Optical Filter Design and Analysis Constant Value Units Speed of light in vacuum c 2.9979 × 108 m/s Permittivity of free space 0 8.8542 × 10–12 F/m Permeability of free space 0 1.2566 × 10–6 H/m Electron charge q 1.6022 × 10–19 C Planck’s constant h 6.6262 × 10–34 J·s Boltzmann’s constant kB 1.3807 × 10–23 J/K Optical Filter Design and Analysis A Signal Processing Approach CHRISTI K MADSEN Bell Laboratories Lucent Technologies JIAN H ZHAO Rutgers University A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS, INC NEW YORK / CHICHESTER / WEINHEIM / BRISBANE / SINGAPORE / TORONTO Designations used by companies to distinguish their products are often claimed as trademarks In all instances where John Wiley & Sons, Inc., is aware of a claim, the product names appear in initial capital or ALL CAPITAL LETTERS Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration Copyright © 1999 by John Wiley & Sons, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic or mechanical, including uploading, downloading, printing, decompiling, recording or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ@WILEY.COM This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold with the understanding that the publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional person should be sought ISBN 0-471-21375-6 This title is also available in print as ISBN 0-471-18373-3 For more information about Wiley products, visit our web site at www.Wiley.com To our families: Eric Lucia, Suqing, and Yumao Contents Preface xi Introduction 1.1 Optical Filters 1.2 Filter Applications in WDM Systems 1.2.1 Bandpass Filters for Multiplexing, Demultiplexing, and Add/Drop 1.2.2 Gain Equalization Filters 1.2.3 Dispersion Compensation Filters 1.3 Scope of the Book References Fundamentals of Electromagnetic Waves and Waveguides 2.1 The Plane Wave 2.1.1 Maxwell’s Equations 2.1.2 The Wave Equation in a Dielectric Medium 2.1.3 Solutions of the Wave Equation 2.1.4 Phase Velocity and Group Velocity 2.1.5 Reflection and Refraction at Dielectric Interfaces 2.2 Slab Waveguides 2.2.1 The Guided Wave Condition 2.2.2 Characteristic Equations for the Slab Waveguide 2.2.3 Waveguide Modes 2.2.4 Orthogonality and Completeness of Modes 2.3.5 Dispersion 2.2.6 Loss and Signal Attenuation 2.3 Rectangular Waveguides 10 13 14 15 19 19 19 20 22 25 27 34 34 37 39 42 44 56 64 vii viii CONTENTS 2.3.1 Wave Equation Analysis 2.3.2 The Effective Index Method 2.3.3 Perturbation Corrections 2.4 Splitters and Combiners 2.4.1 Directional Couplers 2.4.2 Star Couplers 2.4.3 Multi-Mode Interference Couplers 2.5 Material Properties and Fabrication Processes 2.5.1 Materials 2.5.2 Fabrication References Problems Digital Filter Concepts for Optical Filters 3.1 Linear Time-Invariant Systems 3.1.1 Continuous Signals 3.1.2 Discrete Signals 3.2 Digital Filters 3.2.1 The Z-Transform 3.2.2 Poles and Zeros 3.2.3 Stability and Causality 3.2.4 Magnitude Response 3.2.5 Group Delay and Dispersion 3.2.6 Minimum-, Maximum-, and Linear-Phase Filters 3.3 Single-Stage Optical Filters 3.3.1 A Single-Stage MA Filter 3.3.2 A Single-Stage AR Filter 3.3.3 Power Conservation and Reciprocity 3.3.4 Incoherent Optical Signal Processing 3.4 Digital Filter Design 3.4.1 Approximating Functions 3.4.2 Bandpass Filters 3.4.3 The Window Method for MA Bandpass Filters 3.4.4 Classical Filter Designs for ARMA Bandpass Filters 3.4.5 The Least Squares Method for AR Filter Design 3.4.6 Multi-Stage Filter Architectures Appendix References Problems Multi-Stage MA Architectures 4.1 Single-Stage MZI Design 4.1.1 Loss and Fabrication Induced Variations 4.1.2 A Tunable Coupler 4.2 Cascade Filters 65 67 68 69 69 76 78 82 83 86 88 93 95 95 96 102 106 106 108 110 112 114 119 126 129 131 134 136 137 137 139 140 142 148 154 161 162 164 165 165 166 168 171 CONTENTS 4.3 Transversal Filters 4.4 Multi-Port Filters 4.4.1 Diffraction Grating Filters 4.4.2 Waveguide Grating Routers 4.5 Lattice Filters 4.5.1 The Z-Transform Description and Synthesis Algorithm 4.5.2 Generalized Lattice Filters 4.6 Coupled-Mode Filters References Problems Multi-Stage AR Architectures 5.1 Ring Cascade Filter 5.2 Ring Lattice Filter 5.2.1 The Z-Transform Description 5.2.2 Synthesis Algorithm 5.2.3 Sensitivity