of energy transition between energy states will be discussed. When there is an interaction between light and matter, discrete packets of energy (photons) may be released or absorbed by the system. Suppose a photon of energy ðE2  E1 Þ is incident upon an atomic system with two energy levels E1 and E2 along the longitudinal z direction. An electron at the lower energy state E1 may be excited to a higher energy state E2 through the absorption of the incident photon. This process is called induced absorption. If the two-level system is considered a closed system, the induced absorption process results in a net energy loss. Alternatively, an electron found initially at the higher energy level E2 may be induced by the incident photon to jump back to the lower energy state. Such a change of energy will cause the release of a single photon at a frequency f according to Planck’s equation. This process is called stimulated emission. The emitted photon created by stimulated emission has the same frequency as the incident initiator. Furthermore, the incident and stimulated photons share the same phase and polarisation state. In this way, coherent radiation is achieved. Contrary to the absorption process, there is an energy gain for stimulated emissions. Apart from induced absorption and stimulated emissions, an electron may jump from the higher energy state to the lower energy state without the presence of any incident photon. This type of transition is called a spontaneous emission and a net energy gain results at the system output. However, spontaneous emission is a random process and the output photons show variations in phase and polarisation state. This non-coherent radiation created by spontaneous emission is important to the noise characteristics in semiconductor lasers.

2.3.1 Population Inversion in Semiconductor Junctions In gaseous lasers like CO2 or He–Ne lasers, energy transitions occur between two discrete energy levels. In semiconductor lasers, these energy levels cluster together to form energy bands. Energy transitions between these bands are separated from one another by an energy barrier known as an ‘energy gap’ (or forbidden gap). With electrons topping up the ground states, the uppermost filled band is called the valence band and the next highest energy band is denoted the conduction band. The probability of an electronic state at energy E being occupied by an electron is governed by the Fermi–Dirac distribution function, f ðEÞ, such that [3] f ðEÞ ¼

1 ½eðEEf Þ=kT

þ 1

ð2:28Þ

where k is the Boltzmann constant, T is the temperature in Kelvin and Ef is the Fermi level. The concept of the Fermi level is important in characterising the behaviour of semiconductors. By putting E ¼ Ef in the above equation, the Fermi–Dirac distribution function f ðEf Þ becomes 1=2. In other words, an energy state at the Fermi level has half the chance of being occupied. The basic properties of an equilibrium p–n junction will not be covered here as they can be found in almost any solid state electronics textbook [4]. Only some important characteristics of the p–n junction will be discussed here.

BASIC PRINCIPLES OF SEMICONDUCTOR LASERS

39

According to Einstein’s relationship on the two-level system, the population of electrons in the higher energy state needs to far exceed that of electrons found in the lower energy state before any passing wave can be amplified. Such a condition is known as population inversion. At thermal equilibrium, however, this condition cannot be satisfied. To form a population inversion along a semiconductor p–n junction, both the p and n type materials must be heavily doped (degenerate doping) so that the doping concentrations exceed the density of states of the band. The doping is so heavy that the Fermi level is forced into the energy band. As a result, the upper part of the valence band of the p type material (from the Fermi level Ef to the valence band edge Ev) remains empty. Similarly, the lower part of the conduction band is also filled by electrons due to heavy doping. Figure 2.2(a) shows the energy band diagram of such a heavily doped p–n junction. At thermal equilibrium, any energy transition between conduction and valence bands is rare.

Figure 2.2 Schematic illustration of a degenerate homojunction. (a) Typical energy level diagram at equilibrium with no biasing voltage; (b) the same homojunction under strong forward bias voltage.

Using an external energy source, the equilibrium can be disturbed. External energy comes in the form of external biasing which enables more electrons to be pumped to the higher energy state and the condition of population inversion is said to be achieved. When a forward bias voltage close to the bandgap energy is applied across the junction, the depletion layer formed across the p–n junction collapses. As shown in Fig. 2.2(b), the quasi-Fermi level in the conduction band, EFc , and that in the valence band EFv are separated from one another under a forward biasing condition. Quantitatively, EFc and EFv could be described in terms of the carrier concentrations such that N ¼ ni eðEFc Ei Þ=kT

ð2:29Þ

P ¼ ni eðEi EFv Þ=kT

ð2:30Þ

and

where Ei is the intrinsic Fermi level, ni is the intrinsic carrier concentration, N and P are the concentration of electrons and holes, respectively. Along the p–n junction, there exists a narrow active region that contains simultaneously the degenerate populations of electrons and holes. Here, the condition of population inversion is satisfied and carrier recombination starts to occur.

40

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

Since the population distribution in a semiconductor follows the Fermi–Dirac distribution function, the probability of an occupied conduction band at energy Ea can be described by fc ðEa Þ ¼

1 1 þ eðEa EFc Þ=kT

where Ea > EFc

ð2:31Þ

Similarly, the probability of an occupied valence band at energy Eb can be written as fv ðEb Þ ¼

1 1 þ eðEb EFv Þ=kT

where Eb < EFv

ð2:32Þ

Since any downward transition implies an electron jumping from the conduction band to the valence band with the release of a single photon, the total downward transition rate, Ra!b , is proportional to the probability that the conduction band is occupied whilst the valence band is vacant. In other words, it can be expressed as Ra!b / fc ðEa Þð1  fv ðEb ÞÞ

ð2:33Þ

Similarly, the total upward transition rate Rb!a becomes Rb!a / fv ðEb Þð1  fc ðEa ÞÞ

ð2:34Þ

As a result, the net effective downward transition rate becomes Ra!b ðnetÞ ¼ Ra!b  Rb!a  fc ðEa Þ  fv ðEb Þ

ð2:35Þ

In order to satisfy the condition of population inversion, the above relationship must remain positive. In other words, it is necessary to have fc ðEa Þ > fv ðEb Þ

ð2:36Þ

Putting Ea  Eb ¼ hf and using the Fermi–Dirac distribution function, the above inequality becomes EFc  EFv > hf

ð2:37Þ

which is known as the Bernard–Duraffourg condition [3]. Since the energy of the radiated photon must exceed or equal that of the energy gap Eg , the final condition for amplification in a semiconductor becomes EFc  EFv > hf  Eg

ð2:38Þ

From a simple two-level system to the semiconductor p–n junction, a necessary condition for light amplification is established. However, this condition is not sufficient to provide lasing as we will discuss in the next section. In order to sustain laser oscillation, certain optical feedback mechanisms are necessary.

BASIC PRINCIPLES OF SEMICONDUCTOR LASERS

41

2.3.2 Principle of the Fabry–Perot Etalon In Chapter 1, the Fabry–Perot laser cavity was briefly mentioned. In this section, the details of this laser diode will be covered. By facing two partially reflected mirrors towards one another, a simple optical resonator is formed. Let L be the distance between the two mirrors. If the spacing between the two mirrors is filled by a medium that processes gain, a Fabry– Perot etalon is formed. As an electric field bounces back and forth between the partially reflected mirrors, the wave is amplified as it passes through the laser medium. If the amplification exceeds other cavity losses due to imperfect reflection from the mirrors or scattering in the laser medium, the field energy inside the cavity will continue to build up. This process will continue until the single pass gain balances the loss. When this occurs, a self-sustained oscillator or a laser cavity is formed. Hence, optical feedback is important in building up the internal field energy so that lasing can be achieved. A simplified FP etalon is shown in Fig. 2.3.

Figure 2.3

A simplified Fabry–Perot cavity.

In Fig. 2.3, ^r1 and ^r2 are, respectively, the amplitude reflection coefficients of the input (left) and output (right) mirrors. Similarly, ^t1 and ^t2 represent the amplitude transmission coefficients of the mirrors. Suppose an incident wave with complex propagation constant k0 enters the etalon from z ¼ 0. After a series of parallel reflections, the total transmitted wave at the output plane ðz ¼ LÞ becomes [5]

0 0 0 Eo ¼ Ei^t1^t2 ejk L 1 þ ^r1^r2 e2jk L þ ^r12^r22 e4jk L þ

ð2:39Þ

Using an infinite sum for a geometric progression (GP) series, the above equation becomes 0

Eo ¼

^t1 ^t2 ejk L 0 Ei 1  ^r1 ^r2 e2jk L

ð2:40Þ

By expanding the propagation constant k0 as in eqn (2.23), eqn (2.40) can also be expressed as 

^t1^t2 ejðkþ
kÞL enet L Eo ¼ Ei 1  ^r1^r2 e2jðkþ
kÞL enet L

 ð2:41Þ

42

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

Where net is the net loss. When net > 0 and the denominator of the above equation becomes very small such that the square bracket term is larger than unity, amplification will occur. To obtain the self-sustained oscillation, the denominator of the above equation must be zero, i.e. 0

^r1 ^r2 e2jk L ¼ 1

ð2:42Þ

This is the threshold condition of a FP laser as the ratio Eo =Ei becomes infinite. Physically, this corresponds to a finite transmitted wave at the output with zero incident wave. With the amplitude and the phase term separated, one will have ^r1 ^r2 enet L ¼ 1

ð2:43Þ

2ðk þ
kÞL ¼ 2mp

ð2:44Þ

and

Equation (2.43) represents a case in which a wave making a round trip inside the resonator will return with the same amplitude at the same plane. Similarly, the phase change after a roundtrip must be an integer multiple of 2 so as to maintain a constructive phase interference. By rearranging eqn (2.43) and (2.24), the threshold gain of the FP laser becomes   2 1 th ¼ 0 þ ln ð2:45Þ with g ¼ th ^r1^r2 L where m ¼

  1 1 ln ^r1^r2 L

ð2:46Þ

is the amplitude mirror loss which accounts for the radiation escaping from the FP cavity due to finite facet reflections. Hence, the threshold gains of FP semiconductor lasers can be determined once the physical structures are known. From eqn (2.44), one can determine the lasing frequency. Due to the dispersive properties shown in section 2.2, the frequency-dependent propagation constant (k þ
k) is replaced by a group refractive index, ng such that Reðk0 Þ ¼ k0 ng ¼ k0 c=vg

ð2:47Þ

where k0 is the free space propagation constant. Replacing k0 with 2pf =c and rearranging eqn (2.44), the cavity resonance frequency fm becomes fm ¼

mc 2ng L

ð2:48Þ

where m is an arbitrary integer. When m increases, it can be seen that there is an infinite number of longitudinal modes. In practice, however, the number of longitudinal modes

BASIC PRINCIPLES OF SEMICONDUCTOR LASERS

43

depends on the width of the material gain spectrum. From the equation shown above, it can also be confirmed that the longitudinal mode spacing is that shown in eqn (1.25) in Chapter 1. The gain values of all probable modes increase with pumping until the threshold condition is finally attained. The mode having the minimum threshold gain becomes the lasing mode whilst others become non-lasing side modes. After the threshold condition is reached, the laser gain spectrum does not clamp to a fixed value as in gaseous lasers. Instead, the lasing gain spectrum keeps changing with the biasing current. Such an inhomogenous broadening effect becomes so complicated that multi-mode oscillation and mode hopping become common in FP semiconductor lasers. The lasing spectrum and the spectral properties of the FP laser cavity are important in the field of semiconductor lasers, since other semiconductor lasers resemble the basic FP design. Simplicity may be an advantage for FP lasers, however, due to broad and unstable spectral characteristics, they have limited application in coherent optical communication systems in which a single longitudinal mode is a requirement.

2.3.3 Structural Improvements in Semiconductor Lasers In section 2.3.1, the condition of population inversion in a heavily doped p–n junction (or diode) was discussed. The so-called homojunction is characterised by having a single type of material found across the p–n junction. When a forward bias voltage is applied across the junction, the contact potential between the p and n regions is lowered. With the energy gap remaining constant throughout the junction, the majority of carriers tend to diffuse across the junction easily. As a result, carrier recombination along the p–n junction becomes less efficient. Typical current density required to achieve lasing in this early diode is of the order of 105 A cm2 [6]. With such a high current density, continuous wave (CW) operation at room temperature is impossible. Pulse mode operation is allowed at extremely low temperature only. With such a low efficiency and high threshold current, the homojunction structure has been replaced by more effective structures.

(a) Improvements in transverse carrier confinement In 1963, it was discovered that the threshold current of semiconductor lasers could be reduced significantly if carriers were confined along the active region. A three-layer structure, which consisted of a thin layer of lower energy gap material sandwiched between two layers of higher energy gap materials, was proposed. However, it was not until 1969 when the liquid phase epitaxy (LPE) growth of AlGaAs on a GaAs homojunction became available. Since two different materials were involved, an additional energy barrier was formed alongside the homogeneous p–n junction. As a result, the chance of carrier diffusion was reduced. The name single heterostructure was given [3] and is shown in Fig. 2.4(a). Apart from the difference in energy gaps, the p-GaAs active layer has a higher refractive index than the n-region. So, with the p-AlGaAs cladding having a considerably lower refractive index, an asymmetric three-layer waveguide was formed within the single heterostructure and the highest refractive index was found along the active region. The asymmetric waveguide confined the optical intensity largely to the active region and so the optical loss due to evanescent mode propagation was reduced. However, the best room

44

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

Figure 2.4 Schematic illustration of a single heterojunction [4]. (a) Typical energy level diagram at equilibrium without biasing voltage; (b) the same heterojunction under strong forward bias voltage.

temperature threshold current density for the single heterostructure device is still too high for CW operation (a typical value would be 8.6 kA cm2). Nevertheless, it is a great improvement on the homostructure. The establishment of CW operation at room temperature was finally achieved in the 1970s. As shown in Fig. 2.5, the thin active layer is now sandwiched between two layers of

Figure 2.5 Schematic illustration of a double heterojunction [4]. (a) Typical energy level diagram at equilibrium without bias voltage; (b) under strong bias voltage.

higher energy gap material, and hence a double heterostructure is formed. Along the boundary where two different materials are used, an energy barrier is formed. Carriers find it so difficult to diffuse across the active region that they are trapped. By using a higher refractive index material at the centre, photons are also confined in a similar way. This type of structure is known as the separate confinement heterostructure (SCH). The combined effects in carrier and optical confinement help bring the threshold current density down to approximately 1.6 kA cm2. Operation at CW becomes feasible provided that the laser itself is mounted on a suitable heat sink.

(b) Improvements in lateral carrier confinement Continuous wave operation at room temperature is a significant achievement and now the double heterostructure design is more or less standard. So far, the structures we have

BASIC PRINCIPLES OF SEMICONDUCTOR LASERS

45

discussed belong to the broad-strip laser family since they do not incorporate any mechanism for the lateral (parallel to the junction plane) confinement of the injected current or the optical mode. By adopting a strip-geometry, carriers are injected over a narrow central region using a strip contact. With carrier recombination restricted to the narrow strip (typical width ranging from 1 to 10 mm), the threshold current is reduced significantly. Such lasers are referred to as gain-guided because it is the lateral variation of the optical gain that confines the optical mode to the strip vicinity. Lasers in which optical modes are confined because of lateral variations of refractive index are known as index-guided lasers. Comparatively, gain-guided lasers are simple to make, but their weak optical confinement limits the beam quality [5]. Moreover, it is difficult to obtain a stable output in single longitudinal mode. As a result, the index-guiding mechanism has become the mainstream in semiconductor laser development and a large number of index-guided structures have been proposed in the past decade. Basically, a lateral variation of refractive indices is used to confine the optical energy. Various index-guided structures like the buried heterostructure (BH), channelled substrate planar (CSP), buried crescent (BC), ridge waveguide (RW) and dual-channel planar buried heterostructure (DCPBH) have been used. A survey of recent research will reveal many other types of laser, but basically they are alternatives of these basic structural designs. The structural improvement in the development of semiconductor lasers has reduced the threshold current density whilst CW single transverse mode operation has become feasible.

2.3.4 Material Gain in Semiconductor Lasers Suppose a medium having complex permittivity "0 is used to build an infinitely long waveguide and an input signal is injected into it. After travelling a distance of L, the power gain of the signal can be defined by an amplifying term, G, such that G ¼ eðgloss ÞL

ð2:49Þ

where g is the material gain (or the power gain coefficient) and loss is the internal cavity loss. It is important that ðg  loss Þ > 0 for an amplified signal. In an index-guided semiconductor laser, the refractive index of the active region ðn1 Þ is higher than the surrounding cladding ðn2 Þ so that a dielectric waveguide is formed. In practice, however, the dielectric waveguide formed is far from ideal. Under the weakly guiding condition where ðn1  n2 Þ n1 , some energy leaks out into the cladding as a result of the evanescent field. To take into account the power leakage, a weighting factor  is introduced into eqn (2.49) such that G ¼ e½ ðga Þð1 Þc þsca  L

ð2:50Þ

where a and c are the absorption losses of the active and cladding layers respectively, and sca is the scattering loss at the heterostructure interface. The weighting factor , known as the optical confinement factor, defines the ratio of the optical power confined in the active region to the total optical power flowing across the structure. In order to determine the optical gain, various approaches have been used. In this section, a phenomenological approach [6] will be introduced, whilst another approach using

46

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

Einstein’s coefficients [7] will be discussed in the next section. The phenomenological approach is based on experimental observations that the peak material gain varies almost linearly with the injected carrier concentration. Such an observation leads to a linear approximation [8] of gpeak ¼ A0 ðN  N0 Þ

ð2:51Þ

where A0 is the differential gain and N0 is the carrier concentration at zero material gain, commonly known as the transparency carrier concentration. The above relation gives only a reasonable approximation in a small biasing range when the carrier concentration is comparable to the transparency carrier concentration. The range of accuracy is extended by adopting a parabolic model [9] such that gpeak ¼ aN 2 þ bN þ c

ð2:52Þ

where a, b and c are constants determined by fitting the available exact solutions using the least squares technique. Due to the dispersive properties of the semiconductor, the actual material gain is also affected by the optical frequency f, and hence the wavelength
. So far, the value of gain has been assumed to be at the resonant frequency, however, if the optical frequency is tuned away from the resonant peak, the exact value of gain becomes different from that of gpeak . Based on experimental observation, Westbrook [10] extended the linear peak gain model further, such that gðN;
Þ ¼ A0 ðN  N0 Þ  A1 ½
 ð
0  A2 ðN  N0 ÞÞ2

ð2:53Þ

where
0 is the wavelength of the peak gain at transparency gain (i.e. g ¼ 0) and A1 governs the base width of the gain spectrum. The wavelength shifting coefficient A2 takes into account the change of the peak wavelength with respect to the carrier concentration. Notice that the negative sign in front of A2 indicates a negative wavelength shift of peak gain wavelength. In semiconductor lasers, energy enters in the form of an external biasing current. In determining the material gain, one must determine the relationship between the carrier concentration N and the injection current I. This is accomplished through the carrier rate equation that includes the generation and decay carriers found in the active region. In its general form, the equation is given as [4,11] vg gðN;
ÞS @N I þ Dðr2 NÞ ¼  RðNÞ  1 þ "S @t qV

ð2:54Þ

where q is the electronic charge and V ¼ dwL is the volume of the active layer with d, w and L being the thickness, the width and the length of the active layer, respectively, I is the injection current, R(N) is the total (i.e. both radiative and non-radiative) carrier recombination process, the term vg gðN;
ÞS=ð1 þ "SÞ shown in the above equation takes into account the carrier loss as a result of stimulated emission. Here, vg is the group velocity and S is the photon density of the lasing mode. The effect of photon non-linearity is included

BASIC PRINCIPLES OF SEMICONDUCTOR LASERS

47

in the non-linear coefficient ". In the above equation, the final term Dðr2 NÞ represents the carrier diffusion with D representing the diffusion coefficient. In RðNÞ shown in equation (2.54), non-radiative carrier recombination implies those processes will not generate any photons. For semiconductor lasers operating at shorter wavelengths ð
< 1 mmÞ, the effects of non-radiative recombination are small. However, non-radiative recombination becomes more important in long-wavelength semiconductor lasers. In quaternary InGaAsP materials operating in the 1.30 and 1.55 mm regions, the total carrier recombination rate can be written as RðNÞ ¼

N þ BN 2 þ CN 3

ð2:55Þ

where is the linear recombination lifetime, B is the radiative spontaneous emission coefficient and C is the Auger recombination coefficient. The linear recombination lifetime

includes recombination at defects or surface recombination at the laser facet. With improvement in fabrication techniques, the number of defects and the chances of surface recombination have been reduced significantly. In long-wavelength semiconductor lasers, the cubic term CN 3 takes into account the non-radiative Auger recombination process. Due to the Coulomb interaction between carriers of the same energy band, each Auger recombination involves four carriers. According to the origins of these carriers, the Auger recombination is classified into band-to-band, photon-assisted and trap-assisted processes. Details on different types of Auger processes are clearly beyond the scope of the present book, though the interested reader may refer to reference [4]. Some typical values of , B and C for the quaternary III–V materials at 1.30 and 1.55 mm are listed in Table 2.1. Based on the simplified carrier rate equation, all of these parameters can be measured simply, as explained in a paper by Chu and Ghafouri-Shiraz [12]. In an index-guided semiconductor laser where the active layer width and thickness are small compared to the carrier diffusion length of 1–3 mm, the diffusion effect becomes of

Table 2.1 (after [4])

Coefficients for the total recombination of quaternary materials at 1.3 mm and 1.55 mm

In1x Gax Asy P1y at
¼ 1:30 mm with y ¼ 0:61, x ¼ 0:28 at T ¼ 300 K

¼ 10 ns B ¼ 1:2  1010 cm3 s1 C ¼ 1:5  1029 cm6 s1 In1x Gax Asy P1y at
¼ 1:55 mm with y ¼ 0:90, x ¼ 0:42 at T ¼ 300 K

¼ 4 ns B ¼ 1:0  1010 cm3 s1 C ¼ 3:0  1029 cm6 s1

48

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

secondary importance and can be neglected hereafter. At the lasing threshold condition, the semiconductor laser begins to lase. With @N=@t ¼ 0, the steady state solution of the carrier rate equation becomes Ith ¼ qV RðNth Þ=i

ð2:56Þ

where Ith is the threshold current and Nth is the threshold carrier density. The internal quantum efficiency i gives the ratio of the radiative recombination to the total carrier recombination. In deriving the above equation, S is assumed to be zero at the lasing threshold condition. Sometimes, rather than the threshold current, a nominal threshold current density Jth (in A m2 ) is used which relates to the threshold current Ith as Ith d ¼ Jth ð2:57Þ V In semiconductors, any change in material gain is accompanied by a change in refractive index as a result of the Kramer–Kroenig relationship [1]. Any change in carrier density will induce changes in the refractive index [13,14] as dn N ð2:58Þ dN where nini is the refractive index of the semiconductor when no current is injected and dn=dN is the differential index of the semiconductor. It should be noted that the value of dn=dN is usually negative. The refractive index becomes smaller as the injection current increases. As we will discuss in a later chapter, any variation in carrier density will affect the spectral behaviour of the laser since the lasing wavelength is so sensitive to variations in refractive index. Both the Fermi–Dirac distribution and the material gain are found to be sensitive to temperature change. In practice, the operating temperature of semiconductor lasers is usually stabilised by a temperature control unit. However, it is also known that the change in optical gain due to the variation of injected carrier is more significant than that due to changes in temperature [15]. As a result, the temperature dependence of the material gain has been neglected in the analysis. nðNÞ ¼ nini þ 

2.3.5 Total Radiative Recombination Rate in Semiconductors The theory for all classes of laser can also be represented by the Einstein relation for absorption, spontaneous emission and stimulated emission rates. In semiconductors, optical transitions are between energy bands whilst other laser transitions are between discrete energy levels. Nevertheless, the Einstein relations are still applicable. The major difference between various material systems is contained in the Einstein coefficient (or transition probabilities) which can only be determined by quantum mechanics. Transitions between any pair of discrete energy levels are separated by hf (or E21 ). The gain coefficient gðE21 Þ and emission rates rspon ðE21 Þ and rrstim ðE21 Þ are related to one another [3,7] by gðE21 Þ ¼

rspon ðE21 Þ ¼

h 3 c2 r ðE21 Þ 2 stim 8pn2g E21 2 8pn2g E21

h3 c2

g21 ðE21 Þ

ð2:59Þ fc ðE2 Þ½1  fv ðE1 Þ fc ðE2 Þ  fv ðE1 Þ

ð2:60Þ

BASIC PRINCIPLES OF SEMICONDUCTOR LASERS

49

and  rstim ðE21 Þ ¼

 1 ½E21 ðEFc EFv Þ e 1 rspon ðE21 Þ kT

ð2:61Þ

where h is Planck’s constant, k is the Boltzmann constant, c is the free space velocity, ng is the group refractive index, fc ðE2 Þ and fv ðE1 Þ are occupation probabilities of electrons in the conduction and valence bands. EFc and EFv being the quasi-Fermi levels. It should be noted that the unit of the gain coefficient is cm1 whilst the units of the emission rate rspon and rstim are number of photons per unit volume per second per energy interval. The expressions from eqns (2.59) to (2.61) demonstrate how gðE21 Þ, rspon ðE21 Þ and rstim ðE21 Þ are related to one another. To evaluate these expressions, one parameter, such as the spontaneous emission rate rspon ðE21 Þ, must be obtained experimentally. Alternatively, they are all related by the Einstein coefficients such that gðE21 Þ ¼ B21 ½ fc ðE2 Þ  fv ðE1 Þng =c

ð2:62Þ

rspon ðE21 Þ ¼ A21 fc ðE2 Þ½1  fv ðE1 Þ

ð2:63Þ

rstim ðE21 Þ ¼ A21 ½ fc ðE1 Þ  fv ðE1 Þ

ð2:64Þ

with A21 ¼ B21

2 8pn3g E21

h3 c 3

ð2:65Þ

at thermal equilibrium. With a known doping concentration, the unknown parameters g, rspon and rstim in eqns (2.62) to (2.64) can then be fixed after determining either A21 or B21. Without any preference, B21 is chosen to be the key parameter. As expected, the coefficient B21 takes into account the interaction between electrons and holes in the presence of electromagnetic radiation. In order to understand the interaction between them, quantum mechanics should be used. Rather than going through the lengthy analysis, some important results will be shown. Starting with the time-dependent Schro¨dinger equation, coefficient B21 is given as [3] B21 ¼

q2 h jM21 j 2m20 "0 n2g E21

ð2:66Þ

4p ng q E21 jM21 j m20 "0 h2 c3

ð2:67Þ

so that A21 ¼

with "0 as the free space permittivity, q the electronic charge, m0 the mass of an electron and M21 the momentum matrix between the initial (subscript 2) and final (subscript 1) electron state. With the actual transition involving various energy states between the conduction band and the valence band of the semiconductor, the analysis will not be complete without the

50

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

inclusion of density of state functions. It is necessary to determine the momentum matrix element as well as the density of states. The density of state function is not difficult for the parabolic band model. From Yariv [1], it is clear that the density of states in the conduction band is c ðE  Ec Þ ¼ 4p

  2mn 3=2 ðE  Ec Þ1=2 h2

ð2:68Þ

where Ec is the conduction band edge and mn is the effective electron mass. Similarly, the density of states in the valence band can be written as 

2mp v ðEv  EÞ ¼ 4p h2

3=2 ðEv  EÞ1=2

ð2:69Þ

with Ev being the valence band edge and mp the effective mass of hole. The momentum matrix element may be determined empirically from the wave function. For the localised state, the wave function of the band is modified by a slowly varying envelope function which represents the influence of impurities. As a result, the momentum matrix becomes M21 ¼ M ¼ Mb Menv

ð2:70Þ

where Mb is the average matrix element of the Bloch state for an intrinsic situation and Menv represents the slowly varying envelope function with impurities present. For III–V quaternary semiconductors, Mb can be expressed as jMb j2 ¼

m20 Eg ðEg þ
Þ  12 mn Eg þ 2=ð3
ÞÞ

ð2:71Þ

where Eg ¼ Ec  Ev is the energy gap,
is the spin orbit splitting. For transitions under the k-selection rule, the wave vector difference between the valence and the conduction band must be equal to that of the emitted photon. In other words, momentum is conserved and the momentum matrix element is given as [3] jMj2 

m20 Eg ðEg þ
Þ ð2pÞ3  V 12mn Eg þ 2=ð3
Þ

ð2:72Þ

where ð2pÞ3 =V is the unit volume in k-space. However, when the semiconductor is biased with high injection current or it is heavily doped, the density of states will be modified. The randomly distributed impurities (from current injection or heavy doping) tend to create an additional continuum of states near the band edge, which is known as the band-tail state. Since momentum will no longer be conserved, one needs to use the relaxed k-selection rule so that the band-tailing effect can be included. Neglecting the band-tailing effect first but including the density of states, the spontaneous emission rate rspon at photon energy E21 can be written as ð1 c ðE  Ec Þ fc ðEÞ v ðEv  EÞ½1  fv ðEÞ dE ð2:73Þ rspon ðE21 Þ ¼ A21 1

COUPLED WAVE EQUATIONS IN DISTRIBUTED FEEDBACK SEMICONDUCTOR LD

51

The integral shown above takes into account various states in the conduction band and the valence band, which are separated by the photon energy E21. With such a clumsy notation, it is common to shift the valence band edge Ev by the photon energy E21. In this way, E0 becomes the energy variable. At the conduction band edge, E0 becomes 0 and so one can define E00 ¼ E0  E21 . As a result, c ðE  Ec Þ becomes c ðE0 Þ while v ðEv  EÞ is shifted to become v ðE00 Þ. By substituting A21 into the above equation, the spontaneous emission becomes rspon ðE21 Þ ¼

4png q2 E21 m20 "0 h2 c3

ð1 1

c ðE0 Þf ðE0 Þ v ðE00 Þ½1  f ðE00 ÞjMj2 dE0

ð2:74Þ

Under the relaxed k-selection rule, the momentum matrix M can be considered as energy independent and so it is taken out of the integration. What remains in the integration is the density of hole (P) and electron (N). Therefore, eqn (2.74) can be simplified to give rspon ðE21 Þ ¼

4p ng q2 E21 jMj2 P N m2o "0 h2 c3

ð2:75Þ

Within a narrow range of photon energy, E21  E0 is fairly constant. As a result, the total spontaneous emission rate ðRsp Þ can be written as ð 4p ng q 2 E0 Rsp rspon ðEÞ dE ¼ 2 2 3 jMj2 P N ¼ B NP m "0 h c

ð2:76Þ

Here, B is commonly known as the radiative recombination coefficient. In most cases, the densities of electrons and holes are the same and hence eqn (2.76) can be written as Rsp ¼ BN 2

2.4

ð2:77Þ

COUPLED WAVE EQUATIONS IN DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

To understand the operational characteristic of a DFB semiconductor laser, it is necessary to consider wave propagation in periodic structures. Grating or corrugation-induced dielectric perturbation leads to a coupling between the forward and backward propagating waves. Historically, various approaches like coupled wave theory [16,17] and Bloch wave analysis [18] have been adopted. Although these methods were proven to be equivalent [19], researchers have been keen on coupled wave theory because of its ease of understanding, and because numerical algorithms can be implemented to solve the equations [20]. In a homogeneous, source-free and lossless medium, any time harmonic electric field must satisfy the vector wave equation [21] E þ k20 n2~ E¼0 r2 ~

ð2:78Þ

52

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

where the time dependence of the electric field is assumed to be ej!t , n is the refractive index and k0 is the free space propagation constant. In a semiconductor laser which has a transversely and laterally confined structure, the electric field must satisfy the one-dimensional homogeneous wave equation such that 

 d2 2 þ k ðzÞ EðzÞ ¼ 0 dz2

ð2:79Þ

Consider a multi-dielectric stack in which periodic corrugations are formed along one boundary as illustrated in Fig. 2.6.

Figure 2.6 General multi-dielectric layers used to show the perturbation of refractive index and amplitude gain. Z1 ðxÞ and Z2 ðxÞ are two corrugated functions.

The material complex permittivity in each layer is denoted as "j while g and  are the height and the period of corrugation, respectively. With corrugations extending along the longitudinal direction, the wave propagation constant, k(z), could be written as k2 ðzÞ ¼ !2 m"0

ð2:80Þ

where ! is the angular frequency and "0 is the complex permittivity. When the radiation frequency is sufficiently close to the resonance frequency, eqn (2.80) becomes [1] k ¼ 2

k20

  2ðzÞ n ðzÞ 1 þ j k0 nðzÞ 2

ð2:81Þ

where nðzÞ and ðzÞ are the refractive index and the amplitude gain coefficient, respectively. Within the grating region ðdx  x  dx þ gÞ, perturbation is introduced so that the refractive index and the amplitude gain coefficient become [16,20]: nðzÞ ¼ n0 þ
n cosð2b0 z þ Þ

ð2:82aÞ

53

COUPLED WAVE EQUATIONS IN DISTRIBUTED FEEDBACK SEMICONDUCTOR LD

and ðzÞ ¼ 0 þ
 cosð2b0 z þ  þ Þ

ð2:82bÞ

Here, n0 and o are the steady-state values of the refractive index and amplitude gain, respectively,
n and
 are the amplitude modulation terms,  is the non-zero residue phase at the z-axis origin and b0 is the propagation constant. In the above equation,  takes into account the relative phase difference between perturbations of the refractive index and amplitude gain. Suppose there is an incident plane wave entering the periodic, lossless waveguide at an angle of  as shown in Fig. 2.7. The propagation constant of the wave is assumed to be b0 .

Figure 2.7

A simple model used to explain Bragg conditions in a periodic waveguide.

At every periodic interval of , the incident wave will experience the same degree of refractive index change so that the incident wave will be reflected in the same direction. For a waveguide that consists of N periodic corrugations, there will be N reflected wavelets. In order that any two reflected wavelets add up in phase or interfere constructively, the phase difference between the reflected wavelets must be a multiple of 2. In other words, b0 ðAB þ BC Þ ¼ b0 ð2 sin Þ ¼ 2mp

ð2:83Þ

where m is an integer. If the incident wave is now approaching more or less at a right angle to the wavefront (i.e.   =2), eqn (2.83) becomes 2b0  ¼ 2mp

ð2:84Þ

This is known as the Bragg condition and b0 becomes the Bragg propagation constant. The integer m shown in the above equation defines the order of Bragg diffraction. Unless otherwise stated, first-order Bragg resonance (m ¼ 1) is assumed. Since a laser forms a selfsustained resonant cavity, the Bragg condition must be satisfied [16]. Rearranging eqn (2.84) gives b0

2n0 n0 !B  ¼ 
B c

ð2:85Þ

54

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

where
B and !B are the Bragg wavelength and the Bragg frequency, respectively. From eqn (2.85), it is clear that the Bragg propagation constant is related to the period of the physical grating, . By altering the grating period , the Bragg wavelength can be shifted according to the specific application. Using small signal analysis, the perturbations of the refractxive index and gain are always smaller than their average values, i.e.
n n0 ;


 0

ð2:86Þ

Substituting eqn (2.82) into (2.81) using the above assumption, generates k2 ðzÞ ¼ k20 n20 þ j2k0 n0 0 þ 2k0 ½k0 n0 þ j0 
n cosð2b0 z þ Þ þ 2jk0 n0
 cosð2b0 z þ  þ Þ With k0 n0 replaced by b and 0 < b, the above equation becomes   p
n
 j jð2b0 zþÞ 2 2 þj e e k ðzÞ  b þ 2jb0 þ 2b
2   p
n
 j jð2b0 zþÞ þj e þ 2b e
2 For the case when  ¼ 0, one can simplify eqn (2.87) to [20]   p
n
 þj k2  b2 þ 2jb0 þ 4b cosð2b0 z þ Þ
2

ð2:87Þ

ð2:88Þ

ð2:89Þ

By collecting all the perturbed terms, one can define a parameter  [16,22,23] such that ¼


n
 þj ¼ i þ jg
2

ð2:90Þ

Here i includes all contributions from the refractive index perturbation whilst g covers all contributions from the gain perturbation. The parameter  introduced in the above equation is known as the coupling coefficient. After a series of simplifications, eqn (2.89) becomes k2  b2 þ 2jb0 þ 4b cosð2b0 z þ Þ

ð2:91Þ

On substituting the above equation back into the wave equation, one ends up with o d2 E n 2 jð2b0 zþÞ jð2b0 zþÞ E¼0 þ b þ 2jb þ 2be þ 2be 0 dz2

ð2:92Þ

where the cosine function shown in eqn (2.91) has been expressed in phasor form. A trial solution of the scalar wave equation could be a linear superposition of two opposing travelling waves such that EðzÞ ¼ AðzÞejkun z þ BðzÞejkun z

ð2:93Þ

COUPLED WAVE EQUATIONS IN DISTRIBUTED FEEDBACK SEMICONDUCTOR LD

55

with k2un ¼ b2 þ 2jb0  ðb þ j0 Þ2

ð, 0 bÞ

ð2:94Þ

In the above equation, AðzÞ and BðzÞ are complex amplitudes of the forward and backward propagating waves [20]. kun is the complex propagation constant for the unperturbed structure with
n ¼
 ¼ 0 (i.e.  ¼ 0). Since kun  b þ j0 , the trial solution of the scalar wave equation can also be expressed in terms of the real propagation constant, b, such that EðzÞ ¼ AðzÞe0 z ejbz þ BðzÞe0 z ejbz ¼ CðzÞejbz þ DðzÞejbz

ð2:95Þ

In order to satisfy the Bragg condition shown earlier in eqn (2.84), the actual propagation constant, b, should be sufficiently close to the Bragg propagation constant, b0 , to make the absolute difference between them much smaller than the Bragg propagation constant. In other words, j b  b 0 j b0

ð2:96Þ

Such a difference between the two propagation constants is commonly known as the detuning factor or detuning coefficient, d, which is defined as d ¼ b  b0

ð2:97Þ

In other words, the trial solution can also be expressed in terms of the Bragg propagation constant, i.e. EðzÞ ¼ CðzÞez ejb0 z þ DðzÞez ejb0 z ¼ RðzÞejb0 z þ SðzÞejb0 z

ð2:98Þ

where RðzÞ and SðzÞ are complex amplitude terms. Since the grating period, , in a DFB semiconductor laser is usually fixed and so is the Bragg propagation constant, it is more convenient to consider eqn (2.98) as the trial solution of the scalar wave equation. By substituting eqn (2.98) into eqn (2.92), one ends up with the following equation  00 R  2jb0 R0  b20 R þ b2 R þ 2jb0 R ejb0 z  þ S00 þ 2jb0 S0  b20 S þ b2 S þ 2jb0 S ejb0 z   ð2:99Þ þ 2b e2jb0 z ej þ e2jb0 z ej R ejb0 z þ Sejb0 z ¼ 0 where R0 and R00 are the first- and second-order derivatives of R. Similarly, S0 and S00 represent the first- and second-order derivatives of S. With a ‘slow’ amplitude approximation, high-order derivatives like R00 and S00 become negligible when compared with their first-order terms. By separating the above equation into two groups, each having similar exponential dependence, one can get the following pair of coupled wave equations 

dR þ ð0  jÞR ¼ jSej dz

ð2:100Þ

dS þ ð0  jÞS ¼ jR ej dz

ð2:101Þ

56

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

Equation (2.100) collects all the expðjb0 zÞ phase terms propagating along the positive z direction, whilst eqn (2.101) gathers all the expð jb0 zÞ phase terms propagating along the negative direction. Since jj b, other rapidly changing phase terms such as expð j3b0 zÞ have been dropped. In deriving the above equations, the following approximation has been assumed b2  b20  b  b0 ¼ d 2b0

ð2:102Þ

Following the above procedures, one ends up with a similar pair of coupled wave equations for a non-zero relative phase difference between the refractive index and the gain perturbation (i.e.  6¼ 0) such that 

dR þ ð0  jÞR ¼ jRS Sej dz

ð2:103Þ

dS þ ð0  jÞS ¼ jSR R ej dz

ð2:104Þ

where RS ¼ i þ jg ej

ð2:105Þ

is the general form [24] known as the forward coupling coefficient and SR ¼ i þ jg ej

ð2:106Þ

is the backward coupling coefficient. From the scalar wave equation, a pair of coupled wave equations have been established. The forward coupling coefficient RS in eqn (2.103) induces the negative travelling electric field SðzÞ to couple in the counter propagating one RðzÞ and vice versa for eqn (2.104). Contrary to FP lasers, where optical feedback is originated from the laser facets, optical feedback in DFB semiconductor lasers occurs continuously along the active layer where corrugations are fabricated. Depending upon the nature of the coupling coefficient, DFB semiconductor lasers are classified into three different groups: purely index-coupled DFB semiconductor lasers; mixed-coupled DFB semiconductor lasers and purely gain- or losscoupled DFB semiconductor lasers.

2.4.1 A Purely Index-coupled DFB Laser Diode Most practical DFB semiconductor lasers belong to this type, where coupling is solely generated by the refractive index perturbation. A single layer of corrugation is fabricated above (or below) the active layer as shown in Fig. 2.8.

COUPLED WAVE EQUATIONS IN DISTRIBUTED FEEDBACK SEMICONDUCTOR LD

Figure 2.8

57

A simplified schematic diagram for a purely index-coupled DFB semiconductor laser.

Since most carrier recombination is confined along the active layer, the amplitude gain of the DFB laser will not be affected. Therefore, with g ¼ 0, the index coupling coefficient i (which is purely real) is related to RS and SR by the expression [20] RS ¼ SR ¼ i

ð2:107Þ

2.4.2 A Mixed-coupled DFB Laser Diode If the corrugation layer is fabricated on the upper part of the active layer as shown in Fig. 2.9, the DFB semiconductor laser will show a mixed coupling characteristic [24–26]. Due to the variation of refractive index along the corrugation layer, index coupling is induced. However, the occurrence of gain coupling needs further explanation. As illustrated in Fig. 2.9, the active layer thickness becomes a periodic function along the longitudinal

Figure 2.9

A simplified schematic diagram showing a mixed-coupled DFB semiconductor laser.

direction and so is the optical confinement factor. Such a periodic modulation of optical confinement factor modifies the amplitude gain along the longitudinal direction and so gain coupling is induced [27]. Since, both refractive index and gain coupling are induced by the same corrugation, the corresponding phases of the i and g are assumed to be equal. For a zero relative phase difference ð ¼ 0Þ, one ends up with the following identity RS ¼ SR ¼ i þ jg

ð2:108Þ

58

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

2.4.3 A Gain-coupled or Loss-coupled DFB Laser Diode With only one single layer of grating, it is difficult to achieve a purely gain-coupled DFB device. However, by fabricating a second layer of grating on top of the original one as shown earlier for the mixed-coupled DFB laser, the effect of index coupling can be cancelled out. As illustrated in Fig. 2.10, the second corrugation demonstrates an inverse corrugation phase with respect to the first layer of grating. Utilising the metal–organic chemical vapour

Figure 2.10 laser.

A simplified schematic diagram showing a purely gain-coupled DFB semiconductor

deposition (MOCVD) technique, the first purely gain-coupled DFB laser based on this double grating structure was made in 1989 by Luo et al. [28]. GaAs was used as the active layer of the laser and the lasing wavelength was about 877 nm. Due to the direct modulation of the active layer thickness, it was found that the actual gain coupling coefficient of this structure may fluctuate according to the strength of the injection current. For such a purely gain-coupled structure having i ¼ 0, both the forward and backward coupling coefficients become purely imaginary and so RS ¼ SR ¼ jg

ð2:109Þ

The relative phase difference between the index and gain coupling becomes insignificant as the index coupling is cancelled. In addition to the possibility shown in Fig. 2.10, a second method of realising purely gain-coupled characteristics is to fabricate a periodic variation of loss as sketched in Fig. 2.11. The optical confinement factor remains constant whilst the

Figure 2.11 laser.

A simplified schematic diagram showing a purely loss-coupled DFB semiconductor

COUPLED WAVE EQUATIONS IN DISTRIBUTED FEEDBACK SEMICONDUCTOR LD

59

cavity loss becomes a periodic function of z. With such a loss-coupling structure, the strength of gain coupling will not be affected by any change in injection current. On the other hand, owing to the additional loss, the loss-coupling structure results in a higher threshold current. Comparatively, the design of purely index-coupled DFB semiconductor lasers has received significant attention in the past decade. There are reasons why the development of mixed- or gain-coupled DFB lasers were hindered. In a mixed-coupled DFB laser, a large number of non-radiative recombination centres were introduced during the fabrication of the corrugation layer. Since the corrugation layer has direct contact with the active layer, such an increase in non-radiative recombination centres implies an enormous increase in the threshold current. The performance of the laser also deteriorates rapidly as temperature increases. Additionally, the change of amplitude gain becomes complicated since the gain always depends on the injection current. Even though the loss-coupling structure proposed by Morthier et al. [29] may ease the problem, it is limited by a higher threshold current. When manufacturing the double grating structure, one must consider the alignment between the first and the second corrugations as this is crucial in the cancellation of the indexcoupling effect. From the pair of coupled wave equations shown earlier in eqns (2.103) and (2.104), one can also obtain the net power change experienced by both counter-running waves travelling along the laser cavity. By considering the conjugate pairs of eqns (2.103) and (2.104), the rate of total power change can be found [30] d ½RR
 SS
 ¼ 20 ½RR
þ SS
 þ Pmut dz

ð2:110Þ

where * is used to represent the complex conjugate and   Pmut ¼ j 
RS  SR RS
ej  j RS  
SR R
Sej

  ¼ 2 Im RS  
SR R
Sej

  ¼ 2 Im 
RS  SR RS
ej

ð2:111Þ

The first term on the right-hand side of eqn (2.110) describes the total power change experienced by each individual coupled wave, whilst the second term, Pmut describes the mutual interaction between the coupled waves. For a purely index-coupled DFB laser, both the forward and backward coupling coefficients are real and equal. As a result, the mutual interaction term becomes negligible. However, one must take into account the mutual interaction term when purely gain-coupled, purely loss-coupled or mixed-coupled DFB lasers are used. The pair of coupled wave equations, which characterise the interaction of electric fields are general. With modification, the pair can be used in other applications which may involve wave or mode interaction. The complex permittivity, i.e. the refractive index and amplitude gain, may change in different ways according to various applications. Based on the coupled wave analysis, equations which describe other physical processes like electro-optic modulation, magneto-optic modulation or non-linear interaction can be found in other references such as Yariv [2].

60

2.5

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

COUPLING COEFFICIENT

2.5.1 A Structural Definition of the Coupling Coefficient for DFB Semiconductor Lasers Depending on the relative position of the corrugation with the active layer, both refractive index and/or gain vary along the longitudinal direction of z. By solving the coupled wave equations, one can solve the threshold conditions of conventional DFB semiconductor lasers. The coupling coefficients RS and SR play an important role as they measure the wave feedback capability due to the presence of the corrugation. So far, the coupling coefficient has been defined with respect to the changes in refractive index and gain such that ¼

p
n
 þj
2

ð2:112Þ

In this section, based upon a general perturbation of the relative permittivity, the coupling coefficient is found to be related to the shape, the depth and the period of the corrugation. To build such a structural definition for the coupling coefficient, one starts again with the time harmonic vector wave equation E þ k2 ~ E¼0 r2 ~

ð2:113Þ

By expanding the propagation constant so as to include the relative permittivity, the above wave equation becomes r2 ~ E þ "ðx; y; zÞk20~ E¼0

ð2:114Þ

where " is the relative permittivity and k0 ¼ !=c is the free space propagation constant. There is a major difference between a normal dielectric planar waveguide and a corrugated waveguide. Provided that the corrugation is extending in the longitudinal direction, one can express the relative permittivity of the corrugated region of the laser as "ðx; y; zÞ ¼ "ini ðx; yÞ þ
"ðx; y; zÞ

ð2:115Þ

where "ini ðx; yÞ is the average relative permittivity of the transverse plane x–y and
" is a perturbation term which is zero everywhere except the corrugated region whose thickness is equivalent to the corrugation depth. Assuming TE mode excitation only, and following eqn (2.95), a general solution of the vector wave equation may take the form ~ E ¼ Uðx; yÞ½Cejbz þ Dejbz  ^j ¼ Ey ^j

ð2:116Þ

where ^j is the unit vector along the junction plane for the TE mode, b is the mode propagation constant and Uðx; yÞ is the field solution along the transverse x–y plane. The trial solution shown above is slightly different from the one we used in the previous section. For a specific waveguiding structure, the field distribution can be obtained by solving

@2U @2V þ 2 þ "ini ðx; yÞk20  b2 U ¼ 0 2 @x @y with appropriate boundary conditions.

ð2:117Þ

61

COUPLING COEFFICIENT

With careful control over the active layer width and the active layer thickness, singlemode oscillation along the transverse plane is assumed. Under the influence of the perturbation term
", the amplitude coefficients C and D become z dependent. Assuming only small perturbation, i.e.
" "ini , it is unlikely that the field distribution Uðx; yÞ will be affected and so it is made separable from the longitudinal component of the field solution. We then substitute eqn (2.116) into eqn (2.114) and allow a ‘slow’ variation of C and D. By multiplying the resulting equation by Uðx; yÞ and integrating over the transverse x–y plane, one ends up with dC jbz dD jbz jk20 e e ¼  2bV dz dz

ðð




"ðx; y; zÞU 2 ðx; yÞ Cejbz þ Dejbz dz dy

ð2:118Þ

where ðð V¼

U 2 ðx; yÞdx dy

ð2:119Þ

is the mode intensity for the unperturbed planar waveguide. For simplicity, only index coupling is assumed so that
" is real. Since the perturbed term
" is a periodic function of z, it could be expanded in Fourier series [31,32] such that
"ðx; y; zÞ ¼

1 X

Aq6¼0 ðx; yÞ exp

q¼1

  2jqpz 

ð2:120Þ

Aq6¼0 ðx; yÞ is the q-th harmonic Fourier coefficient which depends on the shape, the depth and the period of the corrugation. When q ¼ 0, a z-independent function is formed which equals the average relative permittivity term "ini . Now, by substituting the above equation into eqn (2.118) and equating coefficients with exponential terms e jbz , one would get dC ¼ j
Dej2z dz

ð2:121Þ

dD ¼ jCej2z dz

ð2:122Þ

where  ¼ b  b0 ¼ b  mp=

ð2:123Þ

 is the same detuning factor and m is the order of Bragg diffraction. Usually, the smallest detuning factor is found by allowing q ¼ m in eqn (2.120). In the analysis, other higher order phase terms have been neglected.  and 
shown are complex conjugate pairs which can be written as ÐÐ k20 Aq6¼0 ðx; yÞU 2 ðx; yÞdx dy ÐÐ ¼ ð2:124Þ 2b U 2 ðx; yÞdx dy

62

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

When the corrugation is removed or the grating depth is diminished to become zero, a planar waveguide is formed. With the coupling coefficient  vanished, the coupled wave equations could be satisfied by any z-independent function. However, due to the material gain characteristics inside the laser cavity, an additional term is added to eqns (2.121) and (2.122). As a result, the independent solutions C and D, correspond, respectively, to exponentially growing waves along the þz and z directions when  ¼ 0. In order to fulfil the physical requirement, eqns (2.121) and (2.122) become dC ¼ j
Dej2z þ s C dz dD ¼ jCej2z  s D dz

ð2:125Þ ð2:126Þ

where the amplitude gain coefficient s is appended. Unsurprisingly, a similar pair of coupled wave equations has been derived. As compared with eqns (2.100) and (2.101), the pair of equations shown above looks distinctive because a different trial solution has been used in solving the wave equation. By replacing b with b0 þ  in the above equations, the original coupled wave equations can be recovered. The parameter  as shown in eqn (2.124) is the coupling coefficient as defined earlier for purely index-coupled DFB lasers.

2.5.2 The Effect of Corrugation Shape on Coupling Coefficient Since the coupling coefficient is associated with the perturbed relative permittivity, the numerical value of  depends on the shape, the depth and the period of the corrugation. In addition, the composition and the thickness of the active and cladding layers will also affect the coupling coefficient as  involves the calculation of the transverse mode energy. The evaluation of the coupling coefficient and the impact on corrugation shape of three-layer GaAs DFB lasers have been discussed extensively [31,32]. By comparison, little work [33] has been done on five-layer InGaAsP DFB semiconductor lasers. In a buried heterostructure where the active layer thickness is much narrower than its width, one can assume that the corrugation is laterally uniform so that the relative permittivity term
" is independent of y. As a result, eqn (2.115) is simplified to form "ðx; yÞ ¼ "ini ðxÞ þ
"ðx; yÞ and the coupling coefficient for purely index-coupled DFB lasers becomes Ð 2 k20 corrugation Aq6¼0 Ey ðxÞdx Ð ¼ 2b Ey2 ðxÞdx

ð2:127Þ

ð2:128Þ

where Ey ðxÞ is the transverse field component of the TE mode which satisfies the wave equation shown in eqn (2.114). The integral of the numerator is restricted to the corrugated layer only as the perturbed relative permittivity
" (or the Fourier coefficient Aq6¼0 ) is not defined in any other layer. To investigate the effect of different periodic grating shapes, a general multi-layer model as illustrated in Fig. 2.12 is used. This is similar to the one shown in Fig. 2.6, however, the

COUPLING COEFFICIENT

63

Figure 2.12 A general multi-dielectric stack used to evaluate the coupling coefficient of DFB semicondutor lasers.

cosinusoidal corrugation is now replaced by two discontinuous arbitrary functions Z1 ðxÞ þ p and Z2 ðxÞ þ p where p is any integer. For any periodic corrugation shape, it is important that the sum of the corrugation functions Z1 ðxÞ þ p and Z2 ðxÞ þ p will span a longitudinal distance of , the period of the corrugation. For simplicity, integer p is set to zero for the first corrugation function found on the positive x–z plane as the corrugation is extended along the positive z direction. Then, one can express the relative permittivity "ðx; zÞ analytically as "ðx; zÞ ¼ n21 ; ¼

n22 ; .. .

¼ n2l ; ¼

n2lþ1 ; .. .

¼ n2N1 ; ¼

n2N ;

x > d1 d1 > x > d2 .. . dl1 > x > dx þ g dx > x > dlþ1 ... dN2 > x > dN1 dN1 > x

ð2:129Þ

for layers found outside the corrugated layer. For the relative permittivity of the corrugated layer, one can write [31] "ðx; zÞ ¼

1 X

n2l fu½z  Z1 ðzÞ  p  u½z  Z2 ðzÞ  pg

p¼1

þ n2lþ1 fu½z  Z2 ðzÞ  p  u½z  Z1 ðzÞ  ðp þ 1Þg

ð2:130Þ

64

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

where  uðÞ ¼

1; 0;

>0 <0

ð2:131Þ

is the unit step function in . By equating eqn (2.120) with (2.130) and setting p ¼ 0, one can express the harmonic Fourier coefficient as 1 Aq¼0 ¼ A0 ¼ n2lþ1 þ ½Z2 ðxÞ  Z1 ðxÞn2l  n2lþ1 L

ð2:132Þ

for q ¼ 0 and 1 

Aq6¼0 ¼

ð Z1 ðzÞþ Z1 ðzÞ

n2 ðx; zÞe

j2pqz 

dz

i j2pqZ1 ðxÞ n2lþ1  n2l h j2pqZ2 ðxÞ e  e  2jpq

¼

ð2:133Þ

for q 6¼ 0. The last equation is justified as the integral can be separated into ð Z1 ðxÞþ Z1 ðxÞ

¼

ð Z2 ðxÞ Z1 ðxÞ

þ

ð Z1 ðxÞþ Z2 ðxÞ

ð2:134Þ

The relative permittivities at different integral ranges can be expressed as ( n ðx; zÞ ¼ 2

n2l ;

Z2 ðxÞ > z > Z1 ðxÞ

n2lþ1 ;

Z1 ðxÞ þ  > z > Z2 ðxÞ

ð2:135Þ

Since Z1 ðxÞ þ Z2 ðxÞ ¼ , eqn (2.133) is simplified to become [33] Aq6¼0

  n2lþ1  n2l 2mp Z1 ðzÞ ; sin ¼  mp

dx < x < dx þ g

ð2:136Þ

where q ¼ m is assumed for the smallest detuning factor. The corrugation functions Z1 ðxÞ for trapezoidal, rectangular, triangular and sinusoidal corrugations are listed in Table 2.2. For the trapezoidal shape, WT and WB denote the top width and the bottom width of the corrugation. For simplicity, the rising edge and the dropping edge are assumed to occupy the same widths of W. Finally, on combining eqn (2.136) with (2.128), one ends up with the following expression for the coupling coefficients of purely index-coupled DFB LDs [34]  ð      k20 n2lþ1  n2l 2mpZ1 ðxÞ 2  Ð ¼ Ey ðxÞdx sin  2  2mpb Ey ðxÞdx corrugation

ð2:137Þ

where an absolute sign is used to ensure a positive value for the coupling coefficient.

Table 2.2 Various grating shapes and the corresponding corrugation functions Z1 ðxÞ

66

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

2.5.3 Transverse Field Distribution in an Unperturbed Waveguide From eqn (2.137), the coupling coefficients of purely index-coupled DFB LDs were expressed. With a specific corrugation function and a given refractive index distribution, one still needs to determine the propagation constant b and the mode energy of the unperturbed waveguide before the value of the coupling coefficient can be evaluated.

Figure 2.13 A simplified five-layer DFB structure with trapezoidal corrugation.

Figure 2.13 shows the structure of a five-layer separate confinement heterostructure (SCH) DFB LD used in the analysis. It consists of a thin optical confinement region, itself divided into three parts that include the active layer and two waveguiding layers. III–V InGaAsP compounds are used in fabricating the optical confinement region. The refractive indices of the waveguiding layers (n2 and n4 ) are slightly smaller than that of the active layer (n3 ) so that photons are confined in the active layer. Optical feedback provided by a layer of trapezoidal corrugation is fabricated within the upper waveguiding layer. The optical confinement region is bounded by two thick cladding layers. With a higher bandgap material used, the cladding layers act as the optical barriers. Since the active layer thickness remains constant along the longitudinal direction, any gain or loss coupling can be neglected and only pure index coupling is allowed. To compute the unperturbed transverse electric field Ey ðxÞ in the five-layer SCH, the effect due to the presence of corrugations cannot be ignored. One method which was proposed by Handa et al. [35] is to replace the corrugation layer that has an additional layer with an intermediate refractive index. However, the waveguiding properties become more complicated as one extra layer is added. Another method [31,32] that enables one to obtain adequate accuracy without excessive complexity is to choose the unperturbed waveguide boundary at x ¼ t2 < d2 as shown in Fig. 2.13. Since the corrugation depth g is found to be much smaller than the actual lasing wavelength, the unperturbed waveguide boundary at t2 is chosen such that the volume of the n2 material extending into the upper cladding layer equals the volume of n1 extending into the upper waveguiding layer. As a result, the

COUPLING COEFFICIENT

67

contribution due to the refractive index changes above the unperturbed waveguide boundary will be the same as that below. Since the active layer thickness, d3, is usually smaller than the active layer width, the transverse field along the y direction becomes negligible. As a result, the corresponding boundary is determined by equating the area bounded by the upper part of the corrugation (AreaT) to that of the area bounded below (AreaB). This method, which involves a dynamic shifting of one of the waveguide boundaries, is simpler and more effective than the one proposed by Handa et al. [35]. On the other hand, one must take care when choosing the unperturbed waveguide boundary because it is sensitive to the change of corrugation shape. Mathematically, the boundary of the corresponding unperturbed waveguide can be expressed as 1 ^ g ¼ d2  t2 ¼ 

ð d2 d2 g

½Z2 ðxÞ  Z1 ðxÞ dx

ð2:138Þ

where g is the depth of the corrugation and ^ g ¼ d2  t2 is the boundary shift measured from the top of the corrugation to the new boundary of the corresponding unperturbed waveguide. The upper surface of the active layer is fixed at x ¼ 0. Then, Maxwell’s wave equations in each of the five layers may be written as [36] d2 Ey þ hj Ey ¼ 0; dx2

j ¼ 1; . . . ; 5

ð2:139Þ

where j is an integer used to represent different layers and hj is the propagation constant of the j-th layer. For the structure shown, it is assumed that the unperturbed transverse electric field is exponentially decaying in the cladding layers while it is sinusoidal in the others. Then, the propagation constant hj can be written as [6,21] ( hj ¼

b2eff  k20 n2j ; k20 n2j  b2eff ;

j ¼ 1 and 5 j ¼ 2; 3 and 4

ð2:140Þ

where beff ¼ k0 neff ¼ b in eqn (2.137) and neff is denoted as the effective refractive index. As the transverse electric field is propagating along the waveguiding layers as well as the active layer, the refractive index distribution must satisfy the following relation [37] n5  n1 < neff < n4  n2 < n3

ð2:141Þ

For TE mode excitation in the five-layer SCH laser structure, the transverse electric field in each layer can be written as [36] 8 A1 exp½h1 ðt2  xÞ > > > > > < A2 cosðh2 x  a2 Þ Ey ¼ A3 cosðh3 x  a3 Þ > > > A4 cosðh4 x  a4 Þ > > : A5 exp½h5 ðd3 þ d4 þ xÞ

for x  t2 0  x  t2 d3  x  0 ðd3 þ d4 Þ  x  d3 x  ðd3 þ d4 Þ

ð2:142Þ

68

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

where A1 to A5 are the leading coefficients of the transverse electric field in various layers, t2 is the boundary separation for the corresponding unperturbed waveguide and a2 , a3 and a4 are three constants. As the field component Ey and @Ey =@x must be continuous at various boundaries for TE mode excitation, one ends up with the following equations for the TE mode: At

x ¼ t2

At x ¼ 0 At x ¼ d3 At x ¼ ðd3 þ d4 Þ

h2 tanðh2 t2  a2 Þ ¼ h1

ð2:143aÞ

h3 tanða3 Þ ¼ h2 tanða2 Þ h3 tanðh3 d3 þ a3 Þ ¼ h4 tanðh4 d3 þ a4 Þ

ð2:143bÞ ð2:143cÞ

h4 tan½h4 ðd3 þ d4 Þ þ a4  ¼ h5

ð2:143dÞ

where the constants a2 , a3 and a4 can then be determined as   h1 a2 ¼ h2 t2  tan h2   h 2 1 tanða2 Þ a3 ¼ tan h3 1

ð2:144aÞ ð2:144bÞ

and a4 ¼ h4 ðd3 þ d4 Þ þ tan1

  h5 h4

ð2:144cÞ

In order to reduce the number of variables used, it is easier to select one of them as a common variable so that others can then be written in terms of it. From the boundary conditions for the Ey field component, one can express the leading coefficients of the electric field in terms of the common coefficient A1 such that A1 cosðh2 t2  a2 Þ A1 cosða2 Þ A3 ¼ cosðh2 t2  a2 Þ cosða3 Þ A1 cosða2 Þ cosðh3 d3 þ a3 Þ A4 ¼ cosðh4 d3 þ a4 Þ cosðh2 t2  a2 Þ cosða3 Þ A1 cosða2 Þ cosðh3 d3 þ a3 Þ cos½h4 ðd3 þ d4 Þ þ a4  A5 ¼ cosðh4 d3 þ a4 Þ cosðh2 t2  a2 Þ cosða3 Þ A2 ¼

ð2:145aÞ ð2:145bÞ ð2:145cÞ ð2:145dÞ

where constants a2 , a3 and a4 have been fixed in eqns (2.144). By joining all the equations in (2.143), an eigenvalue equation for the even TE is found, which can be expressed [36] as   1=2 h3 d 3 ð1 þ ABÞ þ ½ð1 þ A2 Þð1 þ B2 Þ tan ¼ 2 AB

ð2:146aÞ

COUPLING COEFFICIENT

69

where A ¼ h4 tanðh4 d3 þ a4 Þ=h3

ð2:146bÞ

B ¼ tanða2 Þ ¼ h2 tanða2 Þ=h3

ð2:146cÞ

In finding the effective refractive index, the method of bisection is employed in solving the above eigenequation. The total number of iterative steps used is usually smaller than ten and an error of less than 109 may be achieved. After fixing the effective refractive index, the propagation constants, hj in each layer can be found readily using eqn (2.140). Then, using eqns (2.144), a2 , a3 and a4 are determined. The transverse electric field in the five-layer laser structure can be obtained after solving the eigenequation. However, in evaluating the coupling coefficient, one must determine the total mode energy confined in the five-layer structure. From the proposed transverse electric field distribution, one ends up with [34] I¼

ð1 1 ð0

þ

Ey2 ðxÞdx ¼

d3

ð ðd3 þd4 Þ 1 ð t2

Ey2 ðxÞdx þ

0

Ey2 ðxÞdx þ

Ey2 ðxÞdx þ

ð1 t2

ð d3 ðd3 þd4 Þ

Ey2 ðxÞdx

Ey2 ðxÞdx ¼ I5 þ I4 þ I3 þ I2 þ I1

ð2:147Þ

where A21 2h1   A2 1 1 ¼ 2 t2 þ sinð2h2 t2  2a2 Þ þ sinð2a2 Þ 2h2 2h3 2   2 A 1 1 ¼ 3 d3 þ sinð2h3 d3 þ 2a3 Þ  sinð2a3 Þ 2 2h3 2h3   2 A4 1 1 d4 þ ¼ sin½2h4 ðd3 þ d4 Þ þ 2a4   sinð2h4 d3 þ 2a3 Þ 2h4 2h4 2 A2 ¼ 5 2h5

I1 ¼

ð2:148aÞ

I2

ð2:148bÞ

I3 I4 I5

ð2:148cÞ ð2:148dÞ ð2:148eÞ

By choosing A1 as the common variable, the total mode energy confined in the five-layer waveguide is finalised after the effective refractive index, the propagation constants h1 to h5 and the constants a2 , a3 and a4 are confirmed from the eigenequation for a particular corrugation function. The integral I2, which is a function of t2, depends on the shape of the corrugation.

