Photonic Signal Processing [2nd ed.] 1498769934, 9781498769938

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Photonic Signal Processing

Photonic Signal Processing Second Edition

Le Nguyen Binh

MATLAB ® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB ® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB ® software.

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-4987-6993-8 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Binh, Le Nguyen, author. Title: Photonic signal processing / Le Nguyen Binh. Description: Second edition. | Boca Raton : Taylor & Francis, a CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa, plc, [2019] | Includes bibliographical references. Identifiers: LCCN 2018024823 (print) | LCCN 2018026380 (ebook) | ISBN 9780429436994 (eBook General) | ISBN 9780429792632 (Pdf) | ISBN 9780429792625 (ePub) | ISBN 9780429792618 (Mobipocket) | ISBN 9781498769938 (hardback : acid-free paper) Subjects: LCSH: Optical communications. | Signal processing. | Photonics. | Optoelectronic devices. Classification: LCC TK5103.59 (ebook) | LCC TK5103.59 .B5243 2019 (print) | DDC 621.36/5--dc23 LC record available at https://lccn.loc.gov/2018024823 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Dedication To the memory of my father To my mother, Mrs. Nguyen Thi Huong To Phuong Nguyen and Lam

Contents Preface............................................................................................................................................xvii Author .............................................................................................................................................xix Chapter 1

Introduction ..................................................................................................................1 Acronyms .....................................................................................................................1 1.1 Preamble ............................................................................................................ 1 1.2 Introductory Remarks........................................................................................2 1.3 Organization of Chapters .................................................................................. 4

Chapter 2

Photonic Signal Processing Via Signal-Flow Graph.................................................... 7 2.1 2.2

2.3

2.4 2.5

2.6

2.7

2.8

2.9

Introduction ....................................................................................................... 7 Incoherent Photonic Signal Processing ............................................................. 8 2.2.1 Fiber-Optic Delay Lines ..................................................................... 10 2.2.1.1 Fiber-Optic Directional Couplers ....................................... 10 2.2.2 Fiber-Optic and Semiconductor Ampliiers ....................................... 11 Coherent Integrated-Optic Signal Processing ................................................. 12 2.3.1 Integrated-Optic Delay Lines ............................................................. 15 2.3.2 Integrated-Optic Phase Shifters ......................................................... 16 2.3.3 Integrated-Optic Directional Couplers............................................... 16 2.3.4 Integrated-Optic Ampliiers ............................................................... 18 Remarks ........................................................................................................... 19 Signal-Flow Graph Approach and Photonic Circuits ......................................20 2.5.1 Introductory Remarks ........................................................................ 21 2.5.2 Signal-Flow Graph Theory ................................................................ 21 2.5.3 Deinitions of SFG Elements .............................................................. 22 Rules of SFG.................................................................................................... 23 2.6.1 Rule 1: Transmission Rule.................................................................. 23 2.6.2 Rule 2: Addition Rule ......................................................................... 23 2.6.3 Rule 3: Product Rule ..........................................................................24 Mason’s Gain Formula ....................................................................................25 2.7.1 Analysis of an Incoherent Recursive Fiber-Optic Signal Processor (RFOSP) ............................................................................25 2.7.2 SFG Representation of the Incoherent RFOSP ..................................25 2.7.3 Derivation of the Transfer Functions of the Incoherent RFOSP ........ 27 2.7.4 Stability Analysis of the Incoherent RFOSP...................................... 29 2.7.5 Design of the Incoherent RFOSP ....................................................... 29 2.7.6 Remarks.............................................................................................. 30 Optmason: A Program for Automatic Derivation of the Optical Transfer Functions of Photonic Circuits from Their Connection Graphs ..................... 31 2.8.1 Overview ............................................................................................ 31 2.8.2 Using OPTMASON ........................................................................... 33 2.8.3 Contents of the Input File for above Examples .................................. 35 2.8.4 The OPTMASON Program Structure ............................................... 36 Appendix: Z-Transform ...................................................................................40

vii

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Contents

2.10 2.11 Chapter 3

Appendix: OPTMASON.PAS Program Listing............................................ 42 Appendix: Using “OPTIMASON” the Computer Aided Generator.............66

Bandpass Optical Filters by DSP Techniques ............................................................ 71 3.1

Optical Fixed Bandpass Filter ....................................................................... 71 3.1.1 Introductory Remarks ...................................................................... 71 3.1.2 Chebyshev Optical Filter Speciication and Synthesis Algorithm ..... 71 3.1.3 Basic Characteristics of Chebyshev Lowpass Filters ....................... 72 3.1.3.1 Chebyshev-Type Optical Bandpass Filter Speciication ..................................................................... 72 3.1.3.2 Illustration of a Chebyshev Bandpass Optical Filter ........ 74 3.1.3.3 Optical Components for Chebyshev Filters ...................... 75 3.1.3.4 Realization of the Chebyshev Optical Bandpass Filters ....... 78 3.1.3.5 The COF1 ......................................................................... 78 3.1.3.6 Parallel Realization........................................................... 79 3.1.3.7 The COF2 ......................................................................... 81 3.1.3.8 Discussions ....................................................................... 83 3.1.4 Concluding Remarks ........................................................................ 83 3.2 Tunable Optical Bandpass Waveguide Filters ...............................................84 3.2.1 Introductory Remarks ......................................................................84 3.2.2 Transfer Function of IIR Digital Filters to Be Synthesized ............. 85 3.2.3 Basic Building Blocks of Tunable Optical Filters ............................ 86 3.2.3.1 Tunable Coupler ................................................................ 86 3.2.3.2 All-Pole Filter ................................................................... 87 3.2.3.3 All-Zero Filter................................................................... 89 3.2.4 Tunable Optical Filter....................................................................... 91 3.2.5 Synthesis of Tunable Optical Filters ................................................ 91 3.2.5.1 Design Equations for the Synthesis of Tunable Optical Filters ................................................................... 91 3.2.5.2 Synthesis of Second-Order Butterworth Bandpass and Bandstop Tunable Optical Filters...............................92 3.2.5.3 Designed Parameter Values of the Bandpass and Bandstop Tunable Optical Filters .....................................92 3.2.5.4 Tuning Parameters of the Synthesized Bandpass and Bandstop Tunable Optical Filters...............................94 3.2.6 Synthesis of Bandpass and Bandstop Tunable Optical Filters with Variable Bandwidths and Fixed Center Frequencies ............... 98 3.2.6.1 Synthesis of Tunable Optical Filters with Fixed Bandwidths and Tunable Center Frequencies .................. 98 3.2.6.2 Fabrication Tolerances of Filter Parameters ................... 100 3.2.7 Concluding Remarks ...................................................................... 104 References ................................................................................................................ 105 Chapter 4

Photonic Computing Processors .............................................................................. 107 4.1

Incoherent Fiber-Optic Systolic Array Processors ...................................... 107 4.1.1 Introduction .................................................................................... 107 4.1.2 Digital-Multiplication-by-Analog-Convolution Algorithm and Its Extended Version................................................................ 109 4.1.2.1 Multiplication of Two Digital Numbers.......................... 109

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4.2

4.3

4.4

Chapter 5

4.1.2.2 High-Order Digital Multiplication.................................... 110 4.1.2.3 Sum of Products of Two Digital Numbers........................ 112 4.1.2.4 Two’s Complement Binary Arithmetic ............................. 112 4.1.3 Elemental Optical Signal Processors ............................................... 113 4.1.3.1 Optical Splitter and Combiner .......................................... 113 4.1.3.2 Binary Programmable Incoherent Fiber-Optic Transversal Filter .............................................................. 114 4.1.4 Incoherent Fiber-Optic Systolic Array Processors for Digital Matrix Multiplications ..................................................................... 116 4.1.4.1 Matrix–Vector Multiplication ........................................... 116 4.1.4.2 Matrix–Matrix Multiplication .......................................... 117 4.1.4.3 Cascaded Matrix Multiplication ....................................... 118 4.1.4.4 Performance Comparison ................................................. 121 4.1.4.5 Fiber-Optic Systolic Array Processors Using Non-Binary Data............................................................... 122 4.1.4.6 High-Order Fiber-Optic Systolic Array Processors ......... 123 4.1.5 Remarks............................................................................................ 124 Programmable Incoherent Newton–Cotes Optical Integrator ...................... 125 4.2.1 Introductory Remarks ...................................................................... 125 4.2.2 Newton–Cotes Digital Integrators ................................................... 126 4.2.2.1 Transfer Function .............................................................. 126 4.2.2.2 Synthesis ........................................................................... 127 4.2.2.3 Design of a Programmable Optical Integrating Processor........................................................................... 130 4.2.2.4 Analysis of the FIR Fiber-Optic Signal Processor ........... 132 4.2.2.5 Analysis of the IIR Fiber-Optic Signal Processor ............ 133 4.2.3 Section Remarks............................................................................... 143 Higher-Derivative FIR Optical Differentiators ............................................. 143 4.3.1 Introduction ...................................................................................... 144 4.3.2 Higher-Derivative FIR Digital Differentiators................................. 146 4.3.3 Synthesis of Higher-Derivative FIR Optical Differentiators ........... 147 4.3.4 Computed Differentiators of First and Higher Orders ..................... 150 4.3.4.1 First-Derivative Differentiators ........................................ 150 4.3.4.2 Second-Derivative Differentiators .................................... 152 4.3.4.3 Third-Derivative Differentiators....................................... 154 4.3.4.4 Fourth-Derivative Differentiator ...................................... 158 4.3.5 Remarks............................................................................................ 159 Appendix A: Generalized Theory of the Newton–Cotes Digital Integrators ..................................................................................................... 160 4.4.1 Deinition of Numerical Integration ................................................. 161 4.4.2 Newton’s Interpolating Polynomial .................................................. 162 4.4.3 General Form of the Newton–Cotes Closed Integration Formulas .......................................................................................... 164 4.4.4 Generalized Theory of the Newton–Cotes Digital Integrators........ 164

Optical Dispersion Compensation and Gain Flattening .......................................... 167 5.1 5.2

Introductory Remarks.................................................................................... 167 Dispersion Compensation Using Optical Resonators .................................... 167 5.2.1 Signal-Flow Graph Application in Optical Resonators.................... 170 5.2.2 Stability Test ..................................................................................... 175

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Contents

Frequency and Impulse Responses .................................................. 176 5.2.3.1 Frequency Response ......................................................... 176 5.2.3.2 Impulse and Pulse Responses ........................................... 177 5.2.3.3 Cascade Networks ............................................................ 178 5.2.3.4 Circuits with Bi-directional Flow Path ............................. 178 5.2.3.5 Remarks ............................................................................ 178 5.2.4 Double-Coupler Double-Ring Circuit Under Temporal Incoherent Condition ........................................................................ 178 5.2.4.1 Transfer Function of the DCDR Circuit ........................... 178 5.2.4.2 Circulating-Input Intensity Transfer Functions ................ 181 5.2.4.3 Analysis ............................................................................ 182 5.2.5 DCDR Under Coherence Operation ................................................. 199 5.2.5.1 Field Analysis of the DCDR Circuit ................................. 199 5.2.5.2 Output-Input Field Transfer Function ...............................200 5.2.5.3 Circulating to Input Field Transfer Functions ..................200 5.2.5.4 Resonance of the DCDR Circuit....................................... 201 5.2.5.5 Transient Response of the DCDR Circuit.........................204 5.2.6 DCDR Resonator as a Dispersion Equalizer: Group Delay and Dispersion ..................................................................................209 Optical Eigenilter as Dispersion Compensators ........................................... 220 5.3.1 Introductory Remarks ...................................................................... 220 5.3.2 Formulation and Design ................................................................... 222 5.3.2.1 Dispersive Optical Fiber Channel..................................... 222 5.3.2.2 Formulation of Optical Dispersion Eigencompensation ...... 223 5.3.2.3 Design and IM/DD System Performance .........................224 5.3.2.4 Performance Comparison of Eigenilter and Chebyshev Filter Techniques ............................................ 226 5.3.3 Synthesis of Optical Dispersion Eigencompensators ....................... 227 5.3.3.1 IM/DD Transmission System Model ................................ 228 5.3.3.2 Performance Comparison of Optical Dispersion Eigencompensator and Chebyshev Optical Equalizer.........231 5.3.3.3 Eigencompensated System with Parameter Deviations of the Optical Dispersion Eigencompensator ................... 234 5.3.3.4 Trade-Off Between Transmission Distance and Eigenilter Bandwidth ....................................................... 235 5.3.3.5 Compensation Power of Eigencompensating Technique ....................................................................236 5.3.3.6 Remarks ............................................................................ 238 Photonic Functional Devices ......................................................................... 238 5.4.1 Preamble ........................................................................................... 238 5.4.2 Optical Dispersion Compensation Module (oDCM)........................ 239 5.4.3 Chromatic Dispersion Compensators ...............................................240 5.4.4 Optical Gain Equalizer .................................................................... 242 5.4.4.1 Introductory Remarks ....................................................... 242 5.4.4.2 Dynamic Gain Equalizer .................................................. 243 5.2.3

5.3

5.4

Chapter 6

Optical Dispersion in Guided- Wave FIR and IIR Structures .................................. 245 6.1 6.2

Preamble/Introduction ...................................................................................246 Dispersion Mechanism in Fiber and Waveguide ........................................... 247

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Contents

6.3

6.4 6.5

6.6

6.7 6.8 6.9

Chapter 7

Micro-Ring Resonator (MRR) as an Optical Dispersion Compensator (oDCM).......................................................................................................... 249 6.3.1 Why Resonator? ............................................................................... 249 6.3.2 Transfer Transmittance Function of the Thru Port (Notched Resonant Filter) and Drop Port (Bandpass Filter) ............................ 250 6.3.2.1 Dispersion Characteristics and Dispersion Compensation by MRR .................................................... 250 6.3.2.2 Dispersion Compensating of Multiple DWDM Channels and Slope Dispersion Compensation ................ 251 6.3.3 Tunable Dispersion Compensator..................................................... 253 6.3.4 Length of Fiber Propagation and Dispersion Compensating Module.............................................................................................. 254 6.3.5 Waveguide and Passive MRR Fabrication Technology for oDCM .....255 Active MRR................................................................................................... 255 6.4.1 Structure ........................................................................................... 255 oDCM by Fiber Bragg Grating...................................................................... 256 6.5.1 Motivation ........................................................................................ 256 6.5.2 Analytical Expression of Broadening (Fiber) and Compression (TM-FBG) Factors ........................................................................... 257 6.5.2.1 Dispersion-Induced Pulse Broadening in Optical Fiber..... 257 6.5.2.2 Dispersion-Induced Pulse Broadening in FBG ................ 257 6.5.3 Design Cases ....................................................................................260 6.5.3.1 Design Case I: Finite Uniform Proile Grating ................260 6.5.3.2 Design Case II: Apodized Proile Grating ....................... 263 6.5.3.3 Remarks on FBG–oDCM .................................................264 FIR Discrete Wavelet Transform 2D Dispersion Compensating .................. 265 6.6.1 Introductory Remarks ...................................................................... 265 6.6.2 Analysis and Synthesis ..................................................................... 265 6.6.3 Design Procedures............................................................................ 268 6.6.4 Implementation................................................................................. 270 Concluding Remarks ..................................................................................... 274 Appendix: Dispersion Compensation a Historical View of Development and Why MRR as DCM ................................................................................ 274 SFG and Mason Rules for Photonic Circuit Analysis ................................... 276 6.9.1 SFG and Mason Approach ............................................................... 276 6.9.2 The Gain Formula ............................................................................ 278 6.9.2.1 Procedure .......................................................................... 278 6.9.3 Derivation of Transfer Function of the Micro-Ring Resonator ....... 278 6.9.3.1 Single Ring ....................................................................... 278 6.9.3.2 MRR Incorporating MZDI Structure ...............................280

Photonic Ultra-Short Pulse Generators .................................................................... 283 7.1

Optical Dark-Soliton Generator and Detectors ............................................. 283 7.1.1 Introduction ...................................................................................... 283 7.1.2 Optical Fiber Propagation Model ..................................................... 285 7.1.3 Design and Performance of Optical Dark-Soliton Detectors ........... 286 7.1.4 Design of Optical Dark-Soliton Detectors ....................................... 286 7.1.5 Performance of the Optical Differentiator ....................................... 287 7.1.6 Performance of the Butterworth LPOF ............................................ 288

xii

Contents

Design of the Optical Dark-Soliton Generator ................................. 289 7.1.7.1 Design of the Optical Integrator ....................................... 289 7.1.7.2 Design of an Optical Dark-Soliton Generator .................. 291 7.1.8 Performance of the Optical Dark-Soliton Generator and Detectors .... 293 7.1.8.1 Performance of the Optical Dark-Soliton Generator ........ 293 7.1.8.2 Performance of the Combined Optical Dark-Soliton Generator and Optical Differentiator ............................... 295 7.1.8.3 Performance of the Combined Optical Dark-Soliton Generator and Butterworth LPOF .................................... 295 7.1.9 Remarks............................................................................................ 297 Mode-Locked Ultra-Short Pulse Generators ................................................ 297 7.2.1 Introductory Remarks on Regenerative Mode-Locked Fiber Laser Types ...................................................................................... 298 7.2.2 Ultra-High Repetition-Rate Fiber Mode-Locked Lasers .................302 7.2.2.1 Mode-Locking Techniques and Conditions for Generation of Transform Limited Pulses from a Mode-Locked Laser..........................................................302 7.2.3 MLL and MRLL Experimental Setup and Results.......................... 305 7.2.3.1 40 GHz Regenerative Mode-Locked Laser ......................307 7.2.3.2 Remarks ............................................................................309 7.2.4 Active Mode-Locked Fiber Ring Laser by Rational Harmonic Detuning ........................................................................................... 311 7.2.4.1 Rational Harmonic Mode-Locking .................................. 311 7.2.4.2 Experiment........................................................................ 312 7.2.4.3 Phase Plane Analysis ........................................................ 313 7.2.4.4 Results and Discussion ..................................................... 316 7.2.4.5 Remarks ............................................................................ 319 Rep-Rate Multiplication Ring Laser Using Temporal Diffraction Effects ... 319 7.3.1 GVD Repetition Rate Multiplication Technique .............................. 320 7.3.2 Experiment Setup ............................................................................. 321 7.3.3 Phase Plane Analysis........................................................................ 322 7.3.3.1 Uniform Lasing Mode Amplitude Distribution ................ 322 7.3.3.2 Gaussian Lasing Mode Amplitude Distribution ............... 328 7.3.3.3 Effects of Filter Bandwidth .............................................. 328 7.3.3.4 Nonlinear Effects .............................................................. 328 7.3.3.5 Noise Effects ..................................................................... 328 7.3.4 Demonstration .................................................................................. 328 7.3.5 Remarks............................................................................................ 330 Multi-Wavelength Fiber Ring Lasers............................................................. 331 7.4.1 Theory .............................................................................................. 331 7.4.2 Experimental Results and Discussion .............................................. 333 7.4.3 Multi-wavelength Tunable Fiber Ring Lasers .................................. 336 7.4.4 Remarks............................................................................................ 338 7.1.7

7.2

7.3

7.4

Chapter 8

Multi-Dimensional Photonic Processing by Discrete-Domain Approach ............... 341 8.1

Multi-Dimension (MULTI-D) PSP Design Techniques ................................ 341 8.1.1 An Overview of Photonic Signal Processing ................................... 341 8.1.1.1 Spatial and Temporal Approach ....................................... 342 8.1.1.2 Fiber-Optic or Integrated Optic Delay Line Approach .... 343 8.1.1.3 Motivation .........................................................................344

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Contents

8.1.2

8.2

Multi-Dimensional Signal Processing..............................................344 8.1.2.1 Multi-Dimensional Signal ................................................344 8.1.2.2 Discrete Domain Signals .................................................. 345 8.1.2.3 Multi-Dimensional Discrete Signal Processing................346 8.1.2.4 Separability of 2-D Signals...............................................346 8.1.2.5 Separability of 2-D Signal Processing Operations ...........346 8.1.3 Filter Design Methods for 2-D PSP.................................................. 348 8.1.3.1 2-D Filter Speciications ................................................... 348 8.1.3.2 Mathematical Model of 2-D Discrete Photonic Systems .... 348 8.1.3.3 Filter Design Methods ...................................................... 352 8.1.3.4 Use of Matrix Decomposition .......................................... 352 8.1.4 Direct 2-D Filter Design Methods ................................................... 353 8.1.4.1 FIR and IIR Structures in 2-D Signal Processing ............ 353 8.1.4.2 Frequency Sampling Method............................................ 354 8.1.4.3 Windowing Method .......................................................... 356 8.1.4.4 McClellan Transformation Method .................................. 356 8.1.4.5 2-D Filter Design Using Transformation Method ............ 357 8.1.5 Concluding Remarks ........................................................................ 359 Decomposition Techniques and Implementation Using Fiber Optic Delay Lines .................................................................................................... 359 8.2.1 Introductory Remarks ......................................................................360 8.2.2 Matrix Decomposition Methods ......................................................360 8.2.2.1 Single-Stage Singular Value Decomposition....................360 8.2.2.2 Multiple-Stage Singular Value Decomposition ................ 363 8.2.3 Iterative Singular Value Decomposition .......................................... 365 8.2.3.1 Iterative Singular Value Decomposition ........................... 365 8.2.3.2 A 2-D Filter Design Example Using Iterative Singular Value Decomposition ........................................................ 366 8.2.4 Optimal Decomposition ................................................................... 367 8.2.4.1 Optimal Decomposition.................................................... 367 8.2.4.2 Other 2-D Filter Design Methods Based on Matrix Decomposition .................................................................. 368 8.2.5 2-D Filter Order Reduction Using Balanced Approximation Theory .............................................................................................. 369 8.2.5.1 Motivation for Lower Order Photonic Filters ................... 369 8.2.5.2 Description of 2-D System in State-Space Format........... 369 8.2.5.3 Balanced Approximation Method .................................... 369 8.2.5.4 Filter Order Reduction Using Balanced Approximation: An Example............................................ 372 8.2.6 Fiber-Optic Delay Line Filters ......................................................... 374 8.2.7 Coherent and Incoherent Operation of Photonic Filters .................. 374 8.2.8 Using Optical Fibers to Realize Delayed Line Filter ....................... 375 8.2.8.1 Photonic Realization of Delay .......................................... 375 8.2.8.2 Photonic Realization of Tab Coeficients ......................... 376 8.2.8.3 Photonic Realization of Summer/Splitter ......................... 376 8.2.8.4 Graphical Representation of Photonic Circuits ................ 377 8.2.8.5 Remarks ............................................................................ 379 8.2.9 Concluding Remarks ........................................................................ 379

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8.3

8.4

Realization ..................................................................................................... 380 8.3.1 Introductory Remarks ...................................................................... 380 8.3.2 Photonic Implementation of 2-D Filters ........................................... 381 8.3.2.1 Photonic Filter Structures ............................................... 381 8.3.2.2 Coherent System ............................................................. 381 8.3.2.3 2-D Direct Structure Filter ............................................. 381 8.3.2.4 2-D Separable Structure Filter ........................................ 383 8.3.2.5 Binary Tree Filter ........................................................... 384 8.3.2.6 Photonic Transversal Filter ............................................. 385 8.3.2.7 1-D Direct Structure Photonic Filter .............................. 388 8.3.2.8 Parallel Structure Filters ................................................. 389 8.3.2.9 Other 1-D Filter Structures ............................................. 391 8.3.2.10 Realization of Poles ........................................................ 392 8.3.2.11 Remarks .......................................................................... 393 8.3.3 Design Chart and Discussions.......................................................... 393 8.3.3.1 2-D Photonic Filter Design Flowchart ............................ 393 8.3.3.2 Examples of Photonic 2-D PSP Implementation ............ 393 8.3.3.3 Separable Implementation Using Matrix Decomposition Methods ................................................. 396 8.3.3.4 Non-Separable Implementation Using Direct Methods .....399 8.3.3.5 Comparison of Matrix Decomposition Method Design and Direct Method Design ..................................................402 Concluding Remarks .....................................................................................403

Chapter 9 Generation and Photonic Processing of Radio Waves, Tera-Waves and Multi-Carrier Lightwaves ........................................................................................405 9.1 9.2

9.3

Introduction ...................................................................................................405 Generation of Tera-Hz Waves........................................................................407 9.2.1 Generation of Ultra-High Repetition Rate Pulse Trains.......................408 9.2.2 Necessity of Highly Nonlinear Optical Waveguide Section for Tera-Hz Wave Ultra-High Speed Modulation ..................................409 Photonic Signal Processing of Radio Waves ................................................. 410 9.3.1 Generic Structures ............................................................................ 412 9.3.2 Polarization Dual-Mode Delay Processing Systems ........................ 413 9.3.2.1 Tunable Radio Wave Processing Systems Using Differential Group Display Elements ............................. 413 9.3.2.2 Tunable Multi-Tap Radio Wave Filters Using Higher Order Polarization Mode Dispersion Emulator .............. 416 9.3.3 Integrated Multi-Tap Delay Processing Systems.............................. 416 9.3.3.1 Dual Tunable RW Filters Using Sagnac Loop and CFBGs ............................................................................ 418 9.3.3.2 Wavelength-Division Multiplexing (WDM) Multi-Tap Tunable Radio Wave Filters ............................................ 420 9.3.3.3 Remarks .......................................................................... 423 9.3.4 Buffered Delay Processing Systems................................................. 423 9.3.5 Nonlinear Effects in Photonic Processing Systems of Radio Waves.............................................................................................. 424 9.3.5.1 All Pass Interferometer as Radio Frequency Filter Banks .............................................................................. 425 9.3.5.2 Integrated Radio Frequency and Photonic on Chip........ 428

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Remarks on the Photonic Signal Processing of Radio Waves ...... 430 9.3.6.1 Challenges and Uniqueness of Photonic Processors .................................................................. 430 9.3.6.2 Uniqueness of Tera-Hz Wave Generators ................... 432 Quantum Dot Solitonic Mode-Locked Comb Lasers .................................. 433 9.4.1 Structure and Quantum Optical Gain Waveguide ........................ 433 9.4.1.1 Quantum Dot Growth ................................................. 434 9.4.1.2 QD-BA and BU Structure........................................... 435 9.4.1.3 Lasing in Initial State ................................................. 436 9.4.1.4 Mode Locking and Comb Spectrum Generation ....... 437 9.4.1.5 Absorption Section ..................................................... 438 9.4.2 Performance .................................................................................. 438 9.4.2.1 Measurement Platform................................................ 438 9.4.2.2 Relative Intensity Noise ..............................................440 9.4.2.3 Linewidth of QD-MLL Generated Comb Laser.........440 9.4.3 Optical Frequency Comb in Multiple Radio Wave Channelization ............................................................................. 442 9.4.4 Concluding Remarks on QD-MLL ............................................... 443 9.3.6

9.4

Chapter 10 Optical Devices for Photonic Signal Processing ...................................................... 445 10.1 10.2

10.3

Optical Fiber Communications.................................................................... 445 Photonic Signal Processors ..........................................................................446 10.2.1 Photonic Signal Processing ..........................................................446 10.2.2 Some Processor Optical Components ..........................................446 10.2.2.1 Optical Ampliiers....................................................... 447 10.2.2.2 Pumping Characteristics .............................................448 10.2.2.3 Gain Characteristics ....................................................449 10.2.3 Noise Considerations of EDFAs and Impact on System Performance .................................................................................. 452 10.2.3.1 Noise Considerations................................................... 452 10.2.3.2 Fiber Bragg Gratings ................................................... 454 Optical Modulators ...................................................................................... 456 10.3.1 Introductory Remarks ................................................................... 456 10.3.2 Lithium Niobate Optical Modulators ........................................... 456 10.3.2.1 Optical-Diffused Channel Waveguides ...................... 457 10.3.2.2 Linear Electro-optic Effect .........................................468 10.3.3 Electro-absorption Modulators ..................................................... 472 10.3.3.1 Electro-absorption Effects .......................................... 472 10.3.3.2 Rib Channel Waveguides ............................................ 475 10.3.4 Operational Principles and Transfer Characteristics .................... 482 10.3.4.1 Electro-optic Mach–Zehnder Interferometric Modulator .................................................................... 482 10.3.5 Modulation Characteristics and Transfer Function ...................... 485 10.3.5.1 Transfer Function ........................................................ 485 10.3.5.2 Extinction Ratio for Large Signal Operation .............. 487 10.3.5.3 Small Signal Operation ............................................... 488 10.3.5.4 DC Bias Stability and Linearization ........................... 488

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Chirp in Modulators ..................................................................... 489 10.3.6.1 General Aspects .......................................................... 489 10.3.6.2 Modulation Chirp........................................................ 490 10.3.7 Electro-optic Polymer Modulators ............................................... 492 10.3.8 Modulators for Photonic Signal Processing ................................. 494 10.4 Remarks ....................................................................................................... 495 References ................................................................................................................ 496 10.3.6

Index .............................................................................................................................................. 501

Preface The evolution of optical networking is continuing to progress at tremendous pace. The transmission rate has now reached 200Gbps per wavelength channel by employing discrete modulation format over a limited bandwidth of only 35–45  GHz. The explosion of uses of data center clouds and cloud radio access networking for 5G wireless networks have led to a high possibility of employing photonic signal processing in optical networking for backhaul transporting of ultra-high capacity information. Photonic signal processing (PSP) is the art of manipulating photonic waves in the optical domain, either in coherent or in-coherent states. This is very attractive as it has the potential to overcome the electronic limits for processing ultra-wideband signals. Furthermore, PSP provides signal conditioning that can be integrated in-line with iber optic systems in optic form or iber form modules. Several techniques have been proposed and reported for the implementation of the photonic counter-parts of conventional electronic signal processing systems. Also, signal processing in the photonic domain offers signiicant improvement of signal quality. This book is written as the second edition of its irst version published by CRC Press in 2007,1 to update and address the emerging techniques of processing and manipulating signals propagating in optical domain. That means the pulses or signal envelopes are complex or modulating the optical carriers. Naturally, the applications of such processing techniques in photonics are essential to illustrate their usefulness. The change of the transmission cable from coaxial and metallic waveguide to lexible optically transparent glass iber has allowed the processing of ultra-high-speed signals in the microwave and millimeter-wave domain to the photonic domain in which the delay line can be implemented in the iber lines, which are very lightweight and space eficient. Previously found in Chapter 2 of the irst edition, a generic introduction addressing these fundamental understandings of optical devices is now listed in the last chapter of this second edition. We introduce the subjects of this second edition as follows. Chapter 2 gives a brief historical perspective of PSP and introduces a number of photonic components essential for photonic processing systems, including, but not exclusively, the optical ampliication devices, optical ibers, and optical modulators. Chapter 2 illustrates the representation of photonic circuits using signal-low graph (SFG) techniques, which have been employed in electrical circuits since the 1960s. However, these techniques are now adapted for photonic domain in which the transmittance of a photonic subsystem determines the optical transfer function of a photonic subsystem. The coherence and incoherence of photonic circuits are an important consideration as whether the ield or the power of the lightwaves should be used, as well as if the length of the photonic processors must be less than that of the coherence length. Chapter 3 then illustrates the uses of SFG in the design of optical ilters of ixed and tunable passband. Chapter 4 describes photonic signal processors, such as differentiators and integrators, leading towards their applications for the generation of solitons and their uses in optically-ampliied iber transmission systems. Chapter 5 illustrates the compensation of dispersion using photonic processors. Chapter 6 described the practical implementation of optical dispersion compensators in integrated optic technology. These compensators can offer highly low-cost modules for C- and L-band optical transmission links, which can remove several limitations if they are using the O-band spectral window.

1

Le Nguyen Binh, Photonic Signal Processing: Techniques and Applications, December 17, 2007 by CRC Press, ISBN 9781138746848.

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Chapter 7 gives the design and implementation of generations of ultra-short pulse sequences, and the dark solitons and their behavior under nonlinear instability conditions. Chapter 8 then gives a multi-dimension PSP in which the sampling rates in such processor can have many different rates. Chapter 9 then introduces the techniques for generation and processing in the photonic domain, of radio waves, Tera-waves and multi-carrier lightwaves. Many people have contributed, either directly or indirectly, to this book. Thanks are due to Associate Professor John Ngo of Nanyang Technological University (NTU), Singapore, for the works that he has conducted during his time as a PhD candidate member of my university research group; Professor Shum Ping of the School of Electrical and Electronic Engineering (EEE) of NTU; Dr. W.J. Lai and Miss Anh Le of Australia; my colleagues of Huawei Technologies, Dr. Sun Xu, Bruce Liu Lei, Zhao Zhuang, Dr. Thomas Wang Tao, and Dr. Xie Chang Song are thanked for their kind collaborations and helpful suggestions in developing integrated optical circuits and quantum dot mode-locked lasers. I would like to thank the editors of CRC Press, Taylor and Francis, for their encouragement. Last, but not least, I thank my wife Phuong, and my son Lam (LA, USA) for their care and putting up with my busy writing schedule of this book besides my daily R&D activities at the European Research Center of Huawei Technologies Munich, Germany, which certainly took away a large amount of time that we could have spent together. Le Nguyen Binh Munich, Germany, Spring 2018 MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508 647 7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

Author Le Nguyen Binh holds a BE (Summa Cum Laude) and PhD from the University of Western Australia. He is currently a technical director at Huawei’s European Research Centre in Munich, Germany, and has been awarded three Huawei Technologies Gold Medals for his work on advanced optical communication technologies. He was previously the chair of Commission D (Electronics and Photonics) of the National Committee for Radio Sciences of the Australian Academy of Sciences, and a professorial fellow at Nanyang Technological University, Christian-Albrechts-Universität zu Kiel, and various Australian universities. Widely published, Dr. Binh is the series editor of Photonics and Optics for CRC Press.

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Introduction

ACRONYMS Acronymic Terms ASE DC DMAC DSP e-DSP EDFA E/O FIR FOSAP GBaud Gbps HO-DMAC IC IIR IM/DD INCOI OA OEDC O/E PIC PLC PSP RF RW SBS S-DMAC SOI Tbps or Tera-bps WDM Mux/DeMux

1.1

Full Terms Ampliied spontaneous emission Directional couplers (optical) Digital multiplication by analog convolution (algorithm) Digital signal processing/processor Electronic DSP Erbium-doped iber ampliiers Electro-optic or electrical to optical conversion Finite Impulse Response Fiber-optic systolic array processors Giga-Baud Giga-bits/sec. High order DMAC Integrated circuits Ininite impulse response Intensity modulation/direct detection Incoherent Newton–Cotes optical integrator Optical ampliication/ampliier Optical dispersion eigencompensators Optical to electrical conversion Photonic integrated circuits Planar Lightwave circuit Photonic Signal Processing/processor Radio frequency Radio waves Stimulated Brillouin scattering Serial DMAC Silicon on insulator Tera-bits/sec. Wavelength-division multi/demultiplexers

PREAMBLE

Optical networking has penetrated long-haul, metropolitan, and access networks. Furthermore, data centers and cloud networking has quickly developed to carry information at a capacity reaching several Tera-bits/s. The networking capacity of data centers has increased exponentially. Initially, the transmission rate was low enough for multi-mode use as the transmission medium over ultra-short distances of a few hundred meters to few kilometers. The bit rate has now reached 400 Gbps with 56 GBaud over single-mode ibers for the interconnection of clusters of servers. In connecting different 1

2

Photonic Signal Processing

sections of the sever stacks of large-scale data centers (DCs), a large-scale optical cross-connect system via the processing or switching in the optical domain is preferred to minimize the latency that electronic switching systems are currently facing. The fastest and highest switching capacity by electronic system can only reach a total maximum capacity of not higher than one Tera-bps. Data center networking (DCN) is highly important and inter-data center transmission links have become highly critical with ultra-bit rate and increased capacity over short distances. Photonic signal processing (PSP) can assist in the manipulation of optical channels in the optical domain. The ultimate aim of this book is to show the many ways in photonic processing functions can be implemented.

1.2

INTRODUCTORY REMARKS

The market demand for cost-effective transport of large amounts of information has accelerated the pace for the widespread deployment of optical iber communication systems, which have the capacity to carry large amounts of information. These systems make use of optical signal processors as the basic functional all-optical device for signal processing and transmission. They are used in many areas of telecommunications: undersea cables bridging the oceans, terrestrial cables connecting cities, trunk lines linking metropolitan areas, and subscriber loop systems serving customer premises. Thus, there is an urgent need to design optical signal processors to meet the rapidly growing demands of advanced optical communication systems. Digital signal processing (DSP) has been extensively employed in the coherent1,2 and incoherent3 optical transmission systems to increase the baud rate to 100  GBaud leading to several hundred Giga-bits/s per wavelength channels. Although optical signal processing is an analogue technique by its nature, DSP can be applied and transferred from the electrical domain into the photonic domain as far as we can deine the optical sampling feature so that the sampling z-transform can be used. This is the main feature of the chapters described in this book. In this book, as the second edition of the book Photonic Signal Processing: Techniques and Applications (Optical Science and Engineering),4 two types of linear optical signal processors, which make use of the photon nature of light for performing a wide variety of linear functions, are considered. They are incoherent iber-optic signal processors and coherent integrated-optic signal processors, in which single-mode optical ibers and single-mode optical waveguides, respectively, are employed as a delay-line medium for high-speed processing of broadband signals. The virtually unlimited bandwidth of these processors makes them attractive for processing radio frequency (RF), microwave, and millimeter-wave signals directly in the optical domain. This direct processing eliminates the need for ineficient and costly intermediate “opto-electronic-opto” conversions as in conventional approaches, which eventually lead to electronic bottlenecks. Incoherent iber-optic signal processors employing optical iber, iber-optic directional couplers (DC), and erbium-doped iber ampliiers (EDFAs) have been developed to perform various signal processing functions, such as convolution, correlation, analog matrix operations, frequency iltering, pulse-train generation, data-rate transformation, and code generation. Coherent integrated-optic signal processors employing silica-based waveguides on a silicon substrate have

1

2

3

4

M. Taylor, Phase estimation methods for optical coherent detection using digital signal processing, IEEE J. LightwaveTechnol., 27(7), 901–914, 2009. N. Stojanovic, F. Karinou, B. Mao, Chromatic dispersion estimation method for Nyquist and faster than Nyquist coherent optical systems, Optical Fiber Communications Conference and Exhibition (OFC), San Francisco, CA: OFC’2014, PaperTh2A.19. K. Kikuchi, Electronic polarization-division demultiplexing based on digital signal processing in intensity-modulation direct-detection optical communication systems, Opt. Exp., 22(2), 1971–2001, 2014. doi:10.1364/OE.22.001971. L. N. Binh, Photonic Signal Processing: Techniques and Applications (Optical Science and Engineering), 1st ed., Boca Raton, FL: CRC Press, 2007.

Introduction

3

been demonstrated as optical switches, optical wavelength-division multi/demultiplexers (WDM Mux/DeMux), optical frequency-division multi/demultiplexers, tunable optical ilters, and optical dispersion compensators. Given the signiicant developments of this ield, the overall aims of this book are, therefore, (i) to describe the principal techniques for the design of new incoherent and coherent optical signal processors and (ii) to demonstrate their applications in optical computing and optical communication systems. The speciic aims of this investigation are described as follows. The irst aim of this book is to describe the design of new incoherent iber-optic systolic array processors (FOSAPs), which employ the digital multiplication by analog convolution (DMAC) algorithm, for performing real-valued digital matrix multiplications. The FOSAP multipliers are shown to have massive pipeline capability and higher computational power than the digital electronic multipliers and other optical DMAC multipliers. The second objective of this book is to design a new programmable incoherent Newton–Cotes optical integrator (INCOI) for processing (or integrating) intensity-based signals in the time domain. The new INCOI processor is synthesized using a generalized theory of the Newton–Cotes digital integrators. The third aim of this book is to present the design of the new higher-derivative inite impulse response (FIR) coherent optical differentiators for processing (or differentiating) coherent signals in the time domain. The new optical differentiators are synthesized using a theory of higher-derivative FIR digital differentiators, which are commonly developed for electronic DSP. Although digital integrators and differentiators have received considerable attention in the ield of digital signal processing, the proposed theories of optical integrators and differentiators are believed to be introduced, for the irst time, to the area of optical signal processing. It is hoped that these new theories form the basis for future development with a view to inding potential applications for these new processors. One such application provides the fourth goal of this study, which is to design new optical dark soliton generators and detectors by using the optical integrator and differentiator, respectively, as outlined above. This is because dark solitons, which are dificult to generate and detect, have recently been predicted to offer better stability than the well-known bright solitons against iber loss, interactions between neighboring solitons, and ampliied noise-induced timing jitter. In addition, there are currently so few techniques for dark soliton generation and detection—only three techniques for generating and one technique for detecting dark solitons. The fourth goal of this book is to present the design of new optical dispersion eigencompensators (ODECs) for compensation of the combined effect of laser chirp and iber dispersion at 1550 nm in high-speed, long-haul intensity modulation/direct detection (IM/DD) optical communication systems. Although many optical dispersion compensators have been demonstrated to mitigate this detrimental effect, most of these devices are bandwidth-limited because of a lack of design lexibility. It is found that the proposed ODECs perform better with dispersively chirped signals than with dispersively chirp-free signals; an advantage which is not attainable with other techniques. The ifth and inal aim of this book is to systematically design new tunable optical ilters with the essential features of variable bandwidth and center frequency characteristics as well as general iltering characteristics, such as lowpass, highpass, bandpass, and bandstop characteristics. Although several techniques have been developed to design tunable optical ilters with variable bandwidth and center frequency characteristics, such as bandpass and bandstop, there is still limited design lexibility in these methods; however, the proposed technique provides much greater design lexibility. The ultimate overall clariication in the chapters of this book is also to identify either the incoherence and coherence characteristics of the optical signals under operations, thus either the electromagnetic ield or intensity of the optical waves or signals are to be employed in the PSP (Photonic Signal Processing) systems. The implications are signiicant whether they are processed by coherent or direct detection sub-systems. The design and applications of new incoherent and coherent optical

4

Photonic Signal Processing

signal processors reported in this book constitute a signiicant contribution to the ields of optical computing and optical communications. Digital signal processing (DSP) in the electronic domain by ASIC (Applied Speciic Integrated Circuit) or FPGA (Field programmable gate array) can certainly be used as a hybrid co-processor to enhance the effectiveness of a PSP system. This kind of hybrid processing technique is not presented in this edition of the book.

1.3 ORGANIZATION OF CHAPTERS The chapters of this book are organized in the following order: Chapter 2 presents the fundamental theory of incoherent iber-optic signal processing and coherent integrated-optic signal processing, which provides the basis for Chapters 3–5 and Chapters 6–9, respectively. Chapter 3 gives a framework for the analysis of optical signal processors through the application of the signal-low graph technique, which is developed as a mathematical tool applicable to all chapters. The effectiveness of this approach can be demonstrated by applying it to the analysis and design of an incoherent recursive iber-optic signal processor. Chapter 4 then describes structures and operations of incoherent FOSAPs employing the DMAC, higher order DMAC (HO-DMAC) and serial S DMAC (S-DMAC) algorithms, which are proposed for real-valued digital matrix multiplications. The performances of the FOSAP multipliers are also compared with those of the digital electronic multipliers and other optical DMAC multipliers. Chapter 5 describes a programmable INCOI processor, which is synthesized using a proposed generalized theory of the Newton–Cotes digital integrators. The performance of the processor is also analysed. Chapter 6 gives a theory of higher-derivative FIR optical differentiators developed using a proposed theory of higher-derivative FIR digital differentiators. The performances of the optical differentiators are also analysed. In Chapter 7, the trapezoidal optical integrator described in Chapter 5 is proposed as an optical dark soliton generator and the irst-order irst-derivative optical differentiator outlined in Chapter 6 and a irst-order Butterworth lowpass optical ilter are proposed as optical dark-soliton detectors. The performances of the dark soliton generator and detectors are also analyzed. In Chapter 8, a digital eigenilter approach is employed to design linear ODECs for the compensation of the combined effect of laser chirp and iber dispersion at 1550 nm in high-speed long-haul IM/DD lightwave systems. The performances of the ODECs are also compared with those of the Chebyshev optical equalizers. Chapter 9 presents a digital ilter design method employed to systematically design tunable optical ilters with variable bandwidth and center frequency characteristics, as well as lowpass, highpass, bandpass and bandstop characteristics. The effectiveness of this technique is demonstrated with the design of the second-order Butterworth lowpass, highpass, bandpass and bandstop tunable optical ilters with variable bandwidth and centre frequency characteristics. An experimental development of the irst-order Butterworth lowpass and highpass tunable iber-optic ilters is also described. Comb lasers via the structure embedded with stack layers of quantum dots InAs integrated in a buried heterostructure InGaAsP are also given as an example for a generation of multicarriers in the optical domain. The integrated cavity can then generate soliton under injection of electrons and formation of lightwaves and thence soliton waves till the saturation absorption layer reaches its saturation level and Q-switching happens to thence generate ultra-short pulse train which can then be transformed into wideband comb lines in the frequency domain. Chapter 10 gives a number of principal optical devices required for photonic processing are given. In the irst edition of the book this chapter was described as Chapter 1. In this edition this chapters updated and assigned as last chapter of the book and referred in other chapters, possibly considerable as a combination of appendices of the chapters.

Introduction

5

Major conclusions drawn from the presentations and practical results reported in the chapters and recommendations for future work are presented at the end of each chapter. In this book, it is assumed that all optical systems operate in the C-band of silica iber, which is in the 1550 nm window, the operating wavelength of optical waves at which the signal loss is minimum. Optical signal losses can be compensated with EDFAs integrating optical isolators, and/or a narrow-band optical ilter may be required to suppress the stimulated Brillouin scattering (SBS) and to minimize the ampliied spontaneous emission (ASE), respectively. The effects of iber polarization and waveguide birefringence, which can be overcome in practice, on the performance of the proposed optical signal processors are beyond the scope of this book and are therefore not considered.

2

Photonic Signal Processing Via Signal-Flow Graph

The ields of photonic communications and integrated photonics have progressed signiicantly over the last few decades. Spinning off from such developments, several related areas such as microwave photonics, ultra-broadband optical iber communications, photonic switching, intra- and inter-data center communications, and others have been established. In these systems, photonic signals are propagating and processed at the modulators, through the guided media, at the multiplexers and demultiplexers, through optical ampliiers (Raman, rare-earth doped waveguide, or parametric ampliication). Thus, the necessity of processing of photonic waves has become important and requires fundamental approaches for photonic circuit analysis, syntheses and design, and photonic signal processing. Therefore, this chapter, as the irst of the introductory chapters of photonic processors, presents the fundamental principles and applications of this new ield of photonic signal processing. We state and deine the signiicance of coherence and incoherence of lightwaves propagation through photonic circuits and hence the representation of lightwaves and circuits either by the lightwave intensity or by the photonic electromagnetic ields. The photonic circuits to be processed in coherent or incoherent modes are described and followed by a number of photonic circuit elements. The graphical representation of photonic circuits is described using the signal-low graphs (SFG) and the photonic Mason’s rules. Automatic generation for photonic transfer functions of the input and output lightwave signals between any two nodes of a photonic circuit are presented. Furthermore, this chapter gives a fundamental understanding of incoherent and coherent optical signal processing, which provides the basis for later chapters. The advantages and disadvantages of incoherent and coherent optical systems and means of overcoming their limitations are outlined in Sub-sections 2.1–2.4. The characteristics of the fundamental components of incoherent iber-optic signal processors and coherent integrated-optic signal processors are described in Section 2.8.1 A Borland Pascal source code is included in Section 2.11 as a reference for readers who may want to use it to generate photonic circuits and related transfer functions. Two appendices are given and applied to the very basic z-transform and automatic SFG and transfer function from one node to the other.

2.1

INTRODUCTION

The phase of the optical signal is sensitive to environmental luctuations, such as temperature and pressure changes and acoustic vibrations as well as frequency luctuations of the optical source. The inherently high sensitivity of the optical phase to environmental effects has made it attractive for sensor applications but unattractive for signal processing operations in which stability is essential. Obviously, these effects can be obviated by discarding the optical phase using an incoherent light. Incoherent optical signal processors require the coherence time of the optical source to be much shorter than the basic time delay (or sampling period) to avoid any undesirable effects of optical interference. Hence, incoherent systems use intensity variations on optical carriers for performing signal processing operations. In incoherent systems, single-mode optical ibers can be used as a promising delay-line medium for processing broadband signals because of the large bandwidth of optical ibers. Typically, the basic delay-line length2 of an incoherent iber-optic signal processor is

1

2

Details of the z-transform deinitions and properties can be found in R.D Strum and D.E. Kirk, First Principles of Discrete Systems and Digital Signal Processing, New York: Addison and Wesley, 1989. The basic delay-line length has a delay corresponding to the basic time delay or sampling period of the system. This allows the application of the z-transform, a common feature of DSP applicable in analog photonic processing domain.

7

8

Photonic Signal Processing

in the meter-order and is at least several orders of magnitude greater than the coherence length of the optical source, depending on the frequency of operation. For this reason, changes in the basic delay-line length, due to environmental effects and/or errors in cutting the iber length, can be tolerated without causing signiicant degradation of the system performance. Although incoherent iber-optic signal processors are stable and robust, they can only perform positive-valued signal processing operations, but not bipolar or complex-valued signal processing operations. Hence, incoherent iber-optic signal processors have limited applications. This serious limitation can clearly be overcome with a coherent light if the instability can be found. In contrast to the incoherent case, coherent optical signal processors require the coherence time of the optical source to be much longer than the basic time delay to achieve coherent interference of the delayed signals. Coherent systems are thus capable of performing complex-valued signal processing operations because both the phase and amplitude of the optical signal are retained in the processed information. As pointed out above, coherent systems cannot operate stably unless the frequency luctuations of the optical source and environmental effects can be prevented. The frequency can be stabilized by using highly coherent semiconductor lasers that are commercially available. The environmental effects can be suppressed by using integrated optical waveguides (instead of optical ibers) as a comparatively small delay-line medium for broadband signal processing because of their large bandwidth. Coherent integrated-optic signal processors can operate stably because the waveguide length, which is in the centimeter-order or millimeter-order, can be accurately fabricated to the precision of the wavelength order and the phase of the optical signal can be conveniently controlled to the precision of the wavelength order. In this book, optical ibers and integrated optical waveguides have been considered as the delayline medium of choice for incoherent and coherent optical signal processing, respectively. The fundamental theories of both incoherent iber-optic signal processing and coherent integrated-optic signal processing are presented in the next sections.

2.2

INCOHERENT PHOTONIC SIGNAL PROCESSING

The potentially large bandwidth of optical iber has made it an attractive delay-line medium for incoherent processing of broadband signals. This section describes the theory of incoherent iberoptic signal processing given in references [1] and [2]. In incoherent optical signal processors, the information signal (e.g., RF or microwave) to be processed is modulated as intensity variations onto an optical carrier whose coherence time is much shorter than the basic time delay in the system. The optical source can be a broad-linewidth semiconductor laser diode, which can be directly modulated at speeds up to several gigahertz. In the time domain, the modulated wideband signals do not interfere with each other but are appropriately delayed and incoherently combined at the system output. In the frequency domain, the frequency response of the incoherent system depends on the interference of the modulation frequency (RF or microwave) but not the optical carrier frequency. In other words, an incoherent system is incoherent at the optical carrier frequency but coherent at the modulation frequency. Thus, the phase of the optical carrier can be discarded, and the signals add on an intensity basis. As a result, incoherent systems are stable and robust, but they can only perform positive-valued signal processing operations because intensity cannot be negative. Hence, they have limited applications. Using the theory of positive systems, it has been shown that the impulse response of an incoherent system is real and positive-valued.3 In addition, the magnitude of the frequency response of an incoherent system is maximum at the origin of the frequency axis. Consequently, incoherent systems can only be designed to have a limited number of lowpass characteristics but not highpass or bandpass characteristics.

3

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber-optic lattice signal processing, Proc. IEEE, 72, 909–930, 1984.

Photonic Signal Processing Via Signal-Flow Graph

9

Incoherent iber-optic signal processing was initiated by a research group at Stanford University in the 1980s [3]4. Optical ibers and tunable iber-optic directional couplers have been used in the analysis, as well design and construction of a number of incoherent inite impulse response5 (FIR) and ininite impulse response6 (IIR) iber-optic signal processors that can perform a variety of linear signal processing functions, which include convolution, correlation, analog matrix operations, frequency iltering, pulse-train generation, data-rate transformation, and code generation. Considerable research effort has produced a number of new concepts, techniques, and applications as a result of the advanced development of iber-optic signal processors.7–8 Optical ampliiers, in particular erbium-doped iber ampliiers (EDFAs), have been used to overcome losses as well as to provide greater lexibility in the analysis, synthesis, and construction of incoherent iber-optic signal processors for various iltering applications.9–10 The resulting ampliied iber-optic signal processors have better performances and, hence, more applications than the unampliied processors.11,12,13,14,15 Adaptive techniques have also been proposed to provide dynamic weighting of the ilter coeficients as well as reconiguration of the ilter delays.16,17,18,19 The limitation of the incoherent (or positive) iber-optic signal processors may be reduced by using an electronic differential detection scheme, which can have negative ilter coeficients but at the expense of increased system complexity.20,21 It has been claimed that this synthesis technique can implement not only lowpass ilters but also highpass and bandpass ilters. The performance of the synthesized ilter can only approximate that of the desired ilter because the synthesis method is based on the least squares approach. Nevertheless, impressive performances of the synthesized lowpass and highpass ilters have been experimentally

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13 14

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K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, and H. J. Shaw, Photonic iber delay-line signal processing, IEEE Trans. Microw. Theory Tech., MTT-33, 193–210, 1985. Note that FIR ilters have no feedback loops and are also known as transversal, non-recursive or tapped delay-line ilters. Note that IIR ilters have at least one feedback loops and are also known as recursive or recirculating delay-line ilters. All-pole and all-pass ilters are special types of IIR ilters. C. C. Wang, High-frequency narrow-band single-mode iber-optic transversal ilters, J. Lightwave Technol., LT-5, 77–81, 1987. E. C. Heyde, Theoretical methodology for describing active and passive recirculating delay line systems, Electron. Lett., 31, 2038–2039, 1995. B. Moslehi, Fiber-optic ilters employing photonic ampliiers to provide design lexibility, Electron. Lett., 28, 226–228, 1992. E. C. Heyde, Theoretical methodology for describing active and passive recirculating delay line systems, Electron. Lett., 31, 2038–2039, 1995. K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, and H. J. Shaw, Photonic iber delay-line signal processing, IEEE Trans. Microw. Theory Tech., MTT-33, 193–210, 1985. C. C. Wang, High-frequency narrow-band single-mode iber-optic transversal ilters, J. Lightwave Technol., LT-5, 77–81, 1987. R. I. MacDonald, Switched photonic delay-line signal processors, J. Lightwave Technol., LT-5, 856–861, 1987. D. M. Gookin and M. H. Berry, Finite impulse response ilter with large dynamic range and high sampling rate, Appl. Opt., 29, 1061–1062, 1990. A. Ghosh and S. Frank, Design and performance analysis of iber-optic ininite impulse response ilters, Appl. Opt., 31, 4700–4711, 1992. B. Moslehi, K. K. Chau, and J. W. Goodman, Photonic ampliiers and liquidcrystal shutters applied to electrically reconigurable iber optic signal processors, Opt. Eng., 32, 974–981, 1993. J. Capmany and J. Cascon, Photonic programmable transversal ilters using iber ampliiers, Electron. Lett., 28, 1245– 1246, 1992. J. Capmany and J. Cascon, Discrete time iber-optic signal processors using photonic ampliiers, J. Lightwave Technol., 12, 106–117, 1994. S. Sales, J. Capmany, J. Marti, and D. Pastor, Solutions to the synthesis problem of photonic delay line ilters, Opt. Lett., 20, 2438–2440, 1995. J. Capmany, J. Cascon, J. L. Martin, S. Sales, D. Pastor, and J. Marti, Synthesis of iber-optic delay line ilters, J. Lightwave Technol., 13, 2003–2012, 1994. S. Sales, J. Capmany, J. Marti, and D. Pastor, Experimental demonstration of iber-optic delay line ilters with negative coeficients, Electron. Lett., 31, 1095–1096, 1995.

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demonstrated [21]. However, the synthesis technique can only handle bipolar numbers but not complex numbers, which must be operated by coherent systems. It is well known that the basic elements required for implementation of the FIR and IIR digital signal processors are delays, adders and multipliers.22 As a result, the basic components required for the realization of the FIR and IIR incoherent iber-optic signal processors are: iber-optic delay lines, iber-optic directional couplers, and iber-optic (or semiconductor) ampliiers. These are described in the following sections.

2.2.1 Fiber-Optic Delay lines The low loss and broad bandwidth of optical ibers have made them an attractive delay-line medium for incoherent processing of high-speed broadband signals directly in the optical domain. The loss of optical ibers is about 0.5 dB km at 1300 nm and about 0.2 dB km at 1550 nm. The bandwidth–distance product of optical ibers is about 32 THz ⋅ km at 1300 nm and about 100 GHz ⋅ km at 1550 nm [26]. As a result, the time–bandwidth product of optical ibers exceeds 107 at 1300 nm and exceeds 105 at 1550 nm, assuming a delay per unit length of 5 µs km. 2.2.1.1 Fiber-Optic Directional Couplers One of the fundamental elements in incoherent iber-optic signal processors is a iber-optic directional coupler, which performs signal collection (or addition) or signal distribution (or tapping). The [2 × 2] iber-optic directional coupler (see Figure 2.1) is a symmetrical and reciprocal fourport device, which can be designed to have ixed or tunable coupling coeficient. The underlying principle is based on the interaction of the evanescent ields between two parallel iber cores placed suficiently close to each other.23,24 The coupler exhibits very little dependence on the state of polarization of the input ields, even though the polarization effect can be easily overcome in practice by means of a iber polarization controller. An alternative and more promising approach is to use polarization maintaining ibers in the construction of the couplers. In the incoherent operating regime, the intensity transfer matrix of the 2 × 2 tunable iber-optic directional coupler can be described by [1]  I3  1 − κ   = (γ )   I4   κ

Port 1

κ   I1    1 − κ   I2 

Tunable Directional Coupler

Port 2

(2.1)

Port 3

Port 4

FIGURE 2.1 Schematic diagram of a tunable iber-optic directional coupler. 22 23

24

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1989. S. K. Sheem and T. G. Giallorenzi, Single-mode iber-photonic power divider: Encapsulated etching technique, Opt. Lett., 4, 29–31, 1979. M. J. F. Digonnet and H. J. Shaw, Analysis of a tunable single mode photonic iber coupler, IEEE J. Quantum Electron., QE-18, 746–754, 1982.

Photonic Signal Processing Via Signal-Flow Graph

11

where: { I1, I2 } and { I3 , I4 } are the intensities at the input and output ports, respectively κ (0 ≤ κ ≤ 1) is the cross-coupled intensity coeficient γ (typically 0.95 < γ < 1) is the intensity transmission coeficient Equation (2.1) means that, when the input port {2} is not excited, the signal intensity at the input port is directly coupled to the output port {3} with an intensity coupling coeficient of γ (1− κ ) and cross coupled to the output port {4} with an intensity coupling coeficient of γκ . Similarly, when the input port {1} is not excited, the signal intensity at the input port {2} is directly coupled to the output port {4} with an intensity coupling coeficient of γ (1− κ ) and cross coupled to the output port {3} with an intensity coupling coeficient of γκ .

2.2.2 Fiber-Optic anD semicOnDuctOr ampliFiers An optical ampliier is an important component in incoherent iber-optic signal processors because it can compensate for optical losses as well as providing design lexibility resulting in potential applications. Optical ampliiers can provide signal ampliication directly in the optical domain. The operational principles, characteristics, and performances of the optical ampliiers described here have been obtained from references [22] and25. The gain of an optical ampliier is generated by the processes of stimulated scattering induced by nonlinear scattering in an optical iber, or by the stimulated emission caused by a population inversion in a lasing medium. The former process is utilized by stimulated Raman scattering iber ampliiers and stimulated Brillouin scattering iber ampliiers, which are of little interest in this investigation because they are based on nonlinear effects in ibers. The latter process is employed by semiconductor laser ampliiers (SLAs) or rare-earth doped iber ampliiers. Semiconductor lasers can be designed to act as ampliiers and hence the acronym SLAs. SLAs can be categorized, according to biasing condition and structure, into three types: injection-locked (IL), Fabry-Perot (FP), and travelling-wave (TW) SLAs. IL-SLA and FP-SLA, which are based on resonance effects, require the biasing of the semiconductor laser above and below the lasing threshold, respectively. By contrast, TW-SLA, which exploits single-pass ampliication, requires both facets of the semiconductor laser to have antirelection coating. Considerable research attention was initially paid to IL-SLA and FP-SLA with a view to improving the inferior anti-relection coating methods. However, TW-SLA has recently attracted the most attention because of its superior performance (saturation output, noise, and bandwidth, to mention a few) and considerable progress with coating techniques. However, it is dificult to differentiate between FP-SLA and TW-SLA because complete zero relectivity cannot be easily achieved by actual antirelection coating techniques. Thus, it is generally accepted that TW-SLA and FP-SLA have a relectivity of less than 0.1%–1% and more than 30%, respectively. For these reasons, the TW-SLAs have been chosen in this study for application in incoherent iber-optic signal processors. The rare-earth doped iber ampliiers make use of ions, such as erbium, neodymium, and praseodymium, as the gain medium to provide optical ampliication. In recent years, EDFAs have been the main subject of research simply because they operate near 1550 nm, the wavelength region in which the iber loss is minimum and hence the wavelength window of interest for next-generation lightwave systems. For these reasons, EDFAs have also been considered in this investigation. Because the TW-SLAs and EDFAs have been considered in this research, it is necessary to understand their characteristics and performances, which are summarized in Table 2.1 [25]. Compared

25

S. Shimada and H. Ishio (Eds.), Photonic Ampliiers and Their Applications, New York: John Wiley and Sons, 1994, chapters 1–5.