to Fabrication Variations 5.2.4 Bandpass Filter Design and Experimental Results 5.2.5 Gain Equalizer Design 5.2.6 Dispersion Compensator Design 5.3 Vernier Operation 5.4 Reflective Lattice Filters 5.4.1 Thin-Film Filters 5.4.2 Bragg Gratings References Problems Multi-Stage ARMA Filters 6.1 A Maximally Flat ARMA Filter 6.2 A General ARMA Lattice Architecture 6.2.1 The Z-Transform Description 6.2.2 Synthesis Algorithm 6.2.3 Design Examples 6.3 All-Pass Filters 6.3.1 Optical All-Pass Filters 6.3.2 Fiber Dispersion Compensation 6.3.3 Filter Dispersion Compensation 6.3.4 Wavelength Dependent Delays 6.4 Bandpass Filters 6.4.1 All-Pass Decomposition for × Filters 6.4.2 An N × N Router References Problems ix 177 180 180 184 198 199 216 224 232 236 237 238 241 242 246 258 263 267 270 272 276 277 285 299 302 305 305 310 311 314 317 320 321 323 325 328 330 333 345 352 353 x CONTENTS Optical Measurements and Filter Analysis 7.1 Optical Measurements 7.1.2 Polarization Dependent Loss 7.1.3 Indirect Loss Measurements 7.1.4 Group Delay 7.2 Filter Analysis 7.2.1 Time Domain Measurement 7.2.2 Optical Low-Coherence Interferometry and Fourier Spectroscopy 7.2.3 The AR Analysis Algorithm References Problems Future Directions 8.1 Communication System Applications 8.1.1 Ultra-Dense WDM Systems and Networks 8.1.2 Ultra-Fast TDM and Optical Codes 8.2 Materials and Processing 8.3 Summary References Index 355 355 360 365 372 378 378 380 387 393 394 397 397 397 398 400 402 402 405 394 OPTICAL MEASUREMENTS AND FILTER ANALYSIS 17 D Marcuse, Principles of Optical Fiber Measurements, New York: Academic Press, 1981, p 279 18 C Madsen, “Demonstration of the First General Planar Waveguide Autoregressive Optical Filter,” PhD dissertation New Brunswick, NJ: Rutgers University, 1996 19 G.T Harvey, measurement development (AT&T Bell Laboratories, Holmdel, NJ, 1995) 20 K Yonenaga and N Takachio, “A Fiber Chromatic Dispersion Compensation Technique with an Optical SSB Transmission in Optical Homodyne Detection System,” IEEE Photonics Technol Lett., vol 5, no 18, pp 949–951, 1993 21 G Smith, D Novak, and Z Ahmed, “Technique for optical SSB generation to overcome dispersion penalties in fibre-radio systems,” Electron Lett., vol 33, no 1, pp 74–75, 1997 22 M Abramowitz and A Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables New York: Dover, 1972 23 J Roman, M Frankel, and R Esman, “High resolution technique for characterizing chirped fiber gratings,” in Vol 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), Optical Fiber Communication Conference 1998, pp 6–7 24 K Sasayama, M Okuno, and K Habara, “Coherent Optical Transversal Filter Using Silica-Based Waveguides for High-Speed Signal Processing,” J Lightw Technol., vol 9, no 10, pp 1225–1230, 1991 25 K Takada, H Yamada, and Y Inoue, “Optical Low Coherence Method for Characterizing Silica-Based Arrayed-Waveguide Grating Multiplexers,” J Lightw Technol., vol 14, no 7, pp 1677–1689, 1996 26 H Yamada, K Takada, Y Inoue, Y Hibino, and M Horiguchi, “10 GHz-spaced arrayedwaveguide grating multiplexer with phase-error-compensating thin-film heaters,” Electronics Lett., vol 31, pp 360–361, 1995 27 H Yamada, K Takada, Y Inoue, Y Ohmori, and S Mitachi, “Statically-phase-compensated 10 GHz-spaced arrayed-waveguide grating,” Electronics Lett., vol 32, no 17, pp 1580–1581, 1996 28 E Brinkmeyer, “Simple algorithm for reconstructing fiber gratings from reflectometric data,” Optics Lett., vol 20, no 8, pp 810–912, 1995 PROBLEMS Calculate the noise resulting from the interference of two point reflectors as a function of their separation, their reflectance (assuming equal reflectance values), and the source linewidth Assume that the source has a Lorentzian line shape Derive Eqs (32) and (33) for optical SSB modulation using an MZI driven in quadrature Derive an expression for the power in the carrier and lower sideband as a function of the modulation depth Calculate the interferogram for a channel input with 100 GHz frequency spac- PROBLEMS 395 ing and a center wavelength of 1550 nm Determine the sampling length required for a resolution of 10 GHz Calculate the interference pattern for a third-order AR filter with poles located at 0.9 Є and 0.8 Є ±/20 rad