2.5.4 Results Based upon the Trapezoidal Corrugation In this section, the coupling coefficient values based on the five-layer DFB lasers will be evaluated for the trapezoidal corrugation. From the previous section, the total mode energy

70

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

found inside the five-layer waveguide is expressed in terms of the leading coefficient A1. Before the Fourier coefficient Aq , and hence the coupling coefficients, are determined, the corrugation functions Z1 ðxÞ and Z2 ðxÞ of the trapezoidal corrugation must be defined. According to Fig. 2.13, one ends up with WT W þ ðd2  xÞ g 2 WT W þ 2W þ WB þ ðx  d2 Þ Z2 ðxÞ ¼ 2 g

Z1 ðxÞ ¼

ð2:149aÞ ð2:149bÞ

Therefore, the unperturbed boundary t2 for the equivalent unperturbed planar waveguide can be determined from eqn (2.138) since ð 1 d2 ½Z2 ðxÞ  Z1 ðxÞdx  d2 g WB þ W ¼ g 

^ g ¼ d2  t 2 ¼

ð2:150Þ

where WT (in eqn (2.149)) and WB are the top width and the bottom width of the trapezoidal corrugation. The variable W denotes the longitudinal distance spanned by the rising and the dropping of the corrugation. As a result, the coupling coefficient of the trapezoidal corrugation becomes ¼

 ð d2   k20 ðn22  n21 Þ 2   Ð sin ½ a þ bðd  xÞ E ðxÞdx 2 y   2 2mpb Ey ðxÞdx d2 g

ð2:151Þ

where m is the order of Bragg diffraction, b is the propagation constant determined from the effective refractive index and mpWT  2mp W b¼  g

a¼

ð2:152aÞ ð2:152bÞ

From eqn (2.142), the proposed transverse electric field found along the corrugated layer ðd2  x  d2  gÞ can be written as  Ey ðxÞ ¼

A1 eh1 ðt2 xÞ ; A2 cos½h2 x  a2 ;

d2  x  t2 t 2  x  d2  g

ð2:153Þ

Then, by joining together eqns (2.151), (2.152) and (2.153), and after substantial simplification, the coupling coefficient of the trapezoidal corrugation becomes [33] " " )  (  2 #  2 # k20 n22  n21 h2 Q2 h1 Q 3 h1 þ 1 þ Q4 g Q1 þ 1 þ ¼ 4pbN 2 h1 2 h2 2 h2

ð2:154Þ

COUPLING COEFFICIENT

71

where Q1 ¼ Q2 ¼ Q3 ¼ Q4 ¼

ðh1 g sin c  bÞ  expð2h^gÞ ðh1 g sin a  b cos aÞ ð h1 gÞ 2 þ b2

cos c  cos c b bðcos c  cos c cos vÞ  h2 gðsin c sin vÞ b2  ðh2 gÞ2 h2 gðsin c cos v  sin cÞ  bðcos c sin vÞ b2  ð h2 gÞ 2

ð2:155aÞ ð2:155bÞ ð2:155cÞ ð2:155dÞ

and c ¼ a þ bg c ¼ a þ bg gÞ v ¼ 2h2 ðg  

ð2:156Þ

(a) Trapezoidal Corrugation for First-order Bragg Diffraction (m ¼ 1) In the following calculations, we have considered only the TE0 mode. The structural parameters used are tabulated in Table 2.3. By setting m ¼ 1 in the above equation, the coupling coefficient of the first-order trapezoidal corrugation can be determined. The effects of different corrugation shapes are illustrated in Fig. 2.14, in which the coupling coefficient is plotted against the bottom width of the corrugation, WB, while different values of WT (the top width of the corrugation) are used as a comparison. Both WT and WB are normalised with respect to the corrugation period . As observed in Fig. 2.14, for each selected value of WT =, there exists a peak coupling coefficient where the largest possible optical feedback can be achieved. For instance, the triangular corrugation (with WT = ¼ 0) tends to show a peak coupling coefficient of about 15 mm1 when the normalised bottom width WB = ¼ 0:25. With the normalised top width WT = increasing from 0.0 to 0.9, the associated WB = values of the peak coupling coefficients also increase. The largest coupling

Table 2.3 Structural parameters used in determining the coupling coefficient of the first-order trapezoidal corrugation Parameter

Value

m d2 ¼ d4 d3 n1 ¼ n5 n2 ¼ n4 n3 g

1 200 mm 100 mm 3.17 3.405 3.553 75 nm

72

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

Figure 2.14 The change in the first-order coupling coefficient with WB = for different values of WT =.

coefficient found near WB = ¼ WT = ¼ 0:5 has a value of 20.9 mm1. This is where the symmetrical rectangular corrugation is found. So far, discrete values of WT = have been used. By expanding the results into a contour map as shown in Fig. 2.15, the structural impact due to the variation of corrugation shapes can be examined comprehensively. In fact, the trapezoidal corrugation we examined is so general that other shapes, such as the triangular and rectangular corrugation, are included. Symmetrical triangular corrugation is located at the origin where WT = ¼ WB = ¼ 0. With a corrugation depth g ¼ 75 nm, the coupling coefficient is found to be about 13.0 mm1. Other asymmetric triangular corrugation is included along the WB = axis when WT = is forced to become zero. Similarly, the inverted triangular corrugation (see Table 2.3 for exact corrugation shapes) is found along the WT = axis as WB = drops to zero. For rectangular corrugation, W= ¼ 0 and WT þ WB ¼ . Therefore, by joining the point WT = ¼ 0, WB = ¼ 1 to the opposite diagonal of the contour (i.e. WT = ¼ 1, WB = ¼ 0), one can evaluate the coupling coefficients for the rectangular corrugation. The symmetrical rectangular corrugation which is included as a special case is located at the centre of the contour with WT = ¼ WB = ¼ 0:5. Similarly, by joining the origin with the other extreme end where WT = ¼ WB = ¼ 1, the coupling coefficient of the symmetrical trapezoidal corrugation can be determined. Due to the presence of corrugation and the use of the boundary shifting method, the calculated boundary at x ¼ t2 for the corresponding unperturbed waveguiding structure is expected to be smaller than the actual thickness of the upper waveguiding layer d2. As a result, the DFB waveguiding structure is no longer symmetrical. The thickness of the upper

COUPLING COEFFICIENT

Figure 2.15 corrugation.

73

Contour map showing the coupling coefficient  for the first-order trapezoidal

cladding layer, which depends on the corrugation shape, is found to be thinner than the lower waveguiding layer. Thus, the maximum coupling coefficient is found to be slightly displaced from the centre of the contour where symmetrical rectangular corrugation is located. The maximum coupling coefficient found is about 20.5 mm1.

(b) Trapezoidal Corrugation for Second-order Bragg Diffraction (m ¼ 2) In order to compare the coupling coefficients of DFB LDs with different orders of Bragg diffraction, the effective refractive index, and thus the boundary shift g of the DFB structures, must be identical. For the m-th order Bragg diffraction, the Bragg propagation constant is defined as b0 ¼

mp L

ð2:157Þ

In order to maintain the Bragg propagation constant for the second-order corrugation (m ¼ 2), the second-order grating period must be doubled. Meanwhile, the corrugation depth g is also increased to twice that of the first-order value so that the boundary of the unperturbed waveguide g of eqn (2.150) is maintained. The parameters used to determine the coupling coefficient of the second-order trapezoidal corrugation can be found in Table 2.4.

74

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

Table 2.4 Structural parameters used in determining the coupling coefficient of the second-order trapezoidal corrugation Parameter m d2 ¼ d4 d3 n1 ¼ n5 n2 ¼ n4 n3 g

Value 2 200 mm 100 mm 3.17 3.405 3.553 150 nm

By putting the above parameters into eqns (2.154), (2.155) and (2.156), the effect of corrugation shapes on the second-order coupling coefficient is investigated. This is illustrated in Fig. 2.16. Just as for the first-order plot shown in Fig. 2.14, the coupling coefficient is shown as a function of WB for various values of WT. Both WB and WT are normalised with respect to the second-order corrugation period . As observed in Fig. 2.16, there exist two peak values of coupling coefficients along the WB = axis for each selected value of WT =. In between the peaks, there are places where the coupling coefficient drops to zero value. It is believed that the electric field diffracted by the second-order corrugation is completely out of phase with the incident wave. Therefore, zero coupling coefficients

Figure 2.16 The change in the second-order coupling coefficient with WB = for different values of WT =.

COUPLING COEFFICIENT

75

follow at that particular corrugation shape. The maximum coupling coefficient value of about 26.9 mm1 is located near the point where WB = ¼ 0:25 and WT = ¼ 0:75. When compared with the results from the first-order coupling coefficient, one can see significant changes in the magnitude of the maximum coupling coefficient as well as the normalised corrugation width associated with it.

Figure 2.17 corrugation.

Contour map showing the coupling coefficient  for the second-order trapezoidal

From the contour map shown in Fig. 2.17, one can arrive at a better understanding of the change of coupling coefficient with a continuous change of corrugation shape. Rather than a single peak as seen earlier in the first-order contour map, two outstanding peaks can be observed. For an effective optical feedback in DFB lasers, a device engineer must be aware of specific corrugation shapes in second-order corrugation design that may lead to extremely low values of coupling coefficient. The Fabry–Perot effect due to non-zero facet reflection has to be considered in DFB lasers when the value of the coupling coefficient becomes small. The corrugation of both the first- and second-order grating has been computed for the fundamental TE mode in a five-layer slab waveguide structure. It has been found that a larger value of coupling coefficient can be obtained when second-order corrugation is used. However, the design of the second-order grating requires a precise control of the grating parameters. It is shown in the contour map that certain corrugation shapes may lead to very low values of coupling coefficient. Impacts due to the variation of the lateral electric field [31,38], any misalignment or curvature [39] and any deformation [40] of corrugations which may form during the fabrication process, have been explored in other references and will not be discussed further.

76

2.6

PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

SUMMARY

In this chapter, the operational principles of lasers, in particular the semiconductor laser, were presented. To build a self-sustained oscillator like a laser, it is important that the condition of population inversion is satisfied and an optical resonator is formed. The Fabry– Perot semiconductor laser which forms the simplest optical resonant cavity has limited applications due to its broad gain spectrum. Multi-mode oscillations and unstable mode hopping are common in this type of laser. In optical coherent communication systems, it is important that the optical source generates a single stable longitudinal mode output. With built-in periodic corrugations along the direction of propagation, single longitudinal mode operation becomes feasible in the DFB semiconductor laser. The built-in corrugation acts as an optical bandpass filter so that only frequencies with components near the Bragg frequency are allowed to pass. The operational principles of DFB LDs were explained with the help of the coupled wave equations. From the nature of the coupling coefficient, DFB semiconductor lasers can be classified into purely index-coupled, mixed-coupled and purely gainor loss-coupled structures. The magnitude of the coupling coefficient, and hence the optical feedback, depends on the corrugation. Using the Fourier series technique, the coupling coefficient of a five-layer SCH structure was computed. Contours showing the relationship between the coupling coefficient and the physical dimensions of the corrugation were shown for both first- and second-order Bragg diffraction.

2.7

REFERENCES

1. Yariv, A., Quantum Electronics, 3rd edition. New York: Wiley, 1989. 2. Yariv, A., Optical Electronics, 4th edition. Orlando, FL: Saunders College Publishing, 1991. 3. Casey Jr., H. C. and Panish, M. B., Heterostucture Lasers Part A: Fundamental Principles. New York: Academic Press, 1978. 4. Agrawal, G. P. and Dutta, N. K., Long-Wavelength Semiconductor Lasers. Princeton, NJ: Van Nostrand, 1986. 5. Petermann, K., Laser Diode Modulation and Noise. Tokyo, Japan: KTK Scientific and Kluwer Academic Publishers, 1988. 6. Adams, M. J., Steventon, A. G., Delvin, W. J. and Henning, I. D., Semiconductor Lasers for LongWavelength Optical-Fibre Communications Systems, IEE material & devices series, ed. Dr. N. Parkman and Prof. D. V. Morgan, no. 4. London: Peter Peregrinus, 1987. 7. Lasher, G. and Stern, F., Spontaneous and stimulated recombination radiation in semiconductors, Phys. Rev., 133(2A), 553–563, 1964. 8. Osinski, M. and Adams, M. J., Gain spectra of quaternary semiconductors, IEE Proc., 129(pt. I, no. 3), 229–236, 1982. 9. Ghafouri-Shiraz, H., Temperature, Bandgap-wavelength, and doping dependence of peak-gain coefficient parabolic model parameters for InGaAsP/InP semiconductor lasers, J. Lightwave Technol., LT-6(4), 500–506, 1988. 10. Westbrook, L. D., Measurement of dg/dN and dn/dN and their dependence on photon energy in
¼ 1:5 mm InGaAsP laser diodes, IEE Proc. Pt. J, 133(2), 135–143, 1985. 11. Bissessur, H., Effects of hole burning, carrier-induced losses and the carrier-dependent differential gain on the static characteristics of DFB lasers, J. Lightwave Technol., LT-10(11), 1617–1630, 1992.

REFERENCES

77

12. Chu, C. Y. J. and Ghafouri-Shiraz, H., A simple method to determine carrier recombinations in a semiconductor laser optical amplifier, IEEE Photon. Tech. Lett., 5(10), 1182–1185, 1994. 13. Manning, J., Olshansky, R. and Su, C. B., The carrier-induced index change in AlGaAs and 1.30 mm InGaAsP diode lasers, IEEE J. Quantum Electron., QE-19(10), 1525–1530, 1983. 14. Bennett, B. R., Soref, R. A. and Alamo, A. D., Carrier-induced change in refractive index in InP, GaAs and InGaAsP, IEEE J. Quantum Electron., QE-26(1), 113–122, 1990. 15. Kotaki, Y. and Ishikawa, H., Wavelength tunable DFB and DBR lasers for coherent optical fibre communications, IEE Proc. Pt. J., 138(2), 171–177, 1991. 16. Kogelnik, H. and Shank, C. V., Coupled-wave theory of distributed feedback lasers, J. Appl. Phys., 43(5), 2327–2335, 1972. 17. Kogelnik, H., Coupled wave theory for a thick hologram grating, Bell Syst. Tech. J., 48, 2909–2947, 1969. 18. Wang, S., Principles of distributed feedback and distributed Bragg reflector Lasers, IEEE J. Quantum Electron., QE-10(4), 413–427, 1974. 19. Yariv, A. and Gower, A., Equivalent of the coupled mode and Floquet–Bloch formalism in periodic optical waveguide, Appl. Phys. Lett., 26, 537–539, 1975. 20. Streifer, W., Burnham, R. D. and Scifres, D. R., Effect of external reflectors on longitudinal modes of distributed feedback lasers, IEEE J. Quantum Electron., QE-11(4), 154–161, 1975. 21. Balanis, C. A., Advances in Engineering Electromagnetics. New York: John Wiley & Sons, 1989. 22. David, K., Morthier, G., Vankvikelberge, P., Baets, R., Wolf, T. and Borchert, B., Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: a comparison based on spatial hole burning corrected yield, IEEE J. Quantum Electron., 27(6), 1714–1724, 1991. 23. David, K., Buus, J., Morthier, G. and Baets, R., Coupling coefficient in gain-coupled DFB lasers: Inherent compromise between strength and loss, Photon. Tech. Lett., 3(5), 439–441, 1991. 24. David, K., Buus, J. and Baets, R., Basic analysis of AR-coated, partly gain-coupled DFB lasers: the standing wave effect, IEEE J. Quantum Electron., 28(2), 427–433, 1992. 25. Luo, Y., Nakano, Y., Tada, K., Inoue, T., Homsomatsu, H. and Iwaoka, H., Fabrication and characteristics of gain-coupled distributed feedback semiconductor lasers with a corrugated active layer, IEEE. J. Quantum Electron., QE-27(6), 1724–1732, 1991. 26. Nakano, Y., Luo, Y. and Tada, K., Facet reflection independent, single longitudinal mode oscillation in a GaAlAs/GaAs distributed feedback laser equipped with a gain-coupling mechanism, Appl. Phys. Lett., 55(16), 1606–1608, 1989. 27. David, K., Morthier, G., Vankvikelberge, P. and Baets, R., Yield analysis of non-AR-coated DFB lasers with combined index and gain coupling, Electron. Lett., 26(4), 238–239, 1990. 28. Luo, Y., Nakano, Y. and, Tada, K., Purely gain-coupled distributed feedback semiconductor lasers, Appl. Phys. Lett., 56(17), 1620–1622, 1990. 29. Morthier, G., Vankvikelberge, P., David, K. and Baets, R., Improved performance of AR-coated DFB lasers by the introduction of gain coupling, IEEE Photon. Technol. Lett., 2(3), 170–172, 1990. 30. Kapon, E., Hardy, A. and Katzir, A., The effect of complex coupling coefficients on distributed feedback lasers, IEEE J. Quantum Electron., QE-18(1), 66–71, 1982. 31. Streifer, W., Scifres, D. R. and Burnham, R. D., Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides, IEEE J. Quantum Electron., QE-12(7), 422–428, 1976. 32. Streifer, W., Scifres, D. R. and Burnham, R. D., Coupling coefficient for distributed feedback single- and double-heterostructure diode laser, IEEE J. Quantum Electron., QE-11(11), 867–873, 1975. 33. Correc, P., Coupling coefficient for a trapezoidal grating, IEEE J. Quantum Electron., QE-24(1), 8–10, 1988.

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PRINCIPLES OF DISTRIBUTED FEEDBACK SEMICONDUCTOR LASER DIODES

34. Ghafouri-Shiraz, H. and Lo, B. S. K., Computation of coupling coefficient for a five-layer trapezoidal grating structure, Opt. and Laser Technol., 27(1), 45–48, 1994. 35. Handa, K., Peng, S. T. and Tamir, T., Improved perturbation analysis of dielectric gratings, J. Appl. Phys., 5, 325–328, 1975. 36. Ghafouri-Shiraz, H., Single transverse mode condition in long wavelength SCH semiconductor laser diodes, Trans. IEICE, E 70(2), 130–134, 1987. 37. Adams, M. J. and Wyatt, R., An Introduction to Optical Waveguides. London: Wiley, 1981. 38. Kazarinov, R. F. and Henry, C. H., Second-order distributed feedback lasers with mode selection provided by first-order radiation losses, IEEE J. Quantum Electron., QE-21(2), 144–150, 1985. 39. Streifer, W. and Hardy, A., Analysis of two dimensional waveguides with misaligned or curved gratings, IEEE J. Quantum Electron., QE-14(12), 935–943, 1978. 40. Correc, P., Coupling coefficients for partially meltback trapezoidal gratings, IEEE J. Quantum Electron., QE-24(10), 1963–1965, 1988.

3 Structural Impacts on the Solutions of Coupled Wave Equations: An Overview 3.1

INTRODUCTION

The introduction of semiconductor lasers has boosted the development of coherent optical communication systems. With the built-in wavelength selection mechanism, distributed feedback semiconductor laser diodes with a higher gain margin are superior to the Fabry– Perot laser in that a single longitudinal mode of lasing can be achieved. In this chapter, results obtained from the threshold analysis of conventional and singlephase-shifted DFB lasers will be investigated. In particular, structural impacts on the threshold characteristic will be discussed in a systematic way. The next two sections of this chapter present solutions of the coupled wave equations in DFB laser diode structures. In section 3.4 the concepts of mode discrimination and gain margin are discussed. The threshold analysis of a conventional DFB laser diode is studied in section 3.5, whilst the impact of corrugation phase at the DFB laser diode facets is discussed in section 3.6. By introducing a phase shift along the corrugations of DFB LDs, the degenerate oscillating characteristic of the conventional DFB LD can be removed. In section 3.7, structural impacts due to the phase shift and the corresponding phase shift position (PSP) will be considered. As mentioned earlier in Chapter 2, the introduction of the coupling coefficient  into the coupled wave equations plays a vital role since it measures the strength of feedback provided by the corrugation. In section 3.7, the effect of the selection of corrugation shape on the magnitude of  will be presented. With a =2 phase shift fabricated at the centre of the DFB cavity, the quarterly-wavelength-shifted (QWS) DFB LD oscillates at the Bragg wavelength. However, the deterioration of gain margin limits its use as the current injection increases. This phenomenon induced by the spatial hole burning effect, which is the major drawback of the QWS laser structure, will be examined at the end of this chapter. The limited application of the eigenvalue equation in solving the coupled wave equations will also be considered.

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

80

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

3.2

SOLUTIONS OF THE COUPLED WAVE EQUATIONS

In Chapter 2 it was shown that the characteristics of DFB LDs can be described by a pair of coupled wave equations. The strength of the feedback induced by the perturbed refractive index or gain is measured by the coupling coefficient. Relationships between the forward and the backward coupling coefficients RS and SR were derived for purely index-coupled, mixed-coupled and purely gain-coupled structures. By assuming a zero phase difference between the index and the gain term, the complex coupling coefficient could be expressed as RS ¼ SR ¼ i þ jg ¼ 

ð3:1Þ

where  becomes a complex coupling coefficient. According to eqn (2.98), the trial solution of the coupled wave equation can be expressed in terms of the Bragg propagation constant such that EðzÞ ¼ RðzÞejb0 z þ SðzÞe jb0 z

ð3:2Þ

where the coefficients RðzÞ and SðzÞ are given as [1] RðzÞ ¼ R1 eðgzÞ þ R2 eðgzÞ

ð3:3aÞ

SðzÞ ¼ S1 eðgzÞ þ S2 eðgzÞ

ð3:3bÞ

and

In the above equations, R1 , R2 , S1 and S2 are complex coefficients and g is the complex propagation constant to be determined from the boundary conditions at the laser facets. Without loss of generality, one can assume ReðgÞ > 0. As a result, those terms with coefficients R1 and S2 become amplified as the waves propagate along the cavity. By contrast, those terms with coefficients R2 and S1 are attenuated. By combining the above equations with eqn (3.2), it can be shown easily that the propagation constant of the amplified waves becomes b0  ImðgÞ whilst the decaying waves propagate at b0 þ ImðgÞ. By substituting eqns (3.3a) and (3.3b) into the coupled wave equations, the following relations are obtained by collecting identical exponential terms [2] ^R1 ¼ jej S1

ð3:4aÞ

 R2 ¼ jej S2

ð3:4bÞ

 S1 ¼ je R1 ^S2 ¼ je j R2

ð3:4cÞ ð3:4dÞ

^ ¼ s  jd  g

ð3:5aÞ

 ¼ s  jd þ g

ð3:5bÞ

j

where

By comparing eqns (3.4a) and (3.4c), a non-trivial solution exists if the following equation is satisfied ¼

^ j ¼ j 

ð3:6Þ

81

SOLUTIONS OF THE COUPLED WAVE EQUATIONS

Based on the equation shown above, eqn (3.4) is simplified to become 1 R1 ¼ ej S1 

ð3:7aÞ

R2 ¼ ej S2

ð3:7bÞ

Similarly, by equating eqns (3.4a) and (3.4c), one obtains g2 ¼ ðs  j dÞ2 þ 2

ð3:8Þ

It is important that the dispersion equation shown above is independent of the residue corrugation phase . With a finite laser cavity length L extending from z ¼ z1 to z ¼ z2 (where both z1 and z2 are assumed to be greater than zero), the boundary conditions at the terminating facets become Rðz1 Þ ej b0 z1 ¼ ^r1 Sðz1 Þ e j b0 z1 Sðz2 Þ e

j b0 z 2

¼ ^r2 Rðz2 Þ e

ð3:9aÞ

j b0 z2

ð3:9bÞ

where ^r1 and ^r2 are amplitude reflection coefficients at the laser facets z1 and z2 , respectively. According to eqns (3.3) and (3.4), the above equations could be expanded in such a way that ð1   r1 Þ e2g z1 R1 r1 =  1 ðr2   Þ e2g z2 R2 ¼ R1 1=  r2

R2 ¼

ð3:10aÞ ð3:10bÞ

In the above equation, all RðzÞ and SðzÞ terms are expressed in terms of R1 and R2 , whilst r1 and r2 are the complex field reflectivities of the left and the right facets, respectively such that r1 ¼ ^r1 e2 j b0 z1 e j ¼ ^r1 e j  1 r2 ¼ ^r2 e

2 j b0 z2

j

e

¼ ^r2 e

j 2

ð3:11aÞ ð3:11bÞ

with 1 and 2 being the corresponding corrugation phases at the facets. Equations (3.10a) and (3.10b) are homogeneous in R1 and R2 . In order to have non-trivial solutions, the following condition must be satisfied ð1   r1 Þ e2g z1 ðr2  Þ e2gz2 ¼ r1   1   r2

ð3:12Þ

Then the above equation can be solved for  and 1= whilst employing the relation   j 1  g¼ 2 

ð3:13Þ

82

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

derived from eqns (3.5a) and (3.5b). After some lengthy manipulation [2], one ends up with an eigenvalue equation gL ¼

o j sinhð LÞ n 1 ðr1 þ r2 Þð1  r1 r2 Þ coshðgLÞ ð1 þ r1 r2 Þ
2 D

ð3:14Þ


¼ ðr1  r2 Þ2 sinh2 ðgLÞ þ ð1  r1 r2 Þ2

ð3:15aÞ

where

2

D ¼ ð1 þ r1 r2 Þ  4 r1 r2 cosh ðg LÞ

ð3:15bÞ

r1 ¼ ^r1 e2jb0 z1 e j ¼ ^r1 e j

ð3:15cÞ

r2 ¼ ^r2 e

2jb0 z2 j

e

2

1

¼ ^r2 e

j

2

ð3:15dÞ

By squaring eqn (3.1), and after some simplification, one ends up with a transcendental function ðgLÞ2 þ ðLÞ2 sinh2 ðgLÞ ð1  r12 Þð1  r22 Þ þ 2j L ðr1 þ r2 Þ2 ð1  r1 r2 Þ gL sinhðgLÞ coshðgLÞ ¼ 0

ð3:16Þ

In the above equation, there are four parameters which govern the threshold characteristics of DFB laser structures. These are the coupling coefficient , the laser cavity length L and the complex facet reflectivities r1 and r2 . Due to the complex nature of the above equation, numerical methods like the Newton–Raphson iteration technique can be used, provided that the Cauchy–Riemann condition on complex analytical functions is satisfied. Before starting the Newton–Raphson iteration, an initial value of ð; Þini is chosen from a selected range of ð; Þ values. Usually, the first selected guess will not be a solution of the threshold equation and hence the iteration continues. At the end of the first iteration, a new pair of ð0 ; 0 Þ will be generated and checked to see if it satisfies the threshold equation. The iteration will continue until the newly generated ð0 ; 0 Þ pair satisfies the threshold equation within a reasonable range of error. Starting with different initial guesses of ð; Þini , other oscillating modes can be determined in a similar way. By collecting all ð0 ; 0 Þ pairs that satisfy the threshold equation, the one showing the smallest amplitude gain will then become the lasing mode. The final value ð; Þfinal is then stored up for later use, in which the threshold current and the lasing wavelength of the LD are to be decided. In general, eqn (3.16) characterises all conventional DFB semiconductor LDs with continuous corrugations fabricated along the laser cavity.

3.3

SOLUTIONS OF COMPLEX TRANSCENDENTAL EQUATIONS USING THE NEWTON–RAPHSON APPROXIMATION

All complex transcendental equations can be expressed in a general form such that WðzÞ ¼ UðzÞ þ j VðzÞ ¼ 0

ð3:17Þ

SOLUTIONS OF COMPLEX TRANSCENDENTAL EQUATIONS

83

where z ¼ x þ j y is a complex number and UðzÞ and VðzÞ are, respectively, the real and imaginary parts of the complex function. From the above equation, one can deduce the following equality easily UðzÞ ¼ VðzÞ ¼ 0

ð3:18Þ

By taking the first-order derivative of eqn (3.17) with respect to z, one can obtain @W @U @V @U @V ¼ þj ¼ þj @z @z @z @x @x

ð3:19Þ

The second equality sign can be obtained using the chain rule. Applying the Taylor series, the functions U(z) and V(z) can be approximated about the exact solution ðxreq , yreq Þ such that @U @U ðxreq  xÞ þ ðyreq  yÞ @x @y @V @V ðxreq  xÞ þ ðyreq  yÞ Vðxreq ; yreq Þ ¼ V ðx; yÞ þ @x @y

Uðxreq ; yreq Þ ¼ U ðx; yÞ þ

ð3:20Þ ð3:21Þ

where the values of (x, y) chosen are sufficiently close to the exact solutions. Other higher derivative terms from the Taylor series have been ignored. One then obtains the following equations for xreq and yreq from the above simultaneous equations [2] @U @V  Uðx; yÞ @y @y ¼xþ Det @V @U  Vðx; yÞ Uðx; yÞ @x @x ¼yþ Det Vðx; yÞ

xreq

yreq

ð3:22Þ

ð3:23Þ

where  2  2 @U @V Det ¼ þ @x @y

ð3:24Þ

Terms like @U=@x, @V=@x, @U=@y and @V=@y are the first derivatives of functions U(z) and V(z). For an analytical complex function W(z), the Cauchy–Riemann condition which states that @U @V ¼ ; @x @y must be satisfied [3].

@U @V ¼ @y @x

ð3:25Þ

84

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

On replacing all the @=@y terms with @=@x using the above Cauchy–Riemann condition, eqns (3.22) and (3.24) can be simplified such that 

@U Det ¼ 2 @x xreq ¼ x 

2

Vðx; yÞ

ð3:26Þ @V @U þ Uðx; yÞ @x @x Det

ð3:27Þ

Here, only the first-order derivative terms @U=@x and @V=@x are used. These can be determined from the complex function of eqn (3.19). Given an initial guess of ðx; yÞ, the numerical iteration process then starts. A new guess is generated by following eqns (3.23), (3.26) and (3.27). Unless the new guess is sufficiently close to the exact solution (within 109 , let’s say), the new guess solution formed will become the initial guess of the next iteration. The iteration process continues until approximate solutions of ðxreq , yreq Þ appear. The advantages of this method are its speed and flexibility. In addition, the derivative term @W=@z is found analytically first, before any numerical iteration is started. Using this method, one can avoid any errors associated with other numerical methods such as numerical differentiation.

3.4

CONCEPTS OF MODE DISCRIMINATION AND GAIN MARGIN

At a fixed value of , pairs of ð; Þfinal can be determined following the method discussed in the previous section. Each ð; Þfinal pair, which represents an oscillation mode, is plotted on the – plane. Similarly, pairs of ð; Þfinal values can be obtained by changing the values of . By plotting all ð; Þfinal points on the – plane, the mode spectrum of the DFB LD is formed. A simplified – plot is shown in Fig. 3.1. Different symbols shown represent various longitudinal modes obtained for various coupling coefficients whilst the solid curve shows how longitudinal modes join to form an oscillating mode. When the biasing current increases, the longitudinal mode showing the smallest amplitude gain will reach the threshold condition first and begin to lase. Other modes failing to reach the threshold condition will then be suppressed and become non-lasing side modes. The – plane is split into two halves by the  ¼ 0 line, or the Bragg wavelength. As one moves along the positive -axis, any oscillation modes encountered will be denoted as the þ1, þ2 modes and so on. Similarly, negative values such as 1, 2 are used for the modes found on the negative -axis. The importance of the single longitudinal mode (SLM) in coherent optical communications has been discussed earlier in Chapter 1. To measure the stability of the lasing spectrum, one needs to determine the amplitude gain difference between the lasing mode and the most probable side mode of the DFB laser ½4; 5. A larger amplitude gain difference, better known as the gain margin ð
Þ, implies a better mode discrimination. In other words, the SLM oscillation in the DFB LD involved is said to be more stable. In practice, the actual requirement of
 may vary from one system to another depending on the encoding format

THRESHOLD ANALYSIS OF A CONVENTIONAL DFB LASER

85

Figure 3.1 A simplified – plot showing the mode spectrum and the oscillating mode of a DFB LD. Different symbols are used to show longitudinal modes obtained from various  values.

(return to zero, RZ, or non-return to zero, NRZ), transmission rate, the biasing condition of the laser sources, the length and characteristics of the single-mode fibre (SMF) used. A simulation based on a 20 km dispersive SMF [6] indicated that a
 of 5 cm1 is required for a 2.4 Gb s1 data transmission in order that a bit error rate, BER < 109 can be achieved. A detailed analysis of the requirement of
 under different system configurations is clearly beyond the scope of the present analysis. On the other hand, from the above data one can get some idea of the typical values of gain margin required in a coherent optical communication system. The value of the gain margin, however, is difficult to measure directly from an experiment. An alternative method is to measure the spontaneous emission spectrum. For a stable SLM source, a minimum side mode suppression ratio (SMSR) of 25 dB [7] between the power of the lasing mode and the most probable side mode is necessary.

3.5

THRESHOLD ANALYSIS OF A CONVENTIONAL DFB LASER

For a conventional DFB laser having zero facet reflection, the threshold equation (3.16) becomes j g L ¼ L sinhð LÞ

ð3:28Þ

Using the Newton–Raphson iteration approach, the eigenvalue equation can be solved as a fixed coupling coefficient. Results obtained for the above equation are shown in Fig. 3.2. All

86

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

Figure 3.2 Relationship between the amplitude threshold gain and the detuning coefficient of a mirrorless index-coupled DFB LD.

parameters used have been normalised with respect to the overall cavity length L. Discrete values of L have been selected between 0.25 and 10.0. As shown in the inset of Fig. 3.2, solutions obtained from various L products are shown using different symbols. Oscillation modes are then formed by joining the appropriate solutions together. Solid lines have been used to represent the 4 to þ4 modes. From the figure, it is clear that oscillating modes distribute symmetrically with respect to the Bragg wavelength, whilst no oscillation is found at the Bragg wavelength. Furthermore, it can be seen that the þ1 and 1 modes although having different lasing wavelengths share the same amplitude gain. As a result, degenerate oscillation occurs and these modes will have the same chance to lase once the lasing condition is reached. Figure 3.2 also reveals that the amplitude of the threshold gain decreases with increasing values of L. Since a larger value of  implies a stronger optical feedback, a smaller threshold gain results. Similarly, lasers having a long cavity length help to reduce the amplitude gain since a larger single pass gain can be achieved. With no oscillation found at the Bragg wavelength, a stop band region is formed between the þ1 and 1 modes of the conventional mirrorless DFB LD. From Fig. 3.2, one can conclude that the normalised stop band width is a function of L. Although the change in stop band width becomes less noticeable at lower values of L, the measurement of the stop band width has been used to determine the coupling coefficient of DFB LDs [8]. Figure 3.3 shows the characteristic of a DFB LD having finite facet reflections. It is shown in the figure that the mode distribution is no longer symmetrical and no oscillation is found at the Bragg wavelength. The 1 mode, having the smallest amplitude gain, becomes the lasing mode.

IMPACT OF CORRUGATION PHASE AT LASER FACETS

87

Figure 3.3 Relationship between the amplitude threshold gain and the detuning coefficient of a DFB LD with finite reflectivities.

3.6

IMPACT OF CORRUGATION PHASE AT LASER FACETS

So far, symmetrical laser cavities sharing identical facet reflectivities have been used. In order to understand the effects of the residue phases at facets ½2; 9, asymmetric cavities are now considered. The threshold characteristic of one of these asymmetric DFB LDs is shown in Fig. 3.4. The amplitude reflectivity ^r1 ¼ 0:0343 is assumed whilst the other facet is assumed to be naturally cleaved such that ^r2 ¼ 0:535. Discrete values of L have been chosen. In the figure, the corrugation phase 1 is fixed at  whilst 2 changes in steps of =2. Different symbol markers have been used to represent the different 2 . Solutions obtained from the same L product are joined together as usual to form the oscillation mode. Consider L ¼ 1:0 as an example. It can be seen that the lasing mode changes from the negative to the positive mode as the facet phase 2 changes from =2 to . For L > 1:0, the amplitude gain at the Bragg wavelength remains so high that it never reaches the threshold condition. The 1 mode showing the smallest amplitude gain becomes the dominant lasing mode. If we replace the natural cleaved facet with a highly reflective surface such that ^r2 ¼ 1:0, we change the lasing characteristic to that shown in Fig. 3.5. Various values of L have been used for comparison purposes. In a similar way to Fig. 3.4, the oscillation mode shifts from the 1 to the þ1 mode when 2 changes from =2 to . From both Figs. 3.4 and 3.5, it is clear that SLM operation depends on both the facet reflectivity and the associated phase. On the other hand, due to tolerances inherent during the process of fabrication, it is difficult to control the corrugation phase at the laser facets accurately [10].

88

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

Figure 3.4 The lasing characteristic of a DFB LD having asymmetric facet reflectivities. The corrugation phase 1 is fixed whilst 2 is allowed to change. Results obtained from various L products are compared.

Figure 3.5 The lasing characteristic of a DFB LD having asymmetric facet reflectivities. The corrugation phase 1 is fixed whilst 2 is allowed to change. Results obtained from various L products are compared.

THE EFFECTS OF PHASE DISCONTINUITY ALONG THE DFB LASER CAVITY

89

Various methods have been proposed for adjusting the corrugation phase. One such method is to use the ion beam etching technique ½11; 12. A continuous flux of neutralised argon gas, which acts as an abrasive tool, is targeted at one laser facet. By passing the facet slowly across the beam at a constant rate, a 20–50 nm depth can be etched away at the laser facet in a single process. An annealing process is usually applied afterwards. Experimental results ½11; 12 show that the annealing process does not cause significant variation in the threshold and the external quantum efficiency in DFB lasers. Apart from the extra annealing process required, the ion beam etching technique is effective in adjusting the position of facets and thus the associated corrugation phases. Since the etching depth required may vary from one DFB laser to another, the ion beam etching technique is classified as a chip-by-chip optimisation method. To improve the efficiency, other methods such as the phase control technique [13] can be used. Basically, a multi-layer coating with precise refractive indices and thicknesses is applied to the laser facets so that the overall facet phase and the amplitude reflection can be controlled and determined easily.

3.7

THE EFFECTS OF PHASE DISCONTINUITY ALONG THE DFB LASER CAVITY

In the previous section, the threshold analysis of conventional DFB lasers comprising uniform corrugations was presented. SLM operation can be achieved when different values of facet reflectivity are employed. On the other hand, due to the randomness of the corrugation phase at the laser facet, stable SLM oscillation is not guaranteed. To improve the single-mode performance of DFB lasers, phase discontinuity or phase shift is introduced [14] along the corrugation. As shown in Fig. 3.6, phase shifts along the corrugation can be introduced by two methods. Figure 3.6(a) shows a non-uniform active layer width, whilst the shape and the dimension of the corrugation remain constant ½15; 16. In Fig. 3.6(b), on the other hand, the corrugation shows a phase slip whilst the active layer dimensions remain uniform ½17; 18. Using method (a), the actual phase shift depends on the length of the

Figure 3.6 Phase shift or discontinuity fabricated along the corrugation of a DFB laser. (a) Phase shift formed by uniform corrugation but non-uniform active layer width; (b) phase shift formed by uniform active layer dimension but discontinuous corrugation.

90

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

Figure 3.7 Schematic representation of a single-phase-shifted (1PS) DFB LD. The phase shift is represented by .

phase-adjustment region and the difference in strip width ðW2  W1 Þ. Precise control over the active layer width is required. Using method (b), phase discontinuity is introduced during fabrication in which the slip is written directly along the corrugation. In the analysis, we adopted the latter method in preparing phase shifts in a DFB laser. Consider a single-phase-shifted (1PS) DFB laser as shown in Fig. 3.7. A phase slip of 2 is fabricated along the corrugation at the z origin so that the cavity is subdivided into 2 sections. As one can see, these sections may have different lengths and each resembles a conventional DFB laser cavity with uniform corrugation. In the analysis, zero facet reflectivity is assumed. Following the argument presented earlier in Chapter 2, the refractive index of each section can be written as nð1Þ ðzÞ ¼ n0 þ
n cosð2b0 z þ Þ ð2Þ

n ðzÞ ¼ n0 þ
n cosð2b0 z  Þ

ð3:29aÞ ð3:29bÞ

where superscripts (1) and (2) correspond to sections 1 and 2, respectively. In the above equations, it is assumed that the phase shift is equally split between sections 1 and 2. Based upon the coupled wave theory, counter-running waves are built up in each section such that the following equations can be derived for each section of the laser dRð1Þ þ ð  jÞRð1Þ dz dSð1Þ þ ð  jÞSð1Þ dz dRð2Þ þ ð  jÞRð2Þ  dz dSð2Þ þ ð  jÞSð2Þ dz



¼ jSð1Þ ej

ð3:30aÞ

¼ jRð1Þ e j

ð3:30bÞ

¼ jSð2Þ e j

ð3:30cÞ

¼ jRð2Þ ej

ð3:30dÞ

THE EFFECTS OF PHASE DISCONTINUITY ALONG THE DFB LASER CAVITY

91

where Rð1Þ , Sð1Þ and Rð2Þ , Sð2Þ are the counter-running waves propagating in sections (1) and (2), respectively. In both sections, the corrugation shape and grating depth are assumed to be equal. As a result, the coupling coefficient remains constant throughout. From eqn (2.98), the solution of the coupled wave equations can be written as EðkÞ ðzÞ ¼ RðkÞ ðzÞejb0 z þ SðkÞ ðzÞe jb0 z

ð3:31Þ

where ðkÞ

ðkÞ

RðkÞ ðzÞ ¼ R1 eg z þ R2 e g z ðkÞ

S ðzÞ ¼

ðkÞ S1 e g z

þ

ð3:32aÞ

ðkÞ S2 e  g z

ð3:32bÞ ðkÞ

ðkÞ

ðkÞ

ðkÞ

and k ¼ 1 and 2 for sections (1) and (2), respectively. Here, R1 , R2 , S1 and S2 are the complex coefficients associated with the particular section. Since the discontinuity caused by the phase slip is assumed to be very small, the waves in the two sections can be considered to be continuous at z ¼ 0. In other words, Rð1Þ ðz ¼ 0Þ ¼ Rð2Þ ðz ¼ 0Þ ð1Þ

ð2Þ

S ðz ¼ 0Þ ¼ S ðz ¼ 0Þ

ð3:33aÞ ð3:33bÞ

By allowing ^r1 and ^r2 to be the respective amplitude facet reflection coefficients at the left and right laser facets, the boundary conditions of the 1PS DFB laser become Rð1Þ ðL1 Þe jb0 L1 ¼ ^r1 Sð1Þ ðL1 Þ ejb0 L1 ð2Þ

S ðL2 Þe

jb0 L2

ð2Þ

¼ ^r2 R ðL2 Þ e

jb0 L2

ð3:33cÞ ð3:33dÞ

By matching all the boundary conditions, non-trivial solutions exist if and only if the following eigenvalue equation is satisfied [17] j ð1Þ^r1 þ ð1Þ j ð2Þ^r2 þ ð2Þ ð2Þ ; ¼ e j2 ð1Þ ð1Þ ð2Þ j ^r1 þ T j ^r2 þ T

ð3:34Þ

where  ðkÞ ¼ ^2 þ 2 e2gLðkÞ  ðkÞ ¼ ^ 1  e2gLðkÞ

ð3:35Þ

T ðkÞ ¼ 2 þ ^2 e2gLðkÞ ^ ¼   j  with k ¼ 1 and 2, and Lð1Þ ¼ L1 ;

Lð2Þ ¼ L2

ð3:36Þ

92

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

^r1 and ^r2 in eqn (3.34) can be expressed in terms of the following complex field reflectivities, r1 , and r2 , respectively r1 ¼ ^r1 ejð2b0 L1  Þ r2 ¼ ^r2 ejð2b0 L2  Þ

ð3:37Þ

Compared with the conventional DFB laser, the boundary condition at the phase shift has to be matched for the mirrorless 1PS DFB laser structure. Nevertheless, it was pointed out by Utaka et al. [17] that the use of non-zero facet reflection may not be desirable. This is because the random corrugation phases at the laser facets will cause extra difficulty in controlling the lasing characteristic. Therefore, it is best to have AR coatings applied to both facets of the 1PS DFB laser. For a mirrorless, symmetrical 1PS DFB laser cavity with L1 ¼ L2 ¼ L=2, the phase shift is located at the centre of the cavity. As a result, eqn (3.34) can be simplified further [19] such that "

#2 ^ð1  egL Þ  ¼ e2j  2 þ ^egL

ð3:38Þ

3.7.1 Effects of Phase Shift on the Lasing Characteristics of a 1PS DFB Laser Diode To investigate the effects of phase shifts on the lasing characteristic of 1PS DFB lasers, a symmetrical laser cavity is assumed with a single phase shift fabricated at the centre of the DFB laser. Using a numerical method such as the Newton–Raphson method, the eigenvalue equation (3.38) can be solved numerically for the normalised amplitude threshold gain thL (amplitude gain of the lasing mode) and the lasing wavelength for specific values of  and phase shift. Figure 3.8 illustrates how the variation of phase shift value affects thL for the mirrorless 1PS DFB LD. All parameters used are normalised with respect to the overall cavity length L. Three different L values are plotted in the figure for comparison purposes. All curves in Fig. 3.8 are symmetrical and have a minimum amplitude threshold gain at  ¼ 90 (or =2 in radians) as can be seen. This phase change corresponds to a quarter wavelength shift and so the name single
=4-shifted DFB, or quarterly-wavelength-shifted (QWS) DFB, laser is usually used to represent this laser structure. When the phase shift approaches zero or , the phase-shifted structure is reduced to the conventional, mirrorless DFB laser in which degenerate oscillation results. Figure 3.9 shows the variation of the lasing wavelength with respect to the phase shift. As in Fig. 3.8, results of three sets of L products are shown and compared. In this case, the Bragg wavelength
B is assumed to be 1330 nm and the actual wavelength is shown on the left y-axis. The corresponding normalised detuning coefficient thL is shown on the righthand side. At  ¼ =2, the lasing wavelengths of all three L values coincide at the Bragg wavelength. This reveals an important characteristic of the symmetrical 1PS DFB laser. A QWS DFB LD always oscillates at the Bragg wavelength irrespective of the L chosen.

THE EFFECTS OF PHASE DISCONTINUITY ALONG THE DFB LASER CAVITY

Figure 3.8 DFB LD.

93

The variation of the normalised amplitude threshold gain with the phase shift of a 1PS

Figure 3.9

The variation of the lasing wavelength with the phase shift of a 1PS DFB LD.

94

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

When L increases, the range of lasing wavelengths also increases with varying phase shift. At L ¼ 2:0, the range of wavelengths is found to be 10.8 nm whilst it is about 7.4 nm for L ¼ 0:50. So far, the phase shift has been assumed to be at the centre of the laser cavity. In the next section, effects of the phase shift position on the lasing characteristics of 1PS DFB lasers will be discussed.

3.7.2 Effects of Phase Shift Position (PSP) on the Lasing Characteristics of a 1PS DFB Laser Diode To investigate the effect of the location of the phase shift [20], a parameter known as the phase shift position (PSP) is introduced along the asymmetric laser cavity such that PSP ¼

L1 L

ð3:39Þ

The variation of the amplitude gain and gain margin obtained from a 500 mm long DFB laser cavity with  ¼ 20 cm1 (i.e. L ¼ 1) are shown in Figs. 3.10 and 3.11, respectively. In the analysis, the Bragg wavelength is assumed to be at 1330 nm and the phase shift is fixed at =2. Both Figs. 3.10 and 3.11 show symmetrical distributions of curves at PSP ¼ 0:5, where the phase shift is found. When the phase shift moves from the centre towards the laser facets, the effect of the phase shift becomes less influential. Solutions obtained from the threshold equation indicate that degenerate oscillation begins to occur at PSP ¼ 0:26 and PSP ¼ 0:74. In this situation, the QWS DFB laser is reduced to the conventional one. The dramatic fall of

Figure 3.10 The variation of the amplitude threshold gain with the PSP of a 1PS DFB LD. The phase shift is fixed at /2.

ADVANTAGES AND DISADVANTAGES OF QWS DFB LASER DIODES

95

Figure 3.11 The variation of the gain margin with the PSP of a 1PS DFB LD. The phase shift is fixed at /2.

the gain margin shown in Fig. 3.11 confirms the above argument. When the position of the phase shift moves from the centre ðPSP ¼ 0:5Þ to the laser facets, the gain margin drops from a peak value of 14.7 cm1 to zero value at PSP ¼ 0:26 and PSP ¼ 0:74. As long as the phase shift is fabricated near the centre of the cavity, a QWS DFB laser can operate at the Bragg wavelength.

3.8

ADVANTAGES AND DISADVANTAGES OF QWS DFB LASER DIODES

By introducing a QWS at the centre of the DFB laser cavity, SLM operation at the Bragg wavelength can be achieved. However, as first discussed by Soda et al. [16], for a high L QWS DFB LD, the gain margin drops drastically with increasing biasing current. Multimode oscillation at two distant wavelengths is observed when the optical output power increases. Such a reduction in gain margin is thought to be induced by the longitudinal spatial hole burning effect [21]. When DFB LDs are biased below threshold current, where spontaneous emission is still dominant, the longitudinal carrier and the field intensity distributions are relatively uniform. However, when the bias current exceeds that of the threshold value, the optical field inside the laser cavity becomes intensified at places where corrugation reflections occur [22]. For a QWS DFB LD, the field intensity is so intense at the phase shift position that the rate of spontaneous recombination increases near the phase shift. In order to maintain a round-trip gain of unity, carriers located near to the phase shift will move to fill the carrier-depleted zone. In Chapter 2, it was mentioned that the

96

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

refractive index of the semiconductor depends on the carrier injection. Such a local variation of carrier concentration will result in a non-uniform distribution of refractive indices along the laser cavity. The situation is made worse by the fact that the gain and refractive index are related to one another as a result of the Kramer–Kroenig relationship [23]. When the biasing current changes, the gain of the lasing mode and other non-lasing side modes will change in such a way that the gain margin reduces and consequently, multi-mode oscillation occurs. Due to the deterioration of single-mode stability, the longitudinal spatial hole burning effect limits the QWS DFB LD to a lower power of operation. Although laser structures having a smaller L value are found to be less vulnerable to the longitudinal spatial hole burning effect, these structures are characterised by larger amplitude gain values and relatively large currents. In order to suppress the spatial hole burning whilst improving the maximum single-mode output power available, it was proposed that a laser structure having a flatter field intensity may be used [24]. To optimise the structure with respect to the intensity distribution, a parameter known as the flatness (F) is defined F¼

1 L

Z ðIðzÞ  Iavg Þ2 dz

ð3:40Þ

cavity

where IðzÞ is the local field intensity and Iavg is the average field intensity. In the QWS DFB laser, an optimum value of flatness is found when L ¼ 1:25 [16]. In flattening the field intensity whilst improving the optimum L value that can be used in QWS DFB lasers, a three-electrode QWS DFB laser structure shown in Fig. 3.12 was proposed [25]. By passing a larger biasing current to the central electrode, carriers lost due to spatial hole burning are compensated for [26]. An alternative approach which retains uniform current injection is also used. By introducing more phase shifts along the DFB laser cavity, a multiple-phase-shift (MPS) structure can flatten the field distribution [27]. Figure 3.13 shows a three-phase-shift (3PS) DFB LD.

Figure 3.12 A three-electrode QWS DFB LD (after [25]).

SUMMARY

97

Figure 3.13 Schematic representation of a three-phase-shift (3PS) DFB LD (after [28]). AR: antireflection coating.

Longitudinal spatial hole burning must be considered when the LD operates at the abovethreshold condition. To decide the above-threshold characteristic, one must take into account the local carrier concentration. Using the perturbation method [27] or the quasi-uniform gain assumption [16], characteristics of QWS DFB LDs operating slightly above the threshold current are predicted. However, these methods may not be appropriate when the biasing current becomes high and other non-linear effects such as the gain saturation [28] must be considered. Throughout the analysis, the deviation of the eigenvalue equation becomes tedious as the laser structure becomes more and more complex. The use of numerical analysis, such as the Newton–Raphson method, becomes impractical since the first-order derivative is required. A new model that can cope with different designs of DFB LDs, such as the three-electrode QWS [29] and/or the 3PS DFB LD structures [30], while maintaining a wider range of current injection is necessary. Such a model needs to be capable of considering any local variation and the gain saturation effect with increasing output power.

3.9

SUMMARY

In this chapter, the coupled wave equations have been solved under various structural configurations. By matching all boundary conditions, eigenvalue equations were derived. From the solutions of the eigenvalue equations, the threshold current and the lasing wavelength were determined. Impacts due to the coupling coefficient, the laser cavity length, the facet reflectivities, the residue corrugation phases and phase discontinuities were discussed in a systematic way with regard to the lasing threshold characteristics. With a single QWS fabricated at the centre of the DFB cavity, the QWS DFB LD oscillates at the Bragg wavelength. Due to non-uniform field distribution, however, the single-mode stability

98

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS

is threatened by the spatial hole burning effect. To extend the analysis to the above-threshold operation, a new model is required such that all localised effects and other non-linear effects can be included in the analysis.

3.10

REFERENCES

1. Kogelnik, H. and Shank, C. V., Coupled-wave theory of distributed feedback lasers, J. Appl. Phys., 43(5), 2327–2335, 1972. 2. Streifer, W., Burnham, R. D. and Scifres, D. R., Effect of external reflectors on longitudinal modes of distributed feedback lasers, IEEE J. Quantum Electron., QE-11(4), 154 –161, 1975. 3. Arfken, G., Mathematical Methods for Physicists, 3rd edition. New York: Academic Press, 1985. 4. Ketelsen, L. J. P., Hoshino, I. and Ackerman, D. A., The role of axially nonuniform carrier density in altering the TE-TE gain margin in InGaAsP-InP DFB lasers, IEEE J. Quantum Electron., QE-27(4), 957–964, 1991. 5. Itaya, Y., Matsuoka T., Kuroiwa, K. and Ikegami, T., Longitudinal mode behaviours of 1.5 mm range InGaAsP/InP distributed feedback lasers, IEEE J. Quantum Electron., QE-20(3), 230–235, 1984. 6. Cartledge, J. C. and Elrefaie, A. F., Threshold gain difference requirements for nearly singlelongitudinal-mode lasers, J. Lightwave Technol., LT-8(5), 704–715, 1990. 7. Agrawal, G. P. and Dutta, N. K., Long-Wavelength Semiconductor Lasers. Princeton, NJ: Van Nostrand, 1986. 8. Kinoshita, J., Validity of L evaluation by stopband method for
=4 DFB lasers with low reflecting facets, IEEE J. Quantum Electron., QE-23(5), 499–501, 1987. 9. Chinn, S. R., Effects of mirror reflectivity in a distributed-feedback laser, IEEE J. Quantum Electron., QE-9(6), 574–580, 1973. 10. Buus, J., Mode selectivity in DFB lasers with cleaved facets, Electron. Lett., 21, 179–180, 1985. 11. Itaya, Y., Wakita, K., Motosugi, G. and Ikegami, T., Phase control by coating in 1.5mm distributed feedback lasers, IEEE J. Quantum Electron., QE-21(6), 527–532, 1985. 12. Matsuoka, T., Yoshikuni, Y. and Nagai, H., Verification of the light phase effect at the facet on DFB laser properties, IEEE J. Quantum Electron., QE-21(12), 1880–1886, 1985. 13. Mols, P. P. G., Kuindersma, P. I., Es-spiekman, W. V. and Baele, I. A. G., Yield and device characteristics of DFB lasers: Statistics and novel coating design in theory and experiment, IEEE J. Quantum Electron., QE-25(6), 1303–1312, 1989. 14. Hau, H. and Shank, C., Asymmetric tapers of distributed feedback lasers, IEEE J. Quantum Electron., QE-12, 532–539, 1976. 15. Nakano, Y., Luo Y. and Tada, K., Facet reflection independent, single longitudinal mode oscillation in a GaAlAs/GaAs distributed feedback laser equipped with a gain-coupling mechanism, Appl. Phys. Lett., 55(16), 1606–1608, 1989. 16. Soda, H., Kotaki, Y., Sudo, H., Ishikawa, H., Yamakoshi, S. and Imai, H., Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers, IEEE J. Quantum Electron., QE-23(6), 804–814, 1987. 17. Utaka, K., Akiba, S., Sakai, K. and Matsushima, Y.,
=4-shifted InGaAsP DFB lasers, IEEE J. Quantum Electron., QE-22(3), 1042–1051, 1986. 18. McCall, S. L. and Platzman, P. M., An optimized =2 distributed feedback laser, IEEE J. Quantum Electron., QE-21(12), 1899–1904, 1985. 19. Ghafouri-Shiraz, H. and Chu, C., Effect of phase shift position on spectral linewidth of the =2 distributed feedback laser diode, J. Lightwave Technol., LT-8(7), 1033–1037, 1990. 20. Usami, M., Akiba, S. and Utaka, K., Asymmetric
=4-shifted InGaAsP/InP DFB lasers, IEEE J. Quantum Electron., QE-23(6), 815–821, 1987.

REFERENCES

99

21. Rabinovich, W. S. and Feldman, B. J., Spatial hole burning effects in distributed feedback lasers, IEEE J. Quantum Electron., QE-25(1), 20 –30, 1989. 22. Kinoshita, K. and Matsumoto, K., Transient chirping in distributed feedback lasers: effect of spatial hole-burning along the laser axis, IEEE J. Quantum Electron., QE-24(11), 2160–2169, 1988. 23. Yariv, A., Quantum Electronics, 3rd edition. New York: Wiley, 1989. 24. Kimura, T. and Sugimura, A., Coupled phase-shift distributed-feedback lasers for narrow linewidth operation, IEEE J. Quantum Electron., QE-25(4), 678– 683, 1989. 25. Usami, M. and Akiba, S., Suppression of longitudinal spatial hole-burning effect in
=4-shifted DFB lasers by nonuniform current distribution, IEEE J. Quantum Electron., QE-25(6), 1245–1253, 1989. 26. Kikuchi, K. and Tomofuji, H., Performance analysis of separated-electrode DFB laser diodes, Electron. Lett., 25(2), 162–163, 1989. 27. Kimura, T. and Sugimura, A., Narrow linewidth asymmetric coupled phase-shift DFB lasers, Trans. IEICE., E 79(1), 71–76, 1990. 28. Huang, J. and Casperson, L. W., Gain and saturation in semiconductor lasers, Optical Quantum Electron., QE-27, 369–390, 1993. 29. Kotaki, Y. and Ishikawa, H., Wavelength tunable DFB and DBR lasers for coherent optical fibre communications, IEE Proc. Pt. J, 138(2), 171–177, 1991. 30. Ogita, S., Kotaki, Y., Hatsuda, M., Kuwahara, Y. and Ishikawa, H., Long cavity multiple-phase shift distributed feedback laser diode for linewidth narrowing, J. Lightwave Technol., LT-8(10), 1596–1603, 1990.

4 Transfer Matrix Modelling in DFB Semiconductor Lasers 4.1

INTRODUCTION

In Chapter 3, eigenvalue equations were derived by matching boundary conditions inside DFB laser cavities. From the eigenvalue problem, the lasing threshold characteristic of DFB lasers is determined. The single /2-phase-shifted (PS) DFB laser is fabricated with a phase discontinuity of /2 at or near the centre of the laser cavity. It is characterised by Bragg oscillation and a high gain margin value. On the other hand, the SLM deteriorates quickly when the optical power of the laser diode increases. This phenomenon, known as spatial hole burning, limits the maximum single-mode optical power and consequently the spectral linewidth. Using a multiple-phase-shift (MPS) DFB laser structure, the electric field distribution is flattened and hence the spatial hole burning is suppressed. In dealing with such a complicated DFB laser structure, it is tedious to match all the boundary conditions. A more flexible method which is capable of handling different types of DFB laser structures is necessary. In section 4.2, the transfer matrix method (TMM) [1– 4] will be introduced and explored comprehensively. From the coupled wave equations, it is found that the field propagation inside a corrugated waveguide (e.g. the DFB laser cavity) can be represented by a transfer matrix. Provided that the electric fields at the input plane are known, the matrix acts as a transfer function so that electric fields at the output plane can be determined. Similarly, other structures like the active planar Fabry–Perot (FP) section, the passive corrugated distributed Bragg reflector (DBR) section and the passive planar waveguide (WG) section can also be expressed using the idea of a transfer matrix. By joining these transfer matrices as a building block, a general N-sectioned laser cavity model will be presented. Since the outputs from a transfer matrix automatically become the inputs of the following matrix, all boundary conditions inside the composite cavity are matched. The unsolved boundary conditions are those at the left and right facets. In section 4.3, the threshold equation of the N-sectioned laser cavity model will be determined and the use of TMM in other semiconductor laser devices will be discussed. An adequate treatment of the amplified spontaneous emission spectrum ðPN Þ is very important in the analysis of semiconductor lasers [5], optical amplifiers [6 –8] and optical filters [9–10]. In semiconductor lasers, PN is important for both the estimation of linewidth [11] and the estimation of single-mode stability in DFB laser diodes [12]. In optical

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

102

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

amplifiers and filters, PN has also been used to simulate the bandwidth, tunability and the signal gain characteristic. In section 4.4, the TMM formulation will be extended so as to include the below-threshold spontaneous emission spectrum of the N-sectioned DFB laser structure. Numerical results based on 3PS DFB LDs will be presented.

4.2

BRIEF REVIEW OF MATRIX METHODS

By matching boundary conditions at the facets and the phase-shift position, the threshold condition of the single-phase-shifted DFB LD can be determined from the eigenvalue equation. However, this approach lacks the flexibility required in the structural design of DFB LDs. Whenever a new structural design is involved, a new eigenvalue equation has to be derived by matching all boundary conditions. For a laser with the MPS DFB structure, the formation of eigenvalue equation becomes tedious since it may involve a large number of boundary conditions. One possible approach to simplifying the analysis, whilst improving flexibility and robustness, is to employ matrix methods. Matrices have been used extensively in engineering problems which are highly numerical in nature. In microwave engineering [13], matrices are used to find the electric and magnetic fields inside various microwave waveguides and devices. One major advantage of matrix methods is their flexibility. Instead of repeatedly finding complicated analytical eigenvalue equations for each laser structure, a general matrix equation is derived. Threshold analysis of various laser structures including planar section, corrugated section or a combination of them can be analysed in a systematic way. Since they share the same matrix equation, the algorithm derived to solve the problem can be re-used easily for different laser structures. However, because of the numerical nature of matrix methods, they cannot be used to verify the existence of analytical expressions in a particular problem. In all matrix methods, the structures involved will first be divided into a number of smaller sections. In each section, all physical parameters like the injection current and material gain are assumed to be homogeneous. As a result, the total number of smaller sections used varies and mostly depends on the type of problem. For a problem like the analysis of transient responses in LDs [14], a fairly large number of sections are needed since a highly nonuniform process is involved. On the other hand, only a few sections are required for the threshold analysis of DFB lasers since a fairly uniform process is concerned. For an arbitrary one-dimensional laser structure as shown in Fig. 4.1, the wave propagation is modelled by a 2  2 matrix A such that any electric field leaving (i.e. ER ðziþ1 Þ

Figure 4.1

Wave propagation in a general 1-D laser diode structure.

BRIEF REVIEW OF MATRIX METHODS

103

Table 4.1 Different types of matrix method Name Scattering matrix TLM TMM

U

V

ER ðziþ1 Þ and ES ðzi Þ ER ðziþ1 Þ and ES ðziþ1 Þ ER ðziþ1 Þ and ES ðziþ1 Þ

ER ðzi Þ and ES ðziþ1 Þ ER ðzi Þ and ES ðzi Þ ER ðzi Þ and ES ðzi Þ

Domain frequency time frequency

and ES ðzi Þ) and those entering (i.e. ER ðzi Þ and ES ðziþ1 Þ that section are related to one another by U ¼ AV

ð4:1Þ

where U and V are two column matrices each containing two electric wave components. Depending on the type of matrix method, the contents of U and V may vary. In the scattering matrix method, matrix U includes all electric waves leaving the arbitrary section, whilst matrix V contains those entering the section. In both transmission line matrix (TLM) and transfer matrix methods (TMM), matrix U represents the electric wave components from one side of the section, whilst wave components from the other side are included in matrix V. For analysis of semiconductor laser devices, both TLM and TMM have been used. The difference between TLM and TMM lies in the domain of analysis. TLM is performed in the time domain, whereas TMM works extremely well in the frequency domain. Table 4.1 summarises the characteristics of matrix methods. Using the time-domain-based TLM, transient responses like switching in semiconductor laser devices can be analysed. Steady-state values may then be determined from the asymptotic approximation. However, it is difficult to use TLM to determine noise characteristics, and hence the spectral linewidth, of semiconductor lasers. Due to the fact that most noise-related phenomena are time-averaged stochastic processes, a very long sampling time will be necessary if TLM is used. In general, TLM is not suitable for the analysis of noise characteristics in semiconductor laser devices. In 1987, Yamada and Suematsu first proposed using the TMM for analysing the transmission and reflection gains of laser amplifiers with corrugated structures. This frequency-domain-based method works extremely well for both steady-state and noise analysis [6,9]. In the present study, we are interested in the steady-state and noise characteristics of DFB lasers. Hence, the use of TMM will be more appropriate.

