Download pdf - Planar Lightwave Circuits Employing Coupled Waveguides in … · 2010-02-08 · Abstract Planar Lightwave Circuits Employing Coupled Waveguides in Aluminum Gallium Arsenide Rajiv

Planar Lightwave Circuits

Employing Coupled Waveguides in

Aluminum Gallium Arsenide

by

Rajiv Iyer

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical EngineeringUniversity of Toronto

Copyright c© 2008 by Rajiv Iyer

Abstract

Planar Lightwave Circuits

Employing Coupled Waveguides in


Rajiv Iyer

Doctor of Philosophy

Graduate Department of Electrical Engineering

University of Toronto

2008

This dissertation addresses three research challenges in planar lightwave circuit (PLC)

optical signal processing.

1. Dynamic localization, a relatively new class of quantum phenomena, has not been

demonstrated in any system to date. To address this challenge, the quantum system

was mapped to the optical domain using a set of curved, coupled PLC waveguides in

aluminum gallium arsenide (AlGaAs). The devices demonstrated, for the first time,

exact dynamic localization in any system. These experiments motivate further mappings

of quantum phenomena in the optical domain, leading toward the design of novel optical

signal processing devices using these quantum-analog effects.

2. The PLC microresonator promises to reduce PLC device size and increase optical

signal processing functionality. Microresonators in a parallel cascaded configuration,

called “side coupled integrated spaced sequence of resonators” (SCISSORs), could offer

very interesting dispersion compensation abilities, if a sufficient number of rings is present

to produce fully formed “Bragg” gaps. To date, a SCISSOR with only three rings has

been reported in a high-index material system. In this work, one, two, four and eight-ring

SCISSORs were fabricated in AlGaAs. The eight-ring SCISSOR succeeded in producing

fully formed Bragg peaks, and offers a platform to study interesting linear and nonlinear

ii

phenomena such as dispersion compensators and gap solitons.

3. PLCs are ideal candidates to satisfy the projected performance requirements of

future microchip interconnects. In addition to data routing, these PLCs must provide

over 100-bit switchable delays operating at ∼ 1 Tbit/s. To date, no low loss optical device

has met these requirements. To address this challenge, an ultrafast, low loss, switchable

optically controllable delay line was fabricated in AlGaAs, capable of delaying 126 bits,

with a bit-period of 1.5 ps. This successful demonstrator offers a practical solution for

the incorporation of optics with microelectronics systems.

The three aforementioned projects all employ, in their unique way, the coupling of light

between PLC waveguides in AlGaAs. This central theme is explored in this dissertation

in both its two- and multi-waveguide embodiments.

iii

Acknowledgements

I first and foremost express my sincere gratitude to Professor J. Stewart Aitchison for

his mentorship over the past 5 years. Furthermore, I thank my research colleagues and

mentors: Professor John E. Sipe, Professor Marc M. Dignam, Professor C. Martijn de

Sterke, Professor Marc Sorel, Professor Peter W. E. Smith, Professor Henry M. van Driel,

Professor Arthur L. Smirl, Dr. Jun Wan, Dr. Alan D. Bristow, Dr. Zhenshan Yang, Dr.

Joachim Meier, Dr. Philip Chak, Dr. Francesca Pozzi and all my fellow students and

staff in the Photonics Research Group.

I also thank NSERC, the Ontario Centres of Excellence, the CCPE/Manulife Finan-

cial, SPIE, and the AAPN for funding my research.

I also thank Dr. Henry Lee and Yimin Zhou from the Emerging Communications

Technology Institute for their assistance in fabricating my devices, and to Battista

Calvieri and Steven Doyle from the Microscopy Imaging Lab for the use of their scanning

electron microscopes. I also thank Andrew Bezinger and Dr. Margaret Buchanan from

the Institute for Microstructural Sciences at the NRC for etching many of my devices.

My thanks also extend to James Pond from Lumerical Inc. whose MODE Solutions

served as an essential tool in my research.

This PhD was only made possible thanks to my wife, Deepa, who supported my

decision to return to school to continue my education. Her patience and friendship has

been an invaluable resource of strength and advice.

My sincere thanks to my parents, my niece, Mira, my nephew, Arjun, their parents,

Tara and Vineet, their grandparents, Mana and Mavi, and all my friends, who helped

me through the tough times, and celebrated with me during the high times. I also must

specifically thank Arjun for helping me download my references, and our new baby who

arrived just before final submission!

Above all, I express my deepest gratitude to my Gurumatha Amma, for her unending

guidance, love and support in all my endeavours.

iv

Contents

1 Introduction 1

1.1 Planar Lightwave Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Integrated versus Free-Space Optical Devices . . . . . . . . . . . . 2

1.1.2 PLC Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 PLCs for Spatial Optical Signal Processing: Exact Dynamic Localization 5

1.2.1 Mapping between Quantum and Optical Domains . . . . . . . . . 5

1.2.2 Discrete Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Bloch Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 Dynamic Localization . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.5 The Exact Dynamic Localization Challenge . . . . . . . . . . . . 9

1.2.6 The Exact Dynamic Localization Solution . . . . . . . . . . . . . 9

1.3 PLCs for Spectral Optical Signal Processing: SCISSORs . . . . . . . . . 10

1.3.1 Microresonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2 Serial CROWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.3 Parallel SCISSORs . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.4 The SCISSOR Challenge . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.5 The SCISSOR Solution . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 PLCs for Temporal Optical Signal Processing: Optical Delay Lines . . . 15

1.4.1 PLC Microchip Interconnects . . . . . . . . . . . . . . . . . . . . 15

v

1.4.2 Optical Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.3 PLC optical delays: Resonators versus Differing-Length

Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.4 Optical Delay Switching . . . . . . . . . . . . . . . . . . . . . . . 18

1.4.5 The Optical Delay Line Challenge . . . . . . . . . . . . . . . . . . 19

1.4.6 The Optical Delay Line Solution . . . . . . . . . . . . . . . . . . . 20

1.5 The Challenges and Solutions: Discussion . . . . . . . . . . . . . . . . . 20

1.6 Light Coupling in Waveguides . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6.1 Light Confinement in a Single Waveguide . . . . . . . . . . . . . . 22

1.6.2 Light Coupling in a Two-Waveguide System . . . . . . . . . . . . 23

1.6.3 Light Coupling in a Multi-Waveguide System . . . . . . . . . . . 25

1.6.4 The Nonlinear Directional Coupler Switch . . . . . . . . . . . . . 25

1.7 Aluminum Gallium Arsenide . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.8 Summary and Thesis Organization . . . . . . . . . . . . . . . . . . . . . 28

2 Light Coupling in Waveguides 29

2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Supermodes of the Directional Coupler . . . . . . . . . . . . . . . . . . . 33

2.3 Hamiltonian Formulation of Coupled Mode Equations . . . . . . . . . . . 36

2.3.1 Eigenmodes in a Restricted Basis . . . . . . . . . . . . . . . . . . 36

2.3.2 Effective Fields and the Coupled Mode Equations . . . . . . . . . 39

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Aluminum Gallium Arsenide 42

3.1 AlGaAs Lattice Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 AlGaAs Material Index Dispersion . . . . . . . . . . . . . . . . . . . . . 44

3.3 AlGaAs Optical Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 AlGaAs Wafers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

vi

3.4.1 AlGaAs Wafer: 24/18/24 . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.2 AlGaAs Wafer: 70/20/70 . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Exact Dynamic Localization 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Dynamic Localization in Curved Coupled Optical Waveguide Arrays . . . 52

4.2.1 Paraxial Complex Vector Wave Equation . . . . . . . . . . . . . . 52

4.2.2 Mapping of the Quantum System to Waveguide Arrays . . . . . . 54

4.2.3 Straight Waveguide Array . . . . . . . . . . . . . . . . . . . . . . 57

4.2.4 Bloch Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.5 EDL in CCOW Arrays . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.6 EDL CCOW Design . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 EDL: Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.3 Output Slab Measurement Technique . . . . . . . . . . . . . . . . 66

4.3.4 Staggered Technique for Spatial Mapping . . . . . . . . . . . . . . 67

4.3.5 EDL Measurement Techniques: Validation . . . . . . . . . . . . . 67

4.3.6 EDL: Wavelength Dependence . . . . . . . . . . . . . . . . . . . . 69

4.3.7 EDL: Spatial Map . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Discussion: EDL Tolerance on Discontinuity Smoothing . . . . . . . . . . 73

4.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Side-Coupled Integrated Spaced Sequence of Resonators 77

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 The SCISSOR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

vii

5.3 Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.1 Nanowire Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.2 Nanowire Directional Coupler Design . . . . . . . . . . . . . . . . 82

5.3.3 SCISSOR Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4 Device Fabrication and Characterization . . . . . . . . . . . . . . . . . . 85

5.4.1 Nanowire Directional Coupler Characterization . . . . . . . . . . 87

5.4.2 SCISSOR Characterization . . . . . . . . . . . . . . . . . . . . . . 89

5.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5.1 Nanowires: Future Work . . . . . . . . . . . . . . . . . . . . . . . 91

5.5.2 SCISSORS: Future Work . . . . . . . . . . . . . . . . . . . . . . . 91

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 All-Optical Controllable Delay Line 94

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.1.1 Nonlinear Directional Coupler Behaviour . . . . . . . . . . . . . . 95

6.2 Optical Delay Line: Design and Fabrication . . . . . . . . . . . . . . . . 97

6.2.1 OCDL Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2.2 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3 Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.2 OCDL Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3.3 Self Phase Modulation . . . . . . . . . . . . . . . . . . . . . . . . 110

6.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4.1 New OCDL Design . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5 Extensions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

viii

7 Summary and Conclusions 115

7.1 Exact Dynamic Localization . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.1.1 Summary: Exact Dynamic Localization . . . . . . . . . . . . . . . 116

7.1.2 Future Work: Exact Dynamic Localization . . . . . . . . . . . . . 117

7.2 SCISSORs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.2.1 Summary: SCISSORs . . . . . . . . . . . . . . . . . . . . . . . . 117

7.2.2 Future Work: SCISSORs . . . . . . . . . . . . . . . . . . . . . . . 118

7.3 Optically Controllable Delay Line . . . . . . . . . . . . . . . . . . . . . . 118

7.3.1 Summary: Optically Controllable Delay Line . . . . . . . . . . . . 119

7.3.2 Future Work: Optically Controllable Delay Line . . . . . . . . . . 119

7.4 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Appendices 121

A Nonlinear Polarization Density in AlGaAs 121

A.1 Nonlinear Susceptibility and Polarization Density in AlGaAs . . . . . . . 121

A.1.1 Linear Polarization Density . . . . . . . . . . . . . . . . . . . . . 123

A.1.2 Second Order Nonlinear Polarization Density . . . . . . . . . . . . 123

A.1.3 Third Order Nonlinear Polarization Density . . . . . . . . . . . . 124

A.1.4 Polarization Density of AlGaAs . . . . . . . . . . . . . . . . . . . 126

A.2 Kerr nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

B AlGaAs Photolithography and Etching 129

B.1 AlGaAs Recipe: Anisotropic . . . . . . . . . . . . . . . . . . . . . . . . . 130

B.1.1 Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

B.1.2 Anisotropic Etch Results . . . . . . . . . . . . . . . . . . . . . . . 132

B.2 AlGaAs Recipe: Isotropic . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.2.1 Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.2.2 Isotropic Etch Results . . . . . . . . . . . . . . . . . . . . . . . . 134

ix

B.3 Photoresist Crosslinking and Hard Baking . . . . . . . . . . . . . . . . . 137

C Microresonator Free Spectral Range 138

Bibliography 140

x

List of Figures

1.1 Schematic example of a PLC used for laser modulation. . . . . . . . . . . 2

1.2 Comparison of two JDSU 100 GHz multiplexer/demultiplexer units. . . . 3

1.3 Mapping between quantum and optical domains in the example of wavepacket

spread and optical discrete diffraction. . . . . . . . . . . . . . . . . . . . 6

1.4 Schematic of a straight waveguide array to demonstrate discrete diffraction. 6

1.5 Mapping between quantum and optical domains in the example of Bloch

oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Schematic of a waveguide array with constant curvature to demonstrate

Bloch oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7 Schematic of a waveguide array with an ac curvature to demonstrate dy-

namic localization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.8 PLC Microresonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.9 Cascaded microresonators. . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.10 Measured responses for first-, third-, and 11th-order CROW filters. . . . 12

1.11 Normalized power spectra at the drop-port for 1 and 4 ring SCISSOR. . 13

1.12 Sub-bit delay using the stimulated Brillouin scattering process. . . . . . . 17

1.13 PLCs for optical delays: a comparison. . . . . . . . . . . . . . . . . . . . 18

1.14 Switchable and tunable PLC delay schematic (20 mm x 60 mm chip) using

thermo-optic switches and large ring resonators. . . . . . . . . . . . . . . 19

1.15 A directional coupler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

xi

1.16 Waveguide schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.17 Modal confinement within the waveguide. . . . . . . . . . . . . . . . . . . 23

1.18 Power transfer as a function of propagation distance in a directional coupler. 23

1.19 Normalized transmission of the fabricated nanowire directional coupler

used in the SCISSOR experiments. . . . . . . . . . . . . . . . . . . . . . 24

1.20 Discrete diffraction at the output of a linear waveguide array: measured

(solid line) and simulated (dotted line). . . . . . . . . . . . . . . . . . . . 25

1.21 Nonlinear switching through a directional coupler at low and high intensities. 26

2.1 Coupled Waveguides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 (a) Single-mode waveguide, and (b) its fundamental spatial mode com-

puted using FDMA (“MODE Solutions”). . . . . . . . . . . . . . . . . . 33

2.3 Two-waveguide system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Two lowest-order supermodes of a two-waveguide system. . . . . . . . . . 34

2.5 Dispersion relation for the two lowest-order supermodes of the DC used

in the OCDL of Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 Dispersion relation for the OCDL of Chapter 6. Solid lines are the disper-

sion relation of the two supermodes; the points are the two eigenfrequencies

computed from the restricted basis analysis. . . . . . . . . . . . . . . . . 39

3.1 The primitive cell of a zinc-blende type crystal lattice. . . . . . . . . . . 43

3.2 AlGaAs material dispersion as a function of aluminum content. . . . . . 45

3.3 AlGaAs half bandgap wavelength versus aluminum content. . . . . . . . 46

3.4 The measured AlGaAs values of n2 for TE and TM polarizations. . . . . 46

4.1 Schematic of the CCOW structure . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Simulation of the beam divergence (discrete diffraction) in a straight wave-

guide array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xii

4.3 Simulation of Bloch oscillations in a waveguide array with a constant radius

of curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Simulation of dynamic localization in a waveguide array with an ac square-

wave curvature profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5 Simulation of the lowest order TM supermode of a two-waveguide DC

using the CCOW design parameters. . . . . . . . . . . . . . . . . . . . . 61

4.6 (a) 2D BPM simulation and (b) one-band Schrodinger model of EDL in a

four-period non-NNTB CCOW array. . . . . . . . . . . . . . . . . . . . . 63

4.7 One-band Schrodinger model of ADL breakdown in a four-period non-

NNTB CCOW array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.8 SEM of cleaved end-facet of EDL chip. . . . . . . . . . . . . . . . . . . . 65

4.9 Photomask layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.10 Bloch oscillation experiments using the staggered and output-slab mea-

surement techniques to test beam mapping and accurate beam-relocalization

observation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.11 Bloch oscillation experiment results. . . . . . . . . . . . . . . . . . . . . . 69

4.12 (a) Measured and (b) simulation results of the EDL wavelength depen-

dence at v/Λ = 4 from 1480 nm to 1600 nm. . . . . . . . . . . . . . . . . 70

4.13 (a) Captured image from one of the 21 experiments (plotted on a linear

scale); (b) Measured and (c) one-band Schrodinger simulations of one full

period around the second EDL plane at 1550 nm. . . . . . . . . . . . . . 72

4.14 Smoothing of square wave. . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.15 Deviated square-wave ac-field profiles for future EDL demonstrations. . . 75

5.1 Schematic of a five-ring SCISSOR. . . . . . . . . . . . . . . . . . . . . . 78

5.2 Unit cell of SCISSOR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3 Nanowire loss versus etch depth. . . . . . . . . . . . . . . . . . . . . . . . 81

5.4 Fundamental mode of a 500 nm wide nanowire with a 2.0 µm etch depth. 82

xiii

5.5 Mask design of the 11 µm and 21 µm nanowire directional couplers (wave-

guides are coloured white). . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.6 Dispersion relation for the SCISSOR. Solid lines are the dispersion relation

of the two supermodes; the points are the two eigenfrequencies computed

from the restricted basis analysis. . . . . . . . . . . . . . . . . . . . . . . 84

5.7 Fundamental mode of a bent 500 nm wide nanowire with a 2.0 µm etch

depth and a 5.25 µm bend radius. . . . . . . . . . . . . . . . . . . . . . . 85

5.8 SEM closeup of one racetrack. . . . . . . . . . . . . . . . . . . . . . . . . 86

5.9 SEM of fabricated eight-ring SCISSOR. . . . . . . . . . . . . . . . . . . . 86

5.10 Effective coupling length of nano-DC in the SCISSOR. . . . . . . . . . . 87

5.11 Lowest order TM supermode of a nanowire directional coupler with 570 nm

wide nanowires, separated by 130 nm. . . . . . . . . . . . . . . . . . . . . 88

5.12 Normalized transmission, TA and TB, of the nanowire directional coupler

with 570 nm wide nanowires, separated by 130 nm. . . . . . . . . . . . . 88

5.13 Measured and simulated reflection spectra of the one-, two-, four- and

eight-ring SCISSOR devices. . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.14 Group velocity dispersion versus nanowire width at 1550 nm, illustrating

region of anomalous dispersion for TE polarization. Solid lines are based

on simulations; circles are experimentally measured data points. . . . . . 92

6.1 Behaviour of a nonlinear directional coupler versus propagation distance. 96

6.2 NLDC switching curve for a half-beat length coupler. . . . . . . . . . . . 97

6.3 Schematic of the undelayed-default OCDL. . . . . . . . . . . . . . . . . . 98

6.4 Schematic of the delayed-default OCDL. . . . . . . . . . . . . . . . . . . 98

6.5 Lowest-order TE supermode of the NLDC. . . . . . . . . . . . . . . . . . 99

6.6 SEM top view of the OCDL’s nonlinear directional coupler. . . . . . . . . 100

6.7 SEM of the OCDL waveguide cross section. . . . . . . . . . . . . . . . . 101

xiv

6.8 Calculated switching curves of the fabricated NLDC and a half-beat length

NLDC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.9 The inevitable delay of the low intensity wings of a temporal pulse. . . . 102

6.10 Schematic of the OCDL experimental setup. . . . . . . . . . . . . . . . . 103

6.11 Low-power autocorrelation of the OPO pulse. . . . . . . . . . . . . . . . 104

6.12 Cross-correlation of device output and OPO idler. . . . . . . . . . . . . . 105

6.13 Cross-correlation of 1.5 ps OPO input pulse. . . . . . . . . . . . . . . . . 106

6.14 PD response of 1.5 ps OPO pulse with no device in the optical path. . . 106

6.15 Device output using the PD with a 1550 nm TE launch, 1.5 ps input pulse.108

6.16 OCDL simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.17 Normalized switching ratios from 1530 nm to 1610 nm. . . . . . . . . . . 109

6.18 Wavelength dependency of the 50% crossover intensity. . . . . . . . . . . 109

6.19 Spectral broadening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.20 Independent control pulse to reduce spectral broadening. . . . . . . . . . 111

6.21 Switching curves for low-power 150-ps signal pulse, and high-power 800-ps

control pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.22 Schematic of the second-generation OCDL. . . . . . . . . . . . . . . . . . 113

A.1 AlGaAs crystal planes. Blue cube is the unit-cell. . . . . . . . . . . . . . 125

B.1 SEM of anisotropic etch in 24/18/24 AlGaAs. . . . . . . . . . . . . . . . 133

B.2 Determination of isotropic etch rate. . . . . . . . . . . . . . . . . . . . . 134

B.3 SEM of isotropic etch in 24/18/24 AlGaAs. . . . . . . . . . . . . . . . . . 135

B.4 SEM of isotropically etched waveguide bend. . . . . . . . . . . . . . . . . 135

B.5 Simulation results of lowest order TE supermode of an isotropically etched

directional coupler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.6 Isotropic etch mode around a 2 mm radius bend. . . . . . . . . . . . . . 136

B.7 Hard baked PR with no crosslinking. . . . . . . . . . . . . . . . . . . . . 137

xv

B.8 Hard baked PR with crosslinking. . . . . . . . . . . . . . . . . . . . . . . 137

xvi

List of Tables

1.1 Mapping between the Schrodinger equation and the electromagnetic wave

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Linear and nonlinear indices of AlGaAs, Si, SiO2 and PMMA polymer. . 28

3.1 AlGaAs Sellmeier Coefficients at T = 296 K . . . . . . . . . . . . . . . . 44

xvii

List of Abbreviations

2D Two Dimensional

3D Three Dimensional

ADL Approximate Dynamic Localization

AlGaAs Aluminum Gallium Arsenide

APF All Pass Filter

AWG Arrayed Waveguide Grating

BBO beta-BaB2O4

BER Bit Error Rate

BO Bloch Oscillation

BPM Beam Propagation Method

CCD Charged Coupled Device

CCOW Curved Coupled Optical Waveguides

CMT Coupled Mode Theory

CROW Coupled Resonator Optical Waveguides

CVWE Complex Vector Wave Equation

CW Continuous Wave

DC Directional Coupler

dB Decibels

DL Dynamic Localization

DSW Deviated Square Wave

DI De-ionized

EDL Exact Dynamic Localization

FDMA Finite Difference Modal Analysis

FDTD Finite Difference Time Domain

FSR Free Spectral Range

FWHM Full Width Half Maximum

HCME Hamiltonian Coupled Mode Equations

HCMT Hamiltonian Coupled Mode Theory

IR Infrared

xviii

ITRS International Technology Roadmap for Semiconductors

JDSU JDS Uniphase Corp.

Lc Half-beat Coupling Length

LHS Left Hand Side

MIT Massachusetts Institute of Technology

MPB Multiple Photonic Bands

nano-DC Nanowire Directional Coupler

NCVWE Nonlinear Complex Vector Wave Equation

NL Nonlinear

NLDC Nonlinear Directional Coupler

NNTB Nearest Neighbour Tight Binding

NRC National Research Council

OCDL Optically Controllable Delay Line

OPL Optical Path Length

OPO Optical Parametric Oscillator

OR Optical Rectification

OSlab Output Slab Waveguide

PBG Photonic Bandgap

PECVD Plasma Enhanced Chemical Vapour Deposition

PD Photodetector

PLC Planar Lightwave Circuit

PMMA Poly(methyl Methacrylate)

PR Photoresist

PWE Paraxial Wave Equation

RF Radio Frequency

RHS Right Hand Side

RIE Reactive Ion Etching

SCISSOR Side Coupled Integrated Spaced Sequence of Resonators

QBO Quasi-Bloch Oscillations

SEM Scanning Electron Microscope

SHG Second Harmonic Generation

TE Transverse Electric

TM Transverse Magnetic

TPA Two Photon Absorption

UV Ultraviolet

WDM Wavelength Division Multiplexing

xix

Chapter 1

Introduction

1.1 Planar Lightwave Circuits

Planar lightwave circuits (PLCs) are optical devices that control and route light-signals

along prescribed pathways through a microchip.

PLCs are ideally suited for optical signal generation and processing, which employ

optical waveguides to confine and steer light through on-chip processing elements such as

power splitters, interferometers, switches, and modulators [1, 2]. Because of the ability

to amalgamate these components onto a single substrate, PLC technology is also called

integrated optics. To contrast, traditional free-space optics routes and controls light using

bulk optical components such as lenses, gratings, crystals, prisms and mirrors [2, 3].

An example of a PLC device is shown in Fig. 1.1. This simplified PLC telecommuni-

cations transmitter chip modulates the light emitted from an external continuous-wave

(CW) laser and subsequently couples the signal into an optical fiber after a portion of

the light is tapped and redirected to a photodetector for power monitoring purposes.

This example highlights several important features of PLCs: light remains confined in

the waveguide as it propagates through the device, optical-processing components can

be directly fabricated on the chip, and hybridization with other devices is possible.

1

Chapter 1. Introduction 2

Laser

Photodetector

Optical Fiber

PLC

Substrate

Optical

Waveguide

On-chip

Modulator

Power

Tap

Figure 1.1: Schematic example of a PLC used for laser modulation.

1.1.1 Integrated versus Free-Space Optical Devices

In free-space optical systems, operations such as switching, attenuation, and diffraction

are typically performed in the focal and/or collimated planes, where the beam’s wavefront

is planar [2]. The region between these planes, where the propagating beam diverges or

converges, is seldom used for any optical signal processing. To contrast, in integrated-

optics, the light is literally confined into one (or several) modes of the waveguide. Each

mode, as it propagates through the waveguide, retains its shape, its size, and most

importantly, its planar wavefront; hence, the light is in a “processing-receptive” state

along its entire journey through the PLC.

Furthermore, PLCs offer a robust monolithic platform for optical devices because the

routing and functional components are integrated onto the same wafer. Typically PLC

devices require only temperature control and hermetic sealing. Free-space optical de-

vices, in contrast, suffer from the daunting task of assembling all pre-mounted bulk optic

components into a thermally stable and hermetically sealed optomechanical package.


8 Channel 100 GHz

Mux/Demux

Free-Space Optics

126 x 93 x 10 mm

40 Channel 100 GHz

Mux/Demux

PLC (AWG)

135 x 65 x 14 mm

(a) (b)

Spherical Mirror

LC/

MEMS

Grating

Front End

Figure 1.2: Comparison of two JDSU 100 GHz multiplexer/demultiplexer units: (a) builtusing PLCs, (b) built using free-space optics.

A comparison of two 100 GHz mutiplexer/demultiplexer units (JDS Uniphase) is

shown in Fig. 1.2, built using (a) arrayed waveguide grating (AWG) PLC technology,

and (b) laser-welded free-space optics. Beneath the picture of the PLC package in figure

(a) is a picture of the actual AWG PLC; beneath the free-space optical package in figure

(b) is a schematic of the bulk components and the beam-paths through the device. While

the dimensions of both packages are similar, the AWG PLC unit is capable of handling 40

WDM channels, while the laser-welded free-space unit is only capable of handling eight.

As illustrated by the above example, integrated optical devices provide a robust plat-

form upon which to perform signal processing, allowing for increased functional density,

reduced device sizes, and improved manufacturability.


1.1.2 PLC Applications

PLCs have already found use in many commercial applications, primarily driven by the

growth in the optical telecommunications industry. Although the telecommunications

“bubble-burst” has indeed impeded the deployment of photonics equipment throughout

the world, the need for improved system performance has not abated. In fact, internet

traffic grew 115% per year from 1998 to 2004, and has since grown 45% per year [4].

This growth has pressured telecommunications companies to transition from traditional

free-space optical approaches to integrated architectures, an identical move made decades

ago by the electronics industry: bulk electronic to integrated electronic circuits. These

integrated optic devices have been the focus of intense research and development, result-

ing in practical PLCs now deployed in microchip interconnects, bio and chemical sensing,

microspectroscopy, and medical instrumentation [1]. At the present time, there are many

companies commercially developing PLCs, including JDS Uniphase, Infinera, Gemfire,

IBM, and Intel.