12

Photonic Signal Processing

TABLE 2.1 Comparison of the Characteristics and Performances of TW-SLAs and EDFAs TW-SLAs Various wavelengths 15 ~ 20 dB 0 ~ 3 dBm > 3 THz Yes 6 ~ 9 dB Large loss (9 ~ 10 dB) Fast switching (< 1 ns) Short carrier lifetime Small, ampliier length of < 1 mm

Characteristics/Performances

EDFAs

Signal wavelength Unsaturated gain Saturation output power Bandwidth Polarization dependence Noise igure Fiber coupling loss Switching speed

Currently 1550 nm band only 40 ~ 50 dB 10 ~ 20 dBm 1 ~ 4 THz No 3 ~ 5 dB Low loss (< 0.5 dB) Slow switching ( 0.2 ~ 10 ms) Long carrier lifetime

Size/length

Several meters to several 100 m of iber length

with TW-SLAs, the advantages of EDFAs are: higher unsaturated (or small-signal) gain, higher saturation output power, lower noise igure, lower iber coupling loss, and polarization independence. However, TW-SLAs generally have larger bandwidth and are more compact than EDFAs. The high gain, high saturation output, large bandwidth, low noise, low iber coupling loss, and polarization independence of EDFAs make them an ideal choice for application in the incoherent iber-optic signal processors. In addition to the enormous bandwidth and compactness, the relatively fast switching speed of TW-SLAs makes them more attractive than EDFAs for application in adaptive (or programmable) incoherent iber-optic signal processors, where the gains of the TW-SLAs can be altered by varying the injection current sources driving the semiconductor lasers. Programmable incoherent iber-optic signal processors incorporating TW-SLAs must use polarization-preserving ibers and couplers because of the polarization dependence of TW-SLAs. In this book, EDFAs and TW-SLAs have thus been considered as the ampliiers of choice for non-programmable and programmable incoherent iber-optic signal processors, respectively.

2.3

COHERENT INTEGRATED-OPTIC SIGNAL PROCESSING

Integrated optical waveguides described in Section  2.3.1 can be used as an attractive delay-line medium for coherent processing of high-speed broadband signals because of the high precision (and hence stability) and large bandwidth of the waveguides. In coherent optical signal processors, the information signal (RF, microwave or milli-meter-wave) to be processed is impressed onto an optical carrier whose coherence time is much longer than the basic time delay in the system. The optical source must be a frequency-stabilized highly coherent semiconductor laser with a very narrow linewidth in order to suppress the frequency instability. Thus, the optical source must be externally (rather than directly) modulated so that its high degree of coherence can be maintained. This is because direct current modulation of the injection lasers causes a dynamic shift of the peak emission wavelength resulting in broadening the spectral width [30]. In addition, external modulation (e.g., using titanium-diffused lithium niobate (Ti:LiNbO3  waveguide modulators) of the optical source permits high-speed modulation that is ideal for high-speed signal processing. Thus, the use of external modulators allows the optical source to be optimized for spectral quality as well as obtaining high modulation bandwidth. In the time domain, the modulated signals constructively or destructively interfere with each other, depending on their relative phases. In the frequency domain, the frequency response depends on the interference of the optical carrier frequency. Thus, coherent optical signal processors using integrated-optic waveguides can stably

Photonic Signal Processing Via Signal-Flow Graph

13

perform high-speed complex-valued signal processing operations because both the phase and amplitude of the optical carrier are retained in the processed information. The term “integrated optics” was suggested by Miller in 1969 at a Bell Laboratory as the lightwave equivalent of “integrated electronics.”26,27 Since then research in integrated optics has begun and gained momentum at about the same time as the development of low-loss optical ibers and semiconductor lasers. The concept of integrated optics involves the use of thin-ilm and microfabrication technologies in the development of a large number of individually fabricated miniature optical components, which may be integrated (or individually interconnected) onto a single chip in a similar fashion to that which had taken place with integrated electronic circuits [26,27].28,29,30 Enormous progress has been made in this ield with advances in material developments, design techniques, fabrication processes, and component developments. Developments have now reached the stage where integrated optical circuits can be realized to perform various all-optical signal processing and switching functions. In fact, recent advances in lightwave technology and networks have further accelerated the pace for the development of compact, rugged, stable and economical integrated optical circuits for lexible processing and switching of high-speed broadband signals directly in the optical domain. The integrated optical circuit has several advantages over its counterpart, the integrated electronic circuit, or over conventional bulk-optic systems consisting of relatively large discrete components [31]. When compared to bulk-optic systems, integrated optical circuits share the same advantages as those of the integrated electronic circuits such as smaller size and weight as well as improved reliability and batch fabrication economy. However, the integrated optical circuit, which uses a high carrier frequency for information processing, inherently has a higher processing speed than the integrated electronic circuit. As with any other new technology, a high development cost of integrated-optic technology (e.g., developing new fabrication technology) is initially required. Nevertheless, the great potential of integrated optics will clearly justify the high development cost in the long run. Integrated optical circuits can be fabricated on several different materials, each with its own particular features. The choice of a substrate material depends very much on the function to be performed by the circuit. Substrate materials commonly used are glass, LiNbO3, silicon (Si) , III–V31 semiconductors, such as gallium arsenide (GaAs) and indium phosphide (InP).32,33,34 For example, the InGaAsP/InP material system has been used for the development of InGaAsP/InP 1.55-µm distributed feedback lasers because the InP substrate is capable of emitting light in the 1.3–1.6  µm spectral region that is important for lightwave systems.35 The LiNbO3 dielectric material has been widely used for the development of Ti:LiNbO3 waveguide modulators because of its linear electrooptic or Pockel’s effect, and that the optical waveguide can be easily formed by diffusing a thin ilm of titanium into the LiNbO3 substrate [28–30]. The germanium, InGaAs/InP, and InGaAsP/InP materials have been used for the fabrication of avalanche photodiodes because of their high absorption coeficients (or responsivity) in the 1.1–1.6 µm low-loss wavelength region [35]. 26 27 28 29 30 31

32

33

34

35

S. E. Miller, Integrated optics: an introduction, Bell Syst. Tech. J., 48, 2059–2069, 1969. T. Tamir (Ed.), Integrated Optics, New York: Springer-Verlag, 1975. L. D. Hutcheson, Integrated Photonic Circuits and Components, New York: Marcel Dekker, 1987. S. E. Miller and I. P. Kaminow (Eds.), Photonic Fiber Telecommunications II, San Diego, CA: Academic Press, 1988. R. G. Hunsperger, Integrated Optics: Theory and Technology, New York: Springer-Verlag, third edition, 1991. Note that compounds made of elements (e.g., gallium and arsenic) found in the third and ifth columns of the periodic table are called III–V semiconductors. M. J. F. Digonnet and H. J. Shaw, Analysis of a tunable single mode photonic iber coupler, IEEE J. Quantum Electron., QE-18, 746–754, 1982. S. Shimada and H. Ishio (Eds.), Photonic Ampliiers and Their Applications, New York: John Wiley and Sons, 1994, chapters 1–5. J. C. Cartledge and G. S. Burley, The effect of laser chirping on lightwave system performance, J. Lightwave Technol., 7, 568–573, 1989. J. M. Senior, Photonic Fiber Communications: Principles and Practice, London, UK: Prentice Hall, second edition, 1992.

14

Photonic Signal Processing

The fundamental component of any integrated optical circuit is the waveguide. Compared with waveguides made of other materials, single-mode silica-based waveguides, which have almost the same composition as that of single-mode optical ibers, are more compatible with optical ibers and hence have lower iber coupling loss.36,37,38 Two major processes have been used for the fabrication of single-mode silica-based waveguides on planar silicon substrates: chemical vapor deposition and lame hydrolysis39 deposition. The single-mode waveguide patterns are then deined by photolithographic pattern deinition processes followed by reactive ion etching. The glass systems commonly used for the silica-based waveguides are: phosphorous-doped silica (SiO2 − P2O5 ), which is formed by chemical vapor deposition,40,41 and titanium-doped silica (SiO2 − TiO2 ) and germanium-doped silica (SiO2 − GeO2 ), which are formed by lame hydrolysis deposition.42,43 It has been claimed by research groups at NTT Japan laboratories that the combination of lame hydrolysis deposition and reactive ion etching can produce low-loss silica-based waveguides, which are best matched to optical ibers. A variety of passive integrated optical circuits, also known as planar lightwave circuits (PLCs), using single-mode silica-based waveguides on silicon substrates fabricated by a combination of lame hydrolysis deposition and reactive ion etching have been demonstrated as: splitters, low-speed optical switches,44,45,46,47 optical wavelength-division multi/demultiplexers, optical frequency-division multi/ demultiplexers,48,49,50,51,52 tunable optical ilters,53,54 and optical dispersion compensators [54].55

36

37

38

39 40

41

42

43

44

45

46

47

48

49

50 51

52

53

54

55

M. Kawachi, Silica waveguides on silicon and their application to integrated optic components, Opt. and Quantum Electron., 22, 391–416, 1990. M. Kawachi, Recent progress in planar lightwave circuits, 10th International Conference Integrated Opticsand Photonic Fiber Communication, Hong Kong, Proceedings of the IOOC’95, Vol. 3, pp. 32–33, 1995. C. H. Henry, Silica planar waveguides, Proceedings of the IREE, 19th Australian Conference Optical Fiber Technology, Melbourne, Australia, pp. 326−328, 1994. Flame hydrolysis is a method originally developed for iber preform fabrication. B. H. Verbeek, C. H. Henry, N. A. Olsson, K. J. Orlowsky, R. F. Kazarinov, and B. H. Johnson, Integrated four-channel Mach-Zehnder multi/demultiplexer fabricated with phosphorous doped SiO2 waveguides on Si, J. Lightwave Technol., 6, 1011–1017, 1988. L. Vivien, L. Pavesi, (Ed.), Handbook of Si photonics, CRC Press, Series of Optics and Optoelectronics, Boca Raton, FL: CRC Press, 2012. M. Kawachi, Silica waveguides on silicon and their application to integrated optic components, Opt. and Quantum Electron., 22, 391–416, 1990. M. Kawachi, Recent progress in planar lightwave circuits, 10th International Conf. Integrated Optics and Photonic Fiber Commun., Hong Kong, Proc. IOOC’95, vol. 3, pp. 32–33, 1995. T. Kitoh, N. Takato, K. Jinguji, M.Yasu, and M. Kawachi, Novel broad-band photonic switch using silica-based planar lightwave circuit, IEEE Photon. Technol. Lett., 4, 735–737, 1992. M. Okuno, K. Kato, Y. Ohmori, M. Kawachi, and T. Matsunaga, Improved 88 × integrated photonic matrix switch using silica-based planar lightwave circuits, J. Lightwave Technol., 12, 1597–1606, 1994. R. Nagase, A. Himeno, M. Okuno, K. Kato, K. Yukimatsu, and M. Kawachi, Silica-based 88 × photonic matrix switch module with hybrid integrated driving circuits and its system application, J. Lightwave Technol., 12, 1631–1639, 1994. Y. Hida, N. Takato, and K. Jinguji, Wavelength division multiplexer with passband and stopband for 1.3/1.55 µm using silica-based planar lightwave circuit, Electron. Lett., 31, 1377–1379, 1995. K. Sasayama, M. Okuno, and K. Habara, Photonic FDM multichannel selector using coherent photonictransversal ilter, J. Lightwave Technol., 12, 664–669, 1994. K. Oda, S. Suzuki, H. Takahashi, and H. Toba, A photonicFDM distribution experiment using a high inesse waveguidetypedouble ring resonator, IEEE Photon. Technol. Lett., 6, 1031–1034, 1994. Y. Xiao and S. He, An MMI-based demultiplexer with reduced cross-talk, Opt. Commun., 247(4), 335–339, 2005. S. Suzuki, K. Oda, and Y. Hibino, Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz, J. Lightwave Technol., 13, 1766–1771, 1995. S. Suzuki, M. Yanagisawa, Y. Hibino, and K. Oda, High-density integrated planar lightwave circuits using SiO GeO 22− waveguides with a high-refractive index difference, J. Lightwave Technol., 12, 790–796, 1994. E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, Variable bandwidth and tunable centre frequency ilter using transversal-form programmable photonic ilter, Electron. Lett., 32, 113–114, 1996. K. Okamoto, M. Ishii, Y. Hibino, and Y. Ohmori, Fabrication of variablebandwidth ilters using arrayed-waveguide gratings, Electron. Lett., 31, 1592–1594, 1995. R. Jones, J. Doylend, P. Ebrahimi, S. Ayotte, O. Raday, and O. Cohen., Silicon photonic tunable optical dispersion compensator, Opt. Express 15(24), 15836, 2007.

Photonic Signal Processing Via Signal-Flow Graph

15

Integrated optical circuits can be realized by two different approaches: hybrid integration, where several devices are fabricated on different materials, with each optimized in a given material, and combined on a common substrate, and monolithic integration, where all devices are fabricated on a common substrate  [32–35]. Although the monolithic integration is economically attractive because mass production of the circuit can be achieved by automatic batch processing, there is no single substrate material, which is ideal in all respects, as described above. In recent years, hybrid integration has been considered as a practical and promising approach for combining many desired functions on a common substrate, and silicon is an ideal substrate material [37,38,55–59]. The compatibility of silica-based waveguides on silicon with the optical ibers, the high thermal conductivity of silicon (and hence a good heat sink), and the good mechanical stability of silicon make silicon an attractive substrate not only for passive PLCs, but also as a platform (or motherboard) for hybrid integration of opto-electronic devices. Hybrid integration platforms have been successfully developed to enable the integration of silica-based waveguides and laser diode chips all on the same silicon substrate. The present challenging task is to incorporate integrated-optic waveguide ampliiers (e.g., using erbium-doped silica-based waveguides into the hybrid integration platform to provide greater functionality for next-generation optical networks. In addition, the well-developed silicon technology in the microelectronics industry can be applied to the mass production of hybrid integration of opto-electronic devices, such as optical sources, optical ampliiers, and optical functional circuits, all on a general-purpose silicon platform at a potentially low cost. In this chapter, low-loss single-mode silica-based waveguides embedded on silicon substrates, whose advantages have been described above, have been chosen as the integrated-optic technology for the design of coherent integrated-optic signal processors. However, the methodology and results are applicable to optical signal processors using other waveguide materials. Similar to the incoherent iber-optic signal processors described in Section  2.2, the basic components required for the realization of FIR and IIR coherent integrated-optic signal processors are: integrated-optic delay lines, integrated-optic phase shifters, integrated-optic directional couplers, and integratedoptic ampliiers. These are described in the following section.

2.3.1

integrateD-Optic Delay lines

The high precision and stability of the low-loss large-bandwidth single-mode silica-based waveguide on a silicon substrate have made the waveguide an attractive delay-line medium for processing high-speed broadband signals directly in the optical domain. The minimum curvature (or bending) radius,56 the propagation loss of the waveguides and the waveguide-iber coupling loss are very important characteristics in designing and fabricating PLCs. For high relative refractive index difference between the core and cladding ( ∆), the waveguides have the advantage of having a small curvature radius but at the expense of having a large propagation loss and a large iber coupling loss. In recent years, the SiO2 − GeO2 waveguide has been preferred to the SiO2 − TiO2 waveguide because the former glass system has a lower propagation loss. The discussion here is thus focused on the SiO2 − GeO2 waveguides. Typically, SiO2 − GeO2 waveguides with a core of 8 × 8 ~6 × 6 µm 2 and a low ∆ of 0.25% ~ 0.75% have a minimum curvature radius of 25 ∼ 5 mm and a propagation loss of less than 0.1 dB cm [49]. However, high-∆ waveguides (∆ = 1.5%) with a core of 4.5 × 4.5 µm 2 have a minimum curvature radius of 2 mm, a low propagation loss of 0.073 dB cm and a iber coupling loss of 0.9 dB [49]. The small curvature radius of high-∆ waveguides makes it possible to fabricate circuits with curvatures and hence permits high-density integration of circuits. For example, ring resonators require a small ring radius to have a large free spectral range. However, high-∆ waveguides are disadvantageous in terms of poor coupling with conventional single-mode ibers because of the mismatch between

56

The minimum curvature radius of the waveguide is the radius above which the bending loss is negligible.

16

Photonic Signal Processing

their optical mode ields. This problem can be solved by using mode-ield converters by means of a thermally expanded core technique. For example, the coupling loss was reduced from 2.0 to 0.9 dB when thermally expanded core waveguides were used. The stress-induced birefringence of the waveguide, which is caused by the difference between the thermal expansion coeficients of the silica glass layers and the silicon substrate, is unavoidable in the fabrication of PLCs. The waveguide birefringence can be eliminated by using either polarization mode converter with polyimide half wave-plate or the laser trimming method.

2.3.2

integrateD-Optic phase shiFters

A thermo-optic phase shifter (PS), which utilizes the thermo-optic effect to change the phase of the optical carrier, is an important element in PLCs because it provides an extra degree of freedom in circuit design. The thermo-optic PS, which consists of a thin ilm heater placed on the silica waveguide, is based on the temperature dependence of the refractive index of the waveguide. When an electric voltage is applied to the thin ilm heater, the refractive index of the heated waveguide increases, thus changing the optical path length by (dn dT) L∆T where dn dT = 1 × 10 −5 is the thermo-optic constant of silica waveguide, L is the heated waveguide length and ∆T is the temperature increase. For example, when a 5- mm long waveguide is heated by 30°C, the optical path length changes by 1.5 µm, which corresponds to a phase shift of 2π for a 1.5 - µm lightwave.57

2.3.3

integrateD-Optic DirectiOnal cOuplers

One of the fundamental elements in PLCs is an integrated-optic waveguide directional coupler, which performs signal collection (or addition) or signal distribution (or tapping). It is thus useful to mathematically characterize both the non-tunable and tunable directional coupler. For analytical simplicity, the insertion loss, propagation delay and waveguide birefringence of the directional coupler are not considered. The electric-ield transfer matrix of the lossless non-tunable waveguide directional coupler (the left directional coupler (DC) of Figure 2.2) can be described by58  E1   1 − k  =  E2   − j k

− j k   E1    1 − k   E2 

(2.2)

Phase Shifter E1

DC

E2

k

E1 E2

DC

E3

k

E4

φ

FIGURE 2.2 Schematic diagram of the symmetrical Mach−Zehnder interferometer, which is used as a tunable coupler (TC). DC represents the non-tunable directional coupler.

57

58

N. Takato, K. Jinguji, M. Yasu, H. Toba, and M. Kawachi, Silica-based single mode waveguides on silicon and their application to guided-wave photonic interferometers, J. Lightwave Technol., 6, 1003–1010, 1988. K. Oda, N. Takato, H. Toba, and K. Nosu, A wide-band guided-wave periodic multi/demultiplexer with a ring resonator for photonic FDM transmission systems, J. Lightwave Technol., 6, 1016–1023, 1988.

17

Photonic Signal Processing Via Signal-Flow Graph

where {E1, E2 } and {E1, E2 } are, respectively, the electric-ield amplitudes at the input and output ports of the left DC, k (0 ≤ k ≤ 1) is the cross-coupled intensity coeficient, and j = −1. Equation  (2.2) means that, when the input port {2} is not excited (i.e., E2 = 0 ), the input lightwave E1 is directly coupled to the output port {1} with an amplitude coupling coeficient of 1− k and cross coupled to the output port {2} with an amplitude coupling coeficient of − j k . Similarly, when the input port {1} is not excited (i.e., E1 = 0 ), the input lightwave E2 is directly coupled to the output port {2} with an amplitude coupling coeficient of 1− k and cross coupled to the output port {1} with an amplitude coupling coeficient of − j k . Note that the cross-coupled lightwave experiences a − π 2 phase shift. The ixed coupling coeficient of the non-tunable DC restricts its application in some PLCs. Another disadvantage is that it is dificult to fabricate the non-tunable DC with a precise coupling coeficient. This problem can be overcome by using the symmetrical Mach−Zehnder interferometer (see Figure 2.2), which can be designed to operate as a tunable coupler (TC) or an optical switch [57]. The TC consists of two identical non-tunable DCs interconnected by two waveguide arms of equal length. A thermo-optic PS (see Section 2.3.2) placed on the upper arm induces a phase shift of ϕ . When the input port {2} is not excited (i.e., E2 = 0 ) and using Eq. (2.2), the transfer functions are given by

E3 E1

E2 =0

= γ w  1 − k ⋅ exp( jϕ ) ⋅ 1 − k − j k ⋅ − j k 

(2.3)

= γ w [(1 − k ) exp( jϕ ) − k ] , E4 E1

E2 =0

= γ w  1 − k ⋅ exp( jϕ ) ⋅ − j k − j k ⋅ 1 − k 

(2.4)

= − j γ w k (1 − k ) [1+ exp( jϕ ) ] , where: exp( jϕ ) is the phase shift factor of the PS γ w (typically γ w = 0.89 for an insertion loss59 of 0.5 dB) is the intensity transmission coeficient of the waveguide TC Similarly, when the input port {1} is not excited (i.e., E1 = 0), the transfer functions are given by E3 E = 4 E2 E1 0 E1 = = E4 E2

E1 =0

(2.5) E2 0

= γ w  1 − k ⋅ 1 − k − j k ⋅ exp( jϕ ) ⋅ − j k    = γ w (1 − k ) − k exp( jϕ )  .

59

Note that the insertion loss including iber coupling loss is 0.8 dB [64].

(2.6)

18

Photonic Signal Processing

By a simple algebraic manipulation of Eqs (2.3) through (2.6), the electric-ield transfer matrix of the TC, as proposed by Ngo et al. [66], is given by  1 − K exp( jθ31 )  E3    = γw   E4   K exp( jθ32 )

K exp( jθ32 )   E1    1 − K exp( jθ 42 )   E2 

(2.7)

where: K = 2k (1 − k )(1 + cosϕ )

(2.8)

0 ≤ K ≤ 4k (1 − k ) or 0.5 − 0.5 1 − K ≤ k ≤ 0.5 + 0.5 1 − K ,

(2.9)

  sin ϕ θ31 = tan −1  ,  cos ϕ − k (1 − k ) 

(2.10)

 (1 + cosϕ )  θ32 = − tan −1  ,  sin ϕ 

(2.11)

  sin ϕ θ 42 = tan −1  . ϕ k k − 1 − cos ( )  

(2.12)

In Eqs. (2.11) and (2.12), ( E3 , E4 ) are the output electric-ield amplitudes of the TC, K is the cross-coupled intensity coeficient of the TC, and θ nm is the effective phase shift from the input port m to the output port n of the TC. The maximum value of K is K = 1, which only occurs at k = 0.5 according to Eq. (2.9). It is thus preferable to design both the non-tunable DCs with k ≅ 0.5 in order to maximize the dynamic tuning range of the TC, which is 0 ≤ K ≤ 1 . Note that k = 0.5 results in θ31 = θ32 = θ 42, which implies that the TC is a symmetrical and reciprocal device. Figure 2.3 shows the effective intensity coupling coeficient and the effective phase shifts of the TC for k = 0.5 and 0 ≤ ϕ ≤ 2π . It can be seen that 0 ≤ K ≤ 1 and −π 2 ≤ θ31,θ32 ,θ 42 ≤ + π 2 for 0 ≤ ϕ ≤ 2π , and that the same value of K occurs at two different values of ϕ because of the periodicity of K . Note that it is dificult to precisely fabricate 3-dB (k = 0.5) non-tunable DCs, which are important for optical communication or sensor systems. However, this problem can be overcome by using the TC, which can be made exactly 3 dB ( K = 0.5) provided that 0.1464 ≤ k ≤ 0.8536 (see Eq. [2.9]). The TC can behave as an optical switch when k = 0.5 and ϕ ∈(0, π )  [57]. When no electric power is applied to the PS (ϕ = 0, off state), the input signals are cross-switched according to the paths: (1 → 4 , 2 → 3). With electric power corresponding to a phase shift of ϕ = π is applied (on state), the input signals are direct-switched according to the paths: (1 → 3, 2 → 4). Typically, the power required for switching is about 0.5 W and the response time is about 1 ms. The TC can operate stably against temperature variation because its operating condition is hardly inluenced by environmental temperature change. This is because it is the temperature difference between the two waveguide arms, but not the absolute temperature of each arm, that is important for tunable or switching operation.

2.3.4

integrateD-Optic ampliFiers

Integrated-optic waveguide ampliiers are important active elements for loss compensation as well as for providing design lexibility of PLCs. The resulting ampliied PLCs can perform functions, which are otherwise not available with the unampliied PLCs.

Photonic Signal Processing Via Signal-Flow Graph

19

FIGURE 2.3 Output responses of the tunable coupler (TC) for k = 0.5 and 0 ≤ ϕ ≤ 2π . (a) Intensity coupling coeficient K. (b) Effective phase shifts θ31 = θ32 = θ 42 .

A 50 cm long erbium-doped silica-based waveguide ampliier integrated with a wavelength-division multiplexing (WDM) coupler has been successfully demonstrated with a gain of 27 dB, a low noise igure (NF) of 5 dB, and a saturated output power of 4.4 dBm.60 In order to fully integrate ampliier devices with other components on the same chip, the length of the amplifying waveguide must be as short as possible. This can be achieved by increasing the doping level of erbium concentration as much as possible. However, it is believed that there have been no reports to date of experimental results of waveguide ampliiers integrated on PLCs. With recent success in PLC technology, it would not be surprising that PLCs integrated with waveguide ampliiers (or semiconductor ampliiers) become a reality in the next few years. In this research, PLCs using erbium-doped silica-based waveguide ampliiers have thus been proposed as active functional optical devices for optical communication systems.

2.4 REMARKS The fundamental theories of incoherent iber-optic signal processing and coherent integrated-optic signal processing have been described. The major points that can be drawn from this chapter are given below. Incoherent Fiber-Optic Signal Processing • Incoherent iber-optic signal processors require the coherence time of the optical source to be much shorter than the basic time delay in the system, and can be directly or externally modulated. • In incoherent iber-optic signal processors, the low loss and large bandwidth of the singlemode optical iber have made it an attractive delay-line medium for processing broadband signals directly in the optical domain. 60

R. N. Ghosh, J. Shmulovich, C. F. Kane, M. R. X. de Barros, G. Nykolak, A. J. Bruce, and P. C. Becker, 8-mW threshold Er3+ − doped planar waveguide ampliier, IEEE Photon. Technol. Lett., 8, 518–520, 1996.

20

Photonic Signal Processing

• The characteristics of the fundamental iber-optic elements (such as iber-optic delay lines, iber-optic directional couplers, and iber-optic and semiconductor ampliiers) of the incoherent iber-optic signal processors have been described. • Although incoherent iber-optic signal processors are stable and robust, they can only perform positive-valued signal processing operations and thus have limited applications. They can process RF and microwave signals with speed ranging from hundreds of megahertz to a few gigahertz because the basic ilter length typically ranges from a few meters to tens of meters depending on the frequency of operation. • The fundamental theory of incoherent iber-optic signal processing described in this chapter is used in the analysis and design of incoherent iber-optic signal processors, which are presented in Chapters 3 through 5. Coherent Integrated-Optic Signal Processing • Coherent integrated-optic signal processors require the coherence time of the optical source to be much longer than the basic time delay in the system. The optical source must be a frequency-stabilized highly coherent semiconductor laser, which must be externally modulated. • In coherent integrated-optic signal processors, the high precision and stability of the lowloss large-bandwidth single-mode silica-based waveguide on a silicon substrate have made the waveguide an attractive delay-line medium for high-speed processing of broadband signals directly in the optical domain. • The characteristics of the fundamental integrated-optic elements (such as integrated-optic delay lines, integrated-optic phase shifters, integrated-optic directional couplers, and integrated-optic ampliiers) of the coherent integrated-optic signal processors have been described. • Coherent integrated-optic signal processors can stably perform complex-valued signal processing operations and thus have potential applications in optical communication systems. They can process millimeter-wave signals with speed in the range of tens of gigahertz because the basic ilter length typically ranges from a few millimeters to a few centimeters depending on the frequency of operation. • The fundamental theory of coherent integrated-optic signal processing described in this chapter is used in the analysis and design of coherent integrated-optic signal processors, which are presented in Chapters 6–9. Note that the choice between an incoherent and a coherent optical signal processor depends on the particular application.

2.5

SIGNAL-FLOW GRAPH APPROACH AND PHOTONIC CIRCUITS

The coherent and coherent aspects given in Sections 2.2 and 2.3 indicate that the transmittance of an optical device can be considered as the transfer function of an optical circuit. The low of either the optical ield or optical intensity can be represented by the input and output of a two-port optical network via the transfer transmittance. This section thus gives a framework for the analysis of optical signal processing systems/circuits through application of the SFG technique, which is developed as a mathematical tool for rest of the chapters. The limitations of other techniques for the analysis of these processors are addressed in Section 2.5.1. Section 2.5 describes the fundamental theory of the SFG approach whose effectiveness is then demonstrated by applying it to the analysis of an incoherent recursive iber-optic signal processor (RFOSP), as outlined in Section 2.7.1. Two designs of the incoherent RFOSP and their applications as optical ilters are also presented. The theory of incoherent iber-optic signal processing described in Sections 2.2 and 2.3 is employed in this section where intensity-based signals are considered.

Photonic Signal Processing Via Signal-Flow Graph

2.5.1

21

intrODuctOry remarks

A linear time-invariant optical signal processor is often characterized by its transfer function(s). The analysis and design of optical signal processors thus require the analysis of their transfer functions, which have been obtained by the method of successive substitutions of simultaneous equations61,62,63,64,65 and the transfer matrix method.66 However, these methods can be tedious, timeconsuming, and error-prone especially when dealing with a large-scale system, and do not provide a clear picture of the mechanisms in which an input signal lows through (or relects from) the system. These problems can be overcome by the use of the SFG technique proposed in references [70] and [71], which was applied to optics by Binh et al.67,68,69 References [67–69] were concerned with the application of the SFG technique whose fundamental theory was, however, not given. In this chapter, the fundamental theory as well as the application of the SFG method are presented in a more comprehensive manner so that the underlying principles can be easily understood and applied.

2.5.2

signal-FlOw graph theOry

An SFG is deined as a network of directed branches, connected at nodes [68,71], and is simply a pictorial representation of the simultaneous algebraic equations describing a system and graphically displays the low of signals through a system. The SFG method can be interpreted as a transformation of either the method of successive substitutions of simultaneous equations or the transfer matrix method to a topological approach. Thus, it may be said that “a graph is worth a thousand equations,” which is analogous with the common saying that “a picture is worth a thousand words.” The SFG technique has been widely used with great success in the diverse ields of electronics, and digital signal processing and control systems since its development by Mason in the 1950s.70,71 In general, the SFG theory can be applied to any linear time-invariant systems. The advantages of the SFG technique over conventional methods are: • It yields a pictorial representation of the low of signals through the system, which enhances an understanding of the system operation • It provides an easy and systematic way of manipulating the variables of interests, which allows graphical simulation of the system using a computer program72 • It enables solutions to be easily obtained by direct inspection of simple systems • It permits the identiication of the physical behavior and topological properties of a system 61

62

63

64

65

66

67

68

69

70 71 72

M. C. Vazquez, R. Civera, M. Lopez-Amo, and M. A. Muriel, Analysis of double-parallel ampliied recirculating photonic-delay lines, Appl. Opt., 33, 1015–1021, 1994. C. Vazquez, M. Lopez-Amo, M. A. Muriel, and J. Capmany, Performance parameters and applications of a modiied ampliied recirculating delay line, Fiber Integrated Opt., 14, 347–358, 1995. B. Vizoso, C. Vazquez, R. Civera, M. Lopez-Amo, and M. A. Muriel, Ampliied iber-optic recirculating delay lines, J. Lightwave Technol., 12, 294–305, 1994. B. Vizoso, I. R. Matias, M. Lopez-Amo, M. A. Muriel, and J. M. Lopez-Higuera, Design and application of double ampliied recirculating ring structure for hybrid iber buses, Opt. and Quantum Electron., 27, 847–857, 1995. E. C. Heyde and R. A. Minasian, A solution tothe synthesis problem of recirculating photonic delay line ilters, IEEE Photon. Technol. Lett., 6, 833–835, 1994. K. Oda, N. Takato, and H. Toba, Awide-FSR waveguide double-ring resonator for photonic FDM transmission systems, J. Lightwave Technol., 9, 728–736, 1991. L. N. Binh, N. Q. Ngo, and S. F. Luk, Graphical representation and analysis of the Z-shaped double-coupler photonic resonator, J. Lightwave Technol., 11, 1782–1792, 1993. L. N. Binh, S. F. Luk, and N. Q. Ngo, Ampliied double-coupler double-ring photonic resonators with negative photonic gain, Appl. Opt., 34, 6086–6094, 1995. L. N. Binh, X. T. Nguyen, and N. Q. Ngo, Realisation of Butterworth-type photonic ilters using phase modulators and 3 3 × coupler ring resonators, IEE Proc.-Optoelectron., 143, 126–134, 1996. S. J. Mason, Feedback theory-some properties of signal low graphs, Proc. IRE, 41, 1144–1156, 1953. S. J. Mason, Feedback theory-further properties of signal low graphs, Proc. IRE, 44, 920–926, 1956. L. P. A. Robichaud, M. Boisvert, and J. Robert, Signal Flow Graphs and Applications, Englewood Cliffs, NJ: PrenticeHall, 1962.

22

Photonic Signal Processing

In general, the SFG technique is used to solve a set of linear algebraic equations, which are described by [68] n

xj =

∑t x ,

j = 2, 3… , n,

ij i

(2.13)

i =1

where: x1, the only driving force in the system, is the independent variable, x2 , x3 ,…, n are the dependent variables tij is the transmittance

2.5.3

DeFinitiOns OF sFg elements

To understand the SFG technique, deinitions of the fundamental elements of an SFG [74] are given in Figure 2.4. Deinition 1: A node (•) represents a variable. Deinition 2: A branch is a directed path joining two nodes, its direction is indicated by an arrow and its transmittance is speciied by an attached symbol (or numeral) describing the functional relation between the two nodes. For example, the symbol x1 → x2 represents a branch that has a transmittance of t12. Deinition 3: A source node is a node at which all branches are directed outward. For example, x2 is a source node. Deinition 4: A sink node is a node at which all branches are directed inward. For example, x6 is a sink node. Deinition 5: A feedback loop is a closed path, which starts and terminates at the same node such that the nodes can only be touched once per traversal. For example, x3 → x 4 → x3 is a feedback loop. Deinition 6:A self-loop is a feedback loop consisting of a single branch. For example, x2 → x2 is a self-loop. Deinition 7: Non-touching loops are separated loops, which have no node in common. For example, x2 → x2 and x3 → x 4 → x3 are non-touching loops. Deinition 8: A loopgain is given by the product of all transmittances associated with a feedback loop. For example, t34t43 is the loopgain of the feedback loop x3 → x 4 → x3. t 34

x3

x4

t 43

t 45

t 23 t 12 x1

t 25 x2

t 22

FIGURE 2.4 Example of a SFG.

t 56 x5

x6

23

Photonic Signal Processing Via Signal-Flow Graph Dummy Node

Dummy Node x7

x0

1 t 12 x1

t 25 x2

1 t 12

t56

x1

x6

x5

t25 x2

t56 x6

x5

t 22

t 22

FIGURE 2.5 Example showing the equivalence of two SFGs.

Deinition 9: A forward path is a path that has no feedback loop and consists of at least one branch. For example, x1 → x2 → x5 → x6 and x1 → x2 → x3 → x4 → x5 → x6 are forward paths. Deinition 10: A forward-path gain is given by the product of all transmittances associated with a forward path. For example, t12t25t56 is the forward- path gain of the forward path x1 → x2 → x5 → x6 . For clarity, it is preferable to add an additional branch with a transmittance of unity to the source node and to the sink node. Figure 2.5 shows an example of the equivalence of two graphs where x0 = x1 the source node is and x7 = x6 is the sink node.

2.6

RULES OF SFG

The following basic rules, namely, the transmission, addition and product rules are frequently used in SFG theory.73

2.6.1

rule 1: transmissiOn rule

The value of the variable denoted by a node is transmitted on every branch leaving that node. This can be mathematically described by = x j tij= xi , j 1, 2,⋯, n

(2.14)

and graphically represented by Figure 2.6.

2.6.2

rule 2: aDDitiOn rule

The value of the variable denoted by a node is equal to the sum of all signals entering that node. This can be mathematically described by n

xj =

∑t x

ij i

(2.15)

i =1

and graphically represented by Figure 2.7. 73

J. J. Distefano, A. R. Stubberud, and I. J. Williams, Theory and Problems of Feedback and Control Systems, Singapore: McGraw-Hill, Chapter 8, 1987.

24

Photonic Signal Processing x1 t i1

t i2 xi

t

x2

ik

xk t

in

xn

FIGURE 2.6 SFG representation of the transmission rule.

x1 t 1j x2

t 2j xj

t

kj

t

nj

xk

xn

FIGURE 2.7 SFG representation of the addition rule.

2.6.3

rule 3: prODuct rule

The effective transmittance of a branch is equal to the product of the transmittances of all branches in cascade. This can be mathematically described by xn = (t12t23 ⋯ t( n −1)n ) x1

(2.16)

and graphically represented by Figure 2.8.

t 12 x1

t 12 t 23

t (n-1)n x2

x n-1

xn

FIGURE 2.8 SFG representation of the product rule.

x1

t (n-1)n xn

25

Photonic Signal Processing Via Signal-Flow Graph

2.7 MASON’S GAIN FORMULA The transfer function H between the independent (or source) node j and the dependent (or sink) node k in the SFG is determined using Mason’s gain formula [74]: H=

1 D

N

∑ PD i

(2.17)

i

i =1

where: N = Total number of forward paths from node j to node k Pi = ith forward-path gain of the forward path from node j to node k Pmr = mth possible product of r non-touching loopgains D = SFG determinant or characteristic function = 1 − ( −1) r +1 = 1−

∑ m

∑∑ P

mr

m

r

Pm1 +

∑ m

Pm2 −

∑

Pm3 + ...

(2.18)

m

= 1 − (Sum of all loopgains) + (Sum of all gain-products of 2 non-touching loops) − (sum of all gain-products of 3 non-touching loops) +... Di = Cofactor of the ith forward path = D; Evaluated with all loops touching Pi eliminated. The application of Eq. (2.18) is considerably easier than it appears and is illustrated in Section 2.8.

2.7.1

analysis OF an incOherent recursive Fiber-Optic signal prOcessOr (rFOsp)

As an example, the SFG method is applied to the analysis of the incoherent RFOSP.

2.7.2

sFg representatiOn OF the incOherent rFOsp

Figure 2.9 shows the schematic diagram of the incoherent RFOSP, which consists of an optical iber loop interconnected by two tunable iber-optic directional couplers DC1 and DC2. The couplers are assumed to have the same intensity transmission coeficient γ and cross-coupled intensity coeficients κ1 and κ 2 . The signal intensities at the input and output ports are described by {I1, I8} and {I2, I7}, respectively. The optical transmittances of the lower (Λ1) and upper (Λ 2 ) halves of the ring are deined as Λ1 = G exp( − jωT1 ),

(2.19)

and T1 = delay time of the optical path = equivalent guided propagation length L (2.20) Λ 2 = exp( − jωT2 ),

(2.21)

G = Gɵ exp[ −2α ( L1 + L2 ) − 2(ε1 − ε 2 )].

(2.22)

26

Photonic Signal Processing I7

I1 1

3

DC1 κ1 2

5 Loop Delay T

4

7 DC2 κ2

6

8

I2

I8 ^

EDFA, G

FIGURE 2.9 Schematic diagram of the incoherent RFOSP. Numbers in circles denote the port numbers of the couplers.

ɵ is the intensity gain of the In Eqs. (2.20) and (2.21), G is the effective optical loopgain, G EDFA, which has been described in Section 2.7.2, α is the amplitude attenuation coeficient of the iber, ε1 and ε 2 are the amplitude coeficients of the splice (or connector) losses of the lower and upper halves of the ring, respectively, j = −1, ω is the angular modulation frequency, L1 and L2 are the iber lengths of the lower and upper halves of the ring, respectively, and T1 and T2 are the corresponding time delays of the lower and upper halves of the ring, respectively. Note that the lengths of the lower and upper halves of the ring do not necessarily need to be the same. It is the iber loop length L = L1 + L2 or loop delay T = T1 + T2 that is important for signal processing. As a representative value, the exponential factor in Eq. (2.22), which represents the iber loop loss, is calculated using typical parameter values. The iber loop length is assumed to be L = 100 m. The iber loss is assumed to be 0.2 dB/km at 1550 nm, and this results in α = 0.02303 km −1. Two splices are required for the lower path because of the EDFA, whereas one splice is required for the upper path. The splice loss is assumed to be 0.1 dB/splice, and this results in ε1 = 2 × 0.0115 and ε 2 = 0.0115. Using these values, the iber loop loss is given by exp[ −2α L − 2(ε1 + ε 2 )] = 0.93.

(2.23)

Using the SFG theory presented in Section 2.5 and the iber-optic directional coupler deined in Eq. (2.2), the resulting SFG representation of the incoherent RFOSP is shown in Figure 2.10.

FIGURE 2.10 SFG representation of the incoherent RFOSP. Numbers in circles denote optical nodes, which correspond to the port numbers of the couplers.

Photonic Signal Processing Via Signal-Flow Graph

2.7.3

27

DerivatiOn OF the transFer FunctiOns OF the incOherent rFOsp

The intensity transfer function I2 /I1 of the incoherent RFOSP is derived in detail. Figure 2.10 shows that there are two optical forward paths along which the input signal at node ① can low to the output node ②. Using Mason’s gain formula as deined in Eq. (2.17) where N = 2 , the two optical forwardpath gains can be written as follows: Path 1:

Path 2:

①→② P1 = γ (1− κ1 ),

(2.24)

P2 = γ 3κ12 (1 − κ 2 )Λ1Λ 2 .

(2.25)

①→④→⑥→⑤→③→②

There is only one optical loopgain that can be identiied as follows: Loop: ④→⑥→⑤→③→④ P11 = γ 2 (1 − κ1 )(1 − κ 2 )Λ1Λ 2 .

(2.26)

The SFG determinant is thus given by: D = 1 − P11 = 1 − γ 2 (1 − κ1 )(1 − κ 2 )Λ1Λ 2 .

(2.27)

The cofactors of the forward paths are:

D1 = D because the forward path 1 does not touch the loop, D2 = 1 because the forward path 2 touches the loop.

(2.28) (2.29)

Inserting Eqs. (2.28) and (2.29) into Eqs. (2.26) and (2.27), the intensity transfer function I 2 / I1 of the incoherent RFOSP can be written as I2 P1 D1 + P2 D2 γ (1− κ1 )(1− zzero z −1 ) = = I1 D 1 − zpole z −1

(2.30)

where: z = exp( jωT ) is the well-known z-transform parameter  [25] and T = T1 + T2 is the basic time delay (or sampling period) of the ilter. Furthermore, the zero zzero and the system pole zpole in the z-plane are given by zzero =

γ 2 (1 − 2κ1 )(1 − κ 2 )G , (1 − κ1 )

zpole = γ 2 (1 − κ1 )(1 − κ 2 )G.

(2.31) (2.32)

The intensity transfer function I 7 / I1 can be similarly derived but with less effort than Eq.  (2.30). This is because there is only one forward path from the input node ① to the output node ⑦ [i.e., path ①→④→⑥→⑦], and this path also touches the loop. As a result, the cofactor of this path is equal to unity. By inspection of Figure 2.10, the intensity transfer function I 7 / I1 of the incoherent RFOSP can be written simply as I7 γ 2κ1κ 2G exp(− jωΤ1 ) = I1 1 − zpole z −1

(2.33)

28

Photonic Signal Processing

where the numerator corresponds to the optical forward-path gain. For analytical simplicity, the factor exp(− jωΤ1 ) in Eq. (2.33), which represents the pure propagation delay and only introduces a linear phase term to the phase response, is neglected because it does not alter the essential characteristics of the ilter. It is clear that Eq. (2.33) corresponds to the transfer function of an all-pole optical ilter because the zero is located at the origin. It can thus be stated that:

An optical signal processor will exhibit the characteristics of an all-pole optical ilter if the forward paths touch all optical loops in its SFG representation. Based on the above statement, Figure 2.11a–c show other possible structures of the all-pole optical ilters with transfer functions Y2 /X1. Note that they all have only one forward path, which also touches the optical loop. It is worth mentioning that a single-coupler iber-optic ilter having one loop (or ring) cannot be used as an all-pole optical ilter because there are two forward paths. (a)

X1

Y2

Y1

X2

(b)

X1

Y2

Y1

X2

(c)

X1

Y2

Y1

X2

FIGURE 2.11 Schematic diagrams of other possible all-pole optical ilters with transfer functions Y2 / X1 with the coupling (a) X2 is the main excitation port to the optical ring; (b) Y1 as the main tapping output of the coupler involved Por X1 and Y1; and (c) Y1 is the main output port coupled out from the ring, and X2 is the main excitation port into the optical ring.

29

Photonic Signal Processing Via Signal-Flow Graph

The above analysis shows the advantage of the SFG technique over conventional methods because the intensity transfer functions of the incoherent RFOSP can be easily derived in a systematic manner. In addition, the technique can identify the property of a particular optical system.

2.7.4

stability analysis OF the incOherent rFOsp

Stability is one of the most important requirements in the performance of ampliied recursive optical systems in which optical ampliiers are incorporated. To ensure stable operation of the incoherent RFOSP, a irst-order system, the system pole must be placed within the unit circle so that the following condition holds: G
B, there would be a zero at z = 0 of the multiplicity of (A–B). As it has been found that the transfer functions can be expressed in z-domain, thus the transfer characteristics are dependent on the zero-pole patterns [4] of these transfer functions in the z-plane. The magnitude-frequency response at a particular frequency as the operating point moving on the unit-circle, *z* = 1, of the z-plane is given by B

H ( z ) = H (e

jωτ

b0 )= a0

∏l

zk

k =1 A

∏l

(2.46) pj

j =1

where: z = e jωτ , ω is the angular frequency in radians per second τ is the sampling period of signals in seconds lzk and l pj are the lengths from the operating point to the position of the kth zero and the jth pole of the transfer function respectively. The corresponding phase-frequency response at a particular operating frequency (wavelength) is given by B

arg( H ( z)) =

∑ k =1

A

ϕ zk −

∑ϕ j =1

pj

+ ( A − B)ωτ

(2.47)

42

Photonic Signal Processing

where ϕzk and ϕpj are the phase angles of the zeroes and poles respectively formed by the horizontal real axis and the lines connecting the poles and zeroes to the operating point in the z-plane. Thus, from Eqs. 2.45 and 2.46 we can tailor the magnitude-frequency response by adjusting the pole and zero patterns of the transfer functions. To obtain a maximum magnitude at a A particular operating wavelength (frequency), we require a pole or a very small value of ∏ j =1 l pj at that wavelength. Similarly, in order to obtain a minimum at a particular wavelength, a zero B or an ininitesimal value of ∏ k =1 lzk is required at that wavelength. There are some relationships between the positions of poles in the z-plane to those correspondingly in the s-plane (the continuous frequency domain). One basic property of this relationship is, when recalling that z = e sτ where s = jω , a pole position moves on the imaginary axis of the s-plane, it would move along the unitcircle of the z-plane. In this case, we would have marginal stability and a lossless system. When a pole moves on the imaginary axis towards the left half of the s-plane, the corresponding pole moves inside the unit circle in the z-plane. The system would then become lossy and stable. If one of the system poles lies outside the unit circle in the z-plane, the system becomes unstable. Its temporal response would increase with time. In general, the system would be stable if all the system poles lie inside or on the unit circle in the z-plane. Stability plays an important role in design of photonic circuits. Stability test for the photonic circuit is already introduced in Section 2.7.

2.10 { { { { { { { { { {

APPENDIX: OPTMASON.PAS PROGRAM LISTING

OPTMASON.PAS Written in 1996 by Dean Trower and Le Binh for Photonic Signal Processing This program uses Mason’s rule for signal flow graphs to generate the optical transfer function (in symbolic form) of an arbitrary lumped, linear time-invariant optical circuit, as specified in the input file (file name is a command line parameter). See the file OPTMASON.DOC for more information given in this section}

type term_type = (zterm, anglterm, magterm, magval); link_ptr = ^link_type; node_ptr = ^node_type; path_ptr = ^path_type; loop_ptr = ^loop_type; looplist_ptr = ^looplist_type; pathlist_ptr = ^pathlist_type; nodelist_ptr = ^nodelist_type; expression_ptr = ^expression_type; link_type = record {record for a signal flow graph link} dest:node_ptr; {destination node} val:expression_ptr; {value} next:link_ptr; end; expression_type = record {record for a single term in an expression} format:term_type; sumterm:expression_ptr; prodterm:expression_ptr; num:double; expr:^string; end;

Photonic Signal Processing Via Signal-Flow Graph

43

looplist_type = record {record for a single entry in a list of loops} loop:loop_ptr; next:looplist_ptr; end; pathlist_type = record {record for a single entry in a list of paths} path:path_ptr; next:pathlist_ptr; end; nodelist_type = record {record for a single entry in a list of nodes} node:node_ptr; next:nodelist_ptr; end; node_type = record {record for a node of the s.f. graph} name:string[15]; firstlink:link_ptr; {linked list of OUTPUT links} next:node_ptr; {i.e. next node in s.f. graph} link_taken:link_ptr; {used during search for loops/ paths} touchloop:looplist_ptr; {list of loops containing node} touchpath:pathlist_ptr; {list of paths containing node} end; path_type = record pathgain:expression_ptr; touchloop:looplist_ptr; {list of loops touching path} next:path_ptr; end; loop_type = record loopgain:expression_ptr; touchloop:looplist_ptr; {!!!loops later in list only} nodelist:nodelist_ptr; {circularly linked list of nodes} deactivated:word; {used for calc. of Mason rule deltas} next:loop_ptr; end; var firstnode, sourcenode, sinknode:node_ptr; firstpath:path_ptr; firstloop:loop_ptr; numerator, denominator:expression_ptr; f:text; outfile_name:string; precparam:string; {stores the -d command line parameter} line:longint; {counts lines in the input file} prec:byte; {no. of digits to display reals to in the output} errpos:integer; {used while parsing input and cmnd line params} heap_state:pointer; {so it can be restored when program ends} {the following variables aren’t really global,} path:path_ptr; {but they are here to prevent them being created} loop:loop_ptr; {many times during recursion.} node:node_ptr; pathentry:pathlist_ptr; loopentry, loopentry2:looplist_ptr;

44

Photonic Signal Processing nodeentry:nodelist_ptr; found_duplicate, plus_waiting, addbracket:boolean; tmp_exp, mult_result:expression_ptr;

function singleterm(s:string):boolean; { determines if an "expr" string should be treated as a single variable} var i:integer; b:boolean; begin b:=true; i:=1; while b and (i0) or (pos(copy(s, i,length(s)-i+1),’’’’’’’’’’’’’’’’’’’’’’’’)>0) or ((i>1) and (pos(s[i],'0123456789')>0)); inc(i); end; singleterm:=b; end; procedure writeexpression(var f:text; e:expression_ptr); { recursively procedure to display an expression (or output it to file f)} var t, q:expression_ptr; begin if e=nil then write(f,'0'); t:=e; while tnil do {step through each root-level sumterm} begin {skip over sums of angles inside single exp()} q:=t; {and sums inside brackets in z^(***)... done later} if (t^.format=anglterm) or (t^.format=zterm) then while (q^.prodtermnil) and (q^.prodterm^.sumterm=nil) and (q^.prodterm^.format=t^.format) do q:=q^.prodterm; if q^.prodtermnil then if q^.prodterm^.sumtermnil then begin if plus_waiting then write(f,'+'); plus_waiting:=false; write(f,'('); writeexpression(f, q^.prodterm); write(f,')*'); end else begin if not((q^.prodterm^.format=magval) and {don’t write "1*"} (abs(q^.prodterm^.num)=1)) then {or "-1*"} begin writeexpression(f, q^.prodterm); write(f,'*'); end else if q^.prodterm^.num=-1 then write(f,'-') else if plus_waiting then write(f,'+'); plus_waiting:=false; end; if not((t^.format=magval) and (t^.num=0) then write(f,'+'); if (q^.num1) or (q^.expr=nil) then begin if q^.num=-1 then write(f,'-') else if q^.num=round(q^.num) then write(f, round(q^.num)) else write(f, q^.num:0:prec); if (q^.exprnil) and (q^.num-1) then write(f,'*'); end; if q^.exprnil then if singleterm(q^.expr^) or (q^.num=1) then write(f, q^.expr^) else write(f,'[',q^.expr^,']'); q:=q^.prodterm; until (q=nil) or (q^.sumtermnil) or (q^.formatanglterm); if addbracket then write(f,'))') else write(f,')'); end;

46

Photonic Signal Processing

zterm: begin write(f,'z'); if ((t^.prodterm=nil) or (t^.prodterm^.formatzterm)) and ((t^.expr=nil) or (abs(t^.num)=1)) then addbracket:=false else addbracket:=true; if addbracket or (t^.exprnil) or (t^.num1) then begin if addbracket then write(f,'^(') else write(f,'^'); q:=t; {display an entire list of single-zterm} repeat {products using the same z^(...)} if (qt) and (q^.num>=0) then write(f,'+'); if (q^.num1) or (q^.expr=nil) then begin if q^.num=-1 then write(f,'-') else if q^.num=round(q^.num) then write(f, round(q^.num)) else write(f, q^.num:0:prec); if (q^.exprnil) and (q^.num-1) then write(f,'*'); end; if q^.exprnil then if singleterm(q^.expr^) or (q^.num=1) then write(f, q^.expr^) else write(f,'[',q^.expr^,']'); q:=q^.prodterm; until (q=nil) or (q^.sumtermnil) or (q^.formatzterm); if addbracket then write(f,')'); end; end; end; t:=t^.sumterm; if tnil then plus_waiting:=true; {next term to be added} end; end; procedure dispose_expression(var e:expression_ptr); { deletes all the memory allocated to an expression and sets it to nil (=0)} begin if enil then begin dispose_expression(e^.prodterm); dispose_expression(e^.sumterm); if (e^.formatmagval) and (e^.exprnil) then freemem(e^.expr, length(e^.expr^)+1); dispose(e); e:=nil; end; end; procedure copy_expression(var exp1:expression_ptr; exp2:expression_ptr); { recursively copys the contents of exp2 to exp1; new exp1 pointer created } begin if exp2=nil then exp1:=nil else

Photonic Signal Processing Via Signal-Flow Graph

47

begin new(exp1); exp1^:=exp2^; if (exp2^.formatmagval) and (exp2^.exprnil) then begin getmem(exp1^.expr, length(exp2^.expr^)+1); exp1^.expr^:=exp2^.expr^; end else exp1^.expr:=nil; copy_expression(exp1^.prodterm, exp2^.prodterm); copy_expression(exp1^.sumterm, exp2^.sumterm); end; end; procedure new_scalar(var scalar_expr:expression_ptr; val:double); { creates a new expression pointed to by scalar_expr, with value val } var e:expression_ptr; begin if val=0 then scalar_expr:=nil else begin new(e); scalar_expr:=e; e^.format:=magval; e^.num:=val; e^.expr:=nil; e^.sumterm:=nil; e^.prodterm:=nil; end; end; procedure new_link_val(var link_expr:expression_ptr; mag:double; mag_str:string; mag_pwr:double; angl:double; angl_str:string; zcoeff:double; z_str:string); { creates a new expression pointed to by link_expr, of the form } { mag*[mag_str]^mag_pwr * exp(j*(angl*[angl_str])) * z^(zcoeff*[z_str])} { } { NB this procedure creates the correct expression tree for precedes() } { as it is now; but if precedes() is altered w.r.t. the precedence of} { the different term formats, this procedure MUST be modified also. } var e:expression_ptr; begin if mag=0 then link_expr:=nil else {anything*0 = 0} begin new(e); link_expr:=e; if zcoeff0 then {include a zterm if present} begin e^.format:=zterm; e^.num:=zcoeff; if z_str=''then e^.expr:=nil else begin getmem(e^.expr, length(z_str)+1); e^.expr^:=z_str; end;

48

Photonic Signal Processing e^.sumterm:=nil; new(e^.prodterm); e:=e^.prodterm; end; if angl0 then {include an anglterm if present} begin e^.format:=anglterm; e^.num:=angl; if angl_str=''then e^.expr:=nil else begin getmem(e^.expr, length(angl_str)+1); e^.expr^:=angl_str; end; e^.sumterm:=nil; new(e^.prodterm); e:=e^.prodterm; end; if (mag_str'') and (mag_pwr0) then {include a magterm if present} begin e^.format:=magterm; getmem(e^.expr, length(mag_str)+1); e^.expr^:=mag_str; e^.num:=mag_pwr; e^.sumterm:=nil; new(e^.prodterm); e:=e^.prodterm; end; e^.format:=magval; {the magval term is compulsory for} e^.num:=mag; {nonzero expressions.} e^.expr:=nil; e^.prodterm:=nil; e^.sumterm:=nil;

end; end; function precedes(x, y:expression_ptr):boolean; { establishes an order-of-precedence among expression components} begin if y=nil then precedes:=true else if x=nil then precedes:=false else if (x^.formaty^.format) then precedes:=(x^.format=y^.num) else precedes:=(x^.expr^>y^.expr^); end; function can_add(x, y:expression_ptr):boolean; { true if the expression components pointed to by x, y are directly addable } { they are addable if they are equal or if they are both magvals. } begin if ((x=nil) xor (y=nil)) or (x^.formaty^.format) then can_add:=false else if (x=nil) or (x^.format=magval) then can_add:=true

Photonic Signal Processing Via Signal-Flow Graph

49

else if (x^.numy^.num) or ((x^.expr=nil) xor (y^.expr=nil)) or ((x^.exprnil) and (x^.expr^y^.expr^)) then can_add:=false else can_add:=true; end; procedure add(var e1:expression_ptr; e2:expression_ptr); { computes e1 err!} end; {check next node...} end; procedure getlinkvals; {read the input file to obtain expressions for links (=1 if not listed)} var s:string; node1,node2,tmp_node:node_ptr; link:link_ptr; nm1,nm2:string[15]; oneway:boolean; i, l:byte; mag_val, angl_num, mag_pwr, zcoeff:double; mag_expr, angl_expr, z_expr:string;

56

Photonic Signal Processing

begin repeat {get each remaining line of the input file in turn} readln(f, s); inc(line); killwhitespace(s); if s''then {ignore blanks, comments} begin i:=1; l:=0; if s[1]='*' then {check for * at start of line, denoting} begin {a one-way only link assignment.} oneway:=true; i:=2; if length(s)=1 then error(6,s); end else oneway:=false; {get first node name and the comma} while (i+l15 then error(2,copy(s, i,l)) else nm1:=copy(s, i,l); i:=i+l+1; l:=0; {get second node name} while (i+l15 then error(2,copy(s, i,l)) else nm2:=copy(s, i,l); i:=i+l+1; l:=0; node1:=firstnode; {find both nodes, check that they exist} node2:=node1; while (node1nil) and (node1^.namenm1) do begin if (node2^.namenm2) then node2:=node1; node1:=node1^.next; end; while (node2nil) and (node2^.namenm2) do node2:=node2^.next; if (node1=nil) then error(12,nm1); if (node2=nil) then error(12,nm2); mag_val:=1; {set defaults for any part of the} mag_pwr:=0; {expression not included in the input} angl_num:=0; zcoeff:=0; mag_expr:=''; angl_expr:=''; z_expr:=''; {(1) look for a number} while (i+l0) do inc(l); if l>0 then begin val(copy(s, i,l),mag_val, errpos); if errpos>0 then error(6,copy(s,1,i+errpos)); i:=i+l; l:=0; end; {(2) look for a magterm expr} if (i '+nm2) else {link absent?} begin {set its value} dispose_expression(link^.val); new_link_val(link^.val, mag_val, mag_expr, mag_pwr, angl_num, angl_expr, zcoeff, z_expr); end; if not(oneway) then {if setting value for both directions,} begin {find the reciprocal (other way) link} link:=node2^.firstlink; while (linknil) and (link^.dest^.namenm1) do link:=link^.next; if (link=nil) and (node2^.nextnil) and (node2^.next^. name=nm2)

Photonic Signal Processing Via Signal-Flow Graph

59

then link:=node2^.next^.firstlink; while (linknil) and (link^.dest^.namenm1) do link:=link^.next; if link=nil then error(13,nm2+' -> '+nm1) else {absent?} begin {set its value} dispose_expression(link^.val); new_link_val(link^.val, mag_val, mag_expr, mag_pwr, angl_num, angl_expr, zcoeff, z_expr); end; end; end else {from earlier: names are the same so its a reflection coeff} begin if (node1=sourcenode) or (node1=sinknode) then error(13,nm1+' -> '+nm1); {source or sink not allowed} node2:=node1^.next; {get node and its same-name counterpart} if (node1^.firstlinknil) and {reflection already set?} (node1^.firstlink^.dest=node2) then begin {if so, replace it} dispose_expression(node1^.firstlink^.val); new_link_val(node1^.firstlink^.val, mag_val, mag_expr, mag_pwr, angl_num, angl_expr, zcoeff, z_expr); end else if (node2^.firstlinknil) and {already set, other way?} (node2^.firstlink^.dest=node1) then begin {replace existing link} dispose_expression(node2^.firstlink^.val); new_link_val(node2^.firstlink^.val, mag_val, mag_expr, mag_pwr, angl_num, angl_expr, zcoeff, z_expr); end else if (node1^.firstlink=nil) and (node2^. firstlinknil) then begin {create new link on 1st node and set value} new(link); {if the 1st node is the input node for the} link^.next:=nil; {same-name pair} node1^.firstlink:=link; link^.dest:=node2; new_link_val(link^.val, mag_val, mag_expr, mag_pwr, angl_num, angl_expr, zcoeff, z_expr); end else if (node2^.firstlink=nil) and (node1^. firstlinknil) then begin {otherwise create new link on 2nd node...}