4.2.1 Formulation of Transfer Matrices Based upon the coupled wave equations, one can derive the transfer matrix for a corrugated DFB laser section. From the solution of the coupled wave equations, one can express EðzÞ ¼ ER ðzÞ þ ES ðzÞ ¼ RðzÞejb0 z þ SðzÞe jb0 z

ð4:2Þ

where ER ðzÞ and ES ðzÞ are the complex electric fields of the wave solutions, RðzÞ and SðzÞ are two slow-varying complex amplitude terms and b0 is the Bragg propagation constant. From eqn (3.3), RðzÞ and SðzÞ have proposed solutions of the form RðzÞ ¼ R1 egz þ R2 egz SðzÞ ¼ S1 e þ S2 e gz

gz

ð4:3aÞ ð4:3bÞ

104

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

where R1 , R2 , S1 and S2 are complex coefficients which are found to be related to one another by [15] S1 ¼ e j R1 R2 ¼ e

j

ð4:4aÞ ð4:4bÞ

S2

where  ¼ j=ð  j þ gÞ and  is the residue corrugation phase at the origin. By substituting eqn (4.4) into (4.3), one obtains RðzÞ ¼ R1 egz þ S2 ej egz SðzÞ ¼ R1 e e þ S2 e j gz

gz

ð4:5aÞ ð4:5bÞ

Instead of four variables, the solution of the coupled wave equations is simplified to functions of two coefficients R1 and S2 . Suppose the corrugation inside the DFB laser extends from z ¼ z1 to z ¼ z2 as shown in Fig. 4.2, the amplitude coefficients at the left and the right facets can then be written as Rðz1 Þ ¼ R1 egz1 þ S2 ej egz1 Sðz1 Þ ¼ R1 e e

j gz1

Rðz2 Þ ¼ R1 e

gz2

þ S2 e

þ S2 e

Sðz2 Þ ¼ R1 e e

j gz2

gz1

j gz2

ð4:6aÞ ð4:6bÞ

e

ð4:6cÞ

gz2

ð4:6dÞ

þ S2 e

From eqns (4.6a) and (4.6b), one can express R1 and S2 such that

Figure 4.2

R1 ¼

Sðz1 Þej  Rðz1 Þ ð2  1Þegz1

ð4:7aÞ

S2 ¼

Rðz1 Þej  Sðz1 Þ ð2  1Þegz1

ð4:7bÞ

A simplified schematic diagram for a 1-D corrugated DFB laser diode section.

BRIEF REVIEW OF MATRIX METHODS

105

By substituting the above equations back into eqns (4.6c) and (4.6d), one obtains E  2 E1 ðE  E1 Þej Rðz Þ  Sðz1 Þ 1 1  2 1  2 ðE  E1 Þe j 2 E  E1 Rðz Þ  Sðz1 Þ Sðz2 Þ ¼ 1 1  2 1  2

Rðz2 Þ ¼

ð4:8aÞ ð4:8bÞ

where E ¼ e ðz2 z1 Þ ;

E1 ¼ e ðz2 z1 Þ

ð4:8cÞ

From the above equations, it is clear that the electric fields at the output plane z2 can be expressed in terms of the electric waves at the input plane. By combining the above equations with eqn (4.2) we can relate the output and input electric fields through the following matrix equation [6] 

    ER ðz1 Þ ER ðz2 Þ t ¼ Tðz2 j z1 Þ ¼ 11 t21 ES ðz2 Þ ES ðz1 Þ

   ER ðz1 Þ t12 t22 ES ðz1 Þ

ð4:9Þ

where matrix Tðz2 j z1 Þ represents any wave propagation from z ¼ z1 to z ¼ z2 and its elements tij ði; j ¼ 1; 2Þ are given as t11 ¼

ðE  2 E1 Þ ejb0 ðz2 z1 Þ ð1  2 Þ

ð4:10aÞ

t12 ¼

ðE  E1 Þ ej ejb0 ðz2 þz1 Þ ð1  2 Þ

ð4:10bÞ

t21 ¼

ðE  E1 Þ ej e jb0 ðz2 þz1 Þ ð1  2 Þ

ð4:10cÞ

ð2 E  E1 Þ e jb0 ðz2 z1 Þ ð1  2 Þ

ð4:10dÞ

t22 ¼ 

For convenience, the matrix written in this way is called the forward transfer matrix because the output plane at z ¼ z2 is located further away from the origin. Similarly, waves propagating inside the corrugated structure can also be expressed as the backward transfer matrix such that [16] 

    ER ðz2 Þ ER ðz1 Þ u11 ¼ Uðz1 j z2 Þ ¼ u21 ES ðz1 Þ ES ðz2 Þ

   u12 ER ðz2 Þ u22 ES ðz2 Þ

ð4:11Þ

where matrix Uðz1 j z2 Þ represents any field propagation inside the section from z ¼ z2 to z ¼ z1 . By comparing eqn (4.9) with eqn (4.11), it is obvious that Uðz1 j z2 Þ ¼ ½Tðz2 j z1 Þ1

ð4:12Þ

106

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

where the superscript 1 denotes the inverse of the matrix. Due to conservation of energy, both matrices Tðz2 j z1 Þ and Uðz1 j z2 Þ must satisfy the reciprocity rule such that their determinants always give unity value [4]. In other words, jTj ¼ t11 t22  t12 t21 ¼ 1 jUj ¼ u11 u22  u12 u21 ¼ 1

ð4:13Þ

4.2.2 Introduction of Phase Shift (or Phase Discontinuity) For a single PS DFB laser cavity as shown in Fig. 4.3, the phase shift at z ¼ z2 divides the laser cavity into two sections.

Figure 4.3 Schematic diagram showing a 1PS DFB laser diode section.

The field discontinuity is usually small along the plane of phase shift and any wave travelling across the phase shift is assumed to be continuous. As a result, the transfer matrices are linked up at the phase shift position as:          j2 ER ðz ER ðzþ 0 2Þ 2 Þ ¼ Pð2Þ ER ðz2 Þ ¼ e ð4:14Þ ES ðz ES ðz ES ðzþ 0 ej2 2Þ 2Þ 2Þ  where Pð2Þ is the phase-shift matrix at z ¼ z2 ; zþ 2 and z2 are the greater and lesser values of z2 , respectively, and 2 corresponds to the phase change experienced by the electric waves ER ðzÞ and ES ðzÞ. Alternatively, the physical phase shift of the corrugation may be used [9]. To avoid any confusion, we will use the phase shift of the electric wave hereafter. On combining eqn (4.14) with the transfer matrix shown earlier in eqn (4.9), the overall transfer matrix chain of a single-phase-shifted DFB laser becomes " # 2 ð2Þ ð2Þ 3 " # # 2 ð1Þ ð1Þ 3 " ER ðz1 Þ ER ðz3 Þ t11 t12 t11 t12 e j2 0 5 5 ¼4 4 ð2Þ ð2Þ ð1Þ ð1Þ ES ðz3 Þ ES ðz1 Þ 0 ej2 t21 t22 t21 t22 " # ER ðz1 Þ ð2Þ ð2Þ ð1Þ ¼T P T ð4:15Þ ES ðz1 Þ

107

BRIEF REVIEW OF MATRIX METHODS

Without affecting the results of the above equations, one can multiply a unity matrix I after matrix T(1). This matrix I behaves as if an imaginary phase shift of zero or a multiple of 2 has been introduced. As a result, the above matrix equation can be simplified such that 

   ER ðz3 Þ ER ðz1 Þ ¼ Yðz3 j z1 Þ ES ðz3 Þ ES ðz1 Þ

ð4:16Þ

where

Yðz3 j z1 Þ ¼

1 Y

FðmÞ ¼

m¼2

FðmÞ ¼ TðmÞ PðmÞ P

ð1Þ

 ¼I¼

1

0

0

1



 y11 ðz3 j z1 Þ y12 ðz3 j z1 Þ y21 ðz3 j z1 Þ y22 ðz3 j z1 Þ " ðmÞ ðmÞ # " ðmÞ f11 t11 e jm f12 ¼ ¼ ðmÞ ðmÞ ðmÞ f f22 t21 e jm  21

ð4:17aÞ ðmÞ

t12 ejm ðmÞ

t22 ejm

# ð4:17bÞ ð4:17cÞ

In the above equation, the overall matrix Yðz3 j z1 Þ comprises the characteristics of the field propagation inside the DFB laser cavity, whilst the corrugated matrix TðmÞ and the phaseshift matrix PðmÞ ðm ¼ 1; 2Þ are combined to form the matrix FðmÞ . The use of the transfer matrix method is not restricted to the corrugated DFB laser structure. By modifying the values of  and  in the elements of the transfer matrix, other structures like the planar Fabry–Perot structure, the planar waveguide structure and the corrugated Distributed Bragg Reflector structure can also be represented using the transfer matrix. A DBR structure is different from the DFB structure because DBR structures have no underlying active region. The corrugated DBR structure simply acts as a partially reflecting mirror, the amount of reflection depending on the wavelength. The maximum reflection occurs near the central Bragg wavelength. Table 4.2 summarises all laser structures that can be represented by transfer matrices. The differences between them are also listed. When the grating height g reduces to zero and the grating period  approaches infinity, the feedback caused by the presence of corrugations becomes less important. At g ¼ 0,  becomes zero as does the variable . When  becomes infinite, the detuning coefficient  is reduced to the propagation constant 2n=
. In this case, the DFB corrugated structure becomes a planar structure. Following eqns (4.9) and (4.10), the transfer matrix equation of

Table 4.2 Structure FP WG DFB DBR

Laser structures that can be represented using the TMM Active layer

Corrugation

3 8 3 8

8 8 3 3

Comments  ¼ 0 and  > 0  ¼ 0 and   0 finite  and  > 0 finite  and   0

108

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

the planar structure becomes 

   " ð1Þ ER ðz1 Þ ER ðz2 Þ t ð1Þ ¼T ¼ 11 ð1Þ ES ðz2 Þ ES ðz1 Þ t 21

#   ER ðz1 Þ ð1Þ ES ðz1 Þ t ð1Þ

t12

ð4:18Þ

22

where ð1Þ

t11 ¼ eðz2 z1 Þ ejbðz2 z1 Þ ð1Þ

ð1Þ

ð4:19Þ

t12 ¼ t21 ¼ 0 ð1Þ

t22 ¼ eðz2 z1 Þ e jbðz2 z1 Þ In the above equation, the amplitude gain term  decides the characteristics of the planar structure. For  > 0, the amplitude of the electric wave passing through will be amplified and the structure will behave as if it is a laser amplifier. For   0, the amplitude of the electric wave will either remain constant or be attenuated, as the planar structure becomes a passive waveguide. Similarly, the sign of  will decide whether a corrugated structure belongs to the DFB or DBR type. By joining these matrices together as building blocks, one can extend the idea further to form a general N-sectioned composite laser cavity as shown in Fig. 4.4. Laser

Figure 4.4 Schematic diagram of a general N-section laser cavity. The phase shifts f1 ; 2 ; . . . ; N g are shown. Active regions along the laser cavity are shaded.

structures that comprise different combinations of the sections shown in Table 4.2 can be modelled. By joining these matrices together appropriately, one ends up with 

ER ðzNþ1 Þ ES ðzNþ1 Þ

 ¼F

ðNÞ

F

¼ YðzNþ1

ðN1Þ

ð2Þ

ð1Þ



ER ðz1 Þ

F F ES ð z 1 Þ   ER ð z 1 Þ j z1 Þ ES ð z 1 Þ



ð4:20Þ

109

BRIEF REVIEW OF MATRIX METHODS

where matrix YðzNþ1 j z1 Þ becomes the overall transfer matrix for the N-sectioned laser cavity. Using the backward transfer matrix together with eqns(4.11) and (4.14), one obtains   Y     N ER ðzNþ1 Þ ER ðzNþ1 Þ ER ð z 1 Þ ðmÞ G ¼ ¼ Zðz1 j zNþ1 Þ ð4:21Þ ES ð z 1 Þ ES ðzNþ1 Þ ES ðzNþ1 Þ m¼1 where ðmÞ

G

h

¼ P 

Zðz1 j zNþ1 Þ ¼

ðmÞ

i1

" ðmÞ

U

¼

z11 ðz1 j zNþ1 Þ z21 ðz1 j zNþ1 Þ

ðmÞ

g11

ðmÞ

g12

#

¼ ðmÞ ðmÞ g21 g22  z12 ðz1 j zNþ1 Þ z22 ðz1 j zNþ1 Þ

"

ðmÞ

u12 ejm

ðmÞ

u22 e jm

u11 ejm u21 e jm

ðmÞ

ðmÞ

# ð4:22aÞ ð4:22bÞ



1 is the inverse of the phase shift matrix PðmÞ and Zðz1 j zNþ1 Þ In the above equation, PðmÞ is the overall backward transfer matrix. Comparing eqns (4.20) with (4.21), it is clear that matrices YðzNþ1 j z1 Þ and Zðz1 j zNþ1 Þ are inverse to one another such that Zðz1 j zNþ1 Þ ¼ ½YðzNþ1 j z1 Þ1

ð4:23Þ

where the superscript 1 indicates the inverse of the matrix. From the property of the inverse of matrix products, individual transfer matrices GðmÞ and FðmÞ are related to one another. That is ð4:24Þ GðmÞ ¼ FðmÞ 1 for m ¼ 1 to N The above equation shows the equivalence between the forward and the backward transfer matrices in the general N-sectioned laser cavity. Unless stated otherwise, the forward transfer matrix is assumed hereafter.

4.2.3 Effects of Finite Facet Reflectivities It was discussed in Chapter 3 that the lasing characteristic of the DFB laser depends on the facet reflectivity. In this section, the facet reflectivity will be implemented using the TMM.

Figure 4.5

Schematic diagram showing reflections at the laser facets of a DFB LD.

110

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

In Fig. 4.5, a simplified schematic diagram for the reflections at the facets of the N-sectioned laser cavity is shown. In Fig. 4.5, ^r1 and ^r2 are the amplitude reflections at the left and right facets, respectively and medium 1 is the active region of the LD. In most practical cases, medium 2 is air. Due to the finite thickness of coating on the laser facets, any electric field passing through may suffer a phase change of i ði ¼ 1; 2Þ. Depending on the direction of propagation, all the  outgoing electric fields at the left facet (i.e. ER ðzþ 1 Þ and ES ðz1 Þ) can be expressed in terms of the incoming waves as qffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ^r12 ej1 ER ðz r1 ES ðzþ 1 Þþ^ 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffi ES ðz r1 ER ðz 1  ^r12 e j1 ES ðzþ 1 Þ ¼ ^ 1Þþ 1Þ

ER ðzþ 1Þ ¼

ð4:25aÞ ð4:25bÞ

Rearranging the above equation for the electric fields at z ¼ zþ 1 , one obtains ^r1 ej1 ej1 ffi ER ðz pffiffiffiffiffiffiffiffiffiffiffiffiffi ES ðz Þ þ ER ðzþ 1 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1 1Þ 1  ^r12 1  ^r12

ð4:26aÞ

^r1 ej1 ej1 ffi ER ðz pffiffiffiffiffiffiffiffiffiffiffiffiffi ES ðz ES ðzþ Þ þ 1 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1 1Þ 1  ^r12 1  ^r12

ð4:26bÞ

In matrix form, the above equations can be written as 

  1 ER ðzþ 1 1Þ p ffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ þ 2 ES ðz1 Þ e j1 1  ^r1 ^r1

   ER ðz ^r1 1Þ ES ðz 1 1Þ

Similarly, the reflection at the right facet can be written as " #     ER ðz ER ðzþ 1 ^r2 1 Nþ1 Þ Nþ1 Þ p ffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼ ES ðz ES ðzþ 1 e j2 1  ^r22 ^r2 Nþ1 Þ Nþ1 Þ

ð4:27aÞ

ð4:27bÞ

On combining the propagation matrix YðzNþ1 j z1 Þ with the reflections at the laser facets, the overall transfer function of the N-sectioned DFB laser structure becomes " #      ER ðzþ 1 ^r2 1 ^r1 ER ðz 1 1 1Þ Nþ1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ YðzNþ1 j z1 Þ ES ðzþ e j2 1  ^r22 ^r2 1 e j1 1  ^r12 ^r1 1 ES ðz Nþ1 Þ 1Þ ð4:28Þ It will be easier to simplify the above matrix equation by an overall transfer matrix H such that " #       ER ðzþ ER ðz ER ðz h11 h12 1Þ 1Þ Nþ1 Þ ¼H ¼ ð4:29Þ ES ðzþ ES ðz ES ðz h21 h22 1Þ 1Þ Nþ1 Þ where hi;j ði; j ¼ 1; 2Þ are the elements of the overall transfer matrix H.

111

FORMULATION OF THE AMPLIFIED SPONTANEOUS EMISSION SPECTRUM

4.3

THRESHOLD CONDITION FOR THE N-SECTIONED LASER CAVITY

 Since the laser itself is an oscillatory device, the output waves ER ðzþ Nþ1 Þ and ES ðz1 Þ denoted in the N-sectioned laser cavity should have finite values even though there are no incoming waves [9]. Suppose one of the incoming waves ER ðz 1 Þ becomes zero, then eqn (4.29) is simplified as  ER ðzþ Nþ1 Þ ¼ h12 ES ðz1 Þ

ð4:30Þ

¼

ð4:31Þ

ES ðzþ Nþ1 Þ

h22 ES ðz 1Þ

The transmission gain At of the backward travelling wave can then be expressed as [2] At ¼

ES ðz 1 1Þ ¼ h ES ðzþ Þ 22 Nþ1

ð4:32Þ

When the matrix element h22 approaches zero, the transmission gain becomes infinite and a resonant cavity is formed. Physically, a laser that operates in this condition is said to have the threshold condition satisfied. After substantial manipulation of eqn (4.28), the threshold condition becomes y22 ðzNþ1 j z1 Þ þ ^r1 y21 ðzNþ1 j z1 Þ  ^r2 y12 ðzNþ1 j z1 Þ  ^r1^r2 y11 ðzNþ1 j z1 Þ ¼ 0

ð4:33Þ

For DFB semiconductor lasers having finite facet reflections, one needs to find all the elements of the propagation matrix YðzNþ1 j z1 Þ. For a mirrorless DFB laser cavity where ^r1 ¼ ^r2 ¼ 0, the above threshold equation is simplified such that y22 ðzNþ1 j z1 Þ ¼ 0

ð4:34Þ

In fact, eqn (4.33) is a general expression that can be used to determine the lasing threshold characteristics of semiconductor laser devices. These include FP lasers, conventional DFB lasers (both mirrorless and those having finite facet reflections), single-phase-shifted DFB laser structures, multiple-phase-shifted DFB laser structures [17–19] as well as multiple electrode DFB laser structures [16,20–23]. By increasing the number of sections, TMM can be used to represent a tapered or chirped DFB laser structure [3– 4,24]. Similarly, the transfer matrix method has also been used in surface emitting devices [25–26] which have received worldwide attention in recent years. Table 4.3 summarises the minimum number of transfer matrices and phase shifts required in the threshold analysis of some popular semiconductor laser structures [27].

4.4

FORMULATION OF THE AMPLIFIED SPONTANEOUS EMISSION SPECTRUM USING THE TMM

In the previous section, the threshold equation of the N-sectioned laser cavity was defined using the transfer matrix. In fact, the TMM can also be applied to the below-threshold

112

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

Table 4.3

Semiconductor laser structures that can be analysed using the TMM (After [27])

Laser structure

Number of transfer matrices

Phase shifts

1 1

– –

2 Nþ1 N

2 2 . . . Nþ1 –

3

–

large number

–

b0 varies along the laser cavity

large number

–

 varies along the laser cavity

N

–

–

FP lasers Conventional DFB LD Single PS DFB Multiple PS DFB Multiple electrode DFB –Non-uniform current injection Corrugation-pitch-modulated DFB –Different corrugation period in each section Linear chirped corrugation –continuous change in corrugation period Tapered corrugation –continuous change in corrugation depth g N-layer surface emitting laser

Remarks  ¼ 0,  ! 1 ^r1 ¼ ^r2 ¼ 0 for mirrorless cavity 1 is always zero 1 is always zero Laser medium is homogeneous within a single section only b0 changes in each section

analysis. In semiconductor-based optical amplifiers and filters, the spontaneous emission spectrum has been used to determine the bandwidth, tunability and signal gain characteristics. Based on the use of Green’s function [28] for the noise calculation of the open resonator, the transfer matrix formulation will now be extended to include the output spontaneous emission spectrum taken from the right laser facet.

4.4.1 Green’s Function Method Based on the Transfer Matrix Formulation In this section, we refer once again to the general N-sectioned laser cavity as shown earlier in Fig. 4.4. The amplitude reflections at the left and right laser facets are ^r1 and ^r2 , respectively, and perfect index coupling is assumed. Following Henry [28], the one-dimensional inhomogeneous wave equation of a transversely and laterally confined laser mode in the composite longitudinal structure can be expressed as 

 d2 2 þ bðzÞ E! ðzÞ ¼ F! ðzÞ dz2

ð4:35Þ

where bðzÞ is the propagation constant, E! ðzÞ is the complex Fourier component of the electric wave and F! ðzÞ is the Langevin force term which accounts for the distributed

FORMULATION OF THE AMPLIFIED SPONTANEOUS EMISSION SPECTRUM

113

spontaneous emission noise inside the semiconductor laser [28–29]. In the above equation, bðzÞ includes any spatial variation of the physical systems that may affect the propagation constant. The electric field EðzÞ in the time domain is determined from the inverse Fourier transform such that ð1 EðzÞ ¼ E! ðzÞe j!t þ c:c: ð4:36Þ 0

where c.c. is the complex conjugate. Using Green’s function [30 –31], the general solution of eqn (4.35) can be written as ðz Gðz; z0 ÞF! ðz0 Þ dz0 ð4:37Þ E! ¼ z1

where z0 locates the spontaneous noise source and Gðz; z0 Þ is Green’s function. In the above equation, the integral shown sums up the impulse responses of the spontaneous emission noise source that originates from the left facet at z ¼ z1 to that at the observation point, z. The function Gðz; z0 Þ shown is given as [32] 8 Z1 ðzÞZ2 ðz0 Þ > > ; for z < z0 < W 0 Gðz; z Þ ¼ ð4:38Þ > Z ðz0 ÞZ2 ðzÞ > : 1 ; for z > z0 W where Z1 ðzÞ and Z2 ðzÞ are two independent solutions of the homogeneous wave equation (with F! ðzÞ ¼ 0) satisfying the boundary conditions at z ¼ z1 and z ¼ zNþ1 , respectively. The Wronskian term W is defined as: W ¼ Z1 ðzÞ

dZ2 ðzÞ dZ1 ðzÞ  Z2 ðzÞ dz dz

ð4:39Þ

From the above equation, it is obvious that W is finite if and only if Z1 ðzÞ and Z2 ðzÞ are two independent functions. According to the solution of the coupled wave equations, the normalised electric field at an arbitrary point z1  z0  zNþ1 inside a general N-sectioned DFB laser cavity can be expressed as 0

0

Eðz0 Þ ¼ ER ðz0 Þ þ ES ðz0 Þ ¼ Rðz0 Þejbr0 z þ Sðz0 Þe jbr0 z

ð4:40Þ

where z0 is an arbitrary point lying within the transfer matrix section FðkÞ such that zk  z0  zkþ1 . Using the forward transfer matrix as shown in the previous section, the complex electric field ER ðz0 Þ and ES ðz0 Þ becomes 

       1 Y ER ðz0 Þ ER ðz1 Þ ER ðz1 Þ ER ðz0 j z1 Þ ðkÞ 0 ðjÞ 0 ð z j z Þ F ð j z Þ ¼ ¼ F ¼ Y z k 1 ES ðz0 Þ ES ðz0 j z1 Þ ES ðz1 Þ ES ðz1 Þ j¼k1 ð4:41Þ

where the transfer matrix FðkÞ ðz0 j zk Þ shown has taken into account any wave propagation between z ¼ zk to z ¼ z0 in section k. In simplifying the matrix representation, a matrix

114

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

Yðz0 j zÞ is used to represent any wave change between the input plane at z ¼ z1 and the output at z ¼ z0 . Similarly, the complex electric fields at z ¼ zNþ1 can be expressed in terms of the complex electric field at z ¼ z0 such that 

 Y     kþ1 ER ðzNþ1 Þ ER ðz0 Þ ER ðz0 Þ ðjÞ ðkÞ 0 0 F F ðzkþ1 j z Þ ¼ ¼ YðzNþ1 j z Þ ES ðzNþ1 Þ ES ðz0 Þ ES ðz0 Þ j¼N

ð4:42Þ

Where the same forward transfer matrix technique has been used. In a similar way to eqn (4.41), the matrix representation is simplified by the matrix YðzNþ1 j z0 Þ which corresponds to any wave change between z ¼ z0 and z ¼ zNþ1 . By multiplying both sides of eqn (4.42) by the inverse matrix ½YðzNþ1 j z0 Þ1 , one obtains " # " # h i1 ER ðzNþ1 Þ h i1 ER ðz0 Þ ðkÞ 0 ðkÞ 0 ðkÞ 0 Y ðzNþ1 j z Þ ¼ Y ðzNþ1 j z Þ Y ðzNþ1 j z Þ ES ðzNþ1 Þ ES ðz0 Þ " # " # kþ1 h i1 ER ðzNþ1 Þ Y ER ðz0 Þ ðjÞ ðkÞ 0 F F ðzNþ1 j z Þ ¼ ES ðzNþ1 Þ ES ðz0 Þ j¼N # " # " # " ð4:43Þ N Y ER ðz0 Þ ER ðzNþ1 Þ ðkÞ 0 ðjÞ G ðz j zNþ1 Þ ¼ G ES ðzNþ1 Þ ES ðz0 Þ j¼kþ1 " # " # " # ER ðz0 Þ ER ðz0 j zNþ1 Þ ER ðzNþ1 Þ 0 Zðz j zNþ1 Þ ¼ ¼ ES ðzNþ1 Þ ES ðz0 Þ ES ðz0 j zNþ1 Þ where eqn (4.24) has been used in establishing the last equality. In the equation shown above, the backward transfer matrices have been used, in which complex electric fields at the right laser facets are used as inputs. Following the matrix eqns (4.41) and (4.43), Z1 ðzÞ and Z2 ðzÞ can be written in terms of the elements of the transfer matrix as Z1 ðzÞ ¼ ER ðz j z1 Þ þ ES ðz j z1 Þ ¼ ½^r1 y11 ðz j z1 Þ þ y12 ðz j z1 Þ þ ½^r1 y21 ðz j z1 Þ þ y22 ðz j z1 Þ

ð4:44aÞ

Z2 ðzÞ ¼ ER ðzNþ1 j z1 Þ ½ER ðz j zNþ1 Þ þ ES ðz j zNþ1 Þ ¼ ER ðzNþ1 j z1 Þ f½z11 ðz j zNþ1 Þ þ ^r2 z12 ðz j zNþ1 Þ þ ½z21 ðz j zNþ1 Þ þ ^r2 z22 ðz j zNþ1 Þg ð4:44bÞ In eqn (4.44b), the elements of the backward transfer matrix have been used. Since yij ðz1 j z1 Þ ¼ zij ðzNþ1 j zNþ1 Þ ¼ ij ði; j ¼ 1; 2Þ, the Kronecker delta function [30], it is straightforward to verify that Z1 ðzÞ and Z2 ðzÞ satisfy the boundary conditions at the left and the right laser facet respectively, such that ER ðz1 Þ ER ðz1 j z1 Þ ¼ ¼ ^r1 ES ðz1 Þ ES ðz1 j z1 Þ ES ðzNþ1 Þ ES ðzNþ1 j zNþ1 Þ ¼ ¼ ^r2 ER ðzNþ1 Þ ER ðzNþ1 j zNþ1 Þ

ð4:45Þ ð4:46Þ

FORMULATION OF THE AMPLIFIED SPONTANEOUS EMISSION SPECTRUM

115

Finally, by substituting eqns (4.44a) and (4.44b) into eqn (4.39), the Wronskian term becomes W ¼ 2jbr0 ER ðzNþ1 j z1 Þ fER ðz j z1 ÞES ðz j zNþ1 Þ  ES ðz j z1 ÞER ðz j zNþ1 Þg

ð4:47Þ

where the leading coefficient jb0 is obtained from the exponential expression as shown in eqn (4.40). By expanding the above equation with the matrix equation (4.41) and (4.43), one ends up with W ¼ 2jbr0 ER ðzNþ1 j z1 Þ fz21 ðz j zNþ1 Þy12 ðz j z1 Þ  z11 ðz j zNþ1 Þy22 ðz j z1 Þ þ ^r1 ½z21 ðz j zNþ1 Þy11 ðz j z1 Þ  z11 ðz j zNþ1 Þy12 ðz j z1 Þ þ ^r2 ½z22 ðz j zNþ1 Þy12 ðz j z1 Þ  z12 ðz j zNþ1 Þy22 ðz j z1 Þ þ ^r1^r2 ½z22 ðz j zNþ1 Þy11 ðz j z1 Þ  z12 ðz j zNþ1 Þy12 ðz j z1 Þg

ð4:48Þ

On the other hand, by eqn. (4.43), the elements of matrix Yðz j z1 Þ and those of matrix Zðz j zNþ1 Þ obey the following quality ½Zðz j zNþ1 Þ

1



   z11 ðz j zNþ1 Þ z12 ðz j zNþ1 Þ 1 z22 ðz j zNþ1 Þ z12 ðz j zNþ1 Þ ¼ ¼ z21 ðz j zNþ1 Þ z22 ðz j zNþ1 Þ z21 ðz j zNþ1 Þ z11 ðz j zNþ1 Þ   y11 ðzNþ1 j zÞ y12 ðzNþ1 j zÞ ¼ ð4:49Þ ¼ YðzNþ1 j zÞ y21 ðzNþ1 j zÞ y22 ðzNþ1 j zÞ

As a result, the Wronskain term can be simplified to W ¼ 2jbr0 ER ðzNþ1 j z1 Þ½y22 þ ^r1 y21  ^r2 y12  ^r1 ^r2 y11 

ð4:50Þ

where the bracket term in the above equation is the threshold equation of the N-sectioned laser cavity as shown in eqn (4.33). In deriving the above equation, the following identities were used y22 ¼ y22 ðzNþ1 j z1 Þ ¼ y21 ðzNþ1 j zÞy12 ðz j z1 Þ þ y22 ðzNþ1 j zÞy22 ðz j z1 Þ y21 ¼ y21 ðzNþ1 j z1 Þ ¼ y21 ðzNþ1 j zÞy11 ðz j z1 Þ þ y22 ðzNþ1 j zÞy21 ðz j z1 Þ y12 ¼ y12 ðzNþ1 j z1 Þ ¼ y11 ðzNþ1 j zÞy12 ðz j z1 Þ þ y12 ðzNþ1 j zÞy22 ðz j z1 Þ y11 ¼ y11 ðzNþ1 j z1 Þ ¼ y11 ðzNþ1 j zÞy11 ðz j z1 Þ þ y12 ðzNþ1 j zÞy21 ðz j z1 Þ

ð4:51Þ

When the threshold condition is reached, the bracket term becomes zero and hence does the Wronskian term. In other words, the proposed solutions Z1 ðzÞ and Z2 ðzÞ of the wave equation become dependent upon one another. In fact, it was shown by Makino [2] that they are identical at threshold.

4.4.2 Determination of Below-Threshold Spontaneous Emission Power When a laser diode is biased in the below-threshold regime, there is finite optical power output due to spontaneous emission. From the Poynting vector of the propagating field, the

116

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

spontaneous emission power PN ðzÞ within an angular frequency bandwidth
! can be written as [28] ð ð  c
! 1 0 z  0 PN ðzÞ ¼ dw E! ðzÞH!
0 ðzÞ e jð!! Þt þ c:c: dz ð4:52Þ 4 0 z1 where c.c. is the complex conjugate of the integrand. For a laterally confined structure, the magnetic field H! can be expressed as H! ¼

j @E! ðzÞ ! @z

ð4:53Þ

At the below-threshold biasing condition, there is no stimulated emission. Variables such as the optical gain (g) and the refractive index (n) remain homogeneous along the laser cavity. As a result, by replacing E! with Green’s function and the Wronskian, eqn (4.52) becomes PN ðzÞ ¼

  ðz c2
! 2DFF
dZ2
ðzÞ þ c:c: jZ ðzÞ jZ1 ðzÞj2 dz 2 4! jW j2 dz z1

ð4:54Þ

where DFF
is the diffusion coefficient given as [28] DFF
¼

2!3hngnsp c3

ð4:55Þ

where h ¼ h=2 is the angular Planck constant and nsp is the population inversion factor defined as [33] h i1 nsp ¼ 1  eðh!
EÞ=kT

ð4:56Þ

In the above equation,
E is the energy separation of the quasi-Fermi level between the conduction band and the valence band, k is Boltzmann’s constant and T is the temperature in Kelvin. By replacing the Wronskian term and DFF
with the appropriate transfer matrix elements in the N-sectioned laser cavity, eqn (4.45) finally becomes [6]  ð zNþ1 1  ^r22 nsp g hc
! PN ðzNþ1 Þ ¼ jZ1 ðzÞj2 dz
jy22 þ ^r1 y21  ^r2 y12  ^r1^r2 y11 j2 z1

ð4:57Þ

where Z1 ðzÞ is the solution of the homogenous wave equation as defined in eqn (4.44a). Using the transfer matrix, the equation shown above agrees with the one obtained using multiple reflection inside the DFB laser cavity [7]. To evaluate the integral shown above, a numerical technique such as the trapezoidal rule can be applied. For below-threshold and threshold analysis when g is assumed to be independent of z, an analytical expression has

FORMULATION OF THE AMPLIFIED SPONTANEOUS EMISSION SPECTRUM

117

been proposed [2]. Basically, the integral is first to be broken up first and the contribution from each transfer matrix is then found. In other words, one obtains ð zNþ1 z1

jZ1 ðzÞj2 dz ¼

N ð zkþ1 X k¼1

jZ1 ðz0 Þj dz0 2

ð4:58Þ

zk

where ð zkþ1 zk

  2g Lk    e r 1 1  e2gr Lk 2 þ jbk j2 jZ1 ðz0 Þj dz0 ¼ ð1 þ jj2 Þ jak j2 2gr 2gr   2jg Lk  i e 1 þ 2Re ð þ 
Þak b
k 2j i

ð4:59Þ

where * indicates the complex conjugate and ER ðzk j z1 Þejbr0 zk ej  ES ðzk j z1 Þejbr0 zk 1  2 ES ðzk j z1 Þejbr0 zk  ER ðzk j z1 Þejbr0 zk ej bk ¼ 1  2

ak ¼

ð4:60aÞ ð4:60bÞ

In the above equation, Lk ¼ zkþ1  zk is the length of section k; r and i are the real and imaginary parts of the complex propagation constant , respectively, and  ¼ j=ð  j þ Þ. When the spatial hole burning effect becomes dominant in the above-threshold biasing regime, the carrier distribution along the laser cavity becomes nonuniform. As a result, the refractive index, and consequently the propagation constant, becomes spatially dependent. These variations violate the assumption of the analytical expression in which g is homogeneous along the direction z. Hence, it can be shown that the analytical expression is restricted to the below-threshold condition, when uniformity can still be maintained along the laser cavity. A different technique is required in the above-threshold condition when variables become longitudinally dependent.

4.4.3 Numerical Results from Various DFB Laser Diodes In this section, the below-threshold PN of various DFB laser diodes will be presented. Results obtained from a conventional, a QWS and a 3PS DFB LD will be compared. For all these lasers, a laser cavity length of 500 mm and zero facet reflection are assumed. Therefore, any difference between the values of PN in these lasers is to be caused solely by the structural variation between them. From the threshold equation of the N-sectioned laser cavity, the normalised amplitude gains th L of these lasers are first determined. Under the below-threshold condition, the refractive index and hence the propagation constant are assumed to be constant along the cavity. Figure 4.6 shows a schematic diagram of the 3PS DFB LD used in the analysis. The phase shift 3 shown in the figure has been fixed at the centre of the structure, whilst the positions

118

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

Figure 4.6

Simplified schematic diagram showing a 3PS DFB laser diode structure.

of phase shifts 2 and 4 are allowed to move along the cavity. Their relative positions are defined by a position parameter, , as ¼ It should be noticed that when

2L1 L

ð4:61Þ

¼ 0 or 1, the structure becomes a single-phase-shifted laser.

Figure 4.7 Below-threshold spectra of various DFB semiconductor laser diodes (after [32]).

FORMULATION OF THE AMPLIFIED SPONTANEOUS EMISSION SPECTRUM

119

Figure 4.7 shows the variation of the normalised intensity PN =hfnsp as a function of the detuning coefficient L for three DFB LDs. These include a conventional, a QWS and a 3PS DFB LD. Degenerate oscillation is observed for the conventional case. Compared with the QWS LD, the lasing mode of the 3PS LD is characterised by a detuned oscillation from the Bragg wavelength. Figures 4.8 and 4.9 show results obtained from 3PS DFB LDs under different values of the amplitude threshold gain th L. Other parameters used in the analysis are listed in the figures.

Figure 4.8 Below-threshold spectra of 3PS DFB laser diodes with

¼ 0:5 (after [32]).

The values of phase shift used in the calculations are fixed at =3 whilst is assumed to be 0.50 and 0.75, respectively. By increasing the amplitude gain from 0.5 th L to th L, the normalised intensity shows a substantial increase across the spectrum. The peak showing the largest intensity will become the lasing mode when the threshold condition is reached. Other smaller peaks shown become the non-lasing side modes. From Figs. 4.8 and 4.9, it can be seen that both spectra shown are asymmetric with respect to the Bragg wavelength ð ¼ 0Þ. Such an uneven mode distribution is a well-known characteristic of a 3PS DFB laser [19]. By measuring the intensity difference between the major lasing mode and the next highest potential side mode in the spontaneous emission spectrum, the side mode suppression ratio (SMSR) is used to measure the single-mode stability [5]. At the lasing threshold, an SMSR of 30 dB is achieved in the structure as shown in Fig. 4.8, whilst a 20 dB

120

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

Figure 4.9

Below-threshold spectra of 3PS DFB laser diodes with

¼ 0:75 (after [32]).

SMSR is observed in Fig. 4.9. Comparing these figures, our results reveal that the belowthreshold output spectrum is very sensitive to the phase shift position. Fewer modes are excited in the structure shown in Fig. 4.9 as compared with the one shown in Fig. 4.8, where has changed from 0.5 to 0.75. Other structural variations and impacts induced by the variations of phase shift and PSP can be found in a paper by Ghafouri-Shiraz and Lo [34]. From the spectral measurement of the PN , information such as the coupling coefficient, the reflection at laser facets and the phase shift may also be evaluated [7]. By testing various samples obtained from a wafer, the measurement of PN plays an important role in the quality control of the fabrication process.

4.5

SUMMARY

In this chapter, the idea of the transfer matrix has been introduced and explored. Compared with the boundary matching approach, the TMM is more robust and flexible. By converting the coupled wave equations into a matrix equation, the wave propagating characteristics of the corrugated DFB section can be represented using a transfer matrix. The transfer matrix approach was extended to include phase discontinuity and the residue reflection at the facets.

REFERENCES

121

By modifying the elements of the transfer matrix, it can be used to represent other planar and corrugated structures including passive waveguides, passive DBR and planar FP sections. By joining these transfer matrices as building blocks, a general N-sectioned laser cavity model was constructed and the threshold equation associated with this laser mode was determined. The use of the transfer matrix is not restricted to threshold analysis. Combining the Poynting vector with the Green’s function method, the TMM can also be implemented to evaluate the below-threshold spontaneous emission power spectrum PN . Results obtained from conventional, QWS and 3PS DFB LDs were presented and compared. From the results of the 3PS DFB LDs, it was shown that PN is sensitive to any structural variation. As a result, by comparing PN measured from a batch of LDs, it can be used for quality control over the fabrication process. In the next chapter, the use of the TMM at both the threshold and the below-threshold applications will be explored further. In particular, the structural design and optimisation of DFB LDs will be discussed.

4.6

REFERENCES

1. Makino, T., Transfer-matrix analysis of the intensity and phase noise of multisection DFB semiconductor lasers, IEEE. J. Quantum Electron., QE-27(11), 2404 –2415, 1991. 2. Makino, T., Transfer-matrix formulation of spontaneous emission noise of DFB semiconductor lasers, J. Lightwave Technol., LT-9(1), 84 –91, 1991. 3. Yamada, M. and Sakuda, K., Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach, Appl. Opt., 26(16), 3474–3478, 1987. 4. Yamada, M. and Sakuda, K., Adjustable gain and bandwidth light amplifiers in terms of distributedfeedback structures, J. Opt. Soc. Am. A., 4(1), 69–76, 1987. 5. Whiteaway, J. E. A., Thompson, G. H. B., Collar, A. J. and Armistead, C. J., The design and assessment of a
/4 phase-shifted DFB laser structure, IEEE J. Quantum Electron., QE-25(6), 1261–1279, 1989. 6. Makino, T. and Glinski, J., Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers, IEEE J. Quantum Electron., QE-24(8), 1507–1518, 1988. 7. Soda, H., and Imai, H., Analysis of the spectrum behaviour below the threshold in DFB lasers, IEEE J. Quantum Electron., QE-22(5), 637–641, 1986. 8. Chu, C. Y. J. and Ghafouri-Shiraz, H., Analysis of gain and saturation characteristics of a semiconductor laser optical amplifier using a transfer matrix, J. Lightwave Technol., LT-12(8), 1378–1386, 1994. 9. Numai, T., 1.5-mm wavelength tunable phase-shift-controlled distributed feedback laser, J. Lightwave Technol., LT-10(2), 199–205, 1992. 10. Tan, P. W., and Ghafouri-Shiraz, H. and Lo, B. S. K., Theoretical Analysis of multiple-phase-shift controlled DFB wavelength tunable optical filters, Microwave Opt. Tech. Lett., 8(2), 72–75, 1995. 11. Bjo¨rk, G. and Nilsson, O., A new exact and efficient numerical matrix theory of complicated laser structures: properties of asymmetric phase-shifted DFB lasers, J. Lightwave Technol., LT-5(1), 140 –146, 1987. 12. Okai, M., Tsuji, T., and Chinone, N., Stability of the longitudinal mode in
/4-shifted InGaAsP/InP DFB lasers, IEEE J. Quantum Electron., QE-25(6), 1314–1319, 1989. 13. Ramo, S, Whinnery, J. R. and Van Duzer, T., Fields and Waves in Communication Electronics. New York: John Wiley & Sons, 1984. 14. Lowery, A. J., Amplified spontaneous emission in semicondcutor laser amplifiers: validity of the transmission line model, IEE Proc. Pt. J., 137(4) 241–247, 1990. 15. Streifer, W., Burnham, R. D., and Scifres, D. R., Effect of external reflectors on longitudinal modes of distributed feedback lasers, IEEE J. Quantum Electron., QE-11(4), 154–161, 1975.

122

TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS

16. Agrawal, G. P. and Bobeck, A. H., Modelling of distributed feedback semiconductor lasers with axially-varying parameters, IEEE J. Quantum Electron., QE-24(12), 2407–2414. 1988. 17. Kimura, T., and Sugimura, A., Narrow linewidth asymmetric coupled phase-shift DFB lasers, Trans. IEICE., E 79(1), 71–76, 1990. 18. Ogita, S., Kotaki, Y., Hatsuda, M., Kuwahara, Y., and Ishikawa, H., Long cavity multiple-phase shift distributed feedback laser diode for linewidth narrowing, J. Lightwave Technol., LT-8(10), 1596– 1603, 1990. 19. Yoshikuni, Y. and Motosugi, G., Multielectrode distributed feedback laser for pure frequency modulation and chirping suppressed amplitude modulation, J. Lightwave Technol., LT-5, 516–522, 1987. 20. Davis, M. G. and O’Dowd, R. F., A transfer matrix-based analysis of multielectrode DFB lasers, Photon. Tech. Lett., 3(7), 603–605, 1991. 21. Kawaguchi, H., Magari, K., Yasaka, H., Fukuda, M., and Oe, K., Tunable optical-wavelength conversion using triggerable multielectrode distributed feedback laser diode, IEEE J. Quantum Electron., QE-24(11), 2153–2159, 1988. 22. Kikuchi, K. and Tomofuji, H., Analysis of oscillation characteristics of separated-electrode DFB laser diodes, IEEE J. Quantum Electron., QE-26(10), 1717–1727, 1990. 23. Kuznetsov, M., Theory of wavelength tuning in two-segment distributed feedback lasers, IEEE J. Quantum Electron., QE-24(9), 1837–1844, 1988. 24. Zhou, P. and Lee, G. S., Chirped grating
/4-shifted distributed feedback laser with uniform longitudinal field distribution, Electron. Lett., 26, 1660–1661, 1990. 25. Makino, T., Theoretical analysis of the spectral linewidth of a surface-emitting DFB semiconductor laser, Optics. Comm., 81(2), 71–74, 1991. 26. Makino, T., Transfer-matrix formulation of spontaneous emission noise of DFB semiconductor lasers, J. Lightwave Technol., LT-9(1), 84–91, 1991. 27. Ghafouri-Shiraz, H. and Lo, B., Structural dependence of three-phase-shift distributed feedback semiconductor laser diodes at threshold using the transfer-matrix method (TMM), Semi. Sci. and Technol., 8(5), 1126–1132, 1994. 28. Henry, C. H., Theory of spontaneous emission noise in open resonator and its application to lasers and optical amplifiers, J. Lightwave Technol., LT-4(3), 288–297, 1986. 29. Yamamoto, Y., Coherence, Amplification, and Quantum Effects in Semiconductor Lasers. New York: Wiley, 1991. 30. Arfken, G., Mathematical Methods for Physicists, 3rd edition. New York: Academic Press, 1985. 31. Trombrog, B., Olsen, H. and Pan, X., Theory of linewidth for multielectrode laser diode with spatially distributed noise sources, IEEE J. Quantum Electron., QE-27(2), 178–192, 1991. 32. Morse, P. M. and Feshbach, H., Methods of Theoretical Physics vol. 1, International series in pure and applied physics. New York: McGraw Hill, 1953. 33. Agrawal, G. P. and Dutta, N. K., Long-Wavelength Semiconductor Lasers. Princeton, NJ: Van Nostrand, 1986. 34. Ghafouri-Shiraz, H. and Lo, B., Structural Impact on the below threshold spectral behavior of three phase shift (3PS) distributed feedback (DFB) lasers, Microwave Opt. Tech. Lett., 7(6), 296–299, 1994.

5 Threshold Analysis and Optimisation of Various DFB LDs Using the Transfer Matrix Method 5.1

INTRODUCTION

In the previous chapter, the transfer matrix method (TMM) was introduced to solve the coupled wave equations in DFB laser structures. Its efficiency and flexibility in aiding the analysis of DFB semiconductor LDs has been explored theoretically. A general N-sectioned DFB laser model was built which comprised active/passive and corrugated/planar sections. In this chapter, the N-sectioned laser model will be used in the practical design of the DFB laser. The spatial hole burning effect (SHB) [1] has been known to limit the performance of DFB LDs. As the biasing current of a single quarterly-wavelength-shifted (QWS) DFB LD increases, the gain margin reduces. Therefore, the maximum single-mode output power of the QWS DFB LD is restricted to a relatively low power operation. The SHB phenomenon caused by the intense electric field leads to a local carrier depletion at the centre of the cavity. Such a change in carrier distribution alters the refractive index along the laser cavity and ultimately affects the lasing characteristics. By changing the structural parameters inside the DFB LD, an attempt will be made to reduce the effect of SHB. As a result, a larger single-mode power, and consequently a narrower spectral linewidth, may be achieved. A full structural optimisation will often involve the examination of all possible structural combinations in the above-threshold regime. On the other hand, the analysis of the structural design may be simplified, in terms of time and effort, by optimising the threshold gain margin and the field uniformity. The structural changes and their impacts on the characteristics of DFB LDs will now be presented. By introducing more phase shifts along the laser cavity, a three-phase-shift (3PS) DFB LD will be investigated in section 5.2. In particular, impacts due to the variation of both phase shifts and their positions on the lasing characteristics of the 3PS DFB LD will be discussed. To reduce the SHB effect, it is necessary to have a more uniform field distribution, whilst maintaining a large gain margin ð
LÞ. The optimised structural design for the 3PS DFB laser based on the values of
L and the flatness (F) of the field distribution will be discussed in section 5.3 [2].

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

124

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

By changing the height of the corrugation and thus the coupling coefficient along a DFB laser cavity, a distributed coupling coefficient (DCC) DFB laser can be built. In section 5.4, the threshold characteristics of this structure will be shown. In particular, effects due to the variation of the coupling ratio and the position of the corrugation change will be investigated. To maintain a single-mode oscillation, a single phase shift is introduced at the centre of the cavity. By changing the value of the phase shift, the combined effect with the non-uniform coupling coefficient will be presented. Optimised structural combinations that satisfy both a high gain margin and a low value of flatness will be selected for later use in the above-threshold analysis. In section 5.5, the combined effect of both multiple phase shifts and non-uniform coupling coefficients will be investigated using a DCC þ 3PS DFB laser structure. Finally, a summary will be presented at the end of this chapter.

5.2

THRESHOLD ANALYSIS OF THE THREE-PHASE-SHIFT (3PS) DFB LASER

By introducing more phase shifts along the laser cavity, it has been shown [3–5] that the spatial hole burning effect can be reduced in a 3PS DFB LD which is characterised by a more uniform internal field distribution. Experimental measurement has been carried out [5] using a fixed value of phase shift. However, independent changes in the value of phase shift have not been fully explored. Using the TMM, it was shown in Table 4.1 of Chapter 4 that four transfer matrices are necessary to determine the threshold condition of 3PS DFB lasers. In Fig. 5.1, a

Figure 5.1

Schematic diagram showing a 3PS DFB LD.

schematic diagram of the 3PS DFB laser structure is shown. In the figure, 2 , 3 and 4 represent phase shifts and the length of each smaller section is labelled Lj ð j ¼ 1; 2Þ. In the analysis, zero facet reflection at the laser facets is assumed. Following the formulation of the transfer matrix method, the overall transfer matrix of the 3PS DFB laser becomes:   y11 ðz5 j z1 Þ y12 ðz5 j z1 Þ ð4Þ ð3Þ ð2Þ ð1Þ Y ð z5 j z1 Þ ¼ F F F F ¼ ð5:1Þ y21 ðz5 j z1 Þ y22 ðz5 j z1 Þ where Fð jÞ ð j ¼ 1 to 4) corresponds to the transfer matrix of each smaller section. For a mirrorless cavity, the threshold condition can be found by solving the following equation y22 ðz5 j z1 Þ ¼ 0

ð5:2Þ

THRESHOLD ANALYSIS OF THE THREE-PHASE-SHIFT (3PS) DFB LASER

125

Figure 5.2 Resonance modes of various DFBs that include: (a) a conventional DFB laser diode; (b) a single QWS DFB laser diode; (c) a three /2-phase-shifted DFB laser diode.

Using a numerical approach such as Newton–Raphson’s method [6] for analytical complex equations, the threshold equation above may be solved. Figure 5.2 shows the resonance modes obtained from a symmetrical 3PS DFB laser where 2 ¼ 3 ¼ 4 ¼ =2 and L1 ¼ L2 are assumed. For comparison purposes, results obtained from a mirrorless conventional DFB laser and a single =2 phase-shifted DFB laser are also included. In all three cases, the coupling coefficient and the overall laser cavity length L are fixed at 40 cm1 and 500 mm, respectively. Oscillation modes at the Bragg wavelength are found for both the single =2 and a 3PS DFB structure. However, the Bragg resonance mode of the 3PS DFB laser does not show the smallest amplitude threshold gain. Instead, degenerate oscillation occurs since it is shown that both the 1 and þ1 modes share the same value of amplitude threshold gain. It is interesting to see how a single =2 phase shift enables SLM operation whilst multimode oscillation occurs in the case where there are three phase shifts, i.e. f=2; =2; =2g. The pair of braces f g used hereafter will indicate a phase combination in the 3PS structure, that is f2 ; 3 ; 4 g.

5.2.1 Effects of Phase Shift on the Lasing Characteristics In order that stable SLM operation can be achieved in the 3PS DFB laser, one must change the value or the position of the phase shift. Figure 5.3 shows oscillation modes of various 3PS DFB laser structures. In the analysis, the values of the three phase shifts are assumed to be equal and the phase shift positions are the same as in Fig. 5.2. A shift of resonance mode can be seen when all phase shifts change from =2 to 2=5. The þ1 mode which demonstrates the smallest amplitude threshold gain will become the lasing mode after lasing

126

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

Figure 5.3 Resonance modes in various 3PS DFBs that include: (a) a f=2, =2, =2g 3PS DFB laser; (b) a f2=5; 2=5; 2=5g 3PS DFB laser; (c) a f3=5; 3=5; 3=5g laser.

threshold is reached. On the other hand, the 1 mode will become the lasing mode when the three phase shifts change from =2 to 3=5. With all three phase shifts displaced from the usual =2 values, SLM can be achieved in the 3PS DFB LD.

5.2.2 Effects of Phase Shift Position (PSP) on the Lasing Characteristics The 3PS DFB laser structure we have discussed so far is said to be symmetrical. For a cavity length of L, the position of phase shifts is assumed in such a way that L1 ¼ L2 ¼ L=4. To investigate the effect of the phase shift position (PSP) on the threshold characteristics, a position factor is introduced such that ¼

L1 2L1 ¼ L1 þ L2 L

ð5:3Þ

where 3 is assumed to be located at the centre of the cavity. Using the above equation, it should be noted that both ¼ 0 and ¼ 1 correspond to a single-phase-shifted DFB laser structure. In Fig. 5.4, the variation of the amplitude threshold gain is shown with the position factor for different values of normalised coupling coefficient L. All the phase shifts are fixed at 2 ¼ 3 ¼ 4 ¼ =3. At a fixed value of , the figure shows a decrease in amplitude threshold gain as the L value increases. Along the curve L ¼ 1:0, discontinuities at ¼ 0:12 and ¼ 0:41 indicate possible changes in the oscillation mode.

Figure 5.4 The change of amplitude gain with respect to the phase shift position for different values of coupling coefficient .

Figure 5.5 The variation of detuning coefficient with respect to the phase shift position for coupling coefficient .

128

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

Such a change in oscillation is confirmed when the relationship between the detuning coefficient and the position factor is shown in Fig. 5.5. Along L ¼ 1:0, it can be seen that the 1 mode remains as the oscillation mode when increases from zero. When ¼ 0:12 is reached, however, a sudden change of oscillation mode is observed. Similar mode jumping occurs at ¼ 0:41. When the PSP shifts, there is a continuous change in the resonant cavity formed by the DFB laser such that the actual lasing mode may alter. At ¼ 0:77, it is interesting to see how all L values converge to the same lasing wavelength. It appears that at this particular phase shift position, the effect of the variation of L is irrelevant and the lasing characteristic depends on the presence of the =3 phase shifts.

5.3

OPTIMUM DESIGN OF A 3PS DFB LASER STRUCTURE

A complete structural optimisation of MPS DFB lasers cannot be achieved without analysing the above-threshold performances. This involves solving the carrier rate equation, which is a fairly complex process and needs intensive computation. On the other hand, it is believed that the complexity of the structural design in the 3PS DFB laser can be reduced by optimising the threshold amplitude gain difference and the flatness of field distribution. Hence, we can simply concentrate on those structures satisfying these design criteria. For a high-performance DFB LD, both a stable single-mode oscillation and a uniform field distribution are important to prevent LDs from being affected by the spatial hole burning effect. In our analysis, DFB laser structures having a high gain margin ð
LÞ are considered, whilst the spatial hole burning effect is included by analysing the corresponding effects on field uniformity. Reports by Kimura and Sugimura [3– 4] as well as Ogita et al. [5] suggested that the lasing characteristics are strongly influenced by both and . To maintain a stable SLM oscillation, and consequently improve the performance of the spectral linewidth, these structural parameters need to be optimised.

5.3.1 Structural Impacts on the Gain Margin To achieve a stable laser source that oscillates at a single longitudinal mode, it is important that there is a gain margin
L > 0:25½1. In the analysis, we assumed the length of the laser L to be 500 mm. For a 3PS DFB LD, Fig. 5.6 shows the relationship between the gain margin and the phase shift  in a symmetrical structure for different values of L ranging from 1 to 3. The position factor ¼ 0:5 corresponds to the case where L1 ¼ L2 ¼ L=4. In all cases, the degenerate oscillations occur at  ¼ 0, =2 and , and the distributions of gain margins are symmetrical with respect to  ¼ =2. It is also shown that the variation of L has little effect on the gain margin of the 3PS laser structure. Along the line L ¼ 1, it is found that a stable laser having
L > 0:25 can be obtained provided that 47 <  < 73 or 107 <  < 133 . In Fig. 5.7, a contour map is shown that relates the gain margin to the values of phase shifts in three-phase-shift f2 ; 3 ; 4 g DFB LDs. In the calculations, L ¼ 2 and ¼ 0:5 are assumed. The phase shift 3 introduced at the centre of the cavity is separated from the rest so that its value can be selected independently. Other phase shifts are assumed to be equal as 2 ¼ 4 ¼ side . As stated earlier, to satisfy the requirement of
L > 0:25, side must either be greater than 105 or less than 80 if 3 can be varied freely between 0 and . A maximum

Figure 5.6

Variation of the gain margin versus the phase shift for different coupling coefficients.

Figure 5.7 Relationship between the gain margin
L and phase shifts for a 3PS DFB laser diode.

130

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

Figure 5.8 Variation of the gain margin versus

for various 3PS DFB laser diode structures.

value of
L ¼ 0:73 is obtained at f0; =2; 0g and f; =2; g which corresponds to a single =2-phase-shifted DFB laser. The variation of
L with respect to the position factor is shown in Fig. 5.8. In this figure, the values of phase shifts are equal (i.e. 2 ¼ 3 ¼ 4 ¼ ) and three different sets of results are calculated with  ¼ =2, 2=5 and =3. By changing the values of the phase shifts,
L also changes for each particular value of . At a fixed phase shift  ¼ =2 (solid line), it is shown that a non-zero value of gain margin is observed where < 0:13 and > 0:725. As approaches zero, the phase shifts 2 and 4 move towards the laser facets and their contributions become less influential. Also, as approaches unity, both 2 and 4 move towards the central phase shift 3 . In this case, the 3PS laser structure is reduced to a single-phase-shifted structure and the lasing characteristic is described by an effective phase shift of eff  2 þ 3 þ 4 . Figure 5.9 shows the dependence of
L upon for different values of L. In the analysis, all phase shifts are assumed to be identical (i.e.  ¼ =3). From this figure, it is clear that L has little effect on
L in 3PS DFB lasers.

5.3.2 Structural Impacts on the Uniformity of the Internal Field Distribution In this section, the structural impact on the internal field distribution will be discussed. To quantify the uniformity of the field distribution, it was shown in Chapter 3 that the flatness

OPTIMUM DESIGN OF A 3PS DFB LASER STRUCTURE

Figure 5.9

Variation of the gain margin versus

131

for different coupling coefficients.

(F) of the internal field of a general N-sectioned DFB laser cavity is defined as F¼

1 L

Z

zNþ1

ðIðzÞ  Iavg Þ2 dz

ð5:4Þ

z1

where IðzÞ is the electric field intensity along the longitudinal axis and Iavg is its average value. In the above equation, a zero value of F corresponds to a completely uniform field. In order to minimise the effects of longitudinal spatial hole burning, it has been shown experimentally ½1; 7 that a DFB laser cavity with F < 0:05 is necessary for stable SLM oscillation. To optimise the structural design of 3PS DFB lasers, F < 0:05 will be used as one of the design criteria. In order to evaluate the flatness of the internal field distribution, the threshold equation of the 3PS DFB laser needs to be solved first. The normalised amplitude threshold gain th L and the normalised detuning coefficient thL of the lasing mode are then used to determine the field distribution. In our analysis, a 500 mm long DFB laser is subdivided into a substantial number of small sections with equal length. From the output of each transfer matrix, both the forward and the backward propagating electric fields can be determined, and the electric field intensity at an arbitrary position z0 is found to be Iðz0 Þ ¼ jER ðz0 Þj þ jES ðz0 Þj 2

2

ð5:5Þ

132

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

Figure 5.10

Field distribution in various DFB laser diode structures.

In Fig. 5.10, the internal field distributions of three different structures are shown. These structures include a conventional mirrorless, a single =2-phase-shifted and a three-phaseshift f=3; =3; =3g DFB laser. All the electric field distributions have been normalised so that the intensity at the laser facets is unity. It can be seen that the single =2-phase-shifted DFB laser has a flatness value of F ¼ 0:301. Such a high value of F (which means that the field is highly non-uniform) induces a local carrier escalation near the centre of the cavity after the laser threshold is reached, consequently affecting the single-mode stability of the laser device. With three phase shifts incorporated into the cavity, the intensity distribution spreads out and the overall distribution becomes more uniform (see dashed line with F ¼ 0:012Þ. By optimising the values and the positions of the phase shifts with respect to the flatness, a 3PS DFB laser can maintain a uniform field distribution even at a high value of L, which is necessary to reduce the spectral linewidth of the laser. The effect of on F is shown in Fig. 5.11 for different combinations of L. When small values of L ð< 1:5Þ are used, the field intensity distribution becomes less uniform when the phase shifts 2 and 4 shift towards the laser facets (i.e. as tends to 0). As the optical feedback becomes stronger with increasing L, the field intensity distribution becomes more intense near the centre of the laser cavity where is found to be about 0.77. The contour map shown in Fig. 5.12 can be used to optimise the value of phase shifts with respect to F. In a similar way to Fig. 5.7, the central phase shift 3 is used as the x-axis and other phase shifts are represented in the y-axis. In this figure, all phase combinations with F < 0:05 form a ribbon shape stretching from the lower left-hand corner to the upper righthand corner of the contour. The worst case, which leads to the largest value of F, can be

Figure 5.11

Figure 5.12

Variation of flatness versus

for different coupling coefficients .

Relationship between the flatness and the phase shift for a 3PS DFB LD.

134

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

found at phase combinations of f0; =2; 0g and f; =2; g. For F < 0:05, side must lie in the range 67.5 < side < 112:5 for an unrestricted value of 3 . By comparing the map in Fig. 5.12 with the one shown earlier in Fig. 5.7, it can be seen that a trade-off exists in selecting the appropriate phase shift value for the optimum values of
L and F. On one hand, phases should be chosen such that the gain margin is large enough to avoid mode hopping. On the other hand, the corresponding combination of phase shifts will result in a relatively large value of F. Due to the SHB effect, the associated single-mode stability deteriorates with increasing output power. As a result, a compromise has to be made in selecting the phase shifts in 3PS DFB lasers such that high performance LDs with high
ð> 0:05 cm1 Þ values and small Fð< 0:05Þ can be obtained. Figure 5.13 is a combination of the region
L  0:25 in Fig. 5.8 and the region F  0:05 from Fig. 5.12. In the shaded area of this figure, it can be seen that a phase combination of f=3; =3; =3g will satisfy both the design criteria of gain margin and flatness.

Figure 5.13 Variations of the gain margin and flatness with respect to phase shifts. Shaded area covers all phase combinations that satisfy both
L > 0:25 and F < 0:05 selection criteria for stable single-mode operation.

So far, the value of phase shifts 2 and 4 have been assumed to be identical. By varying 2 and 4 from 0 to 180 , the contour maps of both the gain margin and the flatness of the 3PS DFB LD can be plotted as shown in Figs. 5.14 and 5.15, respectively [8]. In the analysis, the PSP is fixed at 0.3 and 3 is fixed at =9 (or 20 ). Contours shown are for
L > 0:25 and F < 0:05. As expected, both contours show symmetrical distributions along the line where 2 ¼ 4 . From Figs 5.14 and 5.15, it can be seen that most of the region that satisfies
L > 0:25 does not match with the region for F < 0:05. The only area that matches both selection criteria is found when 40 < 2 < 90 and 40 < 4 < 80 .

Figure 5.14 Relationship between the gain margin and the phase shift for a 3PS DFB laser diode. 3 ¼ 20 and ¼ 0:3 are assumed.

Figure 5.15 Relationship between the flatness and the phase shift for a 3PS DFB laser diode. 3 ¼ 20 and ¼ 0:3 are assumed.