1.1.3 Chapter Overview

In this dissertation, we explore PLC research challenges for spatial, spectral and temporal

optical signal processing. In Section 1.2, we investigate the analog between electromag-

netic and quantum theories to demonstrate the unique spatial optical effect called exact

dynamic localization; in Section 1.3, we look at Bragg-featured microring filters using

sub-micron waveguides; in Section 1.4, we study switchable optical delay lines for micro-

chip interconnects. A summary of these challenges (in Section 1.5) identifies a common

theme: the use of coupled waveguides in the aluminum gallium arsenide material sys-

tem. An overview of coupled waveguides is provided in Section 1.6, and a description of

the benefits of the aluminum gallium arsenide material system for PLCs is presented in

Section 1.7. The chapter concludes with the thesis organization in Section 1.8.


1.2 PLCs for Spatial Optical Signal Processing:

Exact Dynamic Localization

1.2.1 Mapping between Quantum and Optical Domains

Analysis of quantum mechanical phenomena is often performed using Schrodinger’s equa-

tion [5], with time as the independent variable. Analysis of electromagnetic wave (optical

beam) propagation through PLC waveguides is often performed using the linear wave

equation, derived from Maxwell’s equations [6], with distance as the independent vari-

able. A comparison of these two formulae reveal a remarkable mathematical correspon-

dence, highlighted in Table 1.1, where the respective parameters can be directly mapped

(via scaling factors) from the quantum domain to the optical domain [7]. This mapping

allows for many interesting quantum effects in periodic superlattices to be demonstrated

in the optical domain using PLCs. This is particularly important in the study of tempo-

rally dynamic quantum systems, which are very difficult to observe empirically because

of electron-electron and electron-phonon interactions (which tend to significantly dete-

riorate the coherence of electron wavepackets [7]), and the necessity of using indirect

measurement techniques (such as four-wave mixing [8, 9] or terahertz emission [10–12]).

Furthermore, the quantum-to-optical analogy gives us new glasses with which to look at

optical phenomena in PLCs.

Table 1.1: Mapping between the Schrodinger equation and the electromagnetic waveequation

Schrodinger’s Equation EM Wave Equation

time to propagation distancepotential well profile (U(u)) to refractive index profile (n(u))

applied electric field (E) to waveguide curvature (1/R)


1.2.2 Discrete Diffraction

To illustrate the mapping between the quantum and optical domains, let us consider

the case of an electron initially localized (at t = 0) in one of an array of coupled finite

quantum wells, as shown in Fig. 1.3(a) (where u is the lateral spatial dimension, and

U(u) is the periodic quantum potential). Over time, the wavepacket spreads to occupy

a spatially wider lateral extent. A schematic of the analogous optical system is shown in

Fig. 1.4. Light, originally localized in the central waveguide of a waveguide array, spreads

via a mechanism called discrete diffraction, over the propagation distance, v [13]. This is

illustrated in Fig. 1.3(b), where n(u) is the lateral periodic profile of the array’s material

index. The coupling of waveguides is described in more detail in Section 1.6.

t

Figure 1.3: Mapping between quantum and optical domains in the example of wavepacketspread and optical discrete diffraction.

Figure 1.4: Schematic of a straight waveguide array to demonstrate discrete diffraction.


1.2.3 Bloch Oscillations

A static dc electric field applied across the quantum array effectively tilts the band

profile of the potential, U(u), as shown in Fig. 1.5(a), consequently causing the beam

to periodically relocalize in time. This effect, called Bloch oscillations [12, 14], can be

observed in the optical domain by applying a curvature to the waveguides (Table 1.1) [15].

This waveguide curvature conformally maps a tilt in the effective index profile, n(u) [16],

resulting in the relocalizing behaviour (as a function of propagation distance, v) shown in

Fig. 1.5(b). A schematic of the constant-curvature waveguide array is shown in Fig. 1.6.

t

v

Figure 1.5: Mapping between quantum and optical domains in the example of Blochoscillations.

Figure 1.6: Schematic of a waveguide array with constant curvature to demonstrate Blochoscillations.


1.2.4 Dynamic Localization

With the application of a periodic alternating (ac) electric field to the quantum system,

the electron wavepacket can also experience periodic relocalizations, called dynamic lo-

calizations (DL) [17]. While Bloch oscillations in the quantum domain have been demon-

strated, quantum observations of DL are extremely challenging because the required ac

fields are very difficult to generate: sub-ps ac periods (i.e., shorter than the decoherence

time of the electron), field amplitudes on the order of 10 kV/cm, and discontinuities at

every sign change (e.g., a square wave field) [7, 18].

Figure 1.7: Schematic of a waveguide array with an ac curvature to demonstrate dynamiclocalization.

A schematic of the equivalent ac system in the optical domain is illustrated in Fig. 1.7,

where the ac electric field is mapped onto a periodic oscillation of the waveguide curvature

along the direction of propagation. While attempts to observe DL in the optical domain

have been recently reported by Longhi et al. in 2006 [19], continuous ac waveguide curva-

tures were used (corresponding to continuous ac fields in the quantum domain), with only

weak waveguide coupling. Dignam et al. have formally stated that in systems without

discontinuities, the beam relocalization formally breaks down after one period (which is


clearly seen in systems with strong coupling), thus claiming that the continuous-system

is therefore only an approximation to true, or exact, dynamic localization [18]. Further-

more, the wavelength dependence of Longhi’s device was only qualitatively characterized,

and the spatial mapping of the dynamic localization was not actually performed.

To date, no exact dynamic localization has been reported in any system.

1.2.5 The Exact Dynamic Localization Challenge

The above discussion in Section 1.2.4 addresses a specific technological challenge that is

formally stated here:

Can a PLC be designed to demonstrate the exact quantum dynamic local-

ization (EDL) effect over several relocalization periods? If so, the spatial

mapping of the beam evolution through the device must be performed, along

with the characterization of the wavelength dependence (with significantly

improved resolution compared with Longhi’s earlier work [19]).

1.2.6 The Exact Dynamic Localization Solution

EDL Requirements

To address the EDL challenge, (i.e., in order to ensure that exact dynamic localization

would be observed in the optical domain), the following PLC design requirements were

defined: discontinuous ac waveguide curvature profiles as defined by Dignam [18], a device

spanning multiple periods, and strong waveguide coupling.

EDL in PLCs could be demonstrated, in principle, in any optical material system,

including silicon, glass, polymer, or compound semiconductors.


EDL Description

The first empirical performance of EDL in any system (e.g., quantum, atomic, or optical)

is reported. The PLC demonstration employed a discontinuous “square-wave” field to

describe the waveguide curvature, and a strongly coupled waveguide design using an

aluminum gallium arsenide PLC wafer. Using newly developed observation techniques,

the devices’ spatial and spectral dynamics were observed with a fine resolution of 250 µm

and 10 nm, respectively, over four full EDL periods.

The devices demonstrated excellent EDL behaviour over the four periods, and in very

close agreement with design targets and simulations. The EDL design details and device

characterization are presented in Chapter 4.

This work was recently published in Optics Express in 2007 [20]1, and presented at

QELS’07 in Baltimore, MD [21].

1.3 PLCs for Spectral Optical Signal Processing:

SCISSORs

1.3.1 Microresonators

Telecommunication optical signal processing functions, such as filtering, switching, wave-

length conversion, and add-dropping were originally performed using free-space optics or

opto-electro-opto conversions [22–25]. These same functions have been recently demon-

strated in all-optical PLCs using four-port microring resonators, also called microres-

onators [26–29], which are simply waveguides patterned into a loop to provide coherent

feedback, evanescently coupled to two straight waveguides, as illustrated in Fig. 1.8.

Akin to the integrated electronic transistor, microresonators have garnered much

recent attention within the past decade because of their compact size (on the order

1Reprinted with permission from the Optical Society of America. Copyright c© 2007.


Figure 1.8: PLC Microresonator.

of tens of microns), their ease of integration with other on-chip components [30–32],

and their cascadability in both parallel and serial configurations. Schematic layouts of

both the parallel configuration (also known as “side coupled integrated spaced sequence

of resonators”, or SCISSORs), and the serial configuration (also known as “coupled

resonator optical waveguides”, or CROWs), are illustrated in Fig. 1.9.

Through

port

Drop portIn port

Through

port

Drop

port

In port

Figure 1.9: Cascaded microresonators: (a) five-ring parallel configuration (SCISSOR),(b) two-ring serial configuration (CROW).


1.3.2 Serial CROWs

Serial (CROW) microresonator devices are ideal for high-order filtering, and have been

the focus of intense study over the past decade [32–42]. Because of the resonant prop-

erties of the microresonator, only the off -resonant wavelengths are transmitted via the

through port, while the on-resonant wavelengths are emitted via the drop port. The filter

shape of the CROW device improves as a function of the number of rings, as illustrated

in Fig. 1.10 [41]. As such, the device possesses a set of band-gaps at the resonant wave-

lengths, defined by the refractive index, the waveguide geometry, and the path-length of

each ring resonator.

Figure 1.10: Measured responses for first-, third-, and 11th-order CROW filters. The re-sponses have been normalized to their 3-dB bandwidths. The dashed curve is the theoreticalfit to the 11th-order filter [41].

1.3.3 Parallel SCISSORs

When configured in a parallel (SCISSOR) configuration, the device’s filter spectrum is

affected by both the ring geometry as well as the ring-to-ring spacing. As in the CROW

devices described above, a set of resonator band-gaps created by the ring geometry is

created. With a large number of rings, a second class of band-gaps is evoked in the


1555 1560 1565 1570 1575 1580 1585 15900

1

Wavelength (nm)

Norm

aliz

ed D

rop P

ort

Pow

er

.

4 ring SCISSOR1 ring SCISSOR

Formation

of the Bragg

features

Figure 1.11: Normalized power spectra at the drop-port for 1 and 4 ring SCISSOR.

SCISSOR spectrum: the Bragg gaps, which are created by the coherent backscattering

from each ring to the drop port. Fig. 1.11 shows the resonator peaks that are observed in

the drop port spectrum (see Fig. 1.9) of a lossless 1-ring SCISSOR, and the formation of

the Bragg features in a 4-ring SCISSOR. With a sufficient number of rings, the intensity

of the Bragg and resonator peaks would be identical.

Careful design of the interplay between the two classes of band-gap resonances would

be extremely useful for optical dispersion engineering (such as dispersion compensators

and slow-light devices) [43,44]. Furthermore, if fabricated in materials with high optical

nonlinearity, these devices could be used to demonstrate optical switching, limiting, tem-

poral solitons, gap solitons, pulse compression [43,45,46] and optical logic [47].

While a 36-period SCISSOR in an effectively linear polymer has been recently re-

ported [48], SCISSORs in highly nonlinear III-V semiconductors have been limited to a

maximum of only three rings [26, 27] due to extremely tight fabrication tolerances (see

Section 1.4).


1.3.4 The SCISSOR Challenge

Can a SCISSOR PLC device with more than three rings be fabricated suc-

cessfully in a highly nonlinear III-V semiconductor? If so, how many rings are

necessary to empirically demonstrate a fully formed Bragg gap (i.e., where

the Bragg and resonator features at the drop port are of equal intensity)?

1.3.5 The SCISSOR Solution

SCISSOR Device Requirements

The multi-ring SCISSOR challenge is primarily a design-and-fabrication effort. High

index-contrast waveguides with sub-micron widths (also known as nanowires) are required

to mitigate bend losses. Substrate leakage losses should be mitigated by an adequately

deep etch. Non-circular rings (e.g., racetracks) could be used to improve the control of

coupling into each ring from the straight bus waveguide [28]. The free-spectral range of

the interlaced Bragg and resonator spectral features should be on the order of tens of nm

to cover as much of the telecommunications band as possible.

Of the many material systems available, aluminum gallium arsenide exhibits both

excellent optical and mechanical properties, well suited for this project. An overview of

these properties is presented in Section 1.7.

SCISSOR Device Description

The nanowire waveguides were designed and fabricated into a set of directional couplers

(i.e., two coupled waveguides) in aluminum gallium arsenide, performing in excellent

agreement with simulations. The SCISSOR devices were designed using one, two, four,

and eight racetrack-shaped ring resonators. The fabricated devices were subsequently

characterized to reveal that eight rings is sufficient for the formation of interlaced Bragg

and resonator peaks in the reflection spectrum of commensurate intensity. The design


details and characterization of the nanowire directional couplers and SCISSOR devices

are presented in Chapter 5.

This work was presented at CLEO’06 in Long Beach, CA [49].

1.4 PLCs for Temporal Optical Signal Processing:

Optical Delay Lines

We have seen that PLCs are useful for optical signal processing in both the spatial domain

(Section 1.2), and the frequency domain (Section 1.3). Here, we discuss signal processing

in the temporal domain.

1.4.1 PLC Microchip Interconnects

Computing systems are placing increasingly heavier demands on their supporting technol-

ogy. By 2013, on-chip data rates are projected to reach between 40 Gbit/s to 1 Tbit/s [50,

51]. The International Technology Roadmap for Semiconductors (ITRS) [52] states,

“For the long term, material innovation with traditional scaling will no longer

satisfy performance requirements. Interconnect innovation with optical, ra-

dio frequency, or vertical integration combined with accelerated efforts in

design and packaging will deliver the solution.”

While free-space optical interconnects can address the ∼ 1 to 10 cm distances required

for board-to-board communications [3], integrated-optic PLCs are well suited for inter-

and intra-chip communications. The use of light instead of electrons for data transmission

is anticipated to significantly improve computer processing performance due to its larger

bandwidth, increased transmission speed, decreased power consumption, and immunity

to electromagnetic noise and temperature changes [53].


1.4.2 Optical Delays

In order to ensure that the optical data streams are properly synchronized, PLC microchip

interconnects must provide multi-bit-switchable optical delays greater than 100 bits [51].

Furthermore, to compensate timing jitter (within 3% of a clock cycle [52]), PLCs must

also provide continuously tunable delays between 0 and 1 bit.

To address pulse timing issues, optical group velocity reduction experiments in atomic

media and fibers have reported large delays on the order of tens of ns [50,51,54–57]. These

systems, however, (1) are physically too large for microchip communciations, (2) provide

delays on the order of approximately one bit and hence are not useful for switchable

buffering applications, and (3) have spectral bandwidths orders of magnitude too small

for even 40 Gbit/s data rates.

An example of a sub-bit delay using the stimulated Brillouin scattering process in

optical fiber with a 63 ns long input pulse is shown in Fig. 1.12 [56]. As illustrated, small

bit-delays with long pulses (i.e., narrow bandwidth) in a large experimental setup do not

meet the requirements for microchip interconnects.

1.4.3 PLC optical delays: Resonators versus Differing-Length

Waveguides

PLC technology is a natural choice for microchip interconnects. Two methods of on-chip

delays have been proposed: exploiting group-delay effects in microresonators, and routing

the data along waveguides of differing length. Resonant components on planar optical

circuits (e.g., microrings (see Section 1.3), microdisks, and photonic crystal microcavities)

are particularly interesting for optical pulse delays, and have been the focus of intense

study over the past decade [31, 33, 35–37, 40, 41, 49, 58–69]. Xia, in 2006, reported an

excellent comparison of three on-chip optical delay devices: an all pass filter (APF)

with 36 ring resonators (Fig. 1.13(a)), a “coupled resonator optical waveguide” (CROW)


Figure 1.12: Sub-bit delay using the stimulated Brillouin scattering process: (a) experi-mental setup; (b) measured delay [56].

configuration with 100 ring resonators (Fig. 1.13(b)), and a simple non-resonant 4 cm

waveguide delay (not shown) [58]. The sub-bit delays achieved from each configuration

for a 1 Gbit/s (i.e., 1 ns) signal are shown in Fig. 1.13(c). The bit error rate (BER) of

each device was measured as a function of data rate, revealing that the CROW device

performed most poorly, while the non-resonant 4 cm waveguide allowed for the highest

data rates below the BER threshold of 10−9 (Fig. 1.13(d)). The poor performance of the

CROW and APF configurations was attributed to attenuation in the rings, producing

device losses over 22 dB, while the 4 cm waveguide approach boasted a device loss of

only 7 dB.

While the resonator-based PLC devices can be made remarkably small (less than


CROW

Bent wg

APF

Figure 1.13: PLCs for optical delays: a comparison: (a) 56-ring all pass filter (APF)configuration; (b) 100-ring CROW configuration; (c) Time resolved measured delays com-paring (i) CROW, (ii) 4 cm long waveguide, and (iii) APF configurations; (d) Bit error ratecomparison [58].

∼ 0.1 mm2), their losses limit practical delays to less than 100 bits [51]. Thus, until

microresonator device losses can be reduced significantly, the non-resonator waveguide

approach promises to offer the best solution for large delays (> 100 bits) in interconnects

operating at the 40 Gbit/s to 1 Tbit/s data rates.

1.4.4 Optical Delay Switching

Optical PLCs require the ability to activate the delay only if desired (i.e., delay switch-

ability) with switching times on the order of a bit-length (to minimize data loss and

latency). In microresonator based delays, switching must be accomplished either by the

non-trivial task of shifting the signal wavelength, or by shifting the material index via

thermo-optical, electro-optical or nonlinear effects.

Recently, Rasras et al. reported a PLC demonstration incorporating both tunable

delays (using ring resonators with a 1 mm radius of curvature) and variable switch-


able delays (using varying-length curved waveguide delay lines and thermo-optic Mach-

Zehnder switches) (see Fig. 1.14) [70]. The device was fabricated in low index, medium

index-contrast waveguides on a relatively large 20 mm x 60 mm chip. While the device

was capable of performing tunable delays between 0 and 320 ps and switchable delays

up to 2.56 ns, the 1 mm ring radii actually places an upper -limit of only 10 Gbit/s on

the data rate (i.e., a lower limit of 100 ps on the pulse length). The device, therefore,

would be capable of delaying, at most, only ∼ 29 bits. The device’s switching speed was

not reported, but thermal switching is a relatively slow process. For example, a recently

reported thermo-optic switch was limited to switching times of only ∼ 10 ms [71].

Figure 1.14: Switchable and tunable PLC delay schematic (20 mm x 60 mm chip) usingthermo-optic switches and large ring resonators [70].

1.4.5 The Optical Delay Line Challenge

Can an PLC switchable optical delay line be fabricated with the following

specifications: greater than 100-bit delay, capable of handling pulse widths

on the order of 1 ps, a chip size of ∼ 10 mm × 10 mm (i.e. the size of

a standard microelectronics chip), relatively low loss (i.e., much less than

20 dB), and a switching time on the order of 1 ps or less?


1.4.6 The Optical Delay Line Solution

Optical Delay Line Device Requirements

Our discussion in Section 1.4.3 stated that the differing-length waveguide approach is,

at present, the best option for low loss optical delays [58]. Ultrafast, low-loss switching

is possible using a material with a strong Kerr nonlinearity [30, 72], while a high-index

material would minimize the size of the PLC [2]; thus, aluminum gallium arsenide is an

ideal material system for this device. This device should easily fit on a 10× 10 mm chip

providing > 100-bit delays for ∼ 1 ps pulses.

The Kerr-based ultrafast switch itself could be implemented using an intensity-dependent

nonlinear directional coupler [73].

Optical Delay Line Device Description

An all-optically self-switchable delay line was built in aluminum gallium arsenide on a

6 mm x 8 mm chip. The device boasted a delay of 126 bits (i.e., a 189-ps delay using

1.5 ps pulses) and losses of only 6.2 dB in the delayed state. The design details and

device characterization are presented in Chapter 6.

This work was presented at OFC’07 in Anaheim CA [74], published in Applied Physics

Letters in 2007 [75]2, and recently highlighted in Nature Photonics in May, 2007 [76].

1.5 The Challenges and Solutions: Discussion

Sections 1.2, 1.3, and 1.4 identified three research challenges in PLC technology for spa-

tial, spectral and temporal optical signal processing, namely exact dynamic localization,

the multi-ring SCISSOR, and the ultrafast, switchable optical delay line, respectively.

The three proposed solutions share a common theme: the use of coupled waveguides in

2Reprinted with permission from the American Institute of Physics. Copyright c© 2007.


the aluminum gallium arsenide material system.

In the following two sections, we review the concepts behind light coupling in wave-

guides, and explore why aluminum gallium arsenide is an excellent material for PLC

design.

1.6 Light Coupling in Waveguides

We begin our discussion of light coupling in waveguides by looking at a two-waveguide

coupled system, otherwise known as a directional coupler (DC) (see Fig. 1.15). DCs are

ubiquitous devices in which light, initially confined in one waveguide, is transferred into

a second adjacent waveguide. Interestingly, and perhaps non-intuitively, although the

second waveguide does not actually touch the first, 100% power transfer is achievable.

Because of their simplicity and elegance, DCs support a host of higher level optical

functions, such as Mach Zehnder interferometers [77], power distribution [78], power mon-

itoring [79], wavelength division multiplexing [80], filtering [81], and all-optical switch-

ing [73]. In the following subsections, we discuss light confinement in a single waveguide,

the coupling of light from one waveguide to another, coupling in a multi-waveguide sys-

tem, and how, in nonlinear materials, this simple device operates as an all-optical switch.

Figure 1.15: A directional coupler.


1.6.1 Light Confinement in a Single Waveguide

We have alluded to the confinement of light in the earlier sections of this chapter and

now investigate this fundamental property of waveguides further.

Fig. 1.16 shows a schematic of a single waveguide. The vertical layer structure (defined

by the layer thicknesses and refractive indices) provides the vertical y-confinement of the

light within the waveguide (in the core layer, surrounded above and below by the cladding

layers), while the ridge width, w, and etch depth, h, provide the horizontal x-confinement.

This two-dimensional confinement can be formally described by Maxwell’s equations [6]

to determine the transverse spatial distributions, or modes, of the light confined within

the waveguide. The two lowest order modes of our waveguide example are shown in

Fig. 1.17, each defined by a unique effective index, n, which is, in effect, a weighted

average of the material indices “seen” by the mode. The number of modes supported by

a waveguide is determined primarily by the ridge width, w, which is typically chosen to

support only the fundamental mode, producing a single-mode waveguide. If there are no

external perturbations, each mode retains its shape and size as it propagates along the

chip.

Lowercladding layer

Core layer

Upper

Cladding layer

Figure 1.16: Waveguide schematic.


21

Figure 1.17: Intensity modal confinement within the waveguide: (a) fundamental mode,(b) first higher-order mode.

1.6.2 Light Coupling in a Two-Waveguide System

Consider light initially confined in a solitary, single-mode waveguide, A. The presence

of a second waveguide, B, in close proximity to A (as illustrated in Fig. 1.15), actually

perturbs A’s supported single mode. The effect of this perturbation is a transfer, or

coupling, of energy from waveguide A to waveguide B, as a function of propagation

distance (see Fig. 1.18). If the waveguides are identical, 100% of the initial power in A

will be transferred to B at odd-multiples of the half-beat length, Lc.

0 Lc 2Lc 3Lc

0

0.2

0.4

0.6

0.8

1

z

No

rma

lize

d P

ow

er i

n

Wa

veg

uid

e A

All power

in A

All power

in B

Figure 1.18: Power transfer as a function of propagation distance in a directional coupler.


A nanowire DC (nano-DC) was tested at varying lengths for use in the SCISSOR

device. The performance of the nano-DC as a function of propagation distance is shown

in Fig. 1.19. A sub-half-beat length nano-DC was used for the SCISSORs to transfer

28% of the on-resonant light from the straight waveguides to each ring.

Directional Coupler Length ( m)

Norm

aliz

ed T

ransm

issio

n

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

Channel A (Experimental)

Channel A (Fit)

Channel B (Experimental)

Channel B (Fit)

Figure 1.19: Normalized transmission of the fabricated nanowire directional coupler usedin the SCISSOR experiments.

The coupling between waveguides can be formally described using several different

methods. In Chapter 2, we investigate a classical approach based on Maxwell’s equations

(the supermode ∆k method), and a Hamiltonian formulation of coupled mode theory.


1.6.3 Light Coupling in a Multi-Waveguide System

The behaviour of light propagating through a coupled multi-waveguide array (see Fig. 1.4)

is an intuitive extension of the description provided above for the two-waveguide system:

light, initially confined in a given waveguide is allowed couple to its immediate neighbours.

These neighbouring waveguides then couple light to their neighbours, etc. This cascading

coupling effect through the array is called discrete diffraction (see Section 1.2.2) [13].

In preparation for the optical EDL demonstration, the discrete diffraction of a beam

through a linear waveguide array was tested. The beam, initially confined in waveguide

position 0, propagated a total distance of 5.4 mm; the resulting beam spread (measured

and simulation) is shown in Fig. 1.20. The EDL demonstration imposes an ac curvature

onto the waveguide array as a function of propagation distance, offering an added degree

of control on the beam evolution.

Figure 1.20: Discrete diffraction at the output of a linear waveguide array: measured(solid line) and simulated (dotted line).

1.6.4 The Nonlinear Directional Coupler Switch

When fabricated in a nonlinear medium, a directional coupler exhibits power-dependent

switching characteristics [73, 82–88]. Beginning with a half-beat length coupler (i.e.

whose length is precisely Lc), low-intensity light initially coupled into waveguide A will

be completely transferred to waveguide B. As the intensity of the light is increased,


an intensity-dependent change of the refractive index is induced, which dephases the

coherent coupling of light into B. With sufficient input intensity, the light coupling into

B can be completely eliminated, such that all the light remains in A. In Fig. 1.21, we see

the operation of the half-beat length nonlinear directional coupler (NLDC) at both low

and high intensities, illustrating its ability to perform as an all-optical switch at L = Lc.

An NLDC switch was used to select either the delayed versus the undelayed path through

the optical delay line device.

0

0.2

0.4

0.6

0.8

1

0 Lc

z

No

rma

lize

d P

ow

er i

n

Wa

veg

uid

e A

Lc / 2

High intensity

Low intensity

All power

in A

All power

in B

Figure 1.21: Nonlinear switching through a directional coupler at low and high intensities.

1.7 Aluminum Gallium Arsenide

PLCs can be manufactured in any optically transparent material: amorphous materials

such as polymer [36], and glass [89], as well as crystalline semiconductors such as sili-

con [58], silicon nitride [90], and indium phosphide [91]. While all PLCs are defined by

the vertical wafer structure and the waveguide geometry (see Section 1.6.1), most mate-

rial systems restrict the designer to a fixed refractive index. If the material’s refractive

index could be varied, the PLC designer would gain an added degree of creative freedom.


Aluminum gallium arsenide, AlxGa1−xAs, is a compound III-V cubic zinc-blende crys-

talline semiconductor material system that allows for the replacement of gallium atoms

with aluminum atoms, at an aluminum concentration of x, where 0 ≤ x ≤ 1 [92]. This

atomic replacement has two outstanding features making AlGaAs an ideal PLC mate-

rial: (1) the extremely large dependence of the refractive index with aluminum concen-

tration (∆n = 0.48 at the telecommunications wavelength of 1550 nm [93]) allows for

the selectivity of the wafer layer indices; and (2) over its entire aluminum concentration

range, AlGaAs remains lattice-matched (i.e. the crystalline lattice constant varies only

0.15% [92]), making AlGaAs particularly robust to delamination and cracking. Hence,

PLCs with layers exhibiting a large aluminum contrast can support nearly arbitrary

mode sizes, allowing, for example, improved coupling efficiency with free-space beams or

optical fiber [94].