60

Photonic Signal Processing new(link); link^.next:=nil; node2^.firstlink:=link; link^.dest:=node1; new_link_val(link^.val, mag_val, mag_expr, mag_pwr, angl_num, angl_expr, zcoeff, z_expr);

end else error(13,nm1+' -> '+nm1); {neither node was the input} end; {of a cut-off fiber end. So reflections aren’t allowed!!!} end; until eof(f); {keep getting lines until end of input file} end; procedure searchfrom(root:node_ptr); { performs a recursive search from "root" node to find loops and paths } begin if root^.link_takennil then {hit a node already passed: loop found} begin found_duplicate:=false; {CHECK IF LOOP ALREADY EXISTS:} loopentry:=root^.touchloop; {compare with existing loops at node} while not(found_duplicate) and (loopentrynil) do {check each one} begin {find the current node in its node list} nodeentry:=loopentry^.loop^.nodelist; while nodeentry^.noderoot do nodeentry:=nodeentry^.next; node:=root; repeat {follow both loops a step at a time until we cycle, or} node:=node^.link_taken^.dest; {we get to 2 different nodes} nodeentry:=nodeentry^.next; until (node=root) or (nodenodeentry^.node); {if we cycled, loop} if node=nodeentry^.node then found_duplicate:=true; {isn’t new} loopentry:=loopentry^.next; {check next loop at node} end; if not(found_duplicate) then {CREATE NEW LOOP:} begin new(loop); {create loop entry} loop^.next:=firstloop; firstloop:=loop; loop^.touchloop:=nil; new_scalar(loop^.loopgain,-1); {-loopgain actually stored} loop^.nodelist:=nil; {will * one link at a time} loop^.deactivated:=0; {activate loop- used later} nodeentry:=nil; node:=root; {along each node on the loop do 5 things:} repeat with node^ do begin new(loopentry); {(1) add loop to list of loops at node} loopentry^.loop:=loop; loopentry^.next:=touchloop; touchloop:=loopentry;

Photonic Signal Processing Via Signal-Flow Graph

61

if nodeentry=nil then {(2) add node to nodelist for loop} begin new(loop^.nodelist); nodeentry:=loop^.nodelist; end else begin new(nodeentry^.next); nodeentry:=nodeentry^.next; end; nodeentry^.node:=node; loopentry:=touchloop; {(3) add all loops at node to} while loopentrynil do {touchloop list for this loop} begin {(but check for duplicates first)} loopentry2:=loop^.touchloop; while (loopentry2nil) and (loopentry2^.looploopentry^.loop) do loopentry2:=loopentry2^.next; if loopentry2=nil then {add loop to list} begin new(loopentry2); loopentry2^.loop:=loopentry^.loop; loopentry2^.next:=loop^.touchloop; loop^.touchloop:=loopentry2; end; loopentry:=loopentry^.next; end; pathentry:=touchpath; {(4) add this loop to touchloop} while pathentrynil do {list of all paths at node} begin loopentry2:=pathentry^.path^.touchloop; while (loopentry2nil) and (loopentry2^.looploop) do loopentry2:=loopentry2^.next; if loopentry2=nil then {check for existing entry} begin {none, so add to list} new(loopentry2); loopentry2^.loop:=loop; loopentry2^.next:=pathentry^.path^.touchloop; pathentry^.path^.touchloop:=loopentry2; end; pathentry:=pathentry^.next; end; {(5) multiply loopgain by value of outgoing link} multiply(mult_result, loop^.loopgain, link_taken^.val); dispose_expression(loop^.loopgain); loop^.loopgain:=mult_result; node:=link_taken^.dest end; until node=root; {cycle through nodes until back at the start} nodeentry^.next:=loop^.nodelist; {nodelist itself should loop} end; {i.e. it’s circularly linked} end else if root=sinknode then {REACHED SINKNODE: NEW PATH FOUND} begin new(path); {create path entry} path^.next:=firstpath;

62

Photonic Signal Processing

firstpath:=path; path^.touchloop:=nil; new_scalar(path^.pathgain,1); node:=sourcenode; {along each node on the path do 3 things:} repeat with node^ do begin new(pathentry); {(1) add path to list of paths at node} pathentry^.path:=path; pathentry^.next:=touchpath; touchpath:=pathentry; loopentry:=touchloop; {(2) add all loops at node to loop} while loopentrynil do {list for this path (but check} begin {for duplicates first)} loopentry2:=path^.touchloop; while (loopentry2nil) and (loopentry2^.looploopentry^.loop) do loopentry2:=loopentry2^.next; if loopentry2=nil then {add loop to list} begin new(loopentry2); loopentry2^.loop:=loopentry^.loop; loopentry2^.next:=path^.touchloop; path^.touchloop:=loopentry2; end; loopentry:=loopentry^.next; end; if link_taken=nil then node:=nil else {(3) multiply pathgain by} begin {value of outgoing link} multiply(mult_result, path^.pathgain, link_taken^.val); dispose_expression(path^.pathgain); path^.pathgain:=mult_result; node:=link_taken^.dest end; end; until node=nil; {step through nodes on the path until none remain} end else with root^ do {continue depth-first search} begin link_taken:=firstlink; {mark the way we went} while link_takennil do begin {don’t follow 'phantom' links TO the source node} if link_taken^.destsourcenode then searchfrom(link_taken^. dest); link_taken:=link_taken^.next; {go a different way...} end; end; end; procedure deletenodedata; { frees memory used for storing node information (no longer needed) } var nextentry:pointer; link:link_ptr; begin loop:=firstloop; {remove each loop’s nodelist} while loopnil do begin nodeentry:=loop^.nodelist;

Photonic Signal Processing Via Signal-Flow Graph while nodeentrynil do begin nextentry:=nodeentry^.next; if nextentry=loop^.nodelist then nextentry:=nil; dispose(nodeentry); nodeentry:=nextentry; end; loop^.nodelist:=nil; loop:=loop^.next; end; node:=firstnode; {remove all data for each node:} while nodenil do begin pathentry:=node^.touchpath; {remove the touchpath list} while pathentrynil do begin nextentry:=pathentry^.next; dispose(pathentry); pathentry:=nextentry; end; loopentry:=node^.touchloop; {remove the touchloop list} while loopentrynil do begin nextentry:=loopentry^.next; dispose(loopentry); loopentry:=nextentry; end; link:=node^.firstlink; {remove the links} while linknil do begin nextentry:=link^.next; dispose_expression(link^.val); dispose(link); link:=nextentry; end; nextentry:=node^.next; {remove the node itself} dispose(node); node:=nextentry; end; end; procedure addloopcombination(var delta:expression_ptr; running_product:expression_ptr; current_loop:loop_ptr); { recursively adds to delta the sum of all possible products of} { nontouching loop gains, down the list from 'current_loop'. } { 'running product' contains the product of those loops already } { used... i.e. the ones up the list from current_loop that were } { taken on the way to getting to current_loop. } var tempexpression:expression_ptr; begin while current_loopnil do {step through all possible branches} begin {and descend to them in turn if active} if current_loop^.deactivated=0 then {skip deactivated ones, i.e.} begin {those already 'touched'} multiply(tempexpression, current_loop^.loopgain, running_product);

63

64

Photonic Signal Processing

add(delta, tempexpression); {add to delta the tree descent so far} loopentry:=current_loop^.touchloop; {deactivate touching loops} while loopentrynil do {further down the list} begin inc(loopentry^.loop^.deactivated); loopentry:=loopentry^.next; end; {recursively add to delta all terms downbranch} addloopcombination(delta, tempexpression, current_loop^.next); dispose_expression(tempexpression); loopentry:=current_loop^.touchloop; {reactivate loops} while loopentrynil do {(or clear deactivations} begin {set by current loop)} dec(loopentry^.loop^.deactivated); loopentry:=loopentry^.next; end; end; current_loop:=current_loop^.next; {do next branch} end; end; procedure calcdelta(var delta:expression_ptr); { calculates the Mason rule delta, using all loops not already deactivated} var running_product:expression_ptr; begin new_scalar(delta,1); new_scalar(running_product,1); addloopcombination(delta, running_product, firstloop); dispose_expression(running_product); end; procedure calctransferfunction; { calculates the numerator and denominator of the signal flow} { graph transfer function using the Mason’s rule equation} var delta:expression_ptr; begin calcdelta(denominator); {calculate denominator = delta} new_scalar(numerator,0); {set numerator = 0} path:=firstpath; while pathnil do {for each path in turn:} begin loopentry:=path^.touchloop; {deactivate each loop that touches it} while loopentrynil do begin loopentry^.loop^.deactivated:=1; loopentry:=loopentry^.next; end; calcdelta(delta); {find delta for this reduced loop set} multiply(mult_result, delta, path^.pathgain); dispose_expression(delta); dispose_expression(path^.pathgain); add(numerator, mult_result); {numerator 3) or ((paramcount=3) and (precparam[1]'-')) or (errpos0) then begin {if the parameters weren’t right, show instructions and quit} writeln('OptMason generates the transfer function of an optical network using Mason''s'); writeln('rule for signal-flow graphs.'); writeln; writeln('Written by Dean Trower, 1996. This program may be freely distributed.'); writeln; writeln; writeln('USAGE: optmason input_file [output_file] [-d]'); writeln; writeln('d is the number of decimal places that numbers are displayed to in the output.'); writeln('If the output file is omitted, output is to the screen (or standard output).'); writeln('If the output file has the same name as the input file, the output is appended'); writeln('to the input file.'); writeln; halt; end; mark(heap_state); {record the state of memory, to restore it when done} firstnode:=nil; {init lists and miscellaneous} firstpath:=nil; firstloop:=nil; plus_waiting:=false; assign(f, paramstr(1)); {attempt to open input file} reset(f); if IOResult0 then error(0,paramstr(1)); getsourcesink; {parse input, output definitions}

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Photonic Signal Processing

getlinks; {parse geometry description} correctlinks; {create signal flow graph geometry, and check} getlinkvals; {parse transmittance expressions} close(f); {close the input file} writeln('SIGNAL FLOW GRAPH CREATED'); {this always goes to the SCREEN} searchfrom(sourcenode); {recursively find all loops and paths} deletenodedata; {get rid signal flow graph, now no longer necessary} writeln('PATH AND LOOP INFORMATION COMPUTED'); {displays to SCREEN} calctransferfunction; {apply Mason’s rule to loop and path info} writeln('TRANSFER FUNCTION T COMPUTED'); {displays to SCREEN} assign(f, outfile_name); {create or append the desired output file} if paramstr(1)=outfile_name then append(f) else rewrite(f); if IOresult0 then error(0,outfile_name); writeln(f); {NB a file name of ''get assigned to DOS standard output} writeln(f,'T = numerator/denominator'); {output the results} writeln(f); write(f,'numerator = '); writeexpression(f, numerator); writeln(f); writeln(f); write(f,'denominator = '); writeexpression(f, denominator); writeln(f); close(f); {close the output file} release(heap_state); {deallocate all memory that got used} end.

2.11

APPENDIX: USING “OPTIMASON” THE COMPUTER AIDED GENERATOR

OPTMASON (Program listed in Section 2.9) is started from the DOS command line by typing: “optmason input_ile [output_ile] [-d]” where “input_ile” and “output_ile” are the corresponding ilenames, and “d” is the number of decimal places to display real numbers to in the output; if omitted, “d” defaults to 3. If the output ilename is omitted, output is to the screen (actually to DOS’s standard output ile, to enable redirection). If the input and output ilenames are the same, the input ile is not overwritten, but is instead appended (i.e., OPTMASON’s output is added to the end of it). The input text ile format for OPTMASON is as follows: $INPUT = nodename $OUTPUT = nodename nodename: nodename, nodename,…; nodename, nodename,… nodename: nodename, nodename,…; nodename, nodename,… nodename: nodename, nodename,…; nodename, nodename,… $TRANSMITTANCES [*]nodename, nodename = expression [*]nodename, nodename = expression [*]nodename, nodename = expression

Here nodename is a label for a node of the photonic connection graph. It may be any string of up to 15 characters, but not containing any of the characters “;:,=$*” (double quotes are OK though). Expression is a mathematical expression (see below). The first line ($INPUT=…) identifies the input (or source) node. The second line ($OUTPUT=…) identifies the output (or sink) node.

Photonic Signal Processing Via Signal-Flow Graph

67

The input and output nodes may be the same node. The next section deines the geometry of the photonic connection graph. A line of the form nodename: nodename, nodename,…; nodename, nodename,…is required for each node in the graph exceptthe output node. (If a line for the output node is included, it will be ignored, unless that node is also the input node.) Each such line begins with the name of the node being deined, followed by a colon “:”. The remainder of the line is a list of all the other nodes that it connects to. Since light may travel independently in bi-directions through a node in a Photonic connection graph, connections on either “side” (photonically speaking) are separated by a semicolon “;”. The deinition for the input node and any node at the free end of a photonic waveguide(relection point) will only include one “side” of this list. Since links in a photonic connection graph are bi-directional, a link from n to min the deinition of a node n must be matched by a corresponding link from m to n in the deinition of node m; links to the output node are an exception. Note that only a single link may join any two nodes. For multiple photonic paths between two nodes, intermediate nodes must be inserted. The geometry deinition section is terminated by the “$TRANSMITTANCES” line. The section following this line deines the values of the links in the photonic connection graph. Since all links have a default value of unity in both directions, only the values of links that differ from this need be deined. The format for specifying the value of a link between two nodes n and m (value given by “expression”) is: “n, m = expression”. To deine the value of the link in the direction n only, place an asterix “*” at the start of the line. Note that if a link value is deined twice, the second deinition replaces the irst (both directions are treated independently). To deine the relection coeficient at a node r, simply write: “r, r = expression” An asterix “*” is optional and has no effect. Relection coeficients may ONLY be deined at nodes that are photonically single-sided (e.g., cut-end of a photonic iber/waveguide), and may not be deined for the input or output nodes. Some other things to note about the input ile format are: • The input is case-sensitive for node names and variables within expressions. • Whitespace (spaces, tabs, and blank lines) are ignored or iltered out. • The start of the ile is ignored up to the line starting with “$INPUT=” so it may be used for a description of the ile contents. • Any line beginning with a semicolon “;” is treated as a comment and ignored. • Input lines are truncated beyond 255 characters. "expression"s have the following form: magnitude {mag_expr} ^power =

Cosh −1 [Cn(ω ) ] Cosh −1(ω ) ω =ω

(3.6) stLp

Eq. (3.5) thus gives the order of the ilter, which is then chosen to be upper nearest integer. • Step 4: Obtaining the lowpass prototype transfer function corresponding to the deined ilter order n above. This transfer function is then normalized, such that H ( j0 ) = 1. Transforming this lowpass transfer function back to the bandpass ilter by using s = s +Bsω0 where s = jω is the Laplace variable. We then obtain an 2n order Chebyshev analog bandpass ilter transfer function 2

H Bp ( s) = H Lp ( s) s= s2 +ω02

2

(3.7)

Bs

• Step 5: Applying the bilinear transformation to obtain required digital ilter transfer function H Bp ( z ) = H Lp ( z ) s= z −1

(3.8)

z +1

where the z variable denotes the z-transform variable of the transfer function. Following this process, converting the digital response to the optical wave length domain can be conducted. • Step 6: From this transfer function, the optical system consists of components that are implementable in iber or integrated optic structures, in particular resonators, interferometers, and others [2].

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Photonic Signal Processing

3.1.3.2 Illustration of a Chebyshev Bandpass Optical Filter For sharper roll-off attenuation at stop band frequencies, one must require a higher order bandpass ilter or more hardware devices in implementation. In this chapter, an example is given for synthesizing a sixth-order Chebyshev bandpass ilter (i.e., a third-order lowpass is designed in the irst step) for an arbitrary center frequency in the useful optical frequency range. In present silica-based optical iber communications, the second and third windows at 1300 and 1550 nm, respectively, are the working spectral regions for optical communications using silica ibers. Supposing that we want to synthesize an optical Chebyshev bandpass ilter with the following speciications: Centre wave-length: Lower cut-off wave length: Upper cut-off wave length: Lower stop band wavelength:

1310 nm 1308 nm 1312 nm 1302 nm

with a passband ripple of 1.0 dB and a −40 dB stopband. It is can be shown easily by using Eqs. 3.1 through 3.9 that for a sixth order Chebyshev bandpass ilter the required transfer function is given by: H BP ( z −1 ) =

−6.792372e − 9(1 − z −2 )3 1 + 3.004723z −2 + 3.009475 z −4 + 1.0004752 z −6

(3.9)

The magnitude and phase responses of the synthesized Chebyshev bandpass ilter, according to the above stringent speciication, is shown in Figure 3.2a and b, and the poles and zeros positions are plotted in the z-plane as shown in Figure 3.2c. This function can be decomposed into a sum of fractions or a multiplier of a number of subsystems, which exhibit only a single root in the numerator or denominator. However, if the roots are complex conjugate then the order of the subsystem

(a)

(b)

(c)

(d)

FIGURE 3.2 Responses of the Chebyshev bandpass ilter (a) and (b) magnitude and phase responses as a function of optical wavelength (c) poles and zeros positions in the z-plane (d) Impulse response at center wavelength of 1550 nm, passband of 0.4 nm and 20 dB roll of with 1 dB ripple in the passband. (a) and (d) are assigned clockwise from top to bottom.

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Bandpass Optical Filters by DSP Techniques

can be quadratic. The partitioning of the transfer function is described in Section 3.1.3.3. In the Section 3.1.3.3, essential optical components for implementing this transfer function are given. 3.1.3.3 Optical Components for Chebyshev Filters Chebyshev ilters can be implemented by using the single pole single zero resonator (SPSZR), which is formed by using a [3 × 3] optical directional coupler with a planar cross section and two optical feedback paths connecting two outputs to two inputs of the coupler. This type of resonator has been described in detail in another article [3], where we outline very briely its main characteristics for the sake of clarity. Further, an all-pole and an all-zero optical circuits (APOC and AZOC), which are required for implementation of the ilters, are described. 3.1.3.3.1 The SPSZR For a planar [3 × 3] optical directional coupler, whose schematic diagram and its signal-low graph are shown in Figure 3.3a and b, respectively, with a direct (or the order of the delay path is zero) shunt feedback from output port 3 to input port 3, the output-input transfer function is given as: E1( d ) = E1(0)

−

1 1 1 j φ +φ j j t1k e φ 1k z −1 − t2 k e φ 2 k + t1k t2 k e ( 1k 2 k ) z −1 2 2 2 1 1 j j (1 − t2 k e φ 2 k )(1 − t1k e φ 1k z −1 ) 2 2

(3.10)

where E1( d ) and E1(0) are the optical ields of the lightwaves at the output and input ports, respectively. The t1k and t2 k are the intensity transmission coeficients of paths 1 and 2, respectively. The φ1k ,φ2 k are the incorporated optical phase modulation in corresponding paths. The coeficients xij ( i , j = 1, 2, 3) in Figure 3.3b are the coupling coeficient of the [3 × 3] coupler matrix. It is assumed that the [3 × 3] coupler has a planar cross-section [3] with a coupling length d and a factor kd = π2 . It can be easily seen that Eq. 3.10 has only one pole and one zero. Therefore, the pole and zero can be independently adjusted by tuning the coeficients tik and _ φik , that is, optical attenuators/ ampliiers or phase modulators placed in-line in the feedback paths. In designing optical Chebyshev ilters, the transfer function (3.10) can provide an arbitrary pair of pole-zero denoted by ( a, b ) given by the roots of its numerator and denominator as: a=

1 t1k e jφ1k 2

or

t1k e jφ1k = 2a

(3.11)

Alternatively, t2 k e jφ2 k = −

ab −1 0.5 − ab −1

(3.12)

with this set of chosen parameters, the transfer function in (3.10) becomes ab −1 (1 − bz −1 ) −1 E1( d ) = 1 − 2ab−1 ab E1(0) (1 − )(1 − az −1 ) 1 − 2ab −1

(3.13)

Again, Eq. 3.13 clearly demonstrates that the resonator would exhibit only one pole and one zero which can be independently adjusted with each other. The gain of the transfer function is dependent on these values of the pole and zero. However, this amplitude gain or loss can be compensated by an in-line optical ampliier.

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Photonic Signal Processing

FIGURE 3.3 The 3 × 3 optical coupler and optical feedback paths as the SPSZR (a) schematic diagram and (b) Graphical signal-low representation.

The circuit of Figure 3.3a can be observed to exhibit only one pole due to the fact that there is only one loop with a single delay line in the graph. According to Mason’s rule the number of poles is the roots of the graph determinant. The graph determinant order is the order of delay of the optical loop. Thus, there is only one delay in the loop and there must be only one pole. The number of zeros of the resonator depends on the number of non-touching loops of the optical circuit. In this case, there are two loops in this resonator and they are non-touching; however, only one unit delay in one loop, thus, ensuring that the order of the numerator is one. It is therefore concluded that this identiication of the double feedback optical resonator leads to implementation of an optical transfer function having a pole and zero pair. We can thus name this type of optical resonance circuit as the single-pole single zero resonator (SPSZR). Since the SPSZR is the core component for designing Chebyshev optical ilters, it is necessary to closely examine the feasibility of a direct optical feedback from the output to the input of the [3 × 3] optical coupler. This type of delay is termed the delay-free feedback and, thus, a delay-free loop is

Bandpass Optical Filters by DSP Techniques

77

formed at the upper part of Figure 3.3a and b [8]. It is stated in Reference [8] that the necessary and suficient condition so the signal-low graph of the structure can be computable for a digital ilter is that there is no delay-free loop. We must make it very clear here that the direct connecting shunt feedback path from the output port 3 to the input port 3 is extremely smaller than the delay of the other loop. Furthermore, the direct connection loop is not operating under resonance at the operating wavelength. Thus, it is reasonable to assume that this loop does not have the same meaning as the delay-free loop deined in Reference [8]. This ensures that our derivation for the transfer function of Eq. 3.10 is valid. In practice, the delay path of the lower loop is much greater than that of the direct loop and that of the coupling length of the 3 × 3 directional coupler. 3.1.3.3.2 The APOC Besides the SPSZR optical circuit described above, there must be optical circuits or components that exhibit pole (or higher order multiple poles) characteristics so that it could form a set of optical components with the SPSZR to simulate the ilter structure. The APOC is in fact an optical resonator using two 2 × 2 optical couplers with an optical feedback from the output of the second coupler to the input of the irst coupler [2]. The schematic diagram of the APOC is given in Figure 3.4. The transmission coeficients are denoted as tip ( i = 1, 2 ) with p and z denoting the all-pole or all-zero, and the phases of each optical paths as γ. The transfer function of the APOC with a irst-order delay in the feedback path can be obtained by using the graphical method given in Chapter 2 as:

H ap (z −1 ) =

(1 − k1 )(1 − k2 )t1z e jϕ1 E7 = E1 1 + t1zt2z k1k2 e j (ϕ21 +ϕ11 ) z −1

(3.14)

Thus the transfer function has a zero at origin and a pole at j φ +φ z = − t1zt2 z k1k2 e ( 1 2 )

(3.15)

This APOC is a quasi-all-pole optical circuit, that is, the zero at the origin can be cancelled by another optical circuit, which would have a inite zero and a pole at the origin such as the AZOC to be considered next.

FIGURE 3.4 Optical resonance loop to obtain a quasi all-pole optical circuit (APOC).

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Photonic Signal Processing

3.1.3.3.3 The AZOC The AZOC is in fact an optical interferometer, which has been studied in detail in [1,2]. The schematic diagram of the AZOC is shown in Figure 3.6. The transmission coeficients are denoted as tip or tiz (i = 1,2) with the subscript z denoting the all-zero type, and the phases of each optical paths as γ ,φ . The transfer functions of the AZOC and its zero position are given in Eqs. 3.16 and 3.17, respectively H az ( z −1 ) =

E7 −1 ( z ) = (1 − k1 )(1 − k2 )t1 e jφ1 − k1k2t2 e jφ2 z −1 E1

(3.16)

which has one pole at the origin, p = 0 and one zero at z=

k1k2t2 e j (φ1 −φ2 ) (1 − k1 )(1 − k2 )t1

(3.17)

Although the transfer function (3.16) of the AZOC contains a pole, it is at the origin in the z-plane. The positions of the zero can be changed to suit the needs for the design of optical systems. It is, thus, a quasi-all-zero optical circuit. It is interesting to note that if an APOC and an AZOC of the same order are cascaded then the overall transfer function exhibit a numerator and denominator of the same pole and zero order because the poles and zeros at the origin cancel each other. This is considered for implementing the Chebyshev ilters in the next section. 3.1.3.4 Realization of the Chebyshev Optical Bandpass Filters In realizing the optical ilters of Chebyshev types, there are two structures that can perform the same iltering functions, namely, the cascaded and the parallel types. The differences between these two types are that one is in tandem and one in parallel combination of each modular optical block. Ideally, each modular optical block must have a single pole and single zero in its transfer function [8]. Two designs for the Chebyshev ilters are considered in this section. One uses a combination of the SPSZRs and the APOC, called Chebyshev Optical Filter-type 1 (COF1), and the other combining the APOCs and AZOCs, called Chebyshev Optical Filter type 2 (COF2). 3.1.3.5 The COF1 3.1.3.5.1 Cascaded form Chebyshev Filters From the transfer function obtained above, the design system should provide the following system of poles and zeros: 3.1.3.5.1.1 Five Zeros at z1 = −1.00567002 + j 0.0000, z2 = −0.99716505 + j 0.0048946, z3 = −0.99761605 − j 0.0048946, z4 = 1.00339252 + j 0.0000, z5 = 0.99830373 + j 0.00294813, z6 = 0.99830373 − j 0.00294813 and

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Bandpass Optical Filters by DSP Techniques

3.1.3.5.1.2 6 Poles at p1 = −0.00000367 + j1.00118581, p2 = −0.00000367 − j1.00118581, p3 = = 0.00231434 + j1.00059005, p4 = 0.00231434 − j1.00059005, p5 = −0.00232166 + j1.00059003, p6 = −0.00232166 − j1.00059003 These poles are quite close to the unit circle, thus the system is marginally stable. However, there are also an equal number of zeros on the unit circle and clustered around the poles. This would generate a stable system as we can observe from the optical response in Figure 3.2a. The poles and zeros of the systems are plotted in the z-plane, as shown in Figure 3.2c. The phase of the systems shown in Figure 3.2b indicate a quasi-linear phase inside the optical passband. The poles and zeros are complex pole pairs. The ilter with the above system of poles and zeros can be implemented by cascading six SPSZR with the chosen parameters shown in Table 3.1. The ilter system thus has the following transfer function: H BP ( z −1 ) =

0.125(1 − z −2 )3 1 + 3.004723z + 3.009475 z −4 + 1.004752 z −6

(3.18)

−2

3.1.3.6 Parallel Realization For parallel realization the transfer function of the Chebyshev bandpass ilter (3.18) can be expressed as H BP ( z −1 ) = −6.7923e − 9 +

8.69874e − 9(1 − 272.8e3z −1 ) (1 − p1z −1 )(1 − p2 z −1 )

(3.19)

−3.00551e − 4(1 + 3.9499 z −1 ) 3.00551e − 4(1 + 3.9477 z −1 ) + + (1 − p3 z −1 )(1 − p4 z −1 ) (1 − p5 z −1 )(1 − p6 z −1 ) The system can be implemented by a parallel realization as shown in Figure 3.5. The sub-systems H1, H 2 , H 3 and H 4 are implemented as followed:

TABLE 3.1 Chosen Parameters for Chebyshev Bandpass Optical Filter SPSZR1 SPSZR2 SPSZR3 SPSZR4 SPSZR5 SPSZR6

t11 = 4.009492 t12 = 4.009492 t13 = 4.004721 t14 = 4.004721 t15 = 4.004743 t16 = 4.004743

t21 = 0.800381366 t22 = 0.800381366 t23 = 0.798711128 t24 = 0.798711128 t25 = 0.801677094 t26 = 0.801677094

ϕ11 = 1.570799992 ϕ12 = −1.570799992 ϕ13 = 1.568483356 ϕ14 = −1.568483356 ϕ14 = 1.573116614 ϕ14 = −1.573116614

ϕ21 = 0.463174471 ϕ22 = 0.463174471 ϕ23 = 0.462948918 ϕ24 = 0.462948918 ϕ25 = 0.463873612 ϕ26 = 0.463873612

p = p1 p = p2 p = p3 p = p4 p = p5 p = p6

z = −1 z = 1 z = −1 z = 1 z = −1 z = 1

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Photonic Signal Processing

FIGURE 3.5

Hardware implementation diagram of Chebyshev ilter.

3.1.3.6.1 Sub-system H1 The sub-system H1 is simply implemented by cascading one optical device with gain 6.7923e-9, and an optical phase modulator with a phase of π. H1 contributes to the system gain factor H1 = −6.7923e-9. 3.1.3.6.2 Sub-system H2 This sub-system H 2 should provide a system of poles and zeros as follows: 2 poles at = p p= p2 1; p and 2 zeros z1 = 272.8e +3 and z2 = 0. This sub-system can be implemented by cascading a SPSZR and an APOC. The chosen parameters for these elements are shown in Table 3.2. The two optical components in Table 3.2 are cascaded with an optical device with gain 0.0122456e-06 and an optical phase modular with phase −0.7848 rads. 3.1.3.6.3 Sub-system H3 This sub-system H3 should provide a system of poles and zeros as follows: 2 poles at p = p3 and p = p4 , and 2 zeros at z = 3.9499 and z = 0. This sub-system can be implemented by cascading a SPSZR and an APOC. The chosen parameters for these elements are shown in Table 3.3. The two optical devices above are cascaded with an optical device with gain 1.254e-04 and an optical phase modular with phase −0.37841 rads. 3.1.3.6.4 Sub-system H 4 This sub-system H 4 should provide a system of poles and zeros as follows: 2 poles at p = p5 and p = p6, and 2 zeros at z = −3.9477 and z = 0. This sub-system can be implemented by cascading

TABLE 3.2 Chosen Parameters for Subsystem H2 SPSZR1 APOC1

t11 = 4.009492 k= k= 0.5 11 21

t21 = 0.0135526

t11 = t21 = 2.00237

φ 11 = 1.570799999

φ 21 = 4.712404286

φ 11 = 0

φ 21 = −1.57079999

ϕ12 = 1.568483356 ϕ12 = 0

ϕ22 = 1.099537311 ϕ22 = −1.568483356

p = p1

p = p2

z = 272.8e03 z = 0

TABLE 3.3 Chosen Parameters for Subsystem H3 SPSZR2 APOC2

t12 = 4.004721 k12 = k22 = 0.5

t22 = 0.2009535 t12 = t22 = 2.00118

p = p3 p = p4

z = −3.9499 z = 0

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Bandpass Optical Filters by DSP Techniques

TABLE 3.4 Chosen Parameters for Subsystem H4 SPSZR3 APOC3

t13 = 4.004743 k= k= 0.5 13 23

t23 = 0.1543989

t13 = t23 = 2.0005926

φ 13 = 1.57311662

φ 23 = 1.205039363

φ 13 = 0

φ 23 = −1.57311662

p = p5 p = p6

z = −3.9477 z = 0

a SPSZR and an AZOC as described in Reference [1]. The chosen parameters for these elements are shown in Table 3.4. The two elements above are cascaded with an optical device with gain 1.152e-04 and an optical phase modular with a phase −0.379212. 3.1.3.7 The COF2 3.1.3.7.1 Cascading Realization The Chebyshev bandpass transfer function with the numbers of poles and zeros given in Section 4.1.1 above can be realized in the cascade coniguration from the combination of all-poles and allzeros basic elements. The overall system transfer function of Eq. (3.18) can be expressed in the more general form that separates the factors containing the poles and zeros as: m

H BP ( z ) =

∏H

azi

( z −1 ) H api ( z −1 )

(3.20)

i =1

where each pole and each zero are realized from one basic element of the AZOC or APOC. The functions H azi ( z −1 ) and H api ( z −1 ) are the irst order function of z−1 numerator and denominator. The schematic diagram showing the cascade realization for this transfer function is shown in Figure 3.6.

FIGURE 3.6 Schematic diagram showing Tandem all-pole and all-zero subsystems using integrated photonic circuits.

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Photonic Signal Processing

TABLE 3.5 Chosen System Parameters for Cascade Realization Subsystem

Coupling Coeficient

Transmission Coeficient

Delay Order

Poles or Zeros

H az1

b11 = b21 = 0.5

t11 = t21 = 1

Φ11 = 0

Phase Shift Modulator Φ21 = 0

2

z = ±1

H az2

b12 = b22 = 0.5

t12 = t22 = 1

Φ12 = 0

Φ22 = 0

2

z = ±1

H az3

b13 = b21 = 0.5

t13 = t23 = 1

Φ13 = 0

Φ23 = 0

2

z = ±1

H ap1

a11 = a21 = 0.5

Φ11 = 0

Φ21 = −1.5684760

1

p1

H ap2

a12 = a22 = 0.5

Φ12 = 0

Φ22 = 1.56847603

1

p2

H ap3

a13 = a23 = 0.5

Φ13 = 0

Φ23 = −1.5731092

1

p3

H ap4

a14 = a24 = 0.5

Φ14 = 0

Φ24 = 1.57310929

1

p4

H ap5

a15 = a25 = 0.5

Φ15 = 0

Φ25 = −1.5707926

1

p5

H ap6

a16 = a26 = 0.5

t11 = 1; t21 = 4.004743245 t12 = 1 t22 = 4.004743245 t13 = 1; t23 = 4.004743245 t14 = 1; t24 = 4.004743245 t15 = 1; t25 = 4.009492105 t16 = 1; t26 = 4.009492105

Φ16 = 0

Φ26 = 1.57079266

1

p6

It is noted that if the poles or zeros appear in conjugate pairs of either real or imaginary, then the numbers of subsystems can be reduced by using more delay coeficients (d ≥ 2) for higher order subsystem realization. From Eqs. 3.13 and 3.15, we can obtain the required parameters with some arbitrary chosen values for each AZOC and APOC subsystems in the cascaded coniguration in Figure 3.6. They are showed in Table 3.4 with parameters X pq that indicate p couplers and q subsystems. It is noted that the accuracy of parameters in the table is necessary and should not be rounded off. Additional ampliier gain may also be required to achieve a unity gain response (Table 3.5). 3.1.3.7.2 Parallel Realization Using the transfer function for the Chebyshev bandpass ilter given by (3.19), the system can be implemented by a parallel realization as shown in Figure 3.5. This structures for the sub-systems H1 to H 4 are given as follows: 3.1.3.7.2.1 Sub-system H1 Sub-system H1 is simply implemented by cascading one optical device with gain 6.7923e-9, and an optical phase modulator with phase biased at π. H1 contributes to the system H1 = −6.7923e−9. 3.1.3.7.2.2 Sub-system H2 This sub-system H 2 should provide a system of poles and zeros of 2 poles at p = p1 and p = p2 and 2 zeros at z = 272.8e03 and z = 0. This sub-system can be implemented by cascading two APOCs and one AZOC with the chosen parameters shown in the Table 3.6. An optical device with gain 1.5011e+6 is required to cascade with these optical components. 3.1.3.7.2.3 Sub-system H3 This sub-system H 3 should provide a system of poles and zeros as follows: 2 poles at p = p3 and p = p4, and 2 zeros at z = −3.9499 and z = 0. This sub-system can be implemented by cascading two APOCs and one AZOC with the chosen parameters shown in Table 3.7. These elements are cascaded with an optical device with gain 1.7882e-3 and an optical phase modulator with phase π.

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Bandpass Optical Filters by DSP Techniques

TABLE 3.6 Chosen Parameters for Subsytem H2 H az11

b11 = b21 = 0.9996335

t21 = 1000; t11 = 0.1

ϕ11 = 0

ϕ21 = 0

z = 272.8e3

H ap11

a11 = a21 = 0.5

t11 = t21 = 2.00237

ϕ11 = 0

ϕ21 = 1.57079999

p = p1

H ap21

a12 = a22 = 0.5

t12 = t22 = 2.00237

ϕ12 = 0

ϕ22 = −1.57079999

p = p2

ϕ21 = 0

z = −3.9499

TABLE 3.7 Chosen Parameters for Subsytem H 3 H az12

b11 = b21 = 0.66385

t21 = 4; t11 = 1

ϕ11 = π

H ap12

a11 = a21 = 0.5

t11 = t21 = 2.00237

ϕ11 = 0

ϕ21 = 1.56848336

p = p3

H ap22

a12 = a22 = 0.5

t12 = t22 = 2.00118

ϕ12 = 0

ϕ22 = −1.56848336

p = p4

TABLE 3.8 Chosen Parameters for Subsystem H4 H az13

b11 = b21 = 0.663735

t21 = 4; t11 = 1

ϕ11 = π

ϕ21 = 0

z = −3.9477

H ap13

a11 = a21 = 0.5

t11 = t21 = 2.0005926

ϕ11 = 0

ϕ21 = 1.57311662

p = p5

H ap23

a12 = a22 = 0.5

t12 = t22 = 2.0005926

ϕ12 = 0

ϕ22 = −1.57311662

p = p6

3.1.3.7.2.4 Sub-system H 4 Similarly, the sub-system H 4 should provide a system of poles and zeros as follows: 2 poles at p = p5 and p = p6, 2 zeros at z = −3.9477 and z = 0. Again this subsystem can be implemented by cascading 2 all-pole and 1 all-zero subsystems with the chosen parameters shown in Table 3.8. These elements are to be cascaded with an optical device with gain 1.787841e-3. 3.1.3.8 Discussions Although the use of the 3  ×  3  optical directional coupler to form the SPSZR would reduce the required number of the couplers for structuring the ilters type COF1, the direct feedback of the optical path delay restrict the operation frequency of the ilter far below the resonance frequency range of the delay-free loop. On the other hand, the COF2 employs the all-poles and all-zeros the optical circuits allowing for better stability, as well as a much wider range of operating frequency or much narrow range of the optical bandpass ilters. These ilters can be implemented in optical iber or integrated optical coniguration. A much better technology would be the use of in-line iber gratings in forward or relection modes. We are currently implementing a number of optical ilters, such the Butterworth or Chebyshev types, using UV written iber gratings or photorefractive gratings in lithium niobate diffused optical waveguides.

3.1.4

cOncluDing remarks

We have demonstrated the synthesizing process for a narrow Chebyshev bandpass ilter according to a given set of speciications. In practice, the hardware implementations of these kind of ilters should be very accurate due to the effects of operational parameters of optical components and the phase modulators used in the network. Because the ilters have a very narrow bandpass centered at a very high frequency region, they are highly sensitive to minute luctuation of these optical elements.

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Photonic Signal Processing

Care should be considered with these ilters implementations. On the other hand, this ilter can be used as a sensitive optical sensor. The employment of the SPSZRs as well as the two quasi all-pole and all-zero optical circuits has been proven to be compact for the implementation of the ilters and the reduction of the required number of the optical couplers. The quasi-all-pole and all-zero optical circuits are also used and integrated into the optical systems for signal processing. We believe that these optical components described in this chapter would ind many applications in optical communications and signal processing networks. However, it is believed that the equalization of optical signals in high-speed data transmission can be implemented using iber-optic resonators with some speciic coniguration to be feasible in the near future [9].

3.2

TUNABLE OPTICAL BANDPASS WAVEGUIDE FILTERS

This section presents a highly versatile synthesis method for the design of a variety of tunable optical waveguide ilters with independently variable bandwidths and tunable center frequencies and arbitrary ininite impulse response (IIR) characteristics. The synthesized Mth-order tunable optical ilter consists of the concatenation of M all-pole ilters (APFs) with M all-zero ilters (AZFs). The bandwidth and center frequency of the designed tunable optical ilter can be independently tuned by applying electric power to thin-ilm heaters loaded on the waveguides of the both APFs and AZFs. One unique advantage of the proposed synthesis technique is that the poles and zeros of the ilter can be adjusted independently of each other to enable the design of tunable optical ilters with arbitrary IIR characteristics. By means of computer simulation, the effectiveness of the synthesis method is demonstrated with the design of the second-order Butterworth bandpass and bandstop tunable optical ilters with variable bandwidths and tunable center frequencies. To study the effects of fabrication tolerances on the ilter performances, the maximum allowable deviations of the ilter parameters from their designed values are also determined. The proposed synthesis method is general and lexible enough to enable the design of a variety of tunable optical ilters with arbitrary IIR characteristics, which include the Chebyshev and elliptic ilter types.

3.2.1

intrODuctOry remarks

Optical ilters with certain spectral characteristics are important devices in a wide variety of applications. For example, in wavelength division multiplexing (WDM) networks, optical ilters have been used as optical demultiplexers, optical add-drop multiplexers, optical dispersion compensators, and optical gain equalizers. Optical ilters are normally designed using complex electromagnetic models, where the ields are determined in the frequency or time domain in a cumbersome manner. As a simpler alternative to ilter design, we have previously proposed the use of digital signal processing technique for the design or synthesis of ixed or non-tunable optical ilters using interferometers and ring resonators as the basic building blocks [10]. Optical ilter design based on the digital signal processing techniques provide unique advantages, such as systematic and lexible approach, scalable to larger and more complex ilters, and great insight into the properties of the ilters. This is because the signal processing techniques that have been well-established over the last several decades can provide a readily available mathematical framework for the design of complex optical ilters. In fact, in recent years, there has been an increasing number of researchers adopting the digital signal processing techniques for the design and analysis of optical ilters, for example, see references [11–13]. In analog to digital ilters [5,14], there are two types of optical ilter architectures, namely, inite impulse response (FIR) and ininite impulse response (IIR) [10]. FIR ilters have only feed-forward paths (i.e., no loops) and are also known as transversal, tapped delay-line, and non-recursive ilters. IIR ilters have feedback paths (or loops) and are also known as recirculating delay-line and recursive ilters. It is well-known in the ield of digital signal processing that the

85

Bandpass Optical Filters by DSP Techniques

IIR ilters, having the same ilter order (or complexity) as that of the FIR ilters, can provide latter passband, sharper roll-off, and greater stopband rejection than the FIR ilters due to the feedback effect [14]. From a practical implementation point of view, it is therefore more advantageous to design (or synthesize) optical ilters based on the IIR topology than the FIR topology. However, the IIR ilters having a rational transfer function are generally more dificult to design than the FIR ilters that only have a polynomial transfer function. Although we have previously employed the digital signal processing technique to design the non-tunable optical ilters with arbitrary IIR characteristics [10], the capability of this technique to enable both the bandwidth and center frequency of the ilter to be independently tuned has not yet been demonstrated. Tunable optical ilters are mostly based on iber Fabry-Perot (FP) interferometer [15], waveguide ring resonator [16], and iber grating [17]. Tunable optical ilters are also important in a wide variety of applications. In a frequency-division multiple access (FDMA) network, a tunable optical ilter is required as an optical demultiplexer at each receiver to select one or more desired frequency channels (or users) at any optical frequency [16]. In a reconigurable WDM network, tunable optical ilters are required to provide dynamic selection of the wavelength channels [18]. In an optical sensor system, a tunable optical ilter is needed to provide interrogation of several iber grating sensors on a single strain of optical iber [19]. In a tunable iber laser system, a tunable optical ilter can be used to select the resonance wavelength in the laser cavity to extract the desired output lasing wavelength [20]. In a soliton transmission system, a tunable optical ilter can be used as a slidingfrequency guiding ilter to reduce the ampliied noise-induced timing jitter [21]. A tunable optical ilter can also be used for spectral iltering in the generation of ultrashort pulses [13,22]. It is noted that, in these tunable optical ilters, only the center frequencies (but not the bandwidths) of the ilters can be tuned. That is, both the bandwidths and center frequencies of these tunable optical ilters cannot be simultaneously tuned. There was no previous report of a synthesis method that can be used for the design of a variety of tunable optical ilters with variable bandwidths and tunable center frequencies and arbitrary IIR characteristics. In this section, an effective and versatile synthesis method based on the digital signal processing technique is proposed for the design of tunable optical waveguide ilters with variable bandwidths and tunable center frequencies and arbitrary IIR characteristics. Section 3.2.2 describes the transfer function of an IIR digital ilter to be synthesized using optical waveguide components. Section  3.2.3 presents the basic building blocks, namely, APF and the AZF of the tunable optical ilter. Section  3.2.4 describes the proposed synthesis method of the tunable optical ilters using the digital signal processing technique, and the design of second-order Butterworth bandpass and bandstop tunable optical ilters. Concluding remarks are given in Sections 3.2.5 and 3.2.6.

3.2.2

transFer FunctiOn OF iir Digital Filters tO be synthesizeD

For clarity, variables with a cap (e.g., Hˆ ( z )) are associated with digital ilters while the corresponding variables without a cap (e.g., H ( z )) are associated with optical ilters. The transfer function of an Mth-order IIR digital ilter to be synthesized can be expressed in a rational form as Hˆ ( z ) = Aˆ

M

( z − zˆ k )

∏ ( z − pˆ ) k =1

k

= Aˆ

( z −zˆ1 ) ( z −zˆ 2 ) ( z − pˆ1 ) ( z − pˆ2 )

⋯ ( z − zˆM ) ⋯ ( z − pˆM )

(3.21)

where Aɵ is a constant and z is the z-transform parameter [5]. Furthermore, pɵk and zɵk are the kth pole and kth zero in the z-plane and they can be expressed in phasor forms as

(

)

pɵk = pɵk e jarg( pɵk ), 0 ≤ pɵk < 1

(3.22)

86

Photonic Signal Processing

zɵk = zɵk e jarg(zɵ ) k

(3.23)

where j = −1 and arg denotes the argument of a complex number. The ilter stability requires all the poles to be located inside the unit circle as described by the condition given in Eq. 3.22. It can be seen from Eq. 3.21 that the Mth-order IIR digital ilter simply consists of the concatenation of M all-pole ilters (with each ilter transfer function given by 1 ( z − pɵk )) with M all-zero ilters (with each ilter transfer function given by ( z − zɵ k )). Hence, an Mth-order IIR optical ilter with a transfer function in the form of Eq. 3.21 can be synthesized from its digital counterpart (i.e., the Mth-order IIR digital ilter). This is presented in Section 3.2.4.

3.2.3

basic builDing blOcks OF tunable Optical Filters

This section describes the basic building blocks, namely, the irst-order all-pole optical ilter (FOAPOF) and the irst-order all-zero optical ilter (FOAZOF) of the tunable optical ilter. 3.2.3.1 Tunable Coupler In this section, we describe the characteristics of a tunable coupler (TC), which is a fundamental component in the APF as described in Section 3.2.3.2. A TC is also one of the fundamental components in many integrated optical circuits because it can provide tuning or switching functions. The coupling coeficient of the TC can be varied from zero to unity, a feature which is not available with a directional coupler (DC) with a ixed coupling coeficient, and can therefore provide great lexibility in the design of integrated optical circuits. The kth-stage TC is shown schematically on the left side of Figure 3.7. It is a symmetrical Mach−Zehnder interferometer, which consists of two identical directional couplers (DCs) (with each having an intensity cross-coupling coeficient of dk ) interconnected by two waveguide arms of equal length, and a phase shifter (PS) (with a phase shift of ϕk ) loaded on one of the arms. The PS, which is used to tune the TC’s coupling coeficient to a desired value as shown below, is a thin-ilm heater loaded on the waveguide and utilizes the thermo-optic effect to change the phase of the optical carrier. When an electric voltage is applied to the thin-ilm heater, the optical path length of the heated waveguide would change because of the temperature dependence of the refractive index. For instance, a change in the optical path length of the heated waveguide by 1.55 µm will correspond to a change in the phase of a 1.55 µm optical carrier by 2π . The TC is stable against temperature variation because it is the temperature difference between

FIGURE 3.7 Schematic diagram of the kth-stage all-pole ilter (APF). The left side of the igure shows the schematic of the tunable coupler (TC).

87

Bandpass Optical Filters by DSP Techniques

the two waveguide arms, not the absolute temperature of each arm, that is important for tuning or switching operation. The transfer matrix of the kth TC (in the electric-ield amplitude domain) can be shown to be given by  1 − ak e jθ13,k  E3  −α w L24 ,k − jωT24 ,k = e e  E   ak e jθ14 ,k  4

ak e jθ23,k   E1    , 1 − ak e jθ24 ,k   E2 

(3.24)

where: ak = 2dk (1 − dk )(1 + cos ϕk ),

(3.25)

0 ≤ ak ≤ 4dk (1 − dk ) or 0 ≤ ak ≤ 1 for dk = 0.5,

(3.26)

  ak − 1 or ϕk = cos −1(2ak − 1) for dk = 0.5, (0 ≤ ϕk ≤ π ) ϕk = cos −1   2dk (1 − dk ) 

(3.27)

  sin ϕk θ13,k = tan −1   , ( − π 2 ≤ θ13,k ≤ π 2 ) , ϕ − d − d 1 cos ( ) k k k  

(3.28)

 1 + cos ϕk  θ14,k = θ 23,k = − tan −1   , ( − π 2 ≤ θ14,k ,θ 23,k ≤ π 2 ) ,  sin ϕk 

(3.29)

  sin ϕk θ 24,k = tan −1   , ( − π 2 ≤ θ 24,k ≤ π 2 ) , ϕ − − d d 1 cos ( ) k k k  

(3.30)

The variables in Eqs. 3.4 through 3.10 are deined as follows. ( E1, E2 ) and ( E3 , E4 ) are the electricield amplitudes at the input ports (1, 2) and output ports (3, 4) of the TC, respectively. e −α w L24,k is the propagation loss of the TC, where α w is the amplitude waveguide loss and L24,k (whose corresponding time delay is T24,k ) is the waveguide length of the TC from the input port 2 to the output port 4. It is valid to assume that L= L= L23,k = L24,k (and hence T= T= T23,k = T24,k ) are due to 13,k 14 ,k 13,k 14 ,k the small dimensions of the waveguide lengths. e − jωT24,k is the propagation delay of the TC, where ω is the relative optical frequency (radial) and T= T= T23,k = T24,k is the time delay of the TC. 13,k 14 ,k ak is the intensity cross-coupling coeficient of the TC (i.e., from port 1 to port 4 or from port 2 to port 3). From Eqs. 3.5 and 3.6, the maximum tuning range of the TC (i.e., 0 ≤ ak ≤ 1) can be achieved by using 3-dB DCs (i.e., dk = 0.5 ) and by tuning the required phase of the phase shifter in the range of 0 ≤ ϕk ≤ π according to Eq. 3.7. For this reason, dk = 0.5 is chosen for ilter synthesis as described in Section 3.2.4. In Eqs. 3.8 through 3.10, θ13,k , θ14,k = θ 23,k , and θ 24,k are the output phases of the TC. It is noted that θ13,k = θ14,k = θ 23,k = θ 24,k when dk = 0.5. This implies that the TC is a symmetrical and reciprocal device. 3.2.3.2 All-Pole Filter Figure 3.7 shows the schematic diagram of the kth-stage all-pole ilter (APF). It consists of a waveguide loop interconnected by a TC (as described in Section 3.2.3.1 above) and a DC with an intensity cross-coupling coupling coeficient of ck. A thermo-optic phase shifter (PS) (as described in Section 3.2.3.1 above) with a phase shift of φk is loaded on the lower section of the loop, and an amorphous-silicon (a-Si) ilm is loaded on the upper section of the loop. The TC and DC are used to provide the required amplitude (or magnitude) of the ilter pole, and the PS to provide the required phase (or angle) of the ilter pole. By laser trimming the a-Si ilm, the stress-induced waveguide

88

Photonic Signal Processing

birefringence can be eliminated [14]. Using the signal-low graph method [1], the transfer function of the kth-stage APF (from the input port 1 to the output port 8) is simply given by H ap,k (ω ) =

forward path gain 1 − loop gain

(3.31)

where the subscript ap denotes all pole. The forward path gain is given by the product of all transmittances associated with a forward path that connects the input port (or node) {1} and the output port (or node) {8} through nodes {4} and {5}. The loop gain is given by the product of all transmittances associated with a feedback loop that connects nodes {2} → {4} → {5} → {7} → {2}. Using the TC transfer matrix as deined in Eq. 3.4, the forward path gain is given by forward path gain ({1} → {4} → {5} → {8}) = e −α w L14 ,k × e − jωT14 ,k × ak e jθ14 ,k  × e −α w L45,k × e − jωT45,k × e jϕk 

(3.32)

× e −α w L58,k × e − jωT58,k × ck e − jπ 2  where the irst term in brackets is the transmittance of path {1} → {4}, the second term in brackets is the transmittance of path {4} → {5}, and the third term in brackets is the transmittance of path {5} → {8}. Similarly, the loop gain is given by loop gain ({2} → {4} → {5} → {7} → {2}) = e −α w L24 ,k × e − jωT24 ,k × 1 − ak e jθ24 ,k  × e −α w L45,k × e − jωT45,k × e jϕk 

(3.33)

× e −α w L57,k × e − jωT57,k × 1 − ck  × e −α w L72,k × e − jωT72,k  It is useful to deine the loop length L, its corresponding loop delay T, and the z-transform parameter z as L = L24,k + L45,k + L57,k + L72,k ,

(3.34)

T = T24,k + T45,k + T57,k + T72,k ,

(3.35)

z = e jωT .

(3.36)

Inserting Eqs. 3.32 through 3.36 into Eq. 3.21 and using L14,k = L24,k and L58,k = L57,k, the z-transform transfer function of the kth-stage APF becomes

H ap,k ( z ) =

Aap,k e j (θ23,k +φk −π 2)e − jω (T24 ,k +T45,k +T57,k ) z , z − pk

(3.37)

where the amplitude Aap,k and the pole location pk in the z-plane are given by Aap,k = ak ck e −α w ( L24 ,k + L45,k + L57,k ) ,

(3.38)

pk = (1 − ak )(1 − ck ) e −α w Le j (θ24 ,k +φk ) .

(3.39)

Bandpass Optical Filters by DSP Techniques

89

where e −α w L is the transmission factor of the loop. It is useful to express Eq. 3.39 in the phasor form as pk = pk e jarg( pk ) , ( 0 ≤ pk < 1) ,

(3.40)

pk = (1 − ak )(1 − ck ) e −α w L ,

(3.41)

arg( pk ) = θ 24,k + φk .

(3.42)

where:

3.2.3.3 All-Zero Filter Figure 3.8 shows the schematic diagram of the kth-stage all-zero ilter (AZF). It is an asymmetrical Mach−Zehnder interferometer, which consists of one tunable coupler (TC) (as described in Section 3.1.3.3) and one directional coupler (DC) (with an intensity cross-coupling coeficient of  bk ) interconnected by two waveguide arms with a differential time delay of T. The TC and DC are used to provide the required amplitude (or magnitude) of the ilter zero. The thermooptic phase shifter (PS) (as described in Section 3.1.3.3 above) with a phase shift of Ψ k loaded on the right arm is used to provide the required phase (or angle) of the ilter zero. The a-Si ilm (as described in Section 3.2.3.2 above) is used to eliminate any polarization dependence of the AZF. Note that the TC uses two 3-dB DCs because, according to Eqs. 3.24 and 3.25, this will allow the TC to have a maximum tuning range of the intensity cross-coupling coeficient, gk (i.e., 0 ≤ gk ≤ 1), which is given by gk = (1 + cos Φ k ) 2 ; ( 0 ≤ gk ≤ 1)

FIGURE 3.8 Schematic diagram of the kth-stage all-zero ilter (AZF).

(3.43)

90

Photonic Signal Processing

From Eq. 3.43, the phase shift, Φ k , of the phase shifter (PS) loaded on one arm of the TC is given by Φ k = cos −1(2 gk − 1); ( 0 ≤ Φ k ≤ π )

(3.44)

From Eq. 3.39, the output phase of the TC is given by  sin Φ k  ξ k = tan −1   , ( − π 2 ≤ ξk ≤ π 2)  cos Φ k − 1 

(3.45)

Using the signal-low graph method [8], the transfer function of the kth-stage AZF (from the input port to the output port) is simply given by H az ,k (ω ) = e −α w L3,k  × e − jωT3,k  ×  1 − gk e jξk  ×  1 − bk  + e −α w L4 ,k  × e − jωT4 ,k  ×  gk e jξk  × e jΨ k  ×  j bk 

(3.46)

where 1− gk e jξk is the direct-coupling coeficient of the TC, gk e jξk is the cross-coupling coeficient of the TC, az denotes all zero, L3,k is the length of the left waveguide arm from the input to the output port, and L4,k is the length of the right waveguide arm from the input to the output port. It is useful to deine the differential length L and its corresponding differential delay T as L = L4,k − L3,k

(3.47)

T = T4,k − T3,k .

(3.48)

Substituting Eqs. 3.35, 3.36, and 3.47, into Eq. 3.45, the z-transform transfer function of the kth-stage AZF becomes H az ,k ( z ) = Aaz ,k e − jωT3,k z −1( z − zk )

(3.49)

where the amplitude and the zero location in the z-plane are given by Aaz ,k = [(1 − gk )(1 − bk ) ] e −α w L3,k 12

(3.50)

12

 −α w L j (Ψ k +3π 2)  gk bk e zk =  .  e g b ( 1 )( 1 ) − − k k  

(3.51)

It is useful to express Eqs. 3.50 and 3.51 in the phasor form as zk = zk e jarg( zk )

(3.52)

where 12

 −α w L  gk bk zk =   e ( 1 )( 1 ) g b − − k k  

(3.53)

arg( zk ) = Ψ k + 3π 2 .

(3.54)

91

Bandpass Optical Filters by DSP Techniques

3.2.4

tunable Optical Filter

The z-transform transfer function of the Mth-order tunable optical ilter, which is the transfer function of the concatenation of M APFs with M AZFs, is given by  H ( z) =  

M

∏ k =1

  H ap,k ( z )  ×   

M

∏H

az ,k

k =1

 ( z) . 

(3.55)

Substituting Eq. 3.36 and 3.38 into Eq. 3.54, the z-transform transfer function of the Mth-order tunable optical ilter becomes

H ( z) = e

M M  Mπ j  ∑ (θ23,k +ϕk +ξ k )− −ω ∑ (T24 ,k +T45 ,k +T57 ,k +T3,k )  2 k =1  k =1 

( z − zk )   ∏ ( z − pk )  k =1

 × A 

M

(3.56)

where the amplitude A is given by 12

M

A=e

− Mα w L

×

∏[a c (1 − g )(1 − b )] k k

k

k

.

(3.57)

k =1

It should be noted in Eq. 3.56 that e −α w L = e −α w ( L24 ,k + L45,k + L57,k + L3,k ) and L72,k = L3,k have been assumed for analytical simplicity. It can be seen from Eq. 3.55 that the Mth-order tunable optical ilter has the unique advantage in that its poles and zeros can be adjusted independently of each other, enabling a particular pole-zero pattern to be easily obtained to allow the design of a variety of ilters with arbitrary IIR characteristics.

3.2.5

synthesis OF tunable Optical Filters

3.2.5.1 Design Equations for the Synthesis of Tunable Optical Filters For analytical simplicity, the exponential factor in Eq. 3.55, which represents a linear phase term, is neglected here because it has no effect on the magnitude response of the ilter. The synthesis of a tunable optical ilter from the characteristics of a digital ilter requires the second factor of Eq. 3.56 to be equal to Eq. 3.21 such that the following equations hold A = Aɵ

(3.58)

pk = pɵk

(3.59)

zk = zɵk .

(3.60)

Substituting Eq. 3.22 and 3.40 through 3.42 into Eq. 3.59 results in 2

pɵk ak = 1 − , (1 − ck )e −2α w L arg( pɵk ) − θ 24,k ,  φk =  arg( pɵk ) − θ 24,k + 2π ,

pɵk < 1 − ck e −α w L arg( pɵk ) − θ 24,k ≥ 0 arg( pɵk ) − θ 24,k < 0

(3.61)

(3.62)

92

Photonic Signal Processing

It is clear from the condition (i.e., pɵk < 1 − ck e −α w L ) given in Eq. 3.61 that the value ck (which can be ixed at the design stage) of the directional coupler (DC) must be designed to be as small as possible and that the transmission factor of the loop, e −α w L , of the all-pole ilter must be designed to be as large as possible (by having a small waveguide propagation loss α w ) in order to achieve the maximum allowable value of the digital ilter pole, pɵk , and hence the maximum tuning range of the ilter bandwidth. This can be achieved by varying the coupling coeficient ak of the tunable coupler (TC) (see Eq. 3.61), which requires tuning of the phase shift φk (see Eq. 3.27) of the TC. It is noted that, for practical purposes, a full-cycle phase shift of 2π has been added, without affecting the ilter performance, to the second equation of Eq. 3.62 so that φk takes a positive value (i.e., 0 ≤ φk ≤ 2π ), which can be realized by applying a positive voltage to the thin ilm heater. Substituting Eqs. 3.23 and 3.52 through 3.54 into Eq. 3.60 results in gk =

1  bk  1+   zɵk  1 − bk 

−2

 arg( zɵk ) − 3π 2 , Ψk =  arg( zɵk ) − 3π 2 + 2π ,

; 0 < gk ≤ 1

(3.63)

e −2α w L arg( zɵk ) − 3π 2 ≥ 0

(3.64)

arg( zɵk ) − 3π 2 < 0

The ilter bandwidth can be varied by changing the magnitude of the ilter pole(s) pɵk , and this can be achieved by varying the coupling coeficient of the TC, ak (see Eq. 3.61 and hence tuning the phase shift of the phase shifter ϕk (see Eq. 3.27 of the TC. It can be seen from the condition given in Eq. 3.61 that the maximum value of the ilter pole and hence the maximum achievable ilter bandwidth is limited by the transmission factor of the loop e −α w L . To change the center frequency of the ilter without affecting its bandwidth, a phase shift of δ 0 (0 < δ 0 < 2π ) must be added to Eqs. 3.62 and 3.64 (i.e., to the phase shifters of both the APFs and the AZFs) to give arg( pɵk ) − θ 24,k + δ 0 ,  φk =  arg( pɵk ) − θ 24,k + δ 0 + 2π , arg( zɵk ) − 3π 2 + δ 0 ,  Ψk =  arg( zɵk ) − 3π 2 + δ 0 + 2π ,

arg( pɵk ) − θ 24,k + δ 0 ≥ 0

(3.65)

arg( pɵk ) − θ 24,k + δ 0 < 0 arg( zɵk ) − 3π 2 + δ 0 ≥ 0

(3.66)

arg( zɵk ) − 3π 2 + δ 0 < 0

In summary, the design equations for the synthesis of the tunable optical ilter are Eqs. 3.27, 3.30, 3.57, and 3.61 through 3.66. 3.2.5.2

Synthesis of Second-Order Butterworth Bandpass and Bandstop Tunable Optical Filters To demonstrate the effectiveness of the design method, this section describes a practical design example of the second-order (M = 2) Butterworth bandpass and bandstop tunable optical ilters with variable bandwidths and tunable center frequencies characteristics. 3.2.5.3 Designed Parameter Values of the Bandpass and Bandstop Tunable Optical Filters This section presents a practical design example of the tunable optical ilters with variable bandwidths and tunable center frequencies using the proposed digital ilter synthesis method. The propagation loss of the waveguide is assumed to be typically 0.1 dB cm and this gives α w = 0.1 dB cm 8.686 = 0.01151 cm −1. As a case study, the loop length L is assumed to be L = 4 cm,

93

Bandpass Optical Filters by DSP Techniques

which results in the loop delay of T = 200 ps assuming the effective refractive index of the waveguide is about 1.5, and hence the free spectral range (FSR) of the ilter is FSR = 1= T 5 GHz . Using α w = 0.01151 cm −1, and L = 4 cm, e −α w L = 0.955 . By choosing c = 0.25 and using e −α w L = 0.955 , the coupling coeficient of the TC, ak , in the APF deined in Eq. 3.41 becomes ak = 1 − 1.462 pɵk , pɵk < 0.827

(3.67)

ϕk = cos −1(2ak − 1) ( 0 ≤ ϕk ≤ π )

(3.68)

 sin ϕk  θ k = tan −1   . ( − π 2 ≤ θk ≤ π 2)  cos ϕk − 1 

(3.69)

2

The value ck = 0.25 is chosen here so that the pole value is pɵk < 0.827, which is adequate to provide a suficient tuning range of the ilter bandwidth as will be shown later. As described in Section 3.2.3.1, the maximum tuning range of the coupling coeficient of the TC (i.e., 0 ≤ ak ≤ 1 ) can be achieved by using 3-dB DCs (i.e., dk = 0.5 ) and by tuning the required PS phase in the range 0 ≤ ϕk ≤ π according to Eq. 3.27. Thus for dk = 0.5 , Eq. 3.27 becomes Let θ13,k = θ14,k = θ 23,k = θ 24,k = θ k and dk = 0.5 , Eq. 3.10 becomes

Using e −α w L = 0.955 and putting bk = 0.5 (i.e., the DC is 3 dB in the TC), the coupling coeficient of the TC, gk , in the AZF deined in Eq. 3.43 becomes gk =

1 1 + (0.955)2 zɵk

−2

(

; zɵk ≥ 0

)

(3.70)

where 0 ≤ gk ≤ 1. Note from Eq. 3.70 that zɵk ≥ 0 means that the zeros can be located anywhere in the z-plane, and this unique feature can provide great lexibility in the design. This is because the proposed method can generally be applied to the design of a variety of tunable ilters, such as the Butterworth, Chebyshev and elliptic ilter types. In particular, the Chebyshev and elliptic ilters can have zero locations outside the unit circle in the z-plane. The maximum tuning range of the coupling coeficient of the TC (i.e., 0 ≤ gk ≤ 1 ) in the AZF deined in Eq. 3.70 can be achieved by tuning the phase shift of the phase shifter (PS) loaded on one arm of the TC in the range of 0 ≤ Φ k ≤ π , which is given as Φ k = cos −1(2 gk − 1); ( 0 ≤ Φ k ≤ π )

(3.71)

Using M = 2, e −α w L = 0.955, ck = 0.25 and bk = 0.5, Eq. 3.37 becomes 12

A = 0.114 [ a1a2 (1 − g1 )(1 − g2 ) ] .