136

5.4

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

THRESHOLD ANALYSIS OF THE DISTRIBUTED COUPLING COEFFICIENT (DCC) DFB LD

By incorporating more phase shifts along the DFB laser cavity, 3PS DFB LDs show an improved performance in maintaining the single-mode stability. With a flatter internal field distribution, the spatial hole burning effect is suppressed. The gain margin of the 3PS DFB laser, however, is reduced as compared with the QWS DFB laser, whilst oscillation at the Bragg wavelength cannot be maintained. To improve the lasing characteristics of the QWS, a novel structure having a non-uniform coupling coefficient has been proposed [9]. Basically, a corrugation having non-uniform depth is fabricated along the laser cavity. Since the coupling coefficient depends on the height of the corrugation, lasers employing this structural configuration are better known as distributed coupling coefficient (DCC) DFB LDs.

Figure 5.16 Schematic diagram of a single-phase-shifted distributed coupling coefficient (DCC) DFB LD.

Figure 5.16 shows the schematic diagram of a DCC DFB LD. The height of the corrugations fabricated near the centre of the cavity is different from those located near the laser facets. As a result, a laser cavity having a longitudinal variation of  is achieved. With a =2 phase shift fabricated at the centre of the cavity, oscillation at the Bragg wavelength is ensured. In the analysis, a constant corrugation period of  and hence a fixed Bragg wavelength
B is assumed. The DCC laser structure used is different from the continuouspitch-modulated (CPM) DFB laser [10] in which  is varying along the laser cavity. In a DCC DFB laser, it is important that the corrugations change smoothly from one height to another, and that there is no change in the corrugation phase along with the position of the corrugation change. To get the best performance out of the DCC DFB structure, it is necessary to investigate the structural impact on the threshold characteristics of DCC LDs.

THRESHOLD ANALYSIS OF THE DISTRIBUTED COUPLING COEFFICIENT DFB LD

137

In the following sections, the structural design and, in particular, effects of the coupling ratio and the position of the corrugation change will be discussed. Both the gain margin and the field uniformity reduce the complexity of the structural design. Optimised structures that satisfy the selection criteria will be presented.

5.4.1 Effects of the Coupling Ratio on the Threshold Characteristics In the structural design of DCC DFB lasers, both the coupling ratio ð1 =2 Þ and the position of the corrugation play a crucial role in the threshold characteristics. To determine the position of the corrugation change, a parameter known as the corrugation position ðCPÞ is defined such that CP ¼

2L1 L

ð5:6Þ

In Fig. 5.17, the variation of both the amplitude threshold gain (thL) and the gain margin ð
LÞ are shown with respect to the coupling coefficient 1 . In the analysis, 2 ¼ 2:0 mm1, 3 ¼ =2 and CP ¼ 0:5 are assumed. When 1 increases, both thL and
L show a monotonic decrease in value. The reduction in thL is obvious since a larger 1 implies a stronger optical feedback and consequently smaller amplitude threshold gain. At 1 ¼ 2:0 mm1 , the corrugation becomes uniform and the DCC laser is reduced to the uniform QWS DFB laser.

Figure 5.17 Variations of both the amplitude threshold gain and the gain margin with respect to the coupling coefficient 1 .

138

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

5.4.2 Effects of the Position of Corrugation Change At a fixed coupling ratio, effects due to the variation of the corrugation position ðCPÞ will be investigated in this section. In Fig. 5.18, the variation of the field uniformity ðFÞ as well as the gain margin are shown for different values of CP. In the analysis, a fixed value of 2 ¼ 2:0 mm1 and 3 ¼ =2 are assumed. Results obtained from various coupling ratios are compared. Other parameters used are listed in the inset of the figure. When CP increases,

Figure 5.18 Variations of the flatness and the gain margin with respect to the corrugation position for various DCC laser diode structures.

with the ratio 1 =2 having the value of 2, it shows a substantial increase in flatness. Structures showing such a high value of flatness are undesirable since they are vulnerable to spatial hole burning. Near to the CP value of 0.78, results from the gain margin reveal that all DCC structures show characteristics similar to that of the uniform QWS structure. It is believed that at this particular value of CP, the characteristics of the non-uniform corrugations become irrelevant and the effect of using DCC becomes less efficient. As far as single-mode stability is concerned, trade-offs exist in selecting the optimum position of corrugation change and the coupling ratio.

5.4.3 Optimisation of the DCC DFB Laser Structure Based on the threshold characteristic, an attempt will be made to optimise the DCC DFB laser structure. In order that results obtained can be compared with QWS and 3PS DFB LDs,

THRESHOLD ANALYSIS OF THE DISTRIBUTED COUPLING COEFFICIENT DFB LD

139

a parameter known as the averaged coupling coefficient avg is introduced into the DCC laser structure such that avg ¼ 1 ðCPÞ þ 2 ð1  CPÞ

ð5:7Þ

where 1 and 2 are the coupling coefficients inside the DCC laser cavity and CP is the position of the corrugation change. For a 500 mm long laser cavity, avg is assumed to be 4:0 mm1 so that the value avg L ¼ 2:0 can be maintained. For a fixed value of avg, other variables like 1 , 2 and CP are allowed to change. At a fixed value of 1 =2 ¼ r, the above equation becomes 1 ¼

r avg 1 þ ðr  1ÞCP

ð5:8Þ

where 1 can be determined. Figure 5.19 demonstrates the variation of the gain margin with the position factor CP. Results obtained from various coupling ratios 1 =2 are compared. For CP ¼ 0 and CP ¼ 1, it can be seen that a uniform QWS DFB laser is formed. For a DCC laser structure with a coupling ratio 1 =2 > 1, the gain margin reduces as CP increases from zero. On the other hand, results obtained from 1 =2 < 1 indicate a significant improvement in the gain margin. At 1 =2 ¼ 1=3, a normalised gain margin value of 1.69 is found at CP ¼ 0:46.

Figure 5.19

Variation of the gain margin versus CP for different coupling ratios.

Figure 5.20 Variation of the flatness versus CP for different coupling coefficients.

Figure 5.21

Variation of the gain margin versus CP for different values of phase shift 3 .

THRESHOLD ANALYSIS OF THE DCC þ 3PS DFB LASER STRUCTURE

141

Figure. 5.20 shows the dependence of the flatness on the variation of CP. Among various 1 =2 ratios used, those having 1 =2 < 1 show an improvement in the field uniformity. At low values of 1 =2 , however, the differences in F due to the variation of 1 =2 become less obvious. Such a phenomenon can be explained by the presence of the =2 phase shift. Due to the intense electric field associated with the phase shift, the change in the distribution of flatness becomes less prominent. From results obtained in Figs 5.19 and 5.20, a DCC DFB with 1 =2 < 1 seems desirable since a high gain margin and a relatively flat field distribution can be achieved. Due to the intense electric field at the centre of the DCC DFB LD, it is difficult to achieve a very low value of flatness (i.e. F < 0:05). On the other hand, the significant improvement in the gain margin should relax the constraints on flatness. Along the line 1 =2 ¼ 1=3, the optimised design is located at CP ¼ 0:46, where the gain margin and the flatness are found to be 1.69 and 0.17, respectively. Throughout the analysis, the phase shift 3 has been fixed at =2. By changing the value of the phase shift, the variation of the gain margin and the flatness are shown in Figs 5.21 and 5.22, respectively. In the analysis, 1 =2 ¼ 1=3 and avg ¼ 4:0 mm1 are assumed.

Figure 5.22

5.5

Variation of the flatness versus CP for different values of phase shift 3 .

THRESHOLD ANALYSIS OF THE DCC þ 3PS DFB LASER STRUCTURE

From the previous section, the single QWS DFB laser with a non-uniform coupling coefficient shows significant improvement in both the threshold gain margin and the flatness

142

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

when compared with the uniform  QWS DFB laser. With 1 =2 ¼ 1=3 and CP ¼ 0:46, oscillation at the Bragg wavelength is achieved when the lasing condition is reached. Despite the fact that there is just a slight improvement in the value of flatness, the large threshold gain margin strengthens the single-mode oscillation in DCC DFB LDs. With more phase shifts introduced along the laser cavity, the 3PS structure has been shown to have an improved field uniformity. In this section, the combined effect of both non-uniform  and 3PS on the threshold characteristics of DFB LDs will be investigated.

Figure 5.23 Schematic diagram of a distributed coupling coefficient DFB LD with three phase shifts (DCC þ 3PS).

A schematic diagram of such a combined DCC þ 3PS DFB laser structure is shown in Fig. 5.23. Bearing in mind that the use of the TMM in the threshold analysis requires all physical parameters to be uniform within each transfer matrix section, six transfer matrices have to be used in this new structure. The three phase shifts shown in the figure have been defined following the N-sectioned laser cavity model discussed in Chapter 4. In the analysis, the overall laser cavity length is assumed to be 500 mm and avg ¼ 4 mm1 . The phase shift 4 is always fixed at the centre of the cavity, whilst the positions of 3 and 5 are allowed to change such that their actual phase shift positions (PSPs) are defined as ¼ PSP ¼

2L L

ð5:9Þ

Similarly, the change of the corrugation position follows the definition used in the DCC þ QWS structure such that CP ¼

2L1 L

ð5:10Þ

To find a suitable value for both the position and the value of phase shift in the DCC þ 3PS laser structure, the optimised design obtained from the uniform  3PS DFB laser was used. From the hatched area shown in Fig. 5.13, all phase shifts are assumed to be =3 and PSP ¼ 0:5 will be used in the analysis of the DCC þ 3PS laser structure.

THRESHOLD ANALYSIS OF THE DCC þ 3PS DFB LASER STRUCTURE

Figure 5.24

143

Variation of the gain margin versus CP for different values of 1 =2 .

The variation of the normalised gain margin
L with respect to CP is shown in Fig. 5.24. In this figure, values of phase shifts are equal ð3 ¼ 4 ¼ 5 ¼ =3Þ and three different sets of results are calculated with 1 =2 ¼ 1=3:5, 1=3 and 1=2:5. In all three cases, the highest gain margin is found at around CP ¼ 0:40. As compared with the uniformly corrugated 3PS DFB laser (where CP ¼ 0 and 1), the improvement in
L corresponds to a positive effect due to the presence of the non-uniform corrugation. At around CP ¼ 0:4, the non-lasing side modes are suppressed to such an extent that an improved value of gain margin ð
LÞ results. There are other regions of CP for which its corresponding gain margin is not as good as the uniformly corrugated 3PS structure. At CP ¼ 0:5, where the place of corrugation change coincides with the position of phase shift, it is interesting to show how
L drops to a local minimum. Figure 5.25 shows the dependence of the flatness upon the corrugation position for different values of 1 =2 . As for Fig. 5.24, avg ¼ 4:0 mm1 and 3 ¼ 4 ¼ 5 are assumed. By changing the value of CP, the flatness of all three DCC þ 3PS LDs used falls within the selection criterion of F < 0:05. There is only a minor change in F when the coupling ratio 1 =2 changes. So far, phase shifts used in the DCC þ 3PS DFB LD have been fixed at =3 and ¼ 0:5 has been assumed. By changing the values of 3 and 5 as well as the PSP, the variation of the normalised gain margin of one of these configurations can be shown (Fig. 5.26). In this configuration, 3 ¼ 5 ¼ =6 and ¼ 0:5 are assumed [11]. For comparison purposes, results obtained from a DCC þ 3PS DFB LD with 3 ¼ 4 ¼ 5 ¼ =3 and ¼ 0:5 are also shown. By altering both the position and the values of 3 and 5 along the laser cavity, DFB lasers show an improvement in the threshold gain margin. A comprehensive optimisation of

144

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

Figure 5.25

Variation of the flatness versus CP for different values of 1 =2 .

the DCC þ 3PS structure is challenging since it involves the optimum design of five variables: namely the corrugation position; phase shift position; phase shifts; 1 =2 and 1 . On the other hand, it is shown here that the N-sectioned laser cavity model can be applied to the design of such a complicated structure. Of course, the use of the TMM has played a crucial role throughout the analysis.

Figure 5.26 Variation of the gain margin for two DCC þ 3PS DFB LDs. (a) 3 ¼ 5 ¼ =3, 4 ¼ =6 and ¼ 0:3; (b) 3 ¼ 4 ¼ 5 ¼ =3 and ¼ 0:5

145

SUMMARY

The DCC þ 3PS DFB laser structure with 3 ¼ 4 ¼ 5 ¼ =3, CP ¼ 0:39, 1 =2 ¼ 1=3 and ¼ 0:5 satisfies the threshold selection criteria on both
L > 0:25 and F < 0:05 for a 500 mm length cavity. This structure will be used in the next chapter for the evaluation of the above-threshold performance.

5.6

SUMMARY

In revealing the potential use of the TMM in the practical design of DFB LDs, the threshold analysis of various DFB laser structures including the 3PS, DCC and DCC þ 3PS have been carried out. In an attempt to minimise the effects of SHB and hence improve the maximum available single-mode output power, it is necessary for a stable SLM LD to show a high normalised gain margin ð
LÞ as well as a uniform field intensity (i.e. small value of F). Based on the lasing performance at threshold, selection criteria were set at
L > 0:25 and F < 0:05 for a 500 mm length laser cavity. With more phase shifts introduced along the laser cavity, a 3PS DFB LD was shown to have an improved field uniformity. By changing the corrugation height along the laser cavity, the DCC þ QWS DFB LD shows an improved threshold gain and field flatness. The combined effect of having 3PS and a non-uniform coupling coefficient was investigated in a novel DCC þ 3PS DFB laser structure. By changing the value of phase shifts, the coupling coefficient and their corresponding positions, the gain margin ð
LÞ and the uniformity of the field distribution ðFÞ of various DFB laser structures were evaluated. Based on the selection criteria
L > 0:25 and F < 0:05 at the lasing threshold condition, optimised structures in the 3PS, DCC þ QWS and DCC þ 3PS DFB lasers were presented. Table 5.1 summarises the results obtained from the threshold characteristics of various DFB laser diodes. For comparison purposes, Fig. 5.27 shows the field distribution of these DFB structures at threshold. A conventional single QWS DFB has been selected as a standard for comparison purposes. This structure is characterised by an intense electric field found at the centre of the cavity. Due to the effects of SHB, the single-mode oscillation deteriorates quickly as the

Table 5.1 Comparison of the threshold characteristics of various DFB LDs DFB LD

PSP

CP

QWS 3PS DCC þ QWS DCC þ 3PS

— 0.5 — 0.5

— — 0.46. 0.39

1 =2

thL

thL


L

F

I avg

— — 1=3 1=3

0.70 0.98 0.93 1.54

0.0 0.91 0.0 0.35

0.73 0.34 1.69 0.49

0.3006 0.0122 0.1678 0.0164

1.43 1.02 1.08 0.65

3PS structure assumed 2 ¼ 3 ¼ 4 ¼ =3. DCC þ 3PS structure assumed 3 ¼ 4 ¼ 5 ¼ =3. Laser structure: QWS ¼ quarterly wavelength shifted; 3PS ¼ three phase shift and DCC ¼ distributed coupling coefficient. CP ¼ corrugation position; PSP ¼ phase shift position; 1 =2 ¼ coupling ratio; thL ¼ normalised amplitude threshold gain; thL ¼ normalised detuning coefficient;
L ¼ threshold gain margin; F ¼ flatness and Iavg ¼ average intensity.

146

THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS

Figure 5.27 The internal field distributions of various DFB LDs as represented in Table 5.1.

biasing current increases. For the 3PS DFB LD, using three =3 phase shifts with a phase shift position factor of 0.5 satisfies the selection criteria of
L and F. For the DCC þ QWS DFB LD, a coupling ratio of 1 =2 ¼ 1=3 and a corrugation change at 0.46 appears to be promising. This structure is characterised by a large threshold gain margin. In using both a non-uniform coupling coefficient and a 3PS laser structure, a DCC þ 3PS DFB LD with CP ¼ 0:39 and PSP ¼ 0:5 satisfies the design criteria at threshold. Throughout the analysis, it has been shown that the N-sectioned laser cavity model derived using the TMM facilitates both at- and below-threshold (Chapter 3) analysis of DFB LDs. However, the TMM becomes inadequate in the above-threshold biasing regime when stimulated emission becomes dominant. For above-threshold analysis of, for example, the single-mode stability and the spectral linewidth, the carrier rate equation must be considered. In the next chapter, a new technique that combines the TMM with the carrier rate equation will be introduced and the above-threshold characteristics of the DFB laser structures summarised in Table 5.1 will be investigated.

5.7

REFERENCES

1. Soda, H., Kotaki, Y., Sudo, H., Ishikawa, H., Yamakoshi, S. and Imai, H., Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers, IEEE J. Quantum Electron., QE-23(6), 804–814, 1987.

REFERENCES

147

2. Ghafouri-Shiraz, H. and Lo, B., Structural dependence of three-phase-shift distributed feedback semiconductor laser diodes at threshold using the transfer-matrix method (TMM), Semi. Sci. and Technol., 8(5), 1126–1132, 1994. 3. Kimura, T. and Sugimura, A., Coupled phase-shift distributed-feedback lasers for narrow linewidth operation, IEEE J. Quantum Electron., QE-25(4), 678–683, 1989. 4. Kimura, T. and Sugimura, A., Narrow linewidth asymmetric coupled phase-shift DFB lasers, Trans. IEICE., E 79(1), 71–76, 1990. 5. Ogita, S., Kotaki, Y., Hatsuda, M., Kuwahara, Y. and Ishikawa, H., Long cavity multiple-phase shift distributed feedback laser diode for linewidth narrowing, J. Lightwave Technol., LT-8(10), 1596– 1603, 1990. 6. Hoffman, J. D., Numerical Methods for Engineers and Scientists. New York: McGraw-Hill, 1992. 7. Ketelsen, L. J. P., Hoshino, I. and Ackerman, D. A., Experimental and theoretical evaluation of the CW suppression of TE side modes in conventional 1.55 mm InP-InGaAsP distributed feedback lasers, IEEE J. Quantum Electron., QE-27(4), 965–975, 1991. 8. Ong, B. S., Ghafouri-Shiraz, H. and Lo, B. S. K., Design proposal of an asymmetric three phase shift distributed feedback laser diode, Microwave and Optic. Technol. Lett., 7(18), 827–831, 1994. 9. Kotaki, Y., Matsuda, M., Fujii, T. and Ishikawa, H., MQW-DFB lasers with nonuniform-depth
=4 shifted gratings, Proc. ECOC/ICOC 91, pp. 137–140, 1991. 10. Okai, M., Tsuchiya T., Uomi, K., Chinone, N. and Harada, T., Corrugation-pitch-modulated MQWDFB laser with narrow spectral linewidth (170 kHz), Photon. Tech. Lett., 2(8), 529–530, 1991. 11. Ong, B., Lo, B. and Ghafouri-Shiraz, H., A proposal for the design of a three-phase-shift (3PS) distributed feedback (DFB) laser diode (LD), Trans. IEICE, Japan National Convention Record, paper no. C-256, pp. 4–256, March 1994.

6 Above-Threshold Characteristics of DFB Laser Diodes: A TMM Approach 6.1

INTRODUCTION

The flexibility of the transfer matrix method allows one to evaluate the spectral behaviour of a corrugated optical filter/amplifier and the threshold characteristic of a laser source. To extend the analysis into the above-threshold biasing regime, the transfer matrix has to be modified so as to include the dominant stimulated emission. Based on a novel numerical technique, the above-threshold DFB laser model will be presented in this chapter. Using a modified transfer matrix, the lasing mode characteristics of DFB LDs will be determined. The new algorithm differs from many other numerical methods in that no first-order derivative of the transfer matrix equation is necessary. As a result, the same algorithm can be applied easily to other DFB laser structures with only minor modification. In section 6.2, the detail of the above-threshold laser model will be presented. Taking into account the carrier rate equation, the dominant stimulated emission will be considered in building the transfer matrix. The numerical algorithm behind the lasing model will be discussed in section 6.3. Using the newly developed laser model, numerical results obtained from various DFB lasers including QWS, 3PS and DCC structures will be shown in section 6.4. Longitudinally varying parameters such as the carrier concentration, photon density, refractive index and the internal field intensity distributions will be presented with respect to biasing current changes. Impacts due to the structural variation in particular will be discussed.

6.2

DETERMINATION OF THE ABOVE-THRESHOLD LASING MODE USING THE TMM

In above-threshold analysis, the lasing wavelength and the optical output power are important. For laser devices to be used in coherent communication systems, the single-mode stability and the spectral linewidth should also be considered. Provided that the longitudinal distributions of the carrier, photons and other parameters are known, one can include the Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

150

ABOVE-THRESHOLD CHARACTERISTICS OF DFB LASER DIODES

spatial hole burning effect as well as the non-linear gain [1] in the above-threshold analysis. From the threshold characteristic of a DFB LD, a quasi-uniform gain model has been proposed using the perturbation technique [2]. However, in the analysis, a uniform gain profile along the cavity and a linear peak gain model were assumed. Using the TMM [3], the uniform gain profile was later improved by introducing a longitudinal dependence of gain along the cavity and an approximated carrier density was obtained for each sub-section under a fixed biasing current. With the laser cavity represented by such a small number of sub-sections, impacts due to the localised SHB effect can only be shown in an approximate manner. For a more realistic laser model, effects of SHB and any other non-linear gain saturation have to be considered. In the last chapter, the flexibility of the TMM allowed us to evaluate a DFB laser design quickly, based on the threshold analysis. However, TMM fails to predict the above-threshold lasing characteristics after the lasing threshold condition is reached and stimulated photons become dominant. To take into account any change of injection current, it is necessary to include the carrier rate equation in the analysis. In this section, the relationship between the injection current (or carrier concentration) and the elements of the transfer matrix (mainly amplitude gain  and detuning factor ) will be presented. From the output electric field obtained from the overall transfer matrix, the optical output power will then be evaluated. To include the localised effect in the TMM, a larger number of transfer matrices have to be used so that the length represented by each transfer matrix becomes much smaller. From the Nsectioned DFB laser model, physical parameters such as the carrier concentration and photon concentration are assumed to be homogeneous within an arbitrary sub-section. As a result, information such as the localised carrier and photon concentrations are obtained from each transfer matrix. Consequently, longitudinal distributions of the lasing mode carrier density, photon density, refractive index and the internal field distribution are obtained. According to Chapter 4, the transfer matrix of an arbitrary section k as shown in Fig. 6.1 can be expressed as         ER ðzkþ1 Þ ER ðzk Þ ER ðzk Þ f f ¼ Fðzkþ1 j zk Þ ¼ 11 12 ð6:1Þ f21 f22 ES ðzkþ1 Þ ES ðzk Þ ES ðzk Þ

Figure 6.1 Schematic diagram showing a general section in a DFB LD cavity. k shows the phase shift between sections k and k  1.

DETERMINATION OF THE ABOVE-THRESHOLD LASING MODE USING THE TMM

151

where Fðzkþ1 j zk Þ is the transfer matrix of the corrugated section between z ¼ zk and zkþ1 whilst its elements fij ði; j ¼ 1; 2Þ are given as f11 ¼

ðE  2 E1 Þ ejb0 ðzkþ1 zk Þ e jk ð1  2 Þ

ð6:2aÞ

f12 ¼

ðE  E1 Þ ej ejb0 ðzkþ1 þzk Þ ejk ð1  2 Þ

ð6:2bÞ

f21 ¼

ðE  E1 Þ e j e jb0 ðzkþ1 þzk Þ e jk ð1  2 Þ

ð6:2cÞ

f22 ¼

ð2 E  E1 Þ e jb0 ðzkþ1 zk Þ ejk ð1  2 Þ

ð6:2dÞ

where  is the residue corrugation phase at z ¼ 0 and k is the phase discontinuity between section k and k  1. Other parameters used are defined as E ¼ egðzkþ1 zk Þ ; ¼

E1 ¼ egðzkþ1 zk Þ

j   j þ g

ð6:3aÞ ð6:3bÞ

For DFB lasers having a fixed cavity length, one must determine both the amplitude gain coefficient  and the detuning coefficient  of the section k in order that each matrix element fij ði; j ¼ 1; 2Þ as shown in eqn (6.2) can be determined. For first-order Bragg diffraction, it was shown in Chapter 2 that  and  can be expressed as:  g  loss 2 2png 2p p n ¼ ð

B Þ 

B


¼

ð6:4Þ ð6:5Þ

where  is the optical confinement factor, g is the material gain, loss includes the absorption in both the active and the cladding layer as well as any scattering loss. In eqn (6.5), n is the refractive index of section k and
B is the Bragg wavelength. To take into account any dispersion due to the difference between the actual wavelength and the Bragg wavelength [4], the group refractive index ng is included in eqn (6.5). In Chapter 2, it was shown that the material gain g of a bulk semiconductor device can be expressed as g ¼ A0 ðN  N0 Þ  A1 ½
 ð
0  A2 ðN  N0 ÞÞ2

ð6:6Þ

where a parabolic model is assumed. In this equation, A0 is the differential gain, N0 is the transparency carrier concentration and
0 is the wavelength of the peak gain at transparency gain (i.e. g ¼ 0). The variable A1 in eqn (6.6) determines the base width of the gain spectrum and A2 corresponds to any change associated with the shift of the peak wavelength. Using a first-order approximation for the refractive index n, we obtain n ¼ nini þ 

@n N @N

ð6:7Þ

152

ABOVE-THRESHOLD CHARACTERISTICS OF DFB LASER DIODES

In the above equation, nini is the effective refractive index at zero carrier injection,  is the optical confinement factor and @n=@N is the differential index. For a symmetrical double heterostructure laser having an active laser width of w and thickness d [5], nini is approximated as 

n2ini  n2act  X log10 1 þ n2act  n2clad =X ð6:8Þ where X¼


2B 2p2 d 2

ð6:9Þ

In eqn (6.8), a single transverse and lateral mode are assumed nact and nclad are the refractive indices of the active and the cladding layer, respectively. From eqns (6.6) and (6.7), it is clear that both g and n are related to the carrier concentration N. As mentioned in Chapter 2, the carrier concentration N and the stimulated photon density S are coupled together through the steady-state carrier rate equation ð@N=@t ¼ 0Þ which is shown here as I ¼ R þ Rst qV

ð6:10Þ

N þ BN 2 þ CN 3

vg gS Rst ¼ 1 þ "S

ð6:11aÞ

where R¼

ð6:11bÞ

In the above equations Rst is the stimulated emission rate per unit volume and R is the rate of other non-coherent carrier recombinations. Other parameters used are as follows: I is the injection current, q is the electronic charge and V is the volume of the active layer, is the linear recombination lifetime, B is the radiative spontaneous emission coefficient, C is the Auger recombination coefficient and vg ¼ c=ng is the group velocity. To include any non-linearity and saturation effects, a non-linear coefficient " has been introduced [6]. For strongly index-guided semiconductor structures like the buried heterostructure, the lasing mode is confined through the total internal reflection that occurs at the active and cladding layer interfaces. Both the active layer width w and thickness d are usually small compared with the diffusion length. As a result, the carrier density does not vary significantly along the transverse plane of the active layer dimensions and the carrier diffusion term in the carrier rate equation has been neglected [7]. In an index-coupled DFB laser cavity, the local photon density inside the cavity can be expressed [8] as SðzÞ 

i 2"0 nðzÞng
2 h c0 jER ðzÞj2 þjES ðzÞj2 hc

ð6:12Þ

where "0 ¼ 8:854  1012 F m1 is the free space electric constant. From the escaping photon density at the output facet, the output power is then determined as Pðzj Þ ¼

dw hc vg Sðzj Þ 


ð6:13Þ

FEATURES OF NUMERICAL PROCESSING

153

According to the general N-sectioned DFB laser cavity model, j ¼ 1 and j ¼ N þ 1 correspond to the power output at the left and right facets, respectively. In eqn (6.12), c0 is a  dimensionless coefficient that determines the total electric field EðzÞ as  EðzÞ ¼ c0 EðzÞ ¼ c0 ½ER ðzÞ þ ES ðzÞ

ð6:14Þ

where ER ðzÞ and ES ðzÞ are the normalised electric field components as shown in eqn (4.40). Using the forward transfer matrix, it is important that both travelling electric fields ER ðzÞ and ES ðzÞ are normalised at the left facet ðz ¼ z1 Þ as jER ðz1 Þj2 þ jES ðz1 Þj2 ¼ 1

ð6:15Þ

Of course, both ER ðz1 Þ and ES ðz1 Þ should satisfy the boundary condition at the left facet such that ER ðz1 Þ ¼ ^r1 ES ðz1 Þ

ð6:16Þ

From the threshold analysis, both the amplitude threshold gain th and detuning coefficient th are determined. With virtually negligible numbers of coherent photons at the laser threshold, the threshold carrier concentration Nth can be determined from eqns (6.4) and (6.6) such that Nth ¼ N0 þ ðloss þ 2th Þ= A0

ð6:17Þ

where peak gain is assumed at threshold with A1 ¼ A2 ¼ 0. Consequently, the refractive index at threshold can be found to be nth ¼ nini þ 

@n Nth @N

ð6:18Þ

By substituting  ¼ th in eqn (6.5) at the threshold condition, the threshold wavelength
th can be obtained  2p
B nth þ ng ð6:19Þ
th ¼ th
B þ 2png þ
B p= Consequently, the peak gain wavelength at zero gain transparency is found from eqn (6.6) to be
0 ¼
th þ A2 ðNth  N0 Þ

ð6:20Þ

In the next section, features of the numerical process that help to determine the abovethreshold characteristics will be discussed in a systematic way.

6.3

FEATURES OF NUMERICAL PROCESSING

To evaluate the longitudinal distribution of the carriers and the photons in the analysis, a large number of transfer matrices must be used. For a 500 mm long QWS DFB laser, at least 5000 transfer matrices have been adopted to evaluate the above-threshold characteristics. To

154

ABOVE-THRESHOLD CHARACTERISTICS OF DFB LASER DIODES

characterise the oscillation mode for such a non-uniform system, a numerical method such as the Newton–Raphson method will not be appropriate since it is almost impossible to find the required first-order derivative. The situation becomes worse when one realises that the oscillation characteristic depends on the laser structure. In the analysis, a novel numerical technique has been developed. Using this numerical technique, it is not necessary to find any first-order derivative. In addition, the algorithm has been designed such that with only minor changes, it can be implemented easily in the design of various DFB laser structures. At a fixed above-threshold current, initial guesses for the lasing wavelength
and the dimensionless coefficient c0 are chosen. By matching the boundary condition at the right facet, lasing characteristics such as the carrier density, photon density, refractive index distribution, optical output power and the lasing wavelength can be evaluated. Consequently, information such as the single-mode stability and the spectral linewidth can be determined. In Fig. 6.2, a flowchart helps to explain the numerical procedure. Features of the novel numerical technique are highlighted as follows [9]: 1.

For a DFB laser diode with a specific structural design (e.g. a QWS, 3PS or a DCC DFB LD), the oscillation condition at the lasing threshold is first determined. Numerical methods like the Newton–Raphson method are applied to determine the threshold characteristic. A reasonable number of roots near the Bragg wavelength are found on the complex plane. Each root ðth ; th Þ that represents an oscillation mode is sorted in rising order of th. The one showing the smallest th will become the lasing mode after the threshold condition is reached.

2.

Using eqns (6.17)–(6.20), Nth, nth,
th and
0 are evaluated from the threshold value of ðth ; th Þ. Since there are virtually no stimulated photons at the lasing threshold condition, the threshold current Ith is determined using eqn (6.10).

3.

The DFB laser cavity is then subdivided into a large number of sections each represented by a transfer matrix.

4.

An injection current that is normalised with respect to the threshold current is specified. To start the iteration, values of
and c0 are given as initial guesses such that a mathematical grid as shown in Fig. 6.3 is built. Each intersection point on the grid (25 points all together) represents a pair of (c0,
) that will be used in the iteration.

5.

Using the forward transfer matrix, the photon density at the inner left facet is first determined. With no information on the carrier concentration, the threshold refractive index nth is assumed. The carrier concentration N at the left facet is then found using eqn (6.10), and subsequently input components ER ðz1 Þ and ES ðz1 Þ are obtained according to eqns (6.15) and (6.16).

6.

At this stage, the photon density at the left facet can be found using eqn (6.12). The carrier concentration is then evaluated by solving the carrier rate equation that includes the multi-carrier recombination. Subsequently, both  and  of the first section and matrix elements fij ði; j ¼ 1; 2Þ of the first matrix are determined.

7.

Using the newly formed transfer matrix, the electric field at the output plane can be evaluated and hence the output photon density found. Both  and  of the following section are then found and a new transfer matrix is formed. The whole process is then

Figure 6.2

Flow chart showing the procedures in the numerical algorithm.

156

ABOVE-THRESHOLD CHARACTERISTICS OF DFB LASER DIODES

Figure 6.3

A 5  5 mathematical grid used in the above-threshold analysis.

repeated until the output plane of the transfer matrix has reached the right facet. The discrepancy with the boundary condition is evaluated and stored (min_err). 8. By repeating the same calculation for all other (c0,
) pairs obtained from the mathematical grid, the pair showing the smallest discrepancy will be selected. Depending on the position of the final point on the mathematical grid (it may be along the boundary, at the corner or near the centre), a new mathematical grid will be created. Possible quantisation error must be considered when forming the new mathematical grid. 9. Procedures (5) to (8) should be repeated until the boundary condition falls within a discrepancy of < 1014 or the iterative change of wavelength ð

Þ falls below 1017 m. The final pair (c0,
)final is then stored. 10. The above-threshold characteristic of the DFB laser is determined by passing the (c0,
)final pair once again through the transfer matrix chain. From the photon density obtained at both facets, the output optical power is obtained. From each transfer matrix, the lasing mode distribution of the carrier concentration N(z), photon density SðzÞ, refractive index nðzÞ, amplitude gain ðzÞ and detuning coefficient ðzÞ can be evaluated. L and L associated with the lasing mode are then obtained from 11. The average values of  the corresponding longitudinal distribution as PN L ¼  L ¼

j¼1

j

N

ð6:21Þ

PN

j¼1 j

N

ð6:22Þ

where N is the total number of transfer matrices used and j and j ( j ¼ 1 to N) are the

NUMERICAL RESULTS

157

amplitude gain and the detuning coefficients obtained from each transfer matrix, respectively. 12. The whole iteration procedure is then repeated for other biasing currents. Using the numerical process described, the above-threshold lasing mode characteristics of various DFB LDs can be obtained. In the analysis, localised effects such as spatial hole burning have been included. With minor modifications, the algorithm shown above can be implemented easily in finding the above-threshold characteristics of various DFB laser designs. In the next section, the lasing characteristics of various DFB laser structures including QWS, 3PS and DCC DFB LDs will be presented using this above-threshold model.

6.4

NUMERICAL RESULTS

The above-threshold model based on the TMM is applicable to various types of DFB laser structures. In this section, results obtained from QWS, 3PS and DCC DFB LDs are Table 6.1

Parameters used in modelling the DFB laser diode

Material parameters Spontaneous emission rate Bimolecular recombination coefficient Auger recombination coefficient Differential gain Gain curvature Differential peak wavelength Internal loss Refractive index at zero injection Carrier concentration at transparency Carrier concentration at threshold Differential index Group velocity at Bragg wavelength Non-linear gain coefficient Peak gain wavelength at transparency Lasing wavelength Lasing wavelength at threshold

Values ———————————

1 ¼ 2:5  1010 s1 B ¼ 1  1016 m3 s1 C ¼ 3  1041 m6 s1 A0 ¼ 2:7  1020 m2 A1 ¼ 1:5  1021 m3 A2 ¼ 2:7  1032 m3 loss ¼ 4  103 m1 n0 ¼ 3:41351524 N0 ¼ 1:5  1024 m3 Nth in m3 dn=dN ¼ 1:8  1026 m3 vg ¼ 3  108 =3:7m s1 " ¼ 1:5  1023 m3
0 ¼ 1:63 mm

th

Structural parameters Active layer width Active layer thickness Coupling coefficient Cavity length Optical confinement factor Grating period Grating phase at the left facet Bragg wavelength Threshold current Threshold current density Injection current

w ¼ 1:5 mm d ¼ 0:12 mm  ¼ 4  103 m1 L ¼ 500 mm  ¼ 0:35  ¼ 227:039 nm  in rad
B ¼ 2=n0 ¼ 1:55 mm Ith in A Jth in A m2 I in A

158

ABOVE-THRESHOLD CHARACTERISTICS OF DFB LASER DIODES

presented. Distributions of the spatially dependent parameters like the photon density and the carrier density will be shown. Table 6.1 summarises both the material and the structural parameters used in the analysis. These parameters are valid for bulk semiconductor lasing at around 1.55 mm. Unless otherwise stated, these parameters will be used throughout the analysis. Other structural parameters associated with each specific design (i.e. the plane of corrugation change, the phase shifts and their positions) will be listed accordingly.

6.4.1 Quarterly-wavelength-shifted (QWS) DFB LDs The QWS DFB LD with uniform coupling coefficient has been used for some time because of its ease of fabrication, and because Bragg oscillation can be achieved readily with a single p=2 phase shift [10]. From the threshold analysis, this DFB laser structure is characterised by a non-uniform field intensity which is vulnerable to the spatial hole burning effect. Experimental results [2] have demonstrated that the gain margin deteriorates quickly when the biasing current increases. For a strongly coupled device (i.e. L  2), the side mode on the shorter wavelength side (þ1 mode) becomes dominant. For a 300 mm length cavity, twomode operation at an output power of around 7.5 mW was observed at a biasing current of 2:25I th. The spatial hole burning effect alters the lasing characteristics of the QWS DFB LD by changing the refractive index along the cavity. Under a uniform current injection, the light intensity inside the laser structure increases with biasing current. For strongly coupled laser devices, most light concentrates at the centre of the cavity. The carrier density at the centre is reduced remarkably as a result of stimulated recombination. Such a depleted carrier concentration induces an escalation of nearby injected carriers and consequently a spatially varying refractive index results. Using the TMM-based model, the above-threshold characteristics of the QWS DFB are to be verified. In the analysis, a 500 mm long laser cavity with L ¼ 2 is assumed and a phase shift of p=2 is located at the centre of the cavity. In Fig. 6.4, the carrier concentration profile is shown with different injection currents. The depleted carrier concentration observed near the centre of the cavity arises from severe spatial hole burning. It is also shown that the dynamic range of the carrier concentration increases with biasing current. Figure 6.5 shows the spatial dependence of the photon density with biasing current changes. The photon distribution is fairly uniform when the biasing current is close to its threshold value. On the other hand, an overall increase in the photon density is observed with increasing biasing current. At the centre of the cavity, in particular, a peak value of the photon density is expected in such a strongly coupled device. An increase in the dynamic range of the photon density is also shown when the biasing current increases. The variation of the spatially distributed refractive index is shown in Fig. 6.6. When the biasing current increases, the longitudinal span of the refractive index also increases. As we will discuss in the next chapter, this phenomenon has a strong impact on the lasing mode characteristics and hence the single-mode stability of the QWS DFB LD. From Fig. 6.6, it can also be seen that the spatially distributed refractive index becomes saturated near the centre of the cavity at high biasing current. As the photon density increases with biasing current, the photon density at the centre of the cavity becomes so high that the non-linear gain coefficient becomes dominant.

Figure 6.4 currents.

Longitudinal distribution of carrier concentration in a QWS DFB LD for different biasing

Figure 6.5 currents.

Longitudinal distribution of photon concentration in a QWS DFB LD for different biasing

Figure 6.6 currents.

Longitudinal distribution of refractive index in a QWS DFB LD for different biasing

Figure 6.7 Longitudinal distribution of the normalized intensity in a QWS DFB LD for different biasing currents.

NUMERICAL RESULTS

161

Table 6.2 Structural parameters used in the analysis of the 3PS DFB LD Parameter Coupling coefficient Phase shifts Phase shift position

Value  ¼ 4 mm1 2 ¼ 3 ¼ 4 ¼ =3 PSP ¼ 0:5

Figure 6.7 shows the dependence of the internal field intensity distribution with the biasing current changes. As the optical power increases, it can be seen that the distribution profile becomes flattened.

6.4.2 Three-phase-shift (3PS) DFB LDs With more than a single phase shift introduced along the laser cavity, a 3PS DFB structure is characterised by a relatively more uniform field distribution. From the threshold analysis, it was shown than a reasonable value of gain margin ð
L > 0:25Þ as well as a relatively low value of flatness ðF < 0:05Þ can be achieved for a 500 mm length cavity. Using the TMMbased above-threshold model, the above-threshold characteristics of 3PS DFB LDs will now be presented. Table 6.2 lists the structural parameters used in the 3PS laser structure analysis. The variation of the carrier density distribution along the laser cavity is shown in Fig. 6.8 for various biasing currents. Compared with the QWS DFB structure, the carrier density profile shown appears to be more uniform. An increase in the biasing current shows little change in spatially distributed carrier distribution. Local minima can be seen at 125, 250 and 375 mm along the cavity, these correspond to the location of the three p=3 phase shifts. The spatial dependence of the photon density, refractive index and the internal field intensity of the 3PS DFB LD is shown in Figs 6.9, 6.10 and 6.11, respectively. As for those obtained from the QWS structures, the average photon density increases with the biasing current. However, it can be seen that the introduction of more phase shifts has flattened out the photon distribution. Rather than a single peak found at the centre of the cavity, local maxima can be seen in Fig. 6.9 along with the phase shift position. The uniform photon distribution also reduces the difference between the central photon density and the escaping photon density at the facet, in particular at high biasing current. Compared with the QWS laser structure, a more uniform distribution can be seen in the case of the 3PS DFB structure. As shown in Fig. 6.10, the refractive index at the phase shift position becomes saturated at high biasing currents. In Fig. 6.11, the internal field intensity shows little change with increasing biasing current. To summarise, the use of an optimised 3PS laser structure appears to be promising. With biasing current increases, the narrower span of the longitudinally distributed refractive index strengthens the single-mode stability of the 3PS structure and hence reduces the threat from the spatial hole burning effect. The details will be discussed in the next chapter.

Figure 6.8 currents.

Longitudinal distribution of the carrier density in a 3PS DFB LD under different biasing

Figure 6.9 Longitudinal distribution of the photon density in a 3PS DFB LD for different biasing currents.

Figure 6.10 Longitudinal distribution of the refractive index inside a 3PS DFB LD under different biasing currents.

Figure 6.11 Longitudinal distribution of the normalised intensity inside a 3PS DFB LD under different biasing currents.

164

ABOVE-THRESHOLD CHARACTERISTICS OF DFB LASER DIODES

Table 6.3 Structural parameters used in the analysis of the DCC + QWS DFB LD Parameter Average coupling coefficient Phase shift Corrugation position Coupling coefficient ratio

Value avg ¼ 4 mm1 3 ¼ =2 CP ¼ 0:46 1 =2 ¼ 1=3

6.4.3 Distributed Coupling Coefficient with Quarterly-Wavelength Shifted (DCC + QWS) DFB LDs From the threshold analysis presented in the previous chapter, it was shown that the introduction of a non-uniform coupling coefficient in the DCC þ QWS structure can improve both the gain margin and the field uniformity when compared with the QWS structure having uniform coupling coefficient. In this section, the above-threshold characteristics of DCC þ QWS laser structures, in particular those longitudinally varying parameters, will be shown. Table 6.3 summarises the structural parameters used. The spatial dependence of the carrier concentration is shown in Fig. 6.12 with respect to the biasing current. Compared with the QWS DFB LD with uniform , the DCC þ QWS

Figure 6.12 Longitudinal distribution of the carrier density in a DCC þ QWS DFB LD under different biasing currents.

NUMERICAL RESULTS

165

Figure 6.13 Longitudinal distribution of the photon density in a DCC þ QWS DFB LD under different biasing currents.

laser structure shows a similar distribution. It can be seen that the carrier density near 115 and 385 mm along the laser cavity is boosted, especially for high injection currents. Such a change in carrier concentration is believed to be caused by the sudden change of coupling coefficient. The distribution of the photon density, refractive index and the internal field intensity is shown in Figs 6.13, 6.14 and 6.15, respectively, against different biasing currents. As expected, all of these figures show a similar distribution to the DFB LD, except for the abrupt change seen near the plane of the corrugation change. In the photon density profile shown in Fig. 6.13, the use of a smaller coupling coefficient near the facet has reduced the dynamic span of the photon density along the laser cavity. Due to the effects of spatial hole burning, it is evident that the refractive index distribution shows a larger dynamic range with increasing biasing current. On the other hand, it is demonstrated in the threshold analysis that the DCC þ QWS laser structure is characterised by an improved threshold gain margin. As a result, the single-mode stability of this structure could be maintained. The above-threshold single-mode stability of this structure will be discussed further in the next chapter.

6.4.4 Distributed Coupling Coefficient with Three-Phase-Shift (DCC + 3PS) DFB LDs We have seen that both the carrier and photon density are flattened when multiple phase shifts are introduced along the corrugation. In an alternative way, the use of non-uniform

Figure 6.14 Longitudinal distribution of the refractive index in a DCC þ QWS DFB LD under different biasing currents.

Figure 6.15 Longitudinal distribution of the normalised intensity in a DCC þ QWS DFB LD under differing biasing currents.

167

NUMERICAL RESULTS

Table 6.4 Structural parameters used in the analysis of the DCC þ 3PS DFB LD Parameter Average coupling coefficient Phase shifts Corrugation position Phase shift position Coupling coefficient ratio

Value avg ¼ 4 mm1 2 ¼ 3 ¼ 4 ¼ =3 CP ¼ 0:39 PSP ¼ 0:5 1 =2 ¼ 1=3

coupling coefficients improves the value of the gain margin, whilst a smaller value of  near the facets reduces the dynamic change of the photon density. In this section, the above-threshold characteristics of the combined DCC with 3PS structure will be investigated. Based on the 3PS laser structure, a longitudinal variation of the coupling coefficient is introduced. Discontinuities associated with both the phase shift and the corrugation change enhance the spontaneous emission and hence a larger threshold current is expected. On the other hand, the potential and the capability of the abovethreshold model can be demonstrated from the design of such a complicated device. Comparatively, the effect of the non-uniform coupling coefficient is expected to be dominant since  is one of the major parameters associated with coupled wave equations. Table 6.4 lists the structural parameters used. Figures 6.16 and 6.17 show the variation of the spatially distributed carrier density and refractive index with biasing current changes, respectively. It can be seen from both figures

Figure 6.16 Longitudinal distribution of the carrier density in a DCC þ 3PS DFB LD under different biasing currents.

168

ABOVE-THRESHOLD CHARACTERISTICS OF DFB LASER DIODES

Figure 6.17 Longitudinal distribution of the refractive index in a DCC þ 3PS DFB LD under different biasing currents.

that the uniform distribution of curves deteriorates quickly when the biasing current increases from 2:5I th.

6.5

SUMMARY

In this chapter, the above-threshold characteristics of various DFB LDs were investigated using a newly developed model based on the transfer matrix method. To take into account any changes in the biasing current, the carrier rate equation was included. In the analysis, multi-carrier recombination and a parabolic gain model have been assumed. To include any gain saturation effects, a non-linear gain coefficient was introduced into the analysis. The algorithm used in the model was developed in such a way that, with minor modifications, it can be applied to various laser structures. The TMM-based above-threshold laser model was applied to several DFB laser structures including QWS, 3PS and DCC DFB LDs. The QWS DFB laser structure, which is characterised by its non-uniform field distribution, was shown to have a large dynamic change of spatially distributed refractive index. Along the carrier concentration profile, a dip was shown at the centre of the cavity where the largest stimulated photon density is found. The field distribution in the QWS DFB LD can be improved by introducing more phase shifts along the corrugation. In the analysis, a 3PS DFB LD with 2 ¼ 3 ¼ 4 ¼ p=3 and

REFERENCES

169

PSP ¼ 0:5 were used. As compared with the QWS structure, uniform distributions were observed in the carrier density, photon density and the refractive index profiles. With an improved threshold gain margin, the above-threshold characteristics of a QWS LD having non-uniform coupling coefficient were presented. As compared with the QWS structure, the introduction of the non-uniform coupling coefficient with 1 =2 ¼ 1=3 and CP ¼ 0:46 has induced an increase of localised carrier concentration near the corrugation change. A significant reduction in the photon density difference between the central peak and the escaping photon density near the facet is also found. Such a reduction improves the single-mode stability of the DCC þ QWS structure and this will be discussed in the next chapter. The above-threshold lasing mode characteristics of a DCC þ 3PS LD were also shown. As compared with the multiple-phase-shift laser structure, it can be seen that the effect of the longitudinally varying coupling coefficient becomes dominant when the biasing current increases. In the next chapter, the lasing mode characteristics obtained will be used to evaluate other characteristics such as the gain margin, spontaneous emission spectrum and the spectral linewidth.

6.6

REFERENCES

1. Pan, X., Olsen, H. and Tromborg, B., Influence of nonlinear gain on DFB laser linewidth, Electron. Lett., 26, 1074–1075, 1990. 2. Soda, H., Kotaki, Y., Sudo, H., Ishikawa, H., Yamakoshi, S. and Imai, H., Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers, IEEE J. Quantum Electron., QE-23(6), 804–814, 1987. 3. Orfanos, I., Sphicopoulos, T., Tsigopoulos, A. and Caroubalos, C., A tractable above-threshold model for the design of DFB and phase-shifted DFB lasers" IEEE J. Quantum Electron., QE-27(4), 946–957, 1991. 4. Westbrook, L. D., Measurement of dg/dN and dn/dN and their dependence on photon energy in
¼ 1:5 mm InGaAsP laser diodes, IEE Proc. Pt. J, 133(2), 135–143, 1985. 5. Chen, K. L. and Wang, S., An approximate expression for the effective refractive index in a symmetric DH laser, IEEE J. Quantum Electron., QE-19(9), 1354–1356, 1983. 6. Huang, J. and Casperson, L. W., Gain and saturation in semiconductor lasers, Optical Quantum Electron., QE-27, 369–390, 1993. 7. Agrawal, G. P. and Dutta, N. K., Long-Wavelength Semiconductor Lasers. Princeton, NJ: Van Nostrand, 1986. 8. Henry, C. H., Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers, J. Lightwave Technol., LT-4(3), 288–297, 1986. 9. Lo, B. S. K. and Ghafouri-Shiraz, H., A method to determine the above threshold characteristics of distributed feedback semiconductor laser diodes, IEEE J. Lightwave Technol., Vol. 13, No. 4, pp. 563–568, April 1995. 10. Whiteaway, J. E. A., Thompson, G. H. B., Collar, A. J. and Armistead, C. J., The design and assessment of a
=4 phase-shifted DFB laser structure, IEEE J. Quantum Electron., QE-25(6), 1261–1279, 1989.

7 Above-Threshold Analysis of Various DFB Laser Structures Using the TMM 7.1

INTRODUCTION

The above-threshold lasing characteristics of DFB semiconductor laser diodes were presented in the previous chapter using a modified transfer matrix. Instead of using an averaged carrier concentration, the inclusion of the actual carrier distribution allows phenomena such as the spatial hole burning effect and non-linear gain to be included. In the analysis, a parabolic gain model and high-order carrier recombination were assumed. Lasing mode characteristics such as the longitudinal distribution of carrier density, photon density, refractive index and the internal field intensity were shown for various laser structures. In this chapter, results obtained from the lasing mode characteristics will be used to determine the mode stability and noise characteristics of DFB LDs. For a coherent optical communication system, it is essential that the LD used oscillates at a stable single mode and that a narrow spectral linewidth is achieved. Using the information obtained for the lasing mode characteristics, a method derived from the above-threshold transfer matrix model will be introduced in section 7.2 which allows the gain margin to be evaluated. By introducing an imaginary wavelength into the transfer matrix equation, characteristics of other non-lasing side modes can be evaluated and hence the single-mode stability can be obtained. Numerical results obtained using this method will be presented in section 7.3. In section 7.4, an alternative method which allows the theoretical prediction of the abovethreshold spontaneous emission will be presented. Based on the Green’s function method, one can use transfer matrices to help determine the single-mode stability of a DFB laser structure by inspecting the spectral components of oscillating modes. The TMM also allows the noise characteristics of the DFB LD to be evaluated. In section 7.5, it will be shown that various contributions to the spectral linewidth can be determined using the information obtained from the above-threshold transfer matrices. In the analysis, the effective linewidth enhancement factor [1] has been used instead of the material-based linewidth enhancement [2]. Using a more realistic effective linewidth enhancement factor, impacts caused by structural changes can be investigated in a systematic way.

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

172

7.2

ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

SINGLE-MODE STABILITY IN DFB LDs

Using the TMM-based above-threshold laser cavity model as presented in the previous chapter, distributions of the carrier density, photon density, refractive index and the normalised internal field intensity were obtained for various DFB laser structures. From the emitting photon density at the facet, the output optical power can be evaluated. Figure 7.1 summarises results obtained for QWS, 3PS and DCC þ QWS LDs with biasing current as a parameter. Within the range of biasing current that we are interested in, no ‘kink’ is observed in any of the three cases. Compared with the QWS structure, it seems that the introduction of multiple phase shifts along the 3PS laser cavity has increased the overall cavity loss. The figure also shows that the 3PS laser structure having the largest amplitude threshold gain has a relatively larger value of threshold current. Among the three cases, both the 3PS and the DCC þ QWS structures appear to have relatively larger output power under the same normalised biasing current.

Figure 7.1

Optical output power of three different types of DFB LD.

Semiconductor lasers having stable single longitudinal outputs and narrow spectral linewidths are indispensable in coherent optical communication systems. With a built-in wavelength selective corrugation, a DFB laser diode has a single longitudinal output. Other oscillation modes failing to reach the threshold condition become the non-lasing side modes. As the biasing current increases, the spatial hole burning effect becomes significant and mode competition between the lasing mode and the most probable non-lasing side mode may occur. Mode competition has been observed for a QWS DFB laser [3–5], which resulted

SINGLE-MODE STABILITY IN DFB LD

173

in multiple mode oscillation as the biasing current increased. In this section, a numerical method that allows theoretical prediction of the above-threshold single-mode stability of DFB LDs will be presented. In the analysis, it is assumed that the detailed lasing mode characteristics have been obtained from the above-threshold transfer matrix model as discussed in the previous chapter. Single-mode stability implies the suppression of non-lasing side modes. There are two possible ways to demonstrate single-mode stability in DFB LDs. The first approach involves the evaluation of the normalised gain margin
L between the lasing mode and the probable non-lasing side modes. The single-mode stability is said to be threatened if the gain margin,
, drops below 5 cm1 for a 500 mm length laser cavity. An alternative method to check the stability of the device involves the measurement of the spectral characteristics. With the help of an optical spectrum analyser, the measured intensity difference between the lasing mode and the side modes will give single-mode stability. The second approach is often used to measure the single-mode stability of DFB LDs. In this section, we will concentrate on the first approach which leads to the evaluation of the above-threshold gain margin. From the numerical method discussed in the previous chapter, oscillation characteristics of the lasing mode were obtained at a fixed biasing current. By dividing the DFB laser into a large number of smaller sections, longitudinal distributions like the carrier and photon densities were determined. Since the laser cavity is now dominated by the lasing mode, the characteristics of other non-lasing side modes should be derived from the lasing mode. In order to evaluate the characteristics of other non-lasing side modes, the dominant lasing mode has to be suppressed in a mathematical way. In the analysis, an imaginary wavelength
i is introduced [6]. As a result, the complex wavelength
c of an unknown side mode becomes
c ¼
þ j
i

ð7:1Þ

where
i takes into account the mathematical gain the side mode may need to reach its threshold value and
becomes the actual wavelength of the side mode. By changing values of both
and
i , wavelengths of other non-lasing side modes can be evaluated. The numerical procedure involved in determining the characteristics of other non-lasing side modes is summarised as follows: 1. A numerical procedure similar to the one discussed in section 6.3 is adopted. To initialise the iteration process, a 5  5 mathematical grid which consists of points (
;
i ) is formed. Each of these (
;
i ) points will be used as an initial guess. Since the most probable non-lasing side mode is usually found near the lasing mode, it is advisable to start with those wavelengths which are close to that of the lasing mode. 2. Lasing mode characteristics like the longitudinal distribution of the carrier, photon density and refractive index are retrieved from the data files obtained earlier. Matrix elements of each transfer matrix are then determined. 3. From the boundary condition at the left facet, electric fields ER ðz1 Þ and ES ðz1 Þ at the left facet are found which serve as the input electric field to the transfer matrix chain. 4. By passing the electric field through the transfer matrix chain, the output electric field at the right laser facet is determined. The discrepancy with the right facet boundary condition is then evaluated and stored.

174

ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

5. Steps (2) to (4) are then repeated with other pairs of (
;
i ) obtained from the 5  5 mathematical grid. By comparing the discrepancy obtained from each of these (
;
i ) pairs, the one showing the minimum discrepancy is then selected. Depending on the position of the selected (
;
i ) on the mathematical grid, a new mathematical grid is formed ready for the next iteration. 6. Procedures shown above are then repeated until the discrepancy with the right facet boundary condition falls below 1014. The final
obtained becomes the non-lasing side mode and distributions of the amplitude gain ðzÞ and the detuning coefficient ðzÞ are stored. SM and SM associated with the side mode are evaluated from the 7. The average values of  corresponding longitudinal distribution as PN j¼1 j SM ¼ ð7:2Þ  N PN j¼1 j  ð7:3Þ SM ¼ N where j and j ð j ¼ 1 to NÞ are the amplitude gain value and the detuning value obtained from each transfer matrix, and N is the total number of transfer matrices. 8. The whole numerical procedure can be repeated for other non-lasing side modes. All SM obtained are then sorted in increasing order. The one showing the smallest value  becomes the most probable side mode. Characteristics of a new dominant lasing mode must be loaded every time a new injection current is used. From the result obtained, the gain margin between the lasing mode and the most probable non-lasing side mode can be evaluated as L   SM
 ¼ 

ð7:4Þ

To maintain a stable single-mode oscillation, one must ensure that
L > 0:25 for a 500 mm length laser cavity.

7.3

NUMERICAL RESULTS ON THE GAIN MARGIN OF DFB LDs

In this section, numerical results obtained for various DFB LDs, including the QWS, 3PS and the DCC þ QWS laser structures, will be presented. Figure  7.2 shows the characteristics  L  plane. In the analysis, an antiof the lasing mode (0) and side modes ( 1) in the L; reflection-coated QWS DFB structure with L ¼ 2 is assumed for a 500 mm length cavity. For each oscillating mode shown in Fig. 7.2, the cross and the black circle correspond to the oscillating mode at threshold and at 5Ith , respectively. When the biasing current increases from the threshold value, an increase in the lasing mode amplitude gain and a corresponding reduction of gain margin between the lasing mode and the þ1 side mode can be seen. Such a phenomenon is well known to be induced by the spatial hole burning effect [5].

NUMERICAL RESULTS ON THE GAIN MARGIN OF DFB LDS

175

 L) Figure 7.2 Lasing characteristics of the QWS DFB laser structure in the (L,  plane showing two values of normalised injection current.

Figure 7.3 shows the normalised amplitude gain change of the QWS DFB LD with respect to the biasing current. The amplitude gain distributions for the lasing mode and the non value of each mode varies in a lasing side modes are shown. It is obvious that the L different way. When the biasing current increases from threshold, it is clear that the amplitude gain of the þ1 mode reduces remarkably and approaches that of the lasing mode. By contrast, the amplitude gain of the 1 mode becomes larger and hence becomes less significant. In Fig. 7.4, for the same QWS structure, the variation of the normalised detuning  is shown. In such a strongly coupled LD, both the lasing mode and the þ1 side coefficient L  with increasing biasing current. mode shift towards the shorter wavelength side (negative L) Among the different modes shown, the shift of the lasing mode is stronger since it is found closer to the gain peak. With multiple phase shifts introduced along the corrugation, the characteristics of the 3PS structure are shown in Fig. 7.5. In the analysis, the 3PS DFB is assumed to be anti-reflection coated. Phase shifts 2 ¼ 3 ¼ 4 ¼ =3 and PSP ¼ 0:5 are assumed for the 500 mm long cavity. Compared with the QWS structure, the 3PS structure shows a smaller shift in mode characteristics. This may be clearer when the variation of both amplitude gain and the detuning coefficient are shown as a function of normalised injection current. From Fig. 7.6 where the amplitude gain change is shown, the injection current alters the oscillating mode in a different way. It can be observed that the gain margin between the lasing mode and the

 of the QWS DFB laser structure versus the normalised Figure 7.3 Average amplitude gain L injection current. Results for both the lasing mode and non-lasing side modes ( 1) are shown.

 of the QWS DFB laser structure versus the normalised Figure 7.4 Average detuning coefficient L injection current. Results for both the lasing mode and non-lasing side modes ( 1) are shown.

 LÞ  plane showing two Figure 7.5 Lasing characteristics of the 3PS DFB laser structure in the ðL; values of normalised injection current.

 of the 3PS DFB laser structure versus the normalised injection Figure 7.6 Average amplitude gain L current. Results for both the lasing mode and non-lasing side modes ( 1) are shown.

178

ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

 of the 3PS DFB laser structure versus the normalised Figure 7.7 Average detuning coefficient L injection current. Results for both the lasing mode and non-lasing side modes ( 1) are shown.

most probable side mode (þ1) shows little change. A similar situation can be seen in Fig. 7.7 where the variation of the detuning coefficient is demonstrated. The lasing mode shown has a milder shift with increasing biasing current. On combining results from both Figs. 7.6 and 7.7, it appears that the 3PS laser structure is not seriously affected by the spatial hole burning effect. No severe reduction of gain margin and a fairly mild shift in detuning coefficient are observed. Results obtained from a DCC þ QWS laser structure for the lasing characteristics, amplitude gain and the detuning coefficient are shown in Figs 7.8 to 7.10, respectively. A 500 mm length cavity is assumed. Other parameters used follow those adopted in the TMMbased laser model as discussed in the previous chapter. These parameters include 1 =2 ¼ 1=3, avg L ¼ 2 and CP ¼ 0:46. Due to effects of spatial hole burning, all figures appear to have a similar trend in the shape of their curves. On the other hand, the singlemode stability in the DCC þ QWS structure is improved due to the presence of the distributed coupling coefficient. As shown in Fig. 7.9, the amplitude gain difference between the lasing mode and that of the þ1 mode remains at a high value even at a high biasing current. Figure 7.11 shows the normalised gain margin (
L) between the lasing mode and the most probable side mode. Results obtained from QWS, 3PS and DCC þ QWS structures are shown. The gain margin of both the QWS and the DCC þ QWS structures reduces when the biasing current increases. From the above-threshold analysis, these structures are

 LÞ  plane Figure 7.8 Lasing characteristics of the DCC þ QWS DFB laser structure in the ðL; showing two values of normalised injection current.

 of the DCC þ QWS DFB laser structure versus the norFigure 7.9 Average amplitude gain L malised injection current. Results for both the lasing mode and non-lasing side modes ( 1) are shown.

 of the DCC þ QWS DFB laser structure versus the Figure 7.10 Average detuning coefficient L normalised injection current. Results for both the lasing mode and non-lasing side modes ( 1) are shown.

Figure 7.11 The variation of gain margin with respect to changes in the injection current for different DFB LD structures.

NUMERICAL RESULTS ON THE GAIN MARGIN OF DFB LDS

181

characterised by an intense electric field located at the centre of the cavity, and hence are affected by spatial hole burning. Nevertheless, the DCC þ QWS laser structure can maintain the gain margin at a sufficiently high level, even under large biasing conditions. By contrast, the gain margin of the 3PS structure shows little change over the range of biasing current. In all three cases, the most dramatic change of gain margin occurs when the biasing current is still close to that of the threshold value. At this biasing condition, the photon density (as shown in the previous chapter) is still fairly uniform and the non-linear gain effect is still far from mature. It is believed to be the dominant spatial hole burning effect which alters the characteristics of oscillating modes. When biasing current increases, the average photon density inside the laser cavity increases and so does its value found along the plane phase or corrugation discontinuities. Under such a high current injection, the non-linear gain effect becomes dominant. The change of lasing wavelength with respect to the injection current change is shown in Fig. 7.12. Results obtained from the QWS, 3PS and the DCC þ QWS structures are shown. Among them, the 3PS structure shows relatively minor changes (0.07 nm) with the biasing current. The introduction of multiple phase shifts along the corrugation has suppressed the spatial hole burning effect to such an extent that the injection current hardly changes the refractive index and hence a less gradual change in lasing wavelength is obtained. With such a stable output, it appears that the 3PS structure has much potential for use as an optical carrier. On the contrary, structures like the DCC þ QWS structure show a larger dynamic change of lasing wavelength. A single-mode continuous tuning of about 0.16 nm is achieved. It appears that this structure has a potential application in the WDM optical network.

Figure 7.12 The variation of the lasing wavelength with respect to the injection current for different DFB LD structures.