The optical path length (OPL) traversed by light through a PLC is defined by L×n,

where L is the device length and n is the effective refractive index. For a given OPL,

PLC devices with higher refractive indices require shorter physical lengths. Therefore,

AlGaAs, with its rather large linear index (which can be chosen between 2.89 and 3.37),

is ideal for the design of small PLCs. A comparison of the linear refractive index of

AlGaAs with other commonly used materials (viz. silicon (Si), silica (SiO2), and doped

PMMA polymer) is presented in the second column of Table 1.2.

Furthermore, AlGaAs possesses high optical nonlinearity, which is particularly useful

for active PLCs (such as switches and modulators) [30, 95]. The Kerr nonlinearity, n2

(which is formally introduced in Appendix A), of AlGaAs, Si, SiO2 and doped-PMMA

are compared in the third column of Table 1.2, where we see that AlGaAs outperforms

its competitors by at least a factor of 3.

In this section, we have seen the beneficial properties of aluminum gallium arsenide for

PLC design. Its high linear refractive index minimizes chip sizes; its lattice-matched crys-

talline structure allows for the uninhibited selection of aluminum concentration, (which


Table 1.2: Linear and nonlinear indices of AlGaAs [93,96], Si [97], SiO2 [98] and PMMApolymer [99]

Material n0 n2 (m2/W)

AlGaAs 2.89 to 3.37 1.5× 10−17

Si 3.46 4.5× 10−18

SiO2 1.46 3× 10−20

doped PMMA (polymer) 1.42 3× 10−18

varies its material index by an astonishing ∆n = 0.48); and its large nonlinear refractive

index permits all-optical devices to be built with significantly lower operating thresholds

compared with other commonly used PLC materials.

1.8 Summary and Thesis Organization

In this chapter, we have explored planar lightwave circuitry from a bird’s-eye view, iden-

tifying a PLC research challenge in spatial, spectral and temporal optical signal process-

ing. A solution for each challenge was introduced by a requirements statement and an

overview of the proposed device, each using coupled waveguides in the AlGaAs material

system to achieve the desired functionality.

The thesis is organized as follows. Chapter 2 introduces the classical formulation of

light confinement in waveguides, followed by two studies on the coupling between the two

waveguides of a directional coupler: the supermode ∆k approach, and the Hamiltonian

formulation of the coupled mode equations. Chapter 3 presents some of the useful optical

parameters of the AlGaAs material system, including a description of the two wafer

designs used in this dissertation. Following these preliminary chapters, the solution to

the three individual challenges are presented in detail: the exact dynamic localization

demonstration in Chapter 4, the SCISSOR device in Chapter 5, and the optical delay

line in Chapter 6. Chapter 7 concludes the thesis.

Chapter 2

Light Coupling in Waveguides

In this chapter, we study the central theme embodied in the scope of this dissertation:

light coupling in a two-waveguide system, otherwise known as the directional coupler

(Fig. 2.1). We study this phenomenon using two different methods: by computing the

supermodes of the two-waveguide system in Section 2.2, and by following a Hamiltonian

formulation of coupled mode theory as laid out by Chak et al. [100] in Section 2.3.

The analysis commences in Section 2.1 with the introduction of Maxwell’s equations to

describe the so-called master equation for the modes in an arbitrary waveguide system.

Figure 2.1: Coupled Waveguides.

29

Chapter 2. Light Coupling in Waveguides 30

2.1 Maxwell’s Equations

The governing equations describing electromagnetic waves were formulated by Maxwell

in 1865 [6]. The four equations in differential form (in SI units) are

∇× ~E = −∂ ~B

∂t(2.1)

∇× ~H = ~J +∂ ~D

∂t(2.2)

∇ · ~B = 0 (2.3)

∇ · ~D = ρ, (2.4)

where ~E is the electric field vector (in V/m), ~D is the electric displacement flux density

(in C/m2), ~H is the magnetic field vector (in A/m), ~B is the magnetic field density (in

W/m2), ~J is the electric current density (in A/m2), and ρ is the electric charge density

(in C/m3). In our development (following Chak et al. [100]), we consider ~D and ~B as

the fundamental fields, and consider ~E and ~H as derived fields.

The flux densities, ~D and ~B, for nonmagnetic media, are related to their respective

field quantities by

~D = εo~E + ~P (2.5)

~B = µo~H (2.6)

where εo, µo, and ~P are the free-space electric permittivity (8.8542×10−12 F/m), the free-

space magnetic permeability (4π × 10−7 H/m), and the polarization density (in C/m2)

of the medium, respectively. The speed of electromagnetic propagation in a vacuum is

defined as c, which, following Maxwell’s equations, is equal to the quantity 1/√

µoεo. The

five fields described above are functions of both space, ~r, and time, t.

The polarization density, ~P , is often expressed as a function of the electric field, ~E


(see Appendix A). However, in this development, we define ~P in terms of ~D, which, for

an inhomogeneous, linear, isotropic medium, can be written as

P i(~r, t) = δij

(1− 1

n2(~r)

)Dj(~r, t) (2.7)

where the superscripts represent the Cartesian components which are to be summed over

if repeated, δij is the Kronecker delta function, and real[n(~r)] describes the local index

of refraction. While this has been written for the linear case, an extension to a more

general equation including nonlinearities can be constructed [100].

Manipulating Eqs. (2.1) through (2.7) for an electromagnetic field in a dielectric

medium (where both ~J = 0 and ρ = 0) in a z-invariant (i.e., z-homogeneous) system, we

derive the following master equation describing the eigenmode, ~Bmk(~r):

∇×[∇× ~Bmk(~r)

n2(~r)

]= ∇×

[∇× ~Bmk(~r)

n2(x, y)

]=

(ωmk

c

)2~Bmk(~r). (2.8)

where the subscript m refers to the eigenmode index, n2(~r) = n2(x, y), and k is the wave

number at which the equation is evaluated. The frequency of each eigenmode, ωmk, in

Eq. (2.8), is related to the wave number via

ωmk =c k

nm(k)(2.9)

where nm(k) is the effective index of the mth eigenmode, which describes its phase ve-

locity, c/nm(k), through the waveguide. From the master equation, we can evaluate the

corresponding electric displacement eigenmode field of the waveguide system, ~Dmk(~r),

using Eq. (2.2):

~Dmk(~r) = i · ∇ × ~Bmk(~r)

µoωmk

. (2.10)

Any arbitrary field supported in the waveguide system can be expanded in terms of its


eigenmodes via

~B(~r, t) =∑m

∫ √~ωmk

2amk

~Bmk(~r)dk + c.c. (2.11)

(and similar for ~D), where amk are the eigenmode amplitudes, and c.c. is the complex

conjugate.

We can describe the eigenmode, ~Bmk(~r), to take the form

~Bmk(~r) =

√1

2π~bmk(x, y)eikz (2.12)

where the spatial mode, ~bmk(x, y), is invariant in z in our z-homogeneous structure, and

bounded in the x and y directions. Identical relations hold for ~Dmk(~r). While this

formulation of the eigenmode is sufficient for our simple z-homogeneous structures, for

periodic photonic crystal waveguides, the reader is asked to follow the development in

Chak et al. [100].

The eigenmode functions satisfy the normalization conditions

∫ ~B∗m′k′(~r) · ~Bmk(~r)

µo

d~r = δmm′δ(k − k′) (2.13)

∫ ~D∗m′k′(~r) · ~Dmk(~r)

εon2(x, y)d~r = δmm′

δ(k − k′) (2.14)

where the integrals are evaluated over all space, and the superscript ∗ refers to the

complex conjugate. From these, the spatial modes, ~bmk(x, y) and ~dmk(x, y), satisfy the

normalization conditions

∫ ~b∗m′k(x, y) ·~bmk(x, y)

µo

dxdy = δmm′(2.15)

∫ ~d∗m′k(x, y) · ~dmk(x, y)

εon2(x, y)dxdy = δmm′

(2.16)

where the integral is evaluated over the entire x-y plane.


Various numerical methods can be used to compute the spatial modes of the wave-

guide system, including finite difference modal analysis (FDMA), finite element, integral-

equation, and series expansion methods [101]. In the work presented in this thesis, the

modes are computed with FDMA using the commercially available program, “MODE

Solutions” (Lumerical Inc.), which is based on the finite-difference algorithm presented

by Zhu, et al. [102]. An example single-mode waveguide and its only supported (fun-

damental) spatial mode computed using MODE Solutions are illustrated in Figs. 2.2(a)

and (b), respectively.

1

x

y

0

0.4

0.8

0.2

0.6

1

Figure 2.2: (a) Single-mode waveguide, and (b) its fundamental spatial mode computedusing FDMA (“MODE Solutions”).

2.2 Supermodes of the Directional Coupler

By defining the index profile of the two-waveguide system (illustrated in Fig 2.3) to be

n2(x, y), the spatial modal solutions, or supermodes, can be described using the master

equation, Eq. (2.8), and solved using FDMA. These supermodes take the form of symmet-

ric (even) and anti-symmetric (odd) functions with different effective indices, neven and

nodd. The two lowest order TE polarized supermodes (i.e., oriented in the x-direction)

of the directional coupler used in the optically controllable delay line (see Chapter 6 for

the design details) are shown in Fig. 2.4.


x

y

Figure 2.3: Two-waveguide system.

Because the two supermodes propagate through the DC with different phase veloc-

ities (defined by c/n) at a given frequency, ω, the interfering supermodes beat as they

propagate along z. The half-beat length, Lc, defines the propagation length where the

two supermodes are 180o out of phase. Lc can be computed directly from the difference

between the effective indices of the two supermodes:

∆φ = π = Lc∆k, (2.17)

xneven

nodd

x

Figure 2.4: Two lowest-order supermodes of a two-waveguide system.


where ∆k = 2π|neven − nodd|/λo. Therefore,

Lc =π

∆k. (2.18)

This direct approach in determining the half-beat length is henceforth referred to as the

“supermode ∆k” approach.

The dispersion relations for the two lowest-order supermodes of the OCDL directional

couplers (see Chapter 6) are plotted in Fig. 2.5, where the ∆k is identified for a particular

frequency, ωref/(2πc) = 6.4516× 105.

1.3206 1.3206 1.3206 1.3207 1.3207 1.3207 1.3207

x 107

6.4508

6.451

6.4512

6.4514

6.4516

6.4518

6.452

6.4522

x 105

Wave number k [m-1

]

Norm

alize

d F

req

uen

cy

/(

2

c)

[m

-1]

ref /(2 c)

k

Figure 2.5: Dispersion relation for the two lowest-order supermodes of the DC used inthe OCDL of Chapter 6.


2.3 Hamiltonian Formulation of Coupled Mode

Equations

In this section, an overview of the Hamiltonian formulation of the coupled mode equations

(HCME) is presented, following Chak et al. [100]. The HCME describe how the complex

amplitude of a mode in a single waveguide is perturbed by the presence of a neigh-

bouring waveguide. The reasons for pursuing this approach are numerous: a quantum

Hamiltonian approach is considered more fundamental than a classical approach (e.g.,

Maxwellian); the classical CME results can be derived from the HCME; the HCME can

be easily generalized to treat nonlinear problems (e.g., second harmonic generation [103]);

and the HCME can handle purely quantum processes (e.g., spontaneous parametric down

conversion [103]).

2.3.1 Eigenmodes in a Restricted Basis

Before commencing the coupled mode analysis, we need to establish that the modes of a

two-waveguide system can be well described by a restricted basis of only the modes in the

individual parent waveguides (either waveguide A or B in Fig. 2.1). In this development,

we assume that the two parent waveguides each only support one (fundamental) parent

mode at each value of k. Each parent mode satisfies the master equation of Eq. (2.8):

∇×[∇× ~B

(j)k (~r)

[n(j)(x, y)]2

]=

(ω

(j)k

c

)2

~B(j)k (~r) (2.19)

where n(j)(x, y) describes the refractive index profile of the jth parent waveguide. The

corresponding electric displacement field, from Eq. (2.10), is

~D(j)k (~r) = i · ∇ × ~B

(j)k (~r)

µoω(j)k

. (2.20)


We can normalize the individual parent modes following Eqs. (2.13) and (2.14) as

∫ ~B∗(j)k′ (~r) · ~B

(j)k (~r)

µo

d~r = δ(k − k′) (2.21)

∫ ~D∗(j)k′ (~r) · ~D

(j)k (~r)

εo[n(j)(x, y)]2d~r = δ(k − k′). (2.22)

The spatial parent modes, ~d(j)k (x, y) and ~b

(j)k (x, y) (introduced as in Eq. (2.12)) are nor-

malized following Eqs. (2.15) and (2.16) to be

∫ ~b∗(j)k (x, y) ·~b(j)

k (x, y)

µo

dxdy = 1 (2.23)

∫ ~d∗(j)k (x, y) · ~d

(j)k (x, y)

εo[n(j)(x, y)]2dxdy = 1. (2.24)

The two-waveguide directional coupler is described by its master equation

∇×[∇× ~Bk(~r)

n2(x, y)

]=

(ωk

c

)2~Bk(~r) (2.25)

where n(x, y) describes the index profile of the two-waveguide system. If the gap between

the waveguides is sufficiently large, each supermode (see Section 2.2) can be equated, to

a good approximation, to an expansion in the restricted basis of a superposition of the

parent modes:

~Bk(~r) = η(1)k

~B(1)k (~r) + η

(2)k

~B(2)k (~r) (2.26)

where η(j)k are the scaling coefficients. Substituting Eq. (2.26) into Eq. (2.25), multiplying

that equation by [ ~B(j)k (~r)]∗, integrating over all space, and subsequently using Eq. (2.20),

(2.21), (2.22), and (2.24), we find the following equation

∑j

M ij(k)η(j)(k) =(ωk

c

)2 ∑j

Sij(k)η(j)(k) (2.27)


with the matrix elements M ij(k) and Sij(k) given by

M ij(k) =ω

(i)k ω

(j)k

c2

∫ ~d∗(i)k (x, y) · ~d

(j)k (x, y)

εo[n(j)(x, y)]2dxdy (2.28)

Sij(k) =

∫ ~b∗(i)k (x, y) ·~b(j)

k (x, y)

µo

dxdy. (2.29)

In matrix form, we have

M(k) · η(k) =(ωk

c

)2

S(k) · η(k), (2.30)

where M(k) and S(k) are 2×2 Hermitian matrices, and η(k) is a 2-element column vector.

We define α(k) ≡ S1/2(k) · η(k), such that the above equation becomes

M · α(k) =(ωk

c

)2

α(k), (2.31)

where the Hermitian matrix M = S−1/2(k) · M(k) · S−1/2(k). Eq. (2.31) is a standard

Hermitian eigenvalue problem whose solution results in a pair of eigenfrequencies, ωmk,

and an associated pair of eigenvectors, αm(k), where m = 1, 2.

To verify that the supermodes can indeed be well approximated by Eq. (2.26), we can

compare the eigenfrequencies with the dispersion relation of the two supermodes of the

two-waveguide structure. In Fig. 2.6, this comparison is performed for the directional

coupler used in the optically controlled delay line (OCDL) (see Chapter 6) where the solid

lines are the dispersion relations for the two supermodes of the two-waveguide structure,

and the points are the two computed eigenfrequencies, evaluated using ωref/(2πc) =

6.4516 × 105 m−1, corresponding to the wavevector, k = 1.3207 × 107 m−1, showing

excellent correspondence (with less than a 0.01% difference).


1.3206 1.3206 1.3206 1.3207 1.3207 1.3207 1.3207

x 107

6.4508

6.451

6.4512

6.4514

6.4516

6.4518

6.452

6.4522

x 105

Wave number k [m-1

]

Norm

alize

d F

req

uen

cy

/(

2

c)

[m

-1]

ref /(2 c)

k

Figure 2.6: Dispersion relation for the OCDL of Chapter 6. Solid lines are the dispersionrelation of the two supermodes; the points are the two eigenfrequencies computed from therestricted basis analysis.

2.3.2 Effective Fields and the Coupled Mode Equations

Following the development in [100], we introduce a pair of scalar effective fields, g1 and

g2,

gn(z, t) =1√2π

∫bnke

i(k−k)dk (2.32)

where the integral is evaluated over all k, and where the operators bnk are related to the

eigenvectors, αm(k), and the mode amplitudes, amk, (introduced in Eq. (2.11)) via

bnk =∑m

α(n)m (k)amk (2.33)

where α(j)m (k) (from Eq. (2.31)) refers to the jth element of the 2-element eigenvector,

αm(k). These effective fields are essentially a way of defining the slowly varying complex

envelope functions of the parent modes, and can, in fact, be used to construct the original

field in the waveguide system (defined by Eq. (2.11)).


The effective fields are based on a canonical formulation of the electromagnetic fields,

which implies that the Hamiltonian and its symmetries are available for studying the

system. Expanding the Hamiltonian of the system defined using the restricted basis (see

Chak et al. [100] for the development details), the following coupled differential equations

result:

∂g1,2

∂t+ vg

∂g1,2

∂z= −iωself(k)g1,2(z, t)− iωcross(k)g2,1(z, t) (2.34)

where the group velocity, vg = ∂ωself(k)/∂(k), evaluated at k = nωref/c, and

ωself = ~−1ωrefk

(1 + ∆1kα

(1)1 (k)[α

(1)1 (k)]∗ + ∆2kα

(1)2 (k)[α

(1)2 (k)]∗

)(2.35)

ωcross = ~−1ωrefk

(∆1kα

(1)1 (k)[α

(2)1 (k)]∗ + ∆2kα

(1)2 (k)[α

(2)2 (k)]∗

)(2.36)

where

∆mk ≡ ωmk − ωrefk

ωrefk

. (2.37)

We can find the half-beat coupling length by finding a stationary solution to Eq. (2.34)

of the form

g1,2(z, t) = g1,2(z)e−iωself(k)t, (2.38)

and finding that when light is input only into one of the two waveguides (i.e., g2(0) = 0),

we have

|g1(z)|2 = |g1(0)|2cos2(κz) ≡ |g1(0)|2σ2 (2.39)

|g2(z)|2 = |g1(0)|2sin2(κz) ≡ |g1(0)|2ϕ2 (2.40)

where

κ ≡ ωcross(k)/vg (2.41)

σ ≡ cos

(ωcross(k)z

vg

)(2.42)


ϕ ≡ sin

(ωcross(k)z

vg

). (2.43)

where we have introduced the unitless amplitude through- and cross-coupling coeffi-

cients, σ and ϕ, which are seen to be sinusoidally varying functions – as expected for a

two-waveguide directional coupler. The half-beat coupling length, Lc, can therefore be

expressed as

Lc =π

2 |κ| =πvg

2|ωcross| . (2.44)

2.4 Conclusion

In this chapter, the nature of light coupling in waveguides was studied. Using Maxwell’s

equations, the master equation was derived describing the propagation of light through

an arbitrary waveguide structure. The half-beat length of a two-waveguide directional

coupler (i.e., the length at which 100% of the light is transferred between waveguides)

was computed in two ways: (1) from the system’s two lowest-order supermodes, and (2)

using the Hamiltonian formulation of coupled mode equations.

The three projects described in this thesis each employ coupled waveguides in their

unique way. Prior to studying the design, simulation, and characterization of each, an

overview of the aluminum gallium arsenide material system is presented in the next

chapter.

Chapter 3


In this chapter, we discuss the benefits of the aluminum gallium arsenide (AlGaAs)

material system for use in planar lightwave circuits. These benefits were listed in the

Introduction, and are presented again here:

• The high linear refractive index minimizes PLC chip sizes.

• Its crystalline structure remains lattice-matched across the aluminum (Al) concen-

tration range (between 0 and 100%).

• The material index can be varied (via Al concentration) by ∆n = 0.48 (at λ =

1550 nm).

• Its large nonlinear refractive index is well suited for all-optically controlled devices.

• Mature processes have been developed for PLC fabrication in AlGaAs.

In this chapter, we first take a look at the AlGaAs crystalline structure in Section 3.1.

Section 3.2 describes the material refractive index of the five different AlGaAs compo-

sitions used in this thesis in terms of Sellmeier coefficients. Section 3.3 presents a brief

discussion on the half bandgap and Kerr nonlinearity of AlGaAs. The final section, Sec-

tion 3.4, describes the AlGaAs wafer layer designs used in the three projects described

in this dissertation.

42

Chapter 3. Aluminum Gallium Arsenide 43

3.1 AlGaAs Lattice Constant

At normal pressure, AlGaAs crystallizes in the linearly-isotropic, non-centrosymmetric

zinc-blende structure (point group 43m(Td)). The lattice constant, a0, for a zinc-blende

crystal is defined by the length of the face-centred cube edge, as illustrated in Fig. 3.1 [92].

At a temperature of 300 K, the stress-free lattice constant, as a function of aluminum

concentration, x, can be written as

a0 = 5.65330 + 0.00809x. (3.1)

The difference in lattice constants between GaAs and AlAs is only 0.14%, making het-

erojunctions comprised of different AlGaAs compositions particularly robust against de-

lamination and cracking.

0

Figure 3.1: The primitive cell of a zinc-blende type crystal lattice: arsenic = white atoms;gallium/aluminum = red atoms.


3.2 AlGaAs Material Index Dispersion

The Gehrsitz model provides an excellent description of the refractive index of AlxGa1−xAs

over temperature, aluminum concentration, and wavelength (below the band-gap) [93].

However, because of its complexity, the model is difficult to use in commercial mode

solvers. As such, it was evaluated at room temperature (T = 296 K), and fit to the

following simplified Sellmeier formula:

n2(λ) = A1 +B1λ

2

λ2 − C1

(3.2)

where the coefficients, A1, B1, and C1, are dependent on the aluminum concentration, x.

The Sellmeier coefficients for the five different AlGaAs compositions used in the projects

described in this dissertation are listed in Table 3.1 (valid between 880 nm and 2100 nm

wavelengths), and the respective dispersion curves are plotted in Fig. 3.2. The results

using these coefficients in the simplified Sellmeier formula were accurate (compared to the

results from the Gehrsitz model) to within a (reduced chi-squared) statistical deviation

of 0.0003.

Table 3.1: AlGaAs Sellmeier Coefficients at T = 296 K

x: 0 0.18 0.20 0.24 0.70A1 7.95747 4.57905 4.41592 4.13405 1.61115B1 2.96499 5.72297 5.82425 5.98733 7.25265C1 0.32166 0.16821 0.16219 0.15176 0.08675

The room-temperature refractive index of GaAs (i.e., x = 0) and AlAs (i.e., x = 1) at

1550 nm evaluates to 3.374 and 2.893, respectively; these large index values (in compari-

son with the ∼ 1.5 indices typically found in glasses) minimize the size of planar lightwave

circuits. The temperature dependence of the refractive index is 2.415×10−4/K [93]. Fur-

thermore, the large variation in the index allows for an added degree of freedom in the

design of the wafer layer structures.


1000 1200 1400 1600 1800 20003

3.1

3.2

3.3

3.4

3.5

.

3.6

x = 0

x = 0.18

x = 0.20

x = 0.24

x = 0.70

Figure 3.2: AlGaAs material dispersion as a function of aluminum content, x (%).

3.3 AlGaAs Optical Nonlinearity

A material’s electronic bandgap is defined by the energy difference between its valence

and conduction bands. Photons with energies greater than the bandgap will be absorbed

by carriers excited from the valence to the conduction band. A similar effect called two-

photon absorption (TPA) occurs with large optical intensities, whereby the energy of two

photons can excite a carrier to the conduction band. For optical devices, any form of

absorption is detrimental, and should be eliminated, if possible [104].

Fortunately, the bandgap of AlGaAs is a function of the aluminum concentration. The

half bandgap wavelength of AlGaAs versus aluminum content, x, is plotted in Fig. 3.3,

which reveals that the aluminum content at 1550 nm must be greater than x = 0.14 to

reduce TPA to a level where it can be ignored. This minimum limit was respected in the

AlGaAs wafer designs used in this dissertation (see Section 3.4).


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

Half B

andgap W

avele

ngth

(nm

)

Aluminum Content, x

Figure 3.3: AlGaAs half bandgap wavelength versus aluminum content.

The Kerr effect originates from the nonlinear polarization generated in a medium,

instantaneously modifying its refractive index (see Appendix A). The Kerr nonlinear

refractive index, n2, was measured for a 5 µm-wide waveguide fabricated in a commonly

used low-index contrast wafer (see Section 3.4.1 for the description of the 24/18/24 wafer).

The wavelength dependence of n2 is plotted in Fig. 3.4 [96]. Because of its instantaneous

nonlinear response, the Kerr effect can be used in the fabrication of ultrafast devices [95].

Figure 3.4: The measured AlGaAs values of n2 for TE (solid circles) and TM (solidtriangles) polarizations [96].


3.4 AlGaAs Wafers

The AlGaAs wafers used in the projects described in this dissertation were grown using

molecular beam epitaxy with aluminum content greater than x = 0.14 (see Section 3.3).

3.4.1 AlGaAs Wafer: 24/18/24

The “24/18/24” AlGaAs wafer was used for the lower index contrast PLCs, whose wave-

guide dimensions were on the order of a few microns. The projects that used this wafer

were the EDL simulator (Chapter 4) and the optically controllable delay line (Chapter 6).

The wafer structure consisted of a 1.5 µm Al0.18Ga0.82As core layer, surrounded by

Al0.24Ga0.76As cladding layers (1.5 µm above, and 4.0 µm below), grown on a GaAs

substrate [105].

3.4.2 AlGaAs Wafer: 70/20/70

The “70/20/70” AlGaAs wafer was used for the high index contrast PLCs, whose wave-

guide dimensions were on the order of 0.5 µm. The SCISSOR device used this wafer to

achieve the tight modal confinement required for the nanowires.

The wafer structure consisted of a 10 nm top layer of GaAs (to prevent aluminum oxi-

dation), a 300 nm top cladding layer of Al0.7Ga0.3As, a 500 nm core layer of Al0.2Ga0.8As,

followed by a 3000 nm lower cladding of Al0.7Ga0.3As, grown on a GaAs substrate [106].

A 300 nm layer of PECVD grown SiO2 was deposited atop the AlGaAs wafer for use as

a hardmask in the e-beam lithography and etching process.


3.5 Conclusion

In this chapter, we quantified the excellent lattice matching properties of AlGaAs at

room temperature. A Sellmeier formulation was presented to compute the wavelength-

dependent refractive index for the five AlGaAs compositions used in this dissertation.

The minimum aluminum concentration required to minimize two-photon absorption at

1550 nm was found to be x = 0.14, and the wavelength dependence of the Kerr nonlinear

index was presented. Finally, the 24/18/24 and 70/20/70 wafer designs used in this

disseration were described.

Chapter 4

Exact Dynamic Localization

In this chapter, we will look into optical signal processing in PLCs using the mathematical

parallels between quantum and optical theory. By successfully employing this parallel to

demonstrate optical exact dynamic localization, we pave the road for PLC devices with

improved and/or new optical functionality, based on a foundation of quantum theory.

4.1 Introduction

The evolution of an electron wavepacket in a one-dimensional periodic spatial potential

under the influence of a time-dependent, periodic electric field has attracted a great deal

of attention over the last 20 years [10,11,17,18,107–109]. Much of the interest arises from

the subtle interplay between the periodic potential and the electric field, which, under

some circumstances, can counteract the tendency of the wavepacket to spread. For cases

when the wavepacket returns to its original localized state, dynamic localization (DL)

occurs, a phenomenon that is particularly interesting, for neither the periodic potential

nor the electric field on their own can inhibit the wavepacket’s delocalization.