for two cases: (a) the unit delay is 10 times the source coherence time and (b) the unit delay is 0.1 times the source coherence time CHAPTER EIGHT Future Directions Optical filters are evolving at a rapid pace This transformation is arising from both a pull by the development needs of optical communication systems and the technology push coming from research in materials and fabrication processes These two areas are discussed with respect to their impact on future directions for optical filters 8.1 COMMUNICATION SYSTEM APPLICATIONS 8.1.1 Ultra-Dense WDM Systems and Optical Networks Dense WDM systems are widely deployed and provide a wealth of applications for optical filters The filter requirements for the bandpass, gain equalization, and dispersion compensating filters become more demanding as the number of channels, the bandwidth utilization, and the system wavelength range increase Different transmission formats are used in WDM systems, for example, non-return-to-zero (NRZ) and solitons The transmission format impacts channel spacing, dispersion compensation and gain equalization; thus, impacting filter requirements The best spectral efficiency of 0.6 bits/s/Hz was demonstrated using duo-binary coding [1] A 2.6 Tb/s WDM system was achieved using 132 channels at 20 Gb/s each over 120 km of standard fiber The channels were separated by 33.3 GHz and demultiplexing was accomplished by using three WGRs with FWHM = 0.32 nm, temperature control for center wavelength alignment, and manually tuned double cavity thin-film filters with 0.3 nm bandwidths To date, point-to-point and fixed add/drop commercial WDM systems have been deployed Optical networks are on the horizon and bring with them a host of challenges The major advantage of networks is to achieve routing flexibility [2] Routing capability is accompanied by added complexity to the overall optical design [3] For example, gain equalization and dispersion compensation are complicated by the 397 398 FUTURE DIRECTIONS fact that channels originate from different locations (thus experiencing a different number of amplifiers and dispersion maps) and their routing can change Optical monitoring is critical, and active gain equalization and dispersion compensation may be necessary Lower bit rates allow more channels to be carried on a network; however, this smaller granularity implies filters with narrower, stable passbands and very precise center wavelengths Other transmission effects also surface in a WDM network In systems with many amplifiers operating in saturation, changing the power into an amplifier (for example, by adding or dropping channels) can induce power transients on the other channels that are extremely fast [4] Dynamic gain control at high speeds may be necessary to suppress this effect A crucial new component for networks is an optical cross-connect where channels on one fiber can be routed onto another fiber and vice-versa Consider a general optical cross-connect for a WDM system with M fibers, each with a capacity of N WDM channels for a total of M × N channels A channel on any fiber must be allowed to switch onto any other fiber For four fibers with 80 channels each, there are a total of 320 channels to route Wavelength conversion is required so that an add channel will not interfere with a channel already on the output fiber Various combinations of demultiplexers, switches, and multiplexers may be used For example, an optical cross-connect consisting of 2M WGRs with an intervening switch is shown in Figure 8-1 Optical micro-electromechanical (MEMs) devices are one option for realizing large switch arrays [5,6] The functionality and cost of an optical cross-connect must be equivalent or better than what can be achieved with electrical cross-connects for it to be commercially viable 8.1.2 Ultra-Fast TDM and Optical Codes Dense WDM systems already employ time division multiplexing (TDM) to combine many lower bit rate signals into a single high bit rate channel of several Gb/s FIGURE 8-1 An optical cross-connect employing WGRs and space switches 8.1 COMMUNICATION SYSTEM APPLICATIONS 399 The ongoing race is to demonstrate the highest bit rate over the longest propagation distance For the propagation