(3.72)

In summary, Eqs. 3.65 through 3.72 are used for the synthesis of the second-order Butterworth bandpass and bandstop tunable optical ilters. In all frequency responses, the transmission (or magnitude) responses are plotted over the normalized FSR of ωT (2π ) = f T , where f is the relative optical frequency. The actual FSR is given by FSR = 1 T (which is 5  GHz in this design example). Note that the transmission responses are periodic with a normalized FSR of f T = 1 or with an FSR of 1 T . Table 3.9 shows the computed designed parameter values of the second-order Butterworth bandpass and bandstop digital ilters with variable normalized 3-dB bandwidths (i.e., ∆ωT (2π ) = ∆f T , where 0.1 ≤ ∆f T ≤ 0.9). For a particular bandwidth, both the bandpass and bandstop ilters have the same values of pɵ1 = pɵ2 , arg( pɵ1), arg( pɵ2 ) = −arg( pɵ1 ) and zɵ1 = zɵ2 . Note that the bandpass digital ilter has

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Photonic Signal Processing

TABLE 3.9 Designed Parameter Values of the Second-Order Butterworth Bandpass and Bandstop Digital Filters with Variable Normalized 3-dB Bandwidths of 0.1 ≤ ≤ ∆f T ≤ 0.9 Poles of Hɵ ap,k (z ), k = 1,2.

Filter Type Bandpass

Bandstop

Normalized Bandwidth ∆f T

Aɵ

pɵ 1 = pɵ 2

arg(pɵ 1) (radian)

arg(pɵ 2 ) = − arg(pɵ 1) (radian)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0201 0.0675 0.1311 0.2066 0.2929 0.3913 0.5050 0.6389 0.8006 0.8006 0.6389 0.5050 0.3913 0.2929 0.2066 0.1311 0.0675 0.0201

0.8008 0.6425 0.5217 0.4425 0.4142 0.4425 0.5217 0.6425 0.8008 0.8008 0.6425 0.5217 0.4425 0.4142 0.4425 0.5217 0.6425 0.8008

−2.9158 −2.6670 −2.3698 −2.0015 −1.5708 −1.1401 −0.7718 −0.4746 −0.2258 −2.9158 −2.6670 −2.3698 −2.0015 −1.5708 −1.1401 −0.7718 −0.4746 −0.2258

2.9158 2.6670 2.3698 2.0015 1.5708 1.1401 0.7718 0.4746 0.2258 2.9158 2.6670 2.3698 2.0015 1.5708 1.1401 0.7718 0.4746 0.2258

Zeros of Hɵ az,k (z ), k = 1,2.

zɵ 1 = zɵ 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

arg(zɵ 1) = − arg( zɵ 2 ) (radian) 0 0 0 0 0 0 0 0 0

π π π π π π π π π

Note: For each bandwidth, both the bandpass and bandstop ilters have the same poles which occur in complex-conjugate pairs. The zeros are located exactly on the unit circle.

arg = ( zɵ1 ) arg = ( zɵ2 ) 0 while the bandstop digital ilter has arg( zɵ1 ) = arg( zɵ2 ) = π . Using Eqs. 3.45 through 3.52, Table 3.10 shows the computed designed parameter values of the corresponding second-order Butterworth bandpass and bandstop tunable optical ilters with variable normalized 3-dB bandwidths of 0.1 ≤ ∆f T ≤ 0.9 and ixed normalized center frequencies which are chosen to be f T = 0.5 in this design example (i.e., δ 0 = 0). For a particular bandwidth, both the bandpass and bandstop tunable optical ilters have the same parameter values of a1 = a2 , ϕ1 = ϕ2 , θ1 = θ 2 and φ1, φ2 of H ap,k ( z ) for k = 1, 2 and the same parameter values of g1 = g2 and Φ1 = Φ 2 of H az ,k ( z ) for k = 1, 2 . Note that the bandpass tunable optical ilter has Ψ1 = Ψ 2 = π /2 while the bandstop tunable optical ilter has Ψ1 = Ψ 2 = 3π / 2. 3.2.5.4 Tuning Parameters of the Synthesized Bandpass and Bandstop Tunable Optical Filters For ilter implementation, it is important to know the range of values of the tuning parameters required for various ilter bandwidths. Figure 3.9 shows the characteristics of the tuning parameters of the bandpass and bandstop tunable optical ilters with variable normalized 3-dB bandwidths and ixed normalized center frequencies of f T = 0.5 (i.e., δ 0 = 0). The values of these parameters are obtained from Table 3.10. Figure 3.9a shows the intensity coupling coeficients of APFs (i.e., a1 = a2 , c= c= 0.25 and d= d= 0.5 ) and the AZFs (i.e., b= b= 0.5 and g= g= 0.523 ) versus the nor1 2 1 2 1 2 1 2 malized ilter bandwidth for both the bandpass and bandstop tunable ilters. The required values of the coupling coeficients of c= c= 0.25 , d= d= 0.5 , b= b= 0.5 , and g= g= 0.523 are ixed 1 2 1 2 1 2 1 2 over the bandwidth range. To tune the ilter bandwidth, the coupling coeficients a1 = a2 of the tunable couplers (TCs) of the all-pole ilters (APFs) must be varied by tuning the phase shifts ϕ1 = ϕ2 of

0.6021 0.7137 0.7492

38.81 × 10 −3

−3

0.4

0.5

0.0625

21.56 × 10 −3

0.8

3.39 × 10

0.3965

32.74 × 10 −3

0.7

0.9

0.6021

38.81 × 10

0.6

−3

0.7137

−3

40.74 × 10

0.3965

21.56 × 10

32.74 × 10 −3

0.3

0.0625

a1 = a2

0.2

Bandpass

−3

−3

3.39 × 10

0.1

Filter Type

Amplitude A

Normalized Bandwidth ∆f T

2.6363

1.7793

1.3651

1.1291

1.0491

1.1291

1.3651

1.7790

2.6363

ϕ 1 = ϕ 2 (radian)

−0.2526

−0.6811

−0.8882

−1.0063

−1.0463

−1.0063

−0.8882

−0.6811

−0.2526

θ 1 = θ 2 (radian)

0.0268

0.2065

0.1164

6.1493

5.7586

5.2879

4.8016

4.2973

3.6200

φ 1 (radian)

Parameters of Hɵ ap,k (z ), k = 1,2. 0.25 and d=1 = d== 0.5) (c=1 = c== 2 2

0.4784

1.1557

1.6600

2.1464

2.6171

3.0078

3.2580

3.3481

3.1684

φ 2 (radian)

0.5230

0.5230

0.5230

0.5230

0.5230

0.5230

0.5230

0.5230

0.5230

g1 = g2

1.5248

1.5248

1.5248

1.5248

1.5248

1.5248

1.5248

1.5248

1.5248

Φ 1 = Φ 2 (radian)

π /2 (Continued )

π /2

π /2

π /2

π /2

π /2

π /2

π /2

π /2

Ψ1 = Ψ 2 (radian)

Parameters of Hɵ az,k (z ), k = 1,2. 0.5) (b=1 = b== 2

TABLE 3.10 Designed Parameter Values of the Second-Order Butterworth Bandpass and Bandstop Tunable Optical Filters with Variable Normalized 3-dB Bandwidths of 0.1 ≤ ≤ ∆f T ≤ 0.9 and Fixed Normalized Center Frequencies at f T = 0.5 (i.e., δ 0 = 0)

Bandpass Optical Filters by DSP Techniques 95

0.6021 0.7137 0.7492

38.81 × 10 −3

−3

0.4

0.5

2.6363

1.7793

1.3651

1.1291

1.0491

1.1291

1.3651

1.7793

2.6363

ϕ 1 = ϕ 2 (radian)

−0.2526

−0.6811

−0.8882

−1.0063

−1.0463

−1.0063

−0.8882

−0.6811

−0.2526

θ 1 = θ 2 (radian)

0.0268

0.2065

0.1164

6.1493

5.7586

5.2879

4.8016

4.2973

3.6200

φ 1 (radian)

Parameters of Hɵ ap,k (z ), k = 1,2. (c=1 = c== 0.25 and d=1 = d== 0.5) 2 2

0.4784

1.1557

1.6600

2.1464

2.6171

3.0078

3.2580

3.3481

3.1684

φ 2 (radian)

0.5230

0.5230

0.5230

0.5230

0.5230

0.5230

0.5230

0.5230

0.5230

g1 = g2

1.5248

1.5248

1.5248

1.5248

1.5248

1.5248

1.5248

1.5248

1.5248

Φ 1 = Φ 2 (radian)

3π / 2

3π / 2

3π / 2

3π / 2

3π / 2

3π / 2

3π / 2

3π / 2

3π / 2

Ψ1 = Ψ 2 (radian)

Parameters of Hɵ az,k (z ), k = 1,2. (b=1 = b== 0.5) 2

Note: For each bandwidth, both the bandpass and bandstop ilters have the same pole values (see Table 3.9) and hence the same parameters of APFs (i.e., the same coupling coeficients of a1 = a2 of the TCs and the same phase shifts of ϕ 1 = ϕ2 , φ 1 and φ 2), and the same zero values (see Table 3.9) and hence the same parameters of AZFs (i.e., the same coupling coeficients of g1 = g2 of the TCs and the same phase shifts of Φ1 = Φ 2 ). For each bandwidth, there is a phase difference of π between the zeros of the bandpass and bandstop ilters, that is, the bandpass ilters have Ψ1 = Ψ 2 = π / 2 and the bandstop ilters to have Ψ1 = Ψ 2 = 3π / 2.

0.0625

21.56 × 10 −3

0.8

3.39 × 10

0.3965

32.74 × 10 −3

0.7

0.9

0.6021

38.81 × 10

0.6

−3

0.7137

−3

40.74 × 10

0.3965

21.56 × 10

32.74 × 10 −3

0.3

0.0625

a1 = a2

0.2

Bandstop

−3

−3

3.39 × 10

0.1

Filter Type

Amplitude A

Normalized Bandwidth ∆f T

TABLE 3.10 (Continued) Designed Parameter Values of the Second-Order Butterworth Bandpass and Bandstop Tunable Optical Filters with Variable Normalized 3-dB Bandwidths of 0.1 ≤ ≤ ∆f T ≤ 0.9 and Fixed Normalized Center Frequencies at f T = 0.5 (i.e., δ 0 = 0)

96 Photonic Signal Processing

Bandpass Optical Filters by DSP Techniques

97

FIGURE 3.9 Characteristics of the tuning parameters versus the normalized 3-dB bandwidth of the bandpass and bandstop tunable optical ilters with variable normalized 3-dB bandwidths and ixed normalized center frequencies at f T = 0.5 (i.e., δ 0 = 0). (a) Intensity coupling coeficients of the all-pole ilters and the allzero ilters of both the bandpass and bandstop tunable ilters. (b) Optical phase shifts of the all-pole ilters and the all-zero ilters of both the bandpass and bandstop tunable ilters.

the TCs (see the dotted-dotted curve in Figure 3.9b). Note that Figure 3.9b also shows that the phase shifts φ1 (dashed curve) and φ2 (dotted-dashed curve) of the APFs of both the bandpass and bandstop tunable ilters must also be tuned in order to tune the ilter bandwidth. However, both the bandpass and bandstop tunable ilters require the same phase shifts of Φ1 = Φ 2 = 1.5248 and of the all-zero ilters (AZFs). The bandpass and bandstop tunable ilters require the phase shifts Ψ1 = Ψ 2 = π /2 and Ψ1 = Ψ 2 = 3π /2 (see the solid curves) of the AZFs, respectively. The tuning parameters of the ilter can be summarized as follows.

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Photonic Signal Processing

• For a particular ilter bandwidth (i.e., for a particular set of phase shift values of ϕ1 = ϕ2, φ1, and φ2 of the APFs and Φ1 = Φ 2 of the AZFs), the phase shifts Ψ1 = Ψ 2 of the AZFs determine the transmission response characteristics of the ilter. That is, a bandpass ilter with a certain bandwidth can be tuned to a bandstop ilter without changing the bandwidth by tuning Ψ1 = Ψ 2 = π / 2 to Ψ1 = Ψ 2 = 3π / 2 of the AZFs and vice versa. • For a particular ilter characteristics (i.e., bandpass where Ψ1 = Ψ 2 = π / 2 or bandstop where Ψ1 = Ψ 2 = 3π / 2), ixing the phase shifts Φ1 = Φ 2 = 1.5248 (of the TCs) of the AZFs and tuning the phase shifts ϕ1 = ϕ2 (of the TCs), and φ1 and φ2 of the APFs will change the locations of the poles and hence changing the bandwidth of the ilter.

3.2.6

synthesis OF banDpass anD banDstOp tunable Optical Filters with variable banDwiDths anD FixeD center Frequencies

Using the designed parameter values shown in Table 3.10 and Figures 3.9 and 3.10a, b show, respectively, the transmission responses of the bandpass and bandstop tunable optical ilters with variable bandwidths (normalized) and ixed center frequencies (normalized) at f T = 0.5 (i.e., δ 0 = 0 in Eqs. 3.65 and 3.66. The 3-dB bandwidth (normalized) of each ilter characteristic (i.e., bandpass or bandstop) can be varied from ∆f T = 0.2 to ∆f T = 0.4 and to ∆f T = 0.6 by tuning the phase shifters of the APFs (i.e., ϕ1 = ϕ2, φ1, and φ2 of H ap,1( z ) and H ap,2 ( z ), and Φ1 = Φ 2 = 1.5248 of H az,1( z ) and H az,2 ( z ), see Figure 3.9b and Table 3.9) and by keeping the phase shifters of the AZFs unchanged (i.e., Ψ1 = Ψ 2 = π / 2 for bandpass and Ψ1 = Ψ 2 = 3π / 2 for bandstop, see Figure 3.9b and Table 3.10). Note that the center frequencies (normalized) here are designed to be at f T = 0.5 for both the bandpass and bandstop tunable ilters. 3.2.6.1

Synthesis of Tunable Optical Filters with Fixed Bandwidths and Tunable Center Frequencies As a design example, the tunable bandpass ilter with a normalized bandwidth of ∆f T = 0.2 and a normalized center frequency of f T = 0.5 is employed as a reference ilter (see the solid-line curve shown in Figure 3.10a where δ 0 = 0 ), and similarly the tunable bandstop ilter with a normalized bandwidth of ∆f T = 0.2 and a normalized center frequency of f T = 0.5 is used as a reference ilter (see the solid-line curve shown in Figure 3.10b where δ 0 = 0 ). The center frequencies of these two types of ilters can be tuned, without affecting their bandwidths (i.e., the normalized bandwidths  will remain unchanged at ∆f T = 0.2), to within one FSR by adding the same phase shift values of δ 0 (0 < δ 0 < 2π ) to the phase shifters of both the APFs (i.e., φ1 and φ 2 in Eq. 3.65) and the AZFs (i.e., Ψ1 and Ψ 2 in Eq. 3.66). In doing so, Figure 3.11a and b show, respectively, the transmission responses of the designed bandpass and bandstop tunable optical ilters with ixed normalized 3-dB bandwidths of ∆f T = 0.2 and variable center frequencies. In both Figure 3.11a and 5b, the normalized center frequency can be varied to within one normalized FSR of between 0 and 1 from f T = 0.2 (where δ 0 = −0.6π , see the dashed-line curves) to f T = 0.5 (where δ 0 = 0 , see the solid-line curves) and to f T = 0.8 (where δ 0 = +0.6π , see the dashed-dotted curves). Note that the designed parameter values here are exactly the same as those shown in Table 3.10 and Figure 3.9, except that appropriate phase shift values of δ 0 (i.e., δ 0 = −0.6π , 0 and +0.6π in this design example) have been added to the phase shifters of the APFs (i.e., φ1 and φ2 in Eq.  3.65) and to the phase shifters of the AZFs (i.e., Ψ1 and Ψ 2 in Eq. 3.66). That is, in Figure 3.11a, for the bandpass tunable ilters with a ixed normalized 3-dB bandwidth of ∆f T = 0.2, the phase shift values of φ 1, φ 2 , Ψ1, and Ψ 2 shown in Table 3.10 and Figure 3.9 have been changed to  φ 1 = 4.2973 − 0.6π = 2.4123, φ 2 = 3.3481 − 0.6π = 1.4631 and Ψ1 = Ψ 2 = π / 2 − 0.6π = −0.1π , respectively; for the new normalized center frequency of fT = 0.2 (see the dashed-line curve), to φ 1 = 4.2973, φ 2 = 3.3481, and Ψ1 = Ψ 2 = π / 2, respectively; for the new normalized center frequency

Bandpass Optical Filters by DSP Techniques

99

FIGURE 3.10 Transmission responses of the designed bandpass and bandstop tunable optical ilters with variable normalized 3-dB bandwidths of ∆f T = 0.2, ∆f T = 0.4 and ∆f T = 0.6, and ixed normalized center frequencies at f T = 0.5 (i.e., δ 0 = 0). (a) Bandpass responses. (b) Bandstop responses.

of f T = 0.5 (see the solid-line curve), and to φ 1 = 4.2973 + 0.6π = 6.1823, φ 2 = 3.3481 + 0.6π = 5.2331, and Ψ1 = Ψ 2 = π / 2 + 0.6π = 1.1π , respectively; for the new normalized center frequency of f T = 0.8 (see the dashed-dotted curve). Similarly, in Figure 3.11b, for the bandstop tunable ilters with a ixed normalized 3-dB bandwidth of ∆f T = 0.2, the phase shift values of φ 1, φ 2 , Ψ1, and Ψ 2 shown in Table  3.10 and Figure 3.9 have been changed to φ 1 = 4.2973 − 0.6π = 2.4123, φ 2 = 3.3481 − 0.6π = 1.4631, and Ψ1 = Ψ 2 = 3π / 2 − 0.6π = 0.9π , respectively; for the new normalized center frequency of f T = 0.2 (see the dashed-line curve), to φ 1 = 4.2973, φ 2 = 3.3481, and Ψ1 = Ψ 2 = 3π / 2, respectively; for the new normalized center frequency of fT = 0.5

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FIGURE 3.11 Transmission responses of the designed bandpass and bandstop tunable optical ilters with a ixed normalized 3-dB bandwidth of ∆f T = 0.2 and tunable normalized center frequencies of f T = 0.2 (i.e., δ 0 = −0.6π ), f T = 0.5 (i.e., δ 0 = 0) and f T = 0.8 (i.e., δ 0 = +0.6π ). (a) Bandpass responses. (b) Bandstop responses.

(see  the solid-line curve) and to φ 1 = 4.2973 + 0.6π = 6.1823, φ 2 = 3.3481 + 0.6π = 5.2331, and Ψ1 = Ψ 2 = 3π / 2 + 0.6π = 2.1π , respectively; for the new normalized center frequency of f T = 0.8 (see the dashed-dotted curve). 3.2.6.2 Fabrication Tolerances of Filter Parameters It is dificult to fabricate ilter parameter values that do not deviate from their designed values. So, it is important to determine the allowable range of the parameter errors for fabrication purposes. The essential parameters that are considered to have fabrication errors are the phase shifts of the

Bandpass Optical Filters by DSP Techniques

101

various phase shifters (i.e., ϕ1, ϕ2 , φ1 , φ2 , Φ1, Φ 2, Ψ1, and Ψ 2 ) and the coupling coeficients of the directional couplers (DCs) (i.e., b1, b2, c1, c2, d1, and d2). It is convenient to consider the coupling angles rather than the coupling coeficients as parameter errors because the phase shifts are considered as parameter errors. The intensity coupling coeficient of the directional coupler y (where = y b= 1, 2) relates to the coupling angle x (where x is the coupling angle of bk , ck , dk ; k , ck , dk ; k k = 1, 2 ) by y = sin 2 x . Thus, the change in the intensity coupling coeficient ∆y (i.e., due to fabrication error) relates to the change in the coupling angle ∆x by ∆y = 2 y(1 − y ) ∆x

(3.73)

The phase shift errors of the phase shifters (i.e., ∆ϕ1, ∆ϕ2, ∆φ1, ∆φ2, ∆Φ1, ∆Φ 2, ∆Ψ1, and ∆Ψ 2 ) and the coupling angle errors of the directional couplers (i.e., ∆x ) are considered to be ±0.001π , ±0.01π , and ±0.1π . Figure 3.12a–c shows the transmission responses of the designed bandpass tunable optical ilter (which has a normalized 3-dB bandwidth of ∆f T = 0.4 and a normalized center frequency of f T = 0.5 in this example) with the phase shift errors and the coupling angle errors of ±0.001π , ±0.01π , and ±0.1π , respectively. It can be seen from these igures that a fabrication accuracy of better than ±0.001π is required for the actual transmission response to be very close to the ideal one. However, Figure 3.12b shows that the transmission response is still suficiently practical even with the random phase errors of ±0.01π . The same conclusion can also be made on the bandstop ilter (see Figure 3.13a–c), because the stopband in the transmission response (see Figure 3.13b which corresponds to the random phase errors of ±0.01π ) is greater than 40 dB. Similar results are also obtained for the designed bandpass and bandstop tunable ilters with other bandwidth values. Thus, the allowable phase shift errors of the phase shifters and the coupling angle errors of the directional couplers must be less than ±0.01π , which is used as a basis to determine if the designed ilters could be fabricated to such an accuracy. This is described as follows. Using the coupling angle errors of ±0.01π in Eq. 3.53, the allowable errors in the intensity coupling

FIGURE 3.12 Transmission responses of the designed bandpass tunable optical ilter with phase shift errors of the phase shifters and coupling angle errors of the directional couplers. (a) ±0.001π , (Continued)

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Photonic Signal Processing

FIGURE 3.12 (Continued) Transmission responses of the designed bandpass tunable optical ilter with phase shift errors of the phase shifters and coupling angle errors of the directional couplers. (b) ±0.01π , and (c) ±0.1π . This particular ilter design has a normalized 3-dB bandwidth of ∆f T = 0.4 and a normalized center frequency of f T = 0.5.

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103

FIGURE 3.13 Transmission responses of the designed bandstop tunable optical ilter with phase shift errors of the phase shifters and coupling angle errors of the directional couplers. (a) ±0.001π , (b) ±0.01π , and (c) ±0.1π . This particular ilter design has a normalized 3-dB bandwidth of ∆f T = 0.4 and a normalized center frequency of f T = 0.5.

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Photonic Signal Processing

coeficients are ∆b1 b1 = ∆b2 b2 = ±0.063 (or ± 6.3%), ∆c1 c1 = ∆c2 c2 = ±0.109 (or ± 10.9%), and ∆d1 d1 = ∆d2 d2 = ±0.063 (or ± 6.3%). Thus, the directional couplers must be fabricated with an accuracy of better than 6% and such an accuracy can be achieved in practice [14]. As discussed in Section 3.2.3.1, when an electric voltage is applied to the thin-ilm heater, the effective refractive index neff of the heated waveguide increases and thus changes the optical path length neff × l by ∆( neff × l ) = ∆Temp

dneff l dTemp

(3.74)

where Temp is the temperature, ∆Temp is the temperature change in Celsius degrees, dneff dTemp = 1× 10 −5 °C is a typical thermo-optic constant of the silica waveguide, and l is the length of the heated waveguide (or the phase shifter). Furthermore, the change in the optical path length ∆( neff × l ) relates to the change in the phase shift of the thin-ilm heater ∆Θ by ∆( neff × l ) =

∆Θ λ 2π

(3.75)

where λ is the wavelength of the optical carrier. When Eq. 3.75 is substituted into Eq. 3.74, the temperature change ∆Temp relates to the phase shift change ∆Θ by ∆Temp =

λ ∆Θ dneff 2π l dTemp

(3.76)

For a λ = 1.55 µ m lightwave, a typical l = 4 mm of the thin-ilm heaters, and the allowable phase errors of ∆Θ = ±0.01π as described above, the allowable temperature change of the ilter must be less than ∆Temp = ±0.19°C according to Eq. 3.76. Thus, the fabricated ilter must be stabilized to within ±0.19°C to ensure that its transmission response is not greatly deteriorated. Such a temperature stability of the ilter can be achieved in practice using a Peltier device, which could stabilize the device temperature to within ±0.1°C [23]. We now discuss means of overcoming the fabrication errors of the waveguide lengths. The error in the waveguide loop length L of the APF (see Eq. 3.34) and the error in the differential waveguide length L of the AZF (see Eq. 3.47) could have an effect on the ilter performance, especially on the FSR. Although the phase shifters are mainly used to change the phase of the optical carrier by means of the thermo-optic effect, as described above, they can also be used to compensate for the waveguide length error because of the change in the optical path length when the phase shifters are heated. Thus, the phase shift of the phase shifter φk of the all-pole ilter can also be used to accurately adjust the waveguide loop length to the designed value of L (see Figure 3.7), and similarly the phase shift of the phase shifter ψ k of the all-zero ilter can also be used to accurately adjust the differential waveguide length to the designed value of L (see Figure 3.8). Nevertheless, it has been found that the waveguide length could be accurately fabricated to within 1% accuracy [23].

3.2.7

cOncluDing remarks

An effective synthesis method has been developed for the design of a variety of tunable optical ilters with independently variable bandwidths and tunable center frequencies and arbitrary IIR characteristics. The synthesized Mth-order tunable optical ilter consists of the cascade of M all-pole ilters (APFs) with M all-zero ilters (AZFs). The bandwidth and center frequency of the designed tunable optical ilter can be independently tuned by applying electric power to thin-ilm heaters

Bandpass Optical Filters by DSP Techniques

105

loaded on the waveguides of the both the APFs and AZFs. The synthesis method has the unique advantage in that the poles and zeros of the tunable optical ilter can be adjusted independently of each other and can thus be used to design tunable optical ilters with arbitrary IIR characteristics. By means of computer simulation, the effectiveness of the synthesis method has been demonstrated with the design of the second-order Butterworth bandpass and bandstop tunable optical ilters with variable bandwidths and tunable center frequencies. In this design, for a ixed center frequency, the ilter bandwidth can be tuned by tuning the phase shifts of the phase shifters of the APFs and by keeping the parameters of the AZFs unchanged; and for a ixed bandwidth, the ilter center frequency can be tuned, to within one FSR, by changing the phase shifts of the phase shifters of both the APFs and AZFs. In terms of fabrication tolerances, the allowable values of the phase shift errors of the phase shifters and the coupling angle errors of the directional couplers have also been determined. In addition to the designed Butterworth ilters as shown in this paper, the synthesis method is general and lexible enough to enable the design of a variety of tunable optical ilters with variable bandwidths and tunable center frequencies and arbitrary IIR characteristics which include the Chebyshev and elliptic ilter types.

REFERENCES 1. E. M. Dowling and D. L. MacFarlane, Lightwave lattice ilters for optically multiplexed communications systems, J. Lightwave Technol., 12, 471–486, 1994. 2. N. Q. Ngo and L. N. Binh, Novel realization of monotonic Butterworth-type low pass, highpass and bandpass ilters using phase-modulated iber-optic interferometers and ring resonators, IEEE J. Lightwave Technol., 12, 827–841, 1994. 3. L. N. Binh, X. T. Nguyen, and N. Q. Ngo, Realization of Butterworth-type optical ilters using 3 × 3 optical directional couplers, IEE Part J. Optoelectronics, 143(2), 126–134, 1996. 4. Y. H. Ja, Analysis of optical iber loop resonators with a collinear 3 × 3 iber coupler, Applied Opt., 33, 6402–6411, 1994. 5. Y. H. Ja and X. Dai, Butterworth-like ilters using an S-shape two coupler optical iber ring resonator, Micorw. Opt. Tech. Lett., 6, 376–378, 1993. 6. R. D. Strum and D. E. Kirk, First Principles of Discrete Systems and Digital Signal Processing, New York: Addison-Wesley, 1989, pp. 676–685. 7. S. K. Mitra and J. F. Kaiser, (Ed.), Handbook of Digital Signal Processing, New York: John Wiley & Sons, 1993, p. 321. 8. S. K. Mitra and J. F. Kaiser, (Ed.), Handbook of Digital Signal Processing, New York: John Wiley & Sons, 1993, pp. 137–142. 9. N. Q. Ngo, L. N. Binh, and X. Dai, Eigenilter approach for designing FIR all-pass optical dispersion compensators for high speed long-haul systems, Proceedings of the Australian Conference on Optical Fibre Technology, Melbourne, Australia: ACOFT, December. 1994, pp. 355–358. 10. N. Q. Ngo and L. N. Binh, Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical ilters using phase-modulated iber-optic interferometers and ring resonators, J. Lightwave Technol., 12, 827–841, 1994. 11. K. Jinguji, Synthesis of coherent two-port optical delay-line circuit with ring waveguides, J. Lightwave Technol., 14, 1882–1898, 1996. 12. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach, New York: John Wiley & Sons, 1999. 13. A. Rostami and G. Rostami, All-optical implementation of tunable lowpass, highpass, and bandpass optical ilters using ring resonators, J. Lightwave Technol., 23, 446–460, 2005. 14. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice Hall, 1989. 15. A. A. M. Saleh and J. Stone, Two-stage Fabry-Perot ilters as demultiplexers in optical FDMA LAN’s, J. Lightwave Technol., 7, 323–330, 1989. 16. K. Oda, S. Suzuki, H. Takahashi, and H. Toba, An optical FDM distribution experiment using a high inesse waveguide-type double ring resonator, IEEE Photonics Technol. Letts., 6, 1031–1034, 1994. 17. S. Y. Li, N. Q. Ngo, S. C. Tjin, P. Shum, and J. Zhang, Thermally tunable narrow bandpass ilter based on linearly chirped iber Bragg grating, Opt. Lett., 29(1), 29−31, 2004.

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18. D. Sadot and E. Boimovich, Tunable optical ilter for dense WDM networks, IEEE Comm. Magaz., 36, 50–55, 1998. 19. M. G. Xu, H. Geiger, and J. P. Dakin, Interrogation of iber-optic interferometer sensors using acoustooptic tunable ilter, Electron. Lett., 31, 1487–1488, 1995. 20. S. H. Yun, D. J. Richardson, D. O. Culverhouse, and B. Y. Kim, Wavelength-swept iber laser with frequency shifted feedback and resonantly swept intra-cavity acoustooptic tunable ilter, IEEE J. Sel. Top. Quantum Electron., 3, 1087–1096, 1997. 21. P. V. Mamyshev and L. F. Mollenauer, Stability of soliton propagation with sliding-frequency guiding ilters, Opt. Lett., 19, 2083–2085, 1994. 22. K. Tamura, E. P. Ippen, and H. A. Haus, Optimization of iltering in soliton iber lasers, IEEE Photonics Technol. Lett., 6, 1433–1435, 1994. 23. S. Suzuki, M. Yanagisawa, Y. Hibino, and K. Oda, High-density integrated planar lightwave circuits using waveguides with a high-refractive index difference, J. Lightwave Technol., 12, 790–796, 1994.

4

Photonic Computing Processors

In this chapter, incoherent iber-optic systolic array processors (FOSAPs), which employ a digital-multiplication-by-analog-convolution (DMAC) algorithm and the extension of the DMAC algorithm, are proposed for real-valued digital matrix computations. The important role of optics in optical computing and a variety of existing optical architectures using the DMAC algorithm are described in Section 4.1.2. Section 4.1.3 presents mathematical formulations of the DMAC algorithm and the two’s complement binary (TCB) arithmetic, while the next section outlines the operational principles of the elemental processors of the FOSAP architectures. The performances of the FOSAP multipliers (Section 4.1.4) are compared with those of digital electronic multipliers and other optical DMAC multipliers. Means of overcoming the limitation of the FOSAP architectures are discussed in Section 4.1.5. The theoretical aspects of incoherent fiber-optic signal processing described in Chapter 2 are applied in this chapter where intensity-based signals are considered. Furthermore, the design of a programmable incoherent Newton–Cotes optical integrator (INCOI) is described. The deinition, existing design techniques, and application of digital integrators are described in Section 4.2. A generalized theory of the Newton–Cotes digital integrators, whose derivation is given in Section 4.4, and their magnitude and impulse responses are given in Section 4.2. Based on this theory, algorithms for the synthesis of the INCOI processor are proposed. These algorithms are then used in the design of the programmable INCOI processor, which essentially consists of a microprocessor, iber-optic architectures, optical switches, optical, and semiconductor ampliiers. Several types of input pulse sequences are chosen as examples for illustration of the incoherent processing accuracy of the programmable INCOI processor. In addition, the incoherent recursive iber-optic signal processor (RFOSP) is used in the design of the programmable INCOI processor. The theory of incoherent iber-optic signal processing described in Chapter 2 is employed in this chapter where intensity-based signals are considered. Section 4.3 then outlines the theoretical development for optical differentiating processors and their implementation.

4.1 4.1.1

INCOHERENT FIBER-OPTIC SYSTOLIC ARRAY PROCESSORS intrODuctiOn

In a digital electronic computer, a co-processor is an important special-purpose processor, which is mainly responsible for performing speciic high-speed arithmetic operations. A high-performance co-processor is necessary for performing massive computational tasks, such as image processing, pattern recognition, and signal processing problems, which would otherwise be performed by very computationally expensive computer software. There has been considerable research interest in developing optical co-processors for general-purpose digital electronic computers because of the massive parallelism, high processing speed, and high spatial- and temporal-bandwidth of optics compared with electronics.

107

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Optical systolic array processors1,2 have been proposed as optical co-processors. Initially, they were proposed for performing analog matrix–vector and matrix–matrix operations using acoustooptic architectures3 and iber-optic architectures.4,5 However, the main drawback of these analog optical processors is their restriction to operating in the low accuracy range (13–16 bits). The accuracy is limited primarily by the linear dynamic range of the devices used in the system. For example, an erbium-doped iber ampliier (EDFA) can provide an intensity gain of 40–50 dB, which translates to 13–16 bits of accuracy. Various algorithms have been described that enable optical systolic array processors to compute with high (or digital) accuracy techniques, such as residue arithmetic,6 modiied signed-digit number representation,7 redundant number representation,8 and symbolic substitution.9 The most commonly used technique is the DMAC algorithm,10 which is employed in this chapter, because convolution can be easily performed in optics. Whitehouse and Speiser11 proposed that the digital multiplication of two binary numbers (originally known as the Swartzlander multiplier12) is equivalent to the analog convolution of these numbers, provided that the analog result of the digital product is represented in the mixed-binary format. This digital-multiplication-by-analog-convolution technique is known as the DMAC algorithm.13 The mixed-binary representation of the digital product can be converted to the standard binary representation of the digital product by an analog-to-digital converter (ADC) and a shift-and-add (S/A) circuit in the post-processing unit. This basic idea has been applied to digital matrix–vector and matrix–matrix operations using time-integrating (TI) and space-integrating (SI) acousto-optic architectures,14,15 magneto-optic spatial light modulators,16 nonlinear optical devices,17,18 and logic counters.19 In this chapter, incoherent FOSAPs employing the DMAC algorithm are proposed for realvalued digital matrix multiplications. Most of the work presented here has been described by Ngo and Binh.19,20 1

2

3 4 5

6

7

8 9 10

11 12 13

14 15

16

17 18 19

20

Although there are various architectures of optical systolic array processors, they all have one general feature in that the input data lows in a pulsating fashion, and hence the name “systolic”, through one-dimensional or two-dimensional identical array processing elements where computation is performed on the data currently present. H. J. Cauield, W. T. Rhodes, M. J. Foster, and S. Horvitz, Optical implementation of systolic array processing, Opt. Commun., 40, 86–90, 1981. R. A. Athale and J. N. Lee, Optical processing using outer-product concepts, Proc. IEEE, 72, 931–941, 1984. B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber-optic lattice signal processing, Proc. IEEE, 72, 909–930, 1984. K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. C. Cutler, J. W. Goodman, and H. J. Shaw, Optical iber delay-line signal processing, IEEE Trans. Microw. Theory Tech., MTT-33, 193–210, 1985. A. Goutzoulis, E. Malarkey, D. K. Davies, J. Bradley, and P. R. Beaudet, Optical processing with residue LED/LD lookup tables, Appl. Opt., 25, 3097–3112, 1986. B. Drake, R. Bocker, M. Lasher, R. Patterson, and W. Miceli, Photonic computing using the modiied signed-digit number representation, Opt. Eng., 25, 38–43, 1986. R. A. Athale, Highly redundant number representation for medium accuracy optical computing, Appl. Opt., 25, 3122–3127, 1986. K. H. Brenner, Programmable optical processor based on symbolic substitution, Appl. Opt., 27, 1687–1691, 1988. H. J. Whitehouse and J. M. Speiser, Linear signal processing architectures, in Aspects of Signal Processing. Part 2, G. Tacconi (Ed.). NATO Advanced Study Institute, Boston, MA, 1976. pp. 669–702. Ibid. E. E. Swartzlander, The quasi-serial multiplier, IEEE Trans. Comp., C-22, 317–321, 1973. D. Psaltis, D. Casasent, D. Neft, and M. Carlotto, Accurate numerical computation by optical convolution, in 1980 International Optical Computing Conference II, W. T. Rhodes (Ed.). Proc Soc Photo-Opt Instrum Eng, 232, 151–156, 1980. Ibid. E. J. Baranoski and D. P. Casasent, High accuracy optical processors: a new performance comparison, Appl. Opt., 28, 5351–5357, 1989. F. T. S. Yu and M. F. Cao, Digital optical matrix multiplication based on a systolic outer-product method, Opt. Eng., 26, 1229–1233, 1987. G. Eichmann, Y. Li, P. P. Ho, and R. R. Alfano, Digital optical isochronous array processing, Appl. Opt., 26, 2726–2733, 1987. Y. Li, G. Eichmann, and R. R. Alfano, Fast parallel optical digital multiplication, Opt. Commun., 64, 99–104, 1987. Y. Li, B. Ha, and G. Eichmann, Fast digital optical multiplication using an array of binary symmetric logic counters, Appl. Opt., 30, 531–539, 1991. N. Q. Ngo and L. N. Binh, Fibre-optic array processors for algebra computations, Proc. IREE, 18th Australian Conf. Opt. Fibre Technol., Wollongong, 356–359, 1993.

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4.1.2

Digital-multiplicatiOn-by-analOg-cOnvOlutiOn algOrithm anD its extenDeD versiOn

The DMAC algorithm, its extended version, and the TCB arithmetic are described in this section. For the sake of hardware simplicity, the binary words are assumed to be of the same length, although the DMAC algorithm is generally valid for binary data of any length. Unless otherwise stated, variables with brackets denote binary sequences. For example, f represents the analog value whose binary sequence is denoted by { f }. 4.1.2.1 Multiplication of Two Digital Numbers Multiplication is a standard operation in matrix computations. It is shown here that the digital multiplication of two binary numbers can be determined by performing the analog convolution of these binary numbers followed by the function of the postprocessor. The standard binary representation of two positive integers f and g are given by the sequences { f } = { f n −1 ⋯ f1 f 0} and {g} = {gn −1 ⋯ g1 g0}, where n is the number of bits in a binary number, fi ,gi ∈ (0,1) for 0 ≤ i ≤ n − 1, f 0 and g0 are the least signiicant bits (LSBs), and f n−1 and gn−1 are the  most signiicant bits (MSBs). The mathematical descriptions of these positive n-bit words are given by n −1 f = ∑ fi 2i , i=0

(4.1)

n −1 j g = ∑ g j2 , j=0

(4.2)

The product of these binary numbers is given by13 f ⋅g =

2n − 2 k ∑ 2 yk , k =0

(4.3)

where k yk = ∑ fi gk − i ,(0 ≤ k ≤ 2n − 2), i=0

(4.4)

with f= g= 0 for i < 0 or i > n −1. i i Eq. (4.4) is easily recognized as the discrete convolution of two sequences, such as { f } ∗ {g} where * designates the convolution operation. The analog result of the discrete convolution is in the mixedbinary format. Note that the digital multiplication of two binary numbers without carries, that is {f}·{g}, is also in the mixed-binary format. Furthermore, the kth analog value yk of the convolution result represents the sum of partial products (without carries) in the kth column of the product {f}·{g}. This process of digital multiplication by analog convolution is known as the DMAC algorithm. Evaluation of Eq. (4.3) requires an optical convolver, an optical detector, an ADC, and a S/A circuit. At a particular discrete time, k, an ADC21 is used to convert the kth analog value yk into its binary representation, which is then up-shifted to the left by k bits by a shift register. This process is equivalent to evaluating the partial product 2k yk of Eq. (4.3). The last step for evaluation of Eq. (4.3) requires the addition of the partial products (with carries) by a binary adder. 21

The ADC requires log 2 n bits accuracy because n is the maximum analog value as a result of the convolution of two n-bit words.

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Photonic Signal Processing

The DMAC algorithm is best illustrated by means of a numerical example of multiplying two positive n-bit words whose standard binary representations are { f } = {110} and { g} = {011}, which correspond to the analog values f = 6 and g = 3. Figure 4.1a and b show the discrete-time representation of these sequences, where f [i ] is the impulse response of the optical convolver, g [i ] represents the modulated laser pulses (rectangular proile assumed) of unity height to be launched into the convolver, T p is the pulse width, and T is the bit period of the input pulse sequence or the sampling period of the optical convolver. The convolver pulse response, which is simply the discrete convolution of its impulse response f [i ] with the input pulse sequence g [i ] is given, from Eq. (4.4), as k y[k ] = ∑ f [i ]g[k − i ], (0 ≤ k ≤ 2n − 2). i=0

(4.5)

Eq. (4.5) can be evaluated graphically as shown in Figure 4.1c–g for n = 3. In Figure 4.1c, g [i ] is folded about i = 0 to become g [ −i ] and is slid past (with the LSB irst) the digits of f [i ]. The sequence g [ k − i ] is simply the folded sequence of g[i] shifted to the right by k units of delay T , as shown in Figure 4.1c–g for k = 0, 1, 2, 3, 4, respectively. The pulse shown by the broken curve corresponds to the LSB of the next word. t0 , t1 denote the starting times of the LSB of the irst and second words, respectively. The word strobe period Tw = t j +1 − t j is the time separation between the LSB of the previous word and the LSB of the next word and is given by Tw = (2n − 1)T . In order to avoid overlow, n −1 zeros are required for padding between the words. The analog result y[k] is shown in Figure 4.1h, where Tconv is the propagation delay of the optical convolver, which has been ignored during the convolution process in Figure 4.1c–g for the sake of clarity. These mixed-binary pulses are converted into their binary representations by a 2-bit ADC and shifted and added by a 2-bit S/A circuit in the post-processing unit, as shown in Figure 4.1. (i). The standard binary representation {010010} corresponds to the integer 18. 4.1.2.2 High-Order Digital Multiplication High-order multiplication is required for high-order matrix computations. It is shown here that the digital multiplication of Nˆ binary numbers can be determined by performing the analog convolution of these Nˆ binary numbers followed by the function of the postprocessor. This process of “HighOrder Digital-Multiplication-by-Analog-Convolution” is referred to as the “HO-DMAC” algorithm. The product of Nˆ positive n-bit words is deined as PNɵ = f (1)f (2) ⋯ f ( Nɵˆ − 1)f ( Nˆ )  n −1 n − 1  n − 1    n −1 =  ∑ f (1,a)2a   ∑ f (2,b)2b  ⋯  ∑ f ( Nˆ − 1,α )2α   ∑ f ( Nˆ ,β )2 β  a=0   b=0  α =0   β =0

 ,  

(4.6)

where  f (1, a), f (2, b),⋯, f ( Nˆ − 1,α ), f ( Nˆ , β ) ∈ (0,1). Substitution of γ = a + b + ⋯ + α + β (or β = γ − a − b − ⋯ − α ) into Eq. (4.6) results in PNɵ =

Nɵ (n − 1) γ ∑ 2 yγ , γ =0

(4.7)

where

γ γ γ yγ = ∑ ∑ ⋯ ∑ f (1,a)f (2,b)⋯ f ( Nˆ − 1,α )f ( Nˆ ,γ − a − b − ⋯ − α ) , a = 0b = 0 α = 0

(4.8)

Photonic Computing Processors

111

FIGURE 4.1 Graphical illustration of the DMAC technique. (a) The input pulse sequence g[i ]. (b) The convolver impulse response f [i ]. (c)–(g) The convolution operation. (h) The convolution output y[k ] is in the mixed-binary format. (i) The analog output is operated by the post-processing unit to obtain the expected standard binary representation of the decimal number 18.

for 0 ≤ γ ≤ Nˆ (n − 1), and f (1,a) = f (2,a) = ⋯ = f ( Nˆ − 1,a) = f ( Nˆ ,a) = 0 for a < 0 or a > n −1. Eq. (4.8) can be recognized as the discrete convolution of Nɵ binary sequences yγ = { f (1)} ∗ { f (2)}⋯{ f ( Nˆ − 1) } ∗ { f (Nˆ )},

(4.9)

The order of performing the convolution in Eq. (4.9) is unimportant because convolution is commutative and associative. In general, there are Nˆ ( n − 1) + 1 mixed-binary points as a result of the convolution of Nˆ n-bit words, the word cycle is Tw = [ Nˆ ( n − 1) + 1]T , and the number of zeros required

112

Photonic Signal Processing

for padding is ( Nˆ − 1)( n − 1). These mixed-binary points can be converted to the standard binary representation by the postprocessor, as described in Section 4.2.1. 4.1.2.3 Sum of Products of Two Digital Numbers The sum of products of two digital numbers, which is a vector inner-product operation, is also a standard operation in matrix computations. It is shown here that the digital summation of products of two binary numbers can be determined by performing the analog summation of convolutions of these binary numbers followed by the function of the postprocessor. This process of “Sum of Digital-Multiplication-by-Analog-Convolution” is referred to as the “S-DMAC” algorithm. The standard binary representations of the pth positive integers f ( p) and g ( p) are given by n −1 n −1 j f ( p) = ∑ fi (p)2i , g (p) = ∑ g j (p)2 , i=0 j =0

(4.10)

ˆ M SMˆ = ∑ f ( p)g ( p), p =1

(4.11)

where fi (p), g j (p) ∈(0,1). The sum of products of these positive n-bit words is given by

ˆ is the number of products of such two n-bit words. Eq. (4.11) can be shown to be94 where M SMˆ =

ˆ  2n − 2 k  M ∑ 2  ∑ yk (p)  , k =0  p = 1 

(4.12)

where k yk (p) = ∑ fi (p)gk − i (p), (0 ≤ k ≤ 2n − 2). i=0

(4.13)

Eq. (4.13) can be recognized as the discrete convolution of two binary numbers. Evaluating Eq. (4.12) ˆ optical convolvers, an optical combiner, an optical detector, an ADC, and a S/A circuit. requires M ˆ At a particular discrete time k, the term in brackets in Eq. (4.12), which represents the sum of M ɵ analog values of yk (p) for 1 ≤ p ≤ M , can be performed by an optical combiner. The analog result of this summation is then passed to the ADC for digitization, followed by the S/A circuit to obtain the standard binary representation. The advantage of the S-DMAC algorithm lies in the fact that a combination of product and summation operations can be performed. 4.1.2.4 Two’s Complement Binary Arithmetic The TCB representation is a powerful encoding scheme that permits both positive and negative numbers to be represented in binary form. The encoding scheme requires a sign-bit (SB) to be attached to the leftmost bit of the binary number, i.e., SB = 0 for positive numbers and SB = 1 for negative numbers. For example, the TCB sequence of the positive number22 +5.5 is {0 1 0 1.1}, and the negative number −13.375 is {1 0 0 1 0 _ 1 0 1}. Based on the DMAC algorithm as described in Section 4.2.1, the multiplication of two real numbers using the TCB arithmetic requires the input numbers to be represented by the same number of bits as the output number. For example, the TCB representation of the product (+5.5) (−13.375)  =  −73.5625 is {101101100111}, which is a 12-bit word. The 12-bit TCB sequence 22

Implementation of the radix (or decimal) point shifting is not discussed here because it is not a hardware issue.

113

Photonic Computing Processors

of the input numbers can thus be obtained by inserting seven zeros to the left of the SB of +5.5 to become {0 0 0 0 0 0 0 0 1 0 11} and four ones to the left of the SB of −13.375 to give {1 1 1 1 1 0 0 1 01 0 1}. The discrete convolution of these two sequences results in the mixedbinary sequence {1 1 1 21 2 0 2 2 2 3 3 2 1 1 0 0 0 0 0 0 0 0} in which the last 13  analog bits are discarded. The standard TCB format of the irst 12  chosen analog bits (the boldface bits) is {1 0 1 1 0 1 1 00 1 1 1} after the ADC and S/A operations, which is expected for the negative number −73.5625. The TCB arithmetic is not only applicable to the DMAC algorithm, but can also be used in conjunction with the HO-DMAC (see Section 4.2.2) and S-DMAC (see Section 4.2.3) algorithms. The main disadvantage of any optical DMAC processor employing the TCB arithmetic is the reduction in the pre-processing speed, and this can be shown as follows. In the following discussion, primes are used to denote the TCB variables. The convolution of Nˆ n′-bit TCB numbers generates [ Nˆ ( n′ − 1) + 1] mixed-binary points, in which only the irst n′ analog bits are useful for decoding into the binary format, and the word cycle is Tw′ = [ Nˆ ( n′ − 1) + 1]T . If the irst useful n′ analog bits of the TCB convolution are equal to the [ Nˆ ( n − 1) + 1] analog bits of the unsigned convolution (see Section 4.2.2), then the following relationship is obtained: ˆ w + (1 − Nˆ )T , Tw′ = NT

(4.14)

in which the irst term is dominant for large word length. Thus, the processing power of the optical DMAC processor incorporating the TCB arithmetic is approximately reduced by a factor of Nˆ (i.e., ˆ w where Nˆ is the number of integers (or matrices) to be multiplied) as compared with its Tw′ ≈ NT unsigned counterpart. Optical DMAC processors based on the TCB arithmetic also suffer from an increase in the resolution bits of the ADC and the S/A circuit because all the bits representing the output number are not fully utilized. This has the effect of reducing the processing speed because a higher-resolution ADC operates at a much lower speed. Nevertheless, optical DMAC processors incorporating the TCB representations can operate on real numbers.

4.1.3

elemental Optical signal prOcessOrs

This section describes three elemental optical signal processors, namely, an optical splitter, an optical combiner, and a binary programmable incoherent iber-optic transversal ilter. These processors are the building blocks of the FOSAP matrix multipliers, which are to be outlined in Section 4.4. It is assumed that the optically encoded signals to be processed are modulated onto an optical carrier whose coherence time T p is very short compared to the basic time delay T of the incoherent iber-optic transversal ilter, i.e., T p   1. If, for example, R = 1 then only one binary multiplication is being performed by the optical system per ADC operation. Because it is about equally dificult to perform multiplications and ADC operations electronically, this example shows that optics offers no advantage over electronics. The maximum value of the matrix–vector convolution, where the elements are base-b n-bit numbers, is given by Nn(b −1)2, which must not exceed the dynamic range of the ADC of 2 N ADC where N ADC is the ADC resolution bits. Thus, the matrix dimension N of the FOSAP matrix–vector multiplier of Figure 4.8 is limited by the dynamic range of the ADC according to N≤

2 N ADC . n(b − 1)2

(4.29)

The FOSAP matrix–vector multiplier, which performs MN operations in time MTw, has the processing speed given by MOPS =

MN N , = MTw (2n − 1)T

(4.30)

where MOPS stands for mega operations per second and 1/T is the clock rate or ADC speed in megahertz. The performance ratio of the FOSAP matrix–vector multiplier is thus given, from Eq. (4.28), as

R=

MOPS N = . 1× 1 T (2n − 1)

(4.31)

Table 4.1 shows the operational parameters for various case studies of the FOSAP matrix–vector multiplier of Figure 4.8 using Eqs. 4.27 through 4.31. Cases 1–3, where the resolution bits NADC and 26

Here, one operation is considered to be equivalent to one multiplication and one addition.

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Photonic Computing Processors

TABLE 4.1 Operational Parameters for Various Case Studies of the Fosap Matrix–Vector Multiplier for 32-bit (m = 32) Multiplications Case

b

n

N

NADC

1/T (MHz)

MOPS

R

Remarks

1 2 3 4 5 6

2 4 8 2 4 8

32 16 11 32 16 11

128 28 7 128 128 128

12 12 12 12 16 18

100 100 100 100 20 0.20

203.2 90.3 33.3 203.2 82.6 1.22

2.032 0.903 0.333 2.032 4.13 6.1

Constant clock rate 1 T , constant ADC bits N ADC Constant matrix dimension N

Note: b is the base used; n, the digits of accuracy; N, the matrix dimension; NADC, the ADC resolution bits; 1/T, the ADC speed or clock rate; MOPS, mega operations per second; and R, the Psaltis-Athale performance ratio.

speed 1/T of the 12-bit 100-MHz ADC are ixed, show that increasing the base b results in decreasing the values of n and N. Consequently, the values of MOPS and R are signiicantly reduced, and the FOSAP matrix–vector multiplier performs best with binary-encoded data (case 1). Cases 4–6 correspond to a ixed matrix dimension N  =  128. Increasing the base b results in increasing NADC, which greatly reduces the processing speed of MOPS because higher-resolution ADCs operate at much lower speeds. However, the FOSAP matrix–vector multiplier still outperforms its digital electronic counterpart because of the large value of R but at the expense of lower accuracy of n. Hence, the overall performance of the FOSAP matrix–vector multiplier is best achieved with binary-encoded data because of the desirable values of MOPS and R, as shown in Case 4, which is in fact Case 1. Similarly, the FOSAP matrix–matrix multiplier of Figure 4.8 also has the same operational parameters as those in Table 4.1 except that it has P times the values of the MOPSs. The analysis here also applies to higher-order FOSAP matrix multipliers, and it can be deduced that the FOSAP matrix multipliers perform better with binary-encoded data. 4.1.4.6 High-Order Fiber-Optic Systolic Array Processors From Section 4.1.4.2, the FOSAP matrix–matrix multiplier achieves the performance  ratio R = 2.032, which indicates its superiority over other non-iber and digital electronic processors. In this section, the performance of the positive-valued high-order FOSAP matrix multiplier is compared with that of its digital electronic counterpart. For analytical simplicity, the matrix is assumed to be square and has dimension M, and its elements are n-bit words. The high-order FOSAP matrix multiplier requires M word cycles to perform the product of x MxM matrices, and its processing time (or the number of clock cycles) is thus given by Tx = M [ x( n − 1) + 1]T .

(4.32)

The number of operations (NO) involved in the product of x MxM matrices is NO = ( x − 1) M 3 .

(4.33)

The number of operations per second (NOPS) performed by the high-order FOSAP matrix multiplier is thus given by NOPS =

( x − 1) M 2 NO . = Tx [ x( n − 1) + 1]T

(4.34)

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Photonic Signal Processing

The number of ADCs required by the high-order FOSAP matrix multiplier is always M . Using Eqs. 4.32 and 4.27, the Psaltis-Athale performance ratio of the positive-valued high-order FOSAP matrix multiplier is thus given by R=

NOPS ( x − 1) M  M   x −1  = ≈    M × 1 T [ x( n − 1) + 1]  n   x 

(4.35)

which is greater than unity if  x  M > n .  x −1 

(4.36)

The performance of the ratio R in Eq. (4.35) is expected to exceed unity because the matrix dimension M is, for many practical high-order matrix operations, usually a few orders of magnitude larger than the word length n. This shows the superiority of the high-order FOSAP matrix multiplier over its digital electronic counterpart. Thus, the high-order FOSAP matrix multiplier may be used to perform various linear algebra operations, such as solutions of algebraic equations, 2-D mathematical transform, matrix-inversions, and pattern recognition, which require high-order matrix operations.

4.1.5

remarks

The FOSAP matrix multipliers may be used to perform several linear algebra operations, which require basic matrix–vector and matrix–matrix operations, for advanced signal processing tasks such as pattern recognition and image processing. For example, the following linear algebra operations involve matrix operations: LU factorization; QR factorization; singular value decomposition; solution of simultaneous algebraic and differential equations; least squares solution; matrix inversion; and solutions to eigenvalue problems. The main limitation of any optical DMAC processor is the slow processing speed of the electronic post-processor in which the ADC is often the slowest component with a bit-time limit of T . This limitation means that the overall performance of any optical DMAC system is highly compromised. It has been predicted that an electronic 8-bit ADC can operate at 1.5  GHz, and that future development of a 6-bit ADC at 6 GHz is feasible.27 Electro-optic 2-bit and 4-bit ADCs were experimentally demonstrated to operate in the gigahertz range.28 However, future development of high-speed and high-resolution optical ADCs, with speed in the gigahertz range and resolution bits greater than 12, will enable the proposed FOSAP architectures to process more information at a faster rate than the digital electronic architectures. • Using the DMAC, HO-DMAC and S-DMAC algorithms, the incoherent FOSAP matrix multipliers have been designed using optical splitters, optical combiners, programmable iber-optic transversal ilters and postprocessors consisting of optical detectors, ADCs and S/A circuits. • The FOSAP multipliers, which perform best with binary-encoded data, can perform real-valued digital matrix–vector, matrix–matrix, triple-matrix, and higher-order matrix computations. • The positive-valued FOSAP matrix–vector and matrix–matrix multipliers have higher computational power than the digital electronic multipliers and other optical DMAC multipliers, showing their massive parallel-processing capability. 27 28

C. A. Liechti, High speed transistors: Directions for the 1990s, Microwave J., 30, 165–177, 1989. R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, Wide-band electrooptic guided-wave analog-todigital converters, Proc. IEEE, 72, 802–819, 1984.

Photonic Computing Processors

125

• The computational power of any real-valued optical DMAC system incorporating TCB arithmetic is reduced by a factor approximately equal to the number of matrices to be multiplied compared with the positive-valued optical DMAC systems. • The positive-valued high-order FOSAP multipliers have higher computational power than the digital electronic multipliers when the matrix dimension is greater than the word length by a few orders of magnitude, which is often the case for many signal processing applications. However, the real-valued high-order FOSAP multipliers using TCB arithmetic are unlikely to offer any advantage over the digital electronic multipliers. • The FOSAP architectures have a number of advantages including: massive pipeline capability; high computational power; modularity (construction of a larger architecture from several smaller architectures); and size scalability (the size of the architecture can be increased with nominal changes in the existing architecture with a comparable increase in performance). • The FOSAP multipliers may be used as high-performance co-processors in a generalpurpose digital electronic computer to perform various linear algebra operations, such as those in pattern recognition and image processing. The optical transversal ilter structure described in this Section will be used in the design of incoherent optical integrators in Section 4.2 and in the design of coherent optical differentiators in Section 4.3.

4.2

PROGRAMMABLE INCOHERENT NEWTON–COTES OPTICAL INTEGRATOR

In this section, the design of a programmable INCOI is described. The deinition, existing design techniques and application of digital integrators are described in Section 4.2.2. A generalized theory of the Newton–Cotes digital integrators, whose derivation is given in Appendix A, and their magnitude and impulse responses are given in Section 4.1.2.1. Based on this theory, algorithms for the synthesis of the INCOI processor are proposed in Section 4.2.2.3. These algorithms are then used in the design of the programmable INCOI processor, which essentially consists of a microprocessor, iber-optic architectures, optical switches, and optical and semiconductor ampliiers. Several types of input pulse sequences are chosen as examples for illustration of the incoherent processing accuracy of the programmable INCOI processor. In addition, the incoherent RFOSP described in Section 4.1.3 is used in the design of the programmable INCOI processor. The theory of incoherent iber-optic signal processing described in Chapter 2 is employed here where intensity-based signals are considered.