Dynamic localization is closely related to Bloch oscillations (BOs) [14, 110], which

occur when the applied field, Fdc, is constant in time, rather than periodic. In a spatially

periodic potential, a localized wavepacket in such a field initially spreads but subsequently

49

Chapter 4. Exact Dynamic Localization 50

reconverges, so that the wavepacket returns to its original state. This motion is periodic

in time, with a Bloch period , τB, that scales as F−1dc . This type of electron localization was

proposed more than 50 years ago by James [111], and was first experimentally verified in

optically-excited semiconductor superlattices in 1992 [8].

The behavior of a wavepacket undergoing DL is similar to BOs: for an ac field with

an appropriately chosen shape, period, and amplitude [18], the wave packet returns to

its initial state. DL was first theoretically shown to arise for sinusoidally varying elec-

tric fields [17, 107]; however, it has been shown that DL can only occur with sinusoidal

fields (or any continuous periodic ac fields) in systems where adjacent spatial wells are

weakly coupled (so that the bandstructure is well described by the nearest-neighbor tight-

binding, NNTB, approximation) [18]. In this approximate dynamic localization (ADL),

the maximum beam divergence (i.e. oscillation amplitude) is small, and the beam does

not relocalize in general (e.g., non-NNTB) structures. In contrast, it has been theoreti-

cally shown that exact dynamic localization (EDL) can occur in these general structures

only if the electric field is discontinuous [18], allowing for rigorous relocalizations, and

with much larger oscillation amplitudes – a phenomenon that has not yet been experi-

mentally demonstrated.

The experimental observation of EDL in electronic systems is very difficult. Large

amplitude, discontinuous ac fields (on the order of 10 kV/cm) in the THz frequency

range (such that the period is shorter than electron decoherence times of ∼ 1 ps) are

extremely difficult to generate. Electronic systems have the additional problem that

electron-electron and electron-phonon interactions significantly alter the dynamics of the

wavepacket evolution. As well, indirect measurement techniques must be used (e.g.,

four-wave mixing [8,9] or terahertz emission [10–12]), since direct imaging of the electron

wavepacket is not possible.

To overcome the difficulties of observing DL in electronic systems, two alternative sys-

tems have been considered to empirically simulate the effect: cooled atomic arrays [112],


and curved coupled optical waveguide (CCOW) arrays [7, 15, 19, 113, 114], the second of

which is investigated in the work presented in this chapter. The spatial propagation of

light in optical waveguide arrays is directly analogous to the temporal evolution of an

electron wavepacket in a superlattice. The effect of the applied electric field on the super-

lattice is obtained in the waveguide structure by inducing a transverse refractive index

gradient across the array (obtained, for example, by a temperature [115], or structural

gradient [105]). Of note, CCOW structures do not suffer the drawbacks of electronic

systems: the 1 − 10 mm period necessary for the waveguide curvature, and the curva-

ture discontinuities are easily patterned using standard photolithographic techniques; the

electron-electron and electron-phonon interactions have no direct linear optical analog;

and the CCOW system permits the spatial monitoring of the evolution of the beam using

standard optical detection methods.

Longhi et al. [19] recently presented the first experimental observations of DL in a

sinusoidally-shaped NNTB CCOW array. However, this was truly only a demonstration

of ADL, because of the continuously varying curvatures employed. At the current time,

there has been no reports of exact dynamic localization in any system.

The challenge: Can a PLC be designed to demonstrate the exact quantum dynamic

localization (EDL) effect over several relocalization periods? If so, the spatial mapping

of the beam evolution through the device must be performed, along with the character-

ization of the wavelength dependence (with significantly improved resolution compared

with Longhi’s earlier work [19]).

In the work presented here, the index gradient was created using CCOW arrays [15,

114], in which the periodically varying waveguide curvature profile served as the optical

analog of the ac electric field. The first experimental demonstration of EDL in CCOWs

with discontinuous waveguide curvature profiles is demonstrated – this being the first

demonstration of EDL in any system. Excellent correspondence of the experimental

results with theory is reported for both DL evolution and wavelength dependence.


This chapter is structured as follows: an overview on the theory of DL in CCOWs,

and details of the device parameters are presented in Section 4.2. Section 4.3 presents the

experimental results, including the wavelength dependence of the beam at the four-period

EDL plane, and spatial field evolution of the beam around the two-period EDL plane.

Section 4.4 discusses the EDL tolerance on discontinuity-smoothing of the fabricated

waveguide curvatures. Discussions of future work and final conclusions are presented in

Sections 4.5 and 4.6, respectively.

4.2 Dynamic Localization in Curved Coupled Optical

Waveguide Arrays

4.2.1 Paraxial Complex Vector Wave Equation

Manipulating Maxwell’s equations, (Eqs. (2.1), (2.2) and (2.6)), for an electromagnetic

field in a dielectric medium (where both ~J = 0 and ρ = 0), the complex vector wave

equation (CVWE) for modes with electric fields tangential to material discontinuities

(such that the electric fields are continuous in space) can be derived as follows [2]:

∇×(∇× ~E

)= −∇× µ

∂

∂t~H (4.1)

= −µ∂

∂t

(∇× ~H

)(4.2)

= −µ∂2

∂t2~D (4.3)

Substituting Eqs. (2.5) and (A.24) into Eq. (4.3), and using the identity ∇×(∇× ~E

)=

∇(∇ · ~E

)−∇2 ~E, the linear CVWE becomes

∇(∇ · ~E

)−∇2 ~E = −µ

∂2

∂t2

[εo

~E + εoχ(1) ~E

](4.4)


which, for complex time-harmonic monochromatic waves (described in Eq. (A.6)) is

∇(∇ · ~Ec

)−∇2 ~Ec = µεoω

2[~Ec + χ(1) ~Ec

](4.5)

= k2o~Ec

(1 + χ(1)

)(4.6)

= k2oεr

~Ec (4.7)

where ~Ec is the complex representation of the electric field (see Appendix A), ko = 2π/λo

is the free-space propagation constant, λo is the free-space wavelength, ω = 2πc/λo, and

εr = 1 + χ(1) (see Eq. (A.25)).

For a dielectric medium with no free-charge density (ρ = 0), Eq. (2.4) can be expanded

as follows:

0 = ∇ · ~Dc (4.8)

0 = ∇ ·(εo

~Ec + ~Pc

)(4.9)

0 = ∇ ·(εo

~Ec + εoχe~Ec

). (4.10)

where the polarization density is defined in Eq. (A.22) as ~Pc = εoχe~Ec. Employing the

identity ∇ ·(α~Ec

)= α∇ · ~Ec + ~Ec · ∇α, we obtain

0 = (1 + χe)∇ · ~Ec + ~Ec · ∇χe. (4.11)

Isolating for ∇ · ~Ec,

∇ · ~Ec =− ~Ec · ∇χe

1 + χe

=− ~Ec · ∇χ(1)

1 + χ(1). (4.12)

We can now rewrite the CVWE as follows:

∇2 ~Ec +∇(

~Ec · ∇εr

εr

)+ k2

oεr~Ec = 0. (4.13)


A 3D waveguide system whose cross-sectional relative electric permittivity profile is

defined as εr(x, y) can be reduced to a 2D system with a cross-sectional permittivity

profile, εr(x), via the effective index method [116, 117]. Applying a mode with fields

oriented in the y-direction (i.e., out of the plane, and hence, tangential with the index

discontinuities) renders the second term of Eq. (4.13) to zero, and reduces the CVWE to

∇2 ~Ec + k2oεr

~Ec = 0. (4.14)

4.2.2 Mapping of the Quantum System to Waveguide Arrays

The CCOW system is a periodic array of curved waveguides. The structure is shown

schematically in Fig. 4.1. The width of the individual waveguides, dw, and the transverse

period of the array, d, are kept constant, while the array as a whole is curved, with a

radius of curvature, R (v) that depends on the propagation distance, v, along the centre

of the central waveguide.

To describe the beam propagation through a curved waveguide array, we assume the

solution to the CVWE (Eq. (4.14)), takes the form

~Ecm = ~Etm(x, y, z)e+jβmz m = 0, 1, 2 · · · . (4.15)

Each ~Etm describes the mth transverse complex spatial supermode of the waveguide

array. The propagation constant, βm, of the mth mode (i.e., its eigenvalue) is defined as

βm = 2πnm/λo, where nm is the effective index of each supermode. Note that the ‘+’

convention is used for the phase term for these equations. Substituting Eq. (4.15) into

the CVWE, we obtain

∇2t~Ecm + 2jβm

∂ ~Ecm

∂z+ (k2

oεr − β2m) ~Ecm = 0, (4.16)


Al24Ga76As

Al18Ga82As

Al24Ga76As

+Ro

-Ro

Fundamental Mode Launch in Central Waveguide

z

Waveguide number: 0 1-1-2-3 Cleaved

end-facet

Cleaved

front-facet

ddw

y

x

Figure 4.1: Schematic of the CCOW structure. The central waveguide is highlighted,defining the origin of the coordinate system, (u, v). Note the dependence of the waveguideradius of curvature, R(v), with respect to the propagation distance, v, where R(v) =+Ro for 1

4Λ < v ≤ 34Λ; R (v) = −Ro for 3

4Λ < v ≤ 54Λ. Also note that u and v are not

drawn to the same scale.

where ∇2t is the transverse Laplacian operator.

Heiblum, et al. [114] have provided the equations to transform a curved waveguide in

the x-z plane with radius, R, to a straight waveguide (in the u-v co-ordinate system) by

a transformation of its relative electric permittivity profile,

ε curvedr (x) = ε straight

r (u)e−2u/R, (4.17)

which can be simplified to a first-order using Taylor expansion to

ε curvedr (x) = ε straight

r (u)

(1− 2u

R

). (4.18)

This first-order approximation is valid for 2u/R < 10−3. For ∼ 4 µm wide waveguides,


the bend-radii must be greater than ∼ 200 µm. As we will see later in this chapter, the

radius required for our device is ∼ 35 mm.

Therefore, the CVWE for a 2D curved waveguide in the u-v co-ordinate system can

be rewritten as

j∂ ~Ecm

∂v=

−1

2kon

∂2 ~Ecm

∂u2+

ko

2n

[−δn2(u) +

2n2

R(v)u

]~Ecm , (4.19)

where δn2 (u) = ε straightr (u) − n2, n is the effective index of the fundamental mode in

a single waveguide, and where we have made the generally valid approximation that

ε straightr ' n2. Eq. (4.19) is the paraxial wave equation (PWE) for a curved waveguide

system.

The PWE, Eq. (4.19), maps onto the 1D time-dependent Schrodinger equation,

j~∂Ψ

∂t=−~2m∗

∂2Ψ

∂u2+ U(u)Ψ. (4.20)

for an electron with mass, m∗, in a potential U(u), in the presence of a time-varying

electric field, F (t), via

t = v/c, m∗ =hn

cλ, U(u) = −hc

δn2(u)

2nλ, eF (t) = hc

n

λR(v),

(4.21)

where t is time, c is the speed of light in a vacuum, h is Planck’s constant, e is the

magnitude of the charge on an electron, and λ is the free space wavelength in the optical

system [7]. The mapping of t onto v, and F (t) onto R (v), allows us to design the CCOW

analog of the quantum system in which DL should arise, where the periodic index profile,

δn2 (u), in the optical domain is constructed by an array of waveguides (corresponding

to the periodic potential, U(u), in the quantum system described by the Kronig-Penney

model [118]), leading to the formation of photonic bands.


4.2.3 Straight Waveguide Array

It is useful to characterize the properties of a CCOW system by first considering an array

with straight waveguides (i.e., the case corresponding to an electronic system with no

external field present). In such a straight waveguide array, the beam diverges uniformly

via discrete diffraction [13], demonstrating that no beam (or electronic wavepacket) relo-

calization occurs in the absence of waveguide curvature (or electric field in the electronic

system). A simulation of the beam divergence through a straight waveguide array is

presented in Fig. 4.2, where the effective index profile is plotted as a function of u/d.

In an array consisting of N single-mode waveguides, there exists a finite number of

N supermodes (per polarization). The initially localized light in the u = 0 waveguide is,

in actuality, a superposition of a weighted sum of these N supermodes. Similar to the

evolution of the light in a two-waveguide directional coupler (as discussed in Chapter 2),

each of the N excited supermodes propagates at a unique phase velocity through the

PLC, thereby producing an evolving superposition at each position along v.

0

2

4

-4

-2

Figure 4.2: Simulation of the beam divergence (discrete diffraction) in a straight wave-guide array.


4.2.4 Bloch Oscillations

As mentioned in Section 4.1, Bloch oscillations (BO) occur in an electronic system under

the influence of a dc electric field [8,111]. We see from Eqs. (4.21), a dc field corresponds

to a constant radius of curvature of the optical waveguide array, which inhibits the beam

divergence and forces the beam to relocalize periodically in space, every optical Bloch

period, ΛB (mapped from the temporal Bloch period, τB, in the electronic system).

This behavior is illustrated in Fig. 4.3, where the curved waveguide array is conformally

mapped to a straight waveguide array by a linear tilt of the index profile [16, 114] (in

direct analogy with the tilting of the band structure of a periodic lattice in the presence

of a dc electric field).

The slight u-asymmetry of the BO is due to two independent effects: on the inner

side of the curvature (i.e., for u > 0), the nature of the diverging beam is to tend toward

the higher index, thereby arresting its divergence and “reflecting” the beam back toward

u = 0; on the outer side of the curvature (i.e., for u < 0), the diverging beam encounters

a transverse Bragg grating which reflects the beam back toward u = 0 [15].

0

2

4

-4

-2

0 41 32

Figure 4.3: Simulation of Bloch oscillations in a waveguide array with a constant radiusof curvature.


4.2.5 EDL in CCOW Arrays

It has been shown that DL of any kind can only occur in the one-band approxima-

tion [119], (i.e., where only the lowest band is considered), and when coupling to ra-

diation modes (i.e., bend loss) is small; the coupling to radiation modes in CCOWs is

analogous to Zener tunneling to higher bands in an electronic system. As mentioned in

Section 4.1, ADL occurs in CCOW structures with continuous curvature profiles only if

the system is weakly coupled (i.e., NNTB) [18]. Longhi, et al. [19] have experimentally

demonstrated ADL in a CCOW structure with continuously varying, sinusoidally-shaped

waveguides, indicating that their structure could be well described using the NNTB ap-

proximation. Experimental investigations on multiband effects in CCOWs under various

excitation geometries have been conducted in a related work [19].

In general CCOW structures (e.g., non-NNTB structures), rigorous DL (i.e., EDL)

is only possible if the radius of curvature profile, R(v), satisfies a number of conditions.

Two important (but not sufficient) conditions are that R(v) must be discontinuous at

every sign change, and R(v) must satisfy

∫ v2

v1

1

R(v)dv =

QλR

nd, (4.22)

where v1 and v2 are locations of adjacent discontinuities, and Q is a positive integer.

Because the mapping from the electric field, F (t), to waveguide radius of curvature,

R(v), depends explicitly on the wavelength (Eq. (4.21)), EDL in the CCOW system only

occurs at resonant wavelengths, λR. This idea has been proposed for use in the design of

optical filters [7, 18].

Of the many different curvature profiles that can produce EDL [7], the simplest is the


square-wave field, with curvature period, Λ, and radius of curvature profile,

R (v) =

+Ro for(m + 1

4

)Λ < v ≤ (

m + 34

)Λ

−Ro for(m + 3

4

)Λ < v ≤ (

m + 54

)Λ

. (4.23)

where m is an integer. The required radius of curvature amplitude, Ro, using Eq. (4.22),

is found to be

Ro =Λnd

2QλR

. (4.24)

In the structures that are investigated in this work, Q = 1 was chosen in order to

maximize Ro and thereby minimize the coupling to radiation modes. The EDL in this

Q = 1 square-wave case is understood by recognizing that the beam experiences a series of

alternating constant-curvature Bloch regions, contributing to the periodic relocalization

behaviour, as illustrated by the simulation results (for Q = 1) in Fig. 4.4.

The measure of the delocalization between relocalization points is defined as the

oscillation amplitude, ΨDL, (in units of transverse periods, d). Therefore, in order to

observe relatively large, unambiguous oscillation amplitudes (i.e., ΨDL ≥ 3d), systems

with strong coupling are required (which, by definition, cannot be described by the NNTB

0

2

4

-4

-2

0 41 32

DL

Figure 4.4: Simulation of dynamic localization in a waveguide array with an ac square-wave curvature profile.


approximation). For this reason, in this work, rigorous EDL in a non-NNTB system was

designed using the discontinuous square-wave curvature profile. Details of the design are

presented in the next subsection.

4.2.6 EDL CCOW Design

The EDL structures were designed using the 24/18/24 AlGaAs wafer (described in Chap-

ter 3; see also Fig. 4.1). The in-plane confinement was achieved by defining waveguides

with an etch depth of h = 1.35 µm and width of dw = 4.0 µm. The waveguides were

designed to be single-mode and polarization independent at 1550 nm, with an effective

refractive index (for both TE and TM polarizations) calculated to be n = 3.261. A two-

waveguide DC using this design was found to have a coupling coefficient, κ = 0.971 mm−1;

the lowest order TM supermode of this DC (computed using full vectorial finite differ-

ence analysis) is shown in Fig. 4.5 for comparison with other DCs presented in this

dissertation.

The structure was designed to satisfy the following two criteria: that (1) the coupling

was strong enough to create a relatively a large oscillation amplitude, such that DL would

not occur with continuous curvature waveguide profiles, and (2) the radius of curvature

was large enough such that radiation loss (i.e., the coupling to higher-band radiation

modes) would be relatively small [7].

0

0.4

0.8

0.2

0.6

1

Figure 4.5: Simulation of the lowest order TM supermode of a two-waveguide DC usingthe CCOW design parameters.


To demonstrate EDL, the square-wave curvature profile was used (Eqs. (4.23) and

(4.24)), with a DL period, Λ = 5 mm, and an amplitude, Ro = 35.24 mm, at the resonant

wavelength, λR = 1550 nm. The spatial evolution of the beam over four periods through

the CCOW structure was calculated using (a) a 2D Beam Propagation Method (BPM)

simulation [120], and (b) Eqs. (4.21) to map the beam evolution onto the Schrodinger

equation and solving the dynamics using an expansion in the basis of the Wannier func-

tions of the lowest band [7,18]. This second method explicitly omits any coupling between

bands and is henceforth referred to as the one-band Schrodinger model. The results are

plotted in Fig. 4.6 on a dB scale (to observe low-intensity features of the beam evolu-

tion) in the (u, v) coordinate system. The relocalized beam profile at v/Λ = 4 is shown

adjacent to both spatial maps. From these simulations, the following observations are

made: (1) DL in the system occurs every v = mΛ (where m is an integer), (2) the DL

persists for a span of ∆vDL ' 300 µm about each localization plane, (3) the oscillation

amplitude, ΨDL = 4d, and (4) the beam propagation is slightly asymmetric in the u

dimension. The radiation modes seen in Fig. 4.6(a) introduce less than 5% loss per DL

period, and create the ripples seen between relocalizations due to interference with the

light still confined in the array. These radiation modes (and ripples) do not appear in

the Schrodinger simulation results (Fig. 4.6(b)), because the higher photonic bands and

Zener tunneling are excluded in the one-band model. Because EDL can only formally

occur in the one-band approximation [119], the one-band Schrodinger model was used to

produce the theoretical results in the remainder of the chapter.

To verify that the CCOW array was designed with sufficiently strong coupling to

render to NNTB approximation (necessary for ADL) invalid, a simulation with con-

tinuous sinusoidal curvature profiles was conducted with a curvature profile R (v) =

1/ (RA sin (2πv/Λ)). It has been shown that for such a curvature profile, DL would

break down for strongly-coupled systems [17, 107, 121, 122]. In Fig. 4.7, the beam evolu-

tion through the continuous CCOW array was modeled using the one-band Schrodinger


0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

2

4

6

-8

-6

-4

-2

8

0

2

4

6

-8

-6

-4

-2

8

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

DL

DL

-25 dB

-15 dB

-5 dB

-20 dB

-10 dB

0 dB

Figure 4.6: (a) 2D BPM simulation and (b) one-band Schrodinger model of EDL ina four-period non-NNTB CCOW array. The beam profiles at v/Λ = 4 show excellentrelocalization. Both spatial maps are plotted on a dB scale, while the beam profiles areplotted on a linear scale.

model. The beam has relocalized reasonably well at v/Λ = 1; however, in this non-NNTB

structure, DL clearly has broken down, the beam evolution has lost its periodicity, and

at v/Λ = 4, the beam has spread to a width of ± ∼ 4d (plotted in the beam profile to

the right of the spatial map).

While other continuous curvature profiles (besides sinusoidal) may lead to improved

ADL performance, true DL cannot be observed without curvature discontinuities, as

previously mentioned [122]. Hence, in the structures presented in this work, discontin-

uous square-wave curvature profiles were employed to demonstrate EDL performance

unambiguously.


0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

-2

-4

-6

8

6

4

2

-8

-25 dB

-15 dB

-5 dB

-20 dB

-10 dB

0 dB

Figure 4.7: One-band Schrodinger model of ADL breakdown in a four-period non-NNTBCCOW array. The beam profile at v/Λ = 4 clearly shows DL breakdown. The spatial mapis plotted on a dB scale, while the beam profile is plotted on a linear scale.

4.3 EDL: Experiment and Results

4.3.1 Fabrication

Fabrication of the optical chips was conducted following the recipe outlined in Ap-

pendix B. Once etched, the chips’ front- and end-facets were cleaved using using stan-

dard diamond scribing techniques. The etch depth was found to hit the design target

of h = 1.35 ± 0.05 µm; the post-etch waveguide width was found to hit the target of

dw = 4.00 ± 0.05 µm; and the waveguide separation was measured to hit the target of

d = 6.70 µm ± 0.05 µm. A scanning electron micrograph of the cross-section of the

cleaved end-facet is shown in Fig. 4.8. Layouts of the photomask designs (which are de-

scribed in detail in the next two subsections) are presented in Fig. 4.9. In these figures,

the waveguide curvature of the arrays is barely visible, because the lateral amplitude of

the curved waveguide path (in the u dimension) is only 22 µm; the curved propagation

distance, v, is therefore effectively equivalent to the linear distance, z, to within 0.02%

(see Fig. 4.1).


dw

d

h

Figure 4.8: SEM of cleaved end-facet of EDL chip.

4.3.2 Experimental Setup

The experimental setup consisted of a collimated CW beam (from a JDSU SWS 15101

tunable laser source) focused into the front-facet of each chip using a Newport FL40B

coupling lens. A half-waveplate + polarizer combination was used to select TM polarized

light for the launch. The beam from the chip’s end-facet was captured using a traditional

waveguide imaging system, using a Hamamatsu C2741 IR camera and a New Focus 5723-

BH aspheric lens for optimal image quality.

An independent single-mode waveguide coupled to the central waveguide of each

51-waveguide CCOW array guaranteed a localized input mode for every experiment (see

Fig. 4.9(a)). A polarization characterization of the experiments revealed that the CCOW

design was polarization independent, as expected (Section 4.2.6). Therefore, only exper-

imental and simulation results for the TM launch are presented below.


Zlen

Figure 4.9: Photomask layout. (a) Launch geometry of the CCOW array consisting of51 waveguides from a single mode waveguide. (b) The four-period EDL CCOW array,terminated on an output slab waveguide. (c) The 21 EDL CCOW experiments, each witha length of 3 DL periods, and offset by ∆z = 250 µm. The aspect ratio is 1 : 1 in all figures.

4.3.3 Output Slab Measurement Technique

Observations of the actual EDL plane for confirmation of relocalization must be done

within the propagation span of ∆vDL ' 300 µm. Since achieving a cleaving tolerance

less than ±0.5 mm is difficult using standard cleaving procedures, a new technique was

proposed, called the output-slab technique that allows the chip to be cleaved at an arbi-

trary z-plane, while still allowing the imaging of any output plane to within an accuracy

of 1 µm.

An example of the output-slab technique is illustrated in the four-period EDL struc-

ture in Fig. 4.9(b): a sufficiently long and wide output-slab waveguide (OSlab), is pho-

tolithographically defined originating at the termination of the desired EDL observation

plane. The width of the OSlab needs to be sufficiently large to avoid reflections from its

side walls, and consequently, multimode interference.


Upon cleaving the chip through the OSlab at an arbitrary z position, the EDL plane

can be brought into transverse focus (in the u dimension) by literally imaging through

the OSlab; whatever beam profile appears at the output of the CCOW array is allowed to

propagate through the OSlab without lateral confinement, as if in a block of transparent

material (with an index equal to the OSlab’s effective-index, nslab). The resulting post-

cleaved length of the OSlab is defined to be Zlen. The imaging system, initially focussed

on the end-facet (i.e., at the output of the OSlab) must be brought closer to the chip

by a travel distance of Zfocus = Zlen/nslab to bring the desired EDL plane into lateral

focus. Zlen must be kept as small as possible to minimize the effect of spherical aberration

on the beam. This imaging scheme allows for the observation of the EDL plane to an

accuracy of ' 1 µm (achievable by photolithography), and limited only by the z-travel

resolution of the imaging system. The results from a test of the output-slab technique

are presented below in Section 4.3.5.

4.3.4 Staggered Technique for Spatial Mapping

To capture the beam progression through the EDL CCOW structures, a staggered multi-

experiment technique was proposed [105], where a number of identical experiments are

deployed side-by-side and staggered by an offset of ∆z with respect to its preceding

neighbor. An example of the staggered technique is shown in the mask layout of a two-

period EDL experiment in Fig. 4.9(c). With an adequate number of experiments, this

staggered approach can sample the beam propagation over an entire DL period. The

results from a test of the staggered multi-experiment technique is presented the next

subsection.

4.3.5 EDL Measurement Techniques: Validation

To validate the performance of both the staggered multi-experiment and the output-

slab measurement techniques, four Bloch oscillation experiments were designed with a


Imaging

System

Imaging

System

Zlen

ab

cd

ab

cd

Zlen

nslab

Figure 4.10: Bloch oscillation experiments using the staggered and output-slab measure-ment techniques to test beam mapping and accurate beam-relocalization observation.

stagger of ∆z = 250 µm, each terminated on the same OSlab waveguide, as shown

in Fig. 4.10. The first (and shortest) experiment, labeled a, was designed to observe

the point of maximum divergence of the beam through the structure; the last (and

longest) experiment, labeled d, was designed to observe the point of Bloch relocalization.

The results are shown in Fig. 4.11, where the convergence of the beam is observed,

thereby validating both the staggered approach (for beam mapping) and the output-slab

technique for accurate observation of the desired dynamic localization plane. Because

the beam remains confined as it propagates within the OSlab, the image obtained at the

focused EDL object plane is necessarily out-of-focus in the vertical dimension.


a

b

c

d

Figure 4.11: Bloch oscillation experiment results: (a) beam at maximum divergence, (b)beam at 2/3rd maximum divergence, (c) beam at 1/3rd maximum divergence, (d) beam atthe Bloch relocalization plane.

4.3.6 EDL: Wavelength Dependence

The wavelength dependence of the four-period EDL CCOW structure using the output

slab technique was characterized between 1480 and 1600 nm. Gaussian beam analy-

sis [123] was used to determine that a 3 mm wide OSlab was required to ensure no side-

wall reflections. After performing the chip’s output-facet cleave, the length of the OSlab,

Zlen, was measured to be approximately 1 mm, resulting in a Zfocus ' 300 microns.

The photomask layout of the four-period wavelength dependence experiment is shown in

Fig. 4.9(b).