of ultra-short pulses, broadband dispersion compensation is required For example, an MA lattice filter has been used to compensate for third-order dispersion and reduce the power penalty by dB for a 200 Gb/s signal transmitted over 100 km [7] The 9-stage filter had a FSR = 300GHz, an insertion loss of 11.4 dB, and a dispersion slope of –7.9ps/nm2 Variable delay structures are critical for TDM applications A delay buffer is needed to multiplex the lower bit rate stream up to the higher one so that it is placed in its correct time slot For demultiplexing, the delay must be chosen to select the desired channel To multiplex a large number of channels, a wide tuning range for the delay is needed The delay must be tuned to within a fraction of the time slot, i.e have high precision A 1024-channel fast tunable delay line has been demonstrated using Mach–Zehnder delay lines with × couplers and fiber delay lines [8] An aggregate rate of 50 Gb/s was experimentally demonstrated using a five stage delay structure with two electro-optic modulators, a 20 ns reconfiguration time, and a loss of 22 dB Transmission on the band-edge of gratings where the dispersion is very high is being explored for delay lines as well A short grating with high index contrast can provide a large group delay For example, a delay of ns was demonstrated using a 10 mm grating in a semiconductor chip [9] Simulation results for a 30 period grating in GaAs/AlAs showed a group index of 13.5, three times the value for the bulk material, with a resonance bandwidth of 2.3 nm and negligible degradation in amplitude and shape of a ps pulse [10] Given the success of code division multiple access (CDMA) modulation format in wireless communication systems, optical systems employing CDMA have been investigated Although users share the same wavelength band, each user’s information is encoded so that an autocorrelation with the matched code is required at the receiver to decode it Various optical filters have been used for code generation including MA transversal [11] and lattice [12] structures, thin-film coupled cavities [13,14], and gratings plus phase masks [15] Two filters are needed for each user, one for multiplexing and demultiplexing Orthogonal codes are required to minimize cross-talk between users The feasibility of optical CDMA systems to handle a large number of users is a topic of research [16] A capacity of Gb/s with four Gb/s channels has been demonstrated [17] The main limiting factor for incoherent systems is optical noise at the square law detector, which is exacerbated over WDM and TDM systems because a large wavelength range is used with multiple users transmitting over the same wavelength range Besides CDMA, codes on optical pulses may find other applications such as routing of channels in the optical level For example, a matched filter for generating an optical code was demonstrated using the fiber-optic delay line shown schematically in Figure 8-2 [18] The experimental versus theoretical autocorrelation output is shown in Figure 8-3 for an bit optical 10 Gb/s packet In another experiment, an 8-bit 100 Gb/s self-synchronizing pattern generator was demonstrated having two 15 ps marker pulses and six 10 ps data pulses using a × silica waveguide splitter, an array of semiconductor laser amplifiers and a silica waveguide circuit of delay lines and a combiner [19] 400 FUTURE DIRECTIONS FIGURE 8-2 Fiber-optic matched filter schematic [18] Reprinted by permission Copyright © 1996 IEEE 8.2 MATERIALS AND PROCESSING While silica has set the standard for low loss optical waveguides, additional functionality is desired for active device control For example, faster and lower power consumption effects for tuning, switching, and modulation are needed than afforded by the thermo-optic effect Nonlinear optical polymers [20], lithium niobate, and semiconductor materials such as InGaAsP and silicon [21,22] offer rapid tuning ef- FIGURE 8-3 Fiber-optic matched filter autocorrelation output comparing the experimental results to the theoretical results shown in the inset [18] Reprinted by permission Copyright © 1996 IEEE 8.2 MATERIALS AND PROCESSING 401 fects Hybridization, or integrating multiple materials, is