182

7.4

ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

ABOVE-THRESHOLD SPONTANEOUS EMISSION SPECTRUM

By measuring the mode intensity difference from the spectrum, single-mode stability can be determined. A minimum side mode suppression ratio of 25 dB is necessary for a stable single mode [7]. With the help of the method using Green’s function as discussed in Chapter 4, the above-threshold spontaneous emission spectrum can be evaluated using the transfer matrices. From the output of an individual transfer matrix, the contribution due to the distributed noise source is found. From eqn (4.42) in Chapter 4, the spontaneous emission power emitted for unity bandwidth ð
! ¼ 1Þ at the right laser facet of an N-sectioned mirrorless DFB laser cavity can be expressed in terms of the elements of the overall transfer matrix as PN ðzNþ1 Þ ¼

hc 1
jy22 ðzNþ1 j z1 Þj2

ð zNþ1

h i nsp g jy22 ðz j z1 Þj2 þ jy12 ðz j z1 Þj2 dz

ð7:5Þ

z1

In this equation, y22 ðzNþ1 j z1 Þ is a matrix element obtained from the overall transfer matrix YðzNþ1 j z1 Þ whilst y22 ðz j z1 Þ and y12 ðz j z1 Þ are elements of the matrix Yðz j z1 Þ at arbitrary z. In the above equation, nsp is the local population inversion factor which is usually approximated as [8] nsp ðzÞ ¼

NðzÞ NðzÞ  N0

ð7:6Þ

with N0 being the carrier concentration at zero gain transparency. In eqn (7.5), the material gain term g becomes longitudinally dependent at the above-threshold biasing regime. In cases of below-threshold and at-threshold biasing conditions, a uniform distribution is expected. Using a sufficiently large number of transfer matrices, a numerical method such as the trapezoidal rule can be applied to evaluate the above integral. As a result, the integral can be approximated as ð zNþ1 z1

N h i
L X nsp g jy22 ðz j z1 Þj2 þ jy12 ðz j z1 Þj2 dz  ½Qð jÞ þ Qð j þ 1Þ 2 j¼1

ð7:7Þ

where Qð jÞ ¼ nsp ðzj Þgðzj Þ½jy22 ðzj j z1 Þj2 þ jy12 ðzj j z1 Þj2 , N is the total number of transfer matrices used and
L ¼ L=N is the length of each transfer matrix represented. From the matrix multiplication, matrix elements y22 zj j z1 and y12 zj j z1 at an arbitrary matrix output plane of z ¼ zj can always be determined from those at z ¼ zj1 as 

y11 y21

  y12 f ¼ 11 y22 ðzj jz1 Þ f21

  f12 y 11 f22 ðzj jzj1 Þ y21

 y12 y22 ðzj1 jz1 Þ

ð7:8Þ

where fij ði; j ¼ 1; 2Þ are the matrix elements of the transfer matrix F. When the biasing current increases, it is shown for the QWS DFB laser structure that the gain margin reduces

ABOVE-THRESHOLD SPONTANEOUS EMISSION SPECTRUM

Figure 7.13

183

Normalised spontaneous emission spectra of a QWS DFB LD.

significantly and the non-lasing side mode becomes significant. Mode competition is also revealed from the above-threshold spontaneous spectrum. Figure 7.13 shows the spectral characteristics of the QWS laser structure with biasing current changes. In the analysis, a unity bandwidth is assumed. Along a fixed biasing current, distinct peaks can be seen along the spectrum which correspond to different oscillating modes. The lasing mode shown near 1546.85 nm becomes the lasing mode after the threshold condition is reached. When the biasing current increases, it can be seen that all peak wavelengths shift towards the shorter wavelength. The so-called ‘blue shift’ in wavelength follows the change of material gain with carrier concentration which has been demonstrated experimentally using a QWS DFB LD [9]. Apart from that, a reduction of the spectral amplitude difference is also shown between the lasing mode and the þ1 mode which is located at the shorter wavelength side. At a biasing current of 5Ith , the side mode suppression ratio (SMSR) is reduced to less than 25 dB. At such an SMSR value, the stability of the single-mode oscillation is weakened and the presence of the þ1 mode becomes significant in the case of a QWS DFB LD [10]. Figure 7.14 shows the spontaneous emission spectrum of a 3PS DFB LD. In the analysis, the 3PS laser structure used has phase shifts 2 ¼ 3 ¼ 4 ¼ =3 and PSP ¼ 0:5 is assumed.

Figure 7.14 Normalised spontaneous emission spectra of a 3PS DFB LD.

Figure 7.15

Normalised spontaneous emission spectra of a DCC þ QWS DFB LD.

SPECTRAL LINEWIDTH

185

Other parameters used are listed in the inset. As with the QWS laser structure, distinct peaks which correspond to different oscillating modes are observed along the spectrum. When the biasing current increases, the spectral amplitude of the dominant lasing mode found near 1546.6 nm shows no sign of reduction and remains at a high value near 106. Compared with the QWS structure, the 3PS laser structure shows no severe mode competition and an SMSR of at least 25 dB is maintained throughout the range of biasing current. The spectral characteristics of a DCC DFB LD with a single QWS are shown in Fig. 7.15 under various biasing conditions. At a fixed biasing current, the central peak shown corresponds to the dominant lasing mode. Due to the introduction of the distributed coupling coefficient, mode competition has been reduced significantly with an improved SMSR between the lasing mode and the most probable non-lasing side mode [11]. Compared with the QWS and the 3PS LD, the DCC þ QWS laser structure shown appears to have a more stable single-mode oscillation and an SMSR of at least 35 dB.

7.5

SPECTRAL LINEWIDTH

In coherent optical communication systems, it is important that the semiconductor LDs used have narrow spectral linewidths. The finite spectral width measured at the output of a semiconductor laser [12] is the result of the spontaneous emission [13] which alters both the intensity and phase components of the lasing mode. The mutual coupling between the intensity and the phase has been included by an intrinsic linewidth enhancement factor, H [13]. Using the self-heterodyne method [14], the spectral linewidths of DFB LDs were measured [15]. Without including any spatial hole burning effect, their formulae, which are based on the H , failed to predict the actual spectral linewidth when the biasing current increased. In order to have a more accurate linewidth prediction and hence a better understanding of the linewidth saturation and rebroadening effects [16–17], formulations based on an equivalent circuit theory were proposed [18–19]. Based on the scattering parameters commonly used in microwave engineering, the spectral linewidths of DFB LDs were determined. However, it is not straightforward to compare this approach with the carrier rate equation method widely adopted for semiconductor lasers. Besides, these analyses are only concerned with the above-threshold spectral linewidth and there is no formulation for any of the other characteristics. Another theory based on the open resonator has also been proposed [20]. Using the method of Green’s function, the spontaneous emission rate and hence the spectral linewidths of DFB LDs were determined [21–24]. In this analysis, the effective linewidth enhancement factor (eff ) was considered, but the analysis was limited to simple DFB laser structures. For complicated designs like the 3PS and DCC laser structures, it would be difficult to find the Green’s function because of the mathematical complexity involved. In this section, results obtained from the TMM will be applied to evaluate noise characteristics of the DFB LD. The linewidth formulae we have adopted are those obtained by Tromborg et al. [21,25]. The spectral linewidth (
) for a single frequency semiconductor laser is given as
 ¼
sp þ
NN þ
NS

ð7:9Þ

186

ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

where
sp is the contribution due to the spontaneous emission,
NN is the contribution due to the fluctuation of local carrier density and
NS is due to the cross correlation term between the fluctuation of the photon number and the carrier density. In the analysis, a nearly-single-mode laser source is assumed and effects due to the presence of other non-lasing side modes [26–27] have been ignored. For spontaneous emission, it is clear that [20]
sp ¼

Rsp ð1 þ 2eff Þ 4Pnum

ð7:10Þ

where Rsp is the spontaneous emission factor, eff is the effective linewidth enhancement factor and Pnum is the total number of photons found inside the DFB laser cavity. According to Henry [20], the spontaneous emission rate Rsp of an open resonator like the DFB LD can be expressed in terms of the field intensity E2 ðzÞ as 4!2 Rsp ¼ 3 c

Ð zNþ1 z1

    Ðz nsp gnE2 ðzÞdz z1Nþ1 ng nE2 ðzÞdz j@W=@ !j2

ð7:11Þ

where nsp is the population inversion factor, g is the material gain, n is the refractive index, ng is the group refractive index and qW=q! is the derivative of the Wronskian term. According to eqn (4.50), @W=@! can be expressed in terms of the transfer matrix such that   qW qy22 qy21 qy12 qy11 þ ^r1  ^r2  ^r1^r2 ¼ 2jb0 ER ðzNþ1 j zÞ q! q! q! q! q!

ð7:12Þ

Due to the fact that matrix YðzNþ1 j z1 Þ ¼ FðNÞ FðN1Þ FðN2Þ . . . Fð2Þ Fð1Þ where Fð jÞ ð j ¼ 1 to NÞ is the transfer matrix of each smaller sub-section, one can define qYðzNþ1 j z1 Þ=q! as  qYðzNþ1 j z1 Þ qy11 =q! ¼ qy21 =q! q!

qy12 =q! qy22 =q!

 ð7:13Þ

where qYðzNþ1 j z1 Þ qFðNÞ ðN1Þ ðN2Þ F ¼ F . . . Fð2Þ Fð1Þ þ þ q! q! qFð1Þ FðNÞ FðN1Þ FðN2Þ . . . Fð2Þ q!

ð7:14Þ

The individual transfer matrix qFðiÞ =q!ði ¼ 1 to NÞ is then obtained as qFðiÞ ¼ q!

"

ðiÞ

ðiÞ

qf11 =q!

qf12 =q!

qf21 =q!

qf22 =q!

ðiÞ

ðiÞ

# ð7:15Þ

SPECTRAL LINEWIDTH

187

where ðiÞ jejb0 ð
zÞ ejðiÞ Kð1Þ qf11 ¼  2 q! vg 1  2ðiÞ gðiÞ

ð7:16aÞ

ðiÞ

jejb0 ð2zðiÞ þ
zÞ ejðiÞ ej Kð2Þ qf12 ¼  2 q! vg 1  2ðiÞ gðiÞ

ð7:16bÞ

ðiÞ

je jb0 ð2zðiÞ þ
zÞ ejðiÞ ej Kð2Þ qf21 ¼  2 q! vg 1  2ðiÞ gðiÞ

ð7:16cÞ

ðiÞ

je jb0 ð
zÞ ejðiÞ Kð3Þ qf22 ¼  2 q! vg 1  2ðiÞ gðiÞ

ð7:16dÞ

with    1 1 Þ ðiÞ  jðiÞ ð
zÞ  22ðiÞ ðEðiÞ  EðiÞ Þ Kð1Þ ¼ 1  2ðiÞ ðEðiÞ þ 2ðiÞ EðiÞ    1 1 Þ ðiÞ  jðiÞ ð
zÞ  ðiÞ ð1 þ 2ðiÞ ÞðEðiÞ  EðiÞ Þ Kð2Þ ¼ ðiÞ 1  2ðiÞ ðEðiÞ þ EðiÞ    1 1 þ 2ðiÞ EðiÞ Þ ðiÞ  jðiÞ ð
zÞ  22ðiÞ ðEðiÞ  EðiÞ Þ Kð3Þ ¼ 1  2ðiÞ ðEðiÞ   ðiÞ ¼ jðiÞ = ðiÞ  jðiÞ þ gðiÞ  2 g2ðiÞ ¼ ðiÞ  jðiÞ þ 2ðiÞ

ð7:17aÞ ð7:17bÞ ð7:17cÞ ð7:17dÞ ð7:17eÞ

EðiÞ ¼ e

jgðiÞ


ð7:17fÞ

1 EðiÞ

jgðiÞ
z

ð7:17gÞ

¼e

In the above equations,
z is the length of each transfer matrix represented. For a mirrorless DFB laser cavity with ^r1 ¼ ^r2 ¼ 0, the spontaneous emission rate Rsp in eqn (7.11) is simplified to become 42 nsp Rsp ¼ vg
2

Ð zNþ1 z1

    Ðz gnE2 ðzÞdz z1Nþ1 nE2 ðzÞdz jy12 @y22 =@!j2

ð7:18Þ

where vg is the group velocity. The denominator jy12 qy22 =q!j2 is determined using the transfer matrices following eqns (7.13) to (7.17). Other integrals shown in the numerator can be found numerically from discrete sets of nðzi Þ, gðzi Þ and E2 ðzi Þði ¼ 1 to N þ 1Þ are obtained from the above-threshold lasing cavity model as discussed in the previous chapter. The effective linewidth enhancement factor eff used in the spontaneous emission linewidth calculation is different from the intrinsic material linewidth enhancement factor

188

ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

[13,28]. By considering the effects due to structural change [29] as well as the photon and carrier distributions [30–32], the effective linewidth enhancement factor is given as [25] eff ¼

IMf X g RefX g

ð7:19Þ

where X¼

 ð zNþ1  qRst  CS dz S CN N qS z1

ð7:20Þ

In the above equations, N ¼ qðR þ Rst Þ=qN is the carrier recombination lifetime that includes spontaneous emission, stimulated emission and other non-radiative recombination; S is the photon distribution of the lasing mode. The parameter qRst =qS is the rate of stimulated emission (sec1) and is defined as: vg g qRst ¼ ð1 þ "S  Sg"Þ qS ð1 þ "SÞ2

ð7:21Þ

The weighted functions CN and CS used in eqn (7.20) are defined as [21] vg E2 ðzÞ CN ðzÞ ¼ Ð zNþ1 2 E dz z1

 1þ

 jH A0  2A1 A2 ½

0 þ A2 ðNðzÞ  N0 Þ 1 þ "SðzÞ 2ð1 þ "SðzÞÞ

vg E2 ðzÞ jgðzÞ" CS ðzÞ ¼ Ð zNþ1 2 E dz 2ð1 þ "SðzÞÞ2 z1

ð7:22aÞ ð7:22bÞ

where the intrinsic linewidth enhancement factor ðH Þ can be expressed as H ¼ 

4 qn=qN
B A0

ð7:23Þ

From the set of discrete values Eðzi Þði ¼ 1 to N þ 1Þ obtained from the TMM, the integral shown in both eqns (7.22a) and (7.22b) is obtained and hence the value of eff can be evaluated. In a similar way, the total number of photons inside the laser cavity can be found as Pnum

wd ¼ 

ð zNþ1 S dz

ð7:24Þ

z1

where the integral is found numerically from the discrete set of Sðzi Þði ¼ 1 to N þ 1Þ. There are other components
NN and
NS which contribute to the spectral linewidth as well. Suppose one defines the local frequency tuning efficiency Ktn as Ktn ðzÞ ¼ N ðzÞfImðCN ðzÞÞ  eff ReðCN ðzÞÞg

ð7:25Þ

SPECTRAL LINEWIDTH

189

With no pump noise suppression,
NN and
NS can be written as ð 1 zNþ1  nsp vg gS þ R þ Rst Ktn2 dz wd z1 ð Rsp eff zNþ1 ¼ SKtn dz Pnum z1


NN ¼

ð7:26Þ


NS

ð7:27Þ

where the integrals shown can be determined numerically using the discrete information provided by the TMM. Based on the lasing mode characteristics obtained from the transfer matrix method for a DFB LD, sets of discrete values obtained on carrier distribution, photon distribution and refractive index distribution are applied in evaluating the spectral linewidth. The use of a transfer matrix enables spatially dependent factors such as the spatial hole burning effect and other longitudinally dependent parameters to be included in the analysis.

7.5.1 Numerical Results on Spectral Linewidth For a QWS laser structure, the variation of the spectral linewidth with respect to the biasing current is shown in Fig. 7.16. Changes of various components including
sp ,
NN and
NS are shown. In the analysis, a 500 mm long cavity and L ¼ 2:0 are assumed. Other

Figure 7.16 The variation of spectral linewidth with respect to injection current for a QWS DFB LD.

190

ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

Figure 7.17

The variation of spectral linewidth with respect to injection current for a 3PS DFB LD.

parameters used are listed in the inset of the figure. When the biasing current is still close to the threshold value, the dominant spontaneous emission remains the major contributor. When the biasing current increases, the value of
sp decreases dramatically due to the dominant stimulated emission and the contribution of
NN becomes significant. Comparatively, the magnitude of
NS remains at a small value and can be neglected throughout the range of biasing currents. By introducing multiple phase shifts along the corrugation of the laser cavity, a 3PS DFB LD is characterised by a relatively uniform field distribution and a reasonably stable singlemode oscillation. Figure 7.17 demonstrates the variation of the spectral linewidth in a 3PS DFB LD. All structural parameters used in the analysis are identical to those for the QWS DFB laser except the phase shifts, which are assumed to be =3, and PSP ¼ 0:5. Compared to the QWS laser structure, the 3PS DFB LD shows a broader spectral linewidth under the same biasing condition. It was revealed in the threshold analysis that the 3PS DFB LD has a relatively larger amplitude threshold gain. As a result, a larger threshold current and hence a larger number of injected carriers are required before reaching the threshold condition. In other words, a larger spontaneous emission rate and, consequently, a larger
sp is expected. With increasing biasing current, the contribution of
sp reduces and
NN becomes influential. By adopting a longitudinally dependent coupling coefficient , the gain margin and the field uniformity of the DCC þ QWS DFB LD are improved. Continuous tuning as far as 0.16 nm can be achieved with fairly stable single-mode oscillations. The variation of the spectral linewidth of such a DCC þ QWS structure is shown in Fig. 7.18. In the analysis, we

SUMMARY

Figure 7.18 DFB LD.

191

The variation of spectral linewidth with respect to injection current for a DCC þ QWS

have used the structural parameters presented in the previous chapter with 1 =2 ¼ 1=3. A cavity length of 500 mm is assumed with avg L ¼ 2. Compared with other structures, the introduction of the longitudinally distributed coupling coefficient is characterised by an overall increase in
sp . The change of the coupling coefficient along the corrugation results in an increase in the amplitude threshold gain and hence the spontaneous emission rate. Compared with the QWS structure, the influence of
NN becomes significant in the case of the DCC structure. When the biasing current increases, it is shown in the carrier density distribution that the carrier density increases near the plane of corrugation change. As a result, a stronger local carrier fluctuation is expected in this structure. With a further increase in biasing current, the saturation of
NN may be a result of the non-linear gain effect. It is useful to mention that the validity of the TMM technique used to evaluate the spectral linewidth has been confirmed with experimental data reported for a single /2 DFB laser structure [32].

7.6

SUMMARY

In this chapter, the TMM-based above-threshold model has been applied to evaluate the performance of various DFB LDs, which include the QWS, 3PS and the DCC þ QWS structures. From the lasing mode distributions of the carrier density, photon density, refractive index and the field intensity, characteristics like the single-mode stability, the

192

ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

Table 7.1 Summary of results obtained from three DFB laser structures at a biasing current of 4Ith QWS Ith (mA) Power (mW)
L SMSR (dB) Tuning range (nm)
vtotal (MHz)
vsp (MHz)
vNN (MHz)

19.79 10.79 0.1863 22 0.14 27.04 12.51 13.19

3PS 21.82 13.13 0.2452 28 0.07 44.57 30.78 12.71

DCC þ QWS 21.41 13.10 0.8129 45 0.16 64.28 26.90 33.55

spontaneous emission spectrum and the spectral linewidth have been investigated. Throughout the analysis, all laser structures adopted are assumed to have laser cavity length 500 mm with L or avg L ¼ 2. Table 7.1 summarises results obtained for all three structures at a fixed biasing current of 4Ith . At a fixed biasing current, it is shown that the QWS structure showing the smallest threshold gain has the smallest spectral linewidth. On the other hand, this structure has a very poor single-mode stability and the þ1 non-lasing side mode becomes influential when the biasing current increases. From Table 7.1, the 3PS structure is shown to have one of the smallest changes in lasing wavelength. With the introduction of multiple phase shifts along the corrugation, the internal field distribution becomes more uniform and hence a stable single-mode oscillation results. Results obtained from the DCC þ QWS structure show the largest gain margin. The introduction of the distributed coupling coefficient has improved the single-mode stability such that the SMSR remains at a high value. A single-mode continuous tuning as wide as 0.16 nm is achieved using the DCC þ QWS laser structure. In this chapter, the complexity in design of the DFB laser is apparent and it may well be that different designs are required for various applications. On the other hand, the TMM has proven to be a useful tool in handling such a problem.

7.7

REFERENCES

1. Pan, X., Olsen, H. and Tromborg, B., Spectral linewidth of DFB lasers including the effect of spatial hole burning and nonuniform current injection, IEEE Photon Technol. Lett., 2(5), 312–315, 1990. 2. Henry, C. H., Theory of the linewidth of semiconductor lasers, IEEE J. Quantum Electron., QE-18(2), 259–264, 1982. 3. Soda, H., Kotaki, Y., Sudo, H., Ishikawa, H., Yamakoshi, S. and Imai, H., Stability in single longitudinal mode operation in GaInAsP/InP phase-adjusted DFB lasers, IEEE J. Quantum Electron., QE-23(6), 804–814, 1987. 4. Ketelsen, L. J. P., Hoshino, I. and Ackerman, D. A., The role of axially nonuniform carrier density in altering the TE–TE gain margin in InGaAsP–InP DFB lasers, IEEE J. Quantum Electron., QE-27(4), 957–964, 1991.

REFERENCES

193

5. Ketelsen, L. J. P., Hoshino, I. and Ackerman, D. A., Experimental and theoretical evaluation of the CW suppression of TE side modes in conventional 1.55 mm InP–InGaAsP distributed feedback lasers, IEEE J. Quantum Electron., QE-27(4), 965–975, 1991. 6. Lo, B.S.K. and Ghafouri-Shiraz, H., A method to determine the above-threshold characteristics of distributed feedback semiconductor laser diodes, IEEE J. Lightwave Technol., Vol. 13, No. 4, pp. 563–568, April 1995. 7. Agrawal, G. P. and Dutta, N. K., Long-Wavelength Semiconductor Lasers. Princeton, NJ: Van Nostrand, 1986. 8. Bissessur, H., Effects of hole burning, carrier-induced losses and the carrier-dependent differential gain on the static characteristics of DFB lasers, J. Lightwave Technol., LT-10(11), 1617–1630, 1992. 9. Ogita, S., Yano, M., Ishikawa, H. and Imai, H., Linewidth reduction in a DFB laser by detuning effect, Electron. Lett., 23, 393–394, 1987. 10. Kru¨ger, U. and Petermann, K., The semiconductor laser linewidth due to the presence of side modes, IEEE J. Quantum Electron., QE-24(12), 2355–2358, 1988. 11. Lo, B.S.K. and Ghafouri-Shiraz, H., Spectral characteristics of distributed feedback laser diodes with distributed coupling coefficient, IEEE J. Lightwave Technol., 13(2), 200–212, 1995. 12. Fleming, M. W. and Mooradian, A., Fundamental line broadening of single-mode (GaAs)As diode lasers, Appl. Phys. Lett., 38(7), 511–513, 1981. 13. Henry, C. H., Theory of the linewidth of semiconductor lasers, IEEE J. Quantum Electron., QE-18(2), 259–264, 1982. 14. Okoshi, T. and Kikuchi, K., Coherent Optical Fiber Communication. Tokyo, Japan: KTK Scientific and Kluwer Academic Publishers, 1988. 15. Kojima, K., Kyuma, K. and Nakayama, T. Analysis of spectral linewidth of distributed feedback laser diodes, J. Lightwave Technol., LT-3(5), 1048–1055, 1985. 16. Amann, M. C. and Schimpe, R., Excess linewidth broadening in wavelength-tunable laser diodes, Electron. Lett., 26(5), 279–280, 1990. 17. Morthier, G., David, K. and Baets, R. Linewidth rebroadening in DFB lasers due to bias dependent dispersion of feedback, Electron. Lett. 27(4), 375–377, 1991. 18. Bjo¨rk, G. and Nilsson, O., A tool to calculate the linewidth of complicated semiconductor lasers, IEEE J. Quantum Electron., QE-23(8), 1303–1313, 1987. 19. Chu, C. Y. J. and Ghafouri-Shiraz, H., Equivalent circuit theory of spontaneous emission power in semiconductor laser optical amplifiers, J. Lightwave Technol., LT-12(2), 760–767, 1994. 20. Henry, C. H., Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers, J. Lightwave Technol., LT-4(3), 288–297, 1986. 21. Pan, X., Olsen, H. and Tromborg, B., Spectral linewidth of DFB lasers including the effect of spatial hole burning and nonuniform current injection, IEEE Photon Technol. Lett., 2(5), 312–315, 1990. 22. Duan, G. H., Gallion, P. and Debarge, G., Analysis of the phase-amplitude coupling factor and spectral linewidth of distributed feedback and composite-cavity semiconductor lasers, IEEE J. Quantum Electron., QE-26(1), 32–44, 1990. 23. Duan, G. H., Gallion, P. and Debarge, G., Analysis of spontaneous emission rate of distributed feedback semiconductor lasers, Electron. Lett., 25(5), 342–343, 1989. 24. Kojima, K. and Kyuma, K., Analysis of the linewidth of distributed feedback laser diodes using Green’s function method, Japan J. Appl. Phys., 27, L1721–1723, 1988. 25. Tromborg, B., Olsen, H. and Pan, X., Theory of linewidth for multielectrode laser diode with spatially distributed noise sources, IEEE J. Quantum Electron., QE-27(2), 178–192, 1991. 26. Miller, S. E., The influence of power level on injection laser linewidth and intensity fluctuations including side-mode contributions, IEEE J. Quantum Electron., QE-24(9), 1837–1876, 1988. 27. Kru¨ger, U. and Petermann, K., The semiconductor laser linewidth due to the presence of side modes, IEEE J. Quantum Electron., QE-24(12), 2355–2358, 1988.

194

ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

28. Osinski, M. and Buus, J., Linewidth broadening factor in semiconductor lasers–an overview, IEEE J. Quantum Electron., QE-23(1), 9–28, 1987. 29. Furuya, K., Dependence of linewidth enhancement factor  on waveguide structure in semiconductor lasers, Electron. Lett., 21, 200–201, 1985. 30. Vahala, K., Chiu, L. C., Margalit, S. and Yariv, A., On the linewidth enhancement factor  in semiconductor injection lasers, Appl. Phys. Lett., 42, 631–633, 1983. 31. Wang, J., Schunk, N. and Petermann, K., Linewidth enhancement factor for DFB lasers due to longitudinal field dependence in the laser cavity, Electron. Lett., 23, 715–717, 1987. 32. Whiteaway, J. E. A., Thompson, G. H. B., Collar, A. J. and Armistead, C. J., The design and assessment of a
/4 phase-shifted DFB laser structure, IEEE J. Quantum Electron., QE-25(6), 1261–1279, 1989.

8 Circuit and Transmission-Line Laser Modelling (TLLM) Techniques 8.1

INTRODUCTION

Although today microwave and optical engineering appear to be separate disciplines, there has been a tradition of interchange of ideas between them. In fact, many traditional microwave concepts have been adapted to yield optical counterparts. The laser, as an optical device that plays a key role in optoelectronics and fibre-optic communications, grew from the work of its microwave predecessor, the maser (microwave amplification by stimulated emission of radiation) [1]. The operating principle behind the laser is very similar to that of the microwave oscillator. In a semiconductor laser, the required feedback may either be provided by the cleaved facets of Fabry–Perot lasers or by a periodic grating in distributed feedback lasers. The optical technique of injection locking of lasers by external light [2] is an idea borrowed from the phenomenon of injection locking of microwave oscillators by an external electronic signal [3]. The close relationship between optical and microwave principles suggests that it may be advantageous to apply microwave circuit techniques in modelling of semiconductor lasers. Engineers work best when using tools they are familiar with. In particular, electrical and electronic engineers are familiar with well-established electrical circuit models as tools to aid themselves in the understanding and prediction of behaviour of electrical machines or electronic devices. Since the early days of radio frequency (RF) and microwave engineering, microwave circuit theory has allowed us to explore fundamental properties of electromagnetic waves by giving us an intuitive understanding of them without the need to invoke detailed and rigorous electromagnetic field theories [4–5]. In the same spirit, microwave circuit formulation of the semiconductor laser diode enhances our understanding of the device, which is otherwise obscured by hard-to-visualise mathematical formulations. Complex mathematical models are too sophisticated to be desirable for engineers, especially those who are not specialists in the field of laser physics but would like to have a quick-todigest method of understanding and designing semiconductor laser devices. It is far more convenient to work in terms of voltages, currents and impedances. In fact, electromagnetic field theory and distributed-element circuits (transmission lines) give identical solutions

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

196

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

when we are dealing with transverse electromagnetic (TEM) fields, where voltages and currents in the transmission lines are uniquely related to the transverse electric and magnetic fields, respectively. The attractiveness of using equivalent circuit models for semiconductor laser devices stems from their ability to provide an analogy of laser theory in terms of microwave circuit principles. In addition, microwave circuit models of laser diodes are compatible with existing circuit models of microwave devices such as heterojunction bipolar transistors (HBTs) and field-effect transistors (FETs) – an attractive feature for optoelectronic integrated circuit (OEIC) design [6]. Equivalent circuit models have effectively helped many to understand, design and optimise integrated circuits (ICs) in the microelectronics industry and they have the potential to do the same for the optoelectronics industry. The main theme of this chapter is microwave circuit modelling techniques applied to semiconductor laser devices. Two types of microwave circuit model for semiconductor lasers have been investigated: the simple lumped-element model based on low-frequency circuit concepts and the more versatile distributed-element model based on transmission-line modelling. The former (lumped-element circuit model) is based on the simplifying assumption that the phase of current or voltage across the dimension of the components has little variation. This is true when considering only the modulated signal instead of the optical carrier signal. In this case, Kirchoff’s law can be applied, which is nothing more than a special case of Maxwell’s equations [7–8]. Strictly speaking, laser devices have dimensions in the order of the operating wavelength, thus lumped-element models may not be suitable in ultrafast applications where propagation plays an important role such as in active mode locking [9]. However, the lumped-element circuit is reasonably accurate for microwave applications if all the important processes and effects are modelled accordingly by the circuit on an equivalence basis. The latter of the two circuit modelling techniques (i.e. transmission-line modelling) is a more powerful circuit model that includes distributed effects, which will be discussed in detail in this chapter. It is worth pointing out that at microwave frequencies and above, voltmeters and ammeters for direct measurement of voltages and currents do not exist, so voltage and current waves are only introduced conceptually in the microwave circuit to make optimum use of the low-frequency circuit concepts.

8.2

THE TRANSMISSION-LINE MATRIX (TLM) METHOD

The transmission-line matrix (TLM) was originally developed to model passive microwave cavities by using meshes of transmission lines [9–10]. The numerical processes involved in TLM resemble the mechanism of wave propagation but they are discretised in both time and space [10–11]. Much work has been carried out using the TLM method for analysis of passive microwave waveguide structures (see [12] and references therein). Most of the work done involved two-dimensional and three-dimensional TLMs, with the exception of the application to lumped networks [13–14], the heat diffusion problem [15] and semiconductor laser modelling [16]. Although the TLM is unconditionally stable when modelling passive devices, the semiconductor laser is an active device and therefore requires more careful consideration. The basics of the one-dimensional (1-D) TLM will be presented in the following section, which forms the basis of the transmission-line laser model (TLLM) [17].

THE TRANSMISSION-LINE MATRIX (TLM) METHOD

197

8.2.1 TLM Link Lines The TLM is a discrete-time model of wave propagation simulated by voltage pulses travelling along transmission lines. The medium of propagation is represented by the transmission lines – a general or lossy transmission line consists of series resistance, shunt admittance, series inductance and shunt capacitance per unit length, whereas an ideal or lossless transmission line has reactive elements only. The transmission line may be described by a set of telegraphist equations [7], which can be shown to be equivalent to Maxwell’s equations. There are two types of TLM element that can be used as the building blocks of a complete TLM network – they are the TLM stub lines and link lines [13]. For a lossless transmission line, the velocity of propagation is expressed by 1
l vp ¼ pffiffiffiffiffiffiffiffiffiffi ¼ Ld Cd
t

ð8:1Þ

where Ld is inductance per unit length, Cd is capacitance per unit length,
l is the unit section length, and
t is the model time step. In Fig. 8.1, it is shown that a lumped series

Figure 8.1

TLM link-lines.

inductor (L) is equivalent to a transmission line with inductance per unit length of Ld , where [13] L ¼ Ld
l The characteristic impedance ðZ0 Þ of the transmission line can be found from rffiffiffiffiffiffi Ld L ¼ Z0 ¼ Cd
t

ð8:2Þ

ð8:3Þ

However, there is a small error associated with the shunt capacitance of the transmission line, which can be expressed as Ce ¼

ð
tÞ2 L

ð8:4Þ

198

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

Similarly, the lumped shunt capacitor (C) is equivalent to a transmission line with capacitance per unit length of Cd (Fig. 8.1) where C ¼ Cd
l

ð8:5Þ

The characteristic impedance ðZ0 Þ of the line can be expressed by [13] Z0 ¼


t C

ð8:6Þ

and the associated error in the form of a series inductor is given by Le ¼

ð
tÞ2 C

ð8:7Þ

The errors Ce and Le are of the order of ð
tÞ2 and can be reduced by using a smaller model time step. In practice, there is no component that is purely inductive nor purely capacitive. The parasitic errors can therefore be adjusted by changing the time step ð
tÞ to model stray inductance or capacitance. If two adjacent reactive elements are required, then the parasitic error from one line can be ‘absorbed’ into its adjacent line so that the parasitic error may be eliminated for at least one of the lines.

8.2.2 TLM Stub Lines In the preceding section, we saw how lumped reactive elements can be simulated by TLM link lines. The lumped reactive elements may also be modelled by TLM stub lines, as shown in Fig. 8.2. The lumped inductor L is also equivalent to a short-circuit stub with a characteristic impedance of [13] Z0 ¼

2L
t

Figure 8.2 TLM stub lines.

ð8:8Þ

SCATTERING AND CONNECTING MATRICES

199

and has a parasitic capacitance expressed by Ce ¼

ð
tÞ2 4L

ð8:9Þ

On the other hand, the lumped capacitor is equivalent to an open-circuit stub with characteristic impedance of [13]
t 2C

ð8:10Þ

ð
tÞ2 4C

ð8:11Þ

Z0 ¼ and has a parasitic inductance expressed by Le ¼

For TLM stub lines, the length of the transmission line is chosen such that it takes half a model time step ð
t=2Þ for the pulse to travel from one end to another (see Fig. 8.2). The reason is to allow the voltage pulses to propagate to the termination of the stub and back again at the scattering node in one complete time iteration ð
tÞ. This way, all incident voltage pulses will arrive at their scattering nodes in exactly the same time, irrespective of stub lines or link lines, i.e. the voltage pulses are synchronised.

8.3

SCATTERING AND CONNECTING MATRICES

The most basic algorithm of TLM involves two main processes: scattering and connecting. When the incident voltage pulses, Vi , arrive at the scattering node, they are operated by a scattering matrix and reflected voltage pulses, Vr , are produced. These reflected pulses then continue to propagate along the transmission lines and become incident pulses at adjacent scattering nodes – this process is described by the connecting matrix. Formally, the TLM algorithm may be expressed as kV

rT

¼ Sk ViT

½Scattering

iT

¼ Ck V

½Connecting

kþ1 V

rT

ð8:12Þ

The terms ViT and VrT are the transpose matrices of the incident and reflected pulses, respectively. The terms k and k þ 1 denote the kth and ðk þ 1Þth time iteration, respectively. The scattering and connecting matrices are denoted by S and C, respectively. As the matrices involved in eqn (8.12) depend on the type of TLM sub-network, a worked example based on the TLM sub-network of Fig. 8.3 follows. The TLM sub-network consists of three ‘branches’ of lossy transmission lines as shown in Fig. 8.3, where scattering and connecting of the voltage pulses are clearly described pictorially. The normalised impedances are unity for the two lines connected to adjacent

200

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

Figure 8.3 The TLM stub line: (a) incident pulses arriving at scattering node; (b) incident pulses scattered into reflected pulses (scattering); (c) reflected pulses arriving at adjacent nodes (connecting).

SCATTERING AND CONNECTING MATRICES

201

nodes and Zs for the remaining branch (open–circuit stub line). The associated normalised resistances are R and Rs , respectively. The matrices ViT and VrT are given as 2

VrT

3r V1 ¼ 4 V2 5 V3

2

3i V1 ViT ¼ 4 V2 5 V3

ð8:13Þ

where V1 , V2 and V3 are the voltage pulses on ports 1, 2, and 3, respectively on each scattering node (see Fig. 8.3). The superscripts r and i denote reflected and incident pulses, respectively. It is convenient to break up the scattering matrix S and express it as [15–19] S¼pqr

ð8:14Þ

where the matrices p, q, and r are defined in the following. For the type of TLM sub-network in Fig. 8.3, we have q ¼ ½ q1

q2

q3 

where

qi ¼

8 2ðRs þ Zs Þ > > > < 1 þ R þ 2ðRs þ Zs Þ > > > :

2ð1 þ RÞ 1 þ R þ 2ðRs þ Zs Þ

i ¼ 1; 2 ð8:15Þ i¼3

The matrix q may be found by replacing the sub-network by its Thevenin-equivalent circuit, which is shown in Fig. 8.4. By definition of the voltage pulse, Vxi [5], its generator or source

Figure 8.4

Thevenin equivalent circuit of the TLM sub-network.

202

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

require a value of 2Vxi , where x denotes the port number (1, 2 or 3). The nodal voltage ðvÞ can be defined as v ¼ qk ViT

ð8:16Þ

This will be explained further by using two simple TLM sub-networks as examples later. The matrix p is found by applying the voltage division rule and is given as p ¼ ½ p1

p2

p3  T

where

pi ¼

8 > > > < > > > :

1 1þR

i ¼ 1; 2 ð8:17Þ

Zs R s þ Zs

i¼3

2

3 0 0 5 r33

and the matrix r is given by r11 r¼4 0 0

0 r22 0

where

rii ¼

8 1R > > > < 1þR

i ¼ 1; 2

> > Z  Rs > : s Rs þ Zs

i¼3

ð8:18Þ

If lossless transmission lines are used, we have R ¼ Rs ¼ 0, and eqns (8.15), (8.17) and (8.18) become q ¼ ½ q1 q2 q3  8 2Zs > > i ¼ 1; 2 < 1 þ 2Zs qi ¼ 2 > > : i¼3 1 þ 2Zs 2 3 1 6 7 p ¼ 415 1 2 3 1 0 0 6 7 r ¼ 40 1 05 0 0 1

ð8:19aÞ ð8:19bÞ

ð8:19cÞ

ð8:19dÞ

203

SCATTERING AND CONNECTING MATRICES

The connecting matrix depends on how the transmission lines are connected, that is, which port(s) of one scattering node is/are connected to which port(s) of other adjacent scattering node(s). The matrix element in the connecting matrix is unity only if a connection allows a pulse to travel from port i of node m to port j of node n. In the example given in Fig. 8.3, the connecting matrix C may be expressed by 3  3 elements, where we have pulses travelling from 1.

port 2 of node n  1 to port 1 of node n

2.

port 3 of node n to port 3 of node n (open-circuit stub)

3.

port 1 of node n þ 1 to port 2 of node n

Thus, the connecting matrices for node n and its adjacent nodes n  1 and n þ 1 are given as 2

Cnðn1Þ

0 ¼ 41 0

0 0 0

3 0 05 0

2

Cnn

0 0 ¼ 40 0 0 0

3 0 05 1

2

Cnðnþ1Þ

0 ¼ 40 0

3 1 0 0 05 0 0

ð8:20Þ

A simple example of another TLM sub-network is shown in Fig. 8.5(a), which consists of a resistor ‘sandwiched’ between two lossless transmission lines. In the Thevenin equivalent circuit shown in Fig. 8.5(b), each transmission line is replaced by its characteristic impedance in series with a voltage generator of twice the incident voltage pulse. The

Figure 8.5

(a) A TLM sub-network and (b) its Thevenin equivalent circuit.

204

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

incident voltage pulses are denoted as V1i and V2i , while the reflected pulses are V1r and V2r . The impedances of the lossless transmission lines are Z1 and Z2 . From Fig. 8.5(b), the nodal voltages (v1 and v2 ) may be expressed as i1 Y1

v2 ¼

i2 Y2

ð8:21Þ

2V1i 2V2i þ Z1 ð R þ Z2 Þ

i2 ¼

2V1i 2V i þ 2 ðR þ Z1 Þ Z2

ð8:22Þ

1 1 þ Z1 R þ Z2

Y2 ¼

1 1 þ Z2 R þ Z1

ð8:23Þ

v1 ¼ where i1 ¼ and Y1 ¼

By substituting eqns (8.22) and (8.23) into (8.21), we have  

1 v1 ¼ 2ðR þ Z2 ÞV1i þ 2Z1 V2i R þ Z1 þ Z2  

1 v2 ¼ 2Z2 V1i þ 2ðR þ Z1 ÞV2i R þ Z1 þ Z2

ð8:24Þ

By the original definition of the nodal voltage [5], we have the relationships v1 ¼ V1i þ V1r

ð8:25Þ

v2 ¼ V2i þ V2r

Now we can write down the reflected voltage pulses in terms of the incident pulses as V1r ¼ v1  V1i  

1 V1r ¼ ðR  Z1 þ Z2 ÞV1i þ 2Z1 V2i R þ Z1 þ Z2

ð8:26Þ

and V2r ¼ v2  V2i  

1 V2r ¼ 2Z2 V1i þ ðR  Z2 þ Z1 ÞV2i R þ Z1 þ Z2 Finally, the complete scattering matrix of the TLM network in Fig. 8.5 at the kth iteration is expressed by [13]  Sk ¼

1 R þ Z1 þ Z2



ðR  Z1 þ Z2 Þ 2Z2

2Z1 ðR  Z2 þ Z1 Þ

 ð8:27Þ

SCATTERING AND CONNECTING MATRICES

Figure 8.6

205

(a) A TLM sub-network and (b) its Thevenin equivalent circuit.

Another simple but useful TLM sub-network is shown in Fig. 8.6(a) and its Thevenin equivalent is given in Fig. 8.6(b). This is similar to one of the sub-networks in Fig. 8.3 but it is lossless in this case and each branch has a different value of line impedance. Now, the common nodal current is defined by i¼

2V1i 2V2i 2V3i þ þ Z1 Z2 Z3

From Fig. 8.6(b), the total admittance at the scattering node is expressed as   1 1 1 þ þ Y¼ Z1 Z2 Z3 Z2 Z3 þ Z1 Z2 þ Z1 Z3 Y¼ Z1 Z2 Z3

ð8:28Þ

ð8:29Þ

206

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

From Millman’s thereom [19] the common nodal voltage can be found from v¼

i 2Z2 Z3 V1i þ 2Z1 Z2 V3i þ 2Z1 Z3 V2i ¼ Z2 Z3 þ Z1 Z2 þ Z1 Z3 Y

ð8:30Þ

Since the sum of incident voltage pulse and reflected pulse gives the nodal voltage, see eqn (8.25), the reflected pulse may again be found by subtracting the incident pulse from the nodal voltage, that is V1r ¼ v  V1i V1r ¼

2Z2 Z3 V1i þ 2Z1 Z2 V3i þ 2Z1 Z3 V2i  V1i Z2 Z3 þ Z1 Z2 þ Z1 Z3

V1r ¼

ðZ2 Z3  Z1 Z2  Z1 Z3 ÞV1i þ 2Z1 Z2 V3i þ 2Z1 Z3 V2i Z2 Z3 þ Z1 Z2 þ Z1 Z3

ð8:31Þ

In a similar manner, the reflected pulses, V2r and V3r , can also be found, and the complete scattering matrix of the sub-network in Fig. 8.6 is expressed by 2

Sk ¼

Z2 Z3  Z1 Z2  Z1 Z3 4 2Z2 Z3 2Z2 Z3

2Z1 Z2 Z1 Z2  Z2 Z3  Z1 Z3 2Z1 Z2 Z1 Z2 þ Z1 Z3 þ Z2 Z3

3 2Z1 Z3 5 2Z1 Z3 Z1 Z3  Z2 Z3  Z1 Z2

ð8:32Þ

The TLM sub-network in Fig. 8.6(a) forms the basis of the matching network. There are many other TLM sub-networks that can be formed by using series and shunt resistors together with the TLM link lines and stub lines. For example, periodically unmatched boundaries of TLM link lines can be used to mimic corrugated gratings [20] and shunt conductances can be included to model gain-coupled DFB lasers [21]. A TLM sub-network which consists of a series resistor and two reactive stub lines will now be used to model the wavelength dependence of semiconductor laser gain in the transmission-line laser model.

8.4

TRANSMISSION-LINE LASER MODELLING (TLLM)

The transmission-line laser model is a wide-bandwidth dynamic laser model that takes into account important considerations such as inhomogeneous effects, multiple longitudinal modes, spectral dependence of gain, carrier-induced refractive index change, and spontaneous emission noise. The TLLM is very flexible and has successfully been used to model a wide range of laser devices including Fabry–Perot lasers [16], DFB (index-coupled and gain-coupled) and DBR lasers [21–23], quantum well (QW) lasers [24], cleavedcoupled-cavity (CCC) lasers [25] external cavity (EC) mode-locked lasers [26], fibre grating lasers [27] and laser amplifiers [28–29]. Electrical parasitics and matching circuits can also be included as part of the model [29]. The TLLM can be thought of as a pedagogical model to aid in the physical understanding of the laser in terms of more familiar circuit techniques. The topology of the model closely

BASIC CONSTRUCTION OF THE MODEL

207

mimics the physical structure of the laser. It was first developed by Lowery [16] and was based on the 1-D TLM concepts discussed in the preceding section. Since the TLLM is a time-domain model its main application is transient analysis, where laser non-linearity is important.

8.5

BASIC CONSTRUCTION OF THE MODEL

In the TLLM, the laser cavity is divided into many smaller sections as shown in Fig. 8.7, hence the longitudinal distributions of carrier and photon density can be modelled. The individual sections are connected to one another by dispersionless transmission lines (i.e. group velocity equal phase velocity) with characteristic impedance of Z0 . Voltage pulses travel along these transmission lines in both forward and backward directions, analogous to

Figure 8.7

The transmission-line laser model (TLLM) and its components.

the propagation of optical waves inside the laser cavity. The model time step, ð
tÞ, describes the time required for the voltage pulse to propagate from one section to the adjacent section. At the centre of each section is the scattering node, where the TLM scattering matrix is placed. When incident voltage pulses arrive at the scattering nodes, the TLM scattering matrix operates on them to produce reflected voltage pulses (see section 8.3). The fundamental processes of the laser such as gain (stimulated emission), loss (absorption), and noise (spontaneous emission) are contained in the scattering matrix.

208

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

Spectral dependence of the material gain, which the optical waves experience, is also included. Coupling between counter-propagating waves that occurs in DFB lasers can also be modelled. At the laser facets, the reflections that provide feedback to achieve lasing action are simulated by unmatched terminal loads. The model assumes single transverse mode operation of the laser diode, and that variations of the carrier and photon densities in the lateral and transverse dimensions are not significant, except for broad-area lasers such as the tapered waveguide structure [30]. Today, advanced fabrication techniques allow single transverse and longitudinal mode laser devices to be achieved, typical of these are strongly index-guided laser structures such as buried heterostructure (BH) lasers. By spatially averaging the lateral x- and transverse y-dependence of the optical field amplitude, the model is simplified into a 1-D model [16]. The 1-D TLLM is a reasonable approach, which has been supported by many other 1-D dynamic laser models such as the transfer matrix model (TMM) [31], time-domain model (TDM) [32–33], and power matrix model (PMM) [33–34]. The reservoir of carriers (Nn where n is the section number) in the carrier density model interacts with the photon density ðSn Þ through the laser rate equations. This dynamic carrier– photon interaction is independently modelled in each and every section. The carriers are supplied by the current sources placed in every section of the model. In this way, electrically-isolated multi-electrode lasers can easily be modelled by injecting different current amplitudes into separate parts of the model. Non-uniform current pumping is one method of ensuring a more evenly distributed carrier concentration in the laser cavity, especially when there is severe spatial hole burning such as in quarterly-wavelength-shifted DFB lasers. In the presence of significant spatial hole burning, the longitudinal carrier diffusion effect can also be included in the model [35]. Therefore, the TLLM is a powerful laser model and yet it is relatively easy to visualise, being a distributed-element circuit model. The outputs of the model are collected as optical field samples, which can be coupled to other TLLM-compatible models such as optical amplifiers, optical filters and photodiodes [36]. The samples of optical field and optical power are readily fast Fourier transformed (FFT) to obtain the optical and RF spectra, respectively. The accuracy of the model can be enhanced by using a smaller model time step ð
tÞ but at the expense of longer computation time.

8.6

CARRIER DENSITY MODEL

The carrier–photon resonance that leads to the transient phenomenon of relaxation oscillations is governed by the laser rate equations. The carrier rate equation is given as dN I N ¼   GðNÞS dt qva n

ð8:33Þ

where N is the carrier density, I is the injected current, va is the active layer volume of the unit section, q is the electronic charge, n is the carrier lifetime, GðNÞ is the stimulated recombination rate, and S is the photon density. The carrier rate equation can be modelled by an equivalent circuit (see Fig. 8.8) at each and every section of the TLLM, where the carrier density is represented by the voltage, V. In the equivalent circuit of Fig. 8.8, the current, Iinj ,

CARRIER DENSITY MODEL

Figure 8.8

209

Equivalent circuit model of the carrier density rate equation.

represents the current injected into the active region, the storage capacitor, C, represents carrier build-up/depletion, the resistor, Rsp , represents the spontaneous emission rate, and the current, Istim , represents the stimulated emission rate. The corresponding circuit equation is given as dQ V ¼ Iinj   Istim dt Rsp

ð8:34Þ

where Q ¼ Nqva ¼ CV Comparing the carrier rate equation from (8.33) and the circuit equation from (8.34), the equivalence between the recombination terms and the circuit variables is found by inspection to be Iinj ¼ I

n Rsp ¼ qva Istim ¼ qva GðNÞS Moreover, the photon density of the local section, Sn , may be found from [17]  2  2  i ng  FM ðnÞ þBiM ðnÞ Sn ¼ Zp hf0 c0 m2

ð8:35Þ

ð8:36Þ

where ng is the effective group index, h is Planck’s constant, f0 is the lasing frequency, m is a i and BiM are the incident voltage pulses on the unity constant with dimension of length, FM main transmission line of section n in opposite propagation directions. The wave impedance is defined by Zp ¼

120png neff

ð8:37Þ

210

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

where neff is the effective refractive index of the semiconductor material. Since the power density may be defined in terms of the transverse fields alone, the wave impedance is used to relate the transverse components of the electric and magnetic fields [5,7]. This enables the power transmitted to be expressed in terms of only one of the transverse fields. Hence, the optical power, P, escaping from the output facet may be defined in terms of the transverse electric field as  r 2 Wd P ¼ F M ðSÞ ð1  R2 Þ Zp

ð8:38Þ

where W is the output width, d is the active layer thickness, R2 is the output facet reflectivity, r is the forward travelling voltage pulse (transverse electric field) on the main and FM transmission line at the output section ðSÞ.

8.7

LASER AMPLIFICATION

The amplification rate of the optical intensity is assumed to be of the form dI ¼ ðGg  sc ÞI dz ð n
L ð In
L dI ¼ ðGg  sc Þdz z¼ðn1Þ
L I¼Iðn1Þ
L I

ð8:39Þ

In
L ¼ Iðn1Þ
L exp½ðGg  sc Þ
L where Iðn1Þ
L is the initial optical intensity, In
L is the intensity after propagating a distance of
L; G is the optical confinement factor, g is the gain coefficient that is frequency dependent, and sc is the loss factor. As the optical field amplitude is proportional to the square root of the intensity, we can write  En
L ¼ Eðn1Þ
L exp

ðGg  sc Þ
L 2

 ð8:40Þ

where E represents the optical field amplitude. The gain term, expðGg
L=2Þ, in eqn (8.40) may be approximated by a Taylor expansion as the first bracketed terms in the following expression [16]    
LGg sc
L En
L ¼ Eðn1Þ
L 1 þ exp  ð8:41Þ 2 2 Meanwhile, the loss term remains as an exponential term in eqn (8.41) and is assumed to be independent of frequency. In modelling the laser gain spectrum, a combination of TLM and digital signal processing techniques are involved. The block diagram equivalent of eqn (8.41) is shown in Fig. 8.9. The frequency (wavelength) dependence of the small amplified

LASER AMPLIFICATION

Figure 8.9

211

Block diagram of the gain filter.

signal term, ðGg
L=2ÞE0 , is modelled by the RLC bandpass filter, which is then added back to the incoming signal ðE0 Þ and attenuated by the factor expðsc
L=2Þ to produce the output amplified signal ðE
L Þ. The lumped circuit representation of the second-order bandpass filter in Fig. 8.9 (rectangular dashed lines) may be converted into its TLM

Figure 8.10

The TLM stub filter.

counterpart, as shown in Fig. 8.10. The bandpass filter has a Lorentzian response, assuming that the laser is a homogeneously broadened two-level system [6,37]. The Theveninequivalent circuit of Fig. 8.10 is shown in Fig. 8.11. The lumped inductance and capacitance of the bandpass filter are modelled as short-circuit and open-circuit TLM stubs, respectively.

Figure 8.11

Thevenin equivalent circuit of the TLM stub filter.

212

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

From the Thevenin-equivalent circuit the common nodal voltage ðvÞ can be expressed as v¼

i ðgVM þ 2VLi YL þ 2VCi YC Þ ð1 þ YL þ YC Þ

ð8:42Þ

where YL ¼

1 ZL

YC ¼

1 ZC

ZL and ZC are the impedance values of the inductive and capacitive stub lines, respectively, VM , VL and VC are the voltage pulses on the main transmission line, inductive stub line (short-circuit), and capacitance stub line (open-circuit), respectively. The superscripts i and r are used to denote incident and reflected pulses, respectively. The reflected voltage pulses are found from the following relationships r i ¼ VM þv VM

ð8:43Þ

VLr ¼ v  VLi VCr ¼ v  VCi From eqn (8.43), the scattering matrix may be found as 3r 2 tðg þ yÞ VM 16 7 6 g 4 VC 5 ¼ 4 y V g L k 2

2tYC 2YC  y 2YC

3i 32 2tYL VM 7 76 2YL 5 4 VC 5 2YL  y k VL

y ¼ 1 þ YC þ Y L   sc
L t ¼ exp 2

ð8:44Þ

The scattering mechanism of eqn (8.44) is implemented for pulses travelling in two opposite propagating directions inside the laser cavity, i.e. forwards and backwards. If the gain coefficient is less than 1010 the gain model may be assumed linear as in the following form g ¼ avg ðN  Ntr Þ

S ð1 þ "ÞS

ð8:45Þ

where vg ¼

c0 ng

where S is the photon density of the local model section, a is the differential gain constant, ng is the effective group index, " is the gain compression factor and c0 is the speed of light

213

LASER AMPLIFICATION

in free space. However, if the gain is greater than 1010 , the gain model is assumed to be logarithmic [38–39] in the form 

 N S g ¼ avg ln Ntr ð1 þ "ÞS

ð8:46Þ

The linear gain model is assumed throughout this book. As the model bandwidth is limited by the sampling rate used, the TLM stub filter response falls to zero at both edges of the model bandwidth. However, there is a good fit between the stub filter response and the actual Lorentzian line shape near the peak, as shown in Fig. 8.12. The stub filter response was found by performing an FFT on the time-domain impulse response.

Figure 8.12

Comparison between the stub filter response and actual Lorentzian shape.

The bandwidth of laser gain spectrum is governed by the Q-factor of the stub filter, which is related to the admittances YC and YL of the stub filter. The Q-factor is defined as the ratio of the centre frequency to its bandwidth. We shall now find the relationship between the Q-factor and the admittances of the stub filter. From transmission-line theory, the input impedance, Zin , is expressed by [5] Zin ¼ Z0

ZðlÞ þ jZ0 tanðk0 lÞ Z0 þ jZðlÞ tanðk0 lÞ

ð8:47Þ

where k0 is the free space wavenumber, l is the distance from load (e.g. open-circuit and short-circuit), and Z0 is the characteristic impedance of the line. When an open-circuit stub is used to simulate a capacitive stub line, ZðlÞ approaches infinity, and we have ZiC ¼

ZC ¼ jZC cotðk0 lÞ j tanðk0 lÞ

ð8:48Þ

214

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

where ZC is the impedance of the capacitive stub line. When a short-circuit stub is used to simulate an inductive stub line, ZðlÞ is zero, and we have ZiL ¼ jZL tanðk0 lÞ

ð8:49Þ

where ZL is the impedance of the inductive stub line. In the laser cavity, the free space wavenumber ðk0 Þ should be replaced by the propagation constant of the semiconductor material, which is given as b ¼ k 0 ng ¼

2f0 ng c0

ð8:50Þ

Since the model is discretised in time ð
tÞ and space ð
lÞ, and to comply with the synchronisation criterion, the length of the stub must be half of a unit section length, i.e.
l=2. Hence, the input impedance of the stubs can be expressed by ZiC ¼ jZC cotðf0
tÞ ZiL ¼ jZL tanðf0
tÞ

ð8:51Þ

In a parallel RLC filter with unity resistance ðR ¼ 1Þ, the Q-factor is given as R ¼ Q¼ !0 L

rffiffiffiffi C L

ð8:52Þ

where 1 !0 ¼ pffiffiffiffiffiffiffi LC By making use of eqn (8.9) and eqn (8.10), eqn. (8.52) can be rewritten in terms of transmission-line impedances rffiffiffiffiffiffiffiffiffiffiffi 1 Q¼ ZC ZL

ð8:53Þ

At resonance, the parallel combination of stub admittance is zero ðYiC þ YiL ¼ 0Þ, leading to the following relationship rffiffiffiffiffiffi ZC ¼ tanðf0
tÞ ZL

ð8:54Þ

From eqn (8.53) and (8.54), we can define the admittances of the stub filter as the following YL ¼ Q tanðf0
tÞ Q YC ¼ tanðf0
tÞ

ð8:55Þ

LASER AMPLIFICATION

215

For the purpose of computational efficiency, the baseband Q-factor ðQdc Þ is used in the model instead of the actual Q-factor. The relationship between them is given as [16]   Bfsamp Qdc ¼ Q 1  ð8:56Þ f0 where fsamp ¼

1
t

f0 is the actual lasing frequency and B is the bandnumber, which will be discussed in section 8.11. For the same reason, the baseband lasing frequency ð fdc Þ replaces the actual lasing frequency ð f0 Þ. The baseband lasing frequency is usually placed at the centre of the model bandwidth, i.e. fdc ¼ 1=4
t. As mentioned earlier, the model bandwidth is determined by the sampling rate used. Therefore, the gain filter response will fall to zero at the edges of the available bandwidth as shown in Fig. 8.13 by the dotted lines. This corresponds to a model time step of
t ¼ 50 fs that gives a total bandwidth of only 10 THz, as shown by the dashed lines. By using a smaller time step
t ¼ 10 fs, the total available bandwidth increased by five times to 50 THz but at the expense of computational speed. The runtime is now 25 times slower because the computational task is proportional to S2 , where S is the total number of sections in the model. One method to increase computational speed and yet achieve reasonably accurate gain spectra is to use first-order TLM stub filters to perform some shaping of the spectral response in addition to some discrete-time signal processing [39]. The improvements are shown as circular symbols in Fig. 8.13. Notice that now the spectral responses do not fall to zero at the

Figure 8.13 Improvement in gain spectrum by using additional TLM stub-filters. Carrier dependence of the gain was included. Solid lines:
T ¼ 10 fs (simple gain stub filter); Dotted lines:
T ¼ 50 fs (simple gain stub filter); Circles:
T ¼ 50 fs (improved gain stub filter including HP and LP stub filters).

216

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

band edges. The dynamic gain peak shift can also be modelled by simply including the carrier dependence of the lasing frequency, fdc [35]. The progression of the amplified and filtered voltage pulses (i.e. connection) along the main transmission lines is described by the following connecting relationship i kþ1 FM ðnÞ

r ¼ k FM ðn  1Þ

i kþ1 BM ðnÞ

¼ k BrM ðn þ 1Þ

ð8:57Þ

where FM ðnÞ and BM ðnÞ are the forward and backward travelling voltage pulses of section n, respectively. At the facets, the connecting stems from the fact that the pulses are reflected into the opposite direction due to the impedance mismatch between transmission line and the terminal load, that is pffiffiffiffiffi r i R1 k BM ð1Þ kþ1 FM ð1Þ ¼ ð8:58Þ pffiffiffiffiffi r i R2 k FM ðSÞ kþ1 BM ðSÞ ¼ where R1 and R2 are the power reflectivities at the left (section 1) and right end (section S) facets, respectively. This is assuming that the voltage pulses travel from left to right for the forward direction and vice versa for the backward direction. Finally, the connecting algorithm for the stub lines is given as, i kþ1 VC ðnÞ i kþ1 VL ðnÞ

¼ k VCr ðnÞ ¼

k VLr ðnÞ

ðopen-circuit stubÞ ðshort-circuit stubÞ

ð8:59Þ

where the voltage pulse, V, is applicable for both the forward and backward directions (F and B).

8.8

CARRIER-INDUCED FREQUENCY CHIRP

The consequence of carrier dependence of the active layer refractive index is a dynamic spectral shift, commonly known as chirping. To simulate this refractive index change, the phase length is altered by phase(adjusting) stubs, which can either be placed in (i) each and every section of the model or (ii) only at the facets of the laser cavity. For the time being, we will only discuss case (ii), which is known as the stub-attenuator model [40]. The former (i.e. case(i)) takes into account inhomogeneous distribution of the carrier density.

Figure 8.14 Stub-attenuator model.

CARRIER-INDUCED FREQUENCY CHIRP

217

Figure 8.15 Three-port circulator.

In the stub-attenuator model of Fig. 8.14, the voltage pulses get reflected back into the opposite direction due to the impedance mismatch between main transmission lines and terminal loads. The voltage pulses then enter the phase stubs to be delayed before they continue propagating along the main transmission line. The three-port circulator at the left facet (section 1) shown in Fig. 8.14 ensures that the delayed pulses from the phase stubs that are incident at port 2 continue to propagate into port 3 and not port 1 (e.g. see Fig. 8.15). The scattering matrix of a three-port circulator is given as [4] 2

3r 2 0 V1 4 V2 5 ¼ 4 1 V3 0

0 0 1

32 3i 1 V1 0 54 V 2 5 V3 0

ð8:60Þ

Taking the left facet (section 1) as an example, the backward travelling wave (i.e. a train of voltage pulses) enters port 3 of the circulator and gets transferred into port 1, where it gets reflected into the phase stubs (port 2). The reflected wave then becomes the incident wave at port 2 of the circulator at the next time step. The delayed wave now entering port 2 gets diverted into port 3 and continues to propagate along the main transmission lines. The scattering matrix that describes the stub-attenuator model at the left facet is given by 2

0 6 V1 ð1Þ 6 2Zs 6 7 6 ð1 þ Z Þ s V ð1Þ ¼ 4 2 5 6 6 4 V3 ð1Þ ðZ  1Þ 2

3r

s

ð1 þ Zs Þ

0 ð1  Zs Þ ð1 þ Zs Þ 2 ð1 þ Zs Þ

13 2 3 7 V1 ð1Þ i 07 76 7 74 V2 ð1Þ 5 7 5 V3 ð1Þ 0

ð8:61Þ

where all impedances are normalised to the characteristic impedance of the main transmission lines ðZ0 Þ, Zs is the phase stub impedance, and R1 is the power reflectivity of the left facet (first section). A similar scattering matrix is needed at the right end facet (Sth section). At first thought, the phase stub length should be varied to adjust the total phase length of the laser cavity. However, the need for synchronisation of pulses in the TLLM methodology only allows us to vary the stub impedance. This is an equivalent approach since the stub impedance is related to the stub length. To find this relationship, the input impedance ðZin Þ of the fixed-length stub (variable line impedance) is equated to the input impedance ðZin Þ of the adjustable-length open-circuit stub (fixed line impedance). The input impedance of the adjustable open-circuit stub is Zin ¼

Z0 j tanðb
lph Þ

ð8:62Þ

218

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

where
lph is the desired extension/contraction. The fixed-length stubs can either be capacitive (open-circuit) or inductive (short-circuit), as defined in eqn (8.51), this depends on the average carrier concentration level. This will be explained as follows. Case 1: When the change in carrier concentration ð
N Þ is negative, the change in phase length is positive, and the fixed-length capacitive stub is used. Comparing eqn (8.62), and eqn (8.51), for the capacitive stub, we have Zin ¼

Z0 ¼ jZs cotðf0
tÞ j tanðb
lph Þ

Z0 tanðf0
tÞ Zs ¼ tanðb
lph Þ

ð8:63Þ

where Zs is the fixed-length stub’s variable impedance. Case 2: On the other hand, when the change in carrier concentration is positive, the change in phase length is negative, and the fixed-length inductive stub is used. Comparing eqns (8.62) and (8.51) for the inductive stub, we have Zin ¼

Z0 ¼ jZs tanðf0
tÞ j tanðb
lph Þ

Z0 Zs ¼  tanðf0
tÞ tanðb
lph Þ

ð8:64Þ

 The variable phase length
lph can have values above and below zero depending on whether the carrier concentration drops below or rises above the reference value ðNref Þ, corresponding to a red and blue chirp, respectively. Since the change in resonant frequency is the same it is due to a change in effective refractive index ð
nÞ or variable phase  whether length
lph , the relationship between
n and
lph can be expressed by [40]
lph ¼

GL
n ng

dn  0 N  Nref
n ¼ dN av

ð8:65Þ

0 where Nav is the carrier density averaged over the entire cavity, and dn=dN is the rate of refractive index change with carrier density.  The dispersionless transmission lines are based on the assumption that group index ng equals phase index ðnÞ. The term Nref is the reference value of carrier density for zero phase shift, usually set to the threshold level. The rate of change of n with N is defined as [40]

dn H c0 a ¼ dN 4f0

ð8:66Þ

where H is Henry’s linewidth enhancement factor [41] and a is the differential gain constant.