As described in Section 4.2.6, dynamic localization in the CCOW device was designed

to occur at λR = 1550 nm. The output images taken over the wavelength range were

combined and expanded using cubic interpolation to provide the data plot shown in

Fig. 4.12(a). Notably, the actual resonant wavelength is seen to be 1555 nm (identified by

the white dashed line), +0.32% higher than the target λR = 1550 nm. Upon performing a

back calculation of Eq. (4.24), the actual waveguide effective-index was found to be 3.271

(+0.32% higher than the target n = 3.261). This effective-index deviation is within the


(c)

1555 nm

Wavelength (nm)

14

80

14

90

15

00

15

10

15

20

15

30

15

40

15

50

15

60

15

70

15

80

15

90

16

00

u/d

u/d

0

2

4

-4

-2

0

2

4

-4

-2

(b) Simulation

(a) Measured Data

Inte

nsity

u/d-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

R = 1555 nm cross section plotted below in (c)

Simulation

Measured

-25 dB

-15 dB

-5 dB

-20 dB

-10 dB

0 dB

0

1

Figure 4.12: (a) Measured and (b) simulation results of the EDL wavelength dependenceat v/Λ = 4 from 1480 nm to 1600 nm. (c) Measured and one-band Schrodinger simulationat 1555 nm of the beam at v/Λ = 4. Both figures (a) and (b) are plotted on a dB scale,while figure (c) is plotted on a linear scale.

cumulative tolerances of its defining parameters: the aluminum percentage of the cladding

and core layers (±0.5%), the thickness of the cladding and core layers (±10 nm), the etch

depth (±0.02 µm), and the waveguide width (±0.05 µm) (see Section 4.3.1).


The simulation results obtained from the one-band Schrodinger model (using the

experimentally obtained values for d, dw, n, and h) are presented in Fig. 4.12(b) (with

the cross-section at λR = 1555 nm identified by the white dashed line). There is excellent

correspondence of the performance of the device with theory, particularly in the spreading

of the beam to ±3d at 1480 nm, and to ±2d at 1600 nm. Both the measurement and

simulation results at 1555 nm (i.e., the cross-sections identified by the white dashed lines

in Figs. 4.12(a) and (b)) are shown in Fig. 4.12(c) revealing excellent dynamic localization

of the beam at the fourth EDL plane in the device.

4.3.7 EDL: Spatial Map

CCD cameras have been used to directly image the beam evolution of optical Bloch os-

cillations through a constant-curvature waveguide array fabricated in Er:Yb doped phos-

phate glass [124]; however, this approach was not applicable for imaging the waveguides

used for the CCOW devices because AlGaAs does not fluoresce, and the waveguides did

not produce enough observable scattered light. Instead, a traditional end-facet imaging

system was employed to capture the light emitted from 21 staggered 3Λ EDL CCOW

experiments (as described in Section 4.3.4).

Each staggered experiment was offset by ∆z = 250 µm with respect to its previous

neighbour. Upon performing the end-facet cleave roughly through the two-period plane of

the middle (i.e. the 11th) experiment, at v/Λ ' 2, 21 cutbacks of the identical experiment

remained at incrementally varying lengths, covering a total span of 5 mm, i.e., one full

EDL period. The capture of the relocalized mode was assured, since the stagger, ∆z,

was less than the predicted DL propagation length ∆vDL ' 300 µm (see Section 4.2.6).

The experiment was conducted at λ = 1550 nm. Vertical slices from each of the 21

output images (e.g., Fig. 4.13(a)) were stacked to produce a single 21-pixel wide image

that was subsequently expanded using cubic interpolation. The resulting plot is shown

in Fig. 4.13(b); the vertical dashed line represents the data slice from Fig. 4.13(a).


v/

u

(b)

u/d 0

1

2

3

-3

-2

-1

-4

4

(c)

u/d 0

1

2

3

-3

-2

-1

-4

4

21.91.81.71.6 2.42.32.22.1 2.5

5

-5

-5

5

2.6

vDL

DL

(a)

-25 dB

-15 dB

-5 dB

-20 dB

-10 dB

0 dB

-25 dB

-15 dB

-5 dB

-20 dB

-10 dB

0 dB

y

Figure 4.13: (a) Captured image from one of the 21 experiments (plotted on a linearscale); (b) Measured and (c) one-band Schrodinger simulations of one full period aroundthe second EDL plane at 1550 nm. Figure (a) is plotted on a linear scale, while both figures(b) and (c) are plotted on a dB scale.

To validate the performance of the CCOW design, the simulation result from the

one-band Schrodinger model over one full period around the 2Λ EDL plane is shown in

Fig. 4.13(c) (using the experimentally obtained parameters listed in Section 4.3.6). Apart

from the presence of some low-intensity scattered and radiated light (neither of which

would appear in the simulation results), the experimental results agree very well with


the theoretical predictions: (1) DL is clearly observed, (2) the measured DL propagation

span is ∆vDL ' 300 nm, (3) the oscillation amplitude, ΨDL ' 4d, and (4) the beam

propagation is seen to be slightly asymmetric in the u dimension.

force space

4.4 Discussion: EDL Tolerance on Discontinuity

Smoothing

As discussed in Section 4.2.5, one of the necessary conditions for EDL is that the ac field

have discontinuities at every sign change. However, since discontinuities are impossible

to create experimentally, EDL is not actually achievable.

For true EDL, no light appears in other than the localized waveguide (u = 0) at

v/Λ = m (where m is a positive integer). This is equivalent in the quantum system of

stating that the probability of finding the electron in other than the u = 0 quantum well

is zero, i.e., Pnot EDL = 0.

Domachuk, et al., have studied the EDL tolerance on the discontinuity-smoothing [122],

where an analytic formula of Pnot EDL is provided for smoothed square-wave fields. The

function f(v, a), describing the smoothing at the zero-crossings, is defined as

f(v, a) = sin( πv

2Λa

), (0 < v/Λ < a) (4.25)

where a (0 < a < 0.25) measures the degree of smoothing as a fraction of the ac period,

Λ. The smoothed square-wave is shown in Fig. 4.14. The reader is encouraged to follow

the derivation of Pnot EDL in [122]. Here, only the resulting formula is provided:

Pnot EDL = 2∑p>1

[βpmΛ32π2I(p2 − 1)a3

]2, (4.26)


10.50 a

0.5 - a 0.5 + a

1 - a

v /

R(v

) /

Ro f (v,a)

1

0

-1

Smoothed

“square”

wave

Square

wave

Figure 4.14: Smoothing of square wave.

where mΛ is the total length of the device, I ' 0.0586 for f(v, a) = sin(

πv2Λa

), and where

βp are the Fourier expansion coefficients of the lowest-band’s propagation constant, β,

defined as

β(ku) =2πn

λ+

∞∑p=−∞

βpejpkud, (4.27)

where p is an integer, and ku is the Bloch wavevector of the mode (−π/d < ku <

π/d) [125]. The band structure is calculated from the solutions to the 1D paraxial

wave equation, Eq. (4.19), using Bloch’s theorem [118]. For the EDL device described

in this chapter, the lowest band’s first five Fourier coefficients were β1 = −9.70 cm−1,

β2 = 1.06 cm−1, β3 = −0.190 cm−1, β4 = 0.04 cm−1, and β5 = −0.01 cm−1.

In the EDL CCOW device presented here, the tolerance of achieving a discontinuous

break in the R (v) curvature profile is limited by the accuracy of PLC fabrication using

photolithography, which is on the order of ∼ 1 µm. Setting a = 1 µm (which is 0.02% of

one period, Λ = 5 mm), Pnot EDL = 3.5× 10−20 for a four-period (m = 4) structure, and

Pnot EDL = 2.2× 10−3 for a one-billion-period (m = 109) structure.

Hence, the nearly-discontinuous ac curvature profiles used in the CCOW devices,

while not truly discontinuous, do effectively demonstrate EDL to an excellent degree.


4.5 Future Work

While the demonstration of exact dynamic localization using an ac square-wave was

indeed a successful first step, more complex discontinuous ac curvature functions should

now be studied. A set of “deviated square-wave” (DSW) ac profiles satisfying the EDL

requirement defined in Eq. (4.22) has been proposed by Wan [126]:

R (v) =

Ro

2

√(1− α)2 + 16αv/Λ for 0 ≤ v ≤ Λ/4

Ro

2

√(1− α)2 + 16α (0.5− v/Λ) for Λ/4 ≤ v ≤ Λ/2

, (4.28)

where Ro is defined in Eq.(4.24). This family of curves is illustrated in Fig. 4.15 for both

positive and negative values of α.

0.0 0.5 1.0

0

-R0/2

R0/2

R

= 0

(square-wave)

= 0.52

= -0.52

0.0 0.5 1.0

0

-R0/2

R0/2

R

= 0

(square-wave)

= 0.52

= -0.52

0.0 0.5 1.0

0

-R0/2

R0/2

R

= 0 (square-wave)

= 0.52

= -0.52

Figure 4.15: Deviated square-wave ac-field profiles for future EDL demonstrations.

Wan has also proposed a set of combined ac+dc fields to produce so-called quasi-

Bloch oscillations (QBO), promising very interesting results in a new class of optical and

quantum behaviour. The DSW and the QBO experiments have not yet been published.

Further work on the optical filtering characteristics of these CCOW devices should

also be investigated. Specifically, preliminary theoretical investigations of finite wave-


guide arrays have shown to improve the performance of traditional straight-waveguide

directional couplers for power tapping and WDM filtering, leading to interesting optical

signal processing devices in the linear and nonlinear regimes.

4.6 Conclusion

This is the first time in any system that (1) the evolution of DL has been spatially

mapped, and (2) exact dynamic localization has been shown experimentally. In this

chapter, optical EDL was demonstrated over four periods in the optical domain, using

non-NNTB CCOW arrays, driven by an ac square-wave field. Novel observation tech-

niques (staggered experiments and the output slab technique) were employed to obtain

the device dynamics demonstrating EDL beam evolution and wavelength dependence at

the two- and four-period planes of the device, respectively. The beam had an oscillation

amplitude of ±4 waveguides, and was resonant at 1555 nm. The devices showed excel-

lent correspondence with theory as predicted by the one-band Schrodinger model, and

in very close agreement with the design targets (within 0.32%). The EDL tolerance of

fabricating the field discontinuities in optical waveguide arrays was within 10−20 for a

four-period device.

Future experiments to extend the scope of this work include deviated square-wave

ac fields for EDL demonstrations, and ac+dc fields for quasi-Bloch oscillation demon-

strations. This work is among a growing body of research aimed toward the use of

quantum theory for the advent of improved and/or brand new optical signal processing

functionality in PLC devices.

Chapter 5

Side-Coupled Integrated Spaced

Sequence of Resonators

5.1 Introduction

Microresonator-based integrated optical devices have attracted much recent attention,

particularly in optical signal processing applications for telecommunications [28]. Config-

ured in a parallel-cascaded array, side-coupled integrated-spaced sequences of resonators

(SCISSORs) have been shown to demonstrate high-order optical filtering [127], and are

promising candidates for fixed and tunable optical delay lines [43, 64]. A schematic of a

SCISSOR is presented in Fig. 5.1, illustrating a series of rings positioned between two

straight bus waveguides. When many rings are present, the behaviour of the SCISSOR

approaches that of an infinite photonic bandgap (PBG) structure. The interplay be-

tween the direct Bragg bandgaps and the indirect resonator bandgaps permit the design

of dispersion compensators and other bandgap-engineered devices [43,44]. Furthermore,

if implemented in materials with high optical nonlinearity, SCISSORs can be used to

demonstrate phenomena such as optical switching, optical limiting, temporal solitons,

gap solitons, pulse compression [43,45,46] and optical logic [47].

77

Chapter 5. Side-Coupled Integrated Spaced Sequence of Resonators 78

Figure 5.1: Schematic of a five-ring SCISSOR.

The fabrication of a many-period SCISSOR would establish “fully” formed Bragg

peaks in the reflection spectrum, with intensities commensurate with the resonator peaks.

Chen et al. recently fabricated a 36-period SCISSOR structure in polymer consisting of

72 rings (2 rings per period) to demonstrate high-order filtering [48]. Similar devices,

with serially coupled rings (called “coupled resonator optical waveguides”, or CROWs),

have been fabricated with up to 100 rings in silicon-on-insulator and polymer (both with

relatively small nonlinearity compared with aluminum gallium arsenide) [36, 58]. To

date, however, experimental demonstrations of microresonator devices in high index and

highly nonlinear III-V semiconductor materials have been limited to a maximum of three

rings [26, 27] due to extremely tight fabrication tolerances and processing limitations.

The challenge: Can a SCISSOR PLC device with more than three rings be fabri-

cated successfully in a highly nonlinear III-V semiconductor? If so, how many rings are

necessary to empirically demonstrate a fully formed Bragg gap (i.e., where the Bragg

and resonator features at the drop port are of equal intensity)?

In this chapter, the performance of one-, two-, four- and eight-ring AlGaAs SCISSORs

are presented. Section 5.2 describes the transfer matrix model used to simulate the

SCISSOR performance. Section 5.3 presents the device design, followed by details of the

device fabrication and characterization in Section 5.4. Future work and conclusions are

presented in Sections 5.5 and 5.6, respectively.


5.2 The SCISSOR Model

The tight bend radii required for the microresonators necessitate the use of high refractive

index contrast waveguides. In order to minimize the number of supported modes in

these waveguides, their widths must be limited to less than 1 µm, and are hence termed

“nanowires”. Because of the SCISSOR’s inherent periodicity, a semi-analytic matrix

approach was used to describe its behaviour, incorporating the Hamiltonian coupled

mode theory approach (described in Chapter 2) to describe the coupling regions.

A racetrack geometry for the microresonator was chosen to relax the fabrication toler-

ances of the gap between the ring and the straight waveguide channel [27]. The SCISSOR

was broken into individual unit cells as shown in Fig. 5.2, where L is the length of each

cell, r is the bend radius of the racetrack, ∆z is the coupling length of the parallel

directional coupler regions, g is the coupling gap, ϕ is the (unitless) amplitude cross-

coupling coefficient, and σ is the (unitless) amplitude through-coupling coefficient (such

that ϕ2 + σ2 = 1) (see Chapter 2). Using these parameters, the relationship between the

electric fields at the upper and lower right side of the cell can be written as a function of

the electric fields at the upper and lower left side of the cell, and expressed in the form

of a 2× 2 transfer matrix, M :

E+low,right

E−up,right

= M ×

E+low,left

E−up,left

. (5.1)

For a system with N periods, the overall transfer function of the system is MN .

Using the Hamiltonian formulas (Eqs. (2.43)) to describe the coupling region between

the straight channel and the racetracks, the final transfer matrix (Eq. (5.1)) of the unit

cell becomes

M =1

2iσsin(

πωωR

)

A ϕ2

−ϕ2 D

(5.2)


E-up,rightE

-up,left

E+

low,left E+

low,right

r

z

L

g

Figure 5.2: Unit cell of SCISSOR.

where

A = eiπωωB

(e

iπωωR − σ2e

−iπωωR

), (5.3)

D = e−iπωωB

(σ2e

iπωωR − e

−iπωωR

), (5.4)

ωB =πc

Lnb(ω), ωR =

πc

nb(ω)Lr/2, (5.5)

where Lr is the circumference of the racetrack, nb is the frequency dependent effective

index of the spatial mode in one waveguide, and ωB and ωR are the Bragg and ring

resonance frequencies, respectively.

5.3 Device Design

In this section, the design details of the SCISSOR devices are presented: the nanowire

design based on FDMA simulations in Section 5.3.1; the nanowire directional couplers

(between each resonator and the bus nanowires) in Section 5.3.2; and the SCISSOR

devices themselves in Section 5.3.3.


5.3.1 Nanowire Design

The computed losses (FDMA) for nanowires (in 70/20/70 AlGaAs, Section 3.4) of varying

widths are plotted against etch depth in Fig. 5.3. As the nanowire width is reduced, the

mode is squeezed downwards into the substrate, resulting in higher losses due to substrate

leakage. For nanowire widths smaller than 500 nm, an etch depth greater than 2.5 µm

is required to keep losses smaller than 1 dB/mm, while for widths 500 nm and larger, an

etch depth of 2.0 µm is sufficient. While these theoretical simulations predict satisfactory

nanowire performance, the nanowire sidewall surface roughness was expected to introduce

much higher losses (on the order of 10s of dB/mm) [106].

The nanowire design selected for the SCISSORs was chosen to have a width of 500 nm,

and a 2.0 µm etch depth. The computed fundamental mode (using FDMA) of this

nanowire is shown in Figs. 5.4(a) and (b) (linear and dB scale, respectively). The dB

plot highlights the energy residing beneath the ridge giving rise to the substrate leakage

losses plotted in Fig. 5.3.

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

Loss (

dB

/mm

)

Etch Depth (microns)

573 nm width

500 nm width

350 nm width

250 nm width

Figure 5.3: Nanowire loss versus etch depth.


0

-40

-80

-120

-20

-60

-100

0

0.4

0.8

0.2

0.6

1(a) (b)

Figure 5.4: Fundamental mode of a 500 nm wide nanowire with a 2.0 µm etch depth: (a)linear scale, (b) dB scale.

5.3.2 Nanowire Directional Coupler Design

The design of stand-alone nanowire directional couplers (employed twice in each SCIS-

SOR racetrack cell) is described in this section. A set of six independent nano-DCs with

varying lengths were designed using the 70/20/70 AlGaAs wafer. The target design pa-

rameters were as follows: waveguide width = 500 nm, etch depth = 2.0 µm, coupling gap

= 200 nm, and coupling lengths ranging from 11 µm to 61 µm in increments of 10 µm.

Fig. 5.5 shows the mask layout of the 11 µm and 21 µm nano-DCs, where the white

regions within the coloured borders define the two adjacent coupled waveguides. Note

the wide 2.5 µm input and output waveguides for improved coupling efficiency with the

free-space focused beam, and the 30 µm long linear adiabatic tapers leading into and

out of the nanowire device region. This taper length (which was computed to have a

transmission efficiency of 74.5% using FDTD (Lumerical Inc. “FDTD Solutions”)) was

chosen to minimize the overall device loss: a compromise between transmission efficiency

and overall device length. The performance of the nano-DC is presented in Section 5.4.1.


11 m

21 m

2.5 m

Directional Coupler

Regions

Linear taper

Input

Waveguide

Input

Waveguide

Waveguide

A

Waveguide

B

Waveguide

A

Waveguide

B

Linear taper

Figure 5.5: Mask design of the 11 µm and 21 µm nanowire directional couplers (wave-guides are coloured white).

Validation of the HCME using a Restricted Basis

To ensure that the supermodes of the nano-DC can indeed be well approximated by

the restricted Hamiltonian basis modes of the individual single waveguides (as outlined

in Section 2.3), we plot in Fig. 5.6 the eigenfrequencies computed using Eq. (2.26) with

respect to k, and compare them with the dispersion relation of the two supermodes of the

two-waveguide structure. In this plot, we observe that the two computed eigenfrequencies

(the points) evaluated at 1585 nm are situated within 0.1% of the dispersion curves of

the two supermodes, showing excellent correspondence with the two-waveguide system.

Using both the supermode ∆k method (Section 2.2) and the HCME (Section 5.3.2),

the half-beat coupling length of the nano-DC was computed to be 22.0 µm.

5.3.3 SCISSOR Design

The target design parameters of the SCISSOR racetracks (see Fig. 5.2) were as follows:

waveguide width = 500 nm, etch depth = 2.0 µm, coupling gap, g = 200 nm, ring radius,

r = 5.25 µm, and a coupling length, ∆z = 5 µm. The resulting resonator circumference

was Lr = 43.0 µm.


1.14 1.15 1.16 1.17 1.18 1.19 1.2 1.21 1.22

x 107

6.2

6.25

6.3

6.35

6.4

6.45

6.5x 10

5

Wave number k [m-1

]

Norm

alize

d F

req

uen

cy

/(

2

c)

[m

-1]

Figure 5.6: Dispersion relation for the SCISSOR. Solid lines are the dispersion relation ofthe two supermodes; the points are the two eigenfrequencies computed from the restrictedbasis analysis.

The separation between the racetracks, L, was 21.65 µm to produce interlaced Bragg

peaks with approximately the same free-spectral-range (FSR) as the ring resonance,

around the 1550 nm telecommunications band (see Appendix C). While the actual

500 nm nanowire width exceeds the single-mode cutoff width of 250 nm, the higher order

modes experience over 41× greater theoretical loss than the fundamental mode due to

substrate leakage and bend losses (computed using FDMA). Etching through the core

layer provides sufficient index contrast to effectively reduce the radiation losses around

the r = 5.25 µm bend to less than 0.001 dB/90o (computed using FDMA). The confined

mode in the bent waveguide is shown in Fig 5.7.

Sets of one-, two-, four-, and eight-ring devices were patterned in the 70/20/70 wafer.

TM polarization was chosen because of low field intensities at the air-AlGaAs boundaries

(not the case for TE polarization), resulting in lower scattering losses arising from sidewall

roughness. Details of the SCISSOR performance are presented in Section 5.4.2.


0

0.4

0.8

0.2

0.6

1

Figure 5.7: Fundamental mode of a bent 500 nm wide nanowire with a 2.0 µm etch depthand a 5.25 µm bend radius.

5.4 Device Fabrication and Characterization

The devices were first patterned into PMMA resist, using e-beam lithography, and sub-

sequently etched (using C2F6) into a 300 nm thick layer of PECVD silica that served as

a hard mask. The pattern was then etched into the AlGaAs wafer using SiCl4. The chips

were fabricated by colleagues at the James Watt Nanofabrication Centre at the University

of Glasgow, Scotland, UK. Due to photoresist adhesion issues, the patterns were slightly

underdeveloped, resulting in larger waveguide widths, and consequently smaller coupling

gaps. The measured post-etch waveguide width, coupling gap and etch depth were found

to be 570±10 nm, 130±10 nm, and 1.83±0.06 µm, respectively. The measurements were

performed from SEM images using the scale defined by the centre-to-centre separation

of the waveguides (i.e. 700 nm). An SEM closeup of one of the SCISSOR rings, and an

SEM of the eight-ring SCISSOR device are shown in Figs. 5.8 and 5.9, respectively.

Using the post-etch measurements, the effective and group indices of the fundamental

TM mode at 1580 nm were computed to be 2.952 and 3.524, respectively, for a straight

channel, and 2.960 and 3.547, respectively, for a 5.25 µm radius bend. The post-cleaved

chip length was measured to be 1.39 ± 0.01 mm. The group index of the wider and


Figure 5.8: SEM closeup of one racetrack.

21.65 mIN

REFLECTION (R)

TRANSMISSION (T)

Figure 5.9: SEM of fabricated eight-ring SCISSOR.

significantly longer input and output waveguides was computed to be 3.419. Based on

these group indices, the average group index of the 180 µm-long eight-ring SCISSOR

device was expected to be 3.431.

Numerical simulations of the z-dependent coupling in the nano-DC using coupled

mode theory show that the effective coupling length was 8.1 µm, (or 62% larger than the

5 µm straight section, as illustrated in Fig. 5.10). This increased effective coupling length

results in a cross-coupling coefficient of ϕ2 = 0.25. Theoretical analyses of z-dependent

coupling through DCs can be found in the following references: [128], [129] and [130],

while experimental observations can be found in [89,130].

The experimental setup consisted of a collimated wavelength-tunable CW beam fo-

cused into the front-facet of each chip using a 60x microscope objective lens. A half-

waveplate + polarizer combination was used to select TM polarized light for the launch.

The beam from the chip’s end-facet was captured using a 40x microscope objective and

imaged onto a the active area of a power meter. Care was taken to factor out the losses


Figure 5.10: Effective coupling length of nano-DC in the SCISSOR.

and wavelength dependence of the experimental system. An iris mounted on a translation

stage was used to select the desired output (i.e., either channel A or B of the nano-DC,

or either transmission or reflection port of the SCISSOR).

5.4.1 Nanowire Directional Coupler Characterization

The TM supermodes of the nano-DC were computed using the post-etched geometries

listed above (Section 5.4); the lowest order TM supermode is shown in Fig. 5.11. Note

the sloping post-etched floor, resulting in a reduced etch depth between the waveguides.

The normalized transmission, TA,B, at the output ports, A and B, of each directional

coupler (see Fig. 5.5) is plotted in Fig. 5.12: TA = PA/(PA + PB), and TB = PB/(PA + PB),

where PA and PB are the powers measured at ports A and B, respectively. Sinusoidal and

cosinusoidal curve-fits illustrate excellent performance of the nanowire directional cou-

pler, whose measured half-beat coupling length was determined to be 21.4 µm (within

2.6% from the predicted value of 22.0 µm).

The average propagation loss (over the chip length, including the wide input and

output waveguides, tapers, and nanowire DC region) was 2.4 ± 0.5 dB/mm, calculated


0

0.4

0.8

0.2

0.6

1

Figure 5.11: Lowest order TM supermode of a nanowire directional coupler with 570 nmwide nanowires, separated by 130 nm.

from the Fabry Perot fringes (not shown in Fig. 5.12) that were established between the

uncoated facets of the 1.39± 0.01 mm long chip [131].

TB TA

Directional Coupler Length ( m)

Norm

aliz

ed T

ransm

issio

n

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

Channel A (Experimental)

Channel A (Fit)

Channel B (Experimental)

Channel B (Fit)

Figure 5.12: Normalized transmission, TA and TB, of the nanowire directional couplerwith 570 nm wide nanowires, separated by 130 nm.


5.4.2 SCISSOR Characterization

The measured and simulated reflection spectra of the one-, two-, four- and eight-ring

SCISSOR devices are shown in Fig. 5.13, where the coupling loss (estimated to be 60.1%

using Gaussian beam analysis [123]) and Fresnel losses (28% per surface over the wave-

length range used in the experiment) have been factored out. The HCMT-based matrix

model (Section 5.2) was used to simulate the devices, taking into account the reflections

from the chip’s input and output facets. In each device, the resonator peaks at 1595 nm

and 1612 nm are clearly distinguished. As the number of rings is increased from one

to eight, the Bragg peaks emerge and grow at 1587 nm, 1603 nm, and 1620 nm, with

a reduction in the overall throughput because of increased propagation losses through

the longer structures. Note that the height of the Bragg peak at 1620 nm is larger than

the resonator peaks for the eight-ring SCISSOR, achieving the desired goal of producing

Bragg and resonator peaks of “equal” intensity.

Analyzing the eight-ring SCISSOR spectra, both resonator and Bragg resonances are

seen to have an FSR of 17±0.5 nm. The high frequency ripple was caused by the Fabry-

Perot cavity established between the chip’s uncoated facets, and used to determine the

average group index of the waveguides to be 3.428 (−0.4%), as expected. The reader

is directed to Appendix C for the relationship of a filter’s spectrum with its group and

effective indices.