attractive for improving device functionality and performance An area for research is the design of filters with gain On the device side, microresonators that combine whispering gallery mode operation [23] and asymmetric cavity resonator designs [24] have been demonstrated The focus has been on lasers so far, but these geometries offer a new coupling mechanism for optical filters as well Another new area is photonic bandgap (PBG) engineering, which employs periodic structures with sub-micron element sizes to make devices The PBGs are created by fabricating a periodic structure with a high index contrast so that it reflects over a large wavelength range, the photonic bandgap [25] One-dimensional PBGs are high index contrast gratings [26], as discussed in Chapter By creating a line defect in a two-dimensional periodic array, waveguiding is achieved Figure 8-4 illustrates a Input Transmission Reflection Backward Transfer FIGURE 8-4 Forward Transfer A 2-D photonic bandgap structure showing a resonant filter [28] 402 FUTURE DIRECTIONS periodic lattice of dielectric rods where two rows have been removed on the top and bottom to act as waveguides Simulations show that 90° bends can be achieved with practically no loss in 2-D PBG waveguides [27] To achieve very small radii bends in conventional waveguides requires very large ⌬’s; however, propagation and scattering losses increase with ⌬ The PBG structures are expected to be robust to random fabrication variations since the field can be localized in air regions away from the high dielectric contrast interfaces A new resonator structure is illustrated by the two defects between the waveguides in Figure 8-4 [28] The resonator has a symmetric and an antisymmetric mode that are degenerate (i.e., resonate at the same frequency) and identical coupling strengths to the input/output waveguides [29] The size or index of the defects between the input/output waveguides and the resonator defects (shown as gray circles) are designed to produce these conditions so that the structure functions as an add/drop filter Many details must be explored for PBG filter applications including coupling from/to a standard waveguide into the PBG, minimizing the polarization dependence, developing and analyzing defect architectures for controlled coupling between waveguides, and methods to change the optical phase or amplitude for tuning and switching applications 8.3 SUMMARY While many of the filters discussed in this book and the technologies for realizing them are quite mature, there are also many new designs and concepts that need experimental investigation With the demands for bandwidth being driven by applications such as the World Wide Web, it is a given that optical communication systems will continue to push the state-of-the-art in optical filter technology We can expect the number of channels and the routing and control functions done at the optical layer to increase When combined with advances in materials, increases in hybridization and integration, and advances in fabrication processes, the field of optical filters has a strong outlook for the future with a large potential for growth REFERENCES Y Yano, T Ono, K Fukuchi, T Ito, H Yamazaki, M Yamaguchi, and K Emura, “2.6 Terabit/s WDM transmission experiment using optical duobinary coding,” 22nd European Conference on Optical Communication Oslo, Norway, 1996, p ThB.3.1 R Wagner, R Alferness, A Saleh, and M Goodman, “MONET: Multiwavelength optical networking,” J Lightw Technol., vol 14, no 6, p 1349, 1996 J Zyskind, J Nagel, and H Kidorf, “Erbium-doped fiber amplifiers for optical communications,” in Optical Fiber Telecommunications IIIB, I Kaminow and T Koch, Eds San Diego: Academic Press, 1997, pp 13–68 J Zyskind, Y Sun, A Srivastava, J Sulhoff, A Lucero, C Wolf, and R Tkach, “Fast power transients in optically amplified multiwavelength optical networks,” in OSA Technical Digest Series, Optical Fiber Communications Conference 1996, p PD31 H Ukita, “Micromechanical Photonics,” Optical Review, vol 4, no 6, p 623–633, 1997 REFERENCES 403 L Lin, “Micromachined free-space matrix switches with submillisecond switching time for large-scale optical crossconnect,” in Vol 1998 OSA Technical