4.2.1

intrODuctOry remarks

While digital signal processing was originally used in the 1950s as a technique for simulating continuous-time systems using discrete-time computations, it has since become a ield of study in its own right. One example is a discrete-time (or digital) integrator, which can be used to simulate the behavior of a continuous-time (or analog) integrator. A digital integrator forms a fundamental part of many practical signal processing systems because the time integral of signals is sometimes required for further use or analysis. For example, digital integrators have been used in the design of compensators for control systems and for measuring the cardiac output, the volume of blood pumped by the heart per unit time. A digital integrator is a processor whose output pulse sequence is obtained by approximating the integral of a continuous-time signal from the samples of that signal. A continuous-time signal x(t ), whose values are known at the discrete time t = nT for n = 0,1, 2,⋯ , where T is the period between successive samples, can be integrated by a digital integrator. The frequency response of an ideal digital integrator is given by29 29

W. J. Tompkins and J. G. Webster, (Eds.), Design of Microcomputers-based Medical Instrumentation, Englewood Cliffs, NJ: Prentice-Hall, 1981.

126

Photonic Signal Processing

 1 0 ≤ ωT (2π ) ≤ 1 2 ,  jωT , H I (ω ) =  1  , 1 2 < ωT (2π ) ≤ 1,  j (2π − ωT )

(4.37)

where j = −1, ω is the angular frequency, and T is the sampling period of the integrator. Digital integrators may be designed by using one of the many classical numerical integration techniques such as the Newton–Cotes, Lagrange, Romberg and Gauss-Legendre formulas.29,30,31,32,33,34 Of these, the Newton–Cotes integration scheme has been extensively used. For example, the wellknown trapezoidal, Simpson’s 1/3, Simpson’s 3/8 and Boole’s integrators all belong to the family of the Newton–Cotes digital integrators.30–34 The underlying principle of the Newton–Cotes integration scheme is to it a continuous-time interpolation polynomial x(t ) to a given input pulse sequence f ( nT ) where f ( nT ) = x( nT ). The continuous-time interpolation polynomial x(t ) is then integrated resulting t in y(t ) = ∫0 x(t )dt . Sampling the integrated continuous-time polynomial y(t ) by a digital integrator at the sampling period yields the output pulse sequence y( nT ). Thus, the output pulse sequence of a digital integrator effectively approximates the integral of a continuous-time signal according to nT

y( nT ) =

∫ x(t )dt.

(4.38)

0

The magnitude responses of the Newton–Cotes digital integrators generally approximate the ideal magnitude response reasonably well over the lower frequency band of 0 ≤ ωT (2π ) ≤ 0.2. The Newton–Cotes digital integrators may thus be referred to as narrow-band integrators. Unlike the well-known Newton–Cotes digital integrators, the concept of optical integration is still new in the area of optical signal processing. In this section, a programmable INCOI processor is described. Most of the work presented here has been described by Ngo and Binh.35 The derivation of a generalized theory of the Newton–Cotes digital integrators, which was not given in reference,36 is shown in Appendix A.

4.2.2 newtOn–cOtes Digital integratOrs 4.2.2.1 Transfer Function The transfer function of the pth-order Newton–Cotes digital integrators can be generally expressed as37 H mp ( z ) =

T 1− z− p

m

∑ C ( p) ∆ D ( z ) k

k

k =0

(4.39)

T C0 ( p) + C1( p)∆D( z ) + C2 ( p)∆ 2 D( z ) + ⋯ + Cm ( p)∆ m D( z )  = 1− z− p  30

31 32

33 34 35 36

37

G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 2nd., Reading, MA: Addison– Wesley, 1990. R. Vich, Z Transform Theory and Applications, Norwell, MA: Kluwer Academic Publishers, 1987. R. Pintelon and J. Schoukens, Real-time integration and differentiation of analog signals by means of digital iltering, IEEE Trans. Intrum. Meas., 39, 923–927, 1990. M. Abramowitz and I. A. Segun, Handbook of Mathematical Function, New York: Dover Publications, 1964. S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 2nd ed., Singapore: McGraw-Hill, 1989. N. Q. Ngo and L. N. Binh, Programmable incoherent Newton–Cotes optical integrator, Opt. Commun., 119, 390–402, 1995. A. Ehrhardt, M. Eiselt, G. Großkoptf, L. Küller, R. Ludwig, W. Pieper, R. Schnabel, and G. Weber, Semiconductor laser ampliier as optical switching gate, J. Lightwave Technol., 11, 1287–1295, 1993. The derivations of Eqs. (4.39) through (4.41) are given in Appendix A.

127

Photonic Computing Processors

where the kth coeficient is given by38 p

η  Ck ( p) =   dη , k  0

∫

(4.40)

∆ k D( z ) = ( −1) k (1 − z −1 ) k ,

(4.41)

and the kth difference equation is given by

where k = 0,1,⋯ , m, 1 ≤ p ≤ m, and z = e jωT is the z-transform parameter.25 Note that these digital integrators are marginally stable because of p poles on the unit circle in the z-plane. Eq. (4.39) can be generally expressed as the product of the transfer function of the FIR (inite impulse response) ilter and the transfer function of the IIR (ininite impulse response) ilter according to m

T⋅ H mp ( z ) =

∑b z

−k

k

k =0

1 − az − p

(4.42)

where a = 1 is the pole value, bk is the tap coeficient of the FIR ilter, bk = bm − k is positive for m = p , and bk is real for m > p . The transfer functions of several families of the Newton–Cotes digital integrators are tabulated in Table 4.1. Note that H11( z ), H 22 ( z ), H 33 ( z ) and H 44 ( z ) are, respectively, the well-known transfer functions of the trapezoidal, Simpson’s 1/3, Simpson’s 3/8 and Boole’s integrators. Figure 4.9 shows a block diagram of programmable INCOI processor. Figure 4.10a–c show the magnitude responses of several families of the Newton–Cotes digital integrators. Figure 4.10a shows that the magnitude response of the trapezoidal integrator H 31( z ) approximates that of the ideal integrator much better than other integrators of the same or a different family. This does not necessarily mean that the trapezoidal integrator is superior to other Newton–Cotes integrators because the time-domain performance, which is described in Section 4.4, needs to be considered. Figure 4.10a–c show that the impulse responses of several families of the Newton–Cotes digital integrators are real and positive and hence characterize incoherent systems (see Section 4.2.2.2). As a result, incoherent Newton–Cotes optical integrators can be synthesized from the Newton–Cotes digital integrators. 4.2.2.2 Synthesis One common approach to the optical synthesis of Eq. (4.42) is to cascade the FIR iber-optic signal processor (FOSP) with the IIR FOSP. This approach can only be used for the synthesis of incoherent FOSPs comprising positive tap coeficients, e.g., the tap coeficients bks of H mp ( z ) m = p are all positive (see Tables 4.1 and 4.2). However, such a technique cannot be used for the case H mp ( z ) m > p where some of the negative tap coeficients cannot be optically implemented with incoherent FOSPs. An alternative optical synthesis method, which enables the negative tap coeficients to be implemented with incoherent FOSPs, is described here. In the following analysis, the transfer function H mp ( z ) and parameters a and bk are associated with any generic digital ilters, while Hˆ mp ( z ), aˆ and ˆbk are the variables of their optical counterparts.

38

η! η  η (η − 1) ⋯ (η − ( k − 1)) . = = k k! (η − k )! k !  

The binomial coeficient is deined as 

128

Photonic Signal Processing

FIGURE 4.9 Block diagram of the proposed programmable INCOI processor.

When p incoherent IIR FOSPs are incorporated into p higher-order delay lines of an incoherent FIR FOSP, the overall transfer function of the pth-order INCOI can be described by m  bˆ k z − k  Hˆ mp ( z ) = T ⋅ (4.43)  ˆ − p , 1 − az  k =0 

∑

0, where aɵ =  a = 1,

for k = 0,1,⋯,m − p, for k = m − ( p − 1),⋯, m.

Eq. (4.43) can be rewritten as m

T⋅ Hˆ mp ( z ) =

∑ ( bˆ − bˆ k

k =0

k− p

aˆ ) z − k

ˆ −p 1 − az

,( bˆ i = 0 for i < 0).

(4.44)

The synthesis technique requires of Eq. (4.44) requires the following necessary and suficient conditions: aˆ= a= 1,

(4.45)

bk = bk > 0,( k = 0,1,⋯, p − 1)

(4.46)

129

Photonic Computing Processors

FIGURE 4.10 Graphical illustration of the performance of the programmable trapezoidal INCOI processor Hˆ 11 ( z ) in processing a rectangular input pulse sequence. (a) TW-SLA gain proile of G0 [n]. (b) TW-SLA gain proile of G1[n]. (c) Rectangular input pulse (rectangular proile assumed) sequence x[n] = 1 for n ≥ 0. (d) Output pulse sequence y[n].

k

bˆ k =

∑b

k − pq

a q > 0, ( k = p, p + 1,⋯, m)

(4.47)

q=0

where bi = 0 for i < 0. The advantage of this synthesis method is evident from Eqs. (4.45) through (4.47), where the new optical tap coeficient ( bˆ k − bˆ k − p aˆ ) can be made positive or negative depending on the design requirements. Table 4.1 shows that the computed optical tap coeficients are positive, showing the effectiveness of the synthesis method. As a result, the requirement for the synthesis of the incoherent INCOI processor is met. Thus, the optical synthesis technique described here provides greater design lexibility than the conventional approach.

130

Photonic Signal Processing

TABLE 4.2 Digital Tap Coeficients of Several Families of the Newton–Cotes Digital Integrators, as Computed From Eqs. (4.43)–(4.47), with Transfer Functions Expressed in the form of Eq. (4.39 or 4.42) Filter Family Trapezoidal

H mp ( z) −1

2T H11 ( z ) −1

12T H 21 ( z )

24T −1 H 31 ( z )

720T −1 H 41 ( z )

1440T −1 H 51 ( z ) Simpson’s 1/3

3T −1 H 22 ( z ) = 

90T −1 H 42 ( z )

b0

b1

b2

b3

b4

b5

1 5 9 251 475

1 8 19 646 1427

0 −1 −5 −264 −798

0 0 1 106 482

0 0 0 −19 −173

0 0 0 0 27

1 29 28

4 124 129

1 24 14

0 4 14

0 −1 −6

0 0 1

90T −1 H 52 ( z )  (8 3)T −1 H 33 ( z )

1

3

3

1

0

0

Boole

( 45 2)T H 44 ( z )

7

32

12

32

7

0

Fifth-order

288T H 55 ( z )

95

375

250

250

375

95

Simpson’s 3/8

−1

−1

4.2.2.3 Design of a Programmable Optical Integrating Processor Based on the optical synthesis technique described in the previous section, the design of a programmable INCOI processor is outlined in this section. 4.2.2.3.1 Adaptive Algorithm for the INCOI Processor One possible approach for overcoming the marginal stability of the INCOI processor is to replace the IIR ilter, as described by Eq. (4.42), by another FIR ilter knowing that 1 = 1 − az − p

∞

∑ (az

−p i

(4.48)

).

i =0

This approach is not practically feasible because of the ininite number of taps involved in the FIR ilter. However, if the order of the INCOI processor is low (e.g., p = 1) and the duration of the input pulse sequence to be processed is short (e.g., a 20-tap INCOI processor is suficient to process a signal with a duration of 20 sampling intervals), then such a technique can be useful for this speciic requirement. The drawback of this method is that only the positive tap coeficients of the two cascaded FIR ilters can be optically implemented with incoherent FOSPs. As a result, the technique is restricted to the INCOI processor with m = p. An adaptive algorithm for overcoming the marginal stability of the INCOI processor is described here. The INCOI impulse response hˆmp [n] is given by the inverse z-transform of Eq. (4.48) as m

T −1 hˆ mp [n] = bˆ 0δ [n] + ⋯ + bˆ m− pδ [n − ( m − p)] +

∑

k = m−( p −1)

∞

bˆ k aˆ ( n−k )

p

∑δ [n − (k + pq)] q =0

(4.49)

131

Photonic Computing Processors

where the unit-sample sequence is deined as25 1, δ [n] =  0,

for n = 0,

(4.50)

for n ≠ 0,

and the delayed unit-sample sequence is deined as 1, δ [n − i ] =  0,

for n = i ,

(4.51)

for n ≠ i ,

where n is the discrete-time index, i.e., n = 1 means one unit-time delay T. Optical implementation of the positive tap coeficients in Eq. (4.49), requires adaptive control of the optical gain values such that m

T −1 hˆ mp [n] = G0 [0] + ⋯ + Gm − p [m − p] +

∑

Gk [n] aˆ ( n − k ) p,

(4.52)

k = m − ( p −1)

where the time-variant optical gains are given by Gk [n] n=k = bˆ k ,( k = 0,1,⋯,m − p),

(4.53)

∞

Gk [n] = bˆ k aˆ −( n−k )

p

∑ δ[n − (k + pq)],(k = m − ( p − 1),⋯, m)

(4.54)

q =0

where 0 < aˆ < 1. The summation term in Eqs. (4.49) and (4.54) shows that the pole value aˆ is not strictly required to be aˆ = 1 but can take any value in the range of 0 < aˆ < 1 while still maintaining the characteristics and overcoming the marginal stability of the INCOI processor. The dynamic range of the time-variant optical gain in Eq. (4.54) can be increased by setting the pole value aɵ as close to unity as possible (e.g., aˆ = 0.95) but not so close to unity that the original problem of marginal stability reappears. 4.2.2.3.2 Analysis of the Programmable INCOI Processor The traveling-wave semiconductor laser ampliiers (TW-SLAs) are considered to be the optical gain  elements in the programmable INCOI processor because of their fast switching speed (see Section 4.2.2). The TW-SLA can operate at either in the 1300 nm O-band or 1550 nm C-band. The TW-SLA gain depends on both the injection current and the injected light power; it increases with increasing injection current but decreases with increasing injected power. Thus, the injected power level must be chosen to within the range over which the TW-SLA gate can be driven into saturation where the gain does not differ signiicantly. With this ixed level of input power at a particular operating wavelength, the required TW-SLA gain can be obtained from the injection current source driving the gate. The effects of polarization sensitivity, arising from different transverse electric (TE) and transverse magnetic (TM) mode coninement factors, and the ampliier spontaneous emission (ASE) noise of the TW-SLAs are ignored here because they can be easily overcome in practice. A polarization insensitive TW-SLA has been demonstrated to be capable of achieving an effective gain of up to 20 dB, with less than 1 dB polarization sensitivity, less than 1 dB spectral gain ripple, and a 3 dB bandwidth of 55 nm. The ASE noise (~7 dB noise igure) can be minimized by means of a tunable optical ilter with suficient narrow bandwidth. Figure 4.9 shows a block diagram of the proposed microprocessor ( µ P ) controlled INCOI processor. The following discussion focuses on the µ P-controlled TW-SLAs. The software-controlled

132

Photonic Signal Processing

µ P chip is used to control the injection current sources driving the TW-SLAs to provide the required time-variant optical gains according to Eq. (4.53). Figure 4.10 shows a graphical illustration of the performance of the programmable trapezoidal INCOI processor Hˆ 11( z ) in processing a rectangular input pulse sequence, x[n] = 1 for n ≥ 0. Figure  4.10a and b show, respectively, the proiles of the time-variant TW-SLA gains G0 [n] and G1[n]. From Eq. (4.53), the time-variant TW-SLA gain Gk [n] n=k is only active at the appropriate discrete time n. This is shown in Figure 4.10a where G0 [n] n=0 = 0.5 is only applied at time n = 0. It is assumed that G0 [n] has already reached its steady-state value when the irst pulse of the input pulse sequence arrives. Figure 4.10b shows that the time-variant TW-SLA gain G1[n] is active at every clock period Tµ of the µ P and takes TSLA, the switching time of the TW-SLA, to reach its steadystate and thereby provides the required gain G1[n] according to Eq. (4.53). From Figure 4.10a–c, the following timing requirements must be met in order to obtain optimum performance of the programmable INCOI processor: τ w T . The next timing requirement assumes that each input pulse sequence is processed by a different family of the INCOI processor. The pulse strobe period, Tpsp, deined as the time separation between the irst pulse of the present input pulse sequence and the irst pulse of the next input pulse sequence, and the programmability period, Tpr , deined as the time separation between the setup of two consecutive programs, must be equal to the INCOI processor time, i.e.

4.2.2.5.5 Remarks This section describes the processing accuracy of the ideal and non-ideal INCOI processor in the time domain. For analytical simplicity, the timer instant n means nT and the sampling period is assumed to be T = 1. The processing accuracy of the INCOI processor is evaluated as Error Response =

True Integral − Output Pulse Sequence × 100 % True Integral

(4.79)

where the True Integral corresponds to the true integral of the input pulse sequence x [ n], and the Output Pulse Sequence corresponds to the convolution of the impulse response hɵ mp [n] of the INCOI processor with the input pulse sequence x [ n]. There are two major error sources in the implementation of the programmable INCOI processor. The irst arises from the deviation of the optical tap coeficient, that is ∆ bˆ k , caused by the inaccuracy in the time-variant TW-SLA gain Gk [n], from its nominal value bˆ k . However, the effect of the deviation of the iber loop gain value (i.e., ∆ aˆ ), caused by the inaccuracy in the TW-SLA gain GR2 [n], from its nominal value aˆ can easily be included in Gk [n]. The second error source is due to the deviation of the unit-time delay from its nominal value T, which is ∆T, caused by the inaccurate cutting of iber lengths. The effects of the detector noise and crosstalks of the optical switches are assumed to be negligible here. The performance of the trapezoidal INCOI processor Hˆ 11( z ) in processing the linear input pulse sequence is now considered in detail. The basic time delay of the INCOI processor is deined as T1 − T0 = T , which is used as a basis for the higher-order delays (i.e., Tk − T0 = kT , k = 2, 3,⋯ , m) of the FIR FOSP for m ≥ 2 and for the iber loop delay T p,IIR = pT of the IIR FOSP. Thus, the trapezoidal integrator requires the nominal value of the iber loop delay T1,IIR of the IIR FOSP to be exactlyT (i.e., T1,IIR = T ) and any deviation from this nominal value is denoted as ∆T1,IIR . Table 4.3 shows three cases of the ±5% deviation of the iber loop delay ∆T1,IIR T1,IIR and their corresponding ±5% deviations of all the possible sets of the optical tap coeficients {∆ bˆ 0 bˆ 0 , ∆ bˆ 1 bˆ 1 }. The percent relative errors (REs) at n = 20 are also shown for each case. Figures 4.11 through 4.13 show the percentage error responses of the trapezoidal integrator for the three cases. Curves Ai , Bi and Ci (i = 1,⋯ , 9) show the error responses of cases A, B, and C with the corresponding ith set of optical tap coeficients. The solid curve A1 in Figure 4.11a shows the performance of the ideal trapezoidal integrator, where there are no deviations of the iber loop length (i.e., ∆T1,IIR = 0 ) and the optical tap coeficients (i.e., ∆bɵ 0 = ∆bɵ 1 = 0) from their nominal values. Curve A1 shows that the trapezoidal integrator has zero processing error at the sampling

137

Photonic Computing Processors

TABLE 4.3 Optical Tap Coeficients of Several Families of the Synthesized INCOI Processor as Computed from Eqs. (3.44b) and (3.44c) INCOI Family Trapezoidal

bɵ 5

1.0

0

0

0

0

0.4167

1.0833

1.0

0

0

0

0.375

1.1667

0.9583

1.0

0

0

0.3486

1.2458

0.8792

1.0264

1.0

0

0.3299

1.3208

0.7667

1.1014

0.9812

1.0

0.3333

1.3333

0.6667

0

0

0

0.3222

1.3778

0.5889

1.4222

0.5778

0

0.3111

1.4333

0.4667

1.5889

0.40

1.6

0.375

1.125

1.125

0.50

0

0

0.3111

1.4222

0.5333

1.4222

0.6222

0

0.3299

1.3021

0.8681

0.8681

1.3021

0.6597

−1

T Hˆ 22 ( z ) = T −1Hˆ 42 ( z ) T −1Hˆ 52 ( z )

Fifth-order

bɵ 4

0.50

T Hˆ 51 ( z )

Boole

bɵ 3

T −1 Hɵ 11 ( z ) T −1 Hɵ 21 ( z )

−1

Simpson’s 3/8

bɵ 2

bɵ 0

T −1 Hɵ 31 ( z ) T −1Hˆ 41 ( z ) Simpson’s 1/3

bɵ 1

Hɵ mp (z )

−1

T Hˆ 33 ( z ) −1

T Hˆ 44 ( z ) −1

T Hˆ 55 ( z )

FIGURE 4.11 Relative error response (%) of the trapezoidal INCOI processor Hɵ 11 ( z) in processing the linear input pulse sequence x[n] = n for n ≥ 0. (a) Cases A1–A4, and (b) Cases A5–A9 correspond to the conditions given in Table 4.3. They are given in the Appendix.

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FIGURE 4.12 Relative error response (%) of the trapezoidal INCOI processor Hɵ 11 ( z) in processing the linear input pulse sequence x[n] = n for n ≥ 0. (a) Cases B1–B4, and (b) Cases B5–B9 correspond to the conditions given in Table 4.3.

FIGURE 4.13 Relative error response (%) of the trapezoidal INCOI processor Hɵ 11 ( z) in processing the linear input pulse sequence x[n] = n for n ≥ 0. (a) Cases C1–C4, and (b) Cases C5–C9 correspond to the conditions given in Table 4.3.

Photonic Computing Processors

139

FIGURE 4.14 Relative error response (%) of the trapezoidal INCOI processor Hˆ 11 ( z ) in processing the linear input pulse sequence x[n] = n for n ≥ 0. (a) Cases A1–A4 (See Appendix), and (b) Cases A5–A9 correspond to the conditions given in Table 4.3.

instants (i.e., n = 0,1, 2,⋯) in processing the linear input pulse sequence. This is as expected from the numerical integration scheme where the trapezoidal rule is equivalent to approximating the area of the trapezoid under the straight line.42 However, Figures 4.14 through 4.16 show that the processing errors are large between the sampling instants and eventually reach their steady states after n = 20 sampling intervals, beyond which they do not differ signiicantly. The steady-state errors of Figures 4.15, and 4.16 are tabulated in Table 4.3, which shows that the cases {A1,A4,A7}, {B1,B4,B7} and {C1,C4,C7}, where ∆ bˆ1 = 0 , have smaller processing errors than other sets belonging to the same cases. Thus, the performance of the trapezoidal integrator is greatly affected by large deviations of bˆ1. In addition, the absolute values of the steady-state errors for all cases in Tables 4.1 and 4.2 are less than 6%. Thus, the ±5% parameter deviation may be considered as an acceptable upper bound value in the implementation of the programmable INCOI processor. Next, the performances of several families of the ideal INCOI processor are analyzed for several types of input pulse sequence. It was found that the ideal Simpson’s 1/3 integrator Hɵ 22 ( z ) has zero processing error of the parabolic input pulse sequence (i.e., x[n] = n2, n ≥ 0) at the sampling instants. This is also expected because the numerical integration scheme was employed.39 From Table 4.4, the Simpson’s 1/3 integrator Hˆ 22 ( z ) has higher processing accuracy of the cubic input pulse sequence (i.e., x[n] = n3 , n ≥ 0) than other families of the INCOI processor, and the error rapidly decreases with time and eventually converges to zero in the steady state. For the fourth-order polynomial input pulse sequence (i.e., x[n] = n4, n ≥ 0), Table 4.5 shows that the integrator Hɵ 52 ( z ) outperforms other families of the INCOI processor and has negligibly small processing error for n ≥ 3. 39

S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 2nd ed., Singapore: McGraw-Hill, 1989.

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FIGURE 4.15 Relative error response (%) of the trapezoidal INCOI processor Hˆ 11 ( z ) in processing the linear input pulse sequence x[n] = n for n ≥ 0. (a) Cases B1–B4, and (b) Cases B5–B9 correspond to the conditions given in Table 4.3.

FIGURE 4.16 Relative error response (%) of the trapezoidal INCOI processor Hˆ 11 ( z ) in processing the linear input pulse sequence x[n] = n for n ≥ 0. (a) Cases C1–C4, and (b) Cases C5–C9 correspond to the conditions given in Table 4.3.

141

Photonic Computing Processors

TABLE 4.4 Relative Errors (REs) (%) at n = 20 of the Trapezoidal INCOI Processor Hɵ 11(z ), with ± 5% % Deviations of the Optical Tap Coeficients and ± 5% % Deviation of the Fiber Loop Delay, in Processing the Linear Input Pulse Sequence x[n] = n for n ≥ 0 Case A

Case B

∆T1,IIR =0 T1,IIR Set 1 2 3 4 5 6 7 8 9

Case C

∆T1,IIR = = + 0.05 T1,IIR

∆T1,IIR = − 0.05 T1,IIR

∆ bɵ 0 bɵ 0

∆ bɵ 1 bɵ 1

RE (%)

∆ bɵ 0 bɵ 0

∆ bɵ 1 bɵ 1

RE (%)

∆ bɵ 0 bɵ 0

∆ bɵ 1 bɵ 1

RE (%)

0 0 0 −0.05 +0.05 +0.05 −0.05 −0.05 −0.05

0 +0.05 −0.05 0 +0.05 −0.05 0 +0.05 −0.05

0 −4.75 +4.75 −0.25 −0.5 +4.5 +0.25 −4.5 +5.0

0 0 0 +0.05 +0.05 −0.05 −0.05 −0.05 −0.05

0 +0.05 −0.05 0 +0.05 −0.05 0 +0.05 −0.05

+0.463 −4.625 +5.19 +0.123 −4.514 +4.491 +0.712 −4.016 5.439

0 0 0 +0.05 +0.05 +0.05 −0.05 −0.05 −0.05

0 +0.05 −0.05 0 +0.05 −0.05 0 +0.05 −0.05

−0.488 −5.261 +4.286 −0.738 −5.512 +4.036 −0.237 −5.011 +4.537

TABLE 4.5 Relative Errors (%) of Several Families of the Ideal INCOI Processor, at the Sampling Instants, in Processing the Cubic Input Pulse Sequence x[n] = n3 for n ≥ 0. Error at Time n = 0 Is Undeined Time, n ˆ mp ( z ) H

1

Hɵ 11 ( z ) Hɵ 21 ( z )

−100

−25

−11.1

−6.25

−4.0

−2.5E−03

−66.7

−10.4

−3.29

−1.43

−0.75

−1.25E−05

Hɵ 31 ( z ) Hɵ 41 ( z )

−50

−4.17

−0.82

−0.26

−0.107

−4.17E−08

−39.4

−0.87

−0.041

−0.013

−0.0053

−2.08E−09

Hɵ 51 ( z )

−31.9

1.007

0.051

−0.013

−0.0053

−2.08E−09

−33.3 −28.9 −24.4

0 1.11 1.94

−0.41 −0.302 −0.41

1.11E–14 0.0694 0.104

−0.053 −0.0391 −0.053

0 1.11E−08 1.67E−08

Hɵ 33 ( z )

−50

−3.13

0

−0.195

−0.08

−3.13E−08

Hɵ 44 ( z )

−24.4

2.22

−0.302

0

−0.0391

0

Hɵ 55 ( z )

−31.9

1.48

0.29

−0.125

−1.82E−14

0

Hɵ 22 ( z ) = Hɵ 42 ( z ) Hɵ 52 ( z )

2

3

4

5

200

It was also found that the overall performance of the ideal Boole’s integrator Hɵ 44 ( z ) is superior to other families of the INCOI processor in processing the rth-order polynomial input pulse sequence (i.e., x[n] = nr , r ≥ 5 for n ≥ 0). From Tables 4.4 and 4.5, as the order of the polynomial input pulse sequence is increased (i.e., r ≥ 2), the higher-order trapezoidal integrator Hɵ m1( z ) m≥2 outperforms the lowest-order trapezoidal integrator Hɵ 11( z ). This shows the advantage of using a higher-order trapezoidal integrator.

142

Photonic Signal Processing

For the exponential input pulse sequence, that is x[n] = exp( − n w ) for n ≥ 0, with the FWHM pulse width w = 50. Table 4.6 shows that the ideal trapezoidal integrator Hɵ 11( z ) has the smallest processing error among other families of the INCOI processor, and its steady-state error converges to −1.006% at n = 300. Its processing error was also found to decrease with increasing pulse width, for example, when w = 100 the steady-state error was reduced to −0.527% at n = 300, which is about half of that for the case w = 50 (Table 4.7). TABLE 4.6 Relative Errors (%) of Several Families of the Ideal INCOI Processor, at the Sampling Instants, in Processing the Fourth-order Polynomial Input Pulse Sequence x[n] = n4 for n ≥ 0. Error at Time n = 0 is Undeined Time, n ˆ mp ( z ) H

1

2

Hˆ 11 ( z ) Hˆ 21 ( z )

−150

−40.6

−18.3

−10.4

−6.64

−4.2E–03

−108

−21.1

−7.17

−3.23

−1.72

−3.1E–05

Hˆ 31 ( z ) Hˆ 41 ( z )

−87.5

−11.9

−2.88

−0.99

−0.427

−1.97E–07

−74.3

−6.62

−0.926

−0.219

−0.072

−7.02E–10

Hˆ 51 ( z )

−64.9

−3.1

−0.0386

1.39E–14

1.82E–14

−7.51E–13

Hˆ 22 ( z ) Hˆ 42 ( z )

−66.7

−4.17

−0.823

−0.26

−0.107

−4.17E–08

−61.1

−2.08

−0.274

−0.065

−0.0213

−2.08E–10

Hˆ 52 ( z )

−55.6

−0.174

0

0

0

−2.03E–13

Hˆ 33 ( z )

−87.5

−11.3

−1.85

−0.525

−0.26

−9.39E–08

Hˆ 44 ( z )

−55.6

0

0.229

−1.39E–14

−0.0178

−1.91E–13

Hˆ 55 ( z )

−64.9

−2.81

0.37

0.0634

0

3

4

5

200

0

TABLE 4.7 Relative Errors (%) of Several Families of the Ideal INCOI Processor, at the Sampling Instants, in Processing the Exponential Input Pulse Sequence x[n] = exp(− n w ) for n ≥ 0 Where w = 50. Error at Time n = 0 is Undeined Time, n ˆ mp ( z ) H

1

2

3

4

5

300

Hˆ 11 ( z ) Hˆ 21 ( z )

−50.51

−25.51

−17.18

−13.01

−10.51

−1.006

−50.67

−25.59

−17.22

−13.05

−10.54

−1.006

Hˆ 31 ( z ) Hˆ 41 ( z )

−54.96

−25.59

−17.23

−13.05

−10.54

−1.006

−60.35

−24.22

−17.23

−13.05

−10.54

−1.006

Hˆ 51 ( z )

−66.06

−21.31

−17.89

−13.05

−10.54

−1.006

Hˆ 22 ( z )= Hˆ 42 ( z )

−67.67

−17.00

−23.01

−8.67

−14.08

−0.668

−71.06

−14.71

−24.55

−7.503

−15.02

−0.578

Hˆ 52 ( z )

−75.57

−10.71

−27.63

−5.168

−16.91

−0.398

Hˆ 33 ( z )

−50.76

−32.01

−12.89

−13.07

−13.19

−0.752

Hˆ 44 ( z )

−74.45

−13.56

−23.79

−8.093

−15.49

−0.624

Hˆ 55 ( z )

−64.17

−25.54

−12.67

−17.52

−6.933

−0.661

Photonic Computing Processors

143

The processing errors of the families of the INCOI processor are large over the initial time interval but eventually converge to a negligibly small, if not zero, value in the steady state. Note that, if the deviation of the basic time delay satisies the condition at which the performance of the INCOI processor is not greatly affected by this factor. It is worth mentioning that the rectangular integrator, whose transfer function is given by Hˆ R ( z ) = z −1 / (1 − z −1 ), has zero processing error of the constant (or rectangular) input pulse sequence, such as that shown in Figure 5.4c, at the sampling instants.

4.2.3 sectiOn remarks • A generalized theory of the Newton–Cotes digital integrators has been developed based on a programmable INCOI processor that has been designed using a microprocessor, iberoptic architectures, optical switches, and optical and semiconductor ampliiers. • The pth-order programmable INCOI processor has high processing accuracy of the pthorder polynomial input pulse sequence (i.e., x[n] = n p , p ≥ 1), while the trapezoidal processor, Hˆ 11( z ), outperforms other higher-order processors in processing the exponential pulse (i.e., x[n] = exp( − n w ) , n ≥ 0). In general, the incoherent processing accuracy of a particular processor order depends very much on the type of input pulse. • Optical integration is a new concept with many potential applications. One example is to apply the trapezoidal optical integrator described here to the design of an optical darksoliton generator, which is outlined in Section 4.3.

4.3

HIGHER-DERIVATIVE FIR OPTICAL DIFFERENTIATORS

In this section, a theory of higher-derivative FIR (inite impulse response) optical differentiators is proposed (see Figure 4.17). Section 4.3.1 describes underlying principle of digital differentiators and their design techniques and applications. Section 4.3.3 presents a theory of the qth-order pthderivative FIR digital differentiator whose derivation is given in Section 4.4. Section 4.3.3 describes the synthesis of a qth-order pth-derivative FIR optical differentiator using an optical transversal ilter structure, as described in Sections 3.2 and 3.3, which consists of integrated-optic components. The theory of coherent integrated-optic signal processing described in Chapter 2 is employed in this chapter where electric-ield amplitude signals are considered.

FIGURE 4.17 Schematic diagram of the proposed (q + 1)-tap FIR coherent optical ilter used to synthesize the qth-order pth-derivative FIR optical differentiator. TC: tunable coupler, PS: phase shifter, and PC: polarization controller.

144

4.3.1

Photonic Signal Processing

intrODuctiOn

Section 4.2 has established that a digital integrator can be used to simulate the behavior of an analog integrator. In this chapter, a digital differentiator is also used to simulate the behavior of an analog differentiator. Like the digital integrator, a digital differentiator also forms an integral part of many practical signal processing systems because the time derivative of signals is sometimes required for further use or analysis. Digital differentiators are useful in modifying the shape of the signal; they can be used to ind positive-going or negative-going slopes, maxima or minima, or point of greatest slope, where the differentiated output would be positive or negative, zero, or maximum, respectively. First-derivative and second-derivative digital differentiators have been used in the design of compensators for control systems,40 for monitoring electrocardiograph (ECG) signals,41 in the study of velocity and acceleration in human locomotion,42 in the analysis of radar signals in radar systems,43 and for the calculation of geometric moments in optical systems.44 A digital differentiator is a processor whose output pulse sequence is obtained by approximating the derivative of a continuous-time signal from the samples of that signal. A continuous-time signal x(t), whose values are known at the discrete time t = nT for n = 0,1,2… where T > 0 is the period between successive samples, can be differentiated by a digital differentiator. The frequency response of an ideal pth-derivative digital differentiator is given by45 p 0 ≤ ωT (2π ) ≤ 1 2 , ( jωT ) , H d (ω ) =  p  [ j (2π − ωT )] , 1 2 < ωT (2π ) ≤ 1,

(4.80)

where j = −1, p = 1, 2,⋯ , ω is the angular frequency and T is the sampling period of the differentiator. The output pulse sequence of the ideal pth-derivative digital differentiator y ( p ) ( nT ) approximates the true pth-derivative of the continuous-time signal x(t ) according to y ( p ) ( nT ) =

d p x (t ) . dt p t = nT

(4.81)

Digital differentiators can be designed in two ways for formulation in either the time or the frequency domains approach as follows. In the time-domain approach, irst-derivative FIR digital differentiators can be designed by using one of the many classical numerical differentiation algorithms such as the Newton, Bessel, Everett, Stirling, and Lagrange formulas.46,47,48,49,50 The underlying principle of these numerical differentiation algorithms is to it a continuous-time interpolation polynomial x(t) to a given input pulse sequence f(nT), where f(nT) = x(nT), which is then differentiated to give y(t ) = dx(t ) / dt . Sampling the differentiated continuous-time polynomial y(t ) (by a digital differentiator) at the discrete time t = nT 40

41

42

43 44

45

46

47 48

49 50

G. F. Franklin, J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 2nd ed., Reading, MA: Addison– Wesley, 1990 W. J. Tompkins and J. G. Webster, (Eds.), Design of Microcomputers-based Medical Instrumentation, Englewood Cliffs, NJ: Prentice-Hall, 1981. S. Usui and I. Amidror, Digital low-pass differentiation for biological signal processing, IEEE Trans. Biomed. Eng., BME-29, 686–693, 1982. M. I. Skolnik, Introduction to Radar Systems, 2nd ed., Boston, MA: McGraw-Hill, 1980. B. V. K. Vijaya Kumar and C. A. Rahenkamp, Calculation of geometric moments using Fourier plane intensities, Appl. Opt., 25, 997–1007, 1986. L. R. Rabiner and R. W. Schafer, On the behavior of minimax relative error FIR digital differentiators, Bell Syst. Tech. J., 53, 333–360, 1974. W. J. Tompkins and J. G. Webster, (Eds.), Design of Microcomputers-based Medical Instrumentation, Englewood Cliffs, NJ: Prentice-Hall, 1981. R. Vich, Z Transform Theory and Applications, Norwell, MA: Kluwer Academic Publishers, 1987. R. Pintelon and J. Schoukens, Real-time integration and differentiation of analog signals by means of digital iltering, IEEE Trans. Intrum. Meas., 39, 923–927, 1990. M. Abramowitz and I. A. Segun, Handbook of Mathematical Function, New York: Dover Publications, 1964. S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 2nd ed., Singapore: McGraw-Hill, 1989.

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yields the output pulse sequence y( nT ) , which effectively approximates the true derivative of a continuous-time signal, that is y( nT ) =

dx(t ) . dt t = nT

(4.82)

The magnitude responses of these digital differentiators generally approximate the ideal magnitude response reasonably well over the lower frequency band 0 ≤ ωT (2π ) ≤ 1 4. Thus, these differentiators are normally referred to as narrow-band digital differentiators. In the frequency-domain approach, irst-derivative and higher-derivative FIR digital differentiators satisfying prescribed speciications of the ideal frequency response can be designed by using the minimum method,51,52 the Fourier series method in conjunction with the Kaiser window function,53,54 the Fourier series method in conjunction with accuracy constraints,55,56 the eigenilter method,57,58 and the least-squares methods.59,60 Because of the constraints imposed on the frequency responses of these digital differentiators, they are normally referred to as frequencyselective differentiating ilters that, in addition to performing the function of signal differentiation, are capable of passing as well as rejecting certain frequency components of the signal. That is, they can be designed to have a narrow-band,61,62 mid-band,63,64 or wide-band,65,66,67,68,69,70,71,72 magnitude 51

52

53 54

55 56

57

58

59

60

61

62

63

64

65 66

67

69

71

68

70

72

L. R. Rabiner and R. W. Schafer, On the behavior of minimax relative error FIR digital differentiators, Bell Syst. Tech. J., 53, 333–360, 1974. C. A. Rahenkamp and B. V. K. Vijaya Kumar, Modiications to the McClellan, Parks, and Rabiner computer program for designing higher order differentiating FIR ilters, IEEE Trans. Acoust. Speech Signal Process., ASSP-34, 1671–1674, 1986. A. Antoniou, Design of digital differentiators satisfying prescribed speciications, Proc. IEE, 127, pt. E, 24–30, 1980. A. Antoniou and C. Charalambous, Improved design method for Kaiser differentiators and comparison with equiripple method, Proc. IEE, 128, pt. E, 190–196, 1981. B. Kumar and S. C. Dutta Roy, Design of digital differentiators for low frequencies, Proc. IEEE, 76, 287–289, 1988. B. Kumar and S. C. Dutta Roy, Maximally linear FIR digital differentiators for high frequencies, IEEE Trans. Circuits and Syst., CAS-36, 890–893, 1989. S. C. Pei and J. J. Shyu, Design of FIR Hilbert transformers and differentiators by eigenilter, IEEE Trans. Circuits and Syst., 35, 1457–1461, 1988. S. C. Pei and J. J. Shyu, Eigenilter design of higher order digital differentiators, IEEE Trans. Acoust. Speech Signal Process., 37, 505–511, 1989. K. Sasayama, M. Okuno, and K. Habara, Coherent optical transversal ilter using silica-based waveguides for highspeed signal processing, J. Light. Technol., 9, 1225–1230, 1991. S. Sunder and R. P. Ramachandran, Least-squares design of higher order nonrecursive differentiators, IEEE Trans. Signal Process., 42, 956–961, 1994. A narrow-band, mid-band, or wide-band digital differentiator has a magnitude response that closely matches the ideal magnitude response over the frequency band of 0 ≤ ωT ( 2π ) ≤ 1 4 , 1 8 ≤ ωT ( 2π ) ≤ 3 8 or 0 ≤ ωT ( 2π ) ≤ 1 2, respectively. R. R. R. Reddy, B. Kumar, and S. C. Dutta Roy, Design of eficient second and higher order FIR digital differentiators for low frequencies, Signal Process., 20, 219–225, 1990. B. Kumar and S. C. Dutta Roy, Maximally linear FIR digital differentiators for midband frequencies, Int. J. Circ. Theor. App., 17, 21–27, 1989. B. Kumar and S. C. Dutta Roy, Design of eficient FIR digital differentiators and Hilbert transformers for midband frequency ranges, Int. J. Circ. Theor. App., 17, 483–488, 1989. G. P. Agrawal, Fiber-Optic Communication Systems, Hoboken, NJ: John Wiley & Sons, 1992. L. R. Rabiner and R. W. Schafer, On the behavior of minimax relative error FIR digital differentiators, Bell Syst. Tech. J., 53, 333–360, 1974. C. A. Rahenkamp and B. V. K. Vijaya Kumar, Modiications to the McClellan, Parks, and Rabiner computer program for designing higher order differentiating FIR ilters, IEEE Trans. Acoust. Speech Signal Process., ASSP-34, 1671–1674, 1986. S. C. Pei and J. J. Shyu, Design of FIR Hilbert transformers and differentiators by eigenilter, IEEE Trans. Circuits Syst., 35, 1457–1461, 1988. S. Sunder, W. S. Lu, A. Antoniou, and Y. Su, Design of digital differentiators satisfying prescribed speciications using optimisation techniques, Proc. IEE, 138, pt. G, 315–320, 1991. A. Antoniou and C. Charalambous, Improved design method for Kaiser differentiators and comparison with equiripple method, Proc. IEE, 128, pt. E, 190–196, 1981. S. C. Pei and J. J. Shyu, Eigenilter design of higher order digital differentiators, IEEE Trans. Acoust. Speech Signal Process., 37, 505–511, 1989. B. Kumar and S. C. Dutta Roy, Maximally linear FIR digital differentiators for high frequencies, IEEE Trans. Circuits Syst., CAS-36, 890–893, 1989.

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frequency response, depending on the application. Although impressive frequency-domain performances can generally be achieved by these frequency-selective differentiating ilters, the required ilter order is generally very high (e.g., 30–40 taps are common). In both the time-domain and frequency-domain approaches, the performances of the digital differentiators have usually been evaluated in the frequency domain by using the ideal frequency response (usually magnitude but not phase response) as a basis. This is because the differentiating accuracy of these digital differentiators in the time domain is dificult to assess as this would depend on a speciic application, and hence the actual shape of the signal. It is obvious that a good digital differentiator must be capable of achieving high differentiation accuracy in the time domain while still able to reject unwanted frequency components. For example, the high frequency components of a digital signal are often corrupted with wide-band noise. A narrow-band digital differentiator would be useful in this case.73,74,75,76 Unlike the digital differentiators, which have been studied for some time, optical differentiation is still a new concept in the area of optical signal processing. Although a 3-tap coherent optical transversal (or FIR) ilter using silica-based waveguides integrated on a silicon substrate has been experimentally demonstrated as a second-derivative optical differentiator, no theoretical background was given.77 In addition, a iber-optic ring resonator has been claimed as a irst-derivative optical differentiator under the resonance condition, but it is not a true differentiator and hence would suffer from low processing accuracy.78 In this section, a theory of higher-derivative FIR optical differentiators using integrated-optic structures is described. Most of the work presented here has been described by Ngo and Binh.79 The derivation of a theory of higher-derivative FIR digital differentiators, which was not presented in reference,79 is given in Appendix A.

4.3.2

higher-Derivative Fir Digital DiFFerentiatOrs

The transfer function of the qth-order pth-derivative FIR digital differentiator [i.e., H q( p ) ( z )] can be generally expressed as80  −a11   −a21  ⋮   −aM 1 Where a pq =

a12 a22 ⋮ aM 2

−a13 −a23 ⋮ − aM 3

⋯ ⋯ ⋯

( −1) M a1M   TH M(1) ( z )   z −1 − 1      ( −1) M a2 M   T 2 H M( 2) ( z )   z −2 − 1  ≅    ⋮  ⋮ ⋮     ( −1) M aMM  T M H M( M ) ( z )   z − M − 1

pq , p, q = 1, 2,⋯ , M , q!

(4.83)

(4.84)

and z = e jωT is the z-transform parameter. The pulse response of the qth-order pth-derivative FIR digital differentiator is deined as yq( p ) ( nT ) = x( nT ) ∗ hq( p ) ( nT ), 73

74

75 76

77

78 79 80

(4.85)

W. J. Tompkins and J. G. Webster, (Eds.), Design of Microcomputers-based Medical Instrumentation, Englewood Cliffs, NJ: Prentice-Hall, 1981. S. Usui and I. Amidror, Digital low-pass differentiation for biological signal processing, IEEE Trans. Biomed. Eng., BME-29, 686–693, 1982. M. I. Skolnik, Introduction to Radar Systems, 2nd ed., Boston, MA: McGraw-Hill, 1980. B. V. K. Vijaya Kumar and C. A. Rahenkamp, Calculation of geometric moments using Fourier plane intensities, Appl. Opt., 25, 997–1007. K. Sasayama, M. Okuno, and K. Habara, Coherent optical transversal ilter using silica-based waveguides for highspeed signal processing, J. Light. Technol., 9, 1225–1230, 1991. G. S. Pandian and F. E. Seraji, Optical pulse response of a iber ring resonator, Proc. IEE, 42, part J, 235–239, 1991. N. Q. Ngo and L. N. Binh, Theory of a FIR optical digital differentiator, Fiber and Integrated Optics, 14, 359–385, 1995. The derivation of Eq. (6.4) is given in Appendix B where Eq. (B.12) corresponds to Eq. (6.4).

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which approximates the true pth-derivative of the input pulse sequence x( nT ) as yq( p ) ( nT ) ≅

d p x (t ) dt p t = nT

(4.86)

where hq( p ) ( nT ) is the impulse response of H q( p ) ( z ). The transfer function of the qth-order pth-derivative FIR digital differentiator takes the general form of H

( p) q

b  ( z ) =  max p  T 

q

∑ b(k ) z

−k

, p, q = 1, 2,⋯ , M ,

(4.87)

k =0

where −1 ≤ b( k ) ≤ 1 is the normalized tap coeficient and bmax ≥ 1 is the normalization factor. The transfer functions of several families of the digital differentiators, as computed from Eq. (4.82) for M = 4, are tabulated in Table 4.8. Note that the signs of the tap coeficients alternate such that even are positive and odd negative. For the special case where q = p , the transfer function of the qth-order pth-derivative FIR digital differentiator is generally given by T p H q( p ) ( z )

4.3.3

q= p

(

= 1 − z −1

)

p

(4.88)

.

synthesis OF higher-Derivative Fir Optical DiFFerentiatOrs

The characteristics of the higher-derivative FIR digital differentiators and the planar lightwave circuit (PLC) technology outlined in Section 4.2.3 are used to synthesize higher-derivative FIR optical differentiators. Coherent integrated-optic signal processing of electric-ield amplitude signals is considered here. The unmodulated signal of the optical source is assumed to be externally modulated by an optical intensity modulator, which minimizes laser chirp as well as permitting high-speed modulation and hence high-speed signal processing. It is also assumed that the optically encoded signals to be processed by the optical differentiator are modulated onto the optical carrier whose coherence time is

TABLE 4.8 Normalized Tap Coeficients and Normalization Factor, as Computed from Eq. (6.4) for M = 4, of the Several Families of the FIR Digital Differentiators with Transfer Functions Expressed in the Form of Eq. (3.77) T p Hq( p ) ( z) (1) 1 (1) 2 (1) 3

TH ( z ) TH ( z ) TH ( z ) TH 4(1) ( z ) 2

T H T 2H T 2H

( 2) 2 ( 2) 3 ( 2) 4

( z) ( z) ( z)

bmax

b(0)

1 2 3 4

1 0.75 0.6111 0.5208

2 5

0.5 0.4

9.5

0.3070

T 3 H 3( 3) ( z ) T 3 H 4( 3) ( z )

3 12

0.3333 0.2083

T 4 H 4( 4 ) ( z )

6

0.1667

b(1) −1 −1 −1 −1

−1 −1 −0.9123 −1 −0.75

−0.6667

b(2)

b(3)

b(4)

0 0.25 0.5 0.75

0 0 −0.1111

0 0 0 0.0625

0.5 0.8 1

−0.2 −0.4912

0

0 0 0.0965

1 1 1

−0.3333

−0.3333 −0.5833 −0.6667

0 0.125 0.1667

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Photonic Signal Processing

much longer than the sampling period T of the optical differentiator. As a result, the pulse response of the optical differentiator depends on the coherent interference of the delayed signals. Figure 4.17 shows the schematic diagram of the proposed ( q +1) tap FIR coherent optical ilter,81 which is used to synthesize the qth-order pth-derivative FIR optical differentiator. The FIR coherent optical ilter essentially consists of a lx( q +1) optical splitter, an ( q +1) xl optical combiner, and ( q +1) waveguide delay lines, into each of which a tunable coupler (TC) and a phase shifter (PS) are incorporated. A coherent optical signal coming into the optical splitter will be evenly distributed to ( q +1) signals, which are then appropriately delayed by the delay lines and weighted by the TCs and PSs. These signals are then coherently collected by the optical combiner to generate the differentiated optical signal. The FIR coherent optical ilter can be constructed using the PLC technology, namely, silicabased waveguides embedded on a silicon substrate as described in Section 4.2.3. The optical splitter and optical combiner can be developed using 3-dB silica-based waveguide directional couplers (DCs), except that no erbium-doped iber ampliiers (EDFAs) are used here. In each delay line, the PS following the TC is a waveguide with a thin-ilm heater deposited on it and utilizes the thermooptic effect to induce a carrier phase change of φ ( k ). The TC is a symmetrical Mach−Zehnder interferometer (see the inset of Figure 4.17 that consists of two 3-dB DCs), two equal waveguide arms, and a thin-ilm heater, with a carrier phase change of φ(κ), attached to one of the arms for controlling the output amplitude (see Section 4.3). Neglecting the insertion loss of the 3-dB DCs, the propagation delay and waveguide birefringence of the TC, the kth TC transfer function, which corresponds to the transfer function E3 / E1 as the output port 3 over the input port 1 ields of the coupler, is given by C ( k ) = C ( k ) exp ( j ∠C ( k ) ) = 0.5 exp ( jϕ ( k ) ) − 1

(4.89)

C ( k ) = 0.5 − 0.5 cos (ϕ ( k ) )

(4.90)

2 ϕ ( k ) = cos −1 1 − 2 C ( k )  ,  

(4.91)

where ∠C ( k ) denotes the argument of C ( k ),

or

for k = 0,1,⋯ , q

and ∠C ( k ) = tan −1 sin(ϕ ( k )) ( cos(ϕ ( k )) − 1) 

(4.92)

Eq. (4.90) indicates that a desired TC amplitude can be obtained by choosing an appropriate PS phase according to Eq. (4.91), and this results in the TC phase as given by Eq. (4.92). The amplitude and phase of the TC can be changed from 0 to 1 and from − π 2 to + π 2, respectively, when ϕ ( k ) is varied from 0 to 2π . Neglecting the propagation delay and waveguide birefringence, the transfer function of the ( q +1) -tap FIR coherent optical ilter is given by ( p) Hɵ q ( z ) = lpath ⋅ ( q + 1) −1 ⋅ G ⋅

q

∑ (−1) ⋅ C (k ) exp( j∠C (k )) ⋅ exp( jφ(k )) ⋅z k

−k

(4.93)

k =0

81

The FIR coherent optical ilter described here has a similar structure to the incoherent iber-optic transversal ilter (see Figure 4.4) outlined in previous section.

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Photonic Computing Processors

where lpath is the intensity path loss that takes into account all the losses associated with each delay line. Such losses include the loss of the straight and bend waveguides, the insertion loss of the 3-dB DCs in the splitter and combiner, and the insertion loss of the TC. Additionally, ( q + 1) −1 is the coupling loss of the splitter and combiner as a result of a 3-dB coupling loss at each stage of the structure, G is the intensity gain of the EDFA, and ( −1) k = exp( jkπ ) is the phase shift factor due to the π /2 cross-coupled phase shift of the 3-dB DCs in the splitter and combiner. Synthesis of the qth-order pth-derivative FIR optical differentiator requires the equality of Eqs. (4.93) and (4.88), i.e., Hˆ q( p ) ( z ) = H q( p ) ( z ), such that G=

2 bmax ( q + 1)2 , lpathT 2 p

(4.94)

C ( k ) = b( k ) ,

(4.95)

φ ( k ) = ∠b( k ) − ∠( −1) k − ∠C ( k ),

(4.96)

φ ( k ) = −∠C ( k ), ∠b( k )= ∠( −1) k ,

(4.97)

where ∠b( k )=∠( −1) k = 0 for even k or ∠b( k )=∠( −1) k = π for odd k . Eq. (4.94) shows that the required gain G is dominated by the small value of the sampling period T, and hence several EDFAs in cascade may be required at both the input and output of the optical differentiator, depending on the application. In each kth delay line, Eq. (4.95) shows that the amplitude of the digital coeficient [i.e., 0 ≤ ωT (2π ) ≤ 1 2] can be optically implemented by the TC amplitude [i.e.,  C ( k ) ], and Eq. (4.97) shows that the PS must provide a phase shift [i.e., φ ( k )] opposite to the TC phase [i.e., ∠C ( k )]. Because of the temperature dependence of the refractive index change of the silica waveguide, the PS can also be used to compensate for the optical path-length difference resulting from imperfect fabrication of the waveguide length. Note that if, for each delay line, a non-tunable DC is used instead of the TC, then the PS is not required. However, it is dificult to practically fabricate a non-tunable DC with a precise coupling coeficient as described in Section 4.2.3.3. Thus, it is preferable to use the TC, which, in addition to implementing the digital coeficient, can accommodate for the deviations of the coupling coeficients of the 3-dB DCs in the splitter and combiner as a result of fabrication errors. Because the temperature of the silicon substrate can be maintained to within a small fraction of a degree to stabilize the refractive index of the waveguides,82 the PS can provide a very accurate phase shift. The temperature stability of the waveguides means that the optical differentiator can operate stably. Since the control of the TC amplitude and PS phase in a particular delay line is independent of those in other delay lines, the amplitude and phase of the digital coeficient can be optically implemented with high accuracy, showing the advantage of the proposed ilter structure. The polarization controller (PC) placed at the input of the optical differentiator is used to counter any birefringence induced in the iber, while the PC placed at its output is used to counter any waveguide birefringence arising from the optical differentiator. Alternatively, the waveguide birefringence may be overcome by inserting the polyamide half waveplates, acting as TE/TM mode converter, into the delay lines.

82

K. Sasayama, M. Okuno, and K. Habara, Coherent optical transversal ilter using silica-based waveguides for highspeed signal processing, J. Light. Technol., 9, 1225–1230, 1991.

150

4.3.4

Photonic Signal Processing

cOmputeD DiFFerentiatOrs OF First anD higher OrDers

The proposed qth-order pth-derivative FIR optical differentiator, Hˆ q( p ) ( z; p, q = 1, 2, 3, 4) , described in Section 4.3.2 is now analyzed. For analytical simplicity, the following assumptions are used in all igures: the discrete-time index m means m = nT , the normalized optical frequency means ωT (2π ), the normalized time t T means m, Hpq means Hˆ q( p ) ( z ), the magnitude response corresponds to Hˆ q( p ) ( z ) , and the sampling period T is set to unity. The processing accuracy of the qth-order pth-derivative FIR optical differentiator is evaluated by means of the Error Response =

True Derivative − Pulse Response × 100 % True Derivative

(4.98)

where True Derivative corresponds to the true derivative of the input pulse sequence x[m] and Pulse Response corresponds to the pulse response of the differentiator. To characterize the performance of the differentiator, the pulse response is deined as the amplitude response at the output of the optical differentiator prior to detection by an optical detector. 4.3.4.1 First-Derivative Differentiators This section analyses the performances of the qth-order irst-derivative differentiators, (1) Hɵ q ( z; q = 1, 2, 3, 4) . Figure 4.18a shows the magnitude responses of the qth-order and ideal differentiators. The magnitude response increases with increasing ilter order q. The magnitudes are zero at ωT (2π ) = 0

FIGURE 4.18 Magnitude responses of the optical differentiators. (a) qth-order irst-derivative differentiators Hˆ q(1) ( z; q = 1, 2, 3, 4) . (b) qth-order second-derivative differentiators Hˆ q( 2) ( z; q = 2, 3, 4). (c) qth-order thirdderivative differentiators Hˆ q( 3) ( z; q = 3, 4). (d) Fourth-order fourth-derivative differentiator Hˆ 4( 4 ) ( z ).

Photonic Computing Processors

151

FIGURE 4.19 Error responses of the qth-order irst-derivative differentiators Hˆ q(1) ( z; q = 1, 2, 3, 4) when processing the qth-order polynomial input pulse. (a) First-order polynomial input pulse: x[m] = m. (b) Secondorder polynomial input pulse: x[m] = m2 . (c) Third-order polynomial input pulse: x[m] = m3. (d) Fourth-order polynomial input pulse: x[m] = m4.

and ωT (2π ) = 1, and maximum at ωT (2π ) = 0.5 because there are at least one zero on the unit circle in the z-plane. Similar accounts can be made for the magnitude responses of the qth-order second-derivative, third-derivative and fourth-derivative differentiators, which are shown in Figure 4.19b–d, respectively. Figure 4.20a–d show the differentiator pulse responses when processing the Gaussian input pulse with various pulse widths83 w, where 2w is the temporal full-width of the intensity pulse at the 1 exp(1) points. The pulse responses closely resemble the true derivatives and the processing accuracy increases with increasing value of 2w. Figure 4.21a and b show the differentiator error responses when processing the Gaussian input pulse with pulse widths w = 5 and w = 20, respectively. For w = 5, Figure 4.22a shows that the second-order, third-order and fourth-order differentiators have lower processing accuracy than the irst-order differentiator for time m > 25. However, for a larger pulse width, w = 20, Figures 4.23 and 4.24b shows that the second-order, third-order and fourth-order differentiators have higher processing accuracy than the irst-order differentiator, but at the expense of having lower processing accuracy over the initial time interval [see the enlarged curves in Figure 4.21c]. A higher-order differentiator requires more hardware components than a lower-order differentiator. Thus, the irstorder differentiator is considered as the optimum ilter for processing a Gaussian pulse because of its structural simplicity and impressive performance. 83

The pulse width w corresponds to the normalized pulse width, i.e., w means wT.

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Photonic Signal Processing

FIGURE 4.20 Pulse responses of the irst-order irst-derivative differentiator Hˆ 1(1) ( z ) when processing the gaussian input pulse (i.e., x[m] = exp[− m2 (2w 2 )]) With various pulse widths w. (A) w = 5. (B) w = 10. (C) w = 15. (D) w = 20.

Figures 4.24 and 4.25 show the differentiator error responses when processing the qth-order polynomial input pulse x[m] = mq. The irst-order, second-order, third-order and fourth-order differentiators have higher processing accuracy of the irst-order, second-order, third-order and fourthorder input pulses, respectively, than other differentiator orders. Thus, for x[m] = mq, the qth-order irst-derivative differentiator has large processing error over the initial time interval 0 ≤ m < q but has zero processing error over the longer time interval m ≥ q. This is because the differentiators have been designed from the polynomial perspective (see Appendix A). 4.3.4.2 Second-Derivative Differentiators This section analyses the performances of the qth-order second-derivative differentiators, Hˆ q( 2) ( z; q = 2, 3, 4), whose magnitude responses were shown in Figure 4.23b. Figure 4.25a–d show the differentiator error responses when processing the qth-order polynomial input pulse. Figures 4.24 through 4.27 show respectively that the second-order, third-order

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Photonic Computing Processors

(a)

(b)

(c)

FIGURE 4.21 Error responses of the qth-order irst-derivative differentiators Hˆ q(1) ( z; q = 1, 2, 3, 4) when processing the Gaussian input pulse with two different pulse widths. (a) w = 5. (b), (c) w = 20 with different time scales.

and fourth-order differentiators have higher processing accuracy of the second-order, third-order and fourth-order pulses, respectively, than other differentiator orders. However, Figure 4.28d shows that the fourth-order differentiator has higher processing accuracy of the ifth-order pulse than the second-order and third-order differentiators. Thus, for x[m] = mq, the qth-order secondderivative differentiator has large processing error over the initial time interval 0 ≤ m < q but has zero processing error over the longer time interval m ≥ q. Figure 4.29 show the differentiator pulse responses when processing the Gaussian input pulse with various pulse widths w. The pulse responses closely resemble the true derivatives and the processing accuracy increases with increasing value of w. The second-order second-derivative differentiator analyzed here can be experimentally demonstrated where a square-type input pulse can be processed.

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Photonic Signal Processing

FIGURE 4.22 Error responses of the qth-order second-derivative differentiators Hˆ q( 2) ( z; q = 2, 3, 4) when processing the qth-order polynomial input pulse. (a) Second-order polynomial input pulse: x[m] = m2 . (b) Third-order polynomial input pulse: x[m] = m3 . (c) Fourth-order polynomial input pulse: x[m] = m4. (d) Fifthorder polynomial input pulse:x[m] = m5 .