The measured Bragg and ring spectral features agree very well with the spectrum

generated using the HCMT matrix model. Adjusting the parameters of the model to

fit the data, the actual effective index of the straight channels at 1586 nm was found to

be 2.931 ± 0.001 (−0.7%), the effective index of the ring waveguides was 2.934 ± 0.001

(−0.6%), and the cross-coupling constant, ϕ2, was 0.28±0.05 (+12%). The parenthetical

percentage values describe the difference of the measurements with respect to the design

targets. The large discrepancy between the measured and expected cross-coupling con-

stant, ϕ2, is attributed to the model’s inaccurate assumption that the 130 nm gap of the


1580 1590 1600 1610 1620 16300

2

4

6

8

10

Th

rou

gh

pu

t (%

)

Wavelength (nm)

4-Ring SCISSOR

Th

rou

gh

pu

t (%

)

Wavelength (nm)

1580 1590 1600 1610 1620 16300

1

2

3

4

5

6

7

Th

rou

gh

pu

t (%

)

Wavelength (nm)

8-Ring SCISSORT

hro

ug

hp

ut

(%)

Wavelength (nm)

1580 1590 1600 1610 1620 16300

2

4

6

8

10

12

Th

rou

gh

pu

t (%

)

Wavelength (nm)

1-Ring SCISSOR

Th

rou

gh

pu

t (%

)

Wavelength (nm)

1580 1590 1600 1610 1620 16300

2

4

6

8

10

12

Thro

ug

hp

ut

(%)

Wavelength (nm)

2-Ring SCISSOR

Th

rou

gh

pu

t (%

)

Wavelength (nm)

Measurement

HCMT Matrix Model

Figure 5.13: Measured and simulated reflection spectra of the one-, two-, four- and eight-ring SCISSOR devices.

nano-DC is large enough that the sum of the two lowest-order supermodes is equivalent

to the mode in an individual waveguide (see Section 2.2).

The loss parameters for the straight channel and ring were 6.5 ± 0.3 dB/mm and

12.2±0.3 dB/mm, respectively. The higher ring loss was attributed to sidewall-roughness

scattering from the spatial mode’s off-centre centroid around the bend. The straight chan-

nel loss was 22% lower than previous reports conducted in the same 70/20/70 AlGaAs

wafer [106]. New equipment and improved processes have yielded losses in straight chan-

nels of less than 1 dB/mm since the fabrication of the devices presented here.


The devices performed significantly better with TM polarization than with TE, as

expected. The simulation runtime using the HCMT was approximately 2 seconds (on a

2 GHz PC with 1 GB of memory), significantly shorter than ∼ 10 hours for an equivalent

2D FDTD simulation.

5.5 Future Work

5.5.1 Nanowires: Future Work

While the work presented in this chapter has dealt primarily with SCISSOR, a set of

detailed experiments should be carried out on the performance of nanowires. A list of

topics are listed below:

• While linear tapers were used in the reported devices, an optimization on taper

shape and length should be experimentally characterized.

• A recent study of nanowire behaviour has discovered a regime of anomalous group

velocity dispersion for the TE polarization for nanowire widths between 290 nm to

670 nm (see Fig. 5.14). This work, which has not yet been published, encourages

further research in the generation of nanowire guiding temporal solitons [106, 132]

and wavelength converters based on four-wave mixing.

• Studies on solitons in nano-DCs should also be conducted.

• Spatial solitons in nanowire waveguide arrays, as well as WDM filtering using

CCOWs (Chapter 4) should also be a rich field for investigation.

5.5.2 SCISSORS: Future Work

Measurements of the group delay, group velocity, and group velocity dispersion of the

SCISSOR structures using an externally-modulated tunable laser and an RF lock-in


0.2 0.4 0.6 0.8 1.0 1.8 2.0-6

-5

-4

-3

-2

-1

0

1

2

3

" [p

s2 m

-1]

Waveguide Width [µm]

TM00

TE00

Region of

anomalous

dispersion

Figure 5.14: Group velocity dispersion versus nanowire width at 1550 nm, illustratingregion of anomalous dispersion for TE polarization. Solid lines are based on simulations;circles are experimentally measured data points.

amplifier [133] are currently underway. The temporal linear and nonlinear response of the

nano-DC and SCISSOR devices will soon be characterized with a long pulse (∼ 100 ps)

laser.

The reduction in losses of all nanowire based devices should be one of the primary

goals in this field. In parallel, SCISSORs built in gain media should be investigated for

the study of micro/nano lasers and other active devices.

5.6 Conclusion

This chapter described the the first report of the fabrication and characterization of an

eight-ring SCISSOR in a highly nonlinear III-V material system, namely AlGaAs. At

least eight rings were found to be necessary to fully form the Bragg gap in a SCISSOR

structure. A matrix-based Hamiltonian coupled mode theory model was used, and found

to be a fast and useful design tool in modeling high-index contrast nanowire structures,

predicting the behaviour of the devices in excellent agreement with experiments. Further


investigation is required to reduce the losses of the device.

The eight-ring SCISSOR presented here is a significant step towards the goal of ob-

serving PBG phenomena in both the linear and nonlinear regime.

Chapter 6

All-Optical Controllable Delay Line

6.1 Introduction

Planar lightwave circuits are ideal candidates to address the issues currently faced by

electronics in microchip data communications [52] due to their small size, wide band-

width, increased transmission speed, decreased power consumption, and immunity to

electromagnetic noise and temperature changes [53]. In order for these PLC intercon-

nects to successfully integrate with their electronic counterparts, they must provide both

multi-bit switchable optical buffering capabilities greater than 100 bits (for packet syn-

chronization) and greater than several hundreds of bits (for packet switching) [51], as

well as continuous sub-bit delay tunability [52].

Novel schemes have been proposed to demonstrate slow light pulse propagation by

changing the effective optical path length at (or near) material resonances or by engineer-

ing the dispersion characteristics of the system [50, 51, 57, 59, 64, 65, 134–137]. Typically,

delays of only several bits have been achieved by these techniques. Alternatively, very

large delays (on the order of 100000s of bits) can be attained by employing optical

switches and long physical path lengths, such as those used in fiber-based buffering ap-

plications [138–140]. These latter systems are both very large (∼ 100 m), and sensitive

94

Chapter 6. All-Optical Controllable Delay Line 95

to thermal effects – rendering them unsuitable for microchip interconnects.

On-chip planar lightwave circuitry provides a stable integrated platform for inter-

mediate delays, ranging from sub-bit to 100s of bits. Microresonators have been stud-

ied to provide on-chip delays, but suffer from narrow bandwidths and extremely high

losses [31, 33, 35–37, 40, 41, 49, 58–69]. An alternative approach using simple differing-

length waveguides has already promised to offer much lower losses with significantly

higher bandwidths [58]. A switchable PLC delay was recently reported with switching

times only on the order of a few ms [70], and thus incapable of handling the next-

generation data rates of 40 Gbit/s to 1 Tbit/s [50, 51].

The challenge: Can an PLC switchable optical delay line be fabricated with the

following specifications: greater than 100-bit delay, capable of handling pulse widths

on the order of 1 ps, a chip size of ∼ 10 mm × 10 mm (i.e. the size of a standard

microelectronics chip), relatively low loss (i.e., much less than 20 dB), and a switching

time on the order of 1 ps or less?

In this chapter, the design and performance of a low-loss all-optical controllable delay

line PLC in AlGaAs is described that uses an ultrafast nonlinear optical switch to select

between a 0-bit or a 126-bit delay, operating at room temperature, over telecommunica-

tion wavelengths, with a bit-length of 1.5 ps.

The design and fabrication of the device is presented in Section 6.2, followed by the

experimental results in Section 6.3. Section 6.4 discusses the design of an improved

second-generation delay line device. Section 6.5 offers a discussion on extending the

PLC delay-line for both discretely-tunable and continuously tunable delays, followed by

concluding statements in Section 6.6.

6.1.1 Nonlinear Directional Coupler Behaviour

The description of the nonlinear behaviour of a two-waveguide directional coupler is

presented in this section, following Jensen [82].


0

0.2

0.4

0.6

0.8

1

Pin = 0.1 Pc

Pin = Pc

Pin = 5 Pc

z

No

rma

lize

d P

ow

er i

n

Wa

veg

uid

e A

Lc 2Lc 3Lc

All power

in A

All power

in B

Figure 6.1: Behaviour of a nonlinear directional coupler versus propagation distance.

The behaviour of the DC is plotted versus propagation length, z, in Fig. 6.1, at three

power levels: 0.1Pc, Pc, and 5Pc. The three curves are normalized to their respective

input powers, and are referenced to the half-beat length, Lc, on the abscissa. At low

powers (i.e. in the linear regime), the DC is seen to efficiently transfer energy completely

from one channel to the other; at the so-called critical power, Pc, the power transfer

asymptotically approaches 50%; and at high powers, the energy remains confined in the

input waveguide, and exhibits only low-amplitude high-frequency oscillations incurred by

dephasing Kerr effects in the input waveguide. The definition of the critical power (in SI

units) is [82],

Pc =κλoAeff

πn2

. (6.1)

The switching curve of a half-beat length NLDC is shown in Fig. 6.2, where the output

versus the input intensity is plotted with respect to the input power (normalized to the

critical power, Pc). At low powers, all the light emerges from waveguide B, while at

sufficiently high powers, all the light emerges from waveguide A.


0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Pin / Pc

All power

in A

All power

in B

No

rma

lize

d P

ow

er

in W

ave

gu

ide

A

Figure 6.2: NLDC switching curve for a half-beat length coupler.

6.2 Optical Delay Line: Design and Fabrication

The self-switched all-optically controllable delay line (OCDL) was implemented using a

nonlinear directional coupler (NLDC) switch, in which a pulse, depending on its intensity,

traverses either the delayed or undelayed path through the chip. The default path tra-

versed by the low-intensity pulses is layout specific; schematic examples of an undelayed-

and delayed-default device are illustrated in Fig. 6.3 and Fig. 6.4, respectively. In both

schematics, the NLDC is a half-beat length coupler (see Section 6.1.1).

For the first-generation device, the delayed-default layout (Fig. 6.4) was chosen to

minimize wafer real-estate. The device functions as follows: at low intensities, the light

fully couples from the straight-channel bus waveguide into the delay-racetrack through

the half-beat length directional coupler. Upon one full cycle through the delay line,

the light fully couples back into the bus, thereby incurring the fixed-time delay. At

high intensities, the instantaneous nonlinear index change induced in the bus waveguide

switches the directional coupler such that the pulse exits the structure undelayed.


NLDC

NLDC Delay Line

Figure 6.3: Schematic of the undelayed-default OCDL.

6.2.1 OCDL Design

The delayed-default OCDL was designed using the 24/18/24 AlGaAs wafer (Section 3.4),

and patterned to produce the straight bus channel (serving as the chip’s input and out-

put), and the racetrack delay line, coupled together using the NLDC. Because of the

instantaneous response of the Kerr nonlinearity in AlGaAs, this device is considered an

ultrafast switch, capable of switching on timescales much shorter than 1 ps. The NLDC

was designed to have a half-beat coupling length, Lc = 2.0 mm. The racetrack, was de-

signed with 2.0 mm straight sections and 2.0 mm bend radii, resulting in a circumference

of L = 16.57 mm.

NLDC

Delay Line

Figure 6.4: Schematic of the delayed-default OCDL.


The single-mode waveguides were designed at a wavelength of 1550 nm. The in-

plane confinement was achieved by photolithography and dry etching of 3.65 µm wide

waveguides. Full vectorial finite difference modal analysis (FDMA) was used to determine

that a 2.0 µm deep etch was required to limit radiation losses around the 2.0 mm radius

racetracks to 0.62 dB per roundtrip. The resulting fundamental mode had a third-order

effective area of 6.25 µm2 [96], an effective refractive index of 3.258, and a group index

of ng = 3.423 (for both TE and TM polarizations). With this waveguide structure, the

target NLDC centre-to-centre waveguide separation was 4.9 µm (i.e., a waveguide gap

of 1.25 µm). Transition losses (incurred from the modal mismatch between the straight

and curved portions of the racetrack), calculated to be only 0.12 dB per transition, were

not compensated in this first-generation device. The lowest order linear TE supermode

of the NLDC is shown in Fig. 6.5. Based on the racetrack-geometry and the computed

group index, the estimated pulse delay was expected to be Lng/c = 189 ps at 1550 nm.

The dependence of group index with respect to wavelength introduces only a ±0.6 ps

variation in the delay.

Because the etch extends 0.5 µm into the wafer’s core layer, a slight polarization

dependence was expected. From the supermode ∆k method (Section 2.2), the half-beat

coupling lengths were determined to be LcTE= 1.85 mm, and LcTM

= 2.11 mm, for

0

0.4

0.8

0.2

0.6

1

Figure 6.5: Lowest-order TE supermode of the NLDC.


the two polarizations, respectively. This corresponds to linear coupling coefficients of

κTE = 0.84 mm−1 and κTM = 0.74 mm−1, respectively.

6.2.2 Device Fabrication

The devices were patterned onto an 8 mm × 6 mm 24/18/24 AlGaAs chip using stan-

dard photolithographic techniques with hard-contact mask alignment (see Appendix B).

The post-etched device geometry parameters were measured using scanning electron mi-

crographs (see Figs. 6.6 and 6.7): waveguide width = 3.63 ± 0.05 µm, waveguide gap

= 1.28 ± 0.05 µm, and etch depth (to the wall-floor corner) = 2.00 ± 0.05 µm. Be-

cause of the floor-tilt, the actual etch depth was slightly deeper, resulting in stronger

confinement, consequently increasing the actual half-beat length of the coupler. The

post-etched device was then remodeled to determine the actual third-order effective area

to be Aeff = 6.05 µm2, and the actual half-beat lengths for TE and TM polarizations to

be LcTE= 2.38 mm and LcTM

= 2.71 mm, respectively. TE polarization was therefore

chosen, since the LcTEwas closer to the target Lc = 2.0 mm.

Because the fabricated half-beat coupling length was longer than the 2.0 mm cou-

pling length defined by the mask layout, the NLDC was actually undercoupled, allowing

only 90% of the light into the delay line. Following Jensen [82] (whose analysis describes

Figure 6.6: SEM top view of the OCDL’s nonlinear directional coupler.


Figure 6.7: SEM of the OCDL waveguide cross section.

continuous-wave light), this undercoupling resulted in a modified switching curve (com-

pared with a half-beat length NLDC), as illustrated in Fig. 6.8. The NLDC critical power

was computed to be Pc = 224 W for TE polarization.

For temporally Gaussian pulses, the OCDL device performance would suffer because

the low-intensity wings of the optical pulses would be inevitably coupled through the di-

rectional coupler into the delay racetrack, as illustrated in Fig. 6.9. This “temporal-wing”

effect consequently reduces the achievable contrast between the switched and unswitched

states, if measured on a time-averaging photodetector [73].


in c

ou

tin

1.8

Figure 6.8: Calculated switching curves of the fabricated NLDC and a half-beat lengthNLDC.

Figure 6.9: The inevitable delay of the low intensity wings of a temporal pulse.


6.3 Experiment and Results

6.3.1 Experimental Setup

The experimental setup is shown in Fig. 6.10. A Ti:sapphire pumped optical parametric

oscillator (OPO) laser at a repetition rate of 78.8 MHz produced nearly transform-limited

pulses, whose centre wavelength was tunable from 1530 to 1610 nm, with a spectral width

of 2 nm. An autocorrelation [141] of the laser pulses is plotted in Fig. 6.11, with a full-

width half-maximum (FWHM) width of ∆τFWHMauto = 2.17 ps. The actual FWHM width

of the pulses (assuming a Gaussian profile) is ∆τFWHM = ∆τFWHMauto /1.41 = 1.54 ps.

The launch power was varied using a motorized continuously rotatable half-waveplate

+ polarizer combination. The pulses were launched into and out of the bus waveguide

using x40 microscope objective lenses (with a focal length of 4.5 mm). Care was taken

to properly factor out the losses and wavelength dependence of the experimental setup.

Fresnel reflection losses of 28% at the front cleaved facet of the chip and a 70% mode

overlap resulted in a 50.4% coupling efficiency into the chip; the maximum average power

launched into the chip (i.e., just inside the front facet) was Pave = 51.7 mW.

NLDC

Racetrack

25 GHz

Photodetector

Data Analyzer

Lens Lens

Delayed &

Undelayed

Output

PulsesHalf-waveplate

Polarizer

OPO

Figure 6.10: Schematic of the OCDL experimental setup.


Figure 6.11: Low-power autocorrelation of the OPO pulse.

Peak Power versus Average Power

The peak power of the light entering the device, Ppeak, is extracted from the measured

average power, Pave, using the following formula (assuming a Gaussian temporal pulse

profile in the form e−t2/τ2):

Ppeak =Pave

τ√

πf(6.2)

where f is the repetition rate (78.8 MHz), and τ is the 1/e temporal half-width of

the Gaussian pulse: τ = ∆τFWHM/1.665 = 0.92 ps [72]. From this calculation, the

peak power that was launched into the device was Ppeak = 402 W, which is 1.8 × Pc

(Pc = 224.1 W). With the post-fabrication effective area, Aeff = 6.05 µm2, the peak

intensity within the waveguide (inside the front facet of the chip) was therefore, Ipeak =

Ppeak/Aeff = 6.65 GW/cm2.

6.3.2 OCDL Measurements

The device was found to perform slightly better for TE polarization launch as expected

(see Section 6.2.2). The linear loss in the waveguides was measured to be 2.6 dB/cm using


the cutback method – on the same order of previous reports using the same 24/18/24

AlGaAs wafer [105]. Thus, the device loss in the delayed state (delay line length, L =

16.57 mm, plus the 8 mm chip length) was 6.4 dB.

To measure the time-resolved response of the OCDL, a BBO-crystal cross-correlator

was built (using the OPO idler pulse, ∆τFWHM = 1.15 ps, as the reference) [141]. The

device output is plotted in Fig. 6.12 at two different input powers, where the undelayed

pulse is used as the time marker, t = 0.

Because the cross-correlator was found to be impractically slow in gathering the re-

quired data, a New Focus 25-GHz InGaAs photodetector (PD) (connected to a digital

data analyzer) was used instead. Measurements of the OPO signal pulse (1.5 ps FWHM)

using the cross-correlator and the PD are compared in Fig. 6.13, revealing that the time

response of the PD broadens the pulse by ∼ 100 ps, and produces a tail extending well

past 300 ps (from its peak). Nonetheless, the PD was acceptable in resolving the rela-

tively large 189-ps delay from the OCDL device. The PD’s response scales linearly, and

exhibits a slight time slip, as a function of power, shown in Fig. 6.14.

0

1

2

3

x 10

-3

2

2

Figure 6.12: Cross-correlation of device output and OPO idler.


-100 0 100 200 3000.0

0.2

0.4

0.6

0.8

1.0

No

rma

lize

d

Inte

nsity (

GW

/cm

2)

Time (ps)

Figure 6.13: Cross-correlation of 1.5 ps OPO input pulse (solid magenta line) comparedwith its impulse response measured on the fast photodetector (dashed blue line).

0 200 400 600

Inte

nsity (

arb

. u

nits)

Time (ps)

Increasing Intensityfrom 0.55 GW/cm2 to

6.65 GW/cm2

Figure 6.14: PD response of 1.5 ps OPO pulse with no device in the optical path.


Because both the delayed and undelayed pulses are detected sequentially by the PD,

the measured response would result in the temporal broadening of both, and a heightening

of the delayed pulse. The measured device response with a 1550-nm TE launch using

the PD is shown in Fig. 6.15, with peak intensities (inside the front surface of the chip)

ranging from 0.55 to 6.65 GW/cm2. For comparison, simulations following Jensen’s

approach [82] are shown in Fig. 6.16 (which incorporate the time response of the PD),

showing very good agreement with the measurements. The simulations included the

temporal-wing effect of the Gaussian pulses (discussed in Section 6.2.2), but did not

include the time slip exhibited by the PD response (Fig. 6.14).

The PD easily resolved the peaks of the two broadened pulses, as expected. In order

to extract the actual peak heights of the undelayed (fast) and delayed (slow) pulses,

(Af and As, respectively) from the measured peak heights (Pf and Ps, respectively), a

deconvolution of the PD response (Fig. 6.14) was required. The normalized switching

ratios, Tf,s = Af,s/(Af + As), were characterized over a wavelength range of 1530 to

1610 nm (plotted in Fig. 6.17). As expected, the switching contrast was limited because

of the temporal-wing effect (see Section 6.2.2). The 50% crossover intensity decreases as

a function of wavelength, as illustrated in Fig. 6.18, due to the wavelength dependence of

the critical power, NLDC half-beat coupling length, coupling efficiency, and losses around

the racetrack.


2

2

Fast

Channel

Slow

Channel

Pf

Ps

Figure 6.15: Device output using the PD with a 1550 nm TE launch, 1.5 ps input pulse.

2

2

Fast

Channel

Slow

Channel

Figure 6.16: OCDL simulations.


0 1 2 30.2

0.3

0.4

0.5

0.6

0.7

0.8

Intensity (GW/cm2)

Norm

aliz

ed S

witchin

g

Ra

tio

1530 nm

1530 nm

1610 nm

1610 nm

Tf

Ts

50% crossover intensity @ 1530 nm

1 2 30 5 64

Figure 6.17: Normalized switching ratios Ts (solid line) and Tf (dashed line) from 1530 nmto 1610 nm.

1530 1550 1570 1590 16100.5

1

1.5

2

2.5

3

3.5

Figure 6.18: Wavelength dependency of the 50% crossover intensity.


6.3.3 Self Phase Modulation

The self phase modulation (SPM) of the waveguide was characterized for a stand-alone,

8 mm long waveguide. The spectral distribution over the power range is plotted in

Fig. 6.19, revealing a phase shift of ∼ 3.5π due to SPM at the maximum peak input

power of 402 W. These results agree well with previous reports of NLDC devices built

in the same 24/18/24 AlGaAs wafer [73].

Because both self- and cross-phase modulation are a function of the temporal shape of

the high-intensity pulse propagating through the waveguide [72], the spectral broadening

of the data signal can be significantly reduced by using an independent control pulse

with a temporal width larger than the signal’s [142]. Following the work presented by

Villeneuve [143], if the short signal pulse temporally lies within the slowly varying high-

power control pulse (as illustrated in Fig. 6.20), the signal pulse would incur minimal

cross-phase modulation as it propagates through the NLDC, and switch almost com-

pletely. Measurements of Villeneuve’s improved switching ratios are shown in Fig. 6.21.

0.5 5 50

1550

1570

1590

1580

1560

1540

1600

Figure 6.19: Spectral broadening.


High intensity

Control pulse

Low intensity

Signal pulse

time

No

rmaliz

ed P

ow

er

Figure 6.20: Independent control pulse to reduce spectral broadening.

Figure 6.21: Switching curves for low-power 150-ps signal pulse, and high-power 800-pscontrol pulse [143].

6.4 Future Work

Further work in this project should focus on both functional and performance improve-

ments:

• Reduce the NLDC critical power by increasing the DC half-beat coupling length.

• Delay only the switched pulse (i.e., default operation: undelayed).

• Bring the NLDC to the PLC input to maximize the power entering the switch.


• Employ an independent control pulse.

• Decrease the polarization dependence of the device.

• Decrease losses by incorporating transitional offsets between bent and straight re-

gions of all waveguide paths [144].

• Improve the NLDC switching-contrast by employing square pulses [145] or soli-

tons [146].

6.4.1 New OCDL Design

The formula for critical power, Eq. (6.1), is rewritten here for convenience:

Pc =κλoAeff

πn2

. (6.1)

In the new OCDL design, the half-beat coupling length was increased from 2 mm to

be 10 mm, which corresponds to a decrease in the linear coupling coefficient, κ, (and

consequently Pc), by a factor of 5. An independent control pulse, operating at 1500 nm,

further reduces the critical power by an additional 19.4%, noting that the Kerr nonlinear

index is a decreasing function of wavelength (see Section 3.3). With these improvements,

the newly designed critical power is expected to be only PGen2c = 36.1 W – i.e., only

0.16× PGen1c .

A schematic of the new OCDL design is shown in Fig. 6.22. Independent waveguides

couple the signal and control pulses into the two input branches of the NLDC. The new

device is configured to pass the signal through the chip undelayed, by default; only upon

the application of the control pulse will the signal be routed through the arbitrarily long

delay line. A control pulse with a temporal duration longer than the signal would reduce

spectral broadening (incurred from cross-phase modulation) and improve the switching

contrast of the device. An efficient Y-branch at the back-end combines the delayed and


the undelayed pulses onto the same output waveguide (with a 3 dB penalty). Alterna-

tively, an output switch would ensure that the signal is always emitted from the same

port with only minimal switch losses (typically less than 1 dB).

24 microns

10 mm

Radius2 mm

Signal

Control

Delay Section

Y-branchNLDC

Figure 6.22: Schematic of the second-generation OCDL.

Maximum Achievable Delay

The maximum delay achievable in our device is limited on the tolerable loss penalty (in

addition to coupling and reflection losses). Using a spiral waveguide design with a gap

between waveguides of 20 µm to maximize the PLC real-estate, an estimate of the bit-

delay (assuming a 1.5 ps bit-length) can be calculated. For a 3 dB tolerable loss penalty,

the achievable bit delay is 1181 bits, while for a 6.8 dB tolerable loss penalty, the bit

delay is 2573 bits.

6.5 Extensions and Applications

Applications such as packet control, mid-range buffering, or optical control of phased

array antennas require discretely tunable delays [147]. By cascading N OCDL stages in

series, each with an independent control, and with delays following a binary sequence

(i.e., D, 2D, 4D, 8D, ...), then any discrete delay (in increments of D) from 0 to (2N−1)D

is achievable.


For timing-jitter compensation, a device that is capable of producing continuously

tunable delays up to ∼ 1 bit-length is required. Independent temperature or electro-

optic tuning of the delay line can provide the necessary control required for the delay

adjustment. For the current device, a tunability of 1.5 ps (i.e., the bit-length) can be

achieved by a temperature change of +112o C (see Section 3.2). While this temperature

change can be achieved using local heaters on the PLC device, the tuning speed would

be limited to timescales much longer than the pulse duration. Studies on the speed of

electro-optic tuning should be conducted for faster jitter control.

6.6 Conclusion

An ultrafast self-switched all-optical AlGaAs PLC racetrack delay line was demonstrated

over the telecommunication bands at room temperature. This proof-of-principle device

achieved a delay of 126 bits using 1.5 ps, with a device loss of only 6.2 dB, on a chip size

of 8 mm × 6 mm. To date, this is the lowest loss and fastest switchable delay reported.

Alternative design layouts would permit the low intensity light to preferentially exit

the device undelayed with or without an independent control, while providing delays

spanning sub-bit to greater than 1000-bits, limited only by device losses.

Chapter 7

Summary and Conclusions

The work presented in this dissertation studied the use of coupled waveguides in three

planar lightwave circuit devices built in the aluminum gallium arsenide material system.

In Chapter 2, light confinement in optical waveguides was investigated, followed by

a study of light coupling in a two-waveguide system using two approaches: (1) the “su-

permode ∆k” method, which analyzes the two-waveguide system as a whole, and (2) the

Hamiltonian formulation of coupled mode theory. The three projects presented herein

employed different methodologies, each specifically suited to the project at hand.

Chapter 3 reviewed the optical and mechanical properties of aluminum gallium ar-

senide, making it an excellent material for planar lightwave circuits:

• The high linear refractive index minimizes PLC chip sizes.

• Its crystalline structure remains lattice-matched across the aluminum concentration

range, x (from 0 and 1).

• The material index can be varied by ∆n = 0.48.

• Its large nonlinear refractive index is well suited for all-optically controlled devices.

• Mature processes have been developed for PLC fabrication in AlGaAs.

115

Chapter 7. Summary and Conclusions 116

To minimize the detrimental effects of two-photon absorption, a lower aluminum concen-

tration limit of 14% (at 1550 nm) was respected in the two AlGaAs wafer designs used

in this work.

The work presented in this dissertation targeted optical signal processing in the spa-

tial, spectral and temporal domains. In the following three sections, a summary is pre-

sented of the solution to an identified research challenge for each project.