SPONTANEOUS EMISSION MODEL

8.9

219

SPONTANEOUS EMISSION MODEL

The TLLM is a stochastic model that takes into account noise, which is inevitable in a real laser device. Consideration of noise in a laser model is important for the following reasons. In ultrashort-pulse generation using gain-switched and mode-locked lasers, the emitted optical pulses do not exactly have regularly spaced intervals but noise-induced timing jitter exists [42–43]. This may lead to increased bit error rates (BER) when used as pulse sources in optical time-division-multiplexed (OTDM) communication systems [44]. Salathe et al. found that spontaneous emission noise in the laser rate equation smoothens the transition between non-lasing and lasing conditions [45]. The presence of spontaneous emission has also been found to dampen relaxation oscillations during pulse code modulation (PCM) [46–47]. In addition, spontaneous emission contributes to phase noise and determines the continuous (CW) linewidth of each longitudinal mode. The CW linewidth of the laser is crucial when it is used as a local oscillator in coherent optical systems [48–49]. The spontaneous emission noise is modelled by current sources in TLLM, which will be shown to be equivalent to the free current density term in Maxwell’s equations [49]. The

Figure 8.16

Equivalent circuit diagram of the lossless transmission line including a current source.

presence of a current source along the lossless transmission lines (see Figure 8.16) can be described mathematically by the telegraphist equations [7], given as 

qI qV ¼C þi qz qt

qV qI ¼ L qz qt

ð8:67Þ

where C is capacitance per unit length, L is inductance per unit length, V is the voltage wave, I is the current wave, and i is the external current source per unit length. For TEM (plane) waves ðEx ¼ Hz ¼ 0Þ, Maxwell’s equations can be written as qHy qEx ¼ n2 " 0 þ jx qz qt

qEx qHy ¼ 0 qz qt

ð8:68Þ

where E is the electric field, H is the magnetic field, n is the index of refraction, "0 is the free space permittivity, 0 is the free space permeability, and j is the free current density. By comparing eqns (8.67) and (8.68), the equivalence between circuit and field quantities can be found as shown in Table  8.1. In a lossless cold cavity of S sections, the total spontaneous emission output power Psp is expressed by Psp ¼ 2WdSI

220

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

Table 8.1 Equivalence between circuit and field quantities Circuit quantities

Field quantities

V=m I=m L C i=m

Ex Hy 0 n2 "0 jx

where   1 V 2 I¼ Zp m

ð8:69Þ

and V ¼ 2

i2rms

  Zp
l 2 2

d is the active layer thickness, W is the active layer width, I is the output intensity of a unit section due to spontaneous emission, and m is a unity constant of unit length to achieve dimensional correctness. With a spontaneous emission current source placed at each and every node of an S-section model and split into two propagating directions (forwards and backwards), the output power per longitudinal mode out of both facets may be expressed by Pm ¼

Wdhi2 iZp
l2 2m2

ð8:70Þ

where hi2 i is the mean square value of the current source. Alternatively, from the photon rate equation, the spontaneous emission output power per longitudinal mode can be written as Pm ¼ WdLhf0 bRðNÞ

ð8:71Þ

where L ¼ S
l is the spontaneous emission coupling factor and RðNÞ is the recombination rate. By comparing eqns (8.70) and (8.71), the mean square current is found to be given as [49] hi2 i ¼

2bRðNÞhf0 m2 S Zp
l

ð8:72Þ

As spontaneous emission is a random process, only its mean square value can be defined. The spontaneous emission current source is simulated by a Gaussian random number

COMPUTATIONAL EFFICIENCY BASEBAND TRANSFORMATION

Figure 8.17

221

One section of the distributed current source model of spontaneous emission.

generator [50] with mean square of hi2 i and mean of zero. Since the mean value of white Gaussian noise is zero, the mean square value is equal to its variance [51]. In order to model the distributive nature of the spontaneous emission noise, the current sources are placed along the TLLM and the mean square current is calculated using local carrier concentration rather than just the average value of the entire cavity. The equivalent circuit diagram of the distributed current source model of spontaneous emission is shown in Fig. 8.17. In each section of the TLLM, the incoming voltage pulses are amplified by wavelength-selective amplifiers (AMP) before spontaneous emission noise is added. Note that the filtered spontaneous emission current must first be converted to voltage before being added to the main voltage pulses. Along the transmission lines, the attenuators (ATTN) simulate the material and scattering losses in the optical cavity. The spontaneous emission spectrum has Gaussian noise statistics and is not white noise (frequency independent) but coloured noise (frequency dependent). Therefore, the TLM stub filter is placed at the output of the spontaneous emission current source to model the frequency dependence. The resonant cavity modes collect all the power that is spontaneously emitted into the frequency intervals between adjacent cavity modes [52].

8.10

COMPUTATIONAL EFFICIENCY BASEBAND TRANSFORMATION

Modelling with digital computers requires discretisation of signals, meaning that real continuous waveforms must be sampled at equally spaced points in time. Since there is a close relationship between time and frequency, the need for sampling in time leads to a signal of finite frequency bandwidth. According to Nyquist’s sampling theorem, an accurate reconstruction of the original signal from the sampled version can only be performed if the

222

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

sampling rate is twice the bandwidth of the original signal. Equivalently, the unit section length,
l, must be at most one half of the group wavelength,
g , in the laser cavity. Therefore, the minimal number of sections S allowed in the laser model with length L is S

2L
g


g ¼

c0 f0 ng

ð8:73Þ

where

Since laser devices operate at the near-infrared region, this will lead to a huge amount of computation if the Nyquist criterion is strictly followed for the optical carrier frequency ð f0 Þ. For example, a 300 mm laser chip with effective group index of 4 operating at 1.55 mm would require at least 1549 sections. This is computationally too intensive considering the amount of processing tasks at each and every section for each time iteration, which may take up to weeks to process on a PentiumTM processor. For reasonable computation times, the total number of sections (S) used in TLLM is usually between 10 and 100. Interestingly, the value of S also corresponds to the number of longitudinal modes that are taken into account by the model [16]. Computation time is proportional to S2 , which is why S must not be too large. Usually, the modulated light signal is at microwave- or millimetre-wave frequencies, and has a bandwidth of at least a few thousand times smaller than its optical carrier frequency. Therefore, the TLLM makes use of a baseband transformation method to shift the optical carrier down to the DC level to speed up the computational time [16], as illustrated in

Figure 8.18

Baseband transformation technique to reduce sampling rate.

Fig. 8.18. Only the positive frequencies are considered here because the model uses real signals even though complex signals can also be used but at the expense of computational efficiency. The baseband transformation method has also been applied to the quantum mechanical model [32].

SIGNAL ANALYSIS – POST-PROCESSING METHODS

223

This baseband transformation method depicted in Fig. 8.18 is acceptable because the semiconductor laser has a relatively narrow signal bandwidth (THz region or less) compared to its optical carrier frequency (109 Hz and above). For example, in Fig. 8.18 the signal bandwidth is given as ð fmax  fmin Þ while the optical frequency is at a much higher value, denoted by f0. The concept here is to neglect the spectral regions that are not of interest lying between DC and the optical carrier frequency ð f0 Þ known as the deadband (the lighter lines in Fig. 8.18). Hence, the optical spectrum can be down-converted to baseband without any signal distortion if Nyquist’s sampling criterion is satisfied for the signal bandwidth ð fband Þ.  Now, the lower limit of the sampling rate fsamp is only restricted by fsamp  2fband

ð8:74Þ

The above restriction ensures that there is no overlap between the alias bands, which would otherwise lead to signal loss and distortion. The degree of scaling down of the actual lasing frequency can be described by the bandnumber (B), which is expressed by  B ¼ int

f0 fsamp





f0 ¼ int 2fband

 ð8:75Þ

where the function ‘int’ rounds off to the nearest but lower integer. In the example of Fig. 8.18, the bandnumber B ¼ 4. The down-converted lasing frequency, fdc , is usually chosen to be at the cavity resonant frequency nearest to the centre of the model bandwidth ð1=
tÞ if a symmetrical gain spectrum is assumed.

8.11

SIGNAL ANALYSIS – POST-PROCESSING METHODS

In order to analyse the performance of the laser device, some form of signal processing must be applied on the laser output. This closely mimics the laboratory instrumentation for signal measurement and analysis such as optical spectrum analysers, sampling oscilloscopes, and streak cameras. The post-processing techniques included in the TLLM are as follows. 1.

The Fourier transform. The output samples of the TLLM are voltage pulses in time domain. In order to find the frequency spectrum of the output signal, the discrete Fourier transform (DFT) or the more computationally efficient fast Fourier transform (FFT) can be used [53]. The information content in the frequency domain will always be less than that of the time-domain data since it must be truncated at some point – a full frequencydomain description can only be obtained if the time samples extend to infinity. The FFT produces uniformly distributed data points over a frequency bandwidth extending from fsamp =2 to fsamp =2. However, only the relative frequency with respect to the lasing optical frequency ðf0 Þ is important. The optical spectrum obtained will be a continuous spectrum displaying linewidth-broadening effects due to noise fluctuations associated with spontaneous emission [54]. This is in contrast to the unrealistic discrete optical spectra produced from multi-mode rate equations [55–56].

2.

Moving average filter. The TLLM is a highly realistic model, where the output samples are voltage pulses that represent optical field amplitudes. The fast-varying field

224

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

amplitudes can be converted to instantaneous power using eqn (8.38). In practical measurements, the time-averaged power or optical intensity is usually observed. In the TLLM, digital filtering can be used to remove the high-frequency oscillations of the optical carrier but at the expense of time resolution. For example, taking the average of a group of ten samples causes the resolution to become ten times coarser (ten points become one averaged point). To solve this problem, the moving average digital filter is used. This is briefly illustrated in Fig. 8.19.

Figure 8.19

Moving average digital filter.

The advantage is that the resolution of the filter does not depend on the size of the moving average window. However, for a window containing a large number of time samples, there is no sharp cutoff – significant attenuation can occur within the frequency range of interest. In order to achieve better cutoff characteristics, a weighted moving average window such as a Gaussian response or binomial series can be used but this results in a longer computation time. 3.

Stable averaging method. When investigating short-pulse generation in TLLM, such as gain switching and mode locking, the optical pulses grow from spontaneous emission. Since spontaneous emission is a random process, this leads to the existence of timing jitter and random sub-structures within the pulse. As a result, the optical spectrum changes with time owing to random fluctuations in turn-on delay time, pulsewidth, and dynamic spectral shift (chirp). Even in CW operation, noise fluctuations are unavoidable unless the noise can be switched off at steady-state [57]. This makes it impossible to define a single value of optical pulsewidth or spectral width without using signal averaging techniques.

SIGNAL ANALYSIS – POST-PROCESSING METHODS

225

In practical measurements, stable pulses can be observed by using digital sampling oscilloscopes in averaging mode. The stable averaging algorithm [58–59] is implemented in TLLM to obtain time-averaged optical pulse and spectrum by filtering out any noise due to spontaneous emission and its beating effects. The stable averaging algorithm can be expressed as [59] Pim ¼ Pim1 þ

pim  Pim1 m

ð8:76Þ

where Pim is the averaged power of the ith time slot at the mth sweep, and pim is the instantaneous power from TLLM. A pre-determined number of time slots in memory is used to store the averaged results. To display the power envelope instead of the fastvarying instantaneous power, the term pim can be replaced by the power that has been averaged by the moving average filter. In this way, a single average value of pulsewidth and spectral width (full width at half maximum, can be found. In addition, the pffiffiffiFWHM) ffi random noise will be reduced by a factor of m for m sweeps used [59]. 4.

Sample and overlay method. Sometimes it is helpful to be able to see how the repetitive optical pulses deviate from one another. By using the stable averaging technique any pulse instability is eliminated and thus cannot be seen in the final averaged pulse. One way of observing the pulse timing jitter is by overlaying samples of sequential pulses in a manner similar to a digital sampling oscilloscope in infinite persistence mode without signal averaging. Any pulse instability will show up as broadening of the traces, since sequential pulses will not lie on top of one another exactly, while a thin trace indicates good pulse stability.

5.

Smoothing algorithm. In order to find the spectral width of the time-averaged spectrum, we need to use both stable averaging and smoothing algorithms. The stable averaging method is used to obtain the time-averaged spectrum. However, in a multimode spectrum with mutiple peaks and sub-structures, especially during shortpulse generation, it is difficult to define the FWHM. Smoothing is thus required before finding the FWHM of the optical spectrum. The smoothing algorithm used in TLLM is based on FFT-filtering [59]. The first step is to set up a smoothing kernel, k, given as k¼ where

1  r2 1  2r cosðtÞ þ r 2

ð8:77Þ

    1 1 t ¼ 2 0 1  in steps of 2Ns Ns

Ns is the total number of samples used (size), and r is the smoothing control (0 < r < 1). Next, we take the FFT of the original time-averaged spectrum and of the smoothing kernel to convert them into time domain, then we multiply them together. In the frequency domain, this is equivalent to a convolution between the original spectrum and the kernel. The result from this multiplication is then transformed back into the frequency domain to obtain the smoothed optical spectrum. Finally, we normalise the smoothed result to its size, Ns . Formally, the smoothing algorithm can be summarised as follows:

226

CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

tk tx td sd tsd

¼ fftðkÞ; ¼ fftðxÞ; ¼ mulcpxðtk ; tx Þ; ¼ invfftðtd Þ; ¼ realðsd Þ=size½realðsd Þ;

FFT of kernel FFT of data convolve kernel and data transform back to frequency domain normalise the data

note: sizeðsd Þ ¼ 2  size½realðsd Þ where x is the data (array) we want smoothed, and tsd is the final smoothed data. The function fft is the FFT, mulcpx is the multiplication in time domain (convolution in frequency domain), invfft is the inverse FFT, real denotes taking real values of the complex data, and size finds the total number of data samples. The above smoothing method can also be used for finding the FWHM of highly unstable pulses but usually the pulsewidth is not of interest outside the stable region of operation in ultrashort-pulse generation.

8.12

SUMMARY

Microwave circuit techniques when applied to semiconductor laser modelling provide us with additional insight into the operation of the device. The transmission-line laser model (TLLM) can be classified as a distributed-element circuit model, which is based on the 1-D transmission-line matrix (TLM) method. The building blocks of a TLM network are the TLM link lines and stub lines. It has been shown how the scattering matrices of several TLM sub-networks may be derived by using Thevenin equivalent circuits. Scattering and connecting are the two main processes that form the basis of TLM. The scattering matrix at a TLM node takes incident voltage pulses and operates on them to produce reflected pulses that travel away from the node. The connecting matrix then directs the reflected pulses from one TLM node to adjacent TLM nodes, where they become incident pulses of the adjacent nodes in the next time iteration. In TLLM, the voltage pulses represent the optical waves that circulate inside the laser cavity. All the important optical processes in the laser are taken into account, such as the spectrally dependent gain of stimulated emission, material and scattering loss, spontaneous emission, carrier–photon interaction, and carrier-dependent phase shift. The microwave circuit elements of TLLM are used to describe these laser processes on an equivalence basis. The baseband transformation method is used to enhance the computational efficiency by down-converting from the true optical carrier frequency to its equivalent baseband value. The TLLM is a stochastic laser model because random noise effects are included, making it a highly realistic model compared to deterministic laser models [60–61]. However, intensive time averaging and smoothing techniques are required to obtain the desired signal, which may otherwise be masked by noise.

8.13

REFERENCES

1. Gordon, E. I., Optical maser oscillators and noise, Bell Syst. Tech. J., 43, 507–539, 1964. 2. Kobayashi, S., Injection-locked semiconductor laser amplifiers, in Coherence, Amplification, and Quantum Effects in Semiconductor Lasers, (Ed. Y. Yamamoto) New York: John Wiley & Sons, Inc., 1991, p. 646.

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3. Adler, R., A study of locking phenomena in oscillators, Proc. I.R.E & Wav. Elec., June, pp. 351– 357, 1946. 4. Altman, J. L., Microwave Circuits. Princeton, New Jersey: D. Van Nostrand Company, Inc., 1964. 5. Collins, R. E., Foundations of Microwave Engineering. New York: McGraw-Hill, 1966. 6. Agrawal, G. P. and Dutta, N. K., Semiconductor Lasers, 2nd edition. New York: Van Nostrand Reinhold, 1993. 7. Ramo, S., Whinnery, J. R. and Van Duzer, T., Fields and Waves in Communication Electronics, 3rd edition. New York: John Wiley & Sons, 1994. 8. Vasil’Ev, P. P., Ultrashort pulse generation in diode lasers, Opt. Quant. Electron., 24, 801–824, 1992. 9. Johns, P. B. and Beurle, R. L., Numerical solution of 2-dimensional scattering problems using a transmission-line matrix, Proc. IEEE, 188(9), 1203–1208, 1971. 10. Johns, P. B., A new mathematical model to describe the physics of propagation, Radio Electron. Eng., 44, 657–666, 1974. 11. Christopoulos, C., The Transmission-Line Modeling Method TLM. New York: The IEEE/OUP Series on Electromagnetic Wave Theory, 1995. 12. Hoefer, W. J. R., The Transmission-Line Matrix Method – Theory and Applications, IEEE Trans. Microwave Theory Tech., MTT-33(10), 882–893, 1985. 13. Bandler, J. W., Johns, P. B. and Rizk, M. R. M., Transmission-Line Modeling and Sensitivity Evaluation for Lumped Network Simulation and Design in the Time Domain, J. Frank. Inst., 304, 15–32, 1977. 14. Johns, P. B. and O’Brien, M. J., Use of transmission-line modelling (t.l.m) method to solve nonlinear lumped networks, Radio Electron. Eng., 50, 59–70, 1980. 15. Johns, P. B., A simple explicit and unconditionally stable numerical routine for the solution of the diffusion equation, Int. J. Num. Methods Eng., 11, 1307–1328, 1977. 16. Lowery, A. J., New dynamic semiconductor laser model based on the transmission-line modelling method, IEE Proceedings Pt. J., 134, 281–289, 1987. 17. Lowery, A. J., Transmission-line modelling of semiconductor lasers: the transmission-line laser model, Int. J. Num. Model., 2, 249–265, 1989. 18. Johns, P. B., On the Relationship Between TLM and Finite-Difference Methods for Maxwell’s Equations, IEEE Trans. Microwave Theory Tech., MTT-35(1), 60–61, 1987. 19. Boctor, S. A., Electric Circuit Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1987. 20. Lowery, A. J., Dynamic Modelling of Distributed-Feedback Lasers Using Scattering Matrices, Electron. Lett., 25(19), 1307–1308, 1989. 21. Lowery, A. J. and Hewitt, D. F., Large-signal dynamic model for gain-coupled DFB lasers based on the transmission-line model, Electron. Lett., 28(21), 1959–1960, 1992. 22. Lowery, A. J., Integrated mode-locked laser design with a distributed Bragg reflector, IEE Proceedings Pt. J., 138(1), 39–46, 1991. 23. Lowery, A. J., New dynamic model for multimode chirp in DFB semiconductor lasers, IEE Proceedings Pt. J., 137(5), 293–300, 1990. 24. Nguyen, L. V. T., Lowery, A. J., Gurney, P. C. R. and Novak, D., A Time Domain Model for HighSpeed Quantum-Well Lasers Including Carrier Transport Effects, IEEE J. Select. Top. Quantum. Electron., 1(2), 494–504, 1995. 25. Lowery, A. J., New dynamic multimode model for external cavity semiconductor lasers, IEE Proceedings Pt. J., 136(4), 229–237, 1989. 26. Lowery, A. J., New time-domain model for active mode locking based on the transmission line laser model, IEE Proceedings Pt. J., 136(5), 264–272, 1989. 27. Zhai, L., Lowery, A. J. and Ahmed, Z., Locking bandwidth of actively mode-locked semiconductor lasers using fiber-grating external cavities, IEEE J. Quantum. Electron., 31(11), 1998–2005, 1995. 28. Lowery, A. J., New inline wideband dynamic semiconductor laser amplifier model, IEE Proceedings Pt. J., 135(3), 242–250, 1988.

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29. Wong, W. M. and Ghafouri-Shiraz, H., Integrated semiconductor laser-transmitter model for microwave-optoelectronic simulation based on transmission-line modelling, IEE Proceedings Pt. J., 146(4), 181–188, 1999. 30. Wong, W. M. and Ghafouri-Shiraz, H., Dynamic model of tapered semiconductor lasers and amplifiers based on transmission line laser modeling, IEEE J. Select. Top. Quantum. Electron., 6(4), 585–593, 2000. 31. Davis, M. G. and Dowd, R. F., A Transfer Matrix Method Based Large-Signal Dynamic Model for Multielectrode DFB Lasers, IEEE J. Quantum. Electron., 30(11), 2458–2466, 1994. 32. Marcenac, D. D. and Carroll, J. E., Quantum-mechanical model for realistic Fabry–Perot lasers, IEE Proceedings Pt. J., 140(3), 157–171, 1993. 33. Tsang, C. F., Marcenac, D. D., Carroll, J. E. and Zhang, L. M., Comparison between ‘power matrix mode’ and ‘time domain model’ in modelling large signal response of DFB lasers, IEE Proceedings Pt. J., 141(2), 89–96, 1994. 34. Zhang, L. M. and Carroll, J. E., Large-Signal Dynamic Model of the DFB Laser, IEEE J. Quantum Electron., 28(3), 604–611, 1992. 35. Lowery, A. J., A Study of the Static and Multigigabit Dynamic Effects of Gain Spectra Carrier Dependence in Semiconductor Lasers Using a Transmission-Line Laser Model, IEEE J. Quantum Electron., 24, 2376–2385, 1988. 36. Lowery, A. J., Gurney, P. C. R., Wang, X. H., Nguyen, L. V. T., Chan, Y. C. and Premaratne, M., Time-domain simulation of photonic devices, circuits, and systems, Proc. SPIE, 2693, 624–635, 1996. 37. Siegman, A. E., Lasers. Mill Valley, California: University Science Books, 1986. 38. Mc Ilroy, P. W., Kurobe, A. and Uematsu, Y., Analysis and application of theoretical gain curves to the design of multi-quantum-well lasers, IEEE J. Quantum Electron., 21, 1958–1963, 1985. 39. Nguyen, L. V. T., Lowery, A. J., Gurney, P. C. R., Novak, D. and Murtonen, C. N., Efficient Material-Gain Models for the Transmission-Line Laser Model, Int. J. Num. Model., 8, 315–330, 1995. 40. Lowery, A. J., Model for multimode picosecond dynamic laser chirp based on transmission line laser model, IEE Proceedings Pt. J., 135(2), 126–132, 1988. 41. Henry, C. H., Theory of the Linewidth of Semiconductor Lasers, IEEE J. Quantum Electron., QE-18(2), 259–264, 1982. 42. Valle, A., Rodriguez, M. and Mirasso, C. R., Analytical calculation of timing jitter in single-mode semiconductor lasers under fast periodic modulation, Opt. Lett., 17(21), 1523–1525, 1992. 43. Derickson, D. J., Morton, P. A., Bowers, J. E. and Thorton, R. L., Comparison of timing jitter in external and monolithic cavity mode-locked semiconductor lasers, Appl. Phys. Lett., 59(26), 3372–3374, 1991. 44. Tucker, R. S., Eisenstein, G. and Korotky, S. K., Optical Time-Division Multiplexing For Very High Bit-Rate Transmission, IEEE J. Lightwave Technol., LT-6(11), 1737–1749, 1988. 45. Salathe, R., Voumard, C. and Weber, H., Rate equation approach for diode lasers. Part 1: Steadystate solutions for a single diode, Opto-electron., 6, 451–456, 1974. 46. Danielson, M., A theoretical analysis for Gb/s pulse code modulation of semiconductor lasers, IEEE J. Quantum Electron., QE-12, 657–659, 1976. 47. Boers, P. M., Vlaardingerbroek, M. T. and Danielsen, M., Dynamic Behaviour of Semiconductor Lasers, Electron. Lett., 11(10), 206–208, 1975. 48. Okoshi, T. and Kikuchi, K., Coherent Optical Fiber Communications. Dordrecht: Kluwer Academic Publishers, 1988. 49. Lowery, A. J., A new time-domain model for spontaneous emission in semiconductor lasers and its use in predicting their transient response, Int. J. Num. Model., 1, 153–164, 1988. 50. Chapter G05 Random Number Generators, NAG Fortran Library (Mark 16). 51. Kreyszig, E., Advanced Engineering Mathematics, 7th edition. New York: John Wiley & Sons, Inc., 1993.

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52. Marcuse, D., Classical derivation of the laser rate equation, IEEE J. Quantum Electron., QE-19, 1228–1231, 1983. 53. Chapter C06 Summation of Series, NAG Fortran Library (Mark 16). 54. Chan, Y. C., Premaratne, M. and Lowery, A. J., Semiconductor laser linewidth from the transmission-line laser model, IEE Proceedings Pt. J., 144(4), 246–252, 1997. 55. Marcuse, D. and Lee, T. P., On Approximate Analytical Solutions of Rate Equations for Studying Transient Spectra of Injection Lasers, IEEE J. Quantum Electron., QE-19(9), 1397–1406, 1983. 56. Lee, T. P., Burrus, C. A., Copeland, J. A., Dentai, A. G. and Marcuse, D., Short-Cavity InGaAsP Injection Lasers: Dependence of Mode Spectra and Single-Longitudinal-Mode Power on Cavity Length, IEEE J. Quantum Electron., QE-18(7), 1101–1113, 1982. 57. Lowery, A. J. and Chan, Y. C., Deterministic spectrum simulation using the transmission line laser model, Electron. Lett., 20(2), 134–136, 1994. 58. Deardorff, J. E. and Trimble, C. R., Calibrated Real-Time Signal Averaging, HP Journal, April, pp. 8–13, 1968. 59. SPSS Inc., Sigmaplot 4.0, smooth.xfm. 60. Marcuse, D., Computer model of an injection laser amplifier, IEEE J. Quantum Electron., QE-19(1), 63–73, 1983. 61. Bendelli, G., Komori, K. and Arai, S., Gain Saturation and Propagation Characteristics of IndexGuided Tapered-Waveguide Traveling-Wave Semiconductor Laser Amplifiers (TTW-SLAs), IEEE J. Quantum Electron., 28(2), 447–457, 1992.

9 Analysis of DFB Laser Diode Characteristics Based on Transmission-Line Laser Modelling (TLLM) 9.1

INTRODUCTION

In Chapter 8 we introduced transmission-line laser modelling (TMLM). In this chapter, TLLM will be modified to allow the study of dynamic behaviour of distributed feedback laser diodes, in particular the effects of multiple phase shifts on the overall DFB LD performance. We can easily model any arbitrary phase-shift value by inserting some phaseshifter stubs into the scattering matrices of TLLM. This helps to make the electric field distribution and hence light intensity of DFB LDs more uniform along the laser cavity and hence minimise the hole burning effect.

9.2

DFB LASER DIODES

As explained in Chapter 2, the feedback necessary for the lasing action in a DFB laser diode is distributed throughout the cavity length. This is achieved through the use of a grating etched in such a way that the thickness of one layer varies periodically along the cavity length. The resulting periodic perturbation of the refractive index provides feedback by means of Bragg diffraction rather than the usual cleaved mirrors in Fabry–Perot laser diodes [1–3]. Bragg diffraction is a phenomenon which couples the waves propagating in the forward and backward directions. Mode selectivity of the DFB mechanism results from the Bragg condition. When the period of the grating, , is equal to m
B =2neff , where
B is the Bragg wavelength, neff is the effective refractive index of the waveguide and m is an integer representing the order of Bragg diffraction induced by the grating, then only the mode near the Bragg wavelength is reflected constructively. Hence, this particular mode will lase whilst the other modes exhibiting higher losses are suppressed from oscillation. The coupling between the forward and backward waves is strongest when m ¼ 1 (i.e. first-order

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

232

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

Bragg diffraction). By choosing  appropriately, a device can be made to provide distributed feedback only at the selected wavelengths. In recent years, DFB LDs have played an important role in the long-span and high-bit-rate optical fibre transmission systems because of their stronger capability of single longitudinal mode operation. To overcome the two modes’ degeneracy and achieve a pure single-mode operation, quarterly-wavelength-shifted (QWS) DFB lasers have been proposed [4]. However, in such QWS DFB lasers, spatial hole burning effects enhance the side modes when the coupling coefficient is large (i.e. L > 3). In order to combat this effect, multiplePhase shift DFB lasers have been proposed [5–8]. It has been shown that side modes can be effectively suppressed and a stable and pure single-mode operation results. With the development of laser structures, efficient and relatively accurate simulation models are becoming more and more important for laser designs and operation optimisation due to the complication and expense involved in laser fabrication processes. Distributed feedback semiconductor lasers have a greater mode selectivity than Fabry– Perot devices and so are preferred as sources for long-haul high-capacity-fibre systems. However, dynamic single-mode (DSM) operation is still difficult. Accurate multi-mode dynamic computer models could help in designing DSM DFB devices. Many DFB models calculate the individual mode threshold gains in an attempt to assess wavelength stability. However, these usually neglect the saturation and inhomogeneity of the gain which occurs at the onset of lasing. Dynamic models are available, but these assume a single oscillating mode, making the study of mode stability impossible. The ideal semiconductor laser model would mimic the operation of the real devices in every detail, simulating all characteristics of the laser while accounting for all variations in device structure, processing, drive electronics and external optical components [9–10]. The model could be connected to other device models to form an optical system model. Such a model would improve the design of photonic devices, circuits and systems. It could also be used for detailed optimisation in particular applications. Limitations in computing resources require that simplifications and assumptions have to be made before a model is developed. Many optoelectronic device models use rate equations to describe the interactions between the average electron and photon populations in the device [9–11]. Numerous adaptations of this technique have been proposed. For example, using a photon rate equation for each longitudinal laser mode gives the laser’s spectrum during modulation [12] and dynamic frequency shifting (chirping) may be estimated from the transient responses of both populations [13]. The laser rate equations may also describe saturation in laser amplifiers [14], the dynamic behaviour of model-locked lasers [15] and the transient response of cleaved-coupled-cavity lasers [16]. The limitation of using photon density as a variable is that it does not contain optical phase information. Optical phase is important when there is a set of coupled optical resonators such as in coupled-cavity lasers, external-cavity lasers, DFB lasers, or even Fabry–Perot lasers with unintentional feedback from external components. In these cases, the output wavelength of the devices and its current to light characteristics are determined by optical interference between the resonators. Although rate equations can be used in simple cases, by calculating effective reflection coefficients at discrete wavelengths [16], finding these wavelengths becomes difficult with multiple resonators exhibiting gain and variable refractive indices, such as in the DFB laser [17]. A development of the rate equation approach is to use a SPICE-compatible equivalent circuit of the laser diode. This may be used to find the time-varying photon density for a given drive current waveform or, alternatively, to find the frequency response of the devices

DFB LASER DIODES

233

[18]. This approach has an advantage in that it includes parasitic components in the laser chirp and mount and can be linked to models of the drive circuit for evaluation of the system’s response to modulation. An alternative variable to photon density is optical field, which contains phase information and thus offers the possibility of dealing with multiple reflections. The optical field within a resonator system may be solved in the frequency domain or in the time domain. Frequencydomain models often use a transfer-matrix description of the laser that may be obtained by multiplying together the transfer matrices describing each individual reflection [19–20]. However, if the spectrum of a modulated laser is required, the multiplication has to be performed for each wavelength at each time step [17]. This is computationally inefficient. Time-domain models using optical fields are better suited to modulated devices with multiple resonators than frequency-domain models because the former are simpler to develop and require less computation. Time-domain optical-field models are commonly based on scattering matrix descriptions of the individual reflections and of the gain medium. The scattering matrices may be connected by delays (transmission lines) so that reflected waves out of one scattering matrix can be connected to each adjacent matrices after the delay. The delays represent the optical propagation time along a portion of the waveguide. A solution for the network is found by iteration, each iteration representing an increase in time equal to the delay. At high-frequency modulation (16–17 GHz) [21], the dynamic characteristics of lasers are important and design methods that can help to predict the chirp and modulation efficiency are needed. The dynamic response of lasers is generally studied by solving a set of rate equations that govern the interaction between the carriers and photons inside the active region of the laser cavity. In the earliest work, the equations are usually linearised to allow solutions to be found for small-signal oscillations. Although this gives insight to the important physical parameters, it has limited applicability. Large-signal dynamics with nonlinear effects such as gain saturation, spatial hole burning and changes of electron and photon densities along the length of lasers are now essential in the study of DFB lasers where these effects are more significant than in Fabry–Perot lasers [22–23]. The transmission-line laser model based on the transmission-line modelling (TLM) method, is being developed to study many of the dynamic effects in lasers. Transmission-line laser modelling, which was developed by Lowery, employs timedomain numerical algorithms for laser simulation [24–33]. This model splits the laser cavity longitudinally into a number of sections. In each section, TLLM uses a scattering matrix to represent the optical process, such as stimulation emission, spontaneous emission and attenuation. The matrices of these sections are then connected by transmission lines, which account for the propagation delays of the waves. From the iterations of scattering and connecting processes, the output electric field in the time domain can be obtained. Then, by applying a Fourier transform, we can easily obtain the laser output spectra. Large-signal dynamics with non-linear effects such as the changes of electron and photon densities along the length of the laser and spatial hole burning are key issues in the analysis of DFB laser diodes. These dynamic effects can be investigated easily by using transmission-line laser modelling. TLLM models have been used to analyse QWS DFB LDs [32]. With the insertion of a zero-reflection interface (identity matrix) half way along the cavity, the effects of QWS on laser operation have been simulated successfully. However, using this method we can only analyse DFB laser structures with one =2 phase shift at the centre of the cavity. We cannot use this technique to analyse other phase shift values or multi-phase-shift (MPS) lasers.

234

9.3

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

TLLM FOR DFB LASER DIODES

In general, two operations, scattering and connecting, are involved in transmission-line laser modelling. The scattering operation takes voltage pulses incident on the nodes, k Ai , and scatters them to give voltage pulses reflected from the nodes, k Ar . The reflected and incident voltage pulses are related together via the following scattering matrix, S which includes stimulation, emission, spontaneous emission and attenuation processes. That is kA

r

¼ S k Ai þ I s

ð9:1Þ

where k is the iteration number and I s is the spontaneous wave. As discussed in Chapter 8, the scattering operation can be derived from a knowledge of the impedances of the transmission lines and associated components, such as resistors at the nodes. Equation (9.1) can be modified to include the source voltage pulses, k As , so kA

r

¼ S k Ai þ k As þ I s

ð9:2Þ

The reflected pulses that propagate to the next scattering nodes become new incident pulses for the next scattering operation. This process can be expressed as kþ1 A

i

¼ C k Ar

ð9:3Þ

In eqn (9.3) C is the connection matrix that can be derived from the topology of the network. It should be noted that for all pulses to arrive at the nodes synchronously, the transmission lines must have equal delay times. Each delay time should also be equal to the iteration time step
t. In the numerical calculation, we need to initialise the value of voltage Ai and then repeat eqns (9.1) and (9.2) to find the time evolution of the voltage Ai or Ar. In transmissionline laser models, the voltage pulses represent the optical fields along the cavity. A chain of transmission lines connects these fields from the laser rear facet via optical cavity to the laser front facet. The scattering matrices represent the optical processes of stimulated emission, spontaneous emission and attenuation. The local carrier density will be updated according to the rate equation model at each time step and the magnitudes of these processes at a particular matrix will also be re-calculated with the new information of the carrier density. It should be noted that the local carrier density should be updated at each time step ð
tÞ accoding to the rate equation model. The updated carrier density will then be used to set the magnitude of the optical processes in the scattering matrix.

9.4

A DFB LASER DIODE MODEL WITH PHASE SHIFT

In a DFB laser diode, the forward and backward waves are coupled along the entire cavity length because of the refractive index modulation along the cavity. This coupling can be

Figure 9.1 The TLLM model for uniform DFB laser diodes.

A DFB LASER DIODE MODEL WITH PHASE SHIFT

Figure 9.2

235

The TLLM model representing a phase shift.

represented by impedance discontinuities placed between the model sections as shown in Fig. 9.1. However, a model for the phase shift is needed to model such DFB laser diodes. In doing so, phase stubs are employed and connected to the main transmission line. In this model circulators are used (see Fig. 9.2) to send the waves out of the stubs in the correct direction. For example, a forward wave will enter the first left-hand circulator (port 1) and is directed to the stub port (port 2). Since the stub presents an impedance mismatch, part of the wave will be reflected back into port 2. The circulator then directs this reflected wave to port 3, where it continues on as a forward wave. The remainder of the wave enters the stub to be delayed before returning to port 2 to be directed to port 3. Backward waves simply pass from port 3 to port 1 of this first circulator. A second set of three-port circulators is used to delay the backward waves. The phase delay caused by a stub can be varied by altering its impedance. For example an infinite stub impedance gives a reflection with zero phase shift; a matched capacitive stub gives a phase shift of ð2p
tf Þ radians; a zero impedance stub gives p radians; a matched inductive (shorted) stub gives ð2p f
tÞ radians where f is the optical frequency. Other phase shifts are available over a limited bandwidth by using other reflection coefficients. A complete DFB laser diode model with phase shift is shown in Fig. 9.3. Here, scattering matrices have been inserted between the circulators of each section. Also, alternate connecting transmission lines have different impedances. This creates impedance mismatches at the section boundaries, which couple the forward and backward waves [28]. Each section has an associated carrier rate equation model to enable the local gain, refractive index and spontaneous noise to be calculated from the injection current and the carrier recombination rates [24].

Figure 9.3 A complete DFB laser diode model with phase shift. p is a phase-shift stub, l and c are gain-filter stubs and i is the injection current.

236

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

The single scattering matrix S shown in Fig. 9.3 represents a section of laser with length
L. This matrix operates on the forward- and backward-travelling incident waves to produce a set of reflected waves. These are then passed along the transmission lines ready to become new incident waves upon adjacent scattering nodes after one iteration time step. If two sections of the model were to be used to represent each period of the DFB grating on the real device, the number of sections and hence the computational task would be excessive. However, it is possible to represent an odd number of grating periods with a single pair of model sections without compromising the model’s accuracy [28]. This technique relies on the model having a square grating modulation. This can be decomposed into a number of sinusoidal gratings at harmonics of the grating period by Fourier techniques. One of these harmonics models the real device’s grating period. Note that the amplitude of each harmonic decreases with the harmonic number, that is, the fifth harmonic produces a coupling of one-fifth of the amplitude of the fundamental harmonic. For this example, the coupling of each period of the square grating has to be increased by a factor of five over the coupling of the real laser’s grating to compensate. A simpler and much neater rule is that the coupling  per unit length must be equal for model and real devices [28]. If a small number of sections is used, the optical field will be sampled less than once per wave period. This under-sampling is essential for realistic computer times. Under-sampling has been used in all TLLMs and does not compromise accuracy if the sampling rate (section length/group velocity) is more than twice the bandwidth of the optical wave [24]. The use of two sections per grating period ensures that the DFB’s spectrum always lies near the centre of the modelled spectrum.

9.5

ANALYSIS OF TLLM FOR DFB LASER DIODES

Once the transmission-line representation of the device has been derived, an algorithm can be produced. One of the advantages of TLLM is that the algorithm is always an exact representation of the transmission-line model; no inaccuracies are introduced once the transmission-line representation has been formulated. This means that all approximations have physical meaning because they are associated with the parameters of the transmission lines. The terms in eqns (9.1) to (9.3) will now be derived for the DFB laser model. Note that the travelling optical electric fields are represented by voltage pulses A (forwards) and B (backwards) in the model. Thus, a unity constant m, with dimension of metres, is used to convert between electric field and voltage to maintain dimensional correctness.

9.5.1 Scattering Matrix for a Uniform DFB LD The scattering matrix can be split into two scattering matrices, one for each wave direction. This is possible as there is no cross coupling between the wave directions in the scattering operation. In a uniform DFB LD, the scattering process for the forward wave, with incident pulses from the previous section Ai ðnÞ, the gain filter’s capacitive stub AiC ðnÞ and the gain filter’s inductive stub AiL ðnÞ, may be expressed as [27] 2

3 2 3 2 3 AðnÞ r AðnÞ i Is Zp =2 S 4 AC ðnÞ 5 ¼ Su 4 AC ðnÞ 5 þ 4 0 5 0 k AL ðnÞ k AL ðnÞ k

ð9:4Þ

ANALYSIS OF TLLM FOR DFB LASER DIODES

237

where 2 16 Su ¼ 4 y

IS ¼

ðg þ yÞ

2 YC

g g

2YC  y 2YC

2 YL

3

7 2YL 5 2YL  y

qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2S ¼ mNðnÞ 2bLhfB=ZP

ð9:5Þ

ð9:6Þ

Zp ¼ 120png =n2eff

ð9:7Þ

y ¼ 1 þ YC þ YL

ð9:8Þ

YL ¼ YC tan2 ðcp
t=
Þ pffiffiffiffiffiffiffiffiffiffiffi Q ¼ YL YC

ð9:9Þ ð9:10Þ


t ¼  neff
L=c

ð9:11Þ

g ¼ exp½a
L ðNðnÞ  N0 Þ=2  1

ð9:12Þ

¼ expðsc
L=2Þ

ð9:13Þ

where Ai ðnÞ, AiC ðnÞ and AiL ðnÞ are the travelling waves in the main transmission line, the capacitive stub and the inductive stub, respectively. The parameters i and r denote incident and reflected pulses to and from the main scattering matrices, respectively. k is the iteration number, Su is the scattering matrix, IS the noise current representing spontaneous emission [26], Zp is the transverse wave impedance for a TE mode in the cavity [24], m is a unit constant with dimension of metres, L is the laser cavity length, hf is the photon energy, B is the radiative recombination coefficient, ng is the group refractive index, neff is the effective mode refractive index, YC is the capacitive admittance of the open-circuit stub, YL is the inductive admittance of the short-circuit stub,
t is the time step, Q is the quality factor of a parallel RLC filter whose R value is unity [34] (see also Fig. 8.9), c and l are, respectively, the light velocity and wavelength in free space, g is the gain coefficient, a is the gain coefficient per unit carrier coefficient,
L the section length,  is the confinement factor, NðnÞ is the carrier density within the nth section and N0 is the carrier density for transparency, g is the attenuation caused by free carrier absorption and scattering across a section and sc is the power attenuation coefficient. It should be noted that, as mentioned in section 2.3.4, due to the dispersive properties of the semiconductor, the actual material gain g given in eqn (9.12) is also affected by the optical frequency f, and hence the wavelength l. So far, the gain has been assumed to be at the resonant frequency. However, if the optical frequency is tuned away from the resonant peak, the exact value of the gain becomes different from the peak value. On the basis of experimental observation, Westbrook [33] extended the linear peak gain model further so gðN h f Þ ¼ a1 ðN  N0 Þ  a2 ½h f  ðE0 þ a3 ðN  N0 ÞÞ2

ð9:14Þ

where h ¼ 6:626  1034 J.s is Planck’s constant, f ¼ c=
 is the optical frequency, a1 is dg=dN at the gain curve peak a1 ¼ 2:7  1016 cm2 , N0 is the transparency carrier density ðN0 ¼ 9  1017 cm3 Þ, a2 is the width parameter of the gain spectrum

238

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM



a2 ¼ 4  105 cm1 eV2 , E0 is the gain peak and a3 is dE0 =dN,  energy at the transparency the gain peak position carrier dependence a3 ¼ 1:4  1020 eVcm3

9.5.2 Scattering Matrix for the DFB Laser Diode with Phase Shift For a DFB LD with phase shift, the scattering process for the forward wave, with incident pulses from the previous section Ai ðnÞ, the gain filter’s capacitive stub AiC ðnÞ, the gain filter’s inductive stub AiL ðnÞ and the phase shifting stub AiP ðnÞ, is given by [30] 3r 3i 2 2 2 3S AðnÞ AðnÞ IS ZC =2 6 A ðnÞ 7 6 A ðnÞ 7 6 7 0 6 C 7 6 C 7 6 7 ð9:15Þ 7 ¼ Sp 6 7þ 6 6 7 4 AL ðnÞ 5 4 AL ðnÞ 5 4 5 0 0 k AP ðnÞ k AP ðnÞ k where 2

ðg þ yÞðZS  1Þ 6 gðZS þ 1Þ 1 6 Sp ¼ 6 yð1 þ Zs Þ 4 gðZS þ 1Þ 2 ðg þ yÞZS

2 YC ðZS  1Þ ð2YC  yÞðZS þ 1Þ

2 YL ðZS  1Þ 2YL ðZS þ 1Þ

2y 0

2YC ðZS þ 1Þ

ð2YL  yÞðZS þ 1Þ

0

4 YC ZS

4 YL ZS

3 7 7 7 5

yð1  ZS Þ ð9:16Þ

where 1  ZS ¼   p neff  tan   l  
L NðnÞ  Np dnr ¼ ng dN

ð9:17Þ

ð9:18Þ

In the above equations n is the number of sections, ZS is the phase-adjusting stub’s impedance normalised to the cavity wave impedance,  is the change in phase length across a section which is due to the dynamic change of the carrier density, neff is the guide’s group effective refractive index, Np is an arbitrary carrier density for zero phase shift and is usually set to the threshold carrier density [25], l is the light wavelength, nr is the refractive index and dnr =dN is the active region’s refractive index carrier dependence which is related to the Henry factor H as [35] dnr
H dg
H a ¼ ¼ dN 4p dN 4p

ð9:19Þ

The scattering process for the backward wave can be obtained by using the above formula with all wave amplitudes A to be replaced by wave amplitudes B. It should be noted that all parameters in the above equations may vary from one section to another, hence they should have subscripts n, also some parameters are time dependent and vary with the iteration number k.

239

CONNECTION MATRIX C

9.6

CONNECTION MATRIX C

The connection equations in TLLM DFB models which describe the cross coupling between the two wave directions occurring at the section interfaces can be expressed as [28]  kþ1

  Aðn þ 1Þ i 1 þ 
L ¼ BðnÞ 
L


L 1  
L

 k

AðnÞ Bðn þ 1Þ

r ð9:20Þ

for a low–high impedance boundary, and  kþ1

  Aðn þ 2Þ i 1  
L ¼ Bðn þ 1Þ 
L


L 1 þ 
L

 k

Aðn þ 1Þ Bðn þ 2Þ

r ð9:21Þ

for a high–low impedance boundary. k is the iteration number and i denotes incident pulses to the main scattering matrices S,  is the coupling coefficient and r denotes reflected pulses from the main scattering matrices. Note that there is a one-iteration time step delay as the optical field samples move along the lines. For a standard DFB device, eqns (9.20) and (9.21) are applied alternately along the device length, i.e. n ¼ ð1; 3; 5; 7; . . .Þ. For QWS grating devices [31] a zero-reflection interface (identity matrix) is inserted half way along the cavity. Figure 9.4 shows the

Figure 9.4

The TLLM model for a QWS DFB laser diode.

schematic diagram of the TLLM for the QWS DFB laser diode. When a facet is placed at a low–high resistive can be used giving: kþ1 Bi ðnÞ ¼ pffiffiffi r impedance boundary a simple pffiffiffi termination i r Rk A ðnÞ at the front facet and kþ1 A ð1Þ ¼ Rk B ð1Þ at the rear facet, where R is the facet power reflectivity and n is the number of sections [24,36].

9.6.1 Connection Matrix C for the Stubs Within a Section There are also equations governing the reflections at the ends of the transmission line stubs. These are half a time step long to ensure that pulses arrive back at the originating scattering

240

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

matrix after a delay of one time step. For the inductive stubs in each section, the reflection coefficient is negative, giving  kþ1

  AL ðnÞ i 1 ¼ BL ðnÞ 0

0 1

 k

AL ðnÞ BL ðnÞ

r ð9:22Þ

For the capacitive stubs in each section, the reflection coefficient is positive, giving  kþ1

  AC ðnÞ i 1 ¼ BC ðnÞ 0

0 1

 k

r

AC ðnÞ BC ðnÞ

ð9:23Þ

For the phase-adjusting stubs, which may be inductive or capacitive when the tangent in eqn (9.17) is negative we have  kþ1

  Ap ðnÞ i 1 ¼ Bp ðnÞ 0

 k

Ap ðnÞ Bp ðnÞ



r

0 1

r ð9:24Þ

and when it is positive we have  kþ1

9.7

  Ap ðnÞ i 1 ¼ Bp ðnÞ 0

0 1

k

Ap ðnÞ Bp ðnÞ

ð9:25Þ

CARRIER DENSITY RATE EQUATION

Neglecting the effect of diffusion along the laser cavity we may apply the carrier density rate equation for each section of the model. However, this assumption may not be accurate for laser diodes with low facet reflectivities, including laser diode amplifiers. Electrons may be injected into the conduction band in the active region of the laser by sandwiching it between two higher-bandgap semiconductor layers and applying a forward bias across the structure. The electrons may leave the region into which they have been injected by diffusion to other regions. The electrons may also recombine to the valence band by stimulated or spontaneous recombination. Electrons involved in these processes may be accounted for using a carrier density rate equation given by dNðnÞ NðnÞ ac IðnÞ ¼  ðNðnÞ  N0 ÞSðnÞ þ  dt

s neff ewdL

ð9:26Þ

Where s is the carrier lifetime, IðnÞ is the component of the injection current injected into section n, the laser diode width, thickness and length are, respectively, denoted by w, d and L. e is the electron charge and the photon density SðnÞ within a section is related to the incident waves from either side by h SðnÞ ¼

2  2 i Ai ðnÞ þ Bi ðnÞ neff m2 h f cZp

ð9:27Þ

RESULTS AND DISCUSSIONS

241

The power exiting the front facet, P, is related to the wave incident on the facet from the section Ar ðnÞ as [24] P¼

ð1  RÞwd ½Ar ðnÞ2 m2 Zp

ð9:28Þ

This power is usually averaged over a number of iterations to remove high-frequency components. To measure the uniformity of the light intensity distribution we define a flatness factor F as follows F¼

1 L

ðL

ðPðzÞ  PÞ2 dz

ð9:29Þ

0

where PðzÞ is the normalised longitudinal field intensity distribution along the laser cavity with length L and P is the average of PðzÞ. To minimise the longitudinal spatial hole burning effect and thus get single longitudinal mode oscillation, it has been proposed that the flatness F should be below 0.05 in DFB laser diodes [24].

9.8

RESULTS AND DISCUSSIONS

Based on the TLLM MPS DFB model introduced in section 9.5, the operation of QWS DFB LDs as well as 3PS DFB LDs have been analysed. Parameters used in our analysis are given in Table 9.1. The reflectivity at both facets is assumed to be 0. To make it easy to understand, a schematic diagram of the 3PS DFB LD structure is shown in Fig. 9.5.

Table 9.1

Parameter values used in the TLLM model

Physical parameter

Symbol and value

Operating wavelength Cavity length Active layer thickness Active layer width Spatial gain constant Optical confinement factor Transparency carrier density Carrier lifetime Spontaneous emission coupling factor Internal attenuation Laser facet reflectivities Monomolecular recombination coefficient Bimolecular recombination coefficient Auger recombination coefficient Number of model sections Grating coupling per unit length Initial carrier density


0 ¼1550 mm L ¼ 200 mm d ¼ 0:15 mm w ¼ 2 mm a ¼ 5  1017 cm2  ¼ 0.3 N0 ¼ 9  1017 cm3

¼ 4.0 ns

¼ 104 t¼0 R¼0 A ¼ 108 s1 B ¼ 8:6  1011 cm3 s1 C ¼ 4:0  1029 cm6 s1 S ¼100  ¼ 160 cm1 Ni ¼ 1:5  1018 cm3

242

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

Figure 9.5

Schematic diagram of a 3PS DFB LD.

9.8.1 Dynamic Characteristics To verify the validity of the TLLM MPS DFB model, QWS DFB LDs have been simulated and the results compared with that of Lowery’s QWS model [28]. It should be noted that Lowery [28] inserted a zero-reflection interface halfway along the cavity whereas in this book we have used a transmission line stub to represent a quarter wave phase shift. Figures 9.6 and 9.7 show the transient response and the power spectral density of QWS DFB LDs using the model proposed in section 9.5 and that of Lowery [28] under 100 mA injection current. It should be noted that the power spectral density given in frequency response is the spectral range of the laser (the laser frequency used in the simulation is that for the 1:55 mm

Figure 9.6

Transient response of QWS DFB LDs by different models.

RESULTS AND DISCUSSIONS

Figure 9.7

243

Frequency response of QWS DFB LDs by different models.

wavelength, which is 193.6 THz). The simulated spectral range is related to the number of sections used in the model [24]. Figures 9.6 and 9.7 clearly indicate that the results obtained based on the model proposed in section 9.5 and those of Lowery [28] are in good agreement. This confirms that the new model proposed in this chapter is accurate enough to be used for analysing single and/or multiple phase shifts in DFB LDs. The dynamic characteristics of 3PS DFB lasers have also been analysed using the TLLM MPS DFB model. In the simulation, we have set L1 ¼ L2 ¼ L3 ¼ L4 and 1 ¼ 2 ¼ 3 ¼ p=3. Both the transient and frequency responses of the 3PS DFB laser diode are shown in Figs. 9.8 and 9.9, respectively. Figure 9.9 clearly shows that this 3PS structure operates under the single-mode condition at this coupling coefficient of  ¼ 180 cm1 .

Figure 9.8

Transient response of a 3PS DFB LD.

244

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

Figure 9.9

Frequency response of a 3PS DFB LD.

Figures 9.10 and 9.11 illustrate the evolution of the longitudinal carrier density profile of QWS DFB LDs and 3PS DFB LDs. The carrier densities in these two structures are initially uniform, but spatial hole burning causes local minima at the phase shift positions on the longitudinal carrier density profiles. Compared with the QWS structure, the 3PS structure produces a more uniform carrier density distribution.

9.8.2 Longitudinal Distribution The DFB laser carrier density, photon density, refractive index and field intensity are different in each section and they play important roles in the laser’s operation. Therefore, it

Figure 9.10

Transient longitudinal carrier density of QWS DFB LDs.

RESULTS AND DISCUSSIONS

Figure 9.11

245

Transient longitudinal carrier density of 3PS DFB LDs.

is necessary to study the longitudinal distributions of these physical parameters in DFB lasers. Figure 9.12 shows variations of the longitudinal carrier density distribution for the QWS DFB under static operation. Simulations have been performed by using three different models, namely Lowery’s QWS model [28], the TLLM MPS model introduced in this chapter and the Advanced Laser Diode Simulator (ALDS) model [37]. The ALDS model is commercial software provided by Apollo Photonics Company, which is a frequency-domain, static model [37]. The figure clearly shows that the results of the model introduced in this chapter closely agree with the other two models. The internal electric field intensity distributions of both the QWS DFB and 3PS DFB LDs are shown in Fig. 9.13. In the simulation it is assumed that L1 ¼ L2 ¼ L3 ¼ L4 and

Figure 9.12 Longitudinal carrier density distribution of QWS DFB LDs by different models.

246

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

Figure 9.13

Longitudinal intensity distribution of QWS DFB LDs and 3PS DFB LDs.

1 ¼ 2 ¼ 3 ¼ p=3. All the electric field distributions have been normalised so that the intensity at the laser facets becomes unity. It can be seen that the internal electric field distribution of the QWS DFB LDs is highly non-uniform. This induces the spatial hole burning effect and thus affects the single-mode stability of the laser. With three phase shifts incorporated into the cavity, the intensity distribution spreads out and the overall electric field distribution becomes more uniform and hence single-mode operation can be realised even under a high coupling coefficient. Figures 9.14 to 9.17 show, respectively, the longitudinal distributions of the photon density, carrier density, refractive index and internal intensity under different injection

Figure 9.14 Photon density distribution of 3PS DFB LDs under different injection currents.

RESULTS AND DISCUSSIONS

Figure 9.15

247

Carrier density distribution of 3PS DFB LDs under different injection currents.

currents. In these calculations the threshold current is set to 20 mA. It has been shown that the introduction of phase shifts can flatten the longitudinal distributions of carrier density, photon density, refractive index and internal field intensity.

9.8.3 Effects of the Number of Phase Shifts To study the effects of the number of phase shifts on the field intensity distribution we have assumed the number of phase shifts varies from 1 to 10 and the value of each

Figure 9.16

Refractive index distribution of 3PS DFB LDs under different injection currents.

248

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

Figure 9.17

Normalised intensity distribution of 3PS DFB LDs under different injection currents.

phase shift changes from 0 to p. Also we have assumed that all phase shifts are distributed evenly along the laser cavity, which means if we have one phase shift it will be located at the centre of the cavity. However, if we have two phase shifts one of them should be located at L=3 and the other one at 2L=3, where L is the laser cavity length. For a given number of phase shifts, we can obtain the minimum flatness by changing the value of each phase shift in the range of 0 to p. Figure 9.18

Figure 9.18

Effects of the number of phase shifts on the flatness.

RESULTS AND DISCUSSIONS

249

shows the relationship of the optimum flatness and the number of phase shifts. The results show that more uniform field distributions can be obtained by increasing the number of phase shifts.

9.8.4 Effects of Phase Position and Value on the 3PS DFB LD’s Characteristics The effects of the phase shift positions on the field intensity distribution will be investigated in this section for 3PS DFB LDs. In the analysis, we will set the position of one of the phase shifts at the centre of the cavity while the positions of the other two phase shifts are set symmetrically with respect to the laser cavity centre point (i.e. L1 ¼ L4 ). To indicate the position of these two phase shifts more accurately we define a parameter called normalised location as ¼ L1 =L. The effect of on flatness F is shown in Fig. 9.19 for three different phase shift values of 1 ¼ 2 ¼ 3 ¼ p=2; p=3 and p=4. It can be seen that when the normalised location changes from ¼ 0:075 to 0.275, the parameter F is very small for all three cases. The effects of the phase shifts on the intensity flatness of 3PS DFB LDs have also been investigated for the case when the normalised position ¼ 0:25 (that is L1 ¼ L2 ¼ L3 ¼ L4 ). The contour map of the flatness for various phase values is shown in Fig. 9.20. This map can be used to optimise the value of the phase shifts with respect to the field intensity flatness F. In Fig. 9.20, the x-axis represents the central phase shift 2 whereas the y-axis represents the other phase shifts (i.e. 1 ¼ 3 ). This figure clearly shows that to meet the requirement of F < 0:05, 1 ¼ 3 ¼ side must lie in the range 65 < side < 110 .

Figure 9.19

Relationship between the flatness and the position of the phase shift.

250

ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

Figure 9.20

9.9

Relationship between the flatness and the phase shift for 3PS DFB LDs.

SUMMARY

In this chapter, we have investigated the characteristic performances of MPS DFB LDs using the TLLM MPS model. First, the validity of the TLLM MPS model has been proved by comparing the results of a QWS DFB structure with those obtained by Lowery [28]. Based on the TLLM MPS model, the characteristics of 3PS DFB laser diodes have been investigated. The dynamic characteristics show that 3PS DFB LDs can operate in singlemode condition even under high coupling coefficients. Also it has been found that the intensity distribution along the cavity of the 3PS DFB LD is smoother than that of the QWS DFB LD. The optimum designs of the multi-phase-shift DFB LD have been discussed and the new model results show that the field uniformity can be improved by increasing the number of phase shifts along the laser cavity. Moreover, the simulation results show that for three-phase-shift (3PS) DFB lasers, if one of the phase shifts is fixed at the centre of the cavity and the other two are located symmetrically beside the fixed one, the optimum normalised locations of the side phase shifts are in the range ¼ 0:075 to 0.275. When ¼ 0:25 the optimum values of the two side phase shifts are in the range 65 < side < 110 .

9.10

REFERENCES

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ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM

26. Lowery, A. J., A new time-domain model for spontaneous emission in semiconductor lasers and its use in predicting their transient response, Int. J. Num. Model., 1, 153–164, 1988. 27. Lowery, A. J., Transmission-line modeling of semiconductor lasers: the transmission-line laser model, Int. J. Num. Model., 2, 249–265, 1989. 28. Lowery, A. J., Dynamic modeling of distributed-feedback lasers using scattering matrices, Electron. Lett., 25(19), 1307–1308, 1989. 29. Lowery, A. J., Amplified spontaneous emission in semiconductor laser amplifiers: validity of the transmission-line laser model, IEE Proceedings Part J, 137(4), 241–247, 1990. 30. Lowery, A. J., New dynamic model for multimode chirp in DFB semiconductor lasers, IEE Proceedings Part J, 137(5), 293–300, 1990. 31. Lowery, A. J., Transmission-line laser modelling of semiconductor laser amplified optical communication systems, IEE Proceedings Part J, 139(3), 180–188, 1992. 32. Lowery, A. J., Keating, A. and Murtonen, C. N., Modelling the Static and Dynamic Behaviour of Quarter-wave Shifted DFB Lasers, IEEE J. Quantum Electron., 28(9), 1874–1883, 1992. 33. Westbrook, L. D., Measurements of dg/dN and dn/dN and their dependence on photon energy in
¼ 1:5 mm InGaAsP laser diodes, IEE proceedings Part J, 133(2), 135–142, 1986. 34. Nguyen, L. V. T., Lowery, A. J., Gurney, P. C. R., Novak, D. and Murtonen, C. N., Efficient Material-Gain Models for the Transmission-Line Lase Model, Int. J. Num. Model. Electron. Networks, Devices and Fields, 8, 315–330, 1995. 35. Lowery, A. J., A new time-domain model for active mode-locking based on the transmission-line laser model, IEE Proceedings Part J, 136, 264–272, 1989. 36. Buss, J., Dynamic single-mode operation of DFB lasers with phase shifted gratings and reflecting mirrors, IEE Proceedings Part J, 133, 163–164, 1986. 37. Zhang, L. M., Yu, S. F., Nowell, M. C., Marcenac, D. D., Carroll, J. E. and Plumb, R. G. S., Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain large-signal traveling wave model, IEEE J. Quantum Electron. 30(6), 1389–1395, 1994.

10 Wavelength Tunable Optical Filters Based on DFB Laser Structures 10.1

INTRODUCTION

In recent years, advances in wavelength division multiplexing (WDM) and dense wavelength division multiplexing (DWDM) technology have enabled the deployment of systems that are capable of providing large amounts of bandwidth [1]. Wavelength tunable optical filters appear to be the key components in realising these WDM/DWDM lightwave systems. Optical filtering for the selection of channels separated by 2 nm is currently achievable, and narrower channel separations will be possible in the near future with improved technology [2–3]. This would give more than 100 broadband channels in the low-loss fibre transmission region of 1.3 mm and/or 1.55 mm wavelength bands, with each wavelength channel having a transmission bandwidth of several gigahertz. In practice, grating-embedded semiconductor wavelength tunable filters are among the most popular active optical filters since they are suitable for monolithic integration with other semiconductor optical devices such as laser diodes, optical switches and photodetectors [4]. As a result,
=4-shifted DFB LDs can be used as semiconductor optical filters when biased below threshold [5–6]. This is a grating-embedded semiconductor optical device, which has the advantages of a high gain and a narrow bandwidth. However, the drawbacks are that the bandwidth and transmissivity will change with the wavelength tuning [5]. Fortunately Magari et al. have solved these problems by using a multi-electrode DFB filter [7–8] in which a wavelength tuning range of 33.3 GHz (0.25 nm) with constant gain and constant bandwidth has been obtained by controlling the injection current. Since then, various DFB LD designs have been developed [9–11]. In this chapter, the wavelength selection mechanism is discussed in detail. Subsequently, the idea of the transfer matrix method (TMM) is again thoroughly explored and the derived solutions from coupled wave equations are also discussed in detail. By converting the coupled wave equations into a matrix equation, these transfer matrices can represent the wave propagating characteristics of DFB structures. Therefore, using this approach, various aspects from different DFB optical filters to enhance the active filter functionality shall be investigated. Finally, we shall compare some of the issues for DFB LDs with those for distributed Bragg reflector (DBR) semiconductor optical filters.

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

254 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Figure 10.1

10.2

Operation principle of wavelength selection.