Figure 4.21a and b show the differentiator error responses when processing the Gaussian pulse with pulse widths w = 5 and w = 20, respectively. For w = 5, Figure 4.26a shows that the third-order and fourth-order differentiators have lower processing accuracy than the second-order differentiator for time m > 25. However, for a larger pulse width, w = 20, Figure 4.28b shows that the third-order and fourth-order differentiators have higher processing accuracy than the second-order differentiator, but at the expense of having lower processing accuracy over the initial time interval [see the enlarged curves in Figure 4.23c]. 4.3.4.3 Third-Derivative Differentiators This section analyses the performances of the qth-order third-derivative differentiators, Hˆ q( 3) ( z; q = 3, 4), whose magnitude responses were shown in Figure 4.25c. Figure 4.25a and b show the differentiator error responses when processing the qth-order polynomial input pulse. Figure 4.25a shows that the third-order differentiator has higher processing

Photonic Computing Processors

155

FIGURE 4.23 Pulse responses of the second-order second-derivative differentiator Hˆ 2( 2) ( z ) when processing : (a ) w 5= . ( b) w 10 = . (c) w 15 = . (d ) w 20.3. the Gaussian input pulse with various pulse widths w=

accuracy of the third-order pulse than the fourth-order differentiator. However, Figure 4.25b shows that the fourth-order differentiator has higher processing accuracy of the fourth-order pulse than the third-order differentiator. Thus, for x[m] = mq, the qth-order third-derivative differentiator has large processing error over the initial time interval 0 ≤ m < q but has zero processing error over the longer time interval m ≥ q. Figure 4.26a–d show the differentiator pulse responses when processing the Gaussian input pulse with various pulse widths w. The pulse responses closely resemble the true derivatives and the processing accuracy increases with increasing value of w. Figure 4.26a and b show the differentiator error responses when processing the Gaussian input pulse with pulse widths = w 5= and w 20, respectively. For w = 5, Figure 4.26a shows that the fourth-order differentiator has lower processing accuracy than the third-order differentiator for time m > 35. However, for a larger pulse width w = 20, Figure 4.28b shows that the fourth-order differentiator has higher processing accuracy than the third-order differentiator, but at the expense of having lower processing accuracy over the initial time interval [see the enlarged curves in Figure 4.26c].

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Photonic Signal Processing

FIGURE 4.24 Error responses of the qth-order second-derivative differentiators Hˆ q( 2) ( z; q = 2, 3, 4) when processing the Gaussian input pulse with two different pulse widths. (a) w = 5. (b), (c) w = 20 with different time scales.

Photonic Computing Processors

157

FIGURE 4.25 Error responses of the qth-order third-derivative differentiators Hˆ q( 3) ( z; q = 3, 4) when processing the qth-order pulse. (a) Third-order polynomial input pulse: x[m] = m3. (b) Fourth-order polynomial input pulse: x[m] = m4.

FIGURE 4.26 Pulse responses of the third-order third-derivative differentiator Hˆ 3( 3) ( z ) when processing the Gaussian input pulse with various pulse widths w. = (a) w 5= . ( b) w 10= . (c) w 15 = . (d) w 20.

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Photonic Signal Processing

ɵ q ( z; q = 3, 4) when proFIGURE 4.27 Error responses of the qth-order third-derivative differentiators H cessing the Gaussian input pulse with two different pulse widths. (a) w = 5. (b), (c) w = 20 with different time scales. ( 3)

4.3.4.4 Fourth-Derivative Differentiator This section analyes the performance of the fourth-order fourth-derivative differentiator Hˆ 4( 4 ) ( z ) whose magnitude response was shown in Figure 4.28d. Figure 4.28a–d show the pulse and error responses of the differentiator when processing the fourth-order and ifth-order polynomial input pulses. For the fourth-order pulse, Figure 4.28a and b show that the fourth-order differentiator has large processing error over the initial time interval 0 ≤ m < 4 but has zero processing error over the longer time interval m ≥ 4. For the ifth-order pulse, Figure 4.29c and d show that the processing error never converges to zero. This is because the order of the differentiator is lower than the order of the input pulse. However, a ifth-order differentiator, which is not considered here, is expected to improve the processing accuracy of the ifth-order pulse. Figure 4.29a–d show the differentiator pulse responses when processing the Gaussian input pulse with various pulse widths w. The pulse responses closely resemble the true derivatives and the processing accuracy increases with increasing value of w.

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(4)

FIGURE 4.28 Pulse and error responses of the fourth-order fourth-derivative differentiator Hɵ 4 ( z ) when processing the fourth-order and ifth-order polynomial input pulses. (a), (b) Fourth-order polynomial input pulse: x[ m ] = m 4 . (c), (d) Fifth-order polynomial input pulse: x[ m ] = m 5 .

4.3.5

remarks

• A theory of the qth-order pth-derivative FIR digital differentiator has been proposed, based on which the qth-order pth-derivative FIR optical differentiator has been synthesized using integrated-optic components. • For a qth-order polynomial input pulse (i.e., x[m] = mq, q ≥ 1), the qth-order pth-derivative FIR optical differentiator, Hˆ q( p ) ( z ), which has higher processing accuracy than other differentiator orders, has large processing error over the initial time interval 0 ≤ m < q but has zero processing error over the longer time interval m ≥ q. • For a Gaussian input pulse (i.e., x[m] = exp[− m2 (2w 2 )]), the qth-order pth-derivative differentiator, Hˆ q( p ) ( z; q = p) , whose processing accuracy increases with increasing pulse width w, is the optimum ilter when compared with the higher-order pth-derivative

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FIGURE 4.29 Pulse responses of the fourth-order fourth-derivative differentiator Hˆ 4( 4 ) ( z ) when processing the Gaussian input pulse with various pulse widths w. = (a ) w 5= . ( b) w 10 = . (c) w 15 = . (d ) w 20.

differentiator, Hˆ q( p ) ( z; q > p) , because of its structural simplicity and impressive performance. The results for the Gaussian pulse are similar to those for the exponential pulse (i.e., x[m] = exp( − m w ), m ≥ 0). • In general, regardless of the type of input pulse, the optical differentiators have large processing error over the initial time interval corresponding to q sampling periods but their processing errors reduce signiicantly over a longer time interval. • Optical differentiation is still a new research area with many potential applications. One example is to apply the irst-order irst-derivative FIR optical differentiator described here  to the design of an optical dark-soliton detector, which is outlined in Section 7.1 of Chapter 7.

4.4

APPENDIX A: GENERALIZED THEORY OF THE NEWTON–COTES DIGITAL INTEGRATORS

In this appendix, a classical numerical integration scheme together with the digital signal processing technique are employed to develop a generalized theory of the Newton–Cotes digital integrators, which is believed to be described for the irst time. This theory has been used in the synthesis of the programmable incoherent Newton–Cotes optical integrator (INCOI) as described in Chapter 5. A deinition of numerical integration is irst given and used as a basis in the derivation process. The Newton’s interpolating polynomial is then described, and based on which a general form of the Newton–Cotes closed integration formula is derived. Finally, a generalized theory of the Newton– Cotes digital integrators is obtained.

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4.4.1

DeFinitiOn OF numerical integratiOn

It is assumed that a continuous-time signal x(t) is given and that its integral t

∫ x(t )dt

y (t ) =

(4.99)

0

is to be determined from a sequence of samples of the continuous-time signal x(t) at the discrete time t = t n where = t n nT = , n 0,1, 2,⋯

(4.100)

with T > 0 being the period between successive samples. Intuitively, the integral y(t ) cannot be obtained for all t, but only for t = t n . Thus, Eq. (4.99) can be written as tn

yn = y (t n ) =

∫ x(t )dt.

(4.101)

0

To simplify the numerical integration algorithm, the integration interval [ 0,t n ] is divided into a number of equal segments, each with a step size of T . The underlying principle of the numerical integration algorithm is shown in Figure 4.30. From Figure 4.30, the integral in Eq. (4.101) can be divided into two integrals as t n− p

yn =

∫

tn

x(t )dt +

∫ x(t )dt = y

n− p

+ ip

(4.102)

t n− p

0

where the partial integral i p, which represents the area of the hatched region of Figure 4.30, is given by tn

ip =

∫ x(t )dt.

t n− p

FIGURE 4.30 Graphical illustration of the numerical integration technique.

(4.103)

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Photonic Signal Processing

The z-transform of Eq. (4.102) is given by Y ( z ) = z − pY ( z ) + I p ( z )

(4.104)

 1  I p ( z ), Y ( z) =  −p  1 − z 

(4.105)

or

where Y ( z ) = Z { yn } and I p ( z ) = Z {i p } with Z {} . being the z-transform of {} . . In Eq. (4.105), the z-transform parameter is deined as z = exp( jωT ) where j = −1, ω is the angular frequency, and T is the sampling period of the integrator. The z-transform of the partial integral I p ( z ) is to be determined in Sections 4.4.2 and 4.4.3.

4.4.2

newtOn’s interpOlating pOlynOmial

For analytical simplicity, the discrete-time variables in Figure 4.30 are re-deined as t k = t n − p , k = n − p,

(4.106)

tk + p = tn ,

(4.107)

tk + m = tm .

(4.108)

Using Eqs. (4.106) and (4.107), Eq. (4.105) and taking the inverse transform we have tk + p

ip =

∫ x(t )dt.

(4.109)

tk

For the time interval [t k , t k + m ] as shown in Figure 4.30, the curve x(t ) can be approximated by the mth-order Newton’s interpolating polynomial, which passes through m +1 data points, as84 x (t ) = x (t k ) +

∆x(t k ) ∆ 2 x (t k ) (t − t k )(t − t k +1 ) (t − t k ) + T 2 !T 2

(4.110)

∆ m x (t k ) + ⋯ + (t − t k )(t − t k +1 ) ⋯ (t − t k + m −1 ), m!T m where the ith discrete-time variable is given by t k + i = t k + iT , i = 0,1,⋯ , m − 1,

(4.111)

and the qth forward difference equation is given by85 q

∆ x (t k ) = q

q

∑ (−1)  i  x(t i

k + q −i

), q = 0,1,⋯ , m.

(4.112)

i =0

84 85

N. Q. Ngo and L. N. Binh, Programmable incoherent Newton–Cotes optical integrator, Opt. Commun., 119, 390–402, 1995. M. Abramowitz and I. A. Segun, Handbook of Mathematical function, New York: Dover Publications, 1964.

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The binomial coeficient in Eq. (4.112) is deined as  q  q( q − 1)⋯ ( q − (i − 1) ) q! =  = ( q − i )!i ! i! i

(4.113)

q q   =   = 1. 0 q

(4.114)

with

Eq. 4.112 can be written in a recursive form as85 ∆ 0 x(t k ) = x(t k ),

(4.115)

∆1x(t k ) = ∆x(t k ) = x(t k +1 ) − x(t k ),

(4.116)

∆ q x(t k ) = ∆ q −1x(t k +1 ) − ∆ q −1x(t k ), q = 2,⋯ , m.

(4.117)

Eqs. (4.110) and (4.117) can be simply expressed as m

x (t ) = x (t k ) +

∑ q =1

∆ q x(t k )   q !T q  

q −1

∏ i =0

 (t − t k + i )  

(4.118)

which can be further simpliied by deining a new quantity

η=

t − tk . T

(4.119)

Substituting Eq. (4.119) into Eq. (4.111), the following equation is obtained: (4.120)

t − t k +i = T (η − i ), i = 0,1,⋯, m − 1. Substituting Eq. (4.120) into Eq. (4.118) results in m

x (t ) = x (t k ) +

∑ q =1

∆ q x(t k )   q!T q 

q −1

∏ i =0

 [T (η − i )] = x(tk ) + 

m

∑ q =1

∆ q x(t k )   q!  

q −1



∏ (η − i) i =0

(4.121)

η (η − 1)⋯(η − ( m − 1) ) η (η − 1) + ∆ m x (t k ) = x(t k ) + ∆x(t k )η + ∆ 2 x(t k ) , 2! m! which can be further simpliied to η  q  ∆ x(t k ) = q q =1   m

x (t ) = x (t k ) +

∑

m

η 

∑  q ∆ x(t ). q

k

(4.122)

q =0

Thus, for the time interval [t k , t k + m ], the mth-order Newton’s interpolating polynomial of the curve x(t ) can be simply described by Eq. (4.122).

164

4.4.3

Photonic Signal Processing

general FOrm OF the newtOn–cOtes clOseD integratiOn FOrmulas

Substituting Eq. (4.122) into Eq. (4.109) results in  m η   q   ∆ x(t k )  dt .  q=0  q  

tk + p

ip =

∫ ∑ tk

(4.123)

From Eq. (4.123), dt = Tdη and the limits of integration are changed from t = t k to η = 0 and from t = t k + p to η = p. Substituting these parameters into Eq. (4.123) results in  m η   q ip =   ∆ x(t k )  ⋅ Tdη   q 0  q =0 p

m

  p η     dη ⋅ ∆ q x(t k )   q   0  

∑∫

(4.124)

∑ C ( p)∆ x(t ),

(4.125)

∫∑

=T

q =0

which can be rearranged to give m

ip = T

q

q

k

q=0

where the qth coeficient Cq ( p) is given by p

η  Cq ( p) =   dη , q = 0,1,⋯, m. q 0

∫

(4.126)

The qth forward difference equation, as described by Eq. (4.118), can be further simpliied by substituting u = q − i or i = q − u into Eq. (4.121) to give q

∆ q x(t k ) = ( −1) q

q

q  q  q ( −1) − u  ( −1)u   x(t k +u )  x(t k +u ) = ( −1) u q −u u =0 u =0

∑

∑

(4.127)

u ∈ integer

(4.128)

Where the following equations are used ( −1) − u = ( −1)u , with  q  q   =  , q − u u

(4.129)

Thus, the general form of the Newton–Cotes closed integration formulas can be simply described by three closed-form formulas, as given in Eqs. (4.127) through (4.129).

4.4.4

generalizeD theOry OF the newtOn–cOtes Digital integratOrs

Taking the z-transform of Eq. (4.125) leads to m

I p ( z) = T

∑ C ( p) ∆ X ( z ) q

q

q=0

(4.130)

165

Photonic Computing Processors

{

}

where ∆ q X ( z ) = Z ∆ q x(t k ) is the z-transform of Eq. (4.127), which is given by q

∆ q X ( z ) = ( −1) q

q

∑ (−1)  u  ⋅ X ( z) z u

−u

(4.131)

u =0

where X ( z ) z − u = Z { x(t k + u )}. Eq. (4.131) can be rearranged to give ∆q X ( z) =  X ( z )( −1) q 

q

q

∑ (−1)  u  z u

−u

=1 − qz −1 +

u =0

q! q( q − 1) −2 z + ⋯ + ( −1) r z − r + ( −1) q z − q 2! ( q − r )! r !

(4.132)

Note that Eq. (4.132) can be recognized as86 ∆q X ( z) = (1 − z −1 ) q  X ( z )( −1) q 

(4.133)

∆ q X ( z ) = ∆ q D( z ) ⋅ X ( z ),

(4.134)

∆ q D( z ) = ( −1) q (1 − z −1 ) q , q = 0,1,⋯ , m.

(4.135)

or

where

Substituting Eq. (4.135) into Eq. (4.130) gives m

I p ( z ) = X ( z )T

∑ C ( p)∆ D( z). q

q

(4.136)

q=0

Substituting Eq. (4.136) into Eq. (4.105), the pth-order transfer function of the Newton–Cotes digital integrators can be generally described by H mp ( z ) = =

Y ( z) T = X ( z) 1 − z − p

m

∑ C ( p) ∆ D ( z ) q

q

q =0

(4.137)

T C0 ( p) + C1( p)∆D( z ) + C2 ( p)∆ 2 D( z ) + ⋯ + Cm ( p)∆ m D( z )  1− z− p 

where the qth coeficient is given, from Eq. (4.126), as p

η  Cq ( p) =   dη , q 0

∫

86

(4.138)

S. Usui and I. Amidror, Digital low-pass differentiation for biological signal processing, IEEE Trans. Biomed. Eng., BME-29, 686–693, 1982.

166

Photonic Signal Processing

and the qth difference equation is given, from Eq. (4.138), as ∆ q D( z ) = ( −1) q (1 − z −1 ) q, for q = 0,1,⋯ , m and 1 ≤ p ≤ m

(4.139)

In summary, a generalized theory of the Newton–Cotes digital integrators has been derived, and Eqs. (4.137) through (4.139), which are referred to in Section 4.2.

5 5.1

Optical Dispersion Compensation and Gain Flattening

INTRODUCTORY REMARKS

This chapter introduces the compensation of dispersion effects in the transmission of lightwave-modulated signals is a very critical task in long haul, ultra-high-speed optical communications. Fast, reliable, and error-free 100G/200G/400G backbone and metro networks are a key ingredient to deliver bandwidth-hungry services, such 4G/5G mobile connectivity, iber to the home (FTTH), and Remote PHY* (for cable operators). Optical functional devices, in such optical transmission systems, mainly are composed of two types: (i) Firstly, dispersion compensators in order to compensate for the dispersion broadening of transmitted pulse sequence; and (ii) secondly, the gain equalizer to equalize the gain over the operating spectral region so that the unevenness of the optical attenuation of iber and that of the optical ampliication can be lattened. Thus even optical powers of all channels at different wavelengths can be achieved over the transmission distance. Variable chromatic dispersion compensators are becoming increasingly important in high speed optical transmission systems with bit rates of 25 or 56Gbaud or more, where it is essential to compensate adaptively for the various dispersions of installed ibers and the dispersion change caused by changes in the environmental temperature or path differences in optical networks. A number of compensation and equalization techniques have been reported. However, this chapter describes only the techniques employing optical ilters and resonators, which can be designed and implemented using photonic signal processing methodology. The generation of highly dispersive effects in resonators that are operated under resonance and eigeniltering are given. Photonic functional devices are described including photonic dispersion compensator, gain equalizers using lattice ilters.

5.2

DISPERSION COMPENSATION USING OPTICAL RESONATORS

In recent years, there have been growing interests in studying the characteristics of the all-iber photonic circuit components, especially the re-circulating delay line1,2,3 and optical resonator.4,5,6,7,8 It has been found that the photonic circuits have great potential in many applications, such as

*

1 2 3 4

5 6

7 8

https://www.cisco.com/c/en/us/solutions/collateral/service-provider/converged-cable-access-platform-ccap-solution/ white-paper-c11-732260.html S. J. Mason, Feedback theory–Some properties of signal low graphs, Proc. IRE, 1144–1156, 1953. S. J. Mason, Feedback theory-Further properties of signal-low graphs, Proc. IRE, 44, 920–926, 1956. A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing, Upper Saddle River, NJ: Prentice-Hall, 1989. B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber lattice optic signal processing, Proc. IEEE, 72, 909–930, 1984. B. Moslehi, Fiber-optic ilters employing optical ampliiers to provide design lexibility, Elect. Lett., 28, 226–228, 1992. B. Moslehi, J. W. Goodman, Novel ampliied iber-optic recirculating delay line processor, J. Light. Technol., 10, 1142–1147, 1992. B. C. Kuo, Digital Control Systems, Chapter 5, pp. 267–296, Hongkong: CBS Pub. Asia, 1987. Y. H. Ja, Single-mode optical iber ring and loop resonators using degenerate two-wave mixing, Appl. Opt., 30, 2424–2426, 1991.

167

168

Photonic Signal Processing

communications,9,10 sensing devices,11,12 and signal processing devices.13 The progress in iber-optic technology also makes the implementation of these circuits easier. For instance, due to the development in optical ampliier technology,14 active optical devices1,3 are a possibility. As the trends of integrated photonic circuits progress, it is expected that the circuits analyzed may consist of more and more elements in them. This surely causes dificulties in the examination of the circuits. Thus, an eficient method is needed to facilitate these problems. As far as the examination of the circuit behavior and its design are concerned, the analysis of the circuit involves the determination of the circuit’s transfer function. This is usually achieved by simultaneously solving a set of linear ield or intensity equations. However, this method becomes error-prone and time-consuming as the complexities of the circuit increase due to the increasing numbers of components and interconnections in the optical circuit network. It makes the formulation of the transfer functions for the circuit and the analysis on the circuits’ characteristics more dificult and tedious. Although a scattering matrix has been used in describing the circuit characteristics15,16 and is shown to be a more systematic approach to the problem, a simple method of determining the circuit’s transfer functions from the set of governing equations has not been proposed, to the best of our knowledge. This restricts our analysis to simple circuits only. The new method employs the use of signal-low graph theory17,18 in optical circuits. The signallow graph theory was irst introduced 80 years ago by S. J. Mason.1,2 The theory has been applied in electrical and electronic circuits for a long time already. This is the irst time, to the best of our knowledge, when the signal-low graph theory is being applied during the analysis of optical circuits. The unique feature of our work is that the optical circuits can be represented in form of a signal-low graph (SFG), and their circuits’ transfer functions can then be determined systematically using the corresponding manipulation rules. The analysis of optical circuits, which employs the new method, has been published recently by our group.19,20,21,22 The new method is found to be much more eficient than the conventional method. The main advantage of our method over the conventional one is that more simple and systematic procedures are used in deriving the circuits’ transfer functions. This certainly lowers the possibility of error. Moreover, the graphical representation of the optical circuits also allows easier examination of recirculating or resonating loops in the circuit. Thus, the locations of loops in the circuits can be identiied with lesser effort and the mechanism

9

10

11 12

13

14 15 16

17 18

19

20

21

22

L. N. Binh, N. Q. Ngo, and S. F. Luk, Graphical representation and analysis of the z-shaped double-coupler optical resonator, J. Light.Technol., 11, 1782–1792, 1993. Y. H. Ja, Generalized theory of optical iber loop and ring resonators with multiple couplers: 1: Circulating and output ields and 2: General characteristics, Appl. Opt., 29, 3517–3529. Y. H. Ja, Optical iber loop resonators with double couplers, Opt. Commun., 75, 239–245, 1990. Y. H. Ja, A double-coupler optical iber ring-loop resonator with degenerate two-wave mixing, Opt. Commun., 81, 113– 120, 1991. Y. H. Ja, On the conigurations of double optical iber loop or ring resonator with double couplers, J. Opt. Commun., 12, 29–32, 1991. B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, Fiber lattice optic signal processing, Proc. IEEE, 72, 909–930, 1984. J. M. Vigoureux and F. Raba, A model for the optical transistor and optical switching, J. Mod. Opt., 38, 2521, 1991. B. Moslehi, J. W. Goodman, Novel ampliied iber-optic recirculating delay line processor, IEEE J. Light. Technol., 10, 1142–1147, 1992. D. Marcuse, Pulse distortion in single-mode ibers, Appl. Opt., 19, 1653–1660, 1980. D. Marcuse, Selected topics in the theory of telecommunications ibers, Optical Fiber Telecommunications II, Chapter 3, Boston, MA: Academic Press, 1988. B. E. A. Saleh and M. I. Irshid, Transmission of pulse sequences through mono-mode ibers, Applied Optics, 21, 4219– 4222, 1982. B. E. A. Saleh and M. I. Irshid, Coherence and inter-symbol interference in digital iber optic communication systems, IEEE J. Quantum Electron., QE-18, 944–951, 1982. K. Joergensen, Transmission of Gaussian pulses through monomode dielectric optical waveguides, Appl. Opt., 16, 22–23, 1977. K. Joergensen, Gaussian pulse transmission through monomode ibers, accounting for source linewidth, Appl. Opt., 17, 2412–2415, 1978.

Optical Dispersion Compensation and Gain Flattening

169

of resonance can then be studied easily. Furthermore, our new method incorporates a stability test, which is useful in the assessment of the photonic circuit operation. The stability of the photonic circuit should be an important part in the design. However, very few studies have been conducted on aspects of stable and unstable states of the photonic processing unit. The stability test should prove to be an important tool in photonic circuit design in the future. The result is signiicant as a systematic method is presented to the analysis of photonic circuit. The new method should lead to a better understanding of the performance of various photonic circuits and may initiate studies on complex photonic circuits on a large scale. Passive resonators have been studied for some simple photonic circuit conigurations.4,5,6,7,8 However, optical ampliiers can be inserted in the iber paths of the circuit to enhance the circuit’s performance. It has been shown3 that optical ampliiers can provide lexibility in the design of the photonic circuit, not only just as compensators for losses in iber paths. An ampliied double-coupler double-ring (DCDR) photonic circuit is studied in detail in this work. The passive operation of this circuit has been examined in the case of a resonator23,24 but its use as re-circulating delay line and the circuit’s temporal response have never been studied. The development of the signal-low graph theory in optical circuits allows us to carry out the generalized study of the DCDR circuit easily and eficiently. We can identify the recirculating or resonant loops in the circuit immediately once its signal-low graph has been drawn. This would signiicantly increase our understanding on the circuit performance, particularly the effects from the physical structure of the circuit. The special feature of the DCDR circuit is that it consists of two loops, which share one common iber path in the circuit. This special feature produces some useful results, like the application as an adder, which are not realizable in other simple conigurations. The DCDR circuit, like most of the optical circuits, is discrete in nature and can be treated as a discrete-time signal system. The discrete behavior of the all-iber optical circuit is due to the delays in the iber paths of the circuit. Hence, z-transform is used for the analysis of discrete-time signal processing in our work. The DCDR circuit system will be represented by transfer functions in the z-domain. The transfer function will contain poles and zeros on the z-plane, which are the roots of the denominator and numerator of the transfer function, respectively. The report will show that the poles and zeros locations, or distributions, of the transfer functions determine the characteristics of the circuit, for instance, the iltering properties. Thus, the circuit responses will be explained in terms of the poles and zeros values of the transfer functions. In this chapter, the circuit is examined under two conditions—illuminated by a temporal incoherent source and a coherent source with inite linewidth. In the former case, the analysis is carried out on the intensity-basis and the circuit is operating as a re-circulating delay line. In the latter case, the use of the circuit as a resonator is investigated together with the transient response of the circuit. The computation of transient response is generalized for optical circuits. Investigation is also carried out on the possibility of employing the DCDR as a iber dispersion equalizer. The use of optical resonators as dispersion equalizers were studied recently,9,10 but they were conined to simple resonators, which may limit the design lexibility. In this work, the DCDR circuit is shown to give more freedom in the design as an equalizer. Section 5.2 is organized as follows. In Section 5.2.3, the graphical technique is used to derive the transfer function of the resonator in association with the Mason’s gain rules. SFG theory and Mason’s rule17,18 are found to be applicable in the analysis of photonic circuits. The unique feature of this method is shown and the advantages of our method over the conventional ones are presented. Section 5.2.4 discusses the DCDR circuit operating under a temporally incoherent source. The circuit is considered on an intensity-basis and it is treated as re-circulating delay lines. Several 23

24

C. Lin and D. Marcuse, Optimum optical pulse width for high bandwidth single-mode iber transmission, Electron. Lett., 17, 54–55, 1981. D. Marcuse and C. Lin, Low dispersion single-mode iber transmission–The question of practical versus theoretical maximum transmission bandwidth, IEEE J. Quantum Electron., QE-17, 869–878, 1981.

170

Photonic Signal Processing

operation modes of the circuit are displayed including passive, active, and active with negative optical gain. Both the frequency and temporal responses are studied. Possible applications of the circuit in signal processing are suggested. Procedures in the design of the circuit are proposed for the case with negative optical gain acting as an illustration of a general design procedure. The DCDR circuit under a coherent source is studied with emphasis on the resonance effects and the transient response of the circuit. The resonance of the DCDR circuit is displayed and its use in certain signal processing applications are mentioned. The effects of the source coherence and the input pulse shape on the transient response of the circuit are studied in detail. Also, algorithms for computations of transient responses for general photonic circuits are given. In Section  5.2.5, one of the applications of the DCDR circuit, a iber dispersion equalizer, is examined speciically. The equalization achieved by the circuit under a different circuit’s parameters is demonstrated. The capability of the DCDR circuit as a dispersion equalizer is shown with different operating points.

5.2.1

signal-FlOw graph applicatiOn in Optical resOnatOrs

The mathematical tools that are used in our analysis are introduced here. They include the z-transform and SFG theory. Z-transform of discrete-time signal processing is useful for the analysis of photonic circuits, especially for those consisting of delay elements. The SFG theory enables us to examine and eficiently investigate the characteristics of photonic circuits. The graph reduction rule and Mason’s rule1,2 are associated with the SFG theory and included. Stability of the circuit is usually an important criterion of evaluating the circuit performance. Therefore, the stability test for the system is also introduced. For the sake of simplicity, a simple photonic circuit—a single-loop resonator circuit—is used as an example to illustrate the manipulation of the signal-low graph theory. The z-transform techniques are used in digital signal processing3 (also called discrete-time signal processing) and employed for the analysis of the photonic circuit in this work. The discrete-time signal can be considered as an equally spaced sampling of a continuous-time signal. The z-transform of a sequence x[n] is deined as ∞

X ( z) =

∑ x[n]z

−n

(5.1)

n =−∞

x[n] can be interpreted as the nth term of the sequence of numbers that describe the discrete-time signal in the time domain. The ratio between the output transform Y ( z ) and the input transform U ( z ) of a system is called the transfer function H ( z), which is given as: H ( z) =

Y ( z) U ( z)

(5.2)

Rearranging (5.2) gives: Y ( z) = H ( z) ⋅U ( z)

(5.3)

This shows that in the z-domain multiplication of the input with the transfer function produces the output. The transfer function given in Eq.  (5.2) is related to the corresponding time domain sequence h[n], indeed it is usually deined as the impulse response of the system. The corresponding operation in the time-domain of Eq. (5.3) is ∞

y[n] =

∑ u[r]h[n − r],

r =−∞

(5.4)

171

Optical Dispersion Compensation and Gain Flattening

which is the convolution sum that can be expressed as

= y[n] u= [n] * h[n] h[n] * u[n].

(5.5)

where the symbol * denotes the convolution. It is to be noted that the convolution operation is commutative. There are two important properties of convolution. The convolution of two functions in the time domain corresponds to the multiplication of their z-transforms. On the other hand, the multiplication of two functions in the time domain corresponds to their convolution in the z-domain. Now writing the numerator and denominator NUM ( z ) as ∑ iB=0 bi z −i and DEN ( z ) as H ( z ), the z-domain transfer function can then be expressed as

∑i =0 ai z −i , A

B

H ( z) =

∑b z

−i

i

NUM ( z ) = DEM ( z )

i =0 A

∑a z

(5.6) −i

i

i =0

which can also be rewritten into product form as B

b H ( z ) = 0 z A− B a0

∏(z − q ) k

k =1 A

∏(z − p )

(5.7)

j

j =1

It is assumed that H ( z) has been expressed in the irreducible form. The values p j are called the poles  of H ( z ) such that H ( p j ) = ∞.; similarly the values qk are the zeroes of H ( z ) for which H ( qk ) = 0. Also, there is a pole at z = 0 of the multiplicity of ( BA) if B > A. If A > B , there would be a zero at z = 0 of the multiplicity of ( A − B). As it has been found that the transfer functions can be expressed in z-domain, thus the transfer characteristics are dependent on the zero-pole patterns4 of these transfer functions in the z-plane. The magnitude-frequency response at a particular frequency as the operating point moving on the unit-circle, z = 1, of the z-plane is given by B

H ( z ) = H ( e jωτ ) =

b0 a0

∏l

zk

k =1 A

∏

(5.8) l pj

j =1

where z = ejωτ, ω is the angular frequency in radians per second and τ is the sampling period of signals in seconds. lzk and l pj are the lengths from the operating point to the position of the kth zero and the jth pole of the transfer function respectively. The corresponding phase-frequency response at a particular operating frequency (wavelength) is given by B

arg( H ( z )) =

∑ k =1

A

φzk −

∑φ

pj

+ ( A − B)ωτ

(5.9)

j =1

where ϕzk and ϕpj are the phase angles of the zeroes and poles respectively formed by the horizontal real axis and the lines connecting the poles and zeroes to the operating point in the z-plane.

172

Photonic Signal Processing

Thus, from Eqs. (5.8) and (5.9) we can design the magnitude-frequency response by adjusting the pole and zero patterns of the transfer functions. To obtain a maximum magnitude at a particular A operating wavelength (frequency), we require a pole or a very small value of ∏ j =1 l pj at that wavelength. Similarly, in order to obtain a minimum at a particular wavelength, a zero or an ininitesimal B value of ∏ k =1 lzk is required at that wavelength. There are some relationships between the positions of poles in the z-plane to those correspondingly in the s-plane (the continuous frequency domain). One basic property of this relationship is that, recall that z = e sτ where s = jω when a pole position moves on the imaginary axis of the s-plane, it would move along the unit-circle of the z-plane. In this case, we would have marginal stability and a lossless system. When a pole moves on the imaginary axis towards the left-half of the s-plane, the corresponding pole moves inside the unit circle in the z-plane. The system would thus become lossy and stable. If one of the system poles lies outside the unit circle in the z-plane, the system becomes unstable. Its temporal response would increase with time. In general, the system would be stable if all the system poles lie inside or on the unit circle in the z-plane. Stability plays an important role in design of photonic circuits. Stability test for the photonic circuit is introduced in Section 5.2.4. The SFG has been well known in the analyses and formulation of massive electrical and electronic circuits. We employ extensively this technique in the analysis of optical circuits here. As mentioned in Chapter 2, SFG is a graphical diagram that comprises directed branches and nodes, in other words, a graph with nodes linked up by directed branches in some way. A branch represents the relationships between two nodes and it is usually associated with a transmittance or transfer function. The signal lows through the branch in a direction indicated by the arrowhead. The transmittance denotes the functional operation that the signal would undergo as it travels through the branch. In general, that functional operation can be linear or non-linear. The rule of engagement of the transmission path is described in Chapter 2. The nodes in the SFG represent the circuit variables. In photonic circuits they are usually optical ields, for coherence case and intensities for incoherence. The value of a node variable is taken to be the sum of all incoming signals entering the node. This node value or signal travels along each outgoing branch connected to it. There are two special kinds of nodes in SFG, source and sink. A source is a node with outgoing branches connected to it. Likewise, a sink is the opposite, it is a node with incoming branches only. There are several terms that are frequently used for describing SFG. For instance, a feed-forward path from node a to node b is a sequence of nodes and branches that the signal passes through from node a to b, in which all nodes are traversed only once. A feedback loop is a path that starts and terminates at the same node such that no node is traversed more than once. A feedback loop that contains only a single node is called a self-loop. Thus, the SFG is a graphical illustration of the relationships among the variables in a photonic circuit. In the following presentation, a single loop resonator circuit is used as a simple example to illustrate different aspects of the SFG theory and associated rules. A schematic diagram of the single loop resonator is shown in Figure 5.1. coupler Input

1

3

Output 2

4

T

FIGURE 5.1 The schematic diagram of the single loop resonator.

173

Optical Dispersion Compensation and Gain Flattening

A single loop resonator comprises a 2  ×  2 directional coupler and a iber path as shown in Figure 5.1. The iber path is connected in such a way that one of the output ports of the coupler is joined to one of the coupler’s input ports through the path. The excess power loss of the coupler is assumed to be zero, i.e. the coupler is lossless. This assumption of the coupler is valid for the whole scope of our analysis. The power coupling coeficient of the coupler is given as k. T ( z ) represents the transmittance of the iber path, it is given as: T ( z ) = t aGz −1 .

(5.10)

where t a is the passive transmission coeficient of the iber path (3)–(2), which is related to the iber’s loss. If ta is equal to its maximum value of 1, the iber is lossless. G is the optical intensity gain factor of the iber optical ampliiers (if any) inserted in the path. G equals to one corresponds to the case where there is no ampliier in the feedback iber path, i.e. passive operation of the circuit. By using the z-transform representation, the delay element in the photonic circuit with a unit delay can be represented by z −1. The unit delay or the basic time delay is deined as the time required for the optical waves to travel along a length l of optical iber such that it is equal to the sampling period of the input optical signal. In this case, z −1 is used to show that the signal at port 2 of the circuit is a unit-delayed version of the signal at port 3, apart from the scaling of the signal’s magnitude. This represents the discrete nature of the signals in the circuit. As the circuit is taken to be an example of the SFG illustration only, we further assume that the source is temporally incoherent so as to simplify the situation. By this assumption, it means that the source coherence length is much shorter than the shortest delay path in the circuit so that the phase change can be ignored. Intensity rather than ield amplitude is thus considered. The directional coupler has a direct coupling of (1− k ) and cross coupling of k of the intensity at the input port. Further discussion of the temporal incoherent source is given in Chapter 2. The SFG in terms of intensity for the single loop resonator circuit is shown in Figure 5.2. It can be seen that the single loop in the circuit and the direction of signal low in the loop can be identiied easily in the SFG in Figure 5.2. In all-iber photonic circuits, the directional couplers and the delay lines usually form the basic building blocks. From Figure 5.2, it can be seen that the SFGs of these two elements are simple. After obtaining the SFG of the photonic circuit, the derivation of the transfer functions between node variables is very straight forward. There are generally two methods to achieve this objective, the graphical reduction of the SFG and Mason’s Rule.

T

k 3

2

1-k

1-k

1

k

FIGURE 5.2 SFG of the single loop resonator circuit.

4

174

Photonic Signal Processing

If the SFG reduction rules (iii), (iv), and (i) are applied in succession to the SFG in Figure 5.2, the output-input intensity transfer function of the single loop circuit is obtained to be: k + (1 − 2k )T 1 − kT

H14 =

(5.11)

It is not necessary to reduce the SFG to ind the desired transfer functions of the circuit. Mason’s Rule1,2 can be applied to give the transfer functions directly. Considering the independent input node a and dependent output node b of an arbitrary photonic graphical network, which are its input and any port. Let the transfer function H ab for the SFG be deined as the overall transmittance between the two nodes a and b, then Mason’s rule can be stated for the circuit as1,2: N

∑ (F

) ( ∆ ab ) q

ab q

H ab =

q =1

(5.12)

,

∆

where N is the total number of optical transmission paths from node a to node b, ( Fab ) q is the optical transmittance of the qth path from node a to node b in the SFG, and ∆ is given by ∆ = 1−

∑T + ∑T T − ∑ T T T lu

u

lu lv

u ,v

lu lv lw

+

(5.13)

u ,v ,w

with Tlu is the loop transmittance and only products of non-touching loops are included in the ∆ expression. Thus Eq. (5.13) can be written in plain word as: ∆ = 1 − ( sum of all loop transmittances ) + (sum of products of all loop transmittances of 2 non-touching optical loops) − ( sum of products of all loop transmittances of 3 non-touching loops ) + with ∆ is deined as the graph determinant. Therefore, it follows that ( ∆ ab ) q is the cofactor of the qth forward transmission path, which is equal to ∆ after all loops touching the qth path have been excluded. Two optical loops are considered as non-touching if they do not share any common node. However, one criterion of using the Mason’s rule is needed to be satisied. The SFG should be able to be represented in the planar form where there are no crossings between the signal-low paths. From the SFG of the single loop photonic circuit, it can be observed that there is only one optical feedback loop in the circuit that is (3)(2)(3). Thereafter, the loops or the paths in the circuit are given by sequences of the node numbers the loop or path follows. Its output-input intensity transfer function can be determined using Mason’s rule of Eq. (5.13). Applying Mason’s Rule, H14 ( z ) of the single loop resonator circuit can then be determined by: 2

∑ (F ) (∆ 14 q

H14 =

)

14 q

q =1

∆

(5.14)

Firstly, the loop transmittance of the single loop circuit is given as: Tl1 = kT

(5.15)

175

Optical Dispersion Compensation and Gain Flattening

Thus, ∆ can be obtained by as ∆ = 1 − Tl1 ⇒ ∆ = 1 − kT .

(5.16)

There are two forward paths going from node 1 to node 4, they are: (i) the direct path (1)(4), which has a transmittance of k, that is ( F14 )1 = k , and ( ∆14 )1 =1 − kT , and (ii) path (1)(3)(2)(4) with the transmittance of ( F14 )2 = (1 − k )2 T and ( ∆14 )2 = 1. Substituting (5)–(8) into (4) yields H14 =

k + (1 − 2k )T 1 − kT

(5.17)

which is the same as the expression obtained in Eq. (5.15) that used the graph reduction rules.

5.2.2

stability test

Before computing the response of the circuit, it is sometimes useful to consider the system’s stability in the design. Stability consideration of the photonic circuit is very important as it is directly related to the performance of the circuit especially in the time-domain applications. Since the photonic circuit can be modeled under the discrete signaling, the Jury’s stability test usually applied for digital control system25 can be employed. Consider a characteristic equation in the form of F ( z ) = an z n + an−1z n−1 +...... + a2 z 2 + a1z + a0 = 0

(5.18)

This characteristic equation is deined as the equation obtained by equating the denominator of the transfer function to zero. For n = 2, that is a second-order system, the necessary and suficient conditions for a stable operation of the circuit requires F (1) > 0, and F ( −1) > 0 and a0 < a2

(5.19)

That is all the roots of the equation must lie inside the unit circle in the z-plane. In this example, n = 1, the stability criterion becomes F(1) > 0 and F( −1) < 0. Now consider Eq. (5.10), the characteristic equation of the single loop circuit is F (z ) = z − kt aG = 0

(5.20)

Thus, the stability criterion is then given by 1 − kt aG > 0 ⇒ kt aG < 1

(5.21)

The expression on the LHS of the inequality in Eq. (5.21) is in fact the “pole” value of the circuit. If the pole value is less than one, the pole will then lie inside the unit circle in the z-plane and stability is achieved. Therefore, it is clearly shown that the Jury’s stability test criterion gives the condition where the poles of the system will lie inside the unit circle in the z-plane.

25

B. C. Kuo, Digital Control Systems, Chapter 5, Asia, Hong Kong: CBS Publications, 1987. pp. 267–296.

176

5.2.3

Photonic Signal Processing

Frequency anD impulse respOnses

5.2.3.1 Frequency Response The magnitude and phase responses of the photonic circuit can be obtained by repeating computations at different values of frequency along the unit circle, i.e. z = 1. The frequency responses of H14 ( z ) of the single loop resonator circuit is plotted against ωτ , as shown in Figure 5.3, together with a plot showing the pole and zero positions. The ω is the angular frequency of the source, and τ is the unit time delay of the iber path in the circuit. The parameters used in this example are = k 0= .4142, t a 1 and G = 2.414 , which gives a pole at +1 and a zero at −1. From the magnitude plot, we can see that the response has minima when the frequency passing a pole and maxima for zeroes. When passing through a pole, the phase changes from −1.57rad ( −90°) to 1.57rad (90°) , likewise, when traveling past a zero, the phase changes from 1.57rad (90°) to −1.57rad ( −90°). At other frequencies, the phase remains constant at either −1.57rad or 1.57rad. This can be explained by examining the phase response expression for the circuit. The phase response of the single loop circuit is given by arg ( H14 ( z ) ) = φz1 − φ p1

(5.22)

Consider geometrically in the z-plane, from Figure 5.4, the difference between φz1 and φ p1 is always 90°. Thus (φz1,φ p1 ), depending on the position of the operating point, takes the value of either +90° or −90°.

FIGURE 5.3 Frequency response for the single loop circuit together with the pole-zero positions with pole at +1 and zero at −1 (pole: x, zero: o).

177

Optical Dispersion Compensation and Gain Flattening z

φz1 -1

φp1 +1

FIGURE 5.4 Representing the phase response obtained in Figure 5.3.

5.2.3.2 Impulse and Pulse Responses The impulse response and the pulse response of the feedback resonator given for Figure  5.1 are shown in Figure 5.5. The pulse input can be obtained by launching a sequence of “111” into the circuit, the “1’s” are delayed from each other by a unit time delay. It can be seen that the impulse response has a steady state value, that means the system is undamped or loseless. From Eq. (5.12) the magnitude of the loop transmittance has a pole at +1. Therefore, the signal maintains its magnitude in each circulation inside the circuit loop, the signals cross-coupled to the circuit’s output after every circulation are thus remaining constant. Using the stability criterion (5.20), kt aG or the pole value equates to unity. Thus, a marginally stable condition is achieved, that is the pole lies exactly on or very close to the unit circle in the z-plane. If the magnitude of pole value is larger than 1, the system becomes unstable and the impulse response will then be increasing as time proceeds.

FIGURE 5.5 Impulse response and pulse response for the single loop circuit with the same circuit parameters as in Figure 5.3.

178

Photonic Signal Processing

5.2.3.3 Cascade Networks This photonic network theory can now be extended to a network of interconnected basic photonic circuits. In fact, we have so far treated the photonic circuits as elements of a signal processing system, in other words, we have systematized the circuit. Thus, the system theory can be applied in the manipulation of the circuit. For instance, there are two photonic circuits with output-input transfer functions H a and H b respectively. If the output of the irst circuit is connected to the input of the second circuit, for example, the two circuits are connected in cascade, then the overall output-input transfer function of the system will then be equal to H a H b, the product of the two transfer functions. In general, if we connect n photonic circuits in cascade, the overall transfer function of the system can be easily determined as the product of all individual transfer functions. 5.2.3.4 Circuits with Bi-directional Flow Path If in a photonic circuit, the signal travels in both directions through a iber path, for instance one signal is transmitted and the other one is relected, the circuit can be represented by two SFGs. If the circuit is linear, the circuit transfer function can be obtained by the superposition of the two transfer functions as obtained individually from each SFG,9 the single ring resonator with two-wave mixing, or reference, the z-shaped double-coupler resonator. 5.2.3.5 Remarks The tools for analyzing general photonic circuit are introduced here. The application of z-transform in the analysis makes both the manipulation and understanding easily. We have developed the application of the powerful signal low graph theory in the study of photonic circuits. This provides an examination of the circuit characteristics visually and it effectively reduces the effort of determining the transfer functions of the circuit. The techniques of handling the signal low graph are demonstrated and include the Mason’s rule and the graphical reduction rule. Together with the stability test provided, the circuit behavior can be examined thoroughly.

5.2.4

DOuble-cOupler DOuble-ring circuit unDer tempOral incOherent cOnDitiOn

Resonator circuits with double or multiple couplers have been studied.9–12 In this section, a DCDR circuit is analyzed. The use of this circuit as a resonator is studied9,10 but its use as recirculating delay lines had not been examined. Here, temporally incoherent of the signal is assumed. Therefore, with this assumption, the circuit is operated as recirculating delay line instead of resonator. Incoherent processing is considered by using the intensities. A detailed study of the circuit is carried out for different sets of circuit parameters including the case where negative optical gain is provided by the optical ampliier in the circuit. The frequency response and impulse response of the circuit are given. From the studies, several applications of the DCDR circuit are illustrated. 5.2.4.1 Transfer Function of the DCDR Circuit The schematic diagram of the DCDR circuit is shown in Figure 5.6 consisting of two 2 × 2 directional couplers interconnected with three optical iber forward and feedback paths. The iber path (3)(6) and (4)(5) are referred as the forward paths of the circuit while path (7)(2) is called the feedback path of the circuit. The circuit can be considered as a two-port device, which incorporates one input port and one output port as visualized Figure 5.6b. In this latter arrangement, the circuit can be viewed as a cascade form of two couplers with an overall feedback loop. The k1 & k2 are the power coupling coeficients of the two couplers 1 and 2, respectively. T1 , T2 , and T3 denote the transmission functions of the forward paths (3)(6) and (4)(5), and the feedback path (7)(2), respectively. The optical transmittances can be represented by Ti = t aiGi z − mi for.....i = 1, 2, 3.

(5.23)

179

Optical Dispersion Compensation and Gain Flattening Input

coupler 1 1

2

coupler 1

Input

3

coupler 2

1

3

T1

6

8

2

4

T2

5

7

Output

4

T3

T2 7

T1

5

Output 8 (a)

6 coupler 2

T3

(b)

FIGURE 5.6 (a) The schematic diagram of the DCDR circuit, (b) the same circuit in a different arrangement.

T

1

3

k1

1-k1

2

T3

k1

6

1-k2

1-k1

1

k2

7

4

T2

5

1-k2

k2

8

FIGURE 5.7 The SFG of the DCDR circuit.

i = 1, 2, 3 corresponds to the three paths as shown in Figure 5.6. tai is the transmission coeficient of the ith optical waveguide path as the same parameter t a deined in Chapter 2. Gi is the optical intensity gain factor, which is provided by the waveguide/iber optical ampliiers incorporated in the paths and mi is the order of the delay path. The SFG of the DCDR circuit is shown in Figure 5.7. The SFG of the DCDR circuit is represented in planar form where, for instance, there is no crossing of optical paths in the graph. The node variable in the SFG represents the optical intensity at that point of the DCDR circuit. From the SFG of the DCDR circuit, it can be easily seen that there are two optical closed loops in the circuit, which can be called feedback loops or the recirculating loops of the circuit. One of the loops is the loop connecting nodes numbered (2), (3), ( 6) and (7), and the other one through nodes (2), ( 4), (5) and (7). Furthermore, these two loops share one common path, the path (7)(2). This demonstrates one of the advantages of representing the photonic circuit with an SFG as we can identify the loops in the circuit without any dificulty. The SFG gives us an insight into the circuit properties. The next step in the analysis is to derive the optical transfer functions of the circuit from the SFG. First of all, the output-input intensity transfer function of the DCDR circuit can be obtained by using either the graph reduction rules or Mason’s rule as introduced in Chapter 2. In deriving the optical transfer functions between nodes a and b , the subscript ab is omitted for the sake of clarity.

180

Photonic Signal Processing

By applying Mason’s rule for the SFG of the DCDR photonic circuit, the output-input intensity transfer function is given by: 4

∑ F∆ q

I H18 = 8 = I1

q

q =1

∆

(5.24)

where I 8 and I1 are the output and input intensity respectively of the DCDR circuit. The path and loop transmittances can be obtained as follows. 1. Two loop transmittances a. Loop 1: Loop 1 is formed by nodes (2), (3), ( 6) and (7), thus Loop 1 can be written as (2 )(3)(6)(7)(2) . The loop optical transmittance of Loop 1 is Tl1 = k1k2T1T3 .

(5.25)

b. Loop 2: Similarly the optical transmittance of Loop 2, which is (2)( 4)(5)(7)(2) and is given by Tl 2 = (1 − k1 )(1 − k2 )T2T3

(5.26)

2. Forward path transmittances From the graphical signal low diagram, it can be observed that there are four optical forward paths connecting nodes 1 and 8. The four forward paths and its related transmittances are: Path 1: (1)(3)(6)(8) F1 = (1 − k1 )(1 − k2 )T1

and... ∆1 = 1 − Tl 2 .

(5.27)

Path 2: (1)(3)(6)(7)(2)(4)(5)(8) 2

F2 = (1 − k1 ) k22T1T2T3

and ∆ 2 = 1

(5.28)

∆2 is equal to unity due to its forward path touching both optical loops. Path 3: (1)(4)(5)(7)(2)(3)(6)(8) F3 = k12 (1 − k2 ) 2 T1T2T3 , and ∆ 3 = 1.

(5.29)

Similar to ∆ 2 , ∆ 3 is equal to unity due to the touching of two loops of the forward path. Path 4: (1)(4)(5)(8) F4 = k1 k 2 T2 , and ∆ 4 = 1 − Tl 1 .

(5.30)

The loop determinant Δ in the denominator is then given by ∆ = 1 − Tl1 − Tl 2

(5.31)

Therefore, the optical transfer function I8/I1 as H18 can be obtained as H18 =

(1 − k1 )(1 − k2 )T1 + k1k2T2 − (1 − 2k1 )(1 − 2k2 )T1T2T3 , DEN DEN = 1 − k1k2T1T3 − (1 − k1 )(1 − k2 )T2T3 .

(5.32) (5.33)

181

Optical Dispersion Compensation and Gain Flattening

(1-k2)T1

8

(1-k2)T1

k2T2

8

k2T2

k1(1-k2)T2T3

1

1-k1

3

(a)

k2T1

7

(1-k1)T3

d1

1-k1 d1

(1-k2)T2

k1T3

1

4

k1

3

(b)

(1-k1)k2T1T3 d2

4

k1 d2

FIGURE  5.8 The reduced signal low graphs for the SFG of the DCDR circuit. (a) First reduction, and (b) Further reduction. d1 = 1 − k1k2T1T3 and d2 = 1 − (1 − k1 )(1 − k2 )T2T3.

Alternatively, the transfer functions of the DCDR circuit can be found by performing a graph reduction of the SFG in Figures  5.7 and 5.8a and b. After reduction to a certain stage, we can see that the transfer function H18 obtained from Figure 5.8b is the same as that given in (5.32). In Figure 5.8b, this can be performed by summing up the transmittances going to port 8, from path (1) (3)(8) and path (1)(4)(8). 5.2.4.2 Circulating-Input Intensity Transfer Functions The same procedure as given in the above Section  5.2.4.1 can be used to get the circulatinginput intensity transfer functions of the DCDR circuit. To obtain the transfer function H13 ( z ), the circulating intensity I3 with respect to the input intensity, we consider the SFG of the DCDR circuit again. There are two optical forward paths traveling from node 1 to node 3, they are the direct path (1)(3) and path (1)(4)(5)(7)(2)(3). Their transmittances are given by: Path 1: (1)(3) : F1 = 1 − k1, ∆1 = 1 − Tl 2

(5.34)

F2 = k12 (1 − k2 )T2T3 , and ∆ 2 = 1

(5.35)

Path 2: (1)( 4)(5)(7)( 2)(3) :

H13 ( z ) is then given as 2

∑ F∆ i

H13 ( z ) =

i

i =1

∆

(5.36)

By substituting (5.34) and (5.35) into (5.36) we have H13 ( z ) =

(1 − k1 ) − (1 − 2k1 )(1 − k2 )T2T3 , DEN ( z )

Similarly, the transfer functions H14 ( z ) and H17 ( z ) can be obtained as

(5.37)

182

Photonic Signal Processing

I1 k + (1 − 2k1 )k2T1T3 = 1 , I4 DEN

(5.38)

I7 (1 − k1 )k2T1 + k1 (1 − k2 )T2 = . I1 DEN

(5.39)

H14 ( z ) =

H17 ( z ) =

5.2.4.3 Analysis In this section, the behavior of the DCDR circuit under incoherent source condition is examined. The responses of the circuit with different circuit parameters are studied. The responses investigated including both frequency response and time response. The frequency responses are magnitude response and phase response, whereas the time responses are impulse and pulse responses of the circuit. The relationships between the pole-zero positions of the transfer functions and the responses of the circuit are considered. This Section mainly focuses on the output response rather than the circulating intensity response as the former one can be easily obtained. 5.2.4.3.1 Case 1: DCDR Resonance with Unity Delay in Each Transmission Path In the irst case the delay introduced by the interconnection between couplers in feedforward and feedback paths is equal to the sampling period of the signal entering the circuit, that means m =1 m= m3 = 1. The denominator DEN of the transfer functions derived in Section 5.2.4.2 becomes: 2 DEN ( z ) = 1 − k1k2t a1t a3G1G3 z −2 − (1 − k1 )(1 − k2 )t a 2t a3G2G3 z −2 ,

(5.40)

DEN ( z ) = 1 − k1k2G1G3 z −2 − (1 − k1 )(1 − k2 )G2G3 z −2

(5.41)

t a= t a3 = 1, we For simplicity, assuming that the iber transmission paths are lossless, that is t= a1 2 would take this assumption throughout this Section unless otherwise speciied. Thus, we have

Rearranging Eq. (5.41) in the form and setting to zero in order to ind the roots, and hence then poles of the transfer function F ( z ) = z 2 − k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 = 0

(5.42)

F (1) > 0 ⇒ 1 − k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 > 0,

(5.43)

F ( −1) > 0 ⇒ 1 − k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 > 0

(5.44)

So, with a2 = 1 and a0 = −k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 Applying Jury’s stability test (5.40) through (5.42), a stable operation of the circuit requires

and a0 < a2 ⇒ −k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 < 1

(5.45)

After some simple algebra, the stability conditions becomes k1k2G1G3 + (1 − k1 )(1 − k2 )G2G3 < 1

(5.46)

If we consider the system stability in terms of the system poles, the necessary and suficient condition for the optical networks to be stable is that all the poles of the system must lie on and/or inside the unit circle of the z-plane. In another word, the magnitude of these poles must be less than or equal to one. Thus, the poles magnitudes given by the characteristics Eq. (5.46) would result in: k1k2G1G3 + (1 − k1 )(1 − k2 )G2G3 < 1

(5.47)

183

Optical Dispersion Compensation and Gain Flattening

It is shown once again that the stability test result is strongly linked to the pole positions in the z-plane. The system poles can be made imaginary when the expression inside the absolute sign on the left-hand-side of (5.47) is set negative. This case would be treated later in the analysis. Several scenarios of Case 1 would now be studied depending on several parameters of the DCDR circuit. 5.2.4.3.2 Passive DCDR Circuit: Case 1(a) Operating the DCDR circuit under passive condition, that is when there is no optical ampliication G= G3 = 1. The stability condition in (5.47) becomes in the circuit implying G= 1 2 k1k2 + (1 − k1 )(1 − k2 ) < 1

(5.48)

The transfer function H18, Eq. (5.38), becomes H18 =

[(1 − k1 )(1 − k2 ) + k1k2 ] z −1 − (1 − 2k1 )(1 − 2k2 )z −3 1 − k1k2 z −2 − (1 − k1 )(1 − k2 ) z −2

(5.49)

where the zeroes are at: z z (1, 2) = ±

(1 − 2k1 )(1 − 2k2 ) (1 − k1 )(1 − k2 ) + k1k2

(5.50)

Let y be the expression of the LHS of (5.48), the plot of y as a function of the coupling coeficients k1 and k2 is shown in Figure  5.9. As the ranges of k1 and k2 fall within [0.5 − 1] only, the stability inequality (5.48) is always satisied. This can also be observed from Figure 5.9. It implies that the passive circuit is always stable. Recall that y is also the square of the system poles’ magnitude, y attains its maximum value of 1 when k= k= k2 1. In either of these two circumstances 1 2 =0 or=k1 the circuit reduces to a single straight through optical path and contains no feedback loop. The system poles can never be positioned on the unit circle of the z-plane.

1 k2 = 0

0.9

0.1

0.8

0.2

0.7

0.3

0.6

0.4 0.5

y 0.5

0.6

0.4

0.7

0.3

0.8 0.9

0.2

1

0.1 0

0

0.1

0.2

0.3

0.4

0.5 k1

0.6

0.7

0.8

0.9

1

FIGURE 5.9 Variation of y (LHS of Eq. (5.48) against the coupling coeficient k1 with k2 as a variable.

184

Photonic Signal Processing

Examining the transfer function H18, it is noted that the zeros of the output-input intensity transfer function for the passive DCDR circuit can be either purely real or purely imaginary depending on the values of k1 and k2. The output-input frequency responses and impulse responses of the passive DCDR circuit have been computed using different values of k1 and k2, they are shown in Figure 5.10. The corresponding responses are given in Figures 5.10 through 5.12.

FIGURE 5.10 From left to right in anticlockwise, frequency responses of the photonic circuit amplitude and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.1 (correspondent row of Figure 5.10).

FIGURE 5.11 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.1 (correspondent row of Figure 5.11).

Optical Dispersion Compensation and Gain Flattening

185

FIGURE 5.12 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.1 (correspondent row of Figure 5.12).

Figure 5.10 shows that the variation in the magnitude plot is a minute and less than 1 dB. This can be clearly observed that the zero and pole pairs are very close to each other and their effects counteract each other. Therefore, the circuit resembles a single pole system, which has the pole at the center of the unit circle in the z-plane, and the magnitude plot is quite uniform with respect to the frequency. The output-input impulse response is mainly an impulse delayed by one-unit delay time from the input, which is similar to the response of a unit delay line with loss. It is noted that there are quasi-linear portions in the phase response, which can be useful. The output-input frequency response in Figure 3.6 shows quite sharp dips. Its impulse response shows a maximum after every circulation in the loop and then decays. In this case, the zeros are purely imaginary, and the poles are purely real. Thus, they would not cancel each other as in the case in Figure 5.10. In Figure 5.12, where k= k= 0.5, it gives the zeros at 0, which is at the center of the unit circle in 1 2 the z-plane. As seen in (5.48) that if any one of k1 and k2 equal to 0.5, then the zeros would be 0 (at the origin in z-plane). Also, it is shown from Figure 3.4 that y (hence the square of system poles’ magnitude) remains constant with different k1 for k2 = 0.5. It is also true when k1 = 0.5, then y is constant when we change k2. In other words, if we set any one of k1 or k2 equal to 0.5, the poles and zeros patterns of the output-input transfer function stays the same when we change the other k. As the output response of the DCDR circuit depends solely on the poles and zeros patterns of the output-input transfer function, the response would stay the same in this case. Moreover, it can be seen from the impulse response in Figure 5.12 that it is a lossy system. It is also found that in the passive circuit the two couplers exhibit k= K is equal to that of k1 = k2 = 1 − K . symmetrical behavior. For instance, the response of k= 1 2 5.2.4.3.3 Case: Active DCDR Circuit with Unit Delay in Each Path 5.2.4.3.3.1 G1 > 1. One Optical Ampliier in Forward Path We investigate here the case where only one optical ampliier is inserted in the DCDR circuit and it is inserted in the path connecting between port 3 and 6, one of the two forward paths. The output-input intensity transfer function H18 ( z ) simpliies to: H18 ( z ) =

[(1 − k1)(1 − k2 )G1 + k1k2 ] z −1 − (1 − 2k1 )(1 − 2k2 )G1z −3 1 − [ k1k2G1 + (1 − k1 )(1 − k2 ) ] z −2

(5.51)

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the zeroes of H18 are thus located at z z ( 1, 2 ) =

(1 − 2k1 )(1 − 2k2 )G1 (1 − k1 )(1 − k2 )G1 + k1k2

(5.52)

The stability condition now becomes | k1 k2G1 + (1 − k1 )(1 − k2 ) | < 1

(5.53)

The pole values of the output intensity transfer function H18 ( z ) and its zero values are plotted against the optical gain G1 with k1 and k2 as parameters are shown in Figure 5.13a–c with different values of k1 and k2 as indicated. To get a pole value close to one and just inside the unit circle, the value of G1 needs to be 1.2, 5.2, and 3 respectively in the cases shown in Figure 5.13a–c. These give us a sharper roll-off in frequency response than the passive counterpart displayed in Figures 5.14 through 5.16. The data for the three igures are listed in Table 5.1. Comparing Figures 5.13 and 5.14 with Figure 5.10, it is observed that the introduction of optical ampliier in the DCDR circuit enhance the performance especially the frequency response. As the pole pairs are pushed very close to the unit circle in the z-plane, it results in sharper response in the magnitude plot. Recall that the pole-zero pattern in the z-plane would play a major part in the sharpness of the resonance peak of photonic circuits. To get a sharper maximum in this particular situation, it is required that the distance between the pole and the unit circle must be less than that between the zero and the unit circle (refer to Eq. (2.2–2.8)). The effect of this active mode operation can be noted by realizing that the peaks in the magnitude plot have values greater than 0 dB, while in the passive operation the corresponding peaks can only come up to 0 dB. This operation can be used as a bandpass ilter with narrow passband.