Digest Series (Optical Society of America, Washington, DC, 1998), Optical Fiber Communication Conference1998, pp 147–148 K Takiguchi, K Jinguji, K Okamoto, and Y Ohmori, “Variable Group-Delay Dispersion Equalizer Using Lattice-Form Programmable Optical Filter on Planar Lightwave Circuit,” IEEE J Selected Areas Commun., vol 2, no 2, pp 270–276, 1996 K Deng, K Kang, I Glask, and P Prucnal, “A 1024-channel fast tunable delay line for ultrafast all-optical TDM networks,” IEEE Photon Technol Lett., vol 9, no 11, pp 1496–1499, 1997 W Stewart, “Optical devices incorporating slow wave structures,” U.S Patent, no 5,311,605, 1994 10 M Scalora, R J Flynn, S Reinhardt, R Fork, M Bloemer, M Tocci, C Bowden, H Ledbetter, J Bendickson, J Dowling, and R Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys Rev E54, pp 1078–1081, 1996 11 P Prucnal, M Santoro, and T Fan, “Spread spectrum fiber-optic local area network using optical processing,” J Lightw Technol., vol 4, no 5, pp 547–554, 1986 12 A Holmes and R Syms, “All-optical CDMA using “quasi-prime” codes,” J Lightw Technol., vol 10, no 2, pp 279–286, 1992 13 V Narayan, E Dowling, and D MacFarlane, “Design of Multi-Mirror Structures for High Frequency Bursts and Codes of Ultrashort Pulses,” IEEE J Quantum Electron., vol 30, no 7, pp 1671–1680, 1994 14 V Narayan, D MacFarlane, and E Dowling, “High-Speed Discrete-Time Optical Filtering,” IEEE Photonics Technol Lett., vol 7, no 9, pp 1042–1044, 1995 15 R Griffin and D Sampson, “Coherence coding of optical pulses for code-division multiple access,” OFC/IOOC OSA Technical Digest Series r, 1993, pp 200–201 16 D Sampson, G Pendock, and R Griffin, “Photonic code-division multiple-access communications,” Fiber and Integrated Optics, vol 16, pp 129–157, 1997 17 G Pendock and D Sampson, “Increasing the transmission capacity of coherence multipled communication systems by using differential detection,” IEEE Photon Technol Lett., vol 7, pp 1504–1507, 1995 18 J Shin, M Jeon, and C Kang, “Fiber-Optic Matched Filters with Metal Films Deposited on Fiber Delay-Line Ends for Optical Packet Address Detection,” IEEE Photonics Technol Lett., vol 8, no 7, pp 941–943, 1996 19 D Rogers, J Collins, C Ford, J Lucek, M Shabeer, G Sherlock, D Cotter, K Smith, C Peed, A Kelly, P Gunning, D Nesset, and I Lealman, “Optical pulse pattern generation for self-synchronizing 100 Gbit/s networks,” Optical Fiber Communications Conference, San Jose, CA, 1996, pp 98–99 20 Wang, D Chen, H Fetterman, Y Shi, W Steier, and L Dalton, “40-GHz polymer electrooptic phase modulators,” IEEE Photon Technol Lett., vol 7, no 6, pp 638–640, 1995 21 P Trinh, S Yegnanarayanan, and B Jalali, “5 × Integrated Optical Star Coupler in Silicon-on-Insulator Technology,” IEEE Photonics Technol Lett., vol 8, no 6, pp 794–796, 1996 22 P D Trinh, S Yegnanarayanan, F Coppinger, and B Jalali, “Silicon-on-insulator (soi) 404 23 24 25 26 27 28 29 FUTURE DIRECTIONS Phased-array Wavelength Multi/demultiplexer With Extremely Low-polarization Sensitivity,” IEEE Photon Technol Lett., p 940, 1997 Y Yamamoto and R Slusher, “Optical processes in microcavities,” Physics Today, June, pp 66–73, 1993 J Nockel, A Stone, G Chen, H Grossman, and R Chang, “Directional emission from asymmetric resonant cavities,” Optics Lett., vol 21, no 19, pp 1609–1611, 1996 J Joannopoulos, R Meade, and J Winn, Photonic Crystals: Molding the Flow of Light Princeton, NJ: Princeton University Press, 1995 J Foresi, P Villeneuve, F Ferrara, E Thoen, G Steinmeyer, S Fan, J Joannopoulos, L Kimerling, H Smith, and E Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature, vol 390, 13 November, pp 143–145, 1997 J Joannopoulos, P Villeneuve, and S Fan, “Photonic crystals: putting a new twist on light,” Nature, vol 386, 13 March, pp 143–149, 1997 S Fan, P Villeneuve, and J Joannopoulos, “Channel Drop Tunneling Through Localized States,” Phys Rev Lett., vol 80, no 5, pp 960–963, 1998 H Haus, S Fan, J Foresi, P Villeneuve, J Joannopoulos, and B Little, “Optical-wavelength-scale filters,” in Vol 2, LEOS 1997, pp 96–97 ONE Index Absorption loss, 56–57 Acousto-optic filter, 229–230 Add/drop filter, 8–10 Aliasing, 103–104 All-pass filter: decomposition, 330–345 digital, 125–126 optical, 