WAVELENGTH SELECTION

Figure 10.1 is a narrowband transmission filter which rejects unwanted channels. If the filter is tunable, the centre wavelength (frequency)
0 (see Fig. 10.1) can be shifted by changing, for example, the voltage or the current applied to the filter. Tunable filters can be classified into three categories: passive, active and tunable LD amplifiers, as shown in Table 10.1 [12–14]. The passive category is composed of those wavelength-selective components that are basically passive and can be made tunable by varying some mechanical elements of the filters, such as mirror position or etalon angle. These include Fabry–Perot etalons, tunable fibre Fabry–Perot filters and tunable Mach–Zehnder (MZ) filters. For Fabry–Perot filters, the number of resolvable wavelengths is related to the value of the finesse F of the filter. One of the advantages of such filters is the very fine frequency resolution that can be achieved. The disadvantages are primarily their tuning speed and losses. The Mach–Zehnder integrated optic interferometer tunable filter is a waveguide device with log2 ðN Þ stages, Table 10.1

A comparison of filtering technologies [12–14] Type

Passive

Etalon (F  200) Fibre Fabry–Perot Waveguide Mach–Zehnder

Active

Fibre Bragg Gratings (FBGs) Electro-optic TE/TM Acousto-optic TE/TM

Laser diode amplifiers

DFB amplifier 2-section DFB amplifier Phase-shift controlled DFB amplifier

Range

No. of channels

Tuning speed

˚ 0.38 A (5 GHz)

˚ 45 A

30 30 128

ms ms ms

˚ –2 A ˚ 1 A ˚ 6A ˚ 10 A

>50 nm ˚ 160 A 400 nm

50 10 100

ms ns 10 ms

˚ 1–2 A ˚ 0.85 A ˚ 0.32 A (4 GHz)

˚ 4–5 A ˚ 6A ˚ 9.5 A (120 GHz)

2–3 8 18

1 ns ns ns

Resolution

SOLUTIONS OF THE COUPLED WAVE EQUATIONS

255

where N is the number of wavelengths. This filter has been demonstrated with 100 wavelengths separated by 10 GHz in optical frequency, and with thermal control of the exact tuning [15]. The number of simultaneously resolvable wavelengths is limited by the number of stages required and the loss incurred in each stage. In the active category, there are two filters based on wavelength-selective polarisation transformation by either electro-optic or acousto-optic means. In both cases, the orthogonal polarisations of the waveguide are coupled together at a specific tunable wavelength. In the electro-optic case, the wavelength selected is tuned by changing the dc voltage on the electrodes; in the acousto-optic case, the wavelength is tuned by changing the frequency of the acoustic drive. A filter bandwidth in full width at half maximum (FWHM) of approximately 1 nm has been achieved by both filters. However, the acousto-optic tunable filter has a much broader tuning range (the entire 1.3 to 1.55 mm range) than the electro-optic type. The third category of filter is LD amplifiers as tunable filters. Operation of a resonant laser structure, such as a DFB or DBR laser, below the threshold results in narrowband amplification. These types of filter offer the following important advantages: electronically controlled narrow bandwidth, the possibility of electronic tuning of the central frequency, net gain (as opposed to loss in passive filters), small size, and integrability. This type of filter is becoming more attractive since only the desired lightwave signal will be passing through the cavity and being amplified simultaneously (thus it is also known as an amplifier filter). We shall investigate the principles and performance of these filters in detail.

10.3

SOLUTIONS OF THE COUPLED WAVE EQUATIONS

In Chapter 2, the derivation of coupled wave equations was discussed in detail. The characteristics of DFB filters can be described by using these coupled wave equations. In the following analysis, we have assumed a zero phase difference between the index and the gain term, hence the complex coupling coefficient can be expressed as RS ¼ SR ¼ i þ jg ¼ 

ð10:1Þ

where  is the complex coupling coefficient. According to eqn (2.98), the complex amplitude terms of the forward, RðzÞ, and backward, SðzÞ, propagating waves can be written as [16] RðzÞ ¼ R1 egz þ R2 egz SðzÞ ¼ S1 egz þ S2 egz

ð10:2Þ ð10:3Þ

where R1 , R2 , S1 and S2 are the complex coefficients and g, known as the complex propagation constant, depends on the boundary conditions at the laser facets. By substituting eqns (10.2) and (10.3) into eqn (2.98), we have R1 ¼ jej S1 ^ 2 ¼ jej S2 R

ð10:4Þ

^ 1 ¼ je j R1 S

ð10:6Þ

S2 ¼ je R2

ð10:7Þ

ð10:5Þ

and

j

256 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

where  ¼ s  j  g ^ ¼ s  j þ g

ð10:8Þ ð10:9Þ

in which s and  are the amplitude gain coefficient and detuning parameter, respectively. If we compare the equations (10.6) and (10.8), a non-trivial solutions exists if the following equation is satisfied ¼

 j ¼ j ^

ð10:10Þ

Similarly, we can obtain the following dispersion equation, which is independent of the residue corrugation phase, . g2 ¼ ðs  jÞ2 þ 2

ð10:11Þ

It is vital to note that in the absence of any coupling effects, the propagation constant is just s  j. With a finite laser cavity length L extending from z ¼ z1 to z ¼ z2 , the boundary conditions at the terminating facets become Rðz1 Þejb0 z1 ¼ ^r1 Sðz1 Þe jb0 z1 Sðz2 Þe jb0 z2 ¼ ^r2 Rðz2 Þejb0 z2

ð10:12aÞ ð10:12bÞ

where ^r1 and ^r2 are the amplitude reflection coefficients at the laser facets z1 and z2, respectively and 0 is the Bragg propagation constant. The above equations could be expanded in such a way that R2 ¼

ð1  r1 Þe2gz1 R1 r1 =  1

ð10:13aÞ

R2 ¼

ðr2  Þe2gz2 R1 1=  r2

ð10:13bÞ

or

In the above equations, r1 and r2 are the complex field reflectivities of the left and right facets, respectively. such that r1 ¼ ^r1 e2jb0 z1 e j ¼ ^r1 e j 1 r2 ¼ ^r2 e

2jb0 z2 j

e

j 2

¼ ^r2 e

ð10:14aÞ ð10:14bÞ

where  1 and  2 are the corresponding corrugation phases at the facets. Equations (10.13a) and (10.13b) are homogeneous in R1 and R2. Hence, in order to obtain a non-trivial solution, we must satisfy ð1  r1 Þe2gz1 ðr2  Þe2gz2 ¼ r1   1  r2

ð10:15Þ

SOLUTIONS OF THE COUPLED WAVE EQUATIONS

257

After further simplification of eqn (10.15), the following eigenvalue equation can be obtained [17] gL ¼

o j sinhðgLÞ n ðr1 þ r2 Þð1  r1 r2 Þ coshðgLÞ ð1 þ r1 r2 Þ1=2 D

ð10:16Þ

where  ¼ ðr1 þ r2 Þ2 sinh2 ðgLÞ þ ð1  r1 r2 Þ2 2

D ¼ ð1 þ r1 r2 Þ  4r1 r2 cosh ðgLÞ 2

ð10:17aÞ ð10:17bÞ

Eventually, we are left with four parameters that govern the threshold characteristics of DFB laser structures – the coupling coefficient, , the laser cavity length, L and the complex facet reflectivities r1 and r2. We have studied the coupling coefficient. Owing to the complex nature of the above equation, numerical methods like the Newton–Raphson iteration technique can be used, provided that the Cauchy–Riemann condition on complex analytical functions is satisfied.

10.3.1 The Dispersion Relationship and Stop Bands As noted in Chapter 2, for a purely index-coupled DFB LD,  ¼ i . For such a case, the dispersion relation of eqn (10.11) is analysed graphically as depicted in Fig. 10.2. At the detuning parameter,  ¼ 0 (Bragg wavelength), the complex propagation constant g is purely imaginary when s <  ðor s = < 1Þ. This indicates evanescent wave propagation in the region known as the stop band [18]. Within this band, any incident wave is reflected efficiently. By contrast, when s >  ðor s = > 1Þ, the propagation constant g will then become a purely real value. As predicted, when s increases, the imaginary part of the propagation constant g decreases appreciably while the real part increases significantly. Consequently, when the waves propagate away from the Bragg wavelength, the imaginary part of the propagation constant g increases at a faster pace than the real part at a given amplitude gain, s . Physically, it means that the wave will be attenuated when it propagates away from the Bragg wavelength. It is paramount to note that we have considered ReðgÞ > 0.

10.3.2 Formulation of the Transfer Matrix From eqns (10.4) to (10.9), we can simply relate the complex coefficients as [17] S1 ¼ e j R1 R2 ¼ e

j

ð10:18Þ ð10:19Þ

S2

And thus eqns (10.2) and (10.3) become RðzÞ ¼ R1 egz þ S2 ej egz gz

SðzÞ ¼ R1 e e þ S2 e j gz

ð10:20Þ ð10:21Þ

258 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Figure 10.2 Normalised dependence of (a) real and (b) imaginary parts of g on  and the amplitude gain s for a purely index-coupled DFB LD.

SOLUTIONS OF THE COUPLED WAVE EQUATIONS

Figure 10.3 section.

259

A simplified schematic diagram for a one-dimensional corrugated DFB laser diode

As shown in Fig. 10.3, the corrugation inside the DFB laser is assumed to extend from z ¼ z1 to z ¼ z2 . The amplitude coefficients at the left and right facets can then be written as Rðz1 Þ ¼ R1 egz1 þ S2 ej egz1 Sðz1 Þ ¼ R1 e e

j gz1

Rð z 2 Þ ¼ R 1 e

gz2

þ S2 e

þ S2 e

Sðz2 Þ ¼ R1 e e

j gz2

gz1

j gz2

ð10:22aÞ ð10:22bÞ

e

ð10:22cÞ

gz2

ð10:22dÞ

þ S2 e

From eqns (10.22a) and (10.22b), R1 and S2 can be expressed as R1 ¼

Sðz1 Þej  Rðz1 Þ ð2  1Þegz1

ð10:23aÞ

S2 ¼

Rðz1 Þe j  Sðz1 Þ ð2  1Þegz1

ð10:23bÞ

Subsequently, by substituting the above equations into eqns (10.22c) and (10.22d), we have   E  E1 ej E  2 E1 Rð z 2 Þ ¼ Rðz1 Þ  Sð z 1 Þ 1  2 1  2   E  E1 ej 2 E  E1 R ð z Þ  Sð z 1 Þ Sð z 2 Þ ¼ 1 1  2 1  2

ð10:24aÞ ð10:24bÞ

where E ¼ egðz2 z1 Þ ;

E1 ¼ egðz2 z1 Þ

ð10:24cÞ

Note that the electric field at the output plane z2 can be expressed in terms of the electric waves at the input plane. Given the solution of the coupled wave equations from eqn (2.98) EðzÞ ¼ RðzÞejb0 z þ SðzÞe jb0 z

ð10:25Þ

260 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Equations (10.24) can then be combined with the solution of the coupled wave equations, the output and input of the electric fields through the matrix approach can therefore be related as         t E R ðz2 Þ ER ð z 1 Þ ER ðz1 Þ t ¼ T ðz2 j z1 Þ ¼ 11 12 ð10:26Þ t21 t22 E S ðz2 Þ ES ð z 1 Þ ES ð z 1 Þ where the matrix Tðz2 j z1 Þ represents any wave propagation from z ¼ z1 to z ¼ z2 and its elements tij ði; j ¼ 1; 2Þ are given as  E  2 E1 ejb0 ðz2 z1 Þ ð10:27aÞ t11 ¼ ð1  2 Þ   E  E1 ej ejb0 ðz2 þz1 Þ ð10:27bÞ t12 ¼ ð1  2 Þ   E  E1 e j e jb0 ðz2 þz1 Þ t21 ¼ ð10:27cÞ ð1  2 Þ  2   E  E1 e jb0 ðz2 z1 Þ ð10:27dÞ t22 ¼ ð1  2 Þ Or from eqn (10.24) in hyperbolic functions [7]      Rð z 2 Þ Rð z 1 Þ f ¼ Fð z 2 j z 1 Þ ¼ 11 f21 Sð z 2 Þ Sð z 1 Þ

   f12 Rð z 1 Þ f22 Sð z 1 Þ

ð10:28Þ

ð  jÞðz2  z1 Þ sinh½gðz2  z1 Þ gð z 2  z 1 Þ

ð10:29aÞ

where f11 ¼ cosh½gðz2  z1 Þ þ

 ðz2  z1 Þ sinh½gðz2  z1 Þ gð z 2  z 1 Þ ðz2  z1 Þ sinh½gðz2  z1 Þ ¼j ðz2  z1 Þ ð  jÞðz2  z1 Þ sinh½gðz2  z1 Þ ¼ cosh½gðz2  z1 Þ  gð z 2  z 1 Þ

f12 ¼ j

ð10:29bÞ

f21

ð10:29cÞ

f22

ð10:29dÞ

Owing to conservation of energy, the determinant of the matrix Tðz2 j z1 Þ must always be unity [19–20]. That is t11 t22  t12 t21 ¼ 1

ð10:30Þ

10.3.3 Solutions of Complex Transcendental Equations using the Newton–Raphson Approximation Transcendental equations will be formed in order to find the threshold gain of DFB LDs [21]. In general these equations can be expressed in complex form such that W ðzÞ ¼ U ðzÞ þ jV ðzÞ ¼ 0

ð10:31Þ

SOLUTIONS OF THE COUPLED WAVE EQUATIONS

261

in which the argument z ¼ x þ jy is a complex number while U ðzÞ and V ðzÞ are the real and imaginary parts of the transcendental equations. If W ðzÞ ¼ 0, the real and imaginary parts will subsequently be zero values. If the firstorder derivative of eqn (10.31) with respect to z is taken as @W ðzÞ @U ðzÞ @V ðzÞ ¼ þj @z @z @z   @U ðzÞ @x @V ðzÞ @x þj ¼ @x @z @x @z @U ðzÞ @V ðzÞ @x þj , ¼1 ¼ @x @x @z

ð10:32Þ

By usingthe Taylor series, the functions U ðzÞ and V ðzÞ can be approximated about the exact solution xapprox ; yapprox such that  @U  @U  xapprox  x þ yapprox  y U xapprox ; yapprox ¼ U ðx; yÞ þ @x @y  @V  @V  xapprox  x þ yapprox  y V xapprox ; yapprox ¼ V ðx; yÞ þ @x @y

ð10:33Þ ð10:34Þ

where ðx; yÞ is the initial guess which is chosen to be sufficiently close to the exact solutions. The other higher-order derivative terms from the above Taylor series have been ignored. Thus, by solving the above simultaneous equations, we have xapprox ¼ x þ yapprox ¼ y þ

@V V ðx; yÞ @U @y  U ðx; yÞ @y

det @U U ðx; yÞ @V @x  V ðx; yÞ @x det

ð10:35Þ ð10:36Þ

where  2  2 @U @V det ¼ þ @x @y

ð10:37Þ

For an analytical complex function W ðzÞ, the Cauchy–Riemann condition must be satisfied [22] @U @V ¼ ; @x @y

@U @V ¼ @y @x

ð10:38Þ

The partial differential with respect to y, @=@y will then be replaced with @=@x using the above Cauchy–Riemann condition  2 @U det ¼ 2 @x xapprox ¼ x 

@U V ðx; yÞ @V @x þ U ðx; yÞ @x det

Only the first-order derivatives @U=@x and @V=@x are used to solve eqn (10.32).

ð10:39Þ ð10:40Þ

262 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Initially, a pair ð x; yÞ is guessed in order to start the numerical iteration process. A new  pair xapprox ; yapprox is then generated until it is sufficiently close to the exact solution. Though there are many other numerical methods to solve transcendental equations, this method is used due to its flexibility and speed. In addition, any errors associated with other numerical methods, such as numerical differentiation, can be avoided. However, the derivative term @W=@z must be solved analytically before any numerical iteration is undertaken. Another numerical method in which the term @W=@z cannot be solved analytically for the case of tapered-structure DFB LDs shall now be discussed.

10.4

THRESHOLD ANALYSIS OF DFB LASER DIODES

For a conventional DFB laser with a zero facet reflection, the threshold eigenvalue eqn (10.16) becomes jgL ¼ L sinhðgLÞ

ð10:41Þ

The above transcendental equation is then solved using the Newton–Raphson iteration approach in which the coupling coefficient is given. The results obtained are shown in Fig. 10.4.

Figure 10.4 The normalised amplitude gain versus the normalised detuning coefficient of a uniform index-coupled DFB LD.

THRESHOLD ANALYSIS OF DFB LASER DIODES

263

Note that all parameters used have been normalised with respect to the overall cavity length L. Different values of normalised coupling coefficient L in the range 0.25 to 5.0 have been set. As predicted mathematically, there exist two pairs of possible solutions for each oscillation mode (complex conjugates). Thus, from the results, we can see that the oscillating modes distribute symmetrically with respect to the Bragg wavelength, where L ¼ 0. In addition, no oscillation can be found at the Bragg wavelength. This region between the þ1 and 1 modes is called the stop band as discussed in section 10.3.1. From Fig. 10.4, it can also be seen that when the coupling strength increases, the normalised amplitude gain will decrease, in other words, the threshold current will be decreasing. This is because a larger value of L indicates a stronger optical feedback along the DFB cavity. Similarly, if the coupling strength is fixed, a longer cavity length will also reduce the threshold gains since a larger single pass gain can be achieved easily. In laser operation, the main (fundamental) mode is large and the sub-modes are sufficiently suppressed because the coupling between the main mode and the sub-modes is large and, as such, the gain concentrates on the main mode. However, if DFB LDs are to be used as amplifier filters, the lasers will then be biased below the lasing threshold, therefore the gain difference between the main mode and the sub-modes is always smaller than in laser operation. As a result, the wavelength tuning range for an optical amplifier filter is smaller than that of a laser.

10.4.1 Phase Discontinuities in DFB LDs The analysis of phase-adjusted DFB LDs is rather similar to the conventional DFB LDs described in the previous section. The only difference is that the boundary conditions at the phase shift position (PSP) have to be matched. Whenever a propagating wave travels past a phase discontinuity along the corrugation, it will experience a phase delay. As noted earlier, TMM is used since it can match the boundary conditions easily by cascading the matrices. Thus, the phase discontinuity along the cavity of the DFB LDs can be best explained by using a two-section DFB structure with a single phase shift at the centre  of the corrugation as depicted in Fig. 10.5. zþ  and z are assumed to be the slight deviations from z .

Figure 10.5

Schematic diagram of a single-phase-shifted DFB LD.

264 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

z 

þ If the distance between z and z  is infinitesimal, we can relate the electric fields at z and as follows "  # " # "  # E R z E R zþ 0 e j   þ ¼   j E S z E S z 0 e  "  #  ER z   ¼ P ð10:42Þ ES z  

where P is the phase discontinuity matrix, which causes the complex electric field delay of  at z ¼ z . By applying the phase discontinuity to eqn (10.26) and following the steps below it, "  #   ER ð z 1 Þ ER z   1  ¼T ð10:43Þ ES z  ES ð z 1 Þ  where T1 is the transfer matrix defined in eqns (10.27a) – (10.27d). "  # "  # ER z ER z þ    þ ¼ P   ES z  ES z  "  #   ER z þ ER ð z 2 Þ   ¼ T2 E S ðz2 Þ ES zþ 

ð10:44Þ ð10:45Þ

If the above concept is employed for N-section (N  1) multiple-phase-shifted (MPS) DFB LDs, the general TMM equation can be expressed as     ER ð z 1 Þ ER ðzNþ1 Þ N  N1  1 ¼ T P T P ...T ES ðzNþ1 Þ ES ð z 1 Þ   kY ¼N ER ð z 1 Þ ¼ T k P ð10:46Þ ES ð z 1 Þ k¼1 Thus far, various numbers of phase discontinuities are being proposed [23–25]. These include the novel multiple-phase-shift DFB LD proposed by Tan et al. and shown in Fig. 10.6 [25].

Figure 10.6

Analytical model for a 3-phase-shift DFB LD structure [25].

THRESHOLD ANALYSIS OF DFB LASER DIODES

265

Figure 10.7 The normalised amplitude gain versus the normalised detuning coefficient of three different index-coupled DFB LD structures: 3-phase-shift, single-phase-shift and conventional.

As shown in Fig. 10.7 for the case of a uniform structure, there are two lowest modes of the same gain, which implies that the gain margin is zero. Therefore, it is not possible to achieve a single longitudinal mode (SLM). Nevertheless, if a single phase shift of /2 is introduced at the middle of the DFB cavity, the lowest mode threshold gain exists at the Bragg wavelength (L ¼ 0). This is an interesting feature in which SLM happens. It is also interesting to consider the electric field intensity in these structures. By solving the threshold condition of eqn (10.16) with appropriate boundary conditions, the threshold gains of the main mode,  and the detuning coefficient,  are obtained. Substituting these parameters back into eqn (10.16), the forward, ER , and backward, ES , electric field intensities can be obtained. The total electric field intensity after the waves propagate along the cavity is then given by Iz ¼ jER j2 þjES j2

ð10:47Þ

Figure 10.8 depicts the electric field intensity plot versus the normalised position along the DFB LD cavity. For the uniform conventional DFB LD structure, the electric field intensity is a smooth curve reaching a maximum point at the centre of the cavity. However, for the case of the single-phase-shift DFB LD structure, the electric field intensity exhibits a peak point at the centre of the cavity. This point tends to lead to instability due to a phenomenon known as spatial hole burning. SHB is the phenomenon in which the depletion of the injected charge carriers is caused by strong stimulated recombination in regions of high photon density. In DFB LDs, the effects of SHB will cause variations in the carrier density, and thus variations in the real refractive index and gain, which in turn cause the magnitude

266 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Figure 10.8 Intensity versus the normalised position along the cavity of three different index-coupled DFB LD structures: 3-phase-shift, single-phase-shift and conventional.

and phase of the feedback from the grating to change. All these factors change the longitudinal mode intensity distribution and alter the gain suppression of the side modes relative to the lasing mode. This brings us to the idea of introducing three phase shifts along the cavity to reduce the discontinuity level in the electric field intensity.

10.4.2 Below-threshold Characteristics The amplification characteristics of conventional DFB LDs are calculated under low-input power saturation. From Fig. 10.3, when z1 ¼ 0 and z2 ¼ L, eqn (10.28) becomes       f RðLÞ f Rð 0 Þ ð10:48Þ ¼ 11 12 f21 f22 SðLÞ Sð 0Þ where f11 ¼ cosh½gL þ

ð  jÞL sinh½gL gL

L sinh½gL gL L f21 ¼ j sinh½gL gL ð  jÞL f22 ¼ cosh½gL  sinh½gL gL 2 ¼  g0  0 f12 ¼ j

ð10:49aÞ ð10:49bÞ ð10:49cÞ ð10:49dÞ ð10:49eÞ

THRESHOLD ANALYSIS OF DFB LASER DIODES

267

In the above equations  is the mode gain per unit length,  is the detuning propagation coefficient from the Bragg condition,  is coupling coefficient, L is cavity length of the DFB LD amplifier,  is confinement factor, g0 is local gain per unit length, and 0 is loss coefficient per unit length. An optical signal injected into DFB LD amplifiers is then amplified by the optical gain. In the calculations, both the facet reflectivities of the laser amplifier are assumed to be zero. Consequently, the power of the injected optical signal, Pin and the amplified transmission output power Pamp are connected respectively to jRð0Þj2 and jRðLÞj2 by the boundary conditions. Using eqns (10.48) and (10.49), the signal gain G, which is defined as the ratio of Pamp to Pin, can be obtained as: 2   Pamp jRðLÞj2  ð  iÞL sinhðgLÞ G¼ ¼ ¼ coshðgLÞ  2 Pin gL j Rð 0 Þ j

ð10:50Þ

The computed results are shown in Fig. 10.9. The parameters used in the calculations are L ¼ 2:9, L ¼ 300 mm and  ¼ 0:3962 mm. In fact, the gain spectrum of the amplifier has two identical peaks that are symmetrical around the Bragg wavelength,
B . It is important to note that the injected light signal is hardly amplified in the vicinity of the Bragg wavelength, and this region is known as the stop band, which has been discussed in section 10.3.1. As the gain  increases, the maximum signal gain Gmax increases and the FWHM bandwidth (the gain bandwidth of full width at

Figure 10.9

Calculated gain profiles of a conventional DFB LD.

268 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

half maximum) becomes narrower and shifts towards the Bragg wavelength through the refractive index change due to the injected carrier, which is given as [7]

B ¼ 2
n

10.5

ð10:51Þ

ACTIVE TUNABILITY DFB LD AMPLIFIER FILTERS

In 1992, Numai proposed a 1.5 mm phase-controlled (PC) DFB LD as depicted in Fig. 10.10 [9]. The PC section does not have an active layer and therefore the gain of the filter and the transmission wavelength can be controlled independently. Although the carrier injection into the PC section causes a small gain change, the gain is compensated almost independently of wavelength by an injection current into the active section [5,7]. In addition, each section is electrically isolated from the others by approximately 20.25 mm wide etched grooves formed on both sides of the central mesa stripe area [9–10].

Figure 10.10

Schematic diagram of the phase-shift-controlled DFB LD [9].

Since there is no active section and no grating in the planar PC section (i.e.  ¼ 0 and  ¼ 0), the transfer matrix for the electric field of this section can be simplified from eqns (10.26) and (10.27) to 

   ER ðziþ1 Þ ER ðzi Þ ¼ Wðziþ1 j zi Þ ES ðziþ1 Þ ES ðzi Þ

ð10:52Þ

where Wðziþ1 j zi Þ is the diagonal transfer matrix given by:  Wðziþ1 j zi Þ ¼ ¼

expð Þ 0 0 expð Þ "  exp gp Lp  jb0 Lp 0



0

 exp  gp Lp  j 0 Lp

# ð10:53Þ

ACTIVE TUNABILITY DFB LD AMPLIFIER FILTERS

269

in which gp is the g value in the PC section while Lp is the PSC section length. In fact, the phase shift, , is given as [26]  ¼ Imð2gp Lp Þ ¼ 2

2pðnactive  nPC Þ Lp
B

ð10:54Þ

where nactive is the effective index for the active section while nPC is the effective index for the PC section. When the current is injected (carrier density increases) into the PC section, nPC decreases due to a phenomenon known as the free carrier plasma effect and the phase shift  increases according to eqn (10.54). The relationship between the refractive index and the presence of free carriers was originally proposed as a mechanism for light confinement in diffused junction lasers [26]. For certain materials such as lithium niobate (LiNbO3) and gallium arsenide (GaAs), the permittivity can be changed through application of an electric field [27]. The change is directly proportional to the applied electric field and independent of the direction of the applied field (isotropic material). The applied current density is given as [28] J ¼ E ¼

 2 ne j E m!

ð10:55Þ

where  is the conductivity, n is the number of free electrons per unit volume, m and ! are the mass of an electron and the optical angular frequency, respectively. E is the electric field. Equation (10.55) clearly shows that the conductivity is a function of carrier density and optical frequency therefore changes in carrier density affects the conductivity. This subsequently affects the permittivity and the refractive index of the waveguide. Hence, the final observation is that the pass band will be shifted towards the desired wavelength, this is known as the tuning capability. Figure 10.11 shows the working prototype of a multi-electrode DFB LD amplifier filter proposed by Magari et al. in 1988 [8]. A continuous wavelength tuning range of 33.3 GHz was demonstrated experimentally in the laboratory. The two electrodes are optically coupled but electrically isolated. The gain and the transmission wavelength are controlled by the respective current injections. Therefore, the overall gain of the DFB LD amplifier is kept constant, when the carrier density of one section increases, the other section’s carrier density decreases. In order to achieve a wider tuning range, the phase-controlled (PC) [9] and the phase-shift-controlled (PSC) [10] DFB LD amplifier filter have been introduced. This is illustrated in Figs. 10.12 and 10.13, respectively. In Fig. 10.12, the active section amplifies the input light signal and controls the gain, while the PC section controls the transmission wavelength using electro-optic effects. A ˚ ) of constant continuous wavelength tuning range of 43 GHz was reported with 5 GHz (0.4 A bandwidth measured at full width at half maximum (FWHM) while maintaining a constant gain of 27 dB. The main mode with the largest gain in DFB LD amplifiers is used for wavelength tuning. However, the tuning range is retracted by the sub-modes whose gain is smaller than the main mode. This is because the gain of the sub-modes increases during the wavelength tuning. Numai proposed another design in order to suppress the intensity of the sub-modes, known

270 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Figure 10.11 Structure of DFB LD and the transmission spectrum for a multi-electrode DFB optical filter proposed by Magari et al. [8].

as the phase-shift-controlled (PSC) DFB LD amplifier filter and shown in Fig. 10.13. This structure has a phase control section in between the two active sections. As noted earlier, the PC section does not have an active layer. Carrier injection into the PC section causes a slight gain change because of the loss in the PC section. However, since the gain change is small, it is then compensated almost independently of the wavelength by an ˚ ) was injection current into the active sections. A tuning range as wide as 120 GHz (9.5 A achieved with a nearly constant bandwidth of 12 to 13 GHz while maintaining a constant gain of 24.5 dB.

10.6

STRUCTURAL IMPACTS ON DFB LD AMPLIFIER FILTERS

In this section, we shall look into the structural impacts on DFB LD amplifier filters. There are several factors which need to be considered in order to obtain a good bandpass filter

STRUCTURAL IMPACTS ON DFB LD AMPLIFIER FILTERS

271

Figure 10.12 Structure of DFB LD and the transmission spectrum for a phase-controlled optical filter proposed by Numai [9].

response with high gain and wide tuning range. Since the electro-optic effects are used in the phase-shift-controlled section to tune the lasers, the tuning speed should be in the range of several nanoseconds [29]. However, the main mode and side mode amplification gain can cause severe problems to researchers and engineers. This can be measured by the side mode suppression ratio (SMSR), which is the ratio of the highest peak to the second highest peak of the transmissivity power. Table 10.2 outlines the relationships among the transmission system parameters, filtering properties and the counter measures to improve the desired system parameters [29–31]. By adopting each counter measure listed in Table 10.2, better amplification and an improved filtering response can be obtained. Nevertheless, the counter measures must be justified since these will influence the desired transmission system parameters. This shall be discussed in a later section.

Figure 10.13 Structure of DFB LD and the transmission spectrum for a phase-shift-controlled optical filter proposed by Numai [10].

272 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES Table 10.2 Relations among desired transmission system parameters, filtering properties and counter measures [29–31] Desired transmission (or switching) system parameters

Filtering properties for better system parameters

Counter measures to improve filtering properties

Large number of WDM/ DWDM light signals

 Large side mode suppression  Cavity mirror loss difference between main mode and subratio (SMSR) mode should be increased  Small channel bandwidth  Coupling strength should be increased

Wide dynamic range of received optical power

 Large saturation gain  Large signal to noise ratio

 Confinement factor should be decreased  Active layer length should be decreased

Good stability

 Small drifts of tuning wavelength and gain due to temperature and injection current drifts

 Temperature and injection current deviation should be decreased

10.6.1 Phase-shift-controlled DFB LD Amplifier Filters The effects of various parameters of the phase-shift-controlled DFB LD amplifier filter proposed by Numai [10] (see Fig. 10.14) will now be discussed. These parameters include the grating pitch and the lengths of the active DFB region and the phase-controlled region. Table 10.3 summarises the characteristics of phase-shift-controlled DFB LD amplifier filters for five different structure parameters. We classify each structure by type A, B, C, D or E. The data is obtained by changing the phase-shift-controlled region in the order of 0.1  from  ¼ 0 to 2. When the grating pitch alters, the effective refractive index for the active region subsequently changes in order to maintain the Bragg wavelength at 1.55 mm since for first, order gratings, the Bragg condition is 2pneff p 
B

ð10:56Þ

where
B refers to the Bragg wavelength.

Figure 10.14 Schematic diagram of the phase-shift-controlled DFB LD amplifier filter proposed by Numai [10].

273

STRUCTURAL IMPACTS ON DFB LD AMPLIFIER FILTERS

Table 10.3 The characteristics of phase-shift-controlled DFB LD amplifier filters for different structure parameters Transmissivity peak (dB)

Tuning range ˚) (A

SMSR (dB)

Bandwidth (GHz)

A. L1 ¼ L3 ¼ 200 mm; L2 ¼ 100 mm;  ¼ 0:2380 mm;  ¼ 6 mm1; neff ¼ 3:7

33.2 to 36.1

9.5 to 9.5

15.9 to 29.2

2.91

B. L1 ¼ L3 ¼ 200 mm; L2 ¼ 150 mm;  ¼ 0:2380 mm;  ¼ 6 mm1; neff ¼ 3:7

33.2 to 36.1

9.5 to 9.5

15.9 to 29.2

2.91

C. L1 ¼ L3 ¼ 150 mm; L2 ¼ 100 mm;  ¼ 0:2380 mm;  ¼ 6 mm1; neff ¼ 3:7

36.0 to 37.2

11.0 to 11.0

13.7 to 27.2

4.8

D. L1 ¼ L3 ¼ 200 mm; L2 ¼ 100 mm;  ¼ 0:2380 mm;  ¼ 8 mm1; neff ¼ 3:7

29.9 to 35.3

11.0 to 11.0

18.6 to 31.0

2.02

L1 ¼ L3 ¼ 200 mm; L2 ¼ 100 mm;  ¼ 0:30 mm;  ¼ 8 mm1; neff ¼ 2:59

34.3 to 36.2

13.6 to 13.6

16.1 to 29.7

3.79

Type description

E.

Figure 10.15 shows the computed transmissivity power in dB for a type A DFB LD amplifier filter [10]. It is obvious that there is no overlapping in the pass band transmission response as in the phase-shift-controlled section,  is tuned from 0p to 2p. The tuning range is counted from  ¼ 0:1p to  ¼ 1:9p since the SMSR for  ¼ 0p and  ¼ 2p is zero and is not suitable for wavelength selection. The transmission spectrum is symmetrical at about zero relative wavelength. As the phase-shift value  increases, the wavelength of the main mode shifts towards the shorter wavelength side (blue shift). When the length of the phase-shift-controlled region increases as described in a type B DFB LD amplifier filter, there will be no significant change in the tuning range, SMSR, bandwidth and the transmissivity peaks. However, based on our simulation results, the threshold gain  increases by approximately 10%. Consequently, for a type C DFB LD amplifier filter in which the length of the active DFB region is reduced while maintaining the length of the phase-shift-controlled region, the tuning range increases but at the expense of increasing the pass band bandwidth.

274 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Figure 10.15 Power transmission spectra for type A wavelength tunable optical filter for the following different values of : (a)  ¼ 0; 2; (b)  ¼ 0:1; (c)  ¼ 0:2; (d)  ¼ 0:3; (e)  ¼ 0:4; (f)  ¼ 0:5; (g)  ¼ 0:6; (h)  ¼ 0:7; ( j)  ¼ 0:8; (k)  ¼ 0:9; (l)  ¼ ; (m)  ¼ 1:1; (n)  ¼ 1:2; (p)  ¼ 1:3; (q)  ¼ 1:4; (r)  ¼ 1:5; (s)  ¼ 1:6; (t)  ¼ 1:7; (u)  ¼ 1:8; (v)  ¼ 1:9.

For a type D DFB LD amplifier filter, the coupling coefficient is altered to 80 cm1. It is found that the tuning range increases by about 16% compared to the type A amplifier filter proposed by Numai. The pass band bandwidth is approximately 2.02 GHz 31% less than the type A. Thus, within the tuning range, more channels can be fitted in. Eventually, when the grating pitch is increased while preserving the other parameters, the tuning range is further increased but at the expense of increasing the pass band bandwidth. The transmissivity peak variability is found to be smaller compared to other types of amplifier filter. In type E all parameters are the same as those in D except the grating period which ˚ with SMSR ranging increased to 0.3 m. This type offers a higher tuning range of 27.2A from 16.1 dB to 29.7 dB and the pass band bandwidth of about 3.79 GHz.

10.7

NEW MULTI-SECTION AND PHASE-SHIFT-CONTROLLED DFB LD OPTICAL FILTER STRUCTURES

In this section, the characteristic performance of the proposed novel multi-section and phase-shift-controlled DFB LD amplifier filter shown in Fig. 10.16 will be analysed according to the structural impacts on DFB LD amplifier filters. In order to obtain a uniform gain, identical pass band bandwidth and mode stability, the injection current I pumping the

NEW MULTI-SECTION AND PSC DFB LD OPTICAL FILTER STRUCTURES

275

Figure 10.16 Analytical model for the multi-section and phase-shift-controlled DFB LD amplifier filter.

DFB active regions should be from the same source. Also, the tuning current, Ituning pumping the phase-shift-controlled regions should be identical. This model is designed such that the total cavity length of the device is 600 mm and thus can monolithically be integrated with the photodiode detector. As depicted in Fig. 10.16, the current injection into the DFB active region is uniform, and the tuning current into the entire phase-controlled (PC) region is identical. The phase shifts, 1 and 2 are fixed to be equal to p/2 and the length of PC sections L2 ¼ L5 ¼ L8 ¼ 50 mm. The lengths of the active sections, which are optimised based on the phase-shift-position (PSP) in order to give maximum tuning range are shown in Table 10.4. In addition, it is assumed that 1 ¼ 2 ¼ . From section 10.3.2, by multiplying the matrices representing the planar phase-shiftcontrolled (PSC) sections, and the corrugated DFB sections together, the overall transfer matrix for the structure shown in Fig. 10.16, becomes: 

ER ðLÞ ES ðLÞ



 ¼

T21

   E R ð 0Þ E S ð 0Þ T22

ð9Þ

ð7Þ

T11

T12

ð6Þ

ð 4Þ

ð3Þ

¼ T PT WT PT WT PT

ð1Þ



ER ð 0 Þ



E S ð 0Þ

Table 10.4 Parameters used in the analysis to obtain maximum tuning range Parameter Phase shifts, 1 and 2 Total cavity length, L Phase-shift-controlled section, L2 ¼ L5 ¼ L8 Active region length, L1 Active region length, L3 ¼ L7 Active region length, L4 ¼ L6 Active region length, L9

Value p/2 600 mm 50 mm 92 mm 26 mm 102 mm 102 mm

ð10:57Þ

276 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

where T is the transfer matrix as shown in eqn (10.26) and its superscripts indicate the section number. P is the transfer matrix in eqn (10.42), representing the phase shifts in between the DFB regions. For the PSC section, the transfer matrix W is represented in eqn (10.53). From eqn (10.57), the power transmissivity T is defined as     ES ðLÞ2  1 2  ¼  T ¼  ER ð0Þ T22 

ð10:58Þ

Hence, the threshold gain th and the detuning parameter  can be obtained by solving the following equation numerically T22 ðth ; Þ ¼ 0

ð10:59Þ

Subsequently, for the computation of the power transmissivity of the amplifier filter, we have used  ¼ 0:98th in order to achieve a higher output power and a smaller bandwidth as discussed in section 10.4.2. Thus, the power transmissivity of the filter is 2    1   T¼ T22 ð ¼ 0:98th ; Þ

ð10:60Þ

Equation (10.59) has been solved numerically to analyse the amplifier filter structure shown in Fig. 10.16. For a given value of , the numerical solution of eqn (10.59) gives oscillation modes of the device. The one having the lowest threshold gain is the main mode whereas sub-modes are the modes where the threshold gain is larger than the main mode. For filtering operation, the gain of the device is biased slightly below the threshold gain of the main mode. The normalised detuning coefficient of the main mode determines the amount of deviation of the oscillation wavelength from the Bragg wavelength. The oscillation wavelength is the central wavelength of the filter. As noted earlier, for a given , the side mode suppression ratio (SMSR) is defined as the ratio of the highest peak to the second highest peak of the filter power transmissivity. It determines the amount of interference from the channel at the side mode wavelength. As the central wavelength drifts away from the Bragg wavelength, the SMSR reduces. If the SMSR is larger than 10 dB, then the adjacent channel interference would be minimal [32]. Figure 10.17 shows the calculated transmission spectra of the filter for various values of the phase shift  ranging from 0 to 2p. The horizontal axis is the relative wavelength defined as

B where
is the operating wavelength of the filter,
B ¼ 2neff  ¼ 1:55 mm is the Bragg wavelength, and neff is the effective refractive index. Note that all the parameters used in the analysis are based on Numai’s [2], and the tuning range has shown an approximate increase of 21.7%. When a grating period and coupling coefficient of 0.21 mm and 6 mm1 ˚ . This gives the filter a wavelength were used, the relative wavelengths were at 16.0 A ˚ . The filter peak gain varies from 34.1 to 36.25 dB with a maximum tuning range of 32 A deviation of 2.15 dB. The SMSR ranges from 12 to 27 dB. To investigate the effect of the grating period  on the filter performance, the value is increased to 0.238 mm while maintaining the other parameters. The result is shown in ˚ ; this gives Fig. 10.18. In this case when  ¼ 0 or 2p the relative wavelengths are 18.15 A ˚ . This shows an increase of 4.3 A ˚ compared to the filter the total filter tuning range of 36.3 A

Figure 10.17 Power transmissivity versus relative wavelength ð

B Þ for the following different values of . The parameters used are  ¼ 6 mm1 , and  ¼ 0:21 mm. (a)  ¼ 0; 2; (b)  ¼ 0:1; (c)  ¼ 0:2; (d)  ¼ 0:3; (e)  ¼ 0:4; (f)  ¼ 0:5; (g)  ¼ 0:6; (h)  ¼ 0:7; ( j)  ¼ 0:8; (k)  ¼ 0:9; (l)  ¼ ; (m)  ¼ 1:1; (n)  ¼ 1:2; (p)  ¼ 1:3; (q)  ¼ 1:4; (r)  ¼ 1:5; (s)  ¼ 1:6; (t)  ¼ 1:7; (u)  ¼ 1:8; (v)  ¼ 1:9.

Figure 10.18 Power transmissivity versus relative wavelength ð

B Þ for the following different values of . The parameters used are  ¼ 6 mm1 , and  ¼ 0:238 mm. (a)  ¼ 0; 2; (b)  ¼ 0:1; (c)  ¼ 0:2; (d)  ¼ 0:3; (e)  ¼ 0:4; (f)  ¼ 0:5; (g)  ¼ 0:6; (h)  ¼ 0:7; ( j)  ¼ 0:8; (k)  ¼ 0:9; (l)  ¼ ; (m)  ¼ 1:1; (n)  ¼ 1:2; (p)  ¼ 1:3; (q)  ¼ 1:4; (r)  ¼ 1:5; (s)  ¼ 1:6; (t)  ¼ 1:7; (u)  ¼ 1:8; (v)  ¼ 1:9.

278 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Figure 10.19 Power transmissivity versus relative wavelength ð

B Þ for the following different values of . The parameters used are  ¼ 8 mm1 , and  ¼ 0:238 mm. (a)  ¼ 0; 2; (b)  ¼ 0:1; (c)  ¼ 0:2; (d)  ¼ 0:3; (e)  ¼ 0:4; (f)  ¼ 0:5; (g)  ¼ 0:6; (h)  ¼ 0:7; ( j)  ¼ 0:8; (k)  ¼ 0:9; (l)  ¼ ; (m)  ¼ 1:1; (n)  ¼ 1:2; (p)  ¼ 1:3; (q)  ¼ 1:4; (r)  ¼ 1:5; (s)  ¼ 1:6; (t)  ¼ 1:7; (u)  ¼ 1:8; (v)  ¼ 1:9.

shown in Fig. 10.17. The filter peak gain varies between 34.51 and 36.61 dB and the filter SMSR ranges from 11.69 to 26.83 dB. The effect of increasing  to 8 mm1 while keeping  ¼ 0:238 mm is shown in Fig. 10.19. ˚ . The filter peak gain varies In this case, the wavelength tuning range has increased to 38.9 A between 31.8 and 35.55 dB and the SMSR ranges from 14 to 28.82 dB. The filter spectra for the case where  ¼ 10 mm1 is shown in Fig. 10.20 where a wavelength tuning range of ˚ has been achieved. The filter peak gain varies between 30.67 and 34.8 dB, which 41.7 A gives a maximum deviation of 4.13 dB. The filter SMSR ranges from 13.95 to 30.21 dB. The effects of grating period and coupling coefficient on the performance characteristics of a multi-section and phase-shift-controlled DFB wavelength tunable optical amplifier filter have been studied. It was found that this filter structure offers a wide tuning range with narrow bandwidth, high gain and large SMSR. The filter has over 30 dB peak gain within the ˚ for  ¼ 10 mm1. The filter ˚ for  ¼ 6 mm1 and 41.7 A tuning range, which is 32.0 A SMSR varies between 12 dB and 30.21 dB. These results show very clearly that with proper design and by adding the correct number of phase-shift-controlled (PSC) sections, it is possible to increase the wavelength tuning range while maintaining a small pass band bandwidth. This is conceptually similar to the thin film filter in which the number of filtered wavelengths depends on the number of layers of thin film. However, in thin film filters, loss is incurred by increasing the number of layers, while in the DFB amplifier filter, the loss of the wavelength passing through the filter can be compensated along the active cavity. There are several measurements which we need to

DFB LDS VERSUS DBR LDS

279

Figure 10.20 Power transmissivity versus relative wavelength ð

B Þ for the following different values of . The parameters used are  ¼ 10 mm1 , and  ¼ 0:238 mm. (a)  ¼ 0; 2; (b)  ¼ 0:1; (c)  ¼ 0:2; (d)  ¼ 0:3; (e)  ¼ 0:4; (f)  ¼ 0:5; (g)  ¼ 0:6; (h)  ¼ 0:7; ( j)  ¼ 0:8; (k)  ¼ 0:9; (l)  ¼ ; (m)  ¼ 1:1; (n)  ¼ 1:2; (p)  ¼ 1:3; (q)  ¼ 1:4; (r)  ¼ 1:5; (s)  ¼ 1:6; (t)  ¼ 1:7; (u)  ¼ 1:8; (v)  ¼ 1:9.

consider during the design process. First, we need to consider the normalised differential gain between the main mode and the side modes,
L in order to maximise the tuning range as discussed in section 10.4. The mode stability along the structure is another critical issue to be considered when adding more sections. This issue can be resolved by designing the length of the sub-sections and the phase shift position (PSP) carefully.

10.8

DFB LDS VERSUS DBR LDS

Distributed Bragg reflector laser diodes can be formed by replacing one or both of the discrete laser mirrors with a passive grating reflector as shown in Fig. 10.21. By definition, the grating reflectors are formed along a passive waveguide section, so one of the issues is to make the transition between the active and passive waveguides without introducing unwanted discontinuities. So far we have just considered the analysis and design of the DFB LD amplifier filter. Unlike the DFB LD amplifier filter, a DBR LD amplifier filter has the possibility of achieving a larger wavelength tuning range. By current injection to the passive DBR section, the refractive index changes due to the free carrier plasma effect (refer to section 10.5), and this in turn changes the effective corrugation period and the Bragg frequency. Since the

280 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Figure 10.21 Schematic illustration of a three-section distributed Bragg reflector (DBR) laser diode.

carrier density is not clamped like an active DFB section, the Bragg wavelength can be varied over a large range, typically several tens of nanometres. Nevertheless, the Auger recombination and free carrier absorption strongly limit the efficiency at high injection current levels. A major problem inherent in DBR LDs is that when the unpumped active material is used to etch the gratings, optical losses inside the Bragg reflector region are high and as a result the DBR reflectivity is poor. The formulation of the transfer matrix of the DBR is rather simple. From section 10.3.2, the DBR region has no active layer, thus the amplitude gain is lossy (i.e.   0) but with finite coupling coefficient . By rearranging the transfer matrix the following reflection coefficient can be obtained easily once the boundary condition is fulfilled [33] r1 ðÞ ¼

j sinhðgLDBR Þ g coshðgLDBR Þ þ ð þ jÞ sinhðgLDBR Þ

ð10:61Þ

where g2 ¼ 2 þ ð þ jÞ2 in which  here represents the field loss coefficient of the gratings and LDBR represents the length of the Bragg reflector section. Assuming that the loss at the interface between the active section and the air is negligible, then the reflection coefficient r2 at the active region is [34] r2 ¼

nair  nactive nair þ nactive

ð10:62Þ

where nair and nactive are the refractive indices of air and the active region, respectively. Hence, we can find the threshold gain th of DBR LDs easily by assuming the structure of Fig. 10.21 as the Fabry–Perot cavity, which is [35]

 th ¼ active þ

  1 1 ln Lactive þ Lphase j r1 ð Þ r2 j

ð10:63Þ

DFB LDS VERSUS DBR LDS

281

Figure 10.22 Normalised amplitude gain th L versus the normalised detuning parameter L for DBR LDs.

where  is the confinement factor of the cavity, Lactive and Lphase represent the length of the active region and phase-controlled region, respectively and active is the gain per unit length of the active section. Since we are interested in the threshold distribution, we assume the Bragg reflector is lossless. Let Lactive ¼ 200 mm, Lphase ¼ 150 mm and LDBR ¼ 150 mm. The average gain in the active region active ¼ 10 cm1. Figure 10.22 depicts the normalised threshold gain versus the normalised detuning parameter for the DBR LD. When the normalised coupling coefficient of the Bragg reflector region LDBR increases, the normalised amplitude gain increases. This is undesirable since it may cost high operating current and the operating temperature may also increase. The main mode in the DBR LD amplifier filter will generally be located close to the Bragg wavelength and within the main lobe of the Bragg reflector. Pan et al. have shown that when the current injection into the Bragg reflector region is varied, the tuning rate of the lasing mode will be slower than that of the Bragg wavelength [36]. Consequently, when the deviation from the Bragg wavelength becomes too large, a mode jump to the opposite side of the main lobe will occur, either because a new mode with a lower threshold gain has been created or one of the existing modes has acquired a lower threshold gain. If the total cavity of the DBR LD is too large, there will be many modes inside the main lobe and mode hopping can easily happen. This causes DBR LDs, which are not they not as attractive as DFB LDs, to be used as optical filters.

282 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

10.9

SUMMARY

An active optical tunable filter with frequency characteristics that can be tailored to a desired response is an enabling technology for exploiting the full potential bandwidth of optical fibre communication systems. Having seen the importance of such active optical tunable filters, it is highly desirable to design a tunable filter which can perform filtering and amplification of the filtered signal simultaneously. In this chapter, active grating-embedded filters have been analysed and designed based on the TMM outlined, and where the coupled wave equations of DFB LD amplifier filters have been solved. In addition, the dispersion relationship and the stop band were discussed in section 10.3.1. From the solutions of the eigenvalue equations, which were derived by matching the boundary conditions, the threshold current and the lasing wavelength were determined in section 10.4. This included analysis of the phase discontinuities and the below-threshold characteristics of DFB LDs. In section 10.5 the principle of the active tunability of a DFB LD amplifier filter was discussed. This was followed by section 10.6 in which the structural impacts on the performance of the filters were explored. Next, in section 10.7, the effects of grating period and coupling coefficient on the performance characteristics of a novel multi-section and phase-shift-controlled DFB wavelength tunable optical filter were studied. It was found that this filter structure offers a wide tuning range with narrow bandwidth, high gain and large SMSR. The filter has a peak ˚ for ˚ for  ¼ 6 mm1 and 42 A gain over 30 dB within the tuning range, which is 32.0 A 1  ¼ 10 mm . The filter SMSR varies between 12 dB and 30.2 dB. Finally, some analyses on the three-section DBR LD amplifier filter were carried out. It was found that this structure is not suitable for a multi-section design since mode hopping occurs in the main lobe, though the tuning range can be increased. Besides, it is very difficult to monolithically integrate the sections without error in practice since the reflectivities of each section will contribute to the threshold amplitude. Furthermore, the Bragg reflector is lossy and thus a higher threshold current will be required.

10.10

REFERENCES

1. Zang, H., Jue, J. P., Sahasrabuddhe, L., Ramamurthy, R. and Mukherjee, B., Dynamic lightpath establishment in wavelength routed WDM networks, IEEE Commun. Mag., 39(9), Sept. 2001. 2. Numai, T., 1.5 mm two-section Fabry–Perot wavelength tunable optical filter, IEEE J. Lightwave Technol., 10(11), 1590–1596, 1992. 3. Ghafouri-Shiraz, H. and Lew, S. W., Analysis of a phase-shifted double phase-shift-controlled distributed feedback wavelength tunable optical filter, IEEE J. Quantum Electron., 33(4), 556–561, 1997. 4. Anderson, D., Optical filters fill many roles, WDM Solutions, pp. 97–99, June 2001. 5. Kawaguchi, H., Magari, K., Oe, K., Nakano, Y. and Motosugi, G., Optical frequency-selective amplification in a distributed feedback type semiconductor laser amplifier, Appl. Phys. Lett., 50, 66– 67, 1987. 6. Numai, T., Fujiwara, M., Shimosaka, N., Kaede, K., Nishio, M., Suzuki, S. and Mito, I., 1.5 mm
=4-shifted DFB LD filter and 100 Mbit/s two-channel wavelength signal switching, Electron. Lett., 24, 236–237, 1988.

REFERENCES

283

7. Magari, K., Kawaguchi, H., Oe, K., Nakano, Y. and Fukuda, M., Optical signal selection with a constant gain and a gain-bandwidth by a multielectrode distributed feedback laser amplifier, Appl. Phys. Lett., 51, 1974–1976, 1987. 8. Magari, K., Kawaguchi, H., Oe, K. and Fukuda, M., Optical narrowband filters using optical amplification with distributed feedback, IEEE J. Quantum Electron., 24, 2178–2190, 1988. 9. Numai, T., 1.5 mm Phase-controlled Distributed Feedback Wavelength Tunable Optical Filter, IEEE J. Quantum Electron., 28(6), 1508–1512, 1992. 10. Numai, T., 1.5 mm Phase-shift-controlled Distributed Feedback Wavelength Tunable Optical Filter, IEEE J. Quantum Electron., 28(6), 1513–1519, 1992. 11. Ghafouri-Shiraz, H., Lew, S.-H., Kobayashi, S., Norihiko Kamata, I. and Yamada, K., Analysis of
/4-Phase-shifted Double Phase-shift-controlled Distributed Feedback Wavelength Tunable Optical Filter, IEEE J. Quantum Electron., 33, 556–561, 1997. 12. Senior, J. M. and Cusworth, S. D., Devices for wavelength multiplexing and demultiplexing, IEE Proc., 136(3), 1989.Q1 13. Kobrinski, H. and Cheung, K. W., Wavelength-tunable optical filters: applications and technologies, IEEE Commun. Mag., pp. 53–63, Oct 1989. 14. Sadot, D. and Boimovich, E., Tunable optical filters for dense WDM networks, IEEE Commun. Mag., pp. 50–55, Dec 1998. 15. Toba, H., et al., 100-channel optical FDM transmission/distribution at 622 Mbit/s over 50 km, Opt. Fiber Commun. Conf., OFC ’90, San Francisco, CA, 1990, postdeadline paper PD1. 16. Kogelnik, H. and Shank, C. V., Coupled-wave theory of distributed feedback lasers, J. Appl. Phys., 43(5), 2327–2335, 1972. 17. Streifer, W., Burnham, R. D. and Scifres, D. R., Effect of external reflectors on longitudinal modes of distributed feedback lasers, IEEE J. Quantum Electron., QE-11(4), 154–161, 1975. 18. Haus, H. A. and Huang, W., Coupled-mode theory, IEEE Proc., 79(10), 1505–1518, 1991. 19. Yamada, M. and Sakuda, K., Analysis of almost-periodic distributed feedback slab waveguide via a fundamental matrix approach, Appl. Opt., 26(16), 1987. 20. Makino, T. and Glinski, J., Transfer matrix analysis of the amplified spontaneous emission of DFB semiconductor laser amplifiers, IEEE J. Quantum Electron., QE-24(8), 1507–1518, 1988. 21. O’Neil, P. V., Advanced Engineering Mathematics, 4th edition, Part 6. London: Thomson, 1995. 22. Arfken, G., Mathematical Methods for Physicists, 3rd edition. New York: Academic Press, 1985. 23. McCall, S. L. and Platzman, P. M., An optimized p/2 distributed feedback laser, IEEE J. Quantum Electron., QE-21(12), 1899–1904, 1985. 24. Ong, B. S., Optimisation of Multiple-phase-shift Distributed Feedback Laser Diodes, B. Eng. Final Year Project Report, no. 68, The University of Birmingham, UK, 1993–94. 25. Tan, P. W., Ghafouri-Shiraz, H. and Lo, B. S. K., Theoretical analysis of multiple-phase-shiftcontrolled distributed feedback wavelength tunable optical filters, Microwave and Optic. Technol. Lett., 8(2), pp. 72–75, 1995. 26. Nash, F. R., Mode guidance parallel to the junction plane of double-heterostructure GaAs lasers, J. Appl. Phys., 44, pp. 4696–4707, 1973. 27. Ramo, S., Whinnery, J. R. and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd edition. New York: Wiley, 1994. 28. Yariv, A., Quantum Electronics, 3rd edition. New York: Wiley, 1989. 29. Cooper, J. M., Dixon, J., Goodman, M. S., Kobrinski, H., Vecchi, M. P. and Tur, M., Nanosecondtunable double-section DFB laser for dynamic wavelength addressing applications, Electron. Lett., 24(19), 1237–1239, 1988. 30. Numai, T., Murata, S. and Mito, I., 1.5 mm wavelength tunable phase-shift controlled distributed feedback laser diode with constant spectral linewidth in tuning operation, Electron. Lett., 24(24), 1526–1528, 1988. 31. Kikushima, K., Nawata, K. and Koga, M., Tunable amplification properties of distributed feedback laser diodes, IEEE J. Quantum Electron., 25(2), 163–169, 1989.

284 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES 32. Numai, T., Murata, S. and Mito, I., Tunable wavelength filters using
/4-shifted waveguide grating resonators, Appl. Phys. Lett., 53, 83–85, 1988. 33. Kazovsky, L. G., Stern, M., Menocal, S. G. and Zah, C. E., DBR active optical filters: transfer function and noise characteristics, IEEE J. Lightwave Technol., 8(10), 1441–1451, 1990. 34. Agrawal, G. P. and Dutta, N. K., Semiconductor Lasers, 2nd edition. New York: Van Nostrand Reinhold, 1993. 35. Komori, K., Arai, S., Suematsu, Y., Arima, I. and Aoki, M., Single-mode properties of distributedreflector lasers, IEEE J. Quantum Electron., 25(6), 1235–1244, 1989. 36. Pan, X., Olesen, H. and Tromborg, B., A theoretical model of multielectrode DBR lasers, IEEE J. Quantum Electron., 24(12), 2423–2432, 1988.

11 Other Wavelength Tunable Optical Filters Based on the DFB Laser Structure 11.1

INTRODUCTION

Optical tunable filters are key components of the future dense wavelength division multiplexed (WDM) optical fibre networks. In such a network a number of information channels are simultaneously transmitted through a single fibre by putting each channel on a different optical carrier wavelength. The wavelength filter allows a single or multiple channel(s) to be isolated at the receiving or routing node. The tunability of the filter allows for dynamic network reconfiguration and increases versatility of the system. Ideally, the wavelength filter should be tunable over the entire system bandwidth and should have no secondary pass bands, or side lobes in its filter function. WDM systems require optical tunable filters not only as channel selectors, but also as post-optical-amplifier filters that reduce amplified spontaneous emission (ASE) noise [1]. Following the recent rapid advances in lightwave technology, wavelength tunable optical filters are now incorporated in wavelength-division-multiplexed transmission systems to increase the line capacity for lightwave telecommunication services. Optical filtering for selection of channels separated by 2 nm is currently achievable, and narrower channel separations may be possible as filter technologies improve. This would give more than a hundred broadband channels in the low-loss fibre transmission region of 1.3 mm and/or 1.55 mm wavelength bands with each wavelength channel having a transmission bandwidth of several gigahertz. Wavelength tunable optical filters have already been built into the receiver for each subscriber in distribution networks [2]. Basically a semiconductor wavelength tunable optical filter is a laser diode which is biased slightly below threshold. When an optical signal of a wavelength close to the oscillation wavelength of the device is incident upon the input, the signal is amplified and emitted at the output. By changing the injection current, the wavelength can be tuned due to free carrier plasma and quantum confined Stark effects. Distributed feedback laser diode amplifiers (DFB LDAs) can be used as tunable wavelength narrowband optical filters. This is because a DFB LDA has two main advantages: single frequency with narrowband amplification and tunability of the lose gain profile

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

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OTHER WAVELENGTH TUNABLE OPTICAL FILTERS

maximum frequency by changing the amplifier’s bias current. DFB LDs have the advantage of a single resonance at the centre of the stop band. Conventional uniform DFB LDs have resonances on both sides of the stop band. This is, in general, a disadvantage since they may oscillate at either of the two frequencies. Furthermore, the grating is less effective outside its stop band. This drawback of index gratings has been overcome by inserting a
=4 phase shift at the centre of the structure [3–4]. In this way a resonance is produced at the centre of the stop band. A passive index grating can perform useful filtering functions [5]. A DFB type filter has the advantages of high gain and narrow bandwidth and disadvantages in that the bandwidth and the transmissivity change with wavelength tuning. Single-electrode wavelength tunable optical filters [6–8] have the problem of a changing transmissivity during tuning. This is because the injection current of a single-electrode device affects both the transmissivity and transmission wavelength. This problem has been solved by employing a multi-electrode DFB filter which has more than one injection current to control the gain and the central wavelength [9]. The tuning range of this filter is 33 GHz with a constant gain and bandwidth. In 1992, Numai [10] reported the phase-controlled (PC) DFB wavelength tunable optical filter. In this device the gain and transmission wavelength were controlled independently by applying different injection currents. For this filter a ˚ ) with constant gain of 27 dB and constant bandwidth of 0.4 A ˚ tuning range of 43 GHz (3.4 A has been reported. The drawback of this filter is its very limited wavelength tuning range. In general, to obtain a wider tuning range, suppression of the sub-modes is essential. To achieve this goal, Numai [11] proposed the phase-shift-controlled (PSC) DFB filter where the side modes were suppressed by the large gain margin when it was tuned around the Bragg ˚ ) with constant gain of wavelength. This filter has a wider tuning range of 120 GHz (9.5 A 24.5 dB and constant bandwidth of 12–13 GHz. In 1994, Tan et al. [12] proposed the multiple-phase-shift-controlled distributed feedback wavelength tunable filter which has a ˚ with side mode suppression ratio of more than 25 dB. wavelength tuning range of about 30 A In this chapter we analyse the performance characteristics of DFB LD-based wavelength tunable optical filters.

11.2

ANALYSIS

The analytical model for the filter structure is shown in Fig. 11.1. This filter consists of two passive PC waveguides which control the transmission wavelength by changing the bias

Figure 11.1 Analytical model for the
=4-phase-shifted double phase-shift-controlled wavelength tunable filter.