FIGURE  5.13 Plot of absolute magnitudes of the pole and zero of H18 against G1 in case 1(b)(i). (a ) k1 = 0.9 and k2 = 0.9, ( b) k1 = 0.2; k2 = 0.8, (c) k1 = 0.5; k2 = 0.5.

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FIGURE 5.14 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.2 (correspondent row of Figure 5.14).

FIGURE 5.15 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.2 (correspondent row of Figure 5.15).

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FIGURE 5.16 From left to right and anticlockwise, frequency responses of the photonic circuit amplitude and phase, the sampled impulse response and the pole-zero plot for the passive DCDR circuit whose parameters are tabulated in Table 5.2 (correspondent row of Figure 5.16).

TABLE 5.1 Parameters Used in the Analysis of the Passive DCDR Response with the Corresponding Poles and Zeros of the Output-Input Intensity Transfer Function H18 Figure 5.10 Figure 5.11 Figure 5.12

k1

k2

Poles

Zeros

0.9 0.2 0.5

0.9 0.8 0.5

0, ±0.905539 0, ±0.565685 0, ±0.707107

±0.883452 ±j1.06066 0, 0

TABLE 5.2 Circuit Parameters Used in the Analysis of the DCDR Response in Case 1(b)(i) with the Corresponding Poles and Zeros of H18 (z ) Figure 5.14 Figure 5.15 Figure 5.16

k1

k2

G1

Poles

Zeros

0.9 0.2 0.5

0.9 0.8 0.5

1.2 5.2 3.0

0, ±0.990959 0, ±0.995992 0, ±1

±0.966595 ±j1.373716 0, 0

Inspecting Figure  5.15, it is noticed that the large ampliier gain has changed the response dramatically as compared to Figure 5.11, since its presence changes the pole-zero pattern greatly. In Figure 5.16 we have a system with a pole at 0, a pole pair at ±1 and zeros at 0 realized by the DCDR circuit. This yields a magnitude plot with very sharp peaks. The lossless system also produces an ininite constant-value impulse response. The impulse response here can be applied to generate continuous sequences of “1” pulses in signal processing system. In other words, a single impulse

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189

fed into the circuit can trigger a continuous stream of impulses of the same magnitude at the output. The causes of “0” in between the ones in the impulse response can be understood by inspecting the geometry of the circuit. As can be seen, the consecutive output signals can only be detected every two sampling times, which, in this case, is the round loop time. Thus, if we want to get a stream of “1” signals, we need to acquire the output every two sampling times. If not, we would get ’0 ’’1’’0 ’’1’’0 ’’1’.... Another application is found with the impulse response generated by the DCDR circuit in the situation of Figure 5.16. We examine the pulse response as shown in Figure 5.17a–f for input sequences consisting of “1” and “0.” For Figure 5.17a, the steady state value of the pulse response is oscillating between 1 and 0 for an input stream with one “1.” For Figure 5.17b, the steady state value of the pulse response is oscillating between 1 and 1 for an input stream with two “1.” In Figure 5.17c, the steady state value of the pulse response is oscillating between 2 and 1 for an input stream with three “1.” In Figure 5.17d, the steady state value of the pulse response is oscillating between 1 and 2 for an input stream with three “1.” Following the above pattern, we can observe that from the steady state magnitude of the pulse response, the number of “1” in the input stream can be detected. It is shown in Figure 5.17e and f for longer input streams. If we look at the stream as sequence of two-digit binary numbers, from the response we can indeed determine the occurrence of “1” in a certain digit position. For instance, in Figure 5.17d, the input streams are “1 1” and “0 1.” The output stream in this case shows one in the irst digit position and two in the second digit position, which corresponds to the occurrence of “1” at that positions. The response, in fact, counts the numbers of “1” at the two digit positions, and this may be used as an adder. This ability of counting arises from the geometry of the circuit and the orders of delay in each path. In Figure 5.17g, m = m= 3, the circuit can count 1 2 numbers of “1” in four-digit numbers as expected.

FIGURE 5.17 Pulse response of the DCDR circuit for different input sequence with k1 = k2 = 0.5, G1 = 3, G2 = G3 = 1 and m1 = m2 = m3 = 1 except (g), which has m1 = m2 = 3 and m3 = 1. Input sequences for each Figure 5.17: ( a) [0 1], (b) [1 1], (c) [111], ( d ) [1101] (Continued )

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FIGURE  5.17 (Continued) Pulse response of the DCDR circuit for different input sequence with k1 = k2 = 0.5, G1 = 3, G2 = G3 = 1 and m1 = m2 = m3 = 1 except (g), which has m1 = m2 = 3 and m3 = 1. Input sequences for each Figure 5.17: (e) [1 0 1 0 1 1 0 1], ( f ) [1 1 1 0 1 0 1 1 0 1], ( g ) [1 0 1 0 1 1 0 1].

5.2.4.3.3.2 G2 > 1. Optical Ampliier in the Other Feed Forward Path This case would be similar to Case 1(b)(i) where only one optical ampliier is placed in the other feedforward path. The output-input intensity H18 becomes H 18 ( z ) =

[(1 − k1)(1 − k2 ) + k1k2G2 ] z −1 − (1 − 2k1 )(1 − 2k2 )G2 z −3 1 − [ k1k2 + (1 − k1 )(1 − k2 )G2 ] z −2

(5.54)

(1 − 2k1 )(1 − 2k2 )G2 (1 − k1 )(1 − k2 )+k1k2G2

(5.55)

with the zeros at:

z z ( 1, 2 ) = ±

The characteristic equation remains similar to those equations given in Section 5.2.2 except that the coupling coeficients are interchanged and G2 is placed appropriately. Applying the Jury’s stability test again we obtain the stability condition as: | k1k2 + (1 − k1 )(1 − k2 )G2 | < 1

(5.56)

The responses of the DCDR circuit in this case are similar to Case 1(b)(i), which has the ampliier inserted in the other forward path of the circuit. To illustrate the duality of the two cases we try k=1 k= 0.1= , G1 1,= G2 1.2 and G3 = 1. The circuit responses are shown to be the same of that in 2 Figure 5.14 for case 1(b)(i) where k= k= 0= .9, G1 1= .2, G2 G3 = 1, since the values of poles and 1 2 zeros are the same for these two sets of parameters.

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191

5.2.4.3.4 G3 > 1. Optical Ampliier in the Feedback Path The output intensity transfer function H18 now becomes: H 18 ( z ) =

[(1 − k1 )(1 − k2 ) + k1k2 ] z −1 − (1 − 2k1 )(1 − 2k2 )G3 z −3 1 − [ k1k2 + (1 − k1 )(1 − k2 ) ]G3 z −2

(5.57)

(1 − 2k1 )(1 − 2k2 )G3 (1 − k1 )(1 − k2 ) + k1k2

(5.58)

with the zeros at: z z1,2 = ±

The stability condition follows immediately that | k1 k2G3 + (1 − k1 )(1 − k2 )G3 | < 1

(5.59)

0.5, G=1 G= In the special case of k=1 k= 2 2 1 and G3 = 2, the output intensity responses are plotted in Figure 5.18. The frequency response of the DCDR circuit is the same as that in Figure 5.16 where the same k values are used there. However, with the optical ampliier situated at the feedback path rather than the feed forward path of the circuit, the optical gain required in the former case (to have the same pole and zero patterns and satisfying the stability condition) is less than that in the latter one. This makes one think to put the ampliier in the feedback path instead of putting it in the feed forward path. Nevertheless, the amplitude of the impulse response is smaller in this case where we put the ampliier in the feedback path. This is because the gain we used in Figure 5.18 is smaller than that in Figure 5.16 and the output is obtained immediately after the ampliier for the case in Figure 5.16. The three cases in case 1(b) have been studied where only one optical ampliier is available for the DCDR circuit. In fact, this is very important in practice when several DCDR circuits are integrated to form a network then the least number of optical ampliiers in the loop is required. Therefore, the most important question remained to be addressed is where should we place the

FIGURE  5.18 Frequency response, impulse response and pole-zero plot for the active DCDR circuit (case 1(b)(iii)) with k1 = k2 = 0.5, G1 = G2 = 1 and G3 = 2.

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Photonic Signal Processing

optical ampliier in the DCDR circuit? Should it be- in the feed forward or feedback path? The answer depends on the speciic applications. Considering the case where the DCDR circuit is applicable as a ilter then the optical ampliier should be placed in the feedback path. As stated above, the ampliier in the feedback path requires less optical gain than that in the feed forward path to achieve the same performance. From another point of view, we can say that with an ampliier of a particular gain, better performance is achieved (as far as the use as a ilter is concerned) if we put it in the feedback path (providing that the stability criterion is met). If we are interested in the impulse response for applications in photonic digital processing systems, the position of the optical ampliier is very important. Also, depending on the output ports where signals are tapped, the optical ampliier must be closed to that port. Thus, both cases where the optical ampliier is placed in feed forward or feedback paths can be used with appropriate applications. 5.2.4.3.5 DCDR Circuit with Multiple Delays: Case 2 Different combinations of delay orders in the two feed forward paths and the feedback path would result in different circuit performances. One useful point to note here is that we can obtain poles, which are evenly and equally spaced around the circle in z-plane by certain choice of delay orders m1 , m2 , and m3. Inspecting the denominator of the circuit’s transfer function, which is DEN in Eq.  (5.33), if m1 = m2 , DEN will always follow: 1− az − b

(5.60)

where a is any real number and b is any positive integer. Thus, we would have multiple poles for the system, and the same stability criterion as outlined in Section 5.2, can be applied as well. The magnitude frequency responses with m1 = m2 , which equals to 2 and 3, respectively, are shown in Figures  5.19 and 5.20. The same k and G values used in Figure  5.16 are used here. In Figure 5.18, the poles’ value is located at 0, 0, −0.5 ± j 0.866 and 1. In Figure 5.20, the poles are located at 0, 0, 0,± j and ± 1. An application of this feature is that optical frequencies or

FIGURE 5.19 Frequency response, impulse response and pole-zero plot for the active DCDR circuit (case 2) with k1 = k2 = 0.5, G1 = 3, G2 = G3 = 1 and m1 = m2 = 2, m3 = 1.

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193

FIGURE 5.20 Frequency response, impulse response and pole-zero plot for the active DCDR circuit (case 2) with k1 = k2 = 0.5, G1 = 3, G2 = G3 = 1 and m1 = m2 = 3, m3 = 1.

wavelengths of input optical signal can be iltered at equal interval, for example as a group of optical carriers at equal intervals. It is expected that if the delays in the two forward paths are different, interesting responses would result. In this case, the intensities in the two forward paths do not always add up at coupler 2 at the same time. A result with= m1 2= , m2 1, m3 = 1 is shown in Figure 5.21. The conjugate pole pairs are now located well inside the unit circle leading to a lossy system and an oscillating time response. The appearance of the impulse response conirms this point.

FIGURE 5.21 Frequency response, impulse response and pole-zero plot for the active DCDR circuit (Case 2) with k=1 k= 0.5= , G1 3= ,= G2 G3 1= and m1 2= , m2 1= , m3 1. 2

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5.2.4.3.6 Special Case with Negative Optical Gain: Case 3 5.2.4.3.6.1 Purely Real or Purely Imaginary Poles, m1 = m2 = m3 = 1 In this section, an optical ampliier with a negative optical small-signal gain is used, this would allow much greater lexibility in tuning the peaks of the magnitude plot corresponding to adjusting the pole-zero pattern of the optical transfer function of the DCDR circuit. The negative optical gain factor can be achieved by using an optical transistor as described in reference. Alternatively, an optical ampliier incorporated with a π-phase shifter such as a LiNbO3 integrated optic phase modulator would allow a negative optical gain. It is mentioned in Section  5.2.4.2 that DEN is of the form as in Eq.  (3.3–19) if m1 m= = 2 . If m1 m= m= 1, DEN is of the form: 2 3 DEN = 1 − az −2

(5.61)

With a = k1k2G1G3 + (1 − k1 )(1 − k2 )G2G3 . The optical system poles would be located at 0, π , 2π ... if a > 0 and at π / 2, 3π / 2, ... if a < 0. As it can be easily seen, the characteristic equation of the photonic DCDR circuit would result in a quadratic equation with roots as purely real or purely complex conjugates in this situation. To achieve the latter case, it requires ampliiers in the DCDR circuit with negative gain. Firstly, we examine a typical example in the above case where k= k= 0.1 and the gain factors 1 2 for the three optical paths are G3 = 1 and G2 = −1.2, −1.25 and − 1.3, respectively, with the gain G1 as a variable. Figure 3.17a shows the magnitudes of the poles (solid line) and that of the zeroes (dashed line) of H18 ( z ) as a function of gain G1 for G3 = 1 and G2 = −1.2. For stability, the values of the poles and zeroes would never reach unity for this particular circumstance. In fact, it can reach closed to unity, however both zero and pole approach unity leading to the cancellation of their effects. When the value of the negative gain G2 is increased to −1.25 and −1.3, the values of the pole and zero are plotted in Figure 5.22b and c respectively. Figure 5.22b exhibits a similar characteristic as in Figure 5.22a except that the pole and zero can now reach unity. When G2 = −1.3 the pole and zero at unity are separated as it can be observed in Figure 5.22c. Thus, we can select G1 = 5.3 and G3 = 1, G2 = −1.3 for the pole equals to unity corresponding to a

FIGURE 5.22 Values of poles (solid line) and zeroes (dashed line) versus gain G1 for DCDR circuit with k1 = k2 = 0.1, G3 = 1 and (a) G2 = −1.2, (b) G2 = −1.25, (c) G2 = −1.3.

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FIGURE  5.23 Magnitude frequency response of H 18 ( z ), H 18 ( z ) with k1 = k2 = 0.1, G1 = 2, G2 = −1.25, G3 = 1 and m1 = m2 = m3 = 1 .

zero value of 1.015. It is also noted here that for two identical couplers the value of zero is always greater than that of the pole. Thus, the amplitudes of the frequency response always have minima according to the analysis of transfer function in the z-domain. Hence, the DCDR with negative gain in one of the forward or feedback paths can only be used as an optical notch ilter at output port 8. For the case where G2 = −1.25 and G1 = 2, the DCDR circuit has altogether three system poles: one at the origin, z p (1) = 0 and one complex pole pair at z p( 2,3) = ± j 0.996243. The transfer function H18 ( z ) has two zeroes at the location z z(1,2) = ± j 0.997663. Clearly the zeroes follow closely the complex pole pair. Figure 5.23 shows the magnitude response of this optical transfer function. The relation in Eq.  (2.2–8) allows us to design photonic circuit with adjustable response by appropriately positioning the poles and zeroes of the optical transfer function. This fact is indeed the novel feature of our analysis using the SFG technique and z-transform. It is also determined that when G1 is increased from 2 to 5 in the case for Figure 5.23 the magnitude H18 ( z ) changes from a ratio of 2336 / 3757 to 4655 / 18929. Figure 5.24 shows the magnitude and phase responses together with the pole-zero position in the z-plane of the circulating transfer function H 14 ( z ) with the same circuit parameters as used in Figure 5.23. This H 14 ( z ) has maxima at an optical frequency corresponding to ωτ equals to odd multiple order of π/2. At these values, the transfer functions H 18 ( z ) and H 13 ( z ) have minima as it can be observed from Eqs. (5.32), (5.49) and (5.57). Thus, the circuit performs a kind of quasi-resonance. Since it is assumed that the source is temporal incoherent here, thus resonance effect should not occur in the circuit. This is because the beams traveling inside the circuit always add up constructively. However, with the application of negative optical gain in one of the paths, destructive interference can occur, which results in a behavior similar to resonance in the coherent case. The resonance condition for the DCDR circuit under coherent source will be determined in Chapter 7. H13 denotes the circulating transfer function in one of the two optical loops. Thus, we can conclude that at this resonance the energy is stored in one of the optical loops only for circuits with two optical loops that share one common path. In addition, with this case of G2 = −1.25 and G3 = 1 and G1 = 2, the loop (3)(6)(7)(2)(3) has a positive transmittance value while the other loop (4)(5)(7)(2) (4) would have a negative loop transmittance. The negative gain in the forward path (4)(5) would interfere with the optical waves in the forward paths (3)(6) in a destructive manner and hence can generate a depletion of the output at a destructive interference. Hence the optical energy is stored only in the loop (4)(5)(7)(2)(4). This helps explaining the quasi-resonant effect mentioned earlier.

196

FIGURE 5.24

Photonic Signal Processing

The responses of H 14 ( z ) with k1 = k2 = 0.1, G1 = 2, G2 = −1.25, G3 = 1 and m1 = m2 = m3 = 1.

This inding is a signiicant development by applying the SFG technique in optical resonators in particular and photonic circuits in general. = = m3 1 and either m1 0 or m2 = 0 5.2.4.3.7 Next, we consider another situation where we have direct connection in either one of the forward paths (3)(6) or (4)(5). m3 is equal to 1. The direct connection would convert the DEN to a quadratic equation of the form: DEN = a + bz −1 + cz −2 ,

(5.62)

The values of the poles are then given by z p( 1, 2 ) =

−b ±

b2 − 4ac , 2a

(5.63)

where a, b, and c are three constants whose values depend on the photonic components and m1 0= , m2 1, m3 = 1. The correspond  with those of Eq.  (5.33). We consider here the case of= expression of DEN in terms of the circuit parameters: 1 − k1k2G1G3 z −1 − (1 − k1 )(1 − k2 )G2G3 z −2 = 0

(5.64)

By applying the Jury’s stability test for digital systems again, with a2 = 1, a1 = −k1k2G1G3 , a0 = (1 − k1 )(1 − k2 )G2G3

(5.65)

Optical Dispersion Compensation and Gain Flattening

197

The stability criteria requires 1 − k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 > 0 1 + k1k2G1G3 − (1 − k1 )(1 − k2 )G2G3 > 0

(5.66)

(1 − k1 )(1 − k2 )G2G3 < 1 Rearranging Eq. (5.66), we obtain k1k2G1G3 + (1 − k1 )(1 − k2 )G2G3 < 1 (1 − k1 )(1 − k2 )G2G3 − k1k2G1G3 < 1

(5.67)

(1 − k1 )(1 − k2 )G2G3 < 1, To obtain a complex conjugate pole pair it then requires from Eq. (5.67) that (k1k2G1G3 )2 + 4(1 − k1 )(1 − k2 )G2G3 < 0,

(5.68)

In order to satisfy Eq. (5.68), the ampliier gain G2 or G3 must take suficient negative values if the couplers are passive i.e. k1, k2 < 1. There are several combinations of photonic circuit parameters of the DCDR resonators to satisfy this equation. A procedure can be established as follows: select a combination of four out of ive of k1, k2 , G1, G2 and G3 , plot Eq. (5.68) against the ifth parameter as a variable, then ind a range of operation so that this inequality is satisied, and inally choose values in this range to design the photonic circuit and re-check the stability condition. An example is given here to illustrate the above procedure where k1 = k2 = 0.3, G1 = 2, G2 = −1 are chosen. Figure 5.25 shows the L.H.S. of Eq. (5.68) as solid line and the other three lines represent the L.H.S. of Eq. (5.68), which are the stability expressions. The optical gain G3 must be chosen so that it satisies Eq. (5.68) as well as the stability conditions. It follows that the maximum value of G3, where the system remains stable, is about 2.

FIGURE 5.25 Selecting optical gain parameters for the design of a DCDR circuit when= m1 0= , m2 m3 = 1.

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FIGURE  5.26 Frequency and discrete-time responses for H18(z) with k1 = k2 = 0.3, G1 = 2, G2 = −1, G3 = 2 and m1 = 0, m2 = m3 = 1. Poles : 0.1800 + j 0.9734, Zeroes : 0.0459 + j 0.8068.

This analytical examination of the stability of the photonic circuits is unique when a z-transform method is employed. Therefore, lexibility in design is achieved. The frequency response of the intensity transfer function H18 ( z ) is shown in Figure 5.26 with the circuit parameters selected as given above. From the pole-zero pattern plotted, we can see that the poles and zeroes of the optical transfer function H18 ( z ) are complex in nature. It is noted here that the poles and zeroes are complex conjugates and are not purely real or purely imaginary. This would allow the peaks of the magnitude plot designed to be closely spaced together. An application of this feature is that two or several wavelengths can be demultiplexed as we can observe from Figure 5.26. A phase change of occurs at each pole position. The periodic variation of the output impulse response of the DCDR shows the circulating of the impulse delay in the re-circulating loops. Another example of the operation of the DCDR circuit is to choose the poles closely spaced. k= 0= .5, G1 1= .9, G2 1 The pole and zero patterns of a DCDR with m1 = 0, m2 = m3 = 1 and k= 1 2 and G3 = −4 is shown in Figure 5.27. The zeroes of the system are now located at the origin and a real negative of −0.5263. The magnitude response shows the feature of demultiplexing of this DCDR coniguration. Furthermore, the impulse response shows the variation of the output pulses with certain pattern different from that in Figure 5.26. 5.2.4.3.8 Remarks In this section, the DCDR circuit is studied under temporal incoherent source for different operation modes. It is found that for active operation, the locations of the ampliiers in the circuit depend on the application required. Greater design lexibility is found with presence of negative optical gain in the circuit. It is discovered that a quasi-resonance behavior is possible with the insertion of negative optical gain in the delay line. Procedures in design of the DCDR circuit are shown in the special case of negative optical gain but in general, this can be applied to other situations. Applications of the circuit in signal processing such as ilters, counters, and adders are realized. One point is needed to be clariied before proceeding to the next chapter. It is more appropriate to call the DCDR photonic circuit a re-circulating delay line rather than a resonator when it is operating at the incoherent source situation. Although the output intensity response can have minimums for certain

Optical Dispersion Compensation and Gain Flattening

199

FIGURE  5.27 (a) Amplitude and (b) phase frequency response and (c) discrete-time impulse response and (d) pole-zero pattern for H18 ( z ) with k1 = k2 = 0.5, G1 = 1.9, G2 = 1, G3 = −4 and m1 = 0, m2 = m3 = 1. Poles: 0.9500 ± j 0.3123, Zeroes : 0, −0.5263.

circuit’s parameters (negative optical gain), the circulating intensity responses do not have maximums at the same condition. It would be shown in the next chapter that for a resonator to have resonance, there are generally two constraints it need to be met. They are the constraints on the circuit’s parameters and the other one is the constraint on the operating point of the resonator. The latter one is usually related to the phase change the signal experiences when it travels along a loop in the circuit. For the source incoherent condition, we have ignored the effect of phase that could have on the signal. The parameter constraint can be easily met but due to the ignorance of the phase in the circuit, the phase constraint cannot be met. The circuit is also very stable due to the ignorance of the phase effect on the circuit’s performance.

5.2.5 DcDr unDer cOherence OperatiOn In the previous sections of this chapter, the DCDR circuit is studied under incoherent source condition. In order to study the resonance effect of the DCDR circuit, the analysis is required to be carried out on the ield-basis. Therefore, in this chapter, the circuit excited by coherent source with inite linewidth will be considered. The DCDR circuit is considered as a resonator here while in the previous chapter it is treated as re-circulating delay line. Previous studies on the effect of source coherence on the performance of resonator circuit, but no one had ever performed the analysis on DCDR circuit or more complicated photonic circuits. Under the current trends of integrated photonics circuits based on Silicon, Si on insulator (SOI) technology photonics circuits will be complex, and the computer aided circuit design will become useful. This mainly due to the complicated manipulations involved. However, with the help of our newly developed SFG theory in optical circuits, these dificulties can be overcome easily. In this chapter, an algorithm is derived to compute the response systematically. The temporal transient response of the circuit is shown with source coherence. 5.2.5.1 Field Analysis of the DCDR Circuit Before considering the case for source with inite linewidth, we irst derive the transfer functions of the DCDR circuit in terms of ield rather than intensity as in the previous chapter. In this section, we derive the ield version of the transfer functions for the DCDR circuit. The analysis in this Section is

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Photonic Signal Processing

different from the preceding intensity-basis analysis in the sense that the phase change encountered by the signal in the circuit would be taken into consideration, thus interferometric effects can take place. Therefore, the resonance effect of the DCDR circuit can be investigated. Firstly, we deine the DCDR circuit parameters with lightwaves represented in the ield amplitude. The subscript f distinguishes the nature of the variables as ield variables from those in terms of intensity. The circuit parameters are deined as: k pcf = − j k p where p = 1,2 as the ield cross-coupling coeficient for the two couplers, k pdf = 1 − k p where p = 1, 2 is the ield direct-coupling coeficient for the two couplers 1 and 2, and Tif = tai Gi z − mi for i = 1, 2 and 3. The −j term in the k pcf expression above accounts for the −π/2 phase shift induced by the coupler during the cross-coupling. 5.2.5.2 Output-Input Field Transfer Function The output-input ield transfer function is obtained by carrying out the same procedure as introduced in Section 5.2.3. The output-input ield transfer function of the DCDR circuit, by using Mason’s rule given in Chapter 2, is given by 4

∑F ∆ qf

E H18 f ( z) 8 = E1

qf

q =1

∆f

(5.69)

E8 and E1 are the output and input ield amplitude respectively of the DCDR circuit. The loop transmittances and the forward path transmittances are stated as follows: 1. Loop transmittances a. Loop 1: (2)(3)(6)(7)(2): The loop optical transmittance of Loop 1 is b. Loop 2: (2)(4)(5)(7)(2) The optical transmittance of Loop 2 is Tl 2 f = k1df k2 df T2 f T3 f 2. Forward path transmittances The four forward paths and their related transmittances are: Path 1: (1)(3)(6)(8) F1 f = k1df k2 df T1 f and ∆1 f = 1 − Tl 2 f Path 2: (1)(3)(6)(7)(2)(4)(5)(8) we have F2 f = k1df 2k2cf 2T1 f T2 f T3 f and ∆ 2 f = 1 where ∆ 2 f is equal to unity due to its forward path touching both optical loops. Path 3: (1)(4)(5)(7)(2)(3)(6)(8) F3 f = k1cf 2k2 df 2T1 f T2 f T3 f , and ∆ 3 f = 1. ∆ 3 f is equal to unity due to the touching of two loops of the forward path. Path 4: (1)(4)(5)(8) F4 f = k1cf k2cf T2 f , and ∆ 4 f = 1 − Tl1 f Furthermore, the loop determinant ∆ f is given by ∆ f = 1 − Tl1 f − Tl 2 f Hence, the output-input ield transfer function, in terms of k1, k2 , T1 f , T2 f and T3 f can be expressed as: H18 f ( z ) =

(1 − k1 )(1 − k2 )T1 f − k1k2 T2 f − T1 f T2 f T3 f 1 + k1k2 T1 f T3 f − (1 − k1 )(1 − k2 )T2 f T3 f

(5.70)

5.2.5.3 Circulating to Input Field Transfer Functions Similarly, the circulating-input ield transfer functions can be derived and is given as follows: H13 f ( z ) =

(1 − k1 ) − (1 − k2 )T2 f T3 f DEN f

(5.71)

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Optical Dispersion Compensation and Gain Flattening

− j k1 − j k2 T1 f T3 f DEN f

(5.72)

− j k1 (1 − k2 )T2 f − j (1 − k1 )k2 T1 f DEN f

(5.73)

H14 f ( z ) =

H17 f ( z ) =

where DEN f = 1 + k1k2 T1 f T3 f − (1 − k1 )(1 − k2 )T2 f T3 f . 5.2.5.4 Resonance of the DCDR Circuit The usual deinition of optical resonance in a photonic circuit is that at a particular frequency or wavelength the optical output takes a minimum value while the optical energy is circulating in the loops of the photonic circuit. Thus, the resonant condition can be found by setting the output transfer function to zero or effectively inding the zeroes of the output-input optical transfer function. In this case, the DCDR circuit can be referred as a resonator. Considering (5.70), if ta1 = ta2 = ta3 = ta , k1 = k2 = k, G1 = G2 = G3 = 1 and m1 = m2 = m3 = 1, the equation simpliies to: H18 f ( z ) =

(1 − 2k ) t a z −1 − t a t a z −3 1 − (1 − 2k )t a z −2

(5.74)

Rearranging, Eq. (5.74) gives H18 f ( z ) =

(1 − 2k ) t a z 2 − t a t a

(5.75)

z 3 − (1 − 2k )t a z

Setting the numerator in Eq.  (5.75) to zero resulting in two resonance conditions: (i) if z 2 = −1, k = (1 + ta ) / 2; and (ii) if z 2 = 1; k = (1 − t a ) / 2; z 2 = −1 means z = ± j , which can be interpreted as ωτ = nπ − π 2 ... with n = 1, 2... Recall that ωτ is the phase change through a iber path in the circuit with unit delay time. Similarly, z 2 = 1, which means z = ±1, can be interpreted as ωτ = n, n = 1, 2.... As ta is close to 1 for low-loss ibers, k for resonance would be close to 1 and 0, respectively, for the two resonant conditions. By inspecting the SFG of the DCDR circuit in Figure  5.28, it can be seen that there are two touching loops. They are Loop 1, which is (2)(3)(6)(7)(2), and Loop 2, which is (2)(4)(5)(7)(2), with their loop transmittances. The frequency response of the output-input intensity transfer function and the circulating-input intensity transfer functions under the resonant conditions given in Eqs.  (5.72) and (5.75) are plotted in Figures 5.29 and 5.30 respectively. The instantaneous optical intensities are

T1f

3

k1df

k1cf

2

k3f

7

k1cf

6

k2df

k2df

k1df 1

k2cf

4

T2f

5

k2cf

FIGURE 5.28 The SFG of the DCDR circuit in terms of the ield variables.

8

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Photonic Signal Processing

FIGURE 5.29 Frequency response of the output-input intensity transfer function and the circulating-input intensity transfer functions of the DCDR resonator under the resonant conditions of ta = 0.99 and k = 0.995, which satisfy the condition listed in Eq. (4.2–4.16).

FIGURE 5.30 Frequency response of the output-input intensity transfer function and the circulating-input intensity transfer functions of the DCDR resonator under the resonant conditions of t a = 0.99 and k = 0.005 , which satisfy the condition listed in Eq. (4.2–4.17).

obtained by taking the square of the modulus of the corresponding ield amplitudes. The vertical scale of the magnitude plot is shown with the absolute magnitude rather than dB as used in the previous chapters. It is found that this scale would give a better illustration of the resonance effect of the circuit. In Figure 5.29, there are maximums for the circulating intensity I3 at ωτ = π / 2, 3π / 2, 5π / 2 and 7π / 2 rad. and correspondingly there are minimums for the output intensity I8 and circulating intensity I4 at these positions. This shows the resonance of Loop 1 of the DCDR resonator. Indeed, the resonance criterion on the phase change for the signal in the circuit is that the round-loop phase

Optical Dispersion Compensation and Gain Flattening

203

change of the resonant loop needed to be 2nπ where n is an integer. Looking at the criterion on ωτ (the phase change per iber path with unit delay) for resonance of Loop 1 in Eq. (5.76), it shows the above point. The round-loop phase change for Loop 1, in this case, is 2(nπ − π /2), plus the phase change encountered across the couplers that is π /2 per coupler. It is obvious that this sum adds up to the value of 2nπ . This is another way of looking at the resonance condition, and it can be determined by examining the circuit’s signal low graph alone. The resonance of Loop 2 of the DCDR resonator is shown in Figure  5.30. There are maximums for the circulating intensity I4 at ωτ = π , 2π , 3π and 4π and correspondingly there are minimums for the output intensity I8 and circulating intensity I4 at these positions. It is also found that the sharpness of the resonance depends on the circuit parameters. In Figure 5.31, resonance behavior of the DCDR resonator is shown with t a = 0.8 and k = 0.1, which corresponds to the resonance of Loop 2 of the resonator. A smaller value of ta is used here, and it indicates the loss in the iber path is larger. It is clearly seen that the maximum value of the circulating intensity I4 is much smaller than the one in Figure 5.30. Thus, in this case, it can be stated that low-loss ibers should be used in building up the resonator in order to have sharp resonant peaks. In the above results, it is observed that the energy can be stored in either of the two loops of the DCDR resonator depending on the circuit parameters. It is also noticed from Figures 5.29 and 5.30 that H17 f has maximums in both cases. Next, we examine the stability of the circuit under the above resonant condition. Applying Jury’s stability test to H18 f results in the stability condition of (1 − 2k )ta < 1

(5.76)

From (5.76), it can be seen that a stable operation of the passive DCDR resonator is always fulilled provided that 0 < k < 1 and 0 < t a < 1. If the DCDR resonator is operated in a different situation as mentioned above, for instance G1 > 1 or k1 ≠ k2, there would be other sets of resonant conditions for the resonator.

FIGURE 5.31 Frequency response of the output-input intensity transfer function and the circulating-input intensity transfer functions of the DCDR resonator under the resonant conditions of ta = 0.8 and k = 0.1, which satisfy the condition listed in Eq. (4.2–4.17).

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Photonic Signal Processing

5.2.5.5 Transient Response of the DCDR Circuit 5.2.5.5.1 Effects of Source Coherence In Section 5.2.4.3, we study the response of the DCDR circuit to a monochromatic light source, i.e. single frequency, which can be called the steady-state response of the circuit. But, practically, a purely monochromatic source is not available. The best we can have is a single frequency source with inite linewidth. In the following analysis, the effect of the source coherence is taken into account in evaluating the transient response of the DCDR circuit. We start the analysis by considering the electric ield of the input light source, which is expressed in the form: E1(t ) = Es (t ) exp ( j (ωot + φ (t ))

(5.77)

H18 f ( z ) = ... + h[1]z1 + h[0] + h[1]z −1 + h[2]z −2 + h[3]z −3 + ...

(5.78)

where Es (t ) is the amplitude of the electric ield at time t, ω0 is the frequency of the source, and φ (t ) is the time-dependent phase that represents the phase luctuation. This is often called the phase noise in optical iber systems. This phase luctuation is the cause of broadening of the optical source spectrum which results in inite linewidth of the source spectrum. If the source amplitude is time invariant and recalling some basic properties of the z-transform, the output-input ield transfer function of the circuit H18 f = H ( z ) can be stated in the form:

An inverse z-transform manipulation would convert H18f from the z-domain to the time domain. For m1 = m2 = m3 = 1, that is, there is unity delay in each path of the DCDR circuit and t= t a= t a3 = 1, a1 2 the expansion of Eq. (5.78): h[n] = 0 for all n < 0,

h[0] = 0,

h[1] = ( (1 − k1 )(1 − k2 )G1 − k1k2G2 ), h[2] = 0 , h[3] = −  k1k2 (1 − k1 )(1 − k2 )G3 (G1 + G2 ) + (k1 + k2 − 2k1k2 ) G1G2G3  ,   Hence, h[n] represents the impulse response of the system at time index n. Although h[n] is calculated for the restricted condition m = m= m3 = 1 and t= t a3 = 1 for simplicity, the 1 2 a1 t a= 2 following analysis is applicable to the general situation. To compute the output of the resonator for an arbitrary input sequence (or pulse shape) x[n], we perform the convolution in the time domain between h[n] and x[n], i.e. the output y[n] is given by: y[n] = h[n]* x[n]

(5.79)

where * represents the convolution operation. This is in fact the theory of operation to get the pulse response of the system. Recall that the basic delay time of the circuit is denoted as τ and it is the sampling time as well. At time t n = n, the time-averaged output intensity I ( tn ) of the DCDR circuit is I8 ( tn ) = E8 ( tn ) E8* ( tn )

(5.80)

where E8* ( tn ) is the complex conjugate of E8 ( tn ). The angular brackets denote ensemble average. From a different point of view, E8 ( tn ), which is the output at time tn = n , can be considered

205

Optical Dispersion Compensation and Gain Flattening

as the nth sampled output and, similarly, the input E1 ( tn ) at time tn = n can be regarded as the nth sampled input into the circuit. So, the former term is in fact y[n] and the latter term corresponds to x[n] in Eq.  (5.79). We can see the relationship between the discrete-time signal representation and the signals in the photonic circuit. Next, the effect of source coherence is considered. In order to include the source coherence contribution to our analysis, the phase luctuation term e( jφ ( t )) of the light source as given in Eq. (5.80) is needed to be examined. This time-varying phase luctuates randomly and is statistical in nature, thus statistical method is used to handle it. Traditionally, it is treated as random signal or process, and it is best described by its correlation function. In general, we consider the autocorrelation function R[( p − s)τ ] of the input optical wave ield and it is given as R[( p − s)τ ] =

E1 (t − pτ ) E1* (t − sτ )

(5.81)

 Es (t − pτ ) Es * (t − sτ )   

where p and s are integers. From Eqs. (5.80), (5.81) becomes: R[( p − s)τ ] = expj[(s − p)ωoτ ] exp[ jφ (t − pτ )  exp  jφ (t − sτ )]

(5.82)

If the spectrum broadening of the laser due to the random phase luctuation ϕ(t) is of the Lorentzian form, the phase function exp[ jφ (t )] would have the following correlation (5.83)

exp [ jφ (t − pτ ) ]exp[ − jφ (t − sτ ) ] = exp [( − s − p)∆ωτ ]

Where ∆ω 2π is the half-width at half-maximum of the Lorentzian spectrum. We also note that e −∆ωτ is the Fourier transform of the Lorentzian spectrum. It is related to the autocorrelation function for the phase. Since the coherence time τ c of the light source is equal to26:

τc =

1 2∆ω

(5.84)

Thus, this can be rewritten as exp[ jφ (t − pτ )  exp  − jφ (t − sτ )] = exp (− s − p )τ / τ c 

(5.85)

Hence, Eq. (5.82) can be rewritten as R[( p − s)τ ] = exp j[(s − p)ωoτ ]exp(− s − p τ / 2τ c ) = exp j[(s − p)ωoτ ]D with

− s− p

(5.86)

D = eτ /2τ c

This gives an expression for the phase autocorrelation function in terms of the source coherence time. When computing I8 ( tn ) in Eq.  (5.83), products of the input ield such as < E1 ( t n ) E1 * ( t n ) >, < E1 ( t n ) E1 * (t n − τ) >,... , E1(t n − pτ) E1 * (t n − sτ) ,... etc. are involved because E8 (t ) is related to E1(t ) via the sampled relation of Eq. (5.79). By using Eqs. (5.86), (5.80), and (5.85), the values of the above products and I 8 ( t n ) can be found. The general algorithms of computing the transient response of the photonic circuit are derived from the following steps: (i) From the transfer function H ( z ) of the optical circuit, obtain the 26

L. N. Binh, Advanced Digital Optical Communications, 2nd ed., Boca Raton, FL: CRC Press, 2017.

206

Photonic Signal Processing

impulse response or indeed the sequence h[n] (Eq. 5.78). (ii) Convolute the impulse response obtained in step 1 with the input sequence to obtain the output of the system for the pulse input (Eq. 5.77). (iii) Obtain the expression for the phase luctuation of the input, for example, the correlation between values of phases at different times. (iv) Compute the output intensity of the resonator by using the result obtained in step 2 in combination with expression in step 3. We can compute the transient response of any photonic circuit with transfer function H ( z ). The results for the DCDR circuit are presented in the following section. It can be seen from Eq.  (5.79) that the transient response of the circuit mainly depends on the ratio of the basic time delay to the source coherence time. For the two extreme cases, monochromatic source (very long coherence time, τ c >> τ ) and temporal incoherent source (very short coherence time, τ c  100 km cases. For distances up to 200  km for the NRZ signal and up to 400  km for the RZ signal, the eigencompensated system performs signiicantly better with chirped than with chirp-free signals. For longer distances, the eigencompensated performance shows little difference for both cases because of the reduced eigenilter bandwidth. For a 1-dB bandwidth-limited penalty, error-free transmission may still be possible for this particular eigencompensated system with distances up to 600  km for both the NRZ and RZ chirped signals. The peaks and dips of these curves are dificult to explain because of the complicated effect of the interaction of the laser chirp, iber dispersion, and eigenilter group delay. However, they provide useful information about the optimum eigencompensated performance for a particular transmission distance, which can be greatly increased without any reduction in bandwidth, by increasing the eigenilter tap according to Eq. (5.116).

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Photonic Signal Processing

FIGURE 5.54 Dependence of bandwidth-limited optical power penalty on transmission distance of an 8-Gbit/s ˆ = −8 π ) for α = 0 and α = 6. (a) NRZ signal. (b) RZ signal. 1550-nm IM/DD system with ODEC (N = 21, D

5.3.3.5 Compensation Power of Eigencompensating Technique The performance of the eigencompensated system is now characterized by means of the compensation power (CP) of the ODEC. Section 5.3.3.4 has shown that, for a 1-dB bandwidth-limited penalty, the maximum eigencompensated distance is about 600 km for both the NRZ and RZ chirped signals. ɵ = −8 π , L = 600 km, T = 50.56 ps, and ∆fmax = 9.89 GHz = 1.24 B, where B In this case, N = 21, D denotes the bit rate. For a 1-dB bandwidth-limited penalty, the product of the square of the bit rate and the eigencompensated distance for both the NRZ and RZ chirped signals is thus given by  c  B2 L ≤ 0.6( N − 4.3)   2 D λ 2   

(5.137)

when ∆fmax ≥ 1.24 B has been substituted into Eq. (5.137). For an uncompensated system, the B2 L value is given by B2 L ≤

π 2(1 + α ) 

2 12

 c   2  + 2α   2 D λ 

(5.138)

which decreases with increasing chirp parameter α. For α = 0, Eq. (5.138) is almost the same and a factor of π/2, where a 1-dB penalty was used as a criterion in both references. Eqs. (5.137) and (5.138) are graphically illustrated in Figure 5.55, where the dispersion limits for both α = 0 and  α = 6 are shown to be signiicantly improved by the ODEC. For example, for B = 8 Gbit/s, the uncompensated distance is limited to only 7.5  km for α  =  6 but is extended to 90  km for α  =  0. However, the presence of the ODEC signiicantly extends the transmission distance to 575 km for N = 21 and B = 8 Gb/s.

Optical Dispersion Compensation and Gain Flattening

237

FIGURE 5.55 Dispersion limits for the uncompensated transmission using external modulation (α = 0) and direct modulation (α = 6), and dispersion improvements for the eigencompensated transmissions using direct modulation (α = 6).

The overall performance of the eigencompensated system can be characterized by the compensation power of the ODEC, which is a measure of the increase in the B2 L value and is deined as the ratio of Eq. (5.138) to the dispersion factor given in Eq. (5.105) as CP = 0.19( N − 4.3) 2(1 + α 2 )1 2 + 2α  .

(5.139)

Figure 5.56 shows that the CP value increases with the eigenilter order as well as with the laser chirp parameter. The above example gives CP = 77 for α = 6, showing that the B2 L value of the eigencompensated direct-modulation system is improved by a factor of 77.

FIGURE 5.56 Compensation power (CP) of the ODEC for various values of the chirp parameter and ilter order.

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Photonic Signal Processing

The reasons for comparing the eigenilter technique with the Chebyshev technique are two-fold. First, the Chebyshev technique58,59 has a greater design lexibility than those of the chirped iber Bragg grating60 (5.54) and the Fabry-Perot equalizer61,62 in modifying the ilter frequency response. Second, the ODEC and the Chebyshev equalizer have common features, such as linearity, nonrecursiveness, and periodic frequency response, and they both can be implemented using the same PLC technology. As briely described in Section 5.3.1, there are advantages as well as disadvantages associated with various techniques, so that it is dificult to make a fair comparison between them. However, the unique feature of the ODEC is its ability to perform better with dispersively chirped signals than with dispersively chirp-free signals, an advantage that has not yet been claimed by other methods. In addition, the ODEC is compact, capable of operating stably and can perform high-speed signal compensation because of its integrated-optic form. Further progress in the PLC technology would make the ODEC even more attractive. 5.3.3.6 Remarks • An effective digital eigenilter approach has been employed to design linear ODECs for compensation of the combined effect of laser chirp and iber dispersion at 1550 nm in highbit-rate long-distance IM/DD lightwave systems. • The ODECs, which have been synthesized using an integrated-optic transversal ilter, are very effective in equalization of a dispersively chirped optical communication channel. • In an 8-Gbit/s 100-km 1550-nm system, the performance of the ODEC is more impressive than that of the Chebyshev equalizer in both the frequency and time domains. The ODEC slightly outperforms the Chebyshev equalizer for external-modulation transmission but signiicantly outperforms the Chebyshev equalizer for direct-modulation transmission. • For a 1-dB power penalty, a 21-tap ODEC may provide error-free transmission of an 8-Gbit/s 1550-nm direct-modulation system for distances up to 600 km for both the NRZ and RZ chirped signals. • The ODEC has a large value of compensation power, which increases with the eigenilter order and also with the laser chirp parameter. • The combined effect of the laser chirp, iber dispersion, and ODEC group delay can re-open the receiver data eye further than that of the ideal eye-opening, resulting in the phenomenon of optical power enhancement, a feature that is not available with other compensation techniques.

5.4 5.4.1

PHOTONIC FUNCTIONAL DEVICES preamble

Photonic functional devices based on a silica planar lightwave circuit. First, lattice-form optical devices are described for chromatic dispersion slope compensation, and the dynamic equalization of chromatic and polarization-mode dispersion and gain non-uniformity, in high-speed wavelength division multiplexing transmissions.

58

59

60 61

62

S. C. Pei and J. J. Shyu, Eigen-approach for designing, FIR ilters and all-pass phase equalizers with prescribed magnitude and phase response, IEEE Trans. Circuits Syst.,39, 137–146, 1992. N. Q. Ngo, L. N. Binh, and X. Dai, Eigenilter approach for designing FIR all-pass optical dispersion compensators for high-speed long-haul systems, Proc. IREE, 19th Australian Conference on Optical Fibre Technology, Melbourne, pp. 355–358, 1994. G. P. Agrawal, Nonlinear iber optics, Boston, MA: Academic Press, 1989. N. Sugimoto, H. Terui, A. Tate, Y. Katoh, Y. Yamada, A. Sugita, A. Shibukawa, and Y. Inoue, A hybrid integrated waveguide isolator on a silica-based planar lightwave circuit, J. Light. Technol., 14, 2537–2546, 1996. D. Marcuse, Single-channel operation in very long nonlinear ibers with optical ampliiers at zero dispersion, J. Light. Technol., 9, 356–361, 1991.

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Advanced optical systems require highly functional optical devices in order to exceed the electrical speed limit. If we are to realize high-speed WDM links, we must develop compensators to deal with the undesirable characteristics of optical ibers, namely, chromatic and polarizationmode dispersion and gain non-uniformity. Lattice-form ilters, which comprise cascades of alternating symmetrical and asymmetrical Mach Zehnder interferometers (MZIs), are suitable for realizing these compensators, because they can achieve various characteristics adaptively and lexibly. Photonic Lightwave Circuits (PLC) based circuits are fabricated on a silica on silicon substrate by a combination of lame hydrolysis deposition and reactive ion etching. The typical waveguide bending radius is around 2 to 25 mm, and large-scale integrated circuit chips can be as large as several centimeters square. This fabrication process can provide a uniform refractive index and core geometry, and the propagation loss throughout a large wafer is low (about 0.01 dB/cm). Alternatively Si integrated photonics can also offer the designed lattice ilters, that these devices can be integrated with electronic circuits and active modulator components. Figure 5.57 shows the cross Section of PLC waveguide. Table 5.7 tabulates the properties of PLC.

5.4.2

Optical DispersiOn cOmpensatiOn mODule (ODcm)

Figure 5.58 shows the basic coniguration of a lattice-form optical ilter, which comprises a cascade of alternating n + 1 symmetrical and n asymmetrical MZIs (n: natural number). The symmetrical and asymmetrical MZIs function as tunable couplers and delay applying parts, respectively. Such ilters are attractive because they can realize several kinds of dynamic compensator for chromatic and polarization-mode dispersion and gain non-uniformity. Since they conine lights in the MZIs without radiation, their properties are superior to those of transversal form devices in that their losses and loss variations are intrinsically smaller. The transfer functions of the lattice ilters are given by a Fourier series with terms identical to the symmetrical MZIs,

FIGURE 5.57 (a) Structure of PLC waveguide and progress of PLC technology, SiON material low-loss is also the technology similar to silica on silicon; (b) Progresses of PLC.

TABLE 5.7 Properties of PLC Index difference (%) Core size (µm) Loss (dB/point) Coupling loss (dB/point)a Bending radius (mm)b a b b

Low-Δ

Medium-Δ

High-Δ

Super high-Δ

0.3 8 × 8 graph determinant ∆ = 1 − loop transmittance = 1 − t 1/ 2 1 − κ C z −1

(6.41)

NUM(z): gain transmittance (Field) paths Path 1: 1-3-10-12-5-7 no touching the loop thus ∆1 = ∆; path1 .field − transmittance = pt1 = 1 − κ1 1 − κ C 1 − κ 2

(6.42)

Path 2: 1-4-6-7: no touching the loop so ∆ 2 = ∆; path 2 .field − transmittance = pt2 = j κ1 t 1/ 2 z −1 j κ 2 = − κ1 κ 2 t 1/ 2 z −1

(6.43)

Path 3: 1-3-10-11-9-12-5-7: touching the loop so ∆ 3 = 1; path3.ield_transmittance = pt3 = 1 − κ1 j κ C t 1/ 2 z −1 j κ C 1 − κ 2 = − 1 − κ1 κ C κ C 1 − κ 2 .t 1/ 2 z −1

(6.44)

Thus, the transfer function can be written by combining Eqs. (6.38) through (6.41) giving E7out ∆1. pt1 + ∆ 2 . pt2 + ∆3 . pt3 = E1in ∆ =

=

=

=

(1 − t

1/ 2

1 − κC z −1

)(

) (

)(

)

1 − κ1 1 − κC 1 − κ2 + 1 − t 1/ 2 1 − κC z −1 − κ1 κ2 t 1/ 2 z −1 + − 1 − κ1 κC 1 − κ2 .t 1/ 2 z −1 1 − t1/ 2 1 − κC z −1

(1− t

1/ 2

1 − κC z −1

)(

(1 − κ1 ) (1 − κC ) (1 − κ2 ) ) + (1 − t1/2

)(

)

1 − κC z −1 − κ1κ2 t 1/ 2 z −1 −

(1 − κ1 ) (1 − κ2 ) κC .t1/2 z −1

1 − t 1/ 2 1 − κC z −1

(

(1 − κ1) (1 − κC ) (1 − κ2 ) − (1 − κC) (1 − κ1) (1 − κ2 ) t1/2 z−1) + ( −

)

κ1κ2 t 1/ 2 z−1 + − κ1κ2 t 1 − κC z−2 − (1 − κ1) (1 − κ2 ) κC .t 1/ 2 z−1

1 − t 1/ 2 1 − κC z −1

(1 − κ1 ) (1 − κC ) (1 − κ2 ) + 

κ1κ2 t 1/ 2 − (1 − κC )

(1 − κ1 ) (1 − κ2 ) t1/2 + (1 − κ1 ) (1 − κ2 ) κC .t1/2  z −1 − (

κ1κ2 t 1 − κC z −2

1 − t 1/ 2 1 − κC z −1

)

(6.45)

If κ1 = κ1 = 0.5 = 3dB_coupler then we can have E7out = E1in

(1 − κ1 ) (1 − κC ) (1 − κ2 ) + 

κ1κ2 t 1/ 2 − (1 − κC )

(1 − κ1 ) (1 − κ2 ) t1/2 + (1 − κ1 ) (1 − κ2 ) κC .t1/2  z −1 − ( 1 − t 1/ 2 1 − κC z −1

1/ 2 1/ 2 1/ 2 −1 −2 E7out 0.5 (1 − κC ) + (0.5t − (1 − κC ) 0.5t + 0.5κC .t ) z − 0.5 1 − κC z = 1/ 2 −1 E1in 1− t 1 − κC z

κ1κ2 t 1 − κC z −2

)= (6.46)

Optical Dispersion in Guided-Wave FIR and IIR Structures

281

Now, if we consider the low loss transmission t = 1.0– and a variable coupling coeficient, then we have the following: • Resonance frequency (pole) at 1 − t 1/ 2 1 − κ C z −1 = 0 → z p 2 = t 1/ 2 1 − κ C _ and _ z p1 = 0

(6.47)

• Zeroes are located at: 0.5 (1 − κ C ) + (0.5t 1/ 2 − (1 − κ C ) 0.5t 1/ 2 + 0.5κ C .t 1/ 2 ) z −1 − 0.5 1 − κ C z −2 = 0 → 0.5 (1 − κ C ) z +2 + (0.5t 1/ 2 − (1 − κ C ) 0.5t 1/ 2 + 0.5κ C .t 1/ 2 ) z +1 − 0.5 1 − κ C  / z 2 = 0   z p1=........... and z p 2=........... and_a doublee poles at origin

(6.48)

So, we have one z = 0 cancelled by one divided z of the denominator, and we have one additional pole at the origin. Thus, the optical circuit MZDI-MRR offers one real pole at … and one pole at the origin. And two real= zeroes z p1 and z p 2 ........... =...........

7

Photonic Ultra-Short Pulse Generators

Ultra-short pulse generators are lightwave sources that emit short pulses of high intensity. This means the energy has been concentrated into a very short time and periodically distributed along the pulses of the sequence. The peak power of these pulses reaches the nonlinear threshold of several materials that are used as the interaction and guided media for photonic signal processing. This chapter presents a number of important and emerging light sources of very narrow pulse width sequences considered for application in soliton communications and soliton logics. Photonic generators for bright and dark solitons as well as bound solitons are given. Mode-locked iber lasers have become well known over the last decades and are emerging as important sources for advanced lightwave communications technology. Practical implementations of these types of lasers will be subsequently discussed after the sections on solitons.

7.1

OPTICAL DARK-SOLITON GENERATOR AND DETECTORS

In this section, the trapezoidal optical integrator described in Chapter 3 is proposed as an optical dark-soliton generator, and the irst-order irst-derivative optical differentiator outlined in Chapter 6 and a irst-order Butterworth Lowpass optical ilter (LPOF) are proposed as optical dark-soliton detectors. A brief review of solitons in optical ibers is presented in Section 7.1. The nonlinear Schroedinger equation describing soliton propagation in a lossless optical iber and the parameters of the dispersion shifted iber and laser source used are given in Section 7.2. The design and performance of the optical dark-soliton detectors are irst investigated in Section 7.3 so that they can be characterized. The design (Section 7.4) and performance (Section 7.5) of the optical dark-soliton generator are then outlined. The performances of the combined dark-soliton generator and detectors are also described in Section 7.5. The theory of coherent integrated-optic signal processing described in Section 7.2.3 is employed in this chapter where electric-ield amplitude signals are considered.

7.1.1

intrODuctiOn

In the near future, optical solitons are believed to be promising candidates as information carriers in ultra-long-distance and/or ultra-high-speed repeater-less, optically-ampliied communication systems. The history of solitons in optical ibers began almost a quarter of a century ago when Hasegawa and Tappert1,2 proposed that optical solitons can propagate without distortion over an ininitely long distance in a lossless single-mode optical iber through the exact balancing of the inherent effects of group velocity dispersion (GVD) and self-phase modulation: bright solitons exist in the negative (or anomalous) GVD regime, while dark solitons appear in the positive (or normal) GVD regime. The pioneering work of Hasegawa and Tappert has led to the ield of optical soliton communications, which has been under intensive worldwide investigation in the last decade.

1

2

A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric ibers: I. Anomalous dispersion, Appl. Phys. Lett., 23, 142–144, 1973. A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric ibers: II. Normal dispersion, Appl. Phys. Lett., 23, 171–173, 1973.

283

284

Photonic Signal Processing

The practical importance of bright solitons in optical ibers was not realized until 1980 when Mollenauer et al.3 reported the irst experimental veriication of the existence of a bright soliton over a 700-m standard single-mode optical iber. Several years after the prediction made by Hasegawa4 that practical propagation of bright solitons over long distances could be made possible by periodically compensating for the iber loss through the use of Raman ampliier gain, Mollenauer and Smith5 exploited this idea and presented the irst bright-soliton transmission experiment ever carried out over 4000 km. One year later, Nakazawa et al.6 reported the irst soliton transmission experiment that used erbium-doped iber ampliier (EDFA) as an optical repeater for the 50-km iber link. Recent technological advances have generated many successful ultra-high-speed and/or ultra-long-reach transmission experiments in which EDFAs were used as optical repeaters; see, for example, the recent papers by Nakazawa7 and Aubin et al.8 Dark solitons have been predicted to offer better stability than bright solitons against iber loss,9 and improved interactions between neighboring solitons10 and ampliied noise-induced timing jitter.11,12 However, Dark-soliton propagation experiments are much more dificult to implement than bright-soliton propagation experiments because of the dificulty of generating and detecting dark solitons. In 1987, Emplit et al.13 experimentally studied the propagation of odd-symmetry optical dark pulses that were generated using amplitude and phase iltering techniques. However, dark pulse propagation was not convincingly observed in their experiment because the iber length was shorter than the soliton period and the low resolution in their pulseshaping and pulse-measurement apparatus. Nevertheless, it was the irst experimental propagation of dark pulses in optical iber. One year later, Krökel et al.14 experimentally demonstrated the evolution of an even-symmetry dark pulse into a pair of low contrast dark pulses. In the same year, Weiner et  al.15 successfully developed a technique for synthesizing femtosecond optical pulses with almost arbitrary shape and phase. They also presented a more convincing experimental observation of soliton-like propagation of odd-symmetry dark pulse superimposed upon a broader Gaussian background pulse.

3

4

5

6

7

8

9 10 11

12

13

14

15

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical ibers, Phys. Rev. Lett., 45, 1095–1098, 1980. A. Hasegawa, Numerical study of optical soliton transmission ampliied periodically by the stimulated Raman process, Appl. Opt., 23, 3302–3309, 1984. L. F. Mollenauer and K. Smith, Demonstration of soliton transmission over more than 4000 km in iber with loss periodically compensated by Raman gain, Opt. Lett., 13, 675–677, 1988. M. Nakazawa, Y. Kimura, and K. Suzuki, Soliton ampliication and transmission with Er3+-doped ibre repeater pumped by GainAsP laser diode, Electron. Lett., 25, 199–200, 1989. M. Nakazawa, Ultrahigh speed optical soliton communication and related technology, IOOC’95 Technical Digest Series, Hong Kong, 4, 96–98, 1995. G. Aubin, E. Jeanney, T. Montalant, J. Moulu, F. Pirio, J.-B. Thomine, and F. Devaux, Electro absorption modulator for a 20 Gbit/s soliton transmission experiment over 1 Million km with a 140 km ampliier span, IOOC’95 Technical Digest Series, Hong Kong, 5, 29–30, Post-deadline paper PD2–5, 1995. W. Zhao and E. Bourkoff, Propagation properties of dark solitons, Opt. Lett., 14, 703–705, 1989. W. Zhao and E. Bourkoff, Interactions between dark solitons, Opt. Lett., 14, 1371–1373, 1989. J. P. Hamaide, P. Emplit, and M. Haelterman, Dark-soliton jitter in ampliied optical transmission systems, Opt. Lett., 16, 1578–1580, 1991. Y. S. Kivshar, M. Haelterman, Ph. Emplit, and J. P. Hamaide, Gordon-Haus effect on dark solitons, Opt. Lett., 19, 19–21, 1994. P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Picosecond steps and dark pulses through nonlinear single mode ibers, Opt. Commun., 62, 374–379, 1987. D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, Dark-pulse propagation in optical ibers, Phys. Rev. Lett., 60, 29–32, 1988. A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Experimental observation of the fundamental dark soliton in optical, ibers, Phys. Rev. Lett., 61, 2445–2448, 1988.

Photonic Ultra-Short Pulse Generators

285

Unlike previous schemes for generating dark pulses with a inite background,16,17 techniques for generating dark pulses with a CW (continuous wave) background (or dark solitons) have also been proposed and experimentally demonstrated.18,19 Richardson et al.16 reported the irst experimental demonstration of the generation of a 100-GHz dark-soliton train by means of nonlinear conversion of a high-intensity beat signal in a positive GVD decreasing iber. They also conirmed the stability of the generated dark-soliton train by propagating it through a 2-km length of positive GVD shifted iber. Zhao and Bourkoff17 proposed the use of an electro-optic intensity modulator driven by square pulses to generate dark solitons. Recently, Nakazawa and Suzuki18 modiied this scheme by using a logic AND circuit and a T-lip-lop circuit to perform data conversion to obtain a pseudo-random dark-soliton train. They also developed the irst dark-soliton detection scheme, based on a one-bit-shifting technique with a Mach–Zehnder interferometer, which converts a darksoliton signal into a modiied return-to-zero (RZ) signal and into an inverted non-return-to-zero (NRZ) signal. Using the developed dark-soliton generation and detection schemes, Nakazawa and Suzuki18 conducted for the irst time a dark-soliton transmission experiment over 1200 km. In this chapter, the trapezoidal optical integrator is proposed as an optical dark-soliton generator, and the irst-order irst-derivative optical differentiator and the irst-order Butterworth LPOF are proposed as optical dark-soliton detectors. Most of the work presented here has been described by Ngo et al.20 The effect of iber loss on dark-soliton propagation is not considered here, so that the underlying principles of dark-soliton generation and detection can be demonstrated.

7.1.2

Optical Fiber prOpagatiOn mODel

To demonstrate the effectiveness of the proposed optical dark-soliton generator and detectors, the nonlinear Schroedinger equation describing pulse propagation in a lossless single-mode optical iber j

∂ A β2 ∂ 2 A 2 − +ξ A A = 0 2 ∂Z 2 ∂t

(7.1)

can be numerically solved using the split-step Fourier method.21 In Eq. (7.1), j = −1 , A( Z , t ) = P0 U ( Z , t ), P0 is the peak power of the incident pulse with normalized amplitude U ( Z , t ), Z the distance of propagation, t the retarded time,22 β 2 = −Dλ 2 (2π c) the GVD parameter, D the iber dispersion parameter, λ the operating wavelength, c the speed of light in vacuum, ξ = 2π n2 ( Aeff λ ) the iber nonlinear coeficient, n2 the nonlinear refractive index, and Aeff is the effective core area. It is useful to write the peak power and soliton period as P0 = N 2 β 2 (ξ T02 ) and Z0 = π T02 (2 β 2 ), respectively, where the integer N is the soliton order and T0 the initial pulse width. The dispersionshifted iber, with zero-dispersion wavelength at 1550 nm, and soliton source are assumed to have the following typical parameters: β 2 = +1.27 ps2 km (for D = −1.0 ps nm km), ξ = 3.2 W −1km −1 (for Aeff = 40 µm 2 and n2 = 3.2 × 10 −20 m 2 W ), P0 = 0.494 mW (for N = 1 and T0 = 28.4 ps) and Z0 = 995 km.