321–323 Amplified spontaneous emission (ASE) source, 358 Apodization, 142, 229, 290–292 Arrayed waveguide grating (AWG), see Waveguide grating router Autocorrelation, 99, 105 Autoregressive filter (AR): digital, 5, 109–110, 113–114, 117–118, 125, 148–154, 158–160 optical, 131–133, 237–285 Autoregressive moving average filter (ARMA): digital, 5, 109, 140, 142–148, 160–161 optical, 133, 244–245, 255–258, 279–282, 284, 305–352 Bend loss, 58–60, 63 Bilinear transform, 143 Birefringence, 54, 83–85, 224 Birefringent filter, 224–225 Blaze angle, 180, 183 Bragg grating, 285–297, 373 Brewster’s angle, 31 Butterworth filter, 142–148, 305–310, 317–318 Cascade filter architecture, 159, 171–177, 238–241, 322 Cauchy’s integral theorem, 51 Causality, 110–112 Characteristic matrix, 281 Chebyshev filter, 142–148, 239, 250 Chebyshev’s identity, 206 Chirped grating, 184–185, 287, 294–295 Circulator, 293–294, 357 Code division multiple access (CDMA), 399 Coherence: interference, 359 length, 381 Convolution, 97 Coupled-mode theory, 70–72, 225–229, 287–288 Coupling ratio, 128 Cross-connect, 398 Degree of polarization, 363 Delay, wavelength-dependent, 328–330; see also Group delay, Unit delay Delta function, 96, 98 Detector, 358 Device under test (DUT), 356 Diffraction, 100–101 Diffraction grating, 180–184 Directional coupler, 69–76, 128 grating-assisted, 227–228 Discrete Fourier transform (DFT), 105, 175 Discrete-time Fourier transform (DTFT), 102–103, 104–105 Dispersion, 27, 44–55 angular, 181, 187 compensation, 13–14 dispersion diagram, 55 filter, 116–119 405 406 INDEX Dispersion (continued) group velocity, 49–50 Inter-modal, 52–53 material, 45–52 polarization mode, 54 waveguide, 53–54 Dispersion compensating filter, 211–217, 225, 270–271, 294–295, 323–328 Distributed feedback (DFB) laser, 359, 379 Effective index, 126 Effective index method, 67–68 Elliptic filter, 142–148, 317–318, 334–335, 337–338 Energy: conservation, 106 density, 24–25 Erbium-doped fiber amplifier (EDFA), 7–8 External cavity laser (ECL), 359 Fabrication variations, 155–157, 166–167, 219–222, 258–263, 309–310, 349–351, 385–387 Fabry-Perot interferometer, 2, 4, 108, 280, 284, 297, 321 Fiber: dispersion compensating fiber (DCF), 13 dispersion shifted fiber (DSF), 48, 53 polarization maintaining fiber (PMF), 361 standard single-mode fiber (SMF), 13, 48, 53, 119 Fiber paddles, 361 Filter analysis, 378–392 Finite impulse response (FIR), see Moving average filter Fourier filter, 216–223 Fourier series, 2, 5, 141, 288 Fourier spectroscopy, 380–382 Fourier transform, 97–98, 100 properties, 99 symmetry, 102 transform pairs, 101 Fraunhofer diffraction, see Diffraction Free spectral range (FSR), 2, 6, 102, 126–127, 132–133, 182, 187 Frequency: absolute, 98, 102 angular, 102 normalized, 102 optical, 126–127 spatial, 100 Frequency response, 98 Frequency selector, 173, 178–179, 339–343 Fresnel diffraction, see Diffraction Fresnel formulas, 29–34 Full width at half maximum (FWHM), 116 Gain equalization filter, 10–13, 208–211, 222–223, 267–270, 319–320 Gaussian mode approximation, 61–62, 191 Gibb’s phenomenon, 112, 138, 141 Gires-Tournois interferometer, 322 Grating: equation, 181, etched, 285 UV-induced, 229, 285–286 see also Chirped grating Group delay, 114–119, 196, 370, 372–378 Group index, 27, 48, 127 Group velocity, 26–27, 37, 49, 54–55 Heaters, 88, 264–265, 385 Helium-neon laser, 359–360 Heterodyne detection, 375 Hilbert transform, 111–112, 372–374 Impulse function, see Delta function Impulse response, 97 Incoherent signal processing, 136–137 Index of refraction, 25 table, 283 Indium gallium arsenide phosphide material system: InGaAs, 358 InGaAsP/InP, 84 Infinite impulse response (IIR), see Autoregressive filter, Autoregressive moving average filter Insertion loss (IL), 356–357 Interference filter, see Thin film filter Isolator, 357 Jones vector/matrix, 224 Kramers-Kronig relations, 50–52, 112 Kronecker delta function, 103 Lagrange multiplier, 364 Laplace transform, 143, 161–162 Lattice filter architecture, 151, 155, 159–161, 198–223, 241–271, 310–320, 322 Least-squares filter design, 148–154 Levinson-Durbin algorithm, 151–154, 247–250, 388–389 Light emitting diode (LED), 358 Linear time-invariant system, 96–97 