7.1 Exact Dynamic Localization

Because of the mathematical equivalence between the paraxial wave equation and the

Schrodinger equation, we can (a) use quantum theory to provide new insight into the

design of novel optical processing devices, and (b) use planar lightwave circuits in the

study of certain quantum effects, especially those too difficult to observe directly in the

quantum domain. One such effect, called exact dynamic localization (EDL), had not yet

been demonstrated in any system.

7.1.1 Summary: Exact Dynamic Localization

In Chapter 4, the one-band Schrodinger model was used to design an optical EDL demon-

stration using a strongly coupled curved optical waveguide array with an effectively dis-

continuous ac square-wave curvature profile.

Exact dynamic localization was experimentally demonstrated for the first time (in

any system) over four periods. Introduction of a new observation technique to accurately

image the desired waveguide plane successfully served to characterize the device’s wave-

length dependence, while a staggered-experiment approach spatially mapped the beam

evolution through the structure – the first-ever beam mapping of exact dynamic local-

ization. The devices performed in very close agreement with the design targets, and in

excellent correspondence with theory.


7.1.2 Future Work: Exact Dynamic Localization

Further work in this area should investigate the empirical performance in PLCs of: (1)

EDL generated using “deviated square-wave” curvature profiles; (2) quasi-Bloch oscilla-

tions comprised of alternating plus constant field curvatures – an effect not yet demon-

strated in any system; and (3) broadband filtering, power tapping, and other optical

processing phenomena using infinite and finite curved waveguide arrays.

The impact of this work is extremely exciting in that it offers optical engineers a

completely new way to approach PLC design, promising improved and/or brand new

optical signal processing functionality.

7.2 SCISSORs

Optical signal processing (such as filtering, switching, and add-dropping) have been per-

formed in PLCs using microresonators. By cascading these microresonators in either

serial (CROW) or parallel (SCISSOR) configurations, slow-light and dispersion control

can be demonstrated. Furthermore, if fabricated in a nonlinear material system, optical

devices can be built to show functionality such as soliton formation and optical logic.

In a SCISSOR configuration, a sufficiently large number of rings would create a fully

formed Bragg gap, with peak intensities of equal strength to the resonator peaks in the

drop spectrum. The interplay between the resonator and Bragg gaps can be engineered to

produce a number of interesting effects in the dispersion characteristics of the device. To

date, however, SCISSORs in highly nonlinear materials had been limited to a maximum

of only three rings due to extremely tight fabrication tolerances.

7.2.1 Summary: SCISSORs

In Chapter 5, the design, fabrication and characterization of a multi-ring SCISSOR de-

vice in AlGaAs were described. A matrix-based Hamiltonian formulation of coupled


mode theory was used to design the SCISSOR structures in one-, two-, four- and eight-

ring configurations. The design of the high-index contrast sub-micron waveguides (or

nanowires) were described; these nanowires were used for the SCISSOR’s directional

couplers (called nano-DCs). The individual nano-DCs demonstrated sinusoidal power

transfer characteristics, as expected.

A SCISSOR with more than three rings in AlGaAs has been demonstrated for the

first time, revealing that at least eight rings are necessary to create fully-formed Bragg

peaks. The work presented here is a significant step towards the goal of observing linear

and nonlinear photonic bandgap phenomena on PLCs.

7.2.2 Future Work: SCISSORs

A full dispersion characterization must be completed on the fabricated SCISSOR struc-

tures (including group delay, group velocity, and group velocity dispersion measure-

ments). Tests in both the linear and nonlinear regime will soon be performed using a

long pulsed laser system. Further improvements to microresonator losses is of paramount

importance for the successful evolution of these devices.

7.3 Optically Controllable Delay Line

The International Technology Roadmap for Semiconductors [52] states that optics should

be investigated to provide the necessary bandwidth and isolation required for next-

generation microchip interconnects approaching data rates of 1 Tbit/s (i.e., bit lengths

on the order of 1 ps) [52]. Based on size alone, PLCs seem best suited to accomplish

the task of data routing, as well as providing the switchable > 100-bit optical delays

needed for data synchronization. To date, no low-loss, fast, switchable PLC delay has

been reported capable of delaying more than a few bit-lengths.

While much current investigation in PLC optical delays is being conducted using


microresonators, large losses (on the order of 20 dB) prohibit their practicality. Differing-

length waveguides, therefore, promise a low-loss alternative.

7.3.1 Summary: Optically Controllable Delay Line

Chapter 6 presented the design details of an optically controllable delay line built in

AlGaAs. An ultrafast nonlinear directional coupler (described in Chapter 2) was used to

switch the optical data stream (with 1.5 ps pulse-lengths) between two paths of differing

lengths. The high AlGaAs index (∼ 3) kept the device size to only 8 mm × 6 mm. The

device was capable of producing a 126-bit delay, and suffered only 6.2 dB of propagation

losses through the delay path.

This is the first report of an > 100-bit ultrafast swithcable optical delay PLC with

losses less than 7 dB.

7.3.2 Future Work: Optically Controllable Delay Line

The requirements for a second-generation device are rewritten here (from Section 6.4):

• Reduce the NLDC critical power by increasing the DC half-beat coupling length.

• Delay only the switched pulse (i.e., default operation: undelayed).

• Bring the NLDC to the PLC’s input to maximize the power entering the switch.

• Employ an independent control pulse.

• Decrease the polarization dependence of the device.

• Decrease losses by incorporating transitional offsets between bent and straight re-

gions of all waveguide paths.

• Improve the NLDC switching-contrast by employing square pulses or solitons.


Further extensions of this device should be investigated, including discretely tun-

able delays using a cascade of binary-sequenced stages, and sub-bit continuously tunable

delays for jitter compensation.

7.4 Final Comments

The goal of this dissertation was to demonstrate the versatility of planar lightwave cir-

cuits for optical signal processing. The three projects described herein examined optical

signal processing in the spatial, spectral and temporal domains, addressing a specific

technological need.

While the scope of the projects was indeed widespread, the commonalities between

them bridged barriers. The contributions presented in this work, both theoretical and

empirical, are extensive and essential to all technological areas that currently use, or may

one day use, planar lightwave circuitry.

Appendix A

Nonlinear Polarization Density in

AlGaAs

In this Appendix, we investigate the nature of the polarization density in AlGaAs. We

begin by defining the polarization density in terms of the electric susceptibility, noting

that for AlGaAs, the linear and nonlinear susceptibility terms reduce to scalars. We then

expand the polarization density in terms of the complex representation of the electric

field. We conclude by relating the third-order nonlinear electric susceptibility to the

Kerr nonlinear refractive index.

A.1 Nonlinear Susceptibility and Polarization

Density in AlGaAs

The (unitless) nonlinear electric susceptibility is a measure of how easily the medium

electrically polarizes in the presence of electric fields, and is defined as

χe = χ(1) + χ(2) ~E + χ(3) ~E ~E + ... (A.1)

121

Appendix A. Nonlinear Polarization Density in AlGaAs 122

where χ(1) is the linear electric susceptibility, and χ(2), χ(3), etc. describe the second,

third, and higher order nonlinear susceptibilities, respectively.

The polarization density, ~P , induced in the medium by the presence of an electric

field, ~E, is defined as

~P = εoχ(1) ~E + εoχ

(2)T : ~E ~E + εoχ

(3)T

... ~E ~E ~E + · · · (A.2)

= ~P (1) + ~P (2) + ~P (3) + · · · (A.3)

where χ(1) is the scalar linear electric susceptibility, χ(2)T , χ

(3)T , etc. are the tensorial

nonlinear electric susceptibilities, and where ~P (1) = εoχ(1) ~E is the linear polarization

density, ~P (2) = εoχ(2)T : ~E ~E, and ~P (3) = εoχ

(3)T

... ~E ~E ~E, are the second and third order

nonlinear polarization densities, respectively. For the purposes of our development, we

shall only include nonlinearities of significant strength found in AlGaAs: up to and

including the third order [96].

The real-valued, monochromatic, time-harmonic electric field, ~Er, can be defined

in terms of its complex representation, ~Ec = ~Eoejωt, and its complex conjugate (c.c.),

~E∗c = ~E∗

oe−jωt,

~E = ~Er (A.4)

= 12

(~Ec + ~E∗

c

)(A.5)

= 12

(~Eoe

jωt + ~E∗oe−jωt

), (A.6)

where ~Eo = ~Ao(x, y, z)e−jkz is the complex spatial electric field, with amplitude, ~Ao(x, y, z).

We will consider linearly-polarized fields: TE polarization where ~Ao = Axx, or TM po-

larization where ~Ao = Ayy, (where x and y are the unit vectors along the x and y axes,

respectively).

The polarization density, ~P , (which is a physically measurable quantity, and hence


real-valued), can also be defined in terms of its complex representation, ~P = 12

(~Pc + ~P ∗

c

).

A.1.1 Linear Polarization Density

The linear polarization density in complex form, ~P(1)c , can be expanded as

~P (1)c = εoχ

(1) ~Ec. (A.7)

A.1.2 Second Order Nonlinear Polarization Density

Second order nonlinearities, defined by ~P (2) = εoχ(2)T : ~E ~E, can only occur in crystals with

no inversion symmetry (non-centrosymmetric), such as AlGaAs. Because of the quadratic

relationship of ~P (2) with ~E, a reversal in the direction of the electric field, ~E → − ~E,

would not change the direction of the polarization density vector ( ~P → ~P ). However, in

centrosymmetric crystals, a change in the direction of the electric field must result in a

change in direction of the polarization vector, by virtue of the physical symmetry. Thus,

χ(2) must be zero in centrosymmetric media to resolve this contradiction [148].

From Eq. (A.2), the second order nonlinear polarization density in complex form,

~P(2)c , for a single frequency can be expanded using Eq. (A.6) as

~P (2)c = 1

2εoχ

(2)T :

(~Eo

~Eoej2ωt + | ~Eo|2

). (A.8)

We see that the second order polarization is comprised of a 2ω term leading to second

harmonic generation (SHG), and a zero-frequency term leading to optical rectification

(OR). The OR term creates a static electric field in the material, and hence, does not lead

to the generation of electromagnetic radiation [148]. If no phase-matching schemes are

intentionally implemented in the PLC design (which is the case for the devices presented

in this thesis), the SHG term is very inefficient [148, 149]. Therefore, for most AlGaAs


PLCs, second order nonlinear effects can be neglected from the analysis, i.e.,

~P (2)c ' 0. (A.9)

A.1.3 Third Order Nonlinear Polarization Density

From Eq. (A.2), the third order nonlinear polarization density in complex form, ~P(3)c , for

a single frequency can be expanded using Eq. (A.6) as

~P (3)c = 1

4εoχ

(3)T

...(

~Eo~Eo

~Eoej3ωt + 3 ~Eo| ~Eo|2ejωt

). (A.10)

Here, we see that the third order polarization is comprised of a 3ω term leading to

third harmonic generation (THG) and a 1ω term that is proportional to the nonlinear

contribution to the refractive index, described in more detail in the next section. As is

the case for SHG, the inefficient THG term can be neglected in the analysis of non phase-

matched devices [72,149].

AlGaAs wafers are normally grown in the [001] direction, and have (110) cleavage

planes (i.e., the input and output facets of the PLC) as shown in Fig. A.1 [150]. Note

that the crystallographic cartesian coordinates, (x, y, z), are oriented differently from the

(x, y, z) coordinates typically used in optics, where the z direction is used to define the

direction of propagation. TM (y-polarized) and TE (x-polarized) light incident normally

upon the (110) plane are defined in the crystallographic coordinate system as ~ETM = zEz,

and ~ETE = xEx/√

2 + yEy/√

2 (|Ex| = |Ey|), respectively.

Confining this discussion to a single frequency, the third-order nonlinear susceptibility

tensor, χ(3)T , for AlGaAs (and all 43mTd crystals), has the following non-zero tensor


z

y

= [001] direction

(110) cleavage plane

(i.e., input and output facets)

Incident light

Directed along the [110] direction

x

z

y

x

Figure A.1: AlGaAs crystal planes. Blue cube is the unit-cell.

elements (where the tetragonal symmetries are explicitly equated) [148]:

I : χ(3)xxxx = χ

(3)yyyy = χ

(3)zzzz

II : χ(3)xxyy = χ

(3)xxzz = χ

(3)yyxx = χ

(3)yyzz = χ

(3)zzxx = χ

(3)zzyy

III : χ(3)xyxy = χ

(3)xzxz = χ

(3)yxyx = χ

(3)yzyz = χ

(3)zxzx = χ

(3)zyzy.

From Boyd [148],the third-order nonlinear polarization can be expressed as

P(3)i = εo

∑

jkl

χ(3)ijklEjEkEl (A.11)

where the indices ijkl refer to the cartesian components of the fields. Applying the

nonzero elements of χ(3)T to Eq. (A.11), we obtain:

P (3)x /εo = χ(3)

xxxxE3x + χ(3)

xxzzExE2z + χ(3)

xxyyExE2y + χ(3)

xzxzExE2z + χ(3)

xyxyExE2y (A.12)

P (3)y /εo = χ(3)

yyyyE3y + χ(3)

yyzzEyE2z + χ(3)

yyxxEyE2x + χ(3)

yzyzEyE2z + χ(3)

yxyxEyE2x (A.13)

P (3)z /εo = χ(3)

zzzzE3z + χ(3)

zzyyEzE2y + χ(3)

zzxxEzE2x + χ(3)

zyzyEzE2y + χ(3)

zxzxEzE2x. (A.14)


Substituting ~ETE = xEx/√

2 + yEy/√

2 (|Ex| = |Ey|) for TE polarization, we obtain:

P (3)xTE

= P (3)yTE

= εoE3x

(χ(3)

xxxx + χ(3)xxyy + χ(3)

xyxy

)= εoE

3xχ

(3)TE eff (A.15)

P (3)zTE

= 0. (A.16)

For TM polarization,

P (3)xTM

= P (3)yTM

= 0 (A.17)

P (3)zTM

= εoE3zχ

(3)zzzz = εoE

3zχ

(3)TM eff . (A.18)

Eq. (A.15) to (A.18) show that the tensorial third-order nonlinear susceptibility for

AlGaAs can be reduced to effective scalar values for TE and TM polarizations, where

χ(3)TE eff = χ

(3)xxxx + χ

(3)xxyy + χ

(3)xyxy, and χ

(3)TM eff = χ

(3)zzzz. Below, we’ll simply use χ(3) to

designate the scalar effective third-order susceptibility, keeping in mind that this is valid

for either TE or TM polarizations. We can rewrite Eq. (A.10), neglecting the THG term

as

~P (3)c = 3

4εoχ

(3) ~Eo| ~Eo|2ejωt = 34εoχ

(3) ~Ec| ~Ec|2. (A.19)

A.1.4 Polarization Density of AlGaAs

Combining Eqs. (A.7), (A.9), and (A.19), the overall polarization density for AlGaAs

may now be written as

~Pc = ~P (1)c + ~P (2)

c + ~P (3)c (A.20)

= εoχ(1) ~Ec + 3

4εoχ

(3)| ~Ec|2 ~Ec (A.21)

= εoχe~Ec, (A.22)


where the electric susceptibility, χe, is

χe = χ(1) + 34χ(3)| ~Ec|2 (A.23)

= χ(1) + χNL (A.24)

where χNL = 34χ(3)| ~Ec|2.

A.2 Kerr nonlinearity

The complex refractive index of the medium, n, is defined to be

n2 = 1 + χe

= 1 + χ(1) + 34χ(3)| ~E|2

= n2o + 3

4χ(3)| ~E|2

= n2o

(1 +

3

4

χ(3)

n2o

| ~E|2)

(A.25)

where n2o = 1 + χ(1) = εr is the complex linear relative electric permittivity. To a

first order approximation, the overall refractive index of the polarizing medium can be

simplified to

n ' no +3

8

χ(3)

no

| ~E|2. (A.26)

The Kerr nonlinear refractive index, n2, is defined as the nonlinear contribution to n,

n = no + n2I (A.27)

where the intensity of the electromagnetic field is I = | ~E|2/η, and η =√

µ/ε =√

µ/(εrεo). Therefore,

n2 =3

8

χ(3)

no

η (A.28)


and

χ(3) =8

3

n2no

η. (A.29)

Appendix B

AlGaAs Photolithography and

Etching

PLCs are fabricated using the same tools and processes used in the manufacture of mi-

croelectronic circuits. Photo or electron-beam (e-beam) lithographic techniques literally

pattern the desired planar designs onto the surface of a semiconductor, polymer or glass

microchip. The patterns are then transferred into the chip using either dry or wet etching.

The AlGaAs PLCs presented in this dissertation were etched using dry anisotropic

reactive-ion etching (RIE) to produce (nearly) vertical sidewalls. The patterning method

is chosen based on the dimensions of the waveguides: photolithography is chosen for

waveguides widths on the order of a several microns (with tolerances of ±0.5 µm), while

e-beam lithography is chosen for waveguide widths < 1 µm (with tolerances on the

order of several 10s of nm). Specifically, the EDL and OCDL structures, presented

in Chapters 4 and 6, were patterned using photolithography, while the nanowire and

SCISSOR experiments presented in Chapter 5 were patterned using e-beam lithography.

Section B.1 presents the recipe used in the photolithography and RIE of the 24/18/24

AlGaAs wafer (Section 3). Section B.2 includes some interesting preliminary results of

isotropic (wet) etching to investigate modal confinement, directional coupling and im-

129

Appendix B. AlGaAs Photolithography and Etching 130

proved sidewall roughness. The final section (Section B.3) presents details of a newly

implemented photoresist crosslinking step. Details of the e-beam lithography and etch-

ing of the nanowires and SCISSOR structures, fabricated by fellow researchers at the

University of Glasgow, are not presented in this appendix.

B.1 AlGaAs Recipe: Anisotropic

The photolithography of the 24/18/24 AlGaAs wafer used in the OCDL and EDL devices

were performed at the Emerging Communications Technology Institute at the University

of Toronto, Canada. The etching was performed at the Institute for Microstructural

Sciences at the National Research Council of Canada. The recipe was optimized for

anisotropic reactive-ion-etching.

B.1.1 Recipe

1. Mask Design

◦ Design chrome-on-quartz mask using L-Edit by Tanner EDA.

◦ Get mask plates fabricated (Compugraphics, Inc.).

2. Preparation

◦ At least four hours prior to procedure, thaw a fresh amount of Shipley SPR511Apositive photoresist dispensed in a small brown bottle (using a 0.2 micron fil-ter).

3. Three-Part Cleaning

◦ Soak all glassware, tools, and samples with acetone for 1 minute.

◦ Blow dry with Ni gun.

◦ Rinse each item with Methanol.


◦ Rinse each item with Isopropanol.


◦ Bake out the solvents on a hotplate set at 120o C for 5 to 10 minutes.


◦ Remove oxide from AlGaAs sample using a 30 sec dip in 1:1 solution of HCL+ water.

4. Spinner Preparation

◦ Set program on the spinner to 6000 rpm (ACL = 005, rpm = 410) for 45seconds.

◦ Ensure that the spinner table is tightly screwed to its axle.

5. Sample Spin

◦ Place the sample on the spinner and set vacuum on.

◦ Dispense HMDS P20 primer onto sample surface using a plastic pipette.

◦ Spin.

◦ Using another plastic pipette, extract SPR511A photoresist from bottle with-out touching the bottle rim.

◦ Dispense SPR511A onto sample surface (no bubbles) in a continuous flow.

◦ Spin.

◦ Perform 0.5 mm edge-bead removal with a semi-dry acetone swab.

◦ Clean interior of spinner with acetone once complete.

6. Soft Bake

◦ Place samples on a hotplate set at 90o C for 3 minutes.

7. Exposure

◦ Follow mask aligner (Karl Suss MA6) setup procedure.

◦ Set mask program to Hard Contact, for 5.0 seconds.

◦ Look away during actual UV exposure.

◦ Need to clean mask with AZ300T and DI water before re-exposing same pat-tern on the mask plate.

8. Development

◦ Fill large sized beaker with DI water.

◦ Fill medium sized beaker with MF-321 developer.

◦ Immerse and swirl sample in developer with tweezers for 45 seconds.

◦ Immediately remove sample from developer, immerse and swirl in DI waterfor 20 seconds.


9. Inspect


◦ Ensure UV filter is set in microscope (to prevent further exposure).

◦ Inspect sample.

◦ Perform further development if required.

10. Crosslinking and Hard Bake

◦ Place samples in UV oven (270 nm, 500 W lamp) for 15 minutes.

◦ Hard bake samples on hotplate set at 100o C.

◦ Again place samples in UV oven for 15 minutes.

◦ Again hard bake samples on hotplate set at 120o C.

11. Etch: Anisotropic

◦ Follow reactive-ion etcher (Trion MiniLock, or Oxford Plasmalab System100)setup instructions.

◦ Open chamber, insert samples, close chamber, run etch recipe (proprietaryrecipe with BCl3+ and Ar+ plasmas).

12. Post-Etch Clean

◦ Bring samples to wetbench.

◦ Put ultrasonic cleaner in wetbench, and turn on heater.

◦ Set hotplate to 80o C.

◦ Fill a medium-sized beaker with DI water.

◦ Pour AZ-300T photoresist stripper into petri dish.

◦ Keep wetbench sash down (for rest of procedure).

◦ Put samples into AZ-300T petri dish (corner first, to break surface tension).

◦ Place petri dish on hotplate set at 80o C for 30 minutes.

◦ Place petri dish into ultrosonic cleaner for 8 minutes.

◦ Remove samples from petri dish, and swirl in DI water for 1 minute.

◦ Dry samples using Ni gun.

13. Inspection

◦ Inspect samples under microscope.

◦ Samples should be free of all photoresist, with clean patterns.

B.1.2 Anisotropic Etch Results

An example of an anisotropic etch using the recipe listed in the previous subsection is

shown in Fig. B.1. The crosslinked and hard baked photoresist is seen sitting atop the

etched waveguides.


Figure B.1: SEM of anisotropic etch in 24/18/24 AlGaAs.

B.2 AlGaAs Recipe: Isotropic

A brief set of isotropic etch tests on the 24/18/24 AlGaAs wafer were conducted to

investigate modal confinement, directional coupling and improved sidewall roughness.

B.2.1 Recipe

The only difference in the isotropic recipe compared with the anisotropic recipe presented

in Section B.1 is step 11:

11. Etch: Isotropic

◦ Prepare a 2:1:1 solution of 48% hydrobromic acid, glacial acetic acid, and

potassium dichromate powder.

◦ Immerse samples in solution for desired time, based on etch rate (see Sec-

tion B.2.2).

◦ Rinse in DI water for 20 seconds.

◦ Dry samples using Ni gun.


B.2.2 Isotropic Etch Results

The isotropic etch rate was determined by conducing three measurements in 10 second

increments. The resulting etch rate was was found to be 0.066 µm/s (or 4.0 µm/min).

The results are plotted in Fig. B.2.

The isotropically etched waveguides exhibit excellent sidewall smoothness, as shown

in Fig. B.3. Additionally, waveguide bends seem to etch quite well, as illustrated in

Fig. B.4.

As with the anisotropic etches, modal confinement is very sensitive to etch depth

(i.e., the duration of the isotropic etch), as well as the actual cross-sectional curvature

profile of the waveguide sidewalls. The lowest-order TE supermode of the DC shown in

Fig. B.3 (with centre-to-centre spacing of 6.3 µm) is shown in Fig. B.5, with a calculated

half-beat coupling length of 5.23 mm. Because the fabricated DC was integrated into a

more complex circuit, it could not be independently measured and compared with the

simulations.

The modal confinement of a 1.95 µm deeply etched waveguide around a 2 mm radius

0 5 10 15 20 25 30

0.0

0.5

1.0

1.5

2.0

Etc

h D

ep

th (

mic

ron

s)

Time (seconds)

Figure B.2: Determination of isotropic etch rate.


Figure B.3: SEM of isotropic etch in 24/18/24 AlGaAs.

Figure B.4: SEM of isotropically etched waveguide bend.


0

0.4

0.8

0.2

0.6

1

Figure B.5: Simulation results of lowest order TE supermode of an isotropically etcheddirectional coupler.

bend was modeled, shown in Fig. B.6. The radiation losses were calculated to be less

than 0.18 dB/90o bend.

0

0.4

0.8

0.2

0.6

1

Figure B.6: Isotropic etch mode around a 2 mm radius bend.

The isotropic etching of AlGaAs seems to be a very promising approach in producing

high quality, low loss waveguides with extremely smooth sidewalls. More work is needed

to characterize the process.

This work was conducted in collaboration with Dr. Alan Bristow, from the Depart-

ment of Physics, University of Toronto, Canada.


B.3 Photoresist Crosslinking and Hard Baking

The UV crosslinking described in line 10 of the AlGaAs recipe (Section B.1) improves

the photoresist cross-section from curved (Fig. B.7) to nearly vertical (Fig. B.8). The

ripples seen in the PR sidewall is caused by UV standing waves during the crosslinking

process.

Figure B.7: Hard baked PR with no crosslinking.

Figure B.8: Hard baked PR with crosslinking.

Appendix C

Microresonator Free Spectral Range

The relationship between the free spectral range and the group index of a microresonator

(or other type of) filter is presented in this appendix.

We denote the effective (phase) index and the group index as n and ng, respectively.

It is well known that

ng = n + ωdn

dω. (C.1)

The cavity modes are determined by the effective index through

n (ωm)ωm

cL = 2πm (C.2)

n (ωm+1)ωm+1

cL = 2π (m + 1) . (C.3)

Assuming that ∆ω = ωm+1 − ωm is small and n (ωm+1) can be approximated as

n (ωm+1) = n (ωm) +dn

dω

∣∣∣∣ωm

∆ω, (C.4)

then Eq. (C.3) can be rewritten as

[n (ωm) +

dn

dω

∣∣∣∣ωm

∆ω

]ωm + ∆ω

cL = 2π (m + 1) . (C.5)

138

Appendix C. Microresonator Free Spectral Range 139

Keeping only terms up to first order in ∆ω on the left of Eq. (C.5),

n (ωm)ωm

cL + n (ωm)

∆ω

cL +

dn

dω

∣∣∣∣ωm

ωm∆ω

cL = 2π (m + 1) , (C.6)

which can be rewritten as

n (ωm)∆ω

cL +

dn

dω

∣∣∣∣ωm

ωm∆ω

cL = 2π (m + 1)− n (ωm)

ωm

cL. (C.7)

Substituting Eq. (C.2) and Eq. (C.1), we obtain

[n (ωm) +

dn

dω

∣∣∣∣ωm

ωm

]∆ω

cL = 2π (m + 1)− 2πm = 2π

ng (ωm)∆ω

cL = 2π. (C.8)

Therefore,

∆ω =2π

ng (ωm)

c

L

∆ν =∆ω

2π=

c

ng (ωm) L. (C.9)

Thus we see that the spectral positions of the modes are determined by the effective

index (Eqs. (C.2) and (C.3)), while the free spectral range is determined by the group

index (Eq. (C.9)).

Bibliography

[1] S. Janz, J. Ctyroky, and S. Tanev, Frontiers in Planar Lightwave Circuit Tech-

nology: Design, Simulation, and Fabrication, vol. 216 of NATO Science Series II.

Mathematics, Physics and Chemistry (Springer, 2006).

[2] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley and Sons,

Inc., New York, 1991).

[3] R. Iyer, Y. S. Liu, G. C. Boisset, D. J. Goodwill, M. H. Ayliffe, B. Robertson, W. M.

Robertson, D. Kabal, F. Lacroix, and D. V. Plant, “Design, implementation, and

characterization of an optical power supply spot-array generator for a four-stage

free-space optical backplane,” Applied Optics 36(35), 9230–9242 (1997).