ANALYSIS

287

current Ip . Each PC section is sandwiched between two corrugated DFB active sections. A
=4 phase shift is also located at the centre of the middle DFB active section. The active sections control the optical gain of the filter through the bias current Ia . In the analysis we have used the transfer matrix method to study the characteristics of this filter [4,13]. In doing so, the filter cavity is divided into seven sections and the wave propagation in each section is represented by a transfer matrix. Let us assume that the device has zero facet reflectivity and the z-axis is along the filter cavity. The electric field EðzÞ within the filter cavity can be expressed as EðzÞ ¼ ER ðzÞ þ ES ðzÞ ¼ RðzÞ expðjbo zÞ þ SðzÞ expðjbo zÞ

ð11:1Þ

where ER ðzÞ and ES ðzÞ are the normalised electric fields that propagate along opposite directions, RðzÞ and SðzÞ are complex amplitudes of the forward and backward electric fields, respectively, bo ¼ p=L is the Bragg frequency of the grating and L is the grating period. Substituting eqn (11.1) into Maxwell’s equations and neglecting the second derivatives of both RðzÞ and SðzÞ with respect to z, as they are slowly varying functions of z, we obtain the following pair of coupled mode equations [4,14] dRðzÞ þ ð  jÞ RðzÞ ¼ j SðzÞ dz dSðzÞ þ ð  jÞ SðzÞ ¼ j RðzÞ dz

ð11:2aÞ ð11:2bÞ

In eqn (11.2) a is the mode gain per unit length, d ¼ b  bo is the detuning of the propagation constant b from the Bragg propagation constant bo , and  is the grating coupling coefficient. The filter structures used in this analysis are shown in Figs 11.1, 11.6, 11.9 and 11.15 where, for example in Fig. 11.1, Ia and Ip are the bias currents for the active and phase-controlled sections, respectively, Li ði ¼ 1; 6Þ is the ith section length and Zj ð j ¼ 1; 7Þ is the jth position. In order to calculate the transmission characteristics of this filter structure it is more convenient to use the transfer matrix method [4,13] where the cavity is divided into seven sections. In each section we assume parameters ;  and  are uniform. From the coupled wave equations, the transfer matrix which describes the propagating electric field in the corrugated section between zi and ziþ1 can be expressed as         f ER ðziþ1 Þ ER ð z i Þ ER ð z i Þ f ¼ 11 12 ¼ FðiÞ ð11:3Þ f21 f22 ES ðziþ1 Þ ES ðzi Þ ES ð z i Þ where the matrix elements of matrix FðiÞ are given as follows f11 f12 f21 f22

  1 2i ¼ Ei  exp½jbo ðziþ1  zi Þ Ei 1  2i   i 1 ¼ Ei  exp½jbo ðziþ1 þ zi Þ Ei 1  2i   i 1 ¼ Ei  exp½jbo ðziþ1 þ zi Þ Ei 1  2i   1 1 2 ¼  i Ei exp½jbo ðziþ1  zi Þ 1  2i Ei

ð11:4aÞ ð11:4bÞ ð11:4cÞ ð11:4dÞ

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OTHER WAVELENGTH TUNABLE OPTICAL FILTERS

with Ei ¼ exp½gi ðziþ1  zi Þ j i ¼ i  ji þ gi

ð11:4eÞ ð11:4fÞ

In the above equations gi is the complex propagation constant that satisfies the following dispersion equation g2i ¼ ði  ji Þ2 þ2

ð11:5Þ

On the other hand, since there is no active section and no grating in the planar phase-shiftcontrolled (PSC) section (i.e. i ¼ 0 and i ¼ 0), the transfer matrix for the electric field of this section is simplified to 

  ER ðziþ1 Þ expð Þ ¼ ES ðziþ1 Þ 0

0 expð Þ



   ER ð z i Þ ðiÞ ER ðzi Þ ¼P ES ð z i Þ E S ðzi Þ

ð11:6Þ

 where ¼ gp L p  j o Lp , gp is the value of gi in the PSC section and Lp is the length of the PSC section. PðiÞ is the corresponding transfer matrix of the PSC section. The amount of phase shift, O, introduced by each PSC section is given by [11]   4 na  np Lp O ¼ Im 2gp Lp ¼
B

ð11:7Þ

where Im means the imaginary part, na and np are the effective indices of the active and PC sections, respectively. The value of np decreases as the current injection into the PC section increases, hence according to eqn (11.7) the value of O increases. The transfer matrix for phase shift in the active section is given by 

      ER ðziþ1 Þ expðjÞ 0 E ðz Þ ER ð z i Þ ¼ ¼S R i ES ðziþ1 Þ ES ðzi Þ 0 expðjÞ ES ðzi Þ

ð11:8Þ

where  is the phase shift in the active section. By multiplying matrices representing the planar phase-control sections, phase-shift section and the corrugated DFB sections together, the overall transfer matrix for the structure shown in Figs 11.1 and 11.6 becomes 

  T11 ER ðLÞ ¼ T21 ES ðLÞ

T12 T22



   E R ð 0Þ ð3Þ ð6Þ ð4Þ ð1Þ ER ð0Þ ¼ F PF SF PF ES ð 0Þ ES ð 0 Þ

ð11:9Þ

For the structures shown in Figs 11.9 and 11.15, respectively, eqn (11.9) becomes 

  T11 ER ðLÞ ¼ ES ðLÞ T21

T12 T22



   ER ð 0 Þ ð4Þ ð1Þ ER ð0Þ ð5Þ ð2Þ ¼ F SF PF SF ES ð 0 Þ E S ð 0Þ

ð11:10Þ

RESULTS AND DISCUSSIONS OF VARIOUS OPTICAL TUNABLE FILTERS

289

and 

  ER ðLÞ T11 ¼ T21 ES ðLÞ

T12 T22



   E R ð 0Þ ER ð 0 Þ ¼ Fð7Þ Fð6Þ Fð5Þ Fð4Þ Fð3Þ Fð2Þ Fð1Þ E S ð 0Þ ES ð0Þ

ð11:11Þ

In the above equation z1 ¼ 0 and in Figs 11.1, 11.6 and 11.15 z7 ¼ L, whereas z6 ¼ L in Fig. 11.9. In an optical filter (such as the ones shown in Figs 11.1, 11.6, 11.9 and 11.15), the power transmissivity, T, is defined as     ES ðLÞ2  1 2  ¼  T ¼  ER ð0Þ T22 

ð11:12Þ

The threshold gain th and the detuning parameter  can be obtained by solving the following equation numerically T22 ðth ; Þ ¼ 0

ð11:13Þ

The power transmissivity of the filter can be calculated by using the following expression 2    1   T ¼ T22 ð ¼ 0:98th ; Þ

ð11:14Þ

In eqn (11.14), we have used  ¼ 0:98th [7] to achieve a higher output power and hence a smaller 10 dB bandwidth.

11.3

RESULTS AND DISCUSSIONS OF VARIOUS OPTICAL TUNABLE FILTERS

In this section we consider three different filter structures and analyse their performances.

11.3.1 A Quarter Wavelength Phase-shifted Double Phase-shift-controlled DFB LD-based Wavelength Tunable Filter In the following analysis we have used the total filter cavity length L ¼ 500 mm and the lengths of PC sections L2 ¼ L5 ¼ 50 mm. The lengths of active sections which are optimised to give maximum tuning range [15] are L1 ¼ 68:5 mm, L3 ¼ 37 mm, L4 ¼ 135 mm and L6 ¼ 159:5 mm. Equation (11.13) has been solved numerically to analyse the filter structure shown in Fig. 11.1. For a given value of , the numerical solution to eqn (11.13) gives various oscillation modes for the device. The one having the lowest threshold gain is the main mode. Sub-modes are the modes with larger threshold gains. The filter operates by biasing the gain of the device slightly below the threshold gain of the main mode. The normalised detuning coefficient of the main mode determines the amount of deviation of the oscillation wavelength from the Bragg wavelength. The oscillation wavelength is the central

290

OTHER WAVELENGTH TUNABLE OPTICAL FILTERS

wavelength of the filter. For a given O the side mode suppression ratio (SMSR) is defined as the ratio of the highest peak to the second highest peak of the filter power transmissivity. It determines the amount of interference from the channel at the side mode wavelength. As the central wavelength drifts away from the Bragg wavelength, the SMSR reduces. If the SMSR is larger than 10 dB then the adjacent channel interference is minimal [7]. Figure 11.2 shows the calculated transmission spectra of the filter for various values of the phase shift O ranging from 0 to 2 p. The horizontal axis is the relative wavelength defined as

B where
is the operating wavelength of the filter,
B ¼ 2 neff Lð¼ 1:55 mm) is the Bragg wavelength and neff is the effective refractive index. The grating period and coupling coefficient of 0:21 mm and 6 mm1 were used in this calculation. The figure clearly indicates that as O increases the wavelength of the main mode shifts towards the shorter wavelength side. The phase shift O can be controlled by changing the injection current Ip of the PC section. For example when Ip increases, the effective refractive index np decreases due to the free carrier plasma effect and hence O increases according to eqn (11.7). When O ¼ 0 or 2p (referred to as the stop band width of the filter, see case (a) in Fig. 11.2), the relative ˚ . This gives the filter wavelength tuning range of 25 A ˚ . The filter wavelengths are at 12.5 A peak gain varies between 34.9 and 36.1 dB with maximum deviation of 1.2 dB. The relative wavelength is zero when O ¼ p (see case (l) in Fig. 11.2) and the filter SMSR ranges from 15.7 to 29.5 dB. To investigate the effect of the grating period L on the filter performance we have increased its value to 0:238 mm while the rest of the parameters remain identical to those in

Figure 11.2 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are L1 ¼ 68:4 mm, L2 ¼ L5 ¼ 50 mm, L3 ¼ 36:98 mm, L4 ¼ 135:02 mm, L6 ¼ 159:6 mm,  ¼ =2,  ¼ 6 mm1 , L ¼ 0:21 mm and N ¼ 3:7. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; ( j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

RESULTS AND DISCUSSIONS OF VARIOUS OPTICAL TUNABLE FILTERS

291

Figure 11.3 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are L1 ¼ 68:4 mm, L2 ¼ L5 ¼ 50 mm, L3 ¼ 36:98 mm, L4 ¼ 135:02 mm, L6 ¼ 159:6 mm,  ¼ =2,  ¼ 6 mm1 , L ¼ 0:238 mm and N ¼ 3:2647. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; ( j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

Fig. 11.2. The result is shown in Fig. 11.3. In this case when O ¼ 0 or 2p the relative ˚ which gives the total filter tuning range of 28.3 A ˚ . This shows an wavelengths are 14.15 A ˚ compared with the filter shown in Fig. 11.2. The filter peak gain varies increase of 3.3 A between 35 and 36.1 dB and the filter SMSR ranges from 15 to 30 dB. The effect of increasing  to 8 mm1 while keeping L ¼ 0:238 mm is shown in Fig. 11.4. ˚ . The filter peak gain varies In this case the wavelength tuning range has increased to 31.1 A between 33 and 35.2 dB and the SMSR ranges from 18.2 to 34.3 dB. These data indicate that the deviation in the filter peak gain has increased to 2.2 dB compared with the previous two cases. The filter spectra for the case where  ¼ 10 mm1 is shown in Fig. 11.5 where a ˚ has been achieved. The filter peak gain varies between wavelength tuning range of 34.3 A 31.1 and 34.6 dB, which gives maximum deviation of 3.5 dB. The filter SMSR ranges from 19.6 to 34.7 dB. We have also studied the performance characteristics of the filter structure shown in Fig. 11.6 where the active sections have different grating coefficients. For example, the result shown in Fig. 11.7 is for the case where 1 ¼ 6 mm1 , 2 ¼ 4 mm1 and L ¼ 0:21 mm. The achieved peak filter gain varies between 35.6 and 36.4 dB, which gives 0.8 dB deviation. The ˚ and its SMSR ranges from 11.5 to 27 dB. wavelength tuning range of the filter is 25.2 A Figure 11.8 shows the case where 1 ¼ 4 mm1 , 2 ¼ 6 mm1 and L ¼ 0:21 mm. This filter ˚ which is 0.8 A ˚ lower than that of Fig. 11.7. gives the wavelength tuning range of 24.4 A Also, the SMSR ranges from 8.2 dB to 28.1 dB where the lower part is less than the

Figure 11.4 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are L1 ¼ 68:4 mm, L2 ¼ L5 ¼ 50 mm, L3 ¼ 36:98 mm, L4 ¼ 135:02 mm, L6 ¼ 159:6 mm,  ¼ =2,  ¼ 8 mm1 , L ¼ 0:238 mm and N ¼ 3:2647. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; ( j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

Figure 11.5 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are L1 ¼ 68:4 mm, L2 ¼ L5 ¼ 50 mm, L3 ¼ 36:98 mm, L4 ¼ 135:02 mm, L6 ¼ 159:6 mm,  ¼ =2,  ¼ 10 mm1 , L ¼ 0:238 mm and N ¼ 3:2647. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; ( j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

RESULTS AND DISCUSSIONS OF VARIOUS OPTICAL TUNABLE FILTERS

293

Figure 11.6 Analytical model for the
=4-phase-shifted double phase-shift-controlled wavelength tunable filter.

minimum required value of 10 dB. The gain of this filter varies between 37 and 37.8 dB. Comparison of Figs 11.7 and 11.8 indicates that when 1 > 2 , both the tuning range and the SMSR of the filter are larger because of better suppression of side modes. In fact with a larger 1, the feedback from both ends (i.e. sections L1 and L6 ) is larger. This results in a stronger effect of the phase-control region and hence a better suppression of the side modes.

Figure 11.7 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are L1 ¼ 68:4 mm, L2 ¼ L5 ¼ 50 mm, L3 ¼ 36:98 mm, L4 ¼ 135:02 mm, L6 ¼ 159:6 mm,  ¼ =2, 1 ¼ 6 mm1 , 2 ¼ 4 mm1 , L ¼ 0:21 mm and N ¼ 3:7. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; ( j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

294

OTHER WAVELENGTH TUNABLE OPTICAL FILTERS

Figure 11.8 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are L1 ¼ 68:4 mm, L2 ¼ L5 ¼ 50 mm, L3 ¼ 36:98 mm, L4 ¼ 135:02 mm, L6 ¼ 159:6 mm,  ¼ =2, 1 ¼ 4 mm1 , 2 ¼ 6 mm1 , L ¼ 0:21 mm and N ¼ 3:7. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; ( j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

11.3.2 A Single-phase-shift-controlled Double-phase-shift DFB Wavelength Tunable Optical Filter The filter structure used in the analysis is shown in Fig. 11.9. It has a passive phase-shiftcontrolled waveguide (O) which is sandwiched between two phase-shifted active sections 1 and 4 . The total length of the filter cavity L ¼ 500 mm. To analyse this filter’s characteristics, eqn (11.13) has been solved numerically. In general, for a given value of

Figure 11.9 Analytical model for a single-phase-shift-controlled double-phase-shift wavelength tunable filter based on the DFB laser diode structure.

RESULTS AND DISCUSSIONS OF VARIOUS OPTICAL TUNABLE FILTERS

295

, the numerical solution to eqn (11.13) gives various oscillation modes for this device. The one having the lowest threshold gain is the main mode. Again, any mode with threshold gain larger than the main mode is referred to as a sub-mode. The filter operates by biasing the gain of the device slightly below the threshold gain of the main mode. The normalised detuning coefficient of the main mode determines the amount of deviation of the oscillation wavelength from the Bragg wavelength. The oscillation wavelength is the central wavelength of the filter. For a given O the side mode suppression ratio (SMSR) is defined as the ratio of the highest peak to the second highest peak of the filter power transmissivity. It determines the amount of interference from the channel at the side mode wavelength. As the central wavelength drifts away from the Bragg wavelength, the SMSR reduces. If the SMSR is larger than 10 dB then the adjacent channel interference is minimal [5]. Figure 11.10 shows the transmission spectra of the filter for various values of O ranging from 0 to 2. The parameters used in this analysis are 1 ¼ 4 ¼ p=3,  ¼ 6 mm1 , L ¼ 0:21 mm and Li ði ¼ 1; 5Þ ¼ 100 mm. The sub-lengths are chosen such that the structure remains symmetrical. The horizontal axis in Fig. 11.10 is the relative wavelength which is defined as

B where
is the operating wavelength of the filter,
B ¼ 2 neff L is the Bragg wavelength and neff is the effective refractive index. The analysis indicates that the filter spectrum has shifted towards the negative side of the horizontal axis which happens when 1;4 < p=2. The stop band of this filter is at O ¼ 0:3p which is very close to p=3 and the filter SMSR at the stop band is 1.9 dB and it varies from 14.1 to 27 dB elsewhere. The tuning ˚ and its peak transmissivity ranges from 34.4 to 36.5 dB. The filter range of the filter is 17 A

Figure 11.10 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are 1 ¼ 4 ¼ =3,  ¼ 6 mm1 and L ¼ 0:21 mm. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; (j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

296

OTHER WAVELENGTH TUNABLE OPTICAL FILTERS

Figure 11.11 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are 1 ¼ 4 ¼ 2=3,  ¼ 6 mm1 and L ¼ 0:21 mm. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; (j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

transmission spectra (FTS) for the case where 1 ¼ 4 ¼ 2p=3 are shown in Fig. 11.11. In this case the FTS is shifted towards the positive side of the horizontal axis which happens when 1;4 > p=2. The filter stop band is at O ¼ 1:7p and its SMSR, tuning range and peak transmissivity are the same as those of Fig. 11.10. Figure 11.12 shows the variation of the filter power transmissivity versus the relative wavelength for different values of O. The parameters used in the calculations are 1 ¼ 4 ¼ p=2,  ¼ 6 mm1 and L ¼ 0:21 mm. As for sub-lengths, we define a parameter named phase-shift-position ¼ L1 =ðL1 þ L2 Þ ¼ L5 =ðL4 þ L5 Þ. In all of the following analysis has been chosen so as to maximise the tuning range while maintaining the SMSR > 10 dB. For Fig. 11.12, ¼ 0:1, L1 ¼ L5 ¼ 20 mm and L3 ¼ 100 mm. Also we have found that the sub-length L3 has no significant effect on the filter tuning range. Figure 11.12 ˚ which gives the indicates that when O ¼ 0 or 2p the transmissivity peaks occur at 11.05 A ˚ . The peak value of transmissivity within this tuning overall filter tuning range of 22.1 A range is almost constant and varies between 37.7 and 38.1 dB. As the value of O increases, the transmissivity peak shifts towards the left side of the horizontal axis (i.e. lower value) and at O ¼ p the relative wavelength value is zero (i.e.
¼
B ). The filter SMSR changes from 11.1 to 25.4 dB. Shown in Fig. 11.13 are the transmission spectra of the filter for the case where 1 ¼ p=3, 4 ¼ 2p=3 and the rest of the parameters are the same as those used in Fig. 11.12. For this ˚ , peak transmissivity ranges filter structure the achieved wavelength tuning range is 21 A

Figure 11.12 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are 1 ¼ 4 ¼ =2,  ¼ 6 mm1 and L ¼ 0:21 mm. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; (j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

Figure 11.13 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are 1 ¼ =3, 4 ¼ 2=3,  ¼ 6 mm1 and L ¼ 0:21 mm. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; (j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

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Figure 11.14 Power transmissivity versus relative wavelength ð

B Þ for the following different values of O. The parameters used are 1 ¼ =4, 4 ¼ 2=4,  ¼ 6 mm1 and L ¼ 0:21 mm. (a) O ¼ 0; 2; (b) O ¼ 0:1; (c) O ¼ 0:2; (d) O ¼ 0:3; (e) O ¼ 0:4; (f) O ¼ 0:5; (g) O ¼ 0:6; (h) O ¼ 0:7; (j) O ¼ 0:8; (k) O ¼ 0:9; (l) O ¼ ; (m) O ¼ 1:1; (n) O ¼ 1:2; (p) O ¼ 1:3; (q) O ¼ 1:4; (r) O ¼ 1:5; (s) O ¼ 1:6; (t) O ¼ 1:7; (u) O ¼ 1:8; (v) O ¼ 1:9.

from 36.4 to 38.1 dB (i.e. 1.7 dB deviation) and the SMSR varies between 14.8 and 27.8 dB. Compared with the filter structure shown in Fig. 11.12, this structure has a better SMSR but ˚ and its maximum deviation in peak transmissivity is larger. its tuning range is less by 1.1 A Figure 11.14 shows the filter spectra for the case of ¼ 0:15, L1 ¼ L5 ¼ 30 mm, ˚ , the SMSR 1 ¼ p=4 and 4 ¼ 3p=4. The wavelength tuning range of this filter is 20 A ranges from 18.6 to 28.3 dB and the peak transmissivity varies between 35.9 and 38.6 dB (maximum deviation is 2.7 dB. From the above analysis it is clear that when 1 < p=2 and 4 > p=2 the filter tuning range increases.

11.3.3 A Multiple-phase-shift Controlled DFB LD-based Wavelength Tunable Optical Filter The analytical model of the device is shown in Fig. 11.15. In the 3PSC structural design, there are four corrugated active sections in which the gain is controlled by the bias current Ia . There is a phase-control section between each corrugated section. As shown in Fig. 11.15, the central phase-control section is controlled by bias current Ip1 , and the other two phasecontrol sections are injected by current Ip2 . The coupling coefficient  for the planar phasecontrol section is assumed to be zero, and it is 6 mm1 for the corrugated active sections. The electrodes of adjacent sections are electrically isolated from each other and zero facet reflection is assumed.

RESULTS AND DISCUSSIONS OF VARIOUS OPTICAL TUNABLE FILTERS

Figure 11.15

299

Analytical model for the transfer matrix method.

In eqn (11.6), the imaginary part of determines the phase shift introduced by each phase-shift-control section. In the phase-shift-control section, the loss is assumed to be so small that it behaves purely as a phase shifter in which the associated phase change is given as (see Fig. 11.15) bp L p ¼

  p b Lp 0 o

ð11:15Þ

where p and 0 are the refractive indices of the planar phase-shift section and the corrugated DFB section, respectively. By changing the carriers injected into the phase-shift-control region, the refractive index is altered as a result of the free carrier plasma effect. The variation of refractive index caused by the free carrier plasma effect is usually small. However, phase variation over a distance of a large number of wavelengths becomes effective for phase control over the counter-running waves. In the 3PSC structure, the phase shifts in the phase-shift-control sections are defined by 2bp Lp. The numerical solutions to eqn (11.15) are the various oscillation modes of the device. The one having the lowest threshold gain is the main mode. The filter operates by biasing the gain of the device slightly below the threshold gain of the main mode. The normalised detuning coefficient of the main mode determines the amount of deviation of the oscillation wavelength from the Bragg wavelength. The oscillation wavelength is the central wavelength of the filter. Figure 11.16 shows the calculated main modes for various values of O1 and O2 (see Fig. 11.15). The normalised gain margin is the normalised threshold gain difference between the main mode and the mode with the second lowest threshold gain. The normalised gain margin determines the side mode suppression ratio of the transmission spectra. The side mode suppression ratio determines the amount of interference from the channel at the side mode

300

OTHER WAVELENGTH TUNABLE OPTICAL FILTERS

Figure 11.16 Fundamental modes for various values of O1 and O2 . (a) O1 ¼ 0; (b) O1 ¼ 0:4; (c) O1 ¼ 0:8; (d) O1 ¼ 1:2 and (e) O1 ¼ 1:6:

wavelength. As the central wavelength drifts away from the Bragg wavelength, the side mode suppression ratio reduces. A side mode suppression ratio of more than 10 dB is adequate for a low level of cross talk. The normalised gain margin of the main modes is plotted against the normalised detuning coefficient in Fig. 11.17. A high gain margin can be

Figure 11.17 Normalised gain margin of various O2 with different O1 .

SUMMARY

301

Figure 11.18 Power transmissivity spectra for a few values of O1 and O2 versus relative wavelength ð

B Þ. (a) O1 ¼ 1:5; O2 ¼ 1:4; (b) O1 ¼ 0; O2 ¼ 1:6; (c) O1 ¼ 0:5; O2 ¼ 1:8; (d) O1 ¼ ; O2 ¼ 1:9; (e) O1 ¼ 1:5; O2 ¼ 0:2; (f) O1 ¼ 0; O2 ¼ 0:4; (g) O1 ¼ 0:5; O2 ¼ 0:6.

maintained over a wide range of normalised detuning coefficient by adjusting the values of O1 and O2 appropriately. This will ensure high side mode suppression. By varying fit alone with zero phase shift for O2 the normalised detuning coefficient of the main mode varies within a certain range centred at the Bragg condition. By increasing the value of O2 , the tuning range of O1 drifts towards a more positive value of the detuning coefficient. When the value of O2 is more than p, the detuning coefficient of the main mode appears in the far negative end, drifting towards a more positive value with an increase in O2 . The power transmissivities of a few values of O1 and O2 are plotted in Fig. 11.18 as a function of the relative wavelength from the Bragg wavelength. The figure demonstrates a spectra of high and consistent peak gain, with small bandwidth and high side mode suppression.

11.4

SUMMARY

The effects of grating period and coupling coefficient on the performance characteristics of a
=4-phase-shifted double phase-shift-controlled DFB wavelength tunable optical filter have been studied in Figs 11.1 to 11.8. It was found that this filter structure offers a wide tuning range with narrow bandwidth, high gain and a large SMSR. The filter has a peak gain of over ˚ for ˚ for  ¼ 6 mm1 and 34.3 A 30 dB within its tuning range, which is 28.3 A 1  ¼ 10 mm . The filter SMSR varies between 15 and 34.7 dB. In Figs 11.9 to 11.14 we investigated the effects of phase shifts 1 and 4 on tuning range, peak transmissivity and SMSR of single PSC DFB filters. It was found that when 1 < p=2

302

OTHER WAVELENGTH TUNABLE OPTICAL FILTERS

and 4 > p=2, the filter tuning range increases. Also, peak transmissivity of more than 35 dB and SMSR of better than15 dB can be achieved. The analytical investigation has shown that the three-phase-shift-controlled DFB wavelength tunable optical filter has a wide tuning range with narrow bandwidth and high gain (see Figs 11.15 to 11.18). However, these two additional waveguide sections contribute to a larger amount of free carrier absorption, which alters the threshold gain slightly. The calculations are based on the assumption that the relationships of the grating phases at the two sides of the wavelength region are the same for the two O2 phase-control regions.

11.5

REFERENCES

1. Brackett, C. A., Dense Wavelength division multiplexing networks: principles and applications, IEEE J. Select. Areas Comm., 8, 948–964, 1990. 2. Ghafouri-Shiraz, H. and Chu, C. Y. J. Distributed Feedback Lasers: An Overview, Fiber Integ. Opt., 10, 23–47, 1991. 3. Okai, M., Tsuji, S. and Chinone, N., Stability of the Longitudinal Mode in
=4-shifted InGaAsP/InP DFB Lasers, IEEE J. Quantum Electron., 25(6), 1314–1319, 1989. 4. Ghafouri-Shiraz, H. and Chu, C. Y. J. Distributed Feedback Lasers: An Overview, Fiber Integ. Opt., 10, 23–47, 1991. 5. Haus, H. A. and Lai, Y., Theory of Cascaded Quarter Wave Shifted Distributed Feedback Resonators, IEEE J. Quantum Electron., 28, 205–213, 1992. 6. Numai, T., Semiconductor Wavelength Tunable Filters. Int. J. Optoelectron., 6(3), 239–252, 1991. 7. Numai, T., Murata, S. and Mito, I., Tunable Wavelength Filters Using
=4-shifted Waveguide Grating Resonators. Appl. Phys. Lett., 53, 83–85, 1988. 8. Kikushima, K. and Nawata, K., Tunable Amplification Properties of Distributed Feedback Laser Diodes. IEEE J. Quantum Electron., QE-25(2), 163–169, 1989. 9. Magari, K., Kawaguchi, H., Oe, K. and Fukuda, M., Optical Narrow Band Filters Using Optical Amplification with Distributed Feedback. IEEE J. Quantum Electron., QE-24, 2178–2190, 1988. 10. Numai, T., 1:5 mm Phase-Controlled Distributed Feedback Wavelength Tunable Optical Filter. IEEE J. Quantum Electron., QE-28(6), 1508–1512, 1992. 11. Numai, T., 1:5 mm Phase-Shift-Controlled Distributed Feedback Wavelength Tunable Optical Filter. IEEE J. Quantum Electron., QE-28(6), 1513–1519, 1992. 12. Tan, P. W., Ghafouri-Shiraz, H. and Lo, B. S. K., Theoretical Analysis of Multiple-phase shift controlled Distributed Feedback Wavelength Tunable Optical Filters. Microwave and Optical Technol. Lett., 8(2), 72–75, 1995. 13. Makino, T., Transfer-matrix Analysis of the Intensity and Phase Noise of Multisection DFB Semiconductor Lasers. IEEE J. Quantum Electron., 27(11), 2404–2414, 1991. 14. Kogelnik, H. and Shank, C. V., Coupled-Wave Theory of Distributed Feedback Lasers. J. Appl. Phys., 43(5), 2327–2335, 1972. 15. Lew, S. H., Design of Wavelength Tunable Optical Filters, Final Year Project Report, School of EE Eng., University of Birmingham, Birmingham, UK, June 1996.

12 Conclusion, Summary and Suggestions 12.1

SUMMARY AND CONCLUSION

In this book, the performance characteristics of distributed feedback semiconductor laser diodes and optical tunable filters based on DFB laser structures have been investigated. As discussed in Chapter 1, these lasers can be used as optical sources and local oscillators in coherent optical communication networks, in which a stable single mode (in both the transverse plane and the longitudinal direction) and narrow spectral linewidth become crucial. Based on the interaction of electromagnetic radiation with a two-energy-band system, the operating principles of semiconductor lasers were reviewed in Chapter 2. With partially reflecting mirrors located at the laser facets, a Fabry–Perot laser forms the simplest type of optical resonator. However, due to the broad gain spectrum, multi-mode oscillations and mode hopping are common for this type of laser. Nevertheless, single longitudinal mode operation becomes feasible with the use of DFB LDs. The characteristics of the DFB laser were explained using the coupled wave equations. With a built-in periodic corrugation, travelling waves are formed along the direction of propagation in which a perturbed refractive index and/or gain are introduced. In fact, DFB lasers act as optical bandpass filters, so that only frequency components near the Bragg frequency are allowed to pass. The strength of optical feedback is measured by the strength of the coupling coefficient. Based on the nature of the coupling coefficient, DFB semiconductor lasers can be classified into purely index-coupled, mixed-coupled and purely gain- or loss-coupled structures. The discussion focused on the coupled wave equations in Chapter 3. In the analysis, eigenvalue equations were derived for various structural configurations and consequently, their threshold currents and lasing wavelengths were determined. From the lasing threshold characteristics, impacts due to the coupling coefficient, the laser cavity length, the facet reflectivities, the residue corrugation phases and phase discontinuities were discussed in a systematic way. With a single /2 phase shift introduced at the centre of the DFB cavity, the quarterly-wavelength-shifted DFB LD oscillates at the Bragg wavelength. Due to nonuniform field distribution, however, the single-mode stability of this structure deteriorates quickly when the biasing current increases. Based on a five-layer separate confinement heterostructure, the coupling coefficient of a trapezoidal corrugation was computed, from

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

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CONCLUSION, SUMMARY AND SUGGESTIONS

which coupling coefficients of other corrugation shapes, like triangular and rectangular gratings, were also evaluated [1]. In Chapter 4, the idea of the transfer matrix was introduced and explored. Compared with the boundary matching approach in deriving the eigenvalue equation, the transfer matrix method (TMM) is more robust and flexible. By converting the coupled wave equations into a matrix formation, the characteristics of a corrugated DFB laser section can be represented by a 2  2 matrix. This approach has been extended to include phase discontinuity and the effect of residue reflection at the facets. By modifying the elements of the transfer matrix, they can also be used to represent other planar and corrugated structures including passive waveguides, the distributed Bragg reflector and planar Fabry–Perot sections. Using these transfer matrices as building blocks, a general N-sectioned laser cavity model was constructed and the threshold analysis for such a laser model was discussed. With perfectly matched boundaries between consecutive transfer matrices, the number of boundary conditions is reduced significantly. Only the boundary condition located at the laser facet remains to be matched. As compared with the eigenvalue equation, the TMM and/or TLLM simplifies the threshold analysis dramatically. In a similar way, the transfer matrix has also been implemented to evaluate the below-threshold spontaneous emission power spectrum PN . By combining the Poynting vector with the method of Green’s function, numerical results obtained from the three-phase-shift DFB LD were presented and the structural impact on the spectral behaviour discussed [2]. In revealing the potential use of the TMM and/or TLLM in the practical design of DFB LDs, the threshold analysis of various DFB laser structures, including the 3PS [3] and distributed coupling coefficient [4], was carried out in Chapter 5. In an attempt to minimise the effect of the SHB and hence improve the maximum available single-mode output power, it is necessary that a stable single longitudinal mode LD shows a high normalised gain margin ð
LÞ and a uniform field intensity (i.e. small value of flatness, F). Based on the lasing performance at threshold, selection criteria were set at
L > 0:25 and F < 0:05 for a 500 mm length laser cavity. Using these optimised structures, complexities with respect to the design of DFB lasers may be reduced. By changing the value of phase shifts, the coupling coefficient and their corresponding positions, results such as the gain margin ð
LÞ and the uniformity of the field distribution (F) were presented. A conventional single QWS DFB was selected for comparison purposes. This structure is characterised by an intense electric field found at the centre of the cavity. With the introduction of multiple phase shifts along the laser cavity, a 3PS DFB LD with three p/3 phase shifts and a position factor of 0.5 falls within the selection criteria of
L and F. In an alternative approach, the introduction of the DCC also appears to be promising. An improvement in the gain margin was shown for a DCC þ QWS DFB laser structure with a coupling ratio of 1 =2 ¼ 1=3 and a corrugation change at 0.46. Despite the fact that the flatness of this design does not match the requirements of the selection criteria, a high gain margin and oscillation at the Bragg wavelength still count as an advantage in the DCC DFB laser design. The N-sectioned laser cavity model has been used to determine both the threshold and the below-threshold performance of DFB LDs. However, the TMM and/or TLLM used has to be modified when the stimulated emission becomes dominant in the above-threshold biasing regime. In Chapter 6, a new technique [5] which combines the TMM and/or TLLM with the carrier rate equation was introduced. In the model, multiple carrier recombination and a parabolic gain model were assumed. To include any gain saturation effects, a non-linear gain coefficient was introduced. The algorithm needs no first-order derivative and has been

SUMMARY AND CONCLUSION

305

developed in such a way that, with minor modification, the same algorithm can be applied to various laser structures. The TMM-based above-threshold laser model was applied to several DFB laser structures including the QWS, 3PS and the DCC DFB LDs. The QWS DFB laser structure, which is characterised by its non-uniform field distribution, was shown to have a large dynamic range of spatially distributed refractive index. Along the carrier concentration profile, a dip was shown at the centre of the cavity where the largest stimulated photon density was found. By introducing more phase shifts along the corrugation, results from a 3PS DFB LD with 2 ¼ 3 ¼ 4 ¼ p=3 and PSP ¼ 0:5 were presented. Uniform distributions were observed in the carrier density, photon density and the refractive index profile. With an improved threshold gain margin, the abovethreshold characteristics of a QWS LD having non-uniform coupling coefficient were also shown. As compared with the QWS structure, the introduction of a non-uniform coupling coefficient with 1 =2 ¼ 1=3 and CP ¼ 0:46 increased the localised carrier concentration near the plane of corrugation change. A significant reduction in the photon density difference between the central peak and the emitting photon density near the facet was also found. Based on the TMM and/or TLLM, the above-threshold model was extended and applied to evaluate spectral and noise properties of DFB LDs in Chapter 7. Based on the lasing mode distributions obtained for the carrier density, photon density, refractive index and the field intensity, characteristics like the single-mode stability, the spontaneous emission spectrum and the spectral linewidth were investigated. At a fixed biasing current, the QWS structure having the smallest threshold gain was shown to have the smallest linewidth. On the other hand, the non-lasing þ1 side mode became stronger with increasing bias current. Comparatively, the 3PS structure was shown to have the smallest change in lasing wavelength. With the introduction of multiple phase shifts along the corrugation, the internal field distribution becomes more uniform and hence a stable single-mode oscillation results. It is shown that the DCC þ QWS structure has the largest gain margin. The introduction of a distributed coupling coefficient improved the single-mode stability in such a way that the side mode suppression ratio remained at a high value. Wavelength tunability is also improved in this structure. From these results, it is apparent that the design of the DFB LD depends much on its applications. On the other hand, the TMM and/or TLLM has proved to be a powerful tool when dealing with such a problem. In Chapter 8 the transmission line laser model was discussed. TLLM can be classified as a distributed-element circuit model, which is based on the 1-D transmission line matrix method. The building blocks of a TLM network are the TLM link lines and stub lines. It has been shown how the scattering matrices of several TLM sub-networks may be derived by using Thevenin-equivalent circuits. Scattering and connecting are the two main processes that form the basis of TLM. The scattering matrix at a TLM node takes incident voltage pulses and operates on them to produce reflected pulses that travel away from the node. The connecting matrix then directs the reflected pulses from one TLM node to adjacent TLM nodes, where they become incident pulses of the adjacent nodes in the next time iteration. In TLLM, the voltage pulses represent the optical waves that circulate inside the laser cavity. All the important optical processes in the laser are taken into account, such as the spectrallydependent gain of stimulated emission, material and scattering loss, spontaneous emission, carrier–photon interaction and carrier-dependent phase shift. The microwave circuit elements of TLLM are used to describe these laser processes on an equivalence basis. The baseband transformation method is used to enhance the computational efficiency by down-converting from the true optical carrier frequency to its equivalent baseband value.

306

CONCLUSION, SUMMARY AND SUGGESTIONS

The TLLM is a stochastic laser model because random noise effects are included, making it a highly realistic model compared to deterministic laser models. However, intensive time averaging and smoothing techniques are required to obtain the wanted signal, which may otherwise be masked by noise. In Chapter 9 TLLM was modified to allow study of dynamic behaviour of distributed feedback laser diodes, in particular the effects of multiple phase shifts on the overall DFB LD performance. We can easily model any arbitrary phase shift value by inserting some phase shifter stubs into the scattering matrices of TLLM. This helps to make the electric field distribution and hence light intensity of DFB LDs more uniform along the laser cavity and hence minimise the hole burning effect. In Chapter 10 optical tunable filters were introduced. An active optical tunable filter with frequency characteristics that can be tailored to a desired response is an enabling technology for exploiting the full potential bandwidth of optical fibre communication systems. Having seen the importance of such active optical tunable filters, it is highly desirable to design a tunable filter which can perform filtering and amplification of the filtered signal simultaneously. In this chapter, active grating-embedded filters were analysed and designed based on the TMM outlined, where the coupled wave equations of the DFB LD amplifier filters were solved. In addition, the dispersion relationship and the stop band were also discussed. From the solutions of the eigenvalue equations, which were derived by matching the boundary conditions, the threshold current and the lasing wavelength were determined. These included the analysis of the phase discontinuities and the below-threshold characteristics of DFB LDs. The principle of the active tunability of DFB LD amplifier filters was discussed and the structural impacts of the performance of the filters were justified. The effects of grating period and coupling coefficient on the performance characteristics of a novel multi-section and phase-shift-controlled DFB wavelength tunable optical filter have also been studied. It was found that this filter structure offers a wide tuning range with narrow bandwidth, high gain and large SMSR. The filter has over 30 dB peak gain within the ˚ for  ¼ 6 mm1 and 42 A ˚ for  ¼ 10 mm1. The filter SMSR tuning range, which is 32.0 A varies between 12 dB and 30.2 dB. Finally, some analyses on the three-section DBR LD amplifier filter were carried out. It was found that this structure is not suitable for a multisection design since mode hopping occurs in the main lobe, though the tuning range can be increased. Besides, it is very difficult to monolithically integrate the sections without error in practice, since the reflectivities of each section will contribute to the threshold amplitude. Furthermore, the Bragg reflector is lossy and thus a higher threshold current will be required. The effects of grating period and coupling coefficient on the performance characteristics of a
=4-phase-shifted double-phase-shift-controlled DFB wavelength tunable optical filter were studied in Chapter 11. It was found that the new laser-based filter structure offers a wide tuning range with narrow bandwidth, high gain and large SMSR. The filter has over ˚ for ˚ for  ¼ 6 mm1 and 34.3 A 30 dB peak gain within its tuning range, which is 28.3 A 1  ¼ 10 mm . The filter SMSR varies between 15 and 34.7 dB. Also, we investigated the effects of phase shifts 1 and 4 on tuning range, peak transmissivity and SMSR of single PSC DFB filters. It was found that when 1 < p=2 and 4 > p=2, the filter tuning range increases. Also, peak transmissivity of more than 35 dB and SMSR of better than 15 dB can be achieved. The analytical investigation showed that the three-phase-shift-controlled DFB wavelength tunable optical filter has a wide tuning range with narrow bandwidth and high gain (see Figs 11.15 to 11.18). However, these two additional waveguide sections

FUTURE RESEARCH

307

contribute to a larger amount of free carrier absorption which alters the threshold gain slightly. The calculations are based on the assumption that the relationships of the grating phases at the two sides of the wavelength region are the same for the two O2 phase-control regions.

12.2

THE TMM AND/OR TLLM ANALYSIS

In the analysis, the most important characteristics, namely the single-mode stability, spectral linewidth and the spectral behaviour of DFB LDs, have been investigated using the QWS, 3PS and DCC laser structures. The TMM and/or TLLM has provided the flexibility one needs in the design of DFB LDs. There are other dynamic characteristics such as AM and FM response [6–9] and the use of multiple electrode configuration [10–11] which are also important in the characterisation of laser devices.

12.3

FUTURE RESEARCH

Based on the TMM and/or TLLM, the characteristics of DFB LDs have been investigated. Detailed analysis covering both below- and above-threshold biasing regimes has been presented. There are at least three possible research directions which may be worth further investigation.

12.3.1 Extension to the Analysis of Quantum Well Devices In this book, we have concentrated on bulk devices only. There is a potential to apply the same TMM and/or TLLM technique to quantum well structures [12]. The major differences between QW lasers and bulk ones, which we have been examining, are the recombination mechanism [13], material gain characteristics, band structure [14] and confinement factor [15]. One can replace some of the equations used in the bulk model with those appropriate for QW structures. The analysis and the algorithm will remain the same as for the bulk devices described earlier.

12.3.2 Extension to Gain-coupled Devices DFB LDs used in this book belong to the group of purely index-coupled devices. The wavelength filtering mechanism is solely caused by the perturbation of refractive index. In recent years, there has been a growing interest in the use of mixed-coupled and purely gaincoupled devices [16–20]. With the coupling coefficient depending on the material gain, it has been shown both in theory [21] and experiment [22] that these devices exhibit stable single-mode oscillation at the Bragg wavelength. Even for a small degree of gain coupling, a mixed-coupled device shows an improvement in the gain margin. By introducing an imaginary term into the coupling coefficient used in the model, the characteristics of these devices can be investigated using the same methodology.

308

CONCLUSION, SUMMARY AND SUGGESTIONS

12.3.3 Further Investigation of Optical Devices to be Used in WDM In this book, much of the emphasis has been on the threshold and above-threshold analysis of various DFB laser structures. With the deployment of WDM techniques in optical communication networks [23], there is a growing demand for different types of optical device. Optical filters which allow end users easy access to various information like television or interactive digital services [24] are important. Recently, a four-channel notch filter based on a DCC DFB laser structure was demonstrated [25]. Channel cross-talk levels between 9 dB and 20 dB were obtained. In this area of application, the flexible and robust TMM and/or TLLM may be used in the design of these devices.

12.3.4 Switching Phenomena In high-speed optical communication networks employing single-mode semiconductor lasers like DFB laser diodes, there is increasing attention towards phenomena associated with high-speed switching [26–28]. One of the system limitations is known to be the chirping effect induced by semiconductor lasers [29]. Due to the strong coupling between gain and refractive index present in the semiconductor, any switching in the form of injection current results in a variation of optical gain and, hence, the refractive index of a semiconductor laser. A dynamic shift in operating wavelength and broadening of spectral linewidth have been observed as a result of frequency chirping [29]. Due to the dispersive nature of optical fibres, such a spectral broadening affects the pulse shape at the fibre output and consequently degrades the overall system performance. To overcome the problem of frequency chirping, a number of methods have been proposed, including the use of an external modulator [30], pre-shaping of electrical signals [30], injection locking [31] and improvements in device structures [5]. Using the flexible TMM and/or TLLM as a design tool, different structural designs of laser diode can be tested systematically, and hence improve the performance of laser devices.

12.4

REFERENCES

1. Ghafouri-Shiraz, H. and Lo, B., Computation of coupling coefficient for a five-layer trapezoidal grating structure, Opt. and Laser Technol., 27(1), 45–48, 1994. 2. Ghafouri-Shiraz, H. and Lo, B., Structural Impact on the below threshold spectral behavior of three phase shift (3PS) distributed feedback (DFB) lasers, Microwave Opt. Tech. Lett., 7(6), 296–299, 1994. 3. Ghafouri-Shiraz, H. and Lo, B. S. K., Structural dependence of three-phase-shift distributed feedback semiconductor laser diodes at threshold using the transfer-matrix method (TMM AND/ OR TLLM), Semi. Sci. Technol., 9(5), 1126–1132, 1994. 4. Lo, B. S. K. and Ghafouri-Shiraz, H., Spectral characteristics of distributed feedback laser diodes with distributed coupling coefficient, IEEE J. Lightwave Technol., 13(2), 200–212, 1995. 5. Lo, B. S. K. and Ghafouri-Shiraz, H., A method to determine the above threshold characteristics of distributed feedback semiconductor laser diodes, IEEE J. Lightwave Technol., in press. 6. Makino, T., Transfer-matrix analysis of the intensity and phase noise of multisection DFB semiconductor lasers, IEEE. J. Quantum Electron., QE-27(11), 2404–2415, 1991. 7. Vankwikelberge, P., Morthier, G. and Baets, R., CLADDISS–A longitudinal multimode model for the analysis of the static, dynamic and stochastic behaviour of diode lasers, IEEE J. Quantum Electron., QE-26(10), 1728–1741, 1990.

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8. Yoshikuni, Y. and Motosugi, G., Multielectrode distributed feedback laser for pure frequency modulation and chirping suppressed amplitude modulation, J. Lightwave Technol., LT-5, 516–522, 1987. 9. Kikuchi, K. and Okoshi, T., Measurement of FM noise, AM noise and field spectra of 1.3 mm InGaAsP/InP DFB lasers and determination of their linewidth enhancement factor, IEEE J. Quantum Electron., QE-21(6), 1814–1818, 1985. 10. Kikuchi, K. and Tomofuji, H., Analysis of oscillation characteristics of separated-electrode DFB laser diodes, IEEE J. Quantum Electron., QE-26(10), 1717–1727, 1985. 11. Kikuchi, K. and Tomofuji, H., Performance analysis of separated-electrode DFB laser diodes, Electron. Lett., 25(2), 162–163, 1989. 12. Zory, Jr., P. S., Quantum Well Lasers. New York: Academic Press, 1993. 13. Agrawal, G. P. and Dutta, N. K., Long-Wavelength Semiconductor Lasers. Princeton, NJ: Van Nostrand, 1986. 14. Yariv, A., Optical Electronics, 4th edition. Orlando, FL: Saunders College Publishing, 1991. 15. Ghafouri-Shiraz, H. and Tsuji, S., Strain effects on refractive index and confinement factor of InxGa(1-x) laser diodes, Microwave Opt. Tech. Lett., 7(3), 113–119, 1994. 16. David, K., Buus, J. and Baets, R., Basic analysis of AR-coated. partly gain-coupled DFB lasers: The standing wave effect, IEEE J. Quantum Electron., 28(2), 427–433, 1992. 17. David, K., Morthier, G., Vankvikelberge, P., Baets, R., Wolf, T. and Borchert, B., Gain-coupled DFB lasers versus index-coupled and phase-shifted DFB lasers: A comparison based on spatial hole burning corrected yield, IEEE J. Quantum Electron., 27(6), 1714–1724, 1991. 18. David, K., Buus, J., Morthier, G. and Baets, R., Coupling coefficient in gain-coupled DFB lasers: Inherent compromise between strength and loss, Photon. Tech. Lett., 3(5), 439–441, 1991. 19. Luo, Y., Nakano, Y., Tada, K., Inoue, T., Homsomatsu, H. and Iwaoka, H., Fabrication and characteristics of gain-coupled distributed feedback semiconductor lasers with a corrugated active layer, IEEE. J. Quantum Electron., QE-27(6), 1724–1732, 1991. 20. Makino, T., Transfer matrix analysis of the spectral linewidth of a partly gain-coupled MQW DFB laser, Optical Quantum Electron., 25, 473–481, 1993. 21. Kogelnik, H. and Shank, C. V., Coupled-wave theory of distributed feedback lasers, J. Appl. Phys., 43(5), 2327–2335, 1972. 22. Luo, Y., Nakano, Y. and Tada, K., Purely gain-coupled distributed feedback semiconductor lasers, Appl. Phys. Lett., 56(17), 1620–1622, 1990. 23. Lee, T. P. and Zah, C. N., Wavelength-tunable and single frequency semiconductor lasers for photonic communications networks, IEEE Communications Magazine, 73, 42–52, 73, 1989. 24. Van Heijnngen, P., Muys, W., Van der Platts, J. and Willems, F., Crosstalk in a fibre access network demonstrator carrying television and interactive digital services, Electron. & Communication Eng. J., 6, 49–55, 1994. 25. Weber, J. P., Stoltz, B., Dasler, M. and Koek, B., Four channel tunable optical notch filter using InGaAsP/InP reflection gratings, IEEE Photon. Techol. Lett. 6(1), 77–82, 1994. 26. Lidoyne, O., Gallion, P., Chabran, C. and Debarge, G., Locking range, phase noise and power spectrum of an injection-locked semiconductor laser, IEE Proc. Pt. J., 137, 147–153, 1990. 27. Cartledge, J. C., Improved transmission performance resulting from the reduced chirp of a semiconductor laser coupled to an external high-Q resonator, J. Lightwave Technol., 8, 716–721, 1990. 28. Mohrdiek, S., Burkhard, H. and Walter, H., Chirp reduction of directly modulated semiconductor lasers at 10 Gb/s by strong CW light injection, J. Lightwave Technol., 12, 418–424, 1994. 29. Linke, R. A., Modulation induced transient chirping in single frequency lasers, IEEE J. Quantum Electron., QE-21, 593–597, 1985. 30. Petermann, K., Laser Diode Modulation and Noise. Tokyo, Japan: KTK Scientific and Kluwer Academic Publishers, 1988. 31. Hui, R., D’Ottavi, D., Mecozzi, A. and Spano, P., Injection locking in distributed feedback semiconductor lasers, IEEE J. Quantum Electron., QE-27, 344–351, 1931.

Index above-threshold analysis DFB LDs 171–94 mathematical grid 156 above-threshold characteristics 3PS DFB LDs 161 DFB LDs 149–94 numerical processing 153–7 QWS DFB LDs 158–61 above-threshold lasing mode, TMM approach 149–53 above-threshold model, numerical results 157–68 above-threshold spontaneous emission spectrum 182–5 absorption of radiation 32–3 active tunability DFB LD amplifier filters 268–70 adjustable-length open-circuit stub 217 AlGaAs 5, 43 AM/FM noise spectrum measurement 21 amplification rate of optical intensity 210 amplified spontaneous emission spectrum formulation 111–20 amplitude coefficients 104, 259 amplitude gain 3PS DFB laser structure 177 and PSP 127 DCC þ QWS DFB laser structure 179 QWS DFB laser structure 176 versus normalised detuning parameter 262, 265, 281 amplitude gain coefficient 52 amplitude mirror loss 42 amplitude reflection coefficients 81 amplitude-shift keying (ASK) 6, 25 amplitude threshold gain 86, 87, 93, 94, 137 asymmetric facet reflectivities 88 atomic transitions, dispersive properties of 36–7 attenuators 221 backward coupling coefficient 56

backward transfer matrix 109 bandnumber 223 baseband transformation technique 221–3 below-threshold characteristics 266–8 below-threshold spontaneous emission power 115–17 numerical results 117–20 Bernard–Duraffourg condition 40 bit error rate (BER) 6, 8, 219 Bragg conditions 53, 55 Bragg diffraction 61, 151, 231 second-order 73–5 Bragg frequency 54, 279, 287 Bragg grating sections 17 Bragg propagation constant 53–5, 80, 103, 256, 287 Bragg reflector 280, 281 Bragg resonance 53 Bragg wavelength 54, 86, 87, 94, 95, 136, 142, 151, 231, 267, 268, 281, 286, 300 buried crescent (BC) 45 buried heterostructure (BH) 45, 208 carrier concentration change in 218 longitudinal distribution 159 carrier density distribution 3PS DFB LD 247 longitudinal 162, 164, 167 carrier density model 208–9 carrier density rate equation 209, 240–1 carrier-induced frequency chirp 216–18 carrier–photon resonance 208 Cauchy–Riemann condition 82–4, 257, 261 channelled substrate planar (CSP) 45 characteristic impedance 197–9 circuit modelling techniques 195–229 cleaved-coupled-cavity (CCC) lasers 206 coherent optical communication system 5–7 schematic diagram 6

Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

312

INDEX

complex field reflectivities 92 complex propagation constant 288 complex transcendental equations 82–4 solutions 260–2 computational efficiency baseband transformation 221–3 connection matrix 199–206, 239–40 for stubs within a section 239–40 continuous-pitch-modulated (CPM) DFB laser 136 continuous radiation regime 11 continuous wave (CW) linewidth 219 continuous wave (CW) operation 44 corrugated DFB LD 259 corrugated section, transfer matrix 151 corrugation phase 87–9 corrugation position, effects of change 138 corrugation shape 62–4 counter-running waves 90–1 coupled wave equations 51–9, 79–99, 103, 104, 253, 287 solution 255–62, 259–60 solutions 80–2 coupled wave theory 31–78 coupling coefficient 79, 127, 131, 133, 137, 139, 140, 257 effect of corrugation shape 62–4 structural definition 60–2 trapezoidal corrugation 69–75 see also DCC coupling ratio 139 effects on threshold characteristics 137 DBR lasers 16, 17, 206 DBR LD amplifier filter 279–81 DCC DFB laser structure, optimisation of 138–41 DCC þ 3PS DFB laser structure, threshold analysis 141–5 DCC þ 3PS DFB LDs 142, 165–8 distributed coupling coefficient 165–8 DCC þ QWS DFB laser structure amplitude gain 179 detuning coefficient 180 lasing characteristics 179 spontateous emission spectra 184 DCC þ QWS DFB LDs 164–5 distributed coupling coefficient 164–5 DCC DFB laser 124 DCC DFB LDs, threshold analysis 136–41 degenerate homojunction 39

dense wavelength division multiplexing (DWDM) 253 density of state function 50 detuning coefficient 3PS DFB laser structure 178 and PSP 127 DCC þ QWS DFB laser structure 180 DFB LD 87 mirrorless index-coupled DFB LD 86 QWS DFB laser structure 176 detuning parameter 289 DFB laser 16 DFB laser diode 231–3 above-threshold analysis 171–94 above-threshold characteristics 149–94 amplifier filters, structural impacts 270–4 characteristics analysis 231–52 gain-coupled or loss-coupled 58–9 index-coupled 56–7 mixed-coupled 57 model with phase shift 234–6 optimisation 123–47 TLLM 231–52 DFB laser diode amplifiers 285 DFB lases 208 DFB semiconductor laser diodes 31–78 DFB semiconductor lasers 21, 232 diagonal transfer matrix 268 differential phase-shift keying (DPSK) 7 digital modulation methods 6 discrete Fourier transform (DFT) 223 dispersion equation 81 dispersion-flattened fibre 3 dispersion relationship 257 dispersion-shifted fibre 3 dispersive properties of atomic transitions 36–7 distributed Bragg reflector (DBR), semiconductor optical filters 253–84 distributed element model 196 double heterostructure 44 dual-channel planar buried heterostructure (DCPBH) 45 dynamic single-mode (DSM) operation 232 effective index method (EIM) 14 effective linewidth enhancement factor 21, 188 effective refractive index 210 at zero carrier injection 152 eigenvalue equation 82, 91, 97, 257 Einstein coefficient of absorption 33 Einstein coefficient of stimulated emission 34

INDEX

Einstein coefficients 48–9 Einstein relations 33–6, 48 emission of radiation 32–3 erbium-doped fibre amplifiers (EDFA) 6 external cavity (EC) mode-locked lasers 206 Fabry–Perot cavity 15, 280 Fabry–Perot etalon 31–2, 254 principle of 41–3 Fabry–Perot laser 23, 195, 206 fast Fourier transform (FFT) 208, 223, 225–6 Fermi–Dirac distribution function 40 fibre grating lasers 206 field-effect transistors (FETs) 196 field-intensity distribution 132 filter transmission spectra (FTS) 295–6 filtering technologies, comparison of 254 finite facet reflectivities 109–10 fixed-length stub 216, 218 free carrier plasma effect 269 frequency-division multiplexing (FDM) 4, 6 frequency-domain models 233 frequency response 3PS DFB LD 244 QWS DFB LD 243 frequency-shift keying (FSK) 6, 25 full width at half maximum (FWHM) 17–18, 225, 255 GaAs 62, 269 gain filter, block diagram 211 gain margin 17, 84–5, 95, 131, 134, 135, 137, 139, 140, 143, 144, 300 numerical results 174–81 structural impacts on 128–30 variation of 180 gain profiles 267 gain spectrum 267 improvement 215 Gaussian noise statistics 221 grating-embedded semiconductor wavelength tunable filters 253 Green’s function method 112–15, 171 group refractive index 42 guided mode 11, 12 Henry’s linewidth enhancement factor 218 heterodyne receiver with coherent postdetection processing (HE/CP) 6

313

with incoherent postdetection processing (HE/IP) 6 heterojunction double 44 single 44 heterojunction bipolar transistors (HBTs) 196 high-speed optical coherent communication, system requirements 7–26 homojunction 43 incident voltage pulses 204 index-coupled DFB LD 257, 258, 262, 266 InGaAsP 3, 62 intensity modulation with direct detection (IM/DD) scheme 5 internal field distribution 132, 146 structural impacts on uniformity 130–4 inter-symbolic interference (ISI) 7 intrinsic linewidth enhancement factor 20, 188 Kirchhoff’s law 196 Kramer–Kroenig relationship 48, 96 Kronecker delta function 114 laser amplification 210–16 laser diode amplifiers 206 as tunable filters 255 lasers, basic principle 32–7 lasing characteristics 88 3PS DFB laser structure 177 DCC þ QWS DFB laser structure 179 QWS DFB laser structure 175 lasing wavelength 93, 149 variation of 181 lateral carrier confinement 44–5 liquid phase epitaxy (LPE) 43 lithium niobate 269 longitudinal correction factor 24 longitudinal distribution 153, 244–7 carrier concentration 159 carrier density 162, 164, 167, 245 normalised intensity 160, 163, 166 photon concentration 159 photon density 162, 165 refractive index 160, 163, 166, 168 longitudinal intensity distribution 3PS DFB LDs 246 QWS DFB LDs 246 Lorentzian line shape 213 lossless transmission lines 219

314

INDEX

lumped element model 196 lumped reactive elements 198 Mach–Zehnder integrated optic interferometer tunable filter 254 material parameters 157 matrix methods 102–10 types 103 maximum laser linewidth 25 Maxwell’s equations 8, 9, 37, 67, 196, 197, 219, 287 metal–organic chemical vapour deposition (MOCVD) 58 microwave circuit models for semiconductor lasers 196 Millman’s theorem 206 mode discrimination 84–5 momentum matrix 50, 51 moving average filter 223–4 multi-dielectric layers 52 multi-electrode DFB LD amplifier filter 269 multi-electrode DFB optical filter, transmission spectrum 270, 271 multi-mode oscillation 32 multiple-phase-shift 96, 101 multiple-phase-shift DFB LD 264 multiple-phase-shift DFB LD-based wavelength tunable optical filter 298–301 multi-section DFB LD amplifier filter 274–9 net amplitude gain coefficient 37 net gain difference 17 Newton–Raphson approximation 82–4, 260–2 Newton–Raphson iteration 82, 85 Newton–Raphson method 97, 154 nodal current 205 nodal voltage 204, 206 nominal threshold current density 48 normalised impedances 199–201 normalised intensity distribution 3PS DFB LDs 248 longitudinal 160, 163, 166 numerical processing, above-threshold characteristics 153–7 Nyquist’s sampling theorem 221 one-dimensional corrugated DFB LD 259 one-dimensional TLM 196 optical communication systems historical progress 1–4 overview 1–30

optical confinement factor 45, 210 optical devices 2 optical fibres 2 communication systems 4–7 components 5 optical field 233 optical field amplitude 210 optical filtering 253 optical output power 149, 172 optical time-division-multiplexed (OTDM) communication systems 219 optical tunable filters see wavelength tunable optical filters optimisation DCC DFB laser structure 138–41 DFB LDs 123–47 optimum design of 3PS DFB laser 128–34 optoelectronic integrated circuit (OEIC) design 196 parasitic capacitance 199 parasitic inductance 199 Petermann’s gain guiding factor 24 phase (adjusting) stubs 216 phase discontinuities effects of 89–95 in DFB LDs 263–6 phase jitters 7 phase noise 17–24 phase position and value effect on 3PS DFB LDs 249 phase shift 118, 119, 133–5, 140, 141, 143 DFB laser 89 DFB laser diode model with 234–6 effect of number of 247–9 effect on lasing characteristics 125–6 1PS DFB laser diode 92 introduction of 106–9 scattering matrix for DFB laser diode with 238 phase-shift-controlled DFB LD 268 phase-shift-controlled DFB LD amplifier filters 269–70, 272–4 characteristics of 273 phase-shift-controlled DFB LD optical filter structures 274–9 phase-shift keying (PSK) 6, 25 phase-shift position (PSP) 79, 263, 278 effect on lasing characteristics 126–8 of 1PS DFB laser diode 94–5

INDEX

photon concentration, longitudinal distribution 159 photon density 209, 233 inside cavity 152 longitudinal distribution of 162, 165 photon density distribution of 3PS DFB LDs 246 photon distribution 158 photons 32 planar structure 108 Planck’s equation 32 population inversion concept 33–6 in semiconductor junctions 38–40 population inversion factor 23 post-processing methods 223–6 power matrix model (PMM) 208 power transmissivity 289 versus relative wavelength 277–8, 290–8, 301 pulse code modulation (PCM) 219 Q-factor 213, 215 QWS DFB LD-based wavelength tunable filter 289–94 QWS DFB LDs 79, 92, 123, 164–5, 232, 233, 241, 242 above-threshold characteristics 158–61 advantages and disadvantages 95–7 detuning coefficient 176 frequency response 243 lasing characteristics 175 longitudinal carrier density distribution 245 longitudinal intensity distribution 246 spontaneous emission spectra 183 transient longitudinal carrier density 244 transient response 242 see also DCC þ QWS DFB LDs radiative recombination coefficient 51 recombination rate 220 reflected voltage pulses 204 refractive index change in 216 distribution of 3PS DFB LDs 247 first-order approximation 151 longitudinal distribution 160, 163, 166, 168 spatially distributed 158 repeater spacing 3 resonance modes 125, 126 ridge waveguide (RW) 45

315

sample and overlay method 225 scattering matrix 103, 199, 199–207, 212 DFB laser diode with phase shift 238 uniform DFB LD 236–8 semiconductor junction diodes 2 semiconductor junctions, population inversion in 38–40 semiconductor laser amplifiers (SLA) 6 semiconductor lasers basic principles 38–51 index-guided 47 material gain 45–8 structural improvements 43–5 separate confinement heterostructure (SCH) 44, 66 side mode suppression ratio (SMSR) 85, 119–20, 183, 271, 276, 278, 290, 291, 293, 295, 296, 298, 300 signal analysis 223–6 silica-based optical fibre 3 single longitudinal mode (SLM) 15–17, 32, 84, 87, 89, 95 single mode along transverse plane 8–15 single-mode fibre (SMF) 6, 85 single-mode stability 96, 172–4 single-phase-shift-controlled double-phase-shift DFB wavelength tunable optical filter 294–8 single-phase-shifted (1PS) DFB LD 90, 263 slab dielectric waveguide 9 smoothing algorithm 225 spatial hole burning effect (SHB) 123, 158 spatially distributed refractive index 158 spectral linewidth 185–91 formula limitations 26 formulation 17–24 measurement 21 numerical results 189–91 requirements 17–26 variation of 189–91 spectral purity requirements 7–17 spontaneous emission coupling factor 220 current source 220 distributed current source model 221 model 219–21 spontaneous emission rate 23, 186 spontaneous emission spectra 3PS DFB LD 184 DCC þ QWS DFB LD 184 QWS DFB LD 183 stable averaging method 224–5

316

INDEX

steady-state carrier rate equation 152 stop bands 257 structural parameters 157, 161, 164, 167 stub-attenuator model 216 stub filter response 213 synchronised voltage pulses 199 system requirements 24–5 system transmission rate 25 telegraphist equations 197 Thevenin equivalent circuit 201, 203, 205, 211, 212 3PS DFB LDs 96, 97, 117–20, 241–2, 264 above-threshold characteristics 161 amplitude gain 177 carrier density distribution 247 detuning coefficient 178 dynamic characteristics 243 frequency response 244 lasing characteristics 177 longitudinal intensity distribution 246 normalised intensity distribution 248 optimum design 128–34 phase position and value effect 249 photon density distribution 246 refractive index distribution 247 spontaneous emission spectra 184 threshold analysis 124–8 transient longitudinal carrier density 245 transient response 243 see also DCC þ 3PS DFB LDs three-port circulator 216 threshold analysis 85, 123–47 3PS DFB laser 124–8 DCC þ 3PS DFB laser structure 141–5 DCC DFB LD 136–41 DFB LDs 262–8 threshold carrier density 48 threshold characteristics coupling ratio effects 137 DFB LDs 145 threshold condition 111 threshold equation 85 threshold gain 23, 276, 289 time-dependent Schro¨dinger equation 49 time-domain model (TDM) 208 time-domain optical-field models 233 time-varying photon density 232 TLLM 196, 206–7, 233 basic construction 207–8 components 207

DFB LDs 234, 236–8 characteristics analysis 231–52 parameter values 241 techniques 195–229 TLLM MPS DFB model 241, 242 TLM 103, 196–9 link lines 197–8 stub filter 211, 215 stub lines 197, 198–9, 200 sub-network 201, 203, 205, 206 TMM 103, 123–47, 208, 253 above-threshold analysis of DFB laser structures 171–94 above-threshold characteristics of laser diodes 149–94 above-threshold lasing mode 149–53 laser structures 107 total radiative recombination rate 48–51 transcendental function 82 transfer matrix 153, 276 arbitrary section 150 corrugated section 151 formulation 103–6, 257–60 transfer matrix chain 106 transfer matrix method see TMM transfer matrix modelling 101–22, 208 transient longitudinal carrier density 3PS DFB LDs 245 QWS DFB LDs 244 transient response 3PS DFB LD 243 QWS DFB LD 242 transmission-line laser modelling see TLLM transmission-line matrix see TLM transmission-line modelling 196 transparency carrier concentration 46 transverse carrier confinement 43–4 transverse electric (TE) mode 9, 11–14, 60, 62, 67 transverse electromagnetic (TEM) field 196 transverse field distribution in unperturbed waveguide 66–9 transverse magnetic fiekd (TM) mode 9, 13, 14 trapezoidal corrugation, coupling coefficient 69–75 tunable fibre Fabry–Perot filters 254 tunable Mach–Zehnder (MZ) filters 254 unperturbed waveguide, transverse field distribution 66–9

INDEX

variable phase length 218 vector wave equation 51 wave impedance 20 wave propagation, in periodic structures 51 wave propagation constant 52 wavelength division multiplexing (WDM) 4, 6, 253, 285 wavelength selection 254–5

mechanism 253 operation principle 254 wavelength selective amplifiers (AMP) 221 wavelength tunable optical filters 253–309 analytical model 286 power transmission spectra for type A 274 zero facet reflection 85

317