Linear-phase, 119–126, 209–210 Lithium niobate, 84 INDEX Logarithmic architecture, 173–177 Long period grating, 229–232 Loss: mechanisms, 56–64 propagation, 127 waveguide, 83–85 Mach-Zehnder interferometer (MZI), 2–3, 129–131, 165–171, 292–294, 305–308, 322, 332 Magnitude response, 98, 112–114 Matched filter, 121, 399–400 Material dispersion, see Dispersion Maximally-flat filter, see Butterworth filter Maximum-phase, 112–113, 117, 119–126, 370 Maxwell’s equations, 19–20 Micro-electro-mechanical (MEMs) devices, 5, 398 Minimum-phase, 112–113, 117, 119–126, 370, 372–373 Modal dispersion, see Dispersion Mode mismatch loss, 61 Modulator: linear response, 344–345 lithium niobate, 84, 375–377 polymer waveguide, 85 Moving average (MA) filter: digital, 2, 109, 115 optical, 129–131, 165–225 Mueller matrix, 363, 365 Multi-mode interference (MMI) coupler, 78–82, 197–198, 345–347 Multiplexer, 8–9 Normalized frequency: waveguide, 41, 67 frequency response, 102, 126–127 Nyquist sampling frequency, 103–104, 382 Optical low coherence interferometry (OLCI), 380–381 Optical spectrum analyzer (OSA), 358, 360 Orthogonality, 42–44 Paley-Weiner theorem, 112 Parseval’s identity, 106 Perturbation waveguide analysis, 68–69 Phase mask, 286–287 Phase response, 98, 115, 279, 307–308, 348 Phase shifter, 128–129; see also Heaters Phase velocity, 25, 54–55 Phased array (PHASAR), see Waveguide grating router 407 Photonic band gap (PBG), 292–293, 297–298, 401–402 Plane wave, 22–24 Polarization, 363 state of polarization (SOP), 54 Polarization dependent loss (PDL), 357, 360–365 Polarization mode dispersion (PMD), 54, 361 Polarization wavelength dependence (PWD), 361 Poles, 108–110 Pole-zero diagram, 113, 148, 174, 367 Polymers, 85 Power conservation, 134 Poynting vector, 24 Propagation constant, 35, 118 Quarter-wave shifted filter, 283, 296–298 Reciprocity, 134–136 Rectangular function, 97, 105 Rectangular waveguide, 64–69 Reflection, 27–34 Refraction, 27–34 Reverse polynomial, 119–120, 130, 133, 199, 244, 279, 312 Ring resonator, 108, 131–134, 238, 241–243, 306, 321, 365–372, 373 Rowland circle, 183–184 Sampled grating, see Superstructure grating Scattering loss, 57–58 Scattering matrix, 133–134 Sellmeir equation, 47–48 Sensitivity to fabrication variations, see Fabrication variations Signal-to-noise ratio (SNR), 10–11, 358 Silica, 83–84 Silicon, 85 Sinc function, 101 Single sideband (SSB) phase measurement, 377–378 Slab coupler, see Star coupler Slab waveguide, 34–39 Snell’s law, 29 Spectral density, 99, 105 Spectral efficiency, 8, 397 Spectral loss, 355 Speed of light, 25 Spot size, 61–62 Stability, 110 Star coupler, 76–78 Stokes vector, 363, 364–365 Sum of all optical paths, 128–129 Superstructure grating, 296 Switch, 197–198, 336, 398 408 INDEX Tapered windows, see Apodization TE, 29–30, 34–39 Thermo-optic effect, 88 Thin film interference filter, 276–285, 373 TM, 29–30, 34–39 Total internal reflection, 31–34 Transfer function, 107 Transfer matrix, 128, 130, 133, 188, 219, 224, 228–229, 244, 281, 290, 311–313 Transition loss, 61 Transversal filter architecture, 157–158, 177–180 Ultraviolet photosensitivity, 83, 229, 286–287 Unit delay, 102, 107–108, 126–127 Unitary matrix, 134 Urbach’s law, 57 Vernier operation, 175, 272–276 Wave vector, 23, 35 Waveguide dispersion, see Dispersion Waveguide grating router (WGR), 184–197, 384–387, 398 Wavelength division multiplexed (WDM) systems, 1, 7, 397 Wavelength meter, 359, 382 Weiner-Klinchine theorem, 106 Weiner-Lee transform, 373 Whispersing gallery mode (WGM), 62–64, 369–372, 401 Window filter design method, 140–142 Zeros, 108–110 Z-transform, 107–108, 126, 182 ... electrical and digital filters For example, passband width, stopband rejection, and the transition width between the passband and stopband are all design parameters for bandpass filters For high bitrate... for a single output port Each channel passband is shaded, L is the minimum passband loss, Xadj and Xnon-adj are the adjacent and non-adjacent channel cross-talk losses, and the filter passband... optical fibers and broadband optical amplifiers, WDM systems have the potential to harness a huge bandwidth, and optical filters are essential to realizing this goal In addition to traditional