[4] E. B. Desurvire, “Capacity demand and technology challenges for lightwave systems

in the next two decades,” Journal of Lightwave Technology 24(12), 4697–4710

(2006).

[5] E. Schrodinger, “An Undulatory Theory of the Mechanics of Atoms and Molecules,”

Physical Review 28(6), 1049–1070 (1926).

[6] J. C. Maxwell, “A Dynamical Theory of the Electromagnetic field,” Philos. Trans.

R. Soc. London 155, 459–512 (1865).

140

Bibliography 141

[7] J. Wan, M. Laforest, C. M. de Sterke, and M. M. Dignam, “Optical filters based

on dynamic localization in curved coupled optical waveguides,” Optical Communi-

cations 247, 353–365 (2005).

[8] J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von

Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, “Optical investigation of

Bloch oscillations in a semiconductor superlattice,” Physical Review B 46(11),

7252–7255 (1992).

[9] R.-B. Liu and B.-F. Zhu, “Degenerate four-wave-mixing signals from a dc- and ac-

driven semiconductor superlattice,” Physical Review B 59(8), 5759–5769 (1999).

[10] W. X. Yan, S. Q. Bao, X. G. Zhao, and J. Q. Liang, “Dynamic localization versus

photon-assisted transport in semiconductor superlattices driven by dc-ac fields,”

Physical Review B 61(11), 7269–7272 (2000).

[11] A. Z. Zhang, L. J. Yang, and M. M. Dignam, “Influence of excitonic effects on

dynamic localization in semiconductor superlattices in combined dc and ac electric

fields,” Physical Review B 67(20), 205,318–1 (2003).

[12] C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, “Co-

herent Submillimeter-Wave Emission from Bloch Oscillations in a Semiconductor

Superlattice,” Physical Review Letters 70(21), 3319–3322 (1993).

[13] S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperg, “Channel

Optical Waveguide Directional Couplers,” Applied Physics Letters 22(1), 46–47

(1973).

[14] F. Bloch, “Uber die Quantenmechanik der Elektronen in Kristallgittern,”

Zeitschrift fur Physik A: Hadrons and Nuclei 52, 555–600.

Bibliography 142

[15] G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an Array of

Curved Optical Waveguides,” Physical Review Letters 83(5), 963–966 (1999).

[16] M. K. Smit, E. C. M. Pennings, and H. Blok, “A Normalized Approach to the

Design of Low-Loss Optical Wave-Guide Bends,” Journal of Lightwave Technology

11(11), 1737–1742 (1993).

[17] D. H. Dunlap and V. M. Kenkre, “Dynamic Localization of a Charged-Particle

Moving under the Influence of an Electric-Field,” Physical Review B 34(6), 3625–

3633 (1986).

[18] M. M. Dignam and C. M. de Sterke, “Conditions for Dynamic Localization in

Generalized ac Electric Fields,” Physical Review Letters 88(4), 046,806–1 (2002).

[19] S. Longhi, M. Lobino, M. Marangoni, R. Ramponi, P. Laporta, E. Cianci, and

V. Foglietti, “Semiclassical motion of a multiband Bloch particle in a time-

dependent field: Optical visualization,” Physical Review B (Condensed Matter

and Materials Physics) 74(15), 155,116–1 (2006).

[20] R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, “Exact

dynamic localization in curved AlGaAs optical waveguide arrays,” Optics Express

15(6), 3212–3223 (2007).

[21] R. Iyer, J. Wan, M. M. Dignam, C. M. De Sterke, and J. S. Aitchison, “Exact

dynamic localization in curved AlGaAs optical waveguide arrays,” in Quantum

Electronics and Laser Science Conference (QELS), p. QTuA6 (Baltimore, MD,

2007).

[22] S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,”

Journal of Lightwave Technology 14(6), 955–966 (1996).

Bibliography 143

[23] X. Wang, H. Masumoto, Y. Someno, L. Chen, and T. Hirai, “Stepwise graded

refractive-index profiles for design of a narrow-bandpass filter,” Applied Optics

40(22), 3746–3752 (2001).

[24] F. N. Timofeev, P. Bayvel, E. G. Churin, and J. E. Midwinter, “1.5 µm free-space

grating multi/demultiplexer and routing switch,” Electronics Letters 32(14), 1307–

1308 (1996).

[25] J. E. Ford and D. J. DiGiovanni, “1 x N fiber bundle scanning switch,” IEEE

Photonics Technology Letters 10(7), 967–969 (1998).

[26] R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V.

Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor microring resonators

for high-order and wide-FSR filters,” Journal of Lightwave Technology 20(5), 872–

877 (2002).

[27] J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher

order filter response in coupled microring resonators,” IEEE Photonics Technology

Letters 12(3), 320–322 (2000).

[28] V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho,

“Optical signal processing using nonlinear semiconductor microring resonators,”

IEEE Journal of Selected Topics in Quantum Electronics 8(3), 705–713 (2002).

[29] T. Barwicz, M. R. Watts, M. A. Popovic, P. T. Rakich, L. Socci, F. X. Kartner,

E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in

the strong confinement limit,” Nature Photonics 1, 57–60 (2007).

[30] J. E. Heebner, N. N. Lepeshkin, A. Schweinsberg, G. W. Wicks, R. W. Boyd,

R. Grover, and P. T. Ho, “Enhanced linear and nonlinear optical phase response

of AlGaAs microring resonators,” Optics Letters 29(7), 769–771 (2004).

Bibliography 144

[31] F. N. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-

insulator photonic wire based racetrack resonators,” Optics Express 14(9), 3872–

3886 (2006).

[32] G. T. Paloczi, Y. Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-

Zehnder interferometer using serially coupled microring resonators,” Optics Express

11(21), 2666–2671 (2003).

[33] A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide:

a proposal and analysis,” Optics Letters 24(11), 711–713 (1999).

[34] J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv,

“Matrix analysis of microring coupled-resonator optical waveguides,” Optics Ex-

press 12(1), 90–103 (2004).

[35] J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator

optical waveguide delay lines,” Journal of the Optical Society of America B: Optical

Physics 21(9), 1665–1673 (2004).

[36] J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay

of microring coupled-resonator optical waveguides,” Optics Letters 31(4), 456–458

(2006).

[37] F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical wave-

guides based on silicon-on-insulator photonic wires,” Applied Physics Letters 89(4),

041,122–1 (2006).

[38] A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass

filters for WDM systems,” Journal of Lightwave Technology 20(2), 296–303 (2002).

Bibliography 145

[39] M. A. Popovic, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X.

Kartner, and H. I. Smith, “Multistage high-order microring-resonator add-drop

filters,” Optics Letters 31(17), 2571–2573 (2006).

[40] T. Barwicz, M. A. Popovic, M. R. Watts, P. T. Rakich, E. P. Ippen, and H. I. Smith,

“Fabrication of add-drop filters based on frequency-matched microring resonators,”

Journal of Lightwave Technology 24(5), 2207–2218 (2006).

[41] B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth,

D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator

filters for WDM applications,” IEEE Photonics Technology Letters 16(10), 2263–

2265 (2004).

[42] O. Schwelb and I. Frigyes, “Vernier operation of series-coupled optical micror-

ing resonator filters,” Microwave and Optical Technology Letters 39(4), 257–261

(2003).

[43] J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and

localized feedback in microresonator sequences for linear and nonlinear optics,”

Journal of the Optical Society of America B: Optical Physics 21(10), 1818–1832

(2004).

[44] G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Disper-

sive properties of optical filters for WDM systems,” IEEE Journal of Quantum

Electronics 34(8), 1390–1402 (1998).

[45] W. Chen and D. L. Mills, “Gap Solitons and the Nonlinear Optical-Response of

Superlattices,” Physical Review Letters 58(2), 160–163 (1987).

[46] S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, “Gap solitons in a two-

channel microresonator structure,” Optics Letters 27(7), 536–538 (2002).

Bibliography 146

[47] S. Pereira, P. Chak, and J. E. Sipe, “All-optical AND gate by use of a Kerr nonlinear

microresonator structure,” Optics Letters 28(6), 444–446 (2003).

[48] W. Y. Chen, V. Van, W. N. Herman, and P. T. Ho, “Periodic microring lattice as

a bandstop filter,” IEEE Photonics Technology Letters 18(19), 2041–2043 (2006).

[49] F. Pozzi, M. Sorel, Z. Yang, R. Iyer, P. Chak, J. E. Sipe, and J. S. Aitchison,

“Integrated high Order filters in AlGaAs waveguides with up to eight side-coupled

racetrack microresonators,” in Conference on Lasers and Electro-Optics (CLEO),

p. CWK2 (Long Beach, CA, 2006).

[50] R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: Ca-

pabilities and fundamental limitations,” Journal of Lightwave Technology 23(12),

4046–4066 (2005).

[51] J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced

transparent media and coupled resonator structures: comparative analysis,” Jour-

nal of the Optical Society of America B: Optical Physics 22(5), 1062–1074 (2005).

[52] ITRS, “International Technology Roadmap for Semiconductors: 2005 Edition,”

(2005).

[53] L. Vivien, F. Grillot, E. Cassan, D. Pascal, S. Lardenois, A. Lupu, S. Laval,

M. Heitzmann, and J. M. Fedeli, “Comparison between strip and rib SOI mi-

crowaveguides for intra-chip light distribution,” Optical Materials 27(5), 756–762

(2005).

[54] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction

to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598

(1999).

Bibliography 147

[55] M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic

medium,” Nature 426(6967), 638–641 (2003).

[56] Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J.

Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin

slow light in an optical fiber,” Physical Review Letters 94(15), 153,902 (2005).

[57] R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time

delay achievable on propagation through a slow-light medium,” Physical Review A

71(2), 023,801 (2005).

[58] F. Xia, L. Sekaric, and Y. A. Vlasov, “Ultracompact optical buffers on a silicon

chip,” Nature Photonics 1, 65–71 (2006).

[59] Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of

slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69

(2005).

[60] J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, en-

hanced nonlinearity, and optical solitons in a resonator-array waveguide,” Physical

Review E 65(3), 036,619 (2002).

[61] M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci,

C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt,

“Ultrashort pulse propagation at the photonic band edge: Large tunable group

delay with minimal distortion and loss,” Physical Review E 54(2), R1078–R1081

(1996).

[62] M. F. Yanik and S. H. Fan, “Stopping light all optically,” Physical Review Letters

92(8), 083,901 (2004).

Bibliography 148

[63] L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, “Tunable delay line

with interacting whispering-gallery-mode resonators,” Optics letters 29(6), 626–

628 (2004).

[64] G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines

based on optical filters,” IEEE Journal of Quantum Electronics 37(4), 525–532

(2001).

[65] Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. H. Fan, and M. Lipson, “Experi-

mental realization of an on-chip all-optical analogue to electromagnetically induced

transparency,” Physical Review Letters 96(12), 123,901 (2006).

[66] B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus,

E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring

resonator optical channel dropping filters,” IEEE Photonics Technology Letters

10(4), 549–551 (1998).

[67] T. Barwicz, M. A. Popovic, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen,

and H. I. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication

and analysis,” Optics Express 12(7), 1437–1442 (2004).

[68] J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer microring coupled-

resonator optical waveguides,” Journal of Lightwave Technology 24(4), 1843–1849

(2006).

[69] J. B. Khurgin, “Dispersion and loss limitations on the performance of optical delay

lines based on coupled resonant structures,” Optics Letters 32(2), 133–135 (2007).

[70] M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J.

Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. Le Grange, and

S. S. Patel, “Integrated resonance-enhanced variable optical delay lines,” IEEE


Bibliography 149

[71] Y. O. Noh, H. J. Lee, Y. H. Won, and M. C. Oh, “Polymer waveguide thermo-

optic switches with - 70 dB optical crosstalk,” Optics Communications 258(1),

18–22 (2006).

[72] G. P. Agrawal, “Nonlinear Fiber Optics,” Academic Press, San Diego (1989).

[73] A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison,

and C. N. Ironside, “Ultrafast All-Optical Switching in Semiconductor Nonlin-

ear Directional-Couplers at Half the Band-Gap,” Applied Physics Letters 61(2),

147–149 (1992).

[74] R. Iyer, A. D. Bristow, Z. Yang, J. S. Aitchison, H. M. Van Driel, J. E. Sipe, and

A. L. Smirl, “Switchable all-optical 188-ps delay line in AlGaAs,” in Optical Fiber

Conference (OFC), p. JWA26 (Anaheim, CA, 2007).

[75] A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, “Switch-

able AlxGa1−xAs all-optical delay line at 1.55 µm,” Applied Physics Letters 90(10),

101,112 (2007).

[76] A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, “Optical

Delays: Changing paths,” Nature Photonics 1, 252 (2007).

[77] F. Sun, J. Z. Yu, and S. W. Chen, “Directional-coupler-based Mach-Zehnder in-

terferometer in silicon-on-insulator technology for optical intensity modulation,”

Optical Engineering 46(2), 025,601 (2007).

[78] K. Kishioka, “A design method to achieve wide wavelength-flattened responses in

the directional coupler-type optical power splitters,” Journal of Lightwave Tech-

nology 19(11), 1705–1715 (2001).

[79] U. Koren, B. I. Miller, M. G. Young, M. Chien, K. Dreyer, R. BenMichael, and

R. J. Capik, “A 1.3-µm wavelength laser with an integrated output power monitor

Bibliography 150

using a directional coupler optical power tap,” IEEE Photonics Technology Letters

8(3), 364–366 (1996).

[80] Y. T. Han, Y. J. Park, S. H. Park, J. U. Shin, D. J. Kim, S. W. Park, S. H. Song,

K. Y. Jung, D. J. Lee, W. Y. Hwang, and H. K. Sung, “1.25-Gb/s bidirectional

transceiver module using 1.5%-Delta silica directional coupler-type WDM,” IEEE


[81] Q. J. Wang, Y. Zhang, and Y. C. Soh, “Flat-passhand 3X3 interleaving filter de-

signed with optical directional couplers in lattice structure,” Journal of Lightwave

Technology 23(12), 4349–4362 (2005).

[82] S. M. Jensen, “The Non-Linear Coherent Coupler,” IEEE Journal of Quantum

Electronics 18(10), 1580–1583 (1982).

[83] Y. Silberberg and G. I. Stegeman, “Nonlinear Coupling of Wave-Guide Modes,”

Applied Physics Letters 50(13), 801–803 (1987).

[84] Y. Silberberg and G. I. Stegeman, “Nonlinear Coupling of Modes - a New Approach

to All-Optical Guided Wave Devices,” Journal of the Optical Society of America

A: Optics Image Science and Vision 3(13), P41–P41 (1986).

[85] F. J. Frailepelaez and G. Assanto, “Coupled-Mode Equations for Nonlinear

Directional-Couplers,” Applied Optics 29(15), 2216–2217 (1990).

[86] F. J. Frailepelaez and G. Assanto, “Improved Coupled-Mode Analysis of Nonlinear

Distributed Feedback Structures,” Optical and Quantum Electronics 23(5), 633–

637 (1991).

[87] S. Trillo and S. Wabnitz, “Nonlinear Nonreciprocity in a Coherent Mismatched

Directional Coupler,” Applied Physics Letters 49(13), 752–754 (1986).

Bibliography 151

[88] S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Soliton Switching in

Fiber Nonlinear Directional-Couplers,” Optics Letters 13(8), 672–674 (1988).

[89] S. M. Eaton, W. Chen, L. Zhang, H. Zhang, R. Iyer, J. S. Aitchison, and P. R.

Herman, “Telecom-band directional coupler written with femtosecond fiber laser,”

IEEE Photonics Technology Letters 18(20), 2174–2176 (2006).

[90] D. H. Yoon, S. G. Yoon, and Y. T. Kim, “Refractive index and etched structure

of silicon nitride waveguides fabricated by PECVD,” Thin Solid Films 515(12),

5004–5007 (2007).

[91] S. Garidel, D. Lauvernier, J. P. Vilcot, M. Francois, and D. Decoster, “Apodized

Bragg filters on InP-materials ridge waveguides using sampled gratings,” Microwave

and Optical Technology Letters 48(8), 1627–1630 (2006).

[92] S. Adachi, GaAs and Related Materials (World Scientific, New Jersey, 1994).

[93] S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg,

“The refractive index of AlxGa1−xAs below the band gap: Accurate determination

and empirical modeling,” Journal of Applied Physics 87(11), 7825–7837 (2000).

[94] J. K. Doylend and A. P. Knights, “Design and simulation of an integrated fiber-to-

chip coupler for silicon-on-insulator waveguides,” IEEE Journal of Selected Topics

in Quantum Electronics 12(6), 1363–1370 (2006).

[95] P. W. E. Smith and S. D. Benjamin, “Materials for All-Optical Devices,” Optical

Engineering 34(1), 189–194 (1995).

[96] J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve,

“The nonlinear optical properties of AlGaAs at the half band gap,” IEEE Journal

of Quantum Electronics 33(3), 341–348 (1997).

Bibliography 152

[97] K. Ikeda and Y. Fainman, “Nonlinear Fabry-Perot resonator with a silicon photonic

crystal waveguide,” Optics Letters 31(23), 3486–3488 (2006).

[98] S. Santran, M. Martinez-Rosas, L. Canioni, L. Sarger, L. N. Glebova, A. Tirpak,

and L. B. Glebov, “Nonlinear refractive index of photo-thermo-refractive glass,”

Optical Materials 28(4), 401–407 (2006).

[99] S. Shettigar, K. Chandrasekharan, G. Umesh, B. K. Sarojini, and B. Narayana,

“Studies on nonlinear optical parameters of bis-chalcone derivatives doped poly-

mer,” Polymer 47(10), 3565–3567 (2006).

[100] P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of

coupled-mode theory in waveguiding structures,” Physical Review E 75(1), 016,608

(2007).

[101] K. S. Chiang, “Review of Numerical and Approximate Methods for the Modal-

Analysis of General Optical Dielectric Wave-Guides,” Optical and Quantum Elec-

tronics 26(3), S113–S134 (1994).

[102] Z. M. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstruc-

tured optical fibers,” Optics Express 10(17), 853–864 (2002).

[103] Z. Yang, P. Chak, A. D. Bristow, H. M. Van Driel, R. Iyer, J. S. Aitchison, A. L.

Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring

resonators,” Optics Letters 32(7), 826–828 (2007).

[104] J. S. Aitchison, M. K. Oliver, E. Kapon, E. Colas, and P. W. E. Smith, “Role

of 2-Photon Absorption in Ultrafast Semiconductor Optical Switching Devices,”

Applied Physics Letters 56(14), 1305–1307 (1990).

Bibliography 153

[105] R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and K. Silberberg, “Ex-

perimental observation of linear and nonlinear optical Bloch oscillations,” Physical

Review Letters 83(23), 4756–4759 (1999).

[106] G. A. Siviloglou, S. Suntsov, R. El-Ganainy, R. Iwanow, G. I. Stegeman, D. N.

Christodoulides, R. Morandotti, D. Modotto, A. Locatelli, C. De Angelis, F. Pozzi,

C. R. Stanley, and M. Sorel, “Enhanced third-order nonlinear effects in optical

AlGaAs nanowires,” Optics Express 14(20), 9377–9384 (2006).

[107] M. Holthaus, “Collapse of Minibands in Far-Infrared Irradiated Superlattices,”

Physical Review Letters 69(2), 351–354 (1992).

[108] X. G. Zhao, “Motion of Bloch Electrons in Time-Dependent Electric-Fields with

Off-Diagonal Effects,” Physics Letters A 167(3), 291–294 (1992).

[109] A. W. Ghosh, A. V. Kuznetsov, and J. W. Wilkins, “Reflection of THz radiation

by a superlattice,” Physical Review Letters 79(18), 3494–3497 (1997).

[110] C. Zener, “A Theory of the Electrical Breakdown of Solid Dielectrics,” Proceedings

of the Royal Society of London, Series A, Containing Papers of a Mathematical and

Physical Character 145, 523–529 (1934).

[111] H. M. James, “Electronic States in Perturbed Periodic Systems,” Physical Review

76(11), 1611–1624 (1949).

[112] K. W. Madison, M. C. Fischer, R. B. Diener, Q. Niu, and M. G. Raizen, “Dynamical

Bloch Band Suppression in an Optical Lattice,” Physical Review Letters 81(23),

5093–5096 (1998).

[113] G. Lenz, R. Parker, M. C. Wanke, and C. M. de Sterke, “Dynamic localization and

AC Bloch oscillations in periodic optical waveguide arrays,” Optical Communica-

tions 218, 87–92 (2003).

Bibliography 154

[114] M. Heiblum and J. H. Harris, “Analysis of Curved Optical-Waveguides by Confor-

mal Transformation,” IEEE Journal of Quantum Electronics 11(2), 75–83 (1975).

[115] T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch os-

cillations in temperature tuned waveguide arrays,” Physical Review Letters 83(23),

4752–4755 (1999).

[116] K. S. Chiang, “Dual Effective-Index Method for the Analysis of Rectangular Di-

electric Wave-Guides,” Applied Optics 25(13), 2169–2174 (1986).

[117] R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through

optical frequency range,” Proceedings MRI Symposium on Submillimeter Waves

pp. 497–516 (1970).

[118] J. P. Mckelvey, Solid State and Semiconductor Physics (New York Harper & Row,

New York, 1966).

[119] K. Drese and M. Holthaus, “Anderson localization in an ac-driven two-band

model,” Journal of Physics: Condensed Matter 8(9), 1193–1206 (1996).

[120] C. Vassallo, “Reformulation for the Beam-Propagation Method,” Journal of the

Optical Society of America A: Optics Image Science and Vision 10(10), 2208–2216

(1993).

[121] S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and

V. Foglietti, “Observation of Dynamic Localization in Periodically Curved Wave-

guide Arrays,” Physical Review Letters 96(24), 243,901 (2006).

[122] P. Domachuk, C. M. de Sterke, J. Wan, and M. M. Dignam, “Dynamic localization

in continuous ac electric fields,” Physical Review B 66(16), 165,313 (2002).

[123] H. Kogelnik and T. Li, “Laser Beams and Resonators,” Proceedings of the Institute

of Electrical and Electronics Engineers 5(10), 1550–1567 (1966).

Bibliography 155

[124] N. Chiodo, G. Della-Valle, R. Osellame, S. Longhi, G. Cerullo, and R. Ramponi,

“Imaging of Bloch oscillations in erbium-doped curved waveguide arrays,” Optics

Letters 31(11), 1651–1653 (2006).

[125] J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman,

M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear wave-

guide arrays,” Optics Express 13(6), 1780–1796 (2005).

[126] J. Wan, “Dynamic localization in electronic systems and optical waveguide arrays,”

Ph.D. thesis, Queen’s University (2005).

[127] B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for

periodically coupled microring resonators,” Optics Letters 25(5), 344–346 (2000).

[128] P. Ganguly, J. C. Biswas, and S. K. Lahiri, “Semi-analytical simulation of titanium-

indiffused lithium niobate-integrated optic directional couplers consisting of curved

waveguides,” Fiber and Integrated Optics 24(6), 511–520 (2005).

[129] R. C. Alferness, “Optical Directional-Couplers with Weighted Coupling,” Applied

Physics Letters 35(3), 260–262 (1979).

[130] P. L. Auger and S. I. Najafi, “New Method to Design Directional Coupler Dual-

Wavelength Multi/Demultiplexer with Bends at Both Extremities,” Optics Com-

munications 111(1-2), 43–50 (1994).

[131] R. Regener and W. Sohler, “Loss in low-finesse Ti:LiNbO3 optical waveguide res-

onators,” Applied Physics B 36(3), 143–147 (1985).

[132] J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet,

“Optical solitons in a silicon waveguide,” Optics Express 15(12), 7682–7688 (2007).

Bibliography 156

[133] C. Madsen and J. H. Zhao, “Optical Filter Design and Analysis,” Wiley Series in

Microwave and Optical Engineering, Kai Chang, Series Editor (1999). John Wiley

and Sons, Canada.

[134] D. J. Gauthier, A. L. Gaeta, and R. W. Boyd, “Slow light: From basics to future

prospects,” Photonics Spectra 40(3), 44–50 (2006).

[135] H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van

Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in

photonic crystal waveguides,” Physical Review Letters 94(7), 073,903 (2005).

[136] Z. J. Deng, D. K. Qing, P. Hemmer, C. H. R. Ooi, M. S. Zubairy, and M. O. Scully,

“Time-bandwidth problem in room temperature slow light,” Physical Review Let-

ters 96(2), 023,602 (2006).

[137] J. E. Sharping, Y. Okawachi, and A. L. Gaeta, “Wide bandwidth slow light using

a Raman fiber amplifier,” Optics Express 13(16), 6092–6098 (2005).

[138] R. Langenhorst, M. Eiselt, W. Pieper, G. Grosskopf, R. Ludwig, L. Kuller, E. Diet-

rich, and H. G. Weber, “Fiber loop optical buffer,” Journal of Lightwave Technology

14(3), 324–335 (1996).

[139] W. D. Zhong and R. S. Tucker, “Wavelength routing-based photonic packet buffers

and their applications in photonic packet switching systems,” Journal of Lightwave

Technology 16(10), 1737–1745 (1998).

[140] T. Sakamoto, K. Noguchi, R. Sato, A. Okada, Y. Sakai, and M. Matsuoka, “Variable

optical delay circuit using wavelength converters,” Electronics Letters 37(7), 454–

455 (2001).

[141] G. Steinmeyer, “A review of ultrafast optics and optoelectronics,” Journal of Optics

A-Pure and Applied Optics 5(1), R1–R15 (2003).

Bibliography 157

[142] A. Braun, S. Kane, and T. Norris, “Compensation of self-phase modulation in

chirped-pulse amplification laser systems,” Optics Letters 22(9), 615–617 (1997).

[143] A. Villeneuve, P. Mamyshev, J. U. Kang, G. I. Stegeman, J. S. Aitchison, and

C. N. Ironside, “Efficient Time-Domain Demultiplexing with Separate Signal and

Control Wavelengths in an AlGaAs Nonlinear Directional Coupler,” IEEE Journal

of Quantum Electronics 31(12), 2165–2172 (1995).

[144] T. Kitoh, N. Takato, M. Yasu, and M. Kawachi, “Bending Loss Reduction in Silica-

Based Wave-Guides by Using Lateral Offsets,” Journal of Lightwave Technology

13(4), 555–562 (1995).

[145] A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J.

Andrejco, and P. W. Smith, “Use of Femtosecond Square Pulses to Avoid Pulse

Breakup in All-Optical Switching,” IEEE Journal of Quantum Electronics 25(12),

2648–2655 (1989).

[146] N. J. Doran and D. Wood, “Soliton Processing Element for All-Optical Switching

and Logic,” Journal of the Optical Society of America B: Optical Physics 4(11),

1843–1846 (1987).

[147] J. Stulemeijer, F. E. van Vliet, K. W. Benoist, D. H. P. Maat, and M. K. Smit,

“Compact photonic integrated phase and amplitude controller for phased-array

antennas,” IEEE Photonics Technology Letters 11(1), 122–124 (1999).

[148] R. W. Boyd, “Nonlinear Optics,” Academic Press, Boston (1992).

[149] A. Yariv and P. Yeh, Photonics : optical electronics in modern communications,

6th ed. (Oxford University Press, New York, 2007).

Bibliography 158

[150] D. C. Hutchings, “Theory of Ultrafast Nonlinear Refraction in Semiconductor Su-

perlattices,” IEEE Journal of Selected Topics in Quantum Electronics 10(5), 1124–

1132 (2004).