16

17

18

19

20 21 22

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, Experimental demonstration of 100 GHz dark soliton generation and propagation using a dispersion decreasing ibre, Electron. Lett., 30, 1326–1327, 1994. W. Zhao and E. Bourkoff, Generation of dark solitons under a cw background using waveguide electro-optic modulators, Opt. Lett., 15, 405–407, 1990. M. Nakazawa and K. Suzuki, Generation of a pseudorandom dark soliton data train and its coherent detection by onebit-shifting with a Mach–Zehnder interferometer, Electron. Lett., 31, 1084–1085, 1995. M. Nakazawa and K. Suzuki, 10 Gbit/s pseudorandom dark soliton data transmission over 1200 km, Electron. Lett., 31, 1076–1077, 1995. N. Q. Ngo, L. N. Binh, and X. Dai, Optical dark-soliton generators and detectors, Opt,. Commun., 132, 389–402, 1996. G. P. Agrawal, Nonlinear Fiber Optics, Boston, MA: Academic Press, 1989. The retarded time is deined as the normalized time measured in a frame of reference moving with the pulse at the group velocity, i.e., t = t − Z vg where t is the actual time and vg the group velocity.

286

7.1.3

Photonic Signal Processing

Design anD perFOrmance OF Optical Dark-sOlitOn DetectOrs

The optical differentiator and Butterworth LPOF are now designed as optical dark-soliton detectors. The fundamental dark-soliton normalized pulse at the iber input is given by10  tanh(t T0 + q0 ), U (0, t ) =  − tanh(t T0 − q0 ),

− ∞ < t T0 < 0 0 ≤ t T0 < ∞

(7.2)

where q0 is a constant, Tb = 2q0T0 the bit time and = Tw1 1= .76T0 50 ps is the full width half mark (FWHM) of the soliton pulse. It is considered that Tb = 200 ps (for q0 = 3.52) corresponds to a bit rate of 1 Tb = 5 Gbit s.

7.1.4

Design OF Optical Dark-sOlitOn DetectOrs

By deinition,23 the derivative of a fundamental dark-soliton pulse with amplitude tanh(t ) is given by a “bright-squared”24 soliton pulse with a temporal amplitude function sech 2 (t ) given by: d [ tanh(t ) ] = sech 2 (t ) dt

(7.3)

The design of an optical differentiator, which can perform the above derivative function, and a irstorder Butterworth LPOF are now described as optical dark-soliton detectors. Figure 7.1 shows the schematic diagram of the asymmetrical Mach–Zehnder interferometer (AMZI) used for dark-soliton detection. The AMZI, which can be implemented using planar lightwave circuit (PLC) technology, namely, silica-based waveguides embedded on a silicon substrate as described in Section 7.2.3, consists of two directional couplers, DC1 and DC2, with cross-coupled intensity coeficients k1 and k2 , which are interconnected by two unequal waveguide arms with a differential time delay of Td . Neglecting the propagation delay, insertion loss and waveguide birefringence,25 the transfer functions of the AMZI are given by26 H 31( z ) = H 41( z ) =

E1 Dark soliton signal

E3 12 = exp( − jωTu ) [(1 − k1 )(1 − k2 ) ] 1 − z31z −1 , E1

{

}

{

}

E4 12 = exp[− j (ωTu + π 2)][(1 − k1 )k2 ] 1 − z41z −1 , E1

DC1

k1

DC2

Td

k2

(7.4) (7.5)

E3 (RZ signal)

E4 (NRZ signal)

FIGURE 7.1 Schematic diagram of the asymmetrical Mach–Zehnder interferometer (AMZI) used for darksoliton detection. All lines are integrated optical waveguides using planar lightwave technology. 23 24

25

26

M. Abramowitz and I. A. Segun, Handbook of Mathematical Function, New York: Dover Publications, 1964. It is well known that the electric-ield amplitude of a bright-soliton pulse is given by a secant pulse sech(t ) . Here, the 2 electric-ield amplitude of a secant-squared pulse sech (t ) is referred to as a “bright-squared” soliton pulse. Thus, the electric-ield amplitude of the “bright-squared” soliton pulse is the square of that of the bright-soliton pulse. The waveguide birefringence of the AMZI can be eliminated by a iber polarisation controller or by inserting polyimide half waveplates into the gap of the waveguide paths.63 These transfer functions have been obtained by using the waveguide directional coupler deined in Eq. (2.2) and the signal-low graph technique described in Chapter 3.

287

Photonic Ultra-Short Pulse Generators

whose zeros in the z-plane are given by 12

  k1k2 z31 =   ,  (1 − k1 )(1 − k2 ) 

(7.6)

12

 k (1 − k2 )  z41 = −  1  ,  (1 − k1 )k2 

(7.7)

where E1 and (E3, E4 ) are the electric-ield amplitudes at the input and output ports, respectively, z = exp( jωTd ) is the z-transform parameter, ω the angular optical frequency, and Td = Tℓ − Tu (the sampling period of the AMZI) is the differential time delay between the lower arm (with delay Tℓ) and the upper arm (with delay Tu ). For k= k= 0.5 and hence z31 = 1, H 31( z ) corresponds to the 1 2 transfer function of the irst-order irst-derivative optical differentiator as described in Chapter 6. Note that H 31( z ) is also the transfer function of the irst-order Butterworth highpass optical ilter with a 3-dB cut-off frequency at ωTd = π 2. While, for k= k= 0.5 and hence z41 = −1, H 41( z ) 1 2 corresponds to the transfer function of the irst-order Butterworth lowpass optical ilter with a 3-dB cut-off frequency also at ωTd = π 2.

7.1.5

perFOrmance OF the Optical DiFFerentiatOr

The sampling period of the AMZI is chosen to be equal to the bit period, i.e. Td = Tb , for reasons to be given below. Figure 7.2 shows the 5 Gb/s bit fundamental dark-soliton signals at Z = 0 and Z = 100 Z0. Compared with the input dark-soliton signal at Z = 0, the initial separation of the darksoliton signal at Z = 100 Z0 is increased by 2.5% as a result of the repulsive force between neighboring dark solitons at a long distance.10 The propagated dark-soliton signal at Z = 100 Z0 is detected by the optical differentiator and the resulting RZ signals are shown in Figure 7.2 for 0.8 ≤ z31 ≤ 1.2. The solid curve corresponds to the performance of the ideal differentiator, which requires k= k= 0.5 and hence z31 = 1. Such 1 2 a requirement is often dificult to achieve in practice because of the dificulty of fabricating directional couplers with the very precise values of the coupling coeficients. From Figure 7.2, as z31 deviates from its nominal value, the power levels of the space signals are raised from the otherwise zero values to P ≤ 0.0125, and this results in decreasing the otherwise ininite mark to space

FIGURE 7.2 5 Gb/s bit fundamental dark-soliton signals at Z = 0 and Z = 100 Z 0 .

288

Photonic Signal Processing

ratio (MTSR) to MTSR ≥ 80. Surprisingly, the middle portion of the adjacent mark signals is not affected by this variation in z31. For MTSR ≥ 80, the allowable values of the coupling coeficients of the AMZI lie in the range of 0.45 ≤ k1, k2 ≤ 0.55 , which is obtained by imposing the condition 0.8 ≤ z31 ≤ 1.2 on Eq. (7.6). Such variation of the coupling coeficients can be easily tolerated using the PLC technology. Note that the detected RZ signals are of the square-type intensity pulse shape rather than the sech 4 (t ) shape expected from the derivative of a tanh(t) amplitude pulse shape. This is because the chosen sampling period is large compared with the dark-soliton pulse width (i.e., Td = 4Tw1). When = Td 0= .05Tw1 2.5 ps, the RZ signals have the expected sech 4 (t ) intensity pulse shape, showing the high processing accuracy of the differentiator. However, the drawbacks of using Td 200 mV. This is also critical for the RMLL set up as the RF signal detected and phase locked via the clock recovery circuitry must be spit to generate this triggering signal. 62

By using the relationship of the quantum noise spectral density of 2qRPav with Pav the average optical power, q the electronic charge and R the responsivity of the detector.

306

Photonic Signal Processing

Typical experimental procedures are: (i) After the connection of all optical components with the ring path broken, ideally at the output of the ibers coupler, a CW optical source can be used to inject optical waves at a speciic wavelength to monitor the optical loss of the ring; (ii) Close the optical ring and monitor the average optical power at the output of the 90:10 ibers coupler and hence estimate the optical power available at the PD is about −3 dBm after a 50:50 ibers coupler; (iii)  Determine whether an optical ampliier is required for detecting the optical pulse train or whether this optical power is suficient for O/E RF feedback condition as stated above; (iv) Set the biasing condition and hence the bias voltage of the optical modulator; (v) Tune the synthesizer or the electrical phase to synchronize the generation and locking of the optical pulse train. One could observe the following: (i) The optical pulse train generated at the output of the MLL or RMML. Experimental set up is shown in Figure 7.14; (ii) Synthesized modulating sinusoidal waveforms can be monitored as shown in Figures 7.16 through 7.18. Figure 7.16 illustrates the mode locking of an MLL operating at around 2 GHz repetition rate with the modulator driven from a pattern generator. Figures 7.17 and 7.18 show the sinusoidal waveforms generating when the MLRL is operating at the self-mode-locking state. (iii) The interference of other super-modes of the MLL without RF feedback for self-locking is indicated in Figure 7.16.; (iv) Observed optical spectrum (not available in electronic form); (v) Electrical spectrum of the generated pulse trains was observed showing a 70-dB super-mode suppression under the locked state of the regenerative MLRL; (vi) Figures 7.17 through 7.19 show that the regenerative MLRL can be operating under the cases when the modulator is biased either at the positive or at the negative going slope of the optical transfer characteristics of the Mach–Zehnder modulator. Optical pulse width is measured using an optical auto-correlator (slow or fast scan mode). Typical pulse width obtained with the slow scan auto-correlator is shown in Figure 7.21. Minimum pulse duration obtained was 5 ps with a time-bandwidth product of about 3.8 showing that the generated pulse is near transform limited. The bit error rate (BER) measurement can be used to monitor the stability of the regenerative MLRL. The BER error detector was then programmed to detect all “1” at the decision level at a tuned amplitude level and phase delay. The clock source used is produced by the laser itself. This set up is shown in Figure 7.15 with an error-free has been achieved for over 20 hours. The O/E detected waveform of the output pulse train for testing the BER is shown in Figure 7.20. Clock signals generated from RMLL Trig input

OPTICAL PULSE TRAINS GENERATED FROM R-MLL

HP-54118A 0.5 - 18 GHz

Output trig

HP-54123A (DC-34 GHz) input Channel inputs

Opt. Att. HP-34GHz If req. dep. pin DECTECTOR on detector or Fermionics HSD-30 HP-54118A sampling head and display unit

FIGURE 7.14 Experimental set up for monitoring the locking of the photonic pulse train.

307

Photonic Ultra-Short Pulse Generators

BER MEASUREMENT OF RMLL-EDF FIGURE 4

BER circuitry and equipment set-up

Optical signals from output port of RMLL

To CLK recovery and feedback to drive MZIM

3 dB FC

PIN bias (9V) HBT TI amp supply (11.5V)

DC blocking cap

Linear power amplifier to provide > 0.2 V signals for BER input port

RMLL Clock

delay

Set @ ALL ‘1’ 16-bit length periodic

Nortel PP-10G 500 Ω TI optical receiver

RF amp AL-7 MA Ltd

BER detector ANRITSU MP1764A

RF amp ERA 10GHz RZ sinusoidal waveform

Data IN CLK

Clk freq: 9.947303 GHz var. 100 MHz

FIGURE 7.15 Experimental set up for monitoring the BER of the photonic pulse train.

After 20 hours operation, the recorded waveform is obtained under ininite persistence mode of the digital oscilloscope. A drift of clock frequency of about 20 kHz over one hour in an open laboratory environment is observed. This is acceptable for a 10 GHz repetition rate. The “clock” recovered waveforms were also monitored at the initial locked state and after the long-term test, as shown in Figures 7.17 and 7.18, respectively. Figure 7.18 was obtained under the ininite persistence mode of the digital oscilloscope. We note the following factors which are related to the above measurements (Figures 7.16 through 7.18): • All the above measurements have been conducted with two distributed optical ampliiers (GTi EDF optical ampliiers) driven at 180 mA and with a speciied output optical power of 16.5 dBm. • Optical pulse trains are detected with 34 GHz 3 dB bandwidth HP pin detector directly coupled to the digital oscilloscope without using any optical pre-ampliier (Figures 7.19 through 7.21). 7.2.3.1 40 GHz Regenerative Mode-Locked Laser Following our initial success of the construction and testing of a regenerative MLRL at 40  GHz repetition rate, a regenerative mode-locked laser was constructed. The schematic arrangement of the 40 G regenerative MLRL is shown in Figure 7.22. Initial observation of the locking and generation of the laser has been observed and progress of this laser design and experiments will be reported in the near future.

308

Photonic Signal Processing Modulation frequency = 2,143,500 kHz

SCOPE 2

2 GHz ML_EDFA_L Competition from other supermodes Neg CLK PULSES

Generated -Detected Laser pulses

0.000501

Generated output pulses 0.000401 @ coupler port 4 0.000301

0.000201

Clock for modulating MZI Modulator in-loop 0.000101 9.99996E-07

00 00 8. 50

19

8.

00

00

00 30 19

8.

00

00 10 19

8.

00

00 90 18

18

70

8.

00 8. 50 18

00

00

00 00 8. 30

18

10

8.

00

00

00 00 18

90

8.

00 17

8. 70 17

17

50

8.

00

00

00

-9.9E-05

Time (picoseconds)

FIGURE 7.16 Detected pulse train at the MLRL output tested at a multiple frequency in the range of 2 GHz repetition frequency.

Regenerative mode-locked generated pulse trains: 10 G rep rates - detected by 34 GHz (3dB BW HP-detector into 50 Ω)

SCOPE 3

31.375 21.375 Generated pulse trains 100 ps spacing - pulse width ~ 12 ps - limited by PD bandwidth

11.375 1.375 -8.625 -18.625

Clock recovery signals for driving MZIM

-28.625 -38.625

00 10 69 9. 16

16

59 9.

00 10

00 10 16

49 9.

00 10 16

39 9.

00 10 29 9. 16

16

19 9.

00 10

-48.625

Time (picoseconds)

FIGURE 7.17 Output pulse trains of the regenerative MLRL and the RF signals as recovered for modulating the MZIM for self-locking.

309

Photonic Ultra-Short Pulse Generators RML laser with V(MZIM bias)=1.9V (more sensitive to competition of supermodes than biased @ 0 V - maximum transmission bias)

SCOPE 4

43 Generated pulse train 33 23 13 3 -7 Clocked regen for modulating MZIM

-17 -27

67

66

01 0.

01 0.

66

65

01 0.

65

01 0.

01 0.

64 01

64 0.

01

63 0.

01

63 0.

01

62 0.

01 0.

0.

01

62

-37

FIGURE 7.18 Detected output pulse trains of the regenerative MLRL and recovered clock signal when the MZIM is biased at a negative going slope of the operating characteristics of the modulator.

RMLLL-09-May-2000 V(bias)= 9.34 Volts 09-May-2000 Regen MLL 10G ps-pulses generated

RF pulses to drive MZIM

Generated optical output pulses

clk & RF signals to MZIM

43 33 23 13 3 -7 -17 -27

0 99

0 16 .6

49 16 .6

99 0 16 .5

49 0 16 .5

0 99 16 .4

49 0 16 .4

0

99 0 16 .3

.3 49 16

16

.2 99

0

0 .2 49 16

16

.1 99

0

-37

Time (nanoseconds)

FIGURE 7.19 Output pulse trains and clock recovered signals of the 10 G regenerative MLRL when the modulator is biased at the positive going slope of the modulator operating transfer curve.

7.2.3.2 Remarks We have successfully constructed a mode-locked laser operating under an open loop condition and with O/E RF feedback providing regenerative mode locking. The O/E feedback can certainly provide a self-locking mechanism under the condition when the polarization characteristics of the ring laser are manageable. This is done by ensuring that all ibers have a path under constant operating condition. The regenerative MLRL can self-lock even under the DC drifting effect of the modulator

310

Photonic Signal Processing BER measurement BER=0 x 10-15 measured for over 20 hours - RF (CLK) frequency varied from 9.954 to 9.952 GHz gradually over measurement period BER measurement - clock signal - infinite persistence non- pulses

clock RF signals while BER measurement

280 180 80 -20 -120 -220

0

0

99 .6

16

.6

16

99 .5

16

49

0

0

0

49 .5 16

0

99 .4

16

16

.4

49

0

0

99

16

.3

0

49 .3

16

16

.2

99

49 .2

.1 16

16

99

0

0

-320

Time (nanoseconds)

FIGURE 7.20 BER measurement – O/E detected signals from the generated output pulse trains for BER test set measurement. The waveform is obtained after 20 hours persistence. Auto-correlated pulse V(bias) = 1.55 volt - phase quadrature neg slop FWHM ∆τ = 4.48 ps autocorrelator set thumbwheel 6 and 0 (100ps range @10ps/s delay rate)

FIGURE 7.21 Auto-correlation trace of output pulse trains of 9.95 GHz regenerative MLRL.

bias voltage (over 20 hours).63 The generated pulse trains of 5 ps duration can be, without dificulty, compressed further to less than 3 ps for 160 Gb/s optical communication systems. The regenerative MLRL can be an important source for all-optical switching of an optical packet switching system. We recommend the following for successful construction and operation of the regenerative MLRL: (i) Eliminating polarization drift through the use of Faraday mirror or all polarization

63

Typically the DC bias voltage of a LiNbO3 intensity modulator is drifted by 1.5 volts after 15 hours of continuous operation.

311

Photonic Ultra-Short Pulse Generators FIGURE 3

40 GHz repetition rate - ps pulse width EFA-MLL SCC 40 GHz MZ modulator Basic fiber 2-STAGE EDFA ring

Fiber connection delay T

-10 or -25 dB RF dir.coupler

40 GHz BP Amp

DC bias Adjustable phase shifter

RF BPF Amp

40 GHz BPF EDFA Section 1

Loop MZI

Re-circulating port

Linear BP amplifier

3nm OF

PC EDFA Section 2

2

90% 3 10%

FIGURE 7.22

NTT 40 GHz det. Opt.att

SYNC HP-det

High Speed Digital Sampling

Set up of a 40 G regenerative MLRL.

maintaining (PM) optical components, for example polarized Er-doped iber ampliiers, PM ibers at the input and output ports of the intensity modulator; (ii) Stabilizing the ring cavity length with appropriate packaging and via piezo/thermal control to improve long term frequency drift; (iii) Controlling and automatic tuning of the DC bias voltage of the intensity modulator; (iv) Developing electronic RF ‘clock’ recovery circuit for regenerative MLRL operating at 40 GHz repetition rate together with appropriate polarization control strategy; (v) Studying the dependence of the optical power circulating in the ring laser by varying the output average optical power of the optical ampliiers under different pump power conditions; and (vi) Incorporating a phase modulator, in lieu of the intensity modulator, to reduce the complexity of polarization dependence of the optical waves propagating in the ring cavity, thus minimizing the bias drift problem of the intensity modulator.

7.2.4

active mODe-lOckeD Fiber ring laser by ratiOnal harmOnic Detuning

In this section, we investigate the system behavior of rational harmonic mode-locking in the iber ring laser using phase plane technique of the nonlinear control engineering. Furthermore, we examine the harmonic distortion contribution to this system performance. We also demonstrate 660x and 1230x repetition rate multiplications on 100 MHz pulse train generated from an active harmonically mode-locked iber ring laser, hence achieving 66 and 123 GHz pulse operations by using rational harmonic detuning, which is the highest rational harmonic order reported to date. 7.2.4.1 Rational Harmonic Mode-Locking In an active harmonically mode-locking iber ring laser, the repetition frequency of the generated pulses is determined by the modulation frequency of the modulator, fm = qfc, where q is the qth harmonic of the fundamental cavity frequency, fc, which is determined by the cavity length of the laser, fc = c/nL, where c is the speed of light, n is the refractive index of the iber, and L is the cavity length. Typically, fc is in the range of kHz or MHz. Hence, in order to generate GHz pulse train, mode-locking is normally performed by modulation in the states of q >> 1, i.e. q pulses circulating within the cavity, which is known as harmonic mode-locking. By applying a slight deviation or a fraction of the fundamental cavity frequency, ∆f = fc/m, where m is the integer, the modulation frequency becomes

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Photonic Signal Processing

f m = qf c ±

fc m

(7.25)

This leads to an m-times increase in the system repetition rate, fr = mf m, where fr is the repetition frequency of the system. When the modulation frequency is detuned by an m fraction, the contributions of the detuned neighboring modes are weakened, only every mth lasing mode oscillates in phase and the oscillation waveform maximums accumulate, hence achieving in m times higher repetition frequency. However, the small but not negligible detuned neighboring modes affect the resultant pulse train, which leads to uneven pulse amplitude distribution and poor long term stability. This is considered as harmonic distortion in our modeling, and it depends on the laser linewidth and amount of detuned, i.e. fraction m. The amount of the allowable detunes or rather the obtainable increase in the system repetition rate by this technique is very limited by the amount harmonic distortion. When the amount of frequency detuned is too small relative to the modulation frequency, i.e. large  m, contributions of the neighboring lasing modes become prominent, thus reducing the repetition rate multiplication capability signiicantly. Another words, no repetition frequency multiplication is achieved when the detuned frequency is unnoticeably small. Often the case, it is considered as the system noise due to improper modulation frequency tuning. In addition, the pulse amplitude luctuation is also determined by this harmonic distortion. 7.2.4.2 Experiment The experimental setup of the active harmonically mode-locked iber ring laser is shown in Figure 7.23. The principal element of the laser is an optical close loop with an optical gain medium, a Mach–Zehnder amplitude modulator (MZM), an optical polarization controller (PC), an optical bandpass ilter (BPF), optical couplers, and other associated optics. The gain medium used in our iber laser system is an erbium doped iber ampliier (EDFA) with a saturation power of 16 dBm. A polarized, independent optical isolator is used to ensure unidirectional lightwave propagation as well as to eliminate back relections from the iber splices and optical connectors. A free space ilter with a 3-dB bandwidth of 4 nm at 1555 nm is inserted into the cavity to select the operating wavelength of the generated signal and to reduce the noise in the system. In addition, it is responsible for the longitudinal modes selection in the mode-locking process. The birefringence of the iber is compensated by a polarization controller, which is also used for the polarization alignment of the linearly polarized lightwave before entering the planar structure modulator for better output eficiency. Pulse operation is achieved by introducing an asymmetric coplanar traveling wave 10  Gb/s Ti:LiNbO3  Mach–Zehnder amplitude modulator into the cavity with a half-wave voltage, Vπ of 5.8 V and an insertion loss of ≤7 dB. The modulator is DC biased

FIGURE 7.23 Schematic diagram of an active mode-locked iber ring laser.

Photonic Ultra-Short Pulse Generators

313

near the quadrature point and not more than the Vπ such that it operates around the linear region of its characteristic curve. The modulator is driven by a 100 MHz, 100 ps step recovery diode (SRD), which is in turn driven by a RF ampliier (RFA)—a RF signal generator. The modulating signal generated by the step recovery diode is a ~1% duty cycle Gaussian pulse train. The output coupling of the laser is optimized using a 10/90 coupler. 90% of the optical ield power is coupled back into the cavity ring loop, while the remaining portion is taken out as the output of the laser and analyzed. 7.2.4.3 Phase Plane Analysis Nonlinear system frequently has more than one equilibrium point. It can also oscillate at ixed amplitude and ixed period without external excitation. This oscillation is called the limit cycle. However, limit cycles in nonlinear systems are different from linear oscillations. First, the amplitude of self-sustained excitation is independent of the initial condition, while the oscillation of a marginally stable linear system has its amplitude determined by the initial conditions. Second, marginally stable linear systems are very sensitive to changes, while limit cycles are not easily affected by parameter changes.53,64 Phase plane analysis is a graphical method of studying second-order nonlinear systems. The result is a family of system motion trajectories on a two-dimensional plane, which allows us to visually observe the motion patterns of the system. Nonlinear systems can display more complicated patterns in the phase plane, such as multiple equilibrium points and limit cycles. In the phase plane, a limit cycle is deined as an isolated closed curve. The trajectory has to be both closed, indicating the periodic nature of the motion, and isolated, indicating the limiting nature of the cycle.53 The system modeling of the rational harmonic mode-locked iber ring laser system is done based on the following assumptions: (i) detuned frequency is perfectly adjusted according to the fraction number required, (ii) small harmonic distortion, (iii) no iber nonlinearity is included in the analysis, (iv) no other noise sources are involved in the system, and (v) Gaussian lasing mode amplitude distribution analysis. The phase plane of a perfect 10 GHz mode-locked pulse train without any frequency detune is shown Figure 7.24 and the corresponding pulse train is shown in Figure 7.25a. The shape of the

FIGURE 7.24 Phase plane of a 10 GHz mode-locked pulse train. (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)). 64

J. J. E. Slotine and W. Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice Hall, 1991.

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Photonic Signal Processing

FIGURE 7.25 Normalized pulse propagation of original pulse (a); detuning fraction of 4, with 0% (b) 5% (c) harmonic distortion noise.

phase plane exposes the phase between the displacement and its derivative. From the phase plane obtained, one can easily observe that the origin is a stable node and the limit cycle around that vicinity is a stable limit cycle, hence leading to stable system trajectory. 4x multiplication pulse trains, i.e. m = 4, without and with 5% harmonic distortion are shown in Figure 7.25b and c. Their corresponding phase planes are shown in Figure 7.26a and b. For the case of zero harmonic distortion,

FIGURE 7.26 Phase plane of detuned pulse train, m = 4, 0% harmonic distortion (a), and 5% harmonic distortion (b); (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).

315

Photonic Ultra-Short Pulse Generators 1.2

% Fluctuation

1 0.8 0.6 0.4 0.2 0 0

0.16

0.32

0.48

0.64

0.8

0.96

% Harmonic Distortion m=2

m=4

m=8

FIGURE 7.27 Relationship between the amplitude luctuation and the percentage harmonic distortion (diamond – m = 2, square – m = 4, triangle – m = 8).

which is the ideal case, the generated pulse train is perfectly multiplied with equal amplitude and the phase plane has stable symmetry periodic trajectories around the origin, as well. However, for the practical case, with 5% harmonic distortion, it is obvious that the pulse amplitude is unevenly distributed, which can be easily veriied with the experimental results obtained in.32 Its corresponding phase plane shows more complex asymmetry system trajectories (Figures 7.27 and 7.29). One may naively think that the detuning fraction, m, could be increased to a very large number, so a very small frequency deviated, ∆f, so as to obtain a very high repetition frequency. This is only true in the ideal world, if no harmonic distortion is present in the system. However, this is unreasonable for a practical mode-locked laser system. We deine the percentage luctuation, %F as follows: %F =

Emax − Emin ×100% Emax

(7.26)

where Emax and Emin are the maximum and minimum peak amplitude of the generated pulse train. For any practical mode-locked laser system, luctuations above 50% should be considered as poor laser system design. Therefore, this is one of the limiting factors in a rational harmonic mode-locking iber laser system. The relationships between the percentage luctuation and harmonic distortion for three multipliers (m = 2, 4 and 8) are shown in Figure 7.26. Thus, the obtainable rational harmonic mode-locking is very much limited by the harmonic distortion of the system. For 100% luctuation, it means no repetition rate multiplication, but with additional noise components; a typical pulse train and its corresponding phase plane are shown in Figure 7.28. (lower plot) and Figure 7.29 with m = 8 and 20% harmonic distortion. The asymmetric trajectories of the phase graph explain the amplitude unevenness of the pulse train. Furthermore, it shows a more complex pulse formation system. Thus, it is clear that for any harmonic mode-locked laser system, the small side pulses generated are

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FIGURE 7.28 10 GHz pulse train (upper plot), pulse train with m = 8 and 20% harmonic distortion (lower plot).

FIGURE 7.29 Phase plane of the pulse train with m = 8 and 20% harmonic distortion.

largely due to improper or not exact tuning of the modulation frequency of the system. An experiment result is depicted in Figure 7.30 for a comparison. 7.2.4.4 Results and Discussion By careful adjustment of the modulation frequency, polarization, gain level and other parameters of the iber ring laser, we manage to obtain the 660th and 1230th order of rational harmonic detuning in the mode-locked iber ring laser with base frequency of 100 MHz, hence achieving 66 and 123 GHz repetition frequency pulse operation. The auto-correlation traces and optical spectrums of the pulse operations are shown in Figure 7.31. With Gaussian pulse assumption, the obtained pulse widths of the operations are 2.5456 ps and 2.2853 ps, respectively. For the 100 MHz pulse operation, i.e. without any frequency detune, the generated pulse width is about 91 ps. Thus, not only do

317

Photonic Ultra-Short Pulse Generators

FIGURE 7.30 Autocorrelation trace (a) and optical spectrum (b) of slight frequency detune in the modelocked iber ring laser.

we achieve an increase in the pulse repetition frequency, but also a decrease in the generated pulse widths. This pulse narrowing effect is partly due to the self-phase modulation effect of the system, as observed in the optical spectrums. Another reason for this pulse shortening is stated by Haus in40,65 where the pulse width is inversely proportional to the modulation frequency, as follows:

τ4 =

2g M ωm2 ω g2

(7.27)

where τ is the pulse width of the mode-locked pulse, ωm is the modulation frequency, g is the gain coeficient, M is the modulation index, and ωg is the gain bandwidth of the system. In addition, the duty cycle of our Gaussian modulation signal is ~1%, which is much less than 50% and leads to a narrow pulse width, as well. Besides the uneven pulse amplitude distribution, a high level of pedestal noise is also observed in the obtained results. For 66 GHz pulse operation, a 4-nm bandwidth ilter is used in the setup, but it is removed for the 123 GHz operation. It is done to allow more modes to be locked during the operation, thus, to achieve better pulse quality. In contrast, this increases the level of dificulty signiicantly in the system tuning and adjustment. As a result, the operation is very much determined by the gain bandwidth of the EDFA used in the laser setup. The simulated phase planes for the above pulse operation are shown in Figure 7.32. They are simulated based on the 100 MHz base frequency, 10 round trips condition and 0.001% of harmonic distortion contribution. There is no stable limit cycle in the phase graphs obtained; hence, the system stability is hardly achievable, which is a known fact in the rational harmonic mode-locking. Asymmetric system trajectories are observed in the phase planes of the pulse operations. This relects the unevenness of the amplitude of the pulses generated. Furthermore, more complex pulse formation process is also revealed in the phase graphs obtained. 65

H. Zmuda, R. A. Soref, P. Payson, S. Johns, and E. N. Toughlian, Photonic beam former for phased array antennas using a iber grating prism, IEEE Photon. Technol. Lett., 9, 241–243, 1997.

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Photonic Signal Processing

FIGURE 7.31 Autocorrelation traces of 66 GHz (a) and 123 GHz (c) pulse operation; optical spectrums of 66 GHz (b) and 123 GHz (d).

(a)

(b)

FIGURE 7.32 Phase plane of the 66 GHz (a) and 123 GHz (b) pulse train with 0.001% harmonic distortion noise.

Photonic Ultra-Short Pulse Generators

319

By a very small amount of frequency deviation, or improper modulation frequency tuning in the general context, we obtain a pulse train with ~100 MHz with small side pulses in between, as shown in Figure 7.10. It is rather similar to the Figure 7.6 (lower plot) shown in the earlier section despite the level of pedestal noise in the actual case. This is mainly because we do not consider other sources of noise in our modeling, except the harmonic distortion. 7.2.4.5 Remarks We have demonstrated 660th and 1230th order of rational harmonic mode locking from a base modulation frequency of 100 MHz in the erbium doped iber ring laser, hence achieving 66 and 123 GHz pulse repetition frequency. To the best of our knowledge, this is the highest rational harmonic order obtained to date. Besides the repetition rate multiplication, we also obtain high pulse compression factor in the system, ~35x and 40x relative to the non-multiplied laser system. In addition, we use phase plane analysis to study the laser system behavior. From the analysis model, the amplitude stability of the detuned pulse train can only be achieved under negligible or no harmonic distortion condition, which is the ideal situation. The phase plane analysis also reveals the pulse forming complexity of the laser system.

7.3

REP-RATE MULTIPLICATION RING LASER USING TEMPORAL DIFFRACTION EFFECTS

The pulse repetition rate of a mode-locked ring laser is usually limited by the bandwidth of the intra-cavity modulator. Hence, a number of techniques have to be used to increase the repetition frequency of the generated pulse train. Rational harmonic detuning32,44 is achieved by applying a slight deviated frequency from the multiple of fundamental cavity frequency. A 40 Ghz repetition frequency has been obtained by32 using a 10 GHz base band modulation frequency with fourth order rational harmonic mode locking. This technique is simple in nature. However, this technique suffers from an inherent pulse amplitude instability, which includes both amplitude noise and inequality in pulse amplitude, and, furthermore, it gives poor long-term stability. Hence, pulse amplitude equalization techniques are often applied to achieve better system performance.31,33,34 Fractional temporal Talbot-based repetition rate multiplication technique uses the interference effect between the dispersed pulses to achieve the repetition rate multiplication. The essential element of this technique is the dispersive medium, such as linearly chirped iber grating (LCFG)56,38 and single mode iber.30,31 This technique will be discussed further in Section 7.3.1. Intra-cavity optical iltering42 uses modulators and a high inesse Fabry-Perot ilter (FFP) within the laser cavity to achieve higher repetition rate by iltering out certain lasing modes in the mode-locked laser. Other techniques used in repetition rate multiplication include higher order FM mode-locking,42 optical time domain multiplexing, and others. The stability of high repetition rate pulse train generated is one of the main concerns for practical multi-Giga bits/sec optical communications system. Qualitatively, a laser pulse source is considered stable if it is operating at a state where any perturbations or deviations from this operating point is not increased but suppressed. Conventionally the stability analyses of such laser systems are based on the linear behavior of the laser in which we can analyze the system behavior in both time and frequency domains. However, when the mode-locked iber laser is operating under nonlinear regime, none of these standard approaches can be used, since direct solution of nonlinear different equation is generally impossible, hence frequency domain transformation is not applicable. Although Talbotbased repetition rate multiplication systems are based on the linear behavior of the laser, there are still some inherent nonlinearities affecting its stability, such as the saturation of the embedded gain medium, non-quadrature biasing of the modulator, nonlinearities in the iber, and so on. Hence, a nonlinear stability approach must be adopted. We investigate the stability and transient analyses of GVD multiplied pulse train using the phase plane analysis of nonlinear control analytical technique.31 This was the irst time that the phase

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Photonic Signal Processing

plane analysis of modern control engineering is being used to study the stability and transient performances of the GVD repetition rate multiplication systems. The stability and the transient response of the multiplied pulses are studied using the phase plane technique of nonlinear control engineering. We also demonstrated four times the repetition rate multiplication on 10 Gb/s pulse train generated from the active harmonically mode-locked iber ring laser, hence achieving 40 Gb/s pulse train by using iber GVD effect. It has been found that the stability of the GVD multiplied pulse train, based on the phase plane analysis is hardly achievable even under the perfect multiplication conditions. Furthermore, uneven pulse amplitude distribution is observed in the multiplied pulse train. In addition to that, the inluences of the ilter bandwidth in the laser cavity, nonlinear effect and the noise performance are also studied in our analyses. In Section 7.3.1, the GVD repetition rate multiplication technique is briely given. Section 7.3.2 describes the experimental setup for the repetition rate multiplication. Section 7.3.4 investigates the dynamic behavior of the phase plane of GVD multiplication system, followed by simulation and experimental results. Finally, some concluding remarks and possible future developments are given.

7.3.1

gvD repetitiOn rate multiplicatiOn technique

When a pulse train is transmitted through an optical iber, the phase shift of kth individual lasing mode due to group velocity dispersion (GVD) is

ϕk =

πλ 2 Dzk 2 f r2 c

(7.28)

where: λ is the center wavelength of the mode-locked pulses D is the iber’s GVD factor z is the iber length fr is the repetition frequency c is the speed of light in vacuum This phase shift induces pulse broadening and distortion. At Talbot distance, zT = 2/∆λ fr /D/ 36 the initial pulse shape is restored, where ∆λ = fr λ2 /c is the spacing between Fourier-transformed spectrum of the pulse train. When the iber length is equal to zT /(2m), (where m = 2,3,4, …), every mth lasing modes oscillates in phase and the oscillation waveform maximums accumulate. However, when the phases of other modes become mismatched, this weakens their contributions to pulse waveform formation. This leads to the generation of a pulse train with a multiplied repetition frequency with m-times. The pulse duration does not change that much, even after the multiplication, because every mth lasing mode dominates in pulse waveform formation of m-times multiplied pulses. The pulse waveform therefore becomes identical to that generated from the mode-locked laser, with the same spectral property. The optical spectrum does not change after the multiplication process, because this technique utilizes only the change of phase relationship between lasing modes and does not use the iber’s nonlinearity. The effect of higher order dispersion might degrade the quality of the multiplied pulses, including pulse broadening, appearance of pulse wings, and pulse-to-pulse intensity luctuation. In this case, any dispersive media to compensate the iber’s higher order dispersion would be required in order to complete the multiplication process. To achieve higher multiplications, the input pulses must have a broad spectrum and the fractional Talbot length must be very precise in order to receive high quality pulses. If the average power of the pulse train induces the nonlinear suppression and experience anomalous dispersion along the iber, solitonic action would occur and prevent the linear Talbot effect from occurring.

Photonic Ultra-Short Pulse Generators

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The highest repetition rate obtainable is limited by the duration of the individual pulses, as pulses start to overlap when the pulse duration becomes comparable to the pulse train period, i.e. mmax = ∆T/∆t, where ∆T is the pulse train period and ∆t is the pulse duration.

7.3.2

experiment setup

GVD repetition rate multiplication is used to achieve 40 Gb/s operation. The input to the GVD multiplier is a 10.217993 Gb/s laser pulse source, obtained from active harmonically mode-locked iber ring laser, operating at 1550.2 nm. The principle element of the active harmonically mode-locked iber ring laser is an optical closed loop with an optical gain medium (i.e., Erbium doped iber with 980 nm pump source), an optical 10 GHz amplitude modulator, optical bandpass ilter, optical iber couplers, and other associated optics. The schematic construction of the active mode-locked iber ring laser is shown in Figure 7.33. The active mode-locked iber laser design is based on a iber ring cavity where the 25 meter EDF with Er3+ ion concentration of 7 × 1024 ions/m3 is pumped by two diode lasers at 980 nm: SDLO-278000-300 and CosetK1116 with maximum forward pump power of 280 mW and backward pump power of 120 mW. The pump lights are coupled into the cavity by the 980/1550 nm WDM couplers; with insertion loss for 980 and 1550 nm signals are about 0.48 and 0.35 dB, respectively. A polarized, independent optical isolator ensures the unidirectional lasing. The birefringence of the iber is compensated by a polarization controller (PC). A tunable FP ilter with a 3-dB bandwidth of 1 nm and a wavelength tuning range from 1530 to 1560 nm is inserted into the cavity to select the center wavelength of the generated signal, as well as to reduce the noise in the system. In addition, it is used for the longitudinal modes selection in the mode-locking process. Pulse operation is achieved by introducing a JDS Uniphase 10 Gb/s (8 GHz 3 dB bandwidth) Ti:LiNbO3 Mach–Zehnder amplitude modulator into the cavity with half wave voltage, Vπ of 5.8 V. The modulator is DC biased near the quadrature point and not more than the Vπ such that it operates on the linear region of its characteristic curve and driven by the sinusoidal signal derived from an Anritsu 68347C Synthesizer Signal Generator. The modulating depth should be less than the unity to avoid signal distortion.

FIGURE 7.33 Schematic diagram for active mode-locked iber ring laser.

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Photonic Signal Processing

FIGURE 7.34 Experiment setup for GVD repetition rate multiplication system.

The modulator has an insertion loss of ≤7dB. The output coupling of the laser is optimized using a 10/90 coupler. Additionally, 90% of the optical ield power is coupled back into the cavity ring loop, while the remaining portion is taken out as the output of the laser and is analyzed using a New Focus 1014B 40 GHz photo-detector, Ando AQ6317B Optical Spectrum Analyzer, Textronix CSA 8000 80E01 50 GHz Communications Signal Analyzer or Agilent E4407B RF Spectrum Analyzer. One rim of about 3.042 km of dispersion compensating iber (DCF), with a dispersion value of −98 ps/nm/km, was used in the experiment; the schematic of the experimental setup is shown in Figure 7.34. The variable optical attenuator used in the setup is to reduce the optical power of the pulse train generated by the mode-locked iber ring laser, hence to remove the nonlinear effect of the pulse. A DCF length for 4x multiplication factor on the ~10 GHz signal is required and estimated to be 3.048173 km. The output of the multiplier (i.e., at the end of DCF) is then observed using Textronix CSA 8000 80E01 50 GHz Communications Signal Analyzer.

7.3.3

phase plane analysis

A nonlinear system frequently has more than one equilibrium point. It can also oscillate at a ixed amplitude and ixed period without external excitation. This oscillation is called the limit cycle. However, limit cycles in nonlinear systems are different from linear oscillations. First, the amplitude of selfsustained excitation is independent of the initial condition, while the oscillation of a marginally stable linear system has its amplitude determined by the initial conditions. Second, marginally stable linear systems are very sensitive to changes, while limit cycles are not easily affected by parameter changes. Phase plane analysis is a graphical method of studying second-order nonlinear systems. The result is a family of system motion trajectories on a two-dimensional plane, which allows us to visually observe the motion patterns of the system. Nonlinear systems can display more complicated patterns in the phase plane, such as multiple equilibrium points and limit cycles. In the phase plane, a limit cycle is deined as an isolated closed curve. The trajectory has to be both closed, indicating the periodic nature of the motion, and isolated, indicating the limiting nature of the cycle. The system modeling for the GVD multiplier is done based on the following assumptions: (i) perfect output pulse from the mode-locked iber ring laser without any timing jitter, (ii) the multiplication is achieved under ideal conditions (i.e., exact iber length for a certain dispersion value), (iii) no iber nonlinearity is included in the analysis of the multiplied pulse, (iv) no other noise sources are involved in the system, and (v) uniform or Gaussian lasing mode amplitude distribution. 7.3.3.1 Uniform Lasing Mode Amplitude Distribution Uniform lasing mode amplitude distribution is assumed at the irst instance, i.e. ideal mode-locking condition. The simulation is done based on the 10 Gb/s pulse train, centered at 1550 nm, with iber dispersion value of −98 ps/km/nm, and a 1-nm lat-top passband ilter is used in the cavity of modelocked iber laser. The estimated Talbot distance is 25.484 km. The original pulse (direct from the mode-locked laser) propagation behavior and its phase plane are shown in Figures 7.35a and 7.36a. From the phase plane obtained, one can observe that the

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323

FIGURE 7.35 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 1 nm ilter bandwidth and equal lasing mode amplitude analysis.

FIGURE 7.36 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 1 nm ilter bandwidth and equal lasing mode amplitude analysis; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).

origin is a stable node and the limit cycle around that vicinity is a stable limit cycle. This agrees very well to our irst assumption: ideal pulse train is at the input of the multiplier. Also, we present the pulse propagation behavior and phase plane for 2×, 4× and 8× GVD multiplication system in Figures 7.33 and 7.34. The shape of the phase graph exposes the phase between the displacement and its derivative (Figures 7.35 through 7.44).

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Photonic Signal Processing

As the multiplication factor increases, the system trajectories are moving away from the origin. As for the 4× and 8× multiplications, there is neither a stable limit cycle nor stable node on the phase planes even with the ideal multiplication parameters. Here, we see the system trajectories spiral out

FIGURE 7.37 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 1 nm ilter bandwidth and Gaussian lasing mode amplitude analysis.

FIGURE 7.38 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 1 nm ilter bandwidth and Gaussian lasing mode amplitude analysis; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).

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FIGURE 7.39 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3 nm ilter bandwidth and Gaussian lasing mode amplitude analysis.

FIGURE 7.40 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3 nm ilter bandwidth and Gaussian lasing mode amplitude analysis; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).

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FIGURE 7.41 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3 nm ilter bandwidth, Gaussian lasing mode amplitude analysis and input power = 1 W.

FIGURE 7.42 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3 nm ilter bandwidth, Gaussian lasing mode amplitude analysis and input power = 1 W; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).

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327

FIGURE 7.43 Pulse propagation of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3 nm ilter bandwidth, Gaussian lasing mode amplitude analysis and 0 dB signal to noise ratio.

FIGURE 7.44 Phase plane of (a) original pulse, (b) 2x multiplication, (c) 4x multiplication, and (d) 8x multiplication with 3 nm ilter bandwidth, Gaussian lasing mode amplitude analysis and 0 dB signal to noise ratio; (solid line – real part of the energy, dotted line – imaginary part of the energy, x-axes – E(t) and y-axes – E′(t)).

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to an outer radius and back to an inner radius again. The change in the radius of the spiral is the transient response of the system. Hence, with the increase in the multiplication factor, the system trajectories become more sophisticated. Although GVD repetition rate multiplication uses only the phase change effect in multiplication process, the inherent nonlinearities still affect its stability indirectly. Despite the reduction in the pulse amplitude, we observe uneven pulse amplitude distribution in the multiplied pulse train. The percentage of unevenness increases with the multiplication factor in the system. 7.3.3.2 Gaussian Lasing Mode Amplitude Distribution In this set of the simulation models, the practical ilter used in the system. It gives us a better insight on the GVD repetition rate multiplication system behavior. The parameters used in the simulation are exactly the same except the ilter of the laser has been changed to 1 nm (125 GHz @ 1550 nm) Gaussian-proile passband ilter. The spirals of the system trajectories and uneven pulse amplitude distribution are more severe than those in the uniform lasing mode amplitude analysis. 7.3.3.3 Effects of Filter Bandwidth Filter bandwidth used in the mode-locked iber ring laser will also affect the system stability of the GVD repetition rate multiplication system. The analysis done above is based on 1 nm ilter bandwidth. The number of modes locked in the laser system increases with the bandwidth of the ilter used, which gives us a better quality of the mode-locked pulse train. The simulation results shown below are based on the Gaussian lasing mode amplitude distribution, a 3-nm ilter bandwidth used in the laser cavity, and other parameters remain unchanged. With wider ilter bandwidth, the pulse width and the percentage pulse amplitude luctuation decreases. This suggests a better stability condition. Instead of spiraling away from the origin, the system trajectories move inward to the stable node. However, this leads to a more complex pulse formation system. 7.3.3.4 Nonlinear Effects When the input power of the pulse train enters the nonlinear region, the GVD multiplier loses its multiplication capability as predicted. The additional nonlinear phase shift due to the high input power is added to the total pulse phase shift and destroys the phase change condition of the lasing modes required by the multiplication condition. Furthermore, this additional nonlinear phase shift also changes the pulse shape and the phase plane of the multiplied pulses. 7.3.3.5 Noise Effects The above simulations are all based on the noiseless situation. However, in the practical optical communication systems, noises are always sources of nuisance which can cause system instability, therefore it must be taken into the consideration for the system stability studies. Since the optical intensity of the m-times multiplied pulse is m-times less than the original pulse, it is more vulnerable to noise. The signal is dificult to differentiate from the noise within the system if the power of multiplied pulse is too small. The phase plane the multiplied pulse is distorted due to the presence of the noise, which leads to poor stability performance.

7.3.4

DemOnstratiOn

The obtained 10  GHz output pulse train from the mode-locked iber ring laser is shown in Figure 7.45. Its spectrum is shown in Figure 7.46. This output was then used as the input to the dispersion compensating iber, which acts as the GVD multiplier in our experiment. The obtained times multiplication by the GVD effect and its spectrum are shown in Figures 7.47 and 7.48.

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FIGURE 7.45 10 GHz pulse train from mode-locked iber ring laser (100 ps/div, 50 mV/div).

FIGURE 7.46 10 GHz pulse spectrum from mode-locked iber ring laser.

The spectrums for both cases (original and multiplied pulse) are exactly the same since this repetition rate multiplication technique utilizes only the change of phase relationship between lasing modes and does not use iber’s nonlinearity. The multiplied pulse suffers an amplitude reduction in the output pulse train; however, the pulse characteristics should remain the same. The instability of the multiplied pulse train is mainly due to the slight deviation from the required DCF length (0.2% deviation). Another reason for the pulse instability, which derived from our analysis; is the divergence of the pulse energy variation in the vicinity around the origin, as the multiplication factor gets higher. The pulse amplitude decreases with the increase in multiplication factor, as the fact of energy conservation, when it reaches certain energy level, is indistinguishable from the noise level in the system, and the whole system will become unstable and noisy.

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FIGURE 7.47 40 GHz multiplied pulse train (20 ps/div, 1 mV/div).

FIGURE 7.48

7.3.5

40 GHz pulse spectrum from GVD multiplier.

remarks

We have demonstrated 4× repetition rate multiplication by using iber GVD effect; hence, 40 GHz pulse train is obtained from 10  GHz mode-locked iber laser source. However, its stability is of great concern for practical use in the optical communications systems. Although the GVD repetition rate multiplication technique is linear in nature, the inherent nonlinear effects in such system may disturb the stability of the system. Hence, any linear approach may not be suitable in deriving the system stability. Stability analysis for this multiplied pulse train has been studied by using the nonlinear control stability theory, which is the irst time, to the best of our knowledge, that phase plane analysis is being used to study the transient and stability performance of the GVD repetition rate multiplication system. Surprisingly, from the analysis model, the stability of the multiplied pulse train can hardly be achieved even under perfect multiplication conditions. Furthermore, we observed uneven pulse amplitude distribution in the GVD multiplied pulse train, which is due to the

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energy variations between the pulses that cause some energy beating between them. Another possibility is the divergence of the pulse energy variation in the vicinity around the equilibrium point that leads to instability. The pulse amplitude luctuation increases with the multiplication factor. Also, with wider ilter bandwidth used in the laser cavity, better stability condition can be achieved. The nonlinear phase shift and noises in the system challenge the system stability of the multiplied pulses. They not only change the pulse shape of the multiplied pulses, they also distort the phase plane of the system. Hence, the system stability is greatly affected by the self-phase modulation (SPM) as well as the system noises. This stability analysis model can further be extended to include some system nonlinearities, such as the gain saturation effect, non-quadrature biasing of the modulator, iber nonlinearities, and so on. The chaotic behavior of the system may also be studied by applying different initial phase and injected energy conditions to the model.

7.4

MULTI-WAVELENGTH FIBER RING LASERS

This section presents the theoretical development and demonstration of a multi-wavelength erbiumdoped iber ring laser with an all-polarization-maintaining iber (PMF) Sagnac loop. The Sagnac loop simply consists of a PMF coupler and a segment of stress-induced PMF, with a single-polarization coupling point in the loop. The Sagnac loop is shown to be a stable comb ilter with equal frequency period which determines the possible output power spectrum of the iber ring laser. The number of output lasing wavelengths is obtained by adjusting the polarization state of the light in the unidirectional ring cavity by means of a polarization controller. This section is organized as follows: Section 7.4.1 presents the theory of the Sagnac PMF loop ilter, which consists of a PMF coupler (instead of a standard single-mode iber coupler used in previous works as described above) and a segment of PMF. Section 7.4.2 presents the experimental results and discussion. Concluding remarks are given.

7.4.1

theOry

In this section, we present a theoretical analysis of the all-PMF Sagnac loop, which is the key component in the iber ring laser. We consider the simplest case (see Figure 7.49a), in which only one polarization mode-coupling point exists. That is, the loop ilter is constructed by splicing the two pigtails (with lengths l1 and l2) of the PMF coupler with a phase difference θ along a certain principal axis at the spliced point. The input light is equally split into two waves by the 3-dB PMF coupler, and the two counter-propagating waves are recombined at the coupler output port after traveling through the loop. The electric components, Eix (ω ) and Eiy (ω ) , of the input light, Ein (ω ) , can be deined as ɶ  Eix (ω )  (7.29) Ein (ω ) =   ɶ  Eiy (ω )  where ω is the angular optical frequency. The PMF with length l can be considered as an ideal waveguide with linear birefringence, which is described by the Jones propagation matrix as66  exp ( j ∆β (ω )l/2 ) J PMF (ω , l ) =  0 

 0  exp ( − j ∆β (ω )l / 2 ) 

(7.30)

where j = −1 and ∆β (ω ) = β x (ω ) − β y (ω ) is the difference between the two propagation constants of a high-birefringence iber, which supports two linearly orthogonal fundamental modes 66

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, Dark-pulse propagation in optical ibers, Phys. Rev. Lett., 60, 29–32, 1988.

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y1

x1

l1

Fre

y2

X2

Fnon-re

y1 y2 l2 x2

X1 x1

θ

(a)

(b)

FIGURE 7.49 (a) Schematic diagram of the proposed all-PMF Sagnac loop ilter with a single coupling point. (b) Representation of the coordinates of the single coupling point.

(i.e., HEx11 and HEy11). It should be noted that a common average phase shift of exp( j β (ω )l ) is omitted in (7.10), because the Sagnac interferometer cannot distinguish the common phase term for the clockwise wave (CW) and counterclockwise wave (CCW). The transfer matrix of the coordinate (i.e., Θ(θ ) ) of the polarization mode-coupling point at the principal axes with a phase difference of θ (see Figure 7.49b) is given as  cos θ Θ (θ ) =   sin θ

sin θ   − cosθ 

(7.31)

The 3-dB PMF coupler is assumed to be ideal so that polarization coupling, polarization-dependent loss, and frequency dependence of the coupler are negligible (i.e., the coupling ratio is 50% in the operating wavelength range). The CW and CCW will experience the same phase shift and a 3-dB loss through the coupler, thus there is no phase difference at the reciprocal port. Hence, the CW’s Jones matrix for the Sagnac loop is given by Gcw (ω ) =

1 J PMF (ω , l1 )Θ(θ ) J PMF (ω , l2 ) 2

(7.32)

where J PMF (ω , l1 ) and J PMF (ω , l2 ) are deined in (7.30). The CCW’s Jones matrix is simply the transpose of the CW’s Jones matrix and is given by T Gccw (ω ) = Gcw (ω )

(7.33)

The electric components of the light at the output ports are given by  Eox (ω )    Eoy (ω ) 

Eout (ω ) = 

(7.34)

The relationship between the electric components at the input and output ports is given by Eout (ω ) = [Gccw (ω ) + Gcw (ω ) ] Ein (ω )

(7.35)

Using Eqs. (7.29) through (7.32), the intensity transfer function, Fre, for the reciprocal port can be derived as Fre = 1 − sin 2 θ ⋅ sin 2 ( ∆β ⋅ ∆l/2 )

(7.36)

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where ∆l = l2 − l1 is the difference between the length of the two PMF segments in the loop. It is noted that (7.16) is independent of the polarization state of the input light due to the fact that the interference terms of the x-component and y-component of the light cancel out with each other at the output port. From (7.36), it can be shown that when θ ≠ 0 or θ ≠ π the spectral peaks of the relection spectrum will have maximum intensity at frequencies according to ∆β (ωm ) ⋅ ∆l = 2π m ( m = 1, 2, ⋅⋅⋅)

(7.37)

Note that light with frequency ωm will disappear at the non-reciprocal port for the case of θ = π 2 because the transfer function of the non-reciprocal port is Fnon−re = 1 − Fre. There are two kinds of highbirefringence ibers, namely, stress-induced birefringent iber and geometry-induced birefringent iber, where the former one has greater birefringence. Here, we only consider the stress-induced birefringent iber, where the effective index of each polarization is inluenced by stress alone. Thus, the modal birefringence B is independent of wavelength over a particular wavelength range, and is given by ∆β (ωm ) =

2π B ( m = 0, 1, 2…….) λ

(7.38)

Using (7.17) and (7.18), the spectral peaks of the relection spectrum will have maximum intensity at frequencies f m given by fm =

mc ( m = 0,1, 2…….) B∆l

(7.39)

where c is the speed of the light in vacuum. From (7.39), the Sagnac loop is a comb ilter whose spectral peaks are separated by frequency spacing given by f m+1 − f m =

c B∆l

(7.40)

It is noted that although the frequency-dependent intensity transfer function of the loop ilter is independent of the state of polarization of the input light, the polarization state of the output light generally depends on the polarization state and frequency of the input light.

7.4.2

experimental results anD DiscussiOn

This section presents the experimental veriication of the theoretical analysis of the all-PM Sagnac loop ilter described in Section 7.4.1 and the experimental results of the iber ring laser. Figure 7.52 shows a typical relective spectrum of the all-PMF Sagnac loop ilter. The loop ilter was constructed by splicing the two pigtails (with lengths l1 and l2) of the PMF coupler in 0° and 90° with respect to their principal axes to form a single coupling point in the loop (i.e., a phase difference of θ = 90°). The loop ilter is highly stable, as expected, because all the components used are allPM components. From Figure 7.50, the frequency period of the ilter is 0.35 nm, which agrees well with the theoretical value as predicted by 20 when the following parameter values, B = 5.2e-4 and ∆l = 13.2 m (in the 1550-nm window) are substituted into the equation. Figure 7.51 shows the schematic diagram of the proposed unidirectional iber ring laser with the all-PMF Sagnac loop ilter. It consists of a 15-m long of Er3+ silica iber doped with ~200 ppm of erbium. The erbium-doped iber has a numerical aperture (NA) of 0.21, a cut-off wavelength of 920 nm, and an absorption coeficient of 12 dB/m at 980 nm. To increase the optical pump eficiency, the erbium-doped iber is pumped by a 980-nm laser diode (LD), which generates 70 mW power in both directions in the ring cavity through the two 980/1550-nm WDM couplers. A polarized, independent iber isolator is used to provide unidirectional ring oscillation so as to avoid spatial hole burning in the gain medium. The coupler-2 is used as the output coupler for the iber laser

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FIGURE 7.50 A typical relective spectrum of the all-PMF Sagnac loop ilter with a single coupling point.

PC WDM2

980nm coupler Er3+ doped fiber

980nm LD

Isolator

WDM1

Output Coupler-2 PMF coupler

coupling point

FIGURE 7.51 Schematic of the proposed unidirectional iber ring laser using the all-PMF Sagnac loop as a stable periodic ilter.

and also to direct the light wave to the Sagnac loop ilter. The periodic spectral peaks of the Sagnac ilter will determine the lasing wavelengths. A polarization controller (PC) is used in the cavity to adjust the polarization state to obtain several lasing wavelengths at the output port. Figure 7.52a and b show the experimental results of the output lasing wavelengths of the iber ring laser under different polarization conditions by adjusting the PC in the cavity. It can be seen that the wavelength spacing is 1.0 nm, which is deined by the 1.0 nm frequency period of the Sagnac loop ilter with parameter values of B = 5.2e^-4 and ∆l = 4.5 m. Figure 7.53 shows the output spectra of the lasing wavelengths of the iber ring laser under a particular polarization condition in the cavity, where the wavelength spacing is 0.50 nm, which is deined by the 0.50 nm frequency period of the Sagnac loop ilter with parameter values of B = 5.2e-4 and ∆l = 9.0 m. It should be noted that the number of output lasing wavelengths of the proposed iber ring laser could be greatly increased by overcoming the large homogeneous broadening of the gain medium of the erbium-doped iber at

335

0.1

0.1

0.01

0.01

Output (dBm)

Output (dBm)

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1E-3

1E-4

1E-4

1556 (a)

1E-3

1558 1560 1562 Wavlength (nm)

1564

1556 (b)

1558 1560 1562 Wavlength (nm)

1564

FIGURE 7.52 (a), (b) Typical output lasing wavelengths of the iber ring laser under different polarization conditions of the PC in the ring cavity. Wavelength spacing is 1.0 nm.

Output (dBm)

1 0.1 0.01 1E-3 1E-4 1552

1554

1556 1558 Wavlength (nm)

1560

1562

FIGURE 7.53 Typical output lasing wavelength of the iber ring laser under a particular polarization condition of the PC in the ring cavity. Wavelength spacing is 0.50 nm.

room temperature.67 This problem can be overcome by cooling the erbium-doped iber to 77 K, but this technique is probably not suitable for practical applications.33,68 A more practical approach is to use an acousto-optic frequency shifter in the ring cavity to prevent the steady-state laser oscillation in order to generate a larger number of stable lasing wavelengths.34 In this section we have demonstrated an Er-doped iber ring laser using an all-polarizationmaintaining-iber (PMF) Sagnac loop ilter for multi-wavelength operation. The theoretical analysis and experimental results of the all-PMF Sagnac loop as a stable comb ilter have been presented. The Sagnac loop ilter is a simple and all-iber device that consists of a PMF coupler and a segment of stress-induced PMF to form the loop. The number of output lasing wavelengths has been obtained by adjusting the polarization state of the light in the ring cavity using a polarization controller. The wavelength spacing is determined by the frequency period of the comb ilter with equal frequency interval.

67

68

S. Arahira, S. Kutsuzawa, Y. Matsui, D. Kunimatsu, and Y. Ogawa, Repetition frequency multiplication of mode-locked using iber dispersion J. Light. Technol., 16, 405–410, 1998. W. J. Lai, P. Shum, and L. N. Binh, Stability and transient analyses of temporal Talbot effect-based repetition-rate multiplication mode-locked laser systems, IEEE Photon. Technol. Lett., 16, 437–439, 2004.

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7.4.3

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multi-wavelength tunable Fiber ring lasers

Furthermore, tunable EDFR lasers57–61,69,70,71,72,73 with a continuous tuning range of 15.5 nm (i.e., from 1546.8 to 1562.3 nm) can be demonstrated. The laser output power is ~9 